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\begin{document}
\title [\texttt{A Novel/Old Modification of the First Zagreb Index}]{A Novel/Old Modification of the First Zagreb Index} \author{Akbar Ali$^{1}$ and Nenad Trinajsti\'{c}$^{2}$}
\address{$^{1}$Department of Mathematics, University of Management and Technology, Sialkot, Pakistan.}
\email{akbarali.maths@gmail.com}
\address{$^{2}$The Rugjer Bo\v{s}kovi\'{c} Institute, P. O. Box 180, HR-10002 Zagreb, Croatia.} \email{trina@irb.hr}
\keywords{topological index, Zagreb indices, chemical tree, vertex connection number}
\begin{abstract}
In the paper [Gutman, I.; Trinajsti\'{c}, N. \textit{Chem. Phys. Lett.} \textbf{1972}, 17, 535], it was shown that total $\pi$-electron energy ($E$) of a molecule $M$ depends on the quantity $\sum_{v\in V(G)}d_{v}^{2}$ (nowadays known as the \textit{first Zagreb index}), where $G$ is the graph corresponding to $M$, $V(G)$ is the vertex set of $G$ and $d_{v}$ is degree of the vertex $v$. In the same paper, the graph invariant $\sum_{v\in V(G)}d_{v}\tau_{v}$ (where $\tau_{v}$ is the connection number of $v$, that is the number of vertices at distance 2 from $v$) was also proved to influence $E$, but this invariant was never restudied explicitly. We call it \textit{modified first Zagreb connection index} and denote it by $ZC_{1}^{*}$. In this paper, we characterize the extremal elements with respect to the graph invariant $ZC_{1}^{*}$ among the collection of all $n$-vertex chemical trees.
\end{abstract}
\maketitle
\section{Introduction}
It is well known fact that chemical compounds can be represented by graphs (known as molecular graphs) in which vertices correspond to the atoms while edges represent the covalent bonds between atoms \cite{trn,gutman1}. In theoretical chemistry, the physicochemical properties of chemical compounds are often modeled by the topological indices \cite{t1,f1}. Topological indices are numerical quantities of molecular graph, which are invariant under graph isomorphism \cite{f1}. In the paper \cite{g3}, it was shown that the following topological indices appears in an approximate formula for total $\pi$-electron energy ($E$) of a molecule $M$: $$M_{1}(G)=\sum_{v\in V(G)}d_{v}^{2} \ , \ \ \ ZC_{1}^{*}(G)=\sum_{v\in V(G)}d_{v}\tau_{v}$$ where $G$ is the graph corresponding to the molecule $M$, $V(G)$ is the vertex set of $G$, $d_{v}$ is degree of the vertex $v$ and $\tau_{v}$ is the connection number of $v$ (that is, the number of vertices at distance 2 from $v$). The topological index $M_{1}$ is known as the \textit{first Zagreb index}. We call the topological index $ZC_{1}^{*}$ as \textit{modified first Zagreb connection index}. The following topological index is called \textit{second Zagreb index}: \cite{g3,gutman75} $$M_{2}(G)=\sum_{uv\in E(G)}d_{u}d_{v},$$ where $E(G)$ is the edge set of the graph $G$ and $uv$ is the edge between the vertices $u,v$. The first Zagreb index and second Zagreb index are among the oldest and most studied topological indices. More than a hundred papers have been devoted to these Zagreb indices, for example see the reviews \cite{Nic03, Das04, Gut04} published on the occasion of their 30th anniversary, recent surveys \cite{g1,Gut14}, very recent papers \cite{Habi16,Reti16,Bor16,wang16,lee16,das16,deng16} and related references cited therein. In contrast, the modified first Zagreb connection index did not attract attention in any of the numerous publications on topological indices till 2016. The main purpose of the present study is to gather some basic properties of the modified first Zagreb connection index, and-especially-to characterize the extremal elements with respect to the aforementioned topological index among the collection of all $n$-vertex chemical trees.
\section{Some Observations on the Modified First Zagreb Connection Index}
In this section, some basic properties of the modified first Zagreb connection index are established. As usual, the path graph, star graph and complete graph on $n$ vertices will be denoted by $P_{n}$, $S_{n}$ and $K_{n}$, respectively. Undefined notations and terminologies from (chemical) graph theory can be found in the books \cite{Ha69,trn,bon76}. Suppose that the graph $G$ has $n_{0}$ vertices with degree zero and let $V_{0}(G)$ be the set of these vertices. In the Reference \cite{Doslic11}, the following identity was derived: \begin{equation}\label{Eq113} \sum_{u\in V(G)\setminus V_{0}(G)}d_{u}f(d_{u})=\sum_{uv\in E(G)}(f(d_{u})+f(d_{v})), \end{equation} where $f$ is a function defined on the set of vertex degrees of the graph $G$. The proof technique used in establishing the identity (\ref{Eq113}) also works for deriving the following more general identity: \begin{equation}\label{Eq114} \sum_{u\in V(G)\setminus V_{0}(G)}d_{u}g(u)=\sum_{uv\in E(G)}(g(u)+g(v)), \end{equation} where $g$ is a function defined on the vertex set $V(G)$. Hence, the modified first Zagreb connection index can be rewritten in the following form by setting $g(u)=\tau_{u}$ in the identity (\ref{Eq114}): \begin{equation}\label{Eq115} ZC_{1}^{*}(G)=\sum_{uv\in E(G)}(\tau_{u}+\tau_{v}). \end{equation}
In 2008, Yamaguchi \cite{Yamaguchi} derived a lower bound (given in the following theorem) for the topological index $M_{2}$ in which the quantity $\sum_{v\in V(G)}d_{v}\tau_{v}$ also appears:
\begin{thm}\label{t555} If $G$ is a connected graph, then \begin{equation}\label{Eq111} M_{2}(G)\geq\frac{1}{2}\left(\sum_{v\in V(G)}d_{v}(d_{v}+\tau_{v})\right), \end{equation} with equality if and only if $G$ is a triangle- and quadrangle-free graph. \end{thm} Hence, if the graph under consideration is triangle- and quadrangle-free then the modified first Zagreb connection index can be written as the linear combination of the topological indices $M_{1}$ and $M_{2}$. \begin{cor}\label{c555} If $G$ is a triangle- and quadrangle-free graph, then \begin{equation}\label{Eq1110} ZC_{1}^{*}(G)=2M_{2}(G)-M_{1}(G). \end{equation} \end{cor}
Here, it needs to be mentioned that Xu \textit{et al.} \cite{Xu-14} considered the linear combination $2M_{2}(G)-M_{1}(G)$ for any connected graph $G$. As there are many (lower and upper) bounds as well as relations for the topological indices $M_{1}$ and $M_{2}$ in the literature. Hence, several bounds for the topological index $ZC_{1}^{*}$ can be easily established using Eq. (\ref{Eq1110}) in case of triangle- and quadrangle-free graphs. For instance, we derive an upper bound for the topological index $ZC_{1}^{*}$ using the following result which was reported in the references \cite{Das04,Das09}:
\begin{thm}\label{t666} If $G$ is an $n$-vertex graph with size $m$ and minimum vertex degree $\delta$ then \[M_{2}(G)\leq2m^{2}-(n-1)m\delta+\frac{1}{2}(\delta-1)M_{1}(G)\] with equality if and only if $G\cong S_{n}$ or $G\cong K_{n}$. \end{thm}
The upper bound for $ZC_{1}^{*}$, given in the following corollary, is a direct consequence of Theorem \ref{t666}:
\begin{cor}\label{cor666} If $G$ is a triangle- and quadrangle-free graph with order $n$, size $m$ and minimum vertex degree $\delta$ then \[ZC_{1}^{*}(G)\leq4m^{2}-2(n-1)m\delta+(\delta-2)M_{1}(G)\] with equality if and only if $G\cong S_{n}$. \end{cor}
\begin{figure}
\caption{All the non-isomorphic chemical trees (together with vertex connection numbers) on (a) four vertices (b) five vertices (c) six vertices.}
\label{f1}
\end{figure}
\section{Modified First Zagreb Connection Index of chemical trees}
Denote by $\mathbb{CT}_{n}$ the collection of all $n$-vertex chemical trees. This section is devoted to characterize the extremal elements with respect to the topological index $ZC_{1}^{*}$ among the collection $\mathbb{CT}_{n}$. Note that the collection $\mathbb{CT}_{n}$ consist of only a single element for $n=1,2,3$. All the non-isomorphic members of of the collections $\mathbb{CT}_{4}$, $\mathbb{CT}_{5}$ and $\mathbb{CT}_{6}$ (together with vertex connection numbers) are depicted in Figure \ref{f1} and their $ZC_{1}^{*}$ values are given in Table \ref{table111}. Note that both non-isomorphic trees on 4 vertices have the same $ZC_{1}^{*}$ value. Hence, the before said problem concerning extremal chemical trees make sense only for $n\geq5$. Firstly, we characterize the chemical tree having minimum $ZC_{1}^{*}$ value among the aforementioned collection for $n\geq5$. For a vertex $u\in V(G)$, denote by $N(u)$ (the neighborhood of $u$) the set of all vertices adjacent with $u$. A vertex having degree 1 is called pendent vertex. A pendent vertex adjacent with a vertex having degree greater than 2 is called star-type pendent vertex.
\begin{table}[p]
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|}\hline Chemical Tree $T_{i}$ shown in Figure \ref{f1} &$T_{1}$&$T_{2}$&$T_{3}$&$T_{4}$&$T_{5}$&$T_{6}$&$T_{7}$&$T_{8}$&$T_{9}$&$T_{10}$\\\hline The $ZC_{1}^{*}$ Value of $T_{i}$&6&6&10&12&12&14&16&18&20&20\\\hline \end{tabular} \caption{The $ZC_{1}^{*}$ values of the chemical trees depicted in Figure \ref{f1}.} \label{table111} \end{table}
\begin{thm}\label{t00} For $n\geq5$, the path graph has minimum $ZC_{1}^{*}$ value among all the members of $\mathbb{CT}_{n}$. \end{thm}
\begin{proof} Simple calculations yield $ZC_{1}^{*}(P_{n})=4n-10$. The result will be proved by induction on $n$. For $n=5$, there are only three non-isomorphic trees and hence the conclusion can be easily verified. Suppose that the result holds for all trees of order $\leq n-1$ where $n\geq6$. Let $T_{n}$ be an $n$-vertex tree and $u$ be its pendent vertex adjacent with the vertex $v$. Set $d_{v}=x$ and $N(v)=\{u=u_{0},u_{1},u_{2},...,u_{r-1},u_{r},...,u_{x-1}\}$ where $d_{u_{i}}=1$ for $0\leq i\leq r-1$ and $d_{u_{i}}\geq2$ for $r\leq i\leq x-1$. As $T_{n}$ is different from the star graph $S_{n}$, so $\tau_{v}\geq1$. Let $T_{n-1}$ be the tree obtained from $T_{n}$ by removing the vertex $u$. Then $$ZC_{1}^{*}(T_{n})=ZC_{1}^{*}(T_{n-1})+\displaystyle\sum_{i=1}^{x-1}d_{u_{i}}\tau_{u_{i}}+d_{v}\tau_{v}+\tau_{u}
-\displaystyle\sum_{i=1}^{x-1}d_{u_{i}}(\tau_{u_{i}}-1)-(d_{v}-1)\tau_{v}$$ which is equivalent to \begin{equation}\label{eq00} ZC_{1}^{*}(T_{n})=ZC_{1}^{*}(T_{n-1})+\displaystyle\sum_{i=1}^{x-1}d_{u_{i}}+d_{v}+\tau_{v}-1 \end{equation} Bearing in mind the facts $\tau_{v}\geq1$, $d_{v}\geq2$, $\sum_{i=1}^{x-1}d_{u_{i}}\geq2$ and inductive hypothesis, from eq. (\ref{eq00}) we have $$ZC_{1}^{*}(T_{n})\geq ZC_{1}^{*}(T_{n-1})+4\geq 4(n-1)-10+4\geq ZC_{1}^{*}(P_{n}).$$ Observe that the equality $ZC_{1}^{*}(T_{n})=ZC_{1}^{*}(P_{n})$ holds if and only if $\tau_{v}=1$, $d_{v}=2$, $d_{u_{1}}=2$ and $T_{n-1}\cong P_{n-1}$. This completes the proof. \end{proof}
It can be easily noted that the proof of Theorem \ref{t00} remains valid if we replace the collection $\mathbb{CT}_{n}$ with the collection of all $n$-vertex general trees different from the star graph $S_{n}$, where $n\geq5$. Moreover, $ZC_{1}^{*}(P_{n})=4n-10<ZC_{1}^{*}(S_{n})=(n-1)(n-2)$ for all $n\geq5$. Hence, one has:
\begin{cor}\label{t6} For $n\geq5$, the path has minimum $ZC_{1}^{*}$ value among all $n$-vertex trees. \end{cor}
Now, we prove some lemmas which will be used to characterize the trees with maximum $ZC_{1}^{*}$ value among the collection $\mathbb{CT}_{n}$. In the remaining part of this section, we will use the alternative formula of $ZC_{1}^{*}$, given in Eq. (\ref{Eq1110}).
\begin{lem}\label{lem1} For $n\geq5$, if the tree $T^{*}\in\mathbb{CT}_{n}$ has the maximum $ZC_{1}^{*}$ value then $T^{*}$ contains at most one vertex of degree 2. \end{lem}
\begin{proof} Suppose to the contrary that $T^{*}$ contains more than one vertex of degree 2. Let $u,v\in V(T^{*})$ such that $d_{u}=d_{v}=2$. Let $N(u)=\{u_{1},u_{2}\}$ and $N(v)=\{v_{1},v_{2}\}$. Suppose that the unique path connecting the vertices $u$ and $v$ contains the vertices $u_{2},v_{2}$. Without loss of generality, we may assume that $d_{u_{1}}+d_{u_{2}}\leq d_{v_{1}}+d_{v_{2}}$. Let $T^{(1)}$ be the tree obtained from $T^{*}$ by removing the edge $u_{1}u$ and adding the edge $u_{1}v$. There are two cases.
\textit{Case 1.} The vertices $u$ and $v$ are adjacent. That is, $u_{2}=v$ and $v_{2}=u$.\\ We calculate the value of the difference $ZC_{1}^{*}(T^{*})-ZC_{1}^{*}(T^{(1)})$. If the vertex $u_{1}$ is pendent then the condition $n\geq5$ guaranties that $d_{v_{1}}\geq2$. Hence, whether the vertex $u_{1}$ is pendent or not, in both cases we have \[ZC_{1}^{*}(T^{*})-ZC_{1}^{*}(T^{(1)})=2(2-d_{u_{1}}-d_{v_{1}})<0,\] which is a contradiction the definition of $T^{*}$. \\ \textit{Case 2.} The vertices $u$ and $v$ are not adjacent.\\ Bearing in mind the inequality $d_{u_{1}}+d_{u_{2}}\leq d_{v_{1}}+d_{v_{2}}$, we have \begin{equation*} ZC_{1}^{*}(T^{*})-ZC_{1}^{*}(T^{(1)})=2(d_{u_{2}}-d_{u_{1}}-d_{v_{1}}-d_{v_{2}}+1)<0, \end{equation*} which contradicts the maximality of $ZC_{1}^{*}(T^{*})$.\\ In both cases, contradiction arises. Hence, $T^{*}$ contains at most one vertex of degree 2. \end{proof}
\begin{lem}\label{lem-1b} For $n\geq5$, if the tree $T^{*}\in\mathbb{CT}_{n}$ has the maximum $ZC_{1}^{*}$ value and $T^{*}$ contains a vertex $u$ of degree 2 then one of the neighbors of $u$ is pendent. \end{lem}
\begin{proof} Suppose to the contrary that both the neighbors of $u$, say $u_{1}$ and $u_{2}$, are non-pendent. Then, Lemma \ref{lem1} suggests that both of the vertices $u_{1},u_{2}$ must have degree at least 3. Observe that all the pendent vertices of $T^{*}$ are star-type.\\
\textit{Case 1.} Neither of the vertices $u_{1},u_{2}$ has a pendent neighbor.\\ Let $w\in V(T^{*})$ be a pendent vertex adjacent with the vertex $t\in V(T^{*})$. Suppose that $T^{(1)}$ is the tree obtained from $T^{*}$ by removing the edges $u_{1}u,u_{2}u,wt$ and adding the edges $u_{1}u_{2},wu,ut$. Observe that both the trees $T^{(1)}$ and $T^{*}$ have same degree sequence. Note that $2d_{u_{1}}+2d_{u_{2}}-d_{u_{1}}d_{u_{2}}\leq3$, which implies that \[ZC_{1}^{*}(T^{*})-ZC_{1}^{*}(T^{(1)})=2(2d_{u_{1}}+2d_{u_{2}}-d_{u_{1}}d_{u_{2}}-2-d_{t})\leq2(1-d_{t})<0.\] This contradicts the maximality of $ZC_{1}^{*}(T^{*})$.
\textit{Case 2.} At least one of the vertices $u_{1},u_{2}$ has a pendent neighbor.\\ Without loss of generality, assume that $u_{1}$ is adjacent with a pendent vertex $u_{11}\in V(T^{*})$. Let $T^{(2)}$ be the tree obtained from $T^{*}$ by removing the edges $u_{1}u,u_{2}u$ and adding the edges $u_{1}u_{2},u_{11}u$. Bearing in mind the inequality $d_{u_{1}}+2d_{u_{2}}-d_{u_{1}}d_{u_{2}}\leq0$, one has \[ZC_{1}^{*}(T^{*})-ZC_{1}^{*}(T^{(2)})=2(d_{u_{1}}+2d_{u_{2}}-d_{u_{1}}d_{u_{2}}-2)<0,\] which is again a contradiction. This completes the proof.
\end{proof}
\begin{lem}\label{lem2} For $n\geq7$, if the tree $T^{*}\in\mathbb{CT}_{n}$ has the maximum $ZC_{1}^{*}$ value then $T^{*}$ contains at most one vertex of degree 3. \end{lem}
\begin{proof} Suppose, contrarily, that $T^{*}$ contains at least two vertices of degree 3. Let $u,v\in V(T^{*})$ such that $d_{u}=d_{v}=3$. Let $N(u)=\{u_{1},u_{2},u_{3}\}$ and $N(v)=\{v_{1},v_{2},v_{3}\}$. Suppose that the unique path connecting the vertices $u$ and $v$ contains the vertices $u_{3},v_{3}$. Without loss of generality, we may also assume that $d_{u_{1}}+d_{u_{2}}+d_{u_{3}}\leq d_{v_{1}}+d_{v_{2}}+d_{v_{3}}$ and $d_{u_{1}}\geq d_{u_{2}}$. Let $T^{(1)}$ be the tree obtained from $T^{*}$ by removing the edge $u_{1}u$ and adding the edge $u_{1}v$. There are two cases.
\textit{Case 1.} The vertices $u$ and $v$ are adjacent. That is, $u_{3}=v$ and $v_{3}=u$.\\ If both the vertices $u_{1}$ and $u_{2}$ are pendent then the assumption $n\geq7$ forces that at least one of the vertices $v_{1}$ and $v_{2}$ must be non-pendent and hence \begin{eqnarray*} ZC_{1}^{*}(T^{*})-ZC_{1}^{*}(T^{(1)})&=& 2(2+d_{u_{2}}-d_{u_{1}}-d_{v_{1}}-d_{v_{2}})<0, \end{eqnarray*} which is a contradiction.\\ If at least one of the vertices $u_{1},u_{2}$ is non-pendent then $d_{u_{1}}\geq2$ (because $d_{u_{1}}\geq d_{u_{2}}$) and hence we have \begin{eqnarray*} ZC_{1}^{*}(T^{*})-ZC_{1}^{*}(T^{(1)})&=& 2(2+d_{u_{2}}-d_{u_{1}}-d_{v_{1}}-d_{v_{2}})\\ &\leq&2(2+d_{u_{1}}-d_{u_{2}}-d_{u_{1}}-d_{u_{2}})=4(1-d_{u_{1}})<0, \end{eqnarray*} again a contradiction.\\ \textit{Case 2.} The vertices $u$ and $v$ are not adjacent.\\ In this case we have: \begin{equation}\label{Eq117} ZC_{1}^{*}(T^{*})-ZC_{1}^{*}(T^{(1)})= 2(1-d_{u_{1}}+d_{u_{2}}+d_{u_{3}}-d_{v_{1}}-d_{v_{2}}-d_{v_{3}}). \end{equation} From the inequality $d_{u_{1}}+d_{u_{2}}+d_{u_{3}}\leq d_{v_{1}}+d_{v_{2}}+d_{v_{3}}$ and eq. \ref{Eq117} it follows that $$ZC_{1}^{*}(T^{*})-ZC_{1}^{*}(T^{(1)})\leq2(1-2d_{u_{1}})<0,$$ which contradicts the maximality of $ZC_{1}^{*}(T^{*})$.\\ In both cases, contradiction arises. Hence, $T^{*}$ contains at most one vertex of degree 3. \end{proof}
\begin{lem}\label{lem3} For $n\geq7$,
if the tree $T^{*}\in\mathbb{CT}_{n}$ has the maximum $ZC_{1}^{*}$ value then $T^{*}$ does not contain vertices of degree 2 and 3 simultaneously. \end{lem}
\begin{proof} Suppose to the contrary that $u,v\in V(T^{*})$ such that $d_{u}=2$ and $d_{v}=3$. Let $N(u)=\{u_{1},u_{2}\}$ and $N(v)=\{v_{1},v_{2},v_{3}\}$. Suppose that the unique path connecting the vertices $u$ and $v$ contains the vertices $u_{2},v_{3}$. Let $T^{(1)}$ be the tree obtained from $T^{*}$ by removing the edge $u_{1}u$ and adding the edge $u_{1}v$. There are two cases.\\
\textit{Case 1.} The vertices $u$ and $v$ are adjacent. That is, $u_{2}=v$ and $v_{3}=u$.\\ Note that the assumption $n\geq7$ implies that at least one of the vertices $u_{1}$, $v_{1}$, $v_{2}$ must be non-pendent and hence \begin{eqnarray*} ZC_{1}^{*}(T^{*})-ZC_{1}^{*}(T^{(1)})&=& 2(4-2d_{u_{1}}-d_{v_{1}}-d_{v_{2}})<0, \end{eqnarray*} which is a contradiction.\\
\textit{Case 2.} The vertices $u$ and $v$ are not adjacent.\\ By the virtue of Lemma \ref{lem1} and Lemma \ref{lem2}, we have $d_{v_{3}}=4$ and hence \begin{equation*} ZC_{1}^{*}(T^{*})-ZC_{1}^{*}(T^{(1)})= 2(2+d_{u_{2}}-2d_{u_{1}}-d_{v_{1}}-d_{v_{2}}-d_{v_{3}})<0, \end{equation*} which contradicts the maximality of $ZC_{1}^{*}(T^{*})$. This completes the proof. \end{proof}
\begin{lem}\label{lem4} For $n\geq7$, let the tree $T^{*}\in\mathbb{CT}_{n}$ has the maximum $ZC_{1}^{*}$ value. If $T^{*}$ contains a vertex $u$ of degree 3 then $u$ has two pendent neighbors. \end{lem}
\begin{proof} Contrarily, suppose that $u$ has at least two non-pendent neighbors. Let $N(u)=\{u_{1},u_{2},u_{3}\}$. Without loss of generality, we may assume that $u_{1},u_{2}$ are non-pendent. Then, due to Lemmas \ref{lem1} - \ref{lem3}, both the vertices $u_{1},u_{2}$ have degree 4 and there must exist a vertex $v\in V(T^{*})$ of degree 4 such that three neighbors of $v$ are pendent. Let $w$ be a pendent neighbor of $v$ and suppose that $T^{(1)}$ is the tree obtained from $T^{*}$ by removing the edge $vw$ and adding the edge $uw$.
\textit{Case 1.} The vertex $v$ doest not belong to the set $N(u)$.\\ Let $v_{1}$ be the unique non-pendent neighbor of $v$. Then, simple calculations yield: \[ZC_{1}^{*}(T^{*})-ZC_{1}^{*}(T^{(1)})= 2(d_{v_{1}}-d_{u_{3}}-6)<0,\] which is a contradiction.
\textit{Case 2.} The vertex $v$ belongs to the set $N(u)$.\\ Without loss of generality, we may assume that $v=u_{1}$. Then, we again have a contradiction as follows: \[ZC_{1}^{*}(T^{*})-ZC_{1}^{*}(T^{(1)})= -2d_{u_{3}}-4<0.\] This completes the proof. \end{proof}
For $n\geq9$ and $n\equiv0$ (mod 3), let $\mathbb{CT}_{n}^{(0)}$ be the collection of those $n$-vertex chemical trees in which one vertex has degree 2 whose one neighbor is pendent, and every other vertex has degree 1 or 4. For $n\geq7$ and $n\equiv1$ (mod 3), let $\mathbb{CT}_{n}^{(1)}$ be the collection of those $n$-vertex chemical trees in which one vertex has degree 3 whose two neighbors are pendent, and every other vertex has degree 4 or 1. For $n\geq8$ and $n\equiv2$ (mod 3), let $\mathbb{CT}_{n}^{(2)}$ be the collection of those $n$-vertex chemical trees which consists of only vertices of degree 1 and 4. Let us take $\mathbb{CT}_{n}^{*}=\mathbb{CT}_{n}^{(0)}\cup\mathbb{CT}_{n}^{(1)}\cup\mathbb{CT}_{n}^{(2)}$.
\begin{thm}\label{thm-1111} For $n\geq7$, the tree $T^{*}\in\mathbb{CT}_{n}$ has maximum $ZC_{1}^{*}$ value if and only if $T^{*}\in\mathbb{CT}_{n}^{*}$. \end{thm}
\begin{proof} The result follows from Lemmas \ref{lem1} - \ref{lem4} and definition of the collection $\mathbb{CT}_{n}^{*}$. \end{proof}
It is interesting to see that the $n$-vertex chemical trees having maximum $M_{2}$ value \cite{vuki14} and maximum $ZC_{1}^{*}$ value among the collection $\mathbb{CT}_{n}$ are same for $n\geq7$ (it is not true for general trees). Denote by $x_{a,b}$ the number of edges in the graph $G$ connecting the vertices of degrees $a$ and $b$. Let $n_{a}$ be the number of vertices of degree $a$ in the graph $G$. The following system of equations holds for all trees of the collection $\mathbb{CT}_{n}$. \begin{equation}\label{Eq555} \sum_{i=1}^{4}n_{i}=n \end{equation} \begin{equation}\label{Eq556} \sum_{i=1}^{4}i\times n_{i}=2(n-1) \end{equation} \begin{equation}\label{Eq557} \sum_{ \substack{ 1\leq i\leq 4, \\
i\neq j}}x_{j,i}+2x_{j,j}=j\times n_{j} \text{ \ } ; \text{ \ \ \ $j=1,2,3,4$.} \end{equation} Eliminating $n_{1}$ from eq. (\ref{Eq555}) and eq. (\ref{Eq556}): \begin{equation}\label{Eq558} \sum_{i=2}^{4}(i-1)\times n_{i}=n-2 \end{equation}
\begin{cor}\label{cor-1111} For $n\geq7$, let $T$ be any member of $\mathbb{CT}_{n}$. Then \[ZC_{1}^{*}(T)\leq \begin{cases} 10(n-4) & \text{if $n\equiv0$ (mod 3) or $n\equiv1$ (mod 3),} \\ 2(5n-19) & \text{otherwise.} \end{cases}\] The equality sign in the first inequality holds if and only if $n\equiv0$ (mod 3) and $T\in\mathbb{CT}_{n}^{(0)}$, or $n\equiv1$ (mod 3) and $T\in\mathbb{CT}_{n}^{(1)}$. The equality sign in the second inequality holds if and only if $n\equiv2$ (mod 3) and $T\in\mathbb{CT}_{n}^{(2)}$. \end{cor}
\begin{proof} Let $T^{*}\in\mathbb{CT}_{n}^{*}$ and $T\not\in\mathbb{CT}_{n}^{*}$. Then, by virtue of Theorem \ref{thm-1111}, we have $ZC_{1}^{*}(T)< ZC_{1}^{*}(T^{*})$. Hence, to obtain the desired result, it is enough to calculate $ZC_{1}^{*}$ value of $T^{*}$. If $n\equiv0$ (mod 3) then $n=3k$ where $k\geq3$. From eq. (\ref{Eq558}), it follows that $n_{2}+2n_{3}\equiv1$ (mod 3) which implies that $n_{2}=1,n_{3}=0$. Hence, from eq. (\ref{Eq555}) and eq. (\ref{Eq556}) we have $n_{1}=2k,n_{4}=k-1$. Also, note that $x_{1,2}=x_{2,4}=1$. Now, from system (\ref{Eq557}) we have $x_{1,4}=2k-1,x_{4,4}=k-2$ and hence $ZC_{1}^{*}(T^{*})=10(n-4)$. In a similar way,
we have: \[ZC_{1}^{*}(T^{*})= \begin{cases} 10(n-4) & \text{if $n\equiv1$ (mod 3),} \\ 2(5n-19) & \text{if $n\equiv2$ (mod 3).} \end{cases}\] From the definitions of the collections $\mathbb{CT}_{n}^{(0)},\mathbb{CT}_{n}^{(1)},\mathbb{CT}_{n}^{(2)},\mathbb{CT}_{n}^{*}$ and from Theorem \ref{thm-1111}, the desired result follows.
\end{proof}
For $n=5,6,$ the trees having maximum $ZC_{1}^{*}$ value among all the members of $\mathbb{CT}_{n}$ can be easily identified from Figure \ref{f1}, using Table \ref{table111}.
\section{Concluding Remarks}
We have fully characterized the extremal chemical trees with fixed number of vertices for the topological index $ZC_{1}^{*}$, which is occurred in an approximate formula for total $\pi$-electron energy, communicated in 1972.
The first Zagreb index $M_{1}$ can also be rewritten as \begin{equation}\label{Eq1115} M_{1}(G)=\sum_{uv\in E(G)}(d_{u}+d_{v}). \end{equation} So, if we replace vertex degree by vertex connection number in eq. (\ref{Eq1115}) we get the topological index $ZC_{1}^{*}$. Hence, it is natural to consider the following connection-number versions of the Zagreb indices $\sum_{v\in V(G)}d_{v}^{2}$ and $\sum_{uv\in E(G)}d_{u}d_{v}$: $$ZC_{1}(G)=\sum_{v\in V(G)}\tau_{v}^{2} \ \ \ \text{and} \ \ \ ZC_{2}(G)=\sum_{uv\in E(G)}\tau_{u}\tau_{v} \ ,$$ respectively. We call $ZC_{1}$ as the first Zagreb connection index and $ZC_{2}$ as the second Zagreb connection index. Moreover, the following bond incident connection-number (BIC) indices may be considered as a generalization of the Zagreb connection indices: \begin{equation}\label{z} BIC(G)=\displaystyle\sum_{0\leq a\leq b\leq n-2}y_{a,b}(G).\phi_{a,b} \end{equation} where $y_{a,b}(G)$ is the number of edges in $G$ connecting the vertices with connection numbers $a$ and $b$, and $\phi_{a,b}$ is a non-negative real valued (symmetric) function which depends on $a$ and $b$.
\end{document} |
\begin{document}
\title[The range of a Ces\`aro-like operator acting on $H^\infty$ ] {Carleson measures and the range of a Ces\`aro-like operator acting on $H^\infty$}
\author{Guanlong Bao, Fangmei Sun and Hasi Wulan} \address{Guanlong Bao\\ Department of Mathematics\\
Shantou University\\
Shantou, Guangdong 515063, China} \email{glbao@stu.edu.cn}
\address{Fangmei Sun\\ Department of Mathematics\\
Shantou University\\
Shantou, Guangdong 515063, China} \email{18fmsun@stu.edu.cn}
\address{Hasi Wulan\\ Department of Mathematics\\
Shantou University\\
Shantou, Guangdong 515063, China} \email{wulan@stu.edu.cn}
\thanks{The work was supported by NNSF of China (No. 11720101003) and NSF of Guangdong Province (No. 2022A1515012117).} \subjclass[2010]{47B38, 30H05, 30H25, 30H35} \keywords{Ces\`aro-like operator, Carleson measure, $H^\infty$, $BMOA$}
\begin{abstract} In this paper, by describing characterizations of Carleson type measures on $[0,1)$, we determine the range of a Ces\`aro-like operator acting on $H^\infty$. A special case of our result gives an answer to a question posed by P. Galanopoulos, D. Girela and N. Merch\'an recently.
\end{abstract}
\maketitle
\section{Introduction}
Let $\D$ be the open unit disk in the complex plane $\c$. Denote by $H(\D)$ the space of functions analytic in $\D$. For $f(z)=\sum_{n=0}^\infty a_nz^n$ in $H(\D)$, the Ces\`aro operator $\C$ is defined by
$$ \C (f)(z)=\sum_{n=0}^\infty\left(\f{1}{n+1}\sum_{k=0}^n a_k\right)z^n, \quad z\in\D. $$ See \cite{BWY, DS, EX, M, S1, S2} for the investigation of the Ces\`aro operator acting on some analytic function spaces.
Recently, P. Galanopoulos, D. Girela and N. Merch\'an \cite{GGM} introduced a Ces\`aro-like operator $\Cu$ on $H(\D)$. For nonnegative integer $n$, let $\mu_n$ be the moment of order $n$ of a finite positive Borel measure $\mu$ on $[0, 1)$; that is, $$ \mu_n=\int_{[0, 1)} t^{n}d\mu(t). $$ For $f(z)=\sum_{n=0}^\infty a_nz^n$ belonging to $H(\D)$, the Ces\`aro-like operator $\Cu$ is defined by $$ \Cu (f)(z)=\sum^\infty_{n=0}\left(\mu_n\sum^n_{k=0}a_k\right)z^n, \quad z\in\D. $$ If $d\mu(t)=dt$, then $\Cu=\C$. In \cite{GGM}, the authors studied the action of $\Cu$ on distinct spaces of analytic functions.
We also need to recall some function spaces. For $0<p<\infty$, $H^p$ denotes the classical Hardy space \cite{D} of those functions $f\in H(\D)$ for which $$ \sup_{0<r<1} M_p(r, f)<\infty, $$ where $$
M_p(r, f)= \left(\f{1}{2\pi}\int_0^{2\pi}|f(re^{i\theta})|^p d\theta \right)^{1/p}. $$ As usual, denote by $H^\infty$ the space of bounded analytic functions in $\D$. It is well known that $H^\infty$ is a proper subset of the Bloch space $\B$ which consists of those functions $f\in H(\D)$ satisfying $$
\|f\|_\B=\sup_{z\in \D}(1-|z|^2)|f'(z)|<\infty. $$
Denote by $\text{Aut}(\D)$ the group of M\"obius maps of $\D$, namely, $$ \text{Aut}(\D)=\{e^{i\theta}\sigma_a:\ \ a\in \D \ \text{and} \ \theta \ \ \text{is real}\}, $$ where $$ \sigma_a(z)=\frac{a-z}{1-\overline{a}z}, \qquad z\in \D. $$
In 1995 R. Aulaskari, J. Xiao and R. Zhao \cite{AXZ} introduced $\qp$ spaces. For $0\leq p<\infty$, a function $f$ analytic in $\D$ belongs to $\qp$ if $$
\|f\|_{\mathcal{Q}_p}^2=\sup_{w\in \D} \int_\D |f'(z)|^2(1-|\sigma_w(z)|^2)^p dA(z)<\infty, $$ where $dA$ is the area measure on $\c$ normalized so that $A(\D)=1$. $\qp$ spaces are M\"obius invariant in the sense that $$
\|f\|_{\qp}=\|f\circ \phi\|_{\qp} $$ for every $f\in \qp$ and $\phi \in \text{Aut}(\D)$. It was shown in \cite{X1} that $\Q_2$ coincides with the Bloch space $\B$. This result was extended in \cite{AL} by showing that $\qp=\B$ for all $1<p<\infty$. The space $\Q_1$ coincides with $BMOA$, the set of analytic functions in $\D$ with boundary values of bounded mean oscillation (see \cite{B, Gir}). The space $\Q_0$ is the Dirichlet space $\mathcal D$. For $0<p<1$, the space $\qp$ is a proper subset of $BMOA$ and has many interesting properties. See J. Xiao's monographs \cite{X2, X3} for the theory of $\qp$ spaces.
For $1\leq p<\infty$ and $0<\alpha\leq 1$, the mean Lipschitz space $\Lambda^p_\alpha$ is the set of those functions $f\in H(\D)$ with a non-tangential limit almost everywhere such that $\omega_p(t, f)=O(t^\alpha)$ as $t\to 0$. Here $\omega_p(\cdot, f)$ is the integral modulus of continuity of order $p$ of the function $f(e^{i\theta})$. It is well known (cf.\cite[Chapter 5]{D}) that $\Lambda^p_\alpha$ is a subset of $H^p$ and $\Lambda^p_\alpha$ consists of those functions $f\in H(\D)$ satisfying $$
\|f\|_{\Lambda^p_\alpha}=\sup_{0<r<1}(1-r)^{1-\alpha}M_p(r, f')<\infty. $$ Among these spaces, the spaces $\Lambda^p_{1/p}$ are of special interest. $\Lambda^p_{1/p}$ spaces increase with $p\in (1, \infty)$ in the sense of inclusion and they are contained in $BMOA$ (cf. \cite{BSS}). By Theorem 1.4 in \cite{ASX}, $\Lambda^p_{1/p}\subseteq \Q_q$ when $1\leq p<2/(1-q)$ and $0<q<1$. In particular, $\Lambda^2_{1/2}\subseteq \Q_q \subseteq \B$ for all $0<q<\infty$.
Given an arc $I$ of the unit circle $\T$ with arclength $|I|$ (normalized such that $|\T|=1$), the
Carleson box $S(I)$ is given by $$
S(I)=\{r\zeta \in \D: 1-|I|<r<1, \ \zeta\in I\}. $$
For $0<s<\infty$, a positive Borel measure $\nu$ on $\D$ is said to be an $s$-Carleson measure if $$
\sup_{I\subseteq\T}\frac{\nu(S(I))}{|I|^s}<\infty. $$ If $\nu$ is a $1$-Carleson measure, we write that $\nu$ is a Carleson measure characterizing $H^p\subseteq L^p(d\nu)$ (cf. \cite{D}). A positive Borel measure $\mu$ on [0, 1) can be seen as a Borel measure on $\D$ by identifying it with the measure $\tilde{\mu}$ defined by $$ \tilde{\mu}(E)=\mu(E \cap [0, 1)), $$ for any Borel subset $E$ of $\D$. Thus $\mu$ is an $s$-Carleson measure on $[0,1)$ if there is a positive constant $C$ such that $$ \mu([t, 1)) \leq C (1-t)^s $$ for all $t\in [0, 1)$. We refer to \cite{BYZ} for the investigation of this kind of measures associated with Hankel measures.
It is known that the Ces\`aro operator $\C$ is bounded on $H^p$ for all $0<p<\infty$ (cf. \cite{M, S1, S2}) but this is not true on $H^\infty$. In fact, N. Danikas and A. Siskakis \cite{DS} gave that $\mathcal{C }(H^\infty)\nsubseteq H^\infty$ but $\mathcal{C }(H^\infty)\subseteq BMOA$. Later M. Ess\'en and J. Xiao \cite{EX} proved that $\mathcal{C }(H^\infty)\subsetneqq \qp$ \ for \ $0<p<1$. Recently, the relation between $\mathcal{C }(H^\infty)$ and a class of M\"obius invariant function spaces was considered in \cite{BWY}.
It is quite natural to study $\Cu(H^\infty)$. In \cite{GGM} the authors characterized positive Borel measures $\mu$ such that $\Cu(H^\infty)\subseteq H^\infty$ and proved that $\Cu(H^\infty)\subseteq \B$ if and only if $\mu$ is a Carleson measure. Moreover, they showed that if $\Cu(H^\infty)\subseteq BMOA$, then $\mu$ is a Carleson measure. In \cite[p. 20]{GGM}, the authors asked whether or not $\mu$ being a Carleson measure implies that $\Cu(H^\infty)\subseteq BMOA$. In this paper, by giving some descriptions of $s$-Carleson measures on $[0,1)$, for $0<p<2$, we show that $\Cu(H^\infty)\subseteq \qp$ if and only if $\mu$ is a Carleson measure, which giving an affirmative answer to their question. We also consider another Ces\`aro-like operator $\C_{\mu, s}$ and describe the embedding $\Cus(H^\infty)\subseteq X$ in terms of $s$-Carleson measures, where $X$ is between $\Lambda^p_{1/p}$ and $\B$ for $\max\{1, 1/s\}<p<\infty$.
Throughout this paper, the symbol $A\thickapprox B$ means that $A\lesssim B\lesssim A$. We say that $A\lesssim B$ if there exists a positive constant $C$ such that $A\leq CB$.
\section{Positive Borel measures on [0, 1) as Carleson type measures }
In this section, we give some characterizations of positive Borel measures on [0, 1) as Carleson type measures.
The following description of Carleson type measures (cf. \cite{Bla} ) is well known. \begin{otherl}\label{S-CM} Suppose $s>0$, $t>0$ and $\mu$ is a positive Borel measure on $\D$. Then $\mu$ is an $s$-Carleson measure if and only if \begin{equation}\label{sCMformula}
\sup_{a\in \D}\int_{\D} \frac{(1-|a|^2)^t}{|1-\overline{a}w|^{s+t}}d\mu(w)<\infty. \end{equation} \end{otherl}
For Carleson type measures on [0, 1), we can obtain some descriptions that are different from Lemma \ref{S-CM}. Now we give the first main result in this section.
\begin{prop}\label{newCM1} Suppose $0<t<\infty$, $0\leq r<s<\infty$ and $\mu$ is a finite positive Borel measure on $[0,1)$. Then the following conditions are equivalent: \begin{enumerate}
\item [(i)] $\mu$ is an $s$-Carleson measure;
\item [(ii)] \begin{equation}\label{1formulaCM}
\sup_{a\in\D}\int_{[0,1)}\frac{(1-|a|)^t}{(1-x)^{r}(1-|a|x)^{s+t-r}}d\mu(x)<\infty; \end{equation}
\item [(iii)] \begin{equation}\label{2formulaCM}
\sup_{a\in\D}\int_{[0,1)}\frac{(1-|a|)^t}{(1-x)^{r}|1-ax|^{s+t-r}}d\mu(x)<\infty. \end{equation} \end{enumerate} \end{prop}
\begin{proof}
$(i)\Rightarrow (ii)$. Let $\mu$ be an $s$-Carleson measure. Fix $a\in \D$ with $|a|\leq 1/2$. If $r=0$, the desired result holds. For $0<r<s$, using a well-known formula about the distribution function(\cite[p.20 ]{Gar}), we get \begin{align}\label{bu1}
&\int_{[0,1)}\frac{(1-|a|)^t}{(1-x)^{r}(1-|a|x)^{s+t-r}}d\mu(x)\nonumber \\ \thickapprox & \int_{[0,1)}\left(\frac{1}{1-x}\right)^rd\mu(x) \nonumber \\ \thickapprox & r \int_0^\infty \lambda^{r-1} \mu(\{x\in [0, 1): 1-\frac{1}{\lambda}<x \})d\lambda \nonumber \\ \lesssim & \int_0^1 \lambda^{r-1} \mu([0, 1))d\lambda + \int_1^\infty \lambda^{r-1} \mu([1-\frac{1}{\lambda}, 1))d\lambda \nonumber \\
\lesssim & 1+ \int_1^\infty \lambda^{r-s-1} d\lambda \lesssim1. \end{align}
Fix $a\in\D$ with $|a|>1/2$ and let \begin{align*}
S_n(a)=\{x\in[0,1): 1-2^n(1-|a|)\leq x<1\}, \ \ n=1, 2, \cdots . \end{align*} Let $n_a$ be the minimal integer such that $1-2^{n_a}(1-a)\leq 0$. Then $S_n(a)=[0, 1)$ when $n\geq n_a$. If $ x\in S_1(a)$, then
\begin{equation}\label{301}
1-|a| \leq 1-|a|x.
\end{equation} Also, for $2\leq n\leq n_a$ and $x\in S_n(a)\backslash S_{n-1}(a)$, we have \begin{equation}\label{302}
1-|a|x \geq |a|-x \geq |a|-(1-2^{n-1}(1-|a|))=(2^{n-1}-1)(1-|a|). \end{equation} We write \begin{align*}
& \int_{[0,1)}\frac{(1-|a|)^t}{(1-x)^r(1-|a|x)^{s+t-r}}d\mu(x)\\
=&\int_{S_1(a)}\frac{(1-|a|)^t}{(1-x)^r(1-|a|x)^{s+t-r}}d\mu(x)\\
&+\sum^{n_a}_{n=2}\int_{S_n(a)\backslash S_{n-1}(a)}\frac{(1-|a|)^t}{(1-x)^r(1-|a|x)^{s+t-r}}d\mu(x)\\
=: &J_1(a)+J_2(a). \end{align*} If $r=0$, bearing in mind that (\ref{301}), (\ref{302}) and $\mu$ is an $s$-Carleson measure, it is easy to check that $J_i(a)\lesssim 1$ for $i=1, 2$. Now consider $0<t<\infty$ and $0< r<s<\infty$. Using (\ref{301}) and some estimates similar to (\ref{bu1}), we have \begin{align*}
J_1(a) \lesssim (1-|a|)^{r-s}\int_{S_1(a)}\left(\frac{1}{1-x}\right)^rd\mu(x) \lesssim 1. \end{align*} Note that (\ref{302}), $0<t<\infty$, $0< r<s<\infty$ and $\mu$ is an $s$-Carleson measure. Then {\small {\small \begin{align*}
&J_2(a)\\
\lesssim& \sum^{n_a}_{n=2}\frac{(1-|a|)^{r-s}}{2^{n(s+t-r)}}\int_{S_n(a)\backslash S_{n-1}(a)}\left(\frac{1}{1-x}\right)^rd\mu(x)\\
\lesssim& \sum^{n_a}_{n=2}\frac{(1-|a|)^{r-s}}{2^{n(s+t-r)}}\int_0^\infty\lambda^{r-1}\mu\big(\big\{x\in[1-2^n(1-|a|),1): 1-\frac{1}{\lambda}<x\big\}\big)d\lambda\\
\thickapprox& \sum^{n_a}_{n=2}\frac{(1-|a|)^{r-s}}{2^{n(s+t-r)}}\bigg(\int_0^{\frac{1}{2^n(1-|a|)}}\lambda^{r-1}\mu\big([1-2^n(1-|a|),1)\big)d\lambda\\
&+\int_{\frac{1}{2^n(1-|a|)}}^\infty\lambda^{r-1}\mu\big(\big[1-\frac{1}{\lambda},1\big)\big)d\lambda\bigg)\\
\lesssim& \sum^{n_a}_{n=2}\frac{(1-|a|)^{r-s}}{2^{n(s+t-r)}}\bigg(2^{ns}(1-|a|)^s\int_0^{\frac{1}{2^n(1-|a|)}}\lambda^{r-1}d\lambda
+\int_{\frac{1}{2^n(1-|a|)}}^\infty\lambda^{r-1-s}d\lambda\bigg)\\ \thickapprox& \sum^{n_a}_{n=2} \frac{1}{2^{tn}}<\infty.
\end{align*}}} Consequently, $$
\sup_{a\in \D}\int_{[0,1)}\frac{(1-|a|)^t}{(1-x)^r(1-|a|x)^{s+t-r}}d\mu(x)<\infty. $$
The implication of $(ii)\Rightarrow (iii)$ is clear.
$(iii)\Rightarrow (i)$. For $r\geq 0$, it is clear that \begin{align*}
\int_{[0,1)}\frac{(1-|a|)^t}{(1-x)^{r}|1-ax|^{s+t-r}}d\mu(x)\geq \int_{[0,1)}\frac{(1-|a|)^t}{|1-ax|^{s+t}}d\mu(x) \end{align*} for all $a\in \D$. Combining this with Lemma \ref{S-CM}, we see that if (\ref{2formulaCM}) holds, then $\mu$ is an $s$-Carleson measure. \end{proof}
\noindent {\bf Remark 1.}\ \ The condition $0\leq r<s<\infty$ in Proposition \ref{newCM1} can not be changed to $r\geq s>0$. For example, let $d\mu_1(x)=(1-x)^{s-1}dx$, $x\in [0, 1)$. Then $\mu_1$ is an $s$-Carleson measure but for $r\geq s>0$, \begin{align*}
&\sup_{a\in\D}\int_{[0,1)}\frac{(1-|a|)^t}{(1-x)^{r}|1-ax|^{s+t-r}}d\mu_1(x)\\ \geq & \int_0^1 (1-x)^{s-1-r}dx=+\infty. \end{align*}
\noindent {\bf Remark 2.}\ \ $\mu$ supported on $[0,1)$ is essential in Proposition \ref{newCM1}. For example, consider $0<t<1$, $0< r<s<1$ and $s=r+t$. Set $d\mu_2(w)=|f'(w)|^2(1-|w|^2)^sdA(w)$, $w\in \D$, where $f\in \Q_s\setminus \Q_t$. Note that for $0<p<\infty$ and $g\in H(\D)$, $|g'(w)|^2(1-|w|^2)^pdA(w)$ is a $p$-Carleson measure if and only if $g\in\Q_p$ (cf. \cite{X2}). Hence $d\mu_2$ is an $s$-Carleson measure. But \begin{align*}
&\sup_{a\in\D}\int_{\D}\frac{(1-|a|)^t}{(1-|w|)^{r}|1-a\overline{w}|^{s+t-r}}d\mu_2(w)\\
&=\sup_{a\in\D}\int_{\D}|f'(w)|^2\frac{(1-|a|)^t(1-|w|)^{s-r}}{|1-a\overline{w}|^{s+t-r}}dA(w)\\
&\thickapprox\sup_{a\in\D}\int_{\D}|f'(w)|^2(1-|\sigma_a(w)|^2)^tdA(w)=+\infty. \end{align*}
Before giving the other characterization of Carleson type measures on $[0,1)$, we need to recall some results.
The following result is Lemma 1 in \cite{Mer}, which generalizes Lemma 3.1 in \cite{GM} from $p=2$ to $1<p<\infty$.
\begin{otherl}\label{inter labda B} Let $f\in H(\D)$ with $f(z)=\sum^\infty_{n=0}a_n z^n$. Suppose $1<p<\infty$ and the sequence $\{a_n\}$ is a decreasing sequence of nonnegative numbers. If $X$ is a subspace of $H(\D)$ with $\Lambda^p_{1/p}\subseteq X\subseteq\B$, then $$
f\in X \iff a_n=O\left(\frac1n\right). $$ \end{otherl}
We recall a characterization of $s$-Carleson measure $\mu$ on [0, 1) as follows (cf. \cite[Theorem 2.1]{BW} or \cite[Proposition 1]{CGP}). \begin{otherp}\label{u n^s} Let $\mu$ be a finite positive Borel measure on [0, 1) and $s>0$. Then $\mu$ is an $s$-Carleson measure if and only if the sequence of moments $\{\mu_n\}_{n=0}^\infty$ satisfies $\sup_{n\geq 0} (1+n)^s \mu_n<\infty$. \end{otherp}
The following characterization of functions with nonnegative Taylor coefficients in $\Q_p$ is Theorem 2.3 in \cite{AGW}.
\begin{otherth}\label{Qp coeff}
Let $0<p<\infty$ and let $f(z)=\sum_{n=0}^\infty a_nz^n$ be an analytic function in $\D$ with $a_n\geq 0$ for all $n$. Then $f\in \Q_p$ if and only if $$ \sup_{0\leq r<1} \sum_{n=0}^\infty \f{(1-r)^p}{(n+1)^{p+1}}\left(\sum_{k=0}^n(k+1)a_{k+1}(n-k+1)^{p-1}r^{n-k}\right)^2<\infty. $$ \end{otherth}
We need the following well-known estimates (cf. \cite[Lemma 3.10]{Zhu}). \begin{otherl}\label{useful estimates}
Let $\beta$ be any real number. Then $$
\int^{2\pi}_0\frac{d\theta}{|1-ze^{-i\theta}|^{1+\beta}}\thickapprox \begin{cases}1 & \enspace \text{if} \ \ \beta<0,\\
\log\frac{2}{1-|z|^2} & \enspace \text{if} \ \ \beta=0,\\
\frac{1}{(1-|z|^2)^\beta} & \enspace \text{if}\ \ \beta>0,
\end{cases} $$ for all $z\in \D$. \end{otherl}
For $0<s<\infty$ and a finite positive Borel measure $\mu$ on $[0, 1)$, set $$ f_{\mu, s}(z)=\sum_{n=0}^\infty \frac{\Gamma(n+s)}{\Gamma(s)n!} \mu_n z^n, \ \ z\in \D. $$ Now we state the other main result in this section which is inspired by Lemma \ref{inter labda B} and Proposition \ref{u n^s}.
\begin{prop} \label{newCM2} Suppose $0<s<\infty$ and $\mu$ is a finite positive Borel measure on $[0, 1)$. Let $1<p<\infty$ and let $X$ be a subspace of $H(\D)$ with $\Lambda^p_{1/p}\subseteq X\subseteq\B$. Then $\mu$ is an $s$-Carleson measure if and only if $f_{\mu, s}\in X$. \end{prop}
\begin{proof}
Let $\mu$ be an $s$-Carleson measure. Clearly,
$$
f_{\mu, s}(z)=\int_{[0, 1)} \frac{1}{(1-tz)^s}d\mu(t)
$$
for any $z\in \D$.
For $p>1$, it follows from the Minkowski inequality and Lemma \ref{useful estimates} that \begin{align*}
M_p(r, f'_{\mu, s})\leq& s \left(\frac{1}{2\pi}\int_0^{2\pi} \left(\int_{[0, 1)}\frac{1}{|1-tre^{i\theta}|^{s+1}}d\mu(t)\right)^pd\theta\right)^{1/p}\\
\leq & s \int_{[0, 1)} \left( \frac{1}{2\pi}\int_0^{2\pi} \frac{1}{|1-tre^{i\theta}|^{(s+1)^p}}d\theta \right)^{1/p} d\mu(t)\\ \lesssim& \int_{[0, 1)} \frac{1}{(1-tr)^{s+1-\frac{1}{p}}} d\mu(t) \end{align*} for all $0<r<1$. Combining this with Proposition \ref{newCM1}, we get $f_{\mu, s}\in \Lambda^p_{1/p}$ and hence $f_{\mu, s}\in X$.
On the other hand, let $f_{\mu, s}\in X$. Then $f_{\mu, s}\in \Q_q$ with $q>1$. By the Stirling formula, $$ \frac{\Gamma(n+s)}{\Gamma(s)n!}\thickapprox (n+1)^{s-1} $$ for all nonnegative integers $n$. Consequently, by Theorem \ref{Qp coeff} we deduce {\small \begin{eqnarray*} \infty&>&\sum_{n=0}^\infty \f{(1-r)^q}{(n+1)^{q+1}}\left(\sum_{k=0}^n(k+2)^{s}\mu_{k+1}(n-k+1)^{q-1}r^{n-k}\right)^2\\ &\gtrsim& \sum_{n=0}^\infty \f{(1-r)^q}{(4n+1)^{q+1}}\left(\sum_{k=0}^{4n}(k+2)^{s}\mu_{k+1}(4n-k+1)^{q-1}r^{4n-k}\right)^2\\ &\gtrsim& \sum_{n=0}^\infty \f{(1-r)^q}{(4n+1)^{q+1}}\left(\sum_{k=n}^{2n}(k+2)^{s}\int_r^1 t^{k+1}d\mu(t) (4n-k+1)^{q-1}r^{4n-k}\right)^2\\ &\gtrsim& \mu^2([r, 1)) (1-r)^q\sum_{n=0}^\infty \f{r^{8n+2}}{(4n+1)^{q+1}}\left(\sum_{k=n}^{2n}(k+2)^{s}(4n-k+1)^{q-1} \right)^2\\ &\gtrsim&\mu^2([r, 1)) (1-r)^q \sum_{n=0}^\infty (4n+2)^{2s+q-1} r^{8n+2}\\ &\thickapprox& \f{\mu^2([r, 1))}{(1-r)^{2s}} \end{eqnarray*} } for all $r\in [0, 1)$ which yields that $\mu$ is an $s$-Carleson measure. The proof is complete. \end{proof}
\section{$\qp$ spaces and the range of $\Cu$ acting on $H^\infty$}
In this section, we characterize finite positive Borel measures $\mu$ on $[0,1)$ such that $\C_\mu(H^\infty)\subseteq \qp$ for $0<p<2$. Descriptions of Carleson measures in Proposition \ref{newCM1} play a key role in our proof.
The following lemma is from \cite{OF}.
\begin{otherl}\label{estiamtes} Suppose $s>-1$, $r>0$, $t>0$ with $r+t-s-2>0$. If $r$, $t<2+s$, then $$
\int_\D \frac{(1-|z|^2)^s}{|1-\overline{a}z|^r|1-\overline{b}z|^t}dA(z)\lesssim \frac{1}{|1-\overline{a}b|^{r+t-s-2}} $$ for all $a$, $b\in \D$. If $t<2+s<r$, then $$
\int_\D \frac{(1-|z|^2)^s}{|1-\overline{a}z|^r|1-\overline{b}z|^t}dA(z)\lesssim \frac{(1-|a|^2)^{2+s-r}}{|1-\overline{a}b|^{t}} $$ for all $a$, $b\in \D$. \end{otherl}
We give our result as follows. \begin{theor}\label{1main} Suppose $0<p<2$ and $\mu$ is a finite positive Borel measure on $[0,1)$. Then $\C_\mu(H^\infty)\subseteq \qp$ if and only if $\mu$ is a Carleson measure. \end{theor}
\begin{proof} Suppose $\C_\mu(H^\infty)\subseteq \qp$. Then $\C_\mu(H^\infty)$ is a subset of the Bloch space. By \cite[Theorem 5]{GGM}, $\mu$ is a Carleson measure.
Conversely, suppose $\mu$ is a Carleson measure and $f\in H^\infty$. Then $f$ is also in the Bloch space $\B$. From Proposition 1 in \cite{GGM}, $$ \C_{\mu}(f)(z)=\int_{[0, 1)} \frac{f(tz)}{1-tz}d\mu(t), \ \ z\in \D. $$ Hence for any $z\in \D$, \begin{align}\label{31}
&\|\Cu (f)\|_{\qp} \nonumber \\
\lesssim & \sup_{a\in\D} \left(\int_{\D}\left(\int_{[0,1)}\frac{|tf'(tz)|}{|1-tz|}d\mu(t)\right)^2(1-|\sigma_a(z)|^2)^p dA(z)\right)^{\frac12} \nonumber \\
& +\sup_{a\in\D}\left(\int_{\D}\left( \int_{[0,1)}\frac{|tf(tz)|}{|1-tz|^2}d\mu(t)\right)^2(1-|\sigma_a(z)|^2)^p dA(z)\right)^{\frac12} \nonumber \\
\lesssim & \|f\|_\B \sup_{a\in\D} \left(\int_{\D}\left(\int_{[0,1)}\frac{1}{(1-|tz|)|1-tz|}d\mu(t)\right)^2(1-|\sigma_a(z)|^2)^p dA(z)\right)^{\frac12} \nonumber \\
& +\|f\|_{H^\infty}\sup_{a\in\D}\left(\int_{\D}\left( \int_{[0,1)}\frac{1}{|1-tz|^2}d\mu(t)\right)^2(1-|\sigma_a(z)|^2)^p dA(z)\right)^{\frac12}.\nonumber \\ \end{align} Let $c$ be a positive constant such that $2c<\min\{2-p, p\}$. Then \begin{equation}\label{32}
(1-|tz|)^2\geq (1-t)^{2-2c} (1-|z|)^{2c} \end{equation}
for all $t\in [0, 1)$ and all $z\in \D$. By the Minkowski inequality, (\ref{32}), Lemma \ref{estiamtes} and Proposition \ref{newCM1}, we get \begin{align}\label{33}
&\sup_{a\in\D} \left(\int_{\D}\left(\int_{[0,1)}\frac{1}{(1-|tz|)|1-tz|}d\mu(t)\right)^2(1-|\sigma_a(z)|^2)^p dA(z)\right)^{\frac12} \nonumber \\
\leq&\sup_{a\in\D} \int_{[0,1)} \left( \int_{\D} \frac{1}{(1-|tz|)^2|1-tz|^2} (1-|\sigma_a(z)|^2)^p dA(z) \right)^{\frac12} d\mu(t)\nonumber \\
\lesssim &\sup_{a\in\D}(1-|a|^2)^{\frac p2}\int_{[0,1)}\frac{1}{(1-t)^{1-c}}d\mu(t)\big(\int_{\D}\frac{(1-|z|^2)^{p-2c}}{|1-tz|^2|1-\bar{a}z|^{2p}} dA(z)\big)^{\frac12} \nonumber \\
\lesssim &\sup_{a\in\D}\int_{[0,1)}\frac{(1-|a|^2)^{\frac p2}}{(1-t)^{1-c}|1-ta|^{\frac{p}{2}+c}}d\mu(t)<\infty. \end{align} Similarly, it follows from Lemma \ref{estiamtes} and Proposition \ref{newCM1} that \begin{align}\label{34}
&\sup_{a\in\D}\left(\int_{\D}\left( \int_{[0,1)}\frac{1}{|1-tz|^2}d\mu(t)\right)^2(1-|\sigma_a(z)|^2)^p dA(z)\right)^{\frac12} \nonumber \\
\leq & \sup_{a\in\D} \int_{[0,1)} \left( \int_{\D} \frac{1}{|1-tz|^4} (1-|\sigma_a(z)|^2)^p dA(z)\right)^{\frac12} d\mu(t)¡¡\nonumber¡¡\\
\lesssim & \sup_{a\in\D}\int_{[0,1)}\frac{(1-|a|^2)^{\frac p2}}{(1-t^2)^{1-\frac p2}|1-at|^p}d\mu(t)<\infty. \end{align} From (\ref{31}), (\ref{33}) and (\ref{34}), we get that $\Cu (f)\in \qp$. The proof is complete, \end{proof} \noindent {\bf Remark 3.}\ \ Set $d\mu_0(x)=dx$ on [0, 1). Then $d\mu_0$ is a Carleson measure and $\C_{\mu_0}(1)(z)=\frac{1}{z}\log\f{1}{1-z}$. Clearly, the function $\C_{\mu_0}(1)$ is not in the Dirichlet space. Thus Theorem \ref{1main} does not hold when $p=0$.
Note that $\qp=\B$ for any $p>1$. Theorem \ref{1main} generalizes Theorem 5 in \cite{GGM} from the Bloch space $\B$ to all $\qp$ spaces. For $p=1$, Theorem \ref{1main} gives an answer to a question raised in \cite[p. 20]{GGM}. The proof given here highlights the role of Proposition \ref{newCM1}. In the next section, we give a more general result where an alternative proof of Theorem \ref{1main} will be provided.
\section{$s$-Carleson measures and the range of another Ces\`aro-like operator acting on $H^\infty$ }
It is also natural to consider how the characterization of $s$-Carleson measures in Proposition \ref{newCM2} can play a role in the investigation of the range of Ces\`aro-like operators acting on $H^\infty$. We consider this topic by another kind of Ces\`aro-like operators.
Suppose $0<s<\infty$ and $\mu$ is a finite positive Borel measure on $[0,1)$. For $f(z)=\sum_{n=0}^\infty a_nz^n$ in $H(\D)$, we define $$ \C_{\mu, s} (f)(z)=\sum^\infty_{n=0}\left(\mu_n\sum^n_{k=0}\frac{\Gamma(n-k+s)}{\Gamma(s)(n-k)!}a_k\right)z^n, \quad z\in\D. $$
Clearly, $\C_{\mu, 1}$ is equal to $\C_{\mu}$.
\begin{limma}\label{intergera repre} Suppose $0<s<\infty$ and $\mu$ is a finite positive Borel measure on $[0,1)$. Then $$ \C_{\mu, s} (f)(z)=\int_{[0,1)}\frac{f(tz)}{(1-tz)^s}d\mu(t) $$ for $f\in H(\D)$. \end{limma} \begin{proof} The proof follows from a simple calculation with power series. We omit it. \end{proof}
We have the following result.
\begin{theor}\label{2main} Suppose $0<s<\infty$ and $\mu$ is a finite positive Borel measure on $[0,1)$. Let $\max\{1, \f{1}{s}\}<p<\infty$ and $X$ is a subspace of $H(\D)$ with $\Lambda^p_{1/p}\subseteq X\subseteq\B$. Then $\C_{\mu, s}(H^\infty)\subseteq X$ if and only if $\mu$ is an $s$-Carleson measure. \end{theor} \begin{proof}
Let $\C_{\mu, s}(H^\infty)\subseteq X$. Then $\C_{\mu, s}(1)\in X$; that is, $f_{\mu, s}\in X$. It follows from Proposition \ref{newCM2} that $\mu$ is an $s$-Carleson measure.
On the other hand, let $\mu$ be an $s$-Carleson measure and $f\in H^\infty$. By Lemma \ref{intergera repre}, we see \begin{align*}
\Cus (f)'(z)=\int_{[0,1)}\frac{tf'(tz)}{(1-tz)^s}d\mu(t)+ \int_{[0,1)}\frac{stf(tz)}{(1-tz)^{s+1}}d\mu(t), \quad z\in\D. \end{align*} Then \begin{align}\label{41}
&\sup_{0<r<1}(1-r)^{1-\frac1p}\left(\f{1}{2\pi}\int_0^{2\pi}|\Cus (f)'(re^{i\theta})|^p d\theta\right)^{\frac1p}\nonumber \\
\lesssim &\|f\|_\B \sup_{0<r<1}(1-r)^{1-\frac1p} \left(\f{1}{2\pi}\int_0^{2\pi}\left(\int_{[0,1)} \frac{1}{|1-tre^{i\theta}|^{s}(1-tr)} d\mu(t)\right)^p d\theta\right)^{\frac1p} \nonumber \\
&+\|f\|_{H^\infty}\sup_{0<r<1}(1-r)^{1-\frac1p} \left(\f{1}{2\pi}\int_0^{2\pi}\left(\int_{[0,1)} \frac{1}{|1-tre^{i\theta}|^{s+1}} d\mu(t)\right)^p d\theta\right)^{\frac1p}. \nonumber \\ \end{align} Note that $ps>1$. By the Minkowski inequality, Lemma \ref{useful estimates} and Lemma \ref{S-CM}, we deduce \begin{align}\label{42}
&\sup_{0<r<1}(1-r)^{1-\frac1p} \left(\f{1}{2\pi}\int_0^{2\pi}\left(\int_{[0,1)} \frac{1}{|1-tre^{i\theta}|^{s}(1-tr)} d\mu(t)\right)^p d\theta\right)^{\frac1p} \nonumber \\
\leq & \sup_{0<r<1}(1-r)^{1-\frac1p} \int_{[0,1)} \left(\f{1}{2\pi}\int_0^{2\pi} \frac{1}{|1-tre^{i\theta}|^{sp}(1-tr)^p} d\theta\right)^{\frac1p} d\mu(t) \nonumber \\ \lesssim & \sup_{0<r<1}(1-r)^{1-\frac1p} \int_{[0,1)} \f{1}{(1-tr)^{s+1-\frac1p}} d\mu(t)<\infty, \nonumber \\ \end{align} and \begin{align}\label{43}
&\sup_{0<r<1}(1-r)^{1-\frac1p} \left(\f{1}{2\pi}\int_0^{2\pi}\left(\int_{[0,1)} \frac{1}{|1-tre^{i\theta}|^{s+1}} d\mu(t)\right)^p d\theta\right)^{\frac1p}\nonumber \\
\lesssim & \sup_{0<r<1}(1-r)^{1-\frac1p} \int_{[0,1)} \left( \f{1}{2\pi}\int_0^{2\pi}\frac{1}{|1-tre^{i\theta}|^{(s+1)p} } d\theta \right)^{\frac1p} d\mu(t) \nonumber \\ \lesssim & \sup_{0<r<1}(1-r)^{1-\frac1p} \int_{[0,1)} \f{1}{(1-tr)^{s+1-\frac1p}} d\mu(t)<\infty. \end{align} From (\ref{41}), (\ref{42}) and (\ref{43}), $\Cus(f)\in \Lambda^p_{1/p}$. Note that $\Lambda^p_{1/p}\subseteq X$. The desired result follows. \end{proof}
\end{document} |
\begin{document}
\title{Strongly vertex-reinforced jump process on a complete graph} \author{Olivier Raimond} \address{(O. Raimond) Mod\'elisation al\'eatoire de l'Universit\'e Paris Nanterre (MODAL'X), 92000 Nanterre, France} \email{olivier.raimond@parisnanterre.fr} \author{Tuan-Minh Nguyen} \address{(T.M. Nguyen) School of Mathematics, Monash University, 3800 Victoria, Australia} \email{tuanminh.nguyen@monash.edu} \date{\today} \keywords{Vertex-reinforced jump processes; nonlinear reinforcement; random walks with memory; stochastic approximation; non convergence to unstable equilibria.} \subjclass[2010]{60J55, 60J75}
\begin{abstract} The aim of our work is to study vertex-reinforced jump processes with super-linear weight function $w(t)=t^{\alpha}$, for some $\alpha>1.$ On any complete graph $G=(V,E)$, we prove that there is one vertex $v\in V$ such that the total time spent at $v$ almost surely tends to infinity while the total time spent at the remaining vertices is bounded.\\
\textbf{Résumé.} Le but de notre travail est d'étudier les processus de sauts renforcés par sites par une fonction de poids sur-linéaire $w(t)= t^{\alpha}$, avec $\alpha>1$. Sur tout graphe complet $G = (V, E)$, on montre qu'il y a un sommet $v \in V$ tel que le temps total passé en $v$ tend presque sûrement vers l'infini tandis que le temps total passé dans les sommets restants est borné. \end{abstract}
\maketitle
\section{Introduction}\label{sec:introduction}
Let $G=(V,E)$ be a finite connected, undirected graph without loops, where $V=\{1,2,...,d\}$ and $E$ respectively stand for the set of vertices and the set of edges. We consider a continuous-time jump process $X$ on the vertices of $G$ such that the law of $X$ satisfies the following condition: \begin{enumerate} \item[i.] at time $t\le 0$, the local time at each vertex $v\in V$ has a positive initial value $\ell^{(v)}_0$, \item[ii.] at time $t>0$, given the $\sigma$-field $\mathcal{F}_t$ generated by $\{X_{s},s\le t\}$, the probability that there is a jump from $X_t$ during $(t,t+h]$ to a neighbour $v$ of $X_t$ (i.e. $\{v,X_t\}\in E$) is given by $$w\left(\ell^{(v)}_0+\int_0^t \mathbf{1}_{\{X_s=v\}}{\rm d} s \right)\cdot h+o(h)$$ as $h\to 0$, where $w:[0,\infty)\to(0,\infty)$ is a weight function. \end{enumerate}
For each vertex $v\in V$, we denote by $L(v,t)=\ell^{(v)}_0+\int_0^t\mathbf{1}_{\{X_s=v\}}{\rm d} s$ the local time at $v$ up to time $t$ and let $$Z_t=\left(\frac{L(1,t)}{\ell_0+t},\frac{L(2,t)}{\ell_0+t},... ,\frac{L(d,t)}{\ell_0+t}\right)$$ stand for the (normalized) occupation measure on $V$ at time $t$, where $\ell_0=\ell^{(1)}_0+\ell^{(2)}_0+\cdots +\ell^{(d)}_0$.
In our work, we consider the weight function $w(t)=t^{\alpha}$, for some $\alpha>0$. The jump process $X$ is called \textit{strongly vertex-reinforced} if $\alpha>1$, \textit{weakly vertex-reinforced} if $\alpha<1$ or \textit{linearly vertex-reinforced} if $\alpha=1$.
The model of discrete time edge-reinforced random walks (ERRW) was first studied by Coppersmith and Diaconis in their unpublished manuscripts \cite{Coppersmith86} and later the model of discrete time vertex-reinforced random walks (VRRW) was introduced by Pemantle in \cite{Pemantle88} and \cite{Pemantle92}. Several remarkable results about localization of ERRW and VRRW were obtained in \cite{Volkov01}, \cite{Tarres04}, \cite{Volkov06}, \cite{Benaim13} and \cite{Cotar2015}. Wendelin Werner then proposed a model in continuous time so-called vertex reinforced jump processes (VRJP) whose linear case was first investigated by Davis and Volkov in \cite{Davis02} and \cite{Davis04}. In particular, these authors showed in \cite{Davis04} that linearly VRJP on any finite graph with $d$ vertices is recurrent, i.e. all local times are almost surely unbounded and the normalized occupation measure process converges almost surely to an element in the interior of the $(d-1)$ dimensional standard unit simplex as time goes to infinity. In \cite{Sabot15}, Sabot and Tarr\`es also obtained the limiting distribution of the centred local times process for linearly VRJP on any finite graph and showed that linearly VRJP is actually a mixture of time-changed Markov jump processes. Many aspects of linearly VRJP as well as its relations to ERRW and the supersymmetric hyperbolic sigma model have been well studied in recent years (see, e.g. \cite{Collevecchio2009}, \cite{Basdevant2012}, \cite{Disertori14}, \cite{Merkl16}, \cite{Sabot15}, \cite{Sabot15b}, \cite{Sabot2019} \cite{Sabot17}, \cite{Zeng16}, and \cite{Lupu2018}).
The main aim of our paper is to prove that strongly VRJP on a complete graph $G=(V,E)$ almost surely have an infinite local time at some vertex $v$, while the local times at the remaining vertices remain bounded. The main technique of our proofs is based on the method of stochastic approximation (see, e.g. \cite{Brandiere96, Benaim96, Benaim97, Benaim99}). We organize the present paper as follows. In Section \ref{sec:outline}, our main Theorem and an outline of its proof are given. In Section \ref{sec:notation}, we give some preliminary notations as well as some results of stochastic calculus being used throughout the paper. We show in Section \ref{sec:Dyn} that the occupation measure process of strongly VRJP on a complete graph is an asymptotic pseudo-trajectory of a flow generated by a vector field. We then prove the convergence towards stable equilibria in Section \ref{sec:convergence} and the non convergence towards unstable equilibria in Section \ref{sec:nonCV}, which yields our above-mentioned main result.
\section{Main result and outline of proof}\label{sec:outline} The main result of our paper is the following theorem: \begin{theorem}\label{thm:mainresult} Assume that $X$ is a strongly VRJP in a complete graph with weight function $w(t)=t^{\alpha}$, for some $\alpha>1$. Then there almost surely exists a vertex such that its local time tends to infinity while the local times at the remaining vertices remain bounded. \end{theorem}
The main technique to prove this theorem is based on the method of stochastic approximation (see, e.g. \cite{Brandiere96, Benaim96, Benaim97, Benaim99}). The core idea of this method is to describe the asymptotic behaviour of stochastic processes (which are stochastic algorithms in the discrete setting) in terms of the behaviour of ordinary differential equations. When the sample path of a stochastic process is asymptotically close to the solution of an autonomous differential equation, it is reasonable to investigate the relation between the limiting set of this process and the set of equilibria of the associated differential equation.
Let us explain how we make use of this idea in the context of VRJP in a complete graph with super-linear weight function $w(t)=t^{\alpha}$, for some $\alpha>1$. We first make the time change: for $t>0$, set $\tilde{Z}_t=Z_{e^t-\ell_0}$ and $\tilde{X}_t=X_{e^t-\ell_0}$. The occupation measure $\tilde{Z}$ satisfies the following equation: $$\frac{{\rm d}\tilde{Z}^i_t}{{\rm d} t}=-\tilde{Z}^i_t+\mathbf{1}_{\{\tilde{X}_t=i\}}.$$ Let now $t$ be a large time. Then, for every fixed $T$, the process $(X_{t+s})_{s\in [0,T]}$ evolves almost like a Markov process with generator $A_t=A(L(\cdot,t))$ (with $A(\lambda \ell)=\lambda^\alpha A(\ell)$). It will be remarked in Section \ref{sec:Dyn} that this diffusion has a unique invariant probability $\pi_t=\pi(Z_t)$. Such properties will allow us to prove Theorem \ref{cvthrm} in which $\tilde{Z}$ is an asymptotic pseudo-trajectory of a semi-flow $\Phi$ generated by the vector field $F(z)=-z+\pi(z)$, i.e. for all $T>0$, the trajectory $(\tilde{Z}_{t+s}:\;s\in [0,T])$ is close as $t\to\infty$ to the trajectory of the semi-flow $(\Phi_s(\tilde{Z}_t):\;s\in [0,T])$.
In Section \ref{sec:convergence}, using Theorem \ref{cvthrm} with the fact that there is a strict Lyapounov function $H$ for the vector field $F$ (i.e. a function such that $\langle F(z),\nabla H(z)\rangle >0$ if and only if $F(z)\ne 0$), we will show that almost surely the limit set of $Z$ is a connected subset of $\mathcal{C}$, the set of equilibria of $F$ (i.e. the set of all $z$ such that $F(z)=0$). Combining with the fact that the set $\mathcal{C}$ is finite, this will prove Theorem \ref{thm:CVeq} stating the a.s. convergence of $Z$ towards an equilibrium. In Section \ref{sec:convergence}, after having remarked that the stable equilibria of $F$ are Dirac measures $\delta_i$, $i\in V$, we will prove Theorem \ref{thm:localization} asserting that a.s. on the event $Z$ converges to $\delta_i$, $X$ eventually localizes at $i$, i.e. $L(i,\infty)=\infty$ and $\sum_{j\ne i} L(j,\infty)<\infty$.
Finally in Section \ref{sec:nonCV} we will prove Theorem \ref{thm:nonCV_VRJP} wherein a.s. $Z$ does not converge towards an unstable equilibrium. In preparation for the proof of this theorem, we will demonstrate Theorem \ref{THM:nonCV} which is a general non convergence theorem for a class of finite variation c\`adl\`ag processes. To do so, we will follow (and correct) arguments from the proof of a theorem by Brandi\`ere and Duflo (see \cite{Brandiere96} or \cite{Duflo1996}), but use a new idea as follows. We will first show that, under additional assumptions, an asymptotic pseudo-trajectory converging towards an unstable equilibrium is attracted exponentially fast towards the unstable manifold of this equilibrium. This will allow the proof of the non convergence theorem to be reduced to the case where the unstable equilibrium has no stable direction. Theorem \ref{thm:nonCV_VRJP} will then permit to conclude the proof of Theorem \ref{thm:mainresult}.
\section{Preliminary notations and remarks}\label{sec:notation} Throughout this paper, we denote by $\Delta$ and $T\Delta$ respectively the $(d-1)$ dimensional standard unit simplex in $\mathbb{R}^d$ and its tangent space, which are defined by \begin{align*} &\Delta=\{z=(z_1,z_2,...,z_d)\in\mathbb{R}^d:z_1+z_2+\cdots +z_d=1, z_j\ge0, j=1,2,\cdots ,d \},\\ &T\Delta=\{z=(z_1,z_2,...,z_d)\in\mathbb{R}^d:z_1+z_2+\cdots +z_d=0\}. \end{align*}
Also, let $\|\cdot\|$ and $\left\langle\cdot,\cdot\right\rangle$ denote the Euclidean norm and the Euclidean scalar product in $\mathbb{R}^d$ respectively.
For a c\`adl\`ag process $Y=(Y_t)_{t\ge0}$, we denote by $Y_{t-}=\lim_{s\to t-}Y_t$ and $\Delta Y_t=Y_t-Y_{t-}$ respectively the left limit and the size of the jump of $Y$ at time $t$. Let $[Y]$ be as usual the \textit{quadratic variation} of the process $Y$. Note that, for a c\`adl\`ag finite variation process $Y$, we have $[Y]_t=\sum_{0<u\le t}(\Delta Y_u)^2$. In the next sections, we will use the following useful well-known results of stochastic calculus (see e.g. \cite{Jacod2003} and \cite{Protter}):
1. \textbf{Change of variables formula.} (see Theorem 31, p.~78 in \cite{Protter}) Let $A=(A^1_t,A^2_t,\dots,A^d_t)_{t\ge0}$ be a c\`adl\`ag finite variation process in $\mathbb{R}^d$ and let $f:\mathbb{R}^d\to \mathbb{R}$ be a $C^1$ function. Then for $ t\ge 0$, $$f(A_t)-f(A_0)=\sum_{i=1}^d\int_0^t \partial_i f(A_{u-}) {\rm d} A^i_u +\sum_{0<u\le t}\left(\Delta f(A_u)-\sum_{i=1}^d\partial _i f(A_{u-})\Delta A_u^i\right).$$
2. Let $M=(M_t)_{t\ge0}$ be a c\`adl\`ag locally square-integrable martingale with finite variation in $\mathbb{R}$. A well-known result is that if ${\mathsf E}[[M]_t]<\infty$ for all $t$, then $M$ is a true martingale (see e.g. Corollary 3, p.~73 in \cite{Protter}). The change of variable formula implies that $$M_t^2=M_0^2+\int_0^t 2M_{s-}{\rm d} M_s+[M]_t.$$ Let $\langle M\rangle$ denote the \textit{angle bracket} of $M$, i.e. the unique predictable non-decreasing process such that $M^2-\langle M\rangle$ is a local martingale. Note that $[M]-\langle M\rangle$ is also a local martingale.
Let $H$ be a locally bounded predictable process and denote by $H\cdot M$ the c\`adl\`ag locally square-integrable martingale with finite variation defined by $(H\cdot M)_t=\int_0^t H_{s} dM_s$. Recall the following rules: $$\langle H\cdot M\rangle_t=\int_0^t H^2_{s}{\rm d}\langle M\rangle_s \quad \text{ and }\quad [H\cdot M]_t=\int_0^t H^2_{s}{\rm d} [M]_s$$ (see Theorem 4.40, p.~48 and the statement 4.54, p.~55 in \cite{Jacod2003}). Recall also that $H\cdot M$ is a square integrable martingale if and only if for all $t>0$, ${\mathsf E}[\langle H\cdot M\rangle_t]<\infty$.
3. \textbf{Integration by part formula.} (see Corollary 2, p.~68 in \cite{Protter}) Let $X=(X)_{t\ge0}$ and $Y=(Y)_{t\ge0}$ be two c\`adl\`ag finite variation processes in $\mathbb{R}$. Then for $t\ge s\ge 0$, $$X_tY_t-X_sY_s=\int_s^t X_{u-}{\rm d} Y_u+\int_s^t Y_{u-}{\rm d} X_u+[X,Y]_t-[X,Y]_s,$$ where we recall that $[X,Y]$ is the \textit{covariation} of $X$ and $Y$, computed as $[X,Y]_t=\sum_{0<u\le t}\Delta X_u \Delta Y_u$.
4. \textbf{Doob's maximal inequality.} (see Theorem 20, p.~11 in \cite{Protter}) Let $X=(X)_{t\ge0}$ be a c\`adl\`ag martingale adapted to a filtration $(\mathcal F_t)_{t\ge0}$. Then for any $p>1$ and $t\ge s\ge 0$,
$$ {\mathsf E}[\sup_{s\le u\le t}|X_u|^p \big|\mathcal F_s ]\le\left(\frac{p}{p-1}\right)^p{{\mathsf E}}[\vert X_t\vert^p\big|\mathcal F_s].$$
5. \textbf{Burkholder-Davis-Gundy inequality.} (see Theorem 48, p.~193 in \cite{Protter}) Let $X=(X)_{t\ge0}$ be a c\`adl\`ag martingale adapted to a filtration $(\mathcal F_t)_{t\ge0}$ such that $X_0=0$. For each $1\le p<\infty$ there exist positive constants $c_p$ and $C_p$ depending on only $p$ such that
$$\displaystyle c_p{{\mathsf E}}\left[ [X]^{p/2}_t\big|\mathcal F_s\right]\le{{\mathsf E}}\left[\sup_{s\le u\le t}|X_u|^p \big|\mathcal F_s\right]\le C_p{{\mathsf E}}\left[ [X]^{p/2}_t \big|\mathcal F_s\right].$$
\section{Dynamics of the occupation measure process\label{sec:Dyn}}
We study in this section the dynamics of the occupation process of VRJP on a complete graph with weight function $w(t)=t^{\alpha},\ \alpha>0$. In particular, we show in Theorem \ref{cvthrm} below that, after a time scaling, the occupation measure process is asymptotically close to the unique solution of an autonomous system of ordinary differential equations. Our approach is inspired by the theory of asymptotic pseudo-trajectories and stochastic approximation techniques introduced in \cite{Benaim99}.
For $t>0$ which is not a jumping time of $X$, we have \begin{equation}\label{ode} \frac{{\rm d} Z_t}{{\rm d} t}=\frac{1}{\ell_0+t}\left(-Z_t+I[{X_t}]\right), \end{equation} where for each matrix $M$, $M[j]$ is the $j$-th row vector of $M$ and $I$ is as usual the identity matrix. Observe that the process $Z=(Z_t)_{t\ge0}$ always takes values in the interior of the standard unit simplex $\Delta$.
For fixed $t\ge0$, let $A_t$ be the $d$-dimensional infinitesimal generator matrix such that the $(i,j)$ element is defined by $$ A^{i,j}_t:=\left\lbrace\begin{matrix}\mathbf{1}_{(i,j)\in E} w_t^{(j)}, \ \ \ \ \ \ i\neq j; \\ \displaystyle - \sum_{k\in V,(k,i)\in E} w_t^{(k)}, i=j,\end{matrix}\right.$$ where we have set $w^{(j)}_t=w(L(j,t))=L(j,t)^{\alpha}$ for each $j\in V$. Also, let $w_t=w^{(1)}_t+w^{(2)}_t+\cdots +w^{(d)}_t$. Note that $$\pi_t:=\left(\frac{w^{(1)}_t}{w_t},\frac{w^{(2)}_t}{w_t},\cdots ,\frac{w^{(d)}_t}{w_t}\right)$$ is the unique invariant probability measure of $A_t$ in the sense that $\pi_tA_t=0$. Since $\pi_t$ can be rewritten as a function of $Z_t$, we will also use the notation $\pi_t=\pi(Z_t)$, where we define the function $\pi: \Delta\to \Delta$, such that for each $z=(z_1,z_2,...,z_d)\in \Delta$, $$\pi(z)=\left( \frac{z_1^{\alpha}}{z_1^{\alpha}+\cdots +z_d^{\alpha}},\cdots , \frac{z_d^{\alpha}}{z_1^{\alpha}+\cdots +z_d^{\alpha}} \right).$$ Now we can rewrite the equation \eqref{ode} as \begin{equation}\label{ode2}\frac{{\rm d} Z_t}{{\rm d} t}=\frac{1}{\ell_0+t}(-Z_t+\pi_t) + \frac{1}{\ell_0+t}(I[X_t]-\pi_t).\end{equation} Changing variable $\ell_0+t=e^u$ and denoting $\tilde{Z}_u=Z_{e^u-\ell_0}$ for $u>0$, we can transform the equation \eqref{ode2} as \begin{equation*}\frac{{\rm d} \tilde{Z}_u}{{\rm d} u}=-\tilde{Z}_u+\pi(\tilde{Z}_u) + (I[X_{e^u-\ell_0}]-\pi_{e^u-\ell_0}).\end{equation*} Taking integral of both sides, we obtain that \begin{equation}\label{ode3}\tilde{Z}_{t+s}-\tilde{Z}_{t}=\int_t^{t+s}\left(-\tilde{Z}_u+\pi(\tilde{Z}_u) \right) {\rm d} u +\int_{e^t-\ell_0}^{e^{t+s}-\ell_0} \frac{ I[X_u]-\pi_u}{\ell_0+u}{\rm d} u.\end{equation}
Let us fix a function $f:\{1,\dots,d\}\to\mathbb{R}$. For $t>0$, define $A_tf:\{1,\dots,d\}\to\mathbb{R}$ by $A_tf(i)=\sum_j A_t^{i,j}f(j)$ and define the process $M^f$ by $$M^f_t=f(X_t)-f(X_0)-\int_0^t A_sf(X_s) {\rm d} s.$$ \begin{lemma} The process $M^f$ is a martingale, with $[M^f]_t=\sum_{0<s\le t} (\Delta f(X_s))^2$ and \begin{equation}\label{eq:anglebracket} \langle M^f\rangle_t = \int_0^t \big(A_sf^2(X_s)-2f(X_s)A_sf(X_s)\big) {\rm d} s. \end{equation} \end{lemma} \begin{proof} Let us first prove that $M^f$ is a martingale. For small $h>0$, we have \begin{align*}
{\mathsf E}[f(X_{t+h})-f(X_t)|\mathcal{F}_t]
&= \sum_{j\sim X_t} (f(j)-f(X_t)) {\mathsf P}[X_{t+h}=j|\mathcal{F}_t]\\ &= \sum_{j\sim X_t} (f(j)-f(X_t)) w^{(j)}_t.h+o(h)\\ &= A_tf(X_t).h + o(h). \end{align*}
Let us fix $0<s<t$ and define $t_j=s+j(t-s)/n$ for $j=0,1,\dots,n$. Note that
\begin{align*} {\mathsf E}\left[ f(X_{t})-f(X_s)|\ \mathcal{F}_s \right] & ={\mathsf E}\left[\left. \sum_{j=1}^n {\mathsf E}[f(X_{t_{j}})-f(X_{t_{j-1}})\ |\ \mathcal{F}_{t_{j-1}} ]\ \right|\ \mathcal{F}_s \right]\\
& ={\mathsf E}\left[ \left.\sum_{j=1}^n A_{t_{j-1}}f(X_{t_{j-1}})(t_j-t_{j-1}) +n\cdot o\left(\frac{t-s}{n}\right) \ \right| \ \mathcal{F}_s \right]. \end{align*} Since the left hand side is independent on $n$, using Lebesgue's dominated convergence theorem and taking the limit of the random sum under the expectation sign on the right hand side, we obtain that
$${\mathsf E}\left[ f(X_{t})-f(X_s)|\ \mathcal{F}_s \right)={\mathsf E}\left[\int_s^t A_uf(X_u) {\rm d} u\ | \ \mathcal{F}_s \right].$$ Thus, ${\mathsf E}[ M^f_t|\ \mathcal{F}_s ]=M_s$.
To prove \eqref{eq:anglebracket}, we calculate (to simplify the calculation, we will suppose that $f(X_0)=0$). \begin{align*} (M^f_t)^2 =&\; f^2(X_t)- 2\left(M^f_t+\int_0^t A_sf(X_s){\rm d} s\right)\int_0^t A_sf(X_s){\rm d} s + \left(\int_0^t A_sf(X_s){\rm d} s \right)^2\\ =&\; M^{f^2}_t+\int_0^t A_sf^2(X_s){\rm d} s - 2M^f_t\int_0^t A_sf(X_s){\rm d} s - \left(\int_0^t A_sf(X_s){\rm d} s \right)^2\\ =&\; N_t + \int_0^t A_sf^2(X_s){\rm d} s\\ &\quad - 2\int_0^t M^f_s A_sf(X_s){\rm d} s - 2\int_0^t A_sf(X_s)\left(\int_0^s A_uf(X_u){\rm d} u\right){\rm d} s \\ =&\; N_t + \int_0^t A_sf^2(X_s){\rm d} s - 2\int_0^t f(X_s) A_sf(X_s){\rm d} s, \end{align*} where the process $N$, defined by $N_t:=M_t^{f^2}-2\int_{0}^t\left(\int_0^sA_uf(X_u){\rm d} u\right){\rm d} M^f_s $, is a martingale. The lemma is proved. \end{proof}
Let $M$ be the process in $\mathbb{R}^d$ defined by $$M_t=I[X_t]-\int_0^t A_s[X_s]{\rm d} s\quad \text{for } t\ge0 .$$ Then for each $j$, $M^j$ is a martingale since $M^j=M^{\delta_j}$, with $\delta_j$ defined by $\delta_j(i)=1$ if $i=j$ and $\delta_j(i)=0$ if $i\neq j$.
We also have that \begin{equation}\label{eq:crochetMj} \langle M^j\rangle_t = \int_0^t \Lambda^j_s {\rm d} s, \end{equation} with $\Lambda^j$ defined by \begin{equation}\label{eq:defLambdaj} \Lambda^j_t= \left\lbrace \begin{array}{ll} w^{(j)}_t & \text{ if } \quad X_t\sim j,\\ \sum_{k\sim X_t} w^{(k)}_t &\text{ if }\quad X_t=j, \\ 0 & \text{ otherwise. } \\ \end{array} \right. \end{equation}
\begin{lemma}\label{noise} Assume that $G=(V,E)$ is a complete graph and $w(t)=t^{\alpha}$ with $\alpha>0$. Then almost surely \begin{equation}
\label{bound}\lim_{t\to\infty} \sup_{1\le c\le C} \left\|\int_{t-\ell_0}^{ct-\ell_0} \frac{I[X_s]-\pi_s}{\ell_0+s}{\rm d} s\right\| =0 \end{equation} for each $C>1$. \end{lemma} \begin{proof} Note that, for $t\ge 0$, $$\pi_t-\displaystyle I[X_t]=\frac{1}{w_t}A_t[X_t].$$ Using the integration by part formula, we obtain the following identity for each $c\in [1,C]$ \begin{align*} \int_{t-\ell_0}^{ct-\ell_0} \frac{\pi_s-I[X_s]}{\ell_0+s}{\rm d} s&=\int_{t-\ell_0}^{ct-\ell_0} A_s[X_s]\frac{{\rm d} s}{(\ell_0+s)w_s}\\ &= \left(\frac{I[X_{ct-\ell_0}]}{ctw_{ct-\ell_0}}-\frac{I[X_{t-\ell_0}]}{tw_{t-\ell_0}}\right)\\ & - \int_{t-\ell_0}^{ct-\ell_0} I[X_s] \frac{{\rm d}}{{\rm d} s}\left(\frac{1}{(s+\ell_0)w_s}\right){\rm d} s\\ & - \int_{t-\ell_0}^{ct-\ell_0} \frac{{\rm d} M_s}{(s+\ell_0)w_s}.
\end{align*}
Observe that for some positive constant $k$, $w_s\ge k s^\alpha$ (which is easy to prove, using the fact that $L(1,t)+L(2,t)+\cdots +L(d,t)=\ell_0+t$). We now estimate the terms in the right hand side of the above-mentioned identity. In the following, the positive constant $k$ may change from lines to lines and only depends on $C$ and $\ell_0$. First,
\begin{equation}\label{first}\left\|\frac{I[X_{ct-\ell_0}]}{ctw_{ct-\ell_0}}-\frac{I[X_{t-\ell_0}]}{tw_{t-\ell_0}}\right\| \le k/ {t^{\alpha+1}}. \end{equation} Second, for $s\in [t,ct]$ which is not a jump time, we have \begin{align*} \frac{{\rm d}}{{\rm d} s}\left(\frac{1}{(\ell_0+s)w_s}\right) = & -\left(\frac{1}{(\ell_0+s)^2w_s}+\frac{1}{(\ell_0+s)w^2_s}\frac{{\rm d} w_s}{{\rm d} s}\right). \end{align*}
When $s$ is not a jump time, it is easy to check that $\left|\frac{{\rm d} w_s}{{\rm d} s}\right|\le \alpha (\ell_0+s)^{\alpha-1}$. Therefore, for $s\in [t,ct]$ which is not a jump time,
$$\left|\frac{{\rm d}}{{\rm d} s}\left(\frac{1}{(\ell_0+s)w_s}\right)\right| \le k/s^{2+\alpha}$$ and thus,
\begin{equation}\label{second}\left\|\int_{t-\ell_0}^{ct-\ell_0} I[X_s] \frac{{\rm d}}{{\rm d} s}\left(\frac{1}{(\ell_0+s)w_s}\right) {\rm d} s\right\| \le k/t^{\alpha+1}. \end{equation} And at last (using Doob's inequality), for $i\in\{1,2,\cdots ,d\}$, \begin{eqnarray*}
{\mathsf E}\left[\sup_{1\le c\le C} \left|\int_{t-\ell_0}^{ct-\ell_0} \frac{{\rm d} M^i_s}{(\ell_0+s)w_s}\right|^2\right] &\le& 4\ {\mathsf E}\left[\left(\int_{t-\ell_0}^{Ct-\ell_0} \frac{{\rm d} M^i_s}{(\ell_0+s)w_s}\right)^2\right]. \end{eqnarray*} Observe that in our setting, for $i\in\{1,2,\cdots ,d\}$, $(\Delta I^i_s)^2=1$ if $s$ is a jump time between $i$ and another vertex. Thus $[M^1]_t+[M^2]_t+\cdots +[M^d]_t$ is just twice the number of jumps up to time $t$ of $X$. So, for $i \in\{1,2,\cdots ,d\}$, \begin{eqnarray*} {\mathsf E}\left[\left(\int_{t-\ell_0}^{Ct-\ell_0} \frac{{\rm d} M^i_s}{(\ell_0+s)w_s}\right)^2\right] &=& {\mathsf E}\left[\int_{t-\ell_0}^{Ct-\ell_0} \frac{{\rm d} [M^i]_s}{(\ell_0+s)^2 w_s^2}\right]\\ &\le& \frac{k}{t^{2(\alpha+1)}} {\mathsf E}\left[ [M^i]_{Ct-\ell_0}-[M^i]_{t-\ell_0}\right]\\ &\le& \frac{k}{t^{2(\alpha+1)}} (Ct)^\alpha (C-1)t, \end{eqnarray*} where in the last inequality, we have used the fact that the number of jumps in $[t-\ell_0,Ct-\ell_0]$ is dominated by the number of jumps of a Poisson process with constant intensity $(C t)^\alpha$ in $[t-\ell_0,Ct-\ell_0]$. Therefore, \begin{eqnarray}\label{third}
{\mathsf E}\left[\sup_{1\le c\le C} \left\|\int_{t-\ell_0}^{ct-\ell_0} \frac{{\rm d} M_s}{(\ell_0+s)w_s}\right\|^2\right] &\le& \frac{k}{t^{\alpha+1}}. \end{eqnarray} From (\ref{first}), (\ref{second}), (\ref{third}) and by using Markov's inequality, we have
\begin{equation}\label{Markov}{\mathsf P}\left[\sup_{1\le c\le C} \left\|\int_{t-\ell_0}^{ct-\ell_0} \frac{I[X_s]-\pi_s}{\ell_0+s}{\rm d} s\right\| \ge \frac{1}{t^{\gamma}}\right]\le \frac{k}{t^{\alpha+1-2\gamma}} \end{equation} for every $0<\gamma\le \frac{\alpha+1}{2}$. Using the Borel-Cantelli lemma, we thus obtain
$$\limsup_{n\to\infty}\sup_{1\le c\le C} \left\|\int_{C^n-\ell_0}^{cC^n-\ell_0} \frac{I[X_s]-\pi_s}{\ell_0+s}{\rm d} s\right\| =0.$$ Moreover, for $C^n\le t\le C^{n+1}$, we have
\begin{align*} \sup_{1\le c\le C} \left\|\int_{t-\ell_0}^{ct-\ell_0} \frac{I[X_s]-\pi_s}{\ell_0+s}{\rm d} s\right\| & \le \left\|\int_{C^n-\ell_0}^{t-\ell_0} \frac{I[X_s]-\pi_s}{\ell_0+s}{\rm d} s\right\| +\sup_{1\le c\le C} \left\|\int_{C^n-\ell_0}^{\min(ct,C^{n+1})-\ell_0} \frac{I[X_s]-\pi_s}{\ell_0+s}{\rm d} s\right\|\\
&+ \sup_{1\le c\le C} \left\|\int^{\max(ct,C^{n+1})-\ell_0}_{C^{n+1}-\ell_0} \frac{I[X_s]-\pi_s}{\ell_0+s}{\rm d} s\right\| \\
& \le 2\sup_{1\le c\le C} \left\|\int_{C^n-\ell_0}^{cC^n-\ell_0} \frac{I[X_s]-\pi_s}{\ell_0+s}{\rm d} s\right\| \\
&+\sup_{1\le c\le C} \left\|\int_{C^{n+1}-\ell_0}^{cC^{n+1}-\ell_0} \frac{I[X_s]-\pi_s}{\ell_0+s}{\rm d} s\right\|. \end{align*} This inequality immediately implies (\ref{bound}). \end{proof}
From now on, we always assume that $w(t)=t^{\alpha}$, $\alpha>1$ and $G=(V,E)$ is a complete graph. Let us define the vector field $F:\Delta\to T\Delta $ such that $F(z)=-z +\pi(z)$ for each $z\in \Delta$. We also remark that for each $z=(z_1,z_2,\cdots,z_d)\in \Delta$, \begin{align}\label{vecF}F(z)=\left(-z_1+ \frac{z_1^{\alpha}}{z_1^{\alpha}+\cdots +z_d^{\alpha}},\cdots ,-z_d+ \frac{z_d^{\alpha}}{z_1^{\alpha}+\cdots +z_d^{\alpha}} \right).\end{align}
A continuous map $\Phi: \mathbb{R}_+\times \Delta \to \Delta$ is called a \textit{semi-flow} if $\Phi(0,\cdot):\Delta \to \Delta$ is the identity map and $\Phi$ has the semi-group property, i.e. $\Phi({t+s},\cdot)=\Phi(t,\cdot)\circ \Phi(s,\cdot)$ for all $s,t\in \mathbb{R}_{+}$.
Now for each $z^0\in \Delta$, let $\Phi_t(z^0)$ be the solution of the differential equation \begin{equation}\label{odeF}\left\lbrace \begin{array}{ll} \displaystyle \frac{{\rm d}}{{\rm d} t}z(t) = & F(z(t)),\ t>0;\\
z(0)\ \ \ = & z^0. \end{array} \right. \end{equation} Note that $F$ is Lipschitz. Thus the solution $\Phi_t(z^0)$ can be extended for all $t\in \mathbb{R}_+$ and $\Phi:\mathbb{R}_+\times \Delta \to \Delta$ defined by $\Phi(t,z)=\Phi_t(z)$ is a semi-flow.
\begin{theorem}\label{cvthrm} $\tilde{Z}$ is an asymptotic pseudo-trajectory of the semi-flow $\Phi$, i.e. for all $T>0$, \begin{equation}\label{pseu}
\lim_{t\to\infty} \sup_{0\le s\le T} \left\| \tilde{Z}_{t+s}- \Phi_s(\tilde{Z}_{t})\right\|=0.\ \text{a.s}. \end{equation} Furthermore, $\tilde{Z}$ is an -$\frac{\alpha+1}{2}$-asymptotic pseudo-trajectory, i.e. for
\begin{equation}\label{pseu2}\limsup_{t\to\infty}\frac{1}{t}\log\left( \sup_{0\le s\le T} \| \tilde{Z}_{t+s}-\Phi_s(\tilde{Z}_{t})\| \right)\le -\frac{\alpha+1}{2} \ \text{a.s}. \end{equation} \end{theorem} \begin{proof} From the definition of $\Phi$, we have $$\Phi_s(\tilde{Z}_{t})-\tilde{Z}_{t}=\int_{0}^{s}F(\Phi_u(\tilde{Z}_{t})){\rm d} u.$$ Moreover, from \eqref{ode3} $$\tilde{Z}_{t+s}-\tilde{Z}_{t}=\int_0^{s}F(\tilde{Z}_{t+u}) {\rm d} u +\int_{e^t-\ell_0}^{e^{t+s}-\ell_0} \frac{ I[X_u]-\pi_u}{\ell_0+u}{\rm d} u.$$ Subtracting both sides of the two above identities, we obtain that $$\tilde{Z}_{t+s}-\Phi_s(\tilde{Z}_{t})=\int_0^{s}\left( F(\tilde{Z}_{t+u}) - F(\Phi_u(\tilde{Z}_{t}))\right) {\rm d} u+\int_{e^t-\ell_0}^{e^{t+s}-\ell_0} \frac{ I[X_u]-\pi_u}{\ell_0+u}{\rm d} u.$$ Observe that $F$ is Lipschitz, hence
$$ \| \tilde{Z}_{t+s}-\Phi_s(\tilde{Z}_{t})\| \le K \int_0^{s}\|\tilde{Z}_{t+u}-\Phi_u(\tilde{Z}_{t})\|{\rm d} u +\left\|\int_{e^t-\ell_0}^{e^{s+t}-\ell_0} \frac{I[X_u]-\pi_u}{\ell_0+u}{\rm d} u\right\|,$$ where $K$ is the Lipschitz constant of $F$. Using Gr\"onwall's inequality, we thus have
\begin{equation}\label{gron}\| \tilde{Z}_{t+s}-\Phi_s(\tilde{Z}_{t})\|\le \sup_{0\le s \le T}\left\|\int_{e^t-\ell_0}^{e^{s+t}-\ell_0} \frac{I[X_u]-\pi_u}{\ell_0+u}{\rm d} u\right\| e^{K s}.\end{equation} On the other hand, from Lemma \ref{noise}, we have \begin{equation}\label{noise2}
\lim_{t\to\infty} \sup_{0\le s\le T} \left\|\int_{e^t-\ell_0}^{e^{s+t}-\ell_0} \frac{I[X_u]-\pi_u}{\ell_0+u}{\rm d} u\right\| =0. \ \ \text{a.s.} \end{equation} The inequality (\ref{gron}) and (\ref{noise2}) immediately imply (\ref{pseu}).
We now prove the second part of the theorem. From (\ref{Markov}), we have
$${\mathsf P}\left[\sup_{0\le s\le T} \left\|\int_{e^t}^{e^{s+t}} \frac{I[X_u]-\pi_u}{\ell_0+u}{\rm d} u\right\| \ge e^{-\gamma t}\right] \le k e^{-(\alpha+1-2\gamma)t},$$ for every $0<\gamma\le\frac{\alpha+1}{2}$. By Borel-Cantelli lemma, it implies that
$$\limsup_{n\to\infty}\frac{1}{nT}\log\left( \sup_{0\le s\le T} \left\|\int_{e^{nT}}^{e^{s+nT}} \frac{I[X_u]-\pi_u}{\ell_0+u}{\rm d} u\right\| \right)\le -\gamma\ \ \text{a.s}.$$ and therefore that (taking $\gamma\to \frac{\alpha+1}{2}$)
$$\limsup_{n\to\infty}\frac{1}{nT}\log\left( \sup_{0\le s\le T} \left\|\int_{e^{nT}}^{e^{s+nT}} \frac{I[X_u]-\pi_u}{\ell_0+u}{\rm d} u\right\| \right)\le -\frac{\alpha+1}{2} \ \text{a.s}.$$ Note that for $nT\le t\le (n+1)T$ and $0\le s\le T$, \begin{align*}
\left\|\int_{e^t}^{e^{s+t}} \frac{I[X_u]-\pi_u}{\ell_0+u}{\rm d} u\right\| & \le 2\sup_{0\le s\le T}\left\|\int_{e^{nT}}^{e^{s+nT}} \frac{I[X_u]-\pi_u}{\ell_0+u}{\rm d} u\right\|\\
& +\sup_{0\le s\le T}\left\|\int_{e^{(n+1)T}}^{e^{s+(n+1)T}} \frac{I[X_u]-\pi_u}{\ell_0+u}{\rm d} u\right\|. \end{align*} Therefore,
\begin{align}\label{logbound2}\limsup_{t\to\infty}\frac{1}{t}\log\left( \sup_{0\le s\le T} \left\|\int_{e^{t}}^{e^{s+t}} \frac{I[X_u]-\pi_u}{\ell_0+u}{\rm d} u\right\| \right)\le -\frac{\alpha+1}{2} \ \text{a.s}. \end{align} Finally, (\ref{pseu2}) is obtained from (\ref{gron}) and (\ref{logbound2}). \end{proof} \section{Convergence to equilibria}\label{sec:convergence} Let $$\mathcal{C}=\{z\in \Delta : F(z)=0\}$$ stand for the \textit{equilibria set} of the vector field $F$ defined in (\ref{vecF}). We say an equilibrium $z\in \mathcal{C}$ is (linearly) \textit{stable} if all the eigenvalues of $DF(z)$, the Jacobian matrix of $F$ at $z$, have negative real parts. If there is one of its eigenvalues having a positive real part, then it is called (linearly) \textit{unstable}.
Observe that $\mathcal{C}=\mathcal{S}\cup \mathcal U$, where we define $$\mathcal S=\{e_1=(1,0,0,\cdots ,0), e_2=(0,1,0,\cdots ,0),\cdots ,e_d= (0,0,\cdots ,0,1)\}$$ as the set of all stable equilibria and $$\mathcal U=\{ z_{j_1,j_2,\cdots, j_k} : 1\le j_1<j_2<\cdots <j_k\le d, k=2,\cdots ,d\}$$ as the set of all unstable equilibria, where $z_{j_1,j_2,\cdots, j_k}$ stands for the point $z=(z_1,\cdots, z_d)\in\Delta$ such that $z_{j_1}=z_{j_2}=\cdots =z_{j_k}=\frac{1}{k}$ and all the remaining coordinates are equal to 0.
Indeed, for each $z\in \mathcal S$, we have that $DF(z)=-I$. Moreover, $$DF\left(\frac{1}{d},\frac{1}{d},\cdots ,\frac{1}{d}\right)=(\alpha-1) I -\frac{\alpha }{d} N,$$ where
$N$ is the matrix such that $N_{m,n}=1$ for all $m,n$ and $DF(z_{j_1,j_2,\cdots,j_k})= (D_{m,n})$ where $$D_{m,n}= \left\lbrace \begin{array}{ll} (\alpha-1)- \frac{\alpha}{k} &\text{if } m=n\in\{j_i:\;i=1,\cdots,k\}; \\ -\frac{\alpha}{k} & \text{if } m\neq n, \hbox{ with } \{m,n\} \subset\{j_i:\;i=1,\cdots,k\};\\ -1 & \text{if } m=n\not\in\{j_i:\;i=1,\cdots,k\};\\ 0 & \text{if } m\neq n, \hbox{ with } \{m,n\} \not\subset\{j_i:\;i=1,\cdots,k\}. \end{array} \right. $$ Therefore, we can easily compute that for each $z\in \mathcal U$, the eigenvalues of $DF(z)$ are $-1$ and $\alpha-1$, having respectively multiplicity $d-k+1$ and $k-1$.
\begin{theorem}\label{thm:CVeq} $Z_t$ converges almost surely to a point in $\mathcal{C}$ as $t\to\infty$. \end{theorem} \begin{proof} Consider the map $H:\Delta\to \mathbb{R}$ such that $$H(z)=z_1^{\alpha}+z_2^{\alpha}+\cdots +z_n^{\alpha}.$$ Note that $H$ is a strict Lyapounov function of $F$, i.e $\langle \nabla H(z), F(z)\rangle$ is positive for all $z\in \Delta\setminus \mathcal{C}$. Indeed, we have \begin{align*} \langle \nabla H(z),F(z)\rangle & =\displaystyle \sum_{i=1}^d \alpha z_i^{\alpha-1} \left(-z_i +\frac{z_i^{\alpha}}{\sum_{j=1}^d z_j^{\alpha}} \right)\\ &=\alpha \left(-\sum_{i=1}^d z_i^{\alpha} +\frac{\sum_{i=1}^d z_i^{2\alpha-1}}{\sum_{i=1}^d z_i^{\alpha}} \right)\\ \ & \displaystyle =\frac{\alpha}{H(z)} \left(-\left( \sum_{i=1}^d z_i^{\alpha}\right)^2 +\sum_{i=1}^d z_i^{2\alpha-1}\sum_{i=1}^dz_i \right)\\ \ & \displaystyle =\frac{\alpha}{H(z)} \sum_{1\le i<j\le d} z_iz_j \left( z_i^{\alpha-1}-z_j^{\alpha-1}\right)^2. \end{align*} For $z\in \Delta\setminus \mathcal{C}$, there exist distinct indexes $j_1, j_2\in \{1,2,...,d\}$ such that $z_{j_1}, z_{j_2}$ are positive and $z_{j_1}\neq z_{j_2}$. Therefore,
$$\langle \nabla H(z),F(z)\rangle \ge \frac{\alpha}{H(z)} z_{j_1}z_{j_2} \left( z_{j_1}^{\alpha-1}-z_{j_2}^{\alpha-1}\right)^2>0.$$
Let $$L(Z)=\bigcap_{t\ge0}\overline{Z([t,\infty))}$$ be limit set of $Z$. Since $\tilde{Z}$ is an asymptotic pseudo-trajectory of $\Phi$, by Theorem 5.7 and Proposition 6.4 in \cite{Benaim99}, we can conclude that $L(Z)=L(\tilde{Z})$ is a connected subset of $\mathcal{C}$. Moreover, $\mathcal{C}$ is actually an isolated set and this fact implies the almost sure convergence of ${Z_t}$ toward an equilibrium $z\in\mathcal{C}$ as $t\to\infty$. \end{proof}
\begin{lemma} \label{cvrate} Let $z^*$ be a stable equilibrium. Then for each small $\epsilon>0$ there exists $\delta_{\epsilon}>0$ such that $z^*$ attracts exponentially $B_{\delta_{\epsilon}}(z^*):=\left\{z\in\Delta : \|z-z^*\|<\delta_{\epsilon} \right\}$ at rate $-1+\epsilon$, i.e.
$$\|\Phi_s(z)-z^*\|\le e^{-(1-\epsilon)s}\|z-z^*\|$$ for all $s>0$ and $z\in B_{\delta_{\epsilon}}(z^*)$. \end{lemma} \begin{proof} We observe that $$F(z)=(z-z^*).DF(z^*)^T+R(z-z^*),$$
where we have set $$R(y)=y.\left(\int_0^1DF(ty+z^*)^T{\rm d} t-DF(z^*)^T\right).$$ Note that $\|R(y)\| \le k \|y\|^{1+\beta},$ where $\beta=\min(1,\alpha-1)$ and $k$ is some positive constant. Therefore, we can transform the differential equation \eqref{odeF} to the following integral form $$z(t)-z^*= (z(0)-z^*)e^{t DF(z^*)^T}+\int_0^t R(z(s)-z^*)e^{(t-s)DF(z^*)^T}{\rm d} s. $$ Note that for $z^*\in \mathcal{S}$, we have $DF(z^*)=-I$. Therefore,
$$\|z(t)-z^*\|\le e^{-t}\| z(0)-z^*\|+\int_0^t e^{-(t-s)}\|R(z(s)-z^*)\|{\rm d} s.$$
For each small $\epsilon>0$, if $\|z(s)-z^*\|\le\left(\frac{\epsilon}{k}\right)^{1/\beta}$ for all $0\le s\le t$, then
$$e^{t}\|z(t)-z^*\|\le \|z(0)-z^*\|+\epsilon\int_0^t e^{s}\|z(s)-z^*\|{\rm d} s. $$
Thus, by Gronwall inequality, if $\|z(s)-z^*\|\le\left(\frac{\epsilon}{k}\right)^{1/\beta}$ for all $0\le s\le t$, then
$$\|z(t)-z^*\| \le \| z(0)-z^*\| e^{-(1-\epsilon)t}.$$
But this also implies that if $\|z(0)-z^*\|\le\left(\frac{\epsilon}{k}\right)^{1/\beta}$ then $\|z(t)-z^*\|\le \left(\frac{\epsilon}{k}\right)^{1/\beta}$ for all $t\ge 0$. Hence, for all $t\ge 0$ and any small $\epsilon>0$ and $z(0)$ such that $\|z(0)-z^*\|\le \left(\frac{\epsilon}{k}\right)^{1/\beta}$, we have
\begin{equation*}\|z(t)-z^*\|\le e^{-(1-\epsilon) t}\|z(0)-z^*\|.\end{equation*} \end{proof}
\begin{lemma}\label{lem:CVspeed} Let $z^*=e_j$ be a stable equilibrium, with $j\in V$. Then, a.s. on the event $\{Z_t\to z^*\}$, for all $\epsilon>0$, $$\sum_{i\ne j} L(i,t)=o(t^{\epsilon}).$$ \end{lemma}
\begin{proof} Let us fix $\epsilon>0$ and let $\delta_{\epsilon}$ be the constant defined in Lemma \ref{cvrate}. Note that on the event $\Gamma(z^*):=\{Z_t\to z^*\}$, there exists $T_{\epsilon}>0$ such that $\tilde{Z}_t\in B_{\delta_{\epsilon}}$ for all $t\ge T_{\epsilon}$. Combining the results in Theorem \ref{cvthrm} with Lemma \ref{cvrate} and using Lemma 8.7 in \cite{Benaim99}, we have a.s. on $\Gamma(z^*),$
$$\limsup_{t\to\infty} \frac{1}{t}\log\|\tilde{Z}_{t}-z^*\| \le -1+\epsilon$$
for arbitrary $\epsilon>0$. This implies that a.s. on $\Gamma(z^*),$ that $\| {Z}_{t}- z^*\|=o(t^{-(1-\epsilon)})$. And the lemma easily follows. \end{proof}
\begin{lemma}\label{boundlm} Let $j\in V$, $\epsilon\in (0,1-1/\alpha)$ and $C$ a finite constant. Set $$A_{j,C,\epsilon}:=\left\{\sum_{i\ne j} L(i,t)\le C t^\epsilon,\;\forall t\ge 1\right\}.$$ Then ${\mathsf E}[\sum_{i\neq j} L(i,\infty) 1_{A_{j,C,\epsilon}}] < \infty$. \end{lemma} \begin{proof}
For each $n\ge 1$, set $\tau_n:=\inf\{t\ge 1:\, L(j,t)= n\}$ and $\gamma_n=\sum_{i\in V\setminus\{j\}} L(i,\tau_n)$. Set also $\tau:=\inf\{t\ge 1:\; \sum_{i\ne j} L(i,t)> C t^\epsilon\}$, $\tau'_n=\tau_n\wedge \tau$ and $\gamma'_n=\sum_{i\in V\setminus\{j\}} L(i,\tau'_n)$. Note that $A_{j,C,\epsilon}=\{\tau=\infty\}$ and on $A_{j,C,\epsilon}$, $\tau_n=\tau'_n<\infty$ and $\gamma_n=\gamma'_n$ for all $n\ge 1$.
During the time interval $[\tau'_n,\tau'_{n+1}]$, the jumping rate to $j$ is larger than $\rho_0= n^{\alpha}$ and the jumping rate from $j$ is smaller than $\rho_1=(C (n+1)^{\epsilon})^{\alpha}$. This implies that on the time interval $[\tau'_n,\tau'_{n+1}]$, the number of jumps from $j$ to $V\setminus\{j\}$ is stochastically dominated by the number of jumps of a Poisson process with intensity $\rho_1$. Since the time spent at $j$ during $[\tau'_n,\tau'_{n+1}]$ is $L(j,\tau'_{n+1})-L(j,\tau'_n)\le 1$, the number of jumps from $j$ is stochastically dominated by a random variable $N\sim \text{Poisson}(\rho_1)$. Therefore, $\gamma'_{n+1}-\gamma'_n$, the time spent at $V\setminus\{j\}$ during $[\tau'_n,\tau'_{n+1}]$, is stochastically dominated by $T:=\sum_{i=1}^N \xi_{i}$, where $\xi_i,i=1,2,...,N$ are independent and exponentially distributed random variables with mean value $1/\rho_0.$ Therefore, $${\mathsf E}[\gamma'_{n+1}-\gamma'_n]\le \frac{\rho_1}{\rho_0}=\frac{C^\alpha(n+1)^{\alpha\epsilon}}{n^\alpha}=O\left(\frac{1}{n^{\alpha(1-\epsilon)}}\right).$$ Since $\lim_{n\to\infty}\gamma'_n=\sum_{i\ne j}L(i,\tau)$, this proves that ${\mathsf E}\left[\sum_{i\ne j}L(i,\tau)\right]<\infty.$ This proves the lemma since $\sum_{i\neq j} L(i,\infty) 1_{A_{j,C,\epsilon}} \le \sum_{i\ne j}L(i,\tau)$. \end{proof}
\begin{theorem}\label{thm:localization} Let $z^*=e_j\in \mathcal{S}$ be a stable equilibrium, with $j\in \{1,2,...,d\}$. Then, a.s. on the event $\{Z_t\to z^*\}$, $$L(j,\infty)=\infty \quad\text{ and }\quad \sum_{i\ne j} L(i,\infty)<\infty.$$ \end{theorem}
\begin{proof} Lemma \ref{lem:CVspeed} implies that for $\epsilon\in (0,1-\frac{1}{\alpha})$, the event $\{Z_t\to z^*\}$ coincides a.s. with $\cup_{C} A_{j,C,\epsilon}$. Lemma \ref{boundlm} states that for all $C>0$, a.s. on $A_{j,C,\epsilon}$, $\sum_{i\neq j} L(i,\infty)<\infty$. Therefore, we have that a.s. on $\{Z_t\to z^*\}$, $\sum_{i\neq j} L(i,\infty)<\infty$. \end{proof}
We will show in the next section that if $z^*$ is an unstable equilibrium, then ${\mathsf P}(Z_t\to z^*)=0$ and therefore this will finish the proof of Theorem \ref{thm:mainresult}.
\section{Non convergence to unstable equilibria}\label{sec:nonCV}
In this section, we prove a general non convergence theorem for a class of finite variation c\`adl\`ag processes. The proof of this theorem follows ideas from the proof of a theorem of Brandi\`ere and Duflo (see \cite{Brandiere96} or \cite{Duflo1996}), but using a new idea presented in Section \ref{sec:dirattract}, where sufficient conditions are given for an asymptotic pseudo-trajectory $Z$ of a dynamical system to be attracted exponentially fast towards the unstable manifold of an equilibrium $z^*$ on the event $Z_t$ converges towards $z^*$. Then, in Section \ref{sec:dirinst}, we prove a non convergence theorem towards an unstable equilibrium that has no stable direction. The proof essentially follows \cite{Brandiere96} and \cite{Duflo1996}. We also point out in Remark \ref{rk:inaccuracy} several inaccuracies in their proof.
The results proved in Sections \ref{sec:dirattract} and \ref{sec:dirinst} are then applied in Section \ref{sec:nonCV_VRJP} to strongly VRJP, showing in particular that the occupation measure process does not converge towards unstable equilibria with probability 1. \subsection{Attraction towards the unstable manifold}\label{sec:dirattract}
In this section, we fix $m\in\{1,2,\dots d\}$, a point $z\in\mathbb{R}^d$ will be written as $z=(x,y)$ where $x\in \mathbb R^m$ and $y\in \mathbb R^{d-m}$. Let $\Pi:\mathbb{R}^d\to\mathbb{R}^m$ be defined by $\Pi(x,y)=x$ (since $\Pi$ is linear, we will often write $\Pi z$ instead of $\Pi(z)$).
We let $F:\mathbb{R}^d\to\mathbb{R}^d$ be a $C^1$ Lipschitz vector field.
Let us consider a finite variation c\`adl\`ag process $Z=(X,Y)$ in $\mathbb{R}^d$, adapted to a filtration $(\mathcal{F}_t)_{t\ge 0}$, satisfying the following equation \begin{align*}Z_t-Z_s=\int_s^t F(Z_u){\rm d} u+\int_s^t {\Psi}_u{\rm d} u + M_t-M_s\end{align*} where $M_t$ is a finite variation c\`adl\`ag martingale w.r.t $(\mathcal{F}_t)$ and $\Psi_t$ is a $(\mathcal{F}_t)$-adapted process.
Let $z^*=(x^*,y^*)$ be an equilibrium of $F$, i.e. $F(z^*)=0$. In the following, $\Gamma$ denotes the event $\{\lim_{t\to\infty} Z_t = z^*\}$.
\begin{hypothesis}\label{hyp:gpt}
There is $\gamma>0$ such that for all $T>0$, there exists a finite constant $C(T)$, such that for all $t>0$, $${\mathsf E}\left[\sup_{0\le h\le T} \left\|\int_{t}^{t+h} (\Psi_u{\rm d} u +{\rm d} M_u)\right\|^2\right]\le C(T) e^{-2\gamma t}.$$ \end{hypothesis} \begin{remark}
Using Doob's inequality, Hypothesis \ref{hyp:gpt} is satisfied as soon as there is a constant $C>0$ such that for all $t>0$, $\|\Psi_t\|\le C e^{-\gamma t}$ and for all $t>s>0$ and all $1\le i\le d$, $\langle M^i\rangle_t-\langle M^i\rangle_s\le Ce^{-2\gamma s}$. \end{remark}
\begin{lemma}\label{lem:gpt} If Hypothesis \ref{hyp:gpt} holds, then $Z$ is a $\gamma$-pseudotrajectory of $\Phi$, the flow generated by $F$, i.e. a.s. for all $T>0$
$$\limsup_{t\to\infty}\frac{1}{t}\log\left(\sup_{0\le h\le T} \|Z_{t+h}-\Phi_h(Z_t)\|\right)\le -\gamma.$$ \end{lemma} \begin{proof} Follow the proof of Proposition 8.3 in \cite{Benaim99}. \end{proof}
\begin{hypothesis} \label{hyp:dirattract} There are $\mu>0$ and $\mathcal{N}=\mathcal{N}_1\times \mathcal{N}_2$ a compact convex neighbourhood of $z^*$ (with $\mathcal{N}_1$ and $\mathcal{N}_2$ respectively neighbourhoods of $x^*\in\mathbb{R}^m$ and of $y^*\in \mathbb{R}^{d-m}$) such that $K:=\{z=(x,y)\in \overline{\mathcal{N}}:y=y^*\}$ attracts exponentially $\overline{\mathcal{N}}$ at rate $-\mu$ (i.e. there is a constant $C$ such that $d(\Phi_t(z),K)\le C e^{-\mu t}$ for all $t>0$). \end{hypothesis}
\begin{lemma}\label{lem:dirattract} If Hypotheses \ref{hyp:gpt} and \ref{hyp:dirattract} hold, then, setting $\beta_0:=\gamma\wedge \mu$, for all $\beta\in (0,\beta_0)$, on the event $\Gamma$, \begin{equation}
\|Y_t-y^*\| = O(e^{-\beta t}). \end{equation} \end{lemma} \begin{proof} This is a consequence of Lemma 8.7 in \cite{Benaim99}. \end{proof}
\begin{hypothesis} \label{hyp:alpha-holder} Suppose there are $\alpha>1$ and $C>0$ such that for all $1\le i\le m$ and all $(x,y)\in \mathcal{N}$,
$$|F_i(x,y)-F_i(x,y^*)|\le C \|y-y^*\|^\alpha.$$ \end{hypothesis}
Set $G:\mathbb{R}^m\to\mathbb{R}^m$ be the $C^1$ vector field defined by $G_i(x)=F_i(x,y^*)$, for $1\le i\le m$ and $x\in\mathbb{R}^m$. For $p>0$, denote $$\Gamma_p:=\Gamma\cap \{\forall t\ge p:\; Z_t\in \mathcal{N}\}.$$ For $1\le i\le m$, set $$\tilde{\Psi}_i(t)=\Psi_i(t) + F_i(X_t,Y_t) - F_i(X_t,y^*).$$ \begin{lemma}\label{lem:reducx} Under Hypotheses \ref{hyp:gpt}, \ref{hyp:dirattract} and \ref{hyp:alpha-holder}, on $\Gamma_p$, it holds that, as $t\to\infty$, $$\tilde{\Psi}_t=\Pi \Psi_t + O(e^{-\alpha\beta t})$$ for all $\beta\in (0,\beta_0)$ and that for all $p<s<t$, $$X_t-X_s=\int_s^t G(X_u){\rm d} u+\int_s^t \tilde{\Psi}_u{\rm d} u + \Pi M_t-\Pi M_s.$$ \end{lemma} \begin{proof} This lemma is a straightforward consequence of Lemma \ref{lem:dirattract}. \end{proof}
\subsection{Avoiding repulsive traps}\label{sec:dirinst} In applications, this subsection will be used for the process $X$ defined in Lemma \ref{lem:reducx}.
In this subsection, we let $F:\mathbb{R}^d\to\mathbb{R}^d$ be a $C^1$ Lipschitz vector field and we consider a finite variation c\`adl\`ag process $Z$ in $\mathbb{R}^d$, adapted to a filtration $(\mathcal{F}_t)_{t\ge 0}$, satisfying the following equation $$Z_t-Z_s=\int_s^t F(Z_u){\rm d} u+\int_s^t {\Psi}_u{\rm d} u + M_t-M_s$$ where $M_t$ is a finite variation c\`adl\`ag martingale w.r.t $(\mathcal{F}_t)$ and $\Psi_t=r_t+R_t$, with $r$ and $R$ two $(\mathcal{F}_t)$-adapted processes.
Let $z^*\in\mathbb{R}^d$ and $\Gamma$ an event on which $\lim_{t\to\infty} Z_t = z^*$. Let $\mathcal{N}$ be a convex neighbourhood of $z^*$. For $p>0$, set $$\Gamma_p:=\Gamma\cap \{\forall t\ge p:\; Z_t\in \mathcal{N}\}.$$ Then, $\Gamma=\cup_{p>0}\Gamma_p$.
We will suppose that \begin{hypothesis}\label{hyp:z*} $z^*$ is a repulsive equilibrium, i.e. $F(z^*)=0$ and all eigenvalues of $DF(z^*)$ have a positive real part. Moreover $DF(z^*)=\lambda I$, with $\lambda>0$ and $I$ the identity $d\times d$ matrix. \end{hypothesis}
For all $z\in\mathbb{R}^d$, \begin{align*} F(z)&=F(z^*)+\int_0^1 DF(z^*+u(z-z^*)).(z-z^*) {\rm d} u\\
&= \lambda (z-z^*) + J(z).(z-z^*) \end{align*} where we have set $$J(z)=\int_0^1 (DF(z^*+u(z-z^*))-DF(z^*)){\rm d} u.$$ Then, for all $t\ge s$, \begin{equation}\label{eq:zl} Z_t-Z_s=\int_s^t \lambda Z_u {\rm d} u+\int_s^t \left[\Psi_u+J(Z_u).(Z_u-z^*)\right]{\rm d} u + M_t-M_s. \end{equation}
Let us fix $p>0$. Note that \eqref{eq:zl} implies that, for all $t\ge p$, \begin{equation}\label{eq:zl2} Z_t=e^{\lambda t} \left(e^{-\lambda p} Z_p +\int_p^t\bar{\Psi}_sds+ \bar M_t-\bar M_p\right) \end{equation} where $\bar M_t= \int_0^t e^{-\lambda s} {\rm d} M_s$ and $\bar{\Psi}_t=\bar{r}_t+\bar{R}_t$, with $$\bar{r}_t:= e^{-\lambda t}r_t \quad \text{ and } \quad \bar{R}_t:=e^{-\lambda t}[R_t+J(Z_t).(Z_t-z^*)] .$$ We assume that the following hypothesis is fulfilled:
\begin{hypothesis} \label{hyp:RM} There is a random variable $K$ finite on $\Gamma$ and there is a continuous function $a:[0,\infty)\to (0,\infty)$ such that $\int_0^\infty a(s){\rm d} s <\infty$,
$\alpha^2(t) :=\int_t^{\infty}a(s){\rm d} s=O\big( \int_t^\infty e^{-2\lambda (s-t)} a(s) {\rm d} s\big)$ as $t\to\infty$ and such that the following items \textit{(i)} and \textit{(ii)} hold. \begin{enumerate}[(i)] \item For each $i$, $\langle M^i\rangle_t = \int_0^t \Lambda^i_s {\rm d} s$, with $\Lambda^i$ a positive $(\mathcal{F}_t)$-adapted process. Setting $\Lambda=\sum_i \Lambda^i$, we have that a.s. on $\Gamma$, for all $t>0$, \begin{align}\label{eq:THM1}
K^{-1}a(t) \le \Lambda(t)\le Ka (t), \end{align} \begin{align}\label{eq:THM2}
\sum_{i=1}^d|\Delta M^i_t|\le K \alpha(t), \end{align} \begin{align} \label{eq:THM4}
\int_0^\infty\frac{\|r_s\|^2}{a(s)} {\rm d} s \le K. \end{align} \item As $t\to\infty$,
\begin{align}\label{eq:THM3}{\mathsf E}\left[1_\Gamma \left(\int_t^\infty \|{R}_s\| {\rm d} s\right)^2\right]=o\big(\alpha^2(t)\big). \end{align} \end{enumerate} \end{hypothesis}
For $p>0$, define $$G_p=\Gamma_p\cap\{\sup_{t\ge p}\|J(Z_t)\|\le \frac{\lambda}{2}\}\cap\{\sup_{t\ge p}\|Z_t\|\le 1\}.$$
\begin{lemma} \label{lem:R} For all $p>0$, as $t\to\infty$, \begin{align}\label{eq:R}
{\mathsf E}\left[1_{G_p} \int_t^\infty \|\bar{R}_s\|{\rm d} s\right]=o(e^{-\lambda t}\alpha(t)). \end{align} \end{lemma} \begin{proof} Fix $p>0$. Since Hypothesis \ref{hyp:RM}-(ii) holds, to prove the lemma it suffices to prove that as $t\to\infty$,
$${\mathsf E}\left[1_{G_p}\int_t^\infty e^{-\lambda s}\|J(Z_s).(Z_s-z^*)\| {\rm d} s\right]= o(e^{-\lambda t}\alpha(t)).$$
To simplify the notation, we suppose $z^*=0$. For $s<t$, (using the convention: $\frac{z}{\|z\|}=0$ if $z=0$) \begin{align*}
\|Z_t\|-\|Z_s\|
=& \;\lambda\int_s^t \|Z_u\| {\rm d} u + \int_s^t \left\langle \frac{Z_u}{\|Z_u\|},J(Z_u)Z_u\right\rangle {\rm d} u\\
& + \int_s^t \left\langle \frac{Z_{u-}}{\|Z_{u-}\|}, {\rm d} M_u\right\rangle + \int_s^t \left\langle \frac{Z_u}{\|Z_u\|}, \Psi_u\right\rangle {\rm d} u\\
&+ \sum_{s<u\le t} 1_{\{Z_{u-}\ne 0\}}\left(\Delta \|Z_u\| - \left\langle \frac{Z_{u-}}{\|Z_{u-}\|},\Delta Z_u\right\rangle \right). \end{align*}
Using the inequality $\|z+\delta\|-\|z\|\ge \langle \frac{z}{\|z\|},\delta \rangle$, we have for all $u>p$,
$$\Delta \|Z_u\| -\left\langle \frac{Z_{u-}}{\|Z_{u-}\|},\Delta Z_u\right\rangle\ge 0.$$ Furthermore, using Cauchy-Schwarz inequality, on the event $G_p$,
$$\left\langle \frac{Z_u}{\|Z_u\|},J(Z_u)Z_u\right\rangle \ge - \|J(Z_u)Z_u\|\ge -\sup_{t\ge p}\|J(Z_t)\|.\|Z_u\|\ge-\frac{\lambda}{2}\|Z_u\|$$ for all $u> p$. From the above it follows that on the event $G_p$, \begin{align*}
\|Z_t\|-\|Z_s\|
\ge& \;\frac{\lambda}{2}\int_s^t \|Z_u\| {\rm d} u + \int_s^t \left\langle \frac{Z_{u-}}{\|Z_{u-}\|}, {\rm d} M_u\right\rangle + \int_s^t \left\langle \frac{Z_u}{\|Z_u\|},\Psi_u\right\rangle {\rm d} u \end{align*} for all $t>s>p$. As a consequence, using Doob's inequality and Hypothesis \ref{hyp:RM}, we obtain that \begin{eqnarray*}
\frac{\lambda}{2}{\mathsf E}\left[1_{G_p}\left(\int_t^\infty \|Z_s\|{\rm d} s\right)^2\right]^\frac{1}{2}
&\le&\; {\mathsf E}\left[1_{G_p}\sup_{T>t}\left|\int_t^T \left\langle \frac{Z_{u-}}{\|Z_{u-}\|}, {\rm d} M_u\right\rangle\right|^2\right]^\frac{1}{2}\\
& &+ \; \alpha(t) {\mathsf E}\left[1_{G_p}\int_t^\infty \frac{\|r_u\|^2}{a(u)} {\rm d} u\right]^\frac{1}{2}\\
& &+ \; {\mathsf E}\left[1_{G_p}\left(\int_t^\infty \|R_u\| {\rm d} u\right)^2\right]^\frac{1}{2}\\ &= &\; O(\alpha(t)). \end{eqnarray*}
Using Cauchy-Schwarz inequality, we have \begin{align*}
{\mathsf E}\left[1_{G_p}\int_t^\infty e^{-\lambda s}\|J(Z_s)Z_s\|{\rm d} s\right]
\le & \; e^{-\lambda t}{\mathsf E}\left[1_{G_p}\sup_{s\ge t} \|J(Z_s)\|^2\right]^{\frac12}
{\mathsf E}\left[1_{G_p}\left(\int_t^\infty \|Z_s\|{\rm d} s\right)^2\right]^{\frac12}. \end{align*}
Note that on $G_p$, $\sup_{s\ge t} \|J(Z_s)\|\le \lambda/2$ and $\lim_{t\to\infty}\sup_{s\ge t} \|J(Z_s)\|=0$ almost surely. Therefore, we conclude that
${\mathsf E}[1_{G_p}\int_t^\infty e^{-\lambda s} \|J(Z_s)Z_s\|{\rm d} s]=o(e^{-\lambda t} \alpha(t))$ as $t\to \infty$. \end{proof}
Hypothesis \ref{hyp:RM} ensures in particular that a.s. on $G_p$, $\int_p^\infty \bar{\Psi}_s{\rm d} s$ and $\bar M_{\infty}$ are well defined and almost surely finite. Let $L$ be a random variable such that $$L=\int_p^\infty \bar{\Psi}_s{\rm d} s+\bar M_{\infty}-\bar M_{p}\ \ \text{ on } G_p.$$ Letting $t\to\infty$ in \eqref{eq:zl2}, $\lambda$ being positive, we have $L=-e^{-\lambda p} Z_p\ \text{\ a.s. on }G_p.$
We now apply Theorem \ref{THM:THMA} to the martingale $\bar M_t$ and to the adapted process $\bar \Psi_t$. We have $\langle \bar{M}^i\rangle_t=\int_0^t \bar{\Lambda}^i_s {\rm d} s,$ with $\bar{\Lambda}^i_s=e^{-2\lambda s}\Lambda^i_s$.
We also have $|\Delta \bar{M}_t|=e^{-\lambda t}|\Delta M_t|$. Hypothesis \ref{hyp:RM}-(i) implies that \eqref{eq:THMA1}, \eqref{eq:THMA2} and \eqref{eq:THMA4} are satisfied with the function $\bar{a}(t)=e^{-2\lambda t}a(t)$. Finally, \eqref{eq:THMA3} follows from Lemma \ref{lem:R}. Therefore, we obtain that
$${\mathsf P}(G_p)={\mathsf P}(G_p\cap\{L=-e^{-\lambda p} Z_p\}]=0.$$
Since ${\mathsf P}(\Gamma)=\lim_{p\to\infty}{\mathsf P}(G_p)=0$, we have proved the following theorem: \begin{theorem}\label{THM:nonCV} Under Hypotheses \ref{hyp:z*} and \ref{hyp:RM}, we have ${\mathsf P}(\Gamma)=0$. \end{theorem}
\subsection{Application to strongly VRJP on complete graphs} \label{sec:nonCV_VRJP}
Recall from Section \ref{sec:Dyn} that the empirical occupation measure process $(Z_t)_{t\ge0}$ satisfies the following equation \begin{align}\label{vrjp} Z_t-Z_s &=\; \int_s^t \frac{1}{u+\ell_0} F(Z_u) {\rm d} u + \frac{I[X_s]}{(s+\ell_0)w_s}- \frac{I[X_t]}{(t+\ell_0)w_t}\\ \nonumber &+ \int_s^t \Psi_u {\rm d} u + \int_s^t \frac{{\rm d} M_u}{(u+\ell_0)w_u}, \end{align} where $$\Psi_t=I[X_t]\frac{{\rm d}}{{\rm d} t}\left(\frac{1}{(t+\ell_0)w_t}\right)\quad \text{ and } \quad M_t=I[X_t]-\int_0^t A_s[X_s] {\rm d} s.$$ Recall that $\langle M^j\rangle_t=\int_0^t \Lambda^j_s {\rm d} s$, where $\Lambda^j$ is defined in \eqref{eq:defLambdaj}.
For $t\ge t_0:=\log(\ell_0)$, let \begin{equation}
\label{def:Zhat}
\widehat{Z}_t=Z_{e^t-\ell_0}+\frac{I[X_{e^t-\ell_0}]}{e^tw_{e^t-\ell_0}}. \end{equation} Equation (\ref{vrjp}) is thus equivalent to \begin{align}\label{mvrjp}\widehat{Z}_t-\widehat{Z}_s=\int_s^t F(\widehat{Z}_u){\rm d} u+\int_s^t \widehat{\Psi}_u{\rm d} u + \widehat{M}_t-\widehat{M}_s, \end{align} where we have set \begin{align*} \widehat{\Psi}_t=e^t \Psi_{e^t-\ell_0} +F(Z_{e^t-\ell_0})-F(\widehat{Z}_t) \qquad \hbox{and} \qquad \widehat{M}_t=\int_0^{e^t-\ell_0}\frac{{\rm d} M_s}{(s+\ell_0)w_s}, \end{align*} which are respectively an adapted process and a martingale w.r.t the filtration $(\widehat{\mathcal{F}}_t)_{t\ge t_0}:=(\mathcal{F}_{e^t-\ell_0})_{t\ge t_0}$. Note that $\langle \widehat{M}^j\rangle_t-\langle \widehat{M}^j\rangle_{t_0}=\int_{t_0}^t \widehat{\Lambda}^j_s {\rm d} s$, with $\widehat{\Lambda}^j_s=\frac{\Lambda^j_{e^s-\ell_0}}{e^sw^2_{e^s-\ell_0}}.$
In this subsection, we will apply the results of Subsection \ref{sec:dirattract} and Subsection \ref{sec:dirinst} to the process $(\widehat{Z}_t)_{t\ge0}$ and thus show that $P[Z_{t}\to z^*]=P[\widehat{Z}_{t}\to z^*]=0$ for each unstable equilibrium $z^*$.
\begin{lemma}\label{lem:togpt}
There exists a positive constant $K$ such that for all $t>t_0$, a.s. \begin{align*}
&\|\widehat{\Psi}_t\|\le K e^{-(\alpha+1)t},\quad \quad \widehat{\Lambda}^j_t \le K e^{-(\alpha+1)t} \hbox{ and } \quad |\Delta \widehat{M}_t^j|\le K e^{-(\alpha+1)t}.
\end{align*} \end{lemma} \begin{proof} Let us first recall that $w_t\ge k(t+\ell_0)^\alpha$ for some constant $k$. Using that $F$ is Lipschitz, we easily obtain the first inequality. To obtain the second inequality, observe that for each $j$, $\Lambda^j_t\le w_t$. Thus for all $t>t_0$, $$\widehat{\Lambda}^j_t\le \frac{1}{e^tw_{e^t-\ell_0}}\le k^{-1} e^{-(\alpha+1)t}.$$ Finally,
$$|\Delta \widehat{M}_t^j|=\frac{|\Delta I[X_{e^t-\ell_0}]|}{e^tw_{e^t-\ell_0}}\le \frac{1}{e^tw_{e^t-\ell_0}} \le k^{-1}e^{-(\alpha+1)t}.$$
\end{proof} \begin{theorem}\label{thm:nonCV_VRJP} Assume that $z^*$ is an unstable equilibrium of the vector field $F$ defined by \eqref{vecF}. Then ${\mathsf P}[Z_t\to z^*]=0$. \end{theorem}
\begin{proof} Note first that Lemma \ref{lem:togpt} implies that Hypothesis \ref{hyp:gpt} holds with $\gamma=\frac{\alpha+1}{2}$.
Let $z^*=(x^*,y^*)$ be an unstable equilibrium, where $y^*=0\in\mathbb{R}^{d-m}$ and $x^*=\left(\frac1m,\frac1m,\dots,\frac1m\right)\in \mathbb{R}^m$, with $m\in\{2,3,\dots, d\}$ (up to a permutation of indices, this describes the set of all unstable equilibria).
Note also that there is a compact convex neighbourhood $\mathcal{N}=\mathcal{N}_1\times \mathcal{N}_2$ of $z^*$ and a positive constant $h$ such that for all $z\in\mathcal{N}$, $H(z)=\sum_i z_i^\alpha \ge h$. Setting $C(z)=\frac{1}{H(z)}$, we have that for all $i\in\{1,2,\dots,d-m\}$, $$ F_{m+i}(x,y)=-y_{i}(1+ C(z)y_{i}^{\alpha-1}).$$ Since $\alpha>1$, it can easily be shown that Hypothesis \ref{hyp:dirattract} holds for all $\mu\in (0,1)$. Hypothesis \ref{hyp:alpha-holder} also holds (with the same constant $\alpha$).
Therefore, Lemma \ref{lem:reducx} can be applied to the process $(\widehat{Z}_t)_{t\ge t_0}$ defined by (\ref{def:Zhat}). Set $\widehat{X}_t:=\Pi\widehat{Z}_t$ and let ${G}:\mathbb{R}^m \to\mathbb{R}^m$ be the vector field defined by $G_i(x)=F_i(x,0)$. Then for all $s<t$, $$\widehat{X}_t-\widehat{X}_s=\int_s^t G(\widehat{X}_u){\rm d} u+\int_s^t \hat{r}_u{\rm d} u + \Pi\widehat{M}_t-\Pi\widehat{M}_s,$$ with $\hat{r}_t=\Pi\widehat{\Psi}_t+O(e^{-\alpha\beta t})$ on $\Gamma$, for all $\beta<\gamma\wedge \mu$. Note that since $\mu$ can be taken as close as we want to $1$ and since $\gamma=\frac{\alpha+1}{2}>1$, $\beta$ can be also taken as close as we want to $1$.
We now apply the result of Section \ref{sec:dirinst}, with $Z$, $F$, $M$, $r$ and $R$ respectively replaced by $\hat{X}$, $G$, $\Pi\hat{M}$, $\hat{r}$ and $0$. The vector field $G$ satisfies Hypothesis \ref{hyp:z*} with $\lambda=\alpha-1$.
Let us now check Hypothesis \ref{hyp:RM} with $a(t)=e^{-(\alpha+1)t}$. Choosing $\beta\in (\frac{\alpha+1}{2\alpha},1)$, we have that $\hat{r}$ satisfies \eqref{eq:THM4}.
Set $\widehat{\Lambda}=\sum_{j=1}^m \widehat{\Lambda}^j$. It remains to verify the inequality (\ref{eq:THM1}) for $\widehat{\Lambda}$. Lemma \ref{lem:togpt} shows that for all $t>0$, $$\widehat{\Lambda}_t\le \frac{m}{k}e^{-(\alpha+1)t} = C_+ e^{-(\alpha+1)t}.$$
Fix $\epsilon\in (0,1)$ and choose the neighbourhood $\mathcal{N}$ sufficiently small such that for all $z\in \mathcal{N}$ and $i\in\{1,\dots,m\}$, $m\pi_i(z)\in (1-\epsilon,1+\epsilon)$. Therefore, if $Z_t\in\mathcal{N}$, we have that for $i\in\{1,\dots,m\}$, $w^{(j)}_t=w_t\pi_j(Z_t)\ge \frac{k(1-\epsilon)}{m} (t+\ell_0)^\alpha$. Therefore, since $m\ge 2$, if $Z_t\in\mathcal{N}$, we have that for all $1\le i\le m$ $$\Lambda^i_t\ge 1_{\{X_t=i\}} \sum_{j\neq i, 1\le j\le m} w^{(j)}_t + 1_{\{X_t\neq i\}} w^{(i)}_t\ge \min_{1\le j \le m}w_u^{(j)}\ge \frac{k(1-\epsilon)}{m}(u+\ell_0)^\alpha.$$ Since $w_t\le d(t+\ell_0)^{\alpha}$, we have that if $Z_t\in\mathcal{N}$, $$\widehat{\Lambda}_t\ge \frac{k(1-\epsilon)e^{\alpha t}}{e^td^2e^{2\alpha t}}=C_- e^{-(\alpha+1)t}.$$ This proves that Hypothesis \ref{hyp:RM} is satisfied.
As a conclusion Theorem \ref{THM:nonCV} can be applied, and this proves that ${\mathsf P}[Z_t\to z^*]={\mathsf P}[\widehat X_t\to x^*]=0$. \end{proof}
\subsection{A theorem on martingales}
In this subsection, we prove a martingale theorem, which is a continuous time version of a theorem by Brandi\`ere and Duflo (see Theorem A in \cite{Brandiere96} or Theorem 3.IV.13 in \cite{Duflo1996}). \begin{theorem}\label{THM:THMA} Let $M$ be a finite variation c\`adl\`ag martingale in $\mathbb{R}^d$ with $M_0=0$, $r$ and $R$ be adapted processes in $\mathbb{R}^d$ with respect to a filtration $(\mathcal{F}_t)_{t\ge0}$. Set $\Psi_t=r_t+R_t$.
Let $\Gamma$ be an event and let $a:[0,\infty)\to (0,\infty)$ be a continuous function such that $\int_0^\infty a(s){\rm d} s <\infty$ and set $\alpha^2(t)=\int_t^\infty a(s) {\rm d} s$. Suppose that for each $i$, $\langle M^i\rangle_t = \int_0^t \Lambda^i_s {\rm d} s$, with $\Lambda^i$ a positive adapted c\`adl\`ag process. Set $\Lambda=\sum_i \Lambda^i$. Suppose that there is a random variable $K$, such that a.s. on $\Gamma$, $1<K<\infty$ and for all $t>0$, \begin{align}\label{eq:THMA1}
K^{-1} a(t) \le \Lambda(t)\le Ka (t). \end{align} \begin{align}\label{eq:THMA2}
\sum_i|\Delta M^i_t|\le K \alpha(t). \end{align} \begin{align}\label{eq:THMA4}
\int_0^\infty \frac{\|r_s\|^2}{a(s)} {\rm d} s \le K \end{align} and as $t\to\infty$, \begin{align}\label{eq:THMA3}
{\mathsf E}\left[1_\Gamma\int_t^\infty \|R_s\| {\rm d} s\right]= o(\alpha(t)).\end{align}
Then, a.s. on $\Gamma$, $S_t:=\int_0^t {\Psi}_s{\rm d} s+ M_t$ converges a.s. towards a finite random variable $L$ and for all $\mathcal{F}_p$-measurable random variable $\eta$, $p>0$, we have $${\mathsf P}[\Gamma\cap\{L=\eta\}]=0.$$ \end{theorem} \begin{remark} \label{rk:inaccuracy} Our theorem here is a continuous-time version of Theorem A by Brandi\`ere and Duflo in \cite{Brandiere96}. Their results is widely applied to discrete stochastic approximation processes, in particular to showing the non convergence to a repulsive equilibrium. Note that there is an inaccuracy in the application of the Burkholder's inequality in their proof. Beside of this, there is also a mistake in the application of their theorem to the proof of Proposition 4 in \cite{Brandiere96} since the process $S_n$ defined in page 406 is not adapted. \end{remark} \begin{proof} \ \\ \textit{Simplification of the hypotheses:} It is enough to prove the Theorem assuming in addition that the random variable $K$ is non-random and that \eqref{eq:THMA1}, \eqref{eq:THMA2} and \eqref{eq:THMA4} are satisfied a.s. on $\Omega$.
Let us explain shortly why: The idea is due to Lai and Wei in \cite{Lai1983} (see also \cite{Duflo1996}, p. 60-61).
For $n\in\mathbb{N}$, let $T_n$ be the first time $t$ such $\Lambda(t)\not\in [n^{-1} a(t),na(t)]$ or $|\Delta M^i_t|>n \alpha(t)$ for some $i$ or $\int_0^t \frac{\|r_s\|^2}{a(s)} {\rm d} s >n$. Then $T_n$ is an increasing sequence of stopping times and a.s. on $\Gamma\cap\{K\le n\}$, $T_n=\infty$.
Possibly extending the probability space, let $N$ be a Poisson process with intensity $a(t)$. For $n\in \mathbb{N}$, $i\in\{1,\dots,d\}$ and $t>0$, set $$\tilde{M}^{i}_t=M^i_{t\wedge T_n} + N_t-N_{t\wedge T_n}\ \text{ and } \ \tilde{r}_t=r_{t\wedge T_n}.$$ Then, $\tilde{M}$ and $\tilde{r}$ satisfy \eqref{eq:THMA1}, \eqref{eq:THMA2} and \eqref{eq:THMA4} a.s. on $\Omega$, with $K=n$, and on the event $\{T_n=\infty\}$, $\tilde{M}=M$ and $\tilde{r}=r$. Now set $$L_n=\int_0^\infty (\tilde{r}_s+R_s) {\rm d} s + \tilde{M}_\infty,$$ which is well defined on $\Gamma$. Then a.s. on the event $\Gamma_n:=\Gamma\cap\{K\le n\}$, we have $L_n=L$.
Suppose now that for all $n$, we have ${\mathsf P}[\Gamma_n\cap \{L_n=\eta\}]=0$, then we also have ${\mathsf P}[\Gamma\cap \{L=\eta\}]=\lim_{n\to\infty}{\mathsf P}[\Gamma_n\cap \{L=\eta\}]=\lim_{n\to\infty}{\mathsf P}[\Gamma_n\cap \{L_n=\eta\}]=0$.
Let $\tilde\Omega$ be the event that \eqref{eq:THMA1}, \eqref{eq:THMA2} and \eqref{eq:THMA4} is satisfied with non-random positive constant $K$. From now on, we suppose that $K$ is non-random and that \eqref{eq:THMA1}, \eqref{eq:THMA2} and \eqref{eq:THMA4} are satisfied a.s. on $\Omega$.
A first consequence is that, $M$, $[M^i]-\langle M^i\rangle$ and $\|M\|^2-A$, with $A=\sum_i \langle M^i\rangle$, are uniformly integrable martingales. Indeed, using Lemma VII.3.34 in \cite{Jacod2003}, p. 423, there are constant $k_1$ and $k_2$ such that
$${\mathsf E}\left[{\sup}_{0\le s\le t}|M_s^i|^4\right]\le k_1\left(\sup_{0\le s\le t,\omega\in\tilde \Omega}|\Delta M_t^i(\omega)|\right)^2\left({\mathsf E}\left[\langle M^i \rangle_t^2\right]\right)^{1/2}+k_2{\mathsf E}\left[\langle M^i \rangle_t^2\right].$$
Recall from (\ref{eq:THMA1}) and (\ref{eq:THMA2}) that $$\langle M^i\rangle_t=\int_0^t\Lambda_s^i{\rm d} s\le K\int_0^t a(s){\rm d} s<K\int_0^{\infty} a(s){\rm d} s,\quad |\Delta M^i_t|\le K\alpha(t)\le K\alpha(0)$$ for all $t\ge0$. It implies that ${\mathsf E}(\|M_t\|^4)$ is uniformly bounded and $M$ is thus uniformly integrable.
Without loss of generality, we also suppose that $p=0$ and $\eta=0$. Otherwise, one can replace $\mathcal{F}_t$, $M_t$, $r_t$ and $R_t$ by $\mathcal{F}_{t+p}$, $M_{t+p}-M_p$, $r_{t+p}+\beta'(t)\left(\eta-\int_0^pr_s{\rm d} s-M_p\right)$ and $R_{t+p}+\beta'(t)\left(\eta-\int_0^pR_s{\rm d} s-M_p\right)$ respectively, where $\beta:[0,\infty)\to (0,\infty)$ is some differentiable function such that $\beta(0)=1$, $\lim_{t\to\infty}\beta(t)=0$ and $\beta(t)=o(\alpha(t))$.
Set $G=\Gamma\cap \{L=0\}$. For $t\ge 0$, define $\rho_t=M_\infty-M_t,\ \tau_t=\int_t^\infty \Psi_s {\rm d} s\ \text{ and } T_t=\rho_t+\tau_t.$ Then $T_t=L-S_t$ and on $G$, $T_t=-S_t$.
Since for all $t>0$, $(\|M_{s}-M_t\|^2-(A_s-A_t),\ s\ge t)$ is a uniformly integrable martingale, we have that for all $t>0$, ${\mathsf E}[\|\rho_t\|^2|\mathcal{F}_t]={\mathsf E}\big[ A_\infty-A_t|\mathcal{F}_t\big]={\mathsf E}\big[ \int_t^{\infty}\Lambda(s){\rm d} s|\mathcal{F}_t\big]$ and therefore
$$K^{-1}\alpha^2(t)\le {\mathsf E}[\|\rho_t\|^2|\mathcal{F}_t]\le K\alpha^2(t).$$
Using Lemma VII.3.34 in \cite{Jacod2003} to the martingale $(M_s-M_t,\ s\ge t)$, we have
\begin{align*}{\mathsf E}\left[|M_s^i-M_t^i|^4|\mathcal{F}_t\right]& \le k_1\left(\sup_{t\le u\le s,\omega\in\tilde \Omega}|\Delta M_u^i(\omega)|\right)^2\left({\mathsf E}\left[\left(\langle M^i \rangle_s-\langle M^i \rangle_t\right)^2|\mathcal{F}_t\right]\right)^{1/2}\\
&+k_2{\mathsf E}\left[\left(\langle M^i \rangle_s-\langle M^i \rangle_t\right)^2|\mathcal{F}_t\right]\\ &\le k_1K^3\alpha^2(t)\int_t^s a(u){\rm d} u+k_2K^2\left(\int_t^s a(u){\rm d} u\right)^2. \end{align*}
Hence, for all $t>0$, there is a constant $k$ such that ${\mathsf E}[\|\rho_t^4\|\mathcal{F}_t]\le k\alpha^4(t).$
Set $c_0=K^{-\frac32} k^{-\frac12}$. Since
${\mathsf E}\big[\|\rho_t\|^2|\mathcal{F}_t\big]
\le {\mathsf E}\big[\|\rho_t\||\mathcal{F}_t\big]^{\frac23}{\mathsf E}\big[\|\rho_t\|^4|\mathcal{F}_t]^\frac13,$ we have that for all $t$, \begin{align*}
{\mathsf E}[\|\rho_t\||\mathcal{F}_t]& \ge c_0 \alpha(t). \end{align*}
Let $U$ be a Borel function from $\mathbb{R}^d\setminus\{0\}$ onto the set of $d\times d$ orthogonal matrices such that $U(a)[a/\|a\|]=e_1$ (with $e_1=(1,0,\dots,0)$). Then on $G$, \begin{align*}
&\|T_t\| e_1+U(S_t) T_t = 0\\
&\big\|\|\rho_t\| e_1+ U(S_t)\rho_t\big\|\le 2\|\tau_t\|. \end{align*}
Set $G_t:=\{{\mathsf P}(G|\mathcal{F}_t)>\frac12\}$. Then for all $t>0$ (using in the second inequality that $S_t$ is $\mathcal{F}_t$-measurable and that ${\mathsf E}[\rho_t|\mathcal{F}_t]=0$) \begin{align*} {\mathsf P}(G_t)
&\le \frac{1}{c_0\alpha(t)}\big\|{\mathsf E}\big[ 1_{G_t}{\mathsf E}[\|\rho_t\|e_1|\mathcal{F}_t]\big]\big\|\\
&\le \frac{1}{c_0\alpha(t)}\big\|{\mathsf E}\big[ 1_{G_t}{\mathsf E}[\|\rho_t\|e_1 + U(S_t)\rho_t|\mathcal{F}_t]\big]\big\|\\
&\le \frac{1}{c_0\alpha(t)}\big\|{\mathsf E}\big[ 1_{G}{\mathsf E}[\|\rho_t\|e_1 + U(S_t)\rho_t|\mathcal{F}_t]\big]\big\|\\
& + \frac{1}{c_0\alpha(t)}\big\|{\mathsf E}\big[ (1_{G_t}-1_{G}){\mathsf E}[\|\rho_t\|e_1 + U(S_t)\rho_t|\mathcal{F}_t]\big]\big\|\\
&\le \frac{2}{c_0\alpha(t)}{\mathsf E}\big[ 1_{G}\|\tau_t\|\big] + \frac{2}{c_0\alpha(t)}\left({\mathsf E}\big[ (1_{G_t}-1_{G})^2\right)^{\frac12} \left({\mathsf E}[\|\rho_t\|^2]\right)^{\frac12}. \end{align*}
Note that $$\lim_{t\to\infty}{\mathsf E}\big[(1_{G_t}-1_{G})^2\big]=0 \quad \text{ and } \quad{\mathsf E}[\|\rho_t\|^2]\le c_+\alpha^2(t).$$ Thus, the second term converges to $0$. For the first term, (using Cauchy-Schwarz inequality to obtain the first term on the right hand side) \begin{align*}
{\mathsf E}\big[ 1_{G}\|\tau_t\|\big]
\le & \; {\mathsf E}\left[ 1_{G}\int_t^\infty \|r_s\| {\rm d} s\right] + {\mathsf E}\left[ 1_{G}\int_t^\infty \|R_s\| {\rm d} s\right]\\
\le & \; \alpha(t) {\mathsf E}\left[ 1_{G}\left(\int_t^\infty \frac{\|r_s\|^2}{a(s)} {\rm d} s\right)^{\frac12}\right] + o(\alpha(t)) = o(\alpha(t)) \end{align*} using Cauchy-Schwarz inequality, Lebesgue's Dominated Convergence Theorem and the hypotheses. We thus obtain that ${\mathsf P}(G)=\lim_{t\to\infty}{\mathsf P}(G_t)=0$. \end{proof}
\end{document} |
\begin{document}
\title[Linear Instability of Sasaki Einstein and nearly parallel ${\rm G}_2$ manifolds]{Linear Instability of Sasaki Einstein and \\ nearly parallel ${\rm G}_2$ manifolds} \author{Uwe Semmelmann} \address{Institut f\"ur Geometrie und Topologie \\ Fachbereich Mathematik\\ Universit{\"a}t Stuttgart\\ Pfaffenwaldring 57 \\ 70569 Stuttgart, Germany} \email{uwe.semmelmann@mathematik.uni-stuttgart.de}
\author{Changliang Wang} \address{School of Mathematical Sciences and Institute for Advanced Study, Tongji University, Shanghai 200092, China} \email{wangchl@tongji.edu.cn}
\author{M. Y.-K. Wang} \address{Department of Mathematics and Statistics, McMaster University, Hamilton, Ontario, L8S 4K1, CANADA} \email{wang@mcmaster.ca}
\date{revised \today}
\begin{abstract} {In this article we study the stability problem for the Einstein metrics on Sasaki Einstein and on complete nearly parallel ${\rm G}_2$ manifolds. In the Sasaki case we show linear instability if the second Betti number is positive. Similarly we prove that nearly parallel $\rm G_2$ manifolds with positive third Betti number are linearly unstable. Moreover, we prove linear instability for the Berger space ${\rm SO}(5)/{\rm SO}(3)_{irr} $ which is a $7$-dimensional homology sphere with a proper nearly parallel ${\rm G}_2$ structure.} \end{abstract}
\maketitle
\noindent{{\it Mathematics Subject Classification} (2000): 53C25, 53C27, 53C44}
\noindent{{\it Keywords:} linear stability, real Killing spinors, nearly parallel ${\rm G}_2$ manifolds, Sasaki Einstein manifolds}
\setcounter{section}{0}
\section{\bf Introduction}
In this article we continue the investigation of the linear instability of Einstein manifolds admitting a non-trivial real Killing spinor. Recall that the linear stability of complete Einstein manifolds admitting a non-trivial parallel or imaginary Killing spinor has been established in \cite{DWW05}, \cite{Kr17}, and \cite{Wan17}. It is therefore of some interest to consider the stability problem for complete Einstein manifolds which admit a real Killing spinor, especially because these manifolds admit more geometric structure than generic Einstein manifolds with positive scalar curvature, and in view of their role in supersymmetric grand unification theories in physics over the years.
We refer the reader to \cite{WW18} for a summary of the various notions of stability under consideration and for a description of different cases of the general problem. Here we only note that in this paper linear instability refers to the second variation at Einstein metrics for the Einstein-Hilbert action. Explicitly, this means that there exists a non-trivial symmetric $2$-tensor $h$ that is {\em transverse traceless} (``TT" in short), i.e., ${\rm tr}_g h = 0, \delta_g h = 0$, such that \begin{equation} \label{instability} \langle \nabla^* \nabla h - 2 \mathring{R} h, h \rangle_{L^2(M, g)} = \langle (\Delta_L - 2E)h, h \rangle_{L^2(M, g)} < 0, \end{equation} where $E$ is the Einstein constant, $\Delta_L$ is the positive Lichnerowicz Laplacian, $\mathring{R}h$ is the action of the curvature tensor on symmetric $2$-tensors, and $\nabla^* \nabla$ is the (positive) rough Laplacian. Condition (\ref{instability}) implies linear instability with respect to Perelman's $\nu$-entropy as well as dynamical instability with respect to the Ricci flow (by a theorem of Kr\"oncke \cite{Kr15}).
Recall that complete spin manifolds with constant positive sectional curvature are stable Einstein manifolds. They are exceptional in the sense that they also admit a maximal family of real Killing spinors. For the sake of a smoother exposition we will henceforth exclude these manifolds from discussion. It then follows that the only even dimension for which there are complete metrics admitting a non-trivial real Killing spinor is six. In this situation the Einstein manifolds are strict nearly K\"ahler or else isometric to round $S^6$. (Recall that a strict, nearly K\"ahler manifold is an almost Hermitian manifold for which the almost complex structure $J$ is non-parallel and satisfies $(\nabla_X J)X = 0$, where $X$ is any tangent vector and $\nabla$ is the Levi-Civita connection.) The round sphere is clearly stable, but it is distinguished by having a maximal family of real Killing spinors. For the first case we showed in \cite{SWW20} that if either the second or third Betti number of the manifold is nonzero, then the nearly K\"ahler metric is linearly unstable. A topological consequence of this fact is that a complete, strict, nearly K\"ahler $6$-manifold that is linearly stable must be a rational homology sphere.
In this paper, we first consider the Sasaki Einstein case, which arises in all odd dimensions. To simplify matters, we will take a Sasaki Einstein manifold to be an odd-dimensional Einstein Riemannian manifold $(M^{2n+1}, g)$ together with \begin{enumerate} \item[(a)] a contact $1$-form $\eta$ whose dual vector field $\xi$ is a unit-length Killing field, (i.e., $\eta \wedge (d\eta)^n \neq 0 $ everywhere on $M$, $\eta(\xi)=1$, and $L_{\xi} g = 0$), ($K$-contact condition) \item[(b)] an endomorphism $\Phi: TM \rightarrow TM$ satisfying, for all tangent vectors $X$ and $Y$, the equations $$ \Phi^2 = - \mbox{${\mathbb I}$} + \eta \otimes \xi, \,\,\,\,\, g(\Phi(X), \Phi(Y)) = g(X, Y) - \eta(X) \eta(Y), \,\, \mbox{\rm and}$$ \item[(c)] $d\eta(X, Y) = 2 g(X, \Phi(Y)).$ \end{enumerate} Systematic expositions of Sasaki Einstein manifolds can be found in \cite{BFGK91} and especially in \cite{BG08}. An equivalent characterization of a Sasaki Einstein manifold is an Einstein manifold whose metric cone has holonomy lying in ${\rm SU}(n+1)$ \cite{Ba93}. Our first result is:
\begin{thm} \label{SE} A complete Sasaki Einstein manifold of dimension $>3$ with non-zero second Betti number $b_2$ must be linearly unstable with respect to the Einstein-Hilbert action and hence dynamically unstable for the Ricci flow. More precisely, the coindex of such an Einstein metric, i.e., the dimension of the maximal subspace of the space of transverse traceless symmetric $2$-tensors on which $($\ref{instability}$)$ holds, is $\geq b_2$. \end{thm}
This result may be viewed as an odd-dimensional analogue of the observation by Cao-Hamilton-Ilmanen \cite{CHI04} that a Fano K\"ahler Einstein manifold with second Betti number $> 1$ is linearly unstable. Note that our result applies to irregular Sasakian Einstein manifolds as well. For dimension $5$, it follows immediately that if a complete Sasaki Einstein $5$-manifold is linearly stable then it must be a rational homology sphere. One can also apply the above result to any of the $2$-sphere's worth of compatible Sasaki Einstein structures on a $3$-Sasakian manifold with nonzero second Betti number to deduce their linear instability. Recall that
$3$-Sasakian manifolds are automatically Einstein.
In dimension seven, a complete simply connected Riemannian manifold admitting a non-trivial real Killing spinor is called a {\it nearly parallel $\mathrm G_2$ manifold}. As we have excluded round spheres, such a manifold falls into one of three classes depending on whether the dimension of the space of Killing spinors is $3, 2$ or $1$. These are respectively the $3$-Sasakian, Sasaki Einstein but not $3$-Sasaki, and the proper nearly parallel $\rm{G}_2$ cases. For this family we obtained the following
\begin{thm} \label{b3} Let $(M^7, g)$ be a complete nearly parallel ${\rm G}_2$-manifold. Then the coindex of the Einstein metric $g$ is at least $b_3$. If the manifold is in addition Sasaki Einstein then the coindex is at least $b_2 + b_3$. Hence in the latter case if such an manifold is linearly stable, then it must be a rational homology sphere. \end{thm}
While there are numerous examples of complete Sasakian Einstein manifolds with nonzero second Betti number \cite{BG08} there are relatively fewer examples with $b_3 \neq 0$. To our knowledge they include fourteen examples due to C. Boyer \cite{Bo08} and ten recent examples by R. G. Gomez \cite{Go19}, all occurring in dimension $7$. In Gomez's examples, $10 \leq b_3 \leq 20$. Note, however, that for complete $3$-Sasakian manifolds Galicki and Salamon \cite{GaS96} showed that all their odd Betti numbers must vanish. We also do not know any example of a {\it proper} nearly parallel ${\mathrm G}_2$ manifold with nonzero third Betti number. Settling the existence question for such an example would be of interest.
We turn next to the special case of a simply connected closed Einstein $7$-manifold that is homogeneous with respect to the isometric action of some semisimple Lie group. In \cite{WW18} it was shown that if the manifold is not locally symmetric then it must be linearly unstable with the possible exception of the isotropy irreducible space ${\rm Sp}(2)/{\rm Sp}(1)\approx {\rm SO}(5)/{\rm SO}(3)$. Here the embedding of ${\rm Sp}(1)$ is given by the irreducible complex $4$-dimensional representation (which is symplectic), while the embedding of ${\rm SO}(3)$ is given by the irreducible complex $5$-dimensional representation (which is orthogonal). The Einstein metric in this unresolved case is known to be of proper nearly parallel $\rm{G}_2$ type.
\begin{thm} \label{Berger} The isotropy irreducible Berger space ${\rm Sp}(2)/{\rm Sp}(1)_{irr} \approx {\rm SO}(5)/{\rm SO}(3)_{irr} $ is linearly unstable. \end{thm}
Recall that the homogeneous Einstein metrics (two up to isometry) on the Aloff-Wallach manifolds $N_{k, l} = {\rm SU}(3)/T_{kl}$, where $T_{kl}$ is a closed circle subgroup, are also of proper nearly parallel $\rm{G}_2$ type, with the exception of one of the Einstein metrics on $N_{1,1}$, which is $3$-Sasakian. It was shown in \cite{WW18} that all these Einstein metrics are linearly unstable. One therefore deduces the conclusion that all compact simply connected homogeneous Einstein $7$-manifolds (many of them admitting a non-trivial real Killing spinor) are linearly unstable.
Interestingly, the Berger space is a rational homology sphere, so the methods based on constructing destabilizing directions via harmonic forms do not apply. Likewise, the Stiefel manifold ${\rm SO}(5)/{\rm SO}(3)$ (the embedding of ${\rm SO}(3)$ is the usual $3$-dimensional vector representation) is also a homology $7$-sphere. The Einstein metric here is instead of regular Sasakian Einstein type (the Stiefel manifold is a circle bundle over the Hermitian symmetric Grassmanian ${\rm SO}(5)/S({\rm O}(3){\rm O}(2))$. It was shown to be linearly unstable in \cite{WW18} by examining the scalar curvature function for homogeneous metrics. But there are also non-homogeneous $\nu$-unstable directions arising from eigenfunctions.
The proof of Theorem \ref{Berger} is based on a result of independent interest: in Section \ref{Berger section} we will show that the Berger space admits a $5$-dimensional space of trace and divergence free Killing $2$-tensors. These are symmetric $2$-tensors with vanishing complete symmetrisation of their covariant derivative. A special motivation for studying Killing tensors stems from the fact that they define first integrals of the geodesic flow, i.e. functions constant on geodesics. On the Berger space the constructed Killing tensors turn out to be eigentensors for the Lichnerowicz Laplacian for an eigenvalue less than the critical $2E$.
\noindent{\bf Acknowledgements:}
The first author would like to thank P.-A. Nagy and G. Weingart for helpful discussions on the topic of this article. He is also grateful for support by the Special Priority Program SPP 2026 "Geometry at Infinity" funded by the DFG. The third author acknowledges partial support by a Discovery Grant of NSERC.
\section{\bf Sasaki Einstein manifolds with $b_2 >0$}
Let $(M^{2n+1}, g, \xi, \eta, \Phi)$ be a Sasaki Einstein manifold as defined in the Introduction. (This definition may be logically redundant but it allows us to keep technicalities to a minimum.) It then follows that $\Phi$ is skew-symmetric, and for all tangent vectors $X$ in $M$ we have $$\nabla_X \xi = - \Phi(X).$$ Also, the Einstein constant $E$ of $g$ is fixed to be $2n$, and this in turn fixes the Killing constant of the Killing spinors to be $\pm \frac{1}{2}$ depending on the orientation.
The Killing field $\xi$ gives rise to a Riemannian foliation structure on $M$, as well as an orthogonal decomposition $$ TM = \mathscr{L} \oplus \mathscr{N} $$ where $\mathscr{L}$ is the line bundle determined by $\xi$ and $\mathscr{N}$ is the normal bundle to the foliation. There is a transverse K\"ahler structure on $\mathscr{N}$ with a transverse Hodge theory associated with the basic de Rham complex. Good references for this material are
\cite{EH86} and sections 7.2, 7.3 in \cite{BG08}. In particular, the transverse almost complex
structure is given by the endomorphism $\Phi|\mathscr{N}.$
Recall that a $k$-form $\omega$ on $M$ is called {\em basic} if $L_\xi \omega = 0 = \xi \lrcorner \, \omega$. These properties are satisfied by $\Phi$ as well by the $K$-contact property. In the transverse Hodge theory, the role of the K\"ahler form is played by $d\eta$, which is $\Phi$-invariant. As usual we can define the adjoint $\Lambda$ of the wedge product with $d\eta$ and the basic elements of the kernel of $\Lambda$ are called the primitive basic forms. By Proposition 7.4.13 in \cite{BG08}, a lemma of Tachibana shows that any harmonic $k$-form on $M$ is horizontal, and this in turn implies that it is basic, and primitive and harmonic for the basic cohomology.
We shall also need the fact that a harmonic $2$-form on $M$ is $\Phi$-invariant, i.e., as a basic $2$-form it is of type $(1, 1)$. This follows from the general fact that there are no nonzero basic harmonic $(0, k)$-forms, $1 \leq k \leq n, $ which is the analogue of Bochner's theorem that on a compact complex manifold with positive first Chern class and complex dimension $n$, there are no non-trivial holomorphic $k$-forms, $1 \leq k \leq n$. A proof of this analogue is indicated on pp. 66-67 of \cite{VC17} (see also \cite{GNT16}). Note that the $\Phi$-invariance of the transverse covariant derivative and transverse curvature tensor, which is the analogue of the K\"ahler condition, is needed in the argument.
Now let $\alpha$ be a harmonic $2$-form on our Sasaki Einstein manifold. The candidate for a destabilizing direction is the symmetric $2$-tensor $h_{\alpha}$ defined by \begin{equation} \label{SE-TT}
h_{\alpha}(X, Y) := \alpha(X, \Phi(Y)) \end{equation} for arbitrary tangent vectors $X, Y$ on $M$.
It turns out that an efficient method to compute the action of the Lichnerowicz Laplacian on $h_{\alpha}$ is to take advantage of the canonical metric connection with skew torsion preserving the Sasakian structure rather than using the Levi-Civita connection. We will also use the fact that the Lichnerowicz Laplacian can be written as $\Delta_L = \nabla^*\nabla + q(R)$, where $q(R)$ is an endomorphism on symmetric tensors fibrewise defined by $ q(R) = \sum (e_i \wedge e_j)_\ast \circ R (e_i \wedge e_j)_\ast $ with an orthonormal basis $\{e_i \}$, where for any $A \in \Lambda^2 \cong \mbox{${\mathfrak s \mathfrak o}$}(n)$ we denote with $A_\ast$ the natural action of $A$ on symmetric tensors. See \cite{SWe19} for the general context of these endomorphisms and their relation with Weitzenb\"ock formulae.
Recall that Sasakian manifolds are equipped with a canonical metric connection $\bar\nabla$ defined by the equation $$ g(\bar\nabla_X Y, Z) \;=\; g(\nabla_X Y, Z) \;+\; \tfrac12 (\eta \wedge d\eta) (X, Y, Z) \ , $$ where the $3$-form $\eta \wedge d\eta$ is precisely the torsion of the connection. Note that the canonical connection $\bar \nabla$ for any tangent vector $X$ can also be written as $\bar\nabla_X = \nabla_X + A_X$ with $A_X:= - \eta(X) \, \Phi + \xi \wedge \Phi(X)$. $\bar \nabla$ preserves the basic forms. The restricted holonomy group of $\bar \nabla$ lies in ${\rm U}(n) \subset {\rm SO}(2n+1)$ and we have $\bar\nabla \Phi = 0$, $\bar \nabla \eta = 0$, and $\bar \nabla d\eta = 0$.
Furthermore, the curvature $\bar R$ of $\bar\nabla$ and its action on tensors is given by $$ \bar R_{X, Y} \;=\; R_{X, Y} \,+\, \tfrac12 \, d\eta(X,Y) \, d\eta \,-\, \Phi(X) \wedge \Phi(Y) \,+\, \xi \wedge (X \wedge Y)\xi, $$ where $(X \wedge Y) \xi := g(X, \xi) Y - g(Y, \xi) X$, and we have identified vectors with covectors as usual via $g$. Hence, we have $\bar R_{X, Y} = R_{X, Y} \,-\, \Phi(X) \wedge \Phi(Y) \,+\, \xi \wedge (X \wedge Y)\xi $ \,for the action of $\bar R_{X, Y} $ on $\Phi$-invariant tensors. As a consequence we obtain for these tensors the formula \begin{equation}\label{diff} q(\bar R) - q(R) \;=\; -\tfrac12 \sum (e_i \wedge e_j)_\ast \circ (\Phi(e_i) \wedge \Phi(e_j))_* \;+\; \sum (\xi \wedge e_j)_\ast \circ (\xi \wedge e_j)_\ast \ , \end{equation} where $q(\bar R)$ denotes the curvature endomorphism with respect to the connection $\bar\nabla$ and its curvature $\bar R$. On specific spaces this difference can be further computed. The result for the present situation is given in the following
\begin{lemma} Let $(M^{2n+1}, g, \xi, \eta, \Phi)$ be a Sasaki Einstein manifold. Then we have $$ q(\bar R) \,-\, q(R) \;=\; \left\{ \begin{array}{ll} \,\;\; 2\, \rm id & \qquad \mbox{\rm on} \quad \Lambda^2 {\rm T} M\\ -2 \, \rm id & \qquad \mbox{\rm on} \quad {\rm Sym}^2_0 {\rm T} M \\ \end{array} \right. $$ for $\Phi$-invariant and basic tensors. \end{lemma} \begin{proof} We start with a remark to understand the action of the curvature term $q(R)$ on $2$-tensors (symmetric or skew-symmetric). Let $A, B \in \Lambda^2 {\rm T} \cong {\rm End}^- {\rm T}$ and let $h$ be a $2$-tensor. Then the composed action of $A$ and $B$ on $h$ is given by $$ (A_* B_* h) (X, Y) \;=\; h(B A X, Y) \,+\, h(A X, B Y) \,+\, h(B X, A Y) \,+\, h(X, B A Y ) \ . $$ Hence, for computing the action of the first summand in \eqref{diff} on $2$-tensors we need the following formula on tangent vectors $X$ $$ -\tfrac12 \sum (\Phi(e_i) \wedge \Phi(e_j))_\ast \,(e_i \wedge e_j)_*X \;=\; - \sum (\Phi(X) \wedge \Phi(e_j))_* e_j \;=\; -\Phi^2(X) \;=\; X \mod \xi $$ where we can neglect any multiples of $\xi$ since in the end we want to apply our difference formula to basic tensors. Similarly we compute the action of the second summand in \eqref{diff} on vector fields. Here we obtain $$ \sum (\xi \wedge e_j)_\ast \, (\xi \wedge e_j)_\ast X \;=\; g(\xi, X) \, \sum (\xi \wedge e_j)_* e_j \,-\, (\xi \wedge X)_* \xi \;=\; - X \mod \xi \ . $$ Next we have to compute $$ - \tfrac12 \sum [h((e_i \wedge e_j)_*X, (\Phi(e_i) \wedge \Phi(e_j))_* Y) \;+\; h( (\Phi(e_i) \wedge \Phi(e_j))_*X, (e_i \wedge e_j)_*Y)] \phantom{xxxxxxxx} $$ \begin{eqnarray*} &=& -\sum [ h(e_j, (\Phi(X) \wedge \Phi(e_j)_*Y) \;+\; h((\Phi(Y) \wedge \Phi(e_j))_*X, e_j )]\\[1ex] &=& \quad \sum [g(\Phi(e_j), Y) h(e_j, \Phi(X)) \;+\; [g(\Phi(e_j), X) h(\Phi(Y), e_j)]\\[.5ex] &=& -2 h(\Phi(Y), \Phi(X)) \;=\; -2 h(Y, X) \;=\; \left\{ \begin{array}{ll} \;\; 2\, h(X, Y)& \qquad \mbox{for} \quad \; h \in \Lambda^2 {\rm T} M\\ -2 \, h(X, Y) & \qquad \mbox{for } \quad h \in {\rm Sym}^2 {\rm T} M \\ \end{array} \right. \end{eqnarray*} Finally, we note $\sum h((\xi \wedge e_j)_*X, (\xi \wedge e_j)_*Y) = g(\xi, X) g(\xi, Y) \sum h(e_j, e_j) = 0$ on basic and trace-free $2$-tensors $h$. Then combining the formulas above finishes the proof of the lemma. \end{proof}
\begin{lemma} On tracefree, divergence-free, $\Phi$-invariant and basic $2$-tensors we have the formula: $\bar\nabla^* \bar\nabla \,-\, \nabla^*\nabla \,=\, - 2 \, \rm id$. \end{lemma} \begin{proof} Let $\{e_i\}$ be a local orthonormal basis with $\nabla_{e_i} e_i = 0 =\bar \nabla_{e_i} e_i $ at an arbitrary but fixed point $p \in M$. Recall that the torsion of $\bar\nabla$ is skew-symmetric. Computing at the point $p$ we get \begin{eqnarray*} \bar\nabla^* \bar\nabla \,-\, \nabla^*\nabla &=& - \sum \bar \nabla_{e_i} \bar \nabla_{e_i} \;+\; \sum \nabla_{e_i} \nabla_{e_i} \;=\; \sum (\bar \nabla_{e_i} - A_{e_i} ) (\bar \nabla_{e_i} - A_{e_i} ) \;-\; \sum \bar \nabla_{e_i} \bar \nabla_{e_i} \\[1ex] &=& \quad \sum A_{e_i }A_{e_i } \;-\; 2 \sum A_{e_i } \bar \nabla_{e_i} \ . \end{eqnarray*} On $\Phi$-invariant and basic tensors the first summand reduces to $ \sum A_{e_i }A_{e_i } = \sum A_{e_i } (\xi \wedge \Phi(e_i))_*$. Computing this first on tangent vectors $X$ (with the factors in reversed order as above) we obtain $$ \sum (\xi \wedge \Phi(e_i))_* A_{e_i } X \;=\; \sum (\xi \wedge \Phi(e_i))_* (-\eta(e_i) \Phi(X) \,+\, (\xi \wedge \Phi(e_i))_*X) \phantom{xxxxxxxx} $$ \begin{eqnarray*} &=& \sum (\xi \wedge \Phi(e_i))_* (\eta(X) \Phi(e_i) \,-\, g(\Phi(e_i), X) \, \xi ) \;=\; - \sum g(\Phi(e_i), X) \Phi(e_i) \mod \xi\\[1ex] &=& \Phi^2(X) \mod \xi \;=\; - X \mod \xi \ . \end{eqnarray*} Here we used $\Phi(\xi)= 0$ and $\xi \perp \mathrm{Im} (\Phi)$. Moreover, for tracefree, $\Phi$-invariant and basic $2$-tensors $h$ we obtain $$ \sum h(A_{e_i} X, (\xi \wedge \Phi(e_i))_*Y ) \;+\; h( (\xi \wedge \Phi(e_i))_* X, A_{e_i} Y) \phantom{xxxxxxxx}\phantom{xxxxxxxxxxxx} $$ $$ \;=\; \sum h( - \eta(e_i) \Phi(X) \,+\, g(\xi, X) \Phi(e_i), \,g(\xi, Y)\, \Phi(e_i)) \;+\; (X \leftrightarrow Y) \phantom{xxxxxxxx} $$ $$ = \; g(\xi, X) \, g(\xi, Y) \, \sum h(\Phi(e_i), \Phi(e_i)) \;+\; (X \leftrightarrow Y) \;=\; 0 \ .\phantom{xxxxxxxxxxxxxx} $$ The last equation holds since $h$ is tracefree and $\Phi$-invariant. The calculation so far shows that $\sum A_{e_i }A_{e_i } h = -2h$. Finally we have to compute the second summand in our formula for $\bar\nabla^* \bar\nabla \,-\, \nabla^*\nabla $. Here we obtain \begin{eqnarray*} \sum A_{e_i } \bar \nabla_{e_i} h &=& \sum(-\eta(e_i)\Phi \,+\, (\eta \wedge \Phi(e_i))_* \bar \nabla_{e_i} h \;=\; \sum (\eta \wedge \Phi(e_i))_* \bar \nabla_{e_i} h
\end{eqnarray*}
since $\bar \nabla_{e_i} h$ is again trace-free, $\Phi$-invariant, and basic. We conclude $\sum A_{e_i } \bar \nabla_{e_i} h = 0$ since
we have $\sum (\eta \wedge \Phi(e_i))_* X = \sum (g(\xi, X) \, \Phi(e_i) \,-\, g(\Phi(e_i), X) \,\xi)$.
Indeed, clearly
$$\sum (\eta \wedge \Phi(e_i))_* \bar \nabla_{e_i} h(X, Y) = 0,$$
if $g(X, \xi)=g(Y, \xi)=0$ or $X=Y=\xi$, since $\bar \nabla_{e_{i}} h$ is basic. Moreover, for $g(\xi, Y)=0$, a straightforward computation gives
$$\sum (\eta \wedge \Phi(e_i))_* \bar \nabla_{e_i} h(\xi, Y)=(\delta h)(\Phi(Y))=0,$$
since $h$ is divergence-free. \end{proof}
We consider the two Laplace type operators $\Delta = \nabla^*\nabla + q(R)$ and $\bar \Delta = \bar\nabla^*\bar \nabla + q(\bar R)$, where the operator $\Delta$ is just the Lichnerowicz operator on tensors. The operator $\bar \Delta$ has the important property that it commutes with parallel bundle maps (see \cite{SWe19}, p. 283). Combining the last two lemmas we obtain
\begin{cor} On $\Phi$-invariant, divergence-free and basic tensors we have $$ \bar \Delta \,-\, \Delta \;=\; \left\{ \begin{array}{ll} \quad 0 & \qquad \mbox{on} \quad \Lambda^2 {\rm T} M\\ -4 \, \rm id & \qquad \mbox{on} \quad {\rm Sym}^2_0 {\rm T} M \\ \end{array} \right. $$ \end{cor}
Let $\alpha \in \Omega^2(M)$ be a harmonic $2$-form, which must be $\Phi$-invariant and basic. Hence we can apply the difference formula above and obtain $\bar \Delta \alpha = 0$. Now the bundle of $\Phi$-invariant $2$-forms can be identified with the bundle of symmetric $2$-tensors by the $\bar\nabla$-parallel bundle map $\alpha \mapsto h_\alpha$, where the symmetric $2$-tensor $h_\alpha$ is given by (\ref{SE-TT}). Then, because $\bar\Delta$ commutes with parallel bundle maps we also have $\bar\Delta h_\alpha = 0$. Using again the difference formula above, now for the case ${\rm Sym}^2 {\rm T} M$, we obtain $\Delta h_\alpha = 4 h_\alpha$. As the Einstein constant is $2n$, $(\Delta - 2E) h_{\alpha} = (4-4n) h_{\alpha}$ and hence the instability condition (\ref{instability}) is satisfied when $n > 1$.
It remains to check that $h_\alpha$ is a TT-tensor. First, we see that ${\rm tr}_g (h_\alpha) = g(\alpha, d\eta) = 0$ since harmonic forms on Sasaki Einstein manifolds are primitive. Moreover, $\delta_g h_\alpha = 0$ follows by an easy calculation from the assumption $d^* \alpha = 0$ and the fact that $\alpha$ is basic. Thus we have proved
\begin{thm} $($Theorem \ref{SE}$)$ Let $(M^{2n+1}, g, \xi, \eta, \Phi)$ be a compact Sasaki Einstein manifold with $n>1$ and $b_2 >0$. Then the Einstein metric $g$ is linearly unstable. \mbox{$\Box$} \end{thm}
\section{\bf Properties of nearly parallel $\mathrm G_2$-manifolds} \label{facts}
In the remainder of the paper we will focus on the dimension $7$ case. In this section we will summarise some properties of nearly parallel ${\rm G}_2$ manifolds that we shall need.
Let $(M^7, g)$ be a nearly parallel $\mathrm G_2$-manifold, i.e., a complete spin manifold with a non-trivial real Killing spinor $\sigma$. For the moment we do not exclude the possibility that the dimension of the space of real Killing spinors is greater than one. We may assume that $\sigma$ has length $1$ and Killing constant $\frac{1}{2}$, i.e., $\nabla_X \sigma = \frac{1}{2} X \cdot \sigma$. In particular the scalar curvature is normalized as ${\rm scal}_g = 42$. Then the Killing spinor $\sigma$ determines a vector cross product by the condition $$ P_{\sigma}(X, Y) \cdot \sigma = X \cdot Y \cdot \sigma + g(X, Y) \sigma = (X \wedge Y) \cdot \sigma, $$ and hence a $3$-form $$ \varphi_{\sigma}(X, Y, Z) = g(P_{\sigma}(X, Y), Z).$$ For details of this construction, see e.g. \cite{FKMS97}. The stabilizers of this $3$-form belong to the conjugacy class ${\rm G}_2 \subset {\rm SO}(7)$ and we obtain a ${\rm G}_2$-structure on $M$, which we regard as a principal ${\rm G}_2$ bundle $Q_{\sigma}$ over $M$. There is a unique metric connection $\bar \nabla$ on this bundle with totally skew torsion. We let $\nabla$ denote the Levi-Civita connection of $g$.
Equivalently, nearly parallel $\mathrm G_2$-manifolds can be defined as Riemannian $7$-manifolds carrying a 3-form $\varphi$ whose stabilizer at each point is isomorphic to the group $\mathrm G_2$ and such that $d\varphi = \lambda \ast \varphi $ for some non-zero real number $\lambda$.
Since the tensor bundles over $M$ are associated fibre bundles of $Q_{\sigma}$, we obtain orthogonal decompositions of these bundles from the decompositions of the corresponding ${\rm SO}(7)$ representations upon restriction to ${\rm G}_2$. The decompositions which we need are \begin{itemize} \item[(i)] $\Lambda^2 {\rm T} = \Lambda^2_7 \oplus \Lambda^2_{14} $ \item[(ii)] ${\rm S}^2 {\rm T} = \mbox{${\mathbb I}$} \oplus {\rm S}^2_{27} $ \item[(iii)] $\Lambda^3 {\rm T} = \mbox{${\mathbb I}$} \oplus \Lambda^3_{7} \oplus \Lambda^3_{27}$ \item[(iv)] the spin bundle $\mbox{${\mathbb S}$} = \mbox{${\mathbb I}$} \oplus {\rm T}$ \end{itemize} where ${\rm T}$ denotes the tangent bundle of $M$, the rank of a sub-bundle is indicated by a subscript, $\mbox{${\mathbb I}$}$ denotes the ($1$-dimensional) trivial bundle, and we have identified orthogonal representations with their duals. We further have equivalences ${\rm S}^2_{27} \cong \Lambda^3_{27},$ and $\Lambda^2_7 \cong \Lambda^3_7 \cong {\rm T}$. Note that the trivial bundle in $\Lambda^3 {\rm T}$ is spanned by $\varphi_{\sigma}$ and that in $\mbox{${\mathbb S}$}$ is spanned by $\sigma$. We also let $\psi = \ast \varphi$, the Hodge dual of $\varphi$, which spans the trivial bundle in $\Lambda^4 {\rm T}$.
More explicitly, it is well-known (see \cite{Br87}) that \begin{itemize} \item[(a)] $ \Lambda^2_7 = \{ X \lrcorner \,\varphi: X \in {\rm T} \} = \{ \omega \in \Lambda^2 {\rm T}: \ast(\varphi \wedge \omega) = -2 \omega\}, $ \item[(b)] $ \Lambda^2_{14} = \{ \omega \in \Lambda^2 {\rm T}: \forall X \in {\rm T}, g(\omega, X \lrcorner \, \varphi)= 0\} =
\{ \omega \in \Lambda^2 {\rm T}: \ast(\varphi \wedge \omega) = \omega \}, $ \item[(c)] $ \Lambda^3_{7} = \{ X \lrcorner \, \psi : X \in {\rm T} \}, $ \item[(d)] $ \Lambda^3_{27} = \{ \omega \in \Lambda^3 {\rm T}: \omega \wedge \varphi = 0 = \omega \wedge \psi \}.$ \end{itemize}
\begin{prop} \label{harmonic-decomp} Let $(M^7,g,\varphi)$ be a closed nearly parallel ${\rm G}_2$ manifold. Then any harmonic $2$-form is a section of $\Lambda^2_{14} $ and any harmonic $3$-form is a section of $\Lambda^3_{27}$. \end{prop} \begin{proof} The Clifford product of a harmonic form with a Killing spinor vanishes, as was shown by O. Hijazi in \cite{Hi86}. Moreover, for a fixed spinor $\sigma$, the map $\omega \mapsto \omega \cdot \sigma: {\rm Cl}(\mbox{${\mathbb R}$}^7) \rightarrow \mbox{${\mathbb S}$}$
is a ${\rm Spin}(7)$-equivariant hence ${\rm G}_2$-equivariant homomorphism. Therefore by Schur's lemma, the components
of a form which correspond to ${\rm G}_2$-representations which do not occur in the spin representation, e.g.,
$\Lambda^2_{14}$ and $\Lambda^3_{27}$, also act trivially on $\sigma$. Hence in order to finish the proof of
the proposition it suffices to show that forms in $\Lambda^2_7, \Lambda^3_1$ and $\Lambda^3_{7}$ act non-trivially on the Killing spinor $\sigma$ of the ${\rm G}_2$ structure.
We will need the following formula for Clifford multiplication of forms: $$ (X\wedge \omega ) \cdot \; =\; X \cdot \omega \cdot \, + \; (X\lrcorner \, \omega) \cdot $$ Here $X$ is a tangent vector, $\omega$ an arbitrary $k$-form, and $\cdot$ denotes Clifford multiplication.
Using the cross product $P$ (where we have suppressed the dependence on $\sigma$) we calculate that $\sum P(e_i, P(e_i, X)) = -6 X$, where $\{e_i\}$ is an orthonormal basis of ${\rm T}$. Moreover, we need the following simple formulas for $\varphi$ and its Hodge dual $\psi=\ast \varphi$
\begin{enumerate} \item\quad $X \lrcorner \, \varphi \;=\; - \tfrac12 \sum e_i \wedge P(e_i, X)$
\item \quad $\psi=\ast \varphi \:=\; - \tfrac{1}{6} \, \sum (e_i \lrcorner \, \varphi ) \wedge (e_i \lrcorner \, \varphi ) $ \quad and \quad $X \, \lrcorner \ast \varphi \;=\; -\tfrac13 \sum P(e_i, X) \wedge (e_i \lrcorner \, \varphi ).$ \end{enumerate}
\noindent First we show that forms in $\Lambda^2_7 $ act non-trivially on the Killing spinor $\sigma$. We have $$ (X \lrcorner \, \varphi ) \cdot \sigma \;=\; - \tfrac12 \sum e_i \cdot P(e_i, X) \cdot \sigma \;=\; - \tfrac12 \sum e_i \cdot (e_i \cdot X + g(e_i, X))\cdot \sigma \;=\; 3 X \cdot \sigma \ . $$ Recall that the map $X \mapsto X \cdot \sigma$ is injective. Next we show the non-trivial action of $\Lambda^3_1$: $$ \varphi \cdot \sigma \;=\; \tfrac13 \sum e_i \wedge (e_i \lrcorner \, \varphi) \cdot \sigma \;=\; \tfrac13 \sum e_i \cdot (e_i \lrcorner \, \varphi) \cdot \sigma \;=\; - 7 \, \sigma \ . $$ Finally we have to show that $\Lambda^3_7 $ acts non-trivial on $\sigma$. Here we compute \begin{eqnarray*} (X \, \lrcorner \,\psi) \cdot \sigma &=& - \tfrac13 \sum (P(e_i, X ) \cdot (e_i \lrcorner \, \varphi) + P(e_i, P(e_i, X))) \cdot \sigma\\ &=&
- \tfrac13 \sum( -3e_i \cdot P(e_i, X) - 6 X ) \cdot \sigma \;=\; - 4 X \cdot \sigma \ . \end{eqnarray*} \end{proof}
\begin{rmk}
The same result was proved in \cite{DS20}, Thm. 3.8, Thm. 3.9. and in the $2$-form case also in \cite{BO19}, Rem. 4.
Note that also the corresponding statement for harmonic forms on $6$-dimensional nearly K\"ahler manifolds
(see \cite{Fos17}, \cite{V11}) can be reproved using Killing spinors and similar arguments as those above. \end{rmk}
\begin{rmk} The above construction and structures clearly depend smoothly on the unit Killing spinor $\sigma$ chosen and can be made in the Sasakian-Einstein (respectively $3$-Sasakian) case using just one of the circle's (resp. two-sphere's) worth of unit Killing spinors. By contrast, the harmonic forms associated to the metric $g$ do not depend on the structures determined by $\sigma$. \end{rmk}
As already mentioned in the introduction, there are three classes of nearly parallel $\mathrm G_2$-manifolds: $3$-Sasakian, Sasaki Einstein and the proper nearly parallel $\mathrm G_2$-manifolds. The homogeneous examples were classified in \cite{FKMS97}. In the proper case we only have the squashed $7$-sphere $S^7_{sq}$, the Aloff-Wallach spaces $N_{k,l}$ and the Berger space ${\rm SO}(5)/{\rm SO}(3)_{irr}$. The only other known class of proper nearly parallel $\mathrm G_2$-structures are given by the second Einstein metric in the canonical variation of the $3$-Sasaki metrics in dimension $7$. In particular $S^7_{sq}$ and $N_{1,1}$ belong to this class. For the purpose of our article we note that it is well-known that both Einstein metrics in the canonical variation are unstable. (See \cite{Be87}, 14.85 together with Fig. 9.72.)
\section{\bf Nearly parallel $G_2$ manifolds with $b_3 >0$}
In this section we prove \begin{thm} \label{dim7-b3} $($Theorem \ref{b3} $)$ Let $(M^7, g, \varphi)$ be a nearly parallel ${\rm G}_2$ manifold admitting a non-trivial harmonic $3$-form, i.e. with $b_3 > 0$. Then the Einstein metric $g$ is linearly unstable. \end{thm} \begin{proof} Let $\beta$ be a harmonic $3$-form on $M$. By Proposition \ref{harmonic-decomp} it is a section of $\Lambda^3_{27}$. Consider the tracefree symmetric $2$-tensor $h := j(\beta)$ defined by the identification map $j: \Lambda^3_{27} \rightarrow {\rm S}^2_0 {\rm T} ,$ which was first studied by Bryant \cite{Br05}. Since $\beta $ is harmonic and in particular closed, Proposition 6.1 in \cite{AS12} immediately implies $\Delta_L h = \tfrac{\tau_0^2}{4} h = \tfrac{2\, {\rm scal}_g}{21} h = 4h$, since we have normalized the scalar curvature to ${\rm scal}_g=42$, which corresponds to choosing $\tau_0 =4$. Hence, $h$ is a $\Delta_L$-eigentensor for the eigenvalue $4< 2E = 12$.
It remains to show that $h$ is also divergence-free and thus a TT-tensor, since it is trace-free by definition. Here an easy calculation, which is essentially also contained in \cite{AS12}, shows that $\delta j(\beta) = -2 P(d^* \beta)$ holds for any section $\beta$ of $\Lambda^3_{27}$, where $P$ is the vector cross product introduced in the previous section. Since a harmonic form $\beta$ is also co-closed we conclude that $h= j(\beta)$ is divergence free. \end{proof}
Note that K. Galicki and S. Salamon showed that the odd Betti numbers of a compact $3$-Sasakian manifold must vanish (see \cite{GaS96}, Theorem A). So the above result does not apply when the nearly parallel ${\rm G}_2$-manifold is $3$-Sasakian. As well, there is no known example of a proper nearly parallel $G_2$-manifold with $b_3>0$. However, there are some examples of Sasaki Einstein $7$-manifolds admitting non-trivial harmonic $3$-forms (see Introduction).
\section{\bf The Berger space} \label{Berger section}
The aim of this section is to show that the nearly parallel ${\rm G}_2$-metric on the Berger space $M^7 = {\rm SO}(5)/{\rm SO}(3)_{irr}$ is linearly unstable. Since the Berger space has no non-trivial harmonic forms we cannot use the arguments of the previous sections. Instead we will prove the instability by showing that there are symmetric Killing tensors which are eigentensors of the Lichnerowicz Laplacian with eigenvalue less than $2E$. We start with recalling a few facts on Killing tensors, referring to \cite{HMS17} for further details.
Recall that a symmetric tensor $h\in \Gamma({\rm Sym}^p {\rm T} M)$ is called a {\it Killing tensor} if the complete symmetrization of $\nabla h $ vanishes, i.e., if $(\nabla_X h)(X, \ldots, X) = 0$ holds for all tangent vectors $X$. Let $d : \Gamma({\rm Sym}^p {\rm T} M) \rightarrow \Gamma({\rm Sym}^{p+1} {\rm T} M)$ be the differential operator defined by $d h = \sum_i e_i \cdot \nabla_{e_i} h$, where $\{e_i\}$ is a local orthonormal basis and $\cdot$ now denotes the symmetric product. Then the Killing condition for the symmetric tensor $h$ is equivalent to $d h = 0$. For a trace-free symmetric Killing tensor $h\in \Gamma({\rm Sym}^p_0 {\rm T} M)$ it is easy to check that $h$ is also divergence-free, i.e., trace-free Killing tensors are TT-tensors (see Corollary 3.10 in \cite{HMS17}).
On compact Riemannian manifolds Killing tensors can be characterized using the Lichnerowicz Laplacian $\Delta_L$ acting on symmetric tensors. Recall that $\Delta_L$ can be written as $\Delta_L = \nabla^*\nabla + q(R)$. Then an easy calculation shows $\Delta_L h \ge 2 q(R) h $ on divergence-free symmetric tensors, with equality exactly for divergence-free Killing tensors $h$, i.e., $h\in \Gamma({\rm Sym}^2_0 {\rm T} M)$ is a Killing tensor if and only if $\Delta_L h = 2 q(R) h$ (see Proposition 6.2 in \cite{HMS17}).
The isotropy representation of the Berger space $ M^7 = {\rm SO}(5)/{\rm SO}(3)_{irr}$ is the unique $7$-dimensional irreducible representation of ${\rm SO}(3)$. It also defines an embedding of ${\rm SO}(3)$ into ${\rm G}_2$ and thus a ${\rm G}_2$-structure on $M^7$ which turns out to be a proper nearly parallel ${\rm G}_2$-structure (see \cite{Br87}, p. 567).
Replacing the groups ${\rm SO}(5)$ and ${\rm SO}(3)$ by their double covers, we can realize the Berger space also as $M^7= {\rm Sp}(2)/{\rm Sp}(1)_{irr}$ where ${\rm Sp}(1)$ is embedded by the unique $4$-dimensional irreducible representation. We write $\mathfrak{sp}(2) = \mathfrak{sp}(1) \oplus \mbox{${\mathfrak m}$}$, where $\mbox{${\mathfrak m}$}$ is the orthogonal complement of $\mathfrak{sp}(1)$ with respect to the Killing form $B$ of $\mathfrak{sp}(2)$. As usual, $\mbox{${\mathfrak m}$}$ is identified with the tangent space at the identity coset and, as mentioned above, is the irreducible $7$-dimensional representation of ${\rm Sp}(1)$. Recall that the irreducible complex representations of ${\rm Sp}(1)$ can be written as the symmetric powers ${\rm Sym}^k E$, where $E = \mbox{${\mathbb C}$}^2$ is the standard representation of ${\rm Sp}(1)$. In particular, we have $\mbox{${\mathfrak m}$}^{\scriptsize\mbox{${\mathbb C}$}} := \mbox{${\mathfrak m}$} \otimes \mbox{${\mathbb C}$} = {\rm Sym}^6 E$. We will need the following decomposition into irreducible summands: \begin{equation}\label{deco1} {\rm Sym}^2_0 \, \mbox{${\mathfrak m}$}^{\scriptsize\mbox{${\mathbb C}$}} \cong \Lambda^3_{27} \mbox{${\mathfrak m}$}^{\scriptsize\mbox{${\mathbb C}$}} \cong {\rm Sym}^4 E \oplus {\rm Sym}^8 E \oplus {\rm Sym}^{12} E \ . \end{equation}
The Peter-Weyl theorem and the Frobenius reciprocity now imply the following decomposition into irreducible summands of the left-regular representation of ${\rm Sp}(2)$ on sections of the vector bundle ${\rm Sym}^2_0 {\rm T} M^{\scriptsize\mbox{${\mathbb C}$}}$ \begin{equation}\label{deco2} \Gamma({\rm Sym}^2_0 \, {\rm T} M^{\scriptsize\mbox{${\mathbb C}$}}) \cong \overline \bigoplus_{k,l} \, V(k,l) \otimes \mathrm{Hom}_{{\rm Sp}(1)} (V(k,l), \, {\rm Sym}^2_0\, \mbox{${\mathfrak m}$}^{\scriptsize\mbox{${\mathbb C}$}}) \end{equation} where the sum goes over all pairs of integer $(k, l)$ with $k \ge l \ge 0$. Here $V(k,l)$ is the irreducible ${\rm Sp}(2)$-representation with highest weight $\gamma = (k,l)$, where $k$ corresponds to the short simple root, and it is easy to compute that the $\mathfrak{sp}(2)$-Casimir operator (with respect to the Killing form) acts on the representation space $V(k,l)$ as $-\tfrac{1}{12}(4k + k^2 + 2l + l^2)\rm id$. In particular, $V(2,0)$ is the adjoint representation of $\mathfrak{sp}(2)$ and the Casimir eigenvalue is $-1$, as it should be.
Interesting for us will be the representation $V(1,1)$ with Casimir eigenvalue $-\tfrac23$. It is easy to check that $V(1,1)$ is $5$-dimensional and that $V(1,1) = {\rm Sym}^4 E$ considered as an ${\rm Sp}(1)$-representation. Moreover, from $\mbox{${\mathfrak m}$}^{\scriptsize\mbox{${\mathbb C}$}} = {\rm Sym}^6 E$ and the decomposition \eqref{deco1} we conclude \begin{equation}\label{hom} \dim \mathrm{Hom}_{{\rm Sp}(1)}(V(1,1), \, {\rm Sym}^2_0 \, \mbox{${\mathfrak m}$}^{\scriptsize\mbox{${\mathbb C}$}}) = 1 \quad \mbox{and} \quad \mathrm{Hom}_{{\rm Sp}(1)}(V(1,1), \,\mbox{${\mathfrak m}$}^{\scriptsize\mbox{${\mathbb C}$}} ) = \{0\} \ . \end{equation} Using the program LiE for the decomposition of ${\rm Sym}^3 {\rm Sym}^6E$ as an ${\rm Sp}(1)$-representation it is also easy to check that $\mathrm{Hom}_{{\rm Sp}(1)}(V(1,1), \,{\rm Sym}^3 \mbox{${\mathfrak m}$}^{\scriptsize\mbox{${\mathbb C}$}}) = \{0\}$. This has the following important consequence:
\begin{lemma} The space $V(1,1) \subset \Gamma({\rm Sym}^2_0 \, {\rm T} M^{\scriptsize \mbox{${\mathbb C}$}} )$ consists of tracefree Killing tensors. \end{lemma} \begin{proof} Killing $2$-tensors are by definition symmetric tensors in the kernel of the differential operator $d :\Gamma({\rm Sym}^2 \, {\rm T} M) \rightarrow \Gamma({\rm Sym}^3 \, {\rm T} M)$ introduced above. In our situation the operator $d$ is an ${\rm Sp}(2)$-invariant differential operator.
Hence, with respect to the decomposition \eqref{deco2} it restricts to an invariant map $$ V(1,1)\otimes \mathrm{Hom}_{{\rm Sp}(1)}(V(1,1), \, {\rm Sym}^2_0 \, \mbox{${\mathfrak m}$}^{\scriptsize\mbox{${\mathbb C}$}}) \longrightarrow V(1,1) \otimes \mathrm{Hom}_{{\rm Sp}(1)}(V(1,1), \,{\rm Sym}^3 \mbox{${\mathfrak m}$}^{\scriptsize\mbox{${\mathbb C}$}}) \ , $$ which has to be zero since the space on right side vanishes. A similar argument shows that also the divergence of elements in $V(1,1)$ has to be zero. However, since elements in $V(1,1)$ are trace-free by definition this was already clear from a remark above. \end{proof}
Since we know that $\Delta_L$ acts by the curvature term $2 q(R)$ on divergence-free Killing tensors it remains to compute the action of $q(R)$ on the space $V(1,1)$. For doing this we will use the comparison formula (5.33) in \cite{AS12}. Let $(M^7, g, \varphi)$ be a nearly parallel $\rm G_2$ manifold, with $d \varphi = \tau_0 \ast \varphi$ and let $\bar R$ denote the curvature of the canonical ${\rm G}_2$ connection
$\bar \nabla$ then this formula states \begin{equation}\label{qr} q(R) \;=\; q(\bar R) \; +\; 3 (\tfrac{\tau_0}{12})^2 \, \mathrm{Cas}^{\mbox{${\mathfrak s \mathfrak o}$}(7)} \,-\;4 (\tfrac{\tau_0}{12})^2 \, S \end{equation} where $ \mathrm{Cas}^{\mbox{${\mathfrak s \mathfrak o}$}(7)} $ is the Casimir operator of $\mbox{${\mathfrak s \mathfrak o}$}(7)$ which acts as $-k(7-k)\rm id$ on forms in $\Lambda^k {\rm T}$ and as $-14\,\rm id$ on ${\rm Sym}^2_0 {\rm T}$. The ${\rm G}_2$ invariant endomorphism $S$ is defined as $S = \sum P_{e_i} \circ P_{e_i}$, where $P_X$ is the skew-symmetric endomorphism $X \lrcorner \, \varphi$. Then a straight forward computation on explicit elements shows that $S = - 6\, \rm id $ on ${\rm T}$ and $S = -14 \,\rm id$ on ${\rm Sym}^2_0 {\rm T}$.
As an example we consider the difference of $q(R)$ and $q(\bar R)$ on ${\rm T} M$. Of course we already know that $q(R) = {\rm Ric} = \tfrac{{\rm scal}}{7} = \tfrac{3\tau^2_0}{8}$. Hence formula \eqref{qr} and the explicit values for the Casimir operator and the endomorphism $S$ yield $ q(\bar R) = q(R) - \tfrac{\tau^2_0}{24} = \tfrac{\tau^2_0}{3}$.
Returning to the Berger space, recall that the induced Einstein metric $g$ defined by the ${\rm G}_2$ structure is given by a multiple of the Killing form $B$ of $\mathfrak{sp}(2)$ restricted to $\mbox{${\mathfrak m}$}$. Moreover, the canonical homogeneous connection coincides with the canonical ${\rm G}_2$ connection $\bar \nabla$. For our calculation we will use the normalization $g= -B$ for which we have $\tau^2_0 = \tfrac{6}{5}$ and ${\rm scal} = \tfrac{63}{20}$ (see \cite{AS12}, Lemma 7.1). In this situation we know that the endomorphism $q(\bar R)$ acts as $ -\mathrm{Cas}^{\mathfrak{sp}(1)}$ (see \cite{AS12}, Lemma 7.2). The Casimir operator of $\mathfrak{sp}(1)$ with respect to the Killing form acts on ${\rm Sym}^k E$ as $-k(k+2)\,\rm id$. Thus $q(\bar R)$ acts on the bundle defined by the ${\rm Sp}(1)$ representation ${\rm Sym}^k E$ as $c k(k+ { 2})\,\rm id$ for some
positive constant $c$, which can be determined from the case $k=6$. Indeed here we have ${\rm Sym}^6E = \mbox{${\mathfrak m}$}^{\scriptsize \mbox{${\mathbb C}$}}$ and we already calculated that $q(\bar R) = \tfrac{\tau^2_0}{3} = \tfrac25$ on $\mbox{${\mathfrak m}$}^{ \scriptsize \mbox{${\mathbb C}$}}$. It follows $c= \tfrac{1}{120}$. In particular we conclude that $q(\bar R) = \tfrac15 \,\rm id$ on ${\rm Sym}^4 E$ and \eqref{qr} in our normalization implies that $q(R) = \tfrac{19}{60}\,\rm id$ on the space $V(1,1)$. Hence, for any Killing tensor
$h \in V(1,1) \subset \Gamma({\rm Sym}^2_0 \, \mbox{${\mathfrak m}$}^{\scriptsize \mbox{${\mathbb C}$}})$ we have $\Delta_L h = 2q(R)\, h = \tfrac{19}{30}\, h$. But $\tfrac{19}{30} < 2E = \tfrac{3\tau^2_0}{4} = \tfrac{9}{10} $, so the instability condition (\ref{instability}) holds.
We have therefore proved the following
\begin{thm} $($Theorem \ref{Berger}$)$ Let $M^7 = {\rm SO}(5)/{\rm SO}(3)_{irr}$ be the Berger space equipped with its homogenous proper nearly parallel ${\rm G}_2$ structure. Then the induced Einstein metric is linearly unstable. Moreover, $M^7$ admits a $5$-dimensional space of divergence and tracefree Killing tensors. \mbox{$\Box$} \end{thm}
\begin{rmk} \label{eigenfunction} The reader might wonder if $\nu$-linear instability for the Berger space can be shown by looking at the spectrum of the Laplacian on functions. It turns out that this is not possible because computations similar to the above show that the smallest nonzero eigenvalue is larger than $2E$, and the corresponding eigenspace is the $35$-dimensional irreducible module $V(4, 0)$. \end{rmk}
\end{document} |
\begin{document}
\title{Bounds on the period of the continued fraction after a M\"obius transformation}
\author{Hanka Řada} \affil{\footnotesize Faculty of Nuclear Sciences and Physical Engineering\\ Czech Technical University in Prague\\ Czech Republic} \author{Štěpán Starosta \thanks{Electronic address: \texttt{stepan.starosta@fit.cvut.cz}}} \affil{\footnotesize Faculty of Information Technology\\ Czech Technical University in Prague\\ Czech Republic}
\date{}
\maketitle
\begin{abstract} We study M\"obius transformations (also known as linear fractional transformations) of quadratic numbers. We construct explicit upper and lower bounds on the period of the continued fraction expansion of a transformed number as a function of the period of the continued fraction expansion of the original number. We provide examples that show that the bound is sharp. \end{abstract}
\ifdraft{
\listoftodos }
\section{Introduction} Eventually periodic continued fraction expansions correspond exactly to quadratic irrational numbers. Some general upper bounds on periods of such an expansion, depending on the number itself, are known, see \cite{Podsypanin,Pohl07}. In some very specific cases, the exact value is known (and so is the expansion), see \cite{cohn1977length,Rockett-1990,Vladimi-2009,BaHru}.
We study such periods after a transformation which preserves eventual periodicity of the expansion. Given a nonsingular matrix $N = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \mathbb{Z}^{2,2}$, we consider the mapping $h_N: \mathbb{R} \setminus \left\{ - \frac{d}{c} \right\} \to \mathbb{R}$ given by \[ h_N(x) = \frac{ax + b}{cx + d}. \] Such a mapping is called \emph{the M\"obius transformation associated with the matrix N}, it is also sometimes referred to as a linear fractional transformation. Given a quadratic irrational number $x$, the number $h_N(x)$ is clearly a quadratic irrational number (in the same field). Our main result is an upper and lower bound on the period of the continued fraction expansion of $h_N(x)$ as a function of the period of the continued fraction expansion of $x$.
To state the main result, we introduce the following notation and definitions. We have $x = [v, \overline{w}]$ with $v \in \mathbb{N}^\ell$ and $w \in \mathbb{N}^k$ for some $\ell, k \in \mathbb{N}, k \neq 0$ (overline denotes infinite repetition of $w$). If such a sequence $w$ is the shortest possible, we say that it is the \emph{repetend} of the continued fraction expansion of $x$ (or simply of $x$). Let $\per(x)$ denote the shortest \emph{period} of a continued fraction of $x$, i.e. the length of the repetend of $x$.
For nonnegative integers $a,c$ not both zero, let $\xi(a,c)$ denote the number of divisions required to compute the $\gcd(a,c)$ using the Euclidean algorithm (ending when $0$ is reached). Thus, for instance, $\xi(7,0) = \xi(0,7) = 0$, $\xi(7,1) = \xi(1,7) = 1$, and $\xi(13,5) = \xi(5,13) = \xi(5,3) + 1 = \xi(3,2) + 2 = \xi(2,1) + 3 = 4$.
Our main result are the following bounds on $\per(h_N(x))$.
\begin{theorem} \label{thm:main}
Let $x$ be a quadratic irrational number, $h_N$ a M\"obius transformation and $n = |\det N|$. We have \[ \frac{1}{S_n} \per(x) \leq \per(h_N(x)) \leq S_n \per(x), \] where \[ S_n = \sum_{\substack{t \in \mathbb{N} \\ t \mid n}} \sum_{\substack{j = t\\ j \not \in J_t }}^{2t-1} \left (2\left\lfloor \frac{\xi(j,t)}{2} \right\rfloor +1 \right) \] with $J_t = \begin{cases} \emptyset & \text{for } \gcd(t,\frac{n}{t}) = 1, \\ \{i \gcd(t,\frac{n}{t}) \colon i \in \mathbb{N}\} & \text{otherwise.} \end{cases}$ \end{theorem}
\begin{example} Let us give an example for $N = \begin{pmatrix} 12 & 1 \\ 17 & 2
\end{pmatrix}$. We have $|\det N| = 7$ and $S_7 = 24$. For $x_1 = [\overline{3}]$, we have $\per(x_1) = 1$ and $\per(h_N(x_1)) = 6 \leq S_7 \per(x_1) = 24$. For $x_2 = [\overline{200}]$, we have $\per(x_2) = 1$ and $\per(h_N(x_2)) = 24 \leq S_7 \per(x_2) = 24$. For \[ x_3 = [-1, 1, 11, \overline{7, 1, 6, 8, 399, 8, 6, 1, 7, 3, 2, 7, 1, 2, 1, 1, 7, 1, 1, 2, 1, 7, 2, 3}], \] we have $\per(x_3) = 24$ and $\per(h_N(x_3)) = 1 \geq \frac{\per(x_3)}{S_7} = 1$ \end{example}
The presented proof of \Cref{thm:main} is based on the famous work of Raney \cite{Raney1973} who described transducers which output the continued fraction expansion of $h_N(x)$ while inputting the continued fraction expansion of $x$.
Note that the action of M\"obius transformations on a number $x$ has been explored for instance in \cite{LaSh97} and \cite{Liu} where the authors give bounds on the value of partial coefficients of the number $x$ after transformation.
The proof of the main result may be considered somewhat technical. In \Cref{sec:prelim}, we first give some necessary notations and recall results of Raney. A number of additional claims and the proof of \Cref{thm:main} are in \Cref{sec:constr}. The last section contains remarks and experiment results.
\section{Preliminaries} \label{sec:prelim}
\subsection{LR representation}
For a more detailed description of the computations with continued fractions, we need another representation of positive real numbers. Before we state it, we introduce the following notation.
\emph{An alphabet} $\mathcal A$ is a finite set of symbols. \emph{A word} over the alphabet $\mathcal A$ is a sequence of symbols from this alphabet. If the sequence is empty, it is \emph{the empty word} and it is denoted by $\varepsilon$. The set of all finite words over an alphabet $\mathcal A$ is denoted by $\mathcal A^*$ and the set of all finite and infinite words by $\mathcal A^{\mathbb{N}}$. If we have $v,w \in \mathcal A^{\mathbb{N}}$, then $vw$ denotes the concatenation of the words $v$ and $w$. If there exists $u \in \mathcal A^{\mathbb{N}}$ such that $w = vu$, we say that $v$ is \emph{a prefix} of $w$. Moreover, if $w \neq v$, we say that $v$ is \emph{a proper prefix} of $w$. Analogously, if $v,w \in \mathcal A^{\mathbb{N}}$ and there exists $u \in \mathcal A^*$ such that $w = uv$, we say that $v$ is \emph{a suffix} of $w$ and moreover, if $w \neq v$, we say that $v$ is \emph{a proper} suffix of $w$. A word $u$ is \emph{primitive} if $u = v^k = \underbrace{v v \cdots v}_{k \text{ times}}$ implies $k=1$.
Let $x \in \mathbb{R}^{+} \setminus \mathbb{Q}$ with its continued fraction expansion equal to $[x_0,x_1,x_2,\ldots]$. Its \emph{LR representation} is the following infinite word over the alphabet $\{L,R\}$: \[ {\bf v} = R^{x_0} L^{x_1} R^{x_2} L^{x_3} \ldots \quad \in \{L,R\}^\mathbb{N}. \] In what follows, we identify a number $x$ with its LR expansion and simply write $x = {\bf v}$. For example, we have $1 + \sqrt{2} = [\overline{2}] = \overline{R^2L^2}$.
\begin{remark} The LR representation is originally connected with the Stern-Brocot tree. The choice of letters $L$ and $R$ also follows from this connection: the two letters stand for ``Left'' and ``Right'' in the tree. For more information about the relation between the Stern-Brocot tree and continued fractions see for instance \cite{niqui}. \end{remark}
Let $V$ be a finite word. A \emph{run} in $V$ is a contiguous subsequence of maximal length which consists of a single letter. That is, it is the longest repetition of one letter, sometimes also called a tandem array. The number $\sigma(V)$ denotes the number of all runs in $V$. For instance, we have $\sigma(LLRRRRL) = 3$.
\subsection{M\"{o}bius transformation and finite state transducers}
In \cite{Gosper}, the author introduces an algorithm that calculates the continued fraction of $h_M(x)$ using the continued fraction of $x$. The general idea of the algorithm is the following: read as many partial coefficients of $x$ so that we are able to decide on the first partial coefficient of $h_M(x)$ and output it. The reading phase is usually called \emph{absorption}, the writing phase \emph{emission}. Then, if needed, continue absorbing the partial coefficients of $x$ and emit the second partial coefficient of $h_M(x)$ when possible. Repeat the whole procedure: if there are no coefficients to absorb, emit the rest of the output. The details and more results on the algorithm were given later by Raney, in \cite{Raney1973}. The main idea of the algorithm is the same but it uses LR representations instead of continued fractions expansions. In this article, we use the latter approach and work with LR representations since they allow capturing more details of the algorithm. In what follows, we sum up the needed results of Raney. We also refer the reader to a more general concept of this idea in \cite[Chapter 5]{Kurka2016}, and for refinements of the results for continued fractions in \cite{LiSta}.
Since for every positive integer $d$ we have $h_{dM}(x) = h_M(x)$, we shall work only with matrices $M$ such that the greatest common divisor of all its elements is $1$.
Following \cite{Raney1973}, we define some special sets of matrices $2 \times 2$ having a key role in the computation of Möbius transformations.
\begin{definition} For $n \in \mathbb{N}$, $n \neq 0$ we set \[ \D{} = \left\{ A \in \mathbb{N}^{2,2} \colon \det(A) = n, \gcd(A) = 1 \right \}, \] where $\gcd(A)$ denotes the greatest common divisor of all elements of $A$.
Furthermore, we define the three following subsets of $\D{}$: \begin{align*} \RB{} & = \left\{ \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \D{} \colon a > c \text{ and } d > b \right \}, \\ \CB{} & = \left\{ \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \D{} \colon a > b \text{ and } d > c \right \}, \\ \DB{}& = \RB{} \cap \CB{}. \end{align*}
\end{definition}
The names of the three above defined sets are abbreviations for ``row-balanced'', ``column-balanced'' and ``double-balanced'', respectively.
For all $n$, the sets $\RB{}$, $\CB{}$, and $\DB{}$ are finite. If $n$ is a prime number, then by Corollary 4.7 of \cite{Raney1973}, we have $\# \DB{}= n$.
We study the period of eventually periodic continued fractions and therefore we do not need the prefix of LR representations of the studied numbers, only the tail is important. In the view of this, the following theorem tells us that we may consider M\"obius transformations associated to a matrix from $\DB{}$.
\begin{theorem}[\cite{Raney1973,LiSta}] \label{thm:dostanu_se_do_Dn}
Let $x$ be an irrational number, $h_M$ a M\"obius transformation, $\gcd(M) = 1$ and $n = \left| \det M \right |$. There exists an algorithm to construct a matrix $N \in \DB{}$ and a positive irrational number $y$ such that the continued fraction expansion of $h_N(y)$ and the continued fraction expansion of $h_M(x)$ have the same tail.
In particular, if $x$ is a quadratic irrational number, we have \[ \per(h_M(x)) = \per(h_N(y)). \] \end{theorem}
\emph{A finite state transducer} is the quadruple $(Q,\mathcal A,\mathcal B,\delta)$ where $Q$ is a finite set of states, $\mathcal A$ is the input alphabet, $\mathcal B$ is the output alphabet and $\delta \subseteq Q \times \mathcal A^* \times\mathcal B^* \times Q$ is the transition relation. The transitions are also called \emph{edges} of this transducer. The first state in the transition relation is the starting state of this edge. The word $v \in \mathcal A^{*}$ is the input label of this edge. The word $w \in \mathcal B^{*}$ is the output label of this edge and the second state in the relation is the ending state of this edge.
Raney shows that once the problem is transformed to involve a M\"obius transformation of a positive number $y$ with a matrix $N \in \DB{}$, there exists a finite state transducer depending on $n$, denoted $\T{}$, that can be used to determine the LR expansion of $h_N(y)$. Namely, the input word of this transducer is the LR expansion of $y$, the initial state is given by $N$, and the output word is the LR expansion of $h_N(y)$.
As we are interested only in the repetend of $h_M(x)$, which is the same for $h_N(y)$, we can focus only on the calculation using the transducer $\T{}$. Thus, we refrain from giving more details on the last theorem and continue with the description of $\T{}$ and its properties.
\subsubsection{Matrices \texorpdfstring{$L$}{L} and \texorpdfstring{$R$}{R} }
We start by identifying the set of all finite words over $\{L,R\}$ with the elements of $\D[1]{}$. Let $\mu: L \mapsto \begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix}, R \mapsto \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$ and for all $V, W \in \{L,R\}^*$, we have $\mu(VW) = \mu(V)\mu(W)$.
\begin{proposition}[\cite{Raney1973}] \label{thm:jednoznacnost_v_D1} The mapping $\mu$ is an isomorphism of $\{L,R\}^*$ and $\D[1]{}$. \end{proposition}
Since $\mu$ is an isomorphism, we shall identify the letters $L$ and $R$ with the two matrices, i.e., we shall consider \[ L = \begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix} \quad \text{ and } \quad R = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}. \]
In what follows, we often need to deduce some claims from matrix equations and these equations include mainly the matrices $L$, $R$ and their inverses. We give the two following lemmas to be used in these cases.
\begin{lemma} Let $i,j \in \mathbb{Z}$. We have \begin{eqnarray} LR^{-j}L^{-1} &=& RL^{j}R^{-1}, \label{eq:bubli_Li_vpravo} \\ R^{-1}L^{-i}R &=& L^{-1}R^iL, \label{eq:bubli_R_vpravo} \\ L \left( L^{i+1}R^{j+1} \right)^{-1} R &=& RL^jR^iL. \label{eq:bubliani_D1_1} \end{eqnarray} \end{lemma}
\begin{proof} The first two equalities may be verified by direct computation. The last equality follows from the combination of \eqref{eq:bubli_R_vpravo} and \eqref{eq:bubli_Li_vpravo} as follows: \[ L \left( L^{i+1}R^{j+1} \right)^{-1} R = L R^{-j-1} L^{-i-1} R \overset{\eqref{eq:bubli_R_vpravo}}{=} L R^{-j}L^{-1}R^{i+1}L \overset{\eqref{eq:bubli_Li_vpravo}}{=} RL^jR^iL. \qedhere \] \end{proof}
\begin{lemma} \label{st:bublani_D1} If $W \in \D[1]{}$, then \[ L \left( LWR \right)^{-1} R \in \D[1]{} \quad \text{ and } \quad R \text{ is a prefix of } L \left( LWR \right)^{-1} R. \] \end{lemma}
\begin{proof} Let \[ LWR = L^{i_0+1}R^{j_0+1}L^{i_1+1}R^{j_1+1} \cdots L^{i_k+1}R^{j_k+1} \] for some $k \geq 0$ and $i_s,j_s \geq 0$ for all $s \in \{0,\dots,k\}$. We have \[ L \left( LWR \right)^{-1} R = \] \[= L \left( L^{i_k+1}R^{j_k+1} \right)^{-1} R R^{-1} L^{-1} L \left( L^{i_{k-1}+1}R^{j_{k-1}+1} \right)^{-1} R R^{-1} \cdots L^{-1} L \left( L^{i_0+1}R^{j_0+1} \right)^{-1} R. \] Using $k$ times \eqref{eq:bubliani_D1_1} on the right side of the last equality, we obtain \[ L \left( LWR \right)^{-1} R = R L^{j_k} R^{i_k} L R^{-1} L^{-1} R L^{j_{k-1}}R^{i_{k-1}} L R^{-1} \cdots L^{-1} R L^{j_0}R^{j_0} L. \] Since by \eqref{eq:bubli_Li_vpravo} we have $L R^{-1} L^{-1} R = RL$, we conclude \[ L \left( LWR \right)^{-1} R = R L^{j_k} R^{i_k} RL L^{j_{k-1}}R^{i_{k-1}} RL \cdots L^{j_0}R^{j_0} L \in \D[1]{}. \qedhere \] \end{proof}
\subsubsection{Transducers \texorpdfstring{$\T{}$}{Tn}}
\begin{theorem}[{\cite[Theorem 5.1]{Raney1973}}] \label{thm:Raney_edge} Let $M \in \RB{}$. For all $V \in \{L,R\}^*$ such that \begin{itemize}
\item $MV \not \in \RB{}$, and
\item $MV_1 \in \RB{}$ for every proper prefix $V_1$ of $V$, \end{itemize} there exists a unique non-empty word $W \in \{L,R\}^*$ and $N \in \DB{}$ such that \begin{equation} \label{eq:Raney_edge} MV = WN. \end{equation} \end{theorem}
Theorem 5.1 in \cite{Raney1973} does not say that the word $W$ is unique, however this property follows directly from the equation $W = MVN^{-1}$.
We have used some statements of \cite{Raney1973} about the sets $\D{}$ which are in \cite{Raney1973} defined without the condition that $\gcd(A) = 1$. The validity of these statements for our definition of $\D{}$ follows from Corollary 8.4 in \cite{Raney1973}.
Based on the last theorem, we may now construct the transducer $\T{}$.
The definition of $\T{}$ is as follows: \begin{enumerate}
\item the set of states of $\T{}$ equals $\DB{}$;
\item the set of transitions between states is given by \Cref{thm:Raney_edge}: there is a transition from $M$ to $N$ if $MV = WN$ for some $V,W \in \D[1]{}$ with $MV \not \in \RB{}$ and $MV_1 \in \RB{}$ for every proper prefix $V_1$ of $V$. The input word of the transition is $V$, the output word is $W$. \end{enumerate}
To ease our notation, the transition from $M$ to $N$ with input $V$ and output $W$ is denoted by \[ \Tedge{M}{V}{W}{N}. \]
Let $M$ and $N$ be two states of $\T{}$ such that there is a sequence of transitions starting at $M$ and ending at $N$ in $\T{}$ with concatenation of respective input words $V$ and output words $W$. We write \[ \Twalk{M}{V}{W}{N}. \] We call this sequence a \emph{walk}. Concatenating the matrix relations of all transition in the walk we obtain the relation $MV = WN$. To ease the notation we also allow $V$ to be the empty word, which implies that $W$ is also empty and $M = N$. If we do not need to know the concrete input or output word, we write $\bullet$ on its position.
Given a walk $\Twalk{M}{V}{W}{N}$, we shall write for instance \[ \Twalk{M}{V}{W}{N} = \Twalk{M}{V_1}{W_1}{\Tedge{M_1}{V_2}{W_2}{\Tedge{M_2}{V_3}{W_3}{N}}} \] to specify some decomposition of the walk. If a walk repeats, we shall also write for instance \[ \Twalk{M}{V}{W}{\Twalk{M}{V}{W}{M}} = \left( \Twalk{M}{V}{W}{M} \right)^2. \]
In what follows, let $A_n = \mat{n}{0}{0}{1}, \assoc{A_n} = \mat{1}{0}{0}{n}$. We have $A_n, \assoc{A_n} \in \DB{}$ for all $n$ and for $n = 2$, we have $\mathcal{DB}_2 = \left\{ A, \assoc{A} \right\}$. See \Cref{fig:RaneyT_2} which depicts $\T[2]{}$ and \Cref{fig:table_T_3} showing the transition labels of $\T[3]$.
\begin{figure}
\caption{Transducer $\T[2]{}$.}
\label{fig:RaneyT_2}
\end{figure}
\begin{table}[!htb]
\centering $\begin{matrix} & A_3 = \begin{pmatrix} 3 & 0 \\ 0 & 1 \\ \end{pmatrix} & B = \begin{pmatrix} 2 & 1 \\ 1 & 2 \\ \end{pmatrix} & \assoc{A_3} = \begin{pmatrix} 1 & 0 \\ 0 & 3 \\ \end{pmatrix} & \\ \begin{pmatrix}3&0\\ 0&1\\
\end{pmatrix} & R|R^{3} , L^{3}|L & LR|R & L^{2}R|RL^{2}\\ \begin{pmatrix}2&1\\ 1&2\\
\end{pmatrix} & L|LR & & R|RL\\ \begin{pmatrix}1&0\\ 0&3\\
\end{pmatrix} & R^{2}L|LR^{2} & RL|L & L|L^{3} , R^{3}|R\\ \end{matrix}$
\caption{Transitions in the transducer $\T[3]{}$.}
\label{fig:table_T_3} \end{table}
\subsubsection{Symmetries of the transducer \texorpdfstring{$\T{}$}{Tn}}
The transducer $\T{}$ possesses some symmetries that we shall use later.
Let $M = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$. The matrix $\assoc{M}$ \emph{associated to} $M$ is given by \[ \assoc{M} = \begin{pmatrix} d & c \\ b & a \end{pmatrix} = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} M \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} . \] Clearly, for all matrices $M$ and $N$, we have $\assoc{MN} = \assoc{M}\assoc{N}$.
Note that since we identified the letters $L$ and $R$ with matrices and due to \Cref{thm:jednoznacnost_v_D1}, this operation is also defined for any word over $\{L,R\}$. The following simple identities are given in \cite[Theorem 7.1]{Raney1973} for $M \in \D[1]{}$, i.e., words over $\{L,R\}$: \begin{align*} M & = L^{i_0}R^{i_1}L^{i_2} \cdots L^{i_\ell}, \\ \assoc{M} & = R^{i_0}L^{i_1}R^{i_2} \cdots R^{i_\ell}, \\ (\assoc{M})^T = \assoc{M^T} & = L^{i_\ell}R^{i_{\ell-1}} \cdots R^{i_1}L^{i_0}, \\ M^T & = R^{i_\ell}L^{i_{\ell-1}} \cdots L^{i_1}R^{i_0}, \end{align*} where $i_0,i_1,\ldots,i_\ell$ are nonnegative integers. These properties imply a symmetry of $\T{}$ in the sense of the following claim.
\begin{proposition} \label{prop:assoc_sym} If the transition $\Tedge{M}{V}{W}{N}$ exists, then the transitions $\Tedge{\assoc{M}}{\assoc{V}}{\assoc{W}}{\assoc{N}}$ and $\Tedge{N^T}{W^T}{V^T}{N^T}$ exist. \end{proposition}
\subsubsection{Relation between repetends and runs}
The following lemma, exhibiting the relation between the number of runs in the repetend of the LR representation of $x$ and the period of its continued fraction, is a direct corollary of the definition of LR representation.
\begin{lemma} \label{le:vypocet_per} Let $V$ be such a repetend of $x$ whose first and last letter are different. We have \[
\per(x) = \begin{cases}
\frac{\sigma(V)}{2} & \text{ if $V = V_1\assoc{V_1}$ for some $V_1 \in \D[1]{}$,} \\
\sigma(V) & \text{ otherwise.}
\end{cases} \] \end{lemma}
To work with the matrix representation of words, we shall need the connection between the number of runs in some word over $\{L,R\}$ and its matrix representation.
\begin{lemma} \label{le:pocet_LRzmen} Let $W = LW'R, W = \mat{a}{b}{c}{d}$ and $W,W' \in \D[1]{}$. We have \[ \sigma(W) = 2 \left\lfloor \frac{\xi(a,c)}{2} \right\rfloor + 2. \] \end{lemma}
\begin{proof} Let $f,e$ be positive integers such that $W = W_1L^eR^f$ for some $W \in \D[1]{}$.
We shall proceed by induction on $\sigma(W)$.
For $\sigma(W) = 2$, we have $W = L^eR^f = \mat{1}{f}{e}{ef+1}$ and since $\xi(1,e)=1$ the claim holds.
Assume now the claim holds for $\sigma(W) = k$. Note that for $W = \mat{a}{b}{c}{d}$, as $W$ starts with $L$ and ends with $R$, we have $c > a \geq 1$ or $c=a=1$. Let $g,h$ be positive integers. We have $\sigma(L^gR^hW) = k+2$ and we have \[
L^gR^hW = \mat{c h + a}{d h + b}{ g(ch + a) + c}{g (dh + b) + d}. \] If $c > a$, then $\xi(c h + a, g(ch + a) + c) = \xi(a,c)+2$ and the claim follows. If $c=a=1$, then in fact $W = LR^f$, $\xi(h + 1, g(h + 1) + 1) = 2$ and $\sigma(L^gR^hLR^f) = 4 = 2 \left\lfloor \frac{\xi(h + 1, g(h + 1) + 1)}{2} \right\rfloor + 2$. \end{proof}
\begin{example} The word $W = L^2RLR^{3}$ is of the form from the above lemma and we have $W = \mat{2}{7}{5}{18}$. Therefore, $ 2 \left\lfloor \frac{\xi(a,c)}{2} \right\rfloor + 2 =2 \left\lfloor \frac{\xi(2,5)}{2} \right\rfloor + 2 = 2\left\lfloor \frac{2}{2} \right\rfloor + 2 = 4 = \sigma(W)$. \end{example}
\subsection{Properties of the transducer \texorpdfstring{$\T{}$}{Tn}}
In what follows, we suppose that $n$ is a fixed integer.
\subsubsection{Input and output words of the edges in the transducer \texorpdfstring{$\T{}$}{Tn}}
First, we investigate the input words on outgoing edges from some state of the transducer $\T{}$.
\begin{lemma} \label{le:vstupni_slova} If $X \in \RB{}$, then $XL \not \in \RB{}$ or $XR \not \in \RB{}$. \end{lemma}
\begin{proof} Let $X = \mat{a}{b}{c}{d} \in \RB{}$. Therefore, $a>c$, $d>b$ and $XL =\mat{a+b}{b}{c+d}{d}, XR = \mat{a}{a+b}{c}{c+d}$. We have $a+b \geq c+d$ and then $XR \not \in \RB{}$, or $a+b \leq c+d$ and then $XL \not \in \RB{}$. \end{proof}
A simple consequence of \Cref{thm:Raney_edge} is that the set of all input word of an outgoing edge of a state of $\T{}$ is a prefix code (no element is a prefix of another). The last lemma implies that the lengths of these input words are $1,2,\ldots,\ell-1,\ell,\ell$ for some $\ell$.
\begin{example} All transitions in the transducer $\mathcal{T}_{14}$ are given in \Cref{tab:T_14}. \end{example}
\begin{sidewaystable}
\centering
\includegraphics[width=\textwidth]{prevodnik_T14_tab.pdf}
\caption[angle=90]{Table of transitions in the transducer $\mathcal{T}_{14}$.} \label{tab:T_14} \end{sidewaystable}
The following lemma shows that there is no edge with input label having suffix $R^{n+1}$ or $L^{n+1}$
\begin{lemma} \label{le:max_n_jednoho_pismene} Let $Q \in \{L, R\}$. Let $X \in \RB{}$ and let $i$ be the minimal positive integer such that $XQ^i \not \in \RB{}$. We have $i \leq n$. \end{lemma}
\begin{proof} Let $Q = L$ and let $X = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \RB{}$. We have \[ XL^i = \begin{pmatrix} a + bi & b \\ c + di & d \end{pmatrix}. \] Assume $i$ is minimal such that $XL^i \not \in \RB{}$. Therefore, $XL^{i-1} \in \RB{}$. Since $d > b$, we have $a + b(i-1) > c + d(i-1)$, which implies \[ i -1 < \frac{a - c}{d - b}. \] Using $a = \frac{n+bc}{d}$, we obtain \[ \frac{a - c}{d - b} = \frac{n}{d(d-b)} - \frac{c}{d} \leq n. \]
The proof for $Q = R$ is analogous. \end{proof}
The next lemma shows a simple link of input and output words.
\begin{lemma}[{\cite[Theorem 5.1]{Raney1973}}] \label{le:stejna_pismena} Let $\Tedge{M}{VQ_1}{Q_2W}{N}$ with $Q_1,Q_2 \in \{L,R\}$. We have $Q_1 = Q_2$. \end{lemma}
\subsubsection{Sets \texorpdfstring{$\LS{}$}{LS} and \texorpdfstring{$\RS{}$}{RS}} \label{sec:LS_n}
In what follows, we show that an important role is played by input words with long runs of the same letter. We now introduce two sets of matrices that are always visited when reading such input words.
\begin{definition} \label{def:LS_a_RS} Let $\LS{}$ respectively $\RS{}$ denote the subset of $\DB{}$ such that $X \in \LS{}$ respectively $X \in \RS{}$ if there exists a walk $\Twalk{X}{L^i}{L^j}{X}$ respectively $\Twalk{X}{R^i}{R^j}{X}$ for some $i,j > 0$. \end{definition}
The names of the sets $\LS{}$ and $\RS{}$ are abbreviations for ``$L$-special'' and ``$R$-special''.
For example, we have $A_n \in \LS{}$ since $\Tedge{A_n}{L^n}{L}{A_n}$ and $A_n \in \RS{}$ since $\Tedge{A_n}{R}{R^n}{A_n}$.
The above definition says that the matrices in set $\LS{}$ ($\RS{}$) can be visited several times while reading the same run of letters. Moreover, the following lemma shows that these matrices are the only ones with such a property.
\begin{lemma} \label{le:char_toceni_na_L} Let $M = \mat{a}{b}{c}{d} \in \DB{}$ and $i > 0$. We have \[ \Twalk{M}{L^i}{W}{M} \] for some $W \in \D[1]{}$ if and only if $M \in \LS{}$ (i.e., $W = L^j$).
Moreover, the minimal value of such integer $i$ equals $\min \left\{ k > 0 \colon \frac{kd}{a} \in \mathbb{N} \right\}$ and is less than or equal to $n$. \end{lemma}
\begin{proof} The walk $\Twalk{M}{L^i}{W}{M}$ exists if and only if $W \in \D[1]{}$. We have \[ W = ML^iM^{-1} = \mat{-\frac{b d i - b c + a d}{b c - a d}}{\frac{b^{2} i}{b c - a d}}{ -\frac{d^{2} i}{b c - a d}}{\frac{b d i + b c - a d}{b c - a d}}. \] As $\frac{b^{2} i}{b c - a d} = \frac{b^2i}{-n} \leq 0$ we have $\frac{b^2i}{bc-ad} \in \mathbb{N} \iff b = 0$. Thus \[ W = \mat{1}{0}{\frac{id}{a}}{1}, \] and we conclude that $W \in \D[1]{}$ if and only if $b = 0$ and $\frac{id}{a} \in \mathbb{N}$. In other words, $W = L^{\frac{id}{a}}$, which is by definition if and only if $M \in \LS{}$.
Let $i$ has the minimal possible value, i.e., $i = \min \left\{ k > 0 \colon \frac{kd}{a} \in \mathbb{N} \right\}$. Since $n = ad$, we have $i \leq a \leq n$. \end{proof}
\begin{example} In the transducer $\mathcal{T}_{14}$, we have $\Twalk{M}{L^7}{W}{M}$ where $M = \mat{7}{0}{0}{2}$ and $W = L^2$. Since $\Twalk{M}{L^7}{L^2}{M} = \Tedge{M}{L^4}{L}{\Tedge{N}{L^3}{L}{M}}$ where $N = \mat{7}{0}{1}{2}$, the minimal $i$ such that $\Twalk{M}{L^i}{W}{M}$ is $i = 7$. This is also the minimal positive integer such that $\frac{id}{a} = \frac{2i}{7} \in \mathbb{N}$. \end{example}
The following characteristic property of the matrices in the set $\LS{}$ follows from the proof of the last lemma.
\begin{corollary} \label{cor:tvar_LS} Let $M = \mat{a}{b}{c}{d} \in \DB{}$. We have $M \in \LS{}$ if and only if $b = 0$. \end{corollary}
\begin{example} We have already seen that the matrices $\mat{7}{0}{0}{2}, \mat{7}{0}{1}{2} \in \mathcal{LS}_{14}$ and that $A_n \in \LS{}$, which corresponds with the fact that these matrices have the element on position 1,2 equal to 0. Moreover, by the symmetric version of this corollary, we have $M \in \RS{}$ if and only if $c = 0$, which corresponds with $A_n \in \RS{}$. According to this condition, we can also see that $\mat{7}{0}{0}{2} \in \RS{}$. Indeed, we have $\Twalk{\mat{7}{0}{0}{2}}{R^2}{R^7}{\mat{7}{0}{0}{2}}$. \end{example}
The last lemma and corollary imply the following statement.
\begin{corollary} \label{le:LS_nacteni_n} For $M \in \LS{}$ there exists $s > 0$ such that $\Twalk{M}{L^n}{L^s}{M}$. \end{corollary}
\begin{example} As already mentioned, we have $\Twalk{M}{L^7}{L^2}{M} $ where $M = \mat{7}{0}{0}{2} \in \mathcal{LS}_{14}$. The last corollary shows that we have also $\Twalk{M}{L^{14}}{L^4}{M} = (\Twalk{M}{L^7}{L^2}{M})^2$.
As stated above, we have $\Twalk{M}{L^7}{L^2}{M} = \Tedge{M}{L^4}{L}{\Tedge{N}{L^3}{L}{M}}$ where $N = \mat{7}{0}{1}{2}$. This means that there is a connection between the matrices $\mat{7}{0}{0}{2}$ and $\mat{7}{0}{1}{2} \in \mathcal{LS}_{14}$. The classes of matrices in $\LS{}$ with this connection are described in the following lemma. \end{example}
\begin{lemma} \label{le:trida_L} Let $M, N \in \LS{}$, $M = \mat{a}{0}{c}{d}$ and $N = \mat{a'}{0}{c'}{d'}$. We have \[ \Twalk{M}{L^i}{W}{N} \text{ for some $i > 0$ and $W \in \D[1]{}$} \quad \Longleftrightarrow \quad a' = a, d' = d, c' \equiv c \pmod{ \gcd(a,d)}. \] \end{lemma}
\begin{proof} Let $\Twalk{M}{L^i}{W}{N}$ for some $i > 0$ and $W \in \D[1]{}$. \Cref{le:char_toceni_na_L} implies that we may take $W = L^j$ for some $j > 0$ and $i \leq n$. We have \[ N = L^{-j} M L^i = \mat{a}{0}{c + id - ja}{d} \in \LS{} \subseteq \DB. \] Since we have $id - ja = k \gcd(a,d)$ for some $k \in \mathbb{Z}$, the first implication is proven.
Assume now $a' = a, d' = d, c' = c + k\gcd(a,d)$. The case $c = c'$ is trivial. For $c \neq c'$ we find $i',j'$ with $i'j' < 0$ such that $k\gcd(a,d) = i'd + j'a $. It implies $ML^{i'} = L^{-j'}N$. If $j' < 0$, we are finished. If $j' > 0$, we multiply by $L^{rn}$ from the right to obtain \[ ML^{i'+rn} = L^{-j'}NL^{rn} = L^{-j'+rs}N, \] where $s$ is the positive integer such that $\Twalk{N}{L^n}{L^s}{N}$. A choice of $r$ such that $-j'+rs > 0$ implies $\Twalk{M}{L^{i'+rn}}{L^{-j'+rs}}{N}$. \end{proof}
The existence of a walk $\Twalk{M}{L^i}{W}{N}$ is in fact an equivalence relation between $M$ and $N$. For each class of this equivalence, we pick a suitable representative in the following definition.
\begin{definition}\label{def:LE_n} Let $\LE{}$ $(\RE{})$ denote the subset of $\LS{}$ $(\RS{})$ such that $M = \mat{a}{0}{c}{d} \in \LE{}$ $(N = \mat{a}{b}{0}{d} \in \RE{})$ if and only if $c < \gcd(a,d)$ $(b< \gcd(a,d))$. \end{definition}
The names $\LE{}$ and $\RE{}$ are abbreviations for ``$L$-exceptional'' and ``$R$-exceptional''.
For instance, $\mat{7}{0}{0}{2} \in \LE[14]{}$ and $\mat{7}{0}{1}{2} \not \in \LE[14]{}$ because $\gcd(a,d) = 1$.
Combining this definition with the last lemma, we immediately obtain the following corollary which says that the suitable representative is unique.
\begin{corollary} \label{cor:jednoznacnost_LE} Let $M \in \LS{}$. There is exactly one $N \in \LE{}$ such that $\Twalk{M}{L^i}{W}{N}$ exists for some $i \geq 0$ and $W \in \D[1]{}$. \end{corollary}
\begin{proof} Let $M = \mat{a}{0}{c}{d}$. \Cref{le:trida_L} implies that the walk $\Twalk{M}{L^i}{W}{N'} $ exists if and only if $N' = \mat{a}{0}{c'}{d}$ and $c' \equiv c \pmod{ \gcd(a,d)}$. As there is exactly one such $c' < \gcd(a,d)$, the claim follows from \Cref{def:LE_n}. \end{proof}
For each state $M \in \LE{}$, we shall need to know the least number $i$ such that we can get from $M$ to $M$ by reading $L^i$ as an input word.
\begin{definition} \label{def:nu} For $M \in \LE{}$ we set \[ \LclassMax{M} = \min \left\{ i > 0 \colon \Twalk{M}{L^i}{L^j}{M} \text{ exists for some } j \right\}. \] For $M \in \RE{}$, we define $\RclassMax{M}$ analogously: \[ \RclassMax{M} = \min \left\{ i > 0 \colon \Twalk{M}{R^i}{R^j}{M} \text{ exists for some } j \right\}. \] \end{definition}
\begin{example} For the state $M = \mat{7}{0}{0}{2} \in \LE[14]{}$, we have $\LclassMax{M} = 7$. Moreover, we have $M \in \RE[14]{}$ and $\RclassMax{M} = 2$. \end{example}
The purpose of the definition of the sets $\LE$ and $\RE$ is in the following lemma.
\begin{lemma} \label{le:spadnu_do_LS} Let $Q \in \DB{}$. There exists exactly one matrix $Z \in \LE{}$ such that $\Twalk{Q}{L^k}{W}{Z}$ for some $k \geq 0$ and $ W \in \D[1]{}$.
Moreover, the integer $k$ can be chosen such that $k \leq n$. \end{lemma}
\begin{proof} \Cref{le:char_toceni_na_L} and $\# \DB{}< + \infty$ imply that there exist $j \geq 0$, $M \in \LS{}$ and $W_1 \in \D[1]{}$ such that $\Twalk{Q}{L^j}{W_1}{M}$. Let $j$ be the least possible. Such $M$ is unique (depending on $Q$ only). \Cref{cor:jednoznacnost_LE} implies that there is exactly one $Z \in \LE{}$ such that $\Twalk{M}{L^{j'}}{W_2}{Z}$ for some $j' \in \mathbb{N}$ and $W_2 \in \D[1]{}$. It follows that $\Twalk{Q}{L^{j+j'}}{W_1W_2}{Z}$. The uniqueness of $Z$ follows from the uniqueness of $M$ and the uniqueness of $Z$ by \Cref{cor:jednoznacnost_LE}.
To prove the second part, we suppose that the walk $\Twalk{Q}{L^{k'}}{W'}{Z}$ exists for some $k' \in \mathbb{N}$ and $W' \in \D[1]{}$. Let $Q = \mat{a}{b}{c}{d}$ and $Z = \mat{a'}{0}{c'}{d'}$. It implies that \[ W' = QL^{k'}Z^{-1} = \mat{\frac{b {{d'}} k' - b {{c'}} + a {{d'}}}{{{a'}} {{d'}}}}{\frac{b}{{{d'}}}}{\frac{d {{d'}} k' - {{c'}} d + c {{d'}}}{{{a'}} {{d'}}}}{\frac{d}{{{d'}}}} \in \D[1]{}. \] Set $k$ such that $k \equiv k' \pmod n$ and $k \in \{1,\ldots,n\}$. Since $\frac{b {{d'}} k' - b {{c'}} + a {{d'}}}{{{a'}} {{d'}}}$ and $\frac{d {{d'}} k' - {{c'}} d + c {{d'}}}{{{a'}} {{d'}}}$ are both integers, $a'd' = n$, and $d' > c'$, we have that $\frac{b {{d'}} k - b {{c'}} + a {{d'}}}{{{a'}} {{d'}}}$ and $\frac{d {{d'}} k - {{c'}} d + c {{d'}}}{{{a'}} {{d'}}}$ are positive integers. Therefore, $W = QL^{k}Z^{-1}$ is a matrix of nonnegative integer elements. We verify by direct calculation that $\det (W) = 1$, and thus $W \in \D[1]{}$ and the walk $\Twalk{Q}{L^k}{W}{Z}$ exists. \end{proof}
The last lemma says that if we start in an arbitrary state of the transducer $\T{}$ and we read a long enough run of $L$'s (at most of length $n$) of the input word, we end in some state from $\LE{}$. Moreover, if we continue to read only the letters $L$, we can attain only the states from $\LS{}$ as follows from \Cref{le:char_toceni_na_L}, and, in particular, we have to return to the same state from $\LE{}$ (\Cref{cor:jednoznacnost_LE}).
\begin{example} If we take $Q = \mat{4}{2}{1}{4} \in \mathcal{DB}_{14}$, we have $\Twalk{Q}{L^5}{LRL}{M}$ where $M = \mat{7}{0}{0}{2} \in \LE{}$ and if we continue to read only $L$'s, we go through the walk $\Twalk{M}{L^7}{L^2}{M} = \Tedge{M}{L^4}{L}{ \Tedge{N}{L^3}{L}{M}}$ where $N = \mat{7}{0}{1}{2} \in \LS{}$. Similarly, we have $\Twalk{Q}{R^{10}}{R^3L}{\assoc{A}_{14}}$ where $\assoc{A}_{14} = \mat{1}{0}{0}{14} \in \RE{}$ and if we continue to read only $R$'s, we go through the edge $\Tedge{\assoc{A}_{14}}{R^{14}}{R}{\assoc{A}_{14}}$. \end{example}
\section{Construction of the bound $S_n$} \label{sec:constr}
In the previous section, we have defined the transducer $\T{}$ and stated its important properties that are used in the construction of the upper bound $S_n$ of \Cref{thm:main}. This section is dedicated to the construction of this bound.
\subsection{Maximalisation of the prolongation} \label{sec:kappa}
We define the mapping $\kappa: \D[1]{} \to \D[1]{}$ which shall be used to modify the runs of a word such that their length is within a suitable interval. For $W = Z_0^{i_0}Z_1^{i_1} \cdots Z_k^{i_k} \in \D[1]{}$ with $i_\ell$ nonzero, $Z_\ell \in \left\{ L,R \right\}$, and $Z_{\ell} \neq Z_{\ell+1}$ we set \[ \kappa(W) = Z_0^{i_0'}Z_1^{i_1'} \cdots Z_k^{i_k'} \] where $i_\ell' \equiv i_\ell \pmod{n}$ and $i_\ell' \in \{4n,4n+1,\ldots,5n-1\}$ for all $\ell \in \{0,\dots,k\}$. For instance, if $n = 10$, we have \[ \kappa(LR^{10}L^{55}) = L^{41}R^{40}L^{45}. \]
First, we show why we do not need runs longer than $5n-1$.
\begin{lemma} \label{le:vyfouknuti_n} Let $\Twalk{M}{V_1L^mV_2}{W}{N}$. If $m \geq 4n$, then $\Twalk{M}{V_1L^{m-n}V_2}{W'}{N}$, $\sigma(W') = \sigma(W)$, and $W'$ starts with the same letter as $W$. \end{lemma}
\begin{proof} \Cref{le:max_n_jednoho_pismene} implies that there exists an integer $t$ with $0 < t \leq n$ and $Q \in \DB{}$ such that \[ \Twalk{M}{V_1L^{t}}{W_1}{Q} \] for some $W_1 \in \D[1]{}$.
By \Cref{le:spadnu_do_LS}, there exists an integer $k$ with $k \leq n$ and $Z \in \LE{}$ such that \[ \Twalk{Q}{L^k}{W_2}{Z} \] for some $W_2 \in \D[1]{}$.
As $Z \in \LE{} \subseteq \LS{}$ we have \[ \Twalk{Z}{L^n}{L^s}{Z} \] for some positive integer $s$.
Let $q$ and $p$ be the integers such that $p = (m-t - k) \bmod{n}, p < n$ and $m-t-k = qn + p$. Since $m \geq 4n \geq 2n + k + t$, we have $q \geq 2$ and there exists the walk \[ \Twalk{Z}{L^pV_2}{W_3}{N} \]
and $W = W_1W_2L^{qs}W_3$.
As $q \geq 2$ implies that the walk $\Twalk{Z}{L^n}{L^s}{Z}$ is used at least twice while reading $V_1L^mV_2$, we conclude that while reading $V_1L^{m-n}V_2$, we take this walk one less time, but at least once. Thus, we have \[ \Twalk{M}{V_1L^{m-n}V_2}{W_1W_2L^{(q-1)s}W_3}{N} \] and $\sigma(W) = \sigma(W_1W_2L^{qs}W_3) = \sigma(W_1W_2L^{(q-1)s}W_3)$. \end{proof}
Similarly, we show that we shall not need the input words with runs shorter than $4n$. We start with a lemma.
\begin{lemma} \label{le:nafouknuti_jedno} Let $\Tedge{M}{VL^iR^j}{W}{N}$ with $i \geq 0$, $j \geq 1$, $V \in \D[1]{}$, and $M,N \in \DB{}$. There exist $Q \in \DB{}$, $W_1, W' \in \D[1]{}$ and an integer $t$ with $0 < t \leq n$ such that for all $r \geq 3$ \[ \Tedge{M}{VL^{i+t}}{W_1}{Q} \quad \text{ and } \quad \Twalk{Q}{L^{rn-t}R^j}{W'}{N} \] with $\sigma(W') > \sigma(W)$. Moreover, the word $W'$ has suffix $R^{-1}W$. \end{lemma}
\begin{proof} The first part of the proof is very similar to the previous proof. By \Cref{le:max_n_jednoho_pismene}, there exists an integer $t$ with $0 < t \leq n$ and $Q \in \DB{}$ such that \begin{equation} \label{pr:nafu_w1} \Tedge{M}{VL^{i+t}}{W_1}{Q} \end{equation} for some $W_1 \in \D[1]{}$.
By \Cref{le:spadnu_do_LS}, there exists an integer $k$ with $k \leq n$ and $Z \in \LE{}$ such that \begin{equation} \label{pr:nafu_w2} \Twalk{Q}{L^k}{W_2}{Z} \end{equation} for some $W_2 \in \D[1]{}$.
Let $r$ be an integer such that $rn \geq t+k$. Let $W_3(r)$ be the matrix given by \begin{equation} \label{pr:nafu_W3} ZL^{rn-t-k}R^j = W_3(r)N. \end{equation} We now show that $W_3(r) \in \D[1]{}$. Notice that this is equivalent to the existence of the following walk: $\Twalk{Z}{L^{rn-t-k}R^j}{W_3(r)}{N}$.
To show $W_3(r) \in \D[1]{}$, we first find $N^{-1}$ using $\Tedge{M}{VL^iR^j}{W}{N}$ as follows \begin{equation} \label{pr:nafu_Ni} N^{-1} = R^{-j}L^{-i}V^{-1}M^{-1}W. \end{equation}
By \eqref{pr:nafu_w1} and \eqref{pr:nafu_w2}, we have $MVL^{i+t+k} = W_1W_2Z$, therefore \begin{equation} \label{pr:nafu_W12i} (W_1W_2)^{-1} = Z L^{-i-t-k} V^{-1} M^{-1}. \end{equation}
Since $Z \in \LE{} \subseteq \LS{}$, we have by \Cref{le:LS_nacteni_n} that $\Twalk{Z}{L^n}{L^s}{Z}$ for some positive integer $s$. We combine this fact with \eqref{pr:nafu_W3} and obtain \[ W_3(r) = ZL^{rn-t-k}R^j N^{-1} = L^{rs} Z L^{-t-k} R^j N^{-1}. \] We continue by using \eqref{pr:nafu_Ni} and then \eqref{pr:nafu_W12i} to obtain \begin{align} \label{eq:W_3} W_3(r) = L^{rs} Z L^{-t-k} R^j N^{-1} & = L^{rs} Z L^{-t-k} R^j R^{-j} L^{-i}V^{-1}M^{-1}W \\ \notag & = L^{rs} Z L^{-i-t-k}V^{-1}M^{-1}W \\ \notag & = L^{rs} (W_1W_2)^{-1} W. \end{align}
We prove now the following claim. \begin{equation} \label{pr:aux_claim} \text{ If $W_3(r) \in \D[1]{}$ for some $r \geq 3$, then $W_3(3) \in \D[1]{}$ and $L^s$ is a prefix of $W_3(3)$. } \end{equation} Indeed, since for $r \geq 3$ we have $rn > 2n \geq t+k$, $W_3(r) \in \D[1]{}$ implies $\Twalk{Z}{L^{rn-t-k}R^j}{W_3(r)}{N}$. Let $rn-t-k = q(r)n + p$ with $0 \leq p < n$, i.e., $p = (-t-k) \bmod n$. Since $\Twalk{Z}{L^n}{L^s}{Z}$, it implies that the walk $\Twalk{Z}{L^{p}R^j}{L^{-q(r)s}W_3(r)}{N}$ exists and $L^{-q(r)s}W_3(r) \in \D[1]{}$. As $2n \geq t+k$, using \eqref{eq:W_3}, we find $W_3(2) = L^{q(2)s - q(r)s}W_3(r)$. Since $0 \leq q(2) < q(r)$, we have $W_3(2) = L^{q(2)s - q(r)s}W_3(r) \in \D[1]{}$. Moreover, we have $q(3)-q(2) = 1$. Therefore $W_3(3) = L^s W_3(2) \in \D[1]{}$ and $L^s$ is a prefix of $W_3(3)$, and thus \eqref{pr:aux_claim} holds.
We have the two following cases:
\begin{enumerate} \item Assume that $W_1W_2$ does not contain $R$. We may choose the integer $r$ large enough so that $L^{rs} (W_1W_2)^{-1} \in \D[1]{}$, and thus $W_3(r) \in \D[1]{}$ and $W$ is suffix of $W_3(r)$. Because $W$ has by \Cref{le:stejna_pismena} prefix $R$, then also $R^{-1}W$ is a suffix of $W_3(r)$ and therefore also of $W'$. By \eqref{pr:aux_claim}, we have $W_3(3) \in \D[1]{}$ and $L^s$ is its prefix. As $R$ is a prefix of $W$, we conclude $\sigma(W_3(3)) = \sigma(W)+1$. Since $W' = W_2W_3(3)$, we conclude $\sigma(W') > \sigma(W)$.
\item Assume that $W_1W_2$ contains $R$, i.e., $RL^h$ is its suffix for some $h \geq 0$. Since by \Cref{le:stejna_pismena} the word $W_1$ starts with $L$, we may write $W_1W_2 = LW_4RL^h$ for some $W_4 \in \D[1]{}$. Using \Cref{st:bublani_D1}, we conclude that $L \left( LW_4R \right)^{-1} R \in \D[1]{}$. We choose $r$ such that $rs-1 \geq h$. As $W$ starts with $R$ we have $R^{-1}W \in \D[1]{}$. We conclude that \[ W_3(r) = L^{rs} (W_1W_2)^{-1} W = L^{rs-1} L \left( LW_4RL^h \right)^{-1} R R^{-1}W = L^{rs-1-h} L \left( LW_4R \right)^{-1} R R^{-1}W \in \D[1]{} \] and $R^{-1}W$ is a suffix of $W_3(r)$ and therefore also of $W'$.
Therefore, by \eqref{pr:aux_claim}, $W_3(3) \in \D[1]{}$ and it starts with $L^s$. By \Cref{st:bublani_D1}, the word $L \left( LW_4R \right)^{-1} R$ starts with $R$. It implies that $\sigma(L^{rs} (W_1W_2)^{-1} R) \geq 2$ and moreover, if $L^{rs} (W_1W_2)^{-1} R$ ends with $L$, then $\sigma(L^{rs} (W_1W_2)^{-1} R)\geq 3$. Together with the facts that if $R^{-1}W$ starts with $R$, then $\sigma(R^{-1}W) = \sigma(W)$ and $\sigma(R^{-1}W) = \sigma(W) - 1$ otherwise, we conclude $\sigma(W_3(3)) > \sigma(W)$. Since $W' = W_2W_3(3)$, we have $\sigma(W') > \sigma(W)$. \end{enumerate}
The proof for $r=3$ is complete. The general case $r \geq 3$ follows from the existence of the walk $\Twalk{Z}{L^n}{L^s}{Z}$. \end{proof}
\begin{example} For instance, for the edge $\Tedge{M}{L^2R}{RL^4}{N}$, where $M = \mat{6}{2}{2}{3}$ and $N = \mat{2}{1}{0}{7}$ in the transducer $\mathcal{T}_{14}$, we have $\Tedge{M}{L^4}{LR^2}{\Tedge{A_{14}}{L^{14}}{L}{A_{14}}}$ and $\Tedge{A_{14}}{L^{12}R}{RL^6}{N}$ with $A_{14} = \mat{14}{0}{0}{1}$. Therefore, using the notation from the last lemma, we have $i = 2, j = 1$, $t = 2$ and there is a walk $\Twalk{A_{14}}{L^{rn-2}R}{L^{r-1}RL^6}{M}$. It means that $W = RL^4$ and $W' = L^{r-1}RL^6$ and so for all $r\geq 3$, $\sigma(W') = 3 > 2 = \sigma(W)$ and moreover, $R^{-1}W = L^4$ is a suffix of $W'$. \end{example}
\begin{corollary} \label{coro:kappa_hrana} Let $\Tedge{M}{VR^j}{W}{N}$ with $j > 0$. If $V$ is empty, then the walk \begin{enumerate}[(a)] \item $\Twalk{M}{L^{4n}R^j}{\widehat{W}}{N}$ with $\sigma(\widehat{W}) > \sigma(W)$ \label{it:kappa_hrana_prazdneV} \end{enumerate} exists. If $V$ ends in $L$, then the walks \begin{enumerate}[resume*]
\item $\Twalk{M}{\kappa(V)R^j}{\widehat{W}}{N}$ with $\sigma(\widehat{W}) > \sigma(W)$, \label{it:kappa_hrana_nic}
\item $\Twalk{M}{L^{4n}\kappa(V)R^j}{\widehat{W}_L}{N}$ with $\sigma(\widehat{W}_L) > \sigma(W)$, and \label{it:kappa_hrana_L}
\item $\Twalk{M}{R^{4n}\kappa(V)R^j}{\widehat{W}_R}{N}$ with $\sigma(\widehat{W}_R) > \sigma(W)$ \label{it:kappa_hrana_R} \end{enumerate} exist. \end{corollary}
\begin{proof} We shall prove the existence of the walk~\ref{it:kappa_hrana_prazdneV} directly and existence of the other walks by induction on $\sigma(V)$.
Assume that $V = L^i$ with $i \geq 0$, i.e., $\sigma(V) \leq 1$. Using \Cref{le:nafouknuti_jedno} with $r=4$, we obtain \[ \Tedge{M}{L^{i+t}}{W_1}{Q} \quad \text{ and } \quad \Twalk{Q}{L^{4n-t}R^j}{W'}{N}. \] with $\sigma(W') > \sigma(W)$ for some $Q \in \DB{}$, $W_1, W' \in \D[1]{}$ and an integer $t$ with $0 < t \leq n$. Therefore, the walk $\Twalk{M}{L^{i+4n}R^j}{W_1W'}{N}$ exists. Applying \Cref{le:vyfouknuti_n} the correct number of times, we obtain $\Twalk{M}{\kappa(L^i)R^j}{\widehat{W}}{N}$ with $\sigma(\widehat{W}) = \sigma(W_1W')\geq \sigma(W') > \sigma(W)$. This proves existence of \ref{it:kappa_hrana_prazdneV} and \ref{it:kappa_hrana_nic} for $\sigma(V) =1 $. Using the symmetric version of \Cref{le:nafouknuti_jedno} (using the symmetry of $L$ and $R$ given by \Cref{prop:assoc_sym}) on $\Tedge{M}{L^{i+t}}{W_1}{Q}$, we obtain that the walk \[ \Twalk{M}{R^{4n}L^{i+t}}{W_2}{Q} \] exists. Therefore, $\Twalk{M}{R^{4n}L^{4n+i}R^j}{W_2W'}{N}$ exists. Considering again \Cref{le:vyfouknuti_n}, we prove \ref{it:kappa_hrana_R}.
To show the existence of \ref{it:kappa_hrana_L}, we proceed as in the case of the walk~\ref{it:kappa_hrana_nic} except for using \Cref{le:nafouknuti_jedno} with $r=8$ and factoring $L^{4n}$ in the input word of the obtained walk.
By \Cref{prop:assoc_sym}, the symmetric version of the claim for $\sigma(V) \leq 1$ holds.
Assume now the claim and its symmetric version hold for $\sigma(V') = k$ and let $V = V'L^i$ with $\sigma(V'L^i) = k+1$. We apply \Cref{le:nafouknuti_jedno} on $\Tedge{M}{V'L^iR^j}{W}{N}$: there exist $Q \in \DB{}$, $W_1, W' \in \D[1]{}$ and an integer $t$ with $0 < t \leq n$ such that \[ \Tedge{M}{V'L^{i+t}}{W_1}{Q} \quad \text{ and } \quad \Twalk{Q}{L^{4n-t}R^j}{W'}{N} \] with $\sigma(W') > \sigma(W)$. By the induction hypothesis and the symmetry of $L$ and $R$ (\Cref{prop:assoc_sym}) on $\Tedge{M}{V'L^{i+t}}{W_1}{Q}$, we obtain $\Twalk{M}{X\kappa(V')L^{i+t}}{W_1'}{Q}$ with $\sigma(W_1') > \sigma(W_1)$ and $X$ being empty, $L^n$, or $R^n$. The situation is illustrated in \Cref{fig:nafukovani}. Therefore, \[ \Twalk{M}{X\kappa(V')L^{i+4n}R^j}{W_1'W'}{N}. \] Using \Cref{le:vyfouknuti_n}, we obtain $\Twalk{M}{X\kappa(V'L^i)R^j}{\widehat{W}}{N}$ with $\sigma(\widehat{W}) = \sigma(W_1'W') \geq \sigma(W') > \sigma(W)$. \end{proof}
\begin{remark} In fact, the mapping $\kappa$ could be defined such that it adjusts the length of each run between $3n$ and $(4n-1)$ and we could prove the same bound in \Cref{thm:main}. Our experiments show that it might also be sufficient to adjust the length of all runs between $2n$ and $(3n-1)$. However, we use the given definition of $\kappa$ since it simplifies the proofs without changing the result. \end{remark}
\begin{figure}
\caption{ An illustration of the idea in the first part of the proof of \Cref{coro:kappa_hrana}. The figure should be read as follows: the top line contains the original edge $\Tedge{M}{VR^j}{W}{N}$ with $\sigma(V) = \ell+1$ even, and we proceed from right to left by constructing a new walk going through the vertices $Q$ given by \Cref{le:nafouknuti_jedno}. We start by finding $Q_0$ and find a walk with the length of the penultimate run in the input word modified. We continue from right to left until we reach the first run, indexed by $\ell$ in the figure, in the current input word. The last steps depends on which item is being shown: \Cref{it:kappa_hrana_L} is depicted by the bottom dashed path on the left with $\omega=4$, \Cref{it:kappa_hrana_nic} is the top dashed path on the left with $\omega=4$, and finally \Cref{it:kappa_hrana_R} is the top dashed path on the left with $\omega=8$. The 3 cases correspond to $X$ being $L^n$, empty or $R^n$, respectively. A figure for $\sigma(V)$ odd is analogous. }
\label{fig:nafukovani}
\end{figure}
The following lemma describes the paths in the transducer $\T{}$ that are taken when the greatest prolongation occurs.
\begin{lemma} \label{le:LS_n_do_RS_n} Let $M = \mat{t_1}{0}{u_1}{m_1} \in \LE{}$ and $i \in \{\LclassMax{M}, \LclassMax{M} + 1, \dots, 2 \LclassMax{M}-1\}$. \begin{enumerate}[(1)]
\item There exist one and only one matrix $N_{L,M,i} \in \RE{}$ such that $\Twalk{M}{L^iR^{j}}{W}{N_{L,M,i}}$ for some $j \in \mathbb{N}$ and $W \in \D[1]{}$.
Moreover, there is exactly one $j_{L,M,i} \in \{3n - \RclassMax{N} + 1, 3n - \RclassMax{N} + 2, \dots, 3n\}$ such that the walk from $M$ to $N_{L,M,i}$ with the input word $L^iR^{j_{L,M,i}}$ and an output word $W_{L,M,i}$ exists.
$N_{L,M,i}, j_{L,M,i}$ and $W_{L,M,i}$ depend only on $M$ and $i$. \label{le:LS_n_do_RS_n:i1}
\item The word $W_{L,M,i}$ starts with $L$, ends with $R$ and satisfies \[ \sigma(W_{L,M,i}) = 2\left\lfloor \frac{\xi_{L,M,i}}{2} \right\rfloor +2, \] where $\xi_{L,M,i} = \xi({im_1+u_1}, {t_1})$. \label{le:LS_n_do_RS_n:i2} \end{enumerate} \end{lemma}
\begin{proof} Assume that we are on a walk which starts in $M$, we input $L^i$, and we start inputting a run of $R$'s. \Cref{le:max_n_jednoho_pismene,le:spadnu_do_LS} imply that after inputting at most $2n$ $R$'s we reach a state $N \in \RE{}$. Moreover, the matrix $N$ is unique and depends only on $M$ and $i$ (and the fact that the walk started by reading $L$'s). By \Cref{def:nu}, there is exactly one $j \in \{3n - \RclassMax{N} + 1, 3n - \RclassMax{N} + 2, \dots, 3n\}$ such that $\Twalk{M}{L^iR^{j}}{W}{N}$, depending only on $M$ and $i$. Therefore, there is also exactly one word $W$, and it depends only on $M$ and $i$, which concludes the proof of \Cref{le:LS_n_do_RS_n:i1}.
Let $N_{L,M,i} = \mat{t_2}{u_2}{0}{m_2}$. We have \[ W_{L,M,i} = ML^iR^{j_{L,M,i}}N_{L,M,i}^{-1} = \mat{\frac{t_1}{t_2}}{f}{\frac{im_1+u_1}{t_2}}{e}. \] for some $e,f \in \mathbb{N}$. Using $W_{L,M,i} \in \D[1]{}$, we conclude that $\frac{im_1+u_1}{t_2}, \frac{t_1}{t_2} \in \mathbb{N}$ and $\gcd{ (\frac{im_1+u_1}{t_2}, \frac{t_1}{t_2})} = 1$. Therefore, $t_2 = \gcd{ ((im_1 + u_1),t_1)}$. As $i \geq \LclassMax{M}$, the walk $\Twalk{M}{L^iR^{j_{L,M,i}}}{W_{L,M,i}}{N_{L,M,i}}$ starts with the walk $\Twalk{M}{L^{\nu}}{L^t}{M}$ with $\nu = \LclassMax{M}$ and some $t>0$, and the walk ends with the walk $\Twalk{N}{R^n}{R^s}{N}$. Thus, the output word of the first walk is a power of $L$, and $L$ is the first letter of $W_{L,M,i}$ and the output word of the last walk is a power of $R$, and $R$ is the last letter of $W_{L,M,i}$.
By \Cref{le:pocet_LRzmen}, $\sigma(W_{L,M,i}) = 2 \left\lfloor \frac{\xi \left(\frac{im_1+u_1}{t_2}, \frac{t_1}{t_2}\right)}{2} \right\rfloor + 2$. By definition of $\xi$, we have $\xi(\frac{im_1+u_1}{t_2}, \frac{t_1}{t_2}) = \xi(im_1+u_1,t_1) = \xi_{L,M,i}$. \end{proof}
In what follows, the notation introduced by the last claim is used. We also use the symmetric version of this notation in the following sense. The symmetric version of~\Cref{le:LS_n_do_RS_n} holds, and given $M \in \RE{}$ and $i \in \{\RclassMax{M}, \RclassMax{M} + 1, \dots, 2 \RclassMax{M}-1\}$, we find $N_{R,M,i} \in \LE{}$, $j_{R,M,i} \in \{3n - \LclassMax{N_{R,M,i}} + 1, 3n - \LclassMax{N_{R,M,i}} + 2, \dots, 3n\}$ and $W_{R,M,i} \in \D[1]{}$. The $L$, resp. $R$, in the subscript is needed for the case $M \in \LS{} \cap \RS{} \neq \emptyset$. In general, we may have $j_{L,M,i} \neq j_{R,M,i}$. Similarly, we also use the notation $\xi_{R,M,i}$.
\subsection{Closed walks}
In this section, we return to the computation of the given M\"{o}bius transformation for a periodic input word. Clearly, the computation in the transducer $\T{}$ ends in a repeating loop, i.e., we end up with some closed walk $\Twalk{M}{V}{W}{M}$. We start with a general lemma that deals with the case when this closed walk is symmetric in the following sense. If the closed walk $\Twalk{M}{V}{W}{M}$ can be decomposed into two parts where the first part is $\Twalk{M}{V_1}{W_1}{\assoc{M}}$ and the second part is $\Twalk{\assoc{M}}{\assoc{V_1}}{W_2}{M}$, then we say that the closed walk is \emph{symmetric}. Note that this implies that $V = V_1\assoc{V_1}$, and by \Cref{prop:assoc_sym} it also implies $W_2 = \assoc{W_1}$, which is stated as the following lemma.
\begin{lemma} \label{le:sym_dvojstav} If $\Twalk{M}{V}{W}{M}$ is a symmetric closed walk, then $W = W_0\assoc{W_0}$ for some $W_0$. \end{lemma}
Note that symmetricity of the closed walk $\Twalk{M}{V}{W}{M}$ is not equivalent to $V = V_1\assoc{V_1}$ and $W = W_1\assoc{W_1}$.
We introduce the following notation, which is due to the fact that we shall require to change the starting vertex of a closed walk. Let $V \in \{L,R\}^*$. We set $\tau(V)$ to be the set of all conjugate words of $V$. A word $W$ is \emph{conjugate} to a word $V$ if $V = V_1V_2$ and $W = V_2V_1$, i.e., $V$ is cyclic shift of $W$. Furthermore, we set \[ {\sigma_{\mathrm{c}}}(V) = \min \left\{ \sigma(V') \colon V' \in \tau(V) \right\}, \] i.e., except for the case $\sigma(V) = 1$, the mapping ${\sigma_{\mathrm{c}}}$ counts the number of runs of a conjugate word of $V$ which starts and ends in a distinct letter, that is ${\sigma_{\mathrm{c}}}(V) = 2 \left \lfloor \frac{\sigma(V)}{2} \right \rfloor$. Note that if $\sigma(V) \geq \sigma(W)$, then ${\sigma_{\mathrm{c}}}(V) \geq {\sigma_{\mathrm{c}}}(W)$.
Let $W \in \D[1]{}$ be non-empty. We set \[ \tau_\kappa(W) = \tau(\kappa(W')) \quad \text{ where } W' \in \tau(W), \sigma(W')={\sigma_{\mathrm{c}}}(W). \] The mapping $\tau_\kappa$ defines the transformation of the input word which yields the upper bound in \Cref{thm:main}. The following theorem expresses its key role.
\begin{restatable}{theorem}{restatableVyfukovani} \label{co:vyfukovani_ze_symetrickeho} Let $\Twalk{M}{V}{W}{M}$. There exists $\widehat{V} \in \tau_\kappa(V)$ such that $\Twalk{\widehat{M}}{\widehat{V}}{\widehat{W}}{\widehat{M}}$ for some $\widehat{M} \in \DB{}$ with ${\sigma_{\mathrm{c}}}(\widehat{W}) \geq {\sigma_{\mathrm{c}}}(W)$. Moreover, \begin{enumerate}[(a)] \item \label{it:1)vyfukovani} if $V = V_1^{m_1}$ for some $V_1, m_1 \geq 2$ and $\Twalk{\widehat{M}}{\widehat{V}}{\widehat{W}}{\widehat{M}} = \left( \Twalk{\widehat{M}}{\widehat{V_1}}{\widehat{W_1}}{\widehat{M}} \right)^{m_1}$ where $\widehat{W_1}^{m_1} = \widehat{W}$ and $\widehat{V_1}^{m_1} = \widehat{V}$, then $\Twalk{M}{V}{W}{M}=(\Twalk{M}{V_2}{W_2}{M})^{m_2}$ for some $m_2 \geq 2$ and $W_2^{m_2} = W$ and $V_2^{m_2} = V$; \item \label{it:2)vyfukovani} if $V = V_1\assoc{V_1}$ for some $V_1$ and $\Twalk{\widehat{M}}{\widehat{V}}{\widehat{W}}{\widehat{M}}$ is symmetric, then $\Twalk{M}{V}{W}{M}$ is symmetric. \end{enumerate} \end{restatable}
\subsubsection{Proof of \Cref{co:vyfukovani_ze_symetrickeho}} \label{subsec:kappa_walks}
Several technical lemmas concerning possible forms of edges in the transducer $\T{}$ follow. These lemmas are used for the proof of \Cref{co:vyfukovani_ze_symetrickeho} given at the end of this subsection.
\begin{lemma} \label{le:vstup_V_1} Let $\Tedge{M}{V}{W}{P}$, where $M,P \in \DB{}$, $V, W \in \D[1]{}$. \begin{enumerate}
\item \label{le:vstup_RLk_1} If $V = \widehat{V}RL^{j}$ for some $ \widehat{V} \in \D[1]{}$ and $j\geq 2$, then \[ W = LR^{i} \quad \text{ and } \quad i \geq j-1. \] \item \label{le:vstup_LRjL_1} If $V = \widehat{V}LR^{j}L$ for some $ \widehat{V} \in \D[1]{}$ and $j\geq 1$, then \[ W = LR^{i} \quad \text{ and } \quad i \geq 1. \] \item \label{le:vstup_L_1} IF $V = L^\ell$ with $\ell \geq 1$, then \[ W = LR^s \quad \text{ or } \quad W = L^t \] where $s \geq 0, t \geq 2$. Moreover, if $W = L^t$, then $\ell = 1$. \end{enumerate} \end{lemma}
\begin{proof} We start with the proof of \Cref{le:vstup_RLk_1}.
Let $M \widehat{V} = \mat{a}{b}{c}{d}$. Then \[ M \widehat{V}RL^{j} = \mat{(j+1)a+jb}{a+b}{(j+1)c+jd}{c+d} \not \in \RB{}. \] Moreover, by \Cref{le:stejna_pismena}, the word $W$ has prefix $L$. Therefore, $c+d>a+b$. Since \[ M \widehat{V}RL^{k} = \mat{(k+1)a+kb}{a+b}{(k+1)c+kd}{c+d} \in \RB{} \] for all $k <j$, we have
\begin{equation} \label{eq:k_RLk} (k+1)a+kb >(k+1)c+kd. \end{equation} For $k = j-1$ we obtain \begin{equation} \label{eq:1_RLk} ja+(j-1)b >jc+(j-1)d. \end{equation} In particular, since $j \geq 2$, we obtain for $k=1$: \begin{equation} \label{eq:2_RLk} 2a+b >2c + d. \end{equation}
If $W$ has prefix $L^2$, then \[ L^{-2}WP = \mat{(j+1)a+jb}{a+b}{(j+1)(c-2a)+j(d-2b)}{c+d-2(a+b)} \in \D{} \] and therefore \[ 0 \leq(j+1)(c-2a)+j(d-2b) \overset{\eqref{eq:1_RLk}}{<} - ja-2a -(j+1)b+c+d \overset{\eqref{eq:2_RLk}}{<} 0 \] which is a contradiction. Therefore, $L^2$ is not a prefix of $W$. Let $i$ be maximal possible such that $LR^i$ is a prefix of $W$. We have \[ R^{-i}L^{-1}WP = R^{-i}L^{-1}M \widehat{V}RL^{j} = \] \[ = \mat{(i+1)[(j+1)a+jb] - i [(j+1)c+jd]}{(i+1)(a+b)-i(c+d)}{j(c+d) - j(a+b)+c-a}{c+d-a-b} \in \D{}. \] It follows that \begin{equation} \label{eq:b_RLk} (i+1)(a+b) -i(c+d)\geq 0. \end{equation}
If $j > i+1$, then \Cref{eq:k_RLk} holds for $k = i+1$ and therefore
\begin{equation} \label{eq:i1_RLk} (i+2)a+(i+1)b >(i+2)c+(i+1)d. \end{equation}
Whereas using that $i$ is maximal possible, we have \[ (R^{-i}L^{-1}WP)_{1,2} \leq (R^{-i}L^{-1}WP)_{2,2} \iff (i+1)(a+b)-i(c+d) \leq c+d-a-b \] \[ \implies \] \[ 0 \leq (i+1)(c+d) - (i+2)(a+b) \overset{\eqref{eq:i1_RLk}}{<} -b -c \leq 0 \] which is a contradiction. It means that $W$ has prefix $LR^i$ for some $i \geq j-1$.
We show that $W$ is also equal to this prefix. Again, we suppose for contradiction that $W$ has prefix $LR^{i}L$. It means that \[ (R^{-i}L^{-1}WP)_{1,1} \leq (R^{-i}L^{-1}WP)_{2,1} \iff (i+1)[(j+1)a+jb] - i [(j+1)c+jd] \leq j(c+d) - j(a+b)+c-a \] \[ \implies \] \[ 0 \geq (i+2)[(j+1)a+jb] - (i+1) [(j+1)c+jd] \overset{\eqref{eq:1_RLk}}{>} (i+2)(a+b)-(i+1)(c+d)+(ja+(j-1)b) \overset{\eqref{eq:b_RLk}}{\geq} \] \[ \overset{\eqref{eq:b_RLk}}{\geq} a+b-(c+d)+(ja+(j-1)b) \overset{\eqref{eq:2_RLk}}{>} 0 \] where the last inequality uses that $j \geq 2$. This is a contradiction and therefore $W = LR^i$ and the first part of this lemma is proven.
We continue with the proof of \Cref{le:vstup_LRjL_1}.
Let again $M\widehat{V} = \mat{a}{b}{c}{d}$. We have \[ M\widehat{V}LR^j = \mat{a+b}{ja+(j+1)b}{c+d}{jc+(j+1)d} \in \RB{}. \] It follows that \begin{equation} \label{eq:ab_LRjL} a+b>c+d. \end{equation} By \Cref{le:stejna_pismena}, we know that $W$ has prefix $L$ and thus \[ L^{-1}WP = L^{-1} M\widehat{V}LR^jL = \] \[ = \mat{(j+1)a+(j+2)b}{ja+(j+1)b}{(j+1)c + (j+2)b-(j+1)a-(j+2)b}{jc+(j+1)d-ja-(j+1)b} \in \D{}. \] Moreover, we have \[ (L^{-1}WP)_{1,2} - (L^{-1}WP)_{2,2} = 2ja+2(j+1)b-jc-(j+1)d \geq (j+1)(a+b-c-d) \overset{\eqref{eq:ab_LRjL}}{>} 0, \] which means that $W$ has prefix $LR$. Let $i\geq 1$ be maximal possible such that $W$ has prefix $LR^i$. We have \[ R^{-i}L^{-1}WP = R^{-i}L^{-1}M \widehat{V}LR^jL = \] \[
\mat{(i+1)[(j+1)a+(j+2)b] - i[(j+1)c+(j+2)d]}{(i+1)[ja+(j+1)b]-i[jc+(j+1)d]}{(j+1)c+(j+2)d-(j+1)a-(j+2)b}{jc+(j+1)d-ja-(j+1)b} \in \D{} \] and therefore \begin{equation} \label{eq:i_LRjL} (i+1)(ja+(j+1)b)-i(jc+(j+1)d) \geq 0. \end{equation} Therefore, \[ (R^{-i}L^{-1}WP)_{1,1} - (R^{-i}L^{-1}WP)_{2,1} = (i+2)[(j+1)a+(j+2)b] - (i+1)[(j+1)c+(j+2)d] \overset{\eqref{eq:i_LRjL}}{\geq} \] \[
\overset{\eqref{eq:i_LRjL}}{\geq} (j+i+2)a+(j+i+3)b- [(j+i+1)c + (j+i+2)d] \geq (j+i+2)(a+b-c-d) \overset{\eqref{eq:ab_LRjL}}{>} 0 \] and finally $W = LR^i$.
To prove the last item of the lemma, let $M = \mata$.
We have \[ ML^k = \mat{a+kb}{b}{c+kd}{d} \in \RB{} \] for all $k <\ell$, which means that $a+kb>c+kd$ and in particular \begin{equation} \label{eq:l1_vstup_L} a+(\ell-1)b>c+(\ell-1)d. \end{equation} By \Cref{le:stejna_pismena}, we know that $W$ has prefix $L$. We first investigate the case in which $L^2$ is not a prefix of $W$. We take $s \geq 0$ maximal possible such that $LR^s$ is a prefix of $W$. If $s = 0$, we have $W = L$ and the claim holds. In the case $s \geq 1$, we have \[ R^{-j}L^{-1}WP = R^{-j}L^{-1}ML^{\ell} = \mat{(a+\ell b)(j+1)-j(c+\ell d)}{(j+1)b-jd}{c+\ell d-(a+\ell b)}{d-b} \in \D{} \] for all $j \leq s$. Therefore, $(j+1)b-jd \geq 0$ and specially for $j = 1$ ($s \geq 1$) \begin{equation} \label{eq:1_vstup_L} 2b-d \geq 0 \end{equation} and for $j = s$ \begin{equation} \label{eq:s1_vstup_L} (s+1)b-sd \geq 0. \end{equation} It follows that \[ (R^{-s}L^{-1}WP)_{1,1} - (R^{-s}L^{-1}WP)_{2,1} = (a+\ell b)(s+2)-(s+1)(c+\ell d) = \] \[
= (s+1)(a-c+(\ell-1)(b-d))+a+\ell b+(s+1)b-(s+1)d \overset{\eqref{eq:s1_vstup_L}}{\geq}(s+1)(a-c+(\ell -1)(b-d)) + a+\ell b-d \overset{\eqref{eq:l1_vstup_L}}{>} \] \[ \overset{\eqref{eq:l1_vstup_L}}{>} a+\ell b-d > 2b-d \overset{\eqref{eq:1_vstup_L}}{\geq} 0. \] Therefore, we have \[ (R^{-s}L^{-1}WP)_{1,1} > (R^{-s}L^{-1}WP)_{2,1} \] and therefore the word $W$ cannot have prefix $LR^sL$. Thus, $W = LR^s$.
Now, we investigate if $W$ can have prefix $L^2$. We have \[ L^{-1}WP = L^{-1}ML^\ell = \mat{a+\ell b}{b}{c+\ell d-(a+\ell b)}{d-b}. \] and for $\ell \geq 2$ we have \[ (L^{-1}WP)_{1,1} - (L^{-1}WP)_{2,1} = 2a+2\ell b-c-\ell d \overset{\eqref{eq:l1_vstup_L}}{>} 2b + c+ (\ell -2)d \geq 0 \] and therefore $W$ cannot have the prefix $L^2$. It remains to deal with the case $\ell = 1$ and $W$ has prefix $L^t$ for some $t \geq 2$ maximal possible. We have \[ L^{-t}WP = L^{-t}ML^ = \mat{a+b}{b}{c+d-t(a+b)}{d-tb} \in \D{}. \] Therefore, we have \begin{equation} \label{eq:t_vstup_L} (c+d)-t(a+b)\geq 0. \end{equation} Further, \[ (L^{-t}WP)_{2,2} - (L^{-t}WP)_{1,2} = d-(t+1)b \overset{\eqref{eq:t_vstup_L}}{\geq} ta-c-b \geq 2a-c-b >0 \] where we have used $t \geq 2$. Therefore, $W$ cannot have prefix $L^tR$, which means $W = L^t$. \end{proof}
\begin{corollary} \label{le:vstup_V_2} Let $\Tedge{M}{V}{W}{P}$ with $M,P \in \DB{}$, $V, W \in \D[1]{}$. \begin{enumerate}
\item \label{le:vstup_RLk_2} If $W = L^{j}R\widehat{W}$ for some $ \widehat{W} \in \D[1]{}$ and $j\geq 2$, then \[ V = R^{i}L \quad \text{ and } \quad i \geq j-1. \] \item \label{le:vstup_L_2} If $W = L^{\ell}$ and $\ell \geq 1$, then \[ V = R^{s}L \quad \text{ or } \quad V = L^{t}, \] where $s \geq 0, t \geq 2$. Moreover, if $V = L^{t}$, then $\ell = 1$. \end{enumerate} \end{corollary}
\begin{proof} The two claims follow directly from \Cref{le:vstup_RLk_1,le:vstup_L_1} of \Cref{le:vstup_V_1} and \Cref{prop:assoc_sym}, namely the fact that $\Tedge{M}{V}{W}{P}$ implies $\Tedge{P^T}{W^T}{V^T}{M^T}$. \end{proof}
\begin{lemma} \label{le:hrany_v_cyklu} Every closed walk in the transducer $\T{}$ either includes at least one edge with a nonempty input which cannot be written as $L^{q}R$ or $R^{q}$ for some $q \geq 1$ or the input word of this walk is $R^r$ for some $r \geq 1$. \end{lemma} \begin{proof} Let $M = \mat{a}{b}{c}{d}$, $N = \mat{e}{f}{g}{h}$ and $\Tedge{M}{L^qR}{W}{N}$. By \Cref{le:stejna_pismena}, $W$ has prefix $R$. By direct computation, we obtain \[ (R^{-1}WN)_{1,1} = a+qb - c - qd \] and therefore \[ e \leq a+ qb-c-qd \leq a + q(b-d) < a, \] where the last inequality follows from $M \in \DB{}$ which implies $b<d$.
Similarly for the transition $\Tedge{M}{R^q}{W}{N}$, we obtain using \Cref{le:vstup_L_1} of \Cref{le:vstup_V_1} that either $W = RL^s$ for some $s \geq 0$, or $W = R^t$ for $t \geq 2$ and $q= 1$. By direct computation, we obtain \[ e = a-c \leq a \] in the first case and \[ e = a-tc \leq a \] in the second case.
This means that the number in the first row and first column of the state matrix after taking the edge with the input word $R^q$ is either the same or it is lessened, and after taking the edge with the input word $L^qR$ it is always lessened. \end{proof}
\begin{lemma} \label{le:hrany_do_P} Let $\Tedge{N}{L^\ell }{LR^s}{P}$, where $N,P \in \DB{}$, $N = \mat{a}{b}{c}{d}$, $\ell \geq 1$ and $s \geq \ell +1$. We have: \[ a <2b. \] \end{lemma} \begin{proof} Let $P = \mat{e}{f}{g}{h}$. By direct computation, we obtain \[ N = LR^sPL^{-\ell } = \mat{e+sg-\ell (f+sh)}{f+sh}{e+(s+1)g-\ell (f+(s+1)h)}{f+(s+1)h} \in \DB{}. \]
Therefore, \begin{equation} \label{eq:cd_hrany_do_P} f+(s+1)h>e+(s+1)g-\ell (f+(s+1)h) \end{equation} and we have \[ 2b-a = 2(f+sh) + \ell (f+sh)-e-sg \overset{\eqref{eq:cd_hrany_do_P}}{>} f+(s-\ell -1)h+g \geq 0 \] where the last inequality holds because $s\geq \ell +1$. It follows that $2b>a$. \end{proof}
\begin{lemma} \label{le:vstup_do_N} Let $\Tedge{M}{V}{W}{N}$, where $N,M \in \DB{}$, $N = \mat{a}{b}{c}{d}$, $V,W \in \D[1]{}$, $V$ has suffix $L$ and $a\leq 2b$. We have: \[ V = R^pL, \] where $p \geq 1$. \end{lemma} \begin{proof}
By \Cref{le:stejna_pismena}, we know that since $V$ has suffix $L$, the word $W$ has prefix $L$. We have by \Cref{le:vstup_V_1} that $W = LR^k$ for some $k\geq 1$ or $W = L^s$ for some $s \geq 1$ or $V = R^pL$ for some $p\geq 1$. In the third case the claim holds. The other two possibilities are discussed separately. \begin{enumerate} \item $W = LR^k$ for some $k \geq 1$.
According to \Cref{prop:assoc_sym}, there exists the edge $\Tedge{N^T}{W^T}{V^T}{M^T}$. The starting vertex of this edge is $N^T = \mat{a}{c}{b}{d}$ and the input word is $W^T = L^kR$. Using \Cref{thm:Raney_edge} we can determine the output word of this edge (the word $V^T$). We already know that the word $V^T$ has prefix $R$.
We have \[ R^{-1}N^TL^kR = \mat{a+kc-b-kd}{a+(k+1)c-b-(k+1)d}{b+kd}{b+(k+1)d}. \] Moreover, because $a \leq 2b$ and $c<d$, we have \[ (R^{-1}N^TL^kR)_{1,1} - (R^{-1}N^TL^kR)_{2,1} = a-2b+k(c-2d) < 0 \] which means that $V^T$ has prefix $RL$.
Let $p \geq 1$ be maximal possible such that $V^T$ has prefix $RL^p$. We have \[ L^{-p}R^{-1}N^TW^T = \mat{a+kc-b-kd}{a+(k+1)c-b-(k+1)d}{(p+1)(b+kd)-p(a+kc)}{(p+1)(b+(k+1)d)-p(a+(k+1)c)} \in \D{} \] and therefore \begin{equation} \label{eq:p_vstup_do_N} (p+1)(b+kd)-p(a+kc) \geq 0. \end{equation} Moreover, \[ (L^{-p}R^{-1}N^TW^T )_{2,2} - (L^{-p}R^{-1}N^TW^T )_{1,2} = (p+2)(b+(k+1)d)-(p+1)(a+(k+1)c) \overset{\eqref{eq:p_vstup_do_N}}{\geq} \] \[ \overset{\eqref{eq:p_vstup_do_N}}{\geq} b+(k+1)d-(a+(k+1)c) +(p+1)d-pc = (k+1)(d-c)+p(d-c)+(b+d-a) > \] \[ > b+d-a >2b-a \geq 0 \] where the last inequality follows from $a \leq 2b$. Therefore, $V^T = RL^p$.
\item $W = L^s$ for some $s \geq 1$.
Because the input word has suffix $L$ we know by \Cref{le:vstup_V_1} that $V = R^pL$ for some $p \geq 1$ or $V = L^t$ for some $t\geq 1$. Now, we suppose for contradiction that the second case holds. We have $M = WNV^{-1} = L^s N L^{-t} = \mat{a-tb}{b}{s(a-tb)+c-td}{sb+d}$. Moreover $M \in \DB{}$ and therefore \[ M_{1,1} >M_{1,2} \iff a-tb>b \implies a>2b \]
which is a contradiction.
\end{enumerate}
\end{proof}
A few lemmas, which are used in the proof of \Cref{prop:hrany_do_P2}, follow.
\begin{lemma} \label{le:1_hrany_do_P2} Let \[ \Tedge{P_1}{V_1L^{k_1}}{LR^{j_1}}{M} \text{ and } \Tedge{P_2}{V_2L^{k_2}}{LR^{j_2}}{M}, \] where $P_1, P_2,M \in \DB{}$, $V_1,V_2 \in \D[1]{}$ and neither of them has suffix $L$, $k_1,k_2 \geq 1,j_1,j_2 \geq 0$ and $j_2 >j_1$. Then $k_2 \leq k_1$. \end{lemma} \begin{proof} We put $M = \mat{a}{b}{c}{d}$. Then \[ P_1V_1 = LR^{j_1}ML^{-k_1} = \mat{a+j_1c-k_1(b+j_1d)}{b+j_1d}{a+(j_1+1)c-k_1(b+(j_1+1)d)}{b+(j_1+1)d}. \] As $V_1$ does not have suffix $L$, we have \[ (P_1V_1)_{2,1} < (P_1V_1)_{2,2} \implies \] \begin{equation} \label{eq:le_1} \implies a+(j_1+1)c-k_1(b+(j_1+1)d) < b+(j_1+1)d. \end{equation} (The case $(P_1V_1)_{2,1} = (P_1V_1)_{2,2} \wedge (P_1V_1)_{1,1} = (P_1V_1)_{1,2}$ is not possible, because $\det(P_1V_1) = n >0$).
Further, we have \[ P_2V_2 = LR^{j_2}ML^{-k_2} = \mat{a+j_2c-k_2(b+j_2d)}{b+j_2d}{a+(j_2+1)c-k_2(b+(j_2+1)d)}{b+(j_2+1)d} \in \D{}. \] Let now suppose for contradiction that $k_2 > k_1$. We have \[ 0 \leq (P_2V_2)_{2,1} = a+(j_2+1)c-k_2(b+(j_2+1)d) \leq a+(j_2+1)c-(k_1 + 1)(b+(j_2+1)d) = \] \[ = a- (k_1 + 1)b + (j_2+1)(c-(k_1 + 1)d) \overset{(k_1+1)d\, \geq \, d\, >\, c ; \;j_1<j_2}{<} a- (k_1 + 1)b + (j_1+1)(c-(k_1 + 1)d) = \] \[ = a + (j_1+1)c - (k_1 + 1)(b +(j_1+1)d)\overset{\eqref{eq:le_1}}{<} 0 \] which is a contradiction. \end{proof}
\begin{lemma} \label{le:2_hrany_do_P2} Let $P_1,M \in \DB{}$, $\ell \geq 2$ and in the transducer $\T{}$ be the edge \[ \Tedge{P_1}{L}{L^\ell }{M}. \] Then every other incoming edge of the state $M$ with the input word which has suffix $L$ is of the form \[ \Tedge{P_2}{R^mL}{W}{M} \] for some $P_2 \in \DB{}, W \in \D[1]{}$ and $m \geq 1$. \end{lemma} \begin{proof} Let $\Tedge{P_2}{V}{W}{M}$ where $V \in \D[1]{}$ and has suffix $L$, be some other incoming edge of the state $M$. By \Cref{le:stejna_pismena}, $W$ has prefix L. By \Cref{prop:assoc_sym}, we know that the edges $\Tedge{P_1}{L}{L^\ell }{M}$, $\Tedge{P_2}{V}{W}{M}$ exist if and only if the edges $\Tedge{M^T}{R^\ell }{R}{P_1^T}$ and $\Tedge{M^T}{W^T}{V^T}{P_2^T}$ exist.
Because $\ell \geq 2$, we know according to \Cref{le:vstupni_slova} that $M^T$ has an outgoing edge with the input word $L$. Further, by \Cref{thm:Raney_edge}, we obtain that every outgoing edge of the state $M^T$, which is not equal to $\Tedge{M^T}{R^\ell }{R}{P_1^T}$ and has input word with suffix $R$, has at least three runs. Therefore, either $W^T = \widehat{W}LR^{j_1}$ or $W^T = \widehat{W} RL^{j_2}R$ for some $j_1 \geq 2,j_2 \geq 1$ and $\widehat{W} \in \D[1]{}$. It means in the first case by the symmetric version of \Cref{le:vstup_RLk_1} of \Cref{le:vstup_V_1} and in the second case by the symmetric version of \Cref{le:vstup_LRjL_1} of \Cref{le:vstup_V_1} that $V^T = RL^m$ for some $m \geq 1$.
Therefore, $\Tedge{P_2}{V}{W}{M} = \Tedge{P_2}{R^mL}{W}{M}$ for some $m \geq 1$. \end{proof}
\begin{lemma} \label{le:3_hrany_do_P2} Let $\Tedge{P}{L^m}{W}{M}$, where $m \geq 1$, and $P= \mat{a}{b}{c}{d}, M = \mat{e}{f}{g}{h} \in \DB{}$. We have: \begin{enumerate} \item If $W = L^\ell$ for some $\ell \geq 1$, then $b=f$ and $a = e-mf$ and if $f \neq 0$, then $\frac{a}{b}< \frac{e}{f}$.
\item If $W = LR^\ell$ for some $\ell \geq 1$, then $b = f + \ell h > f \geq 0 $, $a = e + \ell g - m(f+\ell h) < e-mf$ and if $f \neq 0$, then $\frac{a}{b}< \frac{e}{f}$. \end{enumerate}
\end{lemma} \begin{proof} In the first case, we obtain by direct computation that $b=f$, $a = e-mf$. Because $M \in \DB{}$, we have $f \geq 0$ and therefore $a \leq e$. Moreover, if $f \neq 0$, we have $\frac{a}{b} < \frac{e}{f}$.
We continue with the proof of the second case. By direct computation, we obtain $b = f + \ell h$ and $a = e + \ell g - m(f+\ell h)$. Because $M \in \DB{}$, we have $h > f \geq 0$ and $g<h$. So $e + \ell g - m(f+\ell h) = e-mf +\ell (g-h)m <e-mf$ and $f+\ell h > f \geq 0$. Therefore, $a<e$ and $b>f$, which for $f \neq 0$ means that $\frac{a}{b} <\frac{e}{f}$. \end{proof}
\begin{lemma} \label{le:5_hrany_do_P2} Let \[ \Tedge{P_1}{L^{k_1}}{LR^{j_1}}{M} \text{ and } \Tedge{P_2}{L^{k_2}}{LR^{j_2}}{M}, \] where $P_1 = \mat{e_1}{f_1}{g_1}{h_1}, P_2 = \mat{e_2}{f_2}{g_2}{h_2},M = \mat{a}{b}{c}{d} \in \DB{}$, $k_1,k_2 \geq 1,j_1,j_2 \geq 0$ and $j_2 >j_1$. Then $e_2-(k_1-k_2)f_2<f_2$. \end{lemma} \begin{proof} By direct computation, we obtain $e_2 = a + j_2c-k_2(b+j_2d),f_2 = b+j_2d, g_1 = a+(j_1+1)c-k_1(b+(j_1+1)d)$ and $h_1 = b+(j_1+1)d$. Since $P_1 \in \DB{}$, we have $g_1 <h_1$ and therefore the following series of inequalities holds. \[ a + (j_1+1)c -k_1(b+(j_1+1)d) <b+(j_1+1)d \] \[ \implies \] \[ a -(k_1+1)b <(j_1+1)[d(k_1+1) - c] \leq j_2[d(k_1+1) - c] \] (where the second inequality follows from $j_2 \geq j_1 +1$ and $[d(k_1+1) - c]>0$) \[ \implies \] \[ a + j_2c -k_2(b+j_2d) < (b + j_2 d)(k_1-k_2+1) \] \[ \implies \] \[ e_2 < f_2(k_1-k_2+1), \] which is equivalent to the claim of the lemma. \end{proof}
\begin{lemma} \label{le:6_hrany_do_P2} For every $K = \mat{a}{b}{c}{d} \in \DB{}$ where $a>2b$, there exists an edge $ \Tedge{J}{L^t}{L^\ell }{K}$ where $t,\ell \geq 1$, and every other incoming edge of the state $K$ has $R$ in its output word. \end{lemma} \begin{proof} We have $K^T \in \DB{}$ and by \Cref{thm:Raney_edge}, there exists an edge $\Tedge{K^T}{R^\ell }{W}{J^T}$ for some $\ell \geq 1, W \in \D[1]{}$ and $J \in \DB{}$ and there is no other outgoing edge of the state $K^T$ with input word, which does not contain $L$. We have: \[ K^TR^\ell = \mat{a}{\ell a+c}{b}{\ell b+d}. \] Because $a>2b$, we have either $W = R$ or $W$ has prefix $R^2$. In the second case, we have by the symmetric version of \Cref{le:vstup_L_1} of \Cref{le:vstup_V_1} that $\ell = 1$ and $W = R^t$ for some $t \geq 2$. The claim now follows from \Cref{prop:assoc_sym}. \end{proof}
\begin{lemma} \label{le:7_hrany_do_P2} Every state $J \in \DB{}$ has an incoming edge with the input word, which has suffix $L$. \end{lemma} \begin{proof} We have $J^T \in \DB{}$. By \Cref{thm:Raney_edge}, there is always an edge $\Tedge{J^T}{V}{W}{N^T}$, which has an input word $V$ with suffix $R$. Moreover, by \Cref{le:stejna_pismena}, this edge has output word $W$, which has prefix $R$. It means that by \Cref{prop:assoc_sym}, there is also an edge $\Tedge{N}{W^T}{V^T}{J}$ where $W^T$ has suffix $L$. \end{proof}
\begin{proposition} \label{prop:hrany_do_P2} Let \[ \Tedge{P_1}{L^{k_1}}{LR^{j_1}}{M} \quad \text{ and } \quad \Tedge{P_2}{L^{k_2}}{LR^{j_2}}{M}, \] where $P_1, P_2,M \in \DB{}$, $k_1 > k_2 \geq 1$ and $j_2 >j_1 \geq 0$. Then there is no walk of the form \[ \Twalk{Q}{VL^{(k_1-k_2+1)}}{W}{P_2} \] in the transducer $\T{}$ and for arbitrary walk of the form \[ \Twalk{Q}{VL^{(k_1-k_2)}}{W}{P_2} \] in the transducer $\T{}$ (if there is one) ($Q$ is the first state before reading the start of the run of $L$'s in the input word), we have \[ \Twalk{Q}{VL^{(k_1-k_2)}}{W}{P_2} = \Tedge{Q}{R^pL}{W_1}{ \Twalk{Q_0}{L^{(k_1-k_2-1)}}{W_2}{P_2} }, \] where $W_1W_2 = W$, $Q_0 \in \DB{}$ and $p \geq 1$. \end{proposition} \begin{proof} By \Cref{le:3_hrany_do_P2}, and since $j_2 > 0$, we have $(P_2)_{1,2} \neq 0$. An arbitrary walk of the form \[ \Twalk{Q}{VL^{r}}{W}{P_2} \] in the transducer $\T{}$, where $r \geq 1$ and $Q$ is the first state before reading the start of the run of $L$'s in the input word, can be decomposed in the following way. \[ \Twalk{Q}{VL^{r}}{W}{P_2} = \Tedge{Q}{VL^{u_0}}{W_0}{ \Tedge{Q_0}{L^{u_1}}{W_1}{Q_1} \cdots \Tedge{Q_{m-1}}{L^{u_m}}{W_m}{Q_m} } \] where $Q_m = P_2$, $m \in \mathbb{N}$ and for all $i \in \mathbb{N}, i \leq m$, $W_i \in \D[1]{}, Q_i \in \DB{}$ and $u_i \geq 1$.
By \Cref{le:3_hrany_do_P2}, we have $(Q_i)_{1,2} \neq 0$ for all $i \in \mathbb{N}, i \leq m$ and $\frac{(Q_i)_{1,1}}{(Q_i)_{1,2}} < \frac{(Q_{i+1})_{1,1}}{(Q_{i+1})_{1,2}} $ for all $i \in \mathbb{N}, i \leq {m-1}$.
Let $r$ be maximal possible such that there exists the walk of the form $\Twalk{Q}{VL^{r}}{W}{P_2}$. \Cref{le:7_hrany_do_P2} shows that there is an incoming edge of the state $Q$, which has suffix of its input word equal to $L$. Therefore, $V$ has suffix $R$. We know that $u_0 \geq 1$ and therefore by \Cref{le:stejna_pismena}, $W_0$ has prefix $L$. Now $u_0 = 1$ holds or by \Cref{le:vstup_RLk_1} of \Cref{le:vstup_V_1}, we have $W_0 = LR^s$ for some $s \geq 1$.
We prove by contradiction that $(Q_0)_{1,1} \leq 2 (Q_0)_{1,2}$. We suppose that $(Q_0)_{1,1} > 2 (Q_0)_{1,2}$. It follows from \Cref{le:6_hrany_do_P2} that there exists some edge $\Tedge{J}{L^t}{L^{\ell}}{Q_0}$ where $t, \ell \geq 1$. By \Cref{le:vstup_L_1} of \Cref{le:vstup_V_1} either $\ell = 1$ or $t=1$. By \Cref{le:1_hrany_do_P2} or by \Cref{le:2_hrany_do_P2} or because $t \geq 1$, we have $t \geq u_0$. By \Cref{le:7_hrany_do_P2}, there is an edge, which ends in the state $J$ and has input word with suffix $L$. This is a contradiction with the maximality of $r$.
Therefore, we have $(Q_0)_{1,1} \leq 2 (Q_0)_{1,2}$, which by \Cref{le:vstup_do_N} means that $u_0 =1$ and $V = R^p$ for some $p \geq 1$.
Let $i \geq 1$, $i \leq m$. Further, by \Cref{le:vstup_L_1} of \Cref{le:vstup_V_1} and \Cref{le:3_hrany_do_P2}, we have \[ (Q_{i-1})_{1,1} \leq (Q_i)_{1,1} - u_i (Q_i)_{1,2} \] and \[ (Q_{i-1})_{1,2} \geq (Q_i)_{1,2}. \]
Because $Q_0 \in \DB{}$, we have $(Q_0)_{1,1} > (Q_0)_{1,2}$. It means that \[ (Q_m)_{1,1} - (r-1) (Q_m)_{1,2} = (Q_m)_{1,1} - (u_1 + u_2 + \dots u_m) (Q_m)_{1,2} \geq(Q_0)_{1,1} > (Q_0)_{1,2} \geq (Q_m)_{1,2}. \]
At the same time, we have according to \Cref{le:5_hrany_do_P2} that \[ (Q_m)_{1,1}-(k_1-k_2)(Q_m)_{1,2}<(Q_m)_{1,2}. \] Because $(Q_m)_{1,2} \geq 1$, we have $r-1 < k_1 - k_2$ and the proposition holds. \end{proof}
The symmetry of $\T{}$ can be used for the following observation about the output words.
\begin{lemma} \label{le:vystupy_ruzne} Let $\Tedge{M_1}{V_1}{W_1}{P}$ and $\Tedge{M_2}{V_2}{W_2}{P}$ for some $M_1,M_2,P \in \DB{}$, $V_1,V_2,W_1,W_2 \in \D[1]{}$ be two different edges in the transducer $\T{}$. The word $W_1$ is not a suffix of $W_2$ or vice versa. \end{lemma} \begin{proof} By \Cref{prop:assoc_sym}, the edges $\Tedge{P^T}{W_1^T}{V_1^T}{M_1^T}$ and $\Tedge{P^T}{W_2^T}{V_2^T}{M_2^T}$ exist. The claim follows from \Cref{thm:Raney_edge}. \end{proof}
We may now proceed with a proof of \Cref{co:vyfukovani_ze_symetrickeho}. First, we recall its statement:
\restatableVyfukovani*
\begin{proof}[Proof of \Cref{co:vyfukovani_ze_symetrickeho}] Let $(p_i)_{i=1}^{g}$ be the sequence of all the transitions taken on the walk $\Twalk{M}{V}{W}{M}$, ordered as they appear on this walk. We shall transform each of the transition $p_i$ into a new walk $\widehat{p}_i$ having the same starting state and ending state as $p_i$. Doing that, we shall produce a new walk from $M$ to $M$ given by the sequence $(\widehat{p}_i)_{i=1}^{g}$.
Let $U_iQ_i^{j_i}$ be the input word of the transition $p_i$ with $Q_i \in \{L,R\}$, $j_i > 0$, and $U_i$ empty or ending in a letter distinct from $Q_i$. If $U_i$ is not empty, $F_i$ denotes the first letter of $U_i$.
We proceed from $i = g$ to $i=1$ and replace $p_i$ with $\widehat{p}_i$ using the following rules. In the case $i = 1$, we define $i-1 = g$. The rules use \Cref{coro:kappa_hrana} or its symmetric version and construct walks $\widehat{p}_i$ from the walks given by an appropriate item of \Cref{coro:kappa_hrana}: \begin{enumerate}[I.]
\item if $U_i$ is empty and $Q_i \neq Q_{i-1}$, then apply \cref{it:kappa_hrana_prazdneV} of \Cref{coro:kappa_hrana} if $Q_i = R$ or its symmetric version otherwise; \label{it:nafuk_krok_prazdne}
\item if $U_i$ is empty and $Q_i = Q_{i-1}$, then $\widehat{p}_i = p_i$; \label{it:nafuk_krok_kopie}
\item if $U_i$ is not empty and $F_i = Q_{i-1}$, then apply \cref{it:kappa_hrana_nic} of \Cref{coro:kappa_hrana} if $Q_i = R$ or its symmetric version otherwise;
\item if $U_i$ is not empty and $F_i \neq Q_{i-1}$, then
\begin{enumerate}[label=\Roman{enumi}\alph*.]
\item if $Q_{i-1} = L$ and $Q_i = R$, apply \cref{it:kappa_hrana_L} of \Cref{coro:kappa_hrana};
\item if $Q_{i-1} = R$ and $Q_i = R$, apply \cref{it:kappa_hrana_R} of \Cref{coro:kappa_hrana};
\item if $Q_{i-1} = R$ and $Q_i = L$, apply \cref{it:kappa_hrana_L} of symmetric version of \Cref{coro:kappa_hrana};
\item if $Q_{i-1} = L$ and $Q_i = L$, apply \cref{it:kappa_hrana_R} of symmetric version of \Cref{coro:kappa_hrana}.
\end{enumerate} \end{enumerate}
Let $\Twalk{M}{U}{W_U}{M}$ be the walk composed of the walks $\widehat{p}_1,\widehat{p}_2,\ldots,\widehat{p}_g$. It follows from the above construction that all the runs in $U$, except for the last run, are of length at least $4n$ and $Q_g^{4n}$ is a prefix of $U$. As the new walks $\widehat{p}_i$ are given by \Cref{coro:kappa_hrana} or kept the same, the number of runs in the output words is either always strictly increased or the output word is the same. Since $\sigma(\widehat{A_1}) > \sigma(A_1)$ and $\sigma(\widehat{A_2}) > \sigma(A_2)$ imply $\sigma(\widehat{A_1}\widehat{A_2}) > \sigma(A_1A_2)$ for all $A_1, A_2, \widehat{A_1}, \widehat{A_2} \in \D[1]{}$, we conclude that using the above rules, the number $\sigma(W_U)$ may not be less than $\sigma(W)$. We conclude that ${\sigma_{\mathrm{c}}}(W_U) \geq {\sigma_{\mathrm{c}}}(W)$.
We shall now repeatedly apply \Cref{le:vyfouknuti_n} to the walk $\Twalk{M}{U}{W_U}{M}$ to decrease the length of most of the runs in the input word between $4n$ and $5n-1$, without changing the output words except for decreasing lengths of some of their runs. We end up with a walk \[ \Twalk{M}{Q_g^{e}U'Q_g^{f}}{W_U'}{M} \] where $\sigma(W_U) = \sigma(W_U')$, $U' = \kappa(U')$ and $U'$ starts and ends with the letter distinct from $Q_g$. It remains to deal with the first and the last run, which are both runs of the same letter $Q_g$. In order to do that, we shift the start and the end of the closed walk $\Twalk{M}{Q_g^{e}U'Q_g^{f}}{W_U'}{M}$ to another state, denoted by $\widehat{M}$, on this closed walk such that the run $Q_g^{e+f}$ is inside the input word. This is possible due to the fact that $U'$ contains a run of length at least $4n$ of the letter distinct from $Q$. Let $W'$ be the output word of this shifted closed walk. By the definition, we have ${\sigma_{\mathrm{c}}}(W') = {\sigma_{\mathrm{c}}}(W_U')$. We now apply \Cref{le:vyfouknuti_n} one last time to reduce the length of the run $Q_g^{e+f}$ in the input word of the shifted closed walk. We obtain a new output word $\widehat{W}$ which satisfies \[ {\sigma_{\mathrm{c}}}(\widehat{W}) = {\sigma_{\mathrm{c}}}(W') = {\sigma_{\mathrm{c}}}(W_U') = {\sigma_{\mathrm{c}}}(W_U) \geq {\sigma_{\mathrm{c}}}(W). \] As the input word of this closed walk belongs to $\tau_\kappa(V)$, the first part of the proof is finished.
Now, we prove \Cref{it:1)vyfukovani,it:2)vyfukovani}. The proofs of the two claims of \Cref{it:1)vyfukovani,it:2)vyfukovani} are very similar and therefore we prove them together. In what follows, $\mathcal C$ denotes the closed walk $\Twalk{M}{V}{W}{M}$ and $\widehat{\mathcal C}$ the closed walk $\Twalk{\widehat{M}}{\widehat{V}}{\widehat{W}}{\widehat{M}}$.
The step \ref{it:nafuk_krok_kopie} of the algorithm in the first part of the proof may not be applied to all of the edges $p_i$. In other words, there is an edge $p_i$ such that \Cref{coro:kappa_hrana} is applied to it in the algorithm. Moreover, we may assume that such edge satisfies $p_i = \Tedge{S}{V_3R^j}{\bullet}{T}$ where $V_3$ does not have suffix $R$. It follows from \Cref{le:max_n_jednoho_pismene} that $j \leq n$ and if $V_3$ is empty, then we are in the step \ref{it:nafuk_krok_prazdne}
Let $X \in \{\varepsilon, L^{4n}, R^{4n}\}$. The application of \Cref{coro:kappa_hrana} to $p_i$ produces a walk $\widehat{p}_i = \Twalk{S}{X\kappa(V_3)R^j}{\bullet}{T}$ on the closed walk $\Twalk{\widehat{M}}{\widehat{V}}{\widehat{W}}{\widehat{M}}$. If we change the starting state of this closed walk to $S$, we obtain some walk $\Twalk{S}{B}{\bullet}{S}$. In the first case (the case \ref{it:1)vyfukovani} of \Cref{co:vyfukovani_ze_symetrickeho}), we have $\Twalk{S}{B}{\bullet}{S} = (\Twalk{S}{B_1}{\bullet}{S})^{m_1}$ for $B_1^{m_1} = B$ so there is a walk $\widehat{p}_q$, $i \neq q$ and $\widehat{p}_i = \widehat{p}_q$ such that if we change the starting vertex of the closed walk $\Twalk{S}{B}{\bullet}{S}$ to the starting vertex of $\widehat{p}_q$, we obtain again the closed walk $\Twalk{S}{B}{\bullet}{S}$. In the second case (the case \ref{it:2)vyfukovani} of \Cref{co:vyfukovani_ze_symetrickeho}), the symmetricity of the walk $\Twalk{S}{B}{\bullet}{S}$ implies that we may also change the starting vertex to $\assoc{S}$, and the closed walk is $\Twalk{\assoc{S}}{\assoc{B}}{\bullet}{\assoc{S}}$ where the first walk is $\widehat{p}_q = \assoc{\widehat{p}_i} = \Twalk{\assoc{S}}{\assoc{X\kappa(V_3)}L^j}{\bullet}{\assoc{T}}$. Because we are investigating both cases together, we put $T_0 = T$ for the first case and $T_0 = \assoc{T}$ for the second case.
We shall now investigate the edge $p_{\ell}$ on the original closed walk, which, after application of the algorithm in the first part of this proof, produced the start of the reading of the last run in the input word of $\widehat{p}_q$. This last run is denoted $E^j$, where $E = R$ for the first case and $E = L$ for the second case.
We are again sure that \Cref{coro:kappa_hrana} is applied to $p_{\ell}$ since we are tracking a start of a run.
\begin{enumerate}[A)] \item \label{it:A)vyfukovani}
$p_\ell = \Tedge{S_1}{ZE^k}{\bullet}{T_1}$ and $\widehat{p_\ell} = \Twalk{S_1}{X'\kappa(Z)E^k}{\bullet}{T_1}$ with $k$ maximal possible, $Z \in \D[1]{}$ and $X' \in \{\varepsilon, L^{4n}, R^{4n}\}$.
If $k=j$, then $T_1 =T_0$. Since for the first case $V = (V_1)^{m_1}$, we arrive in the closed walk $\mathcal C$ at the state $T_0= T$ at least two times with the same input and therefore $\mathcal C = (\mathcal C_1)^{m_2}$ for some $m_2 \geq 2$, which means that $\Twalk{M}{V}{W}{M}=(\Twalk{M}{V_2}{W_2}{M})^{m_2}$. In the second case, we have a similar situation. Since $V = V_1\assoc{V_1}$, i.e., the original input word is symmetric itself, we arrive in the closed walk ${\mathcal C}$ at the state $\assoc{T}$ with the input word, which is symmetric to the input word after the edge $p_i$, and so $\Twalk{M}{V}{W}{M}$ is symmetric.
If $k < j$, then the walk $\widehat{p}_q$ ends with the walk $\Twalk{T_1}{E^{j-k}}{\bullet}{T_0}$. The walk $\Twalk{T_1}{E^{j-k}}{\bullet}{T_0}$ is taken after $p_\ell$. Thus, we arrive in the closed walk $\mathcal C$ at the state $T_0$ with the input word, which is either the same (in the first case) or symmetric (in the second case) to the input word after the walk $p_i$, and so either $\Twalk{M}{V}{W}{M}=(\Twalk{M}{V_2}{W_2}{M})^{m_2}$ for some $m_2 \geq 2$ or $\Twalk{M}{V}{W}{M}$ is symmetric, respectively.
If $k > j$, then the walk $\widehat{p}_\ell$ ends with the walk $\Twalk{T_0}{E^{k-j}}{\bullet}{T_1}$. Thus, the walk $\Twalk{T}{R^{k-j}}{\bullet}{T_2}$, where $T_2 = T_1$ in the first case and $T_2 = \assoc{T_1} $ in the second case, exists. Therefore, as the observed run of $R$'s in $p_i$ is of length at least $k$, it is followed by $\Twalk{T}{R^{k-j}}{\bullet}{T_2}$. Again, we either arrive in the closed walk $\mathcal C$ two times at the state $T_1$ with the same input word or we find two states, $\assoc{T_1}$ and $T_1$ that have symmetric input words, and so either $\Twalk{M}{V}{W}{M}=(\Twalk{M}{V_2}{W_2}{M})^{m_2}$ or $\Twalk{M}{V}{W}{M}$ is symmetric, respectively.
\begin{figure}
\caption{ The situation of \Cref{it:co:vyfukovani_ze_symetrickeho_spor} in the proof of \Cref{co:vyfukovani_ze_symetrickeho}, based on \Cref{fig:nafukovani}. On the top line we have the edge $p_\ell$, which is transformed using the procedure given by \Cref{coro:kappa_hrana}. We find an intermediate state $D_g$ connected to the run $E^k$. We identify the state $T_0$ on the bottom line, the new walk $\widehat{p}_\ell$, marked by a triangle. }
\label{fig:nafukovani_vyfukovani_dukaz}
\end{figure}
\item \label{it:co:vyfukovani_ze_symetrickeho_spor}
$p_\ell = \Tedge{S_1}{Z_1E^kZ_2Y^{r_1}}{\bullet}{T_1}$, with $Z_1,Z_2 \in \D[1]{}$, $Y \in \left\{ L,R \right\}, r_1 \geq 1$ and $k\geq 1$ maximal possible. As $V = (V_1)^{m_1}$ or $V = V_1\assoc{V_1}$, we have $k \geq j$.
We have $\widehat{p}_\ell = \Twalk{S_1}{X'\kappa(Z_1E^kZ_2)Y^{r_1}}{\bullet}{T_1}$ where $X' \in \{\varepsilon, L^{4n}, R^{4n}\}$. As $\Twalk{\widehat{M}}{\widehat{V}}{\widehat{W}}{\widehat{M}} = (\Twalk{\widehat{M}}{\widehat{V_1}}{\widehat{W_1}}{\widehat{M}})^{m_1}$ or is symmetric, we know that after reading $E^j$ in the run $\kappa(E^k)$ we arrive at $T_0$. The situation is illustrated in \Cref{fig:nafukovani_vyfukovani_dukaz}. We arrive at the intermediate state $D_g$ by reading $E^{k+t_g}$, where $t_g \geq 1$, which is also read when taking the vertical path to $D_g$. Therefore, $\Tedge{S_1}{Z_1E^{k+t_g}}{W_3}{D_g}$ and $\Twalk{S_1}{Z_1\assoc{E}^{4n}E^{k+t_g}}{W_4}{D_g} = \Twalk{S_1}{Z_1\assoc{E}^{4n}E^j}{\bullet}{ \Twalk{T_0}{E^{k+t_g - j}}{\bullet}{D_g}}$ for some $W_3,W_4 \in \D[1]{}$ and we can find a state $T_2 \in \DB{}$ such that $\Twalk{S_1}{Z_1\assoc{E}^{4n}E^{k+t_g}}{W_4}{D_g} = \Twalk{S_1}{Z_1\assoc{E}^{4n}E^j}{W_5}{\Twalk{T_0}{E^{k+t_g -r - j}}{W_6}{ \Tedge{T_2}{E^r}{W_7}{D_g}}}$ for some $W_5, W_6,W_7\in \D[1]{}$ and $k+t_g-j\geq r \geq 1$.
By \Cref{le:nafouknuti_jedno} or its symmetric version, the word $W_4$ has suffix $E^{-1}W_3$ and because $k+t_g \geq 2$, we have by \Cref{le:vstup_V_1} \Cref{le:vstup_RLk_1} for $Z_1$ nonempty or by \Cref{le:vstup_V_1} \Cref{le:vstup_L_1} otherwise that $W_3 = E\assoc{E}^z$, where $z \geq 0$. And for $Z_1$ nonempty even $z \geq k+t_g - 1 \geq 1$. So $W_4$ has suffix $\assoc{E^z}$. Moreover, by \Cref{le:vstup_V_1} \Cref{le:vstup_L_1} or its symmetric version, we have $W_7 = E\assoc{E}^s$ for some $s \geq 0$ or $W_7 = E^t$ for some $t \geq 2$. If $W_7 = E^t$ for some $t \geq 2$, then $r = 1$ and by \Cref{le:2_hrany_do_P2}, there cannot be the edge $\Tedge{S_1}{Z_1E^{k+t_g}}{W_3}{D_g}$, which is a contradiction. Therefore, $W_7 = E\assoc{E}^s$ for some $s \geq 0$. Further, we know that $W_7$ is a suffix of $W_4$ and therefore $\assoc{E}^z$ is a suffix of $W_7$. Moreover, by \Cref{le:vystupy_ruzne}, $W_3 \neq W_7$, which means that $s>z$.
We distinguish the two following cases:
\begin{enumerate}[1)] \item \label{it:B)vyfukovani} $Z_1$ is not the empty word.
In this case, we have $s \geq z+1 \geq k + t_g \geq k + t_g -j + 1\geq r+1$. Let $T_2 = \mat{a}{b}{c}{d}$. Using \Cref{le:hrany_do_P} or its symmetric version on the edge $\Tedge{T_2}{E^{r}}{E\assoc{E}^s}{D_g}$, we have $a <2b$ for $E = L$ and $a<2c$ for $E=R$.
Moreover, let $\Tedge{N}{V_8}{W_8}{T_2}$, where $N \in \DB{}, V_8,W_8 \in \D[1]{}$ and $V_8$ has suffix $E$, be an edge in the transducer $\T{}$. By \Cref{le:vstup_do_N} or its symmetric version, we have $V_8 = \assoc{E}^yE$ for some $y \geq 1$.
Therefore, the walk $\Twalk{T_0}{E^{k+t_g -r - j}}{W_6}{T_2}$ is empty and $T_0 = T_2$. Therefore also for every edge $\Tedge{N}{V_8}{W_8}{T_0}$, where $V_8$ has suffix $E$, we have $V_8 = \assoc{E}^yE$. Specially we have $p_i = \Tedge{S}{L^yR}{\bullet}{T}$ for some $y \geq 1$ (in the second case, we have used \Cref{prop:assoc_sym}) which means that $j=1$.
Because $Z_1$ is not empty, we can write $p_\ell = \Tedge{S_1}{Z_3\assoc{E^{i_{g+1}}}E^kZ_2Y^{r_1}}{\bullet}{T_1}$, with $i_{g+1} \geq 1$ maximal possible and $Z_3 \in \D[1]{}$.
Now we can have one of the following situations.
\begin{enumerate}[i)] \item \label{it:a)vyfukovani}
The edge $p_{i-1}$ has input word with a suffix $L$ and $Z_3$ is not an empty word.
We find the edge $p_{u}$ on the original closed walk on which starts the reading of the run $L^y$ of the input word of $p_i$ and we apply \Cref{it:co:vyfukovani_ze_symetrickeho_spor} \ref{it:B)vyfukovani} on the edges $p_u$ and $p_{\ell}$.
\item \label{it:b)vyfukovani} The edge $p_{i-1}$ has input word with a suffix $L$, $Z_3$ is empty and the edge $p_{\ell - 1}$ has input word with a suffix $\assoc{E}$.
In this case, we find the edge $p_u$ on the original closed walk on which starts the reading of the run $L^y$ of the input word of $p_i$ and the edge $p_v$ on the original closed walk on which starts the reading of the run $\assoc{E}^{i_{g+1}}$ of the input word of $p_{\ell}$ and we apply \Cref{it:A)vyfukovani} on these two edges.
\item \label{it:c)vyfukovani}
The edge $p_{i-1}$ has input word with a suffix $L$, $Z_3$ is empty and the edge $p_{\ell - 1}$ has input word with a suffix $E$.
In this case, we find the edge $p_u$ on the original closed walk on which starts the reading of the run $L^y$ of the input word of $p_i$ and we apply \Cref{it:co:vyfukovani_ze_symetrickeho_spor} \ref{it:C)vyfukovani} on the edges $p_u$ and $p_{\ell}$.
\item \label{it:d)vyfukovani} The edge $p_{i-1}$ has input word with a suffix $R$ and $Z_3$ has at least two runs.
We find the edge $p_u$ on the original closed walk on which starts the reading of the run of $R$'s, which ends as a suffix of the input word of the edge $p_{i-1}$ and we apply \Cref{it:co:vyfukovani_ze_symetrickeho_spor} \ref{it:B)vyfukovani} on the edges $p_u$ and $p_{\ell}$.
\item \label{it:e)vyfukovani} The edge $p_{i-1}$ has input word with a suffix $R$, $Z_3 = E^{i_{g+2}}$ and the edge $p_{\ell - 1}$ has input word with a suffix $E$.
We find the edge $p_u$ on the original closed walk on which starts the reading of the run of $R$'s, which ends as a suffix of the input word of the edge $p_{i-1}$ and the edge $p_v$ on the original closed walk on which starts the reading of the run $E^{i_{g+2}}$ of the input word of $p_{\ell}$ and we apply \Cref{it:A)vyfukovani} on the edges $p_u$ and $p_v$.
\item \label{it:f)vyfukovani} The edge $p_{i-1}$ has input word with a suffix $R$, $Z_3 = E^{i_{g+2}}$ and the edge $p_{\ell - 1}$ has input word with a suffix $\assoc{E}$.
We find the edge $p_u$ on the original closed walk on which starts the reading of the run of $R$'s, which ends as a suffix of the input word of the edge $p_{i-1}$ and we apply \Cref{it:co:vyfukovani_ze_symetrickeho_spor} \ref{it:C)vyfukovani} on the edges $p_u$ and $p_{\ell}$.
\item \label{it:g)vyfukovani} The edge $p_{i-1}$ has input word with a suffix $R, Z_3$ is empty and the edge $p_{\ell - 1}$ has input word with a suffix $\assoc{E}$.
We find the edge $p_v$ on the original closed walk on which starts the reading of the run $\assoc{E}^{i_{g+1}}$ of the edge $p_\ell$ and we apply \Cref{it:co:vyfukovani_ze_symetrickeho_spor} \ref{it:C)vyfukovani} on the edges $p_v$ and $p_i$.
\item \label{it:h)vyfukovani} The edge $p_{i-1}$ has input word with a suffix $R$, $Z_3$ is empty and the edge $p_{\ell - 1}$ has input word with a suffix $E$.
We find the edge $p_u$ on the original closed walk on which starts the reading of the run of $R$'s, which ends as a suffix of the input word of the edge $p_{i-1}$ and the edge $p_v$ on the original closed walk on which starts the reading of the run $E$, which ends as a suffix of the input word of the edge $p_{\ell - 1}$ and we apply \Cref{it:A)vyfukovani} on the edges $p_u$ and $p_v$.
\end{enumerate}
Since the word $Z_3$ is finite, we are sure that after a finite number of applications of \Cref{it:B)vyfukovani}, one of the possibilities \ref{it:b)vyfukovani},\ref{it:c)vyfukovani},\ref{it:e)vyfukovani},\ref{it:f)vyfukovani},\ref{it:g)vyfukovani} or \ref{it:h)vyfukovani} occurs.
\item \label{it:C)vyfukovani} $Z_1$ is the empty word.
In this case, we have $\Tedge{S_1}{Z_1E^{k+t_g}}{W_3}{D_g} = \Tedge{S_1}{E^{k+t_g}}{E\assoc{E}^z}{D_g}$ and $\Tedge{T_2}{E^{r}}{W_7}{D_g} = \Tedge{T_2}{E^{r}}{E\assoc{E}^s}{D_g}$, where $s>z$. Using \Cref{prop:hrany_do_P2} or its symmetric version on the edges $\Tedge{S_1}{E^{k+t_g}}{E\assoc{E}^z}{D_g}$ and $\Tedge{T_2}{E^r}{E\assoc{E}^s}{D_g}$, we get that for every walk of the form $\Twalk{P}{V_8E^{k+t_g-r}}{W_8}{T_2}$ for some $P \in \DB{}$ and $V_8,W_8 \in \D[1]{}$ we have $\Twalk{P}{V_8E^{k+t_g-r}}{W_8}{T_2} = \Tedge{P}{\assoc{E}^vE}{W_9}{\Twalk{N_1}{E^{k+t_g-r-1}}{W_{10}}{T_2}}$ for some $N_1 \in \DB{}$ and $v \geq 1$. It holds specially for every such walk which end with the walk $\Twalk{T_0}{E^{k+t_g -r - j}}{W_6}{T_2}$ and therefore in the case \Cref{it:1)vyfukovani} we have ($T = T_0$, $E = R$): \[
\Tedge{S}{V_3R^j}{\bullet}{ \Twalk{T}{R^{k+t_g -r - j}}{W_6}{T_2}}= \Tedge{S}{L^vR}{W_9}{ \Twalk{T}{R^{k+t_g -r - 1}}{W_6}{T_2}} \]
and in the case \Cref{it:2)vyfukovani}, we have ($T_0 = \assoc{T}, E = L$): \[
\Tedge{\assoc{S}}{\assoc{V_3}L^j}{\bullet}{ \Twalk{\assoc{T}}{L^{k+t_g -r - j}}{W_6}{T_2}} = \Tedge{\assoc{S}}{R^vL}{W_9}{ \Twalk{\assoc{T}}{L^{k+t_g-r-1}}{W_{10}}{T_2}}. \] Therefore, (in the second case by \Cref{prop:assoc_sym}) we have $j = 1$ and $p_i = \Tedge{S}{L^vR}{\bullet}{T}$ for some $v \geq 1$.
Now we can have two situations.
\begin{enumerate}[i)] \item The edge $p_{i-1}$ has $L$ as a suffix of its input word.
We find the edge $p_{u_1}$ on the original closed walk on which starts the reading of the run of $L$'s, which ends on the edge $p_{i}$ and the edge $p_{u_2}$ on the original closed walk on which starts the reading of the run $\assoc{E}$, which ends as a suffix of the input word of the edge $p_{\ell - 1}$ and we apply \Cref{it:A)vyfukovani} on the edges $p_{u_1}$ and $p_{u_2}$.
\item The edge $p_{i-1}$ has $R$ as a suffix of its input word.
We can find the edge $p_{u_2}$ on the original closed walk on which starts the reading of the run $\assoc{E}$, which ends as a suffix of the input word of the edge $p_{\ell - 1}$ and we apply \Cref{it:co:vyfukovani_ze_symetrickeho_spor} \ref{it:C)vyfukovani} on the edges $p_{u_2}$ and $p_i$.
\end{enumerate} Now we can see that there either exist two edges $p_{u_1}, p_{u_2}$ on which we can apply \Cref{it:A)vyfukovani} or that for all $w$ we have $p_{i_w}= \Tedge{M_{i_w}}{L^{q_{i_w}}R}{\bullet}{N_{i_w}}$ or $p_{i_w}= \Tedge{M_{i_w}}{R^{q_{i_w}}}{\bullet}{N_{i_w}}$ and $p_{\ell_w}= \Tedge{M_{\ell_w}}{{E}^{q_{\ell_w}}\assoc{E}}{\bullet}{N_{\ell_w}}$ or $p_{\ell_w}= \Tedge{M_{\ell_w}}{\assoc{E}^{q_{\ell_w}}}{\bullet}{N_{\ell_w}}$, where $q_{i_w},q_{\ell_w} \geq 1$ and $ i_w \equiv i-w \mod g, \ell_w \equiv \ell-w \mod g$ where $g$ is such that $\mathcal C = (p_i)_{i= 1}^{g}$.
For the final part of the proof, we split the cases according to \Cref{it:1)vyfukovani,it:2)vyfukovani} of the statement. \begin{enumerate}[(a)] \item $p_{i_w}= \Tedge{M_{i_w}}{L^{q_{i_w}}R}{\bullet}{N_{i_w}}$ or $p_{i_w}= \Tedge{M_{i_w}}{R^{q_{i_w}}}{\bullet}{N_{i_w}}$ and $p_{\ell_w}= \Tedge{M_{\ell_w}}{{R}^{q_{\ell_w}}L}{\bullet}{N_{\ell_w}}$ or $p_{\ell_w}= \Tedge{M_{\ell_w}}{L^{q_{\ell_w}}}{\bullet}{N_{\ell_w}}$, for all $w$ and for $q_{i_w}, q_{\ell_w} \geq 1$, which is a contradiction with $\mathcal C = (p_{i_w})_{w= 1}^{g} = (p_{\ell_w})_{w= 1}^{g}$. \item $p_{i_w}= \Tedge{M_{i_w}}{L^{q_{i_w}}R}{\bullet}{N_{i_w}}$ or $p_{i_w}= \Tedge{M_{i_w}}{R^{q_{i_w}}}{\bullet}{N_{i_w}}$ and $p_{\ell_w}= \Tedge{M_{\ell_w}}{{L}^{q_{\ell_w}}R}{\bullet}{N_{\ell_w}}$ or $p_{\ell_w}= \Tedge{M_{\ell_w}}{R^{q_{\ell_w}}}{\bullet}{N_{\ell_w}}$ for all $w$. It means that all the input words of the edges in the closed walk $\mathcal C $ are either $L^{q_{i_w}}R$ or $R^{q_{i_w}}$ for some $i_w$. Moreover, the input word of the closed walk $\mathcal C$ includes at least one run of $L$'s and one run of $R$'s. Therefore, at least one of the input words is $L^{q_{i_w}}R$ for some $i_w \geq 1$. By \Cref{le:hrany_v_cyklu}, this is in contradiction with the fact that $\mathcal C$ is a closed walk. \qedhere \end{enumerate} \end{enumerate} \end{enumerate} \end{proof}
\subsection{The upper bound and the proof of \Cref{thm:main}}
\newcommand\Lf[1]{\widehat{#1}} \newcommand\Lfz[1]{\Lf{#1}_0}
We need one more claim, which is a corollary of a well-known result due to the Fine and Wilf \cite{fine_wilf}.
\begin{theorem}[Fine and Wilf's theorem] \label{the:fine_wilf} If a word $V$ has periods $p$ and $q$ and has length at least $p+q - \gcd(p,q)$, then $V$ has also period $\gcd(p,q)$. \end{theorem}
We use this theorem in the following form.
\begin{corollary} \label{le:mocniny_slovo} Let $\widehat{V_0} ,V_0 \in \D[1]{}$, $V_0 \neq V_1^k$ for some $V_1 \in \D[1]{}$ and $k \geq 2$. If \[ \widehat{V_0}^i = V_0^j \] for some $i,j \geq 1$ then $\frac{j}{i} \in \mathbb{N}$. \end{corollary} \begin{proof}
In this proof, $|V|$ denotes the length of the word $V \in \D[1]{}$.
Let $V = \widehat{V_0}^i = V_0^j$, $m = |V|$ and $p = \frac{m}{i} = |\widehat{V_0}|, q = \frac{m}{j} = |V_0|$. Therefore, the word $V$ has periods $p$ and $q$. Moreover, let $\ell = \gcd(p,q)$ and $p = \ell \widehat{p}$, $q = \ell \widehat{q}$. Thus, $m \geq \ell \widehat{p} \widehat{q} \geq \ell (\widehat{p} + \widehat{q} - 1) = p + q - \ell$. It means that by \Cref{the:fine_wilf}, the word $V$ has also period $\ell$ and because $V_0 \neq V_1^k$, we have $q = \ell$.
Together we obtain the following equation. \[ \frac{j}{i} = \frac{\frac{m}{q}}{\frac{m}{p}} = \frac{p}{q} = \frac{\ell \widehat{p}}{\ell} = \widehat{p} \in \mathbb{N}. \qedhere \] \end{proof}
We recall that we want to find an estimate on the number ${\sigma_{\mathrm{c}}}(W)$, where $W$ is the output word of the closed walk $\mathcal C = \Twalk{M}{V^\gamma}{W}{M}$ with $V$ primitive and \begin{equation} \label{eq:C_ub_assumption} \mathcal C \neq \left( \Twalk{M}{V^{\delta_2}}{W_2}{M} \right)^{m_2} \quad \text{ for some } m_2 \geq 2, \delta_2 \in \mathbb{Z}^+ \text{ and } W_2 \in \D[1]{}. \end{equation}
\begin{remark} \label{re:jednoduchost_nafoukle} \Cref{co:vyfukovani_ze_symetrickeho} implies that there exists a closed walk $\widehat{\mathcal C} = \Twalk{\widehat{M}}{\widehat{V}^\gamma}{\widehat{W}}{\widehat{M}}$ where $\widehat{V} \in \tau_\kappa(V)$ and ${\sigma_{\mathrm{c}}}(\widehat{W}) \geq {\sigma_{\mathrm{c}}}(W)$. If $\widehat{\mathcal C}$ can be decomposed into $m_1 \geq 2$ closed walks with the input word $\widehat{V}^{\delta_1}$, then by \eqref{eq:C_ub_assumption} and $\delta_1 m_1 = \gamma \geq 2$ we obtain a contradiction with \Cref{co:vyfukovani_ze_symetrickeho} \Cref{it:1)vyfukovani}. Thus, \begin{equation} \label{eq:prostota_nafoukleho_cyklu} \widehat{\mathcal C} \neq \left (\Twalk{\widehat{M}}{\widehat{V}^{\delta_1}}{W_1}{\widehat{M}} \right)^{m_1} \end{equation}
for some $\delta_1 \in \mathbb{Z}^+, m_1 \geq 2$ and $W_1 \in \D[1]{}$. Further, let $m \geq 1$ be the largest possible such that \[ \widehat{\mathcal C} = \left( \Twalk{\widehat{M}}{\widehat{V}_0^{\delta}}{\widehat{W}_0}{\widehat{M}} \right)^{m} \] for some $\delta \in \mathbb{Z}^+, \widehat{W}_0, \widehat{V}_0 \in \D[1]{}$, and $\widehat{V}_0$ primitive. It can happen that $m \neq 1$ or $\gamma \neq \delta$ but in all cases we have $\widehat{V}_0^{\delta m} = \widehat{V}^{\gamma}$. By \Cref{le:mocniny_slovo}, we have $\frac{\delta m}{\gamma} \in \mathbb{N}$ and by \eqref{eq:prostota_nafoukleho_cyklu}, the numbers $m$ and $\gamma$ are coprime, which means that $\frac{\delta}{\gamma} \in \mathbb{N}$. Therefore,
\begin{equation} \label{eq:prepocty_cyklu} m {\sigma_{\mathrm{c}}}(\widehat{V}_0) = \frac{\gamma}{\delta} {\sigma_{\mathrm{c}}}(\widehat{V}) \leq {\sigma_{\mathrm{c}}}(\widehat{V}) = {\sigma_{\mathrm{c}}}(V) \quad \text{ and } \quad {\sigma_{\mathrm{c}}}(W) \leq {{\sigma_{\mathrm{c}}}(\widehat{W})} = m{\sigma_{\mathrm{c}}}(\widehat{W}_0). \end{equation}
Therefore, it is sufficient to make an estimate on ${\sigma_{\mathrm{c}}}(\Lfz W)$ only for the closed walks $\widehat{\mathcal C} = \Twalk{\widehat{M}}{{\Lfz V}^\delta}{\Lfz W}{\widehat{M}}$ where $\Lfz{V} \in \tau_\kappa(\Lfz{V})$. \end{remark}
\begin{theorem} Let $\Lfz \mathcal C = \Twalk{\Lf M}{\Lfz V^\delta}{\Lfz W}{\Lf M}$ with $\Lfz V$ primitive, $\Lfz V \in \tau_\kappa(\Lfz V)$, and \[ \Lfz \mathcal C \neq \left( \Twalk{\Lf M}{\Lfz V^{\delta_1}}{W_1}{\Lf M} \right)^{m_1} \quad \text{ for some } m_1 \geq 2, \delta_1 \in \mathbb{Z}^+ \text{ and } W_1 \in \D[1]{}. \] We have \begin{equation} \label{eq:odhad_sigma} {\sigma_{\mathrm{c}}}(\Lfz W) \leq {\sigma_{\mathrm{c}}}(\Lfz V) \sum_{M \in \LE{}} \sum_{i=\LclassMax{M}}^{2 \LclassMax{M}-1} \left( \sigma(W_{L,M,i}) - 1 \right). \end{equation} ($W_{L,M,i}$ is given by \Cref{le:LS_n_do_RS_n}.) \end{theorem}
\begin{proof} We are interested in the value of ${\sigma_{\mathrm{c}}}(\Lfz W)$. Since $\Lfz \mathcal C$ is a closed walk, we can choose any its vertex to be the starting vertex of the closed walk, without changing ${\sigma_{\mathrm{c}}}(\Lfz W)$. In other words, we may add some assumptions on $\Lf M$, to keep the notation simple.
Since $\Lfz V \in \tau_\kappa(\Lfz V)$, while repeatedly reading $\Lfz V$ and looping on $\Lfz \mathcal C$, all the runs we read are of length at least $4n$ and by \Cref{le:max_n_jednoho_pismene,le:spadnu_do_LS,le:LS_nacteni_n}, we are sure to pass through some state from $\LE{}$ at least tree times while reading one run of $L$'s. We select this vertex as the starting vertex of $\mathcal C$, that is $M_0 = \Lf M \in \LE{}$. Moreover, we may assume that $\mathcal C$ starts when we encounter $M_0$ for next-to-last time while reading the current run of $L$'s. In other words, $L^{i_0}R$ is a prefix of $\Lfz V$ for some $i_0\in \{\LclassMax{M_0}, \LclassMax{M_0} + 1, \dots, 2 \LclassMax{M_0}-1\}$.
By \Cref{le:LS_n_do_RS_n}, the walk $\Lfz \mathcal C$ starts with \[ \Twalk{M_0}{L^{i_0}R^{j_{L,M_0,i_0}}}{W_{L,M_0,i_0}}{N_0}, \] where $N_0 = N_{L,M_0,i_0} \in \RE{}$. The next walk on $\Lfz \mathcal C$ is \[ \Twalk{N_0}{R^{q'_0\RclassMax{N_0}}}{R^{q'_0t'_0}}{N_0} \] where $t'_0 >0, q'_0 \geq 0$, $t_0'$ is such that $\Twalk{N_0}{\RclassMax{N_0}}{t_0'}{N_0}$ and $q'_0$ is chosen such that after taking this walk, the input word starts with $R^{i'_0}L$ where $i'_0 \in \{\RclassMax{N_0}, \RclassMax{N_0} + 1, \dots, 2 \RclassMax{N_0}-1\}$. Note that the case $q'_0 = 0$ is also possible since we may have $\RclassMax{N_0} = n$.
The symmetric version of \Cref{le:LS_n_do_RS_n} implies that the next walk that we take on $\Lfz \mathcal C$ is \[ \Twalk{N_0}{R^{i'_0}L^{j_{R,N_0,i'_0}}}{W_{R,N_0,i'_0}}{M_1}, \] followed by \[ \Twalk{M_1}{L^{q_1\LclassMax{M_1}}}{L^{q_1t_1}}{M_1}. \] We continue to decompose $\Lfz \mathcal C$ in this manner. This decomposition allows us to identify the output word of $\mathcal C$. Indeed, if we put $\alpha = \frac{{\sigma_{\mathrm{c}}}(\Lfz V)}{2}$ and recall that the input word is $\Lfz V^\delta$, then we have \begin{equation} \label{eq:decomposition_of_C} \Lfz W = W_{L,M_0,i_0} R^{q'_0t'_0} W_{R,N_0,i'_0}L^{q_1t_1} \cdots W_{L,M_{\alpha \delta - 1},i_{\alpha \delta - 1}} R^{q'_{\alpha \delta - 1}t'_{\alpha \delta - 1}} W_{R,N_{\alpha \delta-1},i'_{\alpha \delta - 1}}L^{q_0t_0}. \end{equation} By \Cref{le:LS_n_do_RS_n}, each $W_{L,M_k,i_k}$ starts with $L$ and ends with $R$, and by the symmetric version of \Cref{le:LS_n_do_RS_n}, each $W_{R,N_k,i'_k}$ starts with $R$ and ends with $L$. Therefore, we obtain \begin{equation} \label{eq:sigma_W_decomposition} \sigma(\Lfz W) = 1 + \sum_{k = 0}^{\alpha\delta - 1} (\sigma(W_{L,M_k,i_k}) -1 + \sigma(W_{R,N_k,i'_k}) - 1) = {\sigma_{\mathrm{c}}}(\Lfz W)+1. \end{equation}
The walk $\Lfz \mathcal C$ inputs $\delta$ times the word $\Lfz V$. We shall now focus on what can happen with a specific run in $\Lfz V$ during those $\delta$ times it is read. Namely, let $\Lfz V = P_1 L^t P_2$, $P_1,P_2 \in \D[1]{}$ with integer $t$ maximal possible, i.e., $L^t$ is a whole run of $L$'s in $\Lfz V$. Let $\ell \in \{1,\ldots,\delta\}$ and $k_\ell$ be the integer such that the $\ell$-th reading of the specific run $L^t$ is associated (ends on it) with the walk from $M_{k_\ell}$ to $N_{k_\ell}$ in the above decomposition of $\Lfz \mathcal C$. (We have $k_\ell = k_1 + \alpha (\ell-1)$.)
Assume that for $\ell \neq \ell'$ we have $M_{k_\ell} = M_{k_{\ell'}}$ and $i_{k_\ell} = i_{k_{\ell'}} $. \Cref{le:LS_n_do_RS_n} implies that $N_{k_\ell} = N_{k_{\ell'}}$ and $j_{L,M_{k_\ell},i_{k_\ell}} = j_{L,M_{k_{\ell'}},i_{k_{\ell'}}}$. Therefore, after the $\ell$-th and $\ell'$-th reading of the run $L^t$ we stumble upon the same state $N_{k_\ell}$ with the same input word. This contradicts the assumptions on $\Lfz \mathcal C$.
As a consequence, we obtain an upper bound on $\delta$ by enumerating all possibilities on $M_{k_\ell}$ and $i_{k_\ell}$, where $i_{k_\ell} \in \{\LclassMax{M_{k_\ell}}, \LclassMax{M_{k_\ell}} + 1, \dots, 2 \LclassMax{M_{k_\ell}}-1\}$. In the case that we select to focus on a run of $R$'s, \Cref{prop:assoc_sym} implies that $W_{L,M,i} = \assoc{W_{R,\assoc{M},i}}$ and therefore the resulting upper bound is the same. Similarly, if we focus on the first run of $L$'s in $V$, which is split in two parts, the very same idea of estimate applies. Overall, we conclude that \[ \delta \leq \sum_{M \in \LE{}} \LclassMax{M} = \sum_{N \in \RE{}} \RclassMax{N}. \] and that the maximum contribution to ${\sigma_{\mathrm{c}}}(\Lfz W)$ in \eqref{eq:sigma_W_decomposition} of the $\delta$ reads of one run equals \[ \sum_{M \in \LE{}} \sum_{i=\LclassMax{M}}^{2 \LclassMax{M}-1} \left( \sigma(W_{L,M,i}) - 1 \right). \] By the symmetry of $R$ and $L$, we need not care if the run is a run of $R$'s or $L$'s as the last number equals $\sum_{N \in \LE{}} \sum_{i=\RclassMax{N}}^{2 \RclassMax{N}-1} \left( \sigma(W_{R,N,i}) - 1 \right)$. Using this estimate for all the runs, we finally obtain \eqref{eq:odhad_sigma}. \end{proof}
\begin{corollary} If $\Lfz V = V_1 \assoc{V_1}$ for some $V_1 \in \D[1]{}$, then either \begin{enumerate}[(a)] \item \label{it:coro_half_sym} the walk $\Lfz \mathcal C$ is symmetric or \item \label{it:coro_half_half} \begin{equation} \label{eq:odhad_sigma_pul} {\sigma_{\mathrm{c}}}(\Lfz W) \leq \frac{1}{2} \left( {\sigma_{\mathrm{c}}}(\Lfz V) \sum_{M \in \LE{}} \sum_{i=\LclassMax{M}}^{2 \LclassMax{M}-1} \left( \sigma(W_{L,M,i}) - 1 \right) \right). \end{equation} \end{enumerate} \end{corollary}
\begin{proof} If the walk $\Lfz \mathcal C$ is not symmetric, then, in the estimate \eqref{eq:odhad_sigma} we do not need to count the symmetric possibilities in the following sense. If when reading a run of $L$'s in $V$, we count the state $M \in \LE$ with $i \in \left\{ \LclassMax{M}, \ldots, 2 \LclassMax{M}-1 \right \}$, then when reading the symmetric run of $R$'s we cannot pass through $\assoc{M} \in \RE$ associated with the integer $i$. The symmetric run of $R$'s exists due to the assumption $\Lfz V = V_1 \assoc{V_1}$. Thus, we can count only half of all the possible states $(M_{k_\ell},i_{k_\ell})$, resp. $(N_{k_\ell},i_{k_\ell}')$, which gives the estimate in \cref{it:coro_half_half}. \end{proof}
We now transform the last corollary into the terms of the period of the continued fraction of $x$ after the transformation.
\begin{theorem} \label{the:odhad_pi} Let $x$ be a quadratic irrational number and $N \in \D{}$. We have \[ \per(h_N(x)) \leq \per(x)\sum_{M \in \LE{}} \sum_{i=\LclassMax{M}}^{2 \LclassMax{M}-1} \left( \sigma(W_{L,M,i}) - 1 \right). \] \end{theorem}
\begin{proof} Let $V$ be the repetend of the LR-representation of $x$. It implies that $V$ is primitive. Using~\Cref{thm:dostanu_se_do_Dn}, we may assume that the calculation of the tail of $h_N(x)$ is given by a closed walk $\mathcal C = \Twalk{M}{V^{\gamma}}{W}{M}$ satisfying \eqref{eq:C_ub_assumption}.
We find the closed walk $\widehat{\mathcal C}_0 = \Twalk{\widehat{M}}{\widehat{V}_0^{\delta}}{\widehat{W}_0}{\widehat{M}}$ with $\widehat{V}_0^{\delta m} \in \tau_\kappa(V^\gamma)$, $\widehat{V_0}$ primitive and $\widehat{M} \in \DB{}$ given by \Cref{co:vyfukovani_ze_symetrickeho} and \Cref{re:jednoduchost_nafoukle}, using the notation therein. According to \eqref{eq:prepocty_cyklu}, we have \begin{equation} \label{eq:pf:nafuk} {\sigma_{\mathrm{c}}}(W) \leq m {\sigma_{\mathrm{c}}}(\widehat{W}_0), \end{equation}
\begin{equation} \label{eq:pf:V} m {\sigma_{\mathrm{c}}}(\widehat{V}_0) \leq {\sigma_{\mathrm{c}}}(V), \end{equation}
from \Cref{le:vypocet_per} it follows that
\begin{equation} \label{eq:pf:A2}
\per(h_N(x)) \leq {\sigma_{\mathrm{c}}}(W)
\end{equation}
and that
\begin{equation} \label{eq:pf:per_sigma}
\frac{1}{2} {\sigma_{\mathrm{c}}}(V) \leq \per(x).
\end{equation}
Set $S = \sum_{M \in \LE{}} \sum_{i=\LclassMax{M}}^{2 \LclassMax{M}-1} \left( \sigma(W_{L,M,i}) - 1 \right)$.
We split the proof into several cases.
\begin{enumerate}[label=(\textbf{\Alph*})]
\item $\widehat{\mathcal C}_0$ is symmetric.
$\widehat{V}_0 = V_1\assoc{V_1}$ for some $V_1 \in \D[1]{}$.
\begin{enumerate}[label=(\textbf{\Alph{enumi}}\arabic*)]
\item $W = W_1\assoc{W_1}$ for some $W_1 \in \D[1]{}$.
By \Cref{le:vypocet_per} we have $\per(h_N(x)) \leq \frac{{\sigma_{\mathrm{c}}}(W)}{2}$. Therefore:
\[
\per(h_N(x)) \leq \frac{{\sigma_{\mathrm{c}}}(W)}{2} \overset{\eqref{eq:pf:nafuk}}{\leq} \frac{1}{2} m{\sigma_{\mathrm{c}}}(\widehat{W}_0) \overset{\eqref{eq:odhad_sigma}}{\leq} \frac{1}{2} m {\sigma_{\mathrm{c}}}(\widehat{V}_0) S \overset{\eqref{eq:pf:V}}{\leq}\frac{1}{2} {\sigma_{\mathrm{c}}}(V)S \overset{\eqref{eq:pf:per_sigma}}{\leq} \per(x)S.
\]
\item $W \neq W_1\assoc{W_1}$ for all $W_1 \in \D[1]{}$
\begin{enumerate}[label=(\textbf{\Alph{enumi}}\arabic{enumii}\roman*)]
\item $V = V_2\assoc{V_2}$ for some $V_2 \in \D[1]{}$
By \Cref{co:vyfukovani_ze_symetrickeho} \Cref{it:2)vyfukovani}, we obtain a contradiction with \Cref{le:sym_dvojstav}.
\item $V \neq V_2\assoc{V_2}$ for all $V_2 \in \D[1]{}$.
By \Cref{le:vypocet_per} we have ${\sigma_{\mathrm{c}}}(V) = \per(x)$. Therefore:
\[
\per(h_N(x)) \overset{\eqref{eq:pf:A2}}{\leq} {\sigma_{\mathrm{c}}}(W) \overset{\eqref{eq:pf:nafuk}}{\leq} m {\sigma_{\mathrm{c}}}(\widehat{W}_0) \overset{\eqref{eq:odhad_sigma}}{\leq} m {\sigma_{\mathrm{c}}}(\widehat{V}_0) S \overset{\eqref{eq:pf:V}}{\leq} {\sigma_{\mathrm{c}}}(V) S \overset{}{=} \per(x)S.
\]
\end{enumerate}
\end{enumerate}
\item $\widehat{\mathcal C}_0$ is not symmetric
\begin{enumerate}[label=(\textbf{\Alph{enumi}}\arabic*)]
\item $\widehat{V}_0 = V_1\assoc{V_1}$ for some $V_1 \in \D[1]{}$. Therefore:
\[
\per(h_N(x)) \overset{\eqref{eq:pf:A2}}{\leq} {\sigma_{\mathrm{c}}}(W) \overset{\eqref{eq:pf:nafuk}}{\leq} m{\sigma_{\mathrm{c}}}(\widehat{W}_0) \overset{\eqref{eq:odhad_sigma_pul}}{\leq} \frac{1}{2} m {\sigma_{\mathrm{c}}}(\widehat{V}_0) S \overset{\eqref{eq:pf:V}}{\leq} \frac{1}{2} {\sigma_{\mathrm{c}}}(V) S \overset{\eqref{eq:pf:per_sigma}}{\leq} \per(x)S.
\]
\item $\widehat{V}_0 \neq V_1\assoc{V_1}$ for all $V_1 \in \D[1]{}$
The fact that $\widehat{V}_0 \neq V_1\assoc{V_1}$ implies $V \neq V_2\assoc{V_2}$ for all $V_2 \in \D[1]{}$. By \Cref{le:vypocet_per} we have ${\sigma_{\mathrm{c}}}(V) = \per(x)$. Therefore:
\[
\per(h_N(x)) \overset{\eqref{eq:pf:A2}}{\leq} {\sigma_{\mathrm{c}}}(W) \overset{\eqref{eq:pf:nafuk}}{\leq} m{\sigma_{\mathrm{c}}}(\widehat{W}_0) \overset{\eqref{eq:odhad_sigma}}{\leq} m {\sigma_{\mathrm{c}}}(\widehat{V}_0) S \overset{\eqref{eq:pf:V}}{\leq} {\sigma_{\mathrm{c}}}(V) S \overset{}{=} \per(x)S. \qedhere
\]
\end{enumerate} \end{enumerate} \end{proof}
As the last theorem holds for all quadratic irrational numbers $x$ and all matrices $N \in \D{}$, the proof of \Cref{thm:main} follows.
\begin{proof}[Proof of \Cref{thm:main}] Trivially, we may assume that $N \in \D{}$.
We shall first prove the upper bound on $\per(h_N(x))$ which follows directly from \Cref{the:odhad_pi} with \[ S_n = \sum_{M \in \LE{}} \sum_{i=\LclassMax{M}}^{2 \LclassMax{M}-1} \left( \sigma(W_{L,M,i}) - 1 \right). \]
By \Cref{def:LE_n}, we have $M \in \LE{} \iff M = M_{t,u}= \mat{t}{0}{u}{m}$ where $mt = n$ and if we put $g_t = \gcd(m,t) = \gcd(t,\frac{n}{t})$, $I_{t} = \begin{cases} \{0\} & \text{ for } g_t = 1 \\ \{1, \dots, g_t - 1\} & \text{ otherwise}\end{cases}$, then $u \in I_{t}$. It follows from \Cref{le:LS_n_do_RS_n} that we have \begin{equation} \label{eq:prepocet_sumy}
S_n
=
\sum_{\substack{ t \in \mathbb{N} \\ t \mid n}} \sum_{u \in I_{t}} \sum_{i=\LclassMax{M_{t,u}}}^{2 \LclassMax{M_{t,u}}-1} \left( 2\left\lfloor \frac{\xi(im+u, t)}{2} \right\rfloor + 1 \right). \end{equation}
We rearrange the two last sums into one. By \Cref{le:char_toceni_na_L}, $\LclassMax{M_{t,u}}$ is the least positive integer such that \begin{equation} \label{eq:ht} m \nu_L(M_{t,u}) = ht \end{equation} for some $h \in \mathbb{N} \setminus \{0\}$ and therefore $\nu_L(M_{t,u}) = \frac{t}{g_t}$. Therefore, we have $i_1m+u \not \equiv i_2m+u \pmod{t}$ for all $i_1, i_2 \in \{\nu_L(M_{t,u}),\nu_L(M_{t,u})+1,\dots,2\nu_L(M_{t,u})-1\}, i_1 \neq i_2$. It means that $\{im+u \pmod{t} \colon i \in \{\nu_L(M_{t,u}),\nu_L(M_{t,u})+1,\dots,2\nu_L(M_{t,u})-1\}\} = \{k \colon k \in \mathbb{N},k<t,k \equiv u \pmod{g_t}\}$ and $\# \{k \colon k \in \mathbb{N},k<t,k \equiv u \pmod{g_t}\} = \nu_L(M_{t,u})$.
Let $J_t = \begin{cases} \emptyset & \text{for } g_t = 1, \\ \{i g_t \colon i \in \mathbb{N}\} & \text{ otherwise.} \end{cases}$. Together with the facts that $u = 0 $ for $g_t = 1$ and $u \in \{1, \dots, g_t - 1\}$ otherwise we have \[ \{im+u \! \! \pmod{t} \colon i \in \{\nu_L(M_{t,u}),\nu_L(M_{t,u})+1,\dots,2\nu_L(M_{t,u})-1\}, u \in I_t\} = \{k \colon k \in \mathbb{N},k<t, k \not \in J_t\} \] and $\# \{k \colon k \in \mathbb{N},k<t, k \not \in J_t\} = \sum_{u \in I_{t}}\nu_L(M_{t,u})$. Now it remains to realize that for all $i \geq \nu_L(M_{t,u})$ we have by \eqref{eq:ht} that $im + u \geq th+u \geq t$ and by definition of $\xi$, we have $\xi(k,t) = \xi(at+k,t)$ for all $a \in \mathbb{N}$ and $k\geq t$. We conclude \[ \sum_{u \in I_{t}} \sum_{i=\LclassMax{M_{t,u}}}^{2 \LclassMax{M_{t,u}}-1} \left( 2\left\lfloor \frac{\xi(im+u, t)}{2} \right\rfloor + 1 \right) = \sum_{\substack{j = t \\ j \not \in J_t }}^{2t-1} \left (2\left\lfloor \frac{\xi(j,t)}{2} \right\rfloor +1 \right) \] which together with \eqref{eq:prepocet_sumy} proves the upper bound.
As the inverse M\"obius transformation preserves the determinant of its associated matrix, i.e, $h_N^{-1} = h_{N'}$ for some $N' \in \D{}$, the lower bound follows from the upper bound. \end{proof}
\section{Concluding remarks and experiment results} \label{sec:conclusion}
We have tested the obtained upper bound of \Cref{thm:main} for various values of $n$. Some of the results can be seen in \Cref{tab:prodlouzeni}. The experiments indicate that the upper bound is sharp for $n=2$, all prime $n$ with $n \equiv 3 \pmod 4$ and for some composite numbers (for example $n = 9,14,27$). For some other values of $n$ the difference between our estimate $S_n$ and the experimentally obtained factor of prolongation (denoted $S_n(x)$ in the table) can be relatively large (for example for $n = 18$).
{ \renewcommand*{\arraystretch}{1.2} \begin{table}[!htb] \centering
\begin{tabular}{c|c|c|c}
$n$ & $S_n$ & $S_n(x)$ & $x$ \\
\hline 7 & 24 & 24.0 & $[\overline{4390}]$ \\ 8 & 36 & 26.0 & $[\overline{4792, 4423}]$ \\ 9 & 36 & 36.0 & $[\overline{4696}]$ \\ 13 & 52 & 51.0 & $[\overline{4771, 4930}]$ \\ 14 & 80 & 80.0 & $[\overline{4693}]$ \\ 15 & 76 & 67.2 & $[2904, 189, \overline{4662, 4147, 4872, 4669, 4875}]$ \\ 18 & 120 & 68.0 & $[\overline{4908, 4057}]$ \\ 20 & 120 & 104.4 & $[4495, 520, \overline{4803, 4060, 4805, 4930, 4643}]$ \\ 24 & 164 & 90.0 & $[\overline{4380, 4843}]$ \\ 27 & 144 & 144.0 & $[\overline{4384}]$ \\ 81 & 538 & 532.0 & $[\overline{4232}]$ \\ \end{tabular}
\caption{Experimental results on the bound of \Cref{thm:main}. $S_n(x)$ denotes an experimental lower bound on $\sup \left\{ \frac{ \per(h_M(x))}{ \per(x)} \colon M \in \D{} \right\}$.}
\label{tab:prodlouzeni} \end{table} }
The difference between $S_n$ and $S_n(x)$ is caused by the fact that in some cases the closed walk $\mathcal C$ cannot go through all of the transitions that we have considered in the estimate \eqref{eq:odhad_sigma}. For composite numbers, a sharper estimate depends on the value of $n$ and its divisors. If $n$ is prime, the sharp bound may be proven to be \begin{equation} \label{eq:n_prime} \begin{aligned} \sup \left\{ \frac{ \per(h_M(x))}{ \per(x)} \colon M \in \D{} \text{ and } x \text{ is quadratic irrational} \right\} = \\ = \begin{cases} 5 & \text{ if } n = 2, \\ \displaystyle 2 + 2\sum_{i=1}^{\frac{n-1}{2}} (\xi(i,n) + 2) & \text{ if } n \equiv 3 \pmod{4}, \\ \displaystyle 1 + 2\sum_{i=1}^{\frac{n-1}{2}} (\xi(i,n) + 2) & \text{ if } n \equiv 1 \pmod{4}. \end{cases} \end{aligned} \end{equation} For $n \equiv 3 \pmod{4}$ the bound in fact equals $S_n$, the formula is only simplified. The bound for $n \equiv 3 \pmod{4}$ equals $S_n-1$, corresponding to the case when $\mathcal C$ cannot pass through all possible vertices. A corresponding experiment is for $n=13$ in \Cref{tab:prodlouzeni}.
We do not give a proof of the formula \eqref{eq:n_prime} as it is only for a very special case and requires some more technical claims.
We do not provide experiment results on the lower bound as the behaviour is completely analogous, one only needs to consider the inverse of the given M\"obius transformation.
\IfFileExists{biblio.bib}{
}{
}
\end{document} |
\begin{document}
\begin{center}
{\bf Asymptotic Analysis for a Nonlinear Reaction-Diffusion System \\ Modeling an Infectious Disease}
Hong-Ming Yin\footnote{Corresponding Author. Email: hyin@wsu.edu}\\ Department of Mathematics and Statistics\\ Washington State University\\ Pullman, WA 99164, USA.\\ and \\ Jun Zou\\ Department of Mathematics\\ The Chinese University of Hong Kong\\ Shatin, N.T., Hong Kong
\end{center}
\begin{abstract}
In this paper we study a nonlinear reaction-diffusion system which models an infectious disease caused by bacteria such as those for cholera. One of the significant features in this model is that a certain portion of the recovered human hosts may lose a lifetime immunity and could be infected again. Another important feature in the model is that the mobility for each species is allowed to be dependent upon both the location and time.
With the whole population assumed to be susceptible with the bacteria, the model is a strongly coupled nonlinear reaction-diffusion system. We prove that the nonlinear system has a unique solution globally in any space dimension under some natural conditions on the model parameters and the given data. Moreover, the long-time behavior and stability analysis for the solutions are carried out rigorously. In particular, we characterize the precise conditions on variable parameters about the stability or instability of all steady-state solutions. These new results provide the answers to several open questions raised in the literature.
\end{abstract} \ \\ {\bf AMS Mathematics Subject Classification:} 35K57 (Primary), 92C60 (Secondary).
\ \\ {\bf Key Words and Phrases:} Infectious disease model; nonlinear reaction-diffusion system; global existence and uniqueness; stability analysis.
\section{Introduction}
In biological, ecological, health and medical sciences, researchers have a great deal of interest to establish a suitable mathematical model for various infectious diseases. The current global pandemic attracts even more scientists to this field. There are many different mathematical models for an infectious disease in the literature. Roughly speaking, these models can be divided by two categories: a data-based discrete model and a continuous model based on a population growth (see \cite{GF2008,SR2013,WMI2018}). Our approach is based on a continuous model which provides a much more convenient tool to analyze the complicated dynamics of the interaction among susceptible, infected and recovered patients. A continuous model is typically governed by a system of ordinary differential equations (ODE model) or a system of partial differential equations (PDE model). For an ODE model, a monumental work was done in 1927 by Kermack and McKendrick \cite{KM1927}. Since then, a significant progress has been made in modeling and analyzing various infectious diseases such as SIR, SEIR models and their various extensions. An ODE model often provides a clear and precise description of physical quantities and their relations. By using an ODE model, one can study detailed dynamical interaction between viruses and various species as well as other qualitative properties such as reproduction numbers. This type of ODE models is widely adopted and used by researchers in all fields, particularly those in biological and health sciences. On the other hand, when one takes the movement of species across different geographical regions into consideration, it is necessary to include a diffusion process in a mathematical model to reflect the movement. This leads to modeling an infectious disease by using a system of partial differential equations (PDEs), often called reaction-diffusion equations. A well-known work \cite{HLBV1994} discussed a number of PDE models arising from biological, ecological and animal sciences and explained why the PDE approach is more appropriate in those areas. There are a large number of research studies, conference proceedings and monograph in both PDE and ODE models in the literature. We list only some of them here as examples, e.g., \cite{AM1979,MA1979,DG1999,DW2002,KP1994} for the SIR ODE models and \cite{ABLN2007,FMWW2019,HLBV1994,JDH1995,LN1996,SLX2019} for the SIR PDE models. Many more references can be found in a SIAM Review paper by Hethcote \cite{H2000} and the monograph by Busenberg and Cooke \cite{BC2012}, Cantres and Cosner \cite{CC1981}, Daley and Gani \cite{DG1999}, Lou and Ni \cite{LN1996}, etc. It is worth noting from the mathematical point of view that the PDE models present significant more challenges for scientists to study the dynamics of the solutions and to analyze qualitative properties of the solutions. Many important mathematical questions such as global existence and uniqueness are still open for some popular PDE models. This is one of the motivations for the current study.
In this paper we consider a mathematical model in a heterogeneous domain for an infectious disease caused by bacteria such as Cholera without lifetime immunity. Without considering the diffusion-process of the population, the ODE models have been studied extensively (see, e.g., \cite{AB2011,BC2012,DW2002,ESTD2002}). The model considered in this work is a direct extension of the ODE model. To describe the mathematical model, we introduce the following variables: \begin{eqnarray*} S(x,t) & = & \mbox{Susceptible population concentration at location $x$ and time $t$} \\ I(x,t) & = & \mbox{ Infected population concentration at location $x$ and time $t$} \\ R(x,t) & = & \mbox{ Recovered population concentration at location $x$ and time $t$} \\ B(x,t) & = & \mbox{Concentration of bacteria at location $x$ and $t$} \end{eqnarray*}.
We assume that the whole population is susceptible to the bacteria. Moreover, the rate of growth for the population, denoted by $b(x,t,S)$, depends on location, time and the population itself.
A classical example for $b$ is that the population growth follows a logistic growth model with a maximum capacity $k_1>0$: \[ b(x,t,s)=b_0s(1-\frac{s}{k_1}),\] where $b_0> 0$ represents the growth rate of the population.
The population reduction caused by infected patients is denoted by a nonlinear function $g_1(x,t,S,I,B)$ which is nonnegative. A typical form of the nonlinear function $g_1$ is given by (see \cite{YW2017,Yin2020}):
\[ g_1(x,t,S,I,B)=\beta_1SI+\beta_2Sh_1(B), ~~h_1(B)=\frac{B}{B+k_2},\]
where $\beta_1, \beta_2$ are positive transmission parameters and $h_1(B)$ represents the maximum saturation rate of bacteria on
human hosts and $k_2>0$.
The bacteria growth follows the same assumption, denoted by $g_2(x,t,s)$ with a maximum capacity $k_3>0$: \[ g_2(x,t,s)=g_0s(1-\frac{s}{k_3}),\] where $g_0> 0$ is the growth rate of the bacteria.
We also assume that the diffusion coefficients depend on location and time. By extending the ODE model (see \cite{BC2012,DW2002,LW2011} etc.,), we obtain the following reaction-diffusion system: \setcounter{section}{1} \setcounter{local}{1}
\begin{eqnarray}
S_t-\nabla\cdot [a_1(x,t)\nabla S] & = & b(x,t,S)-g_1(x,t,S,I,B)-d_1S+\sigma R,\\
I_t-\nabla \cdot [a_2(x,t)\nabla I] & = & g_1(x,t,S,I,B)-(d_2+\gamma)I,\stepcounter{local} \\
R_t-\nabla \cdot [a_3(x,t)\nabla R] & = & \gamma I-(d_3+\sigma)R,\stepcounter{local} \\
B_t-\nabla \cdot [a_4(x,t)\nabla B] & = & \xi I +g_2(x,t,B)-d_4 B.\stepcounter{local}
\end{eqnarray}
The biological meaning of various parameters and functions in the model are given below (see \cite{ESTD2002,WW2015,YW2016}):
\begin{eqnarray*}
a_i & = & \mbox{the diffusion coefficients, $i=1,2,3,4$},\\
\gamma & = & \mbox{the recovery rate of infectious individuals},\\
\sigma & = & \mbox{the rate at which recovered individuals lose immunity},\\
d_i & = & \mbox{the natural death rate of species or bacteria},\\
\xi & = & \mbox{the shedding rate of bacteria by infectious human hosts}.
\end{eqnarray*}
To complete the mathematical model, we assume that the system (1.1)-(1.4) holds in $Q_T=\Omega\times (0,T]$ for any $T>0$, where $\Omega$ is a bounded domain in $R^n$ with $C^2$-boundary $\partial \Omega$. The initial concentrations for all species are known and we assume that no species can cross the boundary $\partial \Omega$. This leads to the following initial and boundary conditions:
\begin{eqnarray}
& & (\nabla_{\nu}S,\nabla_{\nu}I,\nabla_{\nu}R, \nabla_{\nu}B) = 0, \hspace{1cm} (x,t)\in \partial \Omega\times (0,T],\stepcounter{local}\\
& & (S(x,0),I(x,0),R(x,0),B(x,0))=(S_0(x),I_0(x),R_0(x),B_0(x)), x\in \Omega,\stepcounter{local}
\end{eqnarray} where $\nu$ represents the outward unit normal on $\partial \Omega$.
We would like to give a short review about the known results for the above model. For the ODE system corresponding to (1.1)-(1.4), there are many studies for various interesting mathematical problems such as global existences, dynamical interaction between the bacteria and species (see, e.g., \cite{AB2011,DW2002,ESTD2002,TH1992}). The stability analysis is also carried out by several researchers (see \cite{LW2011,SD2012,TW2011} etc.). When the movement of species is considered in the model, the corresponding PDE system is much more complicated to study. This is due to the fact that the maximum principle can not be applied for a system of reaction-diffusion equations.
It is a challenge to establish the global well-posedness for the PDE system (1.1)-(1.6). Nevertheless, when the space dimension is equal to $1$, under certain conditions on $g_1$ and $g_2$, the global existence is established (see \cite{WW2015,YW2016,YW2017,Y2018}). The reason is that the total population is bounded in $L^1(\Omega)$, which implies a global boundedness for $S(x,t)$ by using Sobolev embedding for the space dimension $n=1$. However, this method does not work when the space dimension $n$ is greater than $1$. In a SIAM review article
(\cite{PS2000}), the authors considered the following system
(with $a$ and $b$ being two positive constants):
\begin{eqnarray*}
& & u_t-a\Delta u =f(u,v), \hspace{1cm} x\in \Omega, ~t>0,\\
& & v_t-b\Delta v =g(u,v), \hspace{1cm} x\in \Omega, ~t>0,
\end{eqnarray*}
subject to appropriate initial and boundary conditions.
Suppose $f(0,v), g(u,0)\geq 0$ for all $u,v \geq 0$. Then under the condition that
\[ f(u,v)+g(u,v)\leq 0,\]
the $L^1$-norms of the nonnegative solutions $u$ and $v$ are bounded, i.e.,
\[\sup_{t>0}\int_{\Omega} (u+v)dx\leq C.\]
However, the solution $(u,v)$ may blow up in finite time when the space dimension is greater than 1 if no additional
conditions on $f(u,v)$ and $g(u,v)$ are made. Therefore, as indicated in \cite{PS2000}, one must impose some additional conditions in order to obtain a global bound for a reaction-diffusion system. There are some interesting results for a general reaction-diffusion system when leading coefficients are constants. In 2000, under certain additional conditions, Pierre-Schmitt (\cite{PS2000}) introduced a dual method to establish such a bound for the reaction-diffusion system. In 2007, Desvillettes-Fellner-Pierre-Vovelle introduced in \cite{DFPV2007} an entropy condition originated by Kanel in 1990
(\cite{KA1990}) and extended the dual method to a more general reaction-diffusion system with constant diffusion coefficients and established the global bound with a quadratic-growth reaction as long as a total mass is controlled ($L^1-$boundedness). In 2009, Caputo-Vasseur \cite{CV2009} extended the entropy method to establish a global existence for a reaction-diffusion system where the nonlinear reaction terms grow at most sub-quadratically. One can see an interesting review by M. Pierre in 2010 \cite{PI2010}.
Caceres-Canizo extended in 2017 \cite{CC2017} to the case where the reaction terms grow at most quadratically under certain conditions on the steady-state solutions. In 2018, Souplet \cite{SOU2018} established the global well-posedness for a reaction-diffusion system with quadratic growth in the reaction. Very recently, some considerable progress was made for a reaction-diffusion system by Fellner-Morgan-Tang in 2019 \cite{FMT2019} and Morgan-Tang in 2020 \cite{MT2020}. They are able to derive a global bound for the solution of a reaction-diffusion as long as the diffusion coefficients are smooth and nonlinear reaction terms in the system satisfy a condition called an intermediate growth condition, which replaces the entropy condition. Their approach is based on a combination of the dual method and the entropy method. In 2021, Fitzgibbon-Morgan-Tang-Yin \cite{FMTY2021} studied a very general reaction-diffusion system with a controlled mass and nonsmooth diffusion coefficients.
They established the global well-posedness for the system with at most a polynomial growth for reactions. Moreover, several interesting examples as applications arising from biological, health sciences and chemical reactive-flow were studied in the paper. Those results made a substantial progress for a general reaction-diffusion system with a controlled mass. However, due to the nonlinearity in Eq. (1.1), these results do not cover the nonlinear system (1.1)-(1.4), particularly, we do not have growth conditions here on $g_1$ with respect to $(s_1, s_2, s_3)$ for the global existence (see Theorem 2.1 in section 2).
The purpose of this paper has twofold. The first purpose is to establish the existence of a global solution to the generalized system (1.1)-(1.6) in any space dimension, without any restriction on parameters nor growth conditions with respect to $s_i$ for $g_1$. This extends a result obtained by the first author in his recent work \cite{Yin2020}. Our method in this paper is based on some key ideas developed in \cite{Yin2020}. The special structure of the system (1.1)-(1.4) will also play a key role. We shall also use various techniques from the theories of elliptic and parabolic equations (see \cite{EVANS,LI1996,LSU}). To derive an a priori bound, we use a crucial result for a linear parabolic equation in the Campanato-John-Nirenberg-Morrey space from \cite{Yin1997}, which extends the DiGoigi-Nash's estimate with weaker conditions for nonhomogeneous terms. The other purpose of the current work is to present the stability analysis of all steady-state solutions, which was not addressed in \cite{Yin2020}. In particular, for the following classical choices of the growth model \cite{CC1981}:
\begin{eqnarray}
& & {\ }\hskip-0.8truecm b(x,t,S)=b_0S\left(1-\frac{S}{k_1}\right), ~g_1( x,t,S,I,B)=\beta_1SI+\beta_2Sh_1(B), ~h_1(B)=\frac{B}{B+k_2}\stepcounter{local}\\
& & {\ }\hskip-0.8truecm g_2(B)=g_0B\left(1-\frac{B}{k_2}\right),\stepcounter{local}
\end{eqnarray}
we are able to precisely describe what conditions are needed for a steady-state solution to be stable or unstable. Roughly speaking, we shall demonstrate that under the conditions:
\[ d_1>b_0, ~~d_2\geq 0, ~~d_3\geq 0, ~~d_4>g_0,\]
the steady-state solution is stable. On the other hand, if either $d_1<b_0$ or $d_4<g_0$, then we can choose a set of suitable values for parameters $\sigma, \gamma, \beta_1$ and $\beta_2$ such that the steady-state solution is unstable.
This implies that our stability conditions are optimal. This stability analysis provides some important guidance to practitioners and scientists in biological, ecological and health sciences.
The paper is organized as follows. In Section 2 we first recall some function spaces which are frequently used in the subsequent analysis, and then state our main results. In Section 3, we prove the first part of the main results on global solvability of the system (1.1)-(1.6) (Theorem 2.1 and Corollary 2.1). In Section 4 we focus on a general stability analysis and obtain the sufficient conditions on parameters which ensure the stability of a steady-state solution.
In Section 5, for a set of concrete functions $b(x,t,s), h_1(s)$ and $g_2(x,t,s)$ we give precisely conditions on the model parameters, under which a steady-state solution is stable or unstable. Finally, some concluding remarks are given in Section 6.
Throughout the paper, we shall use $C$, with or without subscript, for a generic constant depending only on the given data in the model, including the upper bound of the terminal time $T$, and it may take a different value at each occurrence.
\section{ Preliminaries and Statement of Main Results}
For reader's convenience, we recall some standard function spaces which will be used frequently in the subsequent analysis.
For $\alpha\in (0,1)$, we denote by $C^{\alpha}(\bar{\Omega})$ (or $C^{\alpha, \frac{\alpha}{2}}(\bar{Q}_{T})$) the H\"older space in which every function is H\"older continuous with respect to $ x$ (or $(x,t))$ with exponent $\alpha$ in $\bar{\Omega}$ (or $(\alpha,\frac{\alpha}{2})$ in $\bar{Q}_{T}$). For $T=\infty$, we write $Q_T=\Omega\times (0, T)$ as $Q=\Omega\times (0,\infty).$
For $p\geq 1$ and a Banach space $V$ with norm $||\cdot||_v$, we define
\[ L^p(0,T; V)=\{ F(t): t\in [0,T]\rightarrow V; ~ ||F||_{L^p(0,T;V)}<\infty\},\]
equipped with the norm
\[ ||F||_{L^{p}(0,T;V)}=\left(\int_{0}^{T} ||F||_v^p dt\right)^{\frac{1}{p}}.\]
When $V=L^{p}(\Omega)$, we simply write
$L^p(Q_{T})=L^{p}(0,T;L^{p}(\Omega))$,
with its norm as $||\cdot||_p$.
Sobolev spaces $W^{k,p}(\Omega)$ and $W_{p}^{k,l}(Q_{T})$ are defined the same as in the classical references (see, e.g., \cite{EVANS}). Let $V_2(Q_T)=\{ u\in C([0,T];W_{2}^{1,0}(\Omega)): ||u||_{V_{2}}<\infty\} $ (see \cite{LSU})
equipped with the norm
\[ ||u||_{V_{2}}=\max_{0\leq t\leq T}||u||_{L^{2}(\Omega)}+\sum_{i=1}^{n}||u_{x_{i}}||_{L^2(Q_{T})}.\]
We will also use the Campanato-John-Nirenberg-Morry space $L^{2,\mu}(Q_T)$, which is defined as a subspace of $L^2(Q_{T})$ with its norm given by
\[||u||_{L^{2, \mu}(Q_{T})}=||u||_{L^{2}(Q_{T})}+[u]_{2, \mu, Q_{T}}<\infty,\] where
\[ [u]_{2, \mu, Q_{T}}=\sup_{\rho>0, z_0\in Q_{T}}\left(\rho^{-\mu}\int_{Q_{\rho}(z_0)}|u-u_Q|^2dxdt\right)^{\frac{1}{2}},\] with $z_0=(x_0,t_0), Q_{\rho}(z_0)=B_{\rho}(x_0)\times (t_0-\rho^2, t_0]$, and $u_Q$ representing the average of $u$ over $Q_{\rho}(z_0)$ for any $Q_{\rho}(z_0)\subset Q_{T}$; see Troianiello $\cite{T1987} $ for its detailed definition and properties. An important fact of the space is that $L^{2, \mu+2}(Q_{T})$ is equivalent to $\ca$ with $\alpha=\frac{\mu-n}{2}$ if $n<\mu\leq n+2$ (Lemma 1.19 in \cite{T1987}). We shall write the norm of $L^{2, \mu}(Q_{T})$ as
$||u||_{2, \mu}$.
We first state the basic assumptions for the diffusion coefficients and the known data involved in our model (1.1)-(1.4).
All other model parameters are assumed to be positive constants throughout this paper.
One can easily extend the well-posedness results to more general system when those parameters are functions of $(x,t)$ as long as the basic structure of the system is preserved.
\ \\
{\bf H(2.1).} Assume that $a_i\in L^{\infty} (Q)$. There exist two positive constants $a_0$ and $A_0$ such that
\[ 0<a_0\leq a_i(x,t)\leq A_0, \hspace{1cm} (x,t)\in Q_{T}, ~i=1,2,3,4.\]
{\bf H(2.2).} Assume that all initial data $U_0(x):=(S_0(x), I_0(x), R_0(x), B_0(x))$ are nonnegative on $\Omega$. Moreover, $\nabla U_0(x)\in L^{2,\mu_0}(\bar{\Omega})^4$ with $\mu_0\in (n-2,n)$.
\ \\
{\bf H(2.3)}. (a) Let $b(x,t,s), d_i(x,t,s)$ and $g_2(x,t,s)$ be measurable in $Q\times R^+$ and locally Lipschitz continuous with respect to $s$, and
$0\leq b(x,t,0), ~d_i(x,t,0)\in L^{\infty}(Q)$. Moreover, it holds
for some $M>0$ that
\[ d_i(x,t,s)\geq d_0\geq 0, ~~b_s(x,t,s)\leq b_0, \hspace{1cm} (x,t,s)\in Q\times[M,\infty).\]
(b) Let $g_1(x,t,s_1,s_2,s_3)$ be measurable in $Q\times (R^+)^3$ and nonnegative, differentiable with respect to $s_1,s_2,s_3$, and
\begin{eqnarray*}
& & g_1(x,t,0,s_2,s_3)\geq 0, \hspace{1cm} s_2, s_3\geq 0,\\
& & g_2(x,t,0)\geq 0, ~~g_{2s}(x,t,s)\leq g_0, \hspace{1cm} (x,t,s)\in Q\times R^+.
\end{eqnarray*} where $k_1, k_2$ and $k_3$ represent the maximum capacity of the general population, the infected population and the bacteria, respectively.
For convenience, we set $U(x,t)=(u_1,u_2,u_3,u_4)$ to be a vector-valued function defined in $Q_T$, with
\[ u_1(x,t)=S(x,t), ~u_2(x,t)=I(x,t), ~u_3(x,t)=R(x,t), ~u_4(x,t)=B(x,t), \, ~(x,t)\in Q_T. \]
The right-hand sides of the equations (1.1)-(1.4) are denoted by $f_1(x,t,U)$, $f_2(x,t,U)$, $f_3(x,t,U)$ and $f_4(x,t,U)$, respectively.
With the new notation, the system (1.1)-(1.6) can be written as the following reaction-diffusion system:
\setcounter{section}{2}
\setcounter{local}{1}
\begin{eqnarray}
& & u_{1t}-\nabla\cdot [a_1(x,t)\nabla u_1]=f_1(x,t,U), \hspace{1cm} (x,t)\in Q_{T},\\
& & u_{2t}-\nabla\cdot [a_2(x,t)\nabla u_2]=f_2(x,t,U), \hspace{1cm} (x,t)\in Q_{T},\stepcounter{local} \\ & & u_{3t}-\nabla\cdot [a_3(x,t)\nabla u_3]=f_3(x,t,U), \hspace{1cm} (x,t)\in Q_{T},\stepcounter{local}\\
& & u_{4t}-\nabla\cdot [a_4(x,t)\nabla u_4]=f_4(x,t,U), \hspace{1cm} (x,t)\in Q_T,\stepcounter{local}
\end{eqnarray}
subject to the initial and boundary conditions:
\begin{eqnarray}
& & U(x,0)=U_0(x):=(S_0(x),I_0(x),R_0(x),B_0(x)), \hspace{1cm} x\in \Omega, \stepcounter{local} \\
& & \nabla_{\nu}U(x,t)=0, \hspace{1cm} (x,t)\in \partial \Omega\times (0,T].\stepcounter{local}
\end{eqnarray}
We define
\[ X=V_2(Q_{T})\bigcap L^{\infty}(Q_{T}).\]
\ \\
{\bf Definition 2.1.} We say $U(x,t)\in X^4$ is a weak solution to the problem (2.1)-(2.6) in $Q_{T}$ if it holds
for all functions $\phi_k\in X$ with $\phi_{kt}\in L^2(Q_{T}),
\phi_k(x,T)=0$ on $\Omega$ for $k=1,2,3,4$:
\begin{eqnarray*}
& & \int_{0}^{T}\int_{\Omega}\left[-u_k\cdot \phi_{kt}+a_k\nabla u_k\cdot \nabla\phi_k\right]dxdt\\ & & = \int_{\Omega} u_k(x,0)\phi _k(x,0)dx+\int_{0}^{T}\int_{\Omega}f_k(x,t,U)\phi_k(x,t)dxdt. \stepcounter{local}
\end{eqnarray*}
\ \\
{\bf Theorem 2.1.} Under the assumptions H(2.1)-H(2.3), the problem (2.1)-(2.6) has a weak solution in $X$ and the weak solution is nonnegative and bounded in $Q_T$ for any $T>0$.
Moreover, it holds that $u_i(x,t)\in C^{\alpha, \frac{\alpha}{2}}(\bar{Q}_{T})$ for $i=1,2,3,4$.
Under some additional conditions on $b$ and $g_2$, we can deduce an uniform bound of the weak solution to the problem (2.1)-(2.6) in $Q$.
We state such a result for the special case
which is needed in the subsequent asymptotic analysis.
\ \\
{\bf Corollary 2.1.} Under the conditions H(2.1)-(2.2), we further assume
\[ b_s(x,t,s)-d \geq \lambda_0>0, ~~g_{2s}(x,s)-d_4\geq \lambda_0>0,\hspace{1cm} (x,t,s)\in Q\times [0,\infty),\]
and
\[ \int_{0}^{\infty}\int_{\Omega}b_0(x,t)dxdt<\infty.\]
Then the weak solution of the problem (2.1)-(2.6) is bounded globally in $Q$.
\ \\
{\bf Remark 2.1.} The weak solution obtained in Theorem 2.1 may grow to infinity as $t\to \infty$ if there is no additional conditions imposed on $b(x,t,S), g_2(x,t,s)$ and $d_1(x,t,s), d_4(x,t,s)$. On the other hand, if one assumes that
$g_1$ and $g_2$ grow at most in a polynomial power with respect to $s_i$, then one can verify that the conditions in \cite{FMTY2021}
hold. Consequently, a global bound in $Q$ can be deduced.
The next theorem states our main stability results for the steady-state solutions to the problem (2.1)-(2.6).
\ \\
{\bf Theorem 2.2.} Under the condition H(4.1) (see Section 4), a steady-state solution
is asymptotically stable if
\[ d_1>B_0, ~~d_4>G_0,\]
and the parameters $\beta_1, \beta_2, \gamma, \sigma$ are appropriately small, where $B_0$
and $G_0$ are constants which depend on the steady-state solution.
It turns out that the conditions in Theorem 2.2 are almost necessary in order to ensure the stability of each steady-state solution. In Section 5, we will see that when $b(x,t,s), g_1$ and $g_2(x,t,s)$ are of the form in (1.7)-(1.8),
then we have a very precise set of conditions for the model parameters to ensure the local stability or instability for each steady-state solution. To avoid repetitions, we state this result in Section 5,
since there are many specific cases we have to consider.
\section{Global Solvability and Proof of Theorem 2.1}
In this section we first derive some a priori estimates for a weak solution to the system (2.1)-(2.6), then show the existence of a unique weak solution. Finally, we establish the global boundedness and the H\"older continuity.
\setcounter{section}{3} \setcounter{local}{1} \ \\ {\bf Lemma 3.1} Under the assumptions H(2.1)-(2.2), a weak solution of the system (2.1)-(2.6) is nonnegative.
This is a well-known result since each $f_i(x,t,u_1,u_2,u_3,u_4)$ is quasi-positive for $i=1,2,3,4$, and is also locally Lipschitz continuous with respect to each $u_k$ for $k=1,2,3,4$. Interested readers may refer to \cite{BLS2002} for a detailed proof.
Next we apply the energy method to derive an a priori estimate in the space $V_2(Q_T)$.
\ \\ {\bf Lemma 3.2} Under the assumptions H(2,1)-(2.3), there exists a constant $C_1$ such that
\[\sum_{k=1}^{4}||u_k||_{V_{2}(Q_T)}\leq C_1.\]
{\bf Proof}. We multiply Eq.(1.1) by $u_1$ and integrate over $\Omega$ to obtain \begin{eqnarray*}
& & \frac{1}{2}\frac{d}{dt}\int_{\Omega}u_1^2dx+a_0\int_{\Omega}|\nabla u_1|^2dx+\int_{\Omega} g_1u_1dx +d_0\int_{\Omega}u_1^2dx \nonumber\\ & & \leq \int_{\Omega}b(x,t,u_1) u_1 dx+\sigma\int_{\Omega}u_1u_3dx\nonumber \\ & & \leq C\int_{\Omega}[1+u_1^2] dx+ C\int_{\Omega}[u_1^2+u_3^2]dx,\stepcounter{local} \end{eqnarray*} where we have used the assumption H(2.3)(a) at the second estimate.
We can perform a similar energy estimate for Eq.(1.3) to deduce \begin{eqnarray*}
& & \frac{1}{2}\frac{d}{dt}\int_{\Omega}u_3^2dx+a_0\int_{\Omega}|\nabla u_3|^2dx
\leq \gamma \int_{\Omega}u_2u_3dx \leq C\int_{\Omega}[u_2^2 + u_3^2]dx.\stepcounter{local} \end{eqnarray*}
In order to derive an estimate for $u_2$, we make use of the special structure of the system (2.1)-(2.4). To do so, we define \[ v(x,t)=u_1(x,t)+u_2(x,t), \hspace{1cm} (x,t)\in Q.\] Then it is easy to see that $v(x,t)$ satisfies \setcounter{section}{3} \setcounter{local}{1} \begin{eqnarray}
v_t-\nabla\cdot [d_2\nabla v]&=&\nabla \cdot [(d_1-d_2)\nabla u_1]+f_1(x,t,U)+f_2(x,t,U), ~~(x,t)\in Q_{T},\\
\nabla_{\nu}v(x,t)&=&0, \hspace{1cm} (x,t)\in \partial \Omega\times (0,T],\stepcounter{local}\\
v(x,0)&=&S_0(x)+I_0(x), \hspace{1cm} x\in \Omega.\stepcounter{local} \end{eqnarray}
We multiply Eq.(3.1) by $ v$ and then integrate over $\Omega$ to obtain \begin{eqnarray*}
& & \frac{1}{2}\frac{d}{dt}\int_{\Omega}v^2dx +a_0\int_{\Omega}|\nabla v|^2dx\\ & & =-\int_{\Omega} [(d_1-d_2)\nabla u_1\cdot \nabla v]dx+\int_{\Omega}v[f_1(x,t,U)+f_2(x,t,U)]dx\\ & & :=J_1+J_2. \end{eqnarray*}
A direct application of the Cauchy-Schwarz's inequality implies
\[ |J_1|\leq \varepsilon\int_{\Omega}|\nabla v|^2dx +C(\varepsilon)\int_{\Omega}|\nabla u_1|^2 dx.\] On the other hand, using the fact that \[ f_1(x,t,U)+f_2(x,t,U)=b(x,t,u_1)-d_1u_1+\sigma u_3-(d_3+\gamma)u_2,\] we readily derive that \begin{eqnarray*}
|J_2| & = & |\int_{\Omega}v[f_1(x,t,U)+f_2(x,t,U)]dx|\\ & \leq & C\int_{\Omega}[v (1+u_1+ u_3)]dx
\leq C+C\int_{\Omega}[v^2+u_1^2+u_3^2]dx. \end{eqnarray*}
Now choosing $\varepsilon=\frac{a_0}{2}$, we can readily derive from the above estimates that \begin{eqnarray*}
& & \frac{d}{dt}\int_{\Omega}v^2dx +a_0\int_{\Omega}|\nabla v|^2dx \leq C+C\int_{\Omega}[v^2+u_1^2+u_3^2]dx. \end{eqnarray*} By combining the above energy estimates for $u_1, v$ and $u_3$, we can further deduce \begin{eqnarray*}
&& \frac{d}{dt}\int_{\Omega}[u_1^2+v^2+u_3^2]dx+\int_{\Omega}[|\nabla u_1|^2+|\nabla v|^2+
|\nabla u_3|^3]dx\\ & & \leq C\int_{\Omega}[u_1^2+v_2^2+u_3^2]dx, \end{eqnarray*} then a direct application of Gronwall's inequality implies \begin{eqnarray*}
& & \sup_{0<t<T}\int_{\Omega}[u_1^2+v^2+u_3^2]dx+\int_{0}^{T}\int_{\Omega}[|\nabla u_1|^2+|\nabla v|^2+
|\nabla u_3|^3]dxdt\\ & & \leq C+C\int_{\Omega}[S_0^2+I_0^2+R_0^2]dx. \end{eqnarray*}
Noting that $v=u_1+u_2$, we can write \begin{eqnarray*}
& & \int_{\Omega}|\nabla v|^2 dx= \int_{\Omega}[|\nabla u_1|^2+|\nabla u_2|^2]dx+2\int_{\Omega} [(\nabla u_1)\cdot (\nabla u_2)]dx. \end{eqnarray*} But using the Cauchy-Schwarz's inequality, we can see \begin{eqnarray*}
\int_{\Omega}
[(\nabla u_1)\cdot (\nabla u_2)]dx & \leq & \varepsilon\int_{\Omega}|\nabla u_2|^2dx+C(\varepsilon)\int_{\Omega}|\nabla u_1|^2dx\\
& \leq & \varepsilon\int_{\Omega}|\nabla u_2|^2dx+C(\varepsilon)\int_{\Omega}[u_1^2+u_3^2]dx. \end{eqnarray*} Using the above estimates and choosing $\varepsilon$ to be sufficiently small, we can obtain \begin{eqnarray*} & & \int_{\Omega}[u_1^2+u_2^2+u_3^2]dx+
+\int\int_{Q_{T}}[|\nabla u_1|^2+|\nabla u_2|^2+|\nabla u_3|^2]dxdt\\ & &\leq C+ C\int_{\Omega}[S_0^2+I_0^2+R_0^2]dx. \end{eqnarray*}
For $u_4$, we note that \[ h_2(x,t,u_4)u_4\leq k_0(u_4^2+1).\] Then we can readily derive from Eq. (2.4) that
\[ \frac{d}{dt}\int_{\Omega}u_4^2dx +a_0\int_{\Omega}|\nabla u_4|^2dx\leq C\int_{\Omega}[u_2^2+u_4^2]dx.\] Now an integration over $(0, T)$ implies \begin{eqnarray*}
& & \sup_{0<t<T}\int_{\Omega} u_4^2dx+\int\int_{Q_{T}}|\nabla u_4|^2dxdt \leq C+C\int_{\Omega}B_0^2dx+C\int\int_{Q_{T}}u_2^2 dxdt\\ & & \leq C+C\int_{\Omega}[S_0^2+I_0^2+R_0^2+B_0^2]dx. \end{eqnarray*}
This proof of Lemma 3.2 is now completed.
Q.E.D.
In order to derive more a priori estimats, we need a crucial result about the Camapanto-John-Nirenberg-Morrey estimate for a general parabolic equation. For reader's convenience, we state the result in detail here (see Lemma 3.3 below). Consider the parabolic equation: \begin{eqnarray} & & u_t-Lu =\sum_{i=1}^{n}f_i(x,t)_{x_i}+f(x,t), \hspace{1cm} (x,t)\in Q_{T},\stepcounter{local}\\ & & u(x,t)=0 ~~\mbox{or} ~~u_{\nu}(x,t)=0, \hspace{1cm} (x,t)\in \partial \Omega \times (0,T],\stepcounter{local}\\ & & u(x,0)=u_0(x), \hspace{1cm} x\in \Omega.\stepcounter{local} \end{eqnarray} where $Lu:=(a_{ij}(x,t)u_{x_{i}})_{x_j}+b_i(x,t)u_{x_i}+c(x,t)$ is an elliptic operator. We assume there are positive constants $A_1,A_2$ and $A_3$ such that $A=(a_{ij}(x,t)_{n\times n}$ is a positive definite matrix that satisfies
\[ A_0|\xi|^2 \leq a_{ij}\xi_i\xi_j \leq A_1|\xi|^2, \hspace{1cm} \xi\in R^n,\] and
\[ \sum_{i=1}^{n}||b_i||_{L^{\infty}(Q_{T})}+||c||_{L^{\infty}(Q_{T})}\leq A_2<\infty.\]
\ \\ {\bf Lemma 3.3}. (\cite{Yin1997}) Let $u(x,t)$ be a weak solution of the parabolic equation (3.8)-(3.10). Let $u_0\in C^{\alpha}(\bar{\Omega})$ with $u_0(x)=0$ on $\partial \Omega$, and $\nabla u_0\in L^{2,\mu_0}(\Omega)$ for some $\mu_0\in (n-2,n)$. Then for any $\mu\in [0,n)$, there exists a constant $C$ such that
\[ ||\nabla u||_{L^{2, \mu}(Q_{T})}\leq C[||\nabla u_0||_{L^{2, (\mu-2)^+}(\Omega)}+||f||_{L^{2, (\mu-2)^+}(\Omega)}+\sum_{i=1}^{n}
||f_i||_{L^{2, \mu}(Q_{T})}].\] Moreover, it holds that $u\in L^{2, \mu+2}(Q_{T})$ and
\[ ||u||_{L^{2, 2+\mu}( Q_{T})}\leq C[||\nabla u_0||_{L^{2, (\mu-2)^+}(\Omega)}+||f||_{L^{2, (\mu-2)^+}(\Omega)}+\sum_{i=1}^{n}||f_i||_{L^{2, \mu}(Q_{T})}\] for a constant $C$ that depends only on $A_0, A_1,A_2, n$ and $Q_{T}$.
\ \\ {\bf Lemma 3.4} Under the assumptions H(2.1)-(2.3), the weak solution of (2.1)-(2.4) satisfies
\[\sum_{k=1}^{4}||u_k||_{\ca}\leq C(T).\]
{\bf Proof}. Let $\mu\in (0, n)$ be arbitrary. By Lemma 3.3, we have \begin{eqnarray}
& & ||\nabla u_3||_{L^{2, \mu}(Q_{T})}\leq C[||\nabla R_0||_{L^{2,(\mu-2)^+}(\Omega)}+||u_2||_{L^{2, (\mu-2)^+}(Q_{T})}+
||u_3||_{L^{2, (\mu-2)^+}(Q_{T})}].\stepcounter{local} \end{eqnarray}
On the other hand, we note that $v(x,t)=u_1(x,t)+u_2(x,t)$ satisfies the system (3.1)-(3.3), so we can apply Lemma 3.3 again to obtain \begin{eqnarray}
& & ||\nabla v||_{L^{2, \mu}(Q_{T})}\leq C[||\nabla v_0||_{L^{2,(\mu-2)^+}(\Omega)}+\sum_{i=1}^{3}||u_i||_{L^{2, (\mu-2)^+}(Q_{T})}].\stepcounter{local} \end{eqnarray}
To derive the $L^{2,\mu}$-estimate for $u_1$, we note that \[ u_{1t}-\nabla [a_1(x,t)\nabla u_1]\leq b(x,t,u_1)-d_1u_1+\sigma u_{3}=[b_s(x,t,\theta)-d_1]u_1+b(x,t,0)+\sigma u_3,\] where $\theta$ is the mean-value between $0$ and $u_1$. Using the facts that $b_s(x,t,s)$ and $b(x,t,0)$ are bounded, we can use the same calculations as in Lemmas 3.2 and 3.3 to obtain
\[ ||\nabla u_1||_{L^{2,\mu}(Q_{T})}\leq C[||\nabla S_0||_{L^{2,\mu}(\Omega)}+||u_3||_{L^{2,\mu}(Q_{T})}].\]
Now we can combine the $L^{2,\mu}(Q_T)$-estimates for $u_1, v$ and $u_3$ and note that $v=u_1+u_2$ to obtain for any $\mu\in [0, n)$ that \begin{eqnarray}
& & \sum_{i=1}^{3}||\nabla u_i||_{L^{2, \mu}(\Omega)} \leq C[||\nabla U_0||_{L^{2,(\mu-2)^+}(\Omega)}
+\sum_{i=1}^{3}||u_i||_{L^{2, (\mu-2)^+}(Q_{T})}]+C.\stepcounter{local} \end{eqnarray} Using the fact that $u_i\in V_2(Q_{T})$, we derive for any $\mu_1\in [0,2)$ that \begin{eqnarray}
\sum_{i=1}^{3}||\nabla u_i||_{L^{2, \mu_1}(Q_{T})}\leq C[\sum_{i=1}^{3}||\nabla u_{i0}||_{L^{2}(\Omega)}+1]. \stepcounter{local} \end{eqnarray}
Now we can apply the interpolation theory for the parabolic equation (2.3) (see Lemma 2.6 in \cite{Yin1997} ) to further deduce
\[ ||u_3||_{L^{2,\mu_1+2}(\Omega)}\leq C[||u_2||_{L^{2}(\Omega)}+||u_3||_{L^{2}(\Omega)}
+||\nabla u_3||_{L^{2}(Q_{T})}]+C.\]
Next we go back to the system (2.1)-(2.3) and apply the same process for $\mu_2=\mu_1+2$ to obtain \begin{eqnarray}
& & \sum_{i=1}^{3}||\nabla u_i||_{2, \mu_2, Q_{T}} \leq C[\sum_{i=1}^{3}||\nabla u_{i0}||_{L^{2,\mu_2}(\Omega)}+\sum_{i=1}^{3}||u_i||_{2, (\mu_2-2)^+,Q_{T}} +C].\stepcounter{local} \end{eqnarray}
Then after a finite number of steps, we can deduce for any $\mu\in (0,n)$ that \begin{eqnarray}
& & \sum_{i=1}^{3}||u_i||_{L^{2, \mu+2}(\Omega)}\leq C[\sum_{i=1}^{3}||u_i||_{L^2(\Omega)}+||\nabla u_{i0}||_{L^{2, (\mu-2)^2}(\Omega)}\nonumber\\
& & \leq C[\sum_{i=1}^{3}||u_i||_{L^{2}(Q_{T})}+\sum_{i=1}^{3}||\nabla u_{i0}||_{L^{2, (\mu-2)^+}]}.\stepcounter{local} \end{eqnarray}
Now we apply the interpolation theory again (see Lemma 2.6 in \cite{Yin1997}) to derive \begin{eqnarray*}
& & \sum_{i=1}^{3}||u_i||_{2, \mu_0+4, Q_{T}}
\leq C[\sum_{i=1}^{3}||u_i||_{L^{2}(Q_{T})}+\sum_{i=1}^{3}||\nabla u_{i0}||_{L^{2, \mu_0}}.\stepcounter{local} \end{eqnarray*}
But noting that $\mu_0\in (n-2, n)$, we can then obtain by Lemma 1.19 in \cite{T1987} that
\[ \sum_{i=1}^{3}||u_i||_{\ca}\leq C,\] for $\alpha =\frac{\mu_0+2-n}{2}$. The proof of Lemma 3.5 is now completed.
Q.E.D.
\ \\ {\bf Proof of Theorem 2.1}. First of all, by using the energy method we see that the weak solution of (2.1)-(2.6) must be unique since the solution is bounded and $f_k$ is locally Lipschitz continuous with respect to $u_i$ for all $k, i\in \{1,2,3,4\}$. With the a priori estimates in Lemmas 3.1-3.4, there are several approaches, such as the truncation method and Galerkin finite element method, to prove the desired result (see, e.g., \cite{BLS2002,FMTY2021, Yin2020}). Here we choose a different approach, the bootstrap argument (see \cite{YCW2017}), for the proof. Let $T\in (0,\infty)$ be any fixed number, it is easy to show that the system (2.1)-(2.6) has a unique local weak solution in $X$ in $Q_{T_0}$ for some small $T_0>0$. Let \[ T^*=sup\{T_0: \mbox{the system (2.1)-(2.6) has a unique weak solution in $Q_{T_0}$}\}.\] Suppose $T^*<T$ (otherwise, nothing is needed to prove). We note that the a priori estimates in Lemmas 3.1 and 3.4 hold for any weak solution. It follows that
\[ \lim_{t\rightarrow T*-}sup [\sum_{k=1}^{4}||u_k||_{V_2(Q_t)}+\sum_{k=1}^{4}||u_k||_{\ca}]<\infty.\] By the compactness, we know that \[ u_k(x,T^*)\in H^1(\Omega), \nabla u_k\in L^{2, (\mu-2)^+}(\Omega) ~~\mbox{for any $\mu\in (n,n+2).$}\]
Now, we use $U(x,T^*)$ as an initial value and consider the system (2.1)-(2.6) for $t\geq T*$. Then the local existence result implies that there exists a small $t_0>0$ such that the problem (2.1)-(2.6) has a unique weak solution in the interval $[T*, T^*+t_0).$ Consequently, we obtain a weak solution to the system (2.1)-(2.6) in the interval $[0,T^*+t_0)$. This is a contradiction with the definition of $T^*$, therefore we have $T^*=T$.
Q.E.D.
\ \\
Next, we prove Corollary 2.1. Assume that there exists a constant $\lambda_0>0$ such that \[ d_1(x,t,s)-b_s(x,t,s)\geq \lambda_0>0, ~~d_4(x,t,s)-g_{2s}(x,t,s)\geq \lambda_0,\hspace{1cm} (x,t,s)\in Q\times [0,\infty).\] With the above assumption, we take the integration over $\Omega$ for Eq. (2.1)-(2.3) to obtain \[ \frac{d}{dt}\int_{\Omega}(u_1+u_2+u_3)dx+\min\{(d_0,\lambda_0\}\int_{\Omega}(u_1+u_2+u_3)dx\leq \int_{\Omega}b(x,t,0)dx.\] Then it is easy to see \[ \sup_{t\geq 0}\int_{\Omega}(u_1+u_2+u_3)dx\leq C.\]
Now we derive a uniform estimate in $L^2(Q)$. By using the energy estimate for Eq.(2.1), we can see that
\[ \frac{d}{dt}\int_{\Omega}u_1^2dx+\int_{\Omega}|\nabla u_1|^2dx \leq C[\int_{\Omega}b(x,t,0)^2dx+C\int_{\Omega}u_3^2dx.]\] For $v(x,t):=u_1(x,t)+u_2(x,t)$, we can derive from Eq.(3.1)-(3.3) that \begin{eqnarray*}
& & \frac{d}{dt}\int_{\Omega}v^2dx+\int_{\Omega}|\nabla v|^2dx
\leq C\int_{\Omega}|\nabla u_1|^2dx+C\int_{\Omega}[(b(x,t,0)^2+u_1^2+u_3^2]dx\\ & & \leq C[\int_{\Omega}(b(x,t,0)^2+u_1^2+u_3^2) dx]. \end{eqnarray*} where we have used the estimate of $u_1$ at the second estimate.
Again, we can use the energy estimate for Eq.(2.3) to obtain
\[ \frac{d}{dt}\int_{\Omega}u_3^2dx+\int_{\Omega}|\nabla u_3|^2dx\leq C\int_{\Omega}u_2^2dx.\] But we know from the Gagliardo-Nirenberg estimate for $p=q=2, s=1, \theta=\frac{n}{n-2}$ and $\varepsilon>0$,
\[ \int_{\Omega} u^2dx\leq \varepsilon \int_{\Omega}|\nabla u|^2 dx+C(\varepsilon) ||u||_{L^1(\Omega)}, \]
then using the uniformly boundedness of $L^1(\Omega)$-norms of $u_1, u_2, u_3$, we get for sufficiently small $\varepsilon$, \begin{eqnarray*}
& & \sup_{t\geq 0}\int_{\Omega}[u_1^2+u_2^2+u_3^2]dx+\int_{0}^{t}\int_{\Omega}[|\nabla u_1|^2+|\nabla u_2|^2+|\nabla u_3|^2dx]\\ & & \leq C_1+C_2\int_{0}^{t}\int_{\Omega}b(x,t,0)^2dxdt \leq C_3\,. \end{eqnarray*}
Next we use the iteration method again as in the proof of Theorem 2.1. From Eq.(3.2) for $v$ and $u_3$, we deduce, respectively,
\[ ||\nabla v||_{2, \mu}\leq C+C||u_1||_{2, \mu}+C||u_3||_{2,\mu}\] and
\[||\nabla u_3||_{2,\mu}\leq C+C||u_2||_{2, \mu}. \]
For $u_1$, we see by noting that $g_1\geq 0$, \[ u_{1t}-\nabla[a_1(x,t)\nabla u_1]\leq [b_0(x,t)-d_1]u_1+\sigma u_3.\] As $u_1\geq 0$ in $Q$, we can follow the same argument as in \cite{Yin1997} to obtain for $\mu\in (n-2, n)$,
\[ ||\nabla u_1||_{2,\mu}\leq C+C[||u_1||_{2, \mu}+||u_3||_{2, \mu}].\]
As $u_1, v, u_3$ are uniformly bounded in $L^2(Q)$, the interpolation for $v$ and $u_3$ with $\mu=0$ yields that
\[ ||v||_{2,2}+||u_3||_{2, 2}\leq C.\]
Hence, we can obtain the $L^{2, \mu}(Q)$-estimate for $\nabla u_1$ with $\mu=2$:
\[ ||\nabla u_1||_{2,2}\leq C+ C[||u_1||_{2, 2}+||u_2||_{2, 2}],\] which is uniformly bounded.
We can now go back to the equations for $v$ and $u_3$ with $\mu=2$ to obtain
\[ ||v||_{2,4}+||u_3||_{2, 4}\leq C[||u_1||_{2,2}+||u_2||_{2,2}+||u_3||_{2,2}].\]
By continuing the above iteration process, after a finite number of steps, we obtain for $\alpha=\frac{\mu_0-n}{2}$ that
\[ ||v||_{\ca}+||u_3||_{\ca}\leq C.\]
Consequently, we get
\[ ||u_1||_{L^{\infty}(Q)}\leq C.\]
Once we know that $u_2$ is uniformly bounded, then from Eq.(2.4), we can apply the maximum principle to obtain
\[\sup_{t\geq 0}||u_4||_{L^{\infty}(\Omega)}\leq C.\]
With the a priori bound for each $u_i$, we can extend the weak solution in $Q_{T}$ to $Q$.
Q.E.D.
\section{Linear Stability Analysis}
To illustrate the main idea, we assume that $b$ and $g_2$ depend only on $x$ and $s$. We also focus on the following model cases: \[ b(x,t,s)=b_0(x)s(1-\frac{s}{k_1}), ~~g_1=\beta_1u_1u_2+\beta_2 \frac{u_1u_4}{u_4+k_2}, ~~g_2=g_0(x)s(1-\frac{s}{k_3}).\] Moreover, we assume that all parameters $\sigma, \gamma, \beta_1, \beta_2, d_i, k_i$ are positive constants. The general case can be carried out similarly as long as the functions are differentiable.
Consider the steady-state problem in $\Omega$: \setcounter{section}{4} \setcounter{local}{1} \begin{eqnarray}
-\nabla\cdot [a_1(x)\nabla u_1] & = & b(x,u_1)-g_1(x,u_1,u_2,u_4)-d_1u_1+\sigma u_3,\\
-\nabla\cdot [a_2(x)\nabla u_2] & = & g_1(x,u_1,u_2,u_4)-(d_2+\gamma)u_2,\stepcounter{local} \\
-\nabla\cdot [a_3(x)\nabla u_3] & = & \gamma u_2-(d_3+\sigma)u_3,\stepcounter{local} \\
-\nabla\cdot [a_4(x)\nabla u_4] & = & \xi u_2 +g_2(x,u_4)-d_4 u_4\stepcounter{local}
\end{eqnarray}
subject to the boundary condition
\begin{eqnarray}
\partial_{\nu}U(x) =0, \hspace{1cm} x\in \partial \Omega,\stepcounter{local}
\end{eqnarray}
where $U(x)=(u_1(x), u_2(x), u_3(x), u_4(x))$.
It is clear that there is a trivial solution $U(x)=(0,0,0,0)$ if $b(x,0)=g_1(x,0)=g_2(x,0)=0.$ But we are interested in nontrivial solutions, and will make the following assumptions.
\ \\
H(4.1). (a) $0<a_0\leq a_i(x)\leq A_0$ on $\Omega$; \\
(b) $b_0(x)\geq b_1>0$ and $g_0(x)\geq g_1>0$, and both are bounded.
\ \\
{\bf Lemma 4.1}. Under the assumptions H(4.1), the elliptic system (4.1)-(4.5) has at least one nonnegative weak solution
$U(x)\in W^{1,2}(\Omega)$. Moreover, the weak solution is H\"older continuous in $\bar{\Omega}$ for any space dimension.\\
{\bf Proof.} Since the argument is very similar to the case for a parabolic system, we only sketch the proof.
The key step is to derive an a priori estimate in H\"older space.
As a first step, we know that a solution of (4.1)-(4.5) must be nonnegative since every right-hand side of (4.1) to (4.4) is quasi-positive. Next we can use the same argument as for the parabolic case to
derive $L^1$-estimate for $u_i(x)\geq 0, i=1,2,3,4$ on $\Omega$.
Indeed, by direct integration we have
\[ \int_{\Omega}[d_2u_2+d_3u_3]dx+\int_{\Omega}b_0(x) u_1^2dx=\int_{\Omega}(b_0-d_1)u_1dx.\]
Then an application of the Cauchy-Schwarz's inequality yields
\[ \int_{\Omega}[u_1^2+d_2u_2+d_3u_3]dx\leq C.\]
On the other hand, we obtain from Eq.(4.1) that
\[ g_0\int_{\Omega}u_4^2dx \leq \xi\int_{\Omega}u_2dx+g_0\int_{\Omega}(g_0-d_4)u_4dx\leq C+\frac{g_0}{2}\int_{\Omega}u_4^2dx,\]
which implies
\[ \int_{\Omega}u_4^2dx \leq C.\]
Next step is to derive the $L^2(\Omega)$-estimate for $u_2$ and $u_3$. The idea is very much similar to the case for a parabolic system.
The energy estimate for Eq.(4.1) yields that, for any $\varepsilon>0$,
\[ \int_{\Omega}|\nabla u_1|^2dx+\int_{\Omega}u_{1}^3dx\leq C(\varepsilon)+\varepsilon\int_{\Omega}u_3^2dx.\]
It is easy to see that, by adding up Eq.(4.1) and Eq.(4.2), $v(x):=u_1(x)+u_2(x)$ satisfies that
\[ -\nabla[a_2(x) \nabla v]=\nabla[(a_1(x)-a_2(x))\nabla u_1]+b(x,u_1)-d_1u_1-(d_2+\gamma)u_2+\sigma u_3.\] Then we can get by the energy estimate that
\[ \int_{\Omega}|\nabla v|^2dx+\int_{\Omega} v^2dx \leq C(\varepsilon)+2\varepsilon\int_{\Omega}u_3^2dx.\]
From Eq.(4.3) we have by using Cauchy-Schwarz's inequality that
\begin{eqnarray*}
& & a_0\int_{\Omega}|\nabla u_3|^2dx+(d_3+\sigma)\int_{\Omega}u_3^2dx\\
& & \leq \gamma \int_{\Omega}u_2u_3dx
\leq \frac{d_3+\sigma}{2}\int_{\Omega} u_3^2dx+\frac{\gamma}{2(d_3+\sigma)}\int_{\Omega} u_2^2dx,
\end{eqnarray*}
which implies
\[ \int_{\Omega}|\nabla u_3|^2dx+\int_{\Omega}u_3^2dx\leq C\int_{\Omega}u_2^2dx.\]
Now we can combine the above estimates for $u_1, v$ and $u_3$ and choose $\varepsilon$ sufficiently small to conclude
\begin{eqnarray}
\sum_{i=1}^{4}||\nabla u_i||_{L^{2}(\Omega)}+\sum_{i=1}^{4}\int_{\Omega}u_i^2dx\leq C. \stepcounter{local}
\end{eqnarray}
To derive a further a priori estimate, we use the Campanato estimate for elliptic equations (\cite{T1987} ) to obtain that $u_i\in C^{\alpha}(\bar{\Omega})$ and
\[ \sum_{i=1}^{4}||u_i||_{ C^{\alpha}(\bar{\Omega})}\leq C.\]
With the above a priori estimates, we can use the Schauder's fixed-point theorem (\cite{GT1987}) to obtain the existence of a weak solution for the system (4.1)-(4.5) and the weak solution is in the space $W^{1,2}(\Omega)\bigcap C^{\alpha}(\bar{\Omega})$. We skip this step here.
Q.E.D.
\ \\
{\bf Remark 4.1} The uniqueness is not expected in general since one can see that there are many nontrivial constant solutions when
$b_1(x,s), g_1, g_2$ have the special forms as stated in the introduction.
Next, we shall consider the steady-state solutions to the system (4.1)-(4.5). Let $Z^*(x)=(u_1^*(x),u_2^*(x),u_3^*(x),u_4^*(x))$ be such a steady-state solution. For $\varepsilon>0$, we consider a small perturbation near $Z^*(x)$ and set \[ Z(x,t)=Z^*(x)+\varepsilon Z_1(x,t), \hspace{1cm} (x,t)\in Q,\] where $Z_1(x,t)=(U_1(x,t), U_2(x,t), U_3(x,t), U_4(x,t))$, with $U_i(x,t)=u_i(x,t)-u_i^*(x)$ for $i=1,2,3,4$.
A direct calculation shows that $Z_1$ satisfies the following linear system: \setcounter{section}{4} \setcounter{local}{6} \begin{eqnarray}
& & U_{1t}-\nabla\cdot [a_1\nabla U_1]=F_1(Z_1), \hspace{1cm} (x,t)\in Q,\\
& & U_{2t}-\nabla\cdot [a_2\nabla U_2]=F_2(Z_1), \hspace{1cm} (x,t)\in Q,\stepcounter{local} \\ & & U_{3t}-\nabla\cdot [a_3\nabla U_3]=F_3(Z_1), \hspace{1cm} (x,t)\in Q,\stepcounter{local}\\
& & U_{4t}-\nabla\cdot [a_4\nabla U_4]=F_4(Z_1), \hspace{1cm} (x,t)\in Q,\stepcounter{local}
\end{eqnarray}
subject to the initial and boundary conditions:
\begin{eqnarray}
& & Z_1(x,0)=Z_1(x,0), \hspace{1cm} x\in \Omega, \stepcounter{local} \\
& & \nabla_{\nu}Z_1(x,t)=0, \hspace{1cm} (x,t)\in \partial \Omega\times (0,\infty),\stepcounter{local}
\end{eqnarray} where the right-hand sides of the system (4.6)-(4.9) are given by \begin{eqnarray*} F_1(Z_1) & = & [b_s(x,u_1^*)-\beta_1u_2^*-\beta_2h_1(u_4^*)-d_1]U_1-\beta_1u_1^*U_2+\sigma U_3-(\beta_2u_1^*h_1^{'}(u_4^*)U_4,\\ F_2(Z_1) & = & (\beta_1u_2^*+\beta_2 h_1(u_2^*0)U_1+[\beta_1u_1^*-(d_2+\gamma)]U_2+\beta_2u_1^*h_1^{'}(u_4^*)U_4,\\ F_3(Z_1) & = & \gamma U_2-(d_3+\sigma)U_3,\\ F_4(Z_1) & = & \xi U_2-h_{2s}(x,u_4^*)U_4. \end{eqnarray*}
\noindent {\bf Theorem 4.1} Under the assumptions H(4.1), the steady-state solution $Z^*(x)$ to the system (4.1)-(4.5) is asymptotically stable if the following conditions hold: \[ d_1-B_0>0, ~~d_4-G_0>0,\] and $\beta_1$ is suitably small, where $B_0$ and $G_0$ are given by
\[ B_0=\max_{x\in \Omega}|b_s(x,u_1^*)|, ~~G_0=\max_{\Omega}|h_{2s}(x,u_4^*)|.\] {\bf Proof}. For any positive integer $k$, we multiply Eq.(4.1) by $U_1^k$ and integrate over $\Omega$ to obtain \begin{eqnarray*} & & \frac{1}{k+1}\frac{d}{dt}\int_{\Omega}U_1^{k+1}dx+\frac{4a_0}{(k+1)^2}\int_{\Omega}
|\nabla U_1^{\frac{k+1}{2}}|^2dx\\ & & +\int_{\Omega}[d_1+\beta_1u_2^*+\beta_2 h_1(u_4^*)-b_s(x,u_1^*)]U_1^{k+1}dx\\
& & \leq |J|, \end{eqnarray*} where $J$ is given by \[ J=-\beta_1\int_{\Omega}u_1^*U_2U_1^k dx +\sigma\int_{\Omega}U_3 U_1^k dx -\beta_2\int_{\Omega}u_1^*h_1^{'}(u_4^{*}) U_4U_1^k dx:=J_1+J_2+J_3.\] Let $U_0=\max_{\Omega}u_1^*(x)$, then we can use the Young's inequality to readily get \begin{eqnarray*}
|J_1|&\leq& \beta_1U_0\int_{\Omega}\left[\frac{ k}{k+1}U_1^{k+1}+\frac{1}{(k+1)} U_2^{k+1}\right] dx,\\
|J_2| & \leq & \sigma\int_{\Omega}\left[\frac{ k}{k+1}U_1^{k+1}+\frac{1}{(k+1)} U_3^{k+1}\right] dx,\\
|J_3| & \leq & \beta_2U_0G_0\int_{\Omega}\left[\frac{ k}{k+1}U_1^{k+1}+\frac{1}{(k+1)} U_4^{k+1}\right] dx. \end{eqnarray*}
Now we can easily see for sufficiently small $\sigma, \beta_1, \beta_2 $ that \begin{eqnarray*} & & \frac{1}{k+1}\frac{d}{dt}\int_{\Omega}U_1^{k+1}dx+\frac{4a_0}{(k+1)^2}\int_{\Omega}
|\nabla U_1^{\frac{k+1}{2}}|^2dx\\ & & +[d_1+\beta_1u_2^*+\beta_2 h_1(u_4^*)-b_s(x,u_1^*)]\int_{\Omega}U_1^{k+1}dx\\ & & \leq \frac{C}{(k+1)}\int_{\Omega}\left[U_2^{k+1}+U_3^{k+1}+U_4^{k+1}\right]dx. \end{eqnarray*}
We can apply the same argument above for $U_2,U_3, U_4$ from Eq.(4.2), Eq.(4.3) and Eq.(4.4), respectively, to obtain \begin{eqnarray*} & & \frac{1}{k+1}\frac{d}{dt}\int_{\Omega}U_2^{k+1}dx+\frac{4a_0}{(k+1)^2}\int_{\Omega}
|\nabla U_2^{\frac{k+1}{2}}|^2dx+(d_2+\gamma-\beta_1U_0)\int_{\Omega}U_2^{k+1} dx\\ & & \leq \frac{C}{(k+1)}\int_{\Omega}\left[U_1^{k+1}+U_4^{k+1}\right]dx;\\ & & \frac{1}{k+1}\frac{d}{dt}\int_{\Omega}U_3^{k+1}dx+\frac{4a_0}{(k+1)^2}\int_{\Omega}
|\nabla U_3^{\frac{k+1}{2}}|^2dx+(d_3+\sigma)\int_{\Omega}U_3^{k+1} dx\\ & & \leq \frac{C}{(k+1)}\int_{\Omega}U_2^{k+1} dx;\\ & & \frac{1}{k+1}\frac{d}{dt}\int_{\Omega}U_4^{k+1}dx+\frac{4a_0}{(k+1)^2}\int_{\Omega}
|\nabla U_4^{\frac{k+1}{2}}|^2dx+(d_4-G_0)\int_{\Omega}U_4^{k+1} dx\\ & & \leq \frac{C}{(k+1)}\int_{\Omega}U_2^{k+1}dx. \end{eqnarray*}
We now look at the quantity \[ Y(t)=\int_{\Omega}\left[ U_1^{k+1}+U_2^{k+1}+U_3^{k+1}+U_4^{k+1}\right]dx.\] Noting from the assumption H(4.1) that there exists a small number, denoted by $\beta_0$, such that \[d_1-B_0\geq \beta_0, ~~d_2+\gamma-\beta_1U_0\geq \beta_0, ~~d_3+\sigma>\beta_0, ~~d_4-G_0\geq \beta_0, \] we can add up the above estimates for $U_i^{k+1}$ to derive for sufficiently large $k$ that \[ \frac{1}{k+1}Y'(t)+\beta_0Y(t)\leq 0.\]
This readily implies \[ Y(t)\leq C (k+1)Y(0).\]
Taking the $k^{th}$-root on both sides, we obtain as $k\rightarrow \infty$ that
\[\sum_{i=1}^{4}\sup_{0<t<\infty}|U_i|_{L^{\infty}(\Omega)}\leq \sup_{\Omega}|Z_1(x,0)|_{L^{\infty}(\Omega)}.\] This implies that the solution $Z_1(x,t)$ is asymptotically stable near the steady-state solution $Z^*(x)$.
Q.E.D.
\section{Further Stability Analysis}
\setcounter{section}{5} \setcounter{local}{1} In this section we investigate the stability of constant steady-state solutions corresponding to the system (1.1)-(1.4). To illustrate the method and physical meaning, we further assume that the diffusion coefficients and the death rate are constants:
\ \\ {\bf H(5.1)}. (a) Let $a_i$ and $d_i$ be positive constants, and \[ a_0=min\{a_1, a_2, a_3, a_4\}, ~~d_0=min \{d_1, d_2, d_3, d_4\}.\]
(b) Functions $b$, $h_1$ and $h_2$ are of the following forms for two constants $b_0$ and $g_0$: \[ b(x,t,s)=b_0s(1-\frac{s}{k_{1}}), ~~h_1(s)=\frac{s}{s+k_2}, ~~h_2(s)=g_0s(1-\frac{s}{k_3}).\]
Consider the corresponding steady-state system in $\Omega$: \setcounter{section}{5} \setcounter{local}{1} \begin{eqnarray}
-\nabla\cdot [a_1(x)\nabla u_1] & = & b(x,u_1)-\beta_1u_1u_2-\beta_2u_1\cdot h_1(u_4)-d_1u_1+\sigma u_3,\\
-\nabla\cdot [a_2(x)\nabla u_2] & = & \beta_1u_1u_2+\beta_2u_1 \cdot h_1(u_4)-(d_2+\gamma)u_2,\stepcounter{local} \\
-\nabla\cdot [a_3(x)\nabla u_3] & = & \gamma u_2-(d_3+\sigma)u_3,\stepcounter{local} \\
-\nabla\cdot [a_4(x)\nabla u_4] & = & \xi u_2 +h_2(x,u_4)-d_4 u_4\stepcounter{local}
\end{eqnarray}
subject to the boundary condition
\begin{eqnarray}
\partial_{\nu}U(x) =0, \hspace{1cm} x\in \partial \Omega,\stepcounter{local}
\end{eqnarray}
where $U(x)=(u_1(x), u_2(x), u_3(x), u_4(x))$.
We can easily derive from (5.1) to (5.4) that \[ (d_1-b_0)\int_{\Omega}u_1dx+d_2\int_{\Omega}u_2dx+d_3\int_{\Omega}u_3dx+\frac{b_0}{K_1}\int_{\Omega}u_1^2dx=0.\] \[(d_4-g_0)\int_{\Omega}u_4dx+\frac{g_0}{K_2}\int_{\Omega}u_4^2dx=\xi\int_{\Omega}u_2dx, \] from which we readily see that there exists one trivial solution, i.e., $u_1=u_2=u_3=u_4=0$ if $b_0\leq d_1$ and $g_0\leq d_4$.
On the other hands, we can also see that there are two sets of steady-state solutions. The first set of constant solutions requires $b_0> d_1$ and $g_0> d_4$: \begin{eqnarray*} & & Z_1=(0,0,0,0); ~~Z_2=(\frac{K_1(b_0-d_1)}{b_{0}}, 0,0, 0); ~~Z_3=(0,0,0,\frac{K_2(g_0-d_4)}{g_{0}}). \end{eqnarray*} There exists another set of constant solutions: \[ Z_4=\left\{(S,I,R,B): R=\frac{\gamma}{d_3+\sigma}I. \right\}, \] where $S,I$ and $B$ are the solutions of the following nonlinear system: \begin{eqnarray} & & \frac{b_0}{K_1}S^2-(b_0-d_1)S+\left(d_2+\gamma-\frac{\sigma \gamma}{d_3+\sigma}\right)I=0,\\ & & \frac{g_0}{K_2}B^2-(g_0-d_4)B-\xi I=0,\stepcounter{local}\\ & & S=\frac{(d_2+\gamma)I}{\beta_1I+\beta_2 h_1(B)}.\stepcounter{local} \end{eqnarray}
\ \\ {\bf Lemma 5.1}. The nonlinear system (5.5)-(5.7) has at least one solution if and only if the following condition holds: \[\frac{K_1(b_0-d_1)}{2b_0}>\frac{d_2+\gamma}{\beta_2}.\] \ \\ {\bf Proof}: We first derive a necessary condition which will ensure the existence of a nontrivial constant solution. By solving the quadratic equation (5.5) for $S$, we obtain \begin{eqnarray*} & & S_1=\frac{(b_0-d_1)+ \sqrt{(b_0-d_1)^2-\frac{4b_0}{K_1}[(d_2+\gamma)-\frac{\sigma \gamma}{d_3+\sigma}]I}}{\frac{2b_0}{K_1}},\\ & & S_2=\frac{(b_0-d_1)- \sqrt{(b_0-d_1)^2-\frac{4b_0}{K_1}[(d_2+\gamma)-\frac{\sigma \gamma}{d_3+\sigma}]I}}{\frac{2b_0}{K_1}}. \end{eqnarray*} Noting that \[ (d_2+\gamma)-\frac{\sigma\gamma}{d_3+\sigma}>0, \] we see that the range of $I$ must satisfy \[ 0\leq I\leq I^*:=\frac{K_1(b_0-d_1)^2}{4b_0[(d_2+\gamma)-\frac{\sigma\gamma}{d_3+\sigma}]}.\] But we can see from Eq.(5.7) that \[S=\frac{(d_2+\gamma)I}{\beta_1I+\beta_2 h_1(B)}=\frac{d_2+\gamma}{\beta_1}[1-\frac{\beta_2 h_1(B)}{\beta_1I+\beta_2h_1(B)}].\] If we consider $S$ as a function of $I$, i.e., $S=S(I)$, we get \[ S(0)=0, ~~S'(I)>0, ~~S(\infty)=\frac{d_2+\gamma}{\beta_1}.\] On the other hand, if we consider $S_1$ as a function of $I$, i.e., $S_1=S_1(I)$, then we have \[ S_1(0)=\frac{K_1(b_0-d_1)}{b_0}, ~~S_1'(I)<0.\] We readily see that \[ \min_{I\in[0,I^*]}S_1(I)=S_1(I^*)=\frac{K_1(b_0-d_1)}{2b_0}, ~~\max_{I\in [0,I^*]}S_1(I)=S_1(0)=\frac{K_1(b_0-d_1)}{b_0}.\] Consequently, $S(I)$ and $S_1(I)$ have an intersection point if and only if \[ \frac{K_1(b_0-d_1)}{2b_0}>\frac{d_2+\gamma}{\beta_2}.\] Moreover, the intersection point is unique since both $S(I)$ and $S_1(I)$ are monotone functions.
Similarly, we see for $S_2$, \[ S_2(0)=0, ~~S_2'(I)>0, ~~S_2''(I)>0.\] Hence we have \[\max_{I\in [0,I^*]}S_2(I)=\frac{K_1(b_0-d_1)}{2b_0}\] The above indicates the existence of an intersection point between $S(I)$ and $S_2(I)$ as long as \[\frac{K_1(b_0-d_1)}{2b_0}>\frac{d_2+\gamma}{\beta_2}.\] Once $I$ and $S$ are determined, one can easily solve for $B$ from Eq.(5.6): \[ B=\frac{K_2\left[(g_0-d_4)+\sqrt{(g_0-d_4)^2+\frac{4g_0\xi}{K_2}I}\,\right]}{2g_0}.\]
Q.E.D.
\ \\ {\bf Proof of Theorem 2.2}. Let $A$ be the diagonal matrix with the diffusion coefficients $a_i$. We can calculate the Jacobian matrix for the nonlinear reaction terms from system (2.1)-(2.4): \[B_1(Z)= \left( \frac{\partial f_i}{\partial u_{i}} \right)_{4\times 4}.\]
For $Z_1=(0,0,0,0)$, it is easy to see the $4 \times 4$~matrix: \[B_1(Z_1)= \left( \begin{array}{cccc} b_0-d_1 & 0 & \sigma & -\frac{\beta_2K_1(b_0-d_1)}{b_0K_2}\\ 0 &\frac{\beta_2K_1(b_0-d_1)}{b_0} -(d_2+\gamma) & 0 & \frac{\beta_2K_1(b_0-d_1)}{K_2b_0} \\ 0 & \gamma & -(d_3+\sigma) & 0\\ 0 & \xi & 0 & g_0-d_4
\end{array} \right).\] Let $0\leq \lambda_1<\lambda_2<\cdots $ be the eighenvalue of the Laplacian operator subject to the homogeneous Neumann boundary condition.
It is easy to calculate the eigenvalues of $A_j(Z_1)=DF(Z_1)-\lambda_jA$:
\[ \mu_{1j}=b_0-d_1-\lambda_j a_1, \mu_{2j}=-(d_2+\gamma)-\lambda_j a_2, \mu_{3j}=-(d_3+\sigma)-\lambda_ja_3, \mu_{4j}=
g_0-d_4-\lambda_j a_4.\]
Since $\lambda_1=0$ is the first eigenvalue and $b_0\geq d_1$ and $g_0\geq d_4$, it follows that
$Z_1=(0,0,0,0)$ is unstable unless $b_0\leq d_1, g_0 \leq d_4$.
Since $\lambda_j\geq 0$, the eigenvalues indicate that the stability of $Z_1$ is not affected by the diffusion processes. This is clear since the birth rate is greater than the death rate. The population must be positive for a long time.
For $Z_2=(\frac{K_1(b_0-d_1)}{b_{0}}, 0,0, 0)$, we can see the $4 \times 4$~matrix:
\[B_1(Z_2)= \left( \begin{array}{cccc} -(b_0-d_1) & -\frac{K_1\beta_1(b_0-d_1)}{b_0} & \sigma & -\frac{\beta_2K_1(b_0-d_1)}{b_0K_2} \\ 0 & \frac{\beta_1K_1(b_0-d_1)}{b_0}-(d_2+\gamma) & 0 & \frac{\beta_2K_1(b_0-d_1)}{b_0K_2} \\ 0 & \gamma & -(d_3+\sigma) & 0\\ 0 & \xi & 0 & g_0-d_4
\end{array} \right).\] Then we consider \[ A_j(Z_2)=DF(Z_2)-\lambda_jA,\] and see its characteristic polynomial, denoted by $P(\mu)$, is equal to \begin{eqnarray*} P(\mu)= & & (b_0-d_1-\lambda_ja_1-\mu)(d_3+\sigma+\lambda_ja_3+\mu)\\ & &\{ [\mu^2-[(g_0-d_4-\lambda_ja_4+m_0-(d_2+\gamma+\lambda_j a_2)\mu\\ & & + [m_0-(d_2+\gamma+\lambda_j a_2)][g_0-d_4-\lambda_j a_4]-\xi m_0\}. \end{eqnarray*} where \[ m_0=\frac{\beta_2K_1(b_0-d_1)}{b_0}.\] We obtain the eigenvalues \begin{eqnarray*} \mu_1 & = & -(b_0-d_1)-\lambda_ja_1,\\ \mu_2 & = & -(d_3+\sigma+\lambda_ja_3),\\ \mu_3 & = & \frac{M_1+\sqrt{M_1^2-4M_2}}{2},\\ \mu_4 & = & \frac{M_1-\sqrt{M_1^2-4M_2}}{2}, \end{eqnarray*} where \begin{eqnarray*} M_1 & = & m_0-(d_2+\gamma+\lambda_ja_2)+(g_0-d_4-\lambda_ja_4);\\ M_2 & = & [m_0-(d_2+\gamma+\lambda_ja_2)][g_0-(d_4+\lambda_ja_4)]-\xi m_0 \end{eqnarray*} It follows that $Z_2$ is locally stable if $M_1<0$ and $M_2>0$ and $Z_2$ is unstable for either $M_1>0$ or $M_2<0$ or $M_1^2-4M_2>0$ when $M_2>0$. On the other hand, we know \[ \lambda_j\rightarrow \infty ~ \mbox{as $j\rightarrow \infty$},\] and $M_1^2-4M_2>0$. Consequently, we conclude that $Z_2$ is an unstable steady-state solution.
Now we calculate $A_j(Z_3)$: \[ A_j(Z_3)=DF(Z_3)-\lambda_j A.\] For $Z_3=(0,0,0,\frac{K_2(g_0-d_4)}{g_{0}})$, we can see the $4 \times 4$~matrix:
\[B_1(Z_3)= \left( \begin{array}{cccc} (b_0-d_1) & 0 & \sigma & 0 \\ \frac{\beta_2(g_0-d_4) }{(2g_0-d_4)} & -(d_2+\gamma) & 0 & 0 \\ 0 & \gamma & -(d_3+\sigma) & 0\\ 0 & \xi & 0 & -( g_0-d_4)
\end{array} \right).\] We know the characteristic polynomial for the matrix $A_j(Z_3)=DF(Z_3)-\mu I_{4\times 4}$ is equal to \begin{eqnarray*}
P(\mu)= & & |A_j(Z_3)|=-[(g_0-d_4+\lambda_j a_4)+\mu]P_0(\mu), \end{eqnarray*} where \[ P_0(\mu)=\ [(b_0-d_1-\lambda_j a_1-\mu)(d_3+\sigma+\lambda_ja_3+\mu)(d_2+\gamma+\lambda_ja_2+\mu)+\frac{\sigma \gamma \beta_2(g_0-d_4)}{2g_0-d_4}.\]
Hence, the first eigenvalue is equal to \begin{eqnarray*} \mu_1 & = & -(g_0-d_1+\lambda_j a_4), \end{eqnarray*} To see the rest of eigenvalues of $P(\mu)$, we use a lemma from Yin-Chen-Wang \cite{YCW2017}. \ \\ {\bf Lemma 5.2} Let $p>0$, $q$ and $h$ be constants, and \[ P_0(\mu)=\mu^3+p\mu^2+q\mu +h=0.\] Then it holds that \\ (a) If $h<0$, there exists a positive root;\\ (b) If $0<h<pq$, all roots have negative real parts;\\ (c) If $pq<h$, there is a root with positive real part;\\ (d) If $pq=h$, the roots are $\mu_1=-p, \mu_2=\sqrt{-q}, \mu_3=-\sqrt{-q}.$
Let \[ P_0(\mu)=\mu^3+p\mu^2+q\mu+h,\] with its coefficients given by \begin{eqnarray*} p & = & (d_2+\gamma+\lambda_j a_2)+(d_3+\sigma+\lambda_ja_3)-(b_0-d_1-\lambda_ja_1);\\ q & = & (d_2+\gamma+\lambda_j a_2)(d_3+\sigma+\lambda_j a_3)-(b_0-d_1-\lambda_j a_1)[(d_2+\gamma+\lambda_j a_2)+(d_3+\sigma+\lambda_ja_3)];\\ h & = & (d_1+\lambda_ja_1-b_0)(d_2+\gamma+\lambda_ja_2)(d_3+\sigma+\lambda_j a_3)-\frac{\sigma\gamma\beta_2(g_0-d_4)}{2g_0-d_4}. \end{eqnarray*}
Since $\lambda_1=0$ is one of the eigenvalues and $d_1-b_0<0, g_0-d_4>0$, we see $h<0$ from the expression of $h$, so $Z_3$ is unstable.
Finally, we study the stability of $Z_4$. Since $u_4$ always has positive solutions as long as $u_2$ is positive, it does not affect the stability of other variables. We only need to focus on the stability of $(u_1,u_2,u_3)$. Furthermore, since $\lambda_1=0$ is the first eigenvalue, the rest of eigenvalues have the same sign with $d_i$ which increases the stability of the solution. Therefore, we only need to find the conditions for the stability when $\lambda_1=0$.
It is easy to calculate the Jacobian matrix \[B_1^*= \left( \begin{array}{ccc} -L_0 & -\beta_1S_0 & \sigma \\ \beta_1I_0+\beta_2h(B_0) & -(d_2+\gamma) & 0 \\ 0 & \gamma & -(d_3+\sigma)
\end{array} \right)\]
where
\[ L_0= (d_1-b_0)+\frac{2b_0S_0}{K_1}+\beta_1I_0+\beta_2h_1(B_0).\] The characteristic polynomial of $B_1^*$ is equal to \[ P(\mu)=\mu^3+p_0\mu^2+q_0\mu+h_0=0.\] where \begin{eqnarray*} p_0 & = & L_0+(d_2+\gamma)+(d_3+\sigma)+L_0;\\ q_0 & = & (d_3+\sigma)(L_0+d_2+\gamma)+L_0(d_2+\gamma) +\beta_1S_0(\beta_1I_0+\beta_2h_1(B_0));\\ h_0 & = & (d_3+\sigma)[L_0(d_2+\gamma)+\beta_1S_0(\beta_1I_0+\beta_2h_1(B_0)] -\sigma\gamma(\beta_1 I_0+\beta_2h_1(B_0)). \end{eqnarray*} By Lemma 5.2, we can see the stability or instability of the steady-state solution precisely when parameters varies. In particular, when $L_0>0$, if $\sigma, \gamma, \beta_1$ and $\beta_2$ are sufficiently small, we see the condition $0<h_0<p_0q_0$ holds. Consequently, the steady-state solution $(S_0,I_0,R_0)$ is stable. This result confirms the result of Theorem 2.2 about the stability analysis of the steady-state solution.
Q.E.D.
\section{Conclusion}
In this paper we have studied a nonlinear mathematical model for an epidemic caused by cholera without life-time immunity.
The diffusion coefficients are different for each species. Moreover, these coefficients are allowed to be dependent upon the concentration as well as the space location and time. The resulting model system is strongly coupled. We established the global well-posedness for the coupled reaction-diffusion system under some very mild conditions on the given data. Moreover,
we have analyzed the linear stability for the steady-state solutions and proved that there is a turing phenomenon when the diffusion coefficients are different. This result indicates that there are some fundamental differences between the ODE model and the corresponding PDE model.
These results show that the mathematical model is well-defined and can be used by other researchers to conduct the field study.
The theoretical results obtained in this paper lays a solid foundation for other scientists in related fields to further study
more constructive qualitative properties of the solutions. The study will provide scientists a deeper understanding of the dynamics of the interaction between bacteria and susceptible, infected and recovered species. We have used many ideas and techniques from the elliptic and parabolic equations, particularly, the energy method and Sobolev's inequalities.
There are some open questions that remain to be answered, and further studies are needed.
\ \\
{\bf Acknowledgements}.
This work was motivated by some open questions raised by Professor K. Yamazaki from Texas Tech University and Professor Jin Wang from University of Tennessee at Chattanooga in WSU biological seminar series. The authors would like to thank them for some helpful discussions about the model.
The work of the second author was substantially supported by Hong Kong RGC General Research Fund (projects 14306921 and 14306719).
\end{document} |
\begin{document}
\title[Highly Degenerate harmonic mean curvature flow]{ Highly Degenerate\\ harmonic mean curvature flow} \author[M.C. Caputo]{M.C. Caputo$^{*}$} \address{Department of Mathematics, University of Texas at Austin, TX} \email{caputo@math.utexas.edu} \thanks{$*$: Partially supported by the NSF grant DMS-03-54639 } \author[P. Daskalopoulos]{P. Daskalopoulos$^{**}$} \address{Department of Mathematics, Columbia University, NY} \email{pdaskalo@math.columbia.edu} \thanks{$**$: Partially supported by the NSF grants DMS-01-02252, DMS-03-54639 and the EPSRC in the UK }
\begin{abstract} We study the evolution of a weakly convex surface $\Sigma_0$ in ${\mathbb R}^3$ with flat sides by the Harmonic Mean Curvature flow. We establish the short time existence as well as the optimal regularity of the surface and we show that the boundaries of the flat sides evolve by the curve shortening flow. It follows from our results that a weakly convex surface with flat sides of class $C^{k,\gamma}$, for some $k\in \mathbb{N}$ and $0 < \gamma \leq 1$, remains in the same class under the flow. This distinguishes this flow from other, previously studied, degenerate parabolic equations, including the porous medium equation and the Gauss curvature flow with flat sides, where the regularity of the solution for $t >0$ does not depend on the regularity of the initial data. \end{abstract}
\maketitle
\section{Introduction}
We consider the motion of a compact, weakly convex two-dimensional surface $\Sigma_0$ in space ${\mathbb R}^3$ under the {\it harmonic mean curvature flow} (HMCF) \begin{equation} \frac {\partial P}{\partial t} = \frac{K}{H}\, N\tag{HMCF} \end{equation} where each point $P$ of $\Sigma_0$ moves in the inward normal direction $N$ with velocity equal to the {\it harmonic mean curvature} of the surface, namely the harmonic mean $$\frac KH= \frac {\lambda_1\,\lambda_2}{\lambda_1 + \lambda_2}$$ of the two principal curvatures $\lambda_1$, $\lambda_2$ of the surface.
The existence of solutions to the HMCF with strictly convex smooth initial data was first shown by Andrews in~\cite{A1}. He also showed that, under the HMCF, strictly convex, smooth surfaces converge to round points in finite time. In~\cite{Di}, Di\"eter established
the short time existence of solutions to the HMCF with weakly convex smooth initial data and mean curvature $H >0$. More precisely, Di\"eter showed that if at time $t=0$ the surface $\Sigma_0$ satisfies $K\geq 0$ and $H>0$, then there exists a unique strictly convex smooth solution $\Sigma_t$ of the HMCF defined on $0<t< \tau$, for some $\tau>0$. By the results of Andrews, this solution exists up to the time where its enclosed volume becomes zero. However, the highly degenerate case where the initial data is weakly convex and both $K$ and $H$ vanish in a region is not studied in~\cite{Di} .
We will consider in this work the evolution of a surface $\Sigma_0$ with flat sides by the HMCF. The parabolic equation describing the motion of the surface becomes degenerate at points where both curvatures $K$ and $H$ become zero. Our main objective is to study the solvability and optimal regularity of the evolving surface for $t >0$, by viewing the flow as a {\it free-boundary} problem. It will be shown that a surface $\Sigma_0$ of class $C^{k,\gamma}$ with $k\in\mathbb{N}$ and $0<\gamma \leq 1$ at $t=0$, will remain in the same class for $t >0$. In addition, we will show that the strictly convex parts of the surface become instantly $C^\infty$ smooth up to the flat sides and the boundaries of the flat sides evolve by the curve shortening flow.
For simplicity we will assume that the surface $\Sigma_0$ has only one flat side, namely $\Sigma =\Sigma_1 \cup \Sigma_2$, with $\Sigma_1$ flat and $\Sigma_2$ strictly convex (both principal curvatures are strictly positive). We may also assume that $\Sigma_1$ lies on the $z=0$ plane and that $\Sigma_2$ lies above this plane since the equation is invariant under rotation and translation. Therefore, the lower part of the surface $\Sigma_0$ can be written as the graph of a function $$z=h(x,y)$$ over a compact domain $\varOmega \subset {{\mathbb R}}^2$ containing the initial flat side $\Sigma_1$. Let $\Gamma$ denote the boundary of the flat side $\Sigma_1$. We define $ g=h^p$, for some $0<p<1 $. Our main assumption on the initial surface $\Sigma_0$ is that it satisfies the following {\it non-degeneracy condition~$(\star)$}:
\begin{equation}
| Dg(P) | \geq \lambda
\qquad \text{and} \qquad g_{\tau\tau}(P)\geq \lambda, \qquad \mbox{for all}\,\,\, P\in\,\Gamma \tag{$\star$} \end{equation} \noindent for some number $\lambda >0$. Here $\tau$ denotes the tangential direction to the level sets of $g$ and $g_{\tau\tau}$ denotes the second order derivative in this direction.
\noindent Under the above conditions, our main results show that for $t\in (0,T)$: \begin{enumerate}
\item The HMCF admits a solution $\Sigma_t=(\Sigma_1)_t\cup (\Sigma_2)_t$ of class $C^{k,\gamma}$, for some $k\in \mathbb{N}$ and $0<\gamma \leq 1$ depending on $p$, which is smooth up to $\Gamma_t =\partial (\Sigma_1)_t $. \item $(\Sigma_1)_t$ is flat and its boundary $\Gamma_t$ evolves by the curve shortening flow. \end{enumerate}
\noindent The fact that the solution $\Sigma_t$ remains in the class $C^{k,\gamma}$ distinguishes this flow from other, previously studied, degenerate free-boundary problems (such as the Gauss curvature flow with flat sides, the porous medium equation and the evolution p-laplacian equation) in which the regularity of the solution for $t >0$ does not depend on the regularity of the initial data.
We define $\mathfrak S$ to be the class of weakly convex compact surfaces $\Sigma_0$ in ${{\mathbb R}}^3$ so that $\Sigma = \Sigma_1 \cup \Sigma_2$, where $\Sigma_1$ is a surface contained in the plane $z=0$ and $\Sigma_2$ is a strictly convex and smooth surface contained in the half-space $z\geq 0$. The main result states as follows:
\noindent {\bf Main Theorem.}\label{thm1} {\em Assume that at time $t=0$, $\Sigma_0$ is a weakly convex compact surface in ${{\mathbb R}}^3$ which belongs to the class $\mathfrak S$ so that the function $g=h^p$ defined as above is smooth up to the interface $\Gamma$ and satisfies ($\star$). Then, there exists a time $T>0$ such that the HMCF admits a solution $\Sigma_t\in \mathfrak S$ on $[0,T)$. Moreover, the function $g=h^p$, defined as above for $\Sigma_t$, is smooth up to the interface $z=0$ and satisfies~($\star$) on $(0,T]$. In particular, the interface $\Gamma_t$ between the flat side and the strictly convex side is a smooth curve for all $t$ in $0<t\leq T$ and it evolves by the curve shortening flow.}
\noindent {\it Sketch of the proof}. A standard computation shows that when $\Sigma_t$ solves the HMCF, the function $h$ evolves by the equation \begin{equation}\label{eqn-hhh}
h_t = \frac{h_{zz}h_{yy}-h_{zy}^2 }{(1+h_y^2)h_{zz}-2\,h_z h_y h_{zy}-(1+h_z^2)h_{yy}}\,\,\,\,\mbox{on}\,\,\,\,\, z>0. \end{equation}
The HMCF can be seen as a free boundary problem arising from the degeneracy near the flat side of the fully nonlinear parabolic PDE which describes the flow. We will show in section \ref{global} that, via a global change of coordinates, this free boundary problem is equivalent to an {\it initial value problem} on $\mathcal D \times [0,T]$, with $D = \{ (u,v) ; u^2+v^2 \leq 1 \}$, namely \begin{equation}\label{eqn-MMM} \left\{ \begin{array}{ll}
Mw =0 & on\,\,{\mathcal{D}}\times [0,T]\\ w=w_0 & at \,\,\,\,\,\,\,t=0 \end{array} \right. \end{equation}
\vskip 0.1 in \noindent The operator $Mw= w_t - F(t,u,v,w,Dw, D^2w)$, is a fully non-linear operator which becomes degenerate at $\partial \mathcal D$, the boundary of $\mathcal D$. We will apply the Inverse Function Theorem between appropriately defined Banach spaces to show that this problem admits a solution.
The linearization of the operator $M$ at a point $\bar w$ close to the initial data $w_0$, can be modeled, after straightening the boundary, on the degenerate equation \begin{equation}\label{eqn-hl} f_t =z^2\, a_{11} f_{zz} + 2 \, z\, a_{12} f_{zy} + a_{22}\, f_{yy} + b_1\, z\, f_z + b_2 \, f_y \end{equation} \vskip 0.05 in
\noindent on $z >0$ with no extra conditions on $f$ along the boundary $z=0$. We observe that the diffusion in the above equation is governed by the Riemannian metric $ds^2=d \bar s^2+|dt|$ where $$d \bar s^2 = {dz^2 \over z^2}+{dy^2}.$$ \vskip 0.01 in \par\noindent The distance (with respect to the singular metric $\bar s$) of an interior point ($z >0$) from the boundary ($z=0$) is hence~{\em infinite}. This fact distinguishes our problem from other, previously studied, degenerate free-boundary problems such as the degenerate Gauss curvature flow and the porous medium equation.
{\em The plan of the paper is the following:} in section \ref{local} we will introduce a local change of coordinates that fixes the free-boundary $\Gamma$ in equation \eqref{eqn-hhh}. We will compute the linearization of
equation \eqref{eqn-hhh} in this new change of coordinates, and
motivate the use of the appropriate Banach spaces $C^{2+\alpha,p}_s$ for our problem.
The detailed definition of these Banach spaces will be given in section
\ref{def:ban}, where we will also present the appropriate Schauder estimates for our problem. In section \ref{deg} we will study the fully-nonlinear degenerate equations \eqref{eqn-MMM} and establish the short time existence for such equations in the Banach spaces $C^{2+\alpha,p}_s$. The global change of coordinates and the proof of the Main Theorem will be given in sections \ref{global} and \ref{sec-main} respectively. In the last section we will establish the comparison principle for equation \eqref{eqn-hhh} and characterize our solutions as viscosity solutions.
{\bf Acknowledgments.} The authors wish to thank G. Huisken for suggesting the problem and R. Hamilton for many stimulating discussions.
\section {Local Change of Coordinates}\label{local} We will assume throughout this section that the surface $\Sigma_0$ belongs to the class $\mathfrak S$. Let $\Sigma_t$ be a solution to the HMCF on $[0,T)$, for some $T>0$ in the sense that $\Sigma_t= (\Sigma_1)_t\cup (\Sigma_2)_t$, with $(\Sigma_1)_t$ flat and $(\Sigma_2)_t$ strictly convex. Let $P_0(x_0,y_0,t_0)$ be a point on the interface $\Gamma_{t_0}$, for $t_0>0$ sufficiently small. Then, the strictly convex part of surface $(\Sigma_{2})_{t_0}$, $t<t_0$ can be expressed locally around $P_0$ as the graph of a function $z=h(x,y,t)$. Let $g$ be defined by $g=h^p$, for $0<p<1$, such that : g is smooth up to the interface and satisfies condition~($\star$). We can assume, by rotating the coordinates, that at the point $P_0$ the normal vector to $\Gamma_{t_0}$ facing outwards the flat side $\Sigma(t_0)$ is parallel to the $x$-axis, so that at $P_0$ we have $$g_x(P_0) > 0 \qquad \text{and} \qquad g_y(P_0) =0.$$ Then we solve locally around the point $P_0$ the equation $z=h(x,y,t)$ with respect to $x$. This yields to a map $x=f(z,y,t)$. The condition on $g$ expressed in terms of $f$ gives the following {\em non-degeneracy condition~($\star \star$)} : \begin{equation}
\left ( \begin{split} -z^{2-p}\,f_{zz}\,\,\, & z^{1-p}\,f_{zy}\\ z^{1-p}\,f_{zy}\,\,\, & \,\, \,\, -f_{yy} \end{split}\right ) \geq \bar \lambda \tag {$\star \star$} \end{equation} in the sense that both eigenvalues of the above matrix are bounded from below by a number $\bar \lambda >0$.
\noindent Since $f$ is the inverse of $h$ and the HMCF is invariant under rotation, the function $f$ satisfies the same equation as $h$ on $z>0$ :
\begin{equation} \label{eqn-oo}
f_t = \frac{f_{zz}f_{yy}-f_{zy}^2 }{(1+f_y^2)f_{zz}-2\,f_z f_y f_{zy}-(1+f_z^2)f_{yy}}. \end{equation} \noindent We will construct a smooth solution to this equation by using the Inverse Function Theorem. To do so, we will define the Banach space $C^{2+\alpha,p}_s$ in the next section. According to our notation, the constants $\alpha$ and $p$ indicate ``how the surface becomes flat'', while $s$ refers to the hyperbolic metric which governs the problem.
\noindent We will prove in the next sections that when ${\it f} \in C^{2+\alpha,p}_s$ and satisfies~condition~($\star \star$), then the equation~(\ref{eqn-oo}) becomes degenerate at $z=0$ implying that: \begin{eqnarray} \label{eqn-f} f_t = \frac{f_{yy}}{1+f_y^2}\,\,\,\mbox{ at the interface}\,\,\, z=0; \end{eqnarray} \noindent This is equivalent to say that the free boundary $\Gamma_t$ evolves by the {\it curve shortening flow}.
\section{The $C^{2+\alpha,p}_s$ space and Schauder estimates}\label{def:ban}
\noindent Let $\mathcal{A}$ be a compact subset of the half space $\{\,(z,y)\in {\mathbb R}^2: \,\, z\ge 0\,\}$ such that $(0,0)\in {\mathcal A}$. Then, we define: $$\begin{array}{lcl} \mathcal{A}^{\circ} & := & \{\,\,y\,\in\,{\mathbb R}\,:(0,y)\,\in\, \mathcal{A}\,\} \\ \tilde{\mathcal{A}} &: = & \{(w,y)\,\in {\mathbb R}^2: w= \ln z,\,(z,y)\,\in\,\mathcal{A},\,z \neq\, 0\}\\ Q_T &: = & {\mathcal A}\times[0,T],\,\, T>0\\
Q_T^{\circ}&:= &{\mathcal A}^{\circ}\times [0,T]\\ \tilde{Q}_T &: = & \tilde{\mathcal A}\times [0,T]. \end{array} $$
\noindent Let $0\,<\,p\,<\,1$. Given a function $f$ on $\mathcal{A}$ we define: $$\begin{array}{lcl} f^{\circ}(y) &: = & f(0,y)\\ \tilde{f}(w,y) & : = & e^{-p\,w}\,(\,f(z,y)-f^{\circ}(y)\,) \end{array}$$
\noindent with \,\,$ w=\ln z,\,\text {for}\,\,z\,>\,0.$
\noindent Analogously, given a function $f$ on $Q_T$ we define: $$\begin{array}{lcl} f^{\circ}(y,t)& : = & f(0,y,t)\\ \tilde{f}(w,y,t)& := & e^{-\,p\,w}\,(\,f(z,y,t)-f^{\circ}(y,t)). \end{array} $$
Given a subspace $\mathcal{A}$ as above, we define the {\it hyperbolic distance} $\bar s(P_1,P_2)$ between two points $P_1=(z_1,y_1)$ and $P_2=(z_2,y_2 ) $ in $\mathcal{A}$ $z_i >0$, $i=1,2$ to be:
\noindent
$$ \bar s(P_1,P_2):= \sqrt{| \ln z_1- \ln z_2|^2+|y_1-y_2|^2},\,\,\,\hbox{ if}\,\,\, 0\,<\, z_1,\,z_2\,\leq 1 $$ \noindent otherwise it is defined to be equivalent to the standard euclidean metric.
We define the {\it parabolic hyperbolic distance} between two points $\tilde P_1=(z_1,y_1,t_1)$ and $\tilde P_2=(z_2,y_2,t_2)$ with $ z_i >\,0$, $i=1,2$ to be:
$$s(\tilde P_1,\tilde P_2):= \bar s(P_1,P_2)+\sqrt{|t_1-t_2|}$$ \noindent where $P_1=(z_1,y_1),\,P_2=(z_2,y_2)$.
\vskip 0.1 in Let $0<\alpha\leq 1$. We define the H\"older space $C^{\alpha,p}_{ \bar s}(\mathcal{A}$) in terms of the above distance.
\noindent We start defining the H\"older semi-norm:
$$\|\,f\,\|_{H^\alpha_{\bar s}(\mathcal{A})}: = \sup\limits_{P_1\not= P_2\in \mathcal{A}\,\cap\,\{(x,y)\in {\mathbb R}^2:z\,>\,0\} } \frac{|\,f(P_1) -
f(P_2)\,|}{ s [P_1, P_2]^{\alpha}}\,$$ and the norm
$$\|\,f\,\|_{C^\alpha_{\bar s}(\mathcal{A})}: = \|\,f\,\|_{C^0 (\mathcal{A})}
+ \|\,f\,\|_{H^\alpha_{\bar s}(\mathcal{A})}$$
\noindent where $||\,f\,||_{C^{0}({\mathcal{A}})}:=\sup\limits_{P\in\mathcal{A}}|\,f(P)\,|$.
\label{def2} \noindent {\it We say that a function $f$ belongs to $C^{\alpha,p}_{\bar s}(\mathcal{A})$ if $f^{\circ}\in C^{\alpha}(\mathcal{A}^{\circ}) \,\, \text{and}\,\, \,\tilde{f}\in C^{\alpha}(\tilde{\mathcal{A}})$}. The norm of $f$ in the space $C^{\alpha,p}_s(\mathcal{A})$ is defined as:
$$||\,f\,||_{C^{\alpha,p}_{\bar s}(\mathcal{A})}:=||\,f^{\circ}\,||_{C^{\alpha}(\mathcal{A}^{\circ})}+||\,\tilde{f}\,||_{C^{\alpha}(\tilde{\mathcal{A}})}.$$
\noindent Moreover, we define: $||\,f\,||_{C^{0,p}(\mathcal A)}:=||\,f^{\circ}\,||_{C^{0}(\mathcal A^{\circ})}+||\,\tilde f\,||_{C^{0}(\tilde{\mathcal A})}$\,. \begin{remark} We observe that $ f(w,y)\in\,C^{\alpha}(\tilde{\mathcal{A}})$ if and only if
$f(z,y)\,\in\,C^{\alpha}_{\bar s}(\mathcal{A})$, where $w= \ln z$. \end{remark}
\hspace{1pt} {\it We say that a continuous function $f$ on $\mathcal{A}$ belongs to $C^{2+p}(\mathcal{A})$ if $f^{\circ}\,\in\,C^2(\mathcal{A}^{\circ})$ and $f$ has continuous derivatives
$$ f_z,\,\, f_y,\,\, f_{zz},\,\, f_{zy},\,\, f_{yy}$$
\noindent in the interior of $\mathcal{A}$, such that
\noindent $$z^{-p}\,(f-f^{\circ}),\,\,z^{1-p}\,f_z,\,z^{-p}\,(\,f_y-f_y^{\circ}\,),\,
z^{2-p}\,f_{zz},\, z^{1-p}\,f_{zy},\,z^{-p}\,(\,f_{yy}-f_{yy}^{\circ}\,)$$ \noindent extend continuously up to the boundary.} The norm of $f$ in the space $C^{2+p}(\mathcal A)$ is defined as follows:
$$||\,f\,||_{C^{2+p}(\mathcal{A})}:=||\displaystyle\sum_{m=0}^2\,D^m_y\,f^{\circ}\,||_{C^{0}(\mathcal{A}^{\circ})}+
\displaystyle\sum_{m+n=0}^2||\,D_z^m\,D_y^n\,\tilde{f}\,||_{C^0(\tilde{\mathcal{A}})}$$
\noindent {\it Given $f\in C^{2+p}(\mathcal{A})$, we say that $f$ belongs to $C^{2+\alpha,p}_{\bar s}(\mathcal{A})$ if $$f^{\circ}\in C^{2+\alpha}(\mathcal{A}^{\circ})\,\,\text{and}\,\, z\, f_z, f_y,\,z^2\,f_{zz}, z\, f_{zy},f_{yy}$$ \vskip 0.1 in \noindent extend continuously up to the boundary, and the extensions are H\"older continuous on $\mathcal{A}$ of class $C_{\bar s}^{{\alpha,p}}(\mathcal{A})$. The norm of $f$ in the space $C^{2+\alpha,p}_{\bar s}(\mathcal{A})$ is defined as:
$$||\,f\,||_{C^{2+\alpha,p}_{ \bar s}(\mathcal{A})}:=
||\,f^{\circ}\,||_{C^{2+\alpha}(\mathcal{A}^{\circ})}+\displaystyle\sum_{m+n=0}^2||\,z^{m}\, D_z^m\,D_y^n \,f\,||_{C^{\alpha,p}_{\bar s}(\mathcal{A})}$$} \begin{remark} It follows by definition that $\tilde{f}_w=-p\,\tilde{f}+z^{1-p}\,f_z$ and $ \tilde{f}_{ww}=-p\,\tilde{f}_z+(1-p)\,z^{2-p}\,f_{zz}$, which implies that:
$$\displaystyle\sum_{m+n=0}^2||\,z^{m}\,D_z^m\,D_y^n
\,f\,||_{C^{\alpha,p}_{ \bar s}(\mathcal{A})}\,\simeq \,||\,\tilde{f}\,||_{ C^{2+\alpha}(\tilde{\mathcal{A}})}.$$ \end{remark}
\begin{remark} The function $f\in C^{2+\alpha,p}_{\bar s}(\mathcal{A})\,$ if and only if $\,f^{\circ}\in C^{2+\alpha}(\mathcal{A}^{\circ})\,$
and $\,\tilde{f}\in C^{2+\alpha}(\tilde{\mathcal{A}})$, therefore, the following norms are equivalent:
$$||\,f\,||_{C^{2+\alpha,p}_{\bar s}(\mathcal{A})}\simeq \,||\,f^{\circ}\,||_{ C^{2+\alpha}(\mathcal{A}^{\circ})}+\,||\,\tilde{f}\,||_{ C^{2+\alpha}(\tilde{\mathcal{A}})}.$$ \end{remark}
\noindent Let $T>0$. The definitions above can be naturally extended on the space-time domain $Q_T$ by using the parabolic distance
$d{s}^2=d\bar s^2+|dt|$. We define the space $C^{\alpha}_s({Q_T})$ to be the standard H{\"o}lder space with respect to the metric $d{s}^2$. {\it We say that a continuous function $f$ on $Q_T$ belongs to $C^{2+p}({Q_T})$ if $f$ has continuous derivatives \vskip 0.01 in
$$ f_t, f_z, f_y, f_{zz}, f_{zy}, f_{yy}$$ \vskip 0.05 in \noindent in the interior of $Q_T$ and $f^{\circ}$ has continuous derivatives that extend continuously up to the boundary and $$z^{-p}\,(f-f^{\circ}),\,\,z^{-p}\,({f}_t-f_t^{\circ}),\,\,z^{1-p} {f}_z,z^{-p}\,{f}_y,\,\, z^{2-p} {f}_{zz},\,\, z^{1-p} {f}_{zy},\,\,z^{-p}\,({f}_{yy}-f_{yy}^{\circ})$$
\noindent extend continuously up to the boundary.} The norm of $f$ in the space $C^{2+p}({Q_T})$ is defined as follows:
\centerline{$||\,f\,||_{C^{2+p}}:=||\,f^{\circ}\,||_{C^{2}}+\displaystyle\sum_{l+m+2j=0}^2||\, D_z^l\,D_y^m\,D_t^j\,\tilde f ||_{C^{\circ}}$}
\noindent
\noindent {\it The function $f$ belongs to $ C^{2+\alpha,p}_{\bar s}(Q_T)$ if $f\in C^{2+p}(Q_T)$, $$ f,f_t,z f_z, f_y\qquad \text{and}\qquad z^2 f_{zz}, z f_{zy}, f_{yy}\,\,\text {belong to}\,\,\, C_{\bar s}^{{\alpha,p}}({Q_T}).$$}
Throughout the paper $k$ will denote a positive integer. We can extend these definitions to spaces of higher order derivatives. We denote by $C^{k,p}(Q_T)$ the space of all functions $f$ whose $k$-th order derivatives $ D^i_z\,D^j_y\,D^l_t\,f $, $ i+j+2l= k$ in the interior of $Q_T$ and $z^{i}\, D^i_z\,D^j_yD^l_t\,( f-f^{\circ}\,)$, $ i+j+2l= k$ exist and belong to the space $C^{0}(Q_T)$. We define $C^{\infty,p}(Q_T)=\cap_{k}\, C^{k,p}(Q_T)$.
We denote by $C^{k+\alpha,p}_s(Q_T)$ the space of all functions $f\in C^{k,p}(Q_T)$ such that
$z^{i}\, D^i_z\,D^j_y\,D^l_t\, f$, for $ i+j+2l= k$ belong to the space $C^{\alpha,p}_s(Q_T)$. The space $C^{k+\alpha,p}_s(Q_T)$ is equipped with the norm:
$$||\,f\,||_{C^{k+\alpha,p}_{\bar s}(Q_T)} : = \sum_{i+j+2l \leq k}
||\,z^{i}\,D^i_zD^j_yD^l_t \,f\,||_{C^{\alpha,p}_{\bar s}(Q_T)}.$$
\begin{remark}\label{rmkfin} \noindent A function $f\,\in\,C^{k+\alpha,p}_s(Q_T)$ iff $f^{\circ}\,\in\,C^{k+\alpha}(Q^{\circ}_T)$ and $\tilde{f}\,\in\,C^{k+\alpha}(\tilde{Q}_T)$. Moreover,
$$||\,f\,||_{C^{k+\alpha,p}_s(Q_T)} \simeq
||\,f^{\circ}\,||_{C^{k+\alpha,[k/2]+\alpha/2}
(Q^{\circ}_T)}+||\,\tilde{f}\,||_{C^{k+\alpha,[k/2]+\alpha/2}(\tilde{Q}_T)}.$$ \end{remark}
\noindent In the next paragraph we denote by ${\mathcal S}_0$ the half space $x \geq 0$ in ${{\mathbb R}}^2$, by $\mathcal S$ the space $\mathcal S = {\mathcal S}_0 \times [0,\infty)$, and by $\mathcal S_T$ the space $\mathcal S \times [0,T]$, for $T>0$. \noindent The operator $L_k: \, C^{k+2+\alpha,p}_s(Q_T) \to C^{k+\alpha,p}_s(Q_T)$ is defined as: \begin{equation}\label{eqn:ope} L_k\,[\,f\,]:= f_t -(\,z^2 a_{11} f_{zz} + 2 \, z\, a_{12}
f_{zy} + a_{22} f_{yy} + b_1 z\, f_z + b_2 f_y+c\,f\,) \end{equation} \noindent where the coefficients $\{a_{ij}\}_{i,j}$ are uniformly elliptic and $\{a_{ij},\,b_i,\,c\}\,\subseteq\,C^{k+2+\alpha}_s(Q_T)$, $\{a_{22},\,b_2,\,c\}\subseteq C^{k+2+\alpha,p}_s(Q_T), \,\,i,j=1,2$.
\begin{theorem} (Existence and Uniqueness) \label{thm:exi}
Let $L_k$ be defined as above. Assume that $\phi \in C^{k+\alpha,p}_s(\mathcal S)$ and $f_0 \in C^{k+2+\alpha,p}_s(\mathcal S_0)$, and that $\phi$, $f_0$ are compactly supported in $\mathcal S$ and $\mathcal S_0$, respectively. Then, for any $T>0$, the initial value problem\,: \begin{equation} \label{eqn:cauchy} \left\{ \begin{array}{cccc}
L_k\,[ f] &=&\phi & in\,\,S_T \\ f(\cdot,0)&= &f_0 & on\,\,S_0 \end{array} \right. \end{equation} admits a unique solution $f \in C^{k+2+\alpha,p}_s(\mathcal
S_T)$. Moreover \begin{eqnarray}\label{ineqg}
||f||_{C^{k+2+\alpha,p}_s(\mathcal S_T)}
\leq C(T) \left ( ||f_0||_{\mathcal C^{k+2+\alpha,p}_s(\mathcal S_0)} + ||\phi||_{\mathcal C^{k+\alpha,p}_s(\mathcal S)} \right )
\end{eqnarray}
\noindent for some constant $C(T)$, depending on $\alpha$, $p$, $k$ and $T$. \end{theorem}
\begin{proof} To solve the above Cauchy problem is equivalent to solve the following Cauchy problems~(\ref{eqn:cauchya}) and (\ref{eqn:cauchyb}).
\noindent The problem~(\ref{eqn:cauchya}) is obtained by evaluating~(\ref{eqn:cauchy}) at $z=0$ and the problem~(\ref{eqn:cauchyb}) is obtained by solving the corresponding one for $\tilde{f}$.
\begin{equation}\label{eqn:cauchya} \left\{ \begin{array}{ll} (L_k)_0\,[ f^{\circ}] =\phi^{\circ} & in\,\,{\mathbb R}\times[0,T]\\ f^{\circ}(\cdot,0)=(f_0)^{\circ} & on\,\,{\mathbb R} \end{array} \right. \end{equation}
\begin{equation}\label{eqn:cauchyb}\left\{ \begin{array}{ll}
\tilde{L_k}\, [\tilde{f}] =\tilde\phi & in\,\,S_T\\ \tilde{f}(\cdot,0)=\tilde{f}_0 & on\,\,S_0 \end{array} \right. \end{equation} where the operators $({L_k})_0$ and $\tilde{L_k}$ are defined respectively as follows: $$({L_k})_0( f^{\circ})= a_{22}^{\circ}\,f_{yy}^{\circ}+b_2^{\circ}\, f_y^{\circ}+c^{\circ} f^{\circ};$$
\begin{equation}\label{eqn:lt} \tilde L_k\,[\tilde{f}] =\tilde{f}_t -( \hat{a}_{11} \tilde{f}_{ww} + 2\, \hat a_{12} \tilde{f}_{wy} + \hat{ a}_{22} \tilde{f}_{yy} + \hat b_1 \,\tilde{f}_w+ \hat b_2 \,\tilde{f}_y +\hat c\,\tilde{f}+\hat G) \end{equation} \par\noindent with $$ \begin{array}{lcl} \hat a_{ij}(w,y,t) & := & a_{ij}(x,y,t)\\ \hat b_1(w,y,t) &:= & (2p-1)\,a_{11}(w,y,t)+b_1(x,y,t)\\
\hat b_2(w,y,t) & := & b_2(x,y,t) \end{array} $$ $$ \begin{array}{lcl} \hat c(w,y,t)& := & e^{-p\,z}[\, p^2\,\hat a_{11}(x,y,t)-2\,p\,\hat a_{12}(w,y,t)+p\, b_1(x,y,t)\, ]\\
\hat{G}(w,y,t) & := & \tilde{b_2}(w,y,t)\,g_y^{\circ}(y,t)+\hat a_{22}(w,y,t)
g_{yy}^{\circ}(y,t). \end{array} $$
By the assumptions on the operator $L_k$ it is clear that the coefficients of the two operators $({L_k})_0$ and $\tilde{L_k}$ satisfy classical conditions. We first find the solution $f^{\circ}$ to~(\ref{eqn:cauchya}), then we solve~(\ref{eqn:cauchyb}). By classical theory both problems have a unique solution. \noindent The function $f$ defined by $f(w,y,t):=f^{\circ}(y,t)+z^p\,\tilde{f}(w,y,t)$ is a solution to~(\ref{eqn:cauchy}).
\phantom{ufveycutyv}
Let $C^{k+\alpha}$ and $C^{k+2+\alpha}$ denote classical parabolic H\"older spaces. Then the following inequalities hold: $$ \begin{small}\begin{array}{rcl}
||f^{\circ}||_{C^{k+2+\alpha}({\mathbb R}^+\times [0,T])} &\leq & C(T) \left (
||f^{\circ}_0||_{C^{k+2+\alpha}({\mathbb R}^+)} + ||g^{\circ}||_{\mathcal C^{k+\alpha}({\mathbb R}^+)} \right )\\ \\
||\tilde{f}||_{C^{k+2+\alpha}(\tilde{\mathcal S}_T)} &\leq & C(T)
\left ( ||\tilde{f}_0
||_{C^{k+2+\alpha}(\tilde{\mathcal S}_0)} +
||\tilde{\phi}||_{C^{k+\alpha}(\tilde{\mathcal S})} \right) \end{array}\end{small} $$ \noindent It follows that the solution to~(\ref{eqn:cauchy}) is unique and satisfies the inequality (\ref{ineqg}). \end{proof}
\noindent Next, we define the boxes in which we prove the Schauder estimates. Let $0<r\leq 1$. We denote by ${\mathcal B}_r(P)$ the box $${\mathcal B}_r(P) = \left\{\left(\begin{smallmatrix}
z\\y\\t\end{smallmatrix}\right) : \begin{smallmatrix} z\geq 0, |x-x_0|\le e^r\\
|y-y_0|\le r\\ t_0 - r^2\le t \le t_0\end{smallmatrix}\right\}\ $$ around the point $P=\left(\begin{smallmatrix} z_0\\y_0\\t_0\end{smallmatrix}\right)$. We set ${\mathcal B}_r$ to be the box around the point $P=\left(\begin{smallmatrix} 0\\0\\1\end{smallmatrix}\right)$. \begin{remark} The choice of the box ${\mathcal B_r}$ is made so that it has the right rescaling. The operators ${L_k}_0$ and $\tilde L_k$ are well understood on the corresponding boxes $\mathcal {B}^{\circ}_r$ and ${\tilde{\mathcal B}}_r$.
\end{remark} \begin{theorem} \label{thm:sc} (\mbox{ Schauder\, Estimate})
Assume that all the coefficients of the operator $$\,L\,f = f_t - (z^2\, a_{11} f_{zz} + 2\, z \, a_{12} f_{zy} + a_{22} f_{yy} +z\, b_1 f_x + b_2 f_y+\,c f)$$
belong to the space $C^{k+\alpha}_s(\mathcal B_1)$ and that the coefficients $a_{22}, b_2$ and $c$ belong to $ C^{k+\alpha,p}_s(Q_T)$ for some numbers $\alpha$, $p$ in $0<p <1$, $0<\alpha\leq 1$ and satisfy
$$a_{ij} \xi^i \xi^j \geq \lambda |\xi|^2, \,\,\, \forall \xi \in {\mathcal R}^2 \setminus \{0 \} \,\,\,\lambda >0$$
$$with \,\,||a_{ij}||_{C^{k+\alpha}_s(Q_T)}, \,||b_i||_{C^{k+\alpha}_s(Q_T)}, \,||a_{22}||_{C^{k+\alpha,p}_s(Q_T)},||b_2||_{C^{k+\alpha,p}_s(Q_T)},
||c||_{C^{k+\alpha,p}_s(Q_T)}
\leq{ 1\over \lambda}.$$
\noindent Then,
there exists a constant $C$ depending only on $\alpha$, $\lambda$ and $p$ such that
$$\|f\|_{ C^{2+ \alpha,p}_s(\mathcal B_{1/2})} \le C\left(\|f\|_{ C^{0,p} (\mathcal B_1)} + \|L[ f]\|_{ C^{ \alpha,p}_s(\mathcal B_1)}\right)$$ for all functions $f \in C^{2+\alpha,p}_s(\mathcal B_1)$. \end{theorem} \begin{proof} The proof follows by the same argument as in Theorem~\ref{thm:exi} and classical Schauder estimates for strictly parabolic operators. \end{proof}
\section{The Degenerate Equation on the disc}\label{deg}
We will extend in this section the existence and uniqueness the Theorem~\ref{thm:exi} to the following class of linear degenerate equations: $$Lw: = w_t - \, (\, a^{ij} w_{ij} + b^i\, w_{i} + c\,w \, ) $$ on the cylinder ${\mathcal D} \times [0,T)$, $T >0$, where ${\mathcal D}$ denotes the unit disk in ${{\mathbb R}}^2$. The sub-indices $i,j\in \{x,y \}$ denote differentiation with respect to the space variables $x,y$.
The matrix $\{a^{ij}\}$ is assumed to be symmetric. Certain assumptions on the coefficients will be made so that this class of equations includes, under appropriate change of coordinates, the operator $L$ given by~({\ref{eqn:ope}).
We define the distance function $s$ in ${\mathcal D}$ as follows: in the interior of ${\mathcal D}$, $\bar s$ it is equivalent to the standard euclidean distance, while around any boundary point $P \in \partial \mathcal D$, $\bar s$ is defined as the pull back of the distance function induced by the metric $$d\bar s^2 = \frac {dz^2}{z^2} + dy^2$$ on the half space $\mathcal S_0 = \{ (z,y) : z \geq 0 \}$, via a map $\varphi: \mathcal S_0 \cap {\mathcal D} \to {\mathcal D}$ that flattens the boundary of the disk ${\mathcal D}$ near $P$.
The parabolic distance is defined by $$s \left [\left(\begin{smallmatrix} P_1\\ t_1\end{smallmatrix}\right), \left(\begin{smallmatrix} P_2\\ t_2\end{smallmatrix}\right) \right ] =
\bar s(P_1,P_2) + \sqrt {|t_1-t_2|}, \,\,\, P_1,P_2 \in {\mathcal D},\,\,0<t_1\leq t_2$$ We define the spaces $C^{k+\alpha,p}_s({\mathcal D})$ and $C^{k+2+\alpha,p}_s({\mathcal D})$. \noindent For a fixed small number $\delta$ in $0 < \delta <1$, we write $${\mathcal D} = {\mathcal D}_{1-\delta/2} \, \cup ({\displaystyle\bigcup_l}\, \left ( {\mathcal D}_{\delta}(P_l) \cap {\mathcal D}) \right )$$ for finite many points $P_l \in \partial {\mathcal D}$, $l \in I$, with ${\mathcal D}_{1-\delta/2}$ denoting the disk centered at the origin of radius $1-\delta/2$ and ${\mathcal D}_{\delta}(P_l)$ denoting the disk of radius $\delta$ centered at $P_l$.
We denote by ${\mathcal D}_+$ the half disk $${\mathcal D}_+ = \{ \, (x,y) \in {\mathcal D} : \,\, x \geq 0 \, \}.$$ We can choose charts $\Upsilon_l : {\mathcal D}_+ \to {\mathcal D}_{\delta}(x_l) \cap {\mathcal D}$ which flatten the boundary of ${\mathcal D}$ and such that $\Upsilon_l (0) = P_l$, $ l\in\,I$. Let $\{\psi$, $\psi_l\}$ be a partition of unity subordinated to the cover $$\{ \, {\mathcal D}_{1-\delta/2},\, ( {\mathcal D}_{\delta}(P_l) \cap {\mathcal D}) \,\} $$ of ${\mathcal D}$, with $l\in\, I$.
We define $C^{k+\alpha,p}_s({\mathcal D})$
to be the space of all functions $w$ on ${\mathcal D}$ such that $w \in C^{k+\alpha} ({\mathcal D}_{1-\delta/2})$ and $w \circ \Upsilon_l \in C^{k+\alpha,p}_s({\mathcal D}_+)$
for all $l \in I$.
Also, we define $C^{k+2+\alpha,p}_s({\mathcal D})$ to be the space of all functions $w$ on ${\mathcal D}$ such that $w \in C^{k+2+\alpha }({\mathcal D}_{1-\delta/2})$ and $w \circ \psi_l \in C^{k+2+\alpha,p}_s({\mathcal D}_+)$ for all $l \in I$. Here $C^{k+\alpha}$ and $C^{k+2+\alpha}$ denote the regular H\"older Spaces, while $C^{k+\alpha,p}_s ({\mathcal D}_+)$ and $ C^{k+2+\alpha,p}_s({\mathcal D}_+)$ denote the H\"older Spaces defined in section~\ref{def:ban}. One can show that both spaces $C^{k+\alpha,p}_s({\mathcal D})$ and $C^{k+2+\alpha,p}_s({\mathcal D})$ are Banach Spaces under the norms
$$ ||w||_{C^{k+\alpha,p}_s({\mathcal D})} = ||\psi \, w||_{C^{k+\alpha} ({\mathcal D}_{1-\delta/2})}
+ \sum_{l} \, ||\psi_l \,( w \circ \Upsilon_l)||_{C^{k+\alpha,p}_s({\mathcal D}_+)}$$ and
$$ ||w||_{C^{k+2+\alpha,p}_s({\mathcal D})} = ||\psi \, w||_{C^{k+2+\alpha}( {\mathcal D}_{1-\delta/2})}
+ \sum_{l} \, ||\psi_l \,( w \circ \Upsilon_l)||_{C^{k+2+\alpha,p}_s({\mathcal D})}.$$ \noindent
The above definitions can be extended in a straight forward manner to the parabolic spaces $C^{\alpha,p}_s(Q)$ and $C^{2+\alpha,p}_s(Q)$ where $Q$ is the cylinder $Q= {\mathcal D} \times [0,T]$, for some $T >0$.
\noindent Before we state the main result in this section, we will give the assumptions on the coefficients of the equation $$w_t =\, a^{ij} w_{ij} + b^i\, w_{i} + c\,w $$ on the cylinder $Q = {\mathcal D} \times [0,T)$, $i,j=1,2$.
We first assume that for any $\delta$ in $0<\delta <1$, the coefficients $\{a^{ij}\}$, $\{b^i\}$ and $c$ belong to the H\"older class $C^{\alpha}(\mathcal D_{1-\delta/2}
\times [0,T])$, which means that the coefficients are of the class $C^{\alpha}$ in the interior of $\mathcal D$. For a number $\delta$ in $0<\delta <1$, let $\Upsilon_l: \mathcal D_+ \to \mathcal D_\delta(P_l) \cap \mathcal D$ be the collection of charts which flatten the boundary of $\mathcal D$, considered above. We assume that there exists a number $\delta$ so that for every $l \in I$, the coordinate change introduced by each of the $\Upsilon_l$ transforms the operator \begin{equation}\label{opel} L[w] = w_t - (\,\, a^{ij}\, w_{ij} + b^i\, w_{i} + c\,w \, )\end{equation} on $\mathcal D_\delta(P_l) \cap \mathcal D$, into an operator $\widetilde L_l$ on $\mathcal D_+$ of the form $$\widetilde L_l \, [\tilde w] = \tilde w_t - (\, x^2\, \tilde a_{11} \, \tilde w_{xx} + 2 \, x \, \tilde a_{12} \, \tilde w_{xy} + \tilde a_{22} \, \tilde w_{yy} + x\,\tilde b_1 \,\tilde w_x + \tilde b_2 \, \tilde w_y \,+\tilde c\,\tilde{w})$$ with the coefficients $\tilde a_{ij}$, $\tilde b_i$ and $\tilde c$ belonging to the class $C^{k+\alpha}_s(\mathcal D_+)$, with $a_{22},\,b_2$ and $c\,\in\,C^{k+\alpha,p}_s$ such that:
$$\tilde a_{ij} \xi^i \xi^j \geq \lambda \, |\xi|^2, \qquad \forall \xi \in {\mathcal R}^2 \setminus \{0\}$$
\noindent for some number $\lambda >0$.\\
\noindent We need the next Lemma to prove the invertibility of the operator $L$: \begin{lemma}\label{holder}{(H\"{o}lder Interpolation)}. For every $\epsilon>0$ there exists a constant $C(\epsilon)$ depending on $\epsilon$, $p$, $k$ and $\alpha$ such that for any $g\in {C^{k+2+\alpha,p}_s (Q_\delta)} $, the following inequality holds: \begin{equation}\label{eqn:hol}
||\vartheta\, D\, g\, ||_{C_s^{k+\alpha,p} (Q_\delta)} \leq
\epsilon \, ||\, g\,||_{C_s^{k+2+\alpha,p} (Q_\delta)} + C(\epsilon)
\, ||\,g\,||_{C^{k,p}(Q_\delta)}. \end{equation} \noindent where $\vartheta $ behaves like distance to the boundary. \end{lemma} \begin{proof} It follows by standard arguments. \end{proof}
The following existence result readily follows from Theorem~\ref{thm:exi} and the above discussion.
\begin{theorem}\label{thm:exi2} Assume that the operator $L$ satisfies all the above conditions on the cylinder $Q={\mathcal{D}} \times [0,T]$. Then, given any function $w_0 \in C^{k+2+ \alpha,p}_s({\mathcal{D}})$ and any function $g \in C^{k+\alpha,p}_s(Q)$ there exists a unique solution $w \in C^{k+ 2+ \alpha,p}_s (Q_T)$ of the initial value problem \[ \left\{ \begin{array}{ll}
Lw =g & in\,\,Q\\ w(\cdot,0)=w_0 & on\,\,\mathcal D \end{array} \right. \] satisfying \begin{equation}\label{ineq:www}
||w||_{C^{k+2+\alpha,p}_s(Q)} \leq C(T) \left ( ||w_0||_{C^{k+2+\alpha,p}_s(\mathcal{D})} + ||g||_{C^{k+\alpha,p}_s(Q)} \right ) \end{equation} The constant $C(T)$ depends only on the numbers $\alpha$, $k$, $\lambda$ and $T$. \end{theorem}
\begin{proof} We can assume, without loss of generality, that $w_0\equiv 0$ and that $g$ is a function in $C_s^{k+\alpha,p}(Q_T)$, which vanishes at $t=0$.
For $\delta >0$, set $Q_\delta = \mathcal{D} \times [0,\delta]$ and denote by $C_{s,0}^{k+2+\alpha,p}(Q_\delta)$ and $C_{s,0}^{k+\alpha,p}(Q_\delta)$ the subspaces of $C_{s}^{k+2+\alpha}(Q_\delta)$ and $C_{s}^{k+\alpha} (Q_\delta)$ respectively, consisting out of all functions which vanish identically at $t=0$. Also, we denote by $I$ the identity operator on $C_{s,0}^{k+\alpha,p}(Q_\delta)$. We will show that, if $\delta$ is sufficiently small, there exists an operator $M\, : \, C_{s,0}^{k+\alpha,p}(Q_\delta) \to C_{s,0}^{k+2+\alpha,p}(Q_\delta)$ such that
$$ || \, L\,M - I \, || \leq \frac 12.$$ This implies that the operator $L\,M : \, C_{s,0}^{k+\alpha,p}(Q_\delta) \to C_{s,0}^{k+\alpha,p}(Q_\delta)$ is invertible and therefore $L: \, C_{s,0}^{k+2+\alpha,p}(Q_\delta) \to C_{s,0}^{k+\alpha,p}(Q_\delta)$ is onto, as desired.
We begin by expressing the compact domain $\mathcal D$ as the finite union
$$\mathcal D = \mathcal D_0 \,\cup{\displaystyle\bigcup_{l\geq 1 }}\, \mathcal D_l$$ of compact domains in such a way that $$\text{dist}\,(\mathcal D_0, \partial \mathcal D) \geq \frac {\rho}2 >0 $$ and for all $l \geq 1$ $$\mathcal D_l = B_\rho (x_l) \cap \mathcal D$$ with $B_\rho (x_l)$ denoting the ball centered at $x_l \in \partial \mathcal D$ of radius $\rho >0$. The number $\rho >0$ will be determined later.
The operator $L$ is non-degenerate when restricted on the interior domain $\mathcal D_0$. Therefore, the classical Schauder theory for linear parabolic equations implies that $L$ is invertible when restricted on functions which vanish outside $\mathcal D_0$.
We denote by $M_0: \, C_{s,0}^{k+\alpha,p}(\mathcal D_0 \times [0,\delta]) \to C_{s,0}^{k+2+\alpha,p}(\mathcal D_0 \times [0,\delta])$ the inverse of the operator $L$ restricted on $\mathcal D_0$. Next, we consider the domains $\mathcal D_l, \, l \geq 1$, close to the boundary of $\mathcal D$, which can be chosen in such a way that the sets $B_{\rho/4}(x_l) \cap \mathcal D $ are disjoint. Denoting by $\overline B$ the half unit ball $$\overline B= \{(x,y) \in B_1(0) \,; \,\,\, x \geq 0 \, \}$$ and by $\overline Q_\delta$ the cylinder $$\overline Q_\delta = \overline B \times [0,\delta]$$ we select smooth charts $\Upsilon_l: \overline B \to \mathcal D_l $, which flatten the boundary of $\mathcal D$, i.e., they map $\overline B \cap \{ x=0 \} $ onto $ \mathcal D_l \cap \partial \mathcal D$ and have $\Upsilon_l (0) = x_l$. This is possible if the number $\rho$ is chosen sufficiently small. Under the change of coordinates induced by the charts $\Upsilon_l$, the operator $L$, restricted on each $\mathcal D_l \times [0,\delta]$,
is transformed to an operator $\bar L_l$ of the form $$\bar L_l[\bar w ] = \bar w_t - (\,x^2 \,\bar a_l^{11}\bar w_{11} + 2\,x \,\bar a_l^{12}\bar w_{12}+\bar a_l^{22}\bar w_{22}+ \,x\bar b_l^1\, \bar w_{1} + \bar b_l^2\, \bar w_{2}+\bar c_l\, \bar w )$$ defined on $\overline B \times [0,\delta]$. Moreover, the charts $\Upsilon_l$ can be chosen appropriately so that the coefficients of $\bar L_l$ satisfy
$$ \bar a_l^{ij} \xi_i \xi_j \geq\bar \lambda |\xi|^2 >0 \qquad \forall \xi \in {\mathbb R}^2 \setminus \{ 0 \}$$ and
$$||\bar a_l^{ij}||_{C^{k+\alpha,p}_s(\bar Q_\delta)} \quad
||\bar b_l^i||_{C^{k+\alpha,p}_s(\bar Q_\delta)} \quad
||\bar c_l||_{C^{k+\alpha,p}_s(\bar Q_\delta)} \leq 1/\,{\bar \lambda}$$
\noindent for some positive constant $\bar \lambda$.
Each of the operators $ \bar L_l$ has the form of the model operators previously studied. Denote by ${S}_0$ the half space $x\geq 0$ in ${\mathbb R}^2$ and by ${S}_\delta$ the space $\mathcal{S}_0 \times [0,\delta]$. Also, consider the subspace $\overline C_{s,0}^{k+\alpha,p}(\mathcal{S}_\delta)$ of $C_{s,0}^{k+\alpha,p}(\mathcal{S}_\delta)$, consisting out of functions which are compactly supported on $\mathcal{S}_\delta$. Then, Theorem~\ref{thm:exi} implies that for every $l=1,2,...$ there is an operator $ \overline{M}_l: \overline{C}_{s,0}^{k+\alpha,p}({S}_\delta) \to {C}_{s,0}^{k+2+\alpha,p}({S}_\delta)$ such that $$ \bar L_l \, \overline{ M}_l = I$$ with $I$ denoting the identity operator on $\overline C_{s,0}^{k+\alpha,p}(\mathcal{S}_\delta)$. Let $M_l$ be the pull back of the operator $ \overline{ M}_l$ via the chart $\Upsilon_l$. Next, choose a nonnegative partition of unity $\,\phi_l; \, l=0,1,...\,$ subordinated to the cover $\,\mathcal D_l; \, l=0,1,...\,$ of $\mathcal D$ and also choose, for each $l \geq 0$, nonnegative, smooth bump functions $\psi_l$, $0 \leq \psi_l \leq 1$, supported in $\mathcal D_l$ with $\psi_l \equiv 1$ on the support of $\phi_l$. Then $\sum_{l \geq 0} \, \phi_l \, = 1$ and $ \psi_l \, \phi_l = \phi_l$ for all
$l$.
We aim to show that the operator $M:\, C_{s,0}^{k+\alpha,p}(Q_\delta) \to C_{s,0}^{k+2+\alpha,p}( Q_\delta)$ defined as $$ M g \,= \, \sum_{} \psi_l M_l\, \phi_l g$$ satisfies
$$||\, LM \, g \,- \,g\, ||_{C_s^{k+\alpha,p} (Q_\delta)} < \frac 12 \,\,
||g||_{C_s^{k+\alpha,p} (Q_\delta)} \qquad \forall g \in C_{s,0}^{k,\alpha,p} (Q_\delta)$$ if the cover $\{ \mathcal D_l \}$ and $\delta$ are chosen appropriately. Indeed, we can write $$ L\,M g - g = \sum_{l} L \, \psi_l M_l\, \phi_l g - \sum_{l} \phi_l g= \sum_{l} \,\psi_l \, (L M_l - I)\, \phi_l g + \sum_{l}\, [\,L,\psi_l \,]\, M_l\, \phi_l g$$ with $[\,L,\psi_l \,]$ denoting the commutator of $L$ and $\psi_l$. The commutator $[\, L, \psi_l \,]$ is only of first order and it can be estimated as
$$ || [\,L,\psi_l \,] \, M_l\phi_l g ||_{C_s^{k+\alpha,p} (Q_\delta)} \leq C \, \left ( \,
|| \vartheta\, D (M_l\phi_l g) ||_{C_s^{k+\alpha,p} (Q_\delta)} +
\, ||M_l\phi_l g||_{C_s^{k+\alpha,p} (Q_\delta)}\, \right ).$$ Let $\epsilon >0$.
It follows via the H\"older spaces interpolation from Lemma~\ref{holder} that
$$|| \vartheta\, D (M_l\phi_l g) ||_{C^{k+\alpha,p}_s(Q_\delta)}
\leq \epsilon \, ||M_l\phi_l g||_{C^{k+2+\alpha,p}_s (Q_\delta)} + C(\epsilon)
\, ||M_l\phi_l g||_{C^{k,p}(Q_\delta)}.$$ However, for each $k$ we have
$$ ||M_l\phi_l\ g||_{C^{k+2+\alpha,p}_s (Q_\delta)} \leq C \, ||g||_{C_s^{k, \alpha,p} (Q_\delta)}$$ and therefore, since $M_l\phi_l g \equiv 0$ at $t=0$,
$$||M_l\phi_l g||_{C^{k,p}_s(Q_\delta)}
\leq C \, \delta\, ||g||_{C^{k+\alpha,p}_s (Q_\delta)}.$$ It follows that if we choose $\delta$ sufficiently small:
$$\sum_{l} \, ||\, [\,L,\psi_l \,] \, M_l\phi_l g\, ||_{C^{k+\alpha,p}_s
(Q_\delta)} \leq \frac 14 \,\, ||g||_{C^{k+\alpha,p}_s(Q_\delta)}.$$ On the other hand we have $(\,L M_0 - I \,)\varphi_0 g=0$,
while for $l \geq 1$, we can make the norm of each of the operators $L M_l - I$ arbitrarily close to zero by choosing the diameters of the domains $\mathcal D_l$ sufficiently small:
$$ ||\, \sum_{l} \psi_l \, (L\,M_l - I)\, \phi_l g \, ||_{C^{k+\alpha,p}_s
(Q_\delta)} < \frac 14\,\, ||g||_{C^{k+\alpha,p}_s(Q_\delta)}$$ for all $g \in C_s^{k+\alpha,p} (Q_\delta)$, if $\rho$ and $\delta$ are both sufficiently small. The above inequalities give:
$$ ||\, L\,M g - g\, || _{C^{k+\alpha,p}_s(Q_\delta)}
\leq \frac 12 \,\, ||g||_{C^{k+\alpha,p}_s(Q_\delta)}$$ for all $g \in C^{k+\alpha,p}_{s,0}(Q_\delta)$. We conclude that for every $g \in C^{k+\alpha,p}_{s,0}(Q_\delta)$ there exists a function $w \in C^{k+2+\alpha}_{s,0}(Q_\delta)$ such that $Lw=g$. In addition \begin{equation}\label{ineq:w}
||w||_{C^{k+2+\alpha,p}_s(Q_\delta)}
\, \leq \, C \, ||g||_{C^{k+\alpha,p}_s(Q_\delta)} \end{equation} with $C$ depending only on $\mathcal D$ and the constants $\alpha$, $k$, $\lambda$ and $T$.
The last inequality implies we can extend the solution on a bigger interval. Hence, one can show that
$$||w||_{C^{k+2+\alpha,p}_s(Q)} \leq C(T) \left ( ||w_0||_{C^{k+2+\alpha,p}_s(\mathcal{D})} + ||g||_{C^{k+\alpha,p}_s(Q)} \right )$$ where the constant $C(T)$ depends only on the numbers $\alpha$, $k$, $\lambda$ and $T$.
\noindent This last inequality implies uniqueness. \end{proof}
Finally, the following existence result follows from Theorem \ref{thm:exi2} and the Inverse Function Theorem between Banach spaces along the line of the proof of Theorem 7.3 in \cite{DH2}:
\begin{theorem}\label{thm:exi3} Let $w_0$ be a function in $C^{k+2+\alpha,p}_s(\mathcal D)$.
Assume that the linearization $DM(\bar w)$ of the fully-nonlinear operator $$Mw = w_t - F(t,u,v,w,Dw, D^2w)$$ defined on the cylinder $Q = \mathcal D \times [0,T]$, satisfies the hypotheses of Theorem~\ref{thm:exi2} at all points $\bar w \in C^{k+2+\alpha,p}_s(Q)$, with
$||\bar w - w_0||_{ C^{k+2+\alpha,p}_s(Q)} \leq \mu$, for some $\mu >0$. Then, there exists a number $\tau_0$ in $0<\tau_0 \leq T$ depending on the constants $\alpha,\,p$, $k$, $\lambda$ and $\mu$, for which the initial value problem $$\left\{\begin{array}{ll} w_t = F(t,u,v,w,Dw, D^2w) \qquad &\text{in $\,\,\mathcal D \times [0,\tau_0]$}\\ w(\cdot,0)=w_0 \qquad &\text{on $\,\, \mathcal D$} \end{array}\right.$$
admits a solution $w$ in the space $C^{k+2+\alpha,p}_s (\mathcal D \times [0,\tau_0])$. Moreover,
$$||w||_{C^{k+2+\alpha,p}_s(\mathcal D \times [0,\tau_0])}
\leq C\, ||w_0||_{C^{k+2+\alpha,p}_s(\mathcal D)}$$ for some positive constant $C$ which depends only on $\alpha$, $p$, $k$, $\lambda$ and $\mu$. \end{theorem}
\section{Global change of coordinates and existence in $C^{k+2+\alpha,p}_s$}\label{global}
In this section we introduce a global change of coordinates which transforms the HMCF for a surface $\Sigma_0$ into a fully-nonlinear degenerate parabolic PDE on $\mathcal D$.
Let $S$ be a smooth surface close to $\Sigma_2$. Let $S : \mathcal D \to \mathbb{R}^3$, indicate a parameterization of $S$ on the unit disk $\mathcal D$.
$$S(u,v)=(x,y,z)\in {\mathbb R}^3\,\,\,\hbox{ which maps }\,\,\,\partial \mathcal D\,\,\, \hbox{ onto}\,\,\, S\cap \{x=0\}$$
\noindent We denote by $x_u, x_v, x_{uu}, x_{uv}, x_{vv}$ the partial derivatives of $x$ with respect to $u$ and $v$. The same notation will be used for the function $y$.
\noindent Let $\eta >0$ be sufficiently small. Let $T = (T_1,T_2,T_3)$ be a smooth vector field transverse to $S$. We define the global change of coordinates $\Phi : \mathcal D \times [-\eta,\eta] \to \mathbb{R}^3$ by
\begin{align} \begin{pmatrix} x\\ y\\ z \end{pmatrix} = \Phi \begin{pmatrix} u\\ v\\ w \end{pmatrix} = S(u,v) + w T(u,v) \end{align}
\noindent or more explicitly
\begin{align} x &= S_1(u,v) + w T_1(u,v)\\ y &= S_2(u,v) + w T_2(u,v)\\ z &=S_3(u,v)+w T_3(u,v) \end{align}
\noindent Let $\delta >0$ be small number, such that $$T_3 (u,v) =0 \qquad \text{on }\,\, {\mathcal D} \setminus {\mathcal D}_{1-\delta}$$
\noindent denoting, as above, the transverse vector field to the surface $\mathcal S$. Notice that by choosing the smooth surface sufficiently close to the surface $z=h(x,y)$, we can make $\delta$ to depend only on the constant $\lambda$ which depends on the initial non-degeneracy conditions on the surface $\Sigma_0$.
We write the
first and second derivatives of $z$ with respect to $x,y$ and $t$ in terms of the first and second derivatives of $w$ with respect to $u,v$ and $t$.
\noindent If $z= h_0(x,y)$ then we compute the first and second partial derivatives of $z$ with respect to $x$ and $y$ in terms of $w = l(u,v)$ seen as functions of $u$ and $v$.
Let $A$ be the Jacobian matrix relative to the transformation of coordinates: $$A = \left ( \begin{split} \frac{\partial u}{\partial x}\,\,\, & \frac{\partial v}{\partial x}\\ \frac{\partial u}{\partial y}\,\,\, & \frac{\partial v}{\partial y} \end{split}\right )= \left ( \begin{split} a\,\,\, & b\\ c\,\,\, & d \end{split}\right ) $$ Let $\nabla z$, $\nabla u$, $\nabla v$ be, respectively the gradients of $z$, $u$ and $v$: $$\nabla z=\left(\begin{smallmatrix} \frac{\partial z}{\partial x}\\ \frac{\partial z}{\partial y}\end{smallmatrix}\right)=\left(\begin{smallmatrix} z_1\\ z_2\end{smallmatrix}\right)\,\,\,\nabla u=\left(\begin{smallmatrix} \frac{\partial u}{\partial x}\\ \frac{\partial u}{\partial y}\end{smallmatrix}\right)=\left(\begin{smallmatrix} u_1\\ u_2\end{smallmatrix}\right)\,\, \nabla v=\left(\begin{smallmatrix} \frac{\partial v}{\partial x}\\ \frac{\partial v}{\partial y}\end{smallmatrix}\right)=\left(\begin{smallmatrix} v_1\\ v_2\end{smallmatrix}\right)$$ \noindent We denote by $D^2 u$ and $D^2 v$ the following matrices:
$$D^2 u = \left ( \begin{split} \frac{\partial^2 u}{\partial x^2}\,\,\, & \frac{\partial^2 u}{\partial x\partial y}\\ \frac{\partial^2 u}{\partial x\partial y}\,\,\, & \frac{\partial^2 u}{\partial y^2} \end{split}\right )= \left ( \begin{split} a_1\,\,\, & c_1\\ c_1\,\,\, & c_2 \end{split}\right ) $$
$$D^2 v = \left ( \begin{split} \frac{\partial^2 v}{\partial x^2}\,\,\,& \frac{\partial^2 v}{\partial x\partial y}\\ \frac{\partial^2 v}{\partial x\partial y}\,\,\, &\frac{\partial^2 v}{\partial y^2} \end{split}\right )= \left ( \begin{split} b_1\,\,\, & d_1\\ d_1\,\,\, & d_2 \end{split}\right ) $$
\noindent Based on the above, we define $a_2:=c_1\,$, $b_2:=d_1$ and denote $e_1=(1,0)\,\,\,e_2=(0,1)$ to be the basis vectors. Let $A^{-1}$ be the inverse matrix of $A$: $$A^{-1}: = \left ( \begin{split} x_u,\,\, & x_v\\ y_u\,\,\, & y_v \end{split}\right )= \left ( \begin{split} x_1\,\,\, & x_2\\ y_1\,\,\, & y_2 \end{split}\right ) $$
\noindent Next, we introduce the matrices $B_1$ and $B_2$ which denote, respectively, the derivative of the inverse matrix of $A$ and $A^{-1}$ with respect to $x$ and $y$; $B_1=\frac{\partial A^{-1}}{\partial x}$, $B_2=\frac{\partial A^{-1}}{\partial y}$ which can be computed as:
$$B_1 = \left ( \begin{split} a\,x_{11}+b\,x_{12}\,\,\,\,\,\,& a\,y_{11}+b\,y_{12}\\ a\,x_{12}+b\,x_{22}\,\,\,\,\,\,& a\,y_{12}+b\,y_{22} \end{split}\right )$$
$$B_2 = \left ( \begin{split} c\,x_{11}+d\,x_{12}\,\,\,\,\,\,& c\,y_{11}+d\,y_{12}\\ c\,x_{12}+d\,x_{22}\,\,\,\,\,\,& c\,y_{12}+d\,y_{22} \end{split}\right )$$
\noindent Note that we are using the following notation: $$x_{11}=x_{uu},\, x_{12}=x_{uv},\,x_{22}=x_{vv},\, y_{11}=y_{uu},\, y_{12}=y_{uv},\,y_{22}=y_{vv}.$$ \noindent The coefficients of the matrices $D^2 u$ and $D^2 v$ are evaluated as follows: $$ \begin{array}{lcl} \left(\begin{smallmatrix} a_1\\ c_1\end{smallmatrix}\right) & = -A\cdot B_1\cdot \nabla u;\,\, \left(\begin{smallmatrix} b_1\\ d_1\end{smallmatrix}\right) & = -A\cdot B_1\cdot \nabla v\\ \\ \left(\begin{smallmatrix} a_2\\ c_2\end{smallmatrix}\right) & =-A\cdot B_2\cdot \nabla u;\,\, \left(\begin{smallmatrix} b_2\\ c_2\end{smallmatrix}\right) & =-A\cdot B_2\cdot \nabla v \end{array} $$
\noindent Since: $\qquad \nabla z=A\cdot \left(\begin{smallmatrix} z_u\\ z_v\end{smallmatrix}\right)+z_w\left(\begin{smallmatrix} \,\frac{\partial w}{\partial x}\\ \frac{\partial w}{\partial y}\end{smallmatrix}\right)$, we obtain that:
$$\frac{\partial}{\partial x}\nabla z=A_x\cdot \left(\begin{smallmatrix} z_u\\ z_v\end{smallmatrix}\right)+A\cdot \left(\begin{smallmatrix} a\,z_{uu}+b\,z_{uv}+z_{uw}\frac{\partial w}{\partial x}\\a\,z_{uv}+b\,z_{vv}+z_{vw}\frac{\partial w}{\partial x}\end{smallmatrix}\right) + (a\,z_{uw}+b\,z_{vw})\left(\begin{smallmatrix} \,\frac{\partial w}{\partial x}\\ \frac{\partial w}{\partial y}\end{smallmatrix}\right)+z_w\,\left(\begin{smallmatrix} \,\frac{\partial^2 w}{\partial^2 x}\\ \frac{\partial^2 w}{\partial x\partial y}\end{smallmatrix}\right).$$
$$\frac{\partial}{\partial y}\nabla z=A_y\cdot \left(\begin{smallmatrix} z_u\\ z_v\end{smallmatrix}\right)+A\cdot \left(\begin{smallmatrix} c\,z_{uu}+d\,z_{uv}+z_{uw}\frac{\partial w}{\partial y}\\ c\,z_{uv}+d\,z_{vv}+z_{vw}\frac{\partial w}{\partial y}\end{smallmatrix}\right) + (c\,z_{uw}+d\,z_{vw})\left(\begin{smallmatrix} \,\frac{\partial w}{\partial x}\\ \frac{\partial w} {\partial y}\end{smallmatrix}\right)+z_w\,\left(\begin{smallmatrix} \,\frac{\partial^2 w}{\partial x\partial y}\\ \frac{\partial^2 w} {\partial^2 y}\end{smallmatrix}\right). $$
\noindent The gradient of the function $w$, as well as its partial derivatives, can be expressed by using the matrix $A$:
$$ \left(\begin{smallmatrix} \,\frac{\partial w}{\partial x}\\ \frac{\partial w}{\partial y}\end{smallmatrix}\right)= \left(\begin{smallmatrix} \,a\,w_u+b\,w_v\\ c\,w_u+d\,w_v\end{smallmatrix}\right)$$ $$ \left(\begin{smallmatrix} \,\frac{\partial^2 w}{\partial^2 x}\\ \frac{\partial^2 w}{\partial x\partial y}\end{smallmatrix}\right)= \left(\begin{smallmatrix} \,a_1\,w_u+a\,(a\,w_{uu}+b\,w_{uv})+b_1\,w_v+b(a\,w_{uv}+b\,w_{vv}) \\ c_1\,w_u+c(a\,w_{uu}+b\,w_{uv})+d_1\,w_v+d(a\,w_{uv}+b\,w_{vv})\end{smallmatrix}\right)$$
$$ \left(\begin{smallmatrix} \,\frac{\partial^2 w}{\partial x\partial y}\\ \frac{\partial^2 w}{\partial^2 y}\end{smallmatrix}\right)= \left(\begin{smallmatrix} \,a_2\,w_u+a\,(c\,w_{uu}+d\,w_{uv})+b_2\,w_v+b(c\,w_{uv}+d\,w_{vv}) \\ c_2\,w_u+c(c\,w_{uu}+d\,w_{uv})+d_2\,w_v+d(c\,w_{uv}+d\,w_{vv})\end{smallmatrix}\right).$$
\noindent
Using the substitutions above we get: $$\frac{\partial^2 z}{\partial^2 x}=A^1_{11}\,w_{11}+A^1_{12}\,w_{12}+A^1_{22}\,w_{22}+B^1_1\,w_1+B^1_2\,w_2+B^1_{12}\,w_1\,w_2+ B_{11}^1\,w_1^2+B_{22}^1\,w_2^2+ C_1$$ $$\frac{\partial^2 z}{\partial^2 y}=A^2_{11}\,w_{11}+A^2_{12}\,w_{12}+A^2_{22}\,w_{22}+B^2_1\,w_1+B^2_2\,w_2+B^2_{12}\,w_1\,w_2+ B_{11}^2\,w_1^2+B_{22}^2\,w_2^2+ C_2$$ $$\frac{\partial^2 z}{\partial x\partial y }=A^{\circ}_{11}\,w_{11}+A^{\circ}_{12}\,w_{12}+A^{\circ}_{22}\,w_{22}+B^{\circ}_1\,w_1+B^{\circ}_2\,w_2+B^{\circ}_{12}\,w_1\,w_2+ B_{11}^{\circ}\,w_1^2+B_{22}^{\circ}\,w_2^2+ C_{\circ}$$
\noindent where the coefficients $A^{k}_{i,j},\,B_i^j,\,C^i$ with $k=0,1,2$, $i,j=1,2$ are defined as follows: $$ \begin{array}{lcl} A_{11}^1: & = & -a^2\,(\,-z_{w}+b\,y_w(z_u+w_1\,z_w)+d\,y_w(z_v+w_2 z_w))\\
A_{22}^1: & = & -b^2\,(\,-z_{w}+b\,y_w(z_u+w_1\,z_w)+d\,y_w(z_v+w_2 z_w))\\
A_{12}^1: & = & -2\,a\,b\,(\,-z_{w}+b\,y_w(z_u+w_1\,z_w)+d\,y_w(z_v+w_2 z_w))\\
A_{11}^2: & = & c^2(z_w - d\,y_w (z_v + w_2\,z_w))\\ A_{22}^2: & = & d^2(z_w - d\,y_w (z_v + w_2\, z_w))\\
A_{12}^2: & = & -2 c d (-z_w + d\, y_w (z_v + w_2 z_w))
\end{array}
$$
$$ \begin{array}{lcl} A_{11}^0: & = & -a c (-z_w + b y_w (z_u + w_1 z_w) + d y_w (z_v + w_2 z_w))\\
A_{22}^0: & = & -b d (-z_w + b y_w (z_u + w_1 z_w) + d y_w (z_v + w_2 z_w))\\
A_{12}^0: & = & -b (2 a c y_{uw} + b c y_{vw} + a d y_{vw}) z_w\\ B_{11}^0: & = & -b (2 a c y_{uw} + b c y_{vw} + a d y_{vw}) z_w\\
B_{22}^0: & = & -d (b c y_{uw} + a d y_{uw} + 2 b d y_{vw}) z_w\\
B_{12}^0 : & = &-((b^2 c + a (b + 2 c) d) y_{uw} + d(b(2 b + c) + a d) y_{vw}) z_w \\
B_1^0 : & = & -a^2(c x_{uu} + d x_{uv}) z_w - b(b c y_{vw} z_u + c d y_{vw} z_v -c z_{vw} + b c y_{uv} z_w +\\ {} & {} & b d y_{vv} z_w) - a (-2 c z_{uw} + 2 c d y_{uw} z_v + d^2 y_{vw} z_v - d z_{vw} + b (2 c y_{uw} z_u + d y_{vw} z_u +\\ {} & {} & c (x_{uv} + y_{uu}) z_w + d (x_{uv} + y_{uv}) z_w))\\
B_2^0 : & = & -b^2 (c y_{uw} + 2 d y_{vw})z_u - b (-c z_{uw} + d (a y_{uw} z_u + c y_{uw} z_v + 2 d y_{vw} z_v - 2 z_{vw}) +\\ {} & {} & (c(c + d) x_{uv} + c d y_{uv} + d^2 y_{vv}) z_w) - a (c^2 x_{uu} z_w + d (-z_{uw} + c (x_{uv} + y_{uu}) z_w +d(y_{uw} z_v + y_{uv} z_w)))\\
C^0 : & = & -a^2 (c x_{uu} + d x_{uv}) z_u - a (b (c (x_{uv} + y_{u}u) + d ( x_{uv} + y_{uv})) z_u - c z_{uu} + c^2 x_{uu} z_v + \\ {} & {} &c d (x_{uv} + y_{uu}) z_v + d (-z_{uv} + d y_{uv} z_v)) - b (b (c y_{uv} + d y_{vv}) z_u - c z_{uv} + c^2 x_{uv}z_v + c \\
{} & {} & d (x_{uv} + y_{uv})z_v + d (d y_{vv}z_v - z_{vv}))
\end{array} $$ $$ \begin{array}{lcl} B_{11}^1: & = & -2 a b ( a y_{uw} + b y_{vw})\\
B_{22}^1: & = & -2 b d ( a y_{uw} + b y_{vw})\\
B_{12}^1 : & = & -2 ( b^2 + a d ) ( a y_{uw} + b y_{vw})\,z_w\\ B_{11}^2: & = & 0\\ B_{22}^2: & = & -2\, d^2(c\, y_{uw} + d\, y_{vw})\, z_w\\ B_{12}^2 : & = & -2 c d (c y_{uw} + d y_{vw}) z_w\\
B_1^1 : & = & -2 a(a(b\,y_{uw}\, z_u - z_{uw} + d\,y_{uw} z_v) + b(b\, y_{vw}\, z_u + d\, y_{vw}\, z_v - z_{vw})) - \\ {} & {} & (a^3\, x_{uu} + a^2 b(2\, x_{uv} + y_{uu}) + a b^2(x_{uv} + 2\, y_{uv}) + b^3 y_{vv}) z_w\\ B_2^1 : & = & -a^2(c x_{uu} + d\ y_{uu})z_w - 2 a b(b y_{uw}z_u - z_{uw} + d y_{uw}z_v + c x_{uv} z_w + d y_{uv} z_w) - \\ {} & {} & b^2 ( 2 b y_{vw} z_u + 2 d y_{vw} z_v - 2 z_{vw} + c x_{uv} z_w + d y_{vv} z_w)
\end{array} $$
$$ \begin{array}{lcl} C_1: & = & -a^3 x_{uu} z_u - a^2(b(2 x_{uv} + y_{uu}) z_u - z_{uu} + c x_{uu} z_v + d y_{uu}z_v) - \\ {} & {} & a b(b (x_{uv} + 2 y_{uv}) z_u - 2 z_{uv} + 2(c x_{uv} + d y_{uv}) z_v) - b^2(b y_{vv}z_u + c x_{uv} z_v + d y_{vv} z_v - z_{vv})\\
B_1^2: & = & 2 c((c(z_{uw} - d y_{uw} z_v) + d(-d y_{vw} z_v + z_{vw})) - a d^2 x_{uv} z_w\\ B_2^2 :& = &-2 d(-c z_{uw} + c d y_{uw} z_v + d^2 y_{vw} z_v - d z_{vw}) - (c^3 x_{uu} + c^2\\
{} & {} & d(2 x_{uv} + y_{uu}) + c d^2(x_{uv} + 2 y_{uv}) + d^3y_{vv}) z_w\\
C_2 : & = & -a d^2 x_{uv} z_u - c^3 x_{uu} z_v + c^2(z_{uu} - d (2 x_{uv} + y_{uu}) z_v)+ \\
{} & {} & c d(2 z_{uv} - d (x_{uv} + 2 y_{uv})z_v) + d^2(-d y_{vv} z_v + z_{vv})
\end{array} $$
\noindent {\it Evolution of $w_t$}. \noindent The evolution equation of $w$ is :
\begin{equation}\label{eqnwt} w_t=\displaystyle{\frac{1}{z_y\,y_w-z_w}\,z_t} \end{equation}
\noindent where the function $z$ satisfies the non linear PDE: \begin{equation} z_t= \frac{z_{xx}z_{yy}-z_{xy}^2 }{(1+z_y^2)z_{xx}-2\,z_x z_y z_{xy}-(1+z_x^2)z_{yy}} \end{equation}
\noindent By replacing the partial derivatives of $z$ in terms of the derivatives of $w$, we find the linearization $L$ to the fully non linear operator given by the Equation~\ref{eqnwt}.
\noindent {\it Linearization}. Let $\tilde w$ be a point close to the initial data $w_0$. Then,
the linearization of the operator given by the Equation~\ref{eqnwt} is given by:
$$L(\tilde{w})=a_{11}\,\tilde{w}_{11}+2\,a_{12}\,\tilde{w}_{12}+a_{22}\,\tilde{w}_{22}+b_1\,\tilde{w}_1+b_2\,\tilde{w}_2+c\,\tilde{w}+d $$
\noindent where its coefficients behave as follows: $$ \begin{array}{ll} & a_{11}\approx \frac{ w_1^2}{w_{11}^2}\, g_{11};\,\, a_{12}\approx \frac{ w_{11}\,w_1}{w_{11}^2}\,g_{12}\\ & a_{11}\approx \frac{ w_{11}^2}{w_{11}^2}\,g_{22};\,\, b_{1}\approx \frac{ w_{11}\,w_1}{w_{11}^2}\,h_1\\ & b_{2}\approx \frac{ w_{11}^2}{w_{11}^2}\,h_2 \end{array} $$ \noindent where $\{g_{ij}\}$, $h_i$, $c$ and $d$ with $i,j=1,2$ are functions which belong to the space $C^{\alpha}_s$. In particular $g_{22}$ and $ b_2$ belong to $C^{\alpha,p}_s$. It follows that the coefficients of the operator $L$ satisfy the same conditions of the operator of Theorem~\ref{thm:exi2}.
\noindent {\it We say that $\Sigma_0$ is of H\"older class $C^{k+2+\alpha,p}_s$ if the function $w_0$ obtained belongs the space $C^{k+2+\alpha,p}_s(\mathcal D)$.}
Based on the above definition, we state the following theorem:
\begin{theorem}\label{thm:exi4} Assume that the initial surface $\Sigma_0$ belongs to the class $C^{k+2+\alpha,p}_s$ and satisfies the non-degeneracy condition~($\star$) . Then, under the coordinate change $\Phi$, the HMCF with initial data the surface $\Sigma_0$ converts into the initial value problem \[ \left\{ \begin{array}{ll}
Mw =0 & on\,\,{\mathcal{D}}\times [0,T]\\ w=w_0 & at \,\,\,\,\,\,\,t=0 \end{array} \right. \] with $w_0 \in C^{k+2+\alpha,p}_s(\mathcal D)$ and $$Mw= w_t - F(t,u,v,w,Dw, D^2w)$$ satisfying the hypotheses of Theorem \ref{thm:exi3}. \end{theorem} \noindent As an immediate Corollary of Theorem \ref{thm:exi3} and Theorem \ref{thm:exi4} we obtain the following existence result:
\begin{thm} Under the same assumptions as in Theorem \ref{thm:exi4}, there exists a number $\tau_k >0$ for which the HMCF with initial data the surface $\Sigma_0$ admits a solution $\Sigma_t$ on $0 \leq t \leq \tau_k$. Moreover, under the coordinate change $\Phi$ the strictly convex part $\Sigma_2(t)$ of $\Sigma_t$ is converted to a function $w(t)$ which belongs to the H\"older class $C^{k+2+\alpha,p}_s(Q_k)$, on $Q_k = \mathcal D \times [0,\tau_k]$. \end{thm}
\begin{theorem}\label{thmreg} Assume that the initial surface $\Sigma_0$ satisfies the assumptions of Theorem~\ref{thm1}. Then, the solution $\Sigma_t$ of the HMCF is converted, via the coordinate change studied in Section~\ref{global}, to a function $w$ which belongs, for any positive integer $k$, to the H\"older class $C^{k+2+\alpha,p}_s(Q)$, on $Q = \mathcal D \times (0,T]$. Moreover, for any $\tau$ in $0<\tau<T$ we have \begin{equation}\label{eqn-w}
||w||_{ C^{k+2+\alpha,p}_s(\mathcal D \times [\tau,T])}
\leq C_k(\tau, ||w_0||_{C^{2+\alpha,p}_s(\mathcal D)}) \end{equation} \end{theorem}
\begin{proof} We omit the proof as it is similar to the one done in~ \cite{DH2}. Also, for more details we invite the reader to look at the Ph.D. thesis~\cite{Cap} of the first author. \end{proof} \section{The proof of the Main Theorem}\label{sec-main}
In this section we will give the proof of the Main Theorem stated in the Introduction. We will actually prove the following stronger result, where we relax the regularity assumptions on the initial surface.
\begin{theorem}\label{thm6}
Assume that the strictly convex part $\Sigma_2$ of the initial surface $\Sigma_0$ belongs to the class $C^{2+\alpha,p}_s$ and satisfies the non-degeneracy conditions~($\star$). Then, the {\it HMCF} $$\frac {\partial P}{\partial t} =\frac{ K}{H}\, \stackrel{\rightarrow}{N}
\quad \qquad t \in [0,T]$$ with initial data the surface $\Sigma_0$ admits a solution $\Sigma_t$ which is smooth up to the interface, for $0<t\leq T$. In particular, the interface $\Gamma_t$ is a smooth curve for every $0<t\leq T$ which moves by the curve shortening Flow. \end{theorem}
\noindent{\em Remark.} It can be easily checked that if the initial surface satisfies the conditions of the Main Theorem, then it will satisfy the weaker conditions of Theorem~\ref{thm6}.
\begin{proof}
Assume that the strictly convex part $\Sigma_2$ of the initial surface $\Sigma_0$ belongs to the class $C^{2+\alpha,p}_s$ and satisfies the non-degeneracy condition~($\star$), then we have proven existence for the {\it HMCF} in Theorem~\ref{thm:exi4}.\\ \noindent From Theorem~\ref{thm:exi4} we have that $w \in C^{k+2+\alpha,p}_s(\mathcal D \times (0,T])$, for all nonnegative integers $k$. In particular this implies that for all integers $k$ we have $w(t) \in C^{k,p}(\mathcal D$,
for all $\tau$ in $0<t<T$. It follows that $w$ is $C^{\infty,p}$ smooth up to the boundary of $\mathcal D$. Going back to the original coordinates, we conclude that the strictly convex part of the surface $\Sigma_t$, $0 < t \leq T$ is smooth up to $z=0$ and that the interface $\Gamma_t$ is smooth.
\end{proof}
\section{Comparison principle}\label{sec-cp} In this final section we will give the proof of the comparison principle for the HMCF and we will show that the solution given in the Main Theorem is a viscosity solution.
\begin{prop} {\bf (Comparison principle)} Let $\Sigma_0$ be a surface of class $C^{2+\alpha,p}_s$ that
satisfies condition~($\star$), and let $\Sigma^{+}$ be a smooth, strictly
convex surface containing $\Sigma_0$ at time $t=0$, then the surface
$\Sigma_t$ obtained by evolving $\Sigma_0$ by the HMCF,
is contained in the surface $\Sigma^{+}_t$ obtained by evolving
$\Sigma^{+}_0$ by the HMCF up to the time of existence
of $\Sigma_t$. Analogously, if $\Sigma_0$ contains a smooth,
strictly convex surface $\Sigma^{-}$ at time $t=0$, then the surface
$\Sigma_t$ contains the surface $\Sigma^{-}_t$ obtained by evolving
$\Sigma^{-}_0$ by the HMCF up to the time of
existence of $\Sigma_t$.
\end{prop}
\begin{proof} We observe that by the the classical maximum principle the surfaces $\Sigma_t$ and $\Sigma^{+}_t$ cannot touch were they are both strictly convex. Next, we suppose that there exists a time $\bar{t}$ where they first touch at a point $\bar{P}$, then this cannot happen in the interior of the flat side. Hence $\bar{P}$ belongs to the boundary of the flat side. Suppose $\Sigma_t$ has the flat side on the $x=0$ plane, then the tangent plane to the surface at the point $\bar P$ would be contained in the $x=0$ plane. Now if the two regions touch, then, because they are of class $C^1$, the tangent to $\Sigma^{+}_{\bar{t}}$ at $\bar P$ would be contained in the $x=0$ plane. But $\Sigma_{\bar{t}}^{+}$ is strictly convex, hence this would imply that a part of $\Sigma_{\bar{t}}^{+}$ is inside $\Sigma_{\bar{t}}$ which leads to a contradiction. The second part of the proof is straightforward since if $\Sigma$ contains a smooth, strictly convex surface $\Sigma^{-}$ at time $t=0$, then the two surfaces cannot touch at the flat side of $\Sigma_t$ because the flat side does not move in its normal direction. Once again, by the classical maximum principle for parabolic equations the two surfaces cannot touch where they are strictly convex either. \end{proof}
\begin{corollary}{\bf (Viscosity solutions)} Let $h\in \mathbb{N}$, $0<\gamma\leq 1$. Let $\Sigma_0$ be a convex surface of class $C^{h,\gamma}$ that satisfies the hypothesis of the Main Theorem. Then the solution $\Sigma_t$ given by Theorem~\ref{thm6} is a viscosity solution of class $C^{h,\gamma}$. Moreover, this solution is unique upon satisfying the non-degeneracy condition~$(\star)$. \end{corollary} \begin{proof} It is a consequence of the comparison principle. \end{proof}
\end{document} |
\begin{document}
\title{Some remarks on biharmonic elliptic problems with a singular nonlinearity \thanks { Mathematics Subject Classification (2000). Primary: 35G30, 35J40.} \thanks {Biharmonic operator, singular nonlinearity, minimal solutions, extremal solutions;} \thanks { This work was supported by the the Natural Science Foundation of China (Grant No: 10971061).}} \author{Baishun Lai } \date{} \maketitle \begin{center} \begin{minipage}{130mm} {\small {\bf Abstract}\ \ \ We study the following semilinear biharmonic equation $$ \left\{ \begin{array}{lllllll} \Delta^{2}u=\frac{\lambda}{1-u}, &\quad \mbox{in}\quad \mathbb B,\\ u=\frac{\partial u}{\partial n}=0, &\quad \mbox{on}\quad \partial\mathbb B,\\
\end{array} \right.
$$ where $\mathbb B$ is the unit ball in $\mathbb R^{n}$ and $n$ is the exterior unit normal vector. We prove the existence of $\lambda^{*}>0$ such that for $\lambda\in (0,\lambda^{*})$ there exists a minimal (classical) solution $\underline{u}_{\lambda}$, which satisfies $0<\underline{u}_{\lambda}<1$. In the extremal case $\lambda=\lambda^{*}$, we prove the existence of a weak solution which is unique solution even in a very weak sense. Besides, several new difficulties arise and many problems still remain to be solved. we list those of particular interest in the final section. } \end{minipage} \end{center} \vskip 0.2in \setcounter{equation}{0}
\setcounter{section}{0} \section{Introduction and results}
\quad \quad In the last forty years a great deal has been written about existence and multiplicity of solutions to nonlinear second order elliptic problems in bounded and unbounded domains of $\mathbb R^{n} (n\geq2)$. Important achievements on this topic have been made by applying various combinations of analytical techniques, which include the variational and topological methods. For the latter, the fundamental tool which has been widely used is the maximum principle. However, for higher order problems, a possible failure of the maximum principle causes several technical difficulties, which attracted the interest of many researchers. In particular, recently fourth order equations with an singular non-linearity have been studied extensively. The motivation for considering these equations stems from a model for the steady states of a simple micro electromechanical system (MEMS) which has the general form (see for example \cite{Pe}) $$ \left\{ \begin{array}{lllllll}
\alpha \Delta^{2}u=(\beta\int_{\Omega}|\nabla u|^{2} dx+\gamma)\Delta u+ \frac{\lambda f(x)}{(1-u)^{2}(1+\chi\int_{\Omega}\frac{dx}{(1-u)^{2}})}& \mbox{in}\ \ \Omega,\\ 0<u<1& \mbox{in}\ \ \Omega,\\ u=\alpha\frac{\partial u}{\partial n}=0 & \mbox{on }\ \partial\Omega, \end{array} \right. \eqno(M_{\lambda}) $$ where $\Delta^{2}(\cdot):=-\Delta(-\Delta)$ denotes the biharmonic operator, $\Omega\subset\mathbb R^{n}$ is a smooth bounded domain, $n$ denotes the outward pointing unit normal to $\partial\Omega$ and
$\alpha,\beta,\gamma,\chi\geq0,$ are physically relevant constants, $f\geq 0$ represents the permittivity profile, $\lambda>0$ is a constant which is increasing with respect to the applied voltage.\vskip 0.1in
Take $\alpha=\beta=\chi=0$ and $\gamma=1$, one obtain a simple approximation of $(M_{\lambda})$ $$ \left\{ \begin{array}{lllllll} -\Delta u=\lambda\frac{f(x)}{(1-u)^{2}} &\ \ \mbox{in}\ \Omega,\\ 0<u<1 &\ \ \mbox{in}\ \Omega,\\ u=0 &\ \ \mbox{on}\ \partial\Omega. \end{array} \right. \eqno(S_{\lambda}) $$ This simple model, which lends itself to the vast literature on second order semilinear eigenvalue problems, is already a rich source of interesting mathematical problems, see e.g. \cite{Es,Es1,Gh} and the references cited therein.
The case where $\gamma=\beta=\chi=0$ and $\alpha=1, f(x)\equiv 1$ in the above model, that is when we replace $(1-u)^{-2}$ with $(1-u)^{-p}$ $$ \left\{ \begin{array}{lllllll} \Delta^{2} u=\frac{\lambda}{(1-u)^{p}} &\ \ \mbox{in}\ \Omega,\\ 0<u<1 &\ \ \mbox{in}\ \Omega,\\ u=\frac{\partial u}{\partial n}=0 &\ \ \mbox{on}\ \partial\Omega. \end{array} \right. \eqno(P_{\lambda}) $$ Because of the lack of a \textquotedblleft maximum principle ", which play such a crucial role in developing the theory for the Laplacian, for $\Delta^{2}$ with Dirichlet boundary condition in general domains (i.e., $\Omega\neq \mathbb B$), very little is known about $(P_{\lambda})$. As far as we are aware, only a paper \cite{Ghe} study this problem for general domains. However, if $p>1$ and the $\Omega$ is a ball, $(P_{\lambda})$ has recently been studied extensively, see e.g. \cite{Ca,Co1,D,Gu4,Li,Mo} and its references. One of the reasons to study $(P_{\lambda})$ in a ball is that a maximum principle holds in this situation, see \cite{Bo}, and so some tools that are well suited for $(S_{\lambda})$ can work for $(P_{\lambda})$. The second reason is that one can easily find a explicit singular radial solution, denoted by $1-
|x|^{\frac{4}{p+1}} (p>1)$, of $(P_{\lambda})$ for $\Omega=\mathbb B$ and a suitable parameter $\lambda$ which satisfy the first boundary condition but not the second. The singular radial solution, called \textquotedblleft ghost" singular solution, play a fundamental role to characterize the \textquotedblleft true" singular solution, see in particular \cite{Co1}.
In this paper, we will focus essentially our attention on the case where $p=1$ and $\Omega$ is a ball, namely $$ \left\{ \begin{array}{lllllll} \Delta^{2}u=\frac{\lambda}{1-u} & \mbox{in}\ \mathbb B,\\ 0<u\leq 1 & \mbox{in}\ \ \mathbb B, \\ u=\frac{\partial u}{\partial n} =0 & \mbox{on}\ \ \partial \mathbb B.\\ \end{array} \right. \eqno(1.1)_{\lambda} $$ For the corresponding second order problem, which is related to the general study of singularities of minimal hypersurfaces of Euclidean space, has been studied by Meadows, see \cite{Mea}. In that case, however, the start point was an explicit singular solution (i.e., $u_{s}(r)=r^{2}$) with parameter $\lambda=n-1$. When turning to the biharmonic problem $(1.1)_{\lambda}$, one can not find any explicit singular solution even \textquotedblleft ghost" singular solution which causes several technical difficulties. The first purpose of the present paper is to extend $(1.1)_{\lambda}$ some well-known results relative to $(P_{\lambda}) $. The second (and perhaps most important) purpose of the present paper is to emphasize some striking differences between $(1.1)_{\lambda}$ and $(P_{\lambda}) $. \vskip0.2in
\subsection{ Preliminaries}
\quad \quad Besides classical solution i.e. $u\in C^{4}(\bar{\mathbb B})$ which satisfy $(1.1)_{\lambda}$, let us introduce the class of weak solutions we will be dealing with. We denote by $H_{0}^{2}(\mathbb B)$ the usual Sobolev space which can be defined by completion as follows: $$
H_{0}^{2}(\mathbb B):=cl\{u\in C_{0}^{\infty}(\mathbb B): \|\Delta u\|_{2}<\infty\} $$ and which is an Hilbert space endowed with the scalar product $$ (u,v)_{H_{0}^{2}(\mathbb B)}:=\int_{\mathbb B}\Delta u\Delta v dx $$
\begin{definition}\label{D1.1}
We say that $u\in L^{1}(\mathbb B)$ is a weak solution of $(1.1)_{\lambda}$ provided $0\leq u\leq 1$ almost everywhere, $\frac{1}{1-u}\in L^{1}(\mathbb B)$ and \begin{equation}\label{Equ1} \int_{\mathbb B} u\Delta^{2}\varphi dx=\lambda \int_{\mathbb B}\frac{\varphi}{(1-u)}dx,\ \ \forall \varphi\in C^{4}(\bar{\mathbb B})\cap H_{0}^{2}(\mathbb B) \end{equation} \end{definition}
When in (\ref{Equ1}) the equality is replaced by the inequality $\geq$ (resp.$\leq$) and $\varphi\geq0$, we say that $u$ is a weak super-solution (resp. weak sub-solution) of (\ref{Equ1}) provided the following boundary conditions are satisfied: $u=0 $ (resp.$=$) and $\frac{\partial u}{\partial n}\leq 0$ (resp.$\geq$) on $\partial \mathbb B$. \vskip 0.1in
\begin{definition}\label{D1.1} We call a solution $u$ of $(1.1)_{\lambda}$ minimal if $u\leq v$ a.e. in $\mathbb B$ for any further solution $v$ of $(1.1)_{\lambda}$ \end{definition}
If $u$ is a classical solution of $(1.1)_{\lambda}$, then it turns out to be well defined the linearized operator at $u$ $$ L_{u}:=\Delta^{2}-\frac{\lambda}{(1-u)^{2}} $$ which yields the following notion of stability \vskip 0.1in
\begin{definition}\label{D1.3}
A classical solution $u$ of $(1.1)_{\lambda}$ is semi-stable provided $$ \mu_{1}(u)=\inf\left\{\int_{\mathbb B}(\Delta \varphi)^{2}-\frac{\lambda\varphi^{2}}{(1-u)^{2}}:
\phi\in H_{0}^{2}(\mathbb B), \|\phi\|_{L^{2}}=1\right\}\geq0 $$ If $\mu_{1}(u)>0$ we say that $u$ is stable. \end{definition}
As far as we are concerned with weak solutions, the linearized operator is no longer well defined,
however we introduce the following weaker notion of stability.\vskip 0.1in
\begin{definition}\label{D1.4}
A weak solution $u$ to $(1.1)_{\lambda}$ is said to be weakly stable if $\frac{1}{(1-u)^{2}}\in L^{1}(\mathbb B)$ and the following holds: $$
\int_{\mathbb B}|\Delta\varphi|^{2}dx\geq \int_{\mathbb B}\frac{\lambda\varphi^{2}}{(1-u)^{2}}, \varphi\in H_{0}^{2}(\mathbb B), \varphi\geq0. $$ \end{definition}
According to the class of solutions which we consider, let us introduce the following values: \begin{equation}\label{Equ2}
\begin{array}{lllllll} \lambda^{*}:=\sup\{\lambda\geq0: (1.1)_{\lambda}\ \ \mbox{ posses a weak solution}\};\\ \lambda_{*}:=\sup\{\lambda\geq0: (1.1)_{\lambda}\ \ \mbox{ posses a classical solution}\}.\\ \end{array}
\end{equation}
\begin{remark}\label{R1.1}
Clearly, a classical solution is also a weak solution, so that one has $\lambda_{*}\leq \lambda^{*}$. Moreover, by standard elliptic regularity theory for the biharmonic operator \cite{Ag}, any weak solution of $(1.1)_{\lambda}$ which satisfies $\|u_{\lambda}\|< 1$ turns out to be smooth. \end{remark}
Besides, we give a notion of $H_{0}^{2}$($\mathbb B$)- weak solutions, which is an intermediate class between classical and weak solutions.
\begin{definition}\label{D1.5}
We say that $u$ is a $H_{0}^{2}$($\mathbb B$)- weak solution of (1.1) if $(1-u)^{-1}\in L^{1}(\mathbb B)$ and if $$ \int_{\mathbb B}\Delta u\Delta\phi=\lambda\int_{\mathbb B}\phi(1-u)^{-1},\ \ \ \forall \phi\in C^{4}(\bar{\mathbb B})\cap H_{0}^{2}(\mathbb B). $$
We say that $u$ is a $H_{0}^{2}$($\mathbb B$)- weak super-solution (resp. $H_{0}^{2}$($\mathbb B$)- weak sub-solution) of $(1.1)_{\lambda}$
if for $\phi\geq0$ the equality is replaced with $\geq$ (resp.$\leq$) and $u\geq 0$ (resp. $\leq$), $\frac{\partial u}{\partial n}\leq 0$
(resp. $\geq$) on $\partial \mathbb B$. \end{definition}
\subsection{Main results}
\quad \quad In order to state our results, we denote by $\nu_{1}$ the first eigenvalue of the biharmonic operator on $\mathbb B$ with Dirichlet boundary conditions, which is characterized variationally as follows: $$
\nu_{1}:=\inf\left\{\int_{\mathbb B}|\Delta u|^{2}dx:\ \ u\in H_{0}^{2}(\mathbb B), \|u\|=1\right\}. $$ It is well known that $\nu_{1}>0$, that it is simple, isolated and that the corresponding eigenfunctions $\psi>0$, spherically symmetric , radially decreasing and do not change sign.
We may now state the following theorem. \vskip 0.1in
\begin{theorem}\label{result1}
There exists $\lambda_{*}>0$ such that for $0<\lambda<\lambda_{*}$, $(1.1)_{\lambda}$ poses a minimal classical solution, denoted by $\underline{u}_{\lambda}$, which is positive and stable. Moreover, $\lambda_{*}$ satisfies the following bounds: $$ \max\{4n(n-2), 2n(n+2)\}\leq \lambda_{*}\leq \frac{\nu_{1}}{4}. $$ \end{theorem}
It is remarkable that at $\lambda_{*}$ there is an immediate switch from existence of regular minimal solutions to nonexistence of any (even singular) solution. The only possibly singular minimal solution corresponds to $\lambda=\lambda_{*}$. This result is known from \cite{Br} for the second order problem($S_{\lambda}$), but the method used there may not be carried over to fourth order problems. Nevertheless, the result extends to biharmonic case in the following theorem.
\begin{theorem}\label{result2} \ \ The following holds: $$ \lambda_{*}=\lambda^{*}. $$ In particular, for $\lambda>\lambda^{*}$ there are no solutions, even in the weak sense. Furthermore, for almost every $x\in \mathbb B$, there exists $$ u^{*}(x):=\lim_{\lambda\to\lambda^{*}}u_{\lambda}(x) $$ and $u^{*}(x)$ is a weakly stable $H_{0}^{2}(\mathbb B)-$ weak solution of $(1.1)_{\lambda}$, which is called the extremal solution.
If $n\leq 4$ then the extremal solution $u^{*}$ of $(1.1)_{\lambda}$ is smooth, i.e., $u^{*}=\lim_{\lambda\to\lambda^{*}}\underline{u}_{\lambda}(x)$ exists in the topology of $C^{4}(\mathbb B)$. It is the unique regular solutions to $(1.1)_{\lambda^{*}}$. \vskip 0.1in \end{theorem}
From the above theorem, we note that the function $u^{*}$ exists in any dimension, dose solve $(1.1)_{\lambda^{*}}$ in the $H_{0}^{2}(\mathbb B)$ weak sense and it is a classical solution in dimensions $1\leq n\leq 4$. This will allow us to start another branch of nonminimal (unstable ) solutions. Besides, inspired by \cite{Ca,Ma,Da} we get the following uniqueness of the extremal solution of $(1.1)_{\lambda}$, which gives Theorem1.3.
\vskip 0.1in
\begin{theorem} \label{result3}
\ \ Let $v$ be a weak super-solution of $(1.1)_{\lambda}$ with parameter $\lambda^{*}$. Then $v=u^{*}$; in particular $(1.1)_{\lambda}$ has a unique weak solution. \vskip 0.1in \end{theorem}
From this theorem, we know that there are no strict super-solutions to equation $(1.1)_{\lambda^{*}}$. \vskip 0.1in
\begin{corollary} \label{C1.1} \ \ Let $u_{\lambda}\in H_{0}^{2}(\mathbb B)$ be a weak solution of
$(1.1)_{\lambda}$ such that $\|u_{\lambda}\|=1$. Then $u_{\lambda}$ is weakly stable if and only if $\lambda=\lambda^{*}$ and $u_{\lambda}=u^{*}$ \end{corollary}
We may also characterize the uniform convergence to 0 of $\underline{u}_{\lambda}$ as $\lambda\to 0$ by giving the precise rate of its extinction.
\begin{theorem} \label{result4}
For all $\lambda\in (0,\lambda^{*})$ let $\underline{u}_{\lambda}$ be the minimal solution of $(1.1)_{\lambda}$ and let $$
V_{\lambda}(x)=\frac{\lambda}{8n(n+2)}\left[1-|x|^{2}\right]^{2}. $$ Then $\underline{u}_{\lambda}>V_{\lambda}(x)$ for all
$\lambda<\lambda^{*}$ and all $|x|<1$, and $$ \lim_{\lambda\to0}\frac{\underline{u}_{\lambda}}{V_{\lambda}(x)}=1\ \mbox{uniformly with respect to}\ \ x\in \mathbb B. $$ \end{theorem}
\subsection{ Key-ingredients}
\quad \quad Now we give some comparison principles which will be used throughout the paper
\begin{lemma}\label{L1.1} \ (Boggio's principle, \cite{Bo}) If $u\in C^{4}(\bar{\mathbb B}_{R})$ satisfies $$ \left\{ \begin{array}{lllllll} \Delta^{2}u\geq 0 & \mbox{in}\ \ \mathbb B_{R},\\ u=\frac{\partial u}{\partial n}=0 & \mbox{on}\ \ \partial \mathbb B_{R}, \end{array} \right. $$ then $u\geq 0$ in $\mathbb B_{R}$. \end{lemma}
\begin{lemma}\label{L1.2} Let $u\in L^{1}(\mathbb B_{R})$ and suppose that $$ \int_{\mathbb B_{R}}u\Delta^{2}\varphi\geq0 $$ for all $\varphi\in C^{4}(\bar{\mathbb B}_{R})$ such that $\varphi\geq0$ in
$\mathbb B_{R}$, $\varphi|_{\partial \mathbb B_{R}}=\frac{\partial
\varphi}{\partial n}|_{\partial \mathbb B_{R}}=0$. Then $u\geq 0$ in $\mathbb B_{R}$. Moreover $u\equiv 0$ or $u>0$ a.e., in $\mathbb B_{R}$. \end{lemma}
\vskip0.1in
For a proof see Lemma 17 in \cite{Ar1}. From this lemma, we know that any solution of $(1.1)_{\lambda}$ is necessarily positive a.e. inside the ball.
\vskip0.1in \noindent \begin{lemma}\label{L1.3} \ If $u\in H^{2}(\mathbb B_{R})$ is radial, $\Delta^{2}u\geq 0$ in $\mathbb B_{R}$ in the weak sense, that is $$ \int_{\mathbb B_{R}}\Delta u\Delta\varphi\geq 0 \ \ \forall \varphi \in C_{0}^{\infty}(\mathbb B_{R}), \ \varphi\geq0 $$
and $u|_{\partial \mathbb B_{R}}\geq0, \frac{\partial u}{\partial n}|_{\partial \mathbb B_{R}}\leq 0$ then $u\geq 0$ in $\mathbb B_{R}$.\vskip0.1in \end{lemma} {\bf Proof.} For the sake of completeness, we include a brief proof here. We only deal with the case $R=1$ for simplicity. Solve $$ \left\{ \begin{array}{lllllll} \Delta^{2}u_{1}=\Delta^{2}u & \mbox{in} \ \ \mathbb B\\ u_{1}=\frac{\partial u_{1}}{\partial n}=0 & \mbox{on}\ \ \partial \mathbb B \end{array} \right. $$ in the sense $u_{1}\in H_{0}^{2}(\mathbb B)$ and $\int_{\mathbb B}\Delta u_{1}\Delta\varphi=\int_{\mathbb B}\Delta u\Delta\varphi$ for all $\varphi \in C_{0}^{\infty}(\mathbb B)$. Then $u_{1}\geq 0$ in $\mathbb B$ by Lemma 2.2.
Let $u_{2}=u-u_{1}$ so that $\Delta^{2}u_{2}=0$ in $\mathbb B$. Define $f=\Delta u_{2}$. Then $\Delta f=0$ in $\mathbb B$ and since $f$ is radial we find that $f$ is a constant. It follows that $u_{2}=ar^{2}+b$. Using the boundary conditions we deduce $a+b\geq 0$ and $a\leq0$, which imply $u_{2}\geq0$.\vskip 0.1in
\begin{lemma} \label{L1.4} Let $f\in L^{1}(\mathbb B_{R}), f\geq0$ almost everywhere. Then there exists a unique $u\in L^{1}(\mathbb B_{R})$ such that $u\geq0$ and \begin{equation}\label{Equ3} \int_{\mathbb B_{R}}u \Delta^{2}\varphi=\int_{\mathbb B_{R}} f \varphi,\ \ \varphi\in C^{4}(\bar{\mathbb B}_{R})\cap H_{0}^{2}(\mathbb B_{R}). \end{equation} Moreover, there exists $C>0$ which does not depend on $f$ such that
$\|u\|_{1}\leq C\|f\|_{1}$. \vskip0.1in \end{lemma}
{\bf Proof.} The proof is standard, see \cite {Ar1}, we give a proof here for the sake of completeness. The uniqueness is clear. Indeed, let $v_{1}$ and $v_{2}$ be two solutions of (\ref{Equ3}). Then $v=v_{1}-v_{2}$ satisfies $$ \int_{\mathbb B}v\Delta\varphi=0 \ \ \varphi\in C^{4}(\bar{\mathbb B}_{R})\cap H_{0}^{2}(\mathbb B_{R}). $$ Given any $\zeta \in C_{0}^{\infty}(\mathbb B)$ let $\varphi$ be the solution of $$ \left\{ \begin{array}{lllllll} \Delta^{2}\varphi= \zeta & \mbox{in}\ \ \mathbb B,\\ \varphi=\frac{\partial \varphi}{\partial n}=0 & \mbox{on}\ \ \partial \mathbb B. \end{array} \right. $$ It follows that $$ \int_{\mathbb B}v\zeta=0. $$ Since $\zeta$ is arbitrary, we deduce that $v=0$.
For the existence, Given an integer $k\geq0$ we set $f_{k}=\min\{f(x),k\}$, so that $f_{k}\rightarrow f $ as $k\to\infty$ in $L^{1}(\mathbb B)$. Let $v_{k}$ be the solution of \begin{equation}\label{Equ4} \left\{ \begin{array}{lllllll} \Delta^{2}v_{k}= f_{k} & \mbox{in}\ \ \mathbb B,\\ v_{k}=\frac{\partial v_{k}}{\partial n}=0 & \mbox{on}\ \ \partial \mathbb B. \end{array} \right.
\end{equation} The sequence $(v_{k})_{k\geq0}$ is clearly monotone nondecreasing. It is also a cauchy sequence in $L^{1}(\mathbb B)$ since $$ \int_{\mathbb B}(v_{k}-v_{l})=\int_{\mathbb B}(f_{k}-f_{l})\zeta_{0}, $$ where $\zeta_{0}$ is defined by $$ \left\{ \begin{array}{lllllll} \Delta^{2}\zeta_{0}= 1 & \mbox{in}\ \ \mathbb B,\\ \zeta_{0}=\frac{\partial \zeta_{0}}{\partial n}=0 & \mbox{on}\ \ \partial \mathbb B. \end{array} \right. $$ Hence $$
\int_{\mathbb B}|v_{k}-v_{l}|\leq C\int_{\mathbb B}|f_{k}-f_{l}|dx. $$ Passing to the limit in (\ref{Equ4}) (after multiplication by $\varphi$) we obtain (\ref{Equ3}) and $u\geq0$ according to the Lemma 1.2. Finally, taking $\varphi=\zeta_{0}$ in (\ref{Equ3}), we obtain $$
\|v\|_{L^{1}}=\int_{\mathbb B}v=\int_{\mathbb B}f\zeta_{0}\leq C\|f\|_{L^{1}} $$ and the proof is completed.
\begin{proposition}\label{P1.1}
Assume the existence of a weak super-solution $U$ of $(1.1)_{\lambda}$. Then there exists a weak solution $u$ of $(1.1)_{\lambda}$ so that $0\leq u\leq U$ a.e in $\mathbb B$. \end{proposition}
{\bf Proof.}\ \ By means of a standard monotone iteration argument, set $u_{0}:= U$ and define recursively $u_{n+1}\in L^{1}(\mathbb B)$ as the unique solution of $$ \int_{\mathbb B}u_{n+1}\Delta^{2}\varphi dx=\lambda\int_{\mathbb B}\frac{\varphi}{(1-u_{n})^{2}}dx, \varphi\in C^{4}(\bar{\mathbb B})\cap H_{0}^{2}(\mathbb B), $$ then we have $$ \int_{\mathbb B}(u_{n}-u_{n+1})\Delta^{2}\varphi dx\geq 0, \varphi\in C^{4}(\bar{\mathbb B})\cap H_{0}^{2}(\mathbb B) $$ and Lemma \ref{L2.2} yields $0\leq u_{n+1}\leq u_{n}<U(x)$ a.e. for all $n\in \mathbb N$. Since $$ (1-u_{n})^{-1}\leq (1-U)^{-1}\ \ \in L^{1}(\mathbb B), $$ and the claim follows from the Lebesgue convergence Theorem. \vskip0.1in
We complete these preliminary results by proving a key lemma which provides a comparison principle.\vskip0.1in
\begin{lemma}\label{L1.5}
\ Assume $u_{1}$ is a weakly stable $H_{0}^{2}$($\mathbb B$)- weak
sub-solution of $(1.1)_{\lambda}$ and $u_{2}$ is $H_{0}^{2}$($\mathbb B$)- weak super-solution of $(1.1)_{\lambda}$. Then,
(1) $u_{1}\leq u_{2}$ almost everywhere in $\mathbb B$.
(2) if $u$ is a classical solution such that $\mu_{1}(u)=0$ and $U$ is any classical super-solution of $(1.1)_{\lambda}$, then $u\equiv U$. \end{lemma}
{\bf Proof.}\ \ (1) Define $\omega:=u_{1}-u_{2}$. Then by the Moreau decomposition \cite{Mo} for the biharmonic operator, there exists $\omega_{1}, \omega_{2}\in H_{0}^{2}(\mathbb B)$, with $\omega=\omega_{1}+\omega_{2}, \omega_{1}\geq0$ a.e., $\Delta^{2}\omega_{2}\leq0$ in the $H_{0}^{2}(\mathbb B)-$ weak sense and $$ \int_{\mathbb B}\Delta \omega_{1}\Delta \omega_{2}=0 $$ By Lemma \ref{L1.1}, we have that $\omega_{2}\leq 0$ a.e. in $\mathbb B$.
Given now $0\leq\varphi\in C_{0}^{\infty}(\mathbb B)$, we have that $$ \int_{\mathbb B}\Delta\omega\Delta\varphi\leq\lambda\int_{\mathbb B}(f(u_{1})-f(u_{2}))\varphi, $$ where $f(u)=(1-u)^{-1}$. Since $u$ is stable, one has $$ \lambda\int_{\mathbb B}f'(u)\omega_{1}^{2}\leq \lambda\int_{\mathbb B}(\Delta\omega_{1})^{2}=\lambda\int_{\mathbb B}\Delta\omega\Delta\omega_{1} \leq\lambda\int_{\mathbb B}(f(u_{1})-f(u_{2}))\omega_{1} $$ Since $\omega_{1}\geq \omega$ one also has $$ \int_{\mathbb B}f'(u)\omega \omega_{1}\leq \int_{\mathbb B}(f(u_{1})-f(u_{2}))\omega_{1} $$ which once re-arrange gives $$ \int_{\mathbb B}\tilde{f}\omega_{1}\geq 0, $$ where $\tilde{f}(u_{1})=f(u_{1})-f(u_{2})-f'(u_{1})(u_{1}-u_{2})$. The strict convexity of $f$ gives $\tilde{f}\leq0$ and $\tilde{f}<0$ whenever $u\neq U$. Since $\omega_{1}\geq0$ a.e. in $\mathbb B$ one sees that $\omega\leq0$ a.e. in $\mathbb B$. The inequality $u_{1}\leq u_{2}$ a.e. in $\mathbb B$ is then established.\vskip0.1in
(2)\ Let $\varphi>0$ be the first eigenfunction of $\Delta^{2}-\lambda f'(u)$ in $H_{0}^{2}(\mathbb B)$, we now, for $0\leq t\leq 1$, define $$ g(t)=\int_{\mathbb B}\Delta(tU+(1-t)u)\Delta\phi-\lambda\int_{\mathbb B}f(tU+(1-t)u)\phi, $$ where $\phi$ is the above first eigenfunction. Since $f$ is convex one sees that $$ g(t)\geq\lambda \int_{\mathbb B}[tf(U)+(1-t)f(u)-f(tU+(1-t)u)]\phi\geq0 $$ for every $t\geq0$. Since $g(0)=0$ and $$ g'(0)=\int_{\mathbb B}\Delta(U-u)\Delta\phi-\lambda f'(u)(U-u)\phi=0, $$ we get that $$ g''(0)=-\lambda\int_{\mathbb B}f''(u)(U-u)^{2}\phi\geq0. $$ Since $f''(u)\phi> 0$ in $\mathbb B$, we finally get that $U=u$ a.e. in $\mathbb B$.\qed \vskip0.1in
\setcounter{equation}{0}
\section{Existence results: proofs of Theorem \ref{result1} and \ref{result2}}
\vskip0.1in
\subsection{ The branch of minimal solutions}
\quad \quad Let us define
$$
\Lambda:=\{\lambda\geq0: (1.1)_{\lambda}\ \mbox{has a classical solution with parameter}\ \lambda \}.
$$
\begin{proposition}\label{P2.1}
For all $0\leq\lambda<\lambda_{*}$, there exists a minimal classical solution $\underline{u}_{{\lambda}}$ of $(1.1)_{\lambda}$ which is smooth and stable. Moreover, \vskip0.1in
(i) The map $\lambda\rightarrow \underline{u}_{{\lambda}},$ for $\lambda\in (0,\lambda_{*})$ is differentiable and strictly increasing;
(ii) The map $\lambda\rightarrow \mu_{1}(\underline{u}_{{\lambda}})$ is decreasing on $(0,\lambda_{*})$;
(iii) Let $\tilde{u}_{\lambda}$ be a regular solution of $(1.1)_{\lambda}$ for $\lambda\in (0,\lambda_{*})$, if $\tilde{u}_{\lambda}$ is not the minimal solution, then $\mu_{1}(\tilde{u}_{\lambda})<0$. \end{proposition}
{\bf Proof.}\ \ First we show that $\Lambda$ dose not consist of just $\lambda=0$. To this end, let $\psi_{R}$ be the first eigenfunction of the biharmonic operator subject to Dirichlet boundary conditions on $\mathbb B_{R}\supset \mathbb B$ which we normalize by $\sup_{\mathbb B_{R}}\Psi_{R}=1$ and let $\nu_{R}>0$ be the corresponding eigenvalue. Next, we are going to prove that for $\theta\in (0,1)$ the function $\psi=\theta\psi_{R}$ is a super-solution of $(1.1)_{\lambda}$ as long as $\lambda$ is sufficiently small. We have $$ 0<1-\theta\psi_{R}<1, \ \ \mbox{in}\ \ \mathbb B $$ and moreover $$ \Delta^{2}\psi=\nu_{R}\theta\psi_{R}\geq \frac{\lambda}{1-\theta\psi_{R}}=\frac{\lambda}{1-\psi} $$ provide that $$ \nu_{R}\theta\psi_{R}(1-\theta\psi_{R})\geq \lambda. $$
Notice that $$ 0<s_{1}:=\inf_{x\in\mathbb B}\psi<s_{2}:=\sup_{x\in\mathbb B}\psi<1 $$ and that $\frac{\partial \psi}{\partial n}<0$ on $\partial\mathbb B$. Thus, looking at the function $g(s)=s(1-s)$, for $s\in [s_{1}, s_{2}]$, it is easily seen that we can choose $\lambda>0$ sufficiently small such that $$ \nu_{R}\inf_{x\in\mathbb B}g(\theta\psi(x))>\lambda. $$ Since $\underline{u}\equiv0$ is a sub-solution of $(1.1)_{\lambda}$, the classical sub-super solution Theorem provides a classical solution $u_{\lambda}$ to $(1.1)_{\lambda}$. With such function $u_{\lambda}$, we can use the Boggio principle to show straightforwardly that the iterative scheme \begin{equation}\label{Equ5} \left\{ \begin{array}{lllllll} \Delta^{2} u_{n,\lambda} =\frac{\lambda}{(1-u_{n,\lambda})}& \ \ \mbox{in}\ \mathbb B, \\\\
u_{n,\lambda}=\frac{\partial u_{n,\lambda}}{\partial n} =0 & \ \ \mbox{in}\ \partial\mathbb B, \\\\
u_{0,\lambda}=0 & \ \ \mbox{in}\ \mathbb B,
\end{array} \right. \end{equation} gives rise to a monotone sequence $\{u_{n,\lambda}\}$ satisfying $$ 0=u_{0,\lambda}\leq u_{1,\lambda} ... u_{n-1,\lambda}\leq. . .\leq u_{\lambda}<1 $$ for all $n\in \mathbb N$. Therefore the minimal solution $\underline{u}_{\lambda}$ is obtained as the increasing limit $$ \underline{u}_{\lambda}(x):=\lim_{n\rightarrow\infty}u_{n,\lambda} $$ Again from the Boggio positivity preserving property (Lemma 1.1) we obtain $0<\underline{u}_{\lambda}<1$; in particular, from standard elliptic regularity theory for the biharmonic operator follows that $\underline{u}_{\lambda}(x)$ is smooth. In order to prove stability, let us argue as follows: set $$ \lambda_{**}:=\sup\{\lambda\in (0,\lambda_{*}): \mu_{1}(\underline{u}_{\lambda})>0\} $$ clearly $\lambda_{**}\leq\lambda_{*}$. Now suppose by contradiction that $\lambda_{**}<\lambda_{*}$ and let $\varepsilon>0$ sufficiently small such that $\lambda_{**}+\varepsilon<\lambda_{*}$ and $v_{\lambda_{**}+\varepsilon}$ be the corresponding minimal solution. By the definition and left continuity of the map $\lambda\to \mu_{1}(\underline{u}_{\lambda})$ we have necessarily $\mu_{1}(\underline{u}_{\lambda_{**}})=0$. Since $v_{\lambda_{**}+\varepsilon}$ is a super-solution of $(1.1)_{\lambda_{**}}$, by Lemma \ref{L1.5} we get $v_{\lambda_{**}+\varepsilon}=\underline{u}_{\lambda_{**}}$ and thus $\varepsilon=0$, a contradiction.
Since each $\underline{u}_{\lambda}$ is stable, then by setting $F(\underline{u}_{\lambda},\lambda):=-\Delta^{2}-\frac{\lambda}{1-\underline{u}_{\lambda}}$, we get that $F_{\underline{u}_{\lambda}}(\underline{u}_{\lambda},\lambda)$ is invertible for $0<\lambda<\lambda_{*}$. It then follows from Implicit Function Theorem that $\underline{u}_{\lambda}(x)$ is differentiable with respect to $\lambda$.
Now we prove the map $\lambda\to \underline{u}_{\lambda}$ is strictly increasing on $(0,\lambda_{*})$. Consider $\lambda_{1}<\lambda_{2}<\lambda_{*}$, their corresponding minimal positive solutions $\underline{u}_{\lambda_{1}}$ and $\underline{u}_{\lambda_{2}}$, and let $u^{*}$ be a solution for $(1.1)_{\lambda_{2}}$. The same as the above iterative scheme, we have $$ \underline{u}_{\lambda_{1}}=\lim_{n\to\infty}u_{n}(\lambda_{1};x)\leq u^{*}\ \ \mbox{in}\ \ \mathbb B, $$ and in particular $\underline{u}_{\lambda_{1}}\leq\underline{u}_{\lambda_{2}}$ in $\mathbb B$. Therefore, $\frac{d\underline{u}_{\lambda}}{d\lambda}\geq0$ for all $x\in \mathbb B$.
Finally, by differentiating $(1.1)_{\lambda}$ with respect to $\lambda$, and since $\lambda\to \underline{u}_{\lambda}$ is nondecreasing, we get $$ -\Delta^{2}\frac{d\underline{u}_{\lambda}}{d\lambda}-\frac{\lambda}{(1-\underline{u}_{\lambda})^{2}}\frac{d\underline{u}_{\lambda}}{d\lambda} =\frac{\lambda}{1-\underline{u}_{\lambda}}\geq 0, \ x\in \mathbb B; \ \ \frac{d\underline{u}_{\lambda}}{d\lambda}=0, \ \ x\in \partial\mathbb B. $$ Applying the strong maximum principle, we conclude that $\frac{d\underline{u}_{\lambda}}{d\lambda}>0$ on $\mathbb B$ for all $0<\lambda<\lambda_{*}$
That $\lambda\to \mu_{1,\lambda}$ is decreasing follow easily from the variational characterization of $\mu_{1,\lambda}$, the monotonicity of $\lambda\to \underline{u}_{\lambda}$, as well as the monotonicity of $(1-\underline{u}_{\lambda})^{-2}$ with respect to $\underline{u}_{\lambda}$ and the proof of the (ii) is completed.
Now we give the proof of (iii). Let $\underline{u}_{\lambda}$ be the minimal solution for $(1.1)_{\lambda}$ so that $\tilde{u}_{\lambda}\geq \underline{u}_{\lambda}$. If the linearization around $\tilde{u}_{\lambda}$ had nonnegative first eigenvalue, then Lemma \ref{L1.5} would also yield $\tilde{u}_{\lambda}\leq \underline{u}_{\lambda}$ so that $\tilde{u}_{\lambda}$ and $\underline{u}_{\lambda}$ necessarily coincide, a contradiction.
\begin{remark} Dose (iii) of Proposition (\ref{P2.1}) extend to weak solutions $u$ as formulated in [21, Theorem 3.1]? \end{remark}
\vskip0.1in
\noindent
\subsection{ Weak solutions versus classical solutions}
\begin{lemma}\label{L2.1}
Let $u_{\mu}$ be a weak solution of $(1.1)_{\mu}$ with $\mu<\lambda^{*}$. Then, for $\varepsilon>0$ sufficiently small, the problem $(1.1)_{(1-\varepsilon)\mu}$ posses a classical solution. \end{lemma} {\bf Proof.} Let $\tilde{u}\in L^{1}(\mathbb B)$ be the unique solution of $$ \int_{\mathbb B}\tilde{u}\Delta^{2}\varphi=\mu\int_{\mathbb B}\frac{(1-\varepsilon)}{1-u_{\mu}}\varphi dx,\ \ \varphi\in C^{4}(\bar{\mathbb B})\cap H_{0}^{2}(\mathbb B) $$ provided by Lemma \ref{L1.4}. By hypothesis we have $$ \int_{\mathbb B}u_{\mu}\Delta^{2}\varphi dx=\mu\int_{\mathbb B}\frac{\varepsilon}{1-u_{\mu}}\varphi dx, \ \ \varphi\in C^{4}(\bar{\mathbb B})\cap H_{0}^{2}(\mathbb B). $$ By uniqueness we get $$ (1-\varepsilon)u_{\mu}=\tilde{u} $$ whereas Lemma \ref{L1.2} yields $\tilde{u}>0$ a.e. in $\mathbb B$ and hence we may assume $$ u_{\mu}>\tilde{u}, \ \ x\in\mathbb B\setminus\{x\in\mathbb B: \tilde{u}=0\} $$ Therefore, $$ \int_{\mathbb B}\tilde{u}\Delta^{2}\varphi=\int_{\mathbb B}\frac{(1-\varepsilon)\mu}{\left(1-\frac{1}{1-\varepsilon}\tilde{u}\right)}dx \geq (1-\varepsilon)\mu \int_{\mathbb B}\frac{1}{1-\tilde{u}}dx,\ \ \varphi\in C^{4}(\bar{\mathbb B})\cap H_{0}^{2}(\mathbb B) $$ thus $\tilde{u}$ is a weak super-solution of $(1.1)_{(1-\varepsilon)\mu}$ and Proposition \ref{P2.1} yields a weak solution $v$ of $(1.1)_{(1-\varepsilon)\mu}$ which satisfies $$ 0\leq v\leq \tilde{u}<u_{\mu}\leq 1 $$ and then classical by Remark \ref{R1.1}.\vskip0.1in
\begin{remark} From this Lemma, we know that $\lambda^{*}=\lambda_{*}$, in what follows, we always denote by $\lambda_{*}$ the largest possible value of $\lambda$ such that $(1.1)_{\lambda}$ has a solution, unless otherwise stated. \end{remark}
\begin{proposition}\label{P2.2} Up to a subsequence, the convergence $$ u^{*}:=\lim_{\lambda\nearrow \lambda_{*}}\underline{u}_{\lambda}(x) $$ holds in $H_{0}^{2}(\mathbb B)$ and the extremal solution $u_{\lambda^{*}}$ satisfies \begin{equation}\label{Equ6} \int_{\mathbb B}\Delta u^{*}\Delta\varphi=\lambda_{*}\int_{\mathbb B}\frac{1}{(1-u^{*})},\ \ \varphi\in C_{0}^{\infty}(\mathbb B). \end{equation} In particular, the extremal solution is weakly stable and if
$\|u^{*}\|_{\infty}<1$ then $\mu_{1}(u^{*})=0$. \end{proposition}
{\bf Proof.} Since $\underline{u}_{\lambda}$ is stable, we have \begin{equation}\label{Equ7}
\lambda\int_{\mathbb B}\frac{\underline{u}_{\lambda}^{2}}{(1-\underline{u}_{\lambda})^{2}}dx\leq\int_{\mathbb B}|\Delta\underline{u}_{\lambda}|^{2}dx =\int_{\mathbb B}\underline{u}_{\lambda}\Delta^{2}\underline{u}_{\lambda}=\lambda\int_{\mathbb B}\frac{\underline{u}_{\lambda}}{1-\underline{u}_{\lambda}}dx. \end{equation} Next, it is easy to check that the following elementary inequality holds: there exists a constant $C>0$ such that $$ (1+C)\frac{s}{(1-s)}\leq\frac{s^{2}}{(1-s)^{2}}+(1+C), \ \ s\in (0,1), $$ which used in (\ref{Equ7}) yields $$ \lambda\int_{\mathbb B}\frac{\underline{u}_{\lambda}}{1-\underline{u}_{\lambda}}\geq\lambda\int_{\mathbb B} \frac{\underline{u}_{\lambda}^{2}}{(1-\underline{u}_{\lambda})^{2}}dx\geq\lambda(1+C) \int_{\mathbb B}\frac{\underline{u}_{\lambda}}{1-\underline{u}_{\lambda}}-C_{1}, $$ where $C_{1}$ is independent of $\lambda$. From the above inequality, we get $$
\|\Delta\underline{u}_{\lambda}\|_{2}^{2}=\lambda\int_{\mathbb B}\frac{\underline{u}_{\lambda}}{1-\underline{u}_{\lambda}}dx\leq C. $$ Therefore, we may assume $\underline{u}_{\lambda}\rightharpoonup u^{*}$ in $H_{0}^{2}(\mathbb B)$ and by monotone convergence theorem (\ref{Equ6}) holds after integration by parts. Since $\mu_{1}(\underline{u}_{\lambda})>0$ for all $\lambda\in (0,\lambda_{*})$, in particular we have $$
\int_{\mathbb B}|\Delta \varphi|^{2}dx\geq \int_{\mathbb B}\frac{\lambda \varphi^{2}}{(1-\underline{u}_{\lambda})^{2}}, \varphi\in C_{0}^{\infty}(\mathbb B) $$ and passing to the limit as $\lambda\nearrow \lambda_{*}$ we obtain that $u_{\lambda_{*}}$ is weakly stable. Finally, if
$\|u_{\lambda_{*}}\|_{\infty}<1$ and hence $u_{\lambda_{*}}$ is a classical solution of $(1.1)_{\lambda_{*}}$, the linearized operator at $u_{\lambda_{*}}$ $$ L(\lambda_{*}, u_{\lambda_{*}}):=\Delta^{2}-\frac{\lambda_{*}}{(1-u_{\lambda_{*}})^{2}} $$ well defined on the space $\mathbb R^{+}\times C^{4,\alpha}(\mathbb B)$. If $\mu_{1}(u_{\lambda_{*}})>0$ then the Implicit Function Theorem applied to the function $$ F(\lambda, \underline{u}_{\lambda}):=\Delta^{2} \underline{u}_{\lambda}-\frac{\lambda}{1-\underline{u}_{\lambda}} $$ would yield a solution for $\lambda>\lambda_{*}$ contradicting the definition of $\lambda_{*}$, thus $\mu_{1}(u^{*})=0$. \vskip0.1in
\begin{corollary}\label{C2.1}
There exists a constant $C$ independent of $\lambda$ such that for each $\lambda\in (0,\lambda_{*})$, the minimal solution $\underline{u}_{\lambda}$ satisfies
$\|(1-\underline{u}_{\lambda})^{-1}\|_{L^{2}}\leq C$. \end{corollary}
{\bf Proof.} From Proposition \ref{P2.2}, we have $$ \int_{\mathbb B}\frac{\underline{u}_{\lambda}^{2}}{(1-\underline{u}_{\lambda})^{2}}dx=\int_{\underline{u}_{\lambda}\geq\frac{1}{2}} \frac{\underline{u}_{\lambda}^{2}}{(1-\underline{u}_{\lambda})^{2}}dx +\int_{\underline{u}_{\lambda}<\frac{1}{2}}\frac{\underline{u}_{\lambda}^{2}}{(1-\underline{u}_{\lambda})^{2}}dx\leq C. $$ So $$ \int_{\underline{u}_{\lambda}\geq\frac{1}{2}} \frac{1}{(1-\underline{u}_{\lambda})^{2}}dx\leq4\int_{\underline{u}_{\lambda}\geq\frac{1}{2}} \frac{\underline{u}_{\lambda}^{2}}{(1-\underline{u}_{\lambda})^{2}}dx\leq C. $$ From this, we easily obtain
$\|(1-\underline{u}_{\lambda})^{-1}\|_{L^{2}}\leq C$, and the proof is completed. \vskip0.2in
\begin{corollary}\label{C2.2}
For dimensions $n\leq 4$, the extremal solution $u^{*}$ is regular, i.e., $u^{*}=\lim_{\lambda\nearrow\lambda_{*}}u_{\lambda}$ exists in the topology of $C^{4}(\mathbb B)$. \end{corollary}
{\bf Proof } Since $u^{*}$ is radial and radially decreasing, we need just to show that $u^{*}(0)<1$ to get the regularity of $u^{*}$. Since $(1-u^{*}(x))\in L^{2}(\mathbb B)$ according to the corollary \ref{C2.1}, we have that $u^{*}(x)\in W^{4,2}(\mathbb B)$ by the standard elliptic regularity theory. And then by the Sobolev imbedding theorem we have $u^{*}(x)\in C^{4-[\frac{n}{8}]-1, [\frac{n}{8}]+1-\frac{n}{8} }(\mathbb B)$. So if $n\leq 4$, one can easy to see that $u^{*}(x)\in C^{2}(\mathbb B)$. As $\nabla u^{*}(0)=0$, we get $$
1-u^{*}(x)=u^{*}(0)-u^{*}(x)\leq C|x|^{2}, $$ hence $$
\infty>\int_{\mathbb B}\frac{dx}{(1-u^{*}(x))^{2}}\geq C\int_{\mathbb B}\frac{dx}{|x|^{4}}=\infty. $$ A contradiction arises, so $u^{*}$ is regular for $n\leq 4$. \qed
\vskip0.2in \subsection{ The upper and lower bounds for $\lambda_{*}$ }\vskip0.2in
\begin{lemma}\label{L2.2} $$ \lambda_{*}\leq \frac{\nu_{1}}{4}, $$ where $\nu_{1}$ is the first eigenvalue of $\Delta^{2}$ in $H_{0}^{2}(\mathbb B)$. \vskip 0.1in \end{lemma} {\bf Proof.} Let $\underline{u}_{\lambda}$ be a solution of $(1.1)_{\lambda}$ and let $(\psi,\nu_{1})$ denote the first eigenpair of $\Delta^{2}$ in $H_{0}^{2}(\mathbb B)$ with $\psi>0$ then, $$ \int_{\mathbb B}\underline{u}_{\lambda}\psi dx=\int_{\mathbb B}\underline{u}_{\lambda} \Delta^{2}\psi dx=\lambda\int_{\mathbb B}\frac{\psi}{1-\underline{u}_{\lambda}} $$ and this implies $$ \int_{\mathbb B}(-\nu_{1}\underline{u}_{\lambda}+\frac{\lambda}{1-\underline{u}_{\lambda}})\psi dx=0. $$ Since $\psi>0$ there must exists a point $\bar{x}\in \mathbb B$ where $$ \frac{\lambda}{1-\underline{u}_{\lambda}}-\nu_{1}\underline{u}_{\lambda}\leq 0, $$ one can conclude that $\lambda_{*}\leq \sup_{0\leq \underline{u}_{\lambda}\leq1}\nu_{1}\underline{u}_{\lambda}(1-\underline{u}_{\lambda})=\frac{\nu_{1}}{4}$. \vskip0.1in
The lower bound for $\lambda_{*}$ is obtained by finding a suitable supersolution . For example, if for some parameter $\tilde{\lambda}_{1}$ there exists a supersolution, then $\lambda_{*}>\tilde{\lambda}_{1}$ by Proposition \ref{P2.1}. \qed \vskip 0.1in
\begin{lemma}\label{L2.3} For $n\geq1$, we have $$ \lambda_{*}\geq \max\{4n(n-2), 2n(n+2)\}. $$ \end{lemma}
{\bf Proof.} For any $\beta>0$ and $C_{0}>0$ let $g_{\beta}(r)=(C_{0}-\log r)^{\beta}, \ r\in (0,1)$. Then, by direct calculation we find the following facts: $$ \Delta g_{\beta}(r)=\beta r^{-2}[(2-n)g_{\beta-1}+(\beta-1)g_{\beta-2}]; $$ and $$ \Delta[r^{2} g_{\beta}]=2n g_{\beta}-\beta(n+2)g_{\beta-1}+\beta(\beta-1)g_{\beta-2}. $$ So we have \begin{eqnarray*} \Delta^{2}(r^{2}g_{\beta})&=&2n\Delta g_{\beta}-\beta(n+2)\Delta g_{\beta-1}(r)+\beta(\beta-1)\Delta g_{\beta-2}(r)\\\\ &=&\beta r^{-2} \times\left\{2n(2-n)g_{\beta-1}(r)+(\beta-1)(2n+n^{2}-4)g_{\beta-2}(r)\right\}\\\\ &+&\beta r^{-2} \times\left\{(\beta-1)(\beta-2)\times(-2n)g_{\beta-3}(r)+(\beta-1)(\beta-2)(\beta-3)g_{\beta-4}\right\}. \end{eqnarray*} Now let $\beta\in (0,1)$ and $n>2$, we have \begin{equation} \label{Equ8} \Delta^{2}(r^{2}g_{\beta})\leq \beta r^{-2}\times 2n(n-2)g_{\beta-1}. \end{equation}
Also for any $A>0$ take $\bar{u}=1-Ar^{2}g_{\beta}$, one conclude from (\ref{Equ8}) that $$ \Delta^{2}u\geq 2n(n-2)A\beta r^{-2}g_{\beta-1}. $$ Set $\beta=\frac{1}{2}$, one can obtain that $$\left\{ \begin{array}{lllllll} \Delta^{2}\bar{u}\geq\frac{n(n-2)A^{2}}{1-\bar{u}} & \mbox{in} \ \ \mathbb B_{1}, \\\\ \bar{u}(r)=1-C_{0}^{\frac{1}{2}}A & \mbox{on}\ \ \partial \mathbb B_{1}, \\\\
\bar{u}'(r) =AC_{0}^{-\frac{1}{2}}(\frac{1}{2}-2C_{0} ) & \mbox{on}\ \ \partial \mathbb B_{1}.
\end{array} \right. $$
Choosing $C_{0}=\frac{1}{4}, A_{0}=2$, one conclude that $\bar{u}(r)$ is a supersolution of $(1.1)_{4n(n-2)}$ and $\lambda_{*}\geq 4n(n-2)$ according to Proposition \ref{P2.1}. Besides, we consider the function $$
\omega_{\alpha}(x):=\alpha(1-|x|^{2})^{2},\ \ \alpha\in (0,1), $$ which satisfies $0\leq \omega_{\alpha}(x)<1$ for $x\in \mathbb B$ and $$ \omega_{\alpha}(x)=0, \frac{\partial\omega_{\alpha}}{\partial n}=0,\ \mbox{for}\ \ x\in \partial \mathbb B;\ \mbox{for all}\ \ \alpha\in (0,1). $$ Now the idea is to obtain from $\omega_{\alpha}(x)$ a super-solution of $(1.1)_{\lambda}$, for a suitable choice of $\alpha$ and for $\lambda$ in a suitable range of the form $0<\lambda\leq \tilde{\lambda}$. For simply calculation, we have \begin{eqnarray*} \Delta^{2}\omega_{\alpha}(r)&=&\frac{d^{4}\omega_{\alpha}}{dr^{4}}+\frac{2(n-1)}{r}+\frac{d^{3}\omega_{\alpha}}{dr^{3}} +\frac{(n-)(n-3)}{r^{2}}\frac{d^{2}\omega_{\alpha}}{dr^{2}}-\frac{(n-1)(n-3)}{r^{3}}\frac{d\omega_{\alpha}}{dr}\\ &=&[8n^{2}+16n]\alpha=:C(n)\alpha, \end{eqnarray*} and thus $$
\Delta^{2}\omega_{\alpha}(r)=\frac{C(n)\alpha (1-\alpha)}{1-\alpha}\geq \frac{C(n)\alpha (1-\alpha)}{[1-\alpha(1-|x|^{4})]} =\frac{C(n)\alpha (1-\alpha)}{1-\omega_{\alpha}} $$ from which we deduce that $$ \lambda_{*}=\lambda^{*}\geq \sup_{\alpha\in (0,1)}C(n)\alpha(1-\alpha)=\frac{1}{4}C(n)=2n(n+2) $$ and the proof is completed.\qed \vskip0.1in
We complete this section by giving proofs of Theorem \ref{result1} and \ref{result2}. \vskip0.1in
{\bf Proofs of Theorem \ref{result1} and \ref{result2}.} The proof of Theorem \ref{result1} follows form Proposition \ref{P2.1}, Lemma \ref{L2.2} and Lemma \ref{L2.3}. For the proof of Theorem \ref{result2}, we only need to prove the uniqueness of the regular extremal solution $u^{*}$, the other parts of Theorem 1.2 follow from Lemma \ref{L2.1} and Corollary \ref{C2.2}. Indeed, if the extremal solution $u^{*}$ is regular, we can easily check that $\mu_{1}(u^{*})=0$ by Implicit Function Theorem, since otherwise, we can continue the minimal branch beyond $\lambda_{*}$. And then the uniqueness follows from the (ii) of the Lemma \ref{L1.5}. \qed\vskip0.2in
\setcounter{equation}{0}
\section{Uniqueness of the extremal solution: proof of Theorem \ref{result3}}
\quad \quad {\bf Proof of Theorem \ref{result3}.} Suppose that $v\in H^{2}(\mathbb B)$ satisfies $$ \left\{ \begin{array}{lllllll} \int_{\mathbb B} v\Delta^{2}\varphi dx\geq\int_{\mathbb B} \frac{\lambda_{*}}{1-v}dx, \forall \varphi\in C_{0}^{\infty}(\bar{\mathbb B}), \varphi\geq0,\\\\
v|_{\partial\mathbb B}=0, \frac{\partial v}{\partial n} |_{\partial\mathbb B} \leq0, \end{array} \right. $$ and $v\not\equiv u^{*}$. Notice that the construction of minimal solutions in Proposition \ref{P2.1} for $\lambda\in (0,\lambda_{*})$, carries over to $\lambda=\lambda_{*}$ but just in the weak sense; precisely, we may assume that for $\lambda=\lambda^{*}$ there exists a minimal weak solution. In other words, it is legitimate to assume $$ v(x)\geq u^{*},\ \ a.e.\ \ \ x\in \mathbb B. $$ The idea of the proof is as follows: first we prove the function $$ u_{0}=\frac{1}{2}(u^{*}+v) $$ is a super-solution to the following perturbation of problem $(1.1)_{\lambda}$ \begin{equation}\label{Equ10} \left\{ \begin{array}{lllllll}
\Delta^{2} u=\frac{\lambda_{*}}{1-u}+\mu\frac{\zeta(x)}{1-u}, & \mbox{in}\ \ \mathbb B;\\
0\leq u\leq1, & \mbox{in} \ \ \mathbb B;\\
u=\frac{\partial u}{\partial n}=0, & \mbox{on} \ \ \partial\mathbb B,\\
\end{array} \right. \end{equation} for a standard cut-off function $\zeta(x)\in C_{0}^{\infty}(\mathbb B)$ and $\mu>0$ to be suitably chosen; besides, a solution is understood in weak sense unless otherwise stated. Second, we construct, for some $\lambda>\lambda_{*}$, a super-solution to $(1.1)_{\lambda}$ by using a solution of (\ref{Equ10}) and this will enable us to build up a weak solution of $(1.1)_{\lambda}$ for $\lambda>\lambda_{*}$ and thus necessarily $v\equiv u^{*}$.
Indeed we first observe that for $0<R<1$ and for some $c_{0}=c_{0}(R)>0$ \begin{equation}\label{Equ11}
v(x)\geq u^{*}+c_{0}\ \ \ |x|\leq R. \end{equation} To prove this we recall the Green's function for $\Delta^{2}$ with Dirichlet boundary conditions $$ \left\{ \begin{array}{lllllll}
\Delta_{x}^{2} G(x,y)=\delta_{y}&\ \ \ x\in\mathbb B; \\
G(x,y)=0 &\ \ \ x\in\partial\mathbb B; \\
\frac{\partial G}{\partial n} (x,y)=0 &\ \ \ x\in\partial\mathbb B,
\end{array} \right. $$ where $\delta_{y}$ is the Dirac mass at $y\in \mathbb B$. Boggio gave an explicit formula for $G(x,y)$ which was used in \cite{Gru} to prove that in dimension $n\geq 5$ \begin{equation}\label{Equ11}
G(x,y)\sim|x-y|^{4-n}\min(1, \frac{d(x)^{2}d(y)^{2}}{|x-y|^{4}}), \end{equation} where $$
d(x)=\mbox{dist}(x,\partial\mathbb B)=1-|x|. $$ Formula (\ref{Equ11}) yields \begin{equation}\label{Equ12} G(x,y)\geq cd(x)^{2}d(y)^{2} \end{equation} for some $c>0$ and this in turn implies that for smooth functions $\bar{v}$ and $\bar{u}$ such that $\bar{v}-\bar{u}\in H_{0}^{2}(\mathbb B)$ and $\Delta^{2}(\bar{v}-\bar{u})\geq0$, \begin{eqnarray*} \tilde{v}-\tilde{u}&=&\int_{\partial B}\left(\frac{\partial\Delta_{x}G}{\partial n_{x}}(x,y)\tilde{v}-\tilde{u} -\Delta_{x}G(x,y)\frac{\partial (\tilde{v}-\tilde{u})}{\partial n}\right)\\ &+&\int_{B}G(x,y)\Delta^{2}(\tilde{v}-\tilde{u})dx\\ &\geq&c d(y)^{2}\int_{B}\Delta^{2}(\tilde{v}-\tilde{u}) d(x)^{2}dx. \end{eqnarray*} Using a standard approximation procedure, we conclude that $$ v(y)-u^{*}(y)\geq cd(y)^{2}\lambda^{*}\int_{\mathbb B}\left(\frac{1}{1-v}-\frac{1}{1-u^{*}}\right)d(x)^{2}dx. $$ Since $v\geq u^{*}, v\not\equiv u^{*}$ we deduce (\ref{Equ11}).
Let $u_{0}=\frac{u^{*}+v}{2}$. Then by Taylar's Theorem \begin{equation}\label{Equ13} \frac{1}{1-v}=\frac{1}{1-u_{0}}+\frac{v-u_{0}}{(1-u_{0})^{2}}+\frac{(v-u_{0})^{2}}{4(1-u_{0})^{3}} +\frac{(v-u_{0})^{3}}{18(1-u_{0})^{3}}+\frac{(v-u_{0})^{4}}{96(1-\varepsilon_{1})^{4}} \end{equation} for some $u_{0}\leq \varepsilon_{1}\leq v$ and \begin{equation}\label{Equ14} \frac{1}{1-u^{*}}=\frac{1}{1-u_{0}}+\frac{u^{*}-u_{0}}{(1-u_{0})^{2}}+\frac{(u^{*}-u_{0})^{2}}{4(1-u_{0})^{3}} +\frac{(u^{*}-u_{0})^{3}}{18(1-u_{0})^{3}}+\frac{(u^{*}-u_{0})^{4}}{96(1-\varepsilon_{2})^{4}} \end{equation} for some $u^{*}\leq\varepsilon_{2}\leq u_{0}$. Adding (\ref{Equ13}) and (\ref{Equ14}) yields \begin{equation}\label{Equ15} \frac{1}{2}(\frac{1}{1-v}+\frac{1}{1-u^{*}})\geq\frac{1}{1-u_{0}}+\frac{1}{16}\frac{(u^{*}-v)^{2}}{(1-u_{0})^{2}} \end{equation} and in turn we obtain, \begin{eqnarray*} \int_{\mathbb B}u_{0}\Delta^{2}\varphi dx\geq \int_{\mathbb B}\left[\frac{\lambda_{*}}{1-u_{0}}+\frac{\lambda_{*}(u^{*}-v)^{2}}{16(1-u_{0})}\right]dx \geq\int_{\mathbb B}\left[\frac{\lambda_{*}}{1-u_{0}}+\frac{\lambda_{*}c_{0}^{2}\zeta(x)}{16(1-u_{0})}\right]dx. \end{eqnarray*} Thus, $u_{0}$ is a weak super-solution of (\ref{Equ10}) with $\mu=\frac{\lambda_{*}c_{0}^{2}}{16}$ and the cut-off $\zeta(x)$ with support in $\mathbb B_{\rho}$. Now reasoning as in Lemma \ref{L2.1}, we may assume for $\varepsilon>0$ sufficiently small, that (\ref{Equ10}) posses a classical solution $0\leq u_{\varepsilon}<1$ with parameter $\lambda_{*}$ replaced by $\lambda_{*}-\varepsilon$. Set $\mu_{\varepsilon}:=[(\lambda_{*}-\varepsilon)c_{0}^{2}]/16$ and let $\psi\in C^{4}(\bar{\mathbb B})$ be the unique classical solution of the following $$ \left\{ \begin{array}{lllllll} \Delta^{2}\psi=\mu_{\varepsilon} \frac{\zeta(x)}{1-u_{\varepsilon}} &\mbox{in}\ \ \mathbb B,\\ \psi=\frac{\partial \psi}{\partial n}=0 &\mbox{on}\ \ \partial\mathbb B. \end{array} \right. $$ We also, by the Boggio principle, have that there exists $M>0$ sufficiently large such that $u_{\varepsilon}\leq M\psi$. Next let $\delta>0$ and set $$ \omega:=\frac{(\lambda_{*}-\varepsilon)+\delta}{\lambda_{*}-\varepsilon}u_{\varepsilon}-\psi $$ and choosing $\delta$ sufficiently small, we obtain $\omega\leq u_{\varepsilon}<1$; moreover, from $$ \left\{ \begin{array}{lllllll} \Delta^{2}(u_{\varepsilon}-\psi)=(\lambda_{*}-\varepsilon)\frac{1}{1-u_{\varepsilon}}\geq0,&\ \ \mbox{in}\ \ \mathbb B,\\
u_{\varepsilon}-\psi=\frac{\partial(u_{\varepsilon}-\psi)}{\partial n}=0 &\ \ \mbox{on} \ \ \partial\mathbb B,
\end{array} \right. $$ we have again by the Boggio principle that $\psi\leq u_{\varepsilon}$ and eventually that $\omega\geq 0$. Finally we have $$ \Delta^{2}\omega=(\lambda_{*}-\varepsilon+\delta)\frac{1}{1-u_{\varepsilon}} +\frac{(\lambda_{*}-\varepsilon+\delta)c_{0}^{2}}{16}\frac{\varepsilon(x)}{1-u_{\varepsilon}} -\mu_{\varepsilon}\frac{\varepsilon(x)}{1-u_{\varepsilon}}\geq (\lambda_{*}-\varepsilon+\delta)\frac{1}{1-\omega} $$ since $\omega\leq u_{\varepsilon}$. Thus it is enough to choose $0<\varepsilon<\delta$ to provide a classical solution to $(1.1)_{\lambda}$ for $\lambda>\lambda_{*}$ which is a contradiction; this completes the proof of Theorem \ref{result3}.
\vskip0.2in \setcounter{equation}{0} \section{Behavior of the minimal solutions as $\lambda\to0$: proof of Theorem \ref{result4}}\vskip0.1in
\quad \quad { \bf Proof of Theorem \ref{result4}.}\ \ We first show that \begin{equation}\label{Eq4.1} \underline{u}_{\lambda}\to 0\ \ \mbox{uniformly as }\ \ \lambda\to 0. \end{equation} Since this standard, we just briefly sketch its proof. By Theorem \ref{result1}, we know that $$ 0<\lambda<\mu<\lambda_{*}\Rightarrow \underline{u}_{\lambda}(x)<\underline{u}_{\mu}(x)
\ \ \mbox{if}\ \ |x|< 1. $$ Then, by multiplying the equation $(1.1)_{\lambda}$ by
$\underline{u}_{\lambda}$ and by integrating by parts, we obtain that $\|\underline{u}_{\lambda}\|_{H_{0}^{2}(\mathbb B)}$ remains bounded. Hence, up to a subsequence, $\{\underline{u}_{\lambda}\}$ converges in the weak $H_{0}^{2}(\mathbb B)$ topology to 0, which is the unique solution of $(1.1)_{0}$. By convergence of the norms, we infer that the convergence is in the norm topology.
Next, note that $U_{\lambda}$ satisfies $$ \left\{ \begin{array}{lllllll} \Delta^{2}U_{\lambda}=\lambda & \mbox{in}\ \mathbb B, \\ U_{\lambda}=\frac{\partial U_{\lambda}}{\partial n} =0 & \mbox{on}\ \partial\mathbb B.
\end{array} \right. $$ Therefore, $\Delta^{2}\underline{u}_{\lambda}>\Delta^{2}U_{\lambda}$, one conclude that $\underline{u}_{\lambda}>U_{\lambda}$ by Lemma 2.1.
In order to prove the last statement of Theorem \ref{result4}, note that from (\ref{Eq4.1}) we know that $$ \mbox{for all}\ \varepsilon>0\ \ \mbox{there exists}\ \lambda_{\varepsilon}>0\ \ \mbox{such that}
\lambda<\lambda_{\varepsilon}\Rightarrow \|\underline{u}_{\lambda}\|_{\infty}<\varepsilon. $$ So, fix $\varepsilon>0$ and let $\lambda<\lambda_{\varepsilon}$. Then $$ \Delta^{2}\underline{u}_{\lambda}=\frac{\lambda}{1-\underline{u}_{\lambda}}<\frac{\lambda}{1-\varepsilon} =\Delta^{2}\frac{U_{\lambda}}{1-\varepsilon} $$ This shows that $\underline{u}_{\lambda}(x)<\frac{U_{\lambda}(x)}{1-\varepsilon}$ for all $x\in \mathbb B$, and the proof is completed according to the arbitrariness of $\varepsilon$. \qed
\setcounter{equation}{0} \section{Further results and open problems} \quad \quad First, we give the following result which is the main tool to guarantee that $u^{*}$ is singular. At the same time, it give a precise estimate for $\lambda_{*}$. The proof of this result is based on an upper estimate of $u^{*}$ by a stable singular subsolution.
\begin{proposition}\label{P4.1} Suppose there exist $\lambda'>0, \beta>0$ and a singular radial function $\omega(r)\in H_{0}^{2}(\mathbb B)$ with $\frac{1}{1-\omega(r)}\in L_{loc}^{\infty}(\bar{\mathbb B}\setminus 0)$ such that \begin{equation}\label{Eq5.1} \left\{ \begin{array}{lllllll}
\Delta^{2} \omega\leq\frac{\lambda'}{1-\omega}& \mbox{for}\ \ 0<r<1,\\
\omega(1)= \omega'(1)=0,
\end{array} \right. \end{equation} and \begin{equation}\label{Eq5.2} \beta\int_{\mathbb B}\frac{\phi^{2}}{(1-\omega)^{2}}\leq \int_{\mathbb B}(\Delta\phi)^{2}\quad \mbox{for all}\quad \phi\in H_{0}^{2}(\mathbb B). \end{equation} If $\beta>\lambda'$, then $\lambda_{*}<\lambda'$ and $u^{*}$ is singular. \end{proposition}
{\bf Proof.}\quad First, note that (\ref{Eq5.2}) and $\frac{1}{1-\omega(r)}\in L_{loc}^{\infty}(\bar{\mathbb B}\setminus 0)$ yield to $\frac{1}{1-\omega}\in L^{1}(\mathbb B)$. (\ref{Eq5.1}) implies that $\omega(r)$ is a $H_{0}^{2}(\mathbb B)-$ weak sub-solution of $(1.1)_{\lambda'}$. If now $\lambda'<\lambda^{*}$, then by Lemma \ref{L1.5}, $\omega(r)$ would necessarily be below the minimal solution $\underline{u}_{\lambda'}$, which is a contradiction since $\omega(r)$ is singular while $\underline{u}_{\lambda'}$ is regular. In the following, we shall prove that $u^{*}$ is singular.
Now let $\frac{\lambda'}{\beta}<\gamma<1$ in such a way that $$ \alpha:=(\frac{\gamma \lambda_{*}}{\lambda'})^{\frac{1}{2}}<1. $$ Setting $\bar{\omega}:=1-\alpha(1-\omega)$, we claim that \begin{equation}\label{Eq5.3} u^{*}\leq \bar{\omega} \ \ \ \mbox{in}\ \ \ \mathbb B. \end{equation} Note that by the choice of $\alpha$ we have $\alpha^{2}\lambda'<\lambda_{*}$, and therefore to prove (\ref{Eq5.3}) it suffices to show that for $\alpha^{2}\lambda'\leq \lambda<\lambda_{*}$, we have $u_{\lambda}\leq \bar{\omega}$ in $\mathbb B$. Indeed, fix such $\lambda$ and note that $$ \Delta^{2}\bar{\omega}=\alpha \Delta^{2}\omega\leq\frac{\alpha \lambda'}{(1-\omega)} =\frac{\alpha^{2}\lambda'}{(1-\bar{\omega})}\leq \frac{\lambda}{(1-\bar{\omega})}. $$ Assume that $\underline{u}_{\lambda}\leq \bar{\omega}$ dose not hold in $\mathbb B$, and consider $$
R_{1}:=\sup\{0\leq R\leq 1| \underline{u}_{\lambda}(R)>\bar{\omega}(R)\}>0. $$ Since $\bar{\omega}(1)=1-\alpha>0=u_{\lambda}(1)$, we then have $$ R_{1}<1, u_{\lambda}(R_{1})=\bar{\omega}(R_{1})\ \mbox{and}\ \underline{u}_{\lambda}'(R_{1})\leq\bar{\omega}'(R_{1}). $$ Now consider the following problem $$ \left\{ \begin{array}{lllllll} \Delta^{2}u=\frac{\lambda}{1-u}& \ \mbox{in}\ \ \mathbb B_{R_{1}},\\ u=u_{\lambda}(R_{1})& \ \ \mbox{on}\ \ \partial \mathbb B_{R_{1}},\\
\frac{\partial u}{\partial n}= u'_{\lambda}(R_{1}) & \ \ \mbox{on}\ \ \partial \mathbb B_{R_{1}}. \end{array} \right. $$ Then $\underline{u}_{\lambda}$ is a solution to above problem while $\bar{\omega}$ is a sub-solution to the same problem. Moreover $\bar{\omega}$ is stable since $\lambda<\lambda_{*}$ and $$ \frac{\lambda}{(1-\bar{\omega})^{2}}\leq\frac{\lambda_{*}}{\alpha^{2}(1-\omega)^{2}} <\frac{\beta}{(1-\omega)^{2}}. $$ By Lemma \ref{L2.1}, we deduce that $\underline{u}_{\lambda}\geq \bar{\omega}$ in $\mathbb B_{R_{1}}$ which is impossible, since $\bar{\omega}$ is singular while $u_{\lambda}$ is regular. This establishes claim (\ref{Eq5.3}) which, combined with the above inequality, yields $$ \frac{\lambda_{*}}{(1-u_{*})^{2}}\leq\frac{\lambda_{*}}{\alpha^{2}(1-\omega)^{2}} <\frac{\beta}{(1-\omega)}, $$ and thus $$ \inf_{\varphi\in C_{0}^{\infty}(\mathbb B)}\frac{\int_{\mathbb B}[(\Delta \varphi)^{2}-\frac{\lambda_{*}\varphi^{2}}{(1-u^{*})^{2}}]dx}{\int_{\mathbb B}\varphi^{2}dx}>0. $$ This is not possible if $u^{*}$ is a smooth function, since otherwise, one could use the Implicit function Theorem
to continue the minimal branch beyond $\lambda_{*}$. The proof is
over. \qed
\begin{itemize} \item $Open\ Problem\ 1$. Dose $(1.1)_{\lambda}$ exist a stable singular subsolution? We know that Cowan etal, with the help of Maple, construct such solution of $(P_{\lambda})$ with $p=2$ by improved Improved Hardy-Rellich Inequalities, see \cite{Mo,Co1} . But the method used there seems invalid. \end{itemize}
We now turn to the extremal solution $u^{*}$. We suggest the following open problems. \begin{itemize} \item $Open\ Problem\ 2$. Dose one find the precise estimate for $u^{*}$ as in \cite{Mo,Co1,Da}, which play a crucial role for investigating the regularity of $u^{*}$. In \cite{Co1}, the precise bound for $u^{*}$ is obtained by finding a stable singular subsolution which relies on the \textquotedblleft ghost" singular solution, as mentioned in introduction. However, in the present paper we can not find any \textquotedblleft ghost" singular solution, so a new trick is needed. \end{itemize}
\begin{itemize} \item $Open\ Problem\ 3$. For the corresponding second equation the extremal solution $u^{*}$ is regular for dimensions $n\leq6$ and singular for dimension $n\geq7$, for details see \cite{Mea}. The threshold $n^{*}=7$ between regular and singular solutions is called the critical dimension. There is a natural question: whether there exists a critical dimension $N^{*}$ for equation $(1.1)_{\lambda}$. We conjecture that $N^{*}=8$. \qed
\end{itemize}
\noindent \textbf{Acknowledgement.} The author is greatly indebted to Prof. Dong Ye for his constructive comments. This research is supported in part by National Natural Science Foundation of China (Grant No. 10971061).
\small {\it 1 Institute of Contemporary Mathematics, Henan University;}\\
\small {\it 2 School of Mathematics and Information Science,Henan University}\\ \small {\it Kaifeng 475004, P. R. China.}
\end{document} |
\begin{document}
\begin{frontmatter}
\title{Virtual Element Method: an equilibrium-based \\stress recovery procedure}
\author{E. Artioli\corref{cor1}$^a$}
\author{S. de Miranda\corref{cor2}$^b$} \author{C. Lovadina\corref{cor4}$^{c,d}$} \author{L. Patruno $^b$ \corref{cor33}} \ead{luca.patruno@unibo.it}
\cortext[cor33]{Corresponding author}
\address{$^a$Department of Civil Engineering and Computer Science, University of Rome Tor Vergata, Via del Politecnico 1, 00133 Rome, Italy} \address{$^b$DICAM, University of Bologna, Viale Risorgimento 2, 40136 Bologna, Italy} \address{$^c$Dipartimento di Matematica, Universit\`a degli Studi di Milano, Via Saldini 50, 20133 Milano, Italy} \address{$^d$IMATI del CNR, Via Ferrata 1, 27100 Pavia, Italy}
\begin{abstract} Within the framework of the displacement-based Virtual Element Method (VEM) for plane elasticity a significant problem is represented by an accurate evaluation of the stress field. In particular, in the classical VEM formulation, a suitable operator which maps to the strain field is introduced in order to allow the calculation of the stiffness matrix. The stress field is then computed using that strian field, by using the constitutive law. Considering for example a first-order formulation for a homogeneous material, strains are locally mapped onto constant functions, and stresses are accordingly piecewise constant. However, the virtual displacements might engender more complex strain fields for polygons which are not triangles. In this paper, Recovery by Compatibility in Patches is used in order to mitigate such an effect and, thus, enhance the accuracy of the recovered stress field. The procedure is simple, efficient and can be readily implemented in existing codes. Numerical tests confirm the soundness of the proposed approach. \end{abstract}
\begin{keyword} Virtual Element Method \sep Stress recovery \sep RCP \end{keyword}
\end{frontmatter}
\section{Introduction} \label{sec:introduction} The Virtual Element Method (VEM) is a relatively new and very powerful discretisation scheme which is rapidly attracting the interest of the scientific community. The technique is well-known for its formal elegance and flexibility, which allows to adopt meshes composed of general polygons/polyhedra as well as allowing the presence of hanging nodes and nonconforming grids \cite{volley,Hitchhikers,BeiraodaVeiga-Brezzi-Marini:2013,nonconforming}. The method, originally proposed in 2012 in its displacement-based formulation and presented for the Laplace operator in a two-dimensional context \cite{volley}, is rapidly developing and has been already applied to numerous physical problems \cite{BeiraodaVeiga-Brezzi-Marini:2013,Antonietti-BeiraodaVeiga-Mora-Verani:20XX,Brezzi-Marini:2012,Artioli2017a,Artioli2017b,WRRH,BCP,TPPM10,ABSV2017,Berrone-VEM,Helmholtz-VEM,Steklov-VEM} as well as extended to three-dimensional cases \cite{Paulino-VEM} and mixed formulations \cite{Brezzi-Falk-Marini,Artioli2017HR}.
Differently from the finite element framework, in the virtual element technique the local polynomial approximation concept is abandoned and the functions used for the discretization procedure are not known pointwise, in general. However, a careful selection of the degrees of freedom and assuming that the unknown fields satisfy appropriate differential equations, make it possible is to compute the stiffness matrix and the load vector.
In particular, for plane elasticity problems, classical VEM formulations assume that displacements are completely known only at the element boundaries while the internal field is unknown and, for high-order schemes, characterised only by means of internal degrees of freedom representing appropriate integral quantities.
Due to the virtual nature of the displacement field, strains (the symmetric gradient of the displacements) are not directly computable and need to be further approximated, in order to compute the stiffness matrix. The standard procedure consists in defining a strain field, polynomial inside the element, which can be determined starting from the displacement degrees od freedom. For example, in the case of the first-order formulation, the aforementioned strategy leads to constant strains inside the elements, irrespective of the number of element edges, and thus irrespective of the number of displacement degrees of freedom. Therefore, applying the constitutive law, the stress field is essentially piecewise constant as well. Especially for meshes consisting of polygons with numerous edges, this often leads to not completely satisfactory results.
In this paper, a modified version of the Recovery by Compatibility in Patches (RCP) \cite{Ubertini2004,Ubertini2006} is used to devise an alternative procedure, which allows to partly alleviate the above drawback and compute accurate stresses in the displacement-based VEM framework. Patch-based stress recovery techniques, initially inspired by the Superconvergent Patch Recovery (SPR) procedure proposed by Zienkiewicz and Zhu \cite{Zienkiewicz1992a,Zienkiewicz1992b}, have quickly developed and found numerous applications, ranging from mesh adaptivity \cite{Boroomand1999b,Castellazzi2010} to the recovery of accurate stresses for crack propagation in XFEM analyses \cite{Prange2012}. In contrast with classical approaches, the common characteristic of such techniques is to recover enhanced stress fields by considering groups of surrounding elements (i.e. patches). In the original SPR procedure this is obtained by calculating stresses as a least-square interpolation of values evaluated at superconvergent points. Later, the procedure has been refined \cite{Blacker1994,Wiberg1997,Lee1997,Park1999,Kvamsdal1998,Okstad1999} introducing, for example, equilibrium constraints in order to lead to more accurate results. Alternative procedures, based on variational principles such as the Recovery by Equilibrium in Patches (REP) \cite{Boroomand1997a,Boroomand1997b,Boroomand1999a}, have been proved to obtain extremely interesting results.
RCP has been originally proposed as a patch-based recovery technique which, by enforcing equilibrium and relaxing compatibility, allows to enhance the stresses obtained from standard displacement-based finite elements schemes.
The key idea underlying RCP is to minimise the complementary energy associated to a patch of elements treated as an isolated system. On each patch, stresses are obtained as a linear combination of self-equilibrated stress modes enriched by an appropriate particular solution, and the explicit knowledge of the displacements is required only along the patch boundaries. Therefore, the RCP approach is naturally well-suited for the virtual element schemes.
In this paper, the application of RCP in the context of VEM is analysed by proposing two approaches. In the first one, a degenerate patch composed of a single polygon is considered. For triangles, this does not significantly improve the results obtained using the classical strain computation. However, the benefit becomes apparent when the number of edges increases. In the second form, RCP is adopted by considering patches of elements in analogy to its original formulation developed in the context of finite elements schemes.
The paper is organised as follows. In Section \ref{sec:vem}, the VEM formulation for plane elasticity problems is briefly recalled. Then, the RCP procedure is introduced in Section \ref{sec:rcp}. Numerical results are presented for a wide selection of test cases in Section \ref{sec:num} and, finally, conclusions are drawn in Section \ref{sec:con}.
\section{Virtual Element Method} \label{sec:vem} Consider a body occupying a region $\Omega$ of the two-dimensional space on which a reference system $(O,x,y)$ is introduced. Let us denote with the symbol $\partial \Omega$ the boundary of such a body and consider the case of homogeneous Dirichlet boundary conditions. Other types of boundary conditions can be treated in the same spirit of finite element schemes. Indicating as $\bm{u}(x,y) = [u,v]^T$, the displacement field, the associated strains are defined as
\begin{equation} \bm{\varepsilon}(\bm{u}) = \bm{D}\bm{u} \qquad \text{with} \qquad \bm{D}=\left[
\begin{array}{cc}
\partial_x & 0 \\
0 & \partial_y \\
\partial_y & \partial_x \\
\end{array}
\right], \end{equation}
\noindent where $\bm{D}$ represents the differential compatibility operator. The plane elasticity problem can be thus expressed as: find $\bm{u} \in \bm{V}$ such that \begin{equation} \int_\Omega \bm{\varepsilon}(\bm{v})^T \bm{C} \bm{\varepsilon}(\bm{u})= \int_\Omega \bm{v}^T \bm{b} \qquad \forall \bm{v} \in \bm{V}, \label{eq:prob} \end{equation} \noindent where $\bm{C}$ is the elastic matrix, $\bm{V}$ is the space of the kinematically admissible displacements, $\bm{b}$ is the applied external load.
In order to build a VEM scheme, it is assumed that $\Omega$ is subdivided into non-overlapping polygons with straight edges. In the following, $E$ indicates one and each of such polygons, $\partial E$ denotes its boundary while $e$ denotes one and each of the edges of $E$. Moreover, we here assume a homogeneous material, but inhomogeneous bodies can be treated as well (cf. for instance \cite{Artioli2017HR, AdMLP_Ho}) We introduce a local approximation space, $\bm{V}_{h|E}$, whose elements are typically not known pointwise, but are determined by means of a suitable set of degrees of freedom (thus, $\bm{V}_{h|E}$ is said to be {\em virtual}).
In the following, reference is made to a first-order VEM discretisation scheme. The space $\bm{V}_{h|E}$ is chosen so that \begin{equation}
\bm{V}_{h|E} = \left\{ \bm{v}_h \in H^1(E) \cap C^0(E): \mathbf{D^\star}(\bm{C} \bm{\varepsilon}(\bm{v}_h))=0,\; \bm{v}_{h|e} \in \bm{P}_{1}(e) \quad \forall e \in \partial E \right\}, \label{eq:space} \end{equation}
\noindent where $\mathbf{D^\star}$ is the formal adjoint of $\mathbf{D}$ while $\bm{P}_{1}(e)$ indicates the space of linear functions on the edges of $E$.
A set of degrees of freedom which can be used in order to describe such a space is represented by the values of the displacements at the element vertices. It must be noticed that, thanks to the requirements expressed in Eq. \eqref{eq:space}, displacements must be linear on edges so that the vertex values completely characterise the displacement field on $\partial E$. A sketch of the virtual field and of the considered degrees of freedom is provided in Fig. \ref{fig:vField}.
\begin{figure}
\caption{Sketch of the virtual function and of the considered degrees of freedom.}
\label{fig:vField}
\end{figure}
Due to the virtual nature of the approximating functions, it is not possible to directly compute the terms of Eq. \eqref{eq:prob}, and especially the left-hand side. It is thus necessary to introduce an operator, $\bm{\Pi}: \bm{V}_{h|E}\to \bm{P}_0(E)$, defined as \begin{equation} \int_E \bm{\Pi}(\bm{v}_h)^T \bm{\varepsilon}^0 = \int_E \bm{\varepsilon}(\bm{v}_h)^T \bm{\varepsilon}^0 \qquad \forall \bm{\varepsilon}^0 \in \bm{P}_0(E). \label{eq:proj} \end{equation} Above, $\bm{P}_0(E)$ denotes the space of constant fields over the element. It can be observed that $\bm{\Pi}$ is an operator which maps the virtual displacements onto constant strain fields. A typical displacement and strain field can be formally represented as \begin{equation} \bm{v}_h = \bm{N}^V \tilde{\bm{v}}_h, \qquad \bm{\varepsilon}^0 = \bm{N}^P \tilde{\bm{\varepsilon}}, \end{equation} \noindent where the vector $\tilde{\bm{v}}_h$ collects the degrees of freedom (i.e. the values of the displacements at the polygon vertices) while $\tilde{\bm{\varepsilon}}$ is a vector collecting the degrees of freedom of the approximated strain field. The term $\bm{N}^V$ collects virtual displacement shape functions and, thus, is not explicitly known, while $\bm{N}^P$, still considering a first order scheme, takes the form \begin{equation} \bm{N}^P = \left[ \begin{array}{ccc}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{array} \right]. \end{equation}
Indicating as $\bm{\Pi}^m$ the $\bm{\Pi}$ operator expressed as a matrix, it is possible to write the discrete strains as \begin{equation} \bm{\Pi}(\bm{v}_h) = \bm{N}^P \bm{\Pi}^m \tilde{\bm{v}}_h. \label{eq:eproj} \end{equation}
The operator $\bm{\Pi}^m$ can be thus evaluated by means of Eq. \eqref{eq:proj}. In particular, substituting Eq. \eqref{eq:eproj} into Eq. \eqref{eq:proj}, we obtain \begin{equation} \int_E (\bm{N}^P\bm{\Pi}^m\tilde{\bm{v}}_h)^T \bm{N}^P \tilde{\bm{\varepsilon}} = \int_E \left[\bm{\varepsilon}(\bm{N}^V \tilde{\bm{v}}_h)\right]^T \bm{N}^P \tilde{\bm{\varepsilon}} \qquad \forall \tilde{\bm{\varepsilon}} \in \mathbb{R}^3, \end{equation}
\noindent which, integrated by parts, leads to \begin{equation} \int_E (\bm{N}^P\bm{\Pi}^m\tilde{\bm{v}}_h)^T \bm{N}^P \tilde{\bm{\varepsilon}} = \int_{\partial E} (\bm{N}^V \tilde{\bm{v}}_h )^T \bm{N}_E\bm{N}^P\tilde{\bm{\varepsilon}} \qquad \forall \tilde{\bm{\varepsilon}} \in \mathbb{R}^3, \label{eq:projeq} \end{equation}
\noindent where, denoting the outward normal as $\bm{n}=[n_x,n_y]^T$, $\bm{N}_E$ is \begin{equation} \bm{N}_E = \left[ \begin{array}{ccc}
n_x & 0 & n_y \\
0 & n_y & n_x
\end{array} \right]. \end{equation}
\noindent Equation \eqref{eq:projeq} can be rewritten as \begin{equation} \tilde{\bm{\varepsilon}}^T \mathcal{\bm{G}} \bm{\Pi}^m \tilde{\bm{v}}_h = \tilde{\bm{\varepsilon}}^T \mathcal{\bm{B}} \tilde{\bm{v}}_h \qquad \forall \tilde{\bm{\varepsilon}} \in \mathbb{R}^3, \label{eq:pim} \end{equation} \noindent where \begin{equation} \mathcal{\bm{G}} = \int_E (\bm{N}^P)^T \bm{N}^P, \qquad \mathcal{\bm{B}} = \int_{\partial E} (\bm{N}_E \bm{N}^P)^T \bm{N}^V. \end{equation}
Equation \eqref{eq:pim} allows to compute $\bm{\Pi}^m$ as the solution of a linear system while matrices $\mathcal{\bm{G}}$ and $\mathcal{\bm{B}}$ are computable because displacements appear only at the boundary where they are explicitly known. This leads to the evaluation of $\bm{\Pi}^m$ as \begin{equation} \bm{\Pi}^m = \mathcal{\bm{G}}^{-1} \mathcal{\bm{B}}, \end{equation} \noindent and, thus, to approximate the l.h.s. of Eq. \eqref{eq:prob} as \begin{equation} \int_\Omega \bm{\varepsilon}(\bm{v})^T \bm{C} \bm{\varepsilon}(\bm{u}) \approx \int_\Omega (\bm{\Pi}^m{\tilde{\bm{v}}})^T \bm{C} \bm{\Pi}^m{\tilde{\bm{u}}}. \end{equation}
\noindent After algebraic manipulations, it is possible to identify the stiffness matrix, $\bm{K}_c$, as \begin{equation} \bm{K}_c = \mathcal{\bm{B}}^T \mathcal{\bm{G}}^{-T} \left[ \int_E (\bm{N}^P)^T \bm{C} \bm{N}^P \right] \mathcal{\bm{G}}^{-1} \mathcal{\bm{B}}. \end{equation}
\noindent In order to ensure that $\bm{K}_c$ has the correct rank, a stabilisation term $\bm{K}_s$ (not here further discussed) must be added to $\bm{K}_c$ so that the stiffness matrix is usually redefined as $\bm{K} = \bm{K}_c + \bm{K}_s$. For further details, we refer to \cite{Hitchhikers,Artioli2017a}, for example.
It is important to notice that the operator $\bm{\Pi}^m$ leads to constant strain fields while the virtual displacements are assumed to be linear over the element edges, and lead to divergence-free stress distributions (cf. Eq. \eqref{eq:space} and Fig. \ref{fig:vField}).
For triangular elements, the virtual (low-order) formulation is equivalent to standard (low-order) triangular finite elements. However, when polygons with a higher number of edges are considered, the degrees of freedom for the displacements could, in principle, give rise to stress/strain fields richer than the constants. A kind of loss of information is thus expected when using operator $\bm{\Pi}^m$ to compute the stresses.
In the following, RCP is introduced as a postprocessing technique for the VEM procedure, with the aim of improving the accuracy of the obtained stresses.
\section{Recovery by Compatibility in Patches} \label{sec:rcp} The key idea of RCP is to minimise the complementary energy associated to a patch of elements treated as an isolated system \cite{Ubertini2004}. Such operation allows to enforce equilibrium, which might be desirable in some applications (see for instance \cite{deMiranda2012}), as well as to increase the accuracy of the obtained stress field. In this work, RCP is presented for degenerate patches composed of a single virtual element. The generalisation of the procedure to the case of patches composed of more than one element does not introduce substantial modifications.
The complementary energy associated to $E$ can be expressed as \begin{equation} J^c(\bm{\sigma}^\ast) = \frac{1}{2} \int_E \bm{\sigma}^{\ast T} \bm{C}^{-1} \bm{\sigma}^\ast - \int_{\partial E} \bm{u}^T \bm{N}_E \bm{\sigma}^\ast, \label{eq:cener} \end{equation}
\noindent where $\bm{\sigma}^\ast$ is the unknown stress field while $\bm{u}^T$ is the known displacement field (i.e. the displacement field computed by the displacement-based VEM analysis). It must be noticed that the displacement field appears only in the boundary term. In order to minimise $J^c$, the stress field is written as the sum of two contributions: \begin{equation} \bm{\sigma}^\ast = \bm{\sigma}^\ast_h + \bm{\sigma}^\ast_p, \end{equation} \noindent which correspond to its divergence-free part and an appropriate particular solution which guarantees equilibrium with the applied external loads. The homogeneous part $\bm{\sigma}^\ast_h$ is approximated as \begin{equation} \bm{\sigma}_h^\ast = \bm{P} \bm{\beta}, \label{eq:stressModes} \end{equation} \noindent where $\bm{P}$ is a matrix of preselected self-equilibrated stress modes, while $\bm{\beta}$ is the vector of the unknown parameters. In particular, if a linear approximation is adopted for $\bm{\sigma}_h$, $\bm{P}$ takes the form \begin{equation} \bm{P} = \left[ \begin{array}{ccccccc}
1 & 0 & 0 & y & 0 & x & 0 \\
0 & 1 & 0 & 0 & x & 0 & y \\
0 & 0 & 1 & 0 & 0 & -y & -x
\end{array} \right]. \label{eq:pmatr} \end{equation}
The particular solutions, $\bm{\sigma}_p^\ast$, appearing in Eq. \eqref{eq:cenersubs} can be calculated as \begin{equation} \bm{\sigma}_p^\ast = \left[\begin{array}{c}
\sigma_x \\
\sigma_y \\
\tau_{xy}
\end{array} \right]^\ast_p =
\left[\begin{array}{c}
- I_x (b_x) \\
- I_y (b_y) \\
0
\end{array} \right]. \end{equation} Above, $b_x$ and $b_y$ are the body force components. Furthermore, $I_x(b_x)$ is an antiderivative of $b_x$ with respect to the coordinate $x$, i.e. it holds $\frac{\partial}{\partial x}I_x(b_x) = b_x$. Analogous definition applies to $I_y(b_y)$ with respect to the coordinate $y$. If the antiderivatives for $b_x$ or $b_y$ are not explicitly known, we first approximate the body force within the element by a constant field, then we take the corresponding antiderivatives.
Substituting Eq. \eqref{eq:stressModes} into Eq. \eqref{eq:cener} leads to \begin{equation} J^c(\bm{\beta}) = \frac{1}{2} \int_E (\bm{P}\bm{\beta} + \bm{\sigma}_p^\ast)^T \bm{C}^{-1} (\bm{P}\bm{\beta} + \bm{\sigma}_p^\ast) - \int_{\partial E} \bm{u}^T \bm{N}_E (\bm{P}\bm{\beta} + \bm{\sigma}_p^\ast). \label{eq:cenersubs} \end{equation}
Imposing null variations of the functional expressed in Eq. \eqref{eq:rcpfun}, the following is obtained \begin{equation} \bm{H} \bm{\beta} = \bm{g}, \label{eq:rcpfun} \end{equation} \noindent where \begin{equation} \bm{H} = \int_E \bm{P}^T \bm{C}^{-1} \bm{P}, \qquad \bm{g} = \int_{\partial E} \bm{P}^T \bm{N}_{E}^T \bm{u} - \int_{E}\bm{P}^T\bm{C}^{-1}\bm{\sigma}_p^\ast. \label{eq:hg} \end{equation}
\noindent In the definition of $\bm{g}$, see Eq. \eqref{eq:hg}, the first term can be integrated by parts and, exploiting that the stress modes are divergence-free, we get \begin{equation} \int_{\partial E} \bm{P}^T \bm{N}_{E}^T \bm{u} = \int_{E} \bm{P}^T \bm{\varepsilon}(\bm{u}). \label{eq:g} \end{equation}
In the finite element framework, the vector $\bm{g}$ can be computed using the left-hand side or the right-hand side of Eq. \eqref{eq:g}, leading to identical results. On the contrary, when virtual element schemes are considered, the left-hand side should be used, since the displacements are pointwise known only at the element boundary.
The adoption of a patch of elements instead of a single element does not introduce additional complexity: simply, the integrals appearing in Eq. \eqref{eq:hg} must be computed over the entire patch. Figure \ref{fig:patches} provides examples of the patches adopted in the following numerical tests. In particular, \textit{Patch 0} indicates the degenerate case of a single element, \textit{Patch 1} is a patch obtained by considering all surrounding elements sharing at least one node with a central element, while \textit{Patch 1B} represents a {\em boundary} patch of the same type as \textit{Patch 1}.
\begin{figure}
\caption{Patch types used in the present study.}
\label{fig:patches}
\end{figure}
\section{Numerical results} \label{sec:num} In this section, numerical tests are used in order to validate the proposed stress recovery procedure. In particular, a square domain characterised by edge length equal to unity is considered. The body is assumed to be homogeneous and isotropic and its mechanical parameters are assigned in terms of the Lam\'e constants $\lambda = 1$ and $\mu = 1$ \cite{BeiraoLovaMora,Artioli2017a}. Plane strain regime is assumed.
Indicating as $\bm{\sigma}^{ex}$ the analytical stress field, the following error norm is used in order to evaluate the accuracy of the obtained results \begin{equation}
\label{eq:stress_err_norm}
E_{\sigma} = \int_\Omega (\bm{\sigma}^{ex} - \bm{\sigma}^\ast)^T \bm{C}^{-1} (\bm{\sigma}^{ex} - \bm{\sigma}^\ast). \end{equation}
As it can be seen in Fig. \ref{fig:meshes}, eight meshes are considered. Four of them are structured and are indicated with the letter 'S' in the following. Those are composed of triangles, quadrilaterals, hexagons and a mixture of convex and concave quadrangles. The remaining four are unstructured meshes composed of triangles, quadrilaterals, random polygons and concave hexagons. Those meshes are indicated with the letter 'U'. The average edge length is indicated as $\bar{h}_e$.
\begin{figure}
\caption{Overview of adopted meshes for convergence assessment numerical tests \cite{Artioli2017HR}.}
\label{fig:meshes}
\end{figure}
Prescribed displacements are assumed and the corresponding body forces are calculated. Such body forces are then applied to the body, together with Dirichlet conditions over the entire boundary. Three tests are considered: \begin{itemize} \item \textbf{Test a}: $u_x = x^3 - 3 x y^2, \; u_y = y^3 - 3 x^2 y$; \item \textbf{Test b}: $u_x = u_y = \sin(\pi x) \sin(\pi y)$; \item \textbf{Test c}: $u_x = xy\sin(\pi x)\sin(\pi y)\; u_y = 0$. \end{itemize}
In particular, \textit{Test a} is characterised by polynomial solution and engenders null body forces with inhomogeneous Dirichlet boundary conditions, while in \textit{Test b} the displacement solution is trigonometric and is characterised by homogeneous Dirichlet boundary conditions, see \cite{Artioli2017HR}. Analogously to \textit{Test b}, \textit{Test c} is characterised by homogeneous Dirichlet boundary conditions, and $u_x$ is a product of polynomial and trigonometric functions, while $u_y$ vanishes.
Three stress recovery procedures are compared and denoted as \textit{VEM}, \textit{RCP0} and \textit{RCP1}. In particular, \textit{VEM} corresponds to the standard procedure based on the adoption of the operator $\bm{\Pi}^m$, together with the use of the constitutive law. Instead, \textit{RCP0} and \textit{RCP1} denote the use of RCP as presented in Eq. \eqref{eq:hg}, for \textit{Patch 0} and \textit{Patch 1}, respectively.
Figures \ref{fig:resuTestA}, \ref{fig:resuTestB} and \ref{fig:resuTestC} report results obtained for \textit{Test a}, \textit{Test b} and \textit{Test c} for all the aforementioned meshes, respectively. It can be clearly seen that RCP always has a favourable effect, leading to more accurate results if compared to the use of the $\bm{\Pi}^m$ operator and, in some cases, to a substantial increase in the convergence rate.
Regarding \textit{RCP0}, we notice that the improvement can be clearly appreciated when the Hex (S) mesh is considered while it tends to disappear for triangular elements. Furthermore, considering now results obtained using \textit{RCP1}, it can be seen that the use of element patches has a significant beneficial effect, in agreement with what has been observed in the context of displacement-based finite elements \cite{Ubertini2004,Ubertini2006}.
\begin{figure}
\caption{$\bar{h}_e-$convergence results for Test $a$: (a) structured and (b) unstructured meshes. Black, red and blue markers indicate \textit{VEM}, \textit{RCP0} and \textit{RCP1}, respectively.}
\label{fig:resuTestA}
\end{figure}
\begin{figure}
\caption{$\bar{h}_e-$convergence results for Test $b$: (a) structured and (b) unstructured meshes. Black, red and blue markers indicate \textit{VEM}, \textit{RCP0} and \textit{RCP1}, respectively.}
\label{fig:resuTestB}
\end{figure}
\begin{figure}
\caption{$\bar{h}_e-$convergence results for Test $c$: (a) structured and (b) unstructured meshes. Black, red and blue markers indicate \textit{VEM}, \textit{RCP0} and \textit{RCP1}, respectively.}
\label{fig:resuTestC}
\end{figure}
Finally, a comparison between the three techniques in terms of von Mises equivalent stress distributions obtained using Hex (S) and Poly (U) meshes is reported for \textit{Test a} and \textit{Test b}. The exact solutions are shown in Fig. \ref{fig:misesAnaSol} while numerical results are reported in Figs. \ref{fig:misesHexaTestA} and \ref{fig:misesHexaTestB}, respectively. The improved compliance with the analytical solution, even for relatively coarse meshes, can be clearly observed, confirming again the effectiveness of the proposed approach.
\begin{figure}
\caption{Exact von Mises equivalent stress distributions: (a) \textit{Test a} and (b) \textit{Test b}.}
\label{fig:misesAnaSol}
\end{figure}
\begin{figure}
\caption{Von Mises equivalent stress distributions for \textit{Test a}: (a) Hex (S) \textit{VEM}, (b) Hex (S) \textit{RCP0}, (c) Hex (S) \textit{RCP1}, (d) Poly (U) \textit{VEM}, (e) Poly (U) \textit{RCP0} and (f) Poly (U) \textit{RCP1}.}
\label{fig:misesHexaTestA}
\end{figure}
\begin{figure}
\caption{Von Mises equivalent stress distributions for \textit{Test b}: (a) Hex (S) \textit{VEM}, (b) Hex (S) \textit{RCP0}, (c) Hex (S) \textit{RCP1}, (d) Poly (U) \textit{VEM}, (e) Poly (U) \textit{RCP0} and (f) Poly (U) \textit{RCP1}.}
\label{fig:misesHexaTestB}
\end{figure}
\section{Conclusions} \label{sec:con}
In the framework of displacement-based virtual element methods, the standard approach to compute stresses is via the constitutive law, using the available approximated strains. When general polygons are considered, such a procedure does not fully exploit the adopted degrees of freedom, thus leading to a quite poor stress field.
In the present paper, the beneficial effect of RCP used in connection with VEM discretisation schemes has been presented. The RCP formulation allows to compute accurate patchwise equilibrated stress fields starting from approximated boundary displacements and from the knowledge of the applied loads. Such a formulation appears to be well-suited for the recovery of stresses in the framework of displacement-based virtual elements, in which such quantities are known only on the elements edges. In its simplest version, the RCP is applied elementwise, so representing an efficient alternative to the standard stress recovery. It has been also shown that further improved effects might be obtained by considering patches of elements, in agreement with the results obtained in the context of finite elements schemes. We finally remark that RCP can be very easily integrated in existing codes without introducing significant computational costs.
\end{document} |
\begin{document}
\frenchspacing \title{Landau's Theorem for $\pi$-blocks\\ of $\pi$-separable groups} \begin{abstract}\noindent Slattery has generalized Brauer's theory of $p$-blocks of finite groups to $\pi$-blocks of $\pi$-separable groups where $\pi$ is a set of primes. In this setting we show that the order of a defect group of a $\pi$-block $B$ is bounded in terms of the number of irreducible characters in $B$. This is a variant of Brauer's Problem 21 and generalizes Külshammer's corresponding theorem for $p$-blocks of $p$-solvable groups. At the same time, our result generalizes Landau's classical theorem on the number of conjugacy classes of an arbitrary finite group. The proof relies on the classification of finite simple groups. \end{abstract}
\textbf{Keywords:} Brauer's Problem 21, $\pi$-blocks, number of characters\\ \textbf{AMS classification:} 20C15
\section{Introduction}
Many authors, including Richard Brauer himself, have tried to replace the prime $p$ in modular representation theory by a set of primes $\pi$. One of the most convincing settings is the theory of $\pi$-blocks of $\pi$-separable groups which was developed by Slattery~\cite{Slattery,Slattery2} building on the work of Isaacs and others (for precise definitions see next section). In this framework most of the classical theorems on $p$-blocks can be carried over to $\pi$-blocks. For instance, Slattery proved versions of Brauer's three main theorems for $\pi$-blocks. Also many of the open conjectures on $p$-blocks make sense for $\pi$-blocks. In particular, \emph{Brauer's Height Zero Conjecture} and the \emph{Alperin--McKay Conjecture} for $\pi$-blocks of $\pi$-separable groups were verified by Manz--Staszewski~\cite[Theorem~3.3]{ManzStaszewski} and Wolf~\cite[Theorem~2.2]{Wolf} respectively. In a previous paper~\cite{SambalePi} the present author proved \emph{Brauer's $k(B)$-Conjecture} for $\pi$-blocks of $\pi$-separable groups. This means that the number $k(B)$ of irreducible characters in a $\pi$-block $B$ is bounded by the order of its defect groups.
In this paper we work in the opposite direction. Landau's classical theorem asserts that the order of a finite group $G$ can be bounded by a function depending only on the number of conjugacy classes of $G$. \emph{Problem~21} on Brauer's famous list~\cite{BrauerLectures} from 1963 asks if the order of a defect group of a block $B$ of a finite group can be bounded by a function depending only on $k(B)$. Even today we do not know if there is such a bound for blocks with just three irreducible characters (it is expected that the defect groups have order three in this case, see \cite[Chapter~15]{habil}). On the other hand, an affirmative answer to Problem~21 for $p$-blocks of $p$-solvable groups was given by Külshammer~\cite{KLandau3}. Moreover, Külshammer--Robinson~\cite{KR} showed that a positive answer in general would follow from the Alperin--McKay Conjecture.
The main theorem of this paper settles Problem~21 for $\pi$-blocks of $\pi$-separable groups.
\begin{ThmA} The order of a defect group of a $\pi$-block $B$ of a $\pi$-separable group can be bounded by a function depending only on $k(B)$. \end{ThmA}
Since $\{p\}$-separable groups are $p$-solvable and $\{p\}$-blocks are $p$-blocks, this generalizes Külshammer's result. If $G$ is an arbitrary finite group and $\pi$ is the set of prime divisors of $|G|$, then $G$ is $\pi$-separable and $\operatorname{Irr}(G)$ is a $\pi$-block with defect group $G$ (see \autoref{facts} below). Hence, Theorem~A also implies Landau's Theorem mentioned above.
Külshammer's proof relies on the classification of finite simple groups and so does our proof. Although it is possible to extract from the proof an explicit bound on the order of a defect group, this bound is far from being optimal. With some effort we obtain the following small values.
\begin{ThmB} Let $B$ be a $\pi$-block of a $\pi$-separable group with defect group $D$. Then \begin{align*} k(B)=1&\Longleftrightarrow D=1,\\ k(B)=2&\Longleftrightarrow D=C_2,\\ k(B)=3&\Longleftrightarrow D\in\{C_3,S_3\} \end{align*} where $C_n$ denotes the cyclic group of order $n$ and $S_n$ is the symmetric group of degree $n$. \end{ThmB}
\section{Notation}
Most of our notation is standard and can be found in Navarro's book~\cite{Navarro}. For the convenience of the reader we collect definitions and crucial facts about $\pi$-blocks. In the following, $\pi$ is any set of prime numbers. We denote the $\pi$-part of an integer $n$ by $n_\pi$. A finite group $G$ is called $\pi$-\emph{separable} if there exists a normal series \[1=N_0\unlhd\ldots\unlhd N_k=G\] such that each quotient $N_i/N_{i-1}$ is a $\pi$-group or a $\pi'$-group. The largest normal $\pi$-subgroup of $G$ is denoted by $\operatorname{O}_{\pi}(G)$.
\begin{Def}
\begin{itemize}
\item A $\pi$-\emph{block} of $G$ is a minimal non-empty subset $B\subseteq\operatorname{Irr}(G)$ such that $B$ is a union of $p$-blocks for every $p\in\pi$ (see \cite[Definition (1.12) and Theorem (2.15)]{Slattery}). In particular, the $\{p\}$-blocks of $G$ are the $p$-blocks of $G$. In accordance with the notation for $p$-blocks we set $k(B):=|B|$ for every $\pi$-block $B$.
\item A \emph{defect group} $D$ of a $\pi$-block $B$ of a $\pi$-separable group $G$ is defined inductively as follows (see \cite[Definition (2.2)]{Slattery2}). Let $\chi\in B$ and let $\lambda\in\operatorname{Irr}(\operatorname{O}_{\pi'}(G))$ be a constituent of the restriction $\chi_{\operatorname{O}_{\pi'}(G)}$ (we say that $B$ \emph{lies over} $\lambda$). Let $G_\lambda$ be the inertial group of $\lambda$ in $G$. If $G_\lambda=G$, then $D$ is a Hall $\pi$-subgroup of $G$ (such subgroups always exist in $\pi$-separable groups). Otherwise there exists a unique $\pi$-block $b$ of $G_\lambda$ lying over $\lambda$ such that $\psi^G\in B$ for any $\psi\in b$ (see \autoref{FR} below). In this case we identify $D$ with a defect group of $b$. As usual, the defect groups of $B$ form a conjugacy classes of $G$. It was shown in \cite[Theorem (2.1)]{Slattery2} that this definition agrees with the usual definition for $p$-blocks.
\item A $\pi$-block $B$ of $G$ \emph{covers} a $\pi$-block $b$ of $N\unlhd G$, if there exist $\chi\in B$ and $\psi\in b$ such that $[\chi_N,\psi]\ne 0$ (see \cite[Definition (2.5)]{Slattery}). \end{itemize} \end{Def}
\begin{Prop}\label{facts} For every $\pi$-block $B$ of a $\pi$-separable group $G$ with defect group $D$ the following holds: \begin{enumerate}[(i)] \item\label{f1} $\operatorname{O}_{\pi}(G)\le D$.
\item\label{f2} For every $\chi\in B$ we have $\frac{|D|\chi(1)_\pi}{|G|_\pi}\in\mathbb{N}$ and for some $\chi$ this fraction equals $1$. \item\label{f3} If the $\pi$-elements $g,h\in G$ are not conjugate, then \[\sum_{\chi\in B}\chi(g)\overline{\chi(h)}=0.\] \item\label{f4} If $B$ covers a $\pi$-block $b$ of $N\unlhd G$, then for every $\psi\in b$ there exists some $\chi\in B$ such that $[\chi_N,\psi]\ne 0$.
\item\label{f5} If $B$ lies over a $G$-invariant $\lambda\in\operatorname{Irr}(\operatorname{O}_{\pi'}(G))$, then $B=\operatorname{Irr}(G|\lambda)$. \end{enumerate} \end{Prop} \begin{proof}
\begin{enumerate}[(i)] \item See \cite[Lemma~(2.3)]{Slattery2}. \item See \cite[Theorems (2.5) and (2.15)]{Slattery2}. \item This follows from \cite[Corollary~8]{Robinsonpi} (by \cite[Remarks on p. 410]{Robinsonpi}, $B$ is really a $\pi$-block in the sense of that paper). \item See \cite[Lemma (2.4)]{Slattery}. \item See \cite[Theorem~(2.8)]{Slattery}.\qedhere \end{enumerate} \end{proof}
The following result allows inductive arguments (see \cite[Theorem~2.10]{Slattery} and \cite[Corollary~2.8]{Slattery2}).
\begin{Thm}[Fong--Reynolds Theorem for $\pi$-blocks]\label{FR} Let $N$ be a normal $\pi'$-subgroup of a $\pi$-separable group $G$. Let $\lambda\in\operatorname{Irr}(N)$ with inertial group $G_\lambda$. Then the induction of characters induces a bijection $b\mapsto b^G$ between the $\pi$-blocks of $G_\lambda$ lying over $\lambda$ and the $\pi$-blocks of $G$ lying over $\lambda$. Moreover, $k(b)=k(b^G)$ and every defect group of $b$ is a defect group of $b^G$. \end{Thm}
Finally we recall $\pi$-special characters which were introduced by Gajendragadkar~\cite{Gajendragadkar}. A character $\chi\in\operatorname{Irr}(G)$ is called $\pi$-\emph{special}, if $\chi(1)=\chi(1)_\pi$ and for every subnormal subgroup $N$ of $G$ and every irreducible constituent $\varphi$ of $\chi_N$ the order of the linear character $\det\varphi$ is a $\pi$-number.
Obviously, every character of a $\pi$-group is $\pi$-special. If $\chi\in\operatorname{Irr}(G)$ is $\pi$-special and $N:=\operatorname{O}_{\pi'}(G)$, then $\chi_N$ is a sum of $G$-conjugates of a linear character $\lambda\in\operatorname{Irr}(N)$ by Clifford theory. Since the order of $\det\lambda=\lambda$ is a $\pi$-number and divides $|N|$, we obtain $\lambda=1_N$. This shows that $N\le\operatorname{Ker}(\chi)$.
\section{Proofs}
At some point in the proof of Theorem~A we need to refer to Külshammer's solution~\cite[Theorem]{KLandau3} of Brauer's Problem 21 for $p$-solvable groups:
\begin{Prop}\label{K}
There exists a monotonic function $\alpha:\mathbb{N}\to\mathbb{N}$ with the following property: For every $p$-block $B$ of a $p$-solvable group with defect group $D$ we have $|D|\le \alpha(k(B))$. \end{Prop}
The following ingredient is a direct consequence of the classification of finite simple groups.
\begin{Prop}[{\cite[Theorem~2.1]{Kohl}}]\label{kohl}
There exists a monotonic function $\beta:\mathbb{N}\to\mathbb{N}$ with the following property: If $G$ is a finite non-abelian simple group such that $\operatorname{Aut}(G)$ has exactly $k$ orbits on $G$, then $|G|\le \beta(k)$. \end{Prop}
In the following series of lemmas, $\pi$ is a fixed set of primes, $G$ is a $\pi$-separable group and $B$ is a $\pi$-block of $G$ with defect group $D$. \autoref{facts}\eqref{f2} guarantees the existence of a height $0$ character in $B$. We need to impose an additional condition on such a character.
\begin{Lem}\label{lemchar}
There exists some $\chi\in B$ such that $\operatorname{O}_{\pi}(G)\le\operatorname{Ker}(\chi)$ and $|D|\chi(1)_\pi=|G|_\pi$. \end{Lem} \begin{proof}
We argue by induction on $|G|$. Let $B$ lie over $\lambda\in\operatorname{Irr}(\operatorname{O}_{\pi'}(G))$. Suppose first that $G_\lambda=G$. Then $|D|=|G|_\pi$ and $B=\operatorname{Irr}(G|\lambda)$ by \autoref{facts}\eqref{f5}. Since $\lambda$ is $\pi'$-special, there exists a $\pi'$-special $\chi\in B$ by \cite[Lemma~(2.7)]{Slattery}. It follows that $\operatorname{O}_\pi(G)\le\operatorname{Ker}(\chi)$ and $|D|\chi(1)_\pi=|D|=|G|_\pi$.
Now assume that $G_\lambda<G$. Let $b$ be the Fong--Reynolds correspondent of $B$ in $G_\lambda$. By induction there exists some $\psi\in b$ such that $\operatorname{O}_\pi(G_\lambda)\le\operatorname{Ker}(\psi)$ and $|D|\psi(1)_\pi=|G_\lambda|_\pi$. Let $\chi:=\psi^G\in B$. Since $[\operatorname{O}_\pi(G),\operatorname{O}_{\pi'}(G)]=1$ we have $\operatorname{O}_\pi(G)\le\operatorname{O}_\pi(G_\lambda)\le\operatorname{Ker}(\psi)$ and $\operatorname{O}_\pi(G)\le\operatorname{Ker}(\chi)$ (see \cite[Lemma~(5.11)]{Isaacs}). Finally, \[|D|\chi(1)_\pi=|D|\psi(1)_\pi|G:G_\lambda|_\pi=|G_\lambda|_\pi|G:G_\lambda|_\pi=|G|_\pi.\qedhere\] \end{proof}
For every $p\in\pi$, the character $\chi$ in \autoref{lemchar} lies in a $p$-block $B_p\subseteq B$ whose defect group has order $|D|_p$. In fact, it is easy to show that every Sylow $p$-subgroup of $D$ is a defect group of $B_p$.
Our second lemma extends an elementary fact on $p$-blocks (see \cite[Theorem (9.9)(b)]{Navarro}).
\begin{Lem}\label{lemquot} Let $N$ be a normal $\pi$-subgroup of $G$. Then $B$ contains a $\pi$-block of $G/N$ with defect group $D/N$. \end{Lem} \begin{proof}
Again we argue by induction on $|G|$. Let $\lambda\in\operatorname{Irr}(\operatorname{O}_{\pi'}(G))$ be under $B$. Suppose first that $G_\lambda=G$. Then $|D|=|G|_\pi$. By \autoref{lemchar}, there exists some $\chi\in B$ such that $N\le\operatorname{O}_\pi(G)\le\operatorname{Ker}(\chi)$ and $\chi(1)_\pi=1$. Hence, we may consider $\chi$ as a character of $\overline{G}:=G/N$. As such, $\chi$ lies in a $\pi$-block $\overline{B}$ of $\overline{G}$. For any $\psi\in\overline{B}$ there exists a sequence of characters $\chi=\chi_1,\ldots,\chi_k=\psi$ such that $\chi_i$ and $\chi_{i+1}$ lie in the same $p$-block of $\overline{G}$ for some $p\in\pi$ and $i=1,\ldots,k-1$. Then $\chi_i$ and $\chi_{i+1}$ also lie in the same $p$-block of $G$. This shows that $\psi\in B$ and $\overline{B}\subseteq B$. For a defect group $P/N$ of $\overline{B}$ we have
\[|P/N|=\max\Bigl\{\frac{|\overline{G}|_\pi}{\psi(1)_\pi}:\psi\in\overline{B}\Bigr\}=\frac{|\overline{G}|_\pi}{\chi(1)_\pi}=|\overline{G}|_\pi=|D/N|\] by \autoref{facts}\eqref{f2}. Since the Hall $\pi$-subgroups are conjugate in $G$, we conclude that $D/N$ is a defect group of $\overline{B}$.
Now let $G_\lambda<G$, and let $b$ be the Fong--Reynolds correspondent of $B$ in $G_\lambda$. After conjugation, we may assume that $D$ is a defect group of $b$. By induction, $b$ contains a block $\overline{b}$ of $G_\lambda/N$ with defect group $D/N$. If we regard $\lambda$ as a character of $\operatorname{O}_{\pi'}(G)N/N\cong\operatorname{O}_{\pi'}(G)$, we see that $\overline{G}_\lambda=G_\lambda/N$. It follows that the Fong--Reynolds correspondent $\overline{B}=\overline{b}^{\overline{G}}$ of $\overline{b}$ is contained in $B$ and has defect group $D/N$. \end{proof}
The next result extends one half of \cite[Proposition]{KLandau1} to $\pi$-blocks.
\begin{Lem}\label{lemsub}
Let $N\unlhd G$, and let $b$ be a $\pi$-block of $N$ covered by $B$. Then $k(b)\le|G:N|k(B)$. \end{Lem} \begin{proof}
By \autoref{facts}\eqref{f4}, $b\subseteq\bigcup_{\chi\in B}{\operatorname{Irr}(N|\chi)}$. For every $\chi\in B$ the restriction $\chi_N$ is a sum of $G$-conjugate characters according to Clifford theory. In particular, $\lvert\operatorname{Irr}(N|\chi)\rvert\le|G:N|$ and
\[k(b)\le\sum_{\chi\in B}\lvert\operatorname{Irr}(N|\chi)\rvert\le |G:N|k(B).\qedhere\] \end{proof}
It is well-known that the number of irreducible characters in a $p$-block $B$ is greater or equal than the number of conjugacy classes which intersect a given defect group of $B$ (see \cite[Problem (5.7)]{Navarro}). For $\pi$-blocks we require the following weaker statement.
\begin{Lem}\label{lemnormal} Let $N$ be a normal $\pi$-subgroup of $G$. Then the number of $G$-conjugacy classes contained in $N$ is at most $k(B)$. \end{Lem} \begin{proof}
Let $R\subseteq N$ be a set of representatives for the $G$-conjugacy classes inside $N$. By \autoref{lemchar}, there exists some $\chi\in B$ such that $\chi(r)=\chi(1)\ne 0$ for every $r\in R$. Thus, the columns of the matrix $M:=(\chi(r):\chi\in B,\,r\in R)$ are non-zero. By \autoref{facts}\eqref{f3}, the columns of $M$ are pairwise orthogonal, so in particular they are linearly independent. Hence, the number of rows of $M$ is at least $|R|$. \end{proof}
We can prove the main theorem now.
\begin{proof}[Proof of Theorem~A] The proof strategy follows closely the arguments in \cite{KLandau3}. We construct inductively a monotonic function $\gamma:\mathbb{N}\to\mathbb{N}$ with the desired property. To this end, let $B$ be a $\pi$-block of a $\pi$-separable group $G$ with defect group $D$ and $k:=k(B)$. If $k=1$, then the unique character in $B$ has $p$-defect $0$ for every $p\in\pi$. It follows from \autoref{facts}\eqref{f2} that this can only happen if $D=1$. Hence, let $\gamma(1):=1$.
Now suppose that $k>1$ and $\gamma(l)$ is already defined for $l<k$. Let $N:=\operatorname{O}_{\pi'}(G)$. By a repeated application of the Fong--Reynolds Theorem for $\pi$-blocks and \autoref{facts}\eqref{f5}, we may assume that $B$ is the set of characters lying over a $G$-invariant $\lambda\in\operatorname{Irr}(N)$. Then $D$ is a Hall $\pi$-subgroup of $G$. By \cite[Problem~(6.3)]{Navarro2}, there exists a character triple isomorphism \[(G,N,\lambda)\to(\widehat{G},\widehat{N},\widehat{\lambda})\]
such that $G/N\cong\widehat{G}/\widehat{N}$ and $\widehat{N}=\operatorname{O}_{\pi'}(\widehat{G})\le\operatorname{Z}(\widehat{G})$. Then $\widehat{B}:=\operatorname{Irr}(\widehat{G}|\widehat{\lambda})$ is a $\pi$-block of $\widehat{G}$ with defect group $\widehat{D}\cong D$ and $k(\widehat{B})=k$. After replacing $G$ by $\widehat{G}$ we may assume that $N\le\operatorname{Z}(G)$. Then \[\operatorname{O}_{\pi'\pi}(G)=N\times P\] where $P:=\operatorname{O}_{\pi}(G)$. If $P=1$, then $G$ is a $\pi'$-group and we derive the contradiction $k=1$. Hence, $P\ne 1$.
Let $M$ be a minimal normal subgroup of $G$ contained in $P$. By \autoref{lemquot}, $B$ contains a $\pi$-block $\overline{B}$ of $G/M$ with defect group $D/M$. Since the kernel of $B$ is a $\pi'$-group (see \cite[Theorem~(6.10)]{Navarro}), we have $k(\overline{B})<k$. By induction, it follows that \begin{equation}\label{DM}
|D/M|\le \gamma(k-1) \end{equation} where we use that $\gamma$ is monotonic. Let $H/M$ be a Hall $\pi'$-subgroup of $G/M$, and let \[K:=\bigcap_{g\in G}gHg^{-1}\unlhd G.\] Then
\[|G:K|\le|G:H|!=|G/M:H/M|!=(|G/M|_\pi)!=|D/M|!\le \gamma(k-1)!\] by \eqref{DM}. Let $b$ be a $\pi$-block of $K$ covered by $B$. By \autoref{lemsub}, \begin{equation}\label{kb}
k(b)\le|G:K|k\le \gamma(k-1)!k. \end{equation} Thus we have reduced our problem to the block $b$ of $K$. Since $K/M\le H/M$ is a $\pi'$-group, $b$ has defect group $M$ by \autoref{facts}\eqref{f1}.
As a minimal normal subgroup, $M$ is a direct product of isomorphic simple groups. Suppose first that $M$ is an elementary abelian $p$-group for some $p\in\pi$. Then $K$ is $p$-solvable and $b$ is just a $p$-block with defect group $M$. Hence, with the notation from \autoref{K} we have \begin{equation}\label{M1}
|M|\le \alpha(k(b))\le \alpha\bigl(\gamma(k-1)!k\bigr) \end{equation} by \eqref{kb}.
Now suppose that $M=S\times\ldots\times S=S^n$ where $S$ is a non-abelian simple group.
Let $x_1,\ldots,x_s\in S$ be representatives for the orbits of $\operatorname{Aut}(S)$ on $S\setminus\{1\}$. Since $\operatorname{Aut}(M)\cong\operatorname{Aut}(S)\wr S_n$ (where $S_n$ denotes the symmetric group of degree $n$), the elements $(x_i,1,\ldots,1)$, $(x_i,x_i,1,\ldots,1),\ldots,(x_i,\ldots,x_i)$ of $M$ with $i=1,\ldots,s$ lie in distinct conjugacy classes of $K$. Consequently, \autoref{lemnormal} yields $ns\le k(b)$. Now with the notation of \autoref{kohl} we deduce that $|S|\le \beta(s+1)$ and \begin{equation}\label{M2}
|M|=|S|^n\le \beta(s+1)^n\le \beta\bigl(k(b)+1\bigr)^{k(b)}\le \beta\bigl(\gamma(k-1)!k+1\bigr)^{\gamma(k-1)!k} \end{equation} by \eqref{kb}.
Setting \[\gamma(k):=\gamma(k-1)\max\bigl\{\alpha\bigl(\gamma(k-1)!k\bigr),\,\beta\bigl(\gamma(k-1)!k+1\bigr)^{\gamma(k-1)!k}\bigr\}\] we obtain
\[|D|=|D/M||M|\le\gamma(k)\] by \eqref{DM}, \eqref{M1} and \eqref{M2}. Obviously, $\gamma$ is monotonic. \end{proof}
\begin{proof}[Proof of Theorem~B] We have seen in the proof of Theorem~A that $k(B)=1$ implies $D=1$. Conversely, \cite[Theorem~3]{SambalePi} shows that $D=1$ implies $k(B)=1$.
Now let $k(B)=2$. Then $B$ is a $p$-block for some $p\in\pi$. By a result of Brandt~\cite[Theorem~A]{Brandt}, $p=2$ and $|D|_2=2$ follows from \autoref{facts}\eqref{f2}. For every $q\in\pi\setminus\{2\}$, $B$ consists of two $q$-defect $0$ characters. This implies $D=C_2$. Conversely, if $D=C_2$, then we obtain $k(B)=2$ by \cite[Theorem~3]{SambalePi}.
Finally, assume that $k(B)=3$. As in the proof of Theorem~A, we may assume that \[\operatorname{O}_{\pi'\pi}(G)=\operatorname{O}_{\pi'}(G)\times P\] with $P:=\operatorname{O}_{\pi}(G)\ne 1$.
By the remark after \autoref{lemchar}, for every $p\in\pi$ there exists a $p$-block contained in $B$ whose defect group has order $|D|_p$. If $|D|_2\ge 4$, we derive the contradiction $k(B)\ge 4$ by \cite[Proposition~1.31]{habil}. Hence, $|D|_2\le 2$. By \autoref{lemquot}, $B$ contains a $\pi$-block $\overline{B}$ of $G/P$ with defect group $D/P$ and $k(\overline{B})<k(B)$. The first part of the proof yields $|D/P|\le 2$. In particular, $P$ is a Hall subgroup of $D$. From \autoref{lemnormal} we see that $P$ has at most three orbits under $\operatorname{Aut}(P)$. If $P$ is an elementary abelian $p$-group, then $B$ contains a $p$-block $B_p$ with normal defect group $P$. The case $p=2$ is excluded by the second paragraph of the proof. Hence, $p>2$ and $k(B_p)=k(B)=3$. Now \cite[Proposition~15.2]{habil} implies $|P|=3$ and $|D|\in\{3,6\}$. A well-known lemma by Hall--Higman states that $\operatorname{C}_G(P)\le \operatorname{O}_{\pi'\pi}(G)$. Hence, $|D|=6$ implies $D\cong S_3$. It remains to deal with the case where $P$ is not elementary abelian. In this case, a result of Laffey--MacHale~\cite[Theorem~2]{LaffeyMacHale} shows that $P=P_1\rtimes Q$ where $P_1$ is an elementary abelian $p$-group and $Q$ has order $q\in\pi\setminus\{p\}$. Moreover, $|P_1|\ge p^{q-1}$. In particular, $p>2$ since $|D|_2\le 2$. Again $B$ contains a $p$-block $B_p$ with normal defect group $P_1$ and $k(B_p)=3$. As before, we obtain $|P_1|=3$ and $q=2$. This leads to $D\cong S_3$.
Conversely, let $D\in\{C_3,S_3\}$. By the first part of the proof, $k(B)\ge 3$. Let $N:=\operatorname{O}_{\pi'}(G)$. Using the Fong--Reynolds Theorem for $\pi$-blocks again, we may assume that $B=\operatorname{Irr}(G|\lambda)$ where $\lambda\in\operatorname{Irr}(N)$ is $G$-invariant and $D$ is a Hall $\pi$-subgroup of $G$. By a result of Gallagher (see \cite[Theorem~5.16]{Navarro2}), we have $k(B)\le k(G/N)$. Moreover, $\operatorname{O}_{\pi}(G/N)\le DN/N$ and \[\operatorname{C}_{G/N}(\operatorname{O}_{\pi}(G/N))\le\operatorname{O}_{\pi}(G/N)\] by the Hall--Higman Lemma mentioned above. It is easy to see that this implies $G/N\le S_3$. Hence, $k(B)\le 3$ and we are done. \end{proof}
\section*{Acknowledgment} The author is supported by the German Research Foundation (\mbox{SA 2864/1-1} and \mbox{SA 2864/3-1}).
\end{document} |
\begin{document}
\title{Modelling calibration uncertainty in networks of environmental sensors}
\begin{abstract} Networks of low-cost sensors are becoming ubiquitous, but often suffer from poor accuracies and drift. Regular colocation with reference sensors allows recalibration but is complicated and expensive. Alternatively the calibration can be transferred using low-cost, mobile sensors. However inferring the calibration (with uncertainty) becomes difficult. We propose a variational approach to model the calibration across the network. We demonstrate the approach on synthetic and real air pollution data, and find it can perform better than the state of the art (multi-hop calibration). We extend it to categorical data produced by citizen-scientist labelling. In Summary: The method achieves uncertainty-quantified calibration, which has been one of the barriers to low-cost sensor deployment and citizen-science research.
\keywords{air pollution, Bayesian modelling, calibration, Gaussian processes, low-cost sensors, variational inference} \end{abstract}
\section{Introduction}
Large networks of low-cost sensors are becoming increasingly common across a range of domains, including weather \citep{van2014trans}, snow-cover \citep{pohl2014potential}, wildlife \citep{dyo2010evolution} and air pollution \citep{khedo2010wireless,liu2020low}. The aim is to collect far more data at low-cost. Unfortunately, across all these domains is the common issue of how to ensure low-cost sensors remain calibrated, while still achieving substantial cost-savings. Citizen science data collection initiatives often have similar issues, necessitating validation by experts, replication and statistical modelling \citep[see][for a review]{kosmala2016assessing}. In particular, the problem of maintaining calibration over time, in low cost sensor networks, typically requires regular colocation recalibration \citep{rai2017end}.
Considerable work has already been invested in developing approaches for the reliable use of low-cost sensors for air pollution monitoring (see Section \ref{background}). \cite{rai2017end} emphasise that the key issue with the use of low-cost sensors is data quality: For the network to have some use for policy makers the sensors need frequent calibration in `\emph{final deployment conditions}' such that sensor calibration drift should be accounted for.
This work is motivated by the need to manage the calibration of a network of low-cost PM2.5, air pollution sensors deployed in Kampala, Uganda. Kampala is a low-income, East African city, associated with a tropical, high-humidity, dusty environment; leading to additional issues with sensor reliability and drift. Calibration in this network is achieved through regular but brief visits to the static (low-cost and reference) sensors, by mobile low-cost sensors mounted on motorbike taxis. The motorbike taxi visits are often opportunistic and usually last less than ninety minutes. This leads to a complex network of in-situ colocation events. Our experience is that drift (due to dust, etc) experienced in Kampala is much greater than similar sensors running in the UK, suggesting that this should be included in any model. We found that most of the research on pollution monitoring were for cities in temperate climates. Very few environmental monitoring networks are on the continent of Africa \citep{mao2019low}. Hopefully the approach described will help guide future deployments across the continent.
\subsection{The task}
We have a set of particulate (PM2.5) sensors. Some are mobile and some are static. Some of the static sensors are `reference' instruments which we assume measure the true pollution. The rest are low-cost sensors. We assume that the true pollution is a parametric function, $\phi(\text{raw measurement}, \text{parameters})$, of the raw observation from the low-cost sensor. \emph{The task is to estimate this calibration function (or rather its vector of parameters) for each sensor, for any given time, to allow us to use the low-cost sensors to estimate the true air pollution in locations where there is no reference instrument.}
The full problem consists of both estimating the calibration functions \emph{and} modelling the pollution over space and time across the city. Modelling these two aspects jointly may have advantages, but we focus in this paper on simply solving the calibration problem, where each sensor-pair colocation occurs at an arbitrary, undefined spatial location.
For a given sensor, $s$ at time, $t$, we wish to find the value of the calibration function's parameter vector, $\bm{f}_{s}(t)$. We will estimate this using a list of colocations. Each colocation records: (a) The time of the colocation (b) The id of the two sensors (c) The raw (uncalibrated) measurements of the two sensors. To illustrate with an example, the calibration function could be a simple linear regression with two parameters (an offset and a gradient), such that: $\text{true pollution} = \text{raw measurement} \times \text{gradient} + \text{offset}$. The gradient and offset make up a vector of two parameters, which we wish to estimate. These we believe can change over time, and differ between sensors.
We want a method that: (1) Can infer the calibration function's parameters even if the associated sensor has not been directly visited by a reference sensor. (2) Can combine calibration information arriving through multiple pathways of calibration. (3) Uses both static and mobile sensor data. (4) Has a reasonable model of how the measurement is related to the true pollution. (5) Provides principled estimates of the uncertainty in the calibration, taking into account time.
\subsection{Background} \label{background} A review by \cite{delaine2019situ} divides the calibration literature in several ways, considering reference instrumentation, sensor mobility, types of calibration (time dependency and complexity) and `grouping strategies'. Calibration can occur prior to deployment or can occur in-situ by finding those times in which pollution is believed to be similar across large distances, for example when few local sources of pollution exist at night \citep[e.g.][]{miskell2018solution}. We are interested in those approaches which use physical post-deployment colocation.
Consider first \cite{arfire2015model}. They have a model-based approach which assumes a low-cost chemical sensor is calibrated by co-location with a reference sensor regularly. This doesn't then handle the more complex network that we are analysing (in which colocations between low-cost sensors also occurs). They consider temporal drift by either a linear or a hyperbolic term with respect to time. There is no indication that they compute or assess the uncertainty in the model's predictions.
Of more interest to our application are studies looking at networks of sensors such that the low-cost sensors are not necessarily directly calibrated by the reference instrument but potentially by intermediate sensors (e.g. mobile sensors). \cite{tsujita2004dynamic} considers one such model. However they make a series of assumptions (e.g. how the sensor calibration drift is a fixed linear number). The calibration is also done in a non-probabilistic approach; with an estimate for the pollution at a location simply the average of all sensors, even though some will, presumably, be more reliable than others (e.g. if they have just visited the reference station). The model also doesn't report its confidence. Setting aside these issues, the paper does suggest that a network approach to calibration might be of benefit. \cite{xiang2012collaborative} use the uncertainty in a given sensor's calibration when combining measurements. The problem this paper considers largely matches ours. One major difference is the error model. In our experience the calibration we are concerned with involves scaling of the true pollution, not just an offset. One potentially could alter their paper by using the log of sensor measurements, which will then make their sum a product. They also require that the calibration computation and data remains local to the device, while we are computing this using all the data from all sensors. \cite{markert2018privacy} assign weights to various estimates but, as with almost all the papers, the correlations between the calibration estimates are not considered. Their treatment is partly Bayesian (for example in how different uncertainties are combined), but has a single value for drift (subtracting the calibrated reading from the raw reading). Also by only looking at previous colocations, it fails to propagate all the calibration information available. We suggest therefore that there are likely to be considerable improvements in calibration accuracy and uncertainty quantification by using a process model of calibration over time, which will be able to leverage prior knowledge around temporal structure in calibration functions. A fully Bayesian treatment will also be able to handle more complex measurement functions.
These criticisms aside, \cite{markert2018privacy} provide a useful foundation and demonstration of the benefits of a network of sensors. Another related work \citep{hasenfratz2012fly} looks at a similar problem, this time using a polynomial fit for the sensor data (rather than just adding an offset). They handle multiple sensors by computing weighted sums, with weights dependent on the time difference (between query and calibration) although not in a particularly probabilistic manner. The paper computes a single ML estimate for the calibration.
Most other papers \citep[e.g.][]{maag2017scan,kizel2018node,bychkovskiy2003collaborative,fonollosa2016calibration} do not consider drift over time. Many, \citep[e.g.][]{maag2017scan,bychkovskiy2003collaborative} do not handle uncertainty in their calibration estimates. Finally note that most of the papers mentioned in this section assume linear/scaling-only calibration. In summary, there is currently no framework for performing calibration across an arbitrary network of sensor-rendezvous events which probabilistically handles the quantification of uncertainty in the calibration with temporal drift. The method described in this paper correctly estimates the uncertainty in calibration across a complex network consisting of mobile, static, low-cost and reference instruments.
\subsubsection{Related work in probabilistic modelling}
We did originally explore a joint model, in which both the calibration parameters \emph{and the pollution} were modelled as Gaussian processes. This had potentially more power as non-colocated (but still correlated) measurements could still inform the estimates of the calibration, but it was found to be somewhat intractable, and hence we focus in this paper on the calibration pair model (in which we treat the colocations independently of their spatial locations) and leave the joint model for future work.
It is however worth considering the necessary tools for inference in the joint model, as it connects with other literature. The joint model involved a calibration function parameterised by a Gaussian process (modelling, e.g. the calibration offset) operating on another Gaussian process (the true pollution over time and space). This is reminiscent, if the calibration function was simply scaling the pollution, of \cite{wilson2012gaussian}, in which a weighted sum of GPs is computed, with the weights consisting of GPs themselves. They experimented with both MCMC (using elliptical slice sampling) and variational EM to perform the inference on the latent variables. We experimented with using MCMC for inference in both models (see supplementary) but found mixing was challenging as the network grew larger. They noted that even with their relatively simple model, Gibbs sampling would `mix poorly because of the tight correlations between the weights and the nodes'. We found that even with whitening, HMC and other tricks, the strong correlations in the posterior made it very difficult to perform inference. \cite{alvarado2017efficient} more recently computed a similarly structured product of Gaussian processes, and used Gauss-Hermite quadrature. In our experiments we used simple Monte Carlo to compute the expectations, following the approaches in \cite{hensman2013gaussian} and \cite{salimbeni2017doubly}.
\subsection{Calibration pair model} \label{calpair} \begin{figure}
\caption{Calibration pair model. \textit{A priori} the vectors of GP random variables in $\bm{F}$ are independent, but become correlated in the posterior due to colocated observations connecting them together in pairs.}
\label{plate}
\end{figure}
\subsubsection{Model definition} Each of the $i=1..N$ observations, $\bm{Y} \in \mathcal{R}^{N \times 2}$, consist of two measurements, $y_i^{(1)}, y_i^{(2)}$, by two sensors with indices $s_i^{(1)},s_i^{(2)}$, colocated at time $t_i$. One could, in principle, have more than two sensors colocated, but for simplicity in representation and likelihood function we have used pair-wise colocation. The model assumes that a deterministic parametric function, $\phi$, describes how a measurement is related to the true pollution:
$\text{true pollution} = \phi(\text{raw measurement}, \text{parameters}) + \text{unstructured noise}$.
For each sensor, $j$, the calibration function's parameter vector consists of $C$ parameters that are each functions of time: $\bm{f}_{j}(t)$. Our model assigns an independent Gaussian process prior (over time) to each of these $c=1..C$ parameters for each of the $j=1..S$ sensors. So, let $\bm{F}(t)$ be an $S \times C$ matrix of \emph{independent} Gaussian processes $[f_{j}]_c(t) \sim GP(0,k_{j,c})$: We are assuming, \textit{a priori}, that the calibration parameters of the sensors are independent between sensors and the parameters are also independent within sensors. In our implementation we have different kernels for sensors of different types and for different parameters, allowing us to model, for example, that one sensor type's offset drifts at a slower rate than its scaling, etc. The aim of the method is to infer the values of $\bm{F}(t)$.
We emphasise that, in this model we are discarding location information and just using information about which sensor pairs are colocated, and when. \newcommand{h_i}{h_i}
\subsubsection{The likelihood}
Here, by likelihood, we mean the probability of the observations, $\bm{Y}$, given the model, its hyper-parameters and any latent variables, $\bm{F}$. To compute $\log p(\bm{Y}|\bm{F})$ we first assume that this can be factorised across observations: $p(\bm{Y}|\bm{F}) = \prod_{i=1}^N p(\bm{y}_i | \bm{F})$. Where $p(\bm{y}_i | \bm{F})$ is the probability of one pair of colocated observations, given our model and latent calibration parameters.
We pick from $\bm{F}$ the two rows associated with the two sensors $s_i^{(1)}$ and $s_i^{(2)}$ that took the two measurements. So the calibrated predicted pollution from the two observations are,
\newcommand{\f}[1]{\bm{f}_{s_i^{(#1)}}(t_i)} \newcommand{\ft}[1]{\bm{f}_{s_i^{(#1)}}(t_i^{(#1)})} \newcommand{\calfn}[1]{\phi \big(y_i^{(#1)},\f{#1} \big)}
$\calfn{1}$ and $\calfn{2}$.
Let $h_i$ be the (latent) true pollution associated with the colocated observation pair $\bm{y}_i$. We assume that these calibration-corrected predictions of the pollution are normally distributed around the true (latent) pollution, i.e.: \begin{equation}
p\Bigg(\begin{bmatrix}\calfn{1}\\ \calfn{2}\end{bmatrix} \Bigg| h_i \Bigg) \sim N\Bigg(\begin{bmatrix}h_i \\ h_i\end{bmatrix}, \begin{bmatrix}\sigma^2 & 0 \\ 0 & \sigma^2 \end{bmatrix}\Bigg). \end{equation} If we place a wide, fairly uninformative, normal prior on $h_i$ (with mean zero, and variance $\gamma^2$), we can integrate out $h_i$ in closed form. Allowing us to write the probability of a given (calibrated) observation pair as a 2d joint multivariate Gaussian, \begin{align} p\Bigg(\begin{matrix}
\calfn{1} \\
\calfn{2} \\
\end{matrix} \Bigg) & = \bigintss p\Bigg(\begin{bmatrix}\calfn{1}\\ \calfn{2}\end{bmatrix} \Bigg| h_i \Bigg)
\times N\Big(h_i \Big| 0, \gamma^2\Big) \;\;\;\text{d}h_i\\
& \sim N\Bigg(0,\begin{bmatrix}
\sigma^2 + \gamma^2 & \gamma^2 \\
\gamma^2 & \sigma^2 + \gamma^2 \\
\end{bmatrix}\Bigg). \label{like3} \end{align}
For inference using \cite{hensman2013gaussian} we need an expression for the conditional probability of the observations $\bm{y}$ given the latent calibration $\bm{f}$. So we need to perform a change of variable, to covert from the distribution of $p(\bm{\phi}(\bm{y},\bm{f}))$ to $p(\bm{y} | \bm{f})$. To do this we multiply the probability of a vector of calibrated observations, $\bm{\phi}$, by the determinant of the vector's Jacobian with respect to $\bm{y}$ (which is diagonal in this case), evaluated at the two observations.
\newcommand{\calfnz}[1]{\phi \big(z_#1,\f{#1} \big)} \begin{equation}p\Bigg( \begin{matrix} y_i^{(1)} \\ y_i^{(2)}
\end{matrix} \Bigg| \bm{F} \Bigg)
=
p\Bigg(\begin{matrix}
\calfn{1} \\
\calfn{2} \\
\end{matrix}\Bigg) \times
\frac{\partial \calfnz{1}}{\partial z_1} \Bigg|_{z_1 = y_i^{(1)}} \times \frac{\partial \calfnz{2}}{\partial z_2} \Bigg|_{z_2 = y_i^{(2)}} \label{likelihood} \end{equation}
For implementation, we use the log probabilities, and so the product becomes the sum of the log probability of $\bm{\phi}$ and the log of the two partial derivatives.
\textbf{Reference Instruments} We define a binary vector $\bm{r}$, which describes which sensors are reference sensors. In those cases, we use the identity function instead of the calibration function, i.e. replacing $\calfn{a}$ with $y_i^{(a)}$.
\subsubsection{Variational inference for the calibration pair model}
We wish to compute the calibration parameters from the observations, $p(\bm{F}|\bm{Y})$. A simple application of Bayes rule, $p(\bm{F}|\bm{Y}) = {p(\bm{Y}|\bm{F}) p(\bm{F})}/{p(\bm{Y})}$, is not soluble or tractable in closed form,
as $N$ may be large and the likelihood function is non-Gaussian. We quickly step through the variational approach, largely based on \citep{hensman2013gaussian}.
We approximate the posterior, $p(\bm{F}|\bm{Y})$ with a variational distribution over $\bm{F}$: $q(\bm{F})$. We want to make this distribution similar to the true posterior. To this end we aim to minimise the KL divergence between the two: \begin{align}
\mathcal{D}_{KL}\Big[q(\bm{F}) || p(\bm{F}|\bm{Y}) \Big] &= - \int{q(\bm{F}) \log \frac{p(\bm{F}|\bm{Y})}{q(\bm{F})} d\bm{F}}\\ &= - \int{q(\bm{F}) \log \frac{p(\bm{F},\bm{Y})}{p(\bm{Y})q(\bm{F})} d\bm{F}}\\ &= - \int{q(\bm{F}) \log \frac{p(\bm{F},\bm{Y})}{q(\bm{F})} d\bm{F}} + \int{q(\bm{F}) \log p(\bm{Y}) d\bm{F}}\\ &= - \underbrace{\int{q(\bm{F}) \log \frac{p(\bm{F},\bm{Y})}{q(\bm{F})} d\bm{F}}}_{\text{The ELBO}} + \log p(\bm{Y}) \end{align} The last term is constant wrt the variational distribution so we can minimise the KL divergence simply by maximimising the integral (the evidence lower bound, ELBO). This is still not tractable without the approximation provided by inducing points.
We introduce an additional vector $\bm{u}$ of values that we assume can describe $\bm{F}$, evaluated at inducing point locations, $\bm{Z}$. Each row of $\bm{Z}$ has a time, and a sensor id/parameter id, to index the associated GP in $\bm{F}$. We found that spacing the inducing points evenly over the sensor domains, rather than being optimised works well for the datasets we experimented with.
We can compute the posterior over both $\bm{F}$ and $\bm{u}$ (we assume that $\bm{Y}$ is conditionally independent of $\bm{u}$ given $\bm{F}$),
$p(\bm{F},\bm{u}|\bm{Y}) \propto p(\bm{Y}|\bm{F}) p(\bm{F}|\bm{u}) p(\bm{u}).$ We don't compute this directly, but instead use our approximation,
$q(\bm{F},\bm{u}|\bm{Y}) = p(\bm{F}|\bm{u})q(\bm{u}).$ Here we have assumed that $\bm{u}$ is a sufficient statistic to determine $\bm{F}$ so that $p(\bm{F}|\bm{u},\bm{Y})=p(\bm{F}|\bm{u})$. We substitute this approximation into the ELBO and augment with the inducing point values, \begin{align} \mathcal{L} &= \int\int{q(\bm{F},\bm{u}) \log \frac{p(\bm{F},\bm{u},\bm{Y})}{q(\bm{F},\bm{u})} d\bm{F} d\bm{u}}\\
&=\int\int{q(\bm{F},\bm{u}) \log \frac{p(\bm{Y}|\bm{F})\cancel{p(\bm{F}|\bm{u})}p(\bm{u})}{\cancel{p(\bm{F}|\bm{u})}q(\bm{u})} d\bm{F} d\bm{u}}\label{deriveelbo}\\
&=\int\int{q(\bm{F},\bm{u}) \log p(\bm{Y}|\bm{F}) d\bm{F} d\bm{u}} + \int\int{q(\bm{F},\bm{u}) \log \frac{p(\bm{u})}{q(\bm{u})} \; d\bm{u}\; d\bm{F}} \end{align} The second term's integral over $\bm{F}$ integrates out, leaving: \begin{align}
\mathcal{L} &=\int\int{q(\bm{F},\bm{u}) \log p(\bm{Y}|\bm{F}) d\bm{F} d\bm{u}} + \int{q(\bm{u}) \log \frac{p(\bm{u})}{q(\bm{u})} d\bm{u}}\\
&=\mathbb{E}_{q(\bm{F},\bm{u})}\Big[\log p(\bm{Y}|\bm{F}) \Big] - \mathcal{D}_{KL}\Big[q(\bm{u})||p(\bm{u})\Big] \end{align} We now maximise $\mathcal{L}$ wrt the parameters of the variational distribution, described by the mean, $\bm{m}$ and triangular matrix $\bm{R}$ (covariance = $\bm{R} \bm{R}^\top$), using stochastic gradient descent. Specifically we approximate the expectation by sampling from $q(\bm{F})$ and then computing the mean of the log likelihoods for all the samples. We use TensorFlow's automatic differentiation to maximise $\mathcal{L}$ wrt the variational parameters, $\bm{m}$ and $\bm{R}$ using this stochastic approximation to the true gradient (stochastic gradient descent). This formulation also allows us to use minibatches, if $N$ becomes intractably large. See Algorithm \ref{vialg}, in the Supplementary Material for the overview of the computation. Finally we can use the estimates of the variational parameters to predict $q(\bm{F})$ for a test point at time $t_*$ for all sensors.
\subsubsection{Optimising hyperparameters} \label{optimise}
To avoid excessive model complexity we used just four hyperparameters: the lengthscale of static sensors; the lengthscale of mobile sensors; likelihood Gaussian noise; and the scale for all kernels.
One could treat these as random variables and integrate over them as with other variables. However, a common approach when using Gaussian processes is to find a point estimate either by maximising the marginal likelihood, or in our case, by optimising an error metric in held out data. We used Bayesian optimisation \citep[using the GPyOpt library, ][]{gpyopt2016} to find such a point estimate, which we used in a model tested on new data. Although we are not now reflecting the uncertainty in the hyperparameters, this point estimate means that the model when deployed in a data processing pipeline, is quicker and more robust. More importantly our lengthscale can reflect our prior belief that a sensor might degrade quickly \emph{even if it hasn't yet done so yet in the training data}. If such a failure doesn't yet exist in the dataset, the model is liable to select very long lengthscales. By manually selecting lengthscales we can incorporate our domain knowledge in sensor quality and performance.
\subsubsection{Sampling}
For the synthetic data example below, we found the algorithm typically optimised the calibration \emph{between} low-cost sensors first, then would very slowly move these distribution means to match the reference sensors. This behaviour is inevitable due to the highly correlated posterior and the imbalance in samples (with most being between non-reference pairs of instruments). Many elements of the approximating distribution mean must all move together in precisely the right direction, to increase the fit to the reference sensor data. As a side note, this might suggest parametrisations of the approximating distribution's mean vector should be investigated that might allow this manipulation using fewer variables. This problem is compounded by the relatively infrequent reference sensor observations: Many mini-batches contain few or no samples from the reference instrument co-locations. The gradient from the reference sensors only directly applies to those sensors that were co-located with it, while the co-locations of these sensors with the other non-reference collection of sensors will exert a gradient in the opposing direction.
We found for some experiments that importance sampling \citep{csiba2018importance} largely solved this issue and led to fast, reliable optimisation. We simply oversampled co-locations involving the reference sensors and corrected for the biased sampling by adjusting the noise variance appropriately.
We did not find this a problem in the experiment with real data from Kampala. This might be due to the shallow nature of the network, making it easier for the reference co-locations to influence the variational distribution across the whole network.
Part of our preprocessing of the real data involved averaging samples of co-located observations (blocks of 10 observations, typically over 15 minutes, were averaged), such averaging is standard practice in the field \citep[e.g.][]{stedman2006review}. This is necessary due to the considerable noise associated with individual observations (each averaging over only 80 seconds). This may have led to some balancing between the reference-sensors (which make measurements averaging over longer periods) and other sensors, which could explain why importance sampling was unnecessary for the real dataset.
\subsubsection{Other variables} OPC PM2.5 sensors are sensitive to environmental factors. Specifically, a high relative humidity can lead to large biases \citep[e.g.][in particular figure 6]{crilley2018evaluation}. In this framework such variables could be included by simply adding additional columns to $\bm{Y}$. These can then be passed to the calibration function, $\phi$, and involved in the parametric expression that computes the `true' pollution.
\subsection{Calibration over categorical data} To demonstrate the flexibility of the approach, we apply the method to another challenge from our research group: How to combine (low accuracy) citizen-science species-labelling of bee videos to predict the true species (with uncertainty quantification). Some of the videos were also labelled by an experienced and trained researcher, which we could consider ground-truth. Can we use the same calibration approach as above, treating each citizen-scientist as a low-cost sensor and the experienced researcher as a reference sensor? Previously (for sensor $j$) we were modelling the parameters, $\bm{f}_j(t_i)$ of a calibration function, $\phi(y,\bm{f}_j(t_i))$, between measured and true pollution - for example the gradient and offset. In the bee-labelling case, the measured observations are the labels guessed by the citizen scientists. The parameters we model are the probabilities in a conditional probability confusion matrix. These are generated using a softmax function applied to a matrix of latent variables which are given Gaussian process priors, allowing each citizen scientist's labelling to vary over time. Maybe people improve over time, with practice, or forget how to distinguish species over the winter (when fewer bees are foraging).
Previously the two observations $y_i^{(1)}$ and $y_i^{(2)}$ were taken at the same time $t_i$. In this categorical example we may need two times (one for each person) - as they may have performed the labelling in different orders and at different times, $t_i^{(1)}$ and $t_i^{(2)}$.
To switch to a categorical dataset, the only part of the algorithm that needs to change is our likelihood function defined previously in \eqref{likelihood}. \newcommand{\psi}{\psi} \newcommand{A}{A}
With number of species = $A$, we can construct a conditional confusion matrix, $\bm{P}$, using $|\bm{f}_j| = A^2$ parameters. Each element, $P_{y,\psi} = p(y|\psi,\bm{f}_j)$, of $\bm{P}$ describes the probability of the observed result \emph{given} the real bee was of (latent) species, $\psi$. We build this matrix from the vector of parameters, $\bm{f}_j$, by simply reshaping the vector into a square $A \times A$ matrix, $\bm{C}$, and use the softmax function to normalise each column, so that $\sum_y P_{y,\psi} = 1$: \begin{equation}P_{y,\psi} = e^{{C}_{y,\psi}(t)} / \sum_z e^{{C}_{z,\psi}(t)}.\end{equation}
The likelihood function takes two observations of the same bee, $y_{i}^{(1)}$ and $y_{i}^{(2)}$ and two parameter vectors $\ft{1}$ and $\ft{2}$, where $s_{i}^{(1)}$ and $s_{i}^{(2)}$ are the identities of the sensors (citizen scientists). We compute $\bm{P}$ for the two sensors/citizen scientists involved and take the relevant column from each, to get $p(y_{i}^{(1)}|\psi, \ft{1})$
and $p(y_{i}^{(2)}|\psi, \ft{2})$
We multiply these together, assuming that there is conditional independence between the citizen scientists, to compute the conditional joint probability, \begin{equation}p(y_{i}^{(1)},y_{i}^{(2)}|\psi, \ft{1}, \ft{2}) = p(y_{i}^{(1)}|\psi, \ft{1}) \; p(y_{i}^{(2)}|\psi, \ft{2}).\end{equation}
Using either the training data, or wider ecological literature, we define a categorical prior distribution over the true species $p(\psi)$. We multiply the conditional joint probability vector with this prior and sum over the resulting matrix, marginalising out the latent species: \begin{equation}
p\Big(y_{i}^{(1)},y_{i}^{(2)}|\ft{1},\ft{2}\Big) = \sum_\psi p\left((y_{i}^{(1)}|\psi, \ft{1}\right) \; p\left(y_{i}^{(2)}|\psi, \ft{2}\right) \; p\Big(\psi\Big). \end{equation}
As before, we sample values of $\bm{f}$ from the variational approximation, $q(\bm{f})$, and compute a Monte Carlo approximation to the likelihood. An autodiff framework, such as tensorflow, again allows us to optimise the variational parameters. In the implementation the only component that changed from the air pollution example was the method for computing the likelihood.
\subsection{Multi-hop instant calibration} \label{hasenfratz} For comparison with our calibration pair method, for the air pollution examples we developed and implemented an algorithm loosely based on the \emph{Multi-hop Instant Calibration} method from \cite{hasenfratz2012fly} and the \emph{rendezvous connection graph} from \cite{saukh2015reducing}, the former paper is a little unclear about combining weights, while the latter one handles sensor drift by only using observations within each time window. In \cite{saukh2015reducing} a shortest path algorithm is applied (personal communication with authors) to convert the network to an acyclic graph.
We augment the original graph in \cite{saukh2015reducing} by adding edges between the same sensor, in neighbouring time windows. The intuition being that the scaling information provided by a long series of hops within the window might be less accurate than a direct reference calibration `carried over' from a different time window. The edges connecting time windows can have a different weight than those connecting sensors in the same time period. The choice of weight ratio (i.e. time-edge-weight/colocation-edge-weight) will decide whether the model leans towards using longer chains of colocations in the same time window (high ratio) vs relying on colocations that occurred a long time ago (low ratio). Figure \ref{graphidea} illustrates. \begin{figure}
\caption{Multi-hop: The ratio of edge weights for neighbouring times (green) over weights for colocated sensors (blue) affects the choice of shortest path to a reference instrument. A low ratio will lead to shorter chains of colocation but relying on colocations from different time windows (dashed grey path), a high ratio will lead to a preference for colocations from nearby times, but with longer chains of colocations pairs (solid grey path).}
\label{graphidea}
\end{figure} We consider the calibration to be a simple scaling factor in the experiments in this paper and so we restrict the calibration function in both the multi-hop and in our more complex variational inference calibration tool to also scaling-only when performing comparisons.
The implementation of the graph building and prediction are detailed in Algorithms \ref{mhga} and \ref{mhgpa} respectively, in the Supplementary Material.
\section{Results}
\subsection{Synthetic example}
\subsubsection{The synthetic data}
Before considering real data we explored the method's response to a representative, but synthetic, dataset. We simulated ten static and four mobile sensors over a 180 day period. Four of the static sensors are assumed to provide noise-free, unbiased, reference measurements, the rest have a time-varying bias and added Gaussian white-noise with variance of $100 \;(\mu \text{g/m}^3)^2$. In the Kampala data discussed later, each mobile sensor is typically localised to one part of the city, with only occasional visits further afield. We simulate this by making the probability of a visit by a mobile sensor to a static sensor, proportional to the inverse distance between a `home' and the static sensor. Simple sinusoidal functions were used to generate both the true pollution and the scaling of the mobile and static sensors. The period of the sinusoids was different for each sensor, but the static sensors had a longer period on average (median = $3070 \;\text{hours}$ vs $1007 \;\text{hours}$). This variation was to (a) emphasise the heterogeneous nature of the sensors and (b) illustrate how the mobile sensors are probably less stable. Figure \ref{results_synth} shows the scalings actually used.
\subsubsection{Variational pair model} We applied the method described in Section \ref{calpair} to the synthetic data. As mentioned in Section \ref{optimise}, we used Bayesian optimisation (30 iterations, optimising the NMSE) to select the model hyperparameters for the variational calibration pair method (kernel variance, likelihood variance and two lengthscales). We optimised using the synthetic data with added noise scale of $10 \mu g m^{-3}$ for all the results. The synthetic data was resampled each iteration of the Bayesian optimisation (and for testing): the sensor locations, colocation events and noise were all resampled; but the frequencies of the synthetic calibration sinusoids remained the same.
Even though each sensor's scaling function has a different period, we use just two lengthscales to model the static and mobile sensors. On the first dataset generated, the Bayesian optimisation chose $2201 \;\text{hours}$ and $581\;\text{hours}$ for the lengthscales of the static and mobile sensors, respectively. This likely reflects both the underlying generating function, but also the strength of evidence for shorter lengthscales. The longer lengthscales assigned to the static sensors is longer than the period of the median sinusoid, but it potentially allows those static sensors with a longer lengthscale to still be modelled well, by allowing more data to be used for any given prediction. It also permits calibration information to be passed over longer periods of time. The likelihood variance and kernel scale (variance) were $6.02 \;(\mu \text{g/m}^3)^2$ and $7.64 \;(\mu \text{g/m}^3)^2$ respectively. These reflect the variance of the scaling factor, so it is hard to compare directly to the noise added to the data. The choice of likelihood variance seems to overestimate the true value a little, and it may be that the longer lengthscales chosen for the static sensors have their inaccuracy `explained away' by this added noise term.
\begin{figure}
\caption{Synthetic sensor example: Colocation visits over a representative 350 hours of the total 4320 hours: Visits to the reference (blue) and low-cost (green) static sensors by the mobile ones (red), indicate by vertical black lines.}
\label{synthetic_sensorplacement}
\end{figure}
\begin{table} \centering
\begin{tabular}{c |c c c| c} Noise Scale / $\mu g\; m{}^{-3}$ & \multicolumn{3}{c}{NMSE} & NLPD \\
& No Method & Multi-hop & Cal. Pair Model & Cal. Pair Model \\ 2 & 0.77 (0.02) & 0.27 (0.01) & 0.18 (0.01) & 42.6 (2.8) \\ 5 & 0.78 (0.02) & 0.44 (0.03) & 0.21 (0.01) & 49.5 (2.3) \\ 10 & 0.97 (0.02) & 0.87 (0.07) & 0.35 (0.01) & 58.2 (2.3) \\ 20 & 1.46 (0.02) & 3.18 (0.69) & 0.84 (0.02) & 87.7 (1.5) \\ \end{tabular} \caption{NMSE using raw data, the multi-hop calibration and the variational calibration pair model, for synthetic data with four levels of added noise. Bracketed values are the standard error of the mean based on predictions generated from 10 synthetic datasets (for each, the multi-hop was optimised on the test data, while the calibration pair model was optimised once on training data with noise scale = 10).} \label{synth_table} \end{table}
\subsubsection{Multi-hop calibration and comparison} We also applied the multi-hop calibration method to the synthetic data. We first performed a grid search to optimise the window size and the ratio of edge weights between those edges representing time and those representing a colocation (again, performed on separately generated training data).
We unfortunately found this model performed very poorly on this data. The best configurations usually had very long window sizes, leading to little or no ability to model the variation over time. We found that this configuration marginally improved on simply using the raw measurements. See Table \ref{synth_table}.
The estimates from the variational method provided considerable improvements over the multi-hop approach. And we can also compute the negative log predictive as we predict a distribution. Figure \ref{results_synth} shows these predictions for the 6 non-reference static sensors and the four mobile, for both the multi-hop method and the calibration pair model. One can see some errors in the calibration pair model. In particular, the high frequency drift of the low-cost mobile sensor has not been properly detected by the model in all the low-cost sensors, leading to this drift appearing in the predictions for the other sensors.
\begin{figure}
\caption{Variational Calibration Pair Model}
\caption{Multi-hop}
\caption{Predictions for the scaling in the synthetic sensor example for both the Variational Pair Model and the Multi-hop method. Noise scale = 10 $\mu g/m^3$. Upper six plots in each, static sensors. Lower four plots, mobile sensors. Black line, posterior mean. Grey lines, 95\% of the posterior density. Blue line, true scaling of each sensor. Green dots, ratio of colocation observation pairs involving this sensor. Red crosses, ratio of colocations observations between this sensor and a reference sensor.}
\label{results_synth}
\end{figure}
The synthetic data was engineered specifically to provide low-quality, noisy, infrequent and challenging data, in an attempt to reflect the problems that are typical in real data. Figure \ref{synthetic_sensorplacement} illustrates how few visits were made to the reference instruments, over a representative sample of the synthetic data. To explore this further, we added different amounts of noise to the data, and reran the analysis, although we kept the variational calibration method's hyperparameters as those trained with a noise scale of 10 $\mu g\; m{}^{-3}$, to see how well it adapts to misspecified noise.
Table \ref{synth_table} reports the NMSE for synthetic data with four added noise-scales for the two methods. The multi-hop method is allowed to optimise its parameters for each noise scale and, as might be expected, its window size reduces as the noise is reduced (for noise scales of $2, 5, 10$ and $20$ $\mu g m^{-3}$ the selected window sizes were $292, 1184, 3010$ and $4800$ hours, respectively). We can also see that the accuracy of the multi-hop method approaches the calibration pair model's, as the noise is reduced. This is probably because the calibration pair model makes use of multiple pathways through multiple colocations and over time to support a prediction, whilst the multi-hop method is constrained to use a single path through the network for a given prediction, making it more vulnerable to the effect of noise. The use of multiple pathways in the pair model gives it considerable robustness against noisy data.
\subsection{Kampala air pollution data} \label{kampaladata}
\begin{figure}
\caption{Upper plots, enlargements of three weeks (highlighted in grey in the middle plot), showing correlation between the low cost sensor and the reference instrument. Middle plot, the full period of measurements. Discontinuity on 18th March, 2020, due to the covid-19 lock down (vertical black line). Lower plot, the ratio of the two sensors. A GP with RBF kernel has been fitted to the log ratio and the exponent of its posterior mean plotted (pre- and post- lock down modelled with independent GPs). Times are in UTC.}
\label{embassy_ratio}
\end{figure} Recently a network of low-cost particulate air pollution sensors has been deployed across Kampala. These consist of pairs of PMS5003 (Plantower) optical particle counters (OPCs) mounted at fixed, static locations and on motorbike taxis, known as boda bodas (Figure \ref{embassydemo}b). Before discussing the network we will briefly inspect the measurements of a static OPC mounted 10m from the US Embassy's air pollution monitor (a regularly calibrated BAM sensor, which provides hourly ground truth measurements of PM2.5 pollution). This brief comparison will give some insight into the calibration problem: What the calibration function might consist of, and how it might vary over time.
Figure \ref{embassy_ratio} illustrates the strong correlation between the two sensors, but also highlights some differences. The earlier plot from September seems to show very similar measurements, with a ratio near one, on average. But later measurements seem to have a greater discrepancy with the low cost OPC sensor significantly underestimating the true pollution. The ratio itself is not enough to explain this change. Figure \ref{embassydemo}a illustrates how an offset can explain some of the difference. In all three the gradient of the linear fit appears to remain between 0.85 and 0.95, however the offset maybe drifts from positive to negative, explaining the increasing disparity in the previous graphs.
\begin{figure}
\caption{Static OPC measurements vs the embassy data}
\caption{A mobile OPC}
\caption{(left) OPC measurements vs the embassy data. Each row corresponds to 90 days of measurements. The left scatter plot shows the raw connection between the two sensors. A black line indicates equality while red lines are sampled from the linear fit's posterior probability distribution. Middle and right plots are the gradient and offset marginal posterior distributions respectively. The gradient distribution remains roughly constant, while the offset appears to `drift' downwards over time. (right) One of the mobile low-cost sensors mounted on the front of a motorbike taxis (`boda boda'), photographed in Kampala by AirQo (\url{www.airqo.net}).}
\label{embassydemo}
\end{figure}
\subsection{Calibration over the network}
\begin{figure}
\caption{Kampala with sensor locations}
\caption{Colocation frequency graph}
\caption{The AirQo air pollution monitoring network (sensors that have been colocated). The four blue nodes are the mobile sensors. The orange node is the reference sensor at Makerere University, the circled node is the reference sensor whose calibration we are testing. (a) Locations of sensors, the latest locations of mobile sensors are plotted, with one not in the city limits. (b) The network of colocations between sensors (from July 2020 to February 2021). The line thicknesses indicates the number of colocations in that time. The green lines indicate two possible paths from the reference to the test sensor.}
\label{network1kampala}
\end{figure}
51 sensors were in the network in the second-half of 2020, consisting of 2 reference sensors (BAM 1022 Particulate Monitor, Met One), 4 mobile OPCs and 45 static OPCs (PMS5003, plantower). Between the 15th July, 2020 and 3rd February, 2021 (203 days) there were 6,118,977 measurements of PM2.5 recorded. We use this dataset for the analysis.
We built a dataset of 433,935 `encounters', consisting of every pair of measurements that occurred within 40m and 30 minutes of each other. We averaged ten minute periods and removed observations outside the $10-300 \mu g/m^3$ range, as these we found to sometimes be associated with faults. This left 40,432 records (approximately 200/day). The majority of these are associated with permanently co-located pairs of static sensors. In Figure \ref{network1kampala}, (a) shows the physical locations of the sensors and (b) the number of colocations between them. The two reference sensors `-24516' (on Makrerere University Campus) and `-24517' (in Nakawa), are approximately 5km apart.
To test the models we only defined -24516 as being a reference sensor and let the models estimate the calibration required of -24517. We gave the models all the colocation data (none was held out) as we are interested in the prediction of the calibration scaling correction for the test sensor. There are co-location observations at the test sensor over a period of 69 days from the 203 days total.
We would expect the test sensor to have a calibration scaling of exactly 1.0 as it's a high quality, carefully maintained and calibrated BAM sensor. Our Gaussian process model has a prior mean of the log-ratio of zero, which is exactly a gain of 1.0, while the multi-hop model doesn't have a prior. To avoid our prior giving an unfair advantage and to simulate the effect of a drifting calibration, as is seen in the low-cost OPCs (Section \ref{kampaladata}), we artificially scaled the measurements of the reference sensor to slowly increase over time (starting at 1.0$\times$ on the first day of observations at the test sensor, and increasing to 1.83$\times$ by the 69th day). In summary: The algorithms being tested must reconstruct this drift using the network of colocations from the known reference sensor.
For our calibration pair model we select hyperparameter values that we believe are appropriate for this dataset (lengthscale for static and mobile sensors = 100 days, RBF kernel variance = 9 $(\log(\mu \text{g/m}^3))^2$, Bias kernel variance = 3 $(\log(\mu \text{g/m}^3))^2$, likelihood (ratio) noise variance = 0.2 $(\log(\mu \text{g/m}^3))^2$). This approach is necessary, as discussed previously, to allow us to include our prior knowledge around lengthscales associated with sensor drift and failure.
We again compare to the multihop method described in Section \ref{hasenfratz}. For comparison with our method, we optimised the parameters of the competing multi-hop method (the window size and weight ratio between time and colocation) on the dataset we also test with. Effectively giving an upper bound on its capability.
\textbf{Results} After running the grid search optimisation, the optimum multi-hop configuration had a window size of just 24 hours and an edge weight ratio of 0.21 (it roughly equally weights a colocation with a reference sensor from 5 days earlier and via an intermediate sensor during the same 24 hours). We computed error metrics between the test values of the reference sensor and the calibration predictions. To review the aim: The two calibration methods used the other reference instrument and the network of colocations to estimate the calibration of the test instrument (with the added synthetic drift). Table \ref{kampala_results} outlines the results for these approaches, while Figure \ref{correcting_kampala} shows the distribution of errors from the different approaches. The `raw measurement' errors are caused by our artificial scaling (without this the raw measurements would have no error).
\begin{table} \centering
\begin{tabular}{c |c c c | c}
Metric & No calibration & Multi-hop calibration & Calibration Pair Model & \\
\hline
MAE & 12.71 & 8.67 & 5.545 (0.47) & $\mu g\; m^{-3}$\\
NMSE & 0.321 & 0.180 & 0.078 (0.011) & \\
\end{tabular} \caption{Kampala experimental results, with synthetic drift added. Using no calibration, the multi-hop calibration and the pair model (bracketed values are standard error from 20 runs. Not required for multi-hop as it's deterministic).}
\label{kampala_results} \end{table}
\begin{figure}
\caption{The distribution of errors in pollution predictions at the test sensor. As we give the three models the measured pollution, in principle the error could be zero if the model correctly estimates the calibration. The left graph is the distribution of predictions if we just use the measurements as the predictions. The added, synthetic, drift causes this to over-estimate. The central graph is using the multi-hop approach, this appears to underestimate. The right graph is for the calibration pair model. Runs of the variational inference optimisation end with slightly varying results here. This is a typical example (with a MAE about the average).}
\label{correcting_kampala}
\end{figure}
\begin{figure}
\caption{Multi-hop scale predictions}
\caption{Variational pair model predictions}
\caption{The predictions of the scaling at the test sensor. Red line, our synthetically generated scaling drift we are trying to predict. Black line, the prediction. Green dots, ratios between the test sensor and other sensor(s) that visit. The variational prediction posterior mean varied quite a lot per run of the algorithm, probably due to high uncertainty, leaving the choice of mean difficult to determine.}
\label{timeseries_kampala}
\end{figure}
We find the multi-hop method does help improve the accuracy of the predictions to an extent, but the variational method performs considerably better. Besides accuracy, the variational pair-model method provides uncertainty estimates for its predictions, and the associated 95\% confidence intervals include the true scaling function. Variational inference is associated with underestimates of the variance of the posterior \citep{blei2017variational} so this result is reassuring.
The co-location data is very noisy. To illustrate this, the green markers in Figure \ref{timeseries_kampala} are the ratios between the test sensor and other sensor(s) that visit. Note that we are not assuming that these are just noisy samples from the true latent scaling - as these OPC sensors will themselves have some scaling associated with their estimates. But plotting them gives the reader a sense as to the scale of the noise in the dataset.
\textbf{Real Calibration} As a final brief experiment we tried calibrating the low cost OPC sensor (1014687) that is colocated with the test reference sensor, then using the calibration combined with the OPC observations to predict the values measured by the test sensor. The sensor was new and we only have 69 days of data associated. We found the actual scaling was the OPC was underestimating the reference sensor by only about 6\% which equates to an error of about 3 $\mu \text{g/m}^3$. The raw data had a MAE of 15.08 $\mu \text{g/m}^3$ while the multi-hop and variational pair model approaches had a MAE of 16.07 and 14.87 respectively. Neither approach really had much effect on the MAE. The earlier demonstration with the more significant artificial scaling was however modelled successfully, giving us confidence in the approach.
\subsection{Categorical data} \subsubsection{Synthetic example}
We demonstrate the approach initially with a synthetic dataset of 300 bees, approximately evenly split between three species (36\%, 30\%, 34\%). We simulated three non-expert (NE) citizen scientists, and an expert (providing effectively the ground-truth reference). The NE capabilities vary over time, with NE A perfectly labelling the first half, then degrading to chance linearly over the remaining half. NE B starting at chance, improving linearly to perfect over the first half and providing perfect predictions for the last half. NE C distinguishes between classes \{1,2\} and 3 initially but, linearly, their classification can distinguish between \{1,3\} and 2 at half way, and then almost chance by the end.
For training, 173 rows had access to the ground truth, leaving the algorithm to predict the remaining 127. The three NEs (A,B and C) made 178, 190, and 200 observations respectively.
To demonstrate we ran the algorithm using either an EQ kernel (scale=25, lengthscale=25\% of the length of the list of images) or a Bias kernel (scale=25). The scale and lengthscales were set \textit{a priori}. The result is a Gaussian process for each of the 27 latent function priors (nine in each confusion matrix for each of the three citizen scientists). For comparison we used a collaborative filtering matrix factorisation approach, in which a binary matrix was first computed for each species (ones are where the species exists in the original dataset). We then used probabilistic matrix factorisation with bias offsets \citep{Hug2020} to predict the missing values in the binary matrices for the reference column, then assign the species to the matrix with the greatest value. We also computed more straightforward baselines: `most guessed' simply does voting, while `trust weighted' assigns a weight to each citizen scientist. The weights are optimised on the training data. We manually optimised the NLPD for these methods to give them the best chance of beating the calibration pair approach (e.g. we found adding 0.2 to the trust weighted sum of guesses for each class, minimised the NLPD for the synthetic data). We also tried weighting the votes by the prior (number of times each bee appears). We finally just report the most common.
Table \ref{cat_synth_table} shows both calibration pair methods achieve higher accuracy and better uncertainty quantification than the others, but the EQ kernel version substantially improves on this uncertainty estimation.
\begin{table} \centering \begin{tabular}{ c r r } \hline Method & Accuracy (\%) & NLPD \\ \hline Calibration (EQ kernel) & 80\% & 53.5 \\ Calibration (Bias kernel) & 77\% & 78.1 \\ Collaborative Filtering & 53\% & 128.1 \\ Most Guessed & 74\% & 81.9 \\ Most Guessed (`trust' weighted) & 76\% & 81.7 \\ Most Guessed (prior weighted) & 75\% & 92.1 \\ Most Common (`chance') & 38\% & 139.5 \\ \hline \end{tabular}
\caption{For synthetic `bee' labelling data, we have data from the labelling of three non-experts and a ground truth expert. } \label{cat_synth_table} \end{table}
\begin{figure}
\caption{For the synthetic categorical dataset: The posteriors for the nine Gaussian processes that describe NE A's predictive ability: Blue line shows the synthetic generating function's `true' conditional probability, in green is the model's estimate with its 95\% CI. The probability refers to the likelihood of observing class $y$, given the true bee is of class $\psi$, with each row for a given \emph{true} bee ($\psi = 0,1,2$ respectively), while each column is the reported observed species. The x-axis in each plot indexes the 300 observations.}
\label{synth_person0}
\end{figure}
\subsubsection{Bumblebee video classification} \begin{figure}
\caption{Parts of two frames from a video of a \textit{Bombus hortorum} foraging. In (a) the yellow bands on \emph{both} its thorax and abdomen are visible. In (b) one can also see the long face and tongue that distinguish it from \textit{Bombus jonellus}. It was sighted in northern England, making it unlikely to be \textit{Bombus ruderatus}. This video was labelled by two non-expert humans who identified it as (1) \textit{Bombus terrestris/lucorum}, (2) \textit{Bombus hortorum} and by the convolutional neural network as \textit{Bombus sylvestris}.}
\label{beevideo}
\end{figure}
We collected a dataset of 103 slow motion bee videos for a separate project \citep{ollett21}. These were labelled by an experienced and trained researcher `ground truth' and by six non-experts (NEs). These included five humans and one deep convolutional neural network (trained on photos from a public dataset). See \cite{ollett21} for details. \textit{Bombus terrestris} and \textit{Bombus lucorum} were combined into a single class as workers of these species are often visually indistinguishable. Those seven species which appear fewer than three times in the data were removed.\footnote{the list of removed bees: \textit{Bombus bohemicus}, \textit{Bombus jonellus}, \textit{Bombus sylvestris}, \textit{Bombus muscorum}, \textit{Bombus campestris}, \textit{Bombus monticola} and the non-bumblebee \textit{Apis mellifera}.} This included five species which only appeared in the NE labels. The result was 93 bees labelled by at least one NE (221 labels) and by ground truth. The NE accuracies and coverage varied considerably.\footnote{The five human NE accuracies (correct/total): 5/5, 56/60, 12/12, 21/25, 16/20 and the CNN NE: 62/89.}
The order the labelling was conducted was not known, so we decided to just use the Bias kernel.
We ran a 5-fold cross validation experiment. In spite of only using the Bias kernel we found the model still performed well. Table \ref{cat_bee_table} shows it achieves higher accuracy and with better uncertainty quantification than the other methods compared.
\begin{table} \centering \begin{tabular}{ c r r } \hline Method & Accuracy (\%) & NLPD \\ \hline Calibration (Bias kernel) & 84\% & 59.9 \\ Collaborative Filtering & 58\% & inf \\ Most Guessed & 80\% & 69.4 \\ Most Guessed (`trust' weighted) & 82\% & 91.7 \\ Most Guessed (prior weighted) & 79\% & 98.4 \\ Most Common (`chance') & 54\% & 130.5 \\ \hline \end{tabular} \caption{Results for the real bee video labelling data. For context the CNN NE (the only NE to label all the videos) achieved 70\% accuracy.} \label{cat_bee_table} \end{table}
\section{Discussion} In this paper we've proposed a method to calibrate a network of sensors using observations consisting of pairs of potentially bias or transformed observations in which a few observers are assumed to provide ground truth. We applied this to our main dataset - of static and mobile air pollution sensors in Kampala, adding a synthetic drift term to illustrate the potential issues in longer term data. We found the calibration pair method models the drift appropriately, making the predictions far more accurate. It also, importantly, provides uncertainty estimates on the calibration. For the real data without the synthetic drift, we found it had only a 1.5\% improvement in the MAE. Potentially over longer periods, we might expect the approach to become more relevant (as demonstrated by its utility when the drift was added). We also demonstrate how, replacing the likelihood function we can apply the method to a categorical dataset, in which the measurements are labels provided by citizen-scientists. We found the method was effective, and had good uncertainty quantification compared to other approaches.
Comparing the results of the multi-hop method and the variational calibration pair method, it seems that, for larger networks, the latter does better on both the synthetic and real data. It was somewhat surprising that it had much of an advantage in the Kampala case (especially with the synthetic drift added). In simulations we typically found the performance was considerably better than the multi-hop method when there were many sources of data to integrate. In the synthetic dataset for example, there were four reference sensors. So, with only one reference sensor, we anticipated the two approaches to perform similarly. However not only is the number of reference sensors relevant here, but also the number of paths between the reference sensor and the test sensor. Figure \ref{network1kampala}b indicates two routes exist from the reference sensor to the test sensor in the real dataset. Finally it isn't simply the path from the reference sensor to the test sensor that supports the variational result. The network graph doesn't show the full complexity of data availability over time, but importantly we envisage that the other sensors visited provide support. By also being visited, they can help constrain the calibration, indirectly.
\subsection{Calibration in context} Many studies exist in which low-cost air pollution sensors are co-located with reference sensors, and calibration functions are then derived to, in principle, allow the low-cost sensor to then be used without the reference sensor \citep{lewis2016evaluating, liu2017performance, zimmerman2018machine, barcelo2018calibrating, crilley2018evaluation, badura2019regression, wang2019calibration, datta2020statistical, lee2020long, ferrer2020multisensor}. This can also extend to calibrating remote observation data (for example \cite{shaddick2018data}). It has been noted many times that on-site field calibration is better than lab-based \citep{rai2017end}, and that sensor drift due to ageing or contamination means that regular recalibration is necessary \citep[][Section 3.2.3; and the results in this paper, Section \ref{kampaladata}]{crilley2018evaluation,marathe2021currentsense,castell2017can}. \cite{rai2017end} wrote, \begin{quote} ...it is necessary to perform a field calibration for each sensor individually. Moreover, the calibration parameters might change over time depending on the meteorological conditions and the location, i.e., once the nodes are deployed it will be difficult to determine if they are under-or over-estimating the pollutant concentrations. \end{quote} Although it is widely agreed regular field recalibration is essential, the complexity of achieving this makes it potentially the main barrier to large-scale, low-cost sensor deployment \citep{rai2017end}. Several attempts have been made to look at how to do field-recalibration using mobile reference sensors \citep{hasenfratz2012fly,arfire2015model}. But using low-cost mobile sensors to act as `intermediaries' \citep{tsujita2004dynamic, hasenfratz2012fly} is far more practical. The low-cost sensors (e.g. the plantower pms5003) are typically smaller and often will be more resilient to being moved, and can be replaced more cheaply if damaged. In the AirQo Project in Kampala, moving the reference instruments regularly would have been very challenging. An alternative would be to swap the static low-cost sensors between reference colocation sites and their static locations. However such access required site permissions and expertise, while the mobile low-cost sensors could visit the static stations simply by the motor-bike taxi drivers parking next to the sensor. This simplicity provided by permanently mobile low-cost sensors is discussed in \cite{zhao2021urban}.
It is arguably only worth deploying, calibrating and maintaining the low-cost sensors if they can provide useful information to policy makers. \cite{castell2017can} suggest that they are limited to non-policy/science applications. They however looked at data from a city with relatively low-pollution. The average PM2.5 in Kampala was roughly 30-60 $\mu g/m^3$. The ten minute-averaged sample error in our experiment for the low cost sensor (with a calibration computed across the network) was about $15\mu g/m^3$. Once averaged to daily or annual estimates this would almost certainly be reduced below the 50\% threshold in the EU Data Quality Objectives (DQO) (for daily PM2.5 estimates). \citep{janssen2017guidance}. This is independent of the calibration approach and is probably possible in Kampala simply due to the high background pollution, reflecting the findings of \cite{castell2017can}, who found their relative accuracy improved at those times and locations with most pollution. This suggests achieving the EU's DQOs may be easier in high-pollution environments. It would be interesting to consider whether, in high-pollution locations, low-cost sensors could provide estimates of sufficient accuracy that they can be used for policy-making decisions. In-field calibration would still be necessary. By using a probabilistic approach to calibration and spatiotemporal modelling, one should also be able to state where (and when) in the domain the model reaches sufficient confidence.
A related field is the detection of sensor failure in a network of sensors. \cite{miskell2016data} used proxies (reference stations in an area with a similar land use) to detect potential sensor failure. Our method might be useful in the detection of drift, or similar signs of hardware failure. For example combining our method for calibration with a spatiotemporal model with leave one-sensor-out cross-validation. Poor prediction accuracy may be due to poor modelling assumptions or inference, e.g. we haven't included some aspect of the local environment; sensor failure, e.g. the inlet is clogged; or training data errors, e.g. if neighbouring sensors are malfunctioning. For more nuance it might be better to follow other approaches to fault detection. \cite{peng2017sensor}, for example, distinguish between types of fault.
Combining sensor estimates using a Bayesian approach is not novel. \cite{talampas2012maximum} take into account each sensor's uncertainty to combine their measurements using a Bayesian approach, and find it performs much better than simply averaging. In a different field \cite{perala2020calibrating} use Gaussian processes for another type of calibration problem: combining expert estimates, potentially with biases using a hierarchical GP. This could be applicable, in particular if different types of sensor can be grouped.
\subsection{The problem of spatially structured bias} \cite{sousan2016evaluation} showed how different sources of particulate pollution lead to different responses by optical particle counters. \cite{castell2017can} and \cite{crilley2018evaluation} also discuss this issue. \cite{sandradewi2008using} propose that one deliberately use the change in the optical properties of the pollution to infer its source. We have assumed that the calibration function remains the same over space, and similar over short periods of time. If the former assumption doesn't hold we will need to colocate the reference instrument at each low-cost sensor location. If the latter assumption doesn't hold, then we can't use low-cost sensors at all. \cite{chu2020spatial} suggest one could model the former, spatial-non stationarity. Presumably the temporal non-stationarity could be addressed in a similar way. They were investigating PM2.5 in a high humidity climate, so potentially applicable to our work in Kampala. Low cost OPC sensors are affected by high humidity, so the variation in humidity over Taiwan lead to changing biases. We could use these relevant features (such as humidity) in the calibration function. Unfortunately the \emph{type} of pollution (road dust, diesel exhaust or charcoal burning) isn't so easily measured, so some sort of spatio-temporal component might need to be introduced to the calibration function.
In the worst case we could imagine that one part of the city has pollution that is underestimated by all the network's low-cost sensors (e.g. a source such as burning tyres), thus all calibration that occurs between low-cost sensors in that area will fail to detect this bias. The only solution is to colocate a reference instruments in those locations. Carefully distributing the small number of reference instruments to sample a broad range of land-use types in the city is therefore prudent.
\subsection{Choice of model and choice of inference} We didn't explore in this paper the potential capabilities of the joint model (in which the pollution is modelled explicitly in time and space), instead we focused on the `pair model' (in which the pollution remains a hidden, latent random variable) as we found it (in unreported work) more reliable than the joint model which had high correlations in the posterior mediated via both the prior \emph{and} each co-location (via the likelihood function). Conversely, the pair model scaled and behaved in a far more robust manner, with reliable optimisation (when using VI).
We have (in unreported work) experimented with using MCMC to approximate the variational distribution \citep{hensman2015mcmc} but found it failed to achieve timely mixing (probably due to the correlations in the posterior) when the network was more than two or three edges deep, so we concentrated on the VI approach.
The variational calibration pair model was quite successful in modelling the posterior variance. One reason might be that the approximating distribution included a full covariance matrix, a necessity due to the strong correlations between sensor parameters in the posterior. By avoiding the mean field approximation we might also have avoided some of the extreme underestimation of the variance \cite[see Figure 1 in][for an explanation]{blei2017variational}. We did (unreported) experiment with using the mean-field approximation in the joint model and found the variance often `collapsed', giving highly confident predictions, supporting this explanation.
One issue that might impact inference in larger networks is how the number of inducing points would roughly scale with $\text{time} (T) \times \text{sensors} (S) \times \text{parameters} (C)$, as we assume the number of inducing points is proportional to time. This would be even worse for the categorial example in which the number of parameters scales by the number of species-squared ($A^2$). This would pose both a computational challenge, with $O((C S T)^3)$ time complexity, but also leads to a huge number of parameters to express the variational distribution's covariance matrix, $\propto (CST)^2$. For the categorical model, we could alter our model to add an assumption of independence between species, making the inverse more tractable and reducing the number of parameters from $( A^2 S T)^2/2$ to $A (A S T)^2/2$.
Using fixed hyperparameters for our model was driven partly by domain knowledge that past stability in calibration does not guarantee future stability, but we also found them somewhat difficult to estimate using black box variational inference, an experience shared by others, e.g. \cite{nguyen2014automated} report that they `have found this to be problematic, ineffectual, and time-consuming.'
\subsection{Hetroscedastic likelihood} In the likelihood used, the variance of the normal distribution was a fixed value. But some sensors (and locations) may have differing amounts of noise, and just as the calibration is expected to vary, so too will this likelihood variance. In the supplementary we explain how to include hetroscedastic noise \citep[see also, for example][]{lazaro2011variational}. We also propose that for the variational approximation of noise, one could use just a Dirac distribution (rather than a multivariate Gaussian) and so just specify the mean. The motivation being that estimating the uncertainty in the noise is unrealistic. Our implementation allows either a fixed likelihood variance, a point estimate `Dirac' approximation, or a full multivariate Gaussian distribution, but in the examples in the paper we used the fixed version, as it seemed to perform sufficiently for our datasets.
\subsection{Summary and future work} In summary, the calibration of a network through co-located observations of pairs of (potentially drifting) sensors can be performed by modelling each sensor's calibration function parameters using one or more Gaussian processes, and providing a likelihood function computed for \emph{two} observations. This provides a principled and robust Bayesian approach to handling this type of data.
This will enable those designing sensor deployments to potentially use pair-wise calibration as part of the design solution, in which a reference sensor does not need to visit every low-cost sensor directly.
To put this paper in context, \cite{cui2021new} describe a four stage calibration process for an air pollution monitoring system: (1 and 2) lab based, (3) field based reference sensor colocation (4) field based with occasional visits of mobile reference sensor. The calibration pair method would support the last of these four stages, allowing complex network structures, involving low-cost mobile sensors.
Future work includes optimising the choice of co-locations to minimise error: specifically choosing the best order to visit the static sensors in Kampala. There might be some sensors that are more useful to calibrate (due to their importance in the spatiotemporal model or location in the network). Cost is also a constraint: In our case, mobile-unit visits to the sensors had different (financial) costs, depending on their distance from the driver's base (known locally as their `stage'). In particular the reference sensors were quite distant.
The categorial bee-labelled problem has its own sampling-optimisation exploitation/exploration questions: deciding which videos to show which citizen-scientist/expert. We might wish to show a video to reduce the uncertainty in their calibration \emph{or} to show them bees that the model believes they can help classify \emph{or} maybe help improve their capabilities through training.
We do not consider uncertainty in the multi-hop model. We anticipate in future work looking at performing probabilistic inference over a simplified graph. We think this could be a good compromise (averaging over longer periods and making a relatively simplified graph network), potentially leading to faster inference than the variational Gaussian process approach, while still leading to good estimates of calibration and uncertainty, albeit in a more constrained modelling space.
The main limitation to future work in developing approaches towards networks of calibration the area of air pollution is that of data. Many low-cost network sensors exist \citep[e.g.][]{parmar2017iot,rai2017end,nyarku2018mobile,feinberg2019examining,zhao2021urban} but few perform regular co-location calibrations. Possibly this is because there are no well developed, easily used, computational tools to perform the in-field calibration discussed in this paper. Hopefully methods, such as the one outlined in this paper, will help solve this chicken-and-egg problem.
The example networks used in this paper (both synthetic and real) were not very deep. We could envisage that, with larger datasets (for example those collected on \url{www.zooniverse.org}) this network could become more complex, both potentially benefiting more from this method, but also providing a more challenging inference problem as the number of GPs grows.
One exciting possibility is to apply the method to the \url{beewalk.org.uk} project, a large, citizen-science initiative in which volunteers walk transects every month and count the number of each bumblebee species. If beewalkers were to occasionally walk other walkers' routes, could the data quality be improved through pair-wise calibration?
\subsection{Conclusion} The problem of calibrating low-cost, large-scale, air pollution sensor networks is quickly becoming acute: A number of large scale deployments of low-cost air pollution networks have happened recently, including one running on 260 cars in Beijing \citep{zhao2021urban}. The difficulty in field-calibration and bounding the accuracy of estimates from these deployments suggests that methods that can provide automatic low-cost recalibration across the network are becoming increasingly necessary.
The calibration pair method described enables sensor deployments with complex networks of colocations. It allows the uncertainty in the predictions from these networks to be quantified, providing guaranteed confidence bounds and robust insights that mean the results from such networks can start being used by policy makers.
\section*{conflict of interest} The authors declare that they have no conflict of interest, and that there is no financial interest to report.
\section*{code and data} The variational and multi-hop calibration methods are available as a python module which can be downloaded and installed from \url{https://github.com/lionfish0/calibration}, which includes a demonstration jupyter notebook, and the notebooks used to make the figures etc in this paper.
Kampala sensors: This data is owned and controlled by AirQo and is not available for public release. In particular, the raw boda boda data can reveal locations of individual journeys and the home locations of the taxi drivers.
Bee videos and labels: we have uploaded these videos and labels to the University of Sheffield’s data repository (ORDA): \url{https://doi.org/10.15131/shef.data.19704538}.
\pagebreak \section*{Supplementary Material} \begin{algorithm}[H] \caption{Variational Inference for calibration. \\ Note: For implementation we structure an input matrix $\bm{X}$ to hold the time, sensor and component as three columns. This matrix is $2C$ times the length of $Y$. Split into $C$ submatrices, each pair of rows in each submatrix is associated with one row of $\bm{Y}$. $\bm{f}$ now becomes a vector with each item associated with one row of $\bm{X}$. The parameters are selected using slice notiation.} \label{vialg}
\hspace*{\algorithmicindent} {\textbf{Inputs}\\Observation time, sensor and parameter, $\bm{X}=\{\bm{x}_i\}_{i=1}^{2CN}$;\\Observation pair values, $\bm{Y} = \{[y_i^{(1)},y_i^{(2)}]\}_{i=1}^N$;\\Inducing point locations (time, sensor and parameter), $\bm{Z} = \{\bm{z}_i\}_{i=1}^M$;\\ Calibration function, $\phi(y,\bm{c})$;\\Number of parameters used by function, $C$;\\Kernel, $k(\cdot,\cdot)$;\\Likelihood variance $\sigma^2$;\\Reference sensors, $\bm{r}=\{0,1\}^S$;\\Number of samples in MC approximation, $P$.\\}
\hspace*{\algorithmicindent}\textbf{Outputs}\\Approximating Gaussian distribution parameters: mean $\bm{m}$ and factor $\bm{R}$ (where distribution covariance is $RR^\top$).\\ \begin{algorithmic}[1] \Procedure{VI}{$\bm{X}, \bm{Y}, \bm{Z}, \phi(\cdot), C, k(\cdot,\cdot), \sigma^2, \bm{r}$} \While{$\bm{m}$ or $\bm{R}$ not converged} \\ \Comment{$q(\bm{f})$ is the approximate posterior over all latent variables defined in $\bm{X}$.} \State $q(\bm{f}) \sim N(K_{xz} K_{zz}^{-1} \bm{m}, K_{xx} - K_{xz}K_{zz}^{-1} K_{zx} + K_{xz}K_{zz}^{-1} RR^\top K_{zz}^{-1} K_{zx})$ \For {$j=1..P$} \Comment {Sample $P$ times}
\For {$i=1..N$}
\\ \Comment{Sample the latent variables relevant to the two sensors.}
\State $\bm{s}^{(1)}_i,\bm{s}^{(2)}_i = \text{sample}[q(\bm{f}_{2i-1::2N}), q(\bm{f}_{2i::2N})]$ \Comment{$\bm{s}_i^{(1)}$ and $\bm{s}_i^{(2)}$ are each of length $C$.}
\State $L_{ij} = \log p\Bigg(\begin{bmatrix}y_{i}^{(1)}\\y_{i}^{(2)}\end{bmatrix} \Bigg| \begin{bmatrix}\bm{s}_i^{(1)} \\ \bm{s}_i^{(2)}\end{bmatrix}\Bigg)$ \Comment{Compute likelihood of sample, using \eqref{likelihood}}
\EndFor \EndFor \State $\mathcal{L} \gets \frac{1}{P} \sum_{j=1}^P \sum_{i=1}^N \left(L_{ij}\right) - D_{KL}\Big(N(\bm{m},RR^\top),N(0,K_{ZZ})\Big)$ \Comment{Compute ELBO} \State compute $\frac{d\mathcal{L}}{\bm{m}}$ and $\frac{d\mathcal{L}}{\bm{R}}$ \Comment{using automatic differentiation.} \State update $\bm{m}$ and $\bm{R}$ using stochastic gradient descent (using Adam). \EndWhile \EndProcedure \end{algorithmic}
\end{algorithm}
\begin{algorithm} \caption{Prediction for Algorithm \ref{vialg}}
\hspace*{\algorithmicindent} {\textbf{Inputs}\\Test time, sensor and component, $\bm{X}_*=\{\bm{x}_{*i}\}_{i=1}^{N_*}$; $\bm{y}_*=\{{y}_{*i}\}_{i=1}^{N_*}$ raw observations (unlike normal regression we need to give uncalibrated observations at test time) \\Inducing point locations (time, sensor and component), $\bm{Z} = \{\bm{z}_i\}_{i=1}^M$;\\ Calibration function, $\phi(y,\bm{c})$;\\Number of components used by function, $C$;\\Kernel, $k(\cdot,\cdot)$;\\Approximating Gaussian distribution parameters: $\bm{m}$ and $\bm{\bm{R}}$.\\Number of samples for each test point, $P$.\\}
\hspace*{\algorithmicindent}\textbf{Outputs}\\An $N_* \times P$ matrix of $P$ samples for each of the $N_*$ test points, $\bm{S}$\\ \begin{algorithmic}[1] \Procedure{Predict}{$\bm{X}_*, \bm{Y}_*, \bm{Z}, \phi(\cdot), C, k(\cdot,\cdot)$} \\ \Comment{$q(\bm{f})$ is the approximate posterior over all latent variables defined in $\bm{X}_*$.} \State $q(\bm{f}) \sim N(K_{x_*z} K_{zz}^{-1} \bm{m}, K_{x_*x_*}-K_{xz}K_{zz}^{-1} K_{zx_*}+K_{x_*z}K_{zz}^{-1} RR^\top K_{zz}^{-1} K_{zx_*})$ \For {$i=1..N_*$}
\\ \Comment{Sample the latent variables relevant to each test point.}
\State $\bm{s}_i = \phi(y_i,\text{sample}[q(\bm{f}_{i::N_*})])$ \Comment Sample $K$ times. \EndFor \EndProcedure \end{algorithmic} \label{alg1pred} \end{algorithm}
\begin{algorithm} \caption{Multi-hop `Graph' algorithm} \label{mhga}
\hspace*{\algorithmicindent} {\textbf{Inputs}\\Observation time, sensor id pair, $\bm{X}=\{\bm{x}_i\}_{i=1}^N$, so $\bm{X}$ is $(N \times 3)$;\\Observation pair values, $\bm{Y} = \{[y_{i1},y_{i2}]\}_{i=1}^N$, so $\bm{Y}$ is $(N \times 2)$;\\Window size, $\delta$;\\Edge `distances', connect by colocation events and over time, $d_\text{colocation}$, $d_\text{time}$;\\Reference sensors, $\bm{r}=\{0,1\}^{|A|}$.\\}
\hspace*{\algorithmicindent}\textbf{Outputs}\\Dictionary of scaling factors for (sensor,window) tuples $\bm{F}$.\\ \begin{algorithmic}[1] \Procedure{BuildGraph}{$\bm{X}, \bm{Y}, \delta, \bm{r}$} \For {$w$ in all time windows $W$ of width $\delta$}
\For {$s_i$,$s_j$ in all pairs of sensors from $X$, where $s_i \neq s_j$}
\State $Y' \gets Y_{X_{:,2}=s_i \land X_{:,3}=s_j \land X_{:,1} \in w}$ \Comment{Selects observation pairs that are in the time window and between sensors $s_i$ and $s_j$}
\If {$|Y'| \geq 5$} \Comment{We don't add edges where there are fewer than five observations}
\State G.addEdge($(s_i,w)\to(s_j,w)$, value=mean($\log(Y'_{:,1})-log(Y'_{:,2})$, distance=$d_{colocation}$))
\EndIf
\EndFor \EndFor \For {$w$ in all time windows $W-1$ of width $\delta$}
\For {$s_i$ in all sensors from $X$}
\State G.addEdge($(s_i,w)\to(s_i,w+1)$, value=0, distance=$d_{time}$)
\State G.addEdge($(s_i,w+1)\to(s_i,w)$, value=0, distance=$d_{time}$)
\EndFor \EndFor \For {$w$ in all time windows $W-1$ of width $\delta$}
\For {$s_i$ in all sensors from $X$}
\State $P \gets \text{Shortest path (using Dijkstra) from}\; (s_i,w)\; \text{to any reference sensor, specified in}\; r.$
\If {Path $P$ exists}
\State $F[s_i,w] = \sum_{p \in P}{p_{value}}$ \Comment{Sum log ratios over path}
\EndIf
\EndFor \EndFor \EndProcedure \end{algorithmic} \end{algorithm}
\begin{algorithm} \caption{Multi-hop `Graph' prediction algorithm} \label{mhgpa}
\hspace*{\algorithmicindent} {\textbf{Inputs}\\Observation time, sensor id, $\bm{X}_*=\{\bm{x}_i\}_{i=1}^N$, so $\bm{X}_*$ is $(N_* \times 2)$;\\Observed raw values, $\bm{y}_{\text{raw}}$, so $\bm{y}_{\text{raw}}$ is an $(N_* \times 1)$ vector;\\Window size, $\delta$;\\Dictionary of scaling factors for (sensor,window) tuples $\bm{F}$, from Algorithm \ref{mhga}.\\}
\hspace*{\algorithmicindent}\textbf{Outputs}\\Predicted calibrated values, $\bm{y}_{*}$, so $\bm{y}_{*}$ is an $(N_* \times 1)$ vector. \begin{algorithmic}[1] \Procedure{Predict}{$\bm{X}_*, \bm{y}_{\text{raw}}, \delta, \bm{r}$, $G$} \For {$\bm{x}_*$ in $\bm{X}_*$; $y_*$ in $\bm{y}_*$; and $y_\text{raw}$ in $\bm{y}_\text{raw}$}
\State $w \gets \text{window computed for} [\bm{x}_*]_1 \text{using window size}\; \delta$
\If{$([\bm{x}_*]_2,w) \notin F$}
\State Raise exception: No path to a reference sensor.
\EndIf
\State $y_* \gets e^{F[[\bm{x}_*]_2,w]} \times y_\text{raw}$
\EndFor \EndProcedure \end{algorithmic} \end{algorithm}
\end{document} |
\begin{document}
\title{f Friedrichs inequality in irregular domains }
\centerline{\scshape Simone Creo, Maria Rosaria Lancia}
{\footnotesize
\centerline{Dipartimento di Scienze di Base e Applicate per l'Ingegneria, Universit\`{a} degli studi di Roma Sapienza, }
\centerline{Via A. Scarpa 16,}
\centerline{00161 Roma, Italy.} }
\begin{abstract} \noindent We prove a generalized version of Friedrichs and Gaffney inequalities for a bounded $(\varepsilon,\delta)$ domain $\Omega\subset\mathbb{R}^n$, $n=2,3$, by adapting the methods of Jones to our framework. \end{abstract}
\noindent\textbf{Keywords:} Friedrichs inequality, Gaffney inequality, $(\varepsilon,\delta)$ domains, Whitney decomposition, coercivity estimates.\\
\noindent{\textbf{2010 Mathematics Subject Classification:} Primary: 35A23. Secondary: 35Q61, 78A25.}
\section{Introduction} \setcounter{equation}{0}
\noindent The aim of this paper is to prove Friedrichs inequality for $(\varepsilon,\delta)$ domains. This inequality has been introduced in different frameworks by Friedrichs \cite{friedrichs} and Gaffney \cite{gaffney}, and in the literature it is known with different names according to the setting where it is used. In the study of Maxwell problems or Navier-Stokes equations, this inequality is a key tool to prove the coercivity of the associated energy forms. From the point of view of applications, it is interesting to study vector BVPs in irregular domains (see e.g. \cite{CHLTV,LVstokes}) and their numerical approximation, hence it is crucial to extend these inequalities to the case of suitable irregular sets. From this perspective, we confine ourselves to two or three-dimensional domains.\\ Gaffney inequality can be deduced from the Friedrichs inequality. To our knowledge, such inequalities hold for convex and Lipschitz domains; among the others, we refer to \cite{bauerpauly,Schw16,NPW15,amrouche}, see also \cite{dacorogna} and the references listed in. In this paper, we first prove Friedrichs inequality for $(\varepsilon,\delta)$ domains, and then prove Gaffney inequality by adapting the methods of \cite{duran} (developed for Korn inequality) to this framework.\\ The class of $(\varepsilon,\delta)$ domains has been introduced by Jones \cite{Jones}, and it is quite general, since the boundary of an $(\varepsilon,\delta)$ domain can be highly non-rectifiable, e.g. fractal or a $d$-set (see Definitions \ref{defepsdelta} and \ref{dset}).\\ In the literature, for $\Omega\subset\mathbb{R}^n$ $(n=2,3)$ sufficiently smooth, the Friedrichs inequality reads as follows: if $v\in W^{1,p}(\Omega)^n$, there exists a positive constant $C$, depending on $\Omega$, $n$ and $p$, such that \begin{equation}\label{Fr1}
\|v\|_{W^{1,p}(\Omega)^n}\leq C(\|v\|_{L^p(\Omega)^n}+\|\dive v\|_{L^p(\Omega)}+\|\curl v\|_{L^p(\Omega)^n}). \end{equation} Gaffney inequality is a direct consequence of Friedrichs inequality \eqref{Fr1} when considering boundary conditions. We introduce the following spaces: \[W^p(\dive,\Omega):=\left\lbrace u\in L^p(\Omega)^n\,:\,\dive u\in L^p(\Omega) \right\rbrace,\] \[W^p_0(\dive,\Omega):=\{u\in W^p(\dive,\Omega)\,:\,\nu\cdot u=0\,\,\text{on}\;\partial\Omega \},\] \[W^p(\curl,\Omega):=\left\lbrace u\in L^p(\Omega)^n\,:\,\curl u\in L^p(\Omega)^n \right\rbrace,\] \[W^p_0(\curl,\Omega):=\{u\in W^p(\curl,\Omega)\,:\,\nu\times u=0\,\,\text{on}\;\partial\Omega \},\] where $\cdot$ and $\times$ denote respectively the usual scalar and cross products between vectors in $\mathbb{R}^n$. The boundary conditions have to be interpreted in a suitable weak sense (see e.g. \cite{temam}).\\ When $v\in W^p(\dive,\Omega)\cap W^p_0(\curl,\Omega)$ or $v\in W^p(\curl,\Omega)\cap W^p_0(\dive,\Omega)$, Gaffney inequality takes the following form: \begin{equation}\label{Fr2}
\|\nabla v\|_{L^p(\Omega)^{n\times n}}\leq C(\|\dive v\|_{L^p(\Omega)}+\|\curl v\|_{L^p(\Omega)^n}). \end{equation} Our aim is to extend Gaffney inequality to those $(\varepsilon,\delta)$ domains for which it is possible to give an interpretation of the boundary conditions. In particular, we consider $(\varepsilon,\delta)$ domains $\Omega$ in $\mathbb{R}^n$ whose boundaries are $d$-sets or arbitrary closed sets in the sense of Jonsson \cite{jonsson91}. In these cases, it can be proved that the spaces $W^p_0(\dive,\Omega)$ and $W^p_0(\curl,\Omega)$ are well defined because generalized Green and Stokes formulas hold. This implies that the normal and tangential traces are well defined as elements of the duals of suitable trace Besov spaces on the boundary (see \cite{LaVe2} and \cite{CHLTV}).
We extend \eqref{Fr1} and \eqref{Fr2} to $(\varepsilon,\delta)$ domains $\Omega\subset\mathbb{R}^n$ for either $v\in W^p(\curl,\Omega)\cap W^p_0(\dive,\Omega)$ (see Section \ref{divergenza}) or $v\in W^p(\dive,\Omega)\cap W^p_0(\curl,\Omega)$ (see Section \ref{rotore}), according to the boundary conditions under consideration. The main results of this paper are Theorems \ref{fridis}, \ref{gaffin}, \ref{fridisrot} and \ref{gaffinrot}.\\ The proof of our results deeply relies on the assumptions on $\Omega$. Since $\Omega$ is an $(\varepsilon,\delta)$ domain, for each $v\in W^p(\curl,\Omega)\cap W^p_0(\dive,\Omega)$ (for each $v\in W^p(\dive,\Omega)\cap W^p_0(\curl,\Omega)$ respectively) we construct a suitable extension $Ev$ by adapting Jones' approach \cite{Jones}. More precisely, we consider a Whitney decomposition of $\Omega$ and we construct an extension operator in terms of suitable linear polynomials which satisfies the crucial estimates \eqref{corol1} and \eqref{corol2} (\eqref{corol1rot} and \eqref{corol2rot} respectively). The thesis is then achieved by density arguments.\\ Throughout the paper, $C$ will denote different positive constant. Sometimes, we indicate the dependence of these constants on some particular parameters in parentheses.
\section{$(\varepsilon,\delta)$ domains and trace results} \setcounter{equation}{0}
We recall the definition of $(\varepsilon,\delta)$ (or Jones) domain.
\begin{definition}\label{defepsdelta} Let $\mathcal{F}\subset\mathbb{R}^n$ be open and connected and $\mathcal{F}^c:=\mathbb{R}^n\setminus\mathcal{F}$. For $x\in\mathcal{F}$, let $\displaystyle d(x):=\inf_{y\in\mathcal{F}^c}|x-y|$. We say say that $\mathcal{F}$ is an $(\varepsilon,\delta)$ domain if, whenever $x,y\in\mathcal{F}$ with $|x-y|<\delta$, there exists a rectifiable arc $\gamma\in\mathcal{F}$ joining $x$ to $y$ such that \begin{center}
$\displaystyle\ell(\gamma)\leq\frac{1}{\varepsilon}|x-y|\quad$ and\quad $\displaystyle d(z)\geq\frac{\varepsilon|x-z||y-z|}{|x-y|}$ for every $z\in\gamma$. \end{center} \end{definition}
As pointed out in the Introduction, we consider two particular classes of $(\varepsilon,\delta)$ domains $\Omega\subset\mathbb{R}^n$: \begin{itemize}
\item[$i)$] $(\varepsilon,\delta)$ domains having as boundary a $d$-set;
\item[$ii)$] arbitrary closed $(\varepsilon,\delta)$ domains in the sense of \cite{jonsson91}. \end{itemize} For the sake of completeness, we recall the definition of $d$-set given in \cite{JoWa}. \begin{definition}\label{dset} A closed nonempty set $\mathcal{M}\subset\mathbb{R}^n$ is a $d$-set (for $0<d\leq n$) if there exist a Borel measure $\mu$ with $\supp\mu=\mathcal{M}$ and two positive constants $c_1$ and $c_2$ such that \begin{equation}\label{defindset} c_1r^{d}\leq \mu(B(P,r)\cap\mathcal{M})\leq c_2 r^{d}\quad\forall\,P \in\mathcal{M}. \end{equation} The measure $\mu$ is called $d$-measure. \end{definition}
In both the cases $i)$ and $ii)$, we can prove trace theorems, i.e. Green and Stokes formulas. For the sake of simplicity, we restrict ourselves to the case in which $\partial\Omega$ is a $d$-set. We recall the definition of Besov space specialized to our case. For generalities on Besov spaces, we refer to \cite{JoWa}. \begin{definition} Let $\mathcal{G}$ be a $d$-set with respect to a $d$-measure $\mu$ and $\alpha=1-\frac{n-d}{p}$. ${B^{p,p}_\alpha(\mathcal{G})}$ is the space of functions for which the following norm is finite: $$
\|u\|_{B^{p,p}_\alpha(\mathcal{G})}=\|u\|_{L^p(\mathcal{G})}+\left(\quad\iint_{|P-P'|<1}\frac{|u(P)-u(P')|^p}{|P-P'|^{d+p\alpha}}\,\mathrm {d}\mu(P)\,\mathrm {d}\mu(P')\right)^\frac{1}{p}. $$ \end{definition}
Throughout the paper, $p'$ will denote the H\"older conjugate exponent of $p$. In the following, we denote the dual of the Besov space on a $d$-set $\mathcal{G}$ with $(B^{p,p}_\alpha(\mathcal{G}))'$; this space coincides with the space $B^{p',p'}_{-\alpha}(\mathcal{G})$ (see \cite{JoWa2}).
\begin{theorem}[Stokes formula]\label{stokes} Let $u\in W^p(\curl,\Omega)$. There exists a linear and continuous operator $l_\tau(u)=u\times\nu$ from $W^p(\curl,\Omega)$ to $((B^{p',p'}_\alpha(\partial\Omega))')^3$.
The following generalized Stokes formula holds for every $v\in W^{1,p'}(\Omega)^n$: \begin{equation}\label{stokesformula} \left\langle u\times\nu,v\right\rangle_{((B^{p',p'}_\alpha(\partial\Omega))')^3, B^{p',p'}_\alpha(\partial\Omega)^3} =\int_\Omega u\cdot \curl v\,\mathrm {d} x +\int_\Omega v\cdot \curl u\,\mathrm {d} x. \end{equation}
Moreover, the operator $u \mapsto l_\tau(u)=u\times\nu$ is linear and continuous on $B^{p',p'}_\alpha(\partial\Omega)^3$.
\end{theorem}
\begin{theorem}[Green formula]\label{green} Let $u\in W^p(\dive,\Omega)$. There exists a linear and continuous operator $l_\nu(u)=u\cdot\nu$ from $W^p(\dive,\Omega)$ to $(B^{p',p'}_\alpha(\partial\Omega))'$.
The following generalized Green formula holds for every $v\in W^{1,p'}(\Omega)$: \begin{equation}\label{greenformula} \left\langle u\cdot\nu,v\right\rangle_{(B^{p',p'}_\alpha(\partial\Omega))', B^{p',p'}_\alpha(\partial\Omega)} =\int_\Omega u\cdot\nabla v\,\mathrm {d} x +\int_\Omega v\dive u\,\mathrm {d} x. \end{equation}
Moreover, the operator $u \mapsto l_\nu(u)=u\cdot\nu$ is linear and continuous on $B^{p',p'}_\alpha(\partial\Omega)$.
\end{theorem}
For the proofs we refer the reader to \cite{LaVe2} and \cite{CHLTV} with small suitable changes. Examples of domains for which Theorems \ref{stokes} and \ref{green} hold are 2D or 3D Koch-type domains. Formulas \eqref{stokesformula} and \eqref{greenformula} give a rigorous meaning of the boundary conditions in $W^p_0(\curl,\Omega)$ and $W^p_0(\dive,\Omega)$ respectively in terms of the dual of suitable Besov spaces.
\section{Friedrichs and Gaffney inequalities} \setcounter{equation}{0}
From now on, let $\Omega\subset\mathbb{R}^n$ be a bounded $(\varepsilon,\delta)$ domain, for $n=2,3$, having as boundary $\partial\Omega$ a $d$-set.
\subsection{The case $v\in W^p(\curl,\Omega)\cap W^p_0(\dive,\Omega)$}\label{divergenza}
We first consider the case $v\in W^p(\curl,\Omega)\cap W^p_0(\dive,\Omega)$. We point out that, since $\nu\cdot v=0$ on $\partial\Omega$, we have that \begin{equation}\label{mediadive} \int_\Omega \dive v\,\mathrm {d} x=0. \end{equation}
Let $S\subset\mathbb{R}^n$ be a measurable subset of $\mathbb{R}^n$; we denote by $\bar x$ its barycenter.\\ We construct the affine vector field $P_S(u)$ associated to $S$ and $u\in W^p(\curl,S)\cap W^p_0(\dive,S)$ in the following way: \begin{equation}\label{Pdef} P_S(u)(x)=a+B(x-\bar x), \end{equation} where $a\in\mathbb{R}^n$ and $B$ is a $n\times n$ matrix with entries $b_{ij}$ defined as \begin{equation}\label{definizioni}
a=\frac{1}{|S|}\int_S u\,\mathrm {d} x\quad\text{and}\quad b_{ij}=\frac{1}{2|S|}\int_S \left(\frac{\partial u_i}{\partial x_j}+\frac{\partial u_j}{\partial x_i}\right)\,\mathrm {d} x. \end{equation}
We point out that, from the definition, $B$ is a symmetric matrix. Moreover, by calculation it follows that $\curl(P_S(u))=0$, \begin{equation}\label{diveP}
\dive (P_S(u))=\frac{1}{|S|}\int_S \dive u\,\mathrm {d} x \end{equation} and \begin{equation}\label{prop2} \int_S (u-P_S(u))\,\mathrm {d} x=0. \end{equation}
By direct computation, it holds that \begin{equation}\label{stimagradiente}
\|\nabla(u-P_S(u))\|_{L^p(S)^{n\times n}}\leq C\|\nabla u\|_{L^p(S)^{n\times n}}, \end{equation}
where $C$ depends only on $|S|$.
Let us now suppose that \eqref{Fr2} holds in $S$ for $u\in W^p(\curl,S)\cap W^p_0(\dive,S)$. \eqref{stimagradiente} infers that \begin{equation}\label{int1}
\|\nabla(u-P_S(u))\|_{L^p(S)^{n\times n}}\leq C\left(\|\curl u\|_{L^p(S)^n}+\|\dive u\|_{L^p(S)}\right). \end{equation} Since $u-P_S(u)$ has vanishing mean value on $S$ from \eqref{prop2}, from Poincar\'e-Wirtinger inequality and \eqref{int1} we have \begin{equation}\label{poincare}
\|u-P_S(u)\|_{L^p(S)^n}\leq C \diam(S)\left(\|\curl u\|_{L^p(S)^n}+\|\dive u\|_{L^p(S)}\right), \end{equation} where $\diam(S)$ is the diameter of $S$. Now, one can easily see that \begin{equation}\label{stimaPinf}
\|\nabla P_S(u)\|_{L^\infty(S)^{n\times n}}\leq\|\nabla u\|_{L^\infty(S)^{n\times n}}; \end{equation} hence, by using again Poincar\'e-Wirtinger inequality (with $p=\infty$), triangle inequality and \eqref{stimaPinf} we get \begin{equation}\label{stimainf}
\|u-P_S(u)\|_{L^\infty(S)^n}\leq C \diam(S)\|\nabla(u-P_S(u))\|_{L^\infty(S)^{n\times n}}\leq 2C\diam(S)\|\nabla u\|_{L^\infty(S)^{n\times n}}. \end{equation} From now on, we choose $v\in W^{1,\infty}(\Omega)^n$. The thesis will then follow by density arguments. We construct the extension $Ev$ following the approach of Jones \cite{Jones} by using the linear polynomials $P_S(v)$.\\ Let us recall that any open set $\Omega\subset\mathbb{R}^n$ admits a so-called \emph{Whitney decomposition} (see \cite{whitney}, \cite{stein}) into dyadic cubes $S_k$, i.e. \begin{center} $\displaystyle\Omega=\bigcup_k S_k$. \end{center} This decomposition is such that \begin{equation}\label{w1} 1\leq\frac{{\rm dist}(S_k,\partial\Omega)}{\ell(S_k)}\leq 4\sqrt{n}\quad\forall\,k, \end{equation} \begin{equation}\label{w2} S_j^0\cap S_k^0=\emptyset\quad\text{if}\,\,j\neq k, \end{equation} \begin{equation}\label{w3} \frac{1}{4}\leq\frac{\ell(S_j)}{\ell(S_k)}\leq 4\quad\text{if}\,\,S_j\cap S_k\neq\emptyset, \end{equation} where $S^0$ denotes the interior of $S$ and $\ell(S)$ is the edgelength of a cube $S$.\\ Let now $W_1=\{S_k\}$ be a Whitney decomposition of $\Omega$ and $W_2=\{Q_j\}$ be a Whitney decomposition of $(\Omega^c)^0$. We set \begin{equation*} W_3=\left\{Q_j\in W_2\,:\,\ell(Q_j)\leq\frac{\varepsilon\delta}{16n}\right\}. \end{equation*} In his paper, Jones has shown that, for every $Q_j\in W_3$, one can choose a $\lq\lq$reflected" cube $Q_j^*=S_k\in W_1$ such that \begin{equation}\label{numero} 1\leq\frac{\ell(S_k)}{\ell(Q_j)}\leq 4\quad\text{and}\quad{\rm dist}(Q_j,S_k)\leq C\ell(Q_j), \end{equation} see Lemma 2.4 and Lemma 2.8 in \cite{Jones}. Moreover, if $Q_j,Q_k\in W_3$ have non-empty intersection, there exists a chain $F_{j,k}=\{Q_j^*=S_1,S_2,\dots,S_m=Q_k^*\}$ of cubes in $W_1$ which connects $Q_j^*$ and $Q_k^*$ such that $S_i\cap S_{i+1}\neq\emptyset$ and $m\leq C(\varepsilon,\delta)$.\\ From \cite{stein}, \cite{whitney} it follows that there exists a partition of unity $\{\phi_j\}$, associated with the Whitney decomposition, such that \begin{center} $\phi_j\in C^\infty(\mathbb{R}^n),\quad\supp\phi_j\subset\frac{17}{16}Q_j,\quad 0\leq\phi_j\leq 1$,\\
$\sum_{Q_j\in W_3}\phi_j=1\,\,\text{on}\,\,\bigcup_{Q_j\in W_3} Q_j\quad$ and $\quad|\nabla\phi_j|\leq C\ell(Q_j)^{-1}\quad\forall\,j$. \end{center}
For $v\in W^{1,\infty}(\Omega)^n$, let $P_j:=P_{Q_j^*}(v)$ be defined as in \eqref{Pdef} and \eqref{definizioni}. We now define the extension $Ev$ of $v$ to $\mathbb{R}^n$ in the following way: \begin{equation*} Ev= \begin{cases} \displaystyle\sum_{Q_j\in W_3} P_j\phi_j\quad &\text{in}\,\,(\Omega^c)^0,\\[2mm] v &\text{in}\,\,\Omega. \end{cases} \end{equation*} We point out that, since the boundary of an $(\varepsilon,\delta)$ domain has zero measure (see Lemma 2.3 in \cite{Jones}), it follows that $Ev$ is defined a.e. in $\mathbb{R}^n$.\\ From now on, if not otherwise specified, in this subsection we assume that $v\in W^p(\curl,\Omega)\cap W^p_0(\dive,\Omega)\cap W^p_0(\dive,S)$ for every $S\in W_1$. We now prove some preliminary lemmas. For the sake of completeness, we recall Lemma 2.1 in \cite{Jones}.
\begin{lemma}\label{lemmajones} Let $Q$ be a cube and let $F,G\subset Q$ be two measurable subsets such that $|F|,|G|\geq\gamma|Q|$ for some $\gamma>0$. If $P$ is a polynomial of degree 1, then \begin{equation*}
\|P\|_{L^p(F)}\leq C(\gamma)\|P\|_{L^p(G)}. \end{equation*} \end{lemma}
\begin{lemma}\label{lemma3.2} Let $F=\{S_1,\dots,S_m\}$ be a chain of cubes in $W_1$. Then \begin{equation}\label{stimaPLp}
\|P_{S_1}(v)-P_{S_m}(v)\|_{L^p(S_1)^n}\leq C(m)\ell(S_1)\left(\|\curl v\|_{L^p(\cup_j S_j)^n}+\|\dive v\|_{L^p(\cup_j S_j)}\right) \end{equation} and \begin{equation}\label{stimaPLinf}
\|P_{S_1}(v)-P_{S_m}(v)\|_{L^\infty(S_1)^n}\leq C(m)\ell(S_1)\|\nabla v\|_{L^\infty(\cup_j S_j)^{n\times n}}. \end{equation} \end{lemma}
\begin{proof} We will use \eqref{poincare}, where $S$ is a cube or a union of two neighboring cubes. From \eqref{w3}, it follows that the number of possible geometries of $S$ is finite; hence, we can find a uniform constant in \eqref{poincare}.\\ By using Lemma \ref{lemmajones}, we get \begin{align*}
&\|P_{S_1}(v)-P_{S_m}(v)\|_{L^p(S_1)^n}\leq\sum_{r=1}^{m-1}\|P_{S_r}(v)-P_{S_{r+1}}(v)\|_{L^p(S_1)^n}\\[2mm]
&\leq c(m)\sum_{r=1}^{m-1}\|P_{S_r}(v)-P_{S_{r+1}}(v)\|_{L^p(S_r)^n}\\[2mm]
&\leq c(m)\sum_{r=1}^{m-1}\left\{\|P_{S_r}(v)-P_{S_r\cup S_{r+1}}(v)\|_{L^p(S_r)^n}+\|P_{S_r\cup S_{r+1}}(v)-P_{S_{r+1}}(v)\|_{L^p(S_{r+1})^n}\right\}\\[2mm]
&\leq c(m)\sum_{r=1}^{m-1}\left\{\|P_{S_r}(v)-v\|_{L^p(S_r)^n}+\|P_{S_{r+1}}(v)-v\|_{L^p(S_{r+1})^n}+\|P_{S_r\cup S_{r+1}}(v)-v\|_{L^p(S_r\cup S_{r+1})^n}\right\}\\[2mm]
&\leq Cc(m)\ell(S_1)\left(\|\curl v\|_{L^p(\cup_j S_j)^n}+\|\dive v\|_{L^p(\cup_j S_j)}\right), \end{align*} where we used the fact that $F$ is a chain, integral properties and finally \eqref{poincare}.\\ The proof of \eqref{stimaPLinf} follows analogously by using \eqref{stimainf}. \end{proof}
For every $Q_j,Q_k\in W_3$ with non-empty intersection, we now choose a chain $F_{j,k}$ which connects $Q_j^*$ and $Q_k^*$ and such that $m\leq C(\varepsilon,\delta)$. We define \begin{equation*} F(Q_j)=\bigcup_{Q_k\in W_3, Q_j\cap Q_k\neq\emptyset} F_{j,k}; \end{equation*} hence \begin{equation}\label{finito}
\left\|\sum_{Q_k\,:\,Q_j\cap Q_k\neq\emptyset} \chi_{\cup F_{j,k}}\right\|_{L^\infty(\mathbb{R}^n)}\leq C\quad\forall\,Q_j\in W_3. \end{equation} We now prove two lemmas which allow us to control the norms of $Ev$, $\dive (Ev)$, $\curl (Ev)$ and $\nabla (Ev)$ in $(\Omega^c)^0$.
\begin{lemma}\label{lemma1} Let $Q_0\in W_3$. We have that: \begin{equation}\label{stima1}
\|Ev\|_{L^p(Q_0)^n}\leq C\left(\|v\|_{L^p(Q_0^*)^n}+\ell(Q_0)(\|\curl v\|_{L^p(F(Q_0))^n}+\|\dive v\|_{L^p(F(Q_0))})\right), \end{equation} \begin{equation}\label{stima2}
\|\curl(Ev)\|_{L^p(Q_0)^n}+\|\dive (Ev)\|_{L^p(Q_0)}\leq C\left(\|\curl v\|_{L^p(F(Q_0))^n}+\|\dive v\|_{L^p(F(Q_0))}\right), \end{equation} \begin{equation}\label{stima3}
\|Ev\|_{L^\infty(Q_0)^n}\leq C\left(\|v\|_{L^\infty(Q_0^*)^n}+\ell(Q_0)\|\nabla v\|_{L^\infty(F(Q_0))^{n\times n}}\right), \end{equation} \begin{equation}\label{stima4}
\|\nabla(Ev)\|_{L^\infty(Q_0)^{n\times n}}\leq C\|\nabla v\|_{L^\infty(F(Q_0))^{n\times n}}. \end{equation}
\end{lemma}
\begin{proof} We recall that, from the definition of $Ev$, on $Q_0$ we have that $\displaystyle Ev=\sum_{Q_j\in W_3} P_j\phi_j$. Moreover, since $\displaystyle\sum_{Q_j\in W_3}\phi_j\equiv 1$ on $\displaystyle\bigcup_{Q_j\in W_3} Q_j$, we get \begin{equation*}
\left\|\sum_{Q_j\in W_3}P_j\phi_j\right\|_{L^p(Q_0)^n}\leq\|P_0\|_{L^p(Q_0)^n}+\left\|\sum_{Q_j\in W_3}(P_j-P_0)\phi_j\right\|_{L^p(Q_0)^n}:=A+B. \end{equation*} We now estimate $A$ and $B$ separately. As to $A$, from Lemma \ref{lemmajones} and \eqref{poincare}, we get \begin{align}\label{stimaAprima}
A&=\|P_0\|_{L^p(Q_0)^n}\leq C\|P_0\|_{L^p(Q_0^*)^n}\leq C(\|P_0-v\|_{L^p(Q_0^*)^n}+\|v\|_{L^p(Q_0^*)^n})\notag\\[2mm]
&\leq C(\ell(Q_0)(\|\curl v\|_{L^p(Q_0^*)^n}+\|\dive v\|_{L^p(Q_0^*)})+\|v\|_{L^p(Q_0^*)^n}), \end{align} where we estimated $\ell(Q_0^*)$ with $\ell(Q_0)$ using \eqref{numero}, since $Q_0\in W_3$. We point out that, thanks to \eqref{finito}, the norms in the right-hand side of \eqref{stimaAprima} can be estimated in terms of the $L^p(\cup_j F_{0,j})$-norms. Hence, we get the following: \begin{equation}\label{stimaA}
A\leq C(\ell(Q_0)(\|\curl v\|_{L^p(\cup_j F_{0,j})^n}+\|\dive v\|_{L^p(\cup_j F_{0,j})})+\|v\|_{L^p(\cup_j F_{0,j})^n}). \end{equation}
As to $B$, from the properties of $\phi_j$ it is sufficient to bound $\|P_j-P_0\|_{L^p(Q_0)^n}$. By using again Lemma \ref{lemmajones}, \eqref{stimaPLp} and proceeding as above, we get \begin{equation}\label{stimaB}
B\leq\|P_j-P_0\|_{L^p(Q_0)^n}\leq C\|P_j-P_0\|_{L^p(Q_0^*)^n}\leq C\ell(Q_0)(\|\curl v\|_{L^p(\cup_j F_{0,j})^n}+\|\dive v\|_{L^p(\cup_j F_{0,j})}). \end{equation} Hence from \eqref{stimaA} and \eqref{stimaB} we get \eqref{stima1}. Estimate \eqref{stima3} follows similarly by using \eqref{stimainf} and \eqref{stimaPLinf}.\\ We now remark that, on $Q_0$, we have that \begin{equation*} Ev=\sum_{Q_j\in W_3} P_j\phi_j=P_0\sum_{Q_j\in W_3}\phi_j+\sum_{Q_j\in W_3} (P_j-P_0)\phi_j=P_0+\sum_{Q_j\in W_3} (P_j-P_0)\phi_j. \end{equation*} Therefore, since $\curl(P_0)=0$, we have that \begin{equation*} \curl(Ev)=\sum_{Q_j\in W_3}\curl((P_j-P_0)\phi_j). \end{equation*} Moreover, from \eqref{mediadive} and \eqref{diveP} it follows that \begin{equation*} \dive(Ev)=\sum_{Q_j\in W_3}\dive((P_j-P_0)\phi_j). \end{equation*}
Since there is a finite number of cubes $Q_j$ such that $\phi_j\neq 0$ in $Q_0$ and having non-empty intersection with $Q_0$, from \eqref{w3} we have that $\ell(Q_j)\geq\frac{1}{4}\ell(Q_0)$. From the properties of $\phi_j$, this implies that $|\nabla\phi_j|\leq\frac{C}{4}\ell(Q_0)^{-1}$.\\ By using vector identities, Lemma \ref{lemmajones} and \eqref{stimaPLp}, we have that \begin{align*}
&\|\curl((P_j-P_0)\phi_j)\|_{L^p(Q_0)^n}=\|(P_j-P_0)\times\nabla\phi_j\|_{L^p(Q_0)^n}\leq C\|P_j-P_0\|_{L^p(Q_0)^n}\|\nabla\phi_j\|_{L^p(Q_0)^n}\\[2mm]
&\leq C\ell(Q_0)^{-1}\|P_j-P_0\|_{L^p(Q_0)^n}\leq C\ell(Q_0)^{-1}\|P_j-P_0\|_{L^p(Q_0^*)^n}\\[2mm]
&\leq C(\|\curl v\|_{L^p(\cup_j F_{0,j})^n}+\|\dive v\|_{L^p(\cup_j F_{0,j})}). \end{align*} As to divergence term, similarly as above we get \begin{align*}
&\|\dive((P_j-P_0)\phi_j)\|_{L^p(Q_0)}=\|(P_j-P_0)\cdot\nabla\phi_j\|_{L^p(Q_0)}\leq\|P_j-P_0\|_{L^p(Q_0)^n}\|\nabla\phi_j\|_{L^p(Q_0)^n}\\[2mm]
&\leq C(\|\curl v\|_{L^p(\cup_j F_{0,j})^n}+\|\dive v\|_{L^p(\cup_j F_{0,j})}). \end{align*} Summing up in $j$ we get \begin{equation*}
\|\curl(Ev)\|_{L^p(Q_0)^n}+\|\dive(Ev)\|_{L^p(Q_0)}\leq C(\|\curl v\|_{L^p(F(Q_0))^n}+\|\dive v\|_{L^p(F(Q_0))}), \end{equation*} i.e. \eqref{stima2}.\\ We are left to prove \eqref{stima4}. Similarly as above, we have that \begin{equation*} \nabla(Ev)=\nabla P_0+\sum_{Q_j\in W_3} \nabla((P_j-P_0)\phi_j). \end{equation*} From Lemma \ref{lemmajones} and \eqref{stimaPinf}, we get \begin{equation*}
\|\nabla P_0\|_{L^\infty(Q_0)^{n\times n}}\leq C\|\nabla P_0\|_{L^\infty(Q_0^*)^{n\times n}}\leq C\|\nabla v\|_{L^\infty(Q_0^*)^{n\times n}}\leq C\|\nabla v\|_{L^\infty(\cup_j F_{0,j})^{n\times n}}. \end{equation*} As above, it follows that \begin{align*}
&\|\nabla(P_j-P_0)\|_{L^\infty(Q_0)^{n\times n}}\leq C\|\nabla(P_j-P_0)\|_{L^\infty(Q_0^*)^{n\times n}}\leq C\|\nabla(P_j-P_0)\|_{L^\infty(Q_0^*\cup Q_j^*)^{n\times n}}\\[2mm]
&\leq C\|\nabla v\|_{L^\infty(Q_0^*\cup Q_j^*)^{n\times n}}\leq C\|\nabla v\|_{L^\infty(\cup_j F_{0,j})^{n\times n}} \end{align*} From these inequalities, \eqref{stima4} follows and the proof is complete. \end{proof} We now prove a result similar to Lemma \ref{lemma1}, which relates to the cubes of $(\Omega^c)^0$ not belonging to $W_3$.
\begin{lemma}\label{lemma2} Let $Q_0\in W_2\setminus W_3$. We have that: \begin{equation}\label{stima1comp}
\|Ev\|_{L^p(Q_0)^n}\leq C\sum_{Q_j\in W_3\,:\,Q_j\cap Q_0\neq\emptyset}\left(\|v\|_{L^p(Q_j^*)^n}+\|\curl v\|_{L^p(Q_j^*)^n}+\|\dive v\|_{L^p(Q_j^*)}\right), \end{equation} \begin{align}
\|\curl(Ev)\|_{L^p(Q_0)^n}&+\|\dive(Ev)\|_{L^p(Q_0)}\notag\\[2mm]
&\leq C\sum_{Q_j\in W_3\,:\,Q_j\cap Q_0\neq\emptyset}\left(\|v\|_{L^p(Q_j^*)^n}+\|\curl v\|_{L^p(Q_j^*)^n}+\|\dive v\|_{L^p(Q_j^*)}\right),\label{stima2comp} \end{align} \begin{equation}\label{stima3comp}
\|Ev\|_{L^\infty(Q_0)^n}\leq C\sum_{Q_j\in W_3\,:\,Q_j\cap Q_0\neq\emptyset}\left(\|v\|_{L^\infty(Q_j^*)^n}+\|\nabla v\|_{L^\infty(Q_j^*)^{n\times n}}\right), \end{equation} \begin{equation}\label{stima4comp}
\|\nabla(Ev)\|_{L^\infty(Q_0)^{n\times n}}\leq C\sum_{Q_j\in W_3\,:\,Q_j\cap Q_0\neq\emptyset}\left(\|v\|_{L^\infty(Q_j^*)^n}+\|\nabla v\|_{L^\infty(Q_j^*)^{n\times n}}\right). \end{equation}
\end{lemma}
\begin{proof} We start by pointing out that, if $\phi_j\neq 0$ on $Q_0$, we have $Q_j\cap Q_0\neq\emptyset$ (since $\supp\phi_j\subset\frac{17}{16}Q_j)$. Therefore, since $Q_0\in W_2\setminus W_3$, we have \begin{equation}\label{stimalunghezza} \ell (Q_j)\geq\frac{1}{4}\ell (Q_0)\geq\frac{\varepsilon\delta}{64n}. \end{equation} On $Q_0$ we have that \begin{equation*}
|Ev|=\left|\sum_{Q_j\in W_3\,:\,Q_j\cap Q_0\neq\emptyset} P_j\phi_j\right|\leq \sum_{Q_j\in W_3\,:\,Q_j\cap Q_0\neq\emptyset} |P_j|. \end{equation*} From Lemma \ref{lemmajones} and triangle inequality, we get \begin{equation}\label{interm}
\|P_j\|_{L^p(Q_0)^n}\leq C\|P_j\|_{L^p(Q_j)^n}\leq C\|P_j\|_{L^p(Q_j^*)^n}\leq C(\|P_j-v\|_{L^p(Q_j^*)^n}+\|v\|_{L^p(Q_j^*)^n}). \end{equation}
From \eqref{interm} and \eqref{poincare}, it follows that \begin{align*}
&\|Ev\|_{L^p(Q_0)^n}\leq\sum_{Q_j\in W_3\,:\,Q_j\cap Q_0\neq\emptyset}\|P_j\|_{L^p(Q_0)^n}\\[3mm]
&\leq C\sum_{Q_j\in W_3\,:\,Q_j\cap Q_0\neq\emptyset}\left(\|v\|_{L^p(Q_j^*)^n}+\diam(Q_j^*)(\|\curl v\|_{L^p(Q_j^*)^n}+\|\dive v\|_{L^p(Q_j^*)})\right). \end{align*} Sine $\Omega$ is bounded, we can estimate $\diam(Q_j^*)$ with a constant depending on $\diam(\Omega)$, thus proving \eqref{stima1comp}.\\ We come to \eqref{stima2comp}. By proceeding as in the proof of Lemma \ref{lemma1} and by using \eqref{stimalunghezza}, the following estimate holds:
\begin{align*}
&\|\curl(Ev)\|_{L^p(Q_0)^n}+\|\dive(Ev)\|_{L^p(Q_0)}=\left\|\sum_{Q_j\in W_3\,:\,Q_j\cap Q_0\neq\emptyset}\curl(P_j\phi_j)\right\|_{L^p(Q_0)^n}\\[2mm]
&+\left\|\sum_{Q_j\in W_3\,:\,Q_j\cap Q_0\neq\emptyset}\dive(P_j\phi_j)\right\|_{L^p(Q_0)}\leq C\sum_{Q_j\in W_3\,:\,Q_j\cap Q_0\neq\emptyset}\|P_j\|_{L^p(Q_0)^n}\|\nabla\phi_j\|_{L^p(Q_0)^{n\times n}}\\[2mm]
&\leq C\ell(Q_0)^{-1}\sum_{Q_j\in W_3\,:\,Q_j\cap Q_0\neq\emptyset}\|P_j\|_{L^p(Q_0)^n}\leq C\left(\frac{\varepsilon\delta}{64n}\right)^{-1}\sum_{Q_j\in W_3\,:\,Q_j\cap Q_0\neq\emptyset}\|P_j\|_{L^p(Q_0^*)^n}. \end{align*} By proceeding as above, we get \eqref{stima2comp}. Estimates \eqref{stima3comp} and \eqref{stima4comp} follow in a similar way by using \eqref{stimainf} and \eqref{stimaPinf}. \end{proof}
From the above lemmas we obtain the following result.
\begin{prop}\label{proposiz} For every $v\in W^{1,\infty}(\Omega)^n$ such that $v\in W^p(\curl,\Omega)\cap W^p_0(\dive,\Omega)$ we have \begin{align}
\|Ev\|_{L^p((\Omega^c)^0)^n}&+\|\dive(Ev)\|_{L^p((\Omega^c)^0)}+\|\curl(Ev)\|_{L^p((\Omega^c)^0)^n}\notag\\[2mm]
&\leq C\left(\|v\|_{L^p(\Omega)^n}+\|\dive v\|_{L^p(\Omega)}+\|\curl v\|_{L^p(\Omega)^n}\right)\label{corol1} \end{align} and \begin{equation}\label{corol2}
\|Ev\|_{W^{1,\infty}((\Omega^c)^0)^n}\leq C\|v\|_{W^{1,\infty}(\Omega)^n}. \end{equation} \end{prop}
\begin{proof} By summing up over every $Q_0\in W_2$, the thesis follows as a direct consequence of Lemma \ref{lemma1} and Lemma \ref{lemma2}. In particular, \eqref{corol1} follows from \eqref{stima1}, \eqref{stima2}, \eqref{stima1comp} and \eqref{stima2comp}, while \eqref{corol2} follows from \eqref{stima3}, \eqref{stima4}, \eqref{stima3comp} and \eqref{stima4comp}. \end{proof}
We now prove the first main result of this paper, which follows from the above lemmas.
\begin{theorem}[Friedrichs inequality]\label{fridis} Let $\Omega\subset\mathbb{R}^n$ be a bounded $(\varepsilon,\delta)$ domain with $\partial\Omega$ a $d$-set. There exists a constant $C=C(\varepsilon,\delta,n,p,\Omega)>0$ such that, for every $v\in W^{1,p}(\Omega)^n$ such that $v\in W^p(\curl,\Omega)\cap W^p_0(\dive,\Omega)$, \begin{equation}\label{friedrichs}
\|v\|_{W^{1,p}(\Omega)^n}\leq C\left(\|v\|_{L^p(\Omega)^n}+\|\curl v\|_{L^p(\Omega)^n}+\|\dive v\|_{L^p(\Omega)}\right). \end{equation} \end{theorem}
\begin{proof} It is sufficient to prove \eqref{friedrichs} for $v\in W^{1,\infty}(\Omega)^n$; the thesis will then follow by density. We recall that the extension $Ev$ is defined a.e. on $\mathbb{R}^n$ since $|\partial\Omega|=0$. Moreover, from the definition of $Ev$ we can suppose that $\supp Ev$ is contained in a ball $B$.\\ Since $Ev\in W^{1,p}(B)^n$, from \eqref{corol1} we have that \begin{equation*}
\|Ev\|_{L^p(B)^n}+\|\curl(Ev)\|_{L^p(B)^n}+\|\dive(Ev)\|_{L^p(B)}\leq C\left(\|v\|_{L^p(\Omega)^n}+\|\curl v\|_{L^p(\Omega)^n}+\|\dive v\|_{L^p(\Omega)}\right). \end{equation*} Hence, from Friedrichs inequality for smooth domains and the above inequality, we get \begin{align*}
\|v\|_{W^{1,p}(\Omega)^n}&=\|Ev\|_{W^{1,p}(\Omega)^n}\leq\|Ev\|_{W^{1,p}(B)^n}\leq C(\|Ev\|_{L^p(B)^n}+\|\curl(Ev)\|_{L^p(B)^n}+\|\dive(Ev)\|_{L^p(B)})\\[2mm]
&\leq C\left(\|v\|_{L^p(\Omega)^n}+\|\curl v\|_{L^p(\Omega)^n}+\|\dive v\|_{L^p(\Omega)}\right), \end{align*} i.e. the thesis. \end{proof}
We conclude this section by proving Gaffney inequality as a direct consequence of Theorem \ref{fridis}.
\begin{theorem}[Gaffney inequality]\label{gaffin} Let $\Omega\subset\mathbb{R}^n$ be a bounded simply connected $(\varepsilon,\delta)$ domain with $\partial\Omega$ a $d$-set. Let $v\in W^{1,p}(\Omega)^n$ be such that $v\in W^p(\curl,\Omega)\cap W^p_0(\dive,\Omega)$. Then there exists $C=C(\varepsilon,\delta,n,p,\Omega)>0$ such that \begin{equation}\label{gaffney}
\|v\|_{W^{1,p}(\Omega)^n}\leq C\left(\|\curl v\|_{L^p(\Omega)^n}+\|\dive v\|_{L^p(\Omega)}\right). \end{equation} \end{theorem}
\begin{proof} We argue by contradiction. Let us suppose that \eqref{gaffney} does not hold; hence, there exists a sequence of vectors $\{v_k\}\subset W^{1,p}(\Omega)^n\cap W^p(\curl,\Omega)\cap W^p_0(\dive,\Omega)$ such that \begin{equation*}
\|v_k\|_{W^{1,p}(\Omega)^n}=1\quad\text{and}\quad\|\curl v_k\|_{L^p(\Omega)^n}+\|\dive v_k\|_{L^p(\Omega)}\xrightarrow[k\to+\infty]{} 0. \end{equation*}
Since $\|v_k\|_{W^{1,p}(\Omega)^n}=1$, there exists a subsequence of $\{v_k\}$ (which we still denote by $v_{k}$) such that \begin{equation*} v_{k}\rightharpoonup v\,\,\text{in}\,\,W^{1,p}(\Omega)^n\quad\text{and}\quad v_{k}\rightarrow v\,\,\text{in}\,\,L^p(\Omega)^n. \end{equation*} Since the distributional limits coincide with the weak limits, it immediately follows that $\dive v=0$ and $\curl v=0$.\\ We now prove that $\{v_k\}$ is a Cauchy sequence in $W^{1,p}(\Omega)^n$. From Friedrichs inequality \eqref{friedrichs}, for every $k,j\in\mathbb{N}$ one has \begin{equation}\label{stimacauchy}
\|v_k-v_j\|_{W^{1,p}(\Omega)^n}\leq C\left(\|v_k-v_j\|_{L^p(\Omega)^n}+\|\dive (v_k-v_j)\|_{L^p(\Omega)}+\|\curl (v_k-v_j)\|_{L^p(\Omega)^n}\right). \end{equation} From the strong convergence of $v_k$ in $L^p(\Omega)^n$, the first term on the right-hand side of \eqref{stimacauchy} vanishes. As to the other two terms, they also vanish since $\curl v_k$ and $\dive v_k$ both tend to 0 in $L^p$ as $k\to+\infty$. Hence $v_k$ is a Cauchy sequence in $W^{1,p}(\Omega)^n$, and $v_k\to v$ strongly in $W^{1,p}(\Omega)^n$.\\ We recall that if $\curl v=0$ in $\Omega$ and $\Omega$ is simply connected, there exists a function $\Phi\in W^{1,p}(\Omega)$ such that $v=\nabla\Phi$. This in turn implies that $\Delta\Phi=\dive\nabla\Phi=\dive v=0$ in $\Omega$. Moreover, since $v\in W^p_0(\dive)$, we also have that \begin{center} $\displaystyle\frac{\partial\Phi}{\partial\nu}=\nu\cdot\nabla\Phi=\nu\cdot v=0$ on $\partial\Omega$. \end{center} Hence $\Phi\in W^{1,p}(\Omega)$ is the unique weak solution of the following problem \begin{equation}\label{probphi} \begin{cases} \Delta\Phi=0\quad &\text{in}\,\,\Omega,\\[2mm] \displaystyle\frac{\partial\Phi}{\partial\nu}=0 &\text{on}\,\,\partial\Omega. \end{cases} \end{equation} This implies that $\Phi$ is constant, and so $v=\nabla\Phi=0$ on $\Omega$. We reached a contradiction, since \begin{equation*}
1=\|v_k\|_{W^{1,p}(\Omega)^n}\xrightarrow[k\to+\infty]{}\|v\|_{W^{1,p}(\Omega)^n}=0. \end{equation*}
\end{proof}
\subsection{The case $v\in W^p(\dive,\Omega)\cap W^p_0(\curl,\Omega)$}\label{rotore}
We now consider the case $v\in W^p(\dive,\Omega)\cap W^p_0(\curl,\Omega)$. We recall that this implies $\nu\times v=0$ on $\partial\Omega$ in the dual of $B^{p',p'}_\alpha(\partial\Omega)$.\\ We approximate $v\in W^p(\dive,\Omega)\cap W^p_0(\curl,\Omega)\cap W^{1,\infty}(\Omega)^n$ by means of the polynomials $P_j$ as in the previous section. We remark that in this case \begin{equation*}
\dive P_j=\frac{1}{|Q_j^*|}\int_{Q_j^*} \dive v\,\mathrm {d} x\neq 0. \end{equation*}
As in the previous subsection, estimates \eqref{int1} and \eqref{poincare} hold, as well as lemmas \ref{lemma3.2}, \ref{lemma1} and \ref{lemma2}, under the hypothesis that $v\in W^p(\dive,\Omega)\cap W^p_0(\curl,\Omega)\cap W^p_0(\curl,S)$ for every $S\in W_1$.\\ For the sake of clarity, we state the analogous of Proposition \ref{proposiz} and Theorem \ref{fridis} in this case.
\begin{prop} For every $v\in W^{1,\infty}(\Omega)^n$ such that $v\in W^p(\dive,\Omega)\cap W^p_0(\curl,\Omega)$ we have \begin{align}
\|Ev\|_{L^p((\Omega^c)^0)^n}&+\|\dive(Ev)\|_{L^p((\Omega^c)^0)}+\|\curl(Ev)\|_{L^p((\Omega^c)^0)^n}\notag\\[2mm]
&\leq C\left(\|v\|_{L^p(\Omega)^n}+\|\dive v\|_{L^p(\Omega)}+\|\curl v\|_{L^p(\Omega)^n}\right)\label{corol1rot} \end{align} and \begin{equation}\label{corol2rot}
\|Ev\|_{W^{1,\infty}((\Omega^c)^0)^n}\leq C\|v\|_{W^{1,\infty}(\Omega)^n}. \end{equation} \end{prop}
\begin{theorem}[Friedrichs inequality]\label{fridisrot} Let $\Omega\subset\mathbb{R}^n$ be a bounded $(\varepsilon,\delta)$ domain with $\partial\Omega$ a $d$-set. There exists a constant $C=C(\varepsilon,\delta,n,p,\Omega)>0$ such that, for every $v\in W^{1,p}(\Omega)^n$ such that $v\in W^p(\dive,\Omega)\cap W^p_0(\curl,\Omega)$, \begin{equation}\label{friedrichsrot}
\|v\|_{W^{1,p}(\Omega)^n}\leq C\left(\|v\|_{L^p(\Omega)^n}+\|\curl v\|_{L^p(\Omega)^n}+\|\dive v\|_{L^p(\Omega)}\right). \end{equation} \end{theorem}
We conclude by proving Gaffney inequality.
\begin{theorem}[Gaffney inequality]\label{gaffinrot} Let $\Omega\subset\mathbb{R}^n$ be a bounded simply connected $(\varepsilon,\delta)$ domain with $\partial\Omega$ a $d$-set. Let $v\in W^{1,p}(\Omega)^n$ be such that $v\in W^p(\dive,\Omega)\cap W^p_0(\curl,\Omega)$. Then there exists $C=C(\varepsilon,\delta,n,p,\Omega)>0$ such that \begin{equation}\label{gaffneyrot}
\|v\|_{W^{1,p}(\Omega)^n}\leq C\left(\|\curl v\|_{L^p(\Omega)^n}+\|\dive v\|_{L^p(\Omega)}\right). \end{equation} \end{theorem}
\begin{proof} We proceed as in the proof of \cite[Corollary 3.51]{monk}; we argue by contradiction and we suppose that \eqref{gaffneyrot} does not hold. As in Theorem \ref{gaffin}, this means that there exists a sequence of vectors $\{v_k\}\subset W^{1,p}(\Omega)^n\cap W^p(\dive,\Omega)\cap W^p_0(\curl,\Omega)$ such that \begin{equation*}
\|v_k\|_{W^{1,p}(\Omega)^n}=1\quad\text{and}\quad\|\curl v_k\|_{L^p(\Omega)^n}+\|\dive v_k\|_{L^p(\Omega)}\xrightarrow[k\to+\infty]{} 0. \end{equation*} This implies that \begin{equation*} v_{k}\rightharpoonup v\,\,\text{in}\,\,W^{1,p}(\Omega)^n\quad\text{and}\quad v_{k}\rightarrow v\,\,\text{in}\,\,L^p(\Omega)^n, \end{equation*} with $\dive v=0$ and $\curl v=0$.\\ From \eqref{friedrichsrot}, for every $k,j\in\mathbb{N}$ we have that \begin{equation}\label{stimacauchyrot}
\|v_k-v_j\|_{W^{1,p}(\Omega)^n}\leq C\left(\|v_k-v_j\|_{L^p(\Omega)^n}+\|\dive (v_k-v_j)\|_{L^p(\Omega)}+\|\curl (v_k-v_j)\|_{L^p(\Omega)^n}\right). \end{equation}
As in the proof of Theorem \ref{gaffin}, all the terms on the right-hand side of \eqref{stimacauchyrot} vanish when $k,j\to+\infty$, hence $\{v_k\}$ is a Cauchy sequence in $W^{1,p}(\Omega)^n$.\\
As in the case $v\in W^p(\curl,\Omega)\cap W^p_0(\dive,\Omega)$, there exists a function $\Phi\in W^{1,p}(\Omega)$ such that $v=\nabla\Phi$ and $\Delta\Phi=0$ in $\Omega$. Since in this case $v\in W^p_0(\curl,\Omega)$, we also have that \begin{center} $\nu\times\nabla\Phi=\nu\times v=0$ on $\partial\Omega$. \end{center} Up to shifting $\Phi$ by a constant, this implies that $\Phi=0$ on $\partial\Omega$ in the trace sense. Hence $\Phi\in W^{1,p}(\Omega)$ is the unique weak solution of the following problem \begin{equation}\label{probphirot} \begin{cases} \Delta\Phi=0\quad &\text{in}\,\,\Omega,\\[2mm] \Phi=0 &\text{on}\,\,\partial\Omega. \end{cases} \end{equation} This implies that $\Phi=0$, therefore $v=0$ on $\Omega$ and we reach the contradiction. \end{proof}
\noindent {\bf Acknowledgements.} The authors have been supported by the Gruppo Nazionale per l'Analisi Matematica, la Probabilit\`a e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).
\end{document} |
\begin{document}
\title{\bf Moment subset sums over finite fields}
\author{Tim Lai\textsuperscript{1}, Alicia Marino\textsuperscript{2}, Angela Robinson\textsuperscript{3}, Daqing Wan\textsuperscript{4}}
\maketitle \thispagestyle{fancy}
\begin{center} \textsuperscript{1}Indiana University, Bloomington\\ \textsuperscript{2}University of Hartford\\ \textsuperscript{3}National Institute of Standards and Technology\\ \textsuperscript{4}University of California, Irvine
\end{center}
{\bf Abstract:} The $k$-subset sum problem over finite fields is a classical NP-complete problem. Motivated by coding theory applications, a more complex problem is the higher $m$-th moment $k$-subset sum problem over finite fields. We show that there is a deterministic polynomial time algorithm for the $m$-th moment $k$-subset sum problem over finite fields for each fixed $m$ when the evaluation set is the image set of a monomial or Dickson polynomial of any degree $n$. In the classical case $m=1$, this recovers previous results of Nguyen-Wang (the case $m=1, p>2$) \cite{WN18} and the results of Choe-Choe (the case $m=1, p=2$) \cite{CC19}.
\section{Introduction} One of the most puzzling problems in theoretical computer science, originally posed in 1971, is to determine whether P = NP \cite{C71}. That is, to determine whether the complexity class of problems which can be solved in deterministic polynomial time is equivalent to the class of problems whose solutions, if any, can be verified in deterministic polynomial time. For a comprehensive survey on this topic, see Widgerson's forthcoming monograph \cite{Wi19}.
All NP-complete problems are equivalent to each other under polynomial time reduction. One approach to proving that P = NP is to find an NP-complete problem and prove (or disprove) that it is deterministically solvable in polynomial time. We choose the $k$-subset sum problem over finite fields \cite{CLRS09}, which is a classical NP-complete problem. Although this problem is out of reach, our aim of this paper is to explore deterministic polynomial time algorithms to this and similar variations of this problem in various interesting special cases.
Let $p$ be a prime, $q=p^s$ for some integer $s>0$, and $\mathbb{F}_q$ the finite field of $q$ elements. Given a subset $D=\{x_1, \ldots, x_d\} \subset \mathbb{F}_q$ and $b \in \mathbb{F}_q$, let \begin{align*} N(D,b)= \#\{S \subseteq D: \sum_{x \in S}x=b\}. \end{align*} The dense input size of $D$ is $d\log q$, since one can simply list all the $d$ elements of $D$ in $\mathbb{F}_q$ where each takes $\log q$ space.
The decision subset sum problem (SSP) over finite fields asks if given $D$ and $b$, can one determine whether $N(D,b)>0$ in polynomial time in terms of the dense input size $d\log q$? If $N(D,b)>0$, then there exists at least one collection $S \subseteq D$ of elements which sum to $b$. This solution, $S$, can be checked by addition of $|S|\leq d$ elements of size $\textnormal{log }q$, thus SSP $\in$ NP for every fixed $p$. When $p=2$, it is a linear algebra problem and thus SSP $\in$ P. It is known SSP is NP-complete for each fixed $p>2$.
Motivated by numerous applications, a more precise version of the SSP is to determine whether there exists a subset $S \subseteq D$ of given size $k$ whose elements sum to $b$ given a set $D$ and target $b$ as above. The decision version of this $k$-subset sum problem ($k$-SSP) is as follows. Given a subset $D=\{x_1, \ldots, x_d\} \subset \mathbb{F}_q, k\in \{1, \ldots, d\}$ and $b \in \mathbb{F}_q$, for \begin{align*}
N_k(D,b)= \#\{S \subseteq D: \sum_{x \in S}x=b, |S|=k\}, \end{align*} determine whether $N_k(D,b)>0$. The decision $k$-SSP problem is NP-hard for every fixed $p$, including the more difficult case $p=2$ which is the main result in \cite{V97} determining that computing the minimum distance of binary codes is NP-hard. In general, the complexity of the $k$-SSP problem depends on the relationship between $d$ and the modulus $q$. When $q=\mathcal{O}(\textnormal{poly}(d))$, dynamic programming solves the problem in polynomial time \cite{GM91, L05}. The trivial exhaustive search algorithm shows that $k$-SSP $\in$ P when $d = \mathcal{O}(\textnormal{log log}\,q)$. It is known that $k$-SSP is NP-hard when $d=(\textnormal{log }q)^c$ for constant $c>0$, see \cite{LO85,GM91}. An explicit formula for $N_k(D,b)$ was presented for the case of $D = \mathbb{F}_q$ \cite{LW08}.
In coding theory, $k$-SSP arises from computing the minimum distance of a linear code and the deep hole problem for Reed-Solomon codes. The set $D$ is called the \textit{evaluation set} as it is exactly the evaluation set of the corresponding Reed-Solomon code. If one moves further to consider the harder problem of computing the error distance of a received word (namely, maximal likelihood decoding) in Reed-Solomon codes, one is naturally lead to the following higher moment $k$-subset sum problem. More formally, given a subset $D=\{x_1, \ldots, x_d\} \subset \mathbb{F}_q, k\in \{1, \ldots, d\}$, $m \in \mathbb{N}$, and $\boldsymbol{b} = (b_1, \ldots, b_m) \in \mathbb{F}_q^m$, determine whether \begin{align*}
N_k(D,\boldsymbol{b},m)= \#\{S \subseteq D: \sum_{y \in S}y^j=b_j, 1\leq j \leq m, |S|=k\}, \end{align*} is positive. This problem is known as the $m$-th moment $k$-SSP and its complexity has been studied recently. It is proven to be NP-hard for general $D$ if $m\leq 3$ \cite{GGG15} or smaller than $\mathcal{O}(\textnormal{log}\,\textnormal{log}\,\textnormal{log}\,q)$ \cite{GGG18}. An explicit combinatorial formula for $N_k(D, \boldsymbol{b}, m)$ is obtained in \cite{Ng19} when $m=2$ and $D=\mathbb{F}_q$.
All the problems and results above are based on a model where we use the dense input $\{D,b\}$ of size $\mathcal{O}(d\, \textnormal{log}\,q)$ by listing all the $d$ elements of $D$. Though improved solutions to the decision $k$-SSP with such dense input are desired, one may also consider an \textit{algebraic input} model wherein $D$ is the set of images under some polynomial map applied to field elements. That is, for some monic polynomial $g(x) \in \mathbb{F}_q [x]$ of degree $n$, \[
D = g(\mathbb{F}_q) = \{g(a) : a \in \mathbb{F}_q\}. \] In this situation, the algebraic input size would be $n\log q$ since it is enough to write down the $n$ coefficients of the input polynomial $g(x)$. A fundamental problem is to ask if the $k$-SSP and the $m$-th moment $k$-SSP can be solved in deterministic polynomial time in terms of the algebraic input size $n\log q$. This appears more difficult as it is not
even clear if the problem is in NP because both $k$ and the set size $d=|D| \geq q/n$ can already be exponential in terms of the algebraic input size $n\log q$. No complexity result is yet known for the algebraic model.
The last author conjectured that $k$-SSP can be solved in deterministic polynomial time in algebraic input size $n\log q$ if the order of the Galois group $G_g$ of $g(x)-t$ over ${\mathbb{F}}_q(t)$ is bounded by a polynomial in $n\log q$. The last condition is trivially satisfied if $$n = O(\log\log q/\log\log\log q)$$
since then $|G_g| \leq n!$ is bounded by a polynomial in $\log q$. This condition is also satisfied when $g(x)$ is a monomial or Dickson polynomial of any degree $n$. Note that this conjecture is highly non-trivial, as it is not even clear whether the problem is in NP since we are using the algebraic (sparse) input size and $d\geq q/n$ is exponential in $n\log q$ for $n=O(\log\log q)$. Thus, we cannot write down all the elements of $D$ as listing all elements of $D$ already takes exponential time. In a sense, our set $D$ is given as a black-box.
As a supporting evidence, this conjecture has been proved to be true in the special case when the evaluation set $D$ is the image of the monomial $x^n$ or Dickson polynomials of degree $n$: see \cite{WN18} for the case $p>2$ and \cite{CC19} for the case $p=2$. The aim of the present paper is to extend these results from $m=1$ ($k$-SSP) to the higher $m$-moment $k$-SSP for each fixed $m$. Namely, our main result is \begin{theorem}\label{THM1} Let the evaluation set $D$ be the image set of a monomial or a Dickson polynomial of degree $n$ over $\mathbb{F}_q$. There is a deterministic algorithm which for any given $m\in \mathbb{N}$, $b\in \mathbb{F}_q^m$ and integer $k\geq 0$, decides if $N_k(D,b,m)>0$ in time $(n\log q)^{C_m}$, where $C_m$ is a constant depending only on $m$. In particular, this is a polynomial time algorithm in the algebraic input size $n\log q$ for each fixed $m$.
\end{theorem}
To prove the above theorem, we will need to combine all the techniques available so far: dynamic programming for large $n>q^{\epsilon}$, Kayal's algorithm \cite{K05} for constant $k$, Brun sieve for medium $k$, the Li-Wan sieve for large $k$ and $p>2$, and the recent Choe-Choe argument \cite{CC19} for large $k$ and $p=2$. In addition, we need to employ the Weil bound to prove a crucial new partial character sum estimate.
\section{Background} One important tool in our proof is character sum estimates. Let $\psi: \mathbb{F}_q \rightarrow \mathbb{C}$ be an additive character.
We know from character theory that for a nontrivial character $\psi$ we have $\sum\limits_{x \in \mathbb{F}_q} \psi (x) = 0$. However, in the case of the trivial character, the sum is the size of the finite field.
Let $G = \mathbb{F}_q$ and let $\widehat{G}$ be the set of all additive characters for $\mathbb{F}_q$. Then we have the following equality \[ \sum_{\psi \in \widehat{G}} \psi(x) = \left\{
\begin{array}{ll}
q & \quad \textnormal{if }x = \textbf{0} \\[1em]
0 & \quad \textnormal{if }x \neq \textbf{0}
\end{array}
\right.. \]
\begin{def1}[Dickson Polynomial] Let $n$ be a positive integer and $a\in \mathbb{F}_q$. The Dickson polynomial of degree $n$ is defined as \begin{equation*} D_n(x, a) = \sum_{i=0}^{\lfloor n/2 \rfloor} \frac{n}{n-i}\binom{n-i}{i} (-a)^i x^{n-2i}. \end{equation*} \end{def1}
\noindent If $n=pn_1$ is divisible by $p$, one checks that $D_{pn_1}(x, a) = D_{n_1}(x, a)^p$. Thus, we can assume that $n$ is not divisible by $p$.
Note that for $a=0$, $D_n(x,0)=x^n$, so we see that Dickson polynomials are generalizations of monomials. Of particular use to us is the size of the image of these polynomials, also known as the \textit{value set}. A simple fact for the monomial $D_n(x,0)=x^n$ is that \[
|D_n(\mathbb{F}_q^\times,0)| =
\begin{cases}
q-1 & \gcd(n, q-1)=1 \\
\frac{1}{\ell}(q-1) & \gcd(n, q-1)=\ell
\end{cases} \]
In the first case, the map is $1$ to $1$; in the latter case, the map is $\ell$ to $1$. It turns out an analogous preimage-counting statement holds when $a\ne 0$. Chou, Mullen, and Wassermann in \cite{CMW} used a character sum argument to calculate the following.
\begin{nots} For $b,c,d \in \mathbb{Z}$, Let $b^c||d$ denote that $b^c$ fully divides $d$ so that $b^{c+1} \nmid d$. \end{nots}
\begin{theorem}\label{dicksonTheorem}
Let $n\geq 2$ and $a\in \mathbb{F}_q^*$. If $q$ is even, then $|D_n^{-1}(D_n(x_0,a))| =$ \begin{equation*}
\left\{
\begin{array}{cl}
\gcd(n, q-1) & \text{ if condition A holds} \\
\gcd(n, q+1) & \text{ if condition B holds} \\
\dfrac{\gcd(n,q-1)+\gcd(n,q+1)}{2} & \text{$D_n(x_0,a)=0$},
\end{array}
\right . \end{equation*} where `condition A' holds $\text{if } x^2+x_0x+a \text{ is reducible over $\mathbb{F}_q$ and } D_n(x_0,a)\ne 0$; `condition B' holds $\text{if } x^2+x_0x+a \text{ is irreducible over $\mathbb{F}_q$ and } D_n(x_0,a)\ne 0$. \newline
\noindent If $q$ is odd, let $\eta$ be the quadratic character of $\mathbb{F}_q$. If $2^r||(q^2-1)$ then $|D_n^{-1}(D_n(x_0,a))|=$ \begin{equation*} \left\{
\begin{array}{cl}
\gcd(n, q-1) & \text{if } \eta(x_0^2-4a)=1 \text{ and } D_n(x_0,a)\ne \pm 2a^{n/2} \\
\gcd(n, q+1) & \text{if } \eta(x_0^2-4a)=-1 \text{ and } D_n(x_0,a)\ne \pm 2a^{n/2} \\
\dfrac{\gcd(n, q-1)}{2} & \text{if } \eta(x_0^2-4a)=1 \text{ and condition C holds} \\
\dfrac{\gcd(n, q+1)}{2} & \text{if } \eta(x_0^2-4a)=-1 \text{ and condition C holds} \\
\dfrac{\gcd(n,q-1)+\gcd(n,q+1)}{2} & \text{otherwise}, \\
\end{array}
\right. \end{equation*} where `condition C' holds if \begin{equation*}
2^t||n \text{ with } 1\leq t\leq r-1, \eta(a)=-1, \text{ and } D_n(x_0,a)=\pm 2a^{n/2} \end{equation*} or \begin{equation*}
2^t||n \text{ with } 1\leq t\leq r-2, \eta(a)=1, \text{ and } D_n(x_0,a)=- 2a^{n/2}. \end{equation*} \end{theorem}
\noindent They also showed an explicit formula for the size of the value set of $D_n(x,a)$, denoted $|V_{D_n(x,a)}|$. We state their result in the odd $q$ case.
\begin{theorem}\label{dicksonTheorem2}
Let $a\in \mathbb{F}_q^*$. If $2^r||(q^2-1)$ and $\eta$ is the quadratic character on $\mathbb{F}_q$ when $q$ is odd, then \begin{equation*}
|V_{D_n(x,a)}| = \frac{q-1}{2\gcd(n,q-1)}+\frac{q+1}{2\gcd(n,q+1)}+\delta \end{equation*} where \begin{equation*} \delta = \left\{
\begin{array}{ll}
1 & \text{if $q$ is odd, $2^{r-1}||n$ and $\eta(a)=-1$} \\
\dfrac{1}{2} & \text{if $q$ is odd, $2^{t}||n$ with $1\leq t\leq r-2$} \\
0 & \text{otherwise}.
\end{array}
\right. \end{equation*} \end{theorem}
As a consequence, for Dickson polynomials
of degree $n$, the value set cardinality $d=|D|$ can be computed in polynomial time in $n\log q$. Note that for a general polynomial $g(x)\in \mathbb{F}_q[x]$ of degree $n$, computing the image size
$|g(\mathbb{F}_q)|$ is a difficult problem, and there is no known polynomial time algorithm in terms of the algebraic input size $n\log q$, see \cite{CHW13} for complexity results and $p$-adic algorithm.
\subsubsection*{Weil's Character Sum Bound}
The following classical case of the Weil bound is well known. We shall give a more general form later.
\begin{theorem} (Weil Bound) Let $f(x) \in \mathbb{F}_q[x]$ be a polynomial of degree $m$, where $(p,m) = 1$ and $\psi$ a non-trivial additive character of $\mathbb{F}_q$. Then \[
\left | \sum_{x \in \mathbb{F}_q} \psi ( f(x)) \right | \le (m-1) \sqrt{q}. \] \end{theorem}
For our purposes it will be important to have a good estimate for certain incomplete character sums, where the sum is not summing over the full field $\mathbb{F}_q$, but over the image set $D$ of another polynomial $g(x)$. This is not available yet for general $g(x)$, but can be proved for monomials and Dickson polynomials. The monomial case is straightforward.
\begin{proposition}\label{MonomialWeil} Let $f(x) \in \mathbb{F}_q[x]$ be a polynomial of degree $m$ such that $p \nmid m$. Let $D = \{x^n : x \in \mathbb{F}_q\}$ where $(n+1)^2 \leq q$. Then \[
\left | \sum_{x \in D} \psi ( f(x)) \right | \le m \sqrt{q}. \] \end{proposition}
\begin{proof}
Without loss of generality, we can assume that $n|(q-1)$. Let $D^\times = \{x^n : x \in \mathbb{F}_q^\times \}$. Using the Weil bound above, \begin{align*}
\left| \sum_{x \in D} \psi(f(x)) \right| &= \left| \psi(f(0)) +\sum_{x \in D^\times} \psi(f(x)) \right| \\
&= \left| \psi(f(0)) + \frac{1}{n} \sum_{x \in \mathbb{F}_q^\times} \psi(f(x^n)) \right| \\
&= \left| \psi(f(0)) + \frac{1}{n} \sum_{x \in \mathbb{F}_q} \psi(f(x^n)) -\frac{1}{n} \psi(f(0))\right| \\
&\leq 1 + \frac{1}{n}(mn-1)\sqrt{q} + \frac{1}{n} \\
&= 1 + m\sqrt{q} - \frac{\sqrt{q}-1}{n}. \end{align*}
If $(n+1)^2 \leq q$ then we conclude $$\left | \sum_{x \in D} \psi ( f(x)) \right | \le m \sqrt{q}.$$ \end{proof}
When $D$ is the image of Dickson polynomials, the corresponding character sum estimate is harder. We need the following version of Weil's bound, which is the case $d=1$ of Theorem 5.6 in \cite{FW}.
\begin{theorem} Let $f_i(t)$ ($1\leq i\leq n$) be polynomials in $\mathbb{F}_q[t]$, let $f_{n+1}(t)$ be a rational function in $\mathbb{F}_q(t)$, let $D_1$ be the degree of the highest square free divisor of $\prod_{i=1}^n f_i(t)$, let \[D_2= \begin{cases}
0 & \deg(f_{n+1})\leq 0 \\
\deg(f_{n+1}) & \deg(f_{n+1})>0,
\end{cases} \] let $D_3$ be the degree of the denominator of $f_{n+1}$, and let $D_4$ be the degree of the highest square free divisor of the denominator of $f_{n+1}(t)$ which is relatively prime to $\prod_{i=1}^n f_i(t)$.
Let $\chi_i:\mathbb{F}_q^* \to \mathbb{C}^*$ $(1\leq i\leq n)$ be multiplicative characters of $\mathbb{F}_q$, and let $\psi=\psi_p \circ \text{Tr}_{\mathbb{F}_q / \mathbb{F}_p}$ for a non-trivial additive character $\psi_p: \mathbb{F}_p \to \mathbb{C}^*$ of $\mathbb{F}_p$. Extend $\chi_i$ to $\mathbb{F}_q$ by setting $\chi_i(0)=0$. Suppose that $f_{n+1}(t)$ is not of the form $r(t)^p-r(t)+c$ in $\mathbb{F}_q(t)$. Then, we have \begin{align*}
&\left| \sum_{a\in \mathbb{F}_{q}, f_{n+1}(a)\ne \infty} \chi_1(f_1(a)) \cdots \chi_n(f_n(a)) \psi(f_{n+1}(a)) \right| \\ &\phantom{\sum}\leq (D_1 + D_2 + D_3 + D_4 - 1)\sqrt{q}, \end{align*} where the sum is taken over those $a\in \mathbb{F}_{q}$ such that $f_{n+1}(a)$ is well-defined. \end{theorem}
\noindent As a consequence, we derive the following character sum bounds.
\begin{cor}\label{weilBounds} Let $\psi_{\text{Tr}}=\psi_p \circ \text{Tr}_{\mathbb{F}_q / \mathbb{F}_p}$ be the canonical additive character, $\psi:\mathbb{F}_q\to \mathbb{C}^*$ any non-trivial additive character of $\mathbb{F}_q$, and $\eta:\mathbb{F}_q^*\to \mathbb{C}^*$ the quadratic character if $q$ is odd. Let $f(x)$ be a polynomial in $\mathbb{F}_q[x]$ of degree $m$ not divisible by $p$.\begin{enumerate} \item For all $q$, we have \begin{equation*}
\left| \sum_{x\in \mathbb{F}_q} \psi(f(D_n(x,a)))\right| \leq (mn-1)\sqrt{q}. \end{equation*} \item If $q$ is odd, then \begin{equation*}
\left| \sum_{x\in \mathbb{F}_q} \eta(x^2-4a)\psi(f(D_n(x,a)))\right| \leq (mn+1)\sqrt{q}. \end{equation*} \item If $q$ is even, then \begin{align*}
\left|\sum_{x\in \mathbb{F}_q^*} \psi_{\text{Tr}}\left( f(D_n(x,a))+a/x^2\right)\right| &= \left| \sum_{x\in \mathbb{F}_q^*} \psi_{\text{Tr}}\left( f(D_n(x,a))+a^{q/2}/x\right) \right| \\ &\leq (mn+1)\sqrt{q}. \end{align*} \end{enumerate}
Note that none of the polynomials in place of $f_{n+1}(x)$ are of the form $r(t)^2-r(t)+c$. This is clear if $n$ is also not divisible by $p$. If $n$ is divisible by $p$, it can be reduced to the case when $n$ is not divisible by $p$ using the identity $D_{pn_1}(x, a) = D_{n_1}(x, a)^p$.
The following lemma is the key character sum estimate we need. The proof follows the method used in \cite{KW16}, where the case $m=1$ is treated.
\end{cor}
\begin{lemma}\label{dicksonWeilEven} Let $f(x)$ be a polynomial in $\mathbb{F}_q[x]$ of degree $m$ not divisible by $p$. Let $D=\{D_n(x,a)\ |\ x\in \mathbb{F}_q\}$, for $a\in \mathbb{F}_q^*$. If $\psi:(\mathbb{F}_q,+)\to \mathbb{C}^*$ is a non-trivial additive character, then the following estimates hold: \begin{equation*}
\left|\sum_{x\in D} \psi(f(x)) \right|\leq (mn+1)\sqrt{q}. \end{equation*} \end{lemma}
\begin{proof} The sum can be rewritten in the following way: \begin{equation*} S_f:= \sum_{y\in D} \psi(f(y)) = \sum_{x\in \mathbb{F}_q} \psi(f(D_n(x,a))) \frac{1}{N_x}, \end{equation*}
where $N_x=|D_n^{-1}(D_n(x,a))|$ is size of the preimage of the value $D_n(x,a)$.
\noindent \textbf{When $q$ is even:}
\noindent By Theorem \ref{dicksonTheorem}, $N_x$ can be quantified. Let $\text{Tr}:\mathbb{F}_q\to \mathbb{F}_2$ denote the absolute trace. Using the fact that $z^2+xz+a$ is reducible over $\mathbb{F}_q$ if and only if $\text{Tr}(a/x^2)=0$, we obtain \begin{align*} S_f &=\sum_{\substack{x\in \mathbb{F}_q^* \\ \text{Tr}(a/x^2)=0}} \frac{1}{\gcd(n, q-1)} \psi(f(D_n(x,a))) \\ &+ \sum_{\substack{x\in \mathbb{F}_q^* \\ \text{Tr}(a/x^2)=1}} \frac{1}{\gcd(n, q+1)} \psi(f(D_n(x,a))) \\ &+ \frac{1}{\gcd(n, q-1)}\psi(f(D_n(0,a))) + O(1), \end{align*} where $O(1)$ is a constant of size at most 1, which we accept by dropping the $D_n(x,a)=0$ case. Denote $\psi_1: \mathbb{F}_2\to \mathbb{C}^*$ as the order two additive character and $\psi_{\text{Tr}}=\psi_1\circ \text{Tr}$, which is an additive character from $\mathbb{F}_q\to \mathbb{C}^*$. Simplifying and rearranging gives \begin{align*} S_f &= \frac{1}{2\gcd(n,q-1)} \sum_{x\in \mathbb{F}_q^*} \psi(f(D_n(x,a)))(1+\psi_{\text{Tr}}(a/x^2)) \\ &\phantom{=}+ \frac{1}{2\gcd(n, q+1)} \sum_{x\in \mathbb{F}_q^*} \psi(f(D_n(x,a)))(1-\psi_{\text{Tr}}(a/x^2)) \\ &\phantom{=}+ \frac{1}{\gcd(n, q-1)}\psi(D_n(0,a)) + O(1) \\ &= \left( \frac{1}{2\gcd(n,q-1)} + \frac{1}{2\gcd(n, q+1)} \right) \sum_{x\in \mathbb{F}_q^*} \psi(f(D_n(x,a))) \\ &\phantom{=}+ \left( \frac{1}{2\gcd(n,q-1)} - \frac{1}{2\gcd(n, q+1)} \right) \sum_{x\in \mathbb{F}_q^*} \psi(f(D_n(x,a)))\psi_{\text{Tr}}(a/x^2) \\ &\phantom{=}+ \frac{1}{\gcd(n, q-1)}\psi(f(D_n(0,a))) + O(1). \end{align*} We add and subtract $\left(\frac{1}{2\gcd(n,q-1)} + \frac{1}{2\gcd(n, q+1)} \right)\psi(D_n(0,a))$ to complete the first sum: \begin{align*} &= \left( \frac{1}{2\gcd(n,q-1)} + \frac{1}{2\gcd(n, q+1)} \right) \sum_{x\in \mathbb{F}_q} \psi(f(D_n(x,a))) \\ &+ \left( \frac{1}{2\gcd(n,q-1)} - \frac{1}{2\gcd(n, q+1)} \right) \sum_{x\in \mathbb{F}_q^*} \psi(f(D_n(x,a)))\psi_{\text{Tr}}(a/x^2) \\ &+ \left( \frac{1}{2\gcd(n,q-1)} -\frac{1}{2\gcd(n, q+1)} \right)\psi(f(D_n(0,a))) + O(1). \end{align*} In order to estimate the sum in second term, take $b\in \mathbb{F}_q^*$ so that $\psi(x)=\psi_{\text{Tr}}(bx)$. Then, \begin{equation*} \sum_{x\in \mathbb{F}_q^*} \psi(f(D_n(x,a)))\psi_{\text{Tr}}(a/x^2) = \sum_{x\in \mathbb{F}_q^*} \psi_{\text{Tr}}(bf(D_n(x,a))+a/x^2). \end{equation*} Applying the bounds in Corollary \ref{weilBounds} with $f$ replaced by $bf$, \begin{align*}
\left|\sum_{y\in D} \psi(f(y))\right| &\leq \left( \frac{1}{2\gcd(n,q-1)} + \frac{1}{2\gcd(n, q+1)} \right)(mn-1)\sqrt{q} \\
&\phantom{=}\phantom{=}+ \left| \frac{1}{2\gcd(n,q-1)} - \frac{1}{2\gcd(n, q+1)} \right| (mn+1)\sqrt{q} + 2 \\ &\leq (mn+1)\sqrt{q}. \end{align*}
\noindent \textbf{When $q$ is odd:}
\noindent We use Theorem \ref{dicksonTheorem} again to calculate $N_x$. Let $\eta$ be the quadratic character of $\mathbb{F}_q$. Then, \begin{align*} S_f &=\sum_{\substack{x\in \mathbb{F}_q \\ \eta(x^2-4a)=1}} \frac{1}{\gcd(n, q-1)} \psi(f(D_n(x,a))) \\ &+ \sum_{\substack{x\in \mathbb{F}_q \\ \eta(x^2-4a)=-1}} \frac{1}{\gcd(n, q+1)} \psi(f(D_n(x,a))) + O(1). \end{align*} The term $O(1)$ is a constant of size at most 2, which we accept by dropping the complicated `condition C' and `otherwise' cases. Simplifying and rearranging gives \begin{align*} &= \frac{1}{2\gcd(n,q-1)} \sum_{x\in \mathbb{F}_q} \psi(f(D_n(x,a)))(1+\eta(x^2-4a)) \\ &\phantom{=}+ \frac{1}{2\gcd(n, q+1)} \sum_{x\in \mathbb{F}_q} \psi(f(D_n(x,a)))(1-\eta(x^2-4a)) + O(1) \\ &= \left( \frac{1}{2\gcd(n,q-1)} + \frac{1}{2\gcd(n, q+1)} \right) \sum_{x\in \mathbb{F}_q} \psi(f(D_n(x,a))) \\ &\phantom{=}+ \left( \frac{1}{2\gcd(n,q-1)} - \frac{1}{2\gcd(n, q+1)} \right) \sum_{x\in \mathbb{F}_q} \psi(f(D_n(x,a)))\eta(x^2-4a) + O(1). \end{align*} Again applying the bounds in Corollary \ref{weilBounds}, \begin{align*}
\left|\sum_{x\in D} \psi(f(x)) \right|&\leq \left( \frac{1}{2\gcd(n,q-1)} + \frac{1}{2\gcd(n, q+1)} \right)(mn-1)\sqrt{q} \\ &\phantom{=}\phantom{=}+ \left| \frac{1}{2\gcd(n,q-1)} - \frac{1}{2\gcd(n, q+1)} \right|(mn+1)\sqrt{q} + 2\\ &\leq (mn+1)\sqrt{q}, \end{align*} which was to be shown. \end{proof}
\section{$k$-MSS($m$)}
We are now ready to consider the $m$-th moment $k$-subset sum problem, called $k$-MSS($m$) in short. Let $m$ be a fixed positive integer, and $g(x) \in \mathbb{F}_q [x]$ a polynomial of degree $n$ with $1\leq n \leq q-1$. Let $D = g ( \mathbb{F}_q)$
and $\boldsymbol{b} = (b_1, b_2, \ldots, b_m) \in \mathbb{F}_q^m$. Since we are working in characteristic $p$, we have that $$(x_1^i + \ldots + x_k^i)^p = x_1^{ip} + \ldots + x_k^{ip}.$$ Thus if $b_i^p \neq b_{ip}$ for some $ip\leq m$, there will be no solutions for $k$-MSS($m$). We may and will assume without loss of generality that $b_i^p = b_{ip}$ for all $ip\leq m$ in the remainder of this paper. Under this assumption, the $j$-th power equation in the $k$-MSS($m$) can and will be dropped for all $j$ divisible by $p$. We introduce the moment subset sum problem over subsets of size $k$ with the value
\begin{align}
N_k(D,\boldsymbol{b},m)=\# \left\{ S \subseteq D : |S| = k, \sum_{y \in S} y^j = b_j , 1 \le j \le m, p \nmid j \right\}. \end{align} Thus, from now on, the index $j$ is not divisible by $p$.
Determining whether $N_k(D,\boldsymbol{b},m) >0$ for given $\{D, \boldsymbol{b}\}$ is the decision version of the $k$-MSS($m$) problem. As indicated before, we shall use the algebraic input size $n\log q$.
A closely related number is the following integer
\begin{align*} M_k (D,\boldsymbol{b},m) = \# \{ (x_1, \dots, x_k) \in D^k : \sum_{i=1}^k x_i^{j} = b_j, \\ x_{i_1} \ne x_{i_2}, \forall \,1 \le i_1 < i_2 \le k, \ p \nmid j \}. \end{align*} It is clear that $M_k (D,\boldsymbol{b},m) = k! N_k (D,\boldsymbol{b},m)$. We deduce \begin{theorem} $M_k(D,\boldsymbol{b},m) > 0$ if and only if $N_k(D,\boldsymbol{b},m) > 0$. \end{theorem} Our problem is then reduced to deciding if $M_k (D,\boldsymbol{b},m)>0$.
We can reduce this further by assuming from duality that $k \le \frac{|D|}{2}$. The strategy to solve this new problem is to combine all established strategy for the original subset sum problem and apply the character sum estimate from the previous section. We shall divide $k$ into three different ranges (constant size, medium size, and large size) and use different methods for each range. The main idea is to use algorithms to solve boundary cases of parameters and to use mathematics to prove that there is a solution when the parameters are in the interior.
If $n>q^{\epsilon}$ for constant $\epsilon>0$, then $q$ is polynomial in $n\log q$, we can list all elements of $D$ and use the dynamic programming algorithm to solve the moment subset sum problem in polynomial time. In the rest of the paper, we can and will assume that $n < q^{\epsilon}$ for whatever positive constant $\epsilon$ we like.
\subsection*{$k$-MSS($m$) for constant size $k$} The main result that we depend on in this case is due to Kayal's solvability algorithm for polynomial systems over $\mathbb{F}_q$ \cite{K05}, which we summarize in this context below. Let $f_1, \dots, f_m \in \mathbb{F}_q[x_1, \dots, x_n]$, where $d$ is the maximum degree of all the polynomials. Let $X = V(f_1, \dots, f_m)$ be the vanishing locus of the polynomials. Then the result of Kayal \cite{K05} states the following. \begin{theorem} The decision problem of $\# X(\mathbb{F}_q) > 0$ can be solved in time $\left(d^{n^{cn}} m \log q\right)^{O(1)}$ for some constant $c>0$. \end{theorem} Most of the conditions in our $k$-MSS($m$) are polynomial equations, with the exception of the condition that the individual elements be distinct. However, we can easily consider this as a polynomial equation at the cost of additional variables. Recall that $D =\{g(x) : x \in \mathbb{F}_q\, , g(x) \in \mathbb{F}_q [x]\}$ for a polynomial $g$ such that $\text{deg}(g)=n$. For the context of the $k$-MSS($m$) problem, we are deciding if the variety determined by the vanishing locus of $$f_j (x_1, \dots, x_k ):=\left(\sum_{i=1}^k g(x_i)^j\right) -b_j, \ 1 \le j \le m, \ p\nmid j$$ and the additional polynomial
\[ \left( \prod_{i_1 \ne i_2} (g(x_{i_1}) - g(x_{i_2})) \right)x_{k+1} - 1 \] have any $\mathbb{F}_q$-rational points. Each $f_j$ has degree at most $mn$ while the latter polynomial has degree $n \binom{k}{2} + 1$.
Now, we assume $k \le 3m+1$. Then, $n\binom{k}{2} + 1 \leq 9nm^2$ and so all the polynomials have degrees bounded by $9nm^2$. Kayal's theorem then states that the decision problem can be solved in time which is bounded by a polynomial in \[
(9nm^2)^{(k+1)^{O((k+1))}}\log q = (9nm^2)^{(3m+2)^{O((3m+2))}}\log q. \] This is $(n \log q)^{O(1)}$ if $m$ is a constant. Thus, we have proved the following \begin{theorem}\label{ThmSmallK} Let $D =\{g(x) : x \in \mathbb{F}_q\}$, where $g(x) \in \mathbb{F}_q [x]$ is any polynomial of degree $n$. Let $m$ be a fixed positive integer. Assume $k \leq 3m+1$. Then $k$-MSS(m) can be solved in time $(n \log q)^{O(1)}$. \end{theorem} The condition $k\leq 3m+1$ is all we need. It can be replaced by any bound $k\leq C$, where $C$ is a positive constant.
\subsection*{$k$-MSS($m$) for medium $k$}
We now consider the moment $k$-subset sum problem for medium-sized values of $k$.
Fix $m \in \mathbb{N}$ and $\boldsymbol{b} = (b_1, \ldots, b_m) \in \mathbb{F}_q^m$. Let $m_p = |\{j : 1 \leq j \leq m, p \nmid j\}| = m - \lfloor \frac{m}{p} \rfloor$. Recall \begin{align*}
M_k(D,\boldsymbol{b},m) = |\{(x_1, \ldots, x_k ) \in D^k :& \sum_{i=1}^k x_i^{j} - b_j =0, \\
&x_{i_1} \neq x_{i_2} \text{ for } i_1 \neq i_2, \\
&1 \leq j \leq m, p \nmid j\}|. \end{align*} and \begin{align*} M_k(D,\boldsymbol{b},m) = k! \cdot N_k(D,\boldsymbol{b},m), \end{align*}
where $$N_k(D,\boldsymbol{b},m) = |\{S \subseteq D : |S| = k, \sum\limits_{y \in S} y^j = b_j, 1 \leq j \leq m, p \nmid j\}|.$$ We wish to decide when $M_k(D,\boldsymbol{b},m) > 0$. The following theorem solves this problem in the medium $k$ case if certain character sum estimate is satisfied.
\begin{theorem}\label{med_k_thm} Let $D = g(\mathbb{F}_q)$ where $g \in \mathbb{F}_q[x]$ with deg$(g) = n$. Let $\psi$ be a non-trivial additive character of $\mathbb{F}_q$. Assume for all $f \in \mathbb{F}_q[x]$ of degree at most $m$ with $p \nmid \text{deg}(f)$, we have
$$\left|\sum_{x \in D} \psi(f(x))\right| \leq (mn+1)\sqrt{q}.$$ Then $M_k(D,\boldsymbol{b},m) > 0$ if $2n(mn+1) < q^\frac{1}{6}$ and $3m_p+1 < k < q^\frac{5}{12}$. \end{theorem}
The first condition $2n(mn+1) < q^\frac{1}{6}$ is already satisfied, since we assumed that $n<q^{\epsilon}$ and $m$ is a constant. The second condition $3m_p+1 < k < q^\frac{5}{12}$ gives the medium range of $k$.
Towards this goal, we define
$$R= |\{(x_1, \ldots, x_k ) \in D^k : \sum_{i=1}^k x_i^{j} - b_j = 0, 1 \leq j \leq m, p \nmid j\}|.$$
We say that $\boldsymbol{x}=(x_1, \ldots, x_k )\in \mathbb{F}_q^k$ \textit{is a solution} if $\boldsymbol{x}$ satisfies the conditions of $R$. Note that $R$ counts solutions allowing for those with repeated entries, while $M_k(D,\boldsymbol{b},m)$ strictly counts solutions with distinct entries. We define a new number to compute the size of $R$ with the added condition that the first two entries of $\boldsymbol{x}$ are equal. Let
$$R_{12} = |\{(x_1, \ldots, x_k ) \in D^k : 2x_2^{j} +\sum_{i=3}^k x_i^{j} - b_j = 0, 1 \leq j \leq m, p \nmid j\}|.$$
Then the Brun sieve tells us that $$M_k(D,\boldsymbol{b},m) \geq R - \sum_{1 \leq i_1 < i_2 \leq k} R_{i_1 i_2} = R - \binom{k}{2}R_{12}.$$ In order to rewrite $R$ and $R_{12}$ and obtain bounds for them we use the theory of characters.
Let $\psi$ be a non-trivial additive character of $\mathbb{F}_q$. Recall that we have the following summation: \[ \sum_{c \in \mathbb{F}_q} \psi(cx) = \left\{
\begin{array}{ll}
q & \quad \text{if } x = 0 \\[1em]
0 & \quad \text{if } x \neq 0
\end{array}
\right. \] We would like to take advantage of this character sum equation and have it evaluate solutions positively and evaluate non-solutions to zero. Thus we have the following identity. \[ \prod_{j=1, p\nmid j}^m\left(\sum_{c \in \mathbb{F}_q} \psi(c(\sum_{i=1}^k x_i^j-b_j))\right) = \left\{
\begin{array}{ll}
q^{m_p} & \quad \text{if } \textbf{$x$} \text{ is a solution}\\[1em]
0 & \quad \text{if } \textbf{$x$} \text{ is not a solution}
\end{array}
\right. \] With this in mind, we can rewrite $R$ as below \begin{align*}
R &= \frac{1}{q^{m_p}}\sum_{x \in D^k}\prod_{j=1, p\nmid j}^m\sum_{c \in \mathbb{F}_q} \psi(c(\sum_{i=1}^k x_i^j-b_j))\\
&= \frac{1}{q^{m_p}} \sum_{\boldsymbol{x} \in D^k} \sum_{\boldsymbol{c} \in \mathbb{F}_q^{m_p}} \prod_{j=1, p\nmid j}^m \psi(c_j (\sum_{i=1}^k x_i^j -b_j)) \\
&= \frac{1}{q^{m_p}} \sum_{\boldsymbol{c} \in \mathbb{F}_q^{m_p}}\sum_{\boldsymbol{x} \in D^k} \prod_{j=1, p\nmid j}^m \psi(c_j (\sum_{i=1}^k x_i^j -b_j)) \\
&= \frac{1}{q^{m_p}} \sum_{\boldsymbol{c} \in \mathbb{F}_q^{m_p}}\sum_{\boldsymbol{x} \in D^k} \psi\left(\sum_{j=1, p\nmid j}^m c_j \left(\sum_{i=1}^k x_i^j -b_j\right)\right) \end{align*} By separating the contribution of the trivial term, we obtain the following. \begin{align*}
R &= \frac{1}{q^{m_p}} \sum_{\boldsymbol{x} \in D^k} \psi(0) + \frac{1}{q^{m_p}}\sum_{0 \neq \boldsymbol{c} \in \mathbb{F}_q^{m_p}} \sum_{\boldsymbol{x} \in D^k} \psi\left(\sum_{j=1, p\nmid j}^m c_j \left(\sum_{i=1}^k x_i^{j}-b_j\right)\right) \\
&= \frac{|D|^k}{q^{m_p}} + \frac{1}{q^{m_p}} \sum_{0 \neq \boldsymbol{c} \in \mathbb{F}_q^{m_p}} S_c, \end{align*} where $$S_c = \sum_{\boldsymbol{x} \in D^k} \psi\left(\sum_{j=1, p\nmid j}^m c_j \left(\sum_{i=1}^k x_i^{j}-b_j\right)\right).$$ Define $$f(x) = \sum\limits_{j=1, p\nmid j}^m c_j x^{j} \in \mathbb{F}_q[x].$$ Note that the degree of $f$ is not divisible by $p$ and at most $m$ if $c\not=0$. We now want to find an upper bound for $S_c$. Notice that \begin{align*}
\psi\left(\sum_{j=1, p\nmid j}^m c_j \left(\sum_{i=1}^k x_i^{j}-b_j\right)\right) &= \psi\left(\sum_{i=1}^k\sum_{j=1, p\nmid j}^m c_j x_i^{j} - \sum_{j=1, p\nmid j}^m c_jb_j\right) \\
&= \psi(f(x_1)) \cdots \psi(f(x_k))\psi(-\sum_{j=1, p\nmid j}^m c_jb_j) \\
&= A \cdot \prod_{i=1}^k \psi(f(x_i)). \end{align*}
Here, $A = \psi(-\sum\limits_{j=1, p\nmid j}^m c_jb_j)$ and so $|A| = \prod\limits_{j=1, p\nmid j}^m|\psi(-c_jb_j)|=1$. Thus $$|S_c| = \left|\sum\limits_{\boldsymbol{x} \in D^k}\prod\limits_{i=1}^k \psi(f(x_i))\right| = \left(\left|\sum\limits_{x \in D} \psi(f(x))\right|\right)^k.$$
By our assumptions, $|S_c| \leq (mn+1)^k(\sqrt{q})^k$. It follows that
$$\left| R - \frac{|D|^k}{q^{m_p}} \right| = \frac{1}{q^{m_p}}\sum_{0 \neq \boldsymbol{c} \in \mathbb{F}_q^{m_p}} |S_c| \leq \frac{q^{m_p}-1}{q^{m_p}}(mn+1)^k q^\frac{k}{2} <(mn+1)^k q^\frac{k}{2}.$$
{\bf Remark}. Igor Shparlinski kindly informed us that the average trick in \cite{Sh15} can be used to improve the above coefficient $(mn+1)^k$ to $(mn+1)^{k-2}$.
The idea is to apply the character sum estimate only to the first $(k-2)$-th power in $|S_c|$, and then compute the remaining quadratic moment over $c$, resulting in a saving of the factor $(mn+1)^2$. This type of improvement is theoretically interesting, but would not significantly improve the lower bound condition $3m_p+1 <k$ in our theorem, which is enough for our algorithmic purpose of this paper.
Now we can rewrite $R_{12}$ in a similar way. \begin{align*}
R_{12} &= \frac{1}{q^{m_p}} \sum_{\boldsymbol{x} \in D^{k-1}} \prod_{j=1, p\nmid j}^m \sum_{c \in \mathbb{F}_q} \psi(c(2x_1^j + \sum_{i=3}^k x_i^j-b_j)) \\
&= \frac{1}{q^{m_p}} \sum_{\boldsymbol{x} \in D^{k-1}} \sum_{\boldsymbol{c} \in \mathbb{F}_q^{m_p}} \prod_{j=1, p\nmid j}^m \psi(c_j (2x_1^j + \sum_{i=3}^k x_i^j -b_j)) \\
&= \frac{1}{q^{m_p}} \sum_{\boldsymbol{c} \in \mathbb{F}_q^{m_p}}\sum_{\boldsymbol{x} \in D^{k-1}} \prod_{j=1, p\nmid j}^m \psi(c_j (2x_1^j + \sum_{i=3}^k x_i^j -b_j)) \\
&= \frac{1}{q^{m_p}} \sum_{\boldsymbol{c} \in \mathbb{F}_q^{m_p}}\sum_{\boldsymbol{x} \in D^{k-1}} \psi\left(\sum_{j=1, p\nmid j}^m c_j \left(2x_1^j + \sum_{i=3}^k x_i^j -b_j\right)\right) \end{align*} By separating the contribution of the trivial character, we obtain the following. \begin{align*}
R_{12} &= \frac{1}{q^{m_p}} \sum_{\boldsymbol{x} \in D^{k-1}} \psi(0) + \frac{1}{q^{m_p}}\sum_{0 \neq \boldsymbol{c} \in \mathbb{F}_q^{m_p}} \sum_{\boldsymbol{x} \in D^{k-1}} \psi\left(\sum_{j=1, p\nmid j}^m c_j \left(2x_1^j + \sum_{i=3}^k x_i^j -b_j\right)\right) \\
&= \frac{|D|^{k-1}}{q^{m_p}} + \frac{1}{q^{m_p}} \sum_{0 \neq \boldsymbol{c} \in \mathbb{F}_q^{m_p}} S_c^{12}, \end{align*} where $$S_c^{12} = \sum\limits_{\boldsymbol{x} \in D^{k-1}}\psi(\sum\limits_{j=1, p\nmid j}^m c_j (2x_1^{j} + \sum\limits_{i=3}^{k} x_i^{j}-b_j)).$$ By a similar manipulation in the previous case, \begin{align*} S_c^{12} &= \sum\limits_{\boldsymbol{x} \in D^{k-1}}\psi(2f(x_1))\psi(f(x_3))\cdots\psi(f(x_{k}))\psi(-\sum\limits_{j=1, p\nmid j}^m c_jb_j) \\ &= A\sum\limits_{\boldsymbol{x} \in D^{k-1}}\psi(2f(x_1))\prod_{i=3}^{k}\psi(f(x_i)). \end{align*} By a rearrangement, we see that \begin{align*}
\left|S_c^{12}\right| &= \left| \sum_{\boldsymbol{x} \in D^{k-1}} \psi(2f(x_1))(\prod_{i=3}^{k} \psi(f(x_i)))\right| \\
&= \left| \left(\sum_{x \in D} \psi(2f(x))\right)\left(\sum_{x \in D} \psi(f(x))\right)^{k-2}\right| \end{align*} By our assumptions, if $p>2$ (and thus $2\not=0$), \begin{align*}
\left|S_c^{12}\right| &\leq (mn+1)\sqrt{q}(mn+1)^{k-2}(\sqrt{q})^{k-2} \\
&= (mn+1)^{k-1}q^\frac{k-1}{2}. \end{align*} The case $p=2$ can be handled in a similar way, and one get the alternate bound
$$\left|S_c^{12}\right| \leq
|D|(mn+1)^{k-2}q^\frac{k-2}{2}.$$ We assume that $p>2$ for simplicity. Now we have that \begin{align*}
\left|R_{12} - \frac{|D|^{k-1}}{q^{m_p}}\right| &= \frac{1}{q^{m_p}} \left|\sum_{\textbf{0} \neq \textbf{c} \in \mathbb{F}_q^{m_p}} S_c^{12}\right| \\
&\leq \frac{1}{q^{m_p}}\sum_{\textbf{0} \neq \textbf{c} \in \mathbb{F}_q^{m_p}} (mn+1)^{k-1}q^\frac{k-1}{2} \\
&= \frac{q^{m_p}-1}{q^{m_p}} (mn+1)^{k-1}q^\frac{k-1}{2} \\
&< (mn+1)^{k-1}q^\frac{k-1}{2}. \end{align*} Since we have the following two inequalities,
$$\left|R_{12} - \frac{|D|^{k-1}}{q^{m_p}}\right| < (mn+1)^{k-1}q^\frac{k-1}{2}$$
$$\left|R - \frac{|D|^{k}}{q^{m_p}}\right| < (mn+1)^k q^\frac{k}{2}$$ we see that
$$\frac{|D|^k}{q^{m_p}}-(mn+1)^k q^\frac{k}{2} < R, \text{ and }$$
$$R_{12} < \frac{|D|^{k-1}}{q^{m_p}} + (mn+1)^{k-1}q^\frac{k-1}{2}.$$ Then \begin{align*}
R &- \binom{k}{2} R_{12} > \frac{|D|^{k}}{q^{m_p}}-(mn+1)^k q^\frac{k}{2} - \binom{k}{2}\left(\frac{|D|^{k-1}}{q^{m_p}} + (mn+1)^{k-1}q^\frac{k-1}{2} \right) \\
&= |D|^{k-1}\frac{1}{q^{m_p}}\left(|D|- \binom{k}{2}\right)-(mn+1)^{k-1}q^\frac{k-1}{2}\left((mn+1)\sqrt{q} + \binom{k}{2}\right) \\
&= \frac{1}{q^{m_p}}\left(|D|^{k-1}\left(|D|-\binom{k}{2}\right)\right) - (mn+1)^{k-1}q^\frac{k-1}{2}\left((mn+1)\sqrt{q} + \binom{k}{2}\right). \end{align*}
We wish to show that $R- \binom{k}{2} R_{12}$ is positive and thus we need to show that $$|D|^{k-1}\left(|D|-\binom{k}{2}\right) \geq q^{m_p}(mn+1)^{k-1}q^\frac{k-1}{2}\left((mn+1)\sqrt{q} + \binom{k}{2}\right).$$
However since $\text{deg}(g) = n$ we know that $|D| \geq \frac{q}{n}$. Thus it is enough to show that $$\left(\frac{q}{n}\right)^{k-1}\left(\frac{q}{n}-\binom{k}{2}\right) \geq q^{m_p+\frac{k-1}{2}}(mn+1)^{k-1}\left((mn+1)\sqrt{q} + \binom{k}{2}\right).$$ Towards this goal, we utilize our assumptions that $2n(mn+1) < q^\frac{1}{6}$ and $3m_p+1 < k < q^\frac{5}{12}$. It is enough to prove $$\left(\frac{q}{n}\right)^{k-1} \geq q^{m_p+\frac{k-1}{2}}(mn+1)^{k-1}, \ \left(\frac{q}{n}-\binom{k}{2}\right) \geq \left((mn+1)\sqrt{q} + \binom{k}{2}\right). $$ For the first inequality, we have \begin{align*}
\left(\frac{q}{n}\right)^{k-1}>q^{m_p+\frac{k-1}{2}}(mn+1)^{k-1}
&\iff q^{k-1-m_p-\frac{k-1}{2}} > (mn+1)^{k-1}n^{k-1} \\
&\iff q^{\frac{k-1}{2}-m_p} > (mn+1)^{k-1}n^{k-1}. \\ \end{align*} Since $2n(mn+1) < q^\frac{1}{6}$,
the right side is bounded by $$(mn+1)^{k-1}n^{k-1}<(n(mn+1))^{k-1} < q^\frac{k-1}{6}.$$ Our problem is now reduced to showing that $q^{\frac{k-1}{6}+m_p} < q^{\frac{k-1}{2}}$. Namely, $$m_p < \frac{k-1}{2}-\frac{k-1}{6} =\frac{k-1}{3}.$$ This is satisfied since $3m_p+1 < k$. Thus we have shown that \begin{equation}\label{med_proof_eq_1}
\left(\frac{q}{n}\right)^{k-1}>q^{m_p+\frac{k-1}{2}}(mn+1)^{k-1}. \end{equation}
For the second inequality, we need to show that $n(mn+1)\sqrt{q} + 2n\binom{k}{2} < q$. Since $k < q^\frac{5}{12}$ and $2n(mn+1) < q^\frac{1}{6}$, we know that $k^2n < q^\frac{5}{6}q^\frac{1}{6}/2 = q/2$.
We deduce that \begin{equation}\label{med_proof_eq_2}
n(mn+1)\sqrt{q} + 2n\binom{k}{2} < \frac{q^{1/6 +1/2}}{2} + \frac{q}{2} < q. \end{equation} The theorem is proved.
\begin{cor} Let $D = \{x^d : x \in \mathbb{F}_q\}$ or $D = \{D_n(x,a) : x \in \mathbb{F}_q\}$ for $a \in \mathbb{F}_q^\times$. Then $M_k(D,b,m) > 0$ if $2n(mn+1) < q^\frac{1}{6}$ and $3m_p+1 < k < q^\frac{5}{12}$. \end{cor}
Let $\psi$ be a non-trivial additive character of $\mathbb{F}_q$. We have shown that all $f \in \mathbb{F}_q[x]$ of degree at most $m$ with $p \nmid \text{deg}(f)$,
$$\left|\sum_{x \in D} \psi(f(x))\right| \leq m \sqrt{q}$$ if $D = \{x^d : x \in \mathbb{F}_q\}$, and
$$|\sum_{x \in D} \psi(f(x))| \leq (mn+1) \sqrt{q}$$ if $D = \{D_n(x,a) : x \in \mathbb{F}_q\}$. Since $m\sqrt{q} \leq (mn+1)\sqrt{q}$, the character sum condition in Theorem \ref{med_k_thm} is satisfied. The medium case is proved.
\subsection*{$k$-MSS($m$) for large $k$} Following established procedures, we use the Li-Wan sieve \cite{LW10} to analyze large values of $k$. This method has been used several times \cite{ZW12, KW16, LW10, LW18, LW19,WN18} and is now standard. So, we will only give an outline and indicate the differences. We begin by discussing the relevant notation and concepts that we will apply in our context. In this section, we assume that $D$ is the image of a monomial or Dickson polynomial of degree $n$. The relevant character sum estimate is then true.
We use the notation $S_k$ to denote the symmetric group on $k$ letters. For a permutation $\tau \in S_k$, its disjoint cycle decomposition is written as \[ \tau = (a_1 a_2 \cdots a_{m_1})(a_{m_1+1} \cdots a_{m_2}) \cdots (a_{m_{k-1}+1} \cdots a_{m_k}). \] We shall refer to $\tau$ interchangeably with its disjoint cycle decomposition, which we fix beforehand.
Denote by $\overline{X} = \{(x_1,\dots,x_k) \in D^k : x_i \ne x_j, \forall i \ne j\}$. For the sake of brevity, we will denote $k$-tuples from such products by $x = (x_1, \dots, x_k)$ when there is no risk of confusion. Let $\psi$ be a fixed non-trivial additive character of $\mathbb{F}_q$. Recall from earlier sections that we are interested in \[ h_c(x_1,\dots,x_k) = \psi \left( \sum_{j=1,p\nmid j}^m c_j \left(\sum_{i=1}^k x_i^{j}-b_j\right)\right), \] where $c$ is not the zero vector. Now define $$F(c) = \sum\limits_{x \in \overline{X}} h_c(x_1,\dots,x_k), \ F_\tau(c) = \sum\limits_{x \in X_\tau} h_c(x_1, \dots x_k),$$ where $X_\tau$ consists of tuples in $\overline{X}$ such that \begin{align*} x_{a_1} = \cdots = x_{a_{m_1}}, x_{m_1+1} = \ldots = x_{m_2}, \ldots, x_{m_{k-1}+1} = \ldots = x_{m_k} \end{align*} and so on. Now, let's think of $\tau$ as having $e_1$ cycles of length $1$, $e_2$ cycles of length $2$, and so on, up until $e_k$ cycles of length $k$. Note that $\sum\limits_{i=1}^k ie_i = k$. This allows us to express $F_\tau(c)$ as: \begin{align*} F_\tau(c) &= \sum_{x \in X_\tau} \psi \left( \sum_{j=1,p\nmid j}^m c_j \left( \sum_{i=1}^k i (x_{i1}^{j} + \cdots + x_{ie_i}^{j}) - b_j \right)\right) \\ &= \sum_{\substack{x_{il} \in D \\ 1 \le i \le k\\ 1 \le l \le e_i}} \psi \left( \sum_{j=1,p\nmid j}^m c_j \left( \sum_{i=1}^k \sum_{l=1}^{e_i} i x_{il}^{j} \right) \right)\psi ( \sum_{j=1,p\nmid j}^m -c_j b_j) \\ &= \sum_{\substack{x_{il} \in D \\ 1 \le i \le k\\ 1 \le l \le e_i}} \prod_{i=1}^k \psi^i \left( \sum_{p\nmid j, l} c_j x_{il}^{j} \right) \psi ( \sum_{j=1,p\nmid j}^m -c_j b_j). \end{align*} Let's consider the inner sum. \[ \sum_{x_{il}\in D}\psi^i(\sum_{p\nmid j} c_j x_{il}^{j}) = \sum_{x \in D} \psi ^i (f(x)) \] where $f(x) = \sum_{j=1,p\nmid j}^m c_j x^{j}$. Hence, if the $c_j$'s are not all zero, we have \[
\left |\sum_{x\in D} \psi(\sum_{j=1,p\nmid j}^m c_j x^{j}) \right | \le (mn+1) \sqrt{q}. \] Now the order of $\psi$ is $p$ so the order of $\psi^i$ is $\frac{p}{(i,p)}$, which is $p$ unless $p \mid i$, in which case it is $1$. Therefore, \begin{align*}
|F_\tau(c)| &= \left | \prod_{i=1}^k \left( \sum_{x \in D} \psi^i \left( \sum_{j=1,p\nmid j}^m c_j x^{j} \right) \right) ^{e_i} \psi(\sum_{j=1,p\nmid j}^m -c_jb_j) \right| \\ & \le \prod_{\substack{i \\ 1 \le i \le k\\ p \nmid i}} ((mn+1) \sqrt{q})^{e_i} \cdot \prod_{\substack{i \\ 1 \le i \le k\\
p \mid i}} |D|^{e_i}. \end{align*}
The Li-Wan sieve says that \[ F(c) = \sum_{\sum ie_i = k} (-1)^{k - \sum e_i} N(e_1, \dots, e_k) F_{e_1, \dots, e_k}(c), \] where $N(e_1,..., e_k)$ denote the number of permutations in $S_k$ with cycle type $(e_1,..., e_k)$, and $F_{e_1, \dots, e_k}(c)$ denotes $F_{\tau}(c)$ for any $\tau$ of cycle type $(e_1,..., e_k)$. Using the above estimates and Lemma 2.1 in \cite{WN18}, one obtains \begin{align*}
|F(c)| &\le \sum_{\sum ie_i = k} N(e_1, \dots, e_k) \prod_{(i,p) =1} ((mn+1) \sqrt{q})^{e_i} \cdot \prod_{p \mid i} |D|^{e_i} \\
& \le \left ((mn+1) \sqrt{q} + k + \frac{|(mn+1)\sqrt{q} - |D||}{p} -1 \right )_k \end{align*} where we define $(x)_k := x(x-1) \cdots (x-k+1)$.
This concludes our discussion of the Li-Wan sieve and the appropriate adaptation to our context. We now return to the framework in the previous sections, with notations as before. Let's see how the above Li-Wan helps. Recall \begin{align*} M_k(D,b,m) &= \sum_{x \in \textbf{X}} \frac{1}{q^{m_p}} \sum_\psi \sum_{c_j \in \mathbf{F}_q} \psi \left( \sum_{j=1,p\nmid j}^m c_j \left( \sum_{i=1}^k x_i^{j} - b_j \right) \right) \\
&= \frac{1}{q^{m_p}} (|D|)_k + \sum_{(\cdots c_j \cdots) \ne 0} \frac{1}{q^{m_p}}F(c). \end{align*} Therefore, \begin{align}
\left | M_k(D,b,m) - \frac{1}{q^{m_p}} (|D|)_k \right | &< \left((mn+1) \sqrt{q} + k + \frac{\left |(mn+1) \sqrt{q} - |D| \right|}{p} - 1\right)_k \\
& \le \left(0.013 |D| + k + \frac{|D|}{p} \right)_k. \end{align} This estimate is the analogue of equation (2.3) in \cite{WN18}, resulting from assuming that \[
(mn+1)\sqrt{q} \le 0.013 |D|. \]
If further, $6m_p \ln q \leq k \leq \frac{|D|}{2}$, the same argument as in the proof of Theorem 2.3 in \cite{WN18} shows that $M_k(D, b, m)>0$. We obtain
\begin{theorem}
Let $D$ be the image of a monomial of Dickson polynomial of degree $n$ .
Assume that $p>2$, $(mn+1)\sqrt{q} \leq 0.013|D|$, and $6m_p \ln q \leq k \leq \frac{|D|}{2}$. Then, $M_k(D,b,m)>0$. \end{theorem}
Note that if $p=2$, the same proof works, but only for $k$ in the shorter range
$6m_p \ln q \leq k \leq \frac{(1-\epsilon)|D|}{2}$. That is, $k$ cannot reach
all the way to $|D|/2$ if $p=2$.
Since $|D|\geq q/n$, the condition
$(mn+1)\sqrt{q} \leq 0.013|D|$ is satisfied if $n(mn+1)\sqrt{q} \leq 0.013q$, which is certainly true since $m$ is fixed and $n<q^{\epsilon}$.
\begin{section}{Case $p=2$} Finally, we examine the $k$-MSS($m)$ over finite fields of characteristic 2. The result of Kayal used for $k$-MSS($m$) for constant $k$ and our proof for medium-sized $k$ still hold in fields of characteristic 2. Thus Theorem \ref{ThmSmallK} and Theorem \ref{med_k_thm} hold for $q=p^s$ for all $p$.
To analyze the case $p=2$ for large $k$, we rely on recent work by Choe and Choe \cite{CC19} which examines the subset sum problem over finite fields of characteristic 2 . We adjust the definitions of this work to fit the higher moment subset sum problem over $D$ which are images of monomials or Dickson polynomials. Note that $p=2$ in this section.
We will prove an analogue of Theorem 2.3 in \cite{CC19}. Let $D \subseteq \mathbb{F}_q$, $k \leq |D|/2$, and $f(x) = \sum\limits_{j=1, p\nmid j}^m c_j x^{j}$, for $c_j \in \mathbb{F}_q$. For a nontrivial additive character $\psi$ of $\mathbb{F}_q$, define
\begin{align*}
S_D(k,\psi,f)=\sum_{\substack{x_{i} \in D \\ x_i \ \textnormal{distinct}}}\psi(f(x_1)+f(x_2)+ \ldots +f(x_k)). \end{align*}
Although $S_D(k,\psi,f)$ sums over distinct $x_i$, there is no assumption that the $f(x_i)$ are distinct. Over finite fields of characteristic 2, however, if $x_i=x_j$, then $f(x_i)=f(x_j)$, and the sum $f(x_i)+f(x_j)$ is equivalent to $2f(x_i)=0$. It follows that
\begin{align*}
S_D(2,\psi,f)&=\sum_{\substack{x_{1},x_{2} \in D \\ x_1 \neq x_2}}\psi(f(x_1)+f(x_2)) \\
&= (\sum_{x \in D}\psi(f(x)))^2 -|D|. \end{align*} By induction , one derives the following recursive formula for $S_D(k,\psi,f)$ for all $k>1$, which is the analogue of Lemma 2.1 \cite{CC19}.
\renewcommand\labelitemi{$\cdot$}
\begin{lemma} Let $D$ be a subset of $\mathbb{F}_q$ with more than 3 elements and $\psi$ be a nontrivial additive character of $\mathbb{F}_q$. Then \begin{itemize}
\item $S_D(1,\psi,f)=\sum\limits_{x\in D}\psi(f(x))$,
\item $S_D(2,\psi,f)= S_D(1,\psi,f)^2-|D|$, and
\item $S_D(k,\psi,f)= S_D(1,\psi,f)S_D(k-1,\psi,f)-(|D|-k+2)(k-1)S_D(k-2,\psi,f)$, where $3 \leq k \leq |D|$. \end{itemize} \end{lemma} This lemma can be applied to prove analogue of Lemma 2.2 \cite{CC19}. The statement is as follows. \begin{lemma} Let $D$ be a subset of $\mathbb{F}_q$ with more than 4 elements and $\psi$ be a nontrivial additive character of $\mathbb{F}_q$. If \begin{align*}
\left|\sum_{x \in D}\psi(f(x))\right| \leq \frac{1}{16}|D|, \end{align*} then \begin{align*}
|S_D(k,\psi,f)| < \left(\frac{9}{16}|D|\right)^k, \textnormal{ for all } k \leq \frac{|D|}{2}. \end{align*} \end{lemma}
From Proposition \ref{MonomialWeil} and Lemma \ref{dicksonWeilEven}, it follows that when $D$ is the image of a polynomial of degree $n$ such that the value set character sum estimate satisfies \begin{align*}
\left|\sum_{x \in D}\psi(f(x))\right| < (mn+1)\sqrt{q}, \end{align*} then the condition $n(mn+1) < \frac{1}{16}\sqrt{q}$ implies that \begin{align*}
\left|\sum_{x \in D}\psi(f(x))\right| < (mn+1)\sqrt{q} < \frac{1}{16}\frac{q}{n} \leq \frac{1}{16}|D|. \end{align*} As in the previous section, a standard character sum argument gives the inequality \begin{align}
\left|M_k(D,b,m) - (\frac{1}{q})^{m_p} (|D|)_k \right| < \max_{c\in \mathbb{F}_q^{m_p}-0}
S(k, \psi, f_c), \end{align} where $f_c = \sum_{j=1, p\nmid j}^m c_j x^{j}$. It follows that \begin{align}
\left|M_k(D,b,m) - (\frac{1}{q})^{m_p} (|D|)_k \right| < \left(\frac{9}{16}|D|\right)^k. \end{align} The same argument as in the proof of Theorem 2.3 in \cite{CC19} shows that if
$$3.05sm_p=3.05 m_p\log_2q < k \leq |D|/2,$$ then \begin{align}
\frac{1}{q^{m_p}} (|D|)_k > \frac{1}{q^{m_p}}\left(\frac{9}{16}|D|\right)^k 2^{sm_p}
= \left(\frac{9}{16}|D|\right)^k, \end{align} Thus, we obtain \begin{theorem} Let $p=2$ and $n(mn+1) < \frac{1}{16}\sqrt{q}$. Then $M_k(D,b,m)>0$ for all
$3.05 m_p\log_2q < k \leq |D|/2$. \end{theorem} We conclude that when $D$ is the image of degree $n$ polynomial satisfying the value set character sum estimate in Lemma \ref{dicksonWeilEven}, the $m$-th moment subset sum problem over $D$ can be solved in deterministic polynomial time in the algebraic input size $n \log q$, for every constant $m$. In particular, this is true when $D$ is the image of a monomial of Dickson polynomial of degree $n$.
\end{section}
\begin{section}{Conclusion} We show that there is a deterministic polynomial time algorithm for the $m$-th moment $k$-subset sum problem over finite fields for each fixed $m$ when the evaluation set is the image set of a monomial or Dickson polynomial of any degree $n$. An open problem is to ask if Theorem \ref{THM1} can be proved for larger range of $m$, say, $m=O(\log\log q)$. The difficulty lies in the small $k$ range such as $k\leq 3m+1$. \end{section}
\begin{center} {\bf Acknowledgements} \end{center}
This work was supported by the Early Career Research Workshop in Coding Theory, Cryptography, and Number Theory held at Clemson University in 2018, under NSF grant DMS-1547399.
\end{document} |
\begin{document}
\title{Orbital angular momentum of the down converted photons} \author{Xi-Feng Ren\thanks{ Electronic address: mldsb@mail.ustc.edu.cn}, Guo-Ping Guo\thanks{ Electronic address: harryguo@mail.ustc.edu.cn}, Bo Yu, Jian Li, and Guang-Can Guo} \address{Key Laboratory of Quantum Information, University of Science and Technology\\ of China, CAS, Hefei 230026, People's Republic of China
} \maketitle
\begin{abstract} We calculate the relative amplitude of orbital angular momentum (OAM) entangled photon pairs from the spontaneous parametric down conversion. The results show that the amplitude depends on both the two Laguerre indices $l,$ $p$. We also discuss the influences of the mostly used holograms and mono-mode fibers for mode analyzation. We conclude that only a few dimensions can be explored from the infinite OAM modes of the down-converted photon pairs.
PACS number(s): 03.67.Mn, 03.65.Ud, 42.50.Dv \end{abstract}
\section{Introduction}
Quantum entanglement is a very important property of quantum mechanics. It is the foundation of quantum teleportation \cite{Benn}, quantum computation \cite{shor84,grov97,eke96}, quantum cryptography \cite{eke91}, superdense coding \cite{ben92}, etc. Up to now most of the theoretical discussions and experiments are focused on quantum states belonging to two-dimensional states, or qubits\cite{Sackett00,Pan00,Pan01,Howell02}. In recent years, the interest in multi-dimensional states, or qudits, is steadily growing for its promise to realize new types of quantum communication protocols\cite {Bartlett00,Bechmann00,Bourennane01}, and its properties in quantum cryptography better than qubits\cite {Bechmann00,Bourennane01,BechmannA00,Guo02}. These theoretical disscussions \cite{Arnaut00,Franke02,Molina02,Barbosa02,Torres03} and experiments\cite {Arlt99,Mair01,Leach02,Vaziri02,Vaziri03} about multi-dimensional states are mostly based on the orbital angular momentum (OAM) of the photons. It has been shown that paraxial Laguerre-Gaussian(LG) laser beams carry a well-defined orbital angular momentum\cite{Allen92}, and that the LG modes form a complete Hilbert set. The down-converted photons from spontaneous parametric down-conversion (SPDC) are entangled in not only polarization, or spin angular momentum, but also OAM\cite{Mair01,Vaziri02}. This provides a promise to explore multi-dimensional quantum state in one photon \cite {Mair01,Molina02}.
In this paper, we calculate the relative amplitude of OAM of the down-converted photons from SPDC and analyze the possible joint detection probability under the influence of the experiment elements. Our results show that the relative amplitude decreases almost exponentially with the growing of OAM. And only a few dimensions can be explored from the infinite OAM modes of the down-converted photon pairs. In Section $2$, we briefly introduce the LG mode, and calculate the relative amplitude of every LG mode of the down converted photons from SPDC in detail. In theory, the relative amplitude determines the joint detection probability. However, in practical experiments, the computer generated holograms and the mono-mode fibres will inevitably influence the joint detection probability. We discuss the mode analysis after the computer generated hologram in Section $3$. Section $4$ analyzes the detection efficiency of mono-mode fibre for every LG\ mode. Section $5$ presents the possible joint detection probability, when the effects of the holograms and mono-mode fibres are both included. The last section is the conclusion.
\section{Spontaneous parametric down conversion and OAM}
It is well known that photons can carry both spin angular momentum and OAM \cite{Allen92}. Spin angular momentum is associated with polarization and OAM with the azimuthal phase of the electric field. The normalized LG mode is given in cylindrical coordinates by
\begin{eqnarray}
LG_p^l(\rho ,\varphi ,z) &=&\sqrt{\frac{2p!}{\pi (\left| l\right| +p)!}}
\frac 1\omega (\frac{\sqrt{2}\rho }\omega )^{\left| l\right| }L_p^{\left|
l\right| }(\frac{2\rho ^2}{\omega ^2}) \nonumber \\ &&\times \exp (-\rho ^2/\omega ^2)\exp (-ik\rho ^2/2R) \nonumber \\
&&\exp (-i[2p+\left| l\right| +1]\psi )\exp (-il\varphi ), \eqnum{1} \end{eqnarray} where $L_p^l(x)$ are the associated Laguerre polynomials,
\begin{equation}
L_p^{|l|}(x)=\sum_{m=0}^p(-1)^m\frac{(|l|+p)!}{(p-m)!(|l|+m)!m!}x^m, \eqnum{2} \end{equation} and the standard definitions for Gaussian beam parameters are used:
$\omega (z)=\omega _0\sqrt{1+(z/z_R)^2}:$ spot size,
$R(z)=z(1+(z_R/z)^2):$ radius of wavefront curvature,
$\psi (z)=\arctan (z/z_R):$ Gouy phase,
$z_R=\frac 12k\omega _0^2:$ Rayleigh range.
$\omega _0$ is the beam width at the beam waist, the index $l$ is referred to as the winding number, and $(p+1)$ is the number of radial nodes. If the mode function is a pure LG mode with winding number $l$ , then every photon of this beam carries an OAM of $l\hbar $. This corresponds to an eigenstate of the OAM operator with eigenvalue $l\hbar $\cite{Allen92}. If the mode function is not a pure LG mode, the state is a superposition state, with the weights dictated by the contribution of the $l$th angular harmonics.
At the beam waist $(z=0)$, the LG mode can be written as
\begin{eqnarray}
LG_p^l(\rho ,\varphi ) &=&\sqrt{\frac{2p!}{\pi (\left| l\right| +p)!}}\frac 1
{\omega _0}(\frac{\sqrt{2}\rho }{\omega _0})^{\left| l\right| }L_p^{\left|
l\right| }(\frac{2\rho ^2}{\omega _0^2}) \nonumber \\ &&\times \exp (-\rho ^2/\omega _0^2)\exp (-il\varphi ), \eqnum{3} \end{eqnarray}
In the following calculation, we use this equation because in the experiment we always manipulate the light at its beam waist.
In the SPDC process, a thin quadratic nonlinear crystal is illuminated by a laser pump beam propagating in the $z$ direction, with wave number $k_p$ and waist $\omega _0$. The generated two-photon quantum state is given by\cite {Torres03}
\begin{equation}
\left| \Psi \right\rangle
=\sum_{l_1,p_1}\sum_{l_2,p_2}C_{p_1,p_2}^{l_1,l_2}\left| l_1,p_1;l_2,p_2\right\rangle , \eqnum{4} \end{equation} where $(l_1,p_1)$ corresponds to the mode of the signal beam and $(l_2,p_2)$ the mode of the idler beam. The probability amplitude $C_{p_1,p_2}^{l_1,l_2}$ is given as\cite{Arnaut00,Franke02,Barbosa02,Torres03}
\begin{equation} C_{p_1,p_2}^{l_1,l_2}\sim \int dr_{\bot }\Phi (r_{\bot })[LG_{p_1}^{l_1}(r_{\bot })]^{*}[LG_{p_2}^{l_2}(r_{\bot })]^{*}, \eqnum{5} \end{equation} where $r_{\bot }$ is the radial coordinate in the transverse $X-Y$ plane, $ \Phi (r_{\bot })$ is the spiral distribution of the pump beam at the input faced of the crystal, $LG_p^l(r_{\bot })$ is the spiral distribution of the LG mode beam at the same plane.
The weights of the quantum superposition are given by $A_{p_1,p_2}^{l_1,l_2}
\sim \left| C_{p_1,p_2}^{l_1,l_2}\right| ^2$. It is the ideal joint detection probability for finding one photon in the signal mode $(l_1,p_1)$ and one photon in the idler mode $(l_2,p_2)$.
Consider the case that the pump beam is in a pure LG mode $LG_{p_0}^{l_0}$ with $p_0=0$. The $LG_0^{l_0}$ mode light at $z=0$ can be written as
\begin{equation} LG_0^{l_0}(\rho ,\varphi )=\sqrt{\frac 2\pi }\frac 1{\omega _0}(\frac{\sqrt{2
}\rho }{\omega _0})^{\left| l_0\right| }\exp (-\rho ^2/\omega _0^2)\exp (-il\varphi ). \eqnum{6} \end{equation} Substitute $LG_0^{l_0}(\rho ,\varphi )$ for $\Phi (r_{\bot })$ into Eq. $(5)$ , and use the OAM conservation law in SPDC\cite{Arnaut00,Mair01}:
\begin{equation} l_1+l_2=l_0, \eqnum{7} \end{equation} where $l_1$,$l_2$,$l_0$ are the winding numbers of signal beam, idler beam and pump beam respectively, we can achieve the probability amplitude
\begin{eqnarray} C_{p_1,p_2}^{l_1,l_2} &\sim &\sum_{m=0}^{p_1}\sum_{n=0}^{p_2}(\frac 23)^{
\frac{2m+2n+\left| l_1\right| +\left| l_2\right| +\left| l_0\right| }2 }(-1)^{m+n} \nonumber \\
&&\frac{\sqrt{p_1!p_2!(\left| l_1\right| +p_1)!(\left| l_2\right| +p_2)!}}{
(p_1-m)!(p_2-n)!(\left| l_1\right| +m)!(\left| l_2\right| +n)!m!n!} \nonumber \\
&&(\frac{2m+2n+\left| l_1\right| +\left| l_2\right| +\left| l_0\right| }2)! \text{.} \eqnum{8} \end{eqnarray}
In the case $l_0=0$, or the input beam is Gaussian mode light, Eq. $(8)$ can be simplified as$(l>0)$
\begin{eqnarray} C_{p_1,p_2}^{l,-l} &\sim &\sum_{m=0}^{p_1}\sum_{n=0}^{p_2}(\frac 23 )^{m+n+l}(-1)^{m+n} \nonumber \\ &&\frac{\sqrt{p_1!p_2!(l+p_1)!(l+p_2)!}(l+m+n)!}{ (p_1-m)!(p_2-n)!(l+m)!(l+n)!m!n!}. \eqnum{9} \end{eqnarray} It can be easily proved that $ C_{p_1,p_2}^{l,-l}=C_{p_1,p_2}^{-l,l}=C_{p_2,p_1}^{l,-l}=C_{p_2,p_1}^{-l,l}$.
Table 1 gives the relative value for $p_1,$ $p_2=0,1,2$ and $l=0,1,2$. We can also illustrate the dependence of the relative probability amplitude $ C_{p_1,p_2}^{l,-l}$ on $p_1,p_2,$ and $l$, with Fig. 1($l=0$ and $ p_1,p_2=0,1,2,3,4$) and Fig. 2($p_1=p_2=0$ and $l=0,1,2,3,4$).
From the above table and figures, we can see that the probability amplitude decreases very rapidly with the growing of $p_1$, $p_2$ and $l$. We then just consider the cases with $p_1,$ $p_2=0,1,2$ and $l=0,1,2$ when $p_0=$ $ l_0=0$. In papers\cite{Mair01,Vaziri02}, they also just consider the cases with $l=0,1,2$.
If additionally assume $p_1=p_2=0$ as in the previous works\cite {Mair01,Vaziri02,Vaziri03,Torres03}, we can obtain:
\begin{equation} C_{0,0}^{l,-l}\sim (\frac 23)^l. \eqnum{10} \end{equation} This result can also be achieved from the Eq. $(14)$ of the paper\cite {Torres03}, when the condition $l_1=-l_2=l$ is assumed.
But till now, this assumption has not been proven either in theoretical discussions\cite{Arnaut00,Torres03} or in experiments\cite{Mair01,Vaziri02}. We will discuss the two cases separately in Section $5$. Before proceed, we analyze the two main experiment elements, computer generated holograms and mono-mode fibre, which will unavoidably affect the detected relative probability amplitude.
\section{Computer generated holograms and the mode analysis}
In most of the recent experiments\cite {Arlt99,Mair01,Leach02,Vaziri02,Vaziri03}, the authors always use computer generated holograms to transform Gaussian mode light into other LG modes light, or change the winding number of LG mode light. It is a kind of transmission holograms with the transmittance function:
\begin{equation} T(\rho ,\varphi )=\exp (i\delta \frac 1{2\pi }
\mathop{\rm mod}
(l\varphi -\frac{2\pi }\Lambda \rho \cos \varphi ,2\pi )), \eqnum{11} \end{equation} where $\delta $ is the amplitude of the phase modulation, $\Lambda =\frac{ 2\pi }{k_x}$ is the period of the grating at a large distance away from the fork, $k_x$ is the $x$ component of the simplest reference beam's wave vector. Corresponding to the diffraction order $m$, the hologram can change the winding number of the input beam by $\Delta l_m=ml$. The diffraction efficiency depends on the phase modulation $\delta $. When $\delta =2\pi $, almost $100\%$ of the incident intensity is diffracted into the first-order.
However, even the input beam is a pure LG mode light, the output beam after the hologram is not a pure LG mode light. The output light will be the superposition of the various LG modes with the same $l$, and different $p$ \cite{Heckenberg92}. In addition, the beam waists of the input and output beam will affect the weights of different components of the output beam. Assume the input beam and the output beam have the same waists $\omega _0$, thus the complex expansion coefficients of the decomposition of the $m$th diffraction order can be calculated as:
\begin{eqnarray} a_{pl} &=&\int \int (LG_{p_1^{\prime }}^{l_1^{\prime }}(\rho ,\varphi )\exp (-im\frac{2\pi }\Lambda r\cos \varphi ))^{*} \nonumber \\ &&T(\rho ,\varphi )E_{in}(\rho ,\varphi )\rho d\rho d\varphi , \eqnum{12} \end{eqnarray} where $E_{in}(\rho ,\varphi )=LG_{p_1}^{l_1}(\rho ,\varphi )$. Consider the first-order diffraction, or $m=1$, Eq. $(12)$ can be then rewritten as
\begin{equation} a_{p_1,p_1^{\prime }}^{l_1,l_1^{\prime }}=\int \int (LG_{p_1^{\prime }}^{l_1^{\prime }}(\rho ,\varphi ))^{*}\exp (-i\Delta l\varphi ))LG_{p_1}^{l_1}(\rho ,\varphi )\rho d\rho d\varphi , \eqnum{13} \end{equation} where $\Delta l=l_1-l_1^{\prime }$ is the winding number changed by the hologram. The relative weight of the output modes is given by
\begin{equation}
P_{p_1,p_1^{\prime }}^{l_1,l_1^{\prime }}=\left| a_{p_1,p_1^{\prime
}}^{l_1,l_1^{\prime }}\right| ^2. \eqnum{14} \end{equation}
As the mono-mode fibres can only detect the photons with $l=0$, we consider the case that the output light is in the modes with $l_1^{\prime }=0$. Then $ P_{p_1,p_1^{\prime }}^{l_1,l_1^{\prime }}$ can be simplified as $ P_{p_1,p_1^{\prime }}^{\Delta l}$, and $P_{p_1,p_1^{\prime }}^{\Delta l}=P_{p_1,p_1^{\prime }}^{-\Delta l}$. In most of the experiments\cite {Mair01,Leach02,Vaziri02,Vaziri03}, only the holograms of $\Delta l=1$ or $2$ are employed. Table 2 and Table 3 give the relative weight of different modes after the computer generated hologram with $\Delta l=1$ and $2$.
From Table 2 and Table 3, we can see most of the input mode $LG_p^{\Delta l}$ is converted into $LG_p^0$ and $LG_{p+1}^0$. Thereby, we only consider the case that the output light is in the modes with $p^{\prime }=0,1,2,3$.
\section{Mono-mode fibre and detection efficiency of the OAM modes}
It is known that only one mode of light can transmit in the mono-mode fibre: $HE_{11}$ mode. And in practical calculation, we always use Gaussian mode to replace the $HE_{11}$ mode. The Gaussian mode is
\begin{equation} E(\rho )=E(0)\exp (-\rho ^2/\omega ^2), \eqnum{15} \end{equation} where $d=2\omega $ is the Mode Field Diameter(MFD) of the fibre, $E(0)$ is amplitude of field at the fibre center. For $LG_p^l$ mode light, the detection efficiency is given as
\begin{equation} Q_{l,p}=\frac{(\int \int (LG_p^l)^{*}E(\rho )\rho d\rho d\varphi )^2}{\int \int (LG_p^l)^{*}LG_p^l\rho d\rho d\varphi \int \int E(\rho )^{*}E(\rho )\rho d\rho d\varphi }. \eqnum{16} \end{equation}
Obviously if $l\neq 0$, then $Q_{l,p}=0$. For the case $l=0$, Eq. $(16)$ can be simplified as:
\begin{equation} Q_p=\frac{(\int \int (LG_p^0)^{*}E(\rho )\rho d\rho d\varphi )^2}{\int \int (LG_p^0)^{*}LG_p^0\rho d\rho d\varphi \int \int E(\rho )^{*}E(\rho )\rho d\rho d\varphi }. \eqnum{17} \end{equation}
To calculate the relative joint detection probability of the down-converted photons from SPDC, we only need the relative detection efficiency of the $ LG_0^0$, $LG_1^0$ , $LG_2^0$ and $LG_3^0$, but not their absolute efficiency. Assume the waist size of the input beam is adjusted equal to $ d/2 $. Then when the detection area is much more larger than the cross-section of the input light , only $LG_0^0$ mode light can be detected. But in practice, the detection area is determined by the fibre diameter. To simplify calculation, we further assume that the detection area is a round area with diameter equal to the MFD. Thus the integral for $\rho $ is from $ 0 $ to $\omega $. With these assumptions, the relative efficiencies for $ p=0,1,2,3$ can be written as
\begin{equation} Q_0:Q_1:Q_2:Q_3=1:0.263:0:0.036 \eqnum{18} \end{equation}
\section{Relative joint detection probabilities of OAM entangled photons from SPDC}
With the above discussions about the influence of computer generated holograms and mono-mode fibres, we now consider the joint detection probability for OAM\ entangled photons generated from an experimental set-up similar to the work\cite{Mair01}. The joint detection probability can be written as
\begin{equation} R_l=\sum_{p_1=0}^2\sum_{p_2=0}^2((C_{p_1,p_2}^{l,-l})^2\sum_{p_1^{^{\prime }}=0}^{p_1+1}\sum_{p_2^{^{\prime }}=0}^{p_2+1}(P_{p_1,p_1^{^{\prime }}}^{-l}P_{p_2,p_2^{^{\prime }}}^lQ_{p_1^{^{\prime }}}Q_{p_2^{^{\prime }}})) \text{.} \eqnum{19} \end{equation} When $l=0$, $R_0$ gives the joint detection probability that there is no hologram in both the signal and idler beam. And for the case $l\neq 0$, $R_l$ represents the joint detection probability with one $\Delta l=-l$ hologram in the signal beam and one $\Delta l=l$ hologram in the idler beam. Obviously, $R_l=R_{-l}$.
Substitute the values of $C$, $P$ and $Q$ calculated in the above sections to Eq. $(19)$, we can get the relative joint detection probability of the three cases $l=0,1,2$ as
\begin{equation} R_0:R_1:R_2=1:0.346:0.101. \eqnum{20} \end{equation}
If we also assume that $p_1=p_2=0$ for the SPDC process as the recent papers \cite{Mair01,Vaziri02,Vaziri03,Torres03}, the joint detection probability can thus be written as \begin{equation} R_l=((C_{0,0}^{l,-l})^2\sum_{p_1^{^{\prime }}=0}^1\sum_{p_2^{^{\prime }}=0}^1(P_{0,p_1^{^{\prime }}}^{-l}P_{0,p_2^{^{\prime }}}^lQ_{p_1^{^{\prime }}}Q_{p_2^{^{\prime }}}))\text{.} \eqnum{21} \end{equation} And the relation for $l=0,1,2$ becomes:
\begin{equation} R_0:R_1:R_2=1:0.311:0.079. \eqnum{22} \end{equation} The difference between Eq. $(20)$ and $(22)$ is caused by the additional assumption that Laguerre index $p_1=p_2=0$. But this assumption has not been proven either in theoretical discussions\cite{Arnaut00,Torres03} or in experiments\cite{Mair01,Vaziri02}. From the above calculations, we can see that this assumption will cause non-trivial influence to the relative joint detection probability. Our results put forward a feasible method to verify the assumption.
If we rule out the influence of the holograms and mono-mode fibres, and make the assumption of $p_1=p_2=0$, the relationship for the relative joint detection probability of the cases $l=0,1,2$ becomes
\begin{equation} R_0:R_1:R_2=1:0.444:0.198, \eqnum{23} \end{equation} Compare Eq. $(22)$ with Eq. $(23)$, we can see that the experimental elements can apparently influence the joint detection probability.
In the experiment by Vaziri and co-workers\cite{Vaziri02}, they found that the state of the OAM entangled photons from SPDC was given by
\begin{equation}
\psi =0.65\left| 0,0\right\rangle +0.60\left| 1,-1\right\rangle +0.47\left| -1,1\right\rangle . \eqnum{24} \end{equation} From this equation we can get the relationship for the relative joint detection probabilities of the cases $l=0,1,-1$ as
\begin{equation} R_0:R_1:R_{-1}=1:0.852:0.523. \eqnum{25} \end{equation} Including the influence of computer generated holograms and mono-mode fibres, and loosening the assumption of $p_1=p_2=0$, we expect this relation be \begin{equation} R_0:R_1:R_{-1}=1:0.346:0.346. \eqnum{26} \end{equation} The reason for the difference between Eq. $(25)$ and $(26)$ might be as follows: in experiment, the diffraction efficiency of the hologram can not be $100\%$ . Generally, different holograms have different diffraction efficiencies. The waists of the input beam and the output beam of holograms will also affect the final experiment detection probabilities. In addition, the fibre diameter and MFD of the practical mono-mode fibre will also affect the detection efficiencies of different modes light. Evidently, the detection efficiency of avalanche detectors has little effect on the relative joint detection probabilities. Thus in the practical experiment, the $P$ and $Q\ $values have to be adjusted according to the particular conditions.
\section{Conclusion}
In conclusion, we have calculated the probability amplitudes of different LG modes of the down converted photons. Our results show that the relative amplitude decreases almost exponentially with growing of OAM. We also discussed the impact of the previous assumption for $p$ on the joint detection probability. In addition, we analyzed the influences of the experiment elements. We concluded that only a few dimensions can be explored from the infinite OAM modes of the down-converted photon pairs. The experiment verification of the present theory is straightforward and is currently in progress in our laboratory.
\begin{center} {\bf Acknowledgments} \end{center}
This work was funded by the National Fundamental Research Program (2001CB309300), the Innovation Funds from Chinese Academy of Sciences, and also by the outstanding Ph. D thesis award and the CAS's talented scientist award rewarded to Lu-Ming Duan.
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{\bf Table 1. }The relative probability amplitude $C_{p_1,p_2}^{l,-l}$ of the down converted photons from SPDC. We let $C_{0,0}^{0,0}$ be unity.
{\bf Fig. 1.} The relative probability amplitude $C$ for $l=0$ and $ p_1,p_2=0,1,2,3,4$. We let $C_{0,0}^{0,0}$ be unity.
{\bf Fig. 2.} The relative probability amplitude $C$ for $p_1=p_2=0$ and $ l=0,1,2,3,4$. We let $C_{0,0}^{0,0}$ be unity.
{\bf Table 2. }The relative weight of different modes after the computer generated hologram with $\Delta l=1$.
{\bf Table 3. }The relative weight of different modes after the computer generated hologram with $\Delta l=2$.
\end{document} |
\begin{document}
\begin{abstract} In order to obtain a closed orientable convex projective four-manifold with small positive Euler characteristic, we build an explicit example of convex projective Dehn filling of a cusped hyperbolic four-manifold through a continuous path of projective cone-manifolds. \end{abstract}
\keywords{Real projective structure, Hyperbolic 4-manifold, Dehn filling, Euler characteristic, Cone-manifold, Hilbert geometry}
\maketitle
\section{Introduction}
Convex projective manifolds form an interesting class of aspherical manifolds, including complete hyperbolic manifolds. We refer to \cite{Bsurvey,survey_ludo,survey_CLM} and \cite{Msurvey} for surveys on convex projective manifolds and hyperbolic 4-manifolds, respectively. This class of geometric manifolds has been studied notably in the context of deformations of geometric structures on manifolds or orbifolds (see the survey \cite{survey_CLM} and the references therein), or for its link to dynamical systems through the notion of Anosov representation \cite{convexe_div_1,CC_Opq,DGK17} (see \cite{labourie_anosov,GW12} for the notion of Anosov representation).
A \emph{convex projective $n$-manifold} is the quotient $\Omega/_\Gamma$ of a properly convex\footnote{A subset $\Omega$ of $\mathbb{R}\mathbb{P}^n$ is \emph{properly convex} if its closure $\overline{\Omega}$ is contained and convex in some affine chart.} domain $\Omega$ in the real projective space $\mathbb{R}\mathbb{P}^n$ by a subgroup $\Gamma$ of the projective linear group $\mathrm{PGL}_{n+1}\mathbb{R}$ acting freely and properly discontinuously on $\Omega$. A \emph{convex projective $n$-orbifold} is defined similarly without requiring the action of $\Gamma$ to be free. If $\Omega$ is endowed with its Hilbert metric, then $\Gamma$ acts on $\Omega$ by isometry, and the manifold (or orbifold) $\Omega/_\Gamma$ inherits a complete Finsler metric (see \cite{intro_constantin} for an introduction to Hilbert geometry). In the case that $\Omega$ is an open ellipsoid, it is isometric to the hyperbolic space $\mathbb{H}^n$,\footnote{This is in fact the projective model of the hyperbolic space, also known as the Beltrami--Cayley--Klein model.} and the quotient $\Omega/_\Gamma$ is a complete hyperbolic manifold (or orbifold). A complete hyperbolic manifold is \emph{cusped} if it is non-compact and of finite volume. A \emph{convex projective} (resp. \emph{complete hyperbolic}) \emph{structure} on a manifold $M$ is a diffeomorphism between $M$ and a convex projective (resp. hyperbolic) manifold $\Omega/_\Gamma$.
The goal of this paper is to prove the following:
\begin{theointro} \label{thm:main} There exists a closed orientable convex projective $4$-manifold $X$ containing 10 disjoint totally geodesic $2$-tori $\Sigma=T_1\sqcup \ldots \sqcup T_{10} \subset X$ such that: \begin{enumerate} \item\label{item:thmA_hyperbolic} The complement $M=X\smallsetminus\Sigma$ admits a complete finite-volume hyperbolic structure. \item\label{item:thmA_Euler} The Euler characteristic of $X$ (and of $M$) is $12$. \item\label{item:thmA_curve} The hyperbolic manifold $M$ has a maximal cusp section in which each filling curve has length $6$. \item\label{item:thmA_rel} The fundamental group $\pi_1 X$ is relatively hyperbolic with respect to the collection of rank-$2$ abelian subgroups $\{\pi_1T_i,\,\pi_1T'_i\}_{i}$, where $\{T'_1,\ldots,T'_{10}\}$ is another collection of disjoint, totally geodesic, $2$-tori such that each $T_i$ is transverse to each $T'_j$. \item\label{item:thmA_cone} The hyperbolic structure $\sigma_0$ on $M$ and the convex projective structure $\sigma_{2\pi}$ on $X$ arise as limits of an analytic path $\theta\mapsto\sigma_\theta$ of projective cone-manifold structures on $X$, singular along $\Sigma$ with cone angle $\theta\in(0,2\pi)$. \item\label{item:thmA_orb} For each integer $m\geq1$, the structure $\sigma_{\nicefrac{2 \pi}{m}}$ is the underlying cone-manifold structure of a convex projective orbifold $\Omega_m/_{\Gamma_m}$. For $m\geq2$, the group $\Gamma_m$ is relatively hyperbolic with respect to the collection of rank-$2$ abelian subgroups $\{\pi_1T_i\}_{i}$. \end{enumerate} \end{theointro}
We refer the reader to Remark \ref{rem:nonsingular} for the meaning of totally geodesic submanifold in the real projective setting and to Section \ref{sec:rel_hyp} for the definition and some facts on relative hyperbolicity.
Note that the manifold $X$ does not admit a hyperbolic structure because $\pi_1 X$ contains $\mathbb{Z}^2$ (the tori $T_i$ and $T'_i$ are indeed $\pi_1$-injective). The hyperbolic manifold $M$ has 10 cusps, each with section a 3-torus. A \emph{filling curve} is a closed geodesic in a cusp section of $M$ (with respect to the induced flat metric) that bounds a disc in $X$.
\subsection*{Cone-manifolds and Dehn filling}
Projective cone-manifolds are singular projective manifolds generalising the more familiar hyperbolic cone-manifolds (see Definition \ref{def:cone-mfd}). The convex projective 4-manifold $X$ of Theorem \ref{thm:main} is obtained from the cusped hyperbolic 4-manifold $M$ by ``projective Dehn filling''. This is in analogy with Thurston's hyperbolic Dehn filling \cite{Tnotes}, where $\theta \mapsto\sigma_\theta$ is a path of hyperbolic cone-manifold structures on a 3-manifold $X$, singular along a link $\Sigma\subset X$. In both (hyperbolic and projective) cases, as the cone angle $\theta$ approaches $2\pi$, the projective cone-manifold structure becomes non-singular, and we get a convex projective structure on $X$. On the other extreme of the path, as $\theta$ tends to $0$, the singular locus $\Sigma$ is drilled away, giving rise to the cusps of the hyperbolic manifold $M$.
The projective cone-manifold structures $\sigma_\theta$ of Theorem \ref{thm:main} are singular along the tori $\Sigma$, and induce (non-singular) projective structures\footnote{A (\emph{real}) \emph{projective structure} on an $n$-manifold is a $(\mathrm{PGL}_{n+1}\mathbb{R},\mathbb{R}\mathbb{P}^n)$-structure.}
on both $M=X\smallsetminus\Sigma$ and $\Sigma$. The path $\theta\mapsto\sigma_\theta$ is \emph{analytic}, meaning that for each $\theta \in (0,2\pi)$, it is possible to choose a holonomy representation $\rho_\theta\in\mathrm{Hom}(\pi_1M,\mathrm{PGL}_5 \mathbb{R})$ of the projective structure $\sigma_\theta|_M$ so that the function $\theta\mapsto\rho_\theta(\gamma)$ is analytic for all $\gamma \in \pi_1M$. Each torus $T_i \subset \Sigma$ has a meridian $\gamma_i\in\pi_1M$ whose holonomy $\rho_\theta(\gamma_i)\in\mathrm{PGL}_5 \mathbb{R}$ is conjugate to a (projective) rotation of angle $\theta$. In addition, the sequence of representations $\{\rho_{\nicefrac{2\pi}{m}}\}_m$ converges algebraically to $\rho_0$ as $m \rightarrow \infty$, and the sequence of convex sets $\{\overline{\Omega_m}\}_m$ converges to $\overline{\mathbb{H}^4}\subset\mathbb{R}\mathbb{P}^4$ in the Hausdorff topology.\footnote{These additional facts can be proved as in \cite[\S 12]{CLM}.}
\subsection*{New features of the result}
Theorem \ref{thm:main} is shown by an explicit construction. Since the convex projective manifold $X$ has non-zero Euler characteristic, it is indecomposable\footnote{A properly convex domain $\Omega$ of $\mathbb{R}\mathbb{P}^n$ is \emph{indecomposable} if it is not a convex hull of lower-dimensional domains. A convex projective manifold or orbifold $\Omega/_\Gamma$ is \emph{indecomposable} if $\Omega$ is indecomposable.} (see Fact \ref{fact:indecomposable}). It seems that, at the time of writing this paper, the literature misses concrete examples of closed indecomposable convex projective $n$-manifolds, $n\geq4$, which do not admit a hyperbolic structure. For the moment, we know only two techniques to obtain such manifolds: \begin{enumerate} \item torsion-free subgroups of some discrete projective reflection groups using Vinberg’s theory \cite{V}, as shown by Benoist \cite{CD4,Benoist_quasi} and Choi and the first two authors \cite{CLM,CLM_ecima}; \item some Gromov--Thurston manifolds, as shown by Kapovich \cite{K}. \end{enumerate}
In contrast with Theorem \ref{thm:main}, the Selberg lemma has an important role to guarantee the existence of such manifolds in all these cases. In particular, very little is known about the topology of closed convex projective manifolds. Note that the techniques involved in the construction of our $X$ are in the spirit of (1) rather than (2).
\begin{remark} There is a clear distinction between the manifolds constructed in \cite{Benoist_quasi,K} and the ones in \cite{CD4,CLM,CLM_ecima}, including our $X$: the fundamental groups of the former are Gromov-hyperbolic, but those of the latter are not. \end{remark}
The Euler characteristic of a closed even-dimensional manifold can be seen as a rough measure of its topological complexity. Note that a well-known conjecture states that closed aspherical 4-manifolds have Euler characteristic $\chi\geq0$, and this is certainly true in the hyperbolic case by the Gau{\ss}--Bonnet theorem. Our manifold $X$ has $\chi(X)=12$, and appears to be the closed orientable indecomposable convex projective 4-manifold with the smallest known Euler characteristic (to the best of our knowledge). In the hyperbolic case, the smallest known value of $\chi$ is $16$ \cite{CM,L}.\footnote{ Similarly to the orientable hyperbolic 4-manifolds that one gets from \cite{CM,L}, our manifold $X$ is built as the orientable double cover of a non-orientable convex projective manifold. So the smallest known value of $\chi$ for a closed indecomposable convex projective (resp. hyperbolic) 4-manifold is currently $6$ (resp. $8$).}
Theorem \ref{thm:main} is an effective version, in dimension four, of a result by Choi and the first two authors \cite[Theorem B]{CLM}. Let us first recall their construction, called ``convex projective generalised Dehn filling''. They build a sequence of discrete projective reflection groups $\{\Gamma_m\}_{m\geqslant m_0}$ of $\mathrm{PGL}_{n+1}\mathbb{R}$, each acting cocompactly on a properly convex domain $\Omega_m\subset\mathbb{R}\mathbb{P}^n$ of dimension $n=4, 5$ or $6$, whose limit as $m\to\infty$ is a discrete hyperbolic reflection group $\Gamma_\infty<\mathrm{Isom}(\mathbb{H}^n)$ of finite covolume. A fundamental domain of the group $\Gamma_m$ is a compact Coxeter polytope $P_m$ in $\Omega_m$ whose combinatorics does not depend on $m$. The hyperbolic Coxeter polytope $P_\infty$, instead, is combinatorially obtained from $P_m$ by substituting a ridge with an ideal vertex. In other words, the cusp of the hyperbolic orbifold $\mathbb{H}^n/_{\Gamma_\infty}$ is ``projectively filled''. By applying a refined version of Selberg’s lemma to $\Gamma_\infty$, they get a statement similar to Theorem \ref{thm:main}.\eqref{item:thmA_orb}. The difference is that $X=\Omega_{m_0}/_{\Gamma'_{m_0}}$ is ``only'' an orbifold (where $\Gamma'_{m_0}$ is a finite-index subgroup of $\Gamma_{m_0}$). To promote $X$ to a manifold, one should then apply again the Selberg lemma, this time to $\Gamma'_{m_0}$. Thus, our improvement is two-fold: we found an $X$ with small Euler characteristic, and a continuous (rather than discrete) family of cone-manifolds.
Another interesting feature of $X$ is the relative hyperbolicity of $\pi_1X$. Indeed, Gromov and Thurston \cite{bleiler_hodgson,A}\footnote{A proof of the Gromov--Thurston $2\pi$ theorem was given by Bleiler and Hodgson \cite{bleiler_hodgson} in the context of $3$-manifolds, and the same proof holds in any dimension, as explained in \cite[Section 2.1]{A}.} have shown that the fundamental group of a Dehn filling of a hyperbolic manifold with torus cusps is relatively hyperbolic with respect to the subgroups associated to the inserted tori, provided that the filling curves are longer than $2\pi$.\footnote{See also \cite{osin,groves_manning,FM2} for the geometric group theoretic generalisation of that statement.} The fundamental group of $X$ is not relatively hyperbolic with respect to the subgroups associated to the inserted tori by Theorem \ref{thm:main}.\eqref{item:thmA_rel} (see Section \ref{sec:rel_hyp}).\footnote{It is relatively hyperbolic with respect to a larger family of abelian groups.} There is no contradiction between the Gromov--Thurston $2\pi$ theorem and Theorem \ref{thm:main}.\eqref{item:thmA_curve} because $2 \pi > 6$.
\subsection*{Divisible convex domains}
Recall that a properly convex domain $\Omega$ of $\mathbb{R}\mathbb{P}^n$ is \emph{divisible \emph{(}by $\Gamma$\emph{)}} if there exists a discrete subgroup $\Gamma$ of $\mathrm{PGL}_{n+1}\mathbb{R}$ acting cocompactly on $\Omega$. A theorem of Benoist \cite{convexe_div_1} implies that the indecomposable divisible convex domains $\Omega_m\subset\mathbb{R}\mathbb{P}^4$ of Theorem \ref{thm:main}.(\ref{item:thmA_orb}) are not strictly convex\footnote{A subset $\Omega$ of $\mathbb{R}\mathbb{P}^n$ is \emph{strictly convex} if it is properly convex and its boundary does not contain any non-trivial projective line segment.} because the groups $\Gamma_m$ of Theorem \ref{thm:main}.(\ref{item:thmA_orb}) are not Gromov-hyperbolic.
There are very few currently known constructions of inhomogeneous indecomposable divisible non-strictly convex domains. For a complete historical account, we refer the reader to the introduction of \cite{CLM}. Here we mention only its essentials.
The first construction of such domains is due to Benoist \cite{CD4}, and has been extended in \cite{ecima_ludo,BDL_3d_convex,CLM_ecima}. In those constructions the compact quotient $\Omega/_\Gamma$ is homeomorphic to the union along the boundaries of finitely many submanifolds, each admitting a complete finite-volume hyperbolic structure on its interior. As a result, if $\Omega$ is of dimension $n$, then $\Gamma$ is relatively hyperbolic with respect to a collection of virtually abelian subgroups of rank $n-1$.
In \cite{CLM}, a different construction of inhomogeneous indecomposable divisible non-strictly convex domains is given by convex projective generalised Dehn filling. In contrast with the previous examples, these are relatively hyperbolic with respect to a collection of virtually abelian subgroups of rank $n-2$. The divisible (by $\Gamma_m$) domains $\Omega_m\subset\mathbb{R}\mathbb{P}^4$ of Theorem \ref{thm:main}.(\ref{item:thmA_orb}) are new examples of this kind.
We point out that at the time of writing there is no example of inhomogeneous indecomposable divisible non-strictly convex domain of dimension $n$, for any $n \geqslant 9$.
We stress that such domains, to all appearances, are linked to the geometrisation problem, i.e. putting a $(G,X)$-structure on a manifold. So far, almost all manifolds geometrised through this process are either obtained by gluing cusped hyperbolic manifolds, or by Dehn filling of a cusped hyperbolic manifold. Here, the goal is to do so with a small manifold and using cone-manifolds. It is especially important that we do not use Selberg's lemma.
\subsection*{Dehn fillings of hyperbolic manifolds}
Let us say that a closed manifold $X$ is a \emph{filling} of a manifold $M$ if there exists a codimension-2 submanifold $\Sigma\subset X$ such that the complement $X\smallsetminus\Sigma$ is diffeomorphic to $M$. Note that the manifold $M$ is diffeomorphic to the interior of a compact manifold $\overline M$ whose boundary $\partial\overline M$ fibres in circles over $\Sigma$. Given $M$ as above, we obtain a filling by attaching to $\overline M$ the total space of a $D^2$-bundle $E\to\Sigma$ through a diffeomorphism $\partial\overline M\to\partial E$. This operation is commonly called a \emph{Dehn filling} of $M$. Any cusped hyperbolic manifold $M$ has a finite covering $M'$ with torus cusps (see e.g. \cite[Theorem 3.1]{MRS}). In other words, $\partial\overline{M'}$ consists of $(n-1)$-tori. The manifold $M'$ has typically infinitely many fillings up to diffeomorphism.
Thurston's hyperbolic Dehn filling theorem states that every filling of a cusped hyperbolic 3-manifold with torus cusps, except for finitely many fillings on each cusp, admits a hyperbolic structure. In dimension $n\geq4$, except for finitely many fillings on each cusp, the fundamental group of a filling of a cusped hyperbolic $n$-manifold is relatively hyperbolic with respect to a collection of subgroups virtually isomorphic to $\mathbb{Z}^{n-2}$, by \cite[Theorem~1.1]{osin}. Since $n\geqslant 4$, those groups contains $\mathbb{Z}^2$ and so those fillings do not admit any hyperbolic structure. The geometry of the remaining fillings is rather unpredictable, but it is expected that they also do not carry a hyperbolic structure. But, Theorem \ref{thm:main}.(\ref{item:thmA_hyperbolic}) and \cite[Theorem B]{CLM} show that some fillings of some cusped hyperbolic $n$-manifolds admit a convex projective structure. This leads to the following:
\begin{question} \label{quest:filling} Which filling of a cusped hyperbolic manifold of dimension $n\geq4$ (with torus cusps) admits a convex projective structure? \end{question}
It is worth mentioning that almost all fillings of any cusped hyperbolic manifold with torus cusps admit a complete Riemannian metric of non-positive sectional curvature by the Gromov--Thurston $2\pi$ theorem \cite{bleiler_hodgson,A}, and an Einstein metric of negative scalar curvature by work of Anderson \cite{A} and Bamler \cite{B}. Both, in some sense, extend Thurston's 3-dimensional theorem to higher dimension (compare also with \cite{S_cuspclosing,FM2,FM}). But those theorems cannot be applied to the manifold $X$ of Theorem \ref{thm:main}, since the filling curves are too short.
Let us also note that there is an opportune version of hyperbolic Dehn filling in dimension 4: one can sometimes fill some cusps of a hyperbolic 4-manifold and get another cusped hyperbolic 4-manifold, at the expense of drilling some totally geodesic surfaces \cite{MR,LMRY}.
\subsection*{Projective flexibility}
Thurston's hyperbolic Dehn filling theorem essentially relies on the flexibility of the complete hyperbolic structure of cusped hyperbolic 3-manifolds. The local rigidity theorem of Garland and Raghunathan \cite{GR} (see also \cite{MR2110758}), on the other hand, says that the holonomy representation $\rho$ of any cusped hyperbolic manifold $M$ of dimension $n\geq4$ has a neighbourhood in $\mathrm{Hom}(\pi_1M,\mathrm{Isom}(\mathbb{H}^n))$ consisting of conjugates of $\rho$. Now, Theorem \ref{thm:main}.(\ref{item:thmA_cone}) shows that the hyperbolic structure $\sigma_0$ on the 4-manifold $M$ is \emph{projectively flexible}, i.e. the conjugacy class of the holonomy representation $\rho_0$ of $\sigma_0$ $$[\rho_0]\in\mathrm{Hom}(\pi_1M,\mathrm{PGL}_5\mathbb{R})\big/_{\mathrm{PGL}_5\mathbb{R}}$$ is not an isolated point. To avoid confusion with the terminology, we mention that in this definition of flexibility there is no restriction on the holonomy of the peripheral subgroups of $\pi_1M$. For instance, all cusped hyperbolic 3-manifolds are projectively flexible by the hyperbolic Dehn filling theorem. It is thus natural to ask the following question, which is a priori different from Question \ref{quest:filling}.
\begin{question} \label{quest:flexibility} Which cusped hyperbolic 4-manifold is projectively flexible? Is every cusped hyperbolic 4-manifold finitely covered by a projectively flexible one?\footnote{Similar considerations were done by Cooper, Long and Thistlethwaite \cite{CLT07} for closed hyperbolic 3-manifolds. } \end{question}
We note that some cusped hyperbolic 4-orbifolds, for example the Coxeter pyramid $[6,3,3,3,\infty]$, are not projectively flexible \cite[Proposition 20]{V}.
\begin{figure}
\caption{\footnotesize The Coxeter diagram of the pyramid $[6,3,3,3,\infty]$ (see Section \ref{subsec:Coxeter_groups} for the basic terminology on Coxeter groups). The associated cusped hyperbolic 4-orbifold is projectively rigid.}
\label{fig:Coxeter_pyramid}
\end{figure}
\subsection*{On the proof}
As already said, the proof of Theorem \ref{thm:main} is constructive. We begin with the ideal hyperbolic rectified 4-simplex $R\subset\mathbb{H}^4$, which is a Coxeter polytope. By applying the techniques introduced in \cite{CLM}, we perform a ``convex projective generalised Dehn filling'' to $R$: the hyperbolic structure on the orbifold $R$ is deformed to projective structures which extend to structures of ``mirror polytope'' (see Section \ref{sec:gen_dehn_fill}) on the bitruncated 4-simplex $Q$ (see Section \ref{sec:truncated_simplex}). Note that $Q$ minus some ridges is stratum-preserving homeomorphic to $R$. This will be translated into the fact that $X$ minus some tori is homeomorphic to $M$.
We build the hyperbolic manifold $M$ as an orbifold covering of $R$, by exploiting a construction by Kolpakov and Slavich \cite{KS}. By lifting the deformation from $R$ to $M$, we get the path $\theta\mapsto\sigma_\theta$. The manifold $X$ covers a Coxeter orbifold based on the bitruncated 4-simplex $Q$.
Since there exist cusped orientable hyperbolic 4-manifolds $M_0$ tessellated by copies of $R$ with $\chi(M_0)<12$ \cite{KS,Sl,RiSl,KRR}, one could wonder why not to build a smaller convex projective manifold $X$ by Dehn filling such an $M_0$. A first obstruction is topological: a cusp section of those $M_0$ does not always fibre in circles. Even when all cusp sections of such an $M_0$ do fibre in circles, $M_0$ does not cover $R$ but covers the quotient of $R$ by its symmetry group. The latter is the Coxeter pyramid $[6,3,3,3,\infty]$ \cite[Lemma 2.2]{RiSl}, which is projectively rigid, so our technique does not apply in these cases.
\subsection*{Structure of the paper}
In Section \ref{sec:preliminaries} we introduce some basic concepts of projective cone-manifolds, mirror polytopes and the truncation process of the 4-simplex, and in Section \ref{sec:proof} we prove Theorem \ref{thm:main}.
\section{Preliminaries} \label{sec:preliminaries}
In this section we introduce some preliminary notions and fix some notation.
\subsection{Projective cone-manifolds} \label{sec:cone-mfds}
Riemannian $(G,X)$ cone-manifolds were introduced by Thurston \cite{Tshapes} (see also \cite{McM}). If the geometry $(G,X)$ locally embeds in real projective geometry $(\mathrm{PGL}_{n+1}\mathbb{R},\mathbb{R}\mathbb{P}^n)$ (such as the constant-curvature geometries), a $(G,X)$ cone-manifold can be thought as a projective cone-manifold. Hyperbolic cone-manifolds of dimension 3 appear in the proofs of Thurston's hyperbolic Dehn filling theorem \cite{Tnotes} (see also \cite[Chapter 15]{Mbook}) and of the orbifold theorem \cite{CHK,BLP}. Projective cone-manifolds were introduced by Danciger \cite{D1,D2} in the context of geometric transition from hyperbolic to Anti-de Sitter 3-dimensional structures. Quite recently, some higher-dimensional cone-manifolds are used, in particular, in dimension 4: for hyperbolic Dehn filling or degeneration \cite{MR,LMRY}, and in the projective context of AdS-hyperbolic transition \cite{RS}.
We now define projective cone-manifolds ``with cone angles along link singularities''. Our definition is in the spirit of Barbot--Bonsante--Schlenker \cite{BBS}.
Let $\mathbb{S}^n=\left(\mathbb{R}^{n+1}\smallsetminus\{0\}\right)/_{\mathbb{R}_{>0}}$ be the projective sphere and $\hat{\mathbb{S}} \colon\mathbb{R}^{n+1}\smallsetminus\{0\}\to\mathbb{S}^n$ the canonical projection. For every subset $U$ of $\mathbb{R}^{n+1}$, let $\mathbb{S}(U)$ denote $\hat{\mathbb{S}}(U \smallsetminus \{0\})$. With a little abuse of notation, we embed the projective spheres $\mathbb{S}^{n-2}$ and $\mathbb{S}^1$ into $\mathbb{S}^n$, $n\geq2$, as follows: $$\mathbb{S}^{n-2}=\mathbb{S}(\{(x_1,\ldots,x_{n-1},0,0)\})\subset\mathbb{S}^n\mbox{\ \ \ and\ \ \ }\mathbb{S}^1=\mathbb{S}(\{(0,\ldots,0,x_n,x_{n+1})\})\subset\mathbb{S}^n.$$ Given an open subset $U\subset\mathbb{S}^1$, we define $$\mathbb{S}^{n-2}*U\ =\bigcup_{p\in\mathbb{S}^{n-2}}\ \bigcup_{q\in U}\ [p,-p]_q\ \subset\ \mathbb{S}^n,$$ where $[p,-p]_q\subset\mathbb{S}^n$ denotes the half circle containing $q$ with endpoints $p$ and its antipode $-p$. For example, $\mathbb{S}^{n-2}*\mathbb{S}^1 = \mathbb{S}^n$.
A \emph{projective circle} is a closed connected $(\mathrm{SL}_2\mathbb{R},\mathbb{S}^1)$-manifold. Let $C$ be a projective circle. We may think of $C$ as $$C= \Big(\coprod_{\alpha \in A }\, U_{\alpha}\Big) \Big/_\sim$$ for a collection $\{ U_{\alpha} \}_{\alpha \in A}$ of open subsets of $\mathbb{S}^1$, a collection $\{ U_{\alpha\beta} \}_{\alpha,\beta \in A}$ of open subsets $U_{\alpha\beta} \subset U_\alpha$, a collection $\{ g_{\alpha \beta} \}_{\alpha,\beta \in A}$ of diffeomorphisms $g_{\alpha \beta} \colon U_{\alpha\beta} \rightarrow U_{\beta\alpha}$ which are restrictions of elements of $\mathrm{SL}_2\mathbb{R}$, and the relation $x\sim g_{\alpha\beta}(x)$ for all $x\in U_{\alpha\beta}$.
We now add the extra requirement that $C=C_\theta$ is an \emph{elliptic circle}, i.e. that the holonomy representation $\rho$ of $C_\theta$ sends a generator $\gamma$ of $\pi_1 C_\theta$ to an elliptic element $\rho(\gamma) \in \mathrm{SL}_2\mathbb{R}$. Passing to the universal covering $\widetilde{\mathbb{S}^1}$ of $\mathbb{S}^1$ and the covering group $\widetilde{\mathrm{SL}_2\mathbb{R}}$ of $\mathrm{SL}_2\mathbb{R}$ which acts on $\widetilde{\mathbb{S}^1}$, we lift $\rho$ to a representation $\widetilde{\rho} \colon \pi_1 C_\theta \rightarrow \widetilde{\mathrm{SL}_2\mathbb{R}}$. To the element $\widetilde{\rho}(\gamma) \in \widetilde{\mathrm{SL}_2\mathbb{R}}$ is naturally associated a unique real number $\theta > 0$ which characterises the elliptic circle $C_\theta$. Note that $\rho(\gamma)$ is conjugate to $\left(\begin{smallmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \phantom{-}\cos\theta \end{smallmatrix}\right)$ (see \cite[Section 5.4]{Gol18}).
By extending $g_{\alpha\beta}\in\mathrm{SL}_2\mathbb{R}$ to $\hat{g}_{\alpha\beta} = \left(\begin{smallmatrix} \mathrm{Id} & 0 \\ 0 & g_{\alpha\beta} \end{smallmatrix}\right) \in\mathrm{SL}_{n+1}\mathbb{R}$ (thus fixing $\mathbb{S}^{n-2}\subset\mathbb{S}^n$ pointwise), we can define $$\mathbb{S}^{n-2}*C_\theta = \Big(\coprod_{\alpha \in A}\, \mathbb{S}^{n-2}*U_{\alpha} \Big)\Big/_\sim$$ with the relation $x\sim \hat{g}_{\alpha\beta}(x)$ for all $x\in \mathbb{S}^{n-2}*U_{\alpha\beta}$.
This space will be the local model for our cone-manifolds. By canonically embedding $\mathbb{S}^{n-2}$ and $C_\theta$ into $\mathbb{S}^{n-2}*C_\theta$, we have that the couple $(\mathbb{S}^{n-2}*C_\theta,\mathbb{S}^{n-2})$ is homeomorphic to $(\mathbb{S}^n,\mathbb{S}^{n-2})$. Moreover, the sphere $\mathbb{S}^{n-2}*C_\theta$ is stratified in two projective manifolds: \begin{itemize} \item the \emph{singular locus} $\mathbb{S}^{n-2}$, and \item the \emph{regular locus} $(\mathbb{S}^{n-2}*C_\theta)\smallsetminus\mathbb{S}^{n-2}$. \end{itemize} The holonomy of a meridian\footnote{Let $N$ be a manifold and $S\subset N$ a connected submanifold of codimension 2. We call \emph{meridian} of $S$ an element $\gamma\in\pi_1(N\smallsetminus S)$ that is represented by a curve which is freely homotopic in $N\smallsetminus S$ to the boundary of a fibre of a tubular neighbourhood of $S$ in $N$.} of the singular locus in the regular locus is the holonomy of a generator of $\pi_1 C_\theta$.
\begin{definition} \label{def:cone-mfd} Let $X$ be an $n$-manifold and $\Sigma\subset X$ a codimension-2 submanifold. A \emph{projective cone-manifold structure} on $X$, singular along $\Sigma$ \emph{with cone angles}, is an atlas for $X$ whose each chart has values into some $\mathbb{S}^{n-2}*C_\theta$ and sends the points of $\Sigma$ to the singular locus $\mathbb{S}^{n-2}$, and whose transition functions restrict to isomorphisms of projective manifolds on each stratum. \end{definition}
The \emph{regular locus} of the cone-manifold $X$ is the complement $X\smallsetminus\Sigma$, while $\Sigma$ is the \emph{singular locus}. Both are (non-singular) projective manifolds. To each connected component $\Sigma_i$ of $\Sigma$ is associated a cone angle $\theta_i>0$.
\begin{remark} \label{rem:nonsingular} A projective cone-manifold $X$ with all cone angles $\theta=2\pi$ is simply a projective manifold with a totally geodesic codimension-2 submanifold $\Sigma\subset X$. Here, by \emph{totally geodesic}, we mean that the preimage of $\Sigma$ under the universal covering of $X$ is locally mapped to codimension-2 projective subspaces of $\mathbb{S}^n$. If moreover $X$ is a convex projective (and so Finsler) manifold, then $\Sigma$ is totally geodesic in the usual sense. \end{remark}
\subsection{Mirror polytopes} \label{sec:gen_dehn_fill}
We now introduce the main objects for our proof of Theorem \ref{thm:main}: mirror polytopes. Roughly speaking, a mirror polytope is a polytope in the projective sphere, together with a choice of a projective reflection along each of the supporting hyperplanes of the facets (satisfying some extra conditions). We refer to \cite{CLM,V} for further details.
A subset $P$ of $\mathbb{S}^n$ is \emph{convex} if there exists a convex cone\footnote{By a \emph{cone}, we mean a subset of $\mathbb{R}^{n+1}$ which is invariant under multiplication by positive scalars.} $U$ of $\mathbb{R}^{n+1}$ such that $P = \mathbb{S}(U)$, and moreover a convex subset $P$ is \emph{properly convex} if its closure $\overline{P}$ does not contain a pair of antipodal points. A \emph{projective $n$-polytope} is a properly convex subset $P$ of $\mathbb{S}^n$ such that $P$ has a non-empty interior and $$P = \bigcap_{i=1}^{N} \mathbb{S}(\{ v \in \mathbb{R}^{n+1} \mid \alpha_i(v) \leqslant 0 \})$$ where $\alpha_i$, $i=1, \dotsc, N$, are linear forms on $\mathbb{R}^{n+1}$. We always assume that the set of linear forms is \emph{minimal}, i.e. none of the half-spaces $\mathbb{S}(\{ v \in \mathbb{R}^{n+1} \mid \alpha_i(v) \leqslant 0 \})$ contains the intersection of all the others, hence the polytope $P$ has $N$ facets. A \emph{facet} (resp. \emph{ridge}) of a polytope is a face of codimension $1$ (resp. $2$).
A \emph{projective reflection} is an element of $\mathrm{SL}^{\pm}_{n+1}\mathbb{R}$ of order 2 which is the identity on a projective hyperplane. Each projective reflection $\sigma$ can be written as: $$\sigma=\mathrm{Id}-\alpha\otimes b,\quad \textrm{in other words} \quad \sigma(v) = v - \alpha(v) b \quad \forall v \in \mathbb{R}^{n+1},$$ where $\alpha$ is a linear form on $\mathbb{R}^{n+1}$ and $b$ is a vector in $\mathbb{R}^{n+1}$ such that $\alpha(b)=2$. The projective hyperplane $\mathbb{S} (\ker(\alpha))$ is called the \emph{support} of $\sigma$. A \emph{projective rotation} is an element of $\mathrm{SL}_{n+1}\mathbb{R}$ which is the identity on a subspace $H \subset \mathbb{R}^{n+1}$ of codimension 2 and whose induced linear map from $\mathbb{R}^{n+1}/_H$ to itself is conjugate to a matrix $\left(\begin{smallmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \phantom{-}\cos\theta \end{smallmatrix}\right)$ with $0 < \theta < 2 \pi$. The real number $\theta$ is called the \emph{angle} of rotation.
\begin{definition}\label{def:mirror_poly} A \emph{mirror polytope} is a pair consisting of a projective polytope $P$ of $\mathbb{S}^n$ and a finite collection of projective reflections $\{ \sigma_s = \mathrm{Id} - \alpha_s \otimes b_s \}_{s \in S}$ with $\alpha_s(b_s)=2$ such that: \begin{itemize} \item The index set $S$ consists of all the facets of $P$. \item For each facet $s \in S$, the support of $\sigma_s$ is the supporting hyperplane of $s$. \item For every pair of distinct facets $s,t$ of $P$, $\alpha_s(b_t)$ and $\alpha_t(b_s)$ are either both negative or both zero. \end{itemize} \end{definition}
\begin{remark} The third item of Definition~\ref{def:mirror_poly} may seem a bit awkward at first glance. In fact, \cite[Proposition~6]{V} shows that the third item holds when the group $\Gamma$ generated by $\{ \sigma_s \}_{s \in S}$ satisfies the condition: $$ \gamma\mathrm{Int}(P) \cap \mathrm{Int} (P) = \varnothing, \quad \forall \gamma \in \Gamma \smallsetminus\{ \mathrm{Id} \}, $$ where $\mathrm{Int}(P)$ denotes the interior of $P$. \end{remark}
Note that the reflections $\sigma_s$ of $P$ determine the couples $\{ (\alpha_s,b_s) \}_{s \in S}$ up to $\alpha_s \mapsto \lambda_s \alpha_s$ and $b_s \mapsto \lambda_s^{-1} b_s$ for $\lambda_s >0$, because $P$ is defined by $\alpha_i \leqslant 0$. Moreover, $\alpha_s(b_t) \alpha_t(b_s) = 4 \cos^2 \theta$ for some $\theta \in (0,\nicefrac{\pi}{2}]$ if and only if the product $\sigma_s \sigma_t$ is a rotation of angle $2\theta$. The \emph{dihedral angle} of a ridge $s \cap t$ of a mirror polytope is $\theta$ if $\sigma_s \sigma_t$ is a rotation of angle $2\theta$, and $0$ otherwise. A \emph{Coxeter polytope} is a mirror polytope whose dihedral angles are submultiples of $\pi$, i.e. each dihedral angle is $\nicefrac{\pi}{m}$ for some integer $m \geqslant 2$ or $m=\infty$.
For any number $N \in \mathbb{N}$, we set $[\![ N ]\!] := \{ 1, \dotsc, N\}$. An $N \times N$ matrix $A = (A_{ij})_{i, j \in [\![ N ]\!]}$ is \emph{Cartan} if $(i)$ $A_{ii} = 2$ for all $i \in [\![ N ]\!]$, $(ii)$ $A_{ij} \leqslant 0$, for all $i \neq j$, and $(iii)$ $A_{ij} =0 \Leftrightarrow A_{ji}=0 $, for all $i \neq j$. A Cartan matrix $A$ is \emph{irreducible} if it is not the direct sum of smaller matrices (after any reordering of the indices). Every Cartan matrix $A$ is obviously the direct sum of irreducible Cartan matrices $A'_1 \oplus \dotsm \oplus A'_k$. Each irreducible Cartan matrix $A'_i$, $i = 1, \dotsc, k$, is called a \emph{component} of $A$. If $x = (x_1, \dotsc, x_N)$ and $y = (y_1, \dotsc, y_N) \in \mathbb{R}^N$, we write $x > y$ if $x_i > y_i$ for all $i \in [\![ N ]\!]$, and $x \geqslant y$ if $x_i \geqslant y_i$ for all $i \in [\![ N ]\!]$.
\begin{proposition}[{Vinberg \cite[Theorem 3]{V}}]\label{prop:cartan_type} If $A$ is an irreducible Cartan matrix of size $N \times N$, then exactly one of the following holds: \begin{itemize} \item[$\mathrm{(\!(P)\!)}$] $(i)$ The matrix $A$ is nonsingular, and $(ii)$ for every $x \in \mathbb{R}^N$, if $Ax \geqslant 0$, then $x>0$ or $x=0$.
\item[$\mathrm{(\!(Z)\!)}$] $(i)$ The rank of $A$ is $N-1$, $(ii)$ there exists a vector $y \in \mathbb{R}^N$ such that $y >0$ and $Ay =0$, and $(iii)$ for every $x \in \mathbb{R}^N$, if $Ax \geqslant 0$, then $Ax=0$.
\item[$\mathrm{(\!(N)\!)}$] $(i)$ There exists a vector $y \in \mathbb{R}^N$ such that $y >0$ and $Ay <0$, and $(ii)$ for every $x \in \mathbb{R}^N$, if $Ax \geqslant 0$ and $x \geqslant 0$, then $x=0$. \end{itemize} We say that $A$ is of \emph{positive, zero} or \emph{negative type} if $\mathrm{(\!(P)\!)}$, $\mathrm{(\!(Z)\!)}$ or $\mathrm{(\!(N)\!)}$ holds, respectively. \end{proposition}
A Cartan matrix $A$ is of \emph{positive} (resp. \emph{zero}) \emph{type} if every component of $A$ is of positive (resp. zero) type. The \emph{Cartan matrix} of a mirror polytope $P$ is the matrix $A_{P} = (\alpha_s(b_t))_{s,t \in S}$. Note that the Cartan matrix of $P$ is well-defined up to conjugation by a positive diagonal matrix because the reflections $\sigma_s$ of $P$ determine the couples $\{ (\alpha_s,b_s) \}_{s \in S}$ up to $\alpha_s \mapsto \lambda_s \alpha_s$ and $b_s \mapsto \lambda_s^{-1} b_s$ for $\lambda_s >0$. Two Cartan matrices $A$ and $B$ are \emph{equivalent} if $A = D B D^{-1}$ for some positive diagonal matrix $D$. We will make essential use of the following:
\begin{theorem}{\cite[Corollary 1]{V}}\label{thm:vinberg_unique} Let $A$ be a Cartan matrix of size $N \times N$. If $A$ is irreducible, of negative type and of rank $n + 1$, then there exists a mirror polytope $P$ of $\mathbb{S}^{n}$ with $N$ facets (unique up to the action of $\mathrm{SL}^{\pm}_{n+1}\mathbb{R}$) such that $A_P = A$. \end{theorem}
\begin{remark} Theorem \ref{thm:vinberg_unique} is not explicitly stated in \cite[Corollary 1]{V} for non-Coxeter polytopes, but it follows from Propositions 13 and 15 of \cite{V} that the consequent Corollary 1 of \cite{V} is valid not only for Coxeter polytopes, but also for mirror polytopes. \end{remark}
To understand the combinatorics\footnote{The \emph{combinatorics} (or \emph{face poset}) of a polytope is the poset of its faces partially ordered by inclusion.} of a mirror polytope $P$ with facets $\{s\}_{s \in S}$, we introduce the poset $\sigma(P) \subset 2^{S}$ partially ordered by inclusion, which is dual\footnote{Two posets $\mathcal{P}_1$ and $\mathcal{P}_2$ are \emph{dual} to each other provided there exists an order-reversing isomorphism between $\mathcal{P}_1$ and $\mathcal{P}_2$.} to the face poset of $P$: $$ \sigma(P) : = \{ T \subset S \mid T = \sigma(f) \textrm{ for some face } f \textrm{ of } P \}, $$ where $\sigma(f) := \{ s \in S \mid f \subset s \}$. For any subset $T \subset S$, we denote by $A_T$ the restriction of the Cartan matrix $A_P$ of $P$ to $T \times T$.
\begin{theorem}{\cite[Theorems 4 and 7]{V}}\label{thm:vinberg} Let $P$ be a mirror $n$-polytope with facets $\{s\}_{s \in S}$ and with irreducible Cartan matrix $A_P$ of negative type. Let $T$ be a proper subset of $S$ (i.e. $T \neq \varnothing, S$). Then: \begin{enumerate} \item\label{thm:vinberg_faces} If $A_T$ is of positive type and $\sharp T = k$, then $T \in \sigma(P)$ and its corresponding face $\cap_{s \in T} s $ is of dimension $n-k$. \item\label{thm:vinberg_faces_zero} If $A_T$ is of zero type and of rank $n-1$, then $T \in \sigma(P)$ and the face $\cap_{s \in T} s $ is of dimension $0$, i.e. a vertex of $P$. \end{enumerate} \end{theorem}
\begin{remark} Theorem \ref{thm:vinberg} is not explicitly stated in \cite[Theorems 4 and 7]{V}, but it is obtained by applying \cite[Theorem 4]{V} as in the proof of \cite[Theorem 7]{V}. \end{remark}
\begin{remark}\label{rem:ridge} Theorem \ref{thm:vinberg}.(\ref{thm:vinberg_faces}) tells us that for any mirror polytope $P$ with facets $\{s\}_{s\in S}$ and reflections $\{ \sigma_s = \mathrm{Id} - \alpha_s \otimes b_s \}_{s \in S}$, if $\alpha_s(b_t) \alpha_t(b_s) < 4$, then the intersection $s \cap t$ is a face of codimension $2$, i.e. a ridge of $P$. \end{remark}
\subsection{Coxeter groups}\label{subsec:Coxeter_groups}
A \emph{Coxeter matrix} $M=(M_{st})_{s,t \in S}$ on a finite set $S$ is a symmetric matrix with the entries $M_{st} \in \{1,2, \dotsc, m, \dotsc,\infty \}$ such that the diagonal entries $M_{ss}=1$ and the others $M_{st} \neq 1$. To any Coxeter matrix $M=(M_{st})_{s,t \in S}$ is associated a \emph{Coxeter group} $W_{S,M}$ given by a presentation $\langle\, S \mid (st)^{M_{st}}=1 \textrm{ for } M_{st} \neq \infty \,\rangle$. We denote the Coxeter group $W_{S,M}$ also simply by $W,W_S$ or $W_M$. The \emph{rank} of $W_{S}$ is the cardinality $\sharp S$ of $S$.
The \emph{Coxeter diagram} of $W_{S,M}$ is a labelled graph $\mathcal{G}_W$ such that (\emph{i}) the set of nodes (i.e. vertices) of $\mathcal{G}_W$ is the set $S$, (\emph{ii}) two nodes $s,t \in S$ are connected by an edge $\overline{st}$ of $\mathcal{G}_W$ if and only if $M_{st} \in \{3,\dotsc, m, \dotsc,\infty \}$, and (\emph{iii}) the edge $\overline{st}$ is labelled by $M_{st}$ if and only if $M_{st} > 3$. A Coxeter group $W$ is \emph{irreducible} if the Coxeter diagram $\mathcal{G}_W$ is connected.
An irreducible Coxeter group $W$ is \emph{spherical} (resp. \emph{affine}) if it is finite (resp. infinite and virtually abelian). For a Coxeter group $W$ (not necessarily irreducible), each connected component of the Coxeter diagram $\mathcal{G}_W$ corresponds to a Coxeter group, called a \emph{component} of $W$. A Coxeter group $W$ is \emph{spherical} (resp. \emph{affine}) if each component of $W$ is spherical (resp. affine). We sometimes refer to Appendix \ref{classi_diagram} for the list of all the irreducible spherical and irreducible affine Coxeter diagrams.
For each $T\subset S$, the subgroup $W'$ of $W$ generated by $T$ is called a \emph{standard subgroup of $W$}. It is well-known that $W'$ identifies with the Coxeter group $W_{T,M_{T}}$, where $M_{T}$ is the restriction of $M$ to $T \times T$. A subset $T \subset S$ is said to be “\emph{something}” if the Coxeter group $W_T$ is “something”. For example, the word “something” can be replaced by “spherical”, “affine” and so on. Two subsets $T, U \subset S$ are \emph{orthogonal} if $M_{tu}=2$ for every $t \in T$ and every $u \in U$. This relationship is denoted by $T \perp U$.
\subsection{Coxeter polytopes}
Recall that a \emph{Coxeter polytope} is a mirror polytope whose dihedral angles are submultiples of $\pi$, i.e. each dihedral angle is $\nicefrac{\pi}{m}$ for some integer $m \geqslant 2$ or $m=\infty$.
Let $P$ be a Coxeter polytope with the set of facets $S$ and the set of reflections $\{ \sigma_s = \mathrm{Id} - \alpha_s \otimes b_s \}_{s\in S}$. The \emph{Coxeter group $W_P$ of $P$} is the Coxeter group $W_{S,M}$ associated to a Coxeter matrix $M = (M_{st})_{s,t \in S}$ satisfying that $M_{st}=m_{st}$ if $\alpha_s(b_t) \alpha_t(b_s) = 4 \cos^2 (\nicefrac{\pi}{m_{st}})$ and $M_{st} = \infty$ if $\alpha_s(b_t) \alpha_t(b_s) \geqslant 4$. For a proper face $f$ of $P$ (i.e. $f \neq \varnothing$, $P$), we write $\sigma (f) = \{ s \in S \mid f \subset s \}$ and $W_f := W_{\sigma(f)}$.
\begin{theorem}[{Tits \cite[Chapter V]{MR0240238} for Tits simplex, and Vinberg \cite{V}}]\label{theo_vinberg} $\,$\\Let $P$ be a Coxeter polytope of $\mathbb{S}^n$ with Coxeter group $W_P$ and let $\Gamma_P$ be the subgroup of $\mathrm{SL}^{\pm}_{n+1}\mathbb{R}$ generated by the reflections $\{ \sigma_s\}_{s \in S}$. Then: \begin{enumerate} \item The homomorphism $\sigma\colon W_P \rightarrow \Gamma_P \subset \mathrm{SL}^{\pm}_{n+1}\mathbb{R}$ defined by $\sigma(s) = \sigma_s$ is an isomorphism.
\item The $\Gamma_P$-orbit of $P$ is a convex subset $\mathcal{C}_P$ of $\mathbb{S}^n$, and $\gamma \,\mathrm{Int}(P) \cap \mathrm{Int}(P) = \varnothing $ for all non-trivial $\gamma \in \Gamma_P$.
\item The group $\Gamma_P$ acts properly discontinuously on the interior $\Omega_P$ of $\mathcal{C}_P$.
\item An open proper face $f$ of $P$ lies in $\Omega_P$ if and only if the Coxeter group $W_f$ is spherical. \end{enumerate} \end{theorem}
Theorem \ref{theo_vinberg} tells us that $\Omega_P$ is a convex domain of $\mathbb{S}^n$, and that if $\Omega_P$ is properly convex then $\Omega_P/_{\Gamma_P}$ is a convex projective Coxeter orbifold.
\subsection{Relative hyperbolicity}\label{sec:rel_hyp}
Let $Y$ be a proper Gromov-hyperbolic space (see e.g. \cite[Section 2]{hruska} for a quick review and \cite[Part III.3]{bridson_haefliger} for details on Gromov-hyperbolic spaces). We recall that for every isometry $\gamma$ of $Y$, exactly one of the following holds: \begin{enumerate}
\item
$\gamma$ fixes a point of $Y$.
\item $\gamma$ fixes exactly one point of the Gromov boundary $\partial Y$ of $Y$.
\item $\gamma$ fixes two points of $\partial Y$. \end{enumerate} We say that $\gamma$ is \emph{parabolic} (resp. \emph{hyperbolic}) if (2) (resp. (3)) holds. Let $\Gamma$ be a subgroup of isometries of $Y$ that acts properly discontinuously. A subgroup of $\Gamma$ is \emph{parabolic} if it is infinite and contains no hyperbolic element. A parabolic subgroup fixes a unique point of $\partial Y$, called a \emph{parabolic point}. The stabiliser of a parabolic point is a maximal parabolic subgroup.
Relative hyperbolicity has many equivalent definitions, see e.g. \cite[Section 3]{hruska}. We recall one of them, named \emph{cusp uniform action}. A group $\Gamma$ is \emph{relatively hyperbolic with respect to a collection $\mathcal{P}$ of subgroups} if there exist a proper Gromov-hyperbolic metric space $Y$ and a properly discontinuous effective action of $\Gamma$ on $Y$ by isometry such that: \begin{itemize}
\item the collection $\mathcal{P}$ is a set of representatives of the conjugacy classes of maximal parabolic subgroups of $\Gamma$,
\item there exists a $\Gamma$-equivariant collection $\mathcal{H}$ of disjoint open horoballs\footnote{We refer to \cite[Section 2]{hruska} for the notion of horoball in Gromov-hyperbolic space.} centred at the parabolic points of $\Gamma$,
\item the action of $\Gamma$ on $Y \smallsetminus U$ is cocompact, where $U$ denotes the union of the horoballs in~$\mathcal{H}$. \end{itemize}
For example, the fundamental group of a cusped hyperbolic $n$-manifold (or $n$-orbifold) is relatively hyperbolic with respect to its cusp subgroups, which are virtually $\mathbb{Z}^{n-1}$ \cite{bowditch,farb}.
For $k\geqslant 2$, any $\mathbb{Z}^k$-subgroup $\Lambda$ of a relatively hyperbolic group $\Gamma$ with respect to a collection $\mathcal{P}$ of subgroups must lie in a conjugate of a subgroup $P \in \mathcal{P}$. Indeed, the centraliser of a hyperbolic element in a discrete subgroup of isometries of $Y$ is virtually $\mathbb{Z}$. Thus $\Lambda$ must contain a parabolic isometry $\delta$ with a unique fixed point $p\in \partial Y$, and any other element $\gamma \in \Lambda$ also has to fix $p$ since it commutes with $\delta$. So $\Lambda$ lies in the stabiliser of $p$.
In particular the fundamental group $\pi_1 X$ of Theorem \ref{thm:main}, which is relatively hyperbolic with respect to the collection $\mathcal{P} = \{\pi_1T_i,\,\pi_1T'_i\}_i$ of rank-$2$ abelian subgroups, is not relatively hyperbolic with respect to any proper sub-collection of $\mathcal{P}$.
We end this section by giving a criterion to determine when a Coxeter group is relatively hyperbolic with respect to a collection of standard subgroups. We will use this criterion in Section \ref{sec:caprace}.
\begin{theorem}[Moussong \cite{moussong} and Caprace \cite{caprace_cox_rel-hyp,caprace_erratum}]\label{moussong_caprace} Let $W_{S}$ be a Coxeter group, and let $\mathcal{T}$ be a collection of subsets of $S$. Then the group $W_S$ is relatively hyperbolic with respect to $\{ W_T \mid T \in \mathcal{T} \}$ if and only if the following hold: \begin{enumerate} \item\label{thm:Caprace1} For every irreducible affine subset $U \subset S$ of rank $\geqslant 3$, there exists $T \in \mathcal{T}$ such that $U \subset T$. \item\label{thm:Caprace2} For every pair of irreducible non-spherical subsets $S_1, S_2$ of $S$ with $S_1 \perp S_2$, there exists $T \in \mathcal{T}$ such that $S_1 \cup S_2 \subset T$. \item\label{thm:Caprace3} For every pair $T, T' \in \mathcal{T}$ with $T \neq T'$, the intersection $T \cap T'$ is spherical. \item\label{thm:Caprace4} For every $T \in \mathcal{T}$ and every irreducible non-spherical subset $U \subset T$, we have $U^{\perp} \subset T$, where $U^{\perp} := \{ s \in S \mid s \perp U\}$. \end{enumerate} \end{theorem}
\subsection{Operation on a simplex} \label{sec:truncated_simplex}
We introduce here three uniform\footnote{A polytope $P$ of dimension $n \geqslant 3$ (in the Euclidean space) is \emph{uniform} if it is a vertex-transitive polytope with uniform facets. A \emph{uniform polygon} is a regular polygon. By \emph{vertex-transitive}, we mean that the symmetry group of $P$ acts transitively on the set of vertices of $P$.} Euclidean 4-polytopes via truncation, rectification and bitruncation of the 4-simplex.
Roughly speaking, by truncation, rectification and bitruncation of a regular polytope $P \subset \mathbb{R}^n$ we mean cutting uniformly $P$ at \emph{every} vertex with a hyperplane orthogonal to the line joining the vertex to the barycentre. This operation is nicely described in the classical book of Coxeter \cite[Section 8.1]{Cox}. Combinatorially, by collapsing some ridges of the bitruncated $P$ to vertices, one gets the rectified $P$. For example, a truncated (resp. rectified, resp. bitruncated) 3-simplex is a truncated tetrahedron (resp. an octahedron, resp. a truncated tetrahedron) in Figure \ref{fig:truncated_3_simplices}.
\begin{figure}
\caption{\footnotesize The truncated (left), rectified (middle), and bitruncated (right) 3-simplex.}
\label{fig:truncated_3_simplices}
\end{figure}
We now explain in detail this operation for the 4-simplex. Consider a regular 4-simplex $\Delta\subset\mathbb{R}^4$ with barycentre the origin and vertices $v_1,\ldots,v_5$. We denote by $F_i$ the facet of $\Delta$ opposite to $v_i$ and by $H_i$ the closed half-space containing the origin with $F_i\subset\partial H_i$. Then $\Delta=H_1\cap\dotsc\cap H_5$.
Let $c :=|v_i|$ and fix a positive parameter $s\leqslant c$. We denote by $H'_i$ the closed half-space (depending on $s$) containing the origin such that $\partial H'_i$ is orthogonal to $ v_i$ and $\frac{s}{c}\, v_i \in\partial H'_i$, and we set $$Q_s= H_1\cap\ldots\cap H_5\cap H'_1\cap\ldots\cap H'_5.$$ Note that $Q_{c}=\Delta$ is the original simplex. There exist some numbers $0 < a < b < c$ such that the combinatorics of the 4-polytope $Q_s$ is constant for $s$ in $(a,b)$ and $(b,c)$, and changes at $s=a$, $b$ and $c$. The polytope $Q_s$ (depicted in Figure \ref{fig:truncated_simplices}) is called: \begin{itemize} \item a \emph{truncated 4-simplex} for $s\in(b,c)$, \item a \emph{rectified 4-simplex} for $s=b$, \item a \emph{bitruncated 4-simplex} for $s\in(a,b)$. \end{itemize}
\begin{figure}
\caption{\footnotesize The Schlegel diagrams of the truncated (left), rectified (middle), and bitruncated (right) 4-simplex. The facets $F'_i$ are coloured darker than $F_i$ (cf. Table \ref{table:comb} below).}
\label{fig:truncated_simplices}
\end{figure}
The rectified simplex $Q_{b}$ is in fact the convex hull of the midpoints of the edges of the regular simplex $\Delta=Q_{c}$, while $Q_a$ is another rectified simplex. For all $s\in[a,c]$, the polytope $Q_s$ is uniform.
\subsection{The combinatorics of the rectified and bitruncated 4-simplices}\label{sec:combi}
We describe the combinatorics of $Q_s$ for $s \in (a,b]$.
The link of a vertex of the rectified (resp. bitruncated) 4-simplex is a triangular prism (resp. a tetrahedron) as in Figure \ref{fig:link_comb}. Each vertex of $Q_s$ is the intersection of facets $F'_{i_1}\cap F'_{i_2}\cap F_{i_3}\cap F_{i_4}\cap F_{i_5}$ for all distinct $i_1,\ldots,i_5$ (resp. $F'_{i_1}\cap F'_{i_2}\cap F_{i_3}\cap F_{i_4}$ for all distinct $i_1,\ldots,i_4$), where $F_i$ and $F'_i$ denote the facets of $Q_s$ whose supporting hyperplanes are $\partial H_i$ and $\partial H'_i$, respectively. The 10 facets of $Q_s$ are divided into 5 octahedra (resp. truncated tetrahedra) $F_i\subset \partial H_i$, and 5 tetrahedra (resp. truncated tetrahedra) $F'_i\subset \partial H'_i$. For all $i\neq j$, the ridge $F_i\cap F_j$ is a triangle, $F'_i\cap F'_j$ is a vertex (resp. triangle), and $F_i\cap F'_j$ is a triangle (resp. hexagon), while $F_i\cap F'_i=\varnothing$ (see Table \ref{table:comb}).
\begin{table}[!h] \begin{center}
\begin{tabular}{c||c|c|c}
& truncated 4-simplex & rectified 4-simplex & bitruncated 4-simplex \\
\hline
$F_i$ & truncated tetrahedron & octahedron & truncated tetrahedron \\
$F'_j$ & tetrahedron & tetrahedron & truncated tetrahedron \\
\hline
$F_i \cap F_j$ & hexagon & triangle & triangle \\
$F_i \cap F'_i$ & $\varnothing$ & $\varnothing$ & $\varnothing$ \\
$F_i \cap F'_j$ & triangle & triangle & hexagon \\
$F'_i \cap F'_j$ & $\varnothing$ & vertex & triangle \\
\hline \end{tabular} \end{center} \caption{\footnotesize Some information on the faces of the truncated, rectified and bitruncated $4$-simplex. The symbols $i,j$ are two distinct indices in $\{1,\ldots, 5 \}$. }\label{table:comb} \end{table}
\begin{figure}
\caption{\footnotesize The vertex link of the rectified (left) and bitruncated (right) 4-simplex $Q_s$. A facet of the link labelled by $i$ (resp $i'$) corresponds to the facet $F_i$ (resp. $F'_i$) of the 4-polytope $Q_s$.}
\label{fig:link_comb}
\end{figure}
The following is a simple observation, but it will be useful later to prove Theorem \ref{thm:main} (more precisely, Proposition \ref{prop:P_theta}).
\begin{lemma}\label{lem:combi} Let $Q_s$ be the rectified or bitruncated $4$-simplex for $s \in (a,b]$. Relabel each facet $F'_i$ of $Q_s$ with $F_{i'}$, and let $S := \{1',\ldots,5',1,\ldots, 5\}$. \begin{enumerate} \item\label{lem:combi_rectified} In the case that $Q_s$ is the rectified $4$-simplex, i.e. $s = b$, each vertex of $Q_s$ corresponds to a subset $\{i',j',k,l,m\} \subset S $ with $\sharp \{ i,j,k,l,m\} = 5$ and each edge of $Q_s$ corresponds to a subset $\{i',j,k\} \subset S $ with $\sharp \{ i,j,k \} = 3$.
\item\label{lem:combi_bitruncated} In the case that $Q_s$ is the bitruncated $4$-simplex, i.e. $a< s < b$, each vertex of $Q_s$ corresponds to a subset $\{i',j',k,l\} \subset S$ with $\sharp \{ i,j,k,l\} = 4$. \end{enumerate} \end{lemma}
\subsection{The ideal hyperbolic rectified 4-simplex}\label{subsec:hyperbolic_rectified}
Every vertex-transitive polytope $Q\subset\mathbb{R}^n$ can be realised as an ideal hyperbolic $n$-polytope, obtained by interpreting the ball in which $Q$ is inscribed as a projective model of the hyperbolic $n$-space $\mathbb{H}^n$. What is nice about the rectified simplex of dimension $n \leqslant 4$ is that its regular ideal hyperbolic realisation $R\subset\mathbb{H}^n$ is a Coxeter polytope.\footnote{It is well-known that the link of the regular ideal hyperbolic rectified $n$-simplex is a Euclidean right simplicial $(n-1)$-prism with regular $(n-2)$-simplicial bases, and the dihedral angle of the Euclidean regular $(n-2)$-simplex is $\arccos(\nicefrac{1}{(n-2)})$. The latter is $\nicefrac{\pi}{m}$ for some integer $m \geqslant 2$ if and only if $n \leqslant 4$. Hence, $R$ is a Coxeter polytope if and only if $n \leqslant 4$.} For instance, the polytope $R$ of dimension 3 is a right-angled hyperbolic octahedron.
Let $R\subset\mathbb{H}^4$ be the ideal rectified 4-simplex. The facets of $R$ are regular ideal Coxeter $3$-polytopes: five right-angled octahedra $F_i$ and five $\nicefrac\pi3$-angled tetrahedra $F'_i$. The horospherical link of any (ideal) vertex of $R$ is a Euclidean right triangular prism with equilateral bases (see the left of Figure \ref{fig:link_comb}). Thus, for all $i\neq j$ the dihedral angle at a ridge $F_i\cap F_j$ (resp. $F_i \cap F'_j$) is $\nicefrac{\pi}{3}$ (resp. $\nicefrac{\pi}{2}$), while $F'_i$ and $F'_j$ are parallel, i.e. the facet $F'_i$ is tangent to $F'_j$ at infinity.
\begin{remark}\label{rem:mathematica} The bitruncated 4-simplex is ``combinatorially'' a ``filling'' of the ideal rectified 4-simplex in the sense that the latter is obtained from the former by collapsing each triangular ridge $F'_i\cap F'_j$ for $i \neq j$ to a point and removing it. We call such triangles the \emph{filling ridges} of the bitruncated 4-simplex. \end{remark}
\subsection{Decomposability and Euler characteristics}
We conclude the section with a remark on convex projective manifolds that has probably been noticed by many experts of the subject, but whose explicit statement seems to miss in the literature. This remark is a simple consequence of works of Vey, Benoist and Gottlieb.
\begin{fact}\label{fact:indecomposable} If a convex projective manifold $\Omega/_\Gamma$ is decomposable, then $\chi(\Omega/_\Gamma)=0$. \end{fact}
\begin{proof} Assume by contradiction that $\Omega/_\Gamma$ is decomposable but $\chi(\Omega/_\Gamma) \neq 0$. By a theorem of Gottlieb \cite[Corollary IV.3]{gottlieb}, the fundamental group of a finite, aspherical polyhedron with non-zero Euler characteristic has trivial centre. But since $\Omega/_\Gamma$ is decomposable, by Proposition 4.4 of Benoist \cite{vey,cd2}, the centre of $\Gamma$ contains a non-trivial free abelian subgroup. \end{proof}
In particular, since $\chi(X)\neq0$, the convex projective manifold $X$ of Theorem \ref{thm:main} is indecomposable.
\section{The proof of Theorem \ref{thm:main}} \label{sec:proof}
In this section, we prove Theorem \ref{thm:main}. In Section \ref{sec:P_alpha}, we perform convex projective generalised Dehn filling to the ideal hyperbolic rectified 4-simplex $R\subset\mathbb{H}^4$, and build mirror polytopes $P_{\alpha}$ combinatorially equivalent\footnote{Two polytopes $Q$ and $Q'$ are \emph{combinatorially equivalent} if the face poset of $Q$ is isomorphic to the face poset of $Q'$.} to the bitruncated 4-simplex. In Section \ref{sec:caprace}, we show that the Coxeter group $W_p=W_{P_{\nicefrac{\pi}{p}}}$ of the Coxeter polytope $P_{\nicefrac{\pi}{p}}$ is relatively hyperbolic. In Sections \ref{sec:M} and \ref{sec:X}, we construct the cusped hyperbolic 4-manifold $M$, by gluing some copies of $R$, and a filling $X$ of $M$, respectively. Finally, in Section \ref{sec:sigma_alpha}, we give $X$ a structure of projective cone-manifold induced by $P_\alpha$, and finish the proof of Theorem \ref{thm:main}.
\subsection{Deforming the rectified 4-simplex} \label{sec:P_alpha}
In this subsection, we obtain a family of projective structures on the orbifold $\mathcal{O}_R$ associated to the ideal hyperbolic rectified 4-simplex $R$ by deforming its original complete hyperbolic structure.
Accordingly, the composition of two projective reflections along the facets $F'_i$ and $F'_j$ of $\mathcal{O}_R$, $i\neq j$, (recall the notation from Section \ref{sec:truncated_simplex}) will deform to be conjugate to a projective rotation of angle $2\alpha > 0$. At the original hyperbolic structure on $\mathcal{O}_R$, the facets $F'_i$ and $F'_j$ are parallel, i.e. $\alpha=0$. But, at the deformed projective structure on $\mathcal{O}_R$, new ridges $F'_i\cap F'_j$ can be added to $\mathcal{O}_R$ to form a mirror polytope $P_{\alpha}$ that is combinatorially equivalent to a bitruncated 4-simplex, where $\alpha$ is the dihedral angle of the ridge $F'_i\cap F'_j$. So, the goal is to prove the following:
\begin{proposition}\label{prop:P_theta} There exists a path $(0,\nicefrac{\pi}{3}]\ni\alpha\mapsto P_\alpha$ of mirror polytopes with the combinatorics of a bitruncated 4-simplex and dihedral angles: \begin{itemize} \item $\alpha$ at the filling ridges $F'_i\cap F'_j$, \item $\nicefrac\pi2$ at the ridges $F_i\cap F'_j$, and \item $\nicefrac\pi3$ at the ridges $F_i\cap F_j$. \end{itemize} Moreover, the limit $P_0$ is the ideal hyperbolic rectified 4-simplex $R$. \end{proposition}
Before proving Proposition \ref{prop:P_theta}, we begin with some auxiliary lemmas. First, let $$t_3 = \frac12\left(11 + 9\sqrt2 - 3\sqrt{31 + 22\sqrt2}\right)\approx0.0422 $$ and for $t\in\left[t_3,1\right]$, $$f(t)=\frac{t(t+2)^3(2t+1)^3}{(t^2+t+1)^2(t^2+7t+1)^2}, \;\; h(t) = 2 + \frac{1}{t} + \frac{ t (t+2)^4 }{(t^2+t+1)(t^2+7t+1)},$$ $$ g_p(t) = \frac{2 t^p (t+2)^p (2t+1)^{3-p}}{(t^2+t+1)(t^2+7t+1)} \;\; \textrm{and} \;\; \overline{g}_p(t) = \frac{4 f(t)}{g_p(t)}, \;\; \textrm{where} \;\; p=0,1,2,3.$$ The number $t_{3}$ is the unique positive solution less than $1$ satisfying the equation $f(t_3) = \nicefrac{1}{4}= \cos^{2} (\nicefrac{\pi}{3})$. Since $t_3$ is positive, it is easy to check:
\begin{lemma} \label{lem:negative_coefficients} The functions $f,h,g_p$ and $\overline g_p\colon\left[t_3,1\right]\to\mathbb{R}$ are well-defined and positive. \end{lemma}
Given $t \in [t_3,1]$, we now consider the matrix $$C_t = \left(
\begin{array}{ccccc | ccccc} 2 & -\overline{g}_0(t) & -\overline{g}_1(t) & -\overline{g}_2(t) & -\overline{g}_3(t) & -h(t) & 0 & 0 & 0 & 0 \\ -g_0(t) & 2 & -\overline{g}_0(t) & -\overline{g}_1(t) & -\overline{g}_2(t) & 0 & -h(t) & 0 & 0 & 0 \\ -g_1(t) & -g_0(t) & 2 & -\overline{g}_0(t) & -\overline{g}_1(t) & 0 & 0 & -h(t) & 0 & 0 \\ -g_2(t) & -g_1(t) & -g_0(t) & 2 & -\overline{g}_0(t) & 0 & 0 & 0 & -h(t) & 0 \\ -g_3(t) & -g_2(t) & -g_1(t) & -g_0(t) & 2 & 0 & 0 & 0 & 0 & -h(t) \\ \hline -2 & 0 & 0 & 0 & 0 & 2 & -\nicefrac1t & -\nicefrac1t & -\nicefrac1t & -\nicefrac1t \\ 0 & -2 & 0 & 0 & 0 & -t & 2 & -\nicefrac1t & -\nicefrac1t & -\nicefrac1t \\ 0 & 0 & -2 & 0 & 0 & -t & -t & 2 & -\nicefrac1t & -\nicefrac1t \\ 0 & 0 & 0 & -2 & 0 & -t & -t & -t & 2 & -\nicefrac1t \\ 0 & 0 & 0 & 0 & -2 & -t & -t & -t & -t & 2 \\ \end{array} \right). $$
\begin{lemma} \label{lem:rank} For every $t\in\left[t_3,1\right]$, the matrix $C_t$ is an irreducible Cartan matrix of negative type and of rank $5$. \end{lemma}
\begin{proof} It easily follows from Lemma \ref{lem:negative_coefficients} that $C_t$ is an irreducible Cartan matrix. A simple but long computation shows that the rank of $C_t$ is $5$, which is less than $9 = 10 - 1$. Hence, the irreducible Cartan matrix $C_t$ is of negative type by Proposition \ref{prop:cartan_type}. \end{proof}
\begin{lemma} \label{lem:alpha} There exists a monotonically decreasing analytic function $\alpha\colon\left[t_3,1\right]\to\mathbb{R}$ satisfying $$\cos^2\alpha(t)=f(t),\quad\alpha(t_3)=\frac\pi3,\quad\alpha(1)=0.$$ \end{lemma}
\begin{figure}
\caption{\footnotesize The graph of $f(t)$ over the interval $[0,1]$}
\label{func:graph_f}
\end{figure}
\begin{proof} The derivative $f'$ of $f$ satisfies: $$f'(t)= - \frac{2(t-1)(t+1)(t+2)^2(2t+1)^2(t^4+2t^3+21t^2+2t+1)}{(t^2+t+1)^3(t^2+7t+1)^3}.$$ Since $f'(t)$ is positive for all $t \in [t_3, 1)$ and is zero for $t=1$, the function $f \colon [t_3,1] \rightarrow \mathbb{R}$ is monotonically increasing with $f(t_3)=\nicefrac{1}{4}$ and $f(1)=1$ (see Figure \ref{func:graph_f}). The lemma now follows easily. \end{proof}
We are finally ready to prove Proposition \ref{prop:P_theta} by applying Theorems \ref{thm:vinberg_unique} and \ref{thm:vinberg}. Recall the combinatorics of the truncated, rectified and bitruncated 4-simplex, described in Table \ref{table:comb}. See Figure \ref{fig:link_geom} for a geometric picture of the vertex link of the mirror polytope $P_\alpha$.
\begin{figure}
\caption{\footnotesize The vertex links of a mirror polytope are also mirror polytopes. For $P_\alpha$ we have: $(i)$ a right triangular prism with equilateral bases for the (ideal) rectified 4-simplex $P_0$ (left), and $(ii)$ a tetrahedron for the bitruncated 4-simplex $P_\alpha$, $\alpha\in(0,\frac\pi3]$ (right). The edges of dihedral angle $\nicefrac{\pi}{2}$ (resp. $\nicefrac{\pi}{3}$, resp. $\alpha$) are drawn in black (resp. red, resp. green).}
\label{fig:link_geom}
\end{figure}
\begin{proof}[Proof of Proposition \ref{prop:P_theta}] Let us call and order the indices of $C_t$ as $S:=\{ 1',\ldots,5',1,\ldots,5 \}$. Lemmas \ref{lem:rank} and \ref{lem:alpha} together with Theorem \ref{thm:vinberg_unique} imply that there exists a path $[0,\nicefrac{\pi}{3}]\ni \alpha \mapsto P_\alpha$ of mirror polytopes in $\mathbb{S}^4$ with facets $\{ F_s\}_{s\in S}$ and with Cartan matrix $C_t$. The facet $F_{i'}$ of $P_\alpha$ ($i = 1, \dotsc, 5$) is also denoted by $F_i'$.
We consider two cases separately: $(i)$ $\alpha \in (0,\nicefrac{\pi}{3}]$ and $(ii)$ $\alpha = 0$.
In the case $(i)$, equivalently, $t \in [t_3,1)$, by Remark \ref{rem:ridge} (or Theorem \ref{thm:vinberg}.(\ref{thm:vinberg_faces})), the intersections $F'_{i} \cap F'_{j}$, $F_i \cap F'_{j}$ and $F_i \cap F_j$, $i \neq j$, are ridges of $P_{\alpha}$ and their dihedral angles are $\alpha$, $\nicefrac{\pi}{2}$ and $\nicefrac{\pi}{3}$, respectively, because for example $$ (C_t)_{i' j'} (C_t)_{j' i'} = g_p(t) \overline{g}_p(t) = 4 f(t) = 4 \cos^2\alpha < 4.$$ We now claim that $P_{\alpha}$ is combinatorially equivalent to the bitruncated $4$-simplex. For every subset $T = \{ i', j', k, l \} \subset S$ with $\sharp \{i, j, k, l \} = 4$, the submatrix $(C_t)_T$ is the direct sum $(C_t)_{\{i', j'\}} \oplus (C_t)_{\{k, l\}}$ of matrices of positive type, hence $\{ F_{s} \}_{s \in T} \in \sigma(P_\alpha)$ \footnote{See the end of Section \ref{sec:gen_dehn_fill} for the definition of the poset $\sigma(P)$.} and $ \cap_{s \in T} F_{s}$ is a vertex of $P_\alpha$ by Theorem \ref{thm:vinberg}.(\ref{thm:vinberg_faces}).
That is, if $\mathcal{V}$ denotes the set of all subsets $\{ F_{i}', F_{j}', F_{k}, F_{l} \}$ with $\sharp \{ i,j,k,l\} = 4$, then $\mathcal{V} \subset \sigma(P_{\alpha})$. Let $\hat{S} = \{ F_s \}_{s \in S}$, and let $\mathcal{F}$ be the subposet of $2^{\hat{S}}$ defined by: $$ \mathcal{F} := \{ \hat{T}' \in 2^{\hat{S}} \mid \hat{T}' \subset \hat{T} \textrm{ for some } \hat{T} \in \mathcal{V} \}. $$
As in the previous argument, Theorem \ref{thm:vinberg}.(\ref{thm:vinberg_faces}) implies that $\mathcal{F}$ is a subposet of $\sigma(P_{\alpha})$. We know, in addition, from Lemma \ref{lem:combi}.(\ref{lem:combi_bitruncated}) and Figure \ref{fig:link_comb} that the poset $\mathcal{F}$ is dual to the face poset of the bitruncated 4-simplex. It is a well-known fact (e.g. \cite[Exercise 1.1.20]{BP15}) that if two polytopes $Q$ and $Q'$ are of same dimension and the face poset of $Q'$ is a subposet of the face poset of $Q$, then $Q$ is combinatorially equivalent to $Q'$. As a result, the polytope $P_{\alpha}$ is combinatorially equivalent to the bitruncated $4$-simplex, as claimed.
In the case $(ii)$, equivalently, $t=1$, we claim that $P_{0}$ is combinatorially equivalent to the rectified 4-simplex. For every subset $U = \{ i', j', k, l, m \} \subset S$ with $\sharp \{i, j, k, l, m \} = 5$, the submatrix $(C_1)_U$ is the direct sum $(C_1)_{\{i', j'\}} \oplus (C_1)_{\{k, l, m\}}$ of matrices of zero type and the rank of $(C_1)_U$ is $3$. Hence $\{ F_{s} \}_{s \in U} \in \sigma(P_0)$ and $ \cap_{s \in U} F_{s}$ is a vertex of $P_0$ by Theorem \ref{thm:vinberg}.(\ref{thm:vinberg_faces_zero}). Furthermore, for every subset $U' = \{ i', j, k \} \subset S$ with $\sharp \{i, j, k \} = 3$, the submatrix $(C_1)_{U'}$ is of positive type. Hence $\{ F_{s} \}_{s \in U'} \in \sigma(P_0)$ and $ \cap_{s \in U'} F_{s}$ is an edge of $P_0$ by Theorem \ref{thm:vinberg}.(\ref{thm:vinberg_faces}). Then, as in the proof of case $(i)$, using Lemma \ref{lem:combi}.(\ref{lem:combi_rectified}) and Figure \ref{fig:link_comb}, we may conclude that the polytope $P_{0}$ is combinatorially equivalent to the rectified $4$-simplex, as claimed. Finally, a simple computation shows that the Cartan matrix $C_1$ is equivalent to a symmetric matrix of signature $(4,1)$, and therefore the polytope $P_{0}$ (without vertices) may be identified with the ideal hyperbolic rectified $4$-simplex $R$. \end{proof}
\begin{remark} \label{rem:symmmetry} The symmetry group of the mirror polytope $P_\alpha$, $\alpha \in [0,\nicefrac{\pi}{3}]$, is of order $\geqslant 5$. For, if we set $\hat{Q} = \bigl(\begin{smallmatrix} Q & 0 \\ 0 & Q \end{smallmatrix} \bigr)$ with $$ Q = \left( \begin{array}{ccccc} 0 & 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ \end{array} \right), $$ then $\hat{Q}$ is a permutation matrix of order $5$ and $\hat{Q} C_t {\hat{Q}}^{-1}$ is equivalent to $C_t$. \end{remark}
\begin{remark} A (bit more complicated) computation reveals that the deformation space of projective structures on the orbifold $\mathcal{O}_R$ associated to the ideal hyperbolic rectified $4$-simplex $R$ is six-dimensional. In other words, one can find a six-parameter family of Cartan matrices (of rank $5$) which correspond to projective structures on $\mathcal{O}_R$. But, we choose a particular one-parameter family of Cartan matrices having symmetry as described in Remark \ref{rem:symmmetry} in order to simplify the computation. \end{remark}
We end the subsection with the following.
\begin{corollary} \label{cor:coxeter} For every integer $p\geq3$, the mirror polytope $P_{\nicefrac{\pi}{p}}$ is a Coxeter polytope. Moreover, if $\Gamma_p$ denotes the subgroup of $\mathrm{SL}^{\pm}_5\mathbb{R}$ generated by the associated reflections, then the $\Gamma_p$-orbit of $P_{\nicefrac{\pi}{p}}$ is a properly convex domain $\Omega_p$ of $\mathbb{S}^4$, i.e. it is divisible by $\Gamma_p$. \end{corollary}
\begin{proof} It is obvious that the mirror polytope $P_{\nicefrac{\pi}{p}}$ is a Coxeter polytope. For any subset $T = \{i',j',k,l\} \subset S$ with $\sharp \{ i,j,k,l\} = 4$, the standard subgroup $W_T$ of the Coxeter group $W_S$ of $P_{\nicefrac{\pi}{p}}$ is isomorphic to $D_p\times D_3$, where $D_m$ is the dihedral group of order $2m$, hence it is a finite group. In the proof of Proposition \ref{prop:P_theta}, we show that every vertex of $P_{\nicefrac{\pi}{p}}$ corresponds to a subset $\{i',j',k,l\} \subset S$ with $\sharp \{ i,j,k,l\} = 4$, so by Theorem \ref{theo_vinberg}, the $\Gamma_p$-orbit of $P_{\nicefrac{\pi}{p}}$ is a divisible convex domain of $\mathbb{S}^4$. \end{proof}
\subsection{The Coxeter group $W_p$}\label{sec:caprace}
The goal of this subsection is to show that the Coxeter group $W_{p}$ of $P_{\nicefrac\pi p}$ is relatively hyperbolic with respect to a collection of virtually abelian subgroups of rank $2$. To do so, we need to analyse the Coxeter diagram of $W_{p}$ in Figure \ref{diag_Wk}, and to use Theorem \ref{moussong_caprace} together with the (complete) list of the irreducible spherical and affine Coxeter groups in Appendix \ref{classi_diagram}.
\begin{figure}
\caption{\footnotesize The Coxeter diagram of the Coxeter group $W_p$ of $P_{\nicefrac\pi p}$}
\label{diag_Wk}
\end{figure}
We denote by $\mathcal{T}_{p}$ the following collection of subsets of $S=\{ 1', \ldots,5', 1, \ldots,5 \}$: \begin{itemize}
\item In the case of $p \geqslant 4$, the collection $\mathcal{T}_{p}$ consists of all subsets $\{i',j',k,l,m\}$ of $S$ with $\sharp \{ i,j,k,l,m\} = 5$, so the cardinality of $\mathcal{T}_p$, $p \geqslant 4$, is $10$.
\item In the case of $p = 3$, the collection $\mathcal{T}_{3}$ consists of all subsets $\{i',j',k,l,m\}$ and $\{i',j',k',l,m\}$ of $S$ with $\sharp \{ i,j,k,l,m\} = 5$, so the cardinality of $\mathcal{T}_3$ is $20$. \end{itemize} For each $U \in \mathcal{T}_p$, the standard subgroup $(W_p)_U$ of $W_p$ is isomorphic to $\widetilde{A}_{2} \times I_2(p)$ (see Appendix \ref{classi_diagram}), hence it is virtually isomorphic to $\mathbb{Z}^{2}$.
\begin{proposition}\label{prop:rel_hyp} The Coxeter group $W_p$ is relatively hyperbolic with respect to the collection of subgroups $ \{ (W_p)_U \mid U \in \mathcal{T}_p \}$, in particular, a collection of virtually abelian subgroups of rank $2$. \end{proposition}
\begin{proof} We only prove it for the case $p=3$; the argument is similar for other cases $p \geqslant 4$. Thanks to Theorem \ref{moussong_caprace}, we just need to carefully analyse the Coxeter diagram of $W_p$ in Figure $\ref{diag_Wk}$, using the list of irreducible spherical and affine Coxeter groups in Appendix \ref{classi_diagram}.
First, the condition (\ref{thm:Caprace1}) holds because all irreducible affine subsets $U \subset S$ of rank $\geqslant 3$ are $\{i',j',k'\}$ and $\{i,j,k\}$ with $\sharp \{i,j,k\} = 3$. Second, the condition (\ref{thm:Caprace2}) holds because there does not exist a pair of irreducible non-spherical subsets $S_{1},S_{2}$ of $S$ with $S_1 \perp S_2$. Third, the condition (\ref{thm:Caprace3}) holds because for every pair $T, T' \in \mathcal{T}_3$ with $T \neq T'$, the intersection $T \cap T'$ is a subset of $\{i',j',k,l\}$ with $\sharp \{i,j,k,l\} =4$. Finally, the condition (\ref{thm:Caprace4}) holds because for every $T \in \mathcal{T}_3$, there exists only one irreducible non-spherical subset $U \subset T$, which is either $\{i',j',k'\}$ or $\{i,j,k\}$ with $\sharp \{i,j,k\} = 3$, and $T = U\, \sqcup \, U^{\perp}$. \end{proof}
\begin{remark} The Coxeter group $W_p$ of $P_{\nicefrac{\pi}{p}}$ is in fact a finite-index subgroup of the following Coxeter group $\hat{W}_p$: \begin{center} \begin{tikzpicture}[thick,scale=0.8, every node/.style={transform shape}] \node[draw, circle, inner sep=1pt, minimum size=4mm] (1) at (0,0) {}; \node[draw, circle, inner sep=1pt, minimum size=4mm] (2) at (2,0) {}; \node[draw, circle, inner sep=1pt, minimum size=4mm] (3) at (4,0) {}; \node[draw, circle, inner sep=1pt, minimum size=4mm] (4) at (6,0) {}; \node[draw, circle, inner sep=1pt, minimum size=4mm] (5) at (8,0) {}; \node[draw, circle, inner sep=1pt, minimum size=4mm] (6) at (10,0) {};
\draw (1) -- (2) node[above,midway] {$6$}; \draw (2) -- (3) node[above,midway] {}; \draw (3) -- (4) node[above,midway] {}; \draw (4) -- (5) node[above,midway] {}; \draw (5) -- (6) node[above,midway] {$2p$}; \end{tikzpicture} \end{center} It is easier to verify the conditions (\ref{thm:Caprace1}) -- (\ref{thm:Caprace4}) of Theorem \ref{moussong_caprace} for the Coxeter group $\hat{W}_p$. \end{remark}
\subsection{The cusped hyperbolic manifold} \label{sec:M}
In this subsection, we build the cusped hyperbolic manifold $M$ of Theorem \ref{thm:main}. A reader who is not familiar with hyperbolic manifolds (with totally geodesic boundary) could consult \cite[Chapter 3]{Mbook}.
\begin{figure}
\caption{\footnotesize The edge-labelled complete graph $K_6$}
\label{fig:K6}
\end{figure}
We first recall the construction of a building block $B$ by Kolpakov and Slavich \cite{KS}. Consider the complete graph $K_6$ on 6 vertices with its edges labelled by numbers in $\{1,\ldots,5\}$ so that adjacent edges have distinct labels (see Figure \ref{fig:K6}). For each vertex of $K_6$, take a copy of the ideal hyperbolic rectified 4-simplex $R\subset\mathbb{H}^4$, described in Section \ref{subsec:hyperbolic_rectified}. If two vertices of $K_6$ are connected by an edge of label $i\in\{1,\ldots,5\}$, then we glue together the facets $F_i$ of the two corresponding copies of $R$ using the identity as a gluing map.
\begin{proposition}[{\cite[Section 3]{KS}}]\label{prop:Building_Block} Let $B$ be the building block constructed above. \begin{itemize}
\item\label{prop:B_mfd} The space $B$ is a non-orientable, complete, finite-volume hyperbolic $4$-manifold with non-compact totally geodesic boundary.
\item\label{prop:B_bdry} The boundary $\partial B$ of $B$ has exactly 5 connected components $\partial_1,\ldots,\partial_5$, each tessellated by the facets $F'_i$ of the copies of $R$ in $B$.
\item\label{prop:B_cusps} The hyperbolic manifold $B$ has exactly 10 cusps $C_{ij}$, $i<j\in\{1,\ldots,5\}$. Each cusp section $S_{ij}$ is diffeomorphic to $K\times [0,1]$, where $K$ denotes the Klein bottle. One boundary component of $S_{ij}$ is contained in $\partial_i$, and the other in $\partial_j$. \end{itemize} \end{proposition}
Now, consider again the edge-labelled graph $K_6$ in Figure \ref{fig:K6}. For each vertex of $K_6$, take a copy of $B$. If two vertices of $K_6$ are joined by an edge of label $i\in\{1,\ldots,5\}$, then we glue together the boundary components $\partial_i$ of the two corresponding copies of $B$ using the identity as a gluing map. Let us call $M'$ the resulting cusped hyperbolic manifold (without boundary), and $M$ its orientable double cover.
\begin{proposition}\label{prop:M} The orientable hyperbolic $4$-manifold $M$ has exactly 10 cusps, each with section diffeomorphic to the $3$-torus. \end{proposition} \begin{proof} We obtain the cusps of $M'$ by gluing together the cusps of the copies of $B$, and the gluing maps are induced by the identity. Hence, each cusp section of $M'$ must be diffeomorphic to $K\times\mathbb{S}^1$. By construction, for each pair $i<j\in\{1,\ldots,5\}$, the simple cycle (of length $6$) in the graph $K_6$ with edges labelled alternately by $i$ and $j$ corresponds to a cusp of $M'$, thus $M'$ has exactly 10 cusps. Since the cusps of $M'$ are non-orientable, each cusp section of $M$ is the orientable double covering of a cusp section of $M'$. In particular, $M$ has precisely 10 cusps, each with $3$-torus section. \end{proof}
\begin{remark} \label{rem:orb_cov_M} The natural map $M\to \mathcal{O}_R$ is an orbifold covering of degree $6\cdot6\cdot2=72$, where $\mathcal{O}_R$ denotes the orbifold associated to $R$. \end{remark}
\subsection{Topology of the filling} \label{sec:X} In this subsection, we build the manifold $X$ of Theorem \ref{thm:main} topologically. The desired (singular) geometric structures on $X$ will be given in the next subsection.
Let $Q \subset \mathbb{R}^4$ be a bitruncated $4$-simplex. We define $X$ in the same way as the manifold $M$ (see Section \ref{sec:M}), but substituting the ideal rectified simplex $R$ with the bitruncated simplex $Q$.
\begin{proposition} The space $X$ is a closed, orientable, smooth $4$-manifold, containing 10 pairwise disjoint embedded $2$-tori whose complement is diffeomorphic to the cusped hyperbolic manifold $M$. \end{proposition}
\begin{proof} To prove that the polyhedral complex $X$ is a smooth manifold, it suffices to show that the link of each vertex is homeomorphic to the 3-sphere. Recall that the polytope $Q$ is vertex transitive. The vertex link $L$ of $Q$ is the tetrahedron in the right of Figure \ref{fig:link_comb}.
When we glue the first 6 copies of $Q$ to form $B$, for each vertex of $Q$, 6 copies of $L$ are glued cyclically around one edge, to form a polyhedral complex $L'$ homeomorphic to the closed 3-disc $D^3$. Thus, $B$ is a 4-manifold with boundary. Note that $R$ is homeomorphic to the complement $Q \smallsetminus \bigcup_{i<j}F'_j\cap F'_j$ of the filling ridges; in particular, $B$ is non-orientable (recall Proposition \ref{prop:Building_Block}). When we glue 6 copies of $B$ to get $X'$, for each vertex of the complex, 6 copies of $L'$ are glued cyclically around a circle $C\subset\partial L'$. The resulting polyhedral complex is clearly homeomorphic to the 3-sphere, so $X'$ is a manifold. Its orientation cover $X$ is thus a manifold.
Now, the cusped hyperbolic manifold $M$ is diffeomorphic to the interior of a compact manifold $\overline M$ with boundary $\partial\overline M$ consisting of 10 3-tori (recall Proposition \ref{prop:M}). Let $\Sigma\subset X$ be the union of the copies of the filling ridges of $Q$. Clearly, $M$ is diffeomorphic to $X\smallsetminus\Sigma$. Moreover, $\partial\overline M$ is diffeomorphic to the boundary of a regular neighbourhood of $\Sigma$ in $X$. Thus, $\Sigma$ consists of 10 2-tori. \end{proof}
We conclude the subsection with some additional information about $X$.
\begin{remark}\label{rem:chi_X} The 4-manifold $X$ has Euler characteristic $\chi(X)=12$. Indeed, $\mathcal O_R$ has orbifold Euler characteristic $\chi^{\mathrm{orb}}(\mathcal O_R)=\frac16$ \cite[Appendix 1]{KS}, and the covering $M\to \mathcal O_R$ has degree 72 (recall Remark \ref{rem:orb_cov_M}), so $\chi(M)=12$. Since $\chi(\partial\overline M)=0$ and $\chi(\Sigma)=0$, we have $\chi(X)=\chi(M)=12$. \end{remark}
\begin{remark}\label{rem:6-filling} The manifold $X$ is a filling of $M$, which has a maximal cusp section $S$ such that each filling curve in $S$ has length $6$. The reason is that the maximal, and maximally symmetric, cusp section of $\mathcal O_R$ consists of 10 Euclidean prisms, each with all edges of length $1$ \cite[Section 3.2]{KS}. Each of the filling curves of $X$ is made of $6$ copies of the height of such prism. Since $6<2\pi$, one cannot apply directly the Gromov-Thurston $2\pi$ theorem to conclude, for instance, that $X$ is aspherical. We will see later that, being convex projective, $X$ is in fact aspherical. \end{remark}
\begin{remark} The 6 theorem, independently obtained by Agol \cite{agol} and Lackenby \cite{lackenby}, shows that a filling of a cusped hyperbolic 3-manifold carries a hyperbolic structure as soon as the filling curves are of length \textit{strictly} greater than $6$. This is thus an improvement of Gromov and Thurston's $2\pi$ theorem in dimension three. It is an open question whether or not it is possible to generalise the 6 theorem to higher dimension as follows: ``The fundamental group of a filling of a cusped hyperbolic $n$-manifold is relatively hyperbolic with respect to the collection of subgroups associated to the inserted $(n-2)$-submanifolds, as soon as the filling curves are of length $>6$.''
Note that in dimension $n=3$ the bound of 6 is sharp, as shown by Agol \cite[Section 7]{agol}. Remark \ref{rem:6-filling} shows that the same bound would be sharp in dimension $n=4$. \end{remark}
\subsection{Cone-manifold structures on the filling} \label{sec:sigma_alpha}
We now conclude the proof of Theorem \ref{thm:main}. Let $\Sigma\subset X$ be the union of the copies of the filling ridges $F'_i\cap F'_j$. We first show item \eqref{item:thmA_cone}, giving a path of projective cone-manifold structures on $X$ with the desired properties.
In Proposition \ref{prop:P_theta}, we built a path of structures $(0,\frac\pi3]\ni\alpha\mapsto P_\alpha$ of mirror polytope on the bitruncated simplex $Q$. Since the manifold $X$ is built by pairing the facets of some copies of $Q$ through the map induced by the identity, for each $\alpha$ we have a well-defined projective structure on the complement in $X$ of the ridges of the copies of $Q$. Indeed, the projective structures of the copies of $P_\alpha\smallsetminus\bigcup(\mbox{ridges})$ match well via the projective reflections associated to the facets of $P_\alpha$. We want to show that this projective structure extends to a projective structure on $X\smallsetminus\Sigma$, and on the whole $X$ as projective cone-manifold structures with cone angle $\theta=6\cdot\alpha$ along $\Sigma$.
The link $L$ of a vertex of $P_\alpha$ is a mirror tetrahedron. Its non-right dihedral angles are $\alpha$ and $\frac\pi3$ along its two opposite edges $\{i'_1,i'_2\}$ and $\{i_3,i_4\}$, respectively (see Figure \ref{fig:link_geom}--right). Recall Section \ref{sec:cone-mfds} about projective cone-manifolds. Some copies of $L$ are glued together in $X$ to form a 3-sphere, which we now show to be the projective cone 3-manifold $\mathbb{S}^1*C_{\theta}$, where $\theta=6\cdot\alpha$.
The manifold $X$ was built in three steps. At the first step (when gluing 6 copies of $Q$ to build the block $B$), 6 copies of $L$ are glued cyclically along its edge $\{i_3,i_4\}$. The resulting space $L'$ is the intersection of two half-spaces of $\mathbb{S}^3$ with \lq\lq dihedral angle\rq\rq\footnote{Since to each hyperplane is associated a reflection, it makes sense to talk about the dihedral angle, even if $L'$ is not properly convex.} $\alpha$. At the second step (when gluing 6 copies of $B$ to build $X'$), 6 copies of $L'$ are glued cyclically along its edge. Indeed, the cycle in the graph $K_6$ with edges labelled alternately by $i_1$ and $i_2$ has length 6. The resulting space is thus $\mathbb{S}^1*C_{\theta}$. The third step is just the orientation double covering $X\to X'$, so the local structure of $X$ is the same as that of $X'$.
The union of the copies of the filling ridge $F'_{i_1}\cap F'_{i_2}$ of $Q$ in $X$ is a component $T$ of $\Sigma$ (a 2-torus). The holonomy of a meridian of $T$ in $X$ is $(\sigma'_{i_1}\sigma'_{i_2})^3$, where $\sigma'_i$ is the projective reflection (depending on $\alpha$) associated to the facet $F'_i$ of the mirror polytope $P_\alpha$.
Since $(X,\Sigma)$ is locally modeled on $(\mathbb{S}^2*C_\theta,\mathbb{S}^2)$, we have a projective cone structure $\sigma_\theta$ for each $\theta\in(0,2\pi]$. Since $\mathbb{S}^2*C_{2\pi}=\mathbb{S}^4$, the projective structure $\sigma_{2 \pi}$ is non-singular, and each component of $\Sigma$ is totally geodesic in $X$ (see Remark \ref{rem:nonsingular}). The associated path of projective cone-manifold structures $\theta\mapsto\sigma_\theta$ on $X$ is analytic because the path of Cartan matrices $t\mapsto C_t$ is analytic.
We have shown item \eqref{item:thmA_cone} of Theorem \ref{thm:main}. Clearly, item \eqref{item:thmA_hyperbolic} follows from Proposition \ref{prop:M}. Item \eqref{item:thmA_Euler} is shown in Remark \ref{rem:chi_X}, and item \eqref{item:thmA_curve} in Remark \ref{rem:6-filling}.
Let us now fix an integer $p\geq3$. By Corollary \ref{cor:coxeter}, the mirror polytope $P_{\nicefrac\pi p}$ is a convex projective orbifold. If moreover $p=3m$ (i.e. $\theta=\nicefrac{2\pi}m$), the natural map $X\to P_{\nicefrac\pi p}$ gives $X$ a structure of convex projective orbifold $\Omega_m/_{\Gamma_m}$ with cone structure $\sigma_{\nicefrac{2\pi}m}$. We have thus shown the first part of item \eqref{item:thmA_orb}.
\begin{remark} We already know that the convex projective manifold $X$ is indecomposable. Another way to see this is as follows. Since $X$ covers the Coxeter polytope $P_{\nicefrac\pi 3}$, and the Coxeter group $W_{3}$ acts strongly irreducibly on $\mathbb{R}^5$ because the Cartan matrix of $P_{\nicefrac\pi 3}$ is indecomposable and $W_3$ is not virtually abelian (see e.g. \cite[Theorem 2.18]{cox_in_hil}), so does $\Gamma_1\cong\pi_1(X)$. \end{remark}
Let $T_1,\ldots,T_{10}$ be the components of $\Sigma$. Being totally geodesic for each $T_i$, the natural map $\pi_1T_i\to\Gamma_m$ induced by the inclusion $T_i\subset X$ is injective. For $m \geqslant 2$, the group $\Gamma_m$ is relatively hyperbolic with respect to $\{\pi_1 T_i\}_i$. Indeed, by Proposition \ref{prop:rel_hyp}, the Coxeter group $W_{3m}$, which is the orbifold fundamental group of $P_{\nicefrac\pi{3m}}$, is relatively hyperbolic with respect to the collection $\{ (W_{3m})_U \mid U \in \mathcal{T}_{3m} \}$ which corresponds to the fundamental groups of the $T_1,\ldots,T_{10}$. The proof of item \eqref{item:thmA_orb} is complete.
It remains to show item \eqref{item:thmA_rel}. For $m=1$ (i.e $\alpha=\nicefrac\pi 3$ and $\theta = 2 \pi$) there is another collection of totally geodesic tori $T'_1,\ldots,T'_{10}$ tiled by the ridges $F_{i} \cap F_{j}$. Being totally geodesic, also $T'_i$ is $\pi_1$-injective. This time, the group $\Gamma_1\cong\pi_1X$ is relatively hyperbolic with respect to $\{\pi_1 T_i,\,\pi_1 T'_i\}_i$. Indeed, by Proposition \ref{prop:rel_hyp}, the Coxeter group $W_{3}$, which is the orbifold fundamental group of $P_{\nicefrac\pi3}$, is relatively hyperbolic with respect to the collection $\{ (W_{3})_U \mid U \in \mathcal{T}_{3} \}$ which corresponds to the fundamental groups of the $T_1,\ldots,T_{10}$ and $T'_1,\ldots,T'_{10}$. Finally, for every $T\in\{T_1,\ldots,T_{10}\}$ and $T'\in\{T'_1,\ldots,T'_{10}\}$, the tori $T$ and $T'$ are transverse (sometimes $T\cap T'=\varnothing$) in $X$, since the ridges $F'_i\cap F'_j$ and $F_k\cap F_\ell$ do so in $\mathbb{S}^4$. Also, in the universal cover $\Omega$, the lifts of $T$ and $T'$ are transverse.
The proof of Theorem \ref{thm:main} is complete.
\appendix
\section{The spherical and affine Coxeter diagrams}\label{classi_diagram}
For the reader’s convenience, we reproduce below the list of the irreducible spherical and irreducible affine Coxeter diagrams.
\newcommand{1.5}{1.5} \begin{table}[ht!] \centering \begin{minipage}[b]{7.7cm} \centering \begin{tikzpicture}[thick,scale=0.55, every node/.style={transform shape}] \node[draw,circle] (A1) at (0,0) {}; \node[draw,circle,right=.8cm of A1] (A2) {}; \node[draw,circle,right=.8cm of A2] (A3) {}; \node[draw,circle,right=1cm of A3] (A4) {}; \node[draw,circle,right=.8cm of A4] (A5) {}; \node[left=.8cm of A1,scale=1.5] {$\Huge A_n\ (n\geqslant 1)$};
\draw (A1) -- (A2) node[above,midway] {}; \draw (A2) -- (A3) node[above,midway] {}; \draw[loosely dotted,thick] (A3) -- (A4) node[] {}; \draw (A4) -- (A5) node[above,midway] {};
\node[draw,circle,below=1.2cm of A1] (B1) {}; \node[draw,circle,right=.8cm of B1] (B2) {}; \node[draw,circle,right=.8cm of B2] (B3) {}; \node[draw,circle,right=1cm of B3] (B4) {}; \node[draw,circle,right=.8cm of B4] (B5) {}; \node[left=.8cm of B1,scale=1.5] {$B_n\ (n\geqslant 2)$};
\draw (B1) -- (B2) node[above,midway] {$4$}; \draw (B2) -- (B3) node[above,midway] {}; \draw[loosely dotted,thick] (B3) -- (B4) node[] {}; \draw (B4) -- (B5) node[above,midway] {};
\node[draw,circle,below=1.5cm of B1] (D1) {}; \node[draw,circle,right=.8cm of D1] (D2) {}; \node[draw,circle,right=1cm of D2] (D3) {}; \node[draw,circle,right=.8cm of D3] (D4) {}; \node[draw,circle, above right=.8cm of D4] (D5) {}; \node[draw,circle,below right=.8cm of D4] (D6) {}; \node[left=.8cm of D1,scale=1.5] {$D_n\ (n\geqslant 4)$};
\draw (D1) -- (D2) node[above,midway] {}; \draw[loosely dotted] (D2) -- (D3); \draw (D3) -- (D4) node[above,midway] {}; \draw (D4) -- (D5) node[above,midway] {}; \draw (D4) -- (D6) node[below,midway] {};
\node[draw,circle,below=1.2cm of D1] (I1) {}; \node[draw,circle,right=.8cm of I1] (I2) {}; \node[left=.8cm of I1,scale=1.5] {$I_2(p)\ (p\geqslant 5)$};
\draw (I1) -- (I2) node[above,midway] {$p$};
\node[draw,circle,below=1.2cm of I1] (H1) {}; \node[draw,circle,right=.8cm of H1] (H2) {}; \node[draw,circle,right=.8cm of H2] (H3) {}; \node[left=.8cm of H1,scale=1.5] {$H_3$};
\draw (H1) -- (H2) node[above,midway] {$5$}; \draw (H2) -- (H3) node[above,midway] {};
\node[draw,circle,below=1.2cm of H1] (HH1) {}; \node[draw,circle,right=.8cm of HH1] (HH2) {}; \node[draw,circle,right=.8cm of HH2] (HH3) {}; \node[draw,circle,right=.8cm of HH3] (HH4) {}; \node[left=.8cm of HH1,scale=1.5] {$H_4$};
\draw (HH1) -- (HH2) node[above,midway] {$5$}; \draw (HH2) -- (HH3) node[above,midway] {}; \draw (HH3) -- (HH4) node[above,midway] {};
\node[draw,circle,below=1.2cm of HH1] (F1) {}; \node[draw,circle,right=.8cm of F1] (F2) {}; \node[draw,circle,right=.8cm of F2] (F3) {}; \node[draw,circle,right=.8cm of F3] (F4) {}; \node[left=.8cm of F1,scale=1.5] {$F_4$};
\draw (F1) -- (F2) node[above,midway] {}; \draw (F2) -- (F3) node[above,midway] {$4$}; \draw (F3) -- (F4) node[above,midway] {};
\node[draw,circle,below=1.2cm of F1] (E1) {}; \node[draw,circle,right=.8cm of E1] (E2) {}; \node[draw,circle,right=.8cm of E2] (E3) {}; \node[draw,circle,right=.8cm of E3] (E4) {}; \node[draw,circle,right=.8cm of E4] (E5) {}; \node[draw,circle,below=.8cm of E3] (EA) {}; \node[left=.8cm of E1,scale=1.5] {$E_6$};
\draw (E1) -- (E2) node[above,midway] {}; \draw (E2) -- (E3) node[above,midway] {}; \draw (E3) -- (E4) node[above,midway] {}; \draw (E4) -- (E5) node[above,midway] {}; \draw (E3) -- (EA) node[left,midway] {};
\node[draw,circle,below=1.8cm of E1] (EE1) {}; \node[draw,circle,right=.8cm of EE1] (EE2) {}; \node[draw,circle,right=.8cm of EE2] (EE3) {}; \node[draw,circle,right=.8cm of EE3] (EE4) {}; \node[draw,circle,right=.8cm of EE4] (EE5) {}; \node[draw,circle,right=.8cm of EE5] (EE6) {}; \node[draw,circle,below=.8cm of EE3] (EEA) {}; \node[left=.8cm of EE1,scale=1.5] {$E_7$};
\draw (EE1) -- (EE2) node[above,midway] {}; \draw (EE2) -- (EE3) node[above,midway] {}; \draw (EE3) -- (EE4) node[above,midway] {}; \draw (EE4) -- (EE5) node[above,midway] {}; \draw (EE5) -- (EE6) node[above,midway] {}; \draw (EE3) -- (EEA) node[left,midway] {};
\node[draw,circle,below=1.8cm of EE1] (EEE1) {}; \node[draw,circle,right=.8cm of EEE1] (EEE2) {}; \node[draw,circle,right=.8cm of EEE2] (EEE3) {}; \node[draw,circle,right=.8cm of EEE3] (EEE4) {}; \node[draw,circle,right=.8cm of EEE4] (EEE5) {}; \node[draw,circle,right=.8cm of EEE5] (EEE6) {}; \node[draw,circle,right=.8cm of EEE6] (EEE7) {}; \node[draw,circle,below=.8cm of EEE3] (EEEA) {}; \node[left=.8cm of EEE1,scale=1.5] {$E_8$};
\draw (EEE1) -- (EEE2) node[above,midway] {}; \draw (EEE2) -- (EEE3) node[above,midway] {}; \draw (EEE3) -- (EEE4) node[above,midway] {}; \draw (EEE4) -- (EEE5) node[above,midway] {}; \draw (EEE5) -- (EEE6) node[above,midway] {}; \draw (EEE6) -- (EEE7) node[above,midway] {}; \draw (EEE3) -- (EEEA) node[left,midway] {};
\draw (0,-18.5) node[]{} ; \end{tikzpicture} \caption{The irreducible spherical Coxeter diagrams} \label{table:spheri_diag} \end{minipage} \begin{minipage}[t]{7.7cm} \centering \begin{tikzpicture}[thick,scale=0.55, every node/.style={transform shape}] \node[draw,circle] (A1) at (0,0) {}; \node[draw,circle,above right=.8cm of A1] (A2) {}; \node[draw,circle,right=.8cm of A2] (A3) {}; \node[draw,circle,right=.8cm of A3] (A4) {}; \node[draw,circle,right=.8cm of A4] (A5) {}; \node[draw,circle,below right=.8cm of A5] (A6) {}; \node[draw,circle,below left=.8cm of A6] (A7) {}; \node[draw,circle,left=.8cm of A7] (A8) {}; \node[draw,circle,left=.8cm of A8] (A9) {}; \node[draw,circle,left=.8cm of A9] (A10) {};
\node[left=.8cm of A1,scale=1.5] {$\widetilde{A}_n\ (n\geqslant 2)$};
\draw (A1) -- (A2) node[above,midway] {}; \draw (A2) -- (A3) node[above,midway] {}; \draw (A3) -- (A4) node[] {}; \draw (A4) -- (A5) node[above,midway] {}; \draw (A5) -- (A6) node[] {}; \draw (A6) -- (A7) node[] {}; \draw (A7) -- (A8) node[] {}; \draw[loosely dotted,thick] (A8) -- (A9) node[] {}; \draw (A9) -- (A10) node[] {}; \draw (A10) -- (A1) node[] {};
\node[draw,circle,below=1.7cm of A1] (B1) {}; \node[draw,circle,right=.8cm of B1] (B2) {}; \node[draw,circle,right=.8cm of B2] (B3) {}; \node[draw,circle,right=1cm of B3] (B4) {}; \node[draw,circle,right=.8cm of B4] (B5) {}; \node[draw,circle,above right=.8cm of B5] (B6) {}; \node[draw,circle,below right=.8cm of B5] (B7) {}; \node[left=.8cm of B1,scale=1.5] {$\widetilde{B}_n\ (n\geqslant 3)$};
\draw (B1) -- (B2) node[above,midway] {$4$}; \draw (B2) -- (B3) node[above,midway] {}; \draw[loosely dotted,thick] (B3) -- (B4) node[] {}; \draw (B4) -- (B5) node[above,midway] {}; \draw (B5) -- (B6) node[above,midway] {}; \draw (B5) -- (B7) node[above,midway] {};
\node[draw,circle,below=1.5cm of B1] (C1) {}; \node[draw,circle,right=.8cm of C1] (C2) {}; \node[draw,circle,right=.8cm of C2] (C3) {}; \node[draw,circle,right=1cm of C3] (C4) {}; \node[draw,circle,right=.8cm of C4] (C5) {}; \node[left=.8cm of C1,scale=1.5] (CCC) {$\widetilde{C}_n\ (n\geqslant 3)$};
\draw (C1) -- (C2) node[above,midway] {$4$}; \draw (C2) -- (C3) node[above,midway] {}; \draw[loosely dotted,thick] (C3) -- (C4) node[] {}; \draw (C4) -- (C5) node[above,midway] {$4$};
\node[draw,circle,below=1cm of C1] (D1) {}; \node[draw,circle,below right=0.8cm of D1] (D3) {}; \node[draw,circle,below left=0.8cm of D3] (D2) {}; \node[draw,circle,right=.8cm of D3] (DA) {}; \node[draw,circle,right=1cm of DA] (DB) {}; \node[draw,circle,right=.8cm of DB] (D4) {}; \node[draw,circle, above right=.8cm of D4] (D5) {}; \node[draw,circle,below right=.8cm of D4] (D6) {}; \node[below=1.5cm of CCC,scale=1.5] {$\widetilde{D}_n\ (n\geqslant 4)$};
\draw (D1) -- (D3) node[above,midway] {}; \draw (D2) -- (D3) node[above,midway] {}; \draw (D3) -- (DA) node[above,midway] {}; \draw[loosely dotted] (DA) -- (DB); \draw (D4) -- (DB) node[above,midway] {}; \draw (D4) -- (D5) node[above,midway] {}; \draw (D4) -- (D6) node[below,midway] {};
\node[draw,circle,below=2.5cm of D1] (I1) {}; \node[draw,circle,right=.8cm of I1] (I2) {}; \node[left=.8cm of I1,scale=1.5] {$\widetilde{A}_1$};
\draw (I1) -- (I2) node[above,midway] {$\infty$};
\node[draw,circle,below=1.2cm of I1] (H1) {}; \node[draw,circle,right=.8cm of H1] (H2) {}; \node[draw,circle,right=.8cm of H2] (H3) {}; \node[left=.8cm of H1,scale=1.5] {$\widetilde{B}_2=\widetilde{C}_2$};
\draw (H1) -- (H2) node[above,midway] {$4$}; \draw (H2) -- (H3) node[above,midway] {$4$};
\node[draw,circle,below=1.2cm of H1] (HH1) {}; \node[draw,circle,right=.8cm of HH1] (HH2) {}; \node[draw,circle,right=.8cm of HH2] (HH3) {}; \node[left=.8cm of HH1,scale=1.5] {$\widetilde{G}_2$};
\draw (HH1) -- (HH2) node[above,midway] {$6$}; \draw (HH2) -- (HH3) node[above,midway] {};
\node[draw,circle,below=1.2cm of HH1] (F1) {}; \node[draw,circle,right=.8cm of F1] (F2) {}; \node[draw,circle,right=.8cm of F2] (F3) {}; \node[draw,circle,right=.8cm of F3] (F4) {}; \node[draw,circle,right=.8cm of F4] (F5) {}; \node[left=.8cm of F1,scale=1.5] {$\widetilde{F}_4$};
\draw (F1) -- (F2) node[above,midway] {}; \draw (F2) -- (F3) node[above,midway] {$4$}; \draw (F3) -- (F4) node[above,midway] {}; \draw (F4) -- (F5) node[above,midway] {};
\node[draw,circle,below=1.2cm of F1] (E1) {}; \node[draw,circle,right=.8cm of E1] (E2) {}; \node[draw,circle,right=.8cm of E2] (E3) {}; \node[draw,circle,right=.8cm of E3] (E4) {}; \node[draw,circle,right=.8cm of E4] (E5) {}; \node[draw,circle,below=.8cm of E3] (EA) {}; \node[draw,circle,below=.8cm of EA] (EB) {}; \node[left=.8cm of E1,scale=1.5] {$\widetilde{E}_6$};
\draw (E1) -- (E2) node[above,midway] {}; \draw (E2) -- (E3) node[above,midway] {}; \draw (E3) -- (E4) node[above,midway] {}; \draw (E4) -- (E5) node[above,midway] {}; \draw (E3) -- (EA) node[left,midway] {}; \draw (EB) -- (EA) node[left,midway] {};
\node[draw,circle,below=3cm of E1] (EE1) {}; \node[draw,circle,right=.8cm of EE1] (EEB) {}; \node[draw,circle,right=.8cm of EEB] (EE2) {}; \node[draw,circle,right=.8cm of EE2] (EE3) {}; \node[draw,circle,right=.8cm of EE3] (EE4) {}; \node[draw,circle,right=.8cm of EE4] (EE5) {}; \node[draw,circle,right=.8cm of EE5] (EE6) {}; \node[draw,circle,below=.8cm of EE3] (EEA) {}; \node[left=.8cm of EE1,scale=1.5] {$\widetilde{E}_7$};
\draw (EE1) -- (EEB) node[above,midway] {}; \draw (EE2) -- (EEB) node[above,midway] {}; \draw (EE2) -- (EE3) node[above,midway] {}; \draw (EE3) -- (EE4) node[above,midway] {}; \draw (EE4) -- (EE5) node[above,midway] {}; \draw (EE5) -- (EE6) node[above,midway] {}; \draw (EE3) -- (EEA) node[left,midway] {};
\node[draw,circle,below=1.8cm of EE1] (EEE1) {}; \node[draw,circle,right=.8cm of EEE1] (EEE2) {}; \node[draw,circle,right=.8cm of EEE2] (EEE3) {}; \node[draw,circle,right=.8cm of EEE3] (EEE4) {}; \node[draw,circle,right=.8cm of EEE4] (EEE5) {}; \node[draw,circle,right=.8cm of EEE5] (EEE6) {}; \node[draw,circle,right=.8cm of EEE6] (EEE7) {}; \node[draw,circle,right=.8cm of EEE7] (EEE8) {}; \node[draw,circle,below=.8cm of EEE3] (EEEA) {}; \node[left=.8cm of EEE1,scale=1.5] {$\widetilde{E}_8$};
\draw (EEE1) -- (EEE2) node[above,midway] {}; \draw (EEE2) -- (EEE3) node[above,midway] {}; \draw (EEE3) -- (EEE4) node[above,midway] {}; \draw (EEE4) -- (EEE5) node[above,midway] {}; \draw (EEE5) -- (EEE6) node[above,midway] {}; \draw (EEE6) -- (EEE7) node[above,midway] {}; \draw (EEE8) -- (EEE7) node[above,midway] {}; \draw (EEE3) -- (EEEA) node[left,midway] {};
\draw (0,-22.5) node[]{} ; \end{tikzpicture} \caption{The irreducible affine Coxeter diagrams} \label{table:affi_diag} \end{minipage} \end{table}
\end{document} |
\begin{document}
\thanks{The first author acknowledges the support of FAPESP grant 2008/11471-6 and the second author the support of NSF grant DMS 0556368} \subjclass[2000]{Primary: 46B03, Secondary 03E15}
\keywords{Tight Banach spaces, Dichotomies, Classification of Banach spaces}
\begin{abstract} We analyse several examples of separable Banach spaces, some of them new, and relate them to
several dichotomies obtained in \cite{FR4}, by classifying them according to which side of the dichotomies they fall. \end{abstract}
\title{Banach spaces without minimal subspaces - Examples}
\tableofcontents
\section{Introduction}\label{intro}
In this article we give several new examples of Banach spaces, corresponding to different classes of a list defined in \cite{FR4}. This paper may be seen as a more empirical continuation of \cite{FR4} in which our stress is on the study of examples for the new classes of Banach spaces considered in that work.
\subsection{Gowers' list of inevitable classes}
In the paper \cite{g:dicho}, W.T. Gowers had defined a program of isomorphic classification of Banach spaces. The aim of this program is a {\em loose classification of Banach spaces up to subspaces}, by producing a list of classes of Banach spaces such that:
(a) if a space belongs to a class, then every subspace belongs to the same class, or maybe, in the case when the properties defining the class depend on a basis of the space, every block subspace belongs to the same class,
(b) the classes are {\em inevitable}, i.e., every Banach space contains a subspace in one of the classes,
(c) any two classes in the list are disjoint,
(d) belonging to one class gives a lot of information about operators that may be defined on the space or on its subspaces.
\
We shall refer to such a list as a {\em list of inevitable classes of
Gowers}. For the motivation of Gowers' program as well as the relation of this program
to classical problems in Banach space theory we refer to \cite{FR4}. Let us just say that the class of spaces $c_0$ and $\ell_p$ is seen as the
nicest or most regular class, and
so, the objective of Gowers' program really is the classification of those spaces (such as Tsirelson's
space $T$) which do not contain a copy of $c_0$ or $\ell_p$. Actually, in \cite{FR4}, mainly spaces without {\em minimal subspaces} are
classified, and so in this article, we shall consider various examples of
Banach spaces without minimal subspaces. We shall first give a summary of the classification obtained in \cite{FR4} and of the results that led to that classification.
After the construction by Gowers and Maurey of a {\em hereditarily indecomposable} (or HI) space $GM$, i.e., a space such that no subspace may be written as the direct sum of infinite dimensional subspaces \cite{GM}, Gowers proved that every Banach space contains either an HI subspace or a subspace with an unconditional basis \cite{g:hi}. This dichotomy is called {\em first dichotomy} of Gowers in \cite{FR4}.
These were the first two examples of inevitable classes. He then refined the list by proving a {\em second dichotomy}: any Banach space contains a subspace with a basis such that either no two disjointly supported block subspaces are isomorphic, or such that any two subspaces have further subspaces which are isomorphic. He called the second property {\em quasi minimality}. Finally, H. Rosenthal had defined a space to be {\em minimal} if it embeds into any of its subspaces. A quasi minimal space which does not contain a minimal subspace is called {\em strictly quasi minimal}, so Gowers again divided the class of quasi minimal spaces into the class of strictly quasi minimal spaces and the class of minimal spaces.
Gowers therefore produced a list of four inevitable classes of Banach spaces, corresponding to classical examples, or more recent couterexamples to classical questions: HI spaces, such as $GM$; spaces with bases such that no disjointly supported subspaces are isomorphic, such as the couterexample $G_u$ of Gowers to the hyperplane's problem of Banach \cite{g:hyperplanes}; strictly quasi minimal spaces with an unconditional basis, such as Tsirelson's space $T$ \cite{tsi} ; and finally, minimal spaces, such as $c_0$ or $\ell_p$, but also $T^*$, Schlumprecht's space $S$ \cite{S1}, or as proved recently in \cite{MP}, its dual $S^*$.
\
\subsection{The three new dichotomies} In \cite{FR4} three dichotomies for Banach spaces were obtained. The first one of these new dichotomies,
the {\em third dichotomy}, concerns the property of minimality defined by Rosenthal. Recall that a
Banach space is minimal if it embeds into any of its infinite dimensional
subspaces. On the other hand, a space $Y$ is {\em tight} in a basic sequence $(e_i)$ if there is a sequence of successive subsets $I_0<I_1<I_2<\ldots$ of $\N$, such that for all infinite subsets $A\subseteq \N$, we have $$ Y\not\sqsubseteq [e_n\del n\notin \bigcup_{i\in A}I_i]. $$
A {\em tight basis} is a basis such that every subspace is tight in it, and a {\em tight space} is a space with a tight basis \cite{FR4}.
The subsets $I_n$ may clearly be chosen to be intervals or even to form a partition of $\N$. However it is convenient not to require this condition in the definition, in view of forthcoming special cases of tightness.
It is observed in \cite{FR4} that the tightness property is hereditary,
incompatible with minimality, and it is proved that:
\begin{thm}[3rd dichotomy, Ferenczi-Rosendal 2007]\label{main} Let $E$ be a Banach space without minimal subspaces. Then $E$ has a tight subspace. \end{thm}
Actual examples of tight spaces in \cite{FR4} turn out to satisfy one of two stronger forms of tightness. The first was called {\em tightness by range}. Here the range, ${\rm range} \ x$, of a vector $x$ is the smallest interval of integers containing its support on the given basis, and the range of a block subspace $[x_n]$ is $\bigcup_n {\rm range} \ x_n$. A basis $(e_n)$ is tight by range when for every block subspace $Y=[y_n]$, the sequence of successive subsets $I_0<I_1<\ldots$ of $\N$ witnessing the tightness of $Y$ in $(e_n)$ may be defined by $I_k={\rm range}\ y_k$ for each $k$. This is equivalent to no two block subspaces with disjoint ranges being comparable, where two spaces are comparable if one embeds into the other.
When the definition of tightness may be checked with $I_k={\rm supp}\ y_k$ instead of ${\rm range}\ y_k$, then a stronger property is obtained which is called tightness by support, and is equivalent to the property defined by Gowers in the second dichotomy that no disjointly supported block subspaces are isomorphic, Therefore $G_u$ is an example of space with a basis which is tight by support and therefore by range.
The second kind of tightness was called {\em tightness with constants}. A basis $(e_n)$ is tight with constants when for for every infinite dimensional space $Y$, the sequence of successive subsets $I_0<I_1<\ldots$ of $\N$ witnessing the tightness of $Y$ in $(e_n)$ may be chosen so that $Y \not\sqsubseteq_K [e_n \del n \notin I_K]$ for each $K$. This is the case for Tsirelson's space $T$ or its $p$-convexified version $T^{(p)}$ \cite{CS}.
As we shall see, one of the aims of this paper is to present various examples of tight spaces of these two forms.
\
In \cite{FR4} it was proved that there are natural dichotomies between each of these strong forms of tightness and respective weak forms of minimality. For the first notion, a space $X$ with a basis $(x_n)$ is said to be {\em subsequentially minimal} if every subspace of $X$ contains an isomorphic copy of a subsequence of $(x_n)$. Essentially this notion had been previously considered by Kutzarova, Leung, Manoussakis and Tang in the context of modified partially mixed Tsirelson spaces \cite{KLMT}.
\begin{thm}[4th dichotomy, Ferenczi-Rosendal 2007]\label{main2} Any Banach space $E$ contains a subspace with a basis that is either tight by range or is subsequentially minimal. \end{thm}
The second case in Theorem \ref{main2} may be improved to the following hereditary property of a basis $(x_n)$, that we call {\em sequential minimality}: $(x_n)$ is quasi minimal and every block sequence of $[x_n]$ has a subsequentially minimal block sequence.
There is also a dichotomy concerning tightness with constants. Recall that given two Banach spaces $X$ and $Y$, we say that $X$ is {\em crudely finitely representable} in $Y$ if there is a constant $K$ such that for any finite-dimensional subspace $F\subseteq X$ there is an embedding $T\colon F\rightarrow Y$ with constant $K$, i.e., $\norm{T}\cdot\norm{T^{-1}}\ensuremath{\leqslant} K$.
A space $X$ is said to be {\em locally minimal} if for some constant $K$, $X$ is $K$-crudely finitely representable in any of its subspaces. \begin{thm}[5th dichotomy, Ferenczi-Rosendal 2007] \label{main3} Any Banach space $E$ contains a subspace with a basis that is either tight with constants or is locally minimal. \end{thm}
\
Finally there exists a sixth
dichotomy theorem due to A. Tcaciuc \cite{T}, stated here in a slightly strengthened form. A space $X$ is
{\em uniformly inhomogeneous} when $$\forall M\ensuremath{\geqslant} 1\; \exists n \in \N\; \forall Y_1,\ldots,Y_{2n} \subseteq X\; \exists y_i \in\ku S_{Y_i}\;(y_i)_{i=1}^n\not\sim_M(y_i)_{i=n+1}^{2n},$$ where $Y_1,\ldots,Y_{2n}$ are assumed to be infinite-dimensional subspaces of $X$. On the contrary, a basis $(e_n)$ is said to be {\em strongly asymptotically $\ell_p$}, $1 \ensuremath{\leqslant} p \ensuremath{\leqslant} +\infty$, \cite{DFKO}, if there exists a constant $C$ and a function $f:\N \rightarrow \N$ such that for any $n$, any family of $n$ unit vectors which are disjointly supported in $[e_k \del k \ensuremath{\geqslant} f(n)]$ is $C$-equivalent to the canonical basis of $\ell_p^n$. Tcaciuc then proves \cite{T} :
\begin{thm}[Tcaciuc's dichotomy, 2005] Any Banach space contains a subspace with a basis which is either uniformly inhomogeneous or strongly asymptotically $\ell_p$ for some $1 \ensuremath{\leqslant} p \ensuremath{\leqslant} +\infty$. \end{thm}
The six dichotomies and the interdependence of the properties involved can be visualised in the following diagram.
\[ \begin{tabular}{ccc} Strongly asymptotic $\ell_p$&$**\textrm{ Tcaciuc's dichotomy }**$& Uniformly inhomogeneous\\
$\Downarrow$& &$\Uparrow$\\
Unconditional basis&$**\textrm{ 1st dichotomy }**$& Hereditarily indecomposable\\
$\Uparrow$& &$\Downarrow$\\
Tight by support & $**\textrm{ 2nd dichotomy }**$ & Quasi minimal \\
$\Downarrow$&&$\Uparrow$\\
Tight by range & $**\textrm{ 4th dichotomy }**$ & Sequentially minimal \\
$\Downarrow$&&$\Uparrow$\\
Tight& $**\textrm{ 3rd dichotomy }**$ & Minimal\\
$\Uparrow$& &$\Downarrow$\\
Tight with constants& $**\textrm{ 5th dichotomy }**$ & Locally minimal\\ \end{tabular} \]
\
Moreover, $${\rm Strongly\ asymptotic\ } \ell_p {\rm\ not\ containing\ } \ell_p. 1 \ensuremath{\leqslant} p <+\infty \Rightarrow {\rm Tight\ with\ constants},$$ and $${\rm Strongly\ asymptotic\ } \ell_{\infty} \Rightarrow {\rm Locally\ minimal}.$$
\
Note that while a basis tight by support must be unconditional, a basis which is tight by range may span a HI space. So tightness by support and tightness by range are two different notions. We would lose this subtle difference if we required the sets $I_n$ to be intervals in the definition of tightness. Likewise a basis may be tight by range without being (nor containing a basis which is) tight with constants, and tight with constants without being (nor containing a basis which is) tight by range. Actually none of the converses of the implications appearing on the left or the right of the list of the six dichotomies holds, even if one allows passing to a further subspace. All the claims of this paragraph are easily checked by looking at the list of examples of Theorem \ref{final}, which is the aim of this paper.
The fact that a strongly asymptotically $\ell_p$ space not containing $\ell_p$ must be tight with constants is proved in \cite{FR4} but is essentially due to the authors of \cite{DFKO}, and
the observation that such bases are unconditional may also be found in \cite{DFKO}. The easy fact that HI spaces are uniformly homogeneous (with $n=2$ in the definition) is observed in \cite{FR4}. That HI spaces are quasi-minimal is due to Gowers \cite{g:dicho}, and that minimal spaces are locally minimal is a consequence of an observation by P. G. Casazza \cite{C} that every minimal space must $K$-embed into all its subspaces for some $K \ensuremath{\geqslant} 1$. The other implications are direct consequences of the definitions, and more explanations and details may be found in \cite{FR4}.
\
\subsection{The list of 19 inevitable classes} Combining the six dichotomies and the relations between them, the following list of 19 classes of Banach spaces contained in any Banach space is obtained in \cite{FR4}:
\begin{thm}[Ferenczi - Rosendal 2007]\label{final} Any infinite dimensional Banach space contains a subspace of one of the types listed in the following chart: \begin{center}
\begin{tabular}{|l|l|l|}
\hline
Type & Properties & Examples \\
\hline
(1a) & HI, tight by range and with constants & ?\\
(1b) & HI, tight by range, locally minimal & $G^*$\\
\hline
(2) & HI, tight, sequentially minimal & ? \\
\hline
(3a) & tight by support and with constants, uniformly inhomogeneous & ? \\
(3b) & tight by support, locally minimal, uniformly inhomogeneous & $G_u^*$ \\
(3c) & tight by support, strongly asymptotically
$\ell_p$, $1 \ensuremath{\leqslant} p <\infty$ & $X_u, X_{abr}$ \\
(3d) & tight by support, strongly asymptotically
$\ell_{\infty}$ & $X_u^*$ \\
\hline
(4) & unconditional basis, quasi minimal, tight by range & ? \\
\hline
(5a) & unconditional basis, tight with constants, sequentially minimal, & ? \\
& uniformly inhomogeneous & \\
(5b) & unconditional basis, tight, sequentially and locally minimal, & ? \\
& uniformly inhomogeneous & \\
(5c) & tight with constants, sequentially minimal, & $T$, $T^{(p)}$ \\
& strongly asymptotically $\ell_p$, $1 \ensuremath{\leqslant} p<\infty$ & \\
(5d) & tight, sequentially minimal, strongly asymptotically $\ell_{\infty}$ & ?\\
\hline
(6a) & unconditional basis, minimal, uniformly inhomogeneous & $S,S^*$ \\
(6b) & minimal, reflexive, strongly asymptotically $\ell_{\infty}$ & $T^*$\\
(6c) & isomorphic to $c_0$ or $l_p$, $1 \ensuremath{\leqslant} p<\infty$ & $c_0$, $\ell_p$\\
\hline
\end{tabular}
\end{center} \end{thm}
\
The class of type (2) spaces may be divided into two subclasses, using the 5th dichotomy, and the class of type (4) into four, using the 5th and the 6th dichotomy, giving a total of 19 inevitable classes. Since we know of no example of a type (2) or type (4) space to begin with, we do not write down the list of possible subclasses of these two classes, leaving this as an exercise to the interested reader.
Note that the tightness property may be used to obtain lower bounds of complexity for the relation of isomorphism between subspaces of a given Banach space. This was initiated by B. Bossard \cite{Bo} who used Gowers' space $G_u$ and its tightness by support. Other results in this direction may be found in \cite{FR4}. We also refer to \cite{ergodic} for a more introductory work to this question.
\
In \cite{FR4} the existence of $X_u$ and the properties of $S$, $G$, $G_u$ and $X_u$ which appear in the chart and are mentioned without proof. It is the main objective of this paper to prove the results about the spaces which appear in the above chart.
\
So in what follows various (and for some of them new) examples of ``pure'' tight spaces are analysed combining some of the properties of tightness or minimality associated to each dichotomy. We shall provide several examples of tight spaces from the two main families of exotic Banach spaces: spaces of the type of Gowers and Maurey \cite{GM} and spaces of the type of Argyros and Deliyanni \cite{AD}. Recall that both types of spaces are defined using a coding procedure to ``conditionalise'' the norm of some ground space defined by induction. In spaces of the type of Gowers and Maurey, the ground space is the space $S$ of Schlumprecht, and in spaces of the type of Argyros and Deliyanni, it is a mixed (in further versions modified or partly modified) Tsirelson space associated to the sequence of Schreier families. The space $S$ is far from being asymptotic $\ell_p$ and is actually uniformly inhomogeneous, and this is the case for our examples of the type of Gowers-Maurey as well. On the other hand, we use a space in the second family, inspired by an example of Argyros, Deliyanni, Kutzarova and Manoussakis \cite{ADKM}, to produce strongly asymptotically $\ell_1$ and $\ell_{\infty}$ examples with strong tightness properties.
\section{Tight unconditional spaces of the type of Gowers and Maurey}
In this section we prove that the dual of the type (3) space $G_u$ constructed by Gowers in \cite{g:hyperplanes} is locally minimal of type (3),
that Gowers' hereditarily indecomposable and asymptotically unconditional
space $G$ defined in \cite{g:asymptotic} is of type (1), and that its dual $G^*$ is locally minimal of type (1). These spaces are
natural variations on Gowers and Maurey's space $GM$, and so familiarity with
that construction will be assumed: we shall not redefine the now classical
notation relative to $GM$, such as the sets of integers $K$ and $L$, rapidly increasing sequences (or R.I.S.), the
set ${\bf Q}$ of functionals, special functionals, etc., instead we shall try to give details on the
parts in which $G_u$ or $G$ differ from $GM$.
The idea of the proofs is similar to \cite{g:hyperplanes}. The HI property for Gowers-Maurey's spaces is obtained as follows. Some vector $x$ is constructed such that $\norm{x}$ is large, but so that if $x'$ is obtained from $x$ by changing signs of the components of $x$, then $x^*(x')$ is small for any norming functional $x^*$, and so $\norm{x'}$ is small. The upper bound for
$x^*(x')$ is obtained by a combination of unconditional estimates (not depending on the signs) and of conditional estimates (i.e., based on the fact that $|\sum_{i=1}^n \epsilon_i|$ is much smaller than $n$ if $\epsilon_i=(-1)^i$ for all $i$).
For our examples we shall need to prove that some operator $T$ is unbounded. Thus we shall construct a vector $x$ such that say $Tx$ has large norm, and such that $x^*(x)$ is small for any norming $x^*$. The upper bound for $x^*(x)$ will be obtained by the same unconditional estimates as in the HI case, while conditional estimates will be trivial due to disjointness of
supports of the corresponding component vectors and functionals. The method will be similar for the dual spaces.
\
Recall that if $X$ is a space with a bimonotone basis, an $\ell_{1+}^n$-average with constant $1+\epsilon$ is a normalised vector of the form $\sum_{i=1}^n x_i$, where $x_1<\cdots<x_n$ and $\norm{x_i} \ensuremath{\leqslant} \frac{1+\epsilon}{n}$ for all $i$. An $\ell_{\infty+}^n$-average with constant $1+\epsilon$ is a normalised vector of the form
$\sum_{i=1}^n x_i$, where $x_1<\cdots<x_n$ and $\norm{x_i} \ensuremath{\geqslant} \frac{1}{1+\epsilon}$ for all $i$. An $\ell_{1+}^n$-vector (resp. $\ell_{\infty+}^n$-vector) is a non zero multiple of an $\ell_{1+}^n$-average (resp. $\ell_{\infty+}^n$-average). The function $f$ is defined by $f(n)=\log_2(n+1)$. The space $X$ is said to satisfy a lower $f$-estimate if for any $x_1<\cdots<x_n$, $$\frac{1}{f(n)}\sum_{i=1}^n \norm{x_i} \ensuremath{\leqslant} \norm{\sum_{i=1}^n x_i}.$$
\begin{lemme}\label{linftyn} Let $X$ be a reflexive space with a bimonotone basis and satisfying a lower $f$-estimate. Let $(y_k^*)$ be a normalised block sequence of $X^*$, $n \in \N$, $\epsilon,\alpha>0$.
Then there exists a constant $N(n,\epsilon)$, successive subsets $F_i$ of
$[1,N(n,\epsilon)]$, $1 \ensuremath{\leqslant} i \ensuremath{\leqslant} n$, and $\lambda>0$ such that if $x_i^*:=\lambda \sum_{k \in F_i} y_k^*$ for all $i$, then $x^*=\sum_{i=1}^n x_i^*$ \ is an $\ell_{\infty +}^n$- average with constant $1+\epsilon$.
Furthermore, if for each $i$, $x_i$ is such that $\norm{x_i} \ensuremath{\leqslant} 1$, ${\rm range}\;x_i \subseteq {\rm range}\;x_i^*$ and $x_i^*(x_i) \ensuremath{\geqslant} \alpha \norm{x_i^*}$, then
$x=\sum_{i=1}^n x_i$ is an $\ell_{1+}^n$-vector with constant
$\frac{1+\epsilon}{\alpha}$ such that $x^*(x) \ensuremath{\geqslant} \frac{\alpha}{1+\epsilon}\norm{x}$. \end{lemme}
\begin{proof} Since $X$ satisfies a lower $f$-estimate, it follows by duality that any sequence of successive functionals $x_1^*<\cdots<x_n^*$ in $G_u^*$ satisfies the following upper estimate: $$1 \ensuremath{\leqslant} \norm{\sum_{i=1}^n x_i^*} \ensuremath{\leqslant} f(n) \max_{1 \ensuremath{\leqslant} i \ensuremath{\leqslant} n}\norm{x_i^*}.$$ Let $N=n^k$ where $k$ is such that $(1+\epsilon)^k>
f(n^k)$. Assume towards a contradiction that the result is false for
$N(n,\epsilon)=N$, then $$y^*=(y_1^*+\ldots+y_{n^{k-1}}^*)+\ldots+(y_{(n-1)n^{k-1}+1}^*+\ldots+y_{n^k}^*)$$ is not an $\ell_{\infty +}^n$-vector with constant $1+\epsilon$, and therefore, for some $i$, $$\norm{y_{i n^{k-1}+1}^*+\ldots+y_{(i+1)n^{k-1}}^*} \ensuremath{\leqslant} \frac{1}{1+\epsilon}\norm{y^*}.$$ Applying the same reasoning to the above sum instead of $y^*$, we obtain, for some $j$, $$\norm{y_{j n^{k-2}+1}^*+\ldots+y_{(j+1)n^{k-2}}^*} \ensuremath{\leqslant} \frac{1}{(1+\epsilon)^2}\norm{y^*}.$$ By induction we obtain that $$1 \ensuremath{\leqslant} \frac{1}{(1+\epsilon)^k} \norm{y^*} \ensuremath{\leqslant} \frac{1}{(1+\epsilon)^k} f(n^k),$$ a contradiction.
Let therefore $x^*$ be such an $\ell_{\infty +}^n$-average with constant $1+\epsilon$ of the form $\sum_i x_i^*$. Let for each $i$, $x_i$ be such that $\norm{x_i} \ensuremath{\leqslant} 1$, ${\rm range}\ x_i \subseteq {\rm range}\ x_i^*$ and $x_i^*(x_i) \ensuremath{\geqslant} \alpha \norm{x_i^*}$.
Then
$$\norm{\sum_i x_i} \ensuremath{\geqslant} x^*(\sum_i x_i) \ensuremath{\geqslant} \alpha (\sum_i \|x_i^*\|) \ensuremath{\geqslant} \frac{\alpha n}{1+\epsilon},$$ and in particular for each $i$, $$\norm{x_i} \ensuremath{\leqslant} 1 \ensuremath{\leqslant} \frac{1+\epsilon}{\alpha n}\norm{\sum_i x_i},$$ so $\sum_i x_i$ is a $\ell_{1+}^n$-vector with constant $\frac{1+\epsilon}{\alpha}$. We also obtain that $$x^*(\sum_i x_i) \ensuremath{\geqslant} \frac{\alpha n}{1+\epsilon} \ensuremath{\geqslant} \frac{\alpha}{1+\epsilon}\norm{\sum_i x_i},$$ as required. \end{proof}
The following lemma is fundamental and therefore worth stating explicitly. It appears for example as Lemma 4 in \cite{g:asymptotic}. Recall that an $(M,g)$-form is a functional of the form $g(M)^{-1}(x_1^*+\ldots+x_M^*)$, with $x_1^*<\cdots<x_M^*$ of norm at most $1$.
\begin{lemme}[Lemma 4 in \cite{g:asymptotic}]\label{fundamental}
Let $f,g\in\mathcal F$ with $g\ensuremath{\geqslant}\sqrt f$, let $X$ be a space with a bimonotone basis satisfying a lower $f$-estimate, let $\epsilon>0$ and $\epsilon'=\min\{\epsilon,1\}$, let $x_1,\ldots,x_N$ be a R.I.S. in X for $f$ with constant $1+\epsilon$ and let $x=\sum_{i=1}^Nx_i$. Suppose that
$$\norm{Ex}\ensuremath{\leqslant}\sup\Bigl\{|x^*(Ex)|:M\ensuremath{\geqslant} 2, x^*\ \hbox{is an $(M,g)$-form} \Bigr\}$$ for every interval $E$ such that $\norm{Ex}\ge 1/3$. Then $\norm x\ensuremath{\leqslant}(1+\epsilon+\epsilon')Ng(N)^{-1}$.
\end{lemme}
\subsection{A locally minimal space tight by support} Let $G_u$ be the space defined in \cite{g:hyperplanes}. This space has a suppression unconditional basis, is tight by support and therefore reflexive, and its norm is given by the following implicit equation, for all $x \in c_{00}$:
$$\norm{x}=\norm{x}_{c_0}\vee\ \sup\Bigl\{f(n)^{-1}\sum_{i=1}^n\norm{E_i x}\Del 2 \ensuremath{\leqslant} n, E_1<\ldots<E_n\Bigr\}$$
$$\vee\ \sup\Bigl\{|x^*(x)|\Del k \in K, x^* \hbox{ special of length } k \Bigr\}$$
where $E_1, \ldots, E_n$ are successive subsets (not necessarily intervals) of $\N$.
\begin{prop}\label{gu} The dual $G_u^*$ of $G_u$ is tight by support and locally minimal. \end{prop}
\begin{proof}
Given $n \in \N$ and $\epsilon=1/10$ we may by Lemma \ref{linftyn} find some $N$ such that there exists in the span of any $x_1^*<\ldots<x_N^*$ an $\ell_{\infty+}^{n}$-average with constant $1+\epsilon$. By unconditionality we deduce that any block-subspace of $G_u^*$ contains $\ell_{\infty}^n$'s uniformly, and therefore $G_u^*$ is locally minimal.
Assume now towards a contradiction that $(x_n^*)$ and $(y_n^*)$ are disjointly supported and equivalent block sequences in $G_u^*$, and let $T: [x_n^*] \rightarrow [y_n^*]$ be defined by $Tx_n^*=y_n^*$.
We may assume that each $x_n^*$ is an $\ell_{\infty +}^n$-average with constant $1+\epsilon$. Using Hahn-Banach theorem, the $1$-unconditionality of the basis, and Lemma \ref{linftyn}, we may also find for each $n$ an $\ell_{1+}^n$-average $x_n$ with constant $1+\epsilon$ such that ${\rm supp}\ x_n \subseteq {\rm supp}\ x_n^*$ and $x_n^*(x_n) \ensuremath{\geqslant} 1/2$. By construction, for each $n$, $Tx_n^*$ is disjointly supported from $[x_k]$, and up to modifying $T$, we may assume that $Tx_n^*$ is in ${\bf Q}$ and of norm at most $1$ for each $n$.
If $z_1,\ldots,z_m$ is a R.I.S. of these ${\ell}_{1+}^n$-averages $x_n$ with constant $1+\epsilon$, with $m \in [\log N, \exp N]$, $N \in L$, and $z_1^*,\ldots,z_m^*$ are the functionals associated to $z_1,\ldots,z_m$, then by \cite{g:hyperplanes} Lemma 7, the $(m,f)$-form $z^*=f(m)^{-1}(z_1^*+\ldots+z_m^*)$ satisfies $$z^*(z_1+\ldots+z_m) \ensuremath{\geqslant} \frac{m}{2f(m)} \ensuremath{\geqslant} \frac{1}{4}\norm{z_1+\ldots+z_m},$$ and furthermore $Tz^*$ is also an $(m,f)$-form. Therefore we may build R.I.S. vectors $z$ with constant $1+\epsilon$ of arbitrary length $m$ in $[\log N, \exp N]$, $N \in L$, so that $z$ is $4^{-1}$-normed by an $(m,f)$-form $z^*$ such that $Tz^*$ is also an $(m,f)$-form. We may then consider a sequence $z_1,\ldots,z_k$ of length $k \in K$ of such R.I.S. vectors of length $m_i$, and some corresponding $(m_i,f)$-forms $z_1^*,\ldots,z_k^*$ (i.e $z_i^*$ $4^{-1}$-norms $z_i$ and $Tz_i^*$ is also an $(m_i,f)$-form for all $i$), such that $Tz_1^*,\ldots,Tz_k^*$ is a special sequence.
Then we let
$z=z_1+\cdots+z_k$ and $z^*=f(k)^{-1/2}(z_1^*+\ldots+z_k^*)$. Since $Tz^*=f(k)^{-1/2}(Tz_1^*+\ldots+Tz_k^*)$ is a special functional it follows that $$\norm{Tz^*} \ensuremath{\leqslant} 1.$$Our aim is now to show that $\norm{z} \ensuremath{\leqslant} 3kf(k)^{-1}$. It will then follow that $$\norm{z^*} \ensuremath{\geqslant} z^*(z)/\norm{z} \ensuremath{\geqslant} f(k)^{1/2}/12.$$
Since $k$ was arbitrary in $K$ this will imply that $T^{-1}$ is unbounded and provide the desired contradiction.
The proof is almost exactly the same as in \cite{g:hyperplanes}. Let $K_0=K \setminus \{k\}$ and let $g$ be the corresponding function given by \cite{g:hyperplanes} Lemma 6. To prove that $\norm{z} \leq3kf(k)^{-1}$ it is enough by \cite{g:hyperplanes} Lemma 8 and Lemma \ref{fundamental} to prove that for any interval $E$ such that $\norm{Ez} \ensuremath{\geqslant} 1/3$, $Ez$ is normed by some $(M,g)$-form with $M \ensuremath{\geqslant} 2$.
By the discussion in the proof of the main theorem in \cite{g:hyperplanes}, the only possible norming functionals apart from $(M,g)$-forms are special functionals of length $k$. So let $w^*=f(k)^{-1/2}(w_1^*+\cdots+w_k^*)$ be a special functional of length $k$, and $E$ be an interval such that $\norm{Ez} \ensuremath{\geqslant} 1/3$. We need to show that $w^*$ does not norm $Ez$.
Let $t$ be minimal such that $w_t^* \neq Tz_t^*$. If $i \neq j$ or $i=j>t$ then by definition of special sequences there exist $M \neq N \in L$, $\min(M,N) \ensuremath{\geqslant} j_{2k}$, such that $w_i^*$ is an $(M,f)$-form and $z_j$ is an R.I.S. vector of size $N$ and constant $1+\epsilon$. By \cite{g:hyperplanes} Lemma~8, $z_j$ is an $\ell_{1+}^{N^{1/10}}$-average with constant $2$. If $M<N$ then
$2M<\log \log \log N$ so, by \cite{g:hyperplanes} Corollary 3, $|w_i^*(Ez_j)| \ensuremath{\leqslant} 6f(M)^{-1}$. If $M>N$ then $\log \log \log M>2N$ so, by
\cite{g:hyperplanes} Lemma 4, $|w_i^*(Ez_j)| \ensuremath{\leqslant} 2f(N)/N$. In both cases it follows that $|w_i^*(Ez_j)| \ensuremath{\leqslant} k^{-2}$.
If $i=j=t$ we have $|w_i^*(Ez_j)| \ensuremath{\leqslant} 1$. Finally if $i=j<t$ then $w_i^*=Tz_i^*$. Since $Tz_i^*$ is disjointly supported from $[x_k]$ and therefore from $z_j$, it follows simply that $w_i^*(Ez_j)=0$ in that case.
Summing up we have obtained that
$$|w^*(Ez)| \ensuremath{\leqslant} f(k)^{-1/2}(k^2. k^{-2}+1)=2f(k)^{-1/2} < 1/3 \ensuremath{\leqslant} \norm{Ez}.$$ Therefore $w^*$ does not norm $Ez$ and this finishes the proof. \end{proof}
\subsection{Uniformly inhomogeneous examples} It may be observed that $G_u^*$ is uniformly inhomogeneous. We state this in a general form which implies the result for $G_u$, Schlumprecht's space $S$ and its dual $S^*$. This is also true for Gowers-Maurey's space $GM$ and its dual $GM^*$, as well as for $G$ and $G^*$, where $G$ is the HI asymptotically unconditional space of Gowers from \cite{g:asymptotic}, which we shall redefine and study later on. As HI spaces are always uniformly inhomogeneous however, we need to observe that a slightly stronger result is obtained by the proof of the next statement to see that Proposition \ref{spaceswithtcaciuc} is not trivial in the case of $GM$, $G$ or their duals - see the three paragraphs after Proposition \ref{spaceswithtcaciuc}.
\begin{prop}\label{spaceswithtcaciuc} Let $f \in {\mathcal F}$ and let $X$ be a space with a bimonotone basis satisfying a lower $f$-estimate. Let $\epsilon_0=1/10$, and assume that for every $n \in [\log N, \exp N], N \in L$, $x_1,\ldots,x_n$ a R.I.S. in $X$
with constant $1+\epsilon_0$ and $x=\sum_{i=1}^Nx_i$,
$$\norm{Ex}\ensuremath{\leqslant}\sup\Bigl\{|x^*(Ex)|:M\ensuremath{\geqslant} 2, x^*\ \hbox{is an $(M,f)$-form} \Bigr\}$$ for every interval $E$ such that $\norm{Ex}\ge 1/3$. Then $X$ and $X^*$ are uniformly inhomogeneous. \end{prop}
\begin{proof} Given $\epsilon>0$, let $m \in L$ be such that $f(m) \ensuremath{\geqslant} 24\epsilon^{-1}$. Let $Y_1,\ldots,Y_{2m}$ be arbitrary block subspaces of $X$. By the classical method for spaces with a lower $f$ estimate, we may find a R.I.S. sequence $y_1<\cdots<y_m$ with constant $1+\epsilon_0$ with $y_i \in Y_{2i-1}, \forall i$. By Lemma \ref{fundamental}, $$\norm{\sum_{i=1}^m y_i} \ensuremath{\leqslant} 2mf(m)^{-1}.$$
Let on the other hand $n \in [m^{10},\exp m]$ and $E_1<\cdots<E_m$ be sets such that $\bigcup_{j=1}^m E_j=\{1,\ldots,n\}$ and $|E_j|$ is within $1$ of $\frac{n}{m}$
for all $j$. We may construct a R.I.S. sequence $x_1,\ldots,x_n$ with
constant $1+\epsilon_0$ such that $x_i \in Y_{2j}$ whenever $i \in E_j$.
By Lemma \ref{fundamental},
$$\norm{\sum_{i \in E_j}x_i} \ensuremath{\leqslant}
(1+2\epsilon_0)(\frac{n}{m}+1)f(\frac{n}{m}-1)^{-1} \ensuremath{\leqslant} 2nf(n)^{-1} m^{-1}.$$ Let $z_j=\norm{\sum_{i \in E_j}x_i}^{-1}\sum_{i \in E_j}x_i$. Then $z_j \in
Y_{2j}$ for all $j$ and $$\norm{\sum_{j=1}^m z_j} \ensuremath{\geqslant} f(n)^{-1}\sum_{j=1}^m \big(\norm{\sum_{i \in E_j}x_i}^{-1}\sum_{i \in E_j}\norm{x_i}\big) \ensuremath{\geqslant} m/2.$$ Therefore $$\norm{\sum_{i=1}^m y_i} \ensuremath{\leqslant} 4f(m)^{-1}\norm{\sum_{i=1}^m z_i} \ensuremath{\leqslant} \epsilon \norm{\sum_{i=1}^m z_i}.$$ Obviously $(y_{i})_{i=1}^m$ is not $\epsilon^{-1}$-equivalent to $(z_{i})_{i=1}^m$, and this means that $X$ is uniformly inhomogeneous.
The proof concerning the dual is quite similar and uses the same notation. Let $Y_{1},\ldots,Y_{2m}$ be arbitrary block subspaces of $X^*$. By Lemma \ref{linftyn}
we may find a R.I.S. sequence $y_1<\cdots<y_m$ with constant $1+\epsilon_0$ and functionals $y_i^* \in Y_{2i-1}$ such that ${\rm range}\ y_i^* \subseteq {\rm range}\ y_i$ and $y_i^*(y_i) \ensuremath{\geqslant} 1/2$ for all $i$. Since $\norm{\sum_{i=1}^m y_i} \ensuremath{\leqslant} 2mf(m)^{-1}$, it follows that $$\norm{\sum_{i=1}^m y_i^*} \ensuremath{\geqslant} \norm{\sum_{i=1}^m y_i}^{-1}\sum_{i=1}^m y_i^*(y_i) \ensuremath{\geqslant} f(m)/4.$$
On the other hand we may construct a R.I.S. sequence $x_1,\ldots,x_n$ with
constant $1+\epsilon_0$ and functionals $x_i^*$ such that ${\rm range}\ x_i^* \subseteq {\rm range}\ x_i$, $x_i^*(x_i) \ensuremath{\geqslant} 1/2$ for all $i$, and such that $x_i^* \in Y_{2j}$ whenever $i \in E_j$. Since
$\norm{\sum_{i \in E_j}x_i} \ensuremath{\leqslant} 2nf(n)^{-1} m^{-1}$, it follows that $$\norm{\sum_{i \in E_j}x_i^*} \ensuremath{\geqslant} \frac{n}{3m}\frac{mf(n)}{2n}=f(n)/6.$$
Let $z_j^*=\norm{\sum_{i \in E_j}x_i^*}^{-1}\sum_{i \in E_j}x_i^*$. Then $z_j^* \in
Y_{2j}$ for all $j$ and $$\norm{\sum_{j=1}^m z_j^*} \ensuremath{\leqslant} \frac{6}{f(n)}f(n)=6.$$ Therefore $$\norm{\sum_{i=1}^m z_i^*} \ensuremath{\leqslant} 24f(m)^{-1}\norm{\sum_{i=1}^m y_i^*} \ensuremath{\leqslant} \epsilon \norm{\sum_{i=1}^m y_i^*}.$$
\end{proof}
\begin{cor} The spaces $S$, $S^*$, $GM$, $GM^*$, $G$, $G^*$, $G_u$, and $G_u^*$ are uniformly inhomogeneous. \end{cor}
\
A slightly stronger statement may be obtained by the proof of Proposition \ref{spaceswithtcaciuc}, in the sense that the vectors $y_i$ in the definition of uniform inhomogeneity may be chosen to be successive. More explicitely, the conclusion may be replaced by the statement that $$ \forall M\ensuremath{\geqslant} 1\; \exists n \in \N\; \forall Y_1,\ldots,Y_{2n} \subseteq X\; \exists y_i \in\ku S_{Y_i}\;(y_i)_{i=1}^n\not\sim_M(y_i)_{i=n+1}^{2n}. $$ where $y_1<\cdots<y_n$ and $y_{n+1}<\cdots<y_{2n}$, and as before $Y_1,\dots,Y_{2n}$ are infinite-dimensional subspaces of $X$.
This property is therefore a block version of the property of uniform inhomogeneity. It was observed in \cite{FR4} that the sixth dichotomy had the following ``block'' version: any Schauder basis of a Banach space contains a block sequence which is either block uniformly inhomogeneous in the above sense or asymptotically $\ell_p$ for some $p \in [1,+\infty]$.
It is interesting to observe that either side of this dichotomy corresponds to one of the two main families of HI spaces, namely spaces of the type of Gowers-Maurey, based on the example of Schlumprecht, and spaces of the type of Argyros-Deliyanni, based on Tsirelson's type spaces. More precisely, spaces of the type of Gowers-Maurey are block uniformly inhomogeneous, while spaces of the type of Argyros-Deliyanni are asymptotically $\ell_1$. Observe that the original dichotomy of Tcaciuc fails to distinguish between these two families, since any HI space is trivially uniformly inhomogeneous, see \cite{FR4}.
\section{Tight HI spaces of the type of Gowers and Maurey}\label{gowers-maurey}
In this section we show that Gowers' space $G$ constructed in \cite{g:asymptotic} and its dual are of type
(1).
The proof is a refinement of the proof that
$G_u$ or $G_u^*$ is of type (3), in which we observe that the hypothesis of
unconditionality may be replaced by asymptotic unconditionality. The idea is
to produce constituent parts of vectors or functionals in Gowers'
construction with sufficient control on their supports (and not just on
their ranges, as would be enough to obtain the HI property for example).
\subsection{ A HI space tight by range} The space $G$ has a norm defined by induction as in $GM$, with the addition of a new term which guarantees that its basis $(e_n)$ is $2$-asymptotically unconditional, that is for any sequence of normalised vectors $N<x_1<\ldots<x_N$, any sequence of scalars $a_1,\ldots,a_N$ and any sequence of signs $\epsilon_1,\ldots,\epsilon_N$, $$\norm{\sum_{n=1}^N \epsilon_n a_n x_n} \ensuremath{\leqslant} 2\norm{\sum_{n=1}^N a_n x_n}.$$ The basis is bimonotone and, although this is not stated in \cite{g:asymptotic}, it may be proved as for $GM$ that $G$ is reflexive. It follows that the dual basis of $(e_n)$ is also $2$-asymptotically unconditional. The norm on $G$ is defined by the implicit equation, for all $x \in c_{00}$:
$$\norm{x}=\norm{x}_{c_0}\vee\ \sup\Bigl\{f(n)^{-1}\sum_{i=1}^n\norm{E_i x}\Del 2 \ensuremath{\leqslant} n, E_1<\ldots<E_n\Bigr\}$$
$$\vee\ \sup\Bigl\{|x^*(Ex)|\Del k \in K, x^* \hbox{ special of length } k, E \subseteq \N\Bigr\}$$ $$\vee\ \sup\Bigl\{\norm{Sx}\Del S \hbox{ is an admissible operator}\Bigr\},$$
\
where $E$, $E_1,\ldots,E_n$ are intervals of integers, and $S$ is an {\em admissible operator} if $Sx=\frac{1}{2}\sum_{n=1}^N \epsilon_n E_n x$ for some sequence of signs $\epsilon_1,\ldots,\epsilon_N$ and some sequence $E_1,\ldots,E_N$ of intervals which is {\em admissible}, i.e. $N<E_1$ and $1+\max E_i=\min E_{i+1}$ for every $i < N$.
{\em R.I.S. pairs} and {\em special pairs} are considered in \cite{g:asymptotic}; first we shall need a more general definition of these. Let $x_1,\ldots,x_m$ be a R.I.S. with constant $C$, $m \in [\log N, \exp N]$, $N \in L$, and let $x_1^*,\ldots, x_m^*$ be successive normalised functionals. Then we call {\em generalised R.I.S. pair with constant $C$} the pair $(x,x^*)$ defined by $x=\norm{\sum_{i=1}^m x_i}^{-1}(\sum_{i=1}^m x_i)$ and $x^*=f(m)^{-1}\sum_{i=1}^m x_i^*$.
Let $z_1,\ldots,z_k$ be a sequence of successive normalised R.I.S. vectors with constant $C$, and let $z_1^*,\ldots, z_k^*$ be a special sequence such that $(z_i,z_i^*)$ is a generalized R.I.S. pair for each $i$. Then we shall call {\em generalised special pair with constant $C$} the pair $(z,z^*)$ defined by $z=\sum_{i=1}^k z_i$ and $z^*=f(k)^{-1/2}(\sum_{i=1}^k z_i^*)$. The pair $(\norm{z}^{-1}z,z^*)$ will be called {\em normalised generalised
special pair}.
\begin{lemme}\label{critical}
Let $(z,z^*)$ be a generalised special pair in $G$, of length $k \in K$, with
constant $2$ and such that ${\rm supp}\ z^* \cap {\rm supp}\ z = \emptyset$. Then $$\norm{z} \ensuremath{\leqslant} \frac{5k}{f(k)}.$$ \end{lemme}
\begin{proof} The proof follows classically the methods of \cite{GM} or \cite{g:hyperplanes}.
Let $K_0=K \setminus \{k\}$ and let $g$ be the corresponding function given by \cite{g:asymptotic} Lemma 5. To prove that $\norm{z} \leq5kf(k)^{-1}$ it is enough by Lemma \ref{fundamental} to prove that for any interval $E$ such that $\norm{Ez} \ensuremath{\geqslant} 1/3$, $Ez$ is normed by some $(M,g)$-form with $M \ensuremath{\geqslant} 2$.
By the discussion in \cite{g:asymptotic} after the definition of the norm, the only possible norming functionals apart from $(M,g)$-forms are of the form $Sw^*$ where $w^*$ is a special functional of length $k$, and $S$ is an ``acceptable'' operator according to the terminology of \cite{g:asymptotic}. We shall not state the definition of an acceptable operator $S$, we shall just need to know that since such an operator is diagonal of norm at most $1$, it preserves support and $(M,g)$-forms, \cite{g:asymptotic} Lemma 6. So let $w^*=f(k)^{-1/2}(w_1^*+\cdots+w_k^*)$ be a special functional of length $k$, $S$ be an acceptable operator, and $E$ be an interval such that $\norm{Ez} \ensuremath{\geqslant} 1/3$. We need to show that $Sw^*$ does not norm $Ez$.
Let $t$ be minimal such that $w_t^* \neq z_t^*$. If $i \neq j$ or $i=j>t$ then by definition of special sequences there exist $M \neq N \in L$, $\min(M,N) \ensuremath{\geqslant} j_{2k}$, such that $w_i^*$ and therefore $Sw_i^*$ is an $(M,f)$-form and $z_j$ is an R.I.S. vector of size $N$ and constant $2$. By \cite{g:asymptotic} Lemma 8, $z_j$ is an $\ell_{1+}^{N^{1/10}}$-average with constant $4$. If $M<N$
then $2M<\log \log \log N$ so, by \cite{g:asymptotic} Lemma 2, $|Sw_i^*(Ez_j)| \ensuremath{\leqslant} 12f(M)^{-1}$. If $M>N$ then $\log \log \log M>2N$ so, by
\cite{g:asymptotic} Lemma 3, $|Sw_i^*(Ez_j)| \ensuremath{\leqslant} 3f(N)/N$. In both cases it follows that $|Sw_i^*(Ez_j)| \ensuremath{\leqslant} k^{-2}$.
If $i=j=t$ we simply have $|Sw_i^*(Ez_j)| \ensuremath{\leqslant} 1$. Finally if $i=j<t$ then $w_i^*=z_i^*$. and
since ${\rm supp}\ Sz^*_i \subseteq {\rm supp}\ z_i^*$ and ${\rm supp}\ Ez_i \subseteq {\rm supp}\ z_i,$ it follows that $Sw_i^*(Ez_j)=0$ in this case.
Summing up we have obtained that
$$|Sw^*(Ez)| \ensuremath{\leqslant} f(k)^{-1/2}(k^2. k^{-2}+1)=2f(k)^{-1/2} < 1/3 \ensuremath{\leqslant} \norm{Ez}.$$ Therefore $Sw^*$ does not norm $Ez$ and this finishes the proof. \end{proof}
The next lemma is expressed in a version which may seem technical but this will make the proof that $G$ is of type (1) more pleasant to read. At first reading, the reader may simply assume that $T=Id$ in its hypothesis.
\begin{lemme}\label{average} Let $n \in \N$ and let $\epsilon>0$. Let $(x_i)_i$ be a normalised block basis in $G$ of length $n^k$ and supported after $2n^k$, where $k=\min\{i \del f(n^i) < (1+\epsilon)^i\}$, and $T:[x_i] \rightarrow G$ be an isomorphism such that $(Tx_i)$ is also a normalised block basis. Then for any $n \in \N$ and $\epsilon>0$, there exist a finite interval $F$ and a multiple $x$ of $\sum_{i \in F}x_i$ such that $Tx$ is an $\ell_{1+}^n$-average with constant $1+\epsilon$, and a normalised functional $x^*$ such that $x^*(x) >1/2$ and ${\rm supp}\ x^* \subseteq \bigcup_{i \in F}{\rm range}\ x_i$. \end{lemme}
\begin{proof} The proof from \cite{g:asymptotic} that the block basis $(Tx_i)$ contains an
$\ell_{1+}^n$-average with constant $1+\epsilon$ is the same as for $GM$, and gives that such a vector exists of the form $Tx=\lambda \sum_{i \in F}Tx_i$, thanks to the condition on the length of $(x_i)$. We may therefore deduce that $2|F|-1<{\rm supp}\ x$. Let $y^*$ be a unit functional which norms $x$ and such that ${\rm range}\; y^* \subseteq {\rm range}\; x$. Let $x^*=Ey^* $ where $E$ is the union of the $|F|$ intervals ${\rm range}\; x_i, i \in F$. Then $x^*(x)=y^*(x)=1$ and by unconditional asymptoticity of $G^*$, $\norm{x^*} \ensuremath{\leqslant} \frac{3}{2}\norm{y^*}<2$.
\end{proof}
The proof that $G$ is HI requires defining ``extra-special sequences'' after having defined special sequences in the usual $GM$ way. However, to prove that $G$ is tight by range, we shall not need to enter that level of complexity and shall just use special sequences.
\begin{prop}\label{type1} The space $G$ is of type (1). \end{prop}
\begin{proof} Assume some normalised block-sequence $(x_n)$ is such that $[x_n]$ embeds into $Y=[e_i, i \notin \bigcup_n {\rm range}\ x_n]$ and look for a contradiction. Passing to a subsequence and by reflexivity we may assume that there is some isomorphism $T:[x_n] \rightarrow Y$ satisfying the hypothesis of Lemma \ref{average}, that is, $(Tx_n)$ is a normalised block basis in $Y$. Fixing $\epsilon=1/10$ we may construct by Lemma \ref{average} some block-sequence of vectors in $[x_n]$ which are $1/2$-normed by functionals in ${\bf Q}$ of support included in $\bigcup_n {\rm range}\; x_n$, and whose images by $T$ form a sequence of increasing length ${\ell}_{1+}^n$-averages with constant $1+\epsilon$. If $Tz_1,\ldots,Tz_m$ is a R.I.S. of these ${\ell}_{1+}^n$-averages with constant $1+\epsilon$, with $m \in [\log N, \exp N]$, $N \in L$, and $z_1^*,\ldots,z_m^*$ are the functionals associated to $z_1,\ldots,z_m$, then by \cite{g:asymptotic} Lemma 7, the $(m,f)$-form $z^*=f(m)^{-1}(z_1^*+\ldots+z_m^*)$ satisfies $$ z^*(z_1+\ldots+z_m) \ensuremath{\geqslant} \frac{m}{2f(m)} \ensuremath{\geqslant} \frac{1}{4}\norm{Tz_1+\ldots+Tz_m} \ensuremath{\geqslant} (4\norm{T^{-1}})^{-1}\norm{z_1+\dots+z_m}. $$ Therefore we may build R.I.S. vectors $Tz$ with constant $1+\epsilon$ of arbitrary length $m$ in $[\log N, \exp N]$, $N \in L$, so that $z$ is $(4\norm{T^{-1}})^{-1}$-normed by an $(m,f)$-form $z^*$ of support included in $\bigcup_n {\rm range}\;x_n$. For such $(z,z^*)$, $(Tz,z^*)$ is a generalised R.I.S. pair. We then consider a sequence $Tz_1,\ldots,Tz_k$ of length $k \in K$ of such R.I.S. vectors, such that there exists some special sequence of corresponding functionals $z_1^*,\ldots,z_k^*$, and finally the pair $(z,z^*)$ where $z=z_1+\cdots+z_k$ and $z^*=f(k)^{-1/2}(z_1^*+\ldots+z_k^*)$: observe that the support of $z^*$ is still included in $\bigcup_n {\rm range}\;x_n$.
Since $(Tz,z^*)$ is a generalised special pair, it follows from Lemma \ref{critical} that $$\norm{Tz} \ensuremath{\leqslant} 5kf(k)^{-1}.$$ On the other hand, $$\norm{z} \ensuremath{\geqslant} z^*(z) \ensuremath{\geqslant} (4\norm{T^{-1}})^{-1}k f(k)^{-1/2}.$$
Since $k$ was arbitrary in $K$ this implies that $T^{-1}$ is unbounded and provides the desired contradiction. \end{proof}
\subsection{A HI space tight by range and locally minimal} As we shall now prove, the dual $G^*$ of $G$ is of type (1) as well, but also locally minimal.
\begin{lemme}\label{linftynbis} Let $(x_i^*)$ be a normalised block basis in $G^*$. Then for any $n \in \N$ and $\epsilon>0$, there exists $N(n,\epsilon)$, a finite interval $F \subseteq [1,N(n,\epsilon)]$, a multiple $x^*$ of $\sum_{i \in F}x_i^*$ which is an $\ell_{\infty +}^n$-average with constant $1+\epsilon$ and an $\ell_{1+}^n$-average $x$ with constant $2$ such that $x^*(x) >1/2$ and ${\rm supp}\ x \subseteq \bigcup_{i \in F}{\rm range}\ x_i^*$. \end{lemme}
\begin{proof} We may assume that $\epsilon<1/6$. By Lemma \ref{linftyn} we may find for each $i \ensuremath{\leqslant} n$
an interval $F_i$, with $|F_i| \ensuremath{\leqslant} 2\min F_i$, and a vector $y_i^*$ of the form $\lambda \sum_{k \in F_i} x_k^*$, such that $y^*=\sum_{i=1}^n y_i^*$ is an $\ell_{\infty +}^n$-average with constant $1+\epsilon$. Let, for each $i$, $x_i$ be normalised such that $y_i^*(x_i)=\norm{y_i^*}$ and ${\rm range}\ x_i \subseteq {\rm range}\ y_i^*$. Let $y_i=E_i x_i$, where $E_i$ denotes the canonical projection on $[e_m, m \in \bigcup_{k \in F_i}{\rm range}\ x_k^*]$. By the asymptotic unconditionality of $(e_n)$, we have that $\norm{y_i} \ensuremath{\leqslant} 3/2$. Let $y_i^{\prime}=\norm{y_i}^{-1}y_i$, then $$y_i^*(y_i^{\prime})=\norm{y_i}^{-1}y_i^*(y_i)=\norm{y_i}^{-1}y_i^*(x_i) \ensuremath{\geqslant} \frac{2}{3}\norm{y_i^*}.$$ By Lemma \ref{linftyn}, the vector $x=\sum_i y_i^{\prime}$ is an $\ell_{1+}^n$-vector with constant $2$, such that $x^*(x) >\norm{x}/2$, and clearly ${\rm supp}\ x \subseteq \bigcup_{i \in F}{\rm range}\ x_i^*$. \end{proof}
\begin{prop} The space $G^*$ is locally minimal and tight by range. \end{prop}
\begin{proof}
By Lemma \ref{linftynbis} we may find in any finite block subspace of $G^*$ of length $N(n,\epsilon)$ and supported after $N(n,\epsilon)$ an $\ell_{\infty+}^n$-average with constant $1+\epsilon$. By asymptotic unconditionality we deduce that uniformly, any block-subspace of $G^*$ contains $\ell_{\infty}^n$'s, and therefore $G^*$ is locally minimal.
We prove that $G^*$ is tight by range. Assume towards a contradiction that some normalised block-sequence $(x_n^*)$ is such that $[x_n^*]$ embeds into $Y=[e_i^*, i \notin \bigcup_n {\rm range}\ x_n^*]$ and look for a contradiction. If $T$ is the associated isomorphism, we may by passing to a subsequence and perturbating $T$ assume that $Tx_n^*$ is successive.
Let $\epsilon=1/10$. By Lemma \ref{linftynbis}, we find in $[x_k^*]$ and for each $n$, an $\ell_{\infty+}^n$-average $y_n^*$ with constant $1+\epsilon$ and an $\ell_{1+}^n$-average $y_n$ with constant $2$, such that $y_n^*(y_n) > 1/2$ and ${\rm supp}\ y_n \subseteq \bigcup_k {\rm range}\ x_k^*$. By construction, for each $n$, $Ty_n^*$ is disjointly supported from $[x_k^*]$, and up to modifying $T$, we may assume that $Ty_n^*$ is in ${\bf Q}$ and of norm at most $1$ for each $n$.
If $z_1,\ldots,z_m$ is a R.I.S. of these ${\ell}_{1+}^n$-averages $y_n$ with constant $2$, with $m \in [\log N, \exp N]$, $N \in L$, and $z_1^*,\ldots,z_m^*$ are the ${\ell}_{\infty+}^n$-averages associated to $z_1,\ldots,z_m$, then by \cite{g:hyperplanes} Lemma 7, the $(m,f)$-form $z^*=f(m)^{-1}(z_1^*+\ldots+z_m^*)$ satisfies $$z^*(z_1+\ldots+z_m) \ensuremath{\geqslant} \frac{m}{2f(m)} \ensuremath{\geqslant} \frac{1}{6}\norm{z_1+\ldots+z_m},$$ and furthermore $Tz^*$ is also an $(m,f)$-form. Therefore we may build R.I.S. vectors $z$ with constant $2$ of arbitrary length $m$ in $[\log N, \exp N]$, $N \in L$, so that $z$ is $6^{-1}$-normed by an $(m,f)$-form $z^*$ such that $Tz^*$ is also an $(m,f)$-form. We may then consider a sequence $z_1,\ldots,z_k$ of length $k \in K$ of such R.I.S. vectors of length $m_i$, and some corresponding functionals $z_1^*,\ldots,z_k^*$ (i.e., $z_i^*$ $6^{-1}$-norms $z_i$ and $Tz_i^*$ is also an $(m_i,f)$-form for all $i$), such that $Tz_1^*,\ldots,Tz_k^*$ is a special sequence. Then we let $z=z_1+\cdots+z_k$ and $z^*=f(k)^{-1/2}(z_1^*+\ldots+z_k^*)$, and observe that $(z,Tz^*)$ is a generalised special pair. Since $Tz^*=f(k)^{-1/2}(Tz_1^*+\ldots+Tz_k^*)$ is a special functional it follows that $$\norm{Tz^*} \ensuremath{\leqslant} 1.$$ But it follows from Lemma \ref{critical} that $\norm{z} \ensuremath{\leqslant} 5kf(k)^{-1}$. Therefore $$\norm{z^*} \ensuremath{\geqslant} z^*(z)/\norm{z} \ensuremath{\geqslant} f(k)^{1/2}/30.$$
Since $k$ was arbitrary in $K$ this implies that $T^{-1}$ is unbounded and provides the desired contradiction. \end{proof}
It remains to check that $G^*$ is HI. The proof is very similar to the one in
\cite{g:asymptotic} that $G$ is HI, and
we shall therefore not give all details. There are two main differences
between the two proofs. In \cite{g:asymptotic} some special vectors and
functionals are constructed, the vectors are taken alternatively in
arbitrary block subspaces $Y$ and $Z$ of $G$, and no condition is imposed on where to pick the functionals. In our case there is no condition on where to choose the vectors but we need to pick the
functionals in arbitrary subspaces $Y$ and $Z$ of $G^*$ instead. This is
possible because of Lemma \ref{linftynbis}. We also need to correct what seems to be a slight imprecision in the proof of
\cite{g:asymptotic} about the value of some normalising factors, and therefore we also get worst constants for our
estimates.
Let $\epsilon=1/10$. Following Gowers we define an {\em R.I.S. pair} of size $N$ to be a generalised R.I.S. pair $(x,x^*)$ with constant $1+\epsilon$ of the form $(\norm{x_1+\ldots+x_N}^{-1}(x_1+\ldots+x_N), f(N)^{-1}(x_1^*+\dots+x_N^*))$, where $x_n^*(x_n)\ensuremath{\geqslant} 1/3$ and ${\rm range}\ x_n^* \subset{\rm range}\ x_n$ for each $n$. A {\em special pair} is a normalised generalised special pair with constant $1+\epsilon$ of the form
$(x,x^*)$ where $x=\norm{x_1+\ldots+x_k}^{-1}(x_1+\ldots+x_k)$ and
$x^*=f(k)^{-1/2}(x_1^*+\dots+x_k^*)$ with ${\rm range}\ x_n^* \subseteq {\rm range}\ x_n$
and for each $n$, $x_n^*\in\bf Q$, $|x_n^*(x_n)-1/2|<10^{-\min{\rm supp}\ x_n}$. By \cite{g:asymptotic} Lemma 8, $z$ is a R.I.S. vector with constant $2$ whenever $(z,z^*)$ is a special pair. We shall also require that
$k \ensuremath{\leqslant} \min{\rm supp}\ x_1$, which will imply by \cite{g:asymptotic} Lemma 9 that for $m<k^{1/10}$, $z$ is a $\ell_{1+}^m$-average with constant 8 (see the beginning of the proof of Proposition \ref{GstarHI}).
Going up a level of ``specialness'', a {\em special R.I.S.-pair} is a generalised R.I.S.-pair with constant 8 of the form $(\norm{x_1+\ldots+x_N}^{-1}(x_1+\ldots+x_N), f(N)^{-1}(x_1^*+\dots+x_N^*))$, where ${\rm range}\ x_n^* \subset{\rm range}\ x_n$ for each $n$, and with
the additional condition that $(x_n,x_n^*)$ is a special pair of length at least $\min {\rm supp}\ x_n$. Finally, an {\em extra-special pair} of size $k$ is a normalised generalised special pair $(x,x^*)$ with constant 8 of the form
$x=\norm{x_1+\ldots+x_k}^{-1}(x_1+\ldots+x_k)$ and
$x^*=f(k)^{-1/2}(x_1^*+\dots+x_k^*)$ with ${\rm range}\ x_n^* \subseteq {\rm
range}\ x_n$, such that, for each $n$, $(x_n,x_n^*)$ is a special R.I.S.-pair of length $\sigma(x_1^*,\dots,x_{n-1}^*)$.
\
Given $Y,Z$ block subspaces of $G^*$ we shall show how to find an extra-special pair $(x,x^*)$ of size $k$, with $x^*$ built out of vectors in $Y$ or $Z$, such that the signs of these constituent parts of $x^*$ can be changed according to belonging to $Y$ or $Z$ to produce a vector $x^{\prime*}$ with $\norm{x^{\prime*}}\ensuremath{\leqslant} 12f(k)^{-1/2}\norm{x^*}$. This will then prove the result.
Consider then an extra-special pair $(x,x^*)$. Then $x$ splits up as $$\nu^{-1}\sum_{i=1}^k\nu_i^{-1}\sum_{j=1}^{N_i}\nu_{ij}^{-1} \sum_{r=1}^{k_{ij}}x_{ijr}$$
and $x^*$ as $$f(k)^{-1/2}\sum_{i=1}^k f(N_i)^{-1}\sum_{j=1}^{N_i}f(k_{ij})^{-1} \sum_{r=1}^{k_{ij}}x^*_{ijr}\,$$ where the numbers $\nu$, $\nu_i$ and $\nu_{ij}$ are the norms of what appears to the right. These special sequences are chosen far enough ``to the right'' so that $k_{ij}\ensuremath{\leqslant}\min{\rm supp}\ x_{ij1}$, and also so that $(\max{\rm supp}\ x_{i\,j-1})^2k_{ij}^{-1}\ensuremath{\leqslant} 4^{-(i+j)}$. We shall also write $x_i$ for $\nu_i^{-1}\sum_{j=1}^{N_i}\nu_{ij}^{-1} \sum_{r=1}^{k_{ij}}x_{ijr}$ and $x_{ij}$ for $\nu_{ij}^{-1} \sum_{r=1}^{k_{ij}}x_{ijr}$.
We define a vector $x'$ by $$\sum_{i=1}^k\nu_i^{\prime -1}\sum_{j=1}^{N_i}\nu_{ij}^{\prime -1} \sum_{r=1}^{k_{ij}}(-1)^rx_{ijr},$$ where the numbers $\nu_i^{\prime}$ and $\nu_{ij}^{\prime}$ are the norms of what appears to the right. We shall write $x'_i$ for $\nu_i^{\prime -1}\sum_{j=1}^{N_i}\nu_{ij}^{\prime -1} \sum_{r=1}^{k_{ij}}(-1)^rx_{ijr}$ and $x'_{ij}$ for $\nu_{ij}^{\prime -1} \sum_{r=1}^{k_{ij}}(-1)^rx_{ijr}$.
Finally we define a functional $x^{\prime *}$ as $$f(k)^{-1/2}\sum_{i=1}^k f(N_i)^{-1}\sum_{j=1}^{N_i}f(k_{ij})^{-1} \sum_{r=1}^{k_{ij}}(-1)^k x^*_{ijr}.$$
\begin{prop}\label{GstarHI} The space $G^*$ is HI.\end{prop}
\begin{proof} Fix $Y$ and $Z$ block subspaces of $G^*$. By Lemma \ref{linftynbis} we may construct an extra-special pair $(x,x^*)$ so that $x^*_{ijr}$ belongs to $Y$ when $r$ is odd and to $Z$ when $r$ is even.
We first discuss the normalisation of the vectors involved in the definition of $x'$. By the increasing condition on $k_{ij}$ and $x_{ijr}$ and by asymptotic unconditionality, we have that $$\norm{\sum_{r=1}^{k_{ij}} (-1)^r x_{ijr}} \ensuremath{\leqslant} 2 \norm{\sum_{r=1}^{k_{ij}} x_{ijr}},$$ which means that $\nu^{\prime}_{ij} \ensuremath{\leqslant} 2 \nu_{ij}$. Furthermore it also follows that
the functional $(1/2)f(k_{ij})^{-1/2}\sum_{r=1}^{k_{ij}}(-1)^rx^*_{ijr}$ is
of norm at most $1$, and therefore we have that
$\norm{\sum_{r=1}^{k_{ij}}(-1)^rx_{ijr}}\ensuremath{\geqslant} (1/2)k_{ij}f(k_{ij})^{-1/2}$.
Lemma 9 from \cite{g:asymptotic} therefore tells us that, for every $i,j$, $x'_{ij}$ is an $\ell_{1+}^{m_{ij}}$-average with constant 8, if $m_{ij}<k_{ij}^{1/10}$. But the $k_{ij}$ increase so fast that, for any $i$, this implies that the sequence $x'_{i1},\dots,x'_{i\,N_i}$ is a rapidly increasing sequence with constant 8. By \cite{g:asymptotic} Lemma 7, it follows that $$\norm{\sum_{j=1}^{N_i} x_{ij}^{\prime}} \ensuremath{\leqslant} 9 N_i/f(N_i).$$ Therefore by the $f$-lower estimate in $G$ we have that $\nu^{\prime}_i \ensuremath{\leqslant} 9\nu_i$.
We shall now prove that $\norm{x'} \ensuremath{\leqslant} 12kf(k)^{-1}$. This will imply that $$\norm{x_*^{\prime}} \ensuremath{\geqslant} \frac{x_*^{\prime}(x')}{\norm{x'}} \ensuremath{\geqslant} \frac{f(k)}{12k}[f(k)^{-1/2} \sum_{i=1}^k f(N_i)^{-1}\nu_i^{\prime -1}\sum_{j=1}^{N_i}f(k_{ij})^{-1}\nu_{ij}^{\prime -1} \sum_{r=1}^{k_{ij}}x_{ijr}^*(x_{ijr})]$$ $$\ensuremath{\geqslant} f(k)^{1/2}(12k)^{-1}.18^{-1} [\sum_{i=1}^k f(N_i)^{-1}\nu_i^{-1}\sum_{j=1}^{N_i}f(k_{ij})^{-1}\nu_{ij}^{-1} \sum_{r=1}^{k_{ij}}x_{ijr}^*(x_{ijr})]$$ $$= f(k)^{1/2} (216k)^{-1} \sum_{i=1}^k x_i^*(x_i) \ensuremath{\geqslant} 648^{-1} f(k)^{1/2}.$$ By construction of $x^*$ and $x^{\prime *}$ this will imply that $$\norm{y^*-z^*} \ensuremath{\geqslant} 648^{-1}f(k)^{1/2}\norm{y^*+z^*}$$ for some non zero $y^* \in Y$ and $z^* \in Z$, and since $k \in K$ was arbitrary, as well as $Y$ and $Z$, this will prove that $G^*$ is HI.
\
The proof that $\norm{x'} \ensuremath{\leqslant} 12kf(k)^{-1}$ is given in three steps:
\
\paragraph{Step 1} {\em The vector $x'$ is a R.I.S. vector with constant 11.} \begin{proof} We already know the sequence $x'_{i1},\dots,x'_{i\,N_i}$ is a rapidly increasing sequence with constant 8. Then by \cite{g:asymptotic} Lemma 8 we get that $x'_i$ is also an $\ell_{1+}^{M_i}$-average with constant 11, if $M_i<N_i^{1/10}$. Finally, this implies that $x'$ is an R.I.S.-vector with constant 11, as claimed.\end{proof}
\paragraph{Step 2} {\em Let $K_0=K\setminus\{k\}$, let $g\in\mathcal F$ be the corresponding function given by \cite{g:asymptotic} Lemma 5. For every interval $E$ such that $\norm{Ex'}\ensuremath{\geqslant} 1/3$, $Ex'$ is normed by an $(M,g)$-form.}
\begin{proof}The proof is exactly the same as the one of Step 2 in the proof
of Gowers concerning $G$, apart from some constants which are modified due to the change of constant in Step 1 and to the normalising constants relating $\nu_i$ and $\nu_{ij}$ respectively to $\nu_i^{\prime}$ and $\nu_{ij}^{\prime}$. The reader is therefore referred to \cite{g:asymptotic}.\end{proof}
\paragraph{Step 3} {\em The norm of $x'$ is at most $12kg(k)^{-1}=12kf(k)^{-1}$} \begin{proof} This is an immediate consequence of Step 1, Step 2 and of
Lemma \ref{fundamental}. \end{proof}
We conclude that the space $G^*$ is HI, and thus locally minimal of type (1). \end{proof}
\section{Unconditional tight spaces of the type of Argyros and Deliyanni}\label{argyros} By Proposition \ref{spaceswithtcaciuc}, unconditional or HI spaces built on the model of Gowers-Maurey's spaces are uniformly inhomogeneous (and even block uniformly inhomogeneous). We shall now consider a space of Argyros-Deliyanni type, more specifically of the type of a space constructed by Argyros, Deliyanni, Kutzarova and Manoussakis \cite{ADKM}, with the opposite property, i.e., with a basis which is strongly asymptotically $\ell_1$. This space will also be tight by support and therefore will not contain a copy of $\ell_1$. By the implication at the end of the diagram which appears just before Theorem \ref{final}, this basis will therefore be tight with constants as well, making this example the ``worst'' known so far in terms of minimality.
Again in this section block vectors will not necessarily be normalized and some familiarity with the construction in \cite{ADKM} will be assumed.
\subsection{A strongly asymptotically $\ell_1$ space tight by support}
In \cite{ADKM} an example of HI space $X_{hi}$ is constructed, based on a ``boundedly modified'' mixed Tsirelson space $X_{M(1),u}$. We shall construct an unconditional version $X_u$ of $X_{hi}$ in a similar way as $G_u$ is an unconditional version of $GM$. The proof that $X_u$ is of type (3) will be based on the proof that $X_{hi}$ is HI, conditional estimates in the proof of \cite{ADKM} becoming essentially trivial in our case due to disjointness of supports.
Fix a basis $(e_n)$ and ${\mathcal M}$ a family of finite subsets of $\N$. Recall that a family $x_1,\ldots,x_n$ is {\em ${\mathcal M}$-admissible} if $x_1<\cdots<x_n$ and $\{\min {\rm supp}\ x_1,\ldots,\min {\rm supp}\ x_n\} \in {\mathcal M}$, and {\em
${\mathcal M}$-allowable} if $x_1,\ldots,x_n$ are vectors with disjoint supports such that $\{\min {\rm supp}\ x_1,\ldots,\min {\rm supp}\ x_n\} \in {\mathcal M}$. Let ${\mathcal S}$ denote the family of Schreier sets, i.e., of subsets $F$ of $\N$ such that $|F| \ensuremath{\leqslant} \min F$, ${\mathcal M}_j$ be the subsequence of the sequence $({\mathcal F}_k)$ of Schreier families associated to sequences of integers $t_j$ and $k_j$ defined in \cite{ADKM} p 70.
We need to define a new notion. For $W$ a set of functionals which is stable under projections onto subsets of $\N$, we let ${\rm conv}_{\Q}W$ denote the set of rational convex combinations of elements of $W$. By the stability property of $W$ we may write any $c^* \in {\rm conv}_{\Q}W$ as a rational convex combination of the form $\sum_i \lambda_i x_i^*$ where $x_i^* \in W$ and ${\rm supp}\ x_i^* \subseteq {\rm supp}\ c^*$ for each $i$. In this case the set $\{x_i^* \}_i$ will be called a $W$-compatible decomposition of $c^*$, and we let $W(c^*) \subseteq W$ be the union of all $W$-compatible decompositions of $c^*$. Note that if ${\mathcal M}$ is a family of finite subsets of $\N$, $(c_1^*,\ldots,c_d^*)$ is ${\mathcal M}$-admissible, and $x_i^* \in W(c_i^*)$ for all $i$, then $(x_1^*,\ldots,x_d^*)$ is also ${\mathcal M}$-admissible.
Let ${\mathcal B}=\{\sum_{n}\lambda_n e_n: (\lambda_n)_n \in c_{00}, \lambda_n \in \Q \cap [-1,1]\}$ and let $\Phi$ be a 1-1 function from ${\mathcal B}^{<\N}$ into $2\N$ such that if $(c_1^*,\ldots,c_k^*) \in {\mathcal B}^{<\N}$, $j_1$ is minimal such that $c_1^* \in {\rm conv}_{\Q}{\mathcal A}_{j_1}$, and $j_l=\Phi(c_1^*,\ldots,c_{l-1}^*)$ for each $l=2,3,\ldots$, then $\Phi(c_1^*,\ldots,c_k^*)>\max\{j_1,\ldots,j_k\}$ (the set ${\mathcal A}_j$ is defined in \cite{ADKM} p 71 by ${\mathcal A}_j=\cup_n(K_j^n \setminus K^0)$ where the $K_j^n$'s are the sets corresponding to the inductive definition of $X_{M(1),u}$).
For $j=1,2,\ldots $, we set $L_j^0=\{ \pm e_n:n\in \N\}$. Suppose that $\{ L_j^n\}_{j=1}^{\infty }$ have been defined. We set $L^n=\cup_{j=1}^{\infty}L^n_j$ and $$L_1^{n+1}=\pm L_1^n\cup\{\frac{1}{2}(x_1^{*}+\ldots +x_d^{*}):d\in \N,x_i^{*}\in L^n,$$
$$(x_1^{*},\ldots,x_d^{*})\;{\rm is}\; \ {\mathcal S}-{\rm allowable}\},$$ \noindent and for $j\ensuremath{\geqslant} 1$, $$L_{2j}^{n+1}=\pm L_{2j}^n\cup\{\frac{1}{m_{2j}}(x_1^{\ast }+\ldots +x_d^{*}):d\in {\N},x_i^{*}\in L^n,$$ $$(x_1^{*},\ldots ,x_d^{*})\;{\rm is}\; {\mathcal M}_{2j}-{\rm admissible }\},$$
$$L_{2j+1}^{\prime\;n+1}=\pm L_{2j+1}^n\cup \{\frac{1}{m_{2j+1}}(x_1^{\ast }+\ldots +x_d^{\ast }):d\in {\N} {\rm\ such\ that}$$ $$\exists (c_1^*,\ldots,c_d^*)\;{\mathcal M}_{2j+1}-{\rm admissible\ and\ } k>2j+1 {\rm\ with\ }c_1^* \in {\rm conv}_{\Q}L_{2k}^n, x_1^*\in L_{2k}^n(c_1^*),$$
$$c_i^* \in {\rm conv}_{\Q}L_{\Phi (c_1^{\ast },\ldots ,c_{i-1}^{\ast })}^n, x_i^{\ast }\in L_{\Phi (c_1^{\ast },\ldots ,c_{i-1}^{\ast })}^n(c_i^*) \;{\rm for}\;1<i\ensuremath{\leqslant} d\},$$
$$L_{2j+1}^{n+1}=\{ Ex^{\ast }:x^{\ast }\in L_{2j+1}^{\prime\;n+1},
E {\rm\ subset\ of\ } \N\}.$$
We set ${\mathcal B}_j=\cup_{n=1}^{\infty }(L_j^n\setminus L^0)$
and we consider the norm on $c_{00}$ defined by the set $L=L^0\cup (\cup_{j=1}^{\infty }{\mathcal B}_j)$. The space $X_u$ is the completion of $c_{00}$ under this norm.
\
In \cite{ADKM} the space $X_{hi}$ is defined in the same way except that $E$ is an {\bf interval} of integers in the definition of $L_{2j+1}^{n+1}$, and the definition of $L_{2j+1}^{\prime\;n+1}$ is simpler, i.e., the coding $\Phi$ is defined directly on ${\mathcal M}_{2j+1}$-admissible families $x_1^*,\ldots,x_d^*$ in $L^{<\N}$ and in the definition each $x_i^*$ belongs to $L_{\Phi (x_1^{\ast},\ldots ,x_{i-1}^{\ast})}^n$. To prove the desired properties for $X_u$ one could use the simpler definition of $L_{2j+1}^{\prime\;n+1}$; however this definition doesn't seem to provide enough special functionals to obtain interesting properties for the dual as well.
The ground space for $X_{hi}$ and for $X_u$ is the space $X_{M(1),u}$ associated to a norming set $K$ defined by the same procedure as $L$, except that $K_{2j+1}^n$ is defined in the same way as $K_{2j}^n$, i.e. $$K_{2j}^{n+1}=\pm K_{2j}^n\cup\{\frac{1}{m_{2j}}(x_1^{\ast }+\ldots +x_d^{*}):d\in {\N},x_i^{*}\in K^n,$$ $$(x_1^{*},\ldots ,x_d^{*})\;{\rm is}\; {\mathcal M}_{2j+1}-{\rm admissible }\}.$$
For $n=0,1,2,\ldots ,$ we see that $L_j^n$ is a subset of $K_j^n$, and therefore $L \subseteq K$.
The norming set $L$ is closed under projections onto {\bf subsets} of $\N$, from which it follows that its canonical basis is unconditional, and has the property that for every $j$ and every ${\mathcal M}_{2j}$--admissible family $f_1, f_2, \ldots f_d$ contained in $L$, $f=\frac{1}{m_{2j}}(f_1+\cdots +f_d)$ belongs to $L$. The {\em weight} of such an $f$ is defined by $w(f)=1/m_{2j}$. It follows that for every $j=1,2,\ldots$ and every
${\mathcal M}_{2j}$--admissible family $x_1<x_2<\ldots<x_n$ in $X_u$,
$$\|\sum_{k=1}^nx_k\|\ensuremath{\geqslant}\frac{1}{m_{2j}}\sum_{k=1}^n\| x_k\|.$$ Likewise, for ${\mathcal S}$--allowable families $f_1,\ldots,f_n$ in $L$, we have $f=\frac{1}{2}(f_1+\cdots+f_d) \in L$, and we define $w(f)=1/2$. The weight is defined similarly in the case $2j+1$.
\begin{lemme} The canonical basis of $X_u$ is strongly asymptotically
$\ell_1$. \end{lemme}
\begin{proof} Fix $n\ensuremath{\leqslant} x_1,\ldots,x_n$ where $x_1,\ldots,x_n$ are normalised and
disjointly supported. Fix $\epsilon>0$ and let for each $i$, $f_i \in L$ be
such that $f_i(x_i) \ensuremath{\geqslant} (1+\epsilon)^{-1}$ and ${\rm supp}\ f_i \subseteq {\rm supp}\ x_i$. The condition on the supports may be imposed because $L$ is
stable under projections onto subsets of $\N$. Then $\frac{1}{2}\sum_{i=1}^n \pm f_i \in L$ and therefore
$$\norm{\sum_{i=1}^n \lambda_i x_i} \ensuremath{\geqslant} \frac{1}{2}\sum_{i=1}^n |\lambda_i|
f_i(x_i) \ensuremath{\geqslant} \frac{1}{2(1+\epsilon)}\sum_{i=1}^n |\lambda_i|,$$ for any $\lambda_i$'s. Therefore $x_1,\ldots,x_n$ is $2$-equivalent to the canonical basis of $\ell_1^n$. \end{proof}
It remains to prove that $X_u$ has type (3). Recall that an analysis $(K^s(f))_s$ of $f \in K$ is a decomposition of $f$ corresponding to the inductive definition of $K$, see the precise definition in Definition 2.3 \cite{ADKM}. We shall combine three types of arguments. First $L$ was constructed so that $L \prec K$, which means essentially that each $f \in L$ has an analysis $(K^s(f))_s$ whose elements actually belong to $L$ (see the definition on page 74 of \cite{ADKM}); so all the results obtained in Section 2 of \cite{ADKM} for spaces defined through arbitrary $\tilde{K} \prec K$ (and in particular the crucial Proposition 2.9) are valid in our case. Then we shall produce estimates similar to those valid for $X_{hi}$ and which are of two forms: unconditional estimates, in which case the proofs from \cite{ADKM} may be applied directly up to minor changes of notation, and thus we shall refer to \cite{ADKM} for details of the proofs; and conditional estimates, which are different from those of $X_{hi}$, but easier due to hypotheses of disjointness of supports, and for which we shall give the proofs.
Recall that if ${\mathcal F}$ is a family of finite subsets of $\N$, then $${\mathcal F}^{\prime}=\{A \cup B: A, B \in {\mathcal F}, A \cap B=\emptyset\}.$$ Given $\varepsilon >0$ and $j=2,3,\ldots $, an $(\varepsilon ,j)$-{\it basic special convex combination ($(\varepsilon ,j)$- basic s.c.c.) (relative to $X_{M(1),u})$} is a vector of the form $\sum_{k\in F}a_ke_k$ such that: $F\in {\mathcal M}_j,a_k\ensuremath{\geqslant} 0, \sum_{k\in F}a_k=1$, $\{a_k\}_{k\in F}$ is decreasing, and, for every $G\in {\mathcal F}^{\prime }_{t_j(k_{j-1}+1)}$, $\sum_{k\in G}a_k< \varepsilon $.
Given a block sequence $(x_k)_{k\in {\bf N}}$ in $X_{u}$ and $j\ensuremath{\geqslant} 2$, a convex combination $\sum_{i=1}^na_ix_{k_i}$ is said to be an $(\varepsilon ,j)$-{\it special convex combination} of $(x_k)_{k\in {\bf N}}$ ($(\varepsilon ,j)$-s.c.c), if there exist $l_1<l_2<\ldots <l_n$ such that $2<{\rm supp}\ x_{k_1}\ensuremath{\leqslant} l_1<{\rm supp}\ x_{k_2}\ensuremath{\leqslant} l_2< \ldots <{\rm supp}\ x_{k_n}\ensuremath{\leqslant} l_n$, and $\sum_{i=1}^na_ie_{l_i}$ is an $(\varepsilon , j)$-basic s.c.c.
An $(\varepsilon ,j)$-s.c.c. $\sum_{i=1}^n a_ix_{k_i}$
is called {\it seminormalised} if $\| x_{k_i}\|=1,\; i=1,\ldots ,n$ and
$$\|\sum_{i=1}^na_ix_{k_i}\|\ensuremath{\geqslant}\frac{1}{2}.$$
Rapidly increasing sequences and $(\varepsilon , j)$--R.I. special convex combinations in $X_u$ are defined by \cite{ADKM} Definitions 2.8 and 2.16 respectively, with $\tilde{K}=L.$
Using the lower estimate for ${\mathcal M}_{2j}$-admissible families in $X_u$ we get as in \cite{ADKM} Lemma 3.1.
\begin{lemme}\label{scc} For $\epsilon>0$, $j=1,2,\ldots$ and every normalised block sequence $\{ x_k\}_{k=1}^{\infty }$ in $X_u$, there exists a finite normalised block sequence $(y_s)_{s=1}^n$ of $(x_k)$ and coefficients $(a_s)_{s=1}^ n$ such that $\sum_{s=1}^na_sy_s$ is a seminormalised $(\epsilon,2j)$--s.c.c.. \end{lemme}
The following definition is inspired from some of the hypotheses of \cite{ADKM} Proposition 3.3.
\begin{defi}
Let $j>100$. Suppose that $\{ j_k\}_{k=1}^n$, $\{ y_k\}_{k=1}^n$, $\{ c_k^{\ast }\}_{k=1}^n$ and $\{b_k\}_{k=1}^n$ are such that
{\rm (i)} There exists a rapidly increasing sequence $$\{ x_{(k,i)}:\; k=1,\ldots ,n,\; i=1,\ldots ,n_k\} $$ with $x_{(k,i)}<x_{(k,i+1)}<x_{(k+1,l)}$ for all $k<n$, $i<n_k$, $l\ensuremath{\leqslant} n_{k+1},$ such that:
\noindent {\rm (a)} Each $x_{(k,i)}$ is a seminormalised $(\frac{1}{m^4_{j_{(k,i)}}}, j_{(k,i)})$--s.c.c. where, for each $k$, $2j_k+2<j_{(k,i)},\; i=1,\ldots n_k.$
\noindent {\rm (b)} Each $y_k$ is a $(\frac{1}{m^4_{2j_k}},2j_k)$--
R.I.s.c.c. of $\{ x_{(k,i)}\}_{i=1}^{n_k}$ of the form $y_k=\sum _{i=1}^{n_k}b_{(k,i)}x_{(k,i)}.$
\noindent {\rm (c)} The sequence $\{ b_k\}_{k=1}^n$ is decreasing and $\sum _{k=1}^nb_ky_k$ is a $(\frac{1} {m^4_{2j+1}}, 2j+1)$--s.c.c.
{\rm (ii)} $c_k^{\ast }\in {\rm conv}_{\Q}L_{2j_k}$, and $\max({\rm supp}\ c_{k-1}^{\ast } \cup {\rm supp}\ y_{k-1}) < \min({\rm supp}\ c_k^* \cup {\rm supp}\ y_k)$, $\forall k$.
{\rm (iii)} $j_1>2j+1$ and $2j_k=\Phi (c_1^{\ast },\ldots ,c_{k-1}^{\ast })$, $k=2,\ldots ,n$.
Then $(j_k,y_k,c_k^*,b_k)_{k=1}^n$ is said to be a {\em $j$-quadruple}. \end{defi}
The following proposition is essential. It is the counterpart of Lemma \ref{critical} for the space $X_u$.
\begin{prop}\label{criticalbis} Assume that $(j_k,y_k,c_k^*,b_k)_{k=1}^n$ is a $j$-quadruple
in $X_u$ such that ${\rm supp}\ c_k^* \cap {\rm supp}\ y_k=\emptyset$ for all $k=1,\ldots,n$. Then $$\norm{\sum_{k=1}^n b_km_{2j_k}y_k} \ensuremath{\leqslant} \frac{75}{m_{2j+1}^2}.$$ \end{prop}
\begin{proof}
Our aim is to show that for every $\varphi\in\cup_{i=1}^{\infty }{\mathcal B}_i$, $$\varphi (\sum_{k=1}^n b_k m_{2j_k}y_k)\ensuremath{\leqslant} \frac{75}{m_{2j+1}^2}.$$ The proof is given in several steps.
{\tt 1st Case}: $w(\varphi)=\frac{1}{m_{2j+1}}$. Then $\varphi$ has the form $\varphi =\frac{1}{m_{2j+1}}(Ey^*_{1}+\cdots +Ey^*_{k_2}+Ey^*_{k_2+1}+\cdots Ey^*_d)$ where $E$ is a subset of $\N$ and where $y_k^* \in L_{2j_k}(c_k^*)\ \forall k \ensuremath{\leqslant} k_2$ and $y_k^* \in L_{2j_k}(d_k^*)\ \forall k \ensuremath{\geqslant} k_2+1$, with $d_{k_2+1}^* \neq c_{k_2+1}^*$ (this is similar to the form of such a functional in $X_{hi}$ but with the integer $k_1$ defined there equal to $1$ in our case).
If $k \ensuremath{\leqslant} k_2$ then $c_s^*$ and therefore $y_s^*$ is disjointly supported from $y_k$, so $Ey_s^*(y_k)=0$ for all $s$, and therefore
$\varphi(y_k)=0$. If $k=k_2+1$ then we simply have $|\varphi(y_k)| \ensuremath{\leqslant} \norm{y_k} \ensuremath{\leqslant} 17m_{2j_k}^{-1}$, \cite{ADKM} Corollary 2.17. Finally if $k>k_2+1$ then since $\Phi$ is 1-1, we have that $j_{k_2+1} \neq j_k$ and for all $s=k_2+1,\ldots,d$, $d_s^*$ and therefore $y_s^*$ belong to ${\mathcal B}_{2t_s}$ with $t_s \neq j_k$. It is then easy to check that we may reproduce the proof of \cite{ADKM} Lemma 3.5, applied to $Ey_1^*,\ldots,Ey_d^*$, to obtain the unconditional estimate
$$|\varphi(m_{2j_k}y_k)| \ensuremath{\leqslant} \frac{1}{m_{2j+1}^2}.$$ In particular instead of \cite{ADKM} Proposition 3.2, which is a reformulation of \cite{ADKM} Corollary 2.17 for $X_{hi}$, we simply use \cite{ADKM} Corollary 2.17 with $\tilde{K}=L$.
Summing up these estimates we obtain the desired result for the 1st Case.
{\tt 2nd Case}: $w(\varphi )\ensuremath{\leqslant} \frac{1}{m_{2j+2}}.$ Then we get an unconditional estimate for the evaluation of $\varphi (\sum _{k=1}^{n} b_k m_{2j_k}y_k)$ directly, reproducing the short proof of \cite{ADKM} Lemma 3.7, using again \cite{ADKM} Corollary 2.17 instead of \cite{ADKM} Proposition 3.2. Therefore
$$|\varphi (\sum_{k=1}^nb_km_{2j_k}y_k)| \ensuremath{\leqslant}\frac{35}{m_{2j+2}} \ensuremath{\leqslant} \frac{35}{m_{2j+1}^2}.$$
{\tt 3rd Case}: $w(\varphi)>\frac{1}{m_{2j+1}}$.
We have $y_k=\sum_{i=1}^{n_k}b_{(k,i)}x_{(k,i)}$ and the sequence $\{ x_{(k,i)},k=1,\ldots n,i=1,\ldots n_k\}$ is a R.I.S. w.r.t. $L$. By \cite{ADKM} Proposition 2.9 there exist a functional $\psi\in K^{\prime }$ (see the definition in \cite{ADKM} p 71) and blocks of the basis $u_{(k,i)}$, $k=1,\ldots ,n$, $i=1,\ldots ,n_k$ with
${\rm supp}\ u_{(k,i)}\subseteq {\rm supp}\ x_{(k,i)}$,
$\| u_k\|_{\ell_1}\ensuremath{\leqslant} 16$ and such that
$$|\varphi (\sum_{k=1}^nb_km_{2j_k}
(\sum_{i=1}^{n_k}b_{(k,i)}x_{(k,i)}))|\ensuremath{\leqslant} m_{2j_1}b_1b_{(1,1)}+\psi (\sum_{k=1}^nb_k m_{2j_k}(\sum_{i=1}^{k_n}b_{(k,i)}u_{(k,i)}))+\frac{1}{m_{2j+2}^2}$$ $$\ensuremath{\leqslant}\psi (\sum_{k=1}^nb_km_{2j_k}(\sum_{i=1}^{k_n} b_{(k,i)}u_{(k,i)}))+\frac{1}{m_{2j+2}}.$$ Therefore it suffices to estimate $$\psi(\sum_{k=1}^nb_km_{2j_k}(\sum_{i=1}^{n_k} b_{(k,i)}u_{(k,i)})).$$
In \cite{ADKM} $\psi$ is decomposed as $\psi_1+\psi_2$ and different estimates are applied to $\psi_1$ and $\psi_2$. Our case is easier as we may simply assume that $\psi_1=0$ and $\psi_2=\psi$. We shall therefore refer to some arguments of \cite{ADKM} concerning some $\psi_2$ keeping in mind that $\psi_2=\psi$.
Let $D_1^k,\ldots,D_4^k$ be defined as in \cite{ADKM} Lemma 3.11 (a). Then as in \cite{ADKM}, $$\bigcup_{p=1}^4D_p^k=\bigcup _{i=1}^{n_k} {\rm supp}\ u_{(k,i)}\cap {\rm supp}\ \psi.$$ The proof that
$$\psi|_{\bigcup_kD_2^k}(\sum_kb_km_{2j_k}(\sum_ib_{(k,i)}u_{(k,i)})) \ensuremath{\leqslant}\frac{1}{m_{2j+2}}, \leqno (1)$$
$$\psi|_{\bigcup_kD_3^k}(\sum_kb_km_{2j_k}(\sum_ib_{(k,i)}u_{(k,i)})) \ensuremath{\leqslant}\frac{16}{m_{2j+2}}, \leqno (2)$$ and
$$\psi|_{\bigcup_kD_1^k}(\sum_kb_km_{2j_k}(\sum_ib_{(k,i)} u_{(k,i)}))\ensuremath{\leqslant}\frac{1}{m_{2j+2}}. \leqno (3)$$ may be easily reproduced from \cite{ADKM} Lemma 3.11. The case of $D_4^k$ is slightly different from \cite{ADKM} and therefore we give more details. We claim
\
\noindent{\em Claim:} Let $D=\bigcup_k D_4^k$. Then
$$\psi|_{D}(\sum_kb_km_{2j_k}(\sum_ib_{(k,i)}u_{(k,i)})) \ensuremath{\leqslant}\frac{64}{m_{2j+2}}, \leqno (4)$$
Once the claim is proved it follows by adding the estimates that the 3rd Case is proved, and this concludes the proof of the Proposition.
\
\noindent{\em Proof of the claim:} Recall that $D_4^k$ is defined by $$D_4^k=\{ m\in\bigcup_{i=1}^{n_k}A_{(k,i)} : {\rm for\ all}\ f\in\bigcup_sK^s(\psi)\ {\rm with}\; m\in {\rm supp}f, w(f) \ensuremath{\geqslant} \frac{1}{m_{2j_k}}\;{\rm and}$$ $${\rm there}\;{\rm exists}\; f\in\bigcup_sK^s(\psi)\;{\rm with}\; m\in {\rm supp}f,
w(f)=\frac{1}{m_{2j_k}}{\rm and}$$ $${\rm for}\;{\rm every}\;g\in\bigcup_sK^s(\psi)\; {\rm with}\; {\rm supp}\ f \subset {\rm supp}\ g {\rm \ strictly}, w(g)\ensuremath{\geqslant}\frac{1}{m_{2j+1}}\}.$$
For every $k=1,\ldots ,n$, $i=1,\ldots ,n_k$ and every $m\in {\rm supp}\ u_{(k,i)}\cap D_4^k,$ there exists a unique functional $f^{(k,i,m)}\in \bigcup _sK^s(\psi)$ with $m\in {\rm supp}\ f$, $w(f)=\frac{1}{m_{2j_k}}$ and such that, for all $g\in \bigcup_sK^s(\psi)$ with ${\rm supp}\ f \subseteq {\rm supp}\ g$ strictly, $w(g)\ensuremath{\geqslant} \frac{1}{m_{2j+1}}.$ By definition, for $k\neq p$ and $i=1,\ldots ,n_k$, $m\in {\rm supp}\ u_{(k,i)}$, we have ${\rm supp}f^{(k,i,m)}\cap D^p_4=\emptyset.$ Also, if $f^{(k,i,m)}\neq f^{(k,r,n)},$ then ${\rm supp}\ f^{(k,i,m)}\cap {\rm supp}\ f^{(k,r,n)}=\emptyset.$
For each $k=1,\ldots ,n$, let $\{ f^{k,t}\}_{t=1}^{r_k}\subseteq \bigcup K^s(\varphi ) $ be a selection of mutually disjoint such functionals with $D^k_4=\bigcup _{t=1}^{r_k} {\rm supp}\ f^{k,t}.$ For each such functional $f^{k,t}$, we set $$a_{f^{k,t}}=\sum_{i=1}^{n_k}b_{(k,i)}\sum_{m\in {\rm supp}\ f^{k,t}} a_m.$$ Then, $$f^{k,t}(b_km_{2j_k}(\sum_ib_{(k,i)}u_{(k,i)}))\ensuremath{\leqslant} b_ka_{f^{k,t}}.\leqno (5)$$ We define as in \cite{ADKM} a functional
$g\in K^{\prime }$ with $|g|_{2j}^{\ast }\ensuremath{\leqslant} 1$ (see definition
\cite{ADKM} p 71), and blocks $u_k$ of the basis so that $\|
u_k\|_{\ell_1}\ensuremath{\leqslant} 16$, ${\rm supp}\ u_k\subseteq \bigcup_i{\rm supp}\ u_{(k,i)}$ and
$$\psi|_{D_4}(\sum_kb_km_{2j_k}(\sum_ib_{(k,i)}u_{(k,i)})) \ensuremath{\leqslant} g(2\sum_kb_ku_k),$$ \noindent hence by \cite{ADKM} Lemma 2.4(b) we shall have the result.
For $f=\frac{1}{m_q}\sum_{p=1}^df_p\in\bigcup_sK^s(\psi|_{D_4})$ we set $$J=\{ 1\ensuremath{\leqslant} p\ensuremath{\leqslant} d: f_p=f^{k,t}\;{\rm for}\;{\rm some}\; k=1\ldots ,n, \; t=1,\ldots , r_k\},$$ $$T=\{ 1\ensuremath{\leqslant} p\ensuremath{\leqslant} d:\;{\rm there}\;{\rm exists}\;f^{k,t}\;{\rm with}\; {\rm supp}f^{k,t} \subseteq {\rm supp}f_p \; {\rm strictly}\}.$$
For every $f\in\bigcup_sK^s(\psi|_{D_4})$ we shall define by induction a functional $g_f$, by $g_f=0$ when $J\cup T=\emptyset$,
while if $J\cup T\neq\emptyset $ we shall construct $g_f$ with the following properties. Let $D_f= \bigcup_{p\in J\cup T}{\rm supp}f_p$ and $u_k=\sum a_{f^{k,t}}e_{f^{k,t}}$, where $e_{f^{k,t}}=e_{\min{\rm supp} f^{k,t}}$, then:
(a) ${\rm supp}\ g_f\subseteq {\rm supp}\ f$.
(b) $g_f\in K^{\prime }$ and $w(g_f)\ensuremath{\geqslant} w(f)$,
(c) $f|_{D_f}(\sum_kb_km_{2j_k}(\sum_ib_{(k,i)}u_{(k,i)})) \ensuremath{\leqslant} g_f(2\sum_kb_ku_k)$.
\
\noindent Let $s>0$ and suppose that $g_f$ have been defined for all
$f\in\bigcup_{t=0}^{s-1}K^t(\psi|_{D_4})$ and let
$f=\frac{1}{m_q}(f_1+\ldots +f_d)\in K^s(\psi|_{D_4})\backslash K^{s-1}(\psi|_{D_4})$ where the family $(f_p)_{p=1}^d$ is ${\mathcal M}_q$-admissible if $q>1$, or ${\mathcal S}$-allowable if $q=1$. The proofs of case (i) ($1/m_q=1/m_{2j_k}$ for some $k \ensuremath{\leqslant} n$) and case (ii) ($1/m_q>1/m_{2j+1}$) are identical with \cite{ADKM} p 106. Assume therefore that case (iii) holds, i.e., $1/m_q=1/m_{2j+1}$. For the same reasons as in \cite{ADKM} we have that $T=\emptyset$.
Summing up we assume that $f \in K^s(\psi|_{D_4})\backslash K^{s-1}(\psi|_{D_4})$ is of the form $$f=\frac{1}{m_{2j+1}}\sum_{p=1}^d
f_p=\frac{1}{m_{2j+1}}(Ey_1^*+\ldots+Ey_{k_2}^*+Ey_{k_2+1}^*+\ldots+Ey_d^*),$$ where $(y_i^*)_i$ is associated to $(c_1^*,\ldots,c_{k_2}^*,d_{k_2+1}^*,\ldots)$ with $d_{k_2+1}^* \neq c_{k_2+1}^*$, that $T=\emptyset$ and $J \neq \emptyset$, and it only remains to define $g_f$ satisfying (a)(b)(c).
Now by the proof of \cite{ADKM} Proposition 2.9, $\psi=\psi_{\varphi}$ was defined through the analysis of $\varphi$, in particular by \cite{ADKM} Remark 2.19 (a), $$\psi=\frac{1}{m_{2j+1}}\sum_{k \in I}\psi_{Ey_k^*}$$ for some subset $I$ of $\{1,\ldots,d\}$. Furthermore, for $l \in I$, $l \ensuremath{\leqslant} k_2$ and $1 \ensuremath{\leqslant} k \ensuremath{\leqslant} d$, ${\rm supp}\ Ey_l^* \cap {\rm supp}\ x_k=\emptyset$, therefore there is no functional in a family of type I and II w.r.t. $\overline{x_k}$ of support included in ${\rm supp}\ Ey_l^*$ (see \cite{ADKM} Definition 2.11 p 77). This implies that $D_{Ey_l^*}=\emptyset$ (\cite{ADKM} Definition p 85), and therefore that $\psi_{Ey_l^*}=0$ (\cite{ADKM} bottom of p 85).
For $l \in I$, $l>k_2+1$, then since $\Phi$ is $1-1$, $w(Ey_l^*)=w(Ed_l^*) \neq 1/m_{2j_k} \forall k$. Therefore $w(\psi_{Ey_l^*}) \neq 1/m_{2j_k} \forall k$,
\cite{ADKM} Remark 2.19 (a). Then by the definition of $D_4^k$, ${\rm supp}\ \psi_{Ey_l^*} \cap D_4^k=\emptyset$ for all $k$.
Finally this means that $\psi_{|D_4}=\frac{1}{m_{2j+1}}\psi_{Ey^*_{k_2+1}|D_4}$ and $J=\{k_2+1\}$, $D_f={\rm supp}\ f_{k_2+1}$. Write then $f_{k_2+1}=f^{k_0,t}$ and set $g_f=\frac{1}{2}e^*_{f_{k_2+1}}$, therefore (a)(b) are trivially verified. It only remains to check (c). But by (5),
$$f|_{D_f}(\sum_kb_km_{2j_k}(\sum_ib_{(k,i)}u_{(k,i)}))\ensuremath{\leqslant} b_{k_0} a_{f_{k_2+1}}$$ $$=b_{k_0} a_{f_{k_2+1}} e^*_{f_{k_2+1}}(e_{f_{k_2+1}}) =g_f(2b_{k_0} a_{f_{k_2+1}}e_{f_{k_2+1}})$$ $$=g_f(2\sum_t b_{k_0} a^{f_{k,t}}e_{f^{k,t}}) =g_f(2\sum_kb_ku_k).$$ So (c) is proved. Therefore $g_f$ is defined for each $f$ by induction, and the Claim is verified. This concludes the proof of the Proposition. \end{proof}
\begin{prop} The space $X_u$ is of type (3). \end{prop}
\begin{proof} Assume towards a contradiction that $T$ is an isomorphism from
some block-subspace $[x_n]$ of $X_u$ into the subspace $[e_i, i \notin \bigcup_n {\rm supp}\ x_n]$. We may assume that $\max({\rm supp}\ x_n,{\rm supp}\ Tx_n) < \min({\rm supp}\ x_{n+1},{\rm supp}\ Tx_{n+1})$ and $\min{\rm supp}\ x_n<\min{\rm supp}\ Tx_n$ for each $n$, and by Lemma \ref{scc}, that each $x_n$ is a $(\frac{1}{m_{2n}^4},2n)$ R.I.s.c.c. (\cite{ADKM} Definition 2.16). We may write $$x_n=\sum_{t=1}^{p_n} a_{n,t}x_{n,t}$$ where $(x_{n,1},\ldots,x_{n,p_n})$ is ${\mathcal M}_{2n}$-admissible. Let for each $n,t$, $x_{n,t}^* \in L$ be such that ${\rm supp}\ x_{n,t}^* \subseteq {\rm supp}\ Tx_{n,t}$ and such that $$x_{n,t}^*(Tx_{n,t}) \ensuremath{\geqslant} \frac{1}{2}\norm{Tx_{n,t}} \ensuremath{\geqslant} \frac{1}{4\norm{T^{-1}}},$$ and let $x_n^*=\frac{1}{m_{2n}}(x_{n,1}^*+\ldots+x_{n,p_n}^*) \in L_{2n}$. Note that ${\rm supp}\ x_n^* \cap {\rm supp}\ x_n=\emptyset$ and that $$x_n^*(Tx_n) \ensuremath{\geqslant} \frac{1}{m_{2n}}\sum_{t=1}^{p_n} \frac{a_{n,t}}{4\norm{T^{-1}}}= (4\norm{T^{-1}}m_{2n})^{-1}.$$ We may therefore for any $j>100$ construct a $j$-quadruple
$(j_k,y_k,c_k^*,b_k)_{k=1}^n$ satisfying the hypotheses of Proposition
\ref{criticalbis} and such that $y_k \in [x_i]_i$ and $c_k^*(Ty_k) \ensuremath{\geqslant}
(4\norm{T^{-1}}m_{2j_k})^{-1}$ for each $k$ (note that we may assume that $c_k^* \in L_{j_{2k}}$ for each $k$). From Proposition
\ref{criticalbis} we deduce $$\norm{\sum_{k=1}^n b_k m_{2j_k}y_k} \ensuremath{\leqslant} \frac{75}{m_{2j+1}^2}.$$ On the other hand $\psi=\frac{1}{m_{2j+1}}\sum_{k=1}^n c_k^*$ belongs to $L$ therefore $$\norm{T(\sum_{k=1}^n b_k m_{2j_k}y_k)} \ensuremath{\geqslant} \psi(\sum_{k=1}^n b_k m_{2j_k}Ty_k) \ensuremath{\geqslant} \frac{1}{4\norm{T^{-1}}m_{2j+1}}.$$ We deduce finally that $$m_{2j+1} \ensuremath{\leqslant} 300 \norm{T}\norm{T^{-1}},$$ which contradicts the boundedness of $T$. \end{proof}
\subsection{A strongly asymptotically $\ell_{\infty}$ space tight by support} Since the canonical basis of $X_u$ is tight and unconditional, it follows that $X_u$ is reflexive. In particular this implies that the dual basis of the canonical basis of $X_u$ is a strongly asymptotically $\ell_{\infty}$ basis of $X_u^*$. It remains to prove that this basis is tight with support.
It is easy to prove by duality that for any ${\mathcal M}_{2j}$-admissible sequence of functionals $f_1,\ldots,f_n$ in $X_u^*$, we have the upper estimate $$\norm{\sum_i f_i} \ensuremath{\leqslant} m_{2j}\sup_i \norm{f_i}.$$ We use this observation to prove a lemma about the existence of s.c.c. normed by functionals belonging to an arbitrary subspace of $X_u^*$. The proof is standard except that estimates have to be taken in $X_u^*$ instead of $X_u$.
\begin{lemme}\label{sccbis} For $\epsilon>0$, $j=1,2,\ldots$ and every normalised block sequence $\{f_k\}_{k=1}^{\infty }$ in $X_u^*$, there exists a normalised functional $f \in [f_k]$ and a seminormalised $(\epsilon,2j)$--s.c.c. $x$ in $X_u$ such that ${\rm supp}\ f \subseteq {\rm supp}\ x$ and $f(x) \ensuremath{\geqslant} 1/2$. \end{lemme}
\begin{proof} For each $k$ let $y_k$ be normalised such that ${\rm supp}\ y_k={\rm supp}\ f_k$ and $f_k(y_k)=1$. Recall that the integers $k_n$ and $t_n$ are defined by $k_1=1$, $2^{t_n}\ensuremath{\geqslant} m_n^2$ and $k_n=t_n(k_{n-1}+1)+1$, and that ${\mathcal M}_j={\mathcal F}_{k_j}$ for all $j$.
Applying Lemma \ref{scc} we find a successive sequence of $(\epsilon,2j)$--s.c.c. of $(y_k)$ of the form $(\sum_{i \in I_k}a_i y_i)_k$ with $\{f_i, i \in I_k\}$ ${\mathcal F}_{k_{2j-1}+1}$-admissible. If $\norm{\sum_{i \in I_k}f_i} \ensuremath{\leqslant} 2$ for some $k$, we are done, for then $$(\sum_{i \in I_k}f_i)(\sum_{i \in I_k}a_i y_i) \ensuremath{\geqslant} \frac{1}{2}\norm{\sum_{i \in I_k}f_i}.$$ So assume $\norm{\sum_{i \in I_k}f_i}>2$ for all $k$, apply the same procedure to the sequence $f_k^1=\norm{\sum_{i \in I_k}f_i}^{-1}\sum_{i \in I_k}f_i$, and obtain a successive sequence of $(\epsilon,2j)$--s.c.c. of the sequence $(y_k^1)_k$ associated to $(f_k^1)_k$, of the form $(\sum_{i \in I_k^1}a_i^1 y_i^1)_k$, with $\{f_l: {\rm supp}\ f_l \subseteq \sum_{i \in I_k^1}f_i^1\}$ a ${\mathcal F}_{k_{2j-1}+1}[{\mathcal F}_{k_{2j-1}+1}]$-admissible, and therefore ${\mathcal M}_{2j}$-admissible set. Then we are done unless $\norm{\sum_{i \in I_k^1}f_i^1}> 2$ for all $k$, in which case we set $$f_k^2=\norm{\sum_{j \in I_k^1}f_j^1}^{-1} \sum_{j \in I_k^1}f_j^1$$ and observe by the upper estimate in $X_u^*$ that $$1=\norm{f_k^2}=\norm{\sum_{j \in I_k^1}\sum_{i \in I_j}\norm{\sum_{j \in I_k^1}f_j^1}^{-1} \norm{\sum_{i \in I_j}f_i}^{-1} f_i} \ensuremath{\leqslant} m_{2j}/4.$$ Repeating this procedure we claim that we are done in at most $t_{2j}$ steps. Otherwise we obtain that the set $$A=\{f_l: {\rm supp}\ f_l \subseteq \sum_{i \in I_k^{t_{2j-1}}}f_i^{t_{2j-1}}\}$$ is ${\mathcal M}_{2j}$-admissible. Since $f_k^{t_{2j}}=\sum_{f_l \in A}\alpha_l f_l$, where the normalising factor $\alpha_l$ is less than $(1/2)^{t_{2j}}$ for each $l$, we deduce from the upper estimate that $$1=\norm{f_k^{t_{2j}}} \ensuremath{\leqslant} 2^{-t_{2j}}m_{2j},$$ a contradiction by definition of the integers $t_i$'s. \end{proof}
\
To prove the last proposition of this section we need to make two observations. First if $(f_1,\ldots,f_n) \in {\rm conv}_{\Q}L$ is ${\mathcal
M}_{2j}$-admissible, then $\frac{1}{m_{2j}}\sum_{k=1}^n f_k \in {\rm conv}_{\Q}L_{2j}.$
Indeed using the stability of $L$ under projections onto subsets of $\N$ we may easily find convex rational coefficients $\lambda_i$
such that each $f_k$ is of the form $$f_k=\sum_i \lambda_i f_i^k,\ f_i^k \in L,\ {\rm supp}\ f_i^k
\subseteq {\rm supp}\ f_k\ \forall i.$$ Then $\frac{1}{m_{2j}}\sum_{k=1}^n f_k=\sum_i \lambda_i
(\frac{1}{m_{2j}}\sum_{k=1}^n f_i^k)$ and each $\frac{1}{m_{2j}}\sum_{k=1}^n f_i^k$ belongs to $L_{2j}$.
Likewise if $\psi=\frac{1}{m_{2j+1}}(c_1^*+\ldots+c_d^*)$, $k>2j+1$, $c_1^* \in {\rm conv}_{\Q} L_{2k}$ and $c_l^* \in {\rm conv}_{\Q} L_{\Phi(c_1^*,\ldots,c_{l-1}^*)}\ \forall l \ensuremath{\geqslant} 2$, then $\psi \in {\rm conv}_{\Q} L$. Indeed as above we may write $$\psi=\sum_i \lambda_i (\frac{1}{m_{2j+1}}\sum_{l=1}^d f_i^l),\ f_i^1 \in L_{2k}, f_i^l \in L_{\Phi(c_1^*,\ldots,c_{i-1}^*)}(c_i^*)\ \forall l \ensuremath{\geqslant} 2,$$ and each $\frac{1}{m_{2j+1}}\sum_{l=1}^d f_i^l$ belongs to $L_{2j+1}^{\prime n+1} \subseteq L$.
\begin{prop} The space $X_u^*$ is of type (3). \end{prop}
\begin{proof} Assume towards a contradiction that $T$ is an isomorphism from
some block-subspace $[f_n]$ of $X_u^*$ into the subspace $[e_i^*, i \notin \cup_n {\rm supp}\ f_n]$. We may assume that $\max({\rm supp}\ f_n,{\rm supp}\ Tf_n) < \min({\rm supp}\ f_{n+1},{\rm supp}\ Tf_{n+1})$ and $\min{\rm supp}\ Tf_n<\min{\rm supp}\ f_n$ for each $n$. Since the closed unit ball of $X_u^*$ is equal to $\overline{{\rm conv}_{\Q}L}$ we may also assume that $f_n \in {\rm conv}_{\Q}L$ for each $n$. Applying Lemma \ref{sccbis}, we may also suppose that each $f_n$ is associated to a $(\frac{1}{m_{2n}^4},2n)$ s.c.c. $x_n$ with $Tf_n(x_n) \ensuremath{\geqslant} 1/3$ and ${\rm supp}\ x_n \subset {\rm supp}\ Tf_n$, and we shall also assume that $\norm{Tf_n}=1$ for each $n$. Build then for each $k$ a $(\frac{1}{m_{2k}^4},2k)$ R.I.s.c.c. $y_k=\sum_{n \in A_k}a_n x_n$ such that $(Tf_n)_{n \in A_k}$ and therefore $(f_n)_{n \in A_k}$ is ${\mathcal M}_{2k}$-admissible. Then note that by the first observation before this proposition, $$c_k^*:=m_{2k}^{-1}\sum_{n \in A_k}f_n \in {\rm conv}_{\Q}L_{2k},$$ and observe that ${\rm supp}\ c_k^* \cap {\rm supp}\ y_k=\emptyset$ and that $Tc_k^*(y_k) \ensuremath{\geqslant} (3m_{2k})^{-1}$.
We may therefore for any $j>100$ construct a $j$-quadruple
$(j_k,y_k,c_k^*,b_k)_{k=1}^n$ satisfying the hypotheses of Proposition
\ref{criticalbis} and such that $c_k^* \in [f_i]_i$ and $Tc_k^*(y_k) \ensuremath{\geqslant}
(3m_{2j_k})^{-1}$ for each $k$. From Proposition
\ref{criticalbis} we deduce $$\norm{\sum_{k=1}^n b_k m_{2j_k}y_k} \ensuremath{\leqslant} \frac{75}{m_{2j+1}^2}.$$ Therefore $$\norm{\sum_{k=1}^d Tc_k^*} \ensuremath{\geqslant} \frac{\sum_{k=1}^d b_k m_{2j_k}Tc_k^*(y_k)}{\norm{\sum_{k=1}^n b_k m_{2j_k}y_k}} \ensuremath{\geqslant} \frac{m_{2j+1}^2}{225},$$ but on the other hand $$\norm{\sum_{k=1}^d c_k^*} \ensuremath{\leqslant} m_{2j+1}$$ since by the second observation the functional $m_{2j+1}^{-1}\sum_{k=1}^d c_k^*$ belongs to ${\rm conv}_{\Q}L$. We deduce finally that $$m_{2j+1} \ensuremath{\leqslant} 225 \norm{T},$$ which contradicts the boundedness of $T$. \end{proof}
\section{Problems and comments}
Obviously the general question one is compelled to ask is whether it is possible to find an example for each of the classes or subclasses appearing in the chart of Theorem \ref{final}. However we wish to be more specific here and concentrate on the classes which either seem particularly interesting, or easier to study, or which are related to one of the spaces considered in this paper.
\
Let us first observe that the examples of locally minimal, tight spaces produced so far could be said to be so for trivial reasons: since they hereditarily contain $\ell_{\infty}^n$'s uniformly, any Banach space is crudely finitely representable in any of their subspaces. It remains open whether there exist other examples. Observing that a locally minimal and tight space cannot be strongly asymptotically $\ell_p, 1 \ensuremath{\leqslant} p<+\infty$, by one of the implications in the diagram before Theorem \ref{final}, and up to the 6th dichotomy, the problem may be summed up as:
\begin{prob} Find
a tight, locally minimal, uniformly inhomogeneous Banach space which does not contain $\ell_{\infty}^n$'s uniformly, or equivalently, which has finite cotype. \end{prob}
It also unknown whether tightness by range and tightness with constants are the only possible forms of tightness, up to passing to subspaces. Equivalently, using the 4th and 5th dichotomy:
\begin{prob} Find a tight Banach space which is sequentially and locally minimal. \end{prob}
Or, since such a space would have to be of type (2), (5b), (5d):
\begin{prob} \
\begin{itemize} \item[(a)] Find a HI space which is sequentially minimal. \item[(b)] Find a space of type (5b). \item[(c)] Find a space of type (5d). Is the dual of some modified mixed Tsirelson's space such a space? \end{itemize} \end{prob}
For the next problem, we observe that the only known examples of spaces tight with constants are strongly asymptotic $\ell_p$ spaces not containing $\ell_p$, where $1 \ensuremath{\leqslant} p <+\infty$.
\begin{prob} Find a space tight with constants and uniformly inhomogeneous. \end{prob}
More specifically, listing two subclasses for which we have a possible candidate:
\begin{prob} \
\begin{itemize} \item[(a)] Find a space of type (1a). Is $G$ or one of its subspaces such a space? \item[(b)] Find a space of type (3a). Is $G_u$ or one of its subspaces such a space? \end{itemize} \end{prob}
\
Recently, S. Argyros, K. Beanland and T. Raikoftsalis \cite{ABR,ABR2} constructed an example $X_{abr}$ with a basis which is strongly asymptotically $\ell_2$ and therefore weak Hilbert, yet every operator is a strictly singular perturbation of a diagonal map, and no disjointly supported subspaces are isomorphic. In our language, $X_{abr}$ is therefore a new space of type (3c), which we include in our chart.
We conclude by mentioning the very recent and remarkable result of S. Argyros and R. Haydon solving the scalar plus compact problem \cite{AH}: there exists a HI space which is a predual of $\ell_1$ and on which every operator is a compact perturbation of a multiple of the identity. To our knowledge nothing is known about the exact position of this space in the chart of Theorem \ref{final}.
\begin{prob} Find whether Argyros-Haydon's space is of type (1) or of type (2). \end{prob}
\
\begin{flushleft}
{\em Address of V. Ferenczi:}\\
Departamento de Matem\'atica,\\
Instituto de Matem\'atica e Estat\' \i stica,\\
Universidade de S\~ao Paulo,\\
rua do Mat\~ao, 1010, \\
05508-090 S\~ao Paulo, SP,\\
Brazil.\\ \texttt{ferenczi@ime.usp.br} \end{flushleft}
\
\begin{flushleft} {\em Address of C. Rosendal:}\\ Department of Mathematics, Statistics, and Computer Science\\ University of Illinois at Chicago,\\ 851 S. Morgan Street,\\ Chicago, IL 60607-7045,\\ USA.\\ \texttt{rosendal@math.uic.edu} \end{flushleft}
\end{document} |
\begin{document}
\title{
{\bf\Large Nonlinear Stability for the Periodic and Non-Periodic Zakharov System }
\begin{abstract} We prove the existence of a smooth curve of periodic traveling wave solutions for the Zakharov system. We also show that this type of solutions are nonlinear stable by the periodic flow generated for the system mentioned before. An improvement of the work of Ya Ping \cite{Wu1} is made, we prove the stability of the solitary wave solutions associated to the Zakharov system. \\
{\bf Key words.} Periodic traveling waves, Solitary waves, Nonlinear Stability, Zakharov System.\\
{\bf AMS subject classifications.} 35Q53; 35B35; 35B10 \end{abstract}
\section{Introduction} In this essay we study the periodic Zakharov system \begin{equation} \label{equaZakha} \left \{ \begin{aligned} iu_{t}+u_{xx}&=uv\\
v_{tt}-v_{xx}&=(|u|^2)_{xx},\\ \end{aligned} \right. \end{equation} where $u=u(x,t)\in\mathbb{C},\ v=v(x,t)\in\mathbb{R}$ and $x,t\in\mathbb{R}.$ This system was introduced by Zakharov in \cite{Zakharov1} to describe the long wave Langmuir turbulence in a plasma. The function $u=u(x,t)$ represents the slowly varying envelope of the highly oscillatory electric field and $v$ denotes the deviation of the ion density from the equilibrium.\\
The goal of this paper is to establish the existence and nonlinear stability of periodic traveling wave solutions for the Zakharov system. More precisely, we are interested in solutions for (\ref{equaZakha}) of the form \begin{equation}\label{solforms} u(x,t)=e^{-i\omega t}e^{i\frac{c}{2}(x-ct)}\phi_{\omega,c}(x-ct)\ \ \ \text{and}\ \ \ v(x,t)=\psi_{\omega,c}(x-ct), \end{equation} where $\omega,c\in\mathbb{R}$ and $\phi_{\omega,c},\psi_{\omega,c}: \mathbb{R}\rightarrow\mathbb{R}$ are periodic smooth functions with the same fundamental period $L>0.$ As far as we know any result of stability for this type of waves has been established before. The first work about existence and nonlinear stability of periodic waves was made by Benjamin in \cite{benjamin3}, where he studied periodic waves of cnoidal type for the Korteweg-de Vries equation. This work had some gaps on central parts of the stability theory that was revised and complemented by Angulo, Bona and Scialom in \cite{anguloBonaScialom}. In the last few years some papers about the nonlinear stability on the periodic case have appeared in the literature, see for instance \cite{AnguloLibro, angulo5, angulo4, AnguloNatali2, anguloNatali, GallayHaragus1, GallayHaragus2, haragus1, NataliPastor, Neves1}. \\
Substituting the type of solutions given in (\ref{solforms}) in the system (\ref{equaZakha}), we get that $\phi=\phi_{\omega,c}$ and $\psi=\psi_{\omega,c}$ have to satisfy the next system of ordinary differential equations, \begin{equation}\label{systemedo} \left \{ \begin{aligned} &(c^2 -1)\psi ''=(\phi^2)''\\ &\phi ''+\left(\omega+\frac{c^2}{4}\right)\phi=\phi\psi. \end{aligned} \right. \end{equation} Integrating the first equation of the system (\ref{systemedo}) and substituting on the second one, we obtain after some algebra that the solution $\phi$ has to satisfy \[\left(\phi'\right)^2=\frac{1}{2(1-c^2)}F(\phi),\] where $F$ is the polynomial given by \[F(t)=-t^4+2(1-c^2)\left(-\omega-\frac{c^2}{4}\right)t^2+4(1-c^2)A_{\phi}\] and $A_{\phi}$ is a constant of integration. It is clear that the solutions of the equation (\ref{equaZakha}) depend of the roots of the polynomial $F.$ Assuming that $F$ has roots $\pm\eta_1$ and $\pm\eta_2$ with $0<\eta_2<\eta_1,$ we obtain the smooth curve of dnoidal waves \[\nu\in\left(\frac{2\pi^2}{L^2},+\infty\right)\longmapsto\left(\psi_{\nu},\phi_{\nu}\right)\in H_{per}^n([0,L])\times H_{per}^n([0,L]),\ \ \text{for all}\ \ n\in \mathbb{N}, \] with $\phi_{\nu}$ and $\psi_{\nu}$ given by \[\phi_{\nu}(\xi)=\eta_1\text{dn}\left(\tfrac{\eta_1\xi}{\sqrt{2(1-c^2)}} ;k\right)\ \ \text{and}\ \ \ \psi_{\nu}(\xi)=-\frac{\eta^2_1}{1-c^2}\text{dn}^2 \left(\tfrac{\eta_1\xi}{\sqrt{2(1-c^2)}};k\right).\] Here, $k^2=\frac{\eta^2_1-\eta^2_2}{\eta^2_1},$ $\nu=-\left(\omega+\frac{c^2}{4}\right)$ and dn denotes the Jacobi elliptic function of dnoidal type. This solutions are constructed with the same fixed minimal period $L>0,$ not necessarily large.\\
With respect to the well-posedness problem for the Zakharov system, on the periodic case, this was studied by Bourgain in \cite{Bourgain2}, where a global well-posedness result was obtained for initial data $(u(0), v(0),v_t(0) )\in H^1_{per}\times L^2_{per}\times H^{-1}_{per}.$ It is worth to note that in the periodic case there exists another \textit{more general} result about well-posedness for the Zakharov system obtained by Takaoka in \cite{Takaoka1}, but for our purpose the result established by Bourgain is good enough. See also Guo and Shen \cite{GuoBoling1}, where the existence of classical periodic solutions for the system (\ref{equaZakha}) is proved. On the continuous case the Cauchy problem associated to the Zakharov system in one and several dimensions have been studied extensively, see for instance \cite{AddedAdded1, bejenaru1, BourgCollia1, Colliander1, GiniTsutVelo1, KenigPonceVega1, OzawaTsut1, Pecher, SchotetWeiste1, SulemSulem1}.\\
In order to establish the spectral properties of some linear operators which appear in the proof of the stability, we use the Floquet theory, more precisely we use the Oscillation Theorem (see Magnus and Winkler \cite{magnus}). Our spectral analysis depends basically of the next periodic and semi-periodic eigenvalue problems associated to the Lam\'e equation, given respectively by \[\left\{ \begin{aligned} y''+&[\lambda-m(m+1)k^2\text{sn}^2(x,k)]y=0\\ y(0)&=y(2K(k)),\ \ y'(0)=y'(2K(k)) \end{aligned} \right.\] and \[\left\{ \begin{aligned} y''+&[\lambda-m(m+1)k^2\text{sn}^2(x,k)]y=0\\ y(0)&=-y(2K(k)),\ \ y'(0)=-y'(2K(k)), \end{aligned} \right.\] where $\lambda\in\mathbb{R},$ $m\in\mathbb{N},$ sn denotes the Jacobi elliptic function of snoidal type and $K$ is the complete elliptic integral of the first type (see Byrd and Friedman \cite{byrdFriedman}). Recently, Neves in \cite{Neves1} proved that is possible to characterize the eigenvalues of the Hill operator $L(y)=-y''+Q(x)y$ in $L^2[0,\pi]$ if we know explicitly one of the eigenfunctions associated to this eigenvalue (in this case, $Q$ is a $C^2$ periodic function with period $\pi$). Unfortunately, we only had access to this work when we already had concluded our spectral results using the associated Lam\'e equation. We are completely sure that this new theory can be use to obtain the spectral properties of the operator studied in this paper.\\
To obtain our result of stability for the dnoidal wave solutions, we rewrite the Zakharov system as \begin{equation}\label{ZakNovoInt} \left \{ \begin{aligned} v_t &=-V_x, \ \int_0^L V(x,t)dx = 0 \\
V_t &= -(v+|u|^2)_x \\ iu_t &+ u_{xx} =uv \end{aligned} \right. \end{equation} and we adapt to the periodic case the ideas established by Benjamin \cite{benjamin1}, Bona \cite{bona2} and Weinstein \cite{weinstein3}, then we impose the restriction \[\int_0^L v_0(x)dx\leq \int_0^L \psi_{\nu}(x)dx,\] where $v(x,0)=v_0(x),$ to obtain that the dnoidal waves with $c\in(-1,1)$ fixed and $\nu>\frac{2\pi^2}{L^2},$ are orbitally stable in \[X:=L^2_{per}([0,L])\times\widetilde{L}^2_{per}([0,L])\times H^1_{per}([0,L])\] by the periodic flow of the system (\ref{ZakNovoInt}). Here, $\widetilde{L}^2_{per}$ is given by \[\widetilde{L}^2_{per}([0,L])=\left\{f \in L^2_{per}([0,L]): \int_0^L f(x) dx=0\right\}.\]
With regard to the existence and stability of solitary wave solutions for the Zakharov system, there exists a result obtained by Ya Ping in \cite{Wu1}, this work is not completely right. In \cite{Wu1} the author considered the \textit{equivalent} system \[ \left \{ \begin{aligned} v_t&=V_{xx},\\
V_t&=v+|u|^2,\\ iu_{t}&+u_{xx}=uv.\\ \end{aligned} \right. \]
Therefore, the solitary wave solution $V(x,t)=\varphi_{\omega,c}(x-ct)$ is given by \begin{equation}\label{solWuvarphi} \varphi_{\omega,c}(\xi)=c\sqrt{-4\omega-c^2}\tanh\left(\frac{\sqrt{-4\omega-c^2}}{2}\xi\right). \end{equation} Observe that this solution is not in any Sobolev space $H^s(\mathbb{R})$ and Ya ping proved stability in $L^2(\mathbb{R})\times H^1(\mathbb{R})\times H^1(\mathbb{R}),$ which is not right because the solution (\ref{solWuvarphi}) is not in the space where the author proves the stability. One of the goals of this paper is to improve the result of stability, for the solitary waves solutions, obtained by Ya Ping. Following the ideas used to establish the stability on the periodic case we prove that the solitary wave solutions \[ \psi_{\omega,c}(\xi)=\left(2\omega+\frac{c^2}{2}\right)\ \text{sech}^2\left(\frac{\sqrt{-4\omega-c^2}}{2}\xi\right), \ \ \ \phi_{\omega,c}(\xi)=\sqrt{\frac{(-4\omega-c^2)(1-c^2)}{2}}\ \text{sech}\left(\frac{\sqrt{-4\omega-c^2}}{2}\xi\right) \] \[ \text{and}\ \ \ \varphi_{\omega,c}(\xi)=c\left(2\omega+\frac{c^2}{2}\right)\ \text{sech}^2\left(\frac{\sqrt{-4\omega-c^2}}{2}\xi\right) \] are orbitally stable in $X=L^2(\mathbb{R})\times L^2(\mathbb{R})\times H^1(\mathbb{R})$ by the flow generated by the Zakharov system if $1-c>0$ and $4\omega+c^2\geq 0.$ It is worth to note that in the continuous case the restriction imposed above for the initial datum $v_0$ is not necessary, because using the property that the solitary wave solutions converges to zero, when $\xi$ goes to infinity, the term that force to impose this condition disappears.\\
The plan of the paper is as follows. The next section is devoted to describe briefly the notation that will be used, and to make a few preliminary remarks regarding periodic and nonperiodic Sobolev spaces. In Section 3 we prove the existence of a smooth curve of dnoidal wave solutions for the system (\ref{equaZakha}). Section 4 contains the spectral analysis of some linear operators necessary to obtain our result of stability. In Section 5 we present the result of nonlinear stability for the dnoidal wave solutions of the system (\ref{equaZakha}). Finally, in Section 6 we present the result of stability of the solitary waves associated to the Zakharov system.
\section{Notation} The $L^2$-based Sobolev spaces of periodic functions are defined as follows (for further details see Iorio and Iorio \cite{ioriolibro}). Let $\mathcal{P}=C^{\infty}_{per}$ denote the collection of all functions $f:\mathbb{R}\rightarrow \mathbb{C}$ which are $C^{\infty}$ and periodic with period $2L>0.$ The collection $\mathcal{P}'$ of all continuous linear functionals from $\mathcal{P}$ into $\mathbb{C}$ is the set of \textit{periodic distributions.} If $\Psi\in \mathcal{P}'$ then we denote the value of $\Psi$ at $\varphi$ by $\Psi(\varphi)=\langle\Psi,\varphi\rangle.$ Define the functions $\Theta_k(x)=\exp(\pi ikx/L), \ k\in \mathbb{Z},\ x\in\mathbb{R}.$ The Fourier transform of $\Psi$ is the function $\widehat{\Psi}:\mathbb{Z}\rightarrow\mathbb{C}$ defined by the formula $\widehat{\Psi}(k)=\frac 1{2L}\langle\Psi,\varphi\rangle, \ k\in\mathbb{Z}.$ So, if $\Psi$ is a periodic function with period $2L,$ we have \[\widehat{\Psi}(k)=\frac 1{2L}\int_{-L}^L \Psi(x)e^{-\frac{ik\pi x}{L}}dx.\]
For $s\in \mathbb{R},$ the Sobolev space of order $s,$ denoted by $H^s_{per}([-L,L])$ is the set of all $f\in \mathcal{P}'$ such that $(1+|k|^{2})^{\frac{s}{2}}\widehat{f}(k)\in l^2(\mathbb{Z}),$ with norm
\[||f||^2_{H^s_{per}}=2L\sum_{k=-\infty}^{\infty}(1+|k|^{2})^s|\widehat{f}(k)|^2.\] We also note that $H^s_{per}$ is a Hilbert space with respect to the inner product
\[(f|g)_s = 2L\sum_{n=-\infty}^{\infty}(1+|k|^2)^{s}\widehat{f}(k)\overline{\widehat{g}(k)}\] In the case $s=0,$ $H^0_{per} $ is a Hilbert space that is isometrically isomorphic to $L^2([-L,L])$ and
\[(f|g)_0 = (f,g) = \int_{-L}^{L} f\overline{g} \ dx.\]
The space $H^0_{per}$ will be denoted by $L^2_{per}$ and its norm will be $\|\cdot\|_{L^2_{per}}.$ Of course $H^s_{per} \subset L^2_{per}$, for any $s \geq 0 $. Moreover, $(H^s_{per})'$, the topological dual of $H^s_{per}$, is isometrically isomorphic to $H^{-s}_{per}$ for all $s \in \mathbb{R}$. The duality is implemented concretely by the pairing \[\langle f,g\rangle_s = 2L\sum_{k=-\infty}^{\infty}\widehat{f}(k)\overline{\widehat{g}(k)}, \ \ \ for \ \ \ f \in H^{-s}_{per}, \ \ g \in H^s_{per}. \] Thus, if $f \in L^2_{per}$ and $g \in H^s_{per} $, with $s\geq 0,$ it follows that $\langle f,g\rangle_s = (f,g)$. Additionally, in the particular case $s=\frac 1{2}$ we will denote the pairing $\langle f,g\rangle_s$ simply by $\langle f,g\rangle.$ One of Sobolev's Lemmas in this context states that if $s>\frac{1}{2}$ and
\[C_{per} = \{f: \mathbb{R} \longrightarrow \mathbb{C} \ | \ f \ \ \text{is continuous and periodic with period} \ \ 2L \},\] then $H^{s}_{per}\hookrightarrow C_{per}$.\\
Let $s\in \mathbb{R}.$ The ($L^2$ type) Sobolev space $H^s(\mathbb{R})$ is the collection of all $f\in \mathcal{S}'(\mathcal{R})$ such that $(1+|\xi|^2)^{\frac s{2}}\widehat{f}\in L^2(\mathbb{R},d\xi),$ that is, $\widehat{f}$ is a measurable function and \[
\| f\|_{s}^2=\int_{\mathbb{R}}(1+|\xi|^2)^s|\widehat{f}(\xi)|^2d\xi<\infty. \]
For more details see Iorio and Iorio \cite{ioriolibro}. Finally, we say that $b\in \widehat{H}^{-1}(\mathbb{R})$ if there exists $V\in L^2(\mathbb{R})$ such that $b=-V'$ and $\|b\|_{\widehat{H}^{-1}}=\|V\|_{L^2}.$
\section{Existence of dnoidal wave solutions} In this section we show the existence of a smooth curve of dnoidal wave solutions, with the same fundamental period, for the Zakharov system. In this case, we are interested in solutions for the system (\ref{equaZakha}) in the form given in (\ref{solforms}). Since $u$ is a periodic function (with period $L$), for $c\neq 0$ we suppose that there exists $m\in\mathbb{N}$ such that $L=\frac{4\pi m}{c}.$ Note that for $c=0$ we obtain immediately that $u$ is a $L$-periodic function. Substituting (\ref{solforms}) in (\ref{equaZakha}), we have that $\phi=\phi_{\omega,c}$ and $\psi=\psi_{\omega,c}$ have to satisfy (\ref{systemedo}). Integrating the first equation in (\ref{systemedo}) we obtain \begin{equation}\label{edo2Zakha} (c^2-1)\psi'=(\phi^2)'+a_0. \end{equation} Using the fact that $\phi^2$ and $\psi$ are periodic we get that $a_0=0.$ Therefore \begin{equation}\label{ecuacorr} (c^2-1)\psi'=(\phi^2)'. \end{equation} Integrating (\ref{ecuacorr}), we have that for all $c\neq 1$ \begin{equation}\label{ecuasegint} \psi=\frac{-\phi^2}{1-c^2}+a_1. \end{equation} We assume in our theory that the constant of integration $a_1$ is zero. Thus, substituting (\ref{ecuasegint}) in the second equation of (\ref{systemedo}) we have that \begin{equation} \label{ecuaordphi} \phi''+\left(\omega+\frac{c^2}{4}\right)\phi+\frac{\phi^3}{1-c^2} = 0. \end{equation} Now, multiplying (\ref{ecuaordphi}) by $\phi '$ and integrating once, we arrived at \[\frac{\left(\phi'\right)^2}{2}+\left(\omega+\frac{c^2}{4}\right)\frac{\phi^2}{2}+\frac{\phi^4}{4(1-c^2)}=A_{\phi},\] where $A_{\phi}$ is a constant of integration. Then, \[\left(\phi'\right)^2=\frac{1}{2(1-c^2)}F(\phi),\]
where $F$ is a polynomial given by \[F(t)=-t^4+2(1-c^2)\left(-\omega-\frac{c^2}{4}-a_1\right)t^2+4(1-c^2)A_{\phi}.\] Suppose that $F$ has roots $\pm\eta_1$ and $\pm\eta_2$ (note that $F$ is even) and without loss of generality that $0<\eta_2<\eta_1$. Thus, we can write \begin{equation} \label{equa9zak} \left(\phi'\right)^2=\frac{1}{2(1-c^2)}(\phi^2-\eta^2_2)(\eta^2_1-\phi^2). \end{equation} Assume also that $1-c^2>0,$ then the left side of (\ref{equa9zak}) is not negative, therefore we have that \[\eta^2_2 \leq \phi^2\leq \eta^2_1.\] Since we are interested in positive solutions, from the last inequality we obtain $\eta_2\leq\phi\leq\eta_1.$ Using (\ref{equa9zak}) we get that the $\eta_{j}$'s satisfy \[\left \{ \begin{aligned} -2(1-c^2)\left(\omega+\frac{c^2}{4}\right)&=\eta^2_1 + \eta^2_2\\ 4(1-c^2)A_{\phi}&=-\eta^2_1\eta^2_2.\\ \end{aligned}\right.\] From the last system, we get the restriction $4\omega+c^2<0.$ \\
Now, define $\varrho(\xi)=\frac{\phi(\xi)}{\eta_1}$, $k^2=\frac{\eta^2_1-\eta^2_2}{\eta^2_1}$ and assume that $\varrho(0)=1$. Thus, we can rewrite the equation (\ref{equa9zak}) as \begin{equation} \label{equa11zak} \left(\varrho'\right)^2=\frac{\eta^2_1}{2(1-c^2)}(1-\varrho^2)(\varrho^2+k^2 -1). \end{equation} Finally, define $\chi$ through the relation $\varrho^2=1-k^2\sin^2\chi$, with $\chi(0)=0,$ then (\ref{equa11zak}) can be reduce to \begin{equation}\label{ecuacorr2} [\chi']^2=\frac{\eta^2_1}{2(1-c^2)}(1-k^2\sin^2\chi). \end{equation} From (\ref{ecuacorr2}) we obtain after some algebra that, \begin{equation}\label{ecuacorr3} \int^{\chi(\xi)}_0 \frac{dt}{\sqrt{1-k^2\sin\ t}}= \frac{\eta_1\xi}{\sqrt{2(1-c^2)}}. \end{equation} Using the identity (\ref{ecuacorr3}), we obtain from the definition of the Jacobi elliptic functions (see Byrd and Friedman \cite{byrdFriedman}) that \[\sin(\chi(\xi))=\text{sn}\left(\frac{\eta_1\xi}{\sqrt{2(1-c^2)}} ;k\right).\] Therefore \begin{align*} \varrho(\xi)&=\sqrt{1-k^2\sin^2(\chi(\xi))}=\sqrt{1-k^2 \text{sn}^2\left(\tfrac{\eta_1\xi}{\sqrt{2(1-c^2)}} ;k\right)} = \text{dn}\left(\tfrac{\eta_1\xi}{\sqrt{2(1-c^2)}} ;k\right), \end{align*} where we use the fact that $k^2\text{sn}^2+\text{dn}^2=1.$ Coming back to the variable $\phi$ we obtain that \begin{equation} \label{equa12zak} \phi_{\omega,c}(\xi) = \eta_1\text{dn}\left(\tfrac{\eta_1\xi}{\sqrt{2(1-c^2)}} ;k\right) \end{equation} and using (\ref{ecuasegint}) we arrive at \begin{equation} \label{equa13zak} \psi_{\omega,c}(\xi)=-\frac{\eta^2_1}{1-c^2}\text{dn}^2 \left(\tfrac{\eta_1\xi}{\sqrt{2(1-c^2)}};k\right). \end{equation}
Now, since dn has fundamental period $2K$, where $K = K(k)$ is the complete elliptic integral of the first type (see Byrd and Friedman \cite{byrdFriedman}), we obtain that $\phi$ e $\psi$ have fundamental period given by \[T_\psi = T_\phi = \frac{2{\sqrt{2(1-c^2)}}}{\eta_1}K(k).\] Fix $\omega$ and $c$ such that $1-c^2 >0$ and $4\omega+c^2<0.$ Additionally, define \[\nu=-\left(\omega+\frac{c^2}{4}\right) \ \ \text{and} \ \ \alpha=1-c^2.\] Then $\eta^2_1 + \eta^2_2 = 2\nu \alpha$ and consequently $0 < \eta_2 < \sqrt{\nu\alpha} < \eta_1 < \sqrt{2\nu\alpha}.$ We express $T_\psi$ and $T_{\phi}$ as functions of the parameter $\eta_2,$ \[T_\psi(\eta_2)= T_\phi(\eta_2)=\frac{2{\sqrt{2\alpha}}}{\sqrt{2\nu\alpha-\eta^2_2 }}K(k(\eta_2)) \ \ \ \ \text{with}\ \ \ \ \ k^2(\eta_2)=\frac{2\nu\alpha-2\eta^2_2}{2\nu\alpha-\eta^2_2}.\] Note that if $ \eta_2\rightarrow 0$, we have that $k(\eta_2)\rightarrow 1^-,$ which implies that $K(k(\eta_2))\rightarrow +\infty $ and consequently $ T_\psi,T_\phi\rightarrow +\infty $. On the other hand, when $\eta_2\rightarrow {\sqrt{\nu\alpha}}$ we get that $k(\eta_2)\rightarrow 0^+$ and then $K(k(\eta_2))\rightarrow \frac{\pi}{2}$. Therefore $T_\psi,T_\phi \rightarrow \frac{\pi{\sqrt{2}}}{\sqrt{\nu}}.$ Since the function $\eta_2 \in (0,\sqrt{\nu\alpha})\mapsto T_\psi(\eta_2)=T_\phi(\eta_2)$ is strictly decreasing (we prove this fact later) we obtain \[T_\phi=T_\psi > \frac{\pi{\sqrt{2}}}{\sqrt{\nu}}.\] Now, for $L>0$ and $1-c^2>0$ fixed, chose $\nu > 0$ such that $\sqrt{\nu}> \frac{\pi{\sqrt{2}}}{L}.$ Then, it follows from the analysis given above that there exists a unique $\eta_2=\eta_2(\nu)\in(0,\sqrt{\nu\alpha})$ such that the dnoidal waves $\phi=\phi(\cdot;\eta_1(\nu),\eta_2(\nu))$ and $\psi=\psi(\cdot;\eta_1(\nu),\eta_2(\nu))$ have fundamental period $L=T_{\psi}(\eta_2)=T_{\phi}(\eta_2).$
\begin{remark} The formula (\ref{equa12zak}) and (\ref{equa13zak}) contains, at least formally, the solitary wave solutions for the system (\ref{equaZakha}) found by Ya Ping in \cite{Wu1}. In fact, if $\eta_2 \rightarrow 0^+$ we obtain that $\eta_1 \rightarrow \sqrt{2\nu\alpha}$, $k(\eta_2) \rightarrow 1^-$ and $dn(x,1^-)=\text{sech}(x)$. Consequently \[\phi_{c,\omega}(x)=\frac{\sqrt{(-4\omega - c^2)(1-c^2)}}{2} \text{sech} \left(\frac{\sqrt{-4\omega - c^2}}{2} \ x\right)\ \ \text{and} \] \[\psi_{c,\omega}(x)=\left(2\omega+\frac{c^2}{2}\right) \text{sech}^2 \left(\frac{\sqrt{-4\omega - c^2}}{2} \ x\right).\] \end{remark}
\begin{theo}\label{TeorDnoidal} Let $L>0$ and $1-c^2>0$ be arbitrarily fixed. Consider $\nu_0>\frac{2\pi^2}{L^2}$ and the unique $\eta_{2,0}=\eta_{2,0}(\nu_0)\in (0,\sqrt{\nu_0\alpha})$ such that $T_{\psi_{\nu_0}}=L=T_{\phi_{\nu_0}}$. Then,
(i) there exist intervals $I(\nu_0)$ and $B(\eta_{2,0})$ around $\nu_0$ and $\eta_{2,0}$ respectively, and a unique smooth function $\Lambda:I(\nu_0)\longrightarrow B(\eta_{2,0})$ such that $\Lambda(\nu_0)=\eta_{2,0}$ and \[\frac{2\sqrt{2\alpha}}{\sqrt{2\nu\alpha - \eta_2^2}}K(k) = L,\] for all $\nu\in I(\nu_0)$, $\eta_2=\Lambda(\nu)$ and \begin{equation}\label{equaZa13c} k^2 = k^2(\nu)=\frac{2\nu\alpha-2\eta_2^2}{2\nu\alpha-\eta_2^2}. \end{equation} Furthermore, we can chose $I(\nu_0)=(\frac{2\pi^2}{L^2},+\infty).$
(ii) The dnoidal waves $\psi(\cdot;\eta_1,\eta_2)$ and $\phi(\cdot;\eta_1,\eta_2)$ given by (\ref{equa12zak}) and (\ref{equa13zak}), and determined by $\eta_1=\eta_1(\nu),$ $\eta_2=\eta_2(\nu)=\Lambda(\nu),$ with $\eta^2_1 + \eta^2_2 = 2\nu\alpha,$ have fundamental period $L$ and satisfy (\ref{ecuasegint}) and (\ref{ecuaordphi}). Furthermore, the map \[\nu \in I(\eta_0)\longmapsto\left(\psi(\cdot;\eta_1(\nu),\eta_2(\nu)),\phi(\cdot;\eta_1(\nu),\eta_2(\nu))\right)\in H_{per}^n([0,L])\times H_{per}^n([0,L]) \] is smooth for all integer $n\geq 1$.
(iii) The map $\Lambda:I(\nu_0)\rightarrow B(\eta_{2,0})$ is strictly decreasing. Therefore, from (\ref{equaZa13c}), $\nu\mapsto k(\nu)$ is a strictly increasing function. \end{theo} \proof The proof of this theorem follows the same ideas of the Theorem 2.1 in Angulo \cite{angulo4}, we will use the Implicit Function Theorem. For this, consider the open set \[\Omega = \left\{(\eta,\nu) \in \mathbb{R}^2: \nu>\frac{2\pi^2}{L^2} \ \text{and} \ \eta \in (0,\sqrt{\nu\alpha}\ )\right\}\] and $\Gamma:\Omega \longrightarrow \mathbb{R}$ defined as \[\Gamma(\eta,\nu) = \frac{2\sqrt{2\alpha}}{\sqrt{2\nu\alpha-\eta^2}} \ K(k(\eta,\nu))- L,\] where \begin{equation}\label{equa13bzak} k^2(\eta,\nu) =\frac{2\nu\alpha-2\eta^2}{2\nu\alpha-\eta^2}. \end{equation} From the hypothesis, we have that $\Gamma(\eta_{2,0},\nu_0) = 0$. We proof that $\frac{d\Gamma}{d\eta}<0$ in $\Omega$. In fact, we use the next relation \begin{equation}\label{equa14zak} \frac{dK(k)}{dk}=\frac{E(k)-k'^2K(k)}{kk'^2}\ \ \text{with}\ \ k \in (0,1), \end{equation} where $E=E(k)$ is the complete elliptic integral of the second type and $k'^2=1-k^{2}$ is the complementary modulus. Deriving (\ref{equa13bzak}) with respect to $\eta,$ we obtain that \begin{equation} \label{equa15zak} \frac{\partial k}{\partial\eta}= -\frac{2\eta\nu\alpha}{k(2\nu\alpha-\eta^2)^2}. \end{equation} Then from (\ref{equa14zak}) and (\ref{equa15zak}) we obtain \[\frac{\partial\Gamma}{\partial\eta}=\frac{2\eta\sqrt{2\alpha}}{(2\nu\alpha-\eta^2)^{\frac{3}{2}}} \ K(k)-\frac{4\eta\nu\alpha\sqrt{2\alpha}}{(2\nu\alpha-\eta^2)^{\frac{5}{2}}}\left[\frac{E(k) - k'^2K(k)}{k^2k'^2}\right].\] Thus, \begin{align*} \frac{\partial \Gamma}{\partial \eta} < 0 & \Leftrightarrow k^2k'^2(2\nu\alpha-\eta^2)K(k) < 2\nu\alpha E(k) - 2\nu\alpha k'^2K(k)\\ & \Leftrightarrow k'^2(2\nu\alpha-2\eta^2)K(k) + 2\nu\alpha k'^2K(k) < 2\nu\alpha E(k)\\ & \Leftrightarrow \frac{2\nu\alpha k'^2}{(1+k'^2)}K(k)< \nu\alpha E(k)\Leftrightarrow (1+k'^2)E(k)- 2k'^2K(k) > 0. \end{align*} Since the last inequality always holds, we obtain that $\frac{\partial \Gamma}{\partial \eta}<0$. By the Implicit Function Theorem we have that there exists an interval $I(\nu_0)$ around $(\nu_0)$, an interval $B(\eta_{2,0})$ around $\eta_{2,0}$ and a smooth function $\Lambda:I(\nu_0)\longrightarrow B(\eta_{2,0})$ such that $\Lambda (\nu_0) = \eta_{2,0}$ and \[\Gamma(\Lambda(\nu),\nu) = 0,\ \ \ \ \ \ \forall \nu \in I(\nu_0).\] Additionally, since $\nu_0$ was chosen arbitrarily in $I=\left(\frac{2\pi^2}{L^2},+\infty\right)$ and from the uniqueness of $\Lambda,$ we extend $\Lambda$ to $I.$ The part $(ii)$ is immediate, using the smoothness of the function involved.\\
Now, we prove that $\Lambda$ is an strictly decreasing function. For this note that $\Gamma(\Lambda(\nu),\nu)=L$ for all $\nu\in I(\nu_0)$ then, using again the Implicit Function Theorem we get that \[\Lambda '(\nu)=-\frac{\partial\Gamma/{\partial\nu}}{\partial\Gamma/\partial\eta}.\] Since $\frac{\partial\Gamma}{\partial\eta}<0,$ we just have to prove that $\frac{\partial\Gamma}{\partial\nu}<0$ in $I(\nu_0)$. In fact, since \[\frac{\partial\Gamma}{\partial\nu}=\frac{2\alpha\sqrt{2\alpha}}{(2\nu\alpha-\eta^2)^{3/2}}\left[ -K+\frac{dK}{dk}\frac{\eta^2}{k(2\nu\alpha-\eta^2)} \right]\] and $\eta^2=(2\nu\alpha-\eta^2)k'^2,$ we obtain \begin{align*} \frac{\partial\Gamma}{\partial\nu}<0&\Leftrightarrow \frac{\eta^2}{\sqrt{2\nu\alpha-\eta^2}\sqrt{2\nu\alpha-2\eta^2}}\frac{dK}{dk}<K\Leftrightarrow k'^2\frac{dK}{dk}-kK<0.\\ \end{align*} From (\ref{equa14zak}), we arrived at \[ \frac{\partial\Gamma}{\partial\nu}<0\Leftrightarrow\frac{E-k'^2K}{k}-kK<0\Leftrightarrow E<K.\] Since the last inequality always holds for any $k\in(0,1)$ (see Byrd and Friedman \cite{byrdFriedman}), we obtain the desired result.\\
Finally, deriving $k$ with respect to $\nu,$ we obtain \[\frac{dk}{d\nu}=\frac{\alpha\eta(\eta-2\eta'\nu)}{k(2\nu\alpha-\eta^2)^2}>0,\] which proves that $\nu \mapsto k(\nu)$ is strictly increasing function, this finishes the proof of the theorem.
$\square$
The next result will be used in the prove of the stability of the dnoidal waves solutions.
\begin{coro}\label{coroDnoidal} Let $L>0$ and $c$ be arbitrarily fixed with $1-c^2 >0.$ Consider the smooth curve of dnoidal waves $\nu \in \left(\frac{2\pi^2}{L^2},+\infty\right) \longmapsto \phi_{\nu}(\cdot ; \eta_1(\nu),\eta_2(\nu))$ determined by Theorem \ref{TeorDnoidal}. Then \[\frac{d}{d\nu}\int_0^L \phi^2_\nu(\xi)d\xi>0.\] \end{coro} \proof Using the facts that $\eta_1L=2\sqrt{2(1-c^2)}K(k)$ and $\int_0^L dn^2 (y) dy = E(k)$ (see Byrd and Friedman \cite{byrdFriedman}) we get that \[\int_0^L \phi^2_{\nu}(\xi) d\xi = 2\eta_1\sqrt{2(1-c^2)} \int_0^K dn^2 (y;k) dy = \frac{8(1-c^2)}{L}K(k)E(k).\] Since $k\mapsto K(k)E(k)$ and $\nu \mapsto k(\nu)$ are strictly increasing functions we obtain \[\frac{d}{d\nu}\int_0^L \phi^2_{\nu}(\xi) d\xi =\frac{8(1-c^2)}{L} \frac{d}{dk }\left[K(k)E(k)\right]\frac{dk}{d\nu}>0\] This finishes the proof of the corollary.
$\square$
\section{Spectral Analysis} In this part of the paper, we study some spectral properties of various operators which will be necessary to obtain our result of stability. First, note that the system (\ref{equaZakha}) can be rewritten as \begin{equation}\label{ZakNovo} \left \{ \begin{aligned} v_t &=-V_x, \ \int_0^L V(x,t)dx = 0 \\
V_t &= -(v+|u|^2)_x \\ iu_t &+ u_{xx} =uv \end{aligned} \right. \end{equation} Therefore, we have the Hamiltonian structure $\frac{\partial U}{\partial t}=JE'(U)$ where $U=(v,V,u)^t,$ $J$ is the linear skew-symmetric operator given by \[J=\left( \begin{array}{crrc}
0 &-\frac{d}{dx} &0 \\
-\frac{d}{dx} &0 &0 \\
0 &0 &-\frac{i}{2} \end{array} \right)\] and $E$ is the energy functional define as \begin{equation} \label{equa2.4Zak}
E(v,V,u)=\frac{1}{2}\int_0^L 2|u_x|^2+v^2+V^2+2v|u|^2 \ dx. \end{equation} We also use the functionals $Q_1$ and $Q_2$ defined as \begin{equation}\label{quantCons2e3}
Q_1(v,V,u)=\int_0^L uV + \text{Im}(u_x\overline{u})\ dx \ \ \ \ \text{and}\ \ \ \ Q_2(v,V,u)= \int_0^L |u|^2 dx. \end{equation} A standard analysis proves that $E,$ $Q_1$ and $Q_2$ are conserved quantities of the system (\ref{ZakNovo}), i.e., \[E(v(t),V(t),u(t))= E(v(0),V(0),u(0)), \ \ \ Q_1(v(t),V(t),u(t))= Q_1(v(0),V(0),u(0))\] \[\text{and}\ \ Q_2(v(t),V(t),u(t))= Q_2(v(0),V(0),u(0)) \] for all $t\in [-T,T],$ where $T$ is the maximal time of existence of solutions.\\
Now, suppose that $V(x,t)=\varphi_{\omega,c}(x-ct),$ with $\varphi_{\omega,c}:\mathbb{R}\rightarrow\mathbb{R}$ a smooth $L-$periodic function, is solution of $v_t=-V_{x}$, then \begin{equation}\label{edo3Zakh} c\psi_{\omega,c}'=\varphi'_{\omega,c}. \end{equation} Therefore, $c\psi= \varphi+d_0$, where $d_0$ is a constant of integration. Since we are interested in $\varphi$ with zero mean, we obtain that $d_0=\frac{c}{L}\int_0^L \psi dx$. Using tha fact that $\int_0^K \text{dn}^2(x,k) dx=E(k)$ we get that $d_0(k)=-\frac{c\eta_1^2}{1-c^2}\frac{E(k)}{K(k)}$. Therefore \begin{equation}\label{varphidnoidal} \varphi(\xi)=-\frac{c\eta_1^2}{1-c^2}\left[\text{dn}^2\left(\frac{\eta_1\xi}{\sqrt{2(1-c^2)}};k\right)-\frac{E(k)}{K(k)}\right]. \end{equation}
It is worth to note that if $\eta_2\rightarrow 0^+,$ then $\eta_1\rightarrow\sqrt{2\alpha\nu}$ and therefore $k\rightarrow 1^-.$ Since dn$(u,1^-)=\text{sech}(u)$, $E(1)=\frac{\pi}{2}$ and $K(1)= +\infty$ we arrive at \[\varphi(\xi)=c\left(2\omega+\frac{c^2}{2}\right)\text{sech}^2\left(\frac{\sqrt{-4\omega-c^2}}{2}\xi \right),\] which is the solitary wave solution for (\ref{edo3Zakh}).\\
Now, using the Theorem \ref{TeorDnoidal} we have that there exist periodic traveling waves for (\ref{ZakNovo}) given by \[\left(\psi_{\omega,c}(x-ct),\varphi_{\omega,c}(x-ct), e^{- i\omega t} e^{i\frac{c}{2}(x-ct)}\phi_{\omega,c}(x-ct)\right),\] where \begin{equation}\label{solZakhpsi} \psi_{\omega,c}(\xi)=\frac{-\eta_{1}^2}{1-c^2}\text{dn}^2\left(\frac{\eta_1\xi}{\sqrt{2(1-c^2)}};k \right), \ \phi_{\omega,c}(\xi)=\eta_1\text{dn}\left(\frac{\eta_1\xi}{\sqrt{2(1-c^2)}};k\right) \end{equation} \begin{equation}\label{solZakhphi} \text{and}\ \ \ \ \varphi_{\omega,c}(\xi)= -\frac {c\eta_1^2}{1-c^2}\left[\text{dn}^2\left(\frac{\eta_1\xi}{\sqrt{2(1-c^2)}};k\right)-\frac{E(k)}{K(k)}\right]. \end{equation} The next operators will be useful in the proof of the stability of the dnoidal wave solutions: \begin{equation}\label{operaDnoidal} \mathcal{L}_{3}=-\frac{d^2}{{dx}^2}-\left(\omega+\frac{c^2}{4}\right)+3\psi\ \ \ \text{and}\ \ \ \ \mathcal{L}_{4}=-\frac{d^2}{{dx}^2}-\left(\omega+\frac{c^2}{4}\right)+\psi. \end{equation} We will study the spectral properties of the operators $\mathcal{L}_{i},\ i=3,4$. Recall that $\sigma(\mathcal{L}_{i})=\sigma_{ess}(\mathcal{L}_{i})\cup\sigma_{disc}(\mathcal{L}_{i})$ where $\sigma_{ess}(\mathcal{L}_{i})$ and $\sigma_{disc}(\mathcal{L}_{i})$ denote, respectively, the essential spectrum and the point spectrum of $\mathcal{L}_{i}$ (see Reed and Simon \cite{ReedSimon1}). Write \[\mathcal{L}_{3}=\left(-\frac{d^2}{{dx}^2}-\left(\omega+\frac{c^2}{4}\right)\right)+3\psi=:\mathcal{L}+M_1,\] \[\mathcal{L}_{4}=\left(-\frac{d^2}{{dx}^2}-\left(\omega+\frac{c^2}{4}\right)\right)+\psi=:\mathcal{L}+M_2,\] where $\mathcal{L}=-\frac{d^2}{dx^2}-(\omega+\frac{c^2}{4}).$ Since $M_1$ and $M_2$ are relatively compact with respect to $\mathcal{L}$, it follows from the Weyl's Essential Spectrum Theorem (see Reed and Simon \cite{ReedSimon1}) that $\sigma_{ess}(\mathcal{L}_i)=\sigma_{ess}(\mathcal{L})=\emptyset,$ with $i= 3,4.$ Thus $\sigma(\mathcal{L}_i)=\sigma_{disc}(\mathcal{L}_i),$ para $i=3,4.$ Therefore we have to analyze the periodic eigenvalue problem on $[0,L]$ \begin{equation}\label{probperiogeral} \left \{ \begin{aligned} \mathcal{L}_{i}\chi&=\lambda\chi\\ \chi(0)&=\chi(L),\ \chi'(0)=\chi'(L). \\ \end{aligned}\right. \end{equation} The problem (\ref{probperiogeral}) determines that the spectrum of $\mathcal{L}_i$ is a countable set of eigenvalues $\{\lambda_n: n=0,1,2,3,...\}$ with \[\lambda_0\leq \lambda_1\leq \lambda_2\leq \lambda_3\leq \cdots, \] where the double eigenvalues are counted twice and $\lambda\rightarrow+\infty$ when $n\rightarrow\infty.$ We denote by $\chi_n$ the eigenfunctions associated to the eigenvalue $\lambda_n.$ It is clear from the conditions $\chi(0)=\chi(L), \chi'(0)=\chi'(L)$ that $\chi_n$ can be extended to all $(-\infty,+\infty)$ as a continuous differentiable function with period $L.$ We know from the Floquet theory that the periodic eigenvalue problem (\ref{probperiogeral}) is related to the study of the next semi-periodic eigenvalue problem consider in $[0,L]$ \[\left \{ \begin{aligned}
\mathcal{L}_{i}\eta&= \mu\eta\\ \eta(0)&=-\eta(L),\ \eta'(0)=-\eta'(L), \\ \end{aligned} \right.\] which also is a self-adjoint problem and therefore determines a sequence of eigenvalues $\{\mu_n: n=0,1,2,3 ...\}$ with \[\mu_0\leq \mu_1\leq \mu_2\leq \mu_3\leq \cdots, \] where the double eigenvalues are counted twice and $\mu_n\rightarrow+\infty$ when $n\rightarrow\infty.$ We denote by $\eta_n$ the eigenfunction associated to the eigenvalue $\mu_n.$
\begin{theo} Let $\phi_{\nu}=\phi$ and $\psi_{\nu} = \psi$ the dnoidal waves given by Theorem \ref{TeorDnoidal}. Then,
(i) the operator $\mathcal{L}_3$ in (\ref{operaDnoidal}) defined in $L_{per}^2([0,L])$ with domain $H_{per}^2([0,L])$ has its fist three eigenvalues simple, where zero is the second one with associated eigenfunction $\phi '$. Furthermore, the rest of the spectrum is constitute by a discrete set of eigenvalues which are double.
(ii) The operator $\mathcal{L}_4$ in (\ref{operaDnoidal}) defined in $L_{per}^2([0,L])$ with domain $H_{per}^2([0,L])$ has zero as its first eigenvalue which is simple with associated eigenfunction $\phi$. Furthermore, the rest of the spectrum is constitute by a discrete set of eigenvalues. \end{theo} {\noindent \bf{Proof:\hspace{4pt}}}$(i)$ The proof is based on the Floquet Theory (see Eastham \cite{eastham}, Mangnus and Winkler \cite{magnus}). Deriving (\ref{ecuaordphi}) and using (\ref{ecuasegint}) we have that $\mathcal{L}_3\phi '=0$. Then zero is an eigenvalue of $\mathcal{L}_3$ with associated eigenfunction $\phi '.$ Since $\phi '$ has exactly two zeros on $[0,L)$, we get that zero is either the second or the third eigenvalue of $\mathcal{L}_3$. We will prove that zero is in fact the second one. For this we have to study the periodic problem \begin{equation} \label{equa2.12Zak} \left \{ \begin{aligned} \mathcal{L}_{3}\chi &=\lambda\chi\\ \chi(0)&=\chi(L),\ \chi'(0)=\chi'(L).\\ \end{aligned} \right. \end{equation} Let $\Lambda(x)=\chi(\eta x)$ where $\eta=\frac{\sqrt{2\alpha}}{\eta_1}$. Then, from the explicit form of $\psi$ and the relation $k^2 \text{sn}^2+\text{dn}^2=1$,we have that the problem (\ref{equa2.12Zak}) is equivalent to \begin{equation} \label{equa2.13Zak} \left \{ \begin{aligned} \Lambda ''+&[\rho-6k^2\text{sn}^2(x; k)]\Lambda=0\\ \Lambda(0)&=\Lambda(2K),\ \Lambda '(0)=\Lambda '(2K), \\ \end{aligned}\right. \end{equation} where \[\rho=\frac{2\alpha}{\eta_{1}^2}\left(\frac{\lambda}{2}+\omega+\frac{c^2}{4}+\frac{3\eta_{1}^2}{\alpha}\right).\] The second order equation given in (\ref{equa2.13Zak}) is called the Jacobian form of the Lam\' e equation. It is well known that such equation determines the existence of exactly three intervals of instability (see Theorem $7.8$ in Mangnus and Winkler \cite{magnus}). We will show that this intervals are the first three. First, observe that $\rho_1= 4+k^2$ and $\Lambda_1(x)=\text{cn}(x;k)\text{sn}(x;k)$ satisfy the problem (\ref{equa2.13Zak}). Furthermore, following Ince \cite{ince} we have that the functions \[\Lambda_0(x)=1-(1+k^2-\sqrt{1+k^2+k^4})\text{sn}^2(x;k),\] \[\Lambda_2(x)=1-(1+k^2+\sqrt{1+k^2+k^4})\text{sn}^2(x;k),\] which have period $2K,$ are the eigenfunctions of (\ref{equa2.13Zak}) with eigenvalues given by \[\rho_0= 2\left(1+k^2 -\sqrt{1+k^2+k^4}\right)\ \ \ \text{and}\ \ \ \rho_2= 2\left(1+k^2 -\sqrt{1+k^2+k^4}\right).\] Since $\Lambda_0$ does not have zeros in $[0,2K]$, it follows that $\rho _0$ is the first eigenvalue of (\ref{equa2.13Zak}). Furthermore, since $\Lambda_2$ has two zeros in $[0,2K)$ and $\rho_1<\rho_2,$ we have that $\rho_1$ is the second eigenvalue of (\ref{equa2.13Zak}) and $\rho_2$ is the third. We also have that $\rho_0,\rho_1$ and $\rho_2$ are simple. Now, since the eigenvalues of (\ref{equa2.12Zak}) and (\ref{equa2.13Zak}) are related as \[\lambda=\frac{\eta_1}{\alpha}(\rho-6)+2\nu,\] we can see $\lambda$ as a function of $\rho$, which is increasing. Since $k^2-2=\frac{2\alpha\nu}{\eta_{1}^2}$, we have that $\lambda(\rho_1) = 0 = \lambda_1$ and since $\lambda_0 < \lambda_1 < \lambda_2$, we obtain that \[\lambda_0<0=\lambda_1<\lambda_2.\] This finishes the proof of the part $(i)$.\\
$(ii)$ Using (\ref{ecuaordphi}) and (\ref{ecuasegint}) we have that $\mathcal{L}_4\phi=0$. Thus, zero is an eigenvalue of $\mathcal{L}_4$ with associated eigenfunction $\phi$. Since $\phi$ does not have zeros in $[0,L]$ we obtain that zero is the first eigenvalue of $\mathcal{L}_4$ and it is simple.
$\square$
\section{Nonlinear Stability for the Dnoidal Wave Solutions} In this section we study the nonlinear stability properties of the periodic traveling wave solution $\Phi(\xi)=(\psi(\xi),\varphi(\xi),\widetilde{\phi}(\xi))$ where $\psi$, $\varphi$ and $\phi$ are given by (\ref{solZakhpsi}), (\ref{solZakhphi}), $\widetilde{\phi}(\xi)=e^{i\frac{c}{2}\xi}\phi(\xi)$ and $1-c^2>0$. First, we define the type of stability in which we are interested: Let $ X:= L^2_{per}([0,L]) \times \widetilde{L}^2_{per}([0,L]) \times H^1_{per}([0,L]),$ where \[\widetilde{L}^2_{per}([0,L])=\left\{f \in L^2_{per}([0,L]): \int_0^L f(x) dx=0\right\}.\] Initially, observe that the system (\ref{ZakNovo}) has two basic symmetries: translations and rotations. This means that if $(v(x,t),V(x,t),u(x,t))$ is a solution of (\ref{ZakNovo}), then the pair of functions \[(v(x+y),V(x+y),u(x+y))\ \ \ \ \ \text{and}\ \ \ \ (v(x,t),V(x,t),e^{-is}u(x,t))\] are also solutions, for any real constants $y$ and $s.$ So, our notion of stability will be modulus these symmetries. More precisely, \begin{defi} We say that the orbit generated by $\Phi(\xi)$, namely \[\mathcal{O}_{\Phi}=\left\{\left(\psi(\cdot + y)),\varphi(\cdot+y)), e^{i\theta}\widetilde{\phi}(\cdot+y)\right):(\theta,y)\in [0,2\pi)\times\mathbb{R}\right\}\] is stable in $X$ by the flow generated by the system (\ref{ZakNovo}), if for all $\epsilon>0$, there exists $\delta>0$ such that for any $(v_0,V_0, u_0)\in X$ satisfying
\[\|v_0-\psi\|_{L^2_{per}}<\delta,\ \ \|V_0-\varphi\|_{L^2_{per}}<\delta \ \ \text{and}\ \ \|u_0-\widetilde{\phi}\|_{H^1_{per}}<\delta,\] we have that the solution $(v, V,u)$ of the system (\ref{ZakNovo}) with $(v(0),V(0),u(0))=(v_0, V_0,u_0)$, satisfies \[(v,V,u) \in C(\mathbb{R}; L^2_{per}([0,L]))\times C(\mathbb{R}; \widetilde{L}^2_{per}([0,L])) \times C(\mathbb{R};H^1_{per}([0,L])),\] \begin{equation}\label{desiImp1}
\inf_{y \in \mathbb{R}} \|v(\cdot+y,t)-\psi\|_{L^2_{per}}<\epsilon, \ \ \ \inf_{y \in \mathbb{R}} \|V(\cdot+y,t)-\varphi\|_{L^2_{per}}<\epsilon \end{equation} \begin{equation}\label{desiImp2}
\text{and}\ \ \ \inf_{\theta \in [0,2\pi),y \in \mathbb{R}} \|e^{i\theta}u(\cdot+y,t)-\widetilde{\phi}\|_{H^1_{per}}<\epsilon. \end{equation} Otherwise, we say that $\Phi$ é $X$-unstable. \end{defi} Next, we present our result of stability for the dnoidal waves.
\begin{theo}\label{teoStabDnoidal} Let $L>0$ and $1-c^2>0$ be fixed numbers. Consider the smooth curve of periodic traveling wave solutions for the system (\ref{ZakNovo}), $\nu\mapsto(\psi_{\nu},\varphi_{\nu},\phi_{\nu}),$ determined by the Theorem \ref{TeorDnoidal} and (\ref{varphidnoidal}). Then, for $\nu>\frac{2\pi^2}{L^2}$ the orbit generated by $\Phi_{\nu}(x,t)=\left(\psi_{\nu}(x),\varphi_{\nu}(x),\widetilde{\phi}_{\nu}(x)\right)$ is stable in $X$ by the periodic flow generated by the system (\ref{ZakNovo}), if the initial datum $(v_0,V_0,u_0)$ satisfies \[\int_0^L v_0(x)dx\leq\int_0^L\psi(x)dx.\] \end{theo} \proof Consider $(\psi_{\nu},\varphi_{\nu},\widetilde{\phi}_{\nu})$ the solution of (\ref{ZakNovo}) given by Theorem \ref{TeorDnoidal}. For $(v_0,V_0,u_0) \in L^2([0,L]) \times \widetilde{L}_{per}^2([0,L]) \times H_{per}^1([0,L])$ and $(v,V,u)$ the global solution for (\ref{ZakNovo}) corresponding to this initial data, we define for $t\geq 0$ and $ \nu > \frac{2\pi^2}{L^2}$
\[\Omega_t(y,\theta)=\|e^{i\theta}(T_cu)'(\cdot + y,t)-\phi_{\nu}'\|_{L_{per}^2}^2+\nu\|e^{i\theta}(T_cu)(\cdot + y,t)-\phi_{\nu}\|_{L_{per}^2}^2,\] where we denote by $T_c$ the bounded linear operator define as \[(T_cu)(x,t)=e^{-ic(x-ct)/2}u(x,t).\] Then, the deviation of the solution $u(t)$ from the orbit generated by $\Phi$ is measure by \begin{equation}\label{equa3} \rho_{\nu}(u(\cdot,t),\phi_{\nu})^2:=\inf\left\{\Omega_t(y,\theta): (y,\theta)\in[0,L]\times[0,2\pi]\right\}. \end{equation} Therefore, from (\ref{equa3}) we have that for each $t$ the $\inf\Omega_t(y,\theta)$ is attained in $(\theta,y)=(\theta(t),y(t)).$ Consider the perturbation of the periodic wave $(\psi,\varphi,\widetilde{\phi})$ \begin{equation} \label{equa4} \left \{ \begin{aligned}
\xi(x,t)&=e^{i\theta}(T_cu)(x+y,t)-\phi_{\nu}(x) \\
\eta(x,t)&= V(x+y,t)-\varphi_{\nu}(x)\\
\gamma(x,t)&= v(x+y,t)-\psi_{\nu}(x). \end{aligned} \right. \end{equation} By the property of minimum of $(\theta,y)=(\theta(t),y(t))$, we obtain from (\ref{equa4}) that $p(x,t)=\text{Re}(\xi(x,t))$ and $q(x,t)=\text{Im}(\xi(x,t))$ satisfy the compatibility relations \begin{equation} \label{equa5} \left \{ \begin{aligned}
\int_0^L q(x,t)\phi_{\nu}(x)\psi_{\nu}(x) dx&= 0 \\ \int_0^L p(x,t)(\phi_{\nu}(x)\psi_{\nu}(x))' dx &= 0. \\ \end{aligned} \right. \end{equation}\\ Now, consider the continuous functional $\mathcal{B}$ defined in $X$ as \[\mathcal{B}(v,V,u) := E (v,V,u)-cQ_1(v,V,u)-\omega Q_2(v,V,u),\] where $E$, $Q_1$ and $Q_2$ were defined in (\ref{equa2.4Zak}) and (\ref{quantCons2e3}). Then, from (\ref{equa4}) and (\ref{equa5}), we get \begin{align*} \Delta\mathcal{B}:=& \ \mathcal{B}(v(t),V(t),u(t))-\mathcal{B}(\psi,\varphi,\widetilde{\phi})\\ =&\left(\mathcal{L}_3p,p\right)+\left(\mathcal{L}_4q,q\right) +\frac{1}{2}\int_0^L \gamma^2+2\gamma(p^2+q^2)-4\psi p^2 + 4\gamma p \phi \ dx \\& + \frac{1}{2} \int_0^L 2\gamma\psi +\eta^2+ 2\eta\varphi + 2\gamma\phi^2 - 2c\gamma\eta -2c\gamma\varphi -2c\psi\eta \ dx \end{align*} where \[\mathcal{L}_3=-\frac{d^2}{dx^2}-\left(\omega+\frac{c^2}{4}\right)+3\psi \ \ \ \text{and} \ \ \ \mathcal{L}_4=-\frac{d^2}{dx^2}-\left(\omega+\frac{c^2}{4}\right)+\psi.\] Using the facts that $c\psi-\varphi=d_0$ and $\int_0^L\eta dx=0,$ we obtain \begin{align*} \Delta\mathcal{B}(t)&=\left(\mathcal{L}_3p,p\right)+\left(\mathcal{L}_4q,q\right) + \frac{1}{2}\int_0^L \left[\sqrt{1-c^2}\gamma+\frac{2\phi p}{\sqrt{1-c^2}}+\frac{p^2+q^2}{\sqrt{1-c^2}}\right]^2 dx \\ &+ \frac{1}{2}\int_0^L(c\gamma-\eta)^2dx - \int_0^L \frac{4\phi p(p^2+q^2)}{1-c^2}+\frac{(p^2+q^2)^2}{1-c^2} dx + \int_0^L(c\gamma-\eta)(c\psi-\varphi)dx\\ & =\left(\mathcal{L}_3p,p\right)+\left(\mathcal{L}_4q,q\right) + \frac{1}{2}\int_0^L \left[\sqrt{1-c^2}\gamma+\frac{2\phi p}{\sqrt{1-c^2}}+\frac{p^2+q^2}{\sqrt{1-c^2}}\right]^2 dx \\ &+\frac{1}{2}\int_0^L(c\gamma-\eta)^2 dx-\int \frac{4\phi p(p^2+q^2)}{1-c^2}+\frac{(p^2+q^2)^2}{1-c^2}dx + cd_0\int_0^L\gamma dx. \end{align*} Since $cd_0\leq 0,$ $\int_0^Lv_0dx\leq\int_0^L\psi(x)dx$ and $\int v(t,x)dx=\int v_0(x)dx,$ we have that $cd_0\int_0^L\gamma dx\geq 0.$ Therefore \begin{align}\label{equa55}
\notag\Delta\mathcal{B}(t)\geq &\left(\mathcal{L}_3p,p\right)+\left(\mathcal{L}_4q,q\right) + \frac{1}{2}\int_0^L \left[\sqrt{1-c^2}\gamma+\frac{2\phi p}{\sqrt{1-c^2}}+\frac{p^2+q^2}{\sqrt{1-c^2}}\right]^2dx \\&+\frac{1}{2}\int_0^L(c\gamma-\eta)^2dx-C_1\|\xi\|_{H^1_{per}}^3-C_2\|\xi\|_{H^1_{per}}^4, \end{align} with $C_i>0$, $i=1,2.$\\
The estimates for $\left(\mathcal{L}_3p,p\right)$ and $\left(\mathcal{L}_4q,q\right)$ will be obtain from the next theorems.
\begin{theo}\label{teoEstForCua1} Let $1-c^2>0$ and $\nu>\frac{2\pi^2}{L^2}$ fixed numbers. Consider $\phi_{\nu}$ the dnoidal wave given by Theorem \ref{TeorDnoidal}. Then \begin{itemize}
\item[(a)] $\inf\{\left(\mathcal{L}_3f,f\right):\|f\|=1\ \text{and}\ \left(f,\phi_{\nu}\right)=0\}=:\alpha_0=0$
\item[(b)] $\inf\{\left(\mathcal{L}_3f,f\right):\|f\|=1,\ \left( f,\phi_{\nu}\right)=0\ \text{and}\ \left(f,(\phi_{\nu}\psi_{\nu})'\right)=0\}=:\alpha>0.$ \end{itemize} \end{theo} \proof
$(a)$ Since $\mathcal{L}_3\left(\frac{d}{dx}\phi_{\nu}\right)=0$ and $\left(\frac{d}{dx}\phi_{\nu},\phi_{\nu}\right)=0$, then $\alpha_0 \leq 0$. We prove that $\alpha_0 \geq 0$ using Lemma E.1 in Weinstein \cite{weinstein2} (which works on the periodic case). We first show that the infimum is attained. In fact, since $\phi_{\nu}$ is bounded we have that $\alpha$ is finite, thus there exists $\{f_j\}\subset H^1_{per}([0,L])$ with $\|f_j\|=1$, $\left( f_j,\phi_{\nu}\right)=0$ and $\lim_{j\rightarrow \infty}\left(\mathcal{L}_3 f_j,f_j\right)=\alpha_0$. Since $\{f_j\}$ is bounded in $H^1_{per}([0,L])$ there exists a subsequence of $\{f_j\}$, that we denote again $f_j,$ such that $f_j \rightharpoonup g$ weakly in $H^1_{per}([0,L]),$ then $f_j \rightarrow g$ in $L^2_{per}([0,L])$. Therefore $(g,\phi_{\nu}) =0$ and $(\phi_c f_j,f_j)\rightarrow (\phi g,g)$ when $j \rightarrow + \infty$. So $g \neq 0$ and $\|g'\|_{L^2_{per}}\leq \liminf \|f_j'\|_{L^2_{per}}$.\\
Now, define $f=g/\|g\|_{L^2_{per}}$, then $(f,\phi_{\nu})=0$, $\|f\|_{L^2_{per}}=1$ and
\[\alpha_0 \leq (\mathcal{L}_3 f,f) \leq\frac{\alpha_0}{\|f\|^2_{L^2_{per}}} =\alpha_0.\] Therefore the infimum is attained. We show now that $\alpha_0 \geq 0$. In fact, $\mathcal{L}_3$ has the spectral properties required to use Lemma E.1, we need to find $\chi$ such that $\mathcal{L}_3\chi=\phi_{\nu}$ and $(\chi,\phi_{\nu})\leq 0$. From Theorem \ref{TeorDnoidal} we have that $\nu\in\left(\frac{2\pi^2}{L^2},+\infty\right)\longmapsto\phi_{\nu}\in H^1_{per}([0,L])$ is of class $C^1$, then differentiating (\ref{ecuaordphi}) with respect to $\nu$ we obtain that $\chi=-\frac{d}{d\nu}\phi_{\nu}$ satisfies $\mathcal{L}_3\chi =\phi_{\nu}$. Using the Corollary \ref{coroDnoidal} we obtain that \[(\chi,\phi_{\nu})=-\frac{1}{2}\frac{d}{d\nu}\int_0^L \phi^2_\nu(\xi)\ d \xi<0\] Therefore $(\chi,\phi_\nu)<0,$ which proves that $\alpha_0 \geq 0$. This finishes the proof of part $(a)$.\\
$(b)$ Using the part $(a)$, we have that $\alpha \geq 0$. Suppose that $\alpha =0$. Using a similar argument as in part $(a)$ we obtain that there exists $f \in H^1_{per}([0,L])$ such that $\|f\|_{L^2_{per}}=1$ and $(f,\phi_\nu)=\left(f,(\phi_\nu\psi)'\right)=0$. Then, from the theory of Lagrange Multipliers, there exists $\lambda,\ \theta$ and $\delta$ such that \[\mathcal{L}_3 f =\lambda f+\theta \phi_\nu + \delta(\phi_\nu\psi_\nu)'.\] Since $\left(\mathcal{L}_3 f,f\right)=0$, we obtain that $\lambda=0$. From the fact that $\mathcal{L}_3\phi'_\nu=0$ we have \[0=\delta \int_0^L \phi_{\nu}'(\phi_\nu\psi_\nu)' d\xi = -\frac{3\delta}{1-c^2}\int_0^L (\phi_{\nu}')^2 \phi^2_{\nu}\ d\xi.\] The last inequality implies $\delta=0,$ thus $\mathcal{L}_3f=\theta\phi_\nu$. Consider $\chi=-\frac{d}{d\nu}\phi_\nu$, then we get that $\mathcal{L}_3(f-\theta\chi)=0$, thus \[0=(f-\theta\chi,\phi_\nu)=-\theta(\chi,\phi_\nu).\] Therefore $\theta=0$ and consequently there exists $s \in \mathbb{R}\setminus\{0\}$ such that $f = s\phi_\nu ',$ which is absurd. Therefore $\alpha>0$, which finishes the proof of the theorem.
$\square$
\begin{theo}\label{teoEstForCua2} Let $1-c^2>0$ and $\nu>\frac{2\pi^2}{L^2}$ be fixed numbers. Consider $\phi_\nu$ and $\psi_\nu$ the dnoidal waves given by Theorem \ref{TeorDnoidal}. Then,
\[\inf \{\left(\mathcal{L}_4 f,f\right):\|f\|_{L^2_{per}}=1\ \ \text{and} \ \ \left(f,\phi_\nu\psi_\nu\right)=0\}=:\beta>0\] \end{theo} \proof From the spectral properties of $\mathcal{L}_4$ is clear that $\mathcal{L}_4$ is a nonnegative operator, therefore $\beta\geq 0$. Suppose that $\beta=0 $. Then, following the same ideas of the proof of Theorem \ref{teoEstForCua1}, we have that the infimum is attained on a admissible function $g\neq 0$ and there exists $(\lambda,\theta)\in\mathbb{R}^2$ such that \[\mathcal{L}_4g =\lambda g + \theta \phi_\nu\psi_\nu.\] Since $\left(g, \phi_\nu\psi_\nu\right) =0$, then $\lambda =0$. Furthermore, \[0=(\mathcal{L}_4\phi,g)=\theta\int_0^L\phi^2_\nu\psi_\nu\ d\xi,\] which implies $\theta =0$. Since zero is a simple eigenvalue of $\mathcal{L}_4$, there exists $s\in\mathbb{R}\setminus\{0\}$ such that $g= s\phi$, which is absurd. This finishes the proof of the theorem.
$\square$
Our goal is to estimate the terms $\left(\mathcal{L}_3 p,p\right)$ and $\left(\mathcal{L}_4 q,q\right)$, where $p$ and $q$ satisfy (\ref{equa5}). Using Theorem \ref{teoEstForCua2} and definition of $\mathcal{L}_4$, we have that there exists $C_0>0$ such that \begin{equation}\label{equa8}
\left(\mathcal{L}_4 q,q\right) \geq C_0 \|q\|^2_{H^1_{per}}. \end{equation}
Now, we estimate $\left(\mathcal{L}_3 p,p\right)$. Suppose with out loos of generality that $\|\phi_\nu\|_{L^2_{per}}=1$. Denote $p_{\perp}=p-p_{\parallel}$, where $p_{\parallel} = (p,\phi_\nu)\phi_\nu$, then from (\ref{equa5}) we obtain that $(p_{\perp},\phi_\nu)=0$ and $(p_{\perp},(\phi_\nu\psi_\nu)')=0$. From Theorem \ref{teoEstForCua1} it follows that $(\mathcal{L}_3 p_{\perp},p_{\perp})\geq \widetilde{C_0}\|p_{\perp}\|^2_{L^2_{per}}$.\\
Also consider the normalization $Q_2(u_0)=Q_2(\phi),$ i.e., $\|u_0\|_{L^2_{per}}=\|\phi_\nu\|_{L^2_{per}}$. Then $\|u(t)\|_{L^2_{per}}=1,$ for all $t\geq 0$, thus $-2(p,\phi_\nu)= \|\xi\|^2_{L^2_{per}}$. Therefore
\[(\mathcal{L}_3p_{\perp},p_{\perp})\geq C_0\|p_{\perp}\|^2_{L^2_{per}} \geq C_0\|p\|^2_{L^2_{per}} -\widetilde{C}_1\|\xi\|^4_{H^1_{per}}.\]
Since $\left(\mathcal{L}_3\phi_\nu,\phi_\nu\right) <0$, it follows that $\left(\mathcal{L}_3 p_{\parallel},p_{\parallel}\right)\geq -\widetilde{C}_2\|\xi\|^4_{H^1_{per}}$. From the Cauchy-Schwarz inequality we get that $\left(\mathcal{L}_3 p_{\parallel},p_{\perp}\right)\geq -\widetilde{C}_3\|\xi\|^4_{H^1_{per}}$. Therefore, from the specific form of the operator $\mathcal{L}_3$ we conclude that \begin{equation}\label{equa9}
\left(\mathcal{L}_3 p,p\right)\geq D_1\|p\|_{H^1_{per}}^2- D_2\|p\|_{H^1_{per}}^3-D_3\|p\|_{H^1_{per}}^4, \end{equation} where $D_j >0$ for $j=1,2,3.$\\
Now, using (\ref{equa55}), (\ref{equa8}) and (\ref{equa9}) we arrive at
\[\Delta\mathcal{B}(t)\geq d_1\|\xi\|_{1,\nu}^2- d_2\|\xi\|_{1,\nu}^3- d_3\|\xi\|_{1,\nu}^4\]
where $d_i>0$, for $i=1,2,3$ and $\|f\|_{1,\nu}^2:=\|f'\|_{L^2_{per}}^2+ \nu\|f\|_{L^2_{per}}^2$. Using a similar argument as in Benjamin \cite{benjamin1}, we obtain that for any $\epsilon>0,$ there exists $\delta(\epsilon)>0$ such that if
\[\|u_0-\widetilde{\phi}\|_{1,\nu}<\delta,\ \ \|V_0-\varphi\|_{L^2_{per}}<\delta \ \ \text{and}\ \ \|v_0-\psi\|_{L^2_{per}}<\delta, \] then \begin{equation}\label{desiFinalZak}
\rho_{\nu}(u(t),\phi_{\nu})^2=\|\xi(t)\|^2_{1,\nu}<\epsilon, \end{equation} for all $t\geq 0.$ Therefore we obtain the inequality (\ref{desiImp2}).\\
Finally, using (\ref{equa55}) and the analysis made above for $\xi$ we obtain that \[\int_0^L \left[\sqrt{1-c^2}\gamma+\frac{2\phi p}{\sqrt{1-c^2}}+\frac{p^2+q^2}{\sqrt{1-c^2}}\right]^2dx\leq\epsilon\ \ \ \text{and}\ \ \ \int_0^L(c\gamma-\eta)^2 dx\leq\epsilon.\] Using the two last inequalities, the Cauchy-Schwarz inequality and (\ref{desiFinalZak}) we arrive at (\ref{desiImp1}), which proves that $(\psi,\varphi,\widetilde{\phi})$ is stable with respect to small perturbations that preserves the $L^2_{per}([0,L])$ norm of $\widetilde{\phi}.$ The general case follows from the continuity of the map \[\nu\in\left(\frac{2\pi^2}{L^2},+\infty\right)\mapsto (\psi, \varphi,\widetilde{\phi}).\] This finishes the proof of the theorem.
$\square$
\section{Stability for the Solitary Wave Solutions} In this section we improve the result established by Ya Ping in \cite{Wu1}, to obtain a correct stability result for the solitary wave solutions associated to the Zakharov system. With regard to the Cauchy problem associate to the system (\ref{equaZakha}) we address the reader to the work of Colliander in \cite{Colliander2}, where is obtained a result of global well-posedness for initial data $(u,v,v_t)(0)\in H^1(\mathbb{R})\times L^2(\mathbb{R})\times \widehat{H}^{-1}(\mathbb{R})$.\\
If $v_t(x,0)\in \widehat{H}^{-1}(\mathbb{R})$ we can rewrite (\ref{equaZakha}) as the equivalent system
\begin{equation}\label{Zaksolit} \left \{ \begin{aligned} v_t &=-V_x, \\
V_t &= -(v+|u|^2)_x \\ iu_t &+ u_{xx} =uv. \end{aligned} \right. \end{equation} The solitary wave solutions for this system are given by
\[ \psi_{\omega,c}(\xi)=\left(2\omega+\frac{c^2}{2}\right)\ \text{sech}^2\left(\frac{\sqrt{-4\omega-c^2}}{2}\xi\right), \ \ \ \phi_{\omega,c}(\xi)=\sqrt{\frac{(-4\omega-c^2)(1-c^2)}{2}}\ \text{sech}\left(\frac{\sqrt{-4\omega-c^2}}{2}\xi\right) \] \[ \text{and}\ \ \ \varphi_{\omega,c}(\xi)=c\left(2\omega+\frac{c^2}{2}\right)\ \text{sech}^2\left(\frac{\sqrt{-4\omega-c^2}}{2}\xi\right). \] As in the periodic case, we have to study the spectral properties of the operators \begin{equation} L_{3}=-\frac{d^2}{{dx}^2}-\left(\omega+\frac{c^2}{4}\right)+3\psi\ \ \ \text{and}\ \ \ \ L_{4}=-\frac{d^2}{{dx}^2}-\left(\omega+\frac{c^2}{4}\right)+\psi. \end{equation} The spectral properties necessary to obtain our result of stability were established by Ya Ping in \cite{Wu1}. See also \cite{Wu1} for the definition of orbital stability for the solitary wave solutions associated to the Zakharon system.\\
The next theorem is the principal result of this section. \begin{theo} Assume that $4\omega+c^2\leq 0$ and $1-c^2>0.$ Then the solitary wave solutions $(\psi_{\omega,c}, \varphi_{\omega,c}, \widetilde{\phi}_{\omega,c}),$ with $ \widetilde{\phi}_{\omega,c}(x)=e^{i\frac{c}{2}x}\phi_{\omega,c}(x),$ are orbitally stable in $X= L^2_{per}([0,L]) \times L^2_{per}([0,L]) \times H^1_{per}([0,L])$ by the flow generated for the system (\ref{Zaksolit}). \end{theo}
\proof
The proof follows the same ideas of Theorem \ref{teoEstForCua2}. We only observe that in this case \begin{align*} \Delta\mathcal{B}(t)& =\left(L_3p,p\right)+\left(L_4q,q\right) + \frac{1}{2}\int_{\mathbb{R}}\left[\sqrt{1-c^2}\gamma+\frac{2\phi p}{\sqrt{1-c^2}}+\frac{p^2+q^2}{\sqrt{1-c^2}}\right]^2 dx \\ &+\frac{1}{2}\int_{\mathbb{R}}(c\gamma-\eta)^2 dx-\int _{\mathbb{R}}\frac{4\phi p(p^2+q^2)}{1-c^2}+\frac{(p^2+q^2)^2}{1-c^2}dx. \end{align*} Note that the constant $d_0$ does not appear because the decaying properties of the solitary wave solutions imply that $d_0=0.$ The rest of the proof follows similarly as the result obtained on the periodic case.
$\square$
\end{document} |
\begin{document}
\title[The $n$ linear embedding theorem]{The $n$ linear embedding theorem} \author[H.~Tanaka]{Hitoshi Tanaka} \address{Graduate School of Mathematical Sciences, The University of Tokyo, Tokyo, 153-8914, Japan} \email{htanaka@ms.u-tokyo.ac.jp} \thanks{ The author is supported by the FMSP program at Graduate School of Mathematical Sciences, the University of Tokyo, and Grant-in-Aid for Scientific Research (C) (No.~23540187), the Japan Society for the Promotion of Science. } \subjclass[2010]{42B20, 42B35 (primary), 31C45, 46E35 (secondary).} \keywords{ multinonlinear discrete Wolff's potential; multilinear positive dyadic operator; multilinear Sawyer's checking condition; $n$ linear embedding theorem. } \date{}
\begin{abstract} Let $\sigma_i$, $i=1,\ldots,n$, denote positive Borel measures on ${\mathbb R}^d$, let ${\mathcal D}$ denote the usual collection of dyadic cubes in ${\mathbb R}^d$ and let $K:\,{\mathcal D}\to[0,\infty)$ be a~map. In this paper we give a~characterization of the $n$ linear embedding theorem. That is, we give a~characterization of the inequality $$ \sum_{Q\in{\mathcal D}}
K(Q)\prod_{i=1}^n\left|\int_{Q}f_i\,d\sigma_i\right| \le C \prod_{i=1}^n
\|f_i\|_{L^{p_i}(d\sigma_i)} $$ in terms of multilinear Sawyer's checking condition and discrete multinonlinear Wolff's potential, when $1<p_i<\infty$. \end{abstract}
\maketitle
\section{Introduction}\label{sec1} The purpose of this paper is to investigate the $n$ linear embedding theorem. We first fix some notations. We will denote by ${\mathcal D}$ the family of all dyadic cubes $Q=2^{-k}(m+[0,1)^d)$, $k\in{\mathbb Z},\,m\in{\mathbb Z}^d$. Let $K:\,{\mathcal D}\to[0,\infty)$ be a~map and let $\sigma_i$, $i=1,\ldots,n$, be positive Borel measures on ${\mathbb R}^d$. In this paper we give a~necessary and sufficient condition for which the inequality \begin{equation}\label{1.1} \sum_{Q\in{\mathcal D}}
K(Q)\prod_{i=1}^n\left|\int_{Q}f_i\,d\sigma_i\right| \le C \prod_{i=1}^n
\|f_i\|_{L^{p_i}(d\sigma_i)}, \end{equation} to hold when $1<p_i<\infty$.
For the bilinear embedding theorem, in the case $\frac1{p_1}+\frac1{p_2}\ge 1$, Sergei Treil gives a~simple proof of the following.
\begin{proposition}[{\rm\cite[Theorem 2.1]{Tr}}]\label{prp1.1} Let $K:\,{\mathcal D}\to[0,\infty)$ be a~map and let $\sigma_i$, $i=1,2$, be positive Borel measures on ${\mathbb R}^d$. Let $1<p_i<\infty$ and $\frac1{p_1}+\frac1{p_2}\ge 1$. The following statements are equivalent:
\begin{itemize} \item[{\rm(a)}] The following bilinear embedding theorem holds: $$ \sum_{Q\in{\mathcal D}}
K(Q)\prod_{i=1}^2\left|\int_{Q}f_i\,d\sigma_i\right| \le c_1 \prod_{i=1}^2
\|f_i\|_{L^{p_i}(d\sigma_i)} <\infty; $$ \item[{\rm(b)}] For all $Q\in{\mathcal D}$, $$ \begin{cases}\displaystyle \left(\int_{Q}\left(\sum_{Q'\subset Q}K(Q')\sigma_1(Q')1_{Q'}\right)^{p_2'}\,d\sigma_2\right)^{1/p_2'} \le c_2 \sigma_1(Q)^{1/p_1} <\infty, \\ \displaystyle \left(\int_{Q}\left(\sum_{Q'\subset Q}K(Q')\sigma_2(Q')1_{Q'}\right)^{p_1'}\,d\sigma_1\right)^{1/p_1'} \le c_2 \sigma_2(Q)^{1/p_2} <\infty. \end{cases} $$ \end{itemize} \noindent Moreover, the least possible $c_1$ and $c_2$ are equivalent. \end{proposition} Here, for each $1<p<\infty$, $p'$ denote the dual exponent of $p$, i.e., $p'=\frac{p}{p-1}$, and $1_{E}$ stands for the characteristic function of the set $E$.
Proposition \ref{prp1.1} was first proved for $p_1=p_2=2$ in \cite{NaTrVo} by the Bellman function method. Later in \cite{LaSaUr}, this was proved in full generality. The checking condition in Proposition \ref{prp1.1} is called \lq\lq the Sawyer type checking condition", since this was first introduced by Eric~T. Sawyer in \cite{Sa1,Sa2}.
To describe the case $\frac1{p_1}+\frac1{p_2}<1$, we need discrete Wolff's potential.
Let $\mu$ and $\nu$ be positive Borel measures on ${\mathbb R}^d$ and let $K:\,{\mathcal D}\to[0,\infty)$ be a~map. For $p>1$, the discrete Wolff's potential ${\mathcal W}^p_{K,\mu}[\nu](x)$ of the measure $\nu$ is defined by $$ {\mathcal W}^p_{K,\mu}[\nu](x) := \sum_{Q\in{\mathcal D}} K(Q)\mu(Q) \left( \frac1{\mu(Q)}\sum_{Q'\subset Q} K(Q')\mu(Q')\nu(Q') \right)^{p-1} 1_{Q}(x), \quad x\in{\mathbb R}^d. $$ The author prove the following.
\begin{proposition}[{\rm\cite[Theorem 1.3]{Ta1}}]\label{prp1.2} Let $K:\,{\mathcal D}\to[0,\infty)$ be a~map and let $\sigma_i$, $i=1,2$, be positive Borel measures on ${\mathbb R}^d$. Let $1<p_i<\infty$ and $\frac1{p_1}+\frac1{p_2}<1$. The following statements are equivalent:
\begin{itemize} \item[{\rm(a)}] The following bilinear embedding theorem holds: $$ \sum_{Q\in{\mathcal D}}
K(Q)\prod_{i=1}^2\left|\int_{Q}f_i\,d\sigma_i\right| \le c_1 \prod_{i=1}^2
\|f_i\|_{L^{p_i}(d\sigma_i)} <\infty; $$ \item[{\rm(b)}] For $\frac1r+\frac1{p_1}+\frac1{p_2}=1$, $$ \begin{cases}\displaystyle
\|{\mathcal W}^{p_2'}_{K,\sigma_2}[\sigma_1]^{1/p_2'}\|_{L^r(d\sigma_1)} \le c_2<\infty, \\[2mm]\displaystyle
\|{\mathcal W}^{p_1'}_{K,\sigma_1}[\sigma_2]^{1/p_1'}\|_{L^r(d\sigma_2)} \le c_2<\infty. \end{cases} $$ \end{itemize} \noindent Moreover, the least possible $c_1$ and $c_2$ are equivalent. \end{proposition}
In his excerent survey of the $A_2$ theorem \cite{Hy}, Tuomas~P. Hyt\"{o}nen introduces another proof of Proposition \ref{prp1.1}, which uses the \lq\lq parallel corona" decomposition. In this paper, following Hyt\"{o}nen's arguments in \cite{Hy}, we shall establish the following theorems (Theorems \ref{thm1.3} and \ref{thm1.4}).
\begin{theorem}\label{thm1.3} Let $K:\,{\mathcal D}\to[0,\infty)$ be a~map and let $\sigma_i$, $i=1,\ldots,n$, be positive Borel measures on ${\mathbb R}^d$. Let $1<p_i<\infty$ and $\sum_{i=1}^n\frac1{p_i}\ge 1$. The following statements are equivalent:
\begin{itemize} \item[{\rm(a)}] The following $n$ linear embedding theorem holds: $$ \sum_{Q\in{\mathcal D}}
K(Q)\prod_{i=1}^n\left|\int_{Q}f_i\,d\sigma_i\right| \le c_1 \prod_{i=1}^n
\|f_i\|_{L^{p_i}(d\sigma_i)} <\infty; $$ \item[{\rm(b)}] For all $j=1,\ldots,n$ and for all $Q\in{\mathcal D}$, $$ \sum_{Q'\subset Q} K(Q')\sigma_j(Q') \prod_{\substack{i=1 \\ i\ne j}}^n
\left|\int_{Q'}f_i\,d\sigma_i\right| \le c_2 \sigma_j(Q)^{1/p_j} \prod_{\substack{i=1 \\ i\ne j}}^n
\|f_i\|_{L^{p_i}(d\sigma_i)} <\infty. $$ \end{itemize} \noindent Moreover, the least possible $c_1$ and $c_2$ are equivalent. \end{theorem}
Let the symmetric group $S_n$ be the set of all permutations of the set $\{1,\ldots,n\}$, that is, the set of all bijections from the set $\{1,\ldots,n\}$ to itself. Let $K:\,{\mathcal D}\to[0,\infty)$ be a~map and let $\sigma_i$, $i=1,\ldots,n$, be positive Borel measures on ${\mathbb R}^d$. Let $1<p_i<\infty$ and $\sum_{i=1}^n\frac1{p_i}<1$.
Let $\phi\in S_n$. Set \begin{align*} &\frac1{r^{\phi}_1}+\frac1{p_{\phi(1)}}=1, \\[2mm] &\frac1{r^{\phi}_2}+\frac1{p_{\phi(1)}}+\frac1{p_{\phi(2)}}=1, \\ &\qquad\vdots \\[2mm] &\frac1{r^{\phi}_{n-1}}+\sum_{i=1}^{n-1}\frac1{p_{\phi(i)}}=1, \\[2mm] &\frac1r+\sum_{i=1}^n\frac1{p_{\phi(i)}}=1. \end{align*} Let, for $Q\in{\mathcal D}$, $$ K^{\phi}_1(Q) := K(Q)\sigma_{\phi(1)}(Q) \left( \frac1{\sigma_{\phi(1)}(Q)} \sum_{Q'\subset Q} K(Q') \prod_{i=1}^n\sigma_{\phi(i)}(Q') \right)^{r^{\phi}_1-1}, $$ let $$ K^{\phi}_2(Q) := K^{\phi}_1(Q)\sigma_{\phi(2)}(Q) \left( \frac1{\sigma_{\phi(2)}(Q)} \sum_{Q'\subset Q} K^{\phi}_1(Q') \prod_{i=2}^n\sigma_{\phi(i)}(Q') \right)^{r^{\phi}_2/r^{\phi}_1-1} $$ and, inductively, for $j=3,\ldots,n-1$, let $$ K^{\phi}_j(Q) := K^{\phi}_{j-1}(Q)\sigma_{\phi(j)}(Q) \left( \frac1{\sigma_{\phi(j)}(Q)} \sum_{Q'\subset Q} K^{\phi}_{j-1}(Q') \prod_{i=j}^n\sigma_{\phi(i)}(Q') \right)^{r^{\phi}_j/r^{\phi}_{j-1}-1}. $$
\begin{theorem}\label{thm1.4} With the notation above, the following statements are equivalent:
\begin{itemize} \item[{\rm(a)}] The following $n$ linear embedding theorem holds: $$ \sum_{Q\in{\mathcal D}}
K(Q)\prod_{i=1}^n\left|\int_{Q}f_i\,d\sigma_i\right| \le c_1 \prod_{i=1}^n
\|f_i\|_{L^{p_i}(d\sigma_i)} <\infty; $$ \item[{\rm(b)}] For all $\phi\in S_n$, $$
\left\|\left( \sum_{Q\in{\mathcal D}} K^{\phi}_{n-1}(Q) 1_{Q} \right)^{1/r^{\phi}_{n-1}}
\right\|_{L^r(d\sigma_{\phi(n)})} \le c_2<\infty. $$ \end{itemize} \noindent Moreover, the least possible $c_1$ and $c_2$ are equivalent. \end{theorem}
Even though Theorems \ref{thm1.3} and \ref{thm1.4} both characterize the same $n$ linear embedding theorem, it seems that the characterizations are very different. In very recent paper \cite{HaHyLi}, Timo~S. H\"{a}nninen, Tuomas~P. Hyt\"{o}nen and Kangwei Li give a~unified approach saying \lq\lq sequential testing" characterization, when $n=2,3$. Especially, our Theorem \ref{thm1.4} with $n=3$ is obtained in \cite[Theorem 1.16]{HaHyLi}. (An alternative form of another unified characterization has been simultaneously obtained by Vuorinen \cite{Vu}.) In \cite{Ta2}, the author gives a~characterization of the trilinear embedding theorem interms of Theorem \ref{thm1.3} and Propositions \ref{prp1.1} and \ref{prp1.2}.
The letter $C$ will be used for constants that may change from one occurrence to another.
\section{Proof of the necessity}\label{sec2} In what follows we shall prove the necessity of theorems. The necessity of Theorem \ref{thm1.3}, that is, (b) follows from (a) at once if we substitute the test function $f_j=1_{Q}$. So, we shall verify the necessity of Theorem \ref{thm1.4}. We need a~lemma (cf. Lemma 2.1 in \cite{Ta1}).
\begin{lemma}\label{lem2.1} Let $\sigma$ be a~positive Borel measure on ${\mathbb R}^d$. Let $1<s<\infty$ and $\{\alpha_{Q}\}_{Q\in{\mathcal D}}\subset[0,\infty)$. Define, for $Q_0\in{\mathcal D}$, \begin{align*} A_1 &:= \int_{Q_0} \left(\sum_{Q\subset Q_0}\frac{\alpha_{Q}}{\sigma(Q)}1_{Q}\right)^s \,d\sigma, \\ A_2 &:= \sum_{Q\subset Q_0} \alpha_{Q}\left(\frac1{\sigma(Q)}\sum_{Q'\subset Q}\alpha_{Q'}\right)^{s-1}, \\ A_3 &:= \int_{Q_0} \sup_{Q\subset Q_0} \left(\frac{1_{Q}(x)}{\sigma(Q)}\sum_{Q'\subset Q}\alpha_{Q'}\right)^s \,d\sigma(x). \end{align*} Then $$ A_1\le c(s)A_2,\quad A_2\le c(s)^{\frac1{s-1}}A_3 \quad\text{and}\quad A_3\le (s')^sA_1. $$ Here, $$ c(s) := \begin{cases}\displaystyle s,\quad 1<s\le 2, \\ \displaystyle \left(s(s-1)\cdots(s-k)\right)^{\frac{s-1}{s-k-1}}, \quad 2<s<\infty, \end{cases} $$ where $k=\lceil s-2 \rceil$ is the smallest integer greater than $s-2$. \end{lemma}
We will use $\fint_{Q}f\,d\sigma$ to denote the integral average $\sigma(Q)^{-1}\int_{Q}f\,d\sigma$. The dyadic maximal operator $M_{{\mathcal D}}^{\sigma}$ is defined by $$ M_{{\mathcal D}}^{\sigma}f(x) := \sup_{Q\in{\mathcal D}}
\frac{1_{Q}(x)}{\sigma(Q)}\int_{Q}|f(y)|\,d\sigma(y). $$
Suppose that (a) of Theorem \ref{thm1.4}. Then, for $\phi\in S_n$, \begin{equation}\label{2.1} \sum_{Q\in{\mathcal D}} K(Q) \prod_{i=1}^n
\left|\int_{Q}f_{\phi(i)}\,d\sigma_{\phi(i)}\right| \le c_1 \prod_{i=1}^n
\|f_{\phi(i)}\|_{L^{p_{\phi(i)}}(d\sigma_{\phi(i)})}. \end{equation} Recall that $\frac1{r^{\phi}_1}+\frac1{p_{\phi(1)}}=1$. By duality, we see that $$ \int_{{\mathbb R}^d}\left( \sum_{Q\in{\mathcal D}} K(Q) \prod_{i=2}^n
\left|\int_{Q}f_{\phi(i)}\,d\sigma_{\phi(i)}\right| 1_{Q} \right)^{r^{\phi}_1} \,d\sigma_{\phi(1)} \le c_1^{r^{\phi}_1} \prod_{i=2}^n
\|f_{\phi(i)}\|_{L^{p_{\phi(i)}}(d\sigma_{\phi(i)})}^{r^{\phi}_1}, $$ which implies by Lemma \ref{lem2.1} \begin{align*} \lefteqn{ \sum_{Q\in{\mathcal D}} K(Q)\sigma_{\phi(1)}(Q) \prod_{i=2}^n
\left|\int_{Q}f_{\phi(i)}\,d\sigma_{\phi(i)}\right| }\\ &\quad\times\left[ \frac1{\sigma_{\phi(1)}(Q)} \sum_{Q'\subset Q} K(Q')\sigma_{\phi(1)}(Q') \prod_{i=2}^n
\left|\int_{Q'}f_{\phi(i)}\,d\sigma_{\phi(i)}\right| \right]^{r^{\phi}_1-1} \\ &\le C c_1^{r^{\phi}_1} \prod_{i=2}^n
\|f_{\phi(i)}\|_{L^{p_{\phi(i)}}(d\sigma_{\phi(i)})}^{r^{\phi}_1}. \end{align*} It follows from this inequality that \begin{align*} \lefteqn{ \sum_{Q\in{\mathcal D}} K^{\phi}_1(Q) \prod_{i=2}^n
\left|\int_{Q}g_{\phi(i)}\,d\sigma_{\phi(i)}\right| }\\ &= \sum_{Q\in{\mathcal D}} K(Q)\sigma_{\phi(1)}(Q) \prod_{i=2}^n
\left|\int_{Q}g_{\phi(i)}\,d\sigma_{\phi(i)}\right| \left[ \frac1{\sigma_{\phi(1)}(Q)} \sum_{Q'\subset Q} K(Q') \prod_{i=1}^n \sigma_{\phi(i)}(Q') \right]^{r^{\phi}_1-1} \\ &= \sum_{Q\in{\mathcal D}} K(Q)\sigma_{\phi(1)}(Q) \prod_{i=2}^n \sigma_{\phi(i)}(Q)
\left|\fint_{Q}g_{\phi(i)}\,d\sigma_{\phi(i)}\right|^{1/r^{\phi}_1} \\ &\quad\times \left[ \frac1{\sigma_{\phi(1)}(Q)} \sum_{Q'\subset Q} K(Q')\sigma_{\phi(1)}(Q') \prod_{i=2}^n \sigma_{\phi(i)}(Q')
\left|\fint_{Q}g_{\phi(i)}\,d\sigma_{\phi(i)}\right|^{1/r^{\phi}_1} \right]^{r^{\phi}_1-1} \\ &\le \sum_{Q\in{\mathcal D}} K(Q)\sigma_{\phi(1)}(Q) \prod_{i=2}^n \int_{Q} \left(M_{{\mathcal D}}^{\sigma_{\phi(i)}}g_{\phi(i)}\right)^{1/r^{\phi}_1} \,d\sigma_{\phi(i)} \\ &\quad\times \left[ \frac1{\sigma_{\phi(1)}(Q)} \sum_{Q'\subset Q} K(Q')\sigma_{\phi(1)}(Q') \prod_{i=2}^n \int_{Q'} \left(M_{{\mathcal D}}^{\sigma_{\phi(i)}}g_{\phi(i)}\right)^{1/r^{\phi}_1} \,d\sigma_{\phi(i)} \right]^{r^{\phi}_1-1} \\ &\le C c_1^{r^{\phi}_1} \prod_{i=2}^n
\|M_{{\mathcal D}}^{\sigma_{\phi(i)}}g_{\phi(i)}\|_{L^{p_{\phi(i)}/r^{\phi}_1}(d\sigma_{\phi(i)})} \\ &\le C c_1^{r^{\phi}_1} \prod_{i=2}^n
\|g_{\phi(i)}\|_{L^{p_{\phi(i)}/r^{\phi}_1}(d\sigma_{\phi(i)})}, \end{align*} where we have used the boundedness of dyadic maximal operators. Thus, we obtain \begin{equation}\label{2.2} \sum_{Q\in{\mathcal D}} K^{\phi}_1(Q) \prod_{i=2}^n
\left|\int_{Q}f_{\phi(i)}\,d\sigma_{\phi(i)}\right| \le C c_1^{r^{\phi}_1} \prod_{i=2}^n
\|f_{\phi(i)}\|_{L^{p_{\phi(i)}/r^{\phi}_1}(d\sigma_{\phi(i)})}. \end{equation} Notice that \begin{equation}\label{2.3} \begin{cases}\displaystyle \frac{r^{\phi}_{i-1}}{r^{\phi}_i} + \frac{r^{\phi}_{i-1}}{p_{\phi(i)}} =1,\quad i=2,\ldots,n-1, \\[4mm]\displaystyle \frac{r^{\phi}_{n-1}}{r} + \frac{r^{\phi}_{n-1}}{p_{\phi(n)}} =1. \end{cases} \end{equation} By the same manner as the above but starting from \eqref{2.2}, instead of \eqref{2.1}, and using \eqref{2.3} with $i=2$, we obtain $$ \sum_{Q\in{\mathcal D}} K^{\phi}_2(Q) \prod_{i=3}^n
\left|\int_{Q}f_{\phi(i)}\,d\sigma_{\phi(i)}\right| \le C c_1^{r^{\phi}_2} \prod_{i=3}^n
\|f_{\phi(i)}\|_{L^{p_{\phi(i)}/r^{\phi}_2}(d\sigma_{\phi(i)})}. $$ By being continued inductively until the $n-1$ step, we obtain $$ \sum_{Q\in{\mathcal D}} K^{\phi}_{n-1}(Q)
\left|\int_{Q}f_{\phi(n)}\,d\sigma_{\phi(n)}\right| \le C c_1^{r^{\phi}_{n-1}}
\|f_{\phi(n)}\|_{L^{p_{\phi(n)}/r^{\phi}_{n-1}}(d\sigma_{\phi(n)})}. $$ Notice that the last equation of \eqref{2.3}. Then by duality $$
\left\| \sum_{Q\in{\mathcal D}} K^{\phi}_{n-1}(Q) 1_{Q}
\right\|_{L^{r/r^{\phi}_{n-1}}(d\sigma_{\phi(n)})} \le C c_1^{r^{\phi}_{n-1}} $$ and, hence, $$
\left\|\left( \sum_{Q\in{\mathcal D}} K^{\phi}_{n-1}(Q) 1_{Q} \right)^{1/r^{\phi}_{n-1}}
\right\|_{L^r(d\sigma_{\phi(n)})} \le C c_1, $$ which completes the necessity of Theorem \ref{thm1.4}.
\section{Proof of the sufficiency}\label{sec3} In what follows we shall prove the sufficiency of theorems.
Let $Q_0\in{\mathcal D}$ be taken large enough and be fixed. We shall estimate the quantity \begin{equation}\label{3.1} \sum_{Q\subset Q_0} K(Q) \prod_{i=1}^n\left(\int_{Q}f_i\,d\sigma_i\right), \end{equation} where $f_i\in L^{p_i}(d\sigma_i)$ is nonnegative and is supported in $Q_0$. We define the collection of principal cubes ${\mathcal F}_i$ for the pair $(f_i,\sigma_i)$, $i=1,\ldots,n$. Namely, $$ {\mathcal F}_i:=\bigcup_{k=0}^{\infty}{\mathcal F}_i^k, $$ where ${\mathcal F}_i^0:=\{Q_0\}$, $$ {\mathcal F}_i^{k+1} := \bigcup_{F\in{\mathcal F}_i^k}ch_{{\mathcal F}_i}(F) $$ and $ch_{{\mathcal F}_i}(F)$ is defined by the set of all \lq\lq maximal" dyadic cubes $Q\subset F$ such that $$ \fint_{Q}f_i\,d\sigma_i > 2\fint_{F}f_i\,d\sigma_i. $$ Observe that \begin{align*} \lefteqn{ \sum_{F'\in ch_{{\mathcal F}_i}(F)}\sigma_i(F') }\\ &\le \left(2\fint_{F}f_i\,d\sigma_i\right)^{-1} \sum_{F'\in ch_{{\mathcal F}_i}(F)} \int_{F'}f_i\,d\sigma_i \\ &\le \left(2\fint_{F}f_i\,d\sigma_i\right)^{-1} \int_{F}f_i\,d\sigma_i = \frac{\sigma_i(F)}{2}, \end{align*} which implies \begin{equation}\label{3.2} \sigma_i(E_{{\mathcal F}_i}(F)) := \sigma_i\left(F\setminus\bigcup_{F'\in ch_{{\mathcal F}_i}(F)}F'\right) \ge \frac{\sigma_i(F)}{2}, \end{equation} where the sets $E_{{\mathcal F}_i}(F)$, $F\in{\mathcal F}_i$, are pairwise disjoint. We further define the stopping parents, for $Q\in{\mathcal D}$, $$ \begin{cases}\displaystyle \pi_{{\mathcal F}_i}(Q) := \min\{F\supset Q:\,F\in{\mathcal F}_i\}, \\\displaystyle \pi(Q) := \left(\pi_{{\mathcal F}_1}(Q),\ldots,\pi_{{\mathcal F}_n}(Q)\right). \end{cases} $$ Then we can rewrite the series in \eqref{3.1} as follows: $$ \sum_{Q\subset Q_0} = \sum_{(F_1,\ldots,F_n)\in({\mathcal F}_1,\ldots,{\mathcal F}_n)} \sum_{\substack{ Q: \\ \pi(Q)=(F_1,\ldots,F_n) }}. $$ We notice the elementary fact that, if $P,R\in{\mathcal D}$, then $P\cap R\in\{P,R,\emptyset\}$. This fact implies, if $\pi(Q)=(F_1,\ldots,F_n)$, then $$ Q\subset F_{\phi(1)}\subset\cdots\subset F_{\phi(n)} \quad\text{for some}\quad \phi\in S_n. $$ Thus, for fixed $\phi\in S_n$, we shall estimate \begin{equation}\label{3.3} \sum_{\substack{ (F_{\phi(i)})\in({\mathcal F}_{\phi(i)}): \\ F_{\phi(1)}\subset\cdots\subset F_{\phi(n)} }} \sum_{\substack{ Q: \\ \pi(Q)=(F_{\phi(i)}) }} K(Q)\prod_{i=1}^n\left(\int_{Q}f_{\phi(i)}\,d\sigma_{\phi(i)}\right). \end{equation}
\paragraph{{\bf Proof of (a) of Theorem \ref{thm1.3}.}} It follows that, for fixed $F_{\phi(n)}\in{\mathcal F}_{\phi(n)}$, \begin{align*} \lefteqn{ \sum_{F_{\phi(1)}\subset\cdots\subset F_{\phi(n)}} \sum_{\substack{ Q: \\ \pi(Q)=(F_{\phi(i)}) }} K(Q)\prod_{i=1}^n\left(\int_{Q}f_{\phi(i)}\,d\sigma_{\phi(i)}\right) }\\ &\le \left(2\fint_{F_{\phi(n)}}f_{\phi(n)}\,d\sigma_{\phi(n)}\right) \sum_{F_{\phi(1)}\subset\cdots\subset F_{\phi(n)}} \sum_{\substack{ Q: \\ \pi(Q)=(F_{\phi(i)}) }} K(Q)\sigma_{\phi(n)}(Q) \prod_{i=1}^{n-1}\left(\int_{Q}f_{\phi(i)}\,d\sigma_{\phi(i)}\right). \end{align*}
We need two observations. Suppose that $F_{\phi(1)}\subset\cdots\subset F_{\phi(n)}$ and $\pi(Q)=(F_{\phi(i)})$. Let $i=1,\ldots,n-1$. If $F'\in ch_{{\mathcal F}_{\phi(n)}}(F_{\phi(n)})$ satisfies $F'\subset Q$. Then \begin{equation}\label{3.4} \pi_{{\mathcal F}_{\phi(n)}}\left(\pi_{{\mathcal F}_{\phi(i)}}(F')\right) = \begin{cases}\displaystyle F_{\phi(n)}, \quad\text{when}\quad f'\notin{\mathcal F}_{\phi(i)}, \\\displaystyle F', \quad\text{when}\quad f'\in{\mathcal F}_{\phi(i)}. \end{cases} \end{equation} By this observation, we define $$ ch_{{\mathcal F}_{\phi(n)}}^{\phi(i)}(F_{\phi(n)}) := \{ F'\in ch_{{\mathcal F}_{\phi(n)}}(F_{\phi(n)}):\, \text{ $F'$ satisfies \eqref{3.4}} \}. $$ We further observe that, when $F'\in ch_{{\mathcal F}_{\phi(n)}}^{\phi(i)}(F_{\phi(n)})$, we can regard $f_{\phi(i)}$ as a~constant on $F'$ in the above integrals, that is, we can replace $f_{\phi(i)}$ by $f_{\phi(i)}^{F_{\phi(n)}}$ in the above integrals, where $$ f_{\phi(i)}^{F_{\phi(n)}} := f_{\phi(i)}1_{E_{{\mathcal F}_{\phi(n)}}(F_{\phi(n)})} + \sum_{F'\in ch_{{\mathcal F}_{\phi(n)}}^{\phi(i)}(F_{\phi(n)})} \left(\fint_{F'}f_{\phi(i)}\,d\sigma_{\phi(i)}\right) 1_{F'}. $$
It follows from (b) of Theorem \ref{thm1.3} that \begin{align*} \lefteqn{ \sum_{F_{\phi(1)}\subset\cdots\subset F_{\phi(n)}} \sum_{\substack{ Q: \\ \pi(Q)=(F_{\phi(i)}) }} K(Q)\sigma_{\phi(n)})(Q) \prod_{i=1}^{n-1} \left(\int_{Q}f_{\phi(i)}^{F_{\phi(n)}}\,d\sigma_{\phi(i)}\right) }\\ &\le c_2 \sigma_{\phi(n)}(F_{\phi(n)})^{1/p_{\phi(n)}} \prod_{i=1}^{n-1}
\|f_{\phi(i)}^{F_{\phi(n)}}\|_{L^{p_{\phi(i)}}(d\sigma_{\phi(i)})}. \end{align*} Thus, we obtain $$ \eqref{3.3} \le C c_2 \sum_{F_{\phi(n)}\in{\mathcal F}_{\phi(n)}} \prod_{i=1}^{n-1}
\|f_{\phi(i)}^{F_{\phi(n)}}\|_{L^{p_{\phi(i)}}(d\sigma_{\phi(i)})} \left(\fint_{F_{\phi(n)}}f_{\phi(n)}\,d\sigma_{\phi(n)}\right) \sigma_{\phi(n)}(F_{\phi(n)})^{1/p_{\phi(n)}}. $$ Since $\sum_{i=1}^n\frac1{p_{\phi(i)}}\ge 1$, we can select the auxiliary parameters $s_{\phi(i)}$, $i=1,\ldots,n-1$, that satisfy $$ \sum_{i=1}^{n-1}\frac1{s_{\phi(i)}} + \frac1{p_{\phi(n)}} =1 \quad\text{and}\quad 1<p_{\phi(i)}\le s_{\phi(i)}<\infty. $$ It follows from H\"{o}lder's inequality with exponents $s_{\phi(1)},\ldots,s_{\phi(n-1)},p_{\phi(n)}$ that \begin{align*} \eqref{3.3} &\le C c_2 \prod_{i=1}^{n-1} \left[ \sum_{F_{\phi(n)}\in{\mathcal F}_{\phi(n)}}
\|f_{\phi(i)}^{F_{\phi(n)}}\|_{L^{p_{\phi(i)}}(d\sigma_{\phi(i)})}^{s_{\phi(i)}} \right]^{1/s_{\phi(i)}} \\ &\quad\times \left[ \sum_{F_{\phi(n)}\in{\mathcal F}_{\phi(n)}} \left(\fint_{F_{\phi(n)}}f_{\phi(n)}\,d\sigma_{\phi(n)}\right)^{p_{\phi(n)}} \sigma_{\phi(n)}(F_{\phi(n)}) \right]^{1/p_{\phi(n)}} \\ &\le C c_2 \prod_{i=1}^{n-1} \left[ \sum_{F_{\phi(n)}\in{\mathcal F}_{\phi(n)}}
\|f_{\phi(i)}^{F_{\phi(n)}}\|_{L^{p_{\phi(i)}}(d\sigma_{\phi(i)})}^{p_{\phi(i)}} \right]^{1/p_{\phi(i)}} \\ &\quad\times \left[ \sum_{F_{\phi(n)}\in{\mathcal F}_{\phi(n)}} \left(\fint_{F_{\phi(n)}}f_{\phi(n)}\,d\sigma_{\phi(n)}\right)^{p_{\phi(n)}} \sigma_{\phi(n)}(F_{\phi(n)}) \right]^{1/p_{\phi(n)}} \\[5mm] &=: C c_2 (I_1)\times\cdots\times(I_n), \end{align*} where we have used
$\|\cdot\|_{l^{p_{\phi(i)}}}\ge\|\cdot\|_{l^{s_{\phi(i)}}}$.
For $(I_n)$, using $\sigma_{\phi(n)}(F_{\phi(n)})\le 2\sigma_{\phi(n)}(E_{{\mathcal F}_{\phi(n)}}(F_{\phi(n)}))$ (see \eqref{3.2}), the fact that $$ \fint_{F_{\phi(n)}}f_{\phi(n)}\,d\sigma_{\phi(n)} \le \inf_{y\in F_{\phi(n)}} M_{{\mathcal D}}^{\sigma_{\phi(n)}}f_{\phi(n)}(y) $$ and the disjointness of the sets $E_{{\mathcal F}_{\phi(n)}}(F_{\phi(n)})$, we have \begin{align*} (I_n) &\le C \left[ \sum_{F_{\phi(n)}\in{\mathcal F}_{\phi(n)}} \int_{E_{{\mathcal F}_{\phi(n)}}(F_{\phi(n)})} \left(M_{{\mathcal D}}^{\sigma_{\phi(n)}}f_{\phi(n)}\right)^{p_{\phi(n)}}\,d\sigma_{\phi(n)} \right]^{1/p_{\phi(n)}} \\ &\le C \left[\int_{Q_0}\left(M_{{\mathcal D}}^{\sigma_{\phi(n)}}f_{\phi(n)}\right)^{p_{\phi(n)}}\,d\sigma_{\phi(n)}\right]^{1/p_{\phi(n)}} \le C
\|f_{\phi(n)}\|_{L^{p_{\phi(n)}}(d\sigma_{\phi(n)})}. \end{align*} It remains to estimate $(I_i)$, $i=1,\ldots,n-1$. It follows that \begin{align*} (I_i)^{p_{\phi(i)}} &= \sum_{F_{\phi(n)}\in{\mathcal F}_{\phi(n)}} \int_{E_{{\mathcal F}_{\phi(n)}}(F_{\phi(n)})}f_{\phi(i)}^{p_{\phi(i)}}\,d\sigma_{\phi(i)} \\ &\quad + \sum_{F_{\phi(n)}\in{\mathcal F}_{\phi(n)}} \sum_{F'\in ch_{{\mathcal F}_{\phi(n)}}^{\phi(i))}(F_{\phi(n)})} \left(\fint_{F'}f_{\phi(i)}\,d\sigma_{\phi(i)}\right)^{p_{\phi(i)}} \sigma_{\phi(i)}(F'). \end{align*} By the pairwise disjointness of the sets $E_{{\mathcal F}_{\phi(n)}}(F_{\phi(n)})$, it is immediate that $$ \sum_{F_{\phi(n)}\in{\mathcal F}_{\phi(n)}} \int_{E_{{\mathcal F}_{\phi(n)}}(F_{\phi(n)})}f_{\phi(i)}^{p_{\phi(i)}}\,d\sigma_{\phi(i)} \le
\|f_{\phi(i)}\|_{L^{p_{\phi(i)}}(d\sigma_{\phi(i)})}^{p_{\phi(i)}}. $$ For the remaining double sum, there holds by the uniqueness of the parent \begin{align*} \lefteqn{ \sum_{F_{\phi(n)}\in{\mathcal F}_{\phi(n)}} \sum_{\substack{ F'\in ch_{{\mathcal F}_{\phi(n)}}(F_{\phi(n)}): \\ \text{$F'$ satisfies \eqref{3.4}} }} \left(\fint_{F'}f_{\phi(i)}\,d\sigma_{\phi(i)}\right)^{p_{\phi(i)}} \sigma_{\phi(i)}(F') }\\ &\le 2 \sum_{F_{\phi(n)}\in{\mathcal F}_{\phi(n)}} \sum_{\substack{ F\in{\mathcal F}_{\phi(i)}: \\ \pi_{{\mathcal F}_{\phi(n)}}(F)=F_{\phi(n)} }} \sum_{\substack{ F'\in ch_{{\mathcal F}_{\phi(n)}}(F_{\phi(n)}): \\ \pi_{{\mathcal F}_{\phi(i)}}(F')=F }} \left(\fint_{F'}f_{\phi(i)}\,d\sigma_{\phi(i)}\right)^{p_{\phi(i)}} \sigma_{\phi(i)}(F') \\ &\le 2 \sum_{F\in{\mathcal F}_{\phi(i)}} \left(2\fint_{F}f_{\phi(i)}\,d\sigma_{\phi(i)}\right)^{p_{\phi(i)}} \sigma_{\phi(i)}(F) \\ &\le C
\|M_{{\mathcal D}}^{\sigma_{\phi(i)}}f_{\phi(i)}\|_{L^{p_{\phi(i)}}(d\sigma_{\phi(i)})}^{p_{\phi(i)}} \le C
\|f_{\phi(i)}\|_{L^{p_{\phi(i)}}(d\sigma_{\phi(i)})}^{p_{\phi(i)}}. \end{align*} Altogether, we obtain $$ \eqref{3.3} \le C c_2 \prod_{i=1}^n
\|f_{\phi(i)}\|_{L^{p_{\phi(i)}}(d\sigma_{\phi(i)})}. $$ This yields (a) of Theorem \ref{thm1.3}.
\paragraph{{\bf Proof of (a) of Theorem \ref{thm1.4}.}} We shall estimate \eqref{3.3} by use of multinonlinear Wolff's potential. We first observe that if $F_{\phi(i)}\in{\mathcal F}_{\phi(i)}$, $i=1,\ldots,n$, satisfy $F_{\phi(1)}\subset\cdots\subset F_{\phi(n)}$ and, for some $Q\in{\mathcal D}$, $\pi(Q)=(F_{\phi(i)})$, then \begin{equation}\label{3.5} \pi_{{\mathcal F}_{\phi(j)}}(F_{\phi(i)}) = F_{\phi(j)} \quad\text{for all}\quad 1\le i<j\le n. \end{equation} Fix $F_{\phi(i)}\in{\mathcal F}_{\phi(i)}$, $i=1,\ldots,n$, that satisfy \eqref{3.5}. Then \begin{align*} \lefteqn{ \sum_{\substack{ Q: \\ \pi(Q)=(F_{\phi(i)}) }} K(Q)\prod_{i=1}^n\left(\int_{Q}f_{\phi(i)}\,d\sigma_{\phi(i)}\right) }\\ &\le \prod_{i=1}^n \left(2\fint_{F_{\phi(i)}}f_{\phi(i)}\,d\sigma_{\phi(i)}\right) \sum_{\substack{ Q: \\ \pi(Q)=(F_{\phi(i)}) }} K(Q)\prod_{i=1}^n\sigma_{\phi(i)}(Q). \end{align*} Recall that \begin{equation}\label{3.6} \begin{cases}\displaystyle \frac1{r^{\phi}_j}+\sum_{i=1}^j\frac1{p_{\phi(i)}}=1, \quad j=1,\ldots,n-1, \\[4mm]\displaystyle \frac1r+\sum_{i=1}^n\frac1{p_{\phi(i)}}=1. \end{cases} \end{equation} In the following estimates, $\sum_{F_{\phi(1)}}$ runs over all $F_{\phi(1)}\in{\mathcal F}_{\phi(1)}$ that satisfy \eqref{3.5} for fixed $F_{\phi(i)}\in{\mathcal F}_{\phi(i)}$, $i=2,\ldots,n$. \begin{align*} \lefteqn{ \sum_{F_{\phi(1)}} \left(\fint_{F_{\phi(1)}}f_{\phi(1)}\,d\sigma_{\phi(1)}\right) \sum_{\substack{ Q: \\ \pi(Q)=(F_{\phi(i)}) }} K(Q)\prod_{i=1}^n\sigma_{\phi(i)}(Q) }\\ &\le \sum_{F_{\phi(1)}} \left(\fint_{F_{\phi(1)}}f_{\phi(1)}\,d\sigma_{\phi(1)}\right) \sum_{Q\subset F_{\phi(1)}} K(Q)\prod_{i=1}^n\sigma_{\phi(i)}(Q) \\ &= \sum_{F_{\phi(1)}} \left(\fint_{F_{\phi(1)}}f_{\phi(1)}\,d\sigma_{\phi(1)}\right) \sigma_{\phi(1)}(F_{\phi(1)})^{1/p_{\phi(1)}} \\ &\quad\times \left( \fint_{F_{\phi(1)}} \left(\sum_{Q\subset F_{\phi(1)}} K(Q)\prod_{i=2}^n\sigma_{\phi(i)}(Q) 1_{Q} \right)\,d\sigma_{\phi(1)}\right) \sigma_{\phi(1)}(F_{\phi(1)})^{1/r^{\phi}_1}, \end{align*} where we have used \eqref{3.6} with $j=1$. By H\"{o}lder's inequality, we have further that \begin{align*} \le& \left[ \sum_{F_{\phi(1)}} \left(\fint_{F_{\phi(1)}}f_{\phi(1)}\,d\sigma_{\phi(1)}\right)^{p_{\phi(1)}} \sigma_{\phi(1)}(F_{\phi(1)}) \right]^{1/p_{\phi(1)}} \\ &\quad\times \left[ \sum_{F_{\phi(1)}} \left( \fint_{F_{\phi(1)}} \left( \sum_{Q\subset F_{\phi(1)}} K(Q)\prod_{i=2}^n\sigma_{\phi(i)}(Q) 1_{Q} \right)\,d\sigma_{\phi(1)} \right)^{r^{\phi}_1} \sigma_{\phi(1)}(F_{\phi(1)}) \right]^{1/r^{\phi}_1}. \end{align*} By the same way as the estimate of $(I_n)$, we see that the last term is majorized by $$ C\left(\int_{F_{\phi(2)}} \left(\sum_{Q\subset F_{\phi(2)}} K(Q)\prod_{i=2}^n\sigma_{\phi(i)}(Q) 1_{Q} \right)^{r^{\phi}_1} \,d\sigma_{\phi(1)} \right)^{1/r^{\phi}_1}. $$ By Lemma \ref{lem2.1}, we have further that $$ \le C \left(\sum_{Q\subset F_{\phi(2)}} K^{\phi}_1(Q)\prod_{i=2}^n\sigma_{\phi(i)}(Q) \right)^{1/r^{\phi}_1}. $$ By \eqref{2.3}, we notice that \begin{equation}\label{3.7} \frac1{r^{\phi}_i} + \frac1{p_{\phi(i)}} = \frac1{r^{\phi}_{i-1}}, \quad i=2,\ldots,n-1. \end{equation} In the following estimates, $\sum_{F_{\phi(2)}}$ runs over all $F_{\phi(2)}\in{\mathcal F}_{\phi(2)}$ that satisfy, for fixed $F_{\phi(i)}\in{\mathcal F}_{\phi(i)}$, $i=3,\ldots,n$, \begin{equation}\label{3.8} \pi_{{\mathcal F}_{\phi(j)}}(F_{\phi(i)}) = F_{\phi(j)} \quad\text{for all}\quad 2\le i<j\le n. \end{equation} There holds \begin{align*} \lefteqn{ \sum_{F_{\phi(2)}} \left(\fint_{F_{\phi(2)}}f_{\phi(2)}\,d\sigma_{\phi(2)}\right) \times \left(\sum_{Q\subset F_{\phi(2)}} K^{\phi}_1(Q)\prod_{i=2}^n\sigma_{\phi(i)}(Q) \right)^{1/r^{\phi}_1} }\\ &\quad\times \left(\sum_{F_{\phi(1)}} \left(\fint_{F_{\phi(1)}}f_{\phi(1)}\,d\sigma_{\phi(1)}\right)^{p_{\phi(1)}} \sigma_{\phi(1)}(F_{\phi(1)}) \right)^{1/p_{\phi(1)}} \\ &= \sum_{F_{\phi(2)}} \left(\fint_{F_{\phi(2)}}f_{\phi(2)}\,d\sigma_{\phi(2)}\right) \sigma_{\phi(2)}(F_{\phi(2)})^{1/p_{\phi(2)}} \\ &\quad\times \left(\fint_{F_{\phi(2)}} \left(\sum_{Q\subset F_{\phi(2)}} K^{\phi}_1(Q)\prod_{i=3}^n\sigma_{\phi(i)}(Q) 1_{Q} \right)\,d\sigma_{\phi(2)} \right)^{1/r^{\phi}_1} \sigma_{\phi(2)}(F_{\phi(2)})^{1/r^{\phi}_2} \\ &\quad\times \left(\sum_{F_{\phi(1)}} \left(\fint_{F_{\phi(1)}}f_{\phi(1)}\,d\sigma_{\phi(1)}\right)^{p_{\phi(1)}} \sigma_{\phi(1)}(F_{\phi(1)}) \right)^{1/p_{\phi(1)}}, \end{align*} where we have used \eqref{3.7} with $i=2$. Recall that \eqref{3.6} with $j=2$. Then H\"{o}lder's inequality gives \begin{align*} \le & \left[\sum_{F_{\phi(2)}} \left(\fint_{F_{\phi(2)}}f_{\phi(2)}\,d\sigma_{\phi(2)}\right)^{p_{\phi(2)}} \sigma_{\phi(2)}(F_{\phi(2)}) \right]^{1/p_{\phi(2)}} \\ &\times \left[ \sum_{F_{\phi(2)}} \sum_{F_{\phi(1)}} \left(\fint_{F_{\phi(1)}}f_{\phi(1)}\,d\sigma_{\phi(1)}\right)^{p_{\phi(1)}} \sigma_{\phi(1)}(F_{\phi(1)}) \right]^{1/p_{\phi(1)}} \\ &\times \left[ \sum_{F_{\phi(2)}} \left( \fint_{F_{\phi(2)}} \left(\sum_{Q\subset F_{\phi(2)}} K^{\phi}_1(Q)\prod_{i=3}^n\sigma_{\phi(i)}(Q) 1_{Q} \right)\,d\sigma_{\phi(2)} \right)^{r^{\phi}_2/r^{\phi}_1} \sigma_{\phi(2)}(F_{\phi(2)}) \right]^{1/r^{\phi}_2}. \end{align*} The last term is majorized by $$ C\left(\int_{F_{\phi(3)}} \left(\sum_{Q\subset F_{\phi(3)}} K^{\phi}_1(Q)\prod_{i=3}^n\sigma_{\phi(i)}(Q) 1_{Q} \right)^{r^{\phi}_2/r^{\phi}_1} \,d\sigma_{\phi(2)} \right)^{1/r^{\phi}_2}. $$ By Lemma \ref{lem2.1}, we have further that $$ \le C \left(\sum_{Q\subset F_{\phi(3)}} K^{\phi}_2(Q)\prod_{i=3}^n\sigma_{\phi(i)}(Q) \right)^{1/r^{\phi}_2}. $$ By being continued inductively until the $n-1$ step, we obtain \begin{align*} \eqref{3.3} &\le C \left[\sum_{F_{\phi(n)}} \left(\fint_{F_{\phi(n)}}f_{\phi(n)}\,d\sigma_{\phi(n)}\right)^{p_{\phi(n)}} \sigma_{\phi(n)}(F_{\phi(n)}) \right]^{1/p_{\phi(n)}} \\ &\quad\times \left[ \sum_{F_{\phi(n)}} \sum_{F_{\phi(n-1)}} \left(\fint_{F_{\phi(n-1)}}f_{\phi(n-1)}\,d\sigma_{\phi(n-1)}\right)^{p_{\phi(n-1)}} \sigma_{\phi(n-1)}(F_{\phi(n-1)}) \right]^{1/p_{\phi(n-1)}} \\ &\quad\times\vdots \\ &\quad\times \left[ \sum_{F_{\phi(n)}} \sum_{F_{\phi(n-1)}} \cdots \sum_{F_{\phi(1)}} \left(\fint_{F_{\phi(1)}}f_{\phi(1)}\,d\sigma_{\phi(1)}\right)^{p_{\phi(1)}} \sigma_{\phi(1)}(F_{\phi(1)}) \right]^{1/p_{\phi(1)}} \\ &\quad\times \left[ \sum_{F_{\phi(n)}} \left(\fint_{F_{\phi(n)}} \left(\sum_{Q\subset F_{\phi(n)}} K^{\phi}_{n-1}(Q) 1_{Q} \right) \,d\sigma_{\phi(n)} \right)^{r/r^{\phi}_{n-1}} \sigma_{\phi(n)}(F_{\phi(n)}) \right]^{1/r}, \end{align*} where $\sum_{F_{\phi(n)}}$ runs over all $F_{\phi(n)}\in{\mathcal F}_{\phi(n)}$ and $\sum_{F_{\phi(k)}}$, $k=3,\ldots,n-1$, runs over all $F_{\phi(k)}\in{\mathcal F}_{\phi(k)}$ that satisfy, for fixed $F_{\phi(i)}$, $i=k+1,\ldots,n$, \begin{equation}\label{3.9} \pi_{{\mathcal F}_{\phi(j)}}(F_{\phi(i)}) = F_{\phi(j)} \quad\text{for all}\quad k\le i<j\le n. \end{equation} The last term is majorized by $$ C\left(\int_{Q_0} \left(\sum_{Q\subset Q_0} K^{\phi}_{n-1}(Q) 1_{Q} \right)^{r/r^{\phi}_{n-1}} \,d\sigma_{\phi(n)} \right)^{1/r} \le c_2. $$ It follows from \eqref{3.5}, \eqref{3.8}, \eqref{3.9} and the uniqueness of the parents that \begin{align*} \lefteqn{ \left[ \sum_{F_{\phi(n)}} \sum_{F_{\phi(n-1)}} \cdots \sum_{F_{\phi(i)}} \left(\fint_{F_{\phi(i)}}f_{\phi(i)}\,d\sigma_{\phi(i)}\right)^{p_{\phi(i)}} \sigma_{\phi(i)}(F_{\phi(i)}) \right]^{1/p_{\phi(i)}} }\\ &\le \left[ \sum_{F_{\phi(n)}} \sum_{\substack{ F_{\phi(i)}\in{\mathcal F}_{\phi(i)}: \\ \pi_{{\mathcal F}_{\phi(n)}}(F_{\phi(i)})=F_{\phi(n)} }} \left(\fint_{F_{\phi(i)}}f_{\phi(i)}\,d\sigma_{\phi(i)}\right)^{p_{\phi(i)}} \sigma_{\phi(i)}(F_{\phi(i)}) \right]^{1/p_{\phi(i)}} \\ &= \left[ \sum_{F_{\phi(i)}\in{\mathcal F}_{\phi(i)}} \left(\fint_{F_{\phi(i)}}f_{\phi(i)}\,d\sigma_{\phi(i)}\right)^{p_{\phi(i)}} \sigma_{\phi(i)}(F_{\phi(i)}) \right]^{1/p_{\phi(i)}} \\[5mm] &\le C
\|f_{\phi(i)}\|_{L^{p_{\phi(i)}}(d\sigma_{\phi(i)})}. \end{align*} Altogether, we obtain $$ \eqref{3.3} \le C c_2 \prod_{i=1}^n
\|f_{\phi(i)}\|_{L^{p_{\phi(i)}}(d\sigma_{\phi(i)})}. $$ This yields (a) of Theorem \ref{thm1.4}.
\end{document} |
\begin{document}
\title{\LARGE \bf
Convergence Rate Estimates for Consensus over Random Graphs
} \thispagestyle{empty} \pagestyle{empty}
\begin{abstract} Multi-agent coordination algorithms with randomized interactions have seen use in a variety of settings in the multi-agent systems literature. In some cases, these algorithms can be random by design, as in a gossip-like algorithm, and in other cases they are random due to external factors, as in the case of intermittent communications. Targeting both of these scenarios, we present novel convergence rate estimates for consensus problems solved over random graphs.
Established results provide asymptotic convergence in this setting, and we provide estimates of the rate of convergence in two forms. First, we estimate decreases in a quadratic Lyapunov function over time to bound how quickly the agents' disagreement decays, and second we bound the probability of being at least a given distance from the point of agreement. Both estimates rely on (approximately) computing eigenvalues of the expected matrix exponential of a random graph's Laplacian, which we do explicitly in terms of the network's size and edge probability, without assuming that any relationship between them holds. Simulation results are provided to support the theoretical developments made. \end{abstract}
\section{Introduction}
Distributed agreement, often broadly referred to as the \emph{consensus} problem, is a canonical problem in distributed coordination and has received attention in diverse fields such as physics \cite{vicsek95}, signal processing \cite{schizas09}, robotics \cite{ren07},
power systems \cite{nozari14}, and communications \cite{mehyar07}. The goal in such problems is to drive all agents in a network to a common final state. A key feature of consensus problems is their distributed nature; consensus is typically carried out across a network of agents in which each agent communicates with some other agents, though generally not all of them. The wide range of fields which study distributed agreement has given rise to corresponding diversity among consensus problem formulations, and a number of variants of consensus have been studied in the literature.
In this paper, we derive convergence rates for consensus over random graphs, studied previously in \cite{hatano05} where asymptotic convergence was shown. In some cases, the motivation for representing a communication network using random graphs comes from agents using an interaction protocol that is randomized by design, such as in a gossip-like algorithm \cite{boyd06}. In other cases, unreliable communications due to poor channel quality, interference, and other factors can be effectively represented by a random communication graph \cite{mesbahi10}, and the work here applies to each of these scenarios. This problem formulation has each agent communicating with a random collection of other agents determined by a random graph. Each agent moves toward the average of its neighbors' states, then a new graph is generated, and each agent moves toward the average state of its new neighbors, with this process repeating until convergence is achieved.
We consider networks of a fixed size, and we examine consensus over random graphs generated by the Erd\H{o}s-R\'{e}nyi model \cite{erdos59}, in which each possible edge in a graph is present with a fixed probability and is independent of all other edges. The Erd\H{o}s-R\'{e}nyi model is used because it accurately captures the behavior both of networks with intermittent and unreliable communications \cite{hatano05} and the behavior of some variants of synchronous gossip algorithms \cite{boyd06}. Our approach consists of estimating the expected matrix exponential of the Laplacian associated with a random graph and then computing its eigenvalues in terms of the graph's size and edge probability. It is shown that the second-largest eigenvalue of this expected matrix exponential is key in estimating convergence rates, and the main contribution of this paper lies in explicitly computing this eigenvalue and using it to derive two novel rates of convergence. The first estimates the rate of decrease in a quadratic Lyapunov function to bound the rate at which agents approach agreement in their state values. The second bounds the probability of the agents' states being at least a given distance away from agreeing on a common state. A well-known related work is \cite{touri14}, which develops a general theory for convergence of consensus over broad classes of time-varying graphs, in addition to providing general-purpose convergence results. The emphasis in this paper is on consensus using the Erd\H{o}s-R\'{e}nyi model due to its prevalence in some work on multi-agent systems \cite{mesbahi10}. Our work differs from that in \cite{touri14} because, by fixing the choice of graph model, we are able to derive concrete convergence rates and implement them numerically.
Both estimates rely in some form on computing eigenvalues of random matrices, and there is an established literature dedicated to doing so \cite{diaconis94,tao12}, including for eigenvalues of random symmetric matrices \cite{alon02,furedi81,wigner58}, and eigenvalues of random graphs' Laplacians specifically \cite{coja07}. A common approach to estimating or computing the eigenvalues of a random symmetric matrix is to let the size of the matrix get arbitrarily large \cite{diaconis94,furedi81,juhasz82,wigner58}. In graph theory, this approach corresponds to letting the number of nodes in a graph grow arbitrarily large and has seen use in spectral graph theory because it allows one to rigorously state results that hold for almost all graphs \cite{bollobas01}.
In the study of multi-agent systems, one is often interested in networks of a fixed, small size, such as in \cite{mehyar07,nozari14,werner06}, and this makes results for asymptotically large networks less applicable in some cases. As a result, we are motivated to derive convergence results in terms of a network's size without taking it to grow asymptotically large. In addition, there are a number of graph theoretic results that estimate eigenvalues of random graphs' Laplacians when edge probabilities bear some known relationship to the size of the network, e.g., \cite{krivelevich03,tran13}. In cases where a random graph is used to model unreliable communications, there is no guarantee that such a relationship will hold as the quality of communication channels can depend upon a wide variety of external factors. Therefore, we allow edge probabilities to take values independent of the network's size and state our results in terms of both a network's size and its edge probability without making assumptions about either. Our results also do not require the common assumption that unions of agents' communication graphs are connected across finite intervals of a prescribed length \cite{jadbabaie03}, which can be difficult to enforce and verify in some practical settings.
The rest of the paper is organized as follows. Section II reviews the necessary background on consensus and graph theory, including that on random graphs. Next, Section III establishes asymptotic convergence of consensus over random graphs. Then Section IV presents our main results on the rate of convergence of consensus over random graphs. Section V then presents simulation results to verify the convergence rates we present. Finally, Section VI concludes the paper.
\section{Review of Graph Theory and Consensus} In this section, we review the basic elements of graph theory required for the remainder of the paper. We introduce unweighted, undirected graphs, then review the consensus problem, and finally review random graphs.
\subsection{Basic Graph Theory} All graphs in this paper are assumed to be unweighted and undirected. Such graphs are defined by pairwise relationships over a finite set of nodes or vertices. Suppose that a graph has a set $V$ of $n$ vertices, with $n \in \mathbb{N}$, and index these vertices over the set $\{1, \ldots, n\}$. We define the \emph{edge set} \begin{equation} E \subset V \times V, \end{equation} and say there is an edge between nodes $i$ and $j$ if $(i, j) \in E$. A graph $G$ is then formally defined as the $2$-tuple $G = (V, E)$. Throughout this paper, all edges are undirected and an edge $(i, j) \in E$ is not distinguished from the edge $(j, i) \in E$. We do not allow self loops and therefore $(i, i) \not\in E$ for all $i$ and all graphs $G$.
The degree associated with node $i$ is defined as the total number of edges that connect node $i$ to some other node. Using $|\cdot|$ to denote the cardinality of a set and using $d_i$ to denote the degree of node $i$, we have \begin{equation}
d_i = \big|\{j \mid (i, j) \in E\}\big|. \end{equation} The $n \times n$ degree matrix associated with a graph $G$ is then defined as \begin{equation}
D(G) = \left(\begin{array}{cccc} d_1 & 0 & \cdots & 0 \\
0 & d_2 & \cdots & 0 \\
\vdots & \vdots & \ddots & \vdots \\
0 & 0 & \cdots & d_n \end{array}\right), \end{equation} which will be written simply as $D$ when the graph $G$ is understood.
The $n \times n$ adjacency matrix associated with $G$, denoted $A(G)$, is defined element-wise as \begin{equation}
a_{i,j} = \begin{cases} 1 & (i, j) \in E \\
0 & \textnormal{otherwise} \end{cases} \end{equation} where $a_{i,j}$ is the $(i,j)$th entry in $A(G)$. Note that, because $(i, j) \in E$ implies $(j, i) \in E$, $A(G)$ is a symmetric matrix. In addition, the absence of self loops results in $A(G)$ having zeroes on its main diagonal for all graphs $G$. We will simply write $A$ when $G$ is clear from context.
Using $D(G)$ and $A(G)$, we define the Laplacian associated with a graph $G$ as \begin{equation} L(G) = D(G) - A(G), \end{equation} and note that the Laplacian of any undirected, unweighted graph is a symmetric, positive semi-definite matrix \cite{godsil01}. In particular, the Laplacians of such graphs have all non-negative eigenvalues. Letting $\lambda_i(M)$ denote the $i^{th}$ smallest eigenvalue of a matrix $M$, for any graph $G$ it is known that $\lambda_1\big(L(G)\big) = 0$ and that $\mathds{1} = (1, 1, \ldots, 1)^T$ is an eigenvector associated with $\lambda_1$ \cite{mesbahi10}, i.e., that $\mathds{1}$ is in the nullspace of the Laplacian of any undirected, unweighted graph. For any graph Laplacian $L$ of size $n \times n$, we find \begin{equation} \label{eq:eiglist} 0 = \lambda_1 \leq \lambda_2 \leq \cdots \leq \lambda_n. \end{equation}
A graph $G$ is said to be \emph{connected} if, for any two nodes $i$ and $j$ in the graph, there exists a sequence of edges one can traverse to travel from node $i$ to node $j$. A seminal result in graph theory provides that $G$ is connected if and only if $\lambda_2 > 0$ \cite{fiedler73}.
\subsection{The Consensus Protocol} \label{ss:reviewcon} A canonical problem in multi-agent control is that of consensus. Consensus consists of having a collection of agents, e.g., robots or mobile sensor nodes, agree on a common value in a distributed way. The term distributed refers to the fact that each agent in a network can only communicate with some other agents in the network, but in general not all other agents. Suppose each agent has a scalar state\footnote{All results in this paper are easily extended to the case of agents having vectors of states by replacing $L$ in the consensus protocol in Equation~\eqref{eq:con1} with $I \otimes L$, where $\otimes$ denotes the Kronecker product of two matrices. We focus on the scalar case to simplify issues of dimensionality.}, with agent $i$'s state denoted $x_i$, and assemble these into the ensemble state vector \begin{equation} x = \left(\begin{array}{c} x_1 \\ \vdots \\ x_n \end{array}\right)^T \in \mathbb{R}^n. \end{equation}
If one represents the agents' communications using a graph $G$ (where edges connect those agents that exchange information), then the continuous-time consensus protocol \cite{olfati-saber04} takes the form \begin{equation} \label{eq:con1} \dot{x} = -Lx, \end{equation} where $L$ is the Laplacian of the graph $G$.
The following well-studied theorem from \cite{olfati-s03} establishes convergence of the consensus protocol in continuous time when $G$ is connected. In it, the symbol \begin{equation} \bar{x}(0) := \frac{1}{n}\sum_{i=1}^{n} x_i(0) \end{equation} is used to denote the centroid of the agents' initial states (which is a scalar) and $\bar{x}(0)\mathds{1}$ is used to denote the vector in $\mathbb{R}^n$ whose entries are all equal to $\bar{x}(0)$. \begin{theorem} \label{thm:consensus} Let $G$ be a connected graph and let an initial ensemble state $x(0)$ be given. Then the consensus protocol \begin{equation} \label{eq:consensus} \dot{x} = -Lx \end{equation} asymptotically converges element-wise to the centroid of the agents' initial states, i.e., $x(t) \to \bar{x}(0)\mathds{1}$. In addition, the rate at which $x(t)$ approaches $\bar{x}(0)\mathds{1}$ is governed by $\lambda_2$. \end{theorem} \emph{Proof}: See \cite{olfati-s03}.
$\blacksquare$
Because the goal in consensus is for a team of nodes to reach a common value, one often refers to the agreement subspace of the agents, defined below as was done in \cite{hatano05}.
\begin{definition} (\cite{hatano05}) \label{def:agree} The agreement subspace is defined as the set of all points at which all agents have the same state value, i.e., $x_i = x_j$ for all $i$ and $j$. Formally, it is defined as \begin{equation} \mathcal{A} = \textnormal{span}\{\mathds{1}\}, \end{equation} where $\mathds{1}$ is the vector of all ones in $\mathbb{R}^n$.
$\triangle$ \end{definition}
In Section~\ref{sec:conv}, we will show asymptotic convergence to $\mathcal{A}$ for consensus over random graphs by showing that the agents' disagreement goes to zero. Next, we introduce the random graph model used in the remainder of the paper.
\subsection{Random Graphs} A common model for random graphs is the Erd\H{o}s-R\'{e}nyi model, originally published in \cite{erdos59}, and we use it here because it accurately captures the behavior of two cases of interest. First, some algorithms are randomized by design, such as gossip algorithms \cite{boyd06}, and Erd\H{o}s-R\'{e}nyi graphs can model the behavior of such algorithms in some cases. Second, members of a network sometimes share information over communication channels which are intermittently lost and regained, and this behavior is well-modeled by Erd\H{o}s-R\'{e}nyi graphs \cite{hatano05}. This model takes two parameters to generate random graphs: a number of nodes $n \in \mathbb{N}$ and an edge probability\footnote{The cases $p = 0$ and $p = 1$ provide edgeless graphs and complete graphs, respectively, and are omitted because their behavior is deterministic.} $p \in (0, 1)$. The Erd\H{o}s-R\'{e}nyi model generates graphs on $n$ nodes whose edge sets contain each possible edge with probability $p$, independent of all other edges. Formally, for each admissible $i$ and $j$, we have \begin{equation} \mathbb{P}[(i, j) \in E] = p. \end{equation} An alternative characterization that we use later can be stated in terms of the elements of the adjacency matrix of a random graph: for $n$ nodes and edge probability $p$, we find \begin{equation} \mathbb{P}[a_{i,j} = 1] = \mathbb{P}[a_{j,i} = 1] = p \textnormal{ and } \mathbb{P}[a_{i,j} = 0] = \mathbb{P}[a_{j,i} = 0] = 1 - p. \end{equation} Thus each $a_{i,j}$ is a Bernoulli random variable.
We use $\mathcal{G}(n, p)$ to denote the sample space of all possible random graphs generated by the Erd\H{o}s-R\'{e}nyi model on $n$ nodes with edge probability $p$, and we use $\mathcal{L}(n, p)$ to denote the set of Laplacians of all such graphs.
One approach to analyzing random graphs that has seen use in the graph theory literature consists of taking the limit as $n$ goes to infinity, with the benefit of this approach being the ability to rigorously determine which properties hold for almost all graphs. However, motivated by the study of multi-agent systems, we are interested in networks of fixed size and therefore develop our results in terms of a fixed value of $n$. Toward doing so, we show asymptotic convergence of consensus over random graphs in the next section.
\section{Consensus over Random Graphs} \label{sec:conv} In this section we examine consensus where the agents' communication graph at each timestep is a random graph. We then show asymptotic convergence of this update law in order to help establish the role of convergence rates in its analysis. The theoretical results of this section are not new, but are presented to contextualize the remainder of the paper. This section closely follows the approach of \cite{hatano05} where these results were originally published.
\subsection{Time-Sampled Consensus} \label{ss:3a}
We assume that all communication graphs hold constant for some positive amount of time $\delta > 0$ and we seek to examine the evolution of the system's state under this condition. We follow the problem setup of \cite{hatano05} in which states generated by the consensus protocol in Equation~\eqref{eq:consensus} are sampled in time by defining the state $z(k)$ as \begin{equation} z(k) = x(k\delta), \end{equation} where the communication graph among the agents is assumed to hold constant over the interval $[k\delta, (k+1)\delta)$. We note that this problem is distinct from consensus performed in discrete time; the problem we consider analyzes samples of the continuous-time state $x(t)$ rather than having states actually evolve in discrete time as in \cite{jadbabaie03}. All agents still execute the protocol $\dot{x} = -Lx$ and, as in \cite{hatano05}, $z(k)$ is used only as a theoretical tool for analyzing the behavior of the continuous state $x(t)$ over time as the agents' communication graphs change. It is because these communication graphs hold constant over $[k\delta, (k+1)\delta)$ that we analyze the agents' states only at these points in time.
At each timestep $k$, the system will generally have a different graph Laplacian than it had at time $k-1$. We denote the communication graph active at time $k$ by $G_k$ and denote its Laplacian by $L_k$. The solution to Equation~\eqref{eq:consensus} is $x(t) = e^{-Lt}x(0)$, and for any times $t_1$ and $t_2$ with $t_1 < t_2$ we find \begin{equation} x(t_2) = e^{-L(t_2 - t_1)}x(t_1). \end{equation} Setting $t_2 = (k+1)\delta$ and $t_1 = k\delta$ then gives \begin{equation} \label{eq:xddef} x\big((k+1)\delta\big) = e^{-\delta L_k}x(k\delta) \end{equation} because the graph $G_k$ is constant over the interval $[k\delta, (k+1)\delta)$. Equation \eqref{eq:xddef} itself is equal to \begin{equation} \label{eq:discon} z(k+1) = e^{-\delta L_k}z(k) \end{equation} and this is the protocol that we analyze in the remainder of the paper.
We emphasize that the agents' states still evolve in continuous time and that we analyze the continuous time signal $x(t)$ by analyzing samples taken every $\delta$ seconds. In Section \ref{sec:rates} we provide convergence rates for Equation \eqref{eq:discon}. Toward doing so, we next show that Equation~\eqref{eq:discon} converges asymptotically to $\mathcal{A}$.
\subsection{Establishing Asymptotic Convergence} \label{ss:asymptotic} We will assess convergence of consensus over random graphs by showing convergence to the agreement set $\mathcal{A}$, defined in Definition~\ref{def:agree}. Define the distance from a point $y \in \mathbb{R}^n$ to $\mathcal{A}$ as \begin{equation} \label{eq:distdef}
\textnormal{dist}(y, \mathcal{A}) = \inf_{z \in \mathcal{A}} \|z - y\|_2. \end{equation} As in \cite{hatano05}, define the orthogonal complement of the agreement subspace as \begin{equation} \A^{\perp} = \{x \mid x^Ta = 0 \textnormal{ for all } a \in \mathcal{A}\}. \end{equation} In particular $x^T\mathds{1} = 0$ for all $x \in \A^{\perp}$. Then define the Euclidean projection of $z(k)$ onto $\A^{\perp}$ as \begin{equation} \hat{z}(k) = \Pi_{\A^{\perp}}[z(k)] = \left(I - \frac{1}{n}J\right)z(k), \end{equation} where $J$ is the $n \times n$ matrix of ones, and where $\hat{z}(k)$ captures the disagreement among agents by excluding the part of $z(k)$ that lies in $\mathcal{A}$.
It was noted in Theorem~\ref{thm:consensus} that the consensus protocol over static graphs converges element-wise to centroid of the agents' initial states. The consensus protocol in Equation \eqref{eq:discon} also converges to the centroid of the agents' initial states, despite being run over random graphs. To see why, first observe that the consensus protocol in Equation \eqref{eq:discon} has converged when $z(k+1) - z(k) = 0$, i.e., when \begin{equation} e^{-\delta L_k}z(k) - z(k) = (e^{-\delta L_k} - I)z(k) = 0 \label{eq:usenull}. \end{equation} Because $\mathcal{A}$ is a subspace, we can decompose $z(k)$ into two parts according to \begin{equation} z(k) = \Pi_{\mathcal{A}}[z(k)] + \Pi_{\A^{\perp}}[z(k)] = \Pi_{\mathcal{A}}[z(k)] + \hat{z}(k). \end{equation} Substituting this decomposition into Equation \eqref{eq:usenull} we find \begin{align}
(e^{-\delta L_k} - I)z(k) &= \Big(-\delta L_k + \frac{1}{2}\delta^2L_k^2 - \frac{1}{3!}\delta^3L_k^3 + \cdots\Big)\big(\Pi_{\mathcal{A}}[z(k)] + \hat{z}(k)\big) \\
&= \left(-\delta L_k + \frac{1}{2}\delta^2L_k^2 - \frac{1}{3!}\delta^3L_k^3 + \cdots\right)\hat{z}(k) \\
&= 0 \label{eq:usenull2}, \end{align} where we have used the fact that $\Pi_{\mathcal{A}}[z(k)]$ is in the nullspace of all $L_k \in \mathcal{L}(n, p)$ by definition.
For Equation \eqref{eq:discon} to have converged, one must therefore have $\hat{z}(k)$ in the nullspace of all $L_k \in \mathcal{L}(n, p)$. Since $\hat{z}(k) \perp \mathcal{A}$, we find $\hat{z}(k) = 0$. Then if Equation \eqref{eq:discon} has converged at time $k$, it has converged to $\Pi_{\mathcal{A}}[z(k)]$.
To see that $\Pi_{\mathcal{A}}[z(k)] = \bar{z}(0)$, consider $\Pi_{\mathcal{A}}[z(1)]$. We have \begin{equation} \Pi_{\mathcal{A}}[z(1)] = z(1) - \hat{z}(1) = z(1) - \left(I - \frac{1}{n}J\right)z(1) = \bar{z}(1)\mathds{1}. \end{equation} Examining $\bar{z}(1)$, we see that \begin{align} \bar{z}(1) &= \frac{1}{n}\mathds{1}^Tz(1) = \frac{1}{n}\mathds{1}^Te^{-\delta L_0}z(0) \\
&= \frac{1}{n}\mathds{1}^T\left(I - \delta L_0 + \frac{\delta^2 L_0^2}{2} - \cdots\right)z(0) \\
&= \frac{1}{n}\mathds{1}^Tz(0) = \bar{z}(0), \end{align} where we have used $\mathds{1}^TL_0 = (L_0^T\mathds{1})^T = (L_0\mathds{1})^T = 0$ because $L_0$ is symmetric and $\mathds{1}$ is in the nullspace of $L_0$. A simple inductive argument shows that $\bar{z}(k) = \bar{z}(0)$ for all $k$.
Therefore, the distance from $z(k)$ to $\mathcal{A}$ is equal to that from $z(k)$ to $\bar{z}(0)\mathds{1}$. Noting that $\bar{z}(0)\mathds{1}$, whose entries are all $\bar{z}(0)$, is equal to $\frac{1}{n}\mathds{1}^Tz(0)\mathds{1}$, we have \begin{align}
\textnormal{dist}\big(z(k), \mathcal{A}\big)^2 &= \left\|z(k) - \frac{1}{n}\mathds{1}^Tz(0)\mathds{1}\right\|^2 \\
&= z(k)^Tz(k) - n\bar{z}(0)^2 \\
&= \frac{1}{n}z(k)^T\hat{L} z(k), \label{eq:vnice} \end{align} where we have used $\hat{L} := nI - J$, with $I$ the $n \times n$ identity matrix and $J$ the $n \times n$ matrix of ones.
In light of the form of $\textnormal{dist}\big(z(k), \mathcal{A}\big)$, we will show convergence of the protocol in Equation~\eqref{eq:discon} using the quadratic Lyapunov function \begin{equation} \label{eq:Vdef} V\big(z(k)\big) = \frac{1}{n}z(k)^T\hat{L} z(k). \end{equation}
We also state the following definition which will be used to characterize stochastic convergence. \begin{definition} \label{def:wp1} A random sequence $\{y(k)\}$ in $\mathbb{R}^n$ converges to $y \in \mathbb{R}^n$ with probability $1$ if, for every $\epsilon > 0$, \begin{equation}
\mathbb{P}\left[\sup_{N \leq k < \infty} \|y(k) - y\|_2 \geq \epsilon\right] \to 0 \textnormal{ as } N \to \infty. \end{equation}
$\triangle$ \end{definition}
We now have the following theorem that proves asymptotic convergence of the consensus protocol in Equation~\eqref{eq:discon}, due originally to \cite{hatano05}.
\begin{theorem} \label{thm:asymptotic} Fix a number of nodes $n \in \mathbb{N}$ and an edge probability $p \in (0, 1)$, and let $z(0)$ be given. The consensus protocol \begin{equation} \label{eq:thmcon} z(k+1) = e^{-\delta L_k}z(k), \end{equation} where $L_k \in \mathcal{L}(n, p)$ for all $k$, converges to $\mathcal{A}$ with probability $1$. In addition, \begin{equation} \hat{z}(k)^T\mathbb{E}[e^{-2\delta L_k} - I]\hat{z}(k) \to 0 \end{equation} with probability $1$ and \begin{equation}
\mathbb{P}\left[\sup_{N \leq k < \infty} \|\hat{z}(k)\|_2^2 \geq \gamma\right] \leq \frac{\hat{z}(0)^T\hat{z}(0)}{\gamma}\lambda_{n-1}\big(\mathbb{E}[e^{-2\delta L_k}]\big)^N \end{equation} where $\lambda_{n-1}(\cdot)$ denotes the second-largest eigenvalue of a matrix. \end{theorem} \emph{Proof:} See \cite{hatano05}.
$\blacksquare$
More details on this theorem can be found in \cite{hatano05}, where its proof originally appeared, and in \cite{mesbahi10}. Having established asymptotic convergence, we derive rates of convergence for consensus over random graphs in the next section.
\section{Convergence Rates for Consensus over Random Graphs} \label{sec:rates} Section~\ref{sec:conv} showed that consensus over random graphs converges asymptotically, and in this section we derive our main results on two rates of convergence for consensus over random graphs. Section \ref{ss:crates} is based on work in \cite{hatano05}, though the form of convergence rate we derive is different from the one derived there. The remainder of the section then presents our main results on novel, explicit convergence rates.
\subsection{Convergence Rates} \label{ss:crates} We now highlight the utility of convergence rates in the context of Theorem \ref{thm:asymptotic}. To do so, we examine the evolution of the disagreement among agents. Using the same Lyapunov function as in Equation \eqref{eq:Vdef}, we find \begin{align} \mathbb{E}[V\big(\hat{z}(k+1)\big) - V\big(\hat{z}(k)\big) \mid \hat{z}(k)] &= \hat{z}(k)^T\mathbb{E}[e^{-2\delta L_k} - I]\hat{z}(k) \\
&= \hat{z}(k)^T\mathbb{E}[e^{-2\delta L_k}]\hat{z}(k) - \hat{z}(k)^T\hat{z}(k). \end{align} By definition, $\hat{z}(k)$ is orthogonal to $\mathds{1}$. $\mathds{1}$ is also the eigenvector associated with the largest eigenvalue of $e^{-2\delta L_k}$ for all $L_k \in \mathcal{L}(n, p)$ (because the eigenvalues of $e^{-2\delta L_k}$ are $e^{-2\delta\lambda_i}$ for each $\lambda_i$ in Equation \eqref{eq:eiglist}). Then we find that \begin{equation}
\hat{z}(k)^T\mathbb{E}\big[e^{-2\delta L_k}\big]\hat{z}(k) \leq \lambda_{n-1}\left(\mathbb{E}\big[e^{-2\delta L_k}\big]\right)\|\hat{z}(k)\|_2^2, \end{equation} where $\lambda_{n-1}(M)$ denotes the second largest eigenvalue of an $n \times n$ matrix $M$. Consequently, we have \begin{equation} \label{eq:lamn-1}
\mathbb{E}[V\big(\hat{z}(k+1)\big) - V\big(\hat{z}(k)\big) \mid \hat{z}(k)] \leq \big(\lambda_{n-1}(\mathbb{E}[e^{-2\delta L_k}]) - 1\big)\|\hat{z}(k)\|_2^2, \end{equation} and by Theorem~\ref{thm:asymptotic} we have \begin{equation} \label{eq:probout}
\mathbb{P}\left[\sup_{N \leq k < \infty} \|\hat{z}(k)\|_2^2 \geq \gamma\right] \leq \frac{\hat{z}(0)^T\hat{z}(0)}{\gamma}\lambda_{n-1}\big(\mathbb{E}[e^{-2\delta L_k}]\big)^N. \end{equation}
Therefore, both convergence rates depend upon $\lambda_{n-1}(\mathbb{E}[e^{-2\delta L_k}])$, and we compute it next.
\subsection{Computing $\lambda_{n-1}\big(\mathbb{E}[e^{-2\delta L_k}]\big)$} We have the following lemma concerning eigenvalues of a matrix of the form $a I + b (J - I)$, which we will use below. \begin{lemma} \label{lem:ab} A matrix $M$ of the form $a I + b (J - I)$, namely \begin{equation} M = \left(\begin{array}{cccc} a & b & \cdots & b \\
b & a & \cdots & b \\
\vdots & \vdots & \ddots & \vdots \\
b & b & \cdots & a \end{array}\right) \in \mathbb{R}^{n \times n}, \end{equation}
has $a + (n-1)b$ as an eigenvalue with multiplicity one and $a - b$ as an eigenvalue with multiplicity $n - 1$. \end{lemma} \emph{Proof:} We proceed using a series of row operations that will preserve the characteristic polynomial of $M$. We see that \begin{equation}
|M - \lambda I| = \left|\left(\begin{array}{cccc} a - \lambda & b & \cdots & b \\
b & a - \lambda & \cdots & b \\
\vdots & \vdots & \ddots & \vdots \\
b & b & \cdots & a - \lambda \end{array}\right)\right|. \end{equation} Next, we add rows $2$ through $n$ to row $1$, giving \begin{equation}
|M - \lambda I| = \big(a + (n-1)b - \lambda\big)\left|\left(\begin{array}{cccc} 1 & 1 & \cdots & 1 \\
b & a - \lambda & \cdots & b \\
\vdots & \vdots & \ddots & \vdots \\
b & b & \cdots & a - \lambda \end{array}\right)\right|. \end{equation}
Subtracting $b$ times row $1$ from each other row, we find \begin{equation}
|M - \lambda I| = \big(a + (n-1)b - \lambda\big)\left|\left(\begin{array}{cccc} 1 & 1 & \cdots & 1 \\
0 & a - b - \lambda & \cdots & 0 \\
\vdots & \vdots & \ddots & \vdots \\
0 & 0 & \cdots & a - b - \lambda \end{array}\right)\right|, \end{equation} where the matrix on the right-hand side is upper-triangular. The determinant on the right-hand side is then the product of the diagonal entries of that matrix, resulting in \begin{equation}
|M - \lambda I| = (a + (n-1)b - \lambda)(a - b - \lambda)^{n-1}, \end{equation} whose roots are indeed $a + (n-1)b$, with multiplicity $1$, and $a - b$ with multiplicity $n-1$.
$\blacksquare$
We now derive the expected value of the first four powers of a random graph's Laplacian, and below we will use these results to approximate the Laplacian's expected matrix exponential. As above, we use the notation $\hat{L} = nI - J$.
\begin{lemma} \label{lem:lpow} Let a number of nodes $n \in \mathbb{N}$ and edge probability $p \in (0, 1)$ be given. The Laplacian $L \in \mathcal{L}(n, p)$ of a graph $G \in \mathcal{G}(n, p)$ satisfies \begin{align} \mathbb{E}[L] &= p\Lhat \\ \mathbb{E}[L^2] &= \left[(n-2)p^2 + 2p\right]\Lhat \\ \mathbb{E}[L^3] &= \left[(n-2)(n-4)p^3 + 6(n-2)p^2 + 4p\right]\Lhat \\ \mathbb{E}[L^4] &= \big[(n-7)(n-3)(n-2)p^4 + 6(2n-7)(n-2)p^3 + 25(n-2)p^2 + 8p\big]\Lhat. \end{align} \end{lemma} \emph{Proof sketch:} We sketch the proof to avoid exposition on many tedious computations and instead elaborate on the core arguments used to derive the above results.
The general form of graph Laplacian for $G \in \mathcal{G}(n, p)$ is \begin{equation} \left(\begin{array}{cccc}
\sum_{\substack{j=1 \\ j \neq 1}}^{n}a_{1, j} & -a_{1, 2} & \cdots & -a_{1, n} \\
-a_{1, 2} & \sum_{\substack{j=1 \\ j \neq 2}}^{n}a_{2, j} & \cdots & -a_{2, n} \\
\vdots & \vdots & \ddots & \vdots \\
-a_{1, n} & -a_{2, n} & \cdots & \sum_{\substack{j=1 \\ j \neq n}}^{n}a_{n, j} \end{array}\right), \end{equation} where each term $a_{i, j}$ is a Bernoulli random variable with expectation equal to $p$. The off-diagonal entries of $L$ have $\mathbb{E}[L_{ij}] = \mathbb{E}[-a_{i,j}] = -p$, while linearity of $\mathbb{E}[\cdot]$ gives $\mathbb{E}[L_{ii}] = (n-1)p$ for diagonal entries of $L$. From this we find $\mathbb{E}[L] = (n-1)pI - p(J - I) = p\hat{L}$.
Computing the expectation of the entries of $L^2$ requires one to consider two cases. The diagonal entries of $L^2$ are formed by the product of row $i$ of $L$ with column $i$ of $L$ (and by symmetry of $L$ these are identical), while the off-diagonal entries result from the product of row $i$ and column $j$ of $L$ (which are not identical when $i \neq j$). It is important to note that $a_{i, j}^2 = a_{i, j}$ because $a_{i, j}$ is a Bernoulli random variable. As a result, when computing expectations one finds that $\mathbb{E}[a_{i,j}^k] = p$ for all $k \in \mathbb{N}$, while products of $\ell$ distinct off-diagonal entries of $A$ have expectation equal to $p^\ell$. In the case of $\mathbb{E}[L^2]$, a diagonal entry takes the form \begin{align} \mathbb{E}\left[(L^2)_{ii}\right] &= \mathbb{E}\left[\sum_{\substack{p = 1 \\ p \neq i}}^{n} a_{i,p}^2 + \left(\sum_{\substack{q = 1 \\ q \neq i}}^{n} a_{i,q}\right)^2\right] \\
&= (n-1)(n-2)p^2 + 2(n-1)p, \end{align} while an off-diagonal entry takes the form \begin{align} \mathbb{E}[(L^2)_{ij}] &= \mathbb{E}\left[\sum_{\substack{k=1 \\ k \neq i, j}}^{n} a_{i,k}a_{k,j}\right] - \mathbb{E}\left[a_{i,j}\sum_{\substack{p = 1 \\ p \neq i}}^{n} a_{i,p}\right] - \mathbb{E}\left[a_{i,j}\sum_{\substack{q = 1 \\ q \neq i}}^{n} a_{q,j}\right] \\
&= -(n-2)p^2 - 2p. \end{align} Then we find \begin{align} \mathbb{E}[L^2] &= \big((n-1)(n-2)p^2 + 2(n-1)p\big)I + \big({-(n-2)}p^2 - 2p\big)(J - I) \\
&= \big((n-2)p^2 + 2p\big)\hat{L}, \end{align} as above.
Computing the general form of $L^3$ is done by multiplying the general form of $L^2$ by that of $L$ and the general form of $L^4$ is found by squaring the general form of $L^2$. Having found these two general forms, one follows the above strategy for computing their expected values: first replace $a_{i, j}^k$ with $a_{i, j}$ for all $k > 1$ and then compute the expectation of the products of $\ell$ distinct off-diagonal entries of $A$ as $p^{\ell}$, resulting in the above.
$\blacksquare$
We use the first four powers of $L$ because that is all that is required for accurate convergence rate estimates, as will be shown in Section~\ref{sec:simulation}. One way in which this accuracy is attained is through the choice of $\delta$. Many choices of $\delta$ are possible and we choose $\delta = 1/n$, for several reasons. First, it was assumed in Section \ref{ss:3a} that the communication graph of the system is constant between samples, i.e., that $G_k$ does not change over the interval $[k\delta, (k+1)\delta)$. As a network grows, the number of possible edges does too and thus a larger network has more ways in which its communication topology may change. As a result, a larger network should use shorter sampling times and $\delta$ should decrease as $n$ increases. The choice of $\delta = 1/n$ provides a simple means of enforcing this condition.
Second, the use of $\delta = 1/n$ is also partly inspired by the same choice made in \cite{jadbabaie03} for discrete-time consensus where it is a necessary condition for stability; though not necessary for stability in the current paper, we make the same choice
to help retain the same broad applicability of the results in \cite{jadbabaie03}. Using this choice of $\delta$, we now give the approximate value of $\lambda_{n-1}(\mathbb{E}[e^{-2\delta L_k}])$ in terms of $n$ and $p$.
\begin{theorem} \label{thm:lambda} Suppose $\delta = 1/n$ and define \begin{align} \kappa_1 &:= p \\ \kappa_2 &:= (n-2)p^2 + 2p \\ \kappa_3 &:= (n-2)(n-4)p^3 + 6(n-2)p^2 + 4p \\ \kappa_4 &:= (n-7)(n-3)(n-2)p^4 + 6(2n-7)(n-2)p^3 + 25(n-2)p^2 + 8p \\ \mu(n, p) &:= -2\frac{\kappa_1}{n} + 2\frac{\kappa_2}{n^2} - \frac{4}{3}\frac{\kappa_3}{n^3} + \frac{2}{3}\frac{\kappa_4}{n^4}. \end{align} Then \begin{equation} E[e^{-2\delta L_k}] \approx I + \mu(n, p)\Lhat, \end{equation} and consequently we have \begin{equation} \lambda_{n-1}(\mathbb{E}[e^{-2\delta L_k}]) \approx 1 + n\mu(n, p). \end{equation} \end{theorem} \emph{Proof:} Taylor expanding the matrix exponential, we find \begin{equation} \label{eq:maineq1} \mathbb{E}[e^{-2\delta L_k}] \approx I - 2\delta\mathbb{E}[L_k] + 2\delta^2\mathbb{E}[L_k^2] - \frac{4}{3}\delta^3\mathbb{E}[L_k^3] + \frac{2}{3}\delta^4\mathbb{E}[L_k^4], \end{equation} where the exact matrix exponential is well-approximated by this truncation in part due to the aforementioned choice of $\delta = 1/n$. Substituting $\delta = 1/n$ and the results of Lemma~\ref{lem:lpow} into Equation~\eqref{eq:maineq1} gives \begin{align} \label{eq:maineq2} \mathbb{E}[e^{-2\delta L_k}] &\approx I + \left(-2\frac{\kappa_1}{n} + 2\frac{\kappa_2}{n^2} - \frac{4}{3}\frac{\kappa_3}{n^3} + \frac{2}{3}\frac{\kappa_4}{n^4}\right)\hat{L} \\
&= I + \mu(n, p)\hat{L}, \end{align} which establishes the first part of the theorem.
Next, we use the definition of $\hat{L}$ as $nI - J$ to find \begin{equation} I + \mu(n, p) \hat{L} = \big(1 + (n-1)\mu(n, p)\big)I - \mu(n, p)\big(J - I\big). \end{equation} Using Lemma~\ref{lem:ab} with $a = 1 + (n-1)\mu(n, p)$ and $b = -\mu(n, p)$, we find that the largest eigenvalue of $\mathbb{E}[e^{-2\delta L_k}]$ is $1$, and the second largest through smallest eigenvalues of $\mathbb{E}[e^{-2\delta L_k}]$ are all approximately $1 + n\mu(n, p)$. In particular, $\lambda_{n-1}\big(\mathbb{E}[e^{-2\delta L_k}]\big) \approx 1 + n\mu(n, p)$, as desired.
$\blacksquare$
\subsection{Explicit Convergence Rates for Consensus over Random Graphs} We now present our unified main convergence rates for consensus over random graphs, stated in terms of network size $n$ and edge probability $p$.
\begin{figure}
\caption{A simulation run showing the expected decrease in $V$ (solid line) and its theoretical upper bound (dashed line) as in Equation~\protect\eqref{eq:sim1}. Here we see that the upper bound indeed holds across the whole time horizon, with the dashed line always above the solid line. In addition, the upper bound becomes more accurate across iterations, with it accurately predicting when the expected decrease in $V$ is near zero. }
\label{fig:vdiff}
\end{figure}
\begin{theorem} \label{thm:main} Let a network size $n \in \mathbb{N}$ and an edge probability $p \in (0, 1)$ be given, and let $\mu(n, p)$ be as defined in Theorem~\ref{thm:lambda}. For sampling constant $\delta = 1/n$ and $\gamma > 0$ we have \begin{equation} \label{eq:mainprob}
\mathbb{P}\left[\sup_{N \leq k < \infty} \|\hat{z}(k)\|_2^2 \geq \gamma\right] \leq \frac{\hat{z}(0)^T\hat{z}(0)}{\gamma}\big(1 + n\mu(n, p)\big)^N \end{equation} for all $N \in \mathbb{N}$. In addition, the expected decrease in the Lyapunov function $V(z) = \frac{1}{n}z^T\hat{L} z$ from time $k$ to time $k+1$ is bounded according to
\begin{equation} \label{eq:nmu}
\mathbb{E}\big[V\big(\hat{z}(k+1)\big) - V\big(\hat{z}(k)\big) \mid \hat{z}(k)\big] \leq n\mu(n, p)\|\hat{z}(k)\|_2^2. \end{equation} \end{theorem} \emph{Proof:} This follows from Equations~\eqref{eq:lamn-1} and \eqref{eq:probout} and Theorem~\ref{thm:lambda}.
$\blacksquare$
\begin{remark} In truncating the Taylor expansion of $\mathbb{E}[e^{-2\delta L_k}]$, some level of error in inevitably introduced into the value of $\lambda_{n-1}\big(\mathbb{E}[e^{-2\delta L_k}]\big)$ that results. By truncating after the fourth term, the error in this case is on the order of $\frac{2^5}{5!}\delta^5\mathbb{E}[L_k^5]$. For the choice of $\delta = 1/n$, this gives error on the order of $2^5/5!$. This error can be further mitigated by other choices of $\delta$, though the choice of $1/n$ will suffice in many cases. In general, the approximations we make are more accurate for smaller values of $p$ because such values cause higher-order terms in the expansion of $\mathbb{E}[e^{-2\delta L_k}]$ to be dominated by the lower-order terms we include in our approximations.
$\triangle$ \end{remark}
The appeal of using these convergence rate estimates together is that one need only compute the constant $\mu(n, p)$ and then both estimates can be used. Both provide information at each timestep because they rely on the current iteration count and can therefore be applied in real time. One can also use Equation \eqref{eq:mainprob} for a range of values of $\gamma$ to obtain the probability of being contained in each member of a family of super-level sets, letting one associate probabilities with all points in state space.
These two estimates also provide information about both individual consensus runs and families of consensus runs. Specifically, Equation \eqref{eq:nmu} applies to single consensus runs and evolves the current expected decrease in $V$ based upon $\hat{z}(k)$, providing a rate estimate specific to that run. On the other hand, Equation \eqref{eq:mainprob} applies to all runs starting from a given initial condition, giving information about how often we should expect one trajectory out of a family to be at least some distance from the point of convergence. Furthermore, they are complementary in that Equation \eqref{eq:nmu} provides an upper bound on the rate of decrease in $V$ and is optimistic in the sense that it over-estimates the expected decrease in disagreement in the system. On the other hand, Equation \eqref{eq:mainprob} over-estimates the probability of the agents' disagreement being a certain size, and is therefore pessimistic.
Taken together, these two convergence rate estimates enable one to probe the behavior of any consensus problem over random graphs, regardless of network size $n$, edge probability $p$, or initial condition $z(0)$, and provide quantitative data on such problems while requiring only the computation of $\mu(n, p)$. In the next section we present numerical results that verify both bounds presented in Theorem~\ref{thm:main}.
\begin{figure}
\caption{A simulation run showing the probability of having
$\|\hat{z}(k)\|_2^2 \geq 3$. The empirical probability is plotted as the solid line, while the upper bound is shown as the dashed line. We see that the upper bound indeed holds and, despite relying only on expected values, provides a close enough approximation to the empirical probability to be useful in a variety of settings. }
\label{fig:prob}
\end{figure}
\section{Simulation Results} \label{sec:simulation}
In this section we present numerical results to support the results in Theorem~\ref{thm:main}. We simulate consensus over random graphs and first numerically examine the expected decrease in $V$, and second bound the probability of being at least some distance away from the point of agreement.
\subsection{Estimating Decreases in Disagreement} We now present simulation results to verify the upper bound on the expected decrease in the Lyapunov function $V(z) = \frac{1}{n}z^T\hat{L} z$ as defined in Equation~\eqref{eq:Vdef}. The consensus problem we ran consists of $n = 50$ agents and edge probability $p = 0.03$. All agents had two states and were initialized to be evenly spaced along a circle of radius $100$ centered on the origin. In this case, to estimate the rate of convergence using Equation~\eqref{eq:nmu}, we compute $n\mu(n, p) = -0.0561$, from which we find \begin{equation} \label{eq:sim1}
\mathbb{E}\big[V\big(\hat{z}(k+1)\big) - V\big(\hat{z}(k)\big) \mid \hat{z}(k)\big] \leq -0.0561\|\hat{z}(k)\|_2^2. \end{equation}
To validate Equation~\eqref{eq:sim1}, a single consensus run was simulated. At each timestep $k$, the value of $\mathbb{E}\big[V\big(\hat{z}(k+1)\big) - V\big(\hat{z}(k)\big) \mid \hat{z}(k)\big]$ was computed numerically by fixing $\hat{z}(k)$, generating $1,000$ random graphs to compute $\tilde{z}(k+1) = e^{-2\delta L(G)}\hat{z}(k)$ for each graph $G$ generated this way, and then computing $V\big(\tilde{z}(k+1)\big) - V\big(\hat{z}(k)\big) \mid \hat{z}(k)$ for each $G$. These values were then averaged to numerically determine $\mathbb{E}\big[V\big(\hat{z}(k+1)\big) - V\big(\hat{z}(k)\big) \mid \hat{z}(k)\big]$ before the algorithm proceeded to run and computed $\hat{z}(k+1)$.
The results of this simulation run are shown in Figure~\ref{fig:vdiff}, where the right-hand side of Equation~\eqref{eq:sim1} is shown as a dashed line and the left-hand side of Equation~\eqref{eq:sim1} is shown as a solid line; though $1,000$ timesteps of consensus were run, only the first $100$ are shown because the two lines are indistinguishable and approximately zero beyond this point. Figure~\ref{fig:vdiff} shows that indeed the upper bound on the decreases in $V$ from Theorem~\ref{thm:main} holds because the dashed line is always above the solid line. Furthermore, as the iteration count increases, the upper bound becomes more accurate, meaning that we not only have an upper bound on the rate of decrease of $V$, but also that this upper bound can accurately predict when decreases in $V$ go to zero, thereby accurately predicting when consensus is achieved.
\subsection{Bounding the Probability of Being away from $\bar{z}(0)\mathds{1}$} We now present a consensus problem to verify the upper bound on the probability that the disagreement among agents will be at least a certain amount after a fixed point in time, stated in Theorem~\ref{thm:main}. In particular, for $n = 10$ agents and edge probability $p = 0.01$, $1,000$ trials were run to find experimentally the value of
$\mathbb{P}\big[\sup_{N \leq k < \infty}\|\hat{z}(k)\|_2^2 \geq \gamma\big]$
for each $N \in \{1, \ldots, 1,000\}$, where we set $\gamma = 3$. All trials were initialized with the agents spaced equally along a circle of radius $100$ whose center was at the origin. The results of these numerical experiments are shown in Figure~\ref{fig:prob} in the solid line, while the theoretical upper bound
$\frac{\hat{z}(0)^T\hat{z}(0)}{\gamma}\big(1 + n\mu(n, p)\big)^N$
is shown in Figure~\ref{fig:prob} as the dashed line.
Figure~\ref{fig:prob} shows that the probability bound in Theorem~\ref{thm:main} indeed holds because the dashed line is always aligned with or above the solid line. In addition, we see that the upper bound's graph over time stays close to that of the empirical probability, indicating that, despite relying only on expected values, the bound on the probability of being at least some distance from consensus provides a useful estimate of the actual probability, enabling one to make predictions about the magnitude of disagreement in a network over time.
\section{Conclusion} Explicit convergence rate bounds were presented for consensus over random graphs. A key feature was that convergence rate estimates were given in terms of the network size and edge probability without making any assumptions about either. Eigenvalues of the expected exponential of random graphs' Laplacians were computed and used to derive approximate convergence rate bounds. Numerical results confirmed that these results accurately capture the behavior of consensus over random graphs.
\section*{Acknowledgments} The authors would like to thank Professor Daniel Spielman for his helpful comments on this work and for steering us to the relevant literature on spectral graph theory.
{}
\end{document} |
\begin{document}
\author{Adrien Dubouloz, David R. Finston, and Imad Jaradat} \address{Adrien Dubouloz\\ CNRS\\ Institut de Math\'{e}matiques de Bourgogne\\ Universit\'{e} de Bourgogne\\ 9 Avenue Alain Savary\\ BP 47870\\ 21078 Dijon Cedex\\ France} \email{Adrien.Dubouloz@u-bourgogne.fr} \address{David R. Finston\\ Mathematics Department\\ Brooklyn College, CUNY\\ 2900 Bedford Avenue\\ Brooklyn, NY 11210} \email{dfinston@brooklyn.cuny.edu} \address{ Imad Jaradat\\ Department of Mathematical Sciences\\ New Mexico State University\\ Las Cruces, New Mexico 88003} \email{imad\_jar@nmsu.edu} \thanks{Research supported in part by NSF Grant OISE-0936691 and ANR Grant "BirPol" ANR-11-JS01-004-01.}
\title{Proper triangular $\mathbb{G}_{a}$-actions on $\mathbb{A}^{4}$ are translations} \begin{abstract} We describe the structure of geometric quotients for proper locally triangulable $\mathbb{G}_{a}$-actions on locally trivial $\mathbb{A}^{3}$-bundles over a n\oe therian normal base scheme $X$ defined over a field of characteristic $0$. In the case where $\dim X=1$, we show in particular that every such action is a translation with geometric quotient isomorphic to the total space of a vector bundle of rank $2$ over $X$. As a consequence, every proper triangulable $\mathbb{G}_{a}$-action on the affine four space $\mathbb{A}_{k}^{4}$ over a field of characteristic $0$ is a translation with geometric quotient isomorphic to $\mathbb{A}_{k}^{3}$. \end{abstract} \maketitle
\section*{Introduction}
The study of algebraic actions of the additive group $\mathbb{G}_{a}=\mathbb{G}_{a,\mathbb{C}}$ on complex affine spaces $\mathbb{A}^{n}=\mathbb{A}_{\mathbb{C}}^{n}$ has a long history which began in 1968 with a pioneering result of Rentschler \cite{Ren68} who established that every such action on the plane $\mathbb{A}^{2}$ is triangular in a suitable polynomial coordinate system. Consequently, every fixed point free $\mathbb{G}_{a}$-action on $\mathbb{A}^{2}$ is a translation, in the sense that the geometric quotient $\mathbb{A}^{2}/\mathbb{G}_{a}$ is isomorphic to $\mathbb{A}^{1}$ and that $\mathbb{A}^{2}$ is equivariantly isomorphic to $\mathbb{A}^{2}/\mathbb{G}_{a}\times\mathbb{G}_{a}$ where $\mathbb{G}_{a}$ acts by translations on the second factor.
Arbitrary $\mathbb{G}_{a}$-actions turn out to be no longer triangulable in higher dimensions \cite{Bass84}. But the question whether a fixed point free $\mathbb{G}_{a}$-action on $\mathbb{A}^{3}$ is a translation or not was settled affirmatively, first for triangulable actions by Snow \cite{Snow88} in 1988, then by Deveney and the second author \cite{DevFin94a} in 1994 under the additional assumption that the action is proper and then in general by Kaliman \cite{Kal04} in 2004. The argument for triangulable actions depends on their explicit form in an appropriate coordinate system which is used to check that the algebraic quotient $\pi:\mathbb{A}^{3}\rightarrow\mathbb{A}^{3}/\!/\mathbb{G}_{a}={\rm Spec}(\Gamma(\mathbb{A}^{3},\mathcal{O}_{\mathbb{A}^{3}})^{\mathbb{G}_{a}})$ is a geometric quotient and that $\mathbb{A}^{3}/\!/\mathbb{G}_{a}$ is isomorphic to $\mathbb{A}^{2}$. For proper actions, the properness implies that the geometric quotient $\mathbb{A}^{3}/\mathbb{G}_{a}$, which a priori only exists as an algebraic space, is separated whence a scheme by virtue of Chow's Lemma. This means equivalently that the $\mathbb{G}_{a}$-action is not only locally equivariantly trivial in the \'etale topology but in fact locally trivial in the Zariski topology, i.e. that $\mathbb{A}^{3}$ is covered by invariant Zariski affine open subsets of the from $V_{i}=U_{i}\times\mathbb{G}_{a}$ on which $\mathbb{G}_{a}$ acts by translations on the second factor. Since $\mathbb{A}^{3}$ is factorial, the open subsets $V_{i}$ can even be chosen to be principal, which implies in turn that $\mathbb{A}^{3}/\mathbb{G}_{a}$ is a quasi-affine scheme, in fact an open subset of $\mathbb{A}^{3}/\!/\mathbb{G}_{a}\simeq\mathbb{A}^{2}$ with at most finite complement. The equality $\mathbb{A}^{3}/\mathbb{G}_{a}=\mathbb{A}^{3}/\!/\mathbb{G}_{a}$ ultimately follows by comparing Euler characteristics. Kaliman's general proof proceeds along a completely different approach, drawing on topological arguments to show directly that the algebraic quotient morphism $\pi:\mathbb{A}^{3}\rightarrow\mathbb{A}^{3}/\!/\mathbb{G}_{a}$ is a locally trivial $\mathbb{A}^{1}$-bundle.
Kaliman's result can be reinterpreted as the striking fact that the topological contractiblity of $\mathbb{A}^{3}$ is a strong enough constraint to guarantee that a fixed point free $\mathbb{G}_{a}$-action on it is automatically proper. This implication fails completely in higher dimensions where non proper fixed point free $\mathbb{G}_{a}$-actions abound, even in the case of triangular actions on $\mathbb{A}^{4}$ as illustrated by Deveney-Finston-Gehrke in \cite{DevFinGe94}. Starting from dimension $5$, it is known that properness and triangulability are no longer enough to imply global equivariant triviality or at least local equivariant triviality in the Zariski topology, as shown by examples of triangular actions on $\mathbb{A}^{5}$ with either strictly quasi-affine geometric quotients or with geometric quotients existing only as separated algebraic spaces constructed respectively by Winkelmann \cite{Win90} and Deveney-Finston \cite{DevFin95}. \\
But the question whether a proper $\mathbb{G}_{a}$-action on $\mathbb{A}^{4}$ is a translation or is at least locally equivariantly trivial in the Zariski topology remains open. Very little progress had been made in the study of these actions during the last decades, and the only existing partial results so far concern triangular actions: Deveney, van Rossum and the second author \cite{DevFinvR04} established in 2004 that a Zariski locally equivariantly trivial triangular $\mathbb{G}_{a}$-action on $\mathbb{A}^{4}$ is a translation. The proof depends on the finite generation of the ring of invariants for such actions established by Daigle-Freudenburg \cite{DaiFreu01} and exploits the very particular structure of these rings. Incidentally, it is known in general that local triviality for a proper action on $\mathbb{A}^{n}$ follows from the finite generation and regularity of the ring of invariants. But even knowing the former for triangular actions on $\mathbb{A}^{4}$, a direct proof of the latter condition remains elusive. The second positive result concerns a special type of triangular $\mathbb{G}_{a}$-actions generated by derivations of $\mathbb{C}[x,y,z,u]$ of the form $r(x)\partial_{y}+q(x,y)\partial_{z}+p(x,y)\partial_{u}$ where $r(x)\in\mathbb{C}[x]$ and $p(x,y),q(x,y)\in\mathbb{C}[x,y,]$. To insist on the fact that $p(x,y)$ belongs to $\mathbb{C}[x,y]$ and not only to $\mathbb{C}[x,y,z]$ as it would be the case for a general triangular situation, these derivations (and the $\mathbb{G}_{a}$-actions they generate) were named \emph{twin-triangular} in \cite{DevFin02}. The case where $r(x)$ has simple roots was first settled in 2002 by Deveney and the second author in \emph{loc. cit.} by explicitly computing the invariant ring $\mathbb{C}[x,y,z,u]^{\mathbb{G}_{a}}$ and investigating the structure of the algebraic quotient morphism $\mathbb{A}^{4}\rightarrow\mathbb{A}^{4}/\!/\mathbb{G}_{a}=\mathrm{Spec}(\mathbb{C}[x,y,z_{1},z_{2}]^{\mathbb{G}_{a}})$. The simplicity of the roots of $r(x)$ was crucial to achieve the computation, and the generalization of the result to arbitrary twin-triangular actions obtained in 2012 by the first two authors \cite{DubFin11} required completely different methods which focused more on the nature of the corresponding geometric quotients $\mathbb{A}_{\mathbb{C}}^{4}/\mathbb{G}_{a}$. The latter a priori exist only as separated algebraic spaces and the crucial step in \emph{loc. cit.} was to show that for twin-triangular actions they are in fact schemes, or, equivalently that proper twin-triangular $\mathbb{G}_{a}$-actions on $\mathbb{A}^{4}$ are not only locally equivariantly trivial in the \'etale topology but also in the Zariski topology. This enabled in turn the use of the aforementioned result of Deveney-Finston-van Rossum to conclude that such actions are indeed translations.
One of the main obstacles to extend the above results to arbitrary triangular actions comes from the fact that in contrast with fixed point freeness, the property for a triangular $\mathbb{G}_{a}$-action on $\mathbb{A}^{4}$ to be proper is in general subtle to characterize effectively in terms of its associated locally nilpotent derivation. A good illustration of these difficulties is given by the following family of fixed point free $\mathbb{G}_{a}$-actions \[ \sigma_{r}:\mathbb{G}_{a}\times\mathbb{A}^{4}\rightarrow\mathbb{A}^{4},\;\left(t,(x,y,z,u)\right)\mapsto(x,y+tx^{2},z+2yt+x^{2}t^{2},u+(1+x^{r}z)t+x^{r}yt^{2}+\frac{1}{3}x^{r+1}t^{3})\quad r\geq1, \] generated by the triangular derivations $\delta_{r}=x^{2}\partial_{y}+2y\partial_{z}+(1+x^{r}z)\partial_{u}$ of $\mathbb{C}[x,y,z,u]$, which are either non proper if $r=1,2$ or translations otherwise. The fact that $\sigma_{r}$ is a translation for every $r\geq4$ follows immediately from the observation that $\delta_{r}$ admits the variable $s=u-x^{r-2}yz+\frac{2}{3}x^{r-4}y^{3}$ as a global slice. The case $r=3$ is slightly more complicated: one can first observe that $\delta_{3}$ is conjugated via the triangular change of variable $\tilde{u}=u-x^{r-2}yz$ to the twin-triangular derivation $x^{2}\partial_{y}+2y\partial_{z}+(1-2xy^{2})\partial_{\tilde{u}}$ of $\mathbb{C}[x,y,z,\tilde{u}]$. The projection $\mathrm{pr}_{x,y,\tilde{u}}:\mathbb{A}^{4}\rightarrow\mathbb{A}^{3}$ is then equivariant for the fixed point free $\mathbb{G}_{a}$-action on $\mathbb{A}^{3}$ generated by the triangular derivation $x^{2}\partial_{y}+(1-2xy^{2})\partial_{\tilde{u}}$ of $\mathbb{C}[x,y,\tilde{u}]$ and it descends to a locally trivial $\mathbb{A}^{1}$-bundle $\rho:\mathbb{A}^{4}/\mathbb{G}_{a}\rightarrow\mathbb{A}^{3}/\mathbb{G}_{a}\simeq\mathbb{A}^{2}$ between the respective geometric quotients. Since $\mathbb{A}^{2}$ is affine and factorial, $\rho$ is a trivial $\mathbb{A}^{1}$-bundle and hence the $\mathbb{G}_{a}$-action generated by $\delta_{3}$ is a translation. On the other hand, the non properness of $\sigma_{2}$ can be seen quickly via the invariant hypersurface method outlined in \cite{DubFin11}, namely, one checks in this case by a direct computation that the induced $\mathbb{G}_{a}$-action on the invariant hypersurface $H=\left\{ x^{2}z=y^{2}-\frac{3}{2}\right\} \subset\mathbb{A}^{4}$ is not proper, with non separated geometric quotient $H/\mathbb{G}_{a}$ isomorphic to the product of the affine line $\mathbb{A}^{1}$ with the affine line with a double origin. The failure of properness in the case where $r=1$ is even more subtle to analyze since in contrast with the previous case, the induced action on every invariant hypersurface of the form $H_{\lambda}=\left\{ x^{2}z=y^{2}-\lambda\right\} $, $\lambda\in\mathbb{C}$, turns out to be proper. Going back to the definition of the properness for the action $\sigma_{1}$, which says that the morphism $\Phi=(\mathrm{pr}_{2},\sigma_{1}):\mathbb{G}_{a}\times\mathbb{A}^{4}\rightarrow\mathbb{A}^{4}\times\mathbb{A}^{4}$ is proper, one can argue that the union of the following sequence of points \[ (p_{n},q_{n})=(p_{n};\mu_{1}(\sqrt{n^{3}},p_{n}))=((\frac{\sqrt[3]{6}}{n},-\frac{\sqrt[3]{36}}{2\sqrt{n}},\frac{1}{\sqrt[3]{6}\sqrt{n}},0);(\frac{\sqrt[3]{6}}{n},\frac{\sqrt[3]{36}}{2\sqrt{n}},\frac{1}{\sqrt[3]{6}\sqrt{n}},1))\in\mathbb{A}^{4}\times\mathbb{A}^{4},\quad n\in\mathbb{N} \] and its limit $(p_{\infty};q_{\infty})=(p_{\infty},\mu_{1}(1,p_{\infty}))=\left((0,0,0,0);(0,0,0,1)\right)$ is a compact subset of $\mathbb{A}^{4}\times\mathbb{A}^{4}$ equipped with the analytic topology whose inverse image by $\Phi$ is unbounded. So $\Phi$ is not proper as an analytic map between the corresponding varieties equipped with their respective underlying structures of analytic manifolds and hence is not proper in the algebraic category either.\\
In this article, we reconsider proper triangular actions on $\mathbb{A}^{4}$ in broader framework and we develop new techniques to overcome the above difficulties. These enable in turn to completely solve the question of global equivariant triviality for such actions. Since a triangular $\mathbb{G}_{a}$-action on $\mathbb{A}^{4}=\mathrm{Spec}(\mathbb{C}[x,y,z,u])$ preserves the variable $x$, it can be considered as an action of the additive group scheme $\mathbb{G}_{a,\mathbb{C}[x]}=\mathbb{G}_{a}\times_{\mathrm{Spec}(\mathbb{C})}\mathrm{Spec}(\mathbb{C}[x])$ on the affine $3$-space $\mathbb{A}_{\mathbb{C}[x]}^{3}$ over $\mathrm{Spec}(\mathbb{C}[x])$ so that the setup is in fact $3$-dimensional over a parameter space. The properties for a $\mathbb{G}_{a,\mathbb{C}[x]}$-action on $\mathbb{A}_{\mathbb{C}[x]}^{3}$ to be proper or triangulable being both local on the parameter space, a cost free generalization is obtained by replacing $\mathrm{Spec}(\mathbb{C}[x])$ by an arbitrary n\oe therian normal scheme $X$ defined over a field of characteristic zero and the trivial $\mathbb{A}^{3}$-bundle $\mathrm{pr}_{x}:\mathbb{A}_{\mathbb{C}[x]}^{3}\rightarrow\mathrm{Spec}(\mathbb{C}[x])$ of $\mathrm{Spec}(\mathbb{C}[x])$ by a Zariski locally trivial $\mathbb{A}^{3}$-bundle $\pi:E\rightarrow X$. Our main result then reads as follows: \begin{thm*} Let $X$ be a n\oe therian normal scheme defined over a field of characteristic zero, let $\pi:E\rightarrow X$ be a Zariski locally trivial $\mathbb{A}^{3}$-bundle equipped with a proper locally triangulable $\mathbb{G}_{a,X}$-action and let $\mathrm{p}:\mathfrak{X}=E/\mathbb{G}_{a,X}\rightarrow X$ be the geometric quotient taken in the category of algebraic $X$-spaces. Then there exists an open sub-scheme $U$ of $X$ with $\mathrm{codim}_{X}(X\setminus U)\geq2$ such that $\mathfrak{X}_{U}=\mathrm{p}^{-1}(U)\rightarrow U$ has the structure of a Zariski locally trivial $\mathbb{A}^{2}$-bundle. \end{thm*} Note in particular that since in the original problem, the base $X=\mathrm{Spec}(\mathbb{C}[x])$ is $1$-dimensional, this Theorem and an appeal to the aforementioned result \cite{DevFinvR04} are enough to settle the question for $\mathbb{A}_{\mathbb{C}}^{4}$. The conclusion of the above Theorem is essentially optimal. Indeed, in the example due to Winkelmann \cite{Win90}, one has $X=\mathrm{Spec}(\mathbb{C}[x,y])$, $\pi=\mathrm{pr}_{x,y}:\mathbb{A}_{X}^{3}=\mathrm{Spec}(\mathbb{C}[x,y][u,v,w])\rightarrow X$ equipped with the proper triangular $\mathbb{G}_{a,X}$-action generated by the $\mathbb{C}[x,y]$-derivation $\partial=x\partial_{u}+y\partial_{v}+(1+xv-yu)\partial_{w}$ of $\mathbb{C}[x,y][u,v,w]$, and the geometric quotient $\mathrm{p}:\mathfrak{X}=\mathbb{A}_{X}^{3}/\mathbb{G}_{a,X}\rightarrow X$ is the strictly quasi-affine complement of the closed subset $\left\{ x=y=z=0\right\} $ in the $4$-dimensional smooth affine quadric $Q\subset\mathbb{A}_{X}^{3}$ with equation $xt_{2}+yt_{1}=z(z+1)$. The structure morphism $\mathrm{p}:\mathfrak{X}\rightarrow X$ is easily seen to be an $\mathbb{A}^{2}$-fibration, which restricts to a locally trivial $\mathbb{A}^{2}$-bundle over the open subset $U=X\setminus\{(0,0)\}$. However, there is no Zariski or \'etale open neighborhood of the origin $(0,0)\in X$ over which $\mathrm{p}:\mathfrak{X}\rightarrow X$ restricts to a trivial $\mathbb{A}^{2}$-bundle for otherwise $\mathrm{p}:\mathfrak{X}\rightarrow X$ would be an affine morphism and so $\mathfrak{X}$ would be an affine scheme. The situation for the $\mathbb{C}[x,y]$-derivation $\partial=x\partial_{u}+y\partial_{v}+(1+xv^{2})\partial_{w}$ of $\mathbb{C}[x,y][u,v,w]$ constructed by Deveney-Finston \cite{DevFin95} is very similar: here the geometric quotient $\mathfrak{X}=\mathbb{A}_{X}^{3}/\mathbb{G}_{a,X}$ is a separated algebraic space which is not a scheme and the structure morphism $\mathrm{p}:\mathfrak{X}\rightarrow X$ is again an $\mathbb{A}^{2}$-fibration restricting to a Zariski locally trivial $\mathbb{A}^{2}$-bundle over $U=X\setminus\{(0,0)\}$ but whose restriction to any Zariski or \'etale open neighborhood of the origin $(0,0)\in X$ is nontrivial. \\
\noindent In contrast, in the case of a $1$-dimensional affine base, we can immediately derive the following Corollaries: \begin{cor*} Let $\pi:E\rightarrow S$ be a rank $3$ vector bundle over an affine Dedekind scheme $S=\mathrm{Spec}(A)$ defined over a field $k$ of characteristic $0$. Then every proper locally triangulable $\mathbb{G}_{a,S}$-action on $E$ is equivariantly trivial with geometric quotient $E/\mathbb{G}_{a,S}$ isomorphic to a vector bundle of rank $2$ over $S$, stably isomorphic to $E$. \end{cor*} \begin{proof} By the previous Theorem, the geometric quotient $\mathrm{p}:E/\mathbb{G}_{a,S}\rightarrow S$ has the structure of a Zariski locally trivial $\mathbb{A}^{2}$-bundle, hence is a vector bundle of rank $2$ by \cite{BCW77}. In particular, $E/\mathbb{G}_{a,S}$ is affine which implies in turn that $\rho:E\rightarrow E/\mathbb{G}_{a,S}$ is a trivial $\mathbb{G}_{a,S}$-bundle. So $E\simeq E/\mathbb{G}_{a,S}\times_{S}\mathbb{A}_{S}^{1}$ as vector bundles over $S$. \end{proof} \begin{cor*} Let $S=\mathrm{Spec}(A)$ be an affine Dedekind scheme defined over a field of characteristic $0$. Then every proper triangular $\mathbb{G}_{a,S}$-action on $\mathbb{A}_{S}^{3}$ is a translation. \end{cor*} \begin{proof} By the previous Corollary, $\mathbb{A}_{S}^{3}/\mathbb{G}_{a,S}$ is a stably trivial vector bundle of rank $2$ over $S$, whence is isomorphic to the trivial bundle $\mathbb{A}_{S}^{2}$ over $S$ by virtue of \cite[IV 3.5]{Bas68}. \end{proof} \noindent Coming back to the original problem for triangular $\mathbb{G}_{a,k}$-actions on $\mathbb{A}_{k}^{4}$, the previous Corollary does in fact eliminate the need for \cite{DevFinvR04} hence the dependency on the fact that the corresponding rings of invariants are finitely generated: \begin{cor*} If $k$ is a field of characteristic $0$, then every proper triangular $\mathbb{G}_{a,k}$-action on $\mathbb{A}_{k}^{4}$ is a translation.\end{cor*} \begin{proof} Letting $\mathbb{A}_{k}^{4}=\mathrm{Spec}(k[x,y,z,u])$, we may assume that the action is generated by a $k$-derivation of the form $\partial=r(x)\partial_{y}+q(x,y)\partial_{z}+p(x,y,z)\partial_{u}$. As explained above, the latter can be considered as a triangular $k[x]$-derivation of $k[x][y,z,u]$ generating a proper $\mathbb{G}_{a,k[x]}$-action on $\mathbb{A}_{k}^{4}=\mathbb{A}_{k[x]}^{3}$ which is, by the previous Corollary, a trivial $\mathbb{G}_{a}$-bundle over its geometric quotient $\mathbb{A}_{k}^{4}/\mathbb{G}_{a,k}\simeq\mathbb{A}_{k[x]}^{3}/\mathbb{G}_{a,k[x]}\simeq\mathbb{A}_{k[x]}^{2}\simeq\mathbb{A}_{k}^{3}$. \end{proof} Let us now briefly explain the general philosophy behind the proof. After localizing at codimension $1$ points of $X$, the Main Theorem reduces to the statement that a proper $\mathbb{G}_{a,S}$-action $\sigma:\mathbb{G}_{a,S}\times_{S}\mathbb{A}_{S}^{3}\rightarrow\mathbb{A}_{S}^{3}$ on the affine affine space $\mathbb{A}_{S}^{3}=\mathrm{Spec}(A[y,z,u])$ over the spectrum of a discrete valuation ring, generated by a triangular $A$-derivation $\partial=a\partial_{y}+q(y)\partial_{z}+p(y,z)\partial_{u}$ of $A[y,z,u]$, where $a\in A\setminus\left\{ 0\right\} $, $q(y)\in A[y]$ and $p(y,z)\in A[y,z]$, is a translation. Triangularity immediately implies that the restriction of $\sigma$ to the generic fiber of $\mathrm{pr}_{S}:\mathbb{A}_{S}^{3}\rightarrow S$ is a translation with $a^{-1}y$ as a global slice. This reduces the problem to the study of neighborhoods of points of the geometric quotient $\mathfrak{X}=\mathbb{A}_{S}^{3}/\mathbb{G}_{a,S}$ supported on the closed fiber of the structure morphism $\mathrm{p}:\mathfrak{X}\rightarrow S$. A second feature of triangularity is that $\sigma$ commutes with the action $\tau:\mathbb{G}_{a,S}\times_{S}\mathbb{A}_{S}^{3}\rightarrow\mathbb{A}_{S}^{3}$ generated by the $A$-derivation $\partial_{u}$ which therefore descends to a $\mathbb{G}_{a,S}$-action $\overline{\tau}$ on the geometric quotient $\mathfrak{X}=\mathbb{A}_{S}^{3}/\mathbb{G}_{a,S}$. On the other hand, $\sigma$ descends via the projection $\mathrm{pr}_{y,z}:\mathbb{A}_{S}^{3}\rightarrow\mathbb{A}_{S}^{2}=\mathrm{Spec}(A[y,z])$ to the action $\overline{\sigma}$ on $\mathbb{A}_{S}^{2}$ generated by the $A$-derivation $\overline{\partial}=a\partial_{y}+q(y)\partial_{z}$ of $A[y,z]$. Even though $\overline{\sigma}$ and $\overline{\tau}$ are no longer fixed point free in general, if we take the quotient of $\mathbb{A}_{S}^{2}$ by the action $\overline{\sigma}$ as an algebraic stack $[\mathbb{A}_{S}^{2}/\mathbb{G}_{a,S}]$ we obtain a cartesian square \[\xymatrix{ \mathbb{A}^3_S \ar[d]_{\mathrm{pr}_{y,z}} \ar[r] & \mathfrak{X}=\mathbb{A}^3_S/\mathbb{G}_{a,S} \ar[d] \\ \mathbb{A}^2_S \ar[r] & [\mathbb{A}^2_S/\mathbb{G}_{a,S}]}\]which simultaneously identifies the quotient stacks $[\mathbb{A}_{S}^{2}/\mathbb{G}_{a,S}]$ for the action $\overline{\sigma}$ and $[\mathfrak{X}/\mathbb{G}_{a,S}]$ for the action $\overline{\tau}$ with the quotient stack of $\mathbb{A}_{S}^{3}$ for the $\mathbb{G}_{a,S}^{2}$-action defined by the commuting actions $\sigma$ and $\tau$. In this setting, the global equivariant triviality of the action $\sigma$ becomes equivalent to the statement that a separated algebraic $S$-space $\mathfrak{X}$ admitting a $\mathbb{G}_{a,S}$-action whose algebraic stack quotient $[\mathfrak{X}/\mathbb{G}_{a,S}]$ is isomorphic to that of a triangular $\mathbb{G}_{a,S}$-action on $\mathbb{A}_{S}^{2}$ is an affine scheme.
While a direct proof of this reformulation seems totally out of reach with existing methods, it turns out that its conclusion holds over a certain $\mathbb{G}_{a,S}$-invariant principal open subset $V$ of $\mathbb{A}_{S}^{2}$ which dominates $S$ and for which the algebraic stack quotient $[V/\mathbb{G}_{a,S}]$ is in fact represented by a locally separated algebraic sub-space of $[\mathbb{A}_{S}^{2}/\mathbb{G}_{a,S}]$. This provides at least an affine open subscheme $V\times_{S}\mathbb{A}_{S}^{1}/\mathbb{G}_{a,S}$ of $\mathfrak{X}$ dominating $S$, and leaves us with a closed subset of codimension at most $2$ of $\mathfrak{X}$, supported on the closed fiber of $\mathrm{p}:\mathfrak{X}\rightarrow S$, in a neighborhood of which no further information is a priori available to decide even the schemeness of $\mathfrak{X}$. But similar to the argument in \cite{DubFin11}, this situation can be rescued for twin-triangular actions: the fact that for such actions $\partial u=p(y,z)$ is actually a polynomial in $y$ only enables the same reasoning with respect to the other projection $\mathrm{pr}_{y,u}:\mathbb{A}_{S}^{3}\rightarrow\mathbb{A}_{S}^{2}=\mathrm{Spec}(A[y,u])$, yielding a second affine open sub-scheme $V'\times_{S}\mathbb{A}_{S}^{1}/\mathbb{G}_{a,S}$ of $\mathfrak{X}$ dominating $S$. This implies at least the schemeness of $\mathfrak{X}$, provided that the open subsets $V$ and $V'$ can be chosen so that the union of the corresponding open subschemes of $\mathfrak{X}$ covers the closed fiber of $\mathrm{p}:\mathfrak{X}\rightarrow S$. \\
The scheme of the article is the following. The first two sections recall basic notions and discuss a couple of preliminary technical reductions. The third section is devoted to establishing an effective criterion for non properness of fixed point free triangular actions from which we deduce the intermediate fact that every proper triangular action is twin-triangulable. Then in the next section, we establish that proper twin-triangular actions are indeed translations. Here, in contrast with the proof for the complex case given in \cite{DubFin11}, our argument is independent of finite generation of rings of invariants and reduces the systematic study of algebraic spaces quotients to a minimum thanks to an appropriate Sheshadri cover trick \cite{Sesh72}.
\section{Recollection on proper, fixed point free and locally triangulable $\mathbb{G}_{a}$-actions }
\subsection{Proper versus fixed point free actions}
\indent\newline\noindent Recall that an action $\sigma:\mathbb{G}_{a,S}\times_{S}E\rightarrow E$ of the additive group scheme $\mathbb{G}_{a,S}=\mathrm{Spec}_{S}(\mathcal{O}_{S}[t])=S\times_{\mathbb{Z}}\mathrm{Spec}(\mathbb{Z}[t])$ on an $S$-scheme $E$ is called proper if the morphism $\Phi=(\mathrm{pr}_{2},\sigma):\mathbb{G}_{a,S}\times_{S}E\rightarrow E\times_{S}E$ is proper.
\begin{parn} If $S$ is moreover defined over a field $k$ of characteristic zero, then the fact that $\mathbb{G}_{a,k}$ is affine and has no nontrivial algebraic subgroups implies that properness is equivalent to $\Phi$ being a closed immersion. In particular, a proper $\mathbb{G}_{a,S}$-action is in this case fixed point free and as such, is equivariantly locally trivial in the \'etale topology on $E$. That is, there exists an affine $S$-scheme $U$ and a surjective \'etale morphism $f:V=U\times_{S}\mathbb{G}_{a,S}\rightarrow E$ which is equivariant for the action of $\mathbb{G}_{a,S}$ on $U\times_{S}\mathbb{G}_{a,S}$ by translations on the second factor. This implies in turn the existence of a geometric quotient $\rho:E\rightarrow\mathfrak{X}=E/\mathbb{G}_{a,S}$ in the form of an \'etale locally trivial principal $\mathbb{G}_{a,S}$-bundle over an algebraic $S$-space $\mathrm{p}:\mathfrak{X}\rightarrow S$ (see e.g. \cite[10.4]{LMB00}). Informally, $\mathfrak{X}$ is the quotient of $U$ by the \'etale equivalence relation which identifies two points $u,u'\in U$ whenever there exists $t,t'\in\mathbb{G}_{a,S}$ such that $f(u,t)=f(u',t')$.
\end{parn}
\begin{parn} \label{par:Properness_charac} Conversely, a fixed point free $\mathbb{G}_{a,S}$-action is proper if and only if the geometric quotient $\mathfrak{X}=E/\mathbb{G}_{a,S}$ is a separated $S$-space. Indeed, by definition $\mathrm{p}:\mathfrak{X}\rightarrow S$ is separated if and only if the diagonal morphism $\Delta:\mathfrak{X}\rightarrow\mathfrak{X}\times_{S}\mathfrak{X}$ is a closed immersion, a property which is local on the target with respect to the fpqc topology \cite[II.3.8]{Knu71} and \cite[VIII.5.5]{SGA1}. Since $\rho:E\rightarrow\mathfrak{X}$ is a $\mathbb{G}_{a,S}$-bundle, taking the fpqc base change by $\rho\times\rho:E\times_{S}E\rightarrow\mathfrak{X}\times_{S}\mathfrak{X}$ yields a cartesian square \[\xymatrix{\mathbb{G}_{a,S} \times_S E \ar[r]^{\Phi} \ar[d]_{\rho \circ \mathrm{pr}_2} & E \times_S E \ar[d]^{\rho\times\rho} \\ \mathfrak{X} \ar[r]^-{\Delta} & \mathfrak{X} \times_S \mathfrak{X} }\]from which we see that $\Delta$ is a closed immersion if and only if $\Phi$ is.
\end{parn}
\subsection{Locally triangulable actions}
\indent\newline\noindent Given an affine scheme $S=\mathrm{Spec}(A)$ defined over a field of characteristic zero, an action $\sigma:\mathbb{G}_{a,S}\times_{S}\mathbb{A}_{S}^{n}\rightarrow\mathbb{A}_{S}^{n}$ generated by a locally nilpotent $A$-derivation $\partial$ of $\Gamma(\mathbb{A}_{S}^{n},\mathcal{O}_{\mathbb{A}_{S}^{n}})$ is called \emph{triangulable} if there exists an isomorphism of $A$-algebras $\tau:\Gamma(\mathbb{A}_{A}^{n},\mathcal{O}_{\mathbb{A}_{A}^{n}})\stackrel{\sim}{\rightarrow}A[x_{1},\cdots,x_{n}]$ such that the conjugate $\delta=\tau\circ\partial\circ\tau^{-1}$ of $\partial$ is triangular with respect to the ordered coordinate system $(x_{1},\ldots,x_{n})$, i.e. has the form \[ \delta=p_{0}\frac{\partial}{\partial x_{1}}+\sum_{i=1}^{n}p_{i-1}(x_{1},\ldots,x_{i-1})\frac{\partial}{\partial x_{i}} \] where $p_{0}\in A$ and where for every $i=1,\ldots,n$, $p_{i-1}(x_{1},\ldots,x_{i-1})\in A[x_{1},\ldots,x_{i-1}]\subset A[x_{1},\ldots,x_{n}]$. By localizing this notion over the base $S$, we arrive at the following definition: \begin{defn} Let $X$ be a scheme defined over a field of characteristic zero and let $\pi:E\rightarrow X$ be a Zariski locally trivial $\mathbb{A}^{n}$-bundle over $X$. An action $\sigma:\mathbb{G}_{a,X}\times_{X}E\rightarrow E$ of $\mathbb{G}_{a,X}$ on $E$ is called \emph{locally triangulable} if there exists a covering of $\mathrm{Spec}(A)$ by affine open sub-schemes $S_{i}=\mathrm{Spec}(A_{i})$, $i\in I$, such that $E\mid_{S_{i}}\simeq\mathbb{A}_{S_{i}}^{n}$ and such that the $\mathbb{G}_{a,S_{i}}$-action $\sigma_{i}:\mathbb{G}_{a,S_{i}}\times_{S_{i}}\mathbb{A}_{S_{i}}^{n}\rightarrow\mathbb{A}_{S_{i}}^{n}$ on $\mathbb{A}_{S_{i}}^{n}$ induced by $\sigma$ is triangulable. \end{defn} A Zariski locally trivial $\mathbb{A}^{1}$-bundle $\pi:E\rightarrow X$ equipped with a fixed point free $\mathbb{G}_{a,X}$-action is nothing but a principal $\mathbb{G}_{a,X}$-bundle. As mentioned in the introduction, the nature of fixed point free locally triangulable $\mathbb{G}_{a,X}$-actions on Zariski locally trivial $\mathbb{A}^{2}$-bundles $\pi:E\rightarrow X$ is classically known. Namely, we have the following generalization of the main theorem of \cite{Snow88}: \begin{prop} \label{prop:Rank2-bundle} Let $X$ be a n\oe therian normal scheme defined over a field of characteristic $0$ and let $\pi:E\rightarrow X$ be a Zariski locally trivial $\mathbb{A}^{2}$-bundle equipped with a fixed point free locally triangulable $\mathbb{G}_{a,X}$-action. Then the geometric quotient $\mathrm{p}:E/\mathbb{G}_{a,X}\rightarrow X$ has the structure of a Zariski locally trivial $\mathbb{A}^{1}$-bundle over $X$. \end{prop} \begin{proof} The assertion being local on the base $X$, we may assume that $X=\mathrm{Spec}(A)$ is the spectrum of a normal local domain containing a field of characteristic $0$ and that $E=\mathbb{A}_{X}^{2}=\mathrm{Spec}(A[y,z])$ is equipped with the $\mathbb{G}_{a,X}$-action generated by a triangular derivation $\partial=a\partial_{y}+q(y)\partial_{z}$ of $A[y,z]$, where $a\in A$ and $q(y)\in A[y]$. The fixed point freeness hypothesis is equivalent to the property that $a$ and $q(y)$ generate the unit ideal in $A[y,z]$. So $q(y)$ has the form $q(y)=b+c\tilde{q}(y)$ where $b\in A$ is relatively prime with $a$, $c\in\sqrt{aA}$ and $\tilde{q}(y)\in A[y]$. Letting $Q(y)=\int_{0}^{y}q(\tau)d\tau=by+c\int_{0}^{y}\tilde{q}(\tau)d\tau$, the polynomial $v=az-Q(y)\in A[y,z]$ belongs to the kernel $\mathrm{Ker}\partial$ of $\partial$ hence defines a $\mathbb{G}_{a,X}$-invariant morphism $v:E\rightarrow\mathbb{A}_{X}^{1}=\mathrm{Spec}(A[t])$. Since $a$ and $b$ generate the unit ideal in $A$, it follows from the Jacobian criterion that $v:E\rightarrow\mathbb{A}_{X}^{1}$ is a smooth morphism. Furthermore, the fibers of $v$ coincide precisely with the $\mathbb{G}_{a,X}$-orbits on $E$. Indeed, over the principal open subset $X_{a}=\mathrm{Spec}(A_{a})$ of $X$, $\partial$ admits $a^{-1}y$ as a slice and we have an equivariant isomorphism $E\mid_{X_{a}}\simeq\mathrm{Spec}(A[a^{-1}v,a^{-1}y])\simeq\mathbb{A}_{X_{a}}^{1}\times_{X}\mathbb{G}_{a,X}$ where $\mathbb{G}_{a,X}$ acts by translations on the second factor. On the other hand, the restriction $E\mid_{Z}$ of $E$ over the closed subset $Z\subset X$ with defining ideal $\sqrt{aA}\subset A$ is equivariantly isomorphic to $\mathbb{A}_{Z}^{2}$ equipped with the $\mathbb{G}_{a,Z}$-action generated by the derivation $\overline{\partial}=\overline{b}\partial_{z}$ of $(A/\sqrt{aA})[y,z]$, where $\overline{b}\in(A/\sqrt{aA})^{*}$ denotes the residue class of $b$. The restriction of $v$ to $E\mid_{Z}$ coincides via this isomorphism to the morphism $\mathbb{A}_{Z}^{2}\rightarrow\mathbb{A}_{Z}^{1}$ defined by the polynomial $\overline{v}=\overline{b}y\in(A/\sqrt{aA})[y,z]$ which is obviously a geometric quotient. The above properties imply that the morphism $\tilde{v}:E/\mathbb{G}_{a,X}\rightarrow\mathbb{A}_{X}^{1}$ induced by $v$ is smooth and bijective. Since it admits \'etale quasi-sections, $\tilde{v}$ is then an isomorphism locally in the \'etale topology on $\mathbb{A}_{X}^{1}$ whence an isomorphism. \end{proof}
\section{preliminary reductions }
\subsection{Reduction to a local base}
The statement of the Main Theorem can be rephrased equivalently as the fact that a proper locally triangulable $\mathbb{G}_{a,S}$-action on a Zariski locally trivial $\mathbb{A}^{3}$-bundle $\pi:E\rightarrow S$ is a translation in codimension $1$. This means that for every point $s\in S$ of codimension $1$ with local ring $\mathcal{O}_{S,s}$, the fiber product $E\times_{S}S'\simeq\mathbb{A}_{S'}^{3}$ of $E\rightarrow S$ with the canonical immersion $S'=\mathrm{Spec}(\mathcal{O}_{S,s})\hookrightarrow S$ equiped with the induced proper triangular action of $\mathbb{G}_{a,S'}=\mathbb{G}_{a,S}\times_{S}S'$ is equivariantly isomorphic to the trivial bundle $\mathbb{A}_{S'}^{2}\times_{S'}\mathbb{G}_{a,S'}$ over $S'$ equipped with the action of $\mathbb{G}_{a,S'}$ by translations on the second factor.
\begin{parn} \label{par:local_notation} So we are reduced to the case where $S$ is the spectrum of a discrete valuation ring $A$ containing a field of characteristic $0$, say with maximal ideal $\mathfrak{m}$ and residue field $\kappa=A/\mathfrak{m}$, and where $\pi=\mathrm{pr}_{S}:E=\mathbb{A}_{S}^{3}=\mathrm{Spec}(A[y,z,u])\rightarrow S=\mathrm{Spec}(A)$ is equipped with a proper triangulable $\mathbb{G}_{a,S}$-action $\sigma:\mathbb{G}_{a,S}\times_{S}\mathbb{A}_{S}^{3}\rightarrow\mathbb{A}_{S}^{3}$. Letting $x\in\mathfrak{m}$ be uniformizing parameter, every such action is equivalent to one generated by an $A$-derivation $\partial$ of $A[y,z,u]$ of the form \[ \partial=x^{n}\partial_{y}+q(y)\partial_{z}+p(y,z)\partial_{u} \] where $n\geq0$, $q(y)\in A[y]$ and $p(y,z)=\sum_{r=0}^{\ell}p_{r}(y)z^{r}\in A[y,z]$, the fixed point freeness of $\sigma$ being equivalent to the property that $x^{n}$, $q(y)$ and $p(y,z)$ generate the unit ideal in $A[y,z,u]$.
\end{parn}
\subsection{\label{sub:Reduction-to-Affineness} Reduction to proving the affineness of the geometric quotient}
With the notation of \S \ref{par:local_notation}, we can already observe that if $n=0$ then $y$ is an obvious global slice for $\partial$ and hence that the action is globally equivariantly trivial with geometric quotient $\mathfrak{X}=\mathbb{A}_{S}^{3}/\mathbb{G}_{a,S}\simeq\mathbb{A}_{S}^{2}$. Similarly, if the residue class of $q(y)$ in $\kappa[y]$ is a non zero constant then the action $\sigma$ is a translation. Indeed, in this case, the $\mathbb{G}_{a,S}$-action $\overline{\sigma}:\mathbb{G}_{a,S}\times_{S}\mathbb{A}_{S}^{2}\rightarrow\mathbb{A}_{S}^{2}$ on $\mathbb{A}_{S}^{2}=\mathrm{Spec}(A[y,z])$ generated by the $A$-derivation $\overline{\partial}=x^{n}\partial_{y}+q(y)\partial_{z}$ of $A[y,z]$ is fixed point free hence globally equivariantly trivial with geometric quotient $\mathbb{A}_{S}^{2}/\mathbb{G}_{a,S}\simeq\mathbb{A}_{S}^{1}$ by virtue of Proposition \ref{prop:Rank2-bundle}. On the other hand, the $\mathbb{G}_{a,S}$-equivariant projection $\mathrm{pr}_{y,z}:\mathbb{A}_{S}^{3}\rightarrow\mathbb{A}_{S}^{2}$ descends to a locally trivial $\mathbb{A}^{1}$-bundle between the geometric quotients $\mathbb{A}_{S}^{3}/\mathbb{G}_{a,S}$ and $\mathbb{A}_{S}^{2}/\mathbb{G}_{a,S}$, and since $\mathbb{A}_{S}^{2}/\mathbb{G}_{a,S}\simeq\mathbb{A}_{S}^{1}$ is affine and factorial, it follows that $\mathbb{A}_{S}^{3}/\mathbb{G}_{a,S}\simeq\mathbb{A}_{S}^{2}/\mathbb{G}_{a,S}\times_{S}\mathbb{A}_{S}^{1}\simeq\mathbb{A}_{S}^{2}.$ The affineness of $\mathbb{A}_{S}^{2}$ implies in turn that the quotient morphism $\mathbb{A}_{S}^{3}\rightarrow\mathbb{A}_{S}^{3}/\mathbb{G}_{a,S}$ is the trivial $\mathbb{G}_{a,S}$-bundle whence that $\sigma:\mathbb{G}_{a,S}\times_{S}\mathbb{A}_{S}^{3}\rightarrow\mathbb{A}_{S}^{3}$ is a translation. Alternatively, one can observe that a global slice $s\in A[y,z]$ for the action $\overline{\sigma}$ is also a global slice for $\sigma$ via the inclusion $A[y,z]\subset A[y,z,u]$
More generally, the following Lemma reduces the question of global equivariant triviality with geometric quotient $\mathfrak{X}=\mathbb{A}_{S}^{3}/\mathbb{G}_{a,S}$ isomorphic to $\mathbb{A}_{S}^{2}$ to showing that $\mathfrak{X}$, which a priori only exists as an algebraic $S$-space, is an affine $S$-scheme: \begin{lem} \label{lem:Reduction_to_affineness} A fixed point free triangular action $\sigma:\mathbb{G}_{a,S}\times_{S}\mathbb{A}_{S}^{3}\rightarrow\mathbb{A}_{S}^{3}$ is a translation if and only if its geometric quotient $\mathfrak{X}=\mathbb{A}_{S}^{3}/\mathbb{G}_{a,S}$ is an affine $S$-scheme. \end{lem} \begin{proof} One direction is clear, so assume that $\mathfrak{X}$ is an affine $S$-scheme. It suffices to show that the structure morphism $\mathrm{p}:\mathfrak{X}\rightarrow S$ is an $\mathbb{A}^{2}$-fibration, i.e. a faithfully flat morphism with all its fibers isomorphic to affine planes over the corresponding residue fields. Indeed, if so, the affineness of $\mathfrak{X}$ implies on the one hand that $\mathfrak{X}$ is isomorphic to the trivial $\mathbb{A}^{2}$-bundle $\mathbb{A}_{S}^{2}$ by virtue of \cite{Sat83} and on the other hand that $\rho:\mathbb{A}_{S}^{3}\rightarrow\mathfrak{X}$ is isomorphic to the trivial $\mathbb{G}_{a,S}$-bundle $\mathfrak{X}\times_{S}\mathbb{G}_{a,S}$ over $S$, which yields $\mathbb{G}_{a,S}$-equivariant isomorphisms $\mathbb{A}_{S}^{3}\simeq\mathfrak{X}\times_{S}\mathbb{G}_{a,S}\simeq\mathbb{A}_{S}^{2}\times_{S}\mathbb{G}_{a,S}$.
To see that $\mathrm{p}:\mathfrak{X}\rightarrow S$ is an $\mathbb{A}^{2}$-fibration, recall that $\mathrm{pr}_{S}:\mathbb{A}_{S}^{3}\rightarrow S$ and the quotient morphism $\rho:\mathbb{A}_{S}^{3}\rightarrow\mathfrak{X}=\mathbb{A}_{S}^{3}/\mathbb{G}_{a,S}$ are both faithfully flat, so that $\mathrm{p}:\mathfrak{X}\rightarrow S$ is faithfully flat too (\cite[II.3.2]{Knu71} and \cite[Corollaire 2.2.13(iii)]{EGA4}). Letting $\mathfrak{m}$ and $\xi$ be the closed and generic points of $S$ respectively, the fibers $\mathrm{pr}_{S}^{-1}(\mathfrak{m})\simeq\mathbb{A}_{\kappa}^{3}$ and $\mathrm{pr}_{S}^{-1}(\xi)\simeq\mathbb{A}_{\kappa(\xi)}^{3}$ coincide with the total spaces of the restriction of the $\mathbb{G}_{a,S}$-bundle $\rho:\mathbb{A}_{S}^{3}\rightarrow\mathfrak{X}$ over the fibers $\mathfrak{X}_{\mathfrak{m}}=\mathrm{p}^{-1}(\mathfrak{m})$ and $\mathfrak{X}_{\xi}=\mathrm{p}^{-1}(\xi)$ respectively. Since the $\mathbb{G}_{a,\kappa(\xi)}$-action induced by $\sigma$ on $\mathrm{pr}_{S}^{-1}(\xi)$ admits $x^{-n}y$ as a global slice, it is a translation with geometric quotient $\mathbb{A}_{\kappa(\xi)}^{3}/\mathbb{G}_{a,\kappa(\xi)}\simeq\mathbb{A}_{\kappa(\xi)}^{2}$ and so $\mathfrak{X}_{\xi}\simeq\mathbb{A}_{\kappa(\xi)}^{2}$. On the other hand, we may assume in view of the above discussion that $n\geq1$ so that the $\mathbb{G}_{a,\kappa}$-action on $\mathrm{pr}_{S}^{-1}(\mathfrak{m})\simeq\mathbb{A}_{\kappa}^{3}$ induced by $\sigma$ coincides with the fixed point free action generated by the $\kappa[y]$-derivation $\overline{\partial}=\overline{q}(y)\partial_{z}+\overline{p}(y,z)\partial_{u}$ of $\kappa[y][z,u]$, where $\overline{q}(y)$ and $\overline{p}(y,z)$ denote the respective residue classes of $q(y)$ and $p(y,z)$ modulo $x$. By virtue of Proposition \ref{prop:Rank2-bundle}, the geometric quotient $\mathbb{A}_{\kappa}^{3}/\mathbb{G}_{a,\kappa}$ has the structure of a Zariski locally trivial $\mathbb{A}^{1}$-bundle over $\mathbb{A}_{\kappa}^{1}=\mathrm{Spec}(\kappa[y])$ hence is isomorphic to $\mathbb{A}_{\kappa}^{2}$. This implies that $\mathfrak{X}_{\mathfrak{m}}\simeq\mathbb{A}_{\kappa}^{3}/\mathbb{G}_{a,\kappa}\simeq\mathbb{A}_{\kappa}^{2}$ as desired. \end{proof} \begin{rem} By exploiting the fact that arbitrary $\mathbb{G}_{a,S}$-actions on the affine $3$-space $\mathbb{A}_{S}^{3}$ over the spectrum $S$ of a discrete valuation ring $A$ containing a field of characteristic $0$ have finitely generated rings of invariants \cite{BhaDa09}, one can derive the following stronger characterization: a fixed point free action $\sigma:\mathbb{G}_{a,S}\times_{S}\mathbb{A}_{S}^{3}\rightarrow\mathbb{A}_{S}^{3}$ is either a translation or its geometric quotient $\mathfrak{X}=\mathbb{A}_{S}^{3}/\mathbb{G}_{a,S}$ is an algebraic space which is not a scheme.
Indeed, the quotient morphism $\rho:\mathbb{A}_{S}^{3}\rightarrow\mathfrak{X}$ is again an $\mathbb{A}^{2}$-fibration thanks to \cite[Theorem 3.2]{DaiKal09} which asserts that for every field $\kappa$ of characteristic $0$ a fixed point free action of $\mathbb{G}_{a,\kappa}$-action on $\mathbb{A}_{\kappa}^{3}$ is a translation, and so the assertion is equivalent to the fact that a Zariski locally equivariantly trivial action $\sigma$ has affine geometric quotient $\mathfrak{X}$. This can be seen in a similar way as in the proof of Theorem 2.1 in \cite{DevFinvR04}. Namely, by hypothesis we can find an open covering of $\mathbb{A}_{S}^{3}$ by finitely many invariant affine open subsets $U_{i}$ on which the induced $\mathbb{G}_{a,S}$-action is a translation with affine geometric quotient $U_{i}/\mathbb{G}_{a,S}$, $i=1,\ldots,n$. Since $U_{i}$ and $\mathbb{A}_{S}^{3}$ are affine, $\mathbb{A}_{S}^{3}\setminus U_{i}$ is a $\mathbb{G}_{a,S}$-invariant Weil divisor on $\mathbb{A}_{S}^{3}$ which is in fact principal as $A$, whence $A[y,z,u]$, is factorial. It follows that there exists invariant regular functions $f_{i}\in A[y,z,u]^{\mathbb{G}_{a}}\simeq\Gamma(\mathfrak{X},\mathcal{O}_{\mathfrak{X}})$ such that $U_{i}=\mathrm{Spec}(A[x,y,z]_{f_{i}})$ coincides with the inverse image by the quotient morphism $\rho:\mathbb{A}_{S}^{3}\rightarrow\mathfrak{X}$ of the principal open subset $\mathfrak{X}_{f_{i}}$ of $\mathfrak{X}$, $i=1,\ldots,n$. Since $\rho:\mathbb{A}_{S}^{3}\rightarrow\mathfrak{X}$ is a $\mathbb{G}_{a,S}$-bundle and $U_{i}\simeq U_{i}/\mathbb{G}_{a,S}\times_{S}\mathbb{G}_{a,S}$ by assumption, we conclude that $\mathfrak{X}$ is covered by the principal affine open subsets $\mathfrak{X}_{f_{i}}\simeq U_{i}/\mathbb{G}_{a,S}$, $i=1,\ldots,n$, whence is quasi-affine. Now since by the aforementioned result \cite{BhaDa09}, $A[y,z,u]^{\mathbb{G}_{a}}$ is an integrally closed finitely generated $A$-algebra, it is enough to check that the canonical open immersion $j:\mathfrak{X}\rightarrow X=\mathrm{Spec}(\Gamma(\mathfrak{X},\mathcal{O}_{\mathfrak{X}}))\simeq\mathrm{Spec}(A[y,z,u]^{\mathbb{G}_{a}})$ is surjective. The surjectivity over the generic point of $S$ follows immediately from the fact the kernel of a locally nilpotent derivation derivation of a polynomial ring in three variables over a field $K$ of characteristic $0$ is isomorphic to a polynomial ring in two variables over $K$ (see e.g. \cite{Miy85}). So it remains to show that the induced open immersion $j_{\mathfrak{m}}:\mathfrak{X}_{m}\simeq\mathbb{A}_{\kappa}^{2}\hookrightarrow X_{\mathfrak{m}}=\mathrm{Spec}(A[y,z,u]^{\mathbb{G}_{a}}\otimes_{A}A/\mathfrak{m})$ between the corresponding fibers over the closed point $\mathfrak{m}$ of $S$ is surjective, in fact, an isomorphism. Since $x\in A[y,z,u]^{\mathbb{G}_{a}}$ is prime, $X_{\mathfrak{m}}\simeq\mathrm{Spec}(A[y,z,u]^{\mathbb{G}_{a}}/(x))$ is an integral $\kappa$-scheme of finite type and Corollary 4.10 in \cite{BhaDa09} can be interpreted more precisely as the fact that $X_{\mathfrak{m}}\simeq C\times_{\kappa}\mathbb{A}_{\kappa}^{1}$ for a certain $1$-dimensional affine $\kappa$-scheme $C$. This implies in turn that $j_{\mathfrak{m}}$ is an isomorphism. Indeed, since $C$ is dominated via $j_{\mathfrak{m}}$ by a general affine line $\mathbb{A}_{\kappa}^{1}\subset\mathbb{A}_{\kappa}^{2}$, its normalization $\tilde{C}$ is isomorphic to $\mathbb{A}_{\kappa}^{1}$ and so $j_{\mathfrak{m}}$ factors through an open immersion $\tilde{j}_{\mathfrak{m}}:\mathbb{A}_{\kappa}^{2}\hookrightarrow\tilde{C}\times_{\kappa}\mathbb{A}_{\kappa}^{1}\simeq\mathbb{A}_{\kappa}^{2}$. The latter is surjective for otherwise the complement of its image would be of pure codimension $1$ hence a principal divisor $\mathrm{div}(f)$ for a non constant regular function $f$ on $\tilde{C}\times_{\kappa}\mathbb{A}_{\kappa}^{1}$. But then $f$ would restrict to a non constant invertible function on the image of $\mathbb{A}_{\kappa}^{2}$ which is absurd. Thus $\tilde{j}_{\mathfrak{m}}:\mathbb{A}_{\kappa}^{2}\hookrightarrow\tilde{C}\times_{\kappa}\mathbb{A}_{\kappa}^{1}\simeq\mathbb{A}_{\kappa}^{2}$ is an isomorphism and since the normalization morphism $\tilde{C}\times_{\kappa}\mathbb{A}_{\kappa}^{1}\rightarrow C\times_{\kappa}\mathbb{A}_{\kappa}^{1}$ is finite whence closed it follows that $j_{\mathfrak{m}}:\mathbb{A}_{\kappa}^{2}\hookrightarrow C\times_{\kappa}\mathbb{A}_{\kappa}^{1}$ is an open and closed immersion hence an isomorphism. \end{rem}
\subsection{Reduction to extensions of irreducible derivations}
In view of the discussion at the beginning of subsection \ref{sub:Reduction-to-Affineness}, we may assume for the $A$-derivation \[ \partial=x^{n}\partial_{y}+q(y)\partial_{z}+p(y,z)\partial_{u} \] that $n>0$ and that the residue class of $q(y)$ in $\kappa[y]$ is either zero or not constant. In the first case, $q(y)\in\mathfrak{m}A[y]$ has the form $q(y)=x^{\mu}q_{0}(y)$ where $\mu>0$ and where $q_{0}(y)\in A[y]$ has non zero residue class modulo $\mathfrak{m}$, so that the derivation $\overline{\partial}=x^{n}\partial_{y}+q(y)\partial_{z}$ induced by $\partial$ on the sub-ring $A[y,z]$ is reducible. On the other hand, the fixed point freeness of the $\mathbb{G}_{a,S}$-action $\sigma$ generated by $\partial$ implies that up to multiplying $u$ by an invertible element in $A$, one has $p(y,z)=1+x^{\nu}p_{0}(y,z)$ for some $\nu>0$ and $p_{0}(y,z)\in A[y,z]$.
If $\mu\geq n$, then letting $Q_{0}(y)=\int_{0}^{y}q_{0}(\tau)d\tau\in A[y]$, the $\mathbb{G}_{a,S}$-invariant polynomial $z_{1}=z-x^{\mu-n}Q_{0}(y)$ is a variable of $A[y,z,u]$ over $A[y,u]$, and so $\partial$ is conjugate to the derivation $x^{n}\partial_{y}+p(y,z_{1}+x^{\mu-n}Q_{0}(y))\partial_{u}$ of the polynomial ring in two variables $A[z_{1}][y,u]$ over $A[z_{1}]$. Since $\sigma$ is fixed point free, Proposition \ref{prop:Rank2-bundle} implies that it is equivariantly trivial with geometric quotient isomorphic to the total space of the trivial $\mathbb{A}^{1}$-bundle over $\mathbb{A}_{S}^{1}=\mathrm{Spec}(A[z_{1}])$ whence to $\mathbb{A}_{S}^{2}$.
Otherwise, if $\mu<n$, then the $\mathbb{G}_{a,S}$-action $\tilde{\sigma}:\mathbb{G}_{a,S}\times_{S}\mathbb{A}_{S}^{3}\rightarrow\mathbb{A}_{S}^{3}$ on $\mathbb{A}_{S}^{3}=\mathrm{Spec}(A[\tilde{y},\tilde{z},\tilde{u}])$ generated by the $A$-derivation \[ \tilde{\partial}=x^{n-\mu}\partial_{\tilde{y}}+q_{0}(\tilde{y})\partial_{\tilde{z}}+(1+x^{\nu}p_{0}(\tilde{y},\tilde{z}))\partial_{\tilde{u}} \] is again fixed point free, hence admits a geometric quotient $\tilde{\rho}:\mathbb{A}_{S}^{3}\rightarrow\tilde{\mathfrak{X}}=\mathbb{A}_{S}^{3}/\mathbb{G}_{a,S}$ in the form of an \'etale locally trivial $\mathbb{G}_{a,S}$-bundle over a certain algebraic $S$-space $\tilde{\mathfrak{X}}$. \begin{lem} The quotient spaces $\mathfrak{X}=\mathbb{A}_{S}^{3}/\mathbb{G}_{a,S}$ and $\tilde{\mathfrak{X}}=\mathbb{A}_{S}^{3}/\mathbb{G}_{a,S}$ for the $\mathbb{G}_{a,S}$-actions $\sigma$ and $\tilde{\sigma}$ on $\mathbb{A}_{S}^{3}$ generated by $\partial$ and $\tilde{\partial}$ respectively are isomorphic. In particular $\sigma$ is proper (resp. equivariantly trivial) if and only if $\tilde{\sigma}$ is proper (resp. equivariantly trivial). \end{lem} \begin{proof} Letting $\tilde{\rho}_{i}:V_{i}=\mathbb{A}_{S}^{3}\rightarrow\tilde{\mathfrak{X}}_{i}=V_{i}/\mathbb{G}_{a,S}$, $i=0,\ldots,\mu$, denote the geometric quotient of $V_{i}=\mathrm{Spec}(A[\tilde{y}_{i},\tilde{z}_{i},\tilde{u}_{i}])$ for the fixed point free $\mathbb{G}_{a,S}$-action $\tilde{\sigma}_{i}$ generated by the $A$-derivation \[ \tilde{\partial}_{i}=\left(1+x^{\nu}p_{0}(\tilde{y}_{i},\tilde{z}_{i})\right)\partial_{\tilde{u}_{i}}+x^{\mu-i}q_{0}(\tilde{y}_{i})\partial_{\tilde{z}_{i}}+x^{n-i}\partial_{\tilde{y}_{i}}, \] the first assertion will follow from the more general fact that $\tilde{\mathfrak{X}}_{i}\simeq\tilde{\mathfrak{X}}_{i+1}$ for every $i=0,\ldots,\mu-1$. Indeed, we first observe that since $\tilde{u}_{i}$ is a slice for $\tilde{\partial}_{i}$ modulo $x$, $\tilde{\mathfrak{X}}_{i,\mathfrak{m}}=\tilde{\mathfrak{X}}_{i}\times_{S}\mathrm{Spec}(\kappa)$ is isomorphic to $\mathbb{A}_{\kappa}^{2}=\mathrm{Spec}((A/\mathfrak{m})[\tilde{y}_{i},\tilde{z}_{i}])$ and the restriction of $\tilde{\rho}_{i}$ over $\tilde{\mathfrak{X}}_{i,\mathfrak{m}}$ is isomorphic to the trivial bundle $\mathrm{pr}_{1}:\tilde{\mathfrak{X}}_{i,\mathfrak{m}}\times_{\kappa}\mathrm{Spec}(\kappa[\tilde{u}_{i}])\rightarrow\tilde{\mathfrak{X}}_{i,\mathfrak{m}}$. Now let $\beta_{i}:V_{i+1}\rightarrow V_{i}$ be the affine modification of the total space of $\tilde{\rho}_{i}:\mathbb{A}_{S}^{3}\rightarrow\tilde{\mathfrak{X}}_{i}$ with center at the zero section of the induced bundle $\mathrm{pr}_{1}:\tilde{\mathfrak{X}}_{i,\mathfrak{m}}\times_{\kappa}\mathrm{Spec}(\kappa[\tilde{u}_{i}])\rightarrow\tilde{\mathfrak{X}}_{i,\mathfrak{m}}$ and with principal divisor $x$. In view of the previous description, $\beta_{i}:V_{i+1}\rightarrow V_{i}$ coincides with the affine modification of $\mathrm{Spec}(A[\tilde{y}_{i},\tilde{z}_{i},\tilde{u}_{i}])$ with center at the ideal $(x,\tilde{u}_{i})$ and principal divisor $x$, that is, with the birational $S$-morphism induced by the homomorphism of $A$-algebra \[ \beta_{i}^{*}:A[\tilde{y}_{i+1},\tilde{z}_{i+1},\tilde{u}_{i+1}]\rightarrow A[\tilde{y}_{i},\tilde{z}_{i},\tilde{u}_{i}],\;(\tilde{y}_{i+1},\tilde{z}_{i+1},\tilde{u}_{i+1})\mapsto(\tilde{y}_{i},\tilde{z}_{i},x\tilde{u}_{i}). \] By construction, $\beta_{i}$ is equivariant for the $\mathbb{G}_{a,S}$-actions $\tilde{\sigma}_{i+1}$ and $\overline{\sigma}_{i}$ generated respectively by the locally nilpotent $A$-derivations $\tilde{\partial}_{i+1}$ of $A[\tilde{y}_{i+1},\tilde{z}_{i+1},\tilde{u}_{i+1}]$ and $\overline{\partial}_{i}=x\tilde{\partial}_{i}$ of $A[\tilde{y}_{i},\tilde{z}_{i},\tilde{u}_{i}]$. Furthermore, since $\tilde{\rho}_{i}:V_{i}\rightarrow\tilde{\mathfrak{X}}_{i}$ is also $\mathbb{G}_{a,S}$-invariant for the action $\overline{\sigma}_{i}$, the morphism $\tilde{\rho}_{i}\circ\beta_{i}:V_{i+1}\rightarrow\tilde{\mathfrak{X}}_{i}$ is $\mathbb{G}_{a,S}$-invariant, whence descends to a morphism $\tilde{\beta}_{i}:\tilde{\mathfrak{X}}_{i+1}\rightarrow\tilde{\mathfrak{X}}_{i}$. Since the latter restricts to an isomorphism over the generic point of $S$, it remains to check that it is also an isomorphism in a neighborhood of every point $p\in\tilde{\mathfrak{X}}_{i}$ lying over the closed point $\mathfrak{m}$ of $S$. Let $f:U=\mathrm{Spec}(B)\rightarrow\tilde{\mathfrak{X}}_{i}$ be an affine \'etale neighborhood of such a point $p\in\tilde{\mathfrak{X}}_{i}$ over which $\tilde{\rho}_{i}:V_{i}\rightarrow\tilde{\mathfrak{X}}_{i}$ becomes trivial, say $V_{i}\times_{\tilde{\mathfrak{X}}_{i}}U\simeq\mathbb{A}_{U}^{1}=\mathrm{Spec}(B[\tilde{v}_{i}])$. The $\mathbb{G}_{a,S}$-action on $V_{i}$ generated by $\overline{\partial}_{i}$ lifts to the $\mathbb{G}_{a,U}$-action on $\mathbb{A}_{U}^{1}$ generated by the locally nilpotent $B$-derivation $x\partial_{\tilde{v}_{i}}$ and since $\beta_{i}:V_{i+1}\rightarrow V_{i}$ is the affine modification of $V_{i}$ with center at the zero section of the restriction of $\tilde{\rho}_{i}:V_{i}\rightarrow\tilde{\mathfrak{X}}_{i}$ over the closed point of $S$, we have a commutative diagram \[\xymatrix@R=12pt@C=11pt{ & V_{i+1} \ar'[d][dd]_(.3){\tilde{\rho}_{i+1}} \ar[dl]_{\beta_i} && \mathbb{A}^1_U \ar[ll] \ar[dd]^{\mathrm{pr}_U} \ar[dl]_{\delta_i} \\ V_i \ar[dd]_{\tilde{\rho}_i} && \mathbb{A}^1_U \ar[ll] \ar[dd]^(.3){\mathrm{pr}_U} \\ & \tilde{\mathfrak{X}}_{i+1} \ar[dl]_{\tilde{\beta}_i} && U \ar@{=}[dl] \ar'[l][ll] \\ \tilde{\mathfrak{X}}_i && U \ar[ll]_{f} }\]in which the top and front squares are cartesian, and where the morphism $\delta_{i}:\mathbb{A}_{U}^{1}=\mathrm{Spec}(B[\tilde{v}_{i+1}])\rightarrow\mathbb{A}_{U}^{1}=\mathrm{Spec}(B[\tilde{v}_{i}])$ is defined by the $B$-algebras homomorphism $B[\tilde{v}_{i}]\rightarrow B[\tilde{v}_{i+1}]$, $\tilde{v}_{i}\mapsto x\tilde{v}_{i+1}$. The latter is equivariant for the action on $\mathrm{Spec}(B[\tilde{v}_{i+1}])$ generated by the locally nilpotent $B$-derivation $\partial_{\tilde{v}_{i+1}}$ and we conclude that $\mathrm{pr}_{2}:\mathbb{A}_{U}^{1}\simeq\mathbb{A}_{U}^{1}\times_{V_{i}}V_{i+1}\rightarrow V_{i+1}$ is an \'etale trivialization of the $\mathbb{G}_{a,S}$-action induced by $\tilde{\sigma}_{i+1}$ on the open sub-scheme $(\tilde{\rho}_{i}\circ\beta_{i})^{-1}(f(U))$ of $V_{i+1}$. This implies in turn that $U\times_{\tilde{\mathfrak{X}}_{i}}\tilde{\mathfrak{X}}_{i+1}\simeq U$, whence that $\tilde{\beta}_{i}:\tilde{\mathfrak{X}}_{i+1}\rightarrow\tilde{\mathfrak{X}}_{i}$ is an isomorphism in a neighborhood of $p\in\tilde{\mathfrak{X}}_{i}$ as desired.
The second assertion is a direct consequence of the fact that properness and global equivariant triviality of $\sigma$ and $\tilde{\sigma}$ are respectively equivalent to the separatedness and the affineness of the geometric quotients $\mathfrak{X}\simeq\tilde{\mathfrak{X}}$. \end{proof} \begin{parn} Summing up, we are now reduced to proving that a proper $\mathbb{G}_{a,S}$-action on $\mathbb{A}_{S}^{3}$ generated by an $A$-derivation \[ \partial=x^{n}\partial_{y}+q(y)\partial_{z}+p(y,z)\partial_{u} \] of $A[y,z,u]$, such that $n>0$ and $q(y)\in A[y]$ has non constant residue class in $\kappa[y]$, has affine geometric quotient $\mathfrak{X}=\mathbb{A}_{S}^{3}/\mathbb{G}_{a,S}$. This will be done in two steps in the next sections: we will first establish that a proper $\mathbb{G}_{a,S}$-action as above is conjugate to one generated by a special type of $A$-derivation called \emph{twin-triangular.} Then we will prove in section \ref{sec:Twin-Triviality} that proper twin-triangular $\mathbb{G}_{a,S}$-actions on $\mathbb{A}_{S}^{3}$ do indeed have affine geometric quotients.
\end{parn}
\section{Reduction to twin-triangular actions}
We keep the same notation as in \S \ref{par:local_notation} above, namely $A$ is a discrete valuation ring containing a field of characteristic $0$, with maximal ideal $\mathfrak{m}$, residue field $\kappa=A/\mathfrak{m}$, and uniformizing parameter $x\in\mathfrak{m}$. We let again $S=\mathrm{Spec}(A)$.
We call an $A$-derivation $\partial$ of $A[y,z,u]$ \emph{twin-triangulable} if there exists a coordinate system $(y,z_{+},z_{-})$ of $A[y,z,u]$ over $A[y]$ in which the conjugate of $\partial$ is \emph{twin-triangular}, that is, has the form $x^{n}\partial_{y}+p_{+}(y)\partial_{z_{+}}+p_{-}(y)\partial_{z_{-}}$ for certain polynomials $p_{\pm}(y)\in A[y]$. This section is devoted to the proof of the following intermediate characterization of proper triangular $\mathbb{G}_{a,S}$-actions: \begin{prop} \label{prop:Main-Local} With the notation above, let $\partial$ by an $A$-derivation of $A[y,z,u]$ of the form \[ \partial=x^{n}\partial_{y}+q(y)\partial_{z}+p(y,z)\partial_{u} \] where $n>0$ and where $q(y)\in A[y]$ has non constant residue class in $\kappa[y]$. If the $\mathbb{G}_{a,S}$-action on $\mathbb{A}_{S}^{3}=\mathrm{Spec}(A[y,z,u])$ generated by $\partial$ is proper, then $\partial$ is twin-triangulable. \end{prop} \noindent The proof given below proceeds in two steps: we first construct a coordinate $\tilde{u}$ of $A[y,z,u]$ over $A[y,z]$ with the property that $\partial\tilde{u}=\tilde{p}(y,z)$ is either a polynomial in $y$ only or its leading term $\tilde{p}_{\ell}(y)$ as a polynomial in $z$ has a very particular form. In the second case, we exploit the properties of $\tilde{p}_{\ell}(y)$ to show that the $\mathbb{G}_{a,S}$-action generated by $\partial$ is not proper.
\subsection{The $\sharp$-reduction of a triangular $A$-derivation}
The conjugate of an $A$-derivation $\partial=x^{n}\partial_{y}+q(y)\partial_{z}+p(y,z)\partial_{u}$ of $A[y,z,u]$ as in Proposition \ref{prop:Main-Local} by an isomorphism of $A[y,z]$-algebras $\psi:A[y,z][\tilde{u}]\stackrel{\sim}{\rightarrow}A[y,z][u]$ is again triangular of the form \[ \psi^{-1}\partial\psi=x^{n}\partial_{y}+q(y)\partial_{z}+\tilde{p}(y,z)\partial_{\tilde{u}} \] for some polynomial $\tilde{p}(y,z)\in A[y,z]$. In particular, we may choose from the very beginning a coordinate system of $A[y,z,u]$ over $A[y,z]$ with the property that the degree of $\partial u\in A[y,z]$ with respect to $z$ is minimal among all possible conjugates $\psi^{-1}\partial\psi$ of $\partial$ as above. In what follows, we will say for short that such a derivation $\partial$ is \emph{$\sharp$-reduced} with respect to the coordinate system $(y,z,u)$. Letting $Q(y)=\int_{0}^{y}q(\tau)d\tau\in A[y]$, this property can be characterized effectively as follows: \begin{lem} Let $\partial=x^{n}\partial_{y}+q(y)\partial_{z}+p(y,z)\partial_{u}$ be a $\sharp$-reduced derivation of $A[y,z,u]$ as in Proposition \ref{prop:Main-Local}. If $\partial$ is not twin-triangular $($i.e. $p(y,z)=p_{0}(y)\in A[y]$$)$ then the leading term $p_{\ell}(y)$, $\ell\geq1$, of $p(y,z)$ as a polynomial in $z$ is not congruent modulo $x^{n}$ to a polynomial of the form $q(y)f(Q(y))$ for some $f(\tau)\in A[\tau]$. \end{lem} \begin{proof} Suppose that $p(y,z)=\sum_{r=0}^{\ell}p_{r}(y)z^{r}$ with $\ell\geq1$ and that $p_{\ell}(y)=q(y)f(Q(y))+x^{n}g(y)$ for some polynomials $f(\tau),g(\tau)\in A[\tau]$. Then letting $G(y)=\int_{0}^{y}g(\tau)d\tau$ and \[ \tilde{u}=u-G(y)z^{\ell}-\sum_{k=0}^{\deg f}\frac{(-1)^{k}}{\prod_{j=0}^{k}(\ell+1+j)}f^{(k)}(Q(y))x^{kn}z^{\ell+1+k}, \] one checks by direct computation that \[ \partial\tilde{u}=\sum_{r=0}^{\ell-2}p_{r}(y)z^{r}+\left(p_{\ell-1}(y)-G(y)q(y)\right)z^{\ell-1}. \] Thus $(y,z,\tilde{u}$) is a coordinate system of $A[y,z,u]$ over $A[y,z]$ in which the image of $\tilde{u}$ by the conjugate of $\partial$ has degree $\leq\ell-1$, a contradiction to the $\sharp$-reducedness of $\partial$. \end{proof} \noindent To prove Proposition \ref{prop:Main-Local}, it remains to show that a proper $\mathbb{G}_{a,S}$-action on $\mathbb{A}_{S}^{3}$ generated by $\sharp$-reduced $A$-derivation of $A[y,z,u]$ is twin-triangular. This is done in the next sub-section.
\subsection{A non-valuative criterion for non-properness}
\indent\newline\noindent To disprove the properness of an algebraic action $\sigma:\mathbb{G}_{a,S}\times_{S}E\rightarrow E$ of $\mathbb{G}_{a,S}$ on an $S$-scheme $E$, it suffices in principle to check that the image of $\Phi=(\mathrm{pr}_{2},\sigma):\mathbb{G}_{a}\times_{S}E\rightarrow E\times_{S}E$ is not closed. However, this image turns out to be complicated to determine in general, and it is more convenient for our purpose to consider the following auxiliary construction: letting $j:\mathbb{G}_{a,S}\simeq\mathrm{Spec}(\mathcal{O}_{S}[t])\hookrightarrow\mathbb{P}_{S}^{1}=\mathrm{Proj}(\mathcal{O}_{S}[w_{0},w_{1}])$, $t\mapsto[t:1]$ be the natural open immersion, the properness of the projection $\mathrm{pr}_{E\times_{S}E}:\mathbb{P}_{S}^{1}\times_{S}E\times_{S}E\rightarrow E\times_{S}E$ implies that $(\mathrm{p}_{2},\sigma)$ is proper if and only if $\varphi=(j\circ\mathrm{pr}_{1},\mathrm{pr}_{2},\sigma):\mathbb{G}_{a,S}\times_{S}E\rightarrow\mathbb{P}_{S}^{1}\times_{S}E\times_{S}E$ is proper, hence a closed immersion. Therefore the non properness of $\sigma$ is equivalent to the fact that the closure of $\mathrm{Im}(\varphi)$ in $\mathbb{P}_{S}^{1}\times_{S}E\times_{S}E$ intersects the ``boundary'' $\{w_{1}=0\}$ in a nontrivial way.
\begin{parn} Now let $\sigma:\mathbb{G}_{a,S}\times_{S}\mathbb{A}_{S}^{3}\rightarrow\mathbb{A}_{S}^{3}$ be the $\mathbb{G}_{a,S}$-action generated by a non twin-triangular $\sharp$-reduced $A$-derivation $\partial=x^{n}\partial_{y}+q(y)\partial_{z}+p(y,z)\partial_{u}$ of $A[y,z,u]$ and let \[ \varphi=(j\circ\mathrm{pr}_{1},\mathrm{pr}_{2},\mu):\mathbb{G}_{a,S}\times_{S}\mathbb{A}_{S}^{3}=\mathrm{Spec}(A[t][y,z,u])\rightarrow\mathbb{P}_{S}^{1}\times_{S}\mathbb{A}_{S}^{3}\times_{S}\mathbb{A}_{S}^{3} \] be the corresponding immersion. To disprove the properness of $\sigma$, it is enough to check that the image by $\varphi$ of the closed sub-scheme $H=\left\{ z=0\right\} \simeq\mathrm{Spec}(A[t][y,u])$ of $\mathbb{G}_{a,S}\times_{S}\mathbb{A}_{S}^{3}$ is not closed in $\mathbb{P}_{S}^{1}\times_{S}\mathbb{A}_{S}^{3}\times_{S}\mathbb{A}_{S}^{3}$. After identifying $A[y,z,u]\otimes_{A}A[y,z,u]$ with the polynomial ring $A[y_{1},y_{2},z_{1},z_{2},u_{1},u_{2}]$ in the obvious way, the image of $H$ by $(\mathrm{pr}_{1},\mathrm{pr}_{2},\sigma):\mathbb{G}_{a,S}\times_{S}\mathbb{A}_{S}^{3}\rightarrow\mathbb{A}_{S}^{1}\times_{S}\mathbb{A}_{S}^{3}\times_{S}\mathbb{A}_{S}^{3}$ is equal to the closed sub-scheme of $\mathrm{Spec}(A[t][y_{1},y_{2},z_{1},z_{2},u_{1},u_{2}])$ defined by the following system of equations \[ \begin{cases} y_{2} & =y_{1}+x^{n}t\\ z_{1} & =0\\ z_{2} & =x^{-n}(Q(y_{1}+x^{n}t)-Q(y_{1}))=(y_{1}-y_{2})^{-1}(Q(y_{2})-Q(y_{1}))t\\ u_{2} & =u_{1}+x^{-n}\int_{0}^{t}p(y_{1}+x^{n}\tau)(Q(y_{1}+x^{n}\tau)-Q(y_{1})))d\tau. \end{cases} \] Letting $p(y,z)=\sum_{r=0}^{\ell}p_{r}(y)z^{r}$ with $\ell\geq1$ and \[ \Gamma_{r}(y_{1},y_{2})=\int_{y_{1}}^{y_{2}}p_{r}(\xi)(Q(\xi)-Q(y_{1}))^{r}d\xi\in A[y_{1},y_{2}],\quad r=0,\ldots,\ell, \] the last equality can be re-written modulo the first ones in the form \begin{eqnarray*} u_{2} & = & u_{1}+\sum_{r=0}^{\ell}x^{-nr}\int_{0}^{t}p_{r}(y_{1}+x^{n}\tau)(Q(y_{1}+x^{n}\tau)-Q(y_{1}))^{r}d\tau\\
& = & u_{1}+t(y_{2}-y_{1})^{-1}\sum_{r=0}^{\ell}x^{-nr}\int_{y_{1}}^{y_{2}}p_{r}(\xi)(Q(\xi)-Q(y_{1}))^{r}d\xi\\
& = & u_{1}+\sum_{r=0}^{\ell}\left((y_{2}-y_{1})^{-r-1}\Gamma_{r}(y_{1},y_{2})\right)t^{r+1}. \end{eqnarray*} It follows that the closure $V$ of $\varphi(H)$ is contained in the closed sub-scheme $W$ of $\mathbb{P}_{S}^{1}\times_{S}\mathbb{A}_{S}^{3}\times_{S}\mathbb{A}_{S}^{3}$ defined by the equations $z_{1}=0$ and \[ \begin{cases} (y_{2}-y_{1})w_{1}-x^{n}w_{0} & =0\\ w_{1}z_{2}-(y_{2}-y_{1})^{-1}(Q(y_{2})-Q(y_{1}))w_{0} & =0\\ w_{1}^{\ell+1}(u_{2}-u_{1})-\sum_{r=0}^{\ell}\left((y_{2}-y_{1})^{-r-1}\Gamma_{r}(y_{1},y_{2})\right)w_{0}^{r+1}w_{1}^{\ell-r} & =0. \end{cases} \] We further observe that $W$ is irreducible, whence equal to $V$, provided that $\Gamma_{\ell}(y_{1},y_{2})\in A[y_{1},y_{2}]$ does not belong to the ideal generated by $x^{n}$ and $Q(y_{2})-Q(y_{1})$. If so, then $W=V$ intersects $\{w_{1}=0\}$ along a closed sub-scheme $Z$ isomorphic to the spectrum of the following algebra:
\[ \left(A[y_{1},y_{2}]/(x^{n},(y_{2}-y_{1})^{-1}(Q(y_{2})-Q(y_{1})),(y_{2}-y_{1})^{-\ell-1}\Gamma_{\ell}(y_{1},y_{2}))\right)[z_{2},u_{1},u_{2}]. \] Since by virtue of the $\sharp$-reducedness assumption $p_{\ell}(y)$ is not of the form $q(y)f(Q(y))+x^{n}g(y)$, the non properness of $\sigma:\mathbb{G}_{a,S}\times_{S}\mathbb{A}_{S}^{3}\rightarrow\mathbb{A}_{S}^{3}$ is then a consequence of the following Lemma which guarantees precisely that $\Gamma_{\ell}(y_{1},y_{2})\not\in(x^{n},Q(y_{2})-Q(y_{1}))A[y_{1},y_{2}]$ and that $Z$ is not empty.
\end{parn} \begin{lem} Let $q(y)\in A[y]$ be a polynomial with non constant residue class in $\kappa[y]$ and let $Q(y)=\int_{0}^{y}q(\tau)d\tau$. For a polynomial $p(y)\in A[y]$ and an integer $\ell\geq1$, the following holds:
a) The polynomial $\Gamma_{\ell}(y_{1},y_{2})=\int_{y_{1}}^{y_{2}}p(y)(Q(y)-Q(y_{1}))^{\ell}dy$ belongs to the ideal $(x^{n},Q(y_{2})-Q(y_{1}))$ if and only if $p(y)$ can be written in the form $q(y)f(Q(y))+x^{n}g(y)$ for certain polynomials $f(\tau),g(\tau)\in A[\tau]$.
b) The polynomial $(y_{2}-y_{1})^{-\ell-1}\Gamma_{\ell}(y_{1},y_{2})$ is not invertible modulo the ideal $(x^{n},(y_{2}-y_{1})^{-1}(Q(y_{2})-Q(y_{1})))$. \end{lem} \begin{proof} For the first assertion, a sequence of $\ell$ successive integrations by parts shows that \begin{eqnarray*} \Gamma_{\ell}(y_{1},y_{2}) & = & \left[E_{1}(y)(Q(y)-Q(y_{1}))^{\ell}\right]_{y_{1}}^{y_{2}}-\ell\int_{y_{1}}^{y_{2}}E_{1}(y)q(y)(Q(y)-Q(y_{1}))^{\ell-1}dy\\
& = & S(y_{1},y_{2})+(-1)^{\ell}\ell!\int_{y_{1}}^{y_{2}}E_{\ell}(y)q(y)dy\\
& = & S(y_{1},y_{2})+(-1)^{\ell}\ell!(E_{\ell+1}(y_{2})-E_{\ell+1}(y_{1})) \end{eqnarray*} where $E_{k}$ is defined recursively by $E_{1}(y)=\int_{0}^{y}p(\tau)d\tau$ and $E_{k+1}(y)=\int_{0}^{y}E_{k}(\tau)q(\tau)d\tau$, and where $S(y_{1},y_{2})\in(Q(y_{2})-Q(y_{1}))A[y_{1},y_{2}]$. So $\int_{y_{1}}^{y_{2}}p(y)(Q(y)-Q(y_{1}))^{r}dy$ belongs to $(x^{n},Q(y_{2})-Q(y_{1}))A[y_{1},y_{2}]$ if and only if $E_{\ell+1}(y_{2})-E_{\ell+1}(y_{1})$ belongs to this ideal.
Since the residue class of $Q(y)\in A[y]$ in $\kappa[y]$ is not constant, it follows from the local criterion for flatness that $A[y]$ is a faithfully flat algebra over $A[Q(y)]$. By faithfully flat descent, this implies in turn that the sequence \[ A[Q(y)]\hookrightarrow A[y]\stackrel{\cdot\otimes1-1\otimes\cdot}{\longrightarrow}A[y]\otimes_{A[\tau]}A[y] \] is exact whence, using the natural identification $A[y]\otimes_{A[\tau]}A[y]\simeq A[y_{1},y_{2}]/(Q(y_{2})-Q(y_{1}))$, that a polynomial $F\in A[y]$ with $F(y_{2})-F(y_{1})$ belonging to the ideal $(Q(y_{2})-Q(y_{1}))A[y_{1},y_{2}]$ has the form $F(y)=G(Q(y))$ for a certain polynomial $G(\tau)\in A[\tau]$. Thus $E_{\ell+1}(y_{2})-E_{\ell+1}(y_{1})$ belongs to $(x^{n},Q(y_{2})-Q(y_{1}))A[y_{1},y_{2}]$, if and only if $E_{\ell+1}(y)$ is of the form $G(Q(y))+x^{n}R_{\ell+1}(y)$ for some $G(\tau),R_{\ell+1}(\tau)\in A[\tau]$. This implies in turn that $E_{\ell}(y)q(y)=G'(Q(y))q(y)+x^{n}R_{\ell+1}'(y)$ whence, since $q(y)\in A[y]\setminus\mathfrak{m}A[y]$ is not a zero divisor modulo $x^{n}$, that $E_{\ell}(y)=G'(Q(y))+x^{n}R_{\ell}(y)$ for a certain $R_{\ell}(\tau)\in A[\tau]$. We conclude by induction that $E_{1}(y)=G^{(\ell+1)}(Q(y))+x^{n}R_{1}(y)$ and finally that $p(y)=G^{(\ell+2)}(Q(y))q(y)+x^{n}R(y)$ for a certain $R(\tau)\in A[\tau]$. This proves a).
The second assertion is clear in the case where $p(y)\in\mathfrak{m}A[y]$. Otherwise, if $p(y)\in A[y]\setminus\mathfrak{m}A[y]$ then reducing modulo $x$ and passing to the algebraic closure $\overline{\kappa}$ of $\kappa$, it is enough to show that if $q(y)\in\overline{\kappa}[y]$ is not constant and $p(y)\in\overline{\kappa}[y]$ is a nonzero polynomial then for every $\ell\geq1$, the affine curves $C$ and $D$ in $\mathbb{A}_{\overline{\kappa}}^{2}=\mathrm{Spec}(\overline{\kappa}[y_{1},y_{2}])$ defined by the vanishing of the polynomials $\Theta(y_{1},y_{2})=(y_{2}-y_{1})^{-\ell-1}\int_{y_{1}}^{y_{2}}p(y)(Q(y)-Q(y_{1}))^{\ell}dy$ and $R(y_{1},y_{2})=(y_{2}-y_{1})^{-1}\int_{y_{1}}^{y_{2}}q(y)dy$ respectively always intersect each other. Suppose on the contrary that $C\cap D=\emptyset$ and let $m=\deg q\geq1$ and $d=\deg p\geq0$. Then the closures $\overline{C}$ and $\overline{D}$ of $C$ and $D$ respectively in $\mathbb{P}_{\overline{\kappa}}^{2}=\mathrm{Proj}(\overline{\kappa}[y_{1},y_{2},y_{3}])$ intersect each others along a closed sub-scheme $Y$ of length $\deg\overline{C}\cdot\deg\overline{D}=m(d+\ell m)$ supported on the line $\{y_{3}=0\}\simeq\mathrm{Proj}(\overline{\kappa}[y_{1},y_{2}])$. By definition, up to multiplication by a nonzero scalar, the top homogeneous components of $R$ and $\Theta$ have the form $\prod_{i=1}^{m}(y_{2}-\zeta^{i}y_{1})$, where $\zeta\in\overline{\kappa}$ is a primitive $(m+1)$-th root of unity, and $(y_{2}-y_{1})^{\ell-1}\int_{y_{1}}^{y_{2}}y^{d}(y^{m+1}-y_{1}^{m+1})^{\ell}dy$ respectively. But on the other hand, we have for every $i=1,\ldots,m$
\[ \overline{\kappa}[y_{2}]/(y_{2}-\zeta^{i},(y_{2}-1)^{-r-1}\int_{1}^{y_{2}}y^{d}(y^{m+1}-1)^{r}dy)\simeq\overline{\kappa}[y_{2}]/(y_{2}-\zeta^{i},(\zeta^{i}-1)^{-r-1}\int_{1}^{\zeta^{i}}\tau^{d}(\tau^{m+1}-1)^{r}d\tau), \] and hence the length of the above algebra is either $1$ or $0$ depending on whether $\int_{1}^{\zeta^{i}}\tau^{d}(\tau^{m+1}-1)d\tau\in\overline{\kappa}$ is zero or not. This implies that the length of $Y$ is at most equal to $m$ and so the only possibility would be that $d=0$ and $\ell=m=1$, i.e. $C$ and $D$ are parallel lines in $\mathbb{A}_{\overline{\kappa}}^{2}$. But since $\int_{1}^{-1}(\tau^{2}-1)d\tau\neq0$, this last possibility is also excluded. \end{proof}
\section{\label{sec:Twin-Triviality} Global equivariant triviality of twin-triangular actions}
By virtue of Proposition \ref{prop:Main-Local}, every proper triangular $\mathbb{G}_{a,S}$-action on $\mbox{\ensuremath{\sigma}:}\mathbb{G}_{a,S}\times_{S}\mathbb{A}_{S}^{3}\rightarrow\mathbb{A}_{S}^{3}$ on $\mathbb{A}_{S}^{3}$ is conjugate to one generated by a twin-triangular $A$-derivation $\partial$ of $A[y,z_{+},z_{-}]$ of the form \[ \partial=x^{n}\partial_{y}+p_{+}(y)\partial_{z_{+}}+p_{-}(y)\partial_{z_{-}} \] for certain polynomials $p_{\pm}(y)\in A[y]$. So to complete the proof of the Main Theorem, it remains to show the following generalization of the main result in \cite{DubFin11}: \begin{prop} \label{prop:Twin-Loc-trivi} Let $S$ be the spectrum of discrete valuation $A$ containing a field of characteristic $0$. Then a proper twin-triangular $\mathbb{G}_{a,S}$-action on $\mathbb{A}_{S}^{3}$ has affine geometric quotient $\mathfrak{X}=\mathbb{A}_{S}^{3}/\mathbb{G}_{a,S}$. \end{prop} \begin{parn} The principle of the proof given below is the following: we exploit the twin triangularity to construct two $\mathbb{G}_{a,S}$-invariant principal open subsets $W_{\Gamma_{+}}$ and $W_{\Gamma_{-}}$ in $\mathbb{A}_{S}^{3}$ with the property that the union of corresponding principal open subspaces $\mathfrak{X}_{\Gamma_{\pm}}=W_{\Gamma_{\pm}}/\mathbb{G}_{a,S}$ of $\mathfrak{X}$ covers the closed fiber of the structure morphism $\mathrm{p}:\mathfrak{X}\rightarrow S$. We then show that $\mathfrak{X}_{\Gamma_{+}}$ and $\mathfrak{X}_{\Gamma_{-}}$ are in fact affine sub-schemes of $\mathfrak{X}$. On the other hand, since $\partial$ admits $x^{-n}y$ as a global slice over $A_{x}$, the generic fiber of $\mathrm{p}$ is isomorphic to the affine plane over the function field $A_{x}$ of $S$. So it follows that $\mathfrak{X}$ is covered by three principal affine open sub-schemes $\mathfrak{X}_{\Gamma_{+}}$, $\mathfrak{X}_{\Gamma_{-}}$ and $\mathfrak{X}_{x}$ corresponding to regular functions $x$, $\Gamma_{+}$, $\Gamma_{-}$ which generate the unit ideal in $\Gamma(\mathfrak{X},\mathcal{O}_{\mathfrak{X}})\simeq A[y,z_{+},z_{-}]^{\mathbb{G}_{a,S}}\subset A[y,z_{+},z_{-}]$, whence is an affine scheme.
\end{parn}
\begin{parn} The fact that the affineness of $\mathrm{p}:\mathfrak{X}=\mathbb{A}_{S}^{3}/\mathbb{G}_{a,S}\rightarrow S=\mathrm{Spec}(A)$ is a local property with respect to the fpqc topology on $S$ \cite[VIII.5.6]{SGA1} enables a reduction to the case where the discrete valuation ring $A$ is Henselian or complete. Since it contains a field of characteristic zero, an elementary application of Hensel's Lemma implies that a maximal subfield of such a local ring $A$ is a field of representatives, i.e. a subfield which is mapped isomorphically by the quotient projection $A\mapsto A/\mathfrak{m}$ onto the residue field $\kappa=A/\mathfrak{m}$. This is in fact the only property of $A$ that we will use in the sequel. So from now on, $(A,\mathfrak{m},\kappa)$ is a discrete valuation ring containing a field $\kappa$ of characteristic $0$ and with residue field $A/\mathfrak{m}\simeq\kappa$.
\end{parn}
\subsection{Twin-triangular actions in general position and associated invariant covering}
Here we construct a pair of principal $\mathbb{G}_{a,S}$-invariant open subsets $W_{\pm}=W_{\Gamma_{\pm}}$ of $\mathbb{A}_{S}^{3}$ associated with a twin-triangular $A$-derivation of $A[y,z_{+},z_{-}]$ whose geometric quotients will be studied in the next sub-section. We begin with a technical condition which will be used to guarantee that the union of $W_{+}$ and $W_{-}$ covers the closed fiber of the projection $\mathrm{pr}_{S}:\mathbb{A}_{S}^{3}\rightarrow S$. \begin{defn} \label{def:general_position} Let $(A,\mathfrak{m},\kappa)$ be a discrete valuation valuation ring containing a field of characteristic $0$ and let $x\in\mathfrak{m}$ be a uniformizing parameter. A twin-triangular $A$-derivation $\partial=x^{n}\partial_{y}+p_{+}(y)\partial_{z_{+}}+p_{-}(y)\partial_{z_{-}}$ of $A[y,z_{+},z_{-}]$ is said to be in \emph{general position} if it satisfies the following properties:
a) The residue classes $\overline{p}_{\pm}\in\kappa[y]$ of the polynomials $p_{\pm}\in A[y]$ modulo $\mathfrak{m}$ are both non zero and relatively prime,
b) There exist integrals $\overline{P}_{\pm}\in A[y]$ of $\overline{p}_{\pm}$ with respect to $y$ for which the inverse images of the branch loci of the morphisms $\overline{P}_{+}:\mathbb{A}_{\kappa}^{1}\rightarrow\mathbb{A}_{\kappa}^{1}$ and $\overline{P}_{-}:\mathbb{A}_{\kappa}^{1}\rightarrow\mathbb{A}_{\kappa}^{1}$ are disjoint. \end{defn} \begin{lem} \label{lem:Bad-Plane-Removal} With the notation above, every twin-triangular $A$-derivation $\partial$ of $A[y,z_{+},z_{-}]$ generating a fixed point free $\mathbb{G}_{a,S}$-action on $\mathbb{A}_{S}^{3}$ is conjugate to one in general position. \end{lem} \begin{proof} A twin-triangular derivation $\partial=x^{n}\partial_{y}+p_{+}(y)\partial_{z_{+}}+p_{-}(y)\partial_{z_{-}}$ generates a fixed point free $\mathbb{G}_{a,S}$-action if and only if $x^{n}$, $p_{+}(y)$ and $p_{-}(y)$ generate the unit ideal in $A[y,z_{+},z_{-}]$. So the residue classes $\overline{p}_{+}$ and $\overline{p}_{-}$ of $p_{+}$ and $p_{-}$ are relatively prime and at least one of them, say $\overline{p}_{-}$, is nonzero. If $\overline{p}_{+}=0$ then $p_{-}$ is necessarily of the form $p_{-}(y)=c+x\tilde{p}_{-}(y)$ for some $c\in A^{*}$ and so changing $z_{+}$ for $z_{+}+z_{-}$ yields a twin-triangular derivation conjugate to $\partial$ for which the corresponding polynomials $p_{\pm}(y)$ both have non zero residue classes modulo $x$. More generally, changing $z_{-}$ for $az_{-}+bz_{+}$ for general $a\in A^{*}$ and $b\in A$ yields a twin-triangular derivation conjugate to $\partial$ and still satisfying condition a) in Definition \ref{def:general_position}. So it remains to show that up to such a coordinate change, condition b) in the Definition can be achieved.
This can be seen as follows : we consider $\mathbb{A}_{\kappa}^{2}$ embedded in $\mathbb{P}_{\kappa}^{2}={\rm Proj}(\kappa[u,v,w])$ as the complement of the line $L_{\infty}=\left\{ w=0\right\} $ so that the coordinate system $\left(u,v\right)$ on $\mathbb{A}^{2}$ is induced by the projections from the $\kappa$-rational points $\left[0:1:0\right]$ and $\left[1:0:0\right]$ respectively. We let $C$ be the closure in $\mathbb{P}^{2}$ of the image of the morphism $j=(\overline{P}_{+},\overline{P}_{-}):\mathbb{A}_{\kappa}^{1}={\rm Spec}(\kappa[y])\rightarrow\mathbb{A}_{\kappa}^{2}$ defined by the residue classes $\overline{P}_{+}$ and $\overline{P}_{-}$ in $\kappa[y]$ of integrals $P_{\pm}(y)\in A[y]$ of $p_{\pm}(y)$, and we denote by $Z\subset C$ the image by $j$ of the inverse image of the branch locus of $\overline{P}_{+}:\mathbb{A}_{\kappa}^{1}\rightarrow\mathbb{A}_{\kappa}^{1}$. Note that $Z$ is a finite subset of $C$ defined over $\kappa$. Since the condition that a line through a fixed point in $\mathbb{P}_{\kappa}^{2}$ intersects transversally a fixed curve is Zariski open, the set of lines in $\mathbb{P}_{\kappa}^{2}$ passing through a point of $Z$ and tangent to a local analytic branch of $C$ at some point is finite. Therefore, the complement of the finitely many intersection points of these lines with $L_{\infty}$ is a Zariski open subset $U$ of $L_{\infty}$ with the property that for every $q\in U$, the line through $q$ and every arbitrary point of $Z$ intersects every local analytic branch of $C$ transversally at every point. By construction, the rational projections from $\left[0:1:0\right]$ and an arbitrary $\kappa$-rational point in $U\setminus\{\left[0:1:0\right]\}$ induce a new coordinate system on $\mathbb{A}_{\kappa}^{2}$ of the form $\left(u,av+bu\right)$, $a\neq0$, with the property that $Z$ is not contained in the inverse image of the branch locus of the morphism $a\overline{P}_{-}+b\overline{P}_{+}:\mathbb{A}_{\kappa}^{1}\rightarrow\mathbb{A}_{\kappa}^{1}$. Changing $z_{-}$ for $az_{-}+bz_{+}$ for a pair $(a,b)\in\kappa^{*}\times\kappa\subset A^{*}\times A$ corresponding to a general point in $U$ yields a twin-triangular derivation conjugate to $\partial$ and satisfying simultaneously conditions a) and b) in Definition \ref{def:general_position}. \end{proof} \begin{parn} \label{par:finite_etale_restriction} Now let $\partial=x^{n}\partial_{y}+p_{+}(y)\partial_{z_{+}}+p_{-}(y)\partial_{z_{-}}$ be a twin-triangular $A$-derivation of $A[y,z_{+},z_{-}]$ generating a proper whence fixed point free $\mathbb{G}_{a,S}$-action $\sigma:\mathbb{G}_{a,S}\times_{S}\mathbb{A}_{S}^{3}\rightarrow\mathbb{A}_{S}^{3}$. By virtue of Lemma \ref{lem:Bad-Plane-Removal} above, we may assume up to a coordinate change preserving twin-triangularity that $\partial$ is in general position. Property a) in Definition \ref{def:general_position} then guarantees in particular that the triangular derivations $\partial_{\pm}=x^{n}\partial_{y}+p_{\pm}(y)\partial_{z_{\pm}}$ of $A[y,z_{\pm}]$ are both irreducible. Furthermore, given any integral $P_{\pm}(y)\in A[y]$ of $p_{\pm}(y)$, the morphism $\overline{P}_{\pm}:\mathbb{A}_{\kappa}^{1}\rightarrow\mathbb{A}_{\kappa}^{1}$ obtained by restricting $P_{\pm}:\mathbb{A}_{S}^{1}={\rm Spec}(A[y])\rightarrow\mathbb{A}_{S}^{1}={\rm Spec}(A[t])$ to the closed fiber of $\mathrm{pr}_{S}:\mathbb{A}_{S}^{3}\rightarrow S$ is not constant. The branch locus of $\overline{P}_{\pm}$ is then a principal divisor $\mathrm{div}(\alpha_{\pm}(t))$ for a certain polynomial $\alpha_{\pm}(t)\in\kappa[t]\subset A[t]$ generating the kernel of the homomorphism $\kappa[t]\rightarrow\kappa[y]/(\overline{p}_{\pm}(y))$, $t\mapsto\overline{P}_{\pm}(y)+(\overline{p}_{\pm}(y))$. Property b) in Definition \ref{def:general_position} guarantees that we can choose $P_{+}$ and $P_{-}$ in such a way that the polynomial $\alpha_{+}(\overline{P}_{+}(y))$ and $\alpha_{-}(\overline{P}_{-}(y))$ generate the unit ideal in $\kappa[y]$. Up to a diagonal change of coordinates on $A[y,z_{+},z_{-}]$, we may further assume without loss of generality that $\overline{P}_{+}$ and $\overline{P}_{-}$ are monic.
\end{parn}
\begin{parn} \label{par:open_cover_def} We let $R_{\pm}=A[t]_{\alpha_{\pm}}$ and we let $U_{\pm}=\mathrm{Spec}(R_{\pm})$ be the principal open subset of $\mathbb{A}_{S}^{1}=\mathrm{Spec}(A[t])$ where $\alpha_{\pm}$ does not vanish. The polynomial $\Phi_{\pm}=-x^{n}z_{\pm}+P_{\pm}(y)\in A[y,z_{+},z_{-}]$ belongs to the kernel of $\partial$ hence defines a $\mathbb{G}_{a,S}$-invariant morphism $\Phi_{\pm}:\mathbb{A}_{S}^{3}=\mathrm{Spec}(A[y,z_{+},z_{-}])\rightarrow\mathbb{A}_{S}^{1}=\mathrm{Spec}(A[t])$. We let \begin{eqnarray*} W_{\pm} & = & \Phi_{\pm}^{-1}(U_{\pm})\simeq\mathrm{Spec}(R_{\pm}[y,z_{+},z_{-}]/(-x^{n}z_{\pm}+P_{\pm}(y)-t)) \end{eqnarray*} Note that $W_{\pm}$ is a $\mathbb{G}_{a,S}$-invariant open subset of $\mathbb{A}_{S}^{3}$ which can be identified with the principal open subset where the $\mathbb{G}_{a,S}$-invariant regular function $\Gamma_{\pm}=\alpha_{\pm}\circ\Phi_{\pm}$ does not vanish. Since $\alpha_{+}(\overline{P}_{+}(y))$ and $\alpha_{-}(\overline{P}_{-}(y))$ generate the unit ideal in $\kappa[y]$, it follows that the union of $W_{+}$ and $W_{-}$ covers the closed fiber of the projection $\mathrm{pr}_{S}:\mathbb{A}_{S}^{3}\rightarrow S$.
\end{parn}
\subsection{Affineness of geometric quotients}
With the notation of \S \ref{par:open_cover_def} above, the geometric quotient $\mathfrak{X}_{\pm}=W_{\pm}/\mathbb{G}_{a,S}$ for the action induced by $\sigma:\mathbb{G}_{a,S}\times_{S}\mathbb{A}_{S}^{3}\rightarrow\mathbb{A}_{S}^{3}$ can be identified with the principal open sub-space $\mathfrak{X}_{\Gamma_{\pm}}$ of $\mathfrak{X}=\mathbb{A}_{S}^{3}/\mathbb{G}_{a,S}$ where the invariant function $\Gamma_{\pm}\in A[y,z_{+},z_{-}]^{\mathbb{G}_{a,S}}\simeq\Gamma(\mathfrak{X},\mathcal{O}_{\mathfrak{X}})$ does not vanish. The properness of $\sigma$ implies that $\mathfrak{X}$, whence $\mathfrak{X}_{+}$ and $\mathfrak{X}_{-}$, are separated algebraic spaces, and the construction of $W_{+}$ and $W_{-}$ guarantees that the closed fiber of the structure morphism $\mathrm{p}:\mathfrak{X}\rightarrow S$ is contained in the union of $\mathfrak{X}_{+}$ and $\mathfrak{X}_{-}$. So to complete the proof of Proposition \ref{prop:Twin-Loc-trivi}, it remains to show that $\mathfrak{X}_{\pm}$ is an affine scheme. In fact, since $\mathfrak{X}_{\pm}$ is by construction an algebraic space over the affine scheme $U_{\pm}=\mathrm{Spec}(R_{\pm})$, its affineness is equivalent to that of the structure morphism $q_{\pm}:\mathfrak{X}_{\pm}\rightarrow U_{\pm}$, a property which can be checked locally with respect to the \'etale topology on $U_{\pm}$.
\begin{parn} In our situation, there is a natural finite \'etale base change $\varphi_{\pm}:\tilde{U}_{\pm}\rightarrow U_{\pm}$ which is obtained as follows: By construction, see \S \ref{par:finite_etale_restriction} above, the morphism $\overline{P}_{\pm}:\mathbb{A}_{\kappa}^{1}=\mathrm{Spec}(\kappa[y])\rightarrow\mathrm{Spec}(\kappa[t])$, restricts to a finite \'etale covering $h_{0,\pm}:C_{1,\pm}={\rm Spec}(\kappa[y]_{\alpha_{\pm}(\overline{P}_{\pm}(y))})\rightarrow C_{\pm}={\rm Spec}(\kappa[t]_{\alpha_{\pm}(t)})$ of degree $r_{\pm}=\deg_{y}(\overline{P}_{\pm}(y))$. Letting $\tilde{C}_{\pm}=\mathrm{Spec}(B_{\pm})$ be the normalization of $C_{\pm}$ in the Galois closure $L_{\pm}$ of the field extension $i_{\pm}:\kappa(t)\hookrightarrow\kappa(y)$, the induced morphism $h_{\pm}:\tilde{C}_{\pm}\rightarrow C_{\pm}$ is an \'etale Galois cover with Galois group $G_{\pm}=\mathrm{Gal}(L_{\pm}/\kappa(t))$, which factors as \[ h_{\pm}:\tilde{C}_{\pm}=\mathrm{Spec}(B_{\pm})\stackrel{h_{1,\pm}}{\longrightarrow}C_{1,\pm}={\rm Spec}(\kappa[y]_{\alpha_{\pm}(\overline{P}_{\pm}(y))})\stackrel{h_{0,\pm}}{\longrightarrow}C_{\pm}={\rm Spec}(\kappa[t]_{\alpha_{\pm}(t)}) \] where $h_{1,\pm}:\tilde{C}_{\pm}\rightarrow C_{1,\pm}$ is an \'etale Galois cover for a certain subgroup $H_{\pm}$ of $G_{\pm}$ of index $r_{\pm}$. Letting $\tilde{R}_{\pm}=A\otimes_{\kappa}B_{\pm}\simeq A[t]_{\alpha_{\pm}(t)}\otimes_{\kappa[t]_{\alpha_{\pm}(t)}}B_{\pm}$ and $\tilde{U}_{\pm}=\mathrm{Spec}(\tilde{R}_{\pm})$, the morphism $\varphi_{\pm}=\mathrm{pr}_{1}:\tilde{U}_{\pm}\simeq U_{\pm}\times_{C_{\pm}}\tilde{C}_{\pm}\rightarrow U_{\pm}$ is an \'etale Galois cover with Galois group $G_{\pm}$, in particular a finite morphism. Since $\mathfrak{X}_{\pm}$ is separated, the algebraic space $\tilde{\mathfrak{X}}_{\pm}=\mathfrak{X}_{\pm}\times_{U_{\pm}}\tilde{U}_{\pm}$ is separated and, by construction, isomorphic to the geometric quotient of the scheme \begin{eqnarray*} \tilde{W}_{\pm}=W_{\pm}\times_{U_{\pm}}\tilde{U}_{\pm} & \simeq & \mathrm{Spec}(\tilde{R}_{\pm}[y,z_{+},z_{-}]/(-x^{n}z_{\pm}+P_{\pm}(y)-t)) \end{eqnarray*} by the proper $\mathbb{G}_{a,\tilde{U}_{\pm}}$-action generated by the locally nilpotent $\tilde{R}_{\pm}$-derivation $x^{n}\partial_{y}+p_{+}(y)\partial_{z_{+}}+p_{-}(y)\partial_{z_{-}}$ of $\tilde{R}_{\pm}[y,z_{+},z_{-}]//(-x^{n}z_{\pm}+P_{\pm}(y)-t)$, which commutes with the action of $G_{\pm}$. The following Lemma completes the proof of Proposition \ref{prop:Twin-Loc-trivi} whence of the Main Theorem.
\end{parn} \begin{lem} The geometric quotient $\tilde{\mathfrak{X}}_{\pm}=\tilde{W}_{\pm}/\mathbb{G}_{a,\tilde{U}_{\pm}}$ is an affine $\tilde{U}_{\pm}$-scheme. \end{lem} \begin{proof} Since $\tilde{U}_{\pm}$ is affine, the assertion is equivalent to the affineness of $\tilde{\mathfrak{X}}_{\pm}$. From now on, we only consider the case of $\tilde{\mathfrak{X}}_{+}=\tilde{W}_{+}/\mathbb{G}_{a,\tilde{U}_{+}}$, the case of $\tilde{\mathfrak{X}}_{-}$ being similar. To simplify the notation, we drop the corresponding subscript ``$+$'', writing simply $\tilde{W}=\mathrm{Spec}(\tilde{R}[y,z,z_{-}]/(-x^{n}z+P(y)-t))$. We denote $x\otimes1\in\tilde{R}=A\otimes_{\kappa}B$ by $x$ and we further identify $B$ with a sub-$\kappa$-algebra of $\tilde{R}$ via the homomorphism $1\otimes\mathrm{id}_{B}:B\rightarrow\tilde{R}$ and with the quotient $\tilde{R}/x\tilde{R}$ via the composition $1\otimes\mathrm{id}_{B}:B\rightarrow A\otimes_{\kappa}B\rightarrow A\otimes_{\kappa}B/((x\otimes1)A\otimes_{\kappa}B)=\kappa\otimes_{\kappa}B\simeq B$.
By construction of $B$, the monic polynomial $\overline{P}(y)-t\in B\left[y\right]$ splits as $\overline{P}(y)-t=\prod_{\overline{g}\in G/H}(y-t_{\overline{g}})$ for certain elements $t_{\overline{g}}\in B$, $\overline{g}\in G/H$, on which the Galois group $G$ acts by permutation $g'\cdot t_{\overline{g}}=t_{\overline{(g')^{-1}\cdot g}}$. Furthermore, since $h_{0}:C_{1}\rightarrow C$ is \'etale, it follows that for distinct $\overline{g},\overline{g}'\in G/H$, $t_{\overline{g}}-t_{\overline{g'}}\in B$ is an invertible regular function on $\tilde{C}$ whence on $\tilde{U}=S\times_{\mathrm{Spec}(\kappa)}\tilde{C}$ via the identifications made above. This implies in turn that there exists a collection of elements $\sigma_{\overline{g}}\in\tilde{R}$ with respective residue classes $t_{\overline{g}}\in B=\tilde{R}/x\tilde{R}$ modulo $x$, $\overline{g}\in G/H$, on which $G$ acts by permutation, a $G$-invariant polynomial $S_{1}\in\tilde{R}\left[y\right]$ with invertible residue class modulo $x$ and a $G$-invariant polynomial $S_{2}\in\tilde{R}\left[y\right]$ such that in $\tilde{R}\left[y\right]$ one can write \[ P(y)-t=S_{1}(y)\prod_{\overline{g}\in G/H}(y-\sigma_{\overline{g}})+x^{n}S_{2}(y). \] Concretely, the elements $\sigma_{\overline{g}}=\sigma_{\overline{g},n-1}\in\tilde{R}$, $\overline{g}\in G/H$, can be constructed by induction via a sequence of elements $\sigma_{\overline{g},m}\in\tilde{R}$, $\overline{g}\in G/H$, $m=0,\ldots,n-1$, starting with $\sigma_{\overline{g},0}=t_{\overline{g}}\in B\subset\tilde{R}$ and culminating in $\sigma_{\overline{g},n-1}=\sigma_{\overline{g}}$, and characterized by the property that for every $m=0,\ldots,n-1$, there exists $\mu_{\overline{g},m}\in\tilde{R}$ such that $P(\sigma_{\overline{g},m})-t=x^{m+1}\mu_{\overline{g},m}$, $\overline{g}\in G/H$. Indeed, writing $P(y)-t=\prod_{\overline{g}\in G/H}(y-t_{\overline{g}})+x\tilde{P}(y)$ for a certain $\tilde{P}(y)\in\tilde{R}[y]$ and assuming that the $\sigma_{\overline{g},m}$, $\overline{g}\in G/H$, have been constructed up to a certain index $m<n-1$, we look for elements $\sigma_{\overline{g},m+1}\in\tilde{R}$ written in the form $\sigma_{\overline{g},m}+x^{m+1}\lambda_{\overline{g}}$ for some $\lambda_{\overline{g}}\in\tilde{R}$. For a fixed $\overline{g}_{0}\in G/H$, the conditions impose that
\begin{eqnarray*} P(\sigma_{\overline{g}_{0},m+1})-t & = & \prod_{\overline{g}\in G/H}(\sigma_{\overline{g}_{0},m}+x^{m+1}\lambda_{\overline{g}_{0}}-t_{\overline{g}})+x\tilde{P}(\sigma_{\overline{g}_{0},m}+x^{m+1}\lambda_{\overline{g}_{0}})\\
& = & x^{m+1}\lambda_{\overline{g}_{0}}\prod_{\overline{g}\in(G/H)\setminus\{\overline{g}_{0}\}}(t_{\overline{g}_{0}}-t_{\overline{g}})+P(\sigma_{\overline{g}_{0},m})-t+x^{m+2}\nu_{\overline{g}_{0},m}\\
& = & x^{m+1}\lambda_{\overline{g}_{0}}\prod_{\overline{g}\in(G/H)\setminus\{\overline{g}_{0}\}}(t_{\overline{g}_{0}}-t_{\overline{g}})+x^{m+1}\mu_{\overline{g}_{0,}m}+x^{m+2}\nu_{\overline{g}_{0},m} \end{eqnarray*} for some $\nu_{\overline{g}_{0},m}\in\tilde{R}$, and since $\prod_{\overline{g}\in(G/H)\setminus\{\overline{g}_{0}\}}(t_{\overline{g}_{0}}-t_{\overline{g}})\in\tilde{R}^{*}$, we conclude that \[ \lambda_{\overline{g}_{0}}=\frac{\mu_{\overline{g}_{0},m}}{\prod_{\overline{g}\in(G/H)\setminus\{\overline{g}_{0}\}}(t_{\overline{g}_{0}}-t_{\overline{g}})}\quad\textrm{and}\quad\mu_{\overline{g}_{0},m+1}=\nu_{\overline{g}_{0},m}. \] A direct computation shows further that $g'\cdot\sigma_{\overline{g},m+1}=\sigma_{\overline{(g')^{-1}\cdot g},m+1}$ and that $g'\cdot\mu_{\overline{g},m+1}=\mu_{\overline{(g')^{-1}\cdot g},m+1}$. Iterating this procedure $n-1$ times yields the desired collection of elements $\sigma_{\overline{g}}=\sigma_{\overline{g},n-1}\in\tilde{R}$. By construction, $\prod_{\overline{g}\in G/H}(y-\sigma_{\overline{g}})\in\tilde{R}[y]$ is then an invariant polynomial which divides $P(y)-t$ modulo $x^{n}\tilde{R}$, which implies in turn the existence of the $G$-invariant polynomials $S_{1}(y),S_{2}(y)\in\tilde{R}[y]$.
The closed fiber of the induced morphism $\tilde{W}\rightarrow S$ consists of a disjoint union of closed sub-schemes $D_{\overline{g}}\simeq\mathrm{Spec}(\tilde{R}[z,z_{-}])\simeq\mathbb{A}_{\tilde{C}}^{2}$ with defining ideals $(x,y-\sigma_{\overline{g}})$, $\overline{g}\in G/H$. The open sub-scheme $\tilde{W}_{\overline{g}}=\tilde{W}\setminus\bigcup_{\overline{g}'\in(G/H)\setminus\{\overline{g}\}}D_{\overline{g}'}$ of $\tilde{W}$ is $\mathbb{G}_{a,\tilde{U}}$-invariant and one checks using the above expression for $P(y)-t$ that the rational map \[ \tilde{W}\dashrightarrow\mathrm{Spec}(\tilde{R}[u_{\overline{g}},z_{-}]),\quad(y,z,z_{-})\mapsto(u_{\overline{g}},z_{-})=(\frac{y-\sigma_{\overline{g}}}{x^{n}}=\frac{z-S_{2}(y)}{S_{1}(y)\prod_{\overline{g}'\in(G/H)\setminus\{\overline{g}\}}(y-\sigma_{\overline{g}'})},z_{-}) \] induces a $\mathbb{G}_{a,\tilde{U}}$-equivariant isomorphism $\tau_{g}:\tilde{W}_{\overline{g}}\stackrel{\sim}{\rightarrow}\mathbb{A}_{\tilde{U}}^{2}=\mathrm{Spec}(\tilde{R}[u_{\overline{g}},z_{-}])$ for the $\mathbb{G}_{a,\tilde{U}}$-action on $\mathbb{A}_{\tilde{U}}^{2}$ generated by the locally nilpotent $\tilde{R}$-derivation $\partial_{u_{\overline{g}}}+p_{-}(x^{n}u_{\overline{g}}+\sigma_{\overline{g}})\partial_{z_{-}}$ of $\tilde{R}[u_{\overline{g}},z_{-}]$. The latter is a translation with $u_{\overline{g}}$ as a global slice and with geometric quotient $\tilde{W}_{\overline{g}}/\mathbb{G}_{a,\tilde{U}}$ isomorphic to $\mathrm{Spec}(\tilde{R}[v_{\overline{g}}])$ where \[ v_{\overline{g}}=z_{-}-x^{-n}(P_{-}(x^{n}u_{\overline{g}}+\sigma_{\overline{g}})-P_{-}(\sigma_{\overline{g}}))\in\tilde{R}[u_{\overline{g}},z_{-}]^{\mathbb{G}_{a,\tilde{U}}}. \] By construction, for distinct $\overline{g},\overline{g}'\in G/H$, the rational functions $\tau_{\overline{g}}^{*}v_{\overline{g}}$ and $\tau_{\overline{g}'}^{*}v_{\overline{g}'}$ on $\tilde{W}$ differ by the addition of the element \[ f_{\overline{g},\overline{g}'}=x^{-n}(P_{-}(\sigma_{\overline{g}})-P_{-}(\sigma_{\overline{g}'}))\in\tilde{R}_{x}\in\Gamma(\tilde{W}_{\overline{g}}\cap\tilde{W}_{\overline{g}'},\mathcal{O}_{\tilde{W}}). \] This implies that $\tilde{\mathfrak{X}}=\tilde{W}/\mathbb{G}_{a,\tilde{U}}$ is isomorphic to the $\tilde{U}$-scheme obtained by gluing $r$ copies $\tilde{\mathfrak{X}}_{g}=\mathrm{Spec}(\tilde{R}[v_{\overline{g}}])$ of $\mathbb{A}_{\tilde{U}}^{1}$ along the principal open subsets $\tilde{\mathfrak{X}}_{\overline{g},x}\simeq\mathrm{Spec}(\tilde{R}_{x}[v_{\overline{g}}])$ via the isomorphisms induced by the $\tilde{R}_{x}$-algebra isomorphisms \[ \xi_{\overline{g},\overline{g}'}^{*}:\tilde{R}_{x}[v_{\overline{g}}]\rightarrow\tilde{R}_{x}[v_{\overline{g}'}],v_{\overline{g}}\mapsto v_{\overline{g}'}+f_{\overline{g},\overline{g}'},\quad\overline{g},\overline{g}'\in G/H,\;\overline{g}\neq\overline{g}'. \] Since by assumption $\tilde{\mathfrak{X}}$ is separated, it follows from \cite[I.5.5.6]{EGA1} that for every pair of distinct elements $\overline{g},\overline{g}'\in G/H$, the sub-ring $\tilde{R}[v_{\overline{g}'},f_{\overline{g},\overline{g}'}]$ of $\tilde{R}_{x}[v_{\overline{g}'}]$ generated by the union of $\tilde{R}[v_{\overline{g}'}]$ and $\xi_{\overline{g},\overline{g}'}^{*}(\tilde{R}[v_{\overline{g}}])$ is equal to $\tilde{R}_{x}[v_{\overline{g}'}]$. This holds if and only if $\tilde{R}[f_{\overline{g},\overline{g}'}]=\tilde{R}_{x}$ whence if and only if $f_{\overline{g},\overline{g}'}\in\tilde{R}_{x}$ has the form $f_{\overline{g},\overline{g}'}=x^{-m_{\overline{g},\overline{g}'}}F_{\overline{g},\overline{g}'}$ for a certain $m_{\overline{g},\overline{g}'}>1$ and an element $F_{\overline{g},\overline{g}'}\in\tilde{R}$ with invertible residue class modulo $x$.
This additional information enables a proof of the affineness of $\tilde{\mathfrak{X}}$ by induction on $r$ as follows: given a pair of distinct elements $\overline{g},\overline{g}'\in G/H$ such that $m_{\overline{g},\overline{g}'}=m>0$ is maximal, we let $\theta_{\overline{g}}=0$ and $\theta_{\overline{g}''}=x^{m-m_{\overline{g},\overline{g}''}}F_{\overline{g},\overline{g}''}\in\tilde{R}$ for every $\overline{g}''\in(G/H)\setminus\{\overline{g}\}$. The choice of the elements $\theta_{\overline{g}''}\in\tilde{R}$ guarantees that the local sections \[ \psi_{\overline{g}''}=x^{m}v_{\overline{g}''}+\theta_{\overline{g}''}\in\Gamma(\tilde{\mathfrak{X}}_{\overline{g}''},\mathcal{O}_{\tilde{\mathfrak{X}}}),\quad\overline{g}''\in G/H \] glue to a global regular function $\psi\in\Gamma(\tilde{\mathfrak{X}},\mathcal{O}_{\tilde{\mathfrak{X}}})$. Since $\theta_{\overline{g}'}=F_{\overline{g},\overline{g}'}$ is invertible modulo $x$, the regular functions $x$, $\psi$ and $\psi-\theta_{\overline{g}'}$ generate the unit ideal in $\Gamma(\tilde{\mathfrak{X}},\mathcal{O}_{\tilde{\mathfrak{X}}})$. The principal open subset $\tilde{\mathfrak{X}}_{x}$ of $\tilde{\mathfrak{X}}$ is isomorphic to $\tilde{\mathfrak{X}}_{\overline{g},x}\simeq\mathrm{Spec}(\tilde{R}_{x}[v_{\overline{g}}])$ for every $\overline{g}\in G/H$, hence is affine. On the other hand, $\tilde{\mathfrak{X}}_{\psi}$ and $\tilde{\mathfrak{X}}_{\psi-\theta_{\overline{g}'}}$ are contained respectively in the open sub-schemes $\tilde{\mathfrak{X}}(\overline{g})$ and $\tilde{\mathfrak{X}}(\overline{g}')$ obtained by gluing only the $r-1$ open subsets $\tilde{\mathfrak{X}}_{\overline{g}''}$ corresponding to the elements $\overline{g}''$ in $\left(G/H\right)\setminus\{\overline{g}\}$ and $\left(G/H\right)\setminus\{\overline{g}'\}$ respectively. By the induction hypothesis, the latter are both affine and hence $\tilde{\mathfrak{X}}_{\psi}$ and $\tilde{\mathfrak{X}}_{\psi-\theta_{\overline{g}'}}$ are affine as well. This shows that $\tilde{\mathfrak{X}}$ is an affine scheme and completes the proof. \end{proof}
\end{document} |
\begin{document}
\begin{abstract}
In this text, we generalize Cech cohomology to sheaves $\mathcal F$ with values in blue $B$-modules where $B$ is a blueprint with $-1$. If $X$ is an object of the underlying site, then the cohomology sets $H^l(X,\mathcal F)$ turn out to be blue $B$-modules. For locally free $\mathcal O_X$-module $\mathcal F$ on a monoidal scheme $X$, we prove that $H^l(X,\mathcal F)^+=H^l(X^+,\mathcal F^+)$ where $X^+$ is the scheme associated with $X$ and $\mathcal F^+$ is the locally free $\mathcal O_{X^+}$-module associated with $\mathcal F$.
In an appendix, we show that the naive generalization of cohomology as a right derived functor is infinite-dimensional for the projective line over $\mathbb F_1$. \end{abstract}
\title{\v Cech cohomology over $\Funsq$}
\section*{Introduction} \label{section: introduction}
\noindent
While many standard methods in algebraic geometry carry over readily to ${\F_1}$-geometry, other methods withstand a straightforward generalization since essential properties from usual algebraic geometry fail to be true or produce unusual results.
Sheaf cohomology with values in categories over ${\F_1}$ belongs to the latter class of theories. Though methods from homological algebra generalize without great difficulties to injective resolutions of sheaves on ${\F_1}$-schemes (see \cite{Deitmar11b}), the derived cohomology sets are larger than one would expect. For instance, the first cohomology set $H^1(X,{\mathcal O}_{X})$ of the projective line $X={\mathbb P}^1_{\F_1}$ over ${\F_1}$ is of infinite rank over ${\F_1}$, cf.\ Appendix \ref{appendix: cohomology of p1}.
There have been some ad hoc observations for the projective line ${\mathbb P}^1_{\F_1}$ in \cite{Connes-Consani10a}, for which \v Cech cohomology works well as long as the chosen covering consists of at most two open sets. For larger coverings, however, it is not clear how to make sense of the alternating sums in the definition of \v Cech cohomology.\footnote{During the time of writing, Jaiung Jun has published his preprint \cite{Jun15} on \v Cech cohomology for semirings. His method of double complexes might be applicaple to the setting of this paper.}
This problem resolves naturally for sheaves over ${\F_{1^2}}$, since ${\F_{1^2}}$ contains an additive inverse $-1$ of $1$, i.e.\ it bears a relation $1+(-1)=0$. This leads naturally to the theory of blueprints, which deals with multiplicative monoids that come together with certain additive relations that might be weaker than an addition.
The aim of this paper is to define \v Cech cohomology for sheaves with values in blue $B$-modules where $B$ is a blueprint with $-1$, and to show that this leads to a meaningful theory.
We calculate the cohomology of a monoidal scheme $X$ in terms of a comparison with the cohomology of their associated scheme $X^+$, which is also denoted by ``$X\otimes_{\F_1}{\mathbb Z}$'' in the literature. For this comparison, we assume the following mild technical assumption on an open covering $\{U_i\}_{i\in I}$ of $X$.
\noindent\textbf{Hypothesis (H):} For all finite subsets $J\subset I$ of ${\mathcal I}$, the restriction map
\[
\res_{U_J,U_I}:{\mathcal O}_X(U_J)\to{\mathcal O}_X(U_I)
\]
is injective.
The following is Theorem \ref{thm: comparison of cech cohomology} of the main text.
\begin{thmA}
Given a monoidal scheme $X$ over $B$ that admits a finite covering $\{U_i\}$ with Hypothesis (H) such that ${\mathcal O}_X(U_i)$ are monoid blueprints over $B$. Then we have for every locally free sheaf ${\mathcal F}$ on $X$ that
\[
H^l(X,{\mathcal F})^+ \ = \ H^l(X^+,{\mathcal F}^+).
\] \end{thmA}
Note that the class of monoidal schemes with a covering satisfying Hypothesis (H) contains, in particular, a model for every toric variety. Therefore the results of this paper might be helpful for calculations of sheaf cohomology for toric varieties, cf.\ Remark \ref{rem: cohomology for toric varieties}.
In the first part of this paper, we define \v Cech cohomology for sheaves of blue $B$-modules on an arbitrary site. We choose this general formulation because it is applicable to arithmetic questions like the \'etale cohomology of the compactification $\overline{\Spec {\mathbb Z}}$ of the arithmetic line; see \cite{L14} for a model of $\overline{\Spec {\mathbb Z}}$ and some ideas towards such a theory.
In the second part of this paper, we introduce the notion of monoidal schemes over a blueprint $B$, which extends the notion of monoidal schemes from ${\F_1}$ to any blueprint, and we discuss the notion of locally free sheaves. In a final section, we formulate and prove our main result Theorem A.
Since there are several introduction to blueprints and blue schemes, we do not provide another one in this text, but provide the reader with a reference where this is necessary. As a general reference, we suggest the overview paper \cite{L13}. In particular, the reader will find the definition of a blueprint in section II.1.1, and the definition of a blue $B$-module in section II.6.1 of this paper.
\part{\v Cech cohomology over ${\F_{1^2}}$} \label{part: cech cohomology}
\section{Definition for a fixed covering} \label{section: definition for a fixed covering}
\noindent In this part of the paper, we consider a site ${\mathcal T}$ and an object $X$ of this site. We assume that ${\mathcal T}$ contains fibre products, so that we have a notion of covering families ${\mathcal U}=\{U_i\}_{i\in {\mathcal I}}$ of $X$. We will define \v Cech cohomology for $X$ with values in sheaves in blue $B$-modules where $B$ is a blueprint with $-1$, which can also be thought of as an ${\F_{1^2}}$-algebra.
Throughout this part of the paper, we fix the site ${\mathcal T}$ and the object $X$. For this section, we also fix the covering family ${\mathcal U}$ and aim for defining the \v Cech cohomology $H^l(X,{\mathcal F};{\mathcal U})$ w.r.t.\ ${\mathcal U}$.
A blueprint with $-1$ is a blueprint that has an element $-1$ that satisfies the additive relation $1+(-1)\=0$. This element is necessarily unique, which means that there is a unique blueprint morphism ${\F_{1^2}}\to B$ from \[
{\F_{1^2}} \ = \ \bpgenquot{\{0,1,-1\}} {1+(-1)\=0} \] to $B$. By multiplying the defining relation for $-1$ with an arbitrary element $a$ of $B$, we see that $-a=(-1)\cdot a$ is an \emph{additive inverse of $a$}, i.e.\ it satisfies the relation $a+(-a)\=0$.
Let $\Mod_B^{\textup{bl}}$ be the category of blue $B$-modules and ${\mathcal F}$ a sheaf on ${\mathcal T}$ with values in $\Mod_B^{\textup{bl}}$. Let ${\mathcal U}=\{U_i\}_{i\in{\mathcal I}}$ be a covering family of $X$ where ${\mathcal I}$ is a totally ordered index set.
\begin{df}
For $l\geq 0$, we denote by ${\mathcal I}_l$ the family of all subsets $I$ of ${\mathcal I}$ with cardinality $l+1$. For such a subset, we write $I=(i_0,\dotsb,i_l)$ if $I=\{i_0,\dotsc,i_l\}$ and $i_0<\dotsb<i_l$. We define
\[
U_I \ = \ U_{i_0}\times_X\dotsc\times_X U_{i_l} \qquad\text{and}\qquad {\mathcal F}_I \ = \ {\mathcal F}(U_I),
\]
which is a blue $B$-module. Given $I\in{\mathcal I}_l$ and $k\in\{0,\dotsc,l\}$, we denote by $I^k$ the set $\{i_0,\dotsc,\widehat{i_k},\dotsc,i_l\}$. The canonical projection $U_I\to U_{I^k}$ onto all factors but $U_k$ defines a morphism
\[
\partial_{k,I}^{(l)}: \ {\mathcal F}_{I^k} \ \longrightarrow \ {\mathcal F}_I.
\]
If we define
\[
{\mathcal C}^l \ = \ \prod_{I\in{\mathcal I}_l}{\mathcal F}_I
\]
the morphisms $\partial_{k,I}^{(l)}$ for varying $I$ define a morphism
\[
\partial_k^{(l)}: \ {\mathcal C}^{l-1} \ \longrightarrow \ {\mathcal C}^{l}
\]
for every $k=0,\dotsc,l$. The \emph{\v Cech complex of ${\mathcal U}$ with values in ${\mathcal F}$} is the cosimplicial blue $B$-module
\[
{\mathcal C}^\bullet \ = \ {\mathcal C}^\bullet(X,{\mathcal F};{\mathcal U}) \ = \ \Biggl( \xymatrix{{\mathcal C}^0 \ar@<0.5ex>[r]^{\partial_0^{(1)}}\ar@<-0.5ex>[r]_{\partial_1^{(1)}} & {\mathcal C}^1 \ar@<1ex>[r]^{\partial_0^{(2)}}\ar[r]\ar@<-1ex>[r]_{\partial_2^{(2)}} & {\mathcal C}^2 \ar@<1.5ex>[r]\ar@<0.5ex>[r]\ar@<-0.5ex>[r]\ar@<-1.5ex>[r] & {\mathcal C}^3 \quad\dotsb \quad} \Biggr).
\] \end{df}
\begin{rem} \label{rem: total ceh complex}
In practice, the index set ${\mathcal I}$ is often finite. Then the \v Cech complex is finite since ${\mathcal C}^l$ is the empty product, i.e.\ ${\mathcal C}^l=0$, if $l\geq\#{\mathcal I}$.
This cosimplicial set is often called the \emph{ordered \v Cech complex} in literature, in contrast to the \emph{total \v Cech complex ${\mathcal C}_\textup{tot}^\bullet$} with ${\mathcal C}_{\textup{tot}}^l=\prod{\mathcal F}_{\{i_0,\dotsc,i_l\}}$ where the product is taken over all elements $(i_0,\dotsc,i_l)\in{\mathcal I}^{l+1}$ without any assumption on the ordering or distinctness of the $i_k$'s. \end{rem}
\begin{df}
Let ${\mathcal C}^\bullet$ be a cosimplicial blue $B$-module. The \emph{set of $l$-cocycles of ${\mathcal C}^\bullet$} is \[
{\mathcal Z}^l \ = \ {\mathcal Z}^l({\mathcal C}^\bullet) \ = \ \biggl\{\ x\in{\mathcal C}^l \ \biggl|\ \sum_{k=0}^{l+1} (-1)^k\partial_k^{(l+1)}(x) \equiv 0 \ \bigg\} \] which we consider as a full blue $B$-submodule of ${\mathcal C}^l$, i.e.\ the pre-addition of ${\mathcal Z}^l$ is the restriction of the pre-addition of ${\mathcal C}^l$ to ${\mathcal Z}^l$. The \emph{set of $l$-coboundaries} is \[
{\mathcal B}^l \ = \ {\mathcal B}^l({\mathcal C}^\bullet) \ = \ \biggl\{\ x\in{\mathcal C}^l \ \biggl|\ \exists y=\sum_i y_i \in({\mathcal C}^{l-1})^+\text{ such that }x \equiv \sum_i\sum_{k=0}^{l} (-1)^k\partial_k^{(l)}(y_i) \ \bigg\}, \] which is considered as a full blue $B$-submodule of ${\mathcal C}^l$. For the case $l=0$, we use ${\mathcal C}_{-1}=\{0\}$.
If ${\mathcal C}^\bullet={\mathcal C}^\bullet(X,{\mathcal F};{\mathcal U})$, then we also write ${\mathcal Z}^l(X,{\mathcal F};{\mathcal U}) \ = \ {\mathcal Z}^l({\mathcal C}^\bullet)$ and ${\mathcal B}^l(X,{\mathcal F};{\mathcal U}) \ = \ {\mathcal B}^l({\mathcal C}^\bullet)$. In this case, we have \[
{\mathcal Z}^l(X,{\mathcal F};{\mathcal U}) \ = \ \biggl\{\ (a_I)\in \prod_{I\in{\mathcal I}_l}{\mathcal F}_I \ \biggl|\ \forall J\in{\mathcal I}_{l+1},\quad \sum_{k=0}^{l+1} (-1)^k\partial_{k,I}^{(l+1)}(a_{J^k}) \equiv 0 \ \bigg\} \] and \[
{\mathcal B}^l(X,{\mathcal F};{\mathcal U}) \ = \ \biggl\{\ (a_I)\in \prod_{I\in{\mathcal I}_l}{\mathcal F}_I \ \biggl|\ \exists (b_J)\in \prod_{J\in{\mathcal I}_{l-1}}{\mathcal F}_J^+,\quad \forall I\in{\mathcal I}_{l},\ \ a_I \equiv \sum_{k=0}^{l} (-1)^k\partial_{k,I}^{(l)}(b_{I^k})\ \bigg\}. \] where we define $\delta_{k,I}^{(l)}(b_{J})=\sum_j\delta_{k,I}^{(l)}(b_{J,j})$ for $b_{J}=\sum_j b_{J,j}\in{\mathcal F}_J^+$ and $J=I^k$. \end{df}
\begin{lemma}
For every $l\geq0$, we have ${\mathcal B}^l({\mathcal C}^\bullet)\subset {\mathcal Z}^l({\mathcal C}^\bullet)$. \end{lemma}
\begin{proof}
Given $(a_i)\in{\mathcal B}^l({\mathcal C}^\bullet)$, i.e.\ there is an element $(b_J)\in({\mathcal C}^{l-1})^+$ such that
\[
a_I \ \equiv \ \sum_{k=0}^{l} (-1)^k\partial_k^{(l)}(b_{I^k})
\]
for all $I\in{\mathcal I}_l$, then we have for every $L\in{\mathcal I}_{l+1}$
\[
\sum_k (-1)^k\partial_k^{(l+1)}(a_{L^k}) \ \equiv \ \sum_{k'\neq k} (-1)^k \ \partial_{k}^{l+1}\circ\partial_{k'}^l\,\Bigl((-1)^{k'+\epsilon}b_{L^{k,k'}}\Bigr)
\]
where $\epsilon=0$ if $k'<k$ and $\epsilon=1$ if $k'>k$, and $L^{k,k'}=L-\{k,k'\}$. Since $\partial_{k}^{l+1}\circ\partial_{k'}^l=\partial_{k'}^{l+1}\circ\partial_{k}^l$, the above sum equals
\[
\sum_{k'< k} (-1)^{k+k'}\ \partial_{k}^{l+1}\circ\partial_{k'}^l\,\bigl(b_{L^{k,k'}}\bigr) \quad + \quad \sum_{k< k'} (-1)^{k+k'+1}\ \partial_{k}^{l+1}\circ\partial_{k'}^l\,\bigl(b_{L^{k,k'}}\bigr) \quad \equiv \quad 0,
\]
which shows that $(a_I)\in{\mathcal Z}^l({\mathcal C}^\bullet)$. \end{proof}
\begin{df}
The \emph{$l$-th \v Cech cohomology of $X$ w.r.t\ ${\mathcal U}$ and with values in ${\mathcal F}$} is defined as the quotient
\[
H^l({\mathcal C}^\bullet) \ = \ {\mathcal Z}^l({\mathcal C}^\bullet)\ / \ {\mathcal B}^l({\mathcal C}^\bullet)
\]
of blue $B$-modules. If ${\mathcal C}^\bullet={\mathcal C}^\bullet(X,{\mathcal F};{\mathcal U})$, then we also write $H^l(X,{\mathcal F};{\mathcal U}) \ = \ H^l({\mathcal C}^\bullet)$. \end{df}
Recall that a morphism $\Psi: {\mathcal C}^\bullet\to{\mathcal D}^\bullet$ of cosimplicial blue $B$-modules is a collection of morphisms $\psi_l:{\mathcal C}^l\to{\mathcal D}^l$ of blue $B$-modules for all $l\geq0$ that commute with the respective coboundary maps $\partial_k^{(l)}$ of ${\mathcal C}^\bullet$ and ${\mathcal D}^\bullet$, i.e.\ $\partial_{k}^{(l)}\circ\psi_{l-1}=\psi_l\circ\partial_{k}^{(l)}$ for all $l\geq 0$ and $0\leq k\leq l$.
\begin{lemma}\label{lemma: morphisms induce maps between cohomology}
Let $\Psi: {\mathcal C}^\bullet\to{\mathcal D}^\bullet$ be a morphism of cosimplicial blue $B$-modules. Then
\[
\psi_l({\mathcal Z}^l({\mathcal C}^\bullet))\subset{\mathcal Z}^l({\mathcal D}^\bullet) \qquad \text{and} \qquad \psi_l({\mathcal B}^l({\mathcal C}^\bullet))\subset{\mathcal B}^l({\mathcal D}^\bullet).
\]
Consequently, $\Psi$ induces a morphism
\[
H^l({\mathcal C}^\bullet) \ = \ {\mathcal Z}^l({\mathcal C}^\bullet) \ / \ {\mathcal B}^l({\mathcal C}^\bullet) \quad \longrightarrow \quad {\mathcal Z}^l({\mathcal D}^\bullet) \ / \ {\mathcal B}^l({\mathcal D}^\bullet) \ = \ H^l({\mathcal D}^\bullet)
\]
for every $l\geq0$. \end{lemma}
\begin{proof}
Let $x\in{\mathcal Z}^l({\mathcal C}^\bullet)$, i.e.\ $\sum(-1)^k\partial_k^{(l+1)}(x)\=0$. Then $\psi_l(x)\in{\mathcal D}^l$ satisfies
\[
\sum_{k=0}^{l+1} \ (-1)^k \ \partial_k^{(l+1)}(\psi(x)) \quad \equiv \quad \sum_{k=0}^{l+1} \ (-1)^k \ \psi_{l+1}\partial_k^{(l+1)}(x) \quad \equiv \quad \psi_{l+1}(0) \quad \equiv \quad 0.
\]
This shows that $\psi(x)\in{\mathcal Z}^l({\mathcal D}^\bullet)$. Let $x\in{\mathcal B}^l({\mathcal C}^\bullet)$, i.e.\ there exists an $y \in{\mathcal C}^{l-1}$ with $x\equiv\sum(-1)^k\partial_k^{(l)}(y)$. Then we have
\[
\psi_l(x) \quad \equiv \quad \sum(-1)^k\psi_l\Bigl(\partial_k^{(l)}(y)\Bigr) \quad \equiv \quad \sum(-1)^k\partial_k^{(l)}\bigl(\psi_{l-1}(y)\bigl),
\]
which shows that $\psi_l(x)\in{\mathcal B}^l({\mathcal D}^\bullet)$. \end{proof}
Next, we prove that the \v Cech cohomology w.r.t.\ ${\mathcal U}$ does not depend on the ordering of the index set ${\mathcal I}$. Note that the definition of the \v Cech complex ${\mathcal C}^\bullet$ is independent of the ordering of ${\mathcal I}$.
\begin{prop}
For $l\geq0$, the subsets ${\mathcal B}^l$ and ${\mathcal Z}^l$ of ${\mathcal C}^l$ are independent of the ordering of ${\mathcal I}$. Consequently, $H^l(X,{\mathcal F};{\mathcal U})$ does not depend on the ordering of ${\mathcal I}$. \end{prop}
\begin{proof}
The usual argument works in this context: we show that ${\mathcal C}^\bullet$ is isomorphic to the \emph{alternating \v Cech complex ${\mathcal C}_\textup{alt}^\bullet$} as a cosimplicial blue $B$-module. The blue $B$-modules ${\mathcal C}^l_\textup{alt}$ are defined as all elements $a_I$ of ${\mathcal C}^l_\textup{tot}$ that satisfy
\[
a_I \ = \ 0
\]
if $I=(i_0,\dotsc,i_l)$ with $i_k=i_{k'}$ for some $k\neq k'$, and
\[
a_{\sigma I} \ = \ \sign(\sigma) \, a_{I}
\]
for a permutation $\sigma\in S_{l+1}$ and $\sigma I=(i_{\sigma(0)},\dotsc,i_{\sigma(l)})$. This defines a cosimplicial subset ${\mathcal C}^\bullet_\textup{alt} $ of ${\mathcal C}^\bullet_\textup{tot}$. Consider the following morphisms of blue $B$-modules
\[
\begin{array}{cccc}
\pi: & {\mathcal C}^l_\textup{alt} & \longrightarrow & {\mathcal C}^l. \\
& (a_I)_{I\in{\mathcal I}^{l+1}} & \longmapsto & (a_I)_{I\in{\mathcal I}_{l}}
\end{array}
\]
and
\[
\begin{array}{cccc}
\iota: & {\mathcal C}^l & \longrightarrow & {\mathcal C}^l_\textup{alt} \\
& (a_I)_{I\in{\mathcal I}_l} & \longmapsto & (\widetilde{a_I})_{I\in{\mathcal I}^{l+1}}
\end{array}
\]
with $\widetilde{a_I}=\sign(\sigma)a_{\sigma I}$ if $\sigma I\in{\mathcal I}_l$ and $\widetilde{a_I}=0$ if $I=(i_0,\dotsc,i_l)$ with $i_k=i_{k'}$ for some $k\neq k'$. As in the usual case of \v Cech cohomology with values in abelian categories, it is easily verified that $\iota$ and $\pi$ are mutually inverse isomorphisms.
If $\widetilde{\mathcal I}$ is the index set ${\mathcal I}$ with a different ordering and $\tilde\pi:{\mathcal C}^\bullet_\textup{alt}\to{\mathcal C}^\bullet$ is the isomorphism with respect to this ordering, then the automorphism $\tilde\pi\circ\iota:{\mathcal C}^\bullet\to{\mathcal C}^\bullet$ sends the set ${\mathcal Z}^l$ of $l$-cocycles w.r.t.\ to the ordering of ${\mathcal I}$ to the set $\widetilde{\mathcal Z}^l$ of $l$-coboundaries w.r.t.\ the ordering of $\widetilde{\mathcal I}$. More precisely, $\tilde\pi\circ\iota$ sends $a_I$ to $\sign(\sigma)a_{\sigma I}$ where $\sigma$ is the permutation such that $\sigma I$ is ordered w.r.t.\ to the ordering of ${\mathcal I}$. Since $B$ is with $-1$, we see that $\widetilde{\mathcal Z}^l={\mathcal Z}^l$.
Similarly, $\tilde\pi\circ\iota$ restricts to an automorphism of ${\mathcal B}^l$. This shows the claim of the proposition. \end{proof}
\section{Refinements} \label{section: refinements}
\noindent In this section, we show that the \v Cech cohomology $H^l(X,{\mathcal F};{\mathcal U})$ is functorial in refinements, so that we form the colimit $H^l(X,{\mathcal F})=\colim H^l(X,{\mathcal F};{\mathcal U})$, which does not depend on the choice of a covering family of $X$ anymore.
\begin{df}
A \emph{refinement of a covering family ${\mathcal U}=\{U_i\}_{i\in{\mathcal I}}$} is a covering family ${\mathcal V}=\{V_j\}_{j\in{\mathcal J}}$ together with a map $\varphi:{\mathcal J}\to{\mathcal I}$ and a morphism $\varphi_j:V_j\to U_{\varphi(i)}$ for every $j\in{\mathcal J}$. We write $\Phi:{\mathcal V}\to {\mathcal U}$ for such a refinement. \end{df}
Given a refinement $\Phi:{\mathcal V}\to {\mathcal U}$ of ${\mathcal U}$, we get induced maps $\varphi:{\mathcal J}_l\to{\mathcal I}_l$ that send $J=\{j_0,\dotsc,j_l\}$ to $\varphi(J)=\{\varphi(j_0),\dotsc,\varphi(j_l)\}$ and morphisms \[
\varphi_J: \quad V_J \ = \ V_{j_0}\times_X \dotsb \times_X V_{j_l} \quad \longrightarrow \quad U_{\varphi(j_0)}\times_X \dotsb \times_X U_{\varphi(j_l)} \ = \ U_{\varphi(J)} \] for every $J\in{\mathcal J}_l$ and $l\geq 0$. This defines, in turn, a morphism $\psi_l:{\mathcal C}^l(X,{\mathcal F};{\mathcal U})\to{\mathcal C}^l(X,{\mathcal F};{\mathcal V})$ for every $l\geq0$. The morphisms $\psi_l$ commute with the respective coboundary morphisms $\partial_k^{(l)}$ of ${\mathcal C}^\bullet(X,{\mathcal F};{\mathcal V})$ and ${\mathcal C}^\bullet(X,{\mathcal F};{\mathcal U})$. Thus $\Phi:{\mathcal V}\to{\mathcal U}$ induces a morphism $\Psi:{\mathcal C}^\bullet(X,{\mathcal F};{\mathcal U})\to{\mathcal C}^\bullet(X,{\mathcal F};{\mathcal V})$ of cosimplicial blue $B$-modules, which maps cocycles to cocycles and coboundaries to coboundaries. This means that we get a morphism \[
\Psi: \quad H^l(X,{\mathcal F};{\mathcal U}) \quad \longrightarrow \quad H^l(X,{\mathcal F};{\mathcal V}) \] from the \v Cech cohomology w.r.t.\ ${\mathcal U}$ to the \v Cech cohomology w.r.t.\ ${\mathcal V}$.
\begin{df}
The \emph{\v Cech cohomology of $X$ with values in ${\mathcal F}$} is defined as the colimit
\[
H^l(X,{\mathcal F}) \ = \ \colim\ H^l(X,{\mathcal F};{\mathcal U})
\]
over the system of all covering families ${\mathcal U}$ of $X$ together with all refinements $\Phi:{\mathcal V}\to{\mathcal U}$ of covering families. \end{df}
\part{Cohomology of monoidal schemes} \label{part: monoidal schemes}
\section{Monoidal schemes over a blueprint} \label{section: monoidal schemes}
\noindent Monoidal schemes a.k.a.\ monoid schemes a.k.a.\ ${\F_1}$-schemes (in the sense of Deitmar, \cite{Deitmar05}, or To\"en and Vaqui\'e, \cite{Toen-Vaquie09}) form the core of ${\F_1}$-geometry in the sense that they appear as a natural subclass in every approach towards ${\F_1}$-schemes.
In this section, we introduce monoidal schemes over a blueprint $B$ as certain blue schemes over $B$. Note that refer to the notion of blue schemes from \cite{L15}, which can be seen as an improvement of the original definition in terms of prime ideals, as contained in \cite{L13}. If $B$ happens to be a global blueprint, e.g.\ a monoid, a ring or a blue field, then both definitions give rise to an equivalent theory. In this case, one can also adopt the viewpoint of To\"en and Vaqui\'e in \cite{Toen-Vaquie09}, which yields yet another theory in general.
To start with, we adapt the concept of a semigroup ring to the context of blueprints. By a \emph{monoid}, we mean a commutative and associative semigroup with neutral element $1$ and absorbing element $0$. All monoids will be written multiplicatively. A monoid morphism is a multiplicative map that sends $1$ to $1$ and $0$ to $0$.
Let $B=\bpquot{A}{{\mathcal R}}$ be a blueprint and $M$ a monoid. The \emph{monoid blueprint of $M$ over $B$} is the blueprint $B[M]=\bpquot{A_M}{{\mathcal R}_M}$ that is defined as follows. The monoid $A_M$ is the smash product $A\wedge M$, which is the quotient of $A\times M$ by the equivalence relation that is generated by the relations $(a,0)\sim (b,0)$ and $(0,m)\sim(0,n)$ with $a,b\in A$ and $m,n\in M$. The pre-addition ${\mathcal R}$ is generated by the set of additive relations \[
\bigl\{ \ \sum(a_i,1)\equiv\sum(b_j,1) \ \bigl| \ \sum a_j\equiv\sum b_j\text{ in }B \ \bigr\}. \] Note that $B[M]$ has the universal property that any pair of a blueprint morphism $B\to C$ and a monoid morphism $M\to C$ extends uniquely to a blueprint morphism $B[M]\to C$. Note further that as a blue $B$-module, $B[M]$ is isomorphic to \mbox{$\bigvee_{m\in M-\{0\}} B\cdot m$}.
\begin{df}
Let $B$ be a blueprint. A \emph{monoidal scheme over $B$} is a blue scheme $X$ that has an open affine covering $\{U_i\}_{i\in{\mathcal I}}$ such that
\begin{enumerate}
\item for every $i\in{\mathcal I}$, there is a monoid $M_i$ and an isomorphism ${\mathcal O}_X(U_i)\simeq B[M_i]$ of blueprints;
\item for every $i,j\in{\mathcal I}$, the intersection $U_i\cap U_j$ is covered by affine opens of the form $V_{i,j,k}=\Spec B[N_{i,j,k}]$ for some monoids $N_{i,j,k}$ such that the restriction map $\res:B[M_i]\to B[N_{i,j,k}]$ is the localization at some multiplicative subset $S_{i,k}$ of $M_i$, i.e.\ $N_{i,j,k}=S^{-1}_{i,k} M_i$; and the same holds true for the restriction map $\res:B[M_j]\to B[N_{i,j,k}]$.
\end{enumerate} \end{df}
\begin{rem}
Note that a monoidal scheme over ${\F_1}$ is nothing else than a monoidal scheme in the usual sense. One can extend the method from \cite{Deitmar08} to show that $X^+_{\mathbb Z}$ is a toric variety over the ring $B^+_{\mathbb Z}$ if $X$ is connected separated integral torsion-free monoidal scheme of finite type over $B$. \end{rem}
\begin{prop}\label{prop: monoidal schemes are defined over f1}
Let X be a blue scheme over $B$. Then $X$ is monoidal over $B$ if and only if there is a monoidal scheme $X_{\F_1}$ over ${\F_1}$ such that $X$ is isomorphic to $X_{\F_1}\times_{\Spec{\F_1}}\Spec B$. \end{prop}
\begin{proof}
Since $B[M]\simeq{\F_1}[M]\otimes_{\F_1} B$, it is clear that the base extension $X_{\F_1}\times_{\Spec{\F_1}}\Spec B$ of a monoidal scheme $X_{\F_1}$ to $B$ is monoidal over $B$.
To prove the other direction of the equivalence, assume that $X$ has a covering $\{U_i\}$ with $U_i=\Spec B[M_i]$ for certain monoids $M_i$. The pairwise intersections $U_i\cap U_j$ have coverings $\{V_{i,j,k}\}$ with $V_{i,j,k}=\Spec B[N_{i,j,k}]$ where each monoid $N_{i,j,k}$ is a localization of both $M_i$ and $M_j$, i.e.\
\[
B[N_{i,j,k}] \quad = \quad B[S_{i,k}^{-1}M_i]\quad = \quad B[S_{j,k}^{-1}M_j]
\]
for certain multiplicative subsets $S_{i,k}$ of $M_i$ and $S_{j,k}$ of $M_j$. If ${\mathcal D}$ is the diagram of all $U_i$ and $V_{i,j,k}$ together with the inclusions $V_{i,j,k}\to U_i$ and $V_{i,j,k}\to U_j$, then $X$ is the colimit of ${\mathcal D}$.
We define the affine monoidal schemes $U_{i,{\F_1}}=\Spec{\F_1}[M_i]$ and $V_{i,j,k}=\Spec{\F_1}[N_{i,j,k,{\F_1}}]$. The colimit of resulting diagram ${\mathcal D}_{\F_1}$ defines a blue scheme $X_{\F_1}$, which is monoidal over ${\F_1}$ since $\{U_{i,{\F_1}}\}$ is a covering of $X_{\F_1}$. It is clear from the construction that $X\simeq X_{\F_1}\times_{\Spec{\F_1}}\Spec B$. This finishes the proof of the proposition. \end{proof}
Recall that a blue scheme $X$ is \emph{separated over ${\F_1}$} if the diagonal morphism $\Delta: X\to X\times X$ is a closed immersion. An important consequence is that the intersection of two affine subschemes of a separated blue scheme is affine.
\begin{cor}\label{cor: intersections of opens in monoidal schemes}
Let $X_{\F_1}$ be a separated monoidal scheme over ${\F_1}$ and $X_B=X\otimes_{\F_1} B$ its base extension to $B$. Consider two open affine subsets $U_1$ and $ U_2$ of $X_B$ such that ${\mathcal O}_{X_B}(U_1)$ and ${\mathcal O}_{X_B}(U_2)$ are monoid blueprints over $B$. Then ${\mathcal O}_{X_B}(U_1\cap U_2)$ is a monoid blueprint over $B$. \end{cor}
\begin{proof}
Let $M_1$ and $M_2$ be monoids such that ${\mathcal O}_{X_B}(U_i)\simeq B[M_i]$ for $i=1,2$. By Proposition \ref{prop: monoidal schemes are defined over f1}, there are open affine subschemes $V_1$ and $V_2$ of $X$ such that ${\mathcal O}_X(V_i)\simeq{\F_1}[M_i]$ for $i=1,2$. We have
\[
U_1 \ \cap \ U_2 \ = \ U_1 \, \times_{X_B} \, U_2 \ = \ \bigl( V_1 \, \times_X \, V_2 \bigr) \, \otimes_{\F_1} \, B \ = \ \bigl( V_1 \, \cap \ V_2 \bigr) \, \otimes_{\F_1} \ B.
\]
Since $X$ is separated over ${\F_1}$, the intersection $V_0=V_1\cap V_2$ is affine. By \cite[Thm.\ 30]{Vezzani12}, there is a monoid $M_0$ that is a localization of both $M_1$ and $M_2$ such that ${\mathcal O}_X(V_0)\simeq{\F_1}[M_0]$. Since the intersection $U_0=U_1\cap U_2$ is isomorphic to the base extension of $V_0$ to $B$, we have ${\mathcal O}_{X_B}(U_0)\simeq B[M_0]$. This proves the corollary. \end{proof}
For monoidal schemes over blue fields we can conclude the following.
\begin{prop}\label{prop: monoidal schemes over blue fields}
If $B$ is a blue field and $X$ is a monoidal scheme over $B$, then every open subset $U$ of $X$ has an open affine covering $\{U_i\}$ with $U_i=\Spec B[M_i]$ for certain monoids $M_i$. \end{prop}
\begin{proof}
Let $U$ be an open subset of $X$ and $\{V_j\}$ an open affine covering with ${\mathcal O}_X(V_j)=B[N_j]$. Then the intersections $U\cap V_j$ can be covered by subsets $W_{j,k}$ such that ${\mathcal O}_X(W_{j,k})$ is a localization of $B[N_j]$. Since $B$ is a blue field, we have for every multiplicative subset $S$ of $B[N_j]$ that
\[
S^{-1}B[N_j] \ = \ T^{-1}B[N_j] \ = \ B[T^{-1}N_j] \qquad \text{where}\qquad T \ = \ S\,\cap\, \{ \,1\cdot a\,|\, a\in N_j\,\}.
\]
Thus $U$ is covered by the $W_{j,k}$ and ${\mathcal O}_X(W_{j,k})$ are monoid blueprints over $B$, which proves the proposition. \end{proof}
\begin{ex}
The proposition is not true over an arbitrary blueprint $B$ since the localizations of $B$ are in general not monoid blueprints. For instance consider the integers $B={\mathbb Z}$. Then $\Spec{\mathbb Z}$ is a monoidal scheme over ${\mathbb Z}$, but every proper open subset is of the form $\Spec {\mathbb Z}[d^{-1}]$ for some integer $d\geq 2$, and ${\mathbb Z}[d^{-1}]$ is not a monoid blueprint over ${\mathbb Z}$ since we have
\[
\underbrace{d^{-1}+\dotsb+d^{-1}}_{d\text{ times}} \ \equiv \ 1.
\]
Note further that even if $B$ is a blue field, not every open subset $U$ of a monoidal scheme $X$ over $B$ satisfies that it is isomorphic to the spectrum of a monoid blueprint over $B$. For instance consider $X=\Spec(B\times B)=\Spec B\amalg\Spec B$, which is monoidal over $B$ since it is covered by two copies of $\Spec B$. However, $B\times B$ is not a monoid blueprint over $B$ since it contains the additive relation
\[
(1,0)+(0,1) \ \equiv \ (1,1).
\] \end{ex}
\section{Locally free sheaves} \label{section: locally free sheaves}
\noindent Let $B$ be a blueprint and $M$ a blue $B$-module.
\begin{df}
A \emph{basis for $M$} is a subset $\beta$ of $M$ such that
\begin{enumerate}
\item for all $m\in M$, there are elements $a_1,\dotsc,a_r\in B$ and pairwise distinct elements $b_1,\dotsc,b_r\in \beta$ such that
\[
m \ \equiv \ \sum_{i=1}^r \ a_i b_i.
\]
\item If
\[
\sum_{i=r}^r \ a_i b_i \ \equiv \ \sum_{j=s}^r \ a'_j b'_j
\]
for $a_1,\dotsc,a_r,a'_1,\dotsc,a'_s\in B$ and pairwise distinct elements $b_1,\dotsc,b_r,b'_1,\dotsc,b'_s\in M$, then $a_1=\dotsb=a_r=a'_1=\dotsb=a'_s=0$.
\end{enumerate}
A blue $B$-module is \emph{freely generated} if it has a basis. A blue $B$-module is \emph{free} if is isomorphic to $\bigvee_{b\in\beta}B\cdot b$ for a subset $\beta$ of $B$. \end{df}
Note that $\beta$ is a basis for the free blue module $\bigvee_{b\in\beta}B\cdot b$. Thus a free module is freely generated. The larger class of freely generated modules can be classified as follows.
\begin{lemma}
Let $M$ be a blue $B$-module with basis $\beta$.
\begin{enumerate}
\item There is a unique isomorphism from $M$ onto a blue submodule of
\[
\bigoplus_{b\in\beta} \ B \cdot b \quad = \quad \bigl\{ \ (m_b)\in\prod_{b\in\beta}B \cdot b \ \bigl| \ m_b=0\text{ for all but finitely many }b\ \bigr\}
\]
that maps $b\in\beta$ to $1\cdot b$. Conversely, any blue $B$-submodule of $\bigoplus_{b\in\beta}B\cdot b$ that contains $\beta$ is freely generated by $\beta$.
\item Any two bases of $M$ have the same cardinality.
\end{enumerate} \end{lemma}
\begin{proof}
Since every $m\in M$ is a unique linear combination $m\equiv\sum m_b b$ of the basis elements $b\in\beta$ where all but finitely many $m_b\in B$ are $0$, the only possible morphism $\Phi:M\to\bigoplus_{b\in\beta}B\cdot b$ sends $m$ to $(m_b b)$. Since there are no additive relations between the different basis elements by \eqref{part2} of the definition of a basis, the map $\Phi$ is indeed a morphism of blue $B$-modules. It is clearly injective and thus defines an isomorphism onto its image. The latter claim of \eqref{part1} of the lemma is obvious.
The embedding $\Phi:M\to \bigoplus_{b\in\beta}B\cdot b$ defines an isomorphism $\Phi^+:M^+\to \bigoplus_{b\in\beta}B^+\cdot b$ of blue $B^+$-modules, which have the same basis $\beta$. Thus we obtain an isomorphism $\Phi_{\mathbb Z}^+:M^+_{\mathbb Z}\to \bigoplus_{b\in\beta}B_{\mathbb Z}^+\cdot b$ of free $B_{\mathbb Z}^+$-modules with basis $\beta$. Since any two bases of a free module over a ring have the same cardinality, we obtain the same result for the freely generated blue $B$-module $M$. \end{proof}
\begin{df}
Let $M$ be a freely generated blue $B$-module with basis $\beta$. The \emph{rank ${\textup{rk}}_B M$ of $M$ over $B$} is defined as the cardinality of $\beta$. \end{df}
Let $X$ be a blue scheme over $B$ and $\beta$ a (possibly infinite) set of cardinality $r$. In this part of the paper, a sheaf on $X$ is a sheaf on the small Zariski site of $X$.
\begin{df}
A \emph{locally free sheaf of rank $r$} on $X$ is a sheaf ${\mathcal F}$ on $X$ in blue $B$-modules that has an open affine covering $\{U_i\}_{i\in{\mathcal I}}$ with the following properties:
\begin{enumerate}
\item\label{part1} if $B_i={\mathcal O}_X(U_i)$, then for every $i\in{\mathcal I}$,
\[
{\mathcal F}(U_i) \quad \simeq \quad \bigvee_{b\in\beta} B_i\cdot b\,;
\]
\item\label{part2} for every $i\in{\mathcal I}$ and every open subset $V$ of $U_i$ with ${\mathcal O}_X(V)=S_V^{-1}B_i$ for some multiplicative subset $S_V$ of $B_i$, there is an isomorphism ${\mathcal F}(V)\simeq\bigvee_{b\in\beta} S_V^{-1}B_i\cdot b$ such that the restriction map
\[
\res_{U_i,V}:\quad \bigvee_{b\in\beta} B_i\cdot b \quad \longrightarrow \quad \bigvee_{b\in\beta} S_V^{-1}B_i\cdot b
\]
corresponds to the localization of each component $B_i\cdot b$ at $S_V$.
\end{enumerate}
We call a covering $\{U_i\}$ of $X$ that satisfies properties \eqref{part1} and \eqref{part2} a \emph{trivialization of ${\mathcal F}$}. \end{df}
Note that the \emph{localizations} $V$ of the $U_i$, i.e.\ open subsets of the form $V=\Spec S_V^{-1} B_i$ for some multiplicative subset $S_V$ of $B_i$, form a basis for the topology of $X$. Thus a locally free sheaf ${\mathcal F}$ is uniquely determined by a trivialization $\{U_i\}$ together with the restriction maps to subsets of the form $V=\Spec S_V^{-1} B_i$.
\begin{rem}
There is an obvious notion of a quasi-coherent sheaf on $X$ (cf.\ \cite{CLS12} for the case of monoidal schemes). Property \eqref{part2} is automatically satisfied if ${\mathcal F}$ is quasi-coherent. In other words, a quasi-coherent sheaf is locally free if and only if there are a set $\beta$ and an open affine covering $\{U_i\}$ of $X$ such that ${\mathcal F}(U_i) \simeq \bigvee_{b\in\beta} B_i\cdot b$ for all $i$. \end{rem}
\begin{ex}
The sheaf $\bigvee_{b\in\beta}{\mathcal O}_X$ that sends an open subset $U$ of $X$ to $\bigvee_{b\in\beta}{\mathcal O}_X(U)$, together with the obvious restriction maps, is locally free of rank $r=\#\beta$. It is called the \emph{trivial locally free sheaf of rank $r$}. \end{ex}
We construct the base extension of a locally free sheaf to rings. Let ${\mathcal F}$ be a locally free sheaf on $X$ of rank $r$ and ${\mathcal U}$ the family of all open affine subsets of $X$ such that ${\mathcal F}(U)\simeq\bigvee_{b\in\beta} {\mathcal O}_X(U)\cdot b$ together with all inclusion maps. By properties \eqref{part1} and \eqref{part2}, $X$ is the colimit of ${\mathcal U}$. If ${\mathcal U}_{\mathbb Z}^+$ denotes the family of all $U_{\mathbb Z}^+$ for $U$ in ${\mathcal U}$ and all inclusion maps $U_{\mathbb Z}^+\to V_{\mathbb Z}^+$ whenever $U\to V$ is in ${\mathcal U}$, then $X_{\mathbb Z}^+$ is the colimit of ${\mathcal U}_{\mathbb Z}^+$.
We define ${\mathcal F}_{\mathbb Z}^+(U_{\mathbb Z}^+)=\bigl({\mathcal F}(U)\bigr)_{\mathbb Z}^+$ for all $U$ in ${\mathcal U}$ and we obtain restriction morphisms $\res:{\mathcal F}(U)_{\mathbb Z}^+\to{\mathcal F}(V)_{\mathbb Z}^+$ for every inclusion $V\to U$ in ${\mathcal U}$. Since localizations commute with base extensions to rings, i.e.\ $(S^{-1}B)^+_{\mathbb Z} =S^{-1}(B_{\mathbb Z}^+)$, the values ${\mathcal F}_{\mathbb Z}^+(U_{\mathbb Z}^+)$ for $U$ in ${\mathcal U}$ glue together to a uniquely determined sheaf ${\mathcal F}_{\mathbb Z}^+$ on $X_{\mathbb Z}^+$. Since \[
\Bigl( \ \bigvee_{b\in\beta} B_U \cdot b \Bigr)^+_{\mathbb Z} \quad = \quad \bigoplus_{b\in\beta} \ B_{U,{\mathbb Z}}^+ \cdot b, \] the sheaf ${\mathcal F}_{\mathbb Z}^+$ is locally free on $X$ as a sheaf with values in $B_{\mathbb Z}^+$-modules.
\section{{\v C}ech cohomology of monoidal schemes} \label{section: cech cohomology of monoidal schemes}
\noindent In this section, we prove the comparison result for the cohomology of locally free sheaves on monoidal schemes with the cohomology of its base extension to rings.
Let $B$ be a blueprint with $-1$ and $X$ a monoidal scheme over $B$. Let $\beta$ be a set of cardinality $r$ and ${\mathcal F}$ a locally free sheaf of rank $r$ on $X$. A trivialization ${\mathcal U}=\{U_i\}_{i\in{\mathcal I}}$ of ${\mathcal F}$ is \emph{finite} if ${\mathcal I}$ is a finite set. It is \emph{monoidal} if the coordinate blueprints $B_i={\mathcal O}_X(U_i)$ are monoid blueprints of the form $B[M_i]$ over $B$.
We employ the notation from Part \ref{part: cech cohomology} of the paper. We assume that ${\mathcal I}$ is totally ordered and denote by ${\mathcal I}_l$ the set of cardinality $l+1$-subsets $I$ of ${\mathcal I}$, which inherits an ordering from ${\mathcal I}$. We write $I=(i_0,\dotsc,i_l)$ if $I=\{i_0,\dotsc,i_l\}$ and $i_0<\dotsb<i_l$. For $I\in{\mathcal I}_l$, we define \[
U_I \ = \ \bigcap_{i\in I} \ U_{i} \ , \qquad B_I \ = \ {\mathcal O}_X(U_I) \qquad \text{and} \qquad {\mathcal F}_I \ = \ {\mathcal F}(U_I). \] Let ${\mathcal C}^l=\prod_{I\in{\mathcal I}_l}{\mathcal F}_I$ and ${\mathcal C}^\bullet={\mathcal C}^\bullet(X,{\mathcal F};{\mathcal U})$ the \v Cech complex of $X$ w.r.t.\ ${\mathcal U}$ and values in ${\mathcal F}$. We denote the coboundary maps as usual by $\partial_k^{(l)}:{\mathcal C}^{l-1}\to{\mathcal C}^l$. We state the following hypothesis on $X$ and ${\mathcal U}=\{U_i\}_{i\in{\mathcal I}}$.
\noindent\textbf{Hypothesis (H):} For all finite subsets $J\subset I$ of ${\mathcal I}$, the restriction map
\[
\res_{U_J,U_I}:{\mathcal O}_X(U_J)\to{\mathcal O}_X(U_I)
\]
is injective.
\begin{rem} Recall that a blueprint $B$ is \emph{integral} if every non-zero element $a\in B$ acts injectively on $B$ by multiplication. A blue scheme is \emph{integral} if the coordinate blueprint of every open affine subscheme $U$ of $X$ is integral. If $X$ is integral, then (H) is satisfied for all open affine coverings ${\mathcal U}$ of $X$. \end{rem}
Since $B$ is with $-1$, we have that $B^+$ is a ring, i.e.\ $B^+=B^+_{\mathbb Z}$. Similarly, $X^+=X^+_{\mathbb Z}$ is a scheme over $B^+$ and ${\mathcal F}^+$ is a locally free sheaf on $X^+$ in $B^+$-modules. Defining ${\mathcal U}^+$ as the collection of all affine opens $U_i^+$ of $X^+$, we obtain a trivialization of ${\mathcal F}^+$ and can form the \v Cech complex ${\mathcal C}^\bullet(X^+,{\mathcal F}^+;{\mathcal U}^+)$ of $X^+$ w.r.t.\ ${\mathcal U}^+$ and values in ${\mathcal F}^+$. Then the subsets ${\mathcal Z}^l(X^+,{\mathcal F}^+;{\mathcal U}^+)$ and ${\mathcal B}^l(X^+,{\mathcal F}^+;{\mathcal U}^+)$ are $B^+$-modules, and so is $H^l(X^+,{\mathcal F}^+;{\mathcal U}^+)$.
There is a canonical morphism ${\mathcal C}^\bullet(X,{\mathcal F};{\mathcal U}) \longrightarrow {\mathcal C}^\bullet(X^+,{\mathcal F}^+;{\mathcal U}^+)$ of cosimplicial blue $B$-modules, which is injective in each degree since all blue $B$-modules ${\mathcal C}^l(X,{\mathcal F};{\mathcal U})$ are with $-1$. Thus we can consider ${\mathcal Z}^l(X,{\mathcal F};{\mathcal U})$ as a subset of ${\mathcal Z}^l(X^+,{\mathcal F}^+;{\mathcal U}^+)$ and ${\mathcal B}^l(X,{\mathcal F};{\mathcal U})$ as a subset of ${\mathcal B}^l(X^+,{\mathcal F}^+;{\mathcal U}^+)$ for every $l\geq 0$. This induces a morphism \[
H^l(X,{\mathcal F};{\mathcal U}) \quad \longrightarrow \quad H^l(X^+,{\mathcal F}^+;{\mathcal U}^+) \] of blue $B$-modules.
\begin{thm}\label{thm: comparison for a fixed covering}
Given a monoidal scheme $X$ over $B$, a locally free sheaf ${\mathcal F}$ on $X$ and a finite monoidal trivialization ${\mathcal U}=\{U_i\}_{i\in{\mathcal I}}$ of ${\mathcal F}$ that satisfies Hypothesis (H). Then
\[
{\mathcal Z}^l(X,{\mathcal F},{\mathcal U})^+ \ = \ {\mathcal Z}^l(X^+,{\mathcal F}^+,{\mathcal U}^+) \qquad \text{and} \qquad {\mathcal B}^l(X,{\mathcal F},{\mathcal U})^+ \ = \ {\mathcal B}^l(X^+,{\mathcal F}^+,{\mathcal U}^+)
\]
for every $l\geq0$. Consequently, we have
\[
H^l(X,{\mathcal F},{\mathcal U})^+ \ = \ H^l(X^+,{\mathcal F}^+,{\mathcal U}^+).
\] \end{thm}
\begin{proof} We will establish the following two lemmas in order to prove Theorem \ref{thm: comparison for a fixed covering}. In the proofs of these lemmas, we will make use of the usual \v Cech chain complex \begin{multline*}
{\mathcal C}^0(X^+,{\mathcal F}^+;{\mathcal U}^+) \quad \stackrel{d^1}\longrightarrow \quad {\mathcal C}^1(X^+,{\mathcal F}^+;{\mathcal U}^+) \quad \longrightarrow \quad \dotsb \\
\dotsb \quad \longrightarrow \quad {\mathcal C}^{l-1}(X^+,{\mathcal F}^+;{\mathcal U}^+) \quad \stackrel{d^l}\longrightarrow \quad {\mathcal C}^l(X^+,{\mathcal F}^+;{\mathcal U}^+) \quad \longrightarrow \quad \dotsb \end{multline*} where the differentials $d^l=\sum_{i=0}^l (-1)^i \partial_k^{(l)}$ are the alternating sums of the respective restriction maps.
\begin{lemma}\label{lemma: base extension of coboundary blueprints}
${\mathcal B}^l(X,{\mathcal F},{\mathcal U})^+ \ = \ {\mathcal B}^l(X^+,{\mathcal F}^+,{\mathcal U}^+)$. \end{lemma}
\begin{proof}
Since ${\mathcal U}$ is finite, a set of generators for ${\mathcal B}^l(X^+,{\mathcal F}^+,{\mathcal U}^+)$ is given by the images of the vectors $x_{a,b,J}=(0,\dotsc,0,a\cdot b,0,\dotsc,0)$ with $a\in B_J$ and $b\in\beta$. The image of such a vector is of the form $(d^l(x_{a,b,J})_I)_{I\in{\mathcal I}_l}$. Since $x_{a,b,J}$ has only one non-trivial component, we have $d^l(x_{a,b,J})_I=\partial_k^{(l)}(x_{a,b,J})_I$ for some $k$. Therefore, the image of $x$ in ${\mathcal C}^{l,+}$ is contained in ${\mathcal C}^l$. Since ${\mathcal B}^l(X,{\mathcal F},{\mathcal U})={\mathcal B}^l(X^+,{\mathcal F}^+,{\mathcal U}^+)\cap{\mathcal C}^l$, the lemma follows. \end{proof}
\begin{lemma}\label{lemma: base extension of cocycle blueprints}
${\mathcal Z}^l(X,{\mathcal F},{\mathcal U})^+ \ = \ {\mathcal Z}^l(X^+,{\mathcal F}^+,{\mathcal U}^+)$. \end{lemma}
\begin{proof}
Let $B_\eta=\colim B_I$ be the colimit of the blueprints $B_I$ for finite subsets $I$ of ${\mathcal I}$. By Hypothesis (H), the canonical inclusions $B_I\to B_\eta$ are injective for all finite $I\subset{\mathcal I}$. Since $\partial_k^{(l)}$ extends to a $B_{I^k}^+$-linear map
\[
\partial_k^{(l),+}: \quad {\mathcal F}_{I^k}^+ \ \simeq \ \bigoplus_{b\in\beta} \ B_{I^k}^+\cdot b \quad \longrightarrow \quad \bigoplus_{b\in\beta} \ B_I^+\cdot b \ \simeq \ {\mathcal F}_I^+,
\]
the \v Cech chain complex
\[
{\mathcal C}^0(X^+,{\mathcal F}^+;{\mathcal U}^+) \quad \stackrel{d^1}\longrightarrow \quad {\mathcal C}^1(X^+,{\mathcal F}^+;{\mathcal U}^+) \quad \longrightarrow \quad \dotsb
\]
with ${\mathcal C}^l(X^+,{\mathcal F}^+;{\mathcal U}^+)=\prod_{i\in{\mathcal I}_l}\bigoplus_{b\in\beta}B_I^+\cdot b$ defines a chain complex
\[
{\mathcal C}^{0,+}_\eta \quad \stackrel{d^1}\longrightarrow \quad {\mathcal C}^{1,+}_\eta \quad \longrightarrow \quad \dotsb
\]
with ${\mathcal C}^{l,+}_\eta=\prod_{i\in{\mathcal I}_l}\bigoplus_{b\in\beta}B_\eta^+\cdot b$. This chain complex is the \v Cech chain complex of the affine scheme $X_\eta^+=\Spec B_\eta^+$ w.r.t.\ the covering ${\mathcal U}_\eta^+=\{U_{i,\eta}^+\}_{i\in{\mathcal I}}$ where $U_{i,\eta}^+=X_\eta^+$ and with values in the locally free sheaf ${\mathcal F}_\eta^+$ associated to the $B_\eta^+$-module $F_\eta^+=\colim {\mathcal F}^+(U_I^+)$ that is the colimit over all finite $I\subset{\mathcal I}$.
Since the cohomology of coherent sheaves on affine schemes is concentrated in degree $0$, we have
\begin{align*}
{\mathcal Z}^0(X_\eta^+,{\mathcal F}^+_\eta;{\mathcal U}_\eta^+) \quad &= \quad F_\eta^+ &&\text{and} \\
{\mathcal Z}^l(X_\eta^+,{\mathcal F}^+_\eta;{\mathcal U}_\eta^+) \quad &= \quad {\mathcal B}^l(X_\eta^+,{\mathcal F}^+_\eta;{\mathcal U}_\eta^+) &&\text{for}\quad l>0.
\end{align*}
Since $F_\eta^+=\bigl(\bigvee_{b\in\beta} B_\eta\cdot b\bigl)^+$ is generated by elements in ${\mathcal Z}^0(X,{\mathcal F};{\mathcal U})=\bigvee_{b\in\beta} B\cdot b$ as a blue $B_\eta$-module, the claim of the lemma follows for $l=0$.
For $l>0$, we can apply Lemma \ref{lemma: base extension of coboundary blueprints} to $X_\eta=\Spec B_\eta$, the locally free sheaf associated with $F_\eta=\colim {\mathcal O}_X(U_I)$ and ${\mathcal U}_\eta=\{U_{i,\eta}\}_{i\in{\mathcal I}}$ with $U_{i,\eta}=X_\eta$ and get
\[
{\mathcal B}^l(X_\eta^+,{\mathcal F}^+_\eta;{\mathcal U}_\eta^+) \quad = \quad {\mathcal B}^l(X_\eta,{\mathcal F}_\eta;{\mathcal U}_\eta)^+.
\]
Since
\[
{\mathcal B}^l(X_\eta,{\mathcal F}_\eta;{\mathcal U}_\eta) \quad \subset \quad {\mathcal Z}^l(X_\eta,{\mathcal F}_\eta;{\mathcal U}_\eta) \quad \subset \quad {\mathcal Z}^l(X_\eta^+,{\mathcal F}^+_\eta;{\mathcal U}_\eta^+),
\]
we conclude that ${\mathcal Z}^l(X_\eta,{\mathcal F}_\eta;{\mathcal U}_\eta)^+={\mathcal Z}^l(X_\eta^+,{\mathcal F}^+_\eta;{\mathcal U}_\eta^+)$. Therefore
\begin{align*}
{\mathcal Z}^l(X,{\mathcal F};{\mathcal U})^+ \quad &= \quad \Bigl( \ {\mathcal C}^l(X,{\mathcal F};{\mathcal U}) \quad \cap \quad {\mathcal Z}^l(X_\eta,{\mathcal F}_\eta;{\mathcal U}_\eta) \ \Bigr)^+ \\
\quad &= \quad {\mathcal C}^l(X,{\mathcal F};{\mathcal U})^+ \quad \cap \quad {\mathcal Z}^l(X_\eta,{\mathcal F}_\eta;{\mathcal U}_\eta)^+ \\
\quad &= \quad {\mathcal C}^l(X^+,{\mathcal F}^+;{\mathcal U}^+) \quad \cap \quad {\mathcal Z}^l(X_\eta^+,{\mathcal F}_\eta^+;{\mathcal U}_\eta^+) \\
\quad &= \quad {\mathcal Z}^l(X^+,{\mathcal F}^+;{\mathcal U}^+)
\end{align*}
as desired. \end{proof}
Since taking quotients commutes with the base extension to rings, we have that \begin{multline*}
H^l(X,{\mathcal F};{\mathcal U})^+ \ = \ {\mathcal Z}^l(X,{\mathcal F};{\mathcal U})^+ / {\mathcal B}^l(X,{\mathcal F};{\mathcal U})^+ \ = \\ {\mathcal Z}^l(X^+,{\mathcal F}^+;{\mathcal U}^+) / {\mathcal B}^l(X^+,{\mathcal F}^+;{\mathcal U}^+) \ = \ H^l(X^+,{\mathcal F}^+;{\mathcal U}^+), \end{multline*} which proves Theorem \ref{thm: comparison for a fixed covering}. \end{proof}
\begin{thm}\label{thm: comparison of cech cohomology}
Given a monoidal scheme $X$ over $B$ that admits a finite covering $\{U_i\}$ with Hypothesis (H) such that ${\mathcal O}_X(U_i)$ are monoid blueprints over $B$. Then we have for every locally free sheaf ${\mathcal F}$ on $X$ that
\[
H^l(X,{\mathcal F})^+ \ = \ H^l(X^+,{\mathcal F}^+).
\] \end{thm}
\begin{proof}
Let ${\mathcal U}$ be a covering of $X$ with Hypothesis (H) and ${\mathcal F}$ a locally free sheaf on $X$. Then there is a finite refinement ${\mathcal V}$ of ${\mathcal U}$ that satisfies all conditions of Theorem \ref{thm: comparison for a fixed covering}. Since we can choose ${\mathcal U}$ itself arbitrary fine, the coverings ${\mathcal V}$ that satisfy the hypotheses of Theorem \ref{thm: comparison for a fixed covering} form a cofinal system in the category of all finite coverings of $X$ together with refinements. Since $X$ is quasi-compact, the ${\mathcal V}$ are cofinal in the category of all coverings of $X$.
Therefore the colimit of the cohomology blueprints $H^l(X,{\mathcal F};{\mathcal V})$ over all coverings ${\mathcal V}$ that satisfy Theorem \ref{thm: comparison for a fixed covering} equals $H^l(X,{\mathcal F})$. For the same reasons, the colimit of the cohomology groups $H^l(X^+,{\mathcal F}^+,{\mathcal V}^+)$ over all such ${\mathcal V}$ equals $H^l(X^+,{\mathcal F}^+)$. Since $(-)^+$ commutes with filtered colimits, this establishes the claim of the theorem. \end{proof}
\begin{ex}[Line bundles on projective space]
Let $B$ be a blueprint with $-1$ and ${\mathcal O}(d)$ the twisted sheaf on ${\mathbb P}^n_B$. If $d\geq0$, then the cohomology $H^\ast({\mathbb P}^{n,+}_B,{\mathcal O}(d)^+)$ is concentrated in degree $0$. Therefore $H^0({\mathbb P}^n_B,{\mathcal O}(d))$ is the only non-trivial cohomology of ${\mathbb P}^n_B$ with values in ${\mathcal O}(d)$. It is clear that $H^0({\mathbb P}^n_B,{\mathcal O}(d))$ equals the blue $B$-module of global sections of ${\mathcal O}(d)$, which is a free $B$-module of rank ${\textup{rk}}\; H^0({\mathbb P}^{n,+}_B,{\mathcal O}(d)^+)$.
For $-n\leq d\leq -1$, the cohomology $H^\ast({\mathbb P}^n_B,{\mathcal O}(d))$ is trivial. If $d\leq -n-1$, then the cohomology $H^\ast({\mathbb P}^{n,+}_B,{\mathcal O}(d)^+)$ is concentrated in degree $n$. Therefore $H^n({\mathbb P}^n_B,{\mathcal O}(d))$ is the only non-trivial cohomology of $X$ with values in ${\mathcal O}(d)$. If ${\mathcal U}=\{U_i\}_{i\in{\mathcal I}}$ is the canonical atlas of ${\mathbb P}^n_B$, then we have $H^n({\mathbb P}^n_B,{\mathcal O}(d))=H^n({\mathbb P}^n_B,{\mathcal O}(d);{\mathcal U})$ by comparison with the compatible situation for ${\mathbb P}^{n,+}_B$ and the canonical covering ${\mathcal U}^+$. Therefore we have ${\mathcal Z}^l({\mathbb P}^n_B,{\mathcal O}(d))={\mathcal O}(d)(U_{\mathcal I})$, and ${\mathcal B}^l({\mathbb P}^n_B,{\mathcal O}(d))$ is generated by the images $d(x_{a,b,J})\in{\mathcal O}(d)(U_{\mathcal I})$ (cf.\ the proof of Lemma \ref{lemma: base extension of coboundary blueprints}). Therefore $H^n({\mathbb P}^n_B,{\mathcal O}(d))$ is a free blue $B$-module of rank ${\textup{rk}}\; H^n({\mathbb P}^{n,+}_B,{\mathcal O}(d)^+)$. \end{ex}
Also in more complicated examples, we found that the cohomology blueprints are free over the base blueprint. Therefore we pose the following problem.
\begin{question*}
Let $B$ be a blueprint with $-1$ and $X$ a quasi-compact monoidal scheme over $B$ that admits an open affine covering satisfying Hypothesis (H). Is it true that $H^l(X,{\mathcal F})$ is a free blue $B$-module for every locally free sheaf ${\mathcal F}$? \end{question*}
\begin{rem}[Sheaf cohomology for toric varieties] \label{rem: cohomology for toric varieties}
We conlcude this text with the following remark on possible applications to the computation of sheaf cohomology for toric varieties.
Every toric variety ${\mathcal X}$ over the ring $B^+$ admits a monoidal model $X$ over $B$, i.e.\ a monoidal scheme $X$ over $B$ such that ${\mathcal X}\simeq X^+$ as a $B^+$-scheme. The maximal open affine covering of $X$ satisfies Hypothesis (H) since the restriction maps correpond to inclusions of subsemigroups of the ambient character lattice of the toric variety.
Since the \v Cech cohomology for monoidal schemes is amenable to explicit calculation due to their rigid structure, Theorem \ref{thm: comparison of cech cohomology} yields an application for calculations of sheaf cohomology over toric varieties.
The drawback is, however, that only a very limited class of locally free sheaves over toric varieties can be defined over a monoidal model. Namely, the rigid structure of the wedge product implies that every locally free sheaf ${\mathcal F}$ on a monoidal scheme $X$ over a blueprint $B$ decomposes into the wedge product $\bigvee {\mathcal L}_i$ of line bundles.
This means that the only locally free sheaves of toric varieties for which our methods apply are (direct sums of) line bundles. There exists an algorithm to calculate the cohomology of toric line bundles, as conjectured in \cite{BJRR10} and proven independently in \cite{Roschy-Rahn10} and \cite{Jow11}. The method of this algorithm seems to be quite different from the perspective of our text, but it would be interesting to understand the precise relationship. \end{rem}
\appendix
\section{Cohomology of {${\mathbb P}^1$} via injective resolutions} \label{appendix: cohomology of p1}
\noindent In this section, we mimic the methods of homological algebra and injective resolutions to calculate the cohomology $H^i_\textup{hom}(X,{\mathcal O}_X)$ of the projective line $X={\mathbb P}^1_{\F_1}$. While $H_\textup{hom}^0(X,{\mathcal O}_X)$ equals the global sections of ${\mathcal O}_X$, it turns out that $H_\textup{hom}^1(X,{\mathcal O}_X)$ is of infinite rank over ${\F_1}$.
Note that the following calculations apply also to the projective line over ${\F_{1^2}}$, which shows that $H^i_\textup{hom}(X,{\mathcal F})$ differs from the cohomology blueprints $H^i(X,{\mathcal F})$, as considered in the main text of this paper.
Deitmar has given in \cite{Deitmar11b} a rigorous treatment of cohomology via injective resolutions for sheaves in so called \emph{belian} categories. This applies, in particular, to sheaves on ${\mathbb P}^1_{\F_1}$ in pointed ${\F_1}$-modules (also known as pointed ${\F_1}$-sets). Note that the general hypotheses of \cite{Deitmar11b} are not satisfied by the category of blue $B$-modules.
To emphasize that we abandon any additive structure in the discussion that follows, we avoid mentioning blueprints, but employ the language of monoids and monoidal schemes.
Let $A={\F_1}[T]$ be the coordinate monoid of ${\mathbb A}^1_{\F_1}$. All of the $A$-modules in the following are pointed $A$-modules (following the terminology of \cite{Deitmar11b}), and we denote the base point generally by $\ast$. Let $F=\{T^i\}_{i\geq0}\cup\{\ast\}$ be the free module over $A$ of rank $1$, $I=\{T^i\}_{i\in{\mathbb Z}}\cup\{\ast\}$ and $J=\{T^i\}_{i<0}\cup\{\ast\}$. Then both $I$ and $J$ are injective $A$-modules. Let $G={\F_1}[T^{\pm1}]$ be the ``quotient monoid'' of $A$. Then the corresponding localizations of $I$ and $J$ are $I$ itself resp.\ $0=\{\ast\}$, which are both injective $G$-modules.
The topological space of $X={\mathbb P}^1_{\F_1}$ has three points; namely, two closed points $x_1,x_2$ and one generic point $x_0$. It can be covered by two opens $U_i=\{x_0,x_i\}$ ($i=1,2$), which are both isomorphic to ${\mathbb A}^1_{\F_1}$ and which intersect in $U_0=\{x_0\}$. The coordinate monoids of these opens are respectively ${\mathcal O}_X(U_1)\simeq {\mathcal O}_X(U_2)\simeq A$ and ${\mathcal O}_X(U_0)\simeq G$, where ${\mathcal O}_X$ is the structure sheaf of $X$.
We define the injective sheaf ${\mathcal I}_0$ over $X$ by ${\mathcal I}_0(U_i)=I$ for $i=0,1,2$ together with the identity maps $\textup{id}:I\to I$ as restriction maps. We define the injective sheaf ${\mathcal I}_1$ over $X$ by ${\mathcal I}_0(U_i)=J$ for $i=1,2$ and $I_1(U_0)=0$ together with the trivial maps $0:J\to 0$ as restriction maps.
It is easily seen that the structure sheaf ${\mathcal O}_X$ of $X$ has an injective resolution of the form \[
0 \ \longrightarrow \ {\mathcal O}_X \ \longrightarrow \ {\mathcal I}_0 \ \longrightarrow \ {\mathcal I}_1 \ \longrightarrow \ 0. \] Taking stalks at $x_1$ or at $x_2$ yields the exact sequence \[
0 \ \longrightarrow \ F \ \longrightarrow \ I \ \longrightarrow \ J \ \longrightarrow \ 0 \] of $A$-modules. Talking stalks at $x_0$ yields the exact sequence \[
0 \ \longrightarrow \ I \ \longrightarrow \ I \ \longrightarrow \ 0 \ \longrightarrow \ 0 \] of $H$-modules.
The next step is to apply $\Hom({\mathcal O}_X,-)$ to the given injective resolution of ${\mathcal O}_X$. A morphism $\varphi:{\mathcal O}_X\to {\mathcal I}_0$ is determined by the image of $\varphi_{x_0}(T^0)\in {\mathcal I}_{1,x_0}=I$ of $T^0\in {\mathcal I}_{0,x_0}=I$. Thus $\Hom({\mathcal O}_X,{\mathcal I}_0)\simeq I$ (as $A$-module, or even $H$-module).
A morphism $\psi:{\mathcal O}_X\to {\mathcal I}_1$ is given by two $A$-module maps \[\psi_i: {\mathcal O}_{X,x_i}=F \to J={\mathcal I}_{1,x_i}\] ($i=1,2$), which do not have to satisfy any relation since the restriction maps of ${\mathcal I}_1$ are trivial. Thus $\Hom({\mathcal O}_X,{\mathcal I}_1)\simeq J\times J$ (as $A$-module). Note that, a priori, these homomorphism sets are merely ${\F_1}$-modules, but the richer structure as $A$-modules makes it easier to study the induced morphism $\Phi:\Hom({\mathcal O}_X,{\mathcal I}_0)\to \Hom({\mathcal O}_X,{\mathcal I}_1)$, which is the only non-trivial map in the complex \[
0 \ \longrightarrow \ \Hom({\mathcal O}_X,{\mathcal I}_0) \ \stackrel\Phi\longrightarrow \ \Hom({\mathcal O}_X,{\mathcal I}_1) \ \longrightarrow \ 0. \] We define the cohomology groups $H_\textup{hom}^i({\mathbb P}^1_{\F_1},{\mathcal O}_X)$ as the cohomology groups of this complex.
The kernel of $\Phi$ consists of the trivial morphism and the morphism $\varphi:{\mathcal O}_X\to{\mathcal I}_0$ that is characterized by $\varphi_{x_0}(T^0)=T^0$. Thus $H_\textup{hom}^0({\mathbb P}^1_{\F_1},{\mathcal O}_X)=\{\ast,\varphi\}$ is an $1$-dimensional ${\F_1}$-vector space, in accordance with the analogous result for sheaf cohomology of ${\mathbb P}^1$ over a ring.
The image of $\Phi$ are morphisms $\psi:{\mathcal O}_X\to{\mathcal I}_1$ such that either $\psi_1$ or $\psi_2$ is trivial. Thus $\im\Phi=J\vee J\subset J\times J$ (as $A$-modules). Consequently $H_\textup{hom}^1({\mathbb P}^1,{\mathcal O}_X)=(J\times J)/(J\vee J)$ is an infinite-dimensional ${\F_1}$-vector space. This result is not at all in coherence with the situation over a ring where $H_\textup{hom}^1({\mathbb P}^1,{\mathcal O}_X)=0$.
\begin{rem}
The above calculation can also be used to calculate $H_\textup{hom}^i({\mathbb P}^1,{\mathcal O}(n))$ for the twists ${\mathcal O}(n)$ of the structure sheaf, which yields the expected outcome for $H_\textup{hom}^0$, namely, an ${\F_1}$-vector space of dimension $n+1$ if $n\geq1$ and $0$ if $n<0$, but which yields, again, an infinite-dimensional ${\F_1}$-vector space $H_\textup{hom}^1({\mathbb P}^1,{\mathcal O}(n))$. \end{rem}
\begin{rem}
As explained to the second author by Anton Deitmar, this does not contradict Theorem 2.7.1 in \cite{Deitmar11b}, which implies that the rank of the cohomology over ${\F_1}$ is at most the rank of the corresponding cohomology over ${\mathbb Z}$. The reason is that the base extension of the twisted sheaf ${\mathcal O}(n)$ to ${\mathbb Z}$ (in the sense of \cite{Deitmar11b}) is not the twisted sheaf on the projective line over ${\mathbb Z}$, but a sheaf on ${\mathbb P}^1_{\mathbb Z}$ that is not of finite type.
To explain, the definition of the base extension of a sheaf ${\mathcal F}$ on an ${\F_1}$-scheme $X$ to the associated scheme $X_{\mathbb Z}$ in \cite{Deitmar11b} is the pullback $\pi^\ast{\mathcal F}$ along the base extension map $\pi: X_{\mathbb Z}\to X$, not tensored with the structure sheaf of $X$. This differs from the sheaf ${\mathcal F}^+_{\mathbb Z}$ considered in this text. \end{rem}
\end{document} |
\begin{document}
\baselineskip=17pt
\title{Rotated Odometers and Actions on Rooted Trees}
\begin{abstract} A rotated odometer is an infinite interval exchange transformation (IET) obtained as a composition of the von Neumann-Kakutani map and a finite IET of intervals of equal length. In this paper, we consider rotated odometers for which the finite IET is of intervals of length $2^{-N}$, for some $N \geq 1$. We show that every such system is measurably isomorphic to a ${\mathbb Z}$-action on a rooted tree, and that the unique minimal aperiodic subsystem of this action is always measurably isomorphic to the action of the adding machine. We discuss the applications of this work to the study of group actions on binary trees. \end{abstract}
\section{Introduction}\label{sec:intro}
In this paper, we consider infinite interval exchange transformations (IETs) obtained by precomposing the von Neumann-Kakutani map of an interval with a finite IET of equal length intervals, and study the dynamics of such systems.
Let $\am$ be the von Neumann-Kakutani map, represented on the half-open unit interval $[0,1)$ as \begin{align}\label{eq-odometer} \am(x) = x - (1-3 \cdot 2^{-n}) \qquad \text{ if } x \in [1-2^{1-n}, 1-2^{-n}),\ n \geq 1. \end{align} For $q \in {\mathbb N}$, divide the interval $I = [0,1)$ into $q$ half-open subintervals of length $\frac{1}{q}$. Let $\pi$ be a permutation of $q$ symbols and let $R_\pi$ be the corresponding piecewise continuous map of the subintervals. The infinite IET $F_\pi:I \to I$ defined by $F_\pi = \am \circ R_\pi$ is called the \emph{rotated odometer}. This generalizes the case when $R_\pi:x \mapsto x+ p/q \mod 1$ is a circle rotation, and we keep the name for the general case.
It was shown in \cite{BL2021} that every rotated odometer $(I,F_\pi,\lambda)$ with Lebesgue measure $\lambda$ is measurably isomorphic to the first return map of a flow of rational slope on a certain infinite-type translation surface. The translation surfaces in question have interesting properties: they are non-compact surfaces of finite area, infinite genus and with a finite number of ends. The closure of such a surface contains a single \emph{wild} singularity and possibly a finite number of cone angle singularities, see \cite{DHP,Rbook} for definitions and details about translation surfaces of infinite type. On the other hand, one can consider $(I,F_\pi,\lambda)$ as a perturbation of the von Neumann-Kakutani system $(I,\am,\lambda)$. A natural question is, what dynamical properties of $(I,\am, \lambda)$ are preserved under such perturbation? For the case $q \ne 2^N$, $N \geq 1$, this question was partially answered in \cite{BL2021}.
Let $I_{per}$ the set of periodic points in $I$ and $I_{np} = I \setminus I_{per}$ be the non-periodic points. It was shown in \cite{BL2021} that the aperiodic subsystem $(I_{np},F_\pi)$ of the rotated odometer $(I,F_\pi)$ can be embedded into the Bratteli-Vershik system on a suitable Bratteli diagram, which can be constructed using coding partitions. The ergodic measures and the spectrum of the Koopman operator for $(I_{np}, F_\pi)$ can then be studied using the methods developed in the literature for stationary Bratteli diagrams, see \cite{BKMS2010,Fogg2002}. In \cite{BL2021} we investigated these questions for the case $q \ne 2^N$, $N \geq 1$. In particular, it was shown that $(I_{np},F_\pi)$ may be non-minimal with unique minimal set, and that it admits at most $q$ invariant ergodic measures (examples of rotated odometers with $2$ invariant ergodic measures are given too).
In this paper, we consider the case $q = 2^N$, $N \geq 1$, where it is possible to construct a different, simpler Cantor model for the dynamical system of a rotated odometer than in \cite{BL2021}. More precisely, we show that the rotated odometer $(I,F_\pi,\lambda)$ is measurably isomorphic to a ${\mathbb Z}$-action on a rooted binary tree, and, using this model, we study the dynamical and ergodic properties of the system. We also discuss the applications of our results to the study of group actions on binary trees.
We now give an overview of the main steps in the procedure which builds a measurable isomorphism between $(I,F_\pi,\lambda)$ and a ${\mathbb Z}$-action on a tree.
As a first step, we embed $(I,F_\pi)$ into a dynamical system given by a homeomorphism of a Cantor set, that is, there exists a Cantor set $I^*$, a homeomorphism $F_\pi^*:I^* \to I^*$ and an injective map $\iota: I \to I^*$, such that the image $\iota(I)$ is dense in $I^*$ and $\iota \circ F_\pi = F_\pi^* \circ \iota$. This procedure has an important difference with an embedding of $(I, F_\pi)$ into a compact space $(I^*,F_\pi)$ constructed in \cite{BL2021}.
Indeed, to define the compact space $I^*$ in \cite{BL2021} we employ a technique standard in the study of finite IETs, see for instance \cite{Keane1975}. Namely, we create gaps in $I$ by doubling points in the orbits of discontinuities of $F_\pi$. Periodic points in \cite{BL2021} have half-open neighborhoods where each point is periodic with the same period as $x$, and no points in this neighborhood get doubled. Consequently $I^*$ is not totally disconnected. However, the closure of $\iota(I_{np})$ is always a Cantor set.
In this paper $I^*$ is constructed by simply doubling every dyadic rational $p/2^m$, $m \geq 1$, $0 < p < 2^m$, thus repeating the construction of the middle-third Cantor set, if we think of the middle interval as collapsed to a point. The compact space $I^*$ obtained this way is always totally disconnected. The discontinuity points of $(I,F_\pi)$ are among the doubled points, which implies that $F_\pi$ extends to a homeomorphism $F_\pi^*$ of $I^*$. The embedding $\iota$ is a measurable map with respect to the Lebesgue measure $\lambda$ on $I$ and the measure $\mu$ on $I^*$ defined in Section~\ref{subsec-embedding}.
We next build a tree model.
\begin{definition}\label{def-binary} A \emph{rooted binary tree} $T$ consists of the set $V = \bigsqcup_{i \geq 0} V_i$ of vertices and the set $E = \bigsqcup_{i \geq 1}E_i$ of edges, which satisfy the following properties for all $i \geq 0$: \begin{enumerate}
\item The cardinality $|V_i| = 2^i$. \item Every vertex in $V_i$ is connected by edges to precisely two vertices in $V_{i+1}$. \item Every vertex in $V_{i+1}$ is connected by an edge to precisely one vertex in $V_i$. \end{enumerate} \end{definition}
We modify the binary tree to obtain a \emph{grafted} binary tree as follows.
\begin{definition}\label{def-grafted} For $N \geq 1$, a \emph{grafted} binary tree $T_N$ consists of the set $V = \bigsqcup_{i \geq 0} V_i$ of vertices and the set $E = \bigsqcup_{i \geq 1}E_i$ of edges, such that: \begin{enumerate}
\item $|V_0|=1$, $|V_1| = 2^N$ and for $i \geq 2$ we have $|V_i| = 2^{N+i-1}$. \item The root $v_0 \in V_0$ is connected by edges to $2^N$ vertices in $V_1$. \item For $i \geq 1$, every vertex in $V_i$ is connected by edges to precisely $2$ vertices in $V_{i+1}$, and to a single vertex in $V_{i-1}$. \end{enumerate} \end{definition}
In the notation of Definition~\ref{def-grafted}, we have $T_1 = T$, where $T$ is the binary tree of Definition~\ref{def-binary}.
We introduce a labelling of vertices in $V$. Write $\cA_{k} = \{0,1,\ldots,2^k-1\}$, for $k \geq 1$, and consider the tree $T_N$. The root $v_0 \in V_0$ is not labelled, vertices in $V_1$ are labelled by digits in $\cA_{N}$, and for $i \geq 1$, if $v \in V_i$ is labelled by a word $w_1w_2 \cdots w_i$ where $w_1 \in \cA_{N}$ and $w_i \in \cA_1$ for $i \geq 2$, then the two vertices in $V_{i+1}$ connected to $v$ are labelled by $w_1 \cdots w_i 0$ and $w_1 \cdots w_i 1$.
\begin{definition} An \emph{infinite path} in the tree $T_N$ is an infinite sequence in the product space
\begin{align}\label{eq-pathspaceproduct}\partial T_N = \{(w_i) = w_1 w_2 \ldots \mid w_1 \in \cA_{N}, w_i \in \cA_1, i\geq 2\} = \cA_{N} \times \prod_{i \geq 2} \cA_{1,i}, & & \cA_{1,i} = \cA_1 \textrm{ for } i\geq 2. \end{align}
The space $\partial T_N$ is called the \emph{boundary} of the tree $T_N$. \end{definition}
Since $N$ is finite and the cardinality of $\cA_1$ is two, $\partial T_N$ is a Cantor set.
\begin{definition}\label{def-auto} An automorphism $g: T_N \to T_N$ is a map of $T_N$ which restricts to bijective maps on the sets $V$ and $E$ of vertices and edges respectively, and which preserves the structure of the tree. That is, if $v_1 \cdots v_i \in V_i$ is a vertex, then for any vertex $v_1 \cdots v_i w \in V_{i+1}$, where $w \in \{0,1\}$, we have that $g(v_1 \cdots v_i)$ is a subword of $g (v_1 \cdots v_i w_i)$. In other words, two vertices in $V_i$ and $V_{i+1}$ are joined by an edge if and only if their images under $g$ are joined by an edge. \end{definition}
We denote by $Aut(T_N)$ the group of automorphisms of $T_N$. It is straightforward to see that every automorphism $g \in Aut(T_N)$ induces a homeomorphism of the boundary $\partial T_N$.
A \emph{cylinder}, or a \emph{cylinder set} $[w_1w_2 \ldots w_i]$ in $T_N$, $i \geq 1$, is the set of all infinite paths starting with the finite sequence $w_1 w_2 \ldots w_i$. The {\em Bernoulli measure} $\mu_N$ on $\partial T_N$ is the standard measure in which every cylinder $[w_1w_2\dots w_i]$ has the mass $2^{-N-i+1}$. It is straightforward that $\mu_N$ is preserved under every automorphism of $T_N$.
\begin{theorem}\label{thm-main3} Let $q = 2^N$, let $\pi$ be a permutation on $q$ symbols and let $(I,F_\pi,\lambda)$ be a rotated odometer with Lebesgue measure $\lambda$. Then there exists an automorphism $\widetilde F_\pi \in Aut(T_N)$ and a measurable isomorphism
$${\phi}: (I,F_\pi,\lambda) \to (\partial T_N,\widetilde F_\pi, \mu_N)$$ such that $\widetilde F_\pi \circ \phi = \phi \circ F_\pi$. \end{theorem}
Theorem~\ref{thm-main3} is proved in Section~\ref{subsec-treemodel}.
A consequence of Theorem~\ref{thm-main3} is the following description of the dynamics of $(I,F_\pi,\lambda)$ in the case $q=2^N$, $N \geq 1$, which is more precise than the result of \cite{BL2021}.
\begin{theorem}\label{thm-main1} Let $q = 2^N$ for some $N \geq 1$, and let $(I,F_\pi)$ be a rotated odometer. There exists a decomposition $I = I_{per} \cup I_{np}$ with the following properties: \begin{enumerate} \item[(i)] Every point in $I_{per}$ is periodic, the restriction $F_\pi: I_{per} \to I_{per}$ is well-defined and invertible. \item[(ii)] If $I_{per}$ is non-empty, then $I_{per}$ is a finite union of half-open maximal periodic intervals $[x,y)$, $x,y \in I$. Thus the set of periods of points in $(I,F_\pi)$ is finite. \item[(iii)] The set $I_{np}$ contains $0$ and $F_\pi: I_{np} \to I_{np}$ is well-defined and invertible at every point in $I_{np} \setminus \{ 0 \}$. \item[(iv)] The aperiodic system $(I_{np}, F_\pi)$ is minimal. \end{enumerate} \end{theorem}
The difference with the general case $q \geq 2$ in \cite{BL2021} is that there $I_{per}$ can be an infinite union of half-open intervals, while for $q = 2^N$, $I_{per}$ is at most a \emph{finite} union of half-open intervals. It follows that the set of periods which occur in $(I,F_\pi)$ is finite, which need not be the case in \cite{BL2021}. Another difference is that for $q = 2^N$ the aperiodic subsystem $(I_{np},F_\pi)$ is always minimal, while this need not hold for $q \ne 2^N$. Theorem~\ref{thm-main1} is proved in Section~\ref{subsec-treemodel}.
Since $I_{per}$ is a finite union of half-open intervals, $I_{np}$ is also a finite union of half-open intervals, and its Lebesgue measure $\lambda(I_{np}) > 0$. We normalise $\lambda_{np}(U) = \lambda(U)/\lambda(I_{np})$ for every $U \subset I_{np}$. The \emph{ dyadic adding machine} $\am: \{0,1\}^{\mathbb N} \to \{0,1\}^{\mathbb N}$ is a well-known example of a minimal ${\mathbb Z}$-action on the space of one-sided infinite sequences of $0$'s and $1$'s. For a finite set $S = \{0,\ldots,r-1\}$, $r \geq 1$, we define the adding machine $\am_S: S \times \{0,1\}^{\mathbb N} \to S \times \{0,1\}^{\mathbb N} $ as the addition of $1$ in $S$ with infinite carry to the right. In other words, \begin{align}\label{addingS} \am_S(s,x) = \begin{cases}
(s+1,x) & \text{ if } s < r-1,\\[1mm]
(0,\am(x)) & \text{ if } s=r-1, \text{ where $\am$ is the dyadic adding machine},
\end{cases} \end{align} The adding machine $\am_S$ preserves the obvious Bernoulli measure $\mu_S$.
Since $\{0,1\}^{\mathbb N}$ is homeomorphic to $\partial T_1$, there is a conjugate action on $\partial T_1$ which we also call the adding machine and denote by $a$. A recursive definition of the adding machine on the boundary $\partial T_N$ of the grafted tree $T_N$ is given in Example~\ref{ex-addmach1}.
\begin{corollary}\label{cor-addmach}
Let $q = 2^N$ for some $N \geq 1$, and let $(I,F_\pi)$ be a rotated odometer. The aperiodic system $(I_{np},F_\pi,\lambda_{np})$ is measurably isomorphic to the action of the adding machine on $S \times \{0,1\}^{{\mathbb N}}$, for $|S| \leq 2^N$, with Bernoulli measure $\mu_S$. \end{corollary}
A rotated odometer need not be conjugate to an automorphism of the binary tree $T$, since $F_\pi$ may be such that the permutation $\pi$ does not respect the structure of the binary tree, see Remark~\ref{rem-period3}. Therefore $T_N$ cannot be substituted by $T_1$ in Theorem~\ref{thm-main3}.
Theorem~\ref{thm-main3} has applications in the study of group actions on binary trees. Infinite IETs and actions of self-similar groups on binary trees are related. For instance, the famous Grigorchuk group was initially defined as a group of infinite IETs of the unit interval, see \cite[Section 2]{Grig2011}. Actions of self-similar groups on binary rooted trees are an active topic of research in Geometric Group Theory \cite{BN2008,Grig2011,Nekr2005}, and they also have applications in the study of arboreal representations of absolute Galois groups of number fields \cite{Jones2013,Lukina2018}. We now present a corollary of Theorem~\ref{thm-main3} for the actions of groups on binary trees.
To this end, let $T_1 = T$ be the binary tree. An automorphism $g \in Aut(T)$ is of finite order if $g^m = id$ for some $m \geq 1$. For instance, if $g_i$ interchanges $0$'s and $1$'s in the $i$-th coordinate $w_i$, then $g_i$ has order $2$. Another example of an element of order $2$ is $g_{even}$, which interchanges $0$ and $1$ in $w_i$ for every even $i$, and of course one can construct many more examples. The adding machine \eqref{addingS} is an automorphism of $T$ of infinite order.
Let $G \subset Aut(T)$ be a profinite group such that $G$ acts transitively on $\partial T$. Given $g \in Aut(T)$, the restriction $g|V_n$ is a permutation of a finite set $V_n$, and so it can be written as a product of cycles. Let $(x_i) = x_1 x_2 \cdots \in \partial T$, then $x_1 \cdots x_n$ is a vertex in $V_n$. Denote by $g_{n,x_1 \cdots x_n}$ the cycle containing $x_1 \cdots x_n$, then one can ask how the sequence of cycles $\{g_{n,x_1 \cdots x_n}\}$ behaves as $n$ increases. It is conjectured in \cite{BJ2007}, that when $G$ is a representation of the absolute Galois group of a number field, elements with a certain type of cycle structure are dense in $G$. To the best of our knowledge, this conjecture is solved only in a few cases.
As a rule, given $g \in Aut(T)$, it is not immediate to determine the cycle structure of $g$, except in a few simple cases when $g$ is periodic or when $g$ acts transitively on every level $V_n$, $n \geq 1$. The theorem below allows us to determine the cycle structure for compositions of the adding machine and some periodic elements of $Aut(T)$.
\begin{theorem}\label{theorem-appl} Let $\mu_1$ be the Bernoulli measure on $\partial T$, and let $\lambda$ be Lebesgue measure on the half-open unit interval $I$. Let $g\in Aut(T)$ be such that there exists $m \geq 1$ such that for every $i > m$ and every sequence $w_1 w_2 \cdots \in \partial T$ the action of $g$ leaves $w_i$ unchanged (which implies that $g$ has finite order). Let $a \in Aut(T)$ be the adding machine. Then the following is true: \begin{enumerate} \item For some permutation $\pi$ on $2^m$ intervals, there exists a rotated odometer $(I,F_\pi)$ and an injective measure-preserving map $ \phi: (I,\lambda) \to (\partial T,\mu)$, such that $\phi \circ F_\pi = (a \circ g)\circ \phi$.
\item Consequently, $a \circ g$ has infinite order, there is a clopen subset $U \subset \partial T$ such that the restriction $\langle a \circ g \rangle|U$ is minimal, and there is an $n_0 \geq 0$ such that every $x \in \partial T \setminus U$ is periodic of period $2^k$ for some $k \leq n_0$. \end{enumerate} \end{theorem}
The realization of a tree automorphism as an interval exchange transformation in Theorem~\ref{theorem-appl} relies on the fact that, under the hypotheses of the theorem, $g$ respects the embedding of an interval into the boundary of a tree $T$ in Theorem~\ref{thm-main1}. This means, in particular, that the orbits of points which do not have preimages under $\phi$ consist of points which also do not have preimages under $\phi$. This condition need not hold for a general finite order automorphism of $T$. We discuss this and the possibility of generalizing Theorem~\ref{theorem-appl} to a larger class of tree automorphisms in Remark~\ref{remark-generalize}.
\begin{remark} {\rm In the literature, an \emph{odometer} in $Aut(T)$ is sometimes defined as any $h \in Aut(T)$ such that the action of the cyclic group $\langle h \rangle$ is transitive on each $V_n$, $n \geq 1$. Every such $h$ is conjugate to the adding machine in Example~\ref{ex-addmach1} by some $g \in Aut(T)$ \cite{Pink13}. We stress that Theorem~\ref{theorem-appl} only holds for the adding machine and need not hold for an odometer $h$. To this end we show in Remark~\ref{remark-notminimal} that it is possible to find $h \in Aut(T)$ such that the action of the cyclic subgroup $\langle h \rangle$ on $\partial T$ is minimal, and a periodic $g \in Aut(T)$, such that the product $h \circ g$ has finite order. There exists an infinite IET that is measurably isomorphic to the action of such $\langle h \rangle$ on $\partial T$, but this IET will not be the rotated odometer of the form defined at the beginning of the introduction. } \end{remark}
We finish with a sample open question motivated by applications to actions on binary trees. Consider compositions of the adding machine with a periodic element which does not satisfy the hypotheses of Theorem~\ref{theorem-appl} but which respects the embedding of $I$ in Theorem~\ref{thm-main1}, see Remark~\ref{remark-generalize} for the justification of such an assumption. It may be possible to solve the following problem by considering a sequence $\{F_{\pi_i}\}_{i \geq 1}$ of rotated odometers, where each $\pi_i$ is a (possibly different) permutation of a finite number of symbols.
\begin{problem}\label{prob-compositions} Let $g \in Aut(T)$ be periodic such that for any $i \geq 1$ there is $j>i$ and $w_1 \cdots w_j \cdots \in \partial T$ such that $g(w_j) \ne w_j$, and such that $g$ preserves the embedding $\phi$ in Theorem~\ref{thm-main1}. Find a model for the action of the product $a \circ g$, where $a$ is the adding machine, in terms of rotated odometers. What are the topological properties of infinite translation surfaces, which admit flows whose first return map is measurably isomorphic to such systems? \end{problem}
The paper is organized as follows. In Section~\ref{sec:general} we develop a tree model for rotated odometers and prove Theorem~\ref{thm-main3}. In Section~\ref{subsec-dynamics} we discuss the dynamics of rotated odometers and prove Theorems~\ref{thm-main1} and~\ref{theorem-appl} and Corollary~\ref{cor-addmach}.
\section{The tree model}\label{sec:general}
In this section we build a tree model for a rotated odometer with $q = 2^N$, and prove Theorem~\ref{thm-main3}.
\subsection{Embedding into a Cantor set}\label{subsec-embedding}
Set $C = \{p 2^{-n} \mid n\geq 1, 0 < p < 2^n\}$; these dyadic rationals are used as cut-points. For each point $x \in C$ we add a double point $x^-$ to $I$, and define
$$I^* = I \cup \{x^- \mid x \in C\} \cup \{1\}.$$ The subset $I \cup \{1\}$ of $ I^*$ has total order $<$ induced from ${\mathbb R}$. We extend this order to $I^*$ by defining $x^- < x$ if $x \in C$, and $y < x^-$ if $y \in I \setminus C$, $x \in C$ and $y < x$. Since there are no points between $x^-$ and $x$ in $I^*$, adding $x^-$ to $I$ can be thought of as creating a gap. We give $I^*$ an order topology with open sets
\begin{align*} \cB = \{(a,b) \mid a,b \in I^*\} \bigsqcup \{[0,b) \mid b \in I^*\} \cup \{(a,1] \mid a \in I^*\}. \end{align*} It is straightforward that the sets $\{[x,y^-] \mid x,y \in C\}$ are clopen in this topology. Since $C$ is dense in $I$, every point $z \in I^*$ has a system of decreasing clopen neighborhoods
$$C(z,n) = \{[p_n2^{-n},(p_n+1)2^{-n}] \mid n \geq 0, \, 0 \leq p_n < 2^n\}.$$
Recall that a metric $d$ on a space $X$ is an \emph{ultrametric} if it satisfies the following stronger form of the triangle inequality, $$ d(x,y) = \max\{d(x,z), d(z,y)\} \ \textrm{ for all }x,y,z \in X. $$ We put an ultrametric on $I^*$ by declaring that
\begin{align*} d(z_1,z_2) = \frac{1}{2^r}, \quad r = \max\{n \geq 0 \mid C(z_1,n) = C(z_2,n) \}. \end{align*} Then $I^*$ is a compact totally disconnected perfect metric space, that is, $I^*$ is a Cantor set.
Define a measure $\mu$ on $I^*$ by setting for each clopen set $\{[x,y^-] \mid x,y \in C\}$
$$\mu([x,y^-]) = y - x,$$ and denote by $\iota: I \to I^*$ the inclusion map. Clearly $\mu(I^*) = 1$. Since $C$ is countable, the following is straightforward.
\begin{lemma}\label{lemma-iota} The map $\iota: (I,\lambda) \to (I^*,\mu)$ measurable. \end{lemma}
Denote by $D_0 = \{1 - 2^{-k} \mid k \geq 0\}$ the set of discontinuities of the von Neumann-Kakutani map $\am$, and let $D^+$ and $D^-$ be the sets of forward and backward (whenever defined) orbits of points in $D_0$. Since $\am$ is continuous on the intervals $I_k = [1 - 2^{-(k-1)},1-2^{-k})$, $k \geq 1$, and, moreover, the restriction $\am|I_k$ for each $k \geq 1$ is a translation by $\pm p 2^{-s}$ for some $p,s \in {\mathbb N}$, the set $D_0 \cup D^+ \cup D^-$ of forward and backward orbits of the points of discontinuity of $\am$ is contained in $C$.
We can extend $\am: I \to I$ to a continuous map $\am^*:I^* \to I^*$ by setting $\am^*(x) = \am(x)$ if $x \in I$, and
$$\am^*(x^-) = \lim_{y \nearrow x} \am(y), \quad \textrm{for all } x \in C \cup \{1\}.$$ Every point $x \in I$ except $0$ has a two-sided orbit, and it follows that $\iota(x)$ has a two-sided orbit in $I^*$.
For any sequence $y \nearrow 1$ the sequence of images $\am(y) \searrow 0$, so $\am^*(1) = 0$ and $0$ has a two-sided orbit in $I^*$ under $\am^*$. It follows that $\am^*$ is a homeomorphism.
It is immediate that $\am^* \circ \iota(x) = \iota \circ \am (x)$ for all $x \in I$.
Note that the finite IET $R_\pi: I \to I$ extends in a similar manner to a periodic homeomorphism $R_\pi^*:I^* \to I^*$, which satisfies $R_\pi^* \circ \iota (x) = \iota \circ R_\pi(x)$ for all $x \in I$. Then for the composition $F_\pi^* = \am^* \circ R_\pi^*$ it follows that $F_\pi^* \circ \iota(x) = \iota \circ F_\pi(x)$ for all $x \in I$, and thus $(I,F_\pi,\lambda)$ is measurably isomorphic to $(I^*,F_\pi^*,\mu)$ via the embedding $\iota$.
\subsection{Actions on trees}
The binary tree and the grafted binary trees were defined in Definitions~\ref{def-binary} and~\ref{def-grafted}, and automorphisms of trees were defined in Definition~\ref{def-auto}. We now introduce a description of elements in $Aut(T_N)$ convenient for computations. This approach is a slight modification of the one routinely used in Geometric Group Theory to study actions on binary trees, see for instance \cite{Nekr2005}. The purpose of this modification is to take into account the fact that in the grafted binary tree the vertex set $V_1$ has more than $2$ vertices.
Let $T = T_1$ be the binary tree with the labelling of vertices by finite words in $\cA_1$ as defined in the Introduction. Let $w = w_1 \cdots w_k \in \prod_{i=1}^k\cA_1$ and denote by $T(w)$ the subtree of $T$ consisting of all paths starting with the finite word $w$. All such paths pass through the vertex in $V_k$ labelled by $w$. Then there is an isomorphism of trees
\begin{align}\label{eq-binaryhomeo}
\kappa_w: T(w) \to T, \qquad w_1 \cdots w_k v_{k+1} \cdots \mapsto v_{k+1} \cdots, \, v_i \in \{0,1\} \textrm{ for }i > k.\end{align}
For every $g \in Aut(T)$, the restriction $g|V_n$ is a permutation of a set of $2^n$ elements.
\begin{definition}\label{def-section} Given an automorphism $g \in Aut(T)$, and a finite word $w$, we define a \emph{section} at $w$ by
\begin{align}\label{eq-section}g_w = \kappa_{g(w)} \circ g \circ \kappa_w^{-1} \in Aut(T).\end{align}
\end{definition}
Let $g|V_n = \tau$. Then we can write $g$ as a composition (we compose the maps on the left)
\begin{align}\label{eq-recur}
g = (g_{\tau^{-1}(0^n)},g_{\tau^{-1}(0^{n-1}1)}, \ldots, g_{\tau^{-1}(1^n)}) \tau,
\end{align} where $g_{\tau^{-1}(w)}$ are sections, for finite words $w$ of $n$ letters. Equation \eqref{eq-recur} means that to compute $g$, we first apply $\tau$ on $V_n$, and then we apply a section $g_{\tau^{-1}(w)}$ to the subtree $T(w)$, for all $w \in V_n$.
\begin{example}\label{ex-addmach1} {\rm Using sections, we can write automorphisms of $T$ recursively. Recall that a generator of the adding machine action on a Cantor space $\{0,1\}^{{\mathbb N}}$ is given by
\begin{align}\label{eq-addingmach}a (w_1 w_2 \cdots) = \left\{ \begin{array}{ll} (w_1+1)\, w_2 \cdots & \textrm{ if }w_1 = 0, \\ 0 \, 0 \cdots 0 \, (w_{k+1}+1) \, w_{k+2} \cdots &\textrm{ if } w_i= 1 \textrm{ for } 1 \leq i \leq k, \, w_{k+1} =0, \\ 0 \, 0 \cdots & \textrm{ if }w_k = 1 \textrm{ for all }k \geq 1. \end{array}\right. \end{align} Recall that for the binary tree $T$ we have $\partial T \cong \{0,1\}^{\mathbb N}$. Let $\sigma$ be the non-trivial permutation of $\cA_1$. Then using \eqref{eq-recur} we can write
$$a = (a,1)\sigma,$$ where $1$ is the identity map in $Aut(T)$.
Here $\sigma$ performs the addition of $1$ modulo two in the first entry of the sequence, interchanging $0$ and $1$, while $(a,1)$ implements the recursive procedure of infinite carry to the right. For example, if $w = 10^\infty$, then applying $\sigma$ to $w$ interchanges $1$ to $0$ in the first component, so $\sigma(w) = 0^\infty$, and we must compute $(a,1)(0^\infty)$ next. The sequence $0^\infty$ belongs to the subtree $T(0)$ which means that we must apply the section $a_0=a$ to $0^\infty$. That is, we apply $a$ to $0^\infty$ starting from the second entry. Since $a|V_1 = \sigma$, we must interchange $0$ and $1$ in the second entry, obtaining $010^\infty \in T(01)$. We have for the sections $a_1 = 1$, then also $a_{01} = 1$, and the computation stops with the result $a(10^\infty) = 010^\infty$. } \end{example}
Using \eqref{eq-recur} we can compute the compositions of elements in $Aut(T)$. The following statement is obtained by a straightforward computation.
\begin{lemma}\label{lemma-product} Let $g,h \in Aut(T)$, and suppose $g = (g_0,\ldots,g_{2^n-1}) \tau $ and $h = (h_0,\ldots,h_{2^n-1}) \nu $, where $\tau, \nu$ are permutations of $2^n$ symbols and $g_i,h_i \in Aut(T)$ for $0 \leq i < 2^n$. Then
\begin{align}\label{eq-compose} gh = (g_0,\ldots,g_{2^n-1}) \tau (h_0,\ldots,h_{2^n-1}) \nu = (g_0 h_{\tau^{-1}(0)},\ldots,g_{2^n-1}h_{\tau^{-1}(2^n-1)}) \tau \nu.\end{align}
\end{lemma}
Now let $T_N$ be the grafted binary tree. Similarly to \eqref{eq-binaryhomeo}, for any $w \in V_k$, $k \geq 1$ we define a map
\begin{align}\label{eq-kappaw}\kappa_w: T_N(w) \to T_1, \qquad w_1 \cdots w_k v_{k+1} \cdots \mapsto v_{k+1} \cdots.\end{align} The difference with \eqref{eq-binaryhomeo} is that the range of $\kappa_w$ is not the grafted tree $T_N$ but the binary tree $T$. A section $g_w$ of the grafted tree $T_N$ at $w$ is defined by \eqref{eq-section} with $\kappa_w$ given by \eqref{eq-kappaw}. Again, the difference with the setting of the binary tree is that for the grafted binary tree $T_N$ sections are elements of $Aut(T)$ and not of $Aut(T_N)$.
\begin{lemma}\label{lemma-1} Given an automorphism $g \in Aut(T)$ of the binary tree $T$, there is always an automorphism $\widehat g \in Aut(T_N)$ of the grafted tree $T_N$, such that the induced homeomorphisms on the boundaries of the corresponding trees are conjugate. \end{lemma}
\begin{proofof}{Lemma \ref{lemma-1}} Vertices in the vertex level set $V_N$ of $T$ are labelled by words of length $N$ in the alphabet $\cA_1$. Define the map
$$\kappa_N: \cA_1^N \to \cA_{N}, \qquad w_1 \cdots w_N \mapsto \sum_{i=1}^N 2^{N-i} w_i.$$ Using the identification \eqref{eq-pathspaceproduct} of the path spaces $\partial T$ and $\partial T_N$ with products of finite sets we obtain a homeomorphism
$$\kappa_\infty: \partial T \to \partial T_N, \qquad w_1 w_2 \cdots \mapsto \kappa_N(w_1\cdots w_N)w_{N+1} \cdots.$$ It follows that the map $\widetilde g = \kappa_\infty \circ g \circ \kappa_\infty^{-1} : \partial T_N \to \partial T_N$ is a homeomorphism. Moreover, by construction if two paths $(w_i),(v_i) \in \partial T$ coincide up to level $m \geq N$, then their images under $\kappa_\infty$ coincide up to level $m-N$, so every subtree $T(w)$ for $w \in V_N$ is mapped isomorphically onto a subtree $T_N(\kappa_N(w))$. It follows that $\widetilde g $ defines an automorphism $\widehat g$ of $T_N$. \end{proofof}
Given a recursive definition of $g \in Aut(T)$ as in \eqref{eq-recur}, we can obtain a recursive definition of $\widehat g \in Aut(T_N)$. Indeed, let $g|V_N = \tau$ be a permutation of $V_N$ induced by $g$. Then $\tau_N = \kappa_N \circ \tau \circ \kappa_N^{-1}$ is a permutation of the level set $V^N_1$ of $T_N$, and if $g = (g_0,\ldots,g_{2^N-1})\tau$, then $\widehat g =(g_0,\ldots,g_{2^N-1})\tau_N $.
\begin{example}\label{ex-addmach2} {\rm Let $a = (a,1)\sigma$ be the standard adding machine as in Example~\ref{ex-addmach1}, and let $N \geq 2$. We can compute that
\begin{align}\label{eq-tauodometer}\tau_N = \kappa_N \circ (a|V_N) \circ \kappa_N^{-1} = (0, \, 2^{N-1}, \, 2^{N-2}, \, 2^{N-1} + 2^{N-2}, \, \ldots ,2^N-1),\end{align} and $\widehat a = (a,1,\ldots,1)\tau_N \in Aut(T_N)$.
More generally, given a finite set $S \subset V_N$, we can consider a subtree $T_S = \bigcup_{w \in S} T_N(w) \subset T_N$. Let $\eta$ be a transitive permutation of $S$, and consider the map $a_S = (a,1,\ldots,1)\eta$ on $T_S$. Then $a_S$ is transitive on $V_n \cap T_S$, for any $n \geq 1$, so $a_S$ is the adding machine on $T_S$. } \end{example}
We note that, given $h \in Aut(T_N)$, the composition $\kappa_\infty^{-1} \circ h \circ \kappa_\infty$ need not define an automorphism of $T$. Indeed, let $N=2$, so $T_N$ has $4$ vertices at the first level, and let $h|V_2 = \tau_2 = (012)$, so the vertex $3$ is fixed. We have $\kappa_2^{-1}(3) = 11 \in V_2$ and $\kappa_2^{-1}(2) = 10 \in V_2$. At the same time
$$ \kappa_2^{-1} \circ \tau_2 \circ \kappa_2(10) = \kappa_2^{-1}(\tau_2(2)) = \kappa_2^{-1}(0) = 00.$$ Thus $\kappa_\infty^{-1} \circ h \circ \kappa_\infty$ maps paths starting with $1$ in $\partial T$ to paths starting with either $1$ or $0$ depending on the second symbol in the sequence. This means that $ \kappa_2^{-1} \circ \tau_2 \circ \kappa_2$ is incompatible with the structure of the binary tree $T$, and so $\kappa_\infty^{-1} \circ h \circ \kappa_\infty$ does not define an automorphism of $T$.
\subsection{Tree models for rotated odometers}\label{subsec-treemodel} In this section we prove Theorem~\ref{thm-main3}.
\begin{proofof}{Theorem~\ref{thm-main3}} Recall that $\pi$ is a permutation of $2^N$ symbols, and $\iota: (I,\lambda) \to (I^*,\mu)$ is a measurable embedding into a Cantor set. Write $x_{n,p} = p2^{-n}$ for points in $C$, and $x_{n,p}^-$ for the corresponding double points in $I^*$. For each $n \geq 1$, set $x_{n,2^n}^- = 1$.
Note that for any $n \geq 0$ we have
$$I^* = \bigcup \{[x_{n,p}, x_{n,p+1}^-] \mid 0 \leq p < 2^n-1 \}. $$
Consider the grafted tree $T_N$, and recall that $|V_1| = 2^N$. We are going to construct a homeomorphism $\widetilde \phi: I^* \to \partial T_N $ inductively as follows.
Define $\widetilde \phi_1: I^* \to \cA_{N}$ by setting
$$\widetilde \phi_1(z) = p \quad \textrm{ if and only if } \quad z \in [x_{N,p}, x_{N,p+1}^-].$$ For $n \geq 2$, there is a unique $0 \leq m < 2^{n}-1$ such that $z \in [x_{n,m}, x_{n,m+1}^-]$. Set
$$w_n = \widetilde \phi_{n}(z) = m \mod 2.$$ Then define
$$\widetilde \phi_\infty: I^* \to \partial T_N, \qquad z \mapsto (\widetilde \phi_1(z), \widetilde \phi_2(z), \ldots).$$ This mapping is bijective, since every point in $I^*$ has a system of clopen neighborhoods of the form $\{[x_{n,p},x_{n,p+1}^-] \mid n \geq 0\}$, and every clopen neighborhood $[x_{n,p}, x_{n,p+1}^-]$ is non-empty. The mapping $\widetilde \phi_\infty$ is clearly continuous and so it is a homeomorphism. Note that by construction the inclusions of clopen sets in $I^*$ correspond to vertices in $T_N$ joined by finite paths.
The measure $\mu$ assigns equal weight to each interval $\{[x_{n,p},x_{n,p+1}^-] \mid 0 \leq p <2^n\}$ in the partition of $I^*$, and $\mu(I^*) = 1$. By construction each $[x_{n,p},x_{n,p+1}^-] $ is mapped onto a unique vertex in $V_{n-N+1}$. The Bernoulli measure $\mu_N$ assigns equal weight to every set $\partial T_N(w)$, where $w \in V_{n-N+1}$, and $\mu_N(\partial T) = 1$. It follows that $\widetilde \phi_\infty$ is measure-preserving.
Every map of $I^*$ which for all $n \geq 1$ induces a permutation of clopen sets $\{[x_{n,p}, x_{n,p+1}] \mid 0 \leq p < 2^n-1\}$, induces a family of permutations of the vertex level sets $V_{n-N+1}$, $n \geq 1$ of $T_N$. Since paths in $T_N$ correspond to inclusions of clopen sets in $I^*$, such permutations are compatible with the structure of the tree $T_N$ and induce an automorphism of $T_N$. We note that the maps $\am^*: I^* \to I^*$ and $R_\pi^*:I^* \to I^*$ described in Section~\ref{subsec-embedding} satisfy this condition. Therefore, the composition $F_\pi^* = \am^* \circ R_\pi^*: I^* \to I^*$ induces an automorphism of $T_N$. The proof of Theorem~\ref{thm-main3} is completed by composing $\phi = \widetilde \phi_\infty \circ \iota: I \to \partial T_N$ with the measurable isomorphism $\iota: (I,F_\pi,\lambda) \to (I^*,F_\pi^*,\mu)$. \end{proofof}
\begin{remark}\label{remark-addedpoints} {\rm Consider the set of added points $\{x_{m,p}^- \mid x_{m,p} \in C\}$. Suppose $x_{m,p} = p2^{-m}$ is an irreducible fraction, that is, $p$ is odd. Then for $n > m$ we have that $\widetilde \phi_n(x_{m,p}^-) = 1$ since in that case $x_{m,p}^-$ corresponds to a right endpoint of a clopen interval in the partition $\{[x_{n,r},x_{n,r+1}^-] \mid 0 \leq r < 2^n\}$, and it is always contained in the second interval of the subdivision of $[x_{n,r},x_{n,r+1}^-]$ into two intervals. Then the image of $x_{m,p}^-$ in $\partial T_N$ is a sequence which is eventually constant with entries equal to $1$. } \end{remark}
\section{Dynamics of rotated odometers} \label{subsec-dynamics}
Using the tree model obtained in Theorem~\ref{thm-main3} we study the dynamics of rotated odometers and prove Theorems~\ref{thm-main1} and~\ref{theorem-appl} and Corollary~\ref{cor-addmach}.
\subsection{Periodic and non-periodic points}\label{subsec-periodic} For the von Neumann-Kakutani map $\am^*:I^* \to I^*$ denote by $A = \widetilde\phi_\infty \circ \am^* \circ \widetilde\phi_\infty^{-1}: \partial T_N \to \partial T_N$ the induced map of the binary tree $T$. We want to describe $A$ using the recursive formula \eqref{eq-recur}.
\begin{proofof}{Theorem~\ref{thm-main1} and Corollary~\ref{cor-addmach}}
In what follows $n \geq N$. Let $L_n = [0, 2^{-n})$ and $M_n = [1-2^{-n},1)$, so that $L_n$ is the first and $M_n$ is the last set of the partition of $I$ into $2^n$ sets of equal lengths. Then $\iota(L_n) \subset [x_{n,0},x_{n,1}^-] \subset I^*$ and $\iota(M_n) \subset [x_{n,2^{n}-1},1] \subset I^*$. The definition of the von Neumann-Kakutani map in \eqref{eq-odometer} implies that $\am(x) \in L_n$ if and only if $x \in M_n$, and for any $[x_{n,p}, x_{n,p+1})$ except $M_n$ the restriction $\am|[x_{n,p}, x_{n,p+1})$ is a translation. Thus it preserves the order $\leq $ on the points in $[x_{n,p}, x_{n,p+1})$ induced from ${\mathbb R}$. The relation $\leq$ is not preserved by the restriction $\am: M_n \to L_n$, where the order of two halves of $M_n$ is interchanged, and the intervals inside the image of the second half of $M_n$ are further interchanged. The second half of $M_n$ is the set $M_{n+1}$, and we have $\am(M_{n+1}) = L_{n+1}$. Thus the restriction of $\am$ to the set of intervals $\{[x_{n,p}, x_{n,p+1}) \mid 0 \leq p < 2^n\}$, and therefore of $\am^*$ to the set of intervals $\{[x_{n,p}, x_{n,p+1}^-] \mid 0 \leq p < 2^n\}$, defines a permutation of $2^n$ symbols, which is transitive since $\am$ is minimal on $I$.
It follows that $A|V_{n-N+1}$ is a transitive permutation of $V_{n-N+1}$. Since further permutations of subintervals, which do not respect the order $<$, only happen for the interval, mapped onto $L_{n-N+1}$, for any $w \ne 0^{n-N+1} \in V_{n-N+1}$,
the section $A_w \in Aut(T)$ is the identity map. The restriction of the section $A_{0^{n-N+1}}$ to $V_{n-N+2}$ is a non-trivial permutation of two symbols, since $\am$ permutes two subintervals of $L_{n-N+1}$. For $n = N$, we have $A|V_{N-N+1} = A|V_1= \tau_N$, for $\tau_N$ given by \eqref{eq-tauodometer}, and so $A = (a,1,\ldots,1)\tau_N$, where $a = (a,1) \sigma$ is described in Example~\ref{ex-addmach1}.
Similarly, given a permutation $\pi$ of $2^N$ symbols, and the corresponding finite IET $R_\pi: I \to I$, we deduce that the induced map $R = \widetilde\phi_\infty \circ R_\pi^* \circ \widetilde\phi_\infty^{-1}$ is given by $R = (1,1,\ldots,1)\pi$, with $R|V_1 = \pi$.
Now using the law for composition of tree automorphisms \eqref{eq-compose} we can easily understand the dynamics of the system $(\partial T_N, A \circ R)$. In particular,
$$A \circ R = (a,1,\ldots,1)\tau \pi,$$ which leads to the following conclusions: \begin{enumerate} \item[(i)] Consider the decomposition of $\tau \pi$ into cycles, and suppose $c$ is a cycle containing $0$. Let $\cO \subset \cA_{N}$ be the set of symbols in $c$. Then
$$\partial T_N(\cO) = \bigcup \{\partial T(s) \mid s \in \cO \}$$ is a clopen subset of $\partial T$ and the restriction of $A \circ R$ to this set satisfies
$$A\circ R|\partial T_N(\cO) = (a,1,\ldots,1)c,$$
which shows that this system is the addition of $1$ in the first component with infinite carry to the right, and so it is minimal. Set $S=\{0,\ldots,|c|-1\}$, then $A \circ R| \partial T_N(\cO)$ is isomorphic to the adding machine on $S \times \{0,1\}^{\mathbb N}$ defined in Example~\ref{ex-addmach2}, and Corollary~\ref{cor-addmach} follows. Here $|c|$ denotes the length of the cycle $c$. In particular, the system $(\partial T_N, A \circ R)$ is minimal if and only if $\tau\pi$ is a transitive permutation. \item[(ii)] In the cycle decomposition of $\tau \pi$, let $c'$ be a cycle not containing $0$, and let $\cO' \subset \cA_{N}$ be the set of symbols in $c'$. Then $\partial T_N(\cO') = \bigcup \{ \partial T(s) \mid s \in \cO' \} $ is a clopen subset of $\partial T$, and we have
$$A\circ R|\partial T_N(\cO') = (1,1,\ldots,1)c'.$$
Thus every point in $T_N(\cO')$ has period $|c'|$. \item[(iii)] Since $\tau \pi$ contains a finite number of cycles, the set of periods of periodic points in $(\partial T_N, A \circ R)$, and so in $(I,F_\pi)$, is finite. Also, it follows that a point $x \in \partial T_N$ is periodic if and only if $x \in \partial T_N(\cO')$ for some cycle $c'$ not containing $0$. There is at most a finite number of such cycles $c'$ in $\tau \pi$, and so there is a finite number of half-open intervals in $I$ whose image under the inclusion map $\phi = \widetilde \phi_\infty \circ \iota$ is contained in $\partial T_N\setminus \partial T_N(\cO)$. It follows that the set of periodic points in $(I,F_\pi)$ is at most a finite union of half-open intervals. \end{enumerate} These prove Theorem~\ref{thm-main1} and Corollary~\ref{cor-addmach}. \end{proofof}
\begin{remark}\label{rem-period3} {\rm We note that the periods of points in $(I,F_\pi)$ need not be powers of $2$. Let $N = 2$, then $A = (a,1,1,1)(0213)$. Let $\pi = (03)$. Then
$$A \circ R = (a,1,1,1)(0213)(03) = (a,1,1,1)(0)(321),$$ so the orbit of every infinite sequence in $\partial T_2$ starting with $1$, $2$ or $3$ is periodic with period $3$.
It follows from this example that there exist rotated odometers whose action is not measurably isomorphic to the action of an automorphism of the binary tree $T$. Indeed, if $g \in Aut(T)$ is an automorphism and $x \in \partial T$ is a periodic point, then the period of $x$ is a power of $2$. To see this, consider $x = (x_0,x_1,\ldots)$, and let $r_k$ be the period of $x_k$ in $V_k$. Then the period $x_{k+1}$ in $V_{k+1}$ is either $r_k$ or $2r_k$. Since the period $r_1$ of $x_1$ in $V_1$ is either $1$ or $2$, the statement follows. } \end{remark}
\subsection{Applications} We prove Theorem~\ref{theorem-appl}.
\begin{proofof}{Theorem~\ref{theorem-appl}} Suppose that $g \in Aut(T)$ is of finite order such that there is $m \geq 1$ such that for every $i >m$ the action of $g$ leaves $w_i$ unchanged. We need to show that there exists a rotated odometer $(I,F_\pi)$ for some permutation $\pi$ on $2^m$ intervals, such that $(I,F_\pi,\lambda)$ is measurably isomorphic to $(\partial T, a \circ g , \mu_1)$, where $\lambda$ is Lebesgue measure, $\mu_1$ is the Bernoulli measure on the binary tree $T$ and $a$ is the adding machine described in Example~\ref{ex-addmach1}.
Consider the partition of $\partial T$ into clopen sets $\partial T(w)$, where $w= w_1 \cdots w_n$. Also, consider a partition of $I^*$ into subintervals $\{[x_{n,p},x_{n,p+1}) \mid 0 \leq p < 2^n\}$. By construction every such subinterval is mapped under $\phi = \widetilde \phi_\infty \circ \iota$ into a distinct clopen set $\partial T(w)$, and $\phi$ is injective on $I$. Define $$\widetilde{g}:I \to I, \qquad x \mapsto (\widetilde \phi_\infty \circ \iota)^{-1} \circ g \circ (\widetilde \phi_\infty \circ \iota)(x).$$
The map $\widetilde{g}$ is well-defined. Indeed, by Remark~\ref{remark-addedpoints} the points in $I^*$ which do not have preimages in $I$ under $\iota$ are mapped into sequences which are eventually constant with entries equal to $1$. Since $g$ does not change $w_i$ for $i \geq m$, $w \in \partial T$ is eventually a sequence of $1$'s if and only if $g(w)$ is eventually a sequence of $1$'s. Therefore, the map $\phi$ is invertible at $g \circ \phi(x)$. Since $g$ does not change $w_i$ for $i \geq m$, $\widetilde g$ preserves the order of points in the sets $\{[x_{n,p},x_{n,p+1}) \mid 0 \leq p < 2^n\}$, and it follows that the restriction of $\widetilde{g}$ to every interval $\{[x_{m,p},x_{m,p+1}) \mid 0 \leq p <2^m\}$ is a translation. We conclude that $\widetilde g: I \to I$ is a finite IET.
It is proved in Section~\ref{subsec-periodic} that the von Neumann-Kakutani map $(I, \am,\lambda)$ is measurably isomorphic to $(\partial T, a, \mu_1)$, where $a = (a,1)\sigma$ is the standard adding machine. Set $F_\pi = \am \circ \widetilde g $, then $(I, F_\pi, \lambda)$ is measurably isomorphic to $(\partial T, a \circ g, \mu_1)$. The second statement of Theorem~\ref{theorem-appl} follows from Theorem~\ref{thm-main1}. \end{proofof}
\begin{remark}\label{remark-notminimal} {\rm In the literature a transformation $g$ such that the cyclic group $\langle h \rangle$ acts transitively on every level $V_n$, $n \geq 1$, of the tree $T$, is sometimes called an \emph{odometer}. Every odometer is conjugate to the adding machine in Example~\ref{ex-addmach1} by a tree automorphism \cite{Pink13}. We note that Theorem~\ref{theorem-appl} only holds for the adding machine, but need not hold for its conjugates.
Indeed, define $a_1 = \sigma$ and $a_2 = (a_1,a_2)$ using the recursive notation, then $\langle a_1,a_2\rangle$ is the dihedral group. Both $a_1$ and $a_2$ have order two, and $h = a_1a_2$ generates an infinite cyclic group whose action on every $V_n$, $n \geq 1$, is transitive. The element $h$ is conjugate to $a = (a,1)\sigma$ but it is not equal to $a$. Recall that $\sigma$ acts on $(w_1,w_2,\ldots) \in \partial T$ by interchanging $0$ and $1$ in the first entry, and keeps the remaining entries fixed. We compute that $h \sigma = \sigma (a_1,a_2) \sigma = (a_2,a_1)$ has order two.
} \end{remark}
\begin{remark}\label{remark-generalize} {\rm We have seen in the proof of Theorem~\ref{theorem-appl} that, under its hypotheses, a finite order element $g \in Aut(T)$ respects the embedding of $I$ into $\partial T$. More precisely, by Remark~\ref{remark-addedpoints} points which do not have preimages under this embedding correspond to sequences which are eventually constant with entries equal to $1$, and if $g \in Aut(T)$ satisfies the hypotheses of Theorem~\ref{theorem-appl}, then it preserves the set of such sequences. Suppose $g \in Aut(T)$ does not satisfy the hypotheses of Theorem~\ref{theorem-appl}, that is, for any $n \geq 1$ there is $(w_i) \in \partial T$ and $m_n \geq n$ such that $g(w_{m_n}) \ne w_{m_n}$. Then the action of $g$ on $\partial T$ may or may not respect the embedding of $I$. If such $g \in Aut(T)$ respects the embedding of $I$ into $\partial T$ ($h$ in Remark~\ref{remark-notminimal} is an example), then $g$ induces an IET of \emph{infinite} number of intervals. At the moment we do not have a unified way of describing the dynamics of a composition of such an IET with the von Neumann-Kakutani map, and we pose this as an open question in Problem~\ref{prob-compositions}. An example of an element which does not respect the embedding is, for instance, $g_{even}$ given for any $(w_i) \in \partial T$ by
$$g_{even}(w_{2n}) = w_{2n}+1 \mod 2, \quad g_{even}(w_{2n+1}) = w_{2n+1}, \quad n \geq 1.$$ It is not clear whether such elements induce IETs on the interval $I$, even if one discards the measure $0$ set of orbits in $\partial T$ which do not have preimages under the embedding, and/or allows reflections of subintervals.} \end{remark}
\end{document} |
\begin{document}
\title{The stability of extended Floater-Hormann interpolants} \author{Andr\'{e} Pierro de Camargo and Walter F. Mascarenhas}
\institute{Andr\'{e} Pierro de Camargo at Centro de Matem\'{a}tica, Computa\c{c}\~{a}o e Cogni\c{c}\~{a}o, Universidade Federal do ABC – UFABC, Rua Santa Ad\'{e}lia, 166, bairro Bangu, CEP 09210-170 Santo Andr\'{e}, SP, Brazil, and Walter F. Mascarenhas at Instituto de Matem\'{a}tica e Estat\'{i}stica, Universidade de S\~{a}o Paulo,
Cidade Universit\'{a}ria, Rua do Mat\~{a}o 1010, S\~{a}o Paulo SP, Brazil. CEP 05508-090
Tel.: +55-11-3091 5411, Fax: +55-11-3091 6134, \email{walter.mascarenhas@gmail.com}.
Andr\'{e} is supported by grant 14225012012-0 from
Conselho Nacional de Desenvolvimento Cient\'{i}fico e Tecnol\'{o}gico, CNPq.
Walter is supported by grant 2013/10916-2 from Funda\c{c}\~{a}o de Amparo \`{a}
Pesquisa do Estado de S\~{a}o Paulo (FAPESP.) }
\maketitle
\begin{abstract} We present a new analysis of the stability of extended Floater-Hormann interpolants, in which both noisy data and rounding errors are considered. Contrary to what is claimed in the current literature, we show that the Lebesgue constant of these interpolants can grow exponentially with the parameters that define them, and we emphasize the importance of using the proper interpretation of the Lebesgue constant in order to estimate correctly the effects of noise and rounding errors. We also present a simple condition that implies the backward instability of the barycentric formula used to implement extended interpolants. Our experiments show that extended interpolants mentioned in the literature satisfy this condition and, therefore, the formula used to implement them is not backward stable. Finally, we explain that the extrapolation step is a significant source of numerical instability for extended interpolants based on extrapolation. \end{abstract}
\maketitle
\section{Introduction} Given nodes $\wvec{x} = \wlr{x_0,\dots,x_n}$, an integer $\delta$ with $0 \leq \delta \leq n$ and function values $\wvec{y} = \wlr{y_0,\dots,y_n}$, the Floater-Hormann interpolation formula is defined as \begin{equation} \label{fhdef} \wfc{r_\delta}{t,\wvec{x},\wvec{y}} := \frac{\sum_{i = 0}^{n-\delta} \wfc{\lambda_i}{t, \wvec{x}} \wfc{p_i}{t, \wvec{x}, \wvec{y}}} {\sum_{i = 0}^{n-\delta} \wfc{\lambda_i}{t,\wvec{x}}}, \end{equation} where $\wfc{p_i}{t,\wvec{x},\wvec{y}}$ is the unique polynomial of degree at most $\delta$ which interpolates $y_i, y_{i+1}, \dots, y_{i+\delta}$ at $x_i, x_{i+1}, \dots, x_{i+\delta}$, and the weights $\lambda_i$ are defined as \[ \wfc{\lambda_i}{t,\wvec{x}} := \frac{\wlr{-1}^i}{\wlr{t - x_i} \wlr{t - x_{i+1}} \dots \wlr{t - x_{i+\delta}}}, \hspace{1cm} \wrm{for} \ i = 0, \dots, n - \delta. \]
In exact arithmetic, when $y_i = \wfc{f}{x_i}$ for a smooth function $f$, the error incurred by the Floater-Hormann interpolant defined by $\wvec{x}$ and $\delta$ is of order $h^{\delta+1}$, where \[ h := \max_{0 \leq k < n} x_{k+1} - x_k. \] Unfortunately, when the nodes are equally spaced the Lebesgue constant of the Floater-Hormann interpolant defined by $\wvec{x}$ and $\delta$ grows exponentially with $\delta$ (see \cite{Bos}). Therefore, $\delta$ must be chosen carefully in order to balance the high order of approximation $h^{\delta+1}$ with the numerical errors due to large Lebesgue constants.
In an attempt to reduce the effects of the large Lebesgue constants for equally spaced nodes, Klein \cite{Klein} introduced the {\it extended Floater-Hormann} interpolants. These interpolants are defined in terms of an integer parameter $d \geq 0$, extended nodes $\tilde{x}_{-d}, \tilde{x}_{1 - d}, \dots, \tilde{x}_0, \dots \tilde{x}_n, \dots, \tilde{x}_{n + d}$, with $\tilde{x}_i = x_0 + i h$, and extended function values $\tilde{y}_{-d}, \dots \tilde{y}_0, \dots \tilde{y}_n, \dots \tilde{y}_{n + d}$. Each \[ \tilde{\wvec{y}} = \wfc{Y}{d,\wvec{x},\wvec{y}} \in \wrn{n + 2 d + 1} \] in combination with $r_d$ in \wref{fhdef} leads to an extended interpolant given by \begin{equation} \label{defrdy} \wfc{\tilde{r}_{d, Y}}{t, \wvec{x}, \wvec{y}} :=
\wfc{r_d}{t, \tilde{\wvec{x}}, \wfc{Y}{d,\wvec{x},\wvec{y}}}. \end{equation} The choice of the extended function values is a crucial point regarding the stability and accuracy of the extended interpolants. Usually, we do not have information outside of the interpolation interval and in practice the $\tilde{y}$ must be estimated, and they will not be exact. To the best of our knowledge, the only concrete way for choosing the $\tilde{y}$ mentioned in the literature prior to our writing of this article is the one outlined in the fifth page of Klein and Berrut \cite{BerrutKleinCAM}, which is based on two additional parameters $\tilde{d}$ and $\tilde{n}$: \begin{quote} ``More precisely, $2d$ extra nodes $x_{-d},\dots,x_{-1}$ are considered, $d$ on each side of the interval, and approximate values $\tilde{f}_i$ of $f$ at these nodes are computed by a discrete Taylor polynomial with derivatives approximated by (linear rational) finite differences (see Section 8) {\it using only the given values of $f$ in $[a,b]$.} These finite differences are the derivatives of the Floater-Hormann family with parameters $\tilde{d}$ in the nodes $x_0,\dots,x_{\tilde{n}}$, resp. $x_{n-\tilde{n}}, \dots, x_n$, for an $\tilde{n}$ much smaller than $n$. At the original nodes, $x_j$, $j = 0,\dots,n$, the given $f_f$ are used.'' \end{quote}
Klein \cite{Klein} shows that the order of approximation of the extended interpolant above is $h^{\mu + 1}$, where $\mu := \min\{d, \tilde{d}\}$. Usual Floater-Hormann interpolants have order of approximation $\delta$, and $\mu$ is the analogous to the parameter $\delta$ used to define usual Floater-Hormann interpolants. Therefore, it is important to distinguish $\delta$ from $d$. In fact, when choosing the parameters in practice, one must be aware that the order of approximation $\mu$ will be unaffected by increasing the parameter $d$ once this parameter is already larger than $\tilde{d}$. This argument and our practical experience with extended interpolants suggest that as first choice one should pick $d = \tilde{d} = \delta$ (see also Fig. \ref{figure_conjecture} below.) For this reasons, our theory pays special attention to the case $d = \delta$, but we do address more general cases in our experiments.
The articles Klein and Berrut \cite{BerrutKleinCAM} and Klein \cite{Klein} also claim that the Lebesgue constant of Klein's extended Floater-Hormann interpolant grows logarithmically with $n$ and $d$, regardless of $\tilde{n}$ and $\tilde{d}$. Theorem 5.1 in page 6 of \cite{BerrutKleinCAM} summarizes this and other claims from \cite{Klein} (see, in particular, Theorems 2.1 and 3.1 of \cite{Klein} and the remark following the latter.) We cite:\\[0.02cm]
{\bf Theorem 5.1}(from \cite{BerrutKleinCAM}) {\em Suppose $n$, $d$, $\tilde{n}$ and $\tilde{d}$ are positive integers, $\tilde{d} \leq \tilde{n} < n$ and assume that $f \in C^{d+2}[a - d h, d + d h] \cap \wfc{C^{2 d + 1}}{[a, a + \tilde{n} h] \cup [b - \tilde{n} h,b]}$ is sampled at $n + 1$ equispaced nodes in $[a,b]$. Then \begin{itemize} \item[(i)] $\tilde{r}_n[f]$ has no real poles; \item[(ii)] For a constant $K$ independent of $n$, $ \wnorm{\tilde{r}_n[f] - f} \leq K h^{\min\wset{d,\tilde{d}} + 1}$; \item[(iii)] The associated Lebesgue constant $\tilde{\Lambda}_n$ grows logarithmically with $n$ and $d$: \[ \tilde{\Lambda}_n \leq 2 + \wfc{\ln}{n + 2 d}. \] \end{itemize} } Here we show that, in general, the third item in this theorem is false if, as in page 2 of Berrut and Klein \cite{BerrutKleinCAM}, we assume the standard definition of the Lebesgue constant as the norm of the interpolation operator, and take into account the unavoidable errors in the extended function values $\tilde{y}$. We prove that the traditional Lebesgue constant grows exponentially with $d$ when $d = \tilde{n} = \tilde{d}$ and $\tilde{\wvec{y}}$ is outlined in \cite{BerrutKleinCAM}. In the version of Theorem 5.1 stated in Klein \cite{Klein} the reader is informed that actually the logarithmic bound assumes a peculiar interpretation of the Lebesgue constant, namely, essentially that the mentioned approximate function values have no errors; see the paragraph above Theorem 3.1 in \cite{Klein}. However, this limitation of the result (iii) is not mentioned in \cite{BerrutKleinCAM}, and neither \cite{BerrutKleinCAM} nor \cite{Klein} point out that the theorem does not apply to the choices of $\tilde{\wvec{y}}$ proposed for the extended Floater-Hormann interpolants and used in the experiments.
Rigorously, our proof applies only to the case $d = \tilde{d} = \tilde{n}$. It suffices as a counterexample to Theorem 5.1, but it is unsatisfactory from a broader practical perspective. However, we emphasize that, in practice, Theorem 5.1 gives a misleading impression regarding the Lebesgue constant of Extended Floater-Hormann interpolants for broader classes of parameters. We do not have a formal theory supporting this claim, but Sections \ref{section_practice} and \ref{section_reduced} and Figure \ref{figure_conjecture} below present strong experimental evidence of its validity.
\begin{figure}
\caption{Log10 of the correct Lebesgue constant as a function of $3 \leq d, \tilde{d} \leq 20$, for $n = 100$ and $\tilde{n} = \tilde{d}$. Note that the diagonal $d = \tilde{d}$ highlighted in this figure crosses the lines of constant $\tilde{d}$ in places in which the Lebesgue constant is near the minimal value along such lines. Therefore, it makes little sense to choose $d$ much less than $\tilde{d}$, and the case considered in our counterexample is quite relevant in practice. We have observed similar pictures for other values of $n$ and our experiments indicate that the Lebesgue constant increases as $\tilde{n}$ gets larger than $\tilde{d}$.}
\label{figure_conjecture}
\end{figure}
This article presents an stability analysis of extended Floater-Hormann interpolants based on the appropriate interpretation of the Lebesgue constant. Formally, when the function $Y$ which yields the extended function values $\tilde{\wvec{y}}$ is linear in $\wvec{y}$, we consider the linear operator $\wvec{y} \mapsto r_{\wvec{x},d,Y}[\wvec{y}] \in C^0[a,b]$ given by \[ \wfc{r_{\wvec{x},d,Y}[\wvec{y}]}{t} := \wfc{\tilde{r}_{d, Y}}{t, \wvec{x}, \wvec{y}}, \] for $\tilde{r}_{d, Y}$ defined in \wref{defrdy}, and the Lebesgue constant can be defined either as the norm of this linear operator with respect to the supremum norm in $\wrn{n+1}$ and $C^0[x_0,x_n]$, or as the supremum of the Lebesgue function \begin{equation} \label{defleby} \wfc{\Lambda_{\wvec{x}, d,Y}}{t} := \sup_{\wvec{y} \ \wrm{with} \ \wnorm{\wvec{y}}_\infty = 1 } \wabs{\wfc{r_{\wvec{x},d,Y}[\wvec{y}]}{t}} \end{equation} in $[x_0,x_n]$. These two definitions are equivalent, and lead to a concept which has a fundamental role in theory and in practice, provided that it is interpreted correctly.
By considering the correct Lebesgue constant, we gain a more realistic view of the stability of extended Floater-Hormann interpolants. For instance, we learn that the version of these interpolants mentioned in Klein and Berrut \cite{BerrutKleinCAM} and Klein \cite{Klein} should not be used with large $\tilde{d}$. Since the order of approximation of these interpolants is $h^\mu$ for $\mu =\min \wset{d,\tilde{d}}$, this limits their accuracy in practice.
The articles \cite{BerrutKleinCAM} and \cite{Klein} say nothing about the disastrous effect that a large $\tilde{d}$ may have on extended interpolants. Instead, they emphasize that these interpolants can be used with large $d$. This is illustrated by Figures 6 in \cite{BerrutKleinCAM} and \cite{Klein}, which explore only the case $\tilde{d} = 7$ and compare the resulting extended interpolants with usual interpolants with $d = \delta$ as large as $50$. If instead of considering $d = \delta$ as large as 50 in their Figure 6, they had focused on the more modest case $1 \leq d = \delta \leq \tilde{d} = 7$, as in their other experiments, then they would have a more realistic argument in favor of extended interpolants. Indeed, for this range of $d = \delta$, Figures 5 to 10 in \cite{Klein} show that extended interpolants are better than usual ones in cases of practical interest. Therefore, extended interpolants have merit and are a relevant topic for research. However, our experiments in Section \ref{section_instability} show that there are also cases in which extended interpolants are worse than usual ones, and here we aim at a balanced view of their properties and limitations. In particular, we discuss the role played by each one of their parameters and the ranges in which they should be used.
In sections \ref{section_reduced} and \ref{section_lebesgue} we analyze the Lebesgue constants of extended interpolants from a theoretical perspective. We present an exponential lower bound on the Lebesgue constant when $d = \tilde{n} = \tilde{d}$. We also present experimental data showing that the dependency of the Lebesgue function on these three parameters is not accurately described by Theorem 5.1 in more general settings. Section \ref{section_backward} discusses the backward stability of extended Floater-Hormann interpolants in the general case in which the function $\wfc{Y}{d,\wvec{x},\wvec{y}}$ that defines $\tilde{\wvec{y}}$ is linear in $\wvec{y}$. We present a simple condition that implies the backward instability of the barycentric formula used to implement extended interpolants in this case, and we show experimentally that this condition for backward instability is satisfied by an extended interpolant mentioned in \cite{Klein}.
Section \ref{section_instability} presents an empirical analysis of the stability of the extended interpolants outlined in Berrut and Klein \cite{BerrutKleinCAM}. We explain that the extrapolation step may lead to numerical instability, and due to this instability the overall error incurred by these interpolants can be much larger than $\epsilon n \tilde{\Lambda}_{\wvec{x},d,Y}{} \, \wnorm{\wvec{y}}_\infty$, where $\tilde{\Lambda}_{\wvec{x},d,Y}{}$ is its Lebesgue constant and $\epsilon$ is the machine precision. In order to illustrate this fact, we present the results of experiments in which the accuracy of the extended interpolants is much worse than the accuracy of the usual interpolants.
On the positive side, once we become aware of the problems caused by the extrapolation step, we may consider ways to reduce them. When evaluating the interpolants for many values of $t \in [x_0,x_n]$, it is worth computing the relatively few extrapolated function values in multiple precision. Numerical experiments show that this strategy leads to more accurate extended interpolants.
Finally, the appendix considers the difficulties involved in the construction of a general stability theory for extended interpolants. This appendix is at a more abstract level than the rest of the article: we argue about the arguments one would use to discuss the stability of extended interpolants. We hope that people interested in an in depth analysis of the stability of these interpolants will appreciate our remarks regarding the difficulties in formulating realistic hypotheses and theorems about this subject.
\section{The rounding errors and the Lebesgue constant in practice} \label{section_practice} The Lebesgue constant is a fundamental concept in approximation theory. It is also fundamental in practice, because it measures the sensitivity of the interpolants to perturbations (or noise) in the data. In its proper interpretation, the Lebesgue constant is equivalent to what numerical analysts call {\it condition number}, and use to evaluate the numerical stability of algorithms.
This section shows that that rounding errors and noisy data have devastating effects on interpolants for which theorems in \cite{BerrutKleinCAM} and \cite{Klein} claim that the Lebesgue constant is small. Therefore, such claims may lead readers to believe that these interpolants are much less affected by noise and rounding errors than they really are. Figure \ref{figure_practice} considers the approximation of $\wfc{f}{t} = \wfc{\sin}{2t}$ for $t \in [-1,1]$. The plot on the left of Figure \ref{figure_practice} shows that by implementing the extrapolation procedure proposed in \cite{BerrutKleinCAM} and \cite{Klein} with the usual IEEE 754 double precision arithmetic we may have numerical errors of order $10^{14}$ in circumstances in which Theorem 5.1 yields a bound smaller than $8$ on the constant which Berrut and Klein call by Lebesgue's name. These errors are several orders of magnitude larger than the ones reported in \cite{BerrutKleinCAM} and \cite{Klein} for the same kind of extended interpolant, because we do not restrict ourselves to the same small values of $\tilde{d}$ as \cite{BerrutKleinCAM} and \cite{Klein}.
\begin{figure}
\caption{The function $\wfc{f}{t} = \wfc{\sin}{2 t}$ (in red) and the approximation obtained following the procedure proposed in \cite{BerrutKleinCAM} with $n = 200$ (in blue). We use $d = \tilde{n} = \tilde{d} = 40$, whereas \cite{BerrutKleinCAM} and \cite{Klein} consider $0 \leq d \leq 50$ and smaller values of $\tilde{n}$ and $\tilde{d}$ in their experiments. The $\tilde{d}$ and $\tilde{n}$ in this figure satisfy the hypothesis of Theorem 5.1 and $d$ is within the range considered in the experiments in \cite{BerrutKleinCAM} and \cite{Klein}.}
\label{figure_practice}
\end{figure}
The plot on the right of Figure \ref{figure_practice} illustrates the sensitivity of extended interpolants to noise in the function values. It was obtained by adding random values of order $10^{-10}$ to $y_0, \dots, y_n$. The effects of rounding errors in this plot are negligible because we used the high precision arithmetic provided by the MPFR library \cite{MPFR}, with a mantissa of 640 bits. The experiment on the right indicates that in this case the condition number is about $10^{10}$, and not $8$ as suggested by Theorem 5.1 of Berrut and Klein \cite{BerrutKleinCAM}.
Figure \ref{figure_practice} raises an interesting question: why does the plot on the
left display errors of order $10^{14}$ while the plot on the right shows errors of order one? This question is intriguing because the IEEE 754's double precision machine epsilon is of order $10^{-16}$ and is much smaller than the $\wfc{O}{10^{-10}}$ perturbations used to generate the plot on the right. As we explain in the rest of the article, the answer to this question lies in the instabilities in the extrapolation process proposed by Klein \cite{Klein} and this is one more reason why, in practice, the logarithmic bound presented in \cite{BerrutKleinCAM} and \cite{Klein} underestimates the effects of noise and rounding errors.
\section{The barycentric and reduced forms and the Lebesgue function} \label{section_reduced} In this section we show how to write extended interpolants in barycentric form and introduce another way to describe them, which we call {\it reduced form}. This form is numerically unstable and we do not advocate its use in practice. Its purpose is to help us to deduce an expression for the Lebesgue function of extended interpolants.
The Lebesgue function measures the sensitivity of the output of the complete interpolation process to perturbations in its input, and we emphasize that the input to the interpolation process are the original function values $y_i$; not the extrapolated function values $\tilde{y}_i$. Therefore, we can not ignore how changes in $\wvec{y}$ affect $\wvec{\tilde{y}}$ as suggested by Equation (3.2) in \cite{Klein}.
We recall that extended Floater-Hormann interpolants are defined only for equally spaced nodes, and in this section focus on the interpolants with $\tilde{\wvec{y}}$ as in the fifth section of \cite{BerrutKleinCAM}, ie., $\tilde{\wvec{y}}$ is defined using extrapolation. When the nodes are equally spaced, \cite{Floater} shows that usual Floater-Hormann interpolants can be written in the barycentric form \begin{equation} \label{bary_fh} \wfh{t} = \ \ \left. \sum\limits_{i = 0}^{n} \frac{w_{n,\delta,i} \, y_i}{t - x_i} \right/ \sum\limits_{i = 0}^{n} \frac{w_{n,\delta,i}}{t - x_i}, \end{equation} with weights \begin{equation} \label{bary_w} w_{n,\delta,i} = (-1)^{i-\delta} \sum_{j = \max\{0, \, i - \delta\}}^{\min\{n-\delta, \, i\}} \left( \begin{array}{c} \delta \\ i-j \end{array} \right), \end{equation} where the $y_i$ are the interpolated function values. In the last paragraph of page 5 of \cite{BerrutKleinCAM}, extended interpolants are defined by extrapolating $\wvec{y}$ according to the following Taylor series, which are defined in terms of the parameters $\tilde{d}$ and $\tilde{n}$: \begin{eqnarray} \label{ytila} \tilde{y}_i & := & y_0 + \sum_{k=1}^{\tilde{d}} \wfc{r_{\underline{\wvec{x}},\tilde{d}}^{(k)}[\underline{\wvec{y}}]}{x_0} \frac{\wlr{\tilde{x}_i - x_0}^k}{k!} \ \hspace{1.5cm} \wrm{for} \ -d \leq i < 0, \\ \label{ytilb} \tilde{y}_i & := & y_i \hspace{5.82cm} \wrm{for} \ 0 \leq i \leq n, \\ \label{ytilc} \tilde{y}_i & := & y_n + \sum_{k=1}^{\tilde{d}} \wfc{r_{\overline{\wvec{x}},\tilde{d}}^{(k)}[\overline{\wvec{y}}]}{x_n} \frac{\wlr{\tilde{x}_i - x_n}^k}{k!} \ \hspace{1.4cm} \wrm{for} \ n < i \leq n + d, \end{eqnarray} where $\tilde{x}_i = x_0 + i h$ for $-d \leq i \leq n + d$ and $\wfhk{t}$ is the $k$th derivative of the Floater-Hormann interpolant $\wfc{r_{\wvec{x},d}[\wvec{y}]}{t}$ in \wref{bary_fh} and $\underline{\wvec{x}} := \wlr{x_0,\dots, x_{\tilde{n}} }$, $\underline{\wvec{y}} := \wlr{y_0,\dots, y_{\tilde{n}} }$, $\overline{\wvec{x}} := \wlr{x_{n - \tilde{n}},\dots, x_n}$ and $\overline{\wvec{y}} := \wlr{y_{n - \tilde{n}},\dots, y_n}$. Therefore, the extended interpolant is specified once we define $\wvec{x}$, $d$, $\tilde{n}$ and $\tilde{d}$, and we can write it in the following barycentric form: \begin{equation} \label{bary_xfh} \wxfh{t} = \ \ \left. \sum\limits_{i = -d}^{n + d} \frac{\tilde{w}_{n,d,i} \, \tilde{y}_i}{t - \tilde{x}_i} \right/ \sum\limits_{i = -d}^{n + d} \frac{\tilde{w}_{n,d,i}}{t - \tilde{x}_i}, \end{equation} with weights \[ \tilde{w}_{n,d,i} := w_{n + 2 d,d,i + d} = (-1)^{i} \sum_{j = \max\{0, \, i\}}^{\min\{n +d, \, i + d\}} \left( \begin{array}{c} d \\ i + d -j \end{array} \right). \] It is difficult to derive the Lebesgue function of the extended interpolant $\tilde{r}_{\wvec{x},d, \tilde{n},\tilde{d}}{}$ directly from Equation \wref{bary_xfh}, because this equation depends on $\tilde{\wvec{y}}$, which is not part of the original interpolation problem. To derive an expression for this Lebesgue function, it is helpful to write the extended interpolant only in terms of the original $\wvec{y}$. The next lemma explains how to achieve this goal when $2 \tilde{n} < n$: \begin{lemma} \label{lem_clean} Given extended nodes $\tilde{x}_i = x_0 + i h$ for $-d \leq i \leq n + d$, the extrapolated function values $\tilde{y}_i$ used to define the extended interpolant $\tilde{r}_{\wvec{x},d, \tilde{n},\tilde{d}}{}$ can be written as \begin{eqnarray} \label{tilyi_clean_a} \tilde{y}_i & = & \sum_{j = 0}^{\tilde{n}} a_{ij} y_j, \hspace{2.25cm} \wrm{for} \hspace{0.1cm} - d \leq i < 0, \\ \label{tilyi_clean_b} \tilde{y}_i & = & \sum_{j = n - \tilde{n}}^n b_{\wlr{i-n}\wlr{j - n}} y_j, \hspace{0.6cm} \wrm{for} \hspace{0.1cm} n < i \leq n + d, \end{eqnarray} where the numbers \[ \wset{a_{ij}, \ -d \leq i < 0, \ 0 \leq j \leq \tilde{n}} \hspace{0.5cm} \wrm{and} \hspace{0.5cm} \wset{b_{ij}, \ 0 < i \leq d, \ - \tilde{n} \leq j \leq 0} \] depend on $i$,$j$,$\tilde{n}$ and $\tilde{d}$ but do not depend on $h$, $d$ or $n$, in the sense that there exist functions $\alpha, \beta: \mathds Z{}^4 \rightarrow \mathds R{}$ such that $a_{ij} = \wfc{\alpha}{i,j,\tilde{n},\tilde{d}}$ and $b_{ij} = \wfc{\beta}{i,j,\tilde{n},\tilde{d}}$. When $2 \tilde{n} < n$ the extended interpolant can be written in the reduced form \begin{equation} \label{reduced_form} \wxfh{t} = \left. \sum_{j = 0}^n \wfc{c_j}{t} y_j \right/ \sum_{j = -d}^{n+d} \frac{\tilde{w}_{n,d,j}}{t - \tilde{x}_j}, \end{equation} where the functions $c_j$ are given by \begin{eqnarray} \label{ca} \hspace{0.4cm} \wfc{c_{j}}{t} & = & \frac{\tilde{w}_{n,d,j}}{t - x_{j}} + \sum_{i = -d}^{-1} \frac{\tilde{w}_{n,d,i} a_{ij}}{t - \tilde{x}_{i}} \hspace{2.35cm} \wrm{for} \ 0 \leq j \leq \tilde{n},\\ \label{cb} \hspace{0.4cm} \wfc{c_{j}}{t} & = & \frac{\tilde{w}_{n,d,j}}{t - x_{j}} \hspace{4.9cm} \wrm{for} \ \tilde{n} < j < n - \tilde{n}, \\ \label{cc} \hspace{0.4cm} \wfc{c_{j}}{t} & = & \frac{\tilde{w}_{n,d,j}}{t - x_{j}} + \sum_{i = n + 1}^{n + d} \frac{\tilde{w}_{n,d,i} b_{\wlr{i-n}\wlr{j-n}}}{t - \tilde{x}_{i}} \hspace{1.0cm} \wrm{for} \ n - \tilde{n} \leq j \leq n . \end{eqnarray} \end{lemma} In the end of this section we prove Lemma \ref{lem_clean} and present explicit expressions for $a_{ij}$ and $b_{ij}$. We also provide formulae analogous to \wref{ca}--\wref{cc} for the case $2 \tilde{n} \geq n$.
The Lebesgue functions of the interpolants $r_{\wvec{x},\delta}{}$ and $\tilde{r}_{\wvec{x},d, \tilde{n},\tilde{d}}{}$ are defined as \[ \wlebffh{t} := \sup_{\wnorm{\wvec{y}}_\infty = 1} \wabs{ \wfh{t}} \hspace{1cm} \wrm{and} \hspace{1cm} \wlebfxfh{t} := \sup_{\wnorm{\wvec{y}}_\infty = 1} \wabs{ \wxfh{t}}, \] and using Equation \wref{bary_fh} and the reduced form \wref{reduced_form} it is easy to show that \begin{equation} \label{leb_fun_fh} \wlebffh{t} = \ \ \left. \sum\limits_{j = 0}^{n} \wabs{\frac{w_{n,\delta,j}}{t - x_j}} \right/ \wabs{\sum\limits_{j = 0}^{n} \frac{w_{n,\delta,j}}{t - x_j}} \end{equation} and \begin{equation} \label{leb_fun_xfh} \wlebfxfh{t} = \ \ \left. \sum_{j = 0}^n \wabs{\wfc{c_j}{t} } \right/ \wabs{\sum_{j = -d}^{n+d} \frac{\tilde{w}_{n,d,j}}{t - \tilde{x}_j}}. \end{equation} We emphasize that, in general, \begin{equation} \label{wrong} \wlebfxfh{t} \neq \ \left. \sum\limits_{j = -d}^{n + d} \wabs{\frac{\tilde{w}_{n,d,j}}{t - \tilde{x}_j}} \right/ \wabs{\sum\limits_{j = -d}^{n + d} \frac{\tilde{w}_{n,d,j}}{t - \tilde{x}_j}}, \end{equation} that is, we can deduce \wref{leb_fun_fh} from \wref{bary_fh}, but $\wlebfxfh{t}$ is not equal to, or even bounded by, the right hand side of \wref{wrong}. This is why Equation (3.2){} in \cite{Klein} is misleading. The fact that this equation refers to the peculiar interpretation of the Lebesgue constant used in \cite{Klein}, and not to the actual Lebesgue constant, becomes evident when we plot the right and the left hand side of \wref{wrong} for
$n = 50$, $d = 3$, $\tilde{n} = 11$ and $\tilde{d} = 7$, as in Figure \ref{figure_kleins_blunder} below.
\begin{figure}
\caption{The correct Lebesgue function for $n = 50$, $d = 3$, $\tilde{n} = 11$ and $\tilde{d} = 7$.}
\label{figure_kleins_blunder}
\end{figure}
The dependency of the Lebesgue function on the parameters $d$, $\tilde{n}$ and $\tilde{d}$ is subtle. For instance, Figure \ref{figure_lebesgue_functions} shows that the Lebesgue function may decrease as we increase $d$ and keep the other parameters fixed. Moreover, the Lebesgue constant for $n = 50$ and $d = \tilde{n} = \tilde{d} = 7$ is $12$ times smaller than the Lebesgue constant if $n = 50,$ $d = 3$, $\tilde{n} = 11$ and $\tilde{d} = 7$, as considered in \cite{Klein}.
\begin{figure}
\caption{The dependency of the Lebesgue function on $d$, $\tilde{n}$ and $\tilde{d}$.}
\label{figure_lebesgue_functions}
\end{figure}
\subsection{Proof of Lemma \ref{lem_clean}} \label{subsection_lem_clean} In this subsection we prove Lemma \ref{lem_clean} and show that when $n \leq 2 \tilde{n}$ we have an analogous result with \begin{eqnarray} \label{cjba} \hspace{0.4cm} \wfc{c_{j}}{t} & = & \frac{\tilde{w}_{n,d,j}}{t - x_{j}} + \sum_{i = -d}^{-1} \frac{\tilde{w}_{n,d,i} \, a_{i j}}{t - \tilde{x}_{i}} \hspace{2.7cm} \wrm{for} \ 0 \leq j < n - \tilde{n}, \\ \hspace{0.4cm} \nonumber \wfc{c_{j}}{t} & = & \frac{\tilde{w}_{n,d,j}}{t - x_{j}} + \sum_{i = -d}^{-1} \frac{\tilde{w}_{n,d,i}\, a_{i j}}{t - \tilde{x}_{i}}\\ \label{cjbb} & + & \sum_{i = n + 1}^{n + d} \frac{\tilde{w}_{n, d, i} \, b_{\wlr{i-n}\wlr{j-n}}}{t - \tilde{x}_i}
\hspace{2.6cm} \wrm{for} \ n - \tilde{n} \leq j \leq \tilde{n}, \\
\label{cjbc} \wfc{c_{j}}{t} & = & \frac{\tilde{w}_{n,d,j}}{t - x_{j}} + \sum_{i = n + 1}^{n + d} \frac{\tilde{w}_{n,d,i} \, b_{\wlr{i-n} \wlr{j-n}}}{t - \tilde{x}_i} \hspace{1.27cm} \wrm{for} \ \tilde{n} < j \leq n . \end{eqnarray} Our proof involves the numbers $D^{(k)}_{ij}$ mentioned in the third section of \cite{KleinBerrutDerivatives}, which represent the $k$th derivative at the node $x_i$ of the $j$th Lagrange fundamental rational function. In order to simplify the notation, we use the following lemma. \begin{lemma} \label{lem_e} Consider nodes $x_0 < x_1 < \dots < x_{\tilde{n}}$, weights $w_0, w_1,\dots, w_{\tilde{n}} \in \mathds R{} - \wset{0}$, and the numbers $E^{(k)}_{ij}$, with $k \geq 0$ and $0 \leq i, j \leq \tilde{n}$, defined inductively in $k$ by \begin{eqnarray} \label{eijkA} E^{(0)}_{ii} & := & 1 \hspace{1cm} \wrm{and} \hspace{1cm} E^{(0)}_{ij} := 0 \hspace{1.42cm} \wrm{for} \ i \neq j. \\ \label{eijkB} E^{(k)}_{ij} & := & \frac{k}{x_i - x_j} \wlr{\frac{w_j}{w_i} E_{ii}^{\wlr{k-1}} - E_{ij}^{\wlr{k-1}}} \hspace{0.9cm} \wrm{for} \ i \neq j, \\ \label{eijkC} E_{ii}^{(k)} & := & - \sum_{j = 0, j \neq i}^{\tilde{n}} E_{ij}^{(k)}. \end{eqnarray} If $k > 0$ then $E^{(k)}_{ij}$ is equal to the number $D^{(k)}_{ij}$ in Equations (3.1) and (3.2) in \cite{KleinBerrutDerivatives} with $n$ replaced by $\tilde{n}$. \end{lemma} In order to prove this lemma, replace $k$ by $1$ in Equations \wref{eijkB} and \wref{eijkC}, use \wref{eijkA} to simplify the result and Equation (3.2) of \cite{KleinBerrutDerivatives} to verify that $E^{(1)}_{ij} = D^{(1)}_{ij}$. The case $k > 1$ follows by induction from \wref{eijkB} and \wref{eijkC} and Equation (3.3) in \cite{KleinBerrutDerivatives}.
Since we are assuming that the nodes are equally spaced, $x_i - x_j = \wlr{i - j} h$ and it is convenient to consider the normalized numbers \[ \overline{E}_{ij}^{(k)} := h^k E_{ij}^{(k)} / k!. \] Replacing $E_{ij}^{(k)}$ by $\overline{E}_{ij}^{(k)} k! \, h^{-k}$ in \wref{eijkA}--\wref{eijkC} we obtain \begin{eqnarray} \label{final_d0} \hspace{0.4cm} \overline{E}^{(0)}_{ii} & = & 1 \hspace{1cm} \wrm{and} \hspace{1cm} \overline{E}^{(0)}_{ii} = 0 \hspace{1.73cm} \wrm{for} \ i \neq j, \\ \label{final_d1} \hspace{0.4cm} \overline{E}^{(k)}_{ij} & = & \frac{1}{i- j} \wlr{\frac{w_j}{w_i} \overline{E}_{ii}^{\wlr{k-1}} - \overline{E}_{ij}^{\wlr{k-1}}} \hspace{1.47cm} \wrm{for} \ i \neq j \hspace{0.2cm} \wrm{and} \hspace{0.2cm} k \geq 1, \\ \label{final_d2} \hspace{0.4cm} \overline{E}_{ii}^{(k)} & = & - \sum_{j = 0, j \neq i}^{\tilde{n}} \overline{E}_{i j}^{(k)}. \end{eqnarray} Equations \wref{final_d0}--\wref{final_d2} show that $\overline{E}^{(k)}_{ij}$ depends on $i$, $j$, $k$, and $\tilde{n}$, but it can depend on $h$, $d$, $n$ or $\tilde{d}$ only via the weights $w_i$, because there is no mention to $h$, $d$, $n$ and $\tilde{d}$ in \wref{final_d0} -- \wref{final_d2}. In the case that concerns us, the weights $w_i$ correspond to the usual Floater-Hormann interpolants in equally spaced nodes $x_0,\dots, x_{\tilde{n}}$ with parameter $\delta = \tilde{d}$. These weights are given by \wref{bary_w}, with $n$ and $\delta$ replaced by $\tilde{n}$ and $\tilde{d}$, and depend on $\tilde{n}$ and $\tilde{d}$, but not on $h$, $n$ or $d$. Therefore, in the case relevant to our discussion, $\overline{E}^{(k)}_{ij}$ does not depend on $h$, $n$ or $d$.
Equation (3.1) in \cite{KleinBerrutDerivatives} and the identities $D_{ij}^{(k)} = E^{(k)}_{ij}$ for $k > 0$ imply that \begin{eqnarray} \nonumber \wfc{r^{(k)}_{\underline{\wvec{x}},\tilde{d}}[\underline{\wvec{y}}]}{x_0} & = & \sum_{j = 0}^{\tilde{n}} D_{0j}^{(k)} y_j = \sum_{j = 0}^{\tilde{n}} E_{0j}^{(k)} y_j = \frac{k!}{h^k} \sum_{j = 0}^{\tilde{n}} \overline{E}_{0j}^{(k)} y_j, \\ \nonumber \wfc{r^{(k)}_{\overline{\wvec{x}},\tilde{d}}[\overline{\wvec{y}}]}{x_n} & = & \sum_{j = n - \tilde{n}}^{n} D_{\tilde{n}\wlr{j -n + \tilde{n}}}^{(k)} y_j = \sum_{j = n - \tilde{n}}^{n} E_{\tilde{n}\wlr{j -n + \tilde{n}}}^{(k)} y_j = \frac{k!}{h^k} \sum_{j = n - \tilde{n}}^{n} \overline{E}_{\tilde{n}\wlr{j - n + \tilde{n}}}^{(k)} y_j, \end{eqnarray} for $k > 0$. Combining the last two equations with the identities $\overline{E}^{(0)}_{ii} = 1$ and $\overline{E}^{(0)}_{ij} = 0$ for $i \neq j$ we can rewrite \wref{ytila} and \wref{ytilc} as \begin{eqnarray} \nonumber \tilde{y}_i & = & \sum_{k=0}^{\tilde{d}} \sum_{j = 0}^{\tilde{n}} \overline{E}_{0j}^{(k)} i^k y_j \ \hspace{3.4cm} \wrm{for} \ -d \leq i < 0, \\ \nonumber \tilde{y}_i & = & \sum_{k=0}^{\tilde{d}} \sum_{j = n - \tilde{n}}^{n} \overline{E}_{\tilde{n}\wlr{j - n + \tilde{n}}}^{(k)} \wlr{i-n}^k y_j \ \hspace{1.0cm} \wrm{for} \ n < i \leq n + d. \end{eqnarray} These equations are equivalent to \wref{tilyi_clean_a} and \wref{tilyi_clean_b} with \begin{eqnarray} \label{def_aij} \hspace{0.8cm} a_{ij} & := & \sum_{k = 0}^{\tilde{d}} \overline{E}_{0j}^{(k)} i^k \hspace{1.3cm} \wrm{for} \ -d \leq i < 0 \hspace{0.3cm} \wrm{and} \hspace{0.3cm} 0 \leq j \leq \tilde{n}, \\ \label{def_bij} \hspace{0.8cm} b_{i j } & := & \sum_{k = 0}^{\tilde{d}} \overline{E}_{\tilde{n} \wlr{j + \tilde{n}}}^{(k)} i^k \hspace{0.8cm} \wrm{for} \ 0 < i \leq d \hspace{0.3cm} \wrm{and} \hspace{0.3cm} - \tilde{n} \leq j \leq 0. \end{eqnarray} Since $\overline{E}_{ij}^{(k)}$ does not depend on $h$, $d$ or $n$, and there is no mention to $h$, $d$ or $n$ in the right hand side of Equations \wref{def_aij} and \wref{def_bij}, its is clear that $a_{ij}$ and $b_{ij}$ do not depend on $h$, $d$ or $n$, as claimed in Lemma \ref{lem_clean}. Equation \wref{bary_xfh} yields \[ \wxfh{t} = \frac{1}{\wfc{Q}{t}} \sum_{i = -d}^{n + d} \frac{\tilde{w}_{n,d,i} \, \tilde{y}_i}{t - \tilde{x}_i} \hspace{1.0cm} \wrm{with} \hspace{1.0cm} \wfc{Q}{t} = \sum_{i = -d}^{n + d} \frac{\tilde{w}_{n,d,i}}{t - \tilde{x}_i}. \] It follows that \[ \wfc{Q}{t} \wxfh{t} = \sum_{i = -d}^{-1} \frac{\tilde{w}_{n,d,i}}{t - \tilde{x}_i} \sum_{j = 0}^{\tilde{n}} \, a_{i j} \, y_j +
\sum_{j = 0}^{n} \frac{\tilde{w}_{n,d,j} \, y_j}{t - x_j} \ + \] \[
\sum_{i = n + 1}^{n + d} \frac{\tilde{w}_{n,d,i}}{t - \tilde{x}_i} \sum_{j = n - \tilde{n}}^n \, b_{\wlr{i-n} \wlr{j-n}} \, y_j = \] \begin{equation} \label{qform}
\sum_{j = 0}^{\tilde{n}} \sum_{i = -d}^{-1} \frac{\tilde{w}_{n,d,i} \, a_{i j} \, y_j}{t - \tilde{x}_i} +
\sum_{j = 0}^{n} \frac{\tilde{w}_{n,d,j} \, y_j}{t - x_j} +
\sum_{j = n - \tilde{n}}^n \sum_{i = n + 1}^{n + d} \frac{\tilde{w}_{n,d,i} \, b_{\wlr{i-n}\wlr{j-n}} \, y_j}{t - \tilde{x}_i}. \end{equation} We now have two cases: (i) $\tilde{n} < n - \tilde{n}$ and (ii) $\tilde{n} \geq n - \tilde{n}$. In the first case we can rewrite \wref{qform} as \[ \wfc{Q}{t} \wxfh{t} =
\sum_{j = 0}^{\tilde{n}} \wlr{\frac{\tilde{w}_{n,d,j}}{t - x_j} +
\sum_{i = -d}^{-1} \frac{\tilde{w}_{n,d,i} \, a_{i j}}{t - \tilde{x}_i} } y_j \ +
\]
\[
\sum_{j = \tilde{n} + 1}^{n - \tilde{n} - 1} \frac{\tilde{w}_{n,d,j}}{t - x_j} \, y_j +
\sum_{j = n - \tilde{n}}^n \wlr{\frac{\tilde{w}_{n,d,j}}{t - x_j} +
\sum_{i = n + 1}^{n + d} \frac{\tilde{w}_{n,d,i} \, b_{\wlr{i-n}\wlr{j-n}}}{t - \tilde{x}_i}} y_j, \] and this proves \wref{reduced_form}--\wref{cc}. When $\tilde{n} \geq n - \tilde{n}$, we can rewrite \wref{qform} as \[ \wfc{Q}{t} \wxfh{t} =
\sum_{j = 0}^{n - \tilde{n} - 1} \wlr{\frac{\tilde{w}_{n,d,j}}{t - x_j} +
\sum_{i = -d}^{-1} \frac{\tilde{w}_{n,d,i} \, a_{i j}}{t - \tilde{x}_i} } y_j \ +
\]
\[
\sum_{j = n - \tilde{n}}^{\tilde{n}} \wlr{\sum_{i = -d}^{-1} \frac{\tilde{w}_{n,d,i} \, a_{i j}}{t - \tilde{x}_i} +
\frac{\tilde{w}_{n,d,j}}{t - x_j} + \sum_{i = n + 1}^{n + d} \frac{\tilde{w}_{n,d,i} \, b_{\wlr{i-n}\wlr{j-n}}}{t - \tilde{x}_i} } y_j \ +
\]
\[
\sum_{j = \tilde{n} + 1}^n \wlr{\frac{\tilde{w}_{n,d,j}}{t - x_j} +
\sum_{i = n + 1}^{n + d} \frac{\tilde{w}_{n,d,i} \, b_{\wlr{i-n}\wlr{j-n}}}{t - \tilde{x}_i}} y_j. \] This proves \wref{cjba}--\wref{cjbc} and we are done. \qed{}
\section{The Lebesgue constant when $d = \tilde{n} = \tilde{d}$} \label{section_lebesgue} This section presents a proof that the Lebesgue constant of extended interpolants mentioned in \cite{BerrutKleinCAM} and \cite{Klein} grows exponentially with $d = \tilde{n} = \tilde{d}$. This shows that the peculiar interpretation of the Lebesgue constant mentioned in \cite{Klein} does not capture essential points regarding the stability of extended Floater Hormann interpolants in general, because Equation (3.2) in \cite{Klein} does not take properly into account how changes on $\wvec{y}$ affect $\tilde{\wvec{y}}$.
The Lebesgue constant of the extended interpolant $\wxfh{t}$ in \wref{bary_xfh} is \[ \tilde{\Lambda}_{\wvec{x},d,\tilde{n},\tilde{d}}{} := \max_{x_0 \leq t \leq x_n} \wlebfxfh{t}, \] for the Lebesgue function $\wlebfxfh{t}$ in \wref{leb_fun_xfh}, and the Lebesgue constant for polynomial interpolation at $d + 1$ equally spaced nodes is \[ \Lambda_d := \sup_{0 \leq t \leq d, \ \ \wvec{y} \in \wrn{d+1} - \wset{0}} \frac{\wabs{ \wfc{p_{d,\wvec{y}}}{t}}}{\wnorm{\wvec{y}}_\infty}, \] where $p_{d,\wvec{y}}$ is the polynomial with degree less than $d + 1$ such that $\wfc{p_{d,\wvec{y}}}{i} = y_i$ for $i = 0,\dots,d$. In this section we show that $\tilde{\Lambda}_{\wvec{x},d,d,d}{}$ is not much smaller than $\Lambda_d$, by providing a lower bound for $\tilde{\Lambda}_{\wvec{x},d,d,d}{}$ which approaches $\Lambda_d$ exponentially fast as $d$ increases. Formally, we have the following:
\begin{theorem} \label{thm_main} If $n > d + 1 \geq 3$ then $\tilde{\Lambda}_{\wvec{x},d,d,d}{} \geq \kappa_d \Lambda_d$, for \begin{equation} \label{main_thesis} \kappa_d := \wlr{1 - \frac{d}{2^d - 1} - 2^{-d}}. \end{equation} \end{theorem}
We prove Theorem \ref{thm_main} at the end of this section. For now, let us explore its consequences and check them experimentally. As explained in \cite{Trefe}, we have \[ \frac{2^{d-2}}{d^2} < \Lambda_d < \frac{2^{d+3}}{d}. \] Therefore, $\Lambda_d$ grows exponentially with $d$ and Theorem \ref{thm_main} shows that the same applies to $\tilde{\Lambda}_{\wvec{x},d,d,d}{}$. Moreover, Bos et. al. \cite{Bos} show that the Lebesgue constant for the Floater-Hormann interpolant at $\wvec{x}$ with parameter $\delta$ satisfies $\tilde{\Lambda}_{\wvec{x},\delta}{} \leq 2^{\delta-1} \wlr{2 + \log n}$. Theorem \ref{thm_main} shows that $\Lambda_d < 1.5 \tilde{\Lambda}_{\wvec{x},d,d,d}{}$ for $d \geq 4$, and combining the two equations above for $d \geq 4$ we conclude that, when $\delta = d$, \[ \tilde{\Lambda}_{\wvec{x},\delta}{} < 2 d^2 \wlr{2 + \log n} \Lambda_d < 3 d^2 \wlr{2 + \log n} \tilde{\Lambda}_{\wvec{x},d,d,d}{}, \] and the ratio $\tilde{\Lambda}_{\wvec{x},\delta}{}/\tilde{\Lambda}_{\wvec{x},d,d,d}{}$ is definitely not as large as claimed in \cite{Klein}. This observation is corroborated by Figure \ref{figure_lebesgue_constants}.
\begin{figure}\label{figure_lebesgue_constants}
\end{figure}
\subsection{Proof of Theorem \ref{thm_main}} We follow the usual convention that a sum of the form $\sum_{i = a}^b u_i$ with $b < a$ is equal to $0$ and a product $\prod_{i = a}^b u_i$ with $b < a$ is equal to $1$. According to \cite{Klein}, \begin{equation} \label{FHinterp2} \wxfh{t} = \frac{\sum\limits_{i = -d}^{n} \tilde{\lambda}_i(t,\tilde{\wvec{x}})
p_i(t,\tilde{\wvec{x}},\wvec{\tilde{y}}) }{\sum\limits_{i = -d}^{n} \tilde{\lambda}_i(t,\tilde{\wvec{x}}) }, \end{equation} where $\tilde{x}_i = x_0 + i h$ for $-d \leq i \leq n + d$, $\tilde{\wvec{y}} = \wlr{\tilde{y}_{-d},\tilde{y}_{1-d}, \, \dots \,, \tilde{y}_{n+d}}$ and \begin{equation} \label{LambdaAnd} \wfc{\tilde{\lambda}_i}{t,\tilde{\wvec{x}}} := \frac{\wlr{-1}^i}{\wlr{t - \tilde{x}_i} \wlr{t - \tilde{x}_{i+1}} \dots \wlr{t - \tilde{x}_{i+d}}} \hspace{1cm} \wrm{for} \ i = 0, \dots, n, \end{equation} and $\wfc{p_{i}}{t,\tilde{\wvec{x}},\tilde{\wvec{y}}}$ is the polynomial with degree less than $d + 1$ such that $\wfc{p_{i}}{\tilde{x}_k,\tilde{\wvec{x}},\tilde{\wvec{y}}} = \tilde{y}_k$ for $k = i,\dots,i + d$. When $\tilde{\wvec{y}}$ is defined as in Equations \wref{ytila}-- \wref{ytilc}, we have \[ \wfc{p_i}{t,\tilde{\wvec{x}},\wvec{\tilde{y}}} = \wfc{p_i}{t,\wvec{x},\wvec{y}} \ \ \ \wrm{for} \ \ \ 0 \leq i \leq n - d, \] and when $d = \tilde{n} = \tilde{d}$ we also have \begin{eqnarray} \nonumber \wfc{p_i}{t,\tilde{\wvec{x}},\wvec{\tilde{y}}} & = & \wfc{p_0}{t,\wvec{x},\wvec{y}} \hspace{1.4cm} \ \wrm{for} \ -d \leq i < 0,\\ \nonumber \wfc{p_i}{t,\tilde{\wvec{x}},\wvec{\tilde{y}}} & = & \wfc{p_{n-d}}{t,\wvec{x},\wvec{y}} \hspace{1cm} \ \wrm{for} \ n-d < i \leq n, \end{eqnarray} because in this case the interpolants $\wfc{r_{\underline{\wvec{x}},\tilde{d}}[\underline{\wvec{y}}]}{t}$ and $\wfc{r_{\overline{\wvec{x}},\tilde{d}}[\overline{\wvec{y}}]\dot{}}{t}$ are polynomials, and the Taylor series of a polynomial is equal to itself. Equation \wref{FHinterp2} leads to \[ \wxfhdiag{t} =
\frac{
\left(\sum_{i = -d}^{0} \wfc{\tilde{\lambda}_i}{t,\tilde{\wvec{x}}}\right)
\wfc{p_0}{t,\wvec{x},\wvec{y}}
+\wfc{\tilde{\lambda}_1}{t,\wvec{x}} \wfc{p_1}{t,\wvec{x},\wvec{y}}} {\sum_{i = -d}^{n} \wfc{\tilde{\lambda}_i}{t,\tilde{\wvec{x}}}} \ + \] \begin{equation} \label{FHinterp3}
\frac{\sum_{i = 2}^{n-d - 1} \wfc{\tilde{\lambda}_i}{t,\wvec{x}} \wfc{p_i}{t,\wvec{x},\wvec{y}}}{\sum_{i = -d}^{n} \wfc{\tilde{\lambda}_i}{t,\tilde{\wvec{x}}}} +
\frac{ \wlr{\sum_{i = n - d}^{n} \frac{\wfc{\tilde{\lambda}_i}{t,\tilde{\wvec{x}}}}{\wfc{\tilde{\lambda}_{n-d}}{t,\wvec{x}}}} \wfc{\tilde{\lambda}_{n-d}}{t,\wvec{x}}\wfc{p_{n-d}}{t,\wvec{x},\wvec{y}} }{\sum_{i = -d}^{n} \wfc{\tilde{\lambda}_i}{t,\tilde{\wvec{x}}} } .
\end{equation} (Since $n > d + 1$ the sums in numerator of the first and last parcels in the expression above do not overlap, even when the sum in the numerator in the middle is empty.) Let $t \ \in \ (x_0, x_1)$ be fixed. We claim that $y^*_0, y^*_1, \dots, y^*_{n} \ \in \ \{-1,1\}$ defined by \[ y^*_0 := y^*_1 := (-1)^d \hspace{0.6cm} \wrm{and} \hspace{0.6cm} y^*_j := (-1)^{d+j-1} \ \wrm{for} \ \ 2 \leq j \leq n \] satisfy \begin{equation} \label{lem2eq}
\wfc{\tilde{\lambda}_i}{t,\wvec{x}} \wfc{p_i}{t,\wvec{x},\wvec{y^*}} = \sum\limits_{j = 0}^{d} \frac{\wabs{u_j}}{\wabs{t - \tilde{x}_{i+j}}}\geq 0
\hspace{0.4cm} \wrm{for} \ i = 0 \ \mbox{or} \ 2 \leq i \leq n - d, \end{equation} where \[ u_j := \frac{(-1)^{d- j}}{d! h^d}\left( \begin{array}{c} d \\ j \end{array} \right). \] In fact, \wref{LambdaAnd} and Equations (3.1), (3.3) and (5.1) of \cite{Ber} show that \begin{equation}\label{And2}
\tilde{\lambda}_i(t,\wvec{x})p_i(t,\wvec{x},\wvec{y^*}) = (-1)^i \sum\limits_{j = 0}^{d} \frac{u_jy^*_{i+j}}{t-x_{i+j}}, \end{equation} and \wref{lem2eq} follows from \[ \frac{u_0y^*_0}{t-x_{0}} = \frac{\wabs{u_0}}{\wabs{t-x_{0}}}, \hspace{1cm} \frac{u_1 y^*_1}{t-x_{1}} = \frac{\wabs{u_1}}{\wabs{t-x_{1}}} \] and \[ \frac{u_j y^*_{i+j}}{t-x_{i+j}} = \frac{(-1)^{d-j}\wabs{u_j}y^*_{i+j}}{t-x_{i+j}} = \frac{(-1)^{i-1}\wabs{u_j}}{t-x_{i+j}} = \frac{(-1)^i \wabs{u_j}}{\wabs{t-x_{i+j}}} \] for $2 \leq i + j \leq n$ and $0 \leq j \leq d$.
Note that \begin{equation} \label{FHinterp4}
\frac{\sum_{i = -d}^{0} \wfc{\tilde{\lambda}_i}{t,\tilde{\wvec{x}}}}
{\tilde{\lambda}_0(t,\wvec{x})} > 0, \end{equation} because, for $-d \leq i < 0$, we have that $d+ i+ 1 \geq 1$ and \wref{LambdaAnd} yields \[ \frac{\wfc{\tilde{\lambda}_i}{t,\tilde{\wvec{x}}}}{\wfc{\tilde{\lambda}_0}{t,\tilde{\wvec{x}}}} = \frac{ \wlr{-1}^{i} \wlr{t - x_0} \dots \wlr{t - x_{d + i}} \wlr{t - x_{d+ i + 1}} \dots \wlr{t - x_{d} }}
{ \wlr{t - \tilde{x}_i} \dots \wlr{t - \tilde{x}_{-1}} \wlr{t - x_0} \dots \wlr{t - x_{d + i}} } \] \begin{equation} \label{same_sign} = \frac{ \wlr{-1}^{i} \wlr{t - x_{d + i + 1}} \dots \wlr{t - x_{d} }}
{ \wabs{t - \tilde{x}_i} \dots \wabs{t - \tilde{x}_{-1}} } = \frac{\wabs{t - x_{d + i + 1}} \dots \wabs{t - x_{d} }}
{ \wabs{t - \tilde{x}_i} \dots \wabs{t - \tilde{x}_{-1}} } \geq 0, \end{equation} and this inequality also holds for $i = 0$.
Moreover, the signs of the numbers $\tilde{\lambda}_{1}(t,\tilde{\wvec{x}})$, $\tilde{\lambda}_{2}(t,\tilde{\wvec{x}})$, $\dots$, $\tilde{\lambda}_{n}(t,\tilde{\wvec{x}})$ alternate, and their magnitude decreases because, for $1 \leq i < n$, \wref{LambdaAnd} yields \begin{equation} \label{alternate} -1 < \frac{\tilde{\lambda}_{i + 1}(t,\tilde{\wvec{x}})}{\tilde{\lambda}_{i}(t,\tilde{\wvec{x}})} = - \frac{\tilde{x}_{i} - t}{\tilde{x}_{d + i + 1} - t} < 0. \end{equation} As a result, \[
\sum_{i = n - d}^{n} \frac{\wfc{\tilde{\lambda}_i}{t,\tilde{\wvec{x}}}}{\wfc{\tilde{\lambda}_{n-d}}{t,\wvec{x}}} \geq 0. \] This inequality and \wref{lem2eq} with $i = n - d$ imply that the numerator of the last parcel in the sum in the right hand side of \wref{FHinterp3} is not negative, and combining \wref{FHinterp3}, \wref{lem2eq} and \wref{FHinterp4} we obtain \begin{equation} \label{FHinterp5} \wabs{\wfc{\tilde{r}_{\wvec{x},d,d,d}[\wvec{y}^*]}{t}} \geq
\frac{\frac{\sum_{i = -d}^{0} \tilde{\lambda}_i(t,\tilde{\wvec{x}})}
{
\tilde{\lambda}_0(t,\wvec{x})} \tilde{\lambda}_0(t,\wvec{x})p_0(t,\wvec{x},\wvec{y^*}) -
\wabs{\tilde{\lambda}_1(t,\wvec{x}) p_1(t,\wvec{x},\wvec{y^*})} }
{\wabs{\sum_{i = -d}^{n} \tilde{\lambda}_i(t,\tilde{\wvec{x}})} }. \end{equation} Moreover, \wref{lem2eq} and \wref{And2} yield \[ \tilde{\lambda}_0(t,\wvec{x})p_0(t,\wvec{x},\wvec{y^*}) - \wabs{\tilde{\lambda}_1(t,\wvec{x})p_1(t,\wvec{x},\wvec{y^*})} \ \ \geq \ \
\sum\limits_{j = 0}^{d} \frac{\wabs{u_j}}{\wabs{t-x_{j}}} - \sum\limits_{j = 0}^{d} \frac{\wabs{u_{j}}}{\wabs{t-x_{j+1}}} \] \begin{equation}\label{posit}
\geq\ \frac{\wabs{u_0}}{\wabs{t-x_0}} +
\wset{\frac{\wabs{u_1}-\wabs{u_0}}{\wabs{t-x_1}} - \frac{\wabs{u_1}}{\wabs{t-x_2}}}
+ \sum\limits_{j = 2}^{d} \left(\frac{\wabs{u_j}}{\wabs{t-x_{j}}} -
\frac{\wabs{u_j}}{\wabs{t-x_{j+1}}} \right). \end{equation}
The last sum in \wref{posit} is positive for all $t \ \in \ (x_0, x_1)$.
The term in brackets is also positive for $t \ \in \ (x_0, x_1)$, because \[
\frac{\wabs{u_1}-\wabs{u_0}}{\wabs{u_1}} = 1 - \frac{1}{d} \geq \frac{1}{2} \geq \frac{\wabs{t-x_1}}{\wabs{t-x_2}}. \] Therefore, \begin{equation} \label{pos_diff} \tilde{\lambda}_0(t,\wvec{x})p_0(t,\wvec{x},\wvec{y^*}) - \wabs{ \tilde{\lambda}_1(t,\wvec{x})p_1(t,\wvec{x},\wvec{y^*}) } \ \ \geq \ \ 0. \end{equation} Equation \wref{same_sign} shows that the numbers $\tilde{\lambda}_{-d}(t,\tilde{\wvec{x}})$, $\tilde{\lambda}_{1-d}(t,\tilde{\wvec{x}})$, $\dots$, $\tilde{\lambda}_{-1}(t,\tilde{\wvec{x}})$ have the same sign as $\tilde{\lambda}_0(t,\tilde{\wvec{x}})$. When $-d < i \leq 0$ we also have \[ \frac{\tilde{\lambda}_{i}(t,\tilde{\wvec{x}})}{\tilde{\lambda}_1(t,\tilde{\wvec{x}})} = \frac{\wlr{-1}^{i + 1}\wlr{t - x_1} \dots \wlr{t - x_{d + i}} \wlr{t - x_{d + i + 1}} \dots \wlr{t - x_{d + 1}}} {\wlr{t - \tilde{x}_i} \dots \wlr{t - x_0} \wlr{t - x_1} \dots \wlr{t - x_{d + i}}} = \] \begin{equation} \label{q01} \frac{\wlr{-1}^{i + 1}\wlr{t - x_{d + i + 1}} \dots \wlr{t - x_{d + 1}}} {\wlr{t - \tilde{x}_i} \dots \wlr{t - x_0}} = \frac{\wlr{x_{d + i + 1} - t} \dots \wlr{x_{d + 1} - t}} {\wlr{t - \tilde{x}_i} \dots \wlr{t - x_0}} \geq 0, \end{equation} and the reader can verify that this inequality also holds for $i = -d$. Therefore, \begin{equation} \label{same_signB} \frac{\tilde{\lambda}_i(t,\tilde{\wvec{x}})}{\tilde{\lambda}_{1}(t,\tilde{\wvec{x}})} > 0 \hspace{1cm} \wrm{for} \hspace{0.2cm} -d \leq i \leq 1. \end{equation} Since all $\wfc{\tilde{\lambda}_i}{t,\wvec{x}}$ for $- d \leq 0 \leq 1$ have the same sign, and for $i \geq 1$ the signs of the $\wfc{\tilde{\lambda}_i}{t,\wvec{x}}$ alternate and their magnitude decreases, we have \begin{equation} \label{denominator} \wabs{\sum_{i = -d}^n \wfc{\tilde{\lambda}_i}{t,\tilde{\wvec{x}}}}
\leq \wabs{\sum_{i = -d}^1 \wfc{\tilde{\lambda}_i}{t,\tilde{\wvec{x}}}}. \end{equation} Equations \wref{FHinterp5}, \wref{pos_diff} and \wref{denominator} lead to \begin{equation} \label{bounda} \wabs{\wfc{\tilde{r}_{\wvec{x},d,d,d}[\wvec{y}^*]}{t}} \geq
\frac{\wabs{\sum_{i = -d}^{-1} \tilde{\lambda}_i(t,\tilde{\wvec{x}})} \wabs{p_0(t,\wvec{x},\wvec{y^*})}}{\wabs{\sum_{i = -d}^{1} \tilde{\lambda}_i(t,\tilde{\wvec{x}})}}. \end{equation} When $-d < i \leq 0$, Equation \wref{same_sign}, and the comment just after it, yield \[ \frac{\wfc{\tilde{\lambda}_i}{t,\tilde{\wvec{x}}}}{\wfc{\tilde{\lambda}_0}{t,\tilde{\wvec{x}}}} \geq \frac{ \wlr{x_{d + i + 1} - x_1} \dots \wlr{x_d - x_1}}
{ \wlr{x_1 - \tilde{x}_i} \dots \wlr{x_1 - \tilde{x}_{-1}} } = \frac{\wlr{d + i} \dots \wlr{d-1} h^{-i}}
{ \wlr{1 - i}! h^{-i} } = \] \[ \frac{\wlr{d + i} \dots \wlr{d-1}}{ \wlr{1 - i}!} = \frac{1}{d} \wvecB{d}{1 - i}. \] Therefore, \begin{equation} \label{sum0} d \times \frac{\sum_{i = -d}^{1} \wabs{\wfc{\tilde{\lambda}_i}{t,\tilde{\wvec{x}}}}}{\wabs{\wfc{\tilde{\lambda}_0}{t,\tilde{\wvec{x}}}}} \geq \sum_{i = 1 - d}^{0} \wvecB{d}{1 - i} = \sum_{j = 1}^{d} \wvecB{d}{j} = 2^d - 1. \end{equation} Moreover, for $-d < i \leq 0$ Equation \wref{q01} yields \[ \frac{\tilde{\lambda}_{i}(t,\tilde{\wvec{x}})}{\tilde{\lambda}_1(t,\tilde{\wvec{x}})} \geq \frac{\wlr{x_{d + i + 1} - x_1} \dots \wlr{x_{d + 1} - x_1}} {\wlr{x_1 - \tilde{x}_i} \dots \wlr{x_1 - x_0} } = \frac{\wlr{d + i} \dots d\, h^{1 - i}} {\wlr{1 - i}! h^{1 - i}} = \wvecB{d}{1 - i}, \] and \begin{equation} \label{sum1} \frac{\sum_{i = -d}^1 \wabs{\tilde{\lambda}_{i}(t,\tilde{\wvec{x}})}}{\wabs{\tilde{\lambda}_1(t,\tilde{\wvec{x}})}} \geq \sum_{i = 1- d}^0 \frac{\wabs{\tilde{\lambda}_{i}(t,\tilde{\wvec{x}})}}{\wabs{\tilde{\lambda}_1(t,\tilde{\wvec{x}})}} + 1 \geq \sum_{i = 1 - d}^1 \wvecB{d}{1 - i} = \sum_{j = 0}^d \wvecB{d}{j} = 2^d. \end{equation} It follows from Equations \wref{main_thesis}, \wref{sum0} and \wref{sum1} that \[ \wabs{\sum_{i = -d}^{-1} \tilde{\lambda}_i(t,\tilde{\wvec{x}})} = \sum_{i = -d}^{-1} \wabs{\tilde{\lambda}_i(t,\tilde{\wvec{x}})} = \sum_{i = -d}^{1} \wabs{\tilde{\lambda}_i(t,\tilde{\wvec{x}})} - \wabs{\tilde{\lambda}_0(t,\tilde{\wvec{x}})} - \wabs{\tilde{\lambda}_1(t,\tilde{\wvec{x}})} \] \[ = \wlr{1 - \frac{\wabs{\tilde{\lambda}_0(t,\tilde{\wvec{x}})}}{\sum_{i = -d}^{1} \wabs{\tilde{\lambda}_i(t,\tilde{\wvec{x}})} } - \frac{\wabs{\tilde{\lambda}_1(t,\tilde{\wvec{x}})}}{\sum_{i = -d}^{1} \wabs{\tilde{\lambda}_i(t,\tilde{\wvec{x}})} }} \sum_{i = -d}^{1} \wabs{\tilde{\lambda}_i(t,\tilde{\wvec{x}})} \geq \kappa_d \wabs{\sum_{i = -d}^{1} \tilde{\lambda}_i(t,\tilde{\wvec{x}})}. \] The inequality in the previous line and \wref{bounda} yield \[ \wabs{\wfc{\tilde{r}_{\wvec{x},d,d,d}[\wvec{y}^*]}{t}} \geq \kappa_d \wabs{p_0(t,\wvec{x},\wvec{y^*})}. \] Equation \wref{lem2eq} shows that, for $t \ \in [x_0, x_1]$, $\wabs{p_0(.,\wvec{x},\wvec{y^*})}$ is identical to the Lebesgue function for polynomial interpolation at $(d+1)$ equally spaced nodes in $[x_0,x_d]$. According to \cite{Brutman}, the Lebesgue function for polynomial interpolation at equally spaced nodes attains its maximum at some $t^* \in (x_0,x_1)$. For this $t^*$ we have \[ \tilde{\Lambda}_{\wvec{x},d,d,d}{} \geq \wlebfxfhdiag{t^*} \ \geq \ \wabs{\wfc{\tilde{r}_{\wvec{x},d,d,d}[\wvec{y^*}]}{t^*}} \ \geq \kappa_d \wabs{ \wfc{p_0}{t^*,\wvec{x},\wvec{y^*}}} =
\kappa_d \Lambda_d, \] and this completes the proof of Theorem \ref{thm_main}. \qed{}
\section{Backward instability} \label{section_backward} In this section we discuss the backward stability of the barycentric formula used to evaluate extended interpolants when $\tilde{\wvec{y}}$ is given by a function $\wfc{Y}{d, \wvec{x}, \wvec{y}}$ which is linear in $\wvec{y}$. Formally, we take $\tilde{y}_i := y_i$ for $0 \leq i \leq n$ and \begin{equation} \label{ytilback} \tilde{y}_i := \wfc{Y_i}{d, \wvec{x}, \wvec{y}} = \sum_{j = 0}^n \wfc{h_{ij}}{d,\wvec{x}} y_j \hspace{1cm} \wrm{for} \hspace{0.2cm} i \in \wset{-d,\dots, n + d} \setminus \wset{0,\dots, n}. \end{equation} The extended function values $\tilde{\wvec{y}}$ are supposed to be evaluated numerically and then to be used to evaluate the barycentric interpolant $\wfc{b}{t, \wvec{y}}$ given by \begin{equation} \label{baryback} \wfc{b}{t, \wvec{y}} := \left. \sum_{i = -d}^{n+d} \frac{\tilde{w}_i \tilde{y}_i}{t - \tilde{x}_i} \right/ \sum_{i = -d}^{n+d} \frac{\tilde{w}_i}{t - \tilde{x}_i} \hspace{0.5cm} \wrm{with} \hspace{0.5cm} \tilde{y}_i = \wfc{Y_i}{d, \wvec{x}, \wvec{y}}. \end{equation} We assume that the weights $\tilde{w}_i$ are such that the denominator of $\wfc{b}{t,\wvec{y}}$ is different from zero for $t \in [x_0,x_n] \setminus \wset{x_0,x_1,\dots x_n}$.
As we have shown in Section 3, the Equation (2.3) in \cite{Klein} and the equation just before Theorem 5.1 in \cite{BerrutKleinCAM} are particular cases of Equation \wref{baryback}. Therefore, by discussing the backward stability of \wref{ytilback}--\wref{baryback} we also cover the the backward stability of the interpolation formulae proposed in the literature.
In order to analyze the backward stability of \wref{ytilback}--\wref{baryback}, it is convenient to proceed as in Section \ref{section_reduced} and rewrite \wref{baryback} as \begin{equation} \label{barybackred} \wfc{b}{t, \wvec{y}} = \left. \sum_{i = 0}^{n} \wfc{d_i}{t} y_i \right/ \sum_{i = -d}^{n+d} \frac{\tilde{w}_i}{t - \tilde{x}_i}, \end{equation} for \begin{equation} \label{dj} \wfc{d_j}{t} := \sum_{i = -d}^{-1} \frac{\tilde{w}_i \wfc{h_{ij}}{d,\wvec{x}}}{t - \tilde{x}_i} + \frac{\tilde{w}_j}{t - x_j}
+ \sum_{i = n + 1}^{n + d} \frac{\tilde{w}_i \wfc{h_{ij}}{d, \wvec{x}}}{t - \tilde{x}_i}. \end{equation} Equations \wref{barybackred}--\wref{dj} can be verified as in the proof of the validity of the reduced form \wref{reduced_form} presented in Section \ref{subsection_lem_clean}.
We adopt the definition of backward stability used by Higham in \cite{HIGHAM_IMA}, that is, the formulae above are backward stable when the value $\wfc{\hat{b}}{t,\wvec{y}}$ obtained by evaluating \wref{ytilback}-\wref{baryback} in inexact arithmetic is equal to the exact value $\wfc{b}{t,\hat{\wvec{y}}}$ for a perturbed vector $\hat{\wvec{y}}$ with $\hat{y}_i = y_i \wlr{1 + \phi_i}$ for $\phi_i$ small. With this in mind, we can summarize this section as follows:\\[-0.2cm] \begin{quote} The barycentric formula \wref{ytilback}-\wref{baryback} is not backward stable in Higham's sense when $\tilde{w}_k \neq 0$ for some $k \in \wset{-d,\dots,n+d} \setminus \wset{0,\dots n}$ and there exists $t \in [x_0,x_n] \setminus \wset{x_0,x_1,\dots x_n}$ and $0 \leq j \leq n$ such that $\wfc{d_j}{t} = 0$.\\[-0.1cm] \end{quote}
In this circumstance, we can prove the backward instability of \wref{ytilback}-\wref{baryback} by considering $\wvec{y} \in \wrn{n+1}$ with $y_j = 1$ and $y_i = 0$ for $i \neq j$ and all $\hat{\wvec{y}} \in\wrn{n+1}$ with $\hat{y}_i = y_i \wlr{1 + \phi_i}$ for some $\phi_i \in \mathds R{}$. On the one hand, we have that $\wfc{d_i}{t} y_i \wlr{1 + \phi_i} = 0$ for all $i$ and $\phi_i \in \mathds R{}$, because $\wfc{d_j}{t} = 0$ and $y_i = 0$ for $i \neq j$. Therefore, Equation \wref{barybackred} shows that when we evaluate \wref{ytilback}-\wref{baryback} in exact arithmetic with $t$ and $\hat{\wvec{y}}$ we obtain $\wfc{b}{t, \hat{\wvec{y}}} = 0$. On the other hand, if the unique rounding error occurs in the evaluation of Equation \wref{ytilback} for $i = k$, so that $\tilde{y}_{k}$ is computed as $\tilde{y}_{k} + \xi$ for $\xi \neq 0$, then equation \wref{barybackred} shows that the computed value $\wfc{\hat{b}}{t,\wvec{y}}$ satisfies \[ \wfc{\hat{b}}{t, \wvec{y}} = \left. \wlr{\frac{\tilde{w}_{k} \xi }{t - \tilde{x}_{k}} + \sum_{i = -d}^{n+d} \frac{\tilde{w}_i \tilde{y}_i}{t - \tilde{x}_i} } \right/ \sum_{i = -d}^{n+d} \frac{\tilde{w}_i}{t - \tilde{x}_i} \] \[ = \left. \frac{\tilde{w}_{k} \xi }{t - \tilde{x}_{k}} \right/ \sum_{i = -d}^{n+d} \frac{\tilde{w}_i}{t - \tilde{x}_i} + \wfc{b}{t, \wvec{y}} = \left. \frac{\tilde{w}_{k} \xi }{t - \tilde{x}_{k}} \right/ \sum_{i = -d}^{n+d} \frac{\tilde{w}_i}{t - \tilde{x}_i} \neq 0 = \wfc{b}{t, \hat{\wvec{y}}}. \] Therefore, the computed value $\wfc{\hat{b}}{t, \wvec{y}}$ differs from all the exact values $\wfc{b}{t, \hat{\wvec{y}}}$ and, according to Higham's definition, \wref{ytilback}-\wref{baryback} is not backward stable in this case.
In practice the $\tilde{w}_k$ are different from zero and the simple condition $\wfc{d_j}{t} = 0$ implies the backward instability of \wref{ytilback}-\wref{baryback}. We conclude this section with Figure \ref{figure_secular}, which shows that there is $t \in [x_0,x_n] \setminus \wset{x_0,\dots,x_n}$ for which $\wfc{d_2}{t} = 0$ for the extended interpolant with $n=50$, $d = 3$, $\tilde{n} = 11$ and $\tilde{d} = 7$ considered by \cite{Klein}. In fact, in this case $h = 2/50 = 0.04$, $x_{2} = -0.92$, $x_{3} = -0.88$ and the function $\wfc{d_2}{t}$ has a zero in the interval $[-0.918,-0.914] \subset (x_2, x_3)$.
\begin{figure}
\caption{The function $\wfc{d_2}{t}$ for the extended interpolant with $n = 50$, $d = 3$, $\tilde{n} = 11$ and $\tilde{d} = 7$ considered in \cite{Klein}.}
\label{figure_secular}
\end{figure}
\section{Sources of numerical instability for extended interpolants} \label{section_instability} This section shows that extended interpolants based on extrapolation are strongly affected by the numerical errors in the extrapolation step when $\tilde{d}$ is large. The current literature pays little attention to this point and presents experimental comparisons of usual and extended interpolants that highlight cases in which $\tilde{d}$ is much smaller than $d$. Such experiments are biased in favor of extended interpolants: increasing $\delta$ for usual interpolants makes as much sense as increasing $\tilde{d}$ for extended interpolants when $d > \tilde{d}$, because for $d > \tilde{d}$ the order of approximation of the extended interpolants is $h^{\tilde{d} + 1}$, and not $h^{d +1}$.
We consider the case $\delta = d = \tilde{d} = \tilde{n}$. In our judgment, this is the most relevant case because it is the minimal one resulting in the same approximation of order $h^{d+1}$ for extended interpolants and usual interpolants. This point is reinforced by Figure \ref{figure_choosing}, which shows that it is pointless to increase $d$ when $d > \tilde{d}$.
\begin{figure}
\caption{Log10 of the error for $\wfc{f}{t} = \wfc{\sin}{20t}$ and $t \in [-1,1]$, with $n = 200$, $\tilde{n} = \tilde{d} = 8$ and $\delta = d$ varying from $1$ to $35$. Note that by simply increasing $d$, with an inappropriate $\tilde{d}$, we may obtain inaccurate results for extended interpolants. In this example, increasing $d$ when $d > \tilde{d} = 8$ has no effect on the accuracy of the extended interpolant, but increasing $\delta$ up to $13$ improves the accuracy of the usual interpolants. This shows that the roles of $d$ and $\delta$ are quite different when $d > \tilde{d}$. }
\label{figure_choosing}
\end{figure}
Figure \ref{figure_choosing} illustrates the importance of choosing appropriate $\tilde{d}$ for extended interpolants and makes clear the distinction between $d$ and $\delta$. Large values of $\delta$ have a devastating effect on usual Floater-Hormann interpolants, and this basic fact is mentioned explicitly in the documentation of libraries that implement these interpolants \cite{ALGLIB}. Consequently, there is little to be learned from comparisons of extended and usual Floater-Hormann interpolants as in Figures 6 of \cite{BerrutKleinCAM} and \cite{Klein}: they fix $\tilde{d}$ at small values for extended interpolants and then raise $d = \delta$ to values as large as 50. Such choices of a large $d = \delta$ have no practical motivation for extended interpolants and are unfavorable to usual Floater-Hormann interpolants.
From this point to the end of this section we consider the interpolation of $\wfc{f}{t} = \wfc{\sin}{20t}$ for $t \in [-1,1]$ with $n = 200$. Figure \ref{figure_rounding} compares usual Floater-Hormann interpolants, extended interpolants with $\tilde{\wvec{y}}$ computed in double precision and extended interpolants with {\it precise} $\tilde{\wvec{y}}$. By {\it precise} we mean that $\tilde{\wvec{y}}$ was computed using the MPFR library \cite{MPFR}, with floating point numbers with a mantissa of 320 bits, from $\wvec{y}$ computed with the same high precision.
\begin{figure}
\caption{Log10 of the error for $\wfc{f}{t} = \wfc{\sin}{20t}$, $t \in [-1,1]$ and $n = 200$.}
\label{figure_rounding}
\end{figure}
By {\it error} in our plots we mean the maximum difference between the numerically evaluated interpolant and the original function at $10^7$ equally spaced points in $[-1,1]$. The barycentric formula \wref{bary_fh} and \wref{bary_xfh} were evaluated in double precision ($\epsilon \approx 10^{-16}$), using straightforward C++ code. The $\wvec{y}$ and $\tilde{\wvec{y}}$ computed in multiple precision were rounded to double precision and the barycentric formula corresponding to them was also evaluated in double precision. In other words, the case {\it precise $\tilde{\wvec{y}}$} differs from the other cases only by the precision of the $\tilde{\wvec{y}}$, and not by the precision used to evaluate the barycentric formulae \wref{bary_fh} and \wref{bary_xfh}.
Figure \ref{figure_rounding} shows that, when the $\tilde{y}_i$ are evaluated in double precision, extended interpolants are not significantly more accurate than usual ones with $\delta= d$, and they become more unstable as $d = \tilde{d} = \tilde{n}$ grows. By contrast, extended interpolants with precise $\tilde{\wvec{y}}$ are remarkably accurate, even for large values of $d = \tilde{d} = \tilde{n}$. This suggests that the inaccuracy of $\tilde{\wvec{y}}$ is the cause of the numerical instability of extended interpolants for large $d = \tilde{d} = \tilde{n}$.
Figure \ref{figure_rounding_leb} considers the ratio ``error divided by Lebesgue constant.'' This ratio is relevant for the understanding of the backward stability of interpolation formulae. As we explain in \cite{Andre}, it is possible to implement the usual Floater-Hormann interpolants so that the backward error is of order $n \epsilon$. Backward errors of this order lead to forward errors of order $n \epsilon \Lambda_{\wvec{x},\delta} \wnorm{\wvec{y}}_\infty$,
as one can verify by looking at Figures \ref{figure_rounding_leb} and \ref{figure_rounding_semi_leb} (in this article we refer to the forward error simply as error.) Therefore, in this case by dividing the error by the Lebesgue constant we obtain an estimate of the backward error. Unfortunately, Figure \ref{figure_rounding_leb} shows that the relation between rounding errors and the Lebesgue constant for large values of $d = \tilde{d} = \tilde{n}$ for extended interpolant is more complex than the analogous relation for usual Floater-Hormann interpolants with $\delta = d$. As a result, the fact that extended interpolants have a smaller Lebesgue constant does not imply that they are more stable for large values of $\delta = d = \tilde{d} = \tilde{n}$ (see Figure 5.) In fact, in this scenario the effects of the large Lebesgue constants are quite different for extended and usual Floater-Hormann interpolants.
The combination of Figures \ref{figure_lebesgue_constants} and \ref{figure_rounding} leads to a surprising observation: according to Figure \ref{figure_lebesgue_constants}, the Lebesgue constant of the usual interpolants in our experiments is about $100$ times larger than the Lebesgue constant of the corresponding extended interpolants for large $\delta = d = \tilde{d} = \tilde{n}$, yet Figure \ref{figure_rounding} shows that the usual interpolants are much more accurate for such large $\delta$, $d$, $\tilde{n}$ and $\tilde{d}$.
\begin{figure}
\caption{Log10 of (forward error divided by the Lebesgue constant) for $\wfc{f}{t} = \wfc{\sin}{20t}$, $t \in [-1,1]$ and $n = 200$.}
\label{figure_rounding_leb}
\end{figure}
The data in Figures \ref{figure_lebesgue_constants} and \ref{figure_rounding} are combined in Figure \ref{figure_rounding_leb}, which highlights important points for the case $d = \tilde{n} = \tilde{d} \geq 15$. Note that this case is covered by the theory in \cite{BerrutKleinCAM} and \cite{Klein} and their experiments consider $0 \leq d \leq 50$. Moreover, extended interpolants are claimed to be better than usual ones for allowing larger values of $d$ and, in view of Figure \ref{figure_choosing}, the use of a larger $\tilde{d} \leq \tilde{n}$ is natural in this context. According to Figure \ref{figure_rounding_leb}, \begin{enumerate} \item For fixed $n$, the error incurred by usual Floater-Hormann interpolants is of order $n \epsilon \tilde{\Lambda}_{\wvec{x},\delta}{} \wnorm{f}_\infty$.\\[-0.15cm] \item For fixed $n$, the error incurred by extended interpolants grows faster than $n \epsilon \tilde{\Lambda}_{\wvec{x},d,\tilde{n},\tilde{d}}{} \wnorm{f}_\infty$ when $d = \tilde{d} = \tilde{n}$.\\[-0.15cm] \item The effects of the large Lebesgue constant are reduced when $\tilde{\wvec{y}}$ is precise. In this case, numerical errors occur mostly in the evaluation of the barycentric formula, and the argument following Equation (3.2){} in \cite{Klein} applies and explains the errors of order $\epsilon$ for the extended interpolants with precise $\tilde{\wvec{y}}$ in Figure \ref{figure_rounding}. \end{enumerate}
The data for the extended interpolants with double precision $\tilde{y}_i$ in Figure \ref{figure_rounding_leb} for $d \geq 30$ indicate the existence of another source of numerical instability for these interpolants, in addition to the large Lebesgue constants. This extra source of instability are the enormous entries of the matrices $a_{ij}$ and $b_{ij}$ in Lemma \ref{lem_clean}, which are defined explicitly in \wref{def_aij}--\wref{def_bij}. In fact, Figure \ref{figure_norm_aij} shows that the $a_{ij}$ and $b_{ij}$ grow exponentially with $d = \tilde{d} = \tilde{n}$.
\begin{figure}\label{figure_norm_aij}
\end{figure}
In view of the remarkable accuracy of the extended interpolants with precise $\tilde{\wvec{y}}$, it makes sense to consider the possibility of using $\tilde{\wvec{y}}$ evaluated in multiple precision from the double precision $y_i$ which are usually available. These $\tilde{\wvec{y}}$ are not as precise as the ones obtained from high precision $\wvec{y}$ using multiple precision arithmetic. Figure \ref{figure_roundingsemi} illustrates, however, that this strategy improves the accuracy of extended interpolants, at a relatively low cost when $d$, $\tilde{d}$ and $\tilde{n}$ are small compared to $n$ and we want to evaluate the interpolants for many values of $t$.
\begin{figure}
\caption{Log10 of the forward error for $\wfc{f}{t} = \wfc{\sin}{20t}$, $ t \in [-1,1]$ and $n = 200$.}
\label{figure_roundingsemi}
\end{figure}
In the case considered in this section, Figure \ref{figure_rounding_semi_leb} shows that $\tilde{\wvec{y}}$ evaluated using multiple precision, from double precision $\wvec{y}$, lead to overall numerical errors of order $n \epsilon \tilde{\Lambda}_{\wvec{x},d,\tilde{n},\tilde{d}}{}$, which are about 100 times smaller than the errors incurred by the usual Floater-Hormann interpolants in our experiments for large $\delta = d = \tilde{n} = \tilde{d}$.
\begin{figure}
\caption{Log10 of (forward error divided by the Lebesgue constant) for $\wfc{f}{t} = \wfc{\sin}{20t}$, $t \in [-1,1]$ and $n = 200$.}
\label{figure_rounding_semi_leb}
\end{figure}
\appendix \section{What can we prove about the stability of extended interpolants} \label{section_cancellation}
This appendix illustrates the difficulties in building a general and realistic stability theory for extended interpolants, a theory which would take into account the errors introduced by the current implementations of floating point arithmetic. We explain that the stability of extended interpolants is sensitive to the way we implement the extrapolation step, and that the accuracy of this step depends on the cancellation of the rounding errors. In fact, the errors incurred by extended interpolants can be enormous when we use the extrapolation formula proposed in \cite{BerrutKleinCAM} and \cite{Klein}, and compute $\tilde{\wvec{y}}$ according to the following procedure: \begin{itemize} \item[(a)] If $i$ is even, set the rounding mode upward and evaluate $\tilde{y}_i$ as in \wref{ytila}--\wref{ytilc}. \item[(b)] If $i$ is odd, set the rounding mode downward and evaluate $\tilde{y}_i$ as in \wref{ytila}--\wref{ytilc}. \end{itemize} In this scenario the overall effect of rounding errors can be much larger than what one would expect from the already large Lebesgue constants, as illustrated in Figures \ref{figure_rounding_up} and \ref{figure_rounding_up_leb}. In the plots corresponding to $\tilde{\wvec{y}}$ evaluated as in \wref{tilyi_clean_a}--\wref{tilyi_clean_b} in these figures,
$\tilde{\wvec{y}}$ was obtained by matrix multiplication, with $a_{ij}$ and $b_{ij}$ computed in multiple precision and then rounded to double precision i.e., with $a_{ij}$ and $b_{ij}$ as accurate as possible.
\begin{figure}
\caption{Extended Floater-Hormann. Log10 of the forward error for $\wfc{f}{t} = \wfc{\sin}{20t}$ for $t \in [-1,1]$ and $n = 200$. By ``$\tilde{y}$ by Taylor'' we mean $\tilde{y}$ computed by Taylor series as in \cite{BerrutKleinCAM} and \wref{ytila}--\wref{ytilc}, and by ``$\tilde{y}$ by matrix mult.'' we mean $\tilde{y}$ computed by matrix multiplication of $\wvec{y}$ by the matrices with entries $a_{ij}$ and $b_{ij}$, as in \wref{tilyi_clean_a}--\wref{tilyi_clean_b}.}
\label{figure_rounding_up}
\end{figure}
\begin{figure}
\caption{Extended Floater-Hormann. Log10 of (forward error divided by the Lebesgue constant) for $\wfc{f}{t} = \wfc{\sin}{20t}$, $t \in [-1,1]$ and $n = 200$.}
\label{figure_rounding_up_leb}
\end{figure}
We emphasize that the choices of rounding modes in the steps (a) and (b) above are not frivolous. Their purpose is to help us understand what can be proved about the numerical stability of extended interpolants, so that we do not try to prove something that cannot be proved. It is unlikely that $\tilde{\wvec{y}}$ will be evaluated as in the steps (a) and (b) when rounding to nearest. In this mode, there is a 50\% chance of rounding up in each flop and, under the questionable hypothesis of independence of the rounding errors, there would be a minuscule probability of $2^{-2\wlr{d+1} d}$ of having all the intermediate results in the evaluation of $\tilde{\wvec{y}}$ rounded up when rounding to nearest. Since the set of floating point numbers is finite, such a coincidence may be impossible. However, our experiments indicate that it is difficult to build a realistic theory on the effects of rounding errors on extended interpolants, because the rounding errors induced by our changes of rounding modes would be allowed by usual models of floating point arithmetic, with $\epsilon$ replaced by $2 \epsilon$. More precisely: when evaluating $\tilde{\wvec{y}}$, with our choices of rounding modes, we monitored the relative errors \[ \wabs{\frac{\wrounde{x + y} - \wlr{x + y}}{\wlr{x + y}}} \hspace{1cm} \wrm{and} \hspace{1cm} \wabs{\frac{\wrounde{x * y} - \wlr{x * y}}{\wlr{x * y}}} \] for each operation we performed, and found all of them to be smaller than $1.97 \epsilon$.
In other words, a stability theory for extended interpolants based on the usual models of floating point arithmetic would need to cover the changes of rounding modes in steps (a) and (b) above and, as a result, its predictions would be too pessimistic. Therefore, a realistic stability theory for extended interpolants will require additional hypothesis regarding the floating point arithmetic. By contrast, under the usual models of floating point arithmetic \cite{HIGHAM}, we already have realistic theories bounding the rounding errors in terms of $\epsilon$, $n$ and the Lebesgue constant for other barycentric interpolation schemes, as in \cite{Andre} \cite{HIGHAM_IMA} \cite{Masc} \cite{MascCam} \cite{MascCamB}.
\end{document} |
\begin{document}
\begin{abstract}
Let $M(\alpha)$ denote the Mahler measure of the algebraic number $\alpha$. In a recent paper, Dubickas and Smyth constructed a metric version of the Mahler measure
on the multiplicative group of algebraic numbers. Later, Fili and the author used similar techniques to study a non-Archimedean version. We show how to generalize
the above constructions in order to associate, to each point in $(0,\infty]$, a metric version $M_x$ of the Mahler measure, each having a triangle inequality
of a different strength. We are able to compute $M_x(\alpha)$ for sufficiently small $x$, identifying, in the process, a function $\bar M$ with certain minimality
properties. Further, we show that the map $x\mapsto M_x(\alpha)$ defines a continuous function on the positive real numbers. \end{abstract}
\title{A collection of metric Mahler measures}
\section{Introduction} \label{Intro}
Let $f$ be a polynomial with complex coefficients given by \begin{equation*}
f(z) = a\cdot\prod_{n=1}^N(z-\alpha_n). \end{equation*} We define the {\it (logarithmic) Mahler measure} $M$ of $f$ by \begin{equation*}
M(f) = \log|a|+\sum_{n=1}^N\log^+|\alpha_n|. \end{equation*} If $\alpha$ is a non-zero algebraic number, we define the Mahler measure of $\alpha$ by \begin{equation*}
M(\alpha) = M(\min_\mathbb Z(\alpha)). \end{equation*} In other words, $M(\alpha)$ is simply the Mahler measure of the minimal polynomial of $\alpha$ over $\mathbb Z$. It is well-known that \begin{equation} \label{MahlerInverses}
M(\alpha) = M(\alpha^{-1}) \end{equation} for all algebraic numbers $\alpha$.
It is a consequence of a theorem of Kronecker that $M(\alpha) = 0$ if and only if $\alpha$ is a root of unity. In a famous 1933 paper, D.H. Lehmer \cite{Lehmer} asked whether there exists a constant $c>0$ such that $M(\alpha) \geq c$ in all other cases. He could find no algebraic number with Mahler measure smaller than that of \begin{equation*}
\ell(x) = x^{10}+x^9-x^7-x^6-x^5-x^4-x^3+x+1, \end{equation*} which is approximately $0.16\ldots$. Although the best known general lower bound is \begin{equation*}
M(\alpha) \gg \left(\frac{\log\log\deg\alpha}{\log\deg\alpha}\right)^3, \end{equation*} due to Dobrowolski \cite{Dobrowolski}, uniform lower bounds haven been established in many special cases (see \cite{BDM, Schinzel, Smyth}, for instance). Furthermore, numerical evidence provided by Mossinghoff \cite{Moss, MossWeb} and Mossinghoff, Pinner and Vaaler \cite{MPV} suggests there does, in fact, exist such a constant $c$. This leads to the following conjecture, which we will now call Lehmer's conjecture.
\begin{conj}[Lehmer's conjecture]
There exists a real number $c > 0$ such that if $\alpha\in\alg^{\times}$ is not a root of unity then $M(\alpha) \geq c$. \end{conj}
In an effort to create a geometric structure on the multiplicative group of algebraic numbers $\alg^{\times}$, Dubickas and Smyth \cite{DubSmyth2} constructed a metric version of the Mahler measure. Let us briefly recall this construction. Write \begin{equation} \label{RestrictedProduct}
\mathcal X(\alg^{\times}) = \{(\alpha_1,\alpha_2,\ldots): \alpha_n = 1\ \mathrm{for\ all\ but\ finitely\ many}\ n\} \end{equation} to denote the restricted infinite direct product of $\alg^{\times}$. Let $\tau:\mathcal X(\alg^{\times}) \to \alg^{\times}$ be defined by \begin{equation*}
\tau(\alpha_1,\alpha_2,\cdots) = \prod_{n=1}^\infty \alpha_n \end{equation*} and note that $\tau$ is indeed a group homomorphism. The {\it metric Mahler measure} $M_1$ of $\alpha$ is given by \begin{equation*}
M_1(\alpha) = \inf\left\{ \sum_{n=1}^\infty M(\alpha_n): (\alpha_1,\alpha_2,\ldots)\in \tau^{-1}(\alpha)\right\}. \end{equation*} We note that the infimum in the definition of $M_1(\alpha)$ is taken over all ways of writing $\alpha$ as a product of elements in $\alg^{\times}$. As a result of this construction, the function $M_1$ satisfies that triangle inequality \begin{equation} \label{MahlerTriangle}
M_1(\alpha\beta) \leq M_1(\alpha) + M_1(\beta) \end{equation} for all $\alpha,\beta\in \alg^{\times}$. It can be shown that $M_1(\alpha) = 0$ if and only if $\alpha$ is a root of unity, and moreover, $M_1$ is well-defined on the quotient group $\mathcal G = \alg^{\times}/\mathrm{Tor}(\alg^{\times})$. Using \eqref{MahlerInverses} and \eqref{MahlerTriangle}, we find that the map $(\alpha,\beta)\mapsto M_1(\alpha\beta^{-1})$ is a metric on $\mathcal G$. It is noted in \cite{DubSmyth2} that this map yields the discrete topology if and only if Lehmer's conjecture is true.
Following the strategy of \cite{DubSmyth2}, Fili and the author \cite{FiliSamuels} examined a non-Archimedean version of the metric Mahler measure. That is, define the {\it ultrametric Mahler measure} $M_\infty$ of $\alpha$ by \begin{equation*}
M_\infty(\alpha) = \inf\left\{ \max_{n\geq 1} M(\alpha_n): (\alpha_1,\alpha_2,\ldots)\in\tau^{-1}(\alpha)\right\}, \end{equation*} replacing the sum in the definition of $M_1$ by a maximum. In this case, $M_\infty$ has the strong triangle inequality \begin{equation*}
M_1(\alpha\beta) \leq \max \{M_1(\alpha),M_1(\beta)\} \end{equation*} for all $\alpha,\beta\in \alg^{\times}$. Once again, we are able to verify that $M_\infty$ is well-defined on $\mathcal G$. Here, the map $(\alpha,\beta)\mapsto M_\infty(\alpha\beta^{-1})$ yields a non-Archimedean metric on $\mathcal G$ which induces the discrete topology if and only if Lehmer's conjecture is true.
In view of the definitions of $M_1$ and $M_\infty$, it is natural to define a collection of intermediate metric Mahler measures in the following way. If $x\in (0,\infty]$, we define $M_x:\mathcal X(\alg^{\times})\to [0,\infty)$ by \begin{equation*}
M_x(\alpha_1,\alpha_2,\ldots) = \left\{
\begin{array}{ll}
\displaystyle \left( \sum_{n=1}^{\infty} M(\alpha_n)^x\right)^{1/x} & \mathrm{if}\ x\in (0,\infty) \\
& \\
\displaystyle \max_{n\geq 1}\{M(\alpha_n)\} & \mathrm{if}\ x = \infty.
\end{array}
\right. \end{equation*} In the case that $x \geq 1$, we see that $M_x(\alpha_1,\alpha_2,\ldots)$ is the $L^x$ norm on the vector $(M(\alpha_1),M(\alpha_2),\ldots)$. Then we define the {\it $x$-metric Mahler measure} by \begin{equation} \label{xMetricMahlerDef}
M_x(\alpha) = \inf\{M_x(\bar\alpha): \bar\alpha\in\tau^{-1}(\alpha)\} \end{equation} and note that this definition generalizes those of $M_1$ and $M_\infty$. Indeed, the $1$- and $\infty$-metric Mahler measures are simply the metric and ultrametric Mahler measures, respectively.
In \cite{DubSmyth2}, Dubickas and Smyth showed that if Lehmer's conjecture is true, then the infimum in the definition of $M_1(\alpha)$ must always be achieved. The author \cite{Samuels} was able to verify that the infima in $M_1(\alpha)$ and $M_\infty(\alpha)$ are achieved even without the assumption of Lehmer's conjecture. Moreover, this infimum must always be attained in a relatively simple subgroup of $\alg^{\times}$. In particular, if $K$ is a number field we write \begin{equation*}
\mathrm{Rad}(K) = \left\{\alpha\in\alg^{\times}:\alpha^r\in K\mathrm{\ for\ some}\ r\in\mathbb N\right\}. \end{equation*} For any algebraic number $\alpha$, let $K_\alpha$ denote the Galois closure of $\mathbb Q(\alpha)$ over $\mathbb Q$. We showed in \cite{Samuels} that the infimum in both $M_1(\alpha)$ and $M_\infty(\alpha)$ is always attained by some \begin{equation*}
\bar\alpha \in \tau^{-1}(\alpha) \cap \mathcal X(\mathrm{Rad}(K_\alpha)). \end{equation*} where $\mathcal X(\mathrm{Rad}(K_\alpha))$ is defined similarly to $\mathcal X(\alg^{\times})$ in \eqref{RestrictedProduct}. Not surprisingly, the same argument can be used to establish the analog for all values of $x$.
\begin{thm} \label{Achieved}
Suppose $\alpha$ is a non-zero algebraic number and $x\in (0,\infty]$. Then there exists a point $\bar\alpha\in \tau^{-1}(\alpha) \cap \mathcal X(\mathrm{Rad}(K_\alpha))$ such that
$M_x(\alpha) = M_x(\bar\alpha)$. \end{thm}
We now turn our attention momentarily to the computation of some values of $M_x(\alpha)$. First define \begin{equation*}
C(\alpha) = \inf\{ M(\gamma): \gamma\in K_\alpha\setminus \mathrm{Tor}(\alg^{\times})\} \end{equation*} and note that by Northcott's Theorem \cite{Northcott}, the infimum on the right hand side of this definition is always achieved. In paricular, this means that $C(\alpha) > 0$.
The author \cite{Samuels2} gave a strategy for reducing the computation of $M_\infty(\alpha)$ to a finite set. The method uses the {\it modified Mahler measure} \begin{equation} \label{MBarDef}
\bar M(\alpha) = \inf \{M(\zeta\alpha):\zeta\in \mathrm{Tor}(\alg^{\times})\} \end{equation} and gives the value of $M_\infty$ in terms of $\bar M$. Although $\bar M$ requires taking an infimum over an infinite set, it is often very reasonable to calculate. Indeed, the infimum on the right hand side of \eqref{MBarDef} is always attained at a root of unity $\zeta$ that makes $\deg(\zeta\alpha)$ as small as possible. This function $\bar M$ arises again when computing $M_x(\alpha)$ for small $x$ in a more straightforward way than in \cite{Samuels2}.
\begin{thm} \label{SmallP}
If $\alpha$ is a non-zero algebraic number and $x$ is a positive real number satisfying
\begin{equation} \label{AlwaysX}
x\cdot (\log \bar M(\alpha) - \log C(\alpha)) \leq \log 2
\end{equation}
then $M_x(\alpha) = \bar M(\alpha)$. \end{thm}
As we will discuss in detail in section \ref{AbelianHeights}, the construction given by \eqref{xMetricMahlerDef} is not unique to the Mahler measure. Suppose $\phi:\alg^{\times}\to [0,\infty)$ satisfies \begin{equation} \label{BasicHeightProps}
\phi(1) = 0\quad\mathrm{and}\quad \phi(\alpha) = \phi(\alpha^{-1})\ \mathrm{for\ all}\ \alpha\in\alg^{\times}, \end{equation} and write \begin{equation*}
\phi_x(\alpha_1,\alpha_2,\ldots) = \left\{
\begin{array}{ll}
\displaystyle \left( \sum_{n=1}^{\infty} \phi(\alpha_n)^x\right)^{1/x} & \mathrm{if}\ x\in (0,\infty) \\
& \\
\displaystyle \max_{n\geq 1}\{\phi(\alpha_n)\} & \mathrm{if}\ x = \infty.
\end{array}
\right. \end{equation*} Generalizing the metric Mahler measure, let $\phi_x$ be defined by \begin{equation} \label{xMetricSwitchDef}
\phi_x(\alpha) = \inf\{\phi_x(\bar\alpha): \bar\alpha\in\tau^{-1}(\alpha)\}. \end{equation} We now write $\mathcal S(M)$ to denote the set of all functions $\phi$ satisfying \eqref{BasicHeightProps} such that $\phi_x(\alpha) = M_x(\alpha)$ for all $\alpha\in\alg^{\times}$ and $x\in (0,\infty]$. We are able to show that $\bar M$ belongs to $\mathcal S(M)$. Moreover, it is a consequence of Theorem \ref{SmallP} that $\bar M$ is the minimal element of $\mathcal S(M)$.
\begin{cor} \label{MBarMinimal}
We have that $\bar M\in \mathcal S(M)$. Moreover, if $\psi\in\mathcal S(M)$ then $\psi(\alpha) \geq \bar M(\alpha)$ for all $\alpha\in\alg^{\times}$. \end{cor}
We now ask if the map $x\mapsto M_x(\alpha)$ is continuous on $\mathbb R_{>0}$ for every algebraic number $\alpha$. We recall that Theorem \ref{Achieved} asserts that, for each $x$, there exists a point $\bar\alpha\in \tau^{-1}(\alpha)$ that attains the infimum in the definition of $M_x(\alpha)$. If the infimum is achieved at the same point $(\alpha_1,\alpha_2,\ldots)$ for all real $x$, then we have that \begin{equation*}
M_x(\alpha) = \left( \sum_{n=1}^N M(\alpha_n)^x\right)^{1/x} \end{equation*} which clearly defines a continuous function. Unfortunately, using the example of $M_x(p^2)$ for a rational prime $p$, we see that this is not the case.
\begin{thm} \label{NotUniform}
Let $p$ be a rational prime and assume that $(\alpha_1,\alpha_2,\ldots)\in \tau^{-1}(p^2)$ with $M_x(p^2) = M_x(\alpha_1,\alpha_2,\cdots)$.
\begin{enumerate}[(i)]
\item\label{P2Small} If $x\cdot (\log\log (p^2) - \log\log 2) < \log 2$ then precisely one point $\alpha_n$ differs from a root of unity.
\item\label{P2Large} If $x >1$ then at least two points $\alpha_n$ differ from a root of unity.
\end{enumerate} \end{thm}
Although the infimum in $M_x(\alpha)$ is not achieved at the same point for all $x$, we are able to prove that $x\mapsto M_x(\alpha)$ is continuous for all $\alpha$.
\begin{thm} \label{Continuous}
If $\alpha$ is a non-zero algebraic number then the map $x \mapsto M_x(\alpha)$ is continuous on the positive real numbers. \end{thm}
It is worth noting that continuity appears to be somewhat special to the Mahler measure. That is, we cannot expect an arbitrary function $\phi$ satisfying \eqref{BasicHeightProps} to be such that $x\mapsto \phi_x(\alpha)$ is continuous. Even making a slight modification to the Mahler measure causes continuity to fail. For example, define the {\it Weil height} of $\alpha\in \alg^{\times}$ by \begin{equation*}
h(\alpha) = \frac{M(\alpha)}{\deg\alpha} \end{equation*} and note that, in view of our remarks about the Mahler measure, $h(\alpha) = 0$ if and only if $\alpha$ is a root of unity. In fact, it is well-known that \begin{equation} \label{WeilHeightDefined}
h(\alpha) = h(\zeta\alpha) \end{equation} for all roots of unity $\zeta$. Moreover, we have that $h(\alpha) = h(\alpha^{-1})$ for all $\alpha\in\alg^{\times}$ so that $h$ satisfies \eqref{BasicHeightProps}. Unlike the Mahler measure, we know how to compute $h_x(\alpha)$ for every $x$ and $\alpha$.
\begin{thm} \label{WeilHeightComp}
If $\alpha$ is a non-zero algebraic number then
\begin{equation*}
h_x(\alpha) = \left\{ \begin{array}{ll}
h(\alpha) & \mathrm{if}\ x \leq 1 \\
0 & \mathrm{if}\ x > 1.
\end{array}
\right. \end{equation*} \end{thm}
As we have noted, Theorem \ref{WeilHeightComp} does indeed show that $x\mapsto h_x(\alpha)$ is possibly discontinuous. More specifically, it is continuous if and only if $\alpha$ is a root of unity.
\section{Heights on Abelian groups} \label{AbelianHeights}
In this section, we generalize our $x$-metric Mahler measure construction to a very broad class of functions on an abelian group $G$ by exploring definition \eqref{xMetricSwitchDef} in more detail. We are able to establish some basic properties in this situation that we can use to prove our main results.
Let $G$ be a multiplicatively written abelian group. We say that $\phi:G \to [0,\infty)$ is a {\it (logarithmic) height} on $G$ if \begin{enumerate}[(i)]
\item $\phi(1) = 0$, and
\item $\phi(\alpha) = \phi(\alpha^{-1})$ for all $\alpha\in G$. \end{enumerate} If $\psi$ is another height on $G$, we follow the conventional notation that \begin{equation*}
\phi = \psi \quad \mathrm{or} \quad \phi \leq \psi \end{equation*} when $\phi(\alpha) = \psi(\alpha)$ or $\phi(\alpha) \leq \psi(\alpha)$ for all $\alpha\in G$, respectively. We write \begin{equation*}
Z(\phi) = \{ \alpha\in G: \phi(\alpha) = 0\} \end{equation*} to denote the {\it zero set} of $\phi$.
If $x$ is a positive real number then we say that $\phi$ has the {\it $x$-triangle inequality} if \begin{equation*}
\phi(\alpha\beta) \leq \left (\phi(\alpha)^x + \phi(\beta)^x\right )^{1/x} \end{equation*} for all $\alpha,\beta\in G$. We say that $\phi$ has the {\it $\infty$-triangle inequaltiy} if \begin{equation*}
\phi(\alpha\beta) \leq \max\{\phi(\alpha),\phi(\beta)\} \end{equation*} for all $\alpha,\beta\in G$. For appropriate $x$, we say that these functions are {\it $x$-metric heights}. We observe that the $1$-triangle inequality is simply the classical triangle inequality while the $\infty$-triangle inequality is the strong triangle inequality. We also obtain the following ordering of the $x$-triangle inequalities.
\begin{lem} \label{Intermediates}
Suppose that $G$ is an abelian group and that $x,y\in (0,\infty]$ with $x\geq y$. If $\phi$ is an $x$-metric height on $G$ then
$\phi$ is also a $y$-metric height on $G$. \end{lem} \begin{proof}
If $a,b$ and $q$ are real numbers with $a,b \geq 0$ and $q\geq 1$, then it is easily verified that
\begin{equation} \label{MVTapp}
a^q+b^q \leq (a+b)^q.
\end{equation}
Let us now assume that $\phi$ has the $x$-triangle inequality and that $\alpha,\beta\in G$. If $x=y =\infty$ then the lemma is completely trivial.
If $x = \infty$ and $y <\infty$ then we have that
\begin{equation*}
\phi(\alpha\beta) \leq \max\{\phi(\alpha),\phi(\beta)\} = \max\{\phi(\alpha)^y,\phi(\beta)^y\}^{1/y} \leq (\phi(\alpha)^y + \phi(\beta)^y)^{1/y}
\end{equation*}
so that the result follows easily as well. Hence, we assume now that $\infty > x\geq y$. In this situation, we have that $x/y \geq 1$.
Therefore, by \eqref{MVTapp} we have that
\begin{equation*}
(\phi(\alpha)^y + \phi(\beta)^y)^{x/y} \geq \phi(\alpha)^x + \phi(\beta)^x
\end{equation*}
and it follows that
\begin{equation*}
(\phi(\alpha)^y + \phi(\beta)^y)^{1/y} \geq (\phi(\alpha)^x + \phi(\beta)^x)^{1/x}.
\end{equation*}
Hence, we have that $\phi(\alpha\beta) \leq (\phi(\alpha)^y + \phi(\beta)^y)^{1/y}$ so that $\phi$ has the $y$-triangle inequaity. \end{proof}
We now observe that each $x$-metric height is well-defined on the quotient group $G/Z(\phi)$. In the case that $x\geq 1$, the map $(\alpha,\beta) \mapsto \phi(\alpha\beta^{-1})$ defines a metric on $G/Z(\phi)$.
\begin{thm} \label{MetricProperties}
If $\phi:G\to [0,\infty)$ is an $x$-metric height for some $x\in (0,\infty]$ then
\begin{enumerate}[(i)]
\item\label{Subgroup} $Z(\phi)$ is a subgroup of $G$.
\item\label{WellDefined} $\phi(\zeta \alpha) = \phi(\alpha)$ for all $\alpha\in G$ and $\zeta\in Z(\phi)$. That is, $\phi$ is well-defined on the quotient $G/Z(\phi)$.
\item\label{FancyMetric} If $x\geq 1$, then the map $(\alpha,\beta)\mapsto \phi(\alpha\beta^{-1})$ defines a metric on $G/Z(\phi)$.
\end{enumerate} \end{thm} \begin{proof}
We first establish \eqref{Subgroup}. Obviously, we have that $1\in Z(G)$ by definition of height. Further, if $\phi(\alpha) = 0$
then again by definition of height we know that $\phi(\alpha^{-1}) = 0$.
If $\alpha,\beta\in Z(G)$ then using the $x$ triangle inequality we obtain
\begin{equation*}
\phi(\alpha\beta) \leq (\phi(\alpha)^x + \phi(\beta)^x)^{1/x} = 0.
\end{equation*}
Therefore, $\alpha\beta\in Z(G)$ so that $Z(G)$ forms a subgroup.
To prove \eqref{WellDefined}, we see that the $x$-triangle inequality yields
\begin{align*}
\phi(\alpha) & = \phi(\zeta^{-1}\zeta\alpha) \\
& \leq (\phi(\zeta^{-1})^x + \phi(\zeta\alpha)^x)^{1/x} \\
& = \phi(\zeta\alpha) \\
& \leq (\phi(\zeta)^x + \phi(\alpha)^x)^{1/x} \\
& = \phi(\alpha)
\end{align*}
implying that $\phi(\alpha) = \phi(\zeta\alpha)$.
Finally, if $x\geq 1$ then Lemma \ref{Intermediates} implies that $\phi$ has the triangle inequality. It then follows immediately that
the map $(\alpha,\beta)\mapsto \phi(\alpha\beta^{-1})$ is a metric on $G/Z(\phi)$. \end{proof}
We are careful to note that if $x<1$ then the map $(\alpha,\beta) \mapsto \phi(\alpha\beta^{-1})$ does not, in general, form a metric on $G/Z(\phi)$. In this case, the $x$-triangle inequality is indeed weaker than the triangle inequality, so we cannot expect the above map to form a metric except in trivial cases.
We now follow the method of Dubickas and Smyth for creating a metric from the Mahler measure. Write \begin{equation*}
\mathcal X(G) = \{(\alpha_1,\alpha_2,\ldots): \alpha_n = 1\ \mathrm{for\ almost\ every}\ n\} \end{equation*} and, as before, let $\tau:\mathcal X(G) \to G$ be defined by \begin{equation*}
\tau(\alpha_1,\alpha_2,\cdots) = \prod_{n=1}^\infty \alpha_n \end{equation*} so that $\tau$ is a group homomorphism. For each point $x\in (0,\infty]$ we define the map $\phi_x:\mathcal X(G) \to [0,\infty)$ by \begin{equation*}
\phi_x(\alpha_1,\alpha_2,\ldots) = \left\{
\begin{array}{ll}
\displaystyle \left( \sum_{n=1}^{\infty} \phi(\alpha_n)^x\right)^{1/x} & \mathrm{if}\ x\in (0,\infty) \\
& \\
\displaystyle \max_{n\geq 1}\{\phi(\alpha_n)\} & \mathrm{if}\ x = \infty.
\end{array}
\right. \end{equation*} Then we define the {\it $x$-metric version} of $\phi_x$ of $\phi$ by \begin{equation*}
\phi_x(\alpha) = \inf\{\phi_x(\bar\alpha): \bar\alpha\in\tau^{-1}(\alpha)\}. \end{equation*} It is immediately clear that if $\psi$ is another height on $G$ with $\phi \geq \psi$, then $\phi_x \geq \psi_x$ for all $x$. Among other things, we see that $\phi_x$ is indeed an $x$-metric height on $G$.
\begin{thm} \label{MetricConstruction}
If $\phi:G\to [0,\infty)$ is a height on $G$ and $x\in (0,\infty]$ then
\begin{enumerate}[(i)]
\item\label{MetricHeightConversion} $\phi_x$ is an $x$-metric height on $G$ with $\phi_x\leq\phi$.
\item\label{BestMetricHeight} If $\psi$ is an $x$-metric height with $\psi\leq\phi$ then
$\psi\leq \phi_x$.
\item\label{NoChangeMetric} $\phi = \phi_x$ if and only if $\phi$ is an $x$-metric height. In particular, $(\phi_x)_x = \phi_x$.
\item\label{Comparisons} If $y\in (0,x]$ then $\phi_y \geq \phi_x$.
\end{enumerate} \end{thm} \begin{proof}
For the proofs of \eqref{MetricHeightConversion}-\eqref{NoChangeMetric}, we will assume that $x < \infty$. The proofs for the
case $x = \infty$ are quite similar to the proofs for other cases so we will not include them here. See \cite{FiliSamuels} for detailed proofs when $x=\infty$.
To prove \eqref{MetricHeightConversion}, let $\alpha,\beta\in G$. We observe that if $(\alpha_1,\alpha_2,\ldots)\in \tau^{-1}(\alpha)$ and
$(\beta_1,\beta_2,\ldots)\in \tau^{-1}(\beta)$ then it is obvious that
\begin{equation*}
\alpha\beta = \left(\prod_{n=1}^\infty \alpha_n\right)\left(\prod_{n=1}^\infty \beta_n\right).
\end{equation*}
We may also write
\begin{equation*}
\alpha\beta = \prod_{n=1}^\infty \alpha_n\beta_n
\end{equation*}
implying that $\tau(\alpha_1,\beta_1,\alpha_2,\beta_2,\ldots) = \alpha\beta$. In other words, we have that
\begin{equation} \label{ProductInInverseImage}
(\alpha_1,\beta_1,\alpha_2,\beta_2,\ldots) \in \tau^{-1}(\alpha\beta).
\end{equation}
This yields that
\begin{align} \label{InfIncrease}
\phi_x(\alpha\beta)^x & = \inf \{\phi_x(\gamma_1,\gamma_2,\ldots)^x: (\gamma_1,\gamma_2,\ldots)\in \tau^{-1}(\alpha\beta) \} \nonumber \\
& = \inf\{\phi_x(\alpha_1,\beta_1,\alpha_2,\beta_2,\ldots)^x: \alpha_n,\beta_n\in G,\ (\alpha_1,\beta_1,\ldots)\in \tau^{-1}(\alpha\beta)\} \nonumber \\
& \leq \inf\{\phi_x(\alpha_1,\beta_1,\alpha_2,\beta_2,\ldots)^x: (\alpha_1,\ldots)\in \tau^{-1}(\alpha),\ (\beta_1,\ldots)\in \tau^{-1}(\beta) \}.
\end{align}
We note that
\begin{align*}
\phi_x(\alpha_1,\beta_1,\alpha_2,\beta_2,\ldots)^x & = \sum_{n=1}^\infty \left (\phi(\alpha_n)^x + \phi(\beta_n)^x\right) \\
& = \sum_{n=1}^\infty \phi(\alpha_n)^x + \sum_{n=1}^\infty \phi(\beta_n)^x \\
& = \phi_x(\alpha_1,\ldots)^x + \phi_x(\beta_1,\ldots)^x.
\end{align*}
Then using \eqref{InfIncrease} we find that
\begin{align*}
\phi(\alpha\beta)^x & \leq \inf\{\phi_x(\alpha_1,\ldots)^x + \phi_x(\beta_1,\ldots)^x: (\alpha_1,\ldots)\in \tau^{-1}(\alpha),\ (\beta_1,\ldots)\in \tau^{-1}(\beta) \} \\
& = \inf\{\phi_x(\alpha_1,\ldots)^x: (\alpha_1,\ldots)\in \tau^{-1}(\alpha)\} \\
& \qquad + \inf\{\phi_x(\beta_1,\ldots)^x: (\beta_1,\ldots)\in \tau^{-1}(\beta)\} \\
& = \phi_x(\alpha)^x + \phi_x(\beta)^x
\end{align*}
and it follows that
\begin{equation*}
\phi_x(\alpha\beta) \leq (\phi_x(\alpha)^x + \phi_x(\beta)^x)^{1/x}.
\end{equation*}
To complete the proof of \eqref{MetricHeightConversion}, we observe that $(\alpha,1,1,\ldots) \in \tau^{-1}(\alpha)$ so
that $\phi_x(\alpha) \leq \phi(\alpha)$ for all $\alpha\in G$.
To prove \eqref{BestMetricHeight}, we note that
\begin{align*}
\phi_x(\alpha) & = \inf\left\{ \left(\sum_{n=1}^N\phi(\alpha_n)^x\right)^{1/x}:(\alpha_1,\alpha_2,\ldots)\in \tau^{-1}(\alpha)\right\} \\
& \geq \inf\left\{ \left(\sum_{n=1}^N\psi(\alpha_n)^x\right)^{1/x}:(\alpha_1,\alpha_2,\ldots)\in \tau^{-1}(\alpha)\right\} \\
& \geq \psi(\alpha)
\end{align*}
where the last inequality follows from the fact that $\psi$ has the $x$-triangle inequality.
To prove \eqref{NoChangeMetric}, we first observe that if $\phi = \phi_x$ then clearly $\phi$ is an $x$-metric height. If $\phi$ is already a metric height,
then by \eqref{BestMetricHeight}, we obtain that $\phi\leq \phi_x$. But we always have $\phi_x\leq \phi$ so the result follows. Of course, $\phi_x$ is an $x$-metric height
so this yields immediately $\phi_x = (\phi_x)_x$.
To establish \eqref{Comparisons}, we see that
\begin{align*}
\phi_y(\alpha) & = \inf\left\{ \left(\sum_{n=1}^N\phi(\alpha_n)^y\right)^{1/y}:(\alpha_1,\alpha_2,\ldots)\in \tau^{-1}(\alpha)\right\} \\
& = \inf\left\{ \left(\sum_{n=1}^N\phi(\alpha_n)^y\right)^{\frac{x}{y}\cdot\frac{1}{x}}:(\alpha_1,\alpha_2,\ldots)\in \tau^{-1}(\alpha)\right\}.
\end{align*}
But we have that $x\geq y$ so that $x/y \geq 1$. Therefore, by Lemma \ref{MVTapp} we have that
\begin{equation*}
\left(\sum_{n=1}^N\phi(\alpha_n)^y\right)^{x/y} \geq \sum_{n=1}^N\phi(\alpha_n)^x
\end{equation*}
which yields $\phi_y(\alpha) \geq \phi_x(\alpha)$. \end{proof}
For a given height $\phi$ on $G$, let $\mathcal S(\phi)$ denote the set of all heights $\psi$ on $G$ such that $\psi_x = \phi_x$ for all $x\in (0,\infty]$. Further, define the height $\phi_0$ by \begin{equation} \label{OptimalFunction}
\phi_0(\alpha) = \lim_{x\to 0^+} \phi_x(\alpha). \end{equation} By \eqref{MetricHeightConversion} of Theorem \ref{MetricConstruction}, we know that $\phi_x \leq \phi$ for all $x$. Moreover, \eqref{Comparisons} of the same theorem states that $x\mapsto \phi_x(\alpha)$ is non-increasing. This means that the limit on the right hand side of \eqref{OptimalFunction} does indeed exist and \begin{equation} \label{BoundForOptimal}
\phi_0 \geq \phi_x \end{equation} for all $x\in (0,\infty]$. We now observe that $\phi_0$ is the minimal element of $\mathcal S(\phi)$.
\begin{thm} \label{OptimalMinimal}
If $\phi$ is a height on $G$ then $\phi_0\in\mathcal S(\phi)$. Moreover, if $\psi\in \mathcal S(\phi)$ then $\psi \geq \phi_0$. \end{thm} \begin{proof}
As we have noted, $\phi_0\geq \phi_x$ for all $x$. Hence, we obtain immediately that $(\phi_0)_x \geq (\phi_x)_x = \phi_x$. On the other hand,
we know that $\phi_x \leq \phi$ so that
\begin{equation*}
\phi_0(\alpha) = \lim_{x\to 0^+} \phi_x(\alpha) \leq \phi(\alpha)
\end{equation*}
for all $\alpha\in G$. In other words, we have that $\phi_0 \leq \phi$ so that $(\phi_0)_x \leq \phi_x$ establishing the first statement of the theorem.
To prove the second statement, assume that $\psi\in \mathcal S(\phi)$ so that $\phi_x = \psi_x$ for all $x$. Hence we have that
\begin{equation*}
\phi_0(\alpha) = \lim_{x\to 0^+}\phi_x(\alpha) = \lim_{x\to 0^+}\psi_x(\alpha) \leq \psi(\alpha)
\end{equation*}
for all $\alpha\in G$ verifying the theorem. \end{proof}
We now define the {\it modified version} of $\phi$ by \begin{equation*}
\bar\phi(\alpha) = \inf \{\phi(\zeta\alpha): \zeta\in Z(\phi)\}. \end{equation*} In the case of the Mahler measure, we have stated in the introduction that $\bar\phi = \phi_0$. However, in the general case, we can conclude only that $\bar\phi$ belongs to $\mathcal S(\phi)$.
\begin{thm} \label{BarPhiMetrics}
If $\phi$ is a height on $G$ then $\bar\phi\in\mathcal S(\phi)$. \end{thm} \begin{proof}
We must show that $\bar\phi_x = \phi_x$ for all $x\in (0,\infty]$. Since $1\in Z(\phi)$, we have immediately that $\bar\phi \leq \phi$, which means that
\begin{equation*}
\bar\phi_x \leq \phi_x.
\end{equation*}
Now for any $\alpha\in G$, we have that
\begin{equation*}
\phi_x(\alpha) \leq \inf \{(\phi(\zeta^{-1})^x + \phi(\zeta\alpha)^x)^{1/x}:\zeta\in Z(\phi)\} = \inf \{\phi(\zeta\alpha):\zeta\in Z(\phi)\} = \bar\phi(\alpha)
\end{equation*}
implying that $\phi_x \leq \bar\phi$. Then taking $x$-metric versions and using \eqref{NoChangeMetric} of Theorem \ref{MetricConstruction} we find that
\begin{equation*}
\phi_x = (\phi_x)_x \leq \bar\phi_x
\end{equation*}
completing the proof. \end{proof}
We may now ask what we can say about the map $x\mapsto \phi_x(\alpha)$ for fixed $\phi$ and $\alpha$. As we have noted, this map is non-increasing for all $\alpha$. Since $\phi_x(\alpha)$ is bounded from above and below by constants not depending on $x$, both left and right hand limits exist at every point. Moreover, we always have \begin{equation*}
\lim_{x\to \bar x^-}\phi_x(\alpha) \geq \phi_{\bar x}(\alpha) \geq \lim_{x\to \bar x^+}\phi_x(\alpha) \end{equation*} when $\bar x >0$. We say that a map $f:\mathbb R\to\mathbb R$ is {\it left} or {\it right semi-continuous} at a point $\bar x\in \mathbb R$ if \begin{equation*}
\lim_{x\to \bar x^-}f(x) = f(\bar x)\quad\mathrm{or}\quad \lim_{x\to \bar x^+}f(x) = f(\bar x), \end{equation*} respectively. Indeed, $f$ is continuous at $\bar x$ if and only if $f$ is both left and right semi-continuous at $\bar x$. Although it is a consequence of Theorem \ref{WeilHeightComp} that $x\mapsto \phi_x(\alpha)$ is not continuous in general, we can prove the following partial result.
\begin{thm} \label{LeftSemiContinuous}
If $\phi$ is a height on $G$ and $\alpha\in G$, then the map $x \mapsto \phi_x(\alpha)$ is left semi-continous on the positive real numbers. \end{thm} \begin{proof}
We already know that $\lim_{x\to \bar x^-}\phi_x(\alpha) \geq \phi_{\bar x}(\alpha)$ so we assume that $$\lim_{x\to \bar x^-}\phi_x(\alpha) > \phi_{\bar x}(\alpha).$$
Therefore, there exists $\varepsilon >0$ such that
\begin{equation} \label{EpsilonSqueeze}
\lim_{x\to \bar x^-}\phi_x(\alpha) > \phi_{\bar x}(\alpha) + \varepsilon.
\end{equation}
By definition of $\phi_{\bar x}$, we may choose points $\alpha_1,\ldots,\alpha_N \in G$ such that $\alpha = \alpha_1\cdots\alpha_N$ and
\begin{equation*} \label{CloseEnough}
\phi_{\bar x}(\alpha) + \varepsilon \geq \left( \sum_{n=1}^N \phi(\alpha_n)^{\bar x}\right)^{1/\bar x},
\end{equation*}
and define the function $f_\varepsilon$ by
\begin{equation*}
f_\varepsilon(x) = \left( \sum_{n=1}^N \phi(\alpha_n)^{x}\right)^{1/x}.
\end{equation*}
This yields
\begin{equation} \label{FApprox}
f_\varepsilon(\bar x) \leq \phi_{\bar x}(\alpha) + \varepsilon\quad\mathrm{and}\quad f_\varepsilon(x) \geq \phi_x(\alpha)\ \mathrm{for\ all}\ x.
\end{equation}
Also, since $f_\varepsilon$ is continuous, we have that
\begin{equation} \label{FCont}
f_\varepsilon(\bar x) = \lim_{x\to\bar x^-} f_\varepsilon(x).
\end{equation}
Combining \eqref{EpsilonSqueeze}, \eqref{FApprox} and \eqref{FCont} we obtain that
\begin{equation*}
f_\varepsilon(\bar x) = \lim_{x\to\bar x^-} f_\varepsilon(x) \geq \lim_{x\to\bar x^-} \phi_x(\alpha) > \phi_{\bar x}(\alpha) + \varepsilon \geq f_\varepsilon(\bar x)
\end{equation*}
which is a contradiction. \end{proof}
\section{The Inifimum in $M_x(\alpha)$} \label{AchievedProofs}
Our proof of Theorem \ref{Achieved} will require the use of two results from \cite{Samuels}. The first of these is Theorem 2.1 of \cite{Samuels}, which shows that for any point $\bar\alpha\in \tau^{-1}(\alpha)$, there exists another point $\bar\beta\in\tau^{-1}(\alpha)\cup\mathcal X(\mathrm{Rad}(K_\alpha))$ which has pointwise smaller Mahler measures. We state the Theorem using the notation of \cite{Samuels}.
\begin{thm} \label{Reduction}
If $\alpha,\alpha_1,\ldots,\alpha_N$ are non-zero algebraic numbers with $\alpha = \alpha_1\cdots\alpha_N$ then
there exists a root of unity $\zeta$ and algebraic numbers $\beta_1,\ldots,\beta_N$ satifying
\begin{enumerate}[(i)]
\item $\alpha = \zeta\beta_1\cdots\beta_N$,
\item $\beta_n\in \mathrm{Rad}(K_\alpha)$ for all $n$,
\item $M(\beta_n) \leq M(\alpha_n)$ for all $n$.
\end{enumerate} \end{thm}
In view of Theorem \ref{Reduction}, for each $x$, we need only consider only points $\bar\alpha\in\tau^{-1}(\alpha)\cup\mathcal X(\mathrm{Rad}(K_\alpha))$ in the definition of $M_x(\alpha)$. In other words, in the case of $x <\infty$, the definition of $M_x(\alpha)$ may be rewritten \begin{equation} \label{xMetricAltDef}
M_x(\alpha) = \inf\left\{\left(\sum_{n=1}^\infty M(\alpha_n)^x\right)^{1/x}: (\alpha_1,\alpha_2,\ldots) \in \tau^{-1}(\alpha)\cup\mathcal X(\mathrm{Rad}(K_\alpha))\right\}. \end{equation} Similar remarks apply in the case that $x = \infty$. Therefore, it will be useful to have some control of the Mahler measures in the subgroup $\mathrm{Rad}(K_\alpha)$. For this purpose, we borrow Lemma 3.1 of \cite{Samuels}.
\begin{lem} \label{HeightInK}
Let $K$ be a Galois extension of $\mathbb Q$. If $\gamma\in\mathrm{Rad}(K)$ then there exists a root of unity $\zeta$ and $L,S\in\mathbb N$
such that $\zeta\gamma^L\in K$ and
\begin{equation*}
M(\gamma) = M(\zeta\gamma^L)^S.
\end{equation*}
In particular, the set
\begin{equation*}
\{M(\gamma):\gamma\in\mathrm{Rad}(K),\ M(\gamma) \leq B\}
\end{equation*}
is finite for every $B \geq 0$. \end{lem}
It is an easy consequence of Lemma \ref{HeightInK} that $M(\gamma)$ is bounded below by the Mahler measure of an element in $K$. Indeed, we have that \begin{equation*}
M(\gamma) = M(\zeta\gamma^L)^S \geq M(\zeta\gamma^L) \end{equation*} and $\zeta\gamma^L\in K$. In particular, we recall that $C(\alpha)$ denotes the minimum Mahler measure in the field $K_\alpha$. We now see easily that \begin{equation} \label{RadBound}
M(\gamma) \geq C(\alpha) \end{equation} for all $\gamma\in \mathrm{Rad}(K_\alpha)\setminus\mathrm{Tor}(\alg^{\times})$. We are now prepared to prove Theorem \ref{Achieved}.
\begin{proof}[Proof of Theorem \ref{Achieved}]
By the results of \cite{Samuels}, we know that the theorem holds for $x = \infty$, so we may assume that $x <\infty$. Further, select a real number $B > M_x(\alpha)$.
In view of Theorem \ref{Reduction}, we know that $M_x(\alpha)$ is the infimum of
\begin{equation} \label{FiniteHope}
\left(\sum_{n=1}^N M(\alpha_n)^x\right)^{1/x}
\end{equation}
over the set of all $N\in \mathbb N$ and all points $\alpha_1,\ldots,\alpha_N\in \alg^{\times}$ such that
\begin{enumerate}[(i)]
\item\label{Product} $\alpha = \alpha_1\cdots\alpha_N$,
\item\label{Unity} At most one point $\alpha_n$ is a root of unity,
\item\label{Rad} $\alpha_n\in \mathrm{Rad}(K_\alpha)$ for all $n$, and
\item\label{BBound} $\left(\sum_{n=1}^N M(\alpha_n)^x\right)^{1/x} \leq B$.
\end{enumerate}
We will show that the set of all values of \eqref{FiniteHope} is finite for $\alpha_1,\ldots,\alpha_N$ satisfying conditions \eqref{Product}-\eqref{BBound}.
We must first give an upper bound on $N$. We know that at least $N-1$ of the points $\alpha_1,\ldots,\alpha_N$ are not roots of unity. For all such points, we
have that
\begin{equation*}
M(\alpha_n) \geq C(\alpha)
\end{equation*}
by \eqref{RadBound}. Combining this with \eqref{BBound}, we obtain that
\begin{equation*}
B \geq \left(\sum_{n=1}^N M(\alpha_n)^x\right)^{1/x} \geq (N-1)^{1/x} C(\alpha)
\end{equation*}
which yields
\begin{equation} \label{NUpperBound}
N \leq 1+ \left(\frac{B}{C(\alpha)}\right)^x.
\end{equation}
Also by \eqref{BBound}, it follows that $M(\alpha_n) \leq B$ for all $n$. Moreover, since $\alpha_n\in\mathrm{Rad}(K_\alpha)$, the second statement of Lemma \ref{HeightInK}
implies that there are only finitely many possible values for $M(\alpha_n)$ for each $n$. Since $N$ is bounded above by the right hand side of \eqref{NUpperBound},
it follows that there are only finitely many possible values for \eqref{FiniteHope} with $\alpha_1,\ldots,\alpha_N$ satisfying \eqref{Product}-\eqref{BBound}.
We now know that $M_x(\alpha)$ is an infimum over a finite set, so the infimum must be achieved. \end{proof}
\section{Minimality of $\bar M$} \label{MinimalBarM}
We first give the proof of Theorem \ref{SmallP} showing that $M_x(\alpha) = \bar M(\alpha)$ for sufficiently small values of $x$.
\begin{proof}[Proof of Theorem \ref{SmallP}]
By Theorem \ref{BarPhiMetrics}, we have immediately that $M_x(\alpha) = \bar M_x(\alpha)$ for all $x$, so it follows that
\begin{equation} \label{EasyUpperBound}
M_x(\alpha) \leq \bar M(\alpha).
\end{equation}
Now we must prove the opposite inequality.
We know by Theorem \ref{Achieved} that there exist points $\alpha_1,\ldots,\alpha_N\in \mathrm{Rad}(K_\alpha)$ such that
\begin{equation*}
\alpha = \alpha_1\cdots\alpha_N\quad\mathrm{and}\quad M_x(\alpha) = \left( \sum_{n=1}^N M(\alpha_n)^x\right)^{1/x}.
\end{equation*}
We know that $\alpha$ is not a root of unity, so at least one of $\alpha_1,\ldots,\alpha_N$ is not a root of unity.
We now consider two cases. First, assume that precisely one of $\alpha_1,\ldots,\alpha_N$ is not a root of unity. In other words, there exists
a root of unity $\zeta$ and a point $\beta\in \mathrm{Rad}(K_\alpha)\setminus \mathrm{Tor}(\alg^{\times})$ such that $\alpha = \zeta\beta$ and
\begin{equation*}
M_x(\alpha) = M(\beta).
\end{equation*}
Of course, we also have $\beta = \alpha\zeta^{-1}$ so that
\begin{equation*}
\bar M(\alpha) \leq M(\alpha\zeta^{-1}) = M(\beta) = M_x(\alpha).
\end{equation*}
Combining this inequality with \eqref{EasyUpperBound}, the result follows.
Next, assume that at least two of $\alpha_1,\ldots,\alpha_N$ are not a roots of unity. By Lemma \ref{HeightInK}, we know that
$M(\alpha_n) \geq C(\alpha)$ whenever $\alpha_n$ is not a root of unity. Hence, we obtain that
\begin{equation*}
M_x(\alpha) = \left(\sum_{n=1}^N M(\alpha_n)^x\right)^{1/x} \geq (2C(\alpha)^x)^{1/x}
\end{equation*}
so that
\begin{equation} \label{TwoBound}
M_x(\alpha) \geq 2^{1/x} C(\alpha).
\end{equation}
By our assumption, we have that
\begin{equation*}
\frac{1}{x} \geq \frac{\log \bar M(\alpha) - \log C(\alpha)}{\log 2}
\end{equation*}
which implies that
\begin{align*}
2^{1/x} & \geq 2^{\frac{\log \bar M(\alpha) - \log C(\alpha)}{\log 2}} \\
& = \exp(\log \bar M(\alpha) - \log C(\alpha)) \\
& = \frac{\exp(\log \bar M(\alpha))}{\exp(\log C(\alpha))} \\
& = \frac{\bar M(\alpha)}{C(\alpha)}.
\end{align*}
It now follows from \eqref{TwoBound} that
\begin{equation*}
M_x(\alpha) \geq \bar M(\alpha)
\end{equation*}
completing the proof. \end{proof}
Next, we establish Corollary \ref{MBarMinimal} showing that $\bar M$ is minimal in the set $\mathcal S(M)$.
\begin{proof}[Proof of Corollary \ref{MBarMinimal}]
We observe again by Theorem \ref{BarPhiMetrics} that $\bar M\in \mathcal S(M)$. By Theorem \ref{SmallP}, for all sufficiently small $x$, we have that
$\bar M(\alpha) = M_x(\alpha)$. Hence, it follows that that
\begin{equation*}
\bar M(\alpha) = \lim_{x\to 0^+} M_x(\alpha) = M_0(\alpha)
\end{equation*}
and the result follows from Theorem \ref{OptimalMinimal}. \end{proof}
We begin our proof of Theorem \ref{NotUniform} by giving a slight modification to Theorem \ref{SmallP}. More specifically, it will be useful to consider what happens when the supposed inequality \eqref{AlwaysX} is replaced by a strict inequality.
\begin{lem} \label{StrongSmallP}
Let $\alpha$ be a non-zero algebraic number different from a root of unity and $x$ a positive real number satisfying
\begin{equation*}
x\cdot (\log \bar M(\alpha) - \log C(\alpha)) < \log 2.
\end{equation*}
Then any point $(\alpha_1,\alpha_2,\cdots) \in \tau^{-1}(\alpha)$ that achieves the infimum in the definition of $M_x(\alpha)$ has precisely one component
$\alpha_n$ that is not a root of unity. \end{lem} \begin{proof}
We recall first that
\begin{equation} \label{MexUpper}
M_x(\alpha) \leq \bar M(\alpha)
\end{equation}
by Theorem \ref{BarPhiMetrics}. Next, we note that
\begin{equation} \label{LooseBound}
\frac{1}{x} > \frac{\log \bar M(\alpha) - \log C(\alpha)}{\log 2}.
\end{equation}
Assume that $\alpha_1,\ldots,\alpha_N\in \alg^{\times}$ are such that
\begin{equation} \label{AchievedApplication}
\alpha = \alpha_1\cdots\alpha_N\quad\mathrm{and}\quad M_x(\alpha) = \left( \sum_{n=1}^N M(\alpha_n)^x\right)^{1/x}.
\end{equation}
and at least two of the points $\alpha_1,\ldots,\alpha_N$ are not roots of unity. By Theorem \ref{Reduction}, there exists a root of unity $\zeta$ and
points $\beta_1,\ldots,\beta_N\in \mathrm{Rad}(K_\alpha)$ such that
\begin{equation*}
\alpha = \zeta\beta_1\cdots\beta_N\quad\mathrm{and}\quad M(\beta_n) \leq M(\alpha_n)
\end{equation*}
for all $n$. If for any $n$ we have that $M(\beta_n) < M(\alpha_n)$, then
\begin{equation*}
M_x(\alpha) \leq \left( \sum_{n=1}^N M(\beta_n)^x\right)^{1/x} < \left( \sum_{n=1}^N M(\alpha_n)^x\right)^{1/x}
\end{equation*}
which contradicts the right hand side of \eqref{AchievedApplication}. Therefore, we have that $M(\beta_n) = M(\alpha_n)$ for all
$n$. In particular, at least two of the points $\beta_1,\ldots,\beta_N$ are not roots of unity. Furthermore, since each $\beta_n\in \mathrm{Rad}(K_\alpha)$,
we may apply Lemma \ref{HeightInK} to see that $M(\beta_n) \geq C(\alpha)$ whenever $\beta_n$ is not a root of unity. This yields
\begin{equation*}
M_x(\alpha) = \left(\sum_{n=1}^N M(\beta_n)^x\right)^{1/x} \geq (2C(\alpha)^x)^{1/x}.
\end{equation*}
which implies that
\begin{equation*}
M_x(\alpha) \geq 2^{1/x} C(\alpha).
\end{equation*}
However, we now have the strict inequality \eqref{LooseBound} which gives $2^{1/x} > \bar M(\alpha)/C(\alpha)$ and
\begin{equation*}
M_x(\alpha) > \bar M(\alpha)
\end{equation*}
contradicting \eqref{MexUpper}. Therefore, exactly one point among $\alpha_1,\ldots,\alpha_N$ is not a root of unity. \end{proof}
Before we prove Theorem \ref{NotUniform}, we recall our remark that $\bar M(\alpha)$ is often very reasonable to compute so that Theorem \ref{SmallP} and Lemma \ref{StrongSmallP} are useful in applications. The following proof is a typical example.
\begin{proof}[Proof of Theorem \ref{NotUniform}]
Let $\alpha = p^2$. In order to prove \eqref{P2Small}, we wish to apply Lemma \ref{StrongSmallP}, so we must compute the values of $\bar M(\alpha)$ and $C(\alpha)$.
We begin by observing that
\begin{equation*}
\bar M(\alpha) = \inf\{M(\zeta\alpha):\zeta\in \mathrm{Tor}(\alg^{\times})\} = \inf\{\deg(\zeta\alpha)\cdot h(\zeta\alpha):\zeta\in \mathrm{Tor}(\alg^{\times})\}.
\end{equation*}
Then by \eqref{WeilHeightDefined}, we obtain that
\begin{equation} \label{MBarM}
\bar M(\alpha) = h(\alpha)\cdot \inf\{\deg(\zeta\alpha):\zeta\in \mathrm{Tor}(\alg^{\times})\}.
\end{equation}
It is clear that the infimum on the right hand side of \eqref{MBarM} is achieved since it is an infimum over positive integers. More specifically,
it is achieved by a root of unity $\zeta$ that makes $\deg(\zeta\alpha)$ as small as possible. In our case, $\alpha$ is rational, so this occurs when $\zeta = 1$ leaving
\begin{equation} \label{AlphaUpper}
\bar M(\alpha) = \bar M(p^2) = M(p^2) = \log (p^2).
\end{equation}
In addition, we know that $K_\alpha = \mathbb Q$ so that $C(\alpha) = \log 2$ which now gives
\begin{equation*}
x\cdot (\log \bar M(\alpha) - \log C(\alpha)) = x\cdot (\log \log (p^2) - \log \log 2) < \log 2.
\end{equation*}
By Lemma \ref{StrongSmallP}, we know that any point $(\alpha_1,\alpha_2,\ldots)$ that attains the infimum in $M_x(\alpha) = M_x(p^2)$ must have precisely one point $\alpha_n$
that is not a root of unity. This completes the proof of \eqref{P2Small}.
To prove \eqref{P2Large}, we take $x > 1$ and assume that $(\alpha_1,\alpha_2,\ldots)$ attains the infimum in the definition of $M_x(p^2)$ where are most one
point $\alpha_n$ is different from a root of unity. Therefore, there exists a root of unity $\zeta$ and an algebraic number $\beta$ such that
\begin{equation*}
p^2 = \zeta\beta\quad\mathrm{and}\quad M_x(p^2) = M(\beta).
\end{equation*}
Hence we find immediately that
\begin{equation*}
M(\beta) = M_x(p^2) \leq (M(p)^x + M(p)^x)^{1/x} = 2^{1/x}\log p.
\end{equation*}
Since $x > 1$, this yields that
\begin{equation*} \label{BetaUpper}
M(\beta) < 2\log p.
\end{equation*}
On the other hand, we have that $\beta = \zeta^{-1}p^2$ so that, using \eqref{AlphaUpper}, we obtain
\begin{equation*}
M(\beta) = M(\zeta^{-1}p^2) \geq \bar M(p^2) = 2\log p
\end{equation*}
which is a contradiction. Thus, at least two points among $(\alpha_1,\alpha_2,\ldots)$ must not be roots of unity. \end{proof}
\section{Continuity of $x\mapsto M_x(\alpha)$} \label{ContinuitySection}
We have already proved that, for any height function $\phi$, the map $x\mapsto \phi_x(\alpha)$ is left semi-continuous. In general, we know that such functions are not always right semi-continuous. However, we are able to use Theorem \ref{Achieved} and our observations about the Mahler measure to establish right semi-continuity in this case.
\begin{proof}[Proof of Theorem \ref{Continuous}]
If $\alpha$ is a root of unity, then $M_x(\alpha) = 0$ for all $x$, so we may assume that $\alpha$ is not a root of unity.
Furthermore, we know by Theorem \ref{LeftSemiContinuous} that this map is left semi-continuous at all points, so it remains only to show that it
is right semi-continuous.
Now let $\bar x >0$ be a real number, so we must show that
\begin{equation} \label{RightSemiEnd}
\lim_{y\to \bar x^+} M_y(\alpha) = M_{\bar x}(\alpha).
\end{equation}
Since $x\mapsto M_x(\alpha)$ is decreasing, we know that the left hand side of \eqref{RightSemiEnd} exists. Moreover, we have that
\begin{equation} \label{HalfRightSemi}
\lim_{y\to \bar x^+} M_y(\alpha) \leq M_{\bar x}(\alpha).
\end{equation}
Now we select a point $y\in (\bar x, \bar x +1]$. By Theorem \ref{Achieved}, there must exist points
\begin{equation*}
\alpha_1,\ldots,\alpha_N \in \mathrm{Rad}(K_\alpha) \setminus \mathrm{Tor}(\alg^{\times})
\end{equation*}
and $\zeta\in \mathrm{Tor}(\alg^{\times})$ such that
\begin{equation*}
\alpha = \zeta\alpha_1\cdots\alpha_N\quad\mathrm{and}\quad M_y(\alpha) = \left( \sum_{n=1}^N M(\alpha_n)^y\right)^{1/y}.
\end{equation*}
Since $M_y(\alpha) \leq M(\alpha)$, we may assume without loss of generality that $M(\alpha_n) \leq M(\alpha)$ for all $n$. Furthermore, since $\alpha$ is not a root
of unity, we know that $N\geq 1$. For simplicity, we write now $a_n = M(\alpha_n)$ so that
\begin{equation*}
M_y(\alpha) = \left( \sum_{n=1}^N a_n^y\right)^{1/y},
\end{equation*}
and note that by Lemma \ref{HeightInK}, we have that
\begin{equation} \label{aLower}
a_n \geq C(\alpha)\ \mathrm{for\ all}\ n.
\end{equation}
Next, we define the function $f_y$ by
\begin{equation*}
f_y(x) = \left( \sum_{n=1}^N a_n^x\right)^{1/x}
\end{equation*}
and note that $f_y$ does indeed depend on $y$ because the points $\zeta$ and $\alpha_1,\ldots,\alpha_N$ depend on $y$. We now have immediately that
\begin{equation} \label{fAndM}
f_y(y) = M_y(\alpha).
\end{equation}
Since $\alpha = \zeta\alpha_1\cdots\alpha_N$, we know that
\begin{equation*}
M_{\bar x}(\alpha) \leq \left(\sum_{n=1}^N M(\alpha_n)^{\bar x}\right)^{1/{\bar x}} = \left(\sum_{n=1}^N a_n^{\bar x}\right)^{1/{\bar x}} = f_y(\bar x),
\end{equation*}
and therefore, we obtain that
\begin{equation} \label{fAndM2}
M_{\bar x}(\alpha) \leq f_y(\bar x).
\end{equation}
We know that $a_n> 0$ for all $n$ implying that $f_y(x) > 0$ for all $x$, so we may define the function $g_y(x) = \log f_y(x)$.
Since $f_y$ is differentiable on the positive real numbers, we know that $g_y$ is as well. Therefore, we may apply the Mean Value Theorem to it on $[\bar x, y]$.
Hence, there exists a point $c\in [\bar x,y]$ such that
\begin{equation*}
g_y'(c) = \frac{g_y(y) - g_y(\bar x)}{y-\bar x} = \frac{\log f_y(y) - \log f_y(\bar x)}{y-\bar x}
\end{equation*}
and it follows from \eqref{fAndM} and \eqref{fAndM2} that
\begin{equation} \label{MVTInequality}
g_y'(c) \leq \frac{\log M_y(\alpha) - \log M_{\bar x}(\alpha)}{y-\bar x}.
\end{equation}
We now wish to take limits of both sides of \eqref{MVTInequality} as $y$ tends to $\bar x$ from the right. However, it is possible that the limit of the left hand
side either equals $-\infty$ or does not exist as $y\to \bar x^+$. To solve this problem, we wish to give a lower bound on $g_y'(c)$ that does not depend on $y$.
For any $x >0$, we note that
\begin{align*}
g_y'(x) & = \frac{d}{dx}\log f_y(x) \\
& = \frac{d}{dx} \frac{1}{x}\left( \log \sum_{n=1}^N a_n^x\right) \\
& = \frac{1}{x^2}\left( x\cdot\frac{ \left(\sum_{n=1}^N a_n^x\log a_n\right)}{\left(\sum_{n=1}^N a_n^x\right)}
- \log \sum_{n=1}^N a_n^x \right ).
\end{align*}
Then using \eqref{aLower}, we have that
\begin{equation} \label{FirstLower}
g_y'(x) \geq \frac{1}{x^2}\left( x\cdot \log C(\alpha) - \log \sum_{n=1}^N a_n^x \right ).
\end{equation}
Now we need to give an upper bound on $\sum_{n=1}^N a_n^x$. Recall that we must have $a_n = M(\alpha_n) \leq M(\alpha)$ for all $n$.
Therefore, we have that
\begin{equation*}
\sum_{n=1}^N a_n^x \leq N M(\alpha)^x.
\end{equation*}
But using \eqref{aLower} again, we find that
\begin{equation*}
M(\alpha) \geq M_y(\alpha) = \left(\sum_{n=1}^N a_n^y\right)^{1/y} \geq (N C(\alpha)^y)^{1/y} = N^{1/y}C(\alpha).
\end{equation*}
We also know $C(\alpha) > 0$ and $y\in (\bar x, \bar x+1]$ so that
\begin{equation*} \label{NUpper}
N \leq \left( \frac{M(\alpha)}{C(\alpha)}\right)^y \leq \left( \frac{M(\alpha)}{C(\alpha)}\right)^{\bar x +1},
\end{equation*}
and therefore
\begin{equation*} \label{SumUpper}
\sum_{n=1}^N a_n^x \leq \frac{M(\alpha)^{x + \bar x +1}}{C(\alpha)^{\bar x + 1}}.
\end{equation*}
It now follows that
\begin{equation*}
-\log \sum_{n=1}^N a_n^x \geq -\log \left( \frac{M(\alpha)^{x + \bar x +1}}{C(\alpha)^{\bar x + 1}}\right).
\end{equation*}
Combining this with \eqref{FirstLower}, we obtain that
\begin{equation*}
g_y'(x) \geq\frac{1}{x^2}\left( x\cdot \log C(\alpha) - \log \left( \frac{M(\alpha)^{x + \bar x +1}}{C(\alpha)^{\bar x + 1}}\right) \right),
\end{equation*}
so we have shown that
\begin{equation} \label{SecondLower}
g_y'(x) \geq \frac{x + \bar x +1}{x^2} \log \left( \frac{C(\alpha)}{M(\alpha)} \right).
\end{equation}
For simplicity, we now write $D(\alpha,\bar x, x)$ to denote the right hand side of \eqref{SecondLower}. As a function of $x$, it is obvious that
$D(\alpha,\bar x, x)$ is continuous for all $x > 0$. Hence, we may define
\begin{equation*}
\mathcal D(\alpha,\bar x) = \min \{D(\alpha,\bar x, x): x\in [\bar x, \bar x +1]\}.
\end{equation*}
Now $\mathcal D(\alpha,\bar x)$ is the desired lower bound on $g_y'(c)$ not depending on $y$.
Since $c\in [\bar x,y] \subset [\bar x,\bar x+1]$, we may apply \eqref{MVTInequality} and \eqref{SecondLower} to see that
\begin{equation*}
\mathcal D(\alpha,\bar x) \leq D(\alpha,\bar x, c) \leq g_y'(c) \leq \frac{\log M_y(\alpha) - \log M_{\bar x}(\alpha)}{y-\bar x}.
\end{equation*}
By multiplying through by $y - \bar x$, we find that
\begin{equation} \label{AlmostDone}
(y-\bar x) \mathcal D(\alpha,\bar x) \leq \log M_y(\alpha) - \log M_{\bar x}(\alpha)
\end{equation}
holds for all $y\in (\bar x, \bar x + 1]$.
As we have noted, $\lim_{y\to \bar x^+} M_y(\alpha)$ exists. Since we have assumed that $\alpha$ is not a root of unity, we conclude from Theorem \ref{Achieved}
that $M_y(\alpha) > 0$ for all $y$. It now follows that $\lim_{y\to \bar x^+} \log M_y(\alpha)$ also exists. Moreover, the term $\mathcal D(\alpha,\bar x)$ is a real
number not depending on $y$, so the left hand side of \eqref{AlmostDone} tends to zero as $y$ tends to $\bar x$ from the right. This leaves
\begin{align*}
0 & = \lim_{y\to \bar x^+}((y-\bar x) \mathcal D(\alpha,\bar x)) \\
& \leq \lim_{y\to \bar x^+} (\log M_y(\alpha) - M_{\bar x}(\alpha)) \\
& = \lim_{y\to \bar x^+} \log M_y(\alpha) - \lim_{y\to \bar x^+}\log M_{\bar x}(\alpha) \\
& = \lim_{y\to \bar x^+} \log M_y(\alpha) - \log M_{\bar x}(\alpha),
\end{align*}
which yeilds
\begin{equation*}
\log M_{\bar x}(\alpha) \leq \lim_{y\to \bar x^+} \log M_y(\alpha)
\end{equation*}
so that $M_{\bar x}(\alpha) \leq \lim_{y\to \bar x^+} M_y(\alpha)$ and the result follows by combining this with \eqref{HalfRightSemi}.
\end{proof}
\section{Weil height}
Before we begin our proof of Theorem \ref{WeilHeightComp}, we recall that if $N$ is any integer, then it is well-known that \begin{equation} \label{IntPowers}
h(\alpha^N) = |N|\cdot h(\alpha) \end{equation} for all algebraic numbers $\alpha$. Using this fact, we are able to proceed with our proof.
\begin{proof}[Proof of Theorem \ref{WeilHeightComp}]
First assume that $x \leq 1$. By \eqref{MetricHeightConversion} of Theorem \ref{MetricConstruction}, we have that $h_x(\alpha) \leq h(\alpha)$.
But also, it is well-known that $h$ is already a $1$-metric height. Therefore, \eqref{NoChangeMetric} of Theorem \ref{MetricConstruction} implies that
$h_1(\alpha) = h(\alpha)$. Then by \eqref{Comparisons} of Theorem \ref{MetricConstruction}, we conclude that $h_x(\alpha) \geq h(\alpha)$ verifying the
theorem in the case that $x \leq 1$.
Next, we assume that $x > 1$. Let $N$ be a positive integer and select $\beta\in\alg^{\times}$ such that $\beta^N = \alpha$. Therefore, we have that
\begin{equation*}
h_x(\alpha) \leq \left(\sum_{n=1}^N h(\beta)^x\right)^{1/x} = (N h(\beta)^x)^{1/x} = N^{1/x}\cdot h(\beta).
\end{equation*}
Then using \eqref{IntPowers} we obtain that $h(\alpha) = N\cdot h(\beta)$ which yields
\begin{equation} \label{TightUpper}
h_x(\alpha) \leq N^{\frac{1}{x} - 1}\cdot h(\alpha).
\end{equation}
Since $x > 1$, the right hand side of \eqref{TightUpper} tends to zero as $N\to\infty$ completing the proof. \end{proof}
\section{Acknowledgment}
The author wishes to thank the Max-Planck-Institut f\"ur Mathematik where the majority of this research took place.
\end{document} |
\begin{document}
\preprint{ } \title[Short title for running header]{Determining lower bounds on a measure of multipartite entanglement from few local observables} \author{Jun-Yi Wu, Hermann Kampermann, Dagmar Bru{\ss}} \affiliation{Institut f\"ur Theoretische Physik III, Heinrich-Heine-Universit\"at D\"usseldorf, D-40225 D\"usseldorf, Germany} \author{Claude Kl\"ockl} \affiliation{University of Vienna, Faculty of Mathematics, Nordbergstraße 15, 1090 Wien, Austria} \author{Marcus Huber} \affiliation{University of Bristol, Department of Mathematics, Bristol, BS8 1TW, U.K.} \keywords{entanglement, dimensionality, qudits, multipartite systems} \pacs{03.67.Mn, 03.65.Ud, 03.65.Fd, 03.65.Aa}
\begin{abstract} We introduce a method to lower bound an entropy-based measure of genuine multipartite entanglement via nonlinear entanglement witnesses. We show that some of these bounds are tight and explicitly work out their connection to a framework of nonlinear witnesses that were published recently. Furthermore, we provide a detailed analysis of these lower bounds in the context of other possible bounds and measures. In exemplary cases, we show that only a few local measurements are necessary to determine these lower bounds. \end{abstract} \volumeyear{year} \volumenumber{number} \issuenumber{number} \eid{identifier} \date[Date text]{date} \received[Received text]{date}
\revised[Revised text]{date}
\accepted[Accepted text]{date}
\published[Published text]{date}
\maketitle
\section{Introduction}
Quantum entanglement is central to the field of quantum information theory. Due to its numerous applications in upcoming quantum technology much research has been devoted to its understanding
(for a recent overview consider Ref.~\cite{horodeckiqe}).\newline Especially in systems comprised of many particles entanglement provides numerous challenges and of course potential applications, such as building quantum computers (see Ref.~\cite{qc}), performing quantum algorithms (the connection to multipartite entanglement is demonstrated in Ref.~\cite{qa}) and multi-party cryptography (see e.g. Ref.~\cite{SHH3}).\newline Furthermore, the understanding of the behavior of complex systems seems to be closely linked to the understanding of multipartite entanglement manifestations, demonstrated by the connection to phase transitions and ionization in condensed matter systems (e.g. \cite{cond}), the properties of ground states in relation to entanglement (as shown e.g. in Ref.~\cite{spin,ground}), or potentially even biological systems (such as e.g. bird navigation \cite{bird}).\newline In order to judge the relevance of entanglement in such systems it is crucial to not only detect its presence, but also quantify the amount. The structure of entangled states, especially in multipartite systems \cite{acin}, is very complex and the question whether a given state is entangled is even NP-hard \cite{gurvits}. Thus, in general, it will not be possible to derive a computable measure of entanglement that reveals all entangled states to be entangled and discriminates between different entanglement classes. Furthermore, full information about the state of the system requires a number of measurements that grows exponentially in the size of the system. For the detection of entanglement in multipartite systems most researchers have therefore made it a primary goal to develop entanglement witnesses, which via a limited amount of local measurements can detect the presence of entanglement, even in complex systems (for an overview of multipartite entanglement witnesses consider Ref.~\cite{guehnetoth}).\newline The expectation value of witness-operators are usually expressed in terms of inequalities, which if violated show the presence of entanglement. Nonlinear witnesses (first introduced in Ref.~\cite{horodeckinonlinear} see also early discussions in e.g. Ref.~\cite{nonlin}) provide a generalization that is no longer a linear function of density matrix elements, but a nonlinear one. Thus one cannot reformulate the criteria in terms of an expectation value of a hermitian operator (unless one considers coherent measurements on multiple copies of the state, which out of experimental infeasibility we do not discuss in our manuscript). We will henceforth refer to inequalities that involve nonlinear functions of density matrix elements as nonlinear entanglement witnesses.\\ Recently some authors pointed out a connection between the possible amount of violation of these nonlinear inequalities and quantification of entanglement in multipartite systems (in Ref.~\cite{Guehnetaming} and Ref.~\cite{maetal}).\newline The aims of this paper are twofold. First to systematically show the connection of numerous witnesses to a meaningful measure of genuine multipartite entanglement and second to use this established relation for the development of novel witnesses, which by construction give lower bounds on that measure. To that end we follow and generalize the approach from Ref.~\cite{maetal}.\\ It turns out that only a small number of density matrix elements enters into our lower bounds, making the construction experimentally feasible even in larger systems of high dimensionality.
\section{A measure of multipartite entanglement and its lower bounds}
\subsection{A measure of genuine multipartite entanglement (GME)}
The entropy of subsystems has often been used, in order to quantify entanglement contained in multipartite pure states (e.g. see \cite{horodeckiqe,Milburn,Love,HH2,HHK1}). In this paper we will follow the definition first presented in Ref.\cite{Milburn} and define a measure of GME for multipartite pure states as \begin{align}
E_{m}(|\psi\rangle\langle\psi|):=\min_{\gamma}\sqrt{S_{L}\left( \rho _{\gamma}\right) }=\min_{\gamma}\sqrt{2\left( 1-\text{Tr}(\rho_{\gamma} ^{2})\right) }\,, \label{Def. GME measure} \end{align} where $S_{L}\left( \rho_{\gamma}\right) $ is the linear entropy of the reduced density matrix of subsystem $\gamma$, i.e. $\rho_{\gamma}
:=\text{Tr}_{\bar{\gamma}}(|\psi\rangle\langle\psi|)$. The minimum is taken over all possible reductions $\gamma$ (where the complement is denoted as $\bar{\gamma}$), which corresponds to a bipartite split into $\gamma|\bar{\gamma}$. \newline As any proper measure of multipartite entanglement for pure states can be generalized to mixed states via a convex roof, i.e. \[\label{definitionroof}
E_{m}(\rho):=\inf_{\{p_{i},|\psi_{i}\rangle\}}\sum_{i}p_{i}E_{m}(|\psi _{i}\rangle\langle\psi_{i}|)\,. \] Due to its construction this measure fulfills almost all desirable properties one would expect from measures of GME (see Ref.~\cite{maetal} for details). Because computing all possible pure state decompositions of a density matrix is computationally impossible even if one is given the complete density matrix, we require lower bounds to be calculable for this expression.\\ Also note that a lower bound on the linear entropy directly leads to a lower bound on the R{\'e}nyi $2$-entropy $S_R^{(2)}(\rho_\gamma)$ via the relation $S_R^{(2)}(\rho_\gamma)=-\log_2(\frac{2-S_L(\rho_\gamma)}{2})$, which also provides one of the physical interpretations of this measure. The R{\'e}nyi $2$-entropy in itself is a lower bound to the von Neumann entropy $S(\rho_\gamma)$ and the mutual information can be expressed as $I_{\gamma\bar{\gamma}}:=S(\rho_\gamma)+S(\rho_{\bar{\gamma}})-S(\rho)=2S(\rho_\gamma)$. Thus by our lower bound we gain a lower bound on the average minimal mutual information across all bipartitions of the pure states in the decomposition, minimized over all decompositions.
\subsection{Linear entropy and its convex roof}
The state vector of an $n$-partite qudit state can be expanded in terms of the computational basis \[
|\psi\rangle=\sum_{i_{1},i_{2},\cdots,i_{n}=0}^{d-1}c_{i_{1},i_{2}
,\cdots,i_{n}}|i_{1},i_{2},\cdots,i_{n}\rangle=:\sum_{\eta\in\mathbb{N}
_{d}^{\otimes n}}c_{\eta}|\eta\rangle\,, \] where a basis vector is denoted by $\eta=\left( i_{1},i_{2},\cdots,i_{n}\right) \in\mathbb{N} _{d}^{\otimes n}$.
This vector notation will facilitate the upcoming derivations. A crucial element of the notation in this paper will be the permutation operator acting upon two vectors, exchanging vector components corresponding to the set of indices. E.g. the permutation operator $P_{\{1,3\}}(\eta_1,\eta_2)$ will exchange the first and third component of the vector $\eta_{1}$ with the corresponding component of the vector $\eta_{2}$, i.e. \[ P_{\left\{ \textcolor{red}{1},\textcolor{blue}{3}\right\} } (\mathbf{\textcolor{red}{0}}1\mathbf{\textcolor{blue}{2}} 13,\mathbf{\textcolor{red}{3}}0\mathbf{\textcolor{blue}{1}} 21)=(\mathbf{\textcolor{red}{3}}1\mathbf{\textcolor{blue}{1}} 13,\mathbf{\textcolor{red}{0}}0\mathbf{\textcolor{blue}{2}}21). \] Using this notation one can write down a very simple expression for the linear entropy of a reduced state $\rho_{\gamma}$ (derivation see section \ref{linear entropy of pure state} in the appendix) \begin{equation} S_{L}\left( \rho_{\gamma}\right) =\sum_{\eta_{1}\not =\eta_{2} }\left\vert c_{\eta_{1}}c_{\eta_{2}}-c_{\eta_{1}^{\gamma}}c_{\eta_{2}^{\gamma }}\right\vert ^{2}, \label{result. linear entropy} \end{equation} where $\left( \eta_{1}^{\gamma},\eta_{2}^{\gamma}\right) =P_{\gamma}\left( \eta_{1},\eta_{2}\right) $.
For pure states we can of course find lower bounds on $E_m(|\psi\rangle\langle\psi|)$ by lower bounding the linear entropy for all possible bipartitions. For mixed states we can then provide a lower bound for the convex roof $E_m(\rho)$. We now illustrate our method in one exemplary case and then continue to articulate the main theorem.\\ Note that the linear entropy of subsystems has been widely used for lower bounding measures of entanglement due to the well known and simple structure of eq.(\ref{result. linear entropy}). None of the previous methods, however, work for lower bounding the inherently multipartite measure $E_m(\rho)$, due to the additional minimization over all bipartitions in each decomposition element of the convex roof.
\subsection{W-states\label{sec. W-state}}
In order to demonstrate how our framework works let us start by deriving the explicit lower bound detecting the three-qubit $W$ state $|W\rangle=\frac{1}{\sqrt{3}}(|001\rangle+|010\rangle+|100\rangle)$. For three-qubit states there are three bipartitions ($1|23,2|13,3|12$) and thus we have three linear entropies to look at in order to calculate $E_m(|\psi\rangle\langle\psi|)$, \begin{gather}
\sqrt{S_L(\rho_1)}=\\2\sqrt{|c_{001}c_{100}-c_{101}c_{000}|^2+|c_{010}c_{100}-c_{110}c_{000}|^2+(\cdots)}\,,\nonumber\\
\sqrt{S_L(\rho_2)}=\\2\sqrt{|c_{010}c_{100}-c_{110}c_{000}|^2+|c_{010}c_{001}-c_{011}c_{000}|^2+(\cdots)}\,,\nonumber\\
\sqrt{S_L(\rho_3)}=\\2\sqrt{|c_{001}c_{100}-c_{101}c_{000}|^2+|c_{010}c_{001}-c_{011}c_{000}|^2+(\cdots)}\nonumber\,. \end{gather}
Now using $\sqrt{a^2+b^2}\geq\frac{1}{\sqrt{2}}(a+b)$ (which is a specific case of the inequality \ref{eq. ineq. sqrt} in appendix \ref{appx. ineq.sqrt}) and $|a-b|\geq|a|-|b|$ it is obvious that \begin{align}
\sqrt{S_L(\rho_1)}\geq\frac{2(|c_{001}c_{100}|-|c_{101}c_{000}|+|c_{010}c_{100}|-|c_{110}c_{000}|)}{\sqrt{2}}\,,\\
\sqrt{S_L(\rho_2)}\geq\frac{2(|c_{010}c_{100}|-|c_{110}c_{000}|+|c_{010}c_{001}|-|c_{011}c_{000}|)}{\sqrt{2}}\,,\\
\sqrt{S_l(\rho_3)}\geq\frac{2(|c_{001}c_{100}|-|c_{101}c_{000}|+|c_{010}c_{001}|-|c_{011}c_{000}|)}{\sqrt{2}}\,. \end{align}
Then using $|ab|-\frac{1}{2}(a^2+b^2)\leq 0$ we can add one negative term for each entropy and it will still be a lower bound, i.e. we add $|c_{010}c_{001}|-\frac{1}{2}(|c_{010}|^2+|c_{001}|^2)$ in the first lower bound, $|c_{100}c_{001}|-\frac{1}{2}(|c_{100}|^2+|c_{001}|^2)$ in the second and $|c_{010}c_{100}|-\frac{1}{2}(|c_{010}|^2+|c_{100})|^2$ in the third. Then we can use that $\min[P-N_1,P-N_2,P-N_3]\geq P-N_1-N_2-N_3$ and end up with \begin{align}
E_m(|\psi\rangle\langle\psi|)\geq\sqrt{2}(|c_{001}c_{100}|+|c_{001}c_{010}|+|c_{100}c_{010}|)-\nonumber\\
\frac{\sqrt{2}}{2}(|c_{010}|^2+|c_{100}|^2+|c_{001}|^2)-\nonumber\\
\sqrt{2}(|c_{101}c_{000}|+|c_{110}c_{000}|+|c_{011}c_{000}|)\,. \end{align} Finally we can bound the convex roof using the following two relations \begin{align}
\inf_{\{p_i,|\psi_i\rangle\}}\sum_ip_i|c^i_{\eta_1} c^i_{\eta_2}|&\geq|\langle\eta_1|\rho|\eta_2\rangle|\,,\\
\inf_{\{p_i,|\psi_i\rangle\}}\sum_ip_i|c^i_{\eta_1} c^i_{\eta_2}|&\leq\sqrt{\langle\eta_1|\rho|\eta_1\rangle\langle\eta_2|\rho|\eta_2\rangle}\,, \end{align} and end up with a lower bound for mixed states as
\begin{gather}
E_m(\rho)\geq\sqrt{2}(|\langle 001|\rho|100\rangle|+|\langle 001|\rho|010\rangle|+|\langle 100|\rho|010\rangle|)-\nonumber\\
\frac{\sqrt{2}}{2}(\langle 010|\rho|010\rangle+\langle 100|\rho|100\rangle+\langle 001|\rho|001\rangle)-\nonumber\\
\sqrt{2}\sqrt{\langle 101|\rho|101\rangle\langle 000|\rho|000\rangle}-\nonumber\\
\sqrt{2}\sqrt{\langle 110|\rho|110\rangle\langle 000|\rho|000\rangle}-\nonumber\\
\sqrt{2}\sqrt{\langle 011|\rho|011\rangle\langle 000|\rho|000\rangle}\,. \label{WWitness} \end{gather} Surprisingly this leads directly to the nonlinear entanglement witness inequality presented in Refs.~\cite{Guehnewit,hmgh1} up to a factor of $\sqrt{2}$. Using only simple algebraic relations we have thus shown how to lower bound the convex roof construction. The first apparent strength of this lower bound is the limited number of density matrix elements needed to compute it. E.g. in our exemplary three-qubit case only ten out of possibly sixty-four elements need to be measured. Obviously we can extend the analysis using the same techniques to systems beyond three qubits.
\section{A General Construction of lower bounds on the GME measure $E_{m}$} Now we can generalize the connection of the 3-qubit W state witness and the measure $E_{m}$.
Just as for three qubits we can always get lower bounds by summing the coefficient pairs $c_{\eta_1}c_{\eta_2}$ that belong to a certain target pure state and appear in some or all reduced linear entropies. The construction of such general lower bounds also starts by selecting a subset of coefficient pairs that will be translated into off-diagonal elements $\rho_{\eta_1,\eta_2}$, where $(\eta_1,\eta_2)$ is the vector basis pair denoting the row and column of the element in density matrix $\rho$. We denote the selected vector basis pairs as $R:=\{(\eta_1,\eta_2)\}$. Then we can repeat the steps analogously to eq.(6-11) and arrive at a general lower bound on the measure as the following theorem:
\begin{widetext}
\begin{theorem} [A general lower bound on the GME measure] \label{theorem. lower bound of GME measure}For a set of row-column pairs $R=\{(\eta_1,\eta_2)\}$, the genuine multipartite entanglement measure $E_{m}$ has the following lower bound: \begin{equation} E_{m}\geq 2\sqrt{\frac{1}{\left\vert R\right\vert -N_{R}}}\left[ \sum_{\left( \eta_{1},\eta_{2}\right) \in R}\left( \left\vert \rho_{\eta_{1}\eta_{2} }\right\vert -\sum_{\gamma\in\Gamma(\eta_{1},\eta_{2})}\sqrt{\rho_{\eta _{1}^{\gamma}\eta_{1}^{\gamma}}\rho_{\eta_{2}^{\gamma}\eta_{2}^{\gamma}} }\right) -\left( \frac{1}{2}\sum_{\eta\in I(R)}N_{\eta}\left\vert \rho _{\eta\eta}\right\vert \right) \right] \label{eq. theorem-result}\,. \end{equation}
The right-hand-side of eq.(\ref{eq. theorem-result}) defines a GME witness $W_R(\rho)$, where $\rho_{\eta_{1},\eta_{2}}:=\langle\eta_{1}|\rho|\eta_{2}\rangle$, $(\eta_{1}^{\gamma},\eta_{2}^{\gamma}):=P_{\gamma}(\eta_{1},\eta_{2})$, $\Gamma(\eta_{1},\eta_{2}):=\left\{ \gamma:(\eta_{1}^{\gamma},\eta _{2}^{\gamma})\notin R\right\} $ and $I(R):=\left\{ \eta:\exists\eta ^{\prime}\text{ that }(\eta^{\prime},\eta)\text{ or }(\eta,\eta^{\prime})\in R\right\} $ is the set of basis vectors $\eta$, which appear in the set $R$.
$N_{R}$ is the maximal (or minimal) value of $|R^{\gamma}|$ over all possible bipartitions $\gamma|\bar{\gamma}$, where $R^{\gamma}$ is the set of coefficient pairs $(c_{\eta_{1}},c_{\eta_{2}})\in R$, which do not contribute to the $\gamma$-subsystem entropy.
$N_{\eta}$ are normalization constants given by the maximal value of $n_{\eta}^{\gamma}$ over all possible bipartitions $\gamma|\bar{\gamma}$, where $n_{\eta}^{\gamma}$ is the number of coefficients $c_\eta$ from some pairs in $R$, which are not counted in the $\gamma$-subsystem entropy (and how many are counted depends on whether one chooses $N_R$ to be maximal or minimal). \\ \end{theorem} \begin{proof} See Appendix \ref{appx. proof of lower bound on GME measure} for the full proof. \end{proof} \end{widetext}
It is evident that not every choice of coefficient pairs will yield a useful lower bound, because one really needs to select those that are actually contributing to multipartite entanglement. There is however always an obvious choice. The set of coefficient pairs $R$ must be chosen such that in every subsystem at least one of the elements of $R$ contribute to the linear entropy of the reduced state. E.g. in the case of GHZ states given in a specific basis $|GHZ\rangle=\frac{1}{\sqrt{2}}(|0\rangle^{\otimes n}+|1\rangle^{\otimes n})$ one would choose the pair $({00\cdots0},{11\cdots1})$, which contributes to all reduced entropies. In the general case however there is still some freedom of choice left to get a valid lower bound. For some sets $R$ it can happen, that the coefficients do not contribute to every subsystem entropy equally (which we show in an exemplary case in section \ref{sec. four-qubit-example}). Then one can choose $N_R$ in different ways, but in all considered cases we found that choosing it maximal or minimal will produce the best bounds (where choosing it maximal usually yields the tightest bounds close to pure states, whereas choosing it minimal improves the noise resistance). Since these coefficients are in general basis dependent, so is also our witness construction. The prefactor $\sqrt{\frac{1}{\left\vert R\right\vert-N_R }}$ suggests that the optimal basis for constructing such a lower bound is given by the minimal tensor rank representation of the pure state. \section{Applications and Examples} \subsection{Four-qubit singlet state}\label{sec. four-qubit-example} Let us illustrate how to apply Theorem \ref{theorem. lower bound of GME measure} with an explicit example. In an experimental setting where one expects to produce a four-qubit singlet state (which was e.g. discussed in the context of solving the liar detection problem in Ref.~\cite{cabello}), i.e. \begin{align}
|S_4\rangle=&\frac{1}{2\sqrt{3}}(2|0011\rangle+2|1100\rangle-|0110\rangle\nonumber\\
&-|1001\rangle-|1010\rangle-|0101\rangle)\,, \end{align} one is confronted with the following expected coefficients: $c_{0011},c_{1100},c_{0101},c_{1010},c_{0110},c_{1001}$. Following the recipe of theorem \ref{theorem. lower bound of GME measure} we now select some coefficient pairs. We could choose e.g. $R_1=(0011,0101)$, $R_2=(0011,1010)$, $R_3=(0011,0110)$ and $R_4=(0011,1001)$, such that $R=\{R_1,R_2,R_3,R_4\}$. For this selection we use theorem \ref{theorem. lower bound of GME measure} to bound the GME measure. We see that in every subsystem at least two of these pairs appear naturally. Although there are more coefficient pairs we now choose to only take into account two per subsystem entropy and thus choose $N_R$ to be the minimal number of coefficient pairs in every subsystem which gives $N_R=2$. Thus we need to add negative terms that compensate for the missing terms just as we did in the three-qubit case, but now we need to do it two times in every subsystem. This results in the following individual prefactors $N_\eta$ for the diagonal elements: $N_{0011}=2$ (as this coefficient appears in two missing pairs in every subsystem), $N_{0101}=1$, $N_{1001}=1$, $N_{1010}=1$ and $N_{0110}=1$ (as those appear maximally once per subsystem entropy). Inserting this in theorem \ref{theorem. lower bound of GME measure} we end up with the lower bound as \begin{gather}
E_m(\rho)\geq\frac{2}{\sqrt{2}}(|\rho_{R_1}|+|\rho_{R_2}|+|\rho_{R_3}|+|\rho_{R_4}|\nonumber\\ -\sqrt{\rho_{0111,0111}\rho_{0001,0001}}-\sqrt{\rho_{0111,0111}\rho_{0010,0010}}\nonumber\\ -\sqrt{\rho_{1011,1011}\rho_{0001,0001}}-\sqrt{\rho_{1011,1011}\rho_{0010,0010}}\nonumber\\ -\frac{1}{2}(\rho_{0101,0101}+\rho_{1001,1001}+\rho_{1010,1010}+\rho_{0110,0110})\nonumber\\ -\rho_{0011,0011})\,. \end{gather}
We have thus created a nonlinear witness function that lower bounds our measure. From an experimental point of view this is very favorable as few local measurement settings suffice to ascertain the needed thirteen density matrix elements (especially since the nine diagonal elements can be constructed from a single measurement setting). Of course we could also exploit the connection of our lower bound to the Dicke state witness $Q^{(2)}_2$ (which is discussed in section \ref{Dicke}), which also detects GME in this state (although at the cost of more required measurements). In this case even the resistance to white noise is more favorable with our construction method, as for a state $\rho=p|S_4\rangle\langle S_4|+\frac{1-p}{16}\mathbbm{1}$ this exemplary lower bound detects GME until $p=\frac{21}{29}\approx 0.72$, whereas the old witness construction yields a worse resistance up to $p=\frac{27}{35}\approx 0.77$. This shows the versatility of our general approach. By choosing certain coefficients one can tailor these lower bounds to specific experimental situations. If one is confronted with a low noise system it is always beneficial to choose as few coefficients as possible, such that very few local measurements suffice (even a number that is linear in the size of the system is often sufficient). Every additional measurement can then be included in the lower bound and improves the bound and its noise resistance if necessary.
\subsection{Bipartite witnesses and lower bounds on the measure} Although we have presented our theorem and measures in the general case of $n$-qudits, we can always apply the lower bounds also for $n=2$, as our theorem holds for any $n$ and $d$. Suppose we are given a bipartite qutrit system and want to lower bound the concurrence with only a few local measurements.
If the expected state is e.g. $|\psi\rangle=\frac{1}{\sqrt{3}}(|00\rangle+|11\rangle+|22\rangle)$ we can use the lower bounding procedure outlined above, yielding \begin{align}
E_m(\rho)\geq\frac{2}{\sqrt{3}}(\Re e[\left.\left\langle 00| \rho |11 \right\rangle\right.]-\sqrt{\left.\left\langle 01| \rho| 01 \right\rangle\left\langle| 10 \rho |10 \right\rangle\right.}+\nonumber\\
\Re e[\left.\left\langle 00| \rho |22 \right\rangle\right.]-\sqrt{\left.\left\langle 02| \rho |02 \right\rangle\left\langle 20| \rho| 20 \right\rangle\right.}+\nonumber\\
\Re e[\left.\left\langle 11| \rho |22 \right\rangle\right.]-\sqrt{\left.\left\langle 12| \rho |12\right\rangle\left\langle 21| \rho |21 \right\rangle\right. })\,.
\end{align}
In order to determine the lower bound we have to measure nine different density matrix elements. Of course any density matrix element can always be obtained via local measurements. How these measurements can be performed in a basis consisting of a tensor product of the generalized Gell-Mann matrices we show explicitly in appendix \ref{GellMann}.\\
It turns out that these nine different density matrix elements can be obtained via ten local measurement settings. Let us study the lower bound in the presence of noise. Suppose we have white noise in the system, i.e. $\rho=p|\psi\rangle\langle\psi|+\frac{1-p}{d}\mathbbm{1}$. Calculating the lower bound results in $E_m(\rho)\geq\frac{2(4p-1)}{\sqrt{27}}$, which is equivalent to the analytical expression of Wootter's concurrence for these systems (as proven in Ref.~\cite{hashemi, caves}). In this case we have a necessary and sufficient entanglement criterion and a tight lower bound on the concurrence from ten local measurements for a special class of states. Indeed if one generalizes this example to arbitrary dimension $d$, we find that the bound is always tight for bipartite isotropic states.
\subsection{Dicke States}\label{Dicke} We will now continue to show how this construction relates to an entanglement witness for Dicke-state, which are multi-dimensional generalizations of the $W$ states(which were first introduced in the context of laser emission in Ref.~\cite{Dicke}).
In the original article \cite{maetal}, where this approach was first introduced, the authors connected the violation of a witness suitable for GHZ states (first introduced in Ref.~\cite{Guehnewit} and later presented in a more general framework in Ref.~\cite{hmgh1}) with a lower bound on the measure $E_{m}$. We want to follow this approach and establish a general connection between a set of witnesses suitable for all generalized Dicke states introduced in Ref.~\cite{hesgh1} and generalized in Ref.~\cite{shgh1}. To that end let us first introduce a concise notation for those states.\newline Let
$\alpha$ be a set containing specific subsystems of a multipartite state. We then define the state $|\alpha^{l}\rangle$ as a tensor product of states
$|l\rangle$ for all subsystems not contained in $\alpha$ and excited states $|l+1\rangle$ in the subsystems contained in $\alpha$. E.g. for the four-partite state
$|\{1,3\}^{2}\rangle$ we have $|3232\rangle$. Using this abbreviated notation we can define a generalized set of Dicke states, consisting of $n$ $d$-dimensional subsystems, as \begin{align}
|D_{m}^{d}\rangle=\frac{1}{\sqrt{{n\choose m}(d-1)}}\sum_{l=0}^{d-2}\sum_{\alpha
:|\alpha|=m}|\alpha^{l}\rangle\,, \end{align} where the parameter $m$ denotes the number of excitations, with $0<m<n$.
Since the explicit form of the nonlinear witness from Ref.~\cite{shgh1} will be used in the following considerations we will repeat it in appendix \ref{DickeWitness}. For all biseparable states this witness $Q_{m}^{\left( d\right) }$ is strictly smaller equal zero, i.e. \begin{align*} Q\left( \rho\right) & \leq0\Leftarrow\rho\text{ is biseparable}\\ Q\left( \rho\right) & >0\Rightarrow\rho\text{ is multipartite entangled}\,. \end{align*}
Furthermore, the witness can also detect the ``dimensionality'' of GME, by which we mean the maximal number of degrees of freedom $f_\rho (f_\rho \le d)$ that occurs in the pure states of an ensemble constituting $\rho$, minimised over all ensembles (this is the natural generalization of the concept of Schmidt number \cite{schmidtnumber} to multipartite systems, further explored e.g. in Ref.\cite{shgh1}). I.e. the dimensionality is defined as \begin{align}
f_\rho:=\inf_{\{p_i,|\psi_i\rangle\}}\max_i(\min_\gamma(\text{rank}(\rho_\gamma))) \end{align} Since \begin{align*} Q_{m}^{\left( d\right) }\left( \rho\right) & \leq f_{\rho}-1,\,\forall \rho\,, \end{align*} we can directly infer that \[ Q_{m}^{\left( d\right) }\left( \rho\right) >f-2\Rightarrow f_\rho\geq f \] In fig.\ref{fig. f_dimensional_gme_entanglement} we show how $Q_{m}^{\left( d\right) }$ detects the GME dimensionality. The maximal violation of these inequalities is always achieved for
$m$-excitation Dicke states, i.e. $Q_{m}^{\left( d\right) }(|D_{m}^{d}
\rangle\langle D_{m}^{d}|)=d-1$. \begin{figure}
\caption{(Color online) The witness $Q_{m}^{\left( d\right) }$ can quantify the genuine multipartite entanglement with the GME dimensionality $f_\rho$, where $f_\rho \le d$ . A state is biseparable iff $f_\rho=1$ and GM entangled iff $f_\rho\geq2$. For Dicke states it holds $Q_{m}^{\left( d\right) }\left( \left\vert D_{m}^{f}\right\rangle \left\langle D_{m}^{f}\right\vert \right) =f-1$.}
\label{fig. f_dimensional_gme_entanglement}
\end{figure}
If we can find a proper $R$, as a result of theorem \ref{theorem. lower bound of GME measure} that uses the Dicke state coefficients, we can connect a lower bound of the measure $E_{m}$ with the GME witness $Q_{m}^{\left( d\right) }\,\left( \rho\right) $. Indeed choosing the ordered subset $R_{\sigma}$ of the set of coefficients $\sigma$ used in (\ref{eq.: GME-witness formular}), i.e. \[ R_{\sigma}=\left\{ \left( \alpha^{a},\beta^{b}\right) \in\sigma:a\leq b\right\}\,, \] we immediately arrive at a lower bound on $E_{m}$ as \begin{equation} E_{m}\left( \rho\right) \geq m\sqrt{\frac{1}{\left\vert R_{\sigma} \right\vert - N_{R_{\sigma}} } }Q_{m}^{\left( d\right) }\,\left( \rho\right) \geq m\sqrt{\frac{1}{\left\vert R_{\sigma} \right\vert } }Q_{m}^{\left( d\right) }\,\left( \rho\right) ,\label{result. lower bound on GME measure} \end{equation} where $\left\vert R_{\sigma}\right\vert =\frac{1}{2}\left( d-1\right) ^{2}{\binom{n}{m}m}(n-m)$. In this case $N_{\eta}\leq m\left( n-m-1\right) +\Theta\left( d-3\right) \left( n-m\right) $, where $\Theta$ is a Heaviside step function.
\subsection{PPT-Witness and Our Witness} \begin{figure}
\caption{(Color online) PPT witness compared to our lower bounds (given in terms of the nonlinear witness $W_{R}\left( \rho\right) $): The set of $W_{R}\left( \rho\right) $ undetectable states denotes the set that is not detected by one specific $W_R(\rho)$ and is strictly larger than the set of PPT-states. However the set of states detected by $W_{R}\left( \rho\right) $ is strictly larger than the set detected by any standard PPT-witness. }
\label{fig. ppt_witness_and_vs_witness}
\end{figure}
Using the result on entanglement across bipartitions from the previous section we can explore the relation of our lower bounds to other bipartite entanglement witnesses. In our witness construction, the permutation operator $P_{\gamma}$ acting on a pure state is a $\gamma|\bar{\gamma}$-partial transpose operator, i.e. $P_\gamma|\psi\rangle\langle\psi|=(|\psi\rangle\langle\psi|)^{T_{\gamma|\bar{\gamma}}}$ (in the sense that our permutation operator now acts upon the index pairs of the coefficients of the pure state). It is thus intuitive to believe that there is certain connection between our witness and a PPT-witness \cite{pptwitness}. Indeed our witnesses are related to a standard PPT-witness construction (where the witnesses separate the convex set of states that are positive under partial transpose (PPT) from its complement). E.g. for diagonal GHZ states we can use the standard PPT-witness construction which goes as follows. For $\left\vert \text{GHZ}_{\eta_{1},\eta_{2}}\right\rangle :=\frac{1}{\sqrt{2}}\left( \left\vert \eta_{1}\right\rangle +\left\vert \eta_{2}\right\rangle \right) $
with $\eta_{1}+\eta_{2}=\left( d-1,\cdots,d-1\right) $, we can use the eigenvector belonging to the negative eigenvalue of the $\gamma|\bar{\gamma}$-partial transposed $\left\vert \text{GHZ}_{\eta_{1} ,\eta_{2}}\right\rangle \left\langle \text{GHZ}_{\eta_{1},\eta_{2}}\right\vert ^{T_{\gamma}}$ which we denote as $\left\vert \lambda_{\eta_{1},\eta_{2}}^{-}\right\rangle =\frac{1}{\sqrt{2}}\left( \left\vert \eta_{1}^{\gamma}\right\rangle -\left\vert \eta_{2}^{\gamma}\right\rangle \right) $. One can then construct the PPT-witness and write its expectation value as \begin{equation}
\Omega_{\text{ppt}}^{\gamma|\bar{\gamma}}\left( \rho,\left\vert \lambda_{\eta _{1},\eta_{2}}^{-}\right\rangle \right) =\text{Tr}\left( \left\vert \lambda _{\eta_{1},\eta_{2}}^{-}\right\rangle \left\langle \lambda_{\eta_{1},\eta_{2}
}^{-}\right\vert ^{T_{\gamma|\bar{\gamma}}}\rho\right) ,\label{eq. ppt-witness construction} \end{equation} For instance in the three-qubit case, \begin{align} \left\vert \lambda_{001,110}^{-}\right\rangle
\left\langle \lambda_{001,110}^{-}\right\vert ^{T_{1|23}}=\nonumber \end{align} \begin{align} \frac{1}{2}\left( \begin{array} [c]{cccccccc} 0 & 0 & & \cdots & \cdots & & 0 & 0\\ & 0 & & & & & -1 & \\ & & 1 & & & 0 & & \\ \vdots & & & 0 & 0 & & & \vdots\\ \vdots & & & 0 & 0 & & & \vdots\\ & & 0 & & & 1 & & \\ & -1 & & & & & 0 & \\ 0 & & & \cdots & \cdots & & & 0 \end{array} \right) . \end{align} With the PPT-witness construction in eq.(\ref{eq. ppt-witness construction}) we end up with the following PPT-witness expectation value \begin{align}
\Omega_{\text{ppt}}^{\gamma|\bar{\gamma}}\left( \rho,\left\vert \lambda_{GHZ} ^{-}\right\rangle \right) =\frac{1}{2}\left( \rho_{\eta_{1}^{\gamma}\eta _{1}^{\gamma}}+\rho_{\eta_{2}^{\gamma}\eta_{2}^{\gamma}}\right) -\operatorname{Re}\left( \rho_{\eta_{1}\eta_{2}}\right) . \end{align}
Under the fixed bipartition $\gamma|\bar{\gamma}$, we construct our witness by choosing $R=\left( \eta_{1},\eta_{2}\right) $ as \begin{equation} -W_{\left( \eta_{1},\eta_{2}\right) }^{\gamma}\left( \rho\right) =\sqrt{\rho_{\eta_{1}^{\gamma}\eta_{1}^{\gamma}}\rho_{\eta_{2}^{\gamma} \eta_{2}^{\gamma}}}-\left\vert \rho_{\eta_{1}\eta_{2}}\right\vert\,. \end{equation} It is obvious that $-W_{\left( \eta_{1},\eta_{2}\right) }^{\gamma}\left(
\rho\right) $ $\leq\Omega_{\text{ppt}}^{\gamma|\bar{\gamma}}\left( \rho,\left\vert \lambda_{GHZ}\right\rangle \right) $. Hence we say that the witness
$W_{R}\left( \rho\right) $ is stronger than the PPT-witness $\Omega _{\text{ppt}}^{\gamma|\bar{\gamma}}\left( \rho,\left\vert \lambda_{\eta_{1},\eta _{2}}^{-}\right\rangle \right) $.
The relation between our witness, the PPT-witness and the PPT-convex set is illustrated in fig.\ref{fig. ppt_witness_and_vs_witness}. For clearness we just draw two PPT-witnesses in the figure. For the $n$-qudit case there are $\frac{1}{2}d^{n}$ such eigenvectors $\left\vert
\lambda_{\eta_{1},\eta_{2}}^{-}\right\rangle $, corresponding to negative eigenvalues. Every witness $\Omega _{\text{ppt}}^{\gamma|\bar{\gamma}}\left( \rho,\left\vert \lambda_{\eta_{1},\eta _{2}}^{-}\right\rangle \right) $ is tangent to the set of PPT states (i.e. there exists one PPT state for which the witness yields zero). However also our witness $W_{R}\left( \rho\right)$ is zero for all these PPT states, i.e. our new witness detects more states than the traditional PPT-witness. \section{Conclusions} In conclusion we have presented a method to derive lower bounds on a measure of genuine multipartite entanglement. We show that in experimentally plausible scenarios (i.e. one knows which state one aims to produce) we can derive such lower bounds simply based on coefficients of the corresponding pure states. We also connected the lower bound construction to a framework of nonlinear entanglement witnesses developed in Refs.~\cite{Guehnewit,hmgh1,Dicke,hesgh1,shgh1}. These witnesses are experimentally feasible in terms of required local measurement settings. We provide further evidence in the bipartite case, where we also show that for certain families of mixed states our lower bounds are tight.\\ Some open questions remain, such as whether this general construction method will work for all kinds of states and how it can be generalized beyond just multi- and bipartite entanglement, but anything in between. We want to point out that recently also other authors have used a similar approach to bound this measure in the bipartite case \cite{hashemi} and for multipartite $W$ states \cite{severinima}.
{\em Acknowledgements:} We thank the QCI group in Bristol and Matteo Rossi for discussions. JYW, HK and DB acknowledge financial support of DFG (Deutsche Forschungsgemeinschaft). MH gratefully acknowledges support from the EC-project IP "Q-Essence" and the ERC Advanced Grant "IRQUAT". \begin{thebibliography}{99}
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\appendix*
\section{Proofs\label{appx. proof of result}}
\subsection{The Formulas Used in The Main Article}
\subsubsection{Reduced linear entropy of pure states}\label{linear entropy of pure state} Let $\left\vert \psi\right\rangle =\sum_{\eta\in\mathbb{N}_{d}^{\otimes n}} c_{\eta}\left\vert \eta\right\rangle $ be an n-qudit pure state. The linear entropy of $\left\vert \psi\right\rangle $ can be written as \begin{equation} S_{L}\left( \rho_{\gamma}\right) =\sum_{\eta_{1}\not =\eta _{2},\in\mathbb{N}_{d^{n}}} \left\vert c_{\eta_{1}}c_{\eta_{2}}-c_{\eta_{1}^{\gamma}}c_{\eta_{2}^{\gamma }}\right\vert ^{2}, \end{equation} where $\left( \eta_{1}^{\gamma},\eta_{2}^{\gamma}\right) =P_{\gamma}\left( \eta_{1},\eta_{2}\right) $.
\begin{proof}
The linear entropy regarding a specific partition $\gamma|\bar{\gamma}$ is defined as $S_{L}\left( \rho_{\gamma}\right) =2(1-tr\left( \rho_{\gamma} ^{2}\right))$, where $\rho_{\gamma}$ is the $\gamma$-reduced matrix of $\rho $. The trace of $\rho_{\gamma}$ is $tr\left( \rho_{\gamma}^{2}\right) =\sum_{\alpha_{1},\alpha_{2}\in H_{\gamma}}\left( \rho_{\gamma}\right) _{\alpha_{1}\alpha_{2}}\left( \rho_{\gamma}\right) _{\alpha_{2}\alpha_{1}}$, where $H_{\gamma}$ is the subspace of the reduction $\gamma$. We separate the summation into diagonal and off-diagonal parts. For the diagonal part we use the normalization condition to evaluate its value. \begin{align} & tr\left( \rho_{\gamma}^{2}\right) \nonumber\\ & =\sum_{\alpha_{1}=\alpha_{2}}\left( \rho_{\gamma}\right) _{\alpha _{1}\alpha_{1}}^{2}+\sum_{\alpha_{1}\not =\alpha_{2}}\left( \rho_{\gamma }\right) _{\alpha_{1}\alpha_{2}}\left( \rho_{\gamma}\right) _{\alpha _{2}\alpha_{1}}\nonumber\\ & =\left( \sum_{\alpha}(\rho_{\gamma})_{\alpha\alpha}\right) ^{2} -\sum_{\alpha_{1}\not =\alpha_{2}}\left( \rho_{\gamma}\right) _{\alpha _{1}\alpha_{1}}\left( \rho_{\gamma}\right) _{\alpha_{2}\alpha_{2} }\nonumber\\ & +\sum_{\alpha_{1}\not =\alpha_{2}}\left( \rho_{\gamma}\right) _{\alpha_{1}\alpha_{2}}\left( \rho_{\gamma}\right) _{\alpha_{2}\alpha_{1} }\nonumber\\ & =1-\sum_{\substack{\alpha_{1}\not =\alpha_{2}\in H_{\gamma}\\\beta _{1},\beta_{2}\in H_{\bar{\gamma}}}}\left\vert c_{\alpha_{1}\otimes\beta_{1} }\right\vert ^{2}\left\vert c_{\alpha_{2}\otimes\beta_{2}}\right\vert ^{2}\nonumber\\ & +\sum_{\substack{\alpha_{1}\not =\alpha_{2}\in H_{\gamma}\\\beta_{1} ,\beta_{2}\in H_{\bar{\gamma}}}}c_{\alpha_{1}\otimes\beta_{1}}c_{\alpha _{2}\otimes\beta_{1}}^{\ast}c_{\alpha_{2}\otimes\beta_{2}}c_{\alpha_{1} \otimes\beta_{2}}^{\ast}. \end{align} By exchanging the indices $\alpha_{1}$ and $\alpha_{2}$ one has \begin{align} tr\left( \rho_{\gamma}^{2}\right) & =1-\frac{1}{2}\sum_{\substack{\alpha _{1}\not =\alpha_{2}\in H_{\gamma}\\\beta_{1},\beta_{2}\in H_{\bar{\gamma}} }}\left\vert c_{\alpha_{1}\otimes\beta_{1}}c_{\alpha_{2}\otimes\beta_{2} }-c_{\alpha_{1}\otimes\beta_{2}}c_{\alpha_{2}\otimes\beta_{1}}\right\vert ^{2}\nonumber\\ & =1-\frac{1}{2}\sum_{\eta_{1},\eta_{2}\in\mathbb{N}_{d^{n}}}\left\vert c_{\eta_{1}}c_{\eta_{2}}-c_{\eta_{1}^{\gamma}}c_{\eta_{2}^{\gamma}}\right\vert ^{2}, \end{align} where $\eta=\alpha\otimes\beta$ and $\left( \eta_{1}^{\gamma},\eta _{2}^{\gamma}\right) =P_{\gamma}\left( \eta_{1},\eta_{2}\right) $. The linear entropy is then calculated to \begin{equation} S_{L}\left( \rho_{\gamma}\right) =\sum_{\eta_{1}\not =\eta_{2},\in \mathbb{N}_{d^{n}}}\left\vert c_{\eta_{1}}c_{\eta_{2}}-c_{\eta_{1}^{\gamma} }c_{\eta_{2}^{\gamma}}\right\vert ^{2}. \end{equation}
\end{proof}
\subsubsection{An Important Inequality} \label{appx. ineq.sqrt} The following is an inequality, which is crucial for derivation of the prefactor
$\sqrt{\frac{1}{|R|-N_R}}$ in the theorem \ref{theorem. lower bound of GME measure}: \begin{equation} \left\vert I\right\vert \sum_{i\in I}\left\vert a_{i}\right\vert ^{2} \geq\left\vert \sum_{i\in I}a_{i}\right\vert ^{2}. \label{eq. ineq. sqrt} \end{equation}
\begin{proof}
We prove this inequality by constructing two vectors as follows (using $|I|=n$) \begin{equation} \vec{x}=\left( \begin{array} [c]{c} \left. \begin{array} [c]{c} a_{1}\\ \vdots\\ a_{1} \end{array} \right\} \text{n times}\\ \vdots\\ \left. \begin{array} [c]{c} a_{n}\\ \vdots\\ a_{n} \end{array} \right\} \text{n times} \end{array} \right) ,\vec{y}=\left( \begin{array} [c]{c} \begin{array} [c]{c} a_{1}^{\ast}\\ \vdots\\ a_{n}^{\ast} \end{array} \\ \vdots\\% \begin{array} [c]{c} a_{1}^{\ast}\\ \vdots\\ a_{n}^{\ast} \end{array} \end{array} \right) . \end{equation} The right hand side of \ref{eq. ineq. sqrt} can be written as the scalar product of $\vec{x}$ and $\vec{y}$. \begin{equation} \left\vert \sum_{i\in I}a_{i}\right\vert ^{2}=\sum_{i,j\in I}a_{i}a_{j}^{\ast }=\left\vert \vec{x}\cdot\vec{y}\right\vert . \end{equation} According to the Cauchy-Schwarz inequality, one can derive \begin{equation} \left\vert I\right\vert \sum a_{i}^{2}=\left\vert \vec{x}\right\vert \cdot\left\vert \vec{y}\right\vert \geq\left\vert \vec{x}\cdot\vec {y}\right\vert =\left\vert \sum_{i\in I}a_{i}\right\vert ^{2} . \end{equation}
\end{proof}
\subsection{Proof of Theorem \ref{theorem. lower bound of GME measure} and Approach of Construction of a GME Witness} \label{appx. proof of lower bound on GME measure}
Firstly one can estimate the lower bound on $S_{L}\left( \rho_{\gamma }\right) $ by summing its elements over a selected Region $R$, and dropping the other non-negative summands (i.e. lower bounding them with $0$), \begin{align} S_{L}\left( \rho_{\gamma}^{i}\right) & \geq4\sum_{\left( \eta_{1} ,\eta_{2}\right) \in R}\left\vert c_{\eta_{1}}^{i}c_{\eta_{2}}^{i} -c_{\eta_{1}^{\gamma}}^{i}c_{\eta_{2}^{\gamma}}^{i}\right\vert ^{2} \label{eq.: linear_entropy_lower_bound_1}\\ & =4\sum_{\left( \eta_{1},\eta_{2}\right) \in R\backslash R^{\gamma} }\left\vert c_{\eta_{1}}^{i}c_{\eta_{2}}^{i}-c_{\eta_{1}^{\gamma}}^{i} c_{\eta_{2}^{\gamma}}^{i}\right\vert ^{2}.\nonumber \end{align} Here we add a prefactor $4$ in eq.(\ref{eq.: linear_entropy_lower_bound_1}), since the symmetric factor of all $(\eta_{1},\eta_{2})$ equals $4$. That means for every $(\eta_{1},\eta_{2})$ there are three other $\left( \tilde{\eta
}_{1},\tilde{\eta}_{2}\right) $ having the same value of $|c_{\tilde{\eta }_{1}}c_{\tilde{\eta}_{2}}-c_{\tilde{\eta}_{1}^{\gamma}}c_{\tilde{\eta}
_{2}^{\gamma}}|$ as $(\eta_{1},\eta_{2})$. Here we choose a non-degenerate vector basis set $R$, and therefore need a prefactor $4$ in the lower bound. The set $R^{\gamma}$ is the subset of $R$, whose elements do not contribute to the linear entropy, i.e. $R^{\gamma}:=\left\{ \left( \eta_{1},\eta _{2}\right) \in R:(\eta_{1}^{\gamma},\eta_{2}^{\gamma})=\left( \eta_{1} ,\eta_{2}\right) \text{ or }\left( \eta_{2},\eta_{1}\right) \right\} $. Now we use the inequality (\ref{eq. ineq. sqrt}) to bound the square root of $S_{L}\left( \rho_{\gamma}^{i}\right) $. \begin{align} S_{L}\left( \rho_{\gamma}^{i}\right) & \geq\frac{4}{\left\vert R\backslash R^{\gamma}\right\vert }\left( \sum_{\eta_{1},\eta_{2}\in R}\left\vert c_{\eta_{1}}^{i}c_{\eta_{2}}^{i}-P_{\gamma}c_{\eta_{1}}^{i}c_{\eta} ^{i}\right\vert \right) ^{2}\label{eq. ineq._for_linear_entropy},\\ & \Downarrow\nonumber\\ \sqrt{S_{L}\left( \rho_{\gamma}^{i}\right) } & \geq2\sqrt{\frac {1}{\left\vert R\right\vert -\left\vert R^{\gamma}\right\vert }}\sum_{\left( \eta_{1},\eta_{2}\right) \in R}\left\vert c_{\eta_{1}}^{i}c_{\eta_{2}} ^{i}-c_{\eta_{1}^{\gamma}}^{i}c_{\eta_{2}^{\gamma}}^{i}\right\vert .\label{result. lower bound on linear entropy} \end{align} According to eq.(\ref{definitionroof}) together with eq.(\ref{result. lower bound on linear entropy}), the lower bound reads {\footnotesize { \begin{align} & E_{m}\geq\nonumber\\ & 2\inf_{\left\{ p_{i},\left\vert \psi_{i}\right\rangle \right\} }\sum _{i}p_{i}\left[ \sqrt{\frac{1}{\left\vert R\right\vert -\left\vert R^{\gamma_{i}}\right\vert }}\sum_{\eta_{1},\eta_{2}\in R}\left( \left\vert c_{\eta_{1}}^{i}c_{\eta_{2}}^{i}\right\vert -\left\vert c_{\eta_{1} ^{\gamma_{i}}}^{i}c_{\eta_{2}^{\gamma_{i}}}^{i}\right\vert \right) \right], \end{align} }} where $\gamma_{i}$ is the partition in which the linear entropy $S_{L}\left( \left\vert \psi_{i}\right\rangle \left\langle \psi _{i}\right\vert _{\gamma}\right) $ of $\left\vert \psi_{i}\right\rangle
\left\langle \psi_{i}\right\vert $ has its minimum. By defining the normalization factor $N_{R}:=\min_{\gamma}\left\vert R^{\gamma}\right\vert$ , which is the minimal value of $\left\vert R^{\gamma}\right\vert $ over all possible bipartitions $\left\{ \gamma|\bar{\gamma}\right\} $, we can extract the prefactor from the convex roof summation.{\footnotesize { \begin{align} & E_{m}\geq\nonumber\\ & 2\sqrt{\frac{1}{\left\vert R\right\vert -N_{R}}}\inf_{\left\{ p_{i},\left\vert \psi_{i}\right\rangle \right\} }\sum_{i}p_{i}\left[ \sum_{\eta_{1},\eta_{2}\in R}\left( \left\vert c_{\eta_{1}}^{i}c_{\eta_{2} }^{i}\right\vert -\left\vert c_{\eta_{1}^{\gamma_{i}}}^{i}c_{\eta_{2} ^{\gamma_{i}}}^{i}\right\vert \right) \right]. \label{eq. proof of general lower bound on GME measure} \end{align} }}
The most difficult part of detecting entanglement of mixed states is a result of the mixing of the decomposition coefficients $c_{\eta_{1}}^{i}c_{\eta_{2}}^{i} $. In the lab we have only the information about the mixed density matrix element
$\rho_{\eta_{1}\eta_{2}}$ but not $c_{\eta_{1}}^{i}c_{\eta_{2}}^{i}$, therefore we must exchange the two summations in eq.(\ref{eq. proof of general lower bound on GME measure}), and mix the coefficients $c_{\eta_{1}}^{i}c_{\eta_{2}}^{i}$ into density matrix elements. Therefore we estimate the summands with a bound, which is independent of the specific partition $\gamma_{i}|\bar{\gamma}_{i}$, by adding a summation of non-positive terms $\sum_{R^{\gamma_{i}}}\left[ \left\vert c_{\eta_{1}}^{i}c_{\eta_{2}}^{i}\right\vert -\frac{1}{2}\left( \left\vert c_{\eta_{1}}^{i}\right\vert ^{2}+\left\vert c_{\eta_{2}}^{i}\right\vert ^{2}\right) \right] $ into the summands. {\footnotesize { \begin{align} & \sum_{\left( \eta_{1},\eta_{2}\right) \in R\backslash R^{\gamma_{i}} }\left( \left\vert c_{\eta_{1}}^{i}c_{\eta_{2}}^{i}\right\vert -\left\vert c_{\eta_{1}^{\gamma_{i}}}^{i}c_{\eta_{2}^{\gamma_{i}}}^{i}\right\vert \right) \nonumber\\ \geq & \sum_{\left( \eta_{1},\eta_{2}\right) \in R}\left( \left\vert c_{\eta_{1}}^{i}c_{\eta_{2}}^{i}\right\vert -\sum_{\gamma\in\Gamma\left( \eta_{1},\eta_{2}\right) }\left\vert c_{\eta_{1}^{\gamma}}^{i}c_{\eta _{2}^{\gamma}}^{i}\right\vert \right) \nonumber\\ &-\frac{1}{2}\sum_{R^{\gamma_{i}}} \left( \left\vert c_{\eta_{1}}^{i}\right\vert ^{2}+\left\vert c_{\eta_{2}}^{i}\right\vert ^{2}\right) \nonumber\\ \geq & \sum_{\left( \eta_{1},\eta_{2}\right) \in R}\left( \left\vert c_{\eta_{1}}^{i}c_{\eta_{2}}^{i}\right\vert -\sum_{\gamma\in\Gamma\left( \eta_{1},\eta_{2}\right) }\left\vert c_{\eta_{1}^{\gamma}}^{i}c_{\eta _{2}^{\gamma}}^{i}\right\vert \right) -\frac{1}{2}\sum_{\eta\in I\left( R\right) }n_{\eta}^{\gamma_{i}}\left\vert c_{\eta}^{i}\right\vert^{2}, \label{eq. proof. lower bound on GME measure} \end{align} }} where $I\left( R\right) :=\left\{ \eta\in\mathbb{N}_{d}^{\otimes n}:\exists\left( \eta,\eta^{\prime}\right) \text{ or }\left( \eta^{\prime },\eta\right) \in R\right\} $ is the set of indices contained in the set $R$,
$\Gamma\left( \eta_{1},\eta_{2}\right) =\left\{ \gamma|P\left( \eta _{1},\eta_{2}\right) \not \in R\right\} $ and $n_{\eta}^{\gamma_{i}}$ is the number of vector pairs in $R^{\gamma_{i}}$ containing index $\eta$. In order to eliminate the dependence of the partition $\gamma^{i}$, we define the maximal value of $n_{\eta}^{\gamma}$ over all possible partitions $\left\{ \gamma
|\bar{\gamma}\right\} $ as $N_{\eta}:=\max_{\gamma}n_{\eta}^{\gamma}$. Then one can estimate the GME measure with eq.(\ref{eq. proof of general lower bound on GME measure} and \ref{eq. proof. lower bound on GME measure}) as{\footnotesize { \begin{align} & E_{m}\left( \rho\right) \nonumber\\ \geq & 2\sqrt{\frac{1}{\left\vert R\right\vert -N_{R}}}\sum_{\eta_{1} ,\eta_{2}\in R}\left[ \inf_{\left\{ p_{i},\psi_{i}\right\} }\sum_{i} p_{i}\left( \left\vert c_{\eta_{1}}^{i}c_{\eta_{2}}^{i}\right\vert -\sum_{\gamma\in\Gamma\left( \eta_{1},\eta_{2}\right) }\left\vert c_{\eta_{1}^{\gamma}}^{i}c_{\eta_{2}^{\gamma}}^{i}\right\vert \right) \right. \nonumber\\ & \left. -\frac{1}{2}\sum_{\eta\in I\left( R\right) }N_{\eta}\left\vert c_{\eta}^{i}\right\vert ^{2}\right]. \label{eq. proof of general lower bound on GME measure-1} \end{align} }} Now one can safely exchange the summation in eq.(\ref{eq. proof of general lower bound on GME measure-1}) and lower bound it with the triangle inequality (i.e. $\sum_{p_{i}}p_{i}\left\vert c_{\eta_{1} }^{i}c_{\eta_{2}}^{i}\right\vert \geq\left\vert \rho_{\eta_{1}\eta_{2} }\right\vert $) and the Cauchy-Schwarz inequality (i.e. $\sum_{p_{i}} p_{i}\left\vert c_{\eta_{1}^{\gamma}}^{i}c_{\eta_{2}^{\gamma}}^{i}\right\vert \leq\sqrt{\rho_{\eta_{1}^{\gamma}\eta_{1}^{\gamma}}\rho_{\eta_{2}^{\gamma} \eta_{2}^{\gamma}}}$). Finally we arrive at the result
{\footnotesize { \begin{align} & E_{m}\left( \rho\right) \nonumber\\ \geq & 2\sqrt{\frac{1}{\left\vert R\right\vert -N_{R}}}\left[ \sum_{\eta _{1},\eta_{2}\in R_{\symbol{126}}}\left( \left\vert \rho_{\eta_{1}\eta_{2} }\right\vert -\sum_{\gamma\in\Gamma\left( \eta_{1},\eta_{2}\right) } \sqrt{\rho_{\eta_{1}^{\gamma}\eta_{1}^{\gamma}}\rho_{\eta_{2}^{\gamma}\eta _{2}^{\gamma}}}\right) \right. \nonumber\\ & \left. -\frac{1}{2}\sum_{\eta\in I\left( R\right) }N_{\eta} \rho_{\eta\eta}\right] ,\label{eq. proof of witness constructions} \end{align}
}} where $\rho_{\eta_{1}\eta_{2}}:=\langle\eta_{1}|\rho|\eta_{2}\rangle$.
Above is the proof of theorem \ref{theorem. lower bound of GME measure} in the case of $N_{R}:=\min_{\gamma}\left\vert R^{\gamma}\right\vert$. For the choice of $N_{R}:=\max_{\gamma}\left\vert R^{\gamma}\right\vert$, one just needs to calculate $\max_{\gamma}\left\vert R^{\gamma}\right\vert$ at the first step, i.e. eq.(\ref{eq.: linear_entropy_lower_bound_1}), then pick up $|R|-\max_{\gamma}\left\vert R^{\gamma}\right\vert$ elements from $R\backslash R^{\gamma}$ as summation region in the second line and then repeat the whole proof above. At the end we will attain the same expression for the lower bound on $E_{m}$ as eq.(\ref{eq. proof of witness constructions}), but with different $N_\eta$ from the ones before $N_{R}=\min_{\gamma}\left\vert R^{\gamma}\right\vert$. $N_\eta$ in this maximum choice is greater or equal to the one derived in the minimal-case. In the four-qubit singlet example in sec.\ref{sec. four-qubit-example}, the value of $N_\eta$ is exactly the same for both choices. Therefore we choose the maximum, i.e. $N_R=2$, to get a tighter lower bound on $E_m$.
\subsection{Explicit decomposition of the bipartite witness into local observables}\label{GellMann} The measurements needed to ascertain the relevant density matrix elements in the bipartite scenario can be performed in a basis consisting of a tensor product of the generalized Gell-Mann matrices. We continue to provide for each of the density matrix elements above their respective coefficients. The density matrix elements are either off diagonal elements or diagonal elements.
The off-diagonal elements can be obtained by expectation values of the symmetric and antisymmetric generalized Gell-Mann matrices:
\begin{align}
\Lambda_{s}^{12} & = & \left(\begin{smallmatrix} 0&1&0\\ 1&0&0 \\ 0&0&0 \end{smallmatrix}\right),&
\Lambda_{s}^{13} & = & \left(\begin{smallmatrix} 0&0&1\\ 0&0&0 \\ 1&0&0 \end{smallmatrix}\right),&
\Lambda_{s}^{23} & = & \left(\begin{smallmatrix} 0&0&0\\ 0&0&1 \\ 0&1&0 \end{smallmatrix}\right),&\\
\Lambda_{a}^{12} & = & \left(\begin{smallmatrix} 0&-i&0\\ i&0&0 \\ 0&0&0 \end{smallmatrix}\right),&
\Lambda_{a}^{13} & = & \left(\begin{smallmatrix} 0&0&-i\\ 0&0&0 \\ i&0&0 \end{smallmatrix}\right),&
\Lambda_{a}^{23} & = & \left(\begin{smallmatrix} 0&0&0\\ 0&0&-i \\ 0&-i&0 \end{smallmatrix}\right).&
\end{align}
They can be written as follows:
\begin{align}
\Re e\left[\left\langle 00 |\rho |11 \right\rangle\right] & =
\frac{1}{2}\left\langle\Lambda_{s}^{12} \otimes \Lambda_{s}^{12}-\Lambda_{a}^{12} \otimes \Lambda_{a}^{12}\right\rangle , \\
\Re e\left[\left\langle 00 |\rho| 22 \right\rangle\right] & =
\frac{1}{2}\left\langle\Lambda_{s}^{13} \otimes \Lambda_{s}^{13}-\Lambda_{a}^{13} \otimes \Lambda_{a}^{13}\right\rangle , \\
\Re e\left[\left\langle 11 |\rho| 22 \right\rangle\right] & =
\frac{1}{2}\left\langle\Lambda_{s}^{23} \otimes \Lambda_{s}^{23}-\Lambda_{a}^{23} \otimes \Lambda_{a}^{23}\right\rangle .
\end{align}
We now consider the terms obtained via the diagonal generalized Gell-Mann matrices.
$\Lambda_{d}^{0} = \left(\begin{smallmatrix} 1&0&0\\ 0&1&0 \\ 0&0&1 \end{smallmatrix}\right)$,
$\Lambda_{d}^{1} = \left(\begin{smallmatrix} 1&0&0\\ 0&-1&0 \\ 0&0&0 \end{smallmatrix}\right)$,
$\Lambda_{d}^{2} = \frac{1}{\sqrt{3}}\left(\begin{smallmatrix} 1&0&0\\ 0&1&0 \\ 0&0&-2 \end{smallmatrix}\right)$.
We will expand the soughtafter terms into coefficients, utilizing the following basis:
\begin{align}
b=\left(\begin{matrix}
\Lambda_{d}^{0} \otimes \Lambda_{d}^{0} \\
\Lambda_{d}^{0} \otimes \Lambda_{d}^{1} \\
\Lambda_{d}^{0} \otimes \Lambda_{d}^{2} \\
\Lambda_{d}^{1} \otimes \Lambda_{d}^{0} \\
\Lambda_{d}^{1} \otimes \Lambda_{d}^{1} \\
\Lambda_{d}^{1} \otimes \Lambda_{d}^{2} \\
\Lambda_{d}^{2} \otimes \Lambda_{d}^{0} \\
\Lambda_{d}^{2} \otimes \Lambda_{d}^{1} \\
\Lambda_{d}^{2} \otimes \Lambda_{d}^{2} \\
\end{matrix}\right).
\end{align}
For further reference the coefficients are given as:
\begin{align*}
\left\langle 01| \rho| 01 \right\rangle & = \left\langle b*\left(
\frac{1}{9},\frac{1}{6},\frac{1}{6\sqrt{3}},
-\frac{1}{6},-\frac{1}{4},-\frac{1}{8\sqrt{3}},\frac{1}{6\sqrt{3}},\frac{1}{4\sqrt{3}},\frac{1}{12}
\right)\right\rangle,\\
\left\langle 10 |\rho |10 \right\rangle & = \left\langle b*\left(
\frac{1}{9},-\frac{1}{6},\frac{1}{6\sqrt{3}},\frac{1}{6},-\frac{1}{4},\frac{1}{8\sqrt{3}},\frac{1}{6\sqrt{3}},-\frac{1}{4\sqrt{3}},\frac{1}{12}
\right)\right\rangle,\\
\left\langle 02 |\rho |02 \right\rangle & = \left\langle b*\left(
\frac{1}{9},\frac{1}{6},\frac{1}{6\sqrt{3}},0,0,0,-\frac{1}{3\sqrt{3}},-\frac{1}{2\sqrt{3}},-\frac{1}{6}
\right)\right\rangle,\\
\left\langle 20| \rho| 20 \right\rangle & =
\left\langle b*\left(
\frac{1}{9},0,-\frac{1}{3\sqrt{3}},\frac{1}{6},0,-\frac{1}{4\sqrt{3}},\frac{1}{6\sqrt{3}},0,-\frac{1}{6}
\right)\right\rangle,\\
\left\langle 12 |\rho |12 \right\rangle & = \left\langle b*\left(
\frac{1}{9},-\frac{1}{6},\frac{1}{6\sqrt{3}},0,0,0,-\frac{1}{3\sqrt{3}},\frac{1}{2\sqrt{3}},-\frac{1}{6}
\right)\right\rangle,\\
\left\langle 21| \rho |21 \right\rangle & = \left\langle b*\left(
\frac{1}{9},0,-\frac{3}{3\sqrt{3}},-\frac{1}{6},0,\frac{1}{4\sqrt{3}},\frac{1}{6\sqrt{3}},0,-\frac{1}{6}
\right)\right\rangle.
\end{align*}
\subsection{Explicit form of the GME witness $Q_m^{(d)}$}\label{DickeWitness} Here we recall the explicit form of the nonlinear witness from Ref.~\cite{shgh1}. Using the notation for Dicke states introduced in section \ref{Dicke} we arrive at the following lower bound \begin{widetext} \begin{equation} Q_{m}^{\left( d\right) }=\frac{1}{m}\left[ \sum_{l,l^{\prime}=0}^{d-2} \sum_{\sigma}\left( \left\vert \left\langle \alpha^{l}\left\vert \rho\right\vert \beta^{l^{\prime}}\right\rangle \right\vert -\sum_{\delta \in\Delta}\sqrt{\left\langle \alpha^{l}\right\vert \otimes\left\langle \beta^{l^{\prime}}\right\vert P_{\delta}^{\dagger}\rho^{\otimes2}P_{\delta }\left\vert \alpha^{l}\right\rangle \otimes\left\vert \beta^{l^{\prime} }\right\rangle }\right) -N_{D}\sum_{l=0}^{d-2}\sum_{\alpha}\left\langle \alpha^{l}\left\vert \rho\right\vert \alpha^{l}\right\rangle \right], \label{eq.: GME-witness formular} \end{equation} with \begin{align} m & \in\left\{ 1,\cdots,\left\lfloor n/2\right\rfloor \right\} ,N_{D}=\left( d-1\right) m\left( n-m-1\right) \nonumber,\\ \sigma & :=\left\{ \left( \alpha,\beta\right) :\left\vert \alpha\cap \beta\right\vert =m-1\right\} \nonumber,\\ \Delta & :=\left\{ \begin{array} [c]{cc} \alpha & ,l'=l\\
\left\{ \delta|\delta\subset\overline{\alpha\backslash\beta}\right\} & ,l'<l\\
\left\{ \delta|\delta\subset\overline{\beta\backslash\alpha}\right\} & ,l'>l \end{array} \right. .\label{def. GME witness - permutation set} \end{align} \end{widetext} The properties of this witness are discussed in the main text.
\end{document} |
\begin{document}
\title { About a conjecture for uniformly isochronous polynomial centers }
\author{ Evgenii P. Volokitin \thanks{Partially supported by Grant 05--01--00302 from the Russian Foundation of Basic Research.}\\ Sobolev Institute of Mathematics, Novosibirsk, 630090, Russia\\ e-mail: volok@math.nsc.ru } \date{} \maketitle
\begin{abstract} We study a specific family of uniformly isochronous polynomial systems. Our results permit to solve a problem about centers of such systems. \end{abstract}
Classification: Primary 34C05; Secondary 34C25
Keywords: center conditions; isochronicity; commutativity
\quad
{\bf 1.} Consider the planar autonomous system of ordinary differential equations \refstepcounter{equation} \label{1} $$ \begin{array} {ll} \dot x = -y + x H (x,y),\\ \dot y = x +y H (x,y),\\ \end{array} \eqno{(\ref{1})} $$ where $H(x,y)$ is a polynomial in $x$ and $y$ of degree $n$, and $H(0,0)=0$. This system has only one singular point at $O(0,0)$ which is the center of the linear part of the system. The solutions of this system move around the origin with constant angular speed, and the origin is so a uniformly isochronous singular point.
The problem of characterizing uniformly isochronous centers has attracted attention of several authors; see \cite{1}--\cite{4} and references therein. In particular, the following problem was posed:
{\it It is true that all centers for uniformly isochronous polynomial systems are either reversible or admit a nontrivial polynomial commuting system? }
The problem first appeared in \cite{2} and it was mentioned as one of the open questions in \cite{3}. It was also marked by the reviewers in Zbl. Math. 1037.34024 and in MR1963468 (2004b:34090). We prove the following proposition which permits to give a negative answer to the problem.
{\bf Theorem 1.} Let a uniformly isochronous polynomial system has the form \refstepcounter{equation} \label{2} $$ \begin{array} {ll} \dot x = -y + x Q (x,y) \sum_{i=0}^{m} a_i (x^2+y^2)^i,\\ \dot y = x +y Q (x,y) \sum_{i=0}^{m} a_i (x^2+y^2)^i, \end{array} \eqno{(\ref{2})} $$ where $Q(x,y)$ is a homogeneous polynomial in $x,y$ of degree $k$ and \refstepcounter{equation} \label{3} $$ \int_0^{2\pi} Q(\cos \vartheta, \sin \vartheta) d\vartheta=0. \eqno{(\ref{3})} $$ Then the origin is a center of (\ref{2}). The center is of type $B^\nu$ with $\nu \leq k$, and a <<generic>> center is of type $B^1$ when $k$ is odd or of type $B^2$ when $k$ is even.
{\bf Proof.} System (\ref{2}) can be written as a single separable equation \refstepcounter{equation} \label{4} $$ \frac{d \varrho}{d \vartheta}= \varrho^{k+1} Q(\cos \vartheta, \sin \vartheta) R(\varrho) \eqno{(\ref{4})} $$ with $\varrho, \vartheta$ polar coordinates and $R(\varrho)= \sum_{i=0}^{m} a_i \varrho^{2i}$.
Equation (\ref{4}) has a solution $\varrho \equiv 0$ which is defined for all $\vartheta$. Therefore every solution $\varrho(\vartheta)$ with the initial value $\varrho(0)=\varrho_0$ where $\varrho_0>0$ is small enough is defined for $\vartheta \in [0, 2\pi]$ and satisfies the condition \refstepcounter{equation} \label{5} $$ \int_0^{\vartheta} Q(\cos \varphi, \sin \varphi) d\varphi= \int_{\varrho_0}^{\varrho(\vartheta)} \frac{dr}{r^{k+1} R(r)}. \eqno{(\ref{5})} $$
From (\ref{3}) we conclude that the solution is $2\pi$-periodic, so that the origin is a center. The first part of the theorem is proved.
By \cite{5}, the center of (\ref{2}) is of type $B^{\nu}$, and the boundary of the center region is the union of $\nu$ open unbounded trajectories with $\nu \leq n=k+2m$.
The circles $x^2+y^2=\varrho_i^2$ where $\varrho_i$ are roots of the equations $R(\varrho)=0$ are trajectories of (\ref{2}). All of them lie in the center region. $$ \epsfbox{fig1.eps} \qquad \qquad \qquad \epsfbox{fig2.eps} $$
\hspace{2cm} Fig. 1 \hspace{5.5cm} Fig. 2
Unbounded trajectories of (\ref{2}) correspond to unbounded solutions of (\ref{4}). Studing the behaviour of solution curves of (\ref{4}) for large $\varrho$, it can be shown that for every null isocline $\vartheta=\vartheta^*$ where solutions have a maximum there exist two solutions $\varrho_1^*(\vartheta), \varrho_2^*(\vartheta)$, for which the isocline is a vertical asymptote $$ \lim_{\vartheta \rightarrow \vartheta^*-0} \varrho_1^*(\vartheta)=+\infty, \lim_{\vartheta \rightarrow \vartheta^*+0} \varrho_2^*(\vartheta)=+\infty, $$ (see Fig. 1).
In this situation, there is a relevant equilibrium point at infinity in the intersection of the equator of the Poincar\'e sphere with the ray $x=\varrho \cos \vartheta^*, y=\varrho \sin \vartheta^*, \varrho>0$. The point has one hyperbolic sector with two separtrices which correspond to the solutions $\varrho_1^*(\vartheta), \varrho_2^*(\vartheta)$ (see Fig. 2). The boundary of the center region consists of such separatrices. The number $\nu$ of these equilibrium points coinsides with the number of the null isoclines of direction field (\ref{4}) where solutions has a maximum for large $\varrho$. These isoclines are vertical lines $\vartheta=\vartheta^*_i$, where the values of $\vartheta^*_i$ are determined from the conditions $Q(\cos\vartheta,\sin\vartheta)=0, 0\leq \vartheta <2\pi$. Hence we have the estimate $\nu\leq k$ when we describe type $B^{\nu}$ of the center of system (\ref{2}) \footnote{ If $k$ is even our trigonometric polynomial $Q(\cos\vartheta,\sin\vartheta)$ has a period equal to $\pi$ (but not $2\pi$ as it takes place for odd $k$). Therefore (\ref{4}) has an even number of the blocks discussed above and the relevant equilibrium points are disposed at the diameters of the Poincar\'e sphere. It may be noted that system (\ref{2}) is $O$-symmetric in this case.}. The upper bound $k$ can be attained by $\nu$. As an example we can consider system (\ref{2}) with $Q(\cos\vartheta,\sin\vartheta)=\sin k \vartheta$ and $a_i$ arbitrary real numbers.
In a <<generic>> situation, (\ref{4}) has no solution for which two different null isoclines are its asimptotes. Therefore in such a situation the solution curve which separates bounded and unbounded solutions has a minimum number of discontinuity points within $[0, 2\pi]$, it has one point when $k$ is odd, and it has two points when $k$ is even.
Hence a <<generic>> center is of type $B^1$ when $k$ is odd or type $B^2$ when $k$ is even (see Fig. 3). $$ \epsfbox{fig3.eps} $$ \begin{center} {Fig. 3} \end{center}
The theorem is proved.
{\bf Remark.} Theorem 2.1 from \cite{5} about centers of homogeneous systems is a particular case of our theorem for $m=0, a_0=1$.
The functions $$ f_1(x,y)=x^2+y^2, f_2(x,y)=\sum_{i=0}^{m} a_i (x^2+y^2)^i $$ are invariants for (\ref{2}) with respective cofactors $$ \begin{array}{ll} K_1(x,y)=2Q(x,y)\sum_{i=0}^{m} a_i (x^2+y^2)^i,\\ K_2(x,y)=2Q(x,y)\sum_{i=0}^{m} i a_i (x^2+y^2)^i. \end{array} $$
We have $$ \frac{k+2}{2} K_1(x,y) + K_2(x,y)= div, $$ where $div$ is the divergence of (\ref{2}). In this case the function $\mu(x,y)=f_1^{(k+2)/2}f_2$ is a reciprocal integrating factor of Darboux form\footnote {About algebraic invariants and Darboux's method of integration see \cite{7}, \cite{8}, for example.}. The factor gives information about our system. For instance, it may be used to find a first Darboux integral of (\ref{2}) \cite{6}. A first integral of (\ref{2}) may be found from (\ref{5}) also.
It is obvious that (\ref{2}) commutes with the system \refstepcounter{equation} \label{6} $$ \begin{array}{ll} \dot x = x (x^2+y^2)^{k/2} \sum_{i=0}^{m} a_i (x^2+y^2)^i,\\ \dot y = y (x^2+y^2)^{k/2} \sum_{i=0}^{m} a_i (x^2+y^2)^i. \end{array} \eqno{(\ref{6})} $$
If $k$ is even (\ref{6}) gives a polynomial commuting system without a linear part. If $k$ is odd we have a non-polynomial commuting system. Nevertheless a polinomial commuting system may exist in the case of odd $k$. For example, if (\ref{2}) is homogeneous ($m=0, a_0=1$) then there exists a polynomial system which commmutes with (\ref{2}) \cite{9}.
Using Theorem 1, we may consruct an example of an uniformly isochronous system which is not reversible and commutes with no polynomial system.
If a system is reversible then its trajectories are symmetric with respect a common symmetric line \footnote{ Necessary and sufficient conditions for reversibility of planar analytic vector fields were derived in \cite{10}.}. If a symmetric line of system (\ref{2}) is $x \sin \vartheta^{*} - y \cos \vartheta^{*}=0$, then the vertical line $\vartheta=\vartheta^{*}$ is the symmetric axis of the graph of the trygonometric polynomial $Q(\cos \vartheta, \sin \vartheta)$ and it is the symmetric axis of solution curves of vector field (\ref{4}).
The problem about conditions for the existence of polynomial commuting systems for uniformly isochronous polynomial systems was considered in \cite{2}, \cite{3}. In partucular, it was proved that system (\ref{1}) commutes with a polynomial system if anf only if the function $H(x,y)$ satisfies one of the following two conditions:
\noindent 1) \refstepcounter{equation} \label{7} $$ H(x,y)=P_{2l}(x,y)\sum_{j=0}^r a_j (x^2+y^2)^j \eqno(\ref{7}) $$ where $P_{2l}(x,y)$ is a homogeneous polynomial of degree $2l, l\geq 0$;
\noindent 2) there are homogeneos polynomials $\alpha_l, \beta_l$ of order $l$ ($l\leq n$, $l$ divides $n$), verifying $x \partial_y \beta_l-y \partial_x \beta_l=l \alpha_l$, such that \refstepcounter{equation} \label{8} $$ H(x,y)=\alpha_l \sum_{k=0}^{n/l-1} a_k \beta_l^k. \eqno(\ref{8}) $$
So, to consruct our example it is sufficient to take a system of the form (\ref{2}) where the homogeneous polynomial $Q(x,y)$ is of an odd degree\footnote {In this case (\ref{3}) is fulfilled and the function $H(x,y)$ is not of the form (\ref{7}).}, the graph of the trygonometric polynomial $Q(\cos \vartheta, \sin\vartheta)$ has no symmetric axes and the numbers $m, a_i$ are such that the function $$ H(x,y)=Q(x,y) \sum_{i=0}^{m} a_i (x^2+y^2)^i $$ is not of the form (\ref{8}).
Let us set $$ Q(x,y)=y^3-3x y^2+2x^2 y=y(x-y)(2x-y), m=1, a_0=a_1=1. $$ Then we have \refstepcounter{equation} \label{9} $$ \begin{array}{ll} \dot x=-y+x (y^3-3x y^2+2 x^2 y)(1+x^2+y^2),\\ \dot y=x+y (y^3-3x y^2+2 x^2 y)(1+x^2+y^2). \end{array} \eqno(\ref{9}) $$
According to Theorem 1, (\ref{9}) has a center (isochronous) at the origin.
The function $$ I(x,y)=\frac{r^6}{(1-3r^2-4x^3-3x y^2-3y^3-3r^3\arctan r)^2}, r^2=x^2+y^2. $$ is a first integral of (\ref{9}) obtained from (\ref{5}).
It is evident that the graph of $Q(\cos \vartheta, \sin\vartheta )$ has no symmetric axes \footnote{ It may be shown that if the graph of the homogeneous trygonometric polynomial of degree~3 $$ T_3(\vartheta)=a_1 \cos \vartheta+a_3 \cos 3\vartheta+b_1 \sin \vartheta+ b_3 \sin 3\vartheta $$ has a simmetric axes then its coefficients satisfy the condition $$ a_1 b_3(a_1^2-3b_1^2)=a_3 b_1(3a_1-b_1^2). $$ }, and therefore system (\ref{9}) is non-reversible.
It easy to veryfy that system (\ref{9}) may fail to commute with any nonprortional polynomial systems. This fact follows from the impossibility to present the function $$ H(x,y)=(y^3-3x y^2+2 x^2 y)(1+x^2+y^2)\equiv H_3(x,y)+H_5(x,y) $$ in the form (\ref{8}) but we can also prove it immediately.
Indeed, let system (\ref{9}) commutes with a polynomial system of degree $n$ \refstepcounter{equation} \label{10} $$ \begin{array}{ll} \dot x= R(x,y)\equiv R_1(x,y)+R_2(x,y)+\dots+R_n(x,y),\\ \dot y= S(x,y)\equiv S_1(x,y)+S_2(x,y)+\dots+S_n(x,y), \end{array} \eqno{(\ref{10})} $$ where $R_i(x,y), S_i(x,y)$ are homogeneous polynomials of degree $i$.
Then the Lie bracket between vector fields (\ref{9}), (\ref{10}) is equal to zero: $$ [(-y+x H(x,y), x+y H(x,y))^T, (R(x,y), S(x,y))^T]=(0,0)^T. $$
In particular, we have that terms of highest degree are equal to zero: $$ [(x H_5(x,y), y H_5(x,y))^T, (R_n(x,y), S_n(x,y))^T]=(0,0)^T. $$
After transformations taking into account Euler's theorem for homogeneous functions this equality may be written in the form $$ \begin{array}{ll} (x H_{5x}(x,y)+(1-n)H_5(x,y))R_n(x,y)+x H_{5y}(x,y) S_n(x,y)=0,\\ y H_{5x}(x,y) R_n(x,y)+(y H_{5y}(x,y)+(1-n)H_{5}(x,y)) S_n(x,y)=0. \end{array} $$
The linear system for determing the polynomials $R_n(x,y), S_n(x,y)$ has a nontrivial solution if its determinant $\Delta$ is equal zero: $$ \begin{array}{ll} \Delta\equiv& (x H_{5x}(x,y)+(1-n)H_5(x,y)) (y H_{5y}(x,y)+(1-n)H_{5}(x,y))-\\
&x y H_{5x}(x,y) H_{5y}(x,y)=0, \end{array} $$
Taking into account the fact that $x H_{5x}(x,y) + y H_{5y}(x,y) =5 H_{5}(x,y)$, we have $$ \Delta=(1-n)(6-n)H_{5}^2 (x,y)=0. $$
Therefore we must have that $n=6$ or $n=1$. Straightforward calculations using the software package {\it Mathematica} show that in this case the commuting system (\ref{10}) is proportional to system (\ref{9}) \footnote{According with (\ref{6}) system (\ref{9}) commutes with the system $$ \dot x=x(x^2+y^2)\sqrt{x^2+y^2}(1+x^2+y^2), \dot y=y(x^2+y^2)\sqrt{x^2+y^2}(1+x^2+y^2).\\ $$ }.
Hence system (\ref{9}) has a center but it is non-reversible and it commutes with no polynomial system nonproportional to it.
We derive that the answer to the question from \cite{2}, \cite{3} is negative.
{\bf 2.} We can generalize the first part of Theorem 1.
{\bf Theorem 2.} Let the polynomial $H(x,y)$ in (\ref{1}) has the form $$ H(x,y)=q(x,y) h(x^2+y^2, p(x,y)), $$ where $h(u,v)$ is a polynimial, $p(x,y), q(x,y)$ are homogeneous polynomials of the same degree $k$ and $q(x,y)=c(x p_{y}(x,y)-y p_{x}(x,y))$. Then the origin is a center of (\ref{1}).
{\bf Proof.}
In the case under study system (\ref{1}) can be written as a single equation of the form $$ \frac{d \varrho}{d \vartheta}= c \varrho^{k+1}h(\varrho^2,\varrho^{k} f(\vartheta)) f'(\vartheta) $$ with $\varrho, \vartheta$ polar coordinates and $f(\vartheta)=p(\cos\vartheta, \sin\vartheta)$.
It is clear that solutions of this equation are functions of $f(\vartheta)$. The function $f(\vartheta)$ is 2$\pi$-periodic. Then solutions with initial values which are small enough are 2$\pi$-periodic functions also. So, the origin is a center.
\end{document} |
\begin{document}
\maketitle
\begin{abstract} This article contains several results for $\lambda$-Robertson functions, i.e., analytic functions $f$ defined on the unit disk $\mathbb{D}$ satisfying $f(0) = f'(0)-1=0$ and $\textup{Re}\, e^{-i\lambda}\{1+zf''(z)/f'(z)\} > 0$ in $\mathbb{D}$, where $\lambda \in (-\pi/2,\pi/2)$. We will discuss about conditions for boundedness and quasiconformal extension of Robertson functions. In the last section we provide another proof of univalence for Robertson functions by using the theory of L\"owner chains. \end{abstract}
\section{Introduction}
\par Let $\mathcal{A}$ be the family of functions $f$ analytic in the unit disc $\mathbb{D}=\{z\in \mathbb{C}:\, |z|<1\}$ with the usual normalization $f(0)=f'(0)-1=0$, and $\mathcal{S}$ be the subclass of $\mathcal{A}$ consisting of functions univalent in $\mathbb{D}$.
Let $\lambda$ be a real constant between $-\pi/2$ and $\pi/2$. The curve $\gamma_{\lambda}(t)=\exp(te^{i\lambda})$, $t\in \mathbb{R}$, and its rotations $e^{i\theta}\gamma_{\lambda}(t)$, $\theta\in \mathbb{R}$, are called \textit{$\lambda$-spirals}. A domain $\Omega$ with $0\in \Omega$ is called \textit{$\lambda$-spirallike} (with respect to 0) if for every $w \in \Omega$, the $\lambda$-spiral which connects $w$ and 0 lies in $\Omega$. A function $f\in \mathcal{A}$ is said to be a \textit{$\lambda$-spirallike function} if $f$ maps $\mathbb{D}$ univalently onto a $\lambda$-spirallike domain and the class of such functions is denoted by $\mathcal{SP}(\lambda)$. Spirallike functions are introduced by \v Spa\v cek \cite{Spacek:1932} in 1933. We note that $0$-spirallike functions are precisely starlike functions.
It is known that a necessary and sufficient condition for $f\in \mathcal{A}$ to be in $\mathcal{SP}(\lambda)$ is that \[ \textup{Re}\, \left\{ e^{-i\lambda}\frac{zf'(z)}{f(z)} \right\} >0 \] for all $z \in \mathbb{D}$. In \cite{KimSugawa:pre01}, Kim and Sugawa introduce the notion of \textit{$\lambda$-argument}. Let us set $\theta=\arg_{\lambda} w$ if $w\in e^{i\theta}\gamma_{\lambda}(\mathbb{R})$. We note that $\arg_{0}w=\arg w$. For some more properties of $\lambda$-argument, the reader may be referred to \cite{KimSugawa:pre01}. By utilizing $\lambda$-argument, another equivalence can be obtained \[ f\in \mathcal{SP}(\lambda) \Leftrightarrow \frac{\partial}{\partial\theta}\left(\arg_{\lambda}f(re^{i\theta})\right)>0 \quad (\theta \in \mathbb{R},\,0<r<1). \] For general references about spirallike functions, see e.g. \cite{Duren:1983} or \cite{AhujaSilverman:1991}.
A function $f\in \mathcal{A}$ is said to be a \textit{$\lambda$-Robertson function} \cite{Kulshrestha:1976} if $f$ satisfies \[ \textup{Re}\, \left\{ e^{-i\lambda}\left(1+\frac{zf''(z)}{f'(z)}\right) \right\} >0 \] for all $z \in \mathbb{D}$. Let $\mathcal{R}(\lambda)$ be the set of those functions. The definition of $\lambda$-Robertson functions shows immediately that $\mathcal{R}(0)$ is precisely the class of convex functions which is usually denoted by $\mathcal{K}$. Furthermore in view of the definitions of spirallike and Robertson functions, for a function $f\in \mathcal{A}$ the following relations are true; \begin{eqnarray} \label{relationship} f\in \mathcal{R}(\lambda) &\Leftrightarrow& zf'(z)\in \mathcal{SP}(\lambda)\\[5pt] &\Leftrightarrow& \int_{0}^{z}f'(\zeta)^{\alpha}d\zeta\in \mathcal{K}\nonumber\\[5pt] &\Leftrightarrow& \frac{\partial}{\partial\theta}\left[\arg_{\lambda}\left(\frac{\partial}{\partial\theta}f(re^{i\theta})\right)\right]>0 \quad (\theta\in\mathbb{R},\,0<r<1),\nonumber \end{eqnarray} where $\alpha=e^{-i\lambda}/\cos\lambda$. A distinguished member of $\mathcal{R}(\lambda)$ is \begin{equation}\label{extremal} f_{\lambda}(z)=\frac{(1-z)^{1-2e^{i\lambda}\cos\lambda}-1}{2e^{i\lambda}\cos\lambda -1}. \end{equation}
The class $\mathcal{R}(\lambda)$ was first introduced by Robertson \cite{Robertson:1969}. He showed that all functions in $\mathcal{R}(\lambda)$ are univalent if $0 < \cos \lambda \leq x_{0}$, where $x_{0} =0.2034\cdots$ is the unique positive root of the equation $16x^{3} + 16 x^{2} + x -1 = 0$ (in the original paper $x_{0}$ is evaluated as $0.2315\cdots$ which seems to be erroneous \cite{KimSugawa:2007}). Later Libera and Ziegler \cite{LiberaZiegler:1972} and Chichra \cite{Chichra:1975} gave some improvements on the range of $\lambda$ for which $\mathcal{R}(\lambda) \subset \mathcal{S}$ by estimating the norm of the Schwarzian derivatives for the class $\mathcal{R}(\lambda)$. Finally Pfaltzgraff \cite{Pfaltzgraff:1975} showed that $\mathcal{R}(\lambda) \subset \mathcal{S}$ if $0 < \cos \lambda \leq 1/2$ or $\cos \lambda = 1$. This value is best possible. Indeed, Robertson also presented in \cite{Robertson:1969} a non-univalent function which belongs to $\mathcal{R}(\lambda)$ for each $\lambda$ in the range $1/2 < \cos \lambda < 1$ by making use of Roysters's example \cite{Royster:1965}
$f_{\mu}^{*}(z) = ((1-z)^{-\mu} - 1)/\mu$, where $\mu$ is a number which satisfies $\mu + 1 = |\mu + 1| e^{i\lambda}, |\mu| \leq 1, |\mu+1|>1$ and $|\mu-1|>1$.
The class of $\lambda$-Robertson functions has been investigated by various authors. Recently the class $\mathcal{R}(\lambda)$ is still an interesting topic in geometric function theory (e.g. \cite{PonnuYanagihara:2008}). Actually, under the relationship \eqref{relationship} many properties of Robertson functions follows from those of spirallike functions. For instance the coefficient estimate of $\mathcal{R}(\lambda)$ is an easy consequence of a result of Zamorski \cite{Zamorski:1960} (see also \cite{BajpaiM:1973}). For some more information about Robertson functions, the reader is referred to e.g. \cite[Section 8]{AhujaSilverman:1991}.
In the present paper we would like to give several new results for the $\lambda$-Robertson functions. In section 2 we will show that $\lambda$-Robertson functions are bounded whenever $\cos\lambda<1/\sqrt{2}$ which improves a result of Kim and Sugawa in \cite{KimSugawa:2007}. Quasiconformal extension criteria which are related with Robertson functions are shown in section 3. One of the criteria is also obtained by using the technique of L\"owner's theory. We will discuss this problem in the last section and give an explicit L\"owner chain for Robertson functions.
\section{Boundedness of $\mathcal{R}(\lambda)$}
\subsection{Result and auxiliary lemma}
The boundedness of $\lambda$-Robertson function is analyzed by Kim and Sugawa \cite{KimSugawa:2007}. It can be stated as follows after being translated to our notations.
\begin{knownthm}[\cite{KimSugawa:2007}] $\lambda$-Robertson functions are bounded by a constant depending only on $\lambda$ when $\cos\lambda<1/2$. \end{knownthm}
They remark that the bound $1/2$ cannot be replaced by any number greater than $1/\sqrt{2}$ since the function given by ($\ref{extremal}$) is unbounded when $\cos\lambda>1/\sqrt{2}$. Our next result will verify that the bound $1/\sqrt{2}$ is best possible.
\begin{thm} Let $f\in \mathcal{R}(\lambda)$ with $\cos\lambda<1/\sqrt{2}$, then $f$ is bounded. \end{thm}
In order to prove the above result, the growth theorem of spirallike functions in \cite{Singh:1969} or \cite{AhujaSilverman:1991} is needed. Since those known forms are complicated there, we simplify them as follows.
\begin{lem}\label{lem2}
Let $f\in \mathcal{SP}(\lambda)$, then for $|z|=r<1$, we have \[
\Psi_{1}(r)\leq |f(z)|\leq \Psi_{2}(r) \] where \[
\Psi_{1}(r)=\left|P_{\lambda}(re^{i\theta_{1}}) \right|=\frac{r\exp\left(-\sin2\lambda\arcsin(r\sin\lambda)\right)}{(r\cos\lambda-\sqrt{1-r^2\sin^2\lambda})^{2\cos^2\lambda}} \] and \[
\Psi_{2}(r)=\left|P_{\lambda}(re^{i\theta_{2}}) \right|=\frac{r\exp\left(\sin2\lambda\arcsin(r\sin\lambda)\right)}{(r\cos\lambda-\sqrt{1-r^2\sin^2\lambda})^{2\cos^2\lambda}} \] where \[P_{\lambda}(z)=\frac{z}{(1-z)^{1+e^{2i\lambda}}} \] belongs to $\mathcal{SP}(\lambda)$ and $\theta_{j}$ ($j=1,2$) satisfy \[ \sin(\lambda+\theta_{j})=r\sin\lambda \hspace{15pt}(j=1,2) \] and $\cos(\lambda+\theta_{1})<0$ and $\cos(\lambda+\theta_{2})>0$ respectively. \end{lem}
\subsection{Proof of Theorem 1}
Equivalence (1) and Lemma \ref{lem2} show that \begin{eqnarray*}
|f(z)| &=&\left|\int_{0}^{z}f'(\zeta)d\zeta\right|=\left| \int_{0}^{r} \frac{z}{r}f'(tz/r)dt\right|\\[5pt]
&\leq &\int_{0}^{r}|f'(tz/r)|dt\leq \int_{0}^{r}\frac{\exp(\sin(2\lambda)\arcsin(t\sin\lambda))}{(\sqrt{1-t^2\sin^2\lambda}-t\cos\lambda)^{2\cos^2\lambda}}dt \end{eqnarray*}
where $0<|z|=r<1$.
Since the numerator in the above integrand is bounded over $[0,1]$, it is sufficient to estimate only the denominator.
Upon a change in the variable $s=1-t$, we obtain \begin{eqnarray*} \sqrt{1-t^2\sin^2\lambda}-t\cos\lambda &=& \sqrt{1-(1-s)^2\sin^2\lambda}-(1-s)\cos\lambda\\ &=& \sqrt{\cos^2\lambda+2s\sin^2\lambda-s^2\sin^2\lambda}-(1-s)\cos\lambda\\ &=& \cos\lambda \sqrt{1+2s\tan^2\lambda-s^2\tan^2\lambda}-(1-s)\cos\lambda\\ &=& \cos\lambda[1+1/2(2s\tan^2\lambda-s^2\tan^2\lambda)+O(s^2)]-(1-s)\cos\lambda\\ &=& \frac{s}{\cos\lambda}+O(s^2) \end{eqnarray*} when $s\to 0$.
Therefore $f(z)$ is bounded whenever $2\cos^2\lambda<1$, that is, $\cos\lambda<1/\sqrt{2}$. The example given by $(\ref{extremal})$ ensures the sharpness of the value $1/\sqrt{2}$.
$\square$
\begin{rem} Note that our method is not applicable for the case when $\cos\lambda=1/\sqrt{2}$. Since the function $f_{\lambda}(z)$ given in ($\ref{extremal}$) is bounded when $\cos\lambda=1/\sqrt{2}$, we may expect that $\mathcal{R}(\lambda)$ consists of bounded functions as well in this case. \end{rem}
\section{Quasiconformal Extension}
\subsection{Results}
In this section we would like to discuss about the new quasiconformal extension criteria for Robertson functions. Let us denote by $\mathcal{S}(k)$ the family of functions lie in $\mathcal{S}$ and can be extended to quasiconformal automorphisms of $\mathbb{C}$ so that the complex dilatation $\mu_{f} = (\partial f / \partial \bar{z}) / (\partial f / \partial z)$ satisfies $|\mu_{f}(z)| \leq k < 1$ for almost every $z \in \mathbb{C}$.
We will show the following which is an extension of a result of Ruscheweyh \cite[Corollary 1]{Ruscheweyh:1976};
\begin{thm}\label{result} Let $f \in \mathcal{A},\,k \in [0,1)$ and $\lambda\in(-\pi/2,\pi/2),\,q>-1$ be related by \begin{equation}\label{eq02} 0 < \cos \lambda \leq \left\{ \begin{array}{llc} k/2, & \textit{if}& -1 < q \leq 0 , \\[5pt] k/(2+4q), & \textit{if}& 0 < q. \end{array} \right. \end{equation} If f satisfies \begin{equation}\label{eq01} \textup{Re}\, \left\{ e^{-i\lambda} \left( 1 + \frac{zf''(z)}{f'(z)} + q \frac{zf'(z)}{f(z)} \right) \right\} >0 \end{equation} for all $z \in \mathbb{D}$, then $f \in \mathcal{S}(k)$. If, in addition, $f''(0)=0$, $\eqref{eq02}$ can be replaced by \begin{equation}\label{eq05} 0 < \cos \lambda \leq \left\{ \begin{array}{llc} k, & \textit{if}& -1 < q \leq 0 , \\[5pt] k/(1+2q), & \textit{if}& 0 < q. \end{array} \right. \end{equation}
\end{thm}
We note that when $q=0$ Theorem 3 claims quasiconformal extension of $\lambda$-Robertson functions which can be stated explicitly as follows;
\begin{cor}\label{qccor} Let $f \in \mathcal{R}(\lambda)$ with $\lambda\in (-\pi/2,\pi/2)$ satisfying \begin{equation*}\label{eq06} 0 < \cos \lambda \leq k/2, \end{equation*} then $f\in\mathcal{S}(k)$. If, in addition, $f''(0)=0$ and $\eqref{eq06}$ can be replced by \begin{equation*} 0 < \cos \lambda \leq k, \end{equation*}
then $f\in \mathcal{S}(k)$. \end{cor}
\par We note here that the second case in Corollary 4 also implies that function $f \in \mathcal{R}(\lambda)$ with $f''(0)=0$ for arbitrary $\lambda\in (-\pi/2,\pi/2)$ is univalent which was proved by Singh and Chichra \cite{SinghChichra:1977b} by means of Ahlfors's criterion for univalence as well.
\subsection{Preliminaries} The following several results will be used later in our arguments. Here, set $$ H_{s}(z) = s \left(1 + \frac{zf''(z)}{f'(z)}\right) + (1-s)\frac{zf'(z)}{f(z)}. $$
\begin{knownthm}[\cite{Hotta:2010b}]\label{main} Let $a>0,\,b \in \mathbb{R},\,s = a + i b,\,k \in [0,1)$ and $f \in \mathcal{A}$. Assume that for a constant $c \in \mathbb{C}$ and all $z \in \mathbb{D}$ \begin{equation*}
\left|
c|z|^{2}+s-a(1-|z|^{2})H_{s}(z)
\right| \leq M \end{equation*} with $$ M = \left\{ \begin{array}{ll}
a k |s| + (a-1)|s+c|, &\textit{if}\hspace{10pt} 0 < a \leq 1, \\[5pt]
k |s|, &\textit{if}\hspace{10pt} a>1, \end{array} \right. $$ then $f \in \mathcal{S}(l)$, where \begin{equation*}\label{mainl} l= \frac {
2ka+(1-k^{2}) |b| } {
(1+k^{2})a+(1-k^{2}) |s| } <1. \end{equation*} \end{knownthm}
We note that in the above theorem $l=k$ if and only if $b=0$ (\cite{Hotta:2010b}).
\begin{knownlem}[e.g. \cite{Ruscheweyh:1976}]\label{lemB} Let $p(z) = 1 + a_{n}z^{n} + \cdots$ be analytic and $\textup{Re}\, p(z)>0$ on $\mathbb{D}$. Then \begin{equation*}
\left|
p(z) -1-\frac{2|z|^{2n}}{1-|z|^{2n}}
\right| \leq
\frac{2|z|^{2n}}{1-|z|^{2n}} \end{equation*} for all $z \in \mathbb{D}$. \end{knownlem}
\subsection{Proof of Theorem \ref{result}}
Let $s = 1/(1+q)$ and $f(z)=z+\sum_{n=2}^{\infty}a_{n}z^n$, then for \begin{eqnarray*} p(z) &=& \frac{e^{-i\lambda} H_{s}(z) + i\sin \lambda}{\cos \lambda}\\ &=& 1 + \frac{e^{-i\lambda}}{\cos \lambda} (s+1)a_{2}z + \cdots. \end{eqnarray*} we have $p'(0)=0$ if and only if $f''(0)=0$. Condition \eqref{eq01} implies that $p(z)$ is analytic in $\mathbb{D}$ and fulfills $\textup{Re}\, p(z) > 0$ for all $z \in \mathbb{D}$. With $\displaystyle (c+s) = \frac 2 n se^{i\lambda}\cos\gamma,\,n=1, 2$, we obtain from Lemma $\ref{lemB}$ that $$ \begin{tabular}{llllll} \multicolumn{1}{l}
{$\displaystyle \left|\frac{(c+s)|z|^{2}}{1-|z|^{2}} - s(H_{s}(z) -1)\right|$} \\[12pt]
$\hspace{60pt}\displaystyle \leq s|\cos \lambda| \left\{
\left|\frac{2|z|^{2n}}{1-|z|^{2n}} - (p(z) -1)\right| +
\left|\frac{2|z|^{2n}}{1-|z|^{2n}} - \frac2n \frac{|z|^{2}}{1-|z|^{2}}\right| \right\}$\\[12pt]
$\hspace{60pt}\displaystyle \leq \frac{2s}{n}\frac{|\cos \lambda|}{1-|z|^{2}}$. \end{tabular} $$
\noindent Therefore by Theorem \ref{main} $f$ can be extended to a $k$-quasiconformal automorphism of $\mathbb{C}$ whenever $$
\frac2n s|\cos \lambda| \leq \left\{ \begin{tabular}{llc}
$\displaystyle k s^{2} + \frac2n s |\cos \lambda| (s-1)$, & \textit{if}&$0 < s \leq 1$, \\[5pt] $k s$, & \textit{if}& $1 < s$, \end{tabular} \right. $$ which is equivalent to \eqref{eq02} if $n=1$ and to \eqref{eq05} if $n=2$.
$\square$
\section{L\"owner chain}
We can find another proof for univalency of Robertson functions by making use of the theory of L\"owner chains.
The following theorem is well known. Here, we denote $\partial f / \partial t$ and $\partial f / \partial z$ by $\dot{f}$ and $f'$ respectively.
\begin{knownthm}[\cite{Pom:1965}, see also \cite{Hotta:2010a}] Let $0 < r_{0} \leq 1$. Let $f_{t}(z) = \sum_{n=1}^{\infty}a_{n}(t)z^{n}$, $a_{1}(t) \neq 0$,\, be analytic for each $t \in [0,\infty)$ in $\mathbb{D}_{r_{0}}$ and locally absolutely continuous in $[0,\infty)$, locally uniformly with respect to $\mathbb{D}_{r_{0}}$, where $a_{1}(t)$ is a complex-valued function on $[0,\infty)$. For almost all $t \in [0,\infty)$ suppose \begin{equation}\label{LDE} \dot{f_{t}}(z) =z f_{t}'(z) p_{t}(z) \hspace{20pt} (z \in \mathbb{D}_{r_{0}}) \end{equation}
where $p_{t}(z)$ is analytic in $\mathbb{D}$ and satisfies $\textup{Re}\, p_{t}(z)>0,\,z \in \mathbb{D}$. If $|a_{1}(t)| \to \infty$ for $t \to \infty$ and if $\{f_{t}(z)/a_{1}(t)\}$ forms a normal family in $\mathbb{D}_{r_{0}}$, then for each $t \in [0,\infty)$ $f_{t}(z)$ can be continued analytically to a univalent function on $\mathbb{D}$. \end{knownthm}
The next lemma is needed for our discussion;
\begin{knownlem}[\cite{Wang:preprint}, Theorem 3]\label{lemma3} Suppose that $\lambda \in (-\pi/2,\pi/2)$. Let $p(z)$ be an analytic function defined on $\mathbb{D}$ which satisfies $p(0)=1$ and $\textup{Re}\, e^{-i\lambda}p(z) >0$ for all $z \in \mathbb{D}$. Then we have \begin{equation*}
\left|p(z)-\left(\frac{1}{1-r^2}+\frac{r^2}{1-r^2}e^{2i\lambda}\right)\right|\leq \frac{2r}{1-r^2}\cos \lambda. \end{equation*}
where $r=|z|<1$.
\end{knownlem}
Now we suppose that $|\lambda| \in [\pi/3, \pi/2)$ and $f$ is a $\lambda$-Robertson function. Let us put \begin{equation}\label{LC} f_{t}(z) = f(e^{-t}z) - e^{-2i\lambda}(e^{2t}-1)e^{-t}zf'(e^{-t}z). \end{equation} Here we should note that a more general form of \eqref{LC} appears in \cite{Ruscheweyh:1976}.
It suffices to prove that $p_{t}(z)$ in \eqref{LDE} lies in the right-hand side of the complex plane $\mathbb{C}$ for all $z \in \mathbb{D}$ and a.e. $t \in [0,\infty)$. This is equivalent to $$
\left| \frac{\dot{f_{t}}(z) - zf_{t}'(z)}{\dot{f_{t}}(z) +zf_{t}'(z)}
\right| < 1. $$ Then a calculation shows that \begin{equation}\label{eq03}
\left| e^{-2t} e^{2i\lambda} + 1 - \left( 1 - e^{-2t} \right) \left( 1 + \frac{e^{-t}zf''(e^{-t}z)}{f'(e^{-t}z)} \right)
\right| <1 \end{equation} implies univalency of $f$ and \eqref{eq03} follows from maximum modulus principle and Lemma \ref{lemma3} when $\cos \lambda \leq 1/2$.
\begin{rem} Applying Becker's theorem \cite{Becker:1976} with \eqref{LC}, we also obtain the quasiconformal extension criterion for $\mathcal{R}(\lambda)$ which is in Corollary \ref{qccor}. \end{rem}
\end{document} |
\begin{document}
\title{Quantifying environment non-classicality in dissipative open quantum dynamics} \date{\today } \author{Adri\'{a}n A. Budini} \affiliation{Consejo Nacional de Investigaciones Cient\'{\i}ficas y T\'{e}cnicas (CONICET), Centro At\'{o}mico Bariloche, Avenida E. Bustillo Km 9.5, (8400) Bariloche, Argentina, and Universidad Tecnol\'{o}gica Nacional (UTN-FRBA), Fanny Newbery 111, (8400) Bariloche, Argentina}
\begin{abstract} Open quantum systems are inherently coupled to their environments, which in turn also obey quantum dynamical rules. By restricting to dissipative dynamics, here we propose a measure that quantifies how far the environment action on a system departs from the influence of classical noise fluctuations. It relies on the lack of commutativity between the initial reservoir state and the system-environment total Hamiltonian. Independently of the nature of the dissipative system evolution, Markovian or non-Markovian, the measure can be written in terms of the dual propagator that defines the evolution of system operators. The physical meaning and properties of the proposed definition are discussed in detail and also characterized through different paradigmatic dissipative Markovian and non-Markovian open quantum dynamics. \end{abstract}
\maketitle
\section{Introduction}
Open quantum systems are inherently coupled to their supporting environments~ \cite{breuerbook,vega}. This interaction induces time-irreversible behaviors such as dissipation and decoherence. These phenomena have been studied in a broad class of systems such as for example in quantum optics~\cite {carmichaelbook}, magnetic resonance~\cite{anderson,abragan}, solid state devices~\cite{efenoise}, and quantum sensing~\cite{sensing}.
In a full microscopic description not only\ the system but also the environment obeys quantum dynamical rules. Nevertheless, depending on system and environment properties, as well as on the studied regimes, the system fluctuations induced by the environment influence can be well approximated by the action of classical stochastic fields. In fact, open quantum systems driven by classical noises is a well established physical modelling~\cite {anderson,abragan,efenoise,sensing} that has been characterized from different perspectives. Many specific studies rely on assuming, for example, Gaussian~\cite{GaussianNoise,Gauss,morgado,cialdi,kelly} or telegraphic noises~\cite{dico,dicoAbel,song}.
In the context of open quantum system theory it is of interest to establishing the conditions under which the environment action can be approximated by classical noises. For example, the possibility of representing the system evolution in terms of a statistical superposition of unitary dynamics has been explored recently~\cite{nori,Chen,franco,ChenChen} . When the open system dynamics only leads to dephasing~\cite {strunzTwoQubits,viola,roszak,kata,Cywinski,experK,clerck}, the possibility of detecting quantum entanglement between the system and the environment~ \cite{roszak,kata,Cywinski,experK} gives a solid criteria for determining when a classical noise representation is appropriate or not. In addition many related contributions focused on this problem, providing general or particular conditions under which a classical representation is a valid approximation~\cite {depol,spinbathnoise,shortParis,parislletA,bordone,hala,fabri,liu,Wang,fata,lika,Szanko,sun,liuBis} .
The main goal of this work is to introduce a measure that quantifies how much the environment action departs from the influence of classical noise fluctuations. This result provides an interesting insight and contribution in the described research line. In contrast to previous analysis (see Refs.~ \cite {nori,Chen,franco,ChenChen,strunzTwoQubits,viola,roszak,kata,Cywinski,experK} ), here we are mainly interested in dissipative open quantum dynamics, that is, the environment not only induce decoherence but also is able to induce (energy) transitions between the system states.
The proposed measure has a clear physical motivation related to the quantumness of the environment state and its dynamics. In addition, it is valid independently of the system dynamical regime, that is, Markovian or non-Markovian. In fact, independently of which approach is used to define memory effects, operational~\cite{modi,budiniCPF} or non-operational~\cite {BreuerReview,plenioReview}, the proposed measure can be written in terms of the dual evolution associated to system operators. Thus, it can be defined consistently for Markovian Lindblad equations~\cite{alicki}\ but also, in the same way, outside this regime. The proposal is characterized in detail through its general properties and also through specific dissipative Markovian and non-Markovian dynamics.
The manuscript is outlined as follows. In Sec. II we motivate and formulate the environment non-classicality measure. In Sec. III it is characterized for some general classes of open quantum dynamics. In Sec. IV we study its behavior for specific Markovian and non-Markovian dissipative open system dynamics. In Sec. V we provide the Conclusions.
\section{Measure of environment non-classicality}
In this section we introduce the environment non-classicality measure, providing in addition some of its general properties.
\subsection{Physical motivation and definition}
We consider a system $(s)$ that interacts with its environment $(e).$ Their quantum dynamics is set by a total Hamiltonian $H=H_{s}+H_{e}+H_{I},$ where $ H_{s}$ and $H_{e}$ are the system and environment Hamiltonians respectively, while $H_{I}$ defines their mutual interaction. The system density matrix $ \rho _{t}$\ can be written as \begin{equation} \rho _{t}=\mathbb{G}_{t,0}[\rho _{0}]\equiv \mathrm{Tr}_{e}[e^{-iHt}(\rho _{0}\otimes \sigma _{0})e^{+iHt}]. \label{RhoUni} \end{equation} Here, $\mathrm{Tr}[\cdots ]$ is the trace operation. Furthermore, we assume uncorrelated $s$-$e$ initial conditions, where $\rho _{0}$ and $\sigma _{0}$ are the initial system and environment states respectively.
The quantum nature of system and the environment can be read straightforwardly from Eq.~(\ref{RhoUni}). In fact, all objects appearing in this expression can be written as matrixes that in general do not commutate. Focussing on the environment, we argue that the nature of its influence over the system is inherently quantum because in general its initial state does not commutates with the total Hamiltonian, that is, $[H,\sigma _{0}]\neq 0.$ Supporting this argument, when the initial environment state approaches the identity matrix $\sigma _{0}\simeq \mathrm{I}_{e},$ which implies $[H,\sigma _{0}]\approx 0,$ its action over the system can be represented by classical noises. This result is well known in the context of magnetic resonance~\cite {abragan} and also has been characterized when expressing the system evolution in terms of stochastic wave vectors~\cite{SWF}. For systems coupled to thermal environments the property $[H,\sigma _{0}]\approx 0$ becomes valid in a high temperature limit.
Under the motivation of the previous perspective, we rewrite the system state [Eq.~(\ref{RhoUni})] as \begin{equation} \rho _{t}=\mathrm{Tr}_{e}[(e^{-iHt}\rho _{0}e^{+iHt})\sigma _{0}]+\mathrm{Tr} _{e}[(e^{-iHt}\rho _{0}e^{+iHt})\Delta \sigma _{t}], \label{RhoSplit} \end{equation} where $\Delta \sigma _{t}\equiv e^{-iHt}\sigma _{0}e^{+iHt}-\sigma _{0}.$ We identify the first term with the \textquotedblleft classical\textquotedblright\ contribution of the environment influence. In fact, when $[H,\sigma _{0}]\approx 0$ the first contribution does not vanish while the second one fade out (vanishes) correspondingly. Interestingly, when describing the system evolution in a weak interaction and Markovian limits the previous splitting recovers the structure of quantum master equations~\cite{abragan,goldman,ZM} proposed for dealing with a high temperature approximation~\cite{abragan}.
While the property $\mathrm{Tr}_{s}[\rho _{t}]=1$ is fulfilled, each contribution in Eq.~(\ref{RhoSplit}) does not preserve trace of the system by itself. Thus, for measuring departure of the environment action with respect to classical noises we arrive to the (dimensionless) time-dependent quantumness measure \begin{equation} Q_{t}\equiv \mathrm{Tr}_{se}[(e^{-iHt}\rho _{0}e^{+iHt})\sigma _{0}]. \label{QDef} \end{equation} It corresponds to the trace over the system degrees of freedom of the first term in the splitting~(\ref{RhoSplit}).
\subsection{Degree of environment quantumness}
The definition~(\ref{QDef}) has some desirables properties. For example, when $[H,\sigma _{0}]=0$ it follows $Q_{t}=1.$ Therefore, this value indicates classicality. On the other hand, it is straightforward to obtain \begin{equation} \frac{dQ_{t}}{dt}=-i\mathrm{Tr}_{se}[(e^{-iHt}\rho _{0}e^{+iHt})[H,\sigma _{0}]], \end{equation} while its $n$-time-derivative reads \begin{equation} \frac{d^{n}Q_{t}}{dt^{n}}=(-i)^{n}\mathrm{Tr}_{se}[(e^{-iHt}\rho _{0}e^{+iHt})[H^{(n)},\sigma _{0}]], \end{equation} where $[H^{(1)},\sigma _{0}]=[H,\sigma _{0}],\ [H^{(2)},\sigma _{0}]=[H,[H,\sigma _{0}]],$ and in general $[H^{(n)},\sigma _{0}]=[H,[H^{(n-1)},\sigma _{0}].$ From these expressions we conclude that the time-derivatives of $Q_{t}$ are proportional to the lack of commutativity of $\sigma _{0}$ with higher nested commutators of the total Hamiltonian $H.$
In spite of the previous properties, the definition of $Q_{t}$ [Eq.~(\ref {QDef})] is symmetrical in the initial system and environment states. In particular, when $\rho _{0}=\mathrm{I}_{s}/\dim (\mathcal{H}_{s}),$ where $ \mathrm{I}_{s}$ is the identity matrix and $\dim (\mathcal{H}_{s})$\ is the dimension of the system Hilbert space, it follows that $Q_{t}=1.$
The difference between the roles played by the system and the environment is introduced by defining a \textquotedblleft degree of environment quantumness,\textquotedblright\ denoted as $D_{Q},$ which reads \begin{equation} D_{Q}\equiv \max_{\lbrack \rho _{0}]}\left\vert \lim_{t\rightarrow \infty }\int_{0}^{t}dt^{\prime }\frac{dQ(t^{\prime })}{dt^{\prime }}\right\vert =\max_{[\rho _{0}]}\left\vert \lim_{t\rightarrow \infty }Q_{t}-1\right\vert . \label{Degree} \end{equation} Here, it was used that $Q_{0}=1.$ Furthermore, it is assumed that a stationary regime is achieved. The maximization is over the initial system state $\rho _{0}.$ The equality $D_{Q}=0$ is then associated with environment classicality. In addition, this parameter allows to study $Q_{t}$ \ [Eq.~(\ref{QDef})] by choosing system initial conditions that maximize $ D_{Q}.$
\subsection{Definition in terms of the operator dual evolution}
Given a system operator $A,$ by definition its expectation value reads $ \langle A\rangle _{t}\equiv \mathrm{Tr}_{s}[\rho _{t}A].$ It can alternatively be written as $\langle A\rangle _{t}=\mathrm{Tr}_{s}[\rho _{0}A_{t}]=\mathrm{Tr}_{s}[\rho _{0}\mathbb{G}_{t,0}^{\bigstar }[A_{0}]]$ where the dual propagator $\mathbb{G}_{t,0}^{\bigstar }$ for the operator evolution, from Eq.~(\ref{RhoUni}), is given by \begin{equation} A_{t}=\mathbb{G}_{t,0}^{\bigstar }[A_{0}]\equiv \mathrm{Tr} _{e}[e^{+iHt}Ae^{-iHt}\sigma _{0}]. \label{AtDefinition} \end{equation} Therefore, from this expression and Eq.~(\ref{QDef}) it follows that the quantumness measure $Q_{t}$ can be written as \begin{equation} Q_{t}=\mathrm{Tr}_{s}[\mathbb{G}_{-t,0}^{\bigstar }[\rho _{0}]], \label{QDual} \end{equation} which only depends on the operator dual evolution and the initial system state. From this expression it follows that $Q_{t}/\dim (\mathcal{H}_{s})$ can be read as the expectation value of the \textquotedblleft operator\textquotedblright\ $\rho _{0}$ at time $t$ given that the \textquotedblleft initial system state\textquotedblright\ is $\mathrm{I} _{s}/\dim (\mathcal{H}_{s})$ [see Eqs.~(\ref{AtDefinition}) and (\ref{QDual} )].
Taking into account that $\rho _{0}$ is a positive definite operator, this equivalent interpretation of $Q_{t}$ allow us to obtain \begin{equation} 0\leq Q_{t}\leq \dim (\mathcal{H}_{s}). \label{Bounds} \end{equation} This inequality is valid independently of the system and environment initial conditions and also of the particular microscopic model. On the other hand, it consistently implies that the quantumness of the environment influence is bounded by the dimension of the system Hilbert space. In addition, Eq.~(\ref {Bounds}) implies \begin{equation} 0\leq D_{Q}\leq \dim (\mathcal{H}_{s})-1, \label{DqBounds} \end{equation} where classicality corresponds to $D_{Q}=0.$
\subsection{Optimal initial states}
By analyzing the stationary regime of Eq.~(\ref{QDef}), or alternatively Eq.~(\ref{QDual}), it follows that \begin{equation} \lim_{t\rightarrow \infty }Q_{t}=\dim (\mathcal{H}_{s})\mathrm{Tr}_{s}[ \tilde{\rho}_{\infty }\rho _{0}], \label{QinfExplicito} \end{equation} where the stationary system state is $\tilde{\rho}_{\infty }\equiv \lim_{t\rightarrow \infty }\tilde{\rho}_{t}.$ The upper tilde symbol represents a time-reversal operation, $t\leftrightarrow -t.$ Eqs.~(\ref {Degree}) and~(\ref{QinfExplicito}) implies that \begin{equation} \frac{D_{Q}}{\dim (\mathcal{H}_{s})}=\max_{[\rho _{0}]}\left\vert \mathrm{Tr} _{s}[\tilde{\rho}_{\infty }\rho _{0}]-\frac{1}{\dim (\mathcal{H}_{s})} \right\vert . \label{DqInfinita} \end{equation} Thus, $D_{Q}$ can be seen as a functional of $\rho _{0}$ that is parametrized by the stationary state $\rho _{\infty }.$ The optimal state $ \rho _{0}$ that maximizes $D_{Q}$ is obtained below.
The expression (\ref{DqInfinita}) provides a clear geometric interpretation of $D_{Q}.$ Introducing a basis of vectors $\{|i\rangle \}$ where the stationary system state is a diagonal matrix, $\tilde{\rho}_{\infty
}=\sum_{i}\lambda _{i}|i\rangle \langle i|,$ with $i=1,\cdots \dim (\mathcal{ H}_{s}),$ it follows that $D_{Q}=\max_{\{p_{i}\}}\left\vert \dim (\mathcal{H}
_{s})\sum_{i}\lambda _{i}p_{i}-1\right\vert ,$ where $p_{i}\equiv \langle i|\rho _{0}|i\rangle .$ Therefore, $D_{Q}$ is the maximal (absolute) value assumed by the hyperplane defined by the variables $\{p_{i}\}$ when restricted to the domain $\sum_{i}p_{i}=1.$ It is simple to bound the main contribution to $D_{Q}$ as $\sum_{i}\lambda _{i}p_{i}\leq (\sum_{i}p_{i})\max (\{\lambda _{i}\})=\max (\{\lambda _{i}\}).$ This boundary is always achieved by choosing $\rho _{0}$ as the eigenprojector of $\tilde{\rho}_{\infty }$\ with the maximal eigenvalue. Thus, we conclude that \begin{equation}
D_{Q}=\dim (\mathcal{H}_{s})\max (\{\lambda _{i}\})-1,\ \ \ \ \ \ \rho _{0}=|i_{\max }\rangle \langle i_{\max }|, \label{MaxEigen} \end{equation} where $\max (\{\lambda _{i}\})$ is the largest eigenvalue of the stationary state $\tilde{\rho}_{\infty }\equiv \lim_{t\rightarrow \infty }\tilde{\rho}
_{t},$ while $|i_{\max }\rangle $ is the corresponding eigenstate, $\tilde{
\rho}_{\infty }|i_{\max }\rangle =\max (\{\lambda _{i}\})|i_{\max }\rangle .$
This expression for $D_{Q}$ is valid when the stationary state does not depends on the initial condition. In addition, we notice that in general more than one initial state, $\rho _{0}\neq |i_{\max }\rangle \langle i_{\max }|,$\ may lead to this extreme value (see Sec. IV). On the other hand, it is simple to realize that when the time-reversal operation is equivalent to conjugation Eq.~(\ref{MaxEigen}) is valid with $\tilde{\rho} _{\infty }\rightarrow \rho _{\infty }.$
\section{Classicality for different classes of open system dynamics}
In this Section we characterize the previous proposal for different classes of open quantum system dynamics where the quantumness measure indicates classicality, $Q_{t}=1.$
\subsection{Hamiltonian ensembles}
In Refs.~\cite{nori,Chen,franco,ChenChen} the classicality of the environment action was related to the possibility of representing the open system dynamics in terms of Hamiltonian ensembles, that is, a statistical superposition of different system unitary dynamics. This kind of dynamics is recovered in the present approach after assuming that $[H,\sigma _{0}]=0.$
In fact, introducing a complete basis of environment states $\{|e\rangle \}$
where the initial state is diagonal, $\sigma _{0}=\sum_{e}p_{e}|e\rangle
\langle e|,$ with $p_{e}=\langle e|\sigma _{0}|e\rangle ,$ the system density matrix [Eq.~(\ref{RhoUni})] can be written as \begin{equation} \rho _{t}=\sum_{e}p_{e}e^{-itH_{s}^{(e)}}\rho _{0}e^{+itH_{s}^{(e)}},\ \ \ \ \Rightarrow \ \ \ \ Q_{t}=1, \label{HEnsemble} \end{equation}
where the system Hamiltonians are $H_{s}^{(e)}\equiv H_{s}+\langle e|(H_{e}+H_{I})|e\rangle .$ The equality $Q_{t}=1$ follows from Eq.~(\ref {QDef}) and is valid independently of the system initial condition. The degree of quantumness [Eq.~(\ref{Degree})] also indicates the presence of a classical environment influence, $D_{Q}=0.$
\subsection{Stochastic Hamiltonians}
The coupling of a quantum system to classical noises is usually modelled by (system) stochastic Hamiltonians $H_{st}(t).$ Their time-dependence take into account the action of classical fluctuating external fields. The noises can have arbitrary statistical properties (see for example Refs.~\cite {GaussianNoise,Gauss,morgado,cialdi,kelly,dico,dicoAbel,song}). Even more, their correlation can also be arbitrary, that is, delta correlated (white noises) or color ones (finite correlation times).
For each realization of the noises, we introduce the stochastic propagator $ \mathcal{T}_{st}(t)=\lceil \exp -i\int_{0}^{t}dt^{\prime }H_{st}(t^{\prime })\rceil ,$ where $\lceil \cdots \rceil $ means a time-ordering operation. The system density matrix can then be written as \begin{equation} \rho _{t}=\overline{\mathcal{T}_{st}(t)\rho _{0}\mathcal{T}_{st}^{\dagger }(t)},\ \ \ \ \Rightarrow \ \ \ \ Q_{t}=1, \label{Ruido} \end{equation} where the overline denotes an average over noise realizations. The equality $ Q_{t}=1$ is valid for arbitrary system initial conditions. It can be derived by using the alternative definition in terms of the dual evolution [Eq.~(\ref {QDual})]. Consistently, in this case $D_{Q}=0.$
\subsection{Collisional dynamics}
Open quantum systems dynamics, in Markovian and non-Markovian regimes, can also be modelled through collisional models~\cite {collisional,colisionVacchini}. The underlying stochastic dynamics consists in free propagation with the system Hamiltonian added to the action of an instantaneous transformation that occurs at successive random times. The (stochastic) state of the system conditioned to the occurrence of $n$ -collisional-events can be written as \begin{equation} \rho _{t}^{(n)}=\mathcal{G}_{t-t_{n}}\mathcal{EG}_{t_{n}-t_{n-1}}\cdots \mathcal{EG}_{t_{2}-t_{1}}\mathcal{EG}_{t_{1}}[\rho _{0}]. \end{equation} Here, $\mathcal{G}_{t}$ is the propagator of the free evolution, $\mathcal{G} _{t}[\bullet ]\equiv \exp [-itH_{s}]\bullet \exp [+itH_{s}],$ while $ \mathcal{E}$ is an arbitrary completely positive trace preserving superoperator. The times $\{t_{i}\}_{i=1}^{n}$ are random variables in the interval $(0,t).$ The state of the system follows as \begin{equation} \rho _{t}=\sum_{n=0}^{\infty }\overline{\rho _{t}^{(n)}}, \label{Colision} \end{equation} where the overline means an average over the random collisional times. The dual operator dynamics can be written in a similar way. In the Appendix we develop a formal derivation.
Using the definition~(\ref{QDual}) it is possible to conclude (see Appendix) that \begin{equation} \mathrm{Tr}_{s}[\mathcal{E}^{\bigstar }[A]]=\mathrm{Tr}_{s}[A]\Rightarrow \ \ \ \ \ Q_{t}=1, \label{UnitalCollision} \end{equation} where the dual superoperator is defined from the relation $\mathrm{Tr}_{s}[A \mathcal{E}[\rho ]]=\mathrm{Tr}_{s}[\rho \mathcal{E}^{\bigstar }[A]],$ with $ A$ being an arbitrary system operator. Thus, when the dual superoperator $ \mathcal{E}^{\bigstar }$ preserves trace the quantumness indicator vanishes identically for any system initial condition, which in turn implies $ D_{Q}=0. $ The condition~(\ref{UnitalCollision}) is fulfilled when $\mathcal{ E}$ corresponds to a unitary transformation and in general is fulfilled by unital maps (see below).
\subsection{Lindblad equations}
When the system-environment coupling is weak and the time-correlations of environment operators define the minor time-scale of the problem, a Born-Markov approximation applies. Discarding non-secular terms, the system evolution can be written as a Lindblad equation~\cite{breuerbook,alicki}, \begin{equation} \frac{d\rho _{t}}{dt}=-i[\bar{H}_{,}\rho _{t}]+\sum_{\mu \nu }a_{\mu \nu }(V_{\mu }\rho _{t}V_{\nu }^{\dagger }-\frac{1}{2}\{V_{\nu }^{\dagger }V_{\mu },\rho _{t}\}_{+}). \label{LindbladGen} \end{equation} Here, $\bar{H}$ is an effective system Hamiltonian that may include contributions induced by the interaction with the environment. $\{V_{\mu }\}$ are system operators, while the matrix of rate coefficients $\{a_{\mu \nu }\} $ defines a semi-positive definite matrix. The anticommutator operation is defined as $\{a,c\}_{+}\equiv (ac+ca).$
The quantumness measure $Q_{t}$ can be calculated for the previous quantum master equation by using its definition in terms of the dual evolution [Eq.~( \ref{QDual})]. For an arbitrary operator $A_{t}$ it reads~\cite{alicki} \begin{equation} \frac{dA_{t}}{dt}=+i[\bar{H},A_{t}]+\sum_{\mu \nu }a_{\mu \nu }(V_{\nu }^{\dagger }A_{t}V_{\mu }-\frac{1}{2}\{V_{\nu }^{\dagger }V_{\mu },A_{t}\}_{+}). \label{At} \end{equation}
Consistently, notice that this evolution does not preserve trace. By solving Eq.~(\ref{At}) with initial condition $A_{t}|_{t=0}=\rho _{0},$ the quantumness measure can be written as $Q_{t}=\mathrm{Tr}_{s}[\tilde{A}_{t}],$ where the tilde symbol takes into account the time reversal operation $ t\leftrightarrow -t.$ In this way, the present approach can be applied in the Markovian regime where a Lindblad equation approximate the open system dynamics.
Interestingly, Lindblad equations that are compatible with the influence of classical noises have been characterized from a rigorous mathematical point of view~\cite{viola,mathBiStoch}. The proposed structures were derived as commutative dilations of dynamical semigroups. Specifically, they correspond to Eq.~(\ref{LindbladGen}) written in a diagonal base of operators $(a_{\mu \nu }=\delta _{\mu \nu }a_{\mu })$ with the constraints of Hermitian operators, $V_{\mu }=V_{\mu }^{\dagger },$ or alternatively unitary ones, $ V_{\mu }^{\dagger }V_{\mu }=\mathrm{I}_{s}.$ These solutions can be put in-one-to-one correspondence with models based respectively on stochastic Hamiltonians [Eq.~(\ref{Ruido})] with white-noise fluctuations and collisional models [Eq.~(\ref{Colision}) and~(\ref{UnitalCollision})] with Poisson statistics between collisional events.
\subsection{Unital open system dynamics}
A completely positive open system dynamics can always be written in a Krauss representation as~\cite{breuerbook} \begin{equation} \rho _{t}=\sum_{\alpha }T_{\alpha }\rho _{0}T_{\alpha }^{\dagger },\ \ \ \ \ \ \ \ \ \ \sum_{\alpha }T_{\alpha }^{\dagger }T_{\alpha }=\mathrm{I}_{s}, \end{equation} where the system operators $\{T_{\alpha }\}$ are time-dependent, $T_{\alpha }=T_{\alpha }(t).$ The dynamics is defined as unital when in addition it is fulfilled that \begin{equation} \sum_{\alpha }T_{\alpha }T_{\alpha }^{\dagger }=\mathrm{I}_{s},\ \ \ \ \Rightarrow \ \ \ \ Q_{t}=1. \label{Unital} \end{equation} We notice that the result $Q_{t}=1,$ valid for arbitrary system initial conditions, follows from Eq.~(\ref{QDual}) and after noting that the dual operator evolution can be written as $A_{t}=\sum_{\alpha }T_{\alpha }^{\dagger }A_{0}T_{\alpha },$ which implies that $\mathrm{Tr} _{s}[A_{t}]=\sum_{\alpha }\mathrm{Tr}_{s}[T_{\alpha }T_{\alpha }^{\dagger }A_{0}]=\mathrm{Tr}_{s}[A_{0}].$
In general it is possible to argue that any open quantum dynamics induced by coupling the system with stochastic classical degrees of freedom is always unital, which consistently implies $Q_{t}=1.$ In fact, the dynamics defined by Eqs.~(\ref{Ruido}) and~(\ref{Colision}) can be read as \textquotedblleft non-Markovian\textquotedblright\ extensions of the commutative dilations of dynamical semigroups obtained in Ref.~\cite{mathBiStoch}. With non-Markovian here we mean considering non-white noises or non-Poisson statistics respectively.
On the other hand, the inverse implication is not valid in general, that is, there exist unital dynamics that cannot be obtained by considering the action of classical stochastic fields. This property emerges, for example, in dephasing dynamics with $\dim (\mathcal{H}_{s})\geq 3$~\cite{viola}. In addition, this feature has been related to the break of a time-reversal symmetry~\cite{clerck}. While these cases implies a limitation on the applicability of the indicator $Q_{t},$ the corresponding class of dynamics is well characterized. On the other hand, the examples studied in the next section explicitly demonstrate the consistence of the proposed approach.
\section{Examples}
Here the quantumness indicator $Q_{t}$ is characterized for some specific dissipative Markovian and non-Markovian open quantum dynamics.
\subsection{Two-level system in contact with a thermal environment}
We consider a two-level system interacting with a Bosonic bath at temperature $T.$ It density matrix $\rho _{t}$ evolves as~\cite{breuerbook} \begin{eqnarray} \frac{d\rho _{t}}{dt} &=&\frac{-i\omega _{0}}{2}[\sigma _{z},\rho _{t}]+\kappa (\sigma \rho _{t}\sigma ^{\dagger }-\frac{1}{2}\{\sigma ^{\dagger }\sigma ,\rho _{t}\}_{+}) \notag \\ &&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ +\zeta (\sigma ^{\dagger }\rho _{t}\sigma -\frac{1}{2}\{\sigma \sigma ^{\dagger },\rho _{t}\}_{+}). \label{Thermal} \end{eqnarray} With $\sigma _{z}$ we denote the $z$-Pauli matrix. $\sigma $ and $\sigma ^{\dagger }$ are the standard lowering and raising operators with respect to the eigenvectors of $\sigma _{z}.$ Furthermore, $\kappa =\gamma (n_{th}+1)$ and $\zeta =\gamma n_{th},$ where $\gamma $\ is the natural decay rate and $ n_{th}=\exp (-\beta \hbar \omega _{0})/[1-\exp (-\beta \hbar \omega _{0})]$ is the average number of thermal boson excitations at the natural frequency of the system, with $\beta =1/kT.$
Using the alternative definition~(\ref{QDual}), jointly with the dual evolution~(\ref{At}), for arbitrary system initial conditions it is possible to obtain \begin{equation} Q_{t}=1+\langle \sigma _{z}\rangle _{\infty }\langle \sigma _{z}\rangle _{0}[1-e^{-t(\kappa +\zeta )}], \label{QThermico} \end{equation} where the operator mean values are $\langle \sigma _{z}\rangle _{0}=\mathrm{ Tr}_{s}[\sigma _{z}\rho _{0}]$ and $\langle \sigma _{z}\rangle _{\infty }=\lim_{t\rightarrow \infty }\mathrm{Tr}_{s}[\sigma _{z}\rho _{t}]=(\zeta -\kappa )/(\zeta +\kappa )\leq 0.$ In general, depending on the initial condition, as a function of time $Q_{t}$ decays or grows in a monotonic way. In any of these cases, consistently with Eq.~(\ref{Bounds}), it is fulfilled that $0\leq Q_{t}\leq 2.$
From Eq.~(\ref{QThermico}) it follows that $\lim_{t\rightarrow \infty }Q_{t}=1+\langle \sigma _{z}\rangle _{\infty }\langle \sigma _{z}\rangle _{0}.$ This stationary value has maximal departure from the unity value when $\langle \sigma _{z}\rangle _{0}=\pm 1.$ Thus, the initial conditions that maximize the definition of $D_{Q}$ [Eq.~(\ref{Degree})] are pure states, which in turn are eigenvectors of $\sigma _{z}.$ This result is consistent with Eq.~(\ref{MaxEigen}). The degree of environment quantumness finally reads \begin{equation}
D_{Q}=|\langle \sigma _{z}\rangle _{\infty }|=\left\vert \frac{\zeta -\kappa }{\zeta +\kappa }\right\vert =\tanh \Big{(}\beta \frac{\hbar \omega _{0}}{2} \Big{)}. \label{DqThermal} \end{equation} In the last equality we have used the dependence on temperature of the characteristic rates.
Eq.~(\ref{DqThermal}) defines the degree of environment quantumness corresponding to the evolution~(\ref{Thermal}). As a function of the inverse temperature $\beta $\ it has the expected behaviors. In fact, in the limit of high temperatures it follows $\lim_{\beta \rightarrow 0}D_{Q}=0,$ which correctly means that the environment influence can be represented through classical noises~\cite{abragan,SWF}. In the limit of vanishing temperatures $ D_{Q}$ assumes its maximal value [Eq.~(\ref{DqBounds})], $\lim_{\beta \rightarrow \infty }D_{Q}=1.$
\subsection{Non-Markovian decay at zero temperature}
In contrast to the previous case, here we consider a dynamics where the Born-Markov approximation does not applies in general. The microscopic dynamics is defined by the Hamiltonians $H_{s}=(\omega _{0}/2)\sigma _{z},$ $ H_{e}=\sum_{j}\omega _{k}a_{k}^{\dagger }a_{k},$ while the interaction is sets by $H_{I}=\sum_{k}(g_{k}\sigma ^{\dagger }a_{k}+g_{k}^{\ast }\sigma a_{k}^{\dagger }).$ With $a_{k}$ and $a_{k}^{\dagger }$ we denote the annihilation and creation operators associated to each mode of the Bosonic environment. Memory effects for this open dynamics has been studied from both non-operational~\cite{breuerDecayTLS} and operational~\cite {budiniBrasil} approaches to quantum non-Markovianity
The (two-level) system dynamics can be solved in an exact way by assuming that all modes of the environment begin in their ground states, which is equivalent to a vanishing temperature assumption. The system density matrix reads~\cite{breuerbook} \begin{equation} \rho _{t}=\left( \begin{array}{cc}
\rho _{0}^{++}|c_{t}|^{2} & \rho _{0}^{+-}\ c_{t} \\
\rho _{0}^{-+}\ c_{t}^{\ast } & \ \ \ \rho _{0}^{--}+\rho _{0}^{++}(1-|c_{t}|^{2}) \end{array} \right) . \label{RhoExacta} \end{equation}
Here, $\rho _{0}^{ss^{\prime }}\equiv \langle s|\rho _{0}|s^{\prime }\rangle
,$ where $\{|s\rangle \}=|{\pm \rangle }$ are the eigenvectors of $\sigma _{z}.$ The function $c_{t}$ is defined by $(d/dt)c(t)=-\int_{0}^{t}f(t-t^{
\prime })c(t^{\prime })dt^{\prime },$ where the memory kernel corresponds to the bath correlation function $f(t)\equiv \sum_{k}|g_{k}|^{2}\exp [+i(\omega _{0}-\omega _{k})t].$
Using that $\langle A\rangle _{t}=\mathrm{Tr}_{s}[\rho _{t}A]=\mathrm{Tr} _{s}[\rho _{0}A_{t}],$ from Eq.~(\ref{RhoExacta}) it is possible to obtain the operator dual dynamics, which explicitly reads \begin{equation} A_{t}=\left( \begin{array}{cc}
A_{0}^{++}|c_{t}|^{2}+A_{0}^{--}(1-|c_{t}|^{2})\ \ \ & A_{0}^{+-}\ c_{t}^{\ast } \\ A_{0}^{-+}\ c_{t} & A_{0}^{--} \end{array} \right) , \end{equation}
where $A_{0}^{ss^{\prime }}\equiv \langle s|A_{0}|s^{\prime }\rangle .$ The environment quantumness $Q_{t}$ can be obtained from the relation~(\ref {QDual}), which here delivers \begin{equation}
Q_{t}=1-\langle \sigma _{z}\rangle _{0}[1-|c_{t}|^{2}], \label{QExact} \end{equation} where $\langle \sigma _{z}\rangle _{0}=\mathrm{Tr}_{s}[\sigma _{z}\rho _{0}]. $
From Eq.~(\ref{QExact}) it follows that\ $\lim {}_{t\rightarrow \infty }Q_{t}=1-\langle \sigma _{z}\rangle _{0}.$ This limit assumes extreme values when $\langle \sigma _{z}\rangle _{0}=\pm 1.$ Therefore, the degree of environment quantumness is maximal [Eq.~(\ref{DqBounds})], \begin{equation} D_{Q}=1. \label{DqUnoNoMarkov} \end{equation} Consistently, this value also emerges from the Lindblad modeling Eq.~(\ref {Thermal}) when the environment temperature vanishes [see Eq.~(\ref {DqThermal})]. In addition, the expressions for $Q_{t},$ Eq.~(\ref{QThermico}
) with $\langle \sigma _{z}\rangle _{\infty }=-1$ and Eq.~(\ref{QExact}), assume the same structure. The non-Markovian effects appear through the time behavior of the decay function $|c_{t}|^{2},$ which in contrast to the Markovian case, may develop oscillatory behaviors~\cite{breuerbook}. For example, assuming a Lorentzian spectral density, which implies the exponential correlation $f(t)=(\gamma /2\tau _{c})\exp [-|t|/\tau _{c}],$ it follows $c_{t}=e^{-t/2\tau _{c}}[\cosh (t\chi /2\tau _{c})+\chi ^{-1}\sinh (t\chi /2\tau _{c})],$ where $\chi \equiv \sqrt{1-2\gamma \tau _{c}}.$ In a weak coupling limit $\gamma \ll 1/\tau _{c},$ where the correlation time $ \tau _{c}$\ of the bath is the minor time scale of the problem, an monotonic exponential decay is recovered $c_{t}\simeq \exp (-\gamma t/2)$ [Eq.~(\ref {QThermico})].
\subsection{Resonance fluorescence}
An optical two-level transition submitted to the action of a resonant external laser field can be well approximated through the evolution~\cite {carmichaelbook} \begin{equation} \frac{d\rho _{t}}{dt}=-i\frac{\Omega }{2}[\sigma _{x},\rho _{t}]+\gamma (\sigma \rho _{t}\sigma ^{\dagger }-\frac{1}{2}\{\sigma ^{\dagger }\sigma ,\rho _{t}\}_{+}). \label{Fluor} \end{equation} Here, $\gamma $ is the natural decay rate while the frequency $\Omega $ is proportional to the intensity of the external excitation. With $\sigma _{j}$ $(j=x,y,z)$ we denote the $j$-Pauli matrix. As before, $\sigma $ and $\sigma ^{\dagger }$ are the standard lowering and raising operators. We notice that the effective environment action correspond to a thermal bath a cero temperature. Below we study how the previous result $D_{Q}=1$ [Eqs.~(\ref {DqThermal})] is affected by the presence of the external excitation.
From Eqs.~(\ref{QDual}) and (\ref{At}), in a Laplace domain $ [f(u)=\int_{0}^{\infty }dte^{-ut}f(t)],$\ $Q_{t}$ is defined by the exact expression \begin{eqnarray} Q_{u} &=&\frac{1}{u}-\langle \sigma _{z}\rangle _{0}\frac{\gamma (2u+\gamma ) }{u[(u+\gamma )(2u+\gamma )+2\Omega ^{2}]} \notag \\ &&+\langle \sigma _{y}\rangle _{0}\frac{2\gamma \Omega }{u[(u+\gamma )(2u+\gamma )+2\Omega ^{2}]}, \label{QLaplace} \end{eqnarray} where $\langle \sigma _{j}\rangle _{0}=\mathrm{Tr}_{s}[\sigma _{j}\rho _{0}]. $ Explicitly, \begin{equation} \langle \sigma _{z}\rangle _{0}=(\rho _{0}^{++}-\rho _{0}^{--}),\ \ \ \ \ \ \langle \sigma _{y}\rangle _{0}=i(\rho _{0}^{+-}-\rho _{0}^{-+}). \end{equation} In the time domain, from Eq.~(\ref{QLaplace}) it is possible to write \begin{equation} Q_{t}=1+\int_{0}^{t}\gamma dt^{\prime }e^{-\frac{3}{4}\gamma t^{\prime }}[\langle \sigma _{z}\rangle _{0}\ z(t^{\prime })+\langle \sigma _{y}\rangle _{0}\ y(t^{\prime })], \label{QdeTe} \end{equation} where the auxiliary functions are \begin{subequations} \begin{eqnarray} y(t) &\equiv &4\frac{\Omega }{\Gamma }\sinh \Big{(}\frac{t\Gamma }{4}\Big{)}, \\ z(t) &\equiv &\cosh \Big{(}\frac{t\Gamma }{4}\Big{)}-\frac{\gamma }{\Gamma } \sinh \Big{(}\frac{t\Gamma }{4}\Big{)}, \end{eqnarray} with $\Gamma \equiv \sqrt{\gamma ^{2}-(4\Delta )^{2}}.$
The stationary value of $Q_{t}$ can be obtained straightforwardly from Eq.~( \ref{QLaplace}) as $Q_{\infty }=\lim_{t\rightarrow \infty }Q_{t}=\lim_{u\rightarrow 0}uQ_{u},$ which leads to \end{subequations} \begin{equation} Q_{\infty }=1+\langle \sigma _{z}\rangle _{\infty }\langle \sigma _{z}\rangle _{0}+\langle \sigma _{y}\rangle _{\infty }\langle \sigma _{y}\rangle _{0}, \label{QInfinito} \end{equation} where the stationary mean values are \begin{equation} \langle \sigma _{z}\rangle _{\infty }=-\frac{\gamma ^{2}}{\gamma ^{2}+2\Omega ^{2}},\ \ \ \ \ \langle \sigma _{y}\rangle _{\infty }=\frac{ 2\gamma \Omega }{\gamma ^{2}+2\Omega ^{2}}. \end{equation} Consistently, these expressions also follow as $\langle \sigma _{j}\rangle _{\infty }=\lim_{t\rightarrow \infty }\mathrm{Tr}_{s}[\sigma _{j}\rho _{t}].$
The system initial conditions that lead to extreme values of $(Q_{\infty
}-1) $ can be determine from Eq.~(\ref{QInfinito}). It is found that the initial state must be pure, $\rho _{0}=|\psi _{0}\rangle \langle \psi _{0}|,$
where the state $|\psi _{0}\rangle $ is parametrized in terms of angles $
(\theta _{0},\phi _{0})$ in the Bloch sphere, $|\psi _{0}\rangle =|\psi (\pm ,\theta _{0},\phi _{0})\rangle $ \cite{shankar}. Maximization implies that \begin{subequations} \label{Angles} \begin{equation} \tan (\theta _{0})=-\frac{2\Omega }{\gamma }=\frac{\langle \sigma _{y}\rangle _{\infty }}{\langle \sigma _{z}\rangle _{\infty }},\ \ \ \ \ \ \ \ \ \ \phi _{0}=\frac{\pi }{2}. \end{equation}
We remark that there are two different orthogonal states $\{|\psi (\pm ,\theta _{0},\phi _{0})\rangle \}$ associated to the direction defined by these angles. They correspond the basis where the stationary state $\rho _{\infty }=\lim {}_{t\rightarrow \infty }\rho _{t},$ is a diagonal matrix. In addition, it is found that maximization is achieved with \begin{equation} \tan (\tilde{\theta}_{0})=\frac{2\Omega }{\gamma }=-\frac{\langle \sigma _{y}\rangle _{\infty }}{\langle \sigma _{z}\rangle _{\infty }},\ \ \ \ \ \ \ \ \ \ \tilde{\phi}_{0}=\frac{3\pi }{2}. \end{equation} These angles define the basis where the (time-reversed) stationary system density matrix is a diagonal operator, $\tilde{\rho}_{\infty }=\lim {}_{t\rightarrow \infty }\rho _{t}^{\ast },$ where conjugation is taken in the basis defined by the eigenvectors of $\sigma _{z}.$ These last solutions are consistent with Eq.~(\ref{MaxEigen}).
\begin{figure}
\caption{Left panel, $Q_{t}$ [Eq.~(\protect\ref{QdeTe})] as a function of time for different values of $\Omega /\protect\gamma .$ The system initial conditions fulfill Eq.~(\protect\ref{Angles}). The curves above and below $ Q_{t}=1$ correspond to the lower and upper initial states respectively. Right panel, degree of quantumness $D_{Q}$ [Eq.~(\protect\ref{DQFluor})] (full line) jointly the weak and strong intensity approximations [Eqs.~( \protect\ref{WeakAprox}) and (\protect\ref{DQRabiAlta})] (dotted lines).}
\end{figure}
With the previous election of initial conditions, from the definition~(\ref {Degree}) and Eq.~(\ref{QInfinito}), the degree of environment quantumness associated to the dynamics~(\ref{Fluor}) can be written as \end{subequations} \begin{equation} D_{Q}=\frac{\gamma \sqrt{\gamma ^{2}+4\Omega ^{2}}}{\gamma ^{2}+2\Omega ^{2}} . \label{DQFluor} \end{equation}
In Fig.~1 (left panel) we plot the time-dependence of $Q_{t}$ [Eq.~(\ref {QdeTe})] assuming system initial conditions that maximize the degree of environment quantumness [Eq.~(\ref{Angles})]. When $\Omega /\gamma =0,$ that is in absence of the external excitation, it is obtained a monotonic behavior, $Q_{t}=1\mp \lbrack 1-e^{-t\gamma }].$ Consistently, this case can be recovered from Eq.~(\ref{QThermico}) after taking a vanishing environment temperature $\langle \sigma _{z}\rangle _{\infty }=-1,$ and $\langle \sigma _{z}\rangle _{0}=\pm 1.$ On the other hand, when increasing $\Omega /\gamma $ oscillations in the behavior of $Q_{t}$ emerges. In addition, the asymptotic values $\lim_{t\rightarrow \infty }Q_{t}$ start to approach the unit value. Even more, when $\Omega /\gamma \gg 1$ it follows that $\lim_{t\rightarrow \infty }Q_{t}\approx 1.$
Given that for each value of $\Omega /\gamma $ the initial system state fulfills the condition~(\ref{Angles}), the asymptotic values of $Q_{t}$ shown in Fig.~1 are related to the degree of environment quantumness [Eq.~( \ref{Degree})] as $D_{Q}=\left\vert \lim_{t\rightarrow \infty }Q_{t}-1\right\vert .$ In the right panel of Fig.~1 we plot $D_{Q}$ as a function of amplitude of the external field $\Omega /\gamma .$ When the external excitation is weak, Eq.~(\ref{DQFluor}) can be well approximated by \begin{equation} D_{Q}\simeq 1-2\left( \Omega /\gamma \right) ^{4}\ \ \ \ \ \ \ \ \left( \Omega /\gamma \right) <1. \label{WeakAprox} \end{equation} Thus, in this regime the quantumness of the environment influence is maximal. In fact, the departure from $D_{Q}=1$ depends on the fourth power of $\Omega /\gamma .$ On the other hand, when the external excitation is strong enough, it follows \begin{equation} D_{Q}\simeq 1/(\Omega /\gamma )\rightarrow 0,\ \ \ \ \ \ \ \ \ \ \ (\Omega /\gamma )\gg 1. \label{DQRabiAlta} \end{equation} This result means that, in this extreme regime, the environment influence can be well approximated by classical noises. This is a non-intuitive result. In fact, some quantum features of the dynamics~(\ref{Fluor}) emerge when increasing the external coherent fields~\cite{carmichaelbook}. This apparent contradiction is raised up when realizing that the proposed measure quantifies how much the environment action departs from the influence of classical noise fluctuations by considering the full open quantum dynamics, that is, reservoir and external fields.
By finding the explicit solutions of the matrix elements of $\rho _{t},$ when $\Omega /\gamma \gg 1$ it is possible to approximate the Lindblad evolution~(\ref{Fluor}) by \begin{equation} \frac{d\rho _{t}}{dt}\approx -i\frac{\Omega }{2}[\sigma _{x},\rho _{t}]+ \frac{3}{4}\gamma (\sigma _{z}\rho _{t}\sigma _{z}-\rho _{t}),\ \ \ \ (\Omega /\gamma )\gg 1. \label{Dephasing} \end{equation} Thus, the combined action of the environment (whose effective temperature is cero) and the external excitation can be represented by a dephasing mechanism, that is, $(3\gamma /4)(\sigma _{z}\rho _{t}\sigma _{z}-\rho _{t}). $ This contribution can always be obtained by coupling the system to external white noises such as for example Gaussian noises~\cite {GaussianNoise} or Poisson noises~\cite{viola}. Consequently, the classicality indicated by the result~(\ref{DQRabiAlta}) is completely consistent, which in turn also shows the physical meaning of the developed approach.
\subsection{Optimal states for two-interacting qubits}
We consider two qubits $(a$ and $b)$ whose bipartite density matrix $\rho _{t}^{ab}$\ evolves as \begin{equation} \frac{d\rho _{t}^{ab}}{dt}=-i\frac{\Omega }{2}[\sigma _{x}\otimes \sigma _{x},\rho _{t}^{ab}]+\gamma \mathcal{L}_{a}[\rho _{t}^{ab}]+\gamma \mathcal{L }_{b}[\rho _{t}^{ab}]. \label{TWO} \end{equation} The frequency $\Omega $ scales the Hamiltonian interaction between both systems. In addition, $\mathcal{L}_{a}$ and $\mathcal{L}_{b}$ define the dissipative dynamics of each subsystem. They are defined by the dissipative contribution in Eq.~(\ref{Fluor}), here written in each Hilbert space. We study the relation between the proposed quantumness measure $Q_{t}$ and the optimal initial conditions $\rho _{0}^{ab}$ that lead to its maximal value in the stationary regime.
The stationary state $[\rho _{\infty }^{ab}=\lim_{t\rightarrow \infty }\rho _{t}^{ab}]$ of the dynamics~(\ref{TWO}) can be obtained in an exact analytical way. Introducing the standard base of states $\{|++\rangle
,|+-\rangle ,|-+\rangle ,|--\rangle \},$ it follows \begin{equation} \rho _{\infty }^{ab}=\frac{1}{4\Gamma ^{2}}\left( \begin{array}{cccc} \Omega ^{2} & 0 & 0 & -i2\gamma \Omega \\ 0 & \Omega ^{2} & 0 & 0 \\ 0 & 0 & \Omega ^{2} & 0 \\ i2\gamma \Omega & 0 & 0 & 4\gamma ^{2}+\Omega ^{2} \end{array} \right) , \end{equation} where $\Gamma \equiv \sqrt{\gamma ^{2}+\Omega ^{2}}.$ By using Eq.~(\ref {MaxEigen}), the degree of quantumness can be written in terms of the largest eigenvalue of $\tilde{\rho}_{\infty }^{ab}=\rho _{\infty }^{\ast ab}. $ We get \begin{equation} D_{Q}=\frac{\gamma (\gamma +2\sqrt{\gamma ^{2}+\Omega ^{2}})}{\gamma ^{2}+\Omega ^{2}}. \label{DQTWO} \end{equation} In Fig.~2 (left panel) we plot $D_{Q}$ as a function of $\Omega /\gamma .$ We notice that by increasing the influence of the Hamiltonian contribution classicality is achieved, $\lim_{\Omega /\gamma \rightarrow \infty }D_{Q}=0.$ Similarly to the previous case [Eqs.~(\ref{DQFluor}) and (\ref{Dephasing})], in this limit the combined action of the environment and the subsystems interaction Hamiltonian can be written in terms of dephasing mechanisms, which in turn can be represented by the action of classical noise fluctuations.
\begin{figure}
\caption{Degree of environment quantumness $D_{Q}$ [Eq.~(\protect\ref{DQTWO} )] and concurrence of the initial optimal state $C[\protect\rho
_{0}]=C[|i_{\max }\rangle \langle i_{\max }|]$ [Eq.~(\protect\ref{Imax})], corresponding to the bipartite evolution~(\protect\ref{TWO}).}
\end{figure}
Eq.~(\ref{MaxEigen}) also characterize the initial condition $\rho _{0}^{ab}=|i_{\max }\rangle \langle i_{\max }|$ that leads to maximal stationary values of the quantumness measure $Q_{t}.$ $|i_{\max }\rangle $ is the eigenstate of the stationary state $\tilde{\rho}_{\infty }^{ab}$\ with the largest eigenvalue. It reads \begin{equation}
|i_{\max }\rangle =\frac{1}{\sqrt{2\Gamma (\Gamma -\gamma )}}[i(\Gamma
-\gamma )|++\rangle +\Omega |--\rangle ], \label{Imax} \end{equation}
where as before $\Gamma =\sqrt{\gamma ^{2}+\Omega ^{2}},$ and $\langle i_{\max }|i_{\max }\rangle =1.$
In general $|i_{\max }\rangle $\ is an entangled state. This feature can be quantified through its concurrence~\cite{horodecki} $C[\rho _{0}]=C[|i_{\max
}\rangle \langle i_{\max }|].$ In Fig.~2 (right panel) we plot its dependence with $\Omega /\gamma .$ In the limit of a vanishing unitary coupling, from Eq.~(\ref{Imax}) it follows \begin{equation}
\lim_{\Omega /\gamma \rightarrow 0}|i_{\max }\rangle =|--\rangle . \label{Cero} \end{equation} This is an unentangled state, implying $C[\rho _{0}]=0.$ In contrast, in the limit of strong coupling we get \begin{equation}
\lim_{\Omega /\gamma \rightarrow \infty }|i_{\max }\rangle =\frac{1}{\sqrt{2}
}(i|++\rangle +|--\rangle ), \label{Infinito} \end{equation} which is a maximal entangled state, $C[\rho _{0}]=1.$
The previous behaviors have an interesting physical implication. In the weak coupling limit $[\Omega /\gamma \approx 0],$ an (almost) unentangled initial state leads to the maximal departure from classicality of the environment action (quantified by $Q_{t}).$ When increasing the unitary coupling $ [\Omega /\gamma >0],$ an increasing initial entanglement between both subsystems is necessary to obtain the maximal departure from classicality. In this way, entanglement becomes a necessary resource to detect the quantumness of the environment influence when approaching a limit where a classical noise approximation is valid.
For this model the quantumness measure $Q_{t}$ assumes a simple form [Eq.~( \ref{QDual})]. When maximizing its stationary value with respect to the initial conditions it follows \begin{equation} Q_{t}=1+\frac{\gamma ^{2}(1+e^{-2\gamma t})}{\Gamma ^{2}}+2\frac{\gamma }{ \Gamma }\Big{[}1-\lambda e^{-2\gamma t}\cos (\Omega t)\Big{]}, \label{QBipartito} \end{equation}
where $\lambda \equiv 1+(\gamma /\Gamma ).$ Consistently, the initial state that leads to this expression is $\rho _{0}^{ab}=|i_{\max }\rangle \langle i_{\max }|$ [Eq.~(\ref{Imax})].
The previous results relies on taking both subsystems as the system of interest. One can also deal with the partial dynamics $\rho _{t}^{a}=\mathrm{ Tr}_{b}[\rho _{t}^{ab}],$ or alternatively $\rho _{t}^{b}=\mathrm{Tr} _{a}[\rho _{t}^{ab}].$ The corresponding stationary states read \begin{equation} \rho _{\infty }^{s}=\frac{1}{2\Gamma ^{2}}\left( \begin{array}{cc} \Omega ^{2} & 0 \\ 0 & 2\gamma ^{2}+\Omega ^{2} \end{array} \right) ,\ \ \ \ \ \ s=a,b. \end{equation} Performing similar calculations, the degree of quantumness and the optimal state are \begin{equation}
D_{Q}=\frac{\gamma ^{2}}{\gamma ^{2}+\Omega ^{2}},\ \ \ \ \ \ \ \ |i_{\max
}\rangle =|-\rangle . \end{equation}
Given the symmetry of Eq. (\ref{TWO}), this results applies to both subsystems. Furthermore, assuming $\rho _{0}^{ab}=|i_{\max }\rangle \langle i_{\max }|\otimes \rho _{0}^{b},$ where $\rho _{0}^{b}$ is an arbitrary state, it follows \begin{equation} Q_{t}=1+\frac{\gamma ^{2}}{\Gamma ^{2}}+\frac{\gamma e^{-\gamma t}}{\Gamma ^{2}}\Big{[}\Omega \sin (\Omega t)-\gamma \cos (\Omega t)\Big{]}. \end{equation} The same expression follows from $\rho _{0}^{ab}=\rho _{0}^{a}\otimes
|i_{\max }\rangle \langle i_{\max }|.$ These results differ from those obtained starting from a bipartite representation [Eqs.~(\ref{DQTWO}), (\ref {Imax}) and~(\ref{QBipartito})]. This feature shows that the environment influence over a system cannot in general be related in a simple way with the action over the constitutive subsystems.
\subsection{Quantum harmonic oscillator coupled to a thermal environment}
The developed approach applies consistently to systems with a Hilbert space of finite dimension [see Eqs.~(\ref{Bounds}) and~(\ref{DqBounds})]. Complementarily, here we study the case of a quantum harmonic oscillator coupled to a thermal environment at a finite temperature.
The density matrix evolution can be written as in Eq.~(\ref{Thermal}) under the replacements $\sigma ^{\dagger }\rightarrow a^{\dagger }$ and $\sigma \rightarrow a,$ where $a^{\dag }$ and $a$ are the creation and annihilation Bosonic operators of the system respectively~\cite{breuerbook}. The evolution can alternatively be written through a Wigner function. It is defined as the Fourier transform $W(\alpha ,\alpha ^{\ast },t)\equiv (1/\pi ^{2})\int d^{2}z\chi (z,z^{\ast })e^{-iz^{\ast }\alpha ^{\ast }}e^{-iz\alpha },$ where the characteristic function is$\ \chi (z,z^{\ast })\equiv \mathrm{ Tr}_{s}[\rho _{t}\exp (iz^{\ast }a^{\dag }+iza)].$ Denoting $W_{t}=W(\alpha ,\alpha ^{\ast },t),$ its time evolution reads~\cite{carmichaelbook} \begin{equation} \frac{\partial W_{t}}{\partial t}=\Big{\{}\varphi \frac{\partial }{\partial \alpha }\alpha +\varphi ^{\ast }\frac{\partial }{\partial \alpha ^{\ast }} \alpha ^{\ast }+\Big{(}\frac{\kappa +\zeta }{2}\Big{)}\frac{\partial ^{2}}{ \partial \alpha \partial \alpha ^{\ast }}\Big{\}}W_{t}, \label{Wigner} \end{equation} where $\varphi \equiv i\omega _{0}+(\kappa -\zeta )/2.$ Here, $\omega _{0}$ is the natural frequency of the system. Notice that dissipative contributions (first-order derivatives) are present whenever the underlying rates are different, $\kappa \neq \zeta .$ On the other hand, diffusion (second-order derivatives) always develops, being scaled by $(\kappa +\zeta )/2.$
The operator evolution can be obtained in a similar way from the dual dynamics associated to the system density matrix. Alternatively, it can be deduced by using that operator expectation values can be written as $\langle A\rangle _{t}=\int d\alpha d\alpha ^{\ast }W_{t}A_{0}(\alpha ,\alpha ^{\ast })=\int d\alpha d\alpha ^{\ast }W_{0}A(\alpha ,\alpha ^{\ast },t),$ where $ A_{0}(\alpha ,\alpha ^{\ast })$ is the \textquotedblleft scalar representation\textquotedblright\ of the system operator $A.$ From Eq.~(\ref {Wigner}) we get $[A_{t}=A(\alpha ,\alpha ^{\ast },t)]$ \begin{equation} \frac{\partial A_{t}}{\partial t}=-\Big{\{}\varphi \alpha \frac{\partial }{ \partial \alpha }+(\varphi \alpha )^{\ast }\frac{\partial }{\partial \alpha ^{\ast }}-\Big{(}\frac{\kappa +\zeta }{2}\Big{)}\frac{\partial ^{2}}{ \partial \alpha \partial \alpha ^{\ast }}\Big{\}}A_{t}. \label{DualWigner} \end{equation} Using this representation, the quantumness indicator [Eq.~(\ref{QDual})] can be expressed as $Q_{t}=\int d\alpha d\alpha ^{\ast }A_{t},$ where $A_{t}$ is the solution of the dual evolution with initial condition $A_{0}=W_{0}.$ By integration by parts of Eq.~(\ref{DualWigner}) it is simple to arrive to $ (d/dt)Q_{t}=(\kappa -\zeta )Q_{t},$ which leads to \begin{equation} Q_{t}=\exp [(\kappa -\zeta )t]. \label{QHarmonico} \end{equation} This results is valid independently of the initial system density matrix. Thus, not any maximization procedure is available. The same expression for $ Q_{t}$ follows by using a Glauber-Sudarshan P-representation or Q-representation~\cite{carmichaelbook}, or even from a (diagonal) Fock-number characteristic function approach~\cite{orzag}.
Given that $\kappa \geq \zeta $ $[\kappa =\gamma (n_{th}+1)$ and $\zeta =\gamma n_{th}],$ the indicator $Q_{t}$ develops an exponential divergence in time for any finite temperature of the bath. Only when the reservoir temperature is infinite $(\kappa =\zeta )$ a classical noise representation applies, $Q_{t}=1$ and $D_{Q}=0.$ This behavior has a clear interpretation. In fact, for any finite reservoir temperature, the Wigner function involves dissipative contributions [see Eq.~(\ref{Wigner})]. These (trace preserving) effects develop in the system Hilbert space and cannot be reproduced by any classical external influence. Dissipative contributions only vanishes when $ \kappa =\zeta ,$ which consistently supports the (discontinuous) temperature-dependence of the degree of quantumness in this case.
While the previous result is consistent it strongly differs from the two-level system case [see Eq.~(\ref{DqThermal})], where $D_{Q}$ has a continuous dependence on the reservoir temperature. Interestingly, for Hilbert spaces of infinite dimension the expression~(\ref{DqInfinita}) allows us to define a \textit{renormalized degree of quantumness} as $ D_{Q_{R}}\equiv \max_{\lbrack \rho _{0}]}\mathrm{Tr}_{s}[\tilde{\rho} _{\infty }\rho _{0}].$ In terms of the Wigner function it reads \begin{equation} D_{Q_{R}}=\max_{[W_{0}]}\int d\alpha d\alpha ^{\ast }\tilde{W}_{\infty }W_{0}. \label{DQWigner} \end{equation} Here, the maximization must be performed over all possible (normalized) initial conditions $W_{0}.$ In addition, Eq.~(\ref{Wigner}) implies that $ \tilde{W}_{\infty }=W_{\infty }=\lim_{t\rightarrow \infty }W_{t}=(1/\pi
\sigma _{\infty })\exp (-|\alpha |^{2}/\sigma _{\infty }),$ where $\sigma _{\infty }=(1/2)(\kappa +\zeta )/(\kappa -\zeta )=n_{th}+(1/2)=(1/2)(\tanh [\beta \hbar \omega _{0}/2])^{-1}.$
Given that $D_{Q_{R}}$ is a linear functional of $W_{0}$ the maximization problem cannot be solved by using standard functional derivative techniques. As an ansatz we assume that $W_{0}$ is also a Gaussian function. In such a case, it follows that $W_{0}$ must has the minimal possible wide. Thus, it must be the Wigner function of the ground state of the system, which in turn from Eq.~(\ref{DQWigner}) delivers \begin{equation} D_{Q_{R}}=\frac{1}{\sigma _{\infty }+(1/2)}=1-\exp (-\beta \hbar \omega _{0}). \end{equation} The same result follows by performing a similar ansatz in the energy eigenbasis representation. $D_{Q_{R}}$ has the expected dependence with the environment temperature. In particular, classicality $[D_{Q_{R}}=0]$ is approached in a high temperature limit. In contrast to the two-level system [Eq.~(\ref{DqThermal})], here the renormalized degree of quantumness cannot be associated to the time-dependent quantumness measure $Q_{t}$ [Eq.~(\ref {QHarmonico})].
\section{Summary and Conclusions}
We have developed a consistent proposal that allows quantifying how far the influence of a given environment over an open quantum system departs from the action of classical stochastic fields. Its physical ground relies on associating the quantumness of the environment influence with the lack of commutativity between the reservoir initial state and the total system-environment Hamiltonian. Over this basis we introduced a (time-dependent) quantumness measure [Eq.~(\ref{QDef})]. Its stationary value (long time-regime) when maximized over all possible system initial conditions define a degree of environment non-classicality [Eq.~(\ref{Degree} )]. For dissipative dynamics it can be determine from the largest eigenvalue of the (time reversal) stationary system density matrix [Eq.~(\ref{MaxEigen} )]. Independently of the system dynamical regime, the quantum measure can be written in terms of the operators dual evolution [Eq.~(\ref{QDual})]. This alternative definition provides a powerful tool for characterizing the quantumness measure in both Markovian and non-Markovian regimes.
Consistently the quantumness measure vanishes identically for a wide class of quantum dynamics, which include Hamiltonian ensembles [Eq.~(\ref {HEnsemble})], stochastic Hamiltonians [Eq.~(\ref{Ruido})], and a class of collisional dynamics [Eq.~(\ref{UnitalCollision})].
All of these dynamics can be obtained by considering the action of underlying classical stochastic processes. In spite of the consistence of this result, the quantumness indicator also vanishes when the open system dynamics is defined by a unital map [Eq.~(\ref{Unital})]. Hence, the proposed indicator can also be read as a measure of departure from this dynamical property.
The consistence of the developed approach was supported by studying different dissipative open system dynamics. For two-level systems coupled to a thermal bath, the degree of environment quantumness decreases monotonically with the reservoir temperature [Eq.~(\ref{DqThermal})]. For an optical transition (resonant fluorescence) the amplitude of the external coherent excitation monotonically drives the environment influence to classicality [Eq.~(\ref{DQFluor})]. The consistence of this result follows from the possibility of describing the high intensity regime in terms of a dephasing quantum master equation that can be represented by the action of classical noises. On the other hand, by analyzing two interacting qubits it was found that quantum entanglement may become a necessary resource for detecting the quantumness of the environment influence when approaching a regime where a classical noise representation becomes a valid approximation. Application to systems endowed with a Hilbert space of infinite dimension was also established.
The present formalism lefts open some interesting issues. For example, which dynamical features determine the presence or absence of revivals in the time-behavior of the quantumness indicator is unknown. On the other hand, and operational definition and experimental measurability are also interesting issues that can be tackled from the proposed approach.
\section*{Acknowledgments}
Valuable discussions with Oscar Mensio are gratefully acknowledged. A.A.B. also thanks support from Consejo Nacional de Investigaciones Cient\'{\i} ficas y T\'{e}cnicas (CONICET), Argentina.
\appendix*
\section{Renewal collisional models}
In these models the statistics of the collisional times are defined by a \textquotedblleft waiting time distribution\textquotedblright\ $w(t).$ It gives the probability density for the time interval between consecutive collisional events. The corresponding survival probability is defined as $ P_{0}(t)=1-\int_{0}^{t}dt^{\prime }w(t^{\prime }).$ Poisson statistics corresponds to $w(t)=\gamma \exp (-\gamma t),$ $P_{0}(t)=\exp (-\gamma t),$ assumption that lead to Markovian Lindblad equations for the system dynamics.
In correspondence with Eq.~(\ref{Colision}), the system density matrix can be written in general as~\cite{collisional} \begin{equation} \rho _{t}=\sum_{n=0}^{\infty }\int_{0}^{t}dt^{\prime }\mathcal{P} _{0}(t-t^{\prime })\mathcal{W}^{(n)}(t^{\prime })\rho _{0}. \label{Average} \end{equation} The involved superoperators are written in a Laplace domain $ [f(u)=\int_{0}^{\infty }dte^{-ut}f(t)]$ as \begin{equation} \mathcal{P}_{0}(u)\equiv P_{0}(u-\mathcal{L}_{s}),\ \ \ \ \ \ \mathcal{W} ^{(n)}(u)\equiv \lbrack \mathcal{E}w(u-\mathcal{L}_{s})]^{n}. \label{Woperator} \end{equation} Here, $P_{0}(u)=[1-w(u)]/u.$ Furthermore, the free propagator between events was written as $\mathcal{G}_{t}=\exp (t\mathcal{L}_{s}).$ Notice that in a time domain $\mathcal{W}^{(n)}(t)$ consists in the convolution of free propagation and $n$-collisional events. Consistently, the function $ P_{0}(u)w^{n}(u)$ gives the probability of occurring $n$-events up to time $ t.$
Using that $\langle A\rangle _{t}=\mathrm{Tr}_{s}[A_{0}\rho _{t}]=\mathrm{Tr} _{s}[\rho _{0}A_{t}],$ from Eq.~(\ref{Average}) the operator dual evolution reads \begin{equation} A_{t}=\sum_{n=0}^{\infty }\int_{0}^{t}dt^{\prime }\mathcal{W}^{\bigstar (n)}(t^{\prime })\mathcal{P}_{0}^{\bigstar }(t-t^{\prime })A_{0}. \label{DualAverageA} \end{equation} Here, the involved superoperators are defined as $\mathcal{P}_{0}^{\bigstar }(z)=P_{0}(z-\mathcal{L}_{s}^{\bigstar })$ and $\mathcal{W}^{\bigstar (n)}(z)=[w(z-\mathcal{L}_{s}^{\bigstar })\mathcal{E}^{\bigstar }]^{n}.$ These expressions rely on the definitions $\mathrm{Tr}_{s}[A\mathcal{E}[\rho ]]=\mathrm{Tr}_{s}[\rho \mathcal{E}^{\bigstar }[A]],$ and $\mathrm{Tr} _{s}[A\exp (t\mathcal{L}_{s})[\rho ]]=$ $\mathrm{Tr}_{s}[\rho \exp (t \mathcal{L}_{s}^{\bigstar })[A]].$
The time-dependent quantumness indicator $Q_{t}$ can be calculated as the trace of dual dynamics [Eq.~(\ref{QDual})]. The property $\mathrm{Tr} _{s}[A_{t}]=\mathrm{Tr}_{s}[A_{0}]$ leads to the condition $\mathrm{Tr}_{s}[ \mathcal{E}^{\bigstar }[A]]=\mathrm{Tr}_{s}[A],$ which recovers Eq.~(\ref {UnitalCollision}).
\end{document} |
\begin{document}
\preprint{APS/123-QED}
\title{Protocol for Fermionic Positive-Operator-Valued Measures} \author{D. R. M. Arvidsson-Shukur} \thanks{These authors contributed equally to this paper.} \affiliation{
Cavendish Laboratory, Department of Physics, University of Cambridge, Cambridge CB3 0HE, United Kingdom } \affiliation{ Hitachi Cambridge Laboratory, J. J. Thomson Avenue, CB3 0HE, Cambridge, United Kingdom } \author{H. V. Lepage} \thanks{These authors contributed equally to this paper.} \affiliation{
Cavendish Laboratory, Department of Physics, University of Cambridge, Cambridge CB3 0HE, United Kingdom } \author{E. T. Owen} \thanks{These authors contributed equally to this paper.} \affiliation{ Institute of Photonics and Quantum Sciences, Heriot-Watt University, Edinburgh EH14 4AS, United Kingdom } \affiliation{
Cavendish Laboratory, Department of Physics, University of Cambridge, Cambridge CB3 0HE, United Kingdom } \author{T. Ferrus} \affiliation{ Hitachi Cambridge Laboratory, J. J. Thomson Avenue, CB3 0HE, Cambridge, United Kingdom } \author{C. H. W. Barnes} \affiliation{
Cavendish Laboratory, Department of Physics, University of Cambridge, Cambridge CB3 0HE, United Kingdom }
\date{\today}
\begin{abstract}
In this paper we present a protocol for the implementation of a positive-operator-valued measure (POVM) on massive fermionic qubits. We present methods for implementing non-dispersive qubit transport, spin rotations and spin polarizing beam-splitter operations. Our scheme attains linear optics-like control of the spatial extent of the qubits by considering groundstate electrons trapped in the minima of surface acoustic waves in semiconductor heterostructures. Furthermore, we numerically simulate a high-fidelity POVM that carries out Procrustean entanglement distillation in the framework of our scheme, using experimentally realistic potentials. Our protocol can be applied, not only to pure ensembles with particle pairs of known identical entanglement, but also to realistic ensembles of particle pairs with a distribution of entanglement entropies. This paper provides an experimentally realisable design for future quantum technologies.
\end{abstract}
\pacs{Valid PACS appear here}
\maketitle
\section{Introduction}
In quantum mechanics, the theory of measurement is far from straightforward. Whilst there is considerable debate about the interpretations of quantum mechanics, there remain simple questions about how to formulate a mathematical description of the outcomes of recent experiments. It is often assumed that a measurement apparatus implements von Neumann's projective measurements, whereby a quantum state $\ket{\psi}$ is projected onto the eigenbasis of an observable operator $\hat{A} = \sum_i \ket{A_i} \bra{A_i}$ and the final system is measured to be in the state $\ket{A_i}$ with probability $|\langle A_i | \psi \rangle |^2$. However, in recent years, theory and experiment have shown that this definition of measurement is too restrictive. Projective measurements fail to describe a broad range of fascinating quantum phenomena including non-demolition \cite{Braginsky80}, weak \cite{Aharanov88}, and continuous measurements \cite{Doherty99}.
A crucial component for a generalised theory of quantum measurement, is the positive-operator-valued measure (POVM). These measures consist of a set of semi-definite non-negative operators, each associated with a particular measurement outcome, acting on the relevant Hilbert space. By operating on a system with a POVM, followed by traditional projective measurements, it is possible to access information about a system which cannot be obtained using projective measurements alone (e.g. distinguishing between non-orthogonal states). The uncertainty principle is maintained by allowing a finite probability that no information about the system is collected. POVMs also have a number of applications in quantum technologies\cite{Nielsen11, IVANOVIC1987, PERES1988, Ahnert05}, contributing crucial components to entanglement distillation, quantum cryptography and quantum metrology protocols
Experimental demonstrations of POVMs have been reported in photonic systems \cite{ Kwiat01, Zhao15} but, to date, there have been no realisations of POVMs acting on particles with mass. Whilst photons propagating in free space are non-dispersive, the wavefunction of a massive particle spreads out unless placed in a sufficiently strong confining potential. As POVMs are typically generated with quantum self-interference effects~\cite{Ahnert05}, the dispersion of massive particles is undesirable, as it reduces the fidelity of the interference. The ability to mimic devices from quantum optics, such as the POVM, in systems where quantum information is encoded on massive particles is particularly important for the development of quantum information processing routines in solid state systems~\cite{Bocquillon12, Fletcher13, Waldie15, Kataoka16}. For example, surface acoustic waves (SAWs) propagating on the surface of a piezoelectric semiconductor can both capture and transport electron qubits in electrostatically-defined dynamic quantum dots. Experimentally, beam splitters~\cite{Kataoka09} and polarization readout devices \cite{Elzerman04} have been implemented in GaAs heterostructures and protocols for realising universal quantum computations have been proposed~\cite{Barnes00}. The potential to integrate multiple components on-chip opens the possibility for developing sophisticated quantum optics-like experiments in solid-state devices.
In this paper, we present a protocol for implementing POVMs on massive electron spin-$\frac{1}{2}$ qubits. The protocol is based on the nested polarizing Mach-Zehnder interferometer proposed by Anhert and Payne~\cite{Ahnert05} but adapted for use in a solid state setting. We tailor a Hamiltonian to eliminate the spatial dispersion of electrons when passing through the Mach-Zehnder interferometer and the single qubit gates in order to achieve high fidelity POVMs. The spatial qubit translations are generated by SAW potentials, whilst the single qubit operations are executed with static magnetic fields. Our framework for massive particle POVMs provides a methodology for the implementation of standard optical operations on massive qubits. As an example, we demonstrate a protocol for Procrustean entanglement distillation \cite{Bennett96} of an electron spin-qubit system.
\section{\label{sec:level1} POVM Framework}
A variety of techniques have been proposed \cite{Stephen00, Ahnert05, Ahnert06} and demonstrated \cite{Wu09, Becerra13} for POVMs in optical systems. In this paper, we use the double interferometer device proposed by Ahnert and Payne (AP)~\cite{Ahnert06} as a template from which to develop a POVM for massive particles. Their implementation consists of two nested polarizing Mach-Zehnder interferometers which are joined by polarizing beam splitters. Local operations are performed on the polarization state of the photon qubit in the different arms of the interferometer using electro-optical phase modulators and wave plates as shown in Fig. \ref{fig:POVM}.
A photon entering the system with a polarization state $\ket{\Psi} = \alpha \ket{0} + \beta \ket{1}$ leaves the interferometer in a superposition of spatial states: \begin{equation}
\sum_{j}{\ket{\Psi_j}\left| p_j \right\rangle} = \sum_{j}{\hat{M}_j \ket{\Psi}\left| p_j \right\rangle} , \end{equation} where \begin{align}
\hat{M}_1 &= \cos{(\theta_1)}e^{i\phi_1}\ket{0}\!\!\bra{0} + \cos{(\theta_2)}e^{i\phi_2}\ket{1}\!\!\bra{1} \nonumber \\
\hat{M}_2 &= \sin{(\theta_1)}e^{i\phi_3}\ket{0}\!\!\bra{0} + \sin{(\theta_2)}e^{i\phi_4}\ket{1}\!\!\bra{1} \nonumber \end{align} are the Kraus operators of the POVM. The states $\ket{p_1}$ and $\ket{p_2}$ denote the spatially decoupled output paths such that a specific Kraus operation is performed on the polarisation state of the photon, conditioned on whether the photon exits the interferometer from output $\ket{p_1}$ or $\ket{p_2}$. Non-diagonal Kraus operators can be created by applying unitary operations to the input and outputs of Fig. \ref{fig:POVM}. Note that generally $\hat{M}_1 \hat{M}_2 \ket{\Psi} \neq 0$ and $\hat{M}_1 \hat{M}_1 \ket{\Psi} \neq \hat{M}_1 \ket{\Psi}$. The operators are not necessarily orthogonal and a POVM is different from a projective operation. Whilst the Kraus operators must satisfy: \begin{equation} \sum{\hat{M}^\dagger_i \hat{M}_i} = \hat{1} , \end{equation} the individual Kraus operators, $ \hat{M}_i$, are not necessarily unitary.
\begin{figure}\label{fig:POVM}
\end{figure}
\section{Unitary Evolution of a Massive Particle} \label{sec:MassiveParticleOpticalAnalogues}
The interferometric scheme presented in Sec.~\ref{sec:level1} provides a template for demonstrating POVMs. In order to map the AP POVM to a fermionic system, we will present processes which describes the individual unitary operations shown in Fig.~\ref{fig:POVM} for spin-$\frac{1}{2}$ qubits in semiconductor heterostructures. This provides us with a toolkit allowing us to perform coherent particle propagation, spin rotations and spin-dependent particle translation on massive particles.
The transformation of spatial propagation from photonic to fermionic states is not straightforward. Whilst a photon can pass through free space without dispersing significantly, the wavefunction of a massive particle---such as an ion or an electron---will disperse. As most optical devices, including the polarizing Mach-Zehnder Interferometer (MZI), rely on self-interference of spatially well-defined qubit states, these systems are especially sensitive to dispersion. Fig. \ref{fig:MZIstand} shows a staggered leapfrog \cite{Goldberg67, Askar78, Maestri00, Owen12, ArvShu16} time-evolution for the wavepacket of a massive particles passing through a MZI, where spin-dependent beam-splitters have been inserted at the junctions. The device curvature, wavepacket shape and momentum distribution have been chosen to maximise the output probability density in the upper right port (labelled by $b_1$ in Fig. \ref{fig:MZIstand}) of the polarizing MZI. Nevertheless, over $5 \%$ of the probability density disperses to unwanted locations of the MZI and the shape of the wavepacket is significantly distorted. This places an upper bound of 95\% on the spin-qubit fidelity of a single polarizing MZI. Additionally, the AP POVM relies on the spatial separation of the output states and any distortion of the spatial wavepacket will inhibit optimal control. The dramatic reduction in the fidelity of the qubit operation presents a challenge for the implementation of quantum protocols, highlighting the need for a more sophisticated approach.
\begin{figure}
\caption{Mach-Zehnder interferometer for massive particles at four different time steps. The potential is infinite in the striped area and zero elsewhere. The beam-splitters of the MZI are indicated with grey diagonal lines. }
\label{fig:MZIstand}
\end{figure}
The optical diagram in Fig~\ref{fig:POVM} can be broken down into three separate components: the free dispersiveless propagation of the photon through the interferometer, arbitrary polarization rotation using a combination of birefringent wave plates and the spatial separation of the photon into a pair of polarized modes using polarizing beam splitters. To replicate the AP POVM, we need to find massive particle analogues for each of these processes.
In order to realise high fidelity POVMs on massive particles, the dispersion of the states has to be eliminated. This can be obtained with Gaussian wavepackets in harmonic confining potentials. Such potentials have been used successfully in ion traps to perform coherent diabatic ion transport \cite{Bowler12, Hucul08} but equivalent potentials can be achieved in semiconductors by either electrostatically defining quantum dots using Schottky surface gates~\cite{Davies95, Tilke01, Kataoka09, Veldhorst14} or lithographically confining charges in doped regions separated by tunnel barriers \cite{Ferrus11}.
Our staggered leapfrog simulations confirm that spatial propagation can be obtained in a manner that both preserves the fidelity of the operation and keeps the shape of the wavepacket intact. There are two main ways of realising this. Firstly, the minima of the harmonic potentials can be shifted, displacing the wave packet and generating a coherent state. By imposing a diabatic shift of the ground-state potential of a stationary wavepacket, the particle can be captured when it coherently reaches the other side of the minimum of the intermediary potential. Secondly, by moving the minima of the harmonic potentials in an adiabatic manner it is possible to preserve the structure of the ground state whilst the qubit is moved between the optical component analogues.
However, our simulations show that an optimal way to adiabatically transport electrons non-dispersively is to use propagating SAW potentials. A ground-state electron (near Gaussian) trapped in the minimum of a sinusoidal SAW potential is transported coherently through the device with the propagation speed of the SAW. We suggest the use of surface Schottky gates to impose an overlying potential structure that adiabatically shifts the center of mass of the ground state in the SAW frame of reference. This effectively enables linear-optics like spatial control of the electron qubits. GPU-boosted staggered leapfrog \cite{Goldberg67, Askar78, Maestri00, Owen12} simulations allow for the parameters of the potential to be optimised for the implementation of a specific POVM.
Arbitrary polarization rotations for spin-$\frac{1}{2}$ particles can be described by time-ordered unitary operators: \begin{equation} \hat{R}_{\hat{k}} = {\cal T} \exp{\big[ i \lambda(t) \sigma_{\bm{\hat{k}}} t \big]} , \label{Eq:Rot} \end{equation} where $\lambda(t)$ is some time-dependent strength parameter and $\sigma_{\bm{\hat{k}}}$ are the Pauli matrices. Such unitary operations can be realised using a magnetic field with the Hamiltonian $\hat{H}_{rot} = - \bm{\mu} \cdot \bm{B}(t)$, where $\bm{\mu}$ is the magnetic dipole moment of the particle and the magnetic field $\bm{B}(t)$ is uniform over the particle wavepacket \cite{Furuta04, McNeil10}. Spin-rotations of SAW qubits have been studied in previous works \cite{Barnes00}. Charged qubits moving in a magnetic field will naturally experience a Lorentz force. However, for SAW carried electrons in semiconductor heterostructures, this force is counteracted greatly by the device confinement. Electromagnetic corrections can also be applied as suggested in \cite{Barnes00}. Other techniques for spin rotations include using a DC magnetic field to lift the spin degeneracy and applying an oscillating perpendicular magnetic field set in resonance between the two spin states \cite{Awschalom13}. Yet another technique uses electron spin resonance (ESR), where a pulse of microwaves becomes resonant with the upper and lower Zeeman-split spin states \cite{Kawakami16}.
Although solid-state physics present several possibilities to select the spin of an electron (Pauli blockade \cite{Prati11} or spin filtering \cite{Pla12}), implementing a spin-splitter is difficult in practice, owing to the generally small dimensions of devices and the intrinsic nature of the spin. However, several structures, materials or techniques can be used to channel dedicated spin orientations.
Antidots \cite{Zozoulenko04} or quantum spin hall systems \cite{Kato04} are commonly used to create spin-polarised channels at the edges of structures with a minimum number of gates and simplified geometry. These have been realised in graphene \cite{Tada12} but also in semiconductors. More generally, it is possible to utilise materials with strong spin-orbit interaction to generate spin currents out of charge current. Another approach is to scatter the wave packet off of a narrow magnetic semiconductor barrier, such as EuO~\cite{Santos07}, which will act as a spin filter only transmitting a specific electron spin polarisation. Furthermore, new types of materials, like topological insulators, possess intrinsic properties that allow locking spin states to specific transport directions \cite{Konig07}.
Finally, there exist a number of schemes for the projective measurement of fermion spin \cite{Kane98, Shnirman98, Makhlin99, Gardelis99, Barnes00, Elzerman04, Weber14}. These schemes implement spin-dependent translations of the qubits followed by a single particle charge readout. Technologies for projective spin measurements are based on magnetic readout (utilising the spin-valve effect), double occupation readout (utilising spin-dependent tunneling) or Stern-Gerlach readout.
\section{ Massive Particle POVM}
With this massive particle toolkit, we provide a proof-of-principle simulation of a fermionic POVM. Whilst our protocol can be used to implement any POVM on the massive spin-$\frac{1}{2}$ particle, we use the implementation of an entanglement distilling POVM as a guiding example in this section.
\subsection{Procrustean Entanglement Distillation}
One use of POVMs is found in the implementation of Bennett's Procrustean entanglement distillation.\cite{Bennett96} This protocol allows a subset of pure state qubit pairs to be discarded from a weakly entangled ensemble, such that the remaining particle pairs are more entangled. Significantly, Bennett's method can be { \it local} and {\it non-iterative} as the entanglement distillation is achieved through the application of a single POVM on only one of the particles.
For the arbitrarily entangled state, \begin{equation} \label{eq:in} \ket{\Psi_{\rm{A,B}}} = \alpha \ket{0_{\rm{A}}}\!\ket{0_{\rm{B}}} + \beta \ket{1_{\rm{A}}}\!\ket{1_{\rm{B}}}, \end{equation} shared between say Alice and Bob, Procrustean entanglement distillation can be achieved by applying a POVM to just Alice's particle, creating the maximally entangled Bell state:
\begin{equation} \ket{\Psi_{\rm{A,B}}} = \frac{1}{\sqrt{2}} \Big( \ket{0_{\rm{A}}}\!\ket{0_{\rm{B}}} \pm
\ket{1_{\rm{A}}}\!\ket{1_{\rm{B}}} \Big) . \end{equation}
with probability $P_{dist} = 2(1-\max(|\alpha|^2, |\beta|^2))$.
\subsection{ POVM Parameters for Distillation} \label{subsec:POVMParams}
The parameters for the massive particle POVM can be adjusted to carry out the Procrustean entanglement distillation protocol described above. We introduce two new parameters $\varphi$ and $\gamma$ which, for a known initial state of the form of Eq. \ref{eq:in}, are set such that $\alpha \equiv \cos{(\varphi)}$ and $\beta \equiv \exp({i\gamma})\sin{(\varphi)}$. The POVM parameters are then set according to Table \ref{tab:Param}. Alice inserts a detector at the $\ket{p_2}$ output and passes her particle through the POVM. The wavefunction output at $\ket{p_1}$, is then acted on by the operator $\hat{M}_{1}^A = \tan{(\varphi)}\ket{0}\!\bra{0} + \ket{1}\!\bra{1}$ if $ l\pi - \pi/4 \leq \varphi \leq l\pi + \pi / 4$ (for integer $l$), and $\hat{M}_{1}^A = \ket{0}\!\bra{0} + \cot{(\varphi)}\ket{1}\!\bra{1}$ otherwise. The two-particle state is output as $\ket{\Psi_1}=\frac{1}{\sqrt{2}} (\ket{1_{\rm{A}}}\!\ket{1_{\rm{B}}} + \ket{0_{\rm{A}}}\!\ket{0_{\rm{B}}})$ with probability $P_1=1-|\cos{(2\varphi)}|= 2(1-\max(|\alpha|^2, |\beta|^2))$. The choice of these parameters allows Alice to locally distill the entanglement she shares with Bob, by passing her particle ensemble through the device in Fig. \ref{fig:POVM}. The successful creation of a Bell state at the $p_1$-output can be heralded by the lack of detection of a particle at the $p_2$-output.
\begin{table}[h!] \caption{\label{tab:Param} POVM parameters for the implementation of entanglement distillation of the state in Eq. \ref{eq:in}. } \begin{ruledtabular} \begin{tabular}{cccc}
& $\phi_{1}$ & $0$ & \\
& $\phi_{2}$ & $0$ & \\
& $\phi_{3}$ & $-\gamma$ & \\
& $\phi_{4}$ & $-\gamma$ & \\
& $\theta_1$ & $\Re{[\arccos{(\tan{(\varphi)})}]}$ & \\
& $\theta_2$ & $\Re{[\arccos{(\cot{(\varphi)})}]}$ & \end{tabular} \end{ruledtabular} \end{table}
\subsection{ Simulating the Massive Wavepacket Evolution}
Using the single-qubit operations of Section \ref{sec:MassiveParticleOpticalAnalogues}, the implementation of our POVM for spin-$\frac{1}{2}$ particles in a SAW system can be simulated. By setting the POVM parameters in accordance with Sec.~\ref{subsec:POVMParams}, we implement Procrustean entanglement distillation on a massive wavepacket.
In this section, we demonstrate the implementation of a POVM that distills the entanglement by operating on a single particle from a joint initial state of the form $\ket{\Psi_{A,B}}=\cos{(60\degree)}\ket{\downarrow_A}\ket{\downarrow_B}+i \sin{(60\degree)}\ket{\uparrow_A}\ket{\uparrow_B}$. The spatial degree of freedom is labelled by $\ket{i}$, $\ket{s_{1,2}}$, $\ket{t_{1,2,3,4}}$ and $\ket{p_{1,2}}$, as in Fig. \ref{fig:POVM}.
\onecolumngrid
\begin{figure}
\caption{Simulation of a massive wavepacket travelling through a POVM device. The electrostatic potential of the proposed semiconductor device is represented by grey contour lines. Dashed lines indicate the position of the spin beam-splitters. After each beam-splitter, a magnetic field, represented by the shaded areas, is applied for spin rotations according to Fig. \ref{fig:POVM}. The arrow in the projected Bloch spheres indicate the wavefunction's spin orientation in their respective regions. In the simulation presented, the electron wave function is split with equal probability between the $\ket{p_1}$ and $\ket{p_2}$ outputs.}
\label{fig:POVMSIM}
\end{figure}
\twocolumngrid An overlying potential is necessary in order to achieve the confinement necessary for the double interferometer. It can be implemented with Schottky gates, as described above, or by etching the semiconductor material. The contour lines in Fig. \ref{fig:POVMSIM} show such an electrostatic potential. The sinusoidal SAW potential is not included in the figure.
In Fig. \ref{fig:POVMSIM} the two-dimensional electron wavefunction is traced out in the $x$-dimension, showing the probability distribution in the $y$-dimension as a function of time, $t$. Because of the strong confinement of the SAW potential, the particle distribution and movement in the $x$-direction is minimal. Hence, its $x$-position can be accurately estimated by $x = v \cdot t$, where $v$ is the speed of sound in the material.
The electron initially exists in the ground state of the SAW minimum, in the spatial state $\ket{i}$. The direction of motion is changed, and it is incident on the first polarizing beam-splitter. Here the electron is split into its spin components in a superposition of the spatial states $\ket{s_1}$ and $\ket{s_2}$. Two magnetic fields are applied to the respective components indicated by the shaded areas in Fig. \ref{fig:POVMSIM}. $\ket{s_1}$ and $\ket{s_2}$ are then incident on two beam-splitters forming a new superposition of the states $\ket{t_1}$, $\ket{t_2}$, $\ket{t_3}$ and $\ket{t_4}$ ($\ket{t_4}$ is not occupied for this specific POVM). Again, magnetic fields (shaded areas) are applied to implement local phase shifts and spin-rotations on the individual spatial components of the electron. Following these magnetic fields, the spatial components $\ket{t_2}$ and $\ket{t_3}$ are interfered on a beam-splitter, forming an output component $\ket{p_1}$. Similarly, $\ket{t_1}$ and $\ket{t_4}$ are interfered to form $\ket{p_2}$.
Hence, Fig. \ref{fig:POVMSIM} shows how an input wavefunction $\ket{\psi_A}\ket{i}$ is transformed into a spatial superposition given by $\hat{M}_{1}\ket{\psi_A}\ket{p_1}+\hat{M}_{2}\ket{\psi_A}\ket{p_2}$. In a 2D structure, $\ket{p_1}$ has to be trapped such that $\ket{t_1}$ and $\ket{t_4}$ can evolve around it. However, recent successes in creating rolled-up semiconductor nanotubes \cite{Prinz00, Schmidt01, Brick17} and layered quantum well structures \cite{Laikhtman09,Sivalertporn12,Cohen16} would allow output arms to continue to evolve through space, by enabling “periodic” boundary conditions, and finite 3D movement respectively.
By utilising the stability of a wavepacket carried by a SAW, and by optimising the device parameters, our simulations are able to demonstrate experimentally achievable high fidelity POVMs. Moreover, whilst this subsection has demonstrated a specific implementation, the extension to a general POVM with more than two Kraus operators is straightforward \cite{Ahnert06}. Nested polarizing Mach-Zehnder interferometers can be connected together by inserting the output states at $\ket{p_1}$ and $\ket{p_2}$ into subsequent interferometers in order to generate a POVM with any combination of Kraus operators.
\subsection{ Distillation of Realistic Distributions of Entangled Particle Pairs}
We have shown how a POVM can be implemented on massive spin-$\frac{1}{2}$ qubits. The Procrustean distillation protocol assumes that the initial pure state is known. Experimentally, it is likely that processes which produce entangled massive states produce ensembles of particle pairs with a distribution of entanglement strengths. Whilst there exist theoretical methods for the entanglement distillation and purification of mixed states~\cite{Bennett96-3, Murao98, Pan01}, these methods are iterative and require two-qubit operations. Owing to the experimental difficulties in the application of such operations, it is valuable to investigate the effect of the non-iterative single-qubit protocol on realistic particle pair ensembles.
By selecting a subset of the particles from the ensemble, one can optimise the POVM configuration to maximise the entropy of entanglement of the pairs in the final ensemble. The subset of particles used in the optimisation is consumed. However, the remaining ensemble can pass through the optimised POVM, in order to generate a reduced ensemble of higher pairwise entanglement.
\begin{figure}
\caption{(color online) (a) Difference between the initial and the final ensemble mean entropy of entanglement (contour from color-bar). The horizontal axis shows the POVM parameter, $\varphi$, and the vertical axis shows the initial mean value of the entropy of entanglement. (b) Probability density as a function of $|\alpha|^2$, of two example input distributions, (1) and (2), and their corresponding non-normalized $\ket{p_1}$ output distributions, (1*) and (2*).}
\label{fig:DiffProbs}
\end{figure}
In Fig. \ref{fig:DiffProbs} we show the difference in the von Neumann entanglement entropy distribution for particle pair ensembles before and after the distillation protocol. Fig. \ref{fig:DiffProbs}(a) shows the change in the mean entanglement entropy, $\Delta \overline{\mathcal{S}}$, as a function of initial mean entropy, $\overline{\mathcal{S}}_{\rm{in}}$, and POVM angle, $\varphi$, as previously related to $\theta_1$ and $\theta_2$. We have assumed that the value of $|\alpha|^2$ (the probability of state $\ket{0}$) in the initial particle pairs has a Gaussian profile of width $\sigma=0.01$. The dotted lines show the loci of the optimal POVM angles for ensembles of identical pairs. A lower initial mean entanglement allows for the possibility of a higher increase of mean entanglement. The simulations were carried out using Monte Carlo theory with an ensemble size of $10^5$ particles in each distribution. For well-behaved distributions, the proposed setup will efficiently produce a final ensemble of increased average entanglement. This is true even for wide distributions such as curve (1) in Fig. \ref{fig:DiffProbs}(b).
\section{ Concluding Remarks }
We have developed a methodology for the implementation of massive spin-$\frac{1}{2}$ qubit POVMs. The POVM builds on the framework of the AP double interferometer POVM \cite{Ahnert05}. We have proposed a toolkit for translating the optical components from the AP POVM into processes which are suitable for electrons in surface acoustic wave systems. The use of ground state wavefunctions of SAW minima allows us to virtually eliminate the dispersion of the particle wavepackets, providing the means to replicate the optical POVM with a massive particle analogue. Owing to the difficulty in controlling photon-photon interactions, linear-optics-like processing of massive (more easily interacting) particles will be valuable for quantum computational aspirations or quantum cryptography with hybrid systems.
We demonstrated the effectiveness of the proposed scheme by simulating the evolution of a spin-$\frac{1}{2}$ POVM that performs Procrustean entanglement distillation on a pair of entangled massive qubits. Using a Hamiltonian tailored by GPU-boosted parameter sweeps, our simulation showed a POVM fidelity of $>99.5 \%$. However, this is not an upper bound and additional parameter optimisation can lead to even higher fidelities. Furthermore, our Monte-Carlo based numerical investigation shows how the protocol can increase the average entropy of entanglement of particle pair ensembles with distributions of initial entanglement entropies.
\end{document} |
\begin{document}
\author{Timothy Carson} \title[Asymptotically cylindrical singularities]{Ricci flow from some spaces with asymptotically cylindrical singularities} \email{carstimon@gmail.com} \begin{abstract}
We prove the existence of Ricci flow starting from a class of metrics with unbounded curvature, which are doubly-warped products over an interval with a spherical factor pinched off at an end. These provide a forward evolution from some known and conjectured finite-time local singularities of Ricci flow, generalizing previous examples. The class also includes metrics with non-compact singular ends which become instantaneously compact. Furthermore, we prove local stability of the forward evolution, which allows us to glue it to other manifolds and create a forward evolution from spaces which are not globally warped products. \end{abstract} \maketitle
\section{Introduction} For any complete Riemannnian manifold $(M, g)$ with bounded curvature, there is a smooth solution to the Ricci flow, \begin{align}
\partial_t g(t) = - 2 \Rc[g(t)] \end{align} with $g(0) = g$ \cite{Shi}. The solution exists up to some time $T \in (0, \infty]$. In this paper, we prove the existence of a forward evolution of Ricci flow from a certain class of Riemannian manifolds with unbounded curvature. The initial metrics we consider have singular neighborhoods which are asymptotically cylindrical warped products of spheres.
Our primary motivation for considering this problem is the continuation of Ricci flow after singularities. The forward evolution of a smooth manifold often encounters local singularities in finite time. In some cases, the local singularities can be understood well enough so that Ricci flow with surgery can be implemented, e.g. \cite{h_posiso}, \cite{Perelman2}, \cite{brendlesurg}. In the three dimensional case the body of knowledge is by now quite powerful \cite{singular}, \cite{uniquenessBK}.
All of these surgery examples work by proving that every local singularity encountered has a part close to a shrinking cylinder $\mathbb{R} \times S^{n-1}$. The ideal situation for Ricci flow encountering such a singularity is when the metric is a warped product on $I \times S^{n-1}$ for some interval $I$: \begin{align}
g = a(x)^2dx^2 + \phi(x)^2 g_{S^{n-1}}. \end{align} (Here $g_{S^{n-1}}$ is the standard metric on the $S^{n-1}$ factor.) By choosing $\phi$ correctly, the forward evolution from $g$ encounters a local singularity, named a neckpinch. This was conjectured in \cite{h_formation} and first shown by Simon \cite{simon_pinch}. In \cite{AK}, \cite{AKPrecise} Angenent and Knopf expanded on these singularities and gave a precise asymptotic description. Their description in particular gives a description of the metric at the final time, when the singularity has occurred. In \cite{ACK}, Angenent, Caputo, and Knopf proved the existence of a forward evolution of Ricci flow from these final-time singular metrics.
The first main theorem in the present work provides the forward evolution from a family of singular metrics which includes those explored in \cite{ACK}. It also includes the forward evolution, in the ideal (doubly-warped product) case, from (conjectured) singularities which are modeled on $\mathbb{R}^k \times S^{n-k}$. Our description of the forward evolution is very precise, and we hope to provide a testbed for a more general theory that can deal with these singularities. We hope that our generalizations clarify the role played by various pieces.
The general question of which singular spaces have a forward Ricci flow has received attention from many authors. Particular success has been had with curvature bounds from below \cite{Simon_1}, \cite{Simon_2}, \cite{CRW}, \cite{lowerricci_integralcurvature}, \cite{almostnonneg}. Another work addressing spaces with specific singularity models is \cite{conicalSing}. For some results with low regularity on the initial metric, see \cite{Simon_0} and \cite{kochlamm}. Furthermore, the Ricci flow of warped products lends itself to comparison to reaction-diffusion equations in Euclidean space, where there are quite general existence and uniqueness theories \cite{continuation}.
\subsection{Model Pinches} We now give a definition of the singular metrics, which we call model pinches. Let $q \geq 2$, and let $(S^q, g_{S^q})$ be the round sphere of sectional curvature 1, which satisfies $2\Rc[g_{S^q}] = \mu g_{S^q}$ for $\mu = 2(q-1)$. Also let $(F, g_F)$ be any Einstein manifold with $2 \Rc[g_F] = \mu_F g_F$. The metrics will be metrics on $I \times S^q \times F$ of the form \begin{align}
g_{mp} = dx^2 + \phi(x)^2 g_{S^q} + \psi(x)^2 g_F. \end{align} The main case of interest is $F = S^p$ but $F$ may be zero dimensional (landing us in the singly warped product case) or have negative Ricci curvature. The function $\phi$ will be increasing, so we can use $u = \phi^2$ as a coordinate and write \begin{align}\label{mp_form}
g_{mp} = \frac{du^2}{u V_0(u)} + u g_{S^q} + W_0(u) g_F. \end{align}
Here $V_0(u) = u^{-1}|du|^2_{g_{mp}} = 4 |d\phi|^2_{g_{mp}}$.
For the rest of the paper we fix some $\eta \in (0, \oh)$. For any metric $g$, function $f:M \to \mathbb{R}$, and scale function $\rho: M \to \mathbb{R}_+$ we use the notation $
\twoeta{f}{\rho}{g} : M \to \mathbb{R}_{\geq 0} $ to mean the following. Take any point $p \in M$, scale the metric $g$ to $\hat g = \frac{g}{\rho(p)^2}$, and then take the $C^{2, \eta}$ norm in the ball of radius $1$ around $p$ with respect to $\hat g$.
\begin{definition}\label{definition:model_pinch}
A metric on $M = (0, \infty) \times S^q \times F$ of the form \eqref{mp_form} is a \emph{model pinch} if
\begin{enumerate}[label=(MP\arabic{*}), ref=(MP\arabic{*})]
\item \label{modelpinch_vsmall}As $u \searrow 0$, $V_0(u) \searrow 0$.
\item \label{w_big} If $\mu_F > 0$, there is a $c > 0$ such that $\frac{W_0(u)}{u} \geq (1 + c) \frac{\mu_F}{\mu} $.
\item \label{modelpinch_reg}
For some $C>0$
\begin{align}
\frac{\twoeta{V_0}{u/2}{(du)^2}}{V_0}
+
\frac{\twoeta{W_0}{u/2}{(du)^2}}{W_0}
\leq C
\end{align}
\item For any $u_1 > 0$, on the set $\{u > u_1\}$ the curvature of $g_{mp}$ is strictly bounded, $V_0$ and $W_0$ are $C^{\infty}$ and strictly positive, and $V_0$ is bounded.
\end{enumerate} \end{definition} One way to interpret this definition is that as $u \searrow 0$ the metric is asymptotically some sort of cylinder. At the distance scale given by $\sqrt{u}$, if $W_0(u) \gg u$ the metric is close to the product $(\mathbb{R}, g_{\mathbb{R}})\times (S^q, g_{S^q})\times(\mathbb{R}^p, g_{\mathbb{R}^p})$, and if $W_0(u) \sim au$ it is close to $(\mathbb{R}, g_{\mathbb{R}}) \times (S^q, g_{S^q}) \times (F, ag_F)$. For some precision, see Corollaries \ref{prish_asymptotic_cylinder} and \ref{prish_curvature_control}, which are stated for the forward evolution but in particular hold for the initial metric.
The H\"older condition implies the first and second derivatives of $V_0$ satisfy $|u \partial_u V_0| + |u^2\partial_u^2V_0| \leq CV_0$ (for a different $C$) and similarly for $W_0$. This allows for change by a factor of $(1 + O(\epsilon C))$ in the region where $u$ is $(1 + O(\epsilon))u$.
Our first theorem is the following short-time existence result in the class of warped products. We identify $(0, \infty) \times S^q$ with $\mathbb{R}^{1 + q} \setminus \{0\}$ and write $M \mathrel{\mathop:}= \left( \mathbb{R}^{1+q}\setminus\{0\} \right) \times F \subset \mathbb{R}^{1+q} \times F \mathrel{\mathop=}: \bar M$.
\begin{theorem}\label{theorem:model_pinch_flow}
Let $(M, g_{mp})$ be a model pinch. For some $T_2 \in (0, \infty]$ there is a Ricci flow $(\bar M, g_{wp}(t))$ for $t \in (0, T_2)$. As $t \searrow 0$, $g_{wp}(t) \to g_{mp}$ in $C^{\infty}_{loc}(M)$. There are choices of the parameters of Definitions \ref{productish_barricaded} and \ref{tip_barricaded} such that $g_{wp}$ is controlled in the productish region and in the tip region. \end{theorem} The last sentence of the theorem gives a good description of the forward evolution near the origin, we will give an overview in Section \ref{section:shape_description}. Here we just give a rapid tour of some properties that we think are important. In the forward evolution a small Bryant soliton appears at the origin, and the radius of the $F$ factor is strictly positive for positive time even if it was not for the initial metric $g_{mp}$. The distance-squared scale of the Bryant soliton is on the order of $t V_0(t)$, and the largest Ricci curvature forward in time, which occurs at the origin, is on the order of $1/(t V_0(t))$. At the origin, the distance-squared scale of the $F$ factor is $W_0(t) - \mu_F t$. If $\mu_F \neq 0$ then this is at least order $t$: property \ref{w_big} says $W_0(t) \gtrsim t$ if $\mu_F > 0$, and if $\mu_F < 0$ the term $- \mu_F t$ helps. This is not the case if $\mu_F = 0$ and $W_0(u) = o(u)$, so the largest Riemannian curvature could be on $F$ (and see cancellation in the Ricci curvature). Finally, we can use the control to provide a more precise rate of convergence of $g_{wp}(t)$ to $g_{mp}$; see Corollary \ref{improved_convergence}.
The next theorem removes the global warped product part of the model pinch assumption. For this, we need some additional assumptions on the curvature of the factor $F$, which is inevitable since we allow perturbations of the initial metric in any direction. In particular, we rule out the case $W_0(t) - \mu_F t \lesssim t\nu(t)$. For the metric $g_F$, let $
\Lambda_F = \sup_{p \in F}\max_{h \in Sym_2(T_pF), |h|=1}(\Rm)_{abcd}h^{ac}h^{bd} $. For example, if $F$ has dimension $p$ and constant sectional curvature $k$ then $\Lambda_F = k(p-1)$. In particular, $2\Lambda_{S^q} = \mu = 2(q-1)$. \begin{definition}\label{reasonable_def}
A model pinch is \emph{$\Rm$-permissible} if the following is satisfied.
\begin{enumerate}[label=(RP\arabic{*}), ref=(RP\arabic{*})]
\item \label{lambda_assump}
In the case $\Lambda_F > 0$, we additionally require
$\frac{W_0(u)}{u} \geq \frac{\Lambda_F}{\Lambda_{S^q}}$
\item \label{w_big2}
In the case $\mu_F = 0$ and $\Lambda_F = 0$ (i.e. $(F, g_F)$ is flat) we additionally require $\frac{W_0(u)}{u V_0(u)} \to \infty$ as $u \searrow 0$.
\end{enumerate} \end{definition} \begin{theorem}\label{theorem:unsymmetrical_flow}
Let $g_{mp}$ be an $\Rm$-permissible model pinch. There is an $\epsilon_0$ depending on $g_{mp}$ with the following property.
Let $(N^n, g)$ be a (possibly non-complete) Riemannian manifold. Let $U \subset N$ be open, and assume that $(N \setminus U, g)$ is a complete manifold with boundary, satisfying, for some $r_0 > 0$ and all $p \in N\setminus U$, $|\Rm|(p) \leq r_0^{-2}$ and $\Vol(B(p, r_0)) \geq (1-\epsilon_0) \omega_n$.
Suppose that $u_1>0$ and $\Phi: U \to (0, u_1) \times S^q \times F$ is a diffeomorphism such that in $U$,
\begin{align}
\twoeta{g - \Phi^* g_{mp}}{r_0 |\Rm_{\Phi^*g_{mp}}|}{\Phi^*g_{mp}}
\leq
\epsilon_0 V_0 \circ \Phi.
\end{align}
Let $\bar N \supset N$ be the differential manifold obtained by replacing $U\sim (L, L') \times S^q \times F$ with $\bar U \sim D^{1+q} \times F$. For some $T_* > 0$, there is a Ricci flow $g(t)$, for $t \in [0, T_*]$ on $\bar N$ such that $g(t) \to g$ in $C^{\infty}_{loc}(N)$ as $t \searrow 0$. \end{theorem}
Immediate extensions of our theorems allow for multiple singular neighborhoods of $N$ each close to some model pinch, or for multiple extra warped factors $g_{F_i}$ each satisfying the requirements in the definition of model pinch.
\subsection{Overview of the proofs} Both theorems are proven by constructing smooth mollified initial metrics, which agree with the singular initial metrics outside of a small set, and controlling the forward evolution of the smooth mollified metrics. By sending the size of the mollification to zero, we construct a forward evolution from the singular initial metric.
To prove Theorem \ref{theorem:model_pinch_flow}, we control the relevant functions for the mollified initial metrics in terms of $u$. The advantage of this is that the control is diffeomorphism-invariant-- for example, the value of $w$ or $v = u^{-1}|\nabla u|^2$ at the point where $u = u_1 \in \mathbb{R}_{> 0}$ is a diffeomorphism-invariant property. A usual difficulty in controlling solutions to Ricci flow is that the linearization is only weakly parabolic because of the diffeomorphism invariance of Ricci flow, and this gets around that issue. The most common response is to use Ricci-DeTurck flow, but we were not able to find a sufficiently good background metric to use in our case (and we tried some exotic possibilities). Another option in our case would be to use an arclength coordinate, but that introduces an annoying nonlocal term.
The forward evolution is split into two regions-- the tip region, where a Bryant soliton forms, and the productish region, which includes the initial value and is where the metric continues to look locally like a product metric on $\mathbb{R} \times S^q \times F$. In Lemmas \ref{main_prish_estimates} and \ref{main_tip_estimates}, we obtain local control in the productish and tip regions, assuming a priori boundary control. In Section \ref{section:full_flow} we put this control together. Section \ref{section:buckling} shows that the boundary control needed at the right of the tip region is ensured by the local estimates in the productish region, and the boundary control needed at the left of the productish region is ensured by the local estimates in the tip region. We now have to prove the boundary conditions at the right boundary of the productish region, where the metrics are uniformly smooth. This is accomplished in Section \ref{section:ccc}.
\begin{figure}
\caption{
Map of the tip, productish, and uniformly smooth regions. Here $\nu(t) = V_0(\mu t)$.
}
\label{region_map}
\end{figure}
The local estimates in the productish region use generic estimates for the solution to some reaction-diffusion equations in regions where they are nearly constant, which is dealt with in Appendix \ref{nearly_cnst_pde_sect}. The local estimates in the tip region are more specific.
The proof of Theorem \ref{theorem:unsymmetrical_flow} uses Ricci-DeTurck flow around the already-constructed warped product evolution to control an arbitrary metric. Theorem \ref{local_stability} is the main point in the proof, this gives us control of the Ricci-DeTurck flow in a neighborhood $U$ of the form $\{u < u_*\}$, assuming a priori boundary control. Once we have Theorem \ref{local_stability}, we wrap up by controlling the evolution in the boundary region (where everything is uniformly smooth) in Section \ref{asymmetric}.
Again, the local control in Theorem \ref{local_stability} is split into two parts: control in the productish region, and control in the tip region. As in the warped-product case, in the productish region we control the evolution using the results of Appendix \ref{nearly_cnst_pde_sect}. On the other hand, in the tip region (where the solution is close to a small perturbation of the Bryant soliton), we use a contradiction-compactness argument to move the situation to the Bryant soliton. Then, we use a stability result for the Bryant soliton, Theorem \ref{bry_stabil}. Theorem \ref{bry_stabil} might be compared to results from Section 7 of \cite{uniquenessBK}; see the remark after the statement of the theorem.
\subsection{Infinitely long pinched ends} In an attempt to simplify the initial exposition we have hidden that the left end of a model pinch, $\{u \leq u_1\}$, may have infinite length. An arc length coordinate for the interval factor for a metric of the form \eqref{mp_form} is given by $ds = \frac{1}{\sqrt{u V_0(u)}} du$, so the length is hidden in the integrability of $\frac{1}{\sqrt{uV_0(u)}}$ near $0$. In the case when the left end of the initial model pinch has infinite length, the left end of the evolution on $\bar M$ is compact for positive time.
In two dimensions, Topping \cite{topping_reversecusp} constructed similar examples of noncompact surfaces which immediately become compact. These examples actually have initial metrics with bounded curvature, so it is especially interesting when compared with Shi's existence result, which guarantees that the initial metric has a unique forward complete Ricci flow on the same topology. This means that in two dimensions there is an alternative, perhaps more natural, forward evolution besides the instantaneously compact one. In more than two dimensions, the analysis is different because the $S^q$ factor in the singly warped product has positive curvature and the initial metric must have unbounded curvature. We do not expect a natural forward evolution on the same topology in this case.
\subsection{Some related short-time existence results} Recent work that is close in spirit to ours is \cite{conicalExpanders} and \cite{conicalSing}. In \cite{conicalExpanders}, Deruelle showed that for any cone with positive curvature, i.e. a metric $ds^2 + s^2 g_{X}$ where $\Rm[g_{X}] \geq 1$, there is an expanding Ricci soliton which limits, backwards in time, to the cone. This can be considered as Ricci flow starting from the singular conical space. In \cite{conicalSing}, Gianniotis and Schulze allow us to start Ricci flow from any manifold which has local singularities modeled on these cones, by using local stabilitiy similarly to our Theorem \ref{theorem:unsymmetrical_flow}. Such cones that are especially relevant to us are the singly-warped products $ds^2 + a s^2 g_{S^q}$, for $a \in (0,1)$; these are singly warped products over intervals which are not covered by our theorem.
Alexakis, Chen, and Fournodavlos \cite{singlargerderiv} show the existence of a steady Ricci soliton of the form $ds^2 + \phi(s)^2 g_{S^q}$ with $\phi(s) \sim s^{1/\sqrt{q}}$. They also examine forward evolutions of metrics close to their steady Ricci soliton.
Bamler, Cabezas-Rivas, and Wilking \cite{almostnonneg} examine the Ricci flow of manifolds with a variety of assumptions that curvature is bounded from below. In particular, they deal with complete, bounded curvature manifolds $(M, g)$ satisfying \begin{align}\label{almostnonneg_conditions}
\Rm \geq -1, \quad \Vol_g (B_g(p, 1)) \geq v_0 \text{ for all } p \in M. \end{align} They show that there is a forward evolution for a time which \emph{only} depends on $v_0$ and the dimension. An application is creating forward evolutions from singular spaces which can be approximated by manifolds with curvature bounded from below. This gives an alternative approach to \emph{some} of the initial spaces considered by Gianniotis and Schulze in \cite{conicalSing}.
We wish to remark that we cannot apply the results in \cite{almostnonneg} in our case, but we need to use two different reasons. First note that in the examples with an infinitely long pinched end, the assumption on the volume of balls in \eqref{almostnonneg_conditions} cannot be satisfied by approximating metrics, since the left end has balls of radius one with arbitrarily small volume. We claim that in the compact case the curvature condition in \eqref{almostnonneg_conditions} is not satisfied. Consider just the singly-warped metrics of form \eqref{mp_form}, so $g = \frac{du^2}{u V_0(u)} + u g_{S^q}$. The curvature of such a metric is \begin{align}
\Rm =
L \left( (u g_{S^q}) \mathbin{\bigcirc\mspace{-15mu}\wedge\mspace{3mu}} (u g_{S^q}) \right)
+ K \left( (u g_{S^q}) \mathbin{\bigcirc\mspace{-15mu}\wedge\mspace{3mu}} \frac{du^2}{u V_0(u)} \right) \end{align} where $L = u^{-1} ( 1 - \on4 V_0)$ and $K = - \on2 \partial_u V_0$. The distance between $\{u = 0\}$ and $\{u = u_2\}$ is $
\int_{0}^{u_2}\frac{1}{\sqrt{u V_0(u)}} du
= \int_{0}^{u_2} \frac{1}{u} \sqrt{ \frac{u}{V_0(u)}} du. $ If $K$ is bounded from below, $\partial_u V_0 \leq C$ and so $V_0 \leq C u$ and this integral diverges. So in the compact case $K$ goes to $-\infty$ and \eqref{almostnonneg_conditions} is not satisfied. The other possible conditions of Theorem 2 from \cite{almostnonneg} are also not satisfied: $\Rm$ as an operator on $\bigwedge^2TM$ has the negative eigenvalue $K$ with multiplicity $q \geq 2$ so the curvature is not 2-non-negative, and we can check that the curvature operator is never weakly PIC1, although in the singly-warped case it has positive isotropic curvature.
Note that the model pinches do (in the case when $g_F$ has positive curvature or $W_0(u) \gg u$) satisfy an almost-nonnegativity condition relevant to singularity analysis of Ricci flow, namely $\Rm \geq - f(|\Rm|)|\Rm|$ for a function $f$ satisfying $f(x) \to 0$ as $x \to \infty$. This comes up, for example, in 12.1 of \cite{Perelman}. In this case $f$ is a multiple of $V_0$ and we can use the assumption \ref{modelpinch_reg} to bound $K$, and the assumptions \ref{w_big} and \ref{modelpinch_reg} to bound curvatures involving the other factor.
\subsection{Model Pinches that arise as final-time limits} Here we list some examples of smooth Ricci flows which have a model pinch has final-time limits. \subsubsection{Singly-warped product singularities}
In \cite{AKPrecise}, Angenent and Knopf considered neckpinches occuring on singly warped products over an interval. They proved that the warping function of the final-time limit of a neckpinch satisfies the asymptotics $\phi = \sqrt{u} \sim \frac{s}{\sqrt{|\log s|}}$, where $s$ is the arclength from the singular end. This implies $V_0(u) \sim \frac{1}{\log u}$. Another singularity that may arise in the category of warped products of spheres over an interval is the degenerate neckpinch. In this case, Angenent, Isenberg, and Knopf showed in \cite{AIK} that the final-time limit has the asymptotics $\phi \sim s^{\beta_k}$ where $\beta_k = \frac{2}{2k+1}$, $k \in \mathbb{N} \setminus \{0\}$. Forward evolutions from these specific cases were created in \cite{ACK} and \cite{recoverdegen}, respectively.
\subsubsection{Generalized cylinder singularities}\label{generalized_cylinder} \begin{figure}\label{singularities_phipsigraphs}
\end{figure} For another example of a singularity, consider the doubly-warped product depicted in the top row of Figure \ref{singularities_phipsigraphs}. A more stylized picture of a neighborhood of the singularity is Figure \ref{pancake_pinch}. The metric is a doubly warped product over an interval, with $(F, g_F) = (S^p, g_{S^p})$, and the singularity occurs at the left endpoint of the interval. Before the singular time, the metric satisfies the following boundary conditions at the left endpoint: \begin{align}
\phi > 0, \quad \partial_s \phi = 0, \quad \psi = 0, \quad \partial_s \psi = 1. \end{align} Here $s$ is the distance from the left endpoint. A neighborhood of the left endpoint has topology $S^q \times D^{1+p}$ before the singular time. For the initial metric, the size of the $S^q$ factor has a deep minimum at the center of the $D^{1+p}$.
As time goes on, the $S^q$ factor shrinks drastically, and the metric encounters a singularity which can be rescaled to a generalized cylinder $S^q \times \mathbb{R}^{1+p}$. Without rescaling, at the singular time the metric takes on the topology of the cone over $S^q \times S^p$ (but is not asymptotically a metric cone). This singularity has not been rigorously constructed, but formal calculations suggest that the singular pinched metric should have asymptotics \begin{align}\label{pinched_pancake_asymptotics}
\phi \sim \frac{s}{\sqrt{|\log s|}}, \quad \psi \sim s. \end{align} This is an unsurprising guess. The factor corresponding to the $S^q$ behaves similarly to a standard neckpinch. The $1+p$ dimensional part of the metric, $dx^2 + \psi^2 g_{S^p}$, is close to being a flat $D^{1+p}$, which corresponds to $\psi = x$ exactly. The flat metric is stable enough that the perturbation from the pinching factor does not affect it too much.
In the forward evolution of metrics with asymptotics \eqref{pinched_pancake_asymptotics}, which we do investigate here, the size of the $S^p$ factor expands and the neighborhood takes on the topology $D^{1+q} \times S^p$. \begin{figure}\label{pancake_pinch}
\end{figure}
\subsubsection{Families of neckpinches}\label{submanifold_neckpinches} Here is another example which is a singularity modeled on $\mathbb{R}^{1+p} \times S^q$, but which is qualitatively different from the previous. We can also consider a doubly-warped product over an interval where $\phi$ has a neck somewhere in the interior of the interval. Then we can force a singularity to occur in the interior of the interval modeled on $\mathbb{R}^{1 + p} \times S^q$. Here there is an $S^p$ worth of one-dimensional neckpinches forming. A trivial example of this is when we just cross a standard neckpinch with $S^p$. While the previous example was also modeled on $\mathbb{R}^{1+p} \times S^q$, this one is qualitatively different: for example, the topological change through the singularity is different.
This type of singularity should be stable in the class of doubly warped products; perturbations leave $\phi$ with a local minimum. However, in contrast to the previous example, it should not be stable in the full class of Riemannian metrics. It should not even be stable in the class of singly warped products $g_B + \phi(b)^2 g_{S^q}$ where $B = \mathbb{R} \times S^p$ and $g_B$ is now an arbitrary metric on $B$. (The original metric has $g_B = dx^2 + \psi(x)^2 g_{S^p}$.) Indeed, if we allow $\phi$ to also depend on the $S^p$ factor and perturb it so it has a strict local minimum at some point on that factor, we should approach a singularity at a single point on the $S^p$ factor. (Intuition for this may come from \cite{blowupsinglept_general}, which shows in particular that we can perturb the constant solution of simple reaction-diffusion equations on $\mathbb{R}^n$ to get a single point blowup.) \subsubsection{Scarred neckpinches}\label{scarring} Here is an example which leads to a metric which is not quite a model pinch. First consider a standard singly warped neckpinch with spheres of dimension $S^{q}$: the initial metric is of the form $dx^2 + u(x)g_{S^{q}}$ and the metric at the singular time is a model pinch. This has a forward evolution, which recovers with a smooth disc of dimension $1 + q$ at the tip. So, we have a Ricci flow of a singly warped product, at least on $((-1, 0) \cup (0, 1)) \times S^{q}$, for times $t \in [T_1, T_2]$, $T_1 < 0 < T_2$.
Now, the Ricci flow of warped products with Einstein fibers does not care about the Riemannian curvature tensor of the fiber metric, it only cares about the Ricci curvature. In other words: suppose we have a Ricci flow on $B \times F_1$ of the form $g_B(t) + u(t)g_{F_1}$ (where for each $t$, $u(t):B \to \mathbb{R}_+$) and $\Rc_{g_{F_1}} = \mu g_{F_1}$. Suppose $(F_2, g_{F_2})$ is another Einstein manifold with $\Rc_{g_{F_2}} = \mu g_{F_2}$. Then $g_B(t) + u(t)g_{F_2}$ is also a Ricci flow.
Therefore, in the Ricci flow through a standard neckpinch, we can swap out $g_{S^q}$ with any Einstein manifold $(F_{2}^q, g_{F_2})$ of our choosing, provided it has the same scalar curvature as $g_{S^q}$. The resulting object satisfies Ricci flow wherever $u > 0$, but is not a manifold for $t > 0$. Around the new points at the tip, the result has the topology of the cone over $F_2$. The forward evolution has a scar as a result of its surgery.
A special case of this situation is when $g_{F_2}$ is the standard metric on $F_2 = S^q / \Gamma$ for some group $\Gamma$. This case is important because it cannot be ruled out by a pointwise curvature condition, and so it is relevant to trying to implement Ricci flow with surgery under curvature assumptions. The resulting object after the singularity is an orbifold. This case was dealt with in four dimensions in \cite{completecompactposiso}, and they removed a topological assumption of Hamilton's work in \cite{h_posiso} by considering Ricci flow with orbifold singularities.
Of relevance to us is the case $q = 2k$ and $F_2 = S^k \times S^k$. In this case, the metric at the singular time has the form \footnote{
We are always lazy with writing the lifts of metrics and tensors etc. Here we use the notation $\oplus$ to emphasize that the two terms $g_{S^q}$ which appear are different, one is the lift of the $g_{S^q}$ on the first factor, and the other is the list of the $g_{S^q}$ on the second factor. } \begin{align}
g = dx^2 + u(x) g_{S^k} \oplus u(x) g_{S^k}. \end{align} It satisfies all of the conditions of a model pinch except for \ref{w_big}, since $u = w$ and $\mu_F = \mu$. Since $S^k \times S^k$ is unstable under Ricci flow (we can perturb the size of one of the factors) we thought perhaps there could be two alternative forward evolutions where either of the factors becomes positive after the singular time. We now believe that this is not possible, see Section \ref{w_big_sharpness}.
\subsection{Shape of the forward evolution}\label{section:shape_description} In this section we describe various properties of the forward evolution $g(t)$ of a model pinch. As time goes on, the metric continues to be a doubly warped product: \begin{align}
g(t) = a(x,t) dx^2 + u(x,t) g_{S^q} + w(x,t) g_{F}. \end{align}
Furthermore, we prove that $u$ continues to be increasing in $x$. Therefore we may continue to consider $v = u^{-1}|\nabla u|^2$ and $w$ as functions of $u$, and write the metric as \begin{align}
g(t) = \frac{du^2}{uv} + u g_{S^q} + w g_F. \end{align} For the initial metric, the derivatives of $u$ and $w$ are relatively small. Therefore after investigating the curvature of warped products we see \begin{align}\label{rc_prish}
-2\Rc(X,Y) \approx
- \mu g_{S^q} - \mu_F g_F. \end{align}
Forward in time, this approximation continues to hold for a short time, while the derivatives of $u$ and $w$ continue to be small. We call the region where $v = u^{-1}|\nabla u|^2$ continues to be small the ``productish'' region. Let $\nu(t) = V_0(\mu t)$. The productish region is the set \begin{align}
\Omega_{prish} = \left\{(x,t): \frac{u(x,t)}{t \nu(t)} \geq \sigma_* \text{ and } u < u_* \right\} \end{align} for some sufficiently large $\sigma_*$ and small $u_*$. In this region, we have $v \leq \epsilon$; by choosing $\sigma_*$ and $u_*$ we can have $\epsilon$ as small as we wish.
In the productish region, we get the approximations \begin{align}
v &\approx V_{prish}
\mathrel{\mathop:}= \frac{u + \mu t}{u} V_0\left(u + \mu t \right)
\label{v_approx_prish}
\\
w &\approx W_{prish} \mathrel{\mathop:}= W_0\left( u + \mu t \right) - \mu_F t
\label{w_approx_prish}
. \end{align} Note that these approximations would be exact if the approximation \eqref{rc_prish} were exact so $u(x,t) = u(x,0) - \mu t$, $w(x,t) = w(x,0)-\mu_F t$, and \begin{align}
v(x,t)
= \frac{|\nabla u(x,t)|^2}{u(x,t)}
= \frac{|\nabla u(x,0)|^2}{u(x,0)} \frac{u(x,0)}{u(x,t)}
= V_0(u+\mu t) \frac{u + \mu t}{u}. \end{align}
In Section \ref{corollaries_productish} we give some corollaries of our control in the productish region.
Now we come to a crucial juncture in the calculation of our approximate solution. The approximations \eqref{v_approx_prish} and \eqref{w_approx_prish} work for $u(x,t) \gtrsim t \nu(t)$- in particular they work for $u \ll t$. To understand the approximations for small $u$, put $\nu(t) = V_0(\mu t)$, $\omega(t) = W_0(\mu t)$ and write \begin{align}
V_{prish} &= (1 + \mu t/u)\nu(t)
\frac{V_0\left( \mu t( 1 + \mu^{-1}u/t)\right)}{V_0(\mu t)} \\
W_{prish} + \mu_F t &= \omega(t)
\frac{W_0\left( \mu t( 1 + \mu^{-1}u/t)\right)}{W_0(\mu t)} \end{align} Using our assumptions on $V_0$ and $W_0$, particularly \ref{modelpinch_reg}, we can estimate the quotients for $u/t \ll 1$. Then our approximations say \begin{align}
v &\approx \mu \sigma^{-1}(1 + \mu^{-1}(1 + \dd1{\nu}(t)) \nu(t) \sigma)\label{expose_v_approx}\\
w + \mu_F t
&\approx \omega(t)(1 + \mu^{-1}\dd1{\omega}(t)\nu(t) \sigma) \label{expose_w_approx} \end{align} where $\sigma = u/(t\nu(t))$, $\dd1{\nu} = t\nu'(t)/\nu$, $\dd1{\omega} = t\omega'(t)/\omega$.
If the left end of the manifold is to be smooth and compact, $v$ cannot be small up to $u=0$. In fact, $v \to 4$ is a necessary condition to have a smooth closed disc at the left endpoint. At the left end, on the factor $I \times S^q$, we glue in a steady Bryant soliton of size $\approx t\nu(t) \mathrel{\mathop=}: \alpha(t)$. This is a metric on $\mathbb{R}^{1+q}$ that moves only by diffeomorphisms under Ricci flow. We call the region where $\sigma$ stays small, where we see the Bryant soliton, the ``tip region''. The asymptotics of the Bryant soliton as $u \to \infty$ match with the term $\mu \sigma^{-1}$ in \eqref{expose_v_approx}. A steady soliton is in accordance with the fact that we expect scaling at a rate faster than $t$: as a general principle, if we scaled at rate $t$ we would expect an expanding soliton, whereas if we scale at a faster rate we find a steady soliton.
For the factor $F$, the warping function is approximately constant. Therefore we expect to be able to attach a large $F$ factor to our Bryant soliton. The approximate size of the unrescaled $F$ factor is $\omega(t)-\mu_F t = W(\mu t) - \mu_F t$. Taking for simplicity the case $\mu_F \neq 0$, our assumptions imply that $\omega-\mu_F t \gtrsim t \gg t \nu(t)$. Therefore when we scale by $t \nu(t)$ the size of this factor goes to infinity, and around any point it approaches a Euclidean factor.
Thus, the zeroth order approximation of the metric near the tip (in other words, the expected limit of the rescaled metric as $t \searrow 0$) is $(\text{Bryant Soliton}) \times (\text{Euclidean metric})$. We can get this approximation in a region of the form \begin{align}
\Omega_{tip} = \left\{(x,t): \sigma < \nu^{-1/2} \right\}. \end{align} As $t \searrow 0$ (so $\nu \searrow 0$) this region covers the whole Bryant soliton.
We also need to find the first order approximation near the tip. The perturbation has size $\approx \nu$. The equation we get in space is \begin{align}
(\text{Linearization of Ricci Flow})[g_1] = g_0, \end{align} where $g_0$ and $g_1$ represent the zeroth and first order approximations. This gives us an equation to solve for $g_1$. On the $F$ factor, the solution coincides with the soliton potential, times $g_F$. Our first order approximation matches with all of the terms in \eqref{expose_v_approx}, \eqref{expose_w_approx}.
In Section \ref{corollaries_tip} we give some corollaries of our control in the tip region.
\subsection{Sharpness and further questions} \subsubsection{Regularity conditions \ref{modelpinch_reg}}
Note that an implication of $|u \partial_u W_0| + |u^2 \partial_u^2 W_0| < CW_0$ is that $
\frac{W_0(ru)}{W_0(u)} $ can be bounded for small $r$, independently of $u$. In particular, $W_0(u) = e^{u^{-1}}$ and $W_0(u) = e^{-u^{-1}}$ both do not satisfy our assumptions. We cannot offer any guess as to whether our results hold for these functions.
As examples of wild profiles for $W_0$, consider $W_0(u) = 2 + \sin(\log(u))$ or $W_0(u) = u^{-1}$. Note that if the initial metric has bounded length near $u = 0$, then this may appear bad. Still, around any point where $u = u_\sharp$, rescaling by $u_{\sharp}$ we will see approximately a product metric on a long (length $\approx 1/\sqrt{V_0(u_\sharp)}$) scale. Note in all cases, in the forward evolution $w$ is bounded and positive near the tip for finite time.
We can think of the conditions on $V_0$ in the same way, but it may be more reasonable to look at examples in terms of the arclength coordinate $s$. So, consider the $I \times S^q$ part of the metric written as \begin{align}
ds^2 + u(s)^2 g_{S^q}, \quad s \in (L_0, \infty), \quad L_0 = 0 \text{ or } L_0 - \infty . \end{align} The condition that $\dd1{V_0} \mathrel{\mathop:}= \frac{u \partial_u V_0(u)}{V_0(u)} < C$ actually says, in a sense, that $u$ must be small enough in terms of $s$. (Written in terms of $s$, this condition will involve the functional inverse of $u$.) The following functions satisfy the regularity conditions on $V$: \begin{itemize}
\item $L_0 = 0$ and $u(s) = s^a |\log(s)|^b$, where $a > 2$ and $b \in \mathbb{R}$, or $a = 2$ and $b < 0$.
\item $L_0 = -\infty$ and $u(s) = |s|^{-a}\log(|s|)^{b}$, where $a > 0$ and $b \in \mathbb{R}$.
\item If we write $u(s) = \exp(-f)$ where $f \to \infty$ as $s \searrow L$, then the condition that $|\dd1{V_0}|<C$ is equivalent to $(1/f')' < C$. For example, $u(s) = \exp(-1/s), L_0 = 0$ or $u(s) = \exp(s)$, $L_0 = -\infty$ are both valid model pinches. \end{itemize}
\subsubsection{The profile $\phi(s) = \log(|s|)^{-1}$}
Our results do not provide a forward evolution from the initial metric with $I = (-\infty, \infty)$, and $u(s) \sim \log(|s|)^{-2}$ at $s = -\infty$. Note in that case \begin{align}
v = u^{-1}|\nabla u|^2 = 4 \log(|s|)^{-4}s^{-2} \end{align} so $
V_0(u) = u^2 \exp(-2/u) $. Then $u \partial_uV_0/V_0 = 2u^{-1} + 2$, which violates condition \ref{modelpinch_reg}. It would be interesting to know whether there is a solution to Ricci flow emerging from this example.
In this example, for any $r > 0$, the region which looks approximately like a skinny cylinder of radius $r$ is quite long in comparison to $r$. More precisely, fixing $\epsilon$ there is a $C_\epsilon > 1$ such that for any $r$ we have the following. The region where the radius $\phi$ is within a factor of $(1 \pm \epsilon)$ of $r$ has length $(C_\epsilon)^{1/r^2}$. Maybe this means the cylinder must collapse before anything far away can save it.
\subsubsection{The conditions on the size of $W_0$ \ref{w_big}}\label{w_big_sharpness} For simplicity say $(F, g_F) = (S^q, g_{S^q})$. We find it striking that in the case $W_0(u) = (1 + c)u$, for the initial metric $w$ and $u$ are comparable, but if we rescale the forward evolution to keep the curvature bounded at the origin, the $w$ factor goes to infinity.
We believe that it is possible to relax the condition \ref{w_big} and still have a forward evolution with the same asymptotics. Let's rapidly go through a calculation. Suppose $W_0(u) = (1 + H_0(u))u$, where $H_0(u) \searrow 0$ (violating \ref{w_big}). Calculating from \eqref{w_approx_prish}, in the productish region where $u > C t \nu(t)$, \begin{align}
w
&\approx (1 + H_0(u + \mu t))(u + \mu t) - \mu t\\
&= u + H_0(u + \mu t)(u + \mu t). \end{align} If we write $\eta(t) = H_0(\mu t)$ then for points where $C t \nu(t) < u \ll t$ we have (recall $\sigma \mathrel{\mathop:}= \frac{u}{t\nu(t)}$): \begin{align}
\frac{w}{t\nu(t)} &\approx \sigma + \mu \frac{\eta(t)}{\nu(t)}.\label{speculative_tip_asympts} \end{align}
First consider the case $H_0(u) \gg V_0(u)$ (i.e. $\eta(t) \gg \nu(t)$), which is still a weaker condition than \ref{w_big}. Then scaling $w$ in the same way we scale $u$ sends it to infinity, and $w$ is approximately a constant. We expect this case to behave similarly to the case that is rigorously dealt with in this paper. The major road block in dealing with it, for us, is reproving Lemma \ref{lemma:y_control_tip} which controls the derivative of $w$ and therefore controls the level of interaction between the evolution of $v$ and $w$. Unfortunately our method gives us no more wiggle room in this lemma, but we think that our control on the distance from $w$ to a constant is not optimal.
To continue with our speculation, consider the case when $H_0(u) = c_0 V_0(u)$. Then in \eqref{speculative_tip_asympts} we find $ \frac{w}{t\nu(t)} = \sigma + c_0 \mu$. We still would have the approximation \eqref{expose_v_approx} for $v$. This gives us the asymptotics for an \emph{Ivey soliton} \cite{ivey}, which is a complete soliton on $\mathbb{R}^{1+q} \times F$ of the form $dx^2 + u_{sol}(x)g_{S^q} + w_{sol}(x)g_F$. (The function $u(x)$ goes to zero at $x = 0$, and $w(x)$ stays positive.) So, in this case we expect to see the Ivey soliton in the rescaled limit at the tip. This case should be more difficult, because the system is more strongly coupled.
In the case when $H_0(u) \ll V_0(u)$, we do not think that there is a smooth forward evolution, but there may be a forward evolution with bounded Ricci curvature everywhere. In this forward evolution we glue in a Bryant of dimension $1 + (q + q)$, but with the sphere fibers $S^{q + q}$ replaced with the Einstein manifold $S^q \times S^q$ (with proper scaling to make the scalar curvatures match). The case $H_0(u) = 0$ is the situation discussed in Section \ref{scarring}.
The reason we do not expect a smooth forward evolution is the following: consider $H_0(u) = \epsilon V_0(u)$. Then, we are in the case when we expect the Ivey soliton. The exact asymptotics of the Ivey soliton we get are determined by $\epsilon$, and as $\epsilon \searrow 0$, this family of Ivey solitons approaches the Bryant soliton with $S^{q + q}$ replaced with $S^q \times S^q$. Therefore, even trying to approximate the singular initial metric with smooth ones it seems we are led to the nonsmooth case.
\subsubsection{Pinched sphere warped products over other bases}
Consider a manifold with boundary $(B, \partial B)$ with a metric $g_B$ and a function $u: B \to \mathbb{R}_+$ which tends to zero at the boundary such that $v = u^{-1}|\nabla u|^2$ also goes to zero. Let's stipulate that everywhere $|\Rm_{g_B}| \ll u^{-2}|\nabla u|^2$. Now we want to ask whether there is a forward evolution from the metric $g = g_B + u g_{S^q}$. Note that model pinches with $W_0(u) \gg u$ are a special case, where $g_B = \frac{du^2}{u V_0(u)} + W_0(u)g_F$.
The nice property of the doubly warped products is that the hessian of $u$ is easier to control, because the level sets of $u$ are equidistant. It should be relatively possible to extend to other such cases, like cohomogeneity-one manifolds.
\subsubsection{The closeness required in the asymmetric case} Our condition for Theorem \ref{theorem:unsymmetrical_flow} is that the distance between the asymmetric metric and the model pinch goes to zero near the tip at least as fast as a specific rate. There is a sense in which this is probably not optimal. Our proof technique yields more than is stated in Theorem \ref{theorem:unsymmetrical_flow}: it says that $g(t)$ actually stays close to the forward evolution from $g_{mp}$. We make no attempt to update the approximate model pinch, whereas perhaps the best warped-product forward evolution not the forward evolution from the initial warped-product.
A theorem that we can compare Theorem \ref{theorem:unsymmetrical_flow} to is Theorem 1.3 of \cite{conicalSing}. That theorem constructs forward evolution from metrics close to having conical singularities. There, $g_{c}$ is a cone and the requirement (1.1) is that near the singularity the singular metric $g$ satisfies $|g - \Phi^* g_c| \leq \epsilon_0$. This seems stronger than our theorem, because it makes no exact assumption on the rate at which it approaches the model singularity. On the other hand, the case of a singly-warped cone (which our theorem does not handle) is the case when $V_0$ is constant, so perhaps our condition is not dissimilar.
\subsection{Notation and preliminaries}\label{overview_symmetric} More notation is densely listed in Appendix \ref{appendix_notation}.
Partial derivatives are denoted with $\partial_{\cdot}$. For an arbitrary function $u$ with nonzero derivative, we have $\partial_u = |\nabla u|^{-2}\nabla_{\grad u}$ which is the derivative with respect to $u$, using a metric. We define $\partial_{t;u} = \partial_t - (\partial_t u)\partial_u$ which is the derivative with respect to time along a curve which moves orthogonally to the level sets of $u$ in order to keep $u$ constant.
We adopt the shorthand that when stating hypotheses, the statement ``$x \leq \barr x(y, z)$'' means ``there exists an $\barr x$, depending on $y$ and $z$, such that if $x \leq \barr x$, the following holds.'' This allows us to quickly state ``if $x \leq \barr x(y,z)$ and $w \leq \barr w(x, y)$ then \dots''.
\subsubsection{Equations} We can consider our metrics as singly warped products of spheres over a general base: $g(t) = g_B(t) + u(t)g_{S^q}$ where for each $t$, $u(t):B\to \mathbb{R}_+$. Under Ricci flow, $u$ evolves by \begin{align}
\square_B u = - \mu + \on4 (\mu-2) v,\label{evo_u_in_overview} \end{align} where $\square_B$ is the heat operator $\square_B = \partial_t - \Delta_B$ and $\Delta_B$ is the laplacian for $g_B(t)$. Equivalently, \begin{align}
\square_M u = - \mu - v. \end{align} where $\square_M$ is the heat operator for $g$. Similarly, the function $w$ which controls the size of $g_F$ evolves by \begin{align}
\square_M w = - \mu_F - w^{-1}|\nabla w|^2 = -\mu_F - y, \end{align}
where we have defined $y = w^{-1}|\nabla w|^2$. We use this point of view to find the approximate solutions in the productish region. For an exposition of these equations for Ricci flow on warped products, see Section \ref{warped_product_section}.
For finer control, we need the evolution of $v$ and $w$ as functions of $u$. These are derived in Sections \ref{v_deriving_section} and \ref{additional_wp}. We have \begin{align}
\partial_{t;u} v
&= u v \partial_u^2 v - \oh u (\partial_u v)^2 \label{vevo_basic2}\\
&+ \mu \left(1 - \on4 v \right) u^{-1} v + \mu \partial_u v \\
&- 2 (\kappa^2) v, \end{align} where $\kappa^2 = \on4 (dim(F))w^{-2}u^2 v^2 (\partial_u w)^2$, and \begin{align}
\partial_{t;u}w - u v \partial_u^2 w
= - \mu_F - y + \mu \partial_u w - \mu/2 v\partial_u w \label{evo_w_in_u2}. \end{align}
\subsubsection{Regularity}\label{regularity} We work in $C^{2, \eta}$ H\"older spaces using interior Schauder estimates. Bamler wrote a clean statement of the interior Schauder estimates he needed in \cite{stabilBamler} (Section 2.5). We co-opt this statement, because it is exactly what we need except for standard generalizations. His statement does not allow for the time-dependence of the coefficients that we will have, but in fact the proof carries through exactly; the time dependence enters in the estimate on the $C^{2m-2, 2\alpha; m-1, \alpha}$ norm of $f_i$ in the middle of page 424. Furthermore, his statement does not allow the parabolic ball to hit the initial time, as we will need to. Accounting for this is also standard. In the proof of Lemma 2.6 of \cite{stabilBamler}, one may apply Exercise 9.2.5 of \cite{krylov} rather than Theorem 8.11.1 of \cite{krylov}.
\subsubsection{Ricci-DeTurck flow}\label{section:rdt} We use Ricci-DeTurck flow to control the Ricci flow of metrics near our warped product forward evolutions. For two metrics $(M, g)$ and $(M, \tilde g)$ we define \begin{align}
(V[g,\tilde g])^i = g^{ab}\left(
\left( \Gamma_g\right)_{ab}^i
- \left( \Gamma_{\tilde g} \right)_{ab}^i
\right) \end{align} which is the map Laplacian of the identity map from $(M, g)$ to $(M, \tilde g)$. For a time-dependent metric we define $\Rf[g] = \partial_t g - (-2 \Rc[g])$. The Ricci-DeTurck flow from $g(0)$ with background metric $\tilde g$ is the solution to \begin{align}
g(0)&\quad \text{ given},\\
\Rf[g]
&= \mathcal{L}_{V[g,\tilde g]}g. \label{RDT_def} \end{align} We allow $\tilde g$ to also be time-dependent. It will be useful to consider Ricci flow and Ricci-DeTurck flow modified by a vector field. We set $\Rf_X[g] = \partial_t g - (-2 \Rc[g] - \mathcal{L}_X g)$, and if $\Rf_X[g] = \mathcal{L}_{V[g, \tilde g]}g$ then we say that $g$ is a solution to Ricci-DeTurck flow, modified by $X$, with background metric $\tilde g$.
We will not use the exact form of the evolution of $h$, except to know that we can apply regularity. What we will use is the following evolution of $|h|$ and $|h|^2$. For $p \in M$ We set \begin{align}\label{rmplus_def}
\Lambda_{\Rm}(p) = \max_{h \in Sym_2(T_pM) : |h| = 1}\ip{\Rm[h]}{h}(p). \end{align}
Now, assuming that $\Rf_X[\tilde g] = 0$, and that $|h| \leq \oh$, for $y = |h|^2$ we have (we allow $c_0$ to change from line to line) \begin{align}
\square_{X, \tilde g, g} y
&\leq 4 \Lambda_{\Rm} y - 2(1 - c_0 y^{1/2})|\nabla h|^2 + c_0 |\Rm|y^{3/2}. \label{dtevo_square} \end{align}
For $z = |h|$ we have \begin{align}
\square_{X, \tilde g, g}z
&\leq 2 \Lambda_{\Rm} z + c_0 \left(|\Rm| z^2 + |\nabla h|^2 \right).\label{rdt_norm} \end{align} We show these in Appendix \ref{rcdt}.
\section{Control in the productish region}\label{section:productish} In this section we create some interior estimates for our warped-product forward evolution. We define the productish region as a region of the form \begin{align}\label{omega_prish_def}
\Omega_{prish}=
\left\{
(u, t): u + \mu t < u_* \text{ and } \sigma = \frac{u}{tV_0(\mu t)} > \sigma_*
\right\}. \end{align} In particular, $\Omega_{prish}$ touches an open part of the initial time slice (see Figure \ref{region_map}). All constants and definitions in this section implicitly depend on dimensions, $g_F$, and the chosen functions satisfying the model pinch conditions $V_0$ and $W_0$. We define $\hat w = w + \mu_F t$. In the productish region, we will have approximations of the form \begin{align}
v \approx V
\mathrel{\mathop:}=
\left(\frac{u + \mu t}{u} \right)
V_0(u + \mu t), \quad
\hat w
\approx \hat W
\mathrel{\mathop:}=
W_0(u + \mu t). \end{align} These come directly from the calculations in Appendix \ref{nearly_cnst_pde_sect}. They may be guessed by ignoring all terms in the evolution of $u$ and $w$ which depend on space derivatives of $u$ or $w$. We will prove that $v$ is between $V^-$ and $V^+$, and $\hat w$ is between $\hat W^-$ and $\hat W^+$, where \begin{align}
V^{\pm} = (1 \pm D V)V, \quad \hat W^{\pm} = (1 \pm D V)\hat W. \label{barriers_def} \end{align} We call $V^{\pm}$ and $W^{\pm}$ the barriers.
We make some definitions to state the main result of this section. We will assume that $g(t) = a(x,t) dx^2 + u(x,t) g_{S^q} + w(x,t) g_F$ is a solution to Ricci flow on $[T_1, T_2]$. Our definitions depend on constants $u_*$, $\sigma_*$, controlling the size of the produtish region, and $D$ controlling the separation of the barriers, as well as $c_{safe}$ and $C_{reg}$.
\begin{definition}\label{productish_barricaded}
We say that $g(t)$ is \emph{barricaded} (by the productish barriers)
\footnote{In this section we only say ``barricaded'' but in Section \ref{section:full_flow} we will have to refer to either barricaded by the productish barriers, or barricaded by the tip barriers.}
at a point if it satisfies
$
V^- < v < V^+$ and $\hat W^- < \hat w < \hat W^+$
at that point.
We say that $g(t)$ is \emph{initially controlled in the productish region} if
at $t = T_1$ and for all points satisfying $(1/2)\sigma_* T_1 \nu(T_1) < u < 2u_*$
it is barricaded
and
\begin{align}
\frac{\twoeta{v-V}{u/2}{(du)^2}}{V}
+
\frac{\twoeta{w-W}{u/2}{(du)^2}}{W}
< c_{safe}C_{reg}DV, \label{prish_regularity_ineq_v}
\end{align}
We say that $g(t)$ is \emph{barricaded at the left of the productish region} if it is barricaded for all points satisfying $(1/2)\sigma_* t \nu(t) < u < \sigma_* t \nu(t)$ and $t \in [T_1, T_2]$.
We say that $g(t)$ is \emph{barricaded at the right of the productish region} if it is barricaded for all points satisfying $u_* < u < 2 u_*$ and $t \in [T_1, T_2]$.
We say that $g(t)$ is \emph{controlled in the productish region} if for all points in $\Omega_{prish}$,
\begin{enumerate}[label=(P\arabic{*}), ref=(P\arabic{*})]
\item \label{conc:prish_barrier} The solution is barricaded.
\item \label{conc:prish_reg} We have the inequality
\begin{align}
\frac{\twoeta{v-V}{u/2}{(du)^2}}{V}
+
\frac{\twoeta{w-W}{u/2}{(du)^2}}{W} < C_{reg}DV. \label{prish_regularity_ineq_v}
\end{align}
\end{enumerate} \end{definition}
\begin{lemma}\label{main_prish_estimates}
There is a $c_{safe}$ such that if we let $C_{reg} > \berr C_{reg}$, $D > \berr D$, $u_* < \barr u_*(D, C_{reg})$, and $\sigma_* > \berr \sigma_*(D, C_{reg})$, there is a $T_*$ depending on all other parameters with the following property.
Suppose $0 < T_1 < T_2 < T_*$ and $g(t)$ is defined on $[T_1, T_2]$, initially controlled, and barricaded at the left and the right of the productish region. Then $g(t)$ is controlled in the productish region, for all times in $[T_1, T_2]$. \end{lemma}
In proving the conclusions of Lemma \ref{main_prish_estimates}, we can assume that they hold on the interval $[T_1, T_2)$. This is because they hold strictly at the initial time by assumption, so we can consider $T_2$ to be the infimum of the times at which the fail. This extra assumption is usually useful for controlling terms when we don't care about the exact constant involved, because in any case we can choose our constants $u_*$, $\sigma_*$, and $T_*$ so that it is as small as we want (see e.g. Lemma \ref{lemma:ycontrol_productish}).
With this in mind, Lemma \ref{main_prish_estimates} will be proven by Lemmas \ref{barriers_prish} and \ref{regularity_prish} below, which show items \ref{conc:prish_barrier} and \ref{conc:prish_reg} respectively. First, in Section \ref{examining_section}, we inspect our approximations $V$ and $W$ more closely.
\subsection{Examining our approximate solution}\label{examining_section} We are claiming that $v(p, t) \approx V(u(p,t),t)$ where $V$ is the function \begin{align}
\label{v_expr}
{\uf{V}}(u,t) = \frac{u + \mu t}{u}{\uf{V}}_0(u + \mu t) = \left( 1 + \mu \frac{t}{u} \right){\uf{V}}_0(u + \mu t) \end{align} The effectiveness of the barriers defined in \eqref{barriers_def} is dependent on $V$ staying small. In this section, we prove Lemma \ref{lemma:V_small} which tells us that $V$ does stays small exactly in the productish region $\Omega_{prish}$, and also gives another description of $V$ and $W$. The proof is elementary, but the reformulation of $V$ is key to how the productish region hooks up with the tip region.
We aim to understand where $V$ stays small. An apparent scary term in \eqref{v_expr} is $t/u$. Defining $\rho = u/t$, we can write $
\label{eq:79}
{\uf{V}} = \left( 1 + \mu \rho^{-1} \right) {\uf{V}}_0(u + \mu t) $. If we keep in mind that our main assumption on ${\uf{V}}_0$ is that ${\uf{V}}_0(u) = o(1, u \to 0)$, then the following lemma, which says something about where $V$ is small, is immediately apparent. \begin{lemma}\label{lemma:weaker_V_small}
Let $\epsilon$ be given. For any $\rho_*$ there is $u_*(\epsilon)$ and $T_*(\rho_*, \epsilon)$ so that if $t < T_*$, $u < u_*$, and $u/t > \rho*$ then $V < \epsilon$. \end{lemma}
The discussion is not over: $V$ does not get large if we fix $\rho$ and send $u + \mu t \searrow 0$, as the factor $V_0(u + \mu t)$ helps us. To understand this factor better, let $\nu(t) = V_0(\mu t)$. Then by definition, \begin{align}
\label{eq:81}
V_0(u + \mu t) = \nu(t) \frac{V_0((1 + u/(\mu t)) \mu t)}{V_0(\mu t)}. \end{align} Now we can use the regularity assumption on $V_0$ \ref{modelpinch_reg} to calculate using the Taylor expansion: \begin{align}
V_0(u + \mu t)
&= V_0(\mu t) + u V_0'(\mu t) + u^2 V_0''((1 + r) \mu t) \\
&= \nu(t)
+
\nu(t)^2
\frac{u}{t \nu(t)}
\mu^{-1} \left( t \mu \frac{V_0'(\mu t)}{V_0(\mu t)} \right)
+ u^2 V_0''((1 + r) \mu t). \end{align} Here $r \in [0, u/(\mu t)]$ comes from the remainder term in the Taylor expansion. Now let $\dd1{\nu}(t) = \frac{t \partial_t\nu(t)}{\nu(t)}$ and $\sigma = \frac{u}{t\nu(t)}$, and calculate further: \begin{align}
V_0(u + \mu t)
&= \nu + \nu^2 \sigma \mu^{-1}\dd1{\nu} \\
&+ u^2 V_0''(\mu t) + u^2 \left( V_0''((1 + r)\mu t) - V_0(\mu t) \right) \\
&= \nu + \nu^2 \sigma \mu^{-1}\dd1{\nu} \\
&+ \sigma^2 \nu^3 \frac{t^2 V_0''(\mu t)}{V_0(\mu t)}
+ \sigma^{2+\eta} \nu^{3+\eta}
\left(
(t/u)^{\eta}
\frac{V_0''((1 + r)\mu t) - V_0(\mu t)}{V_0(\mu t)}
\right)\\
&= \nu + \nu^2 \sigma \mu^{-1}\dd1{\nu} + O(\sigma^2 \nu^3). \end{align} Here we used more of the regularity assumption \ref{modelpinch_reg}. Now coming back to our expression for $V(u,t)$ and manipulating it, \begin{align}
\label{eq:84}
V(u, t)
&= (1 + \mu \rho^{-1})V_0(u + \mu t) \\
&= \mu \sigma^{-1}\nu^{-1}(1 + \mu^{-1}\sigma\nu)V_0(u + \mu t) \\
&= \mu \sigma^{-1}\left( 1 + (1 + \dd1 \nu)\mu^{-1}\nu\sigma + O((\nu \sigma)^2) \right). \label{V_expr_smallrho} \end{align} This makes it apparent that if we look at where $\sigma > \sigma_*$ for some large $\sigma_*$, $V$ is still small. We present Lemma \ref{lemma:V_small}. \begin{lemma}\label{lemma:V_small}
Let $\epsilon$ be given. If $\sigma_* > \berr \sigma_*(\epsilon)$ and $u_* < \barr u_*(\epsilon)$, and $T_* < \barr T_*(\sigma_*, u_*, \epsilon)$, then $V < \epsilon$ in the productish region \eqref{omega_prish_def}. \end{lemma} \begin{proof}[Proof. (Lemma \ref{lemma:V_small})]
First, choose $\underline{\sigma}_*$ small enough, and $\barr T_*$ at least small enough, so that $(\sigma^{-1} + \nu) < \epsilon/100$ for all $u, t$ satisfying $\sigma > \underline{\sigma}_*$ and $t < \barr T_*$. Next, by the expression \eqref{V_expr_smallrho}, we can choose $\rho_*$, and decrease $\barr T_*$, so that for $\sigma > \sigma_*$ and $\rho = \nu \sigma < \rho_*$, we have $V < \epsilon/50$. Finally, by Lemma \ref{lemma:weaker_V_small} we can chose $\overline{u}_*$ so that $V < \epsilon$ for all $u,t$ satisfying $\rho > \rho_*$ and $u < \overline{u}_*$. \end{proof}
We also examine the approximate solution for $\hat w$, namely $\hat W = W_0 \circ U_0$ (so $W = W_0 \circ U - \mu_F t$). Similarly to how we handled $V$, we write $\omega(t) = W_0(\mu t)$ and we find \begin{align}
\hat W
&= \omega(t) \left(
1 + \mu^{-1}\nu \sigma \dd1{\omega}(t) + O((\nu \sigma)^2)
\right)\label{Wprish_asymptotic_exp} \end{align}
\subsection{Trapping between barriers}
Recall the equations of warped product Ricci flow (see section \ref{rf_warped_products}). The functions $u$ and $\hat w = w + \mu_F t$ satisfy \begin{align}
\square_M u &= - \mu + c_{v} v, \\
\square_M \hat w &= - y = - \frac{|\nabla \hat w|^2}{\hat w - \mu_F t} \end{align} First, we find bounds given to us by our regularity \ref{conc:prish_reg}. \begin{lemma}\label{lemma:ycontrol_productish}
Suppose we are in the setting of Lemma \ref{main_prish_estimates}. Assume additionally that items \ref{conc:prish_barrier} and \ref{conc:prish_reg} hold on $[T_1, T_2)$. If $\sigma_* > \berr \sigma_*(D, C_{reg})$ and $u_* < \berr u_*(D, C_{reg})$ then
\begin{align}
\frac{uy}{vw} < C, \quad
\frac{|\nabla\nabla u|}{v} \leq C
\end{align}
in $\Omega_{prish}$, where $C$ depends only on the initial data. \end{lemma} \begin{proof}
First we tackle $\frac{uy}{vw}$. Note that
$
\frac{uy}{vw}
= \frac{ u^2 |\nabla w|^2}{|\nabla u|^2 w^2}
= \left( \frac{u \partial_u w}{w} \right)^2
$
so by item \ref{conc:prish_reg}
\begin{align}
\frac{uy}{vw} \leq \left( \frac{u\partial_u W}{w} + C_{reg}DV\frac{W}{w} \right)^2.\label{bnda}
\end{align}
By Lemma \ref{lemma:V_small} we can decrease $u_*$ and increase $\sigma_*$ so that $C_{reg}DV < 1$ and $\frac{W}{W^-} < 2$. Then since $w$ is between its barriers, we can bound $w$ in \eqref{bnda} in terms of $W$.
\begin{align}
\frac{uy}{vw}
&\leq 4 \left( \frac{u\partial_u W}{W} + 1 \right)^2
=
4 \left(
\frac{u \partial_u W_0(u + \mu t)}{W_0(u + \mu t) - \mu_F t} + 1
\right)^2\\
&=
4 \left(
\frac{W_0(u + \mu t)}{W_0(u + \mu t) - \mu_F t}
\dd1{W_0}(u + \mu t)
+ 1
\right)^2.\label{bndb}
\end{align}
By the assumption \ref{w_big} on $W_0$,
\begin{align}
\frac{W_0(u + \mu t)}{W_0(u + \mu t) - \mu_F t}
&= \frac{1}{1 - \frac{\mu_F t}{W_0(u + \mu t)}}
\leq \frac{1}{1 - \frac{1}{1 + c}}
\end{align}
Therefore \eqref{bndb} is bounded by a constant depending only on the initial data, using also our assumption \ref{modelpinch_reg} that $\dd1 W_0 = \frac{u\partial_u W_0}{W_0}$ is bounded.
Now we bound the hessian, which requires the geometry of the warped products. Thinking of the multiply warped product manifold as a family of equidistant hypersurfaces, the norm of the hessian of a function depending only on the hypersurface is given by
\begin{align}
|\nabla \nabla f|^2
&= \on4 |\nabla f|^{-4}\ip{\nabla |\nabla f|^2}{ \nabla f}^2
+ |\nabla f|^2 |A|^2
\end{align}
where $|A|^2$ is the norm of the second fundamental form of the hypersurfaces, which in our case is
\begin{align}
|A|^2 = \on4 q u^{-1} v + \on4 \dim(F) w^{-1}y.
\end{align}
Therefore we find,
\begin{align}
|\nabla \nabla u|^2
&\leq C \left(
|\nabla u|^{-4}|\nabla |\nabla u|^2||\nabla u|^2
+ u^{-1}v |\nabla u|^2 + w^{-1}y |\nabla u|^2
\right) \\
&= C \left(
u^{-1}v^{-1} |\nabla (uv)|^2
+ v^2 + w^{-1}yuv
\right)\\
&\leq C\left(
uv^{-1}|\nabla v|^2 + \left( 1 + \frac{u}{v}\frac{y}{w} \right)v^2
\right) \\
&=C\left(
\left( \frac{u \partial_u v}{v} \right)^2 + \left( 1 + \frac{u}{v}\frac{y}{w} \right)
\right) v^2.
\end{align}
By \ref{conc:prish_reg}, and by the bound on $\dd1{V}$ in Lemma \ref{approx_soln}, we get the desired inequality.
\end{proof}
Now we are in the position to prove that \ref{conc:prish_barrier} continues to hold.
\begin{lemma}\label{barriers_prish}
Suppose we are in the setting of Lemma \ref{main_prish_estimates}, and items \ref{conc:prish_barrier} and \ref{conc:prish_reg} holds on $[T_1, T_2)$.
If $D> \berr D$, $u_* < \barr u_*(D, C_{reg})$, $\sigma_* > \berr \sigma_*(D, C_{reg})$, $T_* < \barr T_*(D, u_*, \sigma_*)$ then \ref{conc:prish_barrier} holds at $t = T_2$. \end{lemma} \begin{proof}
By the evolution equation for $v$, \eqref{EQ:22}, and by our bound on the hessian from Lemma \ref{lemma:ycontrol_productish}, we have
$
|(\square - u^{-1} \mu ) v| \leq C v^2
$
for a constant $C$ depending only on the initial data $V_0$ and $W_0$. Also, by the evolution equation for $w$ and our bound on $y$, we have
$
|\square \hat w | \leq C u^{-1} v \hat w
$.
Therefore, Lemma \ref{sup_solns} shows that, if we chose $D$ sufficiently large, $V^{\pm}$ and $W^{\pm}$ are sub- and supersolutions for the equations satisfied by $v$ and $w$. The maximum principle proves the claim. \end{proof}
\subsection{Regularity} \begin{lemma}\label{regularity_prish}
Suppose we are in the setting of Lemma \ref{main_prish_estimates}. We can choose $\barr c_{safe}$, $\berr C_{reg}$, $\berr u_*$, and $\berr T_*$ such that if \ref{conc:prish_barrier} holds for $t \in [T_1, T_2)$ then \ref{conc:prish_reg} holds for $t\in [T_1, T_2]$. \end{lemma} \begin{proof} We prove this theorem by applying parabolic regularity to the equations solved by $v$ and $w$ in terms of $u$. From \eqref{vevo_basic2} and \eqref{evo_w_in_u2}, we have the equations \begin{align}
\partial_{t; u} v - \mu \partial_u v - \mu u^{-1} v
&= \left( u v \right) \partial^2_u v - \oh u \left( \partial_u v \right)^2 \\
&+ a_1 v \partial_u v + a_2 u^{-1} v^2 + a_3 \left( \frac{vu}{\hat w - \mu_F t} \right)^2 \left( \partial_u \hat w \right)^2, \\
\partial_{t; u} \hat w - \mu \partial_u \hat w
&= \left( u v \right) \partial_u^2 \hat w + b_1 v \partial_u \hat w + b_2 \left( \frac{vu}{\hat w - \mu_F t} \right) \left( \partial_u \hat w \right)^2, \end{align} where $a_1, a_2, a_3$ and $b_1, b_2$ are constants.
We let ${\hat u} = u + \mu t$, $\hat v = {\hat u}^{-1}u v$, and similarly $\hat V = {\hat u}^{-1} u V = V_0({\hat u})$. Calculate, \begin{align}
v &= u^{-1} {\hat u} \hat v, \\
\partial_u v &= - \mu t u^{-2} \hat v + u^{-1} {\hat u} \partial_{\hat u} \hat v, \\
\partial_u^2 v &=
2 \mu t u^{-3} \hat v
- 2 \mu t u^{-2} \partial_{\hat u} \hat v
+ u^{-1} {\hat u} \partial_{\hat u}^2 \hat v. \end{align} Also note that \begin{align}
\partial_{t; u} v - \mu \partial_u v - \mu u^{-1} v
= u^{-1} {\hat u} \partial_{t; {\hat u}} \hat v
\quad \text{and} \quad
\partial_{t;u} w - \mu \partial_u w + \mu_F
= \partial_{t; {\hat u}} \hat w. \end{align} This lets us derive the following equation for $\hat v$: for some constants $c_1, c_2, c_3, c_4$, \begin{align}
\partial_{t;{\hat u}} \hat v
&= {\hat u}\hat v \partial_{\hat u} ^2 \hat v \\
&+ c_1 t u^{-2} \hat v^2
+ c_2 t u^{-1} \hat v \partial_{\hat u} \hat v\\
&+ c_3 {\hat u} (\partial_{\hat u} \hat v)^2
+ c_4 u {\hat u}^{-1}\left( \frac{ v u }{w} \right)^2 (\partial_u \hat w)^2. \end{align} We also derive the evolution for $\hat w$: \begin{align}
\partial_{t;w} \hat w
&= ({\hat u} \hat v) \partial_{\hat u}^2 \hat w
+ b_1 (u^{-1}{\hat u} \hat v) \partial_{\hat u} \hat w
+ b_2 \left( \frac{{\hat u} \hat v}{w} \right) \left( \partial_{\hat u} \hat w\right)^2. \end{align}
Now, let $u_1, t_1$ be any point in the productish region, let ${\hat u}_1 = u_1 + \mu t_1$, $\hat v_1 = \hat v(u_1, t_1)$, and $\hat w_1 = \hat w(u_1, t_1)$. Divide through in both equations by ${\hat u}_1 \hat v_1$. Also divide the equation for $\hat v$ by $\hat V_1 = \hat V(u_1, t_1) = V_0({\hat u}_1)$ and the equation for $\hat w$ by $\hat W_1 = \hat W(u_1, t_1) = W_0({\hat u}_1)$. \begin{align}
\frac{1}{{\hat u}_1 \hat V_1} \partial_{t;{\hat u}} \left( \frac{\hat v}{\hat V_1} \right)
&= \left[ \frac{{\hat u} \hat v}{{\hat u}_1 \hat V_1} \right] \partial_{\hat u}^2 \hat v \\
&+ c_1 \left[ \frac{t}{{\hat u}_1} \frac{\hat v}{\hat V_1}\frac{u_1^2}{u^2} \right]
u_1^{-2} \left( \frac{\hat v}{\hat V_1} \right)
+ c_2 \left[ \frac{t}{{\hat u}_1} \frac{\hat v}{\hat V_1}\frac{u_1}{u} \right]
u_1^{-1}\partial_{\hat u} \left( \frac{\hat v}{\hat V_1} \right) \\
&+ c_3 \left[ \frac{{\hat u}}{{\hat u}_1} \right]
\left(\partial_{\hat u} \left( \frac{\hat v}{\hat V_1}\right) \right)^2
+ c_4 \left[\frac{v}{v_1} \frac{u^2}{{\hat u} {\hat u}_1}\frac{w_1^2}{w^2}\frac{v}{v_1}\right]
\left(\partial_u \left( \frac{w}{w_1} \right) \right)^2 \end{align} \begin{align}
\frac{1}{{\hat u}_1 \hat V_1}\partial_{t;{\hat u}} \left( \frac{\hat w}{\hat W_1} \right)
&= \left[ \frac{{\hat u} \hat v}{{\hat u}_1 \hat V_1} \right]
\partial_{\hat u}^2 \left( \frac{\hat w}{\hat W_1} \right) \\
&+ b_1 \left[ \frac{{\hat u} \hat v}{{\hat u}_1 \hat V_1} \right]
u^{-1}\partial_{\hat u} \left( \frac{\hat w}{\hat W_1} \right)
+ b_2 \left[ \frac{{\hat u}}{{\hat u}_1} \frac{\hat v}{ \hat V_1} \frac{\hat w}{w} \frac{\hat w}{\hat W_1} \right] \left( \partial_{\hat u} \left( \frac{\hat w}{\hat W_1} \right) \right)^2 \end{align} We will apply interior parabolic regularity to these equations, in the region \begin{align}
\Xi = \{({\hat u},t) : ({\hat u}, t) \in
[{\hat u}_1 - \oh u_1, {\hat u}_1 + \oh u_1]
\times
[t_1 - \max(T_1, t_1 - \oh {\hat u}_1^{-1}v_1^{-1}u_1^2), t_1],
\} \end{align} which is a parabolic ball around $({\hat u}_1, t_1)$ of radius $\oh u_1$, if we were to scale time to $\hat t = {\hat u}_1 \hat V_1 t$. We have written the equation so that the factors in square brackets are smooth functions of $u$, $t$, $\frac{\hat v}{\hat V_1}$, and $\frac{\hat w}{\hat W_1}$ in this parabolic ball- this requires the knowledge that $v$ and $w$ are trapped between our barriers, so for example $\frac{w}{w_1}$ is not too far from 1 within $\Xi$. The important thing about this smoothness is that we have bounds on relevant quantities (the $C^{2, \eta}$ norms of the functions) are not dependent on $u_1$ or $t_1$.
All in all, we can apply regularity to bound the ${\hat u}$ derivatives of the functions $\frac{\hat v}{\hat V_1} - \frac{\hat V}{\hat V_1}$ and $\frac{\hat w}{\hat W_1} - \frac{\hat W}{\hat W_1}$. Our barriers tell us that the $C^0$ norm for both of these, in $\Xi$, is bounded by $C D V(u_1, t_1)$, where $C$ depends on on the initial functions only. The regularity theory implies, for some bigger constant $C$ we have, \begin{align}
\frac{\twoeta{ \hat v - \hat V }{r_0u}{(du)^2}}{\hat V}
+
\frac{\twoeta{ \hat w - \hat W }{r_0u}{(du)^2}}{\hat W}
\leq
CDV \end{align}
Now we convert this back to a statement in terms of the hatless functions $v$, and $w$. For instance, just using the definition of the quantities, calculate
\begin{align}
\frac{u}{V}|\partial_u (v-V)|
&= \frac{u}{V}|\partial_u \left( \frac{ u + \mu t}{u} (\hat v - \hat V) \right)| \\
&\leq \frac{u}{V} \mu t u^{-2} |\hat v - \hat V|
+ \frac{u}{V}|\partial_u (\hat v - \hat V)| \\
&=
\mu \frac{t}{{\hat u}}\frac{1}{V} | v - V|
+
\frac{u}{{\hat u}}\frac{u}{\hat V}|\partial_u (\hat v - \hat V) |.
\end{align}
Now using our barriers for the first term, and using the bound for regularity on the second term, as well as $\frac{t}{{\hat u}} \leq 1$ and $\frac{u}{{\hat u}} < 1$,
\begin{align}
\frac{u}{V}|\partial_u (v-V)|
&\leq CDV.
\end{align} Performing similar calculations, we can make the bounds we need. \end{proof} \subsection{Corollaries of control}\label{corollaries_productish} The following corollaries state some precise results which hold for a metric satisfying the conclusions of Lemma \ref{main_prish_estimates}. The corollaries above are just a matter of checking various derivatives and bounds. For Corollary \ref{prish_curvature_control} one can use the calculations of the curvatures for warped products in Appendix \ref{curvatures_our_coordinates}.
First we rephrase our results in terms of how close the metric is to a cylinder. \begin{corollary}\label{prish_asymptotic_cylinder}
There is $C>0$ depending on the initial data and the parameters of control such that the following holds.
Suppose that $g(t)$ is controlled in the productish region at time $t = t_\#$. For $u_\#$ such that $(u_\#, t_\#)$ is in the productish region, let
$$
g_{cyl} = dx^2 + g_{S^q} + \frac{W_{prish}(u_\#, t_\#)}{u_\#} g_{F}.
$$
Let $L$ be given such that $\epsilon = L\sqrt{V_{prish}(u_\#, t_\#)} < 1$. There is a map $\Phi: [-L, L] \times S^q \times F \to M$ which is the identity on the second two factors such that $u(\Phi(0, \cdot, \cdot), t_\#) = u_\#$ and
\begin{align}
\left| g_{cyl} - \Phi^* \left(u_{\#}^{(-1)} g(t_\#) \right) \right|_{C^2([-L, L] \times S^q \times F)} \leq \epsilon C
\end{align} \end{corollary} We also state a result in terms of the curvature of the metrics. \begin{corollary}\label{prish_curvature_control}
Suppose $g(t)$ is controlled in the productish region.
Then there is a constant $C$ depending on the initial data and the parameters of control, such that for all points in the productish region the curvature of $g(t)$ satisfies
\begin{align}
\Rm
&= u^{-1} \left( u g_{S^q} \mathbin{\bigcirc\mspace{-15mu}\wedge\mspace{3mu}} u g_{S^q} \right) + w \Rm_{g_F} + \Rm_{warp} \\
&= u \Rm_{g_{S^q}} + w \Rm_{g_F} + \Rm_{warp}
\end{align}
where $|\Rm_{warp}| \leq C u^{-1} v$. \end{corollary} One further basic statement about the curvature is the following. \begin{corollary}\label{prish_curvature_control_time}
Suppose $g(t)$ is controlled in the productish region. There is a constant $C$ depending on the initial data, such that the following holds for all points in the productish region.
\begin{itemize}
\item
If $\mu_F = 0$, suppose additionally that $(F, g_F)$ is flat. Then $|\Rm| \leq C \sigma_*^{-1}(t \nu(t))^{-1}$.
\item
If $\mu_F \leq 0$, suppose additionally that $(F, g_F)$ is flat or $W_0(u) \geq c' u$ for some $c' > 0$. Then $|\Rm| \leq Cu^{-1}$.
\end{itemize} \end{corollary} \begin{proof}
In Corollary \ref{prish_curvature_control}, since $V$ is uniformly bounded in the productish region, we get $|\Rm_{warp}| \leq C u^{-1}$. We also have $|u \Rm_{g_{S^q}}| \leq C u^{-1}$. (In fact, it is exactly $C_q u^{-1}$ for some constant $C_q$ depending on $q$.) Since in the productish region, we have $u \geq t \nu(t)\sigma_*$, this proves that the terms $|u \Rm_{g_{S^q}}|$ and $|\Rm_{warp}|$ from Corollary \ref{prish_curvature_control} satisfy both of the desired bounds.
We have $|w \Rm_{g_F}| = C_F w^{-1}$. If $\mu_F < 0$, then $W_0(u + \mu t) - \mu_F t \geq (-\mu_F) t$ and so we get the first conclusion. If $\mu_F = 0$ and $(F, g_F)$ is flat, then $C_F = 0$ so we get both conclusions. If $\mu_F \leq 0$ and $W_0(u + \mu t) \geq c' u$, then $W_0(u + \mu t) - \mu_F t \geq c' u$ so we get the second conclusion. If $\mu_F > 0$ then the assumption \ref{w_big} tells us $W_0(u + \mu t) - \mu_F t \geq (1 + c) \mu^{-1}\mu_F u$, so we get both conclusions. \end{proof}
\section{Control in the tip region}\label{section:tip} We are still considering a Ricci flow of model pinches. Recall $\nu(t) = V_0(\mu t)$ and $\omega(t) = W_0(\mu t)$. Section \ref{examining_section} shows that our approximate solutions in the productish region work up to where $\sigma = \frac{u}{t\nu(t)}$ stays very large. In order to examine the solution where $\sigma$ is bounded, we will rescale the metric $g$ by $\alpha = t \nu(t)$: set $\tilde g = \alpha^{-1} g$. Instead of scaling $w$ by $\alpha(t)$ as well, we will work with the function $\bar w = \omega^{-1} (w + \mu_F t) = \omega^{-1}\left(\alpha(t)\tilde w + \mu_F t\right) = \omega^{-1}\hat w$. We also introduce a rescaled time derivative $\partial_{\theta} = \alpha \partial_t$.
The tip region will be, for a constant $\zeta_*$ to be determined, \begin{align}
\Omega_{tip} = \left\{
(u, t) : \frac{u}{t\nu(t)} < \frac{\zeta_*}{\nu^{1/2}}
\right\}. \end{align} In this section, we find the approximate solutions for $v$ and $\bar w$ in the tip region \footnote{We use the same notation $V$ and $V^\pm$ here for different functions than the barriers in Section \ref{section:productish}. In the following section, where we need to refer to both the functions defined here and the functions from Section \ref{section:productish}, we will use e.g. $V_{tip}$ for the function defined here and $V_{prish}$ for the function defined there.} : \begin{align}
V \mathrel{\mathop:}= V_{{Bry}}(\sigma) + \beta V_{{Pert}}(\sigma), \\
\bar W \mathrel{\mathop:}= 1 + (\log \omega)_{\theta} W_{{Pert}}(\sigma), \end{align} where $\beta = \alpha'$, $(\log \omega)_{\theta} = \partial_{\theta}(\log \omega)$, and $V_{{Bry}}$, $V_{{Pert}}$, and $W_{{Pert}}$ are functions which are to be defined. In Lemmas \ref{vsupsoln} and \ref{wsupsoln} we define functions $V^{\pm}$ and $W^{\pm}$, which satisfy $V^- < V < V^+$ and $\bar W^- < \bar W < \bar W^+$, and will serve as barriers for $v$ and $\bar w$. These functions depend on constants $\epsilon_v$, $\epsilon_w$, and $\delta$. The barriers $V^-$ and $V^+$ are carefully defined so that if $V^- < v < V^+$ then $\tilde L = \sigma^{-1}(1 - \on4 v)$ is bounded near $\sigma = 0$.
Here we make definitions similar to Definition \ref{productish_barricaded}. We use the notation $x^{a, b} = x^a(1 + x)^{b-a}$; which is approximately $x^a$ near $x = 0$ and $x^b$ near $x = \infty$. \begin{definition}\label{tip_barricaded}
We say that $g(t)$ is \emph{barricaded (by the tip barriers)} at a point in space-time if it satisfies
\begin{align}
V^- < v < V^+, \quad W^- < w < W^+ \label{tip_barrier_ineq}
\end{align}
at that point.
We say that $g(t)$ is \emph{initially controlled in the tip region}
if at $t = T_1$
for all points satisfying $\sigma \leq 2\zeta_* \nu^{-1/2}(T_1)$
it is barricaded,
and
\begin{align}
\twoeta{v-V}{1}{ (d\sigma^{1/2, 1})^2}
&\leq c_{safe}C_{reg}\left( \delta^{-1}\epsilon_{v} \right)\nu^{1/2}\sigma^{1, -1},
\label{v_reg}\\
\twoeta{w-W}{1}{ (d\sigma^{1/2, 1})^2}
&\leq c_{safe}C_{reg}\epsilon_{w}\nu^{1/2}.
\end{align}
We say that $g(t)$ is \emph{barricaded at the right of the tip region} if it is barricaded for all points satisfying $\zeta_* \nu^{-1/2} < \sigma < 2 \zeta_* \nu^{-1/2}$.
We say that $g(t)$ is \emph{controlled in the tip region} if
\begin{enumerate}[label=(T\arabic{*}), ref=(T\arabic{*})]
\item\label{conc:tip_barrier}
For all points in $\Omega_{tip}$, the solution is barricaded.
\item\label{conc:tip_reg_noncompact}
For all points in $\Omega_{tip}$ with $\sigma \geq 1$ and
\begin{align}
\twoeta{v-V}{1}{ (d\sigma^{1/2, 1})^2}
&\leq C_{reg}\left( \delta^{-1}\epsilon_{v} \right)\nu^{1/2}\sigma^{1, -1},\\
\twoeta{w-W}{1}{ (d\sigma^{1/2, 1})^2}
&\leq C_{reg}\epsilon_{w}\nu^{1/2}.
\end{align}
\end{enumerate} \end{definition} \begin{remark}
$V$ satisfies $(1/C)\sigma^{0,-1} < V < C\sigma^{0,-1}$ and $V^+ - V^-$ satisfies $V^+ - V^- \leq C \sigma^{-1, 0}$. This is the reason for the factor $\sigma^{1, -1}$ in \eqref{v_reg}. To understand $(d \sigma^{1/2,1})^2$, know that for the Bryant soliton, $d(\sigma^{1/2, 1})$ is uniformly comparable to the arc length element- the radius $\sqrt{\sigma}$ grows like distance near $\sigma = 0$ and grows like the square root of distance as $\sigma \to \infty$. So, this metric is comparable to the Bryant soliton metric. Furthermore, $(\delta^{-1} \epsilon_v)$ controls the separation between the barriers for $v$, whereas $\epsilon_w$ controls the separation between the barriers for $w$- this explains is the reason for the appearance of those constants. \end{remark}
The following is the main result of this section.
\begin{lemma}\label{main_tip_estimates}
There is a $c_{safe}$ such that if we let $C_{reg} > \berr C_{reg}$, $\epsilon_v$, $\epsilon_w < \barr \epsilon_w(\epsilon_v)$, $\zeta_*$, and $\delta < \barr \delta(\zeta_*)$, there is a $T_*$ depending on all other constants with the following property.
Suppose $0 < T_1 < T_2 < T_*$ and $g(t)$ is defined on $[T_1, T_2]$, initially controlled in the tip region, and barricaded at the right of the tip region. Then $g(t)$ is controlled in the tip region. \end{lemma}
\subsection{A summary of functions}\label{function_summary} We will be introducing many functions of $\sigma$. Here, we provide the reader with a little cheat sheet to recall the asymptotics of the functions. This makes us feel better about possibly using the asymptotics without warning.
We use the notation $\sigma^{a,b} = \sigma^a (1 + \sigma)^{b-a}$ and $|F|_3 = F + \sigma \partial_\sigma F + \sigma^2 \partial_\sigma^2 F + \sigma^3 \partial_\sigma^3 F$. As usual, $c < C$ are constants depending only on the given model pinch. We use $F \sim G$ if $c G < F < CG$.
We have \begin{align}
V_{{Bry}} \sim \sigma^{0, -1},
&\quad V_{{Pert}} \sim \sigma^{1, 0},
\quad W_{{Pert}} \sim \sigma^{0, 1}, \\
|V_{{Bry}}|_3 \leq C\sigma^{0, -1},
&\quad |V_{{Pert}}|_3 \leq C\sigma^{1, 0},
\quad |W_{{Pert}}|_3 \leq C\sigma^{0, 1}. \end{align} More precisely as $\sigma \to \infty$ we have \begin{align}
V_{{Bry}} = \mu \sigma^{-1} + O(\sigma^{-2}),
\quad V_{{Pert}} = \oh + O(\sigma^{-1}),
\quad W_{{Pert}} = \oh \mu \sigma + O(\log \sigma). \end{align} Our approximate solutions are $V = V_{{Bry}} + \beta V_{{Pert}}$ and $W = 1 + (\log \omega)_{\theta} W_{{Pert}}$. Here are crude bounds on our barriers: for $\nu^{1/2}\sigma < \zeta_*$ \begin{align}
\oh V_{{Bry}} &< V^- < V < V^+ < 2 V_{{Bry}} \\
\oh &< \bar W^- < \bar W < \bar W^+ < 2. \end{align} More precise bounds are given in Lemma \ref{barrier_order_lemma}: \begin{align}
V^+ - V \sim
\left(\delta^{-1}\epsilon_v\right) \nu^{1/2}\sigma^{1,-1},
\quad
W^+ - W \sim \epsilon_{w} \nu^{1/2}, \end{align} and similarly for $V-V^-$ and $W-W^-$.
We also have the following facts about the functions of time $\nu(t) = V_0(\mu t)$, $\omega(t) = W_0(\mu t)$, and $\alpha(t) = t \nu(t)$. We define $\dd{k}{f}(t) = t^k \partial_t^k f/f$, which is bounded for $f = \nu, \omega$ and $k = 1, 2$ by the model pinch assumption \ref{modelpinch_reg}. The following are straightforward calculations. \begin{align}
\beta \mathrel{\mathop:}= \partial_t \alpha = (1 + \dd1 \nu)\nu,
&\quad
\partial_{\theta} \beta = O(\nu^2), \label{beta_bounds}\\
\partial_{\theta} \log \omega = \dd1 \omega \nu,
&\quad
\partial_{\theta}^2 \log \omega = O(\nu^2).\label{omega_bounds} \end{align} \subsection{Type-II rescaling}
Note that $v = u^{-1}|\nabla u|^2_{g} = \sigma^{-1}|\nabla \sigma|^2_{\tilde g}$. Also,
$\tilde \grad \sigma = \grad u$ and $|\nabla \sigma|_{\tilde g}^2 = \alpha |\nabla u|_{g}^2$ so $\partial_u = \alpha^{-1} \partial_\sigma$. So, \begin{align}
\label{eq:85}
\partial_{t;u}
&= \partial_t - (\partial_t u)\partial_u \\
&= \alpha^{-1} \partial_\theta - (\partial_t (\alpha \sigma))(\alpha^{-1}\partial_\sigma) \\
&= \alpha^{-1} \left( \partial_{\theta;\sigma} - \beta \sigma \partial_\sigma \right). \label{partialt_rewrite} \end{align}
We define $\mathcal{Q}_{\sigma}$ and $\mathcal{L}_{\sigma}$ to be $\mathcal{Q}$ and $\mathcal{L}$, from \eqref{V_evo}, with $\partial_u$ replaced with $\partial_\sigma$. Then using our equation for $\partial_{t;u}v$, \eqref{V_evo}, we find \begin{align}
\label{v_evo_theta}
\partial_{\theta;\sigma}v &= \sigma^{-1}\mathcal{Q}_{\sigma}[v,v] + \sigma^{-1}\mathcal{L}_{\sigma}[v] + \beta \sigma \partial_\sigma v - 2 \tilde \kappa^2v \end{align} where $\tilde \kappa = \of \dim(F) \tilde w^{-1}y$. Let \begin{align}
\label{eq:125}
\mathcal{F}_{\sigma}[v, \tilde \kappa] = \left( \sigma^{-1}\mathcal{Q}_\sigma[v,v] + \sigma^{-1}\mathcal{L}_\sigma[v] - 2 \tilde \kappa^2 v \right). \end{align} So \eqref{v_evo_theta} is $\partial_{\theta;\sigma}v - \mathcal{F}_{\sigma}[v, \tilde \kappa] - \beta \sigma \partial_\sigma v= 0$.
\subsection{The Bryant Soliton} The Bryant soliton $({Bry}, g_{{Bry}}, f_{{Bry}})$ is a steady gradient Ricci soliton on the topology ${Bry} = \mathbb{R}^{q+1}$. The metric may be written as \begin{align}
\label{eq:58}
g_{{Bry}} = \frac{d\sigma^2}{\sigma V_{Bry}(\sigma)} + \sigma g_{S^q}. \end{align}
As a steady soliton, under Ricci flow it moves only by diffeomorphisms, which fix the warped product structure. The value of $v = \sigma^{-1}|\nabla \sigma|^2$ at a point where $\sigma = \sigma_*$ is a diffeomorphism-invariant property, so $\partial_{t;\sigma}v_{{Bry}} = 0$. Therefore, \begin{align}
\label{eq:60}
\mathcal{Q}[V_{Bry}, V_{Bry}] + \mathcal{L}[V_{Bry}] = 0. \end{align}
The Bryant soliton has strictly positive sectional curvature, and its scalar curvature has a maximum at $\sigma = 0$. The soliton is defined up to scaling and diffeomorphism, so let's say we have chosen the scaling with maximum scalar curvature $\mu$. As $\sigma \to \infty$, $V_{{Bry}}$ has the asymptotics \begin{align}\label{vbry_infty_asymptotics}
V_{{Bry}}(\sigma) = (1 + O(\sigma^{-1})) \mu \sigma^{-1} \end{align} and as $\sigma \to 0$, $V_{{Bry}}$ has the asymptotics \begin{align}\label{vbry_zero_asymptotics}
V_{{Bry}}(\sigma) = 4 \left( 1 - \frac{\mu}{q(q-1)}\sigma + o(\sigma) \right). \end{align}
For any $k > 0 $ we may scale the metric by $k^{-1}$, resulting in the Bryant soliton with maximum scalar curvature $k \mu$. The corresponding function $V_{k {Bry}}$ is related by $V_{k{Bry}}(\sigma) = V_{{Bry}}(k \sigma)$.
\subsection{Approximation for $v$}
Suppose that $v$ satisfies \eqref{v_evo_theta}, and also converges sufficiently smoothly to a limit $v_0$ as $\theta \searrow -\infty$. Suppose also that $\tilde \kappa^2$ converges to zero as $\theta \searrow 0$. Then we learn, $
\mathcal{Q}_{\sigma}[v_0, v_0] + \mathcal{L}_\sigma[v_0] = 0. $ That is, $v_0$ describes a steady soliton. If the limit metric has $\sigma \in [0, \infty)$, and has nonzero curvature, then we learn that as a function of $u$, $v_0 = V_{k {Bry}}(u)$ for some scaling factor $k$. Comparing the asymptotics \eqref{vbry_infty_asymptotics} with our approximate solution in the parabolic region \eqref{V_expr_smallrho}, we choose $k = 1$.
Now we address the term $\beta \sigma v_\sigma$. This term suggests that our approximation $v \approx v_0$ for small $\theta$ is off by a term of order $\beta$. Write $\tilde v(\sigma,\theta) = v_0(\sigma) + \beta v_1(\sigma)$, and plug into $\partial_{\theta;\sigma} v - \mathcal{F}_\sigma[v, \kappa]-\beta \sigma \partial_\sigma v = 0$. This gives us, \begin{align}
\label{eq:73}
\partial_{\theta;\sigma} v - \mathcal{F}_\sigma[v, \kappa]-\beta \sigma \partial_\sigma v
&= \beta_{\theta} v_1 \\
&- \beta \left( 2 \sigma^{-1}\mathcal{Q}[v_0, v_1] + \sigma^{-1}\mathcal{L}[v_1] + \sigma \partial_\sigma v_0 \right) \\
&+ \left( \dots \right). \end{align} Here the term $(\dots )$ is bounded by \begin{align}
\label{eq:104}
|\dots|
&\leq C\beta^2 \left( \sigma^{-1}|v_1|^2_2 + \beta^{-2}\tilde \kappa^2 |v_0| + \beta^{-1}\tilde \kappa^2 |v_1| \right). \end{align} We also have $\beta_{\theta} = O(\nu^2)$. (See the end of Section \ref{function_summary}.) So, \begin{align}
\label{fop_calcone}
\partial_{\theta;\sigma} v - \mathcal{F}_\sigma[v, \kappa]-\beta \sigma \partial_\sigma v
&=- \beta \left( 2 \sigma^{-1}\mathcal{Q}[v_0, v_1] + \sigma^{-1}\mathcal{L}[v_1] + \sigma \partial_\sigma v_0 \right) \\
&+ \beta^2 E \end{align} where $
E \leq C \left( |v_1| + \sigma^{-1}|v_1|^2_3 + \beta^{-2}\tilde \kappa^2 |v_0| + \beta^{-1}\tilde \kappa^2 |v_1| \right) $. Concerning the equation approximately satisfied by $v_{1}$, we have the following lemma, which is Lemma 4 of \cite{ACK}. Recall $\sigma^{a,b} = \sigma^{a}(1 + \sigma)^{b-a}$. \begin{lemma}\label{pert_fun}
There is a solution $V_{Pert}$ to
\begin{align}
\label{eq:94}
2 \sigma^{-1}\mathcal{Q}_\sigma[V_{{Bry}}, V_{Pert}] + \sigma^{-1}\mathcal{L}[V_{Pert}] = -\sigma \partial_\sigma V_{{Bry}}.
\end{align}
on $[0, \infty)$, which extends to a smooth even function on $(-\infty, \infty)$.
The function $V_{k Pert}(\sigma) = k^{-1}V_{Pert}(k \sigma)$ is a solution to
\begin{align}
\label{eqn_V_pert}
2 \sigma^{-1}\mathcal{Q}_\sigma[V_{k{Bry}}, V_{k{Pert}}] + \sigma^{-1}\mathcal{L}[V_{k{Pert}}] = -\sigma \partial_\sigma V_{k {Bry}}.
\end{align}
As $\sigma \to \infty$, $V_{k {Pert}}$ has the asymptotics
\begin{align}
\label{Vpert_asymptotics}
V_{k {Pert}} = (1 + O(\sigma^{-1})) k^{-1}.
\end{align}
There is a $C > 0$ depending on the dimension such that
\begin{align}
\label{vpert_bnd}
|V_{{Pert}}|_2 < C \sigma^{1,0}
\end{align} \end{lemma}
This invites the choice of approximate solution \begin{align}
V = V_{Bry} + \beta V_{Pert}. \end{align}
\subsection{Approximation for $w$}\label{rescale_w} The expression for our approximation in the productish region \eqref{Wprish_asymptotic_exp} suggests that, in the tip region, $\bar w = \omega^{-1}(w + \mu_F t)$ is approximately constantly $1$. This is a complete solution to the equation satisfied by $\bar w$ (the term $\mu_F t$ takes care of the reaction part of the equation). We derive an equation for $\bar w$, to find the next order term. We can come from the evolution of $\hat w = w + \mu_F t$ in terms of $u$, \eqref{evo_w_in_u}: \begin{align}
\partial_{t;u} \hat w
&= u v \partial_u^2 \hat w - y + \mu \partial_u \hat w - \mu/2 v\partial_u \hat w. \end{align} Multiplying by $\omega^{-1}\alpha$, we have \begin{align}
\alpha \partial_{t;u} \bar w
&= \sigma^{-1}\mathcal{R}[\bar w, v] - (\alpha \omega^{-1})y - (\log \omega)_{\theta} \bar w, \end{align} where \begin{align}
\mathcal{R}[z,v]
= \sigma^2 v \partial_\sigma^2 z + (\mu - (c_v-\oh q)v)\sigma \partial_\sigma z. \end{align} Then using \eqref{partialt_rewrite}, \begin{align}
\partial_{\theta; \sigma} \bar w
&= \sigma^{-1}\mathcal{R}[\bar w, v] - (\alpha \omega^{-1})y - (\log \omega)_{\theta} \bar w - \beta \sigma \partial_\sigma \bar w \\
&= \sigma^{-1}\mathcal{R}[\bar w, v]
- \frac{1}{\bar w - \mu_F \frac{t}{\omega}}v \sigma (\partial_\sigma \bar w)^2
- (\log \omega)_{\theta} \bar w - \beta \sigma \partial_\sigma \bar w. \label{evo_w_in_sigma} \end{align}
Concerning the operator $\mathcal{R}$, we have the following Lemma. \begin{lemma}\label{wpert_exist}
There is a solution $W_{{Pert}}(\sigma)$ to
\begin{align}
\sigma^{-1}\mathcal{R}[W_{{Pert}}, V_{{Bry}}] = 1
\end{align}
which extends to a smooth even function on $(-\infty, \infty)$.
The function $W_{k{Pert}}(\sigma) = W_{{Pert}}(k \sigma)$ is a solution to
\begin{align}
\sigma^{-1}\mathcal{R}[W_{k{Pert}}, V_{k{Bry}}] = 1
\end{align}
As $\sigma \to \infty$, $W_{k{Pert}}$ has the asymptotics
\begin{align}\label{Wpert_asymptotics}
W_{kPert} = (1 + o(1))\oh \mu k\sigma
\end{align} \end{lemma} \begin{proof}
The main idea is that $W_{{Pert}}$ is just a scaling of the gradient potential function $f$. On any steady soliton, the gradient potential function satisfies $\Delta_X f = 1 $ where $\Delta_X$ is the drift laplacian. The operator $\sigma^{-1}\mathcal{R}$ is a recasting of the laplacian in these coordinates. For the derivation see page \pageref{wpert_exist_continuation}. \end{proof}
This suggests the approximate solution $\bar W = 1 + (\log \omega)_{\theta}W_{Pert}$. To find this, plug in $\bar w(\sigma, t) = 1 + \bar w_1(\sigma, t)$ as an initial approximation and assume that $\bar w_1$ goes to zero as $t \searrow 0$. Then taking the highest order terms in the limit $t \searrow 0$ we are left with the equation \begin{align}
- \left( \sigma^{-1} \mathcal{R}[\bar w_1, V_{Bry}] - (\log \omega)_{\theta}\cdot 1 \right) \end{align} for $\bar w_1$. Note $\log (\omega)_{\theta} = O(\nu)$ (see the end of Section \ref{function_summary}).
\subsection{$y$ control} One tricky term which appears in the evolution of $v$, \eqref{v_evo_theta}, is $\tilde \kappa^2 = \on4 \dim(F) w^{-1} y$. This cannot be controlled with simple barrier arguments: at a point where $w$ is trapped between barriers for $w$, and $v$ touches barriers for $v$, we only know that the derivative $v$ matches the derivative of the barrier for $v$, but we do not get a free bound on the derivative of $w$. For this reason we need to use regularity.
\begin{lemma}\label{lemma:y_control_tip}
Suppose we are in the setting of Lemma \ref{main_tip_estimates}, with \ref{conc:tip_barrier} and \ref{conc:tip_reg_noncompact} for $t \in [T_1, T_2)$. Then, if $T_*$ is sufficiently small,
\begin{align}
\tilde \kappa^2 \leq C C_{reg}^2 \epsilon_{w}^2 \sigma^{1,0} \nu
\end{align} \end{lemma} \begin{proof} We rewrite $\tilde \kappa^2$ as, \begin{align}
\tilde \kappa^2
&= C \tilde w^{-1} y
= C v \frac{|\nabla \tilde w|^2_{\tilde g}}{\tilde w^2}\frac{1}{|\nabla \tilde \phi|_{\tilde g}^2} = C v\frac{1}{\tilde w^2} \left( \partial_{\tilde \phi} \tilde w \right)^2 \label{kappa_rewrite} \end{align}
Here, we used that $y$ and $v$ are scale-invariant, and that $v = \on4 |\nabla \tilde \phi|^2_{\tilde g}$. Using the assumption \ref{w_big} on $W_0$, $\omega(t) > (1 + c)\mu t$ (whether $\mu$ is positive or not), so we have \begin{align}
\tilde \kappa^2
&= C v\left( \frac{1}{\bar w - \mu_F \frac{t}{\omega}} \right)^2 \left( \partial_{\tilde \phi} \bar w \right)^2
\leq C v\left( \frac{1}{\bar w - \frac{1}{1+c}} \right)^2 \left( \partial_{\tilde \phi} \bar w \right)^2. \end{align} In the region under consideration, we can take $T_*$ small enough so that $\bar W^- > 1 - \frac{1}{2}\frac{c}{1 + c}$. Therefore since $\bar w > \bar W^-$, increasing $C$, \begin{align}
\tilde \kappa^2
&\leq C v \left( \partial_{\tilde \phi} \bar w \right)^2.\label{basic_tildekappa_ineq} \end{align}
To control $\kappa^2$ in $\{\sigma < 1\}$, we need to use that $\partial_{\tilde \phi} \bar w = 0$ at $\tilde \phi = 0$. Copying \ref{conc:tip_reg_noncompact} for $k = 2$, and writing it in terms of $\tilde \phi$, \begin{align}
\partial_{\tilde \phi}^2 \bar w < \partial_{\tilde \phi}^2 \bar W + 2C_{reg} \epsilon_{w} \nu^{1/2}. \end{align} Since $\partial_{\tilde \phi}^2 \bar W \leq C \nu$, for sufficiently small times the first term dominates, and $\partial_{\tilde \phi}^2 \bar w \leq 4 C_{reg} \epsilon_w \nu^{1/2}$. We can integrate this from $\phi = 0$ to find $
\partial_{\tilde \phi} \bar w < 4C_{reg} \epsilon_{w} \nu^{1/2}\tilde \phi. $ Then \eqref{basic_tildekappa_ineq} proves the claim for $\sigma < 1$.
To control $\kappa^2$ in $\{\sigma > 1\}$, we use \ref{conc:tip_reg_noncompact}: \begin{align}
\partial_{\sigma} \bar w \leq \partial_\sigma \bar W + C_{reg} \epsilon_{w}\nu^{1/2}. \end{align} We have $\partial_\sigma \bar W \leq C \nu$ since $\partial_\sigma W_{pert}$ is bounded. So for small times $\partial_\sigma \bar w \leq 2C_{reg} \epsilon_w \nu^{1/2}$. In terms of $\phi$ this says $\partial_\phi \bar w \leq 4C_{reg} \epsilon_w \phi \nu^{1/2}$. Since $v \leq V^+ \leq C \sigma^{0, -1}$, \eqref{basic_tildekappa_ineq} gives us $\tilde \kappa^2 \leq C C_{reg}^2 \epsilon_w^2 \nu$. \end{proof}
\subsection{Barriers}\label{tip_barriers_section} In this section we define the barriers $V^{\pm}$ and $\bar W^{\pm}$ and prove that item \ref{conc:tip_barrier} continues to hold. The barriers are defined as follows. Let $k(t)^{\pm} = 1 \mp \delta^{-1}\epsilon_{v} \nu^{1/2}$ and then set \begin{align}
V^{\pm} &= V_{k^{\pm}(t) {Bry}}
+ \left(\beta \mp \epsilon_v \nu\right) V_{k^{\pm}(t){Pert}} \label{v_barriers_def}\\
\bar W^{\pm} &= 1 \pm \epsilon_{w}\nu^{1/2}
+ \left( (\log \omega)_{\theta} \mp \delta \epsilon_w \nu \right)W_{{Pert}} .\label{w_barriers_def} \end{align} We will prove that these are sub- and supersolutions to the equations satisfied by $\bar w$ and $v$. The power $\nu^{1/2}$ is a bit mysterious here, but it is the best possible for barriers of this form. We discuss its derivation after Lemma \ref{buckling_lemma}. It is helpful to remember that $\beta \sim \nu$, and $(\log \omega)_{\theta} \lesssim \nu$ (see the end of Section \ref{function_summary}).
The terms $- \epsilon_v \nu V_{k^{\pm}{Pert}}$ and $- \nu \delta \epsilon_w W_{{Pert}}$ are the terms which will give us that $V^+$ and $W^+$ are strict supersolutions to their equations. They are chosen by taking the approximate solution, which is found by starting from a limit at $t = 0$ and adding a perturbation which solves an elliptic equation, and then fiddling with the size of the perturbation.
Because the extra amount of the perturbation needed for a supersolution comes with a negative sign in both cases, we need to add something else to ensure that the supersolution lies above the intended approximate solution. This is the role of $k^{\pm}(t)$ and of $\pm\epsilon_w \nu^{1/2}$. (If it's not clear what's going on with $k^+$, recall that $V_{{Bry}}$ is decreasing so $V_{k^+{Bry}}(\sigma) = V_{{Bry}}(k^+\sigma) > V_{{Bry}}(\sigma)$.) The role of $\delta$ in both equations is to control the ratio of the extra positive term used to make the supersolution bigger than the approximate solution, to the extra negative term used to make the supersolution a supersolution to the equation.
Lemma \ref{barrier_order_lemma} clarifies the role of $\delta$. Recall $\sigma^{a,b} = \sigma^a (1 + \sigma)^{b-a}$. The significance of the factor $\sigma^{1, -1}$ in the inequalities for $V$ in this lemma is the following. At infinity, $V \sim \sigma$ so this is a normalization. At $0$, $V^+ - V^- \sim \sigma$ is necessary to ensure smoothness of a solution with $V$ trapped between $V^-$ and $V^+$. On the other hand $\bar W \sim 1$ everywhere so $\bar W$ requires no normalization. \begin{lemma}\label{barrier_order_lemma}
There are constants $c < C$ depending only on the dimension such that the following holds.
We have, for $V_{diff} = V^+ - V$ or $V_{diff} = V - V^-$,
\begin{align}
c \delta^{-1}\epsilon_{v} \nu^{1/2} \sigma^{1, -1} \left(1 - C\delta \nu^{1/2}\sigma^{0,1} \right)
\leq V_{diff}
\leq C \delta^{-1}\epsilon_{v} \nu^{1/2} \sigma^{1,-1}.
\end{align}
Similarly, for $W_{diff} = \bar W^+ - \bar W$ or $W_{diff} = W - W^-$,
\begin{align}
c \epsilon_{w} \nu^{1/2}\left( 1 - C\delta \nu^{1/2}\sigma^{0, 1} \right)
\leq W_{diff}
\leq C \epsilon_{w} \nu^{1/2}.
\end{align}
In particular, if we choose $\delta < \frac{1}{2C}\zeta_*^{-1}$ then, renaming $c$, for all $\sigma < \zeta_* \nu^{-1/2}$
\begin{align}
c \delta^{-1}\epsilon_{v} \nu^{1/2} \sigma^{1, -1}
&\leq V_{diff}
\leq C \delta^{-1}\epsilon_{v} \nu^{1/2} \sigma^{1,-1},\\
c \epsilon_{w} \nu^{1/2}
&\leq \bar W_{diff}
\leq C \epsilon_{w} \nu^{1/2}.
\end{align} \end{lemma} \begin{proof}
The asymptotics of the $V_{Bry}$ are given in \eqref{vbry_infty_asymptotics} and \eqref{vbry_zero_asymptotics}. Also recall that
$V_{kBry}(\sigma) = V_{Bry}(k\sigma)$.
Using these asymptotics, for small enough $\sigma$,
\begin{align}
c \delta^{-1}\epsilon_v \nu^{1/2} \sigma
< V_{k^+Bry}(\sigma) - V_{Bry}(\sigma)
< C \delta^{-1}\epsilon_v \nu^{1/2}\sigma,
\end{align}
and for large enough $\sigma$,
\begin{align}
c \delta^{-1}\epsilon_v \nu^{1/2}\sigma^{-1}
< V_{k^+Bry}(\sigma) - V_{Bry}(\sigma)
< C \delta^{-1}\epsilon_v \nu^{1/2}\sigma^{-1}.
\end{align}
Furthermore, since $V_{Bry}$ is strictly decreasing in any compact set away from the origin, for any $\sigma_1 < \sigma_2$ there are constants $c_{\sigma_1, \sigma_2}$ and $C_{\sigma_1, \sigma_2}$ such that
\begin{align}
c_{\sigma_1, \sigma_2} \delta^{-1}\epsilon_v \nu^{1/2}
< V_{k^+Bry}(\sigma) - V_{Bry}(\sigma)
< C_{\sigma_1, \sigma_2} \delta^{-1}\epsilon_v \nu^{1/2}.
\end{align}
These three sets of bounds prove that for some $c < C$,
\begin{align}
c \delta^{-1}\epsilon_v \nu^{1/2} \sigma^{1, -1}
< V_{k^+Bry}(\sigma) - V_{Bry}(\sigma)
< C \delta^{-1}\epsilon_v \nu^{1/2} \sigma^{1,-1}.
\end{align}
Putting this together with the bound on $V_{Pert}$, \eqref{vpert_bnd}, which says
\begin{align}
- \epsilon_v \nu V_{pert}(\sigma) > -C \epsilon_v\nu \sigma^{1,0},
\end{align}
and using $\beta \leq C \nu$ (from \eqref{beta_bounds}), we have the claim for $V$.
The proof for $\bar W$ is similar but more straightforward. One needs to use the properties from Lemma \ref{wpert_exist}. \end{proof}
We now prove that $V^{\pm}$ and $\bar W^{\pm}$ are sub- and supersolutions to the equations satisfied by $v$ and $\bar w$. In Lemma \ref{lemma:barrier_wrapup} we will summarize by saying that item \ref{conc:tip_barrier} continues to hold.
\begin{lemma}\label{vsupsoln}
Let
$\zeta_* > 0$, $\epsilon_v > 0$, and $\delta > 0$
be given. Let $V^{\pm}$ be the functions defined in \eqref{v_barriers_def}.
Suppose $\tilde \kappa = \frac{y}{\tilde w} = \alpha \frac{y}{w}$ satisfies $\tilde \kappa^2(\sigma, t) \leq c_{ytip}\epsilon_v \nu \sigma^{1,0}$ where $c_{ytip}$ is a constant (chosen in the proof) depending only on dimensions.
Then there is a $T_*$ depending on all parameters so that for $t < T_*$ and $\sigma < \zeta_* \nu^{-\oh}$ we have, for a constant $c$ depending only on the dimensions,
\begin{align}
\partial_{\theta; \sigma}V^+ - \mathcal{F}_{\sigma}[ V^+, \tilde \kappa] - \beta \sigma \partial_\sigma V^+ \geq c \epsilon_v \nu \sigma^{1,-1} \label{ineq_vplus}
\end{align}
and
\begin{align}
\label{eq:119}
\partial_{\theta; \sigma}V^- - \mathcal{F}_{\sigma}[ V^-, \tilde \kappa] - \beta \sigma \partial_\sigma V^- \leq - c \epsilon_v \nu \sigma^{1,-1}.
\end{align} \end{lemma}
\begin{proof}
Let us first demonstrate the main calculation, implicitly defining error terms $E_1$ and $E_2$. Calculate
\begin{align}
- \mathcal{F}_{\sigma}[V^+, 0]
&= - \left( \sigma^{-1} \mathcal{Q}[V_{k^+{Bry}}, V_{k^+{Bry}} ] + \sigma^{-1}\mathcal{L}[V_{k^+{Bry}}] \right)\\
&- (\beta - \epsilon_v \nu)
\left(
2\sigma^{-1}\mathcal{Q}[V_{k^+{Bry}}, V_{k^+{Pert}}] + \sigma^{-1} \mathcal{L}[V_{k^+{Bry}}]
\right) \\
&- (\beta - \epsilon_v \nu)^2 \sigma^{-1}\mathcal{Q}[V_{k^+{Pert}}, V_{k^+{Pert}}].
\end{align}
The first line vanishes, and the second line can be computed from the equation solved by $V_{k^+{Pert}}$. The last line is error.
\begin{align}
-\mathcal{F}_{\sigma}[V^+,0]
= +(\beta - \epsilon_v\nu) \sigma \partial_\sigma V_{k_+{Bry}} + E_1
\end{align}
Also calculate,
\begin{align}
- \beta \sigma \partial_\sigma V^+
&= - \beta \sigma \partial_\sigma V_{k^+{Bry}}
- (\beta -\epsilon_v\nu)\beta \sigma \partial_\sigma V_{k^+{Pert}} \\
&= - \beta \sigma \partial_\sigma V_{k^+{Bry}} + E_2
\end{align}
Putting these together,
\begin{align}
- \mathcal{F}_{\sigma}[V^+, 0] - \beta \sigma \partial_\sigma V^+
&=
-\epsilon_v \beta \sigma \partial_\sigma V_{k^+{Bry}} + C (\beta \sigma^{1,-1}) \left( \beta \sigma^{0,1}\right) \\
&\geq c \epsilon_v \nu \sigma^{1,-1} + E_1 + E_2
\end{align}
where we used that $\sigma \partial_\sigma V_{k^+ {Bry}} \leq -c\sigma^{1,-1}$ for some $c$. Therefore it remains to bound $E_1$ and $E_2$, as well as the other terms in \eqref{ineq_vplus}, namely
\begin{align}
\partial_{\theta; \sigma}V^+ \quad\text{and}\quad \mathcal{F}_\sigma[V^+, \tilde \kappa] - \mathcal{F}_\sigma[V^+,0] = \tilde \kappa^2 V^+.
\end{align}
For the following, note we can assume that $k(t)$ is in $[1/2,2]$. Using $\beta \sim \nu$, and using the notation $|f|_2 = |f| + \sigma |\partial_\sigma f| + \sigma^2 |\partial_\sigma^2 f|$, we have the bound on $E_1$ and $E_2$,
\begin{align}
|E_1| + |E_2|
&= \left|(\beta-\epsilon\nu)^2 \sigma^{-1}\mathcal{Q} [V_{k^+{Pert}}, V_{k^+{Pert}}]\right|
+
\left|
(\beta - \epsilon \nu)
\beta \sigma \partial_\sigma V_{k^+{Pert}}
\right|
\\
&\leq C\nu^2 \left( \sigma^{-1}|V_{{Pert}}|_2^2 + |V_{{Pert}}|_2 \right)\\
&\leq C\nu^2 \left( \sigma^{-1}\left( \sigma^{1,0} \right)^2 + \sigma^{1,0} \right) \leq C \nu^2 \sigma^{1,0} = C \left( \nu \sigma^{1,-1} \right) \left( \nu \sigma^{0,1} \right)
\end{align}
Now we bound the time term, using \eqref{beta_bounds}.
\begin{align}
\partial_{\theta;\sigma}V^+
&= \partial_{\theta; \sigma}
\left(
V_{{Bry}}(k(t)\sigma)
+
(1-\epsilon)\frac{\beta}{k(t)}V_{{Pert}}(k(t)\sigma)
\right) \\
&\leq C
\left(
\sigma \partial_\sigma V_{{Bry}} \partial_\theta k
+ (\partial_\theta \beta + \beta \partial_\theta k)V_{{Pert}}
+ \beta \sigma \partial_\sigma V_{{Pert}} \partial_\theta k
\right) \\
&\leq C
\left(
\sigma^{1,-1}\nu^{1+1/2}
+ (\nu^2 + \beta \nu^{1+1/2})\sigma^{1,0}
+ \beta \sigma^{1,0}\nu^{1+1/2}
\right) \\
&\leq C \nu \sigma^{1,-1}
\left(
\nu^{1/2} + \nu \sigma^{0,1}
\right)
\end{align}
Finally, we use our assumption on $\tilde \kappa$ to bound the term $\tilde\kappa^2 V^+$ by
\begin{align}
\tilde \kappa^2 V^+
&\leq C
\left(
c_{ytip}\epsilon_v \nu \sigma^{1,0}
\right)
\sigma^{0,-1}
\\
&\leq
C\nu \sigma^{1,-1}
\left( c_{ytip}\epsilon_v \right)
\end{align}
All in all, we find
\begin{align}
\partial_{\theta;\sigma} - \mathcal{F}[V^+, \tilde \kappa ] - \beta \sigma \partial_\sigma V^+
&\geq \nu \sigma^{1,-1}(c\epsilon_v - Cc_{ytip}\epsilon_v - o(1))
\end{align}
Here the term $o(1)$ goes to zero as $t \searrow 0$, in any region where $\sigma < \zeta_* \nu^{-1/2}$. The lemma follows by choosing the $c$ in the statement to be one half of the $c$ above, choosing $c_{ytip}$ to be sufficiently small, and choosing $T_*$ to be small enough so that the $o(1)$ term is sufficiently small. \end{proof}
For arbitrary functions $\bar w$ and $v$ we define \begin{align}
\mathcal{D}(\bar w, v)
\mathrel{\mathop:}=
\partial_{\theta; \sigma} \bar w
-\left(
\sigma^{-1}\mathcal{R}[\bar w, v] - \frac{1}{\bar w - \mu_F \frac{t}{\omega}}v \sigma (\partial_\sigma \bar w)^2 - \beta \sigma \partial_\sigma \bar w - (\log \omega)_{\theta} \bar w
\right). \end{align} The equation solved by $\bar w$ \eqref{evo_w_in_sigma} is therefore $\mathcal{D}(\bar w, v) = 0$.
\begin{lemma}\label{wsupsoln}
Let $\zeta_*>0$, $\epsilon_{v}>0$, $\epsilon_w>0$, and $\delta>0$ be given. Let $V^{\pm}$ and $W^{\pm}$ be the barriers defined in \eqref{v_barriers_def} and \eqref{w_barriers_def}.
There is a $T_*$ depending on all parameters such that for all $t < T_*$ and $\sigma < \zeta_* \nu^{-1/2}$ we have
\begin{align}
\mathcal{D}(\barr W^+,v)
> \oh \delta \epsilon_w \nu
\end{align}
and
\begin{align}
\mathcal{D}(\barr W^-, v)
< - \oh \delta \epsilon_w \nu
\end{align} \end{lemma}
\begin{proof}
The main idea is that
\begin{align}
(\log \omega)_{\theta} \barr W^+ - \sigma^{-1}\mathcal{R}[\barr W^+, v]
&= (1 + \epsilon_w\nu^{1/2})(\log \omega)_{\theta} \\
&+ (\log \omega)_{\theta}\left( (\log \omega)_{\theta} - \delta \epsilon_w \nu \right) W_{Pert}\\
&- (\log \omega)_{\theta}\sigma^{-1}\mathcal{R}( W_{pert}, V_{Bry}) + \delta \epsilon_w \nu \sigma^{-1}\mathcal{R}(W_{Pert}, V_{Bry}) \\
&+ ((\log \omega)_{\theta} - \delta \epsilon_w \nu)
\cdot (\sigma^{-1}\mathcal{R}(W_{pert}, V_{Bry}) - \sigma^{-1}\mathcal{R}(W_{Pert}, v))
\end{align}
We can simplify the first and third lines to find
\begin{align}
(\log \omega)_{\theta} \barr W^+ - \sigma^{-1}\mathcal{R}[\barr W^+, v]
&\geq \epsilon_w (\log \omega)_{\theta} \nu^{1/2} + \delta \epsilon_w \nu \\
&+ (\log \omega)_{\theta}\left( (\log \omega)_{\theta} - \epsilon_w \nu \right) W_{Pert}\\
&+ ((\log \omega)_{\theta} - \epsilon_w \nu)\cdot (\sigma^{-1}\mathcal{R}(W_{pert}, V_{Bry}) - \sigma^{-1}\mathcal{R}(W_{Pert}, v))
\end{align}
The first line has the correct sign, we will use it to bound the other lines and the rest of the terms. First, let's bound the other lines above:
\begin{align}
(\log \omega)_{\theta} \barr W^+ - \sigma^{-1}\mathcal{R}[\barr W^+, v]
&\geq \epsilon_w \nu \\
&- C\nu^2 \sigma^{0, 1} \\
&- C \nu (\delta^{-1}\epsilon_v)\nu^{1/2} \sigma^{1,0}
\end{align}
Here we used the bound $|V_{{Bry}} - v| < c \delta^{-1}\epsilon_v\nu^{p}\sigma^{1,-1}$ together with $|\sigma \partial_\sigma W_{Pert}| + |\sigma^2 \partial_\sigma^2W_{pert}| \leq \sigma^{0,1}$. In the second inequality we also used $(\log \omega)_{\theta} = \nu \dd1{\omega} \leq C \nu$.
Next we find the term $\partial_{\theta;\sigma}\bar W^+$. The term $\partial_{\theta; \sigma}\epsilon_w\nu^{1/2}$ has the correct sign, so we ignore it. For the other time derivatives, we can use $|\partial_{\theta}^2 (\log \omega)| + |\partial_{\theta} \nu| \leq C \nu^2$ (from \eqref{beta_bounds}, \eqref{omega_bounds}):
\begin{align}
\left| \partial_{\theta}\left( (\log \omega)_{\theta} - \epsilon_w\nu \right) W_{Pert} \right|
\leq C \nu^2 W_{pert}
\leq C \nu^{2} \sigma^{0, 1}.
\end{align}
To bound the remaining terms, note
$
|\sigma \partial_\sigma \bar W^+|
\leq \nu \sigma^{1,1}
$
and $v \leq C\sigma^{0, -1}$. Also, as in the proof of Lemma \ref{lemma:y_control_tip} we can bound
$
\frac{1}{\bar W^\pm - \mu_F t} \leq C
$.
So
\begin{align}
\left|
\frac{1}{\bar W^+ - \mu_F \frac{t}{\omega}}v \sigma (\partial_\sigma \bar W^+)^2 + \beta \sigma \partial_\sigma \bar W^+
\right|
&\leq C \left(
\sigma^{-1}v |\sigma \partial_\sigma \bar W^+|^2 +
\nu |\sigma \partial_\sigma \bar W^+|
\right) \\
&\leq C \left( \nu^2 \sigma^{1,0} + \nu^2 \sigma^{1,1} \right)
\leq C \nu \left( \nu \sigma^{1,1} \right)
\end{align}
Putting together all of the inequalities, we have
\begin{align}
\mathcal{D}(\bar W^+, v)
&\geq \nu
\left(
\delta \epsilon_w
- C \nu \sigma^{1,1}-C (\delta^{-1}\epsilon_v)\nu^{1/2} \sigma^{1,0}
\right).
\end{align}
In the space-time region under consideration,
\begin{align}
\mathcal{D}(\bar W^+, v)
&\geq \nu
\left(
\delta \epsilon_w
- C \nu^{1/2}\zeta_*-C (\delta^{-1}\epsilon_v)\nu^{1/2}
\right)
= \nu
\left( \delta \epsilon_w - C(\zeta_* + \delta^{-1}\epsilon_v) \nu^{1/2} \right).
\end{align}
For small enough $T_*$, the positive term dominates. \end{proof}
\begin{lemma}\label{lemma:barrier_wrapup}
Suppose we are in the setting of Lemma \ref{main_tip_estimates}. Suppose $\epsilon_{w} \leq \barr \epsilon_{w}(\epsilon_{v}, C_{reg})$. There is a $T_*$ depending on all parameters such that the following holds.
If items \ref{conc:tip_barrier} and \ref{conc:tip_reg_noncompact} hold for $t \in [T_1, T_2)$, then item \ref{conc:tip_barrier} holds for $t \in [T_1, T_2]$. \end{lemma} \begin{proof}
Choose $\barr \epsilon_{w}$ small enough (i.e. $\lesssim \sqrt{\epsilon_v}$) so that Lemma \ref{lemma:y_control_tip} implies that we have the desired inequality $\tilde \kappa^2 \leq c_{ytip} \epsilon_v \beta \sigma^{1, 0}$
needed to apply Lemma \ref{vsupsoln}.
Now, suppose that $v$ or $w$ touches one of its barriers at time $t = T_2$. By Lemma \ref{vsupsoln} or \ref{wsupsoln}, we get a contradiction to the maximum principle since these lemmas say that $V^\pm$ and $W^{\pm}$ are strict sub- and supersolutions to the corresponding equations. \end{proof}
\subsection{Regularity} We prove the regularity \ref{conc:tip_reg_noncompact} separately for $\sigma \geq 1$ and $\sigma \leq 1$.
\begin{lemma}\label{lemma:tip_reg_noncompact}
Suppose we are in the setting of Lemma \ref{main_tip_estimates}. Suppose $\delta < \barr \delta(\zeta_*)$ so that the conclusion of Lemma \ref{barrier_order_lemma} holds. We can choose $c_{safe}$ and $\berr C_{reg}$ depending only on the dimensions such that the following holds. Suppose item \ref{conc:tip_barrier} holds for $t \in [T_1, T_2)$. Then item \ref{conc:tip_reg_noncompact} holds for $t \in [T_1, T_2]$ and for $\sigma \geq 1.$ \end{lemma} \begin{proof} We copy equation \eqref{v_evo_theta}, using the expression \eqref{kappa_rewrite} for $\tilde \kappa^2$: \begin{align}
\partial_{\theta;\sigma}v
&= \sigma v \partial_\sigma^2 v + c_1 \sigma^{-1}v + c_2 \partial_\sigma v + c_3 \sigma^{-1}v^2 + c_4 \sigma (\partial_\sigma v)^2 \\
&+ \beta \sigma \partial_\sigma v
+ c_5 \sigma v
\left( \frac{1}{\bar w - \mu_F t/\omega } \right)^2
\left(\partial_\sigma \bar w \right)^2 v. \end{align} For $\sigma_1$ arbitrary, we multiply this by $\sigma_1$ to find, \begin{align}
\partial_{\theta;\sigma}\left( \sigma_1 v\right)
&= \left[\sigma v \right]
\partial_\sigma^2 \left(\sigma_1 v\right)
+ c_1 \left[\frac{\sigma_1}{\sigma} \right] \sigma_1^{-1} \left(\sigma_1v\right)
+ c_2 \partial_\sigma \left(\sigma_1 v \right)\\
&+ c_3 \left[ \frac{\sigma_1}{\sigma} \right] \sigma_1^{-2} \left(\sigma_1v\right)^2
+ c_4 \left[ \frac{\sigma}{\sigma_1} \right] \sigma_1^{-1}(\partial_\sigma \left(\sigma_1 v \right))^2 \\
&+ \left[ \beta \sigma \right]\partial_\sigma \left( \sigma_1 v \right)
+c_5
\left[
\frac{\sigma}{\sigma_1} (\sigma_1 v)^2
\left( \frac{1}{\bar w - \mu_F t/\omega } \right)^2
\right]
\left(\partial_\sigma \bar w \right)^2 . \end{align} We also have the equation, from \eqref{evo_w_in_sigma}, \begin{align}
\partial_{\theta; \sigma}\bar w
&= \left[ \sigma v \right] \partial_\sigma^2 w
+ \left[ c_6 - c_8v \right]\partial_\sigma \bar w\\
&- \left[ \frac{1}{\bar w - \mu_F \frac{t}{\omega}}v \right]\sigma (\partial_\sigma \bar w)^2 \\
&- (\log \omega)_{\theta} \bar w - \left[ \beta \sigma \right]\partial_\sigma \bar w. \end{align}
For $\sigma_1$ and $t_1$ arbitrary but satisfying \begin{align}
1 < \sigma_1 < \zeta_* \nu^{-1/2} \end{align} we will apply parabolic regularity to $\sigma_1 v$ and $w$ in the region \begin{align} \Xi = (\sigma, \theta) \in [\sigma_1 - 1/2, \sigma_1 + 1/2] \times [\max(\theta(t_1) - 1/2, \theta(T_1)), \theta(t_1)]. \label{xi_def} \end{align} By Lemma \ref{barrier_order_lemma}, for $\oh < \sigma < \zeta_* \beta^{-1/2}$ we have \begin{align}
\sigma V^+ - \sigma V^- &< C \delta^{-1}\epsilon_v \nu^{1/2}, \\
\bar W^+ - \bar W^- &< C \epsilon_w \nu^{1/2}. \end{align} Also, using that the solution is barricaded, the terms we have written in square brackets are smooth functions of $\sigma$, $\sigma_1 v$, and $w$ within \eqref{xi_def}, independently of the choice of $\sigma_1$ and $t_1$.
Therefore, we may apply regularity to $\sigma_1 (v - V)$ and $\bar w - \bar W$ to find \ref{conc:tip_reg_noncompact} at $(\sigma_1, t_1)$. \end{proof}
The control for $\sigma \leq 1$ requires more delicacy. This requires the knowledge that at $\phi = 0$, we have $v = \sigma^{-1}|\nabla \sigma|_{\tilde g}^2 = 4|\nabla \tilde \phi|_{\tilde g}^2 = 4$ so that as $\tilde \phi$ goes to zero the metric closes off with $\tilde \phi$ behaves likes the radius of polar coordinates near the origin. Here we pay for our choice of using the length-squared warping function (which we chose to minimize the number of square roots), it is much easier to see the equations in terms of $\tilde \phi = \sqrt{\sigma}$. Furthermore, instead of controlling $v$ it is easier to understand the evolution of $\tilde L = (1 - \on4 v)/\sigma = (1-|\nabla \tilde \phi|_{\tilde g}^2)/\tilde \phi^2$ which is a smooth function on the warped product. This is because $v$ naturally satisfies both a Neumann and Dirichlet condition at $\tilde \phi = 0$, whereas $\tilde L$ only satisfies the Neumann condition $\partial_{\tilde \phi}\tilde L = 0$ which comes from it being a rotationally symmetric function.
\begin{lemma}\label{lemma:tip_reg}
Assume that we are in the setting of Lemma \ref{main_tip_estimates}. We can choose $c_{safe}$ and $\berr C_{reg}$ depending only on the dimensions such that the following holds. Suppose additionally that item \ref{conc:tip_barrier} and \ref{conc:tip_reg_noncompact} hold for $t \in [T_1, T_2)$. Then item \ref{conc:tip_reg_noncompact} holds for $t \in [T_1, T_2]$ and for $\sigma \leq 1$. \end{lemma} \begin{proof} We can derive the evolution for $\tilde L$ from \eqref{evo_L_in_phi}.
\begin{align}
\partial_{\theta;\phi} \tilde L
&= \left(1 - \tilde\phi^2 \tilde L\right)\partial^2_{\tilde\phi} \tilde L + \oh \tilde\phi^2 (\partial_{\tilde\phi} \tilde L)^2\\
&+ \tilde\phi^{-1}(\oh \mu + 5 - \tilde\phi^2\tilde L)\partial_{\tilde\phi} \tilde L + (\mu + 2)\tilde L^2 \\
&+ c \tilde \phi^{-2} \frac{\alpha}{\omega} v
(\bar w - \mu_F t/\omega)^{-2} \left( \partial_{\tilde \phi} \bar w \right)^2\\
&+ \beta \tilde L + \on2 \beta \phi \partial_\phi \tilde L.
\end{align}
We can also derive the equation for $\bar w$ in terms of $\tilde \phi$:
\begin{align}
\partial_{\theta;\tilde \phi}\bar w
&= v \partial_{\tilde \phi}^2 \bar w - \frac{\alpha}{\omega}y
+ (\on2\mu - (\on4\mu - 1) v) \tilde \phi^{-1}\partial_{\tilde \phi} \bar w \\
&+ (\log \omega)_{\theta} \bar w + \on2 \beta \phi \partial_\phi \bar w \\
&= v \partial_{\tilde \phi}^2 \bar w
-\on4 v\frac{1}{( \bar w - \mu_F t/\omega)}(\partial_{\tilde \phi}\bar w)^2
+ (\on2\mu - (\on4\mu - 1) v) \tilde \phi^{-1}\partial_{\tilde \phi} \bar w \\
&+ (\log \omega)_{\theta} \bar w + \on2 \beta \phi \partial_\phi \bar w \\
\end{align}
Let $\tilde L_{approx} = \sigma^{-1}( 1 - \on4 V) = \tilde \phi^{-2}(1 - \on4 V)$ which is the approximation for $\tilde L$ given by the approximate solution for $V$. Our barriers tell us that, for $\sigma < 1$, we have
\begin{align}
|\tilde L- \tilde L_{approx}| < c \delta^{-1}\epsilon_{v} \nu^{1/2}, \quad |\bar w - \bar W| < c \epsilon_{w}\nu^{1/2}.
\end{align}
The terms $\tilde \phi^{-1}\partial_{\tilde \phi} \tilde L$ and $\tilde \phi^{-1} \partial_{\tilde \phi} \bar w$ appear with integer coefficients, these are not a problem if we consider $\tilde \phi$ as a radial coordinate from $\phi = 0$, then working with the second derivative $\tilde \phi^2$ they make a laplacian. Furthermore, the term $\tilde \phi^{-2} \left( \partial_{\tilde \phi} \bar w \right)^2$, which appears in the evolution of $\tilde L$, may be controlled as follows. The regularity up to time $T_2$ gives us control on the $C^{0, \eta}$ norm of this term- specifically, that $|\partial_{\tilde \phi} w|_{0, \eta}^2 \leq C_{reg}\epsilon_w \nu$. We have to be careful not to have a circular argument here: since this term with $C_{reg}$ appearing is multiplied by something that goes to zero as $t \searrow 0$, we can restrict $T_*$ depending on $C_{reg}$ and thereby bound this term independently of $C_{reg}$.
So, we can apply regularity to $\tilde L - \tilde L_{approx}$ and $\bar w - \bar W$. Rewriting $\tilde L - \tilde L_{approx} = -\on4 \tilde \phi^{-2}(v - V)$ proves the claim. \end{proof}
\subsection{Corollaries of control}\label{corollaries_tip}
The following corollary follows quickly from the control we have, by checking the curvatures of warped products. \begin{corollary}\label{tip_curvature_control}
Suppose $g_{wp}(t)$ is controlled in the tip region.
If $\mu_F = 0$, suppose $F$ has constant curvature. Then for some $C$, in the tip region,
\begin{align}
|\Rm| \leq \frac{C}{t\nu(t)}
\end{align} \end{corollary}
We now give a specific result about the convergence in tip region as $t \searrow 0$. We assume that $g(t)$ is controlled in the tip region for $t \in (0, T_2)$. For each time, the scaled warping function $\sigma = \frac{u}{t \nu(t)}$ is a function $\sigma : I \to (0, \infty)$ which we extend by the identity to a map $\sigma : M = I \times S^q \times F \to (0, \infty) \times S^q \times F$. For each $t$, $\sigma$ is a bijection if we restrict to some subset of $I$, i.e. we have an inverse \begin{align}
\sigma^{-1}: (0, \sigma_{max}(t)) \times S^q \times F \to I \times S^q \times F. \end{align} By our bounds on $v$, specifically since we keep it positive, $\sigma_{max}(t) \to \infty$ as $t \searrow 0$. We may define \begin{align}
G(t) = \frac{1}{\alpha(t)} \left( \sigma^{-1} \right)^* g(t). \end{align} As $t \searrow 0$ the domain of definition of $G$ exhausts $(0, \infty) \times S^q \times F$. Essentially, we can use $\sigma$ to find the diffeomorphisms such that neighborhoods of the tip converge to the Bryant soliton times a Euclidean factor. \begin{corollary}\label{lemma:barricaded_convergence}
Suppose that $g(t)$ is controlled in the tip region.
The (for each $t$ partially defined) metric $G(t)$, restricted to $(0,\infty) \times S^q$, converges in $C^{\infty}$ as $t \searrow 0$ to the Bryant soliton metric
\begin{align}
\frac{d\sigma_{Bry}^2}{\on4 \sigma_{Bry} v_{Bry}} + \sigma_{Bry} g_{S^q}.
\end{align}
The pullback of the vector field $(\partial_\theta \sigma)\partial_\sigma$,
\begin{align}
X(t) = \left(\sigma^{-1}\right)^* \left( (\partial_\theta \sigma )\partial_\sigma \right)
\end{align}
converges to the soliton vector field for the Bryant soliton.
Put $p = \dim(F)$. Suppose additionally that $g_{mp}$ is $\Rm$-permissible (Definition \ref{reasonable_def}). For any point $P \in (0, \infty) \times S^q \times F$ the pointed manifolds $((0,\infty) \times S^q \times F, G(t), P)$ converge, as $t \searrow 0$, to
\begin{align}
\left(
(0, \infty) \times S^q \times \mathbb{R}^p
,
\frac{d\sigma^2}{\sigma v_{Bry}} + \sigma^2 g_{S^q} + g_{\mathbb{R}^p}
,
\star
\right).
\end{align}
The target point $\star$ doesn't matter since the target manifold is homogeneous. The convergence is in the sense of pointed $C^{\infty}$ Riemannian manifolds, which allows a pullback by a time-dependent diffeomorphism. \end{corollary} \begin{proof}
The convergence to the Bryant soliton in terms of $\sigma$ happens up to some number of derivatives just because of the consequences of Lemma \ref{main_tip_estimates}. To get $C^{\infty}$ convergence, we need extra regularity, i.e. item \ref{conc:tip_reg_noncompact} for larger $k$. To get this, we use interior parabolic regularity in the same way as Lemmas \ref{lemma:tip_reg_noncompact} and \ref{lemma:tip_reg}. In this situation, we no longer need estimates on the initial data. This is because the time variable $\theta$ goes to $-\infty$ as $t \searrow 0$, so the parabolic ball $\Xi$ in \eqref{xi_def} never touches $t=0$, the initial time for $g(t)$.
Note that $\tilde g(t) = \alpha^{-1}g(t)$ satisfies
\begin{align}
\partial_{\theta} \tilde g
= -2 \Rc[\tilde g] - \beta \tilde g.
\end{align}
So $G(t)$ satisfies
\begin{align}
\partial_{\theta} G
=
-2 \Rc[G]
- \mathcal{L}_{\left( \partial_{\theta}\sigma \right) \partial\sigma} G
- \beta \tilde g.
\end{align}
As $t \searrow 0$, we have $\beta \searrow 0$, $G \to G_{Bry}$, and $\partial_{\theta}G \to 0$. This shows the convergence of $\partial_{\theta}\sigma$ to the soliton vector field.
To get the final convergence of the $wg_F$ factor to $g_{\mathbb{R}^{p}}$, note that we have
\begin{align}
w \sim \omega - \mu_F t
\end{align}
so $\alpha^{-1}w \sim \alpha^{-1}\omega \left( 1 - \mu_F t/\omega\right)$. In the case $\mu_F < 0$, this goes to $\infty$ at least as fast as $\frac{t}{\alpha} = \frac{1}{\nu}$ goes to infinity. In the case $\Lambda_F > 0$ or $\mu_F = 0$, this goes to infinity by the assumption that $g_{mp}$ is $\Rm$-permissible. \end{proof}
\section{Full flows of mollified metrics}\label{section:full_flow}
In Sections \ref{section:productish} and \ref{section:tip}, we studied the flow in two regions- the productish region and the tip region. We now want to start from one of our model pinches and create mollified initial metrics. The mollified metrics will exist for a uniform amount of time and satisfy the estimates from Lemmas \ref{main_prish_estimates} and \ref{main_tip_estimates}. We will then take a limit of the mollified flows to construct a forward evolution from the model pinch.
In the previous two sections we constructed functions, which depend on $u$ and time, and serve as barriers of the flow around approximate solutions. Let $V_{prish}$ and $W_{prish}$ be the approximate solutions constructed in Section \ref{section:productish}, and let $V^+_{prish}$, $V^-_{prish}$, $W^+_{prish}$, $W^-_{prish}$ be the functions constructed in Lemma \ref{barriers_prish}. Let $V_{tip}$ and $W_{tip}$ be the approximate solutions constructed in Section \ref{section:tip}, and let $V^+_{tip}$, $V^-_{tip}$, $W^+_{tip}$, $W^-_{tip}$ be the functions constructed in Section \ref{tip_barriers_section}. We remind the reader that $\bar w = \omega^{-1}(w - \mu_F t)$, and we decorate the barriers and approximate solutions with a bar analogously.
As a first step, the following lemma tells us how close the approximate solutions are to each other. Here, $|f|_{2,\eta}$ is the $C^{2, \eta}$ norm for the metric $(d\sigma)^2$, in the ball of radius $1$ around a given point. \begin{lemma}\label{basic_deriv_closeness}
For $\sigma < \epsilon \rho_* \nu^{-1}$,
\begin{align}
\sigma |V_{prish} - V_{tip}|_{2,\eta} &\leq C(\rho_*)
\left( \nu^2 \sigma^2 + \sigma^{-1} + \nu \right) \label{V_approx_diff}\\
|\bar W_{prish} - \bar W_{tip}|_{2,\eta} &\leq C(\rho_*)
\left( \nu^2 \sigma^2 + \nu \log \sigma \right)
\end{align} \end{lemma} \begin{proof}
We claim we can use $V_{common} = \mu \sigma^{-1}\left( 1 + (1 + \dd1{\nu})\mu^{-1}\nu \sigma \right)$ as an approximation for both $V_{prish}$ and $V_{tip}$, and similarly that we can use $\bar W_{common} = (1 + \mu \dd1{\omega} \nu \sigma)$ as an approximation for both $\bar W_{prish}$ and $\bar W_{tip}$.
For $V_{prish}$ and $W_{prish}$, the zeroth order statement follows from the approximations \eqref{V_expr_smallrho} and \eqref{Wprish_asymptotic_exp}. The higher order statements can be found similarly to how we found \eqref{V_expr_smallrho} and \eqref{Wprish_asymptotic_exp}, by estimating the Taylor expansion of the derivatives. We have,
\begin{align}
\sigma |V_{prish}-V_{common}| + |\bar W_{prish} - \bar W_{common}| \leq C \nu^2 \sigma^2.
\end{align}
For $V_{tip}$, the zeroth order statement follows from the asymptotics \eqref{vbry_infty_asymptotics} and \eqref{Vpert_asymptotics} for $V_{{Bry}}$ and $V_{{Pert}}$, and the fact that $\beta = (1 + \dd1{\nu})\nu$. For $W_{tip}$, it follows from the asymptotics \eqref{Wpert_asymptotics} for $W_{{Pert}}$. To get the higher order statements, one needs to use the analyticity of the involved functions. We have,
\begin{align}
\sigma |V_{tip}-V_{common}|_{2, \eta} \leq C(\sigma^{-1} + \nu).
|\bar W_{tip} - \bar W_{common}|_{2, \eta} \leq C \nu \log \sigma
\end{align}
(The term $\sigma^{-1}$ comes from the error in $ V_{{Bry}} \sim \mu \sigma^{-1}$, and the term $\nu$ comes from the error in $\nu V_{pert} \sim \nu$. The term $\nu \log \sigma$ comes from the error in $\nu W_{pert}\sim \nu \sigma$.) \end{proof}
\subsection{Buckling barriers}\label{section:buckling}
In this section, we prove Lemma \ref{buckling_lemma}. This shows that the barriers are ordered in a specific way: see Figure \ref{figure:barrier_switch}. The point is that this ordering means that boundary condition for the tip barriers is guaranteed by the productish barriers, and the left-hand boundary condition for the productish barriers is guaranteed by the tip barriers. We formalize this consequence in Lemma \ref{buckling_wrapup}.
\begin{lemma}\label{buckling_lemma}
Let $\epsilon_v$, $\epsilon_w$, and $\sigma_*$ be given. Assume $D > \berr D$, $\zeta > \berr \zeta_*(D, \epsilon_w)$, $\delta < \barr \delta(\zeta_*, \epsilon_v, D)$, and finally $T_*$ is chosen depending on all other parameters.
Then we have the following inequalities.
For $\zeta_*\nu^{-1/2} \leq \sigma \leq 2\zeta_*\nu^{-1/2}$,
\begin{align}
V_{tip}^+ > V_{prish}^+ &\quad V_{tip}^- < V_{prish}^-, \\
W_{tip}^+ > W_{prish}^+ &\quad W_{tip}^- < W_{prish}^-.
\end{align}
For $\oh \sigma_* \leq \sigma \leq \sigma_*$,
\begin{align}
V_{prish}^+ > V_{tip}^+ &\quad V_{prish}^- < V_{tip}^-, \\
W_{prish}^+ > W_{tip}^+ &\quad W_{prish}^- < W_{tip}^-
\end{align} \end{lemma} \begin{proof}
We note the following inequalities:
\begin{align}
c D \sigma^{-1} &< \sigma V_{prish}^+ - \sigma V_{prish} < C D \sigma^{-1}, \\
c D \sigma^{-1} &< \bar W_{prish}^+ - \bar W_{prish} < C D \sigma^{-1}
\end{align}
This comes from the definition of the barriers $V_{prish}^{\pm} = (1 \pm D V)V$ and $\bar W_{prish}^{\pm} = (1 \pm D V)\bar W$, together with $V \sim \sigma^{-1}$ and $\bar W \sim 1$.
Also, provided we take $\delta < c \zeta_*^{-1}$, by Lemma \ref{barrier_order_lemma} we have
\begin{align}
c \delta^{-1}\epsilon_{v} \nu^{1/2}
&\leq \sigma V_{tip}^+ - \sigma V_{tip}
\leq C \delta^{-1}\epsilon_v \nu^{1/2} ,\\
c \epsilon_w \nu^{1/2}
&\leq \bar W_{tip}^+ - \bar W_{tip}
\leq C \epsilon_w \nu^{1/2}.
\end{align}
We can put all these inequalities, together with \eqref{basic_deriv_closeness}, in terms of $\zeta$: \begin{align}
\sigma |V_{prish} - V_{tip}|
&\leq C(\rho_*)
\left( \nu \zeta^2 + \zeta^{-1} \nu^{1/2} + \nu \right), \label{bar_diff}\\
|\bar W_{prish} - \bar W_{tip}|
&\leq C(\rho_*)
\left( \nu \zeta^2 + \nu |\log\nu| + \nu |\log \zeta| \right), \end{align} \begin{align}
c D \zeta^{-1}\nu^{1/2} &< \sigma V_{prish}^+ - \sigma V_{prish} < C D \zeta^{-1}\nu^{1/2}, \label{prish_bar_sep}\\
c D \zeta^{-1}\nu^{1/2} &< W_{prish}^+ - W_{prish} < C D \zeta^{-1}\nu^{1/2}, \end{align} \begin{align}
c \delta^{-1}\epsilon_{v} \nu^{1/2}
&\leq \sigma V_{tip}^+ - \sigma V_{tip}
\leq C \delta^{-1}\epsilon_{v} \nu^{1/2} ,\label{tip_bar_sep}\\
c \epsilon_w \nu^{1/2}
&\leq \bar W_{tip}^+ - \bar W_{tip}
\leq C \epsilon_w \nu^{1/2}. \end{align}
We now use the inequalities \eqref{bar_diff}, \eqref{prish_bar_sep}, and \eqref{tip_bar_sep} to prove the desired inequalities for the supersolutions. The desired inequalities for the subsolutions are similar.
First we deal with the inequality at $\sigma_*/2 < \sigma < \sigma_*$, where we wish to show that $V^+_{prish} > V_{tip}^+$. By applying \eqref{prish_bar_sep}, then \eqref{bar_diff}, then \eqref{tip_bar_sep} we find \begin{align}
\sigma V^+_{prish}
&\geq \sigma V_{prish} + c D \sigma^{-1} \\
&\geq \sigma V_{tip} + c D \sigma^{-1} \\
&- C \left( \nu^2 \sigma^2 + \sigma^{-1} + \nu \right) \\
&\geq \sigma V_{tip}^+ + c D \sigma^{-1}\\
&- C \left( \nu^2 \sigma^2 + \sigma^{-1} + \nu \right)
- C \delta^{-1}\epsilon_{v} \nu^{1/2}. \end{align} Choosing $D$ such that $c D \geq 2 C$ means that $\delta V^+_{prish} \geq \sigma V^+_{tip}$ at least for short time. Showing that $W^+_{prish} > W_{tip}^+$ is similar.
Now we deal with the inequalities for $\zeta_* \leq \zeta \leq 2 \zeta_*$. First choose $\zeta_* \geq 10\frac{CD}{c \epsilon_w}$, and then chose $\delta \leq \on{10} (CD)^{-1}c \epsilon_v \zeta_*$. Then we have, using \eqref{tip_bar_sep}, \begin{align}
\sigma V^+_{tip}
&\geq \sigma V_{tip} + c \delta^{-1}\epsilon_{v} \nu^{1/2} \\
&\geq \sigma V_{tip} + 10 C D \zeta^{-1}\nu^{1/2}. \end{align} Now using \eqref{bar_diff} and then \eqref{prish_bar_sep}, for $\zeta_* \leq \zeta \leq 2 \zeta_*$, \begin{align}
\sigma V^+_{tip}
&\geq \sigma V_{prish} + 10 C D \zeta^{-1}\nu^{1/2}\\
&- C \left( \nu \zeta^2 + \nu \right)
- C \zeta^{-1}\nu^{1/2} \\
&\geq \sigma V_{prish}^+ + 10 C D \zeta^{-1}\nu^{1/2}\\
&- C \left( \nu \zeta^2 + \nu \right)
- C \zeta^{-1}\nu^{1/2}
- CD \zeta^{-1}\nu^{1/2} \\
&\geq \sigma V_{prish}^+ + 8 CD \zeta^{-1}\nu^{1/2} \end{align} with the last line valid for small enough times. Therefore, for small enough times, $V^+_{tip} \geq V^+_{prish}$ here. The calculation is similar for $W$; since $\zeta \geq 10 \frac{CD}{c \epsilon_w}$ the upper bound $CD \zeta^{-1} \nu^{1/2}$ on $\bar W^+_{prish} - W_{prish}$ is dominated by the lower bound $c \epsilon_w \nu^{1/2}$ on $\bar W_{tip}^+ - \bar W_{tip}$.
\end{proof}
We take a moment here to remark on the design of the tip barriers. To understand the term $\nu^{1/2}$ in the barriers' definitions, consider what would happen in Lemma \ref{barrier_order_lemma} if we replaced $\nu^{1/2}$ with some function $f(\nu) \ll \nu^{1/2}$. We would still have
\begin{align}
V_{diff} = V^+ - V \geq c \delta^{-1}\epsilon_{v} f(\nu) \sigma^{1,-1} - C\epsilon_v \nu \sigma^{1,0}
\end{align}
and upon pulling out the factor $\delta^{-1}\epsilon_vf(\nu)\sigma^{1,-1}$,
\begin{align}
V_{diff} \geq c \delta^{-1}\epsilon_{v} f(\nu) \sigma^{1,-1}\left( 1 - C\delta \frac{\nu}{f(\nu)} \sigma^{0,1}\right).
\end{align}
Since $f(\nu) \ll \nu^{1/2}$, $\frac{\nu}{f(\nu)} \gg f(\nu)$, so the region where $V_{diff} > 0$ is not contained in the region $f(\nu) \sigma \leq \zeta_*$ for any $\zeta_*$.
However, in Lemma \ref{buckling_lemma}, it was important that the region where $V_{diff} > 0$ is contained in the region $f(\nu)\sigma \leq \zeta_*$. The reason is that, in approximating the first term of $V^+_{tip}$, we use the asymptotics of $V_{Bry}$ to say
\begin{align}
V_{k^+{Bry}} =
(\mu + \delta^{-1}\epsilon_v f(\nu))\sigma^{-1}
+ O(\sigma^{-2}).
\end{align}
The term $\mu \sigma^{-1}$ matches with the leading order term of the approximation for $V$ coming from the productish region \eqref{V_expr_smallrho}. The $O(\sigma^{-2})$ term is essentially uncontrollable and falls into the error between $V_{tip}$ and $V_{prish}$ in \eqref{V_approx_diff}. (We could find its sign by studying the Byrant soliton more closely, but that would only help us for either the sub- or supersolution.) Then we need the left over term $f(\nu)\sigma^{-1}$ to cover $O(\sigma^{-2})$- in other words, we need $f(\nu)\sigma \geq C$ for some $C$.
Therefore the $\nu^{1/2}$ is somehow optimal, at least for the technique that we are using.
The point of the inequalities in Lemma \ref{buckling_lemma} is that they immediately imply Lemma \ref{buckling_wrapup} below. This says that we can remove the assumption in Lemma \ref{main_prish_estimates} which assumed that the solution stays within the productish barriers on the left edge of the productish region, and we can remove the assumption from \ref{main_prish_estimates} which assumed that the solution stays within the tip barriers on the right edge of the tip region.
\begin{figure}\label{figure:barrier_switch}
\end{figure}
\begin{lemma}\label{buckling_wrapup}
Let $D > \berr D$, $C_{reg} > \berr C_{reg}$, $u_* < \barr u_*(D, C_{reg})$, $\sigma_* > \berr \sigma_*(D, C_{reg})$, $\epsilon_v$, $\epsilon_w < \berr \epsilon_w(\epsilon_v)$, $\zeta_* \geq \berr \zeta_*(\epsilon_w, D)$, and $\delta < \barr \delta(\epsilon_v, D, \zeta_*)$ be given. There is a $T_*$ depending on all parameters such that if $T_2 < T_*$ we have the following.
Let $0 < T_1 < T_2 < T_*$. Assume that the initial metric is controlled at the initial time in the productish and tip regions, and also controlled at the right of the productish region. Then we have the conclusions \ref{conc:prish_barrier}, \ref{conc:prish_reg} from Lemma \ref{main_prish_estimates} and \ref{conc:tip_barrier}, \ref{conc:tip_reg_noncompact} from Lemma \ref{main_tip_estimates}. \end{lemma} \begin{proof}
Let $T_{bad} > T_1$ be the maximal time such that all the conclusions hold for $g(t)$ on $[T_1, T_{bad})$. By Lemma \ref{buckling_lemma}, the assumption that the solution is barricaded on the right edge of the tip region is satisfied on $[T_1, T_{bad}]$, since the productish region barriers are tighter than the tip region barriers there. Similarly, the assumption that the solution is barricaded on the left edge of the productish region holds on $[T_1, T_{bad}]$. By the assumptions of our lemma, all other assumptions needed to apply Lemmas \ref{main_prish_estimates} and \ref{main_tip_estimates} hold on $[T_1, T_{bad}]$. Therefore all the conclusions still hold at time $t = T_{bad}$. \end{proof}
The assumptions that $g$ is well controlled in the productish and tip regions are all assumptions on the metric at time $T_1$. The only assumption left after Lemma \ref{buckling_wrapup} that is an a priori assumption on the forward evolution is that the metric is barricaded at the right of the productish region.
From now on we consider the constants $D$, $C_{reg}$, $u_*$, $\sigma_*$, $\epsilon_v$, $\epsilon_w$, $\zeta_*$, and $\delta$ to be fixed and satisfying Lemma \ref{buckling_wrapup}.
\subsection{Mollifying metrics}\label{section:mollifying_section} In this section we will define mollified metrics, and prove some basic properties. We introduce a smooth cutoff function $\eta(x):[0, \infty) \to [0, 1]$ which satisfies \begin{align}
\begin{cases}
\eta(x) = 1 & x < 1 \\
\eta(x) \in [0, 1] & 1 \leq x \leq 2\\
\eta(x) = 0 & x > 2
\end{cases} \end{align} and define $\eta_{r}(x) = \eta(x/r)$.
Now, for arbitrary sufficiently small $m$, and $T^{(m)}_1$ to be determined, we define \begin{align}
V_{init}^{(m)}
&=
\begin{cases}
\eta_{2\zeta_*}(\zeta) V_{tip}(u, T^{(m)}_1)
+ (1 - \eta_{2\zeta_*}(\zeta)) V_{prish}(u, T^{(m)}_1)
& \zeta_* \nu^{-1/2} \leq \zeta \leq 4 \zeta_* \nu^{-1/2}\\
V_{prish}(u, T^{(m)}_1)
& 4\zeta_* t\nu^{1/2} \leq u \leq m\\
\eta_m(u) V_{prish}(u, T^{(m)}_1) + (1-\eta_m(u))V_0(u)
& m \leq u \leq \infty
\end{cases} \end{align} and define $W_{init}^{(m)}$ similarly. These functions agree with $V_0$ and $W_0$ for $ u > 2m$, agree with the productish approximation (evaluated at time $T_1^{(m)}$) for $4 \zeta_* t \nu^{1/2} < u \leq m$, and agree with the tip approximation (evaluated at time $T_1^{(m)}$) for $\zeta < 2 \zeta_*$.
So far we have just been dealing with the diffeomorphism invariant considerations of $v$ and $w$ as functions of $u$ and $t$. Now fix a model pinch metric $g_{mp}$ on $M = I \times S^q \times F$, with the corresponding functions $V_0(u)$ and $W_0(u)$. We write $u_0:M \to \mathbb{R}_+$ to be the initial value of $u$ at a given point in $M$. Then \begin{align}
g_{mp} = \frac{du_0^2}{u_0 V_0(u_0)} + u_0 g_{S^q} + W_0(u_0)g_F. \end{align} We also define $M_{[u_1, u_2]} = \{p \in M : u_0(p) \in [u_1, u_2]\}$. Now we define mollifications $g_{init}^{(m)}(t)$. We define them as \begin{align}
g^{(m)}_{init} = \frac{du_0^2}{u_0 V^{(m)}_{init}(u_0)} + u_0^2 g_{S^q} + W^{(m)}_{init}(u_0, t) g_F. \end{align} Note that $g^{(m)}_{init}$ is equal to $g_{mp}$ in $\bar M_{[2m, \infty)}$, and is smooth. It may seem that we have repeated ourselves, since we have already chosen $V^{(m)}_{init}$ and $W^{(m)}_{init}$. The point here is we are also fixing the coordinate of the interval factor.
The following Lemma says that $g_{init}^{(m)}$ satisfies all of the conditions on the initial metric required by Lemmas \ref{main_prish_estimates} and \ref{main_tip_estimates}. \begin{lemma}\label{gm_good_init}
Let $m < \berr m$ and suppose $T^{(m)}_1 < \berr T^{(m)}_1(m) < m$. Let $g_{init}^{(m)} = g^{(m)}(T^{(m)}_1).$ Then for $T_1 = T^{(m)}_1$, the metric $g_{init}^{(m)}$ is initially controlled in the productish and tip regions. \end{lemma} \begin{proof}
That $g_{init}^{(m)}$ is initially controlled in the tip region is immediate, because the functions $v$ and $w$ for $g_{init}^{(m)}$ exactly agree with with the functions $V_{tip}$ and $W_{tip}$ in the tip region.
Where $v$ and $w$ agree with $V_{prish}$ and $W_{prish}$, the assumptions in the productish region are automatic. This is true in $M_{[4\zeta_* T_1^{(m)} \nu^{1/2}, m]}$. What's left is to check the assumptions in $M_{[\sigma_*T_1^{(m)} \nu, 2 \zeta_*T_1^{(m)}\nu^{1/2}]}$ and $M_{[m, 2m]}$.
Both conditions hold for $u_0 \leq \zeta_* t \nu^{1/2}$ by Lemma \ref{basic_deriv_closeness}, the definition of $V_{init}^{(m)}$ and $W_{init}^{(m)}$, and the separation of the barriers. To check the conditions in $M_{[m, 2m]}$, note that they hold strictly in this compact set at time $t = 0$, so for sufficiently small $T_1^{(m)}$ they will continue to hold. \end{proof}
\subsection{Controlling curvature and convergence}\label{section:ccc} Since $g^{(m)}_{init}$ is smooth, there is a solution to Ricci flow $g^{(m)}(t)$ on $[T_1^{(m)}, T_{final}^{(m)})$ with $g^{(m)}(T_1^{(m)}) = g_{init}$. We want to control $g^{(m)}(t)$. By Lemmas \ref{buckling_wrapup} and \ref{gm_good_init}, in order to get the conclusions of Lemmas \ref{main_prish_estimates} and \ref{main_tip_estimates} we just need the condition that the solution is between the barriers for $u_* < u < 2u_*$. Let $T_2^{(m)}$ be the maximal time such that this condition holds on $[T_1^{(m)}, T_2^{(m)})$. In Corollary \ref{T_lower_bound} we will argue that we have a fixed lower bound on $T_2^{(m)}$. In each lemma we may decrease $T_*$.
We do something sort of silly here. For this section, we assume that $(F, g_F)$ has constant sectional curvature. This is so that we can have control on $|\Rm|$ via Corollaries \ref{prish_curvature_control_time} and \ref{tip_curvature_control}. The control on $|\Rm|$ lets us use the full regularity theory for Ricci flow. In the end, we can replace the constant sectional curvature fiber with anything we want, since the Ricci flow of warped products only cares about the Ricci curvature of the fiber. In our case, we could also get this higher regularity by going through the regularity of the involved parabolic PDE on the interval, but it's easier to invoke generic Ricci flow estimates.
\begin{lemma}\label{rm_ctrl_in_x}
Suppose $m < \barr m$. For any $k$, there is a constant $C_k$ depending only on $V_0$, $W_0$, and $u_*$ such that in $M_{[u_*/4, \infty]}$ and for $t \in [T_1^{(m)}, \min(T_*, T_{2}^{(m)})]$,
\begin{align}
|\nabla^k\Rm_{g^{(m)}}| < C_k
\end{align} \end{lemma} \begin{proof}
This is by now standard procedure, see e.g. Corollary A.5 of \cite{topping}.
The curvatures of the metrics $g^{(m)}_{init}$ have a uniform bound on their curvature and the volume of small enough balls in $M_{u_*/4, \infty}$ Therefore we can apply the pseudolocality theorem (Theorem 10.3 of \cite{Perelman}) at any point there, to get control on $|\Rm|$, and then apply local derivative estimates (14.4.1 of \cite{bookanalytic}) to get control on higher derivatives. \end{proof} Since our barrier control is in terms of $u$, we need to be able to transfer the set written in terms of $u$ to being written in terms of $x$. \begin{lemma}\label{u_ctrl_in_x}
Suppose $m < \barr m$. Then for all $t \in [T_1^{(m)}, \min(T_*, T_2^{(m)})]$,
\begin{align}
\{p \in M : u^{(m)}(x,t) \in [u_*, 2u_*]\}
\subset
\bar N_{[u_*/4, 4u_*]}
\end{align} \end{lemma} \begin{proof}
At time $t = T^{(m)}_1$, we have $u_*/4 \leq u^{(m)} \leq 4 u_*{(m)}$ in $M_{[u_*/4, 4u_*]}$ (just by definition). By Lemma \ref{rm_ctrl_in_x}, there is a uniform speed limit on $u$ in $M_{[u_*/4, \infty]}$. Therefore for $x \geq 4u_*$, $u$ cannot decrease too fast and so we can get a time $T_*$ so that $u$ will not go below $u_*$ before time $T_*$.
Also, we can decrease $T_*$ so that $u$ cannot go above $u_*$ in $M_{[0, u_*/4]}$. Since the conclusions of Lemmas \ref{main_prish_estimates} and \ref{main_tip_estimates} hold for $t \in [T_1^{(m)}, T_2^{(m)}]$, $v$ is between its barriers for these times, and is in particular positive for $u \in [0, 2u_*]$. Therefore $u$ is increasing up to the value $2u_*$. Therefore, $u$ is smaller than $u_*$ for $x< u_*/4$. \end{proof}
\begin{lemma}\label{full_curvature_ctrl_time}
For any $k \in \mathbb{N} \cup \{0\}$, there is a constant $C_{time, k}$ such that
\begin{align}
|\nabla^k \Rm_{g^{(m)}}| \leq \frac{C_{time,k}}{t_0^{k/2}t_0 \nu(t_0)}.
\end{align}
for all $t \in \left[\max(t_0, (2-2^{-k})T_1^{(m)}), \min(T_*, T_2^{(m)})\right]$. \end{lemma} \begin{proof}
For $k=0$, this is exactly Lemma \ref{prish_curvature_control_time}, Lemma \ref{tip_curvature_control}, and Lemma \ref{rm_ctrl_in_x}. (Remember that in this section we assume $(F, g_F)$ has constant sectional curvature.) For $k > 0$, we can apply Shi's derivative estimates (Theorem 1.1 of \cite{Shi}), using the result for $k-1$ and for times larger than $\max\left(t_0/2, (2-2^{-k+1})T_1\right)$. The factor in front of $T_1$ ensures us that for any $t_0$ and $k$, and for any point where we want to apply the regularity, there is a uniformly sized (in $m$) parabolic ball of Ricci flow which has the order $k-1$ estimates. \end{proof}
\begin{lemma}\label{T_lower_bound}
$T_{final}^{(m)} > T_2^{(m)} > T_*$. \end{lemma} \begin{proof}
By Lemma \ref{full_curvature_ctrl_time}, Ricci curvature is bounded at time $T_2^{(m)}$. Therefore, $T_{final}^{(m)} > \min(T_*, T_2^{(m)})$.
By Lemmas \ref{rm_ctrl_in_x} and \ref{u_ctrl_in_x} the curvature and its derivatives are bounded for $u^{(m)}(x,t) \in [u_*, 2u_*]$. This implies a speed limit on the functions $v^{(m)}$ and $w^{(m)}$ there. Since the functions are uniformly separated from the barriers are time $t = T^{(m)}_1$, they cannot pass the barriers for some fixed time. Therefore $T_2^{(m)} > T_*$, possibly taking $T_*$ smaller. \end{proof}
We now have all of the conclusions of Lemmas \ref{main_prish_estimates} and \ref{main_tip_estimates}, for each $g^{(m)}(t)$, on $[T_1^{(m)}, T_*]$. Now, we get estimates within fixed subsets of $M$, extending the crude ones from Lemmas \ref{rm_ctrl_in_x} and \ref{u_ctrl_in_x}.
\begin{lemma}\label{u_lower_bnd}
There is a constant $C > 0$ depending only on $g_{mp}$ such that the following holds. Let $u_\# \in [0, u_*]$ and suppose $m < u_{\#}$. Then
in $p \in M_{[u_{\#}, \infty)}$ and for $t < (C+1)^{-1}u_\#$, we have $u^{(m)} \geq u_\# - C t$. \end{lemma}
\begin{proof}
Since $T_1^{(m)} < m$, at the beginning time $T_1^{(m)}$ any point $p \in M_{[u_{\#}, u_*]}$ lies in the productish region, which is defined as the points where $u \geq t \nu(t) \sigma_*$. (By restricting $T_*$, we can assume $\nu(T_1^{(m)}) < \frac{1}{\sigma_*}$.)
The function $u^{(m)}$ satisfies the evolution equation
\begin{align}
\partial_t u^{(m)} = \Delta_M u^{(m)} - 2 (u^{(m)})^{-1}|\nabla u^{(m)}|^2 - \mu
\end{align}
and as long as $u^{(m)}$ is in the productish region, we have the estimate (using the regularity in conclusion \ref{conc:prish_reg})
\begin{align}
|\Delta_M u^{(m)}| + (u^{(m)})^{-1}|\nabla u^{(m)}|^2 \leq C.
\end{align}
Therefore,
\begin{align}
u(p, t) \geq u_\# - (\mu + C)(t - T_1^{(m)}) \geq u_{\#} - (\mu + C) t.
\end{align}
Now, $p$ continues to be in the productish region as long as $u \geq \sigma_* t \nu(t)$, so at least as long as
$
u_{\#} - (\mu + C)t \geq t \nu(t) \sigma_*
$.
Since we assume $\nu(t) < 1/\sigma_*$, this will be implied if $t \leq \frac{u_{\#}}{\mu + C + 1}$. Therefore we have proven the first bullet.
\end{proof}
\begin{lemma}\label{full_ctrl_curvature_x}
For $u_{\#} \in [0, u_*]$ and $k \in \mathbb{N} \cup \{0\}$ there is a constant $C_{space,k}(u_{\#})$ such that the following holds. Suppose $u_\# > (2-2^{-k})m$. Then in $M_{[u_{\#}, \infty)}$ and for $t \in [T_1^{(m)}, T_*]$,
\begin{align}
|\nabla^k \Rm_{g^{(m)}}| \leq C_{space,k}(u_\#).
\end{align} \end{lemma} \begin{proof}
Once we prove the Lemma for $k = 0$, the result follows for $k > 0$ using local derivative estimates and the result for $k' = k-1$ and $x_0' = 2x_0$. (In particular, we have to use the knowledge that all of the metrics $g^{(m)}(T_1^{(m)})$ agree in $M_{[u_{\#}, \infty)}$ and therefore have uniform curvature bounds, and we can use 14.4.1 of \cite{bookanalytic}.)
Now we do $k = 0$. For $t < (C+1)^{-1}u_{\#}$ we an apply \ref{u_lower_bnd} to find that $u^{(m)} > u_{\#} - C t$. So, we stay in the productish region and we can apply Lemma \ref{prish_curvature_control} to get a bound on $|\Rm|$. (This bound will depend on $W_0(u)$, but for example if $W_0(u) \gtrsim u$ we can get $|\Rm| \lesssim u_{\#}^{-1}$.) On the other hand, for $t > (C+1)^{-1}u_{\#}$ we can apply Lemma \ref{full_curvature_ctrl_time} to find $|\Rm| \leq \frac{C_{time, 0}}{(C+1)^{-1}t_\# \nu((C+1)^{-1}t_\#)}$. \end{proof} \begin{lemma}\label{full_ctrl_derivs_x}
For $u_{\#} \in [0, u_*]$ and $k \in \mathbb{N} \cup \{0\}$ there is a constant $C_{k}(u_{\#})$ such that the following holds. Suppose $u_\# > 2m$. Then in $M_{[u_\#, \infty)}$ and for $t \in [T_1^{(m)}, T_*]$,
\begin{align}
|\left( \nabla^{g_{mp}} \right)^k g^{(m)}(x, t)|_{g_{mp}} \leq C(x_0, k).
\end{align} \end{lemma} \begin{proof}
This follows by integrating the Ricci flow equation and derivatives of the Ricci flow equation. \end{proof}
Now we prove Theorem \ref{theorem:model_pinch_flow}. Recall that we identify $M = I \times S^q \times F$ with $(\mathbb{R}^{1+q} \setminus \{0^{1+q}\}) \times F \subset \bar M \mathrel{\mathop:}= \mathbb{R}^{1+q} \times F$, where $0^{1+q}$ is the origin in $\mathbb{R}^{1+q}$. We will construct the Ricci flow $g(t)$ provided by Theorem \ref{theorem:model_pinch_flow} as a limit of our flows of mollified metrics $g^{(m)}(t)$. As a little notational annoyance, we set $g_{shift}^{(m)}(t) = g^{(m)}(t-T_1^{(m)})$, which is a Ricci flow for times at least $[0, T_*/2]$. ($g^{(m)}$ has the nice property that $g^{(m)}$ evaluated at time $t$ is approximately our approximate solution at time $t$, whereas $g_{shift}^{(m)}$ is nice because it always starts at time $0$.) We let $P = \bar M \setminus M$.
\begin{lemma}\label{final_lemma}
There is a sequence $m_k \searrow 0$, and a family of metrics $g_{wp}(t)$, $t \in [0, T_*/2]$ such that
$
g_{shift}^{(m_k)}(t) \to g_{wp}(t)
$
in $C^{\infty}_{loc}\left( \bar M \times [0, T_*/2] \setminus P \times \{0\} \right)$.
$g_{wp}(t)$ is a Ricci flow satisfying all of the conclusions of Theorem \ref{theorem:model_pinch_flow}. \end{lemma}
\begin{proof}
First we get convergence in $C^{\infty}_{loc}(M \times [0, T_*/2])$. For any $u_\# > 0$ and for $m > 2u_\#$ we have $C^{\infty}$ control on the derivatives of $g_{shift}^{(m)}$ in $M_{[u_\#, \infty)} \times [0, T_*/2]$ (Lemma \ref{full_ctrl_derivs_x}). By the Arzel\`a-Ascoli theorem, we get convergence of a subsequence in any such region. By taking a diagonal subsequence, we get convergence to a metric on $g(t)$ as desired. Since the convergence happens in $C^{\infty}$, the Ricci flow equation and all the estimates pass to the limit.
Next we can continue extracting subsequences to get convergence on $C^{\infty}_{loc}(\bar M \times [t_0, T_*/2))$, for any $t_0 > 0$, using Lemma \ref{full_curvature_ctrl_time}. By performing another diagonal argument we can get convergence in the claimed $C^{\infty}_{loc}$ space.
For any $t > 0$, the doubly-warped product metric $g_{wp}(t)$ satisfies the inequalities in the conclusions of Lemma \ref{main_prish_estimates} and \ref{main_tip_estimates}. (Perhaps with non-strict inequalities, but we can make the constants worse to make the inequalities strict.) These imply that the metric has an extension to $\bar M$. \end{proof}
Here, we use the control that we have on $g_{wp}(t)$ to find more precise estimates on the convergence of $g_{wp}(t)$ to $g_{mp}$ as $t \searrow 0$. \begin{corollary}\label{improved_convergence}
Suppose we are in the setting of Theorem \ref{theorem:model_pinch_flow}. Write $g_{mp}$ as $g_{mp} = dx^2 + u_0(x)g_{S^q} + w_0(x) g_{F}$, set $v_0(x) = u_0^{-1}(x)|\nabla u_0(x)|^2$, and set $h(t) = g_{wp}(t) - \left( g_{mp} - t\mu g_{S^q} - t\mu_F g_F\right)$. There are constants $\epsilon_0 = \epsilon_0(g_{mp})$ and $C_0 = C_0(g_{mp})$ such that the following holds. For any $x$ with $u_0(x)<u_*/2$, and for $t < \epsilon_0u(x)$, we have $|\partial_t h|_{g_{mp}}(x, t) \leq C_0\left( u_0(x)^{-1}v_0(x) \right)$.
Let $k \in \mathbb{N}$. If in addition to the regularity assumption \ref{modelpinch_reg} we assume
\begin{align}
\frac{|V_0|_{2+k, \eta; u/2, (du)^2}}{V_0}
+
\frac{|W_0|_{2+k, \eta; u/2, (du)^2}}{W_0}
\leq C
\end{align}
then there are constants $\epsilon_k(g_{mp})$ and $C_k(g_{mp})$ such that for any $x$ with $u(x) < u_*/2$, and for $t < \epsilon_ku_0(x)$, we have
\begin{align}
|\partial_t \left(\nabla^{g_{mp}}\right)^{k}h|_{g_{mp}}(x, t) \leq C_k \left( u_0^{-1}v_0 \right)^{2+k}
\end{align} \end{corollary} \begin{proof}
We first consider $k = 0$. Consider any $x$. Initially $x$ is in the productish region. As long as $x$ is in the productish region, we can use Corollary \ref{prish_curvature_control} to control $\Rc_{g_{wp}(t)} - \left( \mu g_{S^q} + \mu_F g_F\right)$. In particular, for sufficiently small $\epsilon_0$ we will have $u_{g_{wp}}(x, t) > \oh u_0(x)$ for $t < \epsilon_0u_0$, so we stay in the productish region for these times.
To get the higher regularity, note that the extra assumption allows us to improve the degree at which we are allowed to apply all interior Schauder estimates, so we get control on the gradients of $\Rm$ in Corollary \ref{prish_curvature_control} as well. \end{proof}
\section{Stability of the Bryant soliton}\label{stabil_bryant} In this section, we will prove a result that we use for stability of Ricci-DeTurck flow around the Bryant soliton, Theorem \ref{bry_stabil}, assuming a priori control at infinity. This will be used to prove the short-time stability of flows from model pinches. For a complete stability result for the Bryant soliton, see \cite{steady_alix}. That result does not suffice for us because being in the weighted $L^2$ space there requires exponential decay at infinity.
The main result we use from this section is the following. We let $({Bry}, g_{bry})$ be the Bryant steady soliton metric on ${Bry} \sim \mathbb{R}^{1 + q}$ which has soliton vector field $X$, and $({Bry}, g_{bry}) \times (\mathbb{R}^p, g_{\mathbb{R}_p}) = ({Bry} \times \mathbb{R}^p, g_{sol})$ . \begin{theorem}\label{bry_stabil}
For $C_{reg} \geq 0$ there is a constant $\barr \epsilon(C_{reg}) > 0$ depending only on the dimension with the following property. Suppose $\epsilon < \barr \epsilon$ and let $\bar F = \epsilon F$, where $F$ is defined in Lemma \ref{bry_rdt_supsoln}. Suppose that $g(t) = g_{sol} + h(t)$ is a Ricci-DeTurck flow around $g_{sol}$ modified by $X$, on a time interval $I$. Suppose that for all $P \in {Bry} \times \mathbb{R}^p$ and $t \in I$,
\begin{align}
|h(P,t)| \leq \bar F(P).
\end{align}
Suppose that either $I = (-\infty, T]$, or $I = [0, T]$ with the condition at time $t=0$ that
\begin{align}\label{init_nabla_bnds}
u^{0,1/2}|\nabla h| + u^{0,1}|\nabla^2 h| < C_{reg} \bar F.
\end{align}
Then the \emph{strict} inequality $|h(P, t)| < \bar F$ holds for all $P \in {Bry}$ and $t \in I$. \end{theorem}
We wish to compare this result to Section 7 of \cite{uniquenessBK}. Note that the function $F$ here is asymptotic to $1/u$, as is the scalar curvature $R$ on the Bryant soliton. So, our hypothesis implies $|h| \leq \epsilon R$ (for some possibly smaller $\epsilon$). In their Section 7, Bamler and Kleiner find estimates for the perturbation under the assumption that $|h| \leq C R^{1 + \chi}$ for some $\chi > 0$. Indeed, they get an improvement of $\frac{|h|}{R^{1+\chi}}$, where we have only been able to get stability. Both Theorem \ref{bry_stabil} and Section 7 of \cite{uniquenessBK} use the Anderson-Chow estimate; unfortunately our Theorem requires some specific calculations on the Bryant soliton and it's not clear what generalization is available.
\subsection{Anderson-Chow Estimate} We begin with a version of the Anderson-Chow estimate. In \cite{ac_estimate}, Anderson and Chow proved an estimate in three dimensions for solutions to the linearization of Ricci-DeTurck flow, in terms of the scalar curvature. The key inequality for their estimate is \begin{align}\label{ac_inequality}
|\Rc|^2 - R \Lambda_{\Rm} \geq 0 \end{align} valid on any three-dimensional manifold. Recall the definition
$$\Lambda_{\Rm} = \max_{h \in Sym_2(M) : |h| = 1}\ip{\Rm[h]}{h}$$ from Section \ref{section:rdt}. This estimate is useful in classifying solitons \cite{brendlesolitonimportant}, and was also vital in \cite{uniquenessBK}. In \cite{ac_higherdim}, Wu and Chen prove a higher-dimensional version of the Anderson-Chow estimate, assuming that the Weyl tensor vanishes identically along the flow (Claim 2.1 in \cite{ac_higherdim}). For a singly-warped product, the Weyl tensor does vanish identically (since it is conformal to a cylinder) and therefore \cite{ac_higherdim} applies. We also give a proof in the restricted setting we need, because it is more elementary and we need a statement about strictness.
For a singly warped product, $ds^2 + u(s) g_{S^q}$, we let $L = (1-\partial_su^{1/2})/u$ be the sectional curvature of a plane tangent to $S^q$, and $K = - u^{-1/2}\partial_s^2 u^{1/2}$ be the sectional curvature of a plane spanned by $\partial_s$ and a vector from $S^q$. \begin{lemma}
Let $g = ds^2 + u g_{S^q}$ be a warped product metric with nonnegative sectional curvature. Then the Anderson-Chow inequality \eqref{ac_inequality} holds for $g$. Equality is achieved only at points where the sectional curvature is constant or where either $K$ or $L$ is $0$. \end{lemma}
\begin{proof}
Note the calculation below is just done within the vector space $T_PM$ for an arbitrary $P \in M$.
The scalar curvature of $g$ is
$
R = 2qK + q (q-1)L.
$.
The Ricci curvature of $g$ is
$
\Rc = qK ds^2 + (K + (q-1)L)(u g_{S^q})
$
so
$
|\Rc|^2 = q^2 K^2 + q(K + (q-1)L)^2
$.
Writing $\alpha = \frac{K}{(q-1)L}$ we can rewrite these as
\begin{align}
\frac{R}{(q-1)L}
= q(2 \alpha + 1),
\quad
\frac{|\Rc|^2}{((q-1)L)^2} = q\left( q \alpha^2 + (\alpha + 1)^2 \right).
\end{align}
We can deal with $L = 0$ by taking the limit as $\alpha \to \infty$ in the end.
Now let's find $h$ with $|h| = 1$ which maximizes $\Rm[h,h]$. Take an orthonormal basis $V_0 = \partial_s, V_1, \dots, V_q$ for $T_pM$, such that $h$ is diagonal with respect to $V_1 \dots, V_q$, that is for $i$, $j$ nonzero and distinct,
$h_{ii} = \lambda_i$ and $h_{ij} = 0$.
Then,
\begin{align}
\Rm[h,h]
&= \Rm_{aibj}h^{ij}h^{ab} \\
&= \sum_{a = 1}^n h^{00}h^{aa}\Rm_{a0a0}
+ \sum_{i=1}^n h^{ii}h^{00}\Rm_{0i0i}
+ \sum_{i=1}^n \sum_{a=1}^n h^{ii}h^{aa}\Rm_{aiai} \\
&+ \sum_{j=1}^n h^{0j}h^{j0}\Rm_{j00j}
+ \sum_{i=1}^n h^{i0}h^{0i}\Rm_{0ii0}
+ \sum_{i=1}^n \sum_{j=1, j \neq i}^n h^{ij}h^{ji} \Rm_{ijji}.
\end{align}
The first line is the case when $i = j$: the first term is when $i = 0$, the second term is when $a = 0$, and the third term is when neither is 0. The second line is when $i \neq j$: the first term is when $i = 0$, the second term is when $i \neq 0$ but $j = 0$, and the third term is when neither is 0. Note that actually this last term vanishes since $h^{ij} = 0$, and since $\Rm_{0ii0} = \Rm_{j00j} = -L$, the second line is negative. Therefore to optimize $h$ we will take $h^{0i} = 0$. Let $b = h_{00}$. Simplifying, we have
\begin{align}
\Rm[h, h]
&= 2 b (\sum \lambda_i) K
+ \left( \left( \sum \lambda_i \right)^2 - \sum \lambda_i^2 \right) L
\end{align}
We can assume $b > 0$, since negating $h$ does not change $\Rm[h,h]$. Then, to maximize either
\begin{align}
\sum \lambda_i \quad \text{ or } \quad \left( \left( \sum \lambda_i \right)^2 - \sum \lambda_i^2 \right)
\end{align}
we would take the $\lambda_i$ all equal. Since this maximizes either term, and since $K$ and $L$ are positive, it maximizes all of $\Rm[h,h]$. Define $\lambda = \sqrt{q}\lambda_i$, with the motivation that $\lambda$ is the norm of the restriction of $h$ to $TS^q$, so $b^2 + \lambda^2 = 1$.
So, recalling the definition $\alpha = \frac{K}{(q-1)L}$ we arrive at
\begin{align}
\frac{\Rm[h,h]}{(q-1)L}
= 2 \sqrt{q} \alpha (b \lambda) + (\lambda^2).
\end{align}
The positive eigenvalue of the matrix
$\begin{pmatrix}0 & \sqrt{q}\alpha \\ \sqrt{q}\alpha & 1 \end{pmatrix}$ is $\oh (1 + \sqrt{4 q\alpha^2 + 1})$. Therefore, since $b$ and $\lambda$ optimize $2 \sqrt{q}\alpha b \lambda + \lambda^2$ with $b^2 + \lambda^2 = 1$, we have,
\begin{align}
\frac{\Lambda_{\Rm}}{(q-1)L} = \oh (1 + \sqrt{4q \alpha^2 + 1})
\end{align}
Therefore,
\begin{align}
A =
\frac{|\Rc|^2 - R \Rm[h,h]}{q (q-1)^2L^2}
&= q \alpha^2 + (\alpha + 1)^2 - \oh (2 \alpha + 1)(1 + \sqrt{4 q \alpha^2 + 1})
\end{align}
Now, for $q = 2$ and for each $\alpha$, we have $A\geq 0$ by the three dimensional Anderson-Chow estimate. (We could also check by hand.) We claim $A$ doesn't decrease as we increase $q$.
Calculate,
\begin{align}
\frac{dA}{dq}
&= \alpha^2 - (2 \alpha + 1)(4 q \alpha^2 + 1)^{-1/2}\alpha^2 \\
\end{align}
So for $q \geq 2$
\begin{align}
\frac{dA}{dq}
&\geq \alpha^2 \left( 1 - (2 \alpha + 1)(8 \alpha^2 + 1)^{-1/2} \right) \geq 0.
\end{align}
\end{proof}
\begin{corollary}
The Anderson-Chow inequality \eqref{ac_inequality} holds for $g_{sol}$. \end{corollary} \begin{proof}
The extra flat factor does not affect any of the terms in \eqref{ac_inequality}. The Bryant soliton has nonnegative curvature, so the previous lemma applies. \end{proof}
\subsection{Constructing a supersolution} Using the Anderson-Chow estimate, we construct a supersolution to linearized Ricci-DeTurck flow around $g_{sol}$. We write $g_{Bry} = ds^2 + u(s) g_{S^q}$. \begin{lemma}\label{bry_rdt_supsoln}
Let $({Bry} \times \mathbb{R}^p, g_{sol}, X)$ be the Bryant soliton crossed with a euclidean factor. There is a function $F : {Bry} \times \mathbb{R}^p \to \mathbb{R}_{>0}$, which is just a function of $u$, with the following properties.
\begin{enumerate}
\item For some $c > 0$, $\Delta_X F + 2 \Lambda_{\Rm} F \leq - c u^{0,-2}\log( 2 + u )F$.
\item For some $c_1, c_2 > 0$, as $u \to \infty$, $F = c_1u^{-1} \left( 1 + c_{2} \frac{\log u}{u} \right)(1 + o(1)) $.
\end{enumerate} \end{lemma}
\begin{proof}
First recall that if $R_0$ is the maximum scalar curvature, $f$ is the soliton potential, and $\bar f(p) = -\frac{f(p)-f(0)}{R_0}$ then $\bar f$ satisfies
\begin{align}
\label{eq:124}
\Delta_X \bar f = 1,
\quad \bar f(0) = 0,
\quad \nabla \bar f(0) = 0,
\end{align}
and has the asymptotics at $\infty$,
\begin{align}
\bar f = \mu^{-1} u \left( 1 - c_{\bar f} \frac{\log u}{u} \right) (1 + o(1; u \to \infty) )
\end{align}
for some constant $c_{\bar f}$ (see Appendix \ref{bryant_facts} and especially \eqref{bry_second_order_expansion}). Also, $\bar f$ attains its minimum of $0$ at $u=0$.
Now let $F_1 = \left( \bar f + a \right)^{-1}$ for some $a>0$ to be determined. Calculate using $\Delta_X \bar f = 1$ that
$
\Delta_X F_1
= - \left( F_1 - 2 F_1^2 |\nabla \bar f|^2 \right)F_1
$,
so
\begin{align}
\label{eq:151}
-\left( \Delta_X + 2 \Lambda_{\Rm} \right) F_1
&= \left( F_1 - 2 \Lambda_{\Rm} - 2 F_1^2 |\nabla \bar f|^2 \right) F_1
\end{align}
We claim that for large enough $B$, the function $F = F_1 + B R$ satisfies the properties in the lemma. The asymptotics at infinity (i.e. item (2) of the conclusion) are immediate from the asymptotics for $F_1$ and $R = c_1 u^{-1} + O( u^{-2})$. Now calculate,
\begin{align}
\label{eq:156}
-(\Delta_X + 2\Lambda_{\Rm}) F
&= \left( \frac{-(\Delta_X + 2\Lambda_{\Rm}) F_1}{F_1 + B R} + B\frac{ -(\Delta_X +2 \Lambda_{\Rm}) R}{F_1 + B R} \right) F \\
&\mathrel{\mathop=}: \left( T_1 + T_2 \right) F
\end{align}
Note the term $T_2$ is positive everywhere by the singly-warped Anderson-Chow estimate and the equation satisfied by $R$ under Ricci flow:
\begin{align}
\Delta_X R + 2 \Lambda_{\Rm} R
&= - 2 |\Rc|^2 + 2 \Lambda_{\Rm} R \leq 0.
\end{align}
\begin{claim}
Let $K$ be a compact subset of ${Bry}$ not containing the origin. If $B$ is sufficiently large, then there is a $c$ so that $T_1 + T_2 > c$ on $K \times \mathbb{R}^p$.
\end{claim}
\begin{claimproof}
On $K \times \mathbb{R}^p$, the singly-warped Anderson-Chow estimate is not sharp and $R$ is bounded from above, so for some $c_K>0$, $|\Rc|^2 - R \Lambda_{\Rm} > c_KR$ in $K \times \mathbb{R}^p$. Therefore,
\begin{align}
-\left( \Delta_X + 2 \Lambda_{\Rm} \right)R \geq c_K R \quad \text{in} \quad K \times \mathbb{R}^p
\end{align}
By compactness, in $K$, $-(\Delta_X + 2 \Lambda_{\Rm}) F_1$ is bounded from below and $R$ is strictly positive. Therefore, examining the dependence of $T_1$ and $T_2$ on $B$, we can chose $B$ large enough so that $T_1 \geq -c_K/4$ and $T_2 \geq c_K/2$ on $K$.
\end{claimproof}
\begin{claim}
For sufficiently small $a$ in the definition of $F_1$ (independent of $B$), and sufficiently small $u_1$ (independent of $B$), there is a $c$ (which may depend on $B$) such that $T_1>c$ in $\{u < u_1\}$.
\end{claim}
\begin{claimproof}
Choose $a = \frac{1}{4 \Lambda_{\Rm}(0)}$. Then $F_1 - 2 \Lambda_{\Rm} > 0$ in a neighborhood of $0$. Also, $|\nabla \bar f|^2(0) = 0$. The claim follows from \eqref{eq:151} by choosing $u_1$ and $c$ small enough.
\end{claimproof}
\begin{claim}
For sufficiently large $u_2$ (depending on $a$, but independent of $B$) and sufficiently small $c$ (depending on $B$ and $a$), $T_1$ satisfies $T_1 \geq c u^{0,-2}\log(2 + u)$ on the set $\{u > u_2\}$.
\end{claim}
\begin{claimproof}
The Bryant soliton satisfies, as $u \to \infty$,
\begin{align}
\label{eq:152}
\Rm = u^{-1} \left( u g_{S^q} \odot u g_{S^q} \right) + O(u^{-2}|\nabla u|^2) = u^{-1} \left( u g_{S^q} \odot u g_{S^q} \right) + O(u^{-2})
\end{align}
Note that the largest eigenvalue of $u^{-1}\left( u g_{S^q} \odot u g_{S^q} \right)$ is $(q-1) = \oh \mu$.
We can calculate the asymptotics of $F_1$ from the asymptotics of $\bar f$ from \eqref{bry_second_order_expansion} in Section \ref{bryant_facts_nextorder}:
\begin{align}
\label{eq:153}
F_1 = \mu u^{-1} \left( 1 + c_{\bar f} \frac{\log u}{u} \right) ( 1 + o(1; u \to \infty)).
\end{align}
Also, $|\nabla \bar f|^2 = O(1; u \to \infty)$. From this we find,
\begin{align}
\label{eq:154}
(F_1 - 2 \Lambda_{\Rm} - 2 F_1^2 |\nabla \bar f|^2)
&= \mu c_{\bar f} u^{-2} \log u + O(u^{-2}; u \to \infty)
\end{align}
The claim follows by choosing $u_2$ large enough and $c$ small enough.
\end{claimproof}
To prove the lemma, choose $u_1$ and $u_2$ in accordance with the second and third claims above, and then choose $B$ large enough so the conclusion of the first claim holds on the complement of $\{u_1 < u < u_2\}$. Then the conclusion of the lemma holds (taking the minimum over the values of $c$). \end{proof} \subsection{Completion of the proof of Theorem \ref{bry_stabil}} \begin{proof}
In this proof the ever-increasing constant $C$ is chosen independently of $\epsilon$. First, we write the inequality solved by $\bar F$ in terms of the laplacian $\Delta_{X,g_{{Bry}}, g}$. By Lemma \ref{bry_rdt_supsoln}, we have
\begin{align}\label{barF_ineq}
- \left( \Delta_X \bar F + 2 \Lambda_{\Rm} \bar F \right)
\geq c u^{0,-2}\log(2 + u)\bar F.
\end{align}
Since $|\nabla\nabla F| \leq C u^{0,-3} \leq C u^{0,-2}F$, and $|h| \leq \epsilon F$, we have
\begin{align}
|\Delta_{X, g_{{Bry}}, g}F - \Delta_X F| \leq C u^{0,-2}\epsilon F^2
= C\epsilon u^{0,-3}F
\end{align}
and multiplying through by $\epsilon$, $|\Delta_{X, g, \bar g} \bar F - \Delta_X \bar F| \leq C \epsilon u^{0,-3}\bar F$. Therefore, decreasing $c$ and demanding that $\epsilon$ is sufficiently small, we can replace \eqref{barF_ineq} with
\begin{align}
- \left( \Delta_{X, g_{{Bry}}, g} \bar F + 2 \Lambda_{\Rm} \bar F \right)
\geq c u^{0,-2}\log(2 + u)\bar F.\label{barF_ineq_better}
\end{align}
Next we note the regularity available. We claim that for some $C$ (independent of $\epsilon$, $P_*$, and $t_*$, but depending on $C_{reg}$), we have $|\nabla h|(P_*,t_*) < C u^{0,-1/2}\bar F(P_*)$. To see this, let $a = u^{0,-1}(P_*)$ and scale the parabolic system by $a$:
\begin{align}
\tilde g_{{Bry}} = a g, \quad \tilde h = a h, \quad \tilde t = a t, \quad \tilde X = a^{-1} X, \quad \tilde u = a u.
\end{align}
We want to apply regularity in a parabolic neighborhood of some sufficiently small size $r>0$. The Bryant soliton has a bound $|\nabla u|^2 \leq C$ for some $C$. So, for any $r$, for all
$
P \in B_{\tilde g}(P_*, r) = B_{g}(P_*, r/\sqrt{a})
$
we have
\begin{align}
|\tilde u(P) - \tilde u(P_*)|
&= a |u(P)-u(P_*)| \\
&\leq C a \frac{r}{\sqrt{a}} = C r \sqrt{a}
= C r u^{0, -1/2} \leq C r\label{uchangebnd}.
\end{align}
Therefore for sufficiently small $r$, the ball of radius $r$ around $P_*$, with respect to $\tilde g$, is close to a euclidean ball, uniformly in $P_*$.
To continue the regularity argument, in the case when $I = [0,T]$, the parabolic neighborhood of size $r$ around $P_*$ may see the initial condition. We need to check what the bounds on the initial condition \eqref{init_nabla_bnds} says about $\tilde h$. At the initial time,
\begin{align}
|\nabla \tilde h|_{\tilde g}(P,0)
= a^{-1/2}|\nabla h|_{g}(P,0)
\leq c_h\frac{u^{0,-1/2}(P)}{u^{0,-1/2}(P_*)} \bar F
= c_h\frac{u^{0,1/2}(P_*)}{u^{0,1/2}(P)} \bar F
\leq C \bar F
\end{align}
where we used \eqref{uchangebnd} and forced $r$ sufficiently small.
Similarly scaling shows $|\nabla \nabla \tilde h|_{\tilde g}(P,0) \leq C \bar F$.
Therefore, in the parabolic neighborhood of size $r$ we may apply parabolic regularity to find that $|\nabla \tilde h|_{\tilde g_{{Bry}}}$ is bounded by $C\bar F$. Scaling back, we find $|\nabla h|(P_*, t_*) = a^{1/2} |\nabla h|(P_*, t_*) \leq C u^{0,-1/2}(P_*, t_*) \bar F(P_*, t_*)$.
Now, by the bound on the evolution of $|h|$ \eqref{rdt_norm}, we have that $Z = |h|$ satisfies
\begin{align}
\square_{X, g_{{Bry}}, g} Z - 2\Lambda_{\Rm} Z \leq C |\Rm_{g_{{Bry}}}|Z^2 + C |\nabla h|^2.
\end{align}
Or, since we have assumed $Z \leq \bar F$, and also $|\nabla h| < Cu^{0, -\onf2}\bar F$ by the discussion on regularity,
\begin{align}
\square_{X, g_{{Bry}}, g} Z - 2\Lambda_{\Rm} Z
&\leq C u^{0,-1}Z^2 + C u^{0,-1}\bar F^{2}.
\end{align}
Then since $Z \leq \bar F \leq \epsilon C u^{0,-1}$,
\begin{align}
\square_{X,g_{{Bry}}, g} Z - 2\Lambda_{\Rm} Z&\leq \epsilon C u^{0, -2} \bar F.
\end{align}
In particular, we can choose $\epsilon$ sufficiently small so that
\begin{align}
\square_{X,g_{{Bry}}, g} Z - 2\Lambda_{\Rm} Z&\leq (c/2) u^{0, -2} \bar F.
\end{align}
where $c$ is the constant from \eqref{barF_ineq_better}.
Therefore
\begin{align}
\square_{X, g_{{Bry}}, g} (\bar F-Z) - 2 \Lambda_{\Rm} (\bar F - Z) \geq (c/2) u^{0,-2} \bar F > 0
\end{align}
and the lemma follows by the maximum principle. \end{proof}
\section{Local stability of forward evolutions}\label{section:local_stability} Let $g_{mp}$ be an $\Rm$-permissible model pinch and let $(\bar M, g_{wp}(t))$ be a forward evolution from $g_{mp}$ given by Theorem \ref{theorem:model_pinch_flow}. In this section we prove local stability of $(\bar M, g_{wp}(t))$ assuming a priori control at the boundary of a neighborhood of the origin. We let $u_0 = u(p, t)$. For any $u_1, u_2$ we let $\bar M_{[u_1, u_2]} = \{p : u_0(p) \in [u_1, u_2]\}$. Note that while $\{(p,t): u(p,t)\in [0, u_1]\}$ is a subset of space-time which is different for each time-slice, $\bar M_{[0, u_1]}$ is a fixed subset of $M$. However, note that as in Lemma \ref{u_ctrl_in_x} we can argue that for any $u_1$ there is a $T(u_1)$ such that $\bar M_{[0, u_1/2]} \subset \{p : u(p, t) < u_1\} \subset \bar M_{[0, 2u_1]}$ for $t < T(u_1)$.
Let $\hat u = u + \mu t$ and $Q = \hat u / u$, as in Appendix \ref{nearly_cnst_pde_sect}. Also let $F$ be the function defined in Lemma \ref{bry_rdt_supsoln}. For parameters $u_*$, $\sigma_*$, $\sigma_{**}$, $D$, and $b$ to be chosen, we let $F_{full}$ be defined as \begin{align}
F_{full} =
\begin{cases}
(1 + DV)^{1/2}Q V_0(\hat u) & \sigma > \sigma_{**} \\
\min\left( (1 + DV)^{1/2}Q V_0(\hat u), bF(\sigma) \right)
& \sigma_* \leq \sigma < \sigma_{**} \\
bF(\sigma) & \sigma \leq \sigma_*
\end{cases} \end{align}
\begin{theorem}\label{local_stability}
For $r_0 < \barr r_0(g_{mp})$, $\epsilon < \barr \epsilon(g_{mp}, r_0)$, $D > \berr D(g_{mp})$, $u_* < \bar u_*(g_{mp}, D)$, and $\sigma_* = \sigma_*(g_{mp}, D)$, $\sigma_{**} = \sigma_{**}(g_{mp}, D)$, $b=b(g_{mp}, D)$, there is a $T_*(g_{mp}, u_*, r_0)$ and $C(g_{mp}, u_*, r_0)$ with the following property.
Let $0 < T_1 < T_2 < T_*$ be given and suppose that $g(t)$ is a a solution to Ricci-DeTurck flow with background metric $g_{wp}(t)$ on $\bar M_{[0, 2 u_*]}$ for times $[T_1, T_2]$.
Suppose for all $p \in \bar M_{[0, 2u_*]}$,
\begin{align}
|g(T_1) - g_{bg}|_{2, \eta; r_0|\Rm|^{-1/2}}(p, T_1) \leq \epsilon F_{full}(p, T_1),
\end{align}
and for all $(p,t) \in \bar M_{[u_*, 2u_*]} \times [T_1, T_2]$,
\begin{align}
|g(t) - g_{bg}|_{2, \eta; r_0|\Rm|^{-1/2}}(p, t) \leq \epsilon F_{full}(p, t).
\end{align}
Then for all $(p, t) \in \bar M_{[0,u_*]} \times [T_1, T_2]$,
\begin{align}
|g(t) - g_{bg}|_{2, \eta; r_0|\Rm|^{-1/2}}(p, t) \leq C\epsilon F_{full}(p, t).
\end{align}
\end{theorem}
\begin{proof} The $C^{0}$ estimate follows from Lemmas \ref{lemma:asymmetric_prish_barrier}, \ref{lemma:asymmetric_tip_barrier}, and \ref{lemma:gluing_asymmetric}. The $C^{2, \eta}$ estimate follows from interior Schauder estimates, once we have the $C^{0}$ estimate. \end{proof}
\subsection{Control in the productish region}\label{section:asym_productish} In this section we control the Ricci DeTurck flow in the productish region of the warped-product solution $g_{wp}(t)$. This uses the general sub- and supersolutions from Appendix \ref{nearly_cnst_pde_sect}.
Recall the definition
$\Lambda_{\Rm} = \max_{h \in Sym_2(M) : |h| = 1}\ip{\Rm[h]}{h}$ from Section \ref{section:rdt}. Here, and in this section, by default we are taking all inner products, curvatures, and covariant derivatives with respect to $g_{wp}(t)$. First we write down a bound for $\Lambda_{\Rm}$ on our warped product solution $g_{wp}(t)$. Remember we are assuming that $g_{mp} = g_{wp}(0)$ is an $\Rm$-permissible model pinch. \begin{lemma}\label{rmplus_prish}
There is a constant $C(g_{mp})$ such that in the productish region $g_{wp}(t)$ satisfies $
\Lambda_{\Rm} \leq (q-1) u^{-1} + C u^{-1}v
= \oh \mu u^{-1} + C u^{-1}v. $ \end{lemma} \begin{proof}
Note the use of the metric in $\Rm[h]$ to contract tensors. Let us write
\begin{align}
\left(\Rm_{g_1}[h; g_2]\right)_{ef}
= (g_2)^{ab}(g_2)^{cd}\left(\Rm_{g_1}\right)_{acef}h_{bd}
\end{align}
so that $\Rm[h] = \Rm_{g_{wp}}[h; g_{wp}]$. Now we can compute some scaling for the components of $\Rm_{g_{wp}}$ given by Corollary \ref{prish_curvature_control}.
\begin{align}
\left(u\Rm_{g_{S^q}} \right)[h; g_{wp}]
= \left(u\Rm_{g_{S^q}} \right)[h; u g_{S^q}]
&= u^{-1} \left( \Rm_{g_{S^q}} \right)[h; g_{S^q}].
\end{align}
Therefore
\begin{align}
\max_{|h|_{g_{wp}} = 1}
\ip{(u \Rm_{g_{S^q}})[h; g_{wp}]}{h}_{g_{wp}}
&= u^{-1} \max_{|h|_{g_{S^q}} = 1}\ip{\Rm_{g_{S^q}}[h; g_{S^q}]}{h}_{g_{S^q}} \\
&\mathrel{\mathop:}= u^{-1}\Lambda_{S^q} = u^{-1}(q-1). \label{lambdacalc0}
\end{align}
Similarly,
\begin{align}
\max_{|h|_{g_{wp}} = 1}
\ip{(w \Rm_{g_{F}})[h; g_{wp}]}{h}_{g_{wp}}
&= w^{-1}\Lambda_{F}.\label{lambdacalc1}
\end{align}
Let $k = \max\left( \frac{\Lambda_F}{\Lambda_{S^q}}, (1+c)\frac{\mu_F}{\mu}\right)$. By the $\Rm$-permissible assumption, and the assumption \ref{w_big} of all model pinches, $W_0(u + \mu t) \geq k \cdot (u + \mu t)$. Now, we use that $w$ is barricaded in the productish region, i.e. it satisfies \ref{conc:prish_barrier} of Definition \ref{productish_barricaded}:
\begin{align}
\frac{w}{u}
&\geq \frac{W_0(u + \mu t)-\mu_F t}{u} - \frac{DVW_0(u + \mu t)}{u} \\
&\geq \frac{k\cdot (u + \mu t) - \mu_F t}{u} - \frac{kDV\cdot (u + \mu t)}{u}.
\end{align}
Next, first using $k \geq (1+c) \frac{\mu_F}{\mu}$ and then $k \geq \frac{\Lambda_F}{\Lambda_{S^q}}$,
\begin{align}
\frac{w}{u}
&\geq k + \frac{kDV\cdot (u + \mu t)}{u} \geq \frac{\Lambda_F}{\Lambda_{S^q}} - C u^{-1}V
\end{align}
for some $C$ depending only on $g_{wp}$ and the parameter $D$. Therefore, coming back to \eqref{lambdacalc1}, and increasing $C$,
\begin{align}
\max_{|h|_{g_{wp}} = 1}
\ip{(w \Rm_{g_{F}})[h; g_{wp}]}{h}_{g_{wp}}
&\leq u^{-1}\Lambda_{S^{q}} + Cu^{-1}V.\label{lambdacalc2}
\end{align}
Now we put together \eqref{lambdacalc0} and \eqref{lambdacalc2}. Since $\Rm_{g_F}$ and $\Rm_{g_{S^q}}$ act only on the orthogonal components $Sym_2(TF) \subset Sym_2(TM)$ and $Sym_2(TS^q) \subset Sym_2(TM)$ respectively, we can take the maximum of the two pieces to find
\begin{align}
\max_{|h|_{g_{wp}} = 1}
\ip{(u \Rm_{g_S^q} + w \Rm_{g_F})[h; g_{wp}]}{h}_{g_{wp}}
\leq u^{-1}\Lambda_{S^{q}} + Cu^{-1}V.\label{lambdacalc3}
\end{align}
Finally, adding in $\Rm_{warp}$ can only change this result by something proportional to its norm. So, increasing $C$,
\begin{align}
\max_{|h|_{g_{wp}} = 1}
\ip{\Rm_{g_{wp}}[h; g_{wp}]}{h}_{g_{wp}}
&=
\max_{|h|_{g_{wp}} = 1}
\ip{(u \Rm_{g_S^q} + w \Rm_{g_F} + \Rm_{warp})[h; g_{wp}]}{h}_{g_{wp}}\\
&\leq u^{-1}\Lambda_{S^{q}} + Cu^{-1}V.\label{lambdacalc3}
\end{align}
\end{proof}
Consider a solution to Ricci-DeTurck flow around $g_{wp}(t)$ given by $g(t) = g_{wp}(t) + h(t)$. Let $y = |h(t)|^2$. By the equation for the evolution of the norm of the perturbation \eqref{dtevo_square}, in the productish region $y$ satisfies $
\square_{g_{wp}, g} y \leq u^{-1}\left(2\mu + Cv + Cy^{1/2}\right) y $, or, just rewriting, \begin{align}
(\square_{g_{wp}, g} - 2 \mu u^{-1})y\leq C u^{-1}\left(v + y^{1/2}\right) y. \label{eq:54} \end{align} We now use the supersolutions found in Appendix \ref{nearly_cnst_pde_sect} to control $y$ in the productish region. The parameters $D$, $u_*$, $\sigma_*$ here are not the same as the parameters of control of $g_{wp}(t)$. Recall $\hat u = u + \mu t$ and $Q = u^{-1}\hat u$.
\begin{lemma}\label{lemma:asymmetric_prish_barrier}
Suppose $D > \berr D(g_{wp})$, $0<\epsilon<1$, $u_{*} < \barr u_{*}(g_{wp}, D), \sigma_* < \barr \sigma_*(g_{wp}, D)$, and $T_* < \barr T_*(g_{wp}, D)$. Suppose $0 < T_1 < T_2 < T_*$.
Set
\begin{align}
\Omega_{prish, [T_1, T_2]} =
\left\{(p, t) : u < u_*, \sigma = \frac{u}{t \nu(t)} > \sigma_*, t \in [T_1, T_2]\right\}.
\end{align}
Suppose $g(t) = g_{wp}(t) + h(t)$ solves Ricci-DeTurck flow around $g_{wp}(t)$. Let $y = |h|$, $Y^+_{prish} = (1 + D V)Q\left(V_0 \circ \hat u \right) = (1+DV)V$, and $\bar Y^+_{prish} = \epsilon Y^+$. If $y < \bar Y^+$ on the parabolic boundary of $\Omega_{prish, [T_1, T_2]}$, then $y < \bar Y^+$ in $\Omega_{prish, [T_1, T_2]}$. \end{lemma}
\begin{proof}
In the end, we will choose $\barr u_*$, $\berr \sigma_*$, and $\barr T_*$ to ensure that $DV < 1$ and $\bar Y^+$ is smaller than $\oh$ in the region under consideration. (We may do this since we can make $V$ arbitrarily small by Lemma \ref{lemma:V_small}.) Therefore equation \eqref{rdt_norm} is valid.
By Lemma \ref{sup_solns} we have that, for some $c > 0$,
\begin{align}
\label{eq:112}
\left(\square_{g_{wp}} - 2 \mu u^{-1}\right)\bar Y^{+} \geq (c D) u^{-1}v \bar Y^{+}.
\end{align}
Since $(\bar Y^+)^{1/2} \leq CV$, we find by decreasing $c$, \begin{align}
\label{eq:120}
\left(\square_{g_{wp}} - 2 \mu u^{-1}\right) \bar Y^+ \geq (c D) u^{-1} \left( v + (\bar Y^+)^{1/2} \right) \bar Y^+. \end{align}
We can change the $\square_{g_{wp}}$ to $\square_{g_{wp}, g}$. As long as $y < \bar Y^{+}$ we have \begin{align}
|\square_{g_{wp}}\bar Y^{+} - \square_{g_{wp},g}\bar Y^{+}|
\leq C (\bar Y^+)^{1/2}|\nabla \nabla \bar Y^{+}|
\leq Cu^{-1}v(\bar Y^+)^{3/2}. \end{align}
In the second inequality we use Lemma \ref{dd1z_bnd}, and the bound $|\nabla \nabla u| < Cv$ (see the calculation in Lemma \ref{lemma:ycontrol_productish}). Again decreasing $c$, we have \begin{align}
\left( \square_{g_{wp}, g} -2 \mu u^{-1}\right) \bar Y^{+} \geq
(cD) u^{-1} \left( v + (\bar Y^+)^{1/2} \right) \bar Y^+ \label{Yplus_pdi} \end{align} The lemma follows from the maximum principle by comparing \eqref{Yplus_pdi} to the evolution for $y$ \eqref{eq:54} and choosing $D$ large enough. \end{proof}
\subsection{Control in the tip region}\label{assym_tip_section} Recall the rescaled coordinates, $\alpha(t) = t \nu(t)$, $\partial_{\theta} = \alpha \partial_t$, and $\sigma = u/\alpha$.
\begin{lemma}\label{lemma:asymmetric_tip_barrier}
Let $u_*$, $\sigma_*$, $\zeta_*$, $r_0$, and $\epsilon < \barr \epsilon(g_{wp}, r_0)$ be given. Let $F$ be the function from Lemma \ref{bry_rdt_supsoln} and let $\bar F = \epsilon F$. There is a $T_*(u_*, \sigma_*, g_{wp})$ such that we have the following.
Suppose $g(t) = g_{wp}(t) + h(t)$ is a solution to Ricci-DeTurck flow with background metric $g_{wp}(t)$ in $\bar M_{[0, u_*]}$, on a time interval $[T_1, T_2]$, and $T_2 < T_*$. Suppose
\begin{align}
\twoeta{h}{r_0|\Rm|^{-1/2}}{g_{wp}} < \bar F \quad \text{for} \quad t = T_1 \text{ and } \sigma < \nu^{-1/2}(T_1)\zeta_*,
\end{align}
\begin{align}
|h|_{g_{wp}} < \bar F \quad \text{for} \quad t \in [T_1, T_2]\text{ and }\sigma \in [\sigma_*, \nu^{-1/2}\zeta_*].
\end{align}
Then $|h|_{g_{wp}} \leq \bar F$ for $\sigma < \sigma_*$ and $t \in [T_1, T_2]$. \end{lemma}
\begin{proof}
We will choose $\barr \epsilon$ sufficiently small in the end. We use a contradiction-compactness argument to move the situation to Ricci-DeTurck flow around the Bryant soliton crossed with a euclidean factor.
For contradiction, assume that there is no such $T_*$. This means that there is a sequence of counterexamples: there are solutions $g^{(i)} = g_{wp} + h^{(i)}$ to the Ricci-DeTurk flow around $g_{wp}$, defined on intervals $[T_1^{(i)}, T_2^{(i)}]$, satisfying the conditions of the Lemma, but $|h^{(i)}(p^{(i)}, T_2^{(i)})| = \bar F(\sigma(p^{(i)}, T_2^{(i)}))$ for some sequence $p^{(i)}$ with $\sigma(p^{(i)}, T_2^{(i)}) \leq \sigma_*$ and $T_2^{(i)} \searrow 0$. Let $\sigma^{(i)} = \sigma(p^{(i)}, T_2^{(i)})$. We may pass to subsequence so that the $\sigma^{(i)}$ converge to some $\sigma^{(\infty)} \leq \sigma_*$.
Let $\alpha^{(i)} = \alpha(T_1^{(i)})$. We claim that there is a $T_{**}$ depending on $g_{wp}$ and $\sigma_*$ such that $T_2^{(i)} - T_1^{(i)} \geq \alpha^{(i)}T_{**}$. Indeed for $t < T_2^{(i)}$, $|h^{(i)}|$ is bounded by $\bar F$. Therefore we can apply the interior Schauder estimates at the scale $\alpha(T_1^{(i)})$, to get a bound on how far $|h^{(i)}|$ can move. Here we use our bound $\twoeta{h^{(i)}(T_1)}{r_0|\Rm|^{-1/2}}{g_{wp}} < \bar F$, as well as our control on the geometry of $g_{wp}$ at scale $\alpha$.
Now, this says that in terms of the rescaled time coordinate $\theta$, the time difference is bounded from below. Indeed, since by definition $d\theta = \alpha^{-1}dt$:
$$\theta(T_2^{(i)}) - \theta(T_1^{(i)})
= \int_{T_1^{(i)}}^{T_2^{(i)}} \alpha^{-1} dt
\geq \int_{T_1^{(i)}}^{T_1^{(i)}+\alpha^{(i)}T_{**}} \alpha^{-1}(t) dt
\geq T_{**}
\frac
{\alpha\left(T_1^{(i)} \right)}
{\alpha\left(T_1^{(i)} + \alpha(T_{1}^{(i)}) T_{**} \right)}.
$$
For the last inequality we just take the value of the decreasing integrand at the right endpoint and multiply by the length of the interval.
Since $\frac{t|\partial_t\alpha|}{\alpha} + \frac{t^2|\partial_t^2\alpha|}{\alpha}$ is bounded (using assumption \ref{modelpinch_reg} of Definition \ref{definition:model_pinch}) and $\alpha(T_1^{(i)}) = o(T_1^{(i)})$ we can argue by using a Taylor expansion on the denominator that the right hand side is bounded from below by some $\Theta_* > 0$. So, passing to a subsequence, the sequence $\theta(T_2^{(i)}) - \theta(T_1^{(i)})$ either converges to $\infty$ or converges to some $\Theta_1 > 0$.
Let $G_{wp}$ be the family of metrics $G_{wp} = \alpha^{-1}(\sigma^{-1})^*g_{wp}$ which is $g_{wp}$ modified by scaling by $\alpha^{-1}$ and pulling back by $\sigma$. Also let $G^{(i)} = \alpha^{-1}(\sigma^{-1})^* g^{(i)}$ and $H^{(i)} = G^{(i)} - G_{wp} = \alpha^{-1}(\sigma^{-1})^* h^{(i)}$.
Now $G^{(i)}$ satisfies \begin{align}
\partial_{\theta} G^{(i)} = -2 \Rc[G^{(i)}] - \mathcal{L}_{X + V[G^{(i)}, G_{wp}]}G^{(i)} - \beta G^{(i)},\label{rescaled_rcf} \end{align} for $\theta \in [\theta(T_1^{(i)}), \theta(T_2^{(i)})]$. Here $X$ is the vector field $\partial_\theta \sigma$.
Translate the $\theta$ intervals so that the times $\theta(T_2^{(i)})$ all land at time $0$. By Corollary \ref{lemma:barricaded_convergence}, the background metrics $G_{wp}$ converge to the Bryant soliton crossed with $\mathbb{R}^{dim(F)}$, and the vector field $X$ converges to the soliton vector field. Passing to a subsequence, the $H^{(i)}$ converge to a solution $H$ of Ricci-DeTurck flow around the Bryant soliton, modified by the Bryant soliton vector field $X$. (Note that the term $\beta G$ in \eqref{rescaled_rcf} converges to zero.) The time interval is either $\theta \in (-\infty, 0]$, or $\theta \in [-\Theta_1, 0]$. In the second case, we can translate the regularity we assume at time $T_1$, and we find that the bounds in Theorem \ref{bry_stabil} are satisfied for some $C_{reg}$ (independent of $\epsilon$). So, provided we take $\barr \epsilon$ small enough in this lemma to satisfy Theorem \ref{bry_stabil}, we can apply Theorem \ref{bry_stabil}.
However, at time $0$ and at some point $p \in {Bry} \times \mathbb{R}^{dim(F)}$ with $\sigma_{Bry}(p) = \sigma^{(\infty)}$, we will have $|H| = |\bar F|$. This contradicts the strict inequality in the conclusion of Theorem \ref{bry_stabil}. \end{proof}
\subsection{Buckling Barriers} In this lemma, we show that the function $
Y^+ = (1 + D V)Q^2 (V_0 \circ \hat u)^2, $
which we use as a barrier for $|h|^2$ in the productish region, crosses the function $F^2$, which we use as a barrier in the tip region. This shows that they ensure each others' boundary conditions.
The following Lemma deals with the unscaled functions $Y^+$ and $F^2$. Of course, the inequalities \eqref{eq:cross_ineq1} and \eqref{eq:cross_ineq2} also hold for $\bar Y^+ = \epsilon^2 Y$ and $\bar F^2 = \epsilon^2 F^2$.
\begin{lemma}\label{lemma:gluing_asymmetric}
Let the constant $D$, in the definition of $Y^+$ be given. There are $\sigma_* > 0$, $\sigma_2 > 0$, $\zeta_* >0$, and $b \in \mathbb{R}_+$ such that we have the following inequalities.
For $t < T_*$, at $\sigma = \sigma_*$, we have
\begin{align}\label{eq:cross_ineq1}
bF^2 < Y^+.
\end{align}
For $t < T_*$, and $\sigma \in [\sigma_2, \zeta_*\nu^{-1/2}]$, we have
\begin{align}\label{eq:cross_ineq2}
Y^+ < bF^2.
\end{align} \end{lemma} \begin{proof}
Below $c_i$ are positive constants, and all asymptotics are as $\sigma \to \infty$ and $t \searrow 0$. Recall the asymptotics of $F$ from Theorem \ref{bry_stabil}:
\begin{align}
F &= c_1 \sigma^{-1} - c_2 \sigma^{-2}\log\sigma + o(\sigma^{-2}\log \sigma),\\
F^2 &= \sigma^{-2}\left(c_3 - c_4 \sigma^{-1} \log \sigma + o(\sigma^{-1}\log \sigma) \right).
\end{align}
Recall the asymptotics of $V$ from \eqref{lemma:V_small}:
\begin{align}
V &= c_5 \sigma^{-1}\left(1 + O(\nu + \nu^2 \sigma) \right),\\
Y^+
&= (1 + DV)V^2 \\
&= \sigma^{-2}\left(c_6 + c_7D \sigma^{-1} + O(\nu+ \nu^2 \sigma) + O(D\sigma^{-2}) \right).
\end{align}
Letting $d = b c_3 - c_6$, we find,
\begin{align}
\sigma^2 (bF^2 - Y^+) =
d
- \sigma^{-1}\left(c_4 \log \sigma + c_7 D \sigma^{-1} + o(\log \sigma) + O(D \sigma^{-2}) \right) + O(\nu + \nu^2\sigma)
\end{align}
Now choose $\sigma_*$ large enough so that for $\sigma > \sigma_*$ the asymptotic terms $o(\log \sigma)$ and $O(D\sigma^{-2})$ above apply well. Furthermore, since $\sigma \nu^2 < \zeta_* \nu^{3/2}$ in the region $\{\sigma \leq \zeta_* \nu^{-1/2}\}$ under consideration, we can choose $T_*$ small enough so that the $O(\nu + \nu^2 \sigma)$ term is smaller, in absolute value, than $d/2$. Specifically, for $\sigma \in [\sigma_*, \zeta_* \nu^{-p}]$ and $t < T_*$ we have
\begin{align}
&\tfrac{1}{2}d - \tfrac{3}{2} \sigma^{-1}\left( c_4\log \sigma + c_7 D \sigma^{-1} \right) \\
&\leq \sigma^2 (bF^2 - Y^+) \\
&\leq
\tfrac{3}{2}d - \oh \sigma^{-1}\left( c_4 \log \sigma + c_7 D \sigma^{-1} \right).
\end{align}
Now choose $b = b(\sigma_*, D)$ so that $d = b c_3 - c_6$ is positive but small enough that
\begin{align}
\tfrac{3}{2}d - \oh \sigma_*^{-1} \left( c_4\log \sigma_* + c_7 D \sigma_*^{-1} \right) < 0
\end{align}
so the desired inequality holds at $\sigma = \sigma_*$. Then choose $\sigma_1$ large enough so that
\begin{align}
\oh d - \tfrac{3}{2}\sigma^{-1}\left( c_4\log \sigma + c_7 D \sigma^{-1} \right) > 0
\end{align}
for $\sigma > \sigma_1$. Then the desired inequality for $\sigma \in [\sigma_1, \zeta_* \nu^{-1/2}]$ also holds, for small enough times. \end{proof}
\section{Constructing asymmetric forward evolutions}\label{asymmetric} \begin{figure}\label{np_basic}
\end{figure}
In this section we prove Theorem \ref{theorem:unsymmetrical_flow}. We begin by gluing the control from Theorem \ref{local_stability} to a uniformly smooth Ricci flow using pseudolocality and regularity in the region strictly away from $u = 0$. Then we will construct mollified metrics and take a limit.
Assume all the setup of Section \ref{section:local_stability}, including an $\Rm$-permissible model pinch $g_{mp}$ with a forward Ricci flow $(\bar M, g_{wp}(t))$. Fix a compact manifold $\bar N$ with an open subset $U$ and a diffeomorphism $\Phi : U \to \bar M$. Assume that for some time interval $(0, T_{max}]$ and for some $u_{max} > 0$, the image of $\Phi$, $\Phi(U) \subset \bar M$, contains the set $\{(p, t): u(p, t) < u_{max}\}$. Hereafter we suppress the diffeomorphism $\Phi$ and consider all of the functions that we had on $\bar M$ related to $g_{wp}$-- such as $u(p, t)$, $\sigma(p,t) = u(p, t)/\alpha(t)$, $w(p, t)$, and $v(p, t)$-- as functions on $\bar N \cap U$. The function $u_0: \bar N \cap U \to \mathbb{R}_{\geq 0}$ is still the value of $u$ for $g_{mp}$, as a notational convenience we extend the function $u_0$ to all of $\bar N$ so that $u_0 \geq u_{max}$ in $ \bar N \setminus U$. For $u_1, u_2$ we define, similarly to how we defined $\bar M$, $\bar N_{[u_1, u_2]} = \{p: u_0(p) \in [u_1, u_2]\}$; for example $\bar N_{[u_1, \infty)} = \{p \in U: u_0(p) \geq u_1\} \cup \bar N \setminus U$.
\subsection{Flowing complete manifolds near smoothed model pinches}
\begin{lemma}\label{lemma:global_stable}
Suppose $\epsilon < \barr \epsilon(g_{mp})$, $r_0 < \barr r_0(g_{mp})$, $u_* < \bar u_*(g_{mp})$, and $B>0$. Suppose $g_{bg}(t)$ is a complete and smoothly time-dependent metric on $\bar N$ which agrees with $g_{wp}(t)$ in $\bar N_{[0, 4u_*]} \times [0, T_*]$, and $g_{init}$ is a smooth complete metric on $\bar N$. Suppose,
\begin{itemize}
\item $g_{init}$ is close to $g_{\#} = g_{bg}(T_1)$ globally:
\begin{align}
\sup_{p \in \bar N_{[0, 4u_*]}}&|g(T_1) - g_\#|_{2, \eta; r_0|\Rm|^{-1/2}_{g_\#}, g_\# } \leq \epsilon F_{full}(p, T_1).\\
\sup_{p \in \bar N_{[4u_*, \infty)}}&|g(T_1) - g_\#|_{2, \eta; r_0, g_\#} \leq \epsilon
\end{align}
\item $g_{bg}$ does not change much at any point $p \in \bar N_{[u_*/2, \infty)}$:
\begin{align}
\sup_{p \in \bar N_{[u_*/2, \infty)}}\sum_{k=0}^3\left|\partial_t \left(\nabla^{g_{bg}(0)}\right)^kg_{bg}(t)\right|_{g_{bg}(0)} \leq B
\end{align}
\end{itemize}
Then there is a $T_*$ and $C$, depending on $g_{mp}$, $u_*$, $r_0$, and $\epsilon$ as well as $g_{init}$ restricted to the compact set $\bar N_{[u_*/2, \infty)}$, with the following property. If $T_1 < T_*$ then there is a solution $g(t)$ to Ricci-DeTurck flow with background metric $g_{bg}$, which exists at least on the time interval $[T_1, T_*]$, with $g(T_1) = g_{init}$ and for all $t \in [T_1, T_*]$,
\begin{align}
\sup_{p \in \bar N}\twoeta{g(t) - g_{bg}}{r_0|\Rm|^{-1/2}}{g_{bg}}(p, T_1) \leq C\epsilon F_{full}(p, T_1).
\end{align} \end{lemma}
\begin{proof}
By standard theory there is a solution to Ricci flow on some time interval $[T_1, T_{final}]$ with $g(T_1) = g_{init}$.
Let $u_1 = (5/8)u_*$ and $u_2 = (7/8)u_*$ so $u_*/2 < u_1 < u_2 < u_*$.
We can apply pseudolocality followed by regularity (i.e. Lemma A.5 of \cite{topping}) for any point in $\bar N_{[u_1, \infty)}$, which gives us control $|(\nabla^{g_{init}})^k \Rm_{g_{init}}|_{g_{rcf}} < C$ for $k = 0, 1,2,3$, for $t < \min(T_*, T_{final})$. Here $C$ and $T_*$ depend on the things $C$ and $T_*$ are allowed to in the statement of the theorem. We used compactness of $\bar N_{[u_1, \infty)}$ to get a lower bound on the radius at which we can apply pseudolocality. By differentiation the Ricci flow equation, we get that $|\partial_t \left(\nabla^{g_{init}}\right)^k g_{rcf}|_{g_{init}} \leq C$ for $k = 0, 1, 2, 3$.
Now let $\Psi:\bar N \times [T_1, T_{final}) \to \bar N$ be the solution to $\partial_t \Psi = \Delta_{g_{rcf}, g_{bg}} \Psi$ with $\Psi(p, 0) = p$, so that $g = (\Psi^{-1})^*g_{rcf}$ is a solution to Ricci-DeTurck flow with background metric $g_{bg}$. (The existence of $\Psi$, as well as of $g_{rcf}(t)$, is given by Theorem 6.7 of \cite{Shi}.) In local harmonic coordinates, the equation $\partial_t \Psi = \Delta_{g_{rcf}, g_{bg}} \Psi$ has the form
\begin{align}
\partial_t \Psi^l
= g^{ij} \partial_i \partial_{j}\Psi^l
- g^{ij}\left( \Gamma_{g_{rcf}} \right)_{ij}^k \partial_{k}\Psi^l
+ g^{ij} \left(\left(\Gamma_{g_{bg}}\right)^l_{mk} \circ \Psi \right)\partial_i\Psi^m\partial_{j}\Psi^k.
\end{align}
We apply regularity within $\bar N_{[u_2, \infty)}$. We can use compactness again to get a lower bound on the radius at which we can find harmonic coordinates around any point, as well as to get a bound on derivatives of $\Gamma_{g_{rcf}}$ and $\Gamma_{g_{bg}}$ in those coordinates. All in all, we can get that in $\bar N_{[u_2, \infty)}$ and for $t < \min(T_*, T_{final})$ we have $|\partial_t \nabla^k \Psi| \leq C$ for $k = 0, 1, 2, 3$ (say with connection and norms with respect to $g_{init}$, although now we know that all the metrics are comparable).
Now we can take $T_*$ small enough so that for $t < \min(T_*, T_{final})$ we have $\Psi(\bar N_{[u_2, \infty)}, t) \subset \bar N_{[u_*, \infty)}$ and $\Psi^{-1}(\bar N_{[u_2, \infty)}, t) \subset \bar N_{[u_*, \infty)}$. Then we can use our estimates on time derivatives to restrict $T_*$ and get the bound $|(\nabla^{g_{bg}})^k\left( g(t) - g_{bg}(t) \right)| \leq 2\epsilon F_{full}$ in $\bar N_{[u_*, \infty)}$ for $k = 0, 1, 2, 3$. Now we can apply Lemma \ref{local_stability} (with a larger $\epsilon$) to get the desired $C^{2, \eta}$ control in $\bar N_{[0, u_*]}$ for $t < \min(T_*, T_{final})$.
Finally, we can use our $C^2$ bounds on $g(t)-g_{bg}(t)$ to estimate that the curvature at time $\min(T_*, T_{final})$ is bounded, we find $T_{final} > \min(T_*, T_{final})$ and so the Ricci-DeTurck flow exists for $t \in [T_1, T_*]$.
\end{proof}
\subsection{Setup of the background metric}\label{section:bg_setup}
Now, we wish to set up a background metric to use for Ricci-DeTurck flow. We chose constants $u_*$ and $u_{\dagger}$ with $4 u_{*} < u_{\dagger}$ and $4u_{\dagger}<u_{max}$. Let $\eta:[0, \infty) \to \mathbb{R}$ be a fixed smooth cutoff function satisfying $\eta(x) \in [0, 1]$ and \begin{align}
\eta(x) = 1 \text{ for } x < 1
\quad
\eta(x) = 0 \text{ for } x > 2, \end{align} and define $\eta_{r}(x) = \eta(x/r)$. Then define \begin{align}
g_{bg}(t)
=
\eta_{u_\dagger}\left(u_0\right) g_{wp}(t)
+
\left(1-\eta_{u_\dagger}(u_0)\right)g_{init}. \label{eq:gbg_def} \end{align} We define $u_0(p) \geq u_{max}$ for $p \not \in U$. So, $g_{bg}(t)$ is a time-dependent metric which agrees with $g_{wp}(t)$ for points $p \in \bar N_{[0, u_\dagger]}$, and agrees with $g_{init}$ for points $p \in \bar N_{[2u_{\dagger}, \infty)}$.
Note that we can always choose $T_*$ small enough (depending on $u_*$ and $u_\dagger$) so that for $t < T_*$ we have $u_0(p) < u_\dagger$ wherever $u(p,t) < 4u_*$. Therefore $g(p,t) = g_{wp}(t)$ on the set $\{(p,t) : u(p,t) < 4u_*\}$, and $g_{bg}$ will satisfy the hypotheses of Lemma \ref{lemma:global_stable}.
\subsection{Setup of the mollified initial metrics}\label{section:mollified_setup} As in the proof of Theorem \ref{theorem:model_pinch_flow}, we will construct the forward evolution from $g_{init}$ as a limit of mollified flows. A parameter $m \in [0, 1]$ determines the space scale of the mollification. For $T_1^{(m)}$ to be chosen, we define the mollified initial metric $g_{init}^{(m)}$ by \begin{align}
g^{(m)}_{init} = \eta_{m}\left( u_0 \right) g_{wp}(T_1^{(m)}) + \left( 1 - \eta_m\left(u_0 \right) \right) g_{init}. \end{align} Let $h^{(m)}_{init} = g^{(m)}_{init} - g_{bg}(T_1^{(m)})$. We derive bounds on $h_{init}^{(m)}$ and its derivatives.
\begin{lemma}\label{inith_close}
There is a constant $C > 0$ such that for $m < \barr m$ and $T_1^{(m)} < \barr T_1^{(m)}$, we have
\begin{align}
\sup_{p \in \bar N_{[0, 4 u_*]}}&|h^{(m)}_{init}|_{2, \eta; r_0|\Rm|^{-1/2}, g_{bg}(T_1)} \leq C\epsilon_0 F_{full} \\
\sup_{p \in \bar N_{[4u_*, \infty)}}&|h^{(m)}_{init}|_{2, \eta; r_0, g_{bg}(T_1)} \leq C\epsilon_0
\end{align} \end{lemma} \begin{proof}
Note that $g^{(m)}_{init}$ agrees with $g_{bg}(T_1^{(m)})$ in $\bar N_{[0, m]}$ and in $\bar N_{[2u_{\dagger}, \infty)}$ so we just have to worry about the compact set $\bar N_{[m/2, 4u_{\dagger}]}$. In the region $\bar N_{[0, u_{\dagger}/2}$ which is strictly in the interior of where $g_{bg}(T_1)$ agrees with $g_{wp}(T_1)$, we can estimate
\begin{align}
|h^{(m)}_{init}|_{2, \eta; r_0|\Rm|^{-1/2}, g_{bg}(T_1^{(m)})}
&\leq C|g_{init} - g_{mp}|_{2, \eta; r_0|\Rm|^{-1/2}, g_{mp}} \\
&+ C\left(|g_{init}-g_{mp}|_{2, \eta; r_0|\Rm|^{-1/2}, g_{wp}(T_1^{(m)})} - |g_{init} - g_{mp}|_{2, \eta; r_0|\Rm|^{-1/2}, g_{mp}} \right) \\
&+ C\left(|g_{mp} - g_{wp}(T_1^{(m)})|_{2, \eta; r_0|\Rm|^{-1/2}, g_{wp}(T_1^{(m)})}\right).
\end{align}
The constant $C$ comes from estimating terms coming from the cutoff function $\eta$.
The first line is bounded by $C \epsilon_0 F_{full}$ by the assumption on $g_{init}$, and the following lines can be bounded by $C \epsilon_0 F_{full}(m)$ by taking $T_1^{(m)}$ sufficiently small (using the convergence of $g_{wp}(t)$ to $g_{mp}$ as $t \searrow 0$).
In the region $\bar N_{[u_\dagger/2, \infty)}$ we can use compactness similarly. \end{proof}
\subsection{Global control and convergence} The following lemma implies Theorem \ref{theorem:unsymmetrical_flow}. \begin{lemma}
Suppose we choose $\epsilon_0 < \bar \epsilon/C$ where $C$ is the constant from Lemma \ref{inith_close} and $\bar \epsilon$ is the constant from Lemma \ref{lemma:global_stable}.
Let $g^{(m)}(t)$, $t \in [T_1^{(m)}, T_{final}^{(m)})$ be the Ricci-DeTurck flow starting from $g^{(m)}_{init}$. Then $T_{final}^{(m)} > T_*$ for some $T_*$ independent of $m$.
There is a sequence $m_j \searrow 0$ such that the time-dependent metrics $g^{(m_j)}(t)$ converge to a solution $g(t)$ of Ricci-DeTurck flow around $g_{bg}(t)$, with $g(0) = g_{init}.$ The convergence happens in $C^{2, \eta/2}_{loc}\left( \bar M \times [0, T_*]\setminus P\times \{0\} \right)$, where $P = \bar M \setminus M$. Furthermore, the DeTurck vector fields $V[g^{(m)}, g_{bg}]$ converge, in $C^{1, \eta/2}_{loc}$, to $V[g(t), g_{bg}]$. \end{lemma} \begin{proof}
By the previous sections, Lemma \ref{lemma:global_stable} applies to $g_{init}^{(m)}$ with background metric $g_{bg}$. This implies that in any set $K$ compactly contained in $M$ we have $C^{2, \eta}$ control on $g^{(m)}$ for $t \in [0, T_*]$. Also, for any $t_0 > 0$ we have $C^{2, \eta}$ control on $g^{(m)}$ for $t \in [t_0, T_*]$. Therefore, we can apply Arzela-Ascoli and a diagonalization argument to get convergence on $C^{2, \eta/2}_{loc} \left( \bar M \times [0,T_*] \setminus P \times \{0\} \right)$ of a subsequence.
Since the convergence happens in $C^{2, \eta/2}$, the equation passes to the limit. Since $V[g^{(m)}, g_{bg}]$ depends on one derivative of $g^{(m)}$ with respect to $g_{bg}$, we get the convergence of the vector fields. \end{proof}
\appendix \section{Nearly constant regions of reaction-diffusion equations}\label{nearly_cnst_pde_sect} Let $\mu > 0$ and $c_v \in \mathbb{R}$. We study solutions to \begin{align}
\label{eq:29}
\square u
&= -\mu + c_v u^{-1}|\nabla u|^2 \\
&= -\mu + c_v v \label{mypde} \end{align}
where we have defined $v = u^{-1}|\nabla u|^2$. We consider $u:M \times [T_1,T_2] \to \mathbb{R}$ which satisfies \eqref{mypde}, on an evolving Riemannian manifold $(M, g(t))$ which satisfies Ricci flow. (If $(M, g(t))$ does not satisfy Ricci flow, there is another term in \eqref{v_eqn_crude} below.) The value of $c_v$ does not come into play very much here. We are interested investigating regions where $v$ is small, and controlling other functions (and in particular $v$) in terms of $u$. All constants in this section may implicitly depend on $c_v$ and $\mu$.
Applying the parabolic version of the Bochner formula ((1.6) of \cite{weakhaslhofernaber}) yields the following: $v$ satisfies \begin{align}
\square v &= u^{-1}\mu v + E_{error} v^2 \label{EQ:22} \end{align} where $E_{error}:M \times [T_1, T_2] \to \mathbb{R}$ satisfies \begin{align}
\label{v_eqn_crude}
-C \left(1 +\frac{|\nabla \nabla u|^2}{v^2} \right)
\leq E_{error}
\leq C \end{align}
for some constant $C$. Using this equation, Lemma \ref{sup_solns} will allow us to control $v$ from above, and also from below if we obtain a priori that $\frac{|\nabla \nabla u|^2}{v^2}$ is bounded.
We will use functions of $u$ and $t$ to create sub- and supersolutions to other PDE, and in particular to control $v$.
If $F$ depends on $u$ and $t$ alone then \begin{align}
\label{eq:116}
\square F = \left( (\square u)\dd1 F + \dd{t} F - v \dd{2} F \right) \left( u^{-1} F \right). \end{align} Here, we use the notation $\dd{k}{F} = u^k \frac{1}{F}\partial_u^k F$ and $\dd{t} F = u \frac{1}{F} \partial_t F$. These are both invariant under scaling the system or $F$, and in our situation they will always be bounded. Since we have an equation for $\square u$ we can calculate further, \begin{align}
\square F = \left( \dd{t} F - \mu \dd1 F + v\left(c_v \dd{1}F - \dd{2} F\right) \right) \left( u^{-1} F \right). \label{utdep_square_partic} \end{align} This formula tells us that when $v$ is small and $\dd{1}F, \dd{2}F$ are controlled, $\square F$ is approximately the first order linear operator $L[F] \mathrel{\mathop:}= \left( \partial_{t;u} - \mu \partial_u \right) F = \left(\dd{t}F - \mu \dd1 F \right) (u^{-1}F) $.
A relevant function is $\hat u(u,t) \mathrel{\mathop:}= u + \mu t$. We will also use $Q(u,t) \mathrel{\mathop:}= u^{-1}\hat u$. These are related to the linear operator $L[F]$. $\hat u$ gives the characteristic curves of the equation, and $Q$ is a solution to $L[F] = \mu u^{-1}F$ with constant initial data 1.
The following lemma uses these functions to partially solve certain linear equations on the evolving manifold. We claim that for a given smoooth initial function $Z_0 : \mathbb{R}_+ \to \mathbb{R}_+$, $Z \mathrel{\mathop:}= Q^a \cdot \left( Z_0 \circ \hat u \right)$ approximately solves a certain equation. To be explicit, \begin{align}
Z(x, t) = \left( \frac{u(x, t) + \mu t}{u(x, t)} \right)^a Z_0\left( u(x, t) + \mu t \right). \end{align} \begin{lemma}{(Approximate solutions to equations)}\label{approx_soln}
$Z$ satisfies
\begin{align}
\left( \square - a \mu u^{-1} \right) Z = E u^{-1}v {Z} \label{squareZ}\\
{Z}(p, 0) = Z_0(u(p,0))
\end{align}
where $E : M \times [0, T) \to \mathbb{R}$ satisfies $|E| \leq C(1 + |\dd1{Z_0}| + |\dd2{Z_0}|)$, for some constant $C$. Also,
\begin{align}
|\dd1 Z | + |\dd2 Z| \leq C(1 + |\dd1 Z_0| + |\dd2 Z_0|).
\end{align} \end{lemma} \begin{proof}
This is just a calculation, but the following steps give the idea. First, note that $L[\hat u] = 0$ and $L[Q] = \mu u^{-1}Q$. This lets us calculate that $L[Z_0 \circ \hat u] = 0$ and then $L[Z] = a \mu u^{-1}Z$. Therefore by \eqref{utdep_square_partic} all that's left to see is that $|\dd1 Z| + |\dd2 Z| \leq C ( 1 + |\dd1 Z_0| + |\dd2 Z_0|)$, which is just some more calculus. \end{proof}
Now suppose the term $E_{error}$ in \eqref{EQ:22} is bounded. Then we may expect $v$ itself to be approximately given by a solution to $\square v - \mu u^{-1}v = 0$. By Lemma \ref{approx_soln} we find that $v$ should be approximately given by $V \mathrel{\mathop:}= Q \cdot \left( V_0 \circ \hat u \right)$ for some initial data $V_0:M \to \mathbb{R}$. This, in turn, will give us control on the error term $E u^{-1} v Z$ in Lemma \ref{approx_soln}.
Now we create sub- and supersolutions to $\square z = a \mu u^{-1}z$, based off of the approximate solution $Z$, which beat the error in this approximate solution. The supersolution is defined as $Z^+ = (1 + DV)Z$, and the subsolution as $Z^- = (1-DV)Z$, for some sufficiently large $D > 0$. We will assume that \begin{align}
\sup_{\mathbb{R}_+} |\dd1 V_0| + |\dd2 V_0| + |\dd1 Z_0| + |\dd2 Z_0| \leq C_0. \end{align}
\begin{lemma}{(Supersolutions to parabolic equations)}\label{sup_solns}
Suppose $D > \berr D(C_0, a) >0$. There is a $c > 0$ and $\epsilon(D, C_0, a) > 0$ with the following property.
Let $\Omega$ be a subset of space-time where
$v(p,t) \leq 2 V$ and $V < \epsilon$.
Then $Z^-$ and $Z^+$ are sub- and supersolutions to $(\square - a \mu u^{-1})$ on $\Omega$. More precisely,
\begin{align}
\left( \square - a \mu u^{-1} \right) Z^+ &\geq (cD) u^{-1}v Z^+ \label{supsoln_ineq}\\
\left( \square - a \mu u^{-1} \right) Z^- &\leq - (cD) u^{-1}v Z^- \label{subsoln_ineq}
\end{align}
on $\Omega$. \end{lemma} \begin{proof}
Write $Z^+ = Z + Z_2$ with $Z_2 = D V Z = DQ^{p+1}((V_0\cdot Z_0) \circ \hat u)$. Then we can use Lemma \ref{approx_soln} and in particular \eqref{squareZ} to calculate the heat operator applied to $Z_2$:
\begin{align}
\label{eq:52}
\square Z_2 - (a+1) \mu u^{-1}Z_2 = E_2 u^{-1}v Z_2
\end{align}
where $E_2$ is some error which is absolutely bounded depending on $C_0$.
In terms of the linear equation we are interested in, this means
\begin{align}
\label{eq:52}
\left( \square - a \mu u^{-1} \right)Z_2
&=\mu u^{-1} \left(1 + E_2 v \right) Z_2
=\mu Du^{-1} \left(1 + E_2 v \right) VZ.
\end{align}
By choosing $\epsilon$ small enough we can force $1 + E_2 v \geq \on2$ to hold in $\Omega$. Now using again equation \eqref{squareZ} from Lemma \ref{approx_soln}, but now applied to $Z$, we find
\begin{align}
\label{eq:53}
(\square - a \mu u^{-1})(Z^+)
&= (\square - a \mu u^{-1})(Z + Z_2) \\
&\geq \left( E u^{-1}vZ + \on2 \mu u^{-1}D VZ \right) \\
&=
\left( \frac{E}{D} + \on2\mu \frac{V}{v} \right)
\frac{1}{1 + D V} Du^{-1}v Z^+
\end{align}
Here, $E$, another error term of unknown sign, is bounded independently of $D$. The lemma follows by using the assumption that $v \leq \oh V$, choosing $\berr D$ large enough to force $|\frac{E}{D}| \leq \on8 \mu$, and then choosing $\epsilon$ small enough so that $\frac{1}{1 + D V} \geq \oh$. Then we take $c = \on{16} \mu$. \end{proof}
The next lemma claims that the bounds on $|\dd{i}{(Z^+)}|$ carry over to the sub- and supersolutions. \begin{lemma}\label{dd1z_bnd} There is a constant $C$ depending on $C_0$ and $p$, and in particular independent of $D$, such that \begin{align}
|\dd1 {(Z^+)}| + |\dd2{(Z^+)}| \leq C, \end{align} and similarly for the subsolution $Z^-$.
If in addition we assume that $|\nabla \nabla u| \leq C_{hess} v$, then
$|\nabla\nabla Z^+| \leq Cv$ for a constant depending on $C_0$ and $C_{hess}$. \end{lemma} \begin{proof}
First, derive bound for $V = Q V_0 \circ \hat u$ and $Z = Q^a Z_0 \circ \hat u$.
\begin{align}
\dd1 V &= \dd1 Q + \left( \dd1{(V_0)}\circ \hat u \right) \dd1{(\hat u)}\\
&= -(1-Q^{-1}) + \left( \dd1{(V_0)}\circ \hat u \right)Q^{-1},
\end{align}
so $|\dd1 V| \leq 1 + \sup |\dd1 V_0|$. Similarly, we can bound $\dd1 Z$ by $p + \sup |\dd1 Z_0|$.
Now calculate,
\begin{align}
\label{eq:32}
\dd1{(1 + D V)}
&= \frac{u \partial_u (1 + D V)}{1 + DV} \\
&= \frac{DV}{1 + DV} \frac{u \partial_u V}{V} \leq \dd1{V}.
\end{align}
Once we have this, the full bound on $\dd1{(Z^+)}$ follows from
\begin{align}
\dd1{(Z^+)} = \dd1{\left((1 + DV)Z\right)} = \dd1{\left( 1+DV \right)} + \dd1{Z}.
\end{align}
The bound on $\dd2{(Z^+)}$ is similar.
To get the second claim, use the following which is valid for any function $F$ of $u$ and $t$:
\begin{align}
\nabla \nabla F
&= (\partial_u^2F)\nabla u \otimes \nabla u + \partial_u F \nabla \nabla u \\
&= u^{-1}F \cdot \left( (\dd2 F)u^{-1}\nabla u \otimes \nabla u + \dd1 F \nabla \nabla u \right). \end{align} \end{proof}
\section{Calculations on the Bryant soliton}\label{bryant_facts} Let $(Bry, g_{Bry}, X)$ be the Bryant steady soliton with minimum scalar curvature $R_0$. Bryant's original work is \cite{bryant_orig}, see also Section 1.4 of \cite{bookgeometric} for an exposition of the construction. The extra analysis carried out here is generally justified by the analyticity of the solution. Let $
g_{Bry} = ds^2 + u_{Bry}g_{S^q} = ds^2 + \phi^2_{Bry}g_{S^q} $ and $
X = \grad f$.
On any steady soliton we have $R + |\nabla f|^2 = R_0$ (Corollary 1.16 in \cite{bookgeometric}). Taking the trace of the soliton equation we have $R + \Delta f = 0$, so we find $
\Delta_f (-f) = R_0 $. Since the Bryant soliton is a singly warped product, we have more precisely $df = - \sqrt{R_0 - R} ds$.
Either \cite{bryant_orig} or \cite{bookgeometric} show that $\phi_{Bry} = O(\sqrt{s})$ as $s \to \infty$ and $R = O(s^{-1})$. To find the exact coefficient use the equation for $\phi$, \begin{align}
0 = \phi_{ss} - f_s \phi_s - (q-1)\phi^{-1}\left(1 - \phi_s^2\right), \end{align} so $\phi \sim R_0^{-1/4}\sqrt{\mu s}$ and $u \sim R_0^{-1/2}\mu s$ at $\infty$.
\subsection{Next order approximation}\label{bryant_facts_nextorder}
So far we have found as $s \to \infty$ \begin{align}
f &= -(1 + o(1)) R_0^{-1/2} s \\
u &= (1 + o(1)) \mu R_0^{-1/2} s. \end{align} Now we seek the next term in the asymptotic expansion. The function $u$ satisfies \begin{align}
0 = u_{ss} - f_s u_s + c_v u^{-1}u_s^2 - \mu\label{u_eqn_bry1} \end{align} where $c_v = \oh (\oh \mu - 1)$. We also have $\Delta_f (-f) = R_0$ or \begin{align}
0 = (-f)_{ss} - f_s(-f)_s + q \phi^{-1}\phi_s (-f_s) = R_0. \end{align} Strictly in terms of $u$ and $\bar f = -f/R_0$ we have \begin{align}
u_{ss} + R_0 \bar f_s u_s + c_v u^{-1}u_s^2 &= \mu \\
\bar f_{ss} + R_0 \bar f_s^2 + \oh q u^{-1}u_s \bar f_s &= 1 \end{align} Write $G = \bar f_s$. \begin{align}
u_{ss} + R_0 G u_s + c_v u^{-1}u_s^2 &= \mu \label{ug_1}\\
G_{s} + R_0 G^2 + \oh q u^{-1}u_s G &= 1 \label{ug_2} \end{align} Now write $u = \mu R_0^{-1/2} s + u_1$ and $G = R_0^{-1/2} + G_1$. Partially writing out \eqref{ug_1} and \eqref{ug_2}, \begin{align}
u_{1, ss}
+ R_0
\left( R_0^{-1} \mu + \mu R_0^{-1/2} G_1 + R_0^{-1/2}u_{1,s} + u_{1,s}G_1 \right)
+ c_v u^{-1}u_s^2 &= \mu, \\
G_{1,s}
+ R_0
\left( R_0^{-1} + 2 R_0^{-1/2} G_1 + G_1^2 \right)
+ \oh q u^{-1}u_s G
&= 1. \end{align} Simplifying, \begin{align}
u_{1,ss} + \mu R_0^{1/2} G_1 + R_0^{1/2} u_{1,s} + R_0 u_{1,s} G_1 + c_v u^{-1}u_s^2 = 0,\\
G_{1,s} + 2 R_0^{1/2}G_1 + R_0 G_1^2 + \oh q u^{-1}u_s G = 0. \end{align} We have $u^{-1} = \mu^{-1} R_0^{1/2} s^{-1} \left( 1 - u_1 + o(u_1) \right)$. The highest order terms in the equation for $G_1$ are $2 R_0^{1/2}G_1 + \oh q R_0^{-1/2}s^{-1}$, therefore \begin{align}
G_1 = (1 + o(1)) \left( - \on4 R_0^{-1} s^{-1} \right). \end{align} Then the highest order terms in the equation for $u_1$ are $
\mu R_0^{1/2} G_1 + R_0^{1/2}u_{1,s} - c_v \mu R_0^{-1/2}s^{-1} $ which gives \begin{align}
u_1 = (1 + o(1))R_0^{-1}\left(\on4 q \mu + c_v \mu \right) \log s \end{align}
Unravelling definitions, we have found \begin{align}
\bar f &= R_0^{-1/2}s + \on4 q R_0^{-1}\log s + o(\log s), \\
u &= \mu R_0^{-1/2}s + R_0^{-1}\left( \on4 q + c_v \right) \mu \log s + o(\log s), \end{align} so writing $f$ in terms of $u$, \begin{align}\label{bry_second_order_expansion}
\bar f = \mu^{-1} u - \on4 q R_0^{-1} \log u + o(\log u). \end{align}
\subsection{Continuation of the proof of Lemma \ref{wpert_exist}} \label{wpert_exist_continuation}
For the Bryant soliton, we have \eqref{u_eqn_bry1} or equivalently \begin{align}
\Delta_X u - \mu + (c_v - \oh q)v_{{Bry}} = 0\label{u_eqn_bry}. \end{align} (The term $-\oh q v_{{Bry}}$ comes because $\Delta u = u_{ss} + \oh q u^{-1}u_s^2$.) We also have, defining $\bar f = -f/R_0$, $
\Delta_X \bar f = 1 $. (See the introduction of this appendix.) Thinking in terms of $u$ this says, \begin{align}
\partial_uf \Delta_X u + u_s^2\partial_u^2f &= -R_0 \end{align} Then, using \eqref{u_eqn_bry} \begin{align}
( \mu - (c_v - \oh q)v_{{Bry}})\partial_uf + u_s^2\partial_u^2f &= -R_0 \\
( \mu - (c_v - \oh q)v_{{Bry}})\partial_uf + uv_{{Bry}}\partial_u^2f &= -R_0 \end{align} The asymptotics claimed in the Lemma are given in \eqref{bry_second_order_expansion}.
\section{Equations for warped products and Ricci flow}\label{warped_product_section} In this section we review some of the properties of Ricci flow on warped products. The metrics are on the topology $M = B^m \times N^q$, for some manifold $B$ which we call the base. The metrics have the form \begin{align}
g = g_B + \phi^2(b)g_{N}, \end{align} where $g_B$ is a metric on $B$, $g_{N}$ is a metric on $N$, and $\phi: B \to \mathbb{R}_+$. We assume that $g_N$ is an Einstein manifold: $2\Rc[g_N] = \mu_N g_N$.
In this thesis we are mostly concerned with doubly warped products over intervals, i.e. metrics on $I \times S^q \times F$ of the form $
a(x)dx^2 + \phi^2(x) g_{S^q} + \psi^2(x) g_{F} $. These are singly warped products in two ways: with base $I \times S^q$ and fiber $F$ or with base $I \times F$ and fiber $S^q$. Both points of view have been useful for our intuition. A big simplification for a doubly warped product over an interval is that the hessian of a function of $x$ is much simpler than that of a function of a general base.
Everything in this section can be found or derived from Section 7 of \cite{semiriem}.
\subsection{Curvatures} The curvature of a warped product can be described as follows. If $U$ and $V$ are perpendicular unit vectors on the fiber, then \begin{align}
R(U,V,U,V) = \frac{R_N(U,V,U,V) - |\nabla \phi|^2}{\phi^2}. \end{align} In particular, if $(g_N, N)$ is the metric of constant sectional curvature $Sec$, then \begin{align}
R(U,V,U,V) = \frac{Sec - |\nabla \phi|^2}{\phi^2}. \end{align}
For vectors $U$ on the fiber and $X$, $Y$ on the base, we have \begin{align}
R(U,X,U,Y) = - \frac{\nabla_X \nabla_Y \phi}{\phi}, \label{hess_sec} \end{align} and if both $W$, $X$, $Y$, and $Z$ are all vectors on the base, then \begin{align}
R(X,Y,Z,W) = R_B(X,Y,Z,W). \end{align}
From these formulae, we can calculate the Ricci curvature directly from definition. Using $2\Rc[g_N] = \mu_N g_N$, \begin{align}
\Rc(U,V) &= \left( \Rc_{B}(X,Y) - q \phi^{-1}\nabla_X \nabla_Y \phi \right)\\
&+
\left(- \phi\Delta_{B} \phi
+ \oh \mu \left(1 - \frac{2(q-1)}{\mu} |\nabla \phi|^2 \right)
\right) \phi^2g_{N} \end{align}
\subsection{Ricci flow for warped products}\label{rf_warped_products} If $g$ evolves by Ricci flow, then \begin{align}
\Rf[g_B] &= 2 q \phi^{-1}\nabla \nabla \phi \\
\square_B \phi &= - \oh \mu \phi^{-1}\left( 1 - \frac{2(q-1)}{\mu}|\nabla \phi|^2 \right) \\
\square_M \phi &= - \phi^{-1}|\nabla \phi|^2 - \oh \mu \phi^{-1} \end{align}
and $u = \phi^2$ satisfies, setting $v = u^{-1}|\nabla u|^2$, \begin{align}
\square_M u
&= \left( - u^{-1}\mu \right)u - v\label{evo_u_M}\\
\square_B u
&= - \mu + \on4 (2(q-1) - 2) v\label{evo_u_singlywarped}. \end{align} Recall in the case $g_N = g_{S^q}$ we have $\mu = 2(q-1)$.
\subsection{Curvatures for doubly warped products} Now consider a metric of the form \begin{align}
g = a(x) dx^2 + \phi^2 g_{F_1} + \psi^2 g_{F_2}, \quad x \in I. \end{align} We define an arclength coordinate $s$ (up to a constant) by $ds^2 = a dx^2$. We can view $g$ as a warped product with fiber $g_{F_1}$ over base $I \times F_2$, as well as a warped product with fiber $g_{F_2}$ over the base $I \times F_1$. Consider for simplicity the case when $g_{F_1}$ has constant sectional curvature $Sec_1$ and $g_{F_2}$ has constant sectional curvature $Sec_2$. Then there are five special sectional curvatures: \begin{align}
L_1 &= \frac{Sec_1 - |\nabla \phi|^2}{\phi^2} = \frac{Sec_1 - \phi_s^2}{\phi^2},
\quad L_2 = \frac{Sec_2 - \psi_s^2}{\psi^2}, \\
K_1 &= - \frac{\phi_{ss}}{\phi},
\quad K_2 = - \frac{\psi_{ss}}{\psi},
\quad K_{mix} = - \frac{\phi_s \psi_s}{\phi \psi}. \end{align} The curvatures $L_1$ and $L_2$ are those that we get from planes spanned by two perpendicular vectors tangent to the same fiber. $K_1$ and $K_2$ come from planes spanned by $\partial_s$ and a vector on one of the fibers. $K_{mix}$ comes from a plane spanned by a vector on $F_1$ and a vector on $F_2$; this comes from the extra terms (compared to a product) in computing the hessian in \eqref{hess_sec}.
\subsubsection{Curvatures in terms of $u$, $v$ and $w$}\label{curvatures_our_coordinates} We put the curvatures of a doubly warped product in terms of $v$ and $w$, and their $u$ derivatives. Recall the definitions \begin{align}
u = \phi^2, \quad w = \psi^2, \quad v = u^{-1}|\nabla u|^2 = 4 |\nabla \phi|^2 \end{align} First, we have \begin{align}
L_1 = u^{-1}Sec_1 - \on4 u^{-1} v. \end{align} Now calculate, $
\partial_s u \partial_u v = (\partial_s u) 4 (\partial_s \phi) (\partial_u \partial_s \phi) = 4 (\partial_s \phi) (\partial_s^2 \phi) $ so $
2 \phi \partial_u v = 4 (\partial_s^2 \phi) $ and \begin{align}
K_1 =-\on2 \partial_u v = -\frac{\partial_s^2 \phi}{\phi}. \end{align} Now we calculate the curvatures involving $\psi$. \begin{align}
\psi_s = \on2 w^{-1/2}w_s = \on2 w^{-1/2}w_uu_s = \on2 w^{-1/2}u^{1/2}v^{1/2}w_u \end{align} so \begin{align}
L_2 &= \frac{Sec_2}{w} - \on4 u^{-1}v \left( u^2 w^{-2}w_u^2 \right) \\
K_{mix} &= (\on2 u^{-1/2}v^{1/2}) (\on2 w^{-1}u^{1/2}v^{1/2}w_u) = \on4 u^{-1}v \left( uw^{-1}w_u \right) . \end{align} Finally, we calculate \begin{align}
\psi_{ss} &= \on4 \left( w^{-3/2} u^{1/2} v^{1/2}w_u w_s + w^{-1/2}u^{-1/2}v^{1/2}w_uu_s + w^{1/2}u^{1/2}v^{-1/2}w_uv_s + w^{-1/2}u^{1/2}v^{-1/2}w_{us} \right) \\
&= \on4 w^{-1/2} u v (w^{-1}w_u^2 + u^{-1} w_u + v^{-1} w_u v_u + w_{uu}). \end{align} Therefore, \begin{align}
K_2 = - \on4 u^{-1} v (u^2w^{-2}w_u^2 + uw^{-1}w_u + u^2v^{-1} w^{-1}w_u v_u + u^2w^{-1}w_{uu}) \end{align}
\subsection{Deriving the evolution of $v$.}\label{v_deriving_section}
In this Lemma, $\Rf[g_B] = \partial_t g_B - \left( - 2 \Rc_{g_B} \right)$.
\begin{lemma}\label{z_calc_lemma}
Suppose $(B, g_B)$ is an evolving Riemannian manifold and $\phi:B \times [T_1, T_2] \to \mathbb{R}_+$ is an evolving function on $B$. Suppose $g_B$ and $\phi$ satisfy
\begin{align}
\label{eq:70}
\Rf[g_B]
&= 2 c_1 \phi^{-1}\nabla\nabla \phi \\
\square_B \phi &= \oh\phi^{-1}\cdot (-\mu+ c_z z)
\end{align}
where $z = |\nabla \phi|^2$. Let $\kappa(p,t)$ be the norm of the second fundamental form of the level set of $u$ passing through $p$ at time $t$. Then $z$ satisfies
\begin{align}
\label{eq:65}
\square z
&= \phi^{-2}(\mu - c_z z) z \\
&+ (c_z - c_1) \ip{\nabla z}{\nabla \log \phi} - z^{-1}|\nabla z|^2 + \oh \phi^2 z^{-2}\left( \ip{\nabla z}{ \nabla \log \phi} \right)^2\\
&- 2 z \kappa^2
\end{align} \end{lemma}
\begin{proof}
We can apply the parabolic Bochner formula ((1.6) of \cite{weakhaslhofernaber}) to these equations to find \begin{align}
\label{eq:55}
\square |\nabla \phi|^2
&= 2 \ip{\nabla \square \phi }{\nabla \phi} - 2 |\nabla \nabla \phi|_B^2 - \Rf(\nabla \phi, \nabla \phi) \\
&= 2 \ip{\nabla \square \phi }{\nabla \phi} - 2 |\nabla \nabla \phi|_B^2 - 2c_1 \phi^{-1}\nabla_{\nabla \phi}\nabla_{\nabla \phi}\phi \end{align}
We calculate the first term: \begin{align}
\label{eq:61}
2 \ip{\nabla \square \phi}{\nabla \phi}
&= \phi^{-2}(\mu-c_zz) |\nabla \phi|^2 + c_z \phi^{-1}\ip{\nabla z}{\nabla \phi} \\
&= \phi^{-2}(\mu - c_z z)z + c_z \ip{\nabla z}{\nabla \log \phi} \end{align} For the second term, we can change the hessian to \begin{align}
\label{eq:68}
-2 |\nabla \nabla \phi|^2
&= -2 z \kappa^2 - z^{-1}|\nabla z|^2 + \on2 z^{-2}\ip{\nabla z}{\nabla \phi}^2 \\
&= -2 z \kappa^2 - z^{-1}|\nabla z|^2 + \on2 z^{-2}\phi^2\ip{\nabla z}{\nabla \log \phi}^2 \end{align} And for the third term, we can change the hessian using \begin{align}
\label{eq:64}
-2c_1\phi^{-1}\nabla_{\nabla \phi}\nabla_{\nabla \phi}\phi
&= - c_1 \phi^{-1}\ip{\nabla z}{ \nabla \phi} = - c_1 \ip{\nabla z}{\nabla \log \phi} \end{align}
Putting everything together, we find the desired equation. \end{proof}
\begin{corollary}\label{v_calc_lemma}
In the setting of Lemma \ref{z_calc_lemma}, suppose $g_B$ and $u$ satisfy
\begin{align}
\Rf[g_B]
&= 2 c_1 u^{-1/2}\nabla\nabla u^{1/2} \label{base_eqn}\\
\square_B u &= - \mu + c_v v \label{u_eqn}
\end{align}
where $v = u^{-1}|\nabla u|^2$. Define the constants $c_z = (4 c_v + 2)$, $c_{v}' = \on4 c_z$, and $c_3 = \oh \left( c_z - c_1 \right)$. Then $v$ satisfies
\begin{align}
\square v
&= u^{-1}(\mu - c_v' v) v - 2 v \kappa^2 \\
&+ c_3 \ip{\nabla v}{\nabla \log u} - v^{-1}|\nabla v|^2 + \oh u v^{-2}\left( \ip{\nabla v}{ \nabla \log u} \right)^2\\
\end{align} \end{corollary}
\subsubsection{Equidistant Level Sets}
Now, suppose that the level sets of $u$ are equidistant. Then $v$ is dependent on $u$ and $t$ alone so we find $\nabla v = |\nabla u|^{-1}\ip{\nabla v}{\nabla u}$. Then from Corollary \ref{v_calc_lemma}, \begin{align}
\label{square_v_edh1}
\square_B v
&= u^{-1}(\mu - c_v' v)v - 2 \kappa^2 v - u^{-1}\bar T v \\
&+ c_3v(\partial_u v) - \oh u(\partial_u v)^2 \end{align} On the other hand, since the level sets of $u$ are equidistant, we can use that $v$ is a function of $u$ and $t$ to calculate $\square_B v$ in terms of derivatives with respect to $u$, using \eqref{u_eqn}. \begin{align}
\label{square_v_edh}
\square_B v
&= (- \mu + c_vv)\partial_uv + \partial_{t;u}v - uv\partial_u^2v. \end{align} From \eqref{square_v_edh1} and \eqref{square_v_edh} it follows that, \begin{align}
\label{c_4_intro}
\partial_{t;u} v
&= u v \partial_u^2v - \oh u (\partial_uv)^2+ u^{-1}(\mu - c_v' v)v + (c_4 v + \mu )\partial_uv \\
&- 2 \kappa^2 v - u^{-1} \bar T v \end{align} where $c_4 = c_3 - c_v$.
\subsubsection{The case of warped product Ricci flow}\label{wprcf_eqns} In the case of Ricci flow of a metric $g = g_B + u g_{S^q}$, where the Ricci curvature of $g_{S^q}$ is $\mu g_{S^q} = 2(q-1)g_{S^q}$, we have \begin{align}
\Rf[g_B] &= 2 q u^{-1/2}\nabla\nabla u^{1/2} \\
\square_B u &= - \mu + \on4 (\mu-2)v \label{u_eqn_wpsphere} \end{align} Therefore in Lemma \ref{v_calc_lemma} we have $c_v = \on4(\mu-2)$ and $c_1 = q = \oh \mu + 1$. Then we find $c_z = 4c_v + 2 = \mu$, $c_v' = \on4 c_z = \on4 \mu$, $c_3 = \on4 \mu -\on2$, and $c_4 = c_3 - c_v = \on4 \mu - \oh - (\on4 \mu - \oh) = 0$. So, from \eqref{c_4_intro}, \begin{align}
\partial_{t;u} v
&= u v \partial_u^2 v - \oh u (\partial_u v)^2
+ \mu \left(1 - \on4 v \right) u^{-1} v + \mu \partial_u v \label{vevo_basic}\\
&- 2 (\kappa^2) v \end{align} One convenient way to write this is as, \begin{align}
\label{V_evo}
\partial_{t;u}v = u^{-1}\mathcal{Q}[v,v] + u^{-1}\mathcal{L}[v] - 2 (\kappa^2) v \end{align} where $\mathcal{L}$ and $\mathcal{Q}$ are the operators \begin{align}
\label{eq:74}
\mathcal{L}[w]
= L(w, \partial_uw), \quad L(A,B) = \mu A + \mu B. \end{align} and \begin{align}
\label{eq:75}
\mathcal{Q}[w,w]
= Q(w, u\partial_uw, u^2 \partial_u w_{uu}),\quad
Q(A,B,C)
= AC - \oh B^2 - \on4 \mu C^2. \end{align} For $w_1$ and $w_2$ different functions, we define $\mathcal{Q}[w_1,w_2]$ to be the extension of $\mathcal{Q}$ to a symmetric bilinear operator.
\subsubsection{Writing the evolution in terms of $L$ and $\phi$} It is also convenient to consider the evolution of $L = \frac{1 - \on4 v}{u}$. $L$ is a sectional curvature, so it is a geometrically natural quantity to consider. If the metric is smooth near $u=0$ then $L$ will be bounded there, which gives us more information that $v$ being bounded.
Coming from \eqref{vevo_basic}, replace $v = 4(1-uL)$ and divide through by $-4u$ to find \begin{align}
\partial_{t;u} L
&= 4u\left(1 - uL\right)\partial_u^2L
+ 2 u^2 (\partial_u L)^2 \\
&+ (\mu + 8 - 4uL)\partial_uL
+ (\mu + 2)L^2 + \oh u^{-1} \kappa^2 v. \end{align} An important point here is that the terms $u^{-1}L$ cancel. This is expected, since for example the sphere has constant non-zero curvature $L$ despite $u$ going to zero. Let us also put this in terms of derivatives with respect to $\phi = \sqrt{u}$. Note \begin{align}
\partial_u = \oh \phi^{-1}\partial_\phi , \quad
u \partial^2_u = \on4 \left( \partial^2_\phi - \phi^{-1}\partial_\phi \right). \label{u_to_phi_1} \end{align} Since $\phi$ is a function of $u$, $\partial_{t;u} = \partial_{t;\phi}$. So, we have \begin{align}
\partial_{t;\phi} L
&= \left( 1 - \phi^2L \right)(\partial^2_\phi L - \phi^{-1}\partial_\phi L)
+ \on2\phi^2 (\partial_\phi L)^2 \\
&+ \phi^{-1}(\oh \mu + 4 - 2\phi^2L)(\partial_\phi L) + (\mu + 2)L^2
+ \oh \kappa^2 v \\
&= \left(1 - \phi^2 L\right)\partial^2_\phi L + \oh \phi^2 (\partial_\phi L)^2\\
&+ \phi^{-1}(\oh \mu + 5 - \phi^2L)\partial_\phi L + (\mu + 2)L^2
+ \oh \kappa^2 v.\label{evo_L_in_phi} \end{align} The advantage of this is the clear regularity around $\phi = 0$ provided $L$ is bounded.
\subsection{Deriving the evolution of $w$}\label{additional_wp} We continue considering the Ricci flow of a metric of the form $g = g_B + u g_{S^q}$: \begin{align}
\label{eq:10}
\Rf[g_B] &= 2 c_1 u^{-1/2}\nabla \nabla u^{1/2}, \\
\square_B u &= - \mu + c_v v. \end{align}
Here $c_v = \on4(\mu - 2)$. Suppose that $g_B$ itself has a warped product factor: $B = B_2 \times F^{p}$ and $g_B = g_{B_2} + w g_F$. Take $y = w^{-1}|\nabla w|^2$ and suppose that $2\Rc[g_F] = \mu_F g_F$. We make no assumptions on the sign on $\mu_F$.
To quickly derive an equation for $h$ in terms of $\square_B$, go from \eqref{evo_u_M} which says \begin{align}
\square_{B_2 \times F \times S^q} w = - \mu_F - y \end{align}
where $y = w^{-1}|\nabla w|^2$. Since \begin{align}
\square_{B_2 \times F \times S^q} w
&=
\partial_t w - \left( \Delta_{B_2 \times F \times S^q}w \right) \\
&=
\partial_t w -
\left(\Delta_{B}w + \oh q u^{-1}\ip{\nabla u}{\nabla w} \right)\\
&= \square_B w - \oh q u^{-1}\ip{\nabla u}{\nabla w} , \end{align} we find, \begin{align}
\square_B w
&= - \mu_F - y - \oh q u^{-1}\ip{\nabla u}{\nabla w} \\
&= - \mu_F - y - \oh q v \partial_u w. \label{squareb_w} \end{align} Now, using and the fact that $w$ is a function of $u$ and $t$, \begin{align}
\square_B w = (- \mu + c_v v) \partial_u w + \partial_{t;u}w - uv \partial_u^2 w \end{align} so by \eqref{squareb_w} we find \begin{align}
\partial_{t;u}w - u v \partial_u^2 w
&= - \mu_F - y + \mu \partial_u w - c_v v \partial_u w - \oh q v \partial_u w \\
&= - \mu_F - y + \mu \partial_u w - \mu/2 v\partial_u w \label{evo_w_in_u}. \end{align}
Note we may also write $y = w^{-1}|\nabla w|^2 = w^{-1}uv(\partial_u w)^2$.
\subsubsection{Writing the evolution in terms of $\phi$} We also write \eqref{evo_w_in_u} in terms of $\phi$. Using \eqref{u_to_phi_1} we have \begin{align}
\partial_{t;u}w
&= v \left( \partial_\phi^2 w - \phi^{-1}\partial_\phi w \right) \\
&- \mu_F - y + (\mu - \mu/2 v) \on2 \phi^{-1}\partial_\phi w \end{align} or, simplifying, \begin{align}
\partial_{t;\phi}w
&= v \partial_\phi^2 w - \mu_F - y + (\on2\mu - (\on4\mu - 1) v) \phi^{-1}\partial_\phi w. \label{eqn_w} \end{align}
\subsection{Second fundamental form for doubly warped products} Consider the case of a doubly warped product over an interval. The second fundamental form $\kappa$ in Section \ref{v_deriving_section} is the second fundamental form of a surface $(s, p) \times F$, which is \begin{align}
\on4 dim(F) w^{-1}y
&= \on4 dim(F) w^{-2}u^2v (\partial_u w)^2. \end{align} Therefore the term $-2(\kappa^2)v$ is $- \on2 dim(F) w^{-2}u^2 v^2 (\partial_u w)^2 = - \on8 dim(F)w^{-2} v^2 \phi^2(\partial_\phi w)^2$.
\section{Ricci-DeTurck flow}\label{rcdt}
Under Ricci-DeTurck flow with background metric $\tilde g$, the evolution $h(t) = g(t) - \tilde g(t)$ is given by the following lemma. This copies the calculation in Lemma 3.1 of \cite{steady_alix} and generalizes it. \begin{lemma}\label{dtevo}
Let $\tilde g$ be a time-dependent family of metrics, and let $X$ be a time-dependent vector field. Let $g$ be a metric satisfying
\begin{align}
\Rf_X[g] = \mathcal{L}_{V[g, \tilde g]}g. \label{rdt_modified}
\end{align}
Let $g = \tilde g + h$, $g^{-1} = \tilde g^{-1} - \bar h$, and $\hat h^{ij} = \tilde g^{ai}\tilde g^{bj}h_{ab} - \bar h^{ab}$.
Then
\begin{align}
\label{eq:5}
\square_{X, \tilde g, g} h
&= 2\Rm[h] + \UT[h] + Q[h] + \Cov[g, \nabla h]\\
&- \Rf_X[\tilde g] - \left( \Rf_X[\tilde g] \cdot h \right)
\end{align}
where all covariant derivatives and curvatures are with respect to $\tilde g$, and the terms are as follows:
\begin{align}
\label{eq:6}
\left( \Delta_{\tilde g,g} h \right)_{ij} = g^{ab} \nabla_a \nabla_b h_{ij},&\quad
\Delta_{X,\tilde g, g} = \Delta_{\tilde g, g} - \nabla_X ,\quad
\square_{X, \tilde g, g} = \partial_t - \Delta_{X, \tilde g, g}, \\
\Rm[h] = \tilde g^{ac}\tilde g^{bd}\Rm_{ajbi}h_{cd},&\quad
Q[h] = \symme{i}{j}{\Rm^p_{ajb}h^{ab}h_{ip} - \Rm_{ajbi}\hat h^{ab}},\\
(A \cdot B)_{ij} = \oh
\on2\symme{i}{j}{ \tilde g^{ab}A_{ai}B_{bj}}
,&\quad
\UT[B] = \left( (\partial_t \tilde g) \cdot B \right), \\
|\Cov[g, \nabla h]| \leq c_0\left(1 + |h| \right)|\nabla h|^2.&
\end{align}
In the last line $c_0$ is a constant depending only on the dimension. \end{lemma}
\begin{proof}
The convention in this proof is that all curvatures and covariant derivatives are taken with respect to $\tilde g$. By Lemma 2.1 of \cite{Shi} we have
\begin{align}
\label{eq:7}
\partial_t g_{ij} = g^{ab}\nabla_a \nabla_b g_{ij} - \symme{i}{j}{g^{ab}g_{ip}\Rm_{ajb}^p} - \left( \mathcal{L}_X g \right)_{ij}+ \Cov(g, \nabla g)
\end{align}
Since $g = \tilde g + h$ and $\nabla$ is the connection of $\tilde g$,
\begin{align}
\partial_t h_{ij} &= g^{ab}\nabla_a \nabla_b h_{ij} \label{dtevo:shi1}\\
&- \partial_t \tilde g - \symme{i}{j}{g^{ab}g_{ip}\Rm_{ajb}^p} - \left( \mathcal{L}_X g \right)_{ij} \label{dtevo:shi2}\\
&+ \Cov(g, \nabla h) \label{dtevo:shi3}
\end{align}
\textbf{Rewriting the curvature term.} Let $ g^{ij} = \tilde g^{ij} - \bar h^{ij}$. Expand $g^{ab}g_{ip} = (\tilde g^{ab} - \bar h^{ab})(\tilde g_{ip} + h_{ip})$ in the curvature term.
\begin{align}
\label{eq:8}
-g^{ab}g_{ip}\Rm_{ajb}^p
&= \left( -\tilde g^{ab}\tilde g_{ip} + \bar h^{ab}\tilde g_{ip} - \tilde g^{ab}h_{ip} + \bar h^{ab}h_{ip}\right)\Rm_{ajb}^p \\
&= - \Rc_{ij} + \bar h^{ab}\Rm_{ajbi} - \Rc_j^ph_{ip} + \bar h^{ab}h_{ip}\Rm^p_{ajb} \label{dtevo:l1}
\end{align}
Now let
\begin{align}
\label{eq:9}
\hat h^{ij} = \tilde g^{ci}\tilde g^{dj}h_{cd} - \bar h^{ab}
\end{align}
so that
\begin{align}
\label{eq:10}
\bar h^{ab} \Rm_{ajbi} = \tilde g^{ca}\tilde g^{db}\Rm_{cjdi}h_{ab} - \Rm_{ajbi}\hat h^{ab}.
\end{align}
Putting this together with \eqref{dtevo:l1} we have
\begin{align}
\label{eq:11}
-g^{ab}g_{ip}\Rm_{ajb}^p
&= - \Rc_{ij} + \tilde g^{ac}\tilde g^{bd}\Rm_{cjdi}h_{ab} - \Rc_j^ph_{ip} \\
&+ h^{ab}h_{ip}\Rm^p_{ajb} - \Rm_{ajbi}\hat h^{ab}.
\end{align}
Finally taking the symmetrization we find
\begin{align}
\symme{i}{j}{-g^{ab}g_{ip}\Rm_{ajb}^p}
&= - 2 \Rc_{ij} + 2\Rm[h]_{ij} - (\Rc \cdot h)_{ij} + Q(h)_{ij} \label{dtevo:curvrewrite}
\end{align}
\textbf{Rewriting the Lie term}
We have
$
-\mathcal{L}_X g = -\mathcal{L}_X \tilde g - \mathcal{L}_X h
$.
We can relate the lie derivative with the covariant derivative and the lie derivative of the metric,
\begin{align}
\label{eq:14}
\left( -\mathcal{L}_X h \right)_{ij}
&= \left( -\nabla_X h \right)_{ij} - \oh \symme{i}{j}{h_{pi}\tilde g^{pq}(\mathcal{L}_X \tilde g)_{qj}} \\
&= \left( -\nabla_X h \right)_{ij} - \oh \left( \left( \mathcal{L}_X \tilde g \right) \cdot h \right)_{ij}
\end{align}
(The first line is true in general, the second line uses that $h_{ij}$ is symmetric.)
Thus
\begin{align}
-\mathcal{L}_X g = -\mathcal{L}_X \tilde g - \nabla_X h - \oh \left( \mathcal{L}_X \tilde g \right) \cdot h \label{dtevo:lierewrite}
\end{align}
\textbf{Coming back to the evolution.} Using \eqref{dtevo:curvrewrite} and \eqref{dtevo:lierewrite}, the evolution \eqref{dtevo:shi1}-\eqref{dtevo:shi2} becomes
\begin{align}
\label{eq:17}
\partial_t h
&= \hat \Delta h - \nabla_X h \\
&- \partial_t \tilde g - 2 \Rc[\tilde g] - \mathcal{L}_X \tilde g \\
&- \oh \left( \left( 2 \Rc + \mathcal{L}_X \tilde g \right) \cdot h \right) \\
&+ 2\Rm[h] + Q(h) + \Cov(g, h). \\
\end{align}
So unraveling definitions,
\begin{align}
\label{eq:18}
\hat \square_X h
&= - \Rf_X[\tilde g] \\
&+ \oh \left( \left(\partial_t g \right) \cdot h \right) - \oh \left( \left( \partial_t g + 2 \Rc + \mathcal{L}_X \tilde g \right) \cdot h \right) \\
&+ 2\Rm[h] + Q(h) + \Cov(g, h) \\
&\\
&= - \Rf_X[\tilde g] \\
&+ \oh \UT[h] - \oh \left( \Rf_X[\tilde g]\cdot h \right) \\
&+ 2\Rm[h] + Q(h) + \Cov(g, h)
\end{align}
as desired. \end{proof}
What we will use is the following evolution of $|h|$ and $|h|^2$. For $p \in M$ We set $
\Lambda_{\Rm}(p) = \max_{h \in Sym_2(T_pM) : |h| = 1}\ip{\Rm[h]}{h}(p)
$. Now consider the case when $\Rf_X[\tilde g] = 0$. Then, for $y = |h|^2$ we have (we allow $c_0$ to change from line to line) \begin{align}
\square_{X, \tilde g, g} y
&\leq 4 \Lambda_{\Rm} y - 2 g^{ab}\tilde g^{cd}\tilde g^{ef}\nabla_a h_{ce} \nabla_b h_{df} \\
&+ c_0\left( |\Rm|y^{3/2} + |\nabla h|^2y^{1/2} \right). \end{align}
Note that the linear term $\UT[h]$ may be removed using the Uhlenbeck trick, and disappears in the evolution of the norm. The term $-2 g^{ab}\tilde g^{cd}\tilde g^{ef}\nabla_a h_{ce}\nabla_b h_{df}$ is strictly negative. If $|h| < \oh$ we can use $g^{ab}\tilde g^{cd}\tilde g^{ef}\nabla_a h_{ce} \nabla_b h_{df} \geq (1-c_0 y^{1/2})|\nabla h|^2$ and find \begin{align}
\square_{X, \tilde g, g} y
&\leq 4 \Lambda_{\Rm} y - 2(1 - c_0 y^{1/2})|\nabla h|^2 + c_0 |\Rm|y^{3/2}. \label{dtevo_square} \end{align}
Alternatively, we can derive the evolution of $z = |h|$ and use the inequality $g^{ab}\nabla_a |h|^2 \nabla_b |h|^2
\leq
4|h|^2 g^{ab}\tilde g^{cd}\tilde g^{ef}\nabla_a h_{ce}\nabla_b h_{df}$
to find \begin{align}
\square_{X, \tilde g, g}z
&\leq 2 \Lambda_{\Rm} z + c_0 \left(|\Rm| z^2 + |\nabla h|^2 \right).\label{rdt_norm} \end{align}
\section{Notation}\label{appendix_notation} The heat operator is $\square u = \partial_t u - \Delta u$. If $X$ is a vector field then $\Delta_Xu = \Delta u - \nabla_Xu$ and $\square_X = \partial_t - \Delta_X$.
The curvature tensors are $\Rm$ for the full Riemannian $(0, 4)$ tensor, $\Rc$ for the Ricci curvature, and $R$ for the scalar curvature. The indices of $\Rm$ are such that $\Rm_{ijij}$ is a sectional curvature in an orthonormal frame.
The vector field $V[g, \tilde g]$, the operator $\Delta_{g, \tilde g}$, and $\Rf[g]$ are defined in Section \ref{section:rdt}. There we also define $\Rm[h]$ for a symmetric two-tensor $h$, and $\Lambda_{\Rm}:M \to \mathbb{R}$.
Everywhere $g_{S^q}$ is the metric of sectional curvature $1$ on the $q$ dimensional sphere $S^q$. We define $\mu = 2(q-1)$ so that $2\Rc_{g_{S^q}} = \mu g_{S^q}$. We also have a general Einstein manifold $(F, g_F)$ in play, its Ricci curvature satisfies $2\Rc_F = \mu_F g_F$ for some $\mu_F \in \mathbb{R}$.
Usually we have a metric of the form \begin{align}
a dx^2 + u g_{S^q} + w g_F \end{align}
for $x$ in some interval $I$. Here $a$, $u$, and $w$ are functions of $I$. The functions $a$, $u$, and $w$ may also depend on time. On these manifolds we have the derived functions $v = u^{-1}|\nabla u|^2$ and $y = w^{-1}|\nabla w|^2$. Rarely we also use $\phi = \sqrt{u}$ and $\psi = \sqrt{w}$.
We have a lot of scaling. Briefly: \begin{align}
\nu(t) = V_0(\mu t),
\quad \omega(t) = W_0(\mu t),
\quad \alpha(t) = t \nu(t),
\quad \beta(t) = \alpha'(t)\\
\rho = t^{-1}u,
\quad \sigma = (t \nu(t))^{-1}u,
\quad \zeta = t \nu(t)^{-1/2}u = \nu(t)\sigma,\\
\hat u = u + \mu t,
\quad \hat w = w + \mu_F t,
\quad \bar w = \omega(t)^{-1}(w + \mu_F t). \end{align}
We have some functions which are written in terms of $u$. Generally capital letters denote known functions which are written in terms of $u$, whereas lowercase letters denote unknown functions. The functions $V_0$ and $W_0$ are the initial values for $v$ and $w$ in a model pinch. $V_{prish}$ and $W_{prish}$ are our approximations for $v$ and $w$ in the productish region, and $V^{\pm}_{prish}$ and $W^{\pm}_{prish}$ are upper and lower barriers for $v$ and $w$ based on these approximations. Similarly these names with the subscript $tip$ are approximations and barriers in the tip region. In Section \ref{section:productish}, we only refer to the functions for the produtish region, and therefore we drop the subscripts for cleanliness. Similarly in Section \ref{section:tip} we only refer to the tip functions, so we drop the subscript there as well.
Other functions of $u$ are $V_{bry}$, $V_{pert}$, and $W_{pert}$ (introduced in Section \ref{section:tip}, and with an overview in Section \ref{function_summary}).
We define $x^{a,b} = x^a(1 + x)^{b-a}$. The point is that it's a smooth function on $(0, \infty)$ which behaves like $x^a$ at 0 and $x^{b}$ at $\infty$.
\end{document} |
\begin{document}
\title{Ordered Partitions and \\Drawings of Rooted Plane Trees} \author{Qingchun Ren} \date{Jan 27, 2014}
\begin{abstract} We study the bounded regions in a generic slice of the hyperplane arrangement in $\mathbb{R}^n$ consisting of the hyperplanes defined by $x_i$ and $x_i+x_j$. The bounded regions are in bijection with several classes of combinatorial objects, including the ordered partitions of $[n]$ all of whose left-to-right minima occur at odd locations and the drawings of rooted plane trees with $n+1$ vertices. These are sequences of rooted plane trees such that each tree in a sequence can be obtained from the next one by removing a leaf. \end{abstract}
\maketitle{}
\section{Introduction}\label{section-introduction}
We define the combinatorial objects to be studied in this paper. The first one is the following hyperplane arrangement on $\mathbb{R}^n$: \begin{equation*} \mathcal{H}_n = \{x_i,1\leq{}i\leq{}n\}\cup{}\{x_i+x_j,1\leq{}i<j\leq{}n\}. \end{equation*} Let $P$ be the affine hyperplane in $\mathbb{R}^n$ defined by \begin{equation*} P=\{l_1x_1+l_2x_2+\dotsb{}+l_nx_n=1\}, \end{equation*} where $l_1\gg{}l_2\gg{}\dotsb{}\gg{}l_n>0$ (``$\gg{}$" means ``far greater than"). We are interested in the set of bounded regions of the hyperplane arrangement $\mathcal{H}_n\cap{}P = \{H\cap{}P:H\in{}\mathcal{H}_n\}$ in the affine space $P$.
\begin{definition} An {\it ordered partition} of $[n]=\{1,2,\dotsc{},n\}$ (also called a {\it preferential arrangement} by Gross \cite{citation-gross}) is an ordered sequence $(A_1,\dotsc{},A_k)$ of disjoint non-empty subsets whose union is $[n]$. Each $A_i$ is called a {\it block}. A {\it left-to-right minimum} of $(A_1,\dotsc{},A_k)$ is $m_i=\mathrm{min}(A_1\cup{}\dotsb{}\cup{}A_i)$, where $1\leq{}i\leq{}k$. We say that a left-to-right minimum $m_i$ {\it occurs at an odd location} if $m_i\in{}A_j$ for some odd $j$. \end{definition}
\begin{definition} A {\it signed permutation} $((a_1,a_2,\dotsc{},a_n),\sigma{})$ of $[n]$ is a permutation $(a_1,a_2,\dotsc{},a_n)$ of $[n]$ together with a map $\sigma{}\colon{}[n]\to{}\{\pm{}1\}$. $\sigma{}(i)$ is called the {\it sign} of~$i$. It has {\it decreasing blocks} if $a_i>a_{i+1}$ for any two $a_i,a_{i+1}$ with the same sign. A {\it left-to-right minimum} of $((a_1,a_2,\dotsc{},a_n),\sigma{})$ is $m_i=\mathrm{min}(a_1,\dotsc{},a_i)$, where $1\leq{}i\leq{}n$. For simplicity, we indicate the sign of $a_i$ by writing $a_i^+$ or $a_i^-$. \end{definition}
\begin{definition} A {\it build-tree code} is a sequence $c_1c_2\dotsb{}c_n$ of pairs $c_i=(a_i,\sigma{}_i)$ where $0\leq{}a_i\leq{}i-1$ and $\sigma{}_i\in{}\{\pm{}1\}$ such that $(a_i,\sigma{}_i)\neq{}(0,-1)$. For simplicity, we write $a_i^+$ or $a_i^-$ instead of $(a_i,\sigma{}_i)$. \end{definition}
\begin{definition} An {\it increasing labeling} of a rooted plane tree $T$ of $n+1$ vertices, also called a {\it simple drawing} or a {\it heap order}, is a bijection $\lambda{}\colon{}T\to{}\{0,1,\dotsc{},n\}$ such that if $u,v\in{}T$ and $u$ is a child of $v$, then $\lambda{}(u)>\lambda{}(v)$. $\lambda{}(v)$ is called the {\it label} of $v$. An {\it increasingly labeled tree} is a rooted plane tree together with an increasing labeling. For simplicity, we identify a vertex with its label if there is no confusion. \end{definition}
\begin{definition} Let $(T,\lambda{})$ be an increasingly labeled tree. The {\it right associate} of a vertex $v$ is the sibling $u$ to the right of $v$ with the smallest label such that $\lambda{}(u)>\lambda{}(v)$, and $\lambda{}(u)$ is smaller than the labels of all siblings between $u$ and $v$, if such a $u$ exists. $(T,\lambda{})$ is a {\it Klazar tree} if it satisfies the following property: for any vertex $v$ with a right associate $u$, $v$ is not a leaf, and $\lambda{}(u)$ is larger than the minimum of all labels of children of $v$. \end{definition}
Figure 1 shows two different linear extensions of the same tree. The tree in the right of Figure 1 is a Klazar tree. The tree in the left is not a Klazar tree, because the vertex $3$ has a right associate $4$, but $3$ is a leaf.
\begin{figure}
\caption{Two increasingly labeled trees}
\end{figure}
The last object is the set of {\it drawings} of rooted plane trees with $n+1$ vertices:
\begin{definition} A {\it drawing} of a rooted plane tree $T$ is a sequence of rooted plane trees $T_0=\{\text{root}\},T_1,\dotsc{},T_n=T$ such that for each $0\leq{}i\leq{}n-1$, the tree $T_i$ can be obtained from $T_{i+1}$ by removing a leaf together with its pendant edge. \end{definition}
The main result of this paper is
\begin{theorem}\label{theorem-main} The following seven sets are in bijection: \begin{itemize} \item[(1)] The set of bounded regions in the affine hyperplane arrangement $\mathcal{H}_n\cap{}P$. \item[(2)] The set of ordered partitions of $[n]$ all of whose left-to-right minima occur at odd locations. \item[(3)] The set of signed permutations of $[n]$ with decreasing blocks all of whose left-to-right-minima have positive signs. \item[(4)] The set of build-tree codes of length $n$ such that there is a $v^+$ after (but not necessarily adjacent to) each $v^-$. \item[(5)] The set of build-tree codes of length $n$ such that there is a $v^+$ before (but not necessarily adjacent to) each $v^-$. \item[(6)] The set of Klazar trees with $n+1$ vertices. \item[(7)] The set of drawings of rooted plane trees with $n+1$ vertices. \end{itemize} Let $b_n$ be the common cardinality of these sets. Set $b_0=1$. Then, the sequence $\{b_n\}$ has the exponential generating function \begin{equation*} \sum_{n=0}^{\infty{}}b_n\frac{x^n}{n!}=\sqrt{\frac{e^x}{2-e^x}}. \end{equation*} \end{theorem}
The above generating function is due to Klazar \cite{citation-klazar} in a paper that discusses various counting problems of rooted plane trees. The bijections between (5), (6) and (7) are studied by Callan \cite{citation-callan}. Our notations are different from Callan's because we use the top-to-bottom convention for trees in this paper. Callan shows that $b_n$ also equals the number of perfect matchings on the set [2n] in which no even number is matched to a larger odd number. The sequence $\{b_n\}$ begins with \begin{equation*} 1,1,2,7,35,226,1787,16717,\dotsc{}. \end{equation*} This sequence can be found in the On-Line Encyclopedia of Integer Sequences \cite[A014307]{citation-oeis}.
\begin{remark} The number $b_n$ is related to the following {\em urn model}: one starts with $1$ black ball and $0$ white ball in an urn. In each step, one picks a ball randomly in the urn. If the ball is black, one puts that ball back to the urn together with another white ball. Otherwise, one puts that ball back to the urn together with two more black balls. Suppose that all balls are distinguishable. Then, $b_n$ equals the number of possible histories after $n$ steps. A detailed treatment on urn models can be found in \cite{citation-flajolet}. \end{remark}
\begin{remark} The number of bounded regions in $\mathcal{H}_n\cap{}P$ can be obtained by a simple application of the finite field method. However, it takes much more effort to establish a bijective proof. \end{remark}
Our investigation originates from a {\it latent allocation model} in genomics \cite{citation-pachter}. Maximum likelihood estimation for this statistical model involves finding local maxima of the the function \begin{equation*}
\prod_{i=1}^n|x_i|^{u_i}\prod_{1\leq{}i<j\leq{}n}|x_i+x_j|^{u_{ij}} \end{equation*} on the hyperplane $P'=\{x_1+x_2+\dotsb{}+x_n=1\}$, where $u_i,u_{ij}$ are generic positive integers. By a theorem of Varchenko \cite{citation-varchenko}, the ML degree of the statistical model, i.e. the number of local maxima of the above function equals the number of bounded regions in $\mathcal{H}_n\cap{}P'$. Our hyperplane $P$ can be considered as a deformation of $P'$: \begin{equation*} P=\{l_1x_1+l_2x_2+\dotsb{}+l_nx_n=1\}, \end{equation*} where $l_1,\dotsc{},l_n$ are generic parameters. The number of bounded regions in $\mathcal{H}_n\cap{}P$ gives an upper bound on the number of bounded regions in $\mathcal{H}_n\cap{}P'$, and thus gives an upper bound on the maximum likelihood degree of the latent allocation model. Without loss of generality, we will assume that $l_1\gg{}l_2\gg{}\dotsb{}\gg{}l_n>0$.
Our hyperplane arrangement $\mathcal{H}_n$ is refined by the well-studied hyperplane arrangement of type $B_n$: \begin{equation*} \mathcal{B}_n = \{x_i,1\leq{}i\leq{}n\}\cup{}\{x_i\pm{}x_j,1\leq{}i<j\leq{}n\}. \end{equation*}
Section \ref{section-bn} discusses the analogous problem for $\mathcal{B}_n$, and it shows that the bounded regions in $\mathcal{B}_n\cap{}P$ are in bijection with increasingly labeled trees with $n+1$ vertices. Based on this, Section \ref{section-hn} proves our main result, Theorem \ref{theorem-main}.
\section{Bounded regions in a slice of $\mathcal{B}_n$}\label{section-bn}
First, we consider the regions of the central hyperplane arrangement $\mathcal{B}_n$. The hyperplanes $x_i$ in $\mathcal{B}_n$ divide $\mathbb{R}^n$ into $2^n$ orthants. In each orthant, the hyperplanes $x_i\pm{}x_j$ divides the orthant into $n!$ regions, one for each total ordering of $|x_1|,\dotsc{},|x_n|$. Thus, for each signed permutation $((a_1,a_2,\dotsc{},a_n),\sigma{})$, we can associate it with a region $R$ of $\mathcal{B}_n$: \begin{equation*}
R = \{(x_1,\dotsc{},x_n)\in{}\mathbb{R}^n:|x_{a_1}|>|x_{a_2}|>\dotsb{}>|x_{a_n}|,\mathrm{sgn}(x_i)=\sigma{}(i)\}. \end{equation*} Clearly, this is a bijection between regions of $\mathcal{B}_n$ and signed permutations of $[n]$.
\begin{lemma}\label{lemma-bn-regions} Let $R$ be a region of $\mathcal{B}_n$. Let $((a_1,a_2,\dotsc{},a_n),\sigma{})$ be the corresponding signed permutation. Then \begin{itemize} \item[(a)] The polyhedron $R\cap{}P$ is nonempty and bounded if all left-to-right minima of $(a_1,a_2,\dotsc{},a_n)$ have positive signs. \item[(b)] The polyhedron $R\cap{}P$ is empty if all left-to-right minima of $(a_1,a_2,\dotsc{},a_n)$ have negative signs. \item[(c)] The polyhedron $R\cap{}P$ is nonempty and unbounded if neither of the above holds. \end{itemize} \end{lemma}
\begin{proof} Let $e_1,e_2,\dotsc{},e_n$ be the unit vectors $(1,0,\dotsc{},0)$, $(0,1,\dotsc{},0)$, $\dotsc{}$, $(0,0,\dotsc{},1)$ in $\mathbb{R}^n$. Then, the region $R$ is the cone spanned over $\mathbb{R}_{\geq{}0}$ by the following $n$ vectors: \begin{align*} v_1 &= \sigma{}(a_1)e_{a_1},\\ v_2&=\sigma{}(a_1)e_{a_1}+\sigma{}(a_2)e_{a_2},\\ &\dotsb{},\\ v_n&=\sigma{}(a_1)e_{a_1}+\dotsb{}+\sigma{}(a_n)e_{a_n}. \end{align*} Let $L_i$ be the ray $\mathbb{R}_{\geq{}0}(v_i)$. Fix the linear form $f(x) = l_1x_1+\dotsb{}+l_nx_n$ on $\mathbb{R}^n$. Then, \begin{equation*} f(v_i) = \sigma{}(a_1)l_{a_1}+\dotsb{}+\sigma{}(a_i)l_{a_i}. \end{equation*} Since $l_1\gg{}\dotsb{}\gg{}l_n>0$, $f(v_i)$ has the same sign as $\sigma{}(\min{}(a_1,\dotsc{},a_i))$.
(a) By assumption, each left-to-right minima $\min{}(a_1,\dotsc{},a_i)$ has positive sign. Therefore, each $f(v_i)$ is positive. Hence, $P=\{f(x)=1\}$ intersects the ray $L_i$ at $v_i/f(v_i)$. Then, $R\cap{}P$ is the simplex with vertices $v_1/f(v_1),\dotsc{},v_n/f(v_n)$, which is nonempty and bounded.
(b) Similarly, each $f(v_i)$ is negative. Then, $f$ is negative on $R$. Thus, $P=\{f(x)=1\}$ does not intersect $R$.
(c) In this case, some $f(v_i)$ are positive and the others are negative. Say $f(v_i)>0$ and $f(v_j)<0$. Then, $P=\{f(x)=1\}$ intersects $L_i$ at $v_i/f(v_i)$. Moreover, $f(f(v_i)v_j-f(v_j)v_i))=0$. Hence, $R\cap{}P$ contains the affine ray $v_i/f(v_i)+\mathbb{R}^+(f(v_i)v_j-f(v_j)v_i))$. Thus, $R\cap{}P$ is nonempty and unbounded. \end{proof}
\begin{theorem}\label{theorem-bn} The following four sets are in bijection: \begin{itemize} \item[(1)] The set of bounded regions in $\mathcal{B}_n\cap{}P$. \item[(2)] The set of signed permutations of $[n]$ all of whose left-to-right minima have positive signs. \item[(3)] The set of build-tree codes of length $n$. \item[(4)] The set of increasingly labeled trees with $n+1$ vertices. \end{itemize} Moreover, the common cardinality of these sets equals $(2n-1)!!=1\cdot{}3\cdot{}5\dotsb{}(2n-1)$. \end{theorem}
\begin{proof} $(1)\leftrightarrow(2)$. The regions of $\mathcal{B}_n\cap{}P$ are exactly $R\cap{}P$ for region $R$ of $\mathcal{B}_n$ such that $R\cap{}P$ is nonempty. Therefore, it follows from Lemma \ref{lemma-bn-regions} that the cardinalities of the sets (1) and (2) are equal.
$(2)\leftrightarrow(3)$. We construct a bijection between the set of build-tree codes of length $n$ and the set of signed permutations $[n]$ all of whose left-to-right minima have positive signs. Given a build-tree code $c_1c_2\dotsb{}c_n$, we construct a signed permutation. We start from the empty signed permutation. In each step, we look at $c_i$ and add one element to the signed permutation: \begin{itemize} \item[(i)] If $c_i=0^+$, then add $i$ to the beginning with positive sign. \item[(ii)] If $c_i=j^+$ for $j>0$, then add $i$ immediately after $j$ with the opposite sign from $j$. \item[(iii)] If $c_i=j^-$ for $j>0$, then add $i$ immediately after $j$ with the same sign as~$j$. \end{itemize} We obtain a signed permutation of $[n]$ in this way. In each step, if (ii) or (iii) hold, the left-to-right minima stay the same. If (i) holds, then $i$ becomes a new left-to-right minimum, and we construct it to have the positive sign. Thus, the signed permutation we constructed has the property that all of its left-to-right minima have positive signs. On the other hand, given a signed permutation of $[n]$ all of whose left-to-right minima have positive signs, we can reverse the construction and obtain a build-tree code. It is straightforward to verify that this gives a bijection.
$(3)\leftrightarrow{}(4)$. See Callan \cite{citation-callan}. We construct a bijection between the set of build-tree codes of length $n$ and the set of increasingly labeled trees with $n+1$ vertices. Given a build-tree code $c_1c_2\dotsb{}c_n$, we construct an increasingly labeled tree. We start from the rooted plane tree with no non-root vertices. In each step, we look at $c_i$ and add one leaf to the tree with label $i$: \begin{itemize} \item[(i)] If $c_i=0^+$, then add $i$ as the leftmost child of the root. \item[(ii)] If $c_i=j^+$ for $j>0$, then add $i$ as the leftmost child of vertex $j$. \item[(iii)] If $c_i=j^-$ for $j>0$, then add $i$ as the immediate right neighbor of $j$. \end{itemize} On the other hand, given an increasingly labeled tree with $n+1$ vertices, we can reverse the construction and obtain a build-tree code. It is straightforward to verify that this gives a bijection.
It is easy to see that there are $(2n-1)!!$ build-tree codes of length $n$, because each $c_i$ has exactly $2i-1$ independent choices. \end{proof}
\begin{example}\label{example-signed-permutation} Table 1 illustrates how we obtain a signed permutation of $\{1,2,3,4,5,6\}$ from the build-tree code $0^+1^-1^+1^+0^+3^+$ with the construction above.
\begin{table}[h!] \caption{Constructing a signed permutation from a build-tree code}
\begin{tabular}{|c|c|c|c|} \toprule{} Step & Build-tree code & Rule applied & Signed permutation \\ \midrule{} 0 & & & \\ \midrule{} 1 & $0^+$ & (i) beginning, $+$ sign & $1^+$ \\ \midrule{} 2 & $0^+1^-$ & (iii) after $1$, same sign & $1^+2^+$ \\ \midrule{} 3 & $0^+1^-1^+$ & (ii) after $1$, opposite sign & $1^+3^-2^+$ \\ \midrule{} 4 & $0^+1^-1^+1^+$ & (ii) after $1$, opposite sign & $1^+4^-3^-2^+$ \\ \midrule{} 5 & $0^+1^-1^+1^+0^+$ & (i) beginning, $+$ sign & $5^+1^+4^-3^-2^+$ \\ \midrule{} 6 & $0^+1^-1^+1^+0^+3^+$ & (ii) after $3$, opposite sign & $5^+1^+4^-3^-6^+2^+$ \\ \cline{1-4} \end{tabular} \end{table}
\end{example}
\begin{example}\label{example-tree} Table 2 illustrates how we obtain an increasingly labeled tree with $7$ vertices from the build-tree code $0^+1^+1^+1^-0^+3^+$ with the construction above.
\begin{table}[h!] \caption{Constructing an increasingly labeled tree from a build-tree code}
\begin{tabular}{|c|c|c|c|} \toprule{} Step & Build-tree code & Rule applied & Tree \\ \midrule{} \parbox[c]{1cm}{\centering{}0} & & & \parbox[c]{4cm}{\centering{}\includegraphics{figure-example-0}}\\ \midrule{} \parbox[c]{1cm}{\centering{}1} & \parbox[c]{3cm}{\centering{}$0^+$} & \parbox[c]{2.5cm}{\centering{}(i) leftmost child of $0$} & \parbox[c]{4cm}{\centering{}\includegraphics{figure-example-1}}\\ \midrule{} \parbox[c]{1cm}{\centering{}2} & \parbox[c]{3cm}{\centering{}$0^+1^+$} & \parbox[c]{2.5cm}{\centering{}(ii) leftmost child of $1$} & \parbox[c]{4cm}{\centering{}\includegraphics{figure-example-2}}\\ \midrule{} \parbox[c]{1cm}{\centering{}3} & \parbox[c]{3cm}{\centering{}$0^+1^+1^+$} & \parbox[c]{2.5cm}{\centering{}(ii) leftmost child of $1$} & \parbox[c]{4cm}{\centering{}\includegraphics{figure-example-3}}\\ \midrule{} \parbox[c]{1cm}{\centering{}4} & \parbox[c]{3cm}{\centering{}$0^+1^+1^+1^-$} & \parbox[c]{2.5cm}{\centering{}(iii) right neighbor of $1$} & \parbox[c]{4cm}{\centering{}\includegraphics{figure-example-4}}\\ \midrule{} \parbox[c]{1cm}{\centering{}5} & \parbox[c]{3cm}{\centering{}$0^+1^+1^+1^-0^+$} & \parbox[c]{2.5cm}{\centering{}(i) leftmost child of $0$} & \parbox[c]{4cm}{\centering{}\includegraphics{figure-example-5}}\\ \midrule{} \parbox[c]{1cm}{\centering{}6} & \parbox[c]{3cm}{\centering{}$0^+1^+1^+1^-0^+3^+$} & \parbox[c]{2.5cm}{\centering{}(ii) leftmost child of $3$} & \parbox[c]{4cm}{\centering{}\includegraphics{figure-example-6}}\\ \cline{1-4} \end{tabular} \end{table}
\end{example}
\begin{remark} Stanley \cite[Section 5.1]{citation-stanley2} computes the characteristic polynomial for the hyperplane arrangement $\mathcal{B}_n$. The number $(2n-1)!!$ is the signed constant term of the characteristic polynomial. \end{remark}
\section{Bounded regions in a slice of $\mathcal{H}_n$}\label{section-hn}
First, we consider the regions of the central hyperplane arrangement $\mathcal{H}_n$. Let $(A_1,A_2,\dotsc{},A_k)$ be an ordered partition of $[n]$. We define a cone in $\mathbb{R}^n$ by \begin{align*} R^+ = \{(x_1,\dotsc{},x_n)\in{}\mathbb{R}^n\colon{}&x_i>0\text{ for }i\in{}A_j\text{ for odd }j,\\ &x_i<0\text{ for }i\in{}A_j\text{ for even }j,\\
&|x_{i_1}|>|x_{i_2}|\text{ for }i_1\in{}A_{j},i_2\in{}A_{j+1},1\leq{}j\leq{}k-1\}. \end{align*} Equivalently, the $3$rd condition above can be replaced by the condition that $x_{i_1}+x_{i_2}$ has the same sign as $(-1)^{j+1}$. Let $R^-=-R^+$.
\begin{lemma}\label{lemma-two-to-one} There is a $2$ to $1$ map from the set of regions of $\mathcal{H}_n$ to the set of ordered partitions of $[n]$. \end{lemma}
\begin{proof}
First, we notice that $R^+$ and $R^-$ are defined by inequalities involving only the linear forms in $\mathcal{H}_n$. Also, all signs of $x_i$ are implied by the defining inequalities of $R^+$ and $R^-$. These inequalities also imply the order of the $|x_i|$ except those $i$ in the same block. Since the $x_i$ with $i$ in the same block have the same sign, all signs of $x_i+x_j$ are implied by the defining inequalities. Therefore, $R^+$ and $R^-$ are indeed regions of $\mathcal{H}_n$.
On the other hand, given a generic point $(x_1,\dotsc{},x_n)\in{}\mathbb{R}^n$, we claim that it lies in a region of the form $R^+$ or $R^-$. Indeed, we order the $x_i$ by their absolute value: $|x_{p_1}|>|x_{p_2}|>\dotsb{}>|x_{p_n}|$. So, we get a permutation $(p_1,\dotsc{},p_n)$ of $[n]$. Then, we group together consecutive segments of the $p_i$ such that the $x_{p_i}$ has the same sign. In this way, we get an ordered partition of $[n]$. It follows that the point lies in $R^+$ or $R^-$, depending on the sign of $x_{p_1}$. Thus, we get a $2$ to $1$ correspondence from the set of regions of $\mathcal{H}_n$ to the set of ordered partitions of $[n]$. \end{proof}
\begin{lemma}\label{lemma-hn-regions} Let $(A_1,A_2,\dotsc{},A_k)$ be an ordered partition of $[n]$. Let $R^+$, $R^-$ be the two corresponding regions of $\mathcal{H}_n$. Then \begin{itemize} \item[(a)] If $(A_1,A_2,\dotsc{},A_k)$ has all left-to-right minima at odd locations, then $R^+\cap{}P$ is nonempty and bounded, and $R^-\cap{}P$ is empty. \item[(b)] Otherwise, both $R^+\cap{}P$ and $R^-\cap{}P$ are nonempty and unbounded. \end{itemize} \end{lemma}
\begin{proof} Since $\mathcal{B}_n$ refines $\mathcal{H}_n$, a region of $\mathcal{H}_n\cap{}P$ is nonempty (resp. unbounded) if and only if it contains a nonempty (resp. unbounded) region of $\mathcal{B}_n\cap{}P$. From the proof of Lemma \ref{lemma-two-to-one}, $R^+$ (resp. $R^-$) contains exactly the regions in $\mathcal{B}_n$ corresponding to signed permutations $((a_1,\dotsc{},a_n),\sigma{})$ such that $\sigma{}(a_1)=1$, (resp. $\sigma{}(a_1)=-1$) and $(A_1,A_2,\dotsc{},A_k)$ can be obtained from $(a_1,\dotsc{},a_n)$ by grouping together consecutive elements with the same signs.
(a) Let $(x_1,\dotsc{},x_n)\in{}R^+$. Since the $x_i$ for $i\in{}A_j$ have sign $(-1)^{j-1}$, an odd location in $(A_1,A_2,\dotsc{},A_k)$ corresponds to elements in $((a_1,\dotsc{},a_n),\sigma{})$ with positive signs. Therefore, $(A_1,A_2,\dotsc{},A_k)$ has all left-to-right minima at odd locations if and only if all left-to-right minima of $((a_1,\dotsc{},a_n),\sigma{})$ have positive signs. Therefore, $R^+$ contains only regions of type (a) in Lemma \ref{lemma-bn-regions}, which are nonempty and bounded. Thus, $R^+\cap{}P$ is nonempty and bounded. Similarly, $R^-$ contains only regions of type (b) in Lemma \ref{lemma-bn-regions}. Thus, $R^-\cap{}P$ is empty.
(b) Similarly, both $R^+$ and $R^-$ contains only regions of type (c) in Lemma \ref{lemma-bn-regions}. Thus, both $R^+\cap{}P$ and $R^-\cap{}P$ are nonempty and unbounded. \end{proof}
Now we prove our main result.
\begin{proof}[Proof of Theorem \ref{theorem-main}] $(1)\leftrightarrow{}(2)$. The regions of $\mathcal{H}_n\cap{}P$ are exactly $R\cap{}P$ for regions $R$ of $\mathcal{H}_n$ such that $R\cap{}P$ is nonempty. Therefore, it follows from Lemma \ref{lemma-hn-regions} that the cardinalities of the sets (1) and (2) are equal.
$(2)\leftrightarrow{}(3)$. For each ordered partition $(A_1,A_2,\dotsc{},A_k)$, we construct a signed permutation with decreasing blocks by writing elements of each $A_i$ in decreasing order and concatenating them to form a permutation. The signs of the elements of $A_i$ is $(-1)^{i-1}$. For example, the ordered partition $(15,246,3)$ is sent to $5^+1^+6^-4^-2^-3^+$. It is clear that ordered partitions of $[n]$ all of whose left-to-right minima occur at odd locations are in bijection with signed permutations of $[n]$ with decreasing blocks all of whose left-to-right minima have positive signs.
$(3)\leftrightarrow{}(4)$. A signed permutation fails to have decreasing blocks if and only if there are two adjacent elements $u,v$ with the same sign such that $u<v$. In other words, $v$ is added after $u$ with the same sign, and no more element is added after $u$ afterwards. Under the bijection described in the proof of Theorem \ref{theorem-bn}, this translates exactly to the condition that there is no $u^+$ after some $u^-$. Thus, the bijection sends signed permutations with decreasing blocks all of whose left-to-right minima have positive signs to build-tree codes such that there is a $v^+$ after each $v^-$, and vice versa.
$(4)\leftrightarrow{}(5)$. Given a build-tree code, we keep the numerals in the build-tree code, and reverse the order of the signs over each fixed numeral. For example, $0^+1^+1^-2^+2^+2^-$ goes to $0^+1^-1^+2^-2^+2^+$. In this way, the build-tree codes such that there is a $v^+$ before each $v^-$ are sent exactly to the build-tree codes such that there is a $v^+$ after each $v^-$, and vice versa.
$(5)\leftrightarrow{}(6)$. An increasingly labeled tree $(T,\lambda{})$ can be considered as a process of constructing the tree $T$ by adding vertices in the order determined by $\lambda{}$. The right associate of a vertex $v$, if it exists, is the first vertex added as the immediate right neighbor of $v$. Under the bijection described in the proof of Theorem \ref{theorem-bn}, the label of the right associate of $v$ corresponds to the location of the fist appearance of $v^-$ in the build-tree code. The condition that there is a $v^+$ before each $v^-$ translates to the condition that the right associate of $v$, if it exists, is added after at least one child of $v$. This is exactly the defining condition for Klazar trees. Thus, the bijection sends Klazar trees to build-tree codes such that there is a $v^+$ before each $v^-$, and vice versa.
$(6)\leftrightarrow{}(7)$. See Callan \cite{citation-callan}. We elaborate the proof for completion. Given a drawing $T_0,T_1,\dotsc{},T_n=T$, we can reconstruct an increasing labeling of $T$ as follows: suppose we have already constructed an increasing labeling of $T_{n-1}$. By definition, $T_{n-1}$ can be obtained from $T$ by removing a leaf. We label this leaf $n$, and label the rest of the tree in the same way as in $T_{n-1}$. In this way, we get an increasing labeling of $T$. To make the construction unambiguous, if there are multiple leaves in $T$ that can be removed to get $T_{n-1}$, we always choose the leftmost possible one. We claim that the resulting increasingly labeled tree $(T,\lambda{})$ is a Klazar tree. If it is not, then there is a vertex $v$ with a right associate $u$ such that either $v$ is a leaf or $\lambda{}(u)$ is smaller than the labels of all children of $v$. Since $T_{\lambda{}(u)}$ contains exactly the vertices in $T$ with label $\leq{}\lambda{}(u)$, the vertices $u$ and $v$ are adjacent leaves in $T_{\lambda{}(u)}$. Therefore, removing either $u$ or $v$ in $T_{\lambda{}(u)}$ results in the same rooted plane tree. Since $v$ is to the left of $u$, by the construction above, we would choose $v$ rather than $u$ in the $\lambda{}(u)$th step. So we get a contradiction. Thus, $(T,\lambda{})$ is a Klazar tree.
An increasingly labeled tree $(T,\lambda{})$ naturally gives a drawing $T_0,T_1,\dotsc{},T_n=T$, by setting $T_i$ to contain exactly the vertices with labels $\leq{}i$. Clearly this is a left inverse of the construction process above. It suffices to prove that different Klazar trees give different drawings. Assume that two different Klazar trees $(T,\lambda{})$ and $(T',\lambda{}')$ give the same drawing. Let $T_i$ (resp. $T'_i$) be the subtree of $T$ (resp. $T'$) spanned by vertices with labels $\leq{}i$. Then, $T_i$ and $T'_i$ are isomorphic. Thus, we can identify $T$ with $T'$. Let $k$ be the smallest positive integer such that $(T_k,\lambda{}|_{T_k})$ and $(T'_k,\lambda{}'|_{T'_k})$ are not isomorphic increasingly labeled trees. Moreover, both $T_k$ and $T'_k$ are Klazar trees. Without loss of generality, we may assume that $k=n$.
Let $u$ (resp. $u'$) be the vertex of $T$ labeled $n$ in $(T,\lambda{})$ (resp. $(T,\lambda{}')$). Note that $T_{n-1}$ (resp. $T'_{n-1}$) can be obtained from $T$ by removing $u$ (resp. $u'$). Then, both $u$ and $u'$ are leaves of $T$, and $u\neq{}u'$ by the minimality of $k$. Let $v$ be the lowest common ancestor of $u$ and $u'$. Let $v_1,v_2,\dotsc{},v_s$ be the children of $v$, ordered from left to right. Suppose that $u$ (resp. $u'$) is a descendent of $v_j$ (resp. $v_{j'}$). Then $j\neq{}j'$ by the choice of $v$. If neither $u$ nor $u'$ is a child of $v$, then the size of the subtree of $T_{n-1}$ rooted at $v_j$ would be $1$ smaller than the subtree of $T'_{n-1}$ rooted at $v_j$. If exactly one of $u$ or $u'$, say $u$, is a child of $v$, then $v$ would have one more child in $T'_{n-1}$ than in $T_{n-1}$. Both cases contradict our assumption that $T_{n-1}$ is isomorphic to $T'_{n-1}$. Thus, both $u$ and $u'$ are children of $v$. Without loss of generality, we may assume that $u'$ is to the right of $u$. Since $\lambda{}'(u')=n>\lambda{}'(u)$, the vertex $u$ has a right associate in $(T,\lambda{}')$. However, $u$ is a leaf. Thus, the condition for $(T,\lambda{}')$ being a Klazar tree is violated. So we get a contradiction.
It is shown by Klazar \cite{citation-klazar} that the cardinality of the set (7) has the given exponential generating function. \end{proof}
We present an alternative proof by counting the cardinality of the set (2). We say that the {\it type} of an ordered partition $(A_1,\dotsc{},A_k)$ is the set $\{A_1,\dotsc{},A_k\}$, which is a partition of $[n]$. An ordered partition of type $\{ \{1\}, \{2\}, \dotsc{}, \{n\} \}$ is just a permutation of $[n]$.
\begin{lemma} Let $p_n$ be the number of permutations of $[n]$ whose all left-to-right minima occurs at odd locations. Set $p_0=1$. Then \begin{equation*} \sum_{n=0}^{\infty{}}p_n\frac{x^n}{n!} = \sqrt{\frac{1+x}{1-x}}. \end{equation*} \end{lemma}
\begin{proof} The proof is found in a post by Callan in \cite[A000246]{citation-oeis}. For any permutation $(a_1,\dotsc{},a_n)$ of $[n]$ all of whose left-to-right minima occur at odd locations, we can construct a permutation of $[n-1]$ by removing $a_n$ and decrementing all elements greater than $a_n$ by $1$. This new permutation has all left-to right minima at odd locations. On the other hand, for any permutation of $[n-1]$ whose all left-to-right minima occurs at odd locations and any $a_n\in{}[n]$, we can construct a permutation of $[n]$ by incrementing all elements greater than $a_n$ by $1$ and adding $a_n$ to the end. This new permutation has all left-to-right minima at odd locations if and only if $n$ is odd or $n$ is even and $a_n>1$. Therefore, from this correspondence we get $p_n=np_{n-1}$ for odd $n$ and $p_n=(n-1)p_{n-1}$ for even $n$. By induction, $p_{n}=((n-1)!!)^2$ for even $n$ and $p_n=n!!(n-2)!!$ for odd $n$.
Then, \begin{align*} \sum_{n=0}^{\infty{}}p_n\frac{x^n}{n!} &= \sum_{n\text{ even}}((n-1)!!)^2\frac{x^n}{n!}+\sum_{n\text{ odd}}n!!(n-2)!!\frac{x^n}{n!}\\ &=\sum_{m=0}^{\infty{}}((2m-1)!!)^2\frac{x^{2m}}{(2m)!}+\sum_{m=0}^{\infty{}}(2m+1)!!(2m-1)!!\frac{x^{2m+1}}{(2m+1)!}\\ &=\sum_{m=0}^{\infty{}}((2m-1)!!)^2\frac{x^{2m}}{(2m)!}+\sum_{m=0}^{\infty{}}((2m-1)!!)^2\frac{x^{2m+1}}{(2m)!}\\ &=(1+x)\sum_{m=0}^{\infty{}}((2m-1)!!)^2\frac{x^{2m}}{(2m)!}. \end{align*}
On the other hand, \begin{align*} \frac{1}{\sqrt{1-x^2}} &= \sum_{m=0}^{\infty{}}{1/2 \choose m}(-1)^mx^{2m}\\ &=\sum_{m=0}^{\infty{}}\frac{(2m-1)!!}{2^m(m!)}x^{2m}\\ &=\sum_{m=0}^{\infty{}}((2m-1)!!)^2\frac{x^{2m}}{(2m)!}. \end{align*}
So \begin{equation*} \sum_{n=0}^{\infty{}}p_n\frac{x^n}{n!} = \frac{1+x}{\sqrt{1-x^2}} = \sqrt{\frac{1+x}{1-x}}. \end{equation*} \end{proof}
\begin{lemma} The number of ordered partitions of $[n]$ of type $\{A_1,\dotsc{},A_k\}$ all of whose left-to-right minima occurs at odd locations equals $p_k$. \end{lemma}
\begin{proof} We may replace each $A_i$ by $\{\min{}A_i\}$ without affecting the locations of the left-to-right minima. Therefore, we can reduce the problem to the case of ordered partitions of $k$ distinct numbers. Thus, the number is $p_k$. \end{proof}
Let $b_n$ denote the number (2). Set $b_0=0$. Then, it follows from the composition formula \cite[Theorem 5.1.4]{citation-stanley} that \begin{equation*} \sum_{n=0}^{\infty{}}b_n\frac{x^n}{n!} = \sqrt{\frac{1+(e^x-1)}{1-(e^x-1)}} = \sqrt{\frac{e^x}{2-e^x}}. \end{equation*}
\begin{example} The build-tree codes $0^+1^+1^+1^-0^+3^+$ and $0^+1^-1^+1^+0^+3^+$ in Example \ref{example-signed-permutation} and Example \ref{example-tree} can be obtained from each other by reversing the sequence of signs over each fixed numeral in the build-tree code. Therefore, the objects in Table 1 and Table 2 are in bijection. \end{example}
\begin{example}
Figure 2 shows the $7$ bounded regions in $\mathcal{H}_3\cap{}P$.
\begin{figure}
\caption{Seven bounded regions in $\mathcal{H}_3\cap{}P$}
\end{figure}
These $7$ bounded regions are labeled $(1), (2), \dotsc{}, (7)$. They are: \begin{equation*} \begin{array}{cl} (1) & x_1>0,x_2>0,x_3>0 \\
(2) & x_1>0,x_2>0,x_3<0,|x_1|,|x_2|>|x_3| \\
(3) & x_1>0,x_2<0,x_3>0,|x_1|,|x_3|>|x_2| \\
(4) & x_1>0,x_2<0,x_3<0,|x_1|>|x_2|,|x_3| \\
(5) & x_1>0,x_2<0,x_3>0,|x_1|>|x_2|>|x_3| \\
(6) & x_1>0,x_2>0,x_3<0,|x_1|>|x_3|>|x_2| \\
(7) & x_1>0,x_2>0,x_3<0,|x_2|>|x_3|>|x_1| \\ \end{array} \end{equation*}
Table 3 shows various objects that are in bijection with the $7$ bounded regions.
\begin{table}[h!] \caption{The bijections for the $n=3$ case}
\begin{tabular}{|c|c|c|c|c|c|} \toprule{} \parbox[t]{1cm}{Label in Figure 2} & \parbox[t]{1.5cm}{Ordered partition} & \parbox[t]{1.5cm}{Signed permutation with decreasing blocks} & \parbox[t]{1.5cm}{Build-tree code such that there is a $v^+$ after each $v^-$} & \parbox[t]{1.5cm}{Build-tree code such that there is a $v^+$ before each $v^-$} & Klazar tree \\ \midrule{} \parbox[c]{1cm}{\centering{}(1)} & \parbox[c]{1.5cm}{\centering{}$123$} & \parbox[c]{1.5cm}{\centering{}$3^+2^+1^+$} & \parbox[c]{1.5cm}{\centering{}$0^+0^+0^+$} & \parbox[c]{1.5cm}{\centering{}$0^+0^+0^+$} & \parbox[c]{3cm}{\centering{}\includegraphics[scale=0.85]{figure-tree-1}} \\ \midrule{} \parbox[c]{1cm}{\centering{}(2)} & \parbox[c]{1.5cm}{\centering{}$12,3$} & \parbox[c]{1.5cm}{\centering{}$2^+1^+3^-$} & \parbox[c]{1.5cm}{\centering{}$0^+0^+1^+$} & \parbox[c]{1.5cm}{\centering{}$0^+0^+1^+$} & \parbox[c]{3cm}{\centering{}\includegraphics[scale=0.85]{figure-tree-2}} \\ \midrule{} \parbox[c]{1cm}{\centering{}(3)} & \parbox[c]{1.5cm}{\centering{}$13,2$} & \parbox[c]{1.5cm}{\centering{}$3^+1^+2^-$} & \parbox[c]{1.5cm}{\centering{}$0^+1^+0^+$} & \parbox[c]{1.5cm}{\centering{}$0^+1^+0^+$} & \parbox[c]{3cm}{\centering{}\includegraphics[scale=0.85]{figure-tree-3}} \\ \midrule{} \parbox[c]{1cm}{\centering{}(4)} & \parbox[c]{1.5cm}{\centering{}$1,23$} & \parbox[c]{1.5cm}{\centering{}$1^+3^-2^-$} & \parbox[c]{1.5cm}{\centering{}$0^+1^+1^+$} & \parbox[c]{1.5cm}{\centering{}$0^+1^+1^+$} & \parbox[c]{3cm}{\centering{}\includegraphics[scale=0.85]{figure-tree-4}} \\ \midrule{} \parbox[c]{1cm}{\centering{}(5)} & \parbox[c]{1.5cm}{\centering{}$1,2,3$} & \parbox[c]{1.5cm}{\centering{}$1^+2^-3^+$} & \parbox[c]{1.5cm}{\centering{}$0^+1^+2^+$} & \parbox[c]{1.5cm}{\centering{}$0^+1^+2^+$} & \parbox[c]{3cm}{\centering{}\includegraphics[scale=0.85]{figure-tree-5}} \\ \midrule{} \parbox[c]{1cm}{\centering{}(6)} & \parbox[c]{1.5cm}{\centering{}$1,3,2$} & \parbox[c]{1.5cm}{\centering{}$1^+3^-2^+$} & \parbox[c]{1.5cm}{\centering{}$0^+1^-1^+$} & \parbox[c]{1.5cm}{\centering{}$0^+1^+1^-$} & \parbox[c]{3cm}{\centering{}\includegraphics[scale=0.85]{figure-tree-6}} \\ \midrule{} \parbox[c]{1cm}{\centering{}(7)} & \parbox[c]{1.5cm}{\centering{}$2,3,1$} & \parbox[c]{1.5cm}{\centering{}$2^+3^-1^+$} & \parbox[c]{1.5cm}{\centering{}$0^+0^+2^+$} & \parbox[c]{1.5cm}{\centering{}$0^+0^+2^+$} & \parbox[c]{3cm}{\centering{}\includegraphics[scale=0.85]{figure-tree-7}} \\ \cline{1-6} \end{tabular} \end{table}
\end{example}
\noindent \footnotesize {\bf Author's address}: Department of Mathematics, University of California, Berkeley, CA 94720, USA. {\tt qingchun@berkeley.edu}
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\begin{abstract} We examine a transmission problem driven by a degenerate quasilinear operator with a natural interface condition. Two aspects of the problem entail genuine difficulties in the analysis: the absence of representation formulas for the operator and the degenerate nature of the diffusion process. Our arguments circumvent these difficulties and lead to new regularity estimates. For bounded interface data, we prove the local boundedness of weak solutions and establish an estimate for their gradient in ${\rm BMO}-$spaces. The latter implies solutions are of class $C^{0,{\rm Log-Lip}}$ across the interface. Relaxing the assumptions on the data, we establish local H\"older continuity for the solutions. \end{abstract}
\keywords{Transmission problems; $p-$Laplace operator; local boundedness; BMO gradient estimates; Log-Lipschitz regularity.}
\subjclass{35B65; 35J92; 35Q74.}
\maketitle
\section{Introduction}\label{sec_mollybloom}
Transmission problems describe diffusive processes within heterogeneous media that change abruptly across certain interfaces. They find application, for example, in the study of electromagnetic conductivity and composite materials, and their mathematical formulation involves a domain split into sub-regions, where partial differential equations (PDEs) are prescribed. Since the PDEs vary from region to region, the problem may have discontinuities across the interfaces. Consequently, the geometry of these interfaces (which, in contrast to free boundary problems, are fixed and given a priori) and the structure of the underlying equations play a crucial role in analysing transmission problems.
This class of problems first appeared circa 1950, in the work of Mauro Picone \cite{Picone1954}, as an attempt to address heterogeneous materials in elasticity theory. Several subsequent works developed the basics of the theory and generalised it in various directions \cite{Borsuk1968, Campanato1957, Campanato1959, Campanato1959a, Iliin-Shismarev1961, Schechter1960, Sheftel1963, Stampacchia1956}. We refer the interested reader to \cite{Borsuk2010} for a comprehensive account of this literature.
Developments concerning the regularity of the solutions to transmission problems are much more recent. In \cite{Li-Vogelius2000}, the authors study a class of elliptic equations in divergence form, with discontinuous coefficients, modelling composite materials with closely spaced interfacial boundaries, such as fibre-reinforced structures. The main result in that paper is the local H\"older continuity for the gradient of the solutions, with estimates. The findings in \cite{Li-Vogelius2000} are relevant from the applied perspective since the gradient of a solution accounts for the stress of the material, and estimating it shows the stresses remain uniformly bounded, even when fibres are arbitrarily close to each other. The vectorial counterpart of the results in \cite{Li-Vogelius2000} appeared in \cite{Li-Nirenberg2003}, where regularity estimates for higher-order derivatives of the solutions are obtained. See also the developments reported in \cite{Bonnetier2000}.
A further layer of analysis concerns the proximity of sub-regions in limiting scenarios. In \cite{Bao-Li-Yin1}, the authors examine a domain containing two subregions, which are $\varepsilon-$apart, for some $\varepsilon>0$. Within each sub-region, the diffusion process is given by a divergence-form equation with a diffusivity coefficient $A \neq 1$. In the remainder of the domain, the diffusivity is also constant but equal to $1$. By setting $A=+\infty$, the authors examine the case of perfect conductivity. The remarkable fact about this model is that estimates on the gradient of the solutions deteriorate as the two regions approach each other. In \cite{Bao-Li-Yin1}, the authors obtain blow-up rates for the gradient norm in terms of $\varepsilon\to 0$. We also notice the findings reported in \cite{Bao-Li-Yin2} extend those results to the context of multiple inclusions and also treat the case of perfect insulation $A=0$. We also refer the reader to \cite{Briane}.
More recently, the analysis of transmission problems focused on the geometry of the interface. The minimum requirements on the transmission interface yielding regularity properties for the solutions are particularly interesting. In \cite{CSCS2021}, the authors consider a domain split into two sub-regions. Inside each sub-region, the problem's solution is required to be a harmonic function, and a flux condition is prescribed along the interface separating the sub-regions. By resorting to a representation formula for harmonic functions, the authors establish the existence of solutions to the problem and prove that solutions are of class $C^{0,{\rm Log-Lip}}$ across the interface. In addition, under the assumption that the interface is locally of class $C^{1,\alpha}$, they prove the solutions are of class $C^{1,\alpha}$ within each sub-region, \emph{up to the transmission interface}. This fact follows from a new stability result allowing the argument to import information from the case of flat interfaces. In \cite{SCS2022}, the authors extend the analysis in \cite{CSCS2021} to the context of fully nonlinear elliptic operators. Under the assumption that the interface is of class $C^{1,\alpha}$, they prove that solutions are of class $C^{0,\alpha}$ across, and $C^{1,\alpha}$ up to the interface. Furthermore, if the interface is of class $C^{2,\alpha}$, then solutions became $C^{2,\alpha}-$regular, also up to the interface. The findings in \cite{SCS2022} rely on a new Aleksandrov-Bakelman-Pucci estimate and variants of the maximum principle and the Harnack inequality. We also notice the developments reported in \cite{Borsuk2019}. In that paper, the author proves local boundedness in a neighbourhood of boundary points for a transmission problem driven by a $p-$Laplacian type operator.
Our gist in this paper is to extend the results of \cite{CSCS2021} to the case of degenerate quasilinear equations, which are ``\textit{the natural, and, in a sense, the best generalisation of the $p-$Laplace equation}" (cf. \cite{Lieberman_1991}), namely
$$ \textnormal{div}\left(\frac{g\left(| D u|\right)}{| D u|} D u\right)=0,$$ where $g$ is a nonlinearity satisfying appropriate assumptions. We first prove that weak solutions to the transmission problem, properly defined and whose existence follows from well-known methods, are locally bounded. The proof combines delicate inequalities with the careful choice of auxiliary test functions and a cut-off argument to produce a variant of the weak Harnack inequality. Working under a $C^1$ interface geometry, we then obtain an integral estimate for the gradient, leading to regularity in ${\rm BMO}-$spaces. As a corollary, we infer that solutions are of class $C^{0,{\rm Log-Lip}}$ across the fixed interface. For the particular case of the $p-$Laplace operator, this result follows directly from potential estimates obtained in \cite{DM2011, KM2014c}; we also refer to \cite{M2011,M2011a}. Finally, we relax the boundedness assumption on the interface data and derive local H\"older continuity estimates.
This transmission problem driven by a quasilinear degenerate operator presents genuine difficulties compared to the Laplacian's linear case. Firstly, the operator lacks representation formulas, and the strategy developed in \cite{CSCS2021} is no longer available. Secondly, the degenerate nature of the problem rules out the approach put forward in \cite{SCS2022}. Consequently, one must develop new machinery to examine the regularity of the solutions.
Another fundamental question in transmission problems concerns the optimal regularity \emph{up to the interface}. As mentioned before, results of this type appear in the recent works \cite{CSCS2021} and \cite{SCS2022}; see also \cite{dong20}. The issue remains open in the context of quasilinear degenerate problems, particularly for the $p-$Laplace operator. We believe the analysis of the boundary behaviour of $p-$harmonic functions may yield helpful information in this direction.
The remainder of this article is organised as follows. Section \ref{sec_vicosa} contains the precise formulation of the problem, comments on the existence of a unique solution and gathers basic material used in the paper. In Section \ref{sec_alkhawarizmi}, we put forward the proof of the local boundedness. The proof of the BMO--regularity and its consequences is the object of Section \ref{sec_beacon}, where further generalisations are also included.
\section{Setting of the problem and auxiliary results}\label{sec_vicosa}
In this section, we precisely state our transmission problem, introduce the notion of a weak solution and comment on its existence and uniqueness. We then collect several auxiliary results.
\subsection{Problem setting and assumptions} Let $\Omega\subset\mathbb{R}^d$ be a bounded domain and fix $\Omega_1\Subset\Omega$. Define $\Omega_2:=\Omega\setminus\overline{\Omega_1}$ and consider the interface $\Gamma:=\partial\Omega_1$, which we assume is a $(d-1)-$surface of class $C^1$. For a function $u:\overline{\Omega}\to\mathbb{R}$, we set \begin{equation*}
u_1:=u\big|_{\overline{\Omega_1}}\hspace{.3in} \mbox{and}\hspace{.3in} u_2:=u\big|_{\overline{\Omega_2}}. \end{equation*} Note that we necessarily have $u_1=u_2$ on $\Gamma$. Denoting with $\nu$ the unit normal vector to $\Gamma$ pointing inwards to $\Omega_1$, we write $$\frac{\partial u_i}{\partial\nu} = Du_i \cdot \nu, \quad i=1,2.$$
For a nonlinearity $g$, satisfying appropriate assumptions, we consider the quasilinear degenerate transmission problem consisting of finding a function $u:\overline{\Omega}\to\mathbb{R}$ such that \begin{equation}\label{eq_stima118} \begin{cases}
\textnormal{div}\left(\frac{g\left(| D u_1|\right)}{| D u_1|} D u_1\right)=0&\hspace{.2in}\mbox{in}\hspace{.2in}\Omega_1\\
\vspace*{-0.3cm}\\
\textnormal{div}\left(\frac{g\left(| D u_2|\right)}{| D u_2|} D u_2\right)=0&\hspace{.2in}\mbox{in}\hspace{.2in}\Omega_2,\\ \end{cases} \end{equation} with the additional conditions \begin{equation}\label{eq_stima119} \begin{cases} u=0&\hspace{.2in}\mbox{on}\hspace{.2in}\partial\Omega\\ \vspace*{-0.3cm}\\
\frac{g(| D u_1|)}{| D u_1|}\frac{\partial u_1}{\partial\nu}-\frac{g(| D u_2|)}{| D u_2|}\frac{\partial u_2}{\partial\nu}=f&\hspace{.2in}\mbox{on}\hspace{.2in}\Gamma, \end{cases} \end{equation} for a given function $f$.
We assume the function $g\in C^1\left(\mathbb{R}^+_0\right)$ is such that
\begin{equation}
g_0\le\frac{tg'(t)}{g(t)}\le g_1,\quad\forall t>0,
\label{business class}
\end{equation}
for fixed constants $1\le g_0\le g_1$. Moreover, we assume the monotonicity inequality
\begin{equation}\label{monicavitti}
\bigg(\frac{g(|\xi|)}{|\xi|}\xi-\frac{g(|\zeta|)}{|\zeta|}\zeta\bigg)\cdot(\xi-\zeta)\ge C|\xi-\zeta|^p, \quad\forall\xi,\zeta\in\mathbb{R}^d,
\end{equation} holds for a certain $p>2$ and $C>0$.
By choosing $g(t)=t^{p-1}$, with $p>2$, one gets in \eqref{eq_stima118} two degenerate $p-$Laplace equations. A different example of a nonlinearity $g=g(t)$ satisfying \eqref{business class}-\eqref{monicavitti} is
\[
g(t):=t^{p-1}\ln\left(a+t\right)^\alpha,
\]
for $p>2$, $a>1$ and $\alpha>0$.
We now define the primitive of $g$,
\begin{equation*}
G(t)=\int_0^tg(s)\,{\rm d}s, \quad t \geq 0.
\end{equation*} Due to the assumptions on $g$, one concludes that $G:\mathbb{R}\to\mathbb{R}\cup\left\lbrace+\infty\right\rbrace$ is left-continuous and convex, or a \emph{Young function} (see \cite[Definition 3.2.1]{at1}). Before proceeding, we introduce the Orlicz-Sobolev space defined by $G$.
\begin{Definition}[Orlicz-Sobolev space]\label{notto} Let $G$ be a Young function. We define the Orlicz-Sobolev space $W^{1,G}(\Omega)$ as the set of weakly differentiable functions $u\in W^{1,1}(\Omega)$ such that \[
\int_\Omega G\left(|u(x)|\right){\rm d}x+\int_\Omega G\left(|Du(x)|\right){\rm d}x<\infty. \] The space $W^{1,G}_0(\Omega)$ is the closure of $C^{\infty}_c(\Omega)$ in $W^{1,G}(\Omega)$. \end{Definition}
\subsection{Weak solutions} The precise definition of solution we have in mind is the object of the following definition.
\begin{Definition}\label{def_weaksol} A function $u\in W_0^{1,G}(\Omega)$ is a weak solution of \eqref{eq_stima118}-\eqref{eq_stima119} if \begin{equation}
\int_{\Omega}\frac{g\left(| D u|\right)}{| D u|} D u\cdot D v\,{\rm d}x=\int_{\Gamma}f v\,{\rm d}\mathcal{H}^{d-1},\hspace{.2in}\forall\,v\in W^{1,G}_0(\Omega). \label{portia} \end{equation} \end{Definition}
We use the Hausdorff measure $\mathcal{H}^{d-1}$ in the surface integral to emphasise the operator acting on the solution is a measure supported along the interface, and we write \begin{equation}\label{eq_pde}
-\textnormal{div}\left(\frac{g\left(| D u|\right)}{| D u|} D u\right)=f\,{\rm d}\mathcal{H}^{d-1}\big|_{\Gamma}. \end{equation}
To justify \eqref{eq_pde}, we multiply both equations in \eqref{eq_stima118} by a test function $\varphi\in C^\infty_c(\Omega)$, and formally integrate by parts to get \[
\int_{\Omega_1}\frac{g\left(| D u_1|\right)}{| D u_1|} D u_1\cdot D\varphi\,{\rm d}x=-\int_{\Gamma}\left(\frac{g\left(| D u_1|\right)}{| D u_1|} D u_1\cdot \nu\right)\varphi\,{\rm d}\mathcal{H}^{d-1} \] and \[
\int_{\Omega_2}\frac{g\left(| D u_2|\right)}{| D u_2|} D u_2\cdot D\varphi\,{\rm d}x=-\int_{\Gamma}\left(\frac{g\left(| D u_2|\right)}{| D u_2|} D u_2\cdot \nu\right)\varphi\,{\rm d}\mathcal{H}^{d-1} \] Adding and using \eqref{eq_stima119}, we obtain \[
\int_{\Omega}\frac{g\left(| D u|\right)}{| D u|} D u\cdot D \varphi\,{\rm d}x=\int_{\Gamma}f \varphi\,{\rm d}\mathcal{H}^{d-1},\hspace{.2in}\forall\,\varphi\in W^{1,G}(\Omega). \]
\begin{Remark} We notice the integrals in Definition \ref{def_weaksol} are well-defined. Indeed, let $u,v\in W^{1,G}(\Omega)$; we verify that \begin{equation*}
\int_{\Omega}\frac{g(|Du|)}{|Du|}Du\cdot Dv\,{\rm d}x<\infty. \end{equation*} Since $g$ is increasing, we have $tg(t)\le CG(t)$ for $t\ge0$. Also, $G(t+s)\le C\big(G(t)+G(s)\big)$ for $t,s\ge0$. Hence, \begin{align*}
\bigg|\int_{\Omega}\frac{g(|Du|)}{|Du|}Du\cdot Dv\,{\rm d}x\bigg|\le&\int_{\Omega}g(|Du|)|Dv|\,{\rm d}x\\
\le&\int_{\Omega}g(|Du|+|Dv|)(|Du|+|Dv|)\,{\rm d}x\\
\le&C\int_{\Omega}G(|Du|+|Dv|)\,{\rm d}x\\
\le&C\int_{\Omega}G(|Du|)\,{\rm d}x+C\int_{\Omega}G(|Dv|)\,{\rm d}x\\ <&\infty. \end{align*} \end{Remark}
\begin{Remark} Let $u\in W_0^{1,G}(\Omega)$, and suppose that \eqref{monicavitti} is in force. Then one infers $u\in W_0^{1,p}(\Omega)$. Indeed, that inequality yields \begin{align*}
\int_\Omega|Du|^p\,{\rm d}x\le&C\int_\Omega g(|Du|)|Du|\,{\rm d}x\\
\le&C\int_\Omega G(|Du|)\,{\rm d}x. \end{align*} \end{Remark}
\subsection{Existence and uniqueness of weak solutions} To prove the existence of a unique weak solution to \eqref{eq_stima118}-\eqref{eq_stima119}, one can resort to approximation and monotonicity methods. We refer the reader to \cite{baroni2015}; see also \cite{Lieberman_1991}. Additionally, we remark that the weak solution is the global minimiser of the functional $I: W_0^{1,G}(\Omega)\to\mathbb{R}$ defined by \begin{equation}\label{stima117}
I(u)=\int_{\Omega}G\left(| D u|\right)\,{\rm d}x-\int_{\Gamma}fu\,{\rm d}\mathcal{H}^{d-1}, \end{equation} whose Euler-Lagrange equation, in its weak formulation, is precisely \eqref{portia}.
\subsection{Auxiliary results}\label{subsec_prelim} We now collect some auxiliary material which will be instrumental in the proofs of the main results. We start with a technical inequality (c.f. \cite[Lemma 2]{Serrin_1964}).
\begin{Lemma}\label{lemma numerico} Let $p>0$, and $N\in\mathbb{N}$. Let also $a_1,\dots,a_N,q_1,\dots,q_N$ be real numbers such that $0<a_i<\infty$ and $0\le q_i<p$, for every $i=1,\ldots,N$. Suppose that $z,z_1,\ldots,z_N$ are positive real numbers satisfying \begin{equation*} z^p\le\sum_{i=1}^Na_iz_i^{q_i}. \end{equation*} Then there exists $C>0$ such that \begin{equation*} z\le C\sum_{i=1}^Na_i^{\gamma_i} \end{equation*} where $\gamma_i=(p-q_i)^{-1}$, for $i=1,\ldots,N$. Finally, $C=C(N,p,q_1,\ldots,q_N)$. \end{Lemma}
Although standard in the field, the following result lacks detailed proof in the literature. We include it here for completeness and future reference.
\begin{Lemma}\label{lem_stima119} Fix $R_0>0$ and let $\phi:[0,R_0]\to[0,\infty)$ be a non-decreasing function. Suppose there exist constants $C_1,\alpha,\beta>0$, and $C_2,\mu\ge0$, with $\beta<\alpha$, satisfying \begin{equation*} \phi(r)\le C_1\Big[\Big(\frac{r}{R}\Big)^{\alpha}+\mu\Big]\phi(R)+C_2R^{\beta}, \end{equation*} for every $0<r\le R\le R_0$. Then, for every $\sigma\le\beta$, there exists $\mu_0=\mu_0(C_1,\alpha,\beta,\sigma)$ such that, if $\mu<\mu_0$, for every $0<r\le R\le R_0$, we have \begin{equation*} \phi(r)\le C_3\Big(\frac{r}{R}\Big)^{\sigma}\big(\phi(R)+C_2R^{\sigma}\big), \end{equation*} where $C_3=C_3(C_1,\alpha,\beta,\sigma)>0$. Moreover, \begin{equation*} \phi(r)\le C_4r^{\sigma}, \end{equation*} where $C_4=C_4(C_2,C_3,R_0,\phi(R_0),\sigma)$. \end{Lemma} \begin{proof} For clarity, we split the proof into two steps. First, an induction argument leads to an inequality at discrete scales; then, we pass to the continuous case and conclude the argument.
\noindent{\bf Step 1 -} We want to verify that \begin{equation}\label{eq_1} \phi(\theta^{n+1}R)\le\theta^{(n+1)\delta}\phi(R)+C_2\theta^{n\beta}R^{\beta}\sum_{j=0}^n\theta^{j(\delta-\beta)}, \end{equation} for every $n\in\mathbb{N}$. We notice it suffices to prove the estimate for $\sigma=\beta$ and work in this setting. For $0<\theta<1$ and $0<R\le R_0$ the assumption of the lemma yields \[ \phi(\theta R)\le C_1\bigg[\bigg(\frac{\theta R}{R}\bigg)^{\alpha}+\mu\bigg]\phi(R)+C_2R^{\beta}=\theta^{\alpha}C_1(1+\mu\theta^{-\alpha})\phi(R)+C_2R^{\beta}. \] Choose $\theta\in(0,1)$ such that $2C_1\theta^{\alpha}=\theta^{\delta}$ with $\beta<\delta<\alpha$. Notice that $\theta$ depends only on $C_1,\alpha,\delta$. Take $\mu_0>0$ such that $\mu_0\theta^{-\alpha}<1$. For every $R\le R_0$ we then have \begin{equation}\label{eq_induction00} \phi(\theta R)\le\theta^{\delta}\phi(R)+C_2R^{\beta} \end{equation} and the base case follows. Suppose the statement has already been verified for some $k\in\mathbb{N}$, $k\ge2$; then \begin{equation*} \phi(\theta^kR)\le\theta^{k\delta}\phi(R)+C_2\theta^{(k-1)\beta}R^{\beta}\sum_{j=0}^{k-1}\theta^{j(\delta-\beta)}. \end{equation*} Thanks to \eqref{eq_induction00}, we have \[ \begin{split} \phi(\theta^{k+1}R)&=\phi\big(\theta^k(\theta R)\big)\le\theta^{k\delta}\phi(\theta R)+C_2\theta^{(k-1)\beta}(\theta R)^{\beta}\sum_{j=0}^{k-1}\theta^{j(\delta-\beta)}\\ &\le\theta^{k\delta}\big[\theta^{\delta}\phi(R)+C_2R^{\beta}\big]+C_2\theta^{k\beta}R^{\beta}\sum_{j=0}^{k-1}\theta^{j(\delta-\beta)}\\ &=\theta^{(k+1)\delta}\phi(R)+C_2\theta^{k\delta}R^{\beta}+C_2\theta^{k\beta}R^{\beta}\sum_{j=0}^{k-1}\theta^{j(\delta-\beta)}\\ &=\theta^{(k+1)\delta}\phi(R)+C_2\theta^{k\beta}R^{\beta}\sum_{j=0}^k\theta^{j(\delta-\beta)}. \end{split} \] Hence, \eqref{eq_1} holds for every $k\in\mathbb{N}$, and the induction argument is complete.
\noindent{\bf Step 2 -} Next, we pass from the discrete to the continuous case. In particular, we claim that \begin{equation*} \phi(r)\le C_3\Big(\frac{r}{R}\Big)^{\beta}\big(\phi(R)+C_2R^{\beta}\big), \end{equation*} for every $0<r\le R\le R_0$.
Indeed, \begin{align*} \phi(\theta^{k+1}R) \le&\theta^{(k+1)\delta}\phi(R)+C_2\theta^{k\beta}R^{\beta}\frac{1}{1-\theta^{\delta-\beta}}\\ =&\theta^{(k+1)\delta}\phi(R)+C_2R^{\beta}\frac{\theta^{(k+1)\beta}}{\theta^{\beta}-\theta^{\delta}}\\ \le&C_3\theta^{(k+1)\beta}\big(\phi(R)+C_2R^{\beta}\big), \end{align*} for every $k\in\mathbb{N}$. Taking $k\in\mathbb{N}$ such that $\theta^{k+2}R\le r<\theta^{k+1}R$, up to relabeling the constant $C_3$, we get \begin{align*} \phi(r)\le&\phi(\theta^{k+1}R)\le C_3\theta^{(k+1)\beta}\big(\phi(R)+C_2R^{\beta}\big)\\ =&C_3\theta^{(k+2)\beta}\theta^{-\beta}\big(\phi(R)+C_2R^{\beta}\big)\\ \le&C_3\Big(\frac{r}{R}\Big)^{\beta}\big(\phi(R)+C_2R^{\beta}\big). \end{align*} Finally, one notices \[ \phi(r)\le C_3\frac{1}{R_0^{\beta}}\big(\phi(R_0)+C_2R_0^{\beta}\big)r^{\beta}=:C_4r^{\beta}, \] and the proof is complete. \end{proof}
We conclude this section by introducing two functional spaces we resort to in the paper, namely Campanato and Morrey spaces. Indeed, we use embedding properties of these spaces to conclude the H\"older-continuity of weak solutions when the interface data is unbounded.
\begin{Definition}[Campanato spaces] We denote by $L_C^{p,\lambda}(\Omega;\mathbb{R}^d)$, with $1\le p<\infty$ and $\lambda\ge0$, the space of functions $u\in L^p(\Omega;\mathbb{R}^d)$ such that \begin{equation*}
[u]_{L_C^{p,\lambda}(\Omega;\mathbb{R}^d)}^p=\sup_{x^0\in\Omega,\rho>0}\frac{1}{\rho^{\lambda}}\int_{\Omega\cap B(x^0,\rho)}|u-(u)_{\Omega\cap B(x^0,\rho)}|^p\,{\rm d}x<\infty. \end{equation*} \end{Definition}
\begin{Definition}[Morrey spaces] We denote by $L_M^{p,\lambda}(\Omega;\mathbb{R}^d)$, with $1\le p<\infty$ and $\lambda\ge0$, the space of functions $u\in L^p(\Omega;\mathbb{R}^d)$ such that \begin{equation*}
\|u\|_{L_M^{p,\lambda}(\Omega)}^p=\sup_{x^0\in\Omega,\rho>0}\frac{1}{\rho^{\lambda}}\int_{\Omega\cap B(x^0,\rho)}|u|^p\,{\rm d}x<\infty. \end{equation*} \end{Definition}
Notice that $L^{p,\lambda}_{M}$ and $L^{p,\lambda}_{C}$ are isomorphic; see \cite[Proposition 2.3]{giusti}. We recall that a function $u\in W^{1,1}(\Omega)$ such that $Du\in L_M^{p,\lambda}(\Omega;\mathbb{R}^d)$ is H\"older continuous. More precisely, we have $u\in C^{0,\alpha}(\Omega)$ with $\alpha=1-\lambda/p$; see \cite{adamsmorrey}.
\section{Local boundedness}\label{sec_alkhawarizmi}
In this section, we prove the local boundedness for the weak solutions to a particular variant of our problem. Namely, we consider the case $g(t):=t^{p-1}$ and recover the $p-$Laplace operator. Our argument is inspired by the one put forward in \cite{Serrin_1964}.
\begin{Theorem}[Local Boundedness]\label{thm_lb}
Let $u\in W_0^{1,p}(\Omega)$ be the weak solution to \eqref{eq_stima118}-\eqref{eq_stima119}, with $g(t):=t^{p-1}$ and $f \in L^{\infty}(\Gamma)$. Then for any $B_{R}:=B_{R}(x_0)\Subset \Omega$, there exists $C=C\big(d,p,R,\|g\|_{L^{\infty}(\Gamma)}\big)>0$ such that \[
\|u\|_{L^{\infty}(B_{R/2})}\le CR^{-\frac{d}{p}}\big(\|u\|_{L^p(B_R)}+R^{\frac{d}{p}+1}\|f\|_{L^{\infty}(\Gamma)}\big) \] and \[
\| D u\|_{L^p(B_{R/2})}\le CR^{-1}\big(\|u\|_{L^p(B_R)}+R^{\frac{d}{p}+1}\|f\|_{L^{\infty}(\Gamma)}\big). \] \end{Theorem} \begin{proof}
Fix $R>0$ such that $B_R\Subset \Omega$ and set $k:=R\|f\|_{L^{\infty}(\Gamma)}$. Define $\overline{u}:\Omega\to\mathbb{R}$ as \[
\overline{u}(x):=|u(x)|+k \]
for all $x\in\Omega$. Fix $q\ge1$ and $\ell>k$. For $t\in\mathbb{R}$, denote $\overline{t}:=|t|+k$. To ease the presentation, we split the remainder of the proof into four steps.
\noindent{\bf Step 1 -} Let $F:[k,\infty)\to\mathbb{R}$ be defined as \[ F(s):= \begin{cases}
s^q&\hspace{.3in}\mbox{if}\hspace{.3in}k\le s\le \ell\\
q\ell^{q-1}s-(q-1)\ell^q&\hspace{.3in}\mbox{if}\hspace{.3in}\ell<s. \end{cases} \] Then $F\in C^1\big([k,\infty)\big)$ and $F\in C^\infty\big([k,\infty)\setminus{\{\ell\}}\big)$. Let $H:\mathbb{R}\to\mathbb{R}$ be defined as \begin{equation*}
H(t):=\textnormal{sgn}(t)\big(F(\overline{t})F'(\overline{t})^{p-1}-q^{p-1}k^{\beta}\big),\quad\forall t\in\mathbb{R}, \end{equation*} where $\beta=p(q-1)+1>1$. A simple computation yields \begin{align*} H'(t)= \begin{cases}
q^{-1}\beta F'(\overline{t})^p&\hspace{.3in}\mbox{if}\hspace{.3in}|t|<\ell-k\\
F'(\overline{t})^p&\hspace{.3in}\mbox{if}\hspace{.3in}|t|>\ell-k. \end{cases} \end{align*} Notice that \[
|H(u)|\le F(\overline{u})F'(\overline{u})^{p-1} \] and \[
\overline{u}F'(\overline{u})\le qF(\overline{u}). \]
\noindent{\bf Step 2 -} In this step, we introduce auxiliary test functions, which build upon the former inequalities. Fix $0<r<R$. Let $\eta\in C_c^{\infty}(B_R)$, $0\le\eta\le1$, $\eta=1$ in $B_r$, $| D \eta|\le(R-r)^{-1}$. Let $v=\eta^pG(u)$. Since $G\in C^1\big(\mathbb{R}\setminus\{\pm(\ell-k)\}\big)$ is continuous, with bounded derivative, it follows that $G(u)\in W^{1,p}(\Omega)$. Hence $v$ is an admissible test function. We have \[
D v=
\begin{cases}
p\eta^{p-1}H(u) D \eta+\eta^pH'(u) D u&\hspace{.2in}\mbox{if}\hspace{.2in}u\ne\pm(\ell-k)\\
p\eta^{p-1}H(u) D \eta&\hspace{.2in}\mbox{if}\hspace{.2in}u=\pm(\ell-k).
\end{cases} \]
Set $w(x)=F\big(\overline{u}(x)\big)$. Notice that $q^{-1}\beta\ge1$; hence $H'(u)\le q^{-1}\beta F'(\overline{u})^p$. Notice also that $| D u|=| D \overline{u}|$.
Using the trace theorem and the Poincar\'e inequality, we get \begin{equation}\label{stima2}
\int_{\Omega}| D u|^{p-2} D u\cdot D v\,{\rm d}x\le\|f\|_{L^{\infty}(\Gamma)}\int_{\Gamma}|v|\,{\rm d}\mathcal{H}^{d-1}\le C\int_{\Omega}| D v|\,{\rm d}x. \end{equation}
Now we estimate the left-hand side of \eqref{stima2} from below. We get \begin{align}\label{stima5}\notag
\int_{B_1}| D u|^{p-2} D u\cdot D v\,{\rm d}x=&\int_{B_1}| D u|^{p-2} D u\cdot\big(p\eta^{p-1}H(u) D \eta+\eta^pH'(u) D u\big)\,{\rm d}x \notag\\
=&p\int_{B_1}\eta^{p-1}H(u)| D u|^{p-2} D u\cdot D \eta\,{\rm d}x \notag\\
&+\int_{B_1}\eta^pH'(u)| D u|^p\,{\rm d}x \notag\\
\ge&-p\int_{B_1}\eta^{p-1}F(\overline{u})F'(\overline{u})^{p-1}| D \overline{u}|^{p-1}| D \eta|\,{\rm d}x \notag\\
&+\int_{B_1}\eta^pF'(\overline{u})^p| D \overline{u}|^p\,{\rm d}x \notag\\
=&-p\int_{B_1}\eta^{p-1}w| D w|^{p-1}| D \eta|\,{\rm d}x \notag\\
&+\int_{B_1}\eta^p| D w|^p\,{\rm d}x \notag\\
\ge&-p\|w D \eta\|_{L^p(B_1)}\|\eta D w\|_{L^p(B_1)}^{p-1}+\|\eta D w\|_{L^p(B_1)}^p. \end{align} We also control the right-hand side of (\ref{stima2}) by computing \begin{align}\label{stima3}\notag
C\int_{B_1}| D v|\,{\rm d}x=&C\int_{B_1}\frac{\overline{u}^{p-1}}{\overline{u}^{p-1}}|p\eta^{p-1}H(u) D \eta+\eta^pH'(u) D u|\,{\rm d}x \notag\\
\le&Ck^{1-p}p\int_{B_1}\overline{u}^{p-1}\eta^{p-1}|H(u) D \eta|\,{\rm d}x \notag\\
&+Ck^{1-p}\int_{B_1}\overline{u}^{p-1}\eta^pH'(u)| D u|\,{\rm d}x \notag\\
\le&C\int_{B_1}\overline{u}^{p-1}\eta^{p-1}F(\overline{u})F'(\overline{u})^{p-1}| D \eta|\,{\rm d}x \notag\\
&+Cq^{-1}\beta\int_{B_1}\overline{u}^{p-1}\eta^pF'(\overline{u})^p| D u|\,{\rm d}x \notag\\
\le&C\int_{B_1}\eta^{p-1}q^{p-1}F(\overline{u})^{p-1}F(\overline{u})| D \eta|\,{\rm d}x \notag\\
&+Cq^{-1}\beta\int_{B_1}q^{p-1}F(\overline{u})^{p-1}\eta^p F'(\overline{u})| D u|\,{\rm d}x \notag\\
=&Cq^{p-1}\int_{B_1}(\eta w)^{p-1}w| D \eta|\,{\rm d}x \notag\\
&+Cq^{p-2}\beta\int_{B_1}(\eta w)^{p-1}\eta| D w|\,{\rm d}x \notag\\
\le&Cq^{p-1}\|\eta w\|_{L^p(B_1)}^{p-1}\|w D \eta\|_{L^p(B_1)} \notag\\
&+Cq^{p-2}\beta\|\eta w\|_{L^p(B_1)}^{p-1}\|\eta D w\|_{L^p(B_1)}. \end{align} From (\ref{stima2}), combining (\ref{stima3}) with (\ref{stima5}), we get \begin{align}\label{stima6}
\|\eta D w\|_{L^p(\Omega)}^p\le&p\|w D \eta\|_{L^p(\Omega)}\|\eta D w\|_{L^p(\Omega)}^{p-1} \notag\\
&+Cq^{p-1}\|\eta w\|_{L^p(\Omega)}^{p-1}\|w D \eta\|_{L^p(\Omega)} \notag\\
&+Cq^{p-1}\|\eta w\|_{L^p(\Omega)}^{p-1}\|\eta D w\|_{L^p(\Omega)}, \end{align} where we have used \[
\beta=pq-p+1\le pq-p+q\le pq+q=(p+1)q. \]
\noindent{\bf Step 3 -} Set \begin{equation}
z=\frac{\|\eta D w\|_{L^p(\Omega)}}{\|w D \eta\|_{L^p(\Omega)}},\quad\zeta=\frac{\|\eta w\|_{L^p(\Omega)}}{\|w D \eta\|_{L^p(\Omega)}}. \notag \end{equation}
By dividing (\ref{stima6}) for $\|w D \eta\|_{L^p(\Omega)}^p$, we have \begin{align}
z^p\le&pz^{p-1}+Cq^{p-1}\frac{\|\eta w\|_{L^p(\Omega)}^{p-1}}{\|w D \eta\|_{L^p(\Omega)}^{p-1}}+Cq^{p-1}\frac{\|\eta w\|_{L^p(\Omega)}^{p-1}}{\|w D \eta\|_{L^p(\Omega)}^{p-1}}\frac{\|\eta D w\|_{L^p(\Omega)}}{\|w D \eta\|_{L^p(\Omega)}} \notag\\ =&pz^{p-1}+Cq^{p-1}\zeta^{p-1}+Cq^{p-1}\zeta^{p-1}z. \notag \end{align} An application of Lemma \ref{lemma numerico}, implies \begin{equation*}
z\le C\big(p+q^{\frac{p-1}{p}}\zeta^{\frac{p-1}{p}}+q\zeta\big)\le Cq(1+\zeta), \end{equation*} giving \begin{equation}\label{stima7}
\|\eta D w\|_{L^p(\Omega)}\le Cq\big(\|\eta w\|_{L^p(\Omega)}+\|w D \eta\|_{L^p(\Omega)}\big). \end{equation} Using the Sobolev inequality, we get \begin{align}
\|\eta w\|_{L^{p^*}(\Omega)}\le&C\| D (\eta w)\|_{L^p(\Omega)} \notag\\
\le&C\big(\|w D \eta\|_{L^p(\Omega)}+\|\eta D w\|_{L^p(\Omega)}\big) \notag\\
\le&C\Big[\|w D \eta\|_{L^p(\Omega)}+Cq\big(\|\eta w\|_{L^p(\Omega)}+\|w D \eta\|_{L^p(\Omega)}\big)\Big] \notag \end{align} and so \begin{equation}\label{stima8}
\|\eta w\|_{L^{p^*}(\Omega)}\le Cq\big(\|\eta w\|_{L^p(\Omega)}+\|w D \eta\|_{L^p(\Omega)}\big). \end{equation}
Recall that $\eta=1$ in $B_r$ and $| D \eta|\le(R-r)^{-1}$. Hence, (\ref{stima7}) becomes \begin{equation}\label{stima11}
\begin{split}
\| D w\|_{L^p(B_r)}\le& Cq\Bigg[\bigg(\int_{B_R}w^p\,{\rm d}x\bigg)^{\frac{1}{p}}+\frac{1}{R-r}\bigg(\int_{B_R}w^p\,{\rm d}x\bigg)^{\frac{1}{ p}}\Bigg] \notag\\
=&Cq\|w\|_{L^p(B_R)}\bigg(1+\frac{1}{R-r}\bigg) \notag\\
=&Cq\frac{R-r+1}{R-r}\|w\|_{L^p(B_R)} \notag\\
\le&Cq\frac{\textnormal{diam}(B_1)+1}{R-r}\|w\|_{L^p(B_R)} \notag\\
\le&Cq\frac{1}{R-r}\|w\|_{L^p(B_R)}.
\end{split} \end{equation} Similarly, (\ref{stima8}) becomes \begin{equation}\label{stima9}
\|w\|_{L^{p^*}(B_r)}\le Cq\frac{1}{R-r}\|w\|_{L^p(B_R)}. \end{equation} We claim that $F_\ell\le F_{\ell+1}$, for every $\ell\in\mathbb{N}$, $\ell>k$. The only non-trivial case is when $\ell<\overline{t}\le \ell+1$. In this case, we have $$F_\ell(\overline{t})=q\ell^{q-1}\overline{t}-(q-1)\ell^q$$ and $$F_{\ell+1}(\overline{t})=\overline{t}^q.$$ Let $h:(\ell,\ell+1]\to\mathbb{R}$ be defined by $$h(\overline{t})=\overline{t}^q-q\ell^{q-1}\overline{t}+(q-1)\ell^q.$$ We have $h'(\overline{t})=q\overline{t}^{q-1}-q\ell^{q-1}>0$, for every $\overline{t}\in(\ell,\ell+1]$, and hence $h$ is an increasing function. Since $\lim_{\overline{t}\to \ell}h(\overline{t})=0$, we have $h\ge0$ in $(\ell,\ell+1]$, and so $F_\ell\le F_{\ell+1}$. Letting $\ell\to\infty$ in (\ref{stima9}), since $0\le F_\ell\le F_{\ell+1}$ for every $\ell\in\mathbb{N}$, $\ell>k$, by the Monotone Convergence Theorem, we obtain \begin{equation} \bigg(\int_{B_r}\overline{u}^{qp^*}\,{\rm d}x\bigg)^{\frac{1}{p^*}}\le Cq\frac{1}{R-r}\bigg(\int_{B_R}\overline{u}^{qp}\,{\rm d}x\bigg)^{\frac{1}{p}}. \notag \end{equation} Set \[
s:=qp\hspace{.3in}\mbox{and}\hspace{.3in}\gamma:=p^*/p=d/(d-p); \] then \begin{equation*}
\bigg(\int_{B_r}\overline{u}^{s\gamma}\,{\rm d}x\bigg)^{\frac{1}{p\gamma}}\le Cq\frac{1}{R-r}\bigg(\int_{B_R}\overline{u}^{s}\,{\rm d}x\bigg)^{\frac{1}{p}}. \end{equation*} Raising both sides of the previous inequality to $p/s$, one gets \begin{equation}\label{stima10} \bigg(\int_{B_r}\overline{u}^{s\gamma}\,{\rm d}x\bigg)^{\frac{1}{s\gamma}}\le C^{\frac{p}{s}}\bigg(\frac{s}{p}\bigg)^{\frac{p}{s}}\Big(\frac{1}{R-r}\Big)^{\frac{p}{s}}\bigg(\int_{B_R}\overline{u}^{s}\,{\rm d}x\bigg)^{\frac{1}{s}}. \end{equation} Set $s_j=s\gamma^j$ and $r_j=r+2^{-j}(R-r)$, for every $j\in\mathbb{N}_0$. Iterating (\ref{stima10}), which holds for every $s\ge p$, we have \begin{align*} \bigg(\int_{B_{r_{j+1}}}\overline{u}^{s_j\gamma}\,{\rm d}x\bigg)^{\frac{1}{s_j\gamma}}\le&C^{\frac{p}{s_j}}\bigg(\frac{s_j}{p}\bigg)^{\frac{p}{s_j}}2^{\frac{p}{s_j}(j+1)}\Big(\frac{1}{R-r}\Big)^{\frac{p}{s_j}}\bigg(\int_{B_{r_j}}\overline{u}^{s_j}\,{\rm d}x\bigg)^{\frac{1}{s_j}} \notag\\ =&C^{\frac{p}{s_{j-1}\gamma}}\bigg(\frac{s_{j-1}\gamma}{p}\bigg)^{\frac{p}{s_{j-1}\gamma}}2^{\frac{p}{s_{j-1}\gamma}(j+1)}\Big(\frac{1}{R-r}\Big)^{\frac{p}{s_{j-1}\gamma}} \notag\\ &\times\bigg(\int_{B_{r_j}}\overline{u}^{s_{j-1}\gamma}\,{\rm d}x\bigg)^{\frac{1}{s_{j-1}\gamma}} \notag\\ \le& C(j,p,s,d)\Big(\frac{1}{R-r}\Big)^{\frac{p}{s}\sum_{k=0}^j\gamma^{-k}}\bigg(\int_{B_R}\overline{u}^s\,{\rm d}x\bigg)^{\frac{1}{s}}, \end{align*} where \[
C(j,p,s,d):=C^{\frac{p}{s}\sum_{k=0}^j\gamma^{-k}}\bigg(\frac{s}{p}\bigg)^{\frac{p}{s}\sum_{k=0}^j{\gamma^{-k}}}\gamma^{\frac{p}{s}\sum_{k=0}^jk\gamma^{-k}}2^{\frac{p}{s}\sum_{k=0}^j(k+1)\gamma^{-k}}. \] Notice that $r<r_{j}$, for every $j\in\mathbb{N}_0$, the series are convergent and in particular $\sum_{k=0}^\infty\gamma^{-k}=d/p$. By letting $j\to\infty$, we get \begin{equation} \sup_{B_r}\overline{u}\le C\bigg(\frac{1}{(R-r)^d}\int_{B_R}\overline{u}^s\,{\rm d}x\bigg)^{\frac{1}{s}}. \end{equation}
\noindent{\bf Step 4 - }Now, we can choose some parameters in the former inequalities to complete the proof. By choosing $q=1$, setting $r:=R/2$, and recalling that $\overline{u}=|u|+k$, we get \begin{align}
\|u\|_{L^{\infty}(B_{R/2})}\le&\|\overline{u}\|_{L^{\infty}(B_{R/2})}\le CR^{-\frac{d}{p}}\big(\|u\|_{L^p(B_R)}+R^{\frac{d}{p}}k\big). \notag \end{align} The second inequality in the theorem follows by setting $q=1$ and $r:=R/2$ in \eqref{stima11}, obtaining \begin{align}
\| D u\|_{L^p(B_{R/2})}=&\| D \overline{u}\|_{L^p(B_{R/2})}\\
\le & CR^{-1}\|\overline{u}\|_{L^p(B_R)} \notag\\
\le&CR^{-1}\big(\|u\|_{L^p(B_R)}+\|k\|_{L^p(B_R)}\big) \notag\\
\le&CR^{-1}\big(\|u\|_{L^p(B_R)}+R^{\frac{d}{p}}k\big). \notag \end{align} \end{proof}
\section{Gradient regularity estimates in ${\rm BMO}-$spaces}\label{sec_beacon}
In this section, we prove regularity estimates for weak solutions. In case $f\in L^\infty(\Gamma)$, we prove that $Du\in {\rm BMO}_{\rm loc}(\Omega)$. In addition, we allow the interface data to be unbounded, provided it belongs to a Sobolev space $W^{1,p'+\varepsilon}(\Omega)$, where the parameter $\varepsilon>0$ is to be set further. In this case, we verify that $u\in C^{0,\alpha}_{\rm loc}(\Omega)$, for some $\alpha\in(0,1)$ depending only on the dimension, $p$ and $\varepsilon$.
We proceed with an auxiliary lemma. For $w\in W^{1,G}(\Omega)$, let $W^{1,G}_w(\Omega)$ denote the Orlicz-Sobolev space comprising the functions $u\in W^{1,G}(\Omega)$ such that \[
u-w\in W^{1,G}_0(\Omega). \]
\begin{Lemma}\label{stima146}
Let $w\in W^{1,G}(B_R)$. Suppose \eqref{business class}--\eqref{monicavitti} is in force. Suppose further $h\in W_w^{1,G}(B_R)$ is a weak solution to
\begin{equation*}
\textnormal{div}\bigg(\frac{g(| D h|)}{| D h|} D h\bigg)=0\quad\textnormal{in }B_R.
\end{equation*} Then there exists $C>0$ such that
\begin{equation}\label{antonioni}
\int_{B_{R}}G(| D w|)-G(| D h|)\,{\rm d}x\ge C\int_{B_{R}}| D (w-h)|^p\,{\rm d}x.
\end{equation}
\end{Lemma}
\begin{proof}$\\$
Let $\tau\in[0,1]$, define $v_\tau=\tau w+(1-\tau)h$. The monotonicity condition in \eqref{monicavitti} implies
\begin{align*}
\int_{B_{R}}G(| D w|)-G(| D h|)\,{\rm d}x=&\int_0^1\frac{d}{d\tau}\bigg(\int_{B_{R}}G(| D v_{\tau}|)\,{\rm d}x\bigg)\,{\rm d}\tau\\
=&\int_0^1\int_{B_{R}}\frac{d}{d\tau}G(| D v_\tau|)\,{\rm d}x\,{\rm d}\tau\\
=&\int_0^1\int_{B_{R}}\frac{g(| D v_\tau|)}{| D v_\tau|} D v_\tau\cdot D (w-h)\,{\rm d}x\,{\rm d}\tau\\
=&\int_0^1\frac{1}{\tau}\int_{B_{R}}\bigg(\frac{g(| D v_\tau|)}{| D v_\tau|} D v_\tau-\frac{g(| D h|)}{| D h|} D h\bigg)\\
&\cdot D (v_\tau-h)\,{\rm d}x\,{\rm d}\tau\\
\ge&C\int_0^1\frac{1}{\tau}\int_{B_{R}}| D (v_\tau-h)|^p\,{\rm d}x\,{\rm d}\tau\\
=&C\int_{B_{R}}| D (w-h)|^p\,{\rm d}x,
\end{align*}
and the proof is complete.
\end{proof}
\subsection{Regularity estimates in ${\rm BMO}-$spaces}
In this section, we suppose $f\in L^\infty(\Omega)$ and establish ${\rm BMO}-$regularity estimates for the gradient of solutions. We start by recalling a proposition from \cite{baroni2015}.
\begin{Proposition}\label{stima136} Let $h\in W^{1,G}(B_R)$ be a weak solution of
\begin{equation*}
\textnormal{div}\bigg(\frac{g(| D h|)}{| D h|} D h\bigg)=0\quad\textnormal{in }B_R.
\end{equation*} Suppose \eqref{business class}--\eqref{monicavitti} are in force. Then there exist $C>0$ and $\alpha\in(0,1)$ such that, for every $r\in(0,R]$, we have
\begin{equation*}
\int_{B_r}| D h-( D h)_r|\,{\rm d}x\le C\Big(\frac{r}{R}\Big)^{d+\alpha}\int_{B_R}| D h-( D h)_R|\,{\rm d}x.
\end{equation*} \end{Proposition}
For a proof of Proposition \ref{stima136}, we refer the reader to \cite{baroni2015}.
\begin{Proposition}\label{stima161}
Let $w\in W^{1,G}(B_R)$, and suppose $h\in W^{1,G}(B_R)$ is a weak solution of
\begin{equation*}
\textnormal{div}\bigg(\frac{g(| D h|)}{| D h|} D h\bigg)=0\quad\textnormal{in }B_R.
\end{equation*} Suppose \eqref{business class}--\eqref{monicavitti} are in force. Then there exists $C>0$ such that, for every $0<r\le R$, we have
\begin{align}
\int_{B_r}| D w-( D w)_r|\,{\rm d}x\le& C\Big(\frac{r}{R}\Big)^{d+\alpha}\int_{B_R}| D w-( D w)_R|\,{\rm d}x \notag\\
&+C\int_{B_R}| D w- D h|\,{\rm d}x, \notag
\end{align}
where $\alpha$ is given by Proposition \ref{stima136}.
\end{Proposition} \begin{proof}Let $r\in(0,R]$. We have
\begin{align}\label{stima137}
\int_{B_r}| D w-( D w)_r|\,{\rm d}x\le&\int_{B_r}| D w-( D h)_r|\,{\rm d}x \notag\\
&+\int_{B_r}|( D w)_r-( D h)_r|\,{\rm d}x.
\end{align}
Similarly, we have
\begin{align}\label{stima138}
\int_{B_r}| D w-( D h)_r|\,{\rm d}x\le&\int_{B_r}| D w- D h|\,{\rm d}x \notag\\
&+\int_{B_r}| D h-( D h)_r|\,{\rm d}x.
\end{align}
Moreover,
\begin{align}\label{stima139}
\int_{B_r}|( D w)_r-( D h)_r|\,{\rm d}x=&|( D w)_r-( D h)_r|\int_{B_r}\,{\rm d}x \notag\\
=&|B_r|\bigg|\frac{1}{|B_r|}\int_{B_r} D w- D h\,{\rm d}x\bigg| \notag\\
\le&\int_{B_r}| D w- D h|\,{\rm d}x.
\end{align}
Combining \eqref{stima137}, \eqref{stima138} with \eqref{stima139}, we get
\begin{align}\label{stima142}
\int_{B_r}| D w-( D w)_r|\,{\rm d}x\le&\int_{B_r}| D h-( D h)_r|\,{\rm d}x \notag\\
&+2\int_{B_r}| D w- D h|\,{\rm d}x.
\end{align}
Changing the roles of $w$ and $h$ and integrating in the ball $B_R$, we obtsain
\begin{align}\label{stima140}
\int_{B_R}| D h-( D h)_R|\,{\rm d}x\le&\int_{B_R}| D w-( D w)_R|\,{\rm d}x \notag\\
&+2\int_{B_R}| D w- D h|\,{\rm d}x.
\end{align}
Thanks to Proposition \ref{stima136}, we conclude
\begin{align}\label{stima141}
\int_{B_r}| D w-( D w)_r|\,{\rm d}x\le&C\Big(\frac{r}{R}\Big)^{d+\alpha}\int_{B_R}| D h-( D h)_R|\,{\rm d}x \notag\\
&+C\int_{B_R}| D w- D h|\,{\rm d}x.
\end{align}
Combining \eqref{stima142}, \eqref{stima140} with \eqref{stima141} we get
\begin{align}
\int_{B_r}| D w-( D w)_r|\,{\rm d}x\le&C\Big(\frac{r}{R}\Big)^{d+\alpha}\int_{B_R}| D w-( D w)_R|\,{\rm d}x \notag\\
&+C\Big(\frac{r}{R}\Big)^{d+\alpha}\int_{B_R}| D w- D h|\,{\rm d}x \notag\\
&+C\int_{B_R}| D w- D h|\,{\rm d}x \notag\\
\le&C\Big(\frac{r}{R}\Big)^{d+\alpha}\int_{B_R}| D w-( D w)_R|\,{\rm d}x \notag\\
&+C\int_{B_R}| D w- D h|\,{\rm d}x. \notag
\end{align}
The proof is complete.
\end{proof}
We now state and prove the main result in this section.
\begin{Theorem}[Gradient regularity in ${\rm BMO}-$spaces]\label{thm_ll} Let $u\in W^{1,G}_0(\Omega)$ be a weak solution for the transmission problem \eqref{eq_stima118}--\eqref{eq_stima119}. Suppose \eqref{business class}--\eqref{monicavitti} is in force. Then $ D u\in {\rm BMO}_{\rm loc}(\Omega)$. Moreover, for every $\Omega'\Subset\Omega$, \[
\left\|Du\right\|_{{\rm BMO}(\Omega')}\leq C, \]
where $C=C(d,\|f\|_{L^\infty(\Gamma)},{\rm diam}(\Omega),{\rm dist}(\Omega',\partial\Omega))>0$. \end{Theorem} \begin{proof} Let $x^0\in\Gamma$, and let $R>0$ such that $B_R:=B(x^0,R)\Subset\Omega$. Let $h\in W_u^{1,G}(B_R)$ be the weak solution of
\begin{equation*}
\textnormal{div}\bigg(\frac{g(| D h|)}{| D h|} D h\bigg)=0\quad\textnormal{in }B_R.
\end{equation*}
Since $h=u$ on $\partial B_R$ in the trace sense, we can extend $h$ to $\Omega\setminus B_R$ so that $h=u$ in $\Omega\setminus B_R$. This implies that $h\in W_0^{1,G}(\Omega)$ and hence, since $u$ is a global minimizer of \eqref{stima117}, we have
\begin{align}\label{stima144}
\int_{\Omega}G(| D u|)\,{\rm d}x-\int_{\Gamma}fu\,{\rm d}\mathcal{H}^{d-1}\le\int_{\Omega}G(| D h|)\,{\rm d}x-\int_{\Gamma}fh\,{\rm d}\mathcal{H}^{d-1}.
\end{align}
Set $\Gamma_R=B_R\cap\Gamma$. Since $h=u$ in $\Omega\setminus B_R$, \eqref{stima144} becomes
\begin{align}
\int_{B_R}G(| D u|)\,{\rm d}x-\int_{\Gamma_R}fu\,{\rm d}\mathcal{H}^{d-1}\le\int_{B_R}G(| D h|)\,{\rm d}x-\int_{\Gamma_R}fh\,{\rm d}\mathcal{H}^{d-1} \notag
\end{align}
from which, applying the Trace Theorem and Poincaré Inequality, follows
\begin{align}\label{stima145}
\int_{B_R}G(| D u|)\,{\rm d}x-\int_{B_R}G(| D h|)\,{\rm d}x\le&\int_{\Gamma_R}fu\,{\rm d}\mathcal{H}^{d-1}-\int_{\Gamma_R}fh\,{\rm d}\mathcal{H}^{d-1} \notag\\
\le&\|f\|_{L^{\infty}(\Gamma)}\int_{\Gamma_R}|u-h|\,{\rm d}\mathcal{H}^{d-1} \notag\\
\le&C\int_{B_R}|u-h|\,{\rm d}x+C\int_{B_R}| D (u-h)|\,{\rm d}x \notag\\
\le&C\int_{B_{R}}| D (u-h)|\,{\rm d}x.
\end{align}
From Lemma \ref{stima146}, we bound the left-hand side of \eqref{stima145}
\begin{align}\label{stima147}
\int_{B_R}G(| D u|)\,{\rm d}x-\int_{B_R}G(| D h|)\,{\rm d}x\ge&C\int_{B_R}| D (u-h)|^p\,{\rm d}x,
\end{align}
and, combining \eqref{stima145} with \eqref{stima147}, we get
\begin{equation*}
\int_{B_R}| D (u-h)|^p\,{\rm d}x \leq C\int_{B_{R}}| D (u-h)|\,{\rm d}x.
\end{equation*}
Using this and H\"older's inequality, we obtain
\begin{eqnarray*}
\left( \int_{B_R}| D (u-h)|\,{\rm d}x \right)^p & \leq & C' R^{d(p-1)}\int_{B_{R}}| D (u-h)|^p\,{\rm d}x\\
& \leq & C R^{d(p-1)}\int_{B_{R}}| D (u-h)|\,{\rm d}x
\end{eqnarray*}
and thus
\begin{equation*}
\int_{B_R}| D (u-h)|\,{\rm d}x \leq C R^d.
\end{equation*}
From Proposition \ref{stima161}, we get
\begin{equation*}
\int_{B_r}| D u-( D u)_r|\,{\rm d}x\le C\Big(\frac{r}{R}\Big)^{d+\alpha}\int_{B_R}| D u-( D u)_R|\,{\rm d}x+CR^d\
\end{equation*}
for every $0<r\le R$, and, applying Lemma \ref{lem_stima119}, we conclude
\begin{equation*}
\int_{B_r}| D u-( D u)_r|\,{\rm d}x\le Cr^d, \quad\forall r\in(0,R].
\end{equation*}
The proof is complete.
\end{proof}
\begin{Remark}[Potential estimates and the $p$-Laplace operator] If $g(t):=t^{p-1}$, the conclusion of Theorem \ref{thm_ll} has been obtained through the use of potential estimates; see \cite[Corollary 1, item (C9)]{KM2014c}. Indeed, notice that for $B_r\subset\Omega$, we have \[
\int_{B_r}f{\rm d}\mathcal{H}^{d-1}\leq Cr^{d-1}, \] which is precisely the condition in \cite[Corollary 1, item (C9)]{KM2014c}. See also \cite{M2011,M2011a}. \end{Remark}
As a corollary to Theorem \ref{thm_ll}, we obtain a modulus of continuity for the solution $u$ in $C^{0,{\rm Log-Lip}}-$spaces.
\begin{Corollary}[Log-Lipschitz continuity estimates]\label{cor_ll} Let $u\in W_0^{1,G}(\Omega)$ be a weak solution for \eqref{eq_stima118}-\eqref{eq_stima119}. Suppose \eqref{business class}--\eqref{monicavitti} are in force. Then $u\in C^{0,{\rm Log-Lip}}_{\rm loc}(\Omega)$. Moreover, for every $\Omega'\Subset\Omega$, \[
\left\|u\right\|_{C^{0,{\rm Log-Lip}}(\Omega')}\leq C\left(\left\|u\right\|_{L^\infty(\Omega)}+\left\|f\right\|_{L^\infty(\Gamma)}\right), \] where $C=C(p,d,{\rm diam}(\Omega),{\rm dist}(\Omega',\partial\Omega))>0$. \end{Corollary}
Indeed, a function whose partial derivatives are in ${\rm BMO}$ belongs to the Zygmund class (cf. \cite{Zygmund2002}). Because functions in the latter have a $C^{0,{\rm Log-Lip}}$ modulus of continuity, the corollary follows. An alternative argument follows from embedding results for borderline spaces; see \cite[Theorem 3]{Cianchi1996}.
\subsection{H\"older continuity of weak solutions}
Here, we consider unbounded interface data. We work under the condition $f\in W^{1,p'+\varepsilon}(\Omega)$, where $\varepsilon>0$ depends on $p$ and the dimension, and prove a regularity result in H\"older spaces for the weak solutions of \eqref{eq_stima118}-\eqref{eq_stima119}.
\begin{Theorem}\label{thm_c0alpha} Let $u$ be a weak solution to the interface problem \eqref{eq_stima118}--\eqref{eq_stima119}, under assumptions \eqref{business class}--\eqref{monicavitti}. Let $2<p<d$ and $\varepsilon>0$ be such that \[
\frac{d-p}{p-1}<\varepsilon<d-\frac{p}{p-1}, \] and suppose $f\in W^{1,p'+\varepsilon}(\Omega)$. Then $u\in C^{0,\alpha}_{\textnormal{loc}}(\Omega)$, where \[
\alpha=1-\frac{d}{p+\varepsilon(p-1)}, \] with estimates. \end{Theorem} \begin{proof}We split the proof into three steps.
\noindent{\bf Step 1 - }Combining \eqref{stima145} and \eqref{stima147}, one obtains
\begin{equation}\label{stima155}
\| D (u-h)\|_{L^p(B_R)}^p\le C\int_{\Gamma}|f(u-h)|\,{\rm d}\mathcal{H}^{d-1}.
\end{equation}
We proceed by examining the right-hand side of \eqref{stima155}. Using the Trace Theorem, we get
\begin{align}\label{stima156}
\int_{\Gamma_R}|f(u-h)|\,{\rm d}\mathcal{H}^{d-1}\le&C\int_{B_{R}}|f(u-h)|\,{\rm d}x+C\int_{B_{R}}| D \big(f(u-h)\big)|\,{\rm d}x \notag\\
\le&C\int_{B_{R}}|f||u-h|\,{\rm d}x+C\int_{B_{R}}| D f||u-h|\,{\rm d}x \notag \\
&+C\int_{B_{R}}|f|| D (u-h)|\,{\rm d}x \notag\\
=:&I_1+I_2+I_3.
\end{align}
Now, we estimate each of the summands $I_1$, $I_2$ and $I_3$. Concerning $I_1$, we have
\begin{align}\label{stima157}
\int_{B_{R}}|f||u-h|\,{\rm d}x\le&\bigg(\int_{B_{R}}|f|^{p'+\varepsilon}\,{\rm d}x\bigg)^{\frac{1}{p'+\varepsilon}}\bigg(\int_{B_{R}}|u-h|^{\frac{p'+\varepsilon}{p'+\varepsilon-1}}\,{\rm d}x\bigg)^{\frac{p'+\varepsilon-1}{p'+\varepsilon}} \notag\\
\le&C\Bigg[\bigg(\int_{B_R}|u-h|^{\frac{p'+\varepsilon}{p'+\varepsilon-1}\frac{p'+\varepsilon-1}{p'+\varepsilon}p}\,{\rm d}x\bigg)^{\frac{p'+\varepsilon}{p'+\varepsilon-1}\frac{1}{p}} \notag\\
&\times\bigg(\int_{B_{R}}\,{\rm d}x\bigg)^{\frac{\frac{p'+\varepsilon-1}{p'+\varepsilon}p-1}{\frac{p'+\varepsilon-1}{p'+\varepsilon}p}}\Bigg]^{\frac{p'+\varepsilon-1}{p'+\varepsilon}} \notag\\
\le&CR^{d\frac{\frac{p'+\varepsilon-1}{p'+\varepsilon}p-1}{p}}\|u-h\|_{L^p(B_R)} \notag\\
\le&CR^{d\frac{\frac{p'+\varepsilon-1}{p'+\varepsilon}p-1}{p}}\| D (u-h)\|_{L^p(B_R)}.
\end{align}
To estimate $I_2$, one notices that
\begin{align}\label{stima158}
\int_{B_{R}}| D f||u-h|\,{\rm d}x\le&\bigg(\int_{B_{R}}| D f|^{p'+\varepsilon}\,{\rm d}x\bigg)^{\frac{1}{p'+\varepsilon}}\bigg(\int_{B_{R}}|u-h|^{\frac{p'+\varepsilon}{p'+\varepsilon-1}}\,{\rm d}x\bigg)^{\frac{p'+\varepsilon-1}{p'+\varepsilon}} \notag\\
\le&C\Bigg[\bigg(\int_{B_R}|u-h|^{\frac{p'+\varepsilon}{p'+\varepsilon-1}\frac{p'+\varepsilon-1}{p'+\varepsilon}p}\,{\rm d}x\bigg)^{\frac{p'+\varepsilon}{p'+\varepsilon-1}\frac{1}{p}} \notag\\
&\times\bigg(\int_{B_{R}}\,{\rm d}x\bigg)^{\frac{\frac{p'+\varepsilon-1}{p'+\varepsilon}p-1}{\frac{p'+\varepsilon-1}{p'+\varepsilon}p}}\Bigg]^{\frac{p'+\varepsilon-1}{p'+\varepsilon}} \notag\\
\le&CR^{d\frac{\frac{p'+\varepsilon-1}{p'+\varepsilon}p-1}{p}}\|u-h\|_{L^p(B_R)} \notag\\
\le&CR^{d\frac{\frac{p'+\varepsilon-1}{p'+\varepsilon}p-1}{p}}\| D (u-h)\|_{L^p(B_R)}.
\end{align}
Finally, we examine $I_3$. Indeed,
\begin{align}\label{stima159}
\int_{B_{R}}|f|| D (u-h)|\,{\rm d}x\le&\bigg(\int_{B_{R}}|f|^{p'+\varepsilon}\,{\rm d}x\bigg)^{\frac{1}{p'+\varepsilon}}\\
& \times\bigg(\int_{B_{R}}| D (u-h)|^{\frac{p'+\varepsilon}{p'+\varepsilon-1}}\,{\rm d}x\bigg)^{\frac{p'+\varepsilon-1}{p'+\varepsilon}} \notag\\
\le&C\Bigg[\bigg(\int_{B_R}| D (u-h)|^{\frac{p'+\varepsilon}{p'+\varepsilon-1}\frac{p'+\varepsilon-1}{p'+\varepsilon}p}\,{\rm d}x\bigg)^{\frac{p'+\varepsilon}{p'+\varepsilon-1}\frac{1}{p}} \notag\\
&\times\bigg(\int_{B_{R}}\,{\rm d}x\bigg)^{\frac{\frac{p'+\varepsilon-1}{p'+\varepsilon}p-1}{\frac{p'+\varepsilon-1}{p'+\varepsilon}p}}\Bigg]^{\frac{p'+\varepsilon-1}{p'+\varepsilon}} \notag\\
\le&CR^{d\frac{\frac{p'+\varepsilon-1}{p'+\varepsilon}p-1}{p}}\| D (u-h)\|_{L^p(B_R)}.
\end{align}
Because of the role played by the exponents in the previous inequalities, we conclude this step by noticing that
\begin{align*}
\frac{\frac{p'+\varepsilon-1}{p'+\varepsilon}p-1}{p}
=&\frac{\varepsilon(p-1)}{p(p'+\varepsilon)}.
\end{align*}
\noindent{\bf Step 2 -}Now we combine \eqref{stima155}, \eqref{stima156}, \eqref{stima157}, \eqref{stima158}, and \eqref{stima159} to produce
\begin{equation*}
\| D (u-h)\|_{L^p(B_R)}^{p-1}\le CR^{d\frac{\varepsilon(p-1)}{p(p'+\varepsilon)}}.
\end{equation*}
As a consequence, it follows that
\begin{equation}\label{stima160}
\int_{B_{R}}| D (u-h)|^p\,{\rm d}x\le CR^{d\frac{\varepsilon}{p'+\varepsilon}}.
\end{equation}
Hence, Proposition \ref{stima161} builds upon \eqref{stima160} to yield
\begin{equation*}
\int_{B_{r}}| D u-( D u)_r|\,{\rm d}x\le C\Big(\frac{r}{R}\Big)^{d+\alpha}\int_{B_{R}}| D u-( D u)_R|\,{\rm d}x+CR^{d\frac{\varepsilon}{p'+\varepsilon}}.
\end{equation*}
The former inequality, together with Lemma \ref{lem_stima119}, leads to
\begin{equation*}
\int_{B_{r}}| D u-( D u)_r|\,{\rm d}x\le Cr^{d\frac{\varepsilon}{p'+\varepsilon}}\quad\forall r\in(0,R],
\end{equation*}
and one easily concludes
\begin{equation}\label{doron}
r^{(d-d\frac{\varepsilon}{p'+\varepsilon})-d}\int_{B_{r}}| D u-( D u)_r|^p\,{\rm d}x\le C\quad\forall r\in(0,R].
\end{equation}
\noindent{\bf Step 3 - }The inequality in \eqref{doron} implies $Du\in L_C^{p,\lambda}(\Omega;\mathbb{R}^d)$, with
\begin{equation*}
\lambda:=d\bigg(1-\frac{\varepsilon}{p'+\varepsilon}\bigg)=\frac{dp}{p+p\varepsilon-\varepsilon}.
\end{equation*}
Since $\lambda<d$, we have $L_C^{p,\lambda}(\Omega;\mathbb{R}^d)=L_M^{p,\lambda}(\Omega;\mathbb{R}^d)$; as a consequence $u\in C_{\textnormal{loc}}^{0,\alpha}(\Omega)$ with
\begin{equation*}
\alpha=1-\frac{\lambda}{p}
\end{equation*}
if $p>\lambda$. That is, if
\begin{equation*}
\varepsilon>\frac{d-p}{p-1},
\end{equation*}
which holds by assumption.
Since $p'+\varepsilon<d$, we finally get
\begin{equation*}
\frac{d-p}{p-1}<\varepsilon<d-\frac{p}{p-1}.
\end{equation*} \end{proof}
\begin{Remark}[Endpoint-regularity]We conclude by examining the limit behaviour of the modulus of continuity -- encoded by the H\"older exponent $\alpha\in(0,1)$ in Theorem \ref{thm_c0alpha} -- as $\varepsilon$ approaches the endpoints of its interval of definition. Indeed, as \[
\varepsilon\to\bigg(d-\frac{p}{p-1}\bigg)^- \] one gets \[
\alpha\to1-\frac{1}{p-1}. \] On the other hand, as \[
\varepsilon\to\bigg(\frac{d-p}{p-1}\bigg)^+ \] one has \[
\alpha\to0. \] \end{Remark}
{\small \noindent{\bf Acknowledgments.} The authors thank Paolo Baroni and Giuseppe Mingione for insightful comments on the material in the paper. VB is supported by the Centre for Mathematics of the University of Coimbra (UIDB/00324/2020, funded by the Portuguese Government through FCT/MCTES). EP is partially supported by the Centre for Mathematics of the University of Coimbra (UIDB/00324/2020, funded by the Portuguese Government through FCT/MCTES) and by FAPERJ (grants E26/200.002/2018 and E26/201.390/2021). JMU is partially supported by the King Abdullah University of Science and Technology (KAUST) and by the Centre for Mathematics of the University of Coimbra (UIDB/00324/2020, funded by the Portuguese Government through FCT/MCTES).}
\end{document} |
\begin{document}
\draft \title{Canonical Transformations and the Hamilton-Jacobi Theory
in Quantum Mechanics} \author{Jung-Hoon Kim\footnote{e-mail: jhkim@laputa.kaist.ac.kr}
and Hai-Woong Lee} \address{
Department of Physics,
Korea Advanced Institute of Science and Technology,
Taejon, 305-701, Korea }
\maketitle \begin{abstract} Canonical transformations using the idea of quantum generating functions are applied to construct a quantum Hamilton-Jacobi theory, based on the analogy with the classical case. An operator and a c-number forms of the time-dependent quantum Hamilton-Jacobi equation are derived and used to find dynamical solutions of quantum problems. The phase-space picture of quantum mechanics is discussed in connection with the present theory. \end{abstract} \pacs{PACS number(s): 03.65.-w, 03.65.Ca, 03.65.Ge}
\section{Introduction}
Various mechanical problems can be elegantly approached by the Hamiltonian formalism, which not only found well-established ground in classical theories\cite{one}, but also provided much physical insight in the early development of quantum theories\cite{two,three}. It is curious though that the concept of canonical transformations, which plays a fundamental role in the Hamiltonian formulation of classical mechanics, has not attracted as much attention in the corresponding formulation of quantum mechanics. A relatively small quantity of literature is available as of now on this subject [4--10]. The main reason for this is probably that canonical variables in quantum mechanics are not c-numbers but noncommuting operators, manipulation of which is considerably involved. In spite of this difficulty, the great success of canonical transformations in classical mechanics makes it desirable to investigate the possibility of application of the concept of canonical transformations in quantum mechanics at least to the extent allowed in view of the analogy with the classical case.
The usefulness of the classical canonical transformations is most visible in the Hamilton-Jacobi theory where one seeks a generating function that makes the transformed Hamiltonian become identically zero\cite{one}. A quantum analog of the Hamilton-Jacobi theory has previously been considered by Leacock and Padgett\cite{eight} with particular emphasis on the quantum Hamilton's characteristic function and applied to the definition of the quantum action variable and the determination of the bound-state energy levels\cite{one2}. However, the {\em dynamical} aspect of the quantum Hamilton-Jacobi theory appears to remain untouched. In the present study, we concentrate on this aspect of the problem, and derive the time-dependent quantum Hamilton-Jacobi equation following closely the procedure that lead to the classical Hamilton-Jacobi equation.
The analogy between the classical and quantum Hamilton-Jacobi theories can be best exploited by employing the idea of the quantum generating function that was first introduced by Jordan\cite{four} and Dirac\cite{five}, and recently reconsidered by Lee and l'Yi\cite{ten}. The ``well-ordered'' operator counterpart of the quantum generating function is used in constructing our quantum Hamilton-Jacobi equation, which resembles in form the classical Hamilton-Jacobi equation. By means of well-ordering, a unique operator is associated with a given c-number function, thereby the ambiguity in the ordering problem is removed. We identify the quantum generating function accompanying the quantum Hamilton-Jacobi theory as the quantum Hamilton's principal function, and apply this theory to find the dynamical solutions of quantum problems.
The prevailing conventional belief that physical observables should be Hermitian operators invokes in our discussion the unitary transformation that transforms one Hermitian operator to another. This along with the fact that the unitary transformation preserves the fundamental quantum condition for the new canonical variables $[\hat{Q},\hat{P}]=i\hbar$ if the old canonical variables satisfy $[\hat{q},\hat{p}]=i\hbar$ provides a good reason why we call the unitary transformation the quantum canonical transformation. This definition of the quantum canonical transformation is analogous to the classical statement that the classical canonical transformation keeps the Poisson brackets invariant, i.e., $[Q,P]_{PB}=[q,p]_{PB}=1$. In our current discussion of the quantum canonical transformation we will consider exclusively the case of the unitary transformation.
The paper is organized as follows. In Sec.\ II the quantum canonical transformation using the idea of the quantum generating function is briefly reviewed, and the transformation relation between the new Hamiltonian and the old Hamiltonian expressed in terms of the quantum generating function is derived. From this relation, and by analogy with the classical case, we arrive at the quantum Hamilton-Jacobi equation in Sec.\ III. It will be found that the unitary transformation of the special type $\hat{U}(t)=\hat{T}(t)\hat{A}$ where $\hat{T}(t)$ is the time-evolution operator and $\hat{A}$ is an arbitrary time-independent unitary operator satisfies the quantum Hamilton-Jacobi equation. Sec.\ IV is devoted to the discussion of the quantum phase-space distribution function under canonical transformations. The differences between our approach and that of Ref.\cite{one1} are described. Boundary conditions and simple applications of the theory are given in Sec.\ V, where to perceive the main idea easily most of the discussion is developed with the simple case $\hat{A}=\hat{I}$, the unit operator, while keeping in mind that the present formalism is not restricted to this case. Finally, Sec.\ VI presents concluding remarks.
\section{Quantum Canonical Transformations}
Let us begin our discussion by reviewing the theory of the quantum canonical transformations\cite{five,ten}. A quantum generating function that is analogous to a classical generating function is defined in terms of the matrix elements of a unitary operator as follows\cite{five}, \begin{equation}
e^{iF_1(q_1,Q_2,t)/\hbar}\equiv \langle q_1|Q_2\rangle _t
=\langle q_1|\hat{U}(t)|q_2\rangle , \end{equation} where the unitary operator $\hat{U}(t)$ transforms an eigenvector of $\hat{q}$ into an eigenvector of $\hat{Q}=\hat{U}\hat{q}\hat{U}^\dagger$,
i.e., $|Q_1\rangle _t=\hat{U}(t)|q_1\rangle$ (and $|P_1
\rangle _t=\hat{U}(t)|p_1\rangle$).\footnote{
An eigenvalue $X_1$ and an eigenvector $|X_1\rangle$ of an operator $\hat{X}$ are defined by the equation, $\hat{X}|X_1\rangle
=X_1|X_1\rangle$ ($X=q$, $p$, $Q$, and $P$). Different subindices are used to distinguish different eigenvalues or
eigenvectors, e.g., $X_2$, $|X_2\rangle$, etc.; The subscript $t$ on a ket $|\rangle _t$ (bra ${_t\langle}|$) expresses time dependence of the ket $|\rangle _t$ (bra ${_t\langle}|$).} Different types of the quantum generating function can be defined similarly\cite{ten}, i.e.,
$e^{iF_2(q_1,P_2,t)/\hbar}\equiv \langle q_1|P_2\rangle _t
=\langle q_1|\hat{U}(t)|p_2\rangle$,
$e^{iF_3(p_1,Q_2,t)/\hbar}\equiv \langle p_1|Q_2\rangle _t
=\langle p_1|\hat{U}(t)|q_2\rangle$, and
$e^{iF_4(p_1,P_2,t)/\hbar}\equiv \langle p_1|P_2\rangle _t
=\langle p_1|\hat{U}(t)|p_2\rangle$.
The quantum canonical transformation, or the unitary transformation, corresponds to a change of representation or equivalently to a rotation of axes in the Hilbert space. The unitary transformation guarantees that the fundamental quantum condition $[\hat{Q},\hat{P}]
=[\hat{q},\hat{p}]=i\hbar$ holds, the new canonical variables $(\hat{Q},\hat{P})$ are Hermitian operators, and the eigenvectors of $\hat{Q}$ or $\hat{P}$ form a complete basis. One should keep in mind that the eigenvalue $Q_1$ has the same numerical value as the eigenvalue $q_1$ because the unitary transformation preserves the eigenvalue spectrum of an operator\cite{three}. In cases where it is convenient, one is free to interchange $q_1$ $ (p_1)$ with $Q_1$ $(P_1)$.
Transformation relations between $(\hat{q},\hat{p})$ and $(\hat{Q},\hat{P})$ can be expressed in terms of the ``well-ordered'' generating operator $\bar{F}_1(\hat{q},\hat{Q},t)$\cite{five} that is an operator counterpart of the quantum generating function $F_1(q_1,Q_2,t)$ as follows:\footnote{ A well-ordered operator $\bar{G}(\hat{X},\hat{Y})$ is developed from a c-number function $G(X_1,Y_2)$ such that
$\langle X_1|\bar{G}(\hat{X},\hat{Y})|Y_2
\rangle = G(X_1,Y_2)\langle X_1|Y_2\rangle $\cite{five}. For example, if $G(X_1,Y_2)=X_1Y_2+Y_2^2X_1^3$, then $\bar{G}(\hat{X},\hat{Y})= \hat{X}\hat{Y}+\hat{X}^3\hat{Y}^2$.} \begin{equation} \hat{p}=\frac{\partial \bar{F}_1(\hat{q},\hat{Q},t)}{\partial \hat{q}}, \hspace{1cm} \hat{P}=-\frac{\partial \bar{F}_1(\hat{q},\hat{Q},t)}{\partial \hat{Q}}. \end{equation} Similar expressions for other types of the generating operators can be immediately inferred by analogy with the classical relations. For a later reference, we present the relations for $\bar{F}_2(\hat{q},\hat{P},t)$ below, \begin{equation} \hat{p}=\frac{\partial \bar{F}_2(\hat{q},\hat{P},t)}{\partial \hat{q}}, \hspace{1cm} \hat{Q}=\frac{\partial \bar{F}_2(\hat{q},\hat{P},t)}{\partial \hat{P}}. \end{equation}
It is interesting to note that, whereas the four types of the generating functions in classical mechanics are related with each other through the Legendre transformations\cite{one}, the relations between the quantum generating functions of different types can be expressed by means of the Fourier transformations. For example, the transition from $F_1(q_1,Q_2,t)$ to $F_2(q_1,P_2,t)$ can be accomplished by \begin{eqnarray}
e^{iF_2(q_1,P_2,t)/\hbar}&=&\int dQ_2 \langle q_1|Q_2\rangle _t\hspace{0.7mm}
{_t\langle} Q_2|P_2\rangle _t, \nonumber \\ &=&\frac{1}{\sqrt{2\pi \hbar}} \int dQ_2 e^{iF_1(q_1,Q_2,t)/\hbar} e^{iP_2Q_2/\hbar}. \end{eqnarray}
The usefulness of the concept of the quantum generating function can be revealed, for example, by considering the unitary transformation $\hat{U} =e^{ig(\hat{q})/\hbar}$ where $g$ is an arbitrary real function. From the definition of the quantum generating function, we have \begin{eqnarray}
e^{iF_2(q_1,P_2)/\hbar}&=&\langle q_1|e^{ig(\hat{q})/\hbar}|p_2 \rangle , \nonumber \\ &=& \frac{1}{\sqrt{2\pi \hbar}}e^{\frac{i}{\hbar}[g(q_1)+q_1P_2]}. \end{eqnarray} The well-ordered generating operator is then given by \begin{equation} \bar{F}_2(\hat{q},\hat{P})=g(\hat{q})+\hat{q}\hat{P}+i\frac{\hbar}{2} \ln 2\pi \hbar , \end{equation} and Eq.\ (3) yields the transformation relations \begin{eqnarray} \hat{Q}&=& \hat{q}, \\ \hat{P}&=&\hat{p}-\frac{\partial g(\hat{q})}{\partial \hat{q}}. \end{eqnarray} This shows that, in some cases, an introduction of the quantum generating function can provide an effective method of finding the transformation relations between ($\hat{q},\hat{p}$) and ($\hat{Q},\hat{P}$) without recourse to the equations $\hat{Q}=\hat{U}\hat{q}\hat{U}^\dagger$ and $\hat{P}=\hat{U}\hat{p}\hat{U}^\dagger$.
Now we consider the dynamical equations governing the time-evolution of quantum systems. The time-dependent Schr\"{o}dinger equation for the system with the Hamiltonian $H(\hat{q},\hat{p},t)$ is given in terms of a
time-dependent ket $|\psi \rangle _t$ by \begin{equation}
i\hbar \frac{\partial}{\partial t}|\psi \rangle _t
=H(\hat{q},\hat{p},t)|\psi \rangle _t. \end{equation} In $Q$-representation the time-dependent Schr\"{o}dinger equation takes the form \begin{equation} i\hbar \frac{\partial}{\partial t}\psi ^Q(Q_1,t)=K\left( Q_1, -i\hbar \frac{\partial}{\partial Q_1},t\right) \psi ^Q(Q_1,t), \end{equation}
where $\psi ^Q(Q_1,t)={_t\langle}Q_1|\psi \rangle _t$, and \begin{equation} K(\hat{Q},\hat{P},t)=H(\hat{q},\hat{p},t)+i\hbar \hat{U} \frac{\partial \hat{U}^\dagger}{\partial t}. \end{equation} The second term on the right hand side of Eq.\ (11) arises from the fact that we allow the time dependence of the unitary operator $\hat{U}(t)$, which indicates that, even though we adopt here the Schr\"{o}dinger picture where the time dependence associated with the dynamical evolution of a
system is attributed solely to the ket $|\psi \rangle _t$, $\hat{Q}$ and
$|Q_1\rangle _t$ may depend on time also. In terms of the generating operator $\bar{F}_1(\hat{q},\hat{Q},t)$, Eq.\ (11) can be written as \begin{equation} K(\hat{Q},\hat{P},t)=H(\hat{q},\hat{p},t)+\frac{\partial \bar{F} _1 (\hat{q},\hat{Q},t)}{\partial t}. \end{equation} The equivalence of Eqs.\ (11) and (12) can be proved as shown in Appendix A. It is important to note that $K(\hat{Q},\hat{P},t)$ plays the role of the transformed Hamiltonian governing the time-evolution of the system in $Q$-representation. The analogy with the classical theory is remarkable.
\section{Quantum Hamilton-Jacobi Theory}
We are now ready to proceed to formulate the quantum Hamilton-Jacobi theory. One can immediately notice that, if $K(\hat{Q},\hat{P},t)$ of Eq.\ (12) vanishes, the time-dependent Schr\"{o}dinger equation in $Q$-representation yields a simple solution, $\psi ^Q=$\ const. This observation along with Eq.\ (2) naturally leads us to the following quantum Hamilton-Jabobi equation, \begin{equation} H\left( \hat{q},\frac{\partial \bar{S}_1(\hat{q},\hat{Q},t)}{\partial \hat{q}},t\right) +\frac{\partial \bar{S}_1(\hat{q},\hat{Q},t)}{\partial t} =0, \end{equation} where, following the classical notational convention, we denote the generating operator that is analogous to the classical Hamilton's principal function by $\bar{S}_1(\hat{q},\hat{Q},t)$. Eq.\ (13) bears a close formal resemblance to the classical Hamilton-Jacobi equation. It, however, differs from the classical equation in that it is an operator partial differential equation. The procedure of solving dynamical problems is completed if we express the wave function in the original $q$-representation as \begin{eqnarray}
\psi ^q(q_1,t)&=&\int \langle q_1|Q_2 \rangle _t\hspace{0.7mm}
{_t\langle}Q_2| \psi \rangle _t dQ_2, \nonumber \\ &=&\int e^{iS_1(q_1,Q_2,t)/\hbar}\psi ^Q(Q_2)dQ_2, \end{eqnarray} where $S_1(q_1,Q_2,t)$ is the c-number counterpart of $\bar{S}_1(\hat{q},\hat{Q},t)$, and is obtained by replacing the well-ordered $\hat{q}$ and $\hat{Q}$ in $\bar{S}_1$, respectively, with $q_1$ and $Q_2$. In Eq.\ (14), $t$ is dropped from $\psi ^Q$, since
${_t\langle}Q_2| \psi \rangle _t =$ const. As is the case for the classical Hamilton-Jacobi equation, the mission of solving dynamical problems is assigned to the quantum Hamilton-Jacobi equation.
Even though we arrive at the correct form of the quantum Hamilton-Jacobi equation, it seems at first sight quite difficult to attain solutions of it due to its unfamiliar appearance as an operator partial differential equation. Thus it seems desirable to search a corresponding c-number form of the quantum Hamilton-Jacobi equation. For this task, we note that, if the unitary operator $\hat{U}(t)$ is assumed to be separable into $\hat{U}(t)=\hat{T}(t)\hat{A}$, where $\hat{T}(t)$ is the time-evolution operator and $\hat{A}$ is an arbitrary time-independent unitary operator, then
$\psi ^Q(Q_1,t) ={_t\langle}Q_1|\psi \rangle _t=\langle q_1|\hat{A}^\dagger
\hat{T}^\dagger (t)\hat{T}(t)|\psi (t=0) \rangle =\langle q_1|\hat{A}^\dagger
|\psi (t=0)\rangle =$\ const. This means that the left hand side of Eq.\ (10) becomes zero, i.e., the canonical transformation mediated by a separable unitary operator is exactly the one that we seek. Assuming $\hat{U}(t)=\hat{T}(t)\hat{A}$, we rewrite Eq.\ (1) as \begin{equation}
e^{iS_1(q_1,Q_2,t)/\hbar} =\langle q_1|\hat{T}(t)\hat{A}|q_2\rangle . \end{equation} Differentiating this equation with respect to time, we obtain \begin{eqnarray} \frac{i}{\hbar}\frac{\partial S_1}{\partial t}e^{iS_1/\hbar} &=&
\langle q_1 |\frac{\partial \hat{T}}{\partial t}\hat{A}|q_2\rangle
=\frac{1}{i\hbar}\langle q_1|\hat{H}\hat{T}\hat{A}|q_2\rangle , \nonumber \\ &=&\frac{1}{i\hbar}H\left( q_1,-i\hbar \frac{\partial}{\partial q_1},t
\right) \langle q_1|\hat{T}\hat{A}|q_2\rangle ,\nonumber \\ &=&\frac{1}{i\hbar}H\left( q_1,-i\hbar \frac{\partial}{\partial q_1},t \right) e^{iS_1/\hbar}. \end{eqnarray} Eq.\ (16) leads immediately to the desired c-number form of the quantum Hamilton-Jacobi equation \begin{equation} \left[ H\left( q_1,-i\hbar \frac{\partial}{\partial q_1},t\right) +\frac{\partial S_1(q_1,Q_2,t)}{\partial t}\right] e^{iS_1(q_1,Q_2,t)/\hbar}=0. \end{equation} Substitution of $S_2(q_1,P_2,t)$ for $S_1(q_1,Q_2,t)$ generates another c-number form of the quantum Hamilton-Jacobi equation. The equations for the cases of $S_3(p_1,Q_2,t)$ and $S_4(p_1,P_2,t)$ can be derived through a similar process.
Consider a one-dimensional nonrelativistic quantum system whose Hamiltonian is given by \begin{equation} H(\hat{q},\hat{p},t)=\frac{\hat{p}^2}{2}+V(\hat{q},t). \end{equation} The c-number form of the quantum Hamilton-Jacobi equation (17) for this problem becomes \begin{equation} \frac{1}{2}\left(\frac{\partial S_1}{\partial q_1}\right)^2 -i\frac{\hbar}{2}\frac{\partial ^2S_1}{\partial q_1^2}+V(q_1,t) +\frac{\partial S_1}{\partial t}=0. \end{equation} We can see clearly that, in the limit $\hbar \rightarrow 0$, the above equation reduces to the classical Hamilton-Jacobi equation. The second term of Eq.\ (19) represents the quantum effect. We note that it has been known from the early days that substitution of $\psi (q,t)=e^{iS(q,t)/\hbar}$ into the Schr\"{o}dinger equation gives rise to the same Hamilton-Jacobi equation for $S(q,t)$,\footnote{For a stationary state of a system whose Hamiltonian does not depend explicitly on time, one may put $S(q,t) = W(q) -Et$ and obtain a differential equation for $W(q)$. To find a solution to the resulting equation, one may then use the expansion of $W$ in powers of $\hbar$. This approach has been extensively considered in connection with the well-known WKB approximation. In the present paper, the formalism is developed for general nonstationary states (of systems that can possibly have time-dependent Hamiltonians).} where $S(q,t)$ is interpreted merely as the complex-valued phase of the wave function (see, for example, Ref.\cite{schiff}). The present approach more clearly shows the strong analogy between the classical and quantum Hamilton-Jacobi theories emphasizing that the quantum Hamilton's principal function $S_1$ which is related with the wave function via Eq.\ (14) plays the role of the quantum counterpart of the classical generating function. Moreover, as discussed later in Sec.\ V, $e^{iS_1/\hbar}$ defined in Eq.\ (14) can be interpreted as a propagator under a certain choice of $\hat{A}$.
It may be viewed that the Hamilton-Jacobi equation in the form of Eq.\ (19) is no more tractable analytically than the Schr\"{o}dinger equation for general potential problems. Nevertheless, it would be possible at least to obtain an approximate solution of it using a perturbative method as follows. Since the solution of Eq.\ (19) is given by the classical Hamilton's principal function in the limit $\hbar \rightarrow 0$, we can expand the general solution in powers of $\hbar$: \begin{equation} S_1=S_1^{(0)}+\hbar S_1^{(1)}+\hbar ^2S_1^{(2)}+\cdots , \end{equation} where $S_1^{(0)}$ is the classical Hamilton's principal function. Substituting Eq.\ (20) into Eq.\ (19) and collecting coefficients of the same orders in $\hbar$, we can obtain \begin{equation} \frac{1}{2}\left(\frac{\partial S_1^{(0)}}{\partial q_1}\right)^2+V(q_1,t) +\frac{\partial S_1^{(0)}}{\partial t}=0, \end{equation} and \begin{equation} \frac{1}{2}\sum_{k=0}^{n}\frac{\partial S_1^{(k)}}{\partial q_1} \frac{\partial S_1^{(n-k)}}{\partial q_1} -\frac{i}{2}\frac{\partial ^2S_1^{(n-1)}}{\partial q_1^2} +\frac{\partial S_1^{(n)}}{\partial t}=0, \hspace{0.5cm} n\geq 1. \end{equation} Given the solution $S_1^{(0)}$ of the classical Hamilton-Jacobi equation (21), we solve Eq.\ (22) to find $S_1^{(1)}$. $S_1^{(2)}$ can be determined subsequently from the knowledge of $S_1^{(0)}$ and $S_1^{(1)}$, and so forth. We note that Eq.\ (22) is linear in $S_1^{(n)}$ and first-order differential in $q_1$ for $S_1^{(n)}$. Thus, from a practical viewpoint, Eqs.\ (21) and (22) could be more advantageous to deal with than Eq.\ (19) as long as the classical Hamilton's principal function that is the solution of Eq.\ (21) is readily available.
The present formalism provides an encouraging point that the well-ordered operator counterpart of the quantum Hamilton's principal function gives also the solutions of the Heisenberg equations through Eq.\ (2). If we consider the case $\hat{U}(t)=\hat{T}(t)$, we can obtain in the Heisenberg picture the relations $(\hat{q}_H,\hat{p}_H)\equiv (\hat{T}^\dagger \hat{q}_S\hat{T}, \hat{T}^\dagger \hat{p}_S\hat{T})$ and $(\hat{Q}_H,\hat{P}_H)\equiv (\hat{T}^\dagger \hat{Q}_S\hat{T}, \hat{T}^\dagger \hat{P}_S\hat{T})= (\hat{T}^\dagger \hat{T}\hat{q}_S \hat{T}^\dagger \hat{T},\hat{T}^\dagger \hat{T}\hat{p}_S\hat{T}^\dagger \hat{T})=(\hat{q}_S,\hat{p}_S)$, where we attached the subscript $_S$ and $_H$ to operators to explicitly denote, respectively, the Schr\"{o}dinger and the Heisenberg pictures. Thus, when expressed in the Heisenberg picture Eq.\ (2) turns into \begin{equation} \hat{p}_H=\frac{\partial \bar{S}_1(\hat{q}_H,\hat{q}_S,t)}{\partial \hat{q}_H}, \hspace{1cm} \hat{p}_S=-\frac{\partial \bar{S}_1(\hat{q}_H,\hat{q}_S,t)}{\partial \hat{q}_S}, \end{equation} and from these transformation relations we can obtain $\hat{q}_H$ and $\hat{p}_H$ as functions of time and the initial operators $\hat{q}_S$ and $\hat{p}_S$. Obviously, $\hat{q}_H(\hat{q}_S,\hat{p}_S,t)$ and $\hat{p}_H(\hat{q}_S,\hat{p}_S,t)$ obtained in this way evolve according to the Heisenberg equations. \section{Quantum Phase-Space distribution functions and canonical transformations}
Since our theory of the quantum canonical transformations is formulated with the canonical position $\hat{q}$ and momentum $\hat{p}$ variables on an equal footing, it would be relevant to consider the phase-space picture of quantum mechanics, exploiting the distribution functions in relation to the present theory.
\subsection{Distribution functions}
For a given density operator $\hat{\rho}$, a general way of defining quantum distribution functions proposed by Cohen\cite{one5} is that \begin{equation}
F^f(q_1,p_1,t)=\frac{1}{2\pi ^2\hbar}\int \int \int dxdydq_2 \langle q_2+y|\hat{\rho} |q_2-y\rangle f(x,2y/\hbar )e^{ix(q_2-q_1)} e^{-i2yp_1/\hbar}. \end{equation} Various choices of $f(x,2y/\hbar)$ lead to a wide class of quantum distribution functions\cite{one6}. To mention only a few, the choice $f=1$ produces the well-known Wigner distribution function \cite{one7}, while the choice $f(x,2y/\hbar)= e^{-\hbar x^2/4m\alpha -m\alpha y^2/\hbar}$ yields the Husimi distribution function that recently has found its application in nonlinear dynamical problems\cite{one8}. The transformed distribution function is defined in ($Q_1,P_1$) phase space likewise by \begin{equation} G^f(Q_1,P_1,t)=\frac{1}{2\pi ^2\hbar}\int \int \int dXdYdQ_2\hspace{0.7mm}
{_t\langle} Q_2+Y|\hat{\rho} |Q_2-Y\rangle _tf(X,2Y/\hbar )e^{iX(Q_2-Q_1)} e^{-i2YP_1/\hbar}. \end{equation}
Our main objective here is to find a relation between the old and the transformed distribution functions. After a straightforward algebra, which is displayed in Appendix B, it turns out that the transformation relation between the two distribution functions can be expressed as \begin{equation} G^f(Q_1,P_1,t)=\int \int dq_2dp_2\kappa (Q_1,P_1,q_2,p_2,t) F^f(q_2,p_2,t), \end{equation} where the kernel $\kappa$ is given by \begin{eqnarray} \kappa (Q_1,P_1,q_2,p_2,t)=\frac{1}{2\pi ^3\hbar} \int \int \int \int \int \int dXdYdQ_2dxdyd\alpha \frac{f(X,2Y/\hbar )} {f(x,2y/\hbar )} \nonumber \\ \times e^{\frac{i}{\hbar}[F_1(q_2+\alpha -y,Q_2-Y,t)-F_1^*(q_2+\alpha +y, Q_2+Y,t)]} e^{i[X(Q_2-Q_1)-\alpha x]} e^{\frac{2i}{\hbar} [yp_2-YP_1]}. \end{eqnarray} This expression for the kernel can be further simplified if integrations in Eq.\ (27) can be performed with a specific choice of the function $f$. For instance, the simple choice $f=1$ provides the following kernel for the Wigner distribution function, \begin{eqnarray} \kappa (Q_1,P_1,q_2,p_2,t)=\frac{2}{\pi \hbar} \int \int dYdy e^{\frac{i}{\hbar} [F_1(q_2-y,Q_1-Y,t)-F_1^*(q_2+y,Q_1+Y,t)]} e^{\frac{2i}{\hbar}[yp_2-YP_1]}. \end{eqnarray} This equation was first derived by Garcia-Calder\'on and Moshinsky\cite{one9} without employing the idea of the quantum generating function. Curtright {\it et al.}\cite{one10} also obtained an equivalent expression in their recent discussion of the time-independent Wigner distribution functions.
We wish to point out that the quantum canonical transformation described here is basically different from that considered earlier by Kim and Wigner\cite{one1}. While the present approach deals with the transformation between operators ($\hat{q},\hat{p}$) and ($\hat{Q}, \hat{P}$), their approach is about the transformation between c-numbers ($q,p$) and ($Q,P$). For the transformation $Q=Q(q,p,t)$ and $P=P(q,p,t)$, their approach yields for the kernel the expression \begin{equation} \kappa (Q_1,P_1,q_2,p_2,t)=\delta [Q_1-Q(q_2,p_2,t)]\delta [P_1-P(q_2,p_2,t)], \end{equation} where $Q(q,p,t)$ and $P(q,p,t)$ satisfy the classical Poisson brackets relation, $[Q,P]_{PB}=[q,p]_{PB}=1$. The kernels of Eq.\ (28) and Eq.\ (29) coincide with each other for the special case of a linear canonical transformation, as was shown by Garcia-Calder\'on and Moshinsky\cite{one9}. Specifically, for the case of the Wigner distribution function, they showed that the linear transformation for operators, $\hat{Q}=a\hat{q}+b\hat{p}$ and $\hat{P}=c\hat{q}+d\hat{p}$, and that for c-number variables, $Q=aq+bp$ and $P=cq+dp$, yield the same kernel $\kappa (Q_1,P_1,q_2,p_2)=\delta [Q_1-(aq_2+bp_2)]\delta [P_1-(cq_2+dp_2)]$. In general cases, however, Eq.\ (27) and Eq.\ (29) give rise to different kernels. As an example, let us consider the unitary transformation $\hat{U}=e^{ig(\hat{q})/\hbar}$ considered in Sec.\ II. The first-type quantum generating function has the form $e^{iF_1(q_1,Q_2)/\hbar}=e^{ig(q_1)/\hbar} \delta (q_1-Q_2)$. This nonlinear canonical transformation yields for the Wigner distribution function the kernel \begin{equation} \kappa (Q_1,P_1,q_2,p_2)= \frac{\delta (Q_1-q_2)}{\pi \hbar} \int dy e^{\frac{i}{\hbar} [g(q_2-y)-g(q_2+y)]} e^{\frac{2i}{\hbar}(p_2-P_1)y}. \end{equation} It is apparent that the integral of the above equation cannot generally be reduced to the $\delta$-function of Eq.\ (29) except for some trivial cases, e.g., $g=$const, $g=q$, and $g=q^2$. Distribution functions other than the Wigner distribution function do not usually allow the simple expression for the kernel in the form of Eq.\ (29), even if one considers a linear canonical transformation.
\subsection{Dynamics}
In this subsection we describe how the quantum Hamilton-Jacobi theory can lead to dynamical solutions in the phase-space picture of quantum mechanics. For this task, we first consider the time evolution of the transformed distribution function in ($Q_1,P_1$) phase space. Differentiating Eq.\ (25) with respect to time, we can get \begin{eqnarray} \frac{\partial G^f}{\partial t}=\frac{1}{2\pi ^2\hbar} \int \int \int dXdYdQ_2 \left[ \left( \frac{\partial}
{\partial t}{_t\langle} Q_2+Y| \right) \hat{\rho} |Q_2-Y\rangle _t+
{_t\langle}Q_2+Y|\frac{\partial \hat{\rho}}{\partial t}|Q_2-Y\rangle _t \right. \nonumber \\
\left. +{_t\langle}Q_2+Y|\hat{\rho} \left( \frac{\partial}{\partial t}
|Q_2-Y\rangle _t\right) \right] f(X,2Y/\hbar)e^{iX(Q_2-Q_1)} e^{-i2YP_1/\hbar}. \end{eqnarray} We now substitute into Eq.\ (31) the time evolution equations \begin{eqnarray} \frac{\partial \hat{\rho}}{\partial t}&=&-\frac{i}{\hbar} [\hat{H},\hat{\rho}], \\
\frac{\partial}{\partial t}{_t\langle}Q_2+Y|&=&{_t\langle}
Q_2+Y|\hat{U}\frac{\partial \hat{U}^\dagger}{\partial t}, \\
\frac{\partial}{\partial t}|Q_2-Y\rangle _t &=&\frac{\partial \hat{U}}
{\partial t}\hat{U}^\dagger |Q_2-Y\rangle _t=-\hat{U}\frac{\partial
\hat{U}^\dagger}{\partial t}|Q_2-Y\rangle _t, \end{eqnarray} where $\hat{H}=H(\hat{q},\hat{p},t)$ is the Hamiltonian governing the dynamics of the system, and obtain \begin{eqnarray} \frac{\partial G^f}{\partial t}=\frac{1}{2\pi ^2\hbar}
\int \int \int dXdYdQ_2\hspace{0.7mm} {_t\langle}Q_2+Y|\left( -\frac{i}{\hbar}
[\hat{K},\hat{\rho}]\right) |Q_2-Y\rangle _tf(X,2Y/\hbar)e^{iX(Q_2-Q_1)} e^{-i2YP_1/\hbar}, \end{eqnarray} where $\hat{K}=K(\hat{Q},\hat{P},t)$ is just the transformed Hamiltonian already defined in Eq.\ (11). Eq.\ (35) should be compared with the following equation that governs the time evolution of the distribution function in ($q_1,p_1$) phase space, \begin{eqnarray} \frac{\partial F^f}{\partial t}=\frac{1}{2\pi ^2\hbar}
\int \int \int dxdydq_2 \langle q_2+y|\left( -\frac{i}{\hbar}
[\hat{H},\hat{\rho}]\right) |q_2-y\rangle f(x,2y/\hbar)e^{ix(q_2-q_1)} e^{-i2yp_1/\hbar}. \end{eqnarray} We can easily see that, through the quantum canonical transformation, the role played by $\hat{H}$ is turned over to $\hat{K}$.
Just as the wave function has a trivial solution in the representation where the transformed Hamiltonian $K(\hat{Q},\hat{P},t)$ vanishes, so does the distribution function in the corresponding phase space, as can be seen from Eq.\ (35). With the trivial solution $G^f=$\ const., we go back to the original space via the inverse of the transformation equation (26) to obtain $F^f(q_1,p_1,t)$. For example, for the case of the Wigner distribution function the transformation can be accomplished by \begin{equation} F^W(q_1,p_1,t)=\int \int dQ_2dP_2\tilde{\kappa} (q_1,p_1,Q_2,P_2,t) G^W(Q_2,P_2), \end{equation} where $\tilde{\kappa}$ is given in terms of the quantum principal function by \begin{eqnarray} \tilde{\kappa} =\frac{2}{\pi \hbar} \int \int dydY e^{\frac{i}{\hbar} [S_1(q_1+y,Q_2+Y,t)-S_1^*(q_1-y,Q_2-Y,t)]} e^{\frac{2i}{\hbar}[YP_2-yp_1]}. \end{eqnarray} Thus, once the quantum Hamilton-Jacobi equation is solved and the quantum principal function $S_1$ is obtained, the dynamics of the distribution function, as well as that of the wave function, can be determined.
\section{Boundary conditions and Applications}
Up to this point the whole theory has been developed for the case $\hat{U}(t)=\hat{T}(t)\hat{A}$ with $\hat{A}$ taken to be arbitrary unless otherwise mentioned. To see how the quantum Hamilton-Jacobi theory is used to achieve the dynamical solutions of quantum problems, it would be sufficient, though, to consider the case of $\hat{A}=\hat{I}$, the unit operator. This case was considered by Dirac in connection with his action principle (see Sec.\ 32 of Ref.\cite{three}). He showed that $S_1$ defined by Eq.\ (15) equals the classical action function in the limit $\hbar \rightarrow 0$. It should be mentioned that this particular case allows the quantum generating functions to attain the property that $e^{iS_1/\hbar}$ is the propagator in position space and $e^{iS_4/\hbar}$ the propagator in momentum space. We will henceforth work on the case $\hat{U}(t)=\hat{T}(t)$. The general case $\hat{U}(t)=\hat{T}(t)\hat{A}$ will be briefly treated in Appendix C.
Before applying the theory it is necessary to provide some remarks concerning the quantum Hamilton-Jacobi equation (17) and its solution. First, if the Hamiltonian depends only on either $\hat{q}$ or $\hat{p}$, we do not need to solve Eq.\ (17). Instead, since the unitary operator has the simple form $\hat{U}=\hat{T}=e^{-iH(\hat{q})t/\hbar}$ or $e^{-iH(\hat{p})t/\hbar}$, we can obtain $S_1$ directly from Eq.\ (15) by calculating the matrix elements of $\hat{U}$. For example, for a free particle, $\hat{U}=e^{-i\hat{p}^2t/2\hbar}$, it is convenient to calculate
$e^{iS_2/\hbar}=\langle q_1|e^{-i\hat{p}^2t/2\hbar}|p_2\rangle$, and we get $S_2(q_1,P_2,t)=-\frac{P_2^2t}{2}+q_1P_2+i\frac{\hbar}{2} \ln 2\pi \hbar$. Second, in order to solve Eq.\ (17), we need to impose proper boundary conditions on $S_1$. Since here we are dealing with unitary transformations, we immediately get from the definition of $S_1$ the condition \begin{equation} \int dQ_3 e^{i[S_1(q_1,Q_3,t)-S_1^*(q_2,Q_3,t)]/\hbar} =\delta (q_1-q_2), \end{equation} which follows from the calculation of the matrix elements of $\hat{U}(t)\hat{U}^{\dagger}(t)=\hat{I}$. This unitary condition ensures that the well-ordered operator counterpart of $S_1$ yields Hermitian operators for $\hat{Q}$ and $\hat{P}$ from Eq.\ (2). Mathematically, Eq.\ (17) can have several solutions, and there is an arbitrariness in the choice of the new position variable, because any function of the constant of integration of Eq.\ (17) can be a candidate for the new position variable. Not all the possible solutions correspond to the unitary transformations, and from the possible solutions we choose only those which satisfy Eq.\ (39) and thus give Hermitian position and momentum operators that are observables. These solutions correspond to the unitary transformations of the type $\hat{U}(t)=\hat{T}(t)\hat{A}$. Further, from these solutions we single out the one that corresponds to the case $\hat{A}=\hat{I}$ by imposing the condition $e^{iS_1(q_1,Q_2,t=0)/\hbar}=\delta (q_1-Q_2)$ as an initial condition. The appropriate form for $S_2$ corresponding to this condition is that $e^{iS_2(q_1,P_2,t=0)/\hbar}=\frac{1}{\sqrt{2\pi \hbar}} e^{iq_1P_2/\hbar}$. In the limit $\hbar \rightarrow 0$, $S_2$ in this equation reduces to the correct classical generating function for the identity transformation, $S_2=q_1P_2$. In solving the Hamilton-Jacobi equation perturbatively using Eqs.\ (21) and (22), in order to consistently satisfy the initial condition, we start with the classical Hamilton's principal function $S_1^{(0)}$ that gives at initial time the relations $q_1=Q_2$ and $p_1=P_2$ from the classical c-number counterpart of Eq.\ (2). An arbitrary additive constant $c$ to the solution of Eq.\ (17) that always appears in the form $S_1+c$ when we deal with a partial differential equation such as Eq.\ (19) which contains only partial derivatives of $S_1$\cite{one} can also be fixed by the initial condition. Depending whether boundary conditions can readily be expressed in a simple form, one type of the quantum generating function may be favored over another. The existence and uniqueness of the independent solution of Eq.\ (17) satisfying the above conditions can be guaranteed from the consideration of the equation
$e^{iS_1(q_1,Q_2,t)/\hbar}=\langle q_1|\hat{T}(t)|q_2\rangle$, in which $S_1$ is just given by the matrix elements of $T(t)$. It is clear that these matrix elements exist and are uniquely defined.
As illustrations of the application of the theory, we consider the following two simple systems.
{\sl Example 1. A particle under a constant force.}
As a first example, let us consider a particle moving under a constant force of magnitude $a$, for which the Hamiltonian is $\hat{H}=\hat{p}^2/2-a\hat{q}$. We start with the following classical principal function that is the solution of Eq. (21), \begin{equation} S_1^{(0)}=\frac{(q_1-Q_2)^2}{2t} +\frac{at(q_1+Q_2)}{2} -\frac{a^2t^3}{24}. \end{equation} Substituting $S_1^{(0)}$ into Eq.\ (22) and solving the resulting equation, we find that the first order term in $\hbar$ has the general solution \begin{equation} S_1^{(1)}=\frac{i}{2}\ln t +f\left( \frac{q_1-Q_2}{t} -\frac{a}{2}t\right) , \end{equation} where $f$ is an arbitrary differentiable function. To satisfy the proper boundary condition $e^{iS_1(q_1,Q_2,t=0)/\hbar} =\delta (q_1-Q_2)$, $f$ and all higher order terms of $S_1$ are chosen to be zero, and the overall additive constant to be $c=\hbar \frac{i}{2}\ln i2\pi \hbar$. By well-ordering terms, we get the generating operator \begin{equation} \bar{S_1}(\hat{q},\hat{Q},t)=\frac{\hat{q}^2-2\hat{q}\hat{Q} +\hat{Q}^2}{2t}+\frac{at}{2}(\hat{q}+\hat{Q}) -\frac{a^2t^3}{24} +\hbar\frac{i}{2}\ln i2\pi \hbar t. \end{equation} We can easily check that the operator form of the quantum Hamilton-Jacobi equation (13) is satisfied by the above generating operator.
From Eq.\ (14) we obtain the wave function \begin{equation} \psi ^q(q_1,t)=\int \frac{1}{\sqrt{i2\pi \hbar t}} e^{\frac{i}{2\hbar t} [(q_1-Q_2)^2 +at^2(q_1+Q_2)-a^2t^4/12]}\psi ^Q(Q_2)dQ_2. \end{equation} Because $\psi ^Q(Q_2)$ is constant in time, we can express it in terms of the initial wave function. For the present case in which we use the first-type quantum generating function $S_1$ and $\hat{A}=\hat{I}$, we have simply $\psi ^q(q_2=Q_2,t=0)=\psi ^Q(Q_2)$. We note that Eq.\ (43) is in exact agreement with the result of Feynman's path-integral approach\cite{two0}.
For the time evolution of the distribution function, we find from Eq.\ (38) the following kernel for the Wigner distribution function, \begin{eqnarray} \tilde{\kappa} (q_1,p_1,Q_2,P_2,t)&=&\frac{1}{\pi ^2\hbar ^2 t}\int \int dYdy e^{-\frac{2i}{\hbar}\left(Q_2-q_1+p_1t-\frac{at^2}{2}\right) \frac{y}{t}} e^{\frac{2i}{\hbar}\left( P_2-\frac{q_1-Q_2-at^2/2}{t} \right) Y}, \nonumber \\ &=&\delta(Q_2-q_1+p_1t-at^2/2) \delta \left(P_2-\frac{q_1-Q_2-at^2/2} {t}\right) . \end{eqnarray} Substituting Eq.\ (44) into Eq. (37), we obtain \begin{eqnarray} F^W(q_1,p_1,t)=F^W(q_1-p_1t+at^2/2,p_1-at,0), \end{eqnarray} where use has been made of the relation $F^W(q_1,p_1,t=0)=G^W(q_1,p_1)$.
As has been mentioned, the present Hamilton-Jacobi theory also provides the solutions of the Heisenberg equations via the transformation relations between the two sets of canonical operators. From Eqs.\ (2) and (42) we can obtain \begin{eqnarray} \hat{q}_S&=&\hat{Q}_S(t)+\hat{P}_S(t)t+\frac{a}{2}t^2, \\ \hat{p}_S&=&\hat{P}_S(t) +at. \end{eqnarray} In the Heisenberg picture, the above equations become \begin{eqnarray} \hat{q}_H(t)&=&\hat{q}_S+\hat{p}_St+\frac{a}{2}t^2, \\ \hat{p}_H(t)&=&\hat{p}_S +at, \end{eqnarray} which are the solutions of the Heisenberg equations.
By setting $a=0$, we can obtain the free particle solution.
{\sl Example 2. The harmonic oscillator}
For the harmonic oscillator whose Hamiltonian is given by $\hat{H}=\hat{p}^2/2+\hat{q}^2/2$, the classical Hamilton-Jacobi equation (21) can be solved to give the classical principal function \begin{equation} S_1^{(0)}=\frac{1}{2}(q_1^2+Q_2^2)\cot t-q_1Q_2\csc t. \end{equation} With the boundary condition $e^{iS_1(q_1,Q_2,t=0)/\hbar}=\delta (q_1-Q_2)$, Eq.\ (22) can be solved to give \begin{equation} S_1^{(1)}=\frac{i}{2}\ln \sin t, \end{equation} and $S_1^{(2)}=\cdots =0$. The additive constant has the form $c=\hbar \frac{i}{2}\ln i2\pi \hbar$. The well-ordered generating operator is then written as \begin{equation} \bar{S_1}(\hat{q},\hat{Q},t)=\frac{1}{2}(\hat{q}^2+\hat{Q}^2)\cot t -\hat{q}\hat{Q}\csc t +\hbar \frac{i}{2} \ln i2\pi \hbar \sin t. \end{equation}
The wave function takes the form \begin{equation} \psi ^q(q_1,t)=\int \frac{1}{\sqrt{i2\pi \hbar \sin t}} e^{\frac{i}{2\hbar \sin t} [(q_1^2+Q_2^2)\cos t -2q_1Q_2]}\psi ^q(Q_2,0)dQ_2, \end{equation} and the kernel and the distribution function are given respectively by \begin{equation} \tilde{\kappa} (q_1,p_1,Q_2,P_2,t)=\delta (Q_2-q_1\cos t +p_1\sin t ) \delta (P_2+Q_2\cos t -q_1\csc t), \end{equation} and \begin{equation} F^W(q_1,p_1,t)=F^W(q_1\cos t-p_1\sin t,q_1\sin t+p_1\cos t,0). \end{equation} This equation shows that the Wigner distribution function for the harmonic oscillator rotates clockwise in phase space.
The quantum Hamilton-Jacobi equation for other types of generating operators can be solved by a similar technique. For instance, we can obtain the following solution for the second-type generating operator, \begin{equation} \bar{S_2}(\hat{q},\hat{P},t)=-\frac{1}{2}(\hat{q}^2+\hat{P}^2)\tan t +\hat{q}\hat{P}\sec t+\hbar \frac{i}{2} \ln 2\pi \hbar \cos t. \end{equation}
The solutions of the Heisenberg equations can be obtained from Eqs.\ (2) and (52) (or Eqs. (3) and (56)). In the Heisenberg picture we have \begin{eqnarray} \hat{q}_H(t)&=&\hat{q}_S\cos t+\hat{p}_S\sin t, \\ \hat{p}_H(t)&=&-\hat{q}_S\sin t+\hat{p}_S\cos t. \end{eqnarray}
It should be mentioned that, even though we restricted our discussion in this section only to the case $\hat{A}=\hat{I}$ by imposing the special initial condition, it is very probable that another choice of
$\hat{A}$ satisfying the quantum Hamilton-Jacobi equation happens to be more readily obtainable. In that case, the initial condition that is derived from $e^{iS_1(q_1,Q_2,0)/\hbar}=\langle q_1|\hat{A}|q_2\rangle$ is of course different from that described above. As an example, for the harmonic oscillator, we presented a different solution for $S_1$ in Appendix C where the unitary operator $\hat{A}$ corresponds to the transformation that interchanges the position and momentum operators.
\section{Concluding remarks}
We wish to give some final remarks concerning the quantum Hamilton-Jacobi theory. In this approach, the quantum Hamilton-Jacobi equation takes the place of the time-dependent Schr\"{o}dinger equation for solving dynamical problems, and the quantum Hamilton's principal function $S_1$ that is the solution of the former equation gives the solution of the latter equation through Eq.\ (14). As mentioned in Sec.\ V, $e^{iS_1/\hbar}$ becomes the propagator in position space for the case $\hat{A}=\hat{I}$. To find the propagator, Feynman's path-integral approach divides the time difference between a given initial state and a final state into infinitesimal time intervals, and then lets the quantum generating function for the infinitesimal transformation equal the classical action function plus a proper additive constant that vanishes in the limit $\hbar \rightarrow 0$, and finally takes the sum of the infinitesimal transformations. On the other hand, the present approach seeks the quantum generating function that directly transforms the initial state to the final state. The present formalism gives also the solutions of the Heisenberg equations through the transformation relations which in the Heisenberg picture can be expressed as Eq.\ (23). In conclusion, it is clear that the present approach, which has its origin in Dirac's canonical transformation theory, helps better comprehend the interrelations among the existing different formulations of quantum mechanics.
Finally, one more remark may be worth making as to the extent to which the quantum Hamilton-Jacobi theory can stretch the range of its validity. Even though our work here deals with the unitary transformation to ensure that the new operators become Hermitian, and hence observables, the main idea presented in this paper could be extended so as to include the non-unitary transformation that deals with non-Hermitian operators. The theory would then have the form of the quantum Hamilton-Jacobi equation, but it would be associated with different types of transformations, such as $\hat{U}(t)=\hat{T}(t)\hat{B}$ where $\hat{U}(t)$ and $\hat{B}$ are not unitary. However, it may then be necessary to pay particular attention and care to the completeness of the eigenstates of the new operators $\hat{Q}$ and $\hat{P}$, for the property is crucial to several relations derived and has been used implicitly throughout the paper.
\appendix
\section{Proof of the equivalence of equations (11) and (12)} Eq.\ (12) can be derived from Eq.\ (11) by considering the matrix element of the second term on the right hand side of Eq.\ (11) as follows, \begin{eqnarray}
\langle q_1|i\hbar \hat{U}\frac{\partial \hat{U}^\dagger}{\partial t}| q_2
\rangle &=& -\langle q_1|i\hbar \frac{\partial \hat{U}}{\partial t}
\hat{U}^\dagger |q_2 \rangle , \\
&=&-\int dq_3\langle q_1|i\hbar \frac{\partial \hat{U}}{\partial t}
|q_3\rangle \langle q_3|\hat{U}^\dagger |q_2 \rangle , \\
&=&\int dQ_3\left(-i\hbar \frac{\partial}{\partial t}\langle q_1|\hat{U}|q_3\rangle \right) {_t\langle}Q_3|q_2\rangle , \end{eqnarray} where the identity $\hat{U}\hat{U}^\dagger =\hat{I}$ is used to obtain Eq.\ (A1). Using the definition of the quantum generating function (1), we can obtain \begin{eqnarray}
\langle q_1|i\hbar \hat{U}\frac{\partial \hat{U}^\dagger}{\partial t}| q_2 \rangle &=& \int dQ_3 \frac{\partial F_1(q_1,Q_3)}{\partial t}
\langle q_1|Q_3\rangle _t\hspace{0.7mm} {_t\langle}Q_3|q_2\rangle , \\
&=& \int dQ_3\langle q_1|\frac{\partial \bar{F}_1(\hat{q}, \hat{Q},t)}
{\partial t} |Q_3\rangle _t\hspace{0.7mm} {_t\langle} Q_3|q_2\rangle , \\
&=&\langle q_1|\frac{\partial \bar{F}_1(\hat{q},\hat{Q},t)}
{\partial t}|q_2 \rangle . \end{eqnarray} Since $i\hbar \hat{U}\frac{\partial \hat{U}^\dagger}{\partial t}$ and $\frac{\partial \bar{F}_1(\hat{q},\hat{Q},t)}{\partial t}$ have the same matrix element, we conclude that the two operators are identical.
\section{Derivation of the kernel $\kappa$} In order to find the relation between $F^f$ and $G^f$, we first make use of the completeness of the eigenvectors of $\hat{q}$ in Eq.\ (25) and get \begin{eqnarray} G^f(Q_1,P_1,t)&=&\frac{1}{2\pi ^2\hbar}\int \int \int \int \int
dXdYdQ_2dq_3dq_4\hspace{0.7mm} {_t\langle}Q_2+Y|q_3\rangle \langle q_3|
\hat{\rho} |q_4\rangle \langle q_4|Q_2-Y\rangle _t \nonumber \\ &&\times f(X,2Y/\hbar )e^{iX(Q_2-Q_1)} e^{-i2YP_1/\hbar}. \end{eqnarray} Changing variables with $q_5=(q_3+q_4)/2$ and $y=(q_3-q_4)/2$, and using the relation $\int dy'dq_6\delta (y-y')\delta (q_5-q_6) g(y',q_6)=g(y,q_5)$ where the $\delta$-functions are written in the forms \begin{equation} \delta (y-y')=\frac{1}{\pi \hbar}\int dp_3e^{-2ip_3(y-y')/\hbar}, \end{equation} and \begin{equation} \delta (q_5-q_6)=\frac{1}{2\pi}\int dxe^{ix(q_6-q_5)}, \end{equation} we obtain \begin{eqnarray} G^f(Q_1,P_1,t)&=&\frac{1}{2\pi ^4\hbar ^2}\int \int \int \int \int \int \int \int \int dXdYdQ_2dq_5dydy'dp_3dq_6dx
f(X,2Y/\hbar ){_t\langle}Q_2+Y|q_5+y'\rangle \nonumber \\
&& \times \langle q_6+y|\hat{\rho} |q_6-y\rangle \langle
q_5-y'|Q_2-Y\rangle _t e^{iX(Q_2-Q_1)} e^{-i2YP_1/\hbar} e^{-2ip_3(y-y')/\hbar} e^{ix(q_6-q_5)}. \end{eqnarray} Next, we multiply this equation by \begin{equation} \int \int dx''dy'' \frac{f(x,2y/\hbar )}{f(x'',2y''/\hbar )}\delta (x''-x)\delta (y''-y)=1, \end{equation} where \begin{eqnarray} \delta (x''-x)\delta (y''-y)=\frac{1}{2\pi ^2\hbar} \int \int d\alpha d \beta e^{-i\alpha (x''-x)} e^{-2i\beta (y''-y)/ \hbar}. \end{eqnarray} In the resulting equation, we replace the integrations over $q_5$ and $p_3$, respectively, with those over $q_2=q_5-\alpha$ and $p_2=p_3-\beta$, and then integrate over $\beta$ and $y'$. We then obtain \begin{eqnarray} G^f(Q_1,P_1,t)&=&\int \int dq_2dp_2\bigg[ \frac{1}{2\pi ^3\hbar} \int \int \int \int \int \int dXdYdQ_2dx''dy''d\alpha
{_t\langle} Q_2+Y| q_2+\alpha +y''\rangle \nonumber \\
&& \times \langle q_2+\alpha -y''|Q_2-Y \rangle _t \frac{f(X,2Y/\hbar )}{f(x'',2y''/\hbar )} e^{iX(Q_2-Q_1)} e^{-2iYP_1/\hbar} e^{-i\alpha x''} e^{2ip_2y''/\hbar} \bigg] \nonumber \\ && \times \left[ \frac{1}{2\pi ^2\hbar} \int \int \int dxdydq_6
\langle q_6+y|\hat{\rho} |q_6-y\rangle f(x,2y/\hbar ) e^{ix(q_6-q_2)}e^{-2iyp_2/\hbar} \right] , \end{eqnarray} which leads immediately to Eq.\ (26).
\section{Solutions of the quantum Hamilton-Jacobi equation} In Sec.\ V we considered the case $\hat{A}=\hat{I}$ only for convenience, because, as can be noticed from the two examples of Sec.\ V, this case not only gives a simple relation between the initial wave function $\psi ^q(q_1,0)$ (distribution function $F^f(q_1,p_1,0)$) in $q$-representation and the constant wave function $\psi ^Q(Q_2)$ (distribution function $G^f(Q_2,P_2)$) in $Q$-representation, but also makes it easy to express the new operators $(\hat{Q}_H,\hat{P}_H)$ in the Heisenberg picture in terms of the old operators $(\hat{q}_S,\hat{p}_S)$ in the Schr\"{o}dinger picture. In doing so, we required the solution to satisfy the specific initial condition described in the first part of Sec.\ V. This specialization is of course not of necessity, and if this initial condition is discarded with the unitary condition of the transformation still retained, we have a group of general solutions each member of which corresponds to a specific choice of $\hat{A}$. As an illustration, we present below another possible solution of the quantum Hamilton-Jacobi equation (19) for the harmonic oscillator that belongs to unitary transformations of the type $\hat{U}(t)=\hat{T}(t)\hat{A}$, \begin{equation} \bar{S_1}(\hat{q},\hat{Q},t)=-\frac{1}{2}(\hat{q}^2+\hat{Q}^2)\tan t +\hat{q}\hat{Q}\sec t+\hbar \frac{i}{2} \ln 2\pi \hbar \cos t. \end{equation} Equation (C1) satisfies at initial time $e^{iS_1(q_1,Q_2,0)/\hbar}=\frac{1} {\sqrt{2\pi \hbar}}
e^{iq_1Q_2/\hbar}$, which equals $\langle q_1|\hat{A}|q_2\rangle$, i.e., the matrix element of the unitary operator $\hat{A}$. In this case, the operator $\hat{A}$ corresponds to the exchange transformation which generates the transformation relations \begin{eqnarray} \hat{p}&=&\frac{\partial \bar{S_1}(\hat{q},\hat{Q},0)}{\partial \hat{q}} =\hat{Q}, \\ \hat{P}&=&-\frac{\partial \bar{S_1}(\hat{q},\hat{Q},0)}{\partial \hat{Q}} =-\hat{q}. \end{eqnarray}
\begin{references}
\bibitem{one} H. Goldstein. Classical Mechanics. 2nd ed. Addison-Wesley, Reading, MA. 1981. \bibitem{two} W. Heisenberg. The Physical Principles of the Quantum Theory. Dover, New York. 1930. \bibitem{three} P. A. M. Dirac. The Principles of Quantum Mechanics. 4th ed. Oxford University Press, London. 1958. \bibitem{four} P. Jordan. Z. Phys. {\bf 38}, 513 (1926). \bibitem{five} P. A. M. Dirac. Phys. Z. Sowjetunion, Band 3, Heft 1 (1933); Rev. Mod. Phys. {\bf 17}, 195 (1945). \bibitem{six} J. Schwinger. Phys. Rev. {\bf 82}, 914 (1951); {\bf 91}, 713 (1953). \bibitem{seven} M. Moshinski and C. Quesne. J. Math. Phys. {\bf 12}, 1772 (1971); P. A. Mello and M. Moshinski. J. Math. Phys. {\bf 16}, 2017 (1975). \bibitem{eight} R. A. Leacock and M. J. Padgett. Phys. Rev. Lett. {\bf 50}, 3 (1983); Phys. Rev. D {\bf 28}, 2491 (1983). \bibitem{nine} G. I. Ghandour. Phys. Rev. D {\bf 35}, 1289 (1987). \bibitem{ten} H. Lee and W. S. l'Yi. Phys. Rev. A {\bf 51}, 982 (1995). \bibitem{one2} R. S. Bhalla, A. K. Kapoor, and P. K. Panigrahi. Am. J. Phys. {\bf 65}, 1187 (1997). \bibitem{one1} Y. S. Kim and E. P. Wigner. Am. J. Phys. {\bf 58}, 439 (1990). \bibitem{schiff} L. I. Schiff. Quantum Mechanics. 3rd ed. McGraw-Hill, New York. 1968. p. 269. \bibitem{one5} L. Cohen. J. Math. Phys. {\bf 7}, 781 (1966). \bibitem{one6} H. W. Lee. Phys. Rep. {\bf 259}, 147 (1995). \bibitem{one7} E. Wigner. Phys. Rev. {\bf 40}, 749 (1932). \bibitem{one8} K. Takahashi. Prog. Theoret. Phys. Suppl. {\bf 98}, 109 (1989). \bibitem{one9} G. Garcia-Calder\'on and M. Moshinsky. J. Phys. A {\bf 13}, L185 (1980). \bibitem{one10} T. Curtright, D. Fairlie, and C. Zachos. Phys. Rev. D {\bf 58}, 025002 (1998). \bibitem{two0} R. P. Feynman and A. R. Hibbs. Quantum Mechanics and Path Integrals. McGraw-Hill, New York. 1965.
\end{references}
\end{document} |
\begin{document}
\begin{abstract} We consider a contact manifold with a pseudo-Riemannian metric and define a contact vector field intrinsically associated to this pair of structures. We call this new differential invariant the contact Riemannian curl. On a Riemannian manifold, Killing vector fields are those that annihilate the metric; a Killing $1$-form is obtained from a Killing vector field by lowering indices. We show that the contact Riemannian curl vanishes if the metric is of constant curvature and the contact structure is defined by a Killing $1$-form. We also show that the contact Riemannian curl has a strong similarity with the Schwarzian derivative since it depends only on the projective equivalence class of the metric. For the Laplace-Beltrami operator on a contact manifold, the contact Riemannian curl is proportional to the subsymbol defined in~arXiv:1205.6562. We also show that the contact Riemannian curl vanishes on the (co)tangent bundle over a Riemannian manifold. This implies that the corresponding subsymbol of the Laplace-Beltrami operator is identically zero. \end{abstract}
\keywords{Contact geometry, Riemannian geometry, differential invariants}
\maketitle
\section{Introduction} \label{Intro}
The principal object of this paper is related to the notion of {\it invariant differential operator}, i.e., an operator commuting with the action of the group of diffeomorphisms. The notion of {\it differential invariant} is one of the oldest notions of differential geometry. The best known example is perhaps the curvature in all its avatars. The topic to which the present work belongs was initiated by Veblen~\cite{Veb} who started a systematic study of invariant differential operators on smooth manifolds. The theory was intensively studied in the 80's in the context of Gelfand-Fuchs cohomology; see~\cite{Fu86,GLS} and references therein.
We consider a smooth manifold $M$ equipped simultaneously with a contact structure and a pseudo-Riemannian metric. We present a construction of a contact vector field corresponding to these two structures; we call this vector field the {\it contact Riemannian curl}. Our construction is coordinate free and invariant with respect to the action of the group of contact diffeomorphisms, i.e., the contact Riemannian curl is a differential invariant. Moreover, our goal is to define this differential invariant in a ``most symmetric'' way, so that it is also invariant with respect to natural equivalence relations.
One of the equivalence relations we consider is as follows. Two metrics are called {\it projectively equivalent} (or geodesically equivalent) if they have the same non-parametrized geodesics. i.e., their Levi-Civita connections are projectively equivalent. It turns out that the constructed contact Riemannian curl is obtained as contraction of the metric with a certain tensor field invariant with respect to this equivalence relation. This implies, in particular, that the contact Riemannian curl of the pair (a metric of constant scalar curvature, contact structure defined by a Killing $1$-form) vanishes. Projective invariance makes the notion of contact Riemannian curl quite similar to that of classical {\it Schwarzian derivative} (for various multi-dimensional generalizations of the Schwarzian derivative see~\cite{BO,OT,B,OT1} and references therein). We investigate this relation in more details.
Among the main properties of the contact Riemannian curl that we investigate, there is its relation to the Laplace-Beltrami operator. Differential operators on contact manifolds were studied from the geometric point of view in a recent work \cite{CO12}, where the notion of {\it subsymbol} of a differential operator on a contact manifold was introduced. The subsymbol of a differential operator is a tensor field of degree lower than that of the principal symbol. Note that the subsymbol is not well-defined for an arbitrary manifold, one needs a contact structure to obtain an invariant definition. For a given second order differential operator, the subsymbol is just a contact vector field. In the present paper, we consider the {\it Laplace-Beltrami operator} associated to an arbitrary metric on a contact manifold and calculate its subsymbol. It turns out that this subsymbol is proportional to the contact Riemannian curl.
We also apply our general construction to a particularly interesting example of a manifold that has natural contact and Riemannian structures, namely to the spherical (or projectivized) cotangent bundle $ST^*M$ over a Riemannian manifold $(M,g)$. The manifold $ST^*M$ is equipped with the canonical
lift of the metric $g$. We show that the contact Riemannian curl, and therefore the subsymbol of the Laplace-Beltrami operator, is identically zero in this case. Let us mention that the projectivization of the cotangent bundle over a Riemannian manifold $M$, as well as the sphere bundle $ST^*M$, is an example of a ``real-complex'' manifold whose local invariants were recently introduced and computed in~\cite{BGLS}.
At the end of the paper, we provide several concrete examples of the contact Riemannian curls. For instance, we calculate it for the $3{\rm D}$-ellipsoid equipped with the conformally flat metric introduced in \cite{Tab} and intensively used in~\cite{MT,DV}.
We believe that the differential invariants of a pair (a Riemannian metric, a contact structure) is worth a systematic study.
\section{Contact geometry and tensor fields} \label{CODG}
Contact geometry is an old classical subject, that can be viewed as an odd-dimensional version of symplectic geometry. Let $M$ be a contact manifold and $\dim(M)= 2\ell + 1$, we will always assume that $\ell\geq1$. Unlike a symplectic structure in symplectic geometry, a contact structure on $M$ is defined by a differential $1$-form $\theta$, called a {\it contact form}, determined up to a factor (a function), and such that $d\theta$ is a $2$-form of rank $2\ell$. It is important that a contact form is not intrinsically associated with a contact structure.
A contact diffeomorphism (a contact vector field) is a diffeomorphism (a vector field) preserving the contact structure. It preserves a given contact form conformally, up to a factor. The space of all contact vector fields can be identified with the space of smooth functions, but this correspondence depends on the choice of a contact form; see~\cite{Arn}.
In this section, we recall several standard facts of contact geometry --- those of contact structure and contact vector fields --- using somewhat unconventional notation of~\cite{OT,Ovs} which are among our main references. We show that the contact structure can be also described by a special tensor field, which is a {\it weighted contact form}. Contact vector fields are in one-to-one correspondence with weighted densities of weight $-\frac{1}{\ell+1}$.
\subsection{Weighted densities}
A weighted density is a standard object in differential geometry. In~order to make the definitions intrinsic, we recall here this notion.
Let $M$ be a manifold of dimension $n$. For any $\lambda\in\Bbb R$, we denote by $(\Lambda^n T^*M)^{\otimes\lambda}$ the line bundle of homogeneous complex valued functions of weight $\lambda$ on the determinant bundle $\Lambda^n TM$. The space ${\cal F}_\lambda(M)$ of smooth sections of $(\Lambda^n T^*M)^{\otimes\lambda}$ with complex coefficients is called the space of {\it weighted densities} of weight~$\lambda$, (or $\lambda$-{\it densities} for short).
\begin{ex} {\rm If the manifold $M$ is orientable and if $\omega$ is a volume form with constant coefficients, then $\phi\,\omega^\lambda$, where $\phi\in{}C^\infty(M)$, is a $\lambda$-density. } \end{ex}
The space ${\cal F}_\lambda(M)$ has the structure of a module over the Lie algebra $\mathrm{Vect}(M)$ of all smooth vector fields on $M$. We denote by $\mathrm{Div}$ the divergence operator associated with a volume form $\omega$ on $M$. That is, $L_X(\omega)=\mathrm{Div}(X)\,\omega$. The action of a vector fields reads as follows: \begin{equation} \label{LieEq} L_X(\phi\,\omega^\lambda)= \left(X(\phi)+\lambda\mathrm{Div}(X) \phi \right)\omega^\lambda, \end{equation} for every vector field $X$ and $\phi \in C^\infty(M)$.
\subsection{Contact manifolds}
A smooth manifold $M$ is called {\it contact} if it is equipped with a completely non-integrable distribution $$ \mathcal{D}\subset{}TM $$ of codimension~$1$. The distribution $\mathcal{D}$ is called a {\it contact distribution}; the hyperplane $\mathcal{D}_x\subset{}T_xM$ is called a {\it contact hyperplane} for every point $x\in{}M$. A contact structure on $M$ exists only if $\dim M=2\ell+1>1$.
A usual way to define a contact structure is to chose
a (locally defined) differential $1$-form $\theta$ on~$M$ such that
$\mathcal{D}=\ker\theta$. Such a $1$-form is called a {\it contact form}. The complete non-integrability of the distribution $\mathcal{D}$ is equivalent to the fact that \begin{equation} \label{TheVol} \mathrm{vol}:=\theta\wedge(d\theta)^\ell \end{equation} is a (locally defined) volume form; equivalently, the 2-form $d\theta$ is a non-degenerate on the contact hyperplanes $\mathcal{D}_x$ of $\mathcal{D}$. However, there is no canonical choice of a contact form.
A diffeomorphism $f:M\to{}M$ is a {\it contact diffeomorphism} if $f$ preserves $\mathcal{D}$. If $\theta$ is a contact form corresponding to the contact distribution $\mathcal{D}$ and $f$ is a contact diffeomorphism, then $f$ does not necessarily preserve $\theta$, more precisely, $f^*\theta=F_f\theta$, where $F_f$ is a function.
We refer to \cite{Arn,Bla} for excellent textbooks on contact geometry.
\subsection{The contact tensor}
We will be using the notion of a (generalized) tensor field that was suggested in \cite{BL} and goes back to ideas of I. M. Gelfand. Besides the standard tensor fields, i.e., sections of the bundles\footnote{ Throughout this paper, the tensor product is performed over $C^\infty(M)$.} $(TM)^p\otimes(T^*M)^q$, it is often useful to consider {\it weighted} tensor fields that are sections of the bundles $$ (TM)^p\otimes(T^*M)^q\otimes(\Lambda^n T^*M)^{\otimes\lambda}. $$ A wealth of examples of such generalized tensor fields can be found in \cite{Fu86,OT}.
We are ready to introduce the main notion of this section.
\begin{defi} \label{ConTD} {\rm Given a contact form $\theta$, let the {\it contact tensor field} be \begin{equation} \label{CT} \Theta:=\theta\otimes\mathrm{vol}^{-\frac{1}{\ell+1}}, \end{equation} where $\mathrm{vol}$ is as in Eq. (\ref{TheVol}). } \end{defi}
\begin{prop} \label{IPro} The tensor field $\Theta$ is globally defined on a contact manifold $M$, it is independent of the choice of a contact form, and it is invariant with respect to the contact diffeomorphisms. \end{prop}
\begin{proof} Let $F$ be a non-vanishing function and consider the contact form $F\theta$. The corresponding volume form is $F\theta\wedge\left(d(F\theta)\right)^\ell=F^{\ell+1}\theta\wedge(d\theta)^\ell$. Therefore, the contact tensor fields defined by Eq.~(\ref{CT}),
corresponding to the contact forms $\theta$ and $F\theta$, coincide. Hence, $\Theta$ is globally defined and invariant with respect to contact diffeomorphisms. \end{proof}
A contact structure is intrinsically defined by the corresponding contact tensor.
\begin{ex} \label{Dabex} {\rm Local coordinates $(x^1,\ldots,x^\ell,y^1,\ldots,y^\ell,z)$ on $M$ are often called the {\it Darboux coordinates} if the contact structure can be represented by the 1-form \[ \theta_{\mathrm{Dar}}=dz+\frac{1}{2}\,\sum_{i=1}^\ell \left (x^idy^i-y^idx^i \right ). \] The corresponding volume form is then the standard one: $$ \mathrm{vol}= (-1)^{\frac{\ell(\ell-1)}{2}}\,\ell!\, dx^1\wedge\cdots\wedge{}dx^\ell\wedge{}dy^1\wedge\cdots{}dy^\ell\wedge{}dz. $$ A contact structure has no local invariants, therefore Darboux coordinates always exist in the vicinity of every point; see~\cite{Arn} (and~\cite{GL} for a simple algebraic proof). } \end{ex}
\subsection{Contact vector fields}\label{CoHSect}
A {\it contact vector field} on a contact manifold $M$ is a vector field that preserves the contact distribution. This is usually expressed in terms of contact forms: a vector field $X$ is contact if, for every contact form $\theta$, the Lie derivative $L_X\theta$ is proportional to $\theta$: \begin{equation} \label{DivEq}
L_X\theta = {\textstyle \frac{1}{\ell+1}}\mathop{\rm Div}\nolimits(X) \theta. \end{equation} In terms of the contact tensor (\ref{CT}), we have the following corollary of Proposition~\ref{IPro}. \begin{cor} \label{ContV} A vector field $X$ is contact if and only if it preserves the contact tensor: $$ L_X\Theta=0. $$ \end{cor}
Let ${\mathcal K}(M)$ denote the space of all smooth contact vector fields on $M$. This space has a Lie algebra structure, it is also a module over the group of contact diffeomorphisms. The following observation can be found in~\cite{Ovs,CO12}.
\begin{prop} \label{VecPro} As a module over the group of contact diffeomorphisms, the space ${\mathcal K}(M)$ is isomorphic to the space of weighted densities ${\mathcal F}_{-\frac{1}{\ell+1}}(M)$. \end{prop}
\begin{proof} The space of contact forms is isomorphic to ${\mathcal F}_{\frac{1}{\ell+1}}(M)$. Indeed, this follows from Proposition~\ref{IPro} and from Eq.~(\ref{DivEq}). The statement then follows from the fact that there is a natural $C^\infty(M)$-valued pairing between the spaces of contact vector fields and of contact forms: $
(X,\,\theta)\mapsto\theta(X). $ \end{proof}
\begin{rem} {\rm The above proposition means that, unlike the symplectic geometry,
the notion of contact generating function (or ``contact Hamiltonian function'') should be understood as a weighted density and not as a function. However, in the Darboux coordinates, the correspondence between the elements of ${\mathcal K}(M)$ and ${\mathcal F}_{-\frac{1}{\ell+1}}(M)$ becomes the usual correspondence between contact vector fields and functions (see~\cite{Arn}): $$ X_{\phi\,\omega^{-\frac{1}{\ell+1}}}= \sum_{i=1}^\ell\left( \partial_{x^i}(\phi)\,\partial_{y^i}-\partial_{y^i}(\phi)\,\partial_{x^i} \right) \textstyle +\frac{1}{2}\,\partial_{z}(\phi)\,{\mathcal E} +\left(\phi-\frac{1}{2}\,{\mathcal E}(\phi)\right)\partial_{z}, $$ where $$ {\mathcal E}= \sum_{i=1}^\ell \left( x^i\partial_{x^i}+y^i\partial_{y^i} \right) $$ is the Euler vector field. } \end{rem}
\begin{ex} {\rm If $\dim{M}=3$, contact vector fields are identified with $-\frac{1}{2}$-densities; if $\dim{M}=5$, then ${\mathcal K}(M)\cong{\mathcal F}_{-\frac{1}{3}}(M)$, etc. Note also that, in the one-dimensional case, every vector field is contact, one then has $\mathrm{Vect}(M)\cong{\mathcal F}_{-1}(M)$. } \end{ex}
\subsection{Another definition of weighted densities on contact manifolds}\label{Mitia}
In presence of a contact structure defined by a contact form $\theta$, it is natural to express elements of any rank 1 bundle, for example, weighted densities, in terms of powers of $\theta$: $$ \phi\,\mathrm{vol}^{\frac{\lambda}{\ell+1}}\longleftrightarrow\phi\theta^\lambda, $$ where as above $\phi$ is a smooth function. The notation $\phi\theta^\lambda$ is adopted in many works by physicists (see also~\cite{Ovs90,GLS01}). In this notation, many formulas simplify. For instance, if $X$ is a contact vector field, then the corresponding contact Hamiltonian is $\phi\theta^{-1}$, where the function $\phi$ is simply the evaluation $\phi=\theta(X)$.
\subsection{The Poisson algebra of weighted densities}
The space ${\cal F}(M)=\bigoplus_\lambda{\cal F}_\lambda(M)$ of all weighted densities on a contact manifold $M$ can be endowed with a structure of a Poisson algebra (see~\cite{Arn,OT,Ovs}): $$ \{.,.\}:{\mathcal F}_\lambda(M)\times{\mathcal F}_{\mu}(M)\to{\mathcal F}_{\lambda+\mu+\frac{1}{\ell+1}}(M). $$ The explicit formula in Darboux coordinates is as follows: $$ \left\{\phi\,\omega^\lambda,\psi\,\omega^{\mu}\right\}= \left( \sum_{i=1}^n(\partial_{x^i}\phi\,\partial_{y^i}\psi - \partial_{x^i}\psi\,\partial_{y^i}\phi) +\partial_{z}\phi\left(\mu\psi+{\mathcal E}\psi\right) -\partial_{z}\psi\left(\lambda\phi+{\mathcal E}\psi\right) \right)\omega^{\lambda+\mu+\frac{1}{\ell+1}}. $$
The subspace ${\mathcal F}_{-\frac{1}{\ell+1}}(M)$ is a Lie subalgebra of ${\mathcal F}$ isomorphic to ${\mathcal K}(M)$. The Poisson bracket of $-\frac{1}{\ell+1}$-densities precisely corresponds to the Lie derivative: $$ X_{\left\{\Phi,\Psi\right\}}= L_{X_{\Phi}} \left(\Psi\right), $$ where $\Phi=\phi\,\omega^{-\frac{1}{\ell+1}},\,\Psi=\psi\,\omega^{-\frac{1}{\ell+1}}$.
\subsection{The invariant splitting}
The full space of vector fields $\mathrm{Vect}(M)$ splits into direct sum $$ \mathrm{Vect}(M)= {\mathcal K}(M)\oplus{\mathcal Tan}(M), $$ where ${\mathcal Tan}(M)$ is the space of vector fields tangent to the contact distribution, i.e., $\theta(Y)=0$ for every contact form $\theta$ and every $Y\in{\mathcal Tan}(M)$. Such vector fields are called {\it tangent vector fields}. Unlike ${\mathcal K}(M)$, the space ${\mathcal Tan}(M)$ is not a Lie algebra, but a ${\mathcal K}(M)$-module.
The above splitting is invariant with respect to the group of contact diffeomorphisms. In particular, there is an invariant projection \begin{equation} \label{PiPr} \pi:\mathrm{Vect}(M)\to{\mathcal K}(M), \end{equation} that will be very useful.
\section{The contact Riemannian curl and its properties} \label{Definitions}
In this section, we introduce our main notion, a contact vector field corresponding to a metric and a contact structure. We also study its main properties, such as projective invariance and relation to the multi-dimensional Schwarzian derivative.
\subsection{Covariant derivative}
Let us assume now that $M$ is endowed with a pseudo-Riemannian metric $g$. We denote the Levi-Civita connection on $M$ by $\nabla$, and the Christoffel symbols by $\Gamma_{ij}^k$. The {\it covariant derivative}, also denoted by $\nabla$, is the linear map that can be defined for arbitrary space of tensor fields, ${\mathcal T}(M)$: $$ \nabla:{\mathcal T}(M)\to\Omega^1(M)\otimes{\mathcal T}(M), $$ such that $\nabla(fm)=df\otimes m+f\otimes\nabla(m)$ for any $f\in{}C^\infty(M)$ and $m\in T(M)$. It is written in the form $\nabla(t)=\nabla_i(t)\,dx^i$, and therefore it suffices to define the partial derivatives~$\nabla_i$. Here and below summation over repeated indices (one upper, the other one lower) is understood (Einstein's notation); see~\cite{DNF}.
The covariant derivative of vector fields and differential $1$-forms is given, in local coordinates, by the well-known formulas $$ \nabla_i \left(V^j\partial_j\right)= \left(\partial_iV^j+\Gamma_{ik}^jV^k\right)\partial_j, \qquad \nabla_i \left(\beta_jdx^j\right)= \left(\partial_i\beta_j-\Gamma_{ij}^k\beta_k\right)dx^j, $$ respectively, where $\partial_i=\partial/\partial{}x^i$. The covariant derivative then extended to every tensor fields by Leibniz rule.
For instance, the covariant derivative of weighted densities is defined in local coordinates by the following formula: \[ \nabla_i \left( \phi\,\omega^\lambda \right)= \left(\partial_i \phi-\lambda \Gamma_{ij}^j\phi\right)\omega^\lambda, \] that we will extensively use throughout the paper.
\subsection{The main definition}
Let us introduce the main notion of this paper. Recall that the contact tensor field $\Theta$ was introduced in Definition~\ref{ConTD}.
\begin{defi} \label{MainDef} {\rm (a) For every pseudo-Riemannian metric $g$ on a contact manifold $M$, we define a weighted density of degree $-\frac{1}{\ell+1}$: \begin{equation} \label{maindefi} A_{g,\Theta}:= \left\langle {g},\,\nabla\Theta \right\rangle, \end{equation} in local coordinates, $A_{g,\Theta}:= {g}^{ij}\nabla_i\Theta_j$.
(b) We call the contact vector field $X_{A_{g,\Theta}}$ with contact Hamiltonian $A_{g,\Theta}$ the {\it contact Riemannian curl of $g$}. } \end{defi}
Note that the quantity $A_{g,\Theta}$ is, indeed, a weighted density of degree $-\frac{1}{\ell+1}$, so that it has a meaning of contact Hamiltonian; see Proposition~\ref{VecPro}.
\begin{rem} {\rm The tensor field $\nabla\Theta$ is also a differential invariant (that actually contains even more information than $A_{g,\Theta}$). One can obtain a $-\frac{1}{\ell+1}$-density out of $\nabla\Theta$ by contracting with an arbitrary metric, not necessarily with $g$ itself. } \end{rem}
It will be useful to have an explicit expression for $A_{g,\Theta}$ (and of $\nabla\Theta$) in local coordinates.
\begin{prop} \label{LoP} In local coordinates, such that $\Theta=\theta\otimes\mathrm{vol}^{-\frac{1}{\ell+1}}$, one has \begin{equation} \label{ProCurl}
A_{g,\Theta}=
g^{ij}
\left (
\partial_i \theta_j-
\left (\Gamma^k_{ij}-\frac{1}{2(\ell+1)}
\left(\delta^k_i\Gamma_{jr}^r+\delta^k_j\Gamma_{ir}^r\right )\right )\theta_k\right)
\mathrm{vol}^{-\frac{1}{\ell+1}}, \end{equation} where $\delta^k_i$ is the Kronecker symbol. \end{prop}
\begin{proof} This can be obtained directly from Definition \ref{MainDef} and the expression of the covariant derivative of a weighted density. \end{proof}
\begin{rem} {\rm It follows from the intrinsic definition (\ref{maindefi}) that the local expression (\ref{ProCurl}) is actually invariant with respect to the action of the group of contact diffeomorphisms. The formula (\ref{ProCurl}) remains unchanged for any choice of local coordinates. It is also independent of the choice of the contact form. } \end{rem}
\subsection{Projective invariance of $\nabla\Theta$}
Let us recall a fundamental notion of projectively equivalent connections due to Cartan~\cite{Car}.
A {\it projective connection} is an equivalence class of symmetric affine connections giving the same non-parameterized geodesics. The {\it symbol of a projective connection} is given by the expression \[ \textstyle \Pi_{ij}^k:=\Gamma_{ij}^k-\frac{1}{n+1}\left (\delta_i^k \Gamma_{lj}^l+\delta_j^k \Gamma_{il}^l\right ), \] where $n$ is the dimension; see~\cite{KN}. Note that in our case, $n=2\ell+1$.
The simplest properties of a projective connection are as the following.
\begin{enumerate} \item Two affine connections, $\nabla$ and $\tilde \nabla$, are projectively equivalent if and only if $\Pi_{ij}^k=\tilde\Pi_{ij}^k$. \item Equivalently, $\nabla$ and $\tilde \nabla$ are projectively equivalent if and only if there exists a 1-form $\beta$ such that \begin{equation*} \label{assoc} \tilde \Gamma_{ij}^k=\Gamma_{ij}^k+\delta_{j}^k\,\beta_i+\delta_{i}^k\,\beta_j. \end{equation*} \end{enumerate}
The following statement makes the contact Riemannian curl somewhat similar to the Schwarzian derivative.
\begin{thm} \label{CoCuProj} If $g$ and $\tilde g$ are two metrics whose Levi-Civita connections are projectively equivalent, then $\nabla\Theta=\tilde \nabla\Theta$. \end{thm}
\begin{proof} The coordinate formula for $\nabla\Theta$ can be written as follows: $$ \left(\nabla\Theta\right)_{ij}=
\left (
\partial_i \theta_j- \Pi^k_{ij}\,\theta_k\right) \mathrm{vol}^{-\frac{1}{\ell+1}}, $$ see (\ref{ProCurl}). This expression depends only on the projective class of the Levi-Civita connection and implies projective invariance. \end{proof}
Let $[g]$ denote the class of geodesically equivalent metrics, let $[\nabla]$ denote the corresponding projective connection. The above theorem means that the tensor $\nabla\Theta$ actually depends only on $[g]$ and not on the metric itself.
\begin{rem} {\rm Geodesically equivalent metrics is a very classical subject of Riemannian geometry that goes back to Beltrami, Levi-Civita, Weyl, and Cartan. We refer to the classical book~\cite{Eis} for a survey. The subject is still very active, see~\cite{BKM} and references therein. } \end{rem}
\subsection{Projectively flat connections, metrics of constant curvature and Killing contact forms}
It is now natural to investigate projectively flat case.
A connection $\nabla$ on $M$ is called \textit{projectively flat} if, in a neighborhood of every point, there exist local coordinates, often called {\it adapted coordinates}, such that the geodesics are straight lines in these coordinates. If a connection is projectively flat, then $\Pi_{ij}^k\equiv0$ in any system of adapted coordinates.
Note also that projectively flat connections
admit a (local) action of the group $\mathrm{SL}(n+1,\Bbb R)$,
in other words, adapted coordinates admit linear-fractional
changes.
The classical Beltrami theorem states that {\sl the Levi-Civita connection of a Riemannian metric is projectively flat if and only if the metric has a constant sectional curvature}. This fact allows us to obtain an important consequence of Theorem~\ref{CoCuProj}.
Let us recall the notion of Killing differential forms that goes back to Yano~\cite{Yan}. A $1$-form $\beta=\beta_i(x)dx^i$ is said to be a {\it Killing form} if $$ \nabla_i \beta_j+\nabla_j \beta_i=0. $$ Recall also a more common notion of Killing vector field. A vector field $V=V^i(x)\partial_i$ is said to be a {\it Killing vector field} if $$ L_Vg=0 $$ Every Killing $1$-form can be obtained from a Killing vector field by lowering indices: $\beta=\langle g,V\rangle$; i.e., $\beta_i=g_{ij}V^j$ in local coordinates.
\begin{cor} \label{CoCu} If $g$ is a metric of constant sectional curvature and if the contact structure is defined by a contact $1$-form $\theta$ which is a Killing form with respect to a metric from the projective class $[g]$, then $ A_{g,\Theta}=0$. \end{cor}
\begin{proof} Since the Levi-Civita connection corresponding to $g$ is projectively flat, there exist local coordinates for which $\Pi_{ij}^k\equiv0$, and therefore $$ A_{g,\Theta}=g^{ij}\partial_i \theta_j. $$ If, furthermore, $ \partial_i \theta_j+\partial_j \theta_i=0 $ for all $i,j$, then $A_{g,\Theta}$ vanishes identically since the tensor $g^{ij}$ is symmetric. The equation $\partial_i \theta_j+\partial_j \theta_i=0$ means that~$\theta$ is a Killing form with respect to the flat metric which is projectively equivalent to $g$.
The corollary then follows from Theorem \ref{CoCuProj}. \end{proof}
\begin{ex} {\rm The Darboux form in Example \ref{Dabex} is a Killing form with respect to the flat metric. Note that in other works, especially in those on analytical mechanics, another local normal form of the contact form is often used: $dz+\sum_{1\leq{}i\leq{}\ell}x^idy^i$. (Over fields of characteristic $2$, only this latter form can be used, see \cite{Leb}.) However, this is not a Killing form with respect to the flat metric. } \end{ex}
\subsection{Contact equivariance}
Consider the action of the group of contact diffeomorphisms. It immediately follows from the intrinsic (i.e., invariant) definition (\ref{maindefi}) of $A_{g,\Theta}$ of that the map $g\mapsto{}A_{g,\Theta}$ from the space of metrics to that of $-\frac{1}{\ell+1}$-densities commutes with this action: \begin{equation} \label{CoCAct} A_{f^*g, \Theta}= f^*\left(A_{g,\Theta}\right). \end{equation} From this fact and Corollary~\ref{CoCu}, we deduce the following statement.
\begin{cor} \label{SecCor} If a metric $\tilde g $ is contactomorphic to a metric $g $ of constant sectional curvature and if the contact structure is defined by a contact $1$-form $\theta$ which is a Killing form with respect to
$g$, then $A_{\tilde g, \Theta}=0$. \end{cor}
\subsection{Action of the full group of diffeomorphisms}
Let us consider the action of the group of all diffeomorphisms. It turns out that this action is related to a quite remarkable $1$-cocycle.
Recall that the space of connections is an affine space associated with the space of $(2,1)$-tensor fields, i.e., given two connections, $\nabla$ and $\tilde\nabla$, the difference $\nabla-\tilde\nabla$ is a well-defined $(2,1)$-tensor field. This allows one to define a $1$-cocycle on the group of all diffeomorphisms. If $f$ is an arbitrary, not necessarily contact, diffeomorphism, we set: $$ C(f):=f^*\nabla-\nabla, $$ where $\nabla$ is an arbitrary fixed connection, choice of which changes $C$ by a coboundary\footnote{ Note also that the cocycle $C$ provides a universal way to construct representatives of non-trivial classes of the Gelfand-Fuchs cohomology; see~\cite{Gel}.}.
Let $\nabla$ and $\tilde\nabla$ be two connections on~$M.$ The difference of the projective equivalence classes $[\nabla]-[\tilde\nabla]$ can be understood as a traceless $(2,1)$-tensor field. Therefore, a projective connection on $M$ leads to the following $1$-{\it cocycle} on the group of all diffeomorphisms: $$ {\mathfrak T}(f)=f^*[\nabla]-[\nabla] $$ which vanishes on (locally) projective diffeomorphisms. In local coordinates, $$ {\mathfrak T}(f)^k_{ij}:= f^*\Pi_{ij}^k-\Pi_{ij}^k, $$ where $\Pi_{ij}^k$ are the projective Christoffel symbols\footnote{ The $1$-cocycle ${\mathfrak T}$ is often considered as a higher-dimensional analog of the Schwarzian derivative; see~\cite{OT}. If~$\nabla$ is projectively flat, then the group $\mathrm{SL}(n+1,\Bbb R)$ of (local) symmetries of $[\nabla]$ is precisely the kernel of ${\mathfrak T}$.}.
\begin{prop} \label{CoCActThm} If $f:M\to{}M$ is an arbitrary diffeomorphism, then \begin{equation} \label{CoCActArb} f^*\left(A_{g,\Theta}\right)- A_{f^*g, \Theta}=
f^* \left\langle g, \nabla\Theta\right \rangle
-\left\langle f^*g,
\nabla\Theta\right \rangle +\left\langle f^*g\otimes \Theta, {\mathfrak T}(f)\right\rangle. \end{equation} \end{prop}
\begin{proof} Let us first clarify the notation. Since ${\mathfrak T}(f)$ is a $(2,1)$-tensor field, the pairing $\left\langle g\otimes\Theta,\,{\mathfrak T}(f)\right\rangle$ is well-defined. Furthermore, taking into account the weight of the contact tensor $\Theta$, it follows that $\left\langle g\otimes\Theta,\,{\mathfrak T}(f)\right\rangle$ is a weighted density of weight~$-\frac{1}{\ell+1}$.
In local coordinates and using Proposition \ref{LoP}, we have \[ \begin{array}{lcl} A_{f^*g,\Theta}&=&(f^* g)^{ij}\left (\partial_i \theta_j-f^*\Pi^k_{ij}\, \theta_k \right )\\[2mm]
&=& (f^* g)^{ij}\left (\partial_i \theta_j-(f^*\Pi^k_{ij}-\Pi^k_{ij})\, \theta_k \right )- (f^* g)^{ij}\Pi^k_{ij}\theta_k\\[2mm] &= & (f^* g)^{ij}\left (\partial_i \theta_j-{\mathfrak T}(f)^k_{ij}\, \theta_k \right )- (f^* g)^{ij}\Pi^k_{ij}\theta_k\\[2mm] &=& (f^* g)^{ij}\left (\partial_i \theta_j-\Pi^k_{ij}\theta_k \right )-(f^* g)^{ij}\,{\mathfrak T}(f)^k_{ij}\, \theta_k\\[2mm] &=& (f^* g)^{ij}\nabla_i (\Theta_j)-(f^* g)^{ij}\,{\mathfrak T}(f)^k_{ij}\, \theta_k\\[2mm] &=& \left\langle f^*g,
\nabla \left ( \Theta\right )\right \rangle
-\left\langle f^*g\otimes\Theta,\, {\mathfrak T}(f)\right\rangle. \end{array} \] It remains to notice that $f^*\left(A_{g,\Theta}\right)=f^* \left\langle g, \nabla \left ( \Theta \right )\right \rangle$. Proposition~\ref{CoCActThm} is proved. \end{proof}
\section{The subsymbol of the Laplace-Beltrami operator}
In this section, we explain the relation of the Riemannian curl to the classical Laplace-Beltrami operator. Let us mention that study of differential operators on contact manifolds is a classical subject; see a recent work~\cite{vE10} and references therein.
\subsection{Differential operators and diffeomorphism action}
Let $M$ be an arbitrary smooth mani\-fold and ${\cal D}_{\lambda, \mu}(M)$ be the space of linear differential operators acting on the space of weighted densities: $$ T:{\cal F}_\lambda(M)\to{\cal F}_\mu(M). $$ The space ${\cal D}_{\lambda, \mu}(M)$ is naturally a module over the group of diffeomorphisms, the module structure being dependent of the weights $\lambda$ and $\mu$. For $k \in \Bbb N$, let ${\cal D}^k_{\lambda, \mu}(M)$ be the space of linear differential operators of order~$\le k$. The spaces ${\cal D}^k_{\lambda, \mu}(M)$ define a filtration on ${\cal D}_{\lambda, \mu}(M)$ invariant with respect to the group of diffeomorphisms.
Recall the classical notion of {\it symbol} (or the {\it principal symbol}) of a differential operator of order~$k$. It is defined as the image of the projection $$ \sigma:{\cal D}_{\lambda, \mu}(M)\to{\cal D}^k_{\lambda, \mu}(M)/{\cal D}^{k-1}_{\lambda, \mu}(M). $$ Observe that, in the particular case $\lambda=\mu$, the quotient space ${\cal D}^k_{\lambda, \lambda}(M)/{\cal D}^{k-1}_{\lambda, \lambda}(M)$ can be identified with the space of symmetric contravariant tensor fields of degree $k$ on $M$.
We will be especially interested in the space ${\cal D}^2_{\lambda, \lambda}(M)$ of $2$-nd order operators acting on $\lambda$-densities; a systematic study of this space viewed as a module over the group of diffeomorphisms was initiated in~\cite{DO1}.
\subsection{The subsymbol of a second order differential operator}
In~\cite{CO12}, the space of differential operators on a contact manifold was studied as a module over the group of contact diffeomorphisms. It was proved that there exists a notion of {\it subsymbol} which is a tensor field of degree lower than that of the principal symbol.
For a $2$-nd order differential operator, the subsymbol is just a contact vector field. More precisely, for every $\lambda$, there exists a linear map (which is unique up to a constant factor) $$ {\mathrm s}\sigma: {\mathcal D}^2_{\lambda,\lambda}(M) \rightarrow {\mathcal K}(M), $$ invariant with respect to the action of the group of contact diffeomorphisms. The image ${\mathrm s}\sigma(T)$ was called the {\it subsymbol} of the operator $T$. We will need the explicit formula for the subsymbol of a given second order differential operator.
If $M$ is a contact manifold, then every operator $T\in{\cal D}^2_{\lambda, \lambda}(M)$ can be written (in many different ways) in the form: \begin{equation} \label{Rep} T=L_{X_{\phi_1}} \circ L_{X_{\phi_2}}+L_{X_{\phi_3}}\circ L_{Y_1}+L_{Y_2}\circ L_{Y_3} + L_{X_{\phi_4}}+L_{Y_4}+F, \end{equation} where each $Y_i$ is a vector field tangent to the contact distribution, $X_\phi$ is the contact vector field with the contact Hamiltonian $\phi\in{\mathcal F}_{-\frac{1}{\ell+1}}(M)$, the Lie derivative $L$ is defined by Eq.~(\ref{LieEq}), and~$F$ denotes the operator of multiplication by a function.
The explicit expression for the subsymbol of differential operator~(\ref{Rep}) is as follows (see~\cite{CO12}): \begin{equation} \label{vfields} {\mathrm s}\sigma(T)=\textstyle {\textstyle\frac{1}{2}}\bigl[X_{\phi_1},X_{\phi_2}\bigr]- \bigl(\frac{\ell+1}{\ell+2}\bigr)\bigl(\lambda-{\textstyle\frac{1}{2}}\bigr)X_{L_{Y_1}(\phi_3)}+ {\textstyle\frac{1}{2}}\pi\bigl[Y_2,Y_3\bigr]+X_{\phi_4}, \end{equation} where $L_{Y}(\phi)$ denotes the Lie derivative of a $-\frac{1}{\ell+1}$-density $\phi$ along the vector field $Y$, and $\pi:\mathrm{Vect}(M)\to{\mathcal K}(M)$ is defined in (\ref{PiPr}).
\begin{rem} {\rm Although it seems almost impossible, the map ${\mathrm s}\sigma$ defined by~(\ref{vfields}) is well-defined. In other words, it is independent of the choice of the vector fields in the representation (\ref{Rep}) of the operator $T$. This can be checked directly by rewriting it in local coordinates, see formula~(\ref{ExPSS}) below. Since the expression~(\ref{vfields}) is written using invariant terms, it commutes with the action of contact diffeomorphisms. Note also that the existence of such a map is indigenous to contact geometry. There is no similar map commuting with the full group of diffeomorphisms, except for the principal symbol. } \end{rem}
\subsection{The Laplace-Beltrami operator on the space of weighted densities}
The classical Laplace-Beltrami operator acting on the space of smooth functions is defined as follows $$ \Delta_g(f)=d^*df. $$ This operator is completely determined by the metric ${g}$.
We will go to a more general framework and consider the generalized Laplace-Beltrami operator acting on the space of weighted densities: $$ \Delta_g^\lambda: {\mathcal F}_\lambda(M)\to{\mathcal F}_\lambda(M). $$ The explicit formula of this operator
is as follows: \[ \Delta_g^\lambda(\phi\,\omega^\lambda)= \left( {g}^{ij}\nabla_i\nabla_j(\phi)+ \frac{n^2\lambda(\lambda-1)}{(n-1)(n+2)}R\phi \right)\omega^\lambda, \] where $R$ is the scalar curvature (see~\cite{DO}, Proposition 5.2).
\subsection{Calculating the subsymbol of the Laplace-Beltrami operator}
Recall that $M$ is a contact manifold and $n=2\ell+1$. It turns out that the contact Riemannian curl of a given metric $g$ is proportional to the subsymbol of the Laplace-Beltrami operator associated with $g$. This property can be considered as an equivalent definition of the contact Riemannian curl.
\begin{thm} One has \begin{equation} \label{ProPEq} \textstyle \mathrm{s}\sigma(\Delta^\lambda_g)= \left(\frac{\ell+1}{\ell+2}\right) \left(2\lambda-1\right)X_{A_{g,\Theta}}. \end{equation} \end{thm}
\begin{proof} The proof is essentially a direct computation.
Let us choose local Darboux coordinates. Every second order differential operator can be written in these coordinates as: $$ \begin{array}{rcl} T&=&T_{2,0,0}\,\partial_z^2+T_{1,i,0}\,\partial_z\partial_{x_i} +T_{1,0,i}\,\partial_z\partial_{y_i}+T_{0,ij,0}\,\partial_{x_i}\partial_{x_j}+ T_{0,i,j}\,\partial_{x_i}\partial_{y_j}+T_{0,0,ij}\,\partial_{y_i}\partial_{y_j}\\[4pt] &&+T_{1,0,0}\,\partial_z+T_{0,i,0}\,\partial_{x_i}+T_{0,0,i}\,\partial_{y_i}+T_{0,0,0}. \end{array} $$ The coordinate formula of the subsymbol was calculated in \cite{CO12}: \begin{equation} \label{ExPSS} \begin{array}{rcl} \mathrm{s}\sigma(T) &=& \frac{1+2\lambda(\ell+1)}{\ell+2} \Bigl( \partial_z(T_{2,0,0}-{\textstyle\frac{1}{2}}{}y_iT_{1,i,0}+{\textstyle\frac{1}{2}}{}x_iT_{1,0,i})\\[6pt] &&\hskip1.6cm +\partial_{x_i}(T_{1,i,0}-{\textstyle\frac{1}{2}}{}y_jT_{0,ij,0}+{\textstyle\frac{1}{2}}{}x_jT_{0,i,j})\\[6pt] &&\hskip1.6cm +\partial_{y_i}(T_{1,0,i}+{\textstyle\frac{1}{2}}{}x_jT_{0,0,ij}-{\textstyle\frac{1}{2}}{}y_jT_{0,j,i})\Bigr)\\[6pt] && +T_{1,0,0}-{\textstyle\frac{1}{2}}{}y_iT_{0,i,0}+{\textstyle\frac{1}{2}}{}x_iT_{0,0,i}. \end{array} \end{equation} One can check that this is exactly the same formula as (\ref{vfields}).
The expression of the generalized Laplace-Beltrami operator $\Delta^\lambda$ in local coordinates was calculated in~\cite{DO}, the result is: \[ \Delta^\lambda_g= {g}^{ij}\partial_i\partial_j-({g}^{jk}\Gamma^i_{jk}+ 2\lambda {g}^{ij}\Gamma^k_{jk})\partial_i+\mathrm{(0-th\; order \; coefficients)}. \]
Let us combine the above two formulas. We obtain $\mathrm{s}\sigma(\Delta^\lambda_g)=X_{\phi}$, where $\phi$ is a weighted density of the form \begin{equation} \label{sgama} \textstyle \phi= \left(\left(1-\frac{1+2\lambda(\ell+1)}{\ell+2}\right) {g}^{jk}\Gamma^t_{jk}\theta_t+ \left(2\lambda-\frac{1+2\lambda(\ell+1)}{\ell+2}\right) \Gamma^j_{ij}{g}^{it}\theta_t\right)
\mathrm{vol}^{-\frac{1}{\ell+1}}, \end{equation} and $X_\phi$ is the corresponding contact vector field.
Finally, taking into account the fact that ${g}^{ij}\partial_i(\theta_j)=0$, for the Darboux form $\theta$,
the expression (\ref{sgama}), after collecting the terms, coincides with
$\left(\frac{\ell+1}{\ell+2}\right) \left(2\lambda-1\right)A_{g,\Theta}$. \end{proof}
\begin{cor} For a generic metric, $\mathrm{s}\sigma(\Delta^\lambda_g)=0$ if and only if $\lambda={\frac{1}{2}}$. \end{cor}
\begin{rem} {\rm In differential geometry it is known that the space of half-densities and the space of differential operators ${\mathcal D}_{\frac{1}{2},\frac{1}{2}}(M)$ acting on them play a very special role. In our context, the space of half-densities appears naturally. } \end{rem}
\section{Cotangent lift and the geodesic spray} \label{SbundleSec}
In this section, we calculate the contact Riemannian curl on the unit sphere bundle $STM$ over a Riemannian manifold $(M,{g})$. The manifold $STM$ is a classical example of contact manifold, and, furthermore, it is equipped with the canonical lift of the metric. We prove that the contact Riemannian curl vanishes in this case.
Recall that the classical {\it geodesic spray} is the Hamiltonian vector field on $TM$ with Hamiltonian $H(x,y)={g}_{ij}(x)\,y^iy^j,$ where $y^i$ are coordinates on the fibers; the restriction of this vector field to $STM$ is an intrinsically defined contact vector field. It is not reasonable expect existence of another, independent, invariant contact vector field in this case.
\subsection{Statement of the main result}
The Riemannian metric ${g}$ on $M$ has a canonical lift to~$STM$ that will be denoted by $\bar g$. The main result of this section is as follows.
\begin{thm} \label{BigThm} The contact Riemannian curl on $(STM,\bar g)$ is identically zero. \end{thm}
In order to prove this theorem, we will need explicit formulas for the contact structure and the canonical Riemannian metric on $STM$.
\subsection{The coordinates on $STM$}
Let $(M,{g})$ be any Riemannian manifold of dimension $n$. The Riemannian geometry of the sphere bundle $STM$ was studied in~\cite{Taha}, we will be using the notation of that work.
Denote by $(x^1,\ldots,x^n)$ a local coordinate system in $M$ and $(y^1,\ldots,y^n)$ the Cartesian coordinates in the tangent space $T_xM$ at the point $x$ in $M$. The coordinates $(x,y)$ are local coordinates on the tangent bundle on $TM$. The unit sphere bundle $STM$ is a hypersurface of the tangent bundle $T(M)$, singled out as the level surface of the Hamiltonian of the geodesic spray $$ H(x,y)=1 $$ at every point.
\subsection{The contact structure of the sphere bundle $STM$}
The sphere bundle $STM$ is represented by parametric equations: \[ x^h=x^h,\quad x^{\bar h}=y^h=y^h(x^i,u^{\kappa}), \] where $u^{\kappa}$ are local coordinates on the sphere.\footnote{ Following \cite{Taha}, we will adopt the following index gymnastics. Capital Latin letters $A,B,\ldots$ run $1$ to $2n$. Small latin letters $i,j,\ldots$ run $1$ to $n$. Barred Latin indices $\bar i, \bar j,\ldots$ run $n+1$ to $2n$. Some of the Greek letters $\alpha, \beta, \ldots$ run $1$ to $2n-1$. Some other Greek letters $\kappa, \lambda, \ldots$ run $n+1$ to $2n-1$. } The tangent vectors $B^{A}_\alpha=\frac{\partial x^A}{\partial u^\alpha}$ of $STM$ in $T(M)$ are given by \begin{equation} \label{BComp} \begin{array}{lcllcl} B^{h}_i&=&\delta^h_i,& B^{h}_\lambda&=&0,\\[2mm] B^{\bar h}_i&=&\partial_i y^h,& B^{\bar h}_\lambda&=&\partial_\lambda y^h. \end{array} \end{equation} The square matrix $\left ( \begin{array}{c} B^{A}_\alpha\\[2mm]
C^A \end{array} \right )$, where $C^i=0$ and $C^{\bar i}=y^i$, is invertible at each point $x$ in $M$. Its inverse will be the matrix $(B^\alpha_{A}, C_A)$, given by the equations: \begin{equation} \label{BInvComp} \begin{array}{lcllcl} B^{h}_{i}&=&\delta^h_i,& B^{h}_{\bar i}&=&0,\\[2mm] B^{\kappa}_{i}&=&-B^{\bar h}_i B^\kappa_{\bar h},& B^{\kappa}_{\bar i},&& \end{array} \end{equation} and $C_A=\left ( \begin{array}{c} C_i\\[2mm] C_{\bar i} \end{array} \right )$, where $C_{\bar i}={g}_{ih}y^h$ and $C_i=\Gamma^h_{rs} {g}_{hi}\,y^ry^s$. The next formulas can be deduced from Eqs. (\ref{BComp}), (\ref{BInvComp}), and are useful for what follows \[ \begin{array}{lcllcllcl} B^{\bar h}_{\lambda} B^{\kappa}_{\bar h}&=& \delta^\kappa_\lambda,& y^h B^{\kappa}_{\bar h}&=& 0,&B^{\bar h}_{\lambda} B^{\lambda}_{\bar i}+y^h C_{\bar i}&=& \delta^h_i,\\[2mm] B^{\bar h}_{\lambda} C_{\bar h}&=&0,& y^h C_{\bar h},&=&1.&&& \end{array} \] The Riemannian metric indentifies the tangent bundle $T(M)$ and the cotangent bundle $T^*(M),$ and hence induces a 1-form $\theta$ on $T(M)$, called the {\it Liouville form}, which in local coordinates reads as follows: \[ \theta={g}_{ij}y^j \, dx^i, \] Denote by $\bar \theta$ the restriction of the 1-form $\theta$ to the sphere bundle $STM$. It is as follows: \[ \bar \theta_\alpha=\theta_A B^{ A}_\alpha. \] Eq. (\ref{BComp}) imply that $\bar \theta_i={g}_{ij}y^j$ and $\bar \theta_\kappa=0$. \begin{lemma} The form $\bar \theta$ defines a contact structure on $STM$. The volume form associated with it reads (up to a factor) as:
\[ \Omega\; dx^1\wedge \cdots \wedge dx^n\wedge du^{n+1}\wedge...\wedge du^{2n-1}, \] where $ \Omega=\mathrm{det }(B^{A}_\alpha,C^{A})\, \mathrm{det } ({g}_{ij}). $
\end{lemma} \begin{proof} This is well known, see~\cite{Taha}, and can also be checked by a direct computation. \end{proof}
\subsection{The Riemannian metric on $STM$}
The Riemannian metric ${g}$ on $M$ can be extended to a Riemannian metric $\bar{{g}}$ on the sphere bundle $STM$. Explicitly, $\bar{{g}}$ is given by (cf. \cite{Taha}): \[ \begin{array}{lcl} \bar{{g}}_{ji}&=&{{g}}_{ji}+{{g}}_{ts}(\nabla_j y^t)(\nabla_i y^s),\\[2mm] \bar {{g}}_{\mu i}&=&{{g}}_{ts}(\partial_\mu y^t)(\nabla_iy^s),\\[2mm] \bar {{g}}_{\mu \lambda}&=&{{g}}_{ji}(\partial_\mu y^j)(\partial_\lambda y^i). \end{array} \] The inverse of $\bar{{g}}$ is given by \[ \begin{array}{lcl} \bar {{g}}^{ji}&=&{{g}}^{ji},\\[2mm] \bar {{g}}^{\lambda h}&=&-{{g}}^{hl}(\nabla_ly^i) B^\lambda_{\bar i},\\[2mm] \bar {{g}}^{\lambda \kappa}&=&\left ({{g}}^{ih}+ {{g}}^{ts}(\nabla_t y^i)(\nabla_s y^h) \right ) B^\lambda_{\bar i}B^\kappa_{\bar h}. \end{array} \] The Christoffel symbols associated with this metric are given by \[ \begin{array}{ccl} \bar \Gamma^h_{ji}&=&\Gamma^h_{ij}+\frac{1}{2}\left ( R_{r sj}^{ h}y^r\nabla_i y^s+R_{r si}^{h}y^r\nabla_j y^s \right ),\\[2mm] \bar \Gamma^h_{\mu i}&=&\frac{1}{2}R_{r si}^{ h}y^r B_\mu^{\bar s},\\[2mm] \bar \Gamma^h_{\mu \lambda}&=&0,\\[2mm] \bar \Gamma^\kappa_{ji}&=&\left( \nabla_j \nabla_i y^h+\frac{1}{2}R_{r ji}^{h}y^r- \frac{1}{2} R_{r ij}^{ h}y^r - \frac{1}{2} \left ( R_{r sj}^{ l}y^r\nabla_i y^s+ \frac{1}{2} R_{r si}^{l }y^r\nabla_j y^s\right )\nabla_l y^h \right )B^\kappa_{\bar h},\\[2mm] \bar \Gamma^\kappa_{\mu j}&=&\left (\partial_\mu \nabla_i y^h-\frac{1}{2} R_{r si}^{ l}y^r B_\mu^{\bar s} \nabla_l y^h \right )B^\kappa_{\bar h},\\[2mm] \bar \Gamma^\kappa_{\mu \lambda}&=&(\partial_\mu \partial_\lambda y^h)B^\kappa_{\bar h}. \end{array} \]
\subsection{Proof of Theorem \ref{BigThm}}
We are ready to prove the main result of this section.
\begin{lemma} \label{lemma1} We have \begin{equation} \label{CalcEq} \begin{array}{lcl} y^i\partial_{i}(\Omega)&=&- y^i B_\lambda^{\bar h}\partial_i(B^\lambda_{\bar h})\, \Omega+ 2y^i\Gamma_{i}\, \Omega-y^iy^hy^mg_{hm}\Gamma_{ir}^h\, \Omega,\\[2mm] y^l (\nabla_l y^i)B^\lambda_{\bar i}\, \partial_{\lambda}(\Omega)&=& -y^l (\nabla_l y^i) \partial_\lambda(B^\lambda_{\bar i})\, \Omega,\\[2mm] y^l (\nabla_l y^k)(\partial_\mu \partial_\lambda y^j)\,B^\lambda_{\bar k} B^\mu_{\bar j} &=&-y^l (\nabla_l y^k)\partial_\lambda (B^\lambda_{\bar k}),\\[2mm] \partial_\lambda(\nabla_l y^h) B^\lambda_{\bar h}y^l&=& -(\partial_l B^\lambda_{\bar h}) B^{\bar h}_\lambda y^l+ \Gamma^i_{li} \,y^l-\Gamma^h_{rs}\,{g}_{ih}y^iy^ry^s. \end{array} \end{equation} \end{lemma} \begin{proof} The first and the second lines of (\ref{CalcEq}) follow from the fact that \begin{eqnarray*} \label{F1} B^\lambda_{\bar k} \partial_\mu(B_\lambda^{\bar j}) &=& -\partial_\mu(B^\lambda_{\bar k}) B_\lambda^{ j}-\partial_\mu(y^j y^m){g}_{mk},\\[4pt] \label{F2} y^i\partial_i B_\lambda^{\;\; \bar j} &=& -y^iB^{\bar h}_{\lambda}B_\kappa^{\bar j}\partial_i(B^\kappa_{\bar h})-y^iy^jy^mg_{hm}\Gamma_{il}^h\, B_\lambda^{\bar l}, \end{eqnarray*} and the property of the determinant. The third line of (\ref{CalcEq}) follows from the fact that $$ B_\lambda^{\bar h}B^\lambda_{\bar i}+y^h C_{\bar i}=\delta_i^h $$ and applying to it the partial derivative $\partial_\mu$. The fourth line of (\ref{CalcEq}) follows when we substitute the covariant derivative $\nabla_l y^h=\partial_l y^h+\Gamma_{li}^h\,y^i$ and use the third equation. \end{proof}
By definition, \[ A_{\bar {g},\Theta}=\bar {{g}}^{ih}\bar \nabla_i(\theta_h \Omega^{-\frac{1}{n}})+\bar {{g}}^{\lambda h}\bar \nabla_\lambda (\theta_h \Omega^{-\frac{1}{n}})+\bar {{g}}^{h \lambda}\bar \nabla_h (\theta_\lambda \Omega^{-\frac{1}{n}})+\bar {{g}}^{\lambda \kappa}\bar \nabla_\lambda (\theta_\kappa \Omega^{-\frac{1}{n}}). \] The last two summands vanish because $\theta_\kappa=0$. Let us compute the first two summands seperately. Applying the covariant derivative $\bar \nabla$, we get \[ \begin{array}{lcl} \bar {{g}}^{ih}\bar \nabla_i(\theta_h \Omega^{-\frac{1}{n}})&=&\partial_i y^i \; \Omega^{-\frac{1}{n}}+y^i\partial_i ( \Omega^{-\frac{1}{n}})-R_{q r s i} y^qy^r (\nabla_h y^s) {{g}}^{ih}\; \Omega^{-\frac{1}{n}}\\[2mm] &&+(1+\frac{1}{n})\Gamma_{ij}^jy^i\; \Omega^{-\frac{1}{n}}+\frac{1}{n} \left (\frac{1}{2} R_{r sh}^{ h} y^ry^l (\nabla_l y^s)+ \partial_\lambda(\nabla_l y^h)B^\lambda_{\bar h}y^l\right )\; \Omega^{-\frac{1}{n}}. \end{array} \] Similarly, \[ \begin{array}{lcl} \bar {{g}}^{\lambda h}\bar \nabla_\lambda (\theta_h \Omega^{-\frac{1}{n}}) &=& -\partial_i y^i \; \Omega^{-\frac{1}{n}}-\Gamma_{ij}^jy^i \;\Omega^{-\frac{1}{n}}-(\nabla_i y^k) B^\lambda_{\;\; \bar k}y^i\partial_\lambda(\Omega^{-\frac{1}{n}})\\[2mm] &&+\frac{1}{2}R_{rs k h}y^ry^s{{g}}^{hl}(\nabla_l y^k) \; \Omega^{-\frac{1}{n}}- \frac{1}{2n} (\nabla_i y^k)y^i R_{r k h}^{ \bar h} y^r\; \Omega^{-\frac{1}{n}}\\[2mm] &&+ -\frac{1}{n} y^l (\nabla_l y^k)(\partial_\lambda \partial_\mu y^j)B^\lambda_{\bar k}B^\mu_{\bar j}\;\Omega^{-\frac{1}{n}}. \end{array} \] By collecting the terms and using Lemma \ref{lemma1}, we finally obtain: $$ A_{\bar{g},\Theta}= -\frac{1}{2}\left( R_{ilsj}\,y^iy^l(\nabla_h y^s)\,{{g}}^{jh} \right) \mathrm{vol}^{-\frac{1}{n}}\equiv0, $$ since the curvature tensor $R_{ilsj}$ is antisymmetric in two first indices.
Theorem \ref{BigThm} is proved.
\section{Examples} \label{Sphere}
We finish the paper with concrete examples of Riemannian curl for the $3$-dimensional sphere (with two natural metrics) and the $3$-dimensional ellipsoid with the standard metric.
\subsection{The sphere $S^3$}
Consider the sphere $S^3$ in the standard symplectic space $\mathbb R^4$. It is endowed with the natural contact structure that can be defined by the contact form \[ \theta=dz+xdy-ydx, \] where $x,y$ and $z$ are affine coordinates on $S^3$. More precisely, if $p_1,p_2,q^1,q^2$ are symplectic Darboux coordinates on $\mathbb R^4$, then $x=\frac{p_1}{q^2},\,y=\frac{q^1}{q^2},\,z=-\frac{p_2}{q^2}$.
The restriction of the Euclidean metric to the sphere $S^3$ takes the following form: \[ \begin{array}{rcl} {g}_{S^3}&=& \displaystyle F\left(\left( y^2+z^2+1\right) dx^2+(x^2+z^2+1)dy^2+(x^2+y^2+1)dz^2\right . \\[10pt] &&\displaystyle \left . -2\, x y\, dx dy-2\,xz \, dx dz-2\, yz \,dydz \right ), \end{array} \] where $F= \left(\frac{1}{(x^2+y^2+z^2+1 )}\right)^{2}$.
Let us also consider another, conformally equivalent, metric on~$S^3$: \[ \tilde {g}_{S^3}:= \left (\frac{x^2+y^2+z^2+1}{\frac{x^2}{a}+\frac{y^2}{b}+\frac{z^2}{c}+1}\right ){g}_{S^3}, \] which appeared in the context of integrable systems in \cite{Tab,MT}, see also~\cite{DV}.
\begin{prop} The following results hold. \begin{enumerate} \item[(i)] In the case of the ``round'' metric ${g}_{S^3}$, we have $A_{{g}_{S^3}}= 0$; \item[(ii)] In the case of the metric $\tilde {g}_{S^3}$, we have $A_{\tilde {g}_{S^3}}= \frac{5}{2} \left(
\left(\frac{1}{b}-\frac{1}{a}\right) xy+\left(\frac{
1}{c}-1\right) z\right)$. \end{enumerate} \end{prop} \begin{proof} Part (i) follows from Corollary~\ref{CoCu}. Part (ii) can be obtained by a straightforward computation using Eq. (\ref{sgama}). \end{proof}
\subsection{The case of the ellipsoid $E^3(a,b,c)$} \label{Ellipsoid}
Consider the $3$-dimensional ellipsoid endowed with the standard metric \[ \begin{array}{lll}
{g}_{E^3_{a,b,c}}=&{g}_{x,x} dx^2+{g}_{y,y} dy^2+{g}_{z,z} dz^2 + {g}_{x,y} dx\, dy+{g}_{x,z}dx \, dz+ {g}_{y,z} dy\, dz, \end{array} \] where
\[ \begin{array}{ccl} {g}_{x,x}&=&\displaystyle \frac{\left(b^2 y^2+c^2 z^2+1\right)^2+a^4 x^2 \left(y^2+z^2+1\right) }{\left ((ax)^2+(by)^2+(cz)^2+1\right )^2},\\[4mm] {g}_{y,y}&=& \displaystyle \frac{ \left(a^2 x^2+c^2 z^2+1\right)^2+b^4 y^2 \left(x^2+z^2+1\right)}{\left ((ax)^2+(by)^2+(cz)^2+1\right )^{2}},\\[4mm] {g}_{z,z}&=& \displaystyle \frac{\left(a^2 x^2+b^2 y^2+1\right)^2+c^4 z^2 \left(x^2+y^2+1\right)}{\left ((ax)^2+(by)^2+(cz)^2+1\right )^{2}}, \\[4mm] {g}_{x,y}&=&\displaystyle -2 x y \frac{a^4 x^2-a^2 \left(z^2 (b^2-c^2)+b^2 -1\right)+b^2 \left(b^2
y^2+c^2 z^2+1\right)}{ \left ((ax)^2+(by)^2+(cz)^2+1\right )^{2} },\\[4mm] {g}_{x,z}&=& \displaystyle -2 x z \frac{a^4 x^2-a^2 \left(y^2 \left(c^2-b^2\right)+c^2-1\right)+c^2
\left(b^2 y^2+c^2 z^2+1\right)}{\left ((ax)^2+(by)^2+(cz)^2+1\right )^{2}}, \\[4mm] {g}_{y,z}&=& \displaystyle -2 y z \frac{b^4 y^2-b^2 \left(x^2 \left(c^2-a^2\right)+c^2-1\right)+c^2
\left(a^2 x^2+c^2 z^2+1\right)}{\left ((ax)^2+(by)^2+(cz)^2+1\right )^{2}}. \end{array} \] \begin{prop}
We have \[ \begin{array}{rcl} A_{ {g}_{E^3_{a,b,c}}}&=& a^4 (a^2 - b^2) (b^2 + 2 c^2 + 2) x^3 y +
b^4 (a^2 - b^2) (a^2 + 2 c^2 + 2) x y^3 \\[6pt]
&&
+ (a^2 - b^2) c^4 (2 + a^2 + b^2 + c^2) x y z^2 \\[6pt] && - a^4 (c^2 - 1) (a^2 + 2 b^2 + c^2 + 1) x^2 z
- b^4 (c^2 - 1) (2 a^2 + b^2 + c^2 + 1) y^2 z \\[6pt] &&
- c^4 (c^2 - 1) (2 a^2 + 2 b^2 + 1) z^3
+ (a^2 - b^2) (a^2 + b^2 +
2 c^2 + 1) x y \\[6pt] &&- (c^2 - 1) (2 a^2 + 2 b^2 +
c^2) z. \end{array} \]
\end{prop} \begin{proof} Straightforward computation using Eq. (\ref{sgama}). \end{proof}
\noindent \textbf{Acknowledgments}. We thank Dimitry Leites and Christian Duval their interest in this work and careful reading of preliminary versions of it. We are also grateful to Charles Conley, Eugene Ferapontov and Serge Tabachnikov for a number of fruitful discussions. The first author was partially supported by the Grant NYUAD 063. The second author was partially supported by the Grant PICS05974 ``PENTAFRIZ'', of CNRS.
\def\it{\it} \def\bf{\bf}
\end{document} |
\begin{document}
\title[Local rigidity] {Smooth local rigidity for hyperbolic toral automorphisms}
\author[Boris Kalinin$^1$ \and Victoria Sadovskaya$^2$ \and Zhenqi Jenny Wang$^3$ ]{Boris Kalinin$^1$ \and Victoria Sadovskaya$^2$ \and Zhenqi Jenny Wang$^3$ }
\address{Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, USA.} \email{kalinin@psu.edu, sadovskaya@psu.edu}
\address{Department of Mathematics\\
Michigan State University\\
East Lansing, MI 48824,USA} \email{wangzq@math.msu.edu}
\thanks{{\em Key words:} Hyperbolic toral automorphism, conjugacy, local rigidity, linear cocycle, iterative method.}
\thanks{$^1$ Supported in part by Simons Foundation grants 426243 and 855238} \thanks{$^2$ Supported in part by NSF grant DMS-1764216} \thanks{$^3$ Supported in part by NSF grant DMS-1845416}
\begin{abstract} We study the regularity of a conjugacy $H$ between a hyperbolic toral automorphism $A$ and its smooth perturbation $f$. We show that if $H$ is weakly differentiable then it is $C^{1+\text{H\"older}}$ and, if $A$ is also weakly irreducible, then $H$ is $C^\infty$. As a part of the proof, we establish results of independent interest on H\"older continuity of a measurable conjugacy between linear cocycles over a hyperbolic system. As a corollary, we improve regularity of the conjugacy to $C^\infty$ in prior local rigidity results.
\end{abstract}
\maketitle
\section{Introduction and local rigidity results}
Hyperbolic automorphisms of tori are the prime examples of hyperbolic dynamical systems. The action of a matrix $A \in SL(N,\mathbb Z)$ on $\mathbb R^N$ induces an automorphism of the torus $\mathbb T^N=\mathbb R^N/\mathbb Z^N$, which we denote by the same letter. An automorphism $A$ is called {\em hyperbolic}, or {\em Anosov}, if the matrix has no eigenvalues on the unit circle. One of the key properties of hyperbolic systems is {\it structural stability}: any diffeomorphism $f$ of $\mathbb T^N$ sufficiently $C^1$ close to such an $A$ is also hyperbolic and is topologically conjugate to $A$ \cite{A}, that is, there exists a ho\-meo\-morphism $H$ of $\mathbb T^N$, called a {\em conjugacy}, such that \begin{equation} \label{IntrConj} A\circ H= H \circ f. \end{equation} Any two conjugacies differ by an affine automorphisms of $\mathbb T^N$ commuting with $A$ \cite{W}, and hence have the same regularity. Although $H$ is always bi-H\"older continuous, it is usually not even $C^1$, as there are various obstructions to smoothness. This is in sharp contrast with rigidity for actions of larger groups, where often any perturbation, or even any smooth action, is $C^\infty$ conjugate to an algebraic model.
In the classical case of a single system, the problem of establishing smoothness of the conjugacy from some weaker assumptions has been extensively studied. It is often described as rigidity, in the sense that weak equivalence implies strong equivalence.
In dimension two, definitive results were obtained in \cite{L0,LM, L1}. For hyperbolic automorphisms of $\mathbb T^2$, and more generally for Anosov diffeomorphisms of $\mathbb T^2$, $C^\infty$ smoothness of the conjugacy was obtained from absolute continuity of $H$ and from equality of Lyapunov exponents of $A$ and $f$ at the periodic points.
The case of higher dimensional systems is much more complicated. In particular, the problem of the exact level of regularity of $H$ is subtle: for any $k \in \mathbb N$ and any $N\ge 4$ there exists a reducible hyperbolic automorphism $A$ of $\mathbb T^N$ and its analytic perturbation $f$ such that the conjugacy is $C^k$ but is not $C^{k+1}$ \cite{L1}. We recall that $A$ is {\em reducible}\, if it has a nontrivial rational invariant subspace or, equivalently, if its characteristic polynomial is reducible over $\mathbb Q$.
The two-dimensional results were extended in two directions. First, $C^\infty$ conjugacy was obtained for systems that are conformal on full stable and unstable subspaces under various periodic data assumptions which ensured that the perturbed system is also conformal \cite{L2,KS03,L3,KS09}. Second, for some classes of irreducible $A$, equality of Lyapunov exponents or similarity of the periodic data were shown to imply $C^{1+\text{H\"older}}$ smoothness of $H$ \cite{GG,G,GKS11,SY,GKS20,dW}. Irreducibility of $A$ is necessary for these results \cite{L1,L2,G}. Low smoothness of $H$ is due to the method of the proof, which establishes regularity of $H$ along natural one or two-dimensional $f$-invariant foliations of $\mathbb T^N$, whose leaves are typically only $C^{1+\text{H\"older}}$ smooth. Nevertheless, Gogolev conjectured in \cite{G} that the regularity of $H$ should be close to that of $f$, and in particular if $f$ is $C^\infty$ then so is $H$. Until now, the only progress on higher regularity of $H$, outside of the conformal setting, was obtained for automorphisms of $\mathbb T^3$ with real spectrum in \cite{G17}. We refer to \cite{KSW22} for a more detailed account of questions and developments in the area of local rigidity. \vskip.1cm
In this paper we establish general results on bootstrap of regularity of the conjugacy. We show that for {\it any}\, hyperbolic automorphism $A$, if $H$ is weakly differentiable in a certain sense then it is $C^{1+\text{H\"older}}$ and, if in addition $A$ is {\it weakly irreducible}, then $H$ is $C^\infty$. We introduce and discuss the weak irreducibility property, which is weaker than irreducibility and holds for some $A$ with Jordan blocks. Our methods are different from those in the previous local rigidity results. In particular, we prove smoothness of $H$ without establishing it first along invariant foliations. For the $C^\infty$ smoothness of the conjugacy, we use a KAM type iterative scheme. This approach is novel in the setting of hyperbolic systems, and it is substantially different from previous applications of KAM, as we discuss below. As a corollary, we improve the regularity of $H$ from $C^{1+\text{H\"older}}$ to $C^\infty$ in the previous local rigidity results for the irreducible case. \vskip.1cm
Now we formulate our main results. We denote by $W^{1,q}(\mathbb T^N)$ the Sobolev space of $L^q$ functions with $L^q$ weak partial derivatives of first order. We note that Lipschitz functions are in $W^{1,\infty}(\mathbb T^N)$.
The first result holds for an arbitrary hyperbolic automorphism without any irreducibility assumption. We recall that while $H$ satisfying \eqref{IntrConj} is not unique, there is a unique {\em conjugacy $C^0$ close to the identity}. This is $H$ in the homotopy class of the identity with $H(p)=0$, where $p$ is the fixed point of $f$ closest to $0$.
\begin{theorem} \label{HolderConjugacy} Let $A$ be a hyperbolic automorphism of $\mathbb T^N$ and let $f$ be a $C^{1+\text{H\"older}}$ diffeomorphism of $\mathbb T^N$ which is $C^1$ close to $A$. Suppose that for some conjugacy $H$ between $f$ and $A$, either $H$ or $H^{-1}$ is in $W^{1,q}(\mathbb T^N)$ with $q>N$. Then $H$ is a $C^{1+\text{H\"older}}$ diffeomorphism. \vskip.1cm
More precisely, there is a constant $\beta_0=\beta_0(A)$, $0<\beta_0\le1$, so that for any
$0<\beta' <\beta_0$ there exist constants $\delta>0$ and $K>0$ such that for any $0<\beta \le \beta'$ the following holds.
For any ${C^{1+\beta}}$ diffeomorphism $f$ with $\|{f-A}\|_{C^{1}}<\delta$, if some conjugacy between $A$ and $f$, or its inverse, is in $W^{1,q}(\mathbb T^N)$, $q>N$, then any conjugacy is a $C^{1+\beta}$ diffeomorphism. Moreover, for the conjugacy $H$ that is $C^0$ close to the identity, \begin{equation} \label{C1H est}
\|{H-I}\|_{C^{1+\beta}}\leq K\|{f-A}\|_{C^{1+\beta}}. \end{equation} \end{theorem}
\begin{remark} The assumption of being in $W^{1,q}$ with $q>N$ in this and in the next theorem can be replaced with a slightly weaker one that we actually need for the proof: either $H^{-1}$ is in $W^{1,1}$ and its Jacoby matrix of partial derivatives is invertible and gives the differential of $H^{-1}$ for Lebesgue almost every point of $\mathbb T^N$, or the same holds for $H$ and $f$ preserves an absolutely continuous probability measure. \end{remark}
In the next theorem we obtain $C^\infty$ smoothness of the conjugacy assuming that $A$ is weakly irreducible, which we define as follows. Let $\mathbb R^N=\oplus_{\rho_i} E^i$ be the splitting where $E^i$ is the sum of generalized eigenspaces of $A$ corresponding to the eigenvalues of modulus $\rho_i$, and let $\hat E^i=\oplus_{\rho_j\neq \rho_i} E^j.$
We say that $A$ is \emph{weakly irreducible} if each $\hat E^i$ contains no nonzero elements of $\mathbb Z^N$. This condition is weaker than irreducibility, see Section \ref{Weak irred} for details.
\begin{theorem}\label{th:4} Let $A $ be a weakly irreducible hyperbolic automorphism of $\mathbb T^N$. Then there is a constant $\ell=\ell(A)\in\mathbb{N}$ so that for any $C^\infty$ diffeomorphism $f$ which is $C^{\ell}$ close to $A$ the following holds. If for some conjugacy $H$ between $f$ and $A$ either $H$ or $H^{-1}$ is in the Sobolev space $W^{1,q}(\mathbb T^N)$ with $q>N$, then any conjugacy between $f$ and $A$ is a $C^\infty$ diffeomorphism. \end{theorem}
The constant $\ell=\ell(A)$ is chosen sufficiently large to satisfy the inequalities \eqref{for:78}.
\vskip.2cm
Applying Theorem \ref{th:4} we improve the regularity of the conjugacy from $C^{1+\text{H\"older}}$ to $C^\infty$ in the strongest local rigidity results for irreducible toral setting \cite{GKS11,GKS20}:
\begin{corollary} \label{local PD} Let $A:\mathbb T^N\to\mathbb T^N$ be an irreducible Anosov automorphism such that no three of its eigenvalues have the same modulus. Let $f$ be a $C^\infty$ diffeomorphism which is $C^{\ell}$-close to $A$ such that the derivative $D_pf^n$ is conjugate to $A^n$ whenever $p=f^n(p)$. Then $f$ is $C^{\infty}$ conjugate to $A$. \end{corollary}
\begin{corollary} \label{local LS} Let $A:\mathbb T^N\to\mathbb T^N$ be an irreducible Anosov automorphism such that no three of its eigenvalues have the same modulus and there are no pairs of eigenvalues of the form $\lambda, -\lambda$ or $i\lambda, -i\lambda$, where $\lambda\in \mathbb R$. Let $f$ be a volume-preserving $C^\infty$ diffeomorphism of $\mathbb T^N$ sufficiently $C^{\ell}$-close to $A$. If the Lyapunov exponents of $f$ with respect to the volume are the same as the Lyapunov exponents of $A$, then $f$ is $C^{\infty}$ conjugate to $A$. \end{corollary}
Now we briefly discuss our approaches. To prove Theorem \ref{HolderConjugacy}, we first establish results of independent interest on H\"older continuity of a measurable conjugacy between linear cocycles over a hyperbolic system. These results are formulated and discussed in Section \ref{cocycles statements}. In the proof of the theorem we apply them to the conjugacy $DH$ between the derivative cocycles $Df$ and $A$. The methods used yield only H\"older continuity of the conjugacy between the derivative cocycles and hence only $C^{1+\beta}$ regularity of $H$.
We note, however, that existence of {\it some} H\"older conjugacy between the derivative cocycles $Df$ and $A$ does not imply in general that $H$ is $C^1$. Indeed, if all eigenvalues of $A$ are simple with distinct moduli, then conjugacy of $D_pf^n$ and $A^n$, whenever $p=f^n(p)$, always gives H\"older conjugacy of the cocycles, but $H$ may not be $C^1$ if $A$ is reducible. \vskip.1cm
Our approach to proving Theorem \ref{th:4} is different from prior work on this problem. We abandon the geometric arguments which use invariant foliations. Instead, we introduce a new method which combines exponential mixing of the unperturbed system with a KAM type iterative scheme. KAM methods have been extensively used to study local rigidity, primarily for elliptic systems, such as Diophantine translations of a torus. These systems are very different from the hyperbolic ones that we are considering. Closest to our setting, KAM techniques were used in \cite{Damjanovic4} to prove $C^\infty$ local rigidity for some $\mathbb{Z}^2$ actions by partially hyperbolic toral automorphisms. However, our approach is markedly different from the existing work in both main ingredients of the KAM method: a detailed study of the linearized conjugacy equation, and setup and convergence of the iterative process. In particular, the linearized conjugacy equation in our case is a twisted cohomological equation with the twist given by a hyperbolic matrix.
In contrast to the elliptic case, this creates obstructions to solving the equation by sufficiently smooth functions. In \cite{Damjanovic4} the structure of $\mathbb{Z}^2$ action was used in an essential way to show vanishing of the obstructions. In our setting, we instead use the existence of a $C^{1+\beta}$ conjugacy $H$ given by Theorem \ref{HolderConjugacy}. However, two difficulties arise. First, higher regularity is needed for analyzing the obstructions (see Lemma \ref{le:3}(iii)). Second, $C^{1+\beta}$ functions have slow decay of Fourier coefficients. In contrast, super-polynomial decay for $C^\infty$ functions, which yields super-exponential mixing, was crucial in obtaining suitable estimates for convergence in the KAM iteration in \cite{Damjanovic4}. One of the key innovations in our approach is using directional derivatives to ``balance'' the twist. By analyzing differentiated equations in H\"older category we are able to construct a $C^\infty$ approximate solution of the twisted cohomological equation and obtain suitable estimates. This is done in Section \ref{linearized}, see remarks after Theorem \ref{th:3} for details. Relating Fourier coefficients of a function and its directional derivatives is the only place where we use weak irreducibility of $A$. The last issue is that the estimate we obtain for the approximate solution is not tame, in contrast to traditional KAM estimates \cite{Fayad}. This creates problems in establishing convergence of the iterative procedure, which we overcome in Section \ref{proof th:4}.
\vskip.1cm
The paper is structured as follows. In Section \ref{cocycles statements} we formulate our results on continuity of a measurable conjugacy between linear cocycles over a hyperbolic system, Theorems \ref{measurable conjugacy} and \ref{constant cocycle}. These theorems are proved in Sections \ref {cocycle proofs} and \ref{proof of constant}, respectively. In Section \ref{section:notation} we summarize basic notations and facts used throughout the paper. In Section \ref{proof HC} we prove Theorem \ref{HolderConjugacy}. In Section \ref{linearized} we obtain a result on solving a twisted cohomological equation over $A$,
and in Section \ref{proof th:4} we complete the proof of Theorem \ref{th:4}.
\section{Results on continuity of conjugacy between linear cocycles} \label{cocycles statements}
In this section we consider linear cocycles over a transitive Anosov diffeomorphism $f$ of a compact connected manifold $\mathcal{M}$.
We recall that $f$ is {\it Anosov}\, if there exist a splitting of the tangent bundle $T\mathcal{M}$ into a direct sum of two $Df$-invariant continuous subbundles $\tilde E^s$ and $\tilde E^u$, a Riemannian metric on $\mathcal{M}$, and continuous functions $\nu$ and $\hat\nu$ such that \begin{equation}\label{Anosov def}
\|Df_x(\mathbf v^s)\| < \nu(x) < 1 < \hat\nu(x) <\|Df_x(\mathbf v^u)\| \end{equation} for any $x \in \mathcal{M}$ and any unit vectors $\,\mathbf v^s\in \tilde E^s(x)$ and $\,\mathbf v^u\in \tilde E^u(x)$. The diffeomorphism is {\em transitive} if there is a point $x\in \mathcal{M}$ with dense orbit. All known examples satisfy this property.
\vskip.1cm Let $A$ be a map from $\mathcal{M}$ to $GL(N,\mathbb R)$. The $GL(N,\mathbb R)${\em -valued cocycle over $f$ generated by }$A$ is the map $\mathcal{A}:\,X \times \mathbb Z \,\to GL(N,\mathbb R)$ defined by $\,\mathcal{A}(x,0)=\text{Id}\,$ and for $n\in \mathbb N$, $$
\mathcal{A}(x,n)=\mathcal{A}_x^n = A(f^{n-1} x)\circ \cdots \circ A(x) \;\text{ and }\; \mathcal{A}(x,-n)=\mathcal{A}_x^{-n}= (\mathcal{A}_{f^{-n} x}^n)^{-1}. $$
We say that a $GL(d,\mathbb R)$-valued cocycle $\mathcal{A}$ is $\beta$-H\"older continuous if there exists a constant $c$ such that $$ \,d(\mathcal{A}_x, \mathcal{A}_y) \le c\, \text{dist}(x,y)^\beta \quad \text{for all }x,y\in \mathcal{M}, $$ where the metric $d$ on $GL(N,\mathbb R)$ is given by $$
d (A, B) = \| A - B \| + \| A^{-1} - B^{-1} \|, \quad\text{where $\|\,.\,\|$ is the operator norm.} $$
More generally, we consider linear cocycles defined as follows. Let $P : {E} \to \mathcal{M} $ be a finite dimensional $\beta$-H\"older vector bundle over $\mathcal{M}$. A continuous {\em linear cocycle} over $f$ is a homeomorphism $\mathcal{A}:{E}\to{E}$ such that $$P \circ \mathcal{A} = f \circ P \quad\text{and\, $\mathcal{A}_x : {E}_x \to {E}_{fx}$\, is a linear isomorphism for each $x\in \mathcal{M}$.} $$ The linear cocycle $\mathcal{A}$ is called $\beta$-H\"older
if $\mathcal{A}_x$ depends $\beta$-H\"older on $x$, with proper identification of fibers at nearby points. A detailed description of this setting is given in Section 2.2 of \cite{KS13}. \vskip.2cm
The differential of $f$ and its restrictions to invariant sub-bundles of $T\mathcal{M}$, such as
$\tilde E^s$ and $\tilde E^u$, are prime examples of linear cocycles. \vskip.2cm
We say that a $\beta$-H\"older cocycle $\mathcal{A}$ over an Anosov diffeomorphism $f$ is\,
{\em fiber bunched} if there exist numbers $\theta<1$ and $c$ such that for all $x\in\mathcal{M}$ and $n\in \mathbb N$, \begin{equation}\label{fiber bunched}
\| \mathcal{A}_x^n\|\cdot \|(\mathcal{A}_x^n)^{-1}\| \cdot (\nu^n_x)^\beta < c\, \theta^n \;\text{ and}\quad
\| \mathcal{A}_x^{-n}\|\cdot \|(\mathcal{A}_x^{-n})^{-1}\| \cdot (\hat \nu^{-n}_x)^\beta < c\, \theta^n, \end{equation} where $ \;\nu^n_x=\nu(f^{n-1}x)\cdots\nu(x) \,\text{ and }\;
\hat\nu^{-n}_x=(\hat \nu(f^{-n}x))^{-1}\cdots (\hat\nu(f^{-1}x))^{-1}.$ \vskip.2cm
Let $\mu$ be an ergodic $f$-invariant measure on $\mathcal{M}$. We denote by $\lambda_+(\mathcal{A},\mu)$ and $\lambda_-(\mathcal{A},\mu)$ the largest and smallest Lyapunov exponents of $\mathcal{A}$ with respect to $\mu$ given by the Oseledets Multiplicative Ergodic Theorem. For $\mu$ almost all $x\in \mathcal{M}$, they equal the limits \begin{equation} \label{exponents}
\lambda_+(\mathcal{A},\mu)= \lim_{n \to \infty} n^{-1} \ln \| \mathcal{A}_x ^n \| \quad \text{and}\quad
\lambda_-(\mathcal{A},\mu)= \lim_{n \to \infty} n^{-1} \ln \| (\mathcal{A}_x ^n)^{-1} \|^{-1}. \end{equation}
We say that a cocycle $\mathcal{A}$ {\em has one exponent}\, if
for every $f$-periodic point $p$ the invariant measure $\mu_p$ on its orbit satisfies $\lambda_+(\mathcal{A},\mu_p)=\lambda_-(\mathcal{A},\mu_p)$. By Theorem 1.4 in \cite{K11}, this condition is equivalent to $$ \lambda_+(\mathcal{A},\mu)=\lambda_-(\mathcal{A},\mu)\quad\text{for every ergodic $f$-invariant measure.} $$ We note that if $\mathcal{A}$ has one exponent, then it is fiber bunched \cite[Corollary 4.2]{S15}. \vskip.1cm
For $GL(N,\mathbb R)$ cocycles $\mathcal{A}$ and $\mathcal{B}$ over $f$, a (measurable or continuous) function $\mathcal{C}:\mathcal{M}\to GL(N,\mathbb R)$ such that $$
\mathcal{A}_x=\mathcal{C}(fx)\,\mathcal{B}_x\,\mathcal{C}(x)^{-1} \quad\text{for all }x\in \mathcal{M} $$ is called a (measurable or continuous) {\em conjugacy} or {\em transfer map} between $\mathcal{A}$ and $\mathcal{B}$. For linear cocycles $\mathcal{A},\mathcal{B} : E\to E$ a conjugacy is defined similarly with $\mathcal{C}(x) \in GL(E_x)$.
The question whether a measurable conjugacy between two cocycles is continuous has been studied in \cite{PaP97,Pa99,Sch99, S13,S15}. An example in \cite{PW01} shows that a measurable conjugacy between two fiber bunched $GL(2,\mathbb R)$-valued cocycles is not necessarily continuous, moreover, the generators of the cocycles in this example can be chosen arbitrarily close to the identity. Continuity of a measurable conjugacy was proven for cocycles with values in a compact group
\cite{PaP97,Pa99} and, somewhat more generally for cocycles with bounded distortion \cite{Sch99},
for $GL(2, \mathbb R)$-valued cocycles with one exponent \cite{S13}, and for $GL(N, \mathbb R)$-valued cocycles
such that one is fiber bunched and the other one is uniformly quasiconformal \cite{S15}.
The result in \cite{S13} relied on two-dimensionality, and the uniform quasiconformality
assumption in \cite{S15} is much stronger than having one exponent. The next theorem establishes continuity of a measurable conjugacy between a fiber bunched cocycle and a cocycle with one exponent.
\begin{theorem} \label{measurable conjugacy} Let $f$ be a transitive $C^{1+\text{H\"older}}$ Anosov diffeomorphism of a compact manifold $\mathcal{M}$, and let $\mathcal{A}$ and $\mathcal{B}$ be $\beta$-H\"older linear cocycles over $f$. Suppose that $\mathcal{A}$ has one exponent and $\mathcal{B}$ is fiber bunched.
Let $\mu$ be an ergodic $f$-invariant measure on $\mathcal{M}$ with full support and local product structure. Then any $\mu$-measurable conjugacy between $\mathcal{A}$ and $\mathcal{B}$ is $\beta$-H\"older continuous, i.e., coincides with a $\beta$-H\"older continuous conjugacy on a set of full measure.
\end{theorem}
As we mentioned above, continuity of a measurable conjugacy does not hold in general if $\mathcal{A}$ has more than one exponent, however, we prove it in a special case of a constant $\mathcal{A}$. Moreover, we obtain an estimate of the $\beta$-H\"older constant $K_\beta(\mathcal{C})$ of the conjugacy $\mathcal{C}$ in terms of the $\beta$-H\"older constant of $\mathcal{B}$.
\begin{theorem} \label{constant cocycle} Let $f$ and $\mu$ be as in Theorem \ref{measurable conjugacy}, and let $\mathcal{A}$ is be a constant $GL(N,\mathbb R)$-valued cocycle over $f$. Then for any H\"older continuous $GL(N,\mathbb R)$-valued cocycle $\mathcal{B}$ sufficiently $C^0$ close to $\mathcal{A}$, any $\mu$-measurable conjugacy between $\mathcal{A}$ and $\mathcal{B}$ is H\"older continuous. \vskip.05cm
More specifically, there exists a constant $\beta_0(A,f)$ so that the following holds. For any $0<\beta'<\beta_0 (A,f)$ there is $\delta >0$ and $k>0$ such that for any $0<\beta \le \beta'$ and any $\beta$-H\"older $GL(N,\mathbb R)$-valued cocycle $\mathcal{B}$ over $f$ with
$ \|\mathcal{B}_x-A\|_{C^{0}}<\delta$, any $\mu$-measurable conjugacy $\mathcal{C}$ between $\mathcal{A}$ and $\mathcal{B}$ is $\beta$-H\"older and its $\beta$-H\"older constant satisfies \begin{equation} \label {hk C}
K_\beta(\mathcal{C}) \le k \, \| \mathcal{C}\|_{C^0} \, K_\beta (\mathcal{B})
\qquad \text {and} \qquad K_\beta(\mathcal{C}^{-1}) \le k \, \| \mathcal{C}^{-1}\|_{C^0} \, K_\beta (\mathcal{B}).
\end{equation} \end{theorem}
The constant $\beta_0(A,f)$ is explicitly given by \eqref{beta 0} in Section 5.
\section{Basic notations and facts} \label{section:notation}
\subsection{Norms and H\"older constants.}
For $r\in\mathbb{N}\cup\{0\}$ we use $\norm{\cdot}_{C^r}$ for the $C^r$ norm of functions with continuous derivatives up to order $r$ on $\mathbb{T}^N$.
For a $\beta$-H\"{o}lder function $g$, $0<\beta\le1$, we denote its $\beta$-H\"{o}lder constant, or H\"older seminorm, by \begin{align*}
K_\beta(g)=\norm{g}_{C^{0,\beta}}\,\overset{\text{def}}{=}\,\sup \,\{\,|g(x)-g(y)| \, d(x,y)^{-\beta} : \; x\neq y \in \mathbb T^N\,\}\,<\,\infty. \end{align*} We denote by $C^{1,\beta}$ or $C^{1+\beta}$ the space of functions with $\beta$-H\"{o}lder first derivative with norm \begin{align*}
\norm{f}_{C^{1+\beta}} \,\overset{\text{def}}{=} \,\norm{f}_{C^1}+K_\beta (D f). \end{align*}
\subsection{Invariant subspaces}
For $A\in GL(N,\mathbb R)$ let $\rho_1 < \dots < \rho_L$ be the distinct moduli of its eigenvalues and let \begin{equation} \label{splitL} \mathbb R^N = {E}^1 \oplus \dots \oplus {E}^L
\end{equation} be the corresponding $A$-invariant splitting, where ${E}^i$ is the direct sum of generalized eigenspaces corresponding to the eigenvalues with modulus $\rho_i$. We also denote \begin{equation} \label{A_i}
{{\hat E}}^{i}\overset{\text{def}}{=}\oplus_{\rho_j \neq \rho_i} E^{j}, \qquad \ A_i=A|_{ {E}^i}:E^i\to E^i, \qquad \text{and} \qquad N_i=\dim E^i.
\end{equation} For the Euclidean norm on $\mathbb R^N$ there is a constant $K_A$ such that for each $i$ we have
\begin{equation}\label{jordan}
\norm{A_i^m}\leq K_A \,\rho_i^m \,(\abs{m}+1)^N \quad \text{for all } m\in\mathbb{Z}.
\end{equation} Also, for any $\epsilon >0$ there is an ``adapted" inner product on $\mathbb R^N$ such that the direct sum $\oplus E^i$ is orthogonal and for each $1\le i\le L$, \begin{equation} \label{rateA2}
(\rho_i-\epsilon)^m \le \| A_i^m u \| \le (\rho_i+\epsilon)^m \; \text{ for any unit vector }u\in {E}^i \text{ and any } m\in \mathbb Z. \end{equation}
If $A$ is hyperbolic then $\rho_{i_0}<1 <\rho_{i_0+1} $ for some $1\le i_0<L$, and we define
the stable and unstable subspaces of $A$ as \begin{gather*}
E^{s}\overset{\text{def}}{=}\oplus_{\rho_i<1} E^{i}\qquad \text{and} \qquad E^{u}\overset{\text{def}}{=}\oplus_{\rho_i>1} E^{i}.
\end{gather*}
\vskip.3cm
\subsection{Weak irreducibility.} \label{Weak irred} Recall that $GL(N,\mathbb Z)$ denotes the integer matrices with determinant $\pm1$. We say that $A\in GL(N,\mathbb Z)$ is \emph{weakly irreducible} if each $\hat E^i$ contains no nonzero elements of $\mathbb Z^N$.
Irreducibility over $\mathbb Q$ implies weak irreducibility. Indeed, if there is a nonzero integer point $n \in \hat E^i$ then $span \{ A^m n : m\in \mathbb Z \} \subset \hat E^i$ is a nontrivial rational invariant subspace. In fact, weak irreducibility is determined by the characteristic polynomial of $A$ as follows.
\begin{lemma} \label{weak irred} A matrix $A\in GL(N,\mathbb Z)$ is weakly irreducible if and only if there is a set $\Delta \subset \mathbb R$ so that for each irreducible over $\mathbb Q$ factor of the characteristic polynomial of $A$ the set of moduli of its roots equals $\Delta$. \end{lemma}
\begin{proof}
Let $A\in GL(N,\mathbb Z)$, let $p_A$ be its characteristic polynomial, and let $p_A=\prod_{k=1}^K p_k^{d_k}$ be its prime decomposition over $\mathbb Q$. Then we have the corresponding splitting $\mathbb R^N = \oplus V_k$
into rational $A$-invariant subspaces $V_k =\ker p_k^{d_k}(A)$. We also have the (non-rational)
$A$-invariant splitting \eqref{splitL}, and we set $\Delta=\{\rho_1, \dots, \rho_L\}$. We will show that $A$ is weakly irreducible if and only if $\Delta$ is the set of moduli of the roots for each $p_k$.
If for some $\rho_i\in \Delta$ and $k\in \{1,\dots , K\}$ no root of the irreducible polynomial $p_k$ has modulus $\rho_i$, then $V_k \subset {\hat E}^i$. Hence $A$
is not weakly irreducible as $V_k$ is a rational subspace and hence it contains nonzero points of $\mathbb Z^N$.
Conversely, suppose each $p_k$ has $\Delta$ as the set of moduli of its roots.
Suppose that for some $i$ there is $0\ne n \in (\mathbb Z^N \cap {\hat E}^i)$. Then for some $k$ its projection
$n_k$ to $V_k$ is a nonzero rational vector. We note that $n_k \in {\hat E}^i$ as ${\hat E}^i= \oplus_k ({\hat E}^i \cap V_k)$. Then
$$W=span \{ A^m n_k : m\in \mathbb Z \}
$$
is a rational $A$-invariant subspace
contained in ${\hat E}^i \cap V_k$. Then the characteristic polynomial of $A|_W$ is a power of $p_k$
and hence $W$ contains an eigenvector with eigenvalue of modulus $\rho _i \in \Delta$. Thus
$W \cap E^i \ne 0$, contradicting $W \subset {\hat E}^i$. Thus $A$ is weakly irreducible.
\end{proof}
It follows from the lemma that if $A$ is irreducible or weakly irreducible then the following matrices are weakly irreducible $$ \left(\begin{array}{cc}A & 0 \\ 0 & A \end{array}\right) \quad \text{and}\quad \left(\begin{array}{cc}A & \text{I} \\ 0 & A \end{array}\right). $$ These matrices are not irreducible and the latter is not diagonalizable.
\section{Proof of Theorem \ref{measurable conjugacy}} \label{cocycle proofs}
Let $f$ be a transitive $C^{1+\text{H\"older}}$ Anosov diffeomorphism of a compact manifold $\mathcal{M}$, let ${E}$ be a $\beta$-H\"older vector bundle over $\mathcal{M}$, and let $\mathcal{F} :{E} \to {E}$ be a $\beta$-H\"older linear cocycle over $f$.
In Section \ref{holonomy} we recall the definition and properties of holonomies for linear cocycles, in Section \ref{twisted} we prove a preliminary results on twisted cocycles, and in Section \ref{proof of thm measurable} we give a proof of Theorem \ref{measurable conjugacy}.
\subsection{Holonomies of fiber bunched linear cocycles} \label{holonomy}
The notion of {\it holonomies} for linear cocycle was introduced in \cite{BV,V} Existence of holonomies was proved in \cite{V,ASV} under a stronger ``one-step" fiber bunching condition and then extended to bundle setting and weaker fiber bunching \eqref{fiber bunched} in \cite{KS13,S15}.
\begin{proposition} \label{existence of holonomies} Let $\mathcal{F}$ be a $\beta$-H\"older fiber bunched linear cocycle over $(\mathcal{M},f)$. Then for every $x\in \mathcal{M}$ and $y\in W^s(x)$ the limit \begin{equation}\label{hol def}
\mathcal{H}^{s}_{x,y} = \mathcal{H}^{\mathcal{F},s}_{x,y} =\underset{n\to\infty}{\lim} \,(\mathcal{F}^n_y)^{-1} \circ \mathcal{F}^n_x, \end{equation} called the {\em stable holonomy,} exists and satisfies \begin{itemize} \item[($\mathcal{H}$1)] $\mathcal{H}^{s}_{x,y}$ is an invertible linear map from ${E}_x$ to ${E}_y$; \vskip.1cm \item[($\mathcal{H}$2)] $\mathcal{H}^{s}_{x,x}=\text{Id}\,$ and $\,\mathcal{H}^{s}_{y,\,z} \circ \mathcal{H}^{s}_{x,\,y}=\mathcal{H}^{s}_{x,z}$,\,\, and hence $(\mathcal{H}^{s}_{x,y})^{-1}=\mathcal{H}^{s}_{y,x};$ \vskip.1cm \item[($\mathcal{H}$3)] $\mathcal{H}^{s}_{x,y}= (\mathcal{F}^n_y)^{-1}\circ \mathcal{H}^{s}_{f^nx ,\,f^ny} \circ \mathcal{F}^n_x\;$ for all $n\in \mathbb N$; \vskip.1cm
\item[($\mathcal{H}$4)] $\| \mathcal{H}^{s}_{\,x,y} - \text{Id} \,\| \leq c\cdot d (x,y)^{\beta},$
where $c$ is independent of $x$ and $y\in W^s_{\text{loc}}(x).$\\
\end{itemize} \end{proposition}
\subsection{Twisted cocycles} \label{twisted} In this section we study the coboundary equation over $f$ twisted by a $\beta$-H\"older linear cocycle $\mathcal{F} :{E} \to {E}$. We will use its main result, Proposition \ref{twist-meas},
in the inductive process in the proof of Theorem \ref{measurable conjugacy}. \vskip.1cm
Let $\phi,\eta : \mathcal{M} \to {E}$ be sections of the bundle ${E}$ over $\mathcal{M}$. We consider the equation \begin{equation}\label{twisteq} \eta(x)=\phi(x) + (\mathcal{F}_x)^{-1} (\eta(fx))\quad \text{equivalently}\quad \phi(x)=\eta(x) - (\mathcal{F}_x)^{-1} (\eta(fx)). \end{equation} Iterating \eqref{twisteq} and denoting $\mathcal{F}^n_x= \mathcal{F}_{f^{n-1}x} \circ \cdots \circ \mathcal{F}_{fx}\circ \mathcal{F}_x: {E}_x \to {E}_{f^{n}x}$ we obtain $$ \begin{aligned}
\eta(x)& =\phi(x) + (\mathcal{F}_x)^{-1} (\eta(fx))= \phi(x) + (\mathcal{F}_x)^{-1}[\phi (fx)+\mathcal{F}_{fx} (\eta(f^2x))]=...\\
&= \phi(x) + (\mathcal{F}_x)^{-1}(\phi (fx))+\dots + (\mathcal{F}^{n-1}_{x})^{-1}(\phi(f^{n-1}x))+ (\mathcal{F}_{f^{n-1}x})^{-1} (\eta(f^{n}x)). \end{aligned} $$ Thus \begin{equation}\label{twisteq iter} \eta(x) = {\Phi}^n(x)+ (\mathcal{F}_{f^{n-1}x})^{-1} (\eta(f^{n}x)), \quad\text{where} \end{equation} $$ {\Phi}^n(x)= \phi(x) + (\mathcal{F}_x)^{-1}(\phi (fx))+\dots + (\mathcal{F}^{n-1}_{x})^{-1}(\phi(f^{n-1}x)) \in {E}_x. $$
We say that $\mathcal{F}$ is {\em uniformly bounded}\, if there exists $K$
such that $\| \mathcal{F}_x^n \| \le K$ for all $x\in \mathcal{M}$ and $n\in \mathbb Z$. A $\beta$-H\"older bounded cocycle is fiber-bunched and hence it has stable holonomies $\mathcal{H}^s_{x,y}:{E}_x \to {E}_y$ where $y \in W^s(x)$.
\begin{lemma} \label{twist hol} Suppose that $\phi$ is a $\beta$-H\"older section and that $\mathcal{F}$ is a uniformly bounded $\beta$-H\"older cocycle. Then for any $x\in \mathcal{M}$ and $y \in W^s(x)$ the following limit exists $$ {\Phi}_{x,y}^s= \lim _{n \to \infty} ({\Phi}^n(x) - \mathcal{H}_{y,x}^s{\Phi}^n(y) )= \sum_{k=0}^\infty\, [\, (\mathcal{F}^{k}_{x})^{-1}(\phi(f^{k}x))- \mathcal{H}_{y,x}^s (\mathcal{F}^{k}_{y})^{-1}(\phi(f^{k}y))\,] $$
and satisfies $ \|{\Phi}_{x,y}^s\| \le K' d (x,y)^\beta$ with uniform $K'$ for all $x\in \mathcal{M}$ and $y \in W^s_{loc}(x)$. \end{lemma}
The result holds if instead of being uniformly bounded $\mathcal{F}$ satisfies the following. There exist numbers $\theta<1$ and $L$ such that for all $x\in\mathcal{M}$ and $n\in \mathbb N$, $$
\|(\mathcal{F}_x^n)^{-1}\| \cdot (\nu^n_x)^\beta < L\, \theta^n. $$
\begin{proof} For all $x\in \mathcal{M}$ and $y \in W^s_{loc}(x)$ we have $d(f^{k}x,f^{k}y)\le \nu_x^k \,d(x,y)$. As
$\phi$ is $\beta$-H\"older we obtain
$$
\|\phi(f^{k}x)- \phi(f^{k}y)\|\le K_1(\nu^k_x d(x,y))^\beta,
$$
and since $\mathcal{H}_{f^ky,f^kx}^s$ is $\beta$-H\"older close to identity by $(\mathcal{H} 4)$, we have
$$
\|\phi(f^{k}x)- \mathcal{H}_{f^ky,f^kx}^s \phi(f^{k}y)\|\le K_2(\nu^k_x\, d(x,y))^\beta.
$$
By uniformly boundedness of $\mathcal{F}$ we have $\| (\mathcal{F}_x^k)^{-1} \| \le K$, and by continuity of $\phi$ we have $\sup_x \|\phi(x) \| \le K_3$. Therefore, $$ {\Phi}^n(x) - \mathcal{H}_{y,x}^s {\Phi}^n(y) = \sum_{k=0}^{n-1}\,(\mathcal{F}^{k}_{x})^{-1}(\phi(f^{k}x))- (\mathcal{H}_{y,x}^s \circ (\mathcal{F}^{k}_{y})^{-1} \circ \mathcal{H}_{f^kx,f^ky}^s) (\mathcal{H}_{f^ky,f^kx}^s\phi(f^{k}y)) $$ Since $\mathcal{H}_{y,x}^s \circ (\mathcal{F}^{k}_{y})^{-1} \circ \mathcal{H}_{f^kx,f^ky}^s= (\mathcal{F}^k_x)^{-1}$ by ($\mathcal{H}$3),\, the $k^{th}$ term in the sum equals
$$ (\mathcal{F}^{k}_{x})^{-1}(\phi(f^{k}x))- (\mathcal{F}^{k}_{x})^{-1} (\mathcal{H}_{f^ky,f^kx}^s \phi(f^{k}y))= (\mathcal{F}^{k}_{x})^{-1}\,[ \phi(f^{k}x)-\mathcal{H}_{f^ky,f^kx}^s \phi(f^{k}y)], $$ and we estimate
$$
\begin{aligned}
\|(\mathcal{F}^{k}_{x})^{-1}\,&[ \phi(f^{k}x) -\mathcal{H}_{f^ky,f^kx}^s \phi(f^{k}y)] \,\| \le
\|(\mathcal{F}^{k}_{x})^{-1} \| \cdot \|\phi(f^{k}x)-\mathcal{H}_{f^ky,f^kx}^s \phi(f^{k}y) \| \le \\
&\|(\mathcal{F}^{k}_{x})^{-1} \| \cdot K_2(\nu^k_x\, d(x,y))^\beta\le K K_2\,\theta^{k} d(x,y)^\beta \quad\text{for some }\theta<1.
\end{aligned}
$$ Hence the series converges and $$
\| {\Phi}^n(x) - \mathcal{H}_{y,x}^s {\Phi}^n(y) \| \le\, \sum_{k=0}^{n-1}K K_2 \, \theta^{k} d(x,y)^\beta \le K' d(x,y)^\beta, $$
so the limit ${\Phi}_{x,y}^s$ satisfies $ \|{\Phi}_{x,y}^s\| \le K' d (x,y)^\beta$. \end{proof}
\begin{proposition} \label{twist-meas} Let $\mathcal{F}$ be a $\beta$-H\"older uniformly bounded cocycle over an Anosov diffeomorphism $f$ (or a hyperbolic system). Let $\mu$ be an ergodic $f$-invariant measure on $\mathcal{M}$ with full support and local product structure.
Let $\phi : \mathcal{M} \to {E}$ be a $\beta$-H\"older section, and let $\eta:\mathcal{M}\to {E}$ be a $\mu$-measurable section satisfying \eqref{twisteq}. Then $\eta$ is $\beta$-H\"older and $$ \eta(x)=\mathcal{H}_{y,x}^s \,\eta(y)+{\Phi}_{x,y}^s \quad \text{for all $x\in X$ and $y \in W^s(x)$}. $$
\end{proposition}
\begin{proof} Let $x\in \mathcal{M}$ and $y\in W^s(x)$. Using equation \eqref{twisteq iter} for $\eta(x)$ and $\eta(y)$ we obtain $$ \eta(x)-\mathcal{H}_{y,x}^s \,\eta(y)= {\Phi}^n(x) - \mathcal{H}_{y,x}^s {\Phi}^n(y) + \Delta_n, $$ where $$
\Delta_n= (\mathcal{F}_{f^{n-1}x})^{-1} (\eta(f^{n}x)) -\mathcal{H}_{y,x}^s (\mathcal{F}_{f^{n-1}y})^{-1} (\eta(f^{n}y)). $$ By Lemma \ref{twist hol}, $({\Phi}^n(x) - \mathcal{H}_{y,x}^s{\Phi}^n(y) )$ converges to ${\Phi}_{x,y}^s$. \vskip.1cm
Now we show that $\|\Delta_n\|\to 0$ along a subsequence for all $x,y$ in a set of full measure. First we note that by property ($\mathcal{H}$3) we have
$\mathcal{H}_{y,x}^s (\mathcal{F}_{f^{n-1}y})^{-1}=(\mathcal{F}_{f^{n-1}x})^{-1} \circ \mathcal{H}_{f^ny,f^nx}^s$. Hence $$ \Delta_n= (\mathcal{F}_{f^{n-1}x})^{-1} \left( \eta(f^{n}x) -\mathcal{H}_{f^ny,f^nx}^s(\eta(f^{n}y)) \right)= (\mathcal{F}_{f^{n-1}x})^{-1} ( \Delta_n'), $$ where $\Delta_n'=\eta(f^{n}x) -\mathcal{H}_{f^ny,f^nx}^s(\eta(f^{n}y))$. By uniform boundedness of $\mathcal{F}$ we obtain $$
\| \Delta_n\|\le \|(\mathcal{F}_{f^{n-1}x})^{-1} \| \cdot \|\Delta_n'\|\le K \|\Delta_n'\|. $$ Since the section $\eta:\mathcal{M}\to E$ is $\mu$-measurable, by Lusin's theorem there exists a compact set $S\subset \mathcal{M}$ with $\mu(S)>1/2$ such that $\eta$ is uniformly continuous and hence bounded on $S$. Let $Y$ be the set of points in $\mathcal{M}$ for which the frequency of visiting $S$ equals $\mu(S)$. By Birkhoff Ergodic Theorem, $\mu(Y)=1$. \vskip.1cm
If $x,y\in Y$, there exists a subsequence $n_i\to \infty$ such that such that $f^{n_i}x, f^{n_i}y \in S$ for all $i$. Since $y \in W^s(x)$, $\,d(f^{n_i}x, f^{n_i}y)\to 0$
and hence $\Delta_{n_i}' \to 0$ by uniform continuity and boundedness of $\eta$ on $S$ and property ($\mathcal{H}$4) of $\mathcal{H}^s$. Thus $\Delta_{n_i} \to 0$ and we obtain that $$ \eta(x)=\mathcal{H}_{y,x}^s \,\eta(y) +{\Phi}_{x,y}^s \quad\text{ for all $x,y\in Y$ with $y \in W^s(x).$} $$ Since ${\Phi}_{x,y}^s$ is $\beta$-H\"older on $ W^s_{\text{loc}}(x)$ by Lemma \ref{twist hol}, we conclude that $$
\|\eta(x)- \mathcal{H}_{y,x}^s\,\eta(y)\| \le K'd(x,y)^\beta\quad\text{ for all $x,y\in Y$ with $y \in W^s(x).$} $$ Since $\mathcal{H}_{x,y}^s$ is $\beta$-H\"older by property ($\mathcal{H}$4), this means that $\eta$ is essentially $\beta$-H\"older along $ W^s_{\text{loc}}(x)$.
Similar arguments for $y \in W^u_{loc}(x)$ show that $\eta$ is also essentially $\beta$-H\"older along $ W^u_{loc}(x)$. Hence $\eta$ is $\beta$-H\"older by the local product structure of $\mu$ and of the stable and unstable manifolds.
\end{proof}
\subsection{Proof of Theorem \ref{measurable conjugacy}} \label{proof of thm measurable} For convenience, by taking inverse, we will work with a conjugacy $\mathcal{C}$ satisfying \begin{equation} \label{Cback}
\mathcal{B}_x=\mathcal{C}(fx)\,\mathcal{A}_x\,\mathcal{C}(x)^{-1} . \end{equation} First we observe that since $\lambda_+(\mathcal{A},\mu)=\lambda_-(\mathcal{A},\mu)$ and $\mathcal{B}$ is $\mu$-measurably conjugate to $\mathcal{A}$, the following lemma implies that $$ \lambda_+(\mathcal{B},\mu)=\lambda_-(\mathcal{B},\mu). $$
\begin{lemma} \label{equal exp} Let $\mu$ be an ergodic $f$-invariant measure. If $\mathcal{C}$ is a $\mu$-measurable conjugacy between cocycles $\mathcal{A}$ and $\mathcal{B}$, then for $\mu$ a.e. $x$ and for each vector $0\ne u \in {E}_x$ the forward (resp. backward) Lyapunov exponent of $u$ under $\mathcal{A}$ equals that of $\mathcal{C}_x(u)$ under $\mathcal{B}$. \end{lemma} \begin{proof} We fix a set of positive measure $Y\subseteq \mathcal{M}$ such that for some $K$ we have
$\|\mathcal{C}_x\| \le K$ and $\|(\mathcal{C}_x)^{-1}\| \le K$ for all $x\in Y$. Then we choose an $f$-invariant set of full measure $X \subseteq \mathcal{M}$ such that for every $x \in X$ \begin{itemize} \item[(i)] the forward and backward Lyapunov exponents under both $\mathcal{A}$ and $\mathcal{B}$ exist for each non-zero vector $v \in {E}_x$, and \item[(ii)] the frequency of visiting $Y$ under both forward and backward iterates of $f$ equals $\mu(Y)>0$. \end{itemize} For every $x \in X$,\, $0\ne u \in {E}_x$, and $n\in \mathbb Z$ we have $$
n^{-1} \ln \| \mathcal{B}^n_x (\mathcal{C}_x(u)) \| = n^{-1} \ln \| \mathcal{C}_{f^nx} (\mathcal{A}^n_x (u)) \|. $$ The limit of the left hand side as $n \to \infty$ (resp. $n \to -\infty$) is the forward (resp. backward) Lyapunov exponent of $\mathcal{C}_x(u)$ under $\mathcal{B}$. On the other hand, by the choice of $Y$, the limit of the right hand side along a subsequence $n_i \to \infty$ (resp. $n_i \to -\infty$) such that $f^{n_i}x \in Y$ equals the forward (resp. backward) Lyapunov exponent of $u$ under $\mathcal{A}$. \end{proof}
We use the following results from \cite{KS13}. In the three theorems below, $f$ is a transitive $C^{1+\text{H\"older}}$ Anosov diffeomorphism, $\mathcal{A},\mathcal{B}:E\to {E}$ are $\beta$-H\"older linear cocycles over $f$, and $\mu$ is an ergodic $f$-invariant measure with full support and local product structure.
\begin{theorem} \cite[Theorem 3.9]{KS13}\label{reductionH} Suppose that for every $f$-periodic point $p$ the invariant measure $\mu_p$ on its orbit satisfies $\lambda_+(\mathcal{A},\mu_p)=\lambda_-(\mathcal{A},\mu_p)$.
Then there exist a flag of $\beta$-H\"older $\mathcal{A}$-invariant sub-bundles \begin{equation} \label{flagH} \{0\} =U^0 \subset U^1 \subset ... \subset U^{j-1} \subset U^k = {E} \end{equation} and $\beta$-H\"older Riemannian metrics on the quotient bundles $U^{i}/U^{i-1}$, $i=1, ... , k$, so that for some positive $\beta$-H\"older function $\phi : \mathcal{M} \to \mathbb R$ the quotient-cocycles induced by the cocycle $\phi \mathcal{A}$ on $U^{i}/U^{i-1}$ are isometries.
\end{theorem}
\begin{theorem} \cite[Theorem 3.1 and Corollary 3.8]{KS13} \label{structure} If $\mathcal{B}$ is fiber bunched, then any $\mathcal{B}$-invariant $\mu$-measurable conformal structure on ${E}$ coincides $\mu$-a.e. with a H\"older continuous conformal structure. \end{theorem}
If a cocycle has more than one Lyapunov exponent, then the corresponding Lyapunov sub-bundles are invariant and measurable, but not continuous in general. For a fiber bunched cocycle with only one Lyapunov exponent, measurable invariant sub-bundles are continuous.
\begin{theorem} \cite[Theorem 3.3 and Corollary 3.8]{KS13}\label{distribution} Suppose that $\mathcal{B}$\, is fiber bunched and $\lambda_+(\mathcal{B},\mu)=\lambda_-(\mathcal{B},\mu)$. Then any $\mu$-measurable $\mathcal{B}$-invariant sub-bundle of $\mathcal{E}$ coincides $\mu$-a.e. with a H\"older continuous one. \end{theorem}
We consider the flag $U^i$ for $\mathcal{A}$ given by Theorem \ref{reductionH}. Denoting $\mathcal{U}^i_x=\mathcal{C}(x) U^i_x$ we obtain the corresponding flag of measurable $\mathcal{B}$-invariant sub-bundles $$ \{0\} =\mathcal{U}^0 \subset \mathcal{U}^1\subset \mathcal{U}^2 \subset \dots \subset \mathcal{U}^k={E}. $$ By Theorem \ref{distribution} we may assume that the sub-bundles $\mathcal{U}^i$ are H\"older continuous.
The conformal structure $\sigma_1$ on $E^1$ given by the Riemannian metric in Theorem \ref{reductionH} is invariant under $\mathcal{A}$. The push forward of $\sigma_1$ by $\mathcal{C}$ gives a measurable $\mathcal{B}$-invariant conformal structure $\tau_1$ on $\mathcal{U}^1$, which is H\"older continuous by Theorem \ref{structure}.
Similarly, we consider H\"older continuous quotient-bundles ${\tilde V}^i=U^i/U^{i-1}$ and ${\tilde{\mathcal{V}}}^i=\mathcal{U}^i/\mathcal{U}^{i-1}$ over $\mathcal{M}$ with the quotient cocycles $\mathcal{A}^{(i)}$ and $\mathcal{B}^{(i)}$. Since $\mathcal{A}^{(i)}$ preserves a H\"older continuous conformal structure $\sigma_i$ on ${\tilde V}^i$, pushing forward by $\mathcal{C}$ we obtain a measurable conformal structure $\tau_i$ on $\mathcal{U}^i/\mathcal{U}^{i-1}$ invariant under $\mathcal{B}^{(i)}$, which is H\"older continuous by Theorem \ref{structure}. Thus we obtain a ``similar structure" for $\mathcal{B}$.
We fix a $\beta$-H\"older Riemannian metric on ${E}$. We denote by ${V}^i$ the orthogonal complement of $U^{i-1}$ in ${E}_i$, and we denote by $\mathcal{V}^i$ the orthogonal complement of $\mathcal{U}^{i-1}$ in $\mathcal{U}^i$, $i=1,\dots,k$. Thus $U^i={V}^1 \oplus \cdots \oplus {V}^i$ and $\mathcal{U}^i=\mathcal{V}^1 \oplus \cdots \oplus \mathcal{V}^i$. All these sub-bundles are H\"older continuous but for $i>1$ they are not invariant under $\mathcal{A}$ and $\mathcal{B}$, and $\mathcal{C}$ does not necessarily map ${V}^i$ to $\mathcal{V}^i$.
We denote by $P^{j}:{E} \to {V}^j$ the projection to the ${V}^j$ component in the splitting ${E}={V}^1 \oplus \cdots \oplus {V}^k$ and similarly $\mathcal{P}^{j}:\mathcal{E} \to \mathcal{V}^j$.
We denote the restriction of $\mathcal{C}$ to ${V}^i$ by $\mathcal{C}^i$ and we denote by $\mathcal{C}^{j,i}$ its $j$-component
$\mathcal{C}^{j,i}=\mathcal{P}^{j} \circ \mathcal{C}^i : {V}^i \to \mathcal{V}^j$.
Since $\mathcal{U}^i_x=\mathcal{C}(x) U^i_x$, we have $\mathcal{C}^i :{V}^i \to \mathcal{U}^i$ and thus $\mathcal{C}^{j,i}=0$ for $j>i$, that is $\mathcal{C}$ has an upper triangular block structure.
We also define the corresponding blocks $\mathcal{A}^{j,i} : {V}^i \to {V}^j$ and $\mathcal{B}^{j,i} : \mathcal{V}^i \to \mathcal{V}^j$
as $\mathcal{A}^{j,i} =P^j\circ \mathcal{A} |_{ {V}^i}$ and similarly for $\mathcal{B}$. The invariance of the flags also yields
upper triangular block structures for $\mathcal{A}$ and $\mathcal{B}$: $\mathcal{A}^{j,i}=0=\mathcal{B}^{j,i}$ for $j>i$.
We will show inductively that the restriction of $\mathcal{C}$ to $U^i$ is H\"older continuous, $i=1,\dots,k$. The base case $i=1$ follows from the following result from \cite{S15}.
\begin{theorem} \cite[Theorem 2.7]{S15} \label{QC} Let $\mathcal{A},\mathcal{B}:E\to {E}$ be $\beta$-H\"older linear cocycles over a hyperbolic system. Suppose that $\mathcal{A}$ uniformly quasiconformal and $\mathcal{B}$ is fiber bunched. Let $\mu$ be an ergodic invariant measure with full support and local product structure. Then any $\mu$-measurable conjugacy between $\mathcal{A}$ and $\mathcal{B}$ is $\beta$-H\"older continuous, i.e. it coincides with a $\beta$-H\"older continuous conjugacy on a set of full measure. \end{theorem}
Now we describe the inductive step. Assuming that the restriction of $\mathcal{C}$ to $U^{i-1}$ is $\beta$-H\"older continuous we show that so is the restriction to $U^{i}$. Since $U^i={V}^i \oplus U^{i-1}$, it suffices to show that the restriction $\mathcal{C}^{i}$ of $\mathcal{C}$ to ${V}^{i}$ is also $\beta$-H\"older continuous. We will establish this inductively for each of its components $\mathcal{C}^{j,i}$, $j=i,\dots ,1$.
First we observe that $\mathcal{C}^{i,i}$ is H\"older continuous for all $i=1,\dots,k$. For this we identify bundles ${V}^i$ with ${\tilde V}^i$ and $\mathcal{V}^i$ with ${\tilde{\mathcal{V}}}^i$ via the projections. Under these identifications the cocycle $\mathcal{A}^{i,i}:{V}^{i,i} \to {V}^{i,i}$ corresponds to the quotient cocycle $\mathcal{A}^{(i)}$, the cocycle $\mathcal{B}^{i,i}:\mathcal{V}^{i,i} \to \mathcal{V}^{i,i}$ corresponds to $\mathcal{B}^{(i)}$, and the map $\mathcal{C}^{i,i}$ corresponds to the quotient measurable conjugacy $\mathcal{C}^{(i)}$ between $\mathcal{A}^{(i)}$ and $\mathcal{B}^{(i)}$. Since the quotient cocycles $\mathcal{A}^{(i)}$ and $\mathcal{B}^{(i)}$ are conformal, Theorem \ref{QC} shows that
$\mathcal{C}^{(i)}$ is $\beta$-H\"older continuous, and hence so is $\mathcal{C}^{i,i}$.
Now we show that $\mathcal{C}^{i-\ell,i}$ is $\beta$-H\"older assuming that $\mathcal{C}^{i-j,i}$ is $\beta$-H\"older for $j=0,1,\dots \ell-1$. Using the conjugacy equation $$ \mathcal{B}_x \circ \mathcal{C}_x =\mathcal{C}_{fx} \circ \mathcal{A}_x $$ and equating $(i-\ell,i)$ components we obtain $$ \begin{aligned} &\mathcal{B}^{i-\ell,i-\ell}_x \circ \mathcal{C}^{i-\ell,i}_x + \mathcal{B}^{i-\ell,i-\ell+1}_x \circ \mathcal{C}^{i-\ell+1,i}_x + \dots +\mathcal{B}^{i-\ell,i}_x \circ \mathcal{C}^{i,i}_x \\ & =\mathcal{C}^{i-\ell,i-\ell}_{fx} \circ \mathcal{A}^{i-\ell+1,i}_x + \mathcal{C}^{i-\ell,i-\ell+1}_{fx} \circ \mathcal{A}^{i-\ell+1,i}_x + \dots +\mathcal{C}^{i-\ell,i}_{fx} \circ \mathcal{A}^{i,i}_x \end{aligned} $$ and hence \begin{equation} \label{CD} \mathcal{C}^{i-\ell,i}_x = (\mathcal{B}^{i-\ell,i-\ell}_x)^{-1} \circ \mathcal{C}^{i-\ell,i}_{fx} \circ \mathcal{A}^{i,i}_x \, + D_x \end{equation} where $$ \begin{aligned} D_x & = (\mathcal{B}^{i-\ell,i-\ell}_x)^{-1} \circ(\mathcal{C}^{i-\ell,i-\ell}_{fx} \circ \mathcal{A}^{i-\ell+1,i}_x + \dots +\mathcal{C}^{i-\ell,i-1}_{fx} \circ \mathcal{A}^{i-1,i}_x )- \\ &-(\mathcal{B}^{i-\ell,i-\ell}_x)^{-1} \circ ( \mathcal{B}^{i-\ell,i-\ell+1}_x \circ \mathcal{C}^{i-\ell+1,i}_x + \dots +\mathcal{B}^{i-\ell,i}_x \circ \mathcal{C}^{i,i}_x ). \end{aligned} $$ We view $\mathcal{C}^{i-\ell,i}_x$ and $D_x$ as sections of the H\"older bundle $L({V}^i,\mathcal{V}^{i-\ell})$ whose fiber at $x$ is the space of linear maps $L({V}^i_x,\mathcal{V}^{i-\ell}_x)$. Thus equation \eqref{CD} is of the form \eqref{twisteq} with $$ E=L({V}^i,\mathcal{V}^{i-\ell}), \;\;\phi _x =D_x,\;\; \eta_x=\mathcal{C}^{i-\ell,i}_x, \;\;\text{and}\;\; \mathcal{F}_x(\eta_{fx})= (\mathcal{B}^{i-\ell,i-\ell}_x)^{-1} \circ \eta_{fx} \circ \mathcal{A}^{i,i}_x. $$ We note that $D_x$ is $\beta$-H\"older since we inductively know that all its terms are $\beta$-H\"older. Also $\mathcal{F}$ is a linear cocycle on the bundle $L({V}^i,\mathcal{V}^{i-\ell})$ over $f^{-1}$, and it is $\beta$-H\"older since so are $\mathcal{B}^{i-\ell,i-\ell}$ and $ \mathcal{A}^{i,i}$. Moreover, $\mathcal{F}$ is uniformly bounded since cocycles $\mathcal{B}^{i-\ell,i-\ell}$ and $ \mathcal{A}^{i,i}$ are conformal and their normalizations are continuously cohomologous. The latter follows since we know that $\mathcal{B}^{i-\ell,i-\ell}$ and $\mathcal{A}^{i-\ell,i-\ell}$ are continuously cohomologous by $\mathcal{C}^{i-\ell,i-\ell}$ and that the normalizations of all $ \mathcal{A}^{i,i}$ are given by the same function $\phi ^{-1}$ from Theorem \ref{reductionH}. Hence we can apply Proposition \ref{twist-meas} and conclude that $\mathcal{C}^{i-\ell,i}$ is $\beta$-H\"older. \vskip.1cm The argument above applies to $\ell=1, \dots i-1$ and we conclude that all $\mathcal{C}^{1,i}, \dots ,\mathcal{C}^{i,i}$ are H\"older. This proves that the restriction of $\mathcal{C}$ to $U^{i}$ is H\"older and completes the inductive step. We conclude that $\mathcal{C}$ is H\"older, completing the proof of Theorem \ref{measurable conjugacy}.
\section{Proof of Theorem \ref{constant cocycle}} \label{proof of constant}
In this proof we will also work with a conjugacy $\mathcal{C}$ satisfying \eqref{Cback}. First, H\"older continuity of $\mathcal{C}$ is deduced from Theorem \ref{measurable conjugacy} as follows.
Let $A \in GL(N,\mathbb R)$ be the generator of the constant cocycle $\mathcal{A}$. Let $\rho_1 < \dots <\rho_L$ be the distinct moduli of the eigenvalues of $A$ and let \begin{equation} \label{splitA}
\mathbb R^N = {E}^1 \oplus \dots \oplus {E}^L \end{equation} be the corresponding invariant splitting as in \eqref{splitL}. In this section we will use the adapted norm on $\mathbb R^N$ for which we have estimates \eqref{rateA2}. They imply that for any $\beta>0$ the cocycle $\mathcal{A}_i$ generated by $A_i$ is fiber bunched if $\epsilon$ is sufficiently small.
Let $B(x)=\mathcal{B}_x:\mathcal{M}\to GL(N,\mathbb R)$ be the generator of the cocycle $\mathcal{B}$. If $B$ is sufficiently $C^0$ close to $A$, then $\mathcal{B}$ has H\"older continuous invariant splitting $C^0$ close to \eqref{splitA} $$ \mathbb R^N = \mathcal{E}^1_x \oplus \dots \oplus \mathcal{E}^L_x, $$
so that the restrictions $\mathcal{B}_i=\mathcal{B}| \mathcal{E}^i$ satisfy estimates similar to \eqref{rateA2} \begin{equation} \label{rateB}
(\rho_i-2\epsilon)^n \le \| \mathcal{B}_i^n u \| \le (\rho_i+2\epsilon)^n \quad \text{for any unit vector }u\in \mathcal{E}^i. \end{equation} This is well known but also follows from Lemma \ref{Ei est}, which gives explicit estimates of both H\"older exponent and H\"older constant. We conclude that all restrictions $\mathcal{B}_i$ are $\beta$-H\"older and hence are fiber bunched
if $\epsilon$ is sufficiently small.
Let $\mathcal{C}$ be a measurable conjugacy between $\mathcal{A}$ and $\mathcal{B}$. We claim that $\mathcal{C}$ maps ${E}^i$ to $\mathcal{E}^i$, that is $\mathcal{C}_x ({E}^i)=\mathcal{E}^i_x$ for $\mu$ a.e. $x$. Indeed, by Lemma \ref{equal exp}, for $\mu$ a.e. $x$ and for each unit vector $u \in {E}^i$ the forward and backward Lyapunov exponent of $\mathcal{C}_x(u)$ is $\ln \rho_i$. This yields that $\mathcal{C}_x(u) \in \mathcal{E}^i$, as having a non-zero component in another $\mathcal{E}^j$ would imply having forward or backward Lyapunov exponent under $\mathcal{B}$ different from $\ln \rho_i$
if $\epsilon$ is sufficiently small. Then $\mathcal{C}_i=\mathcal{C}|_{ {E}^i}$ is a measurable conjugacy between fiber bunched cocycles $\mathcal{A}_i$ and $\mathcal{B}_i$. By Theorem \ref{measurable conjugacy} each $\mathcal{C}_i$ is H\"older for all $i=1,\dots,L$,
and hence so is $\mathcal{C}$.
\vskip.3cm
Now we prove the more detailed statement. We denote the Lipschitz constants of $f^{-1}$ and $f$ respectively by \begin{equation} \label{alpha}
\alpha_f= \sup_{x \in \mathcal{M}} \| D_xf^{-1}\| >1 \quad\text{and}\quad \alpha_f'=\sup_{x\in \mathcal{M}} \| D_xf\| >1. \end{equation} For $1\le i<L$ we define $$ \beta_i= \text{\small{$\frac{\ln (\rho_{i+1}/\rho_{i})}{ \ln (\alpha_f)}$}} \quad\text{and}\quad \beta_i'= \text{\small{$\frac{\ln (\rho_{i+1}/\rho_{i})}{\ln (\alpha_f')}$}}, $$ and we choose \begin{equation} \label{beta 0} \beta_0= \beta_0(A,f) =\min\,\{1,\, \beta_1, \dots, \beta_{L-1},\, \beta_1', \dots, \beta_{L-1}'\} >0. \end{equation}
Since $\mathcal{B}$ is $\beta$-H\"older with $\beta \le \beta' <\beta_0$, Lemma \ref{Ei est} below shows that the splitting \eqref{rateB} is $\beta$-H\"older and by Lemma \ref{Bi est} so are all restrictions $\mathcal{B}_i$. Then by Theorem \ref{measurable conjugacy} each restriction
$\mathcal{C}_i=\mathcal{C}|_{{E}^i}$ is $\beta$-H\"older and hence so is $\mathcal{C}$. Since $\mathcal{A}_i$ and $ \mathcal{B}_i$ are $\beta$-fiber bunched for any sufficiently small $\epsilon$,
\cite[Proposition 4.5]{S15} yields that $\beta$-H\"older $\mathcal{C}_i$ intertwines their stable holonomies, that is, \begin{equation}\label{intertwines2} \mathcal{H}_{x,y}^{\mathcal{A}_i,s}=\mathcal{C}_i(y)\circ \mathcal{H}_{x,y}^{\mathcal{B}_i,s}\circ \mathcal{C}_i(x)^{-1}\quad \text{for all }x,y \in \mathcal{M} \text{ such that }y\in W^s(x). \end{equation} Since for the constant cocycle $\mathcal{A}_i$ the holonomies are all identity, $\mathcal{H}_{x,y}^{\mathcal{A}_i,s}=\text{Id}$, we get $$\mathcal{C}_i(x)=\mathcal{C}_i(y)\circ \mathcal{H}_{x,y}^{\mathcal{B}_i,s}. $$
Thus using Lemma \ref{Hi est} we obtain that for all $y\in W^s(x)$
$$\| \mathcal{C}_i(x)- \mathcal{C}_i(y)\|=\|\mathcal{C}_i(y) \circ (\mathcal{H}_{x,y}^{\mathcal{B}_i,s}-\text{Id})\| \le \| \mathcal{C}_i\|_{C^0} \cdot k_3 \, K_\beta (\mathcal{B}) \cdot d_{W^s} (x,y)^{\beta}. $$ Combining these estimates for all $i=1,\dots, L$ we conclude that all $y\in W^s(x)$
$$\| \mathcal{C}(x)- \mathcal{C}(y)\| \le \| \mathcal{C}\|_{C^0} \cdot k_4 \, K_\beta (\mathcal{B}) \cdot d_{W^s} (x,y)^{\beta}. $$ Similarly, using the analog of Lemma \ref{Hi est} for unstable holonomies, we obtain
the same estimate for $y\in W^u(y)$. Then the local product structure of stable and unstable
foliations of $f$ implies that the $\beta$-H\"older constant of $\mathcal{C}$ can be estimated as $$
K_\beta(\mathcal{C}) \le k \, \| \mathcal{C}\|_{C^0} \, K_\beta (\mathcal{B}). $$ Now, to complete the proof of the second part of the theorem, we state and prove the lemmas used in the above argument.
\vskip .2cm
\begin{lemma}\label{Ei est} For any $0<\beta'<\beta_0$ there is $\delta >0$ and $k_1>0$ such that for any $0<\beta \le \beta'$
any $\beta$-H\"older $GL(N,\mathbb R)$ cocycle $\mathcal{B}$ with $ \|\mathcal{B}_x-A\|_{C^{0}}<\delta$ preserves $\beta$-H\"older splitting $$ \mathbb R^N = \mathcal{E}^1_x \oplus \dots \oplus \mathcal{E}^L_x $$
which is $C^0$ close to ${E}^1 \oplus \dots \oplus {E}^L$ and for each $1\le i\le L$ the
$\beta$-H\"older constant $K_\beta(\mathcal{E}^i)$ of $\mathcal{E}^i$ satisfies \begin{equation} \label {hkEi}
K_\beta(\mathcal{E}^i) \le k_1\, K_\beta (\mathcal{B})
\end{equation} \end{lemma}
\begin{proof} We deduce this lemma from the one below. We fix $1\le i<L$, and let $$ E'= {E}^1 \oplus \dots \oplus {E}^i\quad\text{and}\quad E={E}^{i+1} \oplus \dots \oplus {E}^L. $$ Lemma \ref{EE' est} below shows that for any $\beta'<\beta_i$ there is $\delta >0$ and
$k'$ such that for any $0<\beta \le \beta'$ any cocycle $\mathcal{B}$ with $ \|\mathcal{B}_x-A\|_{C^{0}}<\delta$ preserves the bundle $\mathcal{E}$ close to ${E}$ with the desired estimate for $\beta$-H\"older constant. Similarly, for any $\beta'<\beta_i'$ using the inverses of $A$ and $f$ we obtain that $\mathcal{B}$ preserves a bundle $\mathcal{E}'$ close to ${E}'$ with a similar estimate for its $\beta$-H\"older constant. Then for each $1\le i\le L$ the bundle $\mathcal{E}^i$ is defined as a suitable intersection and hence is also $C^0$ close to ${E}^i$ and its $\beta$-H\"older constant satisfies \eqref{hkEi}. \end{proof}
\begin{remark} Lemmas \ref{Ei est} and \ref{EE' est} do not rely on hyperbolicity of $f$ and use only that it is bi-Lipschitz. \end{remark}
\begin{lemma}\label{EE' est} Let $A \in GL(N,\mathbb R)$, let $\,\mathbb R^N=E'\oplus E$ be an $A$-invariant splitting, and let
$$
\begin{aligned}
\xi' &= \max \,\{\,\| Av \| : \, v \in E',\; \|v\|=1\,\}=\|A|_{E'}\|\,\,\text{ and } \\
\xi & = \min \,\{\,\|Av \| : \, v \in E,\; \|v\|=1\,\} = \|A^{-1}|_{E}\|^{-1}.
\end{aligned}
$$
Let $\alpha_f=\sup \| Df^{-1}\| >1$ be the Lipschitz constant of $f^{-1}$ and let $\beta'>0$. Suppose that $$\, \xi'<\xi \quad\text{and}\;\quad \text{\small{$\frac{\xi' \alpha_f^{\beta'}}{\xi}$}} <1, \;\text{ that is, }\;\,\beta'< \text{\small{$\frac{\ln (\xi /\xi')}{\ln\alpha_f}$}}. $$ Then there is $\delta >0$ and $k'$ such that for any $0<\beta \le \beta'$
any $\beta$-H\"older $GL(N,\mathbb R)$ cocycle $\mathcal{B}$ with $ \|\mathcal{B}_x-A\|_{C^{0}}<\delta$ preserves a $\beta$-H\"older sub-bundle $\mathcal{E}$ which is $C^0$ close to $E$ and its $\beta$-H\"older constant $K_\beta(\mathcal{E})$ satisfies $$K_\beta(\mathcal{E}) \le k'\, K_\beta (\mathcal{B})$$ \end{lemma}
\begin{proof} The argument is similar to the H\"older version the $C^r$ Section Theorem of M.~Hirsch, C.~Pugh, and M.~Shub (see Theorem 3.8 in \cite{HPS}), but we give the estimate of the H\"older constant.
We consider the space $\mathcal{L}=\mathcal{L}(E,E')$ of linear operators from $E$ to $E'$ and endow it with the standard operator norm. Since $A$ preserves the splitting $E'\oplus E$ it induces the graph transform action $\hat A$ on $\mathcal{L}$ as follows: if $L \in \mathcal{L}$ and $G \subset \mathbb R^N$ is its graph then $\hat A (L) $ is
the operator in $\mathcal{L}$ whose graph is $A (G)$. The map $\hat A$ is linear,
$$
\hat A \,[L] = A |_{E'} \circ L \circ (A |_{E})^{-1},
$$ so we can estimate its norm as $$
\|\hat A \| \le \| A |_{E'}\| \cdot \|(A |_{E})^{-1}\|\le \xi'/\xi <1. $$
Similarly, any linear map $B\in GL(N,\mathbb R)$ sufficiently close to $A$ induces in the same way the graph transform map $\hat B$ on a unit ball $\mathcal{L}_1$ in $\mathcal{L}$. Moreover, $\hat B$ is a contraction
of $\mathcal{L}_1$ with Lipschitz constant $K(\hat B)$ close to $K(\hat A )=\xi'/\xi<1$. Indeed, $B$ induces
an algebraic map on the Grassmannian of $(\dim E)$-dimensional subspaces which, together
with its first derivatives, depends continuously on $B$. Also, it is easy to see that the map
$B \mapsto \hat B$ from a small neighborhood of $A$ to $C^0(\mathcal{L}_1,\mathcal{L}_1)$ is Lipschitz with some
constant $\hat L$.
Now we consider the trivial fiber bundle ${\mathcal{V}}=\mathcal{M} \times \mathcal{L}_1$. Then any $\mathcal{B}_x$ which is $C^0$-close to $A$
induces graph transform maps $\hat \mathcal{B}_x : {\mathcal{V}}_x \to {\mathcal{V}}_{fx}$ and thus the bundle map $\hat \mathcal{B} : {\mathcal{V}} \to {\mathcal{V}}$ covering $f$. We consider the space $S$ of continuous sections of ${\mathcal{V}}$ with the supremum norm, and the induced action $F=F_\mathcal{B}$ on $S$ defined for $s \in S$ as $(Fs)(fx)= \mathcal{B}_x(s(x))$. If $K^{\mathcal{B}}:=\sup _x K({\hat \mathcal{B}_x }) < 1$ then $F$ is a contraction on $S$ and hence has a unique fixed point $s_*=Fs_*$. Let $s_0(x)=0\in \mathcal{L}$ be the zero section, then we can write $s_*= \lim F^ns_0$ and it follows that $s_*$ is $C^0$-close to $s_0$. Denoting the graph of $s(x)$ by $\mathcal{E}_x$ we obtain the unique continuous $\mathcal{B}$-invariant sub-bundle close to $E$.
Now we will show that $s_*$ is $\beta$-H\"older and estimate its $\beta$-H\"older constant. For this we will find $M>0$ such that $K_\beta (s) \le M$ implies $K_\beta (F s) \le M$. Then $K_\beta (F^n (s_0)) \le M$ for all $n$ and since $s_*= \lim F^n(s_0)$ it will follow that $K_\beta (s_*) \le M$.
Fix points $z,z'$ and let $x=f(z)$, $x'= f(z')$. Then for any $\beta$-H\"older $s\in S$ we can estimate, as $\|s(x)\| \le 1$, that $$ \begin{aligned}
&\| Fs (x) - Fs(x')\|=\| \hat \mathcal{B}_z s (z) - \hat \mathcal{B}_{z'}s(z')\| \\
& \le \| \hat \mathcal{B}_z s (z) - \hat \mathcal{B}_{z'}s(z)\| + \| \hat \mathcal{B}_{z'} s (z) - \hat \mathcal{B}_{z'}s(z')\| \\
&\le d_{C^0}(\hat \mathcal{B}_{z} ,\hat \mathcal{B}_{z'} ) + K({\hat \mathcal{B}_{z'} })\| s (z) - s(z')\| \le \hat L \| \mathcal{B}_{z} -\mathcal{B}_{z'}\| + K^{\mathcal{B}}\| s (z) - s(z')\| \\ &\le \hat L\, K_\beta (\mathcal{B}) \,d(z,z')^\beta + K^{\mathcal{B}} K_\beta (s) \,d(z,z')^\beta \le
[\hat L \, K_\beta (\mathcal{B}) + K^{\mathcal{B}} K_\beta (s)]\,(\alpha_f\, d(x,x'))^\beta,
\end{aligned} $$ where $\alpha_f$ is the Lipschitz constant of $f^{-1}$ and $\hat L$ is the Lipschitz constant of the map $B \mapsto \hat B$ on a neighborhood of $A$. Hence $Fs$ is also $\beta$-H\"older and $$ K_\beta (Fs) \le \hat L \, \alpha_f^\beta\, K_\beta (\mathcal{B}) + \alpha_f^\beta \,K^{\mathcal{B}} K_\beta (s). $$ Therefore, $K_\beta (s) \le M$ implies $K_\beta (Fs) \le M$ if we take $$ M= (1-K^{\mathcal{B}}a_f^\beta)^{-1} \hat L \,a_f^\beta\,K_\beta (\mathcal{B}). $$
If $\|\mathcal{B}_x-A\|_{C^0}$ is small then $K^{\mathcal{B}}$ is close to $K(\hat A )=\xi'/\xi$.
Since $\,\xi' \alpha^{\beta'} /\xi <1\,$ and $\beta \le \beta'$ it follows that $1-K^{\mathcal{B}}a_f^\beta>0$
and is separated from $0$. Then there is a constant $k'$ which bounds $ \hat L a_f^\beta\,(1-K^{\mathcal{B}}a_f^\beta)^{-1}$ for all $0<\beta \le \beta'$ and all $\mathcal{B}$ with $\|\mathcal{B}_x-A\|_{C^{0}}<\delta$. Hence,
$$
M \le k' \, K_\beta (\mathcal{B}).
$$ \vskip.1cm
Finally, since $K_\beta (s_0) =0$ it follows that $K_\beta (F^n (s_0)) \le M$ for all $n$ and hence for the limit
we also have $K_\beta (s_*) \le M \le k' K_\beta (\mathcal{B}) $. \end{proof}
\vskip.2cm
Now we estimate the $\beta$-H\"older constants of the restricted cocycles $\mathcal{B}_i=\mathcal{B}|_{ \mathcal{E}^i}$.
\begin{lemma}\label{Bi est} For any $0<\beta'<\beta_0$ there is $\delta >0$ and $k_2>0$ such that for any $0<\beta \le \beta'$ and any $\beta$-H\"older cocycle $\mathcal{B}$ with $\|\mathcal{B}_x-A\|_{C^{0}}<\delta$ the $\beta$-H\"older constant of the cocycle $\mathcal{B}_i$, $i=1,\dots L$, satisfies $$ K_\beta(\mathcal{B}_i) \le k_2\, K_\beta (\mathcal{B}). $$ \end{lemma}
\begin{proof}
Denoting $B(x)=\mathcal{B}_x$ and $B_i(x)=\mathcal{B}_x| _{\mathcal{E}^i}$ we need to estimate the distance between $B_i(x)$ and $B_i(y)$. To do this using their difference, we fix $\beta$-H\"older identifications ${I}_{x,y} :\mathcal{E}^i_x \to \mathcal{E}^i_y$, say by translation from $x$ to $y$ in the trivial bundle $\mathcal{M} \times \mathbb R^N$ followed by an appropriate rotation. Then for a unit vector $u\in \mathcal{E}^i(x)$ we need to estimate $\|(B_i(x)-B_i(y)\circ {I}_{x,y})u\|$. We note that $$
\|u-{I}_{x,y}u\| \le \text{dist} (\mathcal{E}^i_x,\mathcal{E}^i_y)\le K_\beta(\mathcal{E}^i) \,d(x,y)^\beta. $$ Also, since $B(x)$ is $\beta$-H\"older have
$\|B(x)u-B(y)u\| \le K_\beta(\mathcal{B}) \,d(x,y)^\beta$.
Hence we obtain that for a unit vector $u\in \mathcal{E}^i(x)$ $$ \begin{aligned}
\|(B_i(x)-B_i(y)\circ {I}_{x,y})u\|& \le \|B(x)u-B(y)u\|+\|B(y)\| \cdot \|u-{I}_{x,y}u\| \\
& \le K_\beta (\mathcal{B}) \, d(x,y)^\beta+ \|B\|_{C^0} \,K_\beta(\mathcal{E}^i) \, d(x,y)^\beta . \end{aligned} $$ Since
$K_\beta(\mathcal{E}^i) \le k_1\, K_\beta (\mathcal{B})$ by \eqref{hkEi} and
$\|B\|_{C^0} \le \|A\| +\|\mathcal{B}_x-A\|_{C^{0}}\le \|A\| +\delta$ we conclude that $$
\|(B_i(x)-B_i(y)\circ {I}_{x,y})u\| \le k_2 \, K_\beta (\mathcal{B}) \, d(x,y)^\beta. $$ Thus $K_\beta(\mathcal{B}_i) \le k_2\, K_\beta (\mathcal{B}) .$ \end{proof}
In the next lemma we consider the stable holonomies of cocycles $\mathcal{B}_i=\mathcal{B}|_{\mathcal{E}^i}$, $i=1,\dots, L$.
\begin{lemma}\label{Hi est} For any $0<\beta'<\beta_0$ there is $\delta >0$ and $k_3>0$ such that for any $0<\beta \le \beta'$
and a $\beta$-H\"older cocycle $\mathcal{B}$ with $\|\mathcal{B}_x-A\|_{C^{0}}<\delta$ the holonomies of cocycles $\mathcal{B}_i=\mathcal{B}|_{\mathcal{E}^i}$ satisfy $$
\| \mathcal{H}^{s}_{\,x,y} - \text{Id} \,\| \le k_3 \, K_\beta (\mathcal{B}) \, d (x,y)^{\beta} \;\text{ for any } x\in \mathcal{M} \text{ and } y\in W^s_{\text{loc}}(x). $$
\end{lemma}
\begin{proof}
We fix $i$ and denote $\mathcal{F}=\mathcal{B}_i$. The stable holonomies of $\mathcal{F}$ are given by \begin{equation}\label{hol def F}
\mathcal{H}^{\mathcal{F},s}_{x,y} =\underset{n\to\infty}{\lim} \,(\mathcal{F}^n_y)^{-1} \circ \mathcal{F}^n_x. \end{equation} The existence is ensured by fiber bunching of $\mathcal{F}$. Indeed, the contraction along $W^s$ is estimated by \eqref{Anosov def} as $$ d (f^nx, f^ny)\le \nu^n d (x,y) \;\text{ for any } x\in \mathcal{M}, \;\, y\in W^s_{\text{loc}}(x),\;\, n\in \mathbb N, $$ We also obtain from \eqref{rateB} that
\begin{equation}\label{F norm}
\| \mathcal{F}_x^m\|\cdot \|(\mathcal{F}_y^m)^{-1}\| \le \prod_{j=0}^{m-1} \|\mathcal{F}_{x_j}\| \, \|(\mathcal{F}_{y_j})^{-1}\| \le \, \text{\small {$\left( \frac{\rho_i+2\epsilon}{\rho_i-2\epsilon} \right)$}}^m =\sigma^m \quad\text{for all }x,y\in\mathcal{M}, \end{equation} where $\sigma= (\rho_i+2\epsilon) (\rho_i-2\epsilon)^{-1}$ is close to $1$ when $\epsilon$ is small.
It follows that
\begin{equation}\label{F bunch}
\| \mathcal{F}_x^m\|\cdot \|(\mathcal{F}_y^m)^{-1}\| \cdot \nu^{m\beta} \le \sigma^m \cdot \nu^{m\beta} =\theta^m \quad\text{for all }x,y\in\mathcal{M}, \end{equation} where $\,\theta=\sigma \nu^{\beta}<1\,$ if $\delta$ and hence $\epsilon$ are sufficiently small. In particular, $\mathcal{F}$ is fiber bunched so the limit in \eqref{hol def F} exits, though this also follows from the proof. \vskip.1cm
We want to obtain a constant $c$ such that $\| \mathcal{H}^{\mathcal{F},s}_{\,x,y} - \text{Id} \,\| \leq c\,d (x,y)^{\beta}\,$ for all $x\in \mathcal{M}$ and $y\in W^s_{\text{loc}}(x)$.
Denoting $\,x_m=f^m(x)$ and $\,y_m=f^m(y)$, we obtain $$
\begin{aligned}
& (\mathcal{F}^n_y)^{-1}\circ \mathcal{F}^n_x \,= (\mathcal{F}^{n-1}_y)^{-1}\circ \left(
(\mathcal{F}_{y_{n-1}})^{-1} \circ \mathcal{F}_{x_{n-1}}\right) \circ \mathcal{F}^{n-1}_x \\
& = (\mathcal{F}^{n-1}_y)^{-1} \circ (\text{Id}+r_{n-1}) \circ \mathcal{F}^{n-1}_x = (\mathcal{F}^{n-1}_y)^{-1}\circ\mathcal{F}^{n-1}_x+(\mathcal{F}^{n-1}_y)^{-1}\circ r_{n-1}
\circ \mathcal{F}^{n-1}_x \\
&=\dots = \text{Id}+\sum_{m=0}^{n-1} (\mathcal{F}^{m}_y)^{-1}\circ r_{m}\circ \mathcal{F}^m_x ,
\quad \text{where }r_m=(\mathcal{F}_{y_m})^{-1} \circ \mathcal{F}_{x_m}-\text{Id}.
\end{aligned} $$ Since $\mathcal{F}$ is $\beta$-H\"older, denoting $c'= (\rho_i-2\epsilon)^{-1}K_\beta (\mathcal{F})$, we obtain that for every $m\ge 0$ $$
\|r_m\| \,\le\, \|(\mathcal{F}_{y_m})^{-1}\| \cdot \| \mathcal{F}_{x_m} - \mathcal{F}_{y_m}\|
\le \|\mathcal{F}^{-1}\|_{C^0} \, K_\beta (\mathcal{F})\, d(x_m, y_m)^\beta \le c'\,d(x,y)^\beta \nu^{m\beta}. $$ Using \eqref{F bunch} it follows that $$ \begin{aligned}
& \|(\mathcal{F}^{m}_y)^{-1}\circ r_{m}\circ \mathcal{F}^m_x\| \le
\|(\mathcal{F}^{m}_y)^{-1}\| \cdot \|\mathcal{F}^m_x\| \cdot c'\,d(x,y)^\beta \nu^{m\beta} \le \theta ^m \,c'\,d(x,y)^\beta .
\end{aligned} $$
Therefore, for every $n\in \mathbb N$, $$
\|\text{Id}-(\mathcal{F}^n_y)^{-1}\circ \mathcal{F}^n_x\| \,\le\, \sum_{i=0}^{n-1}
\| (\mathcal{F}^{i}_y)^{-1}\circ r_{i}\circ \mathcal{F}^i_x \| \le c'\, d(x,y)^\beta \, \sum_{i=0}^{n-1}
\theta^i \le c \,d(x,y)^\beta,
$$
where
$$
c= \text{\small {$\frac{c'}{1-\theta} $}} \,\le\,
\text{\small {$ \frac { (\rho_i-2\epsilon)^{-1} K_\beta (\mathcal{F})}{1-\sigma \nu^{\beta}}$}} =k_3' \, K_\beta (\mathcal{F})
\quad\text{with}\quad k_3'= (\rho_i-2\epsilon)^{-1} (1-\sigma \nu^{\beta})^{-1}.
$$
By \eqref{hol def F} the sequence $\{(\mathcal{F}^n_y)^{-1}\circ \mathcal{F}^n_x\}$ converges to $\mathcal{H}^{\mathcal{F},s}_{xy}$
(in fact the estimates imply that it is Cauchy) and the limit satisfies
$$
\| \mathcal{H}^{s}_{\,x,y} - \text{Id} \,\| \le c \, d (x,y)^{\beta} \;\text{ for any } x\in \mathcal{M} \text{ and } y\in W^s_{\text{loc}}(x). $$ By Lemma \ref{Bi est} we have $ K_\beta (\mathcal{F}) =K_\beta(\mathcal{B}_i) \le k_2 \, K_\beta (\mathcal{B}) $ and we conclude that
$$
\| \mathcal{H}^{s}_{\,x,y} - \text{Id} \,\| \le k_3 \,K_\beta (\mathcal{B}) \, d (x,y)^{\beta} \;\text{ for any } x\in \mathcal{M} \text{ and } y\in W^s_{\text{loc}}(x). $$ This completes the proof of Lemma \ref{Hi est} \end{proof}
\section{Proof of Theorem \ref{HolderConjugacy}} \label{proof HC} Any two continuous conjugacies between $f$ and $A$ differ by an element of the centralizer of $A$. By \cite[Corollary 1]{W}, any homeomorphism commuting with an ergodic, in particular hyperbolic, automorphism $A$ is an affine automorphism, and hence all conjugacies have the same regularity.
First, using Theorem~\ref{constant cocycle} we will show in Section \ref{C1H} that $H$ is a $C^{1+\text{H\"older}}$ diffeomorphism, and moreover the H\"older constant of its derivative satisfies the estimate \begin{equation} \label{hkDH}
K_\beta(DH) \le k \, \| DH\|_{C^0} \, \|{f-A}\|_{C^{1+\beta}}. \end{equation}
This part does not rely on closeness of $H$ to the identity and the estimate applies to any conjugacy $H$. Then in Section \ref {PGE} we use \eqref{hkDH} and an interpolating inequality to obtain the desired estimate \eqref{C1H est} of $\|{H-I}\|_{C^{1+\beta}}$ for the conjugacy $C^0$ close to the identity.
\subsection{Proving that $H$ is a $C^{1+\text{H\"older}}$ diffeomorphism} \label{C1H} $\;$\\ First we recall some properties of a map $g \in W^{1,q}(\mathbb R^N,\mathbb R^N)$, $q>N$, which also extend to the case when $g \in W^{1,q}(\mathbb T^N,\mathbb T^N)$. It is well known that, as a consequence of Morrey's inequality, for any such $g$ the Jacoby matrix of weak partial derivatives gives the differential $D_x g$
for almost every $x$ with respect to the Lebesgue measure $\mu$. Also, any such $g$ satisfies
{\it Lusin's N-property}\, \cite{MM} that $\mu (E)=0$ implies $\mu (g(E))=0$, as well as
{\it Morse-Sard property}\, \cite{P} that $\mu (g(\EuScript{C}_g))=0$ for the set of critical points of $g$ $$ \EuScript{C}_g=\{x\in \mathbb T^N:\; D_x g \text{ exists but is not invertible}\}, $$ see also \cite{KK} for sharper results and further references.
\vskip.1cm
Now we assume that $H\in W^{1,q}$ with $q>N$, so that the differential $D_x H$ exists $\mu$-a.e., and for the set $$ G_{ H}=\{x\in \mathbb T^N:\; D_x H \text{ exists}\} \;\text{ and its complement }\;E_{H}=\mathbb T^N \setminus G_{ H} $$ we have $\mu(G_H)$=1 and $\mu(E_H)=0$. Further $G_{ H}=\EuScript{C}_{ H}\cup R_{ H}$ is the disjoint union of two measurable sets, the critical set $\EuScript{C}_{ H}$ and the regular set $$ R_{ H}=\{x\in \mathbb T^N: \;D_x H \text{ is invertible}\}. $$ Since $f$ and $A$ are diffeomorphisms, it follows from the conjugacy equation $H\circ f=A\circ H$ that the sets $G_{ H}$, $\EuScript{C}_{H}$, and $R_{ H}$ are $f$-invariant. Further, differentiating the equation on the set $G_H$ we obtain \begin{equation} \label{DH2} D_{fx} H \circ D_x f=A \circ D_x H. \end{equation} Denoting $\mathcal{C}(x)= D_x H$ on the set $R_H$ we obtain the conjugacy equation over $f$ \begin{equation} \label{ABconj2} A=\mathcal{C} (fx) \circ \mathcal{B}_{ x} \circ \mathcal{C}(x)^{-1} \quad \text{for cocycles $\; \mathcal{B}_x=D_{x} f$ and $\mathcal{A}_x=A.$} \end{equation}
Now we show that $ \mu (R_H)=1$ and also that $f$ preserves a measure $\tilde \mu$ equivalent to $\mu$. Since $\mu (E_H)=0$, the Lusin's N-property of $H$ yields $\mu ( H(E_H))=0$. Also, we have
$\mu ( H(\EuScript{C}_H))=0$ by the Morse-Sard property. Hence for $R'_H=H(R_H)$ we
have $\mu(R'_H)=1$.
Now we consider the measure $\tilde \mu=(H^{-1})_*(\mu)$ and note that $\tilde \mu(R_H)=1$ as $\mu(R'_H)=1$. Since $H$ is a topological conjugacy between $f$ and $A$, the measure $\tilde \mu$ is $f$-invariant and, in fact, is the Bowen-Margulis measure of maximal entropy for $f$, since $\mu$ is that for $A$. Indeed, denoting the topological entropy by $\mathbf{h}_{top}$ and metric entropy with respect to $\tilde \mu$ by $\mathbf{h}_{\tilde \mu}$ we get $$
\mathbf{h}_{\tilde \mu}(f)= \mathbf{h}_{\mu}(A) = \mathbf{h}_{top}(A) =\mathbf{h}_{top}(f). $$
In particular, $\tilde \mu$ is ergodic with full support and local product structure.
Since $\mathcal{C}$ is a conjugacy between $\mathcal{B}$ and $A$ on $R_H$ with $\tilde \mu(R_H)=1$, by Lemma \ref{equal exp} we obtain that the Lyapunov exponents $\lambda_i^{f,\tilde \mu}$ of $\tilde \mu$ for the cocycle $\mathcal{B}=D f$ are equal to the Lyapunov exponents $\lambda_i^{A}$ of $A$. Hence
the sum of positive Lyapunov exponents (counted with multiplicities) for $\tilde \mu$ equals its entropy $$
\mathbf{h}_{\tilde \mu}(f)= \mathbf{h}_{\mu}(A)= \sum_{\lambda_i^{A} >0} \lambda_i ^A
=\sum_{\lambda _i^{f,\tilde \mu} >0}\lambda _i^{f,\tilde \mu}. $$ Thus we have equality in the Pesin-Ruelle formula, which implies that $\tilde \mu$ has absolutely continuous conditional measures on the unstable foliation of $f$ \cite{L}. Similarly, equality of the negative Lyapunov exponents yields that $\tilde \mu$ has absolutely continuous conditional measures on the stable foliation of $f$. We conclude that $\tilde \mu$ itself is absolutely continuous. Moreover, the density $\sigma (x)=\frac{d \tilde \mu}{ d \mu}$ is smooth and positive as a measurable solution of the coboundary equation $\sigma (fx)\sigma (x)^{-1}=\det Df(x)$. Thus $\tilde \mu$ is equivalent to $\mu$, so that $\tilde \mu(R_H)=1$ implies $ \mu(R_H)=1$.
Provided that $\|A-\mathcal{B}_x\|_{C^0}=\|A-D_xf\|_{C^0} \le \|A-f\|_{C^1}< \delta$, where $\delta>0$ is from Theorem~\ref{constant cocycle}, we can apply this theorem with $f$ and $\tilde \mu$ to obtain that $$\mathcal{C}(x)= D_x H :\mathbb T^N \to GL(N,\mathbb R)$$
coincides with a H\"older continuous function almost everywhere with respect to $\tilde \mu$ and hence $\mu$. Since $H\in W^{1,q}$ we conclude that
$H$ is $C^{1+\text{H\"older}}$. Also, since $(D_x H)^{-1}= \mathcal{C}(x)^{-1}$ exists and is also H\"older continuous we see that $H$ is $C^{1+\text{H\"older}}$ diffeomorphism. Further, Theorem~\ref{constant cocycle} gives us the estimate \eqref{hkDH}, which we will use to obtain the desired estimate for $\|{H-\text{Id}}\|_{C^{1+\beta}}$ in Section \ref{PGE}. This completes the proof that $H$ is $C^{1+\text{H\"older}}$ diffeomorphism assuming that $H\in W^{1,q}$.
\vskip.15cm
Now we consider the case when $\tilde H = H^{-1}$ is in $W^{1,q}$ and hence
$D_x \tilde H$ exists $\mu$-a.e. We similarly define the sets $G_{\tilde H}$, $E_{\tilde H}$, $\EuScript{C}_{\tilde H}$, and $R_{\tilde H}$, which are measurable and $A$-invariant. Hence by ergodicity of $A$
the set $R_{\tilde H}$ must be null or co-null for $\mu$. If $\mu (R_{\tilde H})=0$ then $\mu ( \tilde H(R_{\tilde H}))=0$ by the Lusin's N-property of $\tilde H$,
but this is impossible since $\mu (\tilde H(E_{\tilde H}))=0$ by the Lusin's N-property and
$\mu (\tilde H(\EuScript{C}_{\tilde H}))=0$ by the Morse-Sard property. Hence $\mu(R_{\tilde H})=1$.
Then for $R_{\tilde H}'= \tilde H (R_{\tilde H} )$ we have $\tilde \mu(R_{\tilde H}')=1$, where
as before $\tilde \mu=\tilde H_*(\mu)$ is the measure of maximal entropy for $f$. Now the Lusin's N-property of $\tilde H$ yields that $\tilde \mu$ is absolutely continuous and then equivalent to $\mu$. Hence we also have $ \mu(R_{\tilde H}')=1$. Since $H= \tilde H^{-1}$ is a homeomorphism, and $D_x \tilde H$ is invertible for $x\in R_{\tilde H}$, it follows that $D_y H=(D_x \tilde H)^{-1}$ is the differential of $H$ for each $y=\tilde H (x)$ in $R_{\tilde H}'$.
Therefore, we can again differentiate $H\circ f=A\circ H$ to obtain \eqref{ABconj2} and then the conjugacy equation \eqref{ABconj2} with $\mathcal{C}(x)= D_x H$ on the set $R_{\tilde H}'$ of full measure for both $\mu$ and $\tilde \mu$. Then by Theorem~\ref{constant cocycle} applied with $f$ and $\tilde \mu$ we obtain that $\mathcal{C}(x)= D_x H $ is H\"older on $\mathbb T^N$ and hence so is $\mathcal{C}(y)^{-1}=D_x \tilde H$. Since $\tilde H = H^{-1}$ is in $W^{1,q}$ we conclude that $H^{-1}$ is $C^{1+\text{H\"older}}$ diffeomorphism. In this case we also get \eqref{hkDH}.
\subsection{Estimating $\|{H-I}\|_{C^{1+\beta}}$} \label{PGE} We showed that any conjugacy $H$ is a $C^{1+\text{H\"older}}$ diffeomorphism satisfying \eqref{hkDH}. Now we prove estimate \eqref{C1H est} for the conjugacy $H$ that is $C^0$ close to the identity.
Any two conjugacies in the homotopy class of the identity differ by a composition with an affine automorphism commuting with $A$, which is translation $T_v(x) = x+v$, where $v\in \mathbb T^N $ is a fixed point of $A$. It is well known that if $f$ is $C^1$-close to $A$, then it has a unique fixed point $p$ which is the perturbation of $0$. More precisely, there are $0<\delta(A),r(A)<1/5$ and
$k(A)$ so that for each $f$ satisfying $\|{f-A}\|_{C^{1}}<\delta(A)$ there is a unique fixed point $p=f(p)$ with $d(p,0)<r(A)$ and it satisfies
$$d(p,0)\le k(A) \|{f-A}\|_{C^{0}}.$$
Since $H$ maps fixed points of $f$ to those of $A$ we see that if $ \|{H-I}\|_{C^{0}}<r(A)$ then it is in the homotopy class of the identity and satisfies $H(p)=0$.
Replacing $f$ by $\tilde f= T_{-p} \circ f \circ T_p$ we can change $p$ to $0$. Since for $\tilde f(x)=f(x+p)-p$ we have that
$$\|D \tilde f -A\|_{C^k}=\|D f -A\|_{C^k}\quad\text{for any $k\ge0$,} $$
and so only $\|{f-A}\|_{C^{0}}$ is affected by this change. Moreover, if we write $f=A+R$,
then $$ \tilde f(x)-A(x)=A(x+p)+R(x+p)-p-A(x)=R(x+p)+A(p)-p $$ and hence $$
\| \tilde f -A\|_{C^0}\le \| R\|_{C^0}+ \|A(p)-p\|=\| f -A\|_{C^0}+ \|A(p)-f(p)\| \le 2\| f -A\|_{C^0}. $$
Thus $\|{\tilde f-A}\|_{C^{1+\beta}}\le 2 \|{f-A}\|_{C^{1+\beta}}$. Also, if $\tilde H $ is the corresponding conjugacy between $\tilde f$ and $A$ then $H(x)=\tilde H (x-p)$ and hence
$$\|{H-\text{Id}}\|_{C^{1+\beta}} \le \|{\tilde H-\text{Id}}\|_{C^{1+\beta}} + d(p,0) \le \|{\tilde H-\text{Id}}\|_{C^{1+\beta}} + k(A) \|{f-A}\|_{C^{0}} $$ Thus the estimate \eqref{hkDH} for $\tilde H$ via $\tilde f$ would yield the corresponding estimate for $H$ via $f$. So without loss of generality we will assume that $$f(0)=0 \;\text{ and }\;H(0)=0. $$
Now we recall how the conjugacy equation $H\circ f=A\circ h$ can be rewritten using lifts. We denote by $\bar f$ and $\bar H$ the lifts of $f$ and $H$ to $\mathbb R^N$ satisfying $\bar f(0)=0 \;\text{ and }\; \bar H(0)=0$ so that we have $\bar H\circ \bar f=A\circ \bar H$ where all maps are $\mathbb R^N \to \mathbb R^N$. Since $H$ is homotopic to the identity and $f$ is homotopic to $A$ we can write $$ \bar H=\text{Id} + h \quad\text{and}\quad \bar f=A+ R, $$ Then the commutation relation on $\mathbb R^N$ $$ (\text{Id} + h)\circ(A + R)=A\circ (\text{Id} + h)\quad\text{yields}\quad
h= A^{-1} ( h\circ \bar f)+A^{-1} R. $$ Since $ h, R:\mathbb R^N \to \mathbb R^N$ are $\mathbb Z^N$-periodic we can view them as $$ h=H-\text{Id} :\; \mathbb T^N \to \mathbb R^N \quad\text{and}\quad R=f-A \; :\; \mathbb T^N \to \mathbb R^N $$ and rewrite the conjugacy equation as one for $\mathbb R^N$-valued functions on $\mathbb T^N$ \begin{equation} \label{conj h} h= A^{-1} (h\circ f)+A^{-1} R. \end{equation}
Using the $A$-invariant splitting $\mathbb R^N= E^u\oplus E^s$ we define the projections $h_*$ and $R_*$ of $h$ and $R$ to $E^*$, where $*=s,u$, and obtain \begin{equation} \label{H_*}
h_*= A_*^{-1} (h_*\circ f)+A^{-1}_* R_*, \quad \text{where } A_*=A|_{E^*}. \end{equation} Thus $h_*$ is a fixed point of the affine operator \begin{equation} \label{T*} T_* (\psi )=A_*^{-1} (\psi \circ f)+A^{-1}_* R_* \end{equation}
Since $\|A_u^{-1}\|<1$, the operator $T_u$ is a contraction on the space $C^0(\mathbb T^d,E^u)$, and thus $h_u$ is its unique fixed point \begin{equation} \label{Hu} h_u = \lim_{m\to \infty} T_u^m (0) \,= \sum_{m=0}^\infty A_u^{-m} (A^{-1}_u R_u \circ f^m). \end{equation} Hence \begin{equation} \label{hu norm}
\|h_u\|_{C^0} \,\le\, \sum_{m=0}^\infty \|A_u^{-1}\|^{m+1} \|R_u\|_{C^0}
\,\le\, k\, \|R_u\|_{C^0} \,\le\, k \, \|A-f\|_{C^0} . \end{equation}
Similarly, $h_s$ is the unique fixed point of contraction $T_s^{-1}$ and hence satisfies a similar estimate. Combining them we conclude that \begin{equation} \label{h norm}
\|H-\text{Id} \|_{C^0}=\|h\|_{C^0} \le k _0 \|R\|_{C^0} = k_0 \|A-f\|_{C^0} . \end{equation}
\vskip .2cm
Now we estimate $\|{H-\text{Id}}\|_{C^{1+\beta}}$ using \eqref{h norm}, \eqref{hkDH}, and the following elementary interpolation lemma. We note that $DH=\text{Id}+Dh$, so that $K_{\beta}({Dh}) =K_{\beta}({DH})$.
\begin{lemma}\label{Int} If $\,h:\mathbb T^N \to \mathbb R^N$ satisfies $K_{\beta}({Dh})\le K$ then $$
\| Dh\|_{C^0} \le 8 \,\| h\|_{C^0}^{\beta/(1+\beta)}K^{1/(1+\beta)}. $$ \end{lemma} \begin{proof}
Denote $b=\| Dh\|_{C^0}$ and choose $x\in \mathbb T^N$ such that $\|D_x h\|=b$. Then for some unit vectors $u,v\in \mathbb R^N$ we have $(D_x h)u=bv.$ For $y\in \mathbb T^N$ let $b_y=\langle (D_y h) u, v \rangle$, so $b_x=b$. Then $$
|b-b_y| \le \|(D_x h) u -(D_y h) u\| \le K \, d(x,y)^\beta \le b/2 \;\; \text{ if } \;\; d(x,y) \le (b/2K)^{1/\beta} $$ and hence $b_y\ge b/2$ for such $y$. Consider $y(t)=x+tu$, with $0\le t \le t_0=(b/2K)^{1/\beta}$, and $g(t)=\langle h(y(t)), v \rangle$. Then $$g'(t)=\langle (D_y h) u, v \rangle=b_{y(t)} \ge b/2, $$
and hence by integrating we get $ bt_0/2 \le g(t_0)-g(0)$. Since $|g(t_0)-g(0)| \le 2\| h\|_{C^0}$
we obtain $bt_0\le 4\| h\|_{C^0}$. Substituting $t_0=(b/2K)^{1/\beta}$ we obtain $$
b(b/2K)^{1/\beta}\le 4\,\| h\|_{C^0} \;\Rightarrow\; b ^{(1+\beta)/\beta} \le 4 \,\| h\|_{C^0} (2K)^{1/\beta}
\;\Rightarrow\; b\le 8 \,\| h\|_{C^0}^{\beta/(1+\beta)}K^{1/(1+\beta)} $$ as $\,4^{\beta/(1+\beta)}2^{1/(1+\beta)}<8$. \end{proof}
\vskip.3cm We denote $a=\|h\|_{C^0}$, $b=\| Dh\|_{C^0}$, and $d=\|{f-A}\|_{C^{1+\beta}}$. Then
$$\| DH\|_{C^0}=\| \text{Id} +Dh\|_{C^0}\le 1+b, $$ and hence \eqref{hkDH} implies that \begin{equation}\label{1+b} K=K_\beta(Dh)=K_\beta(DH)\le k (1+b) d. \end{equation}
Also, by \eqref{h norm} we have $a=\|h\|_{C^0}\le k_0d$. Then Lemma \ref{Int} gives $$b\le 8 (kd)^{\beta/(1+\beta)}(k (1+b) d)^{1/(1+\beta)}< k_1 \,d (1+b)^{1/(1+\beta)}. $$ It follows that $b$ is bounded by some $k_2$ if $d\le1$. Then \eqref{1+b} implies that $$K=K_\beta(Dh)\le k_3 d. $$
With this $K$ Lemma \ref{Int} gives $$b\le 8 (kd)^{\beta/(1+\beta)}(k_3 d)^{1/(1+\beta)} \le k_4 d. $$ We conclude that
$$b=\| Dh\|_{C^0}< k_4 d, \quad a=\|h\|_{C^0}\le k_0\,d, \;\text{ and }\;K_\beta(Dh)\le k_3 d, $$ so that
$$\|{H-\text{Id}}\|_{C^{1+\beta}}=\|h\|_{C^{1+\beta}} \le k_5 d= k_5 \,\|{f-A}\|_{C^{1+\beta}}. $$ This completes the proof of Theorem \ref{HolderConjugacy}.
\section{Linearized conjugacy equation} \label{linearized}
In this section we begin the proof of Theorem \ref{th:4}, and in the next one we will complete it using an iterative process. In these sections we fix a hyperbolic matrix $A\in SL(N,\mathbb Z)$. We will use $K$ to denote any constant that depends only on $A$, and $K_{x}$ to denote a constant that also depends on a parameter $x$.
\subsection{Preliminaries}
Set $\tilde{A}=(A^\tau)^{-1}$ where $A^\tau$ denotes transpose matrix. We call $\tilde{A}$ the dual map on $\mathbb{Z}^N$. Since $A$ is hyperbolic so is $\tilde{A}$, and we denote its stable and unstable subspaces by $\tilde E^s$ and $\tilde E^u$. Thus there is $\rho>1$ ( $\rho < \min \{ \rho_{i_0+1}, \rho_{i_0}^{-1}\}$) such that \begin{align}\label{for:9}
\norm{\tilde{A}^kv}&\geq K\rho^k\norm{v}, \quad k\geq 0,\;v\in \tilde E^u, \\
\norm{\tilde{A}^{-k}v}&\geq K\rho^{k}\norm{v}, \quad k\geq 0,\;v\in \tilde E^s \notag. \end{align}
For a subspace $V$ of $\mathbb{R}^N$, we use $\pi_{V}$ to denote the (orthogonal) projection to $V$. For any integer vector $n\in \mathbb Z^N$ we write $n_s= \pi_{\tilde E^s}n$ and $n_u= \pi_{\tilde E^u}n$. Since $\tilde{A}\in SL(N,\mathbb Z)$ is hyperbolic, for any $0\ne n\in \mathbb Z^N$ both $n_s$ and $n_u$ are nonzero and there is a unique $k_0=k_0(n)\in \mathbb Z$ such that $$ \begin{aligned}
&\| \tilde{A}^kn_s \| \ge \| \tilde{A}^kn_u\| \quad \text{for all } k\le k_0\quad\text{and} \\
&\| \tilde{A}^kn_s \| < \| \tilde{A}^kn_u\| \quad \text{for all } k> k_0. \end{aligned} $$ The corresponding element $\tilde{A}^{k_0(n)} n$ on the orbit of $n$ will be called {\em minimal} and
\begin{equation} \label{Min} M=\{\tilde{A}^{k_0(n)} n:\;0 \ne n\in\mathbb{Z}^N\} \subset \mathbb{Z}^N\backslash 0.
\end{equation}
For any $n\in M$ we have $\|n_s\| \geq\frac 12 \|n\|$ and $\|\tilde{A}n_u\| > \frac 12 \|\tilde{A}n\|$.
For a function $\theta \in L^2(\mathbb T^N,\mathbb C)$ we denote its Fourier coefficients by $\hat \theta_n$, $n\in\mathbb{Z}^N$, so that $$\theta(x)=\sum_{n\in\mathbb{Z}^N}\widehat{\theta}_ne^{2\pi \mathbf{i} \, n\cdot x} \quad \text{in } L^2(\mathbb{T}^N).$$
We say that $\theta$ is \emph{excellent} (for $A$) if $\widehat{\theta}_n=0$ for all $n\notin M$.
\vskip.2cm
To simplify our estimates, instead of the standard Sobolev spaces we will work the spaces $\EuScript{H}^s(\mathbb{T}^N)$, $s>0$, defined as follows. A function $\theta\in L^2(\mathbb{T}^N)$ belongs to $\EuScript{H}^s(\mathbb{T}^N)$ if
\begin{align*}
\norm{\theta}_s\overset{\text{def}}{=}\sup_n\,\abs{\widehat{\theta}_n}\norm{n}^s+\abs{\widehat{\theta}_0}<\infty.
\end{align*} The following relations hold (see, for example, Section 3.1 of \cite{LLAVE}). If $\sigma>N+1$ and $r\in\mathbb{N}$, then for any $\theta\in C^r(\mathbb{T}^N)$ and $\omega\in \EuScript{H}^{r+\sigma}$
we have $\theta\in \EuScript{H}^{r}$ and $\omega\in C^r(\mathbb{T}^N)$ with estimates
\begin{align}\label{for:19}
\norm{\theta}_r\leq K\norm{\theta}_{C^r}\quad \text{and}\quad\norm{\omega}_{C^r}\leq K\norm{\omega}_{r+\sigma}.
\end{align}
For a vector-valued function $\theta :\mathbb T^N \to \mathbb C^m$ we denote its coordinate functions by $\theta _j$, $j=1,\dots,m$. We say that $\theta$ is in $\EuScript{H}^s(\mathbb{T}^N)$ if each $\theta_j$ is in $\EuScript{H}^s(\mathbb{T}^N)$ and set
\begin{align*}
\norm{\theta }_s\overset{\text{def}}{=}\max_{1\leq j\leq m}\,\norm{\theta _j}_s,\quad \widehat{\theta }_n\overset{\text{def}}{=} ((\widehat{\theta _1})_n,\dots,(\widehat{\theta _m})_n) \quad \text{ for any $n\in \mathbb{Z}^N$}
\end{align*} We say that $\theta $ is excellent if $\theta _j$ is excellent for each $j$.
\subsection{Twisted cohomological equation over $A$ in high regularity}$\;$
A crucial step in the iterative process is solving the twisted cohomological equation \begin{equation}
A\omega-\omega\circ A=\theta \end{equation}
over $A$, which can be viewed as the linearized conjugacy equation.
In this section we give preliminary results on solving this equation in high regularity.
We start with a scalar cohomological equation over $A$ twisted by $\lambda\in\mathbb{C}\backslash \{0,1\}$, \begin{align} \label{lambda twist}
\lambda\omega-\omega\circ A=\theta. \end{align} The next lemma shows that the obstructions to solving it in $C^\infty$ category
are sums of Fourier coefficients of $\theta$ along the orbits of $\tilde{A}$. Moreover, for any $C^\infty$
function $\theta$ there is a well behaved splitting $\theta=\theta^\iota+\theta^*$, where $\theta^\iota$ can be view as a projection to the space of twisted coboundaries and $\theta^*$ as the error.
A similar result was proved for ergodic toral automorphisms in \cite{Damjanovic4} and used for establishing $C^\infty$ local rigidity of some partially hyperbolic $\mathbb{Z}^k$ actions. We prove the result for hyperbolic case to keep our exposition self-contained and get a better constant $\sigma (\lambda)$.
\begin{lemma}\label{le:3} For a function $\theta:\mathbb{T}^N\to \mathbb{C}$ in $\EuScript{H}^{a}(\mathbb T^N)$ and $\lambda\in\mathbb{C}\backslash \{0,1\}$ we define
\begin{align*}
D_\theta(n)=\sum_{i=-\infty}^\infty\lambda^{-(i+1)}\widehat{\theta }_{\tilde{A}^{i}n}. \end{align*}
Suppose $a\geq\sigma(\lambda)=\frac{|\log |\lambda| |}{\log\rho}+1$, where $\rho>1$ is the expansion rate of $\tilde{A}$ from \eqref{for:9}. Then \begin{itemize}
\item[{\bf (i)}] The sum $D_\theta(n)$ converges absolutely for any $n\neq0$; moreover the function \begin{align*}
\theta^*\overset{\text{def}}{=}\sum_{n\in M}D_\theta(n)e^{2\pi \mathbf{i} \, n\cdot x}, \end{align*}
where $M$ is from \eqref{Min}, is in $\EuScript{H}^{a}(\mathbb T^N)$ with the estimate $\norm{\theta^*}_{a}\leq K_{a,\lambda}\norm{\theta}_{a}.$
\item[{\bf (ii)}] If $D_\theta(n)=0$ for any $n\neq0$, then the equation \eqref{lambda twist}
has a solution $\omega\in\EuScript{H}^{a}(\mathbb T^N)$ with the estimate \begin{align*}
\norm{\omega}_{a}&\leq K_{r,\lambda}\norm{\theta}_{a}.
\end{align*}
\item[{\bf (iii)}] If the equation \eqref{lambda twist} has a solution $\omega\in\EuScript{H}^{\sigma(\lambda)}(\mathbb T^N)$, then $D_\theta(n)=0$ for any $n\neq0$.
\item[{\bf (iv)}] For $\theta^\iota\overset{\text{def}}{=}\theta-\theta^*$ the equation:
\begin{align*}
\lambda\omega-\omega\circ A=\theta^\iota \end{align*} has a solution $\omega\in\EuScript{H}^{a}(\mathbb T^N)$ with the estimate $\; \norm{\omega}_{a}\leq K_{r,\lambda}\norm{\theta}_{a}. $ \end{itemize}
\end{lemma}
\begin{remark}\label{remark:1} We emphasize that the existence of $\theta^*$ requires a high regularity of $\theta$. In fact, for any $b\leq \sigma (\lambda)$, we have to estimate it as $\norm{\theta^*}_{b}\leq K_{\lambda}\norm{\theta}_{\sigma(\lambda)}.$ \end{remark}
\begin{proof} We define $$ D_\theta(n)_+=\sum_{i\geq 1}\lambda^{-(i+1)}\widehat{\theta }_{\tilde{A}^{i}n}
\quad\text{and}\quad
D_\theta(n)_-=-\sum_{i\leq 0}\lambda^{-(i+1)}\widehat{\theta }_{\tilde{A}^{i}n}. $$
{\bf (i)}. Let $n\in M$. The inequality $\|\pi_{\tilde E^s}(n)\|\ge \frac 12 \|n\|$ we obtain \begin{align}\label{for:26}
|D_\theta(n)_-|&\,\leq\, \norm{\theta}_a\sum_{i\leq 0}\,|\lambda|^{-(i+1)}\,\norm{\tilde{A}^in}^{-a}\leq\, \norm{\theta}_a\sum_{i\leq 0}\,|\lambda|^{-(i+1)} \,\norm{\pi_{\tilde E^s}(\tilde{A}^in)}^{-a}\notag\\
&\leq\,
\norm{\theta}_aC^{-a}\sum_{i\leq 0}\,|\lambda|^{-(i+1)}\rho^{ia}\,\norm{\pi_{\tilde E^s}(n)}^{-a}\overset{\text{(1)}}{\leq} \,K_{a,\lambda}\norm{\theta}_a\,\norm{n}^{-a}. \end{align}
Here in $(1)$ convergence is guaranteed by $a>{\text {\small $\frac{|\log |\lambda| |}{\log\rho}$}}$. The sum $D_\theta(n)_+$ can be estimated similarly using the inequality
$\|\pi_{\tilde E^u}(\tilde{A}n)\|\ge \frac 12 \|\tilde{A}n\|$. Hence we get \begin{align*}
\norm{\theta^*}_{a}&\leq K_{a,\lambda}\norm{\theta}_{a}.
\end{align*} For any $z\in \mathbb{Z}^N$ and $k\in \mathbb{Z}$, we see that \begin{align}\label{for:29}
D_\theta(\tilde{A}^kz)=\lambda^{k}D_\theta(z). \end{align} This shows that $D_\theta(n)$ converges absolutely for any $n\neq0$.
\vskip.2cm
{\bf (ii)} In the dual space the equation $\lambda\omega-\omega\circ A=\theta$ has he form \begin{align*}
\lambda\widehat{\omega}_n-\widehat{\omega}_{\tilde{A}n}=\widehat{\theta}_n,\qquad \forall\,n\in\mathbb{Z}^N. \end{align*} For $n=0$, we let $\widehat{\omega }_0=\frac{\widehat{\theta }_0}{\lambda-1}$. For any $n\neq0$, let $\widehat{\omega }_n=D_\theta(n)_-$. Then $\omega=\sum_{n\in \mathbb{Z}^N}\widehat{\omega }_ne^{2\pi \mathbf{i}}$
is a formal solution. Next, we obtain its Sobolev estimates.
If $\|\pi_{\tilde E^s}(n)\|\ge \frac 12 \|n\|$, then from \eqref{for:26} we have \begin{align}\label{for:27}
|\widehat{\omega }_n|\cdot \norm{n}^{a}\leq K_{a,\lambda}. \end{align}
If $\|\pi_{\tilde E^u}(\tilde{A}n)\|\ge \frac 12 \|\tilde{A}n\|$, then the assumption $D_\theta(n)=0$ implies that $\widehat{\omega }_n=D_\theta(n)_+$. The arguments in {\bf (i)} show that \eqref{for:27} still holds.
\vskip.2cm {\bf (iii)} By {\bf (i)} and \eqref{for:29} we have: for any $n\neq0$ \begin{align*}
D_\theta(n)=D_{\lambda\omega-\omega\circ A}(n)=\lambda D_{\omega}(n)-D_{\omega}(\tilde{A}n)=\lambda D_{\omega}(n)-\lambda D_{\omega}(n)=0. \end{align*}
\vskip.2cm {\bf (iv)} It is clear that $D_{\theta^\iota}(n)=D_{\theta-\theta^*}(n)=D_{\theta}(n)-D_{\theta^*}(n)=0$ for any $n\neq0$. Then the result follows from {\bf (ii)}. \end{proof}
Now we extend Lemma \ref{le:3} to the vector valued case. We consider the equation $$A_i\omega-\omega\circ A=\theta $$
with the twist given by the restriction $A_i=A|E^i$, where $E^i$, $i=1,\dots ,L$,\, is a subspace of the splitting \eqref{splitL}.
We note that any eigenvalue $\lambda$ of $A_i$ satisfies $|\lambda|=\rho_i$.
\begin{lemma}\label{le:2} Let $\rho>1$ be the expansion rate for $\tilde{A}$ from \eqref{for:9} and let \begin{equation}\label{sigma}
\sigma=\max_{i=1,\dots ,L} \left( {\text {\small $\frac{|\log \rho_i |}{\log\rho}$}}+1\right)N+N+2. \end{equation} Then for any $i=1,\dots ,L$ and any $C^\infty$ map $\theta:\mathbb{T}^N\to \mathbb{C}^{N_i}$, there is a splitting of $\theta$
\begin{align*}
\theta=\theta^\iota+\theta^*
\end{align*}
such that the equation:
\begin{align}\label{for:1}
A_i\omega-\omega\circ A=\theta^\iota \end{align} has a $C^\infty$ solution $\omega$ with estimates \begin{align*}
\norm{\omega}_{C^r}\leq K_r\norm{\theta}_{C^{r+\sigma }},\qquad \forall\,r\geq0;
\end{align*}
and $\theta^*:\mathbb{T}^N\to \mathbb{C}^{N_{i}}$ is an excellent $C^\infty$ map so that for any $r\geq0$
\begin{align*}
\norm{\theta^*}_{C^r}\leq K_{r}\norm{\theta}_{C^{r+\sigma}}\qquad \text{and} \qquad
\norm{\theta^*}_{r}\leq K_{r}\norm{\theta}_{r+\sigma-2-N}.
\end{align*} \end{lemma}
\begin{proof} If $A_i$ is semisimple, then the conclusion follows directly from Lemma \ref{le:3} as the equation \eqref{for:1} splits into finitely many equations of the type
\begin{align*}
\lambda_j\omega_j-\omega_j\circ A=(\theta_j)^\iota \end{align*} where $\theta_j$ is a coordinate function of $\theta$ and $\lambda_j$ is the corresponding eigenvalue of $A_i$.
If $A_i$ is not semisimple, we choose a basis in which $A_i$ is in its Jordan normal form with some nontrivial Jordan blocks. We note that the excellency of maps is preserved under the change of basis. Let $J=(J_{l,j})$ to be an $m\times m$ Jordan block of $A_i$ corresponding to an eigenvalue $\lambda$ with $|\lambda|=\rho_i$, that is, $J_{l,l}=\lambda$ for all $1\leq l\leq m$ and $\lambda_{l,l+1}=1$ for all $1\leq l\leq m-1$. Then equation \eqref{for:1} splits into equations of the form \begin{align}\label{for:4}
J\Omega-\Omega\circ A=\Theta^\iota, \end{align} corresponding to the Jordan blocks $J$. Each equation \eqref{for:4} further splits into the following $m$ equations:
\begin{align*}
\lambda\Omega_j-\Omega_j\circ A+\Omega_{j+1}&=(\Theta^\iota)_j, \qquad\text{and} \\
\lambda\Omega_m-\Omega_m\circ A&=(\Theta^\iota)_m=(\Theta_m)^\iota, \end{align*} $1\leq j\leq m-1$. For the $m$-th equation, Lemma \ref{le:3} gives the splitting \begin{align*}
\Theta_m=\lambda\Omega_m-\Omega_m\circ A+(\Theta^*)_m
\end{align*}
where $\Omega_m$, $(\Theta^*)_m=(\Theta_m)^*$, and $(\Theta^\iota)_m=\lambda\Omega_m-\Omega_m\circ A$ are $C^\infty$ functions satisfying the estimates:
\begin{align*}
\max\{\norm{(\Theta^*)_m}_{r},\,\norm{\Omega_m}_{r}\}&\leq K_{r,m}\norm{\Theta}_{r+\sigma(\rho_i)},\qquad \forall\,r\geq0
\end{align*}
and $\Theta_m^*$ is excellent.
Now we proceed by induction. Fix $1\leq k\leq m-1$ and assume that for all $k+1\leq j\leq m$ we already have the splitting \begin{align*}
\Theta_j=\lambda\Omega_j-\Omega_j\circ A+\Omega_{j+1}+(\Theta^*)_j
\end{align*}
where $\Omega_j$, $\Theta_j^*$, and $(\Theta^\iota)_j=\lambda\Omega_j-\Omega_j\circ A+\Omega_{j+1}$ are $C^\infty$ functions satisfying the estimates:
\begin{align}\label{for:2}
\max\{\norm{\Omega_j}_{r},\,\norm{(\Theta^*)_j}_{r}\}&\leq K_{r,j}\norm{\Theta}_{r+(m-j+1)\sigma(\rho_i)},\qquad \forall\,r\geq0
\end{align}
and $(\Theta^*)_j$ is excellent. By Lemma \ref{le:3} we obtain the splitting
\begin{align*}
\Theta_k-\Omega_{k+1}=\lambda\Omega_k-\Omega_k\circ A+(\Theta_k-\Omega_{k+1})^*
\end{align*}
where $\Omega_k$, $(\Theta^*)_k=(\Theta_k-\Omega_{k+1})^*$, and $(\Theta^\iota)_k=\lambda\Omega_k-\Omega_k\circ A+\Omega_{k+1}$ are $C^\infty$ functions satisfying the estimates following from \eqref{for:2}:
\begin{align*}
\max\{\norm{\Omega_k}_{r},\,\norm{(\Theta^*)_k}_{r}\}&\leq K_{r}\norm{\Theta_k-\Omega_{k+1}}_{r+\sigma(\rho_i)}\leq K_{r,k}\norm{\Theta}_{r+(m-k+1)\sigma(\rho_i)}, \quad \forall\,r\geq0
\end{align*}
and $(\Theta^*)_k$ is excellent. Let $\Omega$, $\Theta^\iota$ and $\Theta^*$ be maps with coordinate functions $\Omega_j$, $(\Theta^\iota)_j$ and $(\Theta^*)_j$, $1\leq j\leq m$ respectively. Hence we show that
there is a splitting of $\Theta$
\begin{align*}
\Theta=\Theta^\iota+\Theta^*
\end{align*}
such that the equation \eqref{for:4} has a $C^\infty$ solution $\Omega$ with estimates. \begin{align*}
\max\{\norm{\Theta^*}_{r},\,\norm{\Omega}_{r}\}&\leq K_{r}\norm{\Theta}_{r+m\sigma(\rho_i)}, \quad \forall\,r\geq0
\end{align*}
This can be repeated for all corresponding blocks of $A$. Since the maximal size of a Jordan block is bounded by $N$, we obtain estimates for the $\norm{\cdot}_r$ norms of $\omega$ and $\theta^*$. This implies estimates for the $\norm{\cdot}_{C^r}$ norms as well by \eqref{for:19}. \end{proof}
\subsection{Main result on the linearized equation.}
The next theorem is our main result on solving the linearized equation. It plays the crucial role in the inductive step of the iterative process, Proposition \ref{po:1}. The goal of the inductive step is, given a $C^{1}$ conjugacy $H$ between $A$ and its perturbation $f$, to construct a smaller perturbation $\tilde f$ which is smoothly conjugate to $f$ by $\tilde H$. The conjugacy $\tilde H$ is constructed in the form $\tilde H = I - \omega$, where $\omega$ is a $C^\infty$ approximate solution of the linearized equation given by Theorem \ref{th:3}. The $C^{1}$ conjugacy $H$ is upgraded to $C^{1+a}$ by Theorem \ref{HolderConjugacy}. It yields an {\it approximate} $C^{1+a}$ solution $\mathfrak{h}=H-I$ of the linearized equation \eqref{for:11}. This necessitates the introduction of the error term $\Psi$ in the assumption of the theorem.
\begin{theorem}\label{th:3} Let $A$ be weakly irreducible hyperbolic automorphism of $\mathbb T^N$. Suppose that \begin{align}\label{for:11} A \mathfrak{h}-\mathfrak{h}\circ A=\mathcal{R}+\Psi, \end{align} where maps $\mathfrak{h},\Psi :\mathbb T^N\to \mathbb R^N$ are $C^{1+a}$ and $\mathcal{R}:\mathbb T^N\to \mathbb R^N$ is $C^{\infty}$. \vskip.1cm
Then there exist $C^\infty$ maps $\omega, \Phi :\mathbb T^N\to \mathbb R^N$ satisfying the equation \begin{align}\label{for1}
\mathcal{R}=A \omega-\omega\circ A+\Phi \end{align} and the estimates \begin{gather*}
\norm{\omega}_{C^r}\leq K_r\norm{\mathcal{R}}_{C^{r+\sigma}}\\
\norm{\Phi}_{C^0}\leq K_{l,a}(\norm{\Psi}_{C^{1+a}})^{\frac{l-2-N}{l+N}}(\norm{\mathcal{R}}_{C^{l+\sigma}})^{\frac{2N+2}{l+N}}
\end{gather*} for any $r\geq0$ and $l>N+2$, where $\sigma$ is given by \eqref{sigma}.
\end{theorem} For traditional KAM iteration scheme, the convergence requires the error $\Phi$ in solving the twisted coboundary \eqref{for1} to be small compared with $\mathcal{R}$. This is established by showing that $\Phi$ is tame with respect to $\Psi$, which is almost quadratically small with respect to $\mathcal{R}$. Tameness means that the $C^r$ norm of $\Phi$ can be bounded by the $C^{r+p}$ norm of $\Psi$, where $r$ is arbitrarily large while $p$ is a constant.
One difficulty in our setting is that the estimate of $\Phi$ depends on $\Psi$ and $\mathcal{R}$ rather than on $\Psi$ only. This results in technical issues in proving convergence of the iterative procedure, and so the traditional KAM scheme fails to work. We resolve this issue by introducing a parameter $l$ when estimating $\norm{\Phi}_{C^0}$. If the parameters are well chosen, the constructed approximation behaves as if it were tame.
The main difficulty in estimating $\Phi$ in our setting is that low regularity of $\mathfrak{h}$ yields smallness of $\Psi$ only in $C^{1+\text{H\"{o}lder}}$ norm, see Lemma \ref{le:7} and equation \eqref{for2}. This does not allow us to directly estimate orbit sums of Fourier coefficients and split $R$ into a smooth coboundary $R^\iota=A \omega-\omega\circ A$ and an error term $R^*=\Phi$, see Remark \ref{remark:1}. To overcome this problem we use the splitting $\mathbb R^N=\oplus E^i$ to decompose the equation \eqref{for:11} and then differentiate $i^{th}$ component along directions in $E^i$. This allows us to ``balance" the twist (up to a polynomial growth of Jordan blocks) and
analyze the differentiated equation using H\"older regularity. This is done in the following
Lemma \ref{cor:2}. After that, we establish Lemma \ref{le:4} to relate Fourier coefficients of a function and its directional derivatives. We then complete the proof of Theorem \ref{th:3} in Section \ref{proof th:3}.
\vskip.3cm
Now we begin the analysis of the differentiated equation \eqref{for:11}. For any $1 \le i\le L$ and any unit vector $u_0 \in E^i$, we consider unit vectors $u_k$ and scalars $a_k$, $k\in \mathbb Z$, given by \begin{equation}\label{u_k} u_k= \text{\small $\frac{A_i^ku_0}{\norm{A_i^ku_0}}$} \quad \text{and} \quad a_k=\norm{A_iu_k}= \text{\small $\frac{\norm{A_i^{k+1}u_0}}{\norm{A_i^ku_0}}$}\quad \text{so that} \quad A_iu_k=a_ku_{k+1}. \end{equation} We define a sequence of matrices $P_k\in GL(N_i,\mathbb R)$ which commute with $A_i$ and satisfy the recursive equation \begin{equation}\label{P_k} P_{k+1}=a_kA_i^{-1}P_k. \end{equation} Specifically, we set \begin{align}\label{for:20} P_0=\text{Id} \quad\text{and}\quad P_k=&\left\{\begin{aligned} &\,a_0\cdots a_{k-1}A_i^{-k}= \norm{A_i^{k}u_0}\,A_i^{-k},&\quad &k>0,\\ &\,(a_{-1}\cdots a_{-k})^{-1}A_i^{k}=\norm{A_i^{-k}u_0}\,A_i^{k},&\quad &k<0. \end{aligned}
\right. \end{align}
\begin{lemma}\label{cor:2} Let $\varphi_k: \mathbb{T}^N\to \mathbb R^{N_i}$ be a sequence of maps in $\EuScript{H}^{a}(\mathbb T^N)$, $a>0$, satisfying $\norm{\varphi_k}_a\leq \mathfrak{b}$ for all $k\in\mathbb{Z}$, let $P_k\in GL(N_i,\mathbb R)$ be as in \eqref{for:20}, and let
\begin{align*}
S(n)=\sum_{k\in\mathbb Z}P_k\,(\widehat{\varphi_k})_{\tilde{A}^kn}. \end{align*} \begin{itemize}
\item[{\bf (i)}]\label{for:15} For any $n\in M$ the sum $S(n)$ converges absolutely in $\mathbb C^{N_i}$ with the estimate \\ $\norm{S(n)}\leq K_{a}\mathfrak{b}\,\norm{n}^{-a}$.
\item[{\bf (ii)}]\label{for:16}
If $\mathfrak{h}_k: \mathbb{T}^N\to \mathbb R^{N_i}$ is another sequence in $\EuScript{H}^{a}(\mathbb T^N)$ so that for all $k\in\mathbb{Z}$ we have $\norm{\mathfrak{h}_k}_a\leq \mathfrak{c}$ and \begin{align}\label{for:13} A_i \mathfrak{h}_k-a_k\mathfrak{h}_{k+1}\circ A=\varphi_k, \end{align} then $S(n)=0$ for every $n\in M$. \end{itemize} \end{lemma}
\begin{proof} {\bf (i)}. Since all eigenvalues of $A_i$ have the same modulus $\rho_i$, we have \eqref{jordan}, and so there exists a constant $C$ such that all $P_k$ satisfy the polynomial estimate \begin{align}\label{for:3}
\norm{P_k}\le \norm{A_i^{k}}\cdot \norm{A_i^{-k}}\leq C(\abs{k}+1)^{2N}=:p(|k|),\quad \text{for all }k\in \mathbb Z. \end{align}
Let $n\in M.$ We write $\varphi_k=(\varphi_{k,1},\cdots,\varphi_{k,N_i})$ and set $$ S(n)_+=\sum_{k\geq 1}P_k\,(\widehat{\varphi_k})_{\tilde{A}^kn}
\quad\text{and}\quad
S(n)_-=\sum_{k\leq 0}P_k\,(\widehat{\varphi_k})_{\tilde{A}^kn}. $$
Using the assumption $\norm{\varphi_k}_a\leq \mathfrak{b}$, estimates \eqref{for:3} and \eqref{for:9}, and the inequality $\|\pi_{\tilde E^s}(n)\|\ge \frac 12 \|n\|$ we obtain \begin{align*}
\|S(n)_ -\|&\,\leq\,\sum_{k\leq 0}\,\norm{P_k}\max_{1\leq j\leq m}|(\widehat{\varphi_{k,j}})_{\tilde{A}^kn}|
\,\leq\, \sum_{k\leq 0}\,\norm{\varphi_k}_a\,\norm{P_k}\,\norm{\tilde{A}^kn}^{-a}\\
&\leq\, \mathfrak{b}\sum_{k\leq 0} \,p(|k|)\,\norm{\pi_{\tilde E^s}(\tilde{A}^kn)}^{-a}\,\leq\,
\mathfrak{b}C^{-a}\sum_{k\leq 0}\, p(|k|)\,\rho^{ka}\,\norm{\pi_{\tilde E^s}(n)}^{-a} \\ & \leq \,K_{a}\mathfrak{b}\,\norm{n}^{-a}. \end{align*} The sum $S(n)_+$ can be estimated similarly using the inequality
$\|\pi_{\tilde E^u}(\tilde{A}n)\|\ge \frac 12 \|\tilde{A}n\|$.
\vskip.2cm
{\bf (ii)} Let $n\in M$. From the equation \eqref{for:13} we obtain that for any $k\in \mathbb Z$ $$ P_k \, \varphi_k \circ A^{k}=P_k A_i \, \mathfrak{h}_k \circ A^{k}-a_k P_k \, \mathfrak{h}_{k+1}\circ A^{k+1} $$ Summing from $-m$ to $j$ and observing that the sum on the right is telescoping as $a_k P_k = A_i P_{k+1}=P_{k+1} A_i$ by the choice of $P_k$ in \eqref{P_k}, we obtain \begin{align*} \sum_{k=-m}^j P_k\; \varphi_k\circ A^{k}= A_iP_{-m}\mathfrak{h}_{-m}\circ A^{-m}-a_jP_j \, \mathfrak{h}_{j+1}\circ A^{j+1}. \end{align*} Taking Fourier coefficients and noting that $(\widehat{\theta\circ A^k})_n=\widehat{\theta}_{\tilde{A}^k n}$ we obtain \begin{align*} \sum_{k=-m}^j P_k(\widehat{\varphi_k})_{\tilde{A}^{k}n}= A_iP_{-m}(\widehat{\mathfrak{h}_{-m}})_{\tilde{A}^{-m}n}-a_j P_j(\widehat{\mathfrak{h}_{j+1}})_{\tilde{A}^{j+1}n}. \end{align*} Since the series $\sum_{k\in \mathbb Z} P_k(\widehat{\mathfrak{h}_k})_{\tilde{A}^{k}n}$ converges by part (i), we have $P_k(\widehat{\mathfrak{h}_k})_{\tilde{A}^{k}n}\to 0$ as $k\to \pm \infty$ and hence, as $a_k$ are bounded,
\begin{align*}
&a_j P_j (\widehat{\mathfrak{h}_{j+1}})_{\tilde{A}^{j+1}n}\to 0,\qquad \text{as }j\to\infty;\quad\text{and}\\
&A_iP_m(\widehat{\mathfrak{h}_{m}})_{\tilde{A}^{m}n}\to 0,\qquad \text{as }m\to-\infty.
\end{align*}
We conclude that $S(n)=0$. \end{proof}
\subsection{Directional derivatives}
In this section we establish some estimates for Fourier coefficients of a $C^1$ function $\theta :\mathbb T^N \to \mathbb R$ via Fourier coefficients of its directional derivatives along a subspace $E^i$ of the splitting \eqref{splitL}. This relies on weak irreducibility of $A$.
For any $v\in\mathbb R^N$ with $\norm{v}=1$, we denote the directional derivative of $\theta$ along $v$ by $\theta_v$.
\begin{lemma}\label{le:4} Let $A$ be a weakly irreducible integer matrix and let $v_{i,j}$, $j=1,\dots ,N_i$, be an orthonormal basis of a subspace $E^i$ from \eqref{splitL}. Then there exists a constant $K=K(A)$ such that for any $i=1, \dots, L$ and any $C^1$ function $\theta: \mathbb{T}^N \to \mathbb R$, \begin{align*}
|\hat{\theta}_n|\,\leq\, K\sum_{j=1}^{N_i}|(\widehat{\theta_{v_{i,j}}})_n|\cdot \norm{n}^N\quad \text{for all }\,n\in\mathbb Z^N\backslash 0. \end{align*} \end{lemma}
\begin{proof}
We denote by $\|.\|$ the standard Euclidean norm in $\mathbb R^N$. Since $\theta$ is $C^1$, we have \begin{align*}
2\pi\textrm{i}(n\cdot v_{i,j})\hat{\theta}_n=(\widehat{\theta_{v_{i,j}}})_n,\qquad 1\leq j\leq N_i. \end{align*} Adding over $j$ we obtain that for any $n\in\mathbb{Z}^N\backslash 0$ we have \begin{align*}
|\hat{\theta}_n| \,=\,\frac{\sum_{j=1}^{N_i}|(\widehat{\theta_{v_{i,j}}})_n|}{2\pi\sum_{j=1}^{N_i}|n\cdot v_{i,j}|} \,\le\, \frac{\sum_{j=1}^{N_i}|(\widehat{\theta_{v_{i,j}}})_n|}{2\pi \norm{\pi_{{E}^i}n}}, \end{align*}
since for an orthonormal basis $v_{i,j}$ we have $\sum_{j=1}^{N_i}|n\cdot v_{i,j}| \ge \norm{\pi_{{E}^i}n}$.
Since $\norm{\pi_{{E}^i}n}=d(n,({E}^i)^\perp)$, to complete the proof it remains to show that $d(n,({E}^i)^\perp)\geq K' \norm{n}^{-N}$.
Since $A$ is weakly irreducible, so is the transpose $A^\tau$. This follows from
Lemma \ref{Weak irred} which gives an equivalent condition for weak irreducibility
in terms of the characteristic polynomial. We denote the splitting \eqref{splitL} for $A^\tau$ by $\mathbb R^N=E^1_\tau \oplus \dots \oplus E^L_\tau$ and similarly let $\hat E^i_\tau = \oplus_{j\ne i} E^i_\tau$. Then we obtain $({E}^i)^\perp=\hat E^i_\tau$. Indeed, the polynomial
$$p_i(x)=\prod _{|\lambda |=\rho_i}(x-\lambda)^N,$$ where the product is over all eigenvalues of $A$ of modulus $\rho_i$, is real and $$({E}^i)^\perp=(\ker p_i(A))^\perp = range(p_i(A)^\tau)= range(p_i(A^\tau)) =\hat E^i_\tau,$$ since
$p_i(A^\tau)$ is invertible on $\hat E^i_\tau$. Now the desired inequality $$d(n,({E}^i)^\perp)=d(n,\hat E^i_\tau) \geq K' \norm{n}^{-N}$$
follows from Katznelson's Lemma below. We apply it to $A^\tau$ with the invariant splitting $\mathbb R^N= {\hat E}^i_\tau \oplus E^i_\tau$ and note that ${\hat E}^i_\tau \cap\mathbb Z^N=\{0\}$ by weak irreducibility of $A^\tau$. \end{proof}
\begin{lemma}[Katznelson's Lemma]\label{le:1} Let $A$ be an $N\times N$ integer matrix. Assume that $\mathbb R^N$ splits as $\mathbb R^N=V_1\bigoplus V_2$ with $V_1$ and $V_2$ invariant under $A$
and such that $A|_{V_1}$ and $A|_{V_2}$ have no common eigenvalues. If $V_1\cap\mathbb Z^N=\{0\}$, then there exists a constant $K$ such that \begin{align*} d(n,V_1)\geq K\norm{n}^{-N} \quad\text{for all } 0\ne n\in\mathbb Z^N, \end{align*} where $\norm{v}$ denotes Euclidean norm and $d$ is Euclidean distance. \end{lemma} See e.g. \cite[Lemma 4.1] {Damjanovic4} for a proof.
\subsection{Proof of Theorem \ref{th:3} } \label{proof th:3} Using the splitting $\mathbb R^N=\oplus E^i$ we decompose \eqref{for:11} into equations \begin{align}\label{for:12} A_i \mathfrak{h}_i-\mathfrak{h}_i\circ A=\mathcal{R}_i+\Psi_i,\qquad i=1, \dots, L \end{align} where $\mathfrak{h}_i$, $\mathcal{R}_i$ and $\Psi_i$ are coordinate maps in the of $\mathfrak{h}$, $\mathcal{R}$ and $\Psi$ respectively.
By Lemma \ref{le:2} there is an excellent $C^\infty$ map $\mathcal{R}_i^*$ with estimates
\begin{align}\label{for:23}
\norm{\mathcal{R}_i^*}_{C^r}\leq K_r\,\norm{\mathcal{R}_i}_{C^{r+\sigma}},\quad \norm{\mathcal{R}_i^*}_{r}\leq K_{r}\,\norm{\mathcal{R}_i}_{r+\sigma-N-2}
\end{align}
for any $r\geq0$, such that the equation:
\begin{align}\label{for:18}
A_i\omega_i-\omega_i\circ A=\mathcal{R}_i+\mathcal{R}_i^* \end{align} has a $C^\infty$ solution $\omega_i$ with estimates \begin{align*}
\norm{\omega_i}_{C^r}\leq K_r\norm{\mathcal{R}_i}_{C^{r+\sigma}},\qquad \forall\,r\geq0.
\end{align*}
Let $\omega$ be the map with coordinate maps $\omega_i$.
We obtain from \eqref{for:12} and \eqref{for:18} that $C^{1+a}$ maps $\mathfrak{p}_i=\mathfrak{h}_i-\omega_i$ and $\Lambda_i=-\mathcal{R}_i^*+\Psi_i$ satisfy \begin{align*} A_i \mathfrak{p}_i-\mathfrak{p}_i\circ A=\Lambda_i. \end{align*}
We fix $1 \le i \le L$ and an orthonormal basis $v_{i,j}$ of $E^i$. We fix $1 \le j \le N_i$ and, as in \eqref{u_k}, consider unit vectors $u_0=v_{i,j}$ and $u_k=\frac{A^ku_0}{\norm{A^ku_0}}$, and let $a_k=\norm{Au_k}$, $k\in\mathbb{Z}$. Taking the derivative of the previous equation in the direction of $u_k$ we obtain equations \begin{align*}
A_i (\mathfrak{p}_i)_{u_k}-a_{k}(\mathfrak{p}_i)_{u_{k+1}}\circ A=(\Lambda_i)_{u_k},\qquad \forall\,k\in\mathbb{Z}. \end{align*} We note that for any $k\in\mathbb{Z}$ the maps $(\mathfrak{p}_i)_{u_k}$ and $(\Lambda_i)_{u_k}$ are in $C^{a}$ and hence in $\mathcal{H}^a$, as we recall that for any function $g$ by \eqref{for:19} we have \begin{align}\label{for:21}
\norm{g_{u_k}}_{a}&\leq K\norm{g_{u_k}}_{C^a}\leq K_1\norm{g}_{C^{1+a}}.
\end{align}
Now we use (ii) of Lemma \ref{cor:2} with $\mathfrak{h}_k=(\mathfrak{p}_i)_{u_k}$, $\varphi_k=(\Lambda_i)_{u_k}$, and $P_k$ is as defined in \eqref{for:20} to obtain that for any $n\in \mathcal{M}$ \begin{align*}
\sum_{k\in\mathbb{Z}}P_k\widehat{((\Psi_i)_{u_k})}_{\tilde{A}^kn}-\sum_{k\in\mathbb{Z}}P_k\widehat{((\mathcal{R}_i^*)_{u_k})}_{\tilde{A}^kn}
=\sum_{k\in\mathbb{Z}}P_k\widehat{((\Lambda_i)_{u_k})}_{\tilde{A}^kn}=0. \end{align*} Since $(\mathcal{R}_i^*)_{u_k}$ is excellent, for each $k\in\mathbb{Z}$ we have \begin{align*}
\sum_{k\in\mathbb{Z}}P_k\widehat{((\Psi_i)_{u_k})}_{\tilde{A}^kn}=\sum_{k\in\mathbb{Z}}P_k\widehat{((\mathcal{R}_i^*)_{u_k})}_{\tilde{A}^kn}=\widehat{((\mathcal{R}_i^*)_{u_0})}_{n} \end{align*} for any $n\in M$, which gives \begin{align*}
|\widehat{((\mathcal{R}_i^*)_{u_0})}_{n}|\overset{\text{(1)}}{\leq} K_{a}\max_{k\in\mathbb{Z}}\{\norm{(\Psi_i)_{u_k}}_a\}\norm{n}^{-a}\overset{\text{(2)}}{\leq} K_{a,1}\norm{\Psi_i}_{C^{1+a}}\norm{n}^{-a}. \end{align*}
Here in $(1)$ we use (i) of Lemma \ref{cor:2} and in $(2)$ we use \eqref{for:21}.
We conclude that for any $v_{i,j}$, $1\leq j\leq N_i$, we have \begin{align}\label{for:22}
|\widehat{((\mathcal{R}_i^*)_{v_{i,j}})}_{n}|\leq K_{a}\norm{\Psi_i}_{C^{1+a}}\norm{n}^{-a},\qquad \forall\,n\in M. \end{align}
Finally, using Lemma \ref{le:4} and \eqref{for:22}, we obtain that for any $n\in M$ \begin{align}\label{for:5}
|\widehat{(\mathcal{R}_i^*)}_n|&\leq K\sum_{j=1}^{N_i}|\widehat{((\mathcal{R}_i^*)_{v_{i,j}})}_{n}|\norm{n}^N\leq K_{a}\norm{\Psi_i}_{C^{1+a}}\norm{n}^{N-a}\notag\\
& \leq K_{a}\norm{\Psi_i}_{C^{1+a}}\norm{n}^{N}. \end{align}
Now for any $r>N+2$ and any $n\in M$ we can estimate
splitting the exponent of the first term as $\alpha$ and $1-\alpha$ in the way to get the total
the exponent of $\|n\| $ be zero \begin{align*}
|\widehat{(\mathcal{R}_i^*)}&_n|\norm{n}^{N+2}=|\widehat{(\mathcal{R}_i^*)}_n|^{\frac{l-2-N}{l+N}}|\widehat{(\mathcal{R}_i^*)}_n|^{\frac{2N+2}{l+N}}\norm{n}^{N+2}\\
&\overset{\text{(1)}}{\leq} \big(K_{a}\norm{\Psi_i}_{C^{1+a}}\norm{n}^{N}\big)^{\frac{l-2-N}{l+N}}\big(\norm{n}^{-l}\norm{\mathcal{R}_i^*}_l\big)^{\frac{2N+2}{l+N}}\norm{n}^{N+2}\\
&=K_{a}^{\frac{l-2-N}{l+N}}(\norm{\Psi_i}_{C^{1+a}})^{\frac{l-2-N}{l+N}}(\norm{\mathcal{R}_i^*}_l)^{\frac{2N+2}{l+N}}\\
&\overset{\text{(2)}}{\leq} K_{l,a}(\norm{\Psi_i}_{C^{1+a}})^{\frac{l-2-N}{l+N}}(\norm{\mathcal{R}_i^*}_{C^l})^{\frac{2N+2}{l+N}}\\
& \overset{\text{(3)}}{\leq} K_{l,a}(\norm{\Psi_i}_{C^{1+a}})^{\frac{l-2-N}{l+N}}(\norm{\mathcal{R}_i}_{C^{l+\sigma}})^{\frac{2N+2}{l+N}}.
\end{align*} Here in $(1)$ we use that $\mathcal{R}_i^*$ is $C^\infty$ and \eqref{for:5}; in $(2)$ we use \eqref{for:19}; in $(3)$ we use \eqref{for:23}. Then by \eqref{for:19} we get \begin{align*}
\norm{\mathcal{R}_i^*}_{C^0}\leq C\norm{\mathcal{R}_i^*}_{N+2}\leq K_{l,a}(\norm{\Psi_i}_{C^{1+a}})^{\frac{l-2-N}{l+N}}(\norm{\mathcal{R}_i}_{C^{l+\sigma}})^{\frac{2N+2}{l+N}}. \end{align*} Finally, we denote by $\Phi$ the map with coordinate maps $\mathcal{R}_i^*$.
\section{Proof of Theorem \ref{th:4} } \label{proof th:4}
In this section we complete the proof of Theorem \ref{th:4} using an iterative process. The main part is the inductive step given by Proposition \ref{po:1}. We start with a sufficiently small perturbation $f_n$ of $A$ which is $C^{1}$ conjugate to $A$. We construct a smaller perturbation $f_{n+1}$ which is smoothly conjugate to $f_n$. The conjugacy $\tilde{H}_{n+1}$ between $f_{n}$ and $f_{n+1}$ is obtained using Theorem \ref{th:3}. Then the iterative process is set up so that $f_n$ converges to $A$ and $\tilde{H}_{1}\circ \cdots\circ \tilde{H}_{n+1}$ converge in sufficiently high regularity.
\subsection{Iterative step and error estimate} $\;$
We recall the following results, which will be used the proof of Proposition \ref{po:1}. \begin{lemma} \cite[Propositions 5.5]{dlLO} \label{dlLO lemma} For any $r\ge 1$ there exists a constant $M_r$ such that for any $h,g \in C^r(\mathcal{M})$, $$
\| h\circ g\|_{C^r} \le M_r \left(1+ \| g\|_{C^1}^{r-1} \right)
\left( \|h\|_{C^1}\| g\|_{C^r} + \|h\|_{C^r}\| g\|_{C^1} \right) + \|h\|_{C^0} .
$$ \end{lemma}
\begin{lemma}\label{le:5} \cite[Lemma AII.26.]{La} There is $d>0$ and such that for any $h\in C^r(\mathcal{M})$, if $\norm{h-I}_{C^1}\leq d$ then $h^{-1}$ exists with the estimate $\norm{h^{-1}-I}_{C^r}\leq K_r\norm{h-I}_{C^r}$.
\end{lemma} \begin{proposition}\label{po:1} Let $A$ be a weakly irreducible Anosov automorphism of $\mathbb T^N$. Let $\beta=\frac{\beta_0}{2}$, where $\beta_0$ is as in Theorem \ref{HolderConjugacy}. There exists $0<c<\frac{1}{2}$ such that for any $C^{\infty}$ perturbation $f_n$ of $A$ satisfying $$\norm{f_n-A}_{C^{\sigma+2}}<c, \text{ where $\sigma$ is from Lemma \ref{le:2}},$$ and the conjugacy equation \begin{align}\label{for:31}
H_n\circ f_n=A\circ H_n \;\text{ with a function $H_n\in C^1(\mathbb{T}^N)$ with $\|{H}_{n}-I\|_{C^0}\leq c$} \end{align} the following holds. There exists $\omega_{n+1}\in C^\infty(\mathbb{T}^N)$ so that the functions \begin{equation}\label{functions} \tilde{H}_{n+1}=I-\omega_{n+1}, \quad
H_{n+1}=H_n\circ \tilde{H}_{n+1}, \quad f_{n+1}=\tilde{H}_{n+1}^{-1}\circ f_n\circ \tilde{H}_{n+1}\\ \end{equation} satisfy the new conjugacy equation \begin{align*}
H_{n+1}\circ f_{n+1}=A\circ H_{n+1}, \end{align*} and we have the following estimates. \begin{itemize}
\item[\bf{(i)}] For any $r\geq0$ and any $t>1$
\begin{align*}
\norm{\omega_{n+1}}_{C^r}\leq K_r\min\{t^\sigma\norm{R_n}_{C^{r}}, \norm{R_n}_{C^{r+\sigma}}\}.
\quad\text{where}\quad R_n=f_n-A. \end{align*}
\item[\bf{(ii)}] \label{for:77} For the new error $R_{n+1}=f_{n+1}-A $, we have
\begin{align*}
\norm{R_{n+1}}&_{C^0}\leq Kt^\sigma\norm{R_n}_{C^1}\norm{R_n}_{C^{0}}+K_\ell t^{-\ell}\norm{R_n}_{C^\ell}\\
&+K_{l,\ell}(t^{-\ell+2}\norm{R_n}_{C^\ell}+\norm{R_n}^{1+\frac{\beta}{2}}_{C^2})^{\frac{l-2-N}{l+N}}(t^\sigma\norm{R_n}_{C^{l}})^{\frac{2N+2}{l+N}}
\end{align*}
for any $t>1$, $\ell\geq0$ and $l>N+2$; and also for any $r\geq0$ we have
\begin{align}\label{for:10}
\norm{R_{n+1}}_{C^r}\leq K_rt^\sigma\norm{R_n}_{C^{r}}+K_r. \end{align}
\item[\bf{(iii)}] For the new conjugacy $H_{n+1}$, we have
\begin{align}\label{for:24}
\norm{H_{n+1}-I}&_{C^0}\leq K\norm{R_n}_{C^\sigma}+\norm{H_{n}-I}_{C^0}
\end{align}
\end{itemize} \end{proposition}
\begin{proof}
We denote $h_n=H_n-I$ and $R_n=f_n-A$ and, similarly to \eqref{conj h}, we write the conjugacy equation \eqref {for:31} as \begin{align*}
Ah_n-h_n\circ f_n=R_n \end{align*}
We can assume that $c<\delta$, where $\delta=\delta(\beta)$ is from Theorem \ref{HolderConjugacy}, and that $\|{H}_{n}-I\|_{C^0}\leq c$ yields that $H$ is the conjugacy close to the identity. Then Theorem \ref{HolderConjugacy} gives the estimate \begin{align}\label{for:32} \norm{h_n}_{C^{1+\beta}}\leq K\norm{R_n}_{C^{1+\beta}} . \end{align} We define \begin{align}\label{for:34}
\Omega_{n}=Ah_n-h_n\circ A, \quad\text{and}\quad \Theta_{n}= R_n-\Omega_{n} = h_n\circ A-h_n\circ f_n. \end{align}
\vskip.2cm \begin{lemma} \label{le:7}
$\|\Theta_n \|_{C^{1+\frac \beta 2}}\le K_A \,\norm{R_n}^{1+\frac{\beta}{2}}_{C^{1+\beta}}$. \end{lemma}
\begin{proof} We omit index $n$ in the proof of the lemma. We note that
$$\| R \|_{C^{1+ \beta}}=\| f-A \|_{C^{1+ \beta}}<c<1.$$ Differentiating at $x\in\mathbb{T}^N$ we get \begin{align*}
D\Theta(x)&\overset{\text{*}}{=}Dh(Ax)\circ A-Dh(fx)\circ Df(x)\notag\\
&=Dh(Ax)\circ A-Dh(fx)\circ A+Dh(fx)\circ (A-Df(x)), \end{align*} and hence \begin{align*}
\norm{D\Theta}_{C^0}&\leq \|A\| \, \norm{Dh(Ax)-Dh(fx)}_{C^0}
+\norm{Dh(fx)\circ DR(x)}_{C^0}\\
&\leq \|A\| \, \norm{Dh}_{C^{\beta}}\norm{R}^\beta_{C^0}+\norm{Dh}_{C^{0}}\norm{DR}_{C^0}\\
&\leq \|A\| \,\norm{h}_{C^{1+\beta}}\norm{R}^\beta_{C^0}+\norm{h}_{C^{1}}\norm{R}_{C^1}. \end{align*}
Since we also have $ \norm{\Theta}_{C^0} \leq \norm{h}_{C^1}\norm{R}_{C^0}$, we conclude using \eqref{for:32} and $\| R \|_{C^{1+ \beta}}<1$ that \begin{align} \label{C^1}
\norm{\Theta}_{C^1}\leq \|A\| \, \norm{h}_{C^{1+\beta}}\norm{R}^\beta_{C^0}+\norm{h}_{C^{1}}\norm{R}_{C^1}
\leq K \norm{R}_{C^{1+\beta}}^{1+\beta}. \end{align}
To estimate the H\"older norm of $D\Theta$, using equation $*$ for any $x,\,y\in\mathbb{T}^N$ we have \begin{align*}
&D\Theta(x)-D\Theta(y)\\
&=\big(Dh(Ax)-Dh(Ay)\big)\circ A+Dh(fx)\circ \big(Df(y)-Df(x)\big)\\
&+\big(Dh(fy)-Dh(fx)\big)\circ Df(y), \end{align*} and hence \begin{align*}
&\norm{D\Theta(x)-D\Theta(y)}\\
&\leq \|A\| \,\norm{Dh(Ax)-Dh(Ay)}+\norm{Dh(fx)}\norm{Df(y)-Df(x)}\\
&+\norm{Dh(fy)-Dh(fx)}\norm{Df(y)}\\
&\leq \|A\| \,\norm{Dh}_{C^\beta}\norm{Ax-Ay}^\beta+\norm{h}_{C^1}\norm{Df}_{C^{{\beta}}}\norm{y-x}^\beta\\
&+\norm{f}_{C^{1}}\norm{Dh}_{C^\beta}\norm{fx-fy}^\beta\\
&\leq \|A\|^{1+\beta} \,\norm{h}_{C^{1+\beta}}\norm{x-y}^\beta+\norm{h}_{C^1}\norm{f}_{C^{1+\beta}}\norm{y-x}^\beta\\
&+\norm{f}_{C^{1}}\norm{h}_{C^{1+\beta}}\norm{f}^\beta_{C^{1}}\norm{x-y}^\beta. \end{align*}
We conclude using \eqref{for:32} and $\| f-A \|_{C^{1+ \beta}}<1$ that \begin{align} \label{C^1H}
\norm{D\Theta}_{C^{0,\beta}}\leq \|A\|^{1+\beta} \, \norm{h}_{C^{1+\beta}}+\norm{h}_{C^1}\norm{f}_{C^{1+\beta}}+\norm{h}_{C^{1+\beta}}\norm{f}^{1+\beta}_{C^{1}} \le K \norm{R}_{C^{1+\beta}}. \end{align} Therefore \begin{align}\label{for:35}
\norm{\Theta}_{C^{1+\beta}}\leq\norm{\Theta}_{C^1}+\norm{D\Theta}_{C^{0,\beta}}\le
2K \norm{R}_{C^{1+\beta}}. \end{align} Finally, we complete the proof of the lemma using an interpolation inequality \begin{align}\label{for:36}
\norm{\Theta}_{C^{1+\frac{\beta}{2}}}&\leq K\norm{\Theta}_{C^1}^{\frac{1}{2}}\norm{\Theta}_{C^{1+\beta}}^{\frac{1}{2}}\leq K_A\norm{R}^{1+\frac{\beta}{2}}_{C^{1+\beta}}.
\end{align}
\end{proof}
We recall that there exists a collection of smoothing operators $\mathfrak{s}_t$, $t>0$, such that for any $s\geq s_1\geq0$ and $s_2\geq0$, for any $g\in C^s(\mathbb{T}^N)$ the following holds, see \cite{Damjanovic4} and \cite{Hamilton}: \begin{align} \label{for:28}
\norm{\mathfrak{s}_tg}_{C^{s+s_2}}\leq K_{s,s_2}\,t^{s_2}\,\norm{g}_{C^{s}},\quad \text{and} \quad
\norm{(I-\mathfrak{s}_t)g}_{C^{s-s_1}}\leq K_{s,s'}\,t^{-s_1}\,\norm{g}_{C^{s}}. \end{align} We write \eqref{for:34} as \begin{align}\label{for2} Ah_n-h_n\circ A =\Omega_{n}=R_n-\Theta_{n}=[\mathfrak{s}_{t}R_n]+[(I-\mathfrak{s}_{t})R_n-\Theta_{n}]=:\mathcal R +\Psi \end{align} and apply Theorem \ref{th:3} to get the new splitting and obtain the estimates: \begin{align}\label{for:6}
\mathfrak{s}_{t}R_n=A\omega_{n+1}-\omega_{n+1}\circ A+\Phi_n \end{align} where $\omega_{n+1}$ and $\Phi_n$ are $C^\infty$ maps with the estimates: \begin{align}
\norm{\omega_{n+1}}_{C^r}&\leq K_r\norm{\mathfrak{s}_{t}(R_n)}_{C^{r+\sigma}}\overset{(a)}{\leq} K_r\min\{t^\sigma\norm{R_n}_{C^{r}}, \norm{R_n}_{C^{r+\sigma}}\},\quad\text{and}\label{for:63}\\
\norm{\Phi_{n}}_{C^0}&\leq K_{l}(\norm{(I-\mathfrak{s}_{t})R_n-\Theta_{n}}_{C^{1+\frac{\beta}{2}}})^{\frac{l-2-N}{l+N}}(\norm{\mathfrak{s}_{t}R_n}_{C^{l+\sigma}})^{\frac{2N+2}{l+N}}\notag\\
&\overset{(b)}{\leq} K_{l}(\norm{(I-\mathfrak{s}_{t})R_n}_{C^{2}}+\norm{R_n}^{1+\frac{\beta}{2}}_{C^2})^{\frac{l-2-N}{l+N}}(\norm{\mathfrak{s}_{t}R_n}_{C^{l+\sigma}})^{\frac{2N+2}{l+N}}\notag\\
&\overset{(a)}{\leq} K_{l,\ell}(t^{-\ell+2}\norm{R_n}_{C^\ell}+\norm{R_n}^{1+\frac{\beta}{2}}_{C^2})^{\frac{l-2-N}{l+N}}(t^\sigma\norm{R_n}_{C^{l}})^{\frac{2N+2}{l+N}}\label{for:50}
\end{align} for any $r,\,\ell\geq0$ and any $l>N+2$. Here in $(a)$ we use \eqref{for:28} and in $(b)$ we use \eqref{for:36}.
From equation \eqref{for:6} we obtain a $C^r$ estimate for $\Phi_{n}$ with $r\geq0$ \begin{align*}
\norm{\Phi_{n}}_{C^r}&=\norm{A\omega_{n+1}-\omega_{n+1}\circ A-\mathfrak{s}_{t}R_n}_{C^r}\\
&\leq K\norm{\omega_{n+1}}_{C^r}+\norm{\mathfrak{s}_{t}R_n}_{C^r}\overset{(1)}{\leq} K_r t^\sigma\norm{R_n}_{C^{r}}. \end{align*} Here in $(1)$ we use \eqref{for:28} and \eqref{for:63}.
Let $\tilde{H}_{n+1}=I-\omega_{n+1}$. From \eqref{for:63} we can assume that $\norm{\omega_{n+1}}_{C^1}<\min\{\frac{1}{2},d\}$ (see Lemma \ref{le:5}) if $c$ is sufficiently small. Hence $\tilde{H}_{n+1}$ is invertible. We estimate the new error \begin{align*}
R_{n+1}=f_{n+1}-A \end{align*}
by using \begin{gather*}
f_{n+1}=\tilde{H}_{n+1}^{-1}\circ f_n\circ \tilde{H}_{n+1}\Rightarrow \tilde{H}_{n+1}\circ f_{n+1}=f_n\circ \tilde{H}_{n+1}\\
\Rightarrow (I-\omega_{n+1})\circ f_{n+1}=f_n\circ \tilde{H}_{n+1}\\
\Rightarrow f_{n+1}=\omega_{n+1}\circ f_{n+1}+f_n\circ \tilde{H}_{n+1}. \end{gather*} This gives \begin{align*}
R_{n+1}&=\omega_{n+1}\circ f_{n+1}+f_n\circ \tilde{H}_{n+1}-A\\
&=\omega_{n+1}\circ f_{n+1}+(R_n+A)\circ (I-\omega_{n+1})-A\\
&=\omega_{n+1}\circ f_{n+1}+R_n\circ (I-\omega_{n+1})-A\circ \omega_{n+1}. \end{align*} Hence we see that $R_{n+1}$ has three parts: \begin{align*}
R_{n+1}&=\underbrace {\big( {\omega_{n+1}\circ f_{n+1}-\omega_{n+1}\circ A} \big)}_{\mathcal{E}_1}+\underbrace {\big( {R_n\circ (I-\omega_{n+1})-R_n} \big)}_{\mathcal{E}_2}\\
&+\underbrace {\big( \omega_{n+1}\circ A-{A\circ \omega_{n+1}+R_n} \big)}_{\mathcal{E}_3}. \end{align*} We note that \begin{align*}
\norm{\mathcal{E}_1}_{C^0}&\leq \norm{\omega_{n+1}}_{C^1}\norm{f_{n+1}-A}_{C^0}\overset{(0)}{\leq} \text{\small$\frac{1}{2}$}\norm{R_{n+1}}_{C^0},\\
\norm{\mathcal{E}_2}_{C^0}&\leq K\norm{R_n}_{C^1}\norm{\omega_{n+1}}_{C^0}\overset{(1)}{\leq} Kt^\sigma\norm{R_n}_{C^1}\norm{R_n}_{C^{0}}; \end{align*} and \begin{align*}
\norm{\mathcal{E}_3}_{C^0}&=\norm{\Phi_n+(I-\mathfrak{s}_{t})R_n}_{C^0}\leq \norm{\Phi_n}_{C^0}+\norm{(I-\mathfrak{s}_{t})R_n}_{C^0}\\
&\overset{(2)}{\leq} \norm{\Phi_n}_{C^0}+K_\ell t^{-\ell}\norm{R_n}_{C^\ell} \end{align*} for any $\ell\geq0$. Here in $(0)$ we recall that $\norm{\omega_{n+1}}_{C^1}<\frac{1}{2}$; in $(1)$ we use \eqref{for:63}; and in $(2)$ we use \eqref{for:28}.
Hence it follows that \begin{align*}
\norm{R_{n+1}}_{C^0}&\leq \norm{\mathcal{E}_1}_{C^0}+\norm{\mathcal{E}_2}_{C^0}+\norm{\mathcal{E}_3}_{C^0}\leq \text{\small$\frac{1}{2}$}\norm{R_{n+1}}_{C^0}+\norm{\mathcal{E}_2}_{C^0}+\norm{\mathcal{E}_3}_{C^0}, \end{align*} which gives \begin{align*}
\norm{R_{n+1}}_{C^0}&\leq 2\norm{\mathcal{E}_2}_{C^0}+2\norm{\mathcal{E}_3}_{C^0}\\
&\leq Kt^\sigma\norm{R_n}_{C^1}\norm{R_n}_{C^{0}}+K_rt^{-\ell}\norm{R_n}_{C^\ell}+\norm{\Phi_n}_{C^0}\\
&\overset{(3)}{\leq} Kt^\sigma\norm{R_n}_{C^1}\norm{R_n}_{C^{0}}+K_\ell t^{-\ell}\norm{R_n}_{C^\ell}\\
&+K_{l,\ell}(t^{-\ell+2}\norm{R_n}_{C^\ell}+\norm{R_n}^{1+\frac{\beta}{2}}_{C^2})^{\frac{l-2-N}{l+N}}(t^\sigma\norm{R_n}_{C^{l}})^{\frac{2N+2}{l+N}}\notag
\end{align*} for any $l>N+2$. Here in $(3)$ we use \eqref{for:50}. \vskip.2cm
Now we estimate $\norm{R_{n+1}}_{C^r}$. We note that \begin{align*}
R_{n+1}=(I-\omega_{n+1})^{-1}\circ (R_n+A)\circ (I-\omega_{n+1})-A=(I-\omega_{n+1})^{-1}\circ P-A. \end{align*} By Lemma \ref{dlLO lemma} we have \begin{align*}
\norm{P}_{C^r}&\leq M_r \left(1+ \| I-\omega_{n+1}\|_{C^1}^{r-1} \right)\\
&\cdot\left( \|R_n+A\|_{C^1}\| I-\omega_{n+1}\|_{C^r} + \|R_n+A\|_{C^r}\| I-\omega_{n+1}\|_{C^1} \right) + \|R_n+A\|_{C^0}\\
&\overset{(1)}{\leq} K_rt^\sigma\norm{R_n}_{C^{r}}+K_r,\qquad\text{and}\qquad
\norm{P}_{C^1}\overset{(1)}{\leq} K. \end{align*} Here in $(1)$ we use the fact that $\omega_{n+1}$ satisfies the estimates $\norm{\omega_{n+1}}_{C^r}\leq K_rt^\sigma\norm{R_n}_{C^{r}}$ (see \eqref{for:63}) and $\norm{\omega_{n+1}}_{C^1}<\frac{1}{2}$. Using Lemma \ref{le:5} this also implies that \begin{align*}
\norm{(I-\omega_{n+1})^{-1}}_{C^r}\leq K_r\norm{\omega_{n+1}}_{C^{r}}\leq K_{r,1}t^\sigma\norm{R_n}_{C^{r}} \end{align*}
and $\norm{(I-\omega_{n+1})^{-1}}_{C^1}<2$.
As a direct consequence of Lemma \ref{dlLO lemma} and the above discussion we have \begin{align*}
\norm{R_{n+1}}_{C^r}&\leq M_r \left(1+ \| P\|_{C^1}^{r-1} \right)
\left( \|(I-\omega_{n+1})^{-1}\|_{C^1}\| P\|_{C^r} + \|(I-\omega_{n+1})^{-1}\|_{C^r}\| P\|_{C^1} \right)+K\\
&\leq K_r\| P\|_{C^r}+K_rt^\sigma\norm{R_n}_{C^{r}}+K\\
&\leq K_{r,1}t^\sigma\norm{R_n}_{C^{r}}+K_{r,1}. \end{align*}
To get \eqref{for:24} we have
\begin{align*}
\norm{H_{n+1}-I}_{C^0}&=\norm{H_{n}\circ (I-\omega_{n+1})-I}_{C^0}\leq \norm{H_{n}\circ (I-\omega_{n+1})-H_n}_{C^0}+\norm{H_{n}-I}_{C^0}\\
&\leq \norm{H_n}_{C^1}\norm{\omega_{n+1}}_{C^0}+\norm{H_{n}-I}_{C^0}\\
&\overset{(1)}{\leq} K\norm{R_n}_{C^\sigma}+\norm{H_{n}-I}_{C^0}.
\end{align*} Here in $(1)$ we use \eqref{for:32} and \eqref{for:63}. \end{proof}
\subsection{The iteration scheme} First we note that by \cite[Theorem 6.1]{L1} there exists $\sigma_0=\sigma_0(A)\in \mathbb N$ such that if $H$ and $H^{-1}$ are $C^{\sigma_0}$ then $H$ and $H^{-1}$ are $C^{\infty}$.
To set up the iterative process we take $\ell$ sufficiently large so that the following holds \begin{equation} \label{for:78} \begin{aligned} &\ell\geq \max\left\{ {\text {\small $\frac{3\sigma+10}{1-\frac{\beta}{3}},\,\,\frac{24\sigma}{\beta},\,\,2(5\max\{\sigma_0,\sigma\}+1)$}},\,\,2(2\sigma+5)\right\}, \\ &{\text {\small $\left(1+\frac{\beta}{2}\right) \left(1-\frac{5}{\ell}\right) \left(\frac{\ell-2-N}{\ell+N}\right)-2\frac{2N+2}{\ell+N}$}}\,\geq\, {\text {\small $1+\frac{\beta}{3}$}}. \end{aligned} \end{equation} Now we construct $R_{n}$, $f_{n}$, $\omega_n$ and $H_{n}$ inductively as follows. For $n=0$ we take \begin{align*}
f_0=f,\quad H_0=H,\quad R_0=f-A,\quad \omega_0=0, \quad \text {and define } \;\epsilon_n=\epsilon^{\gamma^n} \end{align*} where $\gamma=1+\frac{\beta}{4}$ and $\epsilon>0$ is sufficiently small so that the following holds \begin{align*}
\norm{R_0}_{C^0} \leq \epsilon_0=\epsilon ,\qquad \norm{R_0}_{C^\ell}\leq \epsilon_0^{-1},\qquad \quad \|H_0-I\|_{C^0}<\epsilon_0^{\frac{1}{2}}. \end{align*} We note that $H_0\in C^1(\mathbb{T}^N)$ by Theorem \ref{HolderConjugacy}. Now we assume inductively that $H_n\in C^1(\mathbb{T}^N)$ satisfies the conjugacy equation \begin{align*}
H_n\circ f_n=A\circ H_n \end{align*} and that $H_n$ and $R_n=f_n-A$ satisfy \begin{align*}
\norm{R_n}_{C^0}\leq \epsilon_n,\qquad \norm{R_n}_{C^\ell}\leq \epsilon_n^{-1},
\quad \|H_n-I\|_{C^0}<\sum_{i=0}^{n-1}\epsilon^{\frac{1}{2}}_i . \end{align*} By interpolation inequalities we have \begin{align} \norm{R_n}_{C^{\sigma+2}}\leq K_\ell\norm{R_n}^{\text{\tiny$\frac{\ell-2-\sigma}{\ell}$}}_{C^0}\norm{R_n}^{\text{\tiny$\frac{2+\sigma}{\ell}$}}_{C^{\ell}}<\epsilon^{1-\frac{5+2\sigma}{\ell}}_n\leq \epsilon^{\frac{1}{2}}_n. \label{for:8} \end{align} provided $\ell\geq2(2\sigma+5)$. Here, and subsequently, we estimate various constants from above by $\epsilon_n^{-\frac 1 \ell}$. This can be done since $\ell$ is fixed, we can take $\epsilon$ small enough. We also have \begin{align}\label{for:25}
\|H_n-I\|_{C^0}<\sum_{i=0}^{n-1}\epsilon^{\frac{1}{2}}_i<\sum_{i=1}^{\infty}(\epsilon^{\frac{1}{4}})^i<2\epsilon^{\frac{1}{4}}. \end{align} Then \eqref{for:8} and \eqref{for:25} allow us to use Proposition \ref{po:1} to obtain the new iterates $R_{n+1}$, $f_{n+1}$, $\omega_{n+1}$ and $H_{n+1}$. Now we show that these iterates satisfy the inductive assumption and establish appropriate convergence.
\subsection{Inductive estimates and convergence} $\;$
\noindent We use Proposition \ref{po:1} with $t_n=\epsilon_n^{-\frac{3}{\ell}}$ and $l=\ell$. \vskip.15cm
$(1)\,$ $C^\ell$ \emph{estimate for} $R_{n+1}$ \begin{align*}
\norm{R_{n+1}}_{C^\ell}&\leq K_\ell t_n^\sigma\norm{R_n}_{C^{\ell}}+K_\ell\leq K_\ell \epsilon_n^{-\frac{3\sigma}{\ell}}(\epsilon_n^{-1}+1)\\
&<\epsilon_n^{-1-\frac{\beta}{8}-\frac{3\sigma}{\ell}}\leq \epsilon_n^{-1-\frac{\beta}{4}}=\epsilon_{n+1}^{-1}, \end{align*} provided $\ell\geq\frac{24\sigma}{\beta}$.
\vskip.15cm
$(2)\,$ $C^0$ \emph{estimate for} $R_{n+1}$ \begin{align*}
\norm{R_{n+1}}_{C^0}&\leq Kt_n^\sigma\norm{R_n}_{C^2}^2+K_\ell t_n^{-\ell}\norm{R_n}_{C^\ell}\\
&+K_{\ell}(t_n^{-\ell+2}\norm{R_n}_{C^\ell}+\norm{R_n}^{1+\frac{\beta}{2}}_{C^2})^{\frac{\ell-2-N}{\ell+N}}(t_n^\sigma\norm{R_n}_{C^{\ell}})^{\frac{2N+2}{\ell+N}}\notag\\
&\overset{\text{(a)}}{\leq} K\epsilon_n^{\text{\tiny$2-\frac{3\sigma+10}{\ell}$}}+K_\ell\epsilon_n^{3}\epsilon_n^{-1}\\
&+K_\ell(\epsilon_n^{\frac{3(\ell-2)}{\ell}}
\epsilon_n^{-1}+\epsilon_n^{\text{\tiny$(1+\frac{\beta}{2})(1-\frac{5}{\ell})$}})^{\frac{\ell-2-N}{\ell+N}}(\epsilon_n^{-\frac{3\sigma}{\ell}}
\epsilon_n^{-1})^{\frac{2N+2}{\ell+N}}\\
&\overset{\text{(b)}}{\leq} K\epsilon_n^{\text{\tiny$2-\frac{3\sigma+10}{\ell}$}}+K_\ell\epsilon_n^{2}\\
&+2K_\ell(\epsilon_n^{\text{\tiny$(1+\frac{\beta}{2})(1-\frac{5}{\ell})$}})^{\frac{\ell-2-N}{\ell+N}}(
\epsilon_n^{-2})^{\frac{2N+2}{\ell+N}}\\
&\overset{\text{(c)}}{<} \epsilon_n^{\gamma}=\epsilon_{n+1}.
\end{align*}
Here in $(a)$ we use interpolation inequalities:
\begin{align}\label{for:68} \norm{R_n}_{C^{2}}\leq C \norm{R_n}^{\text{\tiny$\frac{\ell-2}{\ell}$}}_{C^0}\norm{R_n}^{\text{\tiny$\frac{2}{\ell}$}}_{C^{\ell}} <\epsilon_n^{\text{\tiny$1-\frac{5}{\ell}$}}; \end{align} in $(b)$ we note that \begin{align*}
\text{\small$(1+\frac{\beta}{2})(1-\frac{5}{\ell})$}<\text{\small$2(1-\frac{5}{\ell})$}<2-\frac{6}{\ell}\quad\text{and}\quad\frac{3\sigma}{\ell}<1. \end{align*} Then $\epsilon_n^{-\frac{3\sigma}{\ell}}
\epsilon_n^{-1}<\epsilon_n^{-2}$ and \begin{align*}
\max\{\epsilon_n^{\text{\tiny$(1+\frac{\beta}{2})(1-\frac{5}{\ell})$}}, \epsilon_n^{\frac{3(\ell-2)}{\ell}}
\epsilon_n^{-1}\}= \epsilon_n^{\text{\tiny$(1+\frac{\beta}{2})(1-\frac{5}{\ell})$}}; \end{align*} in $(c)$ we use \begin{gather*}
\epsilon_n^{\text{\tiny$2-\frac{3\sigma+10}{\ell}$}}<\epsilon_n^{1+\text{\tiny$\frac{\beta}{3}$}},\quad (\epsilon_n^{\text{\tiny$(1+\frac{\beta}{2})(1-\frac{5}{\ell})$}})^{\frac{\ell-2-N}{\ell+N}}(
\epsilon_n^{-2})^{\frac{2N+2}{\ell+N}}<\epsilon_n^{1+\text{\tiny$\frac{\beta}{3}$}},
\end{gather*} provided \vskip.15cm $\hskip2cm 2-\frac{3\sigma+10}{\ell}\geq 1+\frac{\beta}{3},\qquad (1+\frac{\beta}{2})(1-\frac{5}{\ell})(\frac{\ell-2-N}{\ell+N})-2\frac{2N+2}{\ell+N}\geq 1+\frac{\beta}{3}$.
\vskip.15cm \noindent By \eqref{for:78} and the assumption all inequalities above are satisfied.
\vskip.2cm
$(3)$\, $C^{\sigma_0}$ \emph{estimate for} $\omega_{n+1}$: By interpolation inequalities we have \begin{align*} \norm{R_{n}}_{C^{\sigma_0}}\leq K_\ell\norm{R_{n}}^{\text{\tiny$\frac{\ell-\sigma_0}{\ell}$}}_{C^0}\norm{R_{n}}^{\text{\tiny$\frac{\sigma_0}{\ell}$}}_{C^{\ell}}<\epsilon_n^{\text{\tiny$1-\frac{2\sigma_0+1}{\ell}$}}. \end{align*} Hence we have \begin{align}\label{for:7}
\norm{\omega_{n+1}}_{C^{\sigma_0}}\leq Kt_n^\sigma\norm{R_n}_{C^{\sigma_0}}\leq K\epsilon_n^{-\frac{3\sigma}{\ell}}\epsilon_n^{\text{\tiny$1-\frac{2\sigma_0+1}{\ell}$}}
<\epsilon_{n}^{\frac{1}{2}},\end{align} provided \begin{align*}
-\frac{3\sigma}{\ell}+1-\frac{2\sigma_0+1}{\ell}>\frac{1}{2}, \end{align*} which is satisfied for $\ell>2(5\max\{\sigma_0,\sigma\}+1)$.
\vskip.2cm
$(4)$\, $C^{0}$ \emph{estimate for} $H_{n+1}$: By \eqref{for:8} we have \begin{align*}
\norm{H_{n+1}-I}&_{C^0}\leq K\norm{R_n}_{C^\sigma}+\norm{H_{n}-I}_{C^0}<K\epsilon^{1-\frac{5+2\sigma}{\ell}}_n+\sum_{i=0}^{n-1}\epsilon^{\frac{1}{2}}_n\leq \epsilon^{\frac{1}{2}}_n+\sum_{i=0}^{n-1}\epsilon^{\frac{1}{2}}_i=\sum_{i=0}^{n}\epsilon^{\frac{1}{2}}_i
\end{align*}
Consequently, we have \begin{align*}
f_{n+1}&=\tilde{H}_{n+1}^{-1}\circ\tilde{H}_{n}^{-1}\circ\cdots\circ \tilde{H}_{1}^{-1}\circ f\circ \tilde{H}_{1}\circ \cdots\circ \tilde{H}_{n+1}\\
&=\mathfrak{L}_{n+1}^{-1}\circ f\circ\mathfrak{L}_{n+1} \end{align*} where $\tilde{H}_{i}=I-\omega_{i}$, $1\leq i\leq n+1$; and $\mathfrak{L}_{n+1}=\tilde{H}_{1}\circ \cdots\circ \tilde{H}_{n+1}$.
Finally, \eqref{for:7} implies that $\mathfrak{L}_n$ converges in $C^{\sigma_0}$ topology to a $C^{\sigma_0}$ diffeomorphism $H$, which is a conjugacy between $f$ and $A$. By \cite[Theorem 6.3]{L1} and the choice of ${\sigma_0}$ we conclude that $H$ is a $C^{\infty}$ diffeomorphism.
\end{document} |
\begin{document}
\title{Penalty Bidding Mechanisms for Allocating Resources and Overcoming Present Bias
hanks{The authors would like to thank Yiling Chen, Ido Erev, Matt Juszczak, Scott Kominers, Jake Marcinek,
and Kyle Pasake for helpful comments and discussions.}
\begin{abstract} From skipped exercise classes to last-minute cancellation of dentist appointments, underutilization of reserved resources abounds. Likely reasons include uncertainty about the future, further exacerbated by present bias. In this paper, we unite resource allocation and commitment devices through the design of contingent payment mechanisms, and propose the {\em two-bid penalty-bidding mechanism}. This extends an earlier mechanism proposed by \citet{ma2019contingent}, assigning the resources based on willingness to accept a no-show penalty, while also allowing each participant to increase her own penalty in order to counter present bias. We establish a simple dominant strategy equilibrium, regardless of an agent's level of present bias or degree of ``sophistication''. Via simulations, we show that the proposed mechanism substantially improves utilization and achieves higher welfare and better equity in comparison with mechanisms used in practice and mechanisms that optimize welfare in the absence of present bias.
\end{abstract}
\section{Introduction} \label{sec:intro}
``It was a disaster,'' recalled Matt Juszczak, co-founder of Turnstyle Cycle and Bootcamp, a fitness company that offers cycling and bootcamp classes across five studios in the Boston area. ``When we opened our first indoor cycling location in Boston's Back Bay, we saw 40 to 50 no-shows and late cancels in an average day--- that's over 15,000 in a year!''\footnote{\href{https://business.mindbody.io/education/blog/tips-reduce-no-shows-and-late-cancels-your-fitness-business}{https://business.mindbody.io/education/blog/tips-reduce-no-shows-and-late-cancels-your-fitness-business}, visited September 1, 2018.}
Like many well-known exercise studios, Turnstyle allowed customers to reserve class spots several days in advance with a first-come-first-serve reservation system. However, ambitious customers, overestimating the amount of time in their schedules or their desire to exercise in the future, often snag a spot only to ultimately cancel last-minute or simply not show up.
\begin{figure}
\caption{Log in page of the squash court reservation system at the Harvard Hemenway Gymnasium.
}
\label{fig:hemenway}
\end{figure}
Similarly, the squash courts at Harvard's Hemenway Gymnasium used to allow members to reserve time-slots to play squash up to seven days ahead of time. Even though the reservation window has since been reduced to three days, the gym operators still feel the need to display the warning shown in Figure~\ref{fig:hemenway} every time someone logs into the reservation system.\footnote{\url{https://recreation.gocrimson.com/recreation/facilities/Hemenway}, visited September 1, 2018.}
For other examples, organizers of free events report to Eventbrite that their no-show rate can be as high as 50\%,\footnote{\url{https://www.eventbrite.com/blog/asset/ultimate-way-reduce-no-shows-free-events/}, visited 5/6/2019.}
and even for prepaid events organized through Doorkeeper, the fraction of no-shows can be 20\%.\footnote{\url{https://www.doorkeeper.jp/event-planning/increasing-participants-decreasing-no-shows?locale=en}, visited May 6, 2019.}
Studies of outpatient clinics report that no-shows can range from 23-34\%, with no-shows costing an estimated 14\% of daily revenue as well as impacting efficiency ~\cite{mehra2018reducing}.
Common to all these examples is the presence of uncertainty, self-interest, and down-stream decisions by participants, together with the interest of the planner (gym manager, event organizer, health clinic) in a resource being used and not wasted. Beyond revenue and efficiency motivations, utilization can have positive externality in and of itself, cycling studio members gaining motivation from fellow bikers, for example.
Complicating the problem is \emph{present bias}, often phrased as the constant struggle between our current and future selves~\citep{laibson1997golden,o1999doing}. It is easy to imagine that at the beginning of the week, someone might prefer a spin class over watching TV on Friday, reserving a spot, but by the time Friday comes around preferring to just watch TV.
Recognizing the problem of low utilization, many reservation systems charge a penalty for no-show. Turnstyle has started to charge a \$20 penalty for missing a class,\footnote{\url{https://kb.turnstylecycle.com/policies/what-is-the-late-cancel-no-show-policy}, visited May 6, 2019.}
patients who miss appointments at hospitals may need to pay a fee that is not covered by insurance, \footnote{\url{https://huhs.harvard.edu/sites/default/files/HDS\%20New\%20Patient\%20Welcome\%20Letter-eps-converted-to.pdf}, visited May 10th, 2018.}
and organizers of some conferences collect a deposit that is returned only to students who actually attended talks.\footnote{\url{https://risingstarsasia2018.ust.hk/guidelines.php}, visited May 10th, 2018.} These approaches can be viewed as ad-hoc, first-come-first serve schemes, for some choice of no-show penalty: a penalty that is too small is not effective, whereas a penalty that is too big will drive away participation in the scheme.
In recent work, \citet{ma2019contingent} model participants' future value from using a resource as a random variable, and propose the \emph{contingent second price mechanism} (CSP). The mechanism elicits from each participant a bid on the highest no-show penalty she is willing accept, assigns the resources to the highest bidders, and charges the highest losing bid as the penalty. The bids provide a good signal for participants' reliability, and the CSP mechanism provably optimizes utilization in dominant strategy equilibrium among a large family of mechanisms. With present bias, however, charging the highest losing bid as penalty no longer guarantees truthfulness: a rational participant always prefers smaller penalties, but a present-biased participant may favor larger penalties when a stronger commitment device is more effective in overcoming myopia.
\subsection{Our results}
In this paper, we unite through contingent payment mechanisms the allocation of scarce resources under uncertainty, and the design of commitment devices--- techniques that aim to overcome present bias and to fulfill a plan for desired future behavior.
We generalize the model proposed in \citet{ma2019contingent}, decomposing an agent's value for a resource into the {\em immediate value} and the {\em future value}. The immediate value is a random variable, and the value experienced at the time of using the resource (modeling for example the opportunity cost and present pain of going to the gym). The future value is not gained until some future time (consider, for example the future benefit from better health).
We incorporate the standard quasi-hyperbolic discounting model for time-inconsistent preferences~\cite{laibson1997golden,o1999doing}, such that when an agent is making a decision on whether to use a resource, the future value is discounted by a present bias factor.
Agents may also have different levels of sophistication in regard to their level of self-awareness, modeled by agents' belief on their own present bias factor--- a {\em naive agent} believes she does not discount the future, a {\em sophisticated agent} knows her bias factor precisely and is able to perfectly forecast her future actions, and a {\em partially naive agent} resides somewhere in between~\cite{o1999doing,o2001choice}.
In period~0, an agent's private information is the distribution of the immediate value, the (fixed) future value, and what she believes to be her present bias factor.
A mechanism elicits information from each agent, assigns each of $m\geq 1$ resources, and may determine both a {\em base payment} that an assigned agent always pays, as well as a {\em penalty} for each assigned agent in the event of a no-show. In period~1, each assigned agent learns her immediate value, and with knowledge of the penalty and future value, decides (under the influence of present bias) whether or not to use the resource.
The \emph{two-bid penalty-bidding mechanism} (2BPB) works as follows. In period~0, the mechanism elicits a bid from each agent, representing the highest penalty she is willing to accept for no show, and assigns the resources to the $m$ highest bidders. To address the non-monotonicity of agent's expected utility in the penalty, the mechanism asks each assigned agent to report a penalty weakly higher than the $m+1^{\mathrm{th}}$ bid, representing the actual amount she would like to be charged in the case of a no-show (thereby operating also as a commitment device).
Given the option to choose an optimal level of commitment weakly above the highest losing bid, it is a dominant strategy under the 2BPB mechanism for each agent to bid her maximum acceptable no-show penalty, regardless of her immediate value distribution, future value, level of present bias, or degree of sophistication (Theorem~\ref{thm:dse_two_bid_mech}).
While naive agents do not see the value of commitment and generally do not take any commitment device when offered~\cite{bryan2010commitment,beshears2011self}, the 2BPB mechanism is still able to help reducing the loss of welfare and utilization due to no show, since a commitment device is designed through the mechanism, and is an integral part of the system.
We also prove that the mechanism satisfies voluntary participation, and runs without a budget deficit.
We show via simulation that the 2BPB mechanism not only improves utilization, but also achieves higher social welfare than the standard $m+1^{\mathrm{th}}$ price auction, which is welfare-optimal for settings without present bias.
The mechanism also outperforms a family of mechanisms widely used in practice, which assign resources first-come-first-served and charge a fixed no-show penalty. Moreover, in a population where agents have different levels of present bias, the more biased agents benefit more than the less biased agents under the 2BPB mechanism. This results in better equity compared with the outcome under the $m+1^{\mathrm{th}}$ price auction, where the most biased agents gain little or no welfare.
\subsection{Related Work} \label{sec:related_work}
To the best of our knowledge, this current paper\rmr{``this paper" or ``the current paper"} is the first to study resource assignment in the presence of uncertainty and present bias. The closest related work is on the design of mechanisms to improve resource utilization where agents have uncertain future values~\cite{ma2019contingent,Ma_ijcai16,Ma_aamas17}.
The proposed mechanisms, however, no longer have dominant strategy equilibrium for present-biased agents.
This present work builds on \citet{ma2019contingent}, generalizing the model to incorporate present bias, and makes use of two-bid penalty bidding to align incentives.
Crucially, the mechanism does not need any knowledge about agents' level of bias or value distributions.
Contingent payments have arisen in the past in the context of oil drilling license auctions~\cite{hendricks1988empirical}, royalties~\cite{caves2003contracts,deb2014implementation}, ad auctions~\cite{varian2007position}, and selling a firm~\cite{ekmekci2016just}. Payments that are contingent on some observable world state also play the role of improving revenue as well as hedging risk~\cite{skrzypacz2013auctions}. In our model, in contrast, payments are contingent on agents' own downstream decisions and serve the role of commitment devices.
In regard to auctions in which actions take place after the time of contracting, \citet{atakan2014auctions} study auctions where the value of taking each action depends on the collective actions by others, but these actions are taken before rather than after observing the world state. \citet{courty2000sequential} study the problem of revenue maximization in selling airline tickets, where passengers have uncertainty about their value for a trip, and may decide not to take a trip after realizing their actual values. The type space considered there is effectively one-dimensional, and present bias is not considered.
\citet{laibson1997golden} introduced the quasi-hyperbolic discounting for modeling time-inconsistent decision making, where in addition to exponential discounting, all future utilities are discounted by an additional present bias factor.
\rmr{this can be slightly more connected to the intro text. E.g. "the distinction we presented above between naive and sophisticated present biased agents, is due to ..., who also ..."} \citet{o1999doing} classify present-biased agents into naive agents (unaware of present bias) and sophisticated agents (fully aware), and find that naive agent procrastinate immediate-cost activities and
do immediate-reward activities too soon, while sophistication lessens procrastination but intensifies the doing-too-soon. \citet{o2001choice} also study how the role of choice affects procrastination, and introduce the idea of a partially naive agent, who is aware of present bias but underestimates the degree of this bias.
Researchers have also attempted to estimate the present bias factor in the real world, however, there has not been consensus about this~\cite{augenblick2015experiment,cohen2016measuring,ericson2018intertemporal}.
Researchers have also examined various kinds of {\em commitment devices} to mitigate present bias.
\citet{gine2010put}, for example, offer smokers a savings account that forfeits deposits to a charity if the they fail a urine test for nicotine.
By bundling a ``want'' activity (listening to one's favorite audio book) with a ``should'' activity (going to the gym), \citet{milkman2013holding} evaluate the effectiveness of temptation bundling as a commitment device to tackle two self-control problems at a time. See also \citet{laibson1997golden} and \citet{beshears2015self}.
In a different setting, \citet{kleinberg2014time} consider how to modify the sequencing of tasks available to individuals in order to help a present-biased agent adopt a more optimal sequence of tasks. This work is later extended to consider sophisticated agents, the interaction between present bias and sunk-cost bias, and agents whose present bias factors are uncertain~\cite{kleinberg2016planning,kleinberg2017planning,gravin2016procrastination}.
There are no uncertain values or costs in these models, and no contention for limited resources.
\section{Preliminaries} \label{sec:preliminaries}
We first introduce the model for the assignment of $m$ homogeneous resources, leaving a discussion of the generalization to heterogeneous resources to Section~\ref{sec:discussion}.
There is a set of agents $N = \{1,~\dots, ~n \}$ and three time periods. In period~$0$, when resources need to be assigned, the value of each agent $i \in N$ for using a resource is uncertain, represented by $V_i = V_i^{(1)} + v_i^{(2)}$.
The period~1 \emph{immediate value} $V_i^{(1)}$ is a random variable with cumulative distribution function $F_i$, whose exact (and potentially negative) value is not realized until period $1$. This models, for example, the opportunity cost and present pain of going to the gym. The period~$2$ {\em future value} $v_i^{(2)} \geq 0$ models the expected future benefit for agent $i$ ( e.g. the future benefit from better health), if she uses a resource in period~$1$.
Agents are present-biased, such that at any point of time, when making decisions, agent $i$ discounts her utility from all future periods by a factor of $\beta_i \in [0,~1]$. Agents may not be fully aware of this bias, however, and agent $i$ believes that when making decisions, her future utility will be discounted by a factor of $\tilde{\beta}_i \in [\beta_i,~1]$.
An agent with $\tilde{\beta}_i = \beta_ 1 = 1$ is \emph{rational} and does not discount her future utility.
An agent with $\tilde{\beta}_i = \beta_i < 1$ is said to be \emph{sophisticated}, and fully aware of the degree of her present bias (thus is able to correctly predict her future decisions). An agent with $\beta_i<1$ and $\tilde{\beta}_i = 1$ is said to be \emph{naive}, believing that she will make rational decisions in the future, and an agent with $\tilde{\beta}_i \in (\beta_i, 1)$ is said to be \emph{partially naive}.
Let $\theta_i = (F_i, v_i^{(2)}, \beta_i, \tilde{\beta}_i)$ denote agent $i$'s \emph{type}, and $\theta = (\theta_1, \dots, \theta_n)$ denote a type profile. The tuple $\tilde{\theta}_i = (F_i, v_i^{(2)}, \tilde{\beta}_i)$ is agent $i$'s private information at period $0$, when the assignment of resources is determined. Each allocated agent privately learns the realization $v_i^{(1)}$ of $V_i^{(1)}$ and then decides whether to use the resource at period $1$. Define $V^+_i \triangleq \max \{V_{i}, \; 0\}$. Following \citet{ma2019contingent}, we make the following assumptions about $V_i$ for each $i \in N$:
\vsq{-0.5em}
\begin{enumerate}[({A}1)]
\item $\E{V_{i}^+}>0$, which means that a rational agent gets positive value from using the resource with non-zero probability, thus the \emph{option} to use the resource has positive value.
\item $\E{V_{i}^+} < +\infty$, which means that agents do not get infinite expected utility from the option to use the resource, thus would not be willing to pay an unboundedly large payment for it.
\item $\E{V_i} < 0$, meaning that being forced to always use the resource regardless of what happens is not favorable for any agent, so that no agent would accept any unboundedly large no-show penalty for the right to use a resource.\footnote{ Regardless of the degree of present bias or sophistication, an agent for which (A3) is violated is willing to accept a 1 billion dollar no-show penalty, (almost) always use the resource, and get a non-negative utility in expectation.} \end{enumerate}
We now provide a few examples of different models for agent types.
\begin{example}[$(c_i,p_i)$ model] \label{ex:vipi} The future value for agent $i$ for using a resource is $v_i^{(2)} = w_i > 0$, however, she is able to do so only with probability $p_i \in (0,1)$, and at a period~$1$ opportunity cost modeled by $V_i^{(1)} = -c_i$. With probability $1 - p_i$, agent $i$ is unable to show up to use the resource. This hard constraint can be modeled as $V_i^{(1)}$ taking value $-\infty$ with probability $1-p_i$.
See Figure~\ref{fig:pmf_vipi}. We have $\E{V_i^+} = (w_i - c_i) p_i > 0$, and $\E{V_i} = -\infty$ thus assumptions (A1)-(A3) are satisfied. \end{example}
\begin{figure}
\caption{Distribution of $V_i^{(1)}$ under the $(c_i,~ p_i)$ type model. }
\label{fig:pmf_vipi}
\end{figure}
\begin{example}[Exponential model] \label{ex:exp_model}
The opportunity cost for an agent to use the resource in period one is an exponentially distributed random variable with parameter $\lambda_i$, (i.e. $-V_i^{(1)} \sim \mathrm{Exp}(\lambda_i)$). If the agent used a resource, she gains a future utility of $v_i^{(2)} = w_i > 0$.
See Figure~\ref{fig:pdf_exp}. The expectation of $\E{V_i^{(1)}}$ is $\lambda_i^{-1}$, thus $\E{V_i} = \lambda_i^{-1}+ w_i $ and (A1)-(A3) are satisfied when $w_i < \lambda_i^{-1}$. \end{example}
\begin{figure}
\caption{Agent period~1 value distribution under the exponential type model. }
\label{fig:pdf_exp}
\end{figure}
\subsection{Two-Period Mechanisms}
We consider {\em two-period mechanisms}, denoted as $\mathcal{M} = (\mathcal{R}, x, s, t)$, and following the timeline suggested by \citet{ma2019contingent}. The mechanisms can, in general, involve both a base payment that an agent will pay irrespective of her utilization decision as well as a penalty for no show. The mechanisms are defined for a general message space $\mathcal{R}$ for reports, and with allocation rule $x$, and with each agent $i$ facing a base payment $s_i(r)$ and a penalty $t_i(r)$.
\if 0 At period 0, each agent makes a report $r_i$ from some set of messages $\mathcal{R}$. Let $r = (r_1, \dots, r_n)\in \mathcal{R}^n$ denote a report profile. Based on the reports, an {\em allocation rule} $x = (x_1, \dots, x_n): \mathcal{R}^n \rightarrow \{0, 1\}^n$ assigns the right to use the resources to a subset $A \subseteq N$ of at most $m$ agents, namely those agents for whom $x_{i}(r) = 1$. $x_i(r) = 0$ for all $i \neq A$.
Each agent is charged $s_i(r)$ in period 1, and the mechanism also determines the penalty $t_i(r)$ for each allocated agent $i \in A$ for no-shows (we set $t_i(r) = 0$ for $i \notin A$).
\fi
The timeline for a two-period mechanism is as follows:
\noindent{\em Period~$0$:}
\begin{enumerate}[$\bullet$]
\setlength\itemsep{0em}
\item Each agent $i \in N$ reports $r_i \in \mathcal{R}$ to the mechanism based on knowledge of $\tilde{\theta}_i$.
\item The mechanism allocates the right to use the resources to a subset of agents, $A \subseteq N $, with $|A| \leq m$, thus $x_i(r) = 1$ for all $i \in A$ and $x_i(r)= 0$ for all $i \notin A$.
\item For each agent $i\in N$, the mechanism determines a base payment $s_i(r)$ that the agent will pay for sure. For each assigned agent $i\in A$, the mechanism determines an additional penalty $t_{i}(r)$ that will be charged for a no show.
\end{enumerate}
\noindent {\em Period~$1$}:
\begin{enumerate}[$\bullet$]
\setlength\itemsep{0em}
\item The mechanism collects base payment $s_i(r)$ from each agent.
\item Each allocated agent $i \in A$ privately observes the realized immediate value $v_{i}^{(1)}$ of $V_i^{(1)}$, and decides whether to use the resource based on this value, the future value $v_i^{(2)}$, and the no-show penalty $t_i(r)$.
\item The mechanism collects the penalty $t_i(r)$ from any agent $i \in A$ who is a no show. \end{enumerate}
\begin{example}[$(m+1)^{\mathrm{th}}$ price auction] The standard $m+1^{\mathrm{th}}$ price auction for assigning $m$ resources can be described as a two-period mechanism, where the report space is $\mathcal{R} =\mathbb R$. Ordering agents in decreasing order of their reports, s.t. $r_1 \geq r_2 \geq \dots r_n$ (breaking ties randomly), the allocation rule is $x_i(r) = 1$ for all $i \leq m$, $x_i(r) = 0$ for $i > m$. Each allocated agent is charged $s_i(r) = r_{m+1}$, and all other payments are zero. The $m+1^{\mathrm{th}}$ price auction does not make use of any penalties. \end{example}
\begin{example}[Generalized contingent second price mechanism] The \emph{generalized contingent second price} (GCSP) mechanism~\cite{ma2019contingent}
for assigning $m$ homogeneous resources collects a single bid from each agent, allocates the right to use resource to the $m$ highest bidders, and charges the $m+1^{\mathrm{th}}$ highest bid, \emph{but only if an allocated agent fails to use the resource}. Formally, $\mathcal{R} =\mathbb R$. Ordering the agents s.t. $r_1 \geq r_2 \geq \dots r_n$ (breaking ties randomly), we have $x_i(r) = 1$ for $i \leq m$, $x_i(r) = 0$ for $i > m$, $t_i(r) = \max_{i' \notin A} r_{i'}$, and all other payments are 0. \end{example}
We assume risk-neutral, expected-utility maximizing agents, but with quasi-hyperbolic discounting for future utilities.
Each assigned agent $i$ faces a \emph{two part payment} $(z,y)$, where $z$ is the \emph{penalty} the agent pays in period~1 in the case of no-show, and $y$ is the \emph{base payment} the agent always pays in period~1.
When period~$1$ arrives and the agent learns the realized immediate value $v_i^{(1)}$, she discounts the future by $\beta_i$, and makes decisions as if that she will gain utility $v_i^{(1)} - y + \beta_i v_i^{(2)} $ from using the resource, and $-y -z$ from not using the resource. Based on this, the agent uses the resource if and only if \begin{align}
v_i^{(1)} - y + \beta_i v_i^{(2)} \geq -y - z \Leftrightarrow v_i^{(1)} \geq - z
- \beta_i v_i^{(2)}, \end{align}
breaking ties in favor of using the resource. Let $\one{\cdot}$ be the indicator function, and define $u_i(z)$, the expected utility of the agent when facing penalty $z$, as \begin{align}
u_i(z) \triangleq \E{ (V_i^{(1)} + v_i^{(2)}) \one{V_i^{(1)} \geq -z - \beta_i v_i^{(2)}}} - z \Pm{V_i^{(1)}< -z - \beta_i v_i^{(2)}}. \label{equ:exp_util_z} \end{align}
The \emph{actual} expected utility of an allocated agent facing a two-part payment $(z,y)$ is $u_i(z) - y$. Under a two-period mechanism $\mathcal{M}$, given report profile $r$, agent $i$'s expected utility is $x_i(r) u_i(t_i(r)) - s_i(r)$.
An agent believes that she will make decisions as if she has present-bias factor $\tilde{\beta}_i$, and will decide to use the resource in period~1 if and only if \begin{align}
v_i^{(1)} \geq - z - \tilde{\beta}_i v_i^{(2)}. \end{align}
Therefore, an agent's {\em subjective expected utility} given penalty $z$ is: \begin{align}
\tilde{u}_i(z) \triangleq \E{ (V_i^{(1)} + v_i^{(2)}) \one{V_i^{(1)} \geq -z - \tilde{\beta}_i v_i^{(2)}}} - z \Pm{V_i^{(1)}< -z - \tilde{\beta}_i v_i^{(2)}}.
\label{equ:exp_util_hat} \end{align}
We call $\tilde{u}_i(z)$ the \emph{subjective expected utility function}. For sophisticated agents who are able to perfectly predict their future decisions (i.e. $\tilde{\beta}_i = \beta_i$), $\tilde{u}_i(z)$ and $u_i(z)$ coincide.
We assume that if allocated, agents' decisions in period~1 are influenced by their present bias, but are otherwise rational. The interesting question is to study an agent's incentives regarding reports in period~$0$, which are made based on subjective expected utility $\tilde{u}_i(z) - y$.
For any vector $g = (g_1, \dots, g_n)$ and any $i \in N$, we denote $g_{-i} \triangleq (g_1, \dots, g_{i-1}, g_{i+1}, \dots, g_n)$.
\begin{definition}[Dominant strategy equilibrium]
A two-period mechanism has a {\em dominant strategy equilibrium} (DSE) if for each agent $i \in N$, for any type $\theta_i$ satisfying (A1)-(A3), there exists a report $r^\ast_i \in \mathcal{R}$ such that $\forall r_{i} \in \mathcal{R}, ~\forall r_{-i} \in \mathcal{R}^{n-1}$,
\begin{align*}
x_i(r^\ast_i, ~ r_{-i}) \tilde{u}_i(t_i(r^\ast_i, ~ r_{-i})) - s_i(r^\ast_i, ~ r_{-i}) \geq x_i(r_i, ~ r_{-i}) \tilde{u}_i(t_i(r_i, ~ r_{-i})) - s_i(r_i, ~ r_{-i}). \end{align*}
\end{definition}
Let $r^\ast(\theta) = (r^\ast_1, \dots, r^\ast_n)$ denote a report profile under a DSE given type profile $\theta$.
\begin{definition}[Voluntary participation]
A two-period mechanism satisfies {\em voluntary participation} (VP) if for each agent $i \in N$, for any type $\theta_i$ satisfying (A1)-(A3), and any report profile $r_{-i} \in \mathcal{R}^{n-1}$,
\begin{align*}
x_i(r^\ast_i, ~ r_{-i}) \tilde{u}_i(t_i(r^\ast_i, ~ r_{-i})) - s_i(r^\ast_i, ~ r_{-i}) \geq 0. \end{align*} \end{definition}
Voluntary participation requires that each agent has non-negative subjective expected utility under her dominant strategy, given that she makes present-biased but otherwise rational decisions in period $1$ if allocated, regardless of the reports made by the rest of the agents.
Voluntary participation allows an agent to have negative utility at the end of period 1.
\if 0 We cannot charge unallocated agents without violating VP, thus $s_i(r) \leq 0$ for all $i \notin A$, for all report profiles $r \in \mathcal{R}^n$. \fi
\if 0 The expected revenue of a two-period mechanism $\mathcal{M}$ from an assigned agent $i \in A$ is the total expected payment made by the agent to the mechanism in DSE, assuming present-biased but otherwise rational decisions of agents in period $1$: \begin{align}
\mathit{rev}_i(\theta) \triangleq &
s_i(r^\ast) + t_{i}(r^\ast) \Pm{ V_{i}^{(1)} < - t_{i}(r^\ast) - \beta_i v_i^{(2)}}. \label{equ:rev} \end{align} \fi
The expected revenue of a two-period mechanism $\mathcal{M}$ is the total expected payment made by the agents in the DSE, assuming present-biased but otherwise rational decisions in period $1$: \begin{align}
\mathit{rev}_\mathcal{M}(\theta) \triangleq &
\sum_{i \in N } \left( s_i(r^\ast) +
x_i(r)t_{i}(r^\ast) \Pm{ V_{i}^{(1)} < - t_{i}(r^\ast) - \beta_i v_i^{(2)}}\right). \end{align}
\begin{definition}[No deficit] A two-period mechanism satisfies {\em no deficit} (ND) if, for any type profile $\theta$ that satisfies (A1)-(A3), the expected revenue is non-negative: $\mathit{rev}_\mathcal{M}(\theta) \geq 0$. \end{definition}
\if 0 We also consider two additional properties:
A mechanism is \emph{anonymous} if the outcome (assignment, payments) is invariant to permuting the identities of agents.
A mechanism is \emph{deterministic} if the outcome is not randomized unless there is a tie.
\fi
The \emph{utilization} achieved by mechanism $\mathcal{M}$ is the expected number of resources used by the assigned agents in the DSE:
\begin{align}
ut_\mathcal{M}(\theta) \triangleq \sum_{i \in N} x_i(r^\ast) \Pm{V_{i}^{(1)} \geq - t_i (r^\ast) - \beta_i v_i^{(2)}}. \end{align}
The expected \emph{social welfare} achieved by mechanism $\mathcal{M}$ is the total expected value derived by agents from using the resources: \begin{align}
sw_\mathcal{M}(\theta) \triangleq \sum_{i \in N} x_i(r^\ast) \E{ (V_i^{(1)} + v_i^{(2)}) \one{V_i^{(1)} \geq - t_i(r^\ast)- \beta_i v_i^{(2)}} }. \label{equ:social_welfare} \end{align}
Our objective is to design mechanisms that maximize expected social welfare. We do not consider monetary transfers as part of the social welfare function. The reason $t_i(r^\ast)$ appears in \eqref{equ:social_welfare} is that it affects decisions of the allocated agents in period $1$.
\section{The Two-Bid Penalty Bidding Mechanism} \label{sec:two_bid_csp}
In this section, we introduce the two-bid penalty bidding mechanism, and prove that agents have simple dominant strategies, regardless of their value distributions, levels of present bias, or degrees of sophistication.
\begin{definition}[Two-bid penalty bidding mechanism] \label{def:two_bid_penalty_bidding} The \emph{two-bid penalty bidding mechanism} (2BPB) collects bids $\bar{b} = (\bar{b}_i, \dots, \bar{b}_n)$ from agents in period~0,
and reorders agents in decreasing order of $\bar{b}_i$ such that $\bar{b}_1 \geq \bar{b}_2 \geq \dots \geq \bar{b}_n$ (breaking ties randomly).
\begin{enumerate}[$\bullet$]
\setlength\itemsep{0em}
\item Allocation rule: $x_{i}(b) = 1$ for $i \leq m$, $x_i(b) = 0$ for $i > m$.
\item Payment rule: the mechanism announces $\bar{b}_{m+1}$, elicits a second bid $\underline{b}_i \geq \bar{b}_{m+1}$ from each assigned agent $i \leq m$, and sets $t_i(b) = \underline{b}_i$. $t_i(b) = 0$ for all $i > m$, and $s_i(b) = 0$ for all $i \in N$. \end{enumerate} \end{definition}
The 2BPB mechanism first asks agents to bid on the maximum penalties they are willing to accept for the option to use the resource for free, and assigns the resources to the highest bidders. The mechanism then asks each assigned agent to bid a penalty that is weakly higher than the $m+1^{\mathrm{th}}$ bid, representing the amount she would like to be charged in case of a no-show.\footnote{Instead of using two rounds of bidding, we may also consider a direct revelation mechanism, where agents report their private information $\tilde{\theta}_i$, with which the mechanism determines the assignment and the contingent payments.
}
To establish the dominant strategy equilibrium under the 2BPB mechanism, we first prove some useful properties of agents' subjective expected utility function $\tilde{u}_i(z)$.
\begin{restatable}{lemma}{lemmaExpUtility} \label{lem:exp_u} Given an agent with any type $\theta_i$ that satisfies (A1)-(A3), the agent's subjective expected utility $\tilde{u}_i(z)$ as a function of the penalty $z$ satisfies: \begin{enumerate}[(i)]
\setlength\itemsep{0em}
\item $\tilde{u}_i(0) \geq 0$, and $\lim_{z \rightarrow +\infty} \tilde{u}_i(z) \leq \E{V_i}$.
\item $\tilde{u}_i(z)$ is right continuous and upper-semi-continuous.
\if 0 i.e. $\lim_{z \downarrow z^\ast} \tilde{u}_i(z) = \tilde{u}_i(z^\ast)$ for all $ z^\ast \geq 0$. Moreover,
$\tilde{u}_i(z)$ is upper semi-continuous, meaning that for all $z^\ast \geq 0$, $\lim_{z \uparrow z^\ast} \tilde{u}_i(z) \leq \tilde{u}_i(z^\ast).$
\fi \end{enumerate} \end{restatable}
\begin{proof}
We first prove part (i). $\tilde{u}_i(0) \geq 0$ holds given \eqref{equ:exp_util_hat} and the fact that $\tilde{\beta}_i \leq 1$ and $w_i \geq 0$. For the limit as $z \rightarrow +\infty$, observe that $\tilde{u}_i(z)$ can be rewritten as: \begin{align}
\tilde{u}_i(z) = & \E{ (V_i^{(1)} + \tilde{\beta}_i v_i^{(2)}) \one{(V_i^{(1)}+\tilde{\beta}_i v_i^{(2)}) \geq -z }} + \notag \\
& (1 - \tilde{\beta}_i) v_i^{(2)} \Pm{(V_i^{(1)}+\tilde{\beta}_i v_i^{(2)}) \geq -z} - z \Pm{(V_i^{(1)} + \tilde{\beta}_i v_i^{(2)})< -z } \notag \\
=& \E{\max\{V_i^{(1)} + \tilde{\beta}_i v_i^{(2)}, ~-z\}} + (1 - \tilde{\beta}_i) v_i^{(2)} \Pm{(V_i^{(1)}+\tilde{\beta}_i v_i^{(2)}) \geq -z}. \label{equ:uhat_rewritten} \end{align}
By the monotone convergence theorem, the first term of \eqref{equ:uhat_rewritten} converges to $ \E{V_i^{(1)} + \tilde{\beta}_i v_i^{(2)}} = \E{V_i^{(1)}} + \tilde{\beta}_i v_i^{(2)}$ as $z \rightarrow +\infty$. The second term is upper bounded by $(1 - \tilde{\beta}_i) v_i^{(2)}$, therefore we get $\lim_{z \rightarrow +\infty} \tilde{u}_i(z) \leq \E{V_i^{(1)}} + v_i^{(2)} = \E{V_i}$.
For part (ii), $\max\{V_i^{(1)} + \tilde{\beta}_i v_i^{(2)}, ~-z\}$ is a continuous function in $z$, therefore its expectation $\E{\max\{V_i^{(1)} + \tilde{\beta}_i v_i^{(2)}, ~-z\}}$ is also continuous in $z$.
$\Pm{(V_i^{(1)}+\tilde{\beta}_i v_i^{(2)}) \geq -z}$ is right continuous, implying the right continuity of $\tilde{u}_i(z)$. The upper semi-continuity (i.e. $\lim_{z \uparrow z^\ast} \tilde{u}_i(z) \leq \tilde{u}_i(z^\ast)$ for all $z^\ast \geq 0$) holds because of the fact that $(1 - \tilde{\beta}_i) v_i^{(2)} \geq 0$, and that $\Pm{(V_i^{(1)}+\tilde{\beta}_i v_i^{(2)}) \geq -z}$ is upper semi-continuous.
\end{proof}
\citet{ma2019contingent} had earlier proved that for a rational agent without present bias, her expected utility as a function of the penalty is continuous, convex, and monotonically decreasing. These properties no longer hold for present-biased agents, since a higher penalty may incentivize an agent to use the resource more optimally, resulting in a higher expected utility.
For any penalty $z$, we define $\tilde{U}_i(z)$ as agent $i$'s highest subjective expected utility for the best choice of penalty, assuming this penalty must be at least $z$:
\begin{align}
\tilde{U}_i(z) = \sup_{z' \geq z} \tilde{u}_i(z'). \label{equ:max_U} \end{align}
The following lemma proves the continuity and monotonicity of $\tilde{U}_i(z)$, together with the existence of a zero-crossing for $\tilde{U}_i(z)$. This zero-crossing point is the maximum penalty an agent will accept, in the case that this agent can choose to be charged any penalty weakly larger than this penalty.
\begin{restatable}{lemma}{lemmaMaxExpUtility} \label{lem:max_exp_u} Given any agent with type $\theta_i$ that satisfies (A1)-(A3), the agent's subjective expected utility $\tilde{U}_i(z)$ as a function of the minimum penalty $z$ satisfies: \begin{enumerate}[(i)]
\setlength\itemsep{0em}
\item $\tilde{U}_i(z)$ is continuous and monotonically decreasing in $z$.
\item There exists a zero-crossing $z^0_i$ s.t. $\tilde{U}_i(z^0_i) = 0$ and $\tilde{U}_i(z) < 0$ for all $z > z^0_i$. \end{enumerate} \end{restatable}
\begin{proof}
For part (i), the monotonicity of $\tilde{U}_i(z)$ is obvious, and the continuity is implied by the right continuity of $\tilde{u}_i(z)$ as shown in Lemma~\ref{lem:exp_u}.
For part (ii), Lemma~\ref{lem:exp_u} and assumption (A3) imply $\lim_{z \rightarrow \infty} \tilde{u}_i(z) \leq \E{V_i} < 0$. Therefore, there exists $Z \in \mathbb{R}$ s.t. $\tilde{u}_i(z) < 0$ for all $z \geq Z$. As a result, $\tilde{U}_i(z) < 0$ holds for all $z \geq Z$,
and the monotonicity and continuity of $\tilde{U}_i(z)$ then imply that the following supreme exists: \begin{align*}
z^0_i \triangleq \sup\{ z \in \mathbb{R} ~|~ \tilde{U}_i(z) \geq 0\}, \end{align*} and that we must have $\tilde{U}_i(z^0_i) = 0$ and $\tilde{U}_i(z) < 0$ for all $z > z^0_i$. \end{proof}
The following example illustrates the expected utility functions of an agent with $(c_i, p_i)$ type (see Example~\ref{ex:vipi}), and shows that there may not exist a DSE under the CSP mechanism.
\begin{example} \label{ex:vipi_expected_utility} Consider a sophisticated agent whose type follow the $(c_i, p_i)$ model,
who is assigned a resource and charged no-show penalty $z$. With probability $1 - p_i$, the agent is not able to use the resource, and has to pay the penalty. With probability $p_i$, the agent is able to use the resource at a cost of $c_i$, but will use the resource if and only if $\beta_i w_i - c_i \geq -z \Leftrightarrow z \geq c_i - \beta_i w_i$. Therefore, the agent's expected utility as a function of the no-show penalty is of the form: \begin{align*}
u_i(z) = \pwfun{-z, & ~\mathrm{if}~ z < c_i - \beta_i w_i, \\
(w_i - c_i)p_i - (1-p_i)z, & ~\mathrm{if}~ z \geq c_i - \beta_i w_i,} \end{align*} and $\tilde{u}_i(z) = u_i(z)$ holds for all $z \geq 0$ since the agent is sophisticated.
Figure~\ref{fig:vipi_util_u} illustrates $\tilde{u}_i(z)$ for an agent with $c_i - \beta_i w_i > 0$. Intuitively, $c_i - \beta_i w_i$ is the minimum penalty the agent needs to be charged so that she will use the resource when she is able to. When $z < c_i - \beta_i w_i$, the agent ends up always paying the penalty, which is too small to incentivize utilization. $\tilde{U}_i(z)$ of this agent is as shown in Figure~\ref{fig:vipi_util_U}. The maximum penalty the agent is willing to accept is $z^0_i = (w_i - c_i)p_i/(1-p_i)$.
\begin{figure}
\caption{$\tilde{u}_i(z)$. }
\label{fig:vipi_util_u}
\caption{$\tilde{U}_i(z)$. }
\label{fig:vipi_util_U}
\caption{Subjective expected utility functions of a sophisticated agent with $(c_i, p_i)$ type, with $c_i - \tilde{\beta}_i w_i > 0$. }
\label{fig:vipi_example_utilities}
\end{figure}
There is no dominant strategy for this agent under the CSP mechanism. Consider the assignment of a single resource. If the highest bid among the rest of the agents satisfies $\max_{i' \neq i} b_{i'} \in [c_i - \tilde{\beta}_i w_i, z^0_i)$, the agent gets positive utility from bidding $b_i = z^0_i$, getting allocated and charged penalty $\max_{i' \neq i} b_{i'}$. However, if $\max_{i' \neq i} b_{i'} < c_i - \tilde{\beta}_i w_i$, bidding $b_i = z^0_i$ results in negative utility--- the agent will be allocated, but charged a penalty that is too small to overcome her present bias. In this case, the agent is better off bidding $b_i = 0$ and get zero utility. \qed \end{example}
We now state and prove the main theorem of this paper.
\begin{restatable}[Dominant strategy equilibrium of the two-bid
penalty bidding mechanism]{theorem}{thmDSE} \label{thm:dse_two_bid_mech} Assuming (A1)-(A3), under the two-bid penalty bidding mechanism, it is a dominant strategy for each agent $i \in N$ to bid $\bar{b}_i^\ast = z^0_i$. If agent $i$ is assigned a resource and given a minimum penalty $\underline{z}$, it is then a dominant strategy to bid $\underline{b}_i^\ast = \arg\max_{z \geq \underline{z}} \tilde{u}_i(z)$. Moreover, the mechanism satisfies voluntary participation and no deficit. \end{restatable}
\begin{proof}
We first consider an agent who is assigned a resource and asked by the mechanism to bid an amount that is at least $\underline{z}$. The right continuity of $\tilde{u}_i(z)$ (see Lemma~\ref{lem:exp_u}) implies that the highest utility $\tilde{U}_i(\underline{z})$ when the agent can choose any penalty weakly higher than $\underline{z}$ is achieved at $\arg \max_{z \geq \underline{z}} \tilde{u}_i(z)$. Since whichever amount an agent bids as $\underline{b}_i$ will be the penalty she is charged by the mechanism, it is a dominant strategy to bid $\underline{b}_i^\ast = \arg \max_{z \geq \underline{z}} \tilde{u}_i(z)$.
Given that an assigned agent will get expected utility $\tilde{U}_i(\underline{z})$ when she is asked to choose a penalty $\underline{b}_i $ that is weakly above $\underline{z}$, $\tilde{U}_i(z)$ is effectively her expected utility function in the first round of bidding. With the monotonicity of $\tilde{U}_i(z)$ and the fact that the minimum penalty is determined by the $m+1^{\mathrm{th}}$ highest bid, it is standard that an agent bids in DSE the highest ``minimum penalty to choose from'' that she is willing to accept, which is $z^0_i$.
\if 0 We first consider an agent who is assigned a resource, and asked by the mechanism to bid an amount that is at least $\underline{z}$. The highest expected utility $\tilde{U}_i(\underline{z})$ when the agent can choose any penalty weakly higher than $\underline{z}$ is achieved at some $\arg \max_{z \geq \underline{z}} \tilde{u}_i(z)$ because of the right continuity of $\tilde{u}_i(z)$. Since whichever amount an agent bids as $\underline{b}_i$ will be the penalty she is charged by the mechanism, it is a dominant strategy to bid $\underline{b}_i^\ast = \arg \max_{z \geq \underline{z}} \tilde{u}_i(z)$.
Given that an assigned agent will get expected utility $\tilde{U}_i(\underline{z})$ when she is asked to bid a penalty at least $\underline{z}$, $\tilde{U}_i(z)$ is effectively her expected utility function in the first round of bidding. With the monotonicity of $\tilde{U}_i(z)$ and the fact that the minimum penalty is determined by the $m+1^{\mathrm{th}}$ highest bid, it is standard that an agent bids in DSE the highest ``minimum penalty to choose from'' that she is willing to accept, which is $z^0_i$.
\fi \end{proof}
The following example shows that the 2BPB mechanism can achieve better social welfare and utilization than the $m+1^{\mathrm{th}}$ price auction by assigning to a ``better'' agent and charging a proper penalty as the commitment device.
\begin{example} \label{ex:two_bid_better_than_sp} Consider the assignment of one resource to two sophisticated agents with $(c_i, p_i)$ types, where: \begin{enumerate}[$\bullet$]
\item $c_1 = 10$, $p_1 = 0.8$, $w_1 = 16$, $\beta_1 = \tilde{\beta}_1 = 0.5$,
\item $c_2 = 6$, $p_2 = 0.5$, $w_2 = 10$, $\beta_2 = \tilde{\beta}_2 = 0.8$. \end{enumerate}
When $z < c_1 - \beta_1 w_1 = 2$, agent $1$ never uses the resource. On the other hand, $c_2 - \beta_2 w_2 < 0$ means that agent~$2$ uses the resource with probability $p_2$ while facing any non-negative penalty. $u_i(z) = \tilde{u}_i(z)$ for $i = 1,2$ since both agents are sophisticated, and the subjective expected utility functions of the two agents are as shown in Figure~\ref{fig:exmp_two_bid_better_than_SP}.
\begin{figure}
\caption{Expected utility functions of two agents in Example~\ref{ex:two_bid_better_than_sp}. }
\label{fig:exmp_two_bid_better_than_SP}
\end{figure}
Under the second price auction, agents bid in DSE $b_{1, {\mathrm{SP}}}^\ast = \tilde{u}_1(0) = 0$ and $b_{2, {\mathrm{SP}}}^\ast = \tilde{u}_2(0) = (w_2 - c_2)p_2 = 2$, the values of the option to use the resource without any penalty (the free option to use the resource has no value to agent $1$ since she knows that she will never show up). Agent $2$ gets assigned the resource and charged no penalty, achieving social welfare $(w_2-c_2)p_2 = 2$ and utilization $p_2 = 0.5$.
Under the 2BPB mechanism, the agents bid in DSE $\bar{b}_1^\ast = z^0_1 = (w_1 - c_1)p_1/(1-p_1) = 24$, and $\bar{b}_2^\ast = z^0_2 = (w_2 - c_2)p_2/(1-p_2) = 4$. Agent~$1$ is therefore assigned and will bid $\underline{b}_1^\ast = 4$ when asked to choose a penalty weakly above $\bar{b}_2^\ast = 4$, since $\tilde{u}_1(z)$ is monotonically decreasing in $z$ for $z \geq c_1 - \tilde{\beta}_1 w_1 = 2$. The 2BPB mechanism achieves social welfare $(w_1-c_1)p_1 = 4.8$ and utilization $p_1 = 0.8$, both are higher than those under the second price auction.
\qed \end{example}
\subsection{Discussion} \label{sec:discussion}
For fully rational agents with $\tilde{\beta}_i = \beta_i = 1$, the subjective expected utility as a function of the penalty $\tilde{u}_i(z)$ is monotonically decreasing in $z$, therefore $\tilde{u}_i(z) = \tilde{U}_i(z)$ for all $z \in \mathbb{R}$. In this case, the equilibrium outcome under the 2BPB mechanism coincides with that under the $m+1^{\mathrm{th}}$-price generalization of the CSP mechanism.
Since $\tilde{u}_i(z)$ is what an agent considers while bidding, in period~0 a naive agent will bid as if she was rational with the same value distribution. In period~1, however, present bias will take effect, and the naive agent may make sub-optimal decisions. The actual expected utility a naive agent gets from participating in the $m+1^{\mathrm{th}}$ price CSP or the 2BPB mechanisms, therefore, may be negative, despite the fact that she is willing to participate and believes she will get non-negative expected utility.
For two agents $i$ and $i'$ who are identical except that $\tilde{\beta}_i > \tilde{\beta}_{i'}$, we can prove that $\tilde{u}_i(z) \geq \tilde{u}_{i'}(z)$ holds for all $z \geq 0$. As a result, $\tilde{u}_i(0) \geq \tilde{u}_{i'}(0)$ and $z^0_i \geq z^0_{i'}$. This implies that an agent who believes that she is less present-biased (i.e. with higher $\tilde{\beta}_i$) will bid higher under both the 2BPB mechanism and the $m+1^{\mathrm{th}}$ price auction. See Proposition~\ref{prop:bid_monotonicity} in Appendix~\ref{appx:bid_monotonicity} for more detailed discussions.
For rational agents without present bias, the CSP mechanism optimizes utilization among a large family of mechanisms with a set of desirable properties~\cite{ma2019contingent}.
The 2BPB mechanism, however, does not provably optimize utilization for present-biased agents. The reason is that the actual present bias factor does not affect a naive agent's bid, and it is still possible for a very biased naive agent to be assigned but rarely show up. On the other hand, the $m+1^{\mathrm{th}}$ auction may not assign the resource to this agent, thus may achieve higher utilization and welfare (see Examples~\ref{exmp:two_bid_suboptimal_utilization_1} and~\ref{exmp:two_bid_suboptimal_utilization_2} in Appendix~\ref{appx:not_utilization_opt}).
The 2BPB mechanism can also be generalized for assigning multiple heterogeneous resources $M = \{a, b, \dots, m\}$,
where each agent $i \in N$ has a random value $V_{i,a} = V_{i,a}^{(1)} + v_{i,a}^{(2)}$ for using each resource $a \in M$. $\tilde{u}_{i,a}(z)$ and $\tilde{U}_{i,a}(z)$ can be defined similarly to \eqref{equ:exp_util_hat} and \eqref{equ:max_U}. The 2BPB mechanism can be generalized through the use of a minimum Walrasian equilibrium price mechanism, which computes the assignment and the minimum penalty each agent faces using $\{\tilde{U}_{i,a}(z)\}_{i \in N, a \in M}$~\cite{demange1985strategy,DBLP:journals/ior/AlaeiJM16,ma2019contingent}. As a second step, each assigned agent is asked to report a weakly higher penalty that she wants to be charged by the mechanism.
\section{Simulation Results} \label{sec:simulations}
In this section, we adopt the exponential model (see Example~\ref{ex:exp_model}), and compare in simulation the social welfare and utilization achieved by different mechanisms and benchmarks. Additional simulation results for the exponential model are presented in Appendix~\ref{appx:additional_simulations}, together with similar results when assuming the $(c_i, p_i)$ type model (see Example~\ref{ex:vipi}) or a uniform type model where agents' period~1 values are uniformly distributed.
For the exponential model, $\E{V_i} = -\lambda_i^{-1}+ w_i $, where $-\lambda_i^{-1}$ is the expected period~1 opportunity cost for using the resource.\footnote{The expected utility functions and dominant strategy bids under various type models are derived in Appendix~\ref{appx:derivations}.}
We consider a type distribution in the population, where the value $w_i$ and the expected opportunity cost $\lambda_{i,a}^{-1}$ are uniformly distributed as $\lambda_{i}^{-1} \sim \mathrm{U}[0,~L]$ and $w_{i} \sim \mathrm{U}[0,~\lambda_{i}^{-1}]$.
$w_{i} < \lambda_{i}^{-1}$ holds almost surely, thus assumptions (A1)-(A3) are satisfied. The results are not sensitive to the choices of $L$ in defining this type distribution, and we fix $L = 20$ for the rest of this section.
\subsection{Varying Resource Scarcity} \label{sec:simulations_scarcity}
Fixing the number of resources at five, we study the impact of varying the scarcity of the resource, by varying the number of agents from $2$ to $30$.
We define the \emph{first best} as the highest achievable social welfare (or utilization) assuming full knowledge of agent types, and without violating voluntary participation or no deficit.
The \emph{first-come-first-serve with fixed penalty mechanism} (FCFS) assumes a random order of arrival, with the effect of assigning to a random subset of at most $m$ agents who are willing to accept the penalty. We consider three levels of penalties for FCFS: 5, 2.5 and 0, where $5$ is equal to the expectation of the future value $w_i$.
\paragraph{Naive Agents} We first consider the scenario where all agents are naive. The present bias factor $\beta_i$ uniformly distributed on $[0,1]$, and all agents believe $\tilde{\beta}_i = 1$. The average social welfare and utilization over 100,000 randomly generated profiles are as shown in Figure~\ref{fig:exp_naive_rand_beta}.
\newcommand{0.85}{0.85}
\begin{figure}
\caption{Social welfare. }
\label{fig:exp_naive_rand_beta_welfare}
\caption{Utilization. }
\label{fig:exp_naive_rand_beta_utilization}
\caption{Social welfare and utilization for naive agents with exponential types. }
\label{fig:exp_naive_rand_beta}
\end{figure}
When the number of agents is small, the outcomes under 2BPB, the $m+1^{\mathrm{th}}$ price auction, and the FCFS without penalty are similar, since all three effectively assign the resources to all agents, without charging any penalty. As the number of agents increases, the 2BPB mechanism achieves higher social welfare and substantially higher utilization than the $m+1^{\mathrm{th}}$ price auction (which optimizes social welfare for rational agents without present bias), and does this without charging any payments from agents who do show up.
The 2BPB mechanism achieves higher welfare and utilization for economies of any size, and does not require any prior knowledge about the number of agents or their bias level or value distributions.
The FCFS mechanism (which are analogous to the reservation system widely used in practice), by comparison, requires careful adjustments of the fixed penalty level. A smaller penalty works fine when the number of agents is small but fails to keep up as the economy becomes more competitive. A larger penalty outperforms the $m+1^{\mathrm{th}}$ price auction for larger economies, but deters participation and leaves resources unallocated when the number of agents is small.
\paragraph{Sophisticated Agents} Consider now fully sophisticated agents with $\tilde{\beta}_i = \beta_i$, whose present-biased factors are distributed as $\beta_i \sim \mathrm{U}[0,1]$.
As the number of agents vary from $2$ to $30$, the average social welfare and utilization over 100,000 randomly generated economies are as shown in Figure~\ref{fig:exp_soph_rand_beta}.
\begin{figure}
\caption{Social welfare. }
\label{fig:exp_soph_rand_beta_welfare}
\caption{Utilization. }
\label{fig:exp_soph_rand_beta_utilization}
\caption{Social welfare and utilization for sophisticated agents with exponential types. }
\label{fig:exp_soph_rand_beta}
\end{figure}
As with the setting with naive agents, we can see that the 2BPB mechanism achieves higher welfare and utilization than the $m+1^{\mathrm{th}}$ price auction, and that the performance of FCFS is very sensitive to the fixed penalty and the competitiveness of the economy.
The $m+1^{\mathrm{th}}$ price auction achieves higher welfare and utilization for sophisticated agents, in comparison to the setting with naive agents. This is because sophisticated agents are able to adjust their bids depending on their present bias level, and avoid the situation where a naive agent bids too much, gets assigned, but rarely show up, resulting in low utilization, welfare, and negative actual expected utility for the naive agent herself.
In Appendix\ref{appx:additional_simu_exp}, we present additional simulation results assuming all agents are fully rational ($\tilde{\beta}_i = \beta_i = 1$) or partially naive (in which case we assume $\tilde{\beta}_i \sim \mathrm{U}[\beta_i,~1]$).
The outcome for partially naive agents is between the outcome for fully naive agents and fully sophisticated agents. For rational agents, the 2BPB mechanism achieves slightly worse welfare than the $m+1^{\mathrm{th}}$ price auction, which is provably optimal for this setting. The 2BPB mechanism, however, still achieves higher utilization and also a significantly better outcome than the FCFS benchmarks.
\subsection{Impact on Agents with Different Degrees of Bias} \label{sec:simulations_degree_of_bias}
In this section, we study the different outcomes for agents with different degrees of present bias. We assume the same type distribution as in the previous section, but fix the present-bias factor of each agent $i$ at $\beta_i = i/n$, where $n$ is the total number of agents--- the smaller an agent's index, the more present-biased an agent.
\paragraph{Naive Agents} We first consider the scenario where all agents are naive. Fixing $n = 30$, for 1 million randomly generated economies, the average per economy welfare and usage (i.e. the probability of being assigned and showing up) of \emph{each agent} is as shown in Figure~\ref{fig:exp_naive_array_beta}.
Under the first-best welfare and the first-best utilization, agents with different degrees of bias achieve the same welfare and utilization. This is because the agents all have the same distribution of $\lambda_i^{-1}$ and $w_i$, and only differ in their bias factor $\beta_i$. The full-information first best knows the exact types of agents, and adjusts the penalties accordingly, so that there is no difference between agents who are more or less biased.
Note that naive agents behave in period~0 as if they were rational, therefore all agents bid in the same way despite their different degrees of bias, and therefore are assigned with the same probability.
\begin{figure}
\caption{Average welfare. }
\label{fig:exp_naive_array_beta_welfare}
\caption{Average usage. }
\label{fig:exp_naive_array_beta_utilization}
\caption{Average welfare and usage for naive agents with exponential types, fixing $\beta_i = i/n$.
}
\label{fig:exp_naive_array_beta}
\end{figure}
Figure~\ref{fig:exp_naive_array_beta_welfare} shows that the less biased agents (higher indices) gain substantially higher welfare than the more biased agents (lower indices) under the $m+1^{\mathrm{th}}$ price auction. By contrast, the 2BPB mechanism helps agents who are more biased to achieve substantially higher welfare than the outcome under the $m+1^{\mathrm{th}}$ price auction, and at the same time slightly reducing the welfare for the least biased agents. This is because the least biased agents are able to make close to optimal decisions in period~$1$ by themselves, and charging a penalty sometimes leads to suboptimal utilization decisions.
From Figure~\ref{fig:exp_naive_array_beta_utilization}, we see that all agents have higher average usage under the 2BPB mechanism,
and agents who are more biased achieve a higher gain in comparison with the $m+1^{\mathrm{th}}$ price auction.
Overall, the outcome under the 2BPB mechanism is subtantially more equitable for agents with all levels of bias. It is also worth noting that while naive agents do not see the value of commitment and do not take any commitment device when offered~\cite{bryan2010commitment,beshears2011self}, the 2BPB mechanism is still able to help, since a commitment device is designed through the mechanism, and it is not an option to not accept a commitment.
\paragraph{Sophisticated Agents} For fully sophisticated agents with varying degrees of present bias, the average welfare and usage per economy of each agent are as shown in Figure~\ref{fig:exp_soph_array_beta}.
\begin{figure}
\caption{Average welfare. }
\label{fig:exp_soph_array_beta_welfare}
\caption{Average usage. }
\label{fig:exp_soph_array_beta_utilization}
\caption{Average welfare and usage for sophisticates with exponential types, fixing $\beta_i=i/n$. }
\label{fig:exp_soph_array_beta}
\end{figure}
The first observation is that under the $m+1^{\mathrm{th}}$ price auction, the welfare and usage for the most biased agents are effectively zero, while the least biased agents achieve better welfare and utilization than the first-best outcome. This is because when the bids of sophisticated agents factor in the level of present bias, the more biased agents bid lower than the less biased agents (see Claim~\ref{prop:bid_monotonicity} in Appendix~\ref{appx:bid_monotonicity}), and therefore get assigned with lower probability.
The more biased sophisticated agents also bid lower under the 2BPB mechanism, and as a result the 2BPB mechanism is not able to achieve the same level of welfare for all agents. Nevertheless, it achieves large improvements for the more biased population compared to the $m+1^{\mathrm{th}}$ price auction, and also higher welfare and better equity than the FCFS benchmarks.
\subsection{Impact on Agents with Different Degrees of Sophistication} \label{sec:simulations_degree_of_naivete}
In this section, we consider a population of agents with the same present-bias factor, but varying degree of sophistication. We assume the same distribution of the expected opportunity cost $\lambda_i^{-1}$ and the future value $w_i$ as in the earlier settings, but fix $\beta_i = 0.5$ and $\tilde{\beta}_i = 1 - 0.5 i/n$ for all agents. The smaller an agent's index, the more naive she is about her present bias: agent $1$ has $\tilde{\beta}_1$ close to $1$ thus is almost fully naive, whereas agent $n$ has $\tilde{\beta}_n = 0.5 = \beta_n$ thus is fully sophisticated.
\begin{figure}
\caption{Average welfare. }
\label{fig:exp_fix_beta_array_betahat_welfare}
\caption{Average usage. }
\label{fig:exp_fix_beta_array_betahat_utilization}
\caption{Welfare and usage for agents with exponential types and varying degrees of naivete.
}
\label{fig:exp_fix_beta_array_betahat}
\end{figure}
Fixing $n = 30$, the per-economy welfare and usage for each agent averaged over 1 million randomly generated economies are as shown in Figure~\ref{fig:exp_fix_beta_array_betahat}. All agents achieve higher welfare and usage under the 2BPB mechanism, in comparison to the $m+1^{\mathrm{th}}$ price auction and the FCFS mechanisms. The outcome under the 2BPB mechanism is again more equitable than that under the $m+1^{\mathrm{th}}$ price auction. It is curious that agents who are less naive (higher indices) have lower welfare and usage. This is because the more sophisticated agents can better predict their future suboptimal decisions, and as a result bid lower and accept fixed penalties with lower probability. It is indeed the case that the more sophisticated agents achieve slightly higher expected utility.
\section{Conclusion} \label{sec:conclusion}
We propose the two-bid penalty-bidding mechanism for resource allocation in the presence of uncertain future values and present bias. We prove the existence of a simple dominant strategy equilibrium, regardless of an agent's value distribution, level of present bias, or degree of sophistication. Simulation results show that the mechanism improves utilization and achieves higher welfare and better equity in comparison with mechanisms broadly used in practice as well as mechanisms that are welfare-optimal for settings without present bias.
In future work, it will be interesting to conduct empirical studies to better understand people's behavior in settings such as exercise studios and events, with the goal of separating the effect on utilization of uncertainty from that of present bias. Another interesting direction is to generalize the model to allow for more than two time periods, where agents may arrive asynchronously, when uncertainty unfolds over time, and where resources can be re-allocated.
\appendix
\noindent{}\textbf{\huge{Appendix}}
\noindent{}Appendix~\ref{appx:additional_simulations} provides additional simulation results. Appendix~\ref{appx:additional_discussion} provides additional discussions and examples omitted from the body of the paper. Appendix~\ref{appx:derivations} derives for various type models the expected utility function and the dominant strategy equilibrium under different mechanisms.
\section{Additional Simulation Results} \label{appx:additional_simulations}
This section presents additional simulation results for the exponential type model that are omitted from the body of the paper, as well as results for the $(c_i, p_i)$ type model and a uniform type model, which we introduce in Appendix~\ref{appx:additional_simu_uniform}.
\subsection{Additional Results for Exponential Model} \label{appx:additional_simu_exp}
We first consider the same setup as analyzed in Section~\ref{sec:simulations}, where agents have exponential types, and there are $m = 5$ homogeneous resources to assign. We present the results as the number of agents varies, for settings where agents are all fully rational, or where agents are partially naive.
\paragraph{Fully Rational Agents}
Figure~\ref{fig:exp_rational_rand_beta} shows the average welfare and utilization of 100,000 randomly generated economies, assuming all agents are fully rational with $\tilde{\beta}_i = \beta_i = 1$. The 2BPB mechanism achieves slightly worse welfare than the $m+1^{\mathrm{th}}$ price auction, which is provably optimal for this setting. The 2BPB mechanism, however, still achieves higher utilization, and also robustly outperforms the FCFS mechanisms.
\begin{figure}
\caption{Social welfare. }
\label{fig:exp_rational_rand_beta_welfare}
\caption{Utilization. }
\label{fig:exp_rational_rand_beta_utilization}
\caption{Social welfare and utilization for rational agents with exponential types. }
\label{fig:exp_rational_rand_beta}
\end{figure}
\paragraph{Partially Naive Agents} We now consider partially naive agents, where each agent has bias factors independently drawn according to $\beta_i \sim \mathrm{U}[0,~ 1]$, and $\tilde{\beta}_i \sim \mathrm{U}[\beta_i,~1]$. The average welfare and utilization over 100,000 random economies are as shown in Figure~\ref{fig:exp_part_rand_beta}. The outcome is in between the fully sophisticated and the fully naive settings discussed in the body of the paper.
\begin{figure}
\caption{Social welfare. }
\label{fig:exp_part_rand_beta_welfare}
\caption{Utilization. }
\label{fig:exp_part_rand_beta_utilization}
\caption{Social welfare and utilization for partially naive agents with exponential types. }
\label{fig:exp_part_rand_beta}
\end{figure}
\subsection{The $(c_i, p_i)$ Type Model}
n this section, we compare via simulation different mechanisms and benchmarks for the $(c_i, p_i)$ type model (see Example~\ref{ex:vipi}).\footnote{Agents' expected utility functions and DSE bids are derived in Appendix~\ref{appx:derivations}.} We consider a type distribution where the future value $w_i$, cost $c_i$, and probability of being able to show up $p_i$ are uniformly distributed: \begin{align*}
w_i &\sim \mathrm{U}[0,~L],\\
c_i & \sim \mathrm{U}[0,~w_i], \\
p_{i} &\sim \mathrm{U}[0,~1]. \end{align*}
With $c_{i} < w_i$ and $p_i \in (0,1)$ with probability~1, assumptions (A1)-(A3) are satisfied almost surely. The results are not sensitive to the choices of parameter $L$, and we fix $L = 10$ for all results presented in the rest of this section. In this case, the expected value of $w_i$ is $5$.
\subsubsection{Varying Resource Scarcity}
Fixing the number of resources at $m = 5$, we first examine the outcomes under different mechanisms and benchmarks as the number of agents varies from $2$ to $30$. When all agents are naive with $\beta_i \sim \mathrm{U}[0,~1]$ and $\tilde{\beta}_i = 1$. The average social welfare and utilization over 100,000 randomly generated profiles are as shown in Figure~\ref{fig:cipi_naive_rand_beta}. For economies with fully sophisticated agents, where present-biased factors are distributed as $\beta_i \sim \mathrm{U}[0,1]$, but $\tilde{\beta}_i = \beta_i$ for all $i \in N$, the average social welfare and utilization are as shown in Figure~\ref{fig:cipi_soph_rand_beta}. We see trends similar to the results under the exponential type model presented in Section~\ref{sec:simulations_scarcity}.
\begin{figure}
\caption{Social welfare. }
\label{fig:cipi_naive_rand_beta_welfare}
\caption{Utilization. }
\label{fig:cipi_naive_rand_beta_utilization}
\caption{Social welfare and utilization for naive agents with $(c_i,p_i)$ types. }
\label{fig:cipi_naive_rand_beta}
\end{figure}
\begin{figure}
\caption{Social welfare. }
\label{fig:cipi_soph_rand_beta_welfare}
\caption{Utilization. }
\label{fig:cipi_soph_rand_beta_utilization}
\caption{Social welfare and utilization for sophisticated agents with $(c_i,p_i)$ types. }
\label{fig:cipi_soph_rand_beta}
\end{figure}
\subsubsection{Impact on Agents with Different Degrees of Bias}
We now consider a population of agents with the same distribution of $c_i$, $p_i$ and $w_i$ as in the previous setting, but where the total number of agents is fixed at $n=30$, and the present bias factor of each agent $i$ is fixed at $\beta_i = i/n$.
Assuming all agents are naive with $\beta_i = 1$, the average welfare and average usage of each agent (over 1 million randomly generated economies) is as shown in Figure~\ref{fig:cipi_naive_array_beta}. For the setting where all agents are fully sophisticated with $\tilde{\beta}_i = \beta_i$, the average welfare and usage of each agent is as shown in Figure~\ref{fig:cipi_soph_array_beta}.
\begin{figure}
\caption{Average welfare. }
\label{fig:cipi_naive_array_beta_welfare}
\caption{Average usage. }
\label{fig:cipi_naive_array_beta_utilization}
\caption{Average welfare and usage for naive agents with $(c_i,p_i)$ types, fixing $\beta_i = i/n$. }
\label{fig:cipi_naive_array_beta}
\end{figure}
\begin{figure}
\caption{Average welfare. }
\label{fig:cipi_soph_array_beta_welfare}
\caption{Average usage. }
\label{fig:cipi_soph_array_beta_utilization}
\caption{Average welfare and usage for sophisticated agents with $(c_i,p_i)$ types, fixing $\beta_i=i/n$. }
\label{fig:cipi_soph_array_beta}
\end{figure}
We can see from Figures~\ref{fig:cipi_naive_array_beta} and~\ref{fig:cipi_soph_array_beta} that for both the naive and the sophisticated settings, the less biased agents (higher indices) achieve significantly higher average welfare and usage under the $m+1^{\mathrm{th}}$ price auction and the FCFS mechanisms. In contrast, this disparity under the 2BPB mechanism is much smaller, and the most biased agents benefit the most under the 2BPB mechanism in comparison to the FCFS mechanisms.
\subsection{Uniform Type Mode} \label{appx:additional_simu_uniform}
In this section, we study the performance of various mechanisms and benchmarks for agents whose period~$1$ values are uniformly distributed as in the following Example~\ref{ex:uniform_model}.
\begin{example}[Uniform model] \label{ex:uniform_model}
In period~$1$, each agent incurs a uniformly distributed opportunity cost for using the resource, i.e. $V_i^{(1)} \sim \mathrm{U}[-\alpha_i, 0]$. If the agent used a resource, she gains a expected future utility of $v_i^{(2)} = w_i > 0$.
See Figure~\ref{fig:pdf_uniform}. $\E{V_i^{(1)}} = -\alpha_i/2 $, thus $\E{V_i} = -\alpha_i/2 + w_i$ and (A1)-(A3) are satisfied as long as $w_i < 1/\alpha_i/2$. \end{example}
\begin{figure}
\caption{Agent period~1 value distribution under the uniform type model. }
\label{fig:pdf_uniform}
\end{figure}
Agents' expected utility functions and DSE bids under the uniform model are derived in Appendix~\ref{appx:derivations}. We consider the following type distribution in the population, where $\alpha_i$ and $w_i$ are both uniformly distributed: \begin{align*}
\alpha_i &\sim \mathrm{U}[0,~L],\\
w_i & \sim \mathrm{U}[0,~\alpha_i/2]. \end{align*}
With $w_i \in (0, \alpha_i/2)$ with probability~1, assumptions (A1)-(A3) are satisfied almost surely. The results are not sensitive to the choices of parameter $L$, and we fix $L = 20$ for all results presented in the rest of this section, in which case the average value of $w_i$ is $5$.
\subsubsection{Varying Resource Scarcity}
Fixing the number of resources at $m = 5$, we first examine the outcomes under various mechanisms and benchmarks as the number of agents varies from $2$ to $30$. For the scenario where all agents are naive, with $\beta_i \sim \mathrm{U}[0,~1]$ and $\tilde{\beta}_i = 1$, the average social welfare and utilization over 100,000 randomly generated profiles are as shown in Figure~\ref{fig:uniform_naive_rand_beta}. For economies with fully sophisticated agents, where present-biased factors are distributed as $\beta_i \sim \mathrm{U}[0,1]$ and $\tilde{\beta}_i = \beta_i$, the average social welfare and utilization are as shown in Figure~\ref{fig:uniform_soph_rand_beta}.
\begin{figure}
\caption{Social welfare. }
\label{fig:uniform_naive_rand_beta_welfare}
\caption{Utilization. }
\label{fig:uniform_naive_rand_beta_utilization}
\caption{Social welfare and utilization for naive agents with uniform types. }
\label{fig:uniform_naive_rand_beta}
\end{figure}
\begin{figure}
\caption{Social welfare. }
\label{fig:uniform_soph_rand_beta_welfare}
\caption{Utilization. }
\label{fig:uniform_soph_rand_beta_utilization}
\caption{Social welfare and utilization for sophisticated agents with uniform types. }
\label{fig:uniform_soph_rand_beta}
\end{figure}
\subsubsection{Impact on Agents with Different Degrees of Bias}
We now consider a population of agents with the same distribution of $\alpha_i$ and $w_i$ as in the previous setting, but where the total number of agents is fixed at $n=30$, and the present bias factor of agent each agent $i$ is fixed at $\beta_i = i/n$. Assuming all agents are naive, the average welfare and average usage of each agent (over 1 million randomly generated economies) is as shown in Figure~\ref{fig:uniform_naive_array_beta}. Assuming that agents are fully sophisticated instead, the average welfare and utilization of each agent is as shown in Figure~\ref{fig:uniform_soph_array_beta}.
\begin{figure}
\caption{Average welfare. }
\label{fig:uniform_naive_array_beta_welfare}
\caption{Average usage. }
\label{fig:uniform_naive_array_beta_utilization}
\caption{Average welfare and usage for naive agents with uniform types, fixing $\beta_i = i/n$. }
\label{fig:uniform_naive_array_beta}
\end{figure}
\begin{figure}
\caption{Social welfare. }
\label{fig:uniform_soph_array_beta_welfare}
\caption{Average usage. }
\label{fig:uniform_soph_array_beta_utilization}
\caption{Average welfare and usage for sophisticated agents with uniform types, fixing $\beta_i=i/n$. }
\label{fig:uniform_soph_array_beta}
\end{figure}
\section{Additional Examples and Discussion} \label{appx:additional_discussion}
In this section, we provide additional discussions and examples that are omitted from the body of this paper.
\subsection{Monotonicity of Bids} \label{appx:bid_monotonicity}
The following proposition shows that the less biased an agent believes she is, the higher she bids in DSE under the 2BPB mechanism and the $m+1^{\mathrm{th}}$ price auction.
\begin{proposition} \label{prop:bid_monotonicity} Under the 2BPB mechanism, or the $m+1^{\mathrm{th}}$ price auction, an agent's bid in dominant strategy is monotonically increasing in her subjective present bias factor $\tilde{\beta}_i$. \end{proposition}
\begin{proof}
We first prove that for any penalty $z$, an agent's subjective expected utility $\tilde{u}_i(z)$ is monotonically increasing in $\tilde{\beta}_i$. Let $i$ and $i'$ be two agents who are identical except that $\tilde{\beta}_i \geq \tilde{\beta}_{i'}$. For any $z \in \mathbb{R}$, we have \begin{align*}
& \tilde{u}_i(z) - \tilde{u}_{i'}(z) \\
= & \E{ (V_i^{(1)} + v_i^{(2)}) \one{V_i^{(1)} \geq -z - \tilde{\beta}_i v_i^{(2)}}} \notag - z \Pm{V_i^{(1)}< -z - \tilde{\beta}_i v_i^{(2)}} \\
& - \E{ (V_i^{(1)} + v_i^{(2)}) \one{V_i^{(1)} \geq -z - \tilde{\beta}_i' v_i^{(2)}}} \notag + z \Pm{V_i^{(1)}< -z - \tilde{\beta}_i' v_i^{(2)}} \\
=& \E{ (V_i^{(1)} + v_i^{(2)} + z) \one{V_i^{(1)} \in [-z - \tilde{\beta}_i v_i^{(2)}, -z - \tilde{\beta}_i' v_i^{(2)})}} \\
\geq & 0. \end{align*}
The last inequality holds since $z \in [-z - \tilde{\beta}_i v_i^{(2)}, -z - \tilde{\beta}_i' v_i^{(2)})$, $V_i^{(1)} + v_i^{(2)} + z \geq -z - \tilde{\beta}_i v_i^{(2)} + v_i^{(2)} + z = (1 - \tilde{\beta}_i) v_i^{(2)} \geq 0$.
This immediately implies the monotonicity of bids under the $m+1^{\mathrm{th}}$ price auction, where agents bid $\tilde{u}_i(0)$ in DSE. Agent $i'$ will also bid higher under the 2BPB mechanism, since the $\tilde{u}_i(z) \geq \tilde{u}_{i'}(z)$ for all $z \in \mathbb{R}$ also implies that the zero-crossings of $\tilde{U}_i(z)$ and $\tilde{U}_{i'}(z)$ also satisfy $z^0_i \geq z^0_{i'}$. \end{proof}
\subsection{The 2BPB Mechanism Does Not Optimize Utilization} \label{appx:not_utilization_opt}
In this section, we provide two examples which illustrate that when agents are not fully rational, the 2BPB mechanism does not necessarily optimize utilization.
The first examples shows that the 2BPB mechanism may end up achieving zero utilization and welfare by allocating to a naive agent and charging a penalty that is too small to incentivize utilization.
\begin{example} \label{exmp:two_bid_suboptimal_utilization_1}
Consider the allocation of one resource to two agents with $(c_i, p_i)$ types, where
\begin{enumerate}[$\bullet$]
\item $c_1 = 5$, $p_1 = 0.8$, $w_1 = 7.5$, $\beta_1 = 0.2$, $ \tilde{\beta}_1 = 1$,
\item $c_2 = 5$, $p_2 = 1/6$, $w_2 = 20$, $\beta_2 = \tilde{\beta}_2 = 1$. \end{enumerate}
Agent $1$ is fully naive and agent $2$ is fully rational. When $z < c_1 - \beta_1 w_1 = 3.5$, agent $1$ never uses the resource.
The expected utility functions and the subjective expected utility functions of the two agents are as shown in Figures~\ref{fig:exmp_not_opt_1_u} and \ref{fig:exmp_not_opt_1_uhat}.
\begin{figure}
\caption{The expected utility functions of two agents in
Example~\ref{exmp:two_bid_suboptimal_utilization_1}.
}
\label{fig:exmp_not_opt_1_u}
\end{figure}
\begin{figure}
\caption{The subjective expected utility functions of two agents in
Example~\ref{exmp:two_bid_suboptimal_utilization_1}.
}
\label{fig:exmp_not_opt_1_uhat}
\end{figure}
Under the second price auction, agents bid in DSE $b_{1, {\mathrm{SP}}}^\ast = \tilde{u}_1(0) = 2$ and $b_{2, {\mathrm{SP}}}^\ast = \tilde{u}_2(0) = (w_2 - c_2)p_2 = 2.5$. Agent $2$ gets assigned the resource and charged no penalty, achieving social welfare $(w_2-c_2)p_2 = 2.5$ and utilization $p_2 = 1/6$.
Under the 2BPB mechanism, the agents bid in DSE $\bar{b}_1^\ast = z^0_1 = (w_1 - c_1)p_1/(1-p_1) = 10$, and $\bar{b}_2^\ast = z^0_2 = (w_2 - c_2)p_2/(1-p_2) = 3$. Agent~$1$ is therefore assigned the resource, and will bid $\underline{b}_1^\ast = 3$ when asked to choose a penalty weakly above $\bar{b}_2^\ast = 3$ (this is because $\tilde{u}_1(z)$ is monotonically decreasing in $z$ due to agent $1$'s naivete). When period $1$ comes, however, agent $1$ never shows up since the utility from using the resource appears to be $\beta_1 v_1^{(2)} - c_1 = -3.5$, which is worse than paying the penalty and get -3. The 2BPB mechanism therefore achieves zero welfare and utilization.
\qed \end{example}
The second example shows that with fully sophisticated agents, it is still possible for the second price auction to achieve higher utilization.
\begin{example} \label{exmp:two_bid_suboptimal_utilization_2}
Consider the allocation of one resource to two agents with $(c_i, p_i)$ types, where
\begin{enumerate}[$\bullet$]
\item $c_1 = 10$, $p_1 = 0.5$, $w_1 = 20$, $\beta_1 = \tilde{\beta}_1 = 0.2$,
\item $c_2 = 5$, $p_2 = 0.6$, $w_2 = 10$, $\beta_2 = \tilde{\beta}_2 = 1$. \end{enumerate}
Agent 1 is fully sophisticated, and agent $2$ is fully rational. $\tilde{u}_i(z) = u_i(z)$ holds for both agents, and the expected utility functions are as shown in Figure~\ref{fig:exmp_not_opt_2_u}.
\begin{figure}
\caption{The expected utility functions of two agents in
Example~\ref{exmp:two_bid_suboptimal_utilization_2}.
}
\label{fig:exmp_not_opt_2_u}
\end{figure}
Under the second price auction, agents bid in DSE $b_{1, {\mathrm{SP}}}^\ast = \tilde{u}_1(0) = 0$ and $b_{2, {\mathrm{SP}}}^\ast = \tilde{u}_2(0) = (w_2 - c_2)p_2 = 3$. Agent $2$ gets assigned the resource and charged no penalty, achieving social welfare $(w_2-c_2)p_2 = 3$ and utilization $p_2 = 0.6$.
Under the 2BPB mechanism, the agents bid in DSE $\bar{b}_1^\ast = z^0_1 = (w_1 - c_1)p_1/(1-p_1) = 10$, and $\bar{b}_2^\ast = z^0_2 = (w_2 - c_2)p_2/(1-p_2) = 7.5$. Agent~$1$ is therefore assigned the resource, and will bid $\underline{b}_1^\ast = 7.5$ when asked to choose a penalty weakly above $\bar{b}_2^\ast = 7.5$. Therefore, the 2BPB mechanism achieves social welfare $(w_1-c_1)p_1 = 5$ and utilization $p_1 = 0.5$.
\qed
\end{example}
\section{Utilities and DSE Bides Under Different Type Models} \label{appx:derivations}
In this section, we derive for various type models the expected utility function and the dominant strategy equilibrium under different mechanisms.
\subsection{$(c_i, p_i)$ Type Model}
Consider an agent with $(c_i, p_i)$ type parametrized by $(c_i, p_i, w_i, \beta_i, \tilde{\beta}_i,)$ who face a no-show penalty $z \in \mathbb{R}$. In period~1, with probability $1-p_i$, the agent cannot show up, therefore gets utility $- z$. When probability $p_i$, the agent can show up at an immediate cost $c_i$. The agent believes that she will show up if and only if \begin{align*}
\tilde{\beta}_i w_i - c_i \geq -z_i \Leftrightarrow z_i \geq c_i - \tilde{\beta}_i w_i. \end{align*}
Therefore, $c_i - \tilde{\beta}_i w_i$ is the ``minimum commitment'' the agent believes that she needs to ever show up to use the resource. When $z < c_i - \tilde{\beta}_i w_i$, the agent never shows up and gets utility $-z$. When $z \geq c_i - \tilde{\beta}_i w_i$, the agent does show up with probability $p_i$. The subjective expected utility of this agent is therefore: \begin{align*}
\tilde{u}_i(z) = \pwfun{ -z, &~\mathrm{if}~ z < c_i - \tilde{\beta}_i w_i,\\
(w_i - c_i)p_i - z_i (1-p_i), & ~\mathrm{if}~ z \geq c_i - \tilde{\beta}_i w_i.} \end{align*}
When $c_i - \tilde{\beta}_i w_i > 0$, the agent believes that she will not show up in a 2nd price auction, in which case she bids zero in DSE. When $c_i - \tilde{\beta}_i w_i \leq 0$, she believes that she will show up with probability $p_i$, and bids her expected utility $(w_i - c_i) p_i$ from using the resource. The DSE bids under SP are therefore: \begin{align*}
b_{i, {\mathrm{SP}}}^\ast = \pwfun{(w_i - c_i) p_i, & ~\mathrm{if}~ c_i - \tilde{\beta}_i w_i \leq 0, \\
0, & ~\mathrm{if}~ c_i - \tilde{\beta}_i w_i > 0.} \end{align*}
The zero-crossing of the curve $(v_i - c_i)p_i - z_i (1-p_i)$ is \begin{align*}
\tilde{\zcmax}_i = \frac{(w_i - c_i) p_i}{1-p_i}, \end{align*}
therefore when $\tilde{\zcmax}_i < c_i - \tilde{\beta}_i w_i$, $\tilde{u}_i(z) < 0$ for any $z > 0$, meaning that the agent will not participate in the 2BPB mechanism.
When $\tilde{\zcmax}_i \geq c_i - \tilde{\beta}_i w_i$, we know that $z^0_i = \tilde{\zcmax}_i$ is the zero-crossing of $\tilde{U}_i(z)$ , therefore the DSE bid on the maximum acceptable penalty is \begin{align*}
\bar{b}_{i}^\ast =& \pwfun{\tilde{\zcmax}_i, & ~\mathrm{if}~ c_i - \tilde{\beta}_i w_i \leq \tilde{\zcmax}_i, \\
0, & ~\mathrm{if}~ c_i - \tilde{\beta}_i w_i > \tilde{\zcmax}_i.} \end{align*}
We also know that $\tilde{u}_i(z)$ is monotonically decreasing when $z \geq c_i - \tilde{\beta}_i w_i$, therefore after given a minimum penalty $\underline{z}$, the agent will bid in DSE her preferred penalty \begin{align*}
\underline{b}_{i}^\ast = \max\{c_i - \tilde{\beta}_i w_i, ~\underline{z} \}. \end{align*}
\if 0 $ c_i - \tilde{\beta}_i w_i$ and $\tilde{\zcmax}_i$ are the agent's minimum and maximum acceptable penalties, respectively. Therefore, the DSE bids are: \begin{align*}
\underline{b}_{i, {\mathrm{CSP}}}^\ast =& \pwfun{c_i - \tilde{\beta}_i w_i, & ~\mathrm{if}~ c_i - \tilde{\beta}_i w_i \leq \tilde{\zcmax}_i, \\
0, & ~\mathrm{if}~ c_i - \tilde{\beta}_i w_i > \tilde{\zcmax}_i.} \end{align*} \fi
If agent $i$ is assigned a resource and charged a penalty $z$, the actual utilization would be \begin{align*}
ut_i(z) = p_i \cdot \one{z \geq c_i - \beta_i w_i}, \end{align*} since when time $1$ comes, she will discount the future utility according to her true discounting factor $\beta_i$. The expected social welfare is therefore \begin{align*}
sw_i(z) = p_i(w_i - c_i) \cdot \one{z \geq c_i - \beta_i w_i}, \end{align*} and the agent's actual expected utility is \begin{align*}
u_i(z) = \pwfun{ -z, &~\mathrm{if}~ z < c_i - \beta_i w_i,\\
(v_i - c_i)p_i - z_i (1-p_i), & ~\mathrm{if}~ z \geq c_i - \beta_i w_i.} \end{align*}
The first-best utilization that can be achieved by this agent is \begin{align*}
ut_i^{\mathrm{FB}} = p_i, \end{align*}
and the first-best welfare is: \begin{align*}
sw_i^{\mathrm{FB}} = (w_i - c_i) p_i \one{c_i - \tilde{\beta}_i w_i \leq \tilde{\zcmax}_i}. \end{align*}
Note that even when $\tilde{\zcmax}_i < c_i - \tilde{\beta}_i w_i$ in which case $\tilde{u}_i(z) < 0$ for any $z > 0$, we may still incentivize the agent to show up with probability $p_i$ by charging a no-show penalty weakly higher than $ \tilde{\beta}_i w_i$, and also making a positive payment to the agent to incentivize participation. It is possible to do this without running a deficit since we achieve a positive welfare $(w_i - c_i)p_i$.
\subsection{Exponential Type Model}
Consider now the exponential type model, where an agent's type is parametrized by $\theta_i = (\lambda_i, w_i, \beta_i, \tilde{\beta}_i)$. In period~$1$, with penalty $z$, the agent will show up to use the resource if and only if \begin{align*}
V_i^{(1)} + \beta_i w_i \geq -z \Leftrightarrow V_i^{(1)} \geq -z - \beta_i w_i. \end{align*}
This happens with probability $\Pm{V_i^{(1)} \geq -z - \beta_i w_i} = 1- e^{-\lambda_i(z + \beta_i w_i)}$, as long as $z + \beta_i w_i \geq 0 \Leftrightarrow z \geq - \beta_i w_i$. Therefore the actual utilization as a function of penalty $z$ is: \begin{align*}
ut_i(z) = \pwfun{ 1- e^{-\lambda_i(z + \beta_i w_i)}, & ~\mathrm{if}~ z \geq - \beta_i w_i\\
0, & ~\mathrm{if}~ z < - \beta_i w_i,} \end{align*} and the expected social welfare is: \begin{align*}
& sw_i(z) \\ = & \pwfun{ w_i - \frac{1}{\lambda_i} +
e^{-\lambda_i (\beta_i w_i + z)} \left( \frac{1}{\lambda_i} - (1 - \beta_i) w_i + z) \right), & ~\mathrm{if}~ z \geq - \beta_i w_i\\
0, & ~\mathrm{if}~ z < - \beta_i w_i.} \end{align*}
With $z < -\beta_i w_i$, the agent never uses the resource, and gets expected utility $u_i(z) = -z$. When $z \geq -\beta_i w_i$, the agent gets expected utility: \begin{align*}
& ut_i(z) \\ = & \E{ (V_i^{(1)} + w_i) \one{V_i^{(1)} + \beta_i w_i \geq -z} } - z \Pm{V_i^{(1)} + \beta_i w_i <-z} \\
= & \int_{0}^{z + \beta_i w_i} (-v + w_i) \lambda_i e^{-\lambda_i v} dv - z e^{-\lambda_i(\beta_i w_i + z)} \\
=& w_i - 1/\lambda_i + e^{-\lambda_i (\beta_i w_i + z)} (1/\lambda_i - (1 - \beta_i) w_i) . \end{align*}
The agent, however, believes that her present bias factor is $\tilde{\beta}_i$, therefore believes that her expected utility as a function of the penalty $z$ is: \begin{align*}
& \tilde{u}_i(z) \\ = & \pwfun{ w_i - 1/\lambda_i + e^{-\lambda_i (\tilde{\beta}_i w_i + z)} (1/\lambda_i - (1 - \tilde{\beta}_i) w_i), & ~\mathrm{if}~ z \geq -\tilde{\beta}_i w_i, \\
-z, & ~\mathrm{if}~ z < -\beta_i w_i.} \end{align*} \if 0 and she believes that her maximum acceptable penalty is: \begin{align*}
\tilde{\zcmax}_i = -\tilde{\beta}_i w_i + \frac{1}{\lambda_i} \log \left( \frac{1 - \lambda_i w_i (1 - \tilde{\beta}_i)}{1 - \lambda_i w_i }\right). \end{align*} \fi
$\tilde{u}_i(0) \geq 0$ always holds, therefore under SP, the agent is going to bid: \begin{align*}
b_{i,{\mathrm{SP}}}^\ast = \tilde{u}_i(0) = w_i - 1/\lambda_i + e^{-\lambda_i \tilde{\beta}_i w_i } (1/\lambda_i - (1 - \tilde{\beta}_i) w_i). \end{align*}
Taking the derivative of $\tilde{u}_i(z)$ w.r.t. $z$ for $z \geq -\tilde{\beta}_i w_i$, we have: \begin{align*}
\frac{d}{dz}\tilde{u}_i(z) = e^{-\lambda_i (\tilde{\beta}_i w_i + z)} (-1 + \lambda_i w_i(1 - \tilde{\beta}_i)). \end{align*} When (A3) holds, $w_i < 1/\lambda_i$ implies $\frac{d}{dz}\tilde{u}_i(z) < 0$, meaning that $\tilde{u}_i(z)$ is monotonically decreasing in $z$, and that $\tilde{U}_i(z)$ and $\tilde{u}_i(z)$ coincide. The zero-crossing (i.e. the maximum acceptable penalty) is therefore equal to \begin{align*}
z^0_i = -\tilde{\beta}_i w_i + \frac{1}{\lambda_i} \log \left( \frac{1 - \lambda_i w_i (1 - \tilde{\beta}_i)}{1 - \lambda_i w_i }\right). \end{align*} Under the 2BPB mechanism, the agent is going to bid in DSE a maximum penalty \begin{align*}
\bar{b}_{i}^\ast = z^0_i, \end{align*} and once given a minimum penalty $\underline{z}$, the agent will then bid the smallest possible $\underline{b}_{i}^\ast = \underline{z}$.
The first-best social welfare for this agent can be achieved by setting $z = (1-\beta_i)w_i$, in which case the agent will use the resource if and only if $V_i^{(1)} + w_i \geq 0$. The first-best welfare is therefore: \begin{align*}
sw_i^{\mathrm{FB}} = w_i + ( e^{-\lambda_i w_i} - 1)/\lambda_i. \end{align*}
The first-best utilization is achieved by charging the highest penalty s.t. $sw_i(z) \geq 0$ still holds, i.e. it is possible for the outcome to be both budget balanced and individually rational. Solving the equation, we get the maximum penalty that we can charge as: \begin{align*}
z_i^{\mathrm{FB}} = -1/\lambda_i + (1-\beta_i) w_i + \frac{1}{\lambda_i}
\mathrm{ProductLog} \left(-1, e^{-1 + \lambda_i w_i} (-1 + \lambda_i w_i)\right). \end{align*} Here, $\mathrm{ProductLog}$ (also called the Lambert $W$ function) is the inverse relation of the function $f(s) = se^s$. The first best utilization achieved at penalty $z_i^{\mathrm{FB}} $ is therefore: \begin{align*}
ut_i^{\mathrm{FB}} = ut_i(z_i^{\mathrm{FB}}) = 1 - e^{1 - \lambda_i w_i +
\mathrm{ProductLog}\left(-1, e^{-1 + \lambda_i w_i} (-1 + \lambda_i w_i) \right)}. \end{align*}
\subsection{Uniform Type Model}
We now consider the uniform type model, where an agent type is parametrized by $(\alpha_i, w_i, \beta_i, \tilde{\beta}_i)$. In period~1, with penalty $z$, the agent will show up to use the resource if and only if \begin{align*}
V_i^{(1)} + \beta_i w_i \geq -z \Leftrightarrow V_i^{(1)} \geq -z - \beta_i w_i. \end{align*}
With $V_i^{(1)} \sim \mathrm{U}[-\alpha_i, ~0]$, we know that there are three cases depending on $z$: \begin{enumerate}[$\bullet$]
\item when $z \leq -\beta_i w_i$, $-z - \beta_i w_i$ is strictly positive, thus the agent never shows up, resulting in utilization and welfare both equal to zero.
\item when $z > \alpha_i -\beta_i w_i$, $-z - \beta_i w_i < -\alpha_i$ so that the agent always shows up. The utilization is therefore equal to $1$, and the welfare is equal to $\E{V_i^{(1)} + w_i} = w_i -\alpha_i/2$.
\item when $z \in (-\beta_i w_i, \alpha_i -\beta_i w_i]$, the agent shows up with probability $(z + \beta_i w_i)/\alpha_i$. \end{enumerate} Putting the three cases together, we know that the utilization as a function of the penalty $z$ is: \begin{align*}
ut_i(z) = \pwfun{0, &~\mathrm{if}~ z < -\beta_i w_i, \\
(z + \beta_i w_i)/\alpha_i, & ~\mathrm{if}~ -\beta_i w_i \leq z < \alpha_i - \beta_i w_i, \\
1, & ~\mathrm{if}~ z > \alpha_i - \beta_i w_i.} \end{align*}
The expected social welfare is: \begin{align*}
sw_i(z) = \pwfun{0, &~\mathrm{if}~ z < -\beta_i w_i, \\
\frac{z + \beta_i w_i}{\alpha_i} \left( w_i - \frac{z + \beta_i w_i}{2}\right), & ~\mathrm{if}~ -\beta_i w_i \leq z < \alpha_i - \beta_i w_i, \\
w_i - \alpha_i/2, & ~\mathrm{if}~ z > \alpha_i - \beta_i w_i,} \end{align*} and the agent's expected utility is: \begin{align*}
& u_i(z) \\ = & \pwfun{-z, &~\mathrm{if}~ z < -\beta_i w_i, \\
\frac{z + \beta_i w_i}{\alpha_i} \left( w_i - \frac{z + \beta_i w_i}{2}\right) - z \frac{\alpha_i - (z + \beta_i w_i)}{\alpha_i}, & ~\mathrm{if}~ -\beta_i w_i \leq z < \alpha_i - \beta_i w_i, \\
w_i - \alpha_i/2, & ~\mathrm{if}~ z > \alpha_i - \beta_i w_i.} \end{align*}
$\tilde{u}_i(z)$ can be obtained simply by replacing $\beta_i$ with $\tilde{\beta}_i$ in the above expression.
$\tilde{u}_i(0) \geq 0$ always holds, Therefore under SP, the agent is going to bid: \begin{align*}
b_{i,{\mathrm{SP}}}^\ast = \tilde{u}_i(0) = \frac{\tilde{\beta}_i w_i}{\alpha_i} \left( w_i - \frac{\tilde{\beta}_i w_i}{2}\right). \end{align*}
Note that for $-\tilde{\beta}_i w_i \leq z < \alpha_i - \tilde{\beta}_i w_i$, $\tilde{u}_i(z)$ can be rewritten in the following quadratic form: \begin{align*}
\tilde{u}_i(z) = \frac{1}{2\alpha_i}\left( z^2 - 2(\alpha_i - w_i)z + w_i^2 \tilde{\beta}_i (2 - \tilde{\beta}_i)\right). \end{align*} The minimum is achieved at $z_i^\ast = \alpha_i - w_i$, which is
\begin{align*}
\tilde{u}_i(z_i^\ast) = \frac{1}{2\alpha_i}\left( - (\alpha_i - w_i)^2 + w_i^2 \tilde{\beta}_i (2 - \tilde{\beta}_i)\right). \end{align*}
When (A3) holds i.e. $\E{V_i^{(1)} + w_i} < 0$, we have $w_i < \alpha_i/2$, which implies $\tilde{u}_i(z^\ast) \leq 0$ for any $\tilde{\beta}_i \in [0,1]$. As a result, $\tilde{u}_i(z)$ is monotonically decreasing in $z$ for $z \leq z_i^*$, monotonically increasing for $z > z_i^*$, and $\tilde{u}_i(z) \leq 0$ holds for all $z < z_i^*$. This implies that
$\tilde{u}_i(z) $ and $\tilde{U}_i(z)$ coincide for all $z$ s.t. $\tilde{u}_i(z) \geq 0$, and that the zero-crossing of $\tilde{u}_i(z)$ and $\tilde{U}_i(z)$ (i.e. the maximum acceptable penalty) is of the form: \begin{align*}
z^0_i = \alpha_i - w_i - \sqrt{\alpha_i^2 - 2 \alpha_i w_i + (-1 + \tilde{\beta}_i)^2 w_i^2}. \end{align*}
The DSE bid on maximum penalty under the 2BPB mechanism is therefore: \begin{align*}
\bar{b}_{i}^\ast = z^0_i, \end{align*} and once allocated given a minimum penalty $\underline{z}$, the agent will then bid $\underline{b}_{i}^\ast = \underline{z}$.
Similar to the exponential model, the first-best welfare is achieved by setting the penalty as $z = (1-\beta_i)w_i$, in which case \begin{align*}
sw_i^{\mathrm{FB}} = \frac{w_i}{\alpha_i} \left( w_i - \frac{w_i}{2}\right) = \frac{w_i^2}{2\alpha}. \end{align*}
The first-best social welfare is achieved at $z = 2 w_i - \beta_i w_i$, in which case: \begin{align*}
ut_i^{\mathrm{FB}} = (2w_i - \beta_i w_i + \beta_i w_i) / \alpha_i = 2w_i/\alpha_i. \end{align*}
\end{document}
\hma{
\section{Additional Discussions and Examples}
\subsection{The First Best Welfare and Utilization} \label{appx:fb}
\subsection{Two-Bid Penalty Bidding Not Optimal} }
\section{A List of Notations}
\begin{itemize}
\setlength\itemsep{0.3em}
\item $N$: the set of agents; $i \in N$; $|N| = n$
\item $m$: the number of items/resources
\item $\theta_i = (F_i^{(1)}, v_i^{(2)}, \beta_i, \tilde{\beta}_i)$ : agent $i$'s type
\begin{itemize}
\item $V_i^{(1)}$: the period~$1$ (random) value of agent $i$ from using a resource
\item $F_i^{(1)}$: the CDF of $V_i^{(1)}$; $f_i^{(1)}$: the pdf or pmf of $V_i^{(1)}$.
\item $v_i^{(2)}$: the period~$2$ value of agent $i$ from using a resource
\item $\beta_i$: the present bias factor of agent $i$
\item $\tilde{\beta}_i$: the present bias factor agent $i$ believes that she has
\item $\tilde{\theta}_i = (F_i^{(1)}, v_i^{(2)}, \tilde{\beta}_i)$: agent $i$'s private information at time~0.
\end{itemize}
\item $\mathcal{M} = (\mathcal{R}, x, s, t)$: a mechanism
\begin{itemize}
\item $\mathcal{R}$: the set of all possible messages (i.e. reports)
\item $r_i \in \mathcal{R}$: the report made by agent $i$
\item $x_i \in \{0, ~1\}$: the allocation rule
\item $t_i^{(0)}$: the period~$0$ payment of agent $i$
\item $t_i^{(1)}$: the period~$1$ penalty payment of agent $i$
\end{itemize}
\item $y_i = s_i$: the ``base payment'' facing agent $i$
\item $z_i= t_i$: the ``penalty payment'' facing agent $i$
\item $u_i(z)$: the actual expected utility of agent $i$, given no base payment and penalty payment $z$
\item $\tilde{u}_i(z)$: the expected utility agent $i$ believes she has, given penalty $z$
\item $(c_i, p_i, w_i, \beta_i, \tilde{\beta}_i)$: the type of an agent with $(c_i, p_i, w_i)$ model type
\item $(\lambda_i, w_i, \beta_i, \tilde{\beta}_i)$: the type of an agent with exponential model type
\item $(\alpha_i, w_i, \beta_i, \tilde{\beta}_i)$: the type of an agent with uniform model type
\item $ut_i^{\mathrm{FB}}$, $sw_i^{\mathrm{FB}}$: the first best utilization and social welfare \end{itemize}
\end{document} |
\begin{document}
\begin{abstract}
In this paper, claims by Lemmens and Seidel in 1973 about equiangular sets of lines with angle $1/5$ are proved by carefully analyzing pillar decomposition, with the aid of the uniqueness of two-graphs on $276$ vertices. The Neumann Theorem is generalized in the sense that if there are more than $2r-2$ equiangular lines in $\mathbb{R}^r$, then the angle is quite restricted. Together with techniques on finding saturated equiangular sets, we determine the maximum size of equiangular sets ``exactly'' in an $r$-dimensional Euclidean space for $r = 8$, $9$, and $10$. \end{abstract}
\title{Equiangular lines and the Lemmens-Seidel conjecture}
\section{Introduction} \label{sec:intro} A set of lines in Euclidean space is called \emph{equiangular} if any pair of lines forms the same angle. For examples, the four diagonal lines of a cube are equiangular in $\mathbb{R}^3$ with the angle $\arccos (1/3)$, and the six diagonal lines of an icosahedron form $6$ equiangular lines with angle $\arccos (1/\sqrt 5)$. The structure of methane \ce{CH4} also contains equiangular lines: carbon-hydrogen chemical bounds form the same angle (about $109.5$ degrees). Equiangular lines in real and complex spaces are related to many beautiful mathematical topics and even quantum physics, such as SIC-POVM~\cite{renes2004symmetric, scott2010symmetric, scott2006tight, zauner2011grundzuge}. First, equiangular lines in real spaces are equivalent to the notion of \emph{two-graphs} which caught much attention in algebra~\cite{godsil2013}. A classical way to construct equiangular lines comes from combinatorial designs. For instance, the 90 equiangular lines in $\mathbb{R}^{20}$ and 72 equiangular lines in $\mathbb{R}^{19}$ can be obtained from the Witt design. The details can be found in Taylor's thesis in 1971~\cite{taylor1971}. The spherical embedding of certain strongly regular graphs can also give arise to equiangular lines ~\cite{cameron2004strongly}; the maximum size of equiangular lines in $\mathbb{R}^{23}$ is $276$ which can be constructed from the strongly regular graphs with parameters $(276,135,78,54)$. Such configuration is the solution to the energy minimizing problems~\cite{saff1997distributing}, also known as the \emph{Thomson Problem}. The Thomson problem, posed by the physicist J.~J.~Thomson in 1904~\cite{thomson1904xxiv}, is to determine the minimum electrostatic potential energy configuration of $N$ electrons constrained to the surface of a unit sphere that repel each other with a force given by Coulomb's law.
The configuration of several maximum equiangular lines would give arise to the minimizer of a large class of energy minimizing problems called the \emph{universal optimal codes}~\cite{cohn2007universally}. Furthermore, if we have $\frac{r(r+1)}{2}$ equiangular lines in $\mathbb{R}^r$ (which is known as the \emph{Gerzon bounds}~\cite{lemmens1973}), then they will offer the construction of tight spherical $5$-designs~\cite{delsarte1977spherical} which are also universal optimal codes. So far, only when $r=2,3,7$, and $23$ can the Gerzon bounds be achieved. The special sets of equiangular lines, called \emph{equiangular tight frames} (ETFs) refer to the optimal line packing problems~\cite{mixon2018short}. ETFs achieve the classical Welch bounds~\cite{welch1974lower} which are the lower bounds for maximum absolute value of inner product values between distinct points on unit sphere, i.e. if we have $M$ points $\{x_i\}_{i=1}^M$ on the unit sphere in $\mathbb{R}^{r}$, then \begin{equation*}
\max_{i \neq j} |\langle x_i,x_j \rangle| \geq \sqrt{\frac{M-r}{r(M-1)}}. \end{equation*} The study of ETFs has numerous references \cite{fickus2018tremain, strohmer2003, fickus2016equiangular, jasper2014kirkman, barg2015finite, waldron2009construction, strohmer2003}.
From another point of view, a set of equiangular lines can be regarded as the collection of points on the unit sphere such that distinct points in the set have mutual inner products either $\alpha$ or $-\alpha$ for some $\alpha \in [0,1)$. Below we formally state its definition. \begin{defn} \label{defn:equiangular} We say that a finite set of unit vector $X = \{ x_1, \dots, x_s \}$ in $\mathbb R^r$ is an \emph{equiangular set} if for some $\alpha \in [0,1)$, \begin{equation} \label{eq:equi-a} \langle x_i, x_j \rangle \in \{ -\alpha, \alpha \} \qquad \text{whenever $i \neq j$}. \end{equation} \end{defn} By abuse of language, we will say that a set of vectors which satisfy the condition~(\ref{eq:equi-a}) are equiangular with \emph{angle} $\alpha$, although the actual angle of intersection is $\arccos \alpha$. A natural question in this context is: what is the maximum size of equiangular sets in $\mathbb{R}^r$? We denote by $M(r)$ for this quantity. The values of $M(r)$ were extensively studied over the last 70 years. It is easy to see that $M(2) = 3$ and the maximum construction is realized by the three diagonal lines of a regular hexagon. In 1948, Haantjes~\cite{haantjes1948} showed that $M(3) = M(4) = 6$. In 1966, van~Lint and Seidel~\cite{vanlint1966} showed that $M(5) = 10$, $M(6) = 16$, and $M(7) \geq 28$. Currently, there are only 35 known values for $M(r)$ and all of them have that $r \leq 43$. To the best of our knowledge, the ranges of $M(r)$ for $2 \leq r \leq 43$ are listed in Table~\ref{tb:smallnd} (see~\cite{azarija2016,barg2014,greaves2016,greaves2018equiangular,greaves2019equiangular,lin2018saturated,yu2015}).
\begin{table}[h]
\centering
\caption{Maximum cardinalities of equiangular lines for small dimensions}
\label{tb:smallnd}
\begin{tabular}{c|ccccccccc}
$r$ & 2 & 3--4 & 5 & 6 & 7--13 & 14 & 15 & 16 & 17 \\ \hline
$M(r)$ & 3 & 6 & 10 & 16 & 28 & 28--29 & 36 & 40--41 & 48--49 \\
\hline\hline
$r$ & 18 & 19 & 20 & 21 & 22 & 23--41 & 42 & 43 \\ \hline
$M(r)$ & 56--60 & 72--75 & 90--95 & 126 & 176 & 276 & 276--288 & 344
\end{tabular} \end{table}
Note that for the dimensions $r=14,16,17,18,19,20$, determining the exact values of $M(r)$ is still an open problem; though we know that the current well-known maximum constructions of equiangular lines are saturated \cite{lin2018saturated}, i.e.\ the current maximum constructions of equiangular lines cannot be added any more line while keeping equiangular. The estimation of upper bounds for equiangular lines can be considered from several different methods. The bounds could be achieved by semidefinite programming method~\cite{barg2014, okuda2016new, glazyrin2018upper}, the analysis of eigenvalues of the Seidel matrices~\cite{greaves2016, greaves2018equiangular, greaves2018equiangular}, polynomial methods~\cite{glazyrin2018upper}, Ramsey theory for asymptotic bounds~\cite{balla2018equiangular}, forbidden subgraphs for graphs of bounded spectral radius~\cite{jiang2017forbidden, jiang2019equiangular}, and algebraic graphs theory~\cite{godsil2013, neumaier1989graph}.
The motivation for the study of equiangular lines can also be various. For instance, Bannai, Okuda and Tagami~\cite{bannai2015spherical} considered the tight harmonic index 4-designs problems and proved that the existence of tight harmonic index 4-designs is equivalent to the existence of $\frac{(r+1)(r+2)}{6}$ equiangular lines with angle $\sqrt{\frac{3}{r+4}}$ in $\mathbb{R}^r$. Later, Okuda-Yu~\cite{okuda2016new} proved such equiangular lines do not exist for all $r > 2$. For more information about harmonic index $t$-designs, please see the references~\cite{bannai2015spherical, zhu2017spherical, bannai2018half, bannai2018classification}.
The main contribution for this paper is that we proved the result which Lemmens-Seidel claimed true in 1973. In \cite{lemmens1973}, Lemmens and Seidel claimed that the following conjecture holds when the base size $K = 2, 3, 5$ (for the definition of base size, see Definition~\ref{defn:base-size}):
\begin{conj}[\cite{lemmens1973}, Conjecture~5.8] \label{conj:LS} The maximum size of equiangular sets in $\mathbb{R}^r$ for angle $\frac{1}{5}$ is $276$ for $23 \leq r \leq 185$, and $\lfloor \frac12 (r-5) \rfloor + r + 1$ for $r \geq 185$. \end{conj}
Although the conjecture was prominent in the study of equiangular lines, no proof was found in the literature for the cases $K=3,5$. Following the discussion of pillar methods, we use techniques from linear algebra, linear programming, and the uniqueness of the two-graphs with $276$ vertices to prove the $K=3,5$ cases, and offer a partial solution for $K=4$. We also offer better upper bounds for the equiangular sets for some special setting on pillar conditions.
There is another interesting phenomenon that receives our attention. It is well known that $M(8) = 28$ (see~Table~\ref{tb:smallnd}), but those $28$ lines always live in a $7$-dimensional subspace of $\mathbb R^8$ (\cite{glazyrin2018upper}, Theorem 4). Glazyrin and Yu~\cite{glazyrin2018upper} asks the maximum size of equiangular sets of general \emph{ranks}. The following theorem essentially states that the angle is restricted when the size of equiangular set is large enough. \begin{thm}[Neumann, cf.~\cite{lemmens1973}] \label{thm:neumann}
Let $X$ be an equiangular set with angle $\alpha$ in $\mathbb R^r$. If $|X| > 2r$, then $\frac{1}{\alpha}$ is an odd integer. \end{thm} We first give a generalization of the Neumann theorem (see Theorem~\ref{thm:general Neu}), then we employ the techniques about saturated equiangular sets in~\cite{lin2018saturated} to determine the maximum size of equiangular sets of ranks $8$, $9$, and $10$.
The organization of the paper is as follows. In Section~\ref{sec:prerequisites} we review the basic notations in the study of equiangular sets and recall the pillar decomposition introduced by Lemmens and Seidel~\cite{lemmens1973}. In Section~\ref{sec:schur} we determine the maximum size of a pillar with orthogonal vectors only. In Section~\ref{sec:a5} we provide a proof for the Lemmen-Seidel conjecture when the base size $K = 3$ or $5$, and also give a new upper bound for $K = 4$. In Section~\ref{sec:max-rank} we discuss the maximum size of equiangular sets of prescribed rank. We close this paper with some discussions and proposing two conjectures based on our computations.
\section{Prerequisites} \label{sec:prerequisites}
Throughout this paper, $\hat{x}$ denotes the unit vector in the same direction as a non-zero vector $x$ in an Euclidean space. We start with some basic definitions for equiangular sets. Let $X$ be an equiangular set with angle $\alpha$ in $\mathbb R^r$. There are a few mathematical objects that could be associated to $X$.
\begin{defn} \label{defn:gramian} Let $X = \{ x_1, \dots, x_s \} \in \mathbb R^r$ be a finite set of vectors. The \emph{Gram matrix} of $X$, denoted by $G(X)$ or $G(x_1, \dots, x_s)$, is the matrix of mutual inner products of $x_1$, \dots, $x_s$; that is, \begin{equation*}
G(X) = X^{\ol{T}} X = \begin{bmatrix} \langle x_i, x_j \rangle \end{bmatrix}_{i,j=1}^s \end{equation*} \end{defn}
When $X$ is equiangular with angle $\alpha$, then its Gram matrix $G(X)$ is symmetric and positive semidefinite, with entries $1$ along its diagonal and $\pm \alpha$ elsewhere. The rank of $G(X)$ is the dimension of the span of vectors in $X$; $X$ is linearly independent if and only if $G(X)$ is of full rank (or equivalently, positive definite).
\begin{defn} \label{defn:seidel-graph} For an equiangular set $X = \{ x_1, \dots, x_s \}$ with angle $\alpha$, the \emph{Seidel graph} of $X$ is a simple graph $S(X)$ whose vertex set is $X$, and two vertices $x_i$ and $x_j$ of $S(X)$ are adjacent if and only if $\langle x_i, x_j \rangle = -\alpha$. \end{defn}
Since we are interested in equiangular lines in $\mathbb R^r$, choices need to be made between two unit vectors that span the same line. However, the choices could affect the signs of their mutual inner products. If two sets of vectors represent the same set of lines, they are called in the same \emph{switching class}. This terminology comes from the graph theory: if we \emph{switch} a vertex $v$ in a simple graph, the resulting graph is obtained by removing all edges that are incident to $v$ but adding edges connecting $v$ to all vertices that were not adjacent to $v$. We also have the freedom to relabel the vertices of the graph. All these actions lead to the following proposition about the switching equivalence for two Gram matrices.
\begin{prop}[\cite{king2016}, Definition~4] \label{prop:switching-equivalent} Two sets of unit vectors $X$, $Y$ in $\mathbb R^r$ are in the same switching class if and only if there are a diagonal $(1,-1)$-matrix $B$ and a permutation matrix $C$ such that \begin{equation*} (CB)^{\ol{T}} \cdot G(X) \cdot (CB) = G(Y). \end{equation*} We would also say that $G(X)$ is switching equivalent to $G(Y)$, and write $G(X) \simeq G(Y)$. \end{prop}
As usual, let $I_s$ (resp.\ $J_s$) denote the identity matrix (resp.\ all-one matrix) of size $s \times s$; the subscript $s$ will sometimes be dropped when the size is clear from the context.
\begin{prop}[\cite{lemmens1973}, Section~4] \label{prop:k-a1} If there are $k \geq 2$ equiangular vectors $p_1$, \dots, $p_k$ such that \begin{equation*} G(p_1, \dots, p_k) \simeq (1 + \alpha) I - \alpha J, \qquad \alpha > 0, \end{equation*} then $k \leq \frac{1}{\alpha} + 1$. Furthermore, if $k < \frac{1}{\alpha} + 1$, then the vectors $p_1$, \dots, $p_k$ are linearly independent; but if $k = \frac{1}{\alpha} + 1$, then the vectors $p_1$, \dots, $p_k$ are linearly dependent. In fact, if $k = \frac{1}{\alpha} + 1$ and $G(p_1, \dots, p_k) = (1 + \alpha) I - \alpha J$, the vectors $p_1$, \dots, $p_k$ form a $k$-simplex in $\mathbb R^{k-1}$. \end{prop}
Under a suitable choice of signs, the vectors $\pm p_1$, \dots, $\pm p_k$ from an equiangular set $X$ will form a $k$-clique in its Seidel graph. Following~\cite{lemmens1973}, we will define two important notions that are associated to an equiangular set $X$ (Definitions~\ref{defn:base-size} and \ref{defn:K-base}).
\begin{defn}[\cite{lemmens1973}] \label{defn:base-size} Let $X$ be an equiangular set in $\mathbb R^r$ with angle $\alpha$. The \emph{base size} of $X$, denoted by $K(X)$, is defined as \begin{equation*} K(X) := \max \{ k \in \mathbb N \colon \text{there exist $p_1$, \dots, $p_k$ in $X$ such that $G(p_1, \dots, p_k) \simeq (1 + \alpha) I - \alpha J$} \}. \end{equation*} In other words, $K(X)$ is the maximum of the clique numbers of Seidel graphs that are switching equivalent to that of $X$. \end{defn}
Note that the clique numbers of Seidel graphs in the switching class of $X$ are not constant, therefore we need to take their maximum. Nevertheless $K(X)$ is always bounded by $\frac{1}{\alpha} + 1$ by Proposition~\ref{prop:k-a1}. Since we are interested in large equiangular sets, we will assume that $\frac{1}{\alpha}$ is an odd integer, thanks to Theorem~\ref{thm:neumann}. The following proposition states that the only meaningful range of base size is $2, 3, \dots, \frac{1}{\alpha} + 1$.
\begin{prop}[\cite{king2016}, Proposition~3] Let $X$ be an equiangular set in $\mathbb R^r$.
If $|X| \geq 2$, then $K(X) \geq 2$. \end{prop}
\begin{proof} If two vertices in the Seidel graph $S(X)$ are independent, then we switch one of the them to form a $2$-clique. \end{proof}
\begin{defn}[\cite{lemmens1973}] \label{defn:K-base} Let $X$ be an equiangular set with angle $\alpha$ and base size $K$. A set of $K$ vectors $p_1$, \dots, $p_K$ is called a \emph{$K$-base} of $X$ if $p_1$, \dots, $p_k$ belong to some set which is switching equivalent to $X$, and $G(p_1, \dots, p_K) = (1 + \alpha) I - \alpha J$. \end{defn}
Let $K$ be the base size of an equiangular set $X$. We will fix a $K$-base $P = \{ p_1, \dots, p_K \}$ that forms a $K$-clique in the Seidel graph of $X$. Now we introduce the \emph{pillar decomposition} of $X$ with respect to $P$, following~\cite{lemmens1973}. (More details can also be found in~\cite{king2016}.)
For each vector $x \in X \setminus P$, there is a $(1,-1)$-vector $\varepsilon(x) \in \mathbb R^K$ such that \begin{equation*}
\bigl( \langle x, p_1 \rangle, \dots, \langle x, p_K \rangle \bigr)
= \alpha \cdot \varepsilon(x). \end{equation*} A vector $x$ in $X$ will be replaced by $-x$ if $\varepsilon(x)$ has more positive entries than $\varepsilon(-x)$, or $\varepsilon(x)$ has the same number of positive entries as $\varepsilon(-x)$ and $\langle x, p_K \rangle = \alpha$; otherwise the vector $x$ stays put.
Let $\Sigma(\varepsilon(x))$ denote the number of positive entries in $\varepsilon(x)$. A \emph{pillar} (with respect to a $K$-base $P$) containing a vector $x \in X \setminus P$, denoted by $\bar{x}$, is the subset of vectors $x' \in X \setminus P$ such that $\varepsilon(x') = \varepsilon(x)$; $\bar{x}$ is called a $(K,n)$ pillar when $\Sigma(\varepsilon(x)) = n$. Thus the vectors in $X \setminus P$ are partitioned into several $(K,n)$ pillars for $1 \leq n \leq \lfloor \frac{K}{2} \rfloor$. The number of different $(K,n)$ pillars is at most $\binom{K}{n}$ when $1 \leq n < \frac{K}{2}$, but is at most $\frac{1}{2} \binom{K}{K/2}$ when $n = \frac{K}{2}$. However, if $K = \frac{1}{\alpha} + 1$, then $p_1, \dots, p_K$ form a $K$-simplex and $\sum_{i=1}^K p_i = 0$. Therefore $\varepsilon(x)$ has the same number of positive entries as negative entries, thus only $(K, \frac{K}{2})$ pillars can exist. The collection of all $(K,n)$ pillars in an equiangular set $X$ will be denoted by $X(K,n)$.
The following fact will be used in many occasions. \begin{prop} \label{prop:K1-indep} Let $X$ be an equiangular set with angle $\alpha$ and base size $K$, and $P = \{ p_1, \dots, p_K \}$ be a $K$-base. If two vectors $x, y$ belong to the same $(K,1)$ pillar with respect to $P$, then $\langle x, y \rangle = \alpha$. \end{prop}
\begin{proof} By definition of $x$ and $y$ being in the same $(K,1)$ pillar, there are $K-1$ vectors in $P$ to which both $x$ and $y$ are adjacent in the Seidel graph $S(X)$ of $X$. If $x$ and $y$ are also adjacent to each other in $S(X)$, $x$ and $y$ together with those $K-1$ vectors that they are connected to form a $(K+1)$-clique in $S(X)$, which contradicts to the definition of the base size $K = K(X)$. Hence there is no edge connecting $x$ and $y$ in $S(X)$, which is equivalent of saying that $\langle x, y \rangle = \alpha > 0$. \end{proof}
\section{Schur decomposition for symmetric positive semidefinite matrices} \label{sec:schur}
In checking a matrix being positive (semi-)definite, we use the Schur decomposition.
\begin{thm}[Schur decomposition~\cite{boyd2004convex}] \label{thm:schur-decomp} Let $M$ be a symmetric real matrix, given by blocks \begin{equation*} M = \begin{bmatrix} A & B \\ B^{\ol{T}} & C \end{bmatrix} \end{equation*} Suppose that $A$ is positive definite. Then $M$ is positive (semi-)definite if and only if $C - B^{\ol{T}} A^{-1} B$ is positive (semi-)definite. \end{thm}
Let $X$ be an equiangular set with angle $\alpha = \frac{1}{(2n+1)}$ and base size $K = K(X) = \frac{1+3\alpha}{2\alpha} = n+2$ in $\mathbb R^r$. The reason for this particular combination of $\alpha$ and $K$ will be clear soon. Let $P = \{ p_1, \dots, p_K \}$ be a $K$-base of $X$, $\Gamma$ be the subspace spanned by $P$, and $\Gamma^\perp$ be the orthogonal complement of $\Gamma$ in $\mathbb R^r$.
For the vectors $x_1, x_2 \in X \setminus P$ belonging to the same $(K,1)$ pillar, let $x_1 = h + c_1$, $x_2 = h + c_2$ be their pillar decomposition, that is, $h \in \Gamma$, and $c_1, c_2 \in \Gamma^\perp$. As $h$ is a linear combination of $p_1, \dots, p_K$, we can write $h = \sum_{i=1}^K c_i p_i$ for some unknown coefficients $c_1, \dots, c_K$. Since $x_1$ belongs to a $(K,1)$ pillar, there is an index $k_0 \in \{1,\dots, K\}$ such that \begin{equation} \label{eq:k1-inner} \langle x, p_k \rangle = \langle h, p_k \rangle = \left\{ \begin{array}{ll} \alpha, & \text{if } k = k_0; \\ -\alpha, & \text{if } k \neq k_0. \end{array} \right. \end{equation} Rewriting (\ref{eq:k1-inner}) as a matrix equation, we see that \begin{equation} \label{eq:k1-inner-matrix-eq} G \cdot \begin{bmatrix} c_1 \\ \vdots \\ c_K \end{bmatrix} = \alpha \cdot \bigl( 2 e_{k_0} -\sum_{i=1}^K e_i \bigr), \end{equation} where $G = G(P) = (1 + \alpha) I - \alpha J$ is the Gram matrix for $P$, and $\{ e_1, \dots, e_K \}$ is the standard orthonormal basis for $\mathbb R^K$. Since $G$ is positive and invertible, we compute \begin{equation*} G^{-1} = \frac{1}{1 + \alpha} I + \frac{\alpha}{(1+\alpha)(1+\alpha-K\alpha)} J. \end{equation*} Hence by (\ref{eq:k1-inner-matrix-eq}) we obtain that \begin{equation*} c_k = \left\{ \begin{array}{ll} 0, & \text{if } k = k_0, \\ -(K-1)^{-1}, & \text{if } k \neq k_0; \end{array} \right. \end{equation*} that is, \begin{equation*} h = \frac{-1}{K-1} \bigl( \sum_{i=1}^K p_i - p_{k_0} \bigr). \end{equation*} From this expression we conclude that $\langle h, h \rangle = \alpha$. Since $\langle x_1, x_2 \rangle = \alpha$ by Proposition~\ref{prop:K1-indep}, we conclude that $\langle \hat{c}_1, \hat{c}_2 \rangle = 0$, that is, the $c$-vectors within a single $(K,1)$ pillar are orthogonal. (The orthogonality condition among the $c$-vectors does not hold for any other combinations of $\alpha$ and $K$.)
\begin{thm} \label{thm:2vec} Let $n$ be a positive integer with $n \geq 2$, and $\alpha = \frac{1}{(2n+1)}$. Let $X$ be an equiangular set with angle $\alpha$ and base size $K = n+2$ in $\mathbb R^r$, and we fix a base $P = \{ p_1, \dots, p_K \}$ for $X$. If there is a $(K,1)$ pillar with at least two vectors, then for any other $(K,1)$ pillar $\bar{x}$, \begin{equation*}
|\bar{x}| \leq \left\{ \begin{array}{ll} 2n^2 (n+1), & \text{if $n \leq 3$}; \\ \frac12 n^2 (n+1)^2, & \text{if $n \geq 3$}. \end{array} \right. \end{equation*} \end{thm}
\begin{proof} Let us look at the situation where two vectors come from different pillars. Suppose that $x = h_1 + c_1$ and $u = h_2 + c_2$ in $X$ belong to distinct $(K,1)$ pillars. Because the Hamming distance of $\varepsilon(x)$ and $\varepsilon(u)$ is $2$, we have \begin{equation*} \langle h_1, h_2 \rangle = \frac{n-1}{(n+1)(2n+1)}. \end{equation*} Therefore \begin{equation*}
\langle \hat{c}_1, \hat{c}_2 \rangle = \frac{\langle x, u \rangle - \langle h_1, h_2 \rangle}{ \| c \|^2 } = \frac{ \pm \frac{1}{2n+1} - \frac{n-1}{(n+1)(2n+1)} }{1 - \frac{1}{2n+1}} = \frac1{n(n+1)}, -\frac{1}{n+1}. \end{equation*}
Now suppose that the pillar $\bar{u}$ contains two vectors $u_1, u_2$, and $\bar{x}$ contains $N$ vectors $x_1, \dots, x_N$. Let $x_i = h_1 + c_i$ and $u_i = h_2 + d_i$ be their pillar decomposition. Then the Gram matrix of $\{ \hat{c}_1, \dots, \hat{c}_N, \hat{d}_1, \hat{d}_2 \}$ has the following form: \begin{equation*} G = G( \hat{c}_1, \dots, \hat{c}_N, \hat{d}_1, \hat{d}_2 ) = \begin{bmatrix} \\ & I_N & & v_1 & v_2 \\ \\ & v_1^{\ol{T}} & & 1 & 0 \\ & v_2^{\ol{T}} & & 0 & 1 \end{bmatrix}, \end{equation*} where $v_1$ and $v_2$ are vectors in $\mathbb R^N$ with entries in $\{ \frac{1}{n(n+1)}, \frac{-1}{n+1} \}$. Let us assume that in $\bar{x}$, \begin{itemize} \item there are $\ell_{11}$ vectors $x$ such that $\langle x, u_1 \rangle = \alpha$, $\langle x, u_2 \rangle = \alpha$; \item there are $\ell_{12}$ vectors $x$ such that $\langle x, u_1 \rangle = \alpha$, $\langle x, u_2 \rangle = -\alpha$; \item there are $\ell_{21}$ vectors $x$ such that $\langle x, u_1 \rangle = -\alpha$, $\langle x, u_2 \rangle = \alpha$; \item there are $\ell_{22}$ vectors $x$ such that $\langle x, u_1 \rangle = -\alpha$, $\langle x, u_2 \rangle = -\alpha$. \end{itemize} Certainly $\ell_{11} + \ell_{12} + \ell_{21} + \ell_{22} = N$. It follows that \begin{align*} \langle v_1, v_1 \rangle &= \frac{\ell_{11} + \ell_{12}}{n^2(n+1)^2} + \frac{\ell_{21}+\ell_{22}}{(n+1)^2}; \\ \langle v_2, v_2 \rangle &= \frac{\ell_{11} + \ell_{21}}{n^2(n+1)^2} + \frac{\ell_{12}+\ell_{22}}{(n+1)^2}; \\ \langle v_1, v_2 \rangle = \langle v_2, v_1 \rangle &= \frac{\ell_{11}}{n^2(n+1)^2} - \frac{\ell_{12}+\ell_{21}}{n(n+1)^2} + \frac{\ell_{22}}{(n+1)^2}. \end{align*} Since the Gram matrix $G$ is positive semidefinite, the following $2 \times 2$ matrix is also positive semidefinite by Theorem~\ref{thm:schur-decomp}: \begin{equation*} M := \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} - \begin{bmatrix} v_1^{\ol{T}} \\ v_2^{\ol{T}} \end{bmatrix} I_N^{-1} \begin{bmatrix} v_1 & v_2 \end{bmatrix} = \begin{bmatrix} 1 - \langle v_1, v_1 \rangle & - \langle v_1, v_2 \rangle \\ - \langle v_2, v_1 \rangle & 1 - \langle v_2, v_2 \rangle \end{bmatrix} \succcurlyeq 0. \end{equation*} Because $M$ is symmetric, $M$ is positive semidefinite if and only if $\op{tr} M \geq 0$ and $\det M \geq 0$. We compute \begin{align} \label{eq:trM} \frac{n^2 (n+1)^2}{2} \op{tr} M &= n^2 (n+1)^2 - \bigl(\ell_{11} + \frac{n^2 + 1}{2} (\ell_{12} + \ell_{21}) + n^2 \ell_{22} \bigr); \\ n^4 (n+1)^4 \det M &= \det \bigl( n^2 (n+1)^2 M \bigr) \nonumber \\ &= n^4 (n+1)^4 - n^2 (n+1)^2 \bigl( 2 (\ell_{11} + n^2 \ell_{22}) + (n^2+1) (\ell_{12} + \ell_{21}) \bigr) \label{eq:detM} \\ &\phantom{= } + (\ell_{11} + n^2 \ell_{22} + \ell_{12} + n^2 \ell_{21}) (\ell_{11} + n^2 \ell_{22} + \ell_{21} + n^2 \ell_{12} ) \nonumber \\ &\phantom{= } - \bigl( \ell_{11} + n^2 \ell_{22} - n (\ell_{12} + \ell_{21}) \bigr)^2. \nonumber \end{align} Keep in mind that we want to maximize $N = \ell_{11} + \ell_{12} + \ell_{21} + \ell_{22}$ subject to $\op{tr} M \geq 0$, $\det M \geq 0$, and the variables $\ell_{ij}$ are all non-negative integers. If we look closely to (\ref{eq:trM}) and (\ref{eq:detM}), the terms $\ell_{11} + n^2 \ell_{22}$ always appear as a pair, and there is no other separate term for $\ell_{11}$ and $\ell_{22}$; as a result, the sum $\ell_{11} + \ell_{22}$ is maximized when $\ell_{22} = 0$. Henceforth we let $\ell_{22} = 0$ and continue the computation from (\ref{eq:detM}): \begin{align} \nonumber n^4 (n+1)^4 \det M &= n^4 (n+1)^4 - n^2 (n+1)^2 \bigl( 2 \ell_{11} + (n^2 + 1)(\ell_{12} + \ell_{21}) \bigr) \\ \nonumber &\phantom{= } + (\ell_{11} + \ell_{12} + n^2 \ell_{21})(\ell_{11} + \ell_{21} + n^2 \ell_{12}) - \bigl( \ell_{11} - n (\ell_{12} + \ell_{21}) \bigr)^2 \\ \label{eq:detM2} &= n^4 (n+1)^4 - n^2 (n+1)^2 \bigl( 2 \ell_{11} + (n^2 + 1)(\ell_{12} + \ell_{21}) \bigr) \\ \nonumber &\phantom{= } + (n + 1)^2 \ell_{11} (\ell_{12} + \ell_{21}) + (n^2 - 1)^2 \ell_{12} \ell_{21}. \end{align} The expressions and (\ref{eq:trM}) and (\ref{eq:detM2}) are symmetric with respect to $\ell_{12}$ and $\ell_{21}$, and if the sum $\ell_{12} + \ell_{21}$ is fixed, (\ref{eq:detM2}) is maximized when $\ell_{12} = \ell_{21}$ by the A.M.-G.M. inequality. So we set $s = \ell_{11}$ and $t = \ell_{12} = \ell_{21}$ and continue the computation: \begin{align*}
n^4 (n+1)^4 \det M &= n^4 (n+1)^4 - 2n^2 (n+1)^2 (s + (n^2+1) t) + 2(n+1)^2 st + (n^2-1)^2 t^2 \\
&= (n+1)^2 (n^2 - t) \bigl( n^2 (n+1)^2 - 2s - (n-1)^2 t \bigr). \end{align*} Therefore the problem becomes \begin{equation} \label{eq:LP} \begin{array}{rl} \text{to maximize} & N = s + 2t \\ \\ \text{subject to} & \left\{ \begin{array}{l} s, t \in \mathbb Z, \quad s, t \geq 0, \\ n^2 (n+1)^2 - s - (n^2 + 1) t \geq 0, \\ (n^2 - t) \bigl( n^2 (n+1)^2 - 2s - (n-1)^2 t \bigr) \geq 0. \end{array} \right. \end{array} \end{equation}
\begin{figure}
\caption{The feasible domain for the linear programming problem (\ref{eq:LP})}
\label{fig:LP}
\end{figure}
This is a standard problem in linear programming, whose feasible domain is shaded in Figure~\ref{fig:LP}. We solve the problem and write the maximum $N_0$ of $N$ as \begin{equation*} N_0 = \left\{ \begin{array}{lll} 2n^2(n+1), & \text{achieved at $(s,t) = (2n^3, n^2)$}, & \text{when $n \leq 3$}; \\ \frac{1}{2} n^2 (n+1)^2, & \text{achieved at $(s,t) = (\frac{1}{2} n^2 (n+1)^2, 0)$}, & \text{when $n \geq 3$}. \end{array} \right. \end{equation*} The proof is now completed. \end{proof}
\noindent{\bf Example.} For $n = 3$, we are looking at the angle $\alpha = \frac{1}{7}$ and the base size $K = 5$. By Theorem~\ref{thm:2vec}, if there is a $(5,1)$ pillar with two or more vectors, then the size of another $(5,1)$ pillars is bounded by $72$. This maximum is achieved in two ways: the quadruple $(\ell_{11}, \ell_{12}, \ell_{21}, \ell_{22})$ defined in the proof of the theorem can be $(72, 0, 0, 0)$ or $(54, 9, 9, 0)$.
\noindent{\bf Remark.} Following the proof of their Lemma~16, King and Tang \cite{king2016} proved that $|\bar{x}| \leq n^2 (n+1)^2$ for a $(K,1)$ pillar $\bar{x}$ if there is another nonempty $(K,1)$ pillar. Theorem~\ref{thm:2vec} cuts their bound by half.
\section{The Lemmens-Seidel conjecture} \label{sec:a5}
Throughout this section we assume that the common angle is $\alpha = \frac{1}{5}$. Let us first recall a theorem in \cite{lemmens1973}.
\begin{thm}[\cite{lemmens1973}, Theorem~5.7]
Any set of unit vectors with inner product $\pm \frac{1}{5}$ in $\mathbb R^r$, which contains $6$ unit vectors with inner product $-\frac{1}{5}$, has maximum cardinality $276$ for $23 \leq r \leq 185$, $\lfloor \frac{1}{2} (r-5) \rfloor + r + 1$ for $r \geq 185$. \end{thm} This theorem corresponds to the case where the common angle $\alpha = \frac{1}{5}$ and base size $K = 6$. Lemmens and Seidel concluded Section~5 of \cite{lemmens1973} with the following remark, which we quote here:
\begin{quote} {\sl It would be interesting to know whether Theorem~5.7 holds true without the requirement of the existence of $6$ unit vectors with inner product $-\frac{1}{5}$. \dots\ The authors have obtained only partial results in this direction. In fact, the cases where [the base size $K$] $=2, 3, 5$ have been proved, but the case [$K=4$] remains unsettled. Yet, there is enough evidence to support the following conjecture. \dots} \end{quote}
So they raised their conjecture (Conjecture~\ref{conj:LS}), but the proofs, even for the cases $K = 3, 5$, have been elusive. Sections~3 and 4 of \cite{king2016} provided some upper bounds for $\alpha = \frac{1}{5}$. It is well known that $|X| \leq r$ if $X \subset \mathbb R^r$ and $K = 2$~(cf.~\cite{king2016}, Corollary~2). In this section we are going to sharpen their results and prove the conjecture when $K = 3, 5$.
\subsection{\texorpdfstring{$K = 3$}{K=3}} \label{sec:a5-k3}
Let $X \subset \mathbb R^r$ be an equiangular set with angle $\frac{1}{5}$ in $\mathbb R^r$, with the base size $K = K(X) = 3$. Let $P = \{ p_1, p_2, p_3 \}$ be a $3$-base in $X$, and the rest of the vectors in $X \setminus P$ are partitioned into three $(3,1)$ pillars. By symmetry, for a unit vector $x \in X \setminus P$ that satisfies \begin{equation*} ( \langle x, p_1 \rangle, \langle x, p_2 \rangle, \langle x, p_3 \rangle ) = \frac{1}{5} (1, -1, -1), \end{equation*} we can decompose $x$ into $x = h + c$, where $h \in \Gamma$ and $c \in \Gamma^\perp$. A little computation shows that \begin{equation*} h = \frac{1}{9} (p_1 - 2p_2 - 2p_3). \end{equation*}
So $\| h \|^2 = \frac{1}{9}$ and $\| c \|^2 = \frac{8}{9}$. If $x_1 = h + c_1$ and $x_2 = h + c_2$ come from the same $(3,1)$ pillar, then $\langle x_1, x_2 \rangle = \frac{1}{5}$ by Proposition~\ref{prop:K1-indep}, henceforth $\langle \hat{c}_1, \hat{c}_2 \rangle = \frac{1}{10}$. If $x = h_1 + c_1$ and $y = h_2 + c_2$ come from different $(3,1)$ pillars, then (by symmetry again) \begin{equation*}
\langle h_1, h_2 \rangle = \langle \frac{1}{9} (p_1 - 2p_2 - 2p_3), \frac{1}{9} (-2p_1 + p_2 - 2p_3) \rangle = -\frac{1}{45}. \end{equation*} Since \begin{equation*} \pm \frac{1}{5} = \langle x, y \rangle = \langle h_1, h_2 \rangle + \langle c_1, c_2 \rangle, \end{equation*} hence $\langle \hat{c}_1, \hat{c}_2 \rangle \in \{ \frac{1}{4}, -\frac{1}{5} \}$.
\begin{lem} \label{lem:two-X31} Suppose that there are two nonempty $(3,1)$ pillars. If one of them has $4$ vectors, then the other has at most $54$ vectors. \end{lem}
\begin{proof} Let \begin{align*} \bar{x} &= \{ h_1 + c_i \colon h_1 \in \Gamma, c_i \in \Gamma^\perp, i = 1, \dots, n \}, \\ \bar{u} &= \{ h_2 + d_i \colon h_2 \in \Gamma, d_i \in \Gamma^\perp, i = 1, 2, 3, 4 \}, \end{align*} be two nonempty $(3,1)$ pillars. Then the Gram matrix of $\hat{c}_i$ and $\hat{d}_i$ has the following form: \begin{equation} \label{eq:G-cd} G = G( \hat{c}_1, \dots, \hat{c}_n, \hat{d}_1, \dots, \hat{d}_4 ) = \begin{bmatrix}
& & & & & & & \\
& & & & & & & \\
& & \frac{9}{10} I_n + \frac{1}{10} J_n & v_1 & v_2 & v_3 & v_4 \\
& & & & & & & \\
& & v_1^{\ol{T}} & & & & & \\
& & v_2^{\ol{T}} & \multicolumn{4}{c}{\frac{9}{10} I_4 + \frac{1}{10} J_4} \\
& & v_3^{\ol{T}} & & & & & \\
& & v_4^{\ol{T}} & & & & & \end{bmatrix}, \end{equation} where $v_1, \dots, v_4$ are column vectors whose entries are $\frac{1}{4}$ or $-\frac{1}{5}$. Since $G$ needs to be positive semidefinite, by Theorem~\ref{thm:schur-decomp} we see that \begin{equation} \label{eq:a5-31-4} M := \Bigl( \frac{9}{10} I_4 + \frac{1}{10} J_4 \Bigr) - V^{\ol{T}} \Bigl( \frac{9}{10} I_n + \frac{1}{10} J_n \Bigr)^{-1} V \succcurlyeq 0, \qquad \text{where } V := \begin{bmatrix} v_1 & v_2 & v_3 & v_4 \end{bmatrix}. \end{equation} The following setup is used to facilitate the computation. Consider the Seidel graph $S'$ generated by the vectors in $\bar{x} \cup \bar{u}$. By Proposition~\ref{prop:K1-indep}, $S'$ is a bipartite graph because every edge must connect a vertex in $\bar{x}$ to a vertex in $\bar{u}$. Let us classify the vectors in $\bar{x}$ by how they are connected to the vectors $u_1, \dots, u_4$ in $\bar{u}$. Let $B_4$ be the set of binary strings of length $4$, and let $B_{4,i}$ denote the subset of $B_4$ consisting of those binary strings $b_1 b_2 b_3 b_4$ such that $\sum_j b_j = i$ for $i = 0, 1, 2, 3, 4$. For $B = b_1 b_2 b_3 b_4 \in B_4$, let $t_B$ denote the number of vectors $h_1 + c$ in the pillar $\bar{x}$ such that \begin{equation*} \langle \hat{c}, \hat{d}_i \rangle = \left\{ \begin{array}{ll} \frac{1}{4}, & \text{if $b_i = 0$}, \\ -\frac{1}{5}, & \text{if $b_i = 1$}, \end{array} \right. \quad i = 1, 2, 3, 4. \end{equation*} In total there are $2^4 = 16$ variables $t_B$, $B \in B_4$, of non-negative integral values. Obviously $n = \sum_{B \in B_4} t_B$, which is the total number of vectors in $\bar{x}$, and $\sum_{B \in B_{4,i}} t_B$ is the number of vertices of degree $i$ in $\bar{x}$, for $i = 0, 1, 2, 3, 4$.
The vectors $v_1, v_2, v_3, v_4$ in the Gram matrix $G$ in (\ref{eq:G-cd}) has the following mutual inner products: \begin{equation*}
\langle v_i, v_j \rangle = \frac{1}{16} \sum_{B \in B^{0,0}_{i,j}} t_B - \frac{1}{20} \sum_{B \in B^{0,1}_{i,j}} t_B + \frac{1}{25} \sum_{B \in B^{1,1}_{i,j}} t_B, \qquad i, j \in \{ 1, 2, 3, 4 \}, \end{equation*} where $B^{k,\ell}_{i,j}$ is the subset of $B_4$ consisting of $B = b_1 b_2 b_3 b_4$ such that $\{b_i, b_j\} = \{k, \ell\}$, for $k, \ell \in \{ 0, 1 \}$. For instance, \begin{align*} \inn{v}{1}{2} &= \frac{1}{16} ( t_{0000} + t_{0001} + t_{0010} + t_{0011} ) \\ &\phantom{= } - \frac{1}{20} ( t_{0100} + t_{0101} + t_{0110} + t_{0111} + t_{1000} + t_{1001} + t_{1010} + t_{1011} ) \\ &\phantom{= } + \frac{1}{25} ( t_{1100} + t_{1101} + t_{1110} + t_{1111} ). \end{align*} We also need \begin{equation*} w_i := \frac{1}{4} \sum_{\substack{ B = b_1 b_2 b_3 b_4 \in B_4 \\ b_i = 0 }} t_B - \frac{1}{5} \sum_{\substack{ B = b_1 b_2 b_3 b_4 \in B_4 \\ b_i = 1 }} t_B, \quad i \in \{ 1, 2, 3, 4 \}. \end{equation*} For example, \begin{align*} w_1 &= \frac{1}{4} ( t_{0000} + t_{0001} + t_{0010} + t_{0011} + t_{0100} + t_{0101} + t_{0110} + t_{0111}) \\ &\phantom{= } - \frac{1}{5} ( t_{1000} + t_{1001} + t_{1010} + t_{1011} + t_{1100} + t_{1101} + t_{1110} + t_{1111}). \end{align*} Since \begin{equation*} \Bigl( \frac{9}{10} I_n + \frac{1}{10} J_n \Bigr)^{-1} = \frac{10}{9} \Bigl( I_n - \frac{1}{9+n} J_n \Bigr), \end{equation*} \begin{equation*} V^{\ol{T}} I_n V = \begin{bmatrix} \inn{v}{i}{j} \end{bmatrix}_{i,j=1}^4, \qquad V^{\ol{T}} J_n V = \begin{bmatrix} w_i w_j \end{bmatrix}_{i,j=1}^4. \end{equation*} we use these informations to expand the left-hand side of (\ref{eq:a5-31-4}) as \begin{equation} \label{eq:a5-31-4-a} M = \frac{9}{10} I_4 + \frac{1}{10} J_4 - \frac{10}{9} V^{\ol{T}} I_n V + \frac{10}{9(9+n)} V^{\ol{T}} J_n V = \begin{bmatrix} m_{ij} \end{bmatrix}_{i,j=1}^4, \end{equation} where the entries $m_{ij}$ are \begin{equation*} m_{ij} = \left\{ \begin{array}{ll} 1 - \frac{10}{9} \inn{v}{i}{i} + \frac{10}{9(9+n)} w_i^2, & \text{if } i = j, \\ \frac{1}{10} - \frac{10}{9} \inn{v}{i}{j} + \frac{10}{9(9+n)} w_i w_j, & \text{if } i \neq j, \end{array} \qquad i, j \in \{1, 2, 3, 4 \}. \right. \end{equation*} Remind that we want to maximize the sum $n = \sum_{B \in B_4} t_B$ subject to the conditions $t_B \in \mathbb Z$, $t_B \geq 0$ for all $B \in B_4$, and $M \succcurlyeq 0$. Notice that when we set some of the variables $t_B$ to be zero, we are focusing on a particular subset of vectors in the pillar $\bar{x}$. We argue that each of the variables $t_B$ has an upper bound as follows: \begin{itemize} \item Set $t_{0000} = n$ and $t_B = 0$ for all $B \neq 0000$. Then \begin{equation*} M = \frac{9}{10} I_4 + \Bigl( \frac{1}{10} - \frac{5n}{8(9+n)} \Bigr) J_4. \end{equation*} By considering its eigenvalues, we see that $M$ is positive semidefinite if and only if \begin{equation*} \frac{9}{10} + 4 \cdot \Bigl( \frac{1}{10} - \frac{5n}{8(9+n)} \Bigr) \geq 0. \end{equation*} Solving this inequality for $n$, we get $-9 \leq n \leq \frac{39}{4}$. Since $n$ only assumes a non-negative integral values, we see that $0 \leq n \leq 9$; this is the range for $t_{0000}$.
\item Set $t_{1000} = n$ and $t_B = 0$ for all $B \neq 1000$. Then \begin{equation*} M = \begin{bmatrix} 1 - \frac{5n}{8(9+n)} & \frac{1}{10} - \frac{5n}{8(9+n)} & \frac{1}{10} - \frac{5n}{8(9+n)} & \frac{1}{10} + \frac{n}{2(9+n)} \\ \frac{1}{10} - \frac{5n}{8(9+n)} & 1 - \frac{5n}{8(9+n)} & \frac{1}{10} - \frac{5n}{8(9+n)} & \frac{1}{10} + \frac{n}{2(9+n)} \\ \frac{1}{10} - \frac{5n}{8(9+n)} & \frac{1}{10} - \frac{5n}{8(9+n)} & 1 - \frac{5n}{8(9+n)} & \frac{1}{10} + \frac{n}{2(9+n)} \\ \frac{1}{10} + \frac{n}{2(9+n)} & \frac{1}{10} + \frac{n}{2(9+n)} & \frac{1}{10} + \frac{n}{2(9+n)} & 1 - \frac{2n}{5(9+n)} \end{bmatrix} \end{equation*} By considering non-negative values for $n$ only, our computation shows that $M$ is positive semidefinite if and only if $0 \leq n \leq 7$. By symmetry, we conclude that $0 \leq t_B \leq 7$ for each $B \in B_{4,1}$.
\item Set $t_{1100} = n$ and $t_B = 0$ for all $B \neq 1100$. Then \begin{equation*} M = \begin{bmatrix} 1 - \frac{5n}{8(9+n)} & \frac{1}{10} - \frac{5n}{8(9+n)} & \frac{1}{10} + \frac{n}{2(9+n)} & \frac{1}{10} + \frac{n}{2(9+n)} \\ \frac{1}{10} - \frac{5n}{8(9+n)} & 1 - \frac{5n}{8(9+n)} & \frac{1}{10} + \frac{n}{2(9+n)} & \frac{1}{10} + \frac{n}{2(9+n)} \\ \frac{1}{10} + \frac{n}{2(9+n)} & \frac{1}{10} + \frac{n}{2(9+n)} & 1 - \frac{2n}{5(9+n)} & \frac{1}{10} - \frac{2n}{5(9+n)} \\ \frac{1}{10} + \frac{n}{2(9+n)} & \frac{1}{10} + \frac{n}{2(9+n)} & \frac{1}{10} - \frac{2n}{5(9+n)} & 1 - \frac{2n}{5(9+n)} \end{bmatrix} \end{equation*} By considering non-negative values for $n$ only, our computation shows that $M$ is positive semidefinite if and only if $0 \leq n \leq 7$. By symmetry, we conclude that $0 \leq t_B \leq 7$ for each $B \in B_{4,2}$.
\item Set $t_{1110} = n$ and $t_B = 0$ for all $B \neq 1110$. Then \begin{equation*} M = \begin{bmatrix} 1 - \frac{5n}{8(9+n)} & \frac{1}{10} + \frac{n}{2(9+n)} & \frac{1}{10} + \frac{n}{2(9+n)} & \frac{1}{10} + \frac{n}{2(9+n)} \\ \frac{1}{10} + \frac{n}{2(9+n)} & 1 - \frac{2n}{5(9+n)} & \frac{1}{10} - \frac{2n}{5(9+n)} & \frac{1}{10} - \frac{2n}{5(9+n)} \\ \frac{1}{10} + \frac{n}{2(9+n)} & \frac{1}{10} - \frac{2n}{5(9+n)} & 1 - \frac{2n}{5(9+n)} & \frac{1}{10} -- \frac{2n}{5(9+n)} \\ \frac{1}{10} + \frac{n}{2(9+n)} & \frac{1}{10} - \frac{2n}{5(9+n)}& \frac{1}{10} - \frac{2n}{5(9+n)} & 1 - \frac{2n}{5(9+n)} \end{bmatrix} \end{equation*} By considering non-negative values for $n$ only, our computation shows that $M$ is positive semidefinite if and only if $0 \leq n \leq 9$. By symmetry, we conclude that $0 \leq t_B \leq 9$ for each $B \in B_{4,3}$.
\item Set $t_{1111} = n$ and $t_B = 0$ for all $B \neq 1111$. Then \begin{equation*} M = \frac{9}{10} I_4 + \Bigl( \frac{1}{10} - \frac{2n}{5(9+n)} \Bigr) J_4. \end{equation*} Hence $M$ is positive semidefinite if and only if $0 \leq n \leq 39$; this is the range for $t_{1111}$. \end{itemize}
Up to this point, we find that there are only a finite number of combinations of $16$-tuples $(t_B : B \in B_4)$ that will make the matrix $M$ positive semidefinite; so far there are $10 \cdot 8^4 \cdot 8^6 \cdot 10^4 \cdot 40 \approx 2.8 \times 10^{15}$ cases to check. To further reduce the computations, we have observed the following\footnote{The {\tt SAGE} script for this part of computations can be downloaded at \url{http://math.ntnu.edu.tw/~yclin/two-31-pillars.sage}.}: \begin{enumerate}[(i)] \item Let us consider the upper bounds on the number of vertices in $\bar{x}$ of each of the degrees in the Seidel graph $S'$ (generated by $\bar{x} \cup \bar{u}$), that is, upper bounds for $\sum_{B \in B_{4,i}} t_B$, $i = 0, 1, 2, 3, 4$. For example, when we only look for vertices of degree $1$, we set $t_B = 0$ whenever $B \in B \setminus B_{4,1}$. Since $0 \leq t_B \leq 7$ for $B \in B_{4,1}$, we only need to pick out those quadruples $(t_{0001}, t_{0010}, t_{0100}, t_{1000}) \in \{ 0, 1, \dots, 7 \}^4$ such that the resulting matrix $M$ in (\ref{eq:a5-31-4-a}) is positive semidefinite (there are only $(7+1)^4 = 4096$ cases to check). Among those quadruples which survive the test, the maximum for the sum $\sum_{B \in B_{4,1}} t_B$ is $16$, which occurs at $t_B = 4$ for each $B \in B_{4,1}$.
The computations for other degrees are similar and we find that \begin{equation*} \sum_{B \in B_{4,1}} t_B \leq 16, \qquad \sum_{B \in B_{4,2}} t_B \leq 13, \qquad \text{and} \quad \sum_{B \in B_{4,3}} t_B \leq 16. \end{equation*} \begin{comment} Combining these inequalities with $t_{0000} \leq 9$ and $t_{1111} \leq 39$, we find that \begin{equation*}
|\bar{x}| = t_{0000} + \sum_{i = 1}^3 \sum_{B \in B_{4,i}} t_B + t_{1111} \leq 9 + 16 + 13 + 16 + 39 = 93, \end{equation*}
that is, the size of a $(3,1)$ pillar $\bar{x}$ when bounded by $93$ if another $(3,1)$ pillar $\bar{u}$ with $|\bar{u}| \geq 4$ is present. \end{comment} This is not good enough to beat the Lemmens-Seidel bound\footnote{When there are two $(3,1)$ pillars with $4$ or more vectors, our computations shows that the size of whole equiangular set is bounded by $3 + 93 \cdot 3 = 282$ (see also the comparison done in Theorem~\ref{thm:a5-k3}). But this is not enough to beat the Lemmens-Seidel's bound of $276$.}, so we proceed further.
\item We fix the value of the variable $t_{1111}$ in the range $0 \leq t_{1111} \leq 39$, and consider the maximum possible value for another variable $t_B$ for $B \in B \setminus B_{4,4}$ subject to that the matrix $M$ in (\ref{eq:a5-31-4-a}) is positive semidefinite. To do this, we set $t_{B'} = 0$ whenever $B' \neq 1111, B' \neq B$. Table~\ref{tbl:tb-upper-bound} lists the upper bounds for $t_B$, $B \in B_{4,i}$, $i = 0, 1, 2, 3$, when the value of $t_{1111}$ is specified.
\begin{table} \centering \caption{Upper bounds for $t_B$ for specified values of $t_{1111}$} \label{tbl:tb-upper-bound}
\begin{tabular}{c|cccc|c} & \multicolumn{4}{c}{Upper bounds for $t_B$} & \\ $t_{1111}$ & $B \in B_{4,0}$ & $B \in B_{4,1}$ & $B \in B_{4,2}$ & $B \in B_{4,3}$ & $M_{\bar{x}}$ \\ \hline 0 & 9 & 7 & 7 & 9 & 54 \\ 1 & 5 & 5 & 6 & 9 & 51 \\ 2 & 3 & 4 & 5 & 9 & 50 \\ 3 & 2 & 3 & 5 & 9 & 46 \\ 4 & 2 & 3 & 4 & 8 & 47 \\ 5 & 1 & 2 & 4 & 8 & 43 \\ 6 & 1 & 2 & 3 & 8 & 44 \\ 7 & 1 & 2 & 3 & 8 & 45 \\ 8 & 1 & 1 & 3 & 7 & 42 \\ 9 & 0 & 1 & 3 & 7 & 42 \\ 10, 11 & 0 & 1 & 2 & 7 & 42, 43 \\ 12, 13 & 0 & 1 & 2 & 6 & 44, 45 \\ 14 & 0 & 0 & 2 & 6 & 42 \\ 15 & 0 & 0 & 1 & 6 & 37 \\ 16--19 & 0 & 0 & 1 & 5 & 38--41 \\ 20--22 & 0 & 0 & 1 & 4 & 42--44 \\ 23 & 0 & 0 & 0 & 4 & 39 \\ 24--27 & 0 & 0 & 0 & 3 & 36--39 \\ 28--31 & 0 & 0 & 0 & 2 & 36--39 \\ 32--35 & 0 & 0 & 0 & 1 & 36--39 \\ 36--39 & 0 & 0 & 0 & 0 & 36--39 \end{tabular} \end{table} \end{enumerate}
Denote the upper bound for $t_B$ for $B \in B_{4,i}$ found in Table~\ref{tbl:tb-upper-bound} by $m_i$, $i = 0, 1, 2, 3$. Since $|B_{4,0}| = 1$, $|B_{4,1}| = 4$, $|B_{4,2}| = 6$, and $|B_{4,3}| = 4$, an upper bound for the size of the pillar $\bar{x}$ is given by \begin{equation*} M_{\bar{x}} = m_0 + \min \{ 4m_1, 16 \} + \min \{ 6m_2, 13 \} + \min \{ 4m_3, 16 \} + t_{1111}. \end{equation*} The values for $M_{\bar{x}}$ are also listed in Table~\ref{tbl:tb-upper-bound}. From here we conclude that the size of a $(3,1)$ pillar cannot exceed 54 when another $(3,1)$ pillar with $4$ or more vectors is present. \end{proof}
\noindent {\bf Remark.} We note here that when a $(3,1)$ pillar $\bar{u}$ has $3$ vectors only, it is possible to have another $(3,1)$ pillar $\bar{x}$ with as many vectors as possible. This occurs when the inner product between any one vector in $\bar{x}$ and any one vector in $\bar{u}$ is $-\frac{1}{5}$.
Assume that $|\bar{x}| = n$. Then the Gram matrix $G = G(\hat{c}_1, \dots, \hat{c}_n, \hat{d}_1, \hat{d}_2, \hat{d}_3)$ is \begin{equation*} G = \begin{bmatrix} & & & & & \\ & \frac{9}{10} I_n + \frac{1}{10} J_n & & v & v & v \\ & & & & & \\ & v^{\ol{T}} & & & & \\ & v^{\ol{T}} & & \multicolumn{3}{c}{\frac{9}{10} I_3 + \frac{1}{10} J_3} \\ & v^{\ol{T}} & & & & \end{bmatrix}, \end{equation*} where $v$ is the vector $(-\frac{1}{5}, -\frac{1}{5}, \dots, -\frac{1}{5})$ in $\mathbb R^n$, and $G$ has the Schur decomposition: \begin{equation*} \frac{9}{10} I_3 + \frac{1}{10} J_3 - \begin{bmatrix} v^{\ol{T}} \\ v^{\ol{T}} \\ v^{\ol{T}} \end{bmatrix} \bigl( \frac{9}{10} I_n + \frac{1}{10} J_n \bigr)^{-1} \begin{bmatrix} v & v & v \end{bmatrix} = \frac{9}{10} I_3 + \Bigl( \frac{1}{10} - \frac{2n}{5(9+n)} \Bigr) J_3,
\end{equation*} which is always positive definite for any $n \in \mathbb N$.
\begin{thm} \label{thm:a5-k3} Let $X$ be an equiangular set with angle $\frac{1}{5}$ and base size $K(X) = 3$ in $\mathbb R^r$. Then \begin{equation*}
|X| \leq \max \{ 165, r+6 \}. \end{equation*} \end{thm}
\begin{proof} The equiangular set $X$ is decomposed as a disjoint union of $P = \{ p_1, p_2, p_3 \}$ and three $(3,1)$ pillars. If there are two $(3,1)$ pillars with four or more vectors, then by Lemma~\ref{lem:two-X31} we have \begin{equation*}
|X| = |P| + |X(3,1)| \leq 3 + 54 \cdot 3 = 165. \end{equation*} Otherwise there is only one big $(3,1)$ pillar and the other two pillars can have at most $3$ vectors each. Since vectors in a single $(3,1)$ pillar is linearly independent of rank $r-3$, we see that in this case \begin{equation*}
|X| = |P| + |X(3,1)| \leq 3 + (r-3) + 3 + 3 = r + 6. \end{equation*} These inequalities finish the proof of the theorem. \end{proof}
Note that $\max \{165, r + 6 \}$ is certainly less than the bound $\max \{ 276, r + 1 + \lfloor \frac{r-5}{2} \rfloor \}$ given in the Lemmens-Seidel conjecture for every $r \geq 23$, hence we have finished the proof when the base size $K(X) = 3$.
\subsection{\texorpdfstring{$K = 4$}{K=4}} \label{sec:a5-k4}
King and Tang (\cite{king2016}, Lemma~16) showed that $|\bar{x}| \leq 36$ for a $(4,1)$ pillar $\bar{x}$ if there is another nonempty $(4,1)$ pillar $\bar{x}$. We get a better upper bound for $|\bar{x}|$ for $\bar{x} \in X(4,1)$ if there is another nonempty $(4,1)$ pillar $\bar{u}$ with two or more vectors by applying Theorem~\ref{thm:2vec}. In the situation $n = 2$, so the maximum of $|\bar{x}|$ is $2n^2 (n+1) = 24$ if there is another $(4,1)$ pillar with two or more vectors. Hence we have the following result. \begin{prop} \label{prop:a5-41} In an equiangular set $X$ with angle $\frac15$ and the base size $K(X) = 4$ in $\mathbb R^r$, the maximum number of vectors that are contained in the four $(4,1)$ pillars is $\max \{ 96, r-1 \}$. \end{prop}
\begin{proof} If there are two $(4,1)$ pillars with two or more vectors, there are at most $24 \times 4 = 96$ vectors in those pillars. Otherwise, there can be one large pillar $\bar{x}$ together with three other pillars each of which contains at most one vector. In the case, since the vectors in $\bar{x}$ are linearly independent in the $(r-4)$-dimensional subspace $\Gamma^\perp$, the number of vectors in these $(4,1)$ pillars is at most $(r-4) + 3 = r - 1$. \end{proof}
\noindent {\bf Remark.} Under computations similar to Theorem~\ref{thm:2vec}, we find that if there are two nonempty $(4,1)$ pillars, then another $(4,1)$ pillar can hold at most $25$ vectors. Hence in the case where there is only one large pillar of size $r-4$ in Proposition~\ref{prop:a5-41}, there can only be one other nonempty $(4,1)$ pillar consisting of one vector when $r - 4 > 25$, i.e., $r \geq 30$.
For each of the three $(4,2)$ pillars, the best known bound of its cardinality is $s(r-4, \frac{1}{13}, -\frac{5}{13})$ obtained in~\cite{king2016}, which denotes the number of vectors in a $2$-distance set in $\mathbb R^{r-4}$ with angles $\frac{1}{13}$ and $-\frac{5}{13}$. With a little improvement under Proposition~\ref{prop:a5-41}, we state the result for $K = 4$.
\begin{prop} \label{prop:a5-412} Let $X$ be an equiangular set with the angle $\frac{1}{5}$ and base size $4$ in $\mathbb R^r$. Then \begin{equation} \label{eq:a5-k4-100}
|X| \leq 100 + 3 \cdot s\bigl(r-4, \frac{1}{13}, -\frac{5}{13} \bigr). \end{equation} \end{prop}
\begin{proof}
The equiangular set $X$ can be partitioned into the following pairwise disjoint subsets: the $4$-base $P$, four $(4,1)$ pillars, and three $(4,2)$ pillars. By Lemma~16 of~\cite{king2016}, any $(4,1)$ pillar $\bar{x}$ will satisfy $| \bar{x} | \leq 39$ if there is a nonempty $(4,2)$ pillar. Since $s(r-4, \frac{1}{13}, -\frac{5}{13}) \geq r - 4$ (which can be realized if all vectors within a $(4,2)$ pillar are linearly independent), we see that \begin{align*}
|X| &\leq |P| + |X(4,1)| + |X(4,2)| \leq 4 + 4 \cdot 24 + 3 \cdot s(r-4, \frac{1}{13}, -\frac{5}{13}) \\ &= 100 + 3 \cdot s(r-4, \frac{1}{13}, -\frac{5}{13}). \end{align*} \end{proof}
Notice that the right-hand side of (\ref{eq:a5-k4-100}) will never beat the Lemmens-Seidel bound. Details will be elaborated in Section~\ref{sec:closing}.
\subsection{\texorpdfstring{$K = 5$}{K=5}} \label{sec:a5-k5}
Let $X \subset \mathbb R^r$ be an equiangular set with angle $\frac{1}{5}$ in $\mathbb R^r$, with the base size $K = K(X) = 5$. Let $P = \{ p_1, p_2, p_3, p_4, p_5 \}$ be a $5$-base in $X$. With respect to $P$, $X \setminus P$ can be partitioned into $5$ possible $(5,1)$ pillars and $10$ possible $(5,2)$ pillars. By carefully analyzing those pillars, we answer affirmatively to the Lemmens-Seidel conjecture for the case $K = 5$.
\begin{thm} \label{thm:a5-k5} Let $X$ be an equiangular set with angle $\frac{1}{5}$ and base size $K(X) = 5$ in $\mathbb R^r$. \begin{enumerate}[$(1)$]
\item If there are two or more nonempty $(5,2)$ pillars, then $|X| \leq 272$.
\item If there is at most one nonempty $(5,2)$ pillar, then $|X| \leq \frac{4}{3} r + 12$. \end{enumerate} \end{thm}
\begin{proof}
By Lemma 18 of~\cite{king2016}, we know that $|X(5,1)| \leq 15$. Let us now consider the rest of the vectors $P \cup X(5,2)$. Note that $Y := P \cup \{ p_6 \} \cup X(5,2)$ is still an equiangular set with $K(Y) = 6$ in $\mathbb R^r$, where $p_6 = - \sum_{i=1}^5 p_i$. Those $(5,2)$ pillars in $X$ will become $(6,3)$ pillars in $Y$, and their classifications have been discussed thoroughly by Lemmens and Seidel~\cite{lemmens1973}. Let us recall a key fact found in the proof of Theorem~5.7 of~\cite{lemmens1973}.
\begin{lem}[\cite{lemmens1973}]
\label{lem:LS-5.7}
Let $Y$ be an equiangular set with angle $\frac{1}{5}$ and base size $K(Y) = 6$.
Let $P_Y$ be a $6$-base in $Y$ and $Y$ be decomposed into $P_Y$ and various $(6,3)$ pillars.
Suppose there are at least two nonempty pillars in $Y$.
\begin{enumerate}[(i)]
\item If there are two distinct pillars each of which contains a pair of adjacent vertices, then $|Y| \leq 276$.
\item If there is only one pillar containing a pair of adjacent vertices and all other pillars contain independent vertices only, then $|Y| \leq 222$.
\item If each of these nonempty pillars contains independent vertices only, then $|Y| \leq 258$.
\end{enumerate} \end{lem}
If there are two or more nonempty $(6,3)$ pillars and $|Y| > 258$, then $Y$ must be a subset of the equiangular set $Z$ with $276$ lines in $\mathbb R^{23}$ with a $6$-base $P \cup \{ p_6 \}$ by Lemma~\ref{lem:LS-5.7}. Goethals and Seidel~\cite{goethals1975} proved that the structure of these $276$ equiangular lines is unique, i.e., there is only one such switching class. Here we need an explicit description of these lines. The following detailed information can be found in~\cite{taylor1971,neumaier1984some}. Let $\mathfrak W$ be the collection of $759$ $8$-subsets of $\{1, 2, \dots, 24 \}$ that comes from the Steiner triple system $S(5,8,24)$ (or the Witt design~\cite{witt1937}), and $\mathfrak W_1$ be the subcollection of $\mathfrak W$ consisting of those $253$ $8$-subsets that contains $1$\footnote{The complete list of these 253 $8$-subsets of $[24]$ can be found at \url{http://math.ntnu.edu.tw/\~yclin/253-8.txt}.}. For any $\sigma \in \mathfrak W_1$, define $w_\sigma$ be the vector in $\mathbb R^{24}$: \begin{equation*} w_\sigma := 4 \sum_{i \in \sigma} e_i - 4 e_1 - \sum_{j=1}^{24} e_j. \end{equation*} For each $k \in \{ 2, 3, \dots, 24 \}$, let $v_k := 4 e_1 + 8 e_k - \sum_{j=1}^{24} e_j$ (with $e_1, \dots, e_{24}$ being the standard basis for $\mathbb R^{24}$). Thus \begin{equation*} Z_0 := \{ w_\sigma \colon \sigma \in \mathfrak W_1 \} \cup \{ v_k \colon k = 2, 3, \dots, 24 \} \end{equation*} gives rise to the $276$ equiangular set with angle $\frac{1}{5}$. Note that all these $276$ vectors lie in the hyperplane $5 x_1 + \sum_{j=2}^{24} x_j = 0$, and it is easy to see that $v_2, \dots, v_{24}$ are linearly independent, so the span of $Z_0$ is of dimension $23$. Consider the following $6$ elements from $\mathfrak W_1$: \begin{equation*} \begin{array}{ll} \sigma_1 = \{ 1, 2, 5, 8, 13, 15, 18, 20 \}, & \sigma_4 = \{ 1, 2, 5, 8, 9, 11, 22, 24 \}, \\ \sigma_2 = \{ 1, 2, 3, 4, 9, 10, 11, 12 \}, & \sigma_5 = \{ 1, 2, 3, 4, 17, 18, 19, 20 \}, \\ \sigma_3 = \{ 1, 3, 5, 7, 17, 19, 22, 24 \}, & \sigma_6 = \{ 1, 3, 5, 7, 10, 12, 13, 15 \}. \end{array} \end{equation*} and define \begin{equation*} p_i = \left\{ \begin{array}{ll} \widehat{w}_{\sigma_i}, & \text{if $i = 1, 2, 3$,} \\ -\widehat{w}_{\sigma_i}, & \text{if $i = 4, 5, 6$.} \end{array} \right. \end{equation*} Then the unit vectors $p_1, \dots, p_6$ have mutual inner products $-\frac{1}{5}$. For the remaining $270$ vectors from $Z_0 \setminus \{ \pm p_i \colon i = 1, \dots, 6 \}$, we normalize them and pick a suitable direction for each vector so that the resulting unit vectors all have inner products $\frac{1}{5}$ with $p_6$. Then these vectors have a pillar decomposition
\begin{equation*} Z = \{ p_1, \dots, p_6 \} \cup \bigcup_{i=1}^{10} \bar{z}_i, \end{equation*}
where each $\bar{z}_i$ is a $(6,3)$ pillar consisting of $27$ unit vectors, whose Seidel graph is a disjoint union of nine $3$-cliques. So there are $90$ $3$-cliques upstairs in the pillars. By uniqueness, we can assume that the set $Y$ above is a subset of $Z$ which contains the base set $p_1, \dots, p_6$. If $|Y| > 258$, then $Y$ misses at most $17$ vectors in the pillars upstairs, therefore $Y$ must contain at least one of those $90$ $3$-cliques. Such a $3$-clique, together with the $3$ vectors in the base $p_1, \dots, p_5$ to which all vertices of this $3$-clique connect, will form a $6$-clique in $X$, that is, $K(X) = 6$, which contradicts to the definition $K(X) = 5$. Therefore if there are two nonempty $(5,2)$ pillars, then $|X| = |Y| - 1 + |X(5,1)| \leq 258 - 1 + 15 = 272$. This finishes the first part of the proof.
Now let us assume that there is exactly one nonempty $(5,2)$ pillar $\bar{x}$.
There is an upper bound for the size of $|\bar{x}|$ in terms of the rank of $\bar{x}$.
\begin{lem} \label{lem:a5k5-one}
Let $\bar{x}$ be a $(5,2)$ pillar. Then $|\bar{x}| \leq \frac{4}{3} (d - 1)$, where $d = \op{rank}(\bar{x})$. \end{lem}
\begin{proof} Let $S$ be the Seidel graph of $\bar{x}$. We first claim that $S$ does not contain any $3$-clique. Let $P$ be the base set of $X$. By the definition of $(5,2)$ pillars, there are three vectors in $P$ such that all vectors in $\bar{x}$ are independent to them in the Seidel graph $S(X)$ of $X$. If there is a $3$-clique in $S$, then together with those three vectors in $P$, they would form a complete bipartite graph $K_{3,3}$ in $S(X)$, which is switching equivalent to $K_6$. This contradicts to the assumption that the base size of $X$ equals $5$.
We consider $Y = P \cup \{ p_6 \} \cup \bar{x}$ again. Now $\bar{x}$ becomes a $(6,3)$ pillar in $Y$. By Theorem~5.1 of~\cite{lemmens1973}, any connected component of the Seidel graph $S$ of $\bar{x}$ is a subgraph of one of the graphs in Figure~\ref{fig:graphtypes}, which are those connected graphs with maximum eigenvalue $2$. Let $S_1, \dots, S_k$ be the connected components of $S$, and $A_i$ be the adjacency matrix of $S_i$ for $i = 1, 2, \dots, k$. The Gram matrix $G$ of $\bar{x}$ assumes the following form: \begin{equation*} G = \frac{1}{5} J + \frac{4}{5} I - \frac{2}{5} A, \qquad \text{where } A = A_1 \oplus A_2 \oplus \cdots \oplus A_k. \end{equation*}
\fivegraphs
Let us now investigate the nullity of $G$. We have \begin{equation} \label{eq:decomp-G} x^{\ol{T}} G x = \frac{1}{5} x^{\ol{T}} J x + \frac{2}{5} x^{\ol{T}} (2I - A) x. \end{equation} The all-one matrix $J$ is already positive semidefinite; so is $2I - A$, because the maximum eigenvalue of $A$ is at most $2$. If $G x = 0$, then $x^{\ol{T}} G x = 0$; using (\ref{eq:decomp-G}) we see that $x^{\ol{T}} J x = 0 = x^{\ol{{T}}} (2I - A) x$ as well. The equation $x^{\ol{T}} J x = 0$ implies that the sum of the coordinates of $x$ vanishes. As $A$ is the direct sum of the $A_i$'s, we will investigate $2I - A_i$ separately. If $S_i$ is a proper subgraph of the five graphs listed in Figure~\ref{fig:graphtypes}, then the largest eigenvalue of $A_i$ is strictly less than $2$ by strict monotonicity~\cite{smith1970some}, and hence $2I - A_i$ is positive definite. Therefore $v^{\ol{T}} (2I - A_i) v = 0$ implies that $v = 0$. After relabeling, assume that $S_1$, \dots, $S_\ell$ are the components among $S_1$, \dots, $S_k$ that are listed in Figure~\ref{fig:graphtypes}. For each $A_j$, $1 \leq j \leq \ell$, let $v_j$ be the (unique) unit eigenvector of $A_j$ with eigenvalue $2$ whose coordinates are all positive. Then that $x^{\ol{T}} (2I - A) x = 0$ implies that $x$ lies in the span of $\tilde{v}_1$, \dots, $\tilde{v}_\ell$, where $\tilde{v}_j$ is the image of $v_j$ under the embedding induced by $A_j \rightarrow A_1 \oplus \cdots \oplus A_k$, $j = 1, \dots, \ell$. The vectors $\tilde{v}_1$, \dots, $\tilde{v}_\ell$ are clearly linearly independent, and since their coordinates are all nonnegative, we conclude that after intersecting with the subspace $x^{\ol{T}} J x = 0$, the nullity of $G$ equals $\ell - 1$.
Except for $K_3 = C_3$, the graphs listed in Figure~\ref{fig:graphtypes} have at least $4$ vertices. Therefore $4\ell \leq m$, where $m = |\bar{x}|$. Denoting the rank of $\bar{x}$ by $d$, we have \begin{equation*} m - d = \op{null}(G) = \ell - 1 \leq \frac{m}{4} - 1 \qquad \Rightarrow \qquad m \leq \frac{4}{3} (d-1), \end{equation*} which is exactly what we want to show. \end{proof}
\begin{comment} \begin{lem} \label{lem:a5k5-one}
Let $\bar{x}$ be a $(5,2)$ pillar. If $|\bar{x}| > \op{rank}(\bar{x})$, then the Seidel graph of $\bar{x}$ contains a $3$-clique. \end{lem} \begin{proof}[Proof of Lemma~\ref{lem:a5k5-one}] Again we consider $Y = P \cup \{ p_6 \} \cup \bar{x}$. Now $\bar{x}$ becomes a $(6,3)$ pillar in $Y$. By Theorem~5.1 of~\cite{lemmens1973}, any connected component of the Seidel graph of $\bar{x}$ is a subgraph of one of the graphs in Figure~\ref{fig:graphtypes}, which are those connected graphs with maximum eigenvalue $2$.
Except for $C_3 = K_3$, we check each case to ensure that $|\bar{x}| = \op{rank}(\bar{x})$: \begin{itemize} \item Type I, $C_n$ with $n \geq 4$. This follows from the property of circulant matrices (for example, see~\cite{meyer2000matrix}). \item Type II: Let $G$ be the Gram matrix for a Type II graph of $n$ vertices, and $x = (x_1, x_2, \dots, x_n)$ be any vector in $\mathbb R^n$. Then \begin{align*} x^{\ol{T}} G x = \frac15 \Bigl( &\Bigl( \sum_{k=1}^n x_k \Bigr)^2 + (2x_1 - x_3)^2 + (2x_2 - x_3)^2 + 2 \sum_{k=3}^{n-3} (x_k - x_{k+1})^2 \\ &+ (2 x_n - x_{n-2})^2 + (2 x_{n-1} - x_{n-2})^2 \Bigr) \geq 0, \end{align*} and the equality holds if and only if $x = 0$. Therefore $G$ is positive definite and of full rank. \item Types III, IV, V: Direct checks. \end{itemize} And the lemma is now proved. \end{proof} \end{comment}
Back to the proof of Theorem~\ref{thm:a5-k5}. Suppose $\bar{x}$ is the unique nonempty $(5,2)$ pillar in $X \subset \mathbb R^r$. Since $\dim \bar{x} = \dim X - 5 = r - 5$, we have $|\bar{x}| \leq \frac{4}{3} (r - 5 - 1) = \frac{4}{3} (r - 6)$ by Lemma~\ref{lem:a5k5-one}.
Together with $P$ and the vectors in $(5,1)$ pillars, we find
\begin{equation*}
|X| = |P| + |X(5,1)| + |X(5,2)| \leq 5 + 15 + \frac{4}{3} (r - 6) = \frac{4}{3} r + 12, \end{equation*} and the proof is now completed. \end{proof}
It is easily verified that $\max \{ 272, \frac{4}{3} r + 12 \} \leq \max \{ 276, r + 1 + \lfloor \frac{r-5}{2} \rfloor \}$ for any $r \in \mathbb N$. Hence we have proven the Lemmens-Seidel conjecture for the case $K = 5$.
\section{Maximum equiangular sets of certain ranks} \label{sec:max-rank}
Besides the maximum cardinality of equiangular sets in $\mathbb R^r$, Glazyrin and Yu considered a similar question in~\cite{glazyrin2018upper}. \begin{defn}
Let $r$ be a positive integer.
We define the number $M^*(r)$ to be the maximum cardinality of equiangular lines of rank $r$. \end{defn}
For example, we know that maximum size of equiangular line in $\mathbb{R}^8$ is 28. However, such $28$ equiangular lines in $\mathbb R^8$ actually live in a $7$-dimensional subspace by the Theorem 4 in~\cite{glazyrin2018upper}, yet $M^*(8)$ is unknown. It is well known that $M^*(7) = 28$ and $M^*(23) = 276$. It seems that $M^*(r)$ is an increasing function on $n$, but Glazyrin and Yu~\cite{glazyrin2018upper} refuted this by showing $M^*(24) < 276 = M^*(23)$. Moreover, not every value of $M^*(r)$ is known even for small $r$ in the literature, for instance $M^*(8)$.
We first deal with $M^*(8)$ and start with the following result. The main technique of identifying saturated equiangular sets can be found in the authors' previous work~\cite{lin2018saturated}.
\begin{prop} \label{prop:8-14}
There are at most $14$ equiangular lines of angle $\frac13$ of rank $8$. \end{prop}
\begin{proof} We first construct $8\times 8$ symmetric matrices whose diagonals are $1$, and $\pm \frac{1}{3}$ elsewhere. By considering their switching classes, we may assume that the entries in the first column and the first rows are all $\frac13$, except that the top-left corner being $1$. Since these matrices are Gram matrices for some bases for $\mathbb R^8$, they are required to be positive definite. The associated graph of such a matrix is a disjoint union of a graph of $7$ vertices and one isolated vertex. By checking all $1044$ such graphs (see \cite{flajolet2009}, Example~II.5), we find that there are only $3$ graphs that satisfy all conditions listed above. For each of those $3$ graphs, we collect all the unit vectors whose mutual inner products with each vector represented by the graph are $\pm \frac{1}{3}$, and transform these vectors as vertices of a new graph in which two vectors are adjacent if and only if their mutual inner products are $\pm \frac{1}{3}$. The clique number of the new graph plus $8$ will be the size of a saturated equiangular set, and we identify the maximum in these clique numbers. Saturated equiangular sets containing these three sets of $8$ basis vectors consist of $8$, $14$, and $14$ lines respectively, from which we conclude that $M_{\frac{1}{3}}(8) = 14$. \end{proof}
\noindent {\bf Remark.} (Uniqueness of maximum equiangular lines of angle $\frac{1}{3}$ of rank $8$) Among the $3$ graphs, found in the proof of Proposition~\ref{prop:8-14}, whose associated Gram matrices are positive definite, one contains $7$ independent vertices; adding another independent vertex to the other two graphs gives two graphs that are switching equivalent: $K_2$ and $6$ independent vertices; $K_{1,6}$ and one more independent vertex. Both of them can only be added a $6$-clique to form $14$ equiangular lines in $\mathbb R^8$ of angle $\frac{1}{3}$, so this is the only isomorphism class of $14$ equiangular lines of angle $\frac{1}{3}$ of rank $8$.
\begin{table}
\centering
\caption{Maximum sizes of equiangular lines with specified angles for small ranks}
\label{tb:smallMr}
\begin{equation*}
\begin{array}{r|cccc}
\text{angle } \alpha & \frac13 & \frac15 & \frac17 & \frac{1}{\sqrt{17}} \\
\hline
M_\alpha(8) & 14 & 10 & 9 \\ \hline
M_\alpha(9) & 16 & 12 & 10 & 18 \\ \hline
M_\alpha(10) & 18 & 16
\end{array}
\end{equation*} \end{table}
\vskip 0.1in
Lemmens and Seidel showed that $M_{\frac{1}{3}}(r) = 2r - 2$ for $r \geq 8$ (cf.~\cite{lemmens1973}, Theorem~4.5). The same technique as in the proof of Proposition~\ref{prop:8-14} is applied to produce Table~\ref{tb:smallMr}. We indicate that the technique in~\cite{lin2018saturated} is more powerful than semidefinite programming method in \cite{barg2014}. For instance, the semidefinite programming bound on equiangular sets with angle $\frac{1}{5}$ in $\mathbb{R}^8$ is $11.2$ and the technique in \cite{lin2018saturated} obtains the bound $10$.
Before we proceed further, we find the following generalization of the Neumann theorem (Theorem~\ref{thm:neumann}) is necessary.
\begin{thm}[Generalization of Neumann Theorem]\label{thm:general Neu} Let $r>3$ be a positive integer. If there are more than $2r-2$ equiangular lines with angle $\alpha$ in $\mathbb{R}^r$, then: \begin{itemize} \item When $r$ is even, $\dfrac{1}{\alpha}$ is an odd integer. \item When $r$ is odd, $\dfrac{1}{\alpha}$ is either an odd integer or $\sqrt{2r-1}$. Moreover, any equiangular set with angle $\frac{1}{\sqrt{2r-1}}$ of size $2r-1$ in $\mathbb R^r$ is a subset of some equiangular tight frame of size $2r$ in $\mathbb R^r$. \end{itemize} \end{thm}
\begin{proof}
It suffices to assume the existence of an equiangular set $X$ with angle $\alpha$ in $\mathbb R^r$ with $|X| = 2r-1$. Consider its Seidel matrix $A=\frac{1}{\alpha} (G(X)-I)$, which is a $(2r-1) \times (2r-1)$ symmetric matrix with integer coefficients and diagonal entries are all zeros, but non-diagonal entries are either $1$ or $-1$. The matrix $A$ will have an eigenvalue $a = \frac{-1}{\alpha}$ with multiplicity at least $r-1$, since $G$ has an eigenvalue zero with multiplicity at least $r-1$. If $a$ is rational, then $a$ must be an odd integer using the same argument as in the proof of the original Neumann theorem (cf.~\cite{lemmens1973}, Theorem~3.4). Otherwise $a$ is irrational, and by degree count $a$ must a zero of an irreducible quadratic polynomial over $\mathbb Z$, which we write as $x^2 - c_1 x + c_2$; let $a^*$ be the other zero of this quadratic polynomial, which must also be an eigenvalue of $A$.
The characteristic polynomial of $A$ assumes the following form: $\op{char}(A) = (x^2-c_1x+c_2)^{r-1}(x-c_3)$, with $c_i$ being all integers. Comparing the coefficients, we get: \begin{equation} \label{eq:rootsum} c_3+(r-1)c_1 = \op{tr} A = 0, \qquad \text{i.e.,} \qquad c_3 = -(r-1) c_1. \end{equation} Next, we see that \begin{equation} \label{eq:squaresum} (r-1) (a^2 + {a^*}^2) + c_3^2 = \op{tr} A^2 = (2r-1) (2r-2). \end{equation} By Vieta's formula, $a^2 + {a^*}^2 = (a + a^*)^2 - 2 a a^* = c_1^2 - 2 c_2$. Plug in this relation and (\ref{eq:rootsum}) back to (\ref{eq:squaresum}), we see that \begin{equation*}
r c_1^2 - 2 c_2 = 4r-2, \qquad \text{i.e.,} \qquad c_2 = \frac{r}{2} (c_1^2 - 4) + 1. \end{equation*} Because $a$ and $a^*$ are distinct real roots of the quadratic equation $x^2 - c_1 x + c_2 = 0$, its discriminant must be positive, that is, \begin{equation*} 0 < \Delta = c_1^2 - 4c_2 = (1-2r)(c_1^2-4). \end{equation*} Since $r \in \mathbb N$, that $\Delta > 0$ implies that $c_1 \in \{ 1, -1, 0 \}$. We look into these three cases separately: \begin{itemize} \item $c_1 = 1$. Then $(c_2, c_3) = (-\dfrac{3r}{2} + 1, -(r-1) )$. However, this case is not allowed, because the matrix $G$, being a Gram matrix, must be positive semidefinite, hence $0$ is the smallest eigenvalue of $G$, and this implies that $a = -\frac{1}{\alpha} = \frac{-1-\sqrt{6r-3}}{2}$ is the smallest eigenvalue of $A$. However, $c_3 = -(r-1)$, which is also an eigenvalue of $A$ by assumption, is always smaller than $a$, and this is a contradiction. \item $c_1 = -1$. Then $(c_2, c_3) = ( -\dfrac{3r}{2} + 1, r - 1 )$. Because $c_2 \in \mathbb Z$, we see that $r$ has to be an even integer; let $r = 2t$ for some $t \in \mathbb N$, $t > 1$. Now we compute the determinant of $A$ as $\det A = c_2^{r-1} \cdot c_3 = (1 - 3t)^{2t-1} \cdot (2t-1)$. If $t$ is even, then $\det A$ is an odd integer; if $t$ is odd, then $\det A$ is a multiple of $2^5$ since $t$ is at least $3$. Both contradict Corollary~3.6 of \cite{greaves2016} which states that $\det A \equiv 1 - (4t-1) = 2 - 4t \equiv 2 \pmod{4}$ when $A$ is a Seidel matrix of order $n = 2r-1 = 4t-1$.
\item $c_1 = 0$. Then $(c_2, c_3) = (-2r+1, 0)$, and the angle is $\alpha = \dfrac{1}{\sqrt{2r-1}}$.
The following proof is taken from~\cite{greaves2017symmetric}. Assume that there exists such a $(2r-1) \times (2r-1)$ symmetric matrix $A$ from the equiangular set $X$. Then the characteristic polynomial of $A$ must be $x (x^2 - 2r + 1)^{2t-1}$. Consider the matrix $M = (2r - 1) I - A^2$. Then the spectrum of $M$ is $\{ [0]^{2r-2}, [2r-1]^1 \}$, which implies that $M$ is positive semidefinite and of rank $1$. Observe that $M$ has diagonal entries only $1$ and has off-diagonal integer entries congruent to $1 \pmod{2}$. Since $M$ is of rank $1$, $M$ is a $(1,-1)$-matrix. By switching $M$ so that the entries in the first column and the first row are all $1$, it follows from the rank of $M$ being $1$ that $M = J$, the all-one matrix. Multiplying the all-one column vector $\bm{1}$ of length $2r-1$ by $(2r-1) I - A^2 = J$, we have $(2r-1) \bm{1} - A^2 \bm{1} = (2r-1) \bm{1}$. This implies that $A \bm{1} = 0$.
Now consider a symmetric matrix $\displaystyle B = \begin{bmatrix} A & \bm{1} \\ \bm{1}^{\ol{T}} & 0 \end{bmatrix}$ of order $2r$ with $0$ on the diagonal and $\pm 1$ otherwise. Then \begin{equation*}
B^2 = \begin{bmatrix}
A^2 + J & A \bm{1} \\ \bm{1}^{\ol{T}} A & \bm{1}^{\ol{T}} \bm{1}
\end{bmatrix}
= \begin{bmatrix}
(2r-1) I & 0 \\ 0 & (2r-1)
\end{bmatrix} = (2r-1) I. \end{equation*} Thus $B$ is a symmetric conference matrix of order $2r$. However, it is known in \cite[Corollary~2.2]{delsarte1971orthogonal} that the order of a symmetric conference matrix must be congruent to $2$ modulo $4$, hence $r$ must be an odd integer in this case. It also follows that $B$ has the eigenvalues $\pm \sqrt{2r-1}$, and thus $G = I - \frac{1}{\sqrt{2r-1}} B$ is positive semidefinite. This matrix $G$ is the Gram matrix of an $r \times 2r$ equiangular tight frame. The upper-left principal submatrix of size $(2r-1)$ of $G$ is the Gram matrix of the equiangular set $X$. Therefore $X$ is in some equiangular tight frame of size $2r$ in $\mathbb R^r$. \end{itemize} \end{proof}
\noindent{\bf Remark.} We also consider possible irrational angles $\alpha$ that could produce $14$ equiangular lines in $\mathbb R^8$. Let $A$ be the Seidel matrix of $14$ equiangular lines $X$ of rank $8$ with angle $\alpha \in \mathbb R \setminus \mathbb Q$. The number $a = -1/\alpha$ is the smallest eigenvalue of $A$ of multiplicity $6$ and hence $\mathbb Q(a)$ is a quadratic number field. We may assume that the characteristic polynomial is of the following form: \begin{equation*}
(x^2 - c_1 x + c_2)^6 (x^2 - c_3 x + c_4), \qquad
c_1, c_2, c_3, c_4 \in \mathbb Z. \end{equation*} The trace conditions on $A$ and $A^2$ imply that \begin{align*}
0 &= \op{tr} A = 6 c_1 + c_3, \\
182 = 14 \cdot 13 &= \op{tr} A^2 = 6 (c_1^2 - 2c_2) + (c_3^2 - 2 c_4). \end{align*} Since all eigenvalues of $A$ are real and $a$ is not rational, we also have $c_1^2 > 4 c_2$ and $c_3^2 \geq 4 c_4$. Putting all these conditions together, we find that $a = -1/\alpha$ could only be the negative root of some quadratic equation $x^2 - c_1 x + c_2 = 0$, where \begin{align*}
(c_1, c_2) \in \{ & (-2, -7), (-2, -6), (-2, -5), (-2, -4), (-2, -2), (-2, -1), (-1, -13), \\
& (-1, -11), (-1, -10), (-1, -9), (-1, -8), (-1, -7), (-1, -5), (-1, -4), \\
& (-1, -3), (-1, -1), (0, -15), (0, -14), (0, -13), (0, -12), (0, -11), \\
& (0, -10), (0, -8), (0, -7), (0, -6), (0, -5), (0, -3), (0, -2), (1, -13) \\
& (1, -11), (1, -10), (1, -9), (1, -8), (1, -7), (1, -5), (1, -4), (1, -3), \\
& (1, -1), (2, -7), (2, -6), (2, -5), (2, -4), (2, -2), (2, -1) \}. \end{align*} We look into each of the possible angles to conclude that $\alpha = \frac{2\sqrt{2}-1}{7}$ is the only irrational angle with which there are $14$ equiangular lines of rank $8$.
\vskip 0.1in
We also need the inequality~(\ref{eq:relative-bound}), which is the so-called \emph{relative bound} for equiangular lines. \begin{thm}[\cite{vanlint1966}, p.342] \label{thm:relative-bound} Let $X$ be an equiangular set with angle $\alpha$ in $\mathbb R^r$. If $r < \frac{1}{\alpha^2}$, then \begin{equation} \label{eq:relative-bound}
|X| \leq \frac{r (1 - \alpha^2)}{1 - r \alpha^2}. \end{equation} \end{thm}
Together with Theorems~\ref{thm:general Neu} and~\ref{thm:relative-bound}, we realize that for each positive integer $r$ there are only a couple of angles to be checked to determine $M^*(r)$.
\begin{thm} \label{thm:m-star-8-9-10} We have \begin{equation*} M^*(8) = 14, \qquad M^*(9) = 18, \qquad \text{and} \qquad M^*(10) = 18. \end{equation*} \end{thm}
\begin{proof} For the case $r=8$, we only need to check the $\alpha$-values for $\{\frac{1}{3}, \frac{1}{5}, \frac{1}{7} \}$ by Theorems~\ref{thm:general Neu} and \ref{thm:relative-bound}. Since $M_{\frac13}(8) = 14$, $M_{\frac15}(8) = 10$, and $M_{\frac17}(8) = 9$, as listed in Table~\ref{tb:smallMr},
we conclude that $M^*(8) = 14$.
For $n = 9$, we read from Table~\ref{tb:smallMr} that $M_{\frac{1}{3}}(9) = 16$ and $M_{\frac{1}{5}}(9) = 12$. By Theorem~\ref{thm:relative-bound} we obtain that $M_{\frac{1}{2n+1}}(9) \leq 10$ for positive integers $n \geq 3$. Finally we find that $M_{\frac{1}{\sqrt{17}}}(9) = 18$, which can be constructed by the Paley graph with cardinality 17 (see \cite{waldron2009construction}). Hence $M^*(9) = 18$.
Table~\ref{tb:smallMr} shows that $M_{\frac{1}{3}}(10) = 18$, and $M_{\frac{1}{2n+1}}(10) \leq 16$ for every positive integer $n \geq 2$ by Theorem~\ref{thm:relative-bound}. According to Theorem~\ref{thm:general Neu}, we do not need to check any other angles. Hence $M^*(10) = 18$. \end{proof}
Notice that Theorem~\ref{thm:general Neu} is universal for every dimension $r$. We may solve for more exact values of $M^*(r)$ if we spend more time on computer calculation. However the work will be repetitious so we stop here.
\section{Closing remarks} \label{sec:closing} We note that the results of Theorem~\ref{thm:2vec} and Lemma~\ref{lem:two-X31} are not optimal in the sense that the upper bound on the cardinality of a pillar can be lowered if more vectors are presents in another pillar. For instance, with the angle $\frac{1}{5}$ and base size $4$, it should not be possible to have four $(4,1)$ pillars with $24$ vectors each (this produces the number $96$ in Proposition~\ref{prop:a5-41}). Nevertheless our bounds are sufficient to beat Lemmens-Seidel's conjecture, so we did not pursue further. On the other hand, these bounds are valid regardless of the dimensions or ranks where the equiangular set lives.
Based on our experiments, we believe that there can only be a large pillar; by this we form the following conjecture.
\begin{conj} There is a constant $C$ that depends on the angle $\alpha$ and the base size $K$, but not to the dimension or rank, of any equiangular set, such that there could not be two pillars of size at least $C$. \end{conj}
This conjecture is coherent to Sudakov's result that when the angle is fixed except for $\frac{1}{3}$, the upper bound for equiangular sets in $\mathbb R^r$ is at most $1.92 r$ asymptotically (see~\cite{balla2018equiangular}). Sudakov had a construction of equiangular sets with angle $\alpha = \frac{1}{2n+1}$ and rank $r$ which concentrates in one pillar whose cardinality is asymptotic to $\frac{(n+1) r}{n}$ for every positive integer $n$~(see \cite{balla2018equiangular}, Conjecture~6.1).
The only unsolved case towards the Lemmens-Seidel conjecture is the $(4,2)$ pillars. King and Tang~\cite{king2016} showed that the unit vectors within one $(4,2)$ pillar form a $2$-distance set of angles $\frac{1}{13}$ and $-\frac{5}{13}$. But the semidefinite linear programming bound $s(r, \frac{1}{13}, -\frac{5}{13})$ cannot be small. Consider the $3\ell \times 3\ell$ matrix of the following block form: \begin{equation*} \begin{bmatrix} B & \frac{1}{13} J_3 & \cdots & \frac{1}{13} J_3 \\ \frac{1}{13} J_3 & B & \cdots & \frac{1}{13} J_3 \\ \vdots & \vdots & \ddots & \vdots \\ \frac{1}{13} J_3 & \frac{1}{13} J_3 & \cdots & B \end{bmatrix}, \qquad \text{where\ } B = \begin{bmatrix} 1 & -\frac{5}{13} & -\frac{5}{13} \\ -\frac{5}{13} & 1 & - \frac{5}{13} \\ -\frac{5}{13} & -\frac{5}{13} & 1 \end{bmatrix}_{3 \times 3} \end{equation*} This matrix has rank $2\ell+1$ and positive semidefinite, so it is the Gram matrix of $3\ell$ vectors of rank $2\ell+1$. On the other hand, the base size of the equiangular set generated from this matrix is $6$, for in such a pillar there are many independent $3$-cliques, and two independent $3$-cliques are switching equivalent to a $6$-clique (by switching all three vertices in one of the $3$-cliques). So we raise another conjecture which relates to Theorem~5.1 of~\cite{lemmens1973}. \begin{conj} In the case where $\alpha = \frac{1}{5}$ and base size $K=4$, there are only a finite number of families of connected graphs $S_i$'s such that the connected components of the Seidel graph of any $(4,2)$ pillar in an equiangular set is either a graph or a subgraph of a graph in $S_i$. \end{conj}
\end{document} |
\begin{document}
\title*{Benchmarking ordering techniques for nonserial dynamic programming\thanks{This research is partly supported by FWF (Austrian Science Funds) under the project P20900-N13.} }
\author{Alexander Sviridenko and Oleg Shcherbina}
\institute{A. Sviridenko \at
Faculty of Mathematics and Computer Science \\
Tavrian National University, Vernadsky Av. 4, Simferopol 95007 Ukraine\\
\email{oleks.sviridenko@gmail.com} \and
O. Shcherbina \at
Faculty of Mathematics, University of Vienna\\
Nordbergstrasse 15, A-1090 Vienna, Austria\\
\email{oleg.shcherbina@univie.ac.at} }
\maketitle
\abstract{Five ordering algorithms for the nonserial dynamic programming algorithm for solving sparse discrete optimization problems are compared in this paper. The benchmarking reveals that the ordering of the variables has a significant impact on the run-time of these algorithms. In addition, it is shown that different orderings are most effective for different classes of problems. Finally, it is shown that, amongst the algorithms considered here, heuristics based on maximum cardinality search and minimum fill-in perform best for solving the discrete optimization problems considered in this paper.} \section{Introduction} \label{intro} The use of discrete optimization (DO) models and algorithms makes it possible to solve many real-life problems in scheduling theory, optimization on networks, routing in communication networks, facility location in enterprize resource planing, and logistics. Applications of DO in the artificial intelligence field include theorem proving, SAT in propositional logic, robotics problems, inference calculation in Bayesian networks, scheduling, and others.\\ Many real-life discrete optimization problems (DOPs) contain a huge number of variables and/or constraints that make the models intractable for currently available DO solvers. Usually, such problems have a special structure, and the matrices of constraints for large-scale problems are sparse. The nonzero elements of the matrices often involve a limited number of blocks. The block form of many DO problems is usually caused by the weak connectedness of subsystems of real-world systems.\\ One of the promising ways to exploit sparsity for solving sparse DOPs is nonserial dynamic programming (NSDP) \cite{BerBri}, \cite{Soa07}. NSDP eliminates variables of DOP using an elimination order which makes significant impact on running time. As finding an optimal ordering is NP-complete \cite{Yanna81}, heuristics are utilized in practice for finding elimination orderings. The literature has reported extensive computational results for the use of different ordering heuristics in the solution of systems of equations \cite{Amestoy96}, \cite{George73}. However, no such experiments have been reported for NSDP to date.\\ Given the increased recent interest in DOPs, the subject of experimental research of NSDP algorithms that utilize heuristic variable orderings is timely. In this paper, we present comparative computational results from the benchmarking of five ordering techniques, namely: minimum degree ordering, nested dissection ordering, maximum cardinality search, minimum fill-in, and lexicographic breadth-first search.
\section{Nonserial Dynamic Programming Algorithm} \label{sec:1} Consider a DOP with constraints: \begin{equation}\label{gen1}
F(x_{1}, x_{2}, \ldots, x_{n})=\sum_{k \in K} f_{k}(Y^{k}) \rightarrow \max \end{equation} subject to the constraints\\ \begin{equation}\label{gen2}
g_{i}(X_{S_i})~ R_{i}~ 0, ~~i \in M = \lbrace 1, 2, \ldots, m \rbrace,
\end{equation}
\begin{equation}\label{gen3} x_{j} \in D_{j}, ~~j \in N= \lbrace 1,\ldots, n \rbrace, \end{equation} where\\ $X = \left\{ x_1,\dots,x_n \right\}$ is a set of discrete variables, $Y^{k} \subseteq \lbrace x_{1}, x_{2}, \ldots, x_{n}\rbrace, k \in K=\left\{1,2,\ldots,t \right\},$ $t$ -- number of components of objective function, $S_{i}\subseteq \{ 1,2, \ldots, n\},~ R_{i} \in \lbrace \leq, = ,\geq \rbrace, i \in M$; $D_j$ is a finite set of admissible values of variable $x_j,~~j \in N$.
The functions $f_k(X^k),~~k \in K$ are called components of the objective function and can be defined in tabular form. We use here a notation: if $S=\{j_1,\ldots,j_q\}$ then $X_S=\{x_{j_1},\ldots,x_{j_q}\}$.\\ \subsection{The structure of sparse discrete optimization problems} \label{sec:11} Let us take a detailed look at an NSDP implementation for solving DO problems for the case when the structural graph is an interaction graph of variables.
\begin{definition} \cite{BerBri}. Variables $x \in X$ and $y \in X$ interact in DOP with constraints (we denote $x \sim y$) if they appear both either in the same component of objective function, or in the same constraint (in other words, if variables are both either in a set $X^k$, or in a set $X_{S_i}$). \end{definition} \begin{definition} \cite{BerBri}. Interaction graph of the DOP is called an undirected graph $G=(X, E)$, such that \begin{enumerate} \item Vertices $X$ of $G$ correspond to variables of the DOP; \item Two vertices of $G$ are adjacent if corresponding variables interact. \end{enumerate} \end{definition} Further, we will use the notion of vertices that correspond one-to-one to variables.
\begin{definition} Set of variables interacting with a variable $x \in X$, is denoted as $Nb(x)$ and called a neighborhood of the variable $x$. For corresponding vertices of $G$ a neighborhood of a vertex $x$ is a set of vertices of interaction graph that are linked by edges with $x$. Denote the latter neighborhood as $Nb_G(x)$. \end{definition} In hypergraph representation of DO problems structure, the set of vertices $H$ of hypergraph, equals to the set of variables $X$ from the DO problems, and hypergraph's hyperedges forms subsets of related variables that are included in constraints, which means the hyperedge defines constraint scope.
\subsection{NSDP (Variable elimination) algorithms} \label{sec:12} Consider a sparse discrete optimization problem (\ref{gen1}) -- (\ref{gen3}) whose structure is described by an undirected interaction graph $G=(X, E)$. Solve this problem with a NSDP.\\
Given ordering of variable indices $\alpha$ the NSDP proceeds in the following way: it subsequently eliminates $x_{\alpha_1},\dots, x_{\alpha_n}$ in the current graph and computes an associated local information $h_i(Nb(x_{\alpha_i}))$ about vertices from $Nb(x_{\alpha_i})$ ($i=1,\dots,n$). This can be described by the combinatorial elimination process: \[ G^0=G,G^1, \ldots, G^{j-1}, G^{j}, \ldots, G^{n}, \] where $G^{j}$ is the $x_{\alpha_j}$-elimination graph of $G^{j-1}$ and $G^n = \emptyset$\\ The process on interaction graph transformation corresponding to the NSDP scheme is known as elimination game which was first introduced by Parter \cite{Part} as a graph analogy of Gaussian elimination. The input of the elimination game is a graph $G$ and an ordering $\alpha$ of $G$. \\
Consider the DOP described above and suppose without loss of generality that variables are eliminated in the order $x_1,\dots, x_n$ . Using the variable elimination scheme eliminate a first variable $x_1$. This $x_1$ is in a set of constraints with the indices of $U_1=\{i~|~x_1 \in X_{S_i}\}$. Together with $x_1$, in constraints $U_1$ are variables from $Nb(x_1)$. To the variable $x_1$ corresponds the following subproblem $P_1$: \[
h_{1}(Nb(x_{1}))=\max_{x_1} \Bigl\{\sum_{k \in K_1} f_{k}(Y^{k})~|~g_{i}(X_{S_i})~ R_{i}~ 0, ~~i \in U_{1},~ x_{j} \in D_j,~ x_{j} \in Nb[x_{1}] \Bigr\}. \] Then the initial DOP can be transformed in the following way: \[
\max_{x_1,\ldots, x_n} \Bigl\{\sum_{k \in K} f_{k}(Y^{k})~|~g_{i}(X_{S_i})~ R_{i}~ 0, ~~i \in M,~ x_{j} \in D_j,~ j \in N \Bigr\}= \] \[ \max_{x_2,\ldots, x_{n}} \Bigl\{ \sum_{k \in K-K_1} f_{k}(Y^{k}) +
h_{1}(Nb(x_{1})~|~ g_{i}(X_{S_i})~ R_{i}~ 0, ~~i \in M-U_{1},~ x_{j} \in D_j,~ j \in N - \{1\}\Bigr\} \] The last problem has $n-1$ variables; from the initial DOP were excluded constraints with the indices from $U_{1}$ and objective function term $\sum_{k \in K_1} f_{k}(Y^{k})$; there appeared a new objective function term $h_{1}(Nb(x_{1}))$. Due to this fact the interaction graph associated with the new problem is changed: a vertex $x_{1}$ is eliminated and its neighbors have become connected.\\ Denote the new interaction graph $G^1$ and find all neighborhoods of variables in $G^1$. NSDP eliminated the remaining variables one by one in an analogous manner.
\section{Elimination Ordering Techniques} \label{sec:2} An efficiency of the NSDP algorithm crucially depends on the interaction graph structure of a DOP. If the interaction graph is rather sparse or, in other words, has a relatively small induced width, then the complexity of the algorithm is reasonable. At the same time an interaction graph leads us to another critical factor such as an elimination order which should be obtained from the interaction graph.\\ From the other side the NSDP algorithm heavily depends on the elimination ordering. A good elimination ordering yields small cliques during variable elimination. There are several successful schemes for finding a good ordering which we will used in this paper: \textit{minimum degree ordering algorithm} (MD), \textit{nested dissection ordering algorithm} (ND), \textit{maximum cardinality search algorithm} (MCS), \textit{minimum fill-in heuristic} (MIN-FILL) and \textit{lexicographic breath-first search algorithm} (LEX-BFS). \subsection{ Minimum degree ordering algorithm} \label{sec:21} The minimum degree (MD) ordering algorithm \cite{Amestoy96} is one of the most widely used in linear algebra heuristic, since it produces factors with relatively low fill-in on a wide range of matrices.\\ In the minimum degree heuristic, a vertex $v$ of minimum degree is chosen. The graph $G'$, obtained by making the neighborhood of $v$ a clique and then removing $v$ and its incident edges, is built. Recursively, a chordal supergraph $H'$ of $G'$ is made with the heuristic. Then a chordal supergraph $H$ of $G$ is obtained, by adding $v$ and its incident edges from $G$ to $H'$. To create an elimination order with help of minimum degree ordering algorithm the \textbf{$minimum\_degree\_ordering()$} function from BOOST library \cite{BOOST} has been used. \subsection{ Nested dissection algorithm} \label{sec:22} To create an elimination order, we recursively partition the elimination graph using nested dissection. More specifically, we use \textbf{$METIS\_EdgeND()$} function from METIS library \cite{Karypis} to find a nested dissection ordering. \subsection{Maximum cardinality search algorithm} \label{sec:23} The Maximum Cardinality Search (MCS) algorithm \cite{Tarjan84} visits the vertices of a graph in an order such that at any point, a vertex is visited that has the largest number of visited neighbors. An MCS-ordering of a graph is an ordering of the vertices that can be generated by the Maximum Cardinality Search algorithm. The visited degree of a vertex $v$ in an MCS-ordering is the number of neighbors of $v$ that are before $v$ in the ordering.\\ To create an elimination ordering the \textbf{$chompack.maxchardsearch()$} function from the \textbf{Chordal Matrix Package} (CHOMPACK) \cite{CHOMPACK} has been used. \subsection{Minimum Fill-in algorithm} \label{sec:24} The minimum fill-in heuristic \cite{Jegou} works similarly with minimum degree heuristic, but now the vertex $v$ is selected such that the number of edges that is added to make a neighborhood of $v$ a clique is as small as possible. \subsection{ Lexicographic breadth-first search algorithm} \label{sec:25} Lexicographic breadth-first search algorithm (LEX-BFS) \cite{Rose76} numbers the vertices from $n$ to 1 in the order that they are selected. This numbering fixes the positions of an elimination scheme. For each vertex $v$, the label of $v$ will consist of a set of numbers listed in decreasing order. The vertices can then be lexicographically ordered according to their labels. \section{Benchmarking} \label{sec:3} \subsection{ NSDP algorithm implementation} \label{sec:31} The NSDP algorithm was implemented by the first author in Python. The ND and MD algorithms were implemented in C and C++, respectively. To work with graph objects was taken class \textbf{networkx.Graph} from the \textbf{networkx} library \cite{NetworkX}. \subsection{ Test problems} \label{sec:32} For benchmarking the DO test problems were generated by using hypergraphs from the CSP\footnote{CSP -- Constraint Satisfaction Problem.} hypergraph library \cite{Musliu}. This collection contains various classes of constraint hypergraphs from industry (DaimlerChrysler, NASA, ISCAS) as well as synthetically generated ones (e.g. Grid or Cliques).\\ The test problems were generated in the following way. The constraints structure of a linear DO problem with binary variables was described by hypergraph from the library \cite{Musliu}. To build constraint $i$ the next hyperedge of hypergraph was taken, which includes a set of variables $X_{S_i}$ for a new building constraint. In the next step, the coefficients for appropriate variables of $A_{S_i}$ were generated using a random number generator. Then the left part of $i$-th constraint had view $A_{S_i}X_{S_i}$, while the right part was $\sigma \sum A_{S_i}$, where $\sigma$ is random number from interval (0, 1). Objective function is linear and includes all variables -- vertices of hypergraph, where coefficients $c_j$ of objective function $\sum_{j=1}^n c_j x_j \rightarrow \max$ where created with help of random number generator.\\ After the test problems were generated, the ordering algorithms MD, ND, MCS, MIN-FILL and LEX-BFS were applied for obtaining an elimination ordering. Then the problems were solved with the NSDP algorithm by utilizing to the specified elimination ordering. \subsection{ Benchmarking ordering analysis} \label{sec:33} The following five groups of 33 test problems have been taken: 'dubois', 'bridge', 'adder', 'pret' and 'NewSystem'. All experimental results were obtained on a machine with Intel Core 2 Duo processor \@ 2.66 GHz, 2 GB main memory and operating system Linux, version 2.6.35-24-generic. The results can be found in table 1, in which $n$ denotes the number of variables, $m$ the number of constraints and the minimal time of problem solving for appropriate heuristics was underlined. We can see that for ND algorithm the minimal run-time of the NSDP algorithm was achieved 0 times (0 \%), for MD 2 times (6,0 \%), LEX-BFS 3 times (9,1 \%), MCS 9 times (27,3 \%) and MIN-FILL 19 times (57,6 \%).\\ \begin{table}[h!]
\caption{Run-time (in seconds) of solving DO problems with help of NSDP algorithm using different orderings.} \label{tab:1}
\begin{tabular}{lrrrrrrr} \hline\noalign{
} Test & $n$ & $m$ & MD & ND & MCS & MIN-FILL & LEX-BFS\\ \hline\noalign{
} dubois20&60&40&1,31&1,43&\underline{1,17}&1,19&1,20\\ dubois21&63&42&1,52&1,67&1,35&\underline{1,31}&1,35\\ dubois22&66&44&\underline{1,37}&1,70&1,51&1,51&1,49\\ dubois23&69&46&1,90&2,02&1,70&\underline{1,68}&1,74\\ dubois24&72&48&2,18&2,17&1,89&\underline{1,80}&1,95\\ dubois25&75&50&2,72&2,50&\underline{2,11}&2,14&2,15\\ dubois26&78&52&2,62&2,82&\underline{2,32}&2,43&2,39\\ dubois27&81&54&\underline{2,58}&3,09&2,60&2,71&2,66\\ dubois28&84&56&3,55&3,43&2,90&\underline{2,84}&2,98\\ dubois29&87&58&3,91&3,84&3,22&\underline{3,21}&3,28\\ dubois30&90&60&4,46&4,14&3,50&3,52&\underline{3,48}\\ dubois50&150&100&17,52&16,31&14,34&\underline{14,00}&14,53\\ dubois100&300&200&126,32&111,64&106,34&\underline{103,23}&106,29\\ adder\_15&106&75&6,20&7,04&\underline{5,25}&5,64&5,38\\ adder\_25&176&125&27,33&32,13&\underline{21,47}&24,27&23,10\\ adder\_50&351&250&326,54&388,74&268,39&276,47&\underline{254,25}\\ adder\_75&526&375&4876,74&5180,46&\underline{3381,28}&3460,32&3435,76\\ bridge\_15&137&135&15,21&15,41&13,90&\underline{12,45}&12,86\\ bridge\_25&227&225&63,14&74,18&62,30&\underline{55,98}&58,62\\ bridge\_50&452&450&900,20&983,40&922,68&\underline{886,91}&1003,89\\ bridge\_75&677&675&4832,55&5049,61&4507,38&\underline{3886,34}&5040,91\\ pret60\_25&60&53&1,89&1,51&1,68&\underline{1,32}&1,59\\ pret60\_40&60&53&1,50&1,48&1,69&\underline{1,26}&1,52\\ pret60\_60&60&53&1,58&1,51&1,83&\underline{1,27}&1,64\\ pret60\_75&60&53&1,44&1,60&1,79&\underline{1,31}&1,54\\ pret150\_25&150&133&21,29&33,06&20,51&\underline{16,25}&23,04\\ pret150\_40&150&133&22,63&32,83&21,47&\underline{16,44}&24,76\\ pret150\_60&150&133&22,09&31,13&23,72&\underline{16,35}&23,99\\ pret150\_75&150&133&21,20&33,23&20,19&\underline{18,57}&23,59\\ NewSystem1&142&85&19,21&16,24&\underline{14,05}&17,33&14,08\\ NewSystem2&345&200&603,83&520,82&\underline{376,33}&489,80&425,35\\ NewSystem3&474&284&1294,86&1352,75&1159,75&1247,81&\underline{1072,58}\\ NewSystem4&718&422&7769,10&8322,58&\underline{6427,85}&7095,17&6845,20\\ &&&&&&&\\ \hline\noalign{
} \end{tabular} \end{table} Let us take a look at the benchmarking results in more detail. Fig. \ref{fig:1} describes results of the experiment for the groups of test problems 'dubois', 'bridge', and 'adder'. These results show to us that MCS, MIN-FILL and LEX-BFS heuristics behave quite similar and give the best result for a given group of problems. At the same time MD and ND show the worst time result. Also for the group 'bridge' we can see the gradual decreasing of the time result for the LEX-BFS heuristics and the obvious domination of MIN-FILL.\\ \begin{figure*}
\caption{Run-time (in seconds) for groups of tests 'dubois', 'adder', 'bridge'.}
\label{fig:1}
\end{figure*} Fig. \ref{fig:2} describes experimental results for the groups of test problems 'pret' and 'NewSystem'. \begin{figure*}
\caption{Run-time (in seconds) for groups of tests 'pret' and 'NewSystem'.}
\label{fig:2}
\end{figure*} Here we can see the importance of the right choice of heuristics for a certain group of problems. In the case of 'pret', we see obvious domination of MIN-FILL algorithm, while MCS and LEX-BFS fall behind. However, the group 'NewSystem' shows completely opposite results, where MIN-FILL runs third while MCS and LEX-BFS take the first two places.\\ In the case of 'pret', we see the obvious domination of the MIN-FILL algorithm, while MCS and LEX-BFS go back. But the group 'NewSystem' shows the completely opposite result, where MIN-FILL goes on the third place while MCS and LEX-BFS take the first places. \section{Conclusion} The goal of this paper to research the role of five variable ordering algorithms and to describe the effect that they play on solving time of sparse DO problems with help of NSDP algorithm. Our computational experiments demonstrate that, for solving DO problems, variable ordering has a significant impact on the run-time for solving the problem. Furthermore, different ordering heuristics were observed to be more effective for different classes of problems. Overall, the MCS and MIN-FILL heuristics have provided the best results for solving DO problems of the problem classes that were considered in this paper. It seems promising to continue this line of research by studying methods of block elimination with suitable partitioning methods.
\end{document} |
\begin{document}
\author{Bego\~na Barrios} \address{ Departamento de Matem\'aticas\\ Universidad Aut\'onoma de Madrid\\ Ciudad Universitaria de Cantoblanco\\ and Instituto de Ciencias Matem\'{a}ticas, (ICMAT, CSIC-UAM-UC3M-UCM)\\ C/Nicol\'{a}s Cabrera 15, 28049-Madrid (Spain) } \email{bego.barrios@uam.es}
\author{Alessio Figalli} \address{ The University of Texas at Austin\\ Mathematics Dept. RLM 8.100\\ 2515 Speedway Stop C1200\\ Austin, TX 78712-1202 (USA) } \email{figalli@math.utexas.edu}
\author{Enrico Valdinoci} \address{ Dipartimento di Matematica\\ Universit\`a degli Studi di Milano\\ Via Cesare Saldini 50\\ 20133 Milano (Italy) } \email{enrico@math.utexas.edu}
\title[Bootstrap regularity and nonlocal minimal surfaces]{Bootstrap regularity\\ for integro-differential operators\\ and its application\\ to nonlocal minimal surfaces} \date{\today}
\begin{abstract} We prove that $C^{1,\alpha}$ $s$-minimal surfaces are of class $C^\infty$. For this, we develop a new bootstrap regularity theory for solutions of integro-differential equations of very general type, which we believe is of independent interest. \end{abstract}
\maketitle
\tableofcontents
\section{Introduction}
Motivated by the structure of interphases arising in phase transition models with long range interactions, in~\cite{CRS} the authors introduced a nonlocal version of minimal surfaces. These objects are obtained by minimizing a ``nonlocal perimeter'' inside a fixed domain $\Omega$: fix $s \in (0,1)$, and given two sets $A,B\subset \mathbb R^n$, let us define the interaction term
$$ L(A,B):=\int_A \int_B \frac{dx\,dy}{|x-y|^{n+s}}.$$ The nonlocal perimeter of $E$ inside $\Omega$ is defined as \begin{multline*} {\rm Per}(E,\Omega,s):=L\big( E\cap\Omega,(\R^n\setminus E)\cap\Omega\big)\\ +L\big( E\cap\Omega,(\R^n\setminus E)\cap(\R^n\setminus \Omega)\big)+L\big( (\R^n\setminus E)\cap\Omega,E\cap(\R^n\setminus \Omega)\big). \end{multline*} Then nonlocal ($s$-)minimal surfaces correspond to minimizers of the above functional with the ``boundary condition'' that $E\cap ({\R^n\setminus }\Omega)$ is prescribed.
It is proved in~\cite{CRS} that ``flat $s$-minimal surface'' are $C^{1,\alpha}$ for all $\alpha<s$, and in~\cite{CV1, ADM, CV2} that, as $s\rightarrow 1^-$, the $s$-minimal surfaces approach the classical ones, both in a geometric sense and in a $\Gamma$-convergence framework, with uniform estimates as~$s\rightarrow1^-$. In particular, when $s$ is sufficiently close to~$1$, they inherit some nice regularity properties from the classical minimal surfaces (see also~\cite{Sou, SV1, SV2} for the relation between $s$-minimal surfaces and the interfaces of some phase transition equations driven by the fractional Laplacian).
On the other hand, all the previous literature only focused on the~$C^{1,\alpha}$ regularity, and higher regularity was left as an open problem. In this paper we address this issue, and we prove that $C^{1,\alpha}$ $s$-minimal surfaces are indeed~$C^\infty$, according to the following result\footnote{ Here and in the sequel, we write~$x\in\mathbb R^n$ as~$x=(x',x_n)\in\mathbb R^{n-1}\times\mathbb R$. Moreover, given~$r>0$ and~$p\in\mathbb R^n$, we define
$$K_r(p):= \{ x\in\mathbb R^n \,:\, |x'-p'|<r
{\mbox{ and }} |x_n-p_n|<r\}.$$ As usual, $B_r(p)$ denotes the Euclidean ball of radius~$r$ centered at~$p$. Given~$p'\in\mathbb R^{n-1}$, we set
$$B^{n-1}_r(p'):=\{ x'\in\mathbb R^{n-1} \,:\, |x'-p'|<r\}.$$ We also use the notation~$K_r:=K_r(0)$, $B_r:=B_r(0)$, $B^{n-1}_r:= B^{n-1}_r(0)$.}:
\begin{theorem}\label{main} Let~$s\in(0,1)$, and~$\partial E$ be a~$s$-minimal surface in~$K_R$ for some~$R>0$. Assume that \begin{equation}\label{XC2} \partial E\cap K_R =\left\{ (x',x_n) \,:\, x' \in B_R^{n-1}{\mbox{ and }} x_n =u(x')\right\} \end{equation} for some
$u:B^{n-1}_R\to \mathbb R$, with $u\in {C^{1,\alpha}} (B^{n-1}_R)$ for any $\alpha<s$ and~$u(0)=0$.
Then $$u\in C^{\infty}(B_{\rho}^{n-1})\quad\forall\,\rho\in (0,R).$$ \end{theorem}
The regularity result of Theorem~\ref{main} combined with \cite[Theorem~6.1]{CRS} and \cite[Theorems 1, 3, 4, 5]{CV2}, implies also the following results (here and in the sequel, $\{ e_1, e_2,\dots, e_n\}$ denotes the standard Euclidean basis):
\begin{corollary} Fix~$s_o\in(0,1)$. Let~$s\in(s_o,1)$ and~$\partial E$ be a~$s$-minimal surface in~$B_R$ for some $R>0$. There exists~$\epsilon_\star>0$, possibly depending on~$n$, $s_o$ and~$\alpha$, but independent of~$s$ and $R$, such that if
$$ \partial E\cap B_R\subseteq \{ |x\cdot e_n|\leqslant \epsilon_\star R\} $$then~$\partial E\cap B_{R/2}$ is a~$C^{\infty}$-graph in the~$e_n$-direction.\end{corollary}
\begin{corollary} There exists~$\epsilon_o\in(0,1)$ such that if~$s\in(1-\epsilon_o,1)$, then: \begin{itemize} \item If~$n\leqslant 7$, any $s$-minimal surface is of class $C^{\infty}$; \item If~$n=8$, any $s$-minimal surface is of class $C^{\infty}$ except, at most, at countably many isolated points. \end{itemize} More generally, in any dimension $n$ there exists~$\epsilon_n\in(0,1)$ such that if~$s\in(1-\epsilon_n,1)$ then any $s$-minimal surface is of class $C^{\infty}$ outside a closed set~$\Sigma$ of Hausdorff dimension $n-8$. \end{corollary}
Also, Theorem~\ref{main} here combined with Corollary~1 in~\cite{SVc} gives the following regularity result in the plane:
\begin{corollary} Let~$n=2$. Then, for any~$s\in(0,1)$, any $s$-minimal surface is a smooth embedded curve of class~$C^{\infty}$. \end{corollary}
In order to prove Theorem~\ref{main} we establish in fact a very general result about the regularity of integro-differential equations, which we believe is of independent interest.
For this, we consider a kernel $K=K(x,w):\mathbb R^n \times(\mathbb R^n\setminus\{0\})\rightarrow(0,+\infty)$ satisfying some general structural assumptions. In the following, $\sigma \in (1,2)$.
First of all, we suppose that $K$ is close to an autonomous kernel of fractional Laplacian type, namely \begin{equation}\label{ass silv} \left\{ \begin{aligned} &{\mbox{there exist~$a_0,r_0>0$ and $\eta \in (0,a_0/4)$ such that}}\\
&\left| \frac{|w|^{n+\sigma}
K(x,w)}{2-\sigma}-a_0\right|\leqslant\eta \qquad \forall \,x\in B_{1},\,w\in B_{r_0}\setminus\{0\}. \end{aligned} \right. \end{equation}
Moreover, we assume that\footnote{Observe that we use~$|\cdot|$
both to denote the Euclidean norm of a vector and, for a multi-index case $\alpha=(\alpha_1,\dots,\alpha_n)\in\mathbb N^n$, to denote $|\alpha|:=\alpha_1+ \dots+\alpha_n$. However, the meaning of $|\cdot|$ will always be clear from the context.} \begin{equation}\label{sm 1} \left\{ \begin{aligned} &{\mbox{there exist $k \in \mathbb N\cup \{0\}$ and $C_k>0$ such that}}\\ &K\in C^{k+1}\big(B_{1}\times (\mathbb R^n\setminus\{0\})\big),\\
&\|\partial^\mu_x \partial^\theta_w K(\cdot, w)\|_{L^\infty(B_{1})} \leqslant \frac{C_k}{|w|^{n+\sigma+|\theta|}}\\ &\qquad
\qquad \forall\,\mu, \theta\in \mathbb N^n,\,|\mu|+|\theta|\leqslant k+1,\,w\in \mathbb R^n\setminus\{0\}.\\ \end{aligned} \right. \end{equation}
Our main result is a ``Schauder regularity theory'' for solutions\footnote{We adopt the notion of viscosity solution used in~\cite{CScpam, CS2011, CS2012}.} of an integro-differential equation. Here and in the sequel we use the notation \begin{equation}\label{delta} \delta u(x,w):= u(x+w)+u(x-w)-2u(x).\end{equation}
\begin{theorem}\label{boot} Let~$\sigma\in (1,2)$, $k\in\mathbb N\cup\{0\}$, and $u\in L^\infty(\mathbb R^n)$ be a viscosity solution of the equation \begin{equation} \label{eq:main} \int_{\mathbb R^{n}}{K(x,w)\,\delta u(x,w)dw}=f(x,u(x))\qquad \text{inside $B_1$,} \end{equation} with $f\in C^{k+1}(B_1\times\mathbb R)$. Assume that $K:B_{1}\times (\mathbb R^n\setminus\{0\})\rightarrow (0,+\infty)$ satisfies assumptions \eqref{ass silv} and \eqref{sm 1} for the same value of $k$.
Then, if $\eta$ in \eqref{ass silv} is sufficiently small (the smallness being independent of $k$), we have $u\in C^{k+\sigma+\alpha}(B_{1/2})$ for any $\alpha<1,$ and \begin{equation}\label{3bis}
\| u\|_{C^{k+\sigma+\alpha}(B_{1/2})} \leqslant C
\left(1+\|u\|_{L^\infty(\mathbb R^n)}+\|f\|_{L^{\infty}(B_{1}\times\mathbb R)}\right) , \end{equation} where\footnote{As customary, when~$\sigma+\alpha\in(1,2)$ (resp. $\sigma+\alpha>2$), by \eqref{3bis} we mean that~$u\in C^{k+1,\sigma+\alpha-1}(B_{1/2})$ (resp. $u\in C^{k+2,\sigma+\alpha-1}(B_{1/2})$). (To avoid any issue, we will always implicitly assume that $\alpha$ is chosen different from $2-\sigma$, so that $\sigma+\alpha\neq 2$.)} $C>0$ depends only on
$n$, $\sigma$, $k$, $C_k$, and~$\|f\|_{C^{k+1}(B_{1}\times\mathbb R)}$. \end{theorem} Let us notice that, since the right hand side in \eqref{eq:main} depends on $u$, there is no uniqueness for such an equation. In particular it is not enough for us to prove a-priori estimates for smooth solutions and then argue by approximation, since we do not know if our solution can be obtained as a limit of smooth solution.
We also note that, if in \eqref{sm 1} one replaces the $C^{k+1}$-regularity of $K$ with the $C^{k,\beta}$-assumption \begin{equation}\label{new condition}
\|\partial^\mu_x
\partial^\theta_w K(\cdot, w)\|_{C^{0,\beta}(B_{1})} \leqslant
\frac{C_k}{|w|^{n+\sigma+|\theta|}}, \end{equation}
for all $|\mu|+|\theta|\leqslant k$, then we obtain the following:
\begin{theorem}\label{boot2} Let~$\sigma\in (1,2)$, $k\in\mathbb N\cup\{0\}$, and $u\in L^\infty(\mathbb R^n)$ be a viscosity solution of equation \eqref{eq:main} with $f\in C^{k, \beta}(B_1\times\mathbb R)$. Assume that $K:B_{1}\times (\mathbb R^n\setminus\{0\})\rightarrow (0,+\infty)$ satisfies assumptions \eqref{ass silv} and \eqref{new condition} for the same value of $k$.
Then, if $\eta$ in \eqref{ass silv} is sufficiently small (the smallness being independent of $k$), we have $u\in C^{k+\sigma+\alpha}(B_{1/2})$ for any $\alpha<\beta,$ and \begin{equation*}
\| u\|_{C^{k+\sigma+\alpha}(B_{1/2})} \leqslant C
\left(1+\|u\|_{L^\infty(\mathbb R^n)}+\|f\|_{L^{\infty}(B_{1}\times\mathbb R)}\right) , \end{equation*}
where $C>0$ depends only on $n$, $\sigma$, $k$, $C_k$, and~$\|f\|_{C^{k,\beta}(B_{1}\times\mathbb R)}$. \end{theorem}
The proof of Theorem \ref{boot2} is essentially the same as the one of Theorem \ref{boot}, the only difference being that instead of differentiating the equations (see for instance the argument in Section \ref{section:uniforml}) one should use incremental quotients. Although this does not introduce any major additional difficulties, it makes the proofs longer and more tedious. Hence, since the proof of Theorem \ref{boot} already contains all the main ideas to prove also Theorem \ref{boot2}, we will show the details of the proof only for Theorem \ref{boot}.
The paper is organized as follows: in the next section we prove Theorem \ref{boot}, and then in Section \ref{section:main} we write the fractional minimal surface equation in a suitable form so that we can apply Theorems~\ref{boot} and~\ref{boot2} to prove Theorem~\ref{main}.\\
\textit{Acknowledgements:} We wish to thank Guido De Philippis and Francesco Maggi for stimulating our interest in this problem. We also thank Guido De Philippis for a careful reading of a preliminary version of our manuscript, and Nicola Fusco for kindly pointing out to us a computational inaccuracy. BB was partially supported by Spanish Grant MTM2010-18128. AF was partially supported by NSF Grant DMS-0969962. EV was partially supported by ERC Grant 277749 and FIRB Grant A\&B.
\section{Proof of Theorem \ref{boot}}
The core in the proof of Theorem \ref{boot} is the step $k=0$, which will be proved in several steps.
\subsection{Toolbox}
We collect here some preliminary observations on scaled H\"older norms, covering arguments, and differentiation of integrals that will play an important role in the proof of Theorem~\ref{boot}. This material is mainly technical, and the expert reader may go directly to Section~\ref{TTT} at page~\pageref{TTT}.
\subsubsection{Scaled H\"older norms and coverings}
Given~$m\in\mathbb N$, $\alpha\in(0,1)$, $x\in\mathbb R^n$, and~$r>0$, we define the $C^{m,\alpha}$-norm of a function~$u$ in~$B_r(x)$ as
$$ \|u\|_{C^{m,\alpha} (B_r(x))}:=
\sum_{|\gamma|\leqslant m} \|D^\gamma u\|_{L^\infty (B_r(x))}
+\sum_{|\gamma|=m}\sup_{y\ne z\in B_r(x)}\frac{|D^\gamma u(y)-
D^\gamma u(z)|}{|y-z|^\alpha}.$$ For our purposes it is also convenient to look at the following classical rescaled version of the norm: \begin{eqnarray*}
\|u\|^*_{C^{m,\alpha} (B_r(x))}&:=&\sum_{j=0}^m \sum_{|\gamma|=j}
r^{j} \|D^\gamma u\|_{L^\infty (B_r(x))}
\\&&+\sum_{|\gamma|=m} r^{m+\alpha}
\sup_{y\ne z\in B_r(x)}\frac{|D^\gamma u(y)- D^\gamma u(z)|}{|y-z|^\alpha}.\end{eqnarray*} This scaled norm behaves nicely under covering, as the next observation points out:
\begin{lemma}\label{co} Let~$m\in\mathbb N$, $\alpha\in(0,1)$, $\rho>0$, and~$x\in\mathbb R^n$. Fix $\lambda \in (0,1)$, and suppose that~$B_{\rho}(x)$ is covered by finitely many balls~$\{B_{\lambda\rho/2}(x_k)\}_{k=1}^N$. Then, there exists~$C_o>0$, depending only on $\lambda$ and~$m$, such that
$$ \|u\|^*_{C^{m,\alpha} (B_\rho (x))}\leqslant C_o
\sum_{k=1}^N \|u\|^*_{C^{m,\alpha} (B_{\lambda\rho}(x_k))}.$$ \end{lemma}
\begin{proof} We first observe that, if~$j\in \{0,\dots,m\}$ and~$|\gamma|=j$, \begin{eqnarray*}
\rho^{j} \|D^\gamma u\|_{L^\infty (B_\rho(x))} &\leqslant& \lambda^{-j}
(\lambda\rho)^{j} \max_{k=1,\dots,N} \|D^\gamma u\|_{L^\infty (B_{\lambda\rho} (x_k))} \\ &\leqslant& \lambda^{-m} \sum_{k=1}^N
(\lambda\rho)^{j} \|D^\gamma u\|_{L^\infty (B_{\lambda\rho} (x_k))} \\ &\leqslant& \lambda^{-m} \sum_{k=1}^N
\|u\|^*_{C^{m,\alpha} (B_{\lambda\rho} (x_k))}. \end{eqnarray*}
Now, let~$|\gamma|=m$: we claim that \begin{equation*}
\rho^{m+\alpha} \sup_{y\ne z\in B_{\rho}(x)}\frac{|D^\gamma u(y)-
D^\gamma u(z)|}{|y-z|^\alpha} \le2 \lambda^{-(m+\alpha)}\sum_{k=1}^N
\|u\|^*_{C^{m,\alpha} (B_{\lambda\rho}(x_k))}. \end{equation*}
To check this, we take~$y, z\in B_{\rho}(x)$ with $y \neq z$ and we distinguish two cases. If~$|y-z|< \lambda\rho/2$ we choose~$k_o\in\{1,\dots,N\}$ such that~$y\in B_{\lambda\rho/2}(x_{k_o})$. Then~$|z-x_{k_o}|\leqslant
|z-y|+|y-x_{k_o}|<\lambda\rho$, which implies~$y,z\in B_{\lambda\rho}(x_{k_o})$, therefore \begin{eqnarray*}
\rho^{m+\alpha} \frac{|D^\gamma u(y)-D^\gamma u(z)|}{|y-z|^\alpha}
&\leqslant& \rho^{m+\alpha} \sup_{\tilde y\ne \tilde z\in B_{\lambda\rho}(x_{k_o})} \frac{|D^\gamma u(\tilde y)-D^\gamma u(\tilde z)|}{
|\tilde y-\tilde z|^\alpha} \\
&\leqslant&\lambda^{-(m+\alpha)}\| u\|^*_{ C^{m,\alpha} (B_{\lambda\rho}(x_{k_o}))}. \end{eqnarray*}
Conversely, if~$|y-z|\geqslant\lambda\rho/2$, recalling that $\alpha \in (0,1)$ we have \begin{eqnarray*}
\rho^{m+\alpha} \frac{|D^\gamma u(y)- D^\gamma u(z)|}{|y-z|^\alpha} &\leqslant& 2 \lambda^{-\alpha} \rho^{m}
{\|D^\gamma u\|_{L^\infty(B_\rho(x))}}\\
&\leqslant& 2 \lambda^{-\alpha} \rho^{m}\sum_{k=1}^{N}{\|D^{\gamma}u\|_{L^{\infty}(B_{\lambda\rho}(x_{k}))}}\\ &\leqslant&2
\lambda^{-(m+\alpha)}\sum_{k=1}^N\|u\|^{*}_{C^{m,\alpha}(B_{\lambda\rho}(x_{k}))}. \end{eqnarray*} This proves the claim and concludes the proof. \end{proof}
Scaled norms behave also nicely in order to go from local to global bounds, as the next result shows:
\begin{lemma}\label{Co2} Let~$m\in\mathbb N$, $\alpha\in(0,1)$, and~$u\in C^{m,\alpha}(B_1)$. Suppose that there exist $\mu \in (0,1/2)$ and $\nu\in (\mu,1]$ for which the following holds:
for any~$\epsilon>0$ there exists~$\Lambda_\epsilon>0$ such that, for any~$x\in B_1$ and any~$r\in (0,1-|x|]$, we have \begin{equation}\label{X0}
\|u\|^*_{C^{m,\alpha}(B_{\mu r}(x))} \leqslant \Lambda_\epsilon +\epsilon
\|u\|^*_{C^{m,\alpha}(B_{\nu r}(x))}. \end{equation} Then there exist constants~$\epsilon_o$, $C>0$, depending only on~$n$, $m$, $\mu$, $\nu$, and $\alpha$, such that
$$ \|u\|_{C^{m,\alpha}(B_{\mu})}\leqslant C\Lambda_{\epsilon_o}.$$ \end{lemma}
\begin{proof} First of all we observe that \begin{equation*}
\|u\|^*_{C^{m,\alpha}(B_{\mu r}(x))}\leqslant
\|u\|_{C^{m,\alpha}(B_{\mu r}(x))} \leqslant \|u\|^*_{C^{m,\alpha}(B_1)} \end{equation*} because~$r\in(0,1)$, which implies that $$
Q:=\sup_{{x\in B_1}\atop{r\in (0,1-|x|]}}
\|u\|^*_{C^{m,\alpha}(B_{\mu r}(x))}<+\infty. $$ We now use a covering argument: pick $\lambda \in (0,1/2]$ to be chosen later, and fixed any~$x\in B_1$ and~$r\in
(0,1-|x|]$ we cover $B_{\mu r}(x)$ with finitely many balls $\{B_{\lambda\mu r/2}(x_k)\}_{k=1}^N$, with $x_k \in B_{\mu r}(x)$, for some~$N$ depending only on $\lambda$ and the dimension $n$. We now observe that, since $\mu<1/2$, \begin{equation} \label{eq:xk}
|x_k|+ {r}/2\leqslant |x_k-x|+|x|+{r}/2\leqslant \mu{r}+
|x|+{r}/2< r+|x|\leqslant 1. \end{equation} Hence, since $\lambda \leqslant 1/2$, we can use~\eqref{X0} (with~$x=x_k$ and $r$ scaled to $\lambda r$) to obtain \begin{eqnarray*}
\|u\|^*_{C^{m,\alpha}(B_{\lambda \mu r}(x_k))} \leqslant \Lambda_\epsilon +\epsilon
\|u\|^*_{C^{m,\alpha}(B_{\lambda \nu r}(x_k))}. \end{eqnarray*} Then, using Lemma~\ref{co} with~$\rho:=\mu r$ and $\lambda=\mu/(2\nu)$, and recalling \eqref{eq:xk} and the definition of $Q$, we get \begin{eqnarray*}
\|u\|^*_{C^{m,\alpha}(B_{\mu r}(x))} &\leqslant & C_o
\sum_{k=1}^N \|u\|_{C^{m,\alpha}(B_{\lambda \mu r}(x_k))}^*\\ &\leqslant & C_o N \Lambda_\epsilon+C_o \epsilon \sum_{k=1}^N
\|u\|^*_{C^{m,\alpha}(B_{\lambda \nu r}(x_k))}\\ &=&C_o N \Lambda_\epsilon+C_o \epsilon \sum_{k=1}^N
\|u\|^*_{C^{m,\alpha}(B_{\mu r/2}(x_k))}\\ &\leqslant & C_oN\Lambda_\epsilon + \epsilon C_oN Q. \end{eqnarray*} Using the definition of $Q$ again, this implies $$ Q \leqslant C_oN\Lambda_\epsilon + \epsilon C_oN Q, $$ so that, by choosing~$\epsilon_o:=1/(2C_oN)$, $$ Q \leqslant 2C_o N \Lambda_{\epsilon_o}.$$ Thus we have proved that $$
\|u\|_{C^{m,\alpha}(B_{\mu r}(x))}^* \leqslant 2C_o N
\Lambda_{\epsilon_o} \qquad \forall\, x\in B_1, \,r\in (0,1-|x|], $$ and the desired result follows setting~$x=0$ and~$r=1$. \end{proof}
\subsubsection{Differentiating integral functions}
In the proof of Theorem~\ref{boot} we will need to differentiate, under the integral sign, smooth functions that are either supported near the origin or far from it. This purpose will be accomplished in Lemmata~\ref{D} and~\ref{E}, after some technical bounds that are needed to use the Dominated Convergence Theorem.
Recall the notation in~\eqref{delta}.
\begin{lemma}\label{D1}
Let~$r>r'>0$, $v\in C^3(B_r)$, $x\in B_{r'}$, $h\in\mathbb R$ with~$|h|<
(r-r')/2$. Then, for any~$w\in\mathbb R^n$ with~$|w|< (r-r')/2$, we have
$$ |\delta v(x+he_1,w)-\delta v(x,w)|\leqslant |h|\, |w|^2
\|v\|_{C^3(B_r)} .$$ \end{lemma}
\begin{proof} Fixed~$x\in B_{r'}$ and~$|w|< (r-r')/2$, for any~$h\in [(r'-r)/2,(r-r')/2]$ we set~$g(h):=v(x+he_1+w)+v(x+he_1-w)-2v(x+he_1)$. Then
\begin{eqnarray*}&& |g(h)-g(0)|\leqslant |h| \sup_{|\xi|\leqslant |h|} |g'(\xi)|
\\ &&\quad\leqslant |h|\sup_{|\xi|\leqslant |h|} \big|\partial_1 v(x+\xi e_1+w)
+\partial_1 v(x+\xi e_1-w)-2 \partial_1 v(x+\xi e_1)\big|. \end{eqnarray*}
Noticing that~$|x+\xi e_1\pm w|\leqslant r'+|h|+|w|<r$, a second order Taylor expansion of~$\partial_1 v$ with respect to the variable~$w$ gives \begin{equation} \label{ed3}
\big|\partial_1 v(x+\xi e_1+w)
+\partial_1 v(x+\xi e_1-w)-2 \partial_1 v(x+\xi e_1)\big| \leqslant
|w|^2 \|\partial_1 v\|_{C^2(B_r)}. \end{equation} Therefore \begin{eqnarray*}
|\delta v(x+he_1,w)-\delta v(x,w)|= |g(h)-g(0)| \leqslant |h|\, |w|^2
\|v\|_{C^3(B_r)}, \end{eqnarray*} as desired. \end{proof}
\begin{lemma}\label{D1bis} Let~$r>r'>0$, $v\in W^{1,\infty}(\mathbb R^n)$, $h\in\mathbb R$. Then, for any~$w\in\mathbb R^n$,
$$ |\delta v(x+he_1,w)-\delta v(x,w)|\leqslant 4|h| \|
\nabla v\|_{L^\infty(\mathbb R^n)}.$$ \end{lemma}
\begin{proof} It sufficed to proceed as in the proof of Lemma~{\ref{D1}}, but replacing~\eqref{ed3} with the following estimate: \begin{eqnarray*}
\big|\partial_1 v(x+\xi e_1+w) +\partial_1 v(x+\xi e_1-w)-2
\partial_1 v(x+\xi e_1)\big| \leqslant 4 \|\partial_1 v\|_{L^\infty (\mathbb R^n)} .\end{eqnarray*} \end{proof}
\begin{lemma}\label{D} Let~$\ell\in\mathbb N$, $r\in(0,2)$,~$K$ satisfy~\eqref{sm 1}, and~$U\in C^{\ell+2}_0(B_r)$. Let~$\gamma=(\gamma_1,\dots,\gamma_n)\in\mathbb N^n$
with~$|\gamma|\leqslant \ell \leqslant k+1$. Then \begin{equation}\label{002}\begin{split} &\partial^\gamma_x \int_{\mathbb R^n} K(x,w)\,\delta U(x,w)\,dw = \int_{\mathbb R^n} \partial^\gamma_x\Big( K(x,w)\,\delta U(x,w)\Big)\,dw \\ &\quad= \sum_{{{1\leqslant i\leqslant n}\atop{0\leqslant \lambda_i\leqslant \gamma_i}}\atop{\lambda=(\lambda_1,\dots,\lambda_n)}} \left( {\gamma_1}\atop{\lambda_1}\right)\dots \left( {\gamma_n}\atop{\lambda_n}\right) \int_{\mathbb R^n} \partial^{\lambda}_x K(x,w)\,\delta (\partial^{\gamma-\lambda}_x U)(x,w)\,dw \end{split}\end{equation} for any~$x\in B_r$. \end{lemma}
\begin{proof} The latter equality follows from the standard product derivation formula, so we focus on the proof of the first identity. The proof is by induction over~$|\gamma|$. If~$|\gamma|=0$ the result is trivially true, so we consider the inductive step. We take~$x$ with~$r':=|x|<r$, we suppose that~$|\gamma|\leqslant \ell-1$ and, by inductive hypothesis, we know that $$g_\gamma(x):= \partial^\gamma_x \int_{\mathbb R^n} K(x,w)\,\delta U(x,w)\,dw= \int_{\mathbb R^n} \theta(x,w)\,dw$$ with $$ \theta(x,w):= \sum_{{{1\leqslant i\leqslant n}\atop{0\leqslant \lambda_i\leqslant \gamma_i}}\atop{\lambda=(\lambda_1,\dots,\lambda_n)}} \left( {\gamma_1}\atop{\lambda_1}\right)\dots \left( {\gamma_n}\atop{\lambda_n}\right) \partial^{\lambda}_x K(x,w)\,\delta
(\partial^{\gamma-\lambda}_x U)(x,w)\,dw.$$ By~\eqref{sm 1}, if~$0<|h|< (r-r')/2$ then \begin{equation}\label{001}
|\partial^{\lambda}_x K(x+he_1,w)-
\partial^{\lambda}_x K(x,w)|\leqslant {C_{|\lambda|+1}} |h|\,
|w|^{-n-\sigma}. \end{equation}
Moreover, if~$|w|<(r-r')/2$, we can apply Lemma~\ref{D1} with~$v:=\partial^{\gamma-\lambda}_x U$ and obtain \begin{equation}\label{v0}
|\delta (\partial^{\gamma-\lambda}_x U)(x+he_1,w)-
\delta(\partial^{\gamma-\lambda}_x U)(x,w)|\\
\leqslant |h|\, |w|^2\|U\|_{C^{|\gamma-\lambda|+3}(B_r)}. \end{equation} On the other hand, by Lemma~\ref{D1bis} we obtain
$$ |\delta (\partial^{\gamma-\lambda}_x U)(x+he_1,w)-\delta
(\partial^{\gamma-\lambda}_x U)(x,w)|\leqslant \,4\,|h|\, \|
\partial^{\gamma-\lambda}_x U\|_{C^1(\mathbb R^n)}.$$ All in all, \begin{equation}\label{eq:1}\begin{split}
&|\delta (\partial^{\gamma-\lambda}_x U)(x+he_1,w)-\delta
(\partial^{\gamma-\lambda}_x U)(x,w)|
\\ &\qquad\leqslant\,|h|\,
\|U\|_{C^{|\gamma-\lambda|+3}(\mathbb R^n)}\min\{4,|w|^2\}. \end{split}\end{equation} Analogously, a simple Taylor expansion provides also the bound \begin{equation}\label{eq:2}
|\delta (\partial^{\gamma-\lambda}_x U)(x,w)|\leqslant\,
\|U\|_{C^{|\gamma-\lambda|+2}(\mathbb R^n)}\min\{4,|w|^2\}. \end{equation} Hence, \eqref{sm 1}, \eqref{001}, \eqref{eq:1}, and \eqref{eq:2} give \begin{eqnarray*}
&& \big| \partial^{\lambda}_x K(x+he_1,w)\,\delta (\partial^{\gamma-\lambda}_x U)(x+he_1,w) - \partial^{\lambda}_x K(x,w)\,\delta (\partial^{\gamma-\lambda}_x U)(x,w)
\big| \\
&\leqslant& \big| \partial^{\lambda}_x K(x+he_1,w)\,\big[\delta (\partial^{\gamma-\lambda}_x U)(x+he_1,w) -\delta
(\partial^{\gamma-\lambda}_x U)(x,w) \big]\big| \\ &&+
\big|\big[ \partial^{\lambda}_x K(x+he_1,w) - \partial^{\lambda}_x K(x,w)
\big]\delta (\partial^{\gamma-\lambda}_x U)(x,w) \big|
\\ &\leqslant& C_1 |h|\,\min\{|w|^{-n-\sigma},|w|^{2-n-\sigma} \}, \end{eqnarray*} with~$C_1>0$ depending only on~$\ell$, $C_\ell$
and~$\|U\|_{C^{\ell+2}(\mathbb R^n)}$. As a consequence,
$$ |\theta(x+he_1,w)-\theta(x,w)|\leqslant C_2 |h|\,\min \{|w|^{-n-\sigma},|w|^{2-n-\sigma} \},$$ and, by the Dominated Convergence Theorem, we get \begin{eqnarray*} \int_{\mathbb R^n}\partial_{x_1}\theta (x,w)\,dw &=& \lim_{h\rightarrow 0} \int_{\mathbb R^n}\frac{\theta (x+he_1,w)-\theta(x,w)}{h}\,dw \\ &=& \lim_{h\rightarrow 0}\frac{g_\gamma(x+he_1)-g_\gamma (x)}{h} \\ &=& \partial_{x_1}g_\gamma(x), \end{eqnarray*} which proves~\eqref{002} with~$\gamma$ replaced by~$\gamma+e_1$. Analogously one could prove the same result with $\gamma$ replaced by~$\gamma+e_i$, concluding the inductive step. \end{proof}
The differentiation under the integral sign in~\eqref{002} may also be obtained under slightly different assumptions, as next result points out:
\begin{lemma}\label{E} Let~$\ell\in\mathbb N$, $R>r>0$. Let~$U\in C^{\ell+1}(\mathbb R^n)$ with~$U=0$ in~$B_{R}$. Let~$\gamma=(\gamma_1,\dots,\gamma_n)\in\mathbb N^n$
with~$|\gamma|\leqslant \ell$. Then~\eqref{002} holds true for any~$x\in B_r$. \end{lemma}
\begin{proof} If~$x\in B_r$, $w\in B_{(R-r)/2}$
and~$|h|\leqslant (R-r)/2$, we have that~$|x+w+h e_1|< R$ and so~$\delta U(x+he_1,w)=0$. In particular $$ \delta U(x+he_1,w)-\delta U(x,w)=0$$ for small~$h$ when~$w\in B_{(R-r)/2}$. This formula replaces~\eqref{v0}, and the rest of the proof goes on as the one of Lemma~\ref{D}. \end{proof}
\subsubsection{Integral computations}
Here we collect some integral computations which will be used in the proof of Theorem \ref{boot}.
\begin{lemma} Let $v:\mathbb R^n \to \mathbb R$ be smooth and with all its derivatives bounded. Let $x\in B_{1/4}$, and $\gamma$, $\lambda\in \mathbb N^n$, with~$\gamma_i\geqslant\lambda_i$ for any~$i\in\{1,\dots,n\}$. Then there exists a constant $C'>0$, depending only on $n$ and $\sigma$, such that \begin{equation}\label{A2 estimate}
\left|\int_{\mathbb R^n} \partial^{\lambda}_x K(x,w)\,\delta
(\partial^{\gamma-\lambda}_x v)(x,w)\,dw\right| \leqslant C' \, C_{|\gamma|} \,\| v\|_{C^{|\gamma-\lambda|+2} (\mathbb R^n)} .\end{equation} Furthermore, if \begin{equation}\label{V001} {\mbox{$v=0$ in $B_{1/2}$}}\end{equation} we have \begin{equation}\label{A3 estimate}
\left|\int_{\mathbb R^n} \partial^{\lambda}_x K(x,w)\,\delta
(\partial^{\gamma-\lambda}_x v)(x,w)\,dw\right|\leqslant C'\, C_{|\gamma|}\, \| v\|_{L^\infty(\mathbb R^n)}. \end{equation} \end{lemma}
\begin{proof} By \eqref{sm 1} and \eqref{eq:2} (with $U=v$), \begin{eqnarray*}
&& \int_{\mathbb R^n} \big|\partial^{\lambda}_x K(x,w)\big|\, \Big|
\,\delta (\partial^{\gamma-\lambda}_x v)(x,w)\Big|\,dw
\\ &&\leqslant C_{|\lambda|} \left( \| v\|_{C^{|\gamma-\lambda|+2}(\mathbb R^n)}
\int_{B_2} |w|^{-n-\sigma+2}\,dw+ 4\|
v\|_{C^{|\gamma-\lambda|}(\mathbb R^n)} \int_{\mathbb R^n\setminus B_2}
|w|^{-n-\sigma}\,dw \right), \end{eqnarray*} which proves \eqref{A2 estimate}.
We now prove \eqref{A3 estimate}. For this we notice that, thanks to~\eqref{V001}, $v(x+w)$ and $v(x-w)$ (and also their derivatives) are equal to zero if $x$ and $w$ lie in $B_{1/4}$. Hence, by an integration by parts we see that \begin{eqnarray*} && \int_{\mathbb R^n} \partial^{\lambda}_x K(x,w)\,\delta (\partial^{\gamma-\lambda}_x v)(x,w)\,dw \\ &=& \int_{\mathbb R^n}\partial^{\lambda}_x K(x,w)\,\partial^{\gamma-\lambda}_w \big[v(x+w)- v(x-w)\big]\,dw \\ &=& \int_{\mathbb R^n\setminus B_{1/4}} \partial^{\lambda}_x K(x,w)\,\partial^{\gamma-\lambda}_w \big[v(x+w)- v(x-w)\big]\,dw
\\ &=& (-1)^{|\gamma-\lambda|} \int_{\mathbb R^n\setminus B_{1/4}} \partial^{\lambda}_x\partial^{\gamma-\lambda}_w K(x,w)\, \big[v(x+w)- v(x-w)\big]\,dw. \end{eqnarray*} Consequently, by~\eqref{sm 1}, \begin{eqnarray*}
&& \left|\int_{\mathbb R^n} \partial^{\lambda}_x K(x,w)\,\delta
(\partial^{\gamma-\lambda}_x v)(x,w)\,dw\right|
\\ &&\leqslant 2C_{|\gamma|}\, \| v\|_{L^\infty(\mathbb R^n)}
\int_{\mathbb R^n\setminus B_{1/4}} |w|^{-n-\sigma-|\gamma-\lambda|} \,dw, \end{eqnarray*} proving \eqref{A3 estimate}. \end{proof}
\subsection{Approximation by nicer kernels} \label{TTT}
In what follows, it will be convenient to approximate the solution $u$ of \eqref{eq:main} with smooth functions $u_\varepsilon$ obtained by solving equations similar to \eqref{eq:main}, but with kernels $K_\varepsilon$ which coincide with the fractional Laplacian in a neighborhood of the origin. Indeed, this will allow us to work with smooth functions, ensuring that in our computations all integrals converge. We will then prove uniform estimates on $u_\varepsilon$, which will give the desired $C^{\sigma+\alpha}$-bound on $u$ by letting $\varepsilon \to 0$.
To simplify the notation, up to multiply both $K$ and $f$ by $1/a_0$, we assume without loss of generality that the constant $a_0$ in \eqref{ass silv} is equal to $1$. \\
Let~$\eta\in C^{\infty}(\mathbb R^n)$ satisfy $$
\eta=\left\{\begin{array}{ll}
1 &\quad\mbox{in } B_{1/2}, \\
0 &\quad\mbox{in } \mathbb R^n\setminus B_{3/4},
\end{array}\right. $$ and for any $\varepsilon,\delta>0$ set $\eta_{\varepsilon}(w):=\eta\big(\frac{w}{\varepsilon}\big)$ for any $\varepsilon>0$, $\hat\eta_{\delta}(x):=\delta^{-n}\eta(x/\delta)$. Then we define \begin{equation}\label{19bis}
K_{\varepsilon}(x,w):=\eta_{\varepsilon}(w)\frac{2-\sigma}{|w|^{n+\sigma}}+(1-\eta_{\varepsilon}(w))\hat K_{\varepsilon}(x,w), \end{equation} where \begin{equation} \label{eq:hat K} \hat K_{\varepsilon}(x,w):=K(x,w)\ast \Big( \hat\eta_{\varepsilon^2}(x)\hat\eta_{\varepsilon^2}(w)\Big), \end{equation} and \begin{equation}\label{17b} L_{\varepsilon}v(x):=\int_{\mathbb R^n}{K_{\varepsilon}(x,w)\,\delta v(x,w) dw}.\end{equation} We also define \begin{equation} \label{eq:f eps} f_{\varepsilon}(x):=f(x,u(x))\ast\hat\eta_{\varepsilon}(x). \end{equation} Note that we get a family $f_\varepsilon\in C^\infty(B_{1})$ such that $$ {\mbox{$\displaystyle\lim_{\varepsilon\to 0^+}{f_{\varepsilon}}=f$ uniformly in~$B_{3/4}$.}}$$ Finally, we define $u_{\varepsilon}\in L^\infty(\mathbb R^{n})\cap C(\mathbb R^n)$ as the unique solution to the following linear problem: \begin{equation}\label{20bis} \begin{array}{llll}\left\{\begin{matrix} L_{\varepsilon}{u_{\varepsilon}}= f_{\varepsilon}(x)&\quad\mbox{in }B_{3/4}\\ u_{\varepsilon}=u&\quad\mbox{in } \mathbb R^{n}\setminus B_{3/4}. \end{matrix}\right. \end{array} \end{equation} It is easy to check that the kernels $K_\varepsilon$ satisfy \eqref{ass silv} and \eqref{sm 1} with constants independent of $\varepsilon$ (recall that, to simplify the notation, we are assuming $a_0=1$). Also, since $K$ satisfies assumption \eqref{sm 1} with $k=0$ and the convolution parameter $\varepsilon^2$ in \eqref{19bis} is much smaller than $\varepsilon$, the operators $L_\varepsilon$ converge to the operator associated to $K$ in the weak sense introduced in~\cite[Definition 22]{CS2011}. Indeed, let $v$ a smooth function satisfying \begin{equation}\label{v condition}
|v|\leqslant M \quad\mbox{in $\mathbb R^{n}$},\qquad |v(w)-v(x)-(w-x)\cdot\nabla v(x)|\leqslant M|x-w|^{2}\quad\forall\,w\in B_{1}(x), \end{equation} for some $M>0$. Then, from \eqref{sm 1} and \eqref{v condition}, it follows that \begin{eqnarray}
&&\int_{\mathbb{R}^{n}}{\left|\eta_{\varepsilon}(w)\frac{2-\sigma}{|w|^{n+\sigma}}+(1-\eta_{\varepsilon}(\omega))\bigl(K(x,w)\ast\hat{\eta}_{\varepsilon^2}(x)\hat{\eta}_{\varepsilon^{2}}(w)\bigr)-K(x,w)\right||\delta v(x,w)|\,dw}\nonumber\\
&\leqslant&\int_{\mathbb{R}^{n}}{\left(\eta_{\varepsilon}(w)\Big|\frac{2-\sigma}{|w|^{n+\sigma}}-K(x,w)\Big|+(1-\eta_{\varepsilon}(\omega))
\Big|K(x,w)\ast\hat{\eta}_{\varepsilon^2}(x)\hat{\eta}_{\varepsilon^{2}}(w)-K(x,w)\Big|\right)}\nonumber\\
&&\qquad\qquad \qquad\qquad\qquad \qquad\qquad\qquad \qquad \times|\delta v(x,w)|\,dw\nonumber\\
&\leqslant&\int_{B_{\varepsilon}}{C|w|^{2-n-\sigma}}+\int_{\mathbb R^n\setminus B_{\varepsilon}}{\bigl|K(x,w)\ast\hat{\eta}_{\varepsilon^2}(x)\hat{\eta}_{\varepsilon^{2}}(w)-K(x,w)\bigr|\,|\delta v(x,w)|\,dw}\nonumber\\ &\leqslant& C\varepsilon^{2-\sigma}+I, \label{ne} \end{eqnarray} with
$$I:=\int_{\mathbb R^n\setminus B_{\varepsilon}}{\bigl|K(x,w)\ast\hat{\eta}_{\varepsilon^2}(x)\hat{\eta}_{\varepsilon^{2}}(w)-K(x,w)\bigr|\,|\delta v(x,w)|\,dw}.$$ By \eqref{sm 1}, \eqref{v condition}, and the fact that $\sigma>1$, we have \begin{eqnarray*}
I&=&\int_{\mathbb R^n\setminus B_{\varepsilon}}{\int_{B_{1}}{\int_{B_{1}}{\left|K(x-\varepsilon^2y,w-\varepsilon^2\tilde{w})\eta(y)\eta(\tilde{w})-K(x,w)\right|\,dy
\,d\tilde{w}\,|\delta v(x,w)|\,dw}}}\\
&\leqslant&\int_{\mathbb R^n\setminus B_{\varepsilon}}{\frac{C\varepsilon^2}{|w|^{n+1+\sigma}}\,|\delta v(x,w)|\,dw}\\
&\leqslant&C\int_{B_{1}\setminus B_{\varepsilon}}{\frac{\varepsilon^2}{|w|^{n-1+\sigma}}\,dw}+C\int_{\mathbb R^n\setminus B_{1}}{\frac{\varepsilon^2}{|w|^{n+1+\sigma}}\,dw}\\ &\leqslant &C(\varepsilon^{3-\sigma}+\varepsilon^{2}). \end{eqnarray*} Combining this estimate with \eqref{ne}, we get
$$\int_{\mathbb{R}^{n}}{\left|\eta_{\varepsilon}(w)\frac{2-\sigma}{|w|^{n+\sigma}}+(1-\eta_{\varepsilon}(\omega))(K(x,w)\ast\hat{\eta}_{\varepsilon^2}(x)\hat{\eta}_{\varepsilon^{2}}(w))-K(x,w)\right|\delta v(x,w)dw}\leqslant C\varepsilon^{2-\sigma},$$ where $C$ depends of $M$ and $\sigma$. Since $\sigma<2$ we conclude that
$$\|L_{\varepsilon}-L\|\to 0 \quad\mbox{as $\varepsilon\to 0$.}$$ Thanks to this fact, we can repeat almost \textit{verbatim}\footnote{In order to repeat the argument in the proof of \cite[Lemma 7]{CS2011} one needs to know that the functions $u_\varepsilon$ are equicontinuous, which is a consequence of \cite[{Lemmata 2 and 3}]{CS2011}. To be precise, to apply
\cite[Lemma 3]{CS2011} one would need the kernels to satisfy the bounds $\frac{(2-\sigma)\lambda}{|w|^{n+\sigma}}\leqslant K_*(x,w)\leqslant
\frac{(2-\sigma)\Lambda}{|w|^{n+\sigma}}$ for all $w \neq 0$, while in our case the kernel $K$ (and so also $K_\varepsilon$) satisfies \begin{equation}\label{Lo}
\frac{(2-\sigma)\lambda}{|w|^{n+\sigma}}\leqslant K(x,w)\leqslant
\frac{(2-\sigma)\Lambda}{|w|^{n+\sigma}}\qquad \forall\,|w|\leqslant r_0 \end{equation} with $\lambda:={a_0}-\eta$, $\Lambda:=a_{0}+\eta$, and $r_0>0$ (observe that, by our assumptions in \eqref{ass silv}, $\lambda \geqslant 3{a_0}/4$).
However this is not a big problem: if $v \in L^\infty(\mathbb R^n)$ satisfies $$ \int_{\mathbb R^n} K_*(x,w)\,\delta v(x,w)\,dw = f(x) \qquad\text{in $B_{3/4}$} $$ for some kernel satisfying {\eqref{sm 1} and}
$\frac{(2-\sigma)\lambda}{|w|^{n+\sigma}}\leqslant K_*(x,w)\leqslant
\frac{(2-\sigma)\Lambda}{|w|^{n+\sigma}}$ only for $|w| \leqslant r_0$, we define $K'(x,w):=\zeta(w)K_*(x,w) +
(2-\sigma)\frac{1-\zeta(w)}{|w|^{n+\sigma}}$, with $\zeta$ a smooth cut-off function supported inside $B_{r_0}$, to get $$ \int_{\mathbb R^n} K'(x,w)\,\delta v(x,w)\,dw = f(x) + \int_{\mathbb R^n}
[1-\zeta(w)]\left({-}K_*(x,w)+ \frac{2-\sigma}{|w|^{n+\sigma}} \right)\,\delta v(x,w)\,dw. $$ Since $1-\zeta(w)=0$ near the origin, {by assumption \eqref{sm 1}}, the second integral is uniformly bounded as a function of $x$, so \cite[Lemma 3]{CS2011} applied to $K'$ gives the desired equicontinuity.
Finally, the uniqueness for the boundary problem $$ \left\{\begin{matrix} \int_{\mathbb R^n} K(x,w) \,\delta v(x,w)\,dw=f(x,u(x))&\quad\mbox{in $B_{3/4}$,}\\ v=u\quad&\mbox{in $\mathbb R^{n}\setminus B_{3/4}$.} \end{matrix}\right. $$ follows by a standard comparison principle argument (see for instance the argument used in the proof of \cite[Theorem 3.2]{BCF}). } the proof of \cite[Lemma 7]{CS2011} to obtain the uniform convergence \begin{equation}\label{Uu} u_{\varepsilon}\to u \quad \text{on $\mathbb R^n$} \qquad\mbox{as $\varepsilon\to 0$.} \end{equation}
\subsection{Smoothness of the approximate solutions}
We prove now that the functions $u_{\varepsilon}$ defined in the previous section are of class $C^{\infty}$ inside a small ball (whose size is uniform with respect to $\varepsilon$): namely, there exists $r\in (0,1/4)$ such that, for any $m\in\mathbb N^{n}$
\begin{equation}\label{LE10} \| D^m u_{\varepsilon}
\|_{L^\infty(B_{r})}\leqslant C \end{equation} for some positive constant $C= C(m, \sigma, \varepsilon,\| u\|_{L^\infty(\mathbb R^n)},
\|f\|_{ L^\infty(B_{1}\times\mathbb R) })$.
For this, we observe that by~\eqref{19bis}
$$ \frac{2-\sigma}{|w|^{n+\sigma}}=K_\varepsilon(x,w)- (1-\eta_{\varepsilon}(w))\hat K_{\varepsilon}(x,w)
+(1-\eta_\varepsilon(w))\,\frac{2-\sigma}{|w|^{n+\sigma}} ,$$ so for any $x\in B_{1/4}$ \begin{eqnarray*} \frac{2-\sigma}{2c_{n,\sigma}}(-\Delta)^{\sigma/2}u_{\varepsilon}(x)&=&
\int_{\mathbb R^n}{\frac{2-\sigma}{|w|^{n+\sigma}}\delta u_{\varepsilon}(x,w)dw}\\ \qquad&=&f_{\varepsilon}(x)-\int_{\mathbb R^n}{(1-\eta_{\varepsilon}(w))\hat K_{\varepsilon}(x,w)\, \delta u_{\varepsilon}(x,w)dw}\\
\quad&&+\int_{\mathbb R^n}{(1-\eta_{\varepsilon}(w))\frac{2-\sigma}{|w|^{n+\sigma}}\, \delta u_{\varepsilon}(x,w)dw} \end{eqnarray*} (here $c_{n,\sigma}$ is the positive constant that appears in the definition of the fractional Laplacian, see e.g. \cite{PhS, guida}). Then, for any $x\in B_{1/4}$ it follows that \begin{eqnarray} && (-\Delta)^{\sigma/2}u_{\varepsilon}(x) \nonumber\\
&=&d_{n,\sigma}\Big[f_{\varepsilon}(x)+\int_{\mathbb R^n}{(1-\eta_{\varepsilon}(w))\Big(\frac{2-\sigma}{|w|^{n+\sigma}}-\hat K_{\varepsilon}(x,w)\Big)\delta u_{\varepsilon}(x,w)dw}\Big]\nonumber\\ &=:&d_{n,\sigma}[f_{\varepsilon}(x)+h_\varepsilon(x)]\label{fund.eq}\\ &=:&d_{n,\sigma}g_\varepsilon(x).\nonumber \end{eqnarray} with $\displaystyle d_{n,\sigma}:=\frac{2c_{n,\sigma}}{2-\sigma}.$
Making some changes of variables we can rewrite $h_\varepsilon$ as follows: \begin{eqnarray}\label{BBB0} \nonumber h_\varepsilon(x)&=&\int_{\mathbb R^n}
(1-\eta_{\varepsilon}(w-x))\Big(\frac{2-\sigma}{|w-x|^{n+\sigma}}- \hat K_{\varepsilon}(x,w-x)\Big)u_{\varepsilon}(w) dw\\ &+&\int_{\mathbb R^n} (1-\eta_{\varepsilon}(x-w))
\Big(\frac{2-\sigma}{|w-x|^{n+\sigma}}-\hat K_{\varepsilon}(x,x-w)\Big)u_{\varepsilon}(w) dw \nonumber \\ &-&2 u_{\varepsilon}(x)\int_{\mathbb R^n}
(1-\eta_{\varepsilon}(w))\Big(\frac{2-\sigma}{|w|^{n+\sigma}}-\hat K_{\varepsilon}(x,w)\Big) dw. \end{eqnarray} We now notice that ``the function $h_{{\varepsilon}}$ is locally as smooth as $u_{\varepsilon}$'', is the sense that for any $m \in \mathbb N$ and $U\subset B_{1/4}$ open we have \begin{equation}\label{BL7}
\|h_{{\varepsilon}}\|_{C^{m}(U)} \leqslant
{C_{\varepsilon,\,m}}\left(1+\|u_{{\varepsilon}}\|_{C^{m}(U)}\right) \end{equation} for some constant $C_{\varepsilon,\,m}>0$. To see this observe that, in the first two integrals, the variable $x$ appears only inside $\eta_\varepsilon$ and in the kernel $\hat K_{\varepsilon}$, and $\eta_{\varepsilon}$ is equal to $1$ near the origin. Hence the first two integrals are smooth functions of $x$ (recall that $\hat K_{\varepsilon}$ is smooth, see \eqref{eq:hat K}). The third term is clearly as regular as $u_{\varepsilon}$ because the third integral is smooth by the same reason as before. This proves \eqref{BL7}.
We are now going to prove that the functions $u_{\varepsilon}$ belong to $C^\infty(B_{1/5})$, with \begin{equation}\label{LE1}
\| u_{\varepsilon} \|_{C^m(B_{1/4-r_{m}})}\leqslant C(r_1,m,{\sigma},\varepsilon, \|
u_{\varepsilon}\|_{L^\infty(\mathbb R^n)},\|f\|_{ L^\infty(B_{1}\times\mathbb R) }) \end{equation}
for any $m\in\mathbb N$, where $r_{m}:=1/20 - 25^{-m}$ (so that $1/4-r_m>1/5$ for any $m$).
To show this, we begin by observing that, since $\sigma\in(1,2)$, by \eqref{fund.eq}, \eqref{BL7}, and \cite[Theorem 61]{CS2011}, we have that $u_{\varepsilon}\in L^\infty(\mathbb R^{n})\cap C^{1,\beta}(B_{1/4-r_1})$ for any $\beta<\sigma-1$ ($r_1=1/100$), and \begin{equation}\label{7B7A}
\|u_{\varepsilon}\|_{C^{1,\beta}(B_{1/4-r_1})}\leqslant
{C_{\varepsilon}}\Big(\|u\|_{L^{\infty}(\mathbb R^n)}+\|f\|_{L^{\infty}(B_{1}\times\mathbb R)}\Big). \end{equation} Now, to get a bound on higher derivatives, the idea would be to differentiate~\eqref{fund.eq} and use again \eqref{BL7} and \cite[Theorem 61]{CS2011}. However we do not have $C^1$ bounds on the function $u_{\varepsilon}$ outside $B_{{1/4-r_1}}$, and therefore we can not apply directly this strategy to obtain the $C^{2,\alpha}$ regularity of the function $u_{\varepsilon}$.
To avoid this problem we follow the localization argument in \cite[Theorem 13.1]{CScpam}: we take $\delta>0$ small (to be chosen) and we consider a smooth cut-off function $$
\vartheta:=\left\{\begin{array}{ll}
1 &\quad\mbox{in } B_{1/4-(1+\delta)r_1}, \\
0 &\quad\mbox{on } \mathbb R^n\setminus B_{1/4-r_1},
\end{array}\right. $$
and for fixed~$e\in S^{n-1}$ and $|h|<\delta r_1$ we define \begin{equation}\label{Dv}
v(x):=\frac{u_{\varepsilon}(x+eh)-u_{\varepsilon}(x)}{|h|}.\end{equation}
{The function $v(x)$ is uniformly bounded in $B_{1/4-(1+\delta)r_1}$ because $u\in C^{1}(B_{1/4-r_1})$.} We now write~$v(x)=v_{1}(x)+v_{2}(x)$, being
$$v_{1}(x):=\frac{\vartheta u_{\varepsilon}(x+eh)-\vartheta u_{\varepsilon}(x)}{|h|}\quad\mbox{and}\quad v_{2}(x):=\frac{(1-\vartheta)u_{\varepsilon}(x+eh)-(1-\vartheta)u_{\varepsilon}(x)}{|h|}.$$ By \eqref{7B7A} it is clear that $$ v_{1}\in L^{\infty}(\mathbb R^n) $$
and that (recall that $|h|<\delta r_{1}$) \begin{equation}\label{v1two} v_1=v\qquad \mbox{inside $B_{1/4-(1+2\delta)r_1}$.} \end{equation} Moreover, for $x\in B_{1/4-(1+2\delta)r_1}$, using \eqref{fund.eq}, \eqref{eq:f eps}, and \eqref{BL7} we get \begin{eqnarray}
\left|(-\Delta)^{\sigma/2}v_1(x)\right|&=&
\left|(-\Delta)^{\sigma/2}v(x)-(-\Delta)^{\sigma/2}v_{2}(x)\right|\nonumber\\
&=&\left|\frac{g_{{\varepsilon}}(x+eh)-g_{{\varepsilon}}(x)}{|h|}-(-\Delta)^{\sigma/2}v_{2}(x)\right|\nonumber\\ &\leqslant&
{C_{\varepsilon}}\Big(1+\|u_{\varepsilon}\|_{C^{1}(B_{1/4-r_1})}\Big)+\left|(-\Delta)^{\sigma/2}v_{2}(x)\right|.\label{M+} \end{eqnarray} Now, let us denote by
$K_o(y):=\frac{c_{n,\sigma}}{|y|^{n+\sigma}}$ the kernel of the fractional Laplacian. Since for $x\in B_{1/4-(1+2\delta)r_1}$ and
$|y|<\delta r_{1}$ we have that $(1-\vartheta)u_{\varepsilon}(x\pm y)=0$, it follows from a change of variable that \begin{eqnarray*}
|(-\Delta)^{\sigma/2}v_{2}(x)|&\leqslant&\Big|\int_{\mathbb R^n}{(v_{2}(x+y)+v_{2}(x-y)-2v_{2}(x))K_o(y)dy}\Big|\\
&\leqslant&\Big|\int_{\mathbb R^n}{\frac{(1-\vartheta)u_{\varepsilon}(x+y+eh)-(1-\vartheta)u_{\varepsilon}(x+y)}{|h|}K_o(y)dy}\Big|\\
&&+\Big|\int_{\mathbb R^n}{\frac{(1-\vartheta)u_{\varepsilon}(x-y+eh)-(1-\vartheta)u_{\varepsilon}(x-y)}{|h|}K_o(y)dy}\Big|\\
&\leqslant&\int_{\mathbb R^n}{(1-\vartheta)|u_{\varepsilon}|(x+y)\Big|\frac{K_o(y-eh)-K_o(y)}{|h|}\Big|dy}\\
&&+\int_{\mathbb R^n}{(1-\vartheta)|u_{\varepsilon}|(x-y)\Big|\frac{K_o(y-eh)-K_o(y)}{|h|}\Big|dy}\\
&\leqslant&\|u_{\varepsilon}\|_{L^{\infty}(\mathbb R^n)}\int_{\mathbb R^n\setminus B_{{\delta r_1}} }{\frac{1}{|y|^{n+\sigma+1}}dy}\\
&\leqslant&C_\delta \|u_{\varepsilon}\|_{L^{\infty}(\mathbb R^n)}. \end{eqnarray*} Therefore, by \eqref{M+} we obtain
$$\big|(-\Delta)^{\sigma/2}v_{1}(x)\big|\leqslant
{C_{\varepsilon,\delta}}\Big(1+ \|u_{\varepsilon}\|_{C^{1}(B_{1/4-r_1})}
+\|u_{\varepsilon}\|_{L^{\infty}(\mathbb R^n)}\Big),\qquad x\in {B_{1/4-(1+2\delta)r_1}},$$ and we can apply \cite[Theorem 61]{CS2011} to get that $v_{1}\in C^{1,\beta}(B_{1/4-r_2})$ for any $\beta<\sigma-1$, with
$$\|v_{1}\|_{C^{1,\beta}(B_{1/4-r_{2}})}\leqslant
{C_{\varepsilon}}\Big(1+\|v_{1}\|_{L^{\infty}(\mathbb R^n)}+
\|u_{\varepsilon}\|_{C^{1}(B_{1/4}-r_1)}
+\|u_{\varepsilon}\|_{L^{\infty}(\mathbb R^n)}\Big),\,$$ provided $\delta>0$ was chosen sufficiently small so that~{$r_2>(1+2\delta)r_1$}. By \eqref{Dv}, \eqref{v1two}, and \eqref{7B7A}, this implies that $u_{\varepsilon}\in C^{2,\beta}(B_{1/4-r_{2}})$, with \begin{eqnarray*}
\|u_{\varepsilon}\|_{C^{2,\beta}(B_{1/4-r_{2}})}&\leqslant& {C_{\varepsilon}}\Big(1+
\|u_{\varepsilon}\|_{C^{1}(B_{1/4-r_1})}
+\|u_{\varepsilon}\|_{L^{\infty}(\mathbb R^n)}\Big) \\ &\leqslant&
{C_\varepsilon}\Big(1 +\|u\|_{L^{\infty}(\mathbb R^n)}
+\|f\|_{L^{\infty}(B_{1}\times\mathbb R)}\Big). \end{eqnarray*} Iterating this argument we obtain~\eqref{LE1}, as desired.
\subsection{Uniform estimates and conclusion of the proof for $k=0$} \label{section:uniforml} Knowing now that the functions $u_\varepsilon$ defined by \eqref{20bis} are smooth inside $B_{1/5}$ (see \eqref{LE1}), our goal is to obtain a-priori bounds independent of $\varepsilon$.
By \cite[Theorem 61]{CS2011} applied\footnote{As already observed in the footnote on page~\pageref{Uu}, the fact that the kernel satisfies \eqref{Lo} only for $w$ small is not a problem, and one can easily check that \cite[Theorem 61]{CS2011} still holds in our setting. } to $u$, we have that $u\in C^{1,\beta}(B_{1-R_{1}})$ for any $\beta<\sigma-1$ and~$R_{1}>0$, with \begin{equation}\label{Be}
\|u\|_{C^{1,\beta}(B_{1-R_1})}\leqslant C\left(\|u\|_{L^{\infty}(\mathbb R^n)}+\|f\|_{L^{\infty}(B_{1}\times\mathbb R)}\right). \end{equation} Then, for any $\varepsilon$ sufficiently small,~$f_{\varepsilon}\in C^1(B_{1/2})$ with
\begin{equation}\label{FE}\begin{split} & \|
f_{\varepsilon}\|_{C^1(B_{1/2})}\leqslant C' \left(1+\| u\|_{C^{1} (B_{1-R_1})}\right)\\
&\qquad\leqslant C' C \left(1+\| u\|_{L^{\infty}(\mathbb R^n)}
+\|f\|_{L^{\infty}(B_{1}\times\mathbb R)}\right), \end{split}\end{equation}
where~$C'>0$ depends on~$\|f\|_{C^1(B_{1}\times\mathbb R)}$ only.
Consider a cut-off function~$\tilde\eta$ which is $1$ inside $B_{1/7}$ and $0$ outside $B_{1/6}$.
Then, recalling~\eqref{20bis}, we write the equation satisfied by $u_{\varepsilon}$ as \begin{equation*} f_{\varepsilon}(x)= \int_{\mathbb R^{n}} K_{\varepsilon}(x,w)\,\delta (\tilde\eta u_{\varepsilon})(x,w)dw +\int_{\mathbb R^{n}} K_{\varepsilon}(x,w)\,\delta ((1-\tilde\eta) u_{\varepsilon})(x,w)dw, \end{equation*} and by differentiating it, say in direction~$e_1$, we obtain (recall Lemmata~\ref{D} and~\ref{E}) \begin{eqnarray*} \partial_{x_1} f_{\varepsilon}(x) &=& \int_{\mathbb R^{n}} K_{\varepsilon}(x,w) \delta (\partial_{x_1}(\tilde\eta u_{\varepsilon}))(x,w)dw \\ &&\quad+\int_{\mathbb R^{n}} \partial_{x_1} \big[ K_{\varepsilon}(x,w)\delta ((1-\tilde\eta) u_{\varepsilon})(x,w)\big] dw\\&&\quad +\int_{\mathbb R^{n}} \partial_{x_1} K_{\varepsilon}(x,w)\delta (\tilde\eta u_{\varepsilon})(x,w)dw \end{eqnarray*}for any~$x\in B_{1/5}$. It is convenient to rewrite this equation as $$ \int_{\mathbb R^{n}} K_{\varepsilon}(x,w) \delta (\partial_{x_1}(\tilde\eta u_{\varepsilon}))(x,w)dw=A_1-A_2-A_3,$$ with \begin{eqnarray*} A_1 &:=& \partial_{x_1} f_{\varepsilon}(x),\\ A_2 &:=& \int_{\mathbb R^{n}} \partial_{x_1} K_{\varepsilon}(x,w)\delta (\tilde\eta u_{\varepsilon})(x,w)dw\\ A_3&:=& \int_{\mathbb R^{n}} \partial_{x_1} \big[ K_{\varepsilon}(x,w)\delta ((1-\tilde\eta) u_{\varepsilon})(x,w)\big] dw. \end{eqnarray*} We claim that \begin{equation}\label{A1233}
\|A_1-A_2-A_3\|_{L^\infty(B_{1/14})} \leqslant C
\left(1+\|u\|_{L^{\infty}(\mathbb R^n)}+\|u_{\varepsilon}\|_{C^{2}(B_{1/6})}\right)\end{equation}
with $C$ depending only on $\|f\|_{C^1(B_1\times\mathbb R)}$. To prove this, we first observe that by~\eqref{FE} $$
\|A_1\|_{L^\infty(B_{1/14})} \leqslant C \left(1+
\|u\|_{L^{\infty}(\mathbb R^n)}\right). $$
Also, since $| \partial_{x_1} \hat {K}_{\varepsilon}(x,w)| \leqslant C|w|^{-(n+\sigma)}$,\footnote{ This can be easily checked using the definition of $\hat {K}_{\varepsilon}$ and \eqref{sm 1}. Indeed, because of the presence of the term $(1-\eta_{\varepsilon}(w))$
which vanishes for $|w|\leqslant \varepsilon/2$, one only needs to check that $$
\int_{\mathbb R^n} |w-z|^{-n-\sigma} \hat{\eta}_{\varepsilon^2} (z) \,dz
\leqslant C|w|^{-n-\sigma} \qquad\text{for } |w|\geqslant \varepsilon/2, $$ which is easy to prove (we leave the details to the reader). } by~\eqref{A2 estimate} (used with $\gamma=\lambda:=(1,0,\dots,0)$ and $v:=\tilde\eta u_{\varepsilon}$) we get $$
\|A_2\|_{L^\infty(B_{1/14})} \leqslant C\|\tilde\eta u_{\varepsilon}\|_{C^2(\mathbb R^n)} \leqslant C
\|u_{\varepsilon}\|_{C^2(B_{1/6})}, $$ where we used that $\tilde\eta$ is supported in $B_{1/6}$.
Moreover, since~$(1-\tilde\eta)u_\varepsilon=0$ inside $B_{1/7}$, we can use \eqref{A3 estimate} with $v:=(1-\tilde\eta)u_{\varepsilon}$ to obtain \begin{eqnarray*}
&& \left|\int_{\mathbb R^{n}} \partial_{x_1} K_{\varepsilon}(x,w)\,\delta
((1-\tilde\eta) u_{\varepsilon})(x,w)\,dw\right| \\&&\qquad+
\left|\int_{\mathbb R^{n}} K_{\varepsilon}(x,w)\,\partial_{x_1}\delta
((1-\tilde\eta) u_{\varepsilon})(x,w)\, dw\right|
\\ &&\qquad\qquad\leqslant C\, C_{k}\, \| (1-\tilde\eta)u_{\varepsilon}\|_{L^\infty(\mathbb R^n)} \end{eqnarray*} for any $x\in B_{1/14}$, which gives (note that, by an easy comparison principle,
$\|u_\varepsilon\|_{L^\infty(\mathbb R^n)} \leqslant C(1+\|u\|_{L^\infty(\mathbb R^n)})$)
$$\|A_3\|_{L^\infty(B_{1/14})} \leqslant C(1+\|u\|_{L^\infty(\mathbb R^n)}).$$ The above estimates imply \eqref{A1233}.
Since $\partial_{x_1}(\tilde{\eta}u_{\varepsilon})$ is bounded on the whole of~$\mathbb R^n$, by \eqref{A1233} and~\cite[Theorem~61]{CS2011} we obtain that~$\partial_{x_1}(\tilde{\eta} u_{\varepsilon})\in C^{1,\beta}(B_{1/14-R_2})$ for any $R_{2}>0$, with $$
\|\partial_{x_1}(\tilde{\eta}
u_{\varepsilon})\|_{C^{1,\beta}(B_{1/14-R_2})} \leqslant C
\left(1+\|u\|_{L^{\infty}(\mathbb R^n)}+\|u_{\varepsilon}\|_{C^{2}(B_{1/6})}\right),$$ which implies \begin{equation}\label{29AA}
\|u_{\varepsilon}\|_{C^{2,\beta}(B_{1/15})} \leqslant C
\left(1+\|u\|_{L^{\infty}(\mathbb R^n)}+\|u_{\varepsilon}\|_{C^{2}(B_{1/6})}\right). \end{equation} To end the proof we need to reabsorb the $C^2$-norm on the right hand side. To do this, we observe that by standard interpolation inequalities (see for instance \cite[Lemma 6.35]{GT}), for any $\delta\in(0,1)$ there exists $C_\delta>0$ such that \begin{equation}\label{29BB}
\|u_{\varepsilon}\|_{C^2(B_{1/6})} \leqslant \delta
\|u_{\varepsilon}\|_{C^{2,\beta}(B_{1/5})}+C_\delta
\|u_{\varepsilon}\|_{L^\infty(\mathbb R^n)}. \end{equation} Hence, by~\eqref{29AA} and~\eqref{29BB} we obtain \begin{equation}\label{29CC}
\|u_{\varepsilon}\|_{C^{2,\beta}(B_{1/15})} \leqslant C_\delta
(1+\|u\|_{L^\infty(\mathbb R^n)})
+C\delta\|u_{\varepsilon}\|_{C^{2,\beta}(B_{1/5})}. \end{equation} To conclude, one needs to apply the above estimates at every point inside $B_{1/5}$ at every scale: for any $x \in B_{1/5}$, let $r>0$ be any radius such that $B_r(x)\subset B_{1/5}$. Then we consider \begin{equation}\label{sc} v_{\varepsilon,r}^x(y):=u_{\varepsilon}(x+ry), \end{equation} and we observe that $v_{\varepsilon,r}^x$ solves an analogous equation as the one solved by~$u_{\varepsilon}$ with the kernel given by $$K^{x}_{\varepsilon,r}(y,z):=r^{n+\sigma}K_{\varepsilon}(x+ry,rz)$$ and with right hand side $$F_{\varepsilon,r}(y):=r^{\sigma}\int_{\mathbb R^n}{f(x+ry-\tilde{x},u(x+ry-\tilde{x}))\hat{\eta}_{\varepsilon}(\tilde{x})d\tilde{x}}.$$
We now observe that the kernels $K^{x}_{\varepsilon,r}$ satisfy assumptions \eqref{ass silv} and \eqref{sm 1} uniformly with respect to $\varepsilon$, $r$, and $x$. Moreover, for $|x|+r\leqslant 1/5$, and $\varepsilon$ small, we have
$$\|F_{\varepsilon,r}\|_{C^{1}(B_{1/2})}\leqslant r^{\sigma}C(1+\|u\|_{C^{1}(B_{3/4})}), $$
with $C>0$ depending on $\|f\|_{C^{1}(B_{1}\times\mathbb R)}$ only. Hence, by \eqref{Be} this implies
$$\|F_{\varepsilon,r}\|_{C^{1}(B_{1/2})}\leqslant r^{\sigma}C\left(1+\|u\|_{L^{\infty}(\mathbb R^n)}+\|f\|_{L^{\infty}(B_{1}\times\mathbb R)}\right).$$ Thus, applying~\eqref{29CC} to $v_{\varepsilon,r}^x$ (by the discussion we just made, the constants are all independent of $\varepsilon$, $r$, and $x$) and scaling back, we get \begin{eqnarray*}
\|u_{\varepsilon}\|^*_{C^{2,\beta}(B_{r/15}(x))} \leqslant C_\delta
\left(1+\|u\|_{L^\infty(\mathbb R^n)}+\|f\|_{L^{\infty}(B_{1}\times\mathbb R)}\right) +C\delta
\|u_{\varepsilon}\|^*_{C^{2,\beta}(B_{r/5}(x))}. \end{eqnarray*} Using now Lemma~\ref{Co2} inside $B_{1/5}$ with $\mu=1/15$, $\nu=1/5$, $m=2$, and $\Lambda_\delta=C_\delta
(1+\|u\|_{L^\infty(\mathbb R^n)}+\|f\|_{L^{\infty}(B_{1}\times\mathbb R)})$, we conclude (observe that $1/15 \cdot 1/5=1/75$)
$$ \|u_{\varepsilon}\|_{C^{2,\beta}(B_{1/75})}\leqslant C\left(1+\|u\|_{L^\infty(\mathbb R^n)}+\|f\|_{L^{\infty}(B_{1}\times\mathbb R)}\right),$$ which implies
$$ \|u\|_{C^{2,\beta}(B_{1/75})}\leqslant C\left(1+\|u\|_{L^\infty(\mathbb R^n)}+\|f\|_{L^{\infty}(B_{1}\times\mathbb R)}\right)$$ by letting $\varepsilon \to 0$ (see \eqref{Uu}). Since $\beta<\sigma-1$, this is equivalent to
$$ \|u\|_{C^{\sigma+\alpha}(B_{1/75})}\leqslant C
\left(1+\|u\|_{L^\infty(\mathbb R^n)}+\|f\|_{L^{\infty}(B_{1}\times\mathbb R)}\right),\,\quad\mbox{for any $\alpha<1$}.$$ A standard covering/rescaling argument completes the proof of Theorem~\ref{boot} in the case $k=0$.
\subsection{The induction argument.}
We already proved Theorem~\ref{boot} in the case $k=0$.
We now show by induction that, for any $k \geqslant 1$, \begin{equation}\label{induction}
\| u\|_{C^{k+\sigma+\alpha}(B_{1/2^{{3k+4}}})} \leqslant C_k
\left(1+\|u\|_{L^\infty(\mathbb R^n)}+\|f\|_{L^{\infty}(B_{1}\times\mathbb R)}\right) , \end{equation} for some constant~$C_k>0$: by a standard covering/rescaling argument, this proves~\eqref{3bis} and so Theorem~\ref{boot}. As we shall see, the argument is more or less identical to the case $k=0$. To be fully rigorous, we should apply the regularization argument with the functions $u_\varepsilon$ as done in the previous step. However, to simplify the notation and make the argument more transparent, we will skip the regularization.
Define $g(x):=f(x,u(x))$, and consider a cut-off function~$\tilde\eta$ which is $1$ inside $B_{1/2^{3k+5}}$ and $0$ outside $B_{1/2^{3k+4}}$.
By Lemmata~\ref{D} and~\ref{E} we differentiate the equation~$k+1$ times according to the following computation: first we observe that, {since \eqref{induction} is true for $k-1$ and we can choose $\alpha \in (2-\sigma,1)$ so that $\sigma+\alpha>2$}, we deduce that $g\in C^{k+{1}}(B_{1/2^{3k+4}})$ with \begin{equation}\label{gg}
\|g\|_{C^{k+{1}}(B_{1/2^{3k+4}})}\leqslant C\left(1+ \| u\|_{C^{k+{1}}
(B_{1/2^{3k+4}})}\right){\leqslant C\left(\|u\|_{L^{\infty}(\mathbb R^n)}+\|f\|_{L^{\infty}(B_1\times\mathbb R^n)}\right)}, \end{equation}
with~$C>0$ depending on~$\|f\|_{C^{k+{1}}(B_{1}\times\mathbb R)}$ only. Now we take~$\gamma\in\mathbb N^n$ with~$|\gamma|=k+{1}$ and we differentiate the equation to obtain \begin{eqnarray*}&& \partial^\gamma g(x) \\ &=& \sum_{{{1\leqslant i\leqslant n}\atop{0\leqslant \lambda_i\leqslant \gamma_i}}\atop{\lambda=(\lambda_1,\dots,\lambda_n)}} \left( {\gamma_1}\atop{\lambda_1}\right)\dots \left( {\gamma_n}\atop{\lambda_n}\right) \int_{\mathbb R^n} \partial^{\lambda}_x K(x,w)\,\delta (\partial^{\gamma-\lambda}_x (\tilde\eta u))(x,w)\,dw \\ &+& \sum_{{{1\leqslant i\leqslant n}\atop{0\leqslant \lambda_i\leqslant \gamma_i}}\atop{\lambda=(\lambda_1,\dots,\lambda_n)}} \left( {\gamma_1}\atop{\lambda_1}\right)\dots \left( {\gamma_n}\atop{\lambda_n}\right) \int_{\mathbb R^n} \partial^{\lambda}_x K(x,w)\,\delta (\partial^{\gamma-\lambda}_x (1-\tilde\eta)u)(x,w)\,dw .\end{eqnarray*} Then, we isolate the term with~$\lambda=0$ in the first sum: $$ \int_{\mathbb R^n} K(x,w)\, \delta (\partial^\gamma_x(\tilde\eta u)) (x,w)\,dw = A_1-A_2-A_3 $$ with \begin{eqnarray*} A_1 &:=& \partial^\gamma g(x),\\ A_2 &:=& \sum_{{{1\leqslant i\leqslant n}\atop{0\leqslant \lambda_i\leqslant \gamma_i}}\atop{\lambda=(\lambda_1,\dots,\lambda_n) \ne0}} \left( {\gamma_1}\atop{\lambda_1}\right)\dots \left( {\gamma_n}\atop{\lambda_n}\right) \int_{\mathbb R^n} \partial^{\lambda}_x K(x,w)\,\delta (\partial^{\gamma-\lambda}_x (\tilde\eta u))(x,w)\,dw\\ A_3&:=& \sum_{{{1\leqslant i\leqslant n}\atop{0\leqslant \lambda_i\leqslant \gamma_i}}\atop{\lambda=(\lambda_1,\dots,\lambda_n)}} \left( {\gamma_1}\atop{\lambda_1}\right)\dots \left( {\gamma_n}\atop{\lambda_n}\right) \int_{\mathbb R^n} \partial^{\lambda}_x K(x,w)\,\delta (\partial^{\gamma-\lambda}_x (1-\tilde\eta)u)(x,w)\,dw \end{eqnarray*} We claim that \begin{equation}\label{R}
\|A_1-A_2-A_3\|_{L^\infty(B_{1/2^{3k+6}})} \leqslant C
\left(1+\|u\|_{L^\infty(\mathbb R^n)}
+\|u\|_{C^{k+{2}}(B_{1/2^{3k+4}})}\right), \end{equation}
Indeed, by the fact that~$|\gamma-\lambda|\leqslant k$ we see that \begin{equation}\label{gg2} \begin{split}
\|A_2\|_{L^\infty(B_{1/2^{3k+6}})}
& \leqslant C \, C_{k} \,\| \tilde\eta u\|_{C^{k+{2}} (\mathbb R^n)}\\
& \leqslant C \, C_{k} \,\| u\|_{C^{k+{2}} (B_{1/2^{3k+4}})}. \end{split}\end{equation} Furthermore, since $(1-\tilde\eta)u=0$ inside $B_{1/2^{3k+5}}$, we can use \eqref{A3 estimate} with $v:=(1-\tilde\eta)u$ to obtain
$$ \|A_3\|_{L^\infty(B_{1/2^{3k+6}})}\leqslant C \|u\|_{L^\infty(\mathbb R^n)}.$$ This last estimate, \eqref{gg}, and \eqref{gg2} allow us to conclude the validity of~\eqref{R}.
Now, by~\cite[Theorem~61]{CS2011} applied to $\partial^\gamma_x(\tilde\eta u)$ we get $$
\|u\|_{C^{\sigma+k+\alpha}(B_{1/2^{3k+7}})} \leqslant C
\left(1+\|u\|_{C^{k+{2}}(B_{1/2^{3k+4}})}+\|u\|_{L^\infty(\mathbb R^n)}\right),$$ which is the analogous of \eqref{29AA} with $\sigma+\alpha=2+\beta$. Hence, arguing as in the case $k=0$ (see the argument after \eqref{29AA}) we conclude that
$$ \|u\|_{C^{\sigma+k+\alpha}(B_{1/2^{{3(2k+1)+5}}})}\leqslant C\left(1+\|u\|_{L^\infty(\mathbb R^n)}+\|f\|_{L^{\infty}(B_{1}\times\mathbb R)}\right).$$ A covering argument shows the validity of \eqref{induction}, comcluding the proof of Theorem~\ref{boot}.
\section{Proof of Theorem~\ref{main}} \label{section:main}
The idea of the proof is to write the fractional minimal surface equation in a suitable form so that we can apply Theorem~\ref{boot}.
\subsection{Writing the operator on the graph of~$u$}
The first step in our proof consists in writing the $s$-minimal surface functional in terms of the function $u$ which (locally) parameterizes the boundary of a set $E$. More precisely, we assume that $u$ parameterizes $\partial E \cap K_R$ and that (without loss of generality) $E \cap K_R$ is contained in the ipograph of $u$. Moreover, since by assumption $u(0)=0$ and $u$ is of class $C^{1,\alpha}$, up to rotating the system of coordinates (so that $\nabla u(0)=0$) and reducing the size of $R$, we can also assume that \begin{equation} \label{eq:small Lip} \partial E \cap K_R\subset B_R^{n-1}\times [-R/8,R/8]. \end{equation} Let $\varphi\in C^\infty(\mathbb R)$ be an even function satisfying $$
\varphi(t)=\left\{\begin{array}{ll}
1 &\quad\mbox{if } |t|\leqslant 1/4, \\
0 &\quad\mbox{if } |t|\geqslant 1/2,
\end{array}\right. $$ and define the smooth cut-off functions $$
\zeta_R(x'):=\varphi(|x'|/R)\qquad\eta_R(x):= \varphi(|x'|/R)
\varphi(|x_n|/R). $$ Observe that $$ \zeta_R= 1 \quad \text{in }B_{R/4}^{n-1},\qquad \zeta_R= 0\quad \text{outside }B_{R/2}^{n-1}, $$ $$ \eta_R= 1\quad \text{in } K_{R/4},\qquad \eta_R= 0 \quad \text{outside } K_{R/2}. $$ We claim that, for any~$x\in\partial E \cap \left(B_{R/2}^{n-1}\times [-R/8, R/8]\right)$, \begin{equation}\label{LHS-1}\begin{split} & \int_{\mathbb R^{n}}\eta_R(y-x)\frac{\chi_E(y)-\chi_{\R^n\setminus E}
(y)}{|x-y|^{n+s}}\,dy \\ &\qquad=2\int_{\mathbb R^{n-1}}
F\left( \frac{u(x'-w')-u(x')}{|w'|}\right)
\frac{\zeta_R(w')}{|w'|^{n-1+s}}\,dw',\end{split} \end{equation} where $$ F(t):=\int_0^t\frac{d\tau}{(1+\tau^2)^{(n+s)/2}}. $$ Indeed, writing $y=x-w$ we have (observe that $\eta_R$ is even) \begin{eqnarray}\label{JW} \nonumber &&\int_{\mathbb R^n}\eta_R(y-x)\frac{\chi_E(y)-\chi_{\R^n\setminus E}
(y)}{|x-y|^{n+s}}\,dy
\\ &=&\int_{\mathbb R^n}\eta_R(w)\frac{\chi_E(x-w)-\chi_{\R^n\setminus E} (x-w)}{|w|^{n+s}}\,dw \\ &=&\int_{\mathbb R^{n-1}}\zeta_R(w') \biggl[ \int_{-R/4}^{R/4} \frac{\chi_E (x-w)-\chi_{\R^n\setminus E}(x-w)}{\Big(
1+(w_n/|w'|)^2\Big)^{(n+s)/2}}\,dw_n
\biggr]\frac{dw'}{|w'|^{n+s}}, \nonumber\end{eqnarray} where the last equality follows from the fact that $\varphi(|w_n|/R)= 1$ for
$|w_n| \leqslant R/4$, and that by \eqref{eq:small Lip} and by symmetry the contributions of $\chi_E (x-w)$ and $\chi_{\R^n\setminus E}(x-w)$ outside $\{|w_n|\leqslant R/4\}$ cancel each other.
We now compute the inner integral: using the change variable~$t:=w_n/|w'|$ we have \begin{eqnarray*} &&\int_{-R/4}^{R/4}\frac{\chi_{E} (x-w)}{\Big(
1+(w_n/|w'|)^2\Big)^{(n+s)/2}}\,dw_n \\ &=& \int_{u(x')-u(x'-w')}^{R/4}\frac{1}{\Big(
1+(w_n/|w'|)^2\Big)^{(n+s)/2}}\,dw_n
\\ &=& |w'| \int_{(u(x')-u(x'-w'))/|w'|}^{R/(4|w'|)}\frac{1}{\big(
1+t^2\big)^{(n+s)/2}}\,dt\\ &=& |w'| \biggl[ F\left(
\frac{R}{4|w|'}\right)-F\left(\frac{u(x')-u(x'-w')}{|w|'}\right)\biggr] .\end{eqnarray*} In the same way, \begin{eqnarray*} && \int_{-R/4}^{R/4}\frac{\chi_{\R^n\setminus E} (x-w)}{\Big(
1+(w_n/|w'|)^2\Big)^{(n+s)/2}}\,dw_n \\ &&\qquad=|w'| \biggl[
F\left(\frac{u(x')-u(x'-w')}{|w|'}\right) -F\left(-
\frac{R}{4|w'|}\right)\biggr] .\end{eqnarray*} Therefore, since~$F$ is odd, we immediately get that \begin{eqnarray*} \int_{-R/4}^{R/4}\frac{\chi_E (x-w)-\chi_{\R^n\setminus E}(x-w)}{\Big(
1+(w_n/|w'|)^2\Big)^{(n+s)/2}}\,dw_n = 2|w'| F\left(
\frac{u(x'-w')-u(x')}{|w'|}\right),\end{eqnarray*} which together with~\eqref{JW} proves~\eqref{LHS-1}.
Let us point out that to justify these computations in a pointwise fashion one would need $u \in C^{1,1}(x)$ (in the sense of \cite[Definition 3.1]{BCF2}). However, by using the viscosity definition it is immediate to check that \eqref{LHS-1} holds in the viscosity sense (since one only needs to verify it at points where the graph of $u$ can be touched with paraboloids).
\subsection{The right hand side of the equation}
Let us define the function \begin{equation}\label{smooth0} \Psi_R(x):=
\int_{\mathbb R^{n}}\left[1-\eta_R(y-x)\right]\frac{\chi_E(y)-\chi_{\R^n\setminus E} (y)}{|x-y|^{n+s}}\,dy. \end{equation} Since $1-\eta_R(y-x)$ vanishes in a neighborhood of $\{x=y\}$, it is immediate to check that the function $ \psi_R(z):=
\displaystyle\frac{1-\eta_R(z)}{|z|^{n+s}}$ is of class $C^\infty$, with $$
|\partial^\alpha \psi_R(z)| \leqslant
\frac{C_{|\alpha|}}{1+|z|^{n+s}}\qquad \forall\,\alpha\in \mathbb N^n. $$
Hence, since $1/(1+|z|^{n+s}) \in L^1(\mathbb R^n)$ we deduce that \begin{equation}\label{R7} {\mbox{$\Psi_R \in C^\infty(\mathbb R^n)$, with all its derivatives uniformly bounded.}} \end{equation}
\subsection{An equation for~$u$ and conclusion}
By~\cite[Theorem 5.1]{CRS} we have that the equation
$$ \int_{\mathbb R^n} \frac{\chi_E(y)-\chi_{\R^n\setminus E}(y)}{|x-y|^{n+s}}\,dy=0$$ holds in viscosity sense for any~$x\in (\partial E)\cap K_R$. Consequently, by~\eqref{LHS-1} and~\eqref{smooth0} we deduce that $u$ is a viscosity solution of $$
\int_{\mathbb R^{n-1}} F\left( \frac{u(x'-w')-u(x')}{|w'|}\right)
\frac{\zeta_R(w')}{|w'|^{n-1+s}}\,dw'=-\frac{\Psi_R(x',u(x'))}{2} $$
inside $B_{R/2}^{n-1}$. Since $F$ is odd, we can add the term $F\left(-\nabla u(x')\cdot \frac{w'}{|w'|}\right)$ inside the integral in the left hand side (since it integrates to zero), so the equation actually becomes \begin{multline}\label{eq1}
\int_{\mathbb R^{n-1}} \biggl[F\left( \frac{u(x'-w')-u(x')}{|w'|}\right)-F\left(-\nabla u(x')\cdot \frac{w'}{|w'|}\right)\biggr]
\frac{\zeta_R(w')}{|w'|^{n-1+s}}\,dw'\\ =-\frac{\Psi_R(x',u(x'))}{2}. \end{multline} We would like to apply the regularity result from Theorem~\ref{boot2}, exploiting~\eqref{R7} to bound the right hand side of~\eqref{eq1}. To this aim, using the Fundamental Theorem of Calculus, we rewrite the left hand side in \eqref{eq1} as \begin{equation} \label{eq:rewrite lhs} \int_{\mathbb{R}^{n-1}}{\bigl(u(x'-w')-u(x')+\nabla u(x')\cdot w'\bigr)
\frac{a(x',-w')\zeta_R(w')}{|w'|^{n+s}}}\,dw', \end{equation} where $$
a(x',-w'):=\int_{0}^{1}\biggl(1+\left(t\frac{u(x'-w')-u(x')}{|w'|}-(1-t)\nabla u(x')\cdot \frac{w'}{|w'|}\right)^{2}\biggr)^{-({n+s})/{2}}\,dt. $$ Now, we claim that \begin{equation}\label{eq2} \begin{split} \int_{\mathbb{R}^{n-1}}\delta u(x',w')\, K_R(x',w')\,dw' =-\Psi_R(x',u(x')) +A_R(x'), \end{split} \end{equation} where $$ K_R(x',w'):=
\frac{[a(x',w')+a(x',-w')]\zeta_R(w')}{2|w'|^{(n-1)+(1+s)}} $$ and $$
A_R(x'):=\int_{\mathbb{R}^{n-1}}[u(x'-w')-u(x')+\nabla u(x')\cdot w'] \frac{[a(x',w')-a(x',-w')]\zeta_R(w')}{|w'|^{n+s}}\,dw' $$ To prove~\eqref{eq2} we introduce a short-hand notation: we define $$
u^{\pm}(x',w'):=u(x'\pm w')-u(x')\mp \nabla u(x')\cdot w',\qquad a^\pm(x',w'):= a(x', \pm w')\frac{\zeta_R (w')}{|w'|^{n+s}},$$ while the integration over~${\mathbb{R}^{n-1}}$, possibly in the principal value sense, will be denoted by~$I[\cdot]$. With this notation, and recalling~\eqref{eq:rewrite lhs}, it follows that \eqref{eq1} can be written \begin{equation}\label{eq:rewrite lhs.2} -\frac{\Psi_R}2= I[u^- a^-].\end{equation} By changing $w'$ with $-w'$ in the integral given by~$I$, we see that $$ I[u^+ a^+]=I[u^- a^-],$$ consequently~\eqref{eq:rewrite lhs.2} can be rewritten as \begin{equation} \label{eq:rewrite lhs2} -\frac{\Psi_R}2= I[u^+ a^+]. \end{equation} Notice also that \begin{equation}\label{extra} u^+ + u^- = \delta u, \qquad I[u^+(a^+-a^-)]=I[u^-(a^--a^+)].\end{equation} {Hence,} adding~\eqref{eq:rewrite lhs.2} and~\eqref{eq:rewrite lhs2}, and {using \eqref{extra}}, we obtain { \begin{eqnarray*} -\Psi_R &=& I[u^+ a^+]+I[u^- a^-] \\ &=& \frac{1}{2}I[(u^+ + u^-) (a^++a^-)]+\frac{1}{2}I[(u^+ - u^-) (a^+-a^-)] \\ &=& \frac{1}{2}I[\delta u\, (a^++a^-)]+\frac{1}{2}I[(u^+ - u^-) (a^+-a^-)] \\ &=& \frac{1}{2}I[\delta u\, (a^++a^-)]-I[u^- (a^+-a^-)], \end{eqnarray*} } which proves~\eqref{eq2}.
Now, to conclude the proof of Theorem~\ref{main} it suffices to apply Theorem~\ref{boot2} iteratively: more precisely, let us start by assuming that $u \in C^{1,\beta}(B_{2r}^{n-1})$ for some $r\leqslant R/2$ and any $\beta<s$. Then, by the discussion above we get that $u$ solves $$ \int_{\mathbb{R}^{n-1}}\delta u(x',w')\, K_{r}(x',w')\,dw' =-\Psi_r(x',u(x')) +A_r(x) \qquad \text{in }B_r^{n-1}. $$ Moreover, one can easily check that the regularity of $u$ implies that the assumptions of Theorem \ref{boot2} {with $k=0$} are satisfied with
$\sigma:=1+s$ and $a_0:= 1/(1-s)$. (Observe that \eqref{new condition} holds since~$\|u\|_{C^{1,\beta}(B^{n-1}_{2r})}$.)
Furthermore, it is not difficult to check that, for $|w'|\leqslant 1$, $$
\left|[u(x'-w')-u(x')+\nabla u(x')\cdot w']\,[a(x',w')-a(x',-w')]\right| \leqslant C|w'|^{2\beta+1}, $$ which implies that the integral defining $A_r$ is convergent by choosing $\beta>s/2$. Furthermore, a tedious computation (which we postpone to Subsection \ref{EEE} below) shows that \begin{equation}\label{END} A_r \in C^{2\beta-s}(B_{r}^{n-1}).\end{equation}
Hence, by Theorem \ref{boot2} with $k=0$ we deduce that $u \in C^{1,2\beta}(B^{n-1}_{r/2})$. But then this implies that $A_{r}\in C^{4\beta-s}(B_{r/4}^{n-1})$ and so by Theorem \ref{boot2} again $u \in C^{1,4\beta}(B^{n-1}_{r/8})$ for all $\beta <s$. Iterating this argument infinitely many times\footnote{Note that, once we know that $\|u\|_{C^{k,\beta}(B^{n-1}_{2r})}$ is bounded for some $k \geqslant 2$ and $\beta \in (0,1]$, for any $|\gamma|\leqslant k-1$ we get $$ \partial_x^\gamma A_r(x)=\int_{\mathbb{R}^{n-1}} \partial_x^\gamma\bigl([u(x'-w')-u(x')+\nabla u(x')\cdot w']\,[a(x',w')-a(x',-w')]\bigr)
\frac{\zeta_r(w')}{|w'|^{n+s}}\,dw', $$ and exactly as in the case $k=0$ one shows that $$
\left|\partial_x^\gamma\bigl([u(x'-w')-u(x')+\nabla u(x')\cdot w']\,[a(x',w')-a(x',-w')]\bigr)\right| \leqslant C|w'|^{2\beta+1}\qquad \forall\,|w'|\leqslant 1, $$ and that $A_r \in C^{k-1,2\beta-s}(B_{r}^{n-1})$.} we get that $u \in C^{m}(B^{n-1}_{\lambda^mr})$ for some $\lambda>0$ small, for any $m\in\mathbb{N}$. Then, by a simple covering argument we obtain that $u \in C^{m}(B^{n-1}_{\rho})$ for any $\rho<R$ and $m\in\mathbb{N}$, that is, $u$ is of class $C^{\infty}$ inside $B_{\rho}$ for any $\rho<R$. This completes the proof of Theorem \ref{main}.
\subsection{H\"older regularity of $A_{R}$.} \label{EEE} We now prove \eqref{END}, i.e., if $u\in C^{1,\beta}(B^{n-1}_{2r})$ then $A_r\in C^{2\beta-s}(B^{n-1}_{r})$ ($r\leqslant R/2$). For this we introduce the following notation:
$$U(x',w'):=u(x'-w')-u(x')+\nabla u(x')\cdot w'$$ and $$p(\tau):=\frac{1}{ (1+\tau^2)^{\frac{n+s}{2}} }.$$ In this way we can write \begin{equation}\label{57a}
a(x',\textcolor{red}{-}w')=\int_{0}^{1}{p\biggl(t\frac{u(x'-w')-u(x')}{|w'|}
-(1-t)\nabla u(x')\cdot \frac{w'}{|w'|}\biggr)\,dt}. \end{equation} Let us define $$\mathscr{A}(x',w'):=a(x',w')-a(x',-w').$$ Then we have
$$A_r(x')=\int_{\mathbb R^{n-1}}{U(x',w')\frac{\mathscr{A}(x',w')}{|w'|^{n+s}}\zeta_r(w')\,dw'}.$$ To prove the desired H\"older condition of the function $A_r(x')$, we first note that $$U(x',w')=\int_0^1\bigl[\nabla u(x')-\nabla u(x'-tw')\bigr] dt\,\cdot w'.$$ Since $u\in C^{1,\beta}(B_{R}^{n-1})$ and $2r\leqslant R$, we get \begin{equation}\label{fin1}
|U(x',w')-U(y',w')|\leqslant C\,
\min\{|x'-y'|^{\beta}|w'|,|w'|^{\beta+1}\},\quad\mbox{for $y'\in B_{r}^{n-1}$} \end{equation} and \begin{equation}\label{fin2}
|U(x',w')|\leqslant C|w'|^{\beta+1}. \end{equation} Therefore, from \eqref{fin1} and \eqref{fin2} it follows that, for any $y'\in B_{r}^{n-1}$, \begin{eqnarray}
|A_{r}(x')-A_{r}(y')|&=&\bigg|\int_{\mathbb R^{n-1}}{\big(U(x',w')
\mathscr{A}(x',w')-U(y',w')\mathscr{A}(y',w')\big)\frac{\zeta_r(w')}{|w'|^{n+s}}\,dw'}\bigg|\nonumber\\
&\leqslant&C\int_{\mathbb R^{n-1}}{\min\{|x'-y'|^{\beta}|w'|,|w'|^{\beta+1}\}
\frac{|\mathscr{A}(x',w')|}{|w'|^{n+s}}\zeta_r(w')\,dw'}\nonumber\\
&+&C\int_{\mathbb R^{n-1}}{|w'|^{\beta+1}
\frac{|\mathscr{A}(x',w')-\mathscr{A}(y',w')|}{|w'|^{n+s}}\zeta_r(w')\,dw'}\nonumber\\ &=:&I_1(x',y')+I_2(x',y')\label{fin3}. \end{eqnarray} To estimate the last two integrals we define $$\mathscr{A_*}(x',w'):=
a(x',w')-p\biggl(\nabla u(x')\cdot\frac{w'}{|w'|}\biggr).$$ With this notation \begin{equation}\label{fin4}
\mathscr{A}(x',w')=\mathscr{A_*}(x',w')-\mathscr{A_*}(x',-w'). \end{equation} By \eqref{57a}
and \eqref{fin2}, since $|p'(t)| \leqslant C$ and $p$ is even, it follows that \begin{eqnarray}\nonumber
&&|\mathscr{A_*}(x',-w')|\\
&\leqslant&\int_0^1\int_0^1 \biggl|\frac{d}{d\lambda} p\biggl(\lambda t\frac{u(x'-w')-u(x')}{|w'|}
-[\lambda (1-t)+(1-\lambda)]\nabla u(x')\cdot\frac{ w'}{|w'|}\biggr)\biggr|\,d\lambda\,dt\nonumber\\
&\leqslant &\int_0^1{t\frac{|U(x',w')|}{|w'|}
\biggl(\int_0^1{\bigg|p'\bigg(\lambda t\frac{U(x',w')}{|w'|}-\nabla u(x')\cdot\frac{w'}{|w'|}\bigg)\bigg|\,d\lambda}\biggr)\,dt}\nonumber\\
&\leqslant& C|w'|^{\beta}\label{fin5} \end{eqnarray}
for all $|w'|\leqslant r$.
Estimating $\mathscr{A_*}(x',w')$ in the same way, by \eqref{fin4} and \eqref{fin5}, we get, for any $\beta>s/2$, \begin{eqnarray}
I_1(x',y')&\leqslant&C\int_{\mathbb R^{n-1}}{\min\{|x'-y'|^{\beta}|w'|,|w'|^{\beta+1}\}|w'|^{\beta-n-s}\zeta_r(w')\,dw'}\nonumber\\
&\leqslant&C|x'-y'|^{\beta}\int_{|x'-y'|}^{r}{t^{\beta-s-1}dt}+\int_{0}^{|x'-y'|}{t^{2\beta-s-1}\,dt}\nonumber\\
&\leqslant&C|x'-y'|^{2\beta-s}.\label{fin6} \end{eqnarray} On the other hand, to estimate $I_2$ we note that \begin{eqnarray}
|\mathscr{A}(x',w')-\mathscr{A}(y',w')|&\leqslant&|\mathscr{A_*}(x',w')-\mathscr{A_*}(y',w')|\nonumber\\
&+&|\mathscr{A_*}(y',-w')-\mathscr{A_*}(x',-w')|.\label{fin7} \end{eqnarray} Hence, arguing as in \eqref{fin5} we have \begin{eqnarray}
&&|\mathscr{A_*}(x',-w')-\mathscr{A_*}(y',-w')|\nonumber\\ &\leqslant &
\int_0^1t\frac{|U(x',w')|}{|w'|}\int_0^1\bigg|p'\bigg(\lambda t\frac{U(x',w')}{|w'|}-\nabla u(x')\cdot\frac{w'}{|w'|}\bigg)\nonumber\\
&&\qquad\qquad\qquad\qquad\qquad\qquad -p'\bigg(\lambda t\frac{U(y',w')}{|w'|}
-\nabla u(y')\cdot\frac{w'}{|w'|}\bigg)\bigg|\,d\lambda\,dt\nonumber\\
&+&\int_0^1{t\frac{|U(x',w')-U(y',w')|}{|w'|}\int_0^1
{\bigg|p'\bigg(\lambda t\frac{U(y',w')}{|w'|}-\nabla u(y')\cdot\frac{w'}{|w'|}\bigg)\bigg|\,d\lambda}\,dt}\nonumber\\ &=:&I_{2,1}(x',y')+I_{2,2}(x',y').\label{fin8} \end{eqnarray}
We bound each of these integrals separately. First, since $|p'(t)|\leqslant C$, it follows immediately from \eqref{fin1} that \begin{eqnarray}
I_{2,2}(x',y')\leqslant C\min\{|x'-y'|^{\beta},|w'|^{\beta}\}.\label{fin9} \end{eqnarray} On the other hand, by \eqref{fin2}, \eqref{fin1}, and the fact that $u\in C^{1,\beta}(B_{R}^{n-1})$ and $p'$ is uniformly Lipschitz, we get \begin{eqnarray}
I_{2,1}(x',y')&\leqslant& C|w'|^{\beta}\bigg(\frac{|U(x',w')-U(y',w')|}{|w'|}+|\nabla u(x')-\nabla u(y')|\bigg)\nonumber\\
&\leqslant&C|w'|^{\beta}\Big(
\min\{|x'-y'|^{\beta},|w'|^{\beta}\}+|x'-y'|^{\beta}\Big)\nonumber\\
&\leqslant&C|w'|^{\beta}|x'-y'|^{\beta}.\label{fin10} \end{eqnarray}
Then, assuming without loss of generality $r \leqslant 1$ (so that also $|x'-y'|\leqslant 1$),
by \eqref{fin8}, \eqref{fin9}, and \eqref{fin10} it follows that \begin{eqnarray}\nonumber
|\mathscr{A_*}(x',-w')-\mathscr{A_*}(y',-w')|&\leqslant&
C\bigg(\min\{|x'-y'|^{\beta},|w'|^{\beta}\}+|w'|^{\beta}|x'-y'|^{\beta}\bigg) \\\label{fin11}
&\leqslant& C\min\{|x'-y'|^{\beta},|w'|^{\beta}\}. \end{eqnarray}
As $|\mathscr{A_*}(y',w')-\mathscr{A_*}(x',w')|$ is bounded in the same way, by \eqref{fin7}, we have
$$|\mathscr{A}(x',w')-\mathscr{A}(y',w')|\leqslant C\min\{|x'-y'|^{\beta},|w'|^{\beta}\}.$$ By arguing as in \eqref{fin6}, we get that, for any $s/2<\beta<s$, \begin{eqnarray} I_{2}(x',y')
&\leqslant &C\int_{\mathbb R^{n-1}}{|w'|^{\beta+1}\frac{\min\{|x'-y'|^{\beta},|w'|^{\beta}\}}{|w'|^{n+s}}\zeta_r(w')dw'}\nonumber\\
&\leqslant& C|x'-y'|^{2\beta-s}.\label{fin12} \end{eqnarray} Finally, by \eqref{fin3}, \eqref{fin6} and \eqref{fin12}, we conclude that
$$|A_{r}(x')-A_{r}(y')|\leqslant C|x'-y'|^{2\beta-s},\quad y'\in B_{r}^{n-1},$$ as desired.
\end{document} |
\begin{document}
\title[QC pseudohermitian normal coordinates]{Quaternionic contact pseudohermitian Normal coordinates} \author{Christopher S. Kunkel}
\begin{abstract} This paper constructs a family of coordinate systems about a point on a quaternionic contact manifold, called quaternionic contact pseudohermitian normal coordinates. Once defined, conformal variations of the quaternionic contact structure induce changes on the coordinates which are studied in an effort to simplify the torsion and curvature at the center point. These normalizations are constructed in the hope that they may help prove the quaternionic contact version of the Yamabe problem. \end{abstract}
\maketitle
\section{Introduction} \label{sec:intro}
Quaternionic contact manifolds, first defined in \cite{Biquard:1999} and \cite{Biquard:2000}, appear naturally as the boundaries at infinity of asymptotically hyperbolic quaternionic manifolds. In this way they generalize to the quaternion algebra the sequence of families of geometric structures that are the boundaries at infinity of real and complex asymptotically hyperbolic spaces. In the real case, these manifolds are simply conformal manifolds, that is, manifolds with a family of Riemannian metrics that are all conformally related by smooth, positive functions. In the complex case, the boundary structure is that of a CR manifold, a manifold with a contact structure and for each choice of contact form, a metric on the contact distribution, satisfying certain additional conditions related to those defining a complex manifold.
A quaternionic contact manifold is similar, having a type of contact structure, defined, not by a single non-vanishing $1$-form as in the CR case, but rather by an $\mathbb{R}^3$-valued $1$-form. On the kernel of this form, there is a metric which, when paired with the exterior derivatives of the three components of the contact form, defines a triple of almost complex structures satisfying the commutation relations of the unit quaternions $i$, $j$ and $k$. Like the CR case, there is not a fixed contact form, but rather a conformal family of them, which of course gives rise to distinct but conformally related metrics on the contact distribution. The almost complex structures defined by this varying family determine a three-dimensional subbundle of the endomorphism bundle of the contact distribution which may be compared to the three dimensional space of imaginary quaternions.
A natural question coming from this conformal freedom is the quaternionic contact Yamabe problem, named for its obvious similarity to the original Yamabe problem which asked if, given any metric on a compact manifold, there is a conformal metric of constant scalar curvature. Many mathematicians have contributed to this problem since it was first posed in 1960; see \cite{LeeParker:1987} for a detailed overview and references.
Because CR manifolds have a similar conformal structure, and it is possible to define a well-adapted linear connection on such manifolds, there is a natural generalization of the Yamabe problem to the CR Yamabe problem. Though more complicated than its conformal counterpart, the CR Yamabe problem has been solved using several of the techniques used in the proof of the conformal case.
In the same way, this paper provides a key step toward the solution of the quaternionic contact Yamabe problem, namely the construction of a coordinate system in which the invariants of the quaternionic contact structure are considerably simplified.
The primary tool for this is a generalized version of Riemannian normal coordinates, better adapted to the study of quaternionic contact manifolds. The main ingredient in the construction of these coordinates is the fact that the tangent space of a quaternionic manifold has a natural \emph{parabolic} dilation, instead of the more common linear dilation seen in Riemannian geometry. This suggests that the curves of interest used to define an exponential map from the tangent space at a point to the base manifold should incorporate this parabolic structure. This gives rise to the following theorem. \begin{thmnn}[pg.~\pageref{thm:coordinates}] Let $(M,\nabla)$ be a manifold with connection whose tangent bundle decomposes as the direct sum of two distributions, $H$ and $V$. Choose any $q\in M$, and let $(X,Y)\in H_q \oplus V_q = T_q M$ be any tangent vector. For any curve $\gamma$ on $M$ and any vector field $X$ along $\gamma$, let $D_t X$ denote the covariant derivative of $X$ along $\gamma$. Define $\gamma_{(X,Y)}$ to be the curve satisfying \begin{equation*} D_t^2 {\dot\gamma}_{(X,Y)} = 0,\quad \gamma_{(X,Y)}(0)=q, \quad {\dot\gamma}_{(X,Y)}(0)=X, \text{ and } D_t {\dot\gamma}_{(X,Y)}(0) = Y. \end{equation*} Then there are neighborhoods $0\in \mathcal{O}\subset T_qM$ and $q\in \mathcal{O}_M \subset M$ so that the function $\Psi : \mathcal{O} \to \mathcal{O}_M : (X,Y)\mapsto \gamma_{(X,Y)}(1)$ is a diffeomorphism, and satisfies the parabolic scaling $\Psi(tX,t^2Y)=\gamma_{(X,Y)}(t)$ wherever either side is defined. \end{thmnn}
This theorem represents a generalization of the work in \cite{JerisonLee:1989}, from which this paper takes its inspiration. There the authors construct a parabolic coordinate system on a CR manifold for the same purpose we have here. Their result requires several other, very specific hypotheses and works only in the case of a CR manifold. This new proof requires nothing more than a linear connection and decomposition of the tangent bundle by complementary distributions.
Using this theorem and a special frame at the center point $q$, I am able to construct a set of parabolic normal coordinates, so named for their similarity to the normal coordinates of Riemannian geometry with a parabolic, rather than linear scaling. In particular we have the following theorem. \begin{thmnn}[pg.~\pageref{thm:paraboliccoordinates}] Let $M$ be a QC manifold with fixed pseudohermitian structure, and let $q\in M$ be any point. Then there exist parabolic normal coordinates $(x^\alpha,t^i)$ about $q$ so that at $q$ the metric is the standard Euclidean metric, the contact forms are the standard forms on the quaternionic Heisenberg group and the connection $1$-forms vanish. Any two such coordinate systems centered at $q$ are related by a unique linear transformation in $Sp(n)Sp(1)$. \end{thmnn}
Once these parabolic normal coordinates are defined, I explore the effect of a conformal change of contact structure on the curvature tensor of the Biquard connection, the standard linear connection of a quaternionic contact manifold. Using the parabolic normal coordinates, I am able to define a function that, when used as the conformal factor, causes the symmetrized covariant derivatives of a certain tensor constructed from the curvature and torsion to vanish, as in the following theorem. \begin{mthm}[pg.~\pageref{thm:symder_vanish}] Let $M$ be a QC manifold. For any $q\in M$ and any $N\geq 2$, there is a choice of pseudohermitian structure such that all the symmetrized covariant derivatives of $Q$ with total order less than or equal to $N$ vanish at $q$. If we express the chosen pseudohermitian structure as a conformal multiple of another pseudohermitian structure, the $1$-jet of the conformal factor at $q$ may be freely chosen. Once this is fixed, the Taylor series of the conformal factor at $q$ is uniquely determined. \end{mthm}
This idea originates in the work of Robin Graham, described in \cite{LeeParker:1987}. Graham developed coordinates in the conformal case which showed that, at a point, the symmetrized covariant derivatives of the Ricci tensor can be made to vanish to high order by carefully choosing the conformal factor for a conformal manifold. In their 1989 paper \cite{JerisonLee:1989} Jerison and Lee showed in a similar fashion that the symmetrized covariant derivatives of a tensor constructed from the CR curvature and torsion could be made to vanish at a point.
The structure of the tensor $Q$ is such that many of the curvature and torsion terms of the Biquard connection vanish at the center of normal coordinates when the symmetrized derivatives of $Q$ vanish to order four (see Theorem \ref{thm:mainthm}). In fact, using invariance theory, I show that the only remaining term of weight less than or equal to $4$ that does not necessarily vanish at the center point is the squared norm of the quaternionic contact version of the Weyl tensor.
Section \ref{sec:background} reviews the relevant background of QC manifolds, including a number of important curvature and torsion identities. Section \ref{sec:coords} defines QC pseudohermitian normal coordinates and the curvature-torsion tensor that can be made to vanish. Finally, section \ref{sec:scal_invariants} shows how to use the normalization of section \ref{sec:coords} to simplify the curvature tensor at the origin of the QC normal coordinates. Throughout the paper many of the results are proved in a similar fashion to the analogous statements in \cite{JerisonLee:1989}, and so the proofs of those results are omitted here. \section{Background} \label{sec:background}
\subsection{Quaternionic contact manifolds}\label{sec:qc}
We begin by defining the focus of this paper, quaternionic contact manifolds. These manifolds were first described in \cite{Biquard:2000}; however, in this paper we will use the definition provided in \cite{Vassilevetal:2006}, since many useful identities come from that paper. In some sense, quaternionic contact manifolds are strictly pseudoconvex CR manifolds working with the quaternions instead of the complex numbers. In fact the similarities are so strong that many of the proofs in section \ref{sec:coords} are almost identical to the corresponding proofs in \cite{JerisonLee:1989}. Formally, we have the following definition. \begin{defn} A \emph{quaternionic contact manifold}, or QC manifold, is a $(4n+3)$-dimensional manifold $M$ with a $4n$-dimensional distribution, $H$, that satisfies the following properties: \begin{itemize} \item $H$ is the kernel of an $\mathbb{R}^3$-valued 1-form $\eta = (\eta^1, \eta^2, \eta^3)$; \item on $H$, there are almost complex structures $I^i$, $i=1,2,3$ that satisfy the commutation relations of the unit quaternions, i.e. \[(I^i)^2=I^1I^2I^3=-Id;\] \item there is a sub-Riemannian metric $g$ on $H$ that satisfies \[2g(I^iX,Y)=d\eta^i(X,Y)\] for all vectors $X,Y$ in $H$ and each $i=1,2,3$. \end{itemize} \end{defn}
On a QC manifold, a choice of contact forms, metric and almost complex structures is called a \emph{QC pseudohermitian structure}. Such a choice is not unique. First note that there is a conformal freedom, given by multiplying $g$ and the $\eta^i$ by the same smooth, positive function and leaving the almost complex structures unchanged. On the other hand, within a given conformal class we have the following lemma from \cite{Vassilevetal:2006}. Given the almost complex structures in the definition above, we let $Q=Span\{I^i\}_{i=1,2,3}$. \begin{lem} \begin{enumerate} \item If $(\eta,I^i, g)$ and $(\eta,\hat{I}^i, \hat{g})$ are two QC structures on $M$, then $I^i = \hat{I}^i$ for $i=1,2,3$ and $g=\hat g$. \item If $(\eta, Q, g)$ and $(\hat\eta, \hat Q, g)$ are two QC structures on $M$ with the same horizontal distribution, then $Q=\hat Q$ and $\hat\eta = \Psi \eta$ for some smooth $SO(3)$-valued function $\Psi$. \end{enumerate} \end{lem}
The presence of the three almost complex structures and their relation to the metric $g$ on $H$ provides an action of $Sp(n)Sp(1)$ on the bundle $H$. An $Sp(n)Sp(1)$ frame for $H$ is an orthonormal frame $\xi_\alpha$, $\alpha=1,\ldots, 4n$ with $\xi_{4k+i+1}=I^i \xi_{4k+1}$ for $k=0,\ldots,n-1$ and $i=1,2,3$.
\subsubsection{The Biquard connection}\label{sec:biquardconn}
In his 2000 paper, Biquard defines a connection well suited to the study of QC manifolds, similar in spirit to the Tanaka-Webster connection on a CR manifold. Let $Q\subset \End{H}$ be the span of the three almost complex structures. \begin{thm}[\cite{Biquard:2000}] \label{thm:biquardconn} Let $(M^{4n+3},\eta,g,Q)$ be a manifold with QC pseudohermitian structure, with $n>1$. Then there exists a unique linear connection $\nabla$ with torsion $T$ and a unique distribution $V\subset TM$, complementary to $H$, such that: \begin{itemize} \item $\nabla$ preserves the decomposition $TM=H\oplus V$ and the metric $g$ on $H$; \item given $X,Y\in H$, $T(X,Y)\in V$; \item $\nabla$ preserves the $Sp(n)Sp(1)$ structure on $H$ (i.e. $Q$ is preserved);
\item given $R\in V$, $T(R,\cdot)|_H$ is an endomorphism of $H$ in $(\mathfrak{sp}(n)\oplus\mathfrak{sp}(1))^\perp\subset \mathfrak{gl}(4n)$; \item there is natural isomorphism $\varphi:V\to Q$ and $\nabla \varphi=0$. \end{itemize} \end{thm}
The distribution $V$ may be described explicitly as the span of the three \emph{Reeb fields}, $R_1,\, R_2,\, R_3$, given by
\[ \eta^i(R_j) = \delta^i_j, \quad R_i\lrcorner d\eta^i|_H = 0,\]
where no summation is implied in the second equation. In fact, more is true. Letting $\tensor{\omega}{\down i \up j} = R_i\lrcorner d\eta^j|_H$, we have $\tensor{\omega}{\down i \up j} = -\tensor{\omega}{\down j \up i}$, and these are the restriction to $H$ of the connection 1-forms for $\nabla$ on $V$. The isomorphism $\varphi$ is then simply $\varphi(R_i) = I_i$, and so these are also connection 1-forms on $Q$.
We can readily extend the metric $g$ to all of $TM$ by requiring that the Reeb fields be orthonormal, and $V \perp H$. This metric depends only on the choice of $g$ and $\eta^i$, and in fact is given by $g\oplus \sum(\eta^i)^2$. Further, from Theorem \ref{thm:biquardconn}, it is apparent that the extended metric is also parallel with respect to the connection.
Because we will be working with a carefully chosen frame in the later sections of the paper, it is now worthwhile to introduce some related notation. For the remainder of this section we will work with a frame $\{ \xi_\alpha,\ R_i\}_{\alpha=1,\ldots,4n; i=1,2,3}$ where $\{\xi_\alpha\}$ is an $Sp(n)Sp(1)$ frame for $H$, and the $R_i$ are the three Reeb fields described above. It is occasionally convenient to have a notation for the entire frame; therefore as necessary we may refer to $R_i$ as $\xi_{4n+i}$. In order to have a consistent index notation we will use different letters for different ranges of indices. In particular \begin{gather*} \alpha, \beta,\gamma,\ldots \in \{ 1, \ldots, 4n\},\\ i,j,k,\ldots \in \{1,2,3\}, \text{ and} \\ a,b,c, \ldots \in \{1,\ldots,4n+3\}. \end{gather*} For the dual basis we use the names $\theta^\alpha$ and $\eta^i$, and as above we may occasionally use $\theta^{4n+i}=\eta^i$. Further, throughout this paper we will obey the Einstein summation convention whenever possible.
Finally we note one last fact of importance, namely that both $H$ and $V$ are orientable. The horizontal bundle is orientable since it admits an $Sp(n)Sp(1)\subset SO(4n)$ structure, and as noted above $Q$ has an $SO(3)$ structure, hence so does $V$. Moreover, the natural volume form on $V$ is given by $\varepsilon = \eta^1 \wedge \eta^2 \wedge \eta^3$, and this tensor provides a handy isomorphism between $V$ and $\bigwedge^2 V$ (or their duals). Though we will not have much occasion to use them, we denote the volume form on $H$ by $\Omega$ and the volume form on $TM$ as $dv = \Omega \wedge \varepsilon$.
Using the volume form $\varepsilon^{ijk}$ and the metrics on $H$ and $V$, there is a convenient way to express composition of the almost complex structures. We will also require a few identities involving contractions of the volume form with itself, and so we list them here. \begin{prop}\label{prop:ACSandVform} The almost complex structures and volume form on $V$ satisfy the following identities: \begin{gather} \tensor{I}{\down i \up \alpha \down \gamma} \tensor{I}{\down j \up \gamma \down \beta} = -g_{ij}\delta^\alpha_\beta + \varepsilon_{ijk}\tensor{I}{\up{k \alpha} \down \beta}; \label{eq:ACS}\\ \varepsilon_{ijk}\varepsilon^{ilm} = \delta_j^l \delta_k^m - \delta_k^l \delta_j^m,\quad \varepsilon_{ijk}\varepsilon^{ijl} = 2\delta_k^l. \label{eq:Vform} \end{gather} \end{prop}
\subsubsection{Curvature and torsion identities} \label{sec:biquardcurv}
Because there are so many relations among the tensors defining a pseudohermitian structure, we can expect that there will be many relations between the curvature and torsion tensors of the Biquard connection. In fact, we will see that there is a very close relation between the quaternionic contact torsion and the horizontal Ricci tensor.
We begin with the torsion. The most basic identity it satisfies is for two vectors in $H$. To wit, since $T(X,Y)\in V$ for $X,Y \in H$, \[ \tensor{T}{\up i \down{\alpha\beta}} = -\eta^i[\xi_\alpha,\xi_\beta] = d\eta^i(\xi_\alpha, \xi_\beta) = 2 g(I^i \xi_\alpha, \xi_\beta) = -2\tensor{I}{\up i \down{\alpha\beta}}, \text{ and } \tensor{T}{\up \alpha \down{\beta\gamma}} = 0.\]
Before we decompose the torsion tensor further, it is handy to introduce the Casimir operator $\Upsilon:\End(H)\to\End(H)$ on a QC manifold, defined in \cite{Vassilevetal:2006}. It is defined locally by \[ \tensor{\Upsilon}{\up {\alpha\gamma} \down {\beta\delta}} = \tensor{I}{\down i \up \alpha \down \beta} \tensor{I}{\up {i\gamma} \down\delta},\] and satisfies the identity $\Upsilon^2 = 2\Upsilon + 3$. It therefore has eigenvalues $3$ and $-1$.
The most interesting part of the torsion is the collection of endomorphisms \[T(R,\cdot)|_H:H\to H\] for a vector $R \in V$. We summarize some results here. \begin{prop}\label{prop:torsionprop} The torsion tensor satisfies \begin{itemize} \item $\tensor{T}{\up i \down{\alpha\beta}} = -2\tensor{I}{\up i \down{\alpha\beta}}$;
\item $\tensor{T}{\up \alpha \down {i\alpha}} = 0= \tensor{T}{\up \alpha \down {i\beta}}\tensor{I}{\up {\beta} \down \alpha} $ for any $I\in Q$;
\item $\tensor{T}{\up \alpha\down {i\beta}} = \tensor{(T^0)}{\down i\up \alpha \down \beta} + \tensor{b}{\down i\up \alpha\down\beta}$ where $T^0$ is symmetric in the horizontal indices and $b$ is antisymmetric in the horizontal indices.
\item $\tensor{b}{\down i \up \alpha \down \beta} = \tensor{I}{\down i \up \alpha \down \gamma}\tensor{\mu}{\up \gamma \down \beta} = \tensor{\mu}{\up \alpha \down \gamma}\tensor{I}{\down i \up \gamma \down \beta}$ where $\mu$ is a symmetric, trace-free endomorphism of $H$ that commutes with any element of $Q$;
\item there is a symmetric, trace-free endomorphism of $H$ called $\tau$ such that \[\tensor{(T^0)}{\down i \up \alpha \down \beta} = \frac{1}{4}(\tensor{\tau}{\up \alpha \down \gamma} \tensor{I}{\down i \up \gamma \down \beta} + \tensor{I}{\down i \up \alpha \down \gamma} \tensor{\tau}{\up \gamma \down \beta});\]
\item $\tensor{\Upsilon}{\down {\alpha\beta}\up{\gamma\delta}} \mu_{\gamma_\delta} = 3\mu_{\alpha\beta}$ and $\tensor{\Upsilon}{\down {\alpha\beta}\up{\gamma\delta}} \tau_{\gamma\delta} = -\tau_{\alpha\beta}$;
\item $\tensor{T}{\up \alpha \down {ij}} = d\theta^\alpha(R_i,R_j)$;
\item $\tensor{T}{\up k \down {ij}} = \lambda\, \tensor{\varepsilon}{\down{ij}\up k}$ for a smooth function $\lambda$; and
\item $\tensor{T}{\up \alpha \down{\beta\gamma}} = \tensor{T}{\up i \down{j\alpha}}=0$. \end{itemize} \end{prop} \begin{proof} These properties are proved in \cite{Biquard:2000} and \cite{Vassilevetal:2006}. \end{proof}
\begin{cor}\label{cor:torsion} The torsion tensor $\tensor{T}{\up \alpha\down {i\beta}}$ satisfies \begin{equation} \tensor{T}{\up \alpha\down {i\beta}} = \frac{1}{4}(\tensor{\tau}{\up \alpha \down \gamma} \tensor{I}{\down i \up \gamma \down \beta} + \tensor{I}{\down i \up \alpha \down \gamma} \tensor{\tau}{\up \gamma \down \beta}) + \tensor{I}{\down i \up \alpha \down \gamma}\tensor{\mu}{\up \gamma \down \beta}.\label{eq:qctorsion} \end{equation} \end{cor}
Throughout this paper we use the index convention $\tensor{R}{\down{abc}\up d} = \theta^d( R(\xi_a,\xi_b)\xi_c)$ for the curvature tensor. The (horizontal) Ricci tensor is $Ric = R_{\alpha\beta} = \tensor{R}{\down{\gamma\alpha\beta}\up\gamma}$ and the scalar curvature is $S=\tensor{R}{\down \alpha\up \alpha}$. We list here some properties of the various curvature tensors.
\begin{prop}\label{prop:curvprop} Let $\lambda$, $\tau$, and $\mu$ be as defined in Proposition \ref{prop:torsionprop}. Then the curvature tensor satisfies \begin{gather} R_{\alpha\beta}= (2n+2)\tau_{\alpha\beta} + 2(2n+5)\mu_{\alpha\beta} + \frac{S}{4n}g_{\alpha \beta}; \label{eq:ricci} \\ S= -8n(n+2)\lambda; \label{eq:scalar}\\ 0= \tensor{\tau}{\down {\alpha\beta,}\up \beta} - 6 \tensor{\mu}{\down {\alpha\beta,}\up \beta} - \frac{4n-1}{2}\varepsilon^{ijk}\tensor{T}{\up \beta \down {jk}}I_{i\beta\alpha} - \frac{3}{16n(n+2)}S_{,\alpha}; \label{eq:taumuS} \\ 0= \tensor{\tau}{\down {\alpha\beta,}\up \beta} - \frac{n+2}{2}\varepsilon^{ijk}\tensor{T}{\up \beta \down {jk}}I_{i\beta\alpha} - \frac{3}{16(n+2)}S_{,\alpha}; \label{eq:tormu}\\ 0 = \tensor{\tau}{\down {\alpha\beta,}\up \beta} - 3\tensor{\mu}{\down {\alpha\beta,}\up \beta} +2\varepsilon^{ijk}\tensor{T}{\up \beta \down {jk}}I_{i\beta\alpha} - \tensor{R}{\down{\gamma i \beta}\up\gamma}\tensor{I}{\up{i\beta}\down \alpha}. \label{eq:VHricci} \end{gather} \end{prop} \begin{proof} These are all proved in \cite{Vassilevetal:2006}. Equations \eqref{eq:ricci} and \eqref{eq:scalar} follow from Theorem 3.12, while equations \eqref{eq:taumuS}, \eqref{eq:tormu} and \eqref{eq:VHricci} follow from Theorem 4.8. \end{proof}
\subsubsection{Conformal changes of QC pseudohermitian structure}\label{sec:confchange}
In the definition of a quaternionic contact manifold is the built-in possibility of a conformal change of structure. By this we mean that there is a natural way to construct a new QC pseudohermitian structure from a given one, namely by multiplying the $\eta^i$ and the metric $g$ by a smooth positive function. In particular, let $u \in C^\infty(M)$ and define $\tilde{\eta}^i = e^{2u}\eta^i$ and $\tilde g = e^{2u}g$. Then $\bigcap_i \ker \tilde{\eta}^i =H$ and \[ d\tilde{\eta}^i = 2e^{2u}du\wedge \eta^i + e^{2u}d\eta^i,\] from which we see that, restricted to $H$, \[d\tilde{\eta}^i(X,Y) = e^{2u}d\eta^i(X,Y) = 2e^{2u}g(I^iX,Y) = 2\tilde{g}(I^iX, Y).\]
From this conformal change, a whole cascade of changes takes place in the connection, torsion and curvature, which we record here for future use. \begin{prop}\label{prop:confchange} Let $M$ be a QC manifold with pseudohermitian structure $\eta$. Let $u\in C^\infty(M)$ and define $\tilde{\eta} = e^{2u}\eta$. Then the Reeb fields for the new structure are given by \begin{equation*} \tilde{R}_i = e^{-2u}(R_i - \tensor{I}{\down i \up \alpha \down \beta} u^\beta \xi_\alpha).\end{equation*} If we define $\tilde{\xi}_\alpha = \xi_\alpha$ and $\tilde{\theta}^\alpha = \theta^\alpha + \tensor{I}{\down i \up \alpha \down \beta}u^\beta \eta^i$, then $\tilde{\theta}^\alpha(\tilde{R}_i)=0$ and $\tilde{\eta}^i(\tilde\xi_\alpha)=0$. Let $P_{-1}$ and $P_3$ denote the projections onto the $(-1)$- and $(3)$-eigenspaces of $\Upsilon$. Then the torsion and curvature tensors change as follows: \begin{gather} \tilde{\tau}_{\alpha\beta} = \tau_{\alpha\beta} + P_{-1}(4u_\alpha u_\beta - 2u_{\alpha\beta}),\label{eq:tauchange}\\ \tilde{\mu}_{\alpha\beta} = \mu_{\alpha\beta} + P_3(-2u_\alpha u_\beta - u_{\alpha\beta}), \label{eq:muchange}\\ \tilde S \tilde{g}_{\alpha\beta} = S g_{\alpha\beta} - 16(n+1)(n+2) u_\gamma u^\gamma g_{\alpha\beta} - 8(n+2)\tensor{u}{\down \gamma \up \gamma}g_{\alpha\beta}.\label{eq:scalarchange} \end{gather} \end{prop} \begin{proof} These are proved in \cite{Vassilevetal:2006}, equations (5.11), (5.12) and (5.14) with the obvious change $h = \frac{1}{2}e^{-2u}$. \end{proof}
\subsubsection{The quaternionic contact conformal curvature tensor} \label{sec:qcconfcurv}
In conformal geometry, the obstruction to conformal flatness is the well-studied Weyl tensor, the portion of the curvature tensor that is invariant under a conformal change of metric. It is this tensor, or rather its vanishing, that determines if a conformal manifold is locally conformally equivalent to the standard sphere.
Likewise, in the CR case, the tensor which determines local CR equivalence to the CR sphere is the Chern tensor, also determined by the curvature of the Tanaka-Webster connection. And just as in the conformal case, it is the key to finding the appropriate bound for the CR Yamabe invariant on a CR manifold.
Something similar appears in the QC case, dubbed the quaternionic contact conformal curvature by Stefan Ivanov and Dimiter Vassilev in \cite{Vassilevetal:2007}. In their paper they define a tensor $W^{qc}$ (which we will hereafter refer to as simply $W$) and prove that it is the conformally invariant portion of the Biquard curvature tensor. Moreover, if it vanishes, they prove that the QC manifold is locally QC equivalent to the quaternionic Heisenberg group. Since the quaternionic Heisenberg group and the QC sphere are locally equivalent, this tensor clearly plays the role of the Weyl or Chern tensors.
In the conformal setting, for a given metric, the full Riemann curvature tensor can be expressed in terms of the Weyl tensor and the Ricci curvature. Analogously, in the QC case, the horizontal Riemann curvature tensor is described in terms of $W$ and a tensor $L$ which is itself determined by the horizontal Ricci tensor. In particular, \begin{equation} \label{eq:defineL} L_{\alpha\beta} = \frac{1}{2} \tau_{\alpha\beta} + \mu_{\alpha\beta} + \frac{S}{32n(n+2)} g_{\alpha\beta},\end{equation} where $\tau$ and $\mu$ are the tensors described in propositions \ref{prop:torsionprop} and \ref{prop:curvprop}. Since $\tau$, $\mu$ and $S$ may be recovered from $L$ using the Casimir operator and the metric, their vanishing is equivalent to the vanishing of $L$.
Equation (4.8) of \cite{Vassilevetal:2007} expresses $W$ in terms of the curvature tensor and the tensor $L$ as \begin{multline} \label{eq:QCconfcurv} W_{\alpha\beta\gamma\delta} = R_{\alpha\beta\gamma\delta} + g_{\alpha\gamma}L_{\beta\delta} - g_{\alpha\delta}L_{\beta\gamma} + g_{\beta\delta}L_{\alpha\gamma} - g_{\beta\gamma}L_{\alpha\delta} \\ + I_{i\alpha\gamma}L_{\beta\rho}\tensor{I}{\up{i\rho}\down\delta} - I_{i\alpha\delta}L_{\beta\rho}\tensor{I}{\up{i\rho}\down\gamma} + I_{i\beta\delta}L_{\alpha\rho}\tensor{I}{\up{i\rho}\down\gamma} - I_{i\beta\gamma}L_{\alpha\rho}\tensor{I}{\up{i\rho}\down\delta}\\ + \frac{1}{2}( I_{i\alpha\beta}L_{\gamma\rho}\tensor{I}{\up{i\rho}\down \delta} - I_{i\alpha\beta}L_{\rho\delta}\tensor{I}{\up{i\rho}\down\gamma} + I_{i\alpha\beta}L_{\rho\sigma}\tensor{I}{\down j \up \rho \down \gamma} \tensor{I}{\down k \up \sigma \down \delta} \varepsilon^{ijk}) \\ + I_{i\gamma\delta}L_{\alpha\rho}\tensor{I}{\up{i\rho}\down \beta} - I_{i\gamma\delta}L_{\rho\beta}\tensor{I}{\up{i\rho}\down\alpha} + \frac{1}{2n} \tensor{L}{\down \rho \up \rho} I_{i\alpha\beta}\tensor{I}{\up i \down{\gamma\delta}}. \end{multline} Therefore, the QC conformal curvature equals the horizontal curvature tensor precisely when the tensor $L$ vanishes.
\subsection{The quaternionic Heisenberg group} \label{sec:quatheis}
We close this section with a brief review of the quaternionic Heisenberg group, a non-compact $4n+3$ manifold with a QC structure. Let $\mathbb{H}$ denote the quaternion algebra and $\mathbb{H}^n$ the right $\mathbb{H}$-module of $n$-tuples of quaternions. Then the quaternionic Heisenberg group, $\mathcal{H}^n$, is diffeomorphic to $\mathbb{H}^n\oplus \textrm{Im} \mathbb{H}$ with group law \[ (p_1,\omega_1)\cdot (p_2,\omega_2) = \big(p_1+p_2, \omega_1 + \omega_2 + 2\textrm{Im} (p_1,p_2)\big),\] where $(p_1,p_2) = \sum_{i=1}^n p_1^i (p_2^i)^\ast$ is the standard hermitian inner product on $\mathbb{H}^n$.
Writing $p=(p^\alpha) = (w^\alpha+x^\alpha i + y^\alpha j + z^\alpha k)$ and $\omega = ri + sj+tk$, the left invariant $1$-forms on $\mathcal{H}^n$ are \begin{gather*} dw^\alpha,\ dx^\alpha,\ dy^\alpha,\ dz^\alpha,\\ \eta^1 = \frac{1}{2}dr - \sum_\alpha x^\alpha dw^\alpha - w^\alpha dx^\alpha + z^\alpha dy^\alpha - y^\alpha dz^\alpha,\\ \eta^2 = \frac{1}{2} ds - \sum_\alpha y^\alpha dw^\alpha - z^\alpha dx^\alpha - w^\alpha dy^\alpha + x^\alpha dz^\alpha,\\ \eta^3 = \frac{1}{2}dt - \sum_\alpha z^\alpha dw^\alpha + y^\alpha dx^\alpha - x^\alpha dy^\alpha - w^\alpha dz^\alpha. \end{gather*} Dual to these are the left invariant vector fields \begin{gather*} W_\alpha = \parfrac{w^\alpha} + 2\sum_\alpha x^\alpha \parfrac{r} + y^\alpha \parfrac{s} + z^\alpha \parfrac{t},\\ X_\alpha = \parfrac{x^\alpha} + 2\sum_\alpha -w^\alpha \parfrac{r} - z^\alpha \parfrac{s} + y^\alpha \parfrac{t},\\ Y_\alpha = \parfrac{y^\alpha} + 2\sum_\alpha z^\alpha \parfrac{r} - w^\alpha \parfrac{s} - x^\alpha \parfrac{t},\\ Z_\alpha = \parfrac{z^\alpha} + 2\sum_\alpha -y^\alpha \parfrac{r} + x^\alpha \parfrac{s} - w^\alpha \parfrac{t},\\
R = 2 \parfrac{r}, \ S= 2 \parfrac{s}, \ T=2 \parfrac{t}. \end{gather*} The QC structure on $\mathcal{H}^n$ is given by declaring the left invariant vector fields above to be orthonormal and using the given $\eta^i$ as the contact forms. The almost complex structures are then given by \begin{gather*} I^1: W_\alpha\mapsto X_\alpha,\quad X_\alpha \mapsto -W_\alpha, \quad Y_\alpha \mapsto Z_\alpha, \quad Z_\alpha \mapsto -Y_\alpha;\\ I^2: W_\alpha \mapsto Y_\alpha, \quad X_\alpha \mapsto -Z_\alpha, \quad Y_\alpha \mapsto -W_\alpha, \quad Z_\alpha \mapsto X_\alpha;\\ I^3=I^1\circ I^2. \end{gather*}
The horizontal and vertical subbundles of $T\mathcal{H}^n$ are given by \[\mathfrak{h} = Span\{W_\alpha,\ X_\alpha,\ Y_\alpha,\ Z_\alpha\} \text{ and } \mathfrak{v}=Span\{R,\ S,\ T\}\] and of course $T\mathcal{H}^n = \mathfrak{h} \oplus \mathfrak{v}$. The Biquard connection is given by declaring these left invariant vector fields to be parallel, and so the flat model of QC geometry is exactly the quaternionic Heisenberg group. Finally we note the important fact that parabolic dilations $\delta_a(X,R) = (aX, a^2R)$ for $a\in \mathbb{R}$ are automorphisms for the Lie group $\mathcal{H}$ and hence also its Lie algebra. \section{Coordinate constructions} \label{sec:coords}
In this section we construct a version of normal coordinates that are adapted to QC geometry in much the same way that standard normal coordinates are adapted to the study of Riemannian geometry.
\subsection{Parabolic normal coordinates}
We begin with a general theorem that constructs ``parabolic geodesics''; that is, curves that satisfy an invariant differential equation with a parabolic-type scaling. By way of motivation, recall that $\mathbb{R}^n$, and also its tangent spaces at every point, come equipped with a natural dilation that sends a vector $v$ to $sv$, for any real scalar $s$. If we consider these vectors to be based at the origin $0\in\mathbb{R}^n$, then by moving from $0$ in the direction of the vector $v$ for time $s$ we arrive at the standard parametrization of a line, $s\mapsto sv$. Further, it is a simple matter to see that this line is uniquely determined by the initial value problem, \[ \ddot{\gamma}=0, \quad \gamma(0)=0, \quad \dot{\gamma}(0)=v.\] Using this as a guide, a geodesic on a manifold with linear connection is a curve satisfying the following initial value problem for a fixed $X\in T_qM$, \[ D_t \dot\gamma_X = 0, \quad \gamma_X(0)=q \in M, \quad \dot\gamma_X(0)=X \in T_qM,\] where for any curve $\gamma$ on $M$ and any vector field $X$ along $\gamma$, we let $D_t X$ denote the covariant derivative of $X$ along $\gamma$. Notice that the standard dilations on $\mathbb{R}^n$ interact naturally with a parametrized geodesic by \[ \gamma_{sX}(t) = \gamma_X(st).\] Further, these dilations are Lie algebra homomophisms for the commutative Lie algebra structure on $\mathbb{R}^n$.
Just as the model for Riemannian geometry is $\mathbb{R}^n$, the model for quaternionic contact geometry is the quaternionic Heisenberg group, described in section \ref{sec:quatheis}. The quaternionic Heisenberg group has a family of parabolic dilations $(x,t)\mapsto (sx,s^2t).$ If we start from a point $0\in\mathcal{H}^n$ and travel along the curve $s\mapsto (sv,s^2a)$, we trace out a parabola. This serves as a guide for our notion of parabolic geodesics on a QC manifold.
As with a line, there is a simple expression for a parabola in terms of a differential equation, namely \[ \dddot{\gamma}_{(v,a)} = 0, \quad \gamma_{(v,a)}(0)=0, \quad \dot{\gamma}_{(v,a)}(0)=v, \quad \ddot{\gamma}_{(v,a)}(0) = a.\] Extending this notion to a manifold with a linear connection produces curves that can rightly be called \emph{parabolic geodesics}, i.e. that satisfy a natural parabolic scaling \[\gamma_{(sv,s^2a)}(t)=\gamma_{(v,a)}(st).\] By appropriately restricting our initial conditions, we can show that there is a parabolic version of the geodesic exponential map called the \emph{parabolic exponential map}.
The following theorem carries out this procedure. It represents a generalization and improvement of the argument given by Jerison and Lee in \cite[Theorem 2.1]{JerisonLee:1989}, which is specific to strictly pseudoconvex CR manifolds. In particular, Theorem \ref{thm:coordinates} requires no assumptions on the manifold except a direct sum decomposition of the tangent bundle by two complementary distributions, which is satisfied by both CR and QC manifolds. In fact this theorem also generalizes the proof of the existence of geodesics on a manifold with connection, simply by assuming that the bundle $V$ is the zero section of $TM$.
\begin{thm} \label{thm:coordinates} Let $(M,\nabla)$ be a manifold with connection whose tangent bundle decomposes as the direct sum of two distributions, $H$ and $V$. Choose any $q\in M$, and let $(X,Y)\in H_q \oplus V_q = T_q M$ be any tangent vector. Define $\gamma_{(X,Y)}$ to be the curve beginning at $q$ satisfying \begin{equation}\label{eq:curveeq} D_t^2 {\dot\gamma}_{(X,Y)} = 0,\quad \gamma_{(X,Y)}(0)=q, \quad {\dot\gamma}_{(X,Y)}(0)=X, \text{ and } D_t {\dot\gamma}_{(X,Y)}(0) = Y. \end{equation} Then there are neighborhoods $0\in \mathcal{O}\subset T_qM$ and $q\in \mathcal{O}_M \subset M$ so that the function $\Psi : \mathcal{O} \to \mathcal{O}_M : (X,Y)\mapsto \gamma_{(X,Y)}(1)$ is a diffeomorphism, and satisfies the parabolic scaling $\Psi(tX,t^2Y)=\gamma_{(X,Y)}(t)$ wherever either side is defined. \end{thm}
\begin{proof} First, note that for any two tangent vectors $X$ and $Y$, there is a unique smooth curve satisfying \eqref{eq:curveeq}. This follows from the standard existence and uniqueness results on systems of ODEs applied to the coordinate form of the equation. In particular, in any local coordinates $\{x^i\}$, centered at $q$, we let $\Gamma_{ij}^k = dx^k(\nabla_{\partial_i}\partial_j)$ be the Christoffel symbols of $\nabla$. Then \begin{equation}\label{eq:coordparabola} (D_t)^2 \dot\gamma = \left( \dddot{\gamma}^k + \ddot{\gamma}^i\dot{\gamma}^j \Gamma_{ij}^k + 2 \dot{\gamma}^i \ddot{\gamma}^j \Gamma_{ij}^k + \dot{\gamma}^i\dot{\gamma}^j\dot{\gamma}^l \partial_l \Gamma_{ij}^k + \dot{\gamma}^i\dot{\gamma}^l\dot{\gamma}^m\Gamma_{lm}^j\Gamma_{ij}^k\right)\partial_k, \end{equation} so \eqref{eq:curveeq} is a third order, nonlinear system of ODEs with smooth coefficients.
Now for any $a\in \mathbb{R}$, define $\sigma(t) = \gamma_{(X,Y)}(at)$. Then $\dot{\sigma}(t) = a \dot{\gamma}(at)$, $D_t \dot\sigma = a^2 D_t\dot\gamma$ and $(D_t)^2\dot\sigma = a^3 (D_t)^2\dot{\gamma}=0$. Thus, by uniqueness of solutions, $\sigma(t) = \gamma_{(X,Y)}(at) = \gamma_{(aX,a^2Y)}(t)$, which shows the parabolic scaling.
Now, let $\pi:E = TM \oplus TM \to M$ be the Whitney sum of $TM$ with itself. Define a vector field $P$ on $E$ by \begin{equation}\label{eq:parabolicvf}
P_{(p,X,Y)}f= \left.\frac{d}{dt}\right|_{t=0} f( \gamma_{(X,Y)}(t), \dot{\gamma}_{(X,Y)}(t), D_t \dot\gamma_{(X,Y)}(t) ), \end{equation} for any function $f\in C^\infty(E,\mathbb{R})$. Let $\{x^i\}$ be coordinates on $M$, and take fiber coordinates $\{\eta^i\}$ and $\{\xi^i\}$ on $E$ where $\eta^i(p,X,Y) = dx^i(X)$ and $\xi^i(p,X,Y) = dx^i(Y)$. Then in these coordinates, if we let $(\gamma_{(X,Y)}, \dot{\gamma}_{(X,Y)}, D_t\dot{\gamma}_{(X,Y)}) =(x^i, \eta^i, \xi^i)$, we can write $P_{(p,X,Y)}f$ as \begin{align} P_{(p,X,Y)}&= \dot{x}^k \frac{\partial}{\partial x^k}f + \dot{\eta}^k \frac{\partial}{\partial \eta^k}f + \dot{\xi}^k \frac{\partial}{\partial \xi^k}f \notag\\
&=\eta^k \frac{\partial}{\partial x^k}f + (\xi^k - \eta^i\eta^j\Gamma_{ij}^k) \frac{\partial}{\partial \eta^k}f - \eta^i\xi^j\Gamma_{ij}^k \frac{\partial}{\partial \xi^k}f \label{eq:parabolicvfcoords} \end{align} In this formula, the $\Gamma_{ij}^k$ are the Christoffel symbols of $\nabla$, lifted to be constant on the fibers of $E$, and we have used the fact that $(D_t)^2\dot\gamma\equiv 0$. This expression shows that $P$ is smooth, that integral curves of $P$ project onto solutions of \eqref{eq:curveeq}, and that solutions of \eqref{eq:curveeq} lift to integral curves of $P$ by $\rho(t) = (\gamma(t), \dot\gamma(t), D_t\dot\gamma(t))$.
The flow $\theta$ of $P$ is defined on some open subset of $\mathcal{O} \subset \mathbb{R}\times E$ containing $\{0\}\times E$. Thus from the above, $\gamma_{(X,Y)}(t)=\pi \circ \theta(t,(q,X,Y))$.
Let $\iota:T_qM \to E$ be the inclusion sending $X \in H_q$ to $(X,0)\in E_q$ and $Y\in V_q$ to $(0,Y)\in E_q$. Then $\Psi(X,Y) = \pi \circ \theta(1, (q, \iota(X+Y)))$, which shows that $\Psi$ is smooth on some open set in $\iota^{-1}(\mathcal{O})$ to $M$.
We will now show that $\Psi$ is a diffeomorphism by showing that $\Psi_*$ is the identity map on $H$ and one half the identity map on $V$. Given $X\in H_q$, we have
\[ \Psi_* X = \left.\frac{d}{dt}\right|_{t=0} \Psi(tX,0) = \left.\frac{d}{dt}\right|_{t=0} \gamma_{(tX,0)}(1) = \left.\frac{d}{dt}\right|_{t=0} \gamma_{(X,0)}(t) = X.\] For $Y\in V_q$, let $\alpha(t)$ be the $\nabla$-geodesic with initial velocity $Y$, and define $\beta(s) = \alpha(s^2t)$. Then $\dot{\beta}(s) = 2st\dot{\alpha}(s^2t)$, $D_s\dot\beta(s) = 2t\dot\alpha(s^2t)$ and $(D_s)^2\dot\beta(s) = 0$. Thus $\beta(s) = \gamma_{(0,2tY)}(s)$ and \begin{multline*}
\Psi_* Y = \left.\frac{d}{dt}\right|_{t=0} \Psi(0,tY) = \left.\frac{d}{dt}\right|_{t=0} \gamma_{(0,tY)}(1) \\= \left.\frac{d}{dt}\right|_{t=0} \gamma_{(0,2tY)}\Big(\frac{\sqrt{2}}{2}\Big)= \left.\frac{d}{dt}\right|_{t=0} \alpha(t/2) =\frac{1}{2}Y. \end{multline*}
Since $\Psi_*$ is invertible, $\Psi$ is a diffeomorphism from some subset of $\iota^{-1}(\mathcal{O})\subset T_q M$ to $M$. Relabeling our open sets if necessary, the theorem is proved. \end{proof}
As a matter of interest, we note here that this proof extends readily to higher order ``polynomial geodesics'' that satisfy a higher order scaling condition, defined by the equations \[
D_t^{n}\dot\gamma = 0,\quad \gamma(0)=p,\quad D_t^{k}\dot\gamma(0)= X_k, \ k=0,\ldots, n-1. \]
Further, if we have a direct sum decomposition $TM = \oplus_{k=0}^{n-1} V^k$, and $X_k \in V^k|_p$, we have a diffeomorphism from a neighborhood of $0 \in TM$ to a neighborhood of $p\in M$ defined in the obvious way.
We will use the diffeomorphism described to provide a useful coordinate system on a QC manifold with a given pseudohermitian structure. Before we do that, we will need the following lemma which allows us to take a frame for $T_qM$ and construct a local frame on a neighborhood of $q$, compatible with the parabolic exponential map.
\begin{lem}\label{lem:smoothvf} If $Z$ is any vector field parallel along all parabolic geodesics beginning at $q$, then $Z$ is smooth. \end{lem} \begin{proof}
Let $\{x^a\}$ be local coordinates centered at $q$, and let \[Z^a(s,X,Y)=dx^a(Z|_{\gamma_{(X,Y)}(s)}).\] Then along every parabolic geodesic $\gamma$, $Z$ satisfies \[ \partial_s Z^c(s,X,Y) + \dot{\gamma}_{(X,Y)}^a(s) Z^b(s,X,Y) \Gamma_{ab}^c(\gamma_{(X,Y)}(s)) = 0.\] Since $\gamma_{(X,Y)}$ depends smoothly on $s$, and $X$, $Y$ and $\Gamma_{ij}^k$ depend smoothly on the coordinates, we see that $Z^a$ is a smooth function of its parameters. Thus $Z_p = Z^a(1,\Psi^{-1}(p))\partial_a$ is a smooth vector field on a neighborhood of $q$. \end{proof}
Now, let us return to the QC case. Let $\{R_i\}$ be an oriented orthonormal frame for $V_q$, and let $\{I_i\}$ be the associated almost complex structures. Choose an orthonormal basis $\{\xi_\alpha\}$ for $H_q$ so that $\xi_{4k+i+1} = I_i \xi_{4k+1}$ for $k=0,\ldots,n-1$. Extending these vectors to be parallel along parabolic geodesics beginning at $q$, we have a smooth local frame for $TM = H\oplus V$. Define the dual $1$-forms $\{\theta^\alpha,\eta^i\}$ by $\theta^\alpha(\xi_\beta) = \delta_\beta^\alpha$, $\theta^\alpha(R_i)=0$, $\eta^i(\xi_\alpha)=0$ and $\eta^i(R_j) = \delta_j^i$. Finally, we extend the almost complex structures by defining $I_i \xi_{4k+1} = \xi_{4k+i+1}$ for each $k$. Then each of these $1$-forms and almost complex structures is also parallel along parabolic geodesics, and all are parallel at $q$. Using this frame and coframe, we have the following lemma. \begin{lem} For the frame, coframe and almost complex structures defined above, and for all vectors $X,Y\in H$, we have \[ d\eta^i(X,Y) = 2g(I^i X, Y).\] Thus the $\eta^i$,$I^i$ and $g$ form a QC pseudohermitian structure on $M$. \end{lem} \begin{proof}
Let $g$, $\tilde{\eta}^i$ and $\tilde{I}^i$ be the metric, contact $1$-forms and almost complex structures defining a QC pseudohermitian structure near $q$. By a constant coefficient rotation, we may assume that $\tilde{\eta}^i|_q = \eta^i|_q$. Since the connection preserves the metric, the orthogonality relations of the $\eta^i$ are preserved by parallel translation. Thus, both $\{\eta^i\}$ and $\{\tilde{\eta}^i\}$ are oriented orthonormal $V$-coframes, and so are related by an orthogonal transformation at each point. Since both frames are smooth, the transformation is smooth, and since the determinant is a continuous function on a connected set with values in $\pm 1$, which equals $1$ at $q$, the transformation actually lies in $SO(3)$. Thus the $\eta^i$ and $I^i$ form a QC pseudohermitian structure with $g$. \end{proof} The frame and coframe constructed above will be called a \emph{special frame} and a \emph{special coframe}.
Given any special frame, we may define a coordinate map on a neighborhood of $q$ by composing the inverse of $\Psi$ with the map $\lambda: T_qM \to \mathbb{R}^{4n+3}:X \mapsto (x^\alpha,t^i)=(\theta^\alpha(X),\eta^i(X))$. These coordinates will be called \emph{QC pseudohermitian normal coordinates}, or pseudohermitian normal coordinates when no confusion can arise.
With these definitions in mind, our index convention defined in section \ref{sec:background} is hereby refined to refer to a special frame and coframe henceforth.
\subsubsection{Parabolic Taylor expansions} \label{sec:parabolictaylor}
Returning to our analogy between scaling operators on $\mathbb{R}^n$ and the quaternionic Heisenberg group, we recall that the generator for the standard dilation $x\mapsto sx$ on $\mathbb{R}^n$ is the Euler vector field $X = x^i \partial_i$, and a tensor field $\varphi$ is called homogeneous of order $m$ if $\mathscr{L}_X \varphi = m \varphi$. In the setting of parabolic dilations on the quaternionic Heisenberg group, the generator of $\delta_s: (x,t)\mapsto(sx, s^2t)$ is the vector field $P = x^\alpha \partial_\alpha + 2 t^i \partial_i$. Note that in the notation of section \ref{sec:quatheis}, we can express $P$ in terms of the left invariant vector fields on $\mathcal{H}$ as \[ P = w^\alpha W_\alpha + x^\alpha X_\alpha + y^\alpha Y_\alpha z^\alpha Z_\alpha + rR + sR + tT.\] As in the Euclidean setting we say a tensor field $\varphi$ is homogeneous of order $m$ if $\mathscr{L}_P \varphi = m\varphi$. For an arbitrary tensor field, we denote by $\varphi_{(m)}$ the part of the tensor that is homogeneous of order $m$.
Given a QC manifold and pseudohermitian coordinates as above centered at a point $q \in M$, we may define the vector $P$ in these coordinates. Then next lemma shows how $P$ is related to the special frame that we have constructed.
\begin{lem} Let $P$ be the vector described in pseudohermitian normal coordinates by $P = x^\alpha \partial_\alpha + 2t^i \partial_i$, and let $\tensor{\omega}{\down a \up b}$ be the connection $1$-forms for the Biquard connection. Then \[ \theta^\alpha(P) = x^\alpha,\ \eta^i(P) = t^i,\text{ and } \tensor{\omega}{\down a \up b}(P)=0.\] Thus $P = x^\alpha \xi_\alpha + t^i R_i$. \end{lem} \begin{proof} The proof is essentially identical to the proof of Lemma 2.4 in \cite{JerisonLee:1989}. \end{proof}
We now use this result to calculate the low order homogeneous terms of the special coframe and the connection $1$-forms. Recall that for a differential form $\varphi$, \begin{equation} \label{eq:ordercalc} \mathscr{L}_P \varphi = P \lrcorner\, d\varphi + d(P\lrcorner\, \varphi). \end{equation} As a result, we have the following proposition. \begin{prop} \label{prop:coframeconnection} In pseudohermitian normal coordinates, the low order homogeneous terms of the special coframe and connection $1$-forms are \begin{multline*} \eta^i_{(2)}=\frac{1}{2}dt^i -\tensor{I}{\up i \down {\alpha \beta}}x^\alpha dx^\beta; \quad \eta^i_{(3)} = 0; \\ \quad \eta^i_{(m)} = \frac{1}{m} ( t^j \tensor{\omega}{\down j \up i} + \tensor{T}{\up i \down{jk}}t^j \eta^k -2 \tensor{I}{\up i \down{\alpha \beta}}x^\alpha \theta^\beta)_{(m)},\quad m\geq 4; \tag{a} \end{multline*} \begin{multline*} \theta^\alpha_{(1)}=dx^\alpha; \quad \theta^\alpha_{(2)} = 0;\\
\quad \theta^\alpha_{(m)} = \frac{1}{m}\big( x^\beta \tensor{\omega}{\down \beta \up \alpha} - \tensor{T}{\up \alpha \down{i\gamma}}x^\gamma \eta^i + \tensor{T}{\up \alpha \down{i\beta}}t^i\theta^\beta + \tensor{T}{\up \alpha \down{ij}} t^i \eta^j\big)_{(m)}, \quad m\geq 3;\tag{b} \end{multline*} \begin{multline*} \tensor{\omega}{\down a \up b}_{(1)}=0; \\
\tensor{\omega}{\down a \up b}_{(m)} = \frac{1}{m}(\tensor{R}{\down{\alpha\beta a}\up b} x^\alpha \theta^\beta + \tensor{R}{\down{\alpha j a}\up b} x^\alpha \eta^j - \tensor{R}{\down{\alpha j a}\up b} t^j \theta^\alpha + \tensor{R}{\down{ija}\up b}t^i \eta^j)_{(m)}, \\ m\geq 2. \tag{c} \end{multline*} \end{prop} \begin{proof} The proof is essentially the same as the proof of Proposition 2.5 in \cite{JerisonLee:1989}. \end{proof}
Let us denote by $\order{m}$ those tensor fields whose Taylor expansions at $q$ contain only terms of order greater than or equal to $m$. For example, from the above proposition, $\eta^i \in \order{2}$ and $\theta^\alpha \in \order{1}$. It is routine to check that if $\varphi \in \order{m}$ and $\psi \in \order{m'}$, then $\varphi \otimes \psi \in \order{m+m'}$. To further extend the utility of this notation we introduce the following: for any index $a$, let $o(a)=1$ if $a \leq 4n$ and $o(a)=2$ if $a>4n$. Given a multiindex $A=(a_1, \ldots a_r)$, we let $\#A = r$ and $o(A)=\sum_i o(a_i)$. Finally, if we have a collection of indexed vector fields, $X_a$, we let $X_A = X_{a_r}\ldots X_{a_1}$, and similarly for similar expressions.
\begin{cor} \label{cor:XT} If we define $X_\alpha = \partial_\alpha+ 2 \tensor{I}{\up i \down{\beta \alpha}}x^\beta \partial_i$ and $T_i = 2 \partial_i$, then $\xi_\alpha = X_\alpha + \order{1}$ and $R_i = T_i+\order{0}$.\end{cor}
This corollary shows that the special frame constructed in these coordinates is particularly close to the standard left-invariant frame on the quaternionic Heisenberg group. In particular, the vector fields $X_\alpha$ and $T_i$ are the standard left-invariant frame on $\mathcal{H}^n$ defined in section \ref{sec:quatheis}, and the given frame on $M$ is expressed as a perturbation of them. Further, as a matter of notation we will occasionally refer to $T_i$ as $X_{4n+i}$, similar to our convention regarding $R_i = \xi_{4n+i}$.
Given any two $Sp(n)Sp(1)$-frames at $q$, they determine distinct parabolic coordinate systems, related by a unique element of $Sp(n)Sp(1)$. Combining Theorem \ref{thm:coordinates} and Proposition \ref{prop:coframeconnection} we have the following theorem. \begin{thm} \label{thm:paraboliccoordinates} Let $M$ be a QC manifold with pseudohermitian structure $\eta$, and let $q\in M$ be any point. Then there exist parabolic normal coordinates $(x^\alpha,t^i)$ about $q$ for which \[ g_{\alpha\beta}(q) = \delta_{\alpha\beta},\quad \eta^i(q) = \frac{1}{2}dt^i(q), \quad \tensor{\omega}{\down b \up b}(q) = 0.\] Further, any two such coordinate systems centered at $q$ are related by a linear transformation in $Sp(n)Sp(1)$. \end{thm}
We close this section with a lemma on the parabolic version of Taylor expansions.
\begin{lem} \label{lem:parabolic_taylor} Let $F$ be a smooth function defined near $q \in M$. Then in pseudohermitian normal coordinates, for any nonnegative integer $m$,
\[ F_{(m)} = \sum_{o(A)=m} \frac{1}{(\#A)!}\Big(\frac{1}{2}\Big)^{o(A)-\# A} x^A (X_A F)|_q.\] \end{lem} \begin{proof} The proof is essentially the same as the proof of Lemma 3.10 in \cite{JerisonLee:1989}. \end{proof}
\subsection[QC normal coordinates]{Quaternionic contact normal coordinates}
Now, using the coordinates constructed above, we will develop a conformal factor $u$ so that the parabolic normal coordinates for the pseudohermitian structure $e^{2u}\eta$ satisfy a number of convenient normalization conditions on the QC curvature and torsion tensors.
We begin with a technical lemma describing the covariant derivative of a tensor field in terms of the action of the vector fields $X_a$ defined in the previous section. \begin{lem} \label{lem:covD_vect_rel} If $\varphi$ is a tensor in $\order{m}$, the components of its covariant derivatives in terms of a special frame satisfy \[ \varphi_{A,B} = X_B \varphi_A + \order{m-o(AB)+2}.\] \end{lem} \begin{proof} The proof is essentially the same as the proof of Lemma 3.2 in \cite{JerisonLee:1989}.\end{proof}
\subsubsection{Parabolic coordinates under a conformal change}
Now let us consider the effect of changing the pseudohermitian structure by a conformal factor. We let $\tilde{\eta}^i = e^{2u}\eta^i$ for some smooth function $u$. Then $H$ remains the kernel of the three $1$-forms, and it is a simple calculation to see that for $\tilde{I}^i=I^i$ and $\tilde{g}=e^{2u}g$ we have \[ d\tilde{\eta}^i(X,Y) = 2\tilde{g}(I^iX,Y), \text{ for all } X,Y \in H.\] As mentioned in Proposition \ref{prop:confchange}, in \cite{Vassilevetal:2006} the authors demonstrate that the Reeb fields for the new structure are given by \begin{equation} \label{eq:reebchange} \tilde{R}_i = e^{-2u}(R_i - \tensor{I}{\down i \up \alpha \down \beta} u^\beta \xi_\alpha).\end{equation} If we define $\tilde{\xi}_\alpha = \xi_\alpha$ and $\tilde{\theta}^\alpha = \theta^\alpha + \tensor{I}{\down i \up \alpha \down \beta}u^\beta \eta^i$, then $\tilde{\theta}^\alpha(\tilde{R}_i)=0$ and $\tilde{\eta}^i(\tilde\xi_\alpha)=0$.
The change of connection $1$-forms under the change of connection is slightly more complicated as shown in the following lemma. \begin{lem} \label{lem:connchange} Suppose the conformal factor $u$ is order $m\geq 2$ with respect to $P$. Then the connection $1$-forms of the Biquard connection transform as follows: \begin{gather*}
\tensor{\tilde\omega}{\down \alpha \up \beta} = \tensor{\omega}{\down \alpha \up \beta} + \order{m},\\
\tensor{\tilde\omega}{\down i \up j} = \tensor{\omega}{\down i \up j} +\order{m}. \end{gather*} \end{lem} \begin{proof} From \cite[Prop 3.5]{Vassilevetal:2006} we can calculate the connection $1$-forms on $V$ directly. In particular, if we write $\grad_H u = u^\alpha \xi_\alpha$, then for $X\in H$, \begin{align*}
\tensor{\tilde\omega}{\down i \up j}(X) &= d\tilde{\eta}^j(\tilde{R}_i, X)\\
&= (2du\wedge \eta^i + d\eta^i)(R_i - I_i \grad_H u, X)\\
&= 2du\wedge \eta^j(R_i,X) -2du\wedge\eta^j(I_i \grad_H u, X)\\
&\qquad + d\eta^j(R_i,X) -d\eta^j(I_i \grad_H u, X)\\
&= -2\delta_i^j du(X) + d\eta^j(R_i,X) - 2g(I^jI_i \grad_H u, X)\\
&= \tensor{\omega}{\down i \up j}(X) -2 \delta_i^j du(X) + 2\delta_i^j du(X) - \tensor{\varepsilon}{\up j \down i \up k} g(I_k \grad_H u, X) \\
&= \tensor{\omega}{\down i \up j}(X) + \tensor{\varepsilon}{\up j \down {ik}}du(I^kX). \end{align*} Since $u\in \order{m}$, so is $du$, and so $ \tensor{\tilde\omega}{\down i \up j}= \tensor{\omega}{\down i \up j}+\order{m}$ acting on $H$. A similar calculation for the action of $\tensor{\omega}{\down i \up j}$ on $V$ shows that $\tensor{\tilde\omega}{\down i \up j} = \tensor{\omega}{\down i \up j} + \order{m}$.
For the connection $1$-forms in the $H$ directions, we refer to equation $(5.5)$ and the equation immediately following equation $(5.12)$ in \cite{Vassilevetal:2006}. They let $S = \tilde{\omega} - \omega$, $X,Y,Z \in H$, and denote the conformal factor by $\frac{1}{2h}=e^{2u}$. Then \begin{multline*} -2h g(S_X Y, Z) = dh(X) g(Y,Z) - \frac{1}{2}dh(I_i X)d\eta^i(Y,Z) + dh(Y)g(Z,X) \\
+ \frac{1}{2}dh(I_i Y)d\eta^i(Z,X) - dh(Z) g(X,Y) + \frac{1}{2}dh(I_i Z) d\eta^i(X,Y), \end{multline*} \begin{multline*} g(S_{\tilde{R}_i} X,Y) =-\frac{1}{4}\big(\nabla dh(I_iX,Y) -\nabla dh(X,I_iY) - \tensor{\varepsilon}{\down i \up {jk}} \nabla dh(I_jX,I_kY)\big) \\
- \frac{1}{2h}\big( \tensor{\varepsilon}{\down i \up {jk}} dh(I_kX)dh(I_j Y)+ dh(I_iX)dh(Y) - dh(I_iY)dh(X)\big) \\
+ \frac{1}{4n}\Big( -\Delta h + \frac{2}{h} \big|dh|_H\big|^2\Big)g(I_iX,Y) - \tensor{\varepsilon}{\down i \up {jk}} dh(R_k)g(I_jX,Y). \end{multline*}
Here we see that $\tensor{\tilde\omega}{\down \alpha \up \beta}$ and $\tensor{\omega}{\down \alpha \up \beta}$ differ by terms involving $dh$ and $\nabla^2 h|_H$. The relation between $u$ and $h$ implies that $dh = -e^{-2u}du$ and \[\nabla^2 h=2e^{-2u}du\otimes du - e^{-2u}\nabla^2 u.\] Since $u$ is order $m$, so is $du$ and from Lemma \ref{lem:covD_vect_rel} \begin{align*}
\nabla^2 u|_H &= u_{\alpha \beta}\, \theta^\alpha \otimes \theta^\beta\\
&= X_{\beta}X_{\alpha}u\,\theta^\alpha \otimes \theta^\beta +\order{m+2} \end{align*} which is also order $m$. The last term, $dh(R_k)g(I_jX,Y)$ is also order $m$ since $dh(R_k)\in\order{m-2}$ and $g\in\order{2}$. \end{proof}
Now we are in a position to relate the covariant derivative of the tilded connection to that of the untilded connection. This will allow us to work only with the original connection by accounting for the orders of the error terms. We have the following lemma. \begin{lem} \label{lem:covderchange} Let $\varphi$ be an $s$-tensor and denote by $\nabla^r \varphi$ and $\tilde\nabla^r\varphi$ its $r$th covariant derivatives with respect to the original and rescaled connections respectively. Let $A$ and $B$ be multiindices with $\#A=s$ and $\#B=r$, and let $\varphi_{A,B}$ and $\tilde\varphi_{A,B}$ denote the components of $\nabla^r \varphi$ and $\tilde\nabla^r \varphi$. For a conformal change as described above with $u\in \order{m}$, $m\geq2$, we have \[ \tilde\varphi_{A,B} = \varphi_{A,B} + \order{m-o(B)-1}.\] Further, if $o(A)=s$ (i.e. $A$ contains no entries greater than $4n$) then \[ \tilde{\varphi}_{A,B} = \varphi_{A,B} + \order{m-o(B)}.\] \end{lem} \begin{proof} The proof is essentially the same as the proof of Lemma 3.5 in \cite{JerisonLee:1989}.\end{proof}
Finally, we will need to know how two sets of parabolic normal coordinates are related for conformally related pseudohermitian structures. This is the content of the next lemma, which also corrects an error in the 1989 paper of Jerison and Lee \cite{JerisonLee:1989}.
\begin{thm} \label{thm:coordchange} Let $\Psi$ and $\tilde\Psi$ denote the parabolic exponential maps based at $q\in M$ of the pseudohermitian structures $\eta$ and $\tilde\eta=e^{2u}\eta$, respectively. Suppose that $u \in \order{m}$ with $m\geq 2$. Then considered as functions on $T_qM$ with the induced pseudohermitian structure, $\tilde\Psi - \Psi$ is order $m+1$. \end{thm} \begin{proof} We will work in parabolic normal coordinates on $M$ given by the original pseudohermitian structure., written as always as $(x^\alpha, t^i)$. Identifying a neighborhood of $0\in T_qM$ with a neighborhood of $q\in M$, we may write $\Psi(x,t)=(x,t)$. Then writing $\tilde\Psi^a(x,t)=x^a + f^a(x,t)$ we need only show that $f^a$ is order $m+1$ for each $a=1,\ldots, 4n+3$. Since these are parabolic normal coordinates defined by the parabolic geodesics of Theorem \ref{thm:coordinates}, this is equivalent to showing the for any particular $(x,t)$, $f^a(sx, s^2t) = O(s^{m+1})$ as $s\to0$. Further, if we write $\gamma$ and $\tilde\gamma$ for the parabolic geodesics with initial data $(X,R)=(x^\alpha,t^i)$ at $q$ for the original and rescaled connections, we are reduced to showing that $\tilde\gamma(s) - \gamma(s) \in O(s^{m+1})$ for small $s$.
Now, let us denote by $\Gamma_{bc}^a$ and $\tilde\Gamma_{bc}^a$ the Christoffel symbols of the two connections in these coordinates, and write $B_{bc}^a=\tilde\Gamma_{bc}^a - \Gamma_{bc}^a$ for the difference tensor. From equation \eqref{eq:curveeq} we see that $\sigma(s) = \tilde\gamma(s)-\gamma(s)$ satisfies the third order equation \begin{multline*}
\dddot{\sigma}^a(s) = \big( \ddot{\gamma}^b(s)\dot\gamma^c(s)\Gamma_{bc}^a(\gamma(s)) - \ddot{\tilde\gamma}^b(s)\dot{\tilde\gamma}^c(s)\tilde\Gamma_{bc}^a(\tilde\gamma(s))\big)\\
+2\big( \dot\gamma^b(s) \ddot\gamma^c(s)\Gamma_{bc}^a(\gamma(s)) - \dot{\tilde\gamma}^b(s) \ddot{\tilde\gamma}^c(s)\tilde\Gamma_{bc}^a(\tilde\gamma(s))\big)\\
+ \big(\dot\gamma^b(s)\dot\gamma^c(s)\dot\gamma^d(s)\partial_d \Gamma_{bc}^a(\gamma(s)) - \dot{\tilde\gamma}^b(s)\dot{\tilde\gamma}^c(s)\dot{\tilde\gamma}^d(s)\partial_d \tilde\Gamma_{bc}^a(\tilde\gamma(s))\big)\\
+ \big( \dot\gamma^b(s)\dot\gamma^d(s)\dot\gamma^e(s)\Gamma_{de}^c(\gamma(s))\Gamma_{bc}^a(\gamma(s)) \\- \dot{\tilde\gamma}^b(s)\dot{\tilde\gamma}^d(s)\dot{\tilde\gamma}^e(s)\tilde\Gamma_{de}^c(\tilde\gamma(s))\tilde\Gamma_{bc}^a(\tilde\gamma(s))\big), \end{multline*} with initial conditions $\sigma^a(0)=0$, $\dot\sigma^a(0)=0$ and $\ddot\sigma^a(0)=-B_{\beta\gamma}^a(0) x^\beta x^\gamma$. From Lemma \ref{lem:connchange} we see that $B_{\beta\gamma}^a$ is order $m-1$ and so $\ddot\sigma(0)^a=0$ as well.
Let us simplify the notation by omitting the dependence on $s$. To that end we write $\Gamma_{bc}^a=\Gamma_{bc}^a(\gamma(s))$, $\tilde\Gamma_{bc}^a = \tilde\Gamma(s)_{bc}^a(\tilde\gamma(s))$ and $\hat\Gamma_{bc}^a = \tilde\Gamma_{bc}^a(\gamma(s))$. Then the equation above becomes much more compact: \begin{multline} \label{eq:sigmadiffeq}
\dddot\sigma^a = (\ddot\gamma^b\dot\gamma^c\Gamma_{bc}^a - \ddot{\tilde\gamma}^b\dot{\tilde\gamma}^c\tilde\Gamma_{bc}^a) + 2(\dot\gamma^b \ddot\gamma^c \Gamma_{bc}^a - \dot{\tilde\gamma}^b \ddot{\tilde\gamma}^c \tilde\Gamma_{bc}^a) \\
+ (\dot\gamma^b\dot\gamma^c \dot\Gamma_{bc}^a - \dot{\tilde\gamma}^b\dot{\tilde\gamma}^c \dot{\tilde\gamma}_{bc}^a) + (\dot{\tilde\gamma}^b\dot{\tilde\gamma}^d\dot{\tilde\gamma}^e\tilde\Gamma_{de}^c\tilde\Gamma_{bc}^a - \dot{\tilde\gamma}^b\dot{\tilde\gamma}^d\dot{\tilde\gamma}^e\tilde\Gamma_{de}^c\tilde\Gamma_{bc}^a) \end{multline}
Our goal is to estimate $\dddot\sigma^a$ and then derive bounds on it to prove the theorem. Thus we shall expand the right-hand side of equation \eqref{eq:sigmadiffeq}. We present one example of this expansion and leave it to the reader to complete the rest. \begin{align*}
\ddot\gamma^b\dot\gamma^c\Gamma_{bc}^a - \ddot{\tilde\gamma}^b\dot{\tilde\gamma}^c\tilde\Gamma_{bc}^a &= (\ddot\gamma^b - \ddot{\tilde\gamma}^b)\dot{\tilde\gamma}^c\tilde\Gamma_{bc}^a + \ddot\gamma^b(\dot\gamma^c-\dot{\tilde\gamma}^c)\tilde\Gamma_{bc}^a \\
& \quad + \ddot\gamma^b\dot\gamma^c(\hat\Gamma_{bc}^a - \tilde\Gamma_{bc}^a) + \ddot\gamma^b \dot\gamma^c (\Gamma_{bc}^a - \hat\Gamma_{bc}^a)\\
&= -\ddot\sigma^b\dot{\tilde\gamma}^c\tilde\Gamma_{bc}^a - \ddot\gamma^b\dot\sigma^c\tilde\Gamma_{bc}^a + \ddot\gamma^b\dot\gamma^c(\hat\Gamma_{bc}^a - \tilde\Gamma_{bc}^a) - \ddot\gamma^b \dot\gamma^c B_{bc}^a. \end{align*}
Using this technique we have the following bound for $|\dddot\sigma^a|$, \begin{multline*}
|\dddot\sigma^a| \leq |\ddot\sigma^b\dot{\tilde\gamma}^c\tilde\Gamma_{bc}^a| + |\ddot\gamma^b\dot\sigma^c\tilde\Gamma_{bc}^a| + |\ddot\gamma^b\dot\gamma^c(\hat\Gamma_{bc}^a - \tilde\Gamma_{bc}^a)| + |\ddot\gamma^b \dot\gamma^c B_{bc}^a| \\
+2|\dot\sigma^b\ddot{\tilde\gamma}^c\tilde\Gamma_{bc}^a| + 2|\dot\gamma^b\ddot\sigma^c\tilde\Gamma_{bc}^a| + 2|\dot\gamma^b\ddot\gamma^c(\hat\Gamma_{bc}^a - \tilde\Gamma_{bc}^a)| + 2|\dot\gamma^b \ddot\gamma^c B_{bc}^a| \\
+|\dot\sigma^b\dot{\tilde\gamma}^c\dot{\tilde\Gamma}_{bc}^a| + |\dot\gamma^b\dot\sigma^c\dot{\tilde\Gamma}_{bc}^a| + |\dot\gamma^b\dot\gamma^c(\dot{\hat\Gamma}_{bc}^a - \dot{\tilde\Gamma}_{bc}^a)| + |\dot\gamma^b \dot\gamma^c \dot{B}_{bc}^a| \\
+|\dot\sigma^b\dot{\tilde\gamma}^d\dot{\tilde\gamma}^e\tilde\Gamma_{de}^c\tilde\Gamma_{bc}^a| + |\dot\gamma^b\dot\sigma^d\dot{\tilde\gamma}^e\tilde\Gamma_{de}^c\tilde\Gamma_{bc}^a|+ |\dot\gamma^b\dot\gamma^d\dot\sigma^e\tilde\Gamma_{de}^c\tilde\Gamma_{bc}^a| \\
+ |\dot\gamma^b\dot\gamma^d\dot\gamma^e B_{de}^c\tilde\Gamma_{bc}^a| + |\dot\gamma^b\dot\gamma^d\dot\gamma^e (\hat\Gamma_{de}^c-\tilde\Gamma_{de}^c)\tilde\Gamma_{bc}^a| \\
+ |\dot\gamma^b\dot\gamma^d\dot\gamma^e \Gamma_{de}^cB_{bc}^a| + |\dot\gamma^b\dot\gamma^d\dot\gamma^e \Gamma_{de}^c(\hat\Gamma_{bc}^a-\tilde\Gamma_{bc}^a)|. \end{multline*}
Since each of $\gamma^a$, $\tilde\gamma^a$, $\Gamma_{bc}^a$ and $\tilde\Gamma_{bc}^a$ is a smooth function, by further shrinking the neighborhood of $q$ we are considering we may bound each of these functions and their derivatives by a uniform constant. Further, from Lemma \ref{lem:connchange} and $B_{bc}^a = dx^a(\tilde\nabla_{\partial_b}\partial_c - \nabla_{\partial_b}\partial_c)$, we have $ B_{bc}^a \in\order{m-o(b)}$. Since \mbox{$\dot\gamma^b(s) = O(s^{o(b)-1})$}, we therefore have \[ B_{bc}^a\dot\gamma^b \in\order{m-1}, \text{ and } B_{bc}^a \ddot\gamma^b,\ \dot{B}_{bc}^a\dot\gamma^b \in \order{m-2}.\] Finally, since $\tilde\Gamma_{bc}^a$ is a smooth function, it satisfies a Lipschitz estimate
\[ |\hat\Gamma_{bc}^a(s) - \tilde\Gamma_{bc}^a(s)| \leq C |\gamma(s)-\tilde\gamma(s)| \leq C \sum_b |\sigma^b(s)|.\]
Now we define $\varphi(s) = \sum_a \big( |\ddot\sigma^a(s)|^2 + |\dot\sigma^a(s)|^2 + |\sigma^a(s)|^2\big)$, so that \begin{align*}
|\dddot\sigma^a(s)| &\leq C\Big( \sum_b \big( |\ddot\sigma^b(s)| +|\dot\sigma^b(s)| + |\sigma^b(s)|\big) + s^{m-2} \Big) \\
&\leq C( \varphi(s)^{1/2} + s^{m-2}). \end{align*} Taking the derivative of $\varphi$ we find \begin{align*}
| \dot\varphi(s)| &= 2 \Big|\sum_b (\dddot\sigma^b(s)\ddot\sigma^b(s) + \ddot\sigma^b(s) \dot\sigma^b(s) + \dot\sigma^b(s) \sigma^b(s)) \Big| \\
& \leq C( \varphi(s) + \varphi(s)^{1/2}s^{m-2}). \end{align*}
It is simple to check that the ODE $ \dot{y}(s) = C(y(s) + y(s)^{1/2}s^{m-2})$ with initial condition $y(0)=0$ has a family of solutions given by \[y_a(s)= \begin{cases}0,&s\leq a \\ \frac{C^2}{4}e^{Cs}\beta^2_{m-2}(s;a),& s\geq a\end{cases}\] where $\beta_k(s;a) = \int_a^s e^{-Ct/2} t^k\, dt$. A routine calculation shows that \[\beta_k(s;a) = O((s-a)^{k+1}),\] so $y_0(s) = O(s^{2m-2})$. Further, $y_0(s)\geq y_a(s)$ for all $a\geq 0$ and hence by Theorem \ref{thm:hartman} below, $\varphi(s) = O(s^{2m-2})$, which implies $\sigma(s) = O(s^{m+1})$. This completes the proof. \end{proof}
Theorem \ref{thm:coordchange} above relies on a technical comparison theorem for ODEs given in \cite[Theorem III.4.1]{Hartman:1973}. The paper which inspired this work, \cite{JerisonLee:1989}, fails to recognize the infinite family of solutions to the ODE $ \dot{y}(s) = C(y(s) + y(s)^{1/2}s^{m-2})$, and so their proof is incorrect as it stands. The argument given above also completes their proof, with the following theorem. \begin{thm}[\cite{Hartman:1973}] \label{thm:hartman} Let $U(t,u)$ be continuous on an open $(t,u)$-set $E$ and $u=u^0(t)$ the maximal solution of \[ \dot{u}=U(t,u), \quad u(t_0)=u_0.\] Let $v(t)$ be a continuous function on $[t_0,t_0+a]$ satisfying the conditions $v(t_0)\leq u_0$, $(t,v(t)) \in E$, and $v(t)$ has a right derivative $D_R v(t)$ on $t_0 \leq t \leq t_0+a$ such that \[ D_R v(t) \leq U(t,v(t)).\] Then, on a common interval of existence of $u^0(t)$ and $v(t)$, \[ v(t) \leq u^0(t).\] \end{thm}
\subsubsection{Curvature and torsion normalizations} \label{sec:curvandtornormalizations}
Now we turn to the coordinate normalization that is the focus of this paper. For each QC pseudohermitian structure, we will construct a $2$-tensor $Q$, defined in such a way that by considering the tensors $Q$ and $\tilde Q$ determined by a conformal change, we may recover the symmetric covariant Hessian of the conformal factor. Since the antisymmetric covariant Hessian is determined by the first derivatives and torsion, this completely determines the Hessian of the conformal factor. In this section we use the common notation $F_{(ab)} = \frac{1}{2}(F_{ab} + F_{ba})$ for the symmetric part of the tensor $F_{ab}$. More generally, $F_{(A)} = \frac{1}{(\#A)!}\sum_{\sigma \in S_{\#A}} F_{\sigma A}$, where the sum is over the permutation group on $\#A$ letters, and $\sigma \in S_{\#A}$ acts on the multiindex $A$ by permuting the indices. That is, $F_{(A)}={Sym}(F)_A$, where $ {Sym}(F)$ is the symmetric part of $F$.
Let $u\in\order{m}$ be a fixed conformal factor. From the transformation rules for $\tau_{\alpha\beta}$, $\mu_{\alpha\beta}$ and $S$ in Proposition \ref{prop:confchange} we know that the tensor $L_{\alpha\beta}= \frac{1}{2} \tau_{\alpha\beta} + \mu_{\alpha\beta} + \frac{S}{32n(n+2)}g_{\alpha\beta}$ transforms as \begin{equation}\label{eq:Lchange} \tilde{L}_{\alpha\beta} = L_{\alpha\beta} - u_{(\alpha\beta)} + \order{m-1},\end{equation} and a routine calculation shows the torsion tensor $\tensor{T}{\up \alpha \down{ij}} = d\theta^\alpha(R_i,R_j)$ changes as \[ \tensor{\tilde T}{\up \alpha \down {ij}} = \tensor{T}{\up \alpha \down {ij}} + (\tensor{I}{\down j \up \alpha \down \beta} \tensor{u}{\up \beta \down i} - \tensor{I}{\down i \up \alpha \down \beta}\tensor{u}{\up \beta \down j}) + \order{m-2}.\] From this and the fact that the volume form on $V$ provides an isomorphism between $V$ and $\bigwedge^2 V$, \begin{equation}\label{eq:torsionchange} \tensor{\tilde T}{\down {\alpha jk}}\tensor{\tilde\varepsilon}{\down i \up {jk}} = \tensor{T}{ \down {\alpha jk}}\tensor{\varepsilon}{\down i \up{jk}} + \tensor{A}{\down {i\alpha} \up{j\beta}}u_{\beta j}+ \order{m-2},\end{equation} where $\tensor{A}{\down{i\alpha} \up {j\beta}}=2\tensor{\varepsilon}{\up {jk} \down i} \tensor{I}{\down {k\alpha}\up \beta}$. The operator $A$ is invertible because its minimal polynomial is $m_A(s) = s^2+2s-8$, which follows from \begin{align*} \tensor{A}{\down{i\alpha}\up{j\beta}}\tensor{A}{\down{j\beta}\up{k\gamma}} &= 4 \tensor{\varepsilon}{\up{jl}\down i} \tensor{\varepsilon}{\up {km}\down j} \tensor{I}{\down {l\alpha}\up\beta}\tensor{I}{\down{m\beta}\up \gamma} \\
&= 4(\delta^{kl}\delta_i^m - \delta^{lm}\delta_i^k)(\tensor{\varepsilon}{\down {lm}\up p} \tensor{I}{\down {p\alpha}\up \gamma} - \delta_{lm} \delta_\alpha^\gamma)\\
&=4(\tensor{\varepsilon}{\up k\down {i}\up p} \tensor{I}{\down {p\alpha}\up \gamma} - \delta_{i}^k \delta_\alpha^\gamma) + 4(3\delta_i^k \delta_\alpha^\gamma)\\
&= -2\tensor{A}{\down {i\alpha} \up {k\gamma}} +8 \delta_i^k\delta_\alpha^\gamma. \end{align*} Because $A$ is constructed from the metric tensors on $H$ and $V$, the almost complex structures and the $V$-volume form, we have $\tilde{A} = A + \order{m}$ and $\tilde{A}^{-1} = A^{-1} + \order{m}$.
Let us now define the tensor \begin{equation} \label{eq:defineB} B_{ij} = R_{kl\alpha\beta}\tensor{\varepsilon}{\up {kl} \down i} \tensor{I}{\down j \up {\alpha\beta}}. \end{equation} Since the connection preserves the decomposition of the tangent bundle, we know that $R_{\alpha i j \beta} = R_{j\alpha i\beta}=0$, and hence the first Bianchi identity \cite[Equations (3.1) and (3.2)]{Vassilevetal:2007} shows \begin{equation} R_{kl\alpha\beta} = T_{\beta l\gamma}\tensor{T}{\up\gamma\down{k\alpha}} - T_{\beta k \gamma}\tensor{T}{\up \gamma \down {l\alpha}} + T_{\beta m \alpha}\tensor{T}{\up m\down {kl}} +T_{\beta kl,\alpha} + T_{\beta l \alpha,k} -T_{\beta k \alpha,l}.\end{equation} Since the curvature is tensorial, by working at a point where the frame is parallel so that $\nabla I_i=0$, we may simplify this using Propositions \ref{prop:torsionprop} and \ref{prop:curvprop} as \begin{multline}\label{eq:firstbianchi} R_{kl\alpha\beta} = T_{\beta l\gamma}\tensor{T}{\up\gamma\down{k\alpha}} - T_{\beta k \gamma}\tensor{T}{\up \gamma \down {l\alpha}} + T_{\beta kl,\alpha}+ (\mu_{\beta\gamma,k} \tensor{I}{\down l \up {\gamma} \down \alpha} - \mu_{\beta\gamma,l} \tensor{I}{\down k \up {\gamma} \down \alpha} ) \\
- \frac{1}{8n(n+2)}S\, \varepsilon_{klm}\Big( \mu_{\beta\gamma} \tensor{I}{\up {m\gamma} \down \alpha} + \frac{1}{4}(\tau_{\beta\gamma} \tensor{I}{\up {m\gamma} \down \alpha} - \tau_{\alpha\gamma}\tensor{I}{\up {m\gamma} \down \beta})\Big) \\
+ \frac{1}{4}(\tau_{\alpha \gamma,l} \tensor{I}{\down k \up \gamma \down \beta} - \tau_{\alpha \gamma,k} \tensor{I}{\down l \up \gamma \down \beta} - \tau_{\beta \gamma,l} \tensor{I}{\down k \up \gamma \down \alpha} + \tau_{\beta \gamma,k} \tensor{I}{\down l \up \gamma \down \alpha}). \end{multline} Because $\mu_{\alpha\beta}$ and $\tau_{\alpha\beta}$ are symmetric and trace-free, contracting this with an almost complex structure on the horizontal indices yields \begin{equation}\label{eq:firstbianchi2} R_{kl\alpha\beta}\tensor{I}{\down j \up {\alpha\beta}} = T_{\beta kl,\alpha}\tensor{I}{\down j \up {\alpha\beta}} + (T_{\beta l\gamma}\tensor{T}{\up\gamma\down{k\alpha}} - T_{\beta k \gamma}\tensor{T}{\up \gamma \down {l\alpha}}) \tensor{I}{\down j \up {\alpha\beta}}. \end{equation} We know from Proposition \ref{prop:confchange} that the terms in parentheses change under a conformal rescaling by terms of order at least $m-2$, which we will be able to ignore below. Thus \[ \tilde{B}_{ij} - B_{ij} = \tilde{T}_{\beta kl,\alpha}\tensor{I}{\down j \up {\alpha\beta}}\tensor{\tilde{\varepsilon}}{\up {kl} \down i} - T_{\beta kl,\alpha}\tensor{I}{\down j \up {\alpha\beta}}\tensor{{\varepsilon}}{\up {kl} \down i} + \order{m-2}.\] Notice that we have already calculated $\tilde{T}_{\beta kl} - T_{\beta kl}$ above and seen that it depended on a second derivative of $u$, one derivative each in the vertical and horizontal directions. Since we are now taking another derivative and $\tensor{I}{\down j \up{\alpha\beta}}$ is antisymmetric in the horizontal indices, we expect that the contraction should result in only a second covariant derivative of $u$ in the vertical direction. Modulo terms of order $m-3$, we have \begin{align*} \tilde{T}_{\beta kl,\alpha}\tensor{I}{\down j \up {\alpha\beta}}\tensor{\tilde{\varepsilon}}{\up {kl} \down i} - T_{\beta kl,\alpha}\tensor{I}{\down j \up {\alpha\beta}}\tensor{{\varepsilon}}{\up {kl} \down i} &= (\tilde{T}_{\beta kl}\tensor{\tilde{\varepsilon}}{\up {kl} \down i} - T_{\beta kl}\tensor{{\varepsilon}}{\up {kl} \down i})_{,\alpha}\tensor{I}{\down j \up {\alpha\beta}} + \ldots \\ &= 2 \tensor{\varepsilon}{\up{kl}\down i} \tensor{I}{\down {l\beta} \up \gamma} \tensor{I}{\down j \up {\alpha\beta}}u_{k\gamma\alpha} + \ldots \\ &= 4 \tensor{I}{\down i \up {\alpha\gamma}} u_{j\gamma\alpha} + \ldots \\ &= 4 \tensor{I}{\down i \up {\alpha\gamma}} u_{\gamma\alpha j} + \ldots \\ &= (16 n) u_{ij} + \ldots . \end{align*} Here we are making use of the fact that commuting vertical covariant derivatives with horizontal ones depends only on terms of order $m-2$ or greater and that \[u_{\gamma\alpha j} = u_{\alpha\gamma j} + 2 \tensor{I}{\up k \down {\alpha\gamma}} u_{kj} + \order{m-3}.\]
This implies that under a conformal change, the symmetric part of the tensor $B$ transforms as \begin{equation}\label{eq:Bchange} \tilde{B}_{(ij)} = B_{(ij)} + (16n) u_{(ij)} + \order{m-3}. \end{equation}
With these identities in mind, we define the tensor $Q$ as \begin{gather*}
Q_{\alpha \beta} = L_{\alpha\beta} +\frac{1}{8(n+2)} S g_{\alpha \beta},\\
Q_{\alpha i} = Q_{i \alpha}= -\tensor{(A^{-1})}{\down {i\alpha}\up{j\beta}} T_{\beta kl}\tensor{\varepsilon}{\down j \up{kl}},\\
Q_{ij} = -\frac{1}{16n}B_{(ij)}. \end{gather*} Then according to \eqref{eq:scalarchange}, \eqref{eq:Lchange}, \eqref{eq:torsionchange} and \eqref{eq:Bchange}, under a conformal change with $u\in \order{m}$, $Q$ changes as \begin{gather*} \tilde{Q}_{\alpha\beta} - Q_{\alpha\beta} = -u_{(\alpha\beta)} + (\Delta_H u)g_{\alpha\beta} + \order{m-1}\\ \tilde{Q}_{i\alpha} - Q_{i\alpha} = -u_{i \alpha} + \order{m-2} \\ \tilde{Q}_{ij} - Q_{ij} = -u_{(ij)} + \order{m-3}. \end{gather*}
Now, from section \ref{sec:parabolictaylor}, we recall that the vector field $P= x^a \xi_a$ is the generator of the parabolic dilations that inspired the parabolic normal coordinate construction. Under the conformal change, by Theorem \ref{thm:coordchange} the new coordinates satisfy $\tilde{x}^a = x^a + \order{m+1}$, and hence $\tilde P = P + \order{m-1}$, since $P$ depends on both $\xi_\alpha \in \order{-1}$ and $R_i\in \order{-2}$. To study the Taylor expansion of $\tilde{Q}$ we define the scalar $\Phi = Q(P,P)= x^a x^b Q_{ab}$. Then by the above comments and the fact that a smooth $2$-tensor is at least order $2$, we have \[ \tilde \Phi = \tilde{Q}(\tilde{P}, \tilde{P}) = \tilde{Q}(P, P) + \order{m+1} = x^a x^b \tilde{Q}_{ab} + \order{m+1}.\]
Recall that the vector fields $X_\alpha$ and $T_i$ defined in Corollary \ref{cor:XT} are the standard left invariant frame on the quaternionic Heisenberg group, and covariant differentiation using the Biquard connection is represented as a perturbation of the derivatives with respect to $X_\alpha$ and $T_i$ according to Lemma \ref{lem:covD_vect_rel}. We let $\mathcal{L}_0 = -\sum_\alpha X_\alpha X_\alpha$ denote the standard sublaplacian on $\mathcal{H}^n$. Combining the above calculations with our definitions of $\Phi$ and $\tilde \Phi$, we have \begin{align*} \tilde \Phi &= \Phi - (u_{(\alpha\beta)}x^\alpha x^\beta + 2u_{i\alpha}t^i x^\alpha + u_{(ij)}t^i t^j) + \Delta_H u g_{\alpha \beta} x^\alpha x^\beta +\order{m+1}\\
&= \Phi - (x^\alpha x^\beta X_\beta X_\alpha u + 2t^i x^\alpha X_\alpha T_i u + t^i t^j T_j T_i u) + |x|^2\mathcal{L}_0 u +\order{m+1} \end{align*}
Now we let $\mathcal{P}_m$ denote the space of homogeneous polynomials in $x$ and $t$ of order $m$. For $u\in \mathcal{P}_m$, it is immediate that
\begin{equation*}\tilde{\Phi}_{(m)} = \Phi_{(m)} - (x^\alpha x^\beta X_\beta X_\alpha u + 2 t^i x^\alpha X_\alpha T_i u + t^i t^j T_j T_i u) + |x|^2\mathcal{L}_0 u .\end{equation*} Further, \begin{align*} m^2u = P^2 u &= (x^\alpha X_\alpha + t^i T_i)^2 u \\
&= x^\alpha X_\alpha u+ x^\alpha x^\beta X_\alpha X_\beta u+ 2t^i x^\alpha X_\alpha T_i u+ 2t^i T_i u + t^i t^j T_i T_ju \\
&= x^\alpha x^\beta X_\beta X_\alpha u + 2t^i x^\alpha X_\alpha T_i u + t^i t^j T_j T_i u + t^i T_i u + Pu\\
&= x^\alpha x^\beta X_\beta X_\alpha u + 2 t^i x^\alpha X_\alpha T_i u + t^i t^j T_j T_i u + t^i T_i u + mu. \end{align*} Combining this with the above calculation for $\tilde{\Phi}_{(m)}$, we have
\begin{equation}\label{eq:Phichange} \tilde{\Phi}_{(m)}= \Phi_{(m)} - m(m-1)u + t^i T_i u + |x|^2 \mathcal{L}_0 u,\end{equation} whenever $u \in \mathcal{P}_m$.
\begin{lem}\label{lem:invertop} The operator $L_m = |x|^2 \mathcal{L}_0 + t^i T_i - m(m-1)$ is invertible on $\mathcal{P}_m$ for $m\geq 3$. For $m=2$, $L_2$ has kernel the subspace of $\mathcal{P}_2$ spanned by $t^i$, $i=1,2,3$, and is invertible on the subspace depending only on $x^\alpha$. \end{lem} \begin{proof} The proof is essentially the same as the proof of Lemma 3.9 in \cite{JerisonLee:1989}. \end{proof}
Using Lemma \ref{lem:parabolic_taylor} we may write
\[ \tilde{\Phi}_{(m)} = \sum_{o(abC)=m} \frac{1}{(\# C)!} \Big(\frac{1}{2}\Big)^{o(C)-\# C} x^a x^b x^C X_C \tilde{Q}_{ab}|_q\] and
\[ \Phi_{(m)} = \sum_{o(abC)=m} \frac{1}{(\# C)!} \Big(\frac{1}{2}\Big)^{o(C)-\# C} x^a x^b x^C X_C Q_{ab}|_q.\] Now if we choose $u\in \mathcal{P}_m$, then by Lemma \ref{lem:covderchange}, the covariant derivatives $\tilde{Q}_{ab,C}$ with \mbox{$o(abC)=m$} may be computed with respect to the original connection with an error of order \mbox{$m-o(C)-1$}, which vanishes at $q$. Furthermore, by our calculations above, $\tilde{Q}-Q \in \order{m}$ and so by Lemma \ref{lem:covD_vect_rel}, at $q$, \[ \tilde{Q}_{ab,C} - Q_{ab,C} = X_C \tilde{Q}_{ab} - X_C Q_{ab}.\] Thus \begin{multline} \label{eq:u_eq} L_m u= \tilde{\Phi}_{(m)} - \Phi_{(m)} \\= \sum_{o(abC)=m} \frac{1}{(\# C)!} \Big(\frac{1}{2}\Big)^{o(C)-\# C}x^ax^b x^C \big(\tilde{Q}_{ab,C}(q) - Q_{ab,C}(q)\big). \end{multline}
The next lemma is the key ingredient in showing that we may force symmetrized covariant derivatives of $Q$ to vanish by appropriately choosing $u$. \begin{lem} \label{lem:order_m_vanish}Let $q\in M$ and $(x^\alpha, t^i)$ be pseudohermitian normal coordinates centered at $q$ for a pseudohermitian structure $\eta$. For any $m\geq 2$, there is a polynomial $u \in \mathcal{P}_m$ in the coordinates $(x,t)$ such that $\tilde{\eta} = e^{2u}\eta$ satisfies \[ \tilde{Q}_{(ab,C)}(q) = 0 \text{ if } o(abC)=m.\] For $m\geq 3$ the polynomial is unique, while for $m=2$, it is unique in $\mathcal{R}_2$. \end{lem} \begin{proof} By Lemma \ref{lem:invertop}, if $m\geq 3$ there is a unique polynomial $u\in \mathcal{P}_m$ such that
\[ L_m u = -\sum_{o(abC)=m} \frac{1}{(\# C)!} \Big(\frac{1}{2}\Big)^{o(C)-\# C} x^a x^b x^C Q_{ab,C}|_q.\] For $m=2$, the right hand side is independent of $t$, and so there is a unique polynomial in $\mathcal{R}_2$. If we now set $\tilde{\eta} = e^{2u}\eta$, it follows from \eqref{eq:u_eq} that
\[ \sum_{o(abC)=m} \frac{1}{(\# C)!} \Big(\frac{1}{2}\Big)^{o(C)-\# C} x^a x^b x^C \tilde{Q}_{ab,C}|_q = 0.\] For any multiindex $abC$, the coefficient of $x^ax^bx^C$ is a nonzero multiple of $\tilde{Q}_{(ab,C)}$. Thus we have determined the required polynomial. \end{proof}
Finally, we come to the proof that symmetrized covariant derivatives of $Q$ can be made to vanish. \begin{thm}[Main Theorem] \label{thm:symder_vanish} Let $M$ be a QC manifold. For any $q\in M$ and any $N\geq 2$, there is a choice of pseudohermitian structure $\eta$ such that all the symmetrized covariant derivatives of $Q$ with total order less than or equal to $N$ vanish at $q$; that is, \[ Q_{(ab,C)}(q) = 0 \text{ if } o(abC)\leq N.\] If we write $\eta = e^{2u} \tilde{\eta}$ for another pseudohermitian structure, we may arbitrarily choose the $1$-jet of $u$ at $q$. Once this is fixed, the Taylor series of $u$ at $q$ is uniquely determined. \end{thm} \begin{proof} We apply Lemma \ref{lem:order_m_vanish} repeatedly for $2\leq m \leq N$. This works because using $u\in \mathcal{P}_m$ as a conformal factor does not change terms of the form $Q_{ab,C}$ with $o(abC)<m$. Choosing the $1$-jet of $u$ allows us to inductively determine higher order parts of the Taylor series. \end{proof}
Using the above normalization for $Q$ and a host of identities from \cite{Vassilevetal:2006} and \cite{Vassilevetal:2007} we show that at the center point $q$, the Ricci tensor, scalar curvature, quaternionic contact torsion and many of their covariant derivatives vanish. \begin{thm} \label{thm:mainthm} Let $M$ be a QC manifold and $\eta$ a pseudohermitian structure for which the symmetrized covariant derivatives of the tensor $Q$ vanish to total order $4$ at a point $q$. The the following curvature and torsion terms vanish at $q$. \begin{gather*} S,\ \tau_{\alpha\beta},\ \mu_{\alpha\beta},\ L_{\alpha\beta},\ R_{\alpha\beta},\ T_{\alpha i \beta},\ T_{ijk} \\ T_{\alpha jk},\ S_{,\beta},\ \tensor{\mu}{\down{\alpha\beta,}\up\alpha},\ \tensor{\tau}{\down {\alpha\beta,} \up \alpha} \\ B_{ij},\ S_{,i},\ \tensor{S}{\down {,\alpha} \up \alpha},\ \tensor{\tau}{\down{\alpha\beta,}\up{\alpha\beta}},\ \tensor{\mu}{\down{\alpha\beta,}\up{\alpha\beta}},\ \tensor{R}{\down{\gamma i \beta}\up\gamma\down {, \alpha}}I^{i\beta\alpha}. \end{gather*} \end{thm} \begin{proof} Considering first the terms of $Q$ with multiindex of order $2$, we take the horizontal trace to find that at $q$, \[ \tensor{Q}{\down \alpha \up \alpha} = \frac{4n+1}{8(n+2)} S = 0,\] form which it is clear that the pseudohermitian scalar curvature vanishes. It follows that at $q$, \[0= Q_{\alpha\beta} = \frac{1}{2}\tau_{\alpha\beta} + \mu_{\alpha\beta}.\] Since $\tau$ and $\mu$ lie in different eigenspaces of the Casimir operator $\Upsilon$ by Proposition \ref{prop:torsionprop}, they must both be zero. Since $L_{\alpha\beta}$ and $R_{\alpha\beta}$ are determined by $\tau_{\alpha\beta}$, $\mu_{\alpha\beta}$ and $S$, they both vanish at $q$.
Now consider the terms of $Q$ with multiindex of order $3$. Looking at $Q_{\alpha i}=0$, we immediately see that $T_{\alpha jk}=0$ since $\tensor{\varepsilon}{\up {jk}\down i}$ is an isomorphism from $V$ to $\bigwedge^2V$.
Next we trace $Q_{(\alpha\beta,\gamma)}$ on any two indices to find \[ 0= \tensor{Q}{\down \alpha \up \alpha \down {,\beta}} + 2\tensor{Q}{\down{\alpha\beta,}\up\alpha} = \frac{(4n+1)(2n+1)}{16n(n+2)}S_{,\beta} + \tensor{\tau}{\down {\alpha\beta,} \up\alpha} + 2\tensor{\mu}{\down{\alpha\beta,} \up \alpha}.\] Since $T_{\alpha jk}=0$, combining this equation with equations \eqref{eq:taumuS} and \eqref{eq:tormu} shows that at $q$ the tensors $\tensor{\tau}{\down{\alpha\beta,}\up \alpha}$, $\tensor{\mu}{\down{\alpha\beta,}\up \alpha}$ and $S_{,\beta}$ satisfy the system of equations \[\left(\begin{array}{ccc}1 & 2 & \frac{(4n+1)(2n+1)}{16n(n+2)} \\1 & -6 & -\frac{3}{16n(n+2)} \\1 & 0 & -\frac{3}{16(n+2)}\end{array}\right)\left(\begin{array}{c}\tensor{\tau}{\down{\alpha\beta,}\up \alpha} \\\tensor{\mu}{\down{\alpha\beta,}\up \alpha} \\S_{,\beta}\end{array}\right) = \left(\begin{array}{c}0 \\0 \\0\end{array}\right).\] The coefficient matrix here is nonsingular and hence $\tensor{\tau}{\down{\alpha\beta,}\up \alpha}$, $\tensor{\mu}{\down{\alpha\beta,}\up \alpha}$ and $S_{,\beta}$ all vanish at $q$.
Moving on to terms of $Q$ with indices of order $4$, we see first that $Q_{ij}= -(1/16n)B_{(ij)}=0$. Next, tracing $Q_{\alpha i,\beta}$ on the horizontal indices gives \[\tensor{Q}{\down{\alpha i,}\up \alpha} = -\tensor{(A^{-1})}{\down{i\alpha}\up{j\beta}} \tensor{T}{\down{\beta kl,}\up\alpha} \tensor{\varepsilon}{\up {kl}\down j}.\] From equation \eqref{eq:firstbianchi2} and the fact that $\tau_{\alpha\beta}$, $\mu_{\alpha\beta}$ and $T_{\alpha i \beta}$ vanish at $q$, we know that \begin{equation} \label{eq:curvandtorsion} R_{kl\alpha\beta}\tensor{I}{\down j \up {\alpha\beta}} =T_{\beta kl,\alpha} \tensor{I}{\down j \up {\alpha\beta}}. \end{equation} Since the automorphism $A$ is parallel at $q$ and has inverse $A^{-1}=(1/8)(A+2)$ and $\tensor{T}{\down{\beta kl,}\up\alpha}$ is antisymmetric in the horizontal indices, we have \begin{align*}
\tensor{Q}{\down {\alpha i,}\up\alpha} &= -\tensor{(A^{-1})}{\down {i\alpha}\up {j\beta}} \tensor{T}{\down{\beta kl,}\up\alpha} \tensor{\varepsilon}{\up {kl}\down j} \\
&= -\frac{1}{4}\tensor{\varepsilon}{\up {jp}\down i}\tensor{I}{\down {p\alpha} \up \beta}\tensor{T}{\down{\beta kl,}\up\alpha} \tensor{\varepsilon}{\up {kl}\down j}\\
&= -\frac{1}{4}\tensor{\varepsilon}{\up {jp}\down i}\tensor{I}{\down {p\alpha} \up \beta}\tensor{R}{\down{kl\alpha}\up\beta} \tensor{\varepsilon}{\up {kl}\down j}\\
&= -\frac{1}{4}\tensor{\varepsilon}{\up{jk}\down i} B_{jk}. \end{align*} Equation (4.6) from \cite{Vassilevetal:2006} tells us that \[ \tensor{\varepsilon}{\up {jk}\down i}B_{jk} = -\frac{1}{4n(n+2)} S_{,i},\] thus tracing $Q_{(\alpha\beta,i)}$ on the horizontal indices yields \[ 0=\tensor{Q}{\down \alpha \up \alpha \down {,i}} + 2 \tensor{Q}{\down {\alpha i,} \up \alpha} = \frac{4n^2+n+1}{8n(n+2)}S_{,i} .\] Since the $V$-volume form is an isomorphism, this also shows that the antisymmetric part of $B_{ij}$ vanishes. We already know the symmetric part vanishes, so $B_{ij}=0$ at $q$.
Finally, using \eqref{eq:curvandtorsion} and the the fact that the almost complex structures and the $V$-volume form are parallel at $q$, equations \eqref{eq:taumuS}, \eqref{eq:tormu} and \eqref{eq:VHricci} become \begin{align*} 0&=\tensor{\tau}{\down {\alpha\beta,}\up {\beta\alpha}} - 6 \tensor{\mu}{\down {\alpha\beta,}\up {\beta\alpha}} + \frac{4n-1}{2}\tensor{B}{\down i \up i} - \frac{3}{16n(n+2)}\tensor{S}{\down{,\alpha}\up\alpha} \\ 0&=\tensor{\tau}{\down {\alpha\beta,}\up {\beta\alpha}} + \frac{n+2}{2}\tensor{B}{\down i \up i} - \frac{3}{16(n+2)}\tensor{S}{\down{,\alpha}\up\alpha}\\ 0 &=\tensor{\tau}{\down {\alpha\beta,}\up {\beta\alpha}} - 3\tensor{\mu}{\down {\alpha\beta,}\up {\beta\alpha}} -2\tensor{B}{\down i \up i} - \tensor{R}{\down{\gamma i \beta}\up\gamma\down , \up \alpha}\tensor{I}{\up{i\beta}\down \alpha}. \end{align*} We know $Q_{(\alpha\beta,\gamma\delta)}=0$ and so tracing on any two pairs of indices yields \[0= 2\tensor{Q}{\down{\alpha\beta,}\up{\alpha\beta}} + \tensor{Q}{\down \alpha \up \alpha \down {,\beta}\up \beta} = \tensor{\tau}{\down{\alpha\beta,}\up{\alpha\beta}}+2\tensor{\mu}{\down{\alpha\beta,}\up{\alpha\beta}} + \frac{(4n+1)(2n+1)}{16n(n+2)}\tensor{S}{\down {,\alpha}\up\alpha}.\] Since $B_{ij}=0$, we therefore have the following system of equations \[\left(\begin{array}{cccc}1 & -6 & -\frac{3}{16n(n+2)} & 0 \\1 & 0 & -\frac{3}{16(n+2)} & 0 \\1 & -3 & 0 & -1 \\1 & 2 & \frac{(4n+1)(2n+1)}{16n(n+2)} & 0\end{array}\right)\left(\begin{array}{c}\tensor{\tau}{\down{\alpha\beta,}\up{\alpha\beta}} \\\tensor{\mu}{\down{\alpha\beta,}\up{\alpha\beta}} \\\tensor{S}{\down{,\alpha}\up\alpha} \\\tensor{R}{\down{\gamma i \beta}\up\gamma\down , \up \alpha}\tensor{I}{\up{i\beta}\down \alpha}\end{array}\right)=\left(\begin{array}{c}0 \\0 \\0 \\0\end{array}\right).\] As before, the coefficient matrix is nonsingular, and hence each of $\tensor{\tau}{\down{\alpha\beta,}\up{\alpha\beta}}$, $\tensor{\mu}{\down{\alpha\beta,}\up{\alpha\beta}}$, $\tensor{S}{\down{,\alpha}\up\alpha}$, and $\tensor{R}{\down{\gamma i \beta}\up\gamma\down , \up \alpha}\tensor{I}{\up{i\beta}\down \alpha}$ vanishes at $q$. This completes the proof.
\end{proof} \section{Scalar polynomial invariants} \label{sec:scal_invariants}
\subsection{More normalizations}
A critical next step in the solution to the Yamabe problem is to consider an asymptotic expansion of the Yamabe functional for a suitable class of test functions. In doing so, we expect to encounter as coefficients certain polynomial tensors in the QC curvature and torsion, to which we are able to assign a weight, defined below. By analogy with the conformal and CR cases, we expect to be required to consider terms that have weight no more than four. Further, the work of the preceding section will allow us to show that, at the origin in QC pseudohermitian normal coordinates, our normalizations imply that the only such tensors of weight at most four are dimensional constants and the square norm of the QC conformal curvature tensor \eqref{eq:QCconfcurv}.
Let us now define the weight of a tensor as follows. \begin{defn} Suppose $F$ is a homogeneous polynomial in $(x,t)$ whose coefficients are polynomial expressions in the curvature, torsion and the covariant derivatives at $q$. We define the \emph{weight} $w(F)$ recursively by \begin{enumerate} \item $w(T_{abc,D}(q)) = o(bcD) - o(a)$, \item $w(R_{abcd,E}(q)) = o(abcE)-o(d) = o(abE)$ since $c$ and $d$ always have the same order, \item $w(F_1F_2) = w(F_1)+w(F_2)$, \item $w(g_{ab}(q)) = w(g^{ab}(q)) = w(I_{i\alpha\beta}(q)) = w(\varepsilon_{ijk}(q)) = w(c) = 0$, \item if $w(F_A)=m$ for all $A$, then $w(\sum_A F_A x^A) = m$. \end{enumerate} Here $c$ denotes an arbitrary constant, independent of the pseudohermitian structure. We also let $w(0)=m$ for all $m$. \end{defn}
We will be interested in the polynomials of weight less than or equal to $4$. To simplify matters, we recall that $R_{abij}$ is determined by $R_{ab\alpha\beta}$ and the only terms of weight $1$ are identically $0$, namely $T_{\alpha\beta\gamma}$ and $T_{ij\alpha}$. Table \ref{tab:weights} lists the remaining curvature and torsion terms, organized by weight. \begin{table}[b] \centering \caption{Curvature and torsion terms of weight less than or equal to $4$.} \begin{tabular}{ccccc} $0$ & $2$ & $3$ & $4$ \\\hline $g_{\alpha\beta}$ & $T_{\alpha i \beta}$ & $T_{\alpha ij}$ & $T_{\alpha ij,\beta}$ \\ $g_{ij}$ & $T_{ijk}$ & $T_{\alpha i \beta,\gamma}$ & $T_{\alpha i \beta, \gamma\delta}$ \\ $I_{i\alpha\beta}$ & $R_{\alpha\beta\gamma\delta}$ & $R_{\alpha\beta\gamma\delta,\rho}$ & $T_{\alpha i \beta, j}$ \\ $\varepsilon_{ijk}$ & & $R_{\alpha i \beta \gamma}$ & $R_{\alpha\beta\gamma\delta,\rho\sigma}$ \\
& & & $R_{\alpha\beta\gamma\delta,i}$ \\
& & & $R_{\alpha i \beta\gamma,\delta}$ \\
& & & $R_{ij\beta\gamma}$ \end{tabular} \label{tab:weights} \end{table} Also, recall that the torsion term $T_{\alpha i \beta}$ is determined by the tensors $\tau_{\alpha\beta}$ and $\mu_{\alpha\beta}$ as in equation \eqref{eq:qctorsion}.
At this point it is convenient to introduce a few more tensors and relations between them from those found in \cite{Vassilevetal:2006}. These are all found by contracting the curvature tensor against the almost complex structures in various ways. We define \begin{equation} \label{eq:ricciforms} \rho_{iab} = \frac{1}{4n}R_{ab \alpha\beta} \tensor{I}{\down i \up {\beta\alpha}}, \ \zeta_{iab} = \frac{1}{4n} R_{\alpha ab \beta}\tensor{I}{\down i \up {\beta\alpha}}, \ \sigma_{iab} = \frac{1}{4n} R_{\alpha\beta ab} \tensor{I}{\down i \up {\beta\alpha}}. \end{equation} These tensors satisfy the following relations, found in \cite[Lemma 3.11]{Vassilevetal:2006} and \cite[Theorem 2.4]{Vassilevetal:2007}. \begin{prop} \label{prop:ricciprop} We may write the tensors $\rho$, $\zeta$, and $\sigma$ in terms of the torsion and scalar curvature as \begin{align} \rho_{i\alpha\beta} &= \frac{1}{2}(\tau_{\alpha\gamma}\tensor{I}{\down i \up \gamma \down \beta} - \tau_{\gamma\beta}\tensor{I}{\down i \up \gamma \down \alpha}) + 2 \mu_{\alpha\gamma}\tensor{I}{\down i \up \gamma\down \beta} - \frac{S}{8n(n+2)}I_{i\alpha\beta}, \label{eq:rho}\\ \zeta_{i\alpha\beta} &= -\frac{2n+1}{4n}\tau_{\alpha\gamma}\tensor{I}{\down i \up \gamma \down \beta} + \frac{1}{4n}\tau_{\gamma\beta}\tensor{I}{\down i \up \gamma \down \alpha} + \frac{2n+1}{2n}\mu_{\alpha\gamma}\tensor{I}{\down i \up \gamma\down \beta} + \frac{S}{16n(n+2)}I_{i\alpha\beta}, \label{eq:zeta}\\ \sigma_{i\alpha\beta} &= \frac{n+2}{2n}(\tau_{\alpha\gamma}\tensor{I}{\down i \up \gamma \down \beta} - \tau_{\gamma\beta}\tensor{I}{\down i \up \gamma \down \alpha}) - \frac{S}{8n(n+2)}I_{i\alpha\beta}. \label{eq:sigma} \end{align} Further, $\rho$ and $\sigma$ are antisymmetric in the horizontal indices, and \begin{equation}\label{eq:rzsscalar} \rho_{i\alpha\beta}I^{i\alpha\beta} = \sigma_{i\alpha\beta}I^{i\alpha\beta}=-\frac{3S}{2(n+2)},\quad \zeta_{i\alpha\beta}I^{i\alpha\beta} = \frac{3S}{4(n+2)}. \end{equation} Finally, the curvature tensor $R_{ab\alpha\beta}$ satisfies \[ R_{ab\alpha\beta} = \mathfrak{R}_{ab\alpha\beta} + \rho_{iab}\tensor{I}{\up i \down {\alpha\beta}},\] where $\mathfrak{R}_{ab\alpha\beta}$ is the $\mathfrak{sp}(n)$ component of $R_{ab\alpha\beta}$, and hence commutes with the almost complex structures in the second pair of indices. \end{prop}
As mentioned above, our interest lies in an asymptotic expansion of the Yamabe functional for which we will need to consider scalar pseudohermitian invariants of weight at most $4$. In particular we would like to know that, for a pseudohermitian structure normalized as in section \ref{sec:coords}, the only interesting terms are constants independent of the structure, and the square norm of the QC conformal curvature tensor. This is the content of the following \begin{thm} \label{thm:noscalarterms} Let $M$ be a QC manifold with pseudohermitian structure $\eta$ normalized according to Theorems \ref{thm:symder_vanish} and \ref{thm:mainthm}. Then, at the center of the normalization, $q$, the only invariant scalar quantities of weight no more than $4$ constructed as polynomials from the invariants listed in Table \ref{tab:weights} are constants independent of the structure and $\norm{W}^2$, the squared norm of the QC conformal curvature tensor; in particular, all other invariant scalar terms vanish at $q$. \end{thm} \begin{proof} We consider the terms by weight. First we notice that the composition law of the almost complex structures puts an upper bound on the number of such factors that appear in any complete contraction. Namely, the number of almost complex structures is no more than half the number of horizontal indices, since otherwise, some of the almost complex structures would contract together, resulting in a reduction to fewer such structures by equation \eqref{eq:ACS}.
Before we begin the consideration of the invariants, it is first useful to recall several of the important identities that we have described in previous sections. In particular we will be using the following equations repeatedly: \begin{gather*} \tensor{T}{\up \alpha\down {i\beta}} = \frac{1}{4}(\tensor{\tau}{\up \alpha \down \gamma} \tensor{I}{\down i \up \gamma \down \beta} + \tensor{I}{\down i \up \alpha \down \gamma} \tensor{\tau}{\up \gamma \down \beta}) + \tensor{I}{\down i \up \alpha \down \gamma}\tensor{\mu}{\up \gamma \down \beta},\tag{\ref{eq:qctorsion}}\\ R_{\alpha\beta}= (2n+2)\tau_{\alpha\beta} + 2(2n+5)\mu_{\alpha\beta} + \frac{S}{4n}g_{\alpha \beta}, \tag{\ref{eq:ricci}}\\ \intertext{and the fact that} \tensor{A}{\up \alpha \down \beta}\tensor{I}{\down i \up \beta \down \gamma} = \tensor{I}{\down i \up \alpha \down \beta} \tensor{A}{\up \beta \down \gamma} \text{ for any } A\in \mathfrak{sp}(n). \end{gather*} We will also need the identities of Theorem \ref{thm:mainthm} and Proposition \ref{prop:ricciprop}.
For the terms of weight $0$ and $1$, the proposition is clear. For terms of weight $2$, we know that $T_{\alpha i \beta}$ and $T_{ijk}$ already vanish at $q$ by Proposition \ref{prop:torsionprop}, since the normalizations of Theorem \ref{thm:mainthm} guarantee that $\tau_{\alpha\beta}$, $\mu_{\alpha\beta}$ and $S$ all vanish there. Then clearly any contractions of these factors also vanish. From the curvature $4$-tensor, the only complete contraction involving only the metric is the scalar curvature which vanishes at $q$. The remaining contractions involve two almost complex structures contracted on their vertical indices and yield contractions of the tensors $\rho_{i\alpha\beta}$, $\sigma_{i\alpha\beta}$ and $\zeta_{i\alpha\beta}$ with the almost complex structures. Then Proposition \ref{prop:ricciprop} shows that these all reduce to multiples of the scalar curvature and hence vanish at $q$.
The terms of weight $3$ are easier to deal with since these factors all have an odd number of horizontal indices. Since these must be contracted in pairs by either $g_{\alpha\beta}$ or $I_{i\alpha\beta}$, there can be no scalars constructed from them.
Finally, the terms of weight $4$ are the most complicated. There are two ways in which we can arrive at a term of weight $4$: by taking a product of two factors of weight $2$, or taking a single factor of weight $4$, and then applying terms of weight $0$ to contract to a scalar.
Let us first consider the case of a product of two factors of weight $2$. In such a case the only possible contractions involve a square of the curvature, since all the torsion terms vanish even before taking contractions. We will break this case into smaller sets based on the number of almost complex structures appearing in the contractions. Also, we note that because the torsion terms vanish, along with $\rho_{i\alpha\beta}$, the curvature tensor satisfies all the standard Riemannian algebraic Bianchi identities, and commutes with the almost complex structures in either the first or second pair of indices. Further, the two horizontal indices in any metric or almost complex structure factor must be split between the two curvature factors; if not, one of the curvature factors becomes either a Ricci tensor or one of $\rho_{i\alpha\beta}$, $\sigma_{i\alpha\beta}$, or $\zeta_{i\alpha\beta}$, all of which vanish at $q$.
\begin{enumerate} \item \emph{No almost complex structures:} In this case all contractions are handled by the metric. By the reasoning above, we may assume the first curvature factor is $R_{\alpha\beta\gamma\delta}$ and that the second is the raised index version, with the indices some permutation of $\alpha\beta\gamma\delta$. Then the antisymmetry in the first and second pair of indices and the ability to switch the first pair with the second pair reduces the 24 possibilities to the following three: \begin{gather*} R_{\alpha\beta\gamma\delta}R^{\alpha\beta\gamma\delta} , \\ R_{\alpha\beta\gamma\delta}R^{\alpha\gamma\beta\delta}, \text{ and } R_{\alpha\beta\gamma\delta}R^{\alpha\delta\beta\gamma}. \end{gather*} The first term is the squared norm of the QC conformal curvature tensor, since by \eqref{eq:QCconfcurv} $R_{\alpha\beta\gamma\delta}=W_{\alpha\beta\gamma\delta}$ when $L_{\alpha\beta}=0$. The last two terms here are negatives of each other since $R_{\alpha\beta\gamma\delta}$ is antisymmetric in $\gamma$ and $\delta$. Then the algebraic Bianchi identity and vanishing of the torsion at $q$ show that \begin{equation}\label{eq:usingalgBianchi}
0= R_{\alpha\beta\gamma\delta}(R^{\alpha\beta\gamma\delta}+R^{\beta\gamma\alpha\delta}+R^{\gamma\alpha\beta\delta}) = \norm{W}^2 -2 R_{\alpha\beta\gamma\delta}R^{\alpha\gamma\beta\delta}. \end{equation} Thus the remaining terms are also multiples of the squared norm of the QC conformal curvature tensor.
\item \emph{One almost complex structure:} Such terms are not possible since they would leave an uncontracted vertical index.
\item \emph{Two almost complex structures:} Two almost complex structures are necessarily contracted on their vertical indices, yielding a Casimir operator. Further, to contract the remaining horizontal indices we require two metric factors. Here we take the first factor to be $R_{\alpha\beta\gamma\delta}$ or $R_{\alpha\gamma\beta\delta}$, so that we are always contracting on the $\alpha$ and $\beta$ indices, and leave the second to be some contraction of curvature and the Casimir operator. By rearranging indices using symmetries, up to a sign, we may assume that the first index of the second curvature factor is contracted against the first index of the first curvature factor. For the second metric contraction, it may either be on the second indices of both curvature factors, on the third indices of both factors, or up to a sign, between the second index of the first factor and the third index of the second factor. Considering these possibilities yields the following list: \begin{subequations} \begin{gather} R_{\alpha\beta\gamma\delta}\tensor{R}{\up {\alpha\beta} \down {\mu \nu}} \tensor{I}{\down i \up {\gamma\mu}}I^{i\delta\nu} \label{eq:ACS2.1},\\ R_{\alpha\beta\gamma\delta}\tensor{R}{\up \alpha \down \mu \up \beta \down \nu} \tensor{I}{\down i \up {\gamma\mu}}I^{i\delta\nu} = - R_{\alpha\beta\gamma\delta}\tensor{R}{\up \alpha \down \nu \up \beta \down \mu} \tensor{I}{\down i \up {\gamma\mu}}I^{i\delta\nu} \label{eq:ACS2.2},\\ R_{\alpha\gamma\beta\delta}\tensor{R}{\up \alpha \down \mu \up \beta \down \nu} \tensor{I}{\down i \up {\gamma\mu}}I^{i\delta\nu} \label{eq:ACS2.3},\\ R_{\alpha\gamma\beta\delta}\tensor{R}{\up \alpha \down \nu \up \beta \down \mu} \tensor{I}{\down i \up {\gamma\mu}}I^{i\delta\nu} \label{eq:ACS2.4}. \end{gather} \end{subequations} The first term is a multiple of $\norm{W}^2$ since $R_{\alpha\beta\gamma\delta}$ is in $\mathfrak{sp}(n)\otimes \mathfrak{sp}(n)$ at $q$ and hence commutes with the almost complex structures. That is, \[ R_{\alpha\beta\gamma\delta}\tensor{R}{\up {\alpha\beta} \down {\mu \nu}} \tensor{I}{\down i \up {\gamma\mu}}I^{i\delta\nu} = R_{\alpha\beta\gamma\delta}\tensor{R}{\up {\alpha\beta\gamma} \down {\mu}} \tensor{I}{\down i \up {\mu}\down \nu}I^{i\delta\nu} = 3R_{\alpha\beta\gamma\delta}R^{\alpha\beta\gamma\delta}.\] The terms \eqref{eq:ACS2.2} are negatives of each other and by an application of the algebraic Bianchi identity as in \eqref{eq:usingalgBianchi} are seen to be proportional to \eqref{eq:ACS2.1}. In \eqref{eq:ACS2.3} we compute \begin{equation} \label{eq:usingsymantisym} \begin{aligned} R_{\alpha\gamma\beta\delta}\tensor{R}{\up \alpha \down \mu \up \beta \down \nu} \tensor{I}{\down i \up {\gamma\mu}}I^{i\delta\nu} &=R_{\alpha\gamma\delta\nu}\tensor{R}{\up \alpha \down \mu \up \beta \down \nu} \tensor{I}{\down i \up {\gamma\mu}}I^{i\beta\delta} &&\text{by commuting}\\ &=R_{\alpha\gamma\beta\delta}\tensor{R}{\up \alpha \down \mu \up \nu \down \delta} \tensor{I}{\down i \up {\gamma\mu}}I^{i\nu\beta} &&\text{renaming indices}\\ &=-R_{\alpha\gamma\delta\beta}\tensor{R}{\up \alpha \down \mu \up \delta \down \nu} \tensor{I}{\down i \up {\gamma\mu}}I^{i\beta\nu} &&\text{by symmetries}\\ &=-R_{\alpha\gamma\beta\delta}\tensor{R}{\up \alpha \down \mu \up \beta \down \nu} \tensor{I}{\down i \up {\gamma\mu}}I^{i\delta\nu}. &&\text{renaming indices} \end{aligned} \end{equation} Therefore \eqref{eq:ACS2.3} vanishes at $q$. Finally, again using the algebraic Bianchi identity as in \eqref{eq:usingalgBianchi}, \eqref{eq:ACS2.4} is seen to be proportional to \eqref{eq:ACS2.1}.
\item \emph{Three almost complex structures:} Three almost complex structures is a viable option, where the three vertical indices are contract together with $\varepsilon^{ijk}$. The remaining horizontal indices are contracted with a metric factor, which by symmetries we may assume to be the first index in each curvature term. Then, using the remaining symmetries we have the following terms: \begin{subequations} \begin{gather} R_{\alpha\beta\gamma\delta} \tensor{R}{\up \alpha \down{\mu\nu\rho}} \tensor{I}{\down i \up {\beta\mu}}\tensor{I}{\down j \up {\gamma\nu}}\tensor{I}{\down k \up {\delta\rho}}\varepsilon^{ijk} \label{eq:ACS3.1},\\ R_{\alpha\beta\gamma\delta} \tensor{R}{\up \alpha \down{\rho\mu\nu}} \tensor{I}{\down i \up {\beta\mu}}\tensor{I}{\down j \up {\gamma\nu}}\tensor{I}{\down k \up {\delta\rho}}\varepsilon^{ijk} \label{eq:ACS3.2},\\ R_{\alpha\beta\gamma\delta} \tensor{R}{\up \alpha \down{\nu\rho\mu}} \tensor{I}{\down i \up {\beta\mu}}\tensor{I}{\down j \up {\gamma\nu}}\tensor{I}{\down k \up {\delta\rho}}\varepsilon^{ijk} \label{eq:ACS3.3}. \end{gather} \end{subequations} Using the fact that the curvature commutes with the almost complex structures in either the first or second pair of indices, an analysis similar to that of equation \eqref{eq:usingsymantisym} shows that \eqref{eq:ACS3.1} vanishes. By commuting the almost complex structures in the remaining terms with the curvature factors and then contracting the almost complex structures together, we reduce to the case of only two almost complex structures, which has already been dealt with. For example \begin{align*} R_{\alpha\beta\gamma\delta} \tensor{R}{\up \alpha \down{\rho\mu\nu}} \tensor{I}{\down i \up {\beta\mu}}\tensor{I}{\down j \up {\gamma\nu}}\tensor{I}{\down k \up {\delta\rho}}\varepsilon^{ijk} &= \tensor{R}{\down{\alpha\beta\gamma}\up\nu} \tensor{R}{\up \alpha \down{\rho\mu\nu}} \tensor{I}{\down i \up {\beta\mu}}\tensor{I}{\down j \up {\gamma}\down\delta}\tensor{I}{\down k \up {\delta\rho}}\varepsilon^{ijk} \\ &= \tensor{R}{\down{\alpha\beta\gamma}\up\nu} \tensor{R}{\up \alpha \down{\rho\mu\nu}} \tensor{I}{\down i \up {\beta\mu}}(-\delta_{jk}g^{\gamma\rho} + \varepsilon_{jkl}I^{l\gamma\rho}) \varepsilon^{ijk}\\ &= 2\tensor{R}{\down{\alpha\beta\gamma}\up\nu} \tensor{R}{\up \alpha \down{\rho\mu\nu}}\tensor{I}{\down i \up {\beta\mu}}I^{l\gamma\rho} , \end{align*} again by Proposition \ref{prop:ACSandVform}. By symmetries of the curvature tensor, this is equivalent to a multiple of \eqref{eq:ACS2.4}.
\item \emph{Four almost complex structures:} Finally, for the case of four almost complex structures we have the following possibilities up to symmetries: \begin{subequations} \begin{gather} R_{\alpha\beta\gamma\delta}R_{\mu\nu\rho\sigma}\tensor{I}{\down i \up {\alpha\mu}}\tensor{I}{\up i \up {\beta\nu}}\tensor{I}{\down j \up {\gamma\rho}}\tensor{I}{\up j \up {\delta\sigma}},\\ R_{\alpha\beta\gamma\delta}R_{\mu\rho\nu\sigma}\tensor{I}{\down i \up {\alpha\mu}}\tensor{I}{\up i \up {\beta\nu}}\tensor{I}{\down j \up {\gamma\rho}}\tensor{I}{\up j \up {\delta\sigma}},\\ R_{\alpha\beta\gamma\delta}R_{\mu\sigma\nu\rho}\tensor{I}{\down i \up {\alpha\mu}}\tensor{I}{\up i \up {\beta\nu}}\tensor{I}{\down j \up {\gamma\rho}}\tensor{I}{\up j \up {\delta\sigma}},\\ R_{\alpha\gamma\beta\delta}R_{\mu\rho\nu\sigma}\tensor{I}{\down i \up {\alpha\mu}}\tensor{I}{\up i \up {\beta\nu}}\tensor{I}{\down j \up {\gamma\rho}}\tensor{I}{\up j \up {\delta\sigma}},\\ R_{\alpha\gamma\beta\delta}R_{\mu\sigma\nu\rho}\tensor{I}{\down i \up {\alpha\mu}}\tensor{I}{\up i \up {\beta\nu}}\tensor{I}{\down j \up {\gamma\rho}}\tensor{I}{\up j \up {\delta\sigma}}. \end{gather} \end{subequations} As in the previous cases, commuting almost complex structures with the curvature factors shows the first term to be a multiple of $\norm{W}^2$, and shows that the remaining terms reduce to the case of only two or three almost complex structures, all of which have been dealt with. \end{enumerate}
Now we turn to the terms of weight $4$ that do not result from a product of terms of weight $2$. From $T_{\alpha ij,\beta}$, the only possible contraction is $T_{\alpha ij, \beta}\varepsilon^{ijk}\tensor{I}{\down k \up {\alpha\beta}}$. But from equations \eqref{eq:defineB} and \eqref{eq:firstbianchi} we see that this expression is $\tensor{B}{\down i \up i} = 0$.
Next, we consider $T_{\alpha i \beta, \gamma\delta}$. Instead of dealing with this directly, we recall equation \eqref{eq:qctorsion} which tells us that this is determined by $\tau_{\alpha\beta}$, $\mu_{\alpha\beta}$, their covariant derivatives, and the almost complex structures and their covariant derivatives. Since the almost complex structures are parallel at $q$ and $\tau$ and $\mu$ vanish there, we need only consider the terms $\tau_{\alpha\beta, \gamma\delta}$ and $\mu_{\alpha\beta,\gamma\delta}$. We deal with $\tau$, since the case for $\mu$ is identical. We know that $\tau_{\alpha\beta}$ is a trace-free tensor, and so $\tensor{\tau}{\down \alpha \up \alpha \down{,\beta} \up \beta}=0$ locally. Further by Theorem \ref{thm:mainthm}, $\tensor{\tau}{\down {\alpha\beta,}\up {\alpha\beta}}=0$ at $q$. Also, tracing $\tau$ against any of the almost complex structures is identically $0$, and so, at $q$ \[ \tau_{\alpha\beta,\gamma\delta} I^{i\alpha\beta}\tensor{I}{\down i \up {\gamma\delta}} = (\tau_{\alpha\beta} I^{i\alpha\beta}\tensor{I}{\down i \up {\gamma\delta}})_{,\gamma\delta} +\ldots\] Here ``$\ldots$'' denotes terms that contain either $\tau_{\alpha\beta}$ undifferentiated, or terms involving a first derivative of the almost complex structures. In either case, these terms are $0$ at $q$, and so $\tau_{\alpha\beta,\gamma\delta} I^{i\alpha\beta}\tensor{I}{\down i \up {\gamma\delta}}$ vanishes there. Finally we may form the trace $\tau_{\alpha\beta,\gamma\delta} I^{i\alpha\gamma}\tensor{I}{\down i \up {\beta\delta}}$. This term is dealt with similarly, by recalling that $\tau$ is in the $(-1)$-eigenspace of the Casimir operator $\Upsilon = I_iI^i$. As mentioned above, the case for $\mu$ is almost identical, since $\mu$ is trace-free, symmetric and in the $3$-eigenspace of $\Upsilon$.
Next in the list is $T_{\alpha i\beta,j}$, which we again deal with by considering $\tau_{\alpha\beta,j}$ and $\mu_{\alpha\beta,j}$. As above, the two tensors are handled almost identically, and so we present only the case for $\tau$. The only trace we may form is $\tau_{\alpha\beta,j}I^{j\alpha\beta}$. But as above, since the almost complex structures are parallel at $q$ we have, at $q$, \[ \tau_{\alpha\beta,j} I^{j\alpha\beta} = (\tau_{\alpha\beta}I^{j\alpha\beta})_{,j} = 0.\]
Now for the curvature terms of weight $4$. Beginning with $R_{ij\alpha\beta}$, the only complete contraction is $R_{ij\alpha\beta}\tensor{I}{\down k \up {\alpha\beta}} \varepsilon^{ijk} = \tensor{B}{\down i\up i} = 0$. Next we consider $R_{\alpha\beta\gamma\delta,i}$. As mentioned above, we need only consider contractions involving no more than two almost complex structures. Further, since there is already a vertical index, we must have at least one. If we have one almost complex structure, we must also have a metric term, and the only possible metric contraction in this case is to $R_{\alpha\beta,i}$. Since the Ricci tensor is symmetric, we see that $R_{\alpha\beta,i}I^{i\alpha\beta}=0$. If we admit two almost complex structures, we need to include an $\varepsilon^{ijk}$ term to contract the vertical indices. Since the almost complex structures are parallel at $q$ we have the following possibilities: \begin{equation} \rho_{j\alpha\beta,i}\tensor{I}{\down k \up {\alpha\beta}} \varepsilon^{ijk},\quad \sigma_{j\alpha\beta,i}\tensor{I}{\down k \up {\alpha\beta}} \varepsilon^{ijk},\quad \zeta_{j\alpha\beta,i}\tensor{I}{\down k \up {\alpha\beta}} \varepsilon^{ijk}. \end{equation} But at $q$, Proposition \ref{prop:ricciprop} shows that modulo terms that vanish at $q$, these three tensors can be expressed in terms of contractions of $\tau_{\alpha\beta,i}$, $\mu_{\alpha\beta,i}$ and $S_{,i}$, all of which have already been dealt with above.
Next, consider $R_{\alpha i \beta \gamma, \delta}$. From \cite[Theorem 3.1]{Vassilevetal:2007} we know that \begin{multline} R_{\alpha i \beta \gamma} = \mu_{\delta\gamma,\alpha}\tensor{I}{\down i \up \delta \down \beta} + \frac{1}{4}(\tau_{\gamma\delta,\beta}\tensor{I}{\down i \up \delta \down \alpha} + \tau_{\delta\alpha,\beta} \tensor{I}{\down i \up \delta\down \gamma} - \tau_{\delta \alpha,\gamma} \tensor{I}{\down i \up \delta\down \beta} - \tau_{\beta\delta,\gamma}\tensor{I}{\down i \up \delta \down \alpha})\\ - I_{j\beta\alpha}\tensor{T}{\down \gamma \up j \down i} + I_{j\gamma\alpha}\tensor{T}{\down \beta \up j \down i} + I_{j\gamma\beta}\tensor{T}{\down \alpha \up j \down i}. \end{multline} Using this and contracting $R_{\alpha i \beta\gamma,\delta}$ against all possible combinations of weight $0$ terms to form scalar quantities yields terms that contain one of the following: \begin{itemize} \item a covariant derivative of an almost complex structure; \item a contraction of $T_{\alpha ij,\beta}$ that reduces to $\tensor{B}{\down i \up i}$ at $q$ (see \eqref{eq:defineB} and \eqref{eq:firstbianchi}); \item the trace of $T_{\alpha i j ,\beta}$ on the vertical indices; \item the trace of the action of the Casimir operator on $\tau$ or $\mu$; or \item a second divergence of $\tau$ or $\mu$. \end{itemize} Each of these terms has been considered already in the above analysis and shown to be $0$ at $q$. Thus, there are no nontrivial scalars to be formed from $R_{\alpha i \beta\gamma,\delta}$.
Finally we come to $R_{\alpha\beta\gamma\delta,\rho\sigma}$. There are six complete contractions that contain a metric contraction, namely \begin{gather*} \tensor{R}{\down{\alpha\beta,} \up{\alpha\beta}},\quad \tensor{S}{\down {,\alpha} \up \alpha},\quad R_{\alpha\beta,\gamma\delta}\tensor{I}{\down i \up {\alpha\beta}}\tensor{I}{\up{i\gamma\delta}},\quad R_{\alpha\beta,\gamma\delta}\tensor{I}{\down i \up {\alpha\gamma}}\tensor{I}{\up{i\beta\delta}},\\ \tensor{R}{\down{\alpha\beta\gamma\delta,\rho}\up \rho} \tensor{I}{\down i \up {\alpha\beta}}\tensor{I}{\up{i\gamma\delta}},\quad \tensor{R}{\down{\alpha\beta\gamma\delta,\rho}\up \rho} \tensor{I}{\down i \up {\alpha\gamma}}\tensor{I}{\up{i\beta\delta}}. \end{gather*} The first two here vanish at $q$ since $\tensor{\tau}{\down{\alpha\beta,}\up {\alpha\beta}}=\tensor{\mu}{\down{\alpha\beta,}\up {\alpha\beta}}=\tensor{S}{\down{,\alpha}\up {\alpha}}=0$ and the Ricci tensor is determined by these tensors via \eqref{eq:ricci}. The third term vanishes because the horizontal Ricci tensor is symmetric, while the fourth vanishes at $q$ since it represents the action of the Casimir operator on the second covariant derivatives of $\tau$ and $\mu$, which has already been considered above. The last two terms may be calculated by first tracing with the almost complex structures and then differentiating, since the differences vanish at $q$. We can then easily compute that these also vanish at $q$. Notice that all terms that contain a second covariant derivative of the almost complex structures also contain factors like $R_{\alpha\beta\gamma\delta}\tensor{I}{\down i\up{\delta\gamma}} = \rho_{i\alpha\beta}=0$.
The remaining terms are to be found from $R_{\alpha\beta\gamma\delta,\rho\sigma}$ by contracting the indices in pairs with three distinct almost complex structures, and then using $\varepsilon^{ijk}$ to contract to a scalar. If we contract the two derivative indices with an almost complex structure, we recover only the antisymmetric part of the second covariant derivative, which may be rewritten in terms of only one covariant derivative in the vertical direction. But notice that we have already taken care of those terms. By the antisymmetry of the curvature in the first and second pairs of indices, it therefore remains to consider the following four terms \begin{subequations}\label{eq:ABCD} \begin{gather} A = R_{\alpha\beta\gamma\delta,\rho\sigma} I^{i\alpha\beta}I^{j\sigma\delta}I^{k\rho\gamma}\varepsilon_{ijk},\\ B= R_{\alpha\beta\gamma\delta,\rho\sigma} I^{i\delta\beta}I^{j\sigma\alpha}I^{k\rho\gamma}\varepsilon_{ijk},\\ C=R_{\alpha\beta\gamma\delta,\rho\sigma} I^{i\beta\gamma}I^{j\sigma\delta}I^{k\rho\alpha}\varepsilon_{ijk}, \\ D=R_{\alpha\beta\gamma\delta,\rho\sigma} I^{i\gamma\delta}I^{j\sigma\beta}I^{k\rho\alpha}\varepsilon_{ijk}. \end{gather} \end{subequations}
To show that these are all zero we recall the algebraic and differential Bianchi identities for the curvature tensor. These may be found, for example, in \cite{KN:1996}. Differentiating the algebraic Bianchi identity twice and the differential Bianchi identity once we have \begin{multline} R_{\alpha\beta\gamma\delta,\rho\sigma} + R_{\beta\gamma\alpha\delta,\rho\sigma} + R_{\gamma\alpha\beta\delta,\rho\sigma} =\\ 2 ( T_{\delta i \gamma} \tensor{I}{\up i \down{\beta\alpha}} + T_{\delta i \alpha}\tensor{I}{\up i \down{\gamma\beta}} + T_{\delta i \beta}\tensor{I}{\up i \down{\alpha\gamma}})_{,\rho\sigma}, \label{eq:bianchione} \end{multline} \begin{multline} R_{\alpha\beta\gamma\delta,\rho\sigma} + R_{\beta\rho\gamma\delta,\alpha\sigma} + R_{\rho\alpha\gamma\delta,\beta\sigma} =\\ 2(\tensor{I}{\up i \down{\beta\alpha}}R_{\rho i \gamma\delta} + \tensor{I}{\up i \down {\rho\beta}}R_{\alpha i \gamma \delta} + \tensor{I}{\up i \down{\alpha\gamma}}R_{\beta i \gamma\delta})_{,\sigma} \label{eq:bianchitwo}. \end{multline} Contracting the right hand sides of \eqref{eq:bianchione} and \eqref{eq:bianchitwo} with the almost complex structures and $\varepsilon$ as in \eqref{eq:ABCD} yields scalar terms constructed of quantities we have already shown to vanish at $q$. Contracting the left hand sides then, we arrive at the following equations for $A$, $B$, $C$ and $D$ at $q$, \[A +2C = 0,\quad B-C-D = 0,\quad A-2C = 0, \quad 2B=0.\] For example, \begin{align*} (R_{\alpha\beta\gamma\delta,\rho\sigma} + R_{\beta\gamma\alpha\delta,\rho\sigma} + &R_{\gamma\alpha\beta\delta,\rho\sigma}) I^{i\alpha\beta}I^{j\sigma\delta}I^{k\rho\gamma}\varepsilon_{ijk} \\ &= A + R_{\beta\gamma\alpha\delta,\rho\sigma} I^{i\alpha\beta}I^{j\sigma\delta}I^{k\rho\gamma}\varepsilon_{ijk} \\ &\qquad+ R_{\gamma\alpha\beta\delta,\rho\sigma} I^{i\alpha\beta}I^{j\sigma\delta}I^{k\rho\gamma}\varepsilon_{ijk} \\ &= A + R_{\alpha\beta\gamma\delta,\rho\sigma}I^{i\gamma\alpha}I^{j\sigma\delta}I^{k\rho\beta}\varepsilon_{ijk}\\ &\qquad + R_{\alpha\beta\gamma\delta,\rho\sigma}I^{i\beta\gamma}I^{j\sigma\delta}I^{k\rho\alpha}\varepsilon_{ijk}\\ &= A - R_{\alpha\beta\gamma\delta,\rho\sigma}I^{i\gamma\beta}I^{j\sigma\delta}I^{k\rho\alpha}\varepsilon_{ijk} + C\\ &= A + R_{\alpha\beta\gamma\delta,\rho\sigma}I^{i\beta\gamma}I^{j\sigma\delta}I^{k\rho\alpha}\varepsilon_{ijk} + C\\ &= A + 2C. \end{align*} Again, this equals zero since the terms on the left-hand side of equation \eqref{eq:bianchione} have already been show to contract to $0$. From these equations, it is clear that $A=B=C=D=0$ at $q$. This concludes the proof. \end{proof}
\end{document} |
\begin{document}
\begin{center} {\Large\bf Twisting the fake monster superalgebra}\\[0.8cm] Nils R. Scheithauer\footnote{Supported by a DAAD grant.},\\ Mathematisches Seminar der Universit\"at Hamburg,\\ Bundesstr. 55, 20146 Hamburg, Germany\\ \end{center} \vspace*{1.5cm}
\noindent We calculate twisted denominator identities of the fake monster superalgebra and use them to construct new examples of supersymmetric generalized Kac-Moody superalgebras. Their denominator identities give new infinite product identities.
\section{Introduction} There are 3 generalized Kac-Moody algebras or superalgebras which represent the physical states of a string moving on a certain variety, namely the monster algebra, the fake monster algebra \cite{B} and the the fake monster superalgebra \cite{NRS}. The no-ghost theorem from string theory can be used to construct actions of finite groups on these algebras. For example the monster group acts on the monster algebra. Applying elements of the finite groups to the Weyl denominator identity of these algebras gives twisted denominator identities. They can be calculated explicitly because the simple roots of the algebras are known. For the monster algebra and the fake monster algebra this has been done in \cite{B}. There Borcherds uses twisted denominator identities of the monster algebra to prove the moonshine conjectures. In this paper we calculate twisted denominator identities of the fake monster superalgebra and use them to construct new examples of supersymmetric generalized Kac-Moody superalgebras. They have similar properties as the fake monster superalgebra. For example they have no real roots and the Weyl vector is zero. Their denominator identities give new infinite product expansions. In a forthcoming paper we show that they define automorphic forms of singular weight.
We describe the sections of this paper.
In the second section we recall some facts about the fake monster superalgebra and describe the action of an extension of the Weyl group of $E_8$ on this algebra.
In the third section we derive the general expression of the twisted denominator identity corresponding to elements in this group of odd order.
In the last section we calculate the twisted denominator identities explicitly for certain elements of order 3 and 7. We construct two supersymmetric generalized Kac-Moody superalgebras of rank 6 and 4 and describe their simple roots and the multiplicities.
\section{The fake monster superalgebra}
In this section we recall some results about the fake monster superalgebra from \cite{NRS} and construct an action of $2^8.2.Aut(E_8)$ on it. The main difference to the bosonic case described in \cite{B} is that we have to work with a double cover of the automorphism group of the light cone lattice rather than with the automorphism group of that lattice.
The fake monster superalgebra $G$ can be constructed as the space of physical states of a chiral N$=$1 superstring moving on a 10 dimensional torus. $G$ is a generalized Kac-Moody superalgebra. The root lattice of $G$ is the 10 dimensional even unimodular Lorentzian lattice $II_{9,1}=E_8 \oplus II_{1,1}$. A nonzero element $\alpha \in II_{9,1}$ is a root of $G$ if and only if $\alpha^2\leq 0$. In particular $G$ has no real roots. This reflects the fact that the superstring has no tachyons. The multiplicity of a root $\alpha$ is given by $mult_0(\alpha)=mult_1(\alpha)=c(-\alpha^2/2)$ where $c(n)$ is the coefficient of $q^n$ in $ 8 \eta(q^2)^8/\eta(q)^{16} = 8+128q+1152q^2+7680q^3+42112q^4+\ldots$ and $\eta(q)$ is the Dedekind eta function. There are 2 cones of negative norm vectors in $II_{9,1}$. We define one of them as the positive cone and denote the closure of the positive cone by $II_{9,1}^+$. Then the positive roots of $G$ are the nonzero vectors in $II_{9,1}^+$ and the simple roots of $G$ are the positive roots of zero norm. The simple roots have multiplicity 8 as even and odd roots. The Cartan subalgebra of $G$ is isomorphic to the vector space generated by $II_{9,1}$. Since $G$ has no real roots the Weyl group is trivial. The Weyl vector of $G$ is zero.
We describe some results about the 8 dimensional real spin group. For more details confer \cite{J}. Let $\mathbb O$ be the real 8 dimensional algebra of octonions, i.e. the unique real alternative division algebra of dimension 8. $\mathbb O$ has a quadratic form $N$ permitting composition. For $b\in {\mathbb O}$ define the left multiplication $L_b$, the right multiplication $R_b$ and the \mbox{operator} $U_b=L_bR_b=R_bL_b$. Let ${\mathbb O}_0$ be the ortho\-gonal complement of ${\mathbb R}1$ and denote $C({\mathbb O},N)$ the Clifford algebra generated by ${\mathbb O}$ with relations $a^2=N(a)1$. There is an isomorphism $\varepsilon$ from the even subalgebra $C^e({\mathbb O},N)$ of $C({\mathbb O},N)$ to the Clifford algebra $C({\mathbb O}_0,-N)$ mapping $1a$, $a\in {\mathbb O}$, to $a$, where we have identified the unit in $C({\mathbb O}_0,-N)$ with the unit in ${\mathbb O}$. $C({\mathbb O}_0,-N)$ acts naturally on $\mathbb O$ by left and right multiplication. An element $u$ of the spin group $\Gamma^e_0({\mathbb O},N)$ can be written as $u=1b_1 \ldots 1b_n$ with $\prod N(b_i)=1$ and $\varepsilon(u)= b_1 \ldots b_n$. The actions $\rho_L(u)=L_{b_1} \ldots L_{b_n}, \, \rho_R(u)=R_{b_1} \ldots R_{b_n}$ and $\rho_V(u)=U_{b_1}\ldots U_{b_n}$ give three irreducible and inequivalent 8 dimensional re\-presentations of the spin group called conjugate spinor, spinor and vector representation. They are related by triality, i.e. $\rho_V(u)(ab)=(\rho_L(u)a)(\rho_R(u)b)$ holds for all $a,b\in {\mathbb O}$. The image of $\Gamma^e_0({\mathbb O},N)$ under each of these re\-presentations is $SO(8)$. The kernel of $\rho_V : \Gamma^e_0({\mathbb O},N) \rightarrow SO(8)$ is $\{1,-1\}$ so that $\Gamma^e_0({\mathbb O},N)$ is a double cover of $SO(8)$. The representations $\rho_L,\rho_R$ and $\rho_V$ of the spin group induce representations of the Lie algebra $so(8)$ with weights $\frac{1}{2}(\pm 1,\ldots, \pm1)$ with an odd number of $-$ signs, $\frac{1}{2}(\pm 1,\ldots, \pm1)$ with an even number of $-$ signs and the permutations of $(\pm1,0,\ldots,0)$.
We embed $E_8$ into ${\mathbb O}$. Let $Aut(E_8)$ be the group of automorphisms of $E_8$ leaving the bilinear form invariant. Then $Aut(E_8)\subset SO(8)$ and the inverse image of $Aut(E_8)$ under $\rho_V$ is a double cover of $Aut(E_8)$.
$Aut(E_8)$ can also be constructed using the ring of integral octonions (cf. \cite{C}). This description implies that $E_8 \subset {\mathbb O}$ is also invariant under the actions $\rho_L$ and $\rho_R$.
Now we construct an action of an extension of the double cover $2.Aut(E_8)$ on the fake monster superalgebra. The extension $2^8.Aut(E_8)$ of $Aut(E_8)$ by $Ho\hspace{0.3mm}m(E_8,{\mathbb Z}_2)$ acts naturally on the vertex algebra $V_{E_8}$ of the lattice $E_8$. The same holds for the extension of $2.Aut(E_8)$ by $2^8$ where $2.Aut(E_8)$ acts by $\rho_V$. The vector space $E_8 \otimes {\mathbb R}[t^{-1}]t^{-\frac{1}{2}}$ is an abelian subalgebra of the Heisenberg algebra $E_8 \otimes {\mathbb R}[t,t^{-1}]t^{\frac{1}{2}}$. The exterior algebra $V_{NS}$ of $E_8 \otimes {\mathbb R}[t^{-1}]t^{-\frac{1}{2}}$ is a vertex superalgebra carrying a representation of the Virasoro algebra. $V_{NS}$ decomposes into eigenspaces of $L_0$ with eigenvalues in $\frac{1}{2}{\mathbb Z}$. $2.Aut(E_8)$ acts on $V_{NS}$ in the vector representation. We define the vertex superalgebra $V_0=V_{NS}\otimes V_{E_8}$. This algebra carries a representation of the Virasoro algebra of central charge $4+8=12$. We write $V_{0,n}$ for the subspace of $L_0$-degree $n\in \frac{1}{2}{\mathbb Z}$. The vector space $E_8 \otimes {\mathbb R}[t^{-1}]t^{-1}$ is an abelian subalgebra of the Heisenberg algebra $E_8 \otimes {\mathbb R}[t,t^{-1}]$. We define $V_R$ as the tensor product of the exterior algebra of $E_8 \otimes {\mathbb R}[t^{-1}]t^{-1}$ with the sum $S \oplus C$ of two $8$ dimensional spaces. The double cover of $Aut(E_8)$ acts in the vector representation on the first tensor factor and in the spinor resp. conjugate spinor representation on the second factor. $V_R$ can be given the structure of a $V_{NS}$-module. We decompose $V_R=V_R^+\oplus V_R^-$ where $V_R^+$ is the subspace generated by vectors $d_{-n_1}\wedge \ldots \wedge d_{-n_k}\otimes v$ where $v$ is in $C$ if $k$ is even and in $S$ if $k$ is odd and analogous for $V_R^-$. The projection of $V_R$ on $V_R^+$ is called GSO projection. We define $V_1=V_{R}^+\otimes V_{E_8}$ and adopt the same notations as for $V_0$. $V_1$ carries a representation the Virasoro algebra of central charge $12$. Now $2^8.2.Aut(E_8)$ acts on the fake monster superalgebra in the following way. We decompose the Cartan subalgebra ${\mathbb R}\otimes II_{9,1}$ by writing $II_{9,1}=E_8 \oplus II_{1,1}$. $2.Aut(E_8)$ acts in the vector representation on the vector space generated by $E_8$. Since $G$ is graded by $II_{9,1}=E_8 \oplus II_{1,1}$ it is also graded by $II_{1,1}$. We denote the corresponding spaces $G_a$ with $a\in II_{1,1}$ and the even resp. odd subspace $G_{0,a}$ resp. $G_{1,a}$. All these spaces are $E_8$-graded. By the no-ghost theorem (cf. \cite{GSW},\cite{P} or \cite{B}) the even subspace $G_{0,a}$ is isomorphic to $V_{0,(1-a^2)/2}$ and $G_{1,a}$ is isomorphic to $V_{1,(1-a^2)/2}$ as $E_8$-graded $2^8.2.Aut(E_8)$-module. This gives us a natural action of $2^8.2.Aut(E_8)$ on the fake monster superalgebra.
\section{The twisted denominator identities}
In this section we calculate twisted denominator identities of the fake monster superalgebra corresponding to elements in $2.Aut(E_8)$ of odd order.
The fake monster superalgebra $G$, like any generalized Kac-Moody superalgebra, can be written as direct sum $E\oplus H \oplus F$ where $H$ is the Cartan subalgebra and $E$ and $F$ are the subalgebras corresponding to the positive and negative roots. We have the standard sequence \[ \ldots \rightarrow \Lambda^2(E) \rightarrow \Lambda^1(E)
\rightarrow \Lambda^0(E) \rightarrow 0 \] with homology groups $H_i(E)$.
Note that $E=E_0\oplus E_1$ is a superspace so that the exterior algebra $\Lambda(E)$ is defined as the tensor algebra of $E$ divided by the two sided ideal generated by $u \otimes v + (-1)^{|u||v|} v\otimes u$.
The Euler-Poincar\'{e} principle implies \[ \Lambda^*(E) = H(E) \] where $\Lambda^* (E)=\oplus_{n\geq 0} (-1)^n \Lambda^n (E)$ is the alternating sum of exterior powers of $E$ and $H(E)=\oplus_{n\geq 0} (-1)^n H_n(E)$ is the alternating sum of homology groups of $E$. Both sides of this identity are graded by the root lattice $II_{9,1}$ of $G$ and the homogeneous subspaces are finite dimensional. The homology groups $H_n(E)$ can be calculated in the same way as for Kac-Moody algebras (cf. \cite{B}). The result is that $H_n(E)$ is the subspace of $\Lambda^n (E)$ spanned by the homogeneous vectors of $\Lambda^n (E)$ of degree $\alpha \in II_{9,1}$ with $\alpha^2=0$. We can work out the homology groups of $G$ explicitly because we know the simple roots. Denote the subspace of $H(E)$ with degree $\alpha \in II_{9,1}$ by $H(E)_{\alpha}$ and analogous for $E$. Let $\lambda$ be a primitive norm zero vector in $II_{9,1}^+$. Then ${\mathbb R} \oplus \oplus_{n>0} H(E)_{n \lambda}
= \Lambda^*(\oplus_{n>0}E_{n\lambda})$. The denominator identity of the fake monster superalgebra now follows easily by calculating the character on both sides of $\Lambda^*(E) = H(E)$.
More generally if $g$ is an automorphism of $G$ then $g$ commutes with the derivations $d_i : \Lambda^i(E) \rightarrow \Lambda^{i-1}(E)$ and we get a sequence\[ \ldots \rightarrow g(\Lambda^2(E)) \rightarrow g(\Lambda^1(E))
\rightarrow g(\Lambda^0(E)) \rightarrow 0 \, . \] $g$ induces an isomorphism between the homology groups of the two complexes above. Hence we can apply $g$ to both sides of the equation $\Lambda^*(E) = H(E)$ and take the trace. This gives a twisted denominator identity. It depends only on the conjugacy class of $g$ in the automorphism group of $G$.
Let $u$ be an element of $2.Aut(E_8)$ of odd order $N$.
The lattice $E_8$ has a unique central extension $\hat{E}_8$ by $ \{1,-1\}$ such that the commutator of any inverse images of $r,v$ in $E_8$ is $(-1)^{(r,v)}$. Since $\rho_V(u)$ has odd order it has a lift to $Aut (\hat{E}_8)=2^8.Aut (E_8)$ such that $\rho_V(u)^n$ fixes all elements of $\hat{E}_8$ which are in the inverse image of the vectors of $E_8$ fixed by $\rho_V(u)^n$ (cf. Lemma 12.1 in \cite{B}). We define $g$ as the automorphism of $G$ induced by this lift.
Let $E_8^u$ be the sublattice of $E_8$ fixed by $\rho_V(u)$. The natural projection $\pi :{\mathbb R} \otimes E_8 \rightarrow
{\mathbb R} \otimes E_8^u$ maps $E_8$ onto the dual lattice $E_8 ^{u*}$ because $E_8$ is unimodular. We define the Lorentzian lattice $L=E_8^u\oplus II_{1,1}$ with dual lattice $L^* = E_8 ^{u*} \oplus II_{1,1}$ and denote the closures of the canonical positive cones by $L^+$ and $L^{*+}$. The fake monster superalgebra has a natural $L^*$-grading. For $\alpha=(r^*,a)\in L^*$ we define \[ \tilde{E}_{0,\alpha} = \oplus_{\pi(r)=r^*} E_{0,(r,a)} \] and analogous for $\tilde{E}_{1,\alpha}$.
Let $\varepsilon_i$ and $\sigma_i$ denote the eigenvalues of $\rho_V(u)$ and $\rho_L(u)$.
Then we have \begin{th1} The twisted denominator identity corresponding to g is given by \[ \prod_{\alpha \in L^{*+}}
\frac{ (1-e^{\alpha})^{ {mult}_0({\alpha}) }}
{ (1+e^{\alpha})^{ {mult}_1({\alpha}) }} =
1 + \sum a(\lambda)e^{\lambda} \] where \begin{eqnarray*} \mbox{mult}_0({\alpha}) &=&
\sum_{ds|(({\alpha},L),N)} \frac{\mu(s)}{ds}\, tr(g^d|\tilde{E}_{0,{\alpha}/ds}) \\ \mbox{mult}_1({\alpha}) &=&
\sum_{ds|(({\alpha},L),N)} \frac{\mu(s)}{ds}\, tr(g^d|\tilde{E}_{1,{\alpha}/ds}) \end{eqnarray*} and $a(\lambda)$ is the coefficient of $q^m$ in \[ \prod_{n\geq 1} \prod_{\: 1\leq i\leq 8}
\frac{(1-\varepsilon_i q^n)}
{(1+\sigma_i q^n)} \] if $\lambda$ is $m$ times a primitive norm zero vector in $L^+$ and zero else. \end{th1} {\em Proof:} We consider both sides of $\Lambda^*(E) = H(E)$ as $L^*$-graded $g$-modules. $\Lambda^*(E)$ is isomorphic to $\Lambda^*(E_0) \otimes S^*(E_1)$ if we forget the superstructure on $E_1$. First we calculate the trace of $g$ on $S^*(E_1)$. For that we recall some formulas. Let $V$ be a finite dimensional vector space. Then we have \[ \sum _{n\geq 0}(-1)^n (dim \, S^n(V)) q^n = (1+q)^{-dim V}
= exp \, \Big\{ \sum_{n>0}(-1)^n (dim \,V) q^n/n \Big\} \, . \] The second equality can be proven by taking logarithms and using the expansion $log (1+q) =- \sum_{n>0} (-1)^n q^n/n$. This can be generalized to
\[ \sum _{n\geq 0}(-1)^n \,tr(g|S^n(V)) q^n
= exp \, \Big\{ \sum_{n>0} (-1)^n tr(g^n|V) q^n/n \Big\} \, . \] Another formula we need is $S^*(V_1 \oplus V_2) = S^*(V_1) \otimes S^*(V_2)$. Using these two formulas we find that the trace of $g$ on $S^*(E_1)$ is given by \[ exp \, \Big\{
\sum_{\beta \in L^{*+}} \sum_{n>0} (-1)^n \,tr(g^n|\tilde{E}_{1,\beta} )
e^{n\beta}/n \Big\} \, . \] We want to express the trace in the form \[ \prod_{\beta \in L^{*+}}(1+e^{\beta})^{ - mult_1({\beta}) }=
\mbox{\it exp} \,
\Big\{ \sum_{\beta \in L^{*+}} \sum_{n>0} (-1)^n mult_1(\beta)e^{n\beta}/n
\Big\} \, . \] Taking logarithms and comparing coefficients at $e^{\alpha }$ this implies \[ \sum_{\beta \in L^{*+} \atop n\beta = \alpha }
(-1)^n \,tr(g^n|\tilde{E}_{1,\beta} )/n =
\sum_{\beta \in L^{*+} \atop n\beta = \alpha }
(-1)^n mult_1(\beta) /n \, . \] The vector $\alpha \in L^*$ is a positive multiple $m$ of a primitive vector in $L^*$. This vector generates a lattice that we can identify with ${\mathbb Z}$. Then the right hand side of the last equation is the convolution product of the arithmetic functions $h(n)=(-1)^n/n$ and $mult_1(n)$. Hence \[
mult_1(\alpha)= \sum_{ds|m} h^{* -1}(s) (-1)^d\, tr(g^d|\tilde{E}_{1,\alpha /ds})/d \, . \] Let $\mu$ be the M\"obiusfunction and $f(n)$ the arithmetic function which is zero if $n$ contains an odd square and $(-1)^k / p_1\ldots p_k$, where the $p_i$ are the different primes dividing $n$, else. Then convolution inverse of $h(n)$ is given by \[ h^{* -1}(n) = \left\{ \begin{array}{cl}
(\mu h) (n) & \quad n \quad \mbox{odd} \\
f(n) & \quad n \quad \mbox{even}
\end{array}
\right. \]
$g$ has the same order as $u$ because $N$ is odd. The trace $tr(g^d|\tilde{E}_{1,\alpha})$ depends only on $\alpha$ and $(d,N)$. This implies that the sum in the expression for $mult_1(\alpha)$ extends only over $ds|(m,N)$. It is easy to see that $m$ is equal to the highest common factor $(\alpha,L)$ of the numbers $(\alpha,\beta),\, \beta \in L$. Hence we get the following formula \[
mult_1(\alpha) = \sum_{ds|((\alpha,L),N)} \mu(s)\, tr(g^d|\tilde{E}_{1,\alpha/ds})/ds \, . \] Note that this formula is wrong for even $N$. This is another reason why we restrict to elements in $2.Aut(E_8)$ of odd order.
If we express the contributions of $\Lambda^*(E_0)$ to the trace in the form \[ \prod_{\alpha \in L^{*+}}(1-e^{\alpha})^{ mult_0(\alpha) } \] then we find
\[ mult_0 (\alpha)= \sum_{ds|((\alpha,L),N)} h^{* -1}(s)\,
tr(g^d|\tilde{E}_{0,\alpha/ds})/d \] with $h(n)=1/n$. This function is strongly multiplicative so that its convolution inverse is given by $\mu h$. An argument as above then gives the expression for $mult_0(\alpha)$.
Next we calculate the trace of $g$ on $H(E)$. We have \[ H(E) = \oplus_{\alpha \in L^{*+}} \tilde{H}(E)_{\alpha} \] with \[ \tilde{H}(E)_{\alpha}=\oplus_{\pi(r)=r^*} H(E)_{(r,a)} \]
for $\alpha=(r^*,a)$ in $L^{*+}$. Clearly $tr(g|\tilde{H}(E)_{\alpha})$ is zero if $\alpha$ is not in $L$ and
$tr(g|\tilde{H}(E)_{\alpha})=tr(g|H(E)_{\alpha})$ for $\alpha\in L$. We can also restrict to $\alpha^2=0$. Let $\alpha\in L^+ \subset II_{9,1}^+$ be m times a primitive norm zero vector $\lambda$ in $II_{9,1}^+$. Then $\lambda$ is also a primitive vector in $L$. $H(E)_{\alpha}$ is determined by \[ {\mathbb R} \oplus \oplus_{n>0} H(E)_{n \lambda}
= \Lambda^*(\oplus_{n>0}E_{0,n\lambda}) \otimes
S^*(\oplus_{n>0}E_{1,n\lambda}) \, . \] g acts by $\rho_V(u)$ on $E_{0,n\lambda}\cong {\mathbb R}\otimes E_8$ and by $\rho_L(u)$ on $E_{1,n\lambda}\cong C$ (cf. section 2).
Going over to complexifications this implies that $tr(g|H(E)_{\alpha})$ is given by the coefficient of $q^m$ in
\[ 1 + \sum_{n \geq 1} tr(g|H(E)_{n \lambda}) q^n =
\prod_{n\geq 1} \prod_{\: 1\leq i\leq 8}
\frac{(1-\varepsilon_i q^n)}{(1+\sigma_i q^n)} \, . \] This finishes the proof of the theorem.
For $u=1$ the theorem gives the denominator identity of the fake monster superalgebra.
If the multiplicities are nonnegative integers then the twisted denominator identity is the untwisted denominator identity of a generalized Kac-Moody superalgebra with the following properties. The root lattice is $L^*$ or a sublattice thereof. The algebra has no real roots so that the Weyl group is trivial. The multiplicities of a root $\alpha$ are given by ${mult}_0({\alpha})$ and ${mult}_1({\alpha})$. There are no real simple roots and the imaginary simple roots are the norm zero vectors in $L^{*+}$. The Weyl vector is zero. We describe the multiplicities of the simple roots. Let $\alpha = n \lambda$ be a simple root where $\lambda$ is a primitive vector in $L^+$. Suppose that $\rho_V(u)$ and $\rho_L(u)$ have cycle shapes $a_1^{b_1}\ldots a_k^{b_k}$ and $c_1^{d_1}\ldots c_l^{d_l}$. Then the multiplicities of $\alpha$ as even and odd root are given by
$\sum_{a_k | n} b_k$ and $\sum_{c_k | n} d_k$.
The formulas for the multiplicities simplify if $u$ has in addition prime order. In this case we only need to calculate the trace of $g$ on $\tilde{E}_{0,\alpha}$ and $\tilde{E}_{1,\alpha}$ with $\alpha \in L^*$ and the dimensions of these spaces. We find \begin{pth1} \label{hel}
For $\alpha \in L^*$ the trace $tr(g|\tilde{E}_{0,\alpha})$ is given by the coefficient of $q^{(1-\alpha^2)/2}$ in \[ \frac{1}{2}\left(
\prod_{n \geq 1} \prod_{\: 1 \leq i \leq 8}
\frac{(1+\varepsilon_i q^{n-1/2})}
{(1-\varepsilon_iq^n)}
- \prod_{n \geq 1} \prod_{\: 1 \leq i \leq 8}
\frac{(1-\varepsilon_i q^{n-1/2})}
{(1-\varepsilon_iq^n)}
\right) \] if $\alpha \in L$ and zero else.
Let $tr (\rho_L(u))=tr (\rho_R(u))$. Then $tr(g|\tilde{E}_{1,\alpha})$ is the coefficient of $q^{(1-\alpha^2)/2}$ in \[ tr (\rho_L(u)) \: q^{1/2}
\prod_{n \geq 1} \prod_{\: 1 \leq i \leq 8}
\frac{(1+\varepsilon_iq^n)}
{(1-\varepsilon_iq^n)} \] if $\alpha \in L$ and zero else. \end{pth1}
{\em Proof:} Clearly the trace of $g$ is zero on $\tilde{E}_{0,\alpha}$ if $\alpha$ is not in $L$. For $\alpha \in L$ we have $tr(g|\tilde{E}_{0,\alpha})=tr(g|E_{0,\alpha})$. Write $\alpha = (r,a)$. Then $E_{0,\alpha}$ is isomorphic as $g$-module to the subspace of $V_{0,(1-a^2)/2}$ of degree $r$ (cf. section 2). This space is generated by products of fermionic and bosonic oscillators and $e^r$. The sum of the $L_0$-contribution of the oscillators and the $L_0$-contribution of $e^r$ is $\frac{1}{2}-\frac{1}{2}a^2$. The vector $e^r$ has $L_0$-eigenvalue $\frac{1}{2}r^2$ so that the $L_0$-contribution of the oscillators is $ \frac{1}{2}-\frac{1}{2}a^2-\frac{1}{2} r^2 = \frac{1}{2}-\frac{1}{2}\alpha^2$. Now we go over to complexifications and choose a basis of ${\mathbb C}\otimes E_8$ in which $\rho_V(u)$ is diagonal. Then the trace of $g$ on $E_{0,\alpha}$ is given by the coefficient of $q^{(1-\alpha^2)/2}$ in \[ \prod_{n \geq 1} \prod_{\: 1 \leq i \leq 8}
\frac{(1+\varepsilon_i q^{n-1/2})}
{(1-\varepsilon_iq^n)} \,. \] Since we only need the half integral exponents of $q$ in this expression we can subtract the integral exponents. This proves the first statement.
The argument for the second statement is similar. Note that there are two types of ground states on which $u$ may act differently. We avoid this problem by assuming that $u$ has in both representations the same trace. This proves the proposition. \begin{pth1} \label{de} Let $\alpha=(r^*,a)\in L^*$ and let $r^{\bot *}\in E_8^{u \bot *}$ such that $r^* + r^{\bot *} \in E_8$. Then the dimension of $\tilde{E}_{0,\alpha}$ and $\tilde{E}_{1,\alpha}$ is given by the coefficient of $q^{(1-\alpha^2)/2}$ in \[ 8 q^{1/2} \frac{\eta(q^2)^8}{\eta(q)^{16}}\:
\theta_{r^{\bot *} +E_8^{u \bot }}(q) \] where $\theta_{r^{\bot *}+E_8^{u \bot }}(q)$ is the theta function of the translated lattice $r^{\bot *}+E_8^{u \bot }$. \end{pth1} {\em Proof:} Clearly $\tilde{E}_{0,\alpha}$ and $\tilde{E}_{1,\alpha}$ have the same dimension. The inverse image of $r^*$ in $E_8$ under $\pi$ is $r^*+(r^{\bot *}+E_8^{u \bot })$. Hence $\tilde{E}_{1,\alpha}=\oplus_{\pi(r)=r^*} E_{1,(r,a)}$ is isomorphic to the direct sum of the $V_{1,(1-a^2)/2}(r^*+s)$ where $s$ is in the translated lattice $r^{\bot *}+E_8^{u \bot}$. As above the sum of the $L_0$-contribution of the bosonic and fermionic oscillators and $\frac{1}{2}s^2$ is $\frac{1}{2}-\frac{1}{2}\alpha^2$. The proposition now follows from simple counting.
We will use the shorter notation $\theta_{r^{\bot *}}(q)$ in the following.
\section{Two supersymmetric algebras}
In this section we calculate explicitly twisted denominator identities corresponding to elements in $2.Aut(E_8)$ of order 3 and 7. These identities are the untwisted denominator identities of 2 supersymmetric generalized Kac-Moody superalgebras.
We choose an orthonormal basis $\{e_0,e_1,\ldots,e_7 \}$ of ${\mathbb R}^8$ and embed the lattice $E_8$ as the set of points $\sum m_i e_i$ where all $m_i$ are in $\mathbb Z$ or all $m_i$ are in $\mathbb Z+\frac{1}{2}$ and $\sum m_i$ is even. In these coordinates the automorphism group of $E_8$ is generated by the permutations of the coordinates, even sign changes and an involution generated by the Hadamard matrix. We also identify ${\mathbb R}^8$ with the alternative algebra ${\mathbb O}$ of octonions by defining $e_0$ as the identity and $e_i e_j= a_{ijk}e_k-\delta_{ij}1$ for $1\leq i,j \leq 7$ where $a_{ijk}$ is the totally antisymmetric tensor with $a_{ijk}=1$ for $ijk=123, 154, 264, 374, 176, 257, 365$.
The element $u=\frac{1}{4}1(e_2-e_3)1(e_1-e_2)1(e_6-e_7)1(e_5-e_6)$ in $2.Aut(E_8)$ has order $3$. It is easy to check that the transformations corresponding to $\rho_V(u),\rho_L(u)$ and $\rho_R(u)$ are all equal and \[ \renewcommand{1.3}{1.3} \begin{array}{llll} \rho_V(u)1=1 & \rho_V(u)e_1 = e_3 & \rho_V(u)e_2 = e_1 & \rho_V(u)e_3 = e_2 \\ \rho_V(u)e_4=e_4 & \rho_V(u)e_5 = e_7 & \rho_V(u)e_6 = e_5
& \rho_V(u)e_7 = e_6 \, . \end{array} \] Hence $\rho_V(u),\rho_L(u)$ and $\rho_R(u)$ all have cycle shape $1^23^2$.
The following proposition collects some results on $E^u_8$. \begin{p1} The sublattice $E^u_8$ of $E_8$ fixed by $\rho_V(u)$ is the 4 dimensional lattice with elements $(m_1,m_2,m_3,m_4)$ where all $m_i$ are in $\mathbb Z$ or all $m_i$ are in $\mathbb Z+\frac{1}{2}$ and $\sum m_i$ is even. The norm is $(m_1,m_2,m_3,m_4)^2=m_1^2+m_2^2+3m_3^2+3m_4^2$. $E^u_8$ has determinant $3^2$ and the quotient $E_8^{u*}/E_8^u$ is ${\mathbb Z}_3\times {\mathbb Z}_3$. The level of $E_8^u$ is $3$ so that $3E_8^{u*}$ is a sublattice of $E^u_8$. The orthogonal complement of $E^u_8$ in $E_8$ is $A_2 \oplus A_2$. \end{p1}
We recall some results about modular forms. The congruence subgroup of $SL_2({\mathbb Z})$ of level $3$ is defined as
$\Gamma(3)=\{ \big( {a \atop c} {b \atop d} \big) \in SL_2({\mathbb Z}) \, |\,
\big( {a \atop c} {b \atop d} \big) = \big( {1 \atop 0} {0 \atop 1} \big)
\mbox{ mod } 3 \, \}$. The vector space of modular forms for $\Gamma(3)$ with even positive weight $n$ has dimension $n+1$. A modular form in this vector space is zero if and only if the coefficients of $q^0,q^{1/3},\ldots,q^{n/3}$ in its Fourier expansion are zero.
For $r\in A_2^*\oplus A_2^*$ we define $\delta(r)=1$ if $r\in A_2 \oplus A_2$ and $0$ else. We have \begin{pp1} \label{tta} Let $r\in A_2^*\oplus A_2^*$. Then \[ \theta_{r+ A_2\oplus A_2}(q) = \frac{1}{4} \, \frac{\eta(q)^{12}}{\eta(q^2)^6} \left\{
\frac{\eta(q^6)^2}{\eta(q^3)^4}\delta(r) + \sum_{j=0}^2 \varepsilon^{-3jr^2/2}
\frac{ \eta\!\left( (\varepsilon^jq^{1/3})^2 \right)^2 }
{ \eta\!\left( \varepsilon^jq^{1/3} \right)^4 } \right\} \] where $\varepsilon = e^{2\pi i/3}$. \end{pp1} {\em Proof:} $A_2\oplus A_2$ has dimension $4$ and level $3$ so that $\theta_{r+ A_2\oplus A_2}$ is a modular form for $\Gamma(3)$ of weight $2$. The same holds for the right hand side of the formula which can be shown by calculating the transformations under a set of generators. The proposition follows from comparing the coefficients at $q^0,q^{1/3}$ and $q^{2/3}$.
The next identity is a twisted version of Jacobi's identity. \begin{pp1} \label{hnel}
Let $|q|<1$. Then \[ \frac{1}{2q^{1/2}}
\left\{
\prod_{n \geq 1}(1+q^{3n-3/2})^2(1+q^{n-1/2})^2
-\prod_{n \geq 1}(1-q^{3n-3/2})^2(1-q^{n-1/2})^2
\right\} \] \[ = 2 \prod_{n \geq 1} (1+q^{3n})^2(1+q^n)^2 \, . \] \end{pp1} {\em Proof:} This is an identity between modular forms and therefore can be proven by comparing sufficiently many coefficients in their Fourier expansions.
The proposition implies that the generating functions for $tr(g|\tilde{E}_{0,\alpha})$ and \linebreak $tr(g|\tilde{E}_{1,\alpha})$ are equal.
Define $c(n)$ by \[ \sum_{n\geq 0} c(n)q^n
= 2\, \prod_{n \geq 1} \frac{(1+q^{3n})^2(1+q^n)^2 }{(1-q^{3n})^2(1-q^n)^2 }
= 2\, \frac{\eta(q^6)^2\eta(q^2)^2}{\eta(q^3)^4\eta(q)^4} \] \[ = 2 + 8q + 24q^2 + 72q^3 + 184q^4 + 432q^5 + 984q^6 + 2112q^7 + \ldots \] Let $g$ be the automorphism induced by $u$. Then we have \begin{thp1} The twisted denominator identity corresponding to $g$ is \[ \prod_{\alpha\in L^{+}}
\frac{ (1-e^{\alpha})^{ c(-\alpha^2/2) }}
{ (1+e^{\alpha})^{ c(-\alpha^2/2) }}
\prod_{\alpha \in L^{+}\cap 3L^*}
\frac{ (1-e^{\alpha})^{ c(-\alpha^2/6) }}
{ (1+e^{\alpha})^{ c(-\alpha^2/6) }} = 1 + \sum a(\lambda)e^{\lambda} \] where $a(\lambda)$ is the coefficient of $q^n$ in \[ \prod_{n \geq 1}\frac{(1-q^{3n})^2(1-q^n)^2}{(1+q^{3n})^2(1+q^n)^2}
= 1 - 4q + 4q^2 - 4q^3 + 20q^4 - 24q^5 + 4q^6 - \ldots \] if $\lambda$ is $n$ times a primitive norm zero vector in $L^+$ and zero else. \end{thp1} {\em Proof:} Recall that $3L^*\subset L$ because $L$ has level 3. Furthermore $3$ divides $(\alpha,L)$ if and only if $\alpha \in 3L^*$. We consider now 4 cases. \\ $\alpha\notin L$. Then $\alpha\notin 3L^*$ and by Proposition \ref{hel} the multiplicities $mult_0(\alpha)$ and $mult_1(\alpha)$ are zero.\\ $\alpha \in L$ and $\alpha\notin 3L^*$.
Then $mult_0(\alpha)=tr(g|\tilde{E}_{0,\alpha})$ and $mult_1(\alpha)=tr(g|\tilde{E}_{1,\alpha})$. Using Propositions \ref{hel} and \ref{hnel} we find $mult_0(\alpha)=mult_1(\alpha)=c(-\alpha^2/2)$.\\ $\alpha\in 3L^*$ and $\alpha \notin 3L$.
Then $mult_0(\alpha)=tr(g|\tilde{E}_{0,\alpha})+tr(1|\tilde{E}_{0,\alpha/3})/3$. The first term gives $c(-\alpha^2/2)$. Write $\alpha/3=(r^*,a)$ where $r^*$ is in $E_8^{u*}$ but not in $E_8^{u}$ and choose $r^{*\bot}$ as in Proposition \ref{de}. Then the dimension of $\tilde{E}_{0,\alpha/3}$ is the coefficient of $q^{-\alpha^2/18}$ in $8 \eta(q^2)^8 \theta_{r^{\bot *}}(q)/{\eta(q)^{16}}$ or equivalently the coefficient of $q^{-\alpha^2/6}$ in $8 \eta(q^6)^8 \theta_{r^{\bot *}}(q^3)/ \eta(q^3)^{16}$. Note that $\alpha^2\in 6 {\mathbb Z}$. We have $\alpha^2/9 = r^{*2} = - r^{\bot*2}$ mod $2$ and $\alpha^2/6 = - 3 r^{\bot*2}/2$ mod $3$ so that by Proposition \ref{tta} \[ \theta_{r^{\bot *}}(q^3) =
\frac{1}{4} \, \frac{\eta(q^3)^{12}}{\eta(q^6)^6}
\sum_{j=0}^2 \varepsilon^{j\alpha^2/6}
\, \frac{ \eta\left( (\varepsilon^j q)^2 \right)^2 }
{ \eta\left( \varepsilon^j q \right)^4 }
\, . \] The coefficient of $q^{-\alpha^2/6}$ in $8 \eta(q^6)^8 \theta_{r^{\bot *}}(q^3)/ \eta(q^3)^{16}$ is equal to the coefficient of $q^{-\alpha^2/6}$ in \[ 6 \, \frac{\eta(q^6)^2\eta(q^2)^2}{\eta(q^3)^4\eta(q)^4}\, . \] This shows that $mult_0(\alpha) = c(-\alpha^2/2) + c(-\alpha^2/6)$. The result for $mult_1(\alpha)$ is clear. \\ $\alpha \in 3L$. Then
$mult_0(\alpha)=tr(g|\tilde{E}_{0,\alpha})-tr(g|\tilde{E}_{0,\alpha/3})/3
+ tr(1|\tilde{E}_{0,\alpha/3})/3$.
Here we have an additional term in $\theta_{r^{\bot *}}$ which cancels exactly with the term from $tr(g|\tilde{E}_{0,\alpha/3})$ so that we get the same result for $mult_0(\alpha)$ as in the case before. The result for $mult_1(r)$ is again clear. \\ This proves the theorem.
Since the multiplicities are all nonnegative integers there is a generalized Kac-Moody superalgebra whose denominator identity is the identity given in the theorem. \begin{cp1} There is a generalized Kac-Moody superalgebra with root lattice $L$ and root multiplicities given by \[ mult_0(\alpha) = mult_1(\alpha) = c(-\alpha^2/2)
\qquad \alpha\in L,\: \alpha \notin 3L^* \] and \[ mult_0(\alpha) = mult_1(\alpha) = c(-\alpha^2/2)+ c(-\alpha^2/6)
\qquad \alpha\in 3L^*\,. \] The simple roots of are the norm zero vectors in $L^+$. Their multiplicities as even and odd roots are equal. Let $\lambda$ be a simple root. Then $mult_0(\lambda)=4$ if $\lambda$ is 3n times a primitive vector in $L^+$ and $mult_0(\lambda)=2$ else. \end{cp1} As the fake monster superalgebra this algebra is supersymmetric and has no real roots. The Weyl group is trivial and the Weyl vector is zero. The supersymmetry is a consequence of the twisted Jacobi identity in Proposition \ref{hnel}. In a forthcoming paper we show that the denominator function of this algebra defines an automorphic form for a discrete subgroup of $O_{6,2}({\mathbb R})$ of weight $2$.
Now we consider the next example. The analysis is similar to the above one. The element $u=\frac{1}{8}1(e_6-e_7)1(e_5-e_6)1(e_4-e_5)1(e_3-e_4)1(e_2-e_3)1(e_1-e_2)$ in $2.Aut(E_8)$ has order $7$. The transformations corresponding to $\rho_V(u),\rho_L(u)$ and $\rho_R(u)$ are all equal and \[ \renewcommand{1.3}{1.3} \begin{array}{llll} \rho_V(u)1=1 & \rho_V(u)e_1 = e_7 & \rho_V(u)e_2 = e_1 & \rho_V(u)e_3 = e_2 \\ \rho_V(u)e_4=e_3 & \rho_V(u)e_5 = e_4 & \rho_V(u)e_6 = e_5
& \rho_V(u)e_7 = e_6 \, . \end{array} \] Hence $\rho_V(u),\rho_L(u)$ and $\rho_R(u)$ have cycle shape $1^17^1$. We have \begin{pp1} The sublattice $E^u_8$ of $E_8$ fixed by $\rho_V(u)$ is the 2 dimensional lattice with elements $(m_1,m_2)$, where either $m_1$ and $m_2$ are in $\mathbb Z$ and $m_1+m_2$ is even or $m_1$ and $m_2$ are in $\mathbb Z+\frac{1}{2}$ and $m_1+m_2$ is odd, and norm $(m_1,m_2)^2=m_1^2+7m_2^2$. The quotient $E_8^{u*}/E_8^u$ is ${\mathbb Z}_7$ and $E^u_8$ has level $7$. The orthogonal complement of $E^u_8$ in $E_8$ is isomorphic to $A_6$. \end{pp1} The theta function $\theta_{r+ A_6}$ of a coset $r+A_6$ of $A_6$ in its dual depends only on $r^2$ mod $2$. \begin{pp1} Let $r\in A_6^*$. Then \[ \theta_{r+ A_6}(q) = \frac{1}{8}\, \frac{\eta(q)^{14}}{\eta(q^2)^7} \left\{
\frac{\eta(q^{14})}{\eta(q^7)^2}\delta(r) + \sum_{j=0}^6 \varepsilon^{-7jr^2/2}
\frac{ \eta\left( (\varepsilon^jq^{1/7})^2 \right)^2 }
{ \eta\left( \varepsilon^jq^{1/7} \right)^4 } \right\} \] where $\varepsilon = e^{2\pi i/7}$. \end{pp1} The following supersymmetry relation holds. \begin{pp1}
Let $|q|<1$. Then \[ \frac{1}{2q^{1/2}}
\left\{
\prod_{n \geq 1}(1+q^{7n-7/2})(1+q^{n-1/2})
-\prod_{n \geq 1}(1-q^{7n-7/2})(1-q^{n-1/2})
\right\} \] \[ = \prod_{n \geq 1} (1+q^{7n})(1+q^n) \] \end{pp1} Here we define the numbers $c(n)$ by \[ \sum_{n\geq 0} c(n)q^n
= \prod_{n \geq 1} \frac{(1+q^{7n})(1+q^n)}{(1-q^{7n})(1-q^n)}
= \frac{\eta(q^{14})\eta(q^2)}{\eta(q^7)^2\eta(q)^2} \] \[ = 1 + 2q + 4q^2 + 8q^3 + 14q^4 + 24q^5 + 40q^6 + 66q^7 + \ldots \] Write $g$ for the automorphism induced by $u$. Then \begin{thp1} The twisted denominator identity corresponding to $u$ is \[ \prod_{\alpha\in L^{+}}
\frac{ (1-e^{\alpha})^{ c(-\alpha^2/2) } }
{ (1+e^{\alpha})^{ c(-\alpha^2/2) } }
\prod_{\alpha \in L^{+}\cap 7L^*}
\frac{ (1-e^{\alpha})^{ c(-\alpha^2/14) } }
{ (1+e^{\alpha})^{ c(-\alpha^2/14) } } = 1 + \sum a(\lambda)e^{\lambda} \] where $a(\lambda)$ is the coefficient of $q^n$ in \[ \prod_{n \geq 1}\frac{(1-q^{7n})(1-q^n)}{(1+q^{7n})(1+q^n)}
= 1 - 2q + 2q^4 - 2q^7 + 4q^8 - 2q^9 - 4q^{11} + 6q^{16} - \ldots \] if $\lambda$ is $n$ times a primitive norm zero vector in $L^+$ and zero else. \end{thp1} Again the multiplicities are all nonnegative integers and we have \begin{cp1} There is a generalized Kac-Moody superalgebra with root lattice $L$ and root multiplicities given by \[ mult_0(\alpha) = mult_1(\alpha) = c(-\alpha^2/2)
\qquad \alpha \in L,\: \alpha \notin 7L^* \] and \[ mult_0(\alpha) = mult_1(\alpha) = c(-\alpha^2/2)+ c(-\alpha^2/14)
\qquad \alpha \in 7L^*\,. \] The simple roots of are the norm zero vectors in $L^+$. Let $\lambda$ be a simple root. Then $mult_0(\lambda)=mult_1(\lambda)=2$ if $\lambda$ is 7n times a primitive vector in $L^+$ and $mult_0(\lambda)=mult_1(\lambda)=1$ else. \end{cp1} The denominator identity of this algebra is given by the identity in the above theorem. It defines an automorphic form for a subgroup of $O_{4,2}({\mathbb R})$ of weight $1$.
\section*{Acknowledgments}
I thank R. E. Borcherds for stimulating discussions.
\end{document} |
\begin{document}
\begin{abstract} We consider electrodiffusion of ions in fluids, described by the Nernst-Planck-Navier-Stokes system, in three dimensional bounded domains, with mixed blocking (no-flux) and selective (Dirichlet) boundary conditions for the ionic concentrations and Robin boundary conditions for the electric potential, representing the presence of an electrical double layer. We prove global existence of strong solutions for large initial data in the case of two oppositely charged ionic species. The result hold unconditionally in the case where fluid flow is described by the Stokes equations. In the case of Navier-Stokes coupling, the result holds conditionally on Navier-Stokes regularity. We use a simplified argument to also establish global regularity for the case of purely blocking boundary conditions for the ionic concentrations for two oppositely charged ionic species and also for more than two species if the diffusivities are equal and the magnitudes of the valences are also equal. \end{abstract} \keywords{electroconvection, ionic electrodiffusion, Nernst-Planck, Navier-Stokes, electrical double layer}
\noindent\thanks{\em{MSC Classification: 35Q30, 35Q35, 35Q92.}} \maketitle \section{Introduction} We study the \textit{Nernst-Planck-Navier-Stokes} (NPNS) system in a connected, bounded domain $\Omega\subset\mathbb{R}^3$ with smooth boundary, which models the electrodiffusion of ions in a fluid in the presence of boundaries. The ions diffuse under the influence of their own concentration gradients and are transported by the fluid and an electric field, which is generated by the local charge density and an externally applied potential. The fluid is forced by the electrical force exerted by the ionic charges. The time evolution of the ionic concentrations is determined by the \textit{Nernst-Planck} equations, \begin{equation} \partial_t c_i+u\cdot\nabla c_i=D_i{\mbox{div}\,}(\nabla c_i+z_ic_i\nabla\Phi),\quad i=1,...,m\label{np} \end{equation} coupled to the Poisson equation \begin{equation} -\epsilon\Delta\Phi=\sum_{i=1}^m z_ic_i=\rho\label{pois} \end{equation} and to the \textit{Navier-Stokes} system, \begin{equation} \partial_t u+u\cdot\nabla u -\nu\Delta u+\nabla p=-K\rho\nabla\Phi,\quad {\mbox{div}\,} u=0\label{nse} \end{equation} or to the \textit{Stokes} system \begin{equation} \partial_t u-\nu\Delta u+\nabla p=-K\rho\nabla\Phi,\quad {\mbox{div}\,} u=0.\label{stokes} \end{equation} In this latter case we refer to the system as the \textit{Nernst-Planck-Stokes} (NPS) system.
The function $c_i$ is the local ionic concentration of the $i$-th species, $u$ is the fluid velocity, $p$ is the pressure, $\Phi$ is a rescaled electrical potential, and $\rho$ is the local charge density. The constant $z_i\in\mathbb{R}$ is the ionic valence of the $i$-th species. The constants $D_i>0$ are the ionic diffusivities, and $\epsilon>0$ is a rescaled dielectric permittivity of the solvent, and it is proportional to the square of the Debye length $\lambda_D$, which is the characteristic length scale of the electrical double layer in a solvent \cite{rubibook}. The constant $K>0$ is a coupling constant given by the product of Boltzmann's constant $k_B$ and the temperature $T_K$. Finally, $\nu>0$ is the kinematic viscosity of the fluid. The dimensional counterparts of $\Phi$ and $\rho$ are given by $(k_BT_K/e)\Phi$ and $e\rho$, respectively, where $e$ is elementary charge.
For the ionic concentrations $c_i$ we consider \textit{blocking} (no-flux) boundary conditions, \begin{equation}
(\partial_n c_i(x,t)+z_ic_i(x,t)\partial_n\Phi(x,t))_{|\partial\Omega}=0 \label{bl} \end{equation} where $\partial_n$ is the outward normal derivative along $\partial\Omega$. This boundary condition represents a surface that is impermeable to the $i$-th ionic species. For regular enough solutions, blocking boundary conditions imply that the total concentration $\int_\Omega c_i\,dx$ is conserved as can formally be seen by integrating (\ref{np}) over $\Omega$. For $c_i$, we also consider \textit{selective} (Dirichlet) boundary conditions,
\begin{equation} c_i(x,t)_{|\partial\Omega}=\gamma_i>0,\label{DI} \end{equation} which, in electrochemistry \cite{davidson,rubishtil}, represents an ion-selective (permselective) membrane that maintains a fixed concentration of ions.
The boundary conditions for the Navier-Stokes (or Stokes) equations are \textit{no-slip},
\begin{equation} u(x,t)_{|\partial\Omega}=0.\label{noslip} \end{equation} The boundary conditions for $\Phi$ are inhomogeneous Robin, \begin{equation}
(\partial_n\Phi(x,t)+\tau\Phi(x,t))_{|\partial\Omega}=\xi(x).\label{robin} \end{equation} This boundary condition represents the presence of an electrical double layer at the interface of a solvent and a surface \cite{prob,rubibook}. The Robin boundary conditions are derived based on the fact that the double layer has the effect of a plate capacitor. The constant $\tau>0$ represents the capacitance of the double layer, and $\xi:\partial\Omega\to\mathbb{R}$ is a smooth function that represents an externally applied potential on the boundary (see also \cite{bothe,fischer,gajewski,lee} where the same boundary conditions are used in similar contexts).
In this paper, we discuss the question of global regularity of solutions of NPNS and NPS. The NPNS system is a semilinear parabolic system, and in general, such systems can blow up in finite time. For example, The Keller-Segel equations, which share some common features with NPNS (e.g. the dissipative structure, Section {\ref{de}}), are known to admit solutions that blow up in finite time, even in two dimensions, for large initial conditions \cite{bedro}. The NPNS system includes the Navier-Stokes equations, where the question of large data global regularity, as is well known, is unresolved in three dimensions \cite{cf}. So for NPNS, we cannot expect at this stage to obtain affirmative results on unconditional global regularity. However, global regularity in three dimensions for the NPS system or even the Nernst-Planck equations, not coupled to fluid flow, is still an open problem except in some special cases, the main obstacle being control of the nonlinear term ${\mbox{div}\,}(c_i\nabla\Phi)$.
In both the physical and mathematical literature, many different boundary conditions are considered for the concentrations $c_i$ and the electric potential $\Phi$, all with different physical meanings. The choice of boundary conditions makes a large difference not only when it comes to determining global regularity, but also in characterizing long time behavior. Global existence and stability of solutions to the uncoupled Nernst-Planck equations is obtained in \cite{biler,biler2,choi,gajewski} for blocking boundary conditions in two dimensions. The full NPNS system is discussed in \cite{schmuck} where the electric potential is treated as a superposition of an internal potential, determined by the charge density $\rho$ and homogeneous Neumann boundary conditions, and an external, prescribed potential. In this case global weak solutions are obtained in both two and three dimensions. The case of blocking and selective (Dirichlet) boundary conditions for the concentrations and Dirichlet boundary conditions for the potential are considered in \cite{ci} for two dimensions, and the authors obtain global existence of strong solutions and, in the case of blocking or \emph{uniformly} selective boundary conditions, unconditional stability. In \cite{np3d}, these results are extended to three dimensions, for initial conditions that are small perturbations of the steady states. In \cite{bothe,fischer} the authors consider blocking boundary conditions for $c_i$ and Robin boundary conditions for $\Phi$, as we do in this paper, and obtain global regularity and stability in two dimensions and global weak solutions in three dimensions. In \cite{cil} global regularity in three dimensions is obtained in the case of Dirichlet boundary conditions for both the concentrations and the potential. In \cite{liu}, the authors establish global existence of weak solutions in the case of no boundaries, $\mathbb{R}^3$.
As established and used effectively in the works referred to in the previous paragraph, the NPNS system, equipped with blocking or uniformly selective (c.f. \cite{ci}) boundary conditions for $c_i$, comes with a dissipative structure (Section \ref{de}), which in particular leads to stable asymptotic behaviors. Deviations from blocking or uniformly selective boundary conditions are known to lead, in general, to instabilities when a large enough electric potential drop is imposed across the spatial domain (e.g. narrow channel). These so-called electrokinetic instabilities (EKI) are observed both experimentally and numerically, and verified analytically through simplified models \cite{davidson,pham,rubinstein,rubizaltz,zaltzrubi}.
\begin{comment} In light of these observations, it may come as somewhat of a surprise that in three dimensions, large data global regularity has been established in this unstable regime of selective boundary conditions \cite{cil}, whereas for blocking or uniformly selective boundary conditions, only nonlinear stability results are currently available \cite{np3d}. This apparent discrepancy is due to the fact that a priori bounds resulting in global regularity are obtained fundamentally differently in these two cases. For selective boundary conditions, the fact that one prescribes the boundary values a priori rules out any potential blow up behavior at the boundary. This only leaves potentially bad behavior in the interior, but this, then, is ruled out by the existence of a strong dissipative term, which roughly speaking corresponds to a charged fluid's tendency to neutralize at regions away from the boundaries (i.e. $\rho\approx 0$) \cite{cil,int}. In the case of blocking boundary conditions, potential blow up behavior at the boundary is not a priori ruled out; instead one may make use of the natural dissipative structure, which does not exist in general for selective boundary conditions. In the case of two dimensions, this dissipative structure is enough to initiate a bootstrapping scheme that yields control of higher regularity norms, thus establishing global regularity \cite{ci}. In three dimensions, if Dirichlet boundary conditions are considered for $\Phi$, the same dissipative structure yields only global regularity for small perturbations from steady states \cite{np3d}. \end{comment}
It is partially these observed instabilities that motivate our current study. One of the simplest configurations for which unstable and complex flow behavior and patterns are observed is when the boundaries exhibit ion-selectivity i.e. many surfaces (membranes) that arise in biology, chemistry, and electrochemistry allow for penetration of certain ions while blocking others \cite{davidson,rubinstein}. Mathematically, this situation can be modelled by mixed boundary conditions wherein, for example, $c_1$ has selective boundary conditions and $c_2$ has blocking boundary conditions. In considering these mixed boundary conditions in three dimensions, the main mathematical difficulties include \begin{enumerate}
\item nonlinear, nonlocal boundary conditions (blocking)
\item supercriticality of the nonlinear, nonlocal flux terms, ${\mbox{div}\,}(c_i\nabla\Phi)$
\item lack of natural dissipative structure. \end{enumerate} We compare our situation with related works \cite{ci}, \cite{bothe}, and \cite{cil}. In \cite{ci} and \cite{bothe} the two dimensional setting allowed for control of the nonlinear term ${\mbox{div}\,}(c_i\nabla\Phi)$ in a large variety of situations, including blocking boundary conditions for $c_i$ and Dirichlet \cite{ci} and Robin \cite{bothe} boundary conditions for $\Phi$. In \cite{ci}, 2D global regularity is shown for mixed boundary conditions, too. However, many important steps of the analysis do not carry over to three dimensions due to the difference in scaling. This is what we mean when we say that ${\mbox{div}\,}(c_i\nabla\Phi)$ is supercritical in three dimensions. On the other hand, in \cite{cil}, the three dimensional setting is considered and global regularity is established when Dirichlet boundary conditions are prescribed for $\Phi$ and also for all $c_i$. The issue of the supercriticality of the nonlinearity is circumvented by, simply put, transferring all the potentially ``bad" nonlinear behavior to the boundary where the behavior is a priori controlled due to the boundary conditions. An important ingredient of the analysis is that the boundary conditions give control of \textit{both} $c_1$ and $c_2$ on the boundary. This is not the case for blocking or mixed boundary conditions.
Our current work culminates in Theorem \ref{gr!!} in Section \ref{mbc}, where we consider the NPNS and NPS systems for two oppositely charged ionic species with mixed boundary conditions for $c_i$ (selective for $c_1$ and blocking for $c_2$) and Robin boundary conditions for $\Phi$. We prove large data global regularity, unconditionally for NPS and conditional on the regularity of the fluid velocity $u$ for NPNS. The general strategy is similar to \cite{cil} in that we transfer all the harmful nonlinearities to the boundary. However, the main difference is that in the mixed boundary conditions scenario, the boundary behavior is not a priori controlled the same way as it is in \cite{cil}. Thus, a careful analysis is necessary to show that the internal dissipative ``forces" of the system are strong enough to counteract the potentially problematic boundary behavior. A novel ingredient used at this stage is control of the quantity $\|c_1(t)\|_{L^1(\Omega)}$. Aside from this control, the Robin boundary conditions for $\Phi$ play an important role - while they do not prescribe the values of $\Phi$ or of $\partial_n\Phi$ on the boundary, they do weaken the nonlinearity at the boundary just enough so that dissipation dominates. A close inspection of the proof reveals that replacing the Robin boundary conditions on $\Phi$ with Dirichlet boundary conditions (as in \cite{ci,np3d,cil}) does not allow for the same proof to go through. Thus the problem of global regularity for blocking and mixed boundary conditions for $c_i$ with Dirichlet boundary conditions for $\Phi$ is, in general, open for three dimensions. On the other hand, the proof also reveals how much the analysis can be simplified if Neumann boundary conditions were chosen for $\Phi$ (see e.g. \cite{schmuck}). Robin boundary conditions are situated appropriately in between Dirichlet and Neumann boundary conditions in such a way that they still allow us to take into account applied electric potentials on the boundary (a feature that makes Robin and Dirichlet boundary conditions appealing for the study of the aforementioned electrokinetic instabilities), while retaining some of the mathematically simplifying features of Neumann boundary conditions. {Ultimately, taking Robin boundary conditions to be a physically suitable description of the electrical field at the boundary, Theorem \ref{gr!!} reveals that, in the physically relevant case of three spatial dimensions and assuming sufficient regularity of the fluid velocity field $u$, the NPNS equations do not admit solutions that blow up in finite time (e.g. Dirac mass type aggregation of ions in finite time), and solutions in fact remain regular for all positive time.}
Leading up to Section \ref{mbc}, in Sections \ref{GR} and \ref{mss}, we consider, respectively, a two species and multiple species setting where all the $c_i$ satisfy blocking boundary conditions {and prove global regularity of solutions, unconditionally for NPS and conditional on the regularity of the fluid velocity $u$ for NPNS. In the latter setting of multiple species, }we require additionally that the diffusivities and magnitudes of ionic valences are equal ($D_1=...=D_m$, $|z_1|=...=|z_m|$) (see also \cite{cil}). Because all the $c_i$ satisfy blocking boundary conditions, there is a natural dissipative structure (Section \ref{de}), which facilitates the analysis, but unlike in the two dimensional case \cite{bothe}, this dissipation alone seems insufficient to prove global regularity. Rather the dissipation must be supplemented by precise control of the boundary behavior using the Robin boundary conditions on $\Phi$. {On one hand, we consider these cases of \textit{only} blocking boundary conditions for $c_i$ because energy estimates in these two sections are more concise relative to the mixed boundary conditions case, and they more clearly illustrate the role played by the Robin boundary conditions and the subsequent estimates of boundary terms. On the other hand, the results of these two sections are nontrivial in their own right and also serve to verify that, despite the impenetrable nature of the boundaries (modelled by blocking boundary conditions), if the fluid velocity field remains regular, then blow up of ions near the boundary (or anywhere in the domain) cannot occur in finite time, regardless of the size of the prescribed data for the electrical potential $\Phi$.}
Prior to the proofs of the main theorems, in Section \ref{prelim}, we introduce the relevant function spaces and and state a local existence theorem, the proof of which we omit but can be found in the references provided.
\section{Preliminaries}\label{prelim} We are concerned with global existence of strong solutions of NPNS and NPS. To define what we mean by a strong solution, we first introduce the relevant function spaces.
We denote by $L^p(\Omega)=L^p$ and $W^{m,p}(\Omega)=W^{m,p}$ the standard Lebesgue spaces and Sobolev spaces, respectively. In the case $p=2$, we denote $W^{m,2}=H^m(\Omega)=H^m$. We also consider Lebesgue spaces on the boundary $\partial\Omega$: $L^p(\partial\Omega)$. In this latter case, we always indicate the underlying domain $\partial\Omega$ to avoid ambiguity. We also denote $L^p_tL^q_x=L^p(0,T;L^q(\Omega))$, $L^p_tW^{m,q}_x=L^p(0,T;W^{m,q}(\Omega)),$ where the time $T$ is made clear from context.
Denoting $\mathcal{V}=\{f\in (C_c^\infty(\Omega))^3\,|\, {\mbox{div}\,} f =0\}$, the spaces $H\subset (L^2(\Omega))^3$ and $V\subset (H^1(\Omega))^3$ are the closures of $\mathcal{V}$ in $(L^2(\Omega))^3$ and $(H^1(\Omega))^3$, respectively. The space $V$ is endowed with the Dirichlet norm $\|f\|_V^2=\int_\Omega |\nabla f|^2\,dx$.
In order to avoid having to explicitly deal with the pressure term in the Navier-Stokes and Stokes equations, we sometimes work with the equations projected onto the space of divergence free vector fields via the Leray projector $\mathbb{P}:L^2(\Omega)^3\to H$, \begin{align}
\partial_t u+B(u,u)+\nu Au=-K\mathbb{P}(\rho\nabla\Phi)\label{nse'}\\
\partial_t u+\nu Au=-K\mathbb{P}(\rho\nabla\Phi)\label{stokes'} \end{align} where $A=-\mathbb{P}\Delta:\mathcal{D}(A)\to H$, $\mathcal{D}(A)=H^2(\Omega)^3\cap V$ is the Stokes operator, and $B(u,u)=\mathbb{P}(u\cdot\nabla u)$ (see \cite{cf} for related theory).
\begin{defi} We say that $(c_i,\Phi,u)$ is a strong solution of NPNS (\ref{np}),(\ref{pois}),(\ref{nse}) or of NPS (\ref{np}),(\ref{pois}),(\ref{stokes}) with boundary conditions (\ref{bl}) (or (\ref{DI})),(\ref{noslip}),(\ref{robin}) on the time interval $[0,T]$ if $c_i\ge 0$, $c_i\in L^\infty(0,T;H^1)\cap L^2(0,T;H^2)$, $u\in L^\infty(0,T;V)\cap L^2(0,T;\mathcal{D}(A))$ and $(c_i,\Phi,u)$ solve the equations in the sense of distributions and satisfy the boundary conditions in the sense of traces.\label{strong} \end{defi}
The NPNS/NPS system is a semilinear parabolic system and local existence and uniqueness of strong solutions have been established by many authors for many different sets of boundary conditions. We refer the reader to \cite{bothe} where local well-posedness is established for dimensions greater than one, arbitrarily many ionic species, blocking boundary conditions for $c_i$, and Robin boundary conditions for $\Phi$. However, as the authors remark, the proof, based on methods of maximal $L^p$ regularity, can be adapted in a straightforward manner for different boundary conditions, including the mixed boundary conditions consiidered later in Section \ref{mbc}. Thus we have the following local existence theorem:
\begin{thm}
For initial conditions $0\le c_i(0)\in H^1$, $u(0)\in V$, there exists $T_0>0$ depending on $\|c_i(0)\|_{H^1},\|u(0)\|_V$, the boundary data $\tau,\xi$ (and $\gamma_i$ if (\ref{DI}) is considered), the parameters of the problem $D_i,z_i,\epsilon,\nu,K$, and the domain $\Omega$ such that NPNS (\ref{np}),(\ref{pois}),(\ref{nse}) (and NPS (\ref{np}),(\ref{pois}),(\ref{stokes})) has a unique strong solution $(c_i,\Phi,u)$ on the time interval $[0,T_0]$ satisfying the boundary conditions (\ref{bl}) (or (\ref{DI})),(\ref{noslip}),(\ref{robin}).\label{local} \end{thm}
\begin{rem} We stress that the nonnegativity of $c_i$ is included in our definition of a strong solution. That nonnegativity is propagated from nonnegative initial conditions $c_i(0)\ge 0$ is not self-evident. Its proof is included in Appendix \ref{pc}. In fact, as proved in Appendix \ref{pc}, more is true: strict positivity is propagated from strictly positive initial conditions $c_i(0)\ge c>0$. \end{rem}
Henceforth, all occurrences of the constant $C>0$, with no subscript, refer to a constant depending only on the parameters of the system, the boundary data, and the domain $\Omega$, and this constant may differ from line to line. For brevity, when a constant, other than $C$, is said to depend on the parameters of the system, we mean this to also include boundary data and the domain.
\section{Global Regularity for Blocking Boundary Conditions (Two Species)}\label{GR} In this section we consider the NPNS and NPS systems for two ($m=2$) oppositely charged ($z_1>0>z_2$) ionic species, both satisfying blocking boundary conditions. In this setting, we prove global existence of strong solutions for the NPS system and the same result, conditional on Navier-Stokes regularity, for the NPNS system.
We begin by proving some a priori bounds that are used for the proof of the global regularity result of this section. We prove some of the estimates in more generality (two or more species) so as to invoke them in Section \ref{mss} as well. In Sections \ref{mss} and \ref{mbc}, for the sake of brevity and fewer repetitive computations, we shall also frequently make references to some estimates from this section that may not hold verbatim but nonetheless hold up to some minor modifications. \subsection{Dissipation Estimate}\label{de} The NPNS and NPS systems come with a dissipative structure when blocking boundary conditions for $c_i$ and Robin boundary conditions for $\Phi$ are considered. We prove the following proposition: \begin{prop} Let $(c_i,\Phi,u)$ be a strong solution of NPNS or NPS on the time interval $[0,T]$, satisfying boundary conditions (\ref{bl}),(\ref{noslip}),(\ref{robin}). Then the functional
\begin{equation} V(t)=\frac{1}{2K}\|u\|_H^2+\sum_{i=1}^m\int_\Omega c_i\log c_i\,dx+\frac{\epsilon}{2}\|\nabla\Phi\|_{L^2(\Omega)}^2+\frac{\epsilon\tau}{2}\|\Phi\|_{L^2(\partial\Omega)}^2 \end{equation} satisfies \begin{equation}
\frac{d}{dt}V+\mathcal{D}+\frac{\nu}{K}\|\nabla u\|_{L^2}^2=0 \end{equation} for $t\in[0,T]$, where \begin{equation}
\mathcal{D}=\sum_{i=1}^m D_i\int_\Omega c_i|\nabla\mu_i|^2\,dx\ge 0 \end{equation} and $\mu_i$ is the electrochemical potential, \begin{equation} \mu_i=\log c_i+z_i\Phi.\label{mu} \end{equation} In particular, $V(t)$ is nonincreasing in time.\label{diss} \end{prop}
\begin{proof} First we note that (\ref{np}) can be written \begin{equation} \partial_t c_i+u\cdot \nabla c_i=D_i{\mbox{div}\,}(c_i\nabla\mu_i).\label{np'} \end{equation} Then we multiply (\ref{np'}) by $\mu_i$ and integrate by parts. This part is somewhat formal as we cannot exclude the possibility that $c_i$ attains the value $0$, in which case $\log c_i$ becomes undefined. Thus, to make this rigorous, we can work instead with the quantity $\mu_i^\delta=\log(c_i+\delta)+z_i\Phi$ and later pass to the limit $\delta\to 0$, as done in \cite{bothe}. For conciseness, we stick to the formal computations involving $\mu_i$. On the right hand side of (\ref{np'}), we obtain after summing in $i$, \begin{equation}
-\sum_{i=1}^m D_i\int_\Omega c_i|\nabla\mu_i|^2\,dx. \end{equation} which is precisely $-\mathcal{D}$. For the terms on the left hand side, we have after summing in $i$ and integrating by parts, \begin{equation} \begin{aligned} \sum_{i=1}^m\int_\Omega \partial_t c_i(\log c_i+z_i\Phi)\,dx&=\frac{d}{dt}\sum_{i=1}^m\int_\Omega c_i\log c_i-c_i\,dx+\int_\Omega(\partial_t\rho)\Phi\,dx\\ &=\frac{d}{dt}\sum_{i=1}^m\int_\Omega c_i\log c_i\,dx-\epsilon\int_\Omega\partial_t(\Delta\Phi)\Phi\,dx\\
&=\frac{d}{dt}\sum_{i=1}^m\int_\Omega c_i\log c_i\,dx-\epsilon\int_{\partial\Omega}\partial_t(\partial_n\Phi)\Phi\,dS+\frac{\epsilon}{2}\frac{d}{dt}\int_\Omega|\nabla\Phi|^2\,dx\\
&=\frac{d}{dt}\sum_{i=1}^m\int_\Omega c_i\log c_i\,dx+\epsilon\tau\int_{\partial\Omega}(\partial_t\Phi)\Phi\,dS+\frac{\epsilon}{2}\frac{d}{dt}\int_\Omega|\nabla\Phi|^2\,dx\\
&=\frac{d}{dt}\sum_{i=1}^m\int_\Omega c_i\log c_i\,dx+\frac{\epsilon\tau}{2}\frac{d}{dt}\int_{\partial\Omega}\Phi^2\,dS+\frac{\epsilon}{2}\frac{d}{dt}\int_\Omega|\nabla\Phi|^2\,dx. \end{aligned}
\end{equation} In the second line, we used the fact that $\|c_i(t)\|_{L^1}=\|c_i(0)\|_{L^1}$ for all time due to blocking boundary conditions. We also used the Poisson equation for $\Phi$. In the fourth line, we used the Robin boundary conditions (\ref{robin}).
Lastly, for the advective term we obtain after summing, \begin{equation} \begin{aligned} \sum_{i=1}^m\int_\Omega u\cdot\nabla c_i(\log c_i+z_i\Phi)\,dx=&\sum_{i=1}^m\int_\Omega u\cdot\nabla(c_i\log c_i-c_i)\,dx+\int_\Omega (u\cdot\nabla\rho)\Phi\,dx\\ =&\int_\Omega (u\cdot\nabla\rho)\Phi\,dx\\ =&-\int_\Omega (u\cdot\nabla\Phi)\rho\,dx \end{aligned} \end{equation} where in the second and third lines we integrated by parts and used the fact that ${\mbox{div}\,} u=0$. Collecting what we have so far, we have \begin{equation}
\frac{d}{dt}\left(\sum_{i=1}^m\int_\Omega c_i\log c_i\,dx+\frac{\epsilon}{2}\|\nabla\Phi\|_{L^2}^2+\frac{\epsilon\tau}{2}\|\Phi\|_{L^2(\partial\Omega)}^2\right)+\mathcal{D}=\int_\Omega(u\cdot\nabla\Phi)\rho\,dx.\label{one} \end{equation} Next we multiply (\ref{nse}) (or (\ref{stokes})) by $u$ and integrate by parts, noticing that the integral corresponding to the nonlinear term for Navier-Stokes vanishes due to the divergence-free condition, \begin{equation}
\frac{1}{2}\frac{d}{dt}\|u\|_{L^2}^2+\nu\|\nabla u\|_{L^2}^2=-K\int_\Omega (u\cdot\nabla\Phi)\rho\,dx.\label{two} \end{equation} Thus, multiplying (\ref{two}) by $K^{-1}$ and adding it to (\ref{one}), we obtain the conclusion of the proposition. \end{proof}
\subsection{Uniform $L^2$ Estimate}\label{UL2}
Using the dissipative estimate from the previous subsection, we obtain uniform in time control of $\|c_i\|_{L^2}$ in the case of two oppositely charged species satisfying blocking boundary conditions.
\begin{prop} Let $(c_i,\Phi,u)$ be a strong solution of NPNS or NPS for two oppositely charged species ($m=2,\, z_1>0>z_2$) on the time interval $[0,T]$, satisfying boundary conditions (\ref{bl}),(\ref{noslip}),(\ref{robin}), and corresponding to initial conditions $0\le c_i(0)\in H^1,\,u(0)\in V$. Then there exists a constant $M_2>0$ independent of time, depending only on the parameters of the system and the initial conditions such that for each $i$ \begin{equation}
\sup_{t\in[0,T]}\|c_i(t)\|_{L^2}<M_2.\label{m2m2} \end{equation}\label{L2'} \end{prop} \begin{proof}
We multiply (\ref{np}) by $\frac{|z_i|}{D_i}c_i$ and integrate by parts: \begin{equation} \begin{aligned}
\frac{|z_i|}{2D_i}\frac{d}{dt}\int_\Omega c_i^2\,dx=&-|z_i|\int_\Omega|\nabla c_i|^2\,dx-z_i|z_i|\int_\Omega c_i\nabla c_i\cdot\nabla\Phi\,dx\\
=&-|z_i|\int_\Omega|\nabla c_i|^2\,dx-z_i|z_i|\frac{1}{2}\int_\Omega \nabla c_i^2\cdot\nabla\Phi\,dx\\
=&-|z_i|\int_\Omega|\nabla c_i|^2\,dx-z_i|z_i|\frac{1}{2\epsilon}\int_\Omega c_i^2\rho\,dx- z_i|z_i|\frac{1}{2}\int_{\partial\Omega}c_i^2\partial_n\Phi\,dS\\
=&-|z_i|\int_\Omega|\nabla c_i|^2\,dx-z_i|z_i|\frac{1}{2\epsilon}\int_\Omega c_i^2\rho\,dx\\
&+z_i|z_i|\frac{\tau}{2}\int_{\partial\Omega}c_i^2\Phi\,dS- z_i|z_i|\frac{1}{2}\int_{\partial\Omega}c_i^2\xi\,dS\\
=&-|z_i|\int_\Omega|\nabla c_i|^2\,dx-z_i|z_i|\frac{1}{2\epsilon}\int_\Omega c_i^2\rho\,dx+I_1^{(i)}+I_2^{(i)}\label{ener} \end{aligned} \end{equation} where in the fourth line, we used the Robin boundary conditions (\ref{robin}). We estimate the two boundary integrals using trace inequalities (Lemma \ref{trace}, Appendix): \begin{equation} \begin{aligned}
|I_1^{(i)}|\le&C\|\Phi\|_{L^4(\partial\Omega)}\|c_i\|_{L^\frac{8}{3}(\partial\Omega)}^2\\
\le&\|\Phi\|_{H^1(\Omega)}(C_\delta\|c_i\|_{L^1(\Omega)}^2+\delta\|\nabla c_i\|_{L^2(\Omega)}^2)\label{I_1} \end{aligned} \end{equation} and similarly, \begin{equation} \begin{aligned}
|I_2^{(i)}|\le&\frac{|z_i|^2\|\xi\|_{L^\infty(\partial\Omega)}}{2}\|c_i\|_{L^2(\partial\Omega)}^2\le C_\delta\|c_i\|_{L^1}^2+\delta\|\nabla c_i\|_{L^2}^2.\label{I_2} \end{aligned}
\end{equation} We recall that $\|c_i\|_{L^1}$ remains constant in time, and since by a generalized Poincaré's inequality we have that \begin{equation}
\|\Phi\|_{L^2(\Omega)}\le C(\|\nabla\Phi\|_{L^2(\Omega)}+\|\Phi\|_{L^2(\partial\Omega)})
\end{equation} we deduce from Proposition \ref{diss} that $\|\Phi\|_{H^1}$ is uniformly bounded in time by initial data. Thus choosing \begin{equation}
\delta=\min\left\{\frac{|z_i|}{4},\frac{|z_i|}{4\sup_t\|\Phi(t)\|_{H^1}}\right\} \end{equation} we obtain from (\ref{ener})-(\ref{I_2}), after summing in $i$ and recalling $z_1>0>z_2$, \begin{equation} \begin{aligned}
\sum_{i=1}^2\frac{|z_i|}{2D_i}\frac{d}{dt}\|c_i\|_{L^2(\Omega)}^2+\sum_{i=1}^2\frac{|z_i|}{2}\|\nabla c_i\|_{L^2(\Omega)}^2&\le C_b-\frac{1}{2\epsilon}\int_\Omega (z_1^2c_1^2-z_2^2c_2^2)\rho\,dx\\
&= C_b-\frac{1}{2\epsilon}\int_\Omega \rho^2(|z_1|c_1+|z_2|c_2)\,dx\\ &\le C_b.\label{cancellation} \end{aligned}
\end{equation} Here, $C_b$ depends on $\sup_t\|\Phi(t)\|_{H^1}, \|c_i(0)\|_{L^1}$ along with the various parameters of the system. Next we use a Gagliardo-Nirenberg inequality to bound \begin{equation}
\|c_i\|_{L^2}^2\le C(\|\nabla c_i\|_{L^2}^2+\|c_i\|_{L^1}^2)\le C'(\|\nabla c_i\|_{L^2}^2+1),\label{poinc}
\end{equation} where $C'$ depends on $\|c_i(0)\|_{L^1}$. Thus, for constants $C'_b, C''_b\ge 0$ depending on $\sup_t\|\Phi(t)\|_{H^1}, \|c_i(0)\|_{L^1}$ and parameters, we obtain from (\ref{cancellation}), \begin{equation}
\frac{d}{dt}\left(\sum_{i=1}^2\frac{|z_i|}{2D_i}\|c_i\|_{L^2}^2\right)\le -C'_b\left(\sum_{i=1}^2\frac{|z_i|}{2D_i}\|c_i\|_{L^2}^2\right)+C''_b.\label{last} \end{equation} Thus from a Grönwall estimate, we find \begin{equation}
\sum_{i=1}^2\frac{|z_i|}{2D_i}\|c_i(t)\|_{L^2}^2\le \left(\sum_{i=1}^2\frac{|z_i|}{2D_i}\|c_i(0)\|_{L^2}^2\right)e^{-C'_bt}+\frac{C''_b}{C'_b}(1-e^{-C'_bt})\le \sum_{i=1}^2\frac{|z_i|}{2D_i}\|c_i(0)\|_{L^2}^2+\frac{C''_b}{C'_b} \end{equation} and (\ref{m2m2}) follows.\end{proof}
\subsection{Higher Order Estimates}\label{he} Now we bootstrap the dissipative and $L^2$ estimates to obtain some higher order estimates. \begin{prop} Let $(c_i,\Phi,u)$ be a strong solution of NPNS or NPS on the time interval $[0,T]$ with $0\le c_i(0)\in H^1\cap L^\infty,\,u(0)\in V$, satisfying boundary conditions (\ref{bl}),(\ref{noslip}),(\ref{robin}). Assume that for each $i$, $c_i$ satisfies a uniform in time $L^2$ bound, \begin{equation}
\|c_i(t)\|_{L^2}<M_2.\label{L2'''} \end{equation} Then there exist constants $M_\infty,M'_2>0$ independent of time, depending only on the parameters of the system, the initial conditions and $M_2$ such that for each $i$ \begin{align}
\sup_{t\in[0,T]}\|c_i(t)\|_{L^\infty}&<M_\infty\label{Mi}\\
\int_{0}^{T}\|\nabla {\tilde{c}_i}(s)\|_{L^2}^2\,ds&< M'_2\label{L2''} \end{align} where ${\tilde{c}_i}=c_ie^{z_i\Phi}$. Specifically for the case of the NPS system, we have \begin{equation}
\sup_{t\in[0,T]}\|u(t)\|_V^2+\frac{1}{T}\int_0^T\|u(s)\|_{H^2}^2\,ds<B\label{Mu} \end{equation} for a constant $B$ independent of time. For the NPNS system, we have instead \begin{equation}
\sup_{t\in[0,T]}\|u(t)\|_V^2+\int_0^T\|u(s)\|_{H^2}^2\,ds<B_T\label{M'u} \end{equation} for a time dependent constant $B_T$ depending also on $U(T)$ where
\begin{equation} U(T)=\int_0^T\|u(s)\|_V^4\,ds. \end{equation}\label{L2} \end{prop} \begin{rem} For two oppositely charged species, $m=2$, $z_1>0>z_2$, the hypothesis (\ref{L2'''}) holds due to Proposition \ref{L2'}. \end{rem} \begin{rem}
Here, and in later theorems, the assumption that $c_i(0)\in L^\infty$ is not, strictly speaking, necessary as the local existence theorem guarantees that $H^1$ initial data is immediately regularized so that on the interval of existence $[0,T]$, we have $c_i\in L^2(0,T;H^2)$. In particular, for some arbitrarily small $\tilde t>0$ we have $c_i(\tilde t)\in H^2\subset L^\infty$. Thus, below, when we derive a priori upper bounds in terms of $\|c_i(0)\|_{L^\infty}$ (c.f. (\ref{sk})), we could instead do so in terms of $\|c_i(\tilde t)\|_{L^\infty}$. To avoid doing this, we include $c_i(0)\in L^\infty$ in the hypothesis. For later theorems (e.g. in Section \ref{mbc}) whose proofs do not invoke $\|c_i(0)\|_{L^\infty},$ we do note include $c_i\in L^\infty$ in the hypothesis. \end{rem} \begin{proof} It follows from (\ref{pois}), (\ref{L2'''}) and Sobolev estimates that for some constants $P_6,\,p_\infty$ independent of time, we have, \begin{align}
\|\nabla\Phi(t)\|_{L^6}&\le P_6\label{P6}\\
\|\Phi(t)\|_{L^\infty}&\le p_\infty\label{Piii}. \end{align}
Now we deduce the uniform in time $L^\infty$ bounds, using a Moser-type iteration (see also \cite{bothe,choi,np3d,schmuck}). For $k=2,3,4...$, we multiply (\ref{np}) by $c_i^{2k-1}$ and integrate by parts to obtain \begin{equation}
\frac{1}{2k}\frac{d}{dt}\|c_i\|_{L^{2k}}^{2k}+\frac{2k-1}{k^2}D_i\|\nabla c_i^k\|_{L^2}^2\le C\frac{2k-1}{k}\|\nabla\Phi\|_{L^6}\|c_i^k\|_{L^3}\|\nabla c_i^k\|_{L^2}.\label{kk} \end{equation} We use (\ref{P6}) and interpolate $L^3$ between $L^1$ and $H^1$ to obtain after a Young's inequality, \begin{equation}
\frac{d}{dt}\|c_i\|_{L^{2k}}^{2k}+D_i\|\nabla c_i^k\|_{L^2}^2\le C_k\|c_i^k\|_{L^1}^2\label{2k} \end{equation} where $C_k$ satisfies, for some $c>0$, for some $m$ large enough and for each $k=2,3,4...$ \begin{equation} C_k\le ck^m.\label{ck} \end{equation} Interpolating $L^2$ between $L^1$ and $H^1$, \begin{equation}
\|c_i^k\|_{L^2}^2\le C(\|\nabla c_i^k\|_{L^2}^2+\|c_i^k\|_{L^1}^2), \end{equation} we obtain from (\ref{2k}), \begin{equation}
\frac{d}{dt}\|c_i\|_{L^{2k}}^{2k}\le -C\|c_i\|_{L^{2k}}^{2k}+C_k\|c_i^k\|_{L^1}^2=-C\|c_i\|_{L^{2k}}^{2k}+C_k\|c_i\|_{L^{k}}^{2k}\label{2k'} \end{equation} for a different $C_k$ still satisfying $(\ref{ck})$ for some $c$. We define
\begin{equation} S_k=\max\{\|c_i(0)\|_{L^\infty},\,\sup_t\|c_i(t)\|_k\}.\label{sk} \end{equation} Applying a Grönwall inequality to (\ref{2k'}), we obtain \begin{equation}
\|c_i\|_{L^{2k}}^{2k}\le\|c_i(0)\|_{L^{2k}}^{2k}+C_kS_k^{2k}\le|\Omega|\|c_i(0)\|_{L^\infty}^{2k}+C_kS_k^{2k}\le Ck^mS_k^{2k}\label{lel} \end{equation} for a possibly different $C_k$ still satisfying $(\ref{ck})$ for some $c$. Assuming without loss of generality that $C\ge 1$ in (\ref{lel}),
\begin{equation} S_{2k}=\max\{\|c_i(0)\|_{L^\infty},\sup_t\|c_i(t)\|_{L^{2k}}\}\le\max\{\|c_i(0)\|_{L^\infty},C^{\frac{1}{2k}}k^{\frac{m}{2k}}S_k\}=C^{\frac{1}{2k}}k^{\frac{m}{2k}}S_k.\label{ss} \end{equation} Setting $k=2^j$, we obtain \begin{equation} S_{2^{j+1}}\le C^{\frac{1}{2^{j+1}}}2^{\frac{jm}{2^{j+1}}}S_{2^j} \end{equation} and thus for all $J\in\mathbb{N}$ \begin{equation} S_{2^J}\le C^a2^bS_2<\infty\label{s2j} \end{equation} where \begin{equation} a=\sum_{j=1}^\infty \frac{1}{2^{j+1}}<\infty,\quad b=\sum_{j=1}^\infty \frac{jm}{2^{j+1}}<\infty. \end{equation} Passing $J\to\infty$ in (\ref{s2j}), we obtain (\ref{Mi}).
Next, (\ref{L2''}) follows from (\ref{Mi}), (\ref{Piii}) and Proposition \ref{diss}. Indeed, from the proposition, using the fact that $\mu_i=\log {\tilde{c}_i}$, we obtain \begin{equation}
\int_0^T\int_\Omega|\nabla {\tilde{c}_i}|^2\,dx\,dt\le C_p\int_0^T\int_\Omega c_i|\nabla\mu_i|^2\,dx\,dt\le M'_2 \end{equation} for $M'_2$ independent of time, and $C_p$ depending on $p_\infty$ and $M_\infty$.
Next, to prove (\ref{Mu}), we multiply the Stokes equations (\ref{stokes'}) by $Au$ and integrate by parts, \begin{equation} \begin{aligned}
\frac{1}{2}\frac{d}{dt}\|u\|_V^2+\frac{\nu}{2}\|A u\|_{L^2}^2\le C\|\rho\nabla\Phi\|_{L^2}^2\le C'M_\infty^2\label{nn} \end{aligned}
\end{equation} where $C'$ depends on $\sup_t\|\nabla\Phi\|_{L^2}$ (Proposition \ref{diss}). Using the elliptic bound $\|u\|_V\le C\|A u\|_{L^2}$, we obtain \begin{equation}
\frac{1}{2}\frac{d}{dt}\|u\|_V^2\le -C''\|u\|_V^2+C'M_\infty^2 \label{unps}
\end{equation} which gives us uniform boundedness of $\|u\|_V$, the first half of (\ref{Mu}). Integrating (\ref{nn}) in time gives us the second half.
Similarly, for NPNS, we multiply the Navier-Stokes equations (\ref{nse'}) by $A u$ and integrate by parts, \begin{equation} \begin{aligned}
\frac{1}{2}\frac{d}{dt}\|u\|_V^2+\frac{\nu}{2}\|A u\|_{L^2}^2\le& C'M_\infty^2+\|u\|_{L^6}\|\nabla u\|_{L^3}\|A u\|_{L^2}\\
\le& C'M_\infty^2 +C\|u\|_V^\frac{3}{2}\|A u\|_{L^2}^\frac{3}{2} \end{aligned} \end{equation} so that after a Young's inequality we obtain \begin{equation} \begin{aligned}
\frac{1}{2}\frac{d}{dt}\|u\|_V^2+\frac{\nu}{4}\|A u\|_{L^2}^2\le& C'M_\infty^2+C\|u\|_V^6\label{nnn} \end{aligned} \end{equation} from which we obtain \begin{equation}
\frac{1}{2}\|u(t)\|_{V}^2+\frac{\nu}{4}\int_0^t\|A u(s)\|_{L^2}^2\,ds\le\left(\frac{1}{2}\|u(0)\|_V^2+C'M_\infty^2t\right)e^{CU(T)}. \end{equation} This gives us (\ref{M'u}) and completes the proof of the proposition. \end{proof}
\subsection{Proof of Global Regularity}\label{gr} Now we prove our main global regularity result of this section. \begin{thm} For initial conditions $0\le c_i(0)\in H^1$, $u(0)\in V$ and for all $T>0$, NPS (\ref{np}),(\ref{pois}),(\ref{stokes}) for two oppositely charged species ($m=2,\,z_1>0>z_2$) has a unique strong solution $(c_i,\Phi,u)$ on the time interval $[0,T]$ satisfying the boundary conditions (\ref{bl}),(\ref{noslip}),(\ref{robin}). NPNS (\ref{np}),(\ref{pois}),(\ref{nse}) for two oppositely charged species has a unique strong solution on $[0,T]$ satisfying the initial and boundary conditions provided
\begin{equation} U(T)=\int_0^T\|u(s)\|_V^4\,ds<\infty. \end{equation} Moreover, the solution to NPS satisfies (\ref{Mu}) in addition to \begin{align}
\sup_{t\in[0,T]}\|c_i(t)\|_{H^1}^2+\frac{1}{T}\int_0^T\|c_i(s)\|_{H^2}^2\,ds&\le M\label{M}\\
\sup_{t\in[0,T]}\|\nabla{\tilde{c}_i}(t)\|_{L^2}^2+\int_0^T\|\Delta{\tilde{c}_i}(s) \|_{L^2}^2\,ds&\le M'\label{M'} \end{align} for ${\tilde{c}_i}=e^{z_i\Phi}$ and constants $M,M'$ depending only on the parameters of the system and the initial conditions but not on $T$. The solution to NPNS satisfies (\ref{M'u}) in addition to \begin{equation}
\sup_{t\in[0,T]}\|c_i(t)\|_{H^1}^2+\int_0^T\|c_i(s)\|_{H^2}^2\,ds\le M_T\label{MU} \end{equation} for a constant $M_T$ depending on the parameters of the system, the initial conditions, $T$, and $U(T)$.\label{gr!} \end{thm}
\begin{proof} We prove the a priori estimates (\ref{M})-(\ref{MU}), which together with the bounds on $u$, (\ref{Mu}) and (\ref{M'u}), and the local existence theorem allow us to uniquely extend a local solution to a global one by virtue of the fact that the strong norms in Definition \ref{strong} do not blow up before time $T$.
Due to (\ref{Mi}), it follows from the embedding $W^{2,\infty}\hookrightarrow W^{1,\infty}$ that \begin{equation}
\|\Phi(t)\|_{W^{1,\infty}}\le P_\infty\label{Pi} \end{equation} for all $t$, where $P_\infty$ depends only on the parameters of the system and uniform $L^p$ bounds on $\rho$.
Next, in order to obtain estimates for $\nabla c_i$, we note that the auxiliary variable \begin{equation} {\tilde{c}_i}=c_ie^{z_i\Phi}\label{cit} \end{equation} satisfies \begin{equation} \partial_t{\tilde{c}_i}+u\cdot\nabla{\tilde{c}_i}=D_i\Delta{\tilde{c}_i}-D_iz_i\nabla{\tilde{c}_i}\cdot\nabla\Phi+z_i((\partial_t+u\cdot\nabla)\Phi){\tilde{c}_i}\label{npt} \end{equation} together with homogeneous Neumann boundary conditions \begin{equation}
{\partial_n{\tilde{c}_i}}_{|\partial\Omega}=0.\label{neu} \end{equation} Multiplying (\ref{npt}) by $-\Delta{\tilde{c}_i}$ and using (\ref{neu}) to integrate by parts, we obtain \begin{equation} \begin{aligned}
\frac{1}{2}\frac{d}{dt}\|\nabla{\tilde{c}_i}\|_{L^2}^2+D_i\|\Delta{\tilde{c}_i}\|_{L^2}^2=&\int_\Omega (u\cdot\nabla{\tilde{c}_i})\Delta{\tilde{c}_i}\,dx+D_iz_i\int_\Omega(\nabla{\tilde{c}_i}\cdot\nabla\Phi)\Delta{\tilde{c}_i}\,dx\\ &-z_i\int_\Omega((\partial_t+u\cdot\nabla)\Phi){\tilde{c}_i}\Delta{\tilde{c}_i}\,dx\\ =& I_1+I_2+I_3.\label{dci} \end{aligned} \end{equation} We estimate using Hölder and Young's inequalities and Sobolev and interpolation estimates, \begin{equation} \begin{aligned}
|I_1|\le&\|u\|_V\|\nabla{\tilde{c}_i}\|_{L^3}\|\Delta{\tilde{c}_i}\|_{L^2}\\
\le& C\|u\|_V\|\nabla{\tilde{c}_i}\|_{L^2}^\frac{1}{2}\|\Delta {\tilde{c}_i}\|_{L^2}^\frac{3}{2}\\
\le&\frac{D_i}{4}\|\Delta {\tilde{c}_i}\|_{L^2}^2+C\|u\|_V^4\|\nabla{\tilde{c}_i}\|_{L^2}^2\label{id} \end{aligned} \end{equation}
\begin{equation} \begin{aligned}
|I_2|\le&C\|\nabla\Phi\|_{L^\infty}\|\nabla {\tilde{c}_i}\|_{L^2}\|\Delta{\tilde{c}_i}\|_{L^2}\\
\le&\frac{D_i}{4}\|\Delta{\tilde{c}_i}\|_{L^2}^2+C_g\|\nabla{\tilde{c}_i}\|_{L^2}^2\label{idd} \end{aligned} \end{equation} where $C_g$ depends on $P_\infty$. Next we split \begin{equation} I_3=-z_i\int_\Omega(u\cdot\nabla\Phi){\tilde{c}_i}\Delta{\tilde{c}_i}\,dx-z_i\int_\Omega(\partial_t\Phi){\tilde{c}_i}\Delta{\tilde{c}_i}\,dx=I_3^1+I_3^2.\label{split} \end{equation} First we estimate $I_3^1$. Noting that \begin{equation}
\|{\tilde{c}_i}\|_{L^3}\le CM_\infty e^{|z_i|P_\infty}=\beta_3 \end{equation} we bound \begin{equation} \begin{aligned}
|I_3^1|\le& C\|u\|_V\|\nabla\Phi\|_{L^\infty}\|{\tilde{c}_i}\|_{L^3}\|\Delta {\tilde{c}_i}\|_{L^2}\\
\le&\frac{D_i}{8}\|\Delta{\tilde{c}_i}\|_{L^2}^2+C'_g\|u\|_V^2 \end{aligned} \end{equation} where $C'_g$ depends on $\beta_3$ and $P_\infty$.
In order to bound $I_3^2$, first we note that the Nernst-Planck equations (\ref{np}) can be written \begin{equation} \partial_tc_i+u\cdot\nabla c_i=D_i{\mbox{div}\,}(e^{-z_i\Phi}\nabla{\tilde{c}_i}) \end{equation} so that in particular we have \begin{equation} \partial_t\rho=\sum_{i=1}^2z_iD_i{\mbox{div}\,}(e^{-z_i\Phi}\nabla {\tilde{c}_i})-u\cdot\nabla\rho.\label{ptrho} \end{equation} We multiply (\ref{ptrho}) by $\epsilon^{-1}\partial_t\Phi$ and integrate by parts. On the left hand side, we have \begin{equation} \begin{aligned}
\int_\Omega \partial_t\rho(\epsilon^{-1}\partial_t\Phi)\,dx=-\int_\Omega\partial_t\Delta\Phi\partial_t\Phi\,dx=&-\int_{\partial\Omega}\partial_t\partial_n\Phi\partial_t\Phi\,dS+\int_\Omega|\nabla\partial_t\Phi|^2\,dx\\
=&\tau\int_{\partial\Omega}|\partial_t\Phi|^2\,dS+\int_\Omega|\nabla\partial_t\Phi|^2\,dx \end{aligned} \end{equation} where in the second line we used the Robin boundary conditions (\ref{robin}). Therefore \begin{equation} \begin{aligned}
\tau\|\partial_t\Phi\|_{L^2(\partial\Omega)}^2+\|\partial_t\nabla\Phi\|_{L^2(\Omega)}^2\le& C\sum_{j=1}^2\left|\int_\Omega{\mbox{div}\,}(e^{-z_j\Phi}\nabla{\tilde{c}_j})\partial_t\Phi\,dx\right|+C\left|\int_\Omega{\mbox{div}\,}(u\rho)\partial_t\Phi\,dx\right|\\
\le&C\sum_{j=1}^2\int_\Omega e^{|z_j|P_\infty}|\nabla {\tilde{c}_j}||\partial_t\nabla\Phi|\,dx+C\int_\Omega M_\infty|u||\partial_t\nabla\Phi|\,dx\\
\le&C_{t}(\sum_{j=1}^2\|\nabla{\tilde{c}_j}\|_{L^2}+\|u\|_V)\|\partial_t\nabla\Phi\|_{L^2}\label{ptg} \end{aligned} \end{equation} where $C_{t}$ depends on $M_\infty,P_\infty$. Therefore, we obtain \begin{equation} (\tau
\|\partial_t\Phi\|_{L^2(\partial\Omega)}^2+\|\partial_t\nabla\Phi\|_{L^2(\Omega)}^2)^\frac{1}{2}\le C_{t}(\sum_{j=1}^2\|\nabla{\tilde{c}_j}\|_{L^2}+\|u\|_V).\label{blaa}
\end{equation} Now we use a generalized Poincaré inequality, $\|f\|_{L^2(\Omega)}\le C(\|\nabla f\|_{L^2(\Omega)}+\|f\|_{L^2(\partial\Omega)})$, which together with Sobolev's inequality $\|f\|_{L^6}\le C\|f\|_{H^1}$ and (\ref{blaa}), gives us \begin{equation}
\|\partial_t\Phi\|_{L^6}\le CC_{t}(\sum_{j=1}^2\|\nabla{\tilde{c}_j}\|_{L^2}+\|u\|_V).\label{pt6} \end{equation} Now we bound $I_3^2$ using (\ref{pt6}) \begin{equation} \begin{aligned}
|I_3^2|\le& C\|\partial_t\Phi\|_{L^6}\|{\tilde{c}_i}\|_{L^3}\|\|\Delta{\tilde{c}_i}\|_{L^2}\\
\le&\frac{D_i}{8}\|\Delta{\tilde{c}_i}\|_{L^2}^2+C_3(\sum_{j=1}^2\|\nabla{\tilde{c}_j}\|_{L^2}^2+\|u\|_V^2)\label{I32} \end{aligned} \end{equation} where $C_3$ depends on $\beta_3$.
Thus, adding the estimates for $I_1,I_2,I_3^1,I_3^2$ and summing in $i$, we obtain from (\ref{dci}), \begin{equation}
\frac{1}{2}\frac{d}{dt}\sum_{i=1}^2\|\nabla {\tilde{c}_i}\|_{L^2}^2+\frac{D_i}{4}\sum_{i=1}^2\|\Delta{\tilde{c}_i}\|_{L^2}^2\le C_F(\|u\|_V^4+1)\frac{1}{2}\sum_{i=1}^2\|\nabla {\tilde{c}_i}\|_{L^2}^2+C'_F\|u\|_V^2\label{cru} \end{equation} where $C_F,C'_F$ depend on $M_\infty,P_\infty$.
In the case of NPNS, we obtain from (\ref{cru}), \begin{equation} \begin{aligned}
\frac{1}{2}\sum_{i=1}^2&\|\nabla{\tilde{c}_i}(t)\|_{L^2}^2+\sum_{i=1}^2\frac{D_i}{4}\int_0^t\|\Delta{\tilde{c}_i}(s)\|_{L^2}^2\,ds\\
\le&\left(\frac{1}{2}\sum_{i=1}^2\|\nabla{\tilde{c}_i}(0)\|_{L^2}^2+C'_F\int_0^t\|u(s)\|_V^2\,ds\right)\exp\left(C_F\int_0^t\|u(s)\|_V^4+1\,ds\right). \end{aligned} \end{equation} This gives us (\ref{MU}) after converting to the original variables $c_i$ using the definition of ${\tilde{c}_i}$ and the uniform bounds on $c_i,\Phi$.
For the NPS system, we recall from Propositions \ref{diss} and \ref{L2} that $\|u\|_V$ is uniformly bounded in time and that $\|u\|_V^2$ and $\|\nabla {\tilde{c}_i}\|_{L^2}^2$ decay and are integrable in time, with a bound that is independent of $T$. Therefore, integrating (\ref{cru}) in time, we find that the right hand side is bounded by a constant independent of time, giving us (\ref{M'}). Converting back to the variables $c_i$ gives us (\ref{M}). \end{proof}
\section{Global Regularity for Blocking Boundary Conditions (Multiple Species)}\label{mss} The results of Sections \ref{de} and \ref{he} hold for more than two species, $m>2$. Therefore the question of whether or not we can extend our global regularity result to a multiple species setting boils down to whether or not we can establish the uniform $L^2$ estimates from Section \ref{UL2} in our current setting. Once such a bound is established, the proof of global regularity for multiple species follows exactly as in the proof of Theorem \ref{gr!}.
In the proof of Proposition \ref{L2'} of Section \ref{UL2}, specifically in (\ref{cancellation}), we use the fact that, due to the assumption of two species and $z_1>0>z_2$, \begin{equation}
(z_1^2c_1^2-z_2^2c_2^2)\rho=(|z_1|c_1-|z_2|c_2)(|z_1|c_1+|z_2|c_2)\rho=\rho^2(|z_1|c_1+|z_2|c_2)\ge 0.\label{sign} \end{equation} Even for the simplest extension of taking 3 species, with $z_1,z_2>0>z_3$, the corresponding leftmost term in (\ref{sign}) is $(z_1^2c_1^2+z_2^2c_2^2-z_3^2c_3^2)\rho$, and in general this term need not be nonnegative. Due to this fact, an analogous proof does not work for more than two species. However, there is one special multiple species setting where we do have global regularity: namely if all the diffusivities are equal $D_1=...=D_m$ and all the valences have the same magnitude (e.g. $z_1=z_2=1=-z_3$) (see also \cite{cil}). Indeed, we have the following theorem,
\begin{thm}
For initial conditions $0\le c_i(0)\in H^1\,(i=1,...,m)$, $u(0)\in V$ and for all $T>0$, if $D_1=...=D_m=D$ for a common value $D$ and $|z_1|=...=|z_m|=z$ for a common value $z$, then NPS (\ref{np}),(\ref{pois}),(\ref{stokes}) has a unique strong solution $(c_i,\Phi,u)$ on the time interval $[0,T]$ satisfying the boundary conditions (\ref{bl}),(\ref{noslip}),(\ref{robin}). For NPNS (\ref{np}),(\ref{pois}),(\ref{nse}), under the same hypotheses, a unique strong solution on $[0,T]$ satisfying the initial and boundary conditions exists provided \begin{equation}
\int_0^T\|u(s)\|_V^4\,ds<\infty. \end{equation} Moreover, in the case of NPS, the solution satisfies the bounds (\ref{Mi}),(\ref{L2''}),(\ref{Mu}),(\ref{M}),(\ref{M'}) for each $i$. In the case of NPNS, the solution satisfies the bounds (\ref{Mi}),(\ref{L2''}),(\ref{M'u}),(\ref{MU}) for each $i$.\label{grm} \end{thm}
\begin{proof} We only prove the result corresponding to Proposition \ref{L2'}. As discussed at the beginning of this section, the other results leading to the proof of global regularity extend naturally from Section \ref{GR}.
This special case of multiple species effectively boils down to a two species setting. The variables $\rho$ and $\sigma=z(c_1+c_2+...+c_m)\ge 0$ satisfy \begin{align} \partial_t\rho+u\cdot\nabla\rho&=D{\mbox{div}\,}(\nabla\rho+z\sigma\nabla\Phi)\label{rhoeq}\\ \partial_t\sigma+u\cdot\nabla\sigma&=D{\mbox{div}\,}(\nabla\sigma+z\rho\nabla\Phi)\label{sigmaeq} \end{align} and boundary conditions \begin{equation} \begin{aligned}
(\partial_n\rho+z\sigma\partial_n\Phi)_{|\partial\Omega}=(\partial_n\sigma+z\rho\partial_n\Phi)_{|\partial\Omega}=0. \end{aligned} \end{equation} Multiplying (\ref{rhoeq}) and (\ref{sigmaeq}) by $\rho$ and $\sigma$ respectively and integrating by parts, we obtain \begin{align}
\frac{1}{2D}\frac{d}{dt}\|\rho\|_{L^2}^2+\|\nabla\rho\|_{L^2}^2&=-z\int_\Omega\sigma\nabla\rho\cdot\nabla\Phi\,dx\label{bla}\\
\frac{1}{2D}\frac{d}{dt}\|\sigma\|_{L^2}^2+\|\nabla\sigma\|_{L^2}^2&=-z\int_\Omega\rho\nabla\sigma\cdot\nabla\Phi\,dx\label{blabla}. \end{align} We estimate the right hand side of (\ref{bla}) using the boundary conditions, \begin{equation} \begin{aligned} -z\int_\Omega\sigma\nabla\rho\cdot\nabla\Phi\,dx=&-z\int_{\partial\Omega}\sigma\rho\partial_n\Phi\,dS+z\int_\Omega\rho\nabla\sigma\cdot\nabla\Phi\,dx-\frac{z}{\epsilon}\int_\Omega\rho^2\sigma\,dx\\ \le&z\tau\int_{\partial\Omega}\sigma\rho\Phi\,dS-z\int_{\partial\Omega}\sigma\rho\xi\,dS+z\int_\Omega\rho\nabla\sigma\cdot\nabla\Phi\,dx\\ =&I_1+I_2+z\int_\Omega\rho\nabla\sigma\cdot\nabla\Phi\,dx.\label{bb} \end{aligned} \end{equation} As in (\ref{I_1}) and (\ref{I_2}), we obtain using Lemma \ref{trace} in the Appendix, \begin{equation} \begin{aligned}
|I_1|\le& C\|\Phi\|_{L^4(\partial\Omega)}\|\rho\|_{L^\frac{8}{3}(\partial\Omega)}\|\sigma\|_{L^\frac{8}{3}(\partial\Omega)}\\
\le&\|\Phi\|_{H^1(\Omega)}\left(C_\delta(\|\rho\|_{L^1}^2+\|\sigma\|_{L^1}^2)+\delta(\|\nabla\rho\|_{L^2}^2+\|\nabla\sigma\|_{L^2}^2)\right) \end{aligned} \end{equation} and similarly \begin{equation}
|I_2|\le C_\delta(\|\rho\|_{L^1}^2+\|\sigma\|_{L^1}^2)+\delta(\|\nabla\rho\|_{L^2}^2+\|\nabla\sigma\|_{L^2}^2).\label{I22} \end{equation} Thus choosing \begin{equation}
\delta=\min\left\{\frac{1}{4},\frac{1}{4\sup_t\|\Phi(t)\|_{H^1(\Omega)}}\right\} \end{equation} we obtain by adding (\ref{bla}) and (\ref{blabla}) and using the estimates (\ref{bb})-(\ref{I22}), \begin{equation}
\frac{1}{2D}\frac{d}{dt}\left(\|\rho\|_{L^2}^2+\|\sigma\|_{L^2}^2\right)+\frac{1}{2}\left(\|\nabla\rho\|_{L^2}^2+\|\nabla\sigma\|_{L^2}^2\right)<R\label{RRRR}
\end{equation} where $R$ is a constant that depends on uniform bounds on $\|\Phi\|_{H^1},\|\rho\|_{L^1}$ and $\|\sigma\|_{L^1}$, along with the parameters of the system. Thus by applying Gagliardo-Nirenberg inequalities to $\|\nabla\rho\|_{L^2}$ and $\|\nabla\sigma\|_{L^2}$ as in (\ref{poinc}), we conclude after a Grönwall estimate on (\ref{RRRR}) that $\|\rho\|_{L^2}$ and $\|\sigma\|_{L^2}$ remain uniformly bounded in time. In particular, because $c_i$ are nonnegative, it follows from the boundedness of $\|\sigma\|_{L^2}$ that $\|c_i\|_{L^2}$ are uniformly bounded in time for each $i$. This result replaces Proposition \ref{L2'}. \end{proof}
\section{Global Regularity for Mixed Boundary Conditions}\label{mbc} In this section, we again consider NPNS and NPS for two oppositely charged species. Suppose $\partial\Omega$ represents a cation selective membrane which allows for permeation of cations but blocks anions. As previously mentioned, ion selectivity is typically modelled by Dirichlet boundary conditions \cite{davidson,rubishtil}. So in this case, the boundary conditions for $c_i$, assuming $z_1>0>z_2$, are \begin{align}
{c_1(x,t)}_{|\partial\Omega}=&\gamma_1>0\label{di}\\
(\partial_n c_2(x,t)+z_2c_2(x,t)\partial_n\Phi(x,t))_{|\partial\Omega}=&0\label{2bl} \end{align} where $\gamma_1$ is \textit{constant}. The boundary conditions for $\Phi$ and $u$ are unchanged (see (\ref{noslip}),(\ref{robin})).
As discussed in the introduction, such configurations are known, in general, to lead to electrokinetic instabilities whereby, for large enough voltage drops $\sup\xi -\inf\xi$, one starts seeing vortical flow patterns, and for even larger drops, even chaotic behavior, resembling fluid turbulence and reminiscent of thermal turbulence in Rayleigh-Bénard convection \cite{davidson}. Mathematically, the instability of the configuration considered in this section is manifested by the lack of a dissipative bound resembling that from Proposition \ref{diss}.
To prove global regularity in this mixed setting, we aim, as in the previous sections, to obtain a priori control of the growth of the strong norms, which characterize strong solutions, namely $c_i\in L^\infty_t H^1_x\cap L^2_tH^2_x,\, u\in L^\infty_t V\cap L^2_tH^2_x.$ Let us remark here that \textit{strong} regularity in fact implies $C^\infty((0,T];C^\infty(\bar\Omega))$ regularity so long as the boundary and boundary conditions are smooth, as is the case here. This equivalence follows from a bootstrapping scheme using standard parabolic theory (see, for example, \cite{evans}). This observation justifies the use of Sard's theorem \cite{jl} in the proof of the following proposition.
\begin{prop} Let $(c_i,\Phi,u)$ be a strong solution of NPNS (\ref{np}),(\ref{pois}),(\ref{nse}) or NPS (\ref{np}),(\ref{pois}),(\ref{stokes}) for two oppositely charged species ($m=2, z_1>0>z_2$) on the time interval $[0,T]$ satisfying boundary conditions (\ref{di}),(\ref{2bl}),(\ref{noslip}),(\ref{robin}) and initial conditions $0\le c_i(0)\in H^1,\,u(0)\in V$. Then there exist ${\tilde M_2},\tilde m_2>0$ depending on the parameters of the system and the initial conditions such that for each $i=1,2$ \begin{equation}
\sup_{t\in[0,T]}\|c_i(t)\|_{L^2}<\tilde M_2 e^{\tilde m_2T^3}=M_2(T).\label{M2t} \end{equation}\label{gL2} \end{prop} \begin{proof}
\textbf{Step 1. $L^\infty_t L^1_x$ bounds on $c_i$.} Integrating (\ref{np}) for $i=2$ over $\Omega$, it follows from (\ref{2bl}) that $\|c_2(t)\|_{L^1}=\|c_2(0)\|_{L^1}$ for all $t\ge 0$. To obtain control of $\|c_1(t)\|_{L^1}$, we proceed as follows. We fix a $C^1$, nondecreasing function $\chi:\mathbb{R}\to[0,\infty)$ such that $\chi(s)=0$ for $s\in (-\infty,0]$, $\chi(s)=s^2$ for $s\in (0,\frac{1}{2})$, $\chi(s)=1$ for $s\in[1,\infty)$, and $\chi'\le 2$ everywhere. We note that $(\chi(s))^\alpha$ converges pointwise to the indicator function ${\bf{1}}_{s>0}(s)$ as $\alpha\to 0^+.$
Now we fix a time $t>0$, and we note that since the mapping $x\in\bar\Omega\mapsto {\bar c}_1(t,x):=c_1(t,x)-\gamma_1$ is smooth, it follows from Sard's theorem that there exists a sequence $\beta_n\to 0$ such that each $\beta_n$ is a regular value of ${\bar c}_1(t):\bar\Omega\mapsto\mathbb{R}$. In particular, it follows that the sets $\{{\bar c}_1(t)>\beta_n\}$ are (possibly empty) smooth submanifolds of $\Omega$ such that if $\{{\bar c}_1(t)>\beta_n\}$ is nonempty, then it has smooth boundary $\{{\bar c}_1(t)=\beta_n\}\subset\Omega$.
Now we multiply (\ref{np}) for $i=1$ by the test function \begin{equation} \psi=\psi_{\alpha,n}=(\chi\circ( {\bar c}_1(t)-\beta_n))^\alpha,\quad 0<\alpha<1 \end{equation} and integrate; then, defining the following primitive, \begin{equation} Q(y)=Q_{\alpha}(y)=\int_0^y (\chi(s))^\alpha\,ds\label{Qa} \end{equation} we have on the left hand side, using ${\mbox{div}\,} u=0$, \begin{equation} \int_\Omega (\partial_t c_1)\psi\,dx+\int_\Omega u\cdot\nabla ( Q\circ({\bar c}_1(t)-\beta_n))\,dx=\int_\Omega (\partial_t c_1)\psi\,dx. \end{equation} On the right hand side, we obtain, using the fact that $\psi=0$ on $\partial\{{\bar c}_1(t)>\beta_n\}=\{ {\bar c}_1(t)=\beta_n\}$ \begin{equation}
\begin{aligned} D_1\int_{ {\bar c}_1(t)>\beta_n}{\mbox{div}\,}(\nabla c_1+c_1\nabla\Phi)\psi\,dx=&-\alpha D_1\int_{ {\bar c}_1(t)>\beta_n}|\nabla c_1|^2\frac{\chi'\circ({\bar c}_1(t)-\beta_n)}{(\chi\circ({\bar c}_1(t)-\beta_n))^{1-\alpha}}\,dx\\ &+D_1\int_{{\bar c}_1(t)>\beta_n}{\mbox{div}\,} (({\bar c}_1(t)-\beta_n)\nabla\Phi)\psi\,dx\\ &-\frac{D_1}{\epsilon}(\gamma_1+\beta_n)\int_{{\bar c}_1(t)>\beta_n}\rho\psi\,dx\\ \le&-\alpha D_1\int_{{\bar c}_1(t)>\beta_n} ({\bar c}_1(t)-\beta_n)\nabla\Phi\cdot\nabla c_1\frac{\chi'\circ({\bar c}_1(t)-\beta_n)}{(\chi\circ({\bar c}_1(t)-\beta_n))^{1-\alpha}}\,dx\\ &-\frac{D_1}{\epsilon}(\gamma_1+\beta_n)\int_{{\bar c}_1(t)>\beta_n}\rho\psi\,dx. \end{aligned} \end{equation} Thus far, we have \begin{equation} \begin{aligned} \int_\Omega (\partial_t c_1)\psi\,dx\le& -\alpha D_1\int_{{\bar c}_1(t)>\beta_n} ({\bar c}_1(t)-\beta_n)\nabla\Phi\cdot\nabla c_1\frac{\chi'\circ({\bar c}_1(t)-\beta_n)}{(\chi\circ({\bar c}_1(t)-\beta_n))^{1-\alpha}}\,dx\\ &-\frac{D_1}{\epsilon}(\gamma_1+\beta_n)\int_{{\bar c}_1(t)>\beta_n}\rho\psi\,dx\label{g1} \end{aligned} \end{equation} Recalling the dependence of $\psi$ on $n$ and taking the limit of (\ref{g1}) as $n\to \infty$, we obtain the following estimate, which holds for each $t>0$ \begin{equation}\begin{aligned} \int_\Omega (\partial_t c_1)(\chi\circ {\bar c}_1(t))^\alpha\,dx\le&-\alpha D_1\int_{{\bar c}_1(t)>0}{\bar c}_1(t)\nabla\Phi\cdot\nabla c_1\frac{\chi'\circ {\bar c}_1(t)}{(\chi\circ {\bar c}_1(t))^{1-\alpha}}\,dx\\ &-\frac{D_1}{\epsilon}\gamma_1\int_{{\bar c}_1(t)>0}\rho(\chi\circ {\bar c}_1(t))^\alpha\,dx.\label{g2} \end{aligned}\end{equation} We note that the sequence $\beta_n$ depended on the fixed time $t$; however upon taking the limit $\beta_n$, the resulting bound (\ref{g2}) holds for all positive time. The limit on the right hand side is justified by dominated convergence along with the fact that, by our choice of $\chi$, we have \begin{equation} \begin{aligned}
{\bf{1}}_{\{{\bar c}_1(t)-\beta_n>0\}}|{\bar c}_1(t)-\beta_n|\frac{\chi'\circ ({\bar c}_1(t)-\beta_n)}{(\chi\circ ({\bar c}_1(t)-\beta_n))^{1-\alpha}}\le&{\bf{1}}_{\{0<{\bar c}_1(t)-\beta_n<\frac{1}{2}\}}|{\bar c}_1(t)-\beta_n|\frac{2|{\bar c}_1(t)-\beta_n|}{|{\bar c}_1(t)-\beta_n|^{2-2\alpha}}\\
&+{\bf{1}}_{\{\frac{1}{2}\le {\bar c}_1(t)-\beta_n\}}\frac{2\|{\bar c}_1(t)-\beta_n\|_{L^\infty}}{(1/2)^{2-2\alpha}}\\
\le&8\max\{\|{\bar c}_1(t)-\beta_n\|_{L^\infty}^{2\alpha},\|{\bar c}_1(t)-\beta_n\|_{L^\infty}\}.\label{g3} \end{aligned} \end{equation} The above bound ensures that the integrand of the first integral on the right hand side of (\ref{g1}) is bounded uniformly in $n$.
We now bound the right hand side of (\ref{g2}). The first integral is bounded as follows \begin{equation}\begin{aligned}
\left|\alpha D_1\int_{{\bar c}_1(t)>0}{\bar c}_1(t)\nabla\Phi\cdot\nabla c_1\frac{\chi'\circ {\bar c}_1(t)}{(\chi\circ {\bar c}_1(t))^{1-\alpha}}\,dx\right|\le& 2\alpha D_1\left|\int_{0<{\bar c}_1(t)<\frac{1}{2}}{\bar c}_1(t)|\nabla\Phi||\nabla c_1|\frac{{\bar c}_1(t)}{({\bar c}_1(t))^{2-2\alpha}}\,dx\right|\\
&+\alpha D_1\left|\int_{\frac{1}{2}\le {\bar c}_1(t)}{\bar c}_1(t)|\nabla\Phi||\nabla c_1|\frac{2}{(1/2)^2}\,dx \right|\\
\le& 2\alpha D_1\|\nabla\Phi\|_{L^\infty}\|\nabla c_1\|_{L^\infty}(1/2)^{2\alpha}|\Omega|\\
&+8\alpha D_1\|\nabla\Phi\|_{L^\infty}\|\nabla c_1\|_{L^\infty}\|{\bar c}_1(t)\|_{L^1}\\
\le&\alpha D_1\|\nabla\Phi\|_{L^\infty}\|\nabla c_1\|_{L^\infty}(2|\Omega|+8\|{\bar c}_1(t)\|_{L^1}).\label{bbb} \end{aligned}\end{equation} Using the nonnegativity of $c_1$, the second integral is bounded by \begin{equation}
-\frac{D_1}{\epsilon}\gamma_1\int_{{\bar c}_1(t)>0}\rho(\chi\circ {\bar c}_1(t))^\alpha\,dx\le \frac{D_1}{\epsilon}\gamma_1\|c_2(t)\|_{L^1}=\frac{D_1}{\epsilon}\gamma_1\|c_2(0)\|_{L^1}\label{bbb'} \end{equation}
Now returning to (\ref{g2}) and writing it as \begin{equation} \frac{d}{dt}\int_\Omega Q\circ {\bar c}_1(t)\,dx\le-\alpha D_1\int_{{\bar c}_1(t)>0}{\bar c}_1(t)\nabla\Phi\cdot\nabla c_1\frac{\chi'\circ {\bar c}_1(t)}{(\chi\circ {\bar c}_1(t))^{1-\alpha}}\,dx-\frac{D_1}{\epsilon}\gamma_1\int_{{\bar c}_1(t)>0}\rho(\chi\circ {\bar c}_1(t))^\alpha\,dx \end{equation} we find, using (\ref{bbb}), (\ref{bbb'}) and integrating in time, \begin{equation} \begin{aligned}
\int_\Omega Q\circ {\bar c}_1(t)\,dx\le& \int_\Omega Q\circ {\bar c}_1(0)\,dx+\alpha D_1\int_0^t \|\nabla\Phi(s)\|_{L^\infty}\|\nabla c_1(s)\|_{L^\infty}(2|\Omega|+8\|{\bar c}_1(s)\|_{L^1})\,ds\\
&+\frac{D_1}{\epsilon}\gamma_1\|c_2(0)\|_{L^1}t.\label{g4} \end{aligned} \end{equation} Now we recall the dependence of $Q$ on $\alpha$ and observe that $Q=Q_\alpha(y)\to y_+:=\max\{y,0\}$ as $\alpha\to 0$. Using these facts, we take the limit of (\ref{g4}) as $\alpha\to 0$ to obtain \begin{equation}\begin{aligned}
\int_\Omega {\bar c}_1(t)_+\,dx\le&\int_\Omega {\bar c}_1(0)_+\,dx+\frac{D_1}{\epsilon}\gamma_1\|c_2(0)\|_{L^1}t=\tilde B(t).\label{g5} \end{aligned}\end{equation} Thus, since, \begin{equation}
\|{c}_1(t)\|_{L^1}\le \int_\Omega {\bar c}_1(t)_++\gamma_1\,dx\le \tilde B(t)+\gamma_1|\Omega|=B(t)\label{B(t)}
\end{equation} we have shown that $\|c_1(t)\|_{L^1}$ grows at most linearly in time.
\textbf{Step 2. Time integrability of $\|\rho\|_{L^2}^2$.} Unlike in Section \ref{UL2}, it is necessary to treat $c_1,c_2$ separately as they satisfy different boundary conditions. We first consider $c_1$. Multiplying (\ref{np}) for $i=1$ by $\frac{1}{D_1}(\log c_1-\log\gamma_1)$ and integrating by parts, we obtain \begin{equation} \begin{aligned}
\frac{1}{D_1}\frac{d}{dt}\left(\int_\Omega c_1\log c_1-c_1-c_1\log\gamma_1\,dx\right)=&-\int_\Omega \frac{|\nabla c_1|^2}{c_1}+z_1\nabla\Phi\cdot\nabla c_1\,dx\\
=&-4\int_\Omega |\nabla \sqrt{c_1}|^2\,dx-z_1\int_\Omega \nabla\Phi\cdot\nabla(c_1-\gamma_1)\,dx\\
=&-4\int_\Omega |\nabla \sqrt{c_1}|^2\,dx+\frac{z_1}{\epsilon}\gamma_1\int_{\Omega}\rho\,dx-\frac{z_1}{\epsilon}\int_\Omega c_1\rho\,dx\label{logc1} \end{aligned} \end{equation} We observe that no boundary terms occur because $c_1=\gamma_1$ on the boundary, and the advective term involving $u$ also vanishes because ${\mbox{div}\,} u=0$ and because $\log\gamma_1$ is a constant.
Similarly, multiplying (\ref{np}) for $i=2$ by $\frac{1}{D_2}\log c_2$ and integrating by parts, we obtain \begin{equation} \begin{aligned}
\frac{1}{D_2}\frac{d}{dt}\left(\int_\Omega c_2\log c_2-c_2\,dx\right)=&-\int_\Omega \frac{|\nabla c_2|^2}{c_2}+z_2\nabla\Phi\cdot\nabla c_2\,dx\\
=&-4\int_\Omega |\nabla \sqrt{c_2}|^2\,dx-z_2\int_{\partial\Omega}c_2\partial_n\Phi\,dS-\frac{z_2}{\epsilon}\int_\Omega c_2\rho\,dx.\label{logc2} \end{aligned} \end{equation} We bound the boundary term using the Robin boundary conditions (\ref{robin}) and Lemma \ref{trace} in the Appendix (specifically (\ref{T1}) with $p=2$), \begin{equation} \begin{aligned}
\left|z_2\int_{\partial\Omega} c_2\partial_n\Phi\,dS\right|\le & \tau|z_2|\left|\int_{\partial\Omega}c_2\Phi\,dS\right|+\|\xi\|_{L^\infty{(\partial\Omega})}|z_2|\int_{\partial\Omega}c_2\,dS\\
\le&C(\|\Phi\|_{L^\infty(\Omega)}+\|\xi\|_{L^\infty(\partial\Omega)})\|\sqrt{c_2}\|_{L^2(\partial\Omega)}^2\\
\le&C(\|\rho\|_{L^\frac{7}{4}}+1)(\|\sqrt{c_2}\|_{L^2}^\frac{1}{2}\|\nabla \sqrt{c_2}\|_{L^2}^\frac{1}{2}+\|\sqrt{c_2}\|_{L^2})^2\\
\le&C(\|\rho\|_{L^1}^\frac{1}{7}\|\rho\|_{L^2}^\frac{6}{7}+1)(\|c_2\|_{L^1}^\frac{1}{2}\|\nabla\sqrt{c_2}\|_{L^2}+\|c_2\|_{L^1})\\
\le&2\|\nabla \sqrt{c_2}\|_{L^2}^2+\frac{1}{2\epsilon}\|\rho\|_{L^2}^2+C(\|\rho\|_{L^1}^2\|c_2\|_{L^1}^7+\|c_2\|_{L^1}+\|\rho\|_{L^1}^\frac{1}{4}\|c_2\|_{L^1}^\frac{7}{4}).\label{c2bdb} \end{aligned} \end{equation} The third line follows from the embedding $W^{2,\alpha}(\Omega)\hookrightarrow L^\infty(\Omega)$ for $\alpha>\frac{3}{2}$ (here, for concreteness we choose $\alpha=\frac{7}{4}$) and the fact that $\Phi$ is related to $\rho$ via the Poisson equation (\ref{pois}) and the boundary conditions (\ref{robin}). The fourth line follows from interpolating $L^\frac{7}{4}$ between $L^1$ and $L^2$. The last line follows from Young's inequalities.
\begin{comment} More precisely, we use the fact that the mapping \begin{equation}
\Phi\in H^2(\Omega)\mapsto (-\epsilon\Delta\Phi,(\partial_n\Phi+\tau\Phi)_{|\partial\Omega})\in L^2(\Omega)\times H^\frac{1}{2}(\partial\Omega) \end{equation} is an isomorphism (we refer the reader to \cite{grisvard} for the relevant elliptic theory). Thus, there exists $C>0$ depending on $\epsilon$ and $\tau$ such that \begin{equation}
\|\Phi\|_{H^2(\Omega)}\le C(\|\rho\|_{L^2(\Omega)}+\|\xi\|_{H^\frac{1}{2}(\partial\Omega)}). \end{equation} Then, from the Hölder embedding $H^2(\Omega)\hookrightarrow C^\alpha(\bar\Omega)\hookrightarrow L^\infty(\Omega)$ ($\alpha>0$), we have that \begin{equation}
\|\Phi\|_{L^\infty(\Omega)}\le C(\|\rho\|_{L^2(\Omega)}+\|\xi\|_{H^\frac{1}{2}(\partial\Omega)}),\label{hol}
\end{equation} which justifies the third line of (\ref{c2bdb}). Also, this fact that $\Phi$ is continuous up to the boundary justifies the bound $\|\Phi\|_{L^\infty(\partial\Omega)}\le\|\Phi\|_{L^\infty(\Omega)}$, implicitly used in the second line of (\ref{c2bdb}). The constant $\delta>0$ is chosen below. \end{comment}
Defining \begin{equation} \mathcal{E}=\frac{1}{D_1}\int_\Omega c_1\log c_1-c_1-c_1\log\gamma_1\,dx+\frac{1}{D_2}\int_\Omega c_2\log c_2-c_2\,dx \end{equation} we obtain by adding (\ref{logc1}) to (\ref{logc2}), using (\ref{c2bdb}), and recalling $\rho=z_1c_1+z_2c_2$, \begin{equation} \begin{aligned}
\frac{d}{dt}\mathcal{E}+2\int_\Omega |\nabla\sqrt{c_1}|^2+|\nabla\sqrt{c_2}|^2\,dx+\frac{1}{2\epsilon}\int_\Omega \rho^2\,dx\le G(t)\label{lgt} \end{aligned} \end{equation} where
\begin{equation} G(t)=C(\|\rho\|_{L^1}^2\|c_2\|_{L^1}^7+\|c_1\|_{L^1}+\|c_2\|_{L^1}+\|\rho\|_{L^1}^\frac{1}{4}\|c_2\|_{L^1}^\frac{7}{4})
\end{equation} Because $\|c_2(t)\|_{L^1}$ is constant and $\|c_1(t)\|_{L^1}$ grows at most linearly in time (\ref{B(t)}, we have that $G(t)$ grows at most quadratically in time and in particular is locally integrable. Thus integrating (\ref{lgt}), we obtain
\begin{equation} 2\epsilon\mathcal{E}(T)+\int_0^T\|\rho(s)\|_{L^2}^2\,ds\le 2\epsilon\left(\mathcal{E}(0)+\int_0^TG(t)\,dt\right)=r(T). \end{equation} Lastly, we observe that because $x\log x$ is superlinear, for any $c>0$ there exists $C>0$ depending only on $c$ such that $-C\le x\log x-cx$ for all $x>0$. It follows that $\mathcal{E}(T)$ is bounded below $\mathcal{E}(T)>-C_0$, with $C_0$ independent of $T$. Therefore \begin{equation}
\int_0^T\|\rho(s)\|_{L^2}^2\,ds\le r(T)+2\epsilon C_0=R(T).\label{RT} \end{equation} We note that $R(T)$ increases at most like $T^3$.
\textbf{Step 3. $L^\infty_tL^2_x$ bounds for $c_i$.} Equipped with this time dependent bound, we proceed to control $c_i$ in $L^2$. Multiplying (\ref{np}) for $i=2$ by $\frac{|z_2|}{D_2}c_2$ and integrating by parts, we obtain exactly as in (\ref{ener}), \begin{equation} \begin{aligned}
\frac{|z_2|}{2D_2}\frac{d}{dt}\int_\Omega c_2^2\,dx=&-|z_2|\int_\Omega|\nabla c_2|^2\,dx-\frac{z_2|z_2|}{2\epsilon}\int_\Omega c_2^2\rho\,dx\\
&+z_2|z_2|\frac{\tau}{2}\int_{\partial\Omega}c_2^2\Phi\,dS- \frac{z_2|z_2|}{2}\int_{\partial\Omega}c_2^2\xi\,dS\\
=&-|z_2|\int_\Omega|\nabla c_2|^2\,dx-\frac{z_2|z_2|}{2\epsilon}\int_\Omega c_2^2\rho\,dx+I_1+I_2.\label{AA} \end{aligned} \end{equation} We bound the boundary integrals using the embedding $H^2\hookrightarrow L^\infty$ and Lemma \ref{trace} in the Appendix, \begin{equation} \begin{aligned}
|I_1|\le& C\|\Phi\|_{L^\infty(\Omega)}\|c_2\|_{L^2(\partial\Omega)}^2\\
\le&C(\|\rho\|_{L^2}+1)(\|c_2\|_{L^2}^\frac{1}{2}\|\nabla c_2\|_{L^2}^\frac{1}{2}+\|c_2\|_{L^2})^2\\
\le&\frac{|z_2|}{4}\|\nabla c_2\|_{L^2}^2+C(\|\rho\|_{L^2}^2+1)\|c_2\|_{L^2}^2.\label{BB} \end{aligned} \end{equation} The last line follows from Young's inequalities. Similarly, \begin{equation} \begin{aligned}
|I_2|\le& C\|\xi\|_{L^\infty(\partial\Omega)}\|c_2\|_{L^2(\partial\Omega)}^2\\
\le&C\|\xi\|_{L^\infty(\partial\Omega)}(\|c_2\|_{L^2}^\frac{1}{2}\|\nabla c_2\|_{L^2}^\frac{1}{2}+\|c_2\|_{L^2})^2\\
\le&\frac{|z_2|}{4}\|\nabla c_2\|_{L^2}^2+C\|c_2\|_{L^2}^2.\label{CC} \end{aligned} \end{equation} From (\ref{AA}), using (\ref{BB}) and (\ref{CC}), we obtain \begin{equation}
\frac{|z_2|}{2D_2}\frac{d}{dt}\|c_2\|_{L^2}^2+\frac{|z_2|}{2}\|\nabla c_2\|_{L^2}^2\le C(\|\rho\|_{L^2}^2+1)\|c_2\|_{L^2}^2-\frac{z_2|z_2|}{2\epsilon}\int_\Omega c_2^2\rho\,dx.\label{DD} \end{equation} Next we obtain the corresponding estimate for $c_1$. We observe that $q_1=c_1-\gamma_1$ satisfies \begin{equation} \begin{aligned} \partial_t q_1+u\cdot\nabla q_1=&D_1{\mbox{div}\,}(\nabla q_1+z_1q_1\nabla\Phi)-\frac{D_1z_1\gamma_1}{\epsilon}\rho.\label{q1} \end{aligned}
\end{equation} Using the fact that $q_1$ vanishes on the boundary, we multiply (\ref{q1}) by $\frac{|z_1|}{D_1}q_1$ and integrate by parts \begin{equation} \begin{aligned}
\frac{|z_1|}{2D_1}\frac{d}{dt}\|q_1\|_{L^2}^2+|z_1|\|\nabla q_1\|_{L^2}^2=&-\frac{z_1|z_1|}{2}\int_\Omega \nabla q_1^2\cdot\nabla\Phi\,dx-\frac{z_1|z_1|\gamma_1}{\epsilon}\int_\Omega\rho q_1\,dx\\
\le&-\frac{z_1|z_1|}{2\epsilon}\int_\Omega q_1^2\rho\,dx+C\|\rho\|_{L^2}\|q_1\|_{L^2}.\label{EE} \end{aligned} \end{equation} Now we add (\ref{DD}) to (\ref{EE}) to obtain \begin{equation} \begin{aligned}
&\frac{d}{dt}\left(\frac{|z_1|}{2D_1}\|q_1\|_{L^2}^2+\frac{|z_2|}{2D_2}\|c_2\|_{L^2}^2\right)+|z_1|\|\nabla q_1\|_{L^2}^2+\frac{|z_2|}{2}\|\nabla c_2\|_{L^2}^2\\
\le&-\frac{1}{2\epsilon}\int_\Omega (z_1|z_1|q_1^2+z_2|z_2|c_2^2)\rho\,dx+C(\|\rho\|_{L^2}^2+1)\|c_2\|_{L^2}^2+C\|\rho\|_{L^2}\|q_1\|_{L^2}\\
\le&-\frac{1}{2\epsilon}\int_\Omega (z_1|z_1|q_1^2+z_2|z_2|c_2^2)\rho\,dx+C(\|\rho\|_{L^2}^2+1)(\|q_1\|_{L^2}^2+\|c_2\|_{L^2}^2)+C.\label{W} \end{aligned}
\end{equation} Next, we observe that because $z_1>0>z_2$ and denoting $\sigma=z_1c_1+|z_2|c_2\ge 0$, \begin{equation} \begin{aligned}
(z_1|z_1|q_1^2+z_2|z_2|c_2^2)\rho=(z_1^2q_1^2-z_2^2c_2^2)\rho=&(z_1q_1+|z_2|c_2)(z_1q_1+z_2c_2)\rho\\
=&(z_1c_1-z_1\gamma_1+|z_2|c_2)(z_1c_1-z_1\gamma_1+z_2c_2)\rho\\ =&(\sigma-z_1\gamma_1)(\rho-z_1\gamma_1)\rho\\ =&\rho^2\sigma-2z_1^2\gamma_1c_1\rho+z_1^2\gamma_1^2\rho\\ \ge&-2z_1^2\gamma_1c_1\rho+z_1^2\gamma_1^2\rho.\label{r3} \end{aligned} \end{equation} We note that in the last line, there is no cubic term. So using (\ref{r3}) together with Young's inequalities, we obtain from (\ref{W}), \begin{equation} \begin{aligned}
\frac{d}{dt}\mathcal{F}\le& C(\|\rho\|_{L^2}^2+1)+C(\|\rho\|_{L^2}^2+1)(\|q_1\|_{L^2}^2+\|c_2\|_{L^2}^2)\\
\le& C(\|\rho\|_{L^2}^2+1)+C(\|\rho\|_{L^2}^2+1)\mathcal{F} \end{aligned} \end{equation} for \begin{equation}
\mathcal{F}=\frac{|z_1|}{2D_1}\|q_1\|_{L^2}^2+\frac{|z_2|}{2D_2}\|c_2\|_{L^2}^2. \end{equation} Thus recalling (\ref{RT}), a Grönwall estimate gives us \begin{equation} \sup_{t\in[0,T]}\mathcal{F}(t)\le \left(\mathcal{F}(0)+C(R(T)+T)\right)\exp\left(C(R(T)+T)\right). \end{equation} This completes the proof of the proposition. \end{proof}
As in the case of blocking boundary conditions, the $L^2$ bounds obtained in the previous proposition are sufficient, in the mixed boundary conditions setting, to obtain control of $c_i$ in the space $L^\infty_tH_x^1\cap L_t^2H_x^2$. Since the $L^2$ bound (\ref{M2t}) is time dependent, all higher regularity bounds are also time dependent. In particular, even for NPS, no time independent bounds of the type (\ref{M'}) are available.
\begin{thm} For initial conditions $0\le c_i(0)\in H^1$, $u(0)\in V$ and for all $T>0$, NPS (\ref{np}),(\ref{pois}),(\ref{stokes}) for two oppositely charged species ($m=2,\,z_1>0>z_2$) has a unique strong solution $(c_i,\Phi,u)$ on the time interval $[0,T]$ satisfying the boundary conditions (\ref{di}),(\ref{2bl}),(\ref{noslip}),(\ref{robin}). Under the same hypotheses, NPNS (\ref{np}),(\ref{pois}),(\ref{nse}) for two oppositely charged species has a unique strong solution on $[0,T]$ satisfying the initial and boundary conditions provided
\begin{equation} U(T)=\int_0^T\|u(s)\|_V^4\,ds<\infty. \end{equation} Moreover, the solution to NPS satisfies \begin{align}
\sup_{t\in[0,T]}\|c_i(t)\|_{H^1}^2+\int_0^T\|c_i(s)\|_{H^2}^2\,ds&\le M(T)\label{M(T)}\\
\sup_{t\in[0,T]}\|u(t)\|_V^2+\int_0^T\|u(s)\|_{H^2}^2\,ds&\le B'(T)\label{B'T} \end{align} for time dependent constants $M(T), B'(T)$ depending also on the parameters of the system and the initial conditions. The solution to NPNS satisfies \begin{align}
\sup_{t\in[0,T]}\|c_i(t)\|_{H^1}^2+\int_0^T\|c_i(s)\|_{H^2}^2\,ds&\le M_U(T)\label{MU(T)}\\
\sup_{t\in[0,T]}\|u(t)\|_V^2+\int_0^T\|u(s)\|_{H^2}^2\,ds&\le B'_U(T)\label{B'UT} \end{align} for time dependent constants $M_U(T),B'_U(T)$ depending also on the parameters of the system, the initial conditions, and $U(T)$.\label{gr!!} \end{thm}
\begin{rem} {We show in the proof that $M(T),B'(T),M_U(T),B'_U(T)$ satisfy the following asymptotic bounds \begin{align}
M(T)\lesssim& \exp\exp\exp\exp(CT^3)\\
B'(T)\lesssim& \exp\exp(CT^3)\\
M_U(T)\lesssim&\exp\left(\exp\exp\exp(CT^3)\exp(CU(T))\right)\\
B'_U(T)\lesssim&\exp\exp(CT^3)\exp(CU(T)). \end{align} These bounds follow from using \eqref{M2t} and bootstrapping.} \end{rem}
\begin{proof} As in the proofs of Theorems \ref{gr!} and \ref{grm}, global existence of strong solutions follows from the local existence theorem and the a priori bounds (\ref{M(T)})-(\ref{B'UT}).
\textbf{Step 1. $L^\infty_tL^4_x$ bounds for $c_i$. $L^\infty_t W^{1,\infty}_x$ bounds for $\Phi$. $L^\infty_tV\cap L^2_tH^2_x$ bounds for $u$.} From (\ref{M2t}) and Sobolev estimates, we obtain as in (\ref{P6}), \begin{equation}
\|\nabla\Phi\|_{L^6}\le P_6(T) \label{P6T} \end{equation} for some time dependent constant $P_6(T)$, {which asymptotically satisfies $P_6(T)\lesssim e^{CT^3}$ (c.f. \eqref{M2t}).} Then multiplying (\ref{np}) for $i=2$ by $c_2^3$, and integrating by parts, we find (cf. (\ref{kk})), \begin{equation} \begin{aligned}
\frac{1}{4}\frac{d}{dt}\|c_2\|_{L^4}^4+\frac{3}{4}D_2\|\nabla c_2^2\|_{L^2}^2\le CP_6(T)\|c_2^2\|_{L^3}\|\nabla c_2^2\|_{L^2} \end{aligned} \end{equation} which, after interpolating $L^3$ between $L^2$ and $H^1$ and using a Young's inequality, gives \begin{equation}
\frac{1}{4}\frac{d}{dt}\|c_2\|_{L^4}^4+\frac{1}{4}D_2\|\nabla c_2^2\|_{L^2}^2\le CP_6(T)^4\|c_2\|_{L^4}^4. \end{equation} Therefore, for a time dependent constant $M'_4(T)$ we have \begin{equation}
\|c_2\|_{L^4}\le M'_4(T)\label{M'4T}, \end{equation} {where $M_4'(T)\lesssim \exp\exp(CT^3)$}. Then multiplying (\ref{q1}) by $q_1^3$ and integrating by parts, we obtain \begin{equation}
\frac{1}{4}\frac{d}{dt}\|q_1\|_{L^4}^4+\frac{3}{4}D_1\|\nabla q_1^2\|_{L^2}^2\le CP_6(T)\|q_1^2\|_{L^3}\|\nabla q_1^2\|_{L^2}+C(1+\|q_1\|_{L^4}^4+\|c_2\|_{L^4}^4) \end{equation} which gives after an interpolation, \begin{equation}
\frac{1}{4}\frac{d}{dt}\|q_1\|_{L^4}^4+\frac{1}{4}D_1\|\nabla q_1^2\|_{L^2}^2\le C(P_6(T)^4+1)\|q_1\|_{L^4}^4+C(1+\|c_2\|_{L^4}^4). \end{equation} This, together with (\ref{M'4T}), allows us to conclude that there exists a time dependent constant $M_4(T)$ such that for each $i$, \begin{equation}
\|c_i\|_{L^4}\le M_4(T)\label{M4T} \end{equation} {with $M_4(T)\lesssim \exp\exp(CT^3)$}. Furthermore, since $W^{2,4}\hookrightarrow W^{1,\infty}$, we have for a time dependent constant $P_\infty(T)$, \begin{equation}
\|\Phi\|_{W^{1,\infty}}\le P_\infty(T)\label{PiT} \end{equation} {with $P_\infty(T)\lesssim \exp\exp(CT^3)$}. At this point, we have established enough bounds to mimic the derivations of (\ref{unps}), (\ref{nnn}) to obtain the bounds (\ref{B'T}), (\ref{B'UT}) {for $u$ with $B'(T)\lesssim \exp\exp(CT^3)$ and $B'_U(T)\lesssim \exp\exp(CT^3)\exp (CU(T))$.}
\textbf{Step 2. $L^\infty_tH^1_x\cap L^2_tH^2_x$ bounds for $c_1$.} Now we bound derivatives of $c_1$. First we multiply (\ref{q1}) by $-\Delta q_1$ and integrate by parts to obtain \begin{equation} \begin{aligned}
\frac{1}{2}\frac{d}{dt}\|\nabla q_1\|_{L^2}^2+D_1\|\Delta q_1\|_{L^2}^2\le& C\|u\|_V\|\nabla q_1\|^\frac{1}{2}_{L^2}\|\Delta q_1\|_{L^2}^\frac{3}{2}+CP_\infty(T)\|\nabla q_1\|_{L^2}\|\Delta q_1\|_{L^2}\\
&+C(M_4(T)^2+1)\|\Delta q_1\|_{L^2}+CM_2(T)\|\Delta q_1\|_{L^2} \end{aligned} \end{equation} so that from Young's inequalities, we obtain \begin{equation}
\frac{1}{2}\frac{d}{dt}\|\nabla q_1\|_{L^2}^2+\frac{D_1}{2}\|\Delta q_1\|_{L^2}^2\le C(T)+C(P_\infty(T)^2+\|u\|_V^4)\|\nabla q_1\|_{L^2}^2\label{naq1} \end{equation} for a time dependent constant {$C(T)\lesssim \exp\exp(CT^3)$}. Using a Grönwall estimate, from (\ref{naq1}), we obtain (\ref{MU(T)}) for $c_1$ in the case of NPNS {with $M_U(T)\lesssim \exp\exp\exp(CT^3)\exp (CU(T))$}. In the case of NPS, (\ref{naq1}) together with (\ref{B'T}) gives us (\ref{M(T)}) for $c_1$ {with $M(T)\lesssim \exp\exp\exp(CT^3)$}.
\textbf{Step 3. $L^\infty_tH^1_x\cap L^2_tH^2_x$ bounds for $c_2$.} To obtain the corresponding bounds for $c_2$, we start with (\ref{dci}), which holds verbatim for $i=2$: \begin{equation} \begin{aligned}
\frac{1}{2}\frac{d}{dt}\|\nabla\tilde c_2\|_{L^2}^2+D_2\|\Delta\tilde c_2\|_{L^2}^2=&\int_\Omega (u\cdot\nabla\tilde c_2)\Delta\tilde c_2\,dx+D_2z_2\int_\Omega(\nabla\tilde c_2\cdot\nabla\Phi)\Delta\tilde c_2\,dx\\ &-z_2\int_\Omega((\partial_t+u\cdot\nabla)\Phi)\tilde c_2\Delta\tilde c_2\,dx\\ =& I_1+I_2+I_3. \end{aligned} \end{equation} The term $I_1$ is bounded identically (see (\ref{id})): \begin{equation}
|I_1|\le\frac{D_2}{4}\|\Delta \tilde c_2\|_{L^2}^2+C\|u\|_V^4\|\nabla\tilde c_2\|_{L^2}^2. \end{equation} The term $I_2$ is also bounded identically (see (\ref{idd})), noting that this time we have a time dependent coefficient $C_g(T)$, depending on $P_\infty(T)$ {so that $C_g(T)\lesssim\exp\exp(CT^3)$}: \begin{equation}
|I_2|\le\frac{D_2}{4}\|\Delta\tilde c_2\|_{L^2}^2+C_g(T)\|\nabla\tilde c_2\|_{L^2}^2. \end{equation} As for the term $I_3=I_3^1+I_3^2$ (see (\ref{split})), we note that \begin{equation}
\|\tilde{c}_2\|_{L^3}\le CM_4(T)e^{|z_2|P_\infty(T)}=\beta_3(T){\lesssim\exp\exp\exp(CT^3)} \end{equation} so that \begin{equation} \begin{aligned}
|I_3^1|\le& C\|u\|_V\|\nabla\Phi\|_{L^\infty}\|\tilde{c}_2\|_{L^3}\|\Delta \tilde{c}_2\|_{L^2}\\
\le&\frac{D_2}{8}\|\Delta\tilde{c}_2\|_{L^2}^2+C'_g(T)\|u\|_V^2 \end{aligned}
\end{equation} where {$C'_g(T)\lesssim \exp\exp\exp(CT^3)$.} Lastly, in order to bound $I_3^2$, we first estimate $\partial_t\nabla\Phi$ as in (\ref{ptrho})-(\ref{ptg}), but this time we work around the fact that ${\partial_n \tilde c_1}_{|\partial\Omega}=0$ does not hold: \begin{equation} \begin{aligned}
\tau\|\partial_t\Phi\|&_{L^2(\partial\Omega)}^2+\|\partial_t\nabla\Phi\|_{L^2(\Omega)}^2\\\le& C\sum_{j=1}^2\left|\int_\Omega{\mbox{div}\,}(e^{-z_j\Phi}\nabla{\tilde{c}_j})\partial_t\Phi\,dx\right|+C\left|\int_\Omega{\mbox{div}\,}(u\rho)\partial_t\Phi\,dx\right|\\
\le&C\int_\Omega e^{|z_2|P_\infty(T)}|\nabla \tilde c_2||\partial_t\nabla\Phi|\,dx+C\left|\int_\Omega {\mbox{div}\,} (e^{-z_1\Phi}\nabla\tilde c_1)\partial_t\Phi\,dx\right|\\
&+C\int_\Omega |\rho||u||\partial_t\nabla\Phi|\,dx\\
\le&C\int_\Omega e^{|z_2|P_\infty(T)}|\nabla \tilde c_2||\partial_t\nabla\Phi|\,dx+CP_\infty(T) e^{|z_1|P_\infty(T)}\int_\Omega |\nabla\tilde c_1||\partial_t\Phi|\,dx\\
&+Ce^{|z_1|P_\infty(T)}\int_\Omega |\Delta\tilde c_1||\partial_t\Phi|\,dx+C\int_\Omega |\rho||u||\partial_t\nabla\Phi|\,dx\\
\le&C_{t}(T)(\sum_{j=1}^2\|\nabla{\tilde{c}_j}\|_{L^2}+\|\Delta\tilde c_1\|_{L^2}+\|u\|_V)(\|\partial_t\nabla\Phi\|_{L^2(\Omega)}^2+\|\partial_t\Phi\|_{L^2(\partial\Omega)}^2)^\frac{1}{2} \end{aligned}
\end{equation} where {$C_t(T)\lesssim \exp\exp\exp(CT^3)$}, and we used $\|\partial_t\Phi\|_{L^2}\le C(\|\partial_t\nabla\Phi\|_{L^2(\Omega)}^2+\|\partial_t\Phi\|_{L^2(\partial\Omega)}^2)^\frac{1}{2}$. Then recalling that we already have the bounds (\ref{M(T)}), (\ref{MU(T)}) for $c_1$, we obtain as in (\ref{pt6}), using the embedding $H^1\hookrightarrow L^6$, \begin{equation}
\|\partial_t\Phi\|_{L^6}\le C_{t}(T,U(T))(\|\nabla\tilde{c}_2\|_{L^2}+\|\Delta\tilde c_1\|_{L^2}+\|u\|_V+1)\label{pt6t} \end{equation} for a time dependent constant {$C_t(T,U(T))\lesssim \exp\exp\exp(CT^3)\exp(CU(T))$ (though in the case of NPS, the dependence on $U(T)$ is redundant and may be dropped)}. Thus we bound $I_3^2$ as in (\ref{I32}), \begin{equation} \begin{aligned}
|I_3^2|\le\frac{D_2}{8}\|\Delta\tilde c_2\|_{L^2}^2+C_3(T,U(T))(\|\nabla\tilde{c}_2\|_{L^2}^2+\|\Delta\tilde c_1\|_{L^2}^2+\|u\|_V^2+1)\label{I32t} \end{aligned} \end{equation} {with $C_3(T,U(T))\lesssim \exp\exp\exp(CT^3)\exp(CU(T))$}. Collecting our estimates for $I_1,I_2,I_3$, we obtain as in (\ref{cru}), \begin{equation} \begin{aligned}
&\frac{1}{2}\frac{d}{dt}\|\nabla \tilde{c}_2\|_{L^2}^2+\frac{D_2}{4}\|\Delta\tilde{c}_2\|_{L^2}^2\\
\le& (C_F(T,U(T))+C\|u\|_V^4)\|\nabla \tilde{c}_2\|_{L^2}^2+C'_F(T,U(T))(\|\Delta\tilde c_1\|_{L^2}^2+\|u\|_V^2+1)\label{crut} \end{aligned} \end{equation} {with $C_F(T,U(T)),C_F'(T,U(T))\lesssim \exp\exp\exp(CT^3)\exp(CU(T))$}. It is straightforward to verify using the chain rule that \begin{equation}
\|\Delta\tilde c_1\|_{L^2}\le C_P(T)(\|\Delta c_1\|_{L^2}+\|\nabla c_1\|_{L^2}+\|c_1\|_{L^2}+\|c_1\rho\|_{L^2}) \end{equation} for a time dependent constant $C_P(T)$ depending on $P_\infty(T)$ {so that $C_P(T)\lesssim \exp\exp\exp(CT^3)$}. So, by the previously established bounds (\ref{M(T)}), (\ref{MU(T)}) for $c_1$ (cf. (\ref{naq1})), together with (\ref{M4T}), we have \begin{equation}
\int_0^T\|\Delta\tilde c_1\|_{L^2}^2\,dx={S(T,U(T))\lesssim \exp\exp\exp(CT^3)\exp(CU(T))} .\label{ST} \end{equation} Then from (\ref{crut}), a Grönwall estimate together with (\ref{ST}) gives us (\ref{MU(T)}) for $i=2$ in the case of NPNS, after we convert back to the variable $c_2$, with $M_U(T)\lesssim\exp(\exp\exp\exp(CT^3)\exp(CU(T)))$. In the case of NPS, (\ref{B'T}) together with (\ref{crut}) and (\ref{ST}) gives us (\ref{M(T)}) for $i=2$ {with $M_U(T)\lesssim \exp\exp\exp\exp(CT^3)$}. This completes the proof. \end{proof}
\appendix \section{Trace Inequalities} \begin{lemma}\label{trace} Let $\Omega\subset\mathbb{R}^3$ be an open, bounded domain with Lipschitz boundary. Then for $p\in[2,4]$, we have the embedding $H^1(\Omega)\hookrightarrow L^p(\partial\Omega).$ Moreover, for $p\in [2,4]$, there exist constants $C_p, C'_p$ depending only on $\Omega$ and $p$ such that \begin{equation}
\|f\|_{L^p(\partial\Omega)}\le C_p\|\nabla f\|_{L^2(\Omega)}^\frac{1}{p}\|f\|_{L^{2(p-1)}(\Omega)}^\frac{p-1}{p}+C'_p\|f\|_{L^p(\Omega)}.\label{T1} \end{equation} In particular, for $p=4$, there exists a constant $c_4$ depending only on $\Omega$ such that \begin{equation}
\|f\|_{L^4(\partial\Omega)}\le c_4\|f\|_{H^1(\Omega)}.\label{T2} \end{equation} And, for $p\in[2,4)$ and any $\gamma>0$ there exists $C_\gamma$ depending on $\Omega,p$ and $\gamma$ such that \begin{equation}
\|f\|_{L^p(\partial\Omega)}^2\le \gamma\|\nabla f\|_{L^2(\Omega)}^2+ C_\gamma\|f\|_{L^1(\Omega)}^2.\label{T3} \end{equation}
\begin{proof} The proof is based on the proof of Theorem 1.5.1.10 in \cite{grisvard}. Because $C^1(\bar\Omega)$ is dense in $H^1(\Omega)$, we assume without loss of generality that $f\in C^1(\bar\Omega).$ Next we use the fact (Lemma 1.5.1.9, \cite{grisvard}) that for bounded Lipschitz domains $\Omega\subset\mathbb{R}^3$, there exists $\mu\in (C^\infty(\bar\Omega))^3$ and a constant $\delta>0$ such that $\mu\cdot n\ge \delta$ almost everywhere on $\partial\Omega$, where $n$ is the outward pointing normal vector on $\partial\Omega$. Then, on one hand, we have \begin{equation}
\int_\Omega \nabla |f|^p\cdot\mu\,dx=p\int_\Omega |f|^{p-2}f\nabla f\cdot\mu\,dx. \end{equation} On the other hand, integrating by parts, we have \begin{equation}
\int_\Omega \nabla |f|^p\cdot\mu\,dx=\int_{\partial\Omega} |f|^p\mu\cdot n\,dS-\int_\Omega |f|^p{\mbox{div}\,}\mu\,dx. \end{equation} Therefore, \begin{equation} \begin{aligned}
\delta\int_{\partial\Omega}|f|^p\,dS\le&\int_{\partial\Omega} |f|^p\mu\cdot n\,dS\\
=&p\int_\Omega |f|^{p-2}f\nabla f\cdot\mu\,dx+\int_\Omega |f|^p{\mbox{div}\,}\mu\,dx\\
\le& p\|\mu\|_{L^\infty(\Omega)}\int_\Omega |f|^{p-1}|\nabla f|\,dx+\|{\mbox{div}\,}\mu\|_{L^\infty(\Omega)}\int_\Omega|f|^p\,dx\\
\le& p\|\mu\|_{L^\infty(\Omega)}\|\nabla f\|_{L^2(\Omega)}\|f\|_{L^{2(p-1)}(\Omega)}^{p-1}+\|{\mbox{div}\,}\mu\|_{L^\infty(\Omega)}\|f\|_{L^p(\Omega)}^p\label{tt} \end{aligned} \end{equation} where the last line follows from a Hölder inequality. Then, (\ref{T1}) follows from taking both sides of (\ref{tt}) to the $p^{-1}$ power. Taking $p=4$ in (\ref{T1}) we have \begin{equation}
\|f\|_{L^4(\partial\Omega)}\le C_4\|\nabla f\|_{L^2(\Omega)}^\frac{1}{4}\|f\|_{L^6(\Omega)}^\frac{3}{4}+C'_4\|f\|_{L^4(\Omega)}, \end{equation} so (\ref{T2}) follows from the embedding $H^1(\Omega)\hookrightarrow L^6(\Omega)$. Lastly, (\ref{T3}) follows from (\ref{T1}) by interpolating the spaces $L^{2(p-1)}(\Omega)$ and $L^p(\Omega)$ between $L^1(\Omega)$ and $H^1(\Omega)$, followed by Young's inequalities. \end{proof} \end{lemma}
\section{Positivity of Concentrations}\label{pc} In this section, we verify that positivity of the ionic concentrations, $c_i$, is propagated in time by the Nernst-Planck equations. \begin{prop} Suppose $(c_1,c_2,u)$ is a strong solution to NPNS (\ref{np}), (\ref{pois}), (\ref{nse}) or NPS (\ref{np}), (\ref{pois}), (\ref{stokes}) with blocking boundary conditions (\ref{bl}), (\ref{noslip}), (\ref{robin}) or mixed boundary conditions (\ref{di}), (\ref{2bl}), (\ref{noslip}), (\ref{robin}) with strictly positive initial conditions $0<c\le c_i(0)\in H^1(\Omega)$. Then, $c_i(t)>0$ for all $t\ge 0$. \end{prop} \begin{proof} We split the proof into two steps. First we show that $c_i(t)\ge 0$ for all $t\ge 0$. This part only uses $c_i(0)\ge 0$ and can be shown as in \cite{ci,cil}. We fix a convex function on the real line that is positive on the negative semiaxis and zero on the positive semiaxis. For example, we take the function \begin{equation} F(y)=\begin{cases} y^4,& y<0\\ 0,&y\ge 0. \end{cases} \end{equation} We observe that $F$ satisfies, for all $y\in\mathbb{R}$, \begin{equation} F''(y)y^2\le 12 F(y). \end{equation} Now we multiply (\ref{np}) by $F'(c_i)$ and integrate by parts, noting that due to the choice of $F$, no boundary terms occur for blocking boundary conditions nor for mixed boundary conditions: \begin{equation} \begin{aligned}
\frac{d}{dt}\int_\Omega F(c_i)\,dx=&-D_i\int_\Omega|\nabla c_i|^2 F''(c_i)\,dx- D_iz_i\int_\Omega c_iF''(c_i)\nabla c_i\cdot\nabla\Phi\,dx\\
\le&-\frac{D_i}{2}\int_\Omega |\nabla c_i|^2F''(c_i)\,dx+\frac{D_iz_i^2}{2}\int_\Omega F''(c_i)c_i^2|\nabla\Phi|^2\,dx\\
\le&6D_i\|\nabla\Phi\|_{L^\infty}^2\int_\Omega F(c_i)\,dx.\label{FF} \end{aligned} \end{equation} We note that the advective term involving $u$ vanishes due to ${\mbox{div}\,} u=0$. It follows from (\ref{FF}) that \begin{equation}
\int_\Omega F(c_i(t))\,dx\le \left(\int_\Omega F(c_i(0))\,dx\right)\exp\left(6D_i\int_0^t\|\nabla\Phi(s)\|_{L^\infty}^2\,ds\right). \end{equation} For strong solutions, the time integral \begin{equation}
\int_0^t\|\nabla\Phi(s)\|_{L^\infty}^2\,ds \end{equation} is finite, and thus since $F(c_i(0))=0$ on $\Omega$, it follows that \begin{equation} \int_\Omega F(c_i(t))=0 \end{equation} which implies $c_i(t)\ge 0$. This proves the nonnegativity of $c_i$.
Improving this result to strict positivity requires an additional argument. We adapt the argument given in \cite{gaj}. We first fix a time interval $[0,T]$, and we assume that $c_i$ satisfy blocking boundary conditions. We then fix $\delta>0$ and multiply (\ref{np}) by $-\frac{1}{(c_i+\delta)^2}$ and integrate by parts. Then on the left hand side, we obtain \begin{equation} \frac{d}{dt}\int_\Omega \frac{1}{c_i+\delta}\,dx. \end{equation} On the right hand side, we have \begin{equation}
-2D_i\int_\Omega \frac{|\nabla c_i|^2}{(c_i+\delta)^3}\,dx+2D_iz_i\int_\Omega c_i\nabla\Phi\cdot\frac{\nabla c_i}{(c_i+\delta)^3}\,dx=I_1+I_2. \end{equation} The integral $I_1$ is nonpositive and the second integral $I_2$ can be estimated as follows, using Young's inequality \begin{equation}
|I_2|\le D_i\int_\Omega \frac{|\nabla c_i|^2}{(c_i+\delta)^3}\,dx+C_T\int_\Omega \frac{1}{c_i+\delta}\,dx
\end{equation} where $C_T$ depends on parameters and on $\sup_{t\in [0,T]}\|\nabla\Phi(t)\|_{L^\infty}$ but not on $\delta$. Thus we have that $L_1:=\int_\Omega \frac{1}{c_i+\delta}\,dx$ satisfies $\frac{d}{dt}L_1\le C_TL_1$, and thus \begin{equation}
\sup_{t\in[0,T]}L_1(t)\le e^{C_T T}\int_\Omega \frac{1}{c_i(0)+\delta}\,dx\le e^{C_T T}\frac{|\Omega|}{c}\label{l1t} \end{equation} where we note that the final upper bound does not depend on $\delta$.
Now the idea is to bootstrap to obtain bounds on $L_{2^k}:=\int_\Omega \frac{1}{(c_i+\delta)^{2^k}}$ for $k=1,2,3,...$ exactly as was done to control $\|c_i\|_{L^\infty}$ in Proposition \ref{L2}.
To this end, we multiply (\ref{np}) by $\frac{-j+1}{(c_i+\delta)^{j}}$ ($j=3,4,...$) and integrate by parts. This yields \begin{equation} \begin{aligned}
\frac{d}{dt}\left\|\frac{1}{(c_i+\delta)^\frac{j-1}{2}}\right\|_{L^2}^2+4D_i\left\|\nabla\frac{1}{(c_i+\delta)^\frac{j-1}{2}}\right\|_{L^2}^2\le& D_i|z_i|(j-1)\left|\int_\Omega c_i\nabla\Phi\cdot\nabla (c_i+\delta)^{-j}\,dx\right|\\
\le& 2D_i|z_i|j\|\nabla\Phi\|_{L^\infty}\int_\Omega|\nabla (c_i+\delta)^\frac{-j+1}{2}|(c_i+\delta)^\frac{-j+1}{2}\,dx\\
\le& 2D_i\left\|\nabla\frac{1}{(c_i+\delta)^\frac{j-1}{2}}\right\|_{L^2}^2\\
&+2D_i|z_i|^2j^2\|\nabla\Phi\|_{L^\infty}^2\left\|\frac{1}{(c_i+\delta)^\frac{j-1}{2}}\right\|_{L^2}^2\label{agag} \end{aligned} \end{equation} Then, using the interpolation estimate \begin{equation}
\left\|\frac{1}{(c_i+\delta)^\frac{j-1}{2}}\right\|_{L^2}\le C\left(\left\|\nabla\frac{1}{(c_i+\delta)^\frac{j-1}{2}}\right\|_{L^2}^\frac{3}{5}\left\|\frac{1}{(c_i+\delta)^\frac{j-1}{2}}\right\|_{L^1}^\frac{2}{5}+\left\|\frac{1}{(c_i+\delta)^\frac{j-1}{2}}\right\|_{L^1}\right)\label{int1} \end{equation} followed by a Young's inequality, we obtain from (\ref{agag}) \begin{equation}
\frac{d}{dt}\left\|\frac{1}{(c_i+\delta)^\frac{j-1}{2}}\right\|_{L^2}^2+D_i\left\|\nabla\frac{1}{(c_i+\delta)^\frac{j-1}{2}}\right\|_{L^2}^2\le \bar{c}j^l\left\|\frac{1}{(c_i+\delta)^\frac{j-1}{2}}\right\|_{L^1}^2
\end{equation} for some $l>0$ large enough and $\bar c$ depending on parameters, the domain, $\sup_{t\in[0,T]}\|\nabla\Phi(t)\|_{L^\infty}$, but not on $j$. Then, again using the interpolation estimate (\ref{int1}) followed by a Young's inequality, we obtain \begin{equation}
\frac{d}{dt}\left\|\frac{1}{(c_i+\delta)^\frac{j-1}{2}}\right\|_{L^2}^2\le -C\left\|\frac{1}{(c_i+\delta)^\frac{j-1}{2}}\right\|_{L^2}^2+ C_j\left\|\frac{1}{(c_i+\delta)^\frac{j-1}{2}}\right\|_{L^1}^2\label{jj} \end{equation} where $C_j\le \tilde c j^l$ for some $\tilde c$ not depending on $j$. Now we set $j=2k+1$ where $k$ is a nonnegative integer, and we rewrite (\ref{jj}) as \begin{equation}
\frac{d}{dt}\left\|\frac{1}{c_i+\delta}\right\|_{L^{2k}}^{2k}\le -C \left\|\frac{1}{c_i+\delta}\right\|_{L^{2k}}^{2k} +C_{2k+1}\left\|\frac{1}{c_i+\delta}\right\|_{L^{k}}^{2k}.\label{2kk} \end{equation} Then, applying Grönwall's inequality to (\ref{2kk}), we obtain as in (\ref{2k'})-(\ref{ss}), for some $\bar C>0$ independent of $k$,
\begin{equation} T_{2k}\le {\bar C}^\frac{1}{2k}k^\frac{l}{2k}T_k \end{equation} where
\begin{equation} T_k=\max\left\{\left\|\frac{1}{c_i(0)+\delta}\right\|_{L^\infty},\sup_{t\in [0,T]}\left\|\frac{1}{c_i(t)+\delta}\right\|_{L^k}\right\}. \end{equation} Now setting $k=2^\kappa$ we have \begin{equation} T_{2^{\kappa+1}}\le {\bar C}^\frac{1}{2^{\kappa+1}}2^\frac{\kappa l}{2^{\kappa+1}}T_{2^\kappa} \end{equation} from which it follows that for all $J\in\mathbb{N}$ \begin{equation} T_{2^J}\le {\bar C}^a2^bT_1\label{t2j} \end{equation} where \begin{equation} a=\sum_{\kappa=0}^\infty \frac{1}{2^{\kappa+1}}<\infty,\quad b=\sum_{\kappa=0}^\infty\frac{\kappa l}{2^{\kappa+1}}<\infty. \end{equation} So passing $J\to\infty$ in (\ref{t2j}) we find that \begin{equation} \begin{aligned}
\sup_{t\in[0,T]}\left\|\frac{1}{c_i(t)+\delta}\right\|_{L^\infty}\le& {\bar C}^a 2^b\max\left\{\left\|\frac{1}{c_i(0)+\delta}\right\|_{L^\infty},\sup_{t\in [0,T]}\left\|\frac{1}{c_i(t)+\delta}\right\|_{L^1}\right\}\\
\le&{\bar C}^a2^b\max\left\{\left\|\frac{1}{c_i(0)+\delta}\right\|_{L^\infty}, e^{C_T T}\frac{|\Omega|}{c}\right\} \end{aligned} \end{equation} where the second inequality follows from (\ref{l1t}. Finally, passing to the limit $\delta\to 0^+$ gives \begin{equation}
\sup_{t\in[0,T]}\left\|\frac{1}{c_i(t)}\right\|_{L^\infty}\le {\bar C}^a2^b\max\left\{\frac{1}{c}, e^{C_T T}\frac{|\Omega|}{c}\right\}, \end{equation} and thus on any finite time interval $c_i(t)$ is uniformly bounded away from $0$ on $\bar\Omega$. This completes the proof of strict positivity of $c_i, i=1,2$ in the case of blocking boundary conditions.
In the case of mixed boundary conditions (\ref{di}), (\ref{2bl}), the strict positivity of $c_2$ is obtained as in the blocking case. For $c_1$, it is possible to obtain strict positivity along same lines as the preceding proof for blocking boundary conditions by choosing appropriate test functions, and in fact, this method generalizes easily to more complex boundary conditions for $c_i$ (e.g. blocking boundary conditions on a nontrivial boundary portion, Dirichlet boundary conditions on the complement, see \cite{ci, np3d}) and also to cases of more than two ionic species. Here, since $c_1$ satisfies purely Dirichlet boundary conditions and because we are considering the case of two oppositely charged ionic species, we argue using a maximum principle argument (see \cite{cil} for a similar argument in the context of upper bounds). As per the remark at the start of Section \ref{mbc}, we assume that $c_i,u,\Phi$ are all smooth in both space and time so that the following considerations are justified.
We know that on the interval $[0,T]$, there exists $c_T>0$ such that \begin{equation} c_T\le c_2(t).\end{equation} Now fix $0<c'<\min\{c, c_T,\gamma_1\}$. Initially, we have, by definition of $c$, that $c_1(0)> c'$ on $\bar\Omega$. Suppose $c_1$ attains the value $c'$ at some time in between $t=0$ and $t=T$. Suppose $t'$ is the \textit{first} time when $c'$ is attained by $c_1$. Then, using (\ref{pois}), we write (\ref{np}) for $i=1$ as \begin{equation} \partial_t c_1+u\cdot\nabla c_1=D_1\Delta c_1+D_1\nabla c_1\cdot\nabla \Phi-\frac{D_1}{\epsilon}c_1(c_1-c_2). \end{equation} Evaluating the above at time $t'$, at the point $x_0\in\Omega$ where the minimal value $c'$ is attained, we find that \begin{equation} \partial_t c_1(t',x_0)\ge -\frac{D_1}{\epsilon}c'(c'-c_T)>0. \end{equation} However, by our choice of $t'$ and $x_0$, we must have $\partial_t c_1(t',x_0)\le 0$, so we have a contradiction. Thus, \begin{equation}\inf_{[0,T]\times\bar\Omega}c_1\ge \min\{c,c_T,\gamma_1\}>0.\end{equation} Since $[0,T]$ is an arbitrary finite time interval, we have shown that $c_1$ is also uniformly bounded away from 0 on every finite time interval. This completes the proof of the strict positivity of $c_i, i=1,2$ in the case of mixed boundary conditions. \end{proof}
{\bf{Acknowledgment.}} The author thanks the anonymous referees for their constructive comments.
\end{document} |
\begin{document}
\def \la{\lambda}
\title{On oriented graphs with minimal skew energy}
\author{Shicai Gong$^{a}$\thanks{Corresponding author. E-mail addresses: scgong@zafu.edu.cn(S. Gong); lxl@nankai.edu.cn(X. Li); ghxu@zafu.edu.cn(G. Xu).} \thanks{ Supported by Zhejiang Provincial Natural Science Foundation of China(No. Y12A010049).}, Xueliang Li$^{b}$\thanks{ Supported by National Natural Science Foundation of China(No. 10831001).} $~$ and Guanghui Xu$^{a}$\thanks{ Supported by National Natural Science Foundation of China(No. 11171373).}\\ \\{\small \it a. Zhejiang A $\&$ F University, Hangzhou, 311300, P. R. China} \\{\small \it b.Center for Combinatorics and LPMC-TJKLC, Nankai University,}\\ {\small \it Tianjin 300071, P. R. China}
} \date{} \maketitle
\begin{abstract} Let $S(G^\sigma)$ be the skew-adjacency matrix of an oriented graph $G^\sigma$. The skew energy of $G^\sigma$ is defined as the sum of all singular values of its skew-adjacency matrix $S(G^\sigma)$. In this paper, we first deduce an integral formula for the skew energy of an oriented graph. Then we determine all oriented graphs with minimal skew energy among all connected oriented graphs on $n$ vertices with $m \ (n\le m < 2(n-2))$ arcs, which is an analogy to the conjecture for the energy of undirected graphs proposed by Caporossi {\it et al.} [G. Caporossi, D. Cvetkovi$\acute{c}$, I. Gutman, P. Hansen, Variable neighborhood search for extremal graphs. 2. Finding graphs with external energy, J. Chem. Inf. Comput. Sci. 39 (1999) 984-996.].
\vskip 0.3cm
\noindent {\bf Keywords}: oriented graph; graph energy; skew energy; skew-adjacency matrix; skew characteristic polynomial.
\noindent {\bf AMS subject classification 2010}: 05C50, 15A18 \end{abstract}
\section{Introduction}
Let $G^\sigma$ be a digraph that arises from a simple undirected graph $G$ with an orientation $\sigma$, which assigns to each edge of $G$ a direction so that $G^\sigma$ becomes an \emph{oriented graph}, or a \emph{directed graph}. Then $G$ is called the \emph{underlying graph} of $G^\sigma$. Let $G^\sigma$ be an undirected graph with vertex set $V(G^\sigma)=\{v_1,v_2,\cdots,v_n\}.$ Denote by $(u, v)$ an arc, of $G^\sigma$, with tail $u$ and head $v$. The {\it skew-adjacency matrix} related to $G^\sigma$ is the $n \times n$ matrix $S(G^\sigma) = [s_{ij} ],$ where the $(i,j)$ entry satisfies: $$s_{ij}=\left \{ \begin{array}{ll} 1, & {\rm if \mbox{ } (v_i, v_j)\in G^\sigma \mbox{ }};\\ -1, & {\rm if \mbox{ } (v_j, v_i)\in G^\sigma \mbox{ } };\\ 0, & {\rm otherwise. \mbox{ }} \end{array}\right.$$ The {\it skew energy} of an oriented graph $G^\sigma$, introduced by Adiga, Balakrishnan and So in \cite{ad} and denoted by $\mathcal{E}_S(G^\sigma)$, is defined as the sum of all singular values of $S(G^\sigma)$. Because the skew-adjacency matrix $S(G^\sigma)$ is skew-symmetric, the eigenvalues $\{\la_1, \la_2,
\cdots, \la_n\}$ of $S(G^\sigma)$ are all purely imaginary numbers. Consequently, the skew energy $\mathcal{E}_S(G^\sigma)$ is the sum of the absolute values of its eigenvalues, {\it i.e.,}$$\mathcal{E}_S(G^\sigma)=\sum_{i=1}^n|\la_i|,$$which has the same expression as that of the energy of an undirected graph with respect to its adjacent matrix; see e.g. \cite{iv}.
The work on the energy of a graph can be traced back to 1970's \cite{g1} when Gutman investigated the energy with respect to the adjacency matrix of an undirected graph, which has a still older chemical origin; see e.g. \cite{cou}. Then much attention has been devoted to
the energy of the adjacency matrix of a graph; see e.g.
\cite{aa,ago,bs,ds,gkm,gkmz,iv1,yp3,ls,lz1,zz}, and the references cited therein. For undirected graphs, Caporossi, Cvetkovi$\acute{c}$, Gutman and Hansen \cite{cc} proposed a conjecture for the minimum energy as follows.
{\bf Conjecture 1.} Let $G$ be the graph with minimum energy among all connected graphs with $n\ge 6$ vertices and $m \ (n\le m \le 2(n-2))$ edges. Then $G$ is $O_{n,m}$ if $m\le n+\lfloor \frac{n-7}{2}\rfloor$; and $B_{n,m}$ otherwise, where $O_{n,m}$ and $B_{n,m}$ are respectively the underlying graphs of the oriented graphs $O^+_{n,m}$ and $B^+_{n,m}$ given in Fig. 1.1.
This conjecture was proved to be true for $m=n-1, 2(n-2)$ by Caporossi et al. (\cite{cc}, Theorem 1), and $m=n$ by Hou \cite{yp1}. In \cite{lz1}, Li, Zhang and Wang confirmed this conjecture for bipartite graphs. Conjecture 1 has not yet been solved completely.
Recently, in analogy to the energy of the adjacency matrix, a few other versions of graph energy were introduced in the mathematical literature, such as Laplacian energy \cite{gz}, signless Laplacian energy \cite{gr} and skew energy \cite{ad}.
In \cite{ad}, Adiga {\it et.al.} obtained the skew energies of directed cycles under different orientations and showed that the skew energy of a directed tree is independent of its orientation, which is equal to the energy of its underlying tree. Naturally, the following question is interesting:
\noindent {\bf Question:} Denote by $M$ a class of oriented graphs. Which oriented graphs have the extremely skew energy among all oriented graphs of $M$ ?
Hou et al. \cite{hou1} determined the oriented unicyclic graphs with the maximal and minimal skew energies. Zhu \cite{Zhu} determined the oriented unicyclic graphs with the first $\lfloor \frac {n-9} 2\rfloor$ largest skew energies. Shen el al. \cite{Shen} determined the bicyclic graphs with the maximal and minimal energies. Gong and Xu \cite{GX} determined the 3-regular graphs with the optimum skew energy, and Tian \cite{Tian} determined the hypercubes with the optimum skew energy. In the following we will study the minimal skew energy graphs of order $n$ and size $m$.
At first, we need some notations. Denote by $K_n$, $S_n$ and $C_n$ the complete undirected graph, the undirected star and the undirected cycle on $n$ vertices, respectively. Let $ O^+_{n,m}$ be the oriented graph on $n$ vertices which is obtained from the oriented star $S^\sigma_n$
by adding $m-n+1$ arcs such that all those arcs have a common vertex; see Fig. 1.1, where $v_1$ is the tail of each arc incident to it and $v_2$ is the head of each arc incident to it, and $ B^+_{n,m}$, the oriented graph obtained from $ O^+_{n,m+1}$ by deleting the arc $(v_1,v_2)$. Denote by $ O_{n,m}$ and $ B_{n,m}$ the underlying graphs of $ O^+_{n,m}$ and $ B^+_{n,m}$, respectively. Notice that both $ O^+_{n,m}$ and $ B^+_{n,m}$ contain $n$ vertices and $m$ arcs.
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\put(90,14){\circle*{2}} \put(90,0){\circle*{2}} \put(90,6){$\vdots$}
\put(100,10){\circle*{2}} \put(98,5){$v_1$}
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\put(110,.5){\vector(1,1){7}}\put(119,9.5){\line(-1,-1){9}}
\put(16,-26) {Fig. 1.1. Two oriented graphs $O^+_{n,m}$ and $B^+_{n,m}$. }
\end{picture} \end{center}
In this paper, we first deduce an integral formula for the skew energy of an oriented graph. Then we study the question above and determine all oriented graphs with minimal skew energy among all connected oriented digraphs on $n$ vertices with $m \ (n\le m < 2(n-2))$ arcs. Interestingly, our result is an analogy to Conjecture 1. \begin{theorem} \label{01} Let $G^\sigma$ be an oriented graph with minimal skew energy among all oriented graphs with $n$ vertices and $m \ (n\le m < 2(n-2))$ arcs. Then, up to isomorphism, $G^\sigma$ is\\ $(1)$ $O^+_{n,m}$ if $m<\frac{3n-5}{2}$; \\ $(2)$ either $B^+_{n,m}$ or $O^+_{n,m}$ if $m=\frac{3n-5}{2}$; and \\ $(3)$ $B^+_{n,m}$ otherwise. \end{theorem}
\vskip 0.5cm
\section{Integral formula for the skew energy}
In this section, based on the formula established by Adiga {\it et al.} \cite{ad}, we deduce an integral formula for the skew energy of an oriented graph, which is an analogy to the Coulson integral formula for the energy of an undirected graph. Firstly, we introduce some notations and preliminary results.
An even cycle $C$ in an oriented graph $G^\sigma$ is called \emph{oddly oriented} if for either choice of direction of traversal around $C$, the number of edges of $C$ directed in the direction of the traversal is odd. Since $C$ is even, this is clearly independent of the initial choice of direction of traversal. Otherwise, such an even cycle $C$ is called as \emph{evenly oriented}. (Here we do not involve the parity of the cycle with length odd. The reason is that it depends on the initial choice of direction of traversal.)
A ``\emph{basic oriented graph}" is an oriented graph whose components are even cycles and/or complete oriented graphs with exactly two vertices.
Denote by $\phi(G^\sigma;x)$ the \emph{ skew characteristic polynomial} of an oriented graph $G^\sigma$, which is defined as \begin{eqnarray*}
\phi(G^\sigma;x)=det(xI_n-S(G^\sigma))=\sum_{i=0}^na_{i}(G^\sigma)x^{n-i},
\end{eqnarray*} where $I_n$ denotes the identity matrix of order $n$. The following result is a cornerstone of our discussion below, which determines all coefficients of the skew characteristic polynomial of an oriented graph in terms of its basic oriented subgraphs; see \cite[Theorem 2.4]{hou} for an independent version.
\begin{lemma} \em{\cite[Corollary 2.3]{gx}} \label{0001} Let $G^\sigma$ be an oriented graph on $n$ vertices, and let the skew characteristic polynomial of $G^\sigma$ be $$\phi(G^\sigma,\la)=\sum_{i=0}^n(-1)^ia_i \la^{n-i}=\la^n-a_1\la^{n-1}+a_2\la^{n-2}+\cdots+(-1)^{n-1}a_{n-1}\la+(-1)^na_n.$$ Then $a_i=0$ if $i$ is odd; and $$a_i=\sum_{\mathscr{H}}(-1)^{c^+}2^c {\rm \mbox{ } \mbox{ } if \mbox{ }} i {\rm \mbox{ } is \mbox{ } even},$$ where the summation is over all basic oriented subgraphs $\mathscr{H}$ of $G^\sigma$ having $i$ vertices and $c^{+}$ and $c$ are respectively the number of evenly oriented even cycles and even cycles contained in $\mathscr{H}$. \end{lemma}
Let $G=(V(G),E(G))$ be a graph, directed or not, on $n$ vertices. Then denote by $\Delta(G)$ the maximum degree of $G$ and set $\Delta(G^\sigma)=\Delta(G)$. An \emph{$r$-matching} in a graph $G$ is a subset of $r$ edges such that every vertex of $V(G)$ is incident with at most one edge in it. Denote by $M(G,r)$ the number of all $r$-matchings in $G$ and set $M(G,0)=1$.
Denote by $q(G)$ the number of quadrangles in a undirected graph $G$. Then as a consequence of Lemma \ref{0001}, we have
\begin{theorem} \label{013} Let $G^\sigma$ be an oriented graph containing $n$ vertices and $m$ arcs. Suppose $$\phi(G^\sigma,\la)=\sum_{i=0}^{n}(-1)^ia_{i}(G^\sigma) \la^{n-i}.$$ Then $a_0(G^\sigma)=1$, $a_2(G^\sigma)=m$ and $a_4(G^\sigma)\ge M(G,2)-2q(G)$ with equality if and only if all oriented quadrangles of $G^\sigma$ are evenly oriented. \end{theorem}
\noindent {\bf Proof.} The result follows from Lemma \ref{0001} and the fact that each arc corresponds a basic oriented graph having $2$ vertices, and each basic oriented graph having $4$ vertices is either a $2$-matching or a quadrangle.
$\blacksquare$
Furthermore, as well-known, the eigenvalues of an arbitrary real skew symmetric matrix are all purely imaginary numbers and
occur in conjugate pairs. Henceforth, Lemma \ref{0001} can be
strengthened as follows, which will provide much convenience for our discussion below.
\begin{lemma} \label{2} Let $G^\sigma$ be an oriented graph of order $n$. Then each coefficient of the skew characteristic polynomial $$\phi(G^\sigma,\la)=\sum_{i=0}^{\lfloor \frac{n}{2} \rfloor}a_{2i}(G^\sigma) \la^{n-2i}$$ satisfies $a_{2i}(G^\sigma)\ge 0$ for each $i(0\le i\le \lfloor \frac{n}{2} \rfloor)$. \end{lemma} {\bf Proof.} Let $\la_1,\la_2,\cdots,\la_n $ be all eigenvalues of the skew adjacency matrix $S(G^\sigma)$ of $G^\sigma$. Because $\la_1,\la_2,\cdots,\la_n $ are all purely imaginary numbers and must occur in conjugate pairs, we suppose, without loss of generality, that there exists an integer number $m(\le \lfloor \frac{n}{2} \rfloor)$ such that $$\la_t=-\la_{n-t+1}=p_t i,\ \ for \ \ t=1,2,\cdots,m,$$ and all other eigenvalues are zero, where each $p_t$ is a positive real number and $i$ satisfies $i^2=-1.$ Then we have \begin{equation*} \begin{array}{lll}
\phi(G^\sigma,\la)
&=&\prod_{t=1}^n(\la-\la_t)\\&=&\la^{n-2m}\prod_{t=1}^m(\la^2+p^2_t), \end{array} \end{equation*} which implies that the result follows.
$\blacksquare$
For an oriented graph $G^\sigma$ on $n$ vertices, an integral formula for the skew energy in terms of the skew characteristic polynomial $\phi(G^\sigma, \la)$ and its derivative is given by \cite{ad} $$ \mathscr{E}_s(G^\sigma) =\frac{1}{\pi}\int_{-\infty}^{+\infty}\left[n+\la \frac{\phi'(G^\sigma, -\la)}{\phi(G^\sigma, -\la)}\right]d\la.\eqno{(2.1)}$$ However, using the above integral, it is by no means easy to calculate the skew energy of an oriented graph. Hence, it is rather important to establish some other more simpler formula.
Applying to (2.1) the fact that the coefficient $a_i=0$ for each odd $i$ from Lemma \ref{0001} and replacing $\la$ by $-\la$, we have $$ \mathscr{E}_s(G^\sigma) =\frac{1}{\pi}\int_{-\infty}^{+\infty}\left[n-\la \frac{\phi'(G^\sigma, \la)}{\phi(G^\sigma, \la)}\right]d\la.$$ Meanwhile, note that $$\frac{\phi'(G^\sigma, \la)}{\phi(G^\sigma, \la)}d\la=d\ln \phi(G^\sigma, \la).$$Then we have \begin{equation*} \begin{array}{lll}
\mathscr{E}_s(G^\sigma) &=&\frac{1}{\pi}\int_{-\infty}^{+\infty}\left[n-\la \frac{\phi'(G^\sigma, \la)}{\phi(G^\sigma, \la)}\right]d\la \\
&=&\frac{1}{\pi}\int_{-\infty}^{+\infty}\left[n-\la (\frac{d}{d\la}) \ln\phi(G^\sigma, \la)\right]d\la. \end{array}\eqno{(2.2)} \end{equation*} Therefore, we have \begin{theorem} \label{04} Let $G^\sigma$ be an oriented graph with order $n$. Then $$ \mathscr{E}_s(G^\sigma)=\frac{1}{\pi}\int_{-\infty}^{+\infty}\la^{-2}\ln \psi(G^\sigma,\la)d\la,\eqno{(2.3)}$$where $$\psi(G^\sigma,\la)=\sum_{i=0}^{\lfloor \frac{n}{2} \rfloor}a_{2i}(G^\sigma) \la^{2i}$$ and $a_{2i}(G^\sigma)$ denotes the coefficient of $\la^{n-2i}$ in the skew characteristic polynomial $\phi(G^\sigma,\la)$. \end{theorem} {\bf Proof.} Let both $G^{\sigma_1}_1$ and $G^{\sigma_2}_2$ be oriented graphs with order $n$. ($G_1$ perhaps equals $G_2$.) Then applying (2.2) we have \begin{eqnarray*}
\mathscr{E}_s(G^{\sigma_1}_1)-\mathscr{E}_s(G^{\sigma_2}_2) =-\frac{1}{\pi}\int_{-\infty}^{+\infty}\la (\frac{d}{d\la}) \ln\left[\frac{\phi(G^{\sigma_1}_1, \la)}{\phi(G^{\sigma_2}_2, \la)}\right]d\la. \end{eqnarray*} Using partial integration, we have $$\mathscr{E}_s(G^{\sigma_1}_1)-\mathscr{E}_s(G^{\sigma_2}_2) =-\frac{\la}{\pi}\ln[ \frac{\phi(G^{\sigma_1}_1, \la)}{\phi(G^{\sigma_2}_2,
\la)}]|_{-\infty}^{+\infty}+\frac{1}{\pi}\int_{-\infty}^{+\infty}\ln\left[ \frac{\phi(G^{\sigma_1}_1, \la)}{\phi(G^{\sigma_2}_2, \la)}\right]d\la. $$ Notice that $$\frac{\la}{\pi}\ln\left[ \frac{\phi(G^{\sigma_1}_1, \la)}{\phi(G^{\sigma_2}_2, \la)}\right]\mid_{-\infty}^{+\infty}=0.$$ Hence $$ \mathscr{E}_s(G^{\sigma_1}_1)-\mathscr{E}_s(G^{\sigma_2}_2)=\frac{1}{\pi}\int_{-\infty}^{+\infty}\ln\left[ \frac{\phi(G^{\sigma_1}_1, \la)}{\phi(G^{\sigma_2}_2, \la)}\right]d\la.$$ Suppose now that $G^{\sigma_2}_2$ is the null oriented graph, an oriented graph containing $n$ isolated vertices. Then $\phi(G^{\sigma_2}_2, \la)=\la^n$ and thus $\mathscr{E}_s(G^{\sigma_2}_2)=0$. After an appropriate change of variables we can derive $$\mathscr{E}_s(G^{\sigma_1}_1)=\frac{1}{\pi}\int_{-\infty}^{+\infty}\la^{-2}\ln \psi(G^{\sigma_1}_1,\la)d\la.$$ Then the result follows.
$\blacksquare$
\section{Proof of Theorem \ref{01}}
From Theorem \ref{04}, for an oriented graph $G^\sigma$ on $n$ vertices, the skew energy $\mathcal{E}_s(G^\sigma)$ is a strictly monotonically increasing function of the coefficients $a_{2k}(G^\sigma) (k = 0, 1, \cdots, \lfloor\frac{n}{2}\rfloor)$, since for each $i$ the coefficient of $\la^{n-i}$ in the characteristic polynomial $\phi(G^\sigma,\la)$, as well as $\psi(G^\sigma,\la)$, satisfies $a_i(G^\sigma)\ge 0$ by Lemma \ref{2}. Thus, similar to comparing two undirected graphs with respect to their energies, we define the quasi-ordering relation $``\preceq"$ of two oriented graphs with respect to their skew energies as follows.
Let $G^{\sigma_1}_1$ and $G^{\sigma_2}_2$ be two oriented graphs of order $n$. ($G_1$ is not necessarily different from $G_2$.) If $a_{2i}(G^{\sigma_1}_1)\le a_{2i}(G^{\sigma_2}_2)$ for all $i$ with $0\le i \le \lfloor\frac{n}{2}\rfloor$, then we write that $G^{\sigma_1}_1 \preceq G^{\sigma_2}_2$.
Furthermore, if $G^{\sigma_1}_1 \preceq G^{\sigma_2}_2$ and there exists at least one index $i$ such that $a_{2i}(G^{\sigma_1}_1)< a_{2i}(G^{\sigma_2}_2)$, then we write that $G^{\sigma_1}_1 \prec G^{\sigma_2}_2$. If $a_{2i}(G^{\sigma_1}_1)= a_{2i}(G^{\sigma_2}_2)$ for all $i$, we write $G^{\sigma_1}_1 \sim G^{\sigma_2}_2$. Note that there are non-isomorphic oriented graphs $G^{\sigma_1}_1$ and $G^{\sigma_2}_2$ such that $G^{\sigma_1}_1 \sim G^{\sigma_2}_2$, which implies that $``\preceq"$ is not a partial ordering in general.
According to the integral formula (2.2), we have, for two oriented graphs $D_1$ and $D_2$ of order $n$, that $$D_1 \preceq D_2\Longrightarrow \mathcal{E}_s(D_1)\le \mathcal{E}_s(D_2)$$ and $$D_1 \prec D_2\Longrightarrow \mathcal{E}_s(D_1)< \mathcal{E}_s(D_2).\eqno{(3.1)}$$
In the following, by discussing the relation ``$\succeq$", we compare the skew energies for two oriented graphs and then complete the proof of Theorem \ref{01}.
Firstly, by a directly calculation we have $$\phi(O^+_{n,m})=\la^n+m\la^{n-2}+(m-n+1)(2n-m-3)\la^{n-4},\eqno{(3.2)}$$and $$\phi(B^+_{n,m})=\la^n+m\la^{n-2}+(m-n+2)(2n-m-4)\la^{n-4}.\eqno{(3.3)}$$
Denote by $G^\sigma(n,m)$ and $G(n,m)$ the sets of all connected oriented graphs and undirected graphs with $n$ vertices and $m$ edges, respectively. The following results on undirected graphs are needed. \begin{lemma} \label{05} Let $n\ge 5$ and $G\in G(n,m)$ be an arbitrary connected undirected graph containing $n$ vertices and $m \ (n\le m <2(n-2))$ edges. Then $q(G)\le \left( \begin{array}{c} m-n+2 \\
2 \end{array}\right) $, where $q(G)$ denotes the number of quadrangles contained in $G$. \end{lemma}
{\bf Proof.} We prove this result by induction on $m$.
The result is obvious for $m= n$. So we suppose that $n< m <2(n-2)$ and the result is true for smaller $m$.
Let $e$ be an edge of $G$ and $q_G(e)$ denote the number of quadrangles containing the edge $e$. Suppose $e=(u,v)$. Let $U$ be the set of neighbors of $u$ except $v$, and $V$ the set of neighbors of $v$ except $u$. Then there are just $q_G(e)$ edges between $U$ and $V$. Let $X$ be the subset of $U$ such that each vertex in $X$
is incident to some of the above $q_G(e)$ edges and $Y$ be the subset of $V$ defined similarly to $X$. Assume $|X| =x$ and $|Y|= y$. Let $G_0$ be the subgraph of $G$ induced by $V(G_0)=u \cup v \cup X \cup Y$. Then there are at least $q_G(e)+x+y+1$ edges and exactly $x + y +2$ vertices in $G_0$. In order for the remaining vertices to connect to $G_0$, the number of remaining edges must be not less than that of the remaining vertices. Thus $$m-(q_G(e)+x+y+1)\ge n-(x+y+2).$$ That is $$q_G(e)\le m-n+1.$$ By induction hypothesis, $q(G-e)\le \left(\begin{array}{c} (m-1)-n+2 \\
2 \end{array}\right)=\left(\begin{array}{c} m-n+1 \\
2 \end{array}\right).$ Then we have $$q(G)=q_G(e)+q(G-e)\le m-n+1+\left(\begin{array}{c} m-n+1 \\
2 \end{array}\right)=\left(\begin{array}{c} m-n+2 \\
2 \end{array}\right).$$ Hence, the result follows.
$\blacksquare$
By a similar method, we can show that \begin{lemma} \label{015} Let $n\ge 5$ and $G\in G(n,m)$ be an arbitrary undirected graph containing $n$ vertices and $m \ (n\le m <2(n-2))$ edges. Suppose $\Delta(G)=n-1$. Then $$q(G)\le \left( \begin{array}{c} m-n+1 \\
2 \end{array}\right).$$ \end{lemma} \begin{lemma} {\em \cite[A part of Theorem 2.6]{gx}}\label{016} Let $G^{\sigma}$ be an oriented graph with an arc $e=(u,v)$. Suppose that $e$ is not contained in any even cycle. Then $$\phi(G^{\sigma}, \la)=\phi(G^{\sigma} -e, \la)+ \phi(G^{\sigma} - u-v,\la).$$ \end{lemma}
As a consequence of Lemma \ref{016}, we have the following result. \begin{lemma}\label{017} Let $G^{\sigma} $ be an oriented graph on $n$ vertices and $(u,v)$ a pendant arc of $G^{\sigma} $ with pendant vertex $v$. Suppose $\phi(G^{\sigma}, \la)=\sum_{i=0}^na_i(G^{\sigma})\la^{n-i}.$ Then $$a_{i}(G^{\sigma})=a_{i}(G^{\sigma}-v)+a_{i-2}(G^{\sigma}-v-u).$$ \end{lemma}
Based on the preliminary results above, we have the following two results. \begin{lemma} \label{06} Let $n\ge 5$ and $G^\sigma\in G^\sigma(n,m)$ be an oriented graph with maximum degree $n-1$. Suppose that $ n \le m < 2(n-2)$ and $G^\sigma \nsim O^+_{n,m}$. Then $G^\sigma \succ O^+_{n,m}$. \end{lemma} {\bf Proof.} By Theorem \ref{013}, it suffices to prove that $a_4(G^\sigma)> a_4(O^+_{n,m})$. Suppose that $v$ is the vertex with degree $n-1$. For convenience, all arcs incident to $v$ are colored as white and all other arcs are colored as black. Then there are $n-1$ white arcs and $m-n+1$ black arcs. We estimate the cardinality of $2$-matchings in $G^\sigma$ as follows. Noticing that all white arcs are incident to $v$, each pair of white arc can not form a $2$-matching of $G^\sigma$. Since $d(v)=n-1$ and each black arc incident to exactly two white arcs, each black arc together with a white arcs except its neighbors forms a $2$-matching of $G^\sigma$, that is, there are $(m-n+1)(n-3)$ black-white $2$-matchings. Moreover, noticing that $G^\sigma\neq O^+_{n,m}$, $G^\sigma-v$ does not contain the directed star $S_{m-n+2}$ as its subgraph, and thus there is at least one $2$-matching formed by a pair of disjoint black arcs, or $G^\sigma$ is an oriented graph of the following graph $F$. \setlength{\unitlength}{1mm} \begin{center} \begin{picture}(50,20) \put(10,20){\circle*{2}} \put(10,14){\circle*{2}} \put(10,0){\circle*{2}} \put(10,6){$\vdots$} \put(20,10){\circle*{2}} \put(18,5){$v_1$} \put(19,11){\line(-1,1){9}} \put(19,10){\line(-2,1){9}} \put(19,9){\line(-1,-1){9}} \put(40,10){\circle*{2}} \put(38,5){$v_2$} \put(21,10){\line(1,0){17}}\put(21,10){\line(1,0){18}} \put(30,18){\circle*{2}} \put(21,10.5){\line(4,3){9}} \put(39,10.5){\line(-4,3){9}} \put(30,1){\circle*{2}} \put(21,9.5){\line(1,-1){9}} \put(30,.5){\line(1,1){9}} \put(30,.5){\line(0,1){18}} \put(8,-9){Fig. 1.2. The graph $F$. } \end{picture} \end{center}
If it is the first case, then the number of $2$-matchings in $G^\sigma$ satisfies $$M(G^\sigma,2)\ge (m-n+1)(n-3)+1.$$ From Lemma \ref{015}, $q(G^\sigma)\le \left( \begin{array}{c} m-n+1 \\
2 \end{array}\right),$ and then by applying Theorem \ref{013} again, we have \begin{eqnarray*} a_4(G^\sigma)&\ge& M(G^\sigma,2)-2q(G^\sigma)\\&\ge& (m-n+1)(n-3)+1- 2\left( \begin{array}{c} m-n+1 \\
2 \end{array}\right)\\&=&a_4(O^+_{n,m})+1 \end{eqnarray*}
by Eq.(3.2).
If it is the second case, clearly $m=n+2$, $q(F)=3$, but the three quadrangles
can not be all evenly oriented. Then
\begin{eqnarray*} a_4(F)\ge M(F,2)-2q(F)\ge (m-n+1)(n-3)-4>a_4(O^+_{n,n+2}). \end{eqnarray*} The result thus follows.
$\blacksquare$
\begin{lemma} \label{07} Let $n\ge 5$ and $G^\sigma\in G^\sigma(n,m)$ be an oriented graph with $n\le m < 2(n-2)$. Suppose that $\Delta(G^\sigma)\le n-2$ and $G^\sigma \nsim B^+_{n,m}$. Then $G^\sigma \succ B^+_{n,m}$. \end{lemma} {\bf Proof.} By Theorem \ref{013} again, it suffices to prove that $a_4(G^\sigma)> a_4(B^+_{n,m})$. We apply induction on $n$ to prove it. By a direct calculation, the result follows if $n=5$, since then $5=m<2(5-2)=6$ and there exists exactly four graphs in $G^\sigma(5,5)$, namely, the oriented cycle $C_3$ together with two pendant arcs attached to two different vertices of the $C_3$, the oddly oriented cycle $C_4$ together with a pendant arc, $B^+_{5,5}$ and the oriented cycle $C_5$. Suppose now that $n\ge 6$ and the result is true for smaller $n$.
\noindent {\bf Case 1.} There is a pendant arc $(u,v)$ in $G^\sigma$ with pendant vertex $v$.
By Lemma \ref{017} we have $$a_4(G^\sigma)=a_4(G^\sigma-v)+a_2(G^\sigma-v-u)=a_4(G^\sigma-v)+e(G^\sigma-v-u).$$ Noticing that $\Delta(G^\sigma)\le n-2$, we have $e(G^\sigma-v-u)\ge m-\Delta(G^\sigma)\ge m-n+2$.
By induction hypothesis, $a_4(G^\sigma-v)\ge a_4(B^+_{n-1,m-1})$ with equality if and only if $G^\sigma-v= B^+_{n-1,m-1}$. Then \begin{equation*} \begin{array}{lll} a_4(G^\sigma)&=&a_4(G^\sigma-v)+a_2(G^\sigma-v-u)\\&\ge& a_4(B^+_{n-1,m-1})+m-n+2\\&=& a_4(B^+_{n-1,m-1})+e(S_{m-n+1})\\ &=& a_4(B_{n,m}) \end{array} \end{equation*} with equality if and only if $G^\sigma= B^+_{n,m}$. The result thus follows.
\noindent {\bf Case 2.} There are no pendant vertices in $G^\sigma$.
Let $$(d)_{G^\sigma}=(d_1,d_2,\cdots,d_i,d_{i+1},\cdots,d_n)$$ be the non-increasing degree sequence of $G^\sigma$. We label the vertices of $G^\sigma$ corresponding to the degree sequence $(d)_{G^\sigma}$ as $v_1,v_2,\cdots,v_n$ such that $d_{G^\sigma}(v_i)=d_i$ for each $i$. Assume $d_1<n-2$. Then there exists a vertex $v_k$ that is not adjacent to $v_1$, but is adjacent to one neighbor, say $v_i$, of $v_1$. Thus $$(d_1+1,d_2,\cdots d_i-1,d_{i+1},\cdots,d_n)$$ is the degree sequence of the oriented graph $D'$ obtained from $G^\sigma$ by deleting the arc $(v_k,v_i)$ and adding the arc $(v_k,v_1)$, regardless the orientation of the arc $(v_k,v_1)$. Rewriting the sequence above such that $$(d)_{D'}=(d'_1,d'_2,\cdots,d'_{i},d'_{i+1},\cdots,d'_n)$$ is also a non-increasing sequence. Then $d_{1}\ge d_i\ge 2$ and thus we have $$\sum_{i=1}^n\left( \begin{array}{c} d_i' \\
2 \end{array}\right)> \sum_{i=1}^n\left( \begin{array}{c} d_i \\
2 \end{array}\right),\eqno{(3.4)}$$since \begin{equation*} \begin{array}{lll}\sum_{i=1}^n\left( \begin{array}{c} d_i' \\
2 \end{array}\right)- \sum_{i=1}^n\left( \begin{array}{c} d_i \\
2 \end{array}\right)&=& \left( \begin{array}{c} d_1+1 \\
2 \end{array}\right)+\left( \begin{array}{c} d_i-1 \\
2 \end{array}\right)-\left( \begin{array}{c} d_1 \\
2 \end{array}\right)-\left( \begin{array}{c} d_i \\
2 \end{array}\right)\\ &=& d_{1}-d_i+1 \\&>&0. \end{array} \end{equation*} Repeating this procedure, we can eventually obtain a non-increasing graph sequence $$(d)_{D''}=(d''_1,d''_2,\cdots,d''_{i},d''_{i+1},\cdots,d''_n)$$such that $\Delta(D'')=d_1''=n-2$ and
$$\sum_{v\in D''}\left( \begin{array}{c} d''(v) \\
2 \end{array}\right)> \sum_{v\in D' }\left( \begin{array}{c} d'(v) \\
2 \end{array}\right)>\cdots > \sum_{v\in G^\sigma}\left( \begin{array}{c} d(v) \\
2 \end{array}\right).\eqno{(3.5)}$$
Similarly, we can assume that there exists a vertex $v_k$ that is not adjacent to $v_i$, but is adjacent to one neighbor, say $v_j$, of $v_i$. Thus $$(d_1,d_2,\cdots d_i+1,d_{i+1},\cdots,d_j-1,d_{j+1},\cdots,d_n)$$ is the degree sequence of the oriented graph $D'''$ obtained from $D''$ by deleting the arc $(v_k,v_j)$ and adding the arc $(v_k,v_i)$, regardless the orientation of the arc $(v_k,v_j)$. By a similar proof, we can get $$\sum_{v\in D'''}\left( \begin{array}{c} d'''(v) \\
2 \end{array}\right)> \sum_{v\in D'' }\left( \begin{array}{c} d''(v) \\
2 \end{array}\right).$$ Then by applying the above procedure repeatedly, we eventually obtain the degree sequence $(d)_{B^+_{n,m}}$, $$(d)_{B^+_{n,m}}=(n-2,m-n+2,2,2,\cdots,2,1,1,\cdots,1),$$ where the number of vertices of degree $2$ is $m-n+2$, and the number of vertices of degree $1$ is $2n-m-4$. Finally, we get $$\sum_{v\in B^+_{n,m}}\left( \begin{array}{c} d^{B^+}(v) \\
2 \end{array}\right)> \sum_{v\in D''' }\left( \begin{array}{c} d'''(v) \\
2 \end{array}\right)> \sum_{v\in D''}\left( \begin{array}{c} d''(v) \\
2 \end{array}\right)>\cdots > \sum_{v\in G^\sigma}\left( \begin{array}{c} d(v) \\
2 \end{array}\right).$$
Then the lemma follows by combining Eq.(3.3) and Lemma \ref{05} with the fact that $M(G,2)=\left( \begin{array}{c} m \\
2 \end{array}\right)-\sum_{v\in G^\sigma}\left( \begin{array}{c} d(v) \\
2 \end{array}\right)$.
$\blacksquare$
Combining Lemma \ref{06} with Lemma \ref{07}, we get the proof of Theorem \ref{01} immediately.
\vskip3mm \noindent {\bf Proof of Theorem \ref{01}.} Combining with Lemmas \ref{06} and \ref{07}, the oriented graph with minimal skew energy among all oriented graphs of $G^\sigma(n,m)$ with $n\le m \le 2(n-2)$ is either $O^+_{n,m}$ or $B^+_{n,m}$. Furthermore, from (3.2) and (3.3), we have $$a_4(O^+_{n,m})=(m-n+1)(2n-m-3)$$ and $$a_4(B^+_{n,m})=(m-n+2)(2n-m-4).$$ Then, by a direct calculation we have $a_4(O^+_{n,m})<a_4(B^+_{n,m})$ if $m<\frac{3n-5}{2}$; $a_4(B^+_{n,m})=a_4(O^+_{n,m})$ if $m=\frac{3n-5}{2}$; and $a_4(O^+_{n,m})>a_4(B^+_{n,m})$ otherwise. The proof is thus complete by (3.1).
$\blacksquare$
\end{document} |
\begin{document}
\title{Bad groups in the sense of Cherlin} \author{Olivier Fr\'econ} \address{Laboratoire de Math\'ematiques et Applications, Universit\'e de Poitiers, T\'el\'eport 2 - BP 30179, Boulevard Marie et Pierre Curie, 86962 Futuroscope Chasseneuil Cedex, France} \email{olivier.frecon@math.univ-poitiers.fr} \subjclass[2010]{20F11, 03C45, 20A15} \date{\today} \keywords{Groups of finite Morley rank, Bad groups, Projective space.} \begin{abstract} There exists no bad group (in the sense of Gregory Cherlin), namely any simple group of Morley rank 3 is isomorphic to ${\rm PSL}_2(K)$ for an algebraically closed field $K$. \end{abstract} \maketitle
\section{Introduction}
Model theory is a branch of mathematical logic concerned with the study of classes of mathematical structures by considering first-order sentences and formulas. The {\em Morley rank} is a model-theoretical notion of abstract dimension. It generalizes the dimension of an algebraic variety (when the ground field is algebraically closed). There are other notions of abstract dimension, the importance of the Morley rank lies on {\em Morley's Categoricity Theorem} below, which \gui can be thought of as the beginning of modern model theory'' (David Marker \cite[p. 2]{Marker}) and the following Baldwin and Zilber Theorems.
We remember that a {\em theory} is a set of first-order $\Lm$-sentences for a language $\Lm$, it is {\em complete} if for any sentence $\phi$, either $\phi$ or $\neg\phi$ belongs to $T$, and a theory is {\em $\kappa$-categorical} for some cardinal $\kappa$ if, up to isomorphism, it has exactly one model of cardinality $\kappa$ (cf. \cite[Chapters 1 and 2]{Marker} for more details).
\bfait Let $T$ be a complete theory in a countable language. \begin{description}[font= $\bullet$ \normalfont \rm] \item {\em (Morley's Categoricity Theorem, \cite{Morley})} If $T$ is {$\kappa$-categorical} for some uncountable $\kappa$, then $T$ is $\kappa$-categorical for every uncountable $\kappa$. \item {\em (Baldwin, \cite{Bal73})} If $T$ is uncountably categorical, then it is {\em of finite Morley rank}. \item {\em (Zilber, \cite{Zil77})} The theory of an {\em infinite simple group} of finite Morley rank is uncountably categorical. \end{description} \efait
In this paper, we are concerned with {\em groups} of finite Morley rank. The main example of such a group is an algebraic group defined over an algebraically closed field in the field language (Zilber, \cite{Zil77}). In the late seventies, Gregory Cherlin \cite[\S 6]{Che79} and Boris Zilber \cite{Zil77} formulated independently the following algebraicity conjecture.
\bconj {\em (Cherlin-Zilber Conjecture {\rm or} Algebraicity Conjecture)} An infinite simple group of finite Morley rank is algebraic over an algebraically closed field. \econj
This is the main conjecture on groups of finite Morley rank, and it is still open. Most of studies on groups of finite Morley rank focus on this conjecture. Actually, the original Cherlin Conjecture concerned simple {\em $\omega$-stable} groups, but the substantial litterature on the Algebraicity Conjecture treats only the finite Morley rank case.
The Algebraicity Conjecture has been proved for several important classes of groups including locally finite groups \cite{Thomas}. The main theorem on groups of finite Morley rank ensures that any simple group of finite Morley rank with an infinite abelian subgroup of exponent 2 satisfies the Cherlin-Zilber Conjecture \cite{ABC}.
However, in despite of numerous papers on the subject, the Cherlin-Zilber Conjecture is still open, even for groups of Morley rank 3. As a matter of fact, in \cite{Che79}, the Algebraicity Conjecture was formulated as a result from an analysis of simple groups of Morley rank 3. The main result of \cite{Che79} can be summarized as follows, where a {\em bad group} is a nonsolvable group of Morley rank 3 containing no definable subgroup of Morley rank 2.
\bfait\label{thcherlin} {\em (Cherlin, \cite{Che79})} Let $G$ be an infinite simple group of Morley rank at most 3. Then $G$ has Morley rank 3, and one of the following two assertions is satisfied: \bi \item there is an algebraically closed field $K$ such that $G\simeq \PSL_2(K)$, \item $G$ is a {bad group}. \ei \efait
Thus bad groups became a major obstacle to the Cherlin-Zilber Conjecture. These groups have been studied in \cite{Che79,Nes89rk3} and \cite{Nes91}, whose results are summarized in Facts \ref{bad} and \ref{factNes} respectively. Later, it was shown that no bad group is existentially closed \cite{JO04} or linear \cite{MusPoi06}. However, these groups appeared very resistant, and very sparse other information was known on bad groups.
Furthermore, Nesin has shown in \cite{Nes91} that a bad group acts on a natural geometry, which is not very far from being a non-Desarguesian projective plane of Morley rank 2. However, Baldwin discovered non-Desarguesian projective planes of Morley rank 2 \cite{Bal94}. Thus, the question of the existence, or not, of a bad group was still fully open. In this paper, we show that bad groups do not exist.
\bmtheo There is no bad group. \emtheo
Note other more general notions of bad groups have been introduced independently by Corredor \cite{Cor89} and by Borovik and Poizat \cite{BP90}, where a {\em bad group} is defined to be a nonsolvable connected group of finite Morley rank all of whose proper connected definable subgroups are nilpotent. Such a bad group has similar properties to original bad groups. Moreover, later Jaligot will introduce a more general notion of bad groups \cite{Jal01}, and he will obtain similar results. However, we recall that, in this paper, a {\em bad group} is defined to be nonsolvable, of Morley rank 3, and containing no definable subgroup of Morley rank 2.
Our proof of Main Theorem goes as follows. First we note that it is sufficient to study {\em simple} bad groups since for any bad group $G$, the quotient group $G/Z(G)$ is a {\em simple} bad group by \cite[\S 4, Introduction]{Nes89rk3}.
Then we fix a simple bad group $G$, and we introduce a notion of lines as cosets of Borel subgroups of $G$ (Definition \ref{defiline}). In \S \ref{secline}, we study their behavior, mainly in regards with conjugacy classes of elements of $G$.
In \S \ref{secplane}, we propose a definition of a plane (Definition \ref{defiplane}). This section is dedicated to prove that $G$ contains a plane (Theorem \ref{thplane}). This result is the key point of our demonstration. Roughly speaking, we show that for each nontrivial element $g$ of $G$ such that $g=[u,v]$ for $(u,v)\in G\times G$, the union of the preimages of $g$, by maps of the form ${\rm ad}_v:G\to G$ defined by ${\rm ad}_v(x)=[x,v]$, is almost a plane, and from this, we obtain a plane.
In last section \S \ref{secfin}, we try to show that our notions of lines and planes provide a structure of projective space over the group $G$. Indeed, such a structure would provide a division ring (see \cite[p. 124, Theorem 7.15]{Harts}), and probably it would be easy to conclude. However, a contradiction occurs along the way, and achieves our proof.
{\em \underline{Note :} in a very recent preprint \cite{PoiWag}, by analyzing the present paper, Poizat and Wagner generalize our main result to other groups, and they eliminate other groups of Morley rank.}
\subsection*{The other simple groups of dimension 3}
\bi \item If $G$ is a non-bad simple group of Morley rank 3, then $G$ is isomorphic to $\PSL_2(K)$ for an algebraically closed field $K$ (Fact \ref{thcherlin}). As in \S \ref{secline}, we may define a {\em line} in $G$ to be a coset of a connected subgroup of dimension 1, and we may define a plane as in \S \ref{secplane}. It is possible to show that two sorts of planes occur: the cosets of Borel subgroups, and the subsets of the form $aJ$ where $J$ is defined to be \bi \item the set of involutions when the characteristic $c$ of $K$ is not 2;
\item $J=\{j\in G~|~j^2=1\}$ when $c=2$. \ei
The plane $J$ is normalized by $G$, and there is no such a plane in a bad group (Lemma \ref{mapalphaaut}). Another important difference between $G$ and a bad group is to be the presence of a Weyl group. Indeed, the first lemma of this paper is not verified in $G$ (Lemma \ref{lem1}), because we have $jT=Tj$ for any torus $T$ and any involution $j\in N_G(T)\setminus T$.
\item The group $\SO_3(\R)$ is not of finite Morley rank, or even stable \cite{Nes89rk3}. However, our definitions of lines and planes naturally extend to $\SO_3(\R)$. Then, as above, the set $J$ of involutions in $\SO_3(\R)$ forms a plane, and the presence of a Weyl group is again a major difference between $\SO_3(\R)$ and bad groups. Moreover, we note that the plane $J$ has a structure of projective plane, whereas this is false in $\PSL_2(K)$ \cite[Fact 8.15]{bn1}. \ei
\section{Background material}
A thorough analysis of groups of finite Morley rank can be found in \cite{bn1} and \cite{ABC}. In this section we recall some definitions and known results.
\subsection{Borovik-Poizat axioms}
Let $(G,\,\cdot\,,^{-1},1,\cdots)$ be a group equipped with additional structure. This group $G$ is said to be {\em ranked} if there is a function \gui {\rm rk}'' which assigns to each nonempty definable set $S$ an integer, its \gui dimension'' $\rk(S)$, and which satisfies the following axioms for every definable sets $A$ and $B$. {\em\begin{description}[font=\normalfont \rm] \item[Definition] For any integer $n$, $\rk(A)>n$ if and only if $A$ contains an infinite family of disjoint definable subsets $A_i$ of rank $n$.
\item[{Definability}] For any uniformly definable family $\{A_b~:~b\in B\}$ of definable sets, and for any $n\in \N$, the set $\{b\in B~:~\rk(A_b)=n\}$ is also definable. \item[{Finite Bounds}] For any uniformly definable family $\Fc$ of finite subsets of $A$, the sizes of the sets in $\Fc$ are bounded. \end{description}} It is shown in \cite{poigrsta} that the groups $(G,\,\cdot\,,\cdots)$ as above satisfy a fourth axiom, namely the {\em additivity axiom}, and they are precisely the groups of finite Morley rank. Moreover, the function $\rk$ assigns to each definable set its Morley rank. In this paper, as in \cite{bn1} and \cite{ABC}, the Morley rank
will be denoted by $\rk$.
\subsection{Morley degree}
A nonempty definable set $A$ is said to have {\em Morley degree 1} if for any definable subset $B$ of $A$, either $\rk B<\rk A$ or $\rk(A\setminus B)<\rk A$. The set $A$ is said to have {\em Morley degree} $d$ if $A$ is the disjoint union of $d$ definable sets of Morley degree 1 and Morley rank $\rk A$.
\bfait \bi \item\cite[Lemmas 4.12 and 4.14]{bn1} Every nonempty definable set has a unique degree. \item \cite[Proposition 4.2]{bn1} Let $X$ and $Y$ be definable subsets of Morley degree $d$ and $d'$ respectively. Then $X\times Y$ has Morley degree $dd'$. \item \cite[\S 2.2]{Che79} A group of finite Morley rank has Morley degree 1 if and only if it is {\em connected}, namely it has no proper definable subgroup of finite index. \ei \efait
Moreover, the following elementary result will be useful for us.
\bfait\label{deg1} Let $f:E\to F$ be a definable map. If the set $E$ has Morley degree 1 and $r=\rk f^{-1}(y)$ is constant for $y\in F$, then the Morley degree of $F$ is 1. \efait
\bpreu Let $B$ be a definable subset of $F$ of Morley rank $\rk F$. We show that $\rk(F\setminus B)<\rk F$. By the additivity axiom, we have $\rk E=r+\rk F$ and $$\rk f^{-1}(B)=r+\rk B=r+\rk F=\rk E$$ Since $E$ has Morley degree 1, we obtain $\rk f^{-1}(F\setminus B)=\rk(E\setminus f^{-1}(B))<\rk E$, and by the additivity axiom again, $$\rk(F\setminus B)=\rk f^{-1}(F\setminus B)-r<\rk E-r=\rk F$$ so $F$ has Morley degree 1. \epreu
\subsection{Bad groups}
Main properties of bad groups are summarized in the following facts, where a {\em Borel subgroup} of a bad group $G$ is defined to be an infinite definable proper subgroup of $G$.
\bfait\label{bad} {\em (\cite[\S 5.2]{Che79} and \cite{Nes89rk3})} Let $G$ be a simple bad group, and $B$ be a Borel subgroup of $G$. \be \item $B=C_G(b)$ for $b\in G\setminus \{1\}$, \item $B$ is connected, abelian, self-normalizing and of Morley rank 1, \item $C_G(x)$ is a Borel subgroup for each nontrivial element $x$ of $G$, \item if $A$ is another Borel subgroup of $G$, then $A$ is conjugate with $B$, and either $A=B$ or $A\cap B=\{1\}$, \item $G=\bigcup_{g\in G}B^g$, \item $G$ has no involution. \ee \efait
\bfait\label{factNes} {\em \cite[Lemma 18]{Nes91}} Let $A$ and $B$ be two distinct Borel subgroups of a simple bad group $G$. Then $\rk(ABA)=3$, $\rk(AB)=2$, and $AB$ has Morley degree 1. \efait
The following result is due to Delahan and Nesin, it is proved for a more general notion of bad groups, and it is used in our final argument.
\bfait\label{invoautobad} {\em \cite[Proposition 13.4]{bn1}} A simple bad group $G$ cannot have an involutive definable automorphism. \efait
\section{Lines}\label{secline}
In this paper, $G$ denotes a fixed {simple bad group}. We fix a Borel subgroup $B$ of $G$ and we denote by $\Bm$ the set of Borel subgroups of $G$.
In this section, we define a {\em line} of $G$, and we provide their basic properties. We note that, by conjugation of Borel subgroups (Fact \ref{bad} (4)), any Borel subgroup is a {\em line} in the following sense.
\bdefi\label{defiline} A {\em line} of $G$ is a subset of the form $uBv$ for two elements $u$ and $v$ of $G$.
We denote by $\Lambda$ the set of lines of $G$. \edefi
We note that, by Fact \ref{bad} (2), each line has Morley rank 1 and Morley degree 1.
\ble\label{lem1} Let $uBv$ and $rBs$ be two lines. Then $uBv=rBs$ if and only if $uB=rB$ and $Bv=Bs$. \ele
\bpreu We may assume that $uBv=rBs$. Then we have $$B=u^{-1}rBsv^{-1}=u^{-1}rsv^{-1}B^{sv^{-1}}$$ so $u^{-1}rsv^{-1}\in B^{sv^{-1}}$ and $B=B^{sv^{-1}}$. Now $sv^{-1}$ belongs to $B$ since $B$ is self-normalizing by Fact \ref{bad}. Hence we obtain $Bv=Bs$, and the equality $uB=rB$ follows from $uBv=rBs$. \epreu
By the above lemma, the set $\Lambda$ identifies with $(G/B)_l\times (G/B)_r$ where $(G/B)_l$ (resp. $(G/B)_r$) denotes the set of left cosets (resp. right cosets) of $B$ in $G$. Then $\Lambda$ is a definable set. Moreover, since $G$ is connected of Morley rank 3 and $B$ has Morley rank 1, the Morley rank of $\Lambda$ is 4 and its Morley degree is 1. In particular, $\Lambda$ is a uniformly definable family.
\ble\label{lem3} Two distinct elements $x$ and $y$ of $G$ lie in one and only one line $l(x,y)$.
Moreover, the map $l:\{(x,y)\in G\times G~|~x\neq y\}\to \Lambda$ is definable. \ele
\bpreu By Fact \ref{bad} (5), there exists $v\in G$ such that $y^{-1}x$ belongs to $B^v$. Then $x$ and $y$ lie in $uBv$ for $u=yv^{-1}$.
Now, if $rBs$ is a line containing $x$ and $y$, then we find two elements $b_1$ and $b_2$ of $B$ such that $x=rb_1s$ and $y=rb_2s$. Thus $y^{-1}x=s^{-1}b_2^{-1}b_1s$ is a nontrivial element of $B^s$. But $y^{-1}x$ belongs to $B^v$ by the choice of $v$, hence we have $B^s=B^v$ (Fact \ref{bad} (4)). Since $B$ is self-normalizing, $sv^{-1}$ belongs to $B$ and we obtain $Bs=Bv$, so there exists $b\in B$ such that $s=bv$. This implies that $u=yv^{-1}=(rb_2s)(s^{-1}b)=rb_2b$ belongs to $rB$, and $rBs=uBv$ is the unique line containing $x$ and $y$.
Moreover, since $\Lambda$ is a uniformly definable family,
the set $\{(x,y)\in G\times G~|~x\neq y\}\times \Lambda$ is definable, and
$$\Gamma=\{((x,y),uBv)\in (G\times G)\times \Lambda~|~ x\neq y,~ x\in uBv,~y\in uBv\}$$
is a definable subset of it.
But $\Gamma$ is precisely the graph of the map $l$, hence $l$ is definable. \epreu
\ble\label{lem2} If $uBv=(uBv)^g$ for $uBv\in \Lambda\setminus\Bm$ and $g\in G$, then $g=1$. \ele
\bpreu We have $uBv=g^{-1}uBvg$, so $uB=g^{-1}uB$ and $Bv=Bvg$ by Lemma \ref{lem1}, and $g$ belongs to the Borel subgroups $B^{u^{-1}}$ and $B^v$. If $g$ is nontrivial, then $B^{u^{-1}}=B^v$ (Fact \ref{bad} (4)), and $vu$ belongs to $N_G(B)=B$. Consequently $u$ belongs to $v^{-1}B$, and we obtain $uBv=B^v$, contradicting $uBv\not\in \Bm$. Thus $g=1$. \epreu
\bdefi For each $g\in G$ and each definable subset $X$ of $G$, we consider the following subsets of $\Lambda$:
$$\Lm(g,X)=\{l(g,x)\in\Lambda~|~x\in X\setminus\{g\}\}$$
$$\Lambda_X=\{\lambda\in \Lambda~|~\lambda\cap X\ {\rm is\ infinite}\}$$ \edefi
Since the map $l$ is definable (Lemma \ref{lem3}), the set $\Lm(g,X)$ is definable for each $g\in G$ and each definable subset $X$ of $G$. Moreover, by Definablity axiom, the set
$\Lambda_X=\{\lambda\in \Lambda~|~\rk(\lambda\cap X)=1\}$ is definable too.
\ble\label{lemdegfin} Let $\lambda_1,\ldots,\lambda_n$ be $n$ lines. Then $\lambda_1\cup\cdots\cup \lambda_n$ is a definable set of Morley rank 1 and Morley degree $n$. \ele
\bpreu For each $i$, the set $A_i=\lambda_i\cap(\bigcup_{j\neq i}\lambda_j)$ has at most $n-1$ elements by Lemma \ref{lem3}, and $\lambda_1\cup\cdots\cup \lambda_n$ is the disjoint union of $\lambda_1\setminus A_1,\ldots,\lambda_n\setminus A_n,\cup_{i=1}^nA_i$. Since each line $\lambda_i$ has Morley rank 1 and Morley degree 1 (Fact \ref{bad} (2)), the result follows. \epreu
\ble\label{lemdef12} If $\Lambda_0$ is a definable subset of $\Lambda$, then $\bigcup\Lambda_0$ is a definable subset of $G$. Moreover, if $\Lambda_0$ is infinite, then $\bigcup\Lambda_0$ has Morley rank at least 2. \ele
\bpreu Since $\Lambda_0$ is a definable subset of the uniformly definable family $\Lambda$,
the set $\bigcup\Lambda_0=\{x\in G~|~\exists \lambda\in \Lambda_0,~x\in \lambda\}$ is definable.
Moreover, if $\Lambda_0$ is infinite, then $\bigcup\Lambda_0$ has Morley rank at least 2 by Lemma \ref{lemdegfin}. \epreu
\bco\label{cordef12} The subset $\bigcup\Lambda_X$ of $G$ is definable for each definable subset $X$ of $G$. \eco
\section{Planes}\label{secplane}
Our aim is to find a definable structure of projective space on our bad group $G$. In this section, we introduce a notion of planes, and we show that $G$ has such a plane (Theorem \ref{thplane}). We fix a definable subset $X$ of $G$, of Morley rank 2.
\bdefi\label{defiplane} The definable subset $X$ of $G$ is said to be a {\em plane} if it satisfies $\ov{X}=X$ where
$$\ov{X}=\{g\in G~|~{\rk}(\Lm(g,X))=1\}$$ \edefi
\ble\label{procomm1} The set $\ov{X}$ is a definable subset of $\bigcup\Lambda_X$. \ele
\bpreu If $g\in G$ does not belong to $\bigcup\Lambda_X$, then $l(g,x)\cap X$ is finite for each $x\in X$, and since $X$ has Morley rank 2, the set $\Lm(g,X)$ has Morley rank 2, so $g\not\in\ov{X}$. Thus $\ov{X}$ is contained in $\bigcup\Lambda_X$.
We show that $\ov{X}$ is definable.
We consider the set $$A=\{(g,\lambda)\in G\times \Lambda~|~g\in \lambda,~\exists x\in X\setminus\{g\},~x\in \lambda\}$$ and the map $f:A\to G$ defined by $f(g,\lambda)=g$. We note that, since $\Lambda$ is a uniformly definable family, $A$ is definable,
and $f$ is definable too. Moreover, the preimage by $f$ of each $g\in G$ is $f^{-1}(g)=\{g\}\times \Lm(g,X)$, and we have $\rk (f^{-1}(g))=\rk\Lm(g,X)$.
Consequently, we obtain $\ov{X}=\{g\in G~|~\rk (f^{-1}(g))=1\}$, and $\ov{X}$ is definable. \epreu
\ble\label{lemrkdm2} The Morley ranks of $\Lambda_X$ and of $\bigcup\Lambda_X$ are at most 2. Moreover, $\Lambda_X$ is infinite if and only if $\rk\bigcup\Lambda_X=2$. \ele
\bpreu We consider the surjective definable map $$l_0:(X\times X)\cap l^{-1}(\Lambda_X)\to \Lambda_X$$ defined by $l_0(x,y)=l(x,y)$. For each $\lambda\in \Lambda_X$, we have
$l_0^{-1}(\lambda)=\{(x,y)\in (\lambda\cap X)\times (\lambda\cap X)~|~x\neq y\}$, and since $\rk \lambda=1$, we obtain $\rk(\lambda\cap X)=1$ and $\rk l_0^{-1}(\lambda)=2$. But we have $$\rk ((X\times X)\cap l^{-1}(\Lambda_X))\leq \rk(X\times X)=2\rk X=4$$ hence $\rk \Lambda_X$ is at most $4-2=2$.
We show that $\rk\bigcup\Lambda_X\leq 2$. We consider the definable set
$$A=\{(x,\lambda)\in G\times \Lambda_X~|~x\in \lambda\setminus X\}$$ and the definable map $l_1:A\to \Lambda_X$ defined by $l_1(x,\lambda)=\lambda$. For each $\lambda\in \Lambda_X$, we have $\rk \lambda=1=\rk(\lambda\cap X)$, so, since each line has Morley degree 1, the preimage $l_1^{-1}(\lambda)$ is finite. Consequently we obtain $\rk A\leq \rk\Lambda_X\leq 2$. But the definable map $l_2:A\to (\bigcup\Lambda_X)\setminus X$, defined by $l_2(x,\lambda)=x$, is surjective, hence the Morley rank of $(\bigcup\Lambda_X)\setminus X$ is at most $\rk A\leq 2$. Since $X$ has Morley rank 2, we obtain $\rk\bigcup\Lambda_X\leq 2$.
Now it follows from Lemmas \ref{lemdegfin} and \ref{lemdef12} that $\Lambda_X$ is infinite if and only if $\rk\bigcup\Lambda_X=2$. \epreu
\bpro\label{lemdm1} For each $g\in \ov{X}$, we have ${\rk}(\Lm(g)\cap \Lambda_X)=1$.
Moreover, if $X$ has Morley degree 1, then $\ov{X}=\{g\in G~|~{\rk}(\Lm(g)\cap \Lambda_X)=1\}$
and $G\setminus \ov{X}=\{g\in G~|~\Lm(g)\cap \Lambda_X~{\rm is\ finite}\}$. \epro
\bpreu First we note that $\Lm(g)\cap \Lambda_X=\Lm(g,X)\cap \Lambda_X$ for any $g\in G$. For each $g\in G$, we consider the definable map $l_g:X\setminus\{g\}\to \Lm(g,X)$ defined by $l_g(x)=l(g,x)$. In particular, the preimage $l_g^{-1}(\lambda)$ of each $\lambda\in \Lm(g,X)$ is $(\lambda\cap X)\setminus\{g\}$.
We show that ${\rk}(\Lm(g,X)\cap \Lambda_X)\leq 1$ for each $g\in G$. We may assume that $\Lambda_X$ is infinite. Then, by Lemma \ref{lemrkdm2}, the set $\cup\Lambda_X$ has Morley rank 2. Let $g\in G$ and $u_g:\cup(\Lm(g)\cap\Lambda_X)\setminus\{g\}\to \Lm(g)\cap\Lambda_X$ be the map defined by $u_g(x)=l(g,x)$. Since each line has Morley rank 1, the preimage of each element of $\Lm(g)\cap\Lambda_X$ has Morley rank 1. Consequently, we have $$\rk (\Lm(g)\cap\Lambda_X)= \rk\cup(\Lm(g)\cap\Lambda_X)-1\leq \rk\cup\Lambda_X-1=1$$
Let $g\in \ov{X}$. We show that ${\rk}(\Lm(g)\cap \Lambda_X)=1$. For each $\lambda\in \Lm(g,X)\setminus \Lambda_X$, the set $l_g^{-1}(\lambda)=(\lambda\cap X)\setminus\{g\}$ is finite, and since $g\in\ov{X}$, we have $\rk\Lm(g,X)=1$. Consequently, $l_g^{-1}( \Lm(g,X)\setminus \Lambda_X)$ has Morley rank at most 1, and $l_g^{-1}( \Lm(g,X)\cap\Lambda_X)$ has Morley rank $\rk X=2$. But the set $l_g^{-1}(\lambda)=(\lambda\cap X)\setminus\{g\}$ is infinite of Morley rank 1 for each $\lambda\in \Lm(g,X)\cap \Lambda_X$. Hence we obtain $\rk (\Lm(g,X)\cap \Lambda_X)=2-1=1$.
Now we assume that $X$ has Morley degree 1. Let $g\in G$ such that ${\rk}(\Lm(g,X)\cap \Lambda_X)=1$. We show that $g\in\ov{X}$. Since the set $l_g^{-1}(\lambda)=(\lambda\cap X)\setminus\{g\}$ is infinite of Morley rank 1 for each $\lambda\in \Lm(g,X)\cap \Lambda_X$, the set $l_g^{-1}( \Lm(g,X)\cap\Lambda_X)$ has Morley rank $$1+{\rk}(\Lm(g,X)\cap \Lambda_X)=2=\rk X$$ Then, since $X$ has Morley degree 1, the preimage of $\Lm(g,X)\setminus \Lambda_X$ has Morley rank at most 1. Moreover, for each $\lambda\in \Lm(g,X)\setminus \Lambda_X$, the preimage $l_g^{-1}(\lambda)=(\lambda\cap X)\setminus\{g\}$ is finite and non-empty, so we obtain $$\rk(\Lm(g,X)\setminus \Lambda_X)=\rk l_g^{-1}(\Lm(g,X)\setminus \Lambda_X)\leq 1$$ This shows that $\rk\Lm(g,X)=1$ and $g\in\ov{X}$.
Furthermore, since ${\rk}(\Lm(g,X)\cap \Lambda_X)\leq 1$ for each $g\in G$,
we obtain $G\setminus \ov{X}=\{g\in G~|~\Lm(g)\cap \Lambda_X~{\rm is\ finite}\}$, as desired. \epreu
\bco\label{corY2} We have $\rk(\ov{X}\setminus X)\leq 1$. \eco
\bpreu We remember that $\ov{X}$ is definable by Lemma \ref{procomm1},
so the sets $Y=\ov{X}\setminus X$ and $A=\{(y,\lambda)\in Y\times \Lambda_X~|~y\in \lambda\}$ are definable too. Let $l_Y:A\to Y$ and $l_D:A\to \Lambda_X$ be the definable maps defined by $l_Y(y,\lambda)=y$ and $l_D(y,\lambda)=\lambda$ respectively. On the one hand, for each $\lambda\in\Lambda_X$, the set $\lambda\cap X$ is infinite, and since $\lambda$ has Morley rank 1 and Morley degree 1 (Fact \ref{bad} (2)), the set $\lambda\cap Y$ is finite and $l_D^{-1}(\lambda)$ has Morley rank at most 0. This implies $\rk A\leq\rk\Lambda_X\leq 2$ (Lemma \ref{lemrkdm2}). On the other hand, for each $y\in Y$, we have ${\rk}(\Lm(y)\cap \Lambda_X)=1$ by Proposition \ref{lemdm1}, so $l_Y^{-1}(y)$ has Morley rank 1, and we obtain $\rk A=1+\rk Y$. Consequently, the Morley rank of $Y$ is at most 1. \epreu
\ble\label{lemmaxdl} For each $g\in G$, the set $\Lm(g,X)$ is infinite. \ele
\bpreu Indeed, $\cup\Lm(g,X)$ is definable (Lemma \ref{lemdef12}) and contains $X$. Since $\rk X=2$, we obtain $\rk(\cup\Lm(g,X))\geq 2$, and $\Lm(g,X)$ is infinite (Lemma \ref{lemdegfin}). \epreu
\bco\label{planedeg1} If the Morley degree of $X$ is not 1, then $\rk\ov{X}<2$. In particular, any plane has Morley degree 1. \eco
\bpreu Let $n$ be the Morley degree of $X$, and $X_1,\ldots,X_n$ be $n$ definable subsets of $X$ of Morley rank 2 and Morley degree 1 such that $X$ is the disjoint union of $X_1,\ldots,X_n$. For each $g\in\ov{X}$, we have $\rk\Lm(g,X)=1$, so we obtain $\rk\Lm(g,X_i)\leq 1$ for each $i$, and $g\in\ov{X_i}$ for each $i$ by Lemma \ref{lemmaxdl}. Thus $\ov{X}$ is contained in $\ov{X_1}\cap\ov{X_2}$. Since $X_1\cap X_2=\emptyset$, the set $\ov{X}$ is contained in $(X_1\cap Y_2)\cup (Y_1\cap X_2)\cup (Y_1\cap Y_2)$ where $Y_1=\ov{X_1}\setminus X_1$ and $Y_2=\ov{X_2}\setminus X_2$. Since $Y_1$ and $Y_2$ have Morley rank at most 1 by Corollary \ref{corY2}, we obtain $\rk\ov{X}<2$. \epreu
\ble\label{plangen1} We assume that $X$ has Morley degree 1, and that $Y$ is another definable subset of $G$ of Morley rank 2 and Morley degree 1. If $X\cap Y$ has Morley rank 2, then $\ov{X}=\ov{Y}$. \ele
\bpreu Let $g\in G$. If $g$ belongs to $\ov{X\cap Y}$, then we have $\rk\Lm(g,X\cap Y)=1$. Since $X$ has Morley degree 1 and $X\cap Y$ has Morley rank 2, the set $X\setminus Y$ has Morley rank at most 1, and the set $\Lm(g,X\setminus Y)$ has Morley rank at most 1. Thus $\Lm(g,X)$ has Morley rank 1, and $g$ belongs to $\ov{X}$.
Conversely, if $g\in\ov{X}$, then $\Lm(g,X)$ has Morley rank 1, so $\Lm(g,X\cap Y)\subseteq \Lm(g,X)$ has Morley rank at most 1. Then Lemma \ref{lemmaxdl} gives $g\in\ov{X\cap Y}$. This shows that $\ov{X\cap Y}=\ov{X}$. By the same way, we obtain $\ov{X\cap Y}=\ov{Y}$, so $\ov{X}=\ov{Y}$. \epreu
For each $a\in G$, let $\Lm(a)=\Lm(a,G)$ be the (definable) set of lines containing $a$.
Moreover, we note that $\Lm(1)=\Bm$.
\ble\label{lemrk21} Let $\Lambda_0$ be a definable subset of $\Lambda$. If $\rk\cup\Lambda_0=2$, then we have $\rk(\Lm(g)\cap\Lambda_0)\leq 1$ for each $g\in G$.
Moreover, if further $\rk\Lambda_0=2$, then
the set $\{g\in G~|~\rk(\Lm(g)\cap\Lambda_0)=1\}$ has Morley rank 2. \ele
\bpreu
We show that $\rk(\Lm(g)\cap\Lambda_0)\leq 1$ for each $g\in G$. Let $g\in G$ and $l_g:\cup(\Lm(g)\cap\Lambda_0)\setminus\{g\}\to \Lm(g)\cap\Lambda_0$ be the map defined by $l_g(x)=l(g,x)$. Since each line has Morley rank 1, the preimage of each element of $\Lm(g)\cap\Lambda_0$ has Morley rank 1. Consequently, we have $$\rk (\Lm(g)\cap\Lambda_0)= \rk\cup(\Lm(g)\cap\Lambda_0)-1\leq \rk\cup\Lambda_0-1=1$$ as desired.
We suppose further that $\rk\Lambda_0=2$, and
we show that $\{g\in G~|~\rk(\Lm(g)\cap\Lambda_0)=1\}$ has Morley rank 2.
Let $U=\cup\Lambda_0$, $A=\{(u,\lambda)\in U\times \Lambda_0~|~u\in \lambda\}$ and $f:A\to \Lambda_0$ be the map defined by $f(u,\lambda)=\lambda$. Then $A$ and $f$ are definable, and the preimage $f^{-1}(\lambda)$ of each $\lambda\in \Lambda_0$ has Morley rank $\rk \lambda=1$, so $\rk A=1+\rk\Lambda_0=3$. Now let $h:A\to U$ be the map defined by $h(u,\lambda)=u$. It is a definable map, and the preimage $h^{-1}(u)$ of each $u\in U$ has Morley rank either 0, or 1 by the previous paragraph.
But the preimage of $U_0=\{u\in U~|~\rk h^{-1}(u)=0\}$ has Morley rank $$\rk h^{-1}(U_0)=\rk U_0\leq \rk U=2<\rk A$$
so the preimage of $U_1=\{u\in U~|~\rk h^{-1}(u)=1\}$ has Morley rank 3. Hence we obtain $\rk U_1=3-1=2$. Moreover, we note that
$$U_1=\{u\in U~|~\rk(\Lm(u)\cap\Lambda_0)=1\}=\{g\in G~|~\rk(\Lm(g)\cap\Lambda_0)=1\}$$
so $\{g\in G~|~\rk(\Lm(g)\cap\Lambda_0)=1\}$ has Morley rank 2. \epreu
\bpro\label{prodeg1} Let $X$ be a definable subset of $G$ of Morley rank 2 and Morley degree 1. Then $\rk\ov{X}=2$ if and only if $\Lambda_X$ has Morley rank 2.
In this case, $\Lambda_X$ and $\ov{X}$ have Morley degree 1, and $\ov{X}$ contains a generic definable subset of $X$. \epro
\bpreu We consider the definable set
$A=\{(x,\lambda)\in \ov{X}\times \Lambda_X~|~x\in \lambda\}$ and the definable maps $l_1:A\to \ov{X}$ and $l_2:A\to \Lambda_X$ defined by $l_1(x,\lambda)=x$ and $l_2(x,\lambda)=\lambda$ respectively. By Proposition \ref{lemdm1}, the preimage $l_1^{-1}(g)$ of each element $g$ of $\ov{X}$ has Morley rank 1, so $\rk A=1+\rk\ov{X}$. Moreover, the preimage $l_2^{-1}(\lambda)$ of each $\lambda\in\Lambda_X$ has Morley rank at most 1, so $\rk A\leq 1+\rk \Lambda_X$. Then we obtain $\rk\ov{X}\leq \rk\Lambda_X$. In particular, it follows from Lemma \ref{lemrkdm2} that if $\rk\ov{X}=2$, then $\rk\Lambda_X=2$. Hence we may assume that $\rk\Lambda_X=2$.
At this stage, Lemma \ref{lemrkdm2} gives $\rk\cup\Lambda_X=2$, and by Lemma \ref{lemrk21} and Proposition \ref{lemdm1}, we obtain $\rk\ov{X}=2$. Moreover, it follows from Corollary \ref{corY2} that $\ov{X}$ has Morley degree 1 and that $X\cap\ov{X}$ is a generic definable subset of $X$ contained in $\ov{X}$.
We show that the Morley degree of $\Lambda_X$ is 1. Let $l_0:\{(x,y)\in X\times X~|~x\neq y\}\to \Lambda$ be the definable map defined by $l_0(x,y)=l(x,y)$. Since the Morley degree of $X$ is 1,
the one of $\{(x,y)\in X\times X~|~x\neq y\}$ is 1 too. For each $\lambda\in\Lambda_X$, we have $\rk l_0^{-1}(\lambda)=\rk((\lambda\cap X)\times (\lambda\cap X))=2$. Since $\rk\Lambda_X=2$, we obtain
$$\rk l_0^{-1}(\Lambda_X)=2+\rk\Lambda_X=4=\rk\{(x,y)\in X\times X~|~x\neq y\}$$
and since the Morley degree of $\{(x,y)\in X\times X~|~x\neq y\}$ is 1, the Morley degree of $l_0^{-1}(\Lambda_X)$ is 1 too. Now the Morley degree of $\Lambda_X$ is 1 by Fact \ref{deg1}. \epreu
\ble\label{linederive} Let $g$ be a nontrivial element such that $g=[u,v]$ for $(u,v)\in G\times G$.
Then we have $\{x\in G~|~[x,v]=g\}=C_G(v)u$ and $\{y\in G~|~[u,y]=g\}=C_G(u)v$. In particular, they are two lines and have Morley rank 1 and Morley degree 1. \ele
\bpreu The equalities are obvious. Moreover, by Fact \ref{bad}, the sets $C_G(v)u$ and $C_G(u)v$ are two lines, and they have Morley rank 1 and Morley degree 1. \epreu
\ble\label{lem4} For each $a\in G$, the set $a^G\cap B$ has exactly one element. \ele
\bpreu We may assume $a\neq 1$. By Fact \ref{bad} (5), there is $g\in G$ such that $a^g$ belongs to $B$. If $a^h\in B$ for $h\in G$, then $a$ is a nontrivial element of $B^{g^{-1}}\cap B^{h^{-1}}$. By Fact \ref{bad} (4), we obtain $B^{g^{-1}}=B^{h^{-1}}$, and $h^{-1}g$ belongs to $N_G(B)=B$. But $B$ is abelian (Fact \ref{bad} (2)), so $h^{-1}g$ centralizes $a^h$, and $a^h=(a^h)^{h^{-1}g}=a^g$. Hence $a^G\cap B=\{a^g\}$. \epreu
The following result isolates a step of the proof of Theorem \ref{thplane}. Its proof and the one of Theorem \ref{thplane} were originally a lot more complicated, and Bruno Poizat provided a simplification.
For each $g\in G$, we consider the following definable subset of $G$:
$$X(g)=\{x\in G~|~\exists y\in G,~[x,y]=g\}$$
\bpro\label{proX2} For each nontrivial element $g$ of $G$, the set $X(g)$ has Morley rank at most 2. \epro
\bpreu We assume toward a contradiction that $X(g)$ has Morley rank 3. Then the Morley rank of $X(g^z)$ is 3 for each $z\in G$. We recall that, by Fact \ref{bad}, the conjugacy class $g^G$ of $g$ has Morley rank $\rk g^G=\rk G-\rk C_G(g)=2$
We consider $V=\{(x,y)\in G\times G~|~[x,y]\in g^G\}$ and the definable surjective map $f:V\to g^G$ defined by $f(x,y)=[x,y]$.
For each $z\in G$, we have $$f^{-1}(g^z)=\{(x,y)\in G\times G~|~[x,y]=g^z\}$$ and by Lemma \ref{linederive}, this set has Morley rank $\rk f^{-1}(g^z)=\rk X(g^z) +1=3 +1=4$, so $\rk V=4+\rk g^G=6$. Since $G\times G$ is a connected group of Morley rank 6, the set $V$ is a definable generic subset of $G\times G$, and there is $(x,y)\in V$ such that $(y,x)$ belongs to $V$. Thus $[x,y]\in g^G$ and its inverse $[y,x]\in g^G$ are conjugate, and they are equal by Lemma \ref{lem4}, contradicting that $G$ has no involution (Fact \ref{bad} (6)). \epreu
\btheo\label{thplane} There is a plane in $G$. \etheo
\bpreu It is sufficient to show that there is a definable subset $X$ of $G$ satisfying the following properties: \be \item its Morley rank is 2 and its Morley degree is 1, \item $\Lambda_X$ has Morley rank 2. \ee Indeed, by Proposition \ref{prodeg1}, for such a subset $X$, the set $\ov{X}$ has Morley rank 2 and Morley degree 1, and it contains a generic definable subset of $X$. At this stage, Lemma \ref{plangen1} shows that $Y=\ov{X}$ is a plane.
We fix a nontrivial element $g$ such that $g=[u,v]$ for $(u,v)\in G\times G$.
{\em 1. For each $x\in X(g)$, there are infinitely many lines containing $x$ and contained in $X(g)$.}
Since $x$ belongs to $X(g)$, there is $y\in G$ such that $[x,y]=g$. We note that, since $g$ is nontrivial, $x$ and $y$ are nontrivial and we have $C_G(x)\neq C_G(y)$. In particular, $C_G(x)y$ is a line, and it does not contain 1. Thus, for each $c\in C_G(x)$, the set $l_c=C_G(cy)x$ is a line, and by Lemma \ref{linederive}, we have $[r,cy]=[x,cy]=[x,y]=g$ for each $r\in l_c$. So $l_c$ is a line containing $x$ and contained in $X(g)$.
If $l_c=l_d$ for two elements $c$ and $d$ of $C_G(x)$, then we have $C_G(cy)=C_G(dy)$, and $C_G(cy)$ is a line containing $cy$ and $dy$. But $C_G(x)y$ is another line containing $cy$ and $dy$, and we have $C_G(x)y\neq C_G(cy)$ because $C_G(x)y$ does not contain $1$. Hence Lemma \ref{lem3} gives $c=d$,
and $\{l_c\in\Lambda~|~c\in C_G(x)\}$ is an infinite family of lines containing $x$ and contained in $X(g)$.
{\em 2. $\rk X(g)=2$.}
By 1., the set $X(g)$ contains infinitely many lines, so it has Morley rank at least 2 (Lemma \ref{lemdegfin}), and by Proposition \ref{proX2}, it has Morley rank 2.
{\em 3. $\rk \Lambda_{X(g)}=2$.}
By Lemma \ref{lemrkdm2}, the set $\Lambda_{X(g)}$ has Morley rank at most 2. Since $X(g)$ is infinite by 2., for each positive integer $n$ we can find $n$ distinct elements $x_1,\ldots,x_n$ in $X(g)$.
By 1., the set $\Lambda_i=\{\lambda\in \Lambda_{X(g)}~|~x_i\in\lambda\}$ is infinite for each $i$. We may assume that its Morley rank is 1 for each $i$. Then, since there are finitely many lines containing two distinct elements among $x_1,\ldots,x_n$ (Lemma \ref{lem3}), the union $\cup_{i=1}^n\Lambda_i$ has Morley rank 1 and Morley degree at least $n$. This implies that $\Lambda_{X(g)}$ does not have Morley rank 1, so $\rk \Lambda_{X(g)}=2$.
{\em 4. Conclusion.}
By 2., the set $X(g)$ has Morley rank 2. Let $d$ be its Morley degree. Then $X(g)$ is the disjoint union of definable subsets $X_1,\ldots,X_d$ of Morley rank 2 and Morley degree 1.
For each element $\lambda$ of $\Lambda_{X(g)}$, since $\lambda\cap X(g)$ is infinite and since $\lambda$ has Morley rank 1 and Morley degree 1, there is a unique $i\in\{1,\ldots,d\}$ such that $\lambda\cap X_i$ is infinite, that is $\lambda\in \Lambda_{X_i}$. Thus, each $\lambda\in\Lambda_{X(g)}$ belongs to a unique definable set $\Lambda_{X_i}$ for $i\in\{1,\ldots,d\}$. Hence $\Lambda_{X(g)}$ is the disjoint union of $\Lambda_{X_1},\ldots,\Lambda_{X_d}$, and there exists $i\in\{1,\ldots,d\}$ such that $\rk\Lambda_{X_i}=2$. Now the set ${X_i}$ satisfies the conditions (1) and (2) of the beginning of our proof, so $\ov{X_i}$ is a plane. \epreu
\section{A projective space ?}\label{secfin}
In this section, we analyze planes. We remember that, by Theorem \ref{thplane}, the group $G$ has a plane, and that by Corollary \ref{planedeg1}, any plane has Morley degree 1. The initial goal of this section was to show that, if $X$ and $Y$ are two distinct planes, then $\Lambda_X\cap \Lambda_Y$ has a unique element. However, along the way, we obtain our final contradiction.
\bdefi For each line $\lambda$, we consider the following subset of $\Lambda$:
$$\Lm(\lambda)=\{m\in \Lambda~|~\lambda\cap m\ {is\ not\ empty}\}$$ \edefi
\ble\label{lemlml} For any line $\lambda$, the set $\Lm(\lambda)$ is definable, it has Morley rank 3 and Morley degree 1. \ele
\bpreu We consider the definable map $f:\lambda\times (G\setminus \lambda)\to \Lm(\lambda)\setminus \{\lambda\}$ defined by $f(x,g)=l(x,g)$. By Lemma \ref{lem3}, for each $m\in \Lm(\lambda)\setminus \{\lambda\}$, there is a unique element $x$ in $\lambda\cap m$. Moreover, for any $g\in G\setminus \lambda$, we have $f(x,g)=m$ if and only if $g\in m\setminus\{x\}$. Consequently we have $\rk f^{-1}(m)=\rk m=1$, and $$\rk\Lm(\lambda)=\rk(\lambda\times (G\setminus \lambda))-1=3$$ Furthermore, since $\lambda$ and $G$ have Morley degree 1, the Morley degree of $\lambda\times G$ and $\lambda\times (G\setminus \lambda)$ is 1, and the Morley degree of $\Lm(\lambda)\setminus \{\lambda\}$ and $\Lm(\lambda)$ is 1 too (Fact \ref{deg1}). \epreu
\ble\label{lemplan1} Let $X$ be a plane, and $\lambda\in\Lambda_X$. Then $\Lm(\lambda)\cap\Lambda_X$ has Morley rank 2. \ele
\bpreu Since $\lambda$ belongs to $\Lambda_X$, the set $\lambda\cap X$ is infinite, and since $\lambda$ is a line, we have $\rk(\lambda\cap X)=1$.
We consider the definable set $$\Am=\{(x,m)\in (\lambda\cap X)\times \Lambda_X~|~m\neq \lambda,~x\in m\}$$ and the definable maps $p:\Am\to \lambda\cap X$ and $q:\Am\to\Lambda_X$ defined by $p(x,m)=x$ and $q(x,m)=m$ respectively. By Proposition \ref{lemdm1}, the set $p^{-1}(x)$ has Morley rank 1 for each $x\in \lambda\cap X$, so $\rk\Am=1+\rk(\lambda\cap X)=2$.
Moreover, each $m\in \Lambda_X\setminus\{\lambda\}$ contains at most one element of $\lambda$ (Lemma \ref{lem3}), so $q$ is an injective map and its image has Morley rank $\rk\Am=2$. But the image of $q$ is contained in $(\Lm(\lambda)\cap\Lambda_X)\setminus\{\lambda\}$, and we have $\rk\Lambda_X\leq 2$ (Lemma \ref{lemrkdm2}), hence $\Lm(\lambda)\cap\Lambda_X$ has Morley rank 2. \epreu
\ble\label{leminterline} Let $\lambda_1$ and $\lambda_2$ be two distinct lines. Then $\Lm(\lambda_1)\cap\Lm(\lambda_2)$ has Morley rank 2 and Morley degree 1. \ele
\bpreu Let $A=\{(x,y)\in \lambda_1\times \lambda_2~|~x\not\in\lambda_1,\, y\not\in\lambda_2\}$, and let $f:A\to (\Lm(\lambda_1)\cap\Lm(\lambda_2))\setminus\{\lambda_1,\lambda_2\}$ be the map defined by $f(x,y)=l(x,y)$. This map is definable and bijective by Lemma \ref{lem3}. Since $\lambda_1$ and $\lambda_2$ are two lines, the sets $\lambda_1\times \lambda_2$ and $A$ have Morley rank 2 and Morley degree 1, and since $f$ is a definable bijection, $\Lm(\lambda_1)\cap\Lm(\lambda_2)$ has Morley rank 2 and Morley degree 1. \epreu
\bpro\label{proDmX} If $X$ and $Y$ are two distinct planes, then $\Lambda_X\cap \Lambda_Y$ has at most one element. \epro
\bpreu Suppose toward a contradiction that $\lambda_1$ and $\lambda_2$ are two distinct elements of $\Lambda_X\cap \Lambda_Y$. By Lemma \ref{lemplan1}, the sets $\Lm(\lambda_1)\cap\Lambda_X$ and $\Lm(\lambda_2)\cap\Lambda_X$ have Morley rank 2. But $\Lambda_X$ has Morley rank 2 and Morley degree 1 by Proposition \ref{prodeg1},
hence $\Lm(\lambda_1)\cap\Lm(\lambda_2)\cap\Lambda_X$ has Morley rank 2. By the same way, $\Lm(\lambda_1)\cap\Lm(\lambda_2)\cap\Lambda_Y$ has Morley rank 2. Thus, since $\Lm(\lambda_1)\cap\Lm(\lambda_2)$ has Morley rank 2 and Morley degree 1 (Lemma \ref{leminterline}), the set $\Lambda_X\cap \Lambda_Y$ has Morley rank 2.
Since $\Lambda_X\cap \Lambda_Y$ is infinite, the set $U=\cup(\Lambda_X\cap \Lambda_Y)$ has Morley rank at least 2 by Lemma \ref{lemdef12}, and since $U$ is contained in $\cup\Lambda_X$, its Morley rank is exactly 2 (Lemma \ref{lemrkdm2}).
Now the set $Z=\{g\in G~|~\rk(\Lm(g)\cap\Lambda_X\cap \Lambda_Y)=1\}$ has Morley rank 2 by Lemma \ref{lemrk21}. But Proposition \ref{lemdm1} says that $Z$ is contained in $X\cap Y$, hence $X\cap Y$ has Morley rank 2 and Lemma \ref{plangen1} gives $X=Y$, a contradiction. \epreu
From now on, we try to show that the set $\Lambda_X\cap \Lambda_Y$ has exactly one element. However, the final contradiction will appear earlier.
\bco\label{coraXb} Let $X$ be a plane and $(a,b)\in G\times G$. Then the following assertions are equivalent: \bi \item $aXb=X$ \item $a\Lambda_X b=\Lambda_X$ \item $aXb\cap X$ has Morley rank 2. \ei \eco
\bpreu We note that $aXb$ is a plane, and that $a\Lambda_X b=\Lambda_{aXb}$. If $aXb\cap X$ has Morley rank 2, then $aXb=X$ by Lemma \ref{plangen1}, and if $aXb=X$, then we have $a\Lambda_X b=\Lambda_{aXb}=\Lambda_X$. Moreover, if $a\Lambda_X b=\Lambda_X$, then we have $\Lambda_{aXb}=\Lambda_X$ and $aXb=X$ by Proposition \ref{proDmX}, so $aXb\cap X=X$ has Morley rank 2. \epreu
By Fact \ref{factNes}, if $A$ is a Borel subgroup distinct from $B$, then $\rk(ABA)=3$. The following result is slightly more general, and its proof is different.
We recall that, if a group $H$ of finite Morley rank acts definably on a set $E$, then the {\em stabilizer} of any definable subset $F$ of $E$ is defined to be
$$\Stab F=\{h\in H~|~\rk((h\cdot F)\Delta F)<\rk (F)\}$$ where $\Delta$ stands for the symmetric difference. It is a definable subgroup of $H$ by \cite[Lemma 5.11]{bn1}.
\ble\label{lemrkABC} Let $A$ and $C$ be two Borel subgroups distinct from $B$. Then $\rk(ABC)=3$. \ele
\bpreu We consider the action of $G$ on itself by left multiplication. Then we have $b\cdot BC=BC$ for each $b\in B$, so $B$ is contained in $\Stab(BC)$.
We assume toward a contradiction that $C$ is contained in $\Stab(BC)$. Since $BC$ has Morley rank 2 and Morley degree 1 (Fact \ref{factNes}), we have $\rk(cBC\setminus BC)\leq 1$ for each $c\in C$, and since $\rk C=1$, we obtain $\rk(CBC\setminus BC)\leq 2$ and $\rk(CBC)=2$, contradicting Fact \ref{factNes}. Consequently, $C$ is not contained in $\Stab(BC)$, and since $\Stab(BC)$ contains $B$, Fact \ref{bad} implies that $\Stab(BC)=B$.
We assume toward a contradiction that $\rk(ABC)\neq 3$. Since $\rk(BC)=2$, we have $\rk(ABC)=2$ and $ABC$ is a disjoint union of finitely many definable subsets $E_1,\ldots,E_k$ of Morley rank 2 and Morley degree 1. For each $a\in A$, the set $aBC$ has Morley rank $\rk (BC)=2$ and Morley degree 1, so there exists a unique $i\in\{1,\ldots,k\}$ such that $\rk(aBC\cap E_i)=2$. Since $A$ is infinite, there are $i\in\{1,\ldots,k\}$ and two distinct elements $a$ and $a'$ of $A$ such that $\rk(aBC\cap E_i)=\rk(a'BC\cap E_i)=2$. Since $E_i$ has Morley degree 1, the Morley rank of $aBC\cap a'BC$ is 2, and we obtain $\rk(a'^{-1}aBC\cap BC)=2$. But $BC$ has Morley degree 1, hence $a'^{-1}a$ belongs to $\Stab(BC)=B$. Thus $a'^{-1}a$ belongs to $A\cap B=\{1\}$ (Fact \ref{bad} (4)), contradicting that $a$ and $a'$ are distinct. So we have $\rk(ABC)=3$, as desired. \epreu
\bco\label{corBABC} Let $A$ and $C$ be two distinct Borel subgroups. Then $\rk(BA\cap BC)=1$. \eco
\bpreu We may assume $A\neq B$ and $C\neq B$. By Fact \ref{factNes}, we have $$1=\rk B\leq \rk(BA\cap BC)\leq \rk(BA)=2$$ We assume toward a contradiction that $\rk(BA\cap BC)=2$. Since $BC$ has Morley rank 2 and Morley degree 1 (Fact \ref{factNes}), the set $E=BC\setminus BA$ has Morley rank at most 1. Consequently, $EA$ has Morley rank at most $\rk E+\rk A=2$, and since $(BA\cap BC)A\subseteq BA$ has Morley rank 2, we obtain $\rk(BCA)=\rk(EA\cup(BA\cap BC)A)=2$, contradicting that $BCA$ has Morley rank 3 (Lemma \ref{lemrkABC}). \epreu
\ble\label{lemborstab} For any plane $X$, we have $BX\neq X$ and $XB\neq X$. \ele
\bpreu We assume toward a contradiction that $BX=X$ for a plane $X$. Let $x\in X$. Since $X$ is a plane, Proposition \ref{lemdm1} gives $\rk(\Lm(x,X)\cap \Lambda_X)=1$, so $\Lm(x,X)\cap \Lambda_X$ is infinite. But each line containing $x$ has the form $B^ux$ for $u\in G$, hence there exist $u\not\in B$ and $v\not\in B$ such that $B^u\neq B^v$, and such that $B^ux$ and $B^vx$ belong to $\Lm(x,X)\cap \Lambda_X$. In particular, there is a co-finite subset $S$ of $B$ such that $S^ux$ and $S^vx$ are contained in $X$.
Now, since $BX=X$, the sets $BS^ux$ and $BS^vx$ are contained in $X$. By Fact \ref{factNes}, the set $BB^{u}$, and so $Bu^{-1}B$, has Morley rank 2, and since $Bu^{-1}(B\setminus S)$ is a finite union of lines, the set $Bu^{-1}(B\setminus S)$ has Morley rank 1 (Lemma \ref{lemdegfin}), and $Bu^{-1}S$ has Morley rank 2. Thus, the sets $BS^ux=Bu^{-1}Sux$ and $BS^vx=Bv^{-1}Svx$ are subsets of $X$ of Morley rank 2, and since the Morley degree of $X$ is 1, the set $BS^ux\cap BS^vx$ has Morley rank 2. This implies that $\rk(BB^u\cap BB^v)=2$, contradicting Corollary \ref{corBABC}. Now we have $BX\neq X$ and by the same way, we show that $XB\neq X$. \epreu
\bco\label{cormultleft} For any plane $X$, the stabilizer of $X$ for the action of $G$ on itself by left multiplication is finite. \eco
\bpreu By Corollary \ref{coraXb}, we have $\Stab X=\{a\in G~|~aX=X\}$. If $\Stab X$ is infinite, then it contains a Borel subgroup, contradicting Lemma \ref{lemborstab}. \epreu
\bpro\label{prodefaXb} Let $X$ be a plane. Then for each plane $Y$, there exist a unique $a\in G$ and a unique $b\in G$ such that $Y=aX=Xb$. \epro
\bpreu We fix $\alpha\in G$, and we consider the following definable subset of $\Lambda$:
$$A=\{(\lambda_1,\lambda_2)\in \Lambda\times\Lambda~|~\alpha\in\lambda_1\cap\lambda_2,\,\lambda_1\neq\lambda_2\}$$ We show that $A$ has Morley rank 4 and Morley degree 1.
Let $U=\{(x,y)\in G\times G~|~y\not\in l(x,\alpha)\}$. Then $U$ is a generic definable subset of $G\times G$, and it has Morley rank 6 and Morley degree 1. Let $f:U\to A$ be the definable surjective map defined by $f(x,y)=(l(x,\alpha),l(y,\alpha))$. Since each line has Morley rank 1, the preimage of each $(\lambda_1,\lambda_2)\in A$ has Morley rank $\rk\lambda_1+\rk\lambda_2=2$, and the set $A$ has Morley rank $\rk U-2=4$ and Morley degree 1 (Fact \ref{deg1}).
For each plane $P$, we consider the following definable set
$$A_P=\{(\lambda_1,\lambda_2)\in \Lambda\times\Lambda~|~ \alpha\in\lambda_1\cap\lambda_2,\,\lambda_1\neq\lambda_2,\, \exists a\in G,\,a^{-1}\lambda_1\in \Lambda_P,\,a^{-1}\lambda_2\in \Lambda_P \}$$ We show that the set $A_X$ is a generic definable subset of $A$. Indeed, for each $a\in \alpha X^{-1}$, we have $\alpha\in aX$ and $\rk(\Lm(\alpha)\cap\Lambda_{aX})=1$ by Proposition \ref{lemdm1}, so the definable set
$$L_{aX}=\{(\lambda_1,\lambda_2)\in \Lambda_{aX}\times\Lambda_{aX}~|~ \alpha\in\lambda_1\cap\lambda_2,\,\lambda_1\neq\lambda_2\}$$ has Morley rank $2\rk(\Lm(\alpha)\cap\Lambda_{aX})=2$. But $\alpha X^{-1}$ has Morley rank $\rk X=2$ and it follows from Proposition \ref{proDmX} that $L_{aX}\cap L_{bX}=\emptyset$ for any two elements $a$ and $b$ of $\alpha X^{-1}$ such that $aX\neq bX$. Moreover, for each $a\in \alpha X^{-1}$, there are finitely many elements $b\in \alpha X^{-1}$ such that $aX=bX$ (Corollary \ref{cormultleft}). Hence the set $A_X=\cup_{a\in \alpha X^{-1}}L_{aX}$ has Morley rank $\rk\alpha X^{-1}+2=4$, and it is a generic definable subset of $A$.
By the same way, $A_Y$ is a generic definable subset of $A$, so there exists $(\lambda_1,\lambda_2)\in A_X\cap A_Y$. Thus there exist two elements $u$ and $v$ of $G$ such that two distinct lines $\lambda_1$ and $\lambda_2$ belong to $\Lambda_{uX}\cap\Lambda_{vY}$, and we obtain $uX=vY$ by Proposition \ref{proDmX}, so $Y=aX$ for $a=v^{-1}u$. By the same way, there exists $b\in G$ such that $Y=Xb$.
We show the uniqueness of $a$ and $b$.
Let $S=\{g\in G~|~gX=X\}$. It is a finite subgroup of $G$ by Corollary \ref{cormultleft}. For each $\alpha\in G$, the previous paragraph gives $\beta\in G$ such that $\alpha X=X\beta$. Then, for each $s\in S$, we have $s(\alpha X)=s(X\beta)=X\beta=\alpha X$, and we obtain $s^\alpha X=X$ and $s^\alpha\in S$. Thus any element $\alpha\in G$ normalizes the finite subgroup $S$, and since $G$ is a simple group, $S$ is trivial. This proves the uniqueness of $a$, and by the same way we obtain the uniqueness of $b$. \epreu
By the previous result, the set of planes is $\Pm=\{aX~|~a\in G\}$, and it identifies with $G$. Thus, the set of planes is uniformly definable and has Morley rank 3.
\ble\label{mapalphaaut} There exists $a\in G$ such that $X^a\neq X$. \ele
\bpreu We assume toward a contradiction that $X^a=X$ for each $a\in G$. Then for each $uBv\in \Lambda_X$ and each $a\in G$, we have $$(uBv)^a\in \Lambda_X^a=\Lambda_{X^a}=\Lambda_X$$ Since $\rk\Lambda_X=2$ (Proposition \ref{prodeg1}), the line $uBv$ is a Borel subgroup (Lemma \ref{lem2}), and by conjugacy of Borel subgroups, we obtain $\Lambda_X=\Bm$. Now we have $\cup\Lambda_X=G$, so $\rk\cup\Lambda_X=3$, contradicting Lemma \ref{lemrkdm2}. \epreu
From now on, we are ready for the final contradiction. Initially, it was more complicated, but Poizat proposed a simplification by introducing the inverted plane.
\bpreu First we note that for each plane $Y$, the set $y^{-1}Y$ is a plane containing 1, and the set $Y^{-1}$ is a plane too. We fix a plane $X$ containing 1. By Proposition \ref{prodefaXb}, there is a bijective map $\mu:G\to G$ defined by $xX=X\mu(x)$, and $\mu$ is definable since the set $\Pm$ of planes is uniformly definable.
Moreover, for each $(a,b)\in G\times G$, we have $X\mu(ab)=abX=aX\mu(b)=X\mu(a)\mu(b)$, so $\mu$ is an automorphism of $G$.
Since $X=\ov{X}$ contains $1$, there are infinitely many Borel subgroups in $\Lambda_X$. Let $B_1$ and $B_2$ be two distinct Borel subgroups belonging to $\Lambda_X$. Then $B_1$ and $B_2$ belong to $\Lambda_{X^{-1}}$ too, and we have $X=X^{-1}$ by Proposition \ref{proDmX}. By the same way, since the plane $x^{-1}X$ contains 1 for each $x\in X$, we have $x^{-1}X=(x^{-1}X)^{-1}=X^{-1}x=Xx$ for each $x\in X$. Thus $\mu(x^{-1})=x$ for each $x\in X$, and since $X=X^{-1}$, we obtain $\mu^2(x)=x$ for each $x\in X$.
But $X$ is a definable subset of $G$ of Morley rank 2, hence $G$ is generated by $X$, and $\mu$ is an involutive automorphism of $G$. Thus $\mu$ is the identity map by Fact \ref{invoautobad}, contradicting Lemma \ref{mapalphaaut}. \epreu
\bre After Lemma \ref{mapalphaaut} we were ready for a new step to provide a structure of projective space over $G$, which was the initial goal of our section. Indeed, in the first version of this paper, we have shown that, if $X$ and $Y$ are two distinct plane, then $\Lambda_X\cap \Lambda_Y$ has a unique element. \ere
\end{document} |
\begin{document}
\begin{centering}
{\huge \textbf{The structure of base phi expansions}}
{\bf \large F.~Michel Dekking}
{CWI, Amsterdam, and DIAM, Delft University of Technology, Faculty EEMCS,\\ P.O.~Box 5031, 2600 GA Delft, The Netherlands.}
{\footnotesize \it Email: F.M.Dekking@TUDelft.nl}
\end{centering}
\begin{abstract}
\noindent In the base phi expansion any natural number is written uniquely as a sum of powers of the golden mean with coefficients 0 and 1, where it is required that the product of two consecutive digits is always 0. We tackle the problem of describing how these expansions look like. We classify the positive parts of the base phi expansions according to their suffices, and the negative parts according to their prefixes, specifying the sequences of occurrences of these digit blocks. Here the situation is much more complex than for the Zeckendorf expansions, where any natural number is written uniquely as a sum of Fibonacci numbers with coefficients 0 and 1, where, again, it is required that the product of two consecutive digits is always 0. In a previous work we have classified the Zeckendorf expansions according to their suffices. It turned out that if we consider the suffices as labels on the Fibonacci tree, then the numbers with a given suffix in their Zeckendorf expansion appear as generalized Beatty sequences in a natural way on this tree.
We prove that the positive parts of the base phi expansions are a subsequence of the sequence of Zeckendorf expansions, giving an explicit formula in terms of a generalized Beatty sequence. The negative parts of the base phi expansions no longer appear lexicographically. We prove that all allowed digit blocks appear, and determine the order in which they do appear. \end{abstract}
\quad {\footnotesize Keywords: Base phi; Zeckendorf expansion; Generalized Beatty sequence, Wythoff sequence }
\section{Introduction}
Let the golden mean be given by $\varphi:=(1+\sqrt{5})/2$.\\
Ignoring leading and trailing zeros, any natural number $N$ can be written uniquely as
$$N= \sum_{i=-\infty}^{\infty} d_i \varphi^i,$$
with digits $d_i=0$ or 1, and where $d_id_{i+1} = 11$ is not allowed.
As usual, we denote the base phi expansion of $N$ as $\beta(N)$, and we write these expansions with a radix point as
$$\beta(N) = d_{L}d_{L-1}\dots d_1d_0\cdot d_{-1}d_{-2} \dots d_{R+1}d_R.$$
We define $$\beta^+(N)=d_{L}d_{L-1}\dots d_1d_0\; {\;\rm and\;}\; \beta^-(N)=d_{-1}d_{-2} \dots d_{R+1}d_R.$$
So $\beta(N)=\beta^+(N)\cdot\beta^-(N)$. For example, $\beta(2)=10\cdot 01$, and $\beta(3)=100\cdot 01$.
This paper deals with the following question: what are the words of 0's and 1's that can occur as digit blocks in the base phi expansion $N$, and for which numbers $N$ do they occur?
In Section \ref{sec:phi}, we perform this task for the suffices of the $\beta^+$-part of the base phi expansions, and in Section \ref{sec:neg} for the complete $\beta^-\!$-part of the base phi expansions, and the prefixes of the $\beta^-\!$-part of length at most 3.
In Section \ref{sec:embed}, we establish in Theorem \ref{th:Zeckphi} a relationship between the base phi expansions and Zeckendorf expansions, also known as Fibonacci representations. This will permit us to exploit the results of the paper \cite{Dekk-Zeck-structure} in Section \ref{sec:phi}. See the paper \cite{Frougny-Saka} for a less direct approach, in terms of two-tape automata.
In Section \ref{sec:RST} we recall the recursive structure of base phi expansions, and derive some tools from this which will be useful in the final two sections.
In Section \ref{sec:closer} we take a closer look at the Lucas intervals.
In Section \ref{sec:GBS} we introduce generalized Beatty sequences, which for the base phi expansion take over the role played by arithmetic sequences in the classical expansions in base $b$, where $b$ is an integer larger than 1.
We end this introduction by pointing out that there is a neat way to obtain $N$ from the $\beta^+(N)$-part of $\beta(N)$, without knowing the $\beta^-(N)$-part. If $\beta(N)=\beta^+(N)\cdot\beta^-(N)$ is the base phi expansion of a natural number $N$, then $N=\lceil \beta^+(N)\rceil$. Here $\lceil \cdot\rceil$ is the ceiling function.
For a proof, add the maximum number of powers corresponding to $\beta^-(N)$, taking into account that no 11 appears. This is bounded by the geometric series starting at $\varphi^{-1}$ with common ratio $\varphi^{-2}$, i.e., by $\varphi^{-1}/(1-\varphi^{-2})=1.$
\section{Embedding base phi into Zeckendorf}\label{sec:embed}
We define the Lekkerkerker-Zeckendorf expansion. Let $(F_n)$ be the Fibonacci numbers. Let $\ddot{F}_0=1, \ddot{F}_1=2, \ddot{F}_2=3,\dots$ be the twice shifted Fibonacci numbers, defined by $\ddot{F}_i=F_{i+2}$.
Ignoring leading and trailing zeros, any natural number $N$ can be written uniquely as
$$N= \sum_{i=0}^{\infty} e_i \ddot{F}_i,$$
with digits $e_i=0$ or 1, and where $e_ie_{i+1} = 11$ is not allowed.
We denote the Zeckendorf expansion of $N$ as $Z(N)$.
Let $V$ be the generalized Beatty sequence (cf. \cite{GBS}) defined by
$$ V(n) = 3\lfloor n\varphi \rfloor + n +1.$$
\noindent Here $\lfloor \cdot \rfloor$ denotes the floor function, and $(\lfloor n\varphi\rfloor)$ is the well known lower Wythoff sequence.
We define the function $S$ by
$$S(n)=\max\{k\in \mathbb{N}: V(k)\le n\}-1.$$
\begin{theorem}\label{th:Zeckphi} \noindent For all $N\ge 0$ $$\beta^+(N)=Z(N+S(N)).$$ \end{theorem}
\noindent This theorem will be proved in the Section \ref{sec:Pf-Zeckphi}.
The basis for the embedding of the $\beta^+(N)$ into the collection of Zeckendorf words is the following analysis.
\subsection{The art of adding 1}\label{sec:add}
It is essential to give ourselves the freedom to write also non-admissible expansions in the form $$\beta(N) = d_{L}d_{L-1}\dots d_1d_0\cdot d_{-1}d_{-2} \dots d_{R+1}d_R.$$
For example, since $\beta(4) =101.01$ and $\beta(2)=10\cdot 01$, we can write \begin{equation}\label{eq:plus1} \beta(5)\doteq \beta(4)+1\doteq 101\cdot01+1\cdot 0\doteq 102\cdot01\doteq 110\cdot02\doteq1000\cdot1001. \end{equation}
Here the $\doteq$-sign indicates that we consider a non-admissible expansion.
It is convenient to generate all Zeckendorf expansions and base phi expansions by repeatedly adding the number 1.
When we compute $\beta(N)+1$ for some number $N$, then, in general, there is a carry both to the left and (two places) to the right. This is illustrated by the example in Equation (\ref{eq:plus1}). Note that there is not only a {\it double carry}, but that we also have to get rid of the 11's, by replacing them with 100's. This is allowed because of the equation $\varphi^{n+2}= \varphi^{n+1}+\varphi^{n}.$ We call this operation a {\it golden mean shift}.
When we compute $Z(N)+1$ for some number $N$, then we have to distinguish between $e_0=0$ and $e_0=1$: $$Z(N)=e_L\dots e_2e_1\,0 {\quad\rm gives\quad} Z(N)+1=e_L\dots e_2e_1\,1$$ and $$Z(N)=e_L\dots e_2e_1\,1 {\quad\rm gives\quad} Z(N)+1\doteq e_L\dots e_2\,10.$$ Here we used the $\doteq$-sign because (several) golden mean shifts might follow, where for the Zeckendorf expansion these are justified by the equation $F_{n+2}=F_{n+1}+F_n$. Note that replacing $e_11+1$ by $10$ follows from 1+1=2 (!).
\noindent For the convenience of the reader we provide a list of the Zeckendorf and base phi expansions of the first 18 natural numbers:
\begin{tabular}{|c|c|c|}
\hline
\; $N^{\phantom{|}}$ & $Z(N)$ & $\beta(N)$ \\[.0cm]
\hline
1\; & \;\;\;\;\;\;\;\,1 & \;${1}\cdot$ \\
2\; & \;\;\;\;\;\;10 & \;\:\,\,$1{0}\cdot01$ \\
3\; &\;\;\; 100 & \;$10{0}\cdot01$ \\
4\; &\;\;\; 101 & \;$10{1}\cdot01$ \\
5\; & \;\;1000 & \;\:\,$100{0}\cdot1001$ \\
6\; & \;\;1001 & \;\:\,$101{0}\cdot0001$ \\
7\; & \;\;1010 & \;$1000{0}\cdot0001$ \\
8\; & 10000 & \;$1000{1}\cdot 0001$ \\
9\; & 10001 & \;$1001{0}\cdot0101$ \\
\hline
\end{tabular}\qquad
\begin{tabular}{|c|c|c|}
\hline
\; $N^{\phantom{|}}$ & $Z(N)$ & $\beta(N)$ \\[.0cm]
\hline
10\; & \;\:10010 & \;$1010{0}\cdot0101$ \\
11\; & \;\:10100 & \;$1010{1}\cdot0101$ \\
12\; & \;\:10101 & \;\,\,$10000{0}\cdot101001$ \\
13\; & 100000 & \;\,\,$10001{0}\cdot001001$ \\
14\; & 100001 & \;\,\,$10010{0}\cdot001001$ \\
15\; & 100010 & \;\,\,$10010{1}\cdot001001$ \\
16\; & 100100 & \;\, $10100{0}\cdot100001$\\
17\; & 100101 & \;\, $101010\cdot000001$\\
18\; & 101000 & \;\, $100000{0}\cdot000001$\\
\hline
\end{tabular}\quad
\subsection{Proof of Theorem \ref{th:Zeckphi}}\label{sec:Pf-Zeckphi}
The essential ingredient of the proof is the following result from \cite{Dekk-phi-FQ}, Theorem 5.1 and Remark 5.4. An alternative, short proof of the first part could be given with the Propagation Principle from Section \ref{sec:RST}.
\begin{proposition}\label{prop:D-numbers} Let $\beta(N)=(d_i(N))$ be the base phi expansion of a natural number $N$. Then:\\[-.3cm]
\hspace*{1.5cm} $d_1d_0\cdot d_{-1}(N)=10\cdot1$ never occurs,\\[.1cm]
\hspace*{1.5cm} $d_1d_0\cdot d_{-1}(N)=00\cdot1$ if and only if $N=3\lfloor n\varphi\rfloor + n + 1$ for some natural number $n$. \end{proposition}
\noindent {\it Proof of Theorem \ref{th:Zeckphi}: } One observes that there are many $\beta(N)$'s such that $\beta^+(N)=Z(N')$ for some $N'$. Moreover, if this is the case, then also $\beta^+(N+1)=Z(N'+1)$, {\it except} if $d_{-1}(N)=1$ in $\beta(N)$. Indeed, as long as $d_{-1}(N)=0$ adding 1 gives the same result for both the Zeckendorf and the positive part of the base phi expansion, as seen in the previous section. However, suppose $$Z(N')=\beta^+(N), {\; \rm and\;} d_{-1}(N)=1.$$ Then, by Proposition \ref{prop:D-numbers}, $d_1d_0\cdot d_{-1}(N)=00\cdot1$, and adding 1 to $N$ gives the expansion $\beta(N+1)$ with digit block $d_1d_0\cdot d_{-1}(N+1)=10\cdot0$. So $\beta^+(N+1)$ ends in exactly the same two digits as $Z(N'+2)$, and in fact $\beta^+(N+1)=Z(N'+2)$. This means that one Zeckendorf expansion has been skipped: that of $N'+1$. Every time a $d_{-1}(N)=1$ occurs, this skipping takes place. Since $Z(0)=\beta^+(0),\dots, Z(5)=\beta^+(5)$, this gives the formula $\beta^+(N)=Z(N+S(N))$, with $S(n)=\max\{k\in \mathbb{N}: 3\lfloor k\varphi\rfloor + k \le n\}$, by the second statement of Proposition \ref{prop:D-numbers}.
$\Box$
\section{The recursive structure of base phi expansions}\label{sec:RST}
The Lucas numbers $(L_n)=(2, 1, 3, 4, 7, 11, 18, 29, 47, 76,123, 199, 322,\dots)$ are defined by $$ L_0 = 2,\quad L_1 = 1,\quad L_n = L_{n-1} + L_{n-2}\quad {\rm for \:}n\ge 2.$$ The Lucas numbers have a particularly simple base phi expansion.
\noindent From the well-known formula $L_{2n}=\varphi^{2n}+\varphi^{-2n}$, and the recursion $L_{2n+1}=L_{2n}+L_{2n-1}$ we have for all $n\ge 1$ \begin{equation}\label{eq:Lm}
\beta(L_{2n}) = 10^{2n}\cdot0^{2n-1}1,\quad \beta(L_{2n+1}) = 1(01)^n\cdot(01)^n. \end{equation} By iterated application of the double carry and the golden mean shift to $\beta(L_{2n+1})+\beta(1)$,\; and a similar operation for $\beta(L_{2n+2}-1)$ (see also the last page of \cite{Dekk-How-to-add}) one finds that for all $n\ge 1$
\begin{equation}\label{eq:Lmplus1}
\beta(L_{2n+1}+1) = 10^{2n+1}\cdot(10)^n01,\quad \beta(L_{2n+2}-1)=(10)^{n+1}\cdot 0^{2n+1}1. \end{equation}
\noindent As in \cite{Dekk-phi-FQ} we partition the natural numbers into Lucas intervals\: $$\Lambda_{2n}:=[L_{2n},\,L_{2n+1}] \quad{\rm and\quad} \Lambda_{2n+1}:=[L_{2n+1}+1,\, L_{2n+2}-1].$$ The basic idea behind this partition is that if
$$\beta(N) = d_{L}d_{L-1}\dots d_1d_0\cdot d_{-1}d_{-2} \dots d_{R+1}d_R,$$ then the left most index $L=L(N)$ and the right most index $R=R(N)$ satisfy
$$L(N)=|R(N)|=2n \;{\rm iff}\; N\in \Lambda_{2n}, \quad L(N)=2n\!+1,\; |R(N)|=2n\!+2 \;{\rm iff}\; N\in \Lambda_{2n+1}.$$ This is not hard to see from the simple expressions we have for the $\beta$-expansions of the Lucas numbers; see also Theorem 1 in \cite{Grabner94}.
To obtain recursive relations, the interval $\Lambda_{2n+1}=[L_{2n+1}+1, L_{2n+2}-1]$ has to be divided into three subintervals. These three intervals are\\[-.8cm]
\begin{align*} I_n:=&[L_{2n+1}+1,\, L_{2n+1}+L_{2n-2}-1],\\ J_n:=&[L_{2n+1}+L_{2n-2},\, L_{2n+1}+L_{2n-1}],\\ K_n:=&[L_{2n+1}+L_{2n-1}+1,\, L_{2n+2}-1]. \end{align*}
\noindent It will be very convenient to use the free group versions of words of 0's and 1's. So, for example, $(01)^{-1}0001=1^{-1}001$.
\begin{theorem}{\bf [Recursive structure theorem]}\label{th:rec}
\noindent{\,\bf I\;} For all $n\ge 1$ and $k=0,\dots,L_{2n-1}$ one has $ \beta(L_{2n}+k) = \beta(L_{2n})+ \beta(k) = 10\dots0 \,\beta(k)\, 0\dots 01.$ \noindent{\bf II} For all $n\ge 2$ and $k=1,\dots,L_{2n-2}-1$ \begin{align*} I_n:&\quad \beta(L_{2n+1}+k) = 1000(10)^{-1}\beta(L_{2n-1}+k)(01)^{-1}1001,\\ K_n:&\quad\beta(L_{2n+1}+L_{2n-1}+k)=1010(10)^{-1}\beta(L_{2n-1}+k)(01)^{-1}0001. \end{align*} Moreover, for all $n\ge 2$ and $k=0,\dots,L_{2n-3}$ $$\hspace*{0.7cm}J_n:\quad\beta(L_{2n+1}+L_{2n-2}+k) = 10010(10)^{-1}\beta(L_{2n-2}+k)(01)^{-1}001001.$$ \end{theorem}
See \cite{Dekk-How-to-add} for a proof of this theorem.
As an illustration of the use of Theorem \ref{th:rec} we shall now prove a lemma that we need in Section \ref{sec:phi}.
\begin{lemma}\label{lem:no} Let $m\ge 1$ be an integer. There are {\bf (a)} no expansions $\beta(N)$ with the digit block $d_{2m}\dots d_0\cdot d_{-1}(N)=10^{2m}\cdot1$, and there are {\bf (b)} no expansions $\beta(N)$ with the digit block $d_{2m+1}\dots d_0\cdot d_{-1}(N)=10^{2m+1}\cdot0$. \end{lemma}
\noindent{\it Proof:} \:{\bf (a)}. The first time $d_{2m}\dots d_0=10^{2m}$ occurs is for $N=L_{2m}$, and then $d_{-1}(N)=0$ (see $\beta(L_{2m})$ formula above). This is also the only occurrence of the digit block $10^{2m}$ at the end of the expansions of the numbers $N$ in $\Lambda_{2m}$. It is also obvious that the digit block $10^{2m}$ will not appear at the end of the expansions of the numbers $N$ in $\Lambda_{2m+1}$.
From part {\bf I} of the Recursive Structure Theorem we see that the digit block $10^{2m}$ at the end of the expansions of the numbers $N$ in $\Lambda_{2m+2}$ only occurs in combination with $d_{-1}(N)=0$.
From part {\bf II} of the Recursive Structure Theorem we will see that the digit block $10^{2m}$ at the end of the expansions of the numbers $N$ in $\Lambda_{2m+3}$ only occurs in combination with $d_{-1}(N)=0$. This is definitely more complicated than this observation for $\Lambda_{2m+2}$. We have to split $\Lambda_{2m+3}$ into the three pieces $I_{m+1}, J_{m+1}$ and $K_{m+1}$. The middle piece $J_{m+1}$ corresponds to numbers in $\Lambda_{2m}$, from which we already know that $d_{2m}\dots d_0(N)=10^{2m}$ implies that $d_{-1}(N)=0$. The numbers $N$ in the first piece, $I_{m+1}$, correspond to numbers in $\Lambda_{2m+1}$ from which the digits $d_{2m+1}d_{2m}=10$ have been replaced by the digits $d_{2m+3}d_{2m+2}d_{2m+1}d_{2m}=1000$. In particular $d_{2m}=0$ excludes any occurrence of $d_{2m}\dots d_0=10^{2m}$. In the same way occurrences of $d_{2m}\dots d_0=10^{2m}$ in $K_{m+1}$ are excluded.
The final conclusion is that both intervals $\Lambda_{2m+2}$ and $\Lambda_{2m+3}$ only contain numbers $N$ for which the occurrence of $10^{2m}$ as end block implies $d_{-1}(N)=0$. In the same way, these properties of $\Lambda_{2m+2}$ and $\Lambda_{2m+3}$ carry over to the two Lucas intervals $\Lambda_{2m+4}$ and $\Lambda_{2m+5}$, and we can finish the proof by induction.
{\bf (b)}. The first time $d_{2m}\dots d_0=10^{2m+1}$ occurs is for $N=L_{2m+1}+1$ in $\Lambda_{2m+1}$ , and then $d_{-1}(N)=1$ (see Equation (\ref{eq:Lmplus1})). This is also the only occurrence of $d_{2m}\dots d_0=10^{2m+1}$ in $\Lambda_{2m+1}$. Moreover, in $\Lambda_{2m+2}$ the word $10^{2m+1}$ does not occur at all as end block. We finish the proof as in Part {\bf (a)}, with the sole difference that now $1010^{2m+1}$ occurring as end block in $\Lambda_{2m+3}$, yields an instance of $10^{2m+1}\cdot 1$ in $\Lambda_{2m+3}$.
$\Box$
It is convenient to have a second version of the Recursive Structure Theorem which involves a higher resemblance between the even Part {\,\bf I\;} case, and the odd Part {\,\bf II\;}. It will also be convenient to have the $\Lambda$-intervals play a more visible role in the recursion. In fact, it is easy to check that the three intervals $I_n, J_n$ and $K_n$ in the Recursive Structure Theorem satisfy $$I_n=\Lambda^{(a)}_{2n-1}:=\Lambda_{2n-1}+L_{2n},\; J_n=\Lambda^{(b)}_{2n-2}:=\Lambda_{2n-2}+L_{2n+1},\; K_n=\Lambda^{(c)}_{2n-1}:=\Lambda_{2n-1}+L_{2n+1}. $$ In this equation we employ the usual notation $A+x:=\{a+x:a\in A\}$ for a set of real numbers $A$ and a real number $x$.
\begin{theorem}{\bf [Recursive structure theorem: 2nd version]}\label{th:rec2}\\ \noindent{\,\bf (i): Odd\;} For all $n\ge 1$ one has\\[-.3cm] $$\Lambda_{2n+1}=\Lambda^{(a)}_{2n-1}\cup\Lambda^{(b)}_{2n-2}\cup\Lambda^{(c)}_{2n-1}, $$ where $\Lambda^{(a)}_{2n-1}=\Lambda_{2n-1}+L_{2n}$,\; $\Lambda^{(b)}_{2n-2}=\Lambda_{2n-2}+L_{2n+1}$, and $\Lambda^{(c)}_{2n-1}=\Lambda_{2n-1}+L_{2n+1}$.\\ We have\\[-.8cm] \begin{subequations} \label{eq:shift-odd} \begin{align}
\beta(N)= & \;1000(10)^{-1}\,\beta(N-L_{2n})\,(01)^{-1}1001& for\; N\in \Lambda^{(a)}_{2n-1}, \\
\beta(N)= & \; 100\,\beta(N-L_{2n+1})\,(01)^{-1}001001& for\; N\in \Lambda^{(b)}_{2n-2},\\
\beta(N)= & \;10\,\beta(N-L_{2n+1})\,(01)^{-1}0001 & for\; N\in \Lambda^{(c)}_{2n-1}. \end{align}\\[-.6cm] \end{subequations} \noindent{\,\bf (ii): Even\;} For all $n\ge 1$ one has\\[-.3cm]
$$\Lambda_{2n+2}=\Lambda^{(a)}_{2n}\cup\Lambda^{(b)}_{2n-1}\cup\Lambda^{(c)}_{2n}, $$
where $\Lambda^{(a)}_{2n}=\Lambda_{2n}+L_{2n+1}$,\; $\Lambda^{(b)}_{2n-1}=\Lambda_{2n-1}+L_{2n+2}$, and $\Lambda^{(c)}_{2n}=\Lambda_{2n}+L_{2n+2}$.\\
We have\\[-.8cm] \begin{subequations} \label{eq:shift-even} \begin{align}
\beta(N)= & \;1000(10)^{-1}\,\beta(N-L_{2n+1})\,(01)^{-1}0001 & for\; N\in \Lambda^{(a)}_{2n},\phantom{x} \label{eq:5a} \\
\beta(N)= & \; 100\,\beta(N-L_{2n+2})\,01 & for\; N\in \Lambda^{(b)}_{2n-1}, \label{eq:5b}\\
\beta(N)= & \;10\,\beta(N-L_{2n+1})\,01 & for\; N\in \Lambda^{(c)}_{2n}.\phantom{x.} \label{eq:5c} \end{align} \end{subequations} \end{theorem}
\noindent{\it Proof:} \:{\bf (i): Odd\;} This is a rephrasing of {Part (II) in Theorem \ref{th:rec}.\\ {\bf (ii): Even\;} We start by showing that the three intervals $\Lambda^{(a)}_{2n},\Lambda^{(b)}_{2n-1},\Lambda^{(c)}_{2n}$ partition $\Lambda_{2n+2}$.
The first number in $\Lambda^{(a)}_{2n}$ is $L_{2n}+L_{2n+1}=L_{2n+2}$, which is the first number of $\Lambda_{2n+2}$. The last number in $\Lambda^{(a)}_{2n}$ is $L_{2n+1}+L_{2n+1}=2L_{2n+1}$.
The first number in $\Lambda^{(b)}_{2n-1}$ is $L_{2n-1}+1+L_{2n+2}=L_{2n-1}+1+L_{2n}+L_{2n+1}=2L_{2n+1}+1$, which indeed, is the successor of the last number in $\Lambda^{(a)}_{2n}$.
The last number in $\Lambda^{(b)}_{2n-1}$ is $L_{2n}-1+L_{2n+2}$, which indeed has successor $L_{2n}+L_{2n+2}$, the first number in $\Lambda^{(c)}_{2n}$. Finally, the last number in $\Lambda^{(c)}_{2n}$ is $L_{2n+1}+L_{2n+2}=L_{2n+3}$, which is the last number in $\Lambda_{2n+2}$.
To prove Equation (\ref{eq:5a}), we first show, using Equation (\ref{eq:Lm}) twice, that this equation is correct for $N=L_{2n+2}$, which is the first number of $\Lambda^{(a)}_{2n}$: \begin{align*} \beta(L_{2n+2}) & = 10^{2n+2}\cdot0^{2n+1}1\\
& = 1000\,0^{2n-1}\cdot0^{2n-2}\,0001 \\
& = 1000(10)^{-1}10^{2n}\cdot0^{2n-1}1\,(01)^{-1}0001 \\
& = 1000(10)^{-1}\beta(L_{2n})\,(01)^{-1}0001 \\
& = 1000(10)^{-1}\beta(L_{2n+2}-L_{2n+1})\,(01)^{-1}0001. \end{align*}
Equation (\ref{eq:5a}) will also be correct for all other $N\in\Lambda^{(a)}_{2n}$, because as above, the digit block $d_Ld_{L-1}d_{L-2}d_{L-3}(N)$ will always be 1000, and the digit block $d_{L-2}d_{L-3}(N-L_{2n+1})$ will always be 10. For the negative digits we have a similar property.
Equation (\ref{eq:5b}) follows directly from the fact that if $N\in \Lambda^{(b)}_{2n-1}$, then \begin{align*} \beta(N-L_{2n+2})+\beta(L_{2n+2}) & = d_{2n-1}\dots d_0\cdot d_{-1}\dots d_{-2n} + 10^{2n+2}\cdot0^{2n+1}1\\
& = d_{2n-1}\dots d_0\cdot d_{-1}\dots d_{-2n} + 100\,0^{2n}\cdot0^{2n}\,01\\
& = 100d_{2n-1}\dots d_0\cdot d_{-1}\dots d_{-2n}01, \end{align*} since the numbers in $\Lambda_{2n-1}$ have a $\beta$-expansion $d_{2n-1}\dots d_0\cdot d_{-1}\dots d_{-2n}$ with $2n$ digits on the left and $2n$ digits on the right. Note that we do not have to use the $\dot{=}$-sign as there are no double carries or golden mean shifts.
Equation (\ref{eq:5c}) follows in the same way.
$\Box$
Lemma \ref{lem:no} is an example of a general phenomenon, which we call the Propagation Principle. It has an extension to combinations of digit blocks which we will give in Lemma \ref{lem:prop}.
The Propagation Principle is closely connected to the following notion. We say an interval $\Gamma$ and a union of intervals $\Delta$ of natural numbers are \emph{$\beta$-congruent modulo $q$} for some natural number $q$ if
$\Delta$ is a disjoint union of translations of $\Lambda$-intervals, such that for all $j=1,\dots,|\Gamma|$,
if $N$ is the $j^{\rm th}$ element of $\Gamma$, and $N'$ is the $j^{\rm th}$ element of $\Delta$, then $$d_{q-1}\dots d_1d_0\cdot d_{-1}\dots d_{-q}(N)=d_{q-1}\dots d_1d_0\cdot d_{-1}\dots d_{-q}(N').$$ We write this as\, $\Gamma\cong \Delta_1\Delta_2\dots\Delta_r \mod q$ when the number of translations of $\Lambda$-intervals in $\Delta$ equals $r$. Note that the definition implies that the $r$ disjoint translations of $\Lambda$-intervals appear in the natural order, and that we refrain from indicating the translations.
Simple examples are $\Lambda_5 \cong \Lambda_3\Lambda_2\Lambda_3 \mod 1$ and $\Lambda_6 \cong \Lambda_4\Lambda_3\Lambda_4 \mod 3$.
Theorem \ref{th:rec2} is a source of many more examples.
An important observation is that if $\Gamma\cong \Delta_1\Delta_2\dots\Delta_r \mod q$ and $\Gamma'\cong \Delta'_1\Delta'_2\dots\Delta'_{r'} \mod q'$, and $\Gamma\cup \Gamma'$ is an interval, then \begin{equation}\label{eq:GG}
\Gamma\Gamma':=\Gamma\cup \Gamma'\cong \Delta_1\Delta_2\dots\Delta_r\Delta'_1\Delta'_2\dots\Delta'_{r'} \mod \min\{q,q'\}. \end{equation}
To keep the formulation and the proof of the following lemma simple, we only formulate it for central digit blocks of length 8 (i.e., $q=4$). In the following, occurrences of digit blocks in $\beta$-expansions have to be interpreted with additional $0$'s added to the left of the expansion.
\begin{lemma}{\bf [Propagation Principle]}\label{lem:prop}
\noindent {\bf (a)}\, Suppose the digit block $d_{3}\dots d_0\cdot d_{-1}\dots d_{-4}$, does not occur in the $\beta$-expansions of the numbers $N=1,2,\dots,17$. Then it does not occur in any $\beta$-expansion.
\noindent {\bf (b)}\, Let $D$ be an integer between 1 and 4. Suppose the digit block $d_{3}\dots d_0\cdot d_{-1}\dots d_{-4}$ occurs in the $\beta$-expansion of $N$ if and only if the digit block $e_{3}\dots e_0\cdot e_{-1}\dots e_{-4}$ occurs in the $\beta$-expansion of the number $N-D$, for $N=D,D+1,\dots,D+17$. Then this coupled occurrence holds for all $N$. \end{lemma}
\noindent{\it Proof:} \:{\bf (a)}\, Let us say that a Lucas interval $\Lambda_m$ satisfies property $\cal D$ if the digit block $d_{3}\dots d_0\cdot d_{-1}\dots d_{-4}$, does not occur in the $\beta$-expansions of the numbers $N$ from $\Lambda_m$. Note that $N=17$ is the last number in $\Lambda_5$, so it is given that the intervals $\Lambda_1,\dots,\Lambda_5$ all satisfy property $\cal D$. Also $\Lambda_6$ satisfies property $\cal D$, by an application of Theorem \ref{th:rec}, Part {\bf I}.
The interval $\Lambda_7= \Lambda^{(a)}_{5}\cup\Lambda^{(b)}_{4}\cup\Lambda^{(c)}_{5}$ satisfies property $\cal D$. For $\Lambda^{(a)}_{5}$, this follows since $\Lambda_5$ satisfies property $\cal D$, and (\ref{eq:5a}) does not change the central block of length 8. The same argument applies to $\Lambda^{(c)}_{5}$. For the interval $\Lambda^{(b)}_{4}$, Equation (\ref{eq:5b}) gives that the positive digit blocks $d_{3}\dots d_0$ are the same as for the corresponding numbers in $\Lambda_4$, and that the negative digit blocks are $d_{-1}\dots d_{-4}(7)(01)^{-1}00=0000$ and $d_{-1}\dots d_{-4}(9)(01)^{-1}00=0100$, which already occurred in the expansions $\beta(0)$ and $\beta(3)$.
The interval $\Lambda_8= \Lambda^{(a)}_{6}\cup\Lambda^{(b)}_{5}\cup\Lambda^{(c)}_{6}$ also satisfies property $\cal D$, since the word transformations in Equation (\ref{eq:shift-even}) do not change the central blocks of length 8 in $\Lambda_{6}$, nor in $\Lambda_{5}$. Another way to put this, is that $\Lambda_8\cong \Lambda_{6}\Lambda_{5}\Lambda_{6} \mod 4$. Since the $\beta$-expansions only get longer, we have in fact that $\Lambda_m\cong \Lambda_{m-2}\Lambda_{m-3}\Lambda_{m-2} \mod 4$ for all $m\ge 8$. Thus it follows by induction that $\Lambda_m$ satisfies property $\cal D$ for all $m\ge 8$.
\noindent\:{\bf (b)}\, Let us say that a Lucas interval $\Lambda_m$, $m\ge 1$ satisfies property $\cal E$ if the numbers $N$ from $\Lambda_m$ have the property that the digit block $d_{3}\dots d_0\cdot d_{-1}\dots d_{-4}$ occurs in the $\beta$-expansion of $N$ if and only if the digit block $e_{3}\dots e_0\cdot e_{-1}\dots e_{-4}$ occurs in the $\beta$-expansion of $N-D$. Then it is given that $\Lambda_1,\dots,\Lambda_5$ all satisfy property $\cal E$. The proof continues as in part {\bf (a)}, but we have to take into account that the numbers $N-D$ and $N$ can be elements of different Lucas intervals. This `boundary' problem is easily solved by induction: it is given for $\Lambda_4\Lambda_{5}$ and $\Lambda_5\Lambda_{6}$, and the equation used for induction is\\[-.4cm] $$\Lambda_{m+1}\Lambda_{m+2}\cong \Lambda_{m-1}\Lambda_{m-2}\Lambda_{m-1}\Lambda_{m}\Lambda_{m-1}\Lambda_{m} \mod 4.$$ This equation is an instance of Equation (\ref{eq:GG}).
$\Box$
\section{A closer look at the Lucas intervals}\label{sec:closer}
Here we say more on the idea of splitting Lucas intervals in unions of translated Lucas intervals. To keep the presentation simple, we start with showing how all the natural numbers can be split into translations of the three Lucas intervals $\Lambda_3, \Lambda_4$ and $\Lambda_5$.
This can of course be done in many ways, but we will consider a way derived from the Recursive Structure Theorem \ref{th:rec2}. One has \begin{align*}
\Lambda_6= & \:\Lambda^{(a)}_{4}\cup\Lambda^{(b)}_{3}\cup\Lambda^{(c)}_{4}=[\Lambda_{4}\!+\!L_5]\cup[\Lambda_{3}\!+\!L_6]\cup[\Lambda_{4}\!+\!L_6], \\
\Lambda_7= &\: \Lambda^{(a)}_{5}\cup\Lambda^{(b)}_{4}\cup\Lambda^{(c)}_{5}=[\Lambda_{5}\!+\!L_5]\cup[\Lambda_{4}\!+\!L_7]\cup[\Lambda_{5}\!+\!L_7], \\
\Lambda_8= & \:\Lambda^{(a)}_{6}\cup\Lambda^{(b)}_{5}\cup\Lambda^{(c)}_{6}=[\Lambda_{6}\!+\!L_7]\cup[\Lambda_{5}\!+\!L_8]\cup[\Lambda_{6}\!+\!L_8]\\
= & \: [\Lambda_{4}\!+\!L_5\!+\!L_7]\cup[\Lambda_{3}\!+\!L_6\!+\!L_7]\cup[\Lambda_{4}\!+\!L_6\!+\!L_7]\cup[\Lambda_{5}\!+\!L_8]\\
& \hspace*{6cm}\cup[\Lambda_{4}\!+\!L_5\!+\!L_8]\cup[\Lambda_{3}\!+\!L_6\!+\!L_8]\cup[\Lambda_{4}\!+\!L_6\!+\!L_8]. \end{align*} Note how the splitting of $\Lambda_6$ was used in the splitting of $\Lambda_8$. Continuing in this fashion we obtain inductively a splitting of all Lucas intervals $\Lambda_n$, which we call the \emph{canonical} splitting.
What is the sequence of translated intervals $\Lambda_3, \Lambda_4$ and $\Lambda_5$ created in this way?
Let the word $C(\Lambda_{n})$ code these successive intervals in $\Lambda_n$ by their indices 3, 4 or 5. Let $\kappa$ be the morphism on the monoid $\{3,4,5\}^*$ defined by $$\kappa(3)=5, \quad \kappa(4)=434 \quad \kappa(5)=545.$$
\begin{theorem}\label{th:L-345} For any $n\ge 3$ the interval $\Lambda_n$ is a union of adjacent translations of the three intervals $\Lambda_3, \Lambda_4$ and $\Lambda_5$. If $C(\cdot)$ is the coding function for the canonical splittings then for $n\ge 0$ $$C(\Lambda_{2n+4})=\kappa^n(4), \quad C(\Lambda_{2n+5})=\kappa^n(5).$$ \end{theorem}
\noindent{\it Proof:} By induction. For $n=0$ this is trivially true.
Suppose it is true for $k =1, \dots n$. Then by Theorem \ref{th:rec2}, \begin{align*}
C(\Lambda_{2n+6})&=C(\Lambda_{2n+4})C(\Lambda_{2n+3})C(\Lambda_{2n+4})=\kappa^n(4)\kappa^{n-1}(5)\kappa^n(4)=\kappa^{n-1}(\kappa(4)5\kappa(4))\\
&=\kappa^{n-1}(4345434)=\kappa^{n-1}(\kappa^2(4))=\kappa^{n+1}(4), \\
C(\Lambda_{2n+7}) &=C(\Lambda_{2n+5})C(\Lambda_{2n+4})C(\Lambda_{2n+5})= \kappa^n(5)\kappa^n(4)\kappa^n(5)=\kappa^{n}(545)=\kappa^{n+1}(5).\quad \Box \end{align*}
We continue this analysis, now focussing on the partition of the natural numbers by the intervals $$\Xi_n:=\Lambda_{2n-1}\cup\Lambda_{2n}=[L_{2n-1}+1,L_{2n+1}].$$ The relevance of the $\Xi_n$ is that these are exactly the intervals where $\beta^-(N)$ has length $2n$, for $n \ge 1$. The results in the sequel of this section will therefore be useful in Section \ref{sec:neg}.
There are three (Sturmian) morphisms $f,g$ and $h$ that play an important role in these results, where it is convenient to look at $a$ and $b$ both as integers and as abstract letters. The morphisms are given by \begin{equation}\label{eq:Fib3} f:\mor{aba}{ab}\,,\qquad g:\mor{baa}{ba}\,,\qquad h:\mor{aab}{ab}. \end{equation}
\begin{theorem}\label{th:d-345} For any $n\ge 2$ the interval $\Xi_n$ is a union of adjacent translations of the three intervals $\Lambda_3, \Lambda_4$ and $\Lambda_5$. If $C(\cdot)$ is the coding function for the canonical splittings, then for $n\ge 0$ $$C(\Xi_{n+2})=\delta(h^n(b)),$$ where $\delta$ is the decoration morphism given by $\delta(a)=54,\:\delta(b)=34$. \end{theorem}
\noindent{\it Proof:} We first establish the commutation relation\;$\kappa\,\delta=\delta \,h.$\\ It suffices to prove this for the generators, and indeed: $$ \kappa(\delta(a))=\kappa(54)=545434=\delta(aab)=\delta(h(a)),\quad \kappa(\delta(b))=\kappa(34)=5434=\delta(ab)=\delta(h(b)). $$ Using Theorem \ref{th:L-345}, and the commutation relation we obtain for $n\ge 1$ \begin{align*}
C(\Xi_{n+2})&=C(\Lambda_{2n+3})C(\Lambda_{2n+4})=\\
&=\kappa^{n-1}(5)\kappa^{n}(4)=\kappa^{n-1}(5434)=\kappa^{n-1}(\delta(ab))=\delta(h^{n-1}(ab))=\delta(h^n(b)). \end{align*} For $n= 0$ we have $\Xi_2=\Lambda_3\cup\Lambda_4$, so $C(\Xi_2)=34=\delta(b)$. \quad $\Box$
\section{Generalized Beatty sequences}\label{sec:GBS}
Let $\alpha$ be an irrational number larger than 1. We call any sequence $V$ with terms of the form $V_n = p\lfloor n \alpha \rfloor + q n +r $, $n\ge 1$ a \emph{generalized Beatty sequence}. Here $p,q$ and $r$ are integers, called the \emph{parameters} of $V$, and we write $V=V(p,q,r)$.
In this paper we will only consider the case $\alpha=\varphi$, the golden mean, so any mention of a generalized Beatty sequence assumes that $\alpha=\varphi$. A prominent role is played by the lower Wythoff sequence $A:=V(1,0,0)$ and the upper Wythoff sequence $B:=V(1,1,0)$. These are complementary sequences, associated to the Beatty pair ($\varphi,\varphi^2)$.
Here is the key lemma that tells us how generalized Beatty sequences behave under compositions. In its statement below, as Lemma \ref{lem:VA}, a typo in its source is corrected.
\begin{lemma}{\bf (\cite{GBS}, Corollary 2) }\label{lem:VA} Let $V$ be a generalized Beatty sequence with parameters $(p,q,r)$. Then $VA$ and $VB$ are generalized Beatty sequences with parameters $(p_{V\!A},q_{V\!A},r_{V\!A})=(p+q,p,r-p)$ and $(p_{V\!B},q_{V\!B},r_{V\!B})=(2p+q,p+q,r)$. \end{lemma}
It will be useful later on to have a sort of converse of this lemma.
If $C$ and $D$ are two ${\mathbb{N}}$-valued sequences, then we denote by $C\sqcup D$ the sequence whose terms give the set $C({\mathbb{N}})\cup D({\mathbb{N}})$, in increasing order.
\begin{lemma}\label{lem:VAconv} Let $V=V(p,q,r)$ be a generalized Beatty sequence. Let $U$ and $W$ be two disjoint sequences with union $V=U\sqcup W$: $$U({\mathbb{N}})\cap W({\mathbb{N}})=\emptyset,\quad U({\mathbb{N}})\cup W({\mathbb{N}})=V({\mathbb{N}}).$$ Suppose $U$ is a generalized Beatty sequence with parameters $(p+q, p, r-p)$. Then $W$ is the generalized Beatty sequence with parameters $(2p+q,p+q,r)$. \end{lemma}
\noindent{\it Proof:} According to Lemma \ref{lem:VA}, we have $U=VA$. Since $A$ and $B$ are disjoint with union ${\mathbb{N}}$, we must have $W=VB$, and Lemma \ref{lem:VA} gives that $W$ is a generalized Beatty sequence with parameters $(2p+q,p+q,r)$.
$\Box$
Here is the key lemma to `recognize' a generalized Beatty sequence, taken from \cite{GBS}. If $S$ is a sequence, we denote its sequence of first order differences as $\Delta S$, i.e., $\Delta S$ is defined by $$\Delta S(n) = S(n+1)-S(n), \quad {\rm for\;} n=1,2\dots.$$
\begin{lemma}\label{lem:diff}{ \rm \bf(\cite{GBS})} Let $V = (V_n)_{n \geq 1}$ be the generalized Beatty sequence defined by $V_n = p\lfloor n \varphi \rfloor + q n +r$, and let $\Delta V$ be the sequence of its first differences. Then $\Delta V$ is the Fibonacci word on the alphabet $\{2p+q, p+q\}$. Conversely, if $x_{a,b}$ is the Fibonacci word on the alphabet $\{a,b\}$, then any $V$ with $\Delta V= x_{a,b}$ is a generalized Beatty sequence $V=V(a-b,2b-a,r)$ for some integer $r$. \end{lemma}
\section{The positive powers of the golden mean}\label{sec:phi}
For any digit block $w$ we will determine the sequence $R_w$ of those numbers $N$ with digit block $w=d_{m-1}\dots d_0$ as suffix of $\beta^+(N)$. We sometimes call $w$ an \emph{end block} of $\beta^+(N)$. More generally, we are also interested in occurrence sequences of numbers $N$ with $d_{m-1}\dots d_0(N)=w$ and $d_{-1}\dots d_{-m'}(N)=v$. We denote these as $R_{w\cdot v}$.
For a couple of small values of $m,m'$, we have the following result from the paper \cite{Dekk-phi-FQ}, Theorem 5.1.
\begin{theorem}\label{th:d0d1} {\bf (\cite{Dekk-phi-FQ})} Let $\beta(N)=(d_i(N))$ be the base phi expansion of a natural number $N$. Then:\\[.1cm]
$R_{1}=V_0(1,2,1)$,\quad $R_{10}=V(1,2,-1)$,\quad $R_{00\cdot 0}=V_0(1,2,0)$,\quad $R_{00\cdot 1}=V(3,1,1)$. \end{theorem}
Here it made sense to add $N=1$ to $V(1,2, 1)$, and $N=0$ to $R_{00\cdot 0}$. We accomplished this by adding the $n=0$ term to the generalized Beatty sequence $V$: we define $V_0$ by $$V_0(p,q,r) := (p\lfloor n \phi \rfloor + q n +r)_{n\ge 0}.$$
The digit blocks $w=d_{m-1}\dots d_1\,0$ behave rather differently from digit blocks $w=d_{m-1}\dots d_1\,1$. We therefore analyse these cases separately, in Section \ref{sec:w0} and \ref{sec:w1} .
\subsection{Digit blocks $w=d_{m-1}\dots d_10$}\label{sec:w0}
We order the digit blocks $w$ with $d_0=0$ in a Fibonacci tree. The first four levels of this tree are depicted below.
\hspace*{-1.1cm} \begin{tikzpicture} [level distance=15mm, every node/.style={fill=orange!20,rectangle,inner sep=1pt}, level 1/.style={sibling distance=72mm,nodes={fill=orange!20}}, level 2/.style={sibling distance=48mm,nodes={fill=orange!20}}, level 3/.style={sibling distance=36mm,nodes={fill=orange!20}}] \node {\footnotesize $\duoZ{=0}{=V_0(-1,3,0)\quad}$} child {node {\footnotesize $\duoZ{=00}{=V_0(1,2,0) \sqcup V(3,1,1)}$}
child {node {\footnotesize $\duoZ{=000}{=V_0(4,3,0) \sqcup V(3,1,1)}$}
child {node {\footnotesize $\duoZ{=0000}{=V_0(4,3,0) \sqcup V(7,4,1)}$}}
child {node {\footnotesize $\duoZ{=1000}{=V(4,3,-2)}$}} }
child {node {\footnotesize $\duoZ{=100}{=V(3,1,-1)}$}
child {node {\footnotesize $\duoZ{=0100}{=V(3,1,-1)}$}}
} } child {node {\footnotesize $\duoZ{=10}{=V(1,2,-1)}$}
child {node {\footnotesize $\duoZ{=010}{=V(1,2,-1)}$}
child {node {\footnotesize $\duoZ{=0010}{=V(3,1,-2)}$}}
child {node {\footnotesize $\duoZ{=1010}{=V(4,3,-1)}$}} } }; \end{tikzpicture}
We start with the short words $w$.
\begin{proposition}\label{prop:small} The sequence of occurrences $R_w$ of numbers $N$ such that the digits $d_{m-1} \dots d_0$ of the base phi expansion of $N$ are equal to $w$, i.e., $d_{m-1}\dots d_0(N)=w$, is given for the words $w$ of length at most 3, and ending in 0 by\\ \mbox{\rm a)} $R_0 = V(-1,3,0)$,\\ \mbox{\rm b)} $R_{00} = V_0(1,2,0) \sqcup V(3,1,1)$,\\ \mbox{\rm c)} $R_{10}=R_{010} = V(1,2,-1)$,\\ \mbox{\rm d)} $R_{000} = V_0(4,3,0) \sqcup V(3,1,1)$,\\ \mbox{\rm e)} $R_{100} = V(3,1,-1)$. \end{proposition}
\noindent{\it Proof:} \noindent {\rm a)} $w=0$: Since the numbers $\varphi+2$ and $3-\varphi$ form a Beatty pair, i.e., $$\frac{1}{\varphi+2}+\frac{1}{3-\varphi}=1,$$ the sequences $V(1,2,0)$ and $V(-1,3,0)$ are complementary in the positive integers. It follows that $R_0=V_0(-1,3,0)$ is the complement of $R_1=V_0(1,2,1)$, by Theorem \ref{th:d0d1}.
\noindent {\rm b)} $w=00$: Theorem \ref{th:d0d1} gives that $R_{00}$ is the union of the two GBS $V_0(1,2,0)$ and $V(3,1,1)$. These two sequences correspond to the numbers $N$ with expansions containing $00\cdot 0$, coded ${\rm B}$ in \cite{Dekk-phi-FQ}, respectively those containing $00\cdot 1$, coded ${\rm D}$ in \cite{Dekk-phi-FQ}.
\noindent {\rm c)} $w=10$ and $w=010$: From Theorem \ref{th:d0d1} we obtain that $R_{10}$ is equal to $V(1,2,-1)$.
\noindent {\rm d)} $w=000$: By Lemma \ref{lem:no} there are no base phi expansions with $d_2d_1d_0d_{-1}(N)=100\cdot 1$. This means that the numbers $N$ from $V(3,1,1)$ in the last part of Theorem \ref{th:d0d1} do exactly correspond with the numbers $N$ with $d_2d_1d_0d_{-1}(N)=000\cdot 1$. This gives one part of the numbers $N$ where $\beta^+(N)$ has suffix 000.
The other part comes from the occurrences of $N$ with $d_2d_1d_0d_{-1}(N)=000\cdot 0$. The trick is to observe that the digit blocks
$1010$ and $000\cdot 0$ always occur in pairs of the expansions of $N-1$ and $N$, for $N=7,\dots 18$. The Propagation Principle (Lemma \ref{lem:prop}, Part {\bf b)}) gives that this coupling will hold for all positive integers $N$.
From Theorem \ref{th:phi-w0} we know that the digit block $1010$ has occurrence sequence $R_{1010}=V(4,3,-1)$. So the coupling implies that the digit block $000\cdot 0$ has occurrence sequence $V_0(4,3,0)$. Here we should mention that Theorem \ref{th:phi-w0} uses the proposition we are on the way of proving (via the formula $R_{1010}=R_{010}\,B$), however, this only uses part {\rm c)}, which we already proved above.
\noindent {\rm e)} $w=100$: We already know that expansions with $100\cdot1$ do not occur, and one checks that an expansion $\beta(N-2)=\dots 100\cdot0\dots$ always occurs coupled to an expansion $\beta(N)=\dots 00\cdot 1\dots$, for $N=2,\dots,19$. The Propagation Principle (Lemma \ref{lem:prop}, Part {\bf b)}) then implies that this coupling occurs for all $N$. This gives that $R_{100}=R_{00\cdot 1}-2=V(3,1,-1)$, using the result of part b).
$\Box$
The sequences $R_{010}$ and $R_{100}$ are examples of what we call Lucas-Wythoff sequences: their parameters are given respectively by $(L_1,L_0,-1)$ and $(L_2,L_1,-1)$.\\ In general, a \emph{ Lucas-Wythoff sequence} $G$ is a generalized Beatty sequence defined for a natural number $m$ by $$G = V(L_{m+1},L_m,r),$$ where $r$ is an integer.
\begin{theorem}\label{th:phi-w0} \noindent For any natural number $m\ge 2$ fix a word $w=w_{m-1}\dots w_0$ of $0$'s and $1$'s, containing no $11$. Let $w_0=0$. Then---except if $w=0^m$---the sequence $R_w$ of occurrences of numbers $N$ such that the digits $d_{m-1} \dots d_0$ of the base phi expansion of $N$ are equal to $w$, i.e., $d_{m-1}\dots d_0(N)=w$, is a Lucas-Wythoff sequence of the form $$R_w= V(L_{m-2},L_{m-3},\gamma_w) \;\;\text{\rm if}\; w_{m-1}=0,\quad R_w=V(L_{m-1},L_{m-2},\gamma_w) \;\;\text{\rm if}\; w_{m-1}=1,$$ where $\gamma_w$ is a negative integer or 0.\\ In case $w$ consists entirely of $0$'s this sequence of occurrences is given by a disjoint union of two Lucas-Wythoff sequences. We have \begin{align*}
R_{0^{2m}} & = V(L_{2m},L_{2m-1},1) \:\sqcup\: V_0(L_{2m-1},L_{2m-2},0),\\
R_{0^{2m+1}} & = V_0(L_{2m+1},L_{2m},0) \, \sqcup \, V(L_{2m},L_{2m-1},1). \end{align*} \end{theorem}
\noindent{\it Proof:} \:Suppose first that $w$ is a word \emph{not} equal to $0^m$ for some $m\ge 2$.
\noindent The proof is by induction on the length $m$ of $w$. For $m=2$ the statement of the theorem holds by Proposition \ref{prop:small}, part c). Next, let $w$ be a word of length $m$ with $w_0=0$.
In the case that $w_{m-1}=1$, $w$ has a unique extension to $0w$, and $R_{0w}=R_w$ is equal to the correct Lucas-Wythoff sequence.
In the case that $w_{m-1}=0$, the induction hypothesis is that $R_w$ is a Lucas-Wythoff sequence $R_w=V(L_{m-2},L_{m-3},\gamma_w)$ .
\noindent By Theorem \ref{th:Zeckphi} the numbers $N$ with a $\beta^+(N)$ ending with the digit block $w$ are in one-to-one correspondence with numbers $N'$ with a $Z(N')$ ending with the digit block $w$, and the same property holds for the digit blocks $0w$, respectively $1w$. Note that the correspondence is one-to-one, since the numbers `skipped' in the Zeckendorf expansions all\footnote{We have to follow a different strategy for the words $w=d_{m-1}\dots d_11$ in the next section.} have $d_0=1$. It therefore follows from Proposition 2.6 in \cite{Dekk-Zeck-structure} that $$R_{0w}=R_wA\quad\text{and\:} R_{1w}=R_wB.$$ By Lemma \ref{lem:VA} these have parameters $$(L_{m-2}+L_{m-3}, L_{m-2}, \gamma_w-L_{m-2})=(L_{m-1},L_{m-2}, \gamma_w-L_{m-2}),$$ respectively $$(2L_{m-2}+L_{m-1}, L_{m-2}+L_{m-3}, \gamma_w)=(L_m,L_{m-1}, \gamma_w).$$ These are indeed the right expressions for the two words $0w$, respectively $1w$ of length $m+1$.
\noindent Next: The words $w=0^m$ for some $m\ge 2$.
We claim that for all $m\ge 1$
\begin{align}
R_{0^{2m}\cdot 0} & = V_0(L_{2m-1},L_{2m-2},0), & \;\;\; R_{0^{2m}\cdot 1} = V(L_{2m},L_{2m-1}, 1)\label{eq:R1} \\
R_{0^{2m+1}\cdot 0} & = V_0(L_{2m+1},L_{2m},0), & R_{0^{2m+1}\cdot 1} = V(L_{2m},L_{2m-1}, 1)\label{eq:R2}. \end{align} The proof is by induction.
We find in the proof of Proposition \ref{prop:small}, part b) that $R_{00\cdot 0}=V_0(1,2,0)$ and $R_{00\cdot 1}=V(3,1,1)$. Since $L_0=2, L_1=1$ and $L_2=3$, this is Equation (\ref{eq:R1}) for $m=1$.
We find in the proof of Proposition \ref{prop:small}, part d) that $R_{000\cdot 0}=V_0(4,3,0)$ and $R_{000\cdot 1}=V(3,1,1)$. This is Equation (\ref{eq:R2}) for $m=1$.
Next we perform the induction step. Suppose that both Equation (\ref{eq:R1}) and Equation (\ref{eq:R2}) hold.
\noindent {\footnotesize \fbox{(\ref{eq:R1})}} Since $10^{2m+1}\cdot 0$ never occurs by Lemma \ref{lem:no}, we must have \begin{equation}\label{eq:R0000.0}
R_{0^{2m+2}\cdot 0} = R_{0^{2m+1}\cdot 0} = V_0(L_{2m+1},L_{2m},0).
\end{equation} This is the left part of Equation (\ref{eq:R1}) for $m+1$ instead of $m$.
That $10^{2m+1}\cdot 0$ never occurs also implies that \begin{equation}\label{eq:R000001.1}
R_{10^{2m+1}\cdot 1}=R_{10^{2m+1}}=V(L_{2m+1},L_{2m},\gamma_{10^{2m+1}})=V(L_{2m+1},L_{2m}, -L_{2m}+1). \end{equation}
Here we used the first part of the proof, determining $\gamma_{10^{2m+1}}$ from the observation that the first occurrence of $d_{2m+1}\dots d_0(N)=10^{2m+1}$ is at $N=L_{2m+1}+1$, the first element of the Lucas interval $\Lambda_{2m+1}$.
Next we take $V=R_{0^{2m+1}\cdot 1}$, $U=R_{10^{2m+1}\cdot 1}$ and $W=R_{0^{2m+2}\cdot 1}$ in Lemma \ref{lem:VAconv}. According to Equation (\ref{eq:R2}), we take $(p,q,r)=(L_{2m},L_{2m-1},1)$. The parameters of the sequence $U$ should be $(p+q,p,r-p)=(L_{2m+1},L_{2m},1-L_{2m})$, which conforms with Equation (\ref{eq:R000001.1}).
The conclusion of Lemma \ref{lem:VAconv} is that $W=R_{0^{2m+2}\cdot 1}$ has parameters $$(2p+q,p+q,r)=(2L_{2m}+L_{2m-1},L_{2m}+L_{2m-1},1)=(L_{2m+2},L_{2m+1},1).$$ This is the right part of Equation (\ref{eq:R1}) for $m+1$.
\noindent {\footnotesize \fbox{(\ref{eq:R2})}} Since $10^{2m+2}\cdot 1$ never occurs by Lemma \ref{lem:no}, we must have, using the final result of {\footnotesize\fbox{(\ref{eq:R1})}}, $$ R_{0^{2m+3}\cdot 1} = R_{0^{2m+2}\cdot 1} = V( L_{2m+2},L_{2m+1},1).$$ This is the left part of Equation (\ref{eq:R2}) for $m+1$ instead of $m$.
That $10^{2m+2}\cdot 1$ never occurs also implies that \begin{equation}\label{eq:R10000.0}
R_{10^{2m+2}\cdot 0}=R_{10^{2m+2}}=V(L_{2m+2},L_{2m+1}, -L_{2m+1}). \end{equation}
Here we used the first part of the proof, determining $\gamma_{10^{2m+2}}$ from the observation that the first occurrence of $d_{2m+3}\dots d_0(N)=10^{2m+2}$ is at $N=L_{2m+2}$, the first element of the Lucas interval $\Lambda_{2m+2}$.
Next we take $V=R_{0^{2m+2}\cdot 0}$, $U=R_{10^{2m+2}\cdot 0}$ and $W=R_{0^{2m+3}\cdot 0}$ in Lemma \ref{lem:VAconv}. According to Equation (\ref{eq:R0000.0}), we take $(p,q,r)=( L_{2m+1},L_{2m},0)$. The parameters of the sequence $U$ should be $(p+q,p,r-p)=(L_{2m+2},L_{2m+1},-L_{2m+1})$, which conforms with Equation (\ref{eq:R10000.0}).
The conclusion of Lemma \ref{lem:VAconv} is that $W=R_{0^{2m+3}\cdot 0}$ has parameters $$(2p+q,p+q,r)=(2L_{2m+1}+L_{2m},L_{2m+1}+L_{2m},0)=(L_{2m+3},L_{2m+2},0).$$ This is the left part of Equation (\ref{eq:R2}) for $m+1$.
$\Box$
\subsection{Digit blocks $w=d_{m-1}\dots d_11$}\label{sec:w1}
Here there are digit blocks that do not occur at all, like $w=1001$. We denote this as $R_{1001}=\emptyset$.
We order the digit blocks $w$ with $d_0=1$ in a tree. The first four levels of this tree (taking into account that the node corresponding to $R_{1001}$ has no offspring) are depicted below.
\begin{tikzpicture} [level distance=15mm, every node/.style={fill=grey!20,rectangle,inner sep=1pt}, level 1/.style={sibling distance=75mm,nodes={fill=grey!20}}, level 2/.style={sibling distance=60mm,nodes={fill=grey!20}}, level 3/.style={sibling distance=36mm,nodes={fill=grey!20}}] \node {\footnotesize $\duoO{=1}{=V_0(1,2,1)\quad}$} child {node {\footnotesize $\duoO{=01}{=V_0(1,2,1)}$}
child {node {\footnotesize $\duoO{=001}{=V_0(4,3,1)}$}
child {node {\footnotesize $\duoO{=0001}{=V_0(4,3,1)}$}
child {node {\footnotesize $\duoO{=00001}{=V_0(11,7,1)}$}}
child {node {\footnotesize $\duoO{=10001}{=V(7,4,-3)}$}}}
child {node {\footnotesize $\duoO{=1001}{=\emptyset}$}
} }
child {node {\footnotesize $\duoO{=101}{=V(3,1,0)}$}
child {node {\footnotesize $\duoO{=0101}{=V(3,1,0)}$}
child {node {\footnotesize $\duoO{=00101}{=V(4,3,-3)}$}}
child {node {\footnotesize $\duoO{=10101}{=V(7,4,0)}$}}} } }; \end{tikzpicture}
Here $R_{01}=R_1=V_0(1,2,1)$ has been given in Theorem \ref{th:d0d1}. The correctness of the other occurrence sequences follows from Theorem \ref{th:phi-w1}.
We next determine an infinite family of excluded blocks.
\begin{lemma}\label{lem:1001} Let $m\ge 2$ be an integer. There are no expansions $\beta^+(N)$ with end block $10^{2m}1$. \end{lemma}
\noindent{\it Proof:} Consider any $N$ such that $\beta^+(N)$ has end block $10^{2m}1$. Such an $N$, of course, has $d_{-1}(N)=0$, so we see that $\beta(N-1)=\dots 10^{2m+1}\cdot 0\dots$. According to Lemma \ref{lem:no} this is not possible.
$\Box$
Next, we establish a connection with the previous section.
\begin{lemma} \label{lem:1to0} Let $m\ge 2$ be an integer. The block $w=d_{m-1}\dots d_11\cdot 0$ is end block of $\beta^+(N)$ if and only if the block $\breve{w}:=d_{m-1}\dots d_10\cdot 0$ occurs in $\beta(N-1)$. \end{lemma}
\noindent{\it Proof:} This follows quickly from the Propagation Principle Lemma \ref{lem:prop} applied to the couple of blocks $00\cdot 0$ and $01\cdot 0$.
$\Box$
\begin{theorem}\label{th:phi-w1} \noindent For any natural number $m\ge 2$ fix a word $w=w_{m-1}\dots w_0$ of $0$'s and $1$'s, containing no $11$. Let $w_0=1$. With exception of the words $w$ with suffix $0^m1$ and $10^m1$, for $m=2,3,\dots$, the sequence $R_w$ of occurrences of numbers $N$ such that the digits $d_{m-1} \dots d_0$ of the base phi expansion of $N$ are equal to $w$, i.e., $d_{m-1}\dots d_0(N)=w$, is a Lucas-Wythoff sequence of the form $$R_w= V(L_{m-2},L_{m-3}, \gamma_w) \;\;\text{\rm if}\; w_{m-1}=0,\quad R_w=V(L_{m-1},L_{m-2}, \gamma_w) \;\;\text{\rm if}\; w_{m-1}=1,$$ where $\gamma_w$ is a negative integer or 0.
In case $w=0^{2m}1$ we have $R_w=V_0(L_{2m+1},L_{2m},1)$, and this is also the sequence of occurrences of $w=0^{2m+1}1$.
In case $w=10^{2m}1$ the word $w$ does not occur at all as digit end block.
In case $w=10^{2m+1}1$ we have $R_w=V(L_{2m+2},L_{2m+1},-L_{2m+1}+1)$. \end{theorem}
\noindent{\it Proof:} It follows from Lemma \ref{lem:1to0} that $R_w=R_{\breve{w}}+1$, if $R_w\ne \emptyset$. So the first part of Theorem \ref{th:phi-w0} yields the statement of the theorem for all $w$ not equal to $0^m1$ or $10^m1$.
In case $w=0^{2m}1\cdot 0$, we have $\breve{w}=0^{2m+1}\cdot 0$, and the result follows from the left part of Equation (\ref{eq:R2}).
In case $w=10^{2m}1$ the word $w$ does not occur as digit end block, according to Lemma \ref{lem:no}.
In case $w=10^{2m+1}1\cdot 0$ we have $\breve{w}=10^{2m+2}\cdot 0$, and now Equation (\ref{eq:R10000.0}) gives that
$R_w=R_{\breve{w}}+1=V(L_{2m+2},L_{2m+1},-L_{2m+1}+1)$.
$\Box$
\section{The negative powers of the golden mean}\label{sec:neg}
Here we discuss what we can say about the words $\beta^-(N)$. These do have an even more intricate structure than the $\beta^+(N)$.
\subsection{The words $\beta^-(N)$}\label{sec:trident}
Here we look at complete $\beta^-(N)$'s. Although at first sight these seem to appear in a random order, there is an order dictated not by a coin toss, but by another dynamical system: the rotation over an angle $\varphi$. Moreover, they appear in singletons, or as triples. This can be proved with the \{${\rm A}{\rm B}{\rm C}$, ${\rm D}$\}--structure found in the paper \cite{Dekk-phi-FQ}.
For a more extensive analysis, partition the natural numbers larger than 1 into intervals $$\Xi_n:=\Lambda_{2n-1}\cup\Lambda_{2n}=[L_{2n-1}+1,L_{2n+1}].$$ The relevance of the $\Xi_n, n=1,2,\dots$ is that these are exactly the intervals where $\beta^-(N)$ has length $2n$. The $\Xi_n$ intervals have length $$L_{2n+1}-L_{2n-1}=L_{2n+1}-L_{2n}+L_{2n}-L_{2n-1}=L_{2n-1}+L_{2n-2}=L_{2n}.$$ Call three consecutive numbers $N,N+1,N+2$ a \emph{trident}, if $\beta^-(N)=\beta^-(N+1)=\beta^-(N+2)$. For example: 2,3,4 and 6,7,8 are tridents. We shall always take the middle number $N\!+\!1$ as the representing number of a trident interval $[N,N\!+\!1,N\!+2]$. We call this number $\Pi$-\emph{essential}. By definition the other $\Pi$-essential numbers are the singletons.
\begin{lemma} {\bf [Trident splitting]} \label{lem:trident} In $\Lambda_{2n-1}\cup\Lambda_{2n}$ the last number of $\Lambda_{2n-1}$ and the first two numbers in $\Lambda_{2n}$ are in
the same trident. \end{lemma}
\noindent {\it Proof:} This is true for $n=1$ and $n=2$: $\Lambda_1\cup\Lambda_2=\{2\}\cup[3,4]$ is a trident, and $\Lambda_3\cup\Lambda_4=[5,6]\cup[7,8,\dots,11]$ contains the trident $[6,7,8]$. The property then follows by induction, using Theorem \ref{th:rec2}.
$\Box$
The following lemma helps to count singletons and tridents.
\begin{lemma} The following relation between Lucas numbers and Fibonacci numbers holds: $F_n+3F_{n+1}=L_{n+2}$ for $n=0,1,2,\dots$. \end{lemma}
For a proof, note that $F_0+3F_1=3=L_2$, and $F_2+3F_3=1+6=L_4$, and then add these two equations, etc.
The lemma describes the fact that the $\Xi_n$ intervals contain $F_{2n-2}$ singletons, and $F_{2n-1}$ tridents, making a total number of $L_{2n}$. The collection of different $\beta^-(N)$-blocks of length $2n$ has thus cardinality $F_{2n-2}+F_{2n-1}=F_{2n}$. This implies that we have proved the following theorem.
\begin{theorem}\label{th:all-beta-min} All Zeckendorf words of even length ending in 1 appear as $\beta^-(N)$-blocks. \end{theorem}
Here we mean by a Zeckendorf word (or golden mean word) all words in which 11 does not occur. We denote by $\mathcal{Z}_{m}$ the set of Zeckendorf words of length $m$, for $m=1,2,\dots$. It is easily proved that the cardinality of $\mathcal{Z}_{m}$ equals $F_{m+2}$. So the cardinality of the set of words from $\mathcal{Z}_{2n}$ ending in 1 is equal to $F_{2n}$, implying the result of Theorem \ref{th:all-beta-min}.
Since all $\beta^-(N)$ have suffix 01, the essential information of these words is contained in
$$ \gamma^-(N):= \beta^-(N)1^{-1}0^{-1}.$$
The words $\gamma^-(N)$ are Zeckendorf words, corresponding one-to-one to the natural numbers $Z^{-1}(\gamma^-(N))$.
Obviously, the $\gamma^-(N)$ have the same ordering as the $\beta^-(N)$.
According to Theorem \ref{th:all-beta-min} we then (after identifying tridents with their middle number) obtain a permutation of length $F_{2n}$ of the $\Pi$-essential elements of $\Xi_n$ by coding these numbers by ${\rm C}(N):=Z^{-1}(\gamma^-(N))$. We denote this permutation by $\Pi^\beta_{2n}$.
The following Zeckendorf words and codes will be important in the sequel.
\begin{lemma} \label{lem:code} For all natural numbers $n$ we have
\begin{equation}\label{eq:bordergammas}
\gamma^-(L_{2n})=0^{2n-2}, \; \gamma^-(L_{2n+1})=[01]^{n-1}, \; \gamma^-(L_{2n+1}+1)=[10]^n,\; \gamma^-(L_{2n+2}-1)=0^{2n}\!.
\end{equation}
\begin{equation}\label{eq:bordercodes}
{\rm C}(L_{2n})=0, \quad {\rm C}(L_{2n+1})=F_{2n-1}-1, \quad {\rm C}(L_{2n+1}+1)=F_{2n+2}-1, \quad {\rm C}(L_{2n+2}-1)=0.
\end{equation}
\end{lemma}
\noindent {\it Proof:} The correctness of Equation (\ref{eq:bordergammas}) follows from Equations (\ref{eq:Lm}) and (\ref{eq:Lmplus1}). So $\gamma^-(L_{2n})$ is the first word in $\mathcal{Z}_{2n-2}$, $\gamma^-(L_{2n+1})$ is 0 followed by the last word in $\mathcal{Z}_{2n-3}$, $\gamma^-(L_{2n+1}+1)$ is the last word in $\mathcal{Z}_{2n}$, and $\gamma^-(L_{2n+2}-1)$ is the first word in $\mathcal{Z}_{2n-2}$. Since $\mathcal{Z}_{m}$ has cardinality $F_{m+2}$, Equation (\ref{eq:bordercodes}) follows.
$\Box$
We have to determine the codings of all natural numbers $N$. For this it is useful to translate Theorem \ref{th:rec2} to the $\gamma^-\!$-blocks.
\begin{theorem}{\bf [Recursive structure theorem: $\gamma^-\!$-version]}\label{th:recg}\\ \noindent{\,\bf (i): Odd\;} For all $n\ge 1$ one has $\Lambda_{2n+1}=\Lambda^{(a)}_{2n-1}\cup\Lambda^{(b)}_{2n-2}\cup\Lambda^{(c)}_{2n-1}, $ where $\Lambda^{(a)}_{2n-1}=\Lambda_{2n-1}+L_{2n}$,\; $\Lambda^{(b)}_{2n-2}=\Lambda_{2n-2}+L_{2n+1}$, and $\Lambda^{(c)}_{2n-1}=\Lambda_{2n-1}+L_{2n+1}$.\\ We have\\[-.8cm] \begin{subequations} \label{eq:shift-odd} \begin{align}
\gamma^-(N)= & \;\gamma^-(N-L_{2n})\,10& for\; N\in \Lambda^{(a)}_{2n-1}, \label{eq:15a}\\
\gamma^-(N)= & \;\gamma^-(N-L_{2n+1})\,0010& for\; N\in \Lambda^{(b)}_{2n-2},\label{eq:15b}\\
\gamma^-(N)= & \;\gamma^-(N-L_{2n+1})\,00 & for\; N\in \Lambda^{(c)}_{2n-1}\label{eq:15c}. \end{align}\\[-.6cm] \end{subequations} \noindent{\,\bf (ii): Even\;} For all $n\ge 1$ one has $\Lambda_{2n+2}=\Lambda^{(a)}_{2n}\cup\Lambda^{(b)}_{2n-1}\cup\Lambda^{(c)}_{2n}, $
where $\Lambda^{(a)}_{2n}=\Lambda_{2n}+L_{2n+1}$,\; $\Lambda^{(b)}_{2n-1}=\Lambda_{2n-1}+L_{2n+2}$, and $\Lambda^{(c)}_{2n}=\Lambda_{2n}+L_{2n+2}$.\\
We have\\[-.8cm] \begin{subequations} \label{eq:shift-even} \begin{align}
\gamma^-(N)= & \;\gamma^-(N-L_{2n+1})\,00 & for\; N\in \Lambda^{(a)}_{2n},\phantom{x} \label{eq:16a} \\
\gamma^-(N)= & \; \gamma^-(N-L_{2n+2})\,01 & for\; N\in \Lambda^{(b)}_{2n-1}, \label{eq:16b}\\
\gamma^-(N)= & \;\gamma^-(N-L_{2n+1})\,01 & for\; N\in \Lambda^{(c)}_{2n}.\phantom{x.} \label{eq:16c} \end{align} \end{subequations} \end{theorem}
\!
We give the situation for $n=2$, where $\Xi_2=\Lambda_3 \cup \Lambda_4=[5,6,\dots,11]$.
\begin{tabular}{|c|c|c|c|c|}
\hline
\; $N^{\phantom{|}}$ & $\Lambda$-int. & $\cdot\beta^-(N)$ & $\cdot\gamma^-(N)$ & ${\rm C}(N)$ \\[.0cm]
\hline
5\; & $\Lambda_3$ & $\cdot1001$ & \; $\cdot10$ & \; 2 \\
6\; & $\Lambda_3$ & $\cdot0001$ & \; $\cdot00$ & \; \grijs{0} \\
\hline
7\; & $\Lambda_4$ & $\cdot0001$ & \; $\cdot00$ & \; 0 \\
8\; & $\Lambda_4$ & $\cdot0001$ & \; $\cdot00$ & \; \grijs{0} \\
9\; & $\Lambda_4$ & $\cdot0101$ & \; $\cdot01$ & \; \grijs{1} \\
10\, & $\Lambda_4$ & $\cdot0101$ & \; $\cdot01$ & \; 1 \\
11\, & $\Lambda_4$ & $\cdot0101$ & \; $\cdot01$ & \; \grijs{1} \\
\hline
\end{tabular}
\noindent We see that $\Pi^\beta_{4}=\big( 2\, 0\, 1\big)$.
Here is the situation for $n=3$, where $\Xi_3=\Lambda_5 \cup \Lambda_6=[12,13,\dots,29]$.
\begin{tabular}{|c|c|c|c|c|}
\hline
\; $N^{\phantom{|}}$ &{\small $\Lambda$-int.}& $\cdot\beta^-(N)$ & $\cdot\gamma^-(N)$ & ${\rm C}(N)$ \\[.0cm]
\hline
12\; & $\Lambda_5$ & $\cdot101001$ & $\cdot1010$ & 7 \\
13\; & $\Lambda_5$ & $\cdot001001$ & $\cdot0010$ & \grijs{2} \\
14\; & $\Lambda_5$ & $\cdot001001$ & $\cdot0010$ & 2 \\
15\; & $\Lambda_5$ & $\cdot001001$ & $\cdot0010$ & \grijs{2} \\
16\; & $\Lambda_5$ & $\cdot100001$ & $\cdot1000$ & 5 \\
17\; & $\Lambda_5$ & $\cdot000001$ & $\cdot0000$ & \grijs{0} \\
\hline
18\; & $\Lambda_6$ & $\cdot000001$ & $\cdot0000$ & 0 \\
19\; & $\Lambda_6$ & $\cdot000001$ & $\cdot0000$ & \grijs{0} \\
20\; & $\Lambda_6$ & $\cdot010001$ & $\cdot0100$ & \grijs{3} \\
\hline
\end{tabular}\quad
\begin{tabular}{|c|c|c|c|c|}
\hline
\; $N^{\phantom{|}}$ &{\small $\Lambda$-int.}& $\cdot\beta^-(N)$ & $\cdot\gamma^-(N)$ & ${\rm C}(N)$ \\[.0cm]
\hline
21\; & $\Lambda_6$ & $\cdot010001$ & $\cdot0100$ & 3 \\
22\; & $\Lambda_6$ & $\cdot010001$ & $\cdot0100$ & \grijs{3} \\
23\; & $\Lambda_6$ & $\cdot100101$ & $\cdot1001$ & 6 \\
24\; & $\Lambda_6$ & $\cdot000101$ & $\cdot0001$ & \grijs{1} \\
25\; & $\Lambda_6$ & $\cdot000101$ & $\cdot0001$ & 1 \\
26\; & $\Lambda_6$ & $\cdot000101$ & $\cdot0001$ & \grijs{1} \\
27\; & $\Lambda_6$ & $\cdot010101$ & $\cdot0101$ & \grijs{4} \\
28\; & $\Lambda_6$ & $\cdot010101$ & $\cdot0101$ & 4 \\
29\; & $\Lambda_6$ & $\cdot010101$ & $\cdot0101$ & \grijs{4} \\
\hline
\end{tabular}\quad
\noindent We see that $\Pi^\beta_{6}=\big(7\,2\,5\,0\,3\,6\,1\,4\big)$.
What are these permutations?
\begin{theorem}\label{th:permut} For all natural numbers $n$ consider the $F_{2n}$ Zeckendorf words of length $2n$ occurring as $\beta^-(N)$ in the $\beta$-expansions of the numbers in $\Xi_n$. Then these occur in an order given by a permutation $\Pi^\beta_{2n}$ which is the orbit of the element $F_{2n}-1$ under the addition by the element $F_{2n-2}$ on the cyclic group $\mathbb{Z}/F_{2n}\mathbb{Z}$. \end{theorem}
\noindent {\it Proof:} We have to show for all $n$ that \begin{equation}\label{eq:IH}
\Pi^\beta_{2n}(1)=F_{2n}-1,\quad \Pi^\beta_{2n}(j+1)=\Pi^\beta_{2n}(j)+F_{2n-2} \!\mod F_{2n}, \; {\rm for}\;j=1,\dots,F_{2n}-1. \end{equation} It is easily checked that the cases $n=2$ and $n=3$ given above conform with this. For $n=3$ one has: $F_6=8$, $F_4=3$, and
$\Pi^\beta_{6}(1)=7, \,\Pi^\beta_{6}(j+1) =\Pi^\beta_{6}(j)+3 \mod 8$ for $j=1,\dots 7$.
The first claim in Equation (\ref{eq:IH}) follows from Lemma \ref{lem:code} for all $n$: since the interval $\Xi_n=[L_{2n-1}+1,L_{2n+1}]$, we have $\Pi^\beta_{2n}(1)=F_{2n}-1$ according to Equation (\ref{eq:bordercodes}).
The proof proceeds by induction, based on Theorem \ref{th:recg}, the $\gamma^-\!$-version of the Recursive Structure Theorem.
For the second part of Equation (\ref{eq:IH}) with $n$ replaced by $n+1$, we have to split the permutation $\Pi_{2n+2}^\beta$ into six pieces, and then we have to glue the expressions together to obtain the full permutation on the set $\Xi_{n+1} = \Lambda_{2n+1} \cup \Lambda_{2n+2} = [L_{2n+1}+1, L_{2n+2}-1] \cup [L_{2n+2},L_{2n+3}]$. According to the Recursive Structure Theorem \begin{equation}\label{eq:six} \Xi_{n+1} = \Lambda^{(a)}_{2n-1}\cup\Lambda^{(b)}_{2n-2}\cup\Lambda^{(c)}_{2n-1}\cup\Lambda^{(a)}_{2n}\cup\Lambda^{(b)}_{2n-1}\cup\Lambda^{(c)}_{2n}. \end{equation} We start with the first interval, $\Lambda^{(a)}_{2n-1}$. From Theorem \ref{th:recg} we have that for $N\in \Lambda^{(a)}_{2n-1}$, \begin{equation}\label{eq:grec1} \gamma^-(N)= \gamma^-(N-L_{2n})\,10. \end{equation}
What does this imply for the codes?
Let $Z({\rm C}(N-L_{2n}))=\gamma^-(N-L_{2n})=d_{2n-3}\dots d_0$, so ${\rm C}(N-L_{2n})= \sum_{i=0}^{2n-3} d_i \ddot{F}_i$. Then Equation (\ref{eq:grec1}) leads to
$$ {\rm C}(N)=\sum_{i=0}^{2n-3} d_i \ddot{F}_{i+2} + 1\cdot \ddot{F}_1 + 0\cdot \ddot{F}_0=\sum_{i=0}^{2n-3} d_i \ddot{F}_{i+2}+2.$$ This implies, in particular, that the differences between the codes of two consecutive $\Pi$-essential numbers within the interval $\Lambda_{2n-1}$ have increased from $F_{2n-2}\!\mod F_{2n}$ to $F_{2n} \!\mod F_{2n+2}$ for the corresponding numbers in the interval $\Lambda^{(a)}_{2n-1}$.
We pass to the second interval, $\Lambda^{(b)}_{2n-2}$. From Theorem \ref{th:recg} we have that for $N$ from $\Lambda^{(b)}_{2n-2}$, \begin{equation}\label{eq:grec2} \gamma^-(N)= \gamma^-(N-L_{2n+1})\,0010. \end{equation}
What does this imply for the codes?
Let $Z({\rm C}(N-L_{2n+1}))=\gamma^-(N-L_{2n+1})=d_{2n-4}\dots d_0$, so ${\rm C}(N-L_{2n+1})= \sum_{i=0}^{2n-4} d_i \ddot{F}_i$. Then Equation (\ref{eq:grec2}) leads to
$${\rm C}(N)=\sum_{i=0}^{2n-4} d_i \ddot{F}_{i+4} + 0\cdot\ddot{F}_3 + 0\cdot \ddot{F}_2 +1\cdot \ddot{F}_1 + 0\cdot \ddot{F}_0=\sum_{i=0}^{2n-4} d_i \ddot{F}_{i+4}+2.$$ This implies that the differences between the codes of two consecutive numbers within the interval $\Lambda_{2n-2}$
have increased from $F_{2n-4}\!\mod F_{2n-2}$ to $F_{2n} \!\mod F_{2n+2}$ for the corresponding numbers in the interval $\Lambda^{(b)}_{2n-2}$.
Similar computations give that for the next 4 intervals $\Lambda^{(c)}_{2n-1}, \Lambda^{(a)}_{2n},\Lambda^{(b)}_{2n-1}$, and $\Lambda^{(c)}_{2n}$ there always is an addition of $F_{2n} \!\mod F_{2n+2}$.
The remaining task is to check that the same holds on the five boundaries between the translated $\Lambda$-intervals.
We number these boundaries with the roman numerals I, II, III, IV, V.
\noindent \fbox{III \& V:}\;For the third and the fifth boundary between respectively the intervals $\Lambda^{(c)}_{2n-1}$ and $\Lambda^{(a)}_{2n}$ and the intervals $\Lambda^{(b)}_{2n-1}$ and $\Lambda^{(c)}_{2n}$ this follows from the Trident Splitting Lemma, Lemma \ref{lem:trident}.
The reason is that if $[N,N+1,N+2]$ is the trident which is splitted, then the difference between ${\rm C}(N-1)$ and ${\rm C}(N)$ is equal to $F_{2n} \!\mod F_{2n+2}$, as these two numbers are both from the first translated $\Lambda$-interval, and not from the same trident. But then the difference between the codes of the last $\Pi$-essential number $N-1$ in the first translated $\Lambda$-interval, and the first $\Pi$-essential number $N+1$ in the second translated $\Lambda$-interval is also equal to $F_{2n} \!\mod F_{2n+2}$.
\noindent \fbox{ I:}\; The last number in the first interval $\Lambda^{(a)}_{2n-1}$ is $2L_{2n}-1$ with associated $\gamma^-\!$-block
$$\gamma^-(2L_{2n}-1)= \gamma^-(2L_{2n}-1-L_{2n})\,10=\gamma^-(L_{2n}-1)\,10=0^{2n-1}\,10.$$
Here we used Equation (\ref{eq:shift-odd}a) in the first, and Equation (\ref{eq:Lmplus1}) in the last step. It follows directly that ${\rm C}(2L_{2n}-1)=2$.
The first number in the second interval $\Lambda^{(b)}_{2n-2}$ is $2L_{2n}$. From Equation (\ref{eq:Lm}) we have $\beta(2L_{2n})\doteq 20^{2n}\cdot 0^{2n-1}2\doteq 20^{2n}\cdot 0^{2n-1}1001$, so $\gamma^-(2L_{2n})=0^{2n-1}10$, giving ${\rm C}(2L_{2n})=2$. It is clear that also the second number $2L_{2n}+1$ in $\Lambda^{(b)}_{2n-2}$ has code ${\rm C}(2L_{2n}+1)=2$. As in the previous case, this implies that the difference between the codes of the last $\Pi$-essential number in the first translated $\Lambda$-interval, and the first $\Pi$-essential number in the second translated $\Lambda$-interval is equal to $F_{2n} \!\mod F_{2n+2}$.
\noindent \fbox{ II:}\; The last number in the second interval $\Lambda^{(b)}_{2n-2}$ is the number $L_{2n-1}+L_{2n+1}$. According to Equation (\ref{eq:shift-odd}b) the associated $\gamma^-\!$-block is
$$\gamma^-(L_{2n-1}+L_{2n+1})= \gamma^-(L_{2n-1}+L_{2n+1}-L_{2n+1})\,0010=\gamma^-(L_{2n-1})\,0010=[01]^{n-2}\,0010.$$ But we know from Lemma \ref{lem:code} that \; $\gamma^-(L_{2n-1})\,0101=[01]^n=\gamma^-(L_{2n+3}).$
By Lemma \ref{lem:code} we have that ${\rm C}(L_{2n+3})=F_{2n+1}-1$. To obtain the code of $N=L_{2n-1}+L_{2n+1}$, we have to subtract the number $F_3+F_1=3$ with Zeckendorf expansion $0101$, and add the number $F_2=2$ with Zeckendorf expansion $0010$. This gives the code
$${\rm C}(L_{2n-1}+L_{2n+1})=F_{2n+1}-1-3+1=F_{2n+1}-3.$$ The first number in the third interval $\Lambda^{(c)}_{2n-1}$ is the number $L_{2n-1}+L_{2n+1}+1$. According to according to Equation (\ref{eq:shift-odd}c) the associated $\gamma^-\!$-block is
$$\gamma^-(L_{2n-1}+L_{2n+1}+1)= \gamma^-(L_{2n-1}+L_{2n+1}+1-L_{2n+1})\,00=\gamma^-(L_{2n-1}+1)\,00.$$ But we know from Lemma \ref{lem:code} that \; $\gamma^-(L_{2n-1}+1)10=[10]^n=\gamma^-(L_{2n+1}+1).$
By Lemma \ref{lem:code} we have that ${\rm C}(L_{2n+1}+1)=F_{2n+2}-1$. To obtain the code of $N=L_{2n-1}+L_{2n+1}+1$, we have to subtract the number $F_2=2$ with Zeckendorf expansion $10$, from this code. This gives the code
$${\rm C}(L_{2n-1}+L_{2n+1}+1)=F_{2n+2}-1-2=F_{2n+2}-3.$$ The conclusion is that $L_{2n-1}+L_{2n+1}$ and $N=L_{2n-1}+L_{2n+1}+1$ are $\Pi$-essential, with difference in codes $F_{2n+2}-3-(F_{2n+1}-3)=F_{2n}.$
\noindent \fbox{ IV:}\; The last number in the fourth interval $\Lambda^{(c)}_{2n}$ is the number $L_{2n+1}+L_{2n+1}=2L_{2n+1}$. According to Equation (\ref{eq:shift-even}a) the associated $\gamma^-\!$-block is
$$\gamma^-(2L_{2n+1})= \gamma^-(2L_{2n+1}-L_{2n+1})\,00=\gamma^-(L_{2n+1})\,00=[01]^{n-1}\,00.$$ But we know from Lemma \ref{lem:code} that \; $\gamma^-(L_{2n+1})\,01=[01]^n=\gamma^-(L_{2n+3}).$
By Lemma \ref{lem:code} we have that ${\rm C}(L_{2n+3})=F_{2n+1}-1$. To obtain the code of $N=2L_{2n+1}$, we have to subtract the number $F_1=1$ with Zeckendorf expansion $01$. This gives the code
$${\rm C}(2L_{2n+1})=F_{2n+1})-1-1=F_{2n+1}-2.$$
The first number in the fifth interval $\Lambda^{(b)}_{2n-1}$ is the number $L_{2n-1}+1+L_{2n+2}$. According to Equation (\ref{eq:shift-even}b) the associated $\gamma^-\!$-block is
$$\gamma^-(L_{2n-1}+1+L_{2n+2})= \gamma^-(L_{2n-1}+1+L_{2n+2}-L_{2n+2})\,01=\gamma^-(L_{2n-1}+1)\,01.$$
But we know from Lemma \ref{lem:code} that \; $\gamma^-(L_{2n-1}+1)10=[10]^n=\gamma^-(L_{2n+1}+1).$
By Lemma \ref{lem:code} we have that ${\rm C}(L_{2n+1}+1)=F_{2n+2}-1$.
To obtain the code of $N=L_{2n-1}+1+L_{2n+2}$, we have to subtract the number $F_2=2$ with Zeckendorf expansion $10$, and add the number $F_1=1$ with Zeckendorf expansion 01 to this code. This gives the code
$${\rm C}(L_{2n-1}+L_{2n+1}+1)=F_{2n+2}-1-2+1=F_{2n+2}-2.$$ The conclusion is that $2L_{2n+1}$ and $L_{2n-1}+1+L_{2n+2}$ are $\Pi$-essential, with difference in codes $F_{2n+2}-2-(F_{2n+1}-2)=F_{2n}.$
$\Box$
We now explain the connection with a rotation on a circle mentioned at the beginning of this section. Note that with this point of view all the cyclic groups of Theorem \ref{th:permut} are represented by a single object: the rotation on the circle.
\begin{theorem}\label{th:rot} For all natural numbers $n$ the permutations $\Pi^\beta_{2n}$ are given by the order in which the first $F_{2n}$ iterates of the rotation $z\rightarrow \exp(2\pi i (z-\varphi))$ occur on the circle. \end{theorem}
We sketch a proof of this result based on the paper \cite{Ravenstein-1988}. In the literature one will not find the rotation $z\rightarrow \exp(2\pi i (z-\varphi))$, but several papers treat the rotation $z\rightarrow \exp(2\pi i (z+\tau))$, where $\tau$ is the algebraic conjugate of $\varphi$. Note that this rotation has exactly the same orbits as $z\rightarrow \exp(2\pi i (z+\varphi))$, and replacing $\varphi$ by $-\varphi$ amounts to reversing the permutation. In the literature the origin is usually added to the orbit. For instance in \cite{Ravenstein-1988}, the $N$ ordered iterates are given by the permutation $\big(u_1\,u_2\,\dots \, u_N\big)$, which for \emph{all} $N$ gives a permutation starting trivially with $u_1=0$.
Lemma 2.1 in \cite{Ravenstein-1988} states that for $j=1,\dots,N$ one has $u_j=(j-1)u_2 \mod N$.\\ Next, Theorem 3.3 in \cite{Ravenstein-1988} states that $u_2=u_2(N)= F_{2n-1}$ in the case that $N=F_{2n}$, $n\ge 1$.
We illustrate this for the case $n=3$.\\ We have $N=F_6=8$, and $0< \{5\tau\}<\{2 \tau\}<\{ 7\tau\}<\{4\tau\}<\{\tau\}<\{6 \tau\}<\{ 3\tau\},$ so $ \big(u_1\,u_2\,\dots \, u_N\big) = \big(0\,5\,2\,7\,4\,1\,6\,3\big)$. As $\{8\tau\}$ is the largest number in the rotation orbit of the first 9 iterations, $\big(u_{N+1}\,u_N\,\dots \, u_2\big)=\big(8\,3\,6\,1\,4\,7\,2\,5\big).$ After subtraction of 1 in all entries, one obtains the permutation $\Pi^\beta_{6}$.
\subsection{Digit blocks $w=d_{-1}\dots d_{-m}(N)$ as prefix of $\beta^-(N)$}
For any digit block $w$ we will try to determine the sequence $R_w$ of those numbers $N$ with $w$ as prefix of $\beta^-(N)$.
The tridents introduced in the previous section give occurrence sequences $R_w$ which are unions of three consecutive generalized Beatty sequences.
We will write for short
$$V(p,q,[r,r+1,r+2]):=V(p,q,r)\sqcup V(p,q,r+1)\sqcup V(p,q,r+2).$$
As before, we order the $w$ in a Fibonacci tree. Here we write $R_{\cdot w}$ for the occurence sequences of words $w$ occurring as a prefix of the words $\beta^-(N)$, to emphasize the positions of these words in the expansion $\beta(N)$. The first four levels of this tree are depicted below.
\hspace*{-0.8cm}\begin{tikzpicture} [level distance=19mm, every node/.style={fill=green!10,rectangle,inner sep=1pt}, level 1/.style={sibling distance=73mm,nodes={fill=green!10}}, level 2/.style={sibling distance=50mm,nodes={fill=green!10}}, level 3/.style={sibling distance=34mm,nodes={fill=green!10}}] \node {\footnotesize $\duoZ{=\Lambda}{=\emptyset\quad}$} child {node {\footnotesize $\duoZ{=0}{=V(1,2,[-1,0,1])}$}
child {node {\footnotesize $\duoZ{=00}{=V(3,1,[2,3,4])}$}
child {node {\footnotesize $\duoZ{=000}{=V(4,3,[-1,0,1])}$}}
child {node {\footnotesize $\duoZ{\!=\! 001}{=V(7,4,[2,3,4])}$}} }
child {node {\footnotesize $\duoZ{=01}{=V_0(4,3,[2,3,4])}$}
child {node {\footnotesize $\duoZ{=010}{=V_0(4,3,[2,3,4])}$}}
} } child {node {\footnotesize $\duoZ{=1}{=V(3,1,1)}$}
child {node {\footnotesize $\duoZ{=10}{=V(3,1,1)}$}
child {node {\footnotesize $\duoZ{=100}{=V(4,3,-2)}$}}
child {node {\footnotesize $\duoZ{=101}{=V(7,4,1)}$}} } }; \end{tikzpicture}
We start with the words $w$ on this tree.
\begin{proposition}\label{prop:smallminus} Let $\beta(N)=\beta^+(N)\cdot\beta^-(N)$ be the base phi expansion of the number $N$.\\
Let $w$ be a word of length $m$. Then the sequence of occurrences $R_w$ of numbers $N$ such that the first $m$ digits of $\beta^-(N)$ are equal to $w$, i.e., $d_{-1}\dots d_{-m}(N)=w$, is given for the words $w$ of length at most 3, by\\ \mbox{\rm a)} $R_{\cdot 0} = V(2,1,-1)\, \sqcup \,V(2,1,0) \, \sqcup \,V(2,1,1)$,\\ \mbox{\rm b)} $R_{\cdot 1}= R_{\cdot 10} = V(3,1,1)$,\\ \mbox{\rm c)} $R_{\cdot 00} = V(3,1,2)\, \sqcup \,V(3,1,3) \, \sqcup \,V(3,1,4)$,\\ \mbox{\rm d)} $R_{\cdot 01} =R_{\cdot 010}= V_0(4,3,2)\, \sqcup\, V_0(4,3,3)\, \sqcup \,V_0(4,3,4)$,\\ \mbox{\rm e)} $R_{\cdot 000} = V(4,3,-1)\, \sqcup\, V(4,3,0)\, \sqcup \,V(4,3,1)$,\\ \mbox{\rm f)} $R_{\cdot 001} = V(7,4,2)\, \sqcup \, V(7,4,3)\, \sqcup \, V(7,4,4)$,\\ \mbox{\rm g)} $R_{\cdot 100} = V(4,3,-2)$,\\ \mbox{\rm h)} $R_{\cdot 101} = V(7,4,1)$. \end{proposition}
\noindent {\it Proof:}
\noindent {\rm a)} $w=\cdot 0$: In Section 5 of the paper \cite{Dekk-phi-FQ} the tridents are coded by triples $({\rm A}, {\rm B}, {\rm C})$. It follows from Theorem 5.1 of \cite{Dekk-phi-FQ} that the first elements (coded ${\rm A}$) of the tridents are all member of $V(2,1,-1)$. This implies the statement in {\rm a)}.
\noindent {\rm b)} $w=\cdot 1$: We already know from Proposition \ref{prop:D-numbers} that $R_{\cdot 1}= V(3,1,1)$.
\noindent {\rm c)} $w=\cdot 00$: Using the Propagation Principle, we see that a digit block $\cdot 10$ is always followed directly by the first element of a trident of $\cdot 00$'s and vice versa. This implies the statement in {\rm c)}, because of {\rm b)}.
\noindent {\rm d)} $w=\cdot 01$: This result is given in Remark 6.2 in the paper \cite{Dekk-phi-FQ}.
\noindent {\rm e)} $w=\cdot 000$: Using the Propagation Principle, we see that a $\cdot 100$ is always followed directly by the first element of a trident of $\cdot 000$'s and vice versa. So {\rm e)} is implied by {\rm g)}.
\noindent {\rm f)} $w=\cdot 001$: Take the first sequence $V(3,1,2)$ of $R_{\cdot 00}$, and put $p=3, q=1, r=2$. Then the first sequence of $R_{\cdot 000}$ is equal to $V(4,3,-1)=V(p+q,p,r-p)$. It then follows from Lemma \ref{lem:VAconv} that the first sequence of $R_{\cdot 001}$ is equal to $V(2p+q,p+q,r)=V(7,4,2)$.
\noindent {\rm g)} $w=\cdot 100$: For the first 17 numbers we check that $\cdot 100$ occurs as prefix of $\beta^-(N)$ if and only if $1000$ occurs as suffix of $\beta^+(N)$.
The result then follows from Theorem \ref{th:phi-w0}: $R_w= V(L_{m-1},L_{m-2},\gamma_w) \;\;\text{\rm if}\; w_{m-1}=0$, where here $m=4$, so $R_{1000} = V(L_{3},L_{2},\gamma_{1000})=V(4,3,-2)$. Here $\gamma_{1000}$ is determined by noting that $N=5$ is the first number in $R_{1000}$.
\noindent {\rm h)} $w= \cdot 101$: Take the sequence $R_{\cdot 10}=V(3,1,1)$, and put $p=3, q=1, r=1$. Then $R_{\cdot 100}$ is equal to $V(4,3,-2)=V(p+q,p,r-p)$. It then follows from Lemma \ref{lem:VAconv} that the sequence $R_{\cdot 101}$ is equal to $V(2p+q,p+q,r)=V(7,4,1)$.
$\Box$
The reader might think that we can now proceed, as we did earlier, from these cases to words $w$ with larger lengths $m$, using the same tools. However, this does not work. The reason is that the $\beta^-(N)$ words do not occur in lexicographical order, in contrast with the $\beta^+(N)$ words. Some occurrence sequences are Lucas-Wythoff, some are not---but still close to Lucas-Wythoff sequences.
Recall the three (Sturmian) morphisms $f,g$ and $h$ from Equation (\ref{eq:Fib3}). Note that $f$ equals the square of the Fibonacci morphism $a\mapsto ab, \, b\mapsto a$, so $f$ has fixed point $x_{\rm \scriptstyle F}$, the Fibonacci word. The fixed points $x_{\rm \scriptstyle G}, x_{\rm \scriptstyle H}$ of $g$ and $h$ are given by $x_{\rm \scriptstyle G}=b\,x_{\rm \scriptstyle F},\,x_{\rm \scriptstyle H}=a\,x_{\rm \scriptstyle F}$ ---see \cite{Berstel-Seebold} Theorem 3.1.
Let $V_{\rm \scriptstyle F}, V_{\rm \scriptstyle G} ,V_{\rm \scriptstyle H}$ denote the families of sequences having $x_{\rm \scriptstyle F}, x_{\rm \scriptstyle G}, x_{\rm \scriptstyle H}$ as first differences, with first element an arbitrary integer. Then, by definition, one example is $V=V_{\rm \scriptstyle F}$, if we take $V_{\rm \scriptstyle F}(1)=p+q+r$. We also already have encountered an $V_{\rm \scriptstyle G}$, since $V_0=V_{\rm \scriptstyle G}$, if we take $V_{\rm \scriptstyle G}(1)=r$. This follows from $V_0(p,q,r)= r, p+q+r, \dots = r, b+r, \dots$, which gives $\Delta V_0 = b x_{\rm \scriptstyle F} =x_{\rm \scriptstyle G}$. We mention that one can show that there do not exist $\alpha, p, q$, and $r$ such that $V_{\rm \scriptstyle H}$ is a generalized Beatty sequence $V = (p\lfloor n \alpha \rfloor + q n +r) $.
We conjecture that the following holds.
\begin{conjecture*} Let $\beta(N)=\beta^+(N)\cdot\beta^-(N)$ be the base phi expansion of the number $N$.\\
Let $w$ be a word of length $m$. Let $R_{\cdot w}$ be the sequence of occurrences of numbers $N$ such that the first $m$ digits of $\beta^-(N)$ are equal to $w$, i.e., $d_{-1}\dots d_{-m}(N)=w$. Then there exist two Lucas numbers $a$ and $b$ such that either $R_{\cdot w} = V_{\rm \scriptstyle F},$ or $ R_{\cdot w} = V_{\rm \scriptstyle G},$ or $R_{\cdot w} = V_{\rm \scriptstyle H}$. A second possibility is that $R_{\cdot w}$ is a union of three of such sequences. \end{conjecture*}
In all cases in Proposition \ref{prop:smallminus} the sequence $ R_{\cdot w}$ is a $V_{\rm \scriptstyle F}$, except $ R_{\cdot 010}$, which is a union of three $V_{\rm \scriptstyle G}$'s, the middle one being $V_{\rm \scriptstyle G}(4,3,-4)$. The first case where a $V_{\rm \scriptstyle G}$ as $ R_{\cdot w}$ occurs, is for $w=\cdot 1001$, where $a=29, b=18$. The first case where $V_{\rm \scriptstyle H}$ as a $R_{\cdot w}$ occurs, is as first element of the trident for the digit block $w=\cdot 0100$, where $a=18, b=11$.
\noindent AMS Classification Numbers: 11D85, 11A63, 11B39
\end{document} |
\begin{document}
\date{}
\title[Dynamics of epidemic models]{Dynamics of epidemic models with asymptomatic infection and seasonal succession}
\author{\sc Yilei Tang $^{\dag}$ \ Dongmei Xiao $^{\dag*}$
\ Weinian Zhang $^{\ddag}$ \ Di Zhu $^{\dag}$}
\thanks{$^*$ Corresponding author.}
\thanks{{\bf Funding}: The first author is partially supported by the National Natural Science Foundation of China (No. 11431008) and the European Union's Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement (No. 655212). The second author is supported by the National Natural Science Foundation of China (No. 11431008 \& 11371248). The third author is supported by the National Natural Science Foundation of China (No. 11521061 \& 11231001). }
\thanks{$^{\dag}$ School of Mathematical Science, Shanghai Jiao Tong University, Shanghai, 200240, P. R. China (xiaodm@sjtu.edu.cn (D. Xiao), mathtyl@sjtu.edu.cn (Y. Tang), di.zhu@auckland.ac.nz (D. Zhu)) }
\thanks{$^{\ddag}$ Yangtze Center of Mathematics and Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, P. R. China (matzwn@163.com (W. Zhang))}
\keywords{Epidemic model, asymptomatic infection, seasonal succession, basic reproduction number, threshold dynamics}
\subjclass{Primary 92D25, 34C23; Secondary 34D23}
\maketitle
{\bf Abstract~~} In this paper, we consider a compartmental SIRS epidemic model with asymptomatic infection and seasonal succession, which is a periodic discontinuous differential system.
The basic reproduction number $\mathcal{R}_0$ is defined and evaluated directly for this model, and the uniformly persistent of the disease
and threshold dynamics are obtained. Specially, global dynamics
of the model without seasonal force are studied. It is shown that the model has only a disease-free equilibrium which is globally stable if $\mathcal{R}_0\le 1$, and as $\mathcal{R}_0>1$ the disease-free equilibrium is unstable and the model has an endemic equilibrium, which is globally stable if the recovering rates of asymptomatic infective and symptomatic infective are close. These theoretical results provide an intuitive basis for
understanding that the asymptomatic infective individuals and the disease seasonal transmission promote the evolution of epidemic, which allow us to predict the outcomes of control strategies during the course of the epidemic.
\section{Introduction}
Since Kermack and McKendrick in \cite{Ker-McK} proposed the classical deterministic compartmental model (called SIR model) to describe epidemic outbreaks and spread, mathematical models have become important tools in analyzing the spread and control of infectious diseases, see \cite{AleM2005, May, Brauer, SeasonalScience, Heth2000, Hsi2014, THRZ08, Tow2012, xiaoruan} and references therein. The number of infected individuals used in these models is usually calculated via data in the hospitals. However, some studies on influenza show that some individuals of the population who are infected never develop symptoms, i.e. being asymptomatic infective. The asymptomatic infected individuals will not go to hospital but they can infect the susceptible by contact, then go to the recovered stage, see for instance \cite{wujh, Long, Feng}. Hence, using the data from hospitals to mathematical models to assess the epidemic, it seems that we will underestimate infection risks in epidemic.
On the other hand, seasonality is very common in ecological and human social systems (cf. \cite{xiao}). For example, variation patterns in climate are repeated every year, birds migrate according to the variation of season, opening and closing of schools are almost periodic, and so on. These seasonal factors significantly influence the survival of pathogens in the environment, host behavior, and abundance of vectors and non-human hosts. A number of papers have suggested that seasonality plays an important role in epidemic outbreaks and the evolution of disease transmissions, see \cite{PeriodWu, periodC, SeasonalD, SeasonalScience, seasonalS, Smi1983, seasonalNature, Tow2012, zhang2007}. However, it is still challenging to understand the mechanisms of seasonality and their impacts on the dynamics of infectious diseases.
Motivated by the studies above on asymptomatic infectivity or seasonality, we develop a compartmental model with asymptomatic infectivity and seasonal factors in this paper. This model is a periodic discontinuous differential system. We try to establish the theoretical analysis on the periodic discontinuous differential systems and obtain the dynamics of the model. This will allow us to draw both qualitative and quantitative conclusions on effect of the asymptomatic infectivity and seasonality on the epidemic.
The rest of the paper is organized as follows.
In section 2, we formulate the SIRS model with asymptomatic infective and seasonal factors, then discuss the existence and regularity of non-negative solutions
for this model.
In section 3, we define the basic reproduction number $\mathcal{R}_0$ for the model, give the evaluation of $\mathcal{R}_0$
and investigate the threshold dynamics of the model (or the uniformly persistent of the disease). It is shown that the length of the season, the transmission rate and the existence of asymptomatic infective affect the basic reproduction number $\mathcal{R}_0$. In section 4, we study the global dynamics of the model ignoring seasonal factor. We prove that there is a unique disease-free equilibrium and the disease always dies out when $\mathcal{R}_0\le 1$;
while when $\mathcal{R}_0> 1$ there is an endemic equilibrium which is global stable if the recovering rates of asymptomatic infective and symptomatic infective are close. A brief discussion is given in the last section.
\section{Model formulation} In this section, we first extend the classic SIRS model to a model which incorporates with the asymptomatic infective and seasonal features of epidemics, and then study the regularity of solutions of the model.
Because there are asymptomatic infectious and symptomatic infectious individuals in the evolution of epidemic, the whole population is divided into four compartments: susceptible, asymptomatic infectious, symptomatic infectious and recovered individuals. More precisely, we let $S$, $I_a$, $I_s$ and $R$ denote the numbers of individuals in the susceptible, asymptomatic, symptomatic and recovered compartments, respectively, and $N$ be the total population size. Let $\mathbb{R}_+=[0, +\infty)$, $\mathbb{Z}_+$ be the set of all nonnegative integers, and $\omega>0$ be given as the period of the disease transmissions. In addition to the assumptions of the classical SIRS model, we list the following assumptions on seasonal factors,
asymptomatic infectivity and symptomatic infectivity.
\begin{itemize} \item[(A1)] Due to the opening and closing of schools or migration of birds, each period of the disease transmission is simply divided into two seasons with high and low transmission rates, which are called high season $J_2$ and low season $J_1$, respectively. The seasonality is described by a piecewise constant function with high transmission rate $\beta_2$ in $J_2$ and low transmission rate $\beta_1$ in $J_1$, respectively, where $J_1=[m\omega, m\omega+(1-\theta)\omega )$ and $J_2=[ m\omega+(1-\theta)\omega, (m+1)\omega)$. Here $m\in \mathbb{Z}_+$, and $0<\theta<1$ which measures the fraction of the high season to the whole infection cycle. \item[(A2)] There are two classes of infective individuals: asymptomatic infective ones and symptomatic infective ones. Both of them are able to infect susceptible individuals by contact.
A fraction $\mu$ of infective individuals proceeds to the asymptomatic infective compartment and the remainder (i.e. a fraction $1-\mu$ of infective individuals) goes directly to the symptomatic infective compartment. And the asymptomatic infective and symptomatic infective individuals recover from disease at rate $r_a$ and $r_s$, respectively. \item[(A3)] The symptomatic infective individuals will get treatment in hospital or be quarantined. Hence, the symptomatic infective individuals reduce their contact rate by a fraction $\alpha$. \end{itemize}
Based on these assumptions, the classical SIRS model can be extended to the following system
\begin{equation}\label{model} \begin{cases} \dot{S}(t)=dN(t)-dS(t)-\beta(t) S(t)(I_{a}(t)+\alpha I_{s}(t))+\sigma R(t), \\ \dot{I_{a}}(t)=\mu\beta(t) S(t)(I_{a}(t)+\alpha I_{s}(t))-(d+r_{a})I_{a}(t), \\ \dot{I_{s}}(t)=(1-\mu)\beta(t) S(t)(I_{a}(t)+\alpha I_{s}(t))-(d+r_{s})I_{s}(t), \\ \dot{R}(t)=r_{a}I_{a}(t)+r_{s}I_{s}(t)-(d+\sigma)R(t), \end{cases} \end{equation} where $N(t)=S(t)+I_a(t)+I_s(t)+R(t)$, all parameters $d$, $\alpha$, $\sigma$, $\mu$, $r_a$ and $r_s$ are nonnegative, and $$ \beta(t)=\left\{ \begin{array}{ll} \beta_1, \ & t\in J_1=[m\omega, m\omega+(1-\theta)\omega ),\\ \beta_2, \ & t\in J_2=[ m\omega+(1-\theta)\omega, (m+1)\omega ).\\ \end{array} \right. $$
Parameters $\beta_2$ and $\beta_1$ are the rates of contact transmission of the disease in high season and low season respectively for which $\beta_2\ge \beta_1\ge 0$.
Besides, $d$ is both birth rate and death rate, $\alpha$, $0\le\alpha\le 1$, is the fraction of the symptomatic infective individuals reducing their contact
rate with susceptible, the fraction of infective individuals becoming asymptomatic infective $\mu$ satisfies $0\le\mu\le 1$,
parameter $\sigma$ is the rate of recovered population loss of the immunity and reentering the susceptible group,
and $r_a$ and $r_s$ are the rates of asymptomatic infective and symptomatic infective recovering with immunity, respectively.
From the biological point of view, we focus on the solutions of system (\ref{model}) with initial conditions \begin{equation}\label{initial}
S(0)=S_0 \ge 0, I_a(0)=I_{a0} \ge 0, I_s(0)=I_{s0}\ge 0, R(0)=R_0 \ge 0
\end{equation} in the first octant $\mathbb{R}_+^4$.
Note that \[ \dot{ N}(t)=\dot{S}(t) + \dot{ I}_a(t) + \dot{I}_s(t) + \dot{ R}(t) \equiv 0, \ t\in J_1 \ \rm{or}\ t\in J_2. \]
Hence, $N(t)=S_0+I_{a0}+I_{s0}+R_0$, which is a constant for almost all $t\in \mathbb{R}_+$. Since the total population does not change by the assumption, we let \[ S(t) + I_a(t) +I_s(t) + R(t) \equiv N \] for almost all $t\in \mathbb{R}_+$. Therefore, system \eqref{model} with the initial condition \eqref{initial} in $\mathbb{R}_+^4$ can be reduced to \begin{equation}\label{SIRS3} \begin{cases} \dot{S}=(d+\sigma)(N-S)-\beta (t) S(I_a+\alpha I_s)-\sigma (I_a+I_s), \\ \dot{I_a}=\mu\beta (t) S(I_a+\alpha I_s)-(d+r_a)I_a, \\ \dot{I_s}=(1-\mu)\beta (t) S(I_a+\alpha I_s)-(d+r_s)I_s, \\ S(0)=S_0, I_a(0)=I_{a0}, I_s(0)=I_{s0},\\ P_0=(S_0, I_{a0}, I_{s0})\in \mathcal{D}_0, \end{cases} \end{equation} where $\mathcal{D}_0\subset \mathbb{R}_+^3$ and \begin{equation}\label{D}
\mathcal{D}_0:=\{(S, I_a, I_s)|\; S\ge 0, I_a\ge 0, I_s\ge 0, ~0\le S+ I_a+I_s\le N \}. \end{equation} Clearly, the right hand side of system \eqref{SIRS3} is not continuous on the domain $\mathbb{R}_+\times\mathcal{D}_0$. We claim that the solution of system \eqref{SIRS3} exists globally on the interval $\mathbb{R}_+=[0, +\infty)$ and is unique.
\begin{theorem}\label{existenceUni} For any $P_0\in \mathcal{D}_0$, system \eqref{SIRS3} has a unique global solution $\varphi(t, P_0)=(S(t,P_0), I_a(t,P_0), I_s(t,P_0))$
in $\mathbb{R}_+$, which is continuous with respect to $t$ and all parameters of this system.
Moreover, $\varphi(t, P_0)\subseteq \mathcal{D}_0$ for any $t\in \mathbb{R}_+$ and the solution $\varphi(t, P_0)$ is differentiable with respect to $P_0$, where
some derivatives are one-sided if $P_0$ is on the domain boundary. \end{theorem}
\begin{proof} Assume that $\varphi(t,P_0)$ is a solution of system \eqref{SIRS3}. We first consider the two systems \begin{equation}\label{SIR-dim32} \begin{cases} \dot{S}=(d+\sigma)(N-S)-\beta_i S(I_a+\alpha I_s)-\sigma (I_a+I_s),\\ \dot{I}_a=\mu\beta_i S(I_a+\alpha I_s)-(d+r_a)I_a,\\ \dot{I}_s=(1-\mu)\beta_i S(I_a+\alpha I_s)-(d+r_s)I_s,\\ S(t_*)=S_*,\; I_{a}(t_*)=I_{a*},\; I_{s}(t_*)=I_{s*},\\ P_*=(S_*, I_{a*},I_{s*})\in \mathbb{R}_+^3 \end{cases} \end{equation} in the domain $\mathbb{R}_+\times\mathbb{R}_+^3$, $i=1, 2,$ respectively.
It is clear that for each $i$ the solution of system \eqref{SIR-dim32} exists and is unique on its maximal interval of existence, and the solution of system \eqref{SIR-dim32} is differentiable with respect to the initial value $P_*$ by the fundamental theory of ordinary differential equations.
Note that the bounded closed set $\mathcal{D}_0$ in $\mathbb{R}_+^3$ is a positive compact invariant set of system \eqref{SIR-dim32} since the vector field of system \eqref{SIR-dim32} on the boundary $\partial\mathcal{D}_0$ of $\mathcal{D}_0$ is directed toward to the interior of $\mathcal{D}_0$ or lies on $\partial\mathcal{D}_0$, where \begin{eqnarray*} \begin{split} \partial\mathcal{D}_0=&\{(S,I_a,I_s):\ (S,I_a,I_s)\in \mathbb{R}_+^3, S=0, \ 0\le I_a+I_s\le N\}\\ &\cup\{(S,I_a,I_s):\ (S,I_a,I_s)\in \mathbb{R}_+^3, I_s =0, \ 0\le S+I_a\le N\} \\
& \cup\{(S,I_a,I_s): \ (S,I_a,I_s)\in \mathbb{R}_+^3, I_a=0,\ 0\le S+I_s\le N\}\\
&\cup \{(S,I_a,I_s): \ (S,I_a,I_s)\in \mathbb{R}_+^3, S+I_s+I_a=N\}. \end{split} \end{eqnarray*} Therefore, the solution of system \eqref{SIR-dim32} exists globally for any $P_*\in \mathcal{D}_0\subset\mathbb{R}_+^3$, and these solutions are in $\mathcal{D}_0$ for all $t> 0$.
Let $\phi_i(t,t_*, P_*)$ for $i=1, 2$ be the solution semiflow of the following system \begin{equation}\label{SIRS1} \begin{cases} \dot{S}=(d+\sigma)(N-S)-\beta_i S(I_a+\alpha I_s)-\sigma (I_a+I_s),\\ \dot{I}_a=\mu\beta_i S(I_a+\alpha I_s)-(d+r_a)I_a,\\ \dot{I}_s=(1-\mu)\beta_i S(I_a+\alpha I_s)-(d+r_s)I_s,\\ \phi_i(t_*,t_*, P_*)=P_*, \ P_*\in \mathcal{D}_0, \end{cases} \end{equation} respectively, that is, $\phi_i(t,t_*, P_*) = (S(t,t_*, P_*), I_a(t,t_*, P_*), I_s(t,t_*, P_*))$ for $t\ge t_*$ is the solution of system \eqref{SIRS1} with the initial condition $\phi_i(t_*,t_*, P_*)=(S_*, I_{a*},I_{s*})\in \mathcal{D}_0$, respectively.
It follows that the solution $\varphi(t,P_0)$ for $t\ge 0$ of system \eqref{SIRS3} can be determined uniquely by induction. For simplicity, we let $s_m=(m-1)\omega$ and $t_m=s_m+(1-\theta)\omega$ for $m\in \mathbb{Z}_+$. Hence, $$ [0,\infty )=\bigcup _{m=1}^\infty [s_m, s_{m+1}] =\bigcup _{m=1}^\infty ([s_{m}, t_m]\cup [t_m, s_{m+1}]), $$ and $\varphi(t,P_0)$ can be written as follows. \begin{equation}\label{solution} \varphi(t,P_0)=\left\{
\begin{array}{ll}
\phi _1(t,s_1, P_0) &\textrm{when}\;t\in [s_1, t_1],
\\[1ex]
\phi _2(t, t_1, \phi_1(t_1, s_1, P_0)) &\textrm{when}\;\;t\in [t_1, s_2], \\[1ex]
...
\\[1ex]
\phi _1(t, s_m, u_m) & \textrm{when}\;\;t\in [s_m , t_m], \\[1ex]
\phi _2(t, t_m, v_m) & \textrm{when}\;\;t\in [t_m, s_{m+1} ],
\end{array} \right. \end{equation} where $u_m$ and $v_m$ are determined by letting $u_1=P_0$, $v_1=\phi_1(t_1, s_1, u_1)$ and $$ u_m=\phi _2(s_m, t_{m-1}, v_{m-1}),\; v_m=\phi _1(t_m, s_m, u_m) \;\;\textrm{for}\;\;m\geq 2. $$
This implies that the solution $\varphi(t,P_0)$ of system \eqref{SIRS3} exits globally in $\mathbb{R}_+$ and is unique for any $P_0\in \mathcal{D}_0$,
and it is continuous with respect to $t$ and all parameters.
By the expression \eqref{solution}, it is easy to see that the solution $\varphi(t,P_0)$ lies in $\mathcal{D}_0$ for all $t\ge 0$ and $\varphi(t,P_0)$ is differentiable with respect to $P_0$. The proof is completed. \end{proof} Theorem \ref{existenceUni} tells us that system \eqref{SIRS3} is $\omega$-periodic with respect to $t$ in $\mathbb{R}_+\times \mathcal{D}_0$, and it suffices to investigate the dynamics of its associated period map $\mathcal{P}$ on $\mathcal{D}_0$ for the dynamics of system \eqref{SIRS3}, where \begin{equation}\label{poincaremap} \begin{split} \mathcal{P}:& \ \mathcal{D}_0 \to \mathcal{D}_0,\\
\mathcal{P}(P_0)&=\varphi(\omega,P_0)=\phi _2(\omega, (1-\theta)\omega, \phi_1((1-\theta)\omega, 0, P_0)),
\end{split} \end{equation} which is continuous in $\mathcal{D}_0$.
\section{ Basic reproduction number and threshold dynamics}
In epidemiology, the basic reproduction number (or basic reproduction ratio) $\mathcal{R}_0$ is an important quantity, defined as the average number of secondary infections produced when an infected individual is introduced into a host population where everyone is susceptible. It is often considered as the threshold quantity that determines whether an infection can invade a new host population and persist. Detailedly speaking, if $\mathcal{R}_0<1$, the disease dies out and the disease cannot invade the population; but if $\mathcal{R}_0>1$, then the disease is established in the population. There have been some successful approaches for the calculations of basic reproduction number for different epidemic models. For example, Diekmann {\it et al} in \cite{Die1990} and van den Driessche and Watmough in \cite{Van2002} presented general approaches of $\mathcal{R}_0$ for autonomous continuous epidemic models. And for periodic continuous epidemic models, Wang and Zhao in \cite{Wang2008} defined the basic reproduction number.
Under some assumptions on the discontinuous states function, Guo, Huang and Zou \cite{Guo2012} determine the basic reproduction number for an SIR epidemic model with discontinuous treatment strategies. To our knowledge, there is no theoretic approach to calculate the basic reproduction number for periodic discontinuous epidemic models such as system \eqref{SIRS3}. In this section, we use the idea and some notations given in \cite{Wang2008} to define and calculate the basic reproduction numbers for system \eqref{SIRS3}, and discuss the uniformly persistent of the disease and threshold dynamics.
We define ${\bf X}$ to be the set of all disease free states of system \eqref{SIRS3}, that is $${\bf X}=\{(S,I_a,I_s):\ 0\le S\le N, I_a=I_s=0\}.$$ Clearly, the disease free subspace ${\bf X}$ is positive invariant for system \eqref{SIRS3}.
It can be checked that the period map $\mathcal{P}(P_0)$ in ${\bf X}$ has a unique fixed point at $(N,0,0)$, which is a unique disease-free equilibrium $(N,0,0)$ of system \eqref{SIRS3}, denoted by $E_0$. We now consider a population near the disease-free equilibrium $E_0$.
For simplicity, we let $\mathbf{x}=(S,I_a,I_s)^T$, and for $i=1,2$ set
\begin{equation*}\label{FV}
\begin{array}{ll}
\ \mathbf{F_i}&=\left(
\begin{array}{rrr} 0 & 0 & 0 \\ 0 & \mu\beta_iN & \alpha\mu\beta_iN \\ 0 & (1-\mu)\beta_iN & \alpha(1-\mu)\beta_iN
\end{array}
\right):=\left(
\begin{array}{rr} 0 & {\bf 0 } \\ {\bf 0} & F_i
\end{array}
\right),\\ \mathbf{V_i}&=\left(
\begin{array}{rrr} d+\sigma & \beta_iN+\sigma & \alpha\beta_iN+\sigma \\ 0 & d+r_a & 0 \\ 0 & 0 & d+r_s
\end{array}
\right):=\left(
\begin{array}{rr} d+\sigma & {\bf b_i } \\ {\bf 0} & V
\end{array}
\right).
\end{array}
\end{equation*}
Then the linearized system of \eqref{SIRS3} at $E_0$ can be rewritten as \begin{equation}\label{real_linear} \frac{d\mathbf{x}}{dt}=(\mathbf{F}(t)-\mathbf{V}(t))\mathbf{x}, \end{equation}
where $\mathbf{F}(t)=\chi_{J_1}(t)\mathbf{F}_{1}\ +\chi_{J_2}(t)\mathbf{F}_{2}\ $, $\mathbf{V}(t)=\chi_{J_1}(t)\mathbf{V}_{1}+\chi_{J_2}(t)\mathbf{V}_{2}$, and
$$
\chi_{J_i}(t)=\left\{\begin{array}{ll}
1 & \ {\textrm as }\ t\in J_i, \\ 0 & \ {\textrm as }\ t\notin J_i.
\end{array}
\right.
$$ System \eqref{real_linear} is a piecewise continuous periodic linear system with period $\omega$ in $t\in \mathbb{R}_+$. In order to determine the fate of a small number of infective individuals introduced into a disease free population, we first extend system \eqref{real_linear} from $t\in \mathbb{R}_+$ to $t\in \mathbb{R}$, and introduce some new notations. When $t\in \cup_{m=-\infty}^{+\infty}(J_1\cup J_2)=(-\infty, +\infty),$ we set $\mathbb{I}(t)=(I_a(t),I_s(t))^T$, and $$ \mathbb{F}(t)= \chi_{J_1}(t){F}_{1}\ +\chi_{J_2}(t){F}_{2}=\left(
\begin{array}{rr}
\mu N\beta(t) & \alpha\mu N\beta(t) \\
(1-\mu) N\beta(t) & \alpha(1-\mu) N\beta(t)
\end{array}
\right), $$
where
\[ \beta(t)=\left\{ \begin{array}{ll} \beta_1, \ & t\in J_1=[m\omega, m\omega+(1-\theta)\omega ),\\ \beta_2, \ & t\in J_2=[ m\omega+(1-\theta)\omega, (m+1)\omega), \ m\in \mathbb{Z}.\\ \end{array} \right.\] Clearly, $\mathbb{F}(t)$ is a $2\times 2$ piecewise continuous periodic matrix with period $\omega$ in $\mathbb{R}$, and it is non-negative. And $$ -V=\left(
\begin{array}{rr}
-(d+r_a) & 0 \\
0 & -(d+r_s)
\end{array}
\right), $$ which is cooperative in the sense that the off-diagonal
elements of $-V$ are non-negative.
Let $Y(t,s)$, $t\ge s$, be the evolution operator of the linear system \begin{equation}\label{vv} \frac{d\mathbb{I}(t)}{dt}=-V\mathbb{I}(t). \end{equation} Since $V$ is a constant matrix, for each $s\in \mathbb{R}$ the matrix $Y(t,s)$ satisfies \begin{equation}\label{v2eq} \frac{d}{dt}Y(t,s)=-V Y(t,s), \ t\ge s, \ Y(s,s)=E^2, \end{equation} where $E^2$ is a $2\times 2$ identity matrix, and $Y(t,s)=e^{-V(t-s)}$. Hence, the monodromy matrix $\Phi_{-V}(t)$ of system \eqref{vv} is $Y(t,0)$, that is, $$ \Phi_{-V}(t)=e^{-Vt}=\left(
\begin{array}{ll} e^{-(d+r_a)t} & 0 \\
0 & e^{-(d+r_s)t}
\end{array}
\right), $$
where $d$, $r_a$ and $r_s$ are positive numbers.
We denote $\|\cdot\|_1$ the $1$-norm of vector and matrix. Thus, there exist $K>0$ and $\kappa >0$ such that $$
\|Y(t,s)\|_1\le Ke^{-\kappa (t-s)}, \ \forall t\ge s, \ s\in \mathbb{R}. $$
And from the boundedness of $\mathbb{F}(t)$, i.e. $\|\mathbb{F}(t)\|_1<K_1$, it follows that there exists a constant $K_1>0$ such that \begin{equation}\label{Lineq}
\|Y(t,t-a)\mathbb{F}(t-a)\|_1\le K K_1e^{-\kappa a},\ \forall t\in \mathbb{R},\ a\in [0, +\infty). \end{equation}
We now consider the distribution of infected individuals in the periodic environment. Assume that $\mathbb{I}(s)$ is the initial distribution of infected individuals in infectious compartments. Then $\mathbb{F}(s)\mathbb{I}(s)$ is the distribution of new infections produced by the infected individuals who were introduced at time $s$.
Given $t\ge s$, then $Y(t,s)\mathbb{F}(s)\mathbb{I}(s)$
is the distribution of those infected individuals which were newly infected at time s and still remain in the infected compartments at time t. Thus, the integration of this distribution from $-\infty$ to $t$ $$ \int_{-\infty}^tY(t,s)\mathbb{F}(s)\mathbb{I}(s)ds =\int_0^{\infty}Y(t,t-a)\mathbb{F}(t-a)\mathbb{I}(t-a)da $$ gives the distribution of cumulative new infections at time t produced by all those infected individuals introduced at times earlier than $t$.
Let $\mathbb{C}_{\omega}=\mathbb{C}(\mathbb{R},\mathbb{R}^2)$ be the ordered Banach space of $\omega$-periodic continuous functions from $\mathbb{R}$ to $\mathbb{R}^2$, which is equipped with the norm $\left \lVert \cdot \right \rVert _c$, $$ \left \lVert \mathbb{I}(s) \right \rVert_c=\max _{s\in [0,\omega ]}\left \lVert\mathbb{I}(s)\right \rVert_1, $$
and the generating positive cone
$$
\mathbb{C}^+_{\omega}=\{\mathbb{I}(s)\in \mathbb{C}_{\omega}:\ \mathbb{I}(s)\ge 0, \ s\in \mathbb{R}\}.
$$
Define a linear operator $\mathcal{L}: \ \mathbb{C}_{\omega}\to \mathbb{C}_{\omega}$ by \begin{equation}\label{operate} (\mathcal{L}\mathbb{I})(t)=\int_{-\infty}^tY(t,s)\mathbb{F}(s)\mathbb{I}(s)ds=\int_0^{\infty}Y(t,t-a)\mathbb{F}(t-a)\mathbb{I}(t-a)da. \end{equation} It can be checked that the linear operator $\mathcal{L}$ is well defined.
\begin{lemma}\label{operatorC} The operator $\mathcal{L}$ is positive, continuous and compact on $\mathbb{C}_{\omega}$. \end{lemma} \begin{proof} Since $Y(t,s)=e^{-V(t-s)}$ and $\mathbb{F}(t)$ is a nonnegative bounded matrix, we get that $\mathcal{L}(\mathbb{C}^+_{\omega})\subset \mathbb{C}^+_{\omega}$. This implies that the linear operator $\mathcal{L}$ is positive.
We now prove the continuity of $\mathcal{L}$. For each $t\in \mathbb{R}$, we have \begin{eqnarray*} \begin{split}
\|\mathcal{L}\mathbb{I}(t)\|_{1}&=\left \|\int_0^{\infty}Y(t,t-a)\mathbb{F}(t-a)\mathbb{I}(t-a)da\right \|_{1}\\
&=\left \|\sum_{j=0}^\infty\int_{j\omega}^{(j+1)\omega}Y(t,t-a)\mathbb{F}(t-a)\mathbb{I}(t-a)da\right \|_{1}\\
&\leq \sum_{j=0}^\infty\int_{j\omega}^{(j+1)\omega}\|Y(t,t-a)\mathbb{F}(t-a)\mathbb{I}(t-a)\|_{1}da\\ &\leq \sum_{j=0}^\infty\int_{j\omega}^{(j+1)
\omega}KK_1e^{-\kappa a}\|\mathbb{I}(t-a)\|_1da\\
&\leq \omega K K_1\sum_{j=0}^\infty e^{-\kappa\omega j}\cdot\|\mathbb{I}\|_c \end{split} \end{eqnarray*} by \eqref{Lineq}. Hence, \begin{equation*}
\|\mathcal{L}\mathbb{I}(t)\|_{c}=\max_{t\in[0,\omega]}\|\mathcal{L}\mathbb{I}(t)\|_{1}\leq \omega K K_1\sum_{j=0}^\infty e^{-\kappa\omega j}\cdot\|\mathbb{I}\|_c, \end{equation*} which implies that $\mathcal{L}$ is continuous and uniformly bounded since $\sum_{j=0}^\infty e^{-\kappa\omega j}$ is convergent.
In the following we prove the compactness of $\mathcal{L}$. We first claim that $\mathcal{L}\mathbb{I}(t)$ is equicontinuous. Consider $\mathbb{I}(t)\in \mathbb{C}_{\omega}$ and $\forall t_1,t_2\in [0,\omega]$ with $t_1<t_2$. Then \begin{eqnarray*} \begin{split}
&\|\mathcal{L}\mathbb{I}(t_2)-\mathcal{L}\mathbb{I}(t_1)\|_1=\left\|\int_{-\infty}^{t_2}Y(t_2,s)\mathbb{F}(s)\mathbb{I}(s)ds-
\int_{-\infty}^{t_1}Y(t_1,s)\mathbb{F}(s)\mathbb{I}(s)ds\right \|_1\\
&=\left \|\int_{-\infty}^{t_2}(Y(t_2,s)-Y(t_1,s))\mathbb{F}(s)\mathbb{I}(s)ds
+\int_{t_1}^{t_2}Y(t_1,s)\mathbb{F}(s)\mathbb{I}(s)ds\right \|_1\\
&\leq \int_{-\infty}^{t_2}\|Y(t_2,s)-Y(t_1,s)\|_1\|\mathbb{F}(s)\|_1\|\mathbb{I}(s)\|_1ds+\int_{t_1}^{t_2}\|Y(t_1,s)\|_1\|\mathbb{F}(s)\|_1\|\mathbb{I}(s)\|_1ds\\
&\leq \int_{-\infty}^{\omega}\|Y(t_2,s)-Y(t_1,s)\|_1\|\mathbb{F}(s)\|_1\|\mathbb{I}(s)\|_1ds+\int_{t_1}^{t_2}Ke^{-\kappa (t_1-s)}
\|\mathbb{F}(s)\|_1\|\mathbb{I}(s)\|_1ds\\
&\leq\|e^{-Vt_2}-e^{-Vt_1}\|_1\sum_{i=-\infty}^0\int_{i\omega}^{(i+1)\omega}K_1\|e^{Vs}\|_1\|\mathbb{I}(s)\|_1ds+\int_{t_1}^{t_2}Ke^{-\kappa (t_1-s)}
K_1\|\mathbb{I}(s)\|_1ds\\
&\leq\sum_{i=-\infty}^0e^{\tilde{d}_1(i+1)\omega}\cdot K_1\|\mathbb{I}\|_c\|e^{-Vt_2}-e^{-Vt_1}\|_1+KK_1e^{\kappa \omega }\|\mathbb{I}\|_c(t_2-t_1), \end{split} \end{eqnarray*} where $\tilde{d}_1=\max\{d+r_a, d+r_s\}$.
Notice that $\sum_{i=-\infty}^0e^{N(i+1)}$ is convergent and $e^{-Vt}$ is continuous on $[0,\omega]$. Thus, if $\{\mathbb{I}(t)\}$ is bounded, for $\forall \epsilon>0$ there exists a $\delta>0$ such that $\|\mathcal{L}\mathbb{I}(t_2)-\mathcal{L}\mathbb{I}(t_1)\|_c<\epsilon$ as $|t_2-t_1|<\delta$. This implies that $\{(\mathcal{L}\mathbb{I})(t)\}$ are equicontinuous. According to Ascoli-Arzela theorem, we know that $\mathcal{L}$ is compact. The proof of this lemma is completed. \end{proof}
$\mathcal{L}$ is called the next infection operator, and the spectral radius of $\mathcal{L}$ can be defined as the basic reproduction number (or ratio) \begin{equation}\label{R_0} \mathcal{R}_0:= \rho(\mathcal{L}) \end{equation} of system \eqref{SIRS3}.
Following \cite{Wang2008}, we consider how to calculate $\mathcal{R}_0$ and whether the basic reproduction ratio (or number) $\mathcal{R}_0$ characterizes the threshold of disease invasion, i.e., the disease-free periodic solution $(N,0,0)$ of system \eqref{SIRS3} is local asymptotically stable if $\mathcal{R}_0 < 1$ and unstable if $\mathcal{R}_0 > 1$.
It is clear that the disease-free periodic solution $(N,0,0)$ of system \eqref{SIRS3} is local asymptotically stable if all characteristic multipliers of periodic system \eqref{real_linear} are less than one, and it is unstable if at least one of characteristic multipliers of periodic system \eqref{real_linear} is greater than one. By straightforward calculation, we obtain that the characteristic multipliers of periodic system \eqref{real_linear} consist of $e^{-(d+\sigma)\omega}$ and the eigenvalues of the following matrix $$ \Phi_{F-V}(\omega)=e^{(F_2-V)\theta\omega}e^{(F_1-V)(1-\theta)\omega}, $$ where \[ F_i-V=\left(
\begin{array}{rr}
\mu\beta_iN -(d+r_a) & \alpha\mu\beta_iN \\
(1-\mu)\beta_iN & \alpha(1-\mu)\beta_iN -(d+r_s)
\end{array}
\right), \ \ i=1,2. \] Note that $e^{-(d+\sigma)\omega}<1$ because $d+\sigma>0$. Therefore, all characteristic multipliers of periodic system \eqref{real_linear} are less than one if and only the largest eigenvalue of $\Phi_{F-V}(\omega)$, denoted by $\rho (\Phi_{F-V}(\omega))$, is less than one (i.e. $\rho (\Phi_{F-V}(\omega))<1$), and at least one of characteristic multipliers of periodic system \eqref{real_linear} is greater than one if and only if $\rho (\Phi_{F-V}(\omega))>1$, here $\rho (\Phi_{F-V}(\omega))$ is called {\it the spectral radius} of matrix $\Phi_{F-V}(\omega)$.
On the other hand, it is easy to check that all assumptions (A2)-(A7) in \cite{Wang2008} are valid for system \eqref{real_linear} except the assumption (A1). Using the notations in \cite{Wang2008}, we define a matrix $V_{\varepsilon}=V-\varepsilon P$, here $P=\left( {\begin{array}{cc}
1 & 1 \\ 1 & 1 \end{array} } \right)$ and $\varepsilon$ is a very small positive number.
Thus, $-V_{\varepsilon}$ is cooperative and irreducible for each $t\in \mathbb R$. Let $Y_{\varepsilon} (t,s)$ be the evolution operator of
the linear system \eqref{v2eq} with $V$ replaced by $V_{\varepsilon}$. For some small $\varepsilon_0$, as $\varepsilon\in [0,\ \varepsilon_0)$
we can define the linear operator
${\mathcal L}_{\varepsilon}$ by replacing $Y (t,s)$ in \eqref{operate} with $Y_{\varepsilon} (t,s)$ such that the operator
${\mathcal L}_{\varepsilon}$ is positive, continuous and compact on $\mathbb{C}_{\omega}$. Let $\mathcal{R}_0^{\varepsilon}:= \rho(\mathcal{L}_{\varepsilon})$
for $\varepsilon\in [0,\ \varepsilon_0)$.
By proof of Theorem \ref{existenceUni}, we know that the solutions of the following system
\begin{equation}\label{linearEq2}
\frac{dx}{dt}=({\mathbb F}(t)-V_{\varepsilon})x
\end{equation}
are continuous with respect to all parameters. Thus,
\[
\lim_{\varepsilon\to 0}\Phi_{F-V_{\varepsilon}}(\omega)=\Phi_{F-V}(\omega),
\] where $\Phi_{F-V_{\varepsilon}}(\omega)$ is the monodromy matrix of system \eqref{linearEq2}, and $\Phi_{F-V}(\omega)$ is the monodromy matrix of system \eqref{linearEq2} as $\varepsilon=0$.
According to the continuity of the spectrum of matrices, we have \[ \lim_{\varepsilon\to 0}\rho(\Phi_{F-V_{\varepsilon}}(\omega))=\rho(\Phi_{F-V}(\omega)). \] From Lemma \ref{operatorC}, we use the similar arguments in \cite{Wang2008} to the two linear operator
${\mathcal L}_{\varepsilon}$ and ${\mathcal L}$, and obtain
\[
\lim_{\varepsilon\to 0}\mathcal{R}_0^{\varepsilon}=\mathcal{R}_0.
\]
We now easily follow the arguments in \cite{Wang2008} to characterize $\mathcal{R}_0$. Let $W_{\lambda}(t, s), t\geq s$ be the fundamental solution matrix of the following linear periodic system \begin{equation*}\label{test} \frac{dw}{dt}=\left(-V+\frac{\mathbb F(t)}{\lambda}\right)w, \end{equation*} where the parameter $\lambda\in (0, +\infty)$. Consider an equation of $\lambda$ \begin{equation}\label{need} \rho(W_{\lambda}(\omega, 0))=1. \end{equation} Then $\mathcal{R}_0$ can be calculated as follows. \begin{theorem}\label{R0Characterize} \begin{itemize} \item[(i)] If equation \eqref{need} has a solution $\lambda_0>0$, then $\lambda_0$ is an eigenvalue of $\mathcal{L}$, which implies that $\mathcal{R}_0>0$; \item[(ii)] If $\mathcal{R}_0>0$, then $\lambda=\mathcal{R}_0$ is the only solution of equation \eqref{need}; \item[(iii)] $\mathcal{R}_0=0$ if and only if $\rho(W_{\lambda}(\omega, 0))<1$ for all positive $\lambda$. \end{itemize} \end{theorem}
Note that $\rho(W_1(\omega,0))=\rho(\Phi_{F-V}(\omega))$. Using similar arguments in \cite{Wang2008}, we can prove that the basic reproduction ratio (or number) $\mathcal{R}_0$ can characterize the threshold of disease invasion. \begin{theorem}\label{threshold} \begin{itemize} \item[(i)]$\mathcal{R}_0>1$ if and only if $\rho (\Phi_{F-V}(\omega))>1$;
\item[(ii)] $\mathcal{R}_0=1$ if and only if $\rho (\Phi_{F-V}(\omega))=1$;
\item[(iii)] $\mathcal{R}_0<1$ if and only if $\rho (\Phi_{F-V}(\omega))<1$.
\end{itemize}
Hence, the disease-free periodic solution $(N,0,0)$ of system \eqref{SIRS3} is local asymptotically stable if $\mathcal{R}_0<1$, and it is unstable if $\mathcal{R}_0>1$. \end{theorem}
To save space, the proofs of the above theorems are omitted. From Theorem \ref{threshold}, we can see that $\mathcal{R}_0$ is a threshold parameter for local stability of the disease-free periodic solution $(N,0,0)$. We next show $\mathcal{R}_0$ is also a threshold parameter for dynamics of system \eqref{SIRS3} in $\mathcal{D}_0$.
\begin{theorem}\label{th-E02} When $\mathcal{R}_{0}<1$, solutions $(S(t),I_{a}(t),I_{s}(t))$ of system (\ref{SIRS3}) with initial points in $\mathcal{D}_0$ satisfies $$ \lim_{t \to +\infty}(S(t),I_{a}(t),I_{s}(t))=(N, 0, 0). $$ And the disease-free periodic solution $(N,0,0)$ of system (\ref{SIRS3}) is global asymptotically stable in $\mathcal{D}_0$. \end{theorem}
\begin{proof} In the invariant pyramid $\mathcal{D}_0$ as shown in \eqref{D}, we consider a subsystem by the last two equations of system (\ref{SIRS3}) \begin{equation} \begin{cases} \label{compare-smaller} \dot{I_{a}}(t)&=\mu\beta(t)S(I_{a}+\alpha I_{s})-(d+r_{a})I_{a} \\ &\le \mu\beta(t) N(I_{a}+\alpha I_{s})-(d+r_{a})I_{a}, \\ \dot{I_{s}}(t)&= (1-\mu)\beta(t)S(I_{a}+\alpha I_{s})-(d+r_{s})I_{s} \\ &\le (1-\mu)\beta(t)N(I_{a}+\alpha I_{s})-(d+r_{s})I_{s}. \end{cases} \end{equation} Thus, the auxiliary system of \eqref{compare-smaller} is
\begin{eqnarray} \label{compare-bigger} \begin{cases} \dot{I_{a}}(t)=\mu\beta(t) N(I_{a}+\alpha I_{s})-(d+r_{a})I_{a}, \\ \dot{I_{s}}(t)=(1-\mu)\beta(t)N(I_{a}+\alpha I_{s})-(d+r_{s})I_{s}, \end{cases} \end{eqnarray} which is a periodic linear discontinuous system with period $\omega$. The periodic map associated with system \eqref{compare-bigger} is defined by $\Phi_{F-V}(\omega)$, which is a linear continuous map.
When $\mathcal{R}_{0}<1$, we have $\rho(\Phi_{F-V}(\omega))<1$, which implies that $(0,0)$ is a global asymptotically stable solution of system \eqref{compare-bigger}.
Note that systems \eqref{compare-smaller} and \eqref{compare-bigger} are cooperative. Using the similar arguments in \cite{Smi1995}, we can prove the comparison principle holds. Hence, $$ \lim_{t \to +\infty}(I_{a}(t),I_{s}(t))=(0, 0). $$
So, for arbitrarily small constant $\varepsilon>0$, there exists $T>0$
such that $I_{a}(t)+\alpha I_{s}(t)<\varepsilon$ as $t>T$. From the first equation of system (\ref{SIRS3}), \begin{equation*} \begin{split} \dot{S}&=dN-dS-\beta(t)S(I_{a}+\alpha I_{s})+\sigma(N-S-I_a-I_s) \\ &> dN-dS-\beta_2S\varepsilon. \end{split} \label{SS1} \end{equation*} Therefore, $\liminf_{t\rightarrow+\infty}S(t)\geqslant\frac{dN}{d+\beta_2\varepsilon}$. Let $\varepsilon\to 0$, we have $$
\liminf_{t\rightarrow+\infty}S(t)\geqslant N. $$ On the other hand, $S(t)\leqslant N$ in $\mathcal{D}_0$, which admits $$ \lim_{t\rightarrow+\infty}S(t)=N. $$ In summary, we have $\lim_{t \to +\infty}(S(t),I_{a}(t),I_{s}(t))=(N, 0, 0)$. Moreover, from Theorem \ref{threshold} we know that $(N,0,0)$ of system \eqref{SIRS3} is global asymptotically stable. \end{proof}
In the following, we show that the disease is uniformly persistent when $\mathcal{R}_0 > 1$.
\begin{theorem} If $\mathcal{R}_{0}>1$, $0<\mu<1$ and $0<\alpha\beta_1$, then there exists a constant $\delta_{0}>0$ such that every solution $(S(t),I_{a}(t),I_{s}(t))$ of system (\ref{SIRS3}) with initial value in $\mathcal{D}_0$ satisfies $$ \liminf_{t \to +\infty}I_{a}(t)\geqslant\delta_{0},\quad \liminf_{t \to +\infty}I_{s}(t)\geqslant\delta_{0}. $$
\end{theorem}
\begin{proof} Since system \eqref{SIRS3} is $\omega$-periodic with respect to $t$ in $\mathbb{R}_+\times \mathcal{D}_0$, it suffices to investigate the dynamics of its associated period map $\mathcal{P}$ defined by \eqref{poincaremap} on $\mathcal{D}_0$ for the dynamics of system \eqref{SIRS3}, where the map $\mathcal{P}$ is continuous. Clearly, $\mathcal{P}(\mathcal{D}_0)\subset \mathcal{D}_0$. Define $$ X_{0}=\{(S,I_a,I_s)\in\mathcal{D}_0: I_a>0,I_s>0\},\ \partial{X_{0}}=\mathcal{D}_0 \backslash X_{0}. $$
Set $$ M_{\partial}=\{P_0\in \partial X_{0} :\mathcal{P}^k(P_0)\in\partial X_{0}, \forall k\ge 0 \}, $$ which is a positive invariant set of $\mathcal{P}$ in $\partial X_{0}$. We claim \begin{equation} \label{M-partial} M_{\partial}=\{(S,0,0):0\leqslant S\leqslant N\}. \end{equation} In fact, $\{(S,0,0):0\leqslant S\leqslant N\}\subset M_{\partial}$ by \eqref{poincaremap}. On the other hand, for any $P_0\in\partial{X_{0}}\setminus \{(S,0,0):0\leqslant S\leqslant N\}$, that is either
$I_{a0}=0, I_{s0}>0, S_0\ge0$ or $I_{a0}>0,I_{s0}=0, S_0\ge0$. In the case $I_{a0}=0, I_{s0}>0, S_0>0$ (resp. $I_{a0}>0, I_{s0}=0, S_0>0$), we calculate by the last two equations of system \eqref{SIRS3} and obtain that \begin{eqnarray*} I_{a}'(0)=\mu\alpha\beta(0) S(0)I_{s}(0)>0\ ({\rm resp.} \ ~I_{s}'(0)=(1-\mu)\beta(0) S(0)I_{a}(0)>0), \end{eqnarray*} if $0<\mu<1$ and $0<\alpha\beta_1$. This implies that $\mathcal{P}^{k_0}(P_0)\not\in\partial{X_{0}}\setminus \{(S,0,0):0\leqslant S\leqslant N\}$ for some $k_0\ge0$ since the subsystem by the last two equations of system \eqref{SIRS3} is cooperative. If $S(0)=0, I_{a0}=0, I_{s0}>0$ (or $S(0)=0, I_{a0}>0,I_{s0}=0$), then $S'(0)=(d+\sigma)N-\sigma I_s(0)>0$ (or $S'(0)=(d+\sigma)N-\sigma I_a(0)>0$), which leads that $\mathcal{P}^{k_1}(P_0)\not\in\partial{X_{0}}\setminus \{(S,0,0):0\leqslant S\leqslant N\}$ for some $k_1\ge 0$. Therefore, \eqref{M-partial} is proved and $M_{\partial}$ is the maximal compact invariant set of $\mathcal{P}$ in $\partial X_{0}$.
Note that $E_0(N, 0, 0)$ is the unique fixed point of $\mathcal{P}$ in $M_{\partial}$ and it is an attractor of $\mathcal{P}$ in $M_{\partial}$ by the first equation of \eqref{SIRS3}. Since $\mathcal{R}_{0}>1$, the stable set $W^s(E_0)$ of $E_0$ satisfies that $W^s(E_0) \cap X_0 =\emptyset$.
Applying \cite[Theorem 1.3.1]{Zhao2003}, we obtain that $\mathcal{P}$ is uniformly persistence with respect to $(X_0, \partial X_0)$. Moreover, from \cite[Theorem 3.1.1]{Zhao2003}, it can see that the conclusion of this theorem is true. The proof is completed. \end{proof}
\section{Global dynamics of system \eqref{SIRS3} without seasonal force}
In this section, we study the effects of asymptomatic infection on dynamics of system \eqref{SIRS3} if there are not seasonal factors, that is,
$\beta_1=\beta_2=\beta$. Then system \eqref{SIRS3} becomes \begin{equation}\label{SIR-dim3} \begin{cases} \dot{S}=(d+\sigma)(N-S)-\beta S(I_a+\alpha I_s)-\sigma (I_a+I_s),\\ \dot{I_a}=\mu\beta S(I_a+\alpha I_s)-(d+r_a)I_a,\\ \dot{I_s}=(1-\mu)\beta S(I_a+\alpha I_s)-(d+r_s)I_s \end{cases} \end{equation} in the domain $\mathbb{R}_+^3$.
By the formula \eqref{R_0}, we let $\beta_1=\beta_2$ and obtain the basic reproduction number
$\mathcal{R}_0$ of system \eqref{SIR-dim3} as follows.
\begin{eqnarray}\label{cons-R0} \mathcal{R}_0=\beta N \left( \frac{\mu}{d+r_{a}}+\frac{\alpha(1-\mu)}{d+r_{s}} \right), \end{eqnarray}
which is consistent with the number calculated using the approach of basic reproduction number in \cite{Die1990} and \cite{Van2002}.
From the expression \eqref{cons-R0}, we can see that there is still the risks of infectious disease outbreaks due to the existence of asymptomatic infection
even if all symptomatic infective individuals has been quarantined, that is, $\alpha=0$. This provides an intuitive basis for
understanding that the asymptomatic infective individuals promote the evolution of epidemic.
In the following we study dynamics of system \eqref{SIR-dim3}. By a straightforward calculation, we obtain the existence of equilibrium for system \eqref{SIR-dim3}.
\begin{lemma}\label{existen} \label{L-equils}(Existence of equilibrium) System (\ref{SIR-dim3}) has the following equilibria in $\mathbb{R}_+^3$. \begin{itemize} \item[(i)] If $\mathcal{R}_0\le 1$, then system (\ref{SIR-dim3}) has a unique equilibrium, which is the disease-free equilibrium $E_0(N,0,0)$. \item[(ii)] If $\mathcal{R}_0>1$ and $0<\mu<1$, then system (\ref{SIR-dim3}) has two equilibria: the disease-free equilibrium $E_0(N,0,0)$ and the endemic equilibrium $E_1(S^*, I_a^*, I_s^*)$ in the interior of $\mathcal{D}_0$, where
$S^* = \frac{N}{\mathcal{R}_0}, I_a^* = \frac{\mu(d+\sigma) (d+r_s)N} {(d+r_a)(d+r_s)+\sigma(d+\mu r_s)+\sigma r_a(1-\mu)} (1-\frac{1}{\mathcal{R}_0}), I_s^* =\frac{(1-\mu)(d+r_a)}{\mu(d+r_s)}I_a^*.$
\item[(iii)] If $\mathcal{R}_0>1$ and $\mu=0$, then system (\ref{SIR-dim3}) has two equilibria: the disease-free equilibrium $E_0(N,0,0)$ and the asymptomatic-free equilibrium $E_2 (S_2^*,0,I_{s2}^*)$, where
$S_2^*= \frac{N}{\mathcal{R}_0}, I_{s2}^*= \frac{d+\sigma}{d+\sigma+r_s}N(1-\frac{1}{\mathcal{R}_0}).$
\item[(iv)] If $\mathcal{R}_0>1$ and $\mu=1$, the system (\ref{SIR-dim3}) has two equilibria: the disease-free equilibrium $E_0(N,0,0)$ and the symptomatic-free equilibrium $E_3 (S_3^*,I_{a3}^*,0)$, where
$S_3^*= \frac{N}{\mathcal{R}_0}, I_{a3}^*= \frac{d+\sigma}{d+\sigma+r_a}N(1-\frac{1}{\mathcal{R}_0}).$
\end{itemize} \end{lemma}
We now discuss the local stability and topological classification of these equilibria in $\mathbb{R}_+^3$, respectively. We first study the disease-free equilibrium $E_0(N, 0, 0)$ and have the following lemma.
\begin{lemma} \label{localstablity} The disease-free equilibrium $E_0(N, 0, 0)$ of system (\ref{SIR-dim3}) in $\mathbb{R}_+^3$ is asymptotically stable if $\mathcal{R}_0<1$; $E_0(N, 0, 0)$ is a saddle-node with one dimensional center manifold and two dimensional stable manifold if $\mathcal{R}_0=1$; and $E_0(N, 0, 0)$ is a saddle with two dimensional stable manifold and one dimensional unstable manifold if $\mathcal{R}_0>1$. \end{lemma}
\begin{proof}
A routine computation shows that the characteristic polynomial of system (\ref{SIR-dim3}) at $E_0$ is \begin{eqnarray}\label{ChEq} f_1(\lambda) = (\lambda+d+\sigma)(\lambda^{2}-a_1\lambda+a_0), \end{eqnarray} where $a_0=(d+r_{a})(d+r_{s})(1-\mathcal{R}_0),$ $$ a_1=(d+r_{a})(\beta N\frac{\mu}{d+r_{a}}-1)+(d+r_{s})(\alpha\beta N\frac{1-\mu}{d+r_{s}}-1) . $$ It is clear that $-(d+\sigma)<0$ is always one root of \eqref{ChEq}. We divide three cases: $\mathcal{R}_0<1$, $\mathcal{R}_0=1$ and $\mathcal{R}_0>1$ to discuss the other roots of \eqref{ChEq}.
If $\mathcal{R}_0<1$, then $a_1<0$ and $a_0>0$ by $\beta N\frac{\mu}{d+r_{a}}<\mathcal{R}_0$ and $\beta N\frac{\alpha(1-\mu)}{d+r_{s}}<\mathcal{R}_0$. Thus, three roots of \eqref{ChEq} have negative real parts, which leads to the local asymptotically stable of the disease-free equilibrium $E_0$.
If $\mathcal{R}_0=1$, then $a_0=0$ and $a_1<0$. Hence, the characteristic equation $f_1(\lambda) =0$ has three roots: $\lambda_1=-(d+\sigma)<0$, $\lambda_2=a_1<0$ and $\lambda_3=0$. For calculating the associated eigenvectors $v_i$ of $\lambda_i$, $i=1,2,3$, we consider $J(E_0)$ with respect to $\mu$ in three cases: (i) $0<\mu<1$, (ii) $\mu=0$ and (iii) $\mu=1$, and we can obtain that $E_0$ is a saddle-node with one dimensional center manifold and two dimensional stable manifold by tedious calculations of normal form.
Summarized the above analysis, we complete proof of this lemma. \end{proof}
From lemma \ref{existen} and lemma \ref{localstablity}, we can see that system \eqref{SIR-dim3} undergoes saddle-node bifurcation in a small neighborhood of $E_0(N,0,0)$ as $\mathcal{R}_0$ increases passing through $\mathcal{R}_0=1.$
About the endemic equilibria, we have the following local stability.
\begin{lemma} \label{L-E1} The endemic equilibrium $E_1(S^*, I_a^*, I_s^*)$ of system (\ref{SIR-dim3}) is asymptotically stable if $\mathcal{R}_0>1$ and $0<\mu<1$; the asymptomatic-free equilibrium $E_2 (S_2^*,0,I_{s2}^*)$ of system (\ref{SIR-dim3}) is asymptotically stable if $\mathcal{R}_0>1$ and $\mu=0$; and the symptomatic-free equilibrium $E_3 (S_3^*,I_{a3}^*,0)$ of system (\ref{SIR-dim3}) is asymptotically stable if $\mathcal{R}_0>1$ and $\mu=1$. \end{lemma}
\begin{proof} Either $\mu=0$ or $\mu=1$, it is easy to compute the eigenvalues of the Jacobian matrix of system (\ref{SIR-dim3}) at $E_2$ or $E_3$, respectively, and find that all eigenvalues have the negative real parts. Hence, $E_2 (S_2^*,0,I_{s2}^*)$ or $E_3 (S_3^*,I_{a3}^*,0)$ is asymptotically stable if $\mathcal{R}_0>1$, respectively.
After here we only prove that $E_1(S^*, I_a^*, I_s^*)$ is asymptotically stable if $\mathcal{R}_0>1$ and $0<\mu<1$. To make the calculation easier, we use the variables change \begin{equation*}\label{change1} S= \frac{ (d+r_s)}{\mu\beta} \hat{S}, ~I_a= \frac{ (d+r_s)}{\beta} \hat{I}_a, ~~I_s= \frac{ (d+r_s)}{\beta}\hat{I}_s, ~dt= \frac{d\tau}{(d+r_s)}, \end{equation*} which reduces system (\ref{SIR-dim3}) into the following system, \begin{equation}\label{SIR-1} \begin{cases} \frac{dS}{d\tau}=N_1-d_1S-\sigma_1 I_a-\sigma_1I_s - S(I_a+\alpha I_s),\\ \frac{dI_a}{d\tau}=-r I_a + S(I_a+\alpha I_s),\\ \frac{dI_s}{d\tau}=-I_s +\mu_1 S(I_a+\alpha I_s), \end{cases} \end{equation} where \begin{eqnarray*}\label{Pchange1} \begin{split} N_1 &=N (d+\sigma) \mu \beta/(d+r_s)^2 , ~d_1=(d+\sigma)/(d+r_s), \\ \sigma_1 &=\sigma\mu/(d+r_s), ~r=(d+r_a)/(d+r_s), ~\mu_1=(1-\mu)/\mu \end{split} \end{eqnarray*} and for simplicity we denote $\hat{S}, \hat{I}_a, \hat{I}_s$ by $S, I_a, I_s$ respectively.
When $\mathcal{R}_0>1$, the disease-free equilibrium $E_0(N, 0, 0)$ and endemic equilibrium $E_1(S^*, I_a^*, I_s^*)$ of system (\ref{SIR-dim3}) are transformed into the disease-free equilibrium $\hat{E}_0(N_1/d_1, 0, 0)$ and endemic equilibrium $\hat{E}_1(\hat{S}^*, \hat{I}_a^*, \hat{I}_s^*)$ of system (\ref{SIR-1}) respectively, where \begin{eqnarray*} \begin{split} \hat{S}^* =\frac{N_1/d_1}{\hat{R}_0}, ~\hat{I}_a^* = \frac{N_1}{\sigma_1+r \sigma_1 \mu_1+r}(1-\frac{1}{\hat{R}_0}), ~\hat{I}_s^* = \mu_1 r I_a^*. \end{split} \end{eqnarray*} Notice that $\hat{R}_0:=\frac{N_1}{d_1}(\frac{1}{r}+\alpha\mu_1)>1$ if and only if $\mathcal{R}_0>1$.
The characteristic equation of system (\ref{SIR-1}) at $\hat{E}_1$ is \begin{eqnarray*}\label{Ch-E1}
f_2(\lambda) ={\rm det} (\lambda I-J(\hat{E}_1) ) = \lambda^3+ \xi_2 \lambda^2 + \xi_1 \lambda +\xi_0,
\end{eqnarray*} where \begin{eqnarray*} \begin{split} \xi_2 &=\{\sigma_1+r \sigma_1 \mu_1+r+r^2 \mu_1 \alpha \sigma_1+r^3 \mu_1^2 \alpha \sigma_1+r^3 \mu_1 \alpha+d_1 \sigma_1 \mu_1 r \alpha+d_1 \sigma_1 \mu_1^2 r^2 \alpha+N_1 \\ ~~ & +d_1 \sigma_1+d_1 r \sigma_1 \mu_1+2 N_1 \mu_1 r \alpha+r^2 \mu_1^2 \alpha^2 N_1\}/\{(\sigma_1+r \sigma_1 \mu_1+r)(r \mu_1 \alpha+1)\}, \\ \xi_1 &= d_1 (1+r^2 \mu_1 \alpha)/(r \mu_1 \alpha+1) +(\sigma_1 \mu_1+1+r+\sigma_1) (r \mu_1 \alpha+1) \hat{I}_a^*, \\ \xi_0 &=N_1\mu_1 r\alpha+N_1-rd_1=rd_1(\hat{R}_0-1). \end{split} \end{eqnarray*} It can be seen that all coefficients $\xi_j$ of polynomial $f_2(\lambda)$ are positive if $\hat{R}_0>1$, where $j=0, 1, 2$. Moreover, we claim that $\xi_2\xi_1-\xi_0>0$. In fact, \begin{eqnarray*} \begin{split} \xi_2\xi_1-\xi_0 =c_0+c_1 \hat{I}_a^* +c_2 (\hat{I}_a^*)^2, \end{split} \end{eqnarray*} where \begin{eqnarray*} \begin{split} c_0= & ~ \frac{d_1(1+r^2\mu_1 \alpha) (r^2 \mu_1 \alpha+ d_1\mu_1 r\alpha+1+d_1)}{(r \mu_1 \alpha+1)^2}, \\ c_1= & ~ d_1 \mu_1^2 r \alpha \sigma_1+r^3 \mu_1 \alpha+r^2 \mu_1 \alpha \sigma_1+2 d_1 r^2 \mu_1 \alpha+d_1 \sigma_1 \mu_1 r \alpha+\sigma_1 \mu_1 \\ &~+d_1 \sigma_1 \mu_1+1+2 d_1+d_1 \sigma_1 +r(d_1-\sigma_1 \mu_1) + \mu_1 r \alpha(d_1-\sigma_1), \\ c_2= & ~ (r \mu_1 \alpha+1)^2 (\sigma_1\mu_1+1+r+\sigma_1). \end{split} \end{eqnarray*} It is easy to see that $c_0>0$ and $c_2>0$. Note that $ d_1-\sigma_1 \mu_1=\frac{d+\sigma \mu}{d+r_s}>0$ and $d_1-\sigma_1=\frac{d+\sigma (1- \mu)}{d+r_s}>0$ since $0<\mu<1$. This implies that $c_1>0$. Moreover, $\hat{I}_a^*>0$ yields that $\xi_2\xi_1-\xi_0>0$ and what we claimed is proved.
By the Routh-Hurwitz Criterion, we know that all eigenvalues of the characteristic polynomial $f_2(\lambda)$ have negative real parts. Thus, endemic equilibrium $\hat{E}_1$ of system (\ref{SIR-1}) is asymptotically stable. This leads that endemic equilibrium $E_1(S^*, I_a^*, I_s^*)$ of system (\ref{SIR-dim3}) is also asymptotically stable. \end{proof}
From lemma \ref{localstablity} and lemma \ref{L-E1}, we can see that $\mathcal{R}_0$ is the threshold quantity of local dynamics of system \eqref{SIR-dim3} in $\mathbb{R}^3_+$.
By Theorem \ref{existenceUni}, we only need to consider system \eqref{SIR-dim3}
for its global dynamics in $\mathcal{D}_0$. The following theorems will show that $\mathcal{R}_0$ is also the threshold quantity of global dynamics of system \eqref{SIR-dim3} in $\mathcal{D}_0$.
\begin{theorem} \label{globalE0} If $\mathcal{R}_0\le 1$, then the disease-free equilibrium $E_0(N, 0, 0)$ of system (\ref{SIR-dim3}) is global asymptotically stable in $\mathcal{D}_0$. \end{theorem}
The proof of this theorem can be finished by constructing a Liapunov function
\begin{equation*}
\label{Lia1} L(S, I_a,I_s) =I_a(t) +\frac{d+r_a}{d+r_s}\alpha I_s(t) \end{equation*} in $\mathcal{D}_0$. For saving space, we bypass it.
\begin{theorem} \label{globalE1} If $\mathcal{R}_0> 1$ and $\mu=0$ (resp. $\mu=1$), then $E_2(S_2^*,0,I_{s2}^*)$ (resp. $E_3(S_3^*,I_{a3}^*,0)$) attracts all orbits of system (\ref{SIR-dim3}) in $\mathcal{D}_0$ except both $E_0(N, 0, 0)$ and a positive orbit $\gamma$ in its two dimensional stable manifold, where $$ \gamma=\{(S,I_a,I_s)\in \mathcal{D}_0:\ I_a=0,\ I_s=0, \ 0<S<N \} . $$ \end{theorem}
\begin{proof} We first prove the case that $\mathcal{R}_0> 1$ and $\mu=0$. When $\mu=0$, system (\ref{SIR-dim3}) becomes \begin{equation}\label{mu0} \begin{cases} \dot{S}=(d+\sigma)(N-S)-\beta S(I_a+\alpha I_s)-\sigma (I_a+I_s),\\ \dot{I_a}=-(d+r_a)I_a,\\ \dot{I_s}=\beta S(I_a+\alpha I_s)-(d+r_s)I_s. \end{cases} \end{equation} It is clear that $\lim_{t\to +\infty}I_a(t)=0$. Hence, the limit system of system \eqref{mu0} in $\mathcal{D}_0$ is \begin{equation}\label{mu0limit} \begin{cases} \dot{S}=(d+\sigma)(N-S)-\alpha\beta S I_s-\sigma I_s,\\ \dot{I_s}=\alpha\beta S I_s-(d+r_s)I_s \end{cases} \end{equation} in $\mathcal{D}_{1}=\{(S,I_s):\ 0\le S\le N,\ 0\le I_s\le N\}$, which has two equilibria: $(N,0)$ and $(S_2^*,I_{s2}^*)$. Equilibrium $(N,0)$ is a saddle and $(S_2^*,I_{s2}^*)$ is locally asymptotically stable if $\mathcal{R}_0>1$.
In the following we prove that $(S_2^*,I_{s2}^*)$ attracts all orbits of system \eqref{mu0limit} in $\mathcal{D}_{1}$ except both $(N,0)$ and its one dimensional stable manifold.
Let $x=S+\frac{\sigma}{\alpha\beta}$ and $y=I_s$. Then system \eqref{mu0limit} becomes \begin{equation}\label{mu0xy} \begin{cases} \dot{x}=(d+\sigma)(N+\frac{\sigma}{\alpha\beta})-(d+\sigma)x-\alpha\beta xy,\\ \dot{y}=\alpha\beta xy-(d+r_s+\sigma)y. \end{cases} \end{equation} Hence, $(x_0,y_0)=(S_2^*+\frac{\sigma}{\alpha\beta},I_{s2}^*)$ is the unique positive equilibrium of system \eqref{mu0xy} if $\mathcal{R}_0>1$. Consider the Liapunov function of system \eqref{mu0xy} $$ V(x,y)=\frac{1}{2}(x-x_0)^2+x_0\left(y-y_0-y_0\ln\frac{y}{y_0}\right) $$ in $\tilde{\mathcal{D}}_{1}=\{(x,y):\ \frac{\sigma}{\alpha\beta}\le x\le N+\frac{\sigma}{\alpha\beta},\ 0\le y\le N\}$. It is clear that $V(x,y)\ge 0$ and $V(x,y)=0$ if and only if $x=x_0$ and $y=y_0$ in $\tilde{\mathcal{D}}_{1}$. And $$
\frac{dV(x(t),y(t))}{dt}|_{\eqref{mu0xy}}=-(x-x_0)^2(\alpha\beta y+d+\sigma)\le 0 $$ in $\tilde{\mathcal{D}}_{1}$.
By LaSalle's Invariance Principle, we know that $(x_0,y_0)$ attracts all orbits of system \eqref{mu0xy} in $\tilde{\mathcal{D}}_{1}$ except both equilibrium $(N+\frac{\sigma}{\alpha\beta},0)$ and its one dimensional stable manifold $\{(x,y):\ y=0, 0<x<N+\frac{\sigma}{\alpha\beta}\}$. This leads to the conclusion, $E_2(S_2^*,0,I_{s2}^*)$
attracts all orbits of system (\ref{SIR-dim3}) in $\mathcal{D}_0$ except both $E_0(N, 0, 0)$ and a positive orbit $\gamma $ if $\mathcal{R}_0> 1$ and $\mu=0$.
Using the similar arguments, we can prove that $E_3(S_3^*,I_{a3}^*,0)$ attracts all orbits of system (\ref{SIR-dim3}) in $\mathcal{D}_0$
except both $E_0(N, 0, 0)$ and a positive orbit $\gamma $ if $\mathcal{R}_0> 1$ and $\mu=1$. \end{proof}
\begin{theorem} \label{th-globE1} If $\mathcal{R}_0> 1$, $0<\mu<1$ and $r_a=r_s$, then the endemic equilibrium $E_1(S^*, I_a^*, I_s^*)$ attracts all orbits of system (\ref{SIR-dim3}) in $\mathcal{D}_0$ except $E_0(N, 0, 0)$.
\end{theorem}
\begin{proof} Let
$I=I_a+\alpha I_s, ~~~N_1=S+I_a+I_s$. Then under the assumption $r_a=r_s=r$, system (\ref{SIR-dim3}) in $\mathcal{D}_0$ can be written as \begin{equation}\label{SIN} \begin{cases} \dot{S}=(d+\sigma)N -\sigma N_1 -dS -\beta SI,\\ \dot{I}=\tilde{\mu}SI-(d+r)I,\\ \dot{N_1}=(d+\sigma)N -(d+r+\sigma) N_1+rS \end{cases} \end{equation}
in $\tilde{\mathcal{D}}_0:=\{(S, I, N_1)|\; S\ge 0, I\ge 0, ~N\ge N_1\ge 0 \}$, where $\tilde{\mu}=(\mu+\alpha(1-\mu))\beta$.
Thus, equilibrium $E_1(S^*, I_a^*, I_s^*)$
of system (\ref{SIR-dim3}) becomes equilibrium $\tilde{E}_1(S^*, I^*, N_1^*)$ of system (\ref{SIN}) and $\tilde{E}_1$ is locally asymptotically stable,
where $I^*=I_a^*+\alpha I_s^*, N_1^*=S^* + I_a^* + I_s^*$.
Applying a typical approach of Liapunov functional, we define \begin{equation*}\label{gx} g(x) = x- 1- \ln x, \end{equation*} and construct a Liapunov functional of system (\ref{SIN}) \begin{equation*}\label{V1} V_1(S, I, N_1)=\frac{\nu_1}{2} (S-S^*)^2+\nu_2 I^*g(\frac{I}{I^*}) + \frac{\nu_3}{2} (N_1-N_1^*)^2, \end{equation*} where arbitrary constant $\nu_1>0$, $\nu_2=\nu_1\beta S^*/\tilde{\mu}$ and $\nu_3=\nu_1\sigma/r$. Note that $g (x)\ge g (1) = 0$ for all $x > 0$ and the global minimum $g (x) = 0$ is attained if and only if $x = 1$. Thus, $V_1(S, I, N_1)\ge 0$ and $V_1(S, I, N_1)=0$ if and only if $S=S^*$, $I=I^*$ and $N_1=N_1^*$ in $\tilde{\mathcal{D}}_0$.
The derivative of $V_1$ along the trajectories of system \eqref{SIN} is \begin{equation*}\label{dV1} \begin{split} \frac{dV_1(S,I, N_1)}{dt}=& -\nu_1 d {S^*}^2(x-1)^2 -\nu_3(d+r+\sigma) {N_1^*}^2(z-1)^2 \\ & - \nu_1\beta {S^*}^2I^*y(x-1)^2\le 0, \end{split} \end{equation*}
where $x=\frac{S}{S^*}, ~~y=\frac{I}{I^*}, ~~ z=\frac{N_1}{N_1^*}$.
Note that the only compact invariant subset of the set $\{(S,I, N_1):\ \frac{dV_1(S,I, N_1)}{dt}=0\}$ is the singleton $\tilde{E}_1(S^*, I^*, N_1^*)$ in $\tilde{\mathcal{D}}_0$. Consequently, we can conclude that $E_1(S^*, I_a^*, I_s^*)$ is globally asymptotically stable and attracts all orbits of system (\ref{SIR-dim3}) in $\mathcal{D}_0$ except $E_0(N, 0, 0)$.
\end{proof}
From Theorem \eqref{th-globE1} and the continuity of solutions with respect to parameters $r_a$ and $r_s$, we obtain
the following results. \begin{theorem} \label{th-globE1-2} If $\mathcal{R}_0> 1$ and $0<\mu<1$, then the endemic equilibrium $E_1(S^*, I_a^*, I_s^*)$
is globally asymptotical stable in the interior of $\mathcal{D}_0$ for $ 0<|r_s-r_a|\ll 1$. \end{theorem}
\section{Discussion}
In this model, we divide the period of the disease transmission into two seasons. In fact, it can be divided into $n$ seasons for any given $n\in\mathbb{Z}_+$. Compared with continuous periodic systems, our piecewise continuous periodic model can provide a straightforward method to evaluate the basic reproduction number $\mathcal{R}_0$, that is to calculate the spectral radius of matrix $\Phi_{F-V}(\omega)=e^{(F_2-V)\theta\omega}e^{(F_1-V)(1-\theta)\omega}$.
It is shown that the length of the season, the transmission rate and the existence of asymptomatic infective affect the basic reproduction number $\mathcal{R}_0$, and there is still the risks of infectious disease outbreaks due to the existence of asymptomatic infection
even if all symptomatic infective individuals has been quarantined, that is, $\alpha=0$. This provides an intuitive basis for
understanding that the asymptomatic infective individuals and the disease seasonal transmission promote the evolution of epidemic.
And theoretical dynamics of the model allow us to predictions of outcomes of control strategies during the course of the epidemic.
\end{document} |
\begin{document}
\title{Nominal Sets in Agda\
A Fresh and Immature Mechanization}
\begin{abstract}
In this paper we present our current development on a new
formalization of nominal sets in Agda. Our first motivation in
having another formalization was to understand better nominal sets
and to have a playground for testing type systems based on nominal
logic. Not surprisingly, we have independently built up the same
hierarchy of types leading to nominal sets. We diverge from other
formalizations in how to conceive finite permutations: in our
formalization a finite permutation is a permutation (i.e. a
bijection) whose domain is finite. Finite permutations have
different representations, for instance as compositions of
transpositions (the predominant in other formalizations) or
compositions of disjoint cycles. We prove that these
representations are equivalent and use them to normalize (up to
composition order of independent transpositions) compositions of
transpositions. \end{abstract}
\section{Introduction} \label{sec:intro}
Nominal sets were introduced to Computer Science by Gabbay and Pitts to give an adequate mathematical universe that permits the definition of inductive sets with binding \cite{Gabbay2002}. Instead of taking equivalence classes of inductively defined sets (as in a formal treatment of, say, the Lambda Calculus) or a particular representation of the variables (as in the de Bruijn approach to Lambda Calculus), nominal sets have a notion of name abstraction that ensures all the properties expected for binders; in particular, alpha-equivalent lambda terms are represented by the same element of the nominal set of lambda terms.
In this paper we present a new mechanization \cite{pagano2022} of nominal sets. Most of the current mechanizations of nominal sets represent finite permutations as compositions of transpositions, where transpositions are represented by pairs of atoms and compositions as lists. In contrast, our starting point is permutations (i.e. bijective functions); finite permutations are permutations that can be represented by composition of transpositions. Moreover they conflate the set of atoms mentioned in a list with the domain of the (represented) permutation. Pondering about this issue, we decided to develop a ``normalization'' procedure for representations of finite permutations; in order to prove its correctness, we were driven to introduce a cycle notation.
The rest of this paper is structured into four sections. In Sect.~\ref{sec:fundamentals} we summarize the fundamentals of Nominal Sets; in Sect.~\ref{sec:perm} we explain the different representations of finite permutations and their equivalence; then, in Sect.~\ref{sec:formalization} we present the most salient aspects of our mechanization in Agda; and finally in Sect.~\ref{sec:conclusion} we conclude by mentioning related works and contrasting them with our approach, indicating also our next steps. We assume some knowledge of Agda, but also hope that the paper can be followed by someone familiar with any other language based on type theory.
\section{Fundamentals of Nominal Sets} \label{sec:fundamentals} In this section we summarize the main concepts underlying the notion of Nominal Sets; for a more complete treatment we refer the reader to \cite{Pitts2013-book}. We repeat the basic definitions of group and group action. A \emph{group} is a set $G$ with a distinguished element ($\epsilon \in G$, the \emph{unit}), a binary operation ($\_\cdot\_\colon G\times G \to G$, the \emph{multiplication}), and a unary operation ($\_^{-1} \colon G\to G$, the \emph{inverse}), satisfying the following axioms: \begin{align*}
\text{Associativity:} && g_1 \cdot (g_2 \cdot g_3) \ &= \ (g_1 \cdot g_2) \cdot g_3 && ,\forall g_1,g_2,g_3 \in G\\
\text{Inverse element:} && g\cdot (g^{-1}) \ &= \ \epsilon \ = \ g^{-1} \cdot g && ,\forall g \in G\\
\text{Identity element:} && \epsilon\cdot g \ &= \ g \ = \ g\cdot \epsilon && ,\forall g \in G \end{align*}
\noindent Although a group is given by the tuple $(G,\epsilon,\_\cdot\_,\_^{-1})$ (and the proofs that these operations satisfy the axioms) we will refer to the group simply by $G$. A sub-group of $G$ is a subset $H\subseteq G$ such that $\epsilon\in H$ and $H$ is closed under the inverse and multiplication.
Let $G$ be a group. A \emph{$G$-set} is a set $X$ with an operation $\gactUn \colon G\times X \to X$ (called the \emph{action}) satisfying: \begin{align*} \text{Identity:} && \gact{\epsilon}{x} & \ = \ x && ,\forall x \in X\\ \text{Compatibility:} && \gact{g_1}{(\gact{g_2}{x})} & \ = \ \gact{(g_1\cdot g_2)}{x} && ,\forall g_1,g_2 \in G, \forall x \in X \end{align*}
A morphism between $G$-sets $X$ and $Y$ is a function $F \colon X \to Y$ that commutes with the actions: \[
F\,(\gact{g}{x}) \ = \ \gact{g}{F\,x}
\quad\qquad ,\forall g \in G, \forall x \in X \] These are called \emph{equivariant} functions. Since $id_{X}$ is equivariant and the composition of equivariant functions yields an equivariant function we can talk of the category of $G$-Sets.
Any set $X$ can be seen as a $G$-set by letting $\gact{g}{x} = x$; such a $G$-set is called the \emph{discrete} $G$-set. Moreover any group acts on itself by the multiplication.
One can form the (in)finitary product of $G$-sets by defining the action of $G$ on a tuple in a pointwise manner: \[
\gact{g}{\tuple{x_1}{\,x_2}} \ = \
\tuple{\gact{g}{x_1}}{\,\gact{g}{x_2}}
\quad\qquad ,\forall g \in G, \forall x_1 \in X_1, \forall x_2 \in X_2 \] The projections and the product morphism $\tuple{F}{H}$ are equivariant, assuming that $F$ and $H$ are also equivariant. $G$-set, as a category, also has co-products.
If $X$ and $Y$ are $G$-sets one can endow the set $Y^X$ of functions from $X$ to $Y$ with the \emph{conjugate} action: \[
(\gact{g}{F})\,x \ = \ \gact{g}{(F\,(\gact{g^{-1}}{x}))}
\quad\qquad ,\forall g \in G, \forall x \in X \enspace . \]
\paragraph{$G$-sets over the Permutation Group} The group of symmetries over a set $X$ consists of $G = \Sym{X}$, where $\Sym{X}$ is the set of bijections on $X$; the multiplication of $\Sym{X}$ is composition, the inverse is the inverse bijection, and the unit is the identity.
Let $\Perm{X}$ be the subset of $\Sym{X}$ of bijections that changes only finitely many elements; i.e., $f\in\Perm{X}$ if $\supp{f} = \{x\in X \mid f\,x\not=x \}$ is finite. It is straightforward to prove that $\Perm{X}$ is a sub-group of $\Sym{X}$. Of course, if $X$ is finite, then $\Perm{X} = \Sym{X}$. Notice that $X$ itself is a $\Perm{X}$-set with the action being function application: $\gact{\pi}{x} \ = \ \mathop{\pi}\,x \enspace$.
In particular, the \emph{transposition} (or \textit{swapping}) of a pair of elements $x, y \in X$ is the finite permutation $\swap x y \in \Perm{X}$ given by \[ \swap x y \, z \ = \ \begin{cases} y & \text{if } z = x \\ x & \text{if } z = y \\ z & \text{otherwise} \end{cases} \]
A basic result (in \cite{Hungerford} is proved as Theorem 6.3 and Corollary 6.5) is that every $\pi \in \Perm{X}$ can be expressed as a composition of \emph{disjoint cycles} \[ \pi \ = \ (x_1\ x_2\ \ldots\ x_n) \circ \cdots \circ (z_1\ z_2\ \ldots\ z_k) \] and every cycle can be expressed as a composition of transpositions \[ (x_1\ x_2\ \ldots\ x_n)\ = \ \swap{x_1}{x_2} \circ \swap{x_2}{x_3} \circ \cdots \circ \swap{x_{n-1}}{x_n} \] Therefore every $\pi \in \Perm{X}$ can be expressed as a composition of transpositions. We elaborate on the equivalence of the representations in Sect.~\ref{sec:perm}. Let us exhibit this with a concrete example.
\begin{example} \label{ex:perm} Let $f\colon \mathbb{N}\to \mathbb{N}$ be defined as \begin{align*}
f\,x \ = \ & \begin{cases}
(x+2) \,\mathop{mod}\, 6 & \text { if } x \leq 5\\
x & \text{ else }
\end{cases} \end{align*} Function $f$ is a finite permutation, because it has finite support: $\{ x \in \mathbb{N} \mid 0 \leq x \leq 5 \}$. Therefore it can be expressed as the composition of two cycles: $(1\ 3\ 5)\circ (0\ 2\ 4)$, or alternatively, it can also be expressed as a composition of four transpositions: $\swap{1}{3}\circ\swap{3}{5}\circ\swap{0}{2}\circ\swap{2}{4}$. \end{example}
\paragraph{Nominal Sets}
If we let $X$ be the set of variables for the lambda calculus, then a permutation on $X$ is a renaming; such a permutation can be lifted to an action over the set of lambda terms (taking care of the bound variables). In the nominal parlance one says that $X$ is the set of \emph{atoms} or that variables are atomic names: an atomic name has no structure in itself. We only assume that a set of atoms is a countable infinite set with decidable equality; from now on we will use $\mathbb{A}$ to refer to a set of atoms.
Let $X$ be a $\Perm{\mathbb{A}}$-set. We say that $x \in X$ \emph{is supported by} $A\subseteq\mathbb{A}$ if \[ \forall\, \pi.\ (\forall\, a\in A.\ \pi\,a\,=\,a) \ \implies \ \gact{\pi}{x} \,=\, x \enspace . \]
\noindent We say that $X$ is a \emph{nominal set} if each element of $X$ is supported by some finite subset of $\mathbb{A}$. Since each finite permutation can be decomposed as a composition of transpositions, then one can prove that the above definition is equivalent to \[ \forall\, a,a'\in\mathbb{A}\setminus A.\ \gact{\swap{a}{a'}}{x}
\,=\, x \enspace . \]
\noindent The following are some examples of nominal sets: \begin{itemize}
\item The discrete $Perm(\mathbb{A})$-set $X$ is nominal, because any $x \in X$ is supported by $\emptyset$.
\item $\mathbb{A}$ itself is nominal once equipped with the action
$\gact{\pi}{a} = \pi\,a$, because any $a\in \mathbb{A}$ is
supported by $\{a\}$. More in general, any
$S \subseteq \mathbb{A}$ containing name $a$ is a support for
$a$.
\item The set $\lambda$\textit{Term} of $\lambda$-calculus terms, inductively defined by $\ t \ ::= \ V(a) \ \mid \ A(t,t) \ \mid \ L(a,t)\ $ where $a \in \mathbb{A}$, equipped with the action $\gactUn \ \colon Perm(\mathbb{A})\times\lambda\textit{Term} \to \lambda\textit{Term}$ such that
\begin{align*}
& \gact{\pi}{V(a)} \ = \ V(\pi\, a) \\
& \gact{\pi}{A(t_1,t_2)} \ = \ A(\gact{\pi}{t_1},\,\gact{\pi}{t_2}) \\
& \gact{\pi}{L(a,t)} \ = \ L(\pi\, a,\,\gact{\pi}{t})
\end{align*}
is nominal because any $t \in \lambda$\textit{Term} is supported by $supp(t)=FreeVars(t)$. \end{itemize}
In his book \cite{Pitts2013-book} Pitts uses classical logic to prove that if $x$ is supported by some finite set $A$, then there exists a least supporting set, called \emph{the} support of $x$. As shown by Swan \cite{Swan2017} one cannot define the least support in a constructive setting; therefore a formalization in a constructive type theory should ask for ``some'' finite support. This affects the notion of freshness: in classical logic we have \[ x \,\textit{ is fresh for }\, y \ \ \Leftrightarrow \ \ \supp{x} \cap \supp{y} = \emptyset, \] with $x\in X$ and $y\in Y$ being elements of different nominal sets; but in a constructive setting one has to limit this relation to atoms, that is \[ a\in\mathbb{A} \,\textit{ is fresh for }\, x \in X \ \ \Leftrightarrow \ \ a\not\in\supp{x}, \] where $\supp{x}$ is the set supporting $x$, not necessarily the least one. Notice that the definition is the same (``there exists some finite support for each element''), but in classical logic that is sufficient to obtain the least support.
\section{Finite Permutations} \label{sec:perm} As we have already said, a finite permutation on a set $A$ can be explicitly given by: \begin{enumerate}
\item a bijection $f : A \to A$ together with its support $\supp{f} \subseteq_{\mathit{fin}} A$; i.e., $a\in \supp{f}$ if and only if $f\,a\not= a$;
\item a composition of disjoint cycles; concretely, we can think of this as a finite set $R \subseteq_{\mathit{fin}} A^*$ of disjoint cycles, each of them without repeated elements;
\item a composition of transpositions; that is, a finite sequence of pairs $p : (A\times A)^*$. \end{enumerate}
We present our proof that these definitions are equivalent. It basically boils down to define a predicate on sequences of elements in $A$ not containing repeated elements ensuring that they are cycles for $f$. We use the usual notation $\swap a b$ to denote the bijection $\{(a,b),(b,a)\}$.
\begin{definition}[List of transpositions from a cycle] We define $\mathit{toFP}\colon A\times A^* \to (A\times A)^*$. \[
\tofp{a}{\rho} = \begin{cases}
[] & \text{ if } \rho=[]\\
(a,b):\tofp{b}{\rho'} & \text{ if } \rho=b:\rho'\\
\end{cases} \] If we know that $\rho=a:\rho'$, then we also write $\tofpUn{\rho}$ to mean $\tofp{a}{\rho'}$. \end{definition}
\begin{definition}[Permutation from a list of transpositions] Let $as : (A\times A)^*$, then $\fromFP{as} : A \to A$ is defined by recursion on $as$: \[
\fromFP{as} = \begin{cases}
\mathit{id} & \text{ if } as=[]\\
\swap{a}{b} \cdot \fromFP{as'} & \text{ if } as =(a,b):as'\\
\end{cases} \] \end{definition}
\begin{definition}[Prefixes] We say that a non-empty sequence $\rho=[a_1,\ldots,a_n] : A^*$ is a \emph{prefix with head $a_0$} \emph{for} bijection $f$ if: \begin{enumerate}
\item $a_0 \in \supp{f}$,
\item $f\,a_i=a_{i+1}$, and
\item $a_0\not\in\rho$. \end{enumerate} A prefix $\rho$ is \emph{closed} if $f\,a_n = a_0$. Since $\rho$ is non-empty, we denote with $\lastEl{\rho}$ its last element. \end{definition}
From this simple definition we can deduce: \begin{lemma}[Properties of prefixes] Let $\rho$ be a prefix with head $a$. \begin{enumerate} \item If $\rho'$ is a prefix with head $\lastEl{\rho}$, then its concatenation $\rho\rho'$ is a prefix with head $a$.
\item $\rho$ has no duplicates.
\item If $\rho$ is closed and $b\in (a:\rho)$, then $f\,b = \fromFP{\tofp{a}{\rho}}\,b$.
\item If $b\not\in(a:\rho)$, then $\fromFP{\tofp{a}{\rho}}\,b = b$.
\end{enumerate} \end{lemma}
We can extend this definition to a sequence of sequences: let $R = [(a_1,\rho_1),\ldots,(a_m,\rho_m)] : (A\times A^*)^*$, then $R$ is a list of prefixes, with its head, if each $\rho_i$ is a prefix and $\rho_i\cap \rho_j = \emptyset$.
\begin{lemma}[Correctness of prefixes] Let $R = [(a_1,\rho_1),\ldots,(a_m,\rho_m)]$ be a list of closed prefixes, then $\fromFP{ \tofp{a_1}{\rho_1}\,\ldots\,\tofp{a_m}{\rho_m}}\,a = f\,a$. \end{lemma}
This proves that from a representation with cycles one can get a representation with transpositions. If we can produce a list of closed prefixes from a finite permutation (as a bijection with its support explicitly given) then we have the equivalence. First we define a function $\mathit{cycle}_f\colon \mathbb{N}\times A\to A^*$ such that $\cycle{f}{n}{a}$ computes a prefix with head $a$ of length at most $n+1$ by recursion on $n$: \[ \begin{aligned}
\cycle{f}{0}{a} &= [f\,a]\\
\cycle{f}{n+1}{a} &= \begin{cases}
\rho & \text{ if } f\,b=a\\
\rho[f\,b] & \text{ otherwise}
\end{cases}\\
\text{ where }& \rho = \cycle{f}{n}{a} \text{ and } b = \lastEl{\rho} \end{aligned} \]
\noindent We can extend this definition to compute a list of prefixes from a list of atoms: \[ \begin{aligned}
\cycles{f}{n}{[]}{R} &= R\\
\cycles{f}{n}{a:as}{R} &= \begin{cases}
\cycles{f}{n}{as}{R} & \text{ if } a\in \bigcup R\\
\cycles{f}{n}{as}{\rho:R} & \text{otherwise}
\end{cases}\\
\text{ where }& \rho = a:\cycle{f}{n}{a} \end{aligned} \]
\begin{lemma}[Correctness of computed cycles] If $f\colon A\to A$ is a bijection and $a\in \supp{f}$, then
$\cycle{f}{n}{a}$ is a prefix with head $a$, for all $n\in\mathbb{N}$. Moreover if $|\supp{f}| \leqslant n$, then $\cycle{f}{n}{a}$ is closed.
If $R$ is a list of prefixes and $as\subseteq\supp{f}$, then $\cycles{f}{n}{as}{R}$ is a list of prefixes; if
$|\supp{f}| \leqslant n$, then $\cycles{f}{n}{as}{R}$ is a list of closed prefixes. \end{lemma}
\begin{theorem}
If $f\colon A\to A$ is a bijection, then $R= \cycles{f}{|\supp{f}|}{\supp{f}}{[]}$ is a list of closed prefixes. Therefore $\fromFP{\tofpUnS{R}}\,a = f\,a$, for all $a\in A$. \end{theorem}
Notice that a composition of transpositions might mention elements that are not in the support of the induced permutation; for example, both $\swap 1 1$ and $\swap 1 2 \swap 2 1$ are equal to the identity permutation. One can get a ``normalized'' representation by composing our functions. As a matter of fact, this was our motivation to formalize cycles. \begin{corollary}[Normalization of transpositions] Let $p$ be a list of transpositions and
$\mathit{ats} = \supp{\tofpUn{p}}$. Moreover, let $R = \cycles{\tofpUn{p}}{|\mathit{ats}|}{\mathit{ats}}{[]}$. Then $\fromFP{\tofpUnS{R}} = \fromFP{as}$; moreover every atom in $\tofpUnS{R}$ is in its support. \end{corollary}
\section{Our Formalization in Agda} \label{sec:formalization} Our formalization is developed on top of the Agda's standard library v1.7 \cite{agdalib}. Figure \ref{fig:proj} shows a high level view of the project. The standard library includes an algebraic hierarchy going beyond groups; it lacks, however, a formalization of group actions. The module \agdainline{GroupAction} includes G-Sets, equivariant functions and constructions like products and co-products. We also have a \agdainline{Permutation} module which includes the concepts of finite permutations, cycles, normalization and the permutation group. And last, in the module \agdainline{Nominal} we formalize the concepts of support, nominal set, equivalence between different notions of support, normalization and again constructions like products and co-products.
\begin{figure}
\caption{High level view of the modular organization in the project.}
\label{fig:proj}
\end{figure}
\noindent We first present the definition of Group in the standard library in order to introduce some terminology and concepts: \begin{minted}{agda}
record Group c ℓ : Set (suc (c ⊔ ℓ)) where
field
Carrier : Set c
_≈_ : Rel Carrier ℓ
_∙_ : Op₂ Carrier
ε : Carrier
_⁻¹ : Op₁ Carrier
isGroup : IsGroup _≈_ _∙_ ε _⁻¹ \end{minted} \noindent A Group is a \emph{bundle} where the components of its definition (the carrier set, the unit, the inverse, the composition) are explicitly mentioned plus a proof, given by \textit{isGroup}, that they satisfy the axioms. Notice that one of the fields is a relation \textit{_≈_}; that relation should be an equivalence relation over the carrier: essentially this amounts to say that the \agdainline{Carrier} has a setoid structure. Setoids allows for greater flexibility as they enable to work with a notion of equality that is not the propositional equality; \agdainline{Func X Y} is the set of functions between setoids \agdainline{X} and \agdainline{Y} that preserve the equality; sometimes these functios are called \emph{respectful}.
\paragraph{G-Sets}
Our first definition is the \emph{structure} that collects the equations required for an action. In the following, we are under a module parameterized by \mintinline{agda}{G : Group}. \begin{minted}{agda}
record IsAction (F : Func (G.setoid ×ₛ X) X) : Set _ where
_●_ : Carrier G → Carrier X → Carrier X
g ● x = Func.f F (g , x)
field
idₐ : ∀ x → ε ∙ₐ x ≈X x
compₐ : ∀ g' g x → g' ● g ● x ≈X (g' ∙ g) ● x \end{minted} \noindent Notice that the record-type \mintinline{agda}{IsAction} is a predicate over respectuful functions from the setoid $G\times X$ to $X$. The definition of G-Set is straightforward and follows the pattern of the standard library \begin{minted}{agda}
record G-Set : Set _ where
field
set : Setoid ℓ₁ ℓ₂
action : Func (G.setoid ×ₛ set) set
isAction : IsAction action \end{minted} In order to introduce the notion of equivariant function\footnote{We note that both Choudhury and Paranhos define equivariant functions only for the group of finite permutations.} we first introduce the predicate \agdainline{IsEquivariant} stating when \agdainline{H : Func X Y} is equivariant for respectful functions \agdainline{FX} and \agdainline{FY}. \begin{minted}{agda}
IsEquivariant :
{X : Setoid ℓ₁ ℓ₂} →
{Y : Setoid ℓ₃ ℓ₄} →
(FX : Func (G.setoid ×ₛ X) X) →
(FY : Func (G.setoid ×ₛ Y) Y) →
(H : Func X Y) → Set (ℓ₁ ⊔ ℓ₄ ⊔ cℓ)
IsEquivariant {Y = Y} FX FY H = ∀ x g → F.f (g ●X x) ≈Y (g ●Y F.f x)
where _●X_ = _●_ {F = FX} ; _●Y_ = _●_ {F = FY} ; _≈Y_ = _≈_ Y
open module F = Func H \end{minted} \noindent Now we pack a respectful function between the setoids of G-Sets together with a proof of it being equivariant. \begin{minted}{agda}
record Equivariant (X : G-Set) (Y : G-Set) : Set _ where
field
F : Func (set X) (set Y)
isEquivariant : IsEquivariant (action X) (action Y) F \end{minted}
In the following snippet we show how to construct binary products of G-Sets (we use copatterns to define record objects). \begin{minted}{agda}
variable X Y : G-Set G
private
open module GX = G-Set X ; open module GY = G-Set Y
G-Set-× : G-Set G
set G-Set-× = GX.set ×ₛ GY.set
f (action G-Set-×) (g , (x , y)) = g GX.● x , g GY.● y
cong (action G-Set-×) (g=g' , (x=x' , y=y')) =
Func.cong GX.action (g=g' , x=x') , Func.cong GY.action (g=g' , y=y')
idₐ (isAction (G-Set-×)) (x , y) = GX.idₐ x , GY.idₐ y
compₐ (isAction (G-Set-×)) g g' (x , y) = GX.compₐ g g' x , GY.compₐ g g' y \end{minted} \noindent We now prove that the first projection is equivariant; notice that \agdainline{G-Set-×} is the product of \agdainline{X} and \agdainline{Y} introduced with the \agdainline{variable} keyword. \begin{minted}{agda}
π₁ : Equivariant G G-Set-× X
f (F π₁) = proj₁
cong (F π₁) = proj₁
isEquivariant π₁ _ _ = refl (set X) \end{minted}
\paragraph{Permutations} Now we focus on the module \agdainline{Permutation}. We start by introducing the group $\Sym{\mathbb{A}}$ using the definitions of inverses from the standard library; notice that the equivalence relation is given by the point-wise (or extensional) equality of functions. \begin{minted}{agda}
-- In this context A-setoid is a Setoid (not necessarily decidable).
A = Carrier A-setoid ; _≈A_ = _≈_ A-setoid
Perm = Inverse A-setoid A-setoid
_≈ₚ_ : Rel Perm _
F ≈ₚ G = (a : A) → f F a ≈A f G a
Sym : Group (ℓ ⊔ ℓ') (ℓ ⊔ ℓ')
Carrier Sym = Perm
_≈_ Sym = _≈ₚ_
_∙_ Sym = _∘ₚ_ -- composition of Perm, from the stdlib
ε Sym = idₚ A-setoid -- identity Perm, from the stdlib
_′ Sym = _⁻¹ -- inverse permutation, from the stdlib
isGroup Sym = record { ... } -- ommited \end{minted}
\noindent If we ask the setoid \agdainline{A-setoid} to be decidable, then we can define the swapping permutation. \begin{minted}{agda}
module Perm (A-setoid : DecSetoid ℓ ℓ') where
open DecSetoid A-setoid renaming (Carrier to A)
transp : A → A → A → A
transp a b c with does (c ≟ a)
... | true = b
... | false with does (c ≟ b)
... | true = a
... | false = c
transp-perm : (a b : A) → Perm
transp-perm a b = record {
f = transp a b ; f⁻¹ = transp a b
; cong₁ = transp-respects-≈ a b ; cong₂ = transp-respects-≈ a b
; inverse = transp-involutive a b , transp-involutive a b
} \end{minted}
Our next goal is to define the group $\Perm{\mathbb{A}}$ of finite permutations of atoms. As we explained before, finite permutation can be given by a bijective map, as a composition of transpositions, or as a composition of disjoint cycles.
In other works the group of finite permutations is explicitly defined as lists of pairs, where each pair represents a transposition and the empty list is the identity permutation: appending a pair $(a,b)$ to a list $p$ amounts to compose the transposition $\swap a b$ to the permutation denoted by $p$. Concatenation of lists $p$ and $p'$ also induces their composition. This choice has the advantage of being explicit and avoids having alternative expressions for composing permutations. On the other hand it still allows different representatives for the same permutation; in fact, $[(a,a)]$, $[(b,a),(a,b)]$, and $[]$ are all representations of \emph{the} identity permutation. It is clear that the setoid of finite permutations should equate those three versions of the identity, therefore the equivalence relation used is that of inducing the same permutation.
We started with the following syntactic representation of Finite Permutations, which is close to that of lists but in terms of $S$-expressions; since we cannot ensure canonicity with lists, why not to be more liberal also on associativity? \begin{minted}{agda}
data FinPerm : Set ℓ where
Id : FinPerm
Swap : (a b : A) → FinPerm
Comp : (p q : FinPerm) → FinPerm \end{minted} The permutation associated with a \agdainline{FinPerm} is given by \begin{minted}{agda}
⟦_⟧ : FinPerm → Perm
⟦ Id ⟧ = idₚ setoid
⟦ Swap a b ⟧ = transp-perm a b
⟦ Comp p q ⟧ = ⟦ q ⟧ ∘ₚ ⟦ p ⟧ \end{minted} Before introducing our concrete formalization of $\Perm{\mathbb{A}}$ let us exploit the fact that we have a decidable setoid of atoms to prove that the equivalence of finite permutation is also decidable. In order to do that, we define a relation \agdainline{_⊆ₛ_} on \agdainline{FinPerm}; \agdainline{p ⊆ₛ q} holds when \agdainline{q} coincides with \agdainline{p} in the support of the latter. Since we can compute the support of \agdainline{FinPerm}s and the equality of atoms is decidable, then we can decide \agdainline{_⊆ₛ_}. \begin{minted}{agda}
_⊆ₛ_ : Rel FinPerm (ℓ ⊔ ℓ')
p ⊆ₛ q = All (λ a → f ⟦ p ⟧ a ≈ f ⟦ q ⟧ a) (support p)
?⊆ₛ : ∀ p q → Dec (p ⊆ₛ q)
?⊆ₛ p q = all? (λ a → f ⟦ p ⟧ a ≟ f ⟦ q ⟧ a) (support p) \end{minted} Moreover we can prove that the mutual containment is equivalent to denoting the same permutation; thus we can decide the equality of finite permutations as given by \agdainline{FinPerm}: \begin{minted}{agda}
_≈ₛ_ : Rel FinPerm (ℓ ⊔ ℓ')
p ≈ₛ q = p ⊆ₛ q × q ⊆ₛ p
≈ₛ-dec : ∀ p q → Dec (p ≈ₛ q)
≈ₛ-dec p q = (?⊆ₛ p q) ×-dec (?⊆ₛ q p)
-- We omit the proofs of these lemmas.
≈ₛ⇒≈ₚ : ∀ p q → p ≈ₛ q → ⟦ p ⟧ ≈ₚ ⟦ q ⟧
≈ₚ⇒≈ₛ : ∀ p q → ⟦ p ⟧ ≈ₚ ⟦ q ⟧ → p ⊆ₛ q
_≟ₚ_ : ∀ p q → Dec (⟦ p ⟧ ≈ₚ ⟦ q ⟧) \end{minted} Furthermore we can normalize a \agdainline{FinPerm} to have an equivalent permutation where every occuring atom is in its support.
Let us first revisit the Example \ref{ex:perm} now in Agda where we see how to encode a finite permutation as a composition of cycles. \begin{minted}{agda}
f : ℕ → ℕ
f x with x ≤? 5
... | yes p = (x + 2) mod 6
... | no ¬p = x \end{minted} We represent cycles simply as lists of atoms; we certainly could also have used fresh lists to represent cycles. A composition of cycles is a list of cycles. \begin{minted}{agda}
Cycle = List A
cycle₀ cycle₁ : Cycle
cycle₀ = 1 ∷ 3 ∷ 5 ∷ []
cycle₁ = 0 ∷ 2 ∷ 4 ∷ []
f-cycles : List Cycle
f-cycles = cycle₀ ∷ cycle₁ ∷ [] \end{minted} Or alternatively, it can also be expressed as a composition of four transpositions: \begin{minted}{agda}
f-swaps : FinPerm
f-swaps = Comp (Comp (Swap 1 3) (Swap 3 5)) (Comp (Swap 0 2) (Swap 2 4)) \end{minted} In Figure \ref{fig:perms} we show the three representations of finite permutations. \begin{figure}
\caption{The mappings between different representations of permutations.}
\label{fig:perms}
\end{figure} The normalization of \agdainline{FinPerm} is simply the composition of the mappings: \begin{minted}{agda}
norm : FinPerm → FinPerm
norm = cycles-to-FP ∘ cycles-from-FP \end{minted} The functions \agdainline{cycles-to-FP} maps lists of disjoint cycles to \agdainline{FinPerm} and \agdainline{cycles-from-FP} goes in the reverse direction, producing a list of disjoint cycles from a \agdainline{FinPerm} (this is the composition of the diagonal arrows in Fig.~\ref{fig:perms}).
The correctness of the normalization follows the proof presented in Sec.~\ref{sec:perm}. Although we do not enforce neither freshness for cycles nor disjointness of cycles we keep that as an invariant when we compute the cycles in \agdainline{to-cycles}. \begin{minted}{agda}
module Thm (p : FinPerm) where
ats = atoms! p -- Fresh list of the atoms in the support of p.
-- from-atom-~* is the proof of Lemma 3.
rel = from-atoms-~* ⟦ p ⟧ ats []* (fp-supp p) (dom⊇atoms! p)
-- the representation as composition of cycles
ρs = to-cycles ⟦ p ⟧ (length ats) ats []
-- This property follows from Lemma 3.
∈-dom⇒∈ρs : (_∈-dom ⟦ p ⟧) ⊆ (_∈ concat ρs)
norm-corr : ⟦ p ⟧ ≈ₚ ⟦ norm p ⟧
norm-corr x with x ∈? concat ρs
... | yes x∈at = ~*-out ⟦ p ⟧ rel x∈at -- Item 3 of Lemma 1.
... | no x∉at = trans -- f ⟦ p ⟧ x = x = f ⟦ norm p ⟧ x
(¬∈-dom⇒∉-dom {⟦ p ⟧} (contraposition ∈-dom⇒∈ρs x∉at))
(~*-out-fresh ⟦ p ⟧ rel x∉at) -- Item 4 of Lemma 1. \end{minted} We also have other correctness result to prove that the \agdainline{FinPerm} obtained from a \agdainline{Perm} and its support is equivalent to it: \begin{minted}{agda}
module Thm’ (F : Perm) {ats : List A} (is-sup : ats is-supp-of F)
(incl : (_∈ ats) ⊆ (_∈-dom F)) where
ρs = to-cycles p (length ats) ats []
norm-corr : F ≈ₚ ⟦ cycles-to-FP ρs ⟧ \end{minted}
Let us remark that \agdainline{FinPerm} is just a representation and the set of finite permutation, \agdainline{PERM}, is the subset of \agdainline{Perm} corresponding to the image of \agdainline{⟦_⟧}: \begin{minted}{agda}
PERM : Set _
PERM = Σ[ p ∈ Perm ] (Σ[ q ∈ FinPerm ] (p ≈ₚ ⟦ q ⟧)) \end{minted} A disadvantage of using this encoding is that we need to deal with triples; for instance, the identity \agdainline{PERM} is represented by \agdainline{Id}. \begin{minted}{agda}
ID : PERM
ID = idₚ setoid , Id , λ _ → refl \end{minted} The group $\Perm{\mathbb{A}}$ is explicity defined as: \begin{minted}{agda}
Perm-A : Group (ℓ ⊔ ℓ') (ℓ ⊔ ℓ')
Carrier Perm-A = PERM
_≈G_ Perm-A = _≈ₚ_ on proj₁
_∙_ Perm-A = _∘P_
ε Perm-A = ID
_′ Perm-A = _⁻¹P
isGroup Perm-A = record { ... } \end{minted}
We alleviate the burden of working with triples by proving lemmas characterizing the action of \agdainline{PERM}s in terms of the finite permutation, for instance for \agdainline{Id}: \begin{minted}{agda}
-- In this context the group acting on G-Sets is Perm-A.
module Act-Lemmas {X-set : G-Set {ℓ₁ = ℓx} {ℓ₂ = ℓx'}} where
_≈X_ = Setoid._≈_ set
id-act : ∀ (π : PERM) (x : X) → proj₁ π ≈ₚ ⟦ Id ⟧ → (π ● x) ≈X x
id-act π x eq = trans (congˡ {π} {ID} x eq) (idₐ x) \end{minted}
\paragraph{Nominal Sets} Remember that a subset $A\subseteq \mathbb{A}$ is a support for $x$ if every permutation fixing every element of $A$ fixes $x$, through the action. A subset of a setoid \agdainline{A} can be defined either as a predicate or as pairs (just as in \agdainline{PERM} where the predicate is \agdainline{λ p → Σ[ q ∈ FinPerm ] (p ≈ₚ ⟦ q ⟧)}) or as another type, say \agdainline{B}, together with an injection \agdainline{ι : Injection B A}. \begin{minted}{agda}
variable
X : G-Set
P : SetoidPredicate A-setoid
is-supp : Pred X _
is-supp x = (π : PERM) → (predicate P ⊆ _∉-dom (proj₁ π)) → (π ● x) ≈X x \end{minted} The predicate \agdainline{λ a → f (proj₁ π) a ≈A a} is \agdainline{_∉-dom (proj₁ π)}; therefore, if \agdainline{P a} iff $a\in A$, then \agdainline{predicate P ⊆ _∉-dom (proj₁ π)} is a correct formalization of $\forall a\in A.\ π\,a=a$.
Our official definition of support is the following: \begin{minted}{agda}
_supports_ : Pred X _
_supports_ x = ∀ {a b} → a ∉ₛ P → b ∉ₛ P → SWAP a b ● x ≈X x \end{minted} Here \agdainline{SWAP} is a \agdainline{PERM}utation equal to \agdainline{⟦Swap a b⟧}. We formally proved that both definitions are equivalent, which is stated by the mutual implications: \begin{minted}[baselinestretch=1]{agda}
is-supp⊆supports : ∀ x → is-supp x → _supports_ x
supports⊆is-supp : _supports_ ⊆ is-supp \end{minted} Let us note that the second implication uses explicitly the normalization of finite permutations and its correctness.
In order to define nominal sets we need to choose how to say that a subset is finite; as explained by Coquand and Spiwak \cite{Coquand2010} there are several possibilities for this. We choose the easiest one: a predicate is finite if there is a list that enumerates all the elements satisfying the predicate. \begin{minted}{agda}
finite : Pred (SetoidPredicate setoid) _
finite P = Σ[ as ∈ List Carrier ] (predicate P ⊆ (_∈ as)) \end{minted} A G-Set is nominal if all the elements of the underlying set are finitely supported. \begin{minted}{agda}
record Nominal (X : G-Set) : Set _ where
field
sup : ∀ x → Σ[ P ∈ SetoidPredicate setoid ] (finite P × P supports x) \end{minted}
It is easy to prove that various constructions are nominals; for instance any discrete G-Set is nominal because every element is supported by the empty predicate \agdainline{⊥ₛ}: \begin{minted}{agda}
Δ-nominal : (S : Setoid _ _) → Nominal (Δ S)
sup (Δ-nominal S) x = ⊥ₛ , ⊥-finite , (λ _ _ → S-refl {x = x})
where open Setoid S renaming (refl to S-refl) \end{minted} We have defined \agdainline{G-Set-⇒ X Y} corresponding to the G-Set of equivariant functions from \agdainline{X} to \agdainline{Y}; now we can prove that \agdainline{G-Set-⇒ X Y} is nominal, again with \agdainline{⊥ₛ} as the support for any \agdainline{F : Equivariant X Y}. \begin{minted}{agda}
→-nominal : Nominal (G-Set-⇒ X Y)
sup (→-nominal) F = ⊥ₛ , ⊥-finite , λ _ _ → supported
where supported : ∀ {a b} x → f ((SWAP a b) ∙→ F) x ≈Y f F x \end{minted}
\section{Conclusion} \label{sec:conclusion} Nominal techniques have been adopted in various developments. We distinguish developments borrowing some concepts from nominal techniques to be applied in specific use cases (e.g. formalization of languages with binders like the $\lambda$ or $\pi$ calculus with their associated meta-theory) \cite{Bengtson2007, Copello2016, Copello2018, Copello2018-2} from more general developments aiming to formalize at least the core aspects of the theory of nominal sets. We are more concerned with the later type.
The nominal datatype package for Isabelle/HOL \cite{Urban2006} developed by Urban and Berghofer implements an infrastructure for defining languages involving binders and for reasoning conveniently about alpha-equivalence classes. This Isabelle/HOL package inspired Aydemir et al.~\cite{Aydemir2007} to develop a proof of concept for the Coq proof assistant, however it had no further development. In his Master thesis~\cite{Chou2015-mt}, Choudhury notes that none of the previous developments following the theory of nominal sets were based on constructive foundations. He showed that a considerable portion (most of the first four chapters of Pitts book \cite{Pitts2013-book}) of the theory of nominal sets can also be developed constructively by giving a formalization in Agda. Pitts original work is based on classical logic, and depends heavily on the existence of the smallest finite support for an element of a nominal set. However, Swan \cite{Swan2017} has shown that in general this existence cannot be constructively guaranteed, as it would imply the law of the excluded middle.
Choudhury works with the notion of \emph{some non-unique support}. In order to formalize the category of Nominal Sets, Choudhury preferred setoids instead of postulating functional extensionality. As far as we know, Choudhury is still the most comprehensive mechanization in terms of instances of constructions having a nominal structure.
Recently Paranhos and Ventura \cite{Paranhos2022} presented a constructive formalization in Coq of the core notions of nominal sets: support, freshness and name abstraction. They follow closely Choudhury’s work in Agda \cite{Chou2015-mt}, acknowledging the importance of working with setoids. They claim that by using Coq’s type class and setoid rewriting mechanism, much shorter and simpler proofs are achieved, circumventing the ``setoid hell'' described by Choudhury. In his master thesis \cite{paranhos-thesis} Paranhos further developed the library.
Both of those two formalizations in type theory take a very pragmatic approach to finite permutations: a finite permutation is a list of pairs of names. In our approach, we start with the more general notion of bijective function from which the finite permutations are obtained as a special case; moreover having different representations allowed us to state and prove some theorems that cannot even be stated in the other formalizations. So far, our main contributions are: the representation of finite permutations and the normalization of composition of transpositions; the equivalence between two definitions of the relation ``$A$ supports the element $x$''; and proving that the extension of every container type can be enriched with a group action (notice that this cover lists, trees, etc.).
Our next steps are the definition of freshness. We are studying an alternative notion of support that would admit having a freshness relation between elements of two nominal sets (in contrast with other mechanization that only consider ``the atom $a$ is fresh for $x$'') and name abstraction. In parallel we hope to be able to prove that extensions of finite containers on nominal sets are also nominal sets. We also hope to streamline further some rough corners of our development.
\subsection*{Acknowledgments} This formalization grew up from discussions with the group of the research project ``Type-checking for a Nominal Type Theory'': Maribel Fernández, Nora Szasz, Álvaro Tasistro, and Sebastián Urciouli. We thank Cristian Vay for discussions about group theory. This work was partially funded by Agencia Nacional de Investigación e Innovación (ANII) of Uruguay.
\end{document} |
\begin{document}
\title{Quantum-speed-limit time for multiqubit open systems} \author{Chen Liu} \email{tarksar@sina.com} \author{Zhen-Yu Xu} \altaffiliation{} \email{zhenyuxu@suda.edu.cn} \author{Shiqun Zhu} \altaffiliation{} \email{szhu@suda.edu.cn} \affiliation{College of Physics, Optoelectronics and Energy, Soochow University, Suzhou 215006, China}
\begin{abstract} Quantum-speed-limit (QSL) time captures the intrinsic minimal time interval for a quantum system evolving from an initial state to a target state. In single qubit open systems, it was found that the memory (non-Markovian) effect of environment plays an essential role in shortening QSL time or, say, increasing the capacity for potential speedup. In this paper, we investigate the QSL time for multiqubit open systems. We find that for a certain class of states the memory effect still acts as the indispensable requirement for cutting the QSL time down, while for another class of states this takes place even when the environment is of no memory. In particular, when the initial state is in product state $\left\vert 111\cdots 1\right\rangle $,\ there exists a sudden transition from no capacity for potential speedup to potential speedup in a memoryless environment. In addition, we also display evidence for the subtle connection between QSL time and entanglement that weak entanglement may shorten QSL time even more. \end{abstract}
\pacs{03.65.Yz} \maketitle
\section{Introduction}
Quantum-speed-limit (QSL) time \cite{MT,ML}, the intrinsic minimal time interval for a quantum system evolving from an initial state to a target state, is of crucial importance in the fields of quantum computation \cite {quantum computer}, quantum control \cite {control1,control2,control3,control4,control5}, quantum metrology \cite {metrology,metrology1,metrology2}, and non-equilibrium thermodynamics \cite {thermo}. Recent decades have witnessed a great deal of research on QSL time both in closed \cite{c1,c2,c3,c4,c5,c6,c7,c8,c9,c10,c11,c12,new1} and open systems \cite{xopen0,xopen1,QSLopen1,QSLopen2,QSLopen3,new2,Xu1,Xu2}. In particular, a QSL time based on the Schatten $p$ norm for an arbitrarily driven open system was presented by Deffner and Lutz \cite{QSLopen3}
\begin{equation} \tau _{QSL}=\frac{\sin ^{2}[B(\rho ,\rho _{\tau })]}{\min \left\{ E_{1}^{\tau },E_{2}^{\tau },E_{\infty }^{\tau }\right\} }, \label{QSL} \end{equation}
where $B(\rho ,\rho _{\tau })=\arccos (\sqrt{\langle \phi |\rho _{\tau
}|\phi \rangle })$ denotes the Bures angle between the initial state $\rho
=\left\vert \phi \right\rangle \langle \phi |$ and the target state $\rho _{\tau }$, which is governed by the time-dependent master equation $\dot{\rho }_{t}=L_{t}\rho _{t}$ ($L_{t}$ is a superoperator). It is reasonable to employ the Bures angle above as a measure for distance between an pure state and a mixed state, since the important Riemannian feature is satisfied under such a circumstance \cite{QSLopen1}. The denominator $E_{p}^{\tau }$ in Eq. (
\ref{QSL}) represents the average of $||L_{t}\rho _{t}||_{p}$ over actual driving time duration $\tau $, i.e., $E_{p}^{\tau }=(1/\tau )\int_{0}^{\tau
}||L_{t}\rho _{t}||_{p}dt$. One important application of the QSL time is to evaluate the speed of quantum evolutions under the following two scenarios:
(I). The fastest evolution appears when the actual driving time $\tau $ achieves the QSL time $\tau _{QSL}$, i.e., $\tau /\tau _{QSL}=1.$ For slower evolutions, $\tau /\tau _{QSL}>1$ and the slower the evolution, the higher ratio $\tau /\tau _{QSL}$ should be. This point of view is clear and unambiguous and has been widely utilized in the field of entanglement assisted speedup of quantum evolutions \cite{c5,en2,en3,en4}.
(II). From another point of view, $\tau /\tau _{QSL}=1$ indicates the evolution is already the fatest, and possesses no potential capacity for further acceleration, while for $\tau /\tau _{QSL}>1,$ the higher ratio $ \tau /\tau _{QSL}$ (or equivalently, the much shorter $\tau _{QSL}$), the greater the capacity for potential speedup will be. This viewpoint has been adopted in exploring the memory effect, characterized by non-Markovianity \cite{nonM-review1,nonM-review2}, on the speed of quantum evolution \cite {QSLopen3,Xu1,Xu2}. It is found that in the damped Jaynes-Cummings model of a single qubit, the transition from no potential capacity of speedup ($\tau /\tau _{QSL}=1$) to potential speedup ($\tau /\tau _{QSL}>1$), or say a reduction of QSL in quantum evolution is just the critical point when the memoryless environment becomes of memory \cite{QSLopen3,Xu1}. Furthermore, if evolution of a\ qubit is already along the QSL time which means its fastest speed in nature environment, the study on it will probably not so urgent as the case of $\tau /\tau _{QSL}>1$ which have speedup potential.
Although the memory effect of environment plays a decisive role in the potential acceleration of quantum evolution in a single qubit case, the question may arise whether it will still be true even in multi-qubit cases. This question is of particular interest, for the QSL time of multi-qubit open systems has now caught increasing attention and several interesting phenomena were discovered. For instances, with the QSL time defined in Ref. ( \cite{QSLopen1}), Taddei \textit{et al}. illustrated that the separable states of a multi-qubit system under Markovian dephasing channels perform the same speedup of quantum evolution as the entangled states when the number of qubits is large enough. On the other hand, del Campo \textit{et al. } demonstrated, with the bound introduced in Ref. (\cite{QSLopen2}), that multi-qubit Greenberger-Horne-Zeilinger (GHZ) states and separable states under Markovian and non-Markovian dephasing are equivalent in metrological parameter estimation \cite{QSLopen2}. In this paper, mainly focused on Scenario 2, we show that for a class of multiqubit states memory effect is a requisite for raising potential ability of speeding up quantum evolution, while for another class of multi-qubit states, memoryless environment is ready to realize the above result. Especially when the open system is initially prepared in product state $\left\vert 111\cdots 1\right\rangle ,$ a transition from no potential capacity for speedup to possess speedup ability is achieved.
The paper is organized as follows: In Sec. II, we present the QSL time for typical two-qubit, three-qubit, and n-qubit states, respectively. Discussion on the role of entanglement in QSL time is performed in Sec. III. Finally, conclusions are drawn in Sec. IV.
\section{Quantum limits to multi-qubit dynamical evolution}
We consider N independent two-level atoms (open system) each locally coupling to a leaky vacuum cavity (environment). The dynamics of the multi-qubit open system is fully determined by each pair of atom cavities \cite{Bellomo} with the following Hamiltonian \cite{Book Open}
\begin{equation} H=\omega _{0}\sigma _{+}\sigma _{-}+\sum_{k}\omega _{k}a_{k}a_{k}^{\dag }+i\sum_{k}g_{k}(a_{k}^{\dag }\sigma _{-}-a_{k}\sigma _{+}), \label{Hamiltonian} \end{equation}
where $\omega _{0}$ is the resonant transition frequency of the atom between the excited state $|1\rangle $ and the ground state $|0\rangle $, and $ \sigma _{\pm }$ are the Pauli raising and lowering operators. $\omega _{k}$ and $a_{k}(a_{k}^{\dag })$ denote the frequency and the annihilation(creation) operators of the $k$th mode of the cavity with $g_{k}$ the corresponding real coupling constant. The master equation for the reduced density matrix of the atom in the Schr\"{o}dinger picture is given by $\dot{\rho}_{t}=L_{t}\rho _{t}$ with \begin{equation} L_{t}\rho _{t}=i\delta _{t}\left[ {\sigma }_{+}{\sigma }_{-},\rho _{t}\right] +\gamma _{t}\left( {\sigma }_{+}{\sigma }_{-}\rho _{t}+\rho _{t}{\sigma }_{+} {\sigma }_{-}-2{\sigma }_{-}\rho _{t}{\sigma }_{+}\right) , \label{master equation} \end{equation} where $\delta _{t}$=Im$(\dot{c}_{t}/c_{t})$ and $\gamma _{t}$=Re$(\dot{c} _{t}/c_{t})$ are the time-dependent Lamb shift and decay rate respectively, and $c_{t}$ is the decoherence function relying on the particular structure of cavity reservoirs \cite{Book Open}. The reduced density matrix of the atom with an initial state $\rho =(\rho _{mn})$ takes the form
\begin{equation} \rho _{t}=\left[ \begin{array}{cc}
\rho _{11}|c_{t}|^{2} & \rho _{10}c_{t} \\
\rho _{01}c_{t}^{\ast } & 1-\rho _{11}|c_{t}|^{2} \end{array} \right] , \label{density matrix} \end{equation}
where the excited state population $|c_{t}|^{2}$ is denoted by $P_{t}$ in the following.
\subsection{Two-qubit cases}
As the exact form of QSL time for a general pure initial state is cumbersome, we only consider two typical Bell-type initial states respectively, i.e., $\left\vert \Psi _{1}\right\rangle =\alpha |01\rangle +
\sqrt{1-\alpha ^{2}}|10\rangle $ and $\left\vert \Psi _{2}\right\rangle
=\alpha |11\rangle +\sqrt{1-\alpha ^{2}}|00\rangle $ with $\alpha \in \lbrack 0,1]$. According to the definition of QSL time in Eq. (\ref{QSL}), we have \begin{equation} \frac{\tau }{\tau _{QSL}}=\frac{\int_{0}^{\tau }\left\vert \overset{\cdot }{P }_{t}\right\vert dt}{1-P_{\tau }} \label{QSL2-1} \end{equation} for state $\left\vert \Psi _{1}\right\rangle $ and
\begin{equation} \frac{\tau }{\tau _{QSL}}=\frac{\int_{0}^{\tau }\max \left\{ \left\vert \overset{\cdot }{P}_{t}(2\alpha P_{t}-\alpha \pm 1)\right\vert \right\} dt}{ \alpha (1-P_{\tau }^{2})} \label{QSL2-2} \end{equation} for state $\left\vert \Psi _{2}\right\rangle $, where $\overset{\cdot }{P} _{t}=\partial _{t}P_{t}.$
Obviously, the QSL time ratio $\tau _{QSL}$ of state $\left\vert \Psi _{1}\right\rangle $ is the same as the exact form in single qubit cases discovered in Refs \cite{QSLopen3,Xu1}, where the reduction in QSL time ($ \tau /\tau _{QSL}>1$) only occurs when the environment is of memory; otherwise the evolution will always be along the QSL time ($\tau /\tau _{QSL}\equiv 1$). In order to illustrate this phenomenon clearly, the QSL time ratio $\tau /\tau _{QSL}$ of initial state $\left\vert \Psi _{1}\right\rangle $ (black dotted line) is depicted in Fig. 1 under a memoryless environment, i.e., $\left\vert \overset{\cdot }{P}_{t}\right\vert =-\overset{\cdot }{P}_{t}$ with the population $P_{t}$ monotonically decreasing from 1 to the target $P_{\tau }$.
However, a complex but interesting phenomenon appears for initial state $ \left\vert \Psi _{2}\right\rangle $: QSL reduction can also take place even when the environment is of no memory. For instances, $\tau /\tau _{QSL}$ versus $P_{\tau }$ of initial states $\left\vert \Psi _{2}(\alpha
=1)\right\rangle =|11\rangle $ (red solid curve) and $\left\vert \Psi _{2}(\alpha =1/\sqrt{2})\right\rangle =$ $(|11\rangle +|00\rangle )/\sqrt{2}$
(blue dashed curve) are depicted, respectively, in Fig. 1, where $\tau /\tau _{QSL}\geq 1$ clearly illustrates the intrinsic acceleration potential of quantum evolution under a memoryless environment. Especially, there exists a sudden change point in QSL time with the critical point $P_{\tau }=1/2$ for initial state $\left\vert \Psi _{2}(\alpha =1)\right\rangle =$ $|11\rangle $
. To explain this phenomenon, we trace back to $||L_{t}\rho _{t}||_{\infty }$ defined in Eq. (\ref{QSL}) with the following expression
\begin{equation}
||L_{t}\rho _{t}||_{\infty }=\left\{ \begin{array}{l} -\overset{\cdot }{P}_{t}(2\alpha ^{2}P_{t}-\alpha ^{2}+\alpha ), \\ \overset{\cdot }{P}_{t}(2\alpha ^{2}P_{t}-\alpha ^{2}-\alpha ), \end{array} \right. \begin{array}{c} P_{t}\geqslant \frac{1}{2}, \\ P_{t}<\frac{1}{2}, \end{array} \label{QSL2-3} \end{equation} where we have employed the condition of memoryless environment $\left\vert \overset{\cdot }{P}_{t}\right\vert =-\overset{\cdot }{P}.$ Therefore, the QSL time ratio $\tau /\tau _{QSL}$ of Eq. (\ref{QSL2-2}) can be conveniently calculated as \begin{equation} \frac{\tau }{\tau _{QSL}}=\left\{ \begin{array}{ll} \frac{1+\alpha P_{\tau }}{\alpha (1+P_{\tau })}, & P_{\tau }\geqslant \frac{1 }{2}, \\ \frac{2(1-P_{\tau })(1-\alpha P_{\tau })+\alpha }{2\alpha (1-P_{\tau }^{2})} & P_{\tau }<\frac{1}{2}. \end{array} \right. \label{QSL2-4} \end{equation}
One may check that $\tau /\tau _{QSL}\geq 1$ is always satisfied. In particular, when the initial state is in $\Psi _{2}(\alpha =1)=|11\rangle $, the QSL time ratio yields \begin{equation} \frac{\tau }{\tau _{QSL}}=\left\{ \begin{array}{ll} 1, & P_{\tau }\geqslant \frac{1}{2}, \\ \frac{2(1-P_{\tau })^{2}+1}{2(1-P_{\tau }^{2})} & P_{\tau }<\frac{1}{2}. \end{array} \right. \label{QSL2-5} \end{equation} The sudden change point of QSL time is therefore justified.
\subsection{Three-qubit cases}
In this subsection, we also consider two typical three-qubit states, i.e., W type state $\left\vert \Psi _{3}\right\rangle =\alpha |001\rangle +\beta
|010\rangle +\sqrt{1-\alpha ^{2}-\beta ^{2}}|100\rangle $ and GHZ type state
$\left\vert \Psi _{4}\right\rangle =\alpha |111\rangle +\sqrt{1-\alpha ^{2}}
|000\rangle .$ According to Eq. (\ref{QSL}), the expressions of the QSL time ratio are obtained: \begin{equation} \frac{\tau }{\tau _{QSL}}=\frac{\int_{0}^{\tau }\left\vert \overset{\cdot }{P }_{t}\right\vert dt}{1-P_{\tau }} \label{QSL3-1} \end{equation} for state $\left\vert \Psi _{3}\right\rangle $ and
\begin{equation} \frac{\tau }{\tau _{QSL}}=\frac{\int_{0}^{\tau }\max \left\{ \left\vert \overset{\cdot }{P}_{t}(\pm \frac{3}{2}X+3\alpha P_{t}-\frac{3}{2}\alpha )\right\vert \right\} dt}{\alpha +\alpha (1-\alpha ^{2})P_{\tau }(3-2P_{\tau }^{\frac{1}{2}}-3P_{\tau })+\alpha (2\alpha ^{2}-1)P_{\tau }^{3}}, \label{QSL3-3} \end{equation} for state $\left\vert \Psi _{4}\right\rangle $, where $X=(4\alpha ^{2}P_{t}^{4}-8\alpha ^{2}P_{t}^{3}+8\alpha ^{2}P_{t}^{2}-5\alpha ^{2}P_{t}+P_{t}+\alpha ^{2})^{\frac{1}{2}}.$
Equation (\ref{QSL3-1}) bears a resemblance to the case of state\ $ \left\vert \Psi _{1}\right\rangle $, implying that the reduction in QSL only occurs in a memory environment. As for Eq. (\ref{QSL3-3}), we consider a special case, i.e., $\left\vert \Psi _{4}(\alpha =1)\right\rangle
=|111\rangle $ under the environment of no memory ($\left\vert \overset{ \cdot }{P}_{t}\right\vert =-\overset{\cdot }{P}$). Therefore, the Eq. (\ref {QSL3-3}) can be simplified as:
\begin{equation} \frac{\tau }{\tau _{QSL}}=\left\{ \begin{array}{ll} 1, & P_{\tau }\geqslant \frac{1}{2}, \\ \frac{(-3P_{\tau }+3P_{\tau }^{2}-P_{\tau }^{3}+\frac{7}{4})}{1-P_{\tau }^{3} } & P_{\tau }<\frac{1}{2}, \end{array} \right. \label{QSL3-5} \end{equation} indicating that there also exists a sudden transition of speedup potential of quantum evolution even the environment is of no memory.
\subsection{N-qubit cases}
In this subsection, we show that the above phenomena are ubiquitous in n-qubit cases (n is an arbitrary positive integer). It is easy to check that if the n-qubit open system is initially prepared in state $\alpha _{1}\left\vert 100\cdots 0\right\rangle +\alpha _{2}\left\vert 010\cdots 0\right\rangle +\cdots +\alpha _{N}\left\vert 000\cdots 1\right\rangle ,$ with $\sum_{j=1}^{N}\alpha _{j}^{2}=1,$ the QSL ratio is exactly the same as Eqs. (\ref{QSL2-1}) and (\ref{QSL3-1}). Therefore, the memory effect of environment becomes the essential condition for the speedup potential emerge in quantum evolution.
However, if the initial state is in $\left\vert 11\cdots 1\right\rangle $, the QSL ratio is given by
\begin{equation} \frac{\tau }{\tau _{QSL}}=\left\{ \begin{array}{ll} 1, & P_{\tau }\geqslant \frac{1}{2}, \\
\frac{\int_{0}^{\tau }\max \{|n\overset{\cdot }{P}_{t}P_{t}^{n-1}|,|-n
\overset{\cdot }{P}_{t}(1-P_{t})^{n-1}|\}dt}{1-P_{\tau }^{n}} & P_{\tau }< \frac{1}{2}. \end{array} \right. \label{QSLN-1} \end{equation} In particular when the environment is memoryless, i.e., $\left\vert \overset{ \cdot }{P}_{t}\right\vert =-\overset{\cdot }{P},$ Eq. (\ref{QSLN-1}) reduces to \begin{equation} \frac{\tau }{\tau _{QSL}}=\left\{ \begin{array}{ll} 1, & P_{\tau }\geqslant \frac{1}{2}, \\ \frac{(1-P_{\tau })^{n}+1-(\frac{1}{2})^{n-1}}{1-P_{\tau }^{n}} & P_{\tau }< \frac{1}{2}. \end{array} \right. \label{QSLN-2} \end{equation} Clearly, $P_{\tau }=1/2$ is the critical point at which the open system experiences the sudden change of QSL time under a memoryless environment.
Equation (\ref{QSLN-2}) also implies that there exists a maximal acceleration potential condition for state $\left\vert 11\cdots 1\right\rangle $ when $P_{\tau }\rightarrow 0$, and the corresponding minimal QSL time ratio is given by:
\begin{equation}
\frac{\tau }{\tau _{QSL}}|_{\max }=\frac{2^{n}-1}{2^{n-1}}. \label{maximal speedup} \end{equation} Especially when $n=2$, Eq. (\ref{maximal speedup}) grows to $3/2$, which is marked in Fig. 1 as the red circle.
\subsection{Memory effect on QSL time}
In this subsection, we intend to show that memory effect of environment is still an important element for quantum acceleration potential for multi-qubit open systems. The memory environment we consider here is characterized by the Lorentzian spectral distribution $J(\omega )=\frac{1}{ 2\pi }\frac{\gamma _{0}\lambda }{(\omega _{0}-\omega )^{2}+\lambda ^{2}}$ , where $\gamma _{0}$ is the Markovian decay rate and $\lambda $ is the spectral width \cite{Book Open}. $P_{t}$ is now written as \cite{Book Open} \begin{equation} P_{t}=e^{-\lambda t}\left\vert \cosh (\frac{dt}{2})+\frac{\lambda }{d}\sinh ( \frac{dt}{2})\right\vert ^{2}, \label{damping} \end{equation} where $d=\sqrt{2\gamma _{0}\lambda -\lambda ^{2}}$. In Fig. 2, we take two-qubit initial states as an example and fix the driving time as $\tau =1.$ The QSL time ratio $\tau /\tau _{QSL}$ versus coupling strength $\gamma _{0}/\omega _{0}$ is plotted with three typical two-qubit states $\left\vert \Psi _{1}\right\rangle $ (black dotted line), $\left\vert \Psi _{2}(\alpha =1)\right\rangle $ (red solid curve), and $\left\vert \Psi _{2}(\alpha =1/ \sqrt{2})\right\rangle $ (blue dashed curve), respectively, with $\lambda =50 $, $\omega _{0}=1$, and we set $\tau _{QSL}=1$. According to Ref. (\cite {Book Open}), we know that $\gamma _{0}=\lambda /2$ is the transient point from memoryless environment ($\gamma _{0}<\lambda /2$) to one of memory ($ \gamma _{0}>\lambda /2$). As is clearly shown in Fig. 2, more capacity for potential speedup will take place when the environment enters the memory region.
\section{Discussion: Entanglement and QSL time}
Considering that the QSL time for multiqubit systems can be shortened in a memoryless environment, it is natural to link this phenomenon to entanglement, which only exists in multiqubit cases. In closed composite systems, entanglement has been taken as a resource in the speedup of quantum evolution [\cite{c5,en2,en3,en4}]. In this subsection, we go a step further to the connection between entanglement and QSL time in bipartite open systems. The initial state we consider here is an arbitrary pure state:
\begin{equation}
\left\vert \phi \right\rangle =\alpha _{1}|11\rangle +\alpha _{2}|10\rangle
+\alpha _{3}|01\rangle +\alpha _{4}|00\rangle , \label{arbitrary state} \end{equation} with $\sum_{j=1}^{4}\alpha _{j}{}^{2}=1,$ which is generated by Monte Carlo method, and the related entanglement is characterized by concurrence $C$ in Ref. \cite{concurrence}, with $C=0$ for a disentangled state and $C=1$ for a maximally entangled state.
In Fig. 3, 20000 random pure states are generated by Monte Carlo sampling
\cite{new3} and their QSL ratios $\tau /\tau _{QSL}$ and their QSL time ratios $\tau /\tau _{QSL}$ under memoryless environments versus concurrence are marked by tiny blue dots. As is clearly displayed in Fig. 3, the lower bound $\tau /\tau _{QSL}=1$ can always be reached by states $\left\vert \Psi _{1}\right\rangle =\alpha |01\rangle +\sqrt{1-\alpha ^{2}}|10\rangle $ (dark red dots in Fig. 3), implying that the concurrence has nothing to do with $ \tau /\tau _{QSL}$ under this circumstance. In addition, the upper bound is taken by a subset of states $\left\vert \Psi _{2}\right\rangle =\alpha
|11\rangle +\sqrt{1-\alpha ^{2}}|00\rangle $ (light green dots in Fig. 3), illustrating that weak entanglement may reduce the QSL time more.
We also note that, if one takes the viewpoint of Scenario 1, the upper subset of light green dots in Fig. 3 implies that entanglement is able to accelerate quantum evolution under such a circumstance.
\section{Conclusion}
In summary, we have explored the quantum-speed-limit time for multi-qubit open systems. For a certain class of initial states, we have demonstrated that the quantum evolution can also be accelerated in a memoryless (Markovian) environment. Moreover, we have found that entanglement plays a subtle role in the speedup of quantum evolution: weak entanglement may be better for speeding up quantum evolution under certain circumstances.
We have only treated non-correlated environments in this paper. It will also be of importance and interest to study the QSL time of multi-qubit systems with the presence of initial correlations among the subsystems of composite environments \cite{EML}.
\section{\textbf{Acknowledgement}}
This work was supported by the National Natural Science Foundation of China (Grants No. 11204196 and No. 11074184), the Specialized Research Fund for the Doctoral Program of Higher Education (Grant No. 20123201120004), and the Priority Academic Program Development of Jiangsu Higher Education Institutions.
\end{document} |
\begin{document}
\title{Correlations and projective measurements in maximally entangled multipartite states}
\author{Arthur Vesperini} \email[Email: ]{arthur_vespe@hotmail.fr} \affiliation{DSFTA, University of Siena,\\ Via Roma 56, 53100 Siena, Italy } \affiliation{Aix Marseille Université, Université de Toulon, CNRS, CPT, 13288 Marseille, France} \date{\today}
\begin{abstract} Multipartite quantum states constitute the key resource for quantum computation. The understanding of their internal structure is thus of great importance in the field of quantum information. This paper aims at examining the structure of multipartite maximally entangled pure states, using tools with a simple and intuitive physical meaning, namely, projective measurements and correlations. We first show how, in such states, a very simple relation arises between post-measurement expectation values and pre-measurement correlations. We then infer the consequences of this relation on the structure of the recently introduced \textit{entanglement metric}, allowing us to provide an upper bound for the \textit{persistency of entanglement}. The dependence of these features on the chosen measurement axis is underlined, and two simple optimization procedures are proposed, to find those maximizing the correlations. Finally, we apply our procedures onto some prototypical examples. \end{abstract}
\maketitle
\section{Introduction}
Entanglement, in addition to be one of the most historically puzzling properties of quantum mechanics, constitutes the main resource for quantum cryptography and computation, and quantum-based technologies. The quantum information community developed, in the past decades, an extensive number of approaches to characterize its abundant phenomenology \cite{GUHNE,horodecki}.
Full characterization of entanglement in multipartite states is a notoriously complex task \cite{ghz,Opt_estimation_ent,PhysRevA.95.062116,PhysRevA.67.022320,horodecki}. The mere definition of a measure of the entanglement of such states, is in itself a great challenge.
Yet, their complete analysis convokes numerous additional notions, as their \textit{$k$-separability}, \textit{$k$-producibility}, \textit{entanglement depth} or \textit{persistency of entanglement} \cite{GUHNE,BRS_persistent,horodecki}.
Measurement processes and their understanding are of a major importance in the field of quantum computing. Measurement based quantum computation, which stands as a universal model of quantum computation, obviously heavily lie on the control of their effects on quantum states. \cite{Nielsen_2003,oneway_BR,meas_based_cluster,NIELSEN2006147} \\
For the sake of simplicity, the present work will be focused on states $\ket{s}\in\mathcal{H}$, with $\mathcal{H}=\bigotimes_{\mu\in Q} \mathcal{H}_\mu$ the Hilbert space of dimension $2^N$ describing a set $Q$ of $N$ qubits. It should be noted that our results are generalizable to hybrid qudits systems with relatively few adjustments.
We will hereafter denote $\sigma_k^\mu$, with $k=1,2,3$ the Pauli matrices acting on qubit $\mu$, $\bm{\sigma}^\mu=\big(\sigma_1^\mu,\sigma_2^\mu,\sigma_3^\mu\big)$ the Pauli vector acting on $\mu$, and $\sigma_{\bm{v}}^\mu=\bm{v}^\mu \cdot\bm{\sigma}^\mu = \sum\limits_{k=1,2,3}v_k^\mu\sigma_k^\mu$
the Pauli observable on $\mu$ oriented in the direction $\bm{v}^\mu$. All of the vectors introduced in the following belong to $\mathbb{R}^3$. \\
As stated before, the ways of quantifying entanglement in multipartite states are manifold. In this work, we will solely refer to \textit{qubit-wise entanglement}, that is entanglement of bipartitions $(\mu,\mu^C)$, with $\mu\in Q$, and $\mu^C$ its complement on the set $Q$.
The reduced state of any qubit $\mu$ can be represented as a vector $\bra{s}\bm{\sigma}^\mu\ket{s}$ in the Bloch ball, i.e. a vector with norm lesser or equal to $1$. For a maximally entangled qubit $\mu$, it is the null vector, and we have \begin{equation}\label{vanishing_expect_max_ent_mu} \forall\, \bm{v}^\mu,\; \bra{s}\sigma_{\bm{v}}^\mu\ket{s}=0, \end{equation} while fully factorizable qubits can always be represented by a well-defined vector on the Bloch sphere (i.e. of unit norm). This is consistent with the fact that a single qubit reduced density matrix $\rho^\mu=\Tr_{\mu^C}\big(\ket{s}\bra{s}\big)$ is maximally mixed, and has a maximal Von Neumann entropy $S(\rho_\mu)=-\Tr\big[\rho_\mu\log(\rho_\mu)\big]=\log(2)$. Yet the latter quantity is in fact the entropy of entanglement of the bipartition $(\mu,\mu^C)$, considered as a paradigmatic measure of bipartite entanglement \cite{Ent_entropy}. \\
The entanglement distance (ED), first defined in Ref. \cite{ED_2020}, is an entanglement measure for general multipartite pure states; it has been adapted in Ref. \cite{vesperini_entanglement_2023} to the more general framework of multipartite mixed states, and it has already found since then some interesting applications \cite{vafafard_nourmandipour,nourmandipour_entanglement_2021}. It finds its theoretical grounds on the Fubini-Study metric associated to the local-unitary invariant projective Hilbert space, called in this context the Entanglement Metric (EM).
An EM of a state $\ket{s}$ is the real symmetric matrix of elements \begin{equation}\label{ent_metric_elements}
g_{\nu\mu}\big(\ket{s},\bm{v}^\mu,\bm{v}^\nu\big)= \bra{s} \sigma_{\bm{v}}^\mu \sigma_{\bm{v}}^\nu\ket{s} - \bra{s} \sigma_{\bm{v}}^\mu\ket{s}\bra{s} \sigma_{\bm{v}}^\nu\ket{s}. \end{equation} $g$ clearly happens to be a covariance matrix. This does not come as a surprise, as the indetermination in pure states, hence the (co-)variance, is always quantum in nature, and intimately linked to entanglement.
In particular, the diagonal elements of the EM, endowed with a minimization procedure, arise as an efficient measure of entanglement, with a behaviour similar to that of the entropy of entanglement. This is due to the fact that \eqref{vanishing_expect_max_ent_mu} implies that a maximally entangled qubit has variance $1$. The single-qubit ED is defined as \begin{equation}\label{single-qubit_ED} \begin{split} E_\mu(\ket{s}):&=\min_{\bm{v}^\mu}\;g_{\mu\mu}\big(\ket{s},\bm{v}^\mu\big)\\
&= 1 - \max_{\bm{v}^\mu}|\bra{s}\sigma_{\bm{v}}^\mu\ket{s}|^2\\
&= 1 - |\bra{s}\bm{\sigma}^\mu\ket{s}|^2, \end{split} \end{equation} which equates $1$ if $\mu$ is maximally entangled with the rest of the system, and $0$ if it is fully factorizable.
We choose here to use the latter definition of entanglement, which possesses the advantage of being very easy to compute, relative to the Von Neumann entropy. We further define the total entanglement of a state as $\sum\limits_{\mu\in Q}E_\mu(\ket{s})$.\\
In the present work, we primarily focus on maximally entangled state in the sense of Eq.\eqref{single-qubit_ED}. We start, in section \ref{sec:Theorems}, demonstrating a few simple theorems, which highlight the strong relationship between pre-measurement correlations and post-measurement average values, and show how the structure of the EM provides an upper bound to the persistency of entanglement. In section \ref{sec:optimization}, we derive two procedures in order to determine sets of measurement axis optimal with respect to pair-wise correlators or with respect to the induced total entanglement breaking; the question of the equivalency of these two optimal sets arises as an interesting open problem, to our best knowledge, yet to be solved. We apply our methods to a few examples in section \ref{sec:examples}. Finally, in section \ref{sec:discussion}, we synthesize our results, and make a few remarks on possible continuations of this work; in particular, we examine the effect of several successive projective measurement on expectation values, and argue that a more thorough study of entanglement breaking in quantum states should investigate the behaviour of higher order correlations.
\section{First order projective measurements}\label{sec:Theorems}
Departing from a generic multipartite quantum state $\ket{s}\in\mathcal{H}$, the state $\ket{s'}$ obtained after a projective measurement of the qubit $\nu$ in the direction $\bm{m}^\nu$ write \begin{equation}\label{proj_meas_sprime}
\ket{s}\longrightarrow \ket{s'}=\frac{P_{\bm{m}}^\nu\ket{s}}{\sqrt{\bra{s}P_{\bm{m}}^\nu\ket{s}}}, \end{equation} with \begin{equation}
P_{\bm{m}}^\nu:=\frac{\mathbb{I}+\sigma_{\bm{m}}^\nu}{2} \end{equation} the single qubit projector onto the eigenstate of $\sigma_{\bm{m}}^\nu$ of eigenvalue $1$.
The expectation value of an arbitrary qubit $\mu$ in the direction $\bm{v}^\mu$ in such a post-measurement state can be expressed as a function of the expectation values and the correlator of $\sigma_{\bm{m}}^\nu$ and $\sigma_{\bm{v}}^\nu$ in the initial state.
\begin{Theorem}\label{thm_1} If $\ket{s}$ is maximally entangled in $\nu$ and $\mu$, and $\ket{s'}$ is the post-measurement state after a projective measure of $\sigma_{\bm{m}}^\nu$, then we have, for any measurement axis $\bm{v}^\mu$ \begin{equation} \bra{s'}\sigma_{\bm{v}}^\mu\ket{s'}=\bra{s}\sigma_{\bm{v}}^\mu \sigma_{\bm{m}}^\nu\ket{s} \end{equation} \end{Theorem}
\begin{proof} From Eq.\eqref{proj_meas_sprime} we draw \begin{equation} \begin{split}
\bra{s'}\sigma_{\bm{v}}^\mu\ket{s'}& = \frac{\bra{s} P_{\bm{m}}^\nu\sigma_{\bm{v}}^\mu P_{\bm{m}}^\nu\ket{s}}{\bra{s}P_{\bm{m}}^\nu\ket{s}}= \frac{\bra{s} P_{\bm{m}}^\nu\sigma_{\bm{v}}^\mu\ket{s}}{\bra{s}P_{\bm{m}}^\nu\ket{s}} \\
&=\frac{\bra{s}\sigma_{\bm{v}}^\mu\ket{s} + \bra{s}\sigma_{\bm{v}}^\mu \sigma_{\bm{m}}^\nu\ket{s}}{1+\bra{s}\sigma_{\bm{m}}^\nu\ket{s}}, \end{split} \end{equation} where we used the fact that $P_{\bm{m}}^\nu$ and $\sigma_{\bm{v}}^\mu$ commute with each other, and that $P_{\bm{m}}^\nu$ is idempotent. By hypothesis, Eq.\eqref{vanishing_expect_max_ent_mu} applies here, hence our Theorem. \end{proof}
\begin{Corollary}
If $\ket{s}$ is maximally entangled in $\nu$ and $\mu$ and $\exists\, \bm{v}^\mu$ such that $|\bra{s}\sigma_{\bm{v}}^\mu \sigma_{\bm{m}}^\nu\ket{s}|=1$, then the measurement of $\nu$ along the axis $\bm{m}^\nu$ completely breaks the entanglement of $\mu$, i.e. $ E_\mu\big(\ket{s'}\big) =0$. \end{Corollary} \begin{proof} If Theorem \ref{thm_1} applies, we can rewrite the post-measurement ED of $\mu$ \begin{equation}\label{Ent_postmeas}
E_\mu\big(\ket{s'}\big) = 1 - \max_{\bm{v}^\mu}|\bra{s}\sigma_{\bm{v}}^\mu \sigma_{\bm{m}}^\nu\ket{s}|^2 , \end{equation} which equates $0$ if the above condition is fulfilled. \end{proof}
\begin{Theorem}\label{thm-2} For any state $\ket{s}$, $\forall \mu,\nu\in Q$ such that $\bra{s}\sigma_{\bm{v}}^\mu \sigma_{\bm{v}}^\nu\ket{s}=1$, the operators $\sigma_{\bm{v}}^\mu$ and $\sigma_{\bm{v}}^\nu$ are equivalent with respect to $\ket{s}$ (they act on it in the same fashion). In particular, this implies, $\forall\eta$, \begin{align} & \bra{s}\sigma_{\bm{v}}^\eta \sigma_{\bm{v}}^\nu\ket{s} = \bra{s}\sigma_{\bm{v}}^\eta \sigma_{\bm{v}}^\mu\ket{s}, \label{correlator_equality}\\ \text{and }&\bra{s}\sigma_{\bm{v}}^\eta \sigma_{\bm{v}}^\mu \sigma_{\bm{v}}^\nu\ket{s}=\bra{s}\sigma_{\bm{v}}^\eta\ket{s}. \end{align} It also results that the measure of $\sigma_{\bm{v}}^\nu$ yields the implicit measure of $\sigma_{\bm{v}}^\mu$. \end{Theorem} \begin{proof} Under the hypothesis, $\ket{s}$ is eigenvector of $\sigma_{\bm{v}}^\mu \sigma_{\bm{v}}^\nu$ of eigenvalue $1$, and the following holds \begin{equation}\label{sigmu-signu_equivalency} \begin{split} \sigma_{\bm{v}}^\mu \sigma_{\bm{v}}^\nu\ket{s}& = \ket{s} \\
\sigma_{\bm{v}}^\nu\ket{s}& = \sigma_{\bm{v}}^\mu\ket{s} \\
P_{\bm{v}}^\nu\ket{s}& = P_{\bm{v}}^\mu\ket{s} \\
P_{\bm{v}}^\nu\ket{s}& = P_{\bm{v}}^\nu P_{\bm{v}}^\mu\ket{s}. \end{split} \end{equation} \end{proof}
Let us now examine the consequences of these results on $g$. For the sake of clarity, we will hereafter drop its dependences and adopt the shortened notation $g_{\nu\mu}\big(\ket{s},\bm{v}^\mu,\bm{v}^\nu\big)=g_{\nu\mu}$.\\
\begin{Theorem}\label{thm-3} Let $\ket{s}$ be a state maximally entangled $\forall\mu\in Q$ and $\{\bm{v}^\mu\}_\mu$ a choice of measurement directions such that $\forall\mu,\nu\in Q,\;g_{\mu\nu}=0$ or $\pm1$. Then:
\begin{itemize}
\item Up to a reordering of its indices (equivalently, a relabelling of the qubits), $g$ is diagonal by blocks filled with $\pm1$.
\item The number $n$ of such blocks provides an upper bound to the persistency of entanglement $P_e$ of the state $\ket{s}$, i.e. $P_e\leq n$. In other words, the minimal number of local measurements necessary to fully break its entanglement is $n$ or less.
\end{itemize} \end{Theorem} \begin{proof} From Eq.\eqref{vanishing_expect_max_ent_mu}, it is clear that, $\forall\mu,\nu\in Q$, Eq.\eqref{ent_metric_elements} simplifies as \begin{equation}\label{gmunu_maxent}
g_{\nu\mu}= \bra{s} \sigma_{\bm{v}}^\mu \sigma_{\bm{v}}^\nu\ket{s}, \end{equation} and, from Theorem \ref{thm-2}, we see that, $\forall\eta\in Q$, the following transitivity relation holds \begin{equation}
|g_{\nu\mu}|=|g_{\nu\eta}|=1 \implies |g_{\mu\eta}|=1 \end{equation}
Added with the symmetry of $g$, this proves the first part of the Theorem.
Using Theorem \ref{thm_1} together with Eq.\eqref{gmunu_maxent}, we can rewrite the diagonal elements of $g$ after a measure of $\sigma_{\bm{m}}^\nu$ as \begin{equation}\label{gmumu_postmeas}
g_{\mu\mu}' = 1 - |g_{\nu\mu}|^2 \end{equation} where $g'=g(\ket{s'})$. This straightforwardly implies that the measure of any qubit belonging to a block will collapse entanglement for the whole block, hence the second part of the Theorem. \end{proof}
The structure of $g$ gives valuable insights on \textit{persistency of entanglement} $P_e$ in such maximally entangled states. First introduced in \cite{BRS_persistent}, it quantifies the minimal number of measurements needed to completely disentangle a quantum state.
The notion of \textit{persistency of entanglement} is distinct from that of \textit{$k$-separability}\footnote{As a counter example, the state $\frac{1}{2}\big(\ket{0000}+\ket{0011}+\ket{1100}-\ket{1111}$ has an EM with $n=2$ blocks when $\forall\mu,\;\bm{v}^\mu=(0,0,1)$, hence has $P_e=2$ and yet is not biseparable.}. They are however related, as the entanglement of a maximally entangled $k$-separable state can clearly be fully broken by $k$ measurements. In other words, the persistency of entanglement provides an upper bound to the separability, and we have $n\geq P_e\geq k$.
Because $g$ only encodes informations on the effects of first order projective measurement (i.e. contains only two-points correlators), it may not capture the actual $P_e$, hence only provides an upper bound. This is due to the fact that new non-vanishing correlation patterns may arise after one or more projective measurements, entailing diminished $P_e$ relative to our first guess. The Briegel Raussendorf state with $N>4$, detailed in section \ref{subsec:BRS}, constitutes an example of such a situation. \\
Let us stress the dependence of $g$ on the set of unit vectors $\{\bm{v}^\mu\}$, representing directions of measurements. This point is of great importance because, as a different $g$ will arise from a different set $\{\bm{v}^\mu\}$, such are also the correlation patterns and subsequent entanglement breakings highlighted in the above. In order to bound $P_e$ as finely as possible, one thus needs to find the appropriate set $\{\bm{v}^\mu\}$.
\section{Optimization of the measure}\label{sec:optimization}
We are now looking for the measurement directions $\bm{v}^\nu$ optimizing the correlators in Eq.\eqref{ent_metric_elements}.\\
First, let us perform the pair-wise optimization of the correlators \begin{equation}\label{optimization_pairwise_corr} \bra{s}\sigma_{\bm{v}}^\mu\sigma_{\bm{v}}^\nu\ket{s} = \sum_{i,j=1}^3 v_i^\mu v_j^\nu \bra{s}\sigma_i^\mu\sigma_j^\nu\ket{s} = (\bm{v}^\mu)^T\bm{C}_s^{\mu\nu}\bm{v}^\nu, \end{equation} where $\bm{C}_s^{\mu\nu}$ is the non-symmetric real matrix of elements \begin{equation} \big(\bm{C}_s^{\mu\nu}\big)_{ij}=\bra{s}\sigma_i^\mu\sigma_j^\nu\ket{s} \end{equation} i.e. the spin correlation matrix. Optimization with respect to both measurement axis yields \begin{equation} \begin{split} \bm{C}_s^{\mu\nu}\bm{v}^\nu&=\lambda\bm{v}^\mu\\ (\bm{v}^\mu)^T\bm{C}_s^{\mu\nu}&=\lambda(\bm{v}^\nu)^T, \end{split} \end{equation} where the superscript $T$ stands for transpose. One can check by easy direct calculation that the eigenvalues indeed coincide. By insertion of the transpose of these two equations, we infer the eigenvalue equations \begin{equation} \begin{split} (\bm{C}_s^{\mu\nu})^T\bm{C}_s^{\mu\nu}\bm{v}^\nu&=\lambda^2\bm{v}^\nu\\ \bm{C}_s^{\mu\nu}(\bm{C}_s^{\mu\nu})^T\bm{v}^\mu&=\lambda^2\bm{v}^\mu, \end{split} \end{equation} of which largest eigenvalue solutions evidently correspond to the measurement directions maximizing Eq.\eqref{optimization_pairwise_corr}. \\
We can instead be interested in finding the measurement axis $\bm{m}^\nu$ that optimally disentangle the entire state $\ket{s}$. Provided the latter is maximally entangled, we have, as a consequence of Theorem \ref{thm_1} \begin{equation}
\big|\bra{s'}\bm{\sigma}^\mu\ket{s'}\big|^2 = \sum_{i=1}^3\big|\bra{s'}\sigma_i^\mu\ket{s'}\big|^2 = \sum_{i=1}^3\big|\bra{s}\sigma_i^\mu\sigma_{\bm{m}}^\nu\ket{s}\big|^2, \end{equation} hence the total entanglement after the measure \begin{equation}\label{optimization_totent} \begin{split}
E\big(\ket{s'}\big)&=\sum_{\mu} E_\mu(\ket{s'})= N - 1 - \sum_{\mu\in \nu^C}|\bra{s'}\bm{\sigma}^\mu\ket{s'}|^2\\
&=N - 1 - \sum_{\mu\in \nu^C}\sum_{i=1}^3\big|\bra{s}\sigma_i^\mu\sigma_{\bm{m}}^\nu\ket{s}\big|^2\\
&=N - 1 - \sum_{\mu\in \nu^C}\sum_{i=1}^3\bra{s}\sigma_{\bm{m}}^\nu\sigma_i^\mu\ket{s}\bra{s}\sigma_i^\mu\sigma_{\bm{m}}^\nu\ket{s}\\
&=N - 1 - \bra{s}\sigma_{\bm{m}}^\nu\Big(\sum_{\mu\in \nu^C}\sum_{i=1}^3\sigma_i^\mu\ket{s}\bra{s}\sigma_i^\mu\Big)\sigma_{\bm{m}}^\nu\ket{s}\\
&=N - 1 - \sum_{j,k=1}^3 m_j^\nu m_k^\nu\bra{s}\sigma_j^\nu\Big(\sum_{\mu\in \nu^C}\sum_{i=1}^3\sigma_i^\mu\ket{s}\bra{s}\sigma_i^\mu\Big)\sigma_k^\nu\ket{s}, \end{split} \end{equation} with $\nu^C=Q\setminus\{\nu\}$. Let us define $\bm{B}_s^{\nu}$, the measurement-induced entanglement breaking matrices (MIEB) of the state $\ket{s}$, of elements \begin{equation}\label{MIEB-matrix}
\Big(\bm{B}_s^{\nu}\Big)_{jk}=\bra{s}\sigma_j^\nu\,\Sigma_s^{\nu^C}\,\sigma_k^\nu\ket{s}, \end{equation} with $\Sigma_s^{\nu^C}=\sum\limits_{\mu\in \nu^C}\sum\limits_{i=1}^3\sigma_i^\mu\ket{s}\bra{s}\sigma_i^\mu$. Its diagonalization straightforwardly yields the results of the optimization problem, as the eigenvectors $\widetilde{\bm{m}}^\nu$ associated to the largest (resp. smallest) eigenvalue of $\bm{B}^{\nu}\big(\ket{s}\big)$ corresponds to the minimum (resp. maximum) of Eq.\eqref{optimization_totent}. The eigenvalues themselves simply equate the total additional loss of entanglement after the corresponding measurement. It results that a comparison of the spectra of the $N$ MIEB matrices allow to easily find the ``weak spots'' of $\ket{s}$, that is the qubits of which the measurement can break entanglement the most.
Interestingly, the largest eigenvalue of $\bm{B}_s^{\nu}$ might have multiplicity greater than one. In such cases, any of the corresponding eigenvectors or linear combination of them maximize Eq.\eqref{optimization_totent}.
This result can straightforwardly be adapted to find the measurement axis optimizing the entanglement breaking of a subset $Q'\subset Q$ with $\nu\notin Q'$, by simply replacing $\Sigma_s^{\nu^C}$ with $\Sigma_s^{Q'}=\sum\limits_{\mu\in Q'}\sum\limits_{i=1}^3\sigma_i^\mu\ket{s}\bra{s}\sigma_i^\mu$.\\
The question naturally arises as to know whether the solutions of Eq.\eqref{optimization_pairwise_corr} correspond in general to those of Eq.\eqref{optimization_totent}.
\begin{Remark} The existence, in the general case, of one (or several) set of measurement axis $\{\bm{v}^\nu\}_\nu^{opt}$ optimizing \textit{simultaneously} every correlator in the EM remains unclear. \footnote{To clarify the meaning of this problem, let us imagine a state $\ket{s}$ such that $\nexists\{\bm{v}^\nu\}_\nu^{opt}$. Then there exists some qubit $\mu$ of which the correlation with $\nu$ is maximal in a direction $v_1^\mu$, and the correlation with $\eta\neq\nu$ is maximal in a direction $v_2^\mu\neq v_1^\mu$.} \begin{itemize} \item If $\{\bm{v}^\nu\}_\nu^{opt}$ indeed always exists, it would be enough, to completely probe the pair-wise correlation patterns of a given state, to solve $\lceil N/2 \rceil$ equations of the form of Eq.\eqref{optimization_pairwise_corr} or, equivalently, to diagonalize $N$ MIEB \eqref{MIEB-matrix}.
\item If, on the contrary, it doesn't, such a complete probing requires to solve all of the $N(N-1)/2$ equations of the form of Eq.\eqref{optimization_pairwise_corr}. \end{itemize} \end{Remark}
Yet, in all of the example we considered, the solutions of Eq.\eqref{optimization_pairwise_corr} equate those of Eq.\eqref{optimization_totent}, hence we were able to determine $\{\bm{v}^\nu\}_\nu^{opt}$. We were not able find any counter-example. We will thus assume from now on that this set exists, in particular in Section \ref{sec:examples}, as this assumption allows us to remain much more concise, by only diagonalizing the $N$ MIEB.
\section{Examples and applications}\label{sec:examples}
\subsection{Briegel-Raussendorf states}\label{subsec:BRS}
The Briegel-Raussendorf states (BRS) form a family of quantum states, introduced in \cite{BRS_persistent}, They are, up to local unitary transformation, equivalent to cluster states (also coined as graph states in the litterature), which were proposed as a model for the measurement-based quantum computer \cite{oneway_BR,NIELSEN2006147,Nielsen_2003,meas_based_cluster}.\\
BRS are defined, for an arbitrary number of qubits $N$ on a $d$-dimensional lattice, as \begin{equation} \ket{\phi(\varphi)} = U(\varphi)\ket{+}^{\otimes N}, \end{equation} where $\ket{+}^\mu$ is the eigenstate of the operator $\sigma_1^\mu$ of eigenvalue $1$, and \begin{equation}\label{UBRS} U(\varphi) = \exp\Big\{-i\varphi\sum_{<\mu,\nu>}P_0^\mu P_1^\nu\Big\}, \end{equation} where the summation runs over all the pairs of nearest neighbours, and $P_{0(1)}^\mu$ is the projector onto the eigenstates of $\sigma_3^\mu$ of eigenvalue $\pm 1$. We are solely interested in the case $\varphi=\pi$, as it results in a maximally entangled state. We will furthermore restrict ourselves to the study of the $1$-dimensional case. \begin{equation} \ket{\phi_N} = \prod_{\mu=0}^{N-2}\frac{1}{2}\big(\mathbb{I}-\sigma_3^\mu+\sigma_3^{\mu+1}+\sigma_3^\mu\sigma_3^{\mu+1}\big)\ket{+}^{\otimes N}. \end{equation}
It has been shown in \cite{ED_2020} that the $E_\mu(\ket{\phi_N})=0$, $\forall\mu$. \\
\paragraph*{$N=3$.}
It is known that the $3$-qubits BR state is local unitary (LU) equivalent to the Greenberger-Horne-Zeilinger state \cite{BRS_persistent}, a prototypical example of maximally entangled and maximally correlated state. We hence expect that $\exists\{\bm{v}^\mu\}$ such that $\forall\mu,\nu,\,g_{\mu\nu}=1$.
We only need to compute the three following MIEB matrices \eqref{MIEB-matrix} \begin{equation} \begin{split} \bm{B}_{\phi_3}^{0} & = \bm{B}_{\phi_3}^{2}= \begin{pmatrix} 2 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix}\text{, hence } \bm{v}^0=\bm{v}^2=(1,0,0) \text{, and }\\ \bm{B}_{\phi_3}^{1} &=\begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 2 \end{pmatrix}\text{, hence } \bm{v}^1=(0,0,1). \end{split} \end{equation} The maximal eigenvalues of these matrices equate $2$ because each qubit is maximally correlated with two others.
This yields the EM \begin{equation} g\big(\ket{\phi_3},\{\bm{v}^\mu\}\big)=\begin{pmatrix} 1 & 1 & -1 \\ 1 & 1 & -1 \\ -1 & -1 &1 . \end{pmatrix} \end{equation} A single measurement is thus sufficient to completely disentangle $\ket{\phi_3}$. We have $P_e(\ket{\phi_3})=1$, hence this state is genuinely entangled.\\
\paragraph*{$N=4$.}
In this case, we have \begin{equation} \begin{split} \bm{B}_{\phi_4}^{0} &=\bm{B}_{\phi_4}^{3} =\begin{pmatrix} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix}\text{, hence } \bm{v}^0=\bm{v}^3=(1,0,0), \text{ and}\\ \bm{B}_{\phi_4}^{1}& =\bm{B}_{\phi_4}^{2} =\begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 1 \end{pmatrix}\text{, hence } \bm{v}^1=\bm{v}^2=(0,0,1), \end{split} \end{equation} thus the EM writes \begin{equation} g(\{\bm{v}^\mu\})=\begin{pmatrix} 1 & 1 & 0 & 0 \\ 1 & 1 & 0 & 0 \\ 0 & 0 & 1 & -1 \\ 0 & 0 & -1 & 1 \end{pmatrix}, \end{equation} hence $P_e(\ket{\phi_4})\leq 2$. \\
\paragraph*{$N>4$.}
In general, for a chain of $N$ qubits, we have \begin{equation} \begin{split} \bm{B}_{\phi_N}^{0} &=\bm{B}_{\phi_N}^{N)} =\begin{pmatrix} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix}, \text{ hence } \bm{v}^0=\bm{v}^{N}=(1,0,0), \text{ and } \\ \bm{B}_{\phi_N}^{1} &=\bm{B}_{\phi_N}^{N-1)} =\begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 1 \end{pmatrix}, \text{ hence } \bm{v}^1=\bm{v}^{N-1}=(0,0,1), \end{split} \end{equation} while $\bm{B}_{\phi_N}^{\nu}=\bm{0},\;\forall\nu\notin \{0,1,N-1,N\}$.
The EM hence contains two $2\times 2$ blocks in its upper left and lower right corners, while the remaining part is just a diagonal filled with ones (which can be seen as trivial blocks of size $1\times1$), hence the number of its blocks is $N-2$. Its persistency of entanglement is however known to be $P_e(\ket{\phi_N})=\lfloor \frac{N}{2} \rfloor$ \citep{BRS_persistent}.
This discrepancy is a paradigmatic example of the fact that Theorem \ref{thm-3} is in general insufficient to capture exactly $P_e$. Measurements can indeed be performed on $\ket{\phi_N}$, after which some elements of the EM will cease to be null. The post-measurement EM thus exhibits some new non trivial blocks, accounting for this discrepancy. \\
\subsection{Supersinglet states}
The supersinglet states, first introduced in \cite{Supersinglets}, form a class of maximally entangled pure states with the property of being invariant under any simultaneous LU operation acting on all qubits, i.e. $U^{\otimes N}\ket{S_N}=e^{i\phi}\ket{S_N}$, with $U$ an arbitrary LU operator. \cite{Supersinglets,Supersinglets2,GUHNE}. \\
From this sole property, a number of facts can be drawn.
First, $\forall U$, $\forall\bm{v}^\nu$, we have \begin{equation} \begin{split} \bra{S_N}\sigma_{\bm{v}}^\nu\ket{S_N} & = \bra{S_N}U^{\dagger\otimes N}\sigma_{\bm{v}}^\nu U^{\otimes N}\ket{S_N} \\ &= \bra{S_N}U^{\nu\dagger}\sigma_{\bm{v}}^\nu U^\nu\ket{S_N} = \bra{S_N}\sigma_{\bm{v}'}^\nu\ket{S_N}, \end{split} \end{equation} yet Pauli expectation values cannot be isotropic unless null, hence Eq.\eqref{vanishing_expect_max_ent_mu} is verified and $E_\nu(\ket{S_N})=1,\,\forall\nu$.
Theorem \ref{thm_1} hence applies here and, choosing $U$ to be a rotation around the axis $\bm{m}^\nu$, hence leaving $\sigma_{\bm{m}}^\nu$ unchanged, we can write \begin{equation} \begin{split} \bra{S_N'}\sigma_{\bm{v}}^\mu\ket{S_N'}&=\bra{S_N}\sigma_{\bm{v}}^\mu\sigma_{\bm{m}}^\nu\ket{S_N}\\ &=\bra{S_N}U^{\dagger\otimes N}\sigma_{\bm{v}}^\mu\sigma_{\bm{m}}^\nu U^{\otimes N}\ket{S_N}\\ &=\bra{S_N}\big(U^{\mu\dagger}\sigma_{\bm{v}}^\mu U^\mu\big)\sigma_{\bm{m}}^\nu \ket{S_N}\\ &=\bra{S_N}\sigma_{\bm{v}'}^\mu\sigma_{\bm{m}}^\nu\ket{S_N}=\bra{S_N'}\sigma_{\bm{v'}}^\mu\ket{S_N'}, \end{split} \end{equation} where $\ket{S_N'}$ is the state post-measurement of $\sigma_{\bm{m}}^\nu$. The same argument as above leads to $\bra{S_N}\sigma_{\bm{v}}^\mu\sigma_{\bm{m}}^\nu\ket{S_N}\neq 0$ if and only if $\bm{v}^\mu=\bm{v}^{\mu\prime}$, that is if $\bm{v}^\mu=\pm\bm{m}^\nu$.
It results $\bm{B}_{S_N}^{\mu}\propto\mathbb{I},\;\forall\mu\in Q$. This means that, provided that the qubits are measured along the same axis, the correlators are always maximal. The optimal set of measurement axis is thus any uniform set $\{\bm{v}^\mu\}^{uni}$. \\
The whole class of supersinglet states of four qubits is spanned by the following two states \begin{equation} \begin{split} \ket{S^{1}_4}&=\frac{1}{\sqrt{3}}\Big(\ket{0011} + \ket{1100} -\frac{1}{2}\big( \ket{0101} + \ket{0110} + \ket{1001} + \ket{1010}\big)\big)\\ \ket{S^{2}_4}&=\frac{1}{2}\Big(\ket{0101} + \ket{1010} - \ket{0110} - \ket{1001} \big)\big). \end{split} \end{equation}
We can hence write a general 4-qubits supersinglet state as $\ket{S_4(a,b)}=a\ket{S^{1}_4}+b\ket{S^{2}_4}$, with $|a|^2+|b|^2$.
We can choose arbitrarily a single measurement axis, for instance $\bm{x}=(1,0,0)$, and straightforwardly compute the optimal EM \begin{equation} g(\{\bm{v}^\mu\}^{uni})=\begin{pmatrix} 1 & \alpha & \gamma & \beta \\ \alpha & 1 & \beta & \gamma \\ \gamma & \beta & 1 & \alpha \\ \beta & \gamma & \alpha & 1 \end{pmatrix}, \end{equation} with \begin{equation} \begin{split}
\alpha&=\frac{|a|^2}{3} - |b|^2,\\
\beta&=\frac{2}{3}(\sqrt{3}\Re(\bar{a}b)-|a|^2), \text{and}\\
\gamma&=-\frac{2}{3}\big(\sqrt{3}\Re(\bar{a}b)+|a|^2\big), \end{split} \end{equation} where the bar over a letter denotes complex conjugation and $\Re(\bar{a}b)$ is the real part of the complex number $\bar{a}b$.\\
This simple expression for the EM allow us to remark some interesting cases arising for specific values of the parameters $a$ and $b$.
Trivially, if $a=0$ and $|b|=1$, we immediately get a block diagonal matrix containing two blocks, as is to be expected, since $\ket{S_4^{(2)}}$ is a tensor product of two Bell states $\ket{\phi_-}=\frac{1}{\sqrt{2}}(\ket{01}-\ket{10})$. In this case, $P_e(\ket{S_4(0,1)})=2$ and the state is of course biseparable.
If $a=\frac{\sqrt{3}e^{i\phi_a}}{2}$ and $b=\frac{e^{i\phi_b}}{2}$, a few calculations lead to \begin{equation} g(\{\bm{v}^\mu\}^{uni})=\begin{pmatrix} 1 & 0 & -c^2 & -s^2 \\ 0 & 1 & -s^2 & -c^2 \\ -c^2 & -s^2 & 1 & 0 \\ -s^2 & -c^2 & 0 & 1 \end{pmatrix}, \end{equation} with $c=\cos(\frac{\phi_a-\phi_b}{2})$ and $s=\sin(\frac{\phi_a-\phi_b}{2})$. Thus, for $\phi_a=\phi_b$ and for $\phi_b=\phi_a+\pi$, the EM of this state, up to qubits permutations, contains two blocks, and $P_e(\ket{S_4(\frac{\sqrt{3}e^{i\phi_a}}{2},\frac{e^{i\phi_a}}{2})})=P_e(\ket{S_4(\frac{\sqrt{3}e^{i\phi_a}}{2},\frac{e^{i(\phi_a+\pi)}}{2})})=2$.
\section{Discussion}\label{sec:discussion}
In this work, we showed how, in pure quantum states, entanglement entails a strong link between correlations and post-measurement expectation values.
The EMs, i.e. covariance matrices, contain valuable informations on the statistics of post-measurement states, and on the patterns that can emerge from projective measurements of entangled states. In particular, its block structure is directly linked to the persistency of entanglement.
We further provided two straightforward procedures of optimization of the Pauli correlators, and observe that they might not, in principle, yield equivalent results. By doing so, we unravel an opened problem, which hasn't, to our best knowledge, been tackled with yet, namely the existence of a set of measurement axis simultaneously optimizing all of these correlators.
These procedures further allow us to recover, if it exists, the optimal EM, along with an upper bound for the persistency of entanglement. \\
Unfortunately, as emphasized above, the information retrieved by the use of the EM is incomplete: since the effect of more than one measurement are not accounted for in this framework, our approach fails to recover a number of important features of some complex entangled states (as BR states or cluster states), amongst which the exact persistency of entanglement.
Multipartite maximally entangled states might indeed possess qubits with only vanishing two-points correlations, regardless of the choice of measurement axis. Somehow counter-intuitively, the measurement of such qubits, though disentangling the concerned qubit from the rest of the system, do not break any entanglement on the latter. Yet it modifies the state, and in particular might bring along some new non-vanishing correlators (i.e. off-diagonal terms in the EM).
This observation motivates the study of higher order measurement schemes.
Let us consider an ordered subset $\mathcal{M}\subset Q$ of $M$ qubits on which successive projective measurements are performed.
The generalization of \eqref{proj_meas_sprime} then yields \begin{equation}
\ket{s}\longrightarrow \ket{s^{_\mathcal{M}}}=\frac{\prod\limits_{\nu\in\mathcal{M}} P_{\bm{m}}^\nu\ket{s}}{\sqrt{\bra{s}\prod\limits_{\nu\in\mathcal{M}}P_{\bm{m}}^\nu\ket{s}}}, \end{equation} and, accordingly, the expectation value of an arbitrary unmeasured qubit $\mu$ after such a series of measurement \begin{equation} \setlength{\jot}{10pt}
\bra{s^{_\mathcal{M}}}\sigma_{\bm{v}}^\mu\ket{s^{_\mathcal{M}}} = \frac{\sum\limits_{k=0}^{M-1}\sum\limits_{X\in[\mathcal{M}]^k}\bra{s}\sigma_{\bm{v}}^\mu \prod\limits_{\nu\in X}\sigma_{\bm{m}}^{\nu}\ket{s}}{\sum\limits_{k=0}^{M-1}\sum\limits_{X\in[\mathcal{M}]^k}\bra{s} \prod\limits_{\nu\in X}\sigma_{\bm{m}}^{\nu}\ket{s}}, \end{equation} where $[\mathcal{M}]^k$ is the set of all the unordered $k$-subsets (i.e. subsets of cardinal $k$) of $\mathcal{M}$.\\
It results that, if all the two-points correlators vanish, one can examine higher order correlators to investigate the breaking of entanglement after a series of measurement rather a single one.
It would be of great interest to pursue this research by a thorough study of higher order covariance tensors, or to devise new procedures and methods, in order to grasp the effects of series of measurements on entangled states, in a more exhaustive fashion.
\begin{acknowledgments} The author acknowledges support from the RESEARCH SUPPORT PLAN 2022 - Call for applications for funding allocation to research projects curiosity driven (F CUR) - Project ”Entanglement Protection of Qubits’ Dynamics in a Cavity” – EPQDC , and the support by the Italian National Group of Mathematical Physics (GNFM-INdAM). The author further acknowledges useful discussions with Roberto Franzosi. \end{acknowledgments}
\end{document} |
\begin{document}
\title{Singular elliptic equation involving the GJMS operator on the standard unit sphere.} \author{Mohammed Benalili and Ali Zouaoui} \maketitle
\begin{abstract} Given a Riemannian compact manifold $\left( M,g\right) $ of dimension $n\geq 5$, we have proven in \cite{1} under some conditions that the equation : \begin{equation} P_{g}(u)=Bu^{2^{\sharp }-1}+\frac{A}{u^{2^{\sharp }+1}}+\frac{C}{u^{p}} \label{E1} \end{equation} where $P_{g}$ is GJMS-operator, $n=\dim (M)>2k$ $(k\in \mathbb{N}^{\star })$ , $A,B$ and $C$ are smooth positive functions on $M$, $p>1$ and $2^{\sharp }= \frac{2n}{n-2k}$ denotes the critical Sobolev of the embedding $H_{k}^{2}(M)$ $\subset $ $L^{2^{\sharp }}(M)$, admits two distinct positive solutions. The proof of this result is essentially based on the given smooth function $ \varphi >0$ with norm $\Vert \varphi \Vert _{P_{g}}=1$ fulfilling some conditions ( see Theorem 3 in \cite{1}). In this note we construct an example of such function on the unit standard sphere $\left( \mathbb{S} ^{n},h\right) $. Consequently the conditions of the Theorem are improved in the case of $\left( \mathbb{S}^{n},h\right) $. \end{abstract}
\section{Construction of the function $\protect\varphi $}
Inspired by the work of F. Robert. (see \cite{4} ) , we construct an example of a smooth function $\varphi >0$ on the Euclidean sphere $\left( \mathbb{S} ^{n},h\right) $ with norm$\Vert \varphi \Vert _{P_{h}}=1$.\newline Indeed let $\lambda >0$ and $x_{0}\in \mathbb{S}^{n}$. To a rotation, we may assume that $x_{0}$ is the north pole i.e. $x_{0}=\left( 0,...0,1\right) $. We consider the transformation \begin{equation*} \phi _{\lambda }:\mathbb{S}^{n}\rightarrow \mathbb{S}^{n} \end{equation*} defined by $\phi _{\lambda }(x)=\psi _{x_{0}}^{-1}\left( \lambda ^{-1}.\psi _{x_{0}}(x)\right) $ if $x\neq x_{0}$ and $\phi _{\lambda }(x_{0})=x_{0}$ where $\psi _{x_{0}}$ is the stereographic projection of $x_{0}$ given by \begin{equation*} \psi _{x_{0}}:\left( \mathbb{S}^{n}\setminus \left\{ x_{0}\right\} ,h\right) \rightarrow \left( \mathbb{R}^{n},\xi \right) , \end{equation*} for any $a=\left( \eta _{1},...,\eta _{n},\zeta \right) $ associates $\psi _{x_{0}}(a)=\left( \frac{\eta _{1}}{1-\zeta },...,\frac{\eta _{n}}{1-\zeta } \right) $ and \begin{equation*} \begin{array}{c} \delta _{\lambda }:\left( \mathbb{R}^{n},\xi \right) \rightarrow \left( \mathbb{R}^{n},\xi \right) \\ x\mapsto \delta _{\lambda }(x)=\frac{1}{\lambda }x \end{array} \end{equation*} is the homothetic mapping. $h$ is the canonical metric on $\mathbb{S}^{n}$ and $\xi $ is the Euclidean one on $\mathbb{R}^{n}$.\newline \newline Note that $\psi _{x_{0}}$ is a conformal, mapping more precisely we have \begin{equation*} \left( \psi _{x_{0}}^{-1}\right) ^{\star }h=U^{\frac{4}{n-2k}}.\xi \end{equation*} where $U(x)=\left( \frac{1+\Vert x\Vert ^{2}}{2}\right) ^{k-\frac{n}{2}}$ . Hence $\phi _{\lambda }$ is conformal i.e. \begin{equation*} \phi _{\lambda }^{\star }h=u_{x_{0},\beta }^{\frac{4}{n-2k}}.h\quad \text{ where}\;\beta =\frac{1+\lambda ^{2}}{\lambda ^{2}-1} \end{equation*} and \begin{equation*} u_{x_{0},\beta }(x)=\left( \dfrac{\sqrt{\beta ^{2}-1}}{\beta -\cos d_{h}(x,x_{0})}\right) ^{\frac{n-2k}{2}}\quad \forall x\in \mathbb{S}^{n}\; \text{with}\;\beta >1. \end{equation*} In particular we have \begin{equation} \int\limits_{\mathbb{S}^{n}}u_{x_{0},\beta }^{2^{\sharp }}dv_{h}=\omega _{n} \label{01} \end{equation} where $\omega _{n}>0$ is the volume of the unit standard sphere $\left( \mathbb{S}^{n},h\right) .$\newline By the conformal invariance of the operator $P_{h}$ on $\left( \mathbb{S} ^{n},h\right) $, we obtain that \begin{equation} P_{h}(u_{x_{0},\beta })=\frac{n-2k}{2}Q_{h}u_{x_{0},\beta }^{2^{\sharp }-1} \label{02} \end{equation} where $Q_{h}$ denotes the $Q$-curvature of $\left( \mathbb{S}^{n},h\right) $ which expresses by the Gover's formula as: \begin{equation*} Q_{h}=\dfrac{2}{n-2k}P_{h}(1)=\dfrac{2}{n-2k}(-1)^{k}\prod_{l=1}^{k}(c_{l} \;Sc) \end{equation*} where $c_{l}=\frac{(n+2l-2)(n-2l)}{4n(n-1)}$, $Sc=n(n-1)$ (the scalar curvature of $\left( \mathbb{S}^{n},h\right) $). So the $Q_{h\text{ \ }}$is a positive constant.\newline Multiplying the two sides of \eqref{02} by $u_{x_{0},\beta }$ and integrating on $\mathbb{S}^{n}$ we get: \begin{equation*} \int\limits_{\mathbb{S}^{n}}u_{x_{0},\beta }P_{h}(u_{x_{0},\beta })dv_{h}= \frac{n-2k}{2}Q_{h}\int\limits_{\mathbb{S}^{n}}u_{x_{0},\beta }^{2^{\sharp }}dv_{h}. \end{equation*} And since \begin{equation*} \int\limits_{\mathbb{S}^{n}}u_{x_{0},\beta }P_{h}(u_{x_{0},\beta })dv_{h}=\Vert u_{x_{0},\beta }\Vert _{P_{h}}^{2} \end{equation*} \eqref{02} writes \begin{equation*} \Vert u_{x_{0},\beta }\Vert _{P_{h}}^{2}=\frac{n-2k}{2}Q_{h}\omega _{n}. \end{equation*} Hence by putting \begin{equation*} \varphi =\left( \frac{n-2k}{2}\omega _{n}Q_{h}\right) ^{\frac{-1}{2} }u_{x_{0},\beta } \end{equation*} we obtain a function satisfying the conditions of Theorem 3 in \cite{1} i.e. $\varphi >0$ smooth on $(\mathbb{S}^{n},h)$ such that $\Vert \varphi \Vert _{P_{g}}=1$
\section{ Existence results on the sphere}
On the standard unit sphere $(\mathbb{S}^{n},h),$ if we take the function $ \varphi $ of Theorem 3 in \cite{1} equals $\left( \frac{n-2k}{2}\omega _{n}Q_{h}\right) ^{\frac{-1}{2}}u_{x_{0},\beta }$ , we obtain
\begin{theorem} \label{th1}{\ }Let $\left( \mathbb{S}^{n},h\right) $ be the unit standard unit sphere of dimension $n>2k,\;k\in \mathbb{N}^{\star }.$ There is a constant $C(n,p,k)>0$ depending only on $n,p,k$ such that \begin{equation} \frac{1}{2^{\sharp }}\left( \frac{n-2k}{2}\omega _{n}Q_{h}\right) ^{\frac{ 2^{\sharp }}{2}}\int_{\mathbb{S}^{n}}\frac{A(x)}{u_{x_{0},\beta }^{2^{\natural }}}dv_{h}\leq C\left( n,p,k\right) \left( S\underset{x\in \mathbb{S}^{n}}{\max }B(x)\right) ^{\frac{2+2^{\sharp }}{2-2^{\sharp }}} \label{2.3} \end{equation} and \begin{equation} \frac{1}{p-1}\left( \frac{n-2k}{2}.\omega _{n}.Q_{h}\right) ^{\frac{p-1}{2} }\int_{\mathbb{S}^{n}}\frac{C(x)}{u_{x_{0},\beta }^{p-1}}dv_{h}\leq C\left( n,p,k\right) \left( S\underset{x\in \mathbb{S}^{n}}{\max }B(x)\right) ^{ \frac{p+1}{2-2^{\sharp }}} \label{2.4} \end{equation} where \begin{equation*} u_{x_{0},\beta }(x)=\left( \dfrac{\sqrt{\beta ^{2}-1}}{\beta -\cos d_{h}(x,x_{0})}\right) ^{\frac{n-2k}{2}}\quad \forall x\in \mathbb{S}^{n}\; \text{and}\;\beta >1. \end{equation*} then the equation \eqref{E1} admits a solution of class $C^{\infty }(\mathbb{ S}^{n})$. If moreover for any $\varepsilon \in \left] 0,\lambda ^{\star } \right[ $ where $\lambda ^{\star }$ is a positive constant the two following conditions are satisfied \begin{equation*} \frac{2}{a}\left( \int\limits_{\mathbb{S}^{n}}\sqrt{A(x)}dv_{h}\right) ^{2}\left( \frac{1}{t_{0}a_{1}}\right) ^{2^{\sharp }}>2^{\sharp }k\frac{ t_{0}^{2}}{4n}(2-a) \end{equation*} and \begin{equation*} \left( \frac{2}{a}\right) ^{\frac{p-1}{2^{\sharp }}}\left( \int\limits_{ \mathbb{S}^{n}}\sqrt{C(x)}dv_{h}\right) ^{2}\left( \frac{1}{t_{0}a_{2}} \right) ^{p-1}>(p-1)k\frac{t_{0}^{2}}{4n}(2-a) \end{equation*} where $a_{1},a_{2}$ are positive constants, $2^{\sharp }=\frac{2n}{n-2k} \;,3<p<2^{\sharp }+1$. Then the equation \eqref{E1} admits a second solution. \end{theorem}
Note that since \begin{equation} \left( \dfrac{\beta -1}{\beta +1}\right) ^{\frac{n-2k}{4}}\leq u_{x_{0},\beta }(x)\leq \left( \dfrac{\beta +1}{\beta -1}\right) ^{\frac{n-2k }{4}} \label{03} \end{equation} we can improve the conditions (\ref{2.3}) and (\ref{2.4}) of Theorem \ref {th1}. Indeed, from (\ref{0.3}) we deduce that \begin{equation*} \varphi (x)\geq \left( \dfrac{\beta -1}{\beta +1}\right) ^{\frac{n-2k}{4} }\left( \frac{n-2k}{2}\omega _{n}Q_{h}\right) ^{-\frac{1}{2}}. \end{equation*} Consequently \begin{equation*} \frac{\Vert \varphi \Vert ^{2^{\sharp }}}{2^{\sharp }}\int_{\mathbb{S}^{n}} \frac{A(x)}{\varphi ^{2^{\natural }}}dv_{h}=\frac{1}{2^{\sharp }}\int_{ \mathbb{S}^{n}}\frac{A(x)}{\varphi ^{2^{\natural }}}dv_{h}\leq \frac{1}{ 2^{\sharp }}\left( \dfrac{\beta +1}{\beta -1}\right) ^{\frac{n}{2}}\left( \frac{n-2k}{2}.\omega _{n}.Q_{h}\right) ^{\frac{2^{\sharp }}{2}}\int_{ \mathbb{S}^{n}}A(x)dv_{h} \end{equation*} So, if \begin{equation*} \frac{1}{2^{\sharp }}\left( \dfrac{\beta +1}{\beta -1}\right) ^{\frac{n}{2} }\left( \frac{n-2k}{2}.\omega _{n}.Q_{h}\right) ^{\frac{2^{\sharp }}{2} }\int_{\mathbb{S}^{n}}A(x)dv_{h}\leq C\left( n,p,k\right) \left( S\underset{ x\in \mathbb{S}^{n}}{\max }B(x)\right) ^{\frac{2+2^{\sharp }}{2-2^{\sharp }}} \end{equation*} then the condition (\ref{2.3}) is fulfilled. Likewise if \begin{equation*} \frac{1}{p-1}\left( \frac{n-2k}{2}\omega _{n}Q_{h}\right) ^{\frac{p-1}{2} }\left( \dfrac{\beta +1}{\beta -1}\right) ^{\frac{n(p-1)}{2.2^{\sharp }} }\int_{\mathbb{S}^{n}}C(x)dv_{h}\leq C\left( n,p,k\right) \left( S\underset{ x\in M}{\max }B(x)\right) ^{\frac{p+1}{2-2^{\sharp }}} \end{equation*} the condition (\ref{2.4}) is also true and we deduce the following result:
\begin{corollary} {\ }Let $\left( \mathbb{S}^{n},h\right) $ be the unit standard unit sphere of dimension $n>2k,\;k\in \mathbb{N}^{\star }.$ There is a constant $ C(n,p,k)>0$ depending only on $n,p,k$ such that \begin{equation} \frac{1}{2^{\sharp }}\left( \dfrac{\beta +1}{\beta -1}\right) ^{\frac{n}{2} }\left( \frac{n-2k}{2}.\omega _{n}.Q_{h}\right) ^{\frac{2^{\sharp }}{2} }\int_{\mathbb{S}^{n}}A(x)dv_{h}\leq C\left( n,p,k\right) \left( S\underset{ x\in \mathbb{S}^{n}}{\max }B(x)\right) ^{\frac{2+2^{\sharp }}{2-2^{\sharp }}} \end{equation} and \begin{equation} \frac{1}{p-1}\left( \frac{n-2k}{2}.\omega _{n}.Q_{h}\right) ^{\frac{p-1}{2} }\left( \dfrac{\beta +1}{\beta -1}\right) ^{\frac{n(p-1)}{2.2^{\sharp }} }\int_{\mathbb{S}^{n}}C(x)dv_{h}\leq C\left( n,p,k\right) \left( S\underset{ x\in M}{\max }B(x)\right) ^{\frac{p+1}{2-2^{\sharp }}} \end{equation} where $\beta >1$. Then the equation\eqref{E1} admits a solution of class $ C^{\infty }(\mathbb{S}^{n})$. If moreover for any $\varepsilon \in \left] 0,\lambda ^{\star }\right[ $, where $\lambda ^{\star }$ is a positive constant, the two following assumptions are satisfied \begin{equation} \frac{2}{a}\left( \int\limits_{\mathbb{S}^{n}}\sqrt{A(x)}dv_{h}\right) ^{2}\left( \frac{1}{t_{0}a_{1}}\right) ^{2^{\sharp }}>2^{\sharp }k\frac{ t_{0}^{2}}{4n}(2-a) \end{equation} and \begin{equation} \left( \frac{2}{a}\right) ^{\frac{p-1}{2^{\sharp }}}\left( \int\limits_{ \mathbb{S}^{n}}\sqrt{C(x)}dv_{h}\right) ^{2}\left( \frac{1}{t_{0}a_{2}} \right) ^{p-1}>(p-1)k\frac{t_{0}^{2}}{4n}(2-a) \end{equation} where $a_{1},a_{2}$ are positive constants, $2^{\sharp }=\frac{2n}{n-2k} \;,3<p<2^{\sharp }+1$. Then the equation \eqref{E1} admits a second solution. \end{corollary}
\end{document} |
\begin{document}
\title{Twin-field quantum digital signatures} \author{Chun-Hui Zhang$^{1,2,3}$} \author{Yu-Teng Fan$^{1,2,3}$} \author{Chun-Mei Zhang$^{1,2,3,4}$} \author{Guang-Can Guo$^{1,2,3,4}$} \author{Qin Wang$^{1,2,3,4}$}\email{qinw@njupt.edu.cn} \address{ $^1$ Institute of quantum information and technology, Nanjing University of Posts and Telecommunications, Nanjing 210003, China \\
$^2$ Broadband Wireless Communication and Sensor Network Technology, Key Lab of Ministry of Education, NUPT, Nanjing 210003, China \\
$^3$ Telecommunication and Networks, National Engineering Research Center, NUPT, Nanjing 210003, China\\
$^4$ Key Laboratory of Quantum Information, CAS, University of Science and Technology of China, Hefei, Anhui 230026, China\\}
\date{\today}
\begin{abstract} Digital signature is a key technique in information security, especially for identity authentications. Compared with classical correspondence, quantum digital signatures (QDSs) provide a considerably higher level of security, i.e., information-theoretic security. At present, its performance is limited by key generation protocols (e.g., BB84 or measurement-device-independent protocols), which are fundamentally limited in terms of channel capacity. Fortunately, the recently proposed twin-field quantum key distribution can overcome this limit. This paper presents a twin-field QDS protocol and details a corresponding security analysis. In its distribution stage, a specific key generation protocol, the sending-or-not-sending twin-field protocol, has been adopted and full parameter optimization method has been implemented. Numerical simulation results show that the new protocol exhibits outstanding security and practicality compared with all other existing protocols. Therefore, the new protocol paves the way toward real-world applications of QDSs. \end{abstract}
\pacs{03.67.Dd, 03.67.Hk, 42.65.Lm}
\maketitle
\section{Introduction} Digital signatures (DSs) possess wide applications in validating the authenticity and integrity of digital documents such as financial transactions and electronic contracts. Present digital signatures, hereafter called classical digital signatures, possess security levels based on computational complexity. For example, the Rivest-Shamir-Adleman protocol \cite{RSA} relies on solving large number factorization problems, and the elliptic-curve-based protocol \cite{ECDSA1,ECDSA2} depends on discrete logarithms. Unfortunately, these protocols would all be cracked through the advancement of mathematical algorithms or the emergence of quantum computers.
In contrast, the security of quantum digital signatures (QDSs) is based on quantum mechanics laws. QDSs have proven to provide information-theoretic security and thus attracted a lot of attention from the scientific world. Since the first QDS protocol was proposed in 2001 \cite{Gottesman} and the first experimental demonstration was accomplished in 2012 \cite{Clarke}, many obstacles to practical applications have been removed (e.g., the demanding of quantum memories; releasing secure quantum channels \cite{Dunjko,Collins2014,Amiri}), such that the quantum key distribution protocols \cite{BB84} can be implemented in the key distribution stage \cite{Wallden}). Furthermore, measurement-device-independent (MDI) protocols \cite{MDIQDS,Roberts} were proposed to solve side-channel attacks on measurements. The passive protocol \cite{Qin1} can avoid information leakage that occurs during the intensity modulating process. To date, many theoretical and experimental studies have been examined this subject matter \cite{Donaldson,Collins2016,Croal,Yin2017,AnXB,Thornton}.
With present QDS protocols, a good balance between security and practical performance is still difficult to achieve. For example, one can obtain a higher signature rate with a lower level of security when using BB84-type QDS protocols. In contrast, MDI-type QDS protocols feature a higher level of security but worse signature rates. Most importantly, both protocol types cannot exceed the fundamental limit of channel capacities without quantum repeaters, as in QKD \cite{TGW,PLOB}. To overcome this fundamental limit in QKD, Lucamarini \emph{et al.} proposed the so-called twin-field quantum key distribution (TF-QKD) protocol \cite{TFQKD} and obtained excellent security and practical performance. Inspired by Lucamarini \emph{et al.}, we for the first time present a twin-field quantum digital signature (TF-QDS) protocol. We first implement a sending-or-not-sending (SNS) protocol \cite{SNSTF1,SNSTF2} into the key distribution stage, and analyze its security against general attacks within QDSs (e.g., forging and repudiation attacks) by taking finite-size-effects into account. Furthermore, we perform corresponding numerical simulations and full parameter optimizations. When compared to other protocols such as BB84-QDS \cite{Amiri} and MDI-QDS \cite{MDIQDS}, our method exhibits outstanding results in terms of signature rates and transmission distances.
\section{Twin-field QDS}\label{sec2} \subsection{Protocol procedure}\label{protocol} A schematic diagram of our protocol is shown in Fig. \ref{fig1}, which consists of two stages, distribution stage and messaging stage. During the distribution stage, Alice-Bob and Alice-Charlie independently implement a twin-field key generation protocol (TF-KGP) to generate correlated bit strings, where Alice-Bob and Alice-Charlie send twin-field states to the untrusted party (Eve), and Eve performs a projection measurement. The messaging stage involves sending and signing classical messages, where Alice is the sender and Bob and Charlie are the two recipients. During the distribution stage, we adopt the SNS protocol \cite{SNSTF1,SNSTF2}.
\begin{figure*}
\caption{Schematic of the TF-QDS protocol. The pairs Alice-Bob and Alice-Charlie perform TF-KGP separately through Eve to generate keys, while Bob and Charlie share a secret channel to Alice to exchange partial keys. In the TF-KGP, Alice-Bob and Alice-Charlie prepare signal and decoy states using a phase modulator (PM) and intensity modulator (IM), after which they send quantum signals to an untrusted party (Eve) to complete the measurement. Finally, Alice's signature is sent to Bob for authentication, and forwarded to Charlie for further verification.}
\label{fig1}
\end{figure*}
\emph{\textbf{Distribution stage:}} (1) Alice-Bob and Alice-Charlie individually generate $N$ photon pulses and code them using a phase modulator (PM) and intensity modulator (IM). During this process, each pulse is randomly chosen as the $X$ (decoy) or $Z$ (signal) window. In the $X$ window, each side randomly prepares and sends out a phase-randomized coherent state with intensity $x$, $x\in\{0,w,v\}$. In the Z window, a phase-randomized coherent state with intensity $u$ is sent with a probability $p_s$, and nothing is sent with probability $1-p_s$.
(2) Eve carries out the projection measurement on the received pulse pairs with a beam-splitter (BS) and two detectors (denoted as $D_0$, $D_1$), and publicly announces the detection results. If one of two detectors clicks, it is recorded as a successful event.
(3) Alice-Bob (Alice-Charlie) publicly announce their windows used for each pulse pair. The only successful measurement results kept are ones in which they use the same windows. Furthermore, if they both use $X$ windows, the phase and decoy states of each pulse should also be disclosed.
(4) Alice-Bob (Alice-Charlie) use the data on $Z$ windows to extract sifted keys and the data on $X$ windows to estimate parameters. In addition, they randomly sacrifice a small number of bits on the $Z$ windows for an error rate test, leaving the remainder as a signature key pool.
(5) For a future possible message $m$ ($m=0$ or $1$), Alice-Bob (Alice-Charlie) choose a length-$L$ block from the key pool to form the signature sequence $A_m^B$ and $B_m^A$ ($A_m^C$ and $C_m^A$), where $A_m^B$ and $A_m^C$ are held by Alice, and $B_m^A$ ($C_m^A$) are held by Bob (Charlie).
(6) Bob and Charlie randomly choose half of their own key bits to exchange through the Bob-Charlie secret channel. The kept half is denoted as $ B_{m,keep}^A$ ($C_{m,keep}^A$ ), and the other half as $B_{m,forward}^A$ ($C_{m,forward}^A$). Bob's and Charlie's symmetrized keys were labeled as $S_m^B=(B_{m,keep}^A,C_{m,forward}^A)$ and $S_m^C=(C_{m,keep}^A, B_{m,forward}^A)$, respectively.
\emph{\textbf{Messaging stage:}} (7) Alice sends the signature, $(m,Sig_m)$, to a recipient (such as Bob), where $Sig_m=( A_m^B, A_m^C)$.
(8) Bob compares his $S_m^B$ with $(m,Sig_m)$ and records the number of mismatches. If the mismatches are fewer than $s_a L/2$ in both key halves, Bob accepts the message and goes to the next step; otherwise, he rejects the message and aborts this round. Here, $s_a$ is the authentication threshold associated with the security level of the QDS protocol.
(9) Bob forwards $(m,Sig_m)$ to Charlie.
(10) Charlie also checks the forwarded message in the same way, but with another threshold $s_v$ ($s_v > s_a$). Charlie accepts the forwarded message if the number of mismatches in both key halves is below $s_v L/2$.
TF-KGP includes steps $(1)-(4)$ in the distribution stage. It is essentially the quantum portion of the SNS TF-QKD scheme but without error correction and privacy amplification. Detailed definitions of TF-KGP are presented in Appendix \ref{App1}, including sifted key size, error test keys, and the key pool as $n_Z$, $n_{test}$ and $n_{pool}$, respectively, and $n_Z=n_{test}+n_{pool}$. In step (6) of the distribution stage, we assume the key exchange (also called key symmetrization) between Bob and Charlie is through the Bob-Charlie secret channel, which can be realized with a TF-QKD process performed by Bob and Charlie.
\subsection{Security analysis}\label{Security} In a QDS, although all components on $Z$ windows are used to generate keys for signature, security still depends on the single-photon components. In TF-QDS, the min-entropy resulting from single-photon components in the half of keys kept by Bob or Charlie at the presence of Eve is \begin{align}\label{Hmin}
H_{\min }^\epsilon (\left. U^A_{m,keep} \right|E) & \geqslant \underline{n}_{L,1}[1 - H_2(\overline{e}_{L,1})], \end{align} where $U\in \{B,C\}$ denotes user Bob or Charlie, and $E$ refers to the system of Eve; $\underline{n}_{L,1}$ and $\overline{e}_{L,1}$ represent the lower bound of single-photon counts and the upper bound of single-photon error rate in $U^A_{m,keep}$, respectively; $H_2(x)=-x\log_2(x)-(1-x)\log_2(1-x)$ is the binary Shannon entropy function. Eq. (\ref{Hmin}) uses a probability of $1-\epsilon$, where $\epsilon$ stands for the failure probability of the estimated parameters. In Appendix \ref{App2}, the derivations of these quantities are explained. With Eq. (\ref{Hmin}), the minimum rate $P_e$ at which Eve can introduce errors in $U^A_{m,keep}$ (length $L/2$) can be evaluated as \begin{align}\label{Pe} H_2(P_e) = \frac{{2\underline{n}_{L,1}}}{L}[1 - H_2(\overline{e}_{L,1})] . \end{align}
When doing security analysis of a TF-QDS protocol, robustness, forging, and repudiation probabilities should be evaluated \cite{Amiri}. Robustness indicates the probability of the QDS aborting when Alice, Bob, and Charlie are all honest, which is caused by an error test failure. Through the error rate of test keys $E_{test}$, we can estimate the error rate in $U^A_{m,keep}$ with the Serfling inequality \cite{Serfling} using \begin{align}\label{PET} E_{keep}^U \leqslant E_{test}^U+\frac{2}{L}\sqrt {\frac{{\left( {\frac{L}{2} + 1} \right)\left( {\frac{L}{2} + {n_{test}}} \right)\ln \left( {\frac{1}{{{\epsilon_{PE}}}}} \right)}}{{2{n_{test}}}}}, \end{align} except with a failure probability $\epsilon_{PE}$ and $\overline{E}_{keep}=\text{max}\{E_{keep}^B,E_{keep}^C\}$. Considering there are failure possibilities for both processes (Alice-Bob and Alice-Charlie), robustness probability can be expressed as \begin{align}\label{PRobust} {\rm{P(Robust)}} \leqslant 2\epsilon_{PE}. \end{align}
The repudiation probability characterizes Alice's signature accepted by Bob but rejected by Charlie. To repudiate, Alice must make the mismatch rate between both elements of $S_m^B$ and the signature $(m,Sig_m)$ lower than $s_a$. In addition, Alice needs the mismatch rate between either element of $S_m^C$ and the signature $(m,Sig_m)$ to be higher than $s_v$ after the key exchange. The best strategy for Alice is to control the error rate of Bob and Charlie as $E_{keep}^B=E_{keep}^C=\frac{1}{2}(s_a+s_v)$ \cite{Amiri}, in which case the repudiation probability is bounded by \begin{align}\label{PRep} {\rm{P}}({\rm{Repudiation}}) \leqslant 2{e^{ - \frac{1}{4}{{\left( {{s_v} - {s_a}} \right)}^2}L}}, \end{align} where ${s_a} = {\overline{E}_{keep}} + {{({P_e} - {\overline{E}_{keep}})} \mathord{\left/{\vphantom {{({P_e} - {\overline{E}_{keep}})} 3}} \right.\kern-\nulldelimiterspace} 3}$, and ${s_v} = { \overline{E}_{keep}} + 2{{({P_e} - {\overline{E}_{keep}})} \mathord{\left/
{\vphantom {{({P_e} - {\overline{E}_{keep}})} 3}} \right.\kern-\nulldelimiterspace} 3}$
The forging indicates that the signature is not signed by Alice but would be accepted by Bob and Charlie. For simplicity, we assume Bob is a forger. In order to forge, Bob must keep the mismatch rate between his declaration $(m,Sigm)$ and Charlie's keys ($S_m^C=(C_{m,keep}^A,B_{m,forward}^A)$) being lower than a given value $s_v$. Considering half of Charlie's string $B_{m,forward}^A$ is forwarded by Bob, Bob needs only to guess the left half $C_{m,keep}^A$. The forging probability includes all the process guessing $C^A_{m,keep}$, which is written as \begin{align}\label{PForge} {\rm{P(Forge)}} \leqslant g + \epsilon_{F} + \epsilon_{PE} + \epsilon_{\underline{n}_{L,1}} + \epsilon_{\overline{e}_{L,1}}, \end{align} where $g$ and $\epsilon_F$ are associated with the probability that Bob finds a signature with an error rate smaller than $s_v$, defined by \begin{align}\label{epsilon} {\epsilon_F}: = \frac{1}{g}\left( {{2^{ - \frac{L}{2}\left\{ {\frac{{2\underline{n}_{L,1}}}{L}[1 - H_2(\overline{e}_{L,1})] - {H_2}({s_v})} \right\}}} + \epsilon} \right). \end{align} $\epsilon_{PE}$, $\epsilon_{\underline{n}_{L,1}}$ and $\epsilon_{\overline{e}_{L,1}}$ are the error probabilities related to the estimation of $\overline{E}_{keep}$, $\underline{n}_{L,1}$, and $\overline{e}_{L,1}$, respectively. To define $\varepsilon$ as the security level of the system, according to \cite{Collins2017} it requires \begin{align}\label{varepsilon}
\text{max}\{ {\rm{P(Robust)}},{\rm{P}}({\rm{Repudiation}}), {\rm{P(Forge)}}\} \leqslant \varepsilon. \end{align}
We now present a simple model to evaluate the performance of a QDS protocol with the desired security level $\varepsilon$. We assume that for each run, the block of photon pulse pairs ($N$) to perform key distribution is known, and set $n_{pool}$ as the corresponding number of keys that can be used for signing. Subsequently we can calculate how many keys are needed to sign a half-bit signature ($L$), and how many bits should be signed ($n_{bits}$) with $n_{pool}$ keys. Finally, the signed bits and signature rate ($bit/pulse$) can be written as \begin{align}\label{BitsRate} n_{bits} = \frac{{{n_{pool}}}}{{2L}}, \\ R = \frac{{{n_{pool}}}}{{2L}}\cdot\frac{1}{N}, \end{align} respectively.
\section{Numerical simulations} In this section, we describe numerical simulations for our proposed TF-QDS protocol, with results shown in Figs. \ref{fig2}-\ref{fig4}. In our simulations, we consider statistical fluctuations using Hoeffdings inequality \cite{Hoeffding} as in Ref. \cite{Amiri}, and the basic system parameters are listed in Table \ref{tab1}. In addition, we set the number of phase slices to $M=16$ \cite{TFQKD} during the TF-KGP processes, where the phase slice is the coding phase difference interval post-selected by Alice-Bob or Alice-Charlie. In addition, we perform full parameter optimization on our TF-QDS, including the value of each light intensity ($w, v, u$), probability of choosing signal window ($p_Z$), probability of sending out the phase-randomized coherent state $u$ ($p_s$), and probabilities of choosing different decoy intensities ($p_{w}$ and $p_{v}$). \begin{table}[htbp]
\caption{The basic system parameters used in our numerical simulations. $\alpha$: the loss coefficient of fiber at telecommunication wavelength; $\eta_d$ and $P_{dc}$: detection efficiency and the dark count rate of detectors, respectively; $e_d$: optical misalignment error; $r_{ET}$: the ratio of keys used for the error test; $\epsilon_{PE}$ and $\epsilon_{SF}$: failure probability of the error test and statistical fluctuation, respectively; $g$: Bob's probability of making $s_vL/2$ errors.} \renewcommand{1.3}{1.3} \begin{tabularx}{\linewidth}{XXXXXXXX} \hline \hline $\alpha$ & $\eta_d$ & $P_{dc}$ & $e_d$ & $r_{ET}$ & $\epsilon_{PE}$ & $\epsilon_{SF}$ & $g$ \\ \hline $0.2dB/km$ & 50\% & $10^{-7}$ & 0.03 & 5.5\% & $10^{-12}$ & $10^{-12}$ & $10^{-12}$ \\ \hline\hline
\end{tabularx}
\label{tab1} \end{table}
In Fig. \ref{fig2}, we illustrate the variation of $n_{pool}$, $L$, and $n_{bits}$ with varying transmission distances, given a data size of $N=10^{13}$ and a security level of $\varepsilon=10^{-5}$, where $n_{bits}$ is shown with the left axis, and $n_{pool}$ and $L$ are indicated by the right axis. The value of $L$ increases rapidly and the value of $n_{bits}$ drops quickly as transmission distance increases, especially after 300 km. These observations can be attributed to the finite-size effect, which is more sensitive at longer transmission distances. \begin{figure}
\caption{The size of key pool ($n_{pool}$), the length for signing message $m$ ($L$), and the number of signed bits ($n_{bits}$) versus the total transmission distance. $n_{pool}$ and $L$ correspond to the right axis while $n_{bits}$ is indicated on the left axis. The data size is $N=10^{13}$ and the security level is $\varepsilon=10^{-5}$ at all transmission distances.}
\label{fig2}
\end{figure}
The signature rate of TF-QDS is plotted in Fig. \ref{fig3}, and compared to two typical QDS protocols, BB84-QDS \cite{Amiri} and MDI-QDS \cite{MDIQDS}. We set the security level as $\varepsilon=10^{-5}$ and the data size as $N=10^{13}$ or $N=10^{15}$ \cite {MDI404}. For fair comparisons, we also perform full parameter optimization on BB84-QDS and MDI-QDS. We can see from Fig. \ref{fig3} that among the three protocols, our TF-QDS protocol exhibits the best performance at longer transmission distances. For example, our TF-QDS can sign signatures at 300 km while the other two protocols stop signing at 230 km and 250 km, respectively. At shorter transmission distances, BB84-QDS exhibits the highest signature rate. However, it possesses the lowest security level among the three protocols. Therefore, when taking security level into account, our protocol exhibits the best performance in terms of both transmission distance and signature rate.
\begin{figure}
\caption{Signature rates of BB84-QDS \cite{Amiri}, MDI-QDS \cite{MDIQDS}, and TF-QDS with a security level of $\varepsilon=10^{-5}$. The dashed lines represent results at data size $N=10^{13}$, and the solid lines at data size $N=10^{15}$. }
\label{fig3}
\end{figure}
\begin{figure}
\caption{Signature rates of TF-QDS versus optical misalignment errors with $\varepsilon=10^{-5}$ or $\varepsilon=10^{-10}$ at 50 km. Here, $N=10^{13}$. }
\label{fig4}
\end{figure} We also investigate the robustness of our TF-QDS protocol by plotting signature rate variations with changes in the misalignment of the optical system in Fig. \ref{fig4}. Here, the transmission distance is set at 50 km and the security level as $\varepsilon=10^{-5}$ or $\varepsilon=10^{-10}$. Fig. \ref{fig4} shows that the signature rate decreases with increasing misalignment error. The maximum tolerable misalignment error is 18\%, which is much larger than values in the BB84 and MDI protocols, and well within current experimental values \cite{Minder,WangS,LiuY,ZhongX}. Moreover, by setting a higher security level, a lower signature rate can be obtained. Therefore, a reasonable security level should be chosen in practical applications of the TF-QDS. In addition, due to the phase sensitivity of TF-KGP, we can use machine learning to achieve phase-modulation stabilization \cite{LiuJY}, enhancing the practical performance of the TF-QDS system.
\section{Conclusions} In this paper, we develop a TF-QDS protocol, which can possess the highest security level among all existing QDS protocols, but also exhibit outstanding performance in terms of both signature rates and secure transmission distances. For example, the TF-QDS protocol can achieve $>$ 100 km longer secure transmission distance than either BB84-QDS or MDI-QDS under the same experimental conditions, and exhibits a higher signature rate than MDI protocol by several orders of magnitude after 200 km. Therefore, our work represents another step towards practical implementation of QDS.
To be noted, this is the first TF-QDS protocol, by adopting Wang \emph{et al.}'s SNS scheme \cite{SNSTF1,SNSTF2}. In principle, other types of TF schemes \cite{CuiC,Curty,MCSTF,AsyTF} and security analysis methods \cite{Maeda,Lorenzo} could also be implemented in QDS, and might show even more interesting characteristics. This will be carried out in our future research work. As for the limitations of the present QDS work, similar to existing TF-QKD protocols, it might pose high challenges for wide applications in the field, e.g., it needs high speed and accurate multi-party synchronization, phase-locking and stabilization techniques. Anyway, with the rapid development of modern technology, all these challenges will be readily solved. Therefore, our work represents another step towards practical implementation of QDS.
\section*{ACKNOWLEDGMENTS} We also acknowledge financial support from the National Key Research and Development Program of China (Grants No. 2018YFA0306400, No. 2017YFA0304100); National Natural Science Foundation of China (NSFC) (Grants No. 11774180, No. 61590932, No. 61705110, No. 11847215); China Postdoctoral Science Foundation (Grant No. 2018M642281).
\section*{COMPETING INTERESTS} The authors declare that there are no competing interests.
\appendix \section{Some detailed notes on the TF-KGP}\label{App1} In this section, we provide detailed notes on the TF-KGP. We start by analyzing the TF-KGP procedure between Alice and Bob, which is the SNS TF-QKD presented in \cite{SNSTF1,SNSTF2} without error correction and privacy amplification.
The phase-randomized coherent state prepared by Alice and Bob, respectively, can be expressed as \begin{align}\label{Coherent}
\left| {\sqrt x_A {e^{i\theta_A }}} \right\rangle = \sum\nolimits_{n = 0}^\infty {\frac{{{e^{ - x_A/2}}{{(\sqrt x_A {e^{i\theta_A }})}^n}}}{{\sqrt n !}}\left| n \right\rangle } , \quad
\left| {\sqrt x_B {e^{i\theta_B }}} \right\rangle = \sum\nolimits_{n = 0}^\infty {\frac{{{e^{ - x_B/2}}{{(\sqrt x_B {e^{i\theta_B }})}^n}}}{{\sqrt n !}}\left| n \right\rangle } , \end{align} where $x_A$ ($x_B$) and $\theta_A$ ($\theta_B$) represent the intensity and phase of coherent state randomly chosen by Alice (Bob), respectively. Here, $x_A, x_B \in \{0, w, v, u\}$ and $\theta_A, \theta_B$ are random in $[0,2\pi)$. Alice (Bob) randomly chooses a vacuum state, decoy states ($w$, $v$), and $Z$ windows with probabilities $p_0, p_w, p_v, p_Z$, respectively, where $p_0+p_w+p_v+p_Z=1$. When a $Z$ window is chosen, Alice (Bob) sends a signal state $u$ with probability $p_s$, and sends nothing with $1-p_s$.
When receiving the pulses from Alice and Bob, Charlie performs measurements and announces the results. During the measurement process, if only one detector clicks, Charlie announces a successful event, recorded as a one-detector heralded event, and announces which detector ($D_0$ or $D_1$) clicks. When the measurement process is complete and results have been announced, Alice and Bob publicly disclose which window was used for each pulse pair. Only the one-detector heralded events for which they both use $X$ or $Z$ windows are kept. When they both use $X$ windows, the phase and decoy state intensity should also be disclosed. However, when they both use $Z$ windows, the phase and SNS operation should be never disclosed. Furthermore, we need to post-select the effective events on $X$ windows; it is deemed an effective event if it is a one-detector heralded event where Alice and Bob both use $X$ windows, the two coherent states of Alice and Bob have the same intensity, and their phases satisfy the following post-selection criterion \begin{align}\label{PhaseSlice}
\left|\theta_{A}-\theta_{B}-\psi_{\mathrm{AB}}-k\pi\right| \leq \frac{\Delta}{2}. \end{align} In Eq. (\ref{PhaseSlice}), $\psi_{\mathrm{AB}}$ is the difference of global phases between Alice-Eve's link and Bob-Eve's link, which results in the optical misalignment error ($e_d$); $k=0,1$ corresponds to in-phase or anti-phase of ${\theta _A}$ and ${\theta _B}$; $\Delta=\frac{{2\pi }}{M}$ represents the size of each slice, and $M$ refers to the total number of phase slices pre-chosen by Alice and Bob. The effective events on $X$ windows are the results of the single-photon interference and a subset of one-detector heralded events on $X$ windows.
For the one-detector heralded events on $Z$ windows, Alice (Bob) denotes it as bit 0 if she (he) sends a vacuum (phased-randomized weak coherent) state and as bit 1 if she (he) sends a phased-randomized weak coherent (vacuum) state. For the effective events on $X$ windows, a right click is the $D_0$ ($D_1$) detector clicking when $k=0$ ($k=1$), and a wrong click is the $D_1$ ($D_0$) detector clicking when $k=0$ ($k=1$). The data on $Z$ windows are defined as the key bits distilled by the one-detector heralded events on $Z$ windows, while the data on $X$ windows are defined as the one-detector heralded and effective events on $X$ windows. The data on $Z$ windows are used for the error test and signature, and finally Alice and Bob form an $n_Z$-length key string $Z_s$ and $Z^\prime_s$, respectively. The data on $X$ windows are used to estimate single-photon contributions, i.e. the counts and error rates of the single-photon components ($\underline{n}_{L,1}, \overline{e}_{L,1}$) on $Z$ windows.
\section{Finite-size estimations of parameters}\label{App2} In this section, we estimate $\underline{n}_{L,1}$ and $\overline{e}_{L,1}$ in finite size. The procedure can be decomposed into three steps.
Firstly, we estimate the lower bound of single-photon counts and upper bound of single-photon error counts on $X$ windows ($\underline{n}_{X,1}$ and $\overline{m}_{X,1}$, respectively) with the observed values taking statistical fluctuations into account. From the data on $X$ windows, we know the counts of one-detector heralded events with various intensity combinations ($n_{ab}$, $a,b \in \{0,w,v\}$), and the counts of error clicks in effective events $m_{aa}$. With these observed values, we obtain \begin{align}\label{n1X} {n_{X,1}} \geqslant \underline{n}_{X,1} = \frac{{{\tau _{X,1}}}}{{2wv(v - w)}}\left[ {\frac{{{v^2}{e^w}({n^-_{0w}} + {n^-_{w0}})}}{{{P_{0w}}}} - \frac{{{w^2}{e^v}({n^+_{0v}} + {n^+_{v0}})}}{{{P_{0v}}}} - \frac{{2({v^2} - {w^2}){n^+_{00}}}}{{{P_{00}}}}} \right], \end{align} \begin{align}\label{m1X} {m_{X,1}} \leqslant {\overline{m}_{X,1}} = \frac{{{\tau _{X,1}}}}{{v - w}}\left[ {\frac{{{e^v}{m^+_{vv}}}}{{P_{vv}^\Delta }} - \frac{{{e^v}{m^-_{ww}}}}{{P_{ww}^\Delta }}} \right], \end{align} and the corresponding single-photon error rate on $X$ windows is $\overline{e}_{X,1}=\overline{m}_{X,1}/\underline{n}_{X,1}$. In Eqs. (\ref{n1X}) and (\ref{m1X}), $\tau _{X,1}$ is the probability of single-photon components with all intensity combinations on $X$ windows, which is ${\tau _{X,1}} = \sum\nolimits_{a,b} {{P_{ab}}(a + b){e^{ - a - b}}}$. $P_{ab}$ is the probability of intensity combination $ab$, and $P_{aa}^\Delta$ is the probability of effective events occurring with intensity combination $aa$, given by $P_{ab}=p_ap_b$ and $P_{aa}^\Delta = 2p_a^2\frac{\Delta }{{2\pi }}$. The $x^-$ and $x^+$ in Eqs. (\ref{n1X}) and (\ref{m1X}) are the observed values when considering the statistical fluctuations by the Hoeffding inequalities \cite{Hoeffding} \begin{align}\label{HoeffdingEq} {\tilde x} \geqslant x^- := x - \delta (x,\epsilon_{SF}) ,\quad {\tilde x} \leqslant x^+ := x + \delta (x,\epsilon_{SF}), \end{align} with failure probability $\epsilon_{SF}$, where \begin{align}\label{HoeffdingDelta} \delta (x,\epsilon_{SF}) = \sqrt {\frac{{x\ln (1/{\epsilon_{SF}})}}{2}}. \end{align}
Secondly, since the single-photon signals on $X$ and $Z$ windows are independent, we use $\underline{n}_{X,1}$ and $\overline{m}_{X,1}$ to estimate the corresponding quantities on $Z$ windows ($n_{Z,1}$ and $m_{Z,1}$) using the Serfling inequality \cite{Serfling}. The population for single-photon preparations on $Z$ windows is lower bounded by \begin{align}\label{Nz1} {\underline{N}_{Z,1}} = 2{p_s}(1 - {p_s})u{e^{ - u}}{N_Z} - \delta ({N_Z},{\epsilon_{SF}}), \end{align} with confidence $1-\epsilon_{SF}$, where $N_Z=p_Z^2N$ represents the runs of Alice and Bob both choosing the $Z$ windows, and $\delta(x,y)$ is the fluctuation in Hoeffding's inequality \cite{Hoeffding}. Similarly, the population for single-photon preparations on $X$ windows is upper bounded by \begin{align}\label{Nx1} {\overline{N}_{X,1}} = \sum\nolimits_{a,b} {\left[ {(a + b){e^{ - a - b}}{N_{ab}} + \delta ({N_{ab}},{\epsilon_{SF}})} \right]}, \end{align} with confidence $1-9\epsilon_{SF}$, where $N_{ab}=P_{ab}N$ represents Alice and Bob choosing an intensity combination $ab$ on the $X$ windows. Subsequently, we can interpret the single-photon contributions on $X$ or $Z$ windows ($n_{X,1}$, $m_{X,1}$, $n_{Z,1}$,$m_{Z,1}$) in the whole population of the single-photon preparations as an operation of sampling without replacement. The Serfling inequality tells us that \begin{align}\label{SerflingEq1} {n_{Z,1}} \geqslant {\underline{n}_{Z,1}} = {\underline{n}_{X,1}}\frac{{{\underline{N}_{Z,1}}}}{{{\overline{N}_{X,1}}}} - \Upsilon ({\underline{N}_{Z,1}},{\overline{N}_{X,1}},{\epsilon_{SF}}),\\ {m_{Z,1}} \leqslant {\overline{m}_{Z,1}} = {\overline{m}_{X,1}}\frac{{{\underline{n}_{Z,1}}}}{{{\underline{n}_{X,1}}}} + \Upsilon ({\underline{n}_{Z,1}},{\underline{n}_{X,1}},{\epsilon_{SF}}), \end{align} where confidence is $1-\epsilon_{SF}$, and $\Upsilon(x,y,z)= \sqrt {(x + 1)(x + y)\ln ({z^{ - 1}})/(2y)}$. The corresponding single-photon error rate on $Z$ windows is \begin{align}\label{eZ1} \overline{e}_{Z,1}=\frac{{{\overline{m}_{Z,1}}}}{{{\underline{n}_{Z,1}}}}. \end{align}
Thirdly, we can use $\underline{n}_{Z,1}$ and $\overline{e}_{Z,1}$ to estimate $n_{L,1}$ and $e_{L,1}$ in $U^A_{m,keep}$ through the Serfling inequality with \begin{align}\label{nL1eL1} {n_{L,1}} &\geqslant {\underline{n}_{L,1}} = {\underline{n}_{Z,1}}\frac{{{L}}}{{{2n_{Z}}}} - \Lambda ({n_{Z}},{\frac{L}{2}},{\epsilon_{SF}}),\\ {e_{L,1}} &\leqslant {\overline{e}_{L,1}} = {\overline{e}_{Z,1}} + {\frac{1}{\underline{n}_{L,1}}}\Lambda ({\underline{n}_{Z,1}},{\underline{n}_{L,1}},{\epsilon_{SF}}), \end{align} where $\Lambda(x,y,z)=\sqrt {(x - y + 1)y\ln ({z^{ - 1}})/(2x)}$. Finally, we obtain $\underline{n}_{L,1}$ and $\overline{e}_{L,1}$.
In addition, we simulate the experimental observed values with the linear model presented in \cite{SNSTF2} and assume symmetric case in the TF-KGP. If the total transmittance of the experiment setups is $\eta=\eta_d 10^{-\frac{\alpha }{{20}}}$, then we have \begin{align*} n_{00}=&2 P_{dc}\left(1-P_{dc}\right) N_{00},\\ n_{0w}=&n_{w0}=2\left[\left(1-P_{dc}\right) e^{\eta w / 2}-\left(1-P_{dc}\right)^{2} e^{-\eta w}\right] N_{0w},\\ n_{0v}=&n_{v0}=2\left[\left(1-P_{dc}\right) e^{\eta w / 2}-\left(1-P_{dc}\right)^{2} e^{-\eta w}\right] N_{0v},\\ m_{w w}=&e_{d}\left[\left(1-P_{d c}\right)\frac{1}{\Delta} \int_{-\frac{\Delta}{2}}^{\frac{\Delta}{2}} e^{-2 \eta w \sin ^{2} \frac{\theta_{AB}}{2}} d \theta_{AB}-\left(1-P_{d c}\right)^{2} e^{-2 \eta w}\right]P_{ww}^\Delta N \\ &+\left(1-e_{d}\right)\left[\left(1-P_{d c}\right)\frac{1}{\Delta} \int_{-\frac{\Delta}{2}}^{\frac{\Delta}{2}} e^{-2 \eta w \cos ^{2} \frac{\theta_{AB}}{2}} d \theta_{A B}-\left(1-P_{d c}\right)^{2} e^{-2 \eta w}\right] P_{ww}^\Delta N, \\ m_{vv}=&e_{d}\left[\left(1-P_{d c}\right)\frac{1}{\Delta} \int_{-\frac{\Delta}{2}}^{\frac{\Delta}{2}} e^{-2 \eta v \sin ^{2} \frac{\theta_{A B}}{2}} d \theta_{AB}-\left(1-P_{d c}\right)^{2} e^{-2 \eta v}\right]P_{vv}^\Delta N \\ &+\left(1-e_{d}\right)\left[\left(1-P_{d c}\right)\frac{1}{\Delta} \int_{-\frac{\Delta}{2}}^{\frac{\Delta}{2}} e^{-2 \eta v \cos ^{2} \frac{\theta_{AB}}{2}} d \theta_{AB}-\left(1-P_{d c}\right)^{2} e^{-2 \eta v}\right] P_{vv}^\Delta N, \\ \end{align*} and \begin{align*} {n_Z} = &2{(1 - {p_s})^2}{P_{dc}}(1 - {P_{dc}}){N_Z} + 4{p_s}(1 - {p_s})\left[ {(1 - {P_{dc}}){e^{ - \eta u/2}} - {{(1 - {P_{dc}})}^2}{e^{ - \eta u}}} \right]{N_Z} \\ &+ 2p_s^2\left[ {(1 - {P_{dc}}){e^{ - \eta u}}\frac{1}{{2\pi }}\int_0^{2\pi } {{e^{\eta u\cos {\theta _{AB}}}}d{\theta_{AB}}} - {{(1 - {P_{dc}})}^2}{e^{ - 2\eta u}}} \right]{N_Z}, \end{align*} where $\theta_{AB} = \theta_{A} - \theta_{B}$.
\end{document} |
\begin{document}
\begin{frontmatter}
\title{
Batch Self Organizing maps for distributional data with automatic weighting of variables and components}
\author{Francisco de A.T. De Carvalho\fnref{myfootnote2}}
\address{Centro de Informatica, Universidade Federal de Pernambuco,
Av. Jornalista Anibal Fernandes s/n - Cidade Universitaria,
CEP 50740-560, Recife-PE, Brazil}
\fntext[myfootnote2]{Corresponding Author: fatc@cin.ufpe.br}
\author{Antonio Irpino, Rosanna Verde and Antonio Balzanella \fnref{myfootnote}}
\address{Universit\'a degli Studi della Campania ``Luigi Vanvitelli'', Dept. of Mathematics and Physics, Viale Lincoln, 5, 81100 Caserta, Italy}
\fntext[myfootnote]{\{antonio.irpino\},\{rosanna.verde\},\{antonio.balzanella\}@unicampania.it}
\begin{abstract} This paper deals with Self Organizing Maps (SOM) for data described by distributional-valued variables. This kind of variables takes as values empirical distributions on the real line or estimates of probability distributions. We propose a Batch SOM strategy (DBSOM) that optimizes an objective function, using a $L_2$ Wasserstein distance that is a suitable dissimilarity measure to compare distributional data, already proposed in different distributional data analysis methods. Moreover, aiming to take into consideration the different contribution of the variables, we propose an adaptive version of the DBSOM algorithm. This adaptive version has an additional step that learns automatically a relevance weight for each distributional-valued variable. Besides, since the $L_2$ Wasserstein distance allows a decomposition into two components: one related to the means and one related to the size and shape of the distributions, also relevance weights are automatically learned for each of the measurement components to emphasize the importance of the different estimated parameters of the distributions. Experiments with real datasets of distributional data corroborate the proposed DBSOM algorithms.
\end{abstract}
\begin{keyword}
Distribution-valued data\sep Wasserstein distance \sep Self-Organizing Maps \sep Relevance weights \sep Adaptive distances
\end{keyword}
\end{frontmatter}
\section{Introduction}
Current big-data age requires the representation of voluminous data by summaries with loss of information as small as possible. Usually, this is achieved by describing data subgroups according to descriptive statistics of their distribution (e.g.: the mean, the standard deviation, etc.) Alternatively, when a dataset is observed with respect to a numerical variable, it can be described either by the estimate of the theoretical distribution that best fits the data or by an empirical distribution.
In these cases, each set of observations is described by a distribution-valued data, and we call \textit{distributional-valued variable} a more general type of variable whose values are one-dimensional empirical or estimated probability or frequency distributions on numeric support.
\emph{Symbolic Data Analysis} \cite{BoDid00} introduced distributional-valued variables as particular set-valued variables, namely, modal variables having numeric support. A particular type of distributional-valued variable is a histogram-valued variable, whose values are histograms.
Such kind of data is arising in many practical situations.
Official statistical institutes collect data about territorial units or administrations and often they carry out them as empirical distributions or histograms.
Similarly, data are often available as aggregates in order to preserve individuals' privacy. For instance, bank transactions or measurements regarding patients of a hospital are often provided as histograms or empirical distributions rather than as individual records.
So far, several methods have been proposed for estimating distributions from a set of classical data, while few methods have been developed for the data analysis of objects described by distribution-valued variables.
Among exploratory methods, Kohonen Self-Organizing Map (SOM) presents both visualization and clustering properties \cite{Kohonen01,Kohonen13}. SOM is based on a map of nodes (neurons) organized on a regular low dimensional grid where the neurons present a priori neighborhood relations. Each neuron is described by a prototype vector (a model) and it is associated with a set of the input data.
In this sense, SOM carries out clustering while preserving the topological order of the prototype vectors on the map: the more the neurons are adjacent on the map, the more they are described by similar prototypes, whereas different prototypes are associated with neurons that are distant on the map. Besides, during the training step, each object must be assigned to a neuron. This can be done in two ways. The classical SOM algorithm assigns an input data to the closest BMU (Best Matching Unit), namely the neuron that is described by the closest prototype. Following the approach considered by Kohonen \cite{Kohonen01} too, it is possible to consider the assignment as a part of an optimization problem. In this case, an objective function is associated with a SOM that is minimized according to the prototypes definition (a representation step) and the data assignment. Thus, the assignment is not merely done accordingly to the closest BMU but according to the BMU that allows a minimization of the objective function. After that each object is assigned to the optimal BMU, the corresponding representative and a subset of representatives of its neighbors on the grid are modified aiming to fit better the data set.
An important property of the SOM is that it preserves at the best the original topological structure of the data: the objects that are similar in the original space have their corresponding representatives similar and located close in the map. Finally, the training of the SOM can be incremental or batch. Kohonen \cite{Kohonen13} states that for practical applications, the batch version of the SOM is the more suitable. However, when the data is presented sequentially, as in stream data, the training has to be incremental.
SOM can be considered as a distance-based clustering method, thus the definition of a distance between the data is essential, mainly in our case where data are one dimensional distributions.
The literature on clustering distributions includes several proposals. In \cite{IrVer06} it is proposed a hierarchical clustering method which uses a Wasserstein distance for comparing distributions estimated by means of histograms. Using the same distance function \cite{IRVERLEC06} propose a method based on the Dynamic Clustering Algorithm (DCA) \cite{DiSIM76}. The latter is a centroid based algorithm which generalizes the classic $k$-means. It optimizes an internal homogeneity criterion by performing, iteratively, a representation and an allocation step until the convergence to a stable value of the optimized criterion. Another centroid based method has been introduced in \cite{Terada10}. It is a $k$-means algorithm which uses empirical joint distributions. Finally, \cite{VracEtAL12} propose a Dynamic Clustering Algorithm based on copula. All the mentioned algorithms require an appropriate dissimilarity measure for comparing distributions. Among these, Wasserstein distances \cite{Rush01} have interesting features, as investigated in \cite{IrpVer2015}.
In classical clustering, usually, it is assumed that variables play the same role in the grouping process. Sometimes a standardization or a rescaling step is performed before running the clustering algorithm but this step does not assure that each variable participates to the clustering process according to its clustering ability. Indeed, there is a wide and unresolved debate about the variable transformation in clustering.
The use of adaptive distances \citep{diday77} in clustering is a valid approach for the identification of the importance of variables in the clustering process. In the framework of Symbolic Data Analysis, several Dynamic Clustering algorithms including adaptive distances providing relevance weights for the variables, have been introduced for interval-valued data \cite{DeCDeS10,DECAYVES09,DECLEC09}, for histogram-valued data \cite{Kim_Bill_2011,Kim_Bill_2013} and for modal symbolic data \cite{Hardy2004,Korenjak-Cerne1998,KorenjakCerne2002}. The use of adaptive distances, based on the $L_2$ Wasserstein metric, has been also proposed in the framework of Dynamic Clustering algorithm by \cite{IrpinoESWA}. Still, a fuzzy version of such algorithm is available in \cite{Irpino2017}.
Some extensions of the SOM have been proposed for interval-valued data \cite{Bock2002,Durso11,Cabanes13,Hajjar11a,Hajjar11b,Hajjar11c,Hajjar13}. Nonetheless, to the best of our knowledge, SOM algorithms for distributional data have not yet been proposed. Moreover, usually SOM algorithms assume that the variables have the same importance for the training of the map, i.e, they assume that the variables have the same relevance. However, in real applications some variables are irrelevant and some are less relevant than others \cite{diday77,friedman04,frigui04,huang05}.
In order to consider the role of the variables in the partitioning structure, different approaches can be adopted.
A first strategy is to set the weight of variables according to the apriori knowledge about the application domain and, then, to perform the SOM procedure to train the map
A suitable alternative is to add a step to the algorithm which computes, automatically, the weights for the variables. This approach has been used in Diday and Govaert \cite{diday77} which propose adaptive distances in which the automatically computed weights have a product to one constraint.
Recently, De Carvalho et al \cite{CarvalhoBS16} proposes batch SOM algorithms that learn a relevance weight for each interval-valued variable during the training phase of the SOM thanks to adaptive distances. In particular, it is proposed to associate a relevance weight to each variable by introducing a weighting step in the algorithm and by modifying the optimized criterion function.
\subsection*{Main contributions} The present paper extends the Batch Self Organized Map (BSOM) algorithm to histogram data. We refer to it as DBSOM, where D stands for Distributional data. Since the DBSOM cost function depends on a distance between data that are distributions, among the different distances for comparing distributions, we propose the use of the Wasserstein distance. It belongs to a family of distance measures defined by different authors in a variety of contexts of analysis and with different norms. In our context, we consider the squared Wasserstein distance, also known as Mallows distance, as defined in \cite{Rush01}, and here named $L_2$ Wasserstein distance. The main motivation of this choice can be found in \cite{VerIRP08did}, where a comparison with other metrics is proposed. It is related to the possibility of defining barycenters of sets of distributions as Fr\'echt means, improving the ease of interpretation of the obtained results. Moreover, it allows of defining the variance over any distribution variables \cite{IrpVer2015}.
Besides, for the $L_2$ Wasserstein distance it has been proved an important decomposition in components related to the location (means) and variability and shape over the compared distributions \cite{IrpinoR07}. Another contribution of the paper is the introduction of a system of weights for considering the different importance of the variables in the clustering process. Indeed, the classical BSOM assumes that each variable has the same role in the map learning, where standardization of the variables is usually performed before the algorithm starts (like in the Principal Component Analysis). However, the effect of teach variable can be relevant in the learning process. To take that into consideration, we introduce an \textit{Adaptive} version of DBSOM, denoted ADBSOM. We call it \textit{Adaptive} since it is based on the use of adaptive distances proposed by \cite{diday77} in clustering. As we will show, the optimization process of ADBSOM allows to compute automatically a system of weights for the variables. Since the $L_2$ Wasserstein distance can be decomposed in two components related to the locations (means), variability and shape, we also propose a system of weights for the two components of each distributional variable. This enriches the interpretation of the components of the variables that are relevant in the learning process and for the results. The procedure is performed through an additional step of the ADBSOM algorithm that automatically learns relevance weights for each variable and/or for each component of the distance. Some preliminary results were presented in \cite{IrpinoVC2012}. In addition to extending the methods mentioned above, we propose new variants of the algorithm which consider new constraints in the optimization process.
The paper is organized as follows. Sec. \ref{SEC_method} introduces details of our proposal focusing on the criterion function optimized by each algorithm; the computation of the relevance weights on data described by distributional-valued variables; the adaptive distances for distributional data. Sec. \ref{SEC_apply} provides an application of the proposed algorithms to real-world datasets. Sec. \ref{SEC_conl} concludes the paper with a discussion on the achieved results on the several datasets.
\section{Batch SOM algorithms for distributional-valued data} \label{SEC_method}
SOM is proposed as an efficient method to address the problem of clustering and visualization, especially for high-dimensional data having as input a topological structure. Since a grid of neurons is chosen for defining the topology of the map, the procedure of mapping a vector from data space onto the map consists in finding the neurons with the closest weighted distance to the data space vector. A correspondence between the input space and the mapping space is built such that, two close data in the input space, should activate the same neuron, or two neighboring neurons, of the SOM. A prototype describes each neuron. Neighboring neurons in the map providing the Best Matching Unit (BMU) of a data update their prototypes to better represent the data.
An extension of SOM to histogram data is suitable to analyze data that are already available in aggregate form, also generated as syntheses of a huge amount of original data. This paper proposes batch SOM algorithms for distributional-valued data that automatically provides a set of relevance weights for the different variables. We present two new batch SOM algorithms, namely DBSOM (Distributional Batch SOM) and ADBSOM (Adaptive Distributional Batch SOM). Both algorithms are based on the $L_2$ Wasserstein distance between distributional-valued data. Specifically, during the training of the map, ADBSOM computes relevance weights for each distributional-valued variable. Such relevance weights are assumed as parameters of the dissimilarity function that compares the prototypes and the data items. Therefore, the computed values of such weights allow selecting the importance of the distributional-valued variables to the training of the map. SOM provides both a visualization (given by the proximity between the neurons) and a partitioning of the input data (by an organization of the data in clusters).
\subsection{DBSOM criterion for distributional-valued data}
This section extends the classical objective function of batch SOM to the case of distributional-valued variables.
Let $E =\{e_1,\ldots,e_N\}$ be a set of $N$ objects described by $P$ distributional-valued variables $Y_j$ ($j=1, \ldots, P$). Each $i$-th object $e_i (1 \leq i \leq N)$ is represented by $P$ distributions (or distributional-valued data) $y_{ij} \, (1 \leq j \leq P)$. These are elements of an \textit{object vector} $\mathbf{y}_i = (y_{i1},\ldots,y_{iP})$. With each one-dimensional distributional data $y_{ij}$ is associated: a one-dimensional estimated density function $f_{ij}$, the corresponding cumulative distribution function (cdf) $F_{ij}$ and the quantile function (qf) $Q_{ij}=F_{ij}^{-1}$ (namely, the inverse of the cdf).
Therefore, the distributional-valued data set $\mathcal{D} = \{\mathbf{y}_1,\ldots,\mathbf{y}_N\}$ is collected in an \textit{object table}:
$$\mathbf{Y} = \big(y_{ij}\big)_{\substack{1 \leq i \leq N\\1 \leq j \leq P}}$$ \noindent where each element is a (one dimensional) distribution $y_{ij}$.
SOM is a low-dimensional (mainly, two-dimensional) regular grid, named {\em map}, that has $M$ nodes named {\em neurons}. A SOM algorithm induces a partition where to each cluster corresponds a unique neuron described by prototype vector. Thus, the neuron $m \, (1 \leq m \leq M)$ is associated with a cluster $C_m$ and a prototype \textit{object vector} $\mathbf{g}_m$.
The assignment function $f: \mathcal{D} \mapsto \{1,\ldots,M\}$ assigns an index $m = f(\textbf{y}_i) \in \{1,\ldots,M\}$ to each distributional-valued data $\mathbf{y}_i$ according to:
\begin{equation} \label{c1}
f(\mathbf{y}_i) = m = \operatornamewithlimits{arg\;min}_{1 \leq r \leq M} d^T(\mathbf{y}_i, \mathbf{g}_r) \ \end{equation}
\noindent where $d^T$ is a suitable dissimilarity function between the object vectors $\mathbf{y}_i$ and the prototype vectors $\mathbf{g}_r$.
The partition $\mathcal{P} = \{C_1,\ldots,C_M\}$ carried out by SOM is obtained according to an assignment function that provides the index of the cluster of the partition to which the object $\mathbf{y}_i$ is assigned: $C_m = \{e_i \in E: f(\textbf{y}_i) = m\}$.
Since the variables are distributional-valued, each prototype $\mathbf{g}_m \, (1 \leq m \leq M)$ is a vector of $P$ distributional data, i.e., $\mathbf{g}_m = (g_{m1}, \ldots, g_{mP})$, where each $g_{mj} \, (1 \leq j \leq P)$ is a distribution. Besides, with each one-dimensional distributional data $g_{mj}$ are associated: an estimate density function $f_{g_{mj}}$, the cdf $G_{mj}$ and the qf $Q_{g_{mj}}$\footnote{ We remark that $g_{mj}$ is a distributional data because the corresponding quantile function $Q_{g_{mj}}$ is a weighted average quantile function (for further details see \cite{Gilchrist} and \cite{IrpVer2015})}. Hereafter, $\mathbf{G}$ is the matrix of the descriptions of each $g_{mj}$ associated with the prototypes: $$\mathbf{G} =
\big(g_{mj}\big)_{\substack{1 \leq m \leq M\\1 \leq j \leq P}}$$ \noindent where each cell contains a (one dimensional) distribution $g_{mj}$.
With the purpose that the obtained SOM represent the data set $\mathcal{D}$ accurately, the prototype matrix $\mathbf{G}$ and the partition $\mathcal{P}$ are computed iteratively according to the minimization of an suitable objective function (also known as \textit{energy function} of the map), hereafter refered as $J_{DBSOM}$, defined as the sum of the dissimilarities between the prototypes (best matching units) and the data unities:
\begin{equation}\label{crit-1} J_{DBSOM}(\mathbf{G}, \mathcal{P}) = \sum_{i = 1}^N d^T(\mathbf{y}_i, \mathbf{g}_{f(\mathbf{y}_i)}) \end{equation}
Dissimilarities between each object and all the prototype vectors are needed to be computed. The best matching unit (BMU) is the winner neuron, i.e., the neuron indexed by $m = f(\mathbf{y}_i)$ with prototype vector $\mathbf{g}_{m}$ of minimum error.
The dissimilarity function, $d^T$, that is used to compare each object $\mathbf{y}_i$ with each prototype $\mathbf{g}_h$, is computed as follows:
\begin{equation}\label{cb-1} d^T(\mathbf{y}_i, \mathbf{g}_m) = \sum_{h = 1}^M \mathcal{K}^T(d(m,h)) \, d^2_W(\mathbf{y}_i,\mathbf{g}_h) \end{equation}
In equation (\ref{cb-1}), $d$ is the distance defined on the set of neurons. Usually, it is computed as the length of the shortest path on the grid between nodes (neurons) $m$ and $h$. $T$ is the {\em neighborhood radius}. $\mathcal{K}^T$ is the neighborhood kernel function that computes the neighborhood influence of winner neuron $m$ on neuron $h$. The neighborhood influence diminishes with $T$ \cite{Badran05}.
Since $\mathbf{y}_i$ is described by $P$ distributional variables as defined above, without any information about the multivariate distribution, we assume that $y_{i1}, \ldots y_{ij},$ $\ldots y_{iP}$ are marginal distributions. Thus, the (standard) L2 Wasserstein distance between the $i$-th object and the prototype $\mathbf{g}_h$ associated with the neuron $h$ is defined as follows: \begin{equation}\label{dimult}
d^2_W\left(\mathbf{y}_i,\mathbf{g}_m\right)=\sum\limits_{j=1}^P d^2_W\left(y_{ij}, g_{mj}\right).
\end{equation}
Following the \citeauthor{Rush01} \cite{Rush01} notation, the squared $L_2$ Wasserstein distance between $y_{ij}$ and $g_{mj}$ is defined as:
\begin{equation}\label{HOMSQ}
d^2_W(y_{ij},g_{mj})=\int\limits_{0}^{1} {\left[ {Q _{ij} (p) - Q_{g_{mj}}(p)} \right] ^2 dp}.
\end{equation}
In \cite{Irpino2017} is shown that the squared $L_2$ Wasserstein distance presents more interpretative properties compered with other distances between distributions. Especially it can be decomposed in two independent distance-components as follows:
\begin{equation} \label{eq:IrpRoma2}
d^{2}_{W}(y_{ij},g_{mj})=(\bar{y}_{ij}-{\bar y}_{g_{mj}})^{2}+\int\limits_{0}^{1} {\left[ {Q^c _{ij} (p) - Q^c_{g_{mj}}(p)} \right] ^2 dp},
\end{equation} where: $\bar{y}_{ij}$ and ${\bar y}_{g_{mj}}$ are the means; $Q^c_{ij}(p)=Q_{ij}(p)-\bar{y}_{ij}$ and $Q^c_{g_{mj}}(p)=Q_{g_{mj}}(p)-\bar{y}_{g_{mj}}$ are the centered quantile functions of $y_{ij}$ and $g_{mj}$, respectively. Briefly, the squared $L_2$ Wasserstein distance is expressed as the sum of the squared Euclidean distance between means and the squared $L_2$ Wasserstein distance between the centered quantile functions corresponding to the two distributions. We rewrite the same equation as follows:
\begin{equation} \label{eq:IrpRoma3}
d^{2}_{W}(y_{ij},g_{mj})=d^{2}(\bar{y}_{ij},{\bar y}_{g_{mj}})+d^{2}_{W}(y_{ij}^c,g_{mj}^c). \end{equation}
Finally, the standard multivariate squared $L_2$ Wasserstein distance between the $i$-th object $\mathbf{y}_i$ and the prototype $\mathbf{g}_m$ is as follows
\begin{equation}
\label{dist-1}
d^2_W(\mathbf{y}_i,\mathbf{g}_m) = \sum\limits^{P}_{j=1}({\bar y}_{ij}-{\bar y}_{g_{mj}})^{2}+ \sum\limits^{P}_{j=1}d^2_W(y^c_{ij},g^c_{mj}).
\end{equation}
For the rest of the paper, we denote with $dM_{im,j}=({\bar y}_{ij}-{\bar y}_{g_{mj}})^{2}$
the squared Euclidean distance between the means of distributional data $y_{ij}$ and ${g_{mj}}$, and with $dV_{im,j}=d^2_W(y^c_{ij},g^c_{mj})$ the squared $L_2$ Wasserstein distance between the centered distributional data. Equation (\ref{dist-1}) can be written in a compact form as follows:
\begin{equation}
\label{dist-1C}
d^2_W(\mathbf{y}_i,\mathbf{g}_m) = \sum\limits^{P}_{j=1} (dM_{im,j}+dV_{im,j}).
\end{equation}
Therefore, the generalized distance $d^T(\mathbf{y}_i, \mathbf{g}_{f(\mathbf{y}_m)})$ (equation \ref{cb-1}) is a weighted sum of the non-adaptive multivariate squared $L_2$ Wasserstein distances $d^2_W$ computed between the vector $\mathbf{y}_i$ and the prototypes of the neighborhood of the winner neuron $f(\textbf{y}_m)$.
\subsection{Adaptive DBSOM criterion for distributional-valued data}
Usually, SOM models assume that the variables have the same importance for the clustering and visualization tasks. Nonetheless, in real applications, some variables are irrelevant and others are more or less relevant. Moreover, each cluster may have its specific set of relevant variables \cite{diday77,friedman04,frigui04,huang05}.
In the framework of clustering analysis, adaptive distances \cite{diday77} have been proposed for solving the issue. Adaptive distances are weighted distances, where, generally, a positive weight is associated with each variable according to its relevance in the clustering process, such that the system of weights satisfies suitable constraints.
Adaptive distances were originally proposed in a k-means-like algorithm in two different ways. In a first algorithm, a weight is associated with each variable for the whole dataset (we call it a Global approach). In a second approach, considering a partition of the dataset into $M$ cluster, a weight is associated with each variable and each cluster (we call it a Cluster-wise approach). In this paper, we remark that each neuron of the SOM is related to a Voronoi set that can be considered as a cluster of input data. Similarly to \cite{IrpinoVC2012}, in this paper, we extend this idea to a dataset of distributional data. Following the same approach, we go beyond by exploiting the decomposition of the squared L2 Wasserstein distance in Eq. (\ref{dist-1C}), proposing to weight the components too. Namely, we provide a method for observing the relevance of the two aspects of a distributional variable related to the two components. The current proposal differs from the one in \cite{IrpinoVC2012} for the constraints on the weights.
In this paper, we denote with $\boldsymbol{\Lambda}$ the matrix of relevance weights. The dimension of $\boldsymbol{\Lambda}$ is $P\times 1$ or, respectively, $2P\times 1$, if relevance weights are associated with each variable or, respectively, each component for the whole dataset;
$P\times M$ or, respectively, $2P\times M$ for each variable or, respectively, each component for each neuron. We recall that adaptive distances rely on relevance weights that are not defined in advance, but they depend on the minimization of the objective function, here denoted with $J_{ADBSOM}$, which measures the dispersion of data around the prototypes. Obviously, the trivial solution of such minimization is obtained when $\mathbf\Lambda$ is a null matrix.
To avoid the trivial solution, a constraint on the relevance weights is necessary. In the literature, a constraint on the product \cite{diday77} or on the sum \cite{Huang_05}, usually to one, is suggested. Even if the latter approach appears more natural, it relies on the tuning of a parameter that must be fixed in advance. So far, a consensus on an optimal value is still missing, we do not discuss its use in this paper.
In the framework of clustering analysis for non-standard data, adaptive squared $L_2$ Wasserstein distances were applied (see \cite{DeCDeS10,DECLEC09}) for deriving relevance weights for each variable and cluster.
\citeauthor{IrpinoESWA} \cite{IrpinoESWA} provided a component-wise adaptive distance approach to clustering, but the relevance weights are related to two independent constraints for each component of the distance, forcing the assignment of high relevance weights to components whose contributes to the clustering process are low too.
In this paper, we propose the approach suggested in \cite{Irpino2017} that solve this issue. Differently to the method proposed by \citeauthor{Kohonen01}\cite{Kohonen01}, we consider the training of the SOM as a set of iterative steps that minimizes a criterion function $J_{ADBSOM}$ \cite{Badran05}. The main difference is the allocation of objects to the Voronoi set of each neuron. Indeed, in the original formulation, that is the widely used one, the allocation is performed according to the minimum distance between the object and the prototype only. That approach does not guarantee a monotonic decreasing of the criterion function along the training step. In our case, at each step of the training of the SOM, a set $\mathbf{G}$ of $M$ prototypes, namely, a prototype for each neuron, the matrix $\mathbf\Lambda$ of relevance weights and the partition $\mathcal{P}$ of the input objects are derived by the the minimization of the error function $J_{ADBSOM}$ (know also as energy function of the map), computed as the following dispersion criterion: \begin{equation}\label{crit-1-1} J_{ADBSOM}(\mathbf{G}, \mathbf{\Lambda}, \mathcal{P}) = \sum_{i = 1}^N d^T_{\mathbf{\Lambda}}(\mathbf{y}_i, \mathbf{g}_{f(\mathbf{y}_i)}) \end{equation}
The dissimilarity function, $d^T_{\mathbf{\Lambda}}$, that compares each data unit $\mathbf{y}_i$ to each prototype $\mathbf{g}_h \, (1 \leq h \leq M)$ is defined as:
\begin{equation}\label{cb-2} d^T_{\mathbf{\Lambda}}(\textbf{y}_i, \textbf{g}_m) = \sum_{h = 1}^M \mathcal{K}^T(d(m,h)) \, d_{\mathbf{\Lambda}}(\mathbf{y}_i,\mathbf{g}_h) \end{equation}
\noindent where $d$, $T$ and $\mathcal{K}^T$ are defined as in equation (\ref{cb-1}), while $d_{\mathbf{\Lambda}}(\mathbf{y}_i,\mathbf{g}_h)$ is one of the four following equations depending on the global or cluster-wise, variable or component assignment of the relevance weights:
\begin{equation}\label{dist-1-1}
d_{\mathbf{\Lambda}}(\mathbf{y}_i,\mathbf{g}_m) = \sum_{j=1}^P \lambda_j \,\left( dM_{im,j}+dV_{im,j}\right)
\end{equation}
\begin{equation}\label{dist-1-3}
d_{\mathbf{\Lambda}}(\mathbf{y}_i,\mathbf{g}_m)
= \sum_{j=1}^P \left(\lambda_{j,\mathcal{M}} \,dM_{im,j} + \lambda_{j,\mathcal{V}} \,dV_{im,j}\right)
\end{equation}
\begin{equation}\label{dist-1-2}
d_{\mathbf{\Lambda}}(\mathbf{y}_i,\mathbf{g}_m)
= \sum_{j=1}^P \lambda_{mj} \, \left(dM_{im,j} + dV_{im,j}\right)
\end{equation}
\begin{equation}\label{dist-1-4}
d_{\mathbf{\Lambda}}(\mathbf{y}_i,\mathbf{g}_m)
= \sum_{j=1}^P \left(\lambda_{mj,\mathcal{M}} \, dM_{im,j} + \lambda_{mj,\mathcal{V}} \,dV_{im,j}\right)
\end{equation}
Thus, the generalized distance $d^T_{\mathbf{\Lambda}}(\textbf{y}_i, \textbf{g}_{f(\textbf{y}_i)})$ is a weighted sum of the adaptive multivariate squared $L_2$ Wasserstein distances $d_{\mathbf{\Lambda}}(\mathbf{y}_i,\mathbf{g}_m)$ computed between the vector $\mathbf{y}_i$ and the prototype of the neighborhood of the winner neuron $f(\textbf{y}_m)$.
For avoiding the trivial solution (namely, null $\lambda$'s), we use the above mentioned product constraint. We suggest four different constraints on the relevance weights, that is:
\begin{description}
\item[(P1) A product constraint for Eq. (\ref{crit-1-1}) and Eq. (\ref{dist-1-1})] See \cite{diday77}:
\begin{equation}\label{Con-P-1}
\prod_{j=1}^P \lambda_{j} = 1, \; \; \;
\lambda_{j} > 0
\end{equation}
\item[(P2) A product constraints for Eq. (\ref{crit-1-1}) and Eq. (\ref{dist-1-3})] See \cite{IrpinoESWA} and \cite{DeCA_IR_VE_IEEE_2015}:
\begin{equation}\label{Con-P-3}
\prod_{j=1}^P \left(\lambda_{j,\mathcal{M}} \cdot \lambda_{j,\mathcal{V}}\right)= 1, \, \; \; \; \lambda_{j,\mathcal{M}} > 0,\, \; \; \; \lambda_{j,\mathcal{V}} > 0
\end{equation}
\item[(P3) $M$ product constraints for Eq. (\ref{crit-1-1}) and Eq. (\ref{dist-1-2})] See \cite{diday77}:
\begin{equation}\label{Con-P-2}
\prod_{j=1}^P \lambda_{mj} = 1, \, \; \; \; \lambda_{mj} > 0\quad m=1,\ldots,M;
\end{equation}
\item[(P4) $M$ product constraints for Eq. (\ref{crit-1-1}) and Eq. (\ref{dist-1-4})]See \cite{DeCA_IR_VE_IEEE_2015} and \cite{IrpinoESWA}:
\begin{equation}\label{Con-P-4}
\prod_{j=1}^P \left( \lambda_{mj,\mathcal{M}}\cdot \lambda_{mj,\mathcal{V}}\right) = 1, \, \; \; \; \lambda_{mj,\mathcal{M}} > 0, \; \; \;
\, \lambda_{mj,\mathcal{V}} > 0\quad m=1,\ldots,M;
\end{equation} \end{description}
\noindent \emph{Remark.} Note that the weights of each variable, or of each component in the cluster-wise scheme, are locally estimated. This means that at each iteration the weights change. Moreover, each neuron (whose Voronoi set represent a cluster) has its specific set of weights. On the other hand, weights defined according a global scheme are the same for all the clusters. In general, note that a relevant variable (or component) has a weight greater than 1 because of the product-to-one constraint.
\subsection{The batch SOM optimization algorithm for distributional-valued data}
In this paper, we use a batch training of the SOM, namely, all the data are presented to the map at the same time. Once $T$ (namely, the neuron radius) is fixed, the training of the DBSOM depends on the minimization the criterion function $J_{DBSOM}$ which is based on classical squared $L_2$ Wasserstein distances. Thus no relevance weights are computed. The training task alternates a representation and an assignment step iteratively. The representation step returns the optimal solution for the prototypes describing the neuron of the map. In the assignment step objects are optimally allocated to the Voronoi sets of each neuron of the map. Differently, the ADBSOM algorithm is trained through the minimization of $J_{ADBSOM}$ function. In that case, three steps are iterated: a representation, a weighting and an assignment one. The representation and assignment steps are performed like in DBSOM. The new weighting step carries out optimal solutions for relevance weights according one of the four proposed schemes.
\subsubsection{Representation step}\label{s-rep}
This section focuses on the optimal solution, for a fixed radius $T$, of the prototype of the cluster associated to each neuron during the training of the DBSOM and ADBSOM algorithms.
In the representation step of DBSOM, for a fixed partition $\mathcal{P}$, the objective function $J_{DBSOM}$ is minimized regarding to the components of the matrix $\mathbf{G}$ of prototypes.
Similarly, in the representation step of ADBSOM, for a fixed partition $\mathcal{P}$ and for a fixed set of weights in the matrix $\mathbf{\Lambda}$, the objective function $J_{ADBSOM}$ is minimized regarding to the components of the matrix $\mathbf{G}$ of prototypes.
DBSOM looks for the prototype $\textbf{g}_m$ of the cluster $C_m \, (1,\leq m \leq M)$ that minimizes the following expression: \begin{equation}\label{crit_1} \sum_{j=1}^{P} \sum_{i=1}^{N} \mathcal{K}^T(d(f(\textbf{y}_i), m)) \; d^2_W(\mathbf{y}_i,\mathbf{g}_m) \end{equation}
ADBSOM looks for the prototype $\textbf{g}_m$ of the cluster $C_m \, (1,\leq m \leq M)$ that minimizes the following expression: \begin{equation}\label{crit_2} \sum_{j=1}^{P} \sum_{i=1}^{N} \mathcal{K}^T(d(f(\textbf{y}_i), m)) \; d_{\mathbf{\Lambda}}(\mathbf{y}_i,\mathbf{g}_m) \end{equation} Since both problems are additive, and depends on quadratic terms, the description of the prototypes $g_{mj}$ as distributions for each variable ($m=1,\ldots,M$, $j=1,\ldots,P$), is obtained as solution of the following minimization problem:
\begin{equation}\label{eq-prot-opt} \sum_{i=1}^{N} \mathcal{K}^T(d(f(\textbf{y}_i), m)) \; \left[({\bar y}_{ij}-{\bar y}_{g_{mj}})^{2} + d^2_W(y^c_{ij},g^c_{mj})\right] \longrightarrow \mbox{ Min .} \end{equation} Thus, for each cluster, setting to zero the partial derivatives w.r.t. ${\bar y}_{g_{mj}}$ and $Q^c_{g_{mj}}$ \cite{IrpVer2015}, the quantile function of the probability density function (\emph{pdf}) describing $g_{mj}$ is obtained as follows:
\begin{equation}\label{prot-1}
Q_{g_{mj}}=Q^{c}_{g_{mj}}+{\bar y}_{{g_{mj}}}=
\frac{
\sum\limits_{i=1}^N \mathcal{K}^T(d(f(\mathbf{y}_i),m) \, Q^{c}_{ij}
}
{
\sum\limits_{i=1}^N \mathcal{K}^T(d(f(\mathbf{y}_i),m)
}
+
\frac{
\sum\limits_{i=1}^N \mathcal{K}^T(d(f(\mathbf{y}_i),m) {\bar y}_{ij}
}
{
\sum\limits_{i=N}^n \mathcal{K}^T(d(f(\mathbf{y}_i),m)
}. \end{equation}
\subsubsection{Weighting step}\label{s-weight}
The aim of this section is to provide, for a fixed radius $T$, the optimal solution of the relevance weights of the distributional-valued variables during the training of the ADBSOM algorithms.
In the weighting step of ADBSOM, fixed the partition $\mathcal{P}$ and the prototypes in the matrix $\mathbf{G}$, the objective function $J_{ADBSOM}$ is minimized w.r.t. to the weights, elements of the matrix of $\mathbf{\Lambda}$.
\begin{proposition}\label{prop-weight-1}
The relevance weights are calculated depending upon the adaptive squared $L_2$ Wasserstein distance:
\begin{description}
\item[(P1)]$$\mathit{If}\;J_{ADBSOM}(\mathbf{G}, \mathbf{\Lambda}, \mathcal{P}) = \sum_{m=1}^M \sum_{i=1}^N \sum_{j=1}^P \mathcal{K}^T(d(f(\mathbf{y}_i),m)) \,\lambda_j \, d^2_W(y_{ij},g_{mj})$$ subject to $\prod_{j=1}^P \lambda_{j} = 1, \, \lambda_{j} > 0$, then $P$ relevance weights are derived as follows:
\begin{equation}
\label{W-Glo-1V-Prod}
\lambda_j=\frac{{{{\left\{ {\prod\limits_{r = 1}^P \left[{\sum\limits_{m = 1}^M {\sum\limits_{i = 1}^N {{\mathcal{K}^T(d(f(\mathbf{y}_i),m))}{d^2_W\left( {{y_{ir}},{g_{mr}}} \right)}} } } \right]} \right\}}^{\frac{1}{P}}}}}{{\sum\limits_{m = 1}^M\sum\limits_{i = 1}^N {{\mathcal{K}^T(d(f(\mathbf{y}_i),m))}{d^2_W \left({y_{ij},g_{mj}} \right)}} }}
\end{equation}
\item[(P2)] $$\mathit{If}\; J_{ADBSOM}(\mathbf{G}, \mathbf{\Lambda}, \mathcal{P}) = \sum_{m=1}^M \sum_{i=1}^N\sum_{j=1}^P \mathcal{K}^T\left(d(f(\mathbf{y}_i),m)\right) (\lambda_{j,\mathcal{M}}dM_{im,j} + \lambda_{j,\mathcal{V}}dV_{im,j})$$
subject to $\prod_{j=1}^P \left(\lambda_{j,\mathcal{M}}\cdot \lambda_{j,\mathcal{V}}\right) = 1$, $\lambda_{j,\mathcal{M}} > 0$ and $\lambda_{j,\mathcal{V}} > 0$, then $2 \times P$ relevance weights are derived as follows:
\begin{eqnarray}
\label{W-Glo-2C-Prod}
\lambda_{j,\mathcal{M}}=\frac{
{{{{
\left\{
{
\prod\limits_{r = 1}^P {
\left[
\sum\limits_{m = 1}^M {
\sum\limits_{i = 1}^N {{\mathcal{K}^T(d(f(\mathbf{y}_i),m))}d{M_{im,r}}}
}
\right]
}
}
{
\left[
\sum\limits_{m = 1}^M {
\sum\limits_{i = 1}^N {{\mathcal{K}^T(d(f(\mathbf{y}_i),m))}d{V_{im,r}}}
}
\right]
}
\right\}
^{\frac{1}{2P}}}}}}
}{{\sum\limits_{m = 1}^M\sum\limits_{i = 1}^N {{\mathcal{K}^T(d(f(\mathbf{y}_i),m))}d{M_{im,j}}} }},\;\text{and}\;\nonumber\\
\lambda_{j,\mathcal{V}}=\frac{
{{{{
\left\{
{
\prod\limits_{r = 1}^P {
\left[
\sum\limits_{m = 1}^M {
\sum\limits_{i = 1}^N {{\mathcal{K}^T(d(f(\mathbf{y}_i),m))}d{M_{im,r}}}
}
\right]
}
}
{
\left[
\sum\limits_{m = 1}^M {
\sum\limits_{i = 1}^N {{\mathcal{K}^T(d(f(\mathbf{y}_i),m))}d{V_{im,r}}}
}
\right]
}
\right\}
^{\frac{1}{2P}}}}}}
}{{\sum\limits_{m = 1}^M\sum\limits_{i = 1}^N {{\mathcal{K}^T(d(f(\mathbf{y}_i),h))}d{V_{im,j}}} }}.
\end{eqnarray}
\item[(P3)] $$\mathit{If} \;J_{ADBSOM}(\mathbf{G}, \mathbf{\Lambda}, \mathcal{P}) = \sum_{m=1}^M \sum_{i=1}^N\sum_{j=1}^P \mathcal{K}^T(d(f(\mathbf{y}_i),m)) \,\lambda_{mj}d^2_W(y_{ij},g_{mj})
$$ subject to $M$ constraints $\prod_{j=1}^P \lambda_{mj} = 1, \, \lambda_{mj} > 0$, then $M\times P$ relevance weights are derived as follows:
\begin{equation}
\label{W-Loc-1V-Prod}
\lambda_{mj}=\frac{{{{ \left\{{\prod\limits_{r = 1}^P \left[{\sum\limits_{i = 1}^N {{\mathcal{K}^T(d(f(\mathbf{y}_i),m))}{d^2_W}\left( {{y_{ir}},{g_{mr}}} \right)} } \right]} \right\} }^{\frac{1}{P}}}}}{{\sum\limits_{i = 1}^N {{\mathcal{K}^T(d(f(\mathbf{y}_i),m))}{d^2_W}\left( {{y_{ij}},{g_{mj}}} \right)} }}
\end{equation}
\item[(P4)]$$\mathit{If}\;
J_{ADBSOM}(\mathbf{G}, \mathbf{\Lambda}, \mathcal{P}) = \sum_{m=1}^M \sum_{i=1}^N\sum_{j=1}^P \mathcal{K}^T(d(f(\mathbf{y}_i),m)) \,[\lambda_{ij,\mathcal{M}}dM_{im,j}+\lambda_{hj,\mathcal{V}} dV_{im,j})]$$ subject to $M$ constraints $\prod_{j=1}^P \left(\lambda_{mj,\mathcal{M}}\cdot\lambda_{mj,\mathcal{V}}\right) = 1$, $\lambda_{mj,\mathcal{M}} > 0$ and $\lambda_{mj,\mathcal{V}} > 0$,
then $2 \times M \times P$ relevance weights are derived as follows:
\begin{eqnarray}
\label{W-Loc-2C-Prod}
\lambda_{mj,\mathcal{M}}=
\frac{
{
{
\left\{
{\prod\limits_{r = 1}^P \left[{\sum\limits_{i = 1}^N {{\mathcal{K}^T(d(f(\mathbf{y}_i),m))} \, dM_{im,r}} } \right]
\left[{\sum\limits_{i = 1}^N {{\mathcal{K}^T(d(f(\mathbf{y}_i),m))} \, dV_{im,r}} } \right]
}
\right\}^{\frac{1}{2P}}}}
}
{{\sum\limits_{i = 1}^N {{\mathcal{K}^T(d(f(\mathbf{y}_i),m))} \, dM_{im,j}} }}\quad \text{and} \,\nonumber\\
\lambda_{mj,\mathcal{V}}=
\frac{
{
{
\left\{
{\prod\limits_{r = 1}^P \left[{\sum\limits_{i = 1}^N {{\mathcal{K}^T(d(f(\mathbf{y}_i),m))} \, dM_{im,r}} } \right]
\left[{\sum\limits_{i = 1}^N {{\mathcal{K}^T(d(f(\mathbf{y}_i),m))} \, dV_{im,r}} } \right]
}
\right\}^{\frac{1}{2P}}}}
}
{{\sum\limits_{i = 1}^N {{\mathcal{K}^T(d(f(\mathbf{y}_i),m))} \, dV_{im,j}} }}.\quad
\end{eqnarray}
\end{description}
\end{proposition}
\begin{proof}
In the weighing step, we assume that the prototypes $\mathbf{G}$ and the partition $\mathcal{P}$ is fixed. The matrix of relevance weights $\boldsymbol{\Lambda}$ are obtained according to one of the four above mentioned constraints.
The minimization of $J_{ADBSOM}$ is done by the Lagrange multipliers method. The four constraints allow for the following Lagrangian equations:
\begin{align}
\mathbf{(P1):}\;{\mathcal L} =& J_{ABSOM}(\mathbf{G}, \mathbf{\Lambda}, \mathcal{P})-\theta \left(\prod_{j=1}^P \lambda_{j} - 1 \right);\\
\mathbf{(P2):}\;{\mathcal L} =& J_{ABSOM}(\mathbf{G}, \mathbf{\Lambda}, \mathcal{P})-\theta \left[\prod_{j=1}^P \left(\lambda_{j,\mathcal{M}}\cdot \lambda_{j,\mathcal{V}}\right) - 1 \right];\\
\mathbf{(P3):}\;{\mathcal L} = &J_{ABSOM}(\mathbf{G}, \mathbf{\Lambda}, \mathcal{P})-\sum_{m=1}^M \theta_m \left(\prod_{j=1}^P \lambda_{ij} - 1\right);\\
\mathbf{(P4):}\;{\mathcal L} =& J_{ABSOM}(\mathbf{G}, \mathbf{\Lambda}, \mathcal{P}) -\sum_{m=1}^M \theta_{m} \left[\prod_{j=1}^P \left(\lambda_{mj,\mathcal{M}}\cdot \lambda_{mj,\mathcal{V}}\right) - 1\right].
\end{align}
Setting to zero the partial derivatives of $\mathcal{L}$ with respect to the {$\lambda$'s} and the {$\theta$'s} respectively, a system of equations of the first order condition is obtained and their solution corresponds to the elements of the matrix $\mathbf{\Lambda}$.
\end{proof}
\subsubsection{Assignment step}\label{s-affect}
The aim of this section is to give the optimal partition of the clusters associated to the neurons of the SOM in the assignment step of the DBSOM and ADBSOM algorithms.
Fixed the prototypes, elements of the matrix $\mathbf{G}$, the objective function $J_{DBSOM}$ is minimized w.r.t. the partition $\mathcal{P}$ and each data unit \textbf{y}$_i$ is assigned to its nearest prototype (BMU).
\begin{proposition}\label{prop-part-1} The objective function $J_{DBSOM}$ is minimized w.r.t. the partition $\mathcal{P}$ when the clusters $C_m \, (m=1,\ldots,M)$ are computed as: \begin{equation} \label{af-l1} C_m = \{\mathbf{y}_i \in {\mathcal{D}}: f(\mathbf{y}_i) = m = \operatornamewithlimits{arg\;min}_{1 \leq r \leq M} d^T(\mathbf{y}_i, \mathbf{g}_r) \} \end{equation} \end{proposition}
\begin{proof} Because the matrix of prototypes $\mathbf{G}$ is fixed, the objective function $J_{DBSOM}$ can be rewrite as: $$ J_{DBSOM}(\mathcal{P}) = \sum_{i = 1}^N d^T(\mathbf{y}_i, \mathbf{g}_{f(\mathbf{y}_i)}) $$ Remark that if, for each $\mathbf{y}_i \in {\mathcal{D}}$, $d^T(\mathbf{y}_i, \mathbf{g}_{f(\mathbf{y}_i)})$ is minimized, then the criterion $J_{DBSOM}(\mathcal{P})$ is also minimized. The matrix of prototypes $\mathbf{G}$ being fixed, $d^T(\mathbf{y}_i, \mathbf{g}_{f(\mathbf{y}_i)})$ is minimized if $f(\mathbf{y}_i) = m = \operatornamewithlimits{arg\;min}_{1 \leq r \leq M} d^T(\mathbf{y}_i, \mathbf{g}_r)$, i.e., if $C_m = \{\mathbf{y}_i \in \mathcal{D}: f(\mathbf{y}_i) = m = \operatornamewithlimits{arg\;min}_{1 \leq r \leq M} d^T(\mathbf{y}_i, \mathbf{g}_r) \} \, (m=1,\ldots,M)$. \end{proof}
In the assignment step of ADBSOM, for a fixed matrix of prototype $\mathbf{G}$ and matrix of weights $\mathbf{\Lambda}$, the objective function $J_{ADBSOM}$ is minimized w.r.t. the partition $\mathcal{P}$ and each data unit $\mathbf{y}_i$ is assigned to its nearest prototype.
\begin{proposition}\label{prop-part-2} The objective function $J_{ADBSOM}$ is minimized w.r.t. the partition $\mathcal{P}$ when the clusters $C_m \, (m=1,\ldots,M)$ are computed as: \begin{equation} \label{af-l2} C_m = \{\textbf{y}_i \in \mathcal{D}: m = f (\mathbf{y}_i)= \operatornamewithlimits{arg\;min}_{1 \leq r \leq M} d^T_{\mathbf{\Lambda}}(\textbf{y}_i, \textbf{g}_r) \} \end{equation} \end{proposition}
\begin{proof} The proof is similar to the proof of the Proposition (\ref{prop-part-1}). \end{proof}
\subsubsection{DBSOM and ADBSOM algorithms} \label{s-algo}
Algorithm \ref{alg1} summarizes the batch SOM algorithms DBSOM and ADBSOM for distributional-valued data. In the initialization step, first the matrix of prototypes $\mathbf{G}$ is initialized by randomly choosing $m$ objects of the distributional-valued data-set and then, the weights of the matrix of relevance weights $\mathbf{\Lambda}$ are set to 1 (each component or each variable are assumed to have the same relevance). Besides, using the initialization of the matrix of prototypes $\mathbf{G}$ and the initialization of the matrix of relevance weights $\mathbf{\Lambda}$, the rest of the objects are assigned to the cluster represented by the nearest representative (BMU) to produce the initial partition $\mathcal{P}$. Finally, from the initialization of $\mathbf{G}$, $\mathbf{\Lambda}$ and $\mathcal{P}$, and with $T \leftarrow T_{max}$, the algorithm calls the ITERATIVE-FUNCTION-1 that provides the initial SOM map and the corresponding new initial values of $\mathbf{G}$, $\mathbf{\Lambda}$ and $\mathcal{P}$.
In the iterative steps, in each iteration a new radius $T$ is computed and for this new radius the algorithm call the ITERATIVE-FUNCTION-1 that alternates once two (DBSOM) or three steps (ADBSOM) aiming to provide the update of the matrix of prototypes $\mathbf{G}$, the matrix of relevance weights $\mathbf{\Lambda}$ and the partition $\mathcal{P}$.
In the final iteration, when the radius $T$ is equal to $T_{min}$, only few neurons belong to the neighborhood of the winner neurons and the SOM algorithm behaves similar to k-means. The algorithm calls the ITERATIVE-FUNCTION-2 that alternates once two (DBSOM) or three steps (ADBSOM) until the assignments no longer change. The final iteration provides the final update of the matrix of prototypes $\mathbf{G}$, the matrix of relevance weights $\mathbf{\Lambda}$ and the partition $\mathcal{P}$.
\begin{algorithm} \caption{DBSOM and ADBSOM algorithms} \label{alg1} \begin{algorithmic}[1] \Require \State the distributional-valued data set $\mathcal{D} = \{\mathbf{y}_1,\ldots,\mathbf{y}_N\}$; the number $M$ of neurons (clusters); the size map; the kernel function $\mathcal{K}^T$; the dissimilarity $d$; the initial radius $T_{max}$ and the final radius $T_{min}$; the number $N_{iter}$ of iterations. \Ensure \State the SOM map; the partition $\mathcal{P}$; the matrix $\mathbf{G}$ of prototypes; the matrix $\mathbf{\Lambda}$ of weights. \State \textbf{INITIALIZATION:} \State Set $t\leftarrow 0$ and $T \leftarrow T_{max}$; \State Initialization the matrix of prototypes $\mathbf{G}^{(0)}$: select randomly $M$ distinct prototypes $\mathbf{g}_r^{(0)} \in \mathcal{D} \; (r=1,\ldots,M)$;
\State Initialization of the matrix of relevance weights: set $\mathbf{\Lambda}^{(0)}=\mathbf{1}$;
\State Initialization of the partition $\mathcal{P}^{(0)}$: assign each object $\mathbf{y}_i$ to the closest prototype $\mathbf{g}_r$ (BMU) according to $r = f^{(0)} (\mathbf{y}_i)= \operatornamewithlimits{arg\;min}_{1 \leq m \leq M} d^T(\mathbf{y}_i, \mathbf{g}^{(0)}_m)$;
\State \Call{ITERATIVE-FUNCTION-1}{$\mathcal{D}$, $t=0$, $\mathbf{G}^{(0)}$, $\mathbf{\Lambda}^{(0)}$, $\mathcal{P}^{(0)}$, $T$} \State \textbf{ITERATIVE STEPS:} \Repeat \State Set $t \leftarrow t + 1$; Compute $T=T_{Max} \left(\frac{T_{min}}{T_{Max}}\right)^{\frac{t}{N_{iter}}}$;
\State Set $\mathcal{P}^{(t)} \leftarrow \mathcal{P}^{(t-1)}$, $\mathbf{\Lambda}^{(t)} \leftarrow \mathbf{\Lambda}^{(t-1)}$, $\mathbf{G}^{(t)} \leftarrow \mathbf{G}^{(t-1)}$; \State \Call{ITERATIVE-FUNCTION-1}{$\mathcal{D}$, $t$, $\mathbf{G}^{(t)}$, $\mathbf{\Lambda}^{(t)}$, $\mathcal{P}^{(t)}$, $T$} \Until{$t == N_{iter}-1$}
\State \textbf{FINAL ITERATION:} \State Set $t \leftarrow N_{iter}$; Set $T \leftarrow T_{min}$; \State Set $\mathcal{P}^{(t)} \leftarrow \mathcal{P}^{(t-1)}$, $\mathbf{\Lambda}^{(t)} \leftarrow \mathbf{\Lambda}^{(t-1)}$, $\mathbf{G}^{(t)} \leftarrow \mathbf{G}^{(t-1)}$; \State \Call{ITERATIVE-FUNCTION-2}{$\mathcal{D}$, $t$, $\mathbf{G}^{(t)}$, $\mathbf{\Lambda}^{(t)}$, $\mathcal{P}^{(t)}$, $T$}\end{algorithmic} \end{algorithm}
\begin{algorithm} \begin{algorithmic}[1] \Function{ITERATIVE-FUNCTION-1}{$\mathcal{D}$, $t$, $\mathbf{G}^{(t)}$, $\mathbf{\Lambda}^{(t)}$, $\mathcal{P}^{(t)}$, $T$} \State \textbf{Step 1:} Representation \State For DBSOM, $\mathcal{P}^{(t)}$ is fixed;
\State For ADBSOM, $\mathbf{\Lambda}^{(t)}$ and $\mathcal{P}^{(t)}$ are both fixed;
\State Compute $\mathbf{G}^{(t)}$ using equation (\ref{prot-1}); \State \textbf{Step 2:} Weighting (only ADBSOM algorithm) \State Compute $\mathbf{\Lambda}^{(t)}$ using the suitable equation from Eq. (\ref{W-Glo-1V-Prod}) to Eq.(\ref{W-Loc-2C-Prod}); \State \textbf{Step 3:} Assignment \State For DBSOM, $\mathbf{G}^{(t)}$ is fixed;
\State For ADBSOM, $\textbf{G}^{(t)}$ and $\mathbf{\Lambda}^{(t)}$ are both fixed;
\For{$1 \leq i \leq N$} \State Let $w = f^{(t)}(y_i)$
\State DBSOM: let $z = \operatornamewithlimits{arg\;min}_{1 \leq m \leq M} d^T(\mathbf{y}_i, \mathbf{g}^{(t)}_m)$
\State ADBSOM: let $z = \operatornamewithlimits{arg\;min}_{1 \leq m \leq M} d_{\mathbf{\Lambda}}^T(\mathbf{y}_i, \mathbf{g}^{(t)}_m)$
\If{$w \neq z$} \State $f^{(t)}(\textbf{y}_i) = z$; \EndIf \EndFor \State \Return $\mathbf{G}^{(t)}$, $\mathbf{\Lambda}^{(t)}$, $\mathcal{P}^{(t)}$ \EndFunction \end{algorithmic} \end{algorithm}
\begin{algorithm} \begin{algorithmic}[1] \Function{ITERATIVE-FUNCTION-2}{$\mathcal{D}$, $t$, $\mathbf{G}^{(t)}$, $\mathbf{\Lambda}^{(t)}$, $\mathcal{P}^{(t)}$, $T$} \Repeat \State $test \leftarrow 0$; \State \textbf{Step 1:} Representation \State For DBSOM, $\mathcal{P}^{(t)}$ is fixed;
\State For ADBSOM, $\mathbf{\Lambda}^{(t)}$ and $\mathcal{P}^{(t)}$ are both fixed;
\State Compute $\mathbf{G}^{(t)}$ using equation (\ref{prot-1}); \State \textbf{Step 2:} Weighting (only ADBSOM algorithm) \State Compute $\mathbf{\Lambda}^{(t)}$ using the suitable equation from Eq. (\ref{W-Glo-1V-Prod}) to Eq.(\ref{W-Loc-2C-Prod}); \State \textbf{Step 3:} Assignment \State For DBSOM, $\mathbf{G}^{(t)}$ is fixed;
\State For ADBSOM, $\textbf{G}^{(t)}$ and $\mathbf{\Lambda}^{(t)}$ are both fixed;
\For{$1 \leq k \leq N$} \State Let $w = f^{(t)}(y_k)$
\State DBSOM: let $z = \operatornamewithlimits{arg\;min}_{1 \leq m \leq M} d^T(\mathbf{y}_i, \mathbf{g}^{(t)}_m)$
\State ADBSOM: let $z = \operatornamewithlimits{arg\;min}_{1 \leq m \leq M} d_{\mathbf{\Lambda}}^T(\mathbf{y}_i, \mathbf{g}^{(t)}_M)$
\If{$w \neq z$} \State $f^{(t)}(\textbf{y}_i) = z$; $test \leftarrow 1$; \EndIf \EndFor \Until{test == 0} \State \Return $\mathbf{G}^{(t)}$, $\mathbf{\Lambda}^{(t)}$, $\mathcal{P}^{(t)}$ \EndFunction \end{algorithmic} \end{algorithm}
\section{Applications} \label{SEC_apply}
In this section, we present an application of the proposed algorithms using
two real-world data-sets.
These real world datasets are used for observing the algorithms' performance whit labeled data and unlabeled ones. All the applications have been executed using the HistDAWass package in R \citep{HistDAWass}.
\subsection{Initialization and choice of the parameters} Before launching a SOM algorithm some choices have to be done. In this paper, we do not propose a strategy for the selection of the best parameters of the SOM. In the literature \cite{Vesanto99}, some rule of thumbs is proposed for the SOM initialization according to some well-known side-effects (for example, the propensity of SOM to push all the objects in the corners of the map because of the kernel weighting impact). Hereafter, we list the choices for the parameters and the topologies chosen before launching the SOM algorithm.
We choice a 2D hexagonal map of $5\sqrt{N}$ neurons \cite{Vesanto99}, where $N$ is the number of input objects. The map is rectangular having ratio (namely, the horizontal side of the map is longer than the vertical one), in order to let the map choice its direction of variability. This, in general, mitigates a SOM side-effect consisting in assigning objects into the Voronoi set of the BMU's associated with the corners of the map. Such an effect is more evident when using a dynamic clustering approach. Indeed, the whole SOM algorithm is based on the minimization of the cost function in Eq. \eqref{crit-1} or Eq. \eqref{crit-1-1}, and these functions are considered also for the assignment of data to the BMU's. To mitigate this effect, we propose to use toroidal maps \cite{Mount2011}. In the following, we present both planar and toroidal map results.
A second set of choices is related to the learning function and to the kernel one. First of all, we use a Gaussian kernel function: \begin{equation}\label{kernel_gau} K^{(t)}(r,m)=\exp{\left(-\frac{d^2(r,m)}{2\cdot T(t)}\right)} \end{equation} where, $d^2(r,m)$ is the squared Euclidean distance in the topological space of the neurons between vertices (clusters) $r$ and $m$, $t$ is the generic epoch and $T$ is the kernel width (radius) at the epoch $t$. Once defined the kernel, it is important choose the number of epochs for learning the map, the initial and the final kernel width (radius), and the rate of decreasing of the width (linear or exponential). In the literature \cite{MATLAB_SOM}, for the batch SOM algorithm, it is suggested that the number of epochs for training the map is lesser or equal than $N_{iter}=50$.
Fixed an initial value of the kernel width $T_{Max}=\sigma(1)$ and ending value with $T_{min}=\sigma(N_{iter})$, we use a power series decreasing function for the width of kernel as follows: \begin{equation}\label{eq_learning_function} T(t)=T_{Max}\left(\frac{T_{min}}{T_{Max}}\right)^\frac{t}{N_{iter}}. \end{equation} About the initial and final value of the kernel width, some heuristics proposed in the literature are mainly related to the classical SOM assignment (namely, the one performed using the distance between objects and BMU without considering the kernel). In \cite{Vesanto99}, it is suggested a value for $T_{Max}$ equal to $1/4$ the diameter of the map, decreasing along the epochs until it reaches 1. In our case, since the assignment is done consistently with the cost function, we experienced that the choices suggested by the literature lead to maps too folded and with a high topographic error. We used a new heuristic as follows. Considering that the kernel function in Eq. \eqref{kernel_gau} ranges in $[0,1]$, we fix $T_{Max}$ such that two neurons having a distance equal to the radius of the map, namely, the half of the maximum topological distance between two neurons, have a kernel value equal to $0.1$, and we fix $T_{min}$ such that two neighboring neurons have a kernel value equal to $0.01$. Denoting the diameter of the map in the topological space with $d_{Max}$, namely, the largest topological distance between two neurons of the map, the initial and the final value of $T$, considering Eq. \eqref{kernel_gau}, are determined as follows: \begin{equation}\label{eq:tvalues} T_{Max}=\sqrt{-\frac{\left(0.5\cdot d_{Max}\right)^2}{2\cdot \log{0.1}}}\;\;;\;\;T_{min}=\sqrt{-\frac{1}{2\cdot \log{0.01}}}. \end{equation} SOM is initialized randomly 20 times and the map with the lower final cost function is considered as the best run.
Considering that the proposed SOM algorithms as particular clustering algorithms, we evaluate the output results using internal and external validity indexes (except for unlabeled data).
\subsection*{Internal validity indexes}
For validating the map results, we consider the topographic error measure of the map, the Silhouette Coefficient as a base validity index for comparing the different algorithms and the Quality of Partition index developed in Ref. \citep{IrpinoESWA}. Further, we propose some extensions of the silhouette index for SOM.
A classical validity index for SOM is the topographic error \citep{topo_err_96} $\mathcal{E}_T$. It is a measure of topology preservation and it is computed as follows: for all input vectors, the respective best and second-best matching units are determined. An error is counted if best and second-best matching units of an input vector are neighbors on the map. The total error is the ratio of the sum of the error counts with respect to the cordiality of the input vectors. It ranges from 0 to 1, where 0 means perfect topology preservation. It is obtained as follows: \begin{equation}\label{eq:topoerr} \mathcal{E}_T = \frac{1}{N}\sum\limits_{i = 1}^N {u(i)}. \end{equation} where $$u(i) = \left\{ {\begin{array}{*{20}{l}} {1,}&{\mathrm{best-\;and\;second-BMU\;not - adjacent}}\\ {0,}&\mathrm{otherwise} \end{array}} \right.$$
We recall that the Silhouette Coefficient was defined for clustering algorithms and it corresponds to the average of the silhouette scores $s(i)$ computed for each input data vector. In particular, $s(i)$ \citep{Silh_87} is obtained as:
$$ s(i)=\frac{b(i)-a(i)}{max[a(i),b(i)]}$$
where $a(i)$ is the average distance between the input vector $i$ and the ones assigned to the same cluster (say A, the cluster of $i$), while $b(i)$ is the minimum among the average distances computed from the $i$ to the input vectors assigned to each cluster except the one to which $i$ belongs (say B, the second best cluster of $i$) . The Silhouette Coefficient is as follows:
\begin{equation}\label{eq:sil}
\mathcal{S}=\frac{1}{N}\sum\limits_{i=1}^N s(i).
\end{equation}
The Silhouette Coefficient ($\mathcal{S}$) ranges from $-1$ to $1$, where $1$ represents the best clustering result since each cluster is compact and well separated from the others. Even if in its original formulation it has a complexity of $\mathcal{O}(N^2)$ for a generic distance, in the case of (squared) Euclidean distances it is possible to show that it is only $\mathcal{O}(N)$ \footnote{See the appendix for the proof.}. For reducing the computational time of the index, a simplified version of the Silhouette Coefficient was also proposed by \cite{Silh_Campello06}, where the distances are computed with respect to the prototypes of the cluster only. We use also this index denoting it with $\mathcal{S}_C$.
Regarding also the SOM solution and the considerations about the topographic error, we remark that the obtained SOM map contains neurons that are represented by prototypes that are generally similar if the neurons are adjacent. This could reduce the Silhouette Coefficient even if the map is a good representation of the cluster structure of the input data. To adapt the Silhouette index to the SOM, we propose to mix together the advantages of the Silhouette Coefficient in revealing a good cluster structure and the ones related to the topographic error. We thus propose to modify the $b(i)$ calculation considering it as the second-best cluster of $i$ as the cluster of input vectors that is not adjacent to the neuron of the $i$ BMU. The Silhouette score of each $i$ is calculated as above, but $b(i)$ is obtained without considering those neurons adjacent to A (namely, the BMU of $i$). We propose the same strategy for the $\mathcal{S}_C$ index and we denote the two new coefficients with $\mathcal{S}_\mathcal{E}$ and $\mathcal{S}_{C\mathcal{E}}$ respectively. We remark that in the case of ADBSOM the distance used for the computation of $a(i)$ and $b(i)$ are the adaptive distances used in the algorithm.
\subsection*{External Validity Indexes}
Let $N$ be the number of instance of a data table, $\mathcal{P}=\{C_1,C_2,\ldots,C_M\}$ the clusters obtained by a clustering algorithm and $\mathcal{P'}=\{C'_1,C'_2,\ldots,C'_K\}$ the classes of the labeled instances. Generally, $M$ is required to be equal to $K$, but in our case we can also consider $M\neq K$. Let $C_m$ (resp. $C'_k$) the instances of cluster $m$ (resp. of class $k$), and $a_m=|C_m|$ (resp., $b_k=|C'_k|$) its cardinality. Let $n_{mk}=|C_m \cap C'_k|$ the number of instances of cluster $m$ being in class $k$.
For evaluating the results of the algorithms, for dataset with labeled instances, we use three external validity indexes: the \emph{Adjusted Rand Index} (ARI) \cite{Hubert1985}, the \emph{purity} (Pur) \cite{Manning_2008} and the \emph{Normalized Mutual Information} ($NMI$) \cite{Meila2007}.
The ARI index \cite{Hubert1985} is widely used for assessing the concordance between apriori partition and the partition provides by the algorithm: the index varies between $-1$ and $1$ where the more the index approaches $1$ the more the two partitions are similar. The ARI shows the ability of the algorithm to recover the original classification. \begin{equation}\label{eq:ARI} ARI=\frac{ \sum_{mk}{\binom{n_{mk}}{2}} - [ \sum_{m} {\binom{a_{m}}{2} } \sum_{k} {\binom{b_{k}}{2} } ] / {\binom{N}{2} } } { \frac{1}{2} [ \sum_{m} { \binom{a_{m}}{2} } + \sum_{k} { \binom{b_{k}}{2} } ] - [\sum_{m} { \binom{a_{m}}{2} } \sum_{k} { \binom{b_{k}}{2} } ] / {\binom{N}{2} } }. \end{equation}
The Normalized Mutual Information (NMI) index between apriori partition and the partition provided by the algorithm, is computed as follows:
\begin{equation}\label{eq:NMI} NMI=\frac{I(\mathcal{P},\mathcal{P'})}{|H(\mathcal{P})+H(\mathcal{P'})|/2}\end{equation} where $I$ is the mutual information $$I(\mathcal{P},\mathcal{P'})=\sum_{m}\sum_{k}\frac{n_{mk}}{N}log \frac{n_{mk}}{a_{m}b_{k}}$$ $H$ are the entropies: $$H(I(\mathcal{P}))=-\sum_{m}\frac{a_m}{N}log\frac{a_m}{N}\;\mathrm{and} \;H(I(\mathcal{P'}))=-\sum_{k}\frac{b_k}{N}log\frac{b_k}{N}.$$ ARI and NMI is maximal when the number of classes is equal to the number of clusters. This can be problematic when evaluating the SOM results. In fact, the considering the SOM as a clustering algorithm it is frequent that the number of clusters is greater than the number of apriori classes. We propose to consider these measures together with the \emph{purity} index, namely, another external validity index that considers also this possibility.
Purity index ($pur$) measures the homogeneity of clusters with respect to apriori partition. The index is calculated as follows: \begin{equation}\label{eq:purit}pur=\frac{1}{N}\sum_{m}\operatornamewithlimits{arg\;max}_k(n_{mk})\end{equation} It consists in evaluating the fraction of labeled instances of the majority class in each cluster for all the clustering. It varies in $[0; 1]$, where $1$ indicates that all clusters are pure, namely, they contain only labeled instances of one class. However, \emph{pur} presents a major drawback. It over estimates the quality of a clustering having a large number of clusters that is the typical situation of a SOM output partition.
Thus, we propose to read together the two sets of external indexes following these guidelines: if a SOM has both a higher value of \emph{pur} and of NMI and ARI, this means that the number of non-empty clusters (namely, neurons that are BMU's for at least one instance) are close to the number of apriori classes, while, if NMI and ARI are relatively low and \emph{pur} index is high it means that the apriori class labeled instances are shared among a set of neurons identifying pure clusters. Obviously, if NMI, ARI and \emph{pur} are low the resulting SOM is less able to recognize the apriori class structure.
In the remainder of the applications, in the tables we denote respectively the algorithms with \textit{St.}, the classic BSOM algorithm with each variable standardized using the standard deviation for distributional variables, \textit{P1}, the ADBSOM algorithm with the automatic detection of the relevance weights on each variable of the whole dataset, \textit{P2}, the ADBSOM algorithm with the automatic detection of relevance weights for the components of each variable included in the dataset, \textit{P3}, the ADBSOM algorithm with the automatic detection of relevance weights for each variable and each neuron, and \textit{P4}, the ADBSOM algorithm with relevance weights automatically detected for the components of each variable and each neuron.
\subsection{Real-world datasets} A first dataset comes from an experiment on activity recognition of people doing \textit{Daily and Sports Activities} that is publicly available from the UCI repository \cite{UCI}. In particular, the raw data consists in the triaxial gyroscope and accelerometers measurements of five sensors (two on the arms and on the legs, and one on the thorax) of eight people performing 19 different activities for 5 minutes \citep{altun_comparative_2010}. Each 5 minutes session of activity is represented by 60 5-seconds time windows described by the histograms of the 125 measurements recorded in that time window. In this case, considering that the records are labeled according to the person and the activity, we show how the proposed maps are able to represent the different activities or people (for a specific activity) using some external validity indexes.
The second dataset describes the population pyramids of 228 countries of the world observed in 2014. Considering that a population pyramid is the description of the age distribution for the male and the female component of a population, the dataset is described by only two distributional variables: the male age distributional variable and the female age one. We will refer to this dataset naming it as the ``AGE PYRAMIDS" dataset. The AGE PYRAMIDS dataset does not contain indications about the cluster structure. In this case, to compare the algorithms we will use some internal validity measures like the silhouette index \citep{Silh_87}, the topographic error of the map \citep{topo_err_96} and the quality of partition index proposed in Ref. \citep{CarvalhoBS16}.
\subsection{Human Behavior Recognition dataset}\label{realdata3}
The dataset that we consider here can be downloaded from the University of California Irvine machine learning repository\footnote{http://archive.ics.uci.edu/ml/}. It collects data on 19 activities performed by 8 different people performed in sessions of 5 minutes. Table \ref{tab: persons} shows the description of the people involved and table \ref{tab: activities} the list of activities monitored.
\begin{table}[htbp] \centering \caption{Description of the 8 individuals in the experiment.} \label{tab: persons}
\begin{tabular}{|l|llll|} \hline ID & gender & age & height & weight \\ \hline 1 & F & 25 & 170 & 63 \\ 2 & F & 20 & 162 & 54 \\ 3 & M & 30 & 185 & 78 \\ 4 & M & 25 & 182 & 78 \\ 5 & M & 26 & 183 & 77 \\ 6 & F & 23 & 165 & 50 \\ 7 & F & 21 & 167 & 57 \\ 8 & M & 24 & 175 & 75 \\ \hline \end{tabular} \end{table}
\begin{table}[htbp] \centering \caption{List of activities in the data set} \label{tab: activities} \begin{tabular}{ p{\dimexpr 0.05\linewidth-2\tabcolsep}
p{\dimexpr 0.45\linewidth-2\tabcolsep}
p{\dimexpr 0.05\linewidth-2\tabcolsep}
p{\dimexpr 0.45\linewidth-2\tabcolsep} }
\hline
$\#$&Action&$\#$&action\\ \hline 1 & sitting &11 & walking on a treadmill at 4km/h in 15ยบ inclined position \\ 2 & standing &12 & running on a treadmill at 8km/h \\ 3 & lying on the back &13 & exercising on a stepper \\ 4 & lying on the right side &14 & exercising on a cross trainer \\ 5 & ascending stairs &15 & cycling on exercise bike in horizontal position \\ 6 & descending stairs &16 & cycling on exercise bike in vertical position \\ 7 & standing still in an elevator &17 & rowing \\ 8 & moving around in an elevator &18 & jumping \\ 9 & walking in a parking lot &19 & playing basketball \\ 10 & walking on a treadmill at 4 km/h in flat position & & \\ \hline \end{tabular} \end{table}
The research group that collected the data set has extensively used it to compare classification algorithms \citep{altun_comparative_2010} and classification software packages \citep{barshan_recognizing_2014}, study inter-subject and inter-activity variability \citep{barshan_investigating_2016}.
The data is collected by means of five MTx3-DOF units, manufactured by Xsens Technologies. Sensor units are calibrated to acquire data at a sampling frequency of 25Hz. Each unit has a tri-axial accelerometer, a tri-axial gyroscope, and a tri-axial magnetometer. Sensor units are placed on the arms, the legs and the thorax of the subject's body. We do not consider the magnetometer sensors. The reason is that the magnetometer recordings reflect the direction of the activity with respect to the Earth's magnetic North and this information somehow contaminates the data set. Thus, the data set has 30 continuous variables (5 units x 2 sensors x 3 axes). For each one of the 30 time series recorded at a sampling frequency of 25Hz during 5 minutes was broken into 60 (no-overlapping) time-windows of 5 seconds each of them containing 125 measures. Each set of 125 measures is aggregated into an equi-depth (or equiprobable) histogram where each bin contains the $10\%$ of the observed values.
As a result of the aggregation, for each activity and person, we have 30 (histogram) variables with 60 histograms. As a result, we have a histogram-data table of 9120 rows (60 time windows $\times$ 8 people $\times$ 19 activities), where each row is a window of 5 seconds of a given person performing one of the activities.
\subsection*{Walking on a treadmill at 4 km/h in flat position} Using the activity recognition dataset, we selected one of the activities and we analyze if it is possible to recover a class structure in data. In particular, we want to test our algorithm about its ability at discriminating people doing the same activity. Among the 19 activities, we did a preliminary exploration and we noted that the activity $\#10$ \textit{Walking on a treadmill at 4 km/h in flat position} shows a particular differentiation among people. In this case, the subset is a table of $14,400$ histograms having $480$ rows, namely, $60$ 5-seconds-windows for each one of the $8$ people, and $30$ columns, namely, the measurements of the $5$ tri-axial sensors recording acceleration (accelerometers) and angular speeds (gyroscope).
In Fig. \ref{Fig_PCA_Walking}, we show the plot of individuals, namely, the time windows recorded for all the people, in the first factorial plane obtained by a PCA performed using the method proposed in \cite{VERDE_PCA_2017}. The first factorial plane explains the $40.50\%$ of the total inertia and the 60 time-windows of each person are contained in a convex polygon that is colored differently for each person. Since the variables are $30$, the percentage of inertia explained by the first factorial plane is not so high. However, we notice that the eight people appear quite separated and, in particular, we can see that females are on the right of the plane, while males are on the left except for person 6 (a woman that presents a particular pattern showing two different ways of performing the activity during the five-minutes session).
\begin{figure}
\caption{Activity Recognition dataset: PCA on people while walking on a treadmill at 4 km/h in flat position.}
\label{Fig_PCA_Walking}
\end{figure}
In this case, we run the five SOM algorithms initializing two types of maps: a planar and a toroidal one. The map is a $16\time 8$ hexagonal grid. The size of sides of the map has been chosen such that the cardinality of the neurons is close to the $5\sqrt{480} \approx110$ and each side has an even number of neurons (this is required for the toroidal map using a hexagonal grid). The main validity indexes are reported in Tab. \ref{TAB_Val_WALKING}. The external validity indexes assume that the reference labels are the ones identifying the people.
\begin{table}[ht] \centering \resizebox{\textwidth}{!}{ \begin{tabular}{lrrrrrrrr}
\multicolumn{9}{c}{$16\times 8$ hexagonal \textsc{planar} map}\B \\
\hline
& \multicolumn{5}{l}{Internal validity indexes}&\multicolumn{3}{l}{External validity indexes}\\
Met. & $\mathcal{S}$ & $\mathcal{S}_\mathcal{E}$ & $\mathcal{S}_{C}$ & $\mathcal{S}_{C\mathcal{E}}$ & $\mathcal{E}_T$ & ARI & NMI & Pur \\
\hline
St. & 0.3065 & 0.4306 & 0.5914 & 0.6888 & 0.1812 & 0.4872 & 0.7391 & 1.0000 \\
P1 & 0.3588 & 0.4769 & 0.6132 & 0.7224 & 0.0750 & 0.5748 & 0.7630 & 1.0000 \\
P2 & 0.3354 & 0.5181 & 0.5823 & 0.7380 & 0.2167 & 0.5646 & 0.7650 & 1.0000 \\
P3 & 0.5519 & 0.6889 & 0.7300 & 0.8358 & 0.1292 & 0.7054 & 0.8305 & 0.9958 \\
P4 & 0.2810 & 0.5384 & 0.5658 & 0.7546 & 0.0521 & 0.5894 & 0.7974 & 1.0000 \\
\hline
\multicolumn{9}{c}{$16\times 8$ hexagonal \textsc{toroidal} map} \T \B\\
\hline
& \multicolumn{5}{l}{Internal validity indexes}&\multicolumn{3}{l}{External validity indexes}\\
Met. & $\mathcal{S}$ & $\mathcal{S}_\mathcal{E}$ & $\mathcal{S}_{C}$ & $\mathcal{S}_{C\mathcal{E}}$ & $\mathcal{E}_T$ & ARI & NMI & Pur \\
\hline
St. & 0.2888 & 0.5886 & 0.5221 & 0.7644 & 0.0229 & 0.4528 & 0.7636 & 1.0000 \\
P1 & 0.3903 & 0.7308 & 0.6045 & 0.8594 & 0.0021 & 0.4979 & 0.7798 & 1.0000 \\
P2 & 0.3954 & 0.6818 & 0.5981 & 0.8180 & 0.0062 & 0.6077 & 0.8194 & 1.0000 \\
P3 & 0.3587 & 0.7154 & 0.5958 & 0.8538 & 0.0062 & 0.5567 & 0.7918 & 0.9958 \\
P4 & 0.4043 & 0.7564 & 0.5823 & 0.8633 & 0.0021 & 0.5322 & 0.8063 & 1.0000 \\
\hline \end{tabular} } \caption{Activity recognition dataset, walking on a treadmill at 4 km/h in flat position: validity indexes. External validity indexes assume people as labels.}\label{TAB_Val_WALKING} \end{table}
In Tab. \ref{TAB_Val_WALKING}, results show that toroidal SOMs have lower topographic errors and are more compact with respect to the planar maps. Looking at the external validity indexes, there are not great substantial differences. Normalized mutual information index appears slightly better for toroidal maps (except for the P3 algorithm), while the obtained non-empty Voronoi sets of each neuron are very pure. The adjusted Rand indexes are not so high, but this is due to the different number of obtained non-empty Voronoi sets with respect to the number of classes.
Internal validity indexes also confirm better compactness of the clusters identified by the Voronoi sets associated with the BMUs for the toroidal map with respect to the planar one.
Fig. \ref{Fig_P4 H_WALK} and \ref{Fig_P3 pla_WALK} show the counts of the Voronoi sets (according to the intensity of the colors) of the neurons of the map obtained using P4 algorithm for the hexagonal map, namely, the map having the lower topographic error and best values of internal validity indexes, and the one obtained from the P3 algorithm and a planar map, namely, the one having the highest ARI and NMI index. Each neuron is labeled according to the labels of the objects contained in its Voronoi set. We observe that the only neuron that has two labels is second on the bottom-left of the planar map. Looking at the maps, we observe how the pushing-toward-the-edges effect is evident for the planar map, while, for the toroidal map, we may appreciate how the cluster of neurons are more evident and separated. This suggests that the topographic error, together with internal validity indexes could be good hints for deciding what map could be more explicative of the class structure of this kind of datasets.
\begin{figure}
\caption{Activity Recognition dataset, walking on a treadmill at 4 km/h in flat position: P4 \textsc{toroidal} SOM count map.}
\label{Fig_P4 H_WALK}
\end{figure}
\begin{figure}
\caption{Activity Recognition dataset, walking on a treadmill at 4 km/h in flat position: P3 \textsc{planar} SOM count map.}
\label{Fig_P3 pla_WALK}
\end{figure}
In the remainder of this paragraph, we continue to describe results for the P4 algorithm with a toroidal map\footnote{The authors can supply the R code and detailed results as supplementary material or on request.}. Since the P4 algorithm assigns a relevance weight to each variable for each neuron, it is interesting to observe what variables are more relevant for SOM. In fig. \ref{Fig_BP_som_wei}, we show the box-plots of the logarithms of the relevance weights for the two maps. We remark that we used the box-plot since each variable may have a different weight for each neuron of the first map. For the sake of readability, we ordered the box-plot according to the median value observed for the relevance weights.
Generally, it is interesting to note that the thorax gyroscope variables assume greater weights (the box-plots on the top left side of the plots), while lower values are generally assumed by accelerometer measures (on the left part). \begin{figure}
\caption{Activity Recognition dataset, walking on a treadmill at 4 km/h in flat position: box-plot of the logarithms of weights for each component of variables sorted according to the median weights: M denotes average component, D the dispersion one, TO, RA, LA, RL and LL are the position of the sensors.}
\label{Fig_BP_som_wei}
\end{figure}
It is interesting to observe on the map what are the neurons where components assume greater relevance with respect the others. In Fig. \ref{Fig_som_wei_map}, are reported the maps of the relevance weights for two (of 60) components, namely the average component of the gyroscope measurements on the $y$ axis of the sensor positioned on the torax ($M.TO\_ygyr$) and the average component of the gyroscope measurements on the $z$ axis of the sensor positioned on the left arm ($M.LA\_ygyr$). The choice is motivated by the fact that, looking at Fig. \ref{Fig_BP_som_wei}, $M.TO\_ygyr$ component is the one having the highest median relevance weights among neurons and $M.LA\_ygyr$ has the highest variability. To allow the reader a more immediate reading of the results, the count map of Fig. \ref{Fig_P4 H_WALK} is replicated at the top of Fig. \ref{Fig_som_wei_map}. It is worth noting that, $M.TO\_ygyr$, that is related to the torsion of the thorax from left to right, is more relevant for male than for female people this because the torso of a male differs from the one of a female and this impact on the rotational change on the $y$ axes. Looking at the map for $M.LA\_ygyr$, the relevance is higher for people 1, 2, 7 and 8, namely three females and one male. This is related to the forward and backward movement of the left arm while walking and the variability in the importance of this component within people may be caused, for example, by their handedness. Indeed, it ranges from positive (namely, highly relevant) log values to negative (namely, lowly relevant) ones. Other interpretations could be done observing the other variables but, for the sake of brevity, we don't go ahead, but we confirmed that the use of relevance weights may enrich the interpretation of the results.
\begin{figure}
\caption{Activity Recognition dataset, walking on a treadmill at 4 km/h in flat position, results of the P4 algorithm using a toroidal map. On the top the count map. At the center, the map showing the logarithms of the relevance weights for the component having the highest median relevance. At the bottom, the map showing the logarithms of the relevance weights for the component having the highest variability of the relevance weights among all the neurons.}
\label{Fig_som_wei_map}
\end{figure}
\subsection{World countries population pyramids dataset}\label{realdata2}
The second application evaluates the effectiveness of the proposed algorithms on a dataset recording the population age-sex pyramids of 228 World countries. The dataset is provided by the Census Bureau of USA in 2014 and it is included in the \texttt{HistDAWass}\footnote{\texttt{https://cran.r-project.org/package=HistDAWass}} package developed in R.
The input dataset consists in the description of 228 countries according to the age relative frequency distributions for the male and the female part of the population, namely, it is a table of $228\times2$ histograms.
The demographic evolution of a population is usually represented by three prototypical pyramids: constrictive, expansive and stationary. These represent the main stages of the demographic evolution of a population and are often used as an indicator of life quality in a country.
In Fig. \ref{Fig_types_pyr}, we report the three prototypical pyramid structures \cite[Ch. 5]{Atlas}.
\begin{figure}
\caption{Types of population pyramid}
\label{Fig_types_pyr}
\end{figure} We remark that data is not labeled according to these three prototypical models. Using BSOM we want to see if such prototypical situations arise in the data and how well the map is able to represent the data structure and the variables importance in the map generation.
Also in this case, we run the five SOM algorithms initializing two types of maps: a planar and a toroidal one. The map is a $16\time 8$ hexagonal grid. The size of sides of the map has been chosen such that the cardinality of the neurons is close to the $5\sqrt{228} \approx110$ and each side has an even number of neurons (this is required for the toroidal map using a hexagonal grid).
Two sized maps, a medium sized hexagonal map ($6\times6$) and a large sized one map ($8\times8$), has been chosen and has been randomly initialized 50 times.
Table \ref{TAB_Val_AGE_Pyr} shows the validity indexes related to the final results of the proposed algorithms.
\begin{table}[htbp] \centering
\begin{tabular}{lrrrrr} \multicolumn{6}{c}{$10\times 8$ hexagonal \textsc{planar} map}\B \\
\hline
& \multicolumn{5}{l}{Internal validity indexes}\\
Met.& $\mathcal{S}$ & $\mathcal{S}_\mathcal{E}$ & $\mathcal{S}_{C}$ & $\mathcal{S}_{C\mathcal{E}}$ & $\mathcal{E}_T$\\
St. & 0.3432 & 0.6134 & 0.6670 & 0.8262 & 0.2368 \\
P1 & 0.2085 & 0.5186 & 0.6176 & 0.7949 & 0.1886 \\
P2 & 0.2688 & 0.5634 & 0.6505 & 0.8181 & 0.1842 \\
P3 & 0.3027 & 0.6060 & 0.6490 & 0.8283 & 0.1360 \\
P4 & 0.2884 & 0.6419 & 0.6572 & 0.8459 & 0.0965 \\
\hline
\multicolumn{6}{c}{$10\times 8$ hexagonal \textsc{toroidal} map} \T \B\\
\hline
& \multicolumn{5}{l}{Internal validity indexes}\\
St. & 0.2820 & 0.6340 & 0.6367 & 0.8371 & 0.1360 \\
P1 & 0.3179 & 0.6252 & 0.6437 & 0.8257 & 0.1228 \\
P2 & 0.3424 & 0.6889 & 0.6474 & 0.8580 & 0.1316 \\
P3 & 0.2129 & 0.6307 & 0.5954 & 0.8312 & 0.0921 \\
P4 & 0.3951 & 0.7417 & 0.6880 & 0.8841 & 0.0570 \\
\hline \end{tabular} \caption{Age Pyramids dataset: validity indexes.}\label{TAB_Val_AGE_Pyr} \end{table}
Considering the topographic errors, the use of adaptive distances provides better results than standard ones. Silhouette indexes $\mathcal{S}_\mathcal{E}$ and $\mathcal{S}_{C\mathcal{E}}$ also show an average better compactness and separation in the clustering structure induced by the SOM with adaptive distances. In particular, for this dataset, P4 algorithm, namely, the one assigning weights to the components of each variable for each cluster, returns a lower topographic error and shows more compactness for the clusters defined by the Voronoi sets of the prototypes associated with the neurons.
Taking the best results in terms of topographic error, in Fig. \ref{Fig_Pyr_counts} are shown the maps with the counts and the ISO 3166-1 alpha-3 Country codes of data attracted by each BMU for the P4 algorithms both in the planar and in the toroidal map case. \begin{figure}
\caption{Age Pyramid dataset.P4 SOM algorithm: map of counts and the codes of countries in each Voronoi set of the map BMU's.}
\label{Fig_Pyr_counts}
\end{figure} As shown in Fig. \ref{Fig_Pyr_counts} planar map tends to push on the corners and the border of the map the data, leaving a central empty zone that is not well justified by the data. Indeed, in this dataset are described population structures of countries that are not so clustered in the reality, but the one which are at one of the stages described in Fig. \ref{Fig_types_pyr}. Thus, we consider the toroidal map description of the data more coherent than the planar one, showing a sort of path of transition from countries with an \emph{Expansive} type of population (namely, the ones in the left top corner) toward \emph{Constrictive} ones (namely, the ones in the mid of the path), and, finally, to \emph{Stationary} ones. This type of pattern is better visualized in Fig. \ref{Fig_P4_tor_protos}, where is represented the population pyramid (namely, a particular type of codebook map for data described by two distributional variables where each prototype is described by two juxtaposed smoothed frequency distributions) associated with the prototype of each BMU of the toroidal SOM. \begin{figure}
\caption{Age Pyramid dataset. Toroidal SOM using P4 algorithm: prototypes map.}
\label{Fig_P4_tor_protos}
\end{figure}
In Fig. \ref{Fig_P4_tor_protos}, the three population structures are almost evident starting from the \emph{expansive} model on the top-left side of the map, passing by the \emph{constrictive} model at the center until the \emph{stationary} model on the bottom right. An interesting zone is the bottom zone of the map. Looking at the Fig.\ref{Fig_Pyr_counts}, we observe that the zone is a representation of the Arab states of the Persian Gulf where the population distributions present some particular patterns related to the story and the economy of the region. What is clear is that a path from the expansive to the constrictive and stationary model arise confirming the demographic theories that assume the models as three phases of an evolutionary path of a population. Finally, in Fig. \ref{Fig Pyr_P4_wei} are shown the map of the logarithms of the relevance weights associated with each Voronoi set of the BMU of the neurons. It is interesting to note that, for example, for Persian gulf countries the variability components of the male and female population age distribution is more relevant, while, in general, the average component of the male age population is the most important in the cluster definition especially for the top-left zone of the map where there is the passage from a \emph{constrictive} to a \emph{stationary} shaped population.
\begin{figure}
\caption{Age Pyramid dataset. Toroidal SOM using P4 algorithm: log of relevance weights plot.}
\label{Fig Pyr_P4_wei}
\end{figure}
\section{Conclusions}\label{SEC_conl} The two Batch Self-Organizing Maps (SOMs) methods proposed DBSOM, to extend classical Batch SOM algorithm for distributional data, and ADBSOM, to innovative Batch SOM algorithms, using adaptive distances.
DBSOM and ADBSOM training algorithms are based on the optimization of different objective functions. that is performed in alternates steps: two for DBSOM and three for ADBSOM.
In the representation step, DBSOM and ADBSOM algorithms compute the prototypes (vectors of distributions) of the clusters related to the neurons. They are achieved consistently with the optimal solution of the objective functions, having used the $L_2$ Wasserstein distance between pairs of distributions vector.
The main contribution to Classical batch SOM algorithm is to overcome the classical assumption of SOM that all the variables have the same relevance for training the SOM. Indeed, it is well known that some variables are less relevant than others in the clustering process.
In particular, using the $L_2$ Wasserstein distance between 1D distributions, we have extended the use of SOM for data described by distributions and we have also introduced adaptive distances in the SOM algorithm according to four different strategies. The particular choice of adaptive distances, differently from other variable weighting schemes, does not require to tune further parameters.
The weighting step of ADBSOM calculates the relevance-weights of the distributional valued variables. This is achieved conforming to the optimal solution provided in the paper. The squared $L_2$ Wasserstein distance between two distributions can be decomposed into two components: a squared Euclidean distance between the means and a $L_2$ Wasserstein distance between the centered quantile functions associated to the distributions. This second component allows taking more into account the variability and shape of the distributions. Relevance weights are automatically learned for each of the measure components to emphasize the importance of the different characteristics (means, variability, and shape) of the distributions. These weights of relevance are calculated for each cluster or for the entire partition such that their product is equal to one. Besides, the ADBSOM algorithm takes into consideration new sets of constraints.
Finally, the second step of DBSOM and the third step of ADBSOM compute the partitions on the neurons of the SOM. This is achieved conforming to the optimal solution provided in the paper.
DBSOM and ADBSOM algorithms were evaluated experimentally on two real distributional-valued data sets. ADBSOM outperformed DBSOM on these data sets, especially using toroidal maps. This corroborates the importance of the weighting step of ADBSOM. We have observed that planar maps, together with the optimized criterion suffer from a pushing-toward-the border effect on the data. This is almost evident in the application on real data. We proposed to overcome such limit of planar maps using a toroidal map that have shown better topology preservation of the map (namely, the topographic error is lower than the one referred to planar maps when all the other map parameters are the same).
From the application on real data, we have observed that the use of adaptive distances reduces generally the topographic error and overcomes the problems related to the scaling of the variables as preprocessing step. Moreover, ADBSOM algorithms that calculate the relevance weights of the distributional variables for each cluster had the best performance in terms of topological preservation and in terms of compactness and separation of clusters induced by the Voronoi sets associated with the neurons.
Moreover, as a supplementary contribution, we introduced two new Silhouette indexes for SOM, taking together the idea behind the topographic error computation and the clustering structure of the SOM. We have also observed how to compute exactly a Silhouette index when data are described in a Euclidean space in a more efficient way.
Finally, the usefulness of DBSOM and ADBSOM algorithms have been highlighted with their applications to the Human Behavior Recognition and World countries population pyramids distributional-valued data sets.
\section*{Acknowledgments}
The first author would like to thanks
to Conselho Nacional de Desenvolvimento Cientifico e Tecnologico - CNPq (303187/2013-1) for its partial financial support.
\section*{APPENDIX: Fast computation of Silhouette index for Euclidean spaces}
Let us consider a table $Y$ containing $N$ numerical data vectors $\mathbf{y}_i=[y_{i1},\ldots,y_{iP}]$ of $P$ components. Without loss of generality, the $N$ vectors are clustered in two groups, namely, A and B, having, respectively, size $N_A$ and $N_B$ such that $N_A+N_B=N$. Let $\bar{y}_{Aj}= n_A^{-1}\sum\limits_{\mathbf{y_i}\in A}y_{ij}$ and $\bar{y}_{Bj}= n_B^{-1}\sum\limits_{\mathbf{y_i}\in B}y_{ij}$, $j=1,\ldots,P$, be the two cluster means, and
$SSE_{Aj}=\sum\limits_{\mathbf{y_i}\in A}{y_{ij}^2}- n_A (\bar{y}_{Aj})^2$ and $SSE_{Bj}=\sum\limits_{\mathbf{y_i}\in B}{y_{ij}^2}- n_B (\bar{y}_{Bj})^2$ the two sum of squares deviations from the respective cluster means.
\subsubsection*{The average Euclidean distance between a point to all the other points of a set where it is contained.}
Let consider that $\mathbf{y}_i\in A$,
the average distance of $\mathbf{y}_i$ with respect all the other vectors in A is computed as follows:
$$(n_A-1)^{-1}\sum\limits_{\mathbf{y}_k\in A}\sum\limits_{j=1}^P{(y_{ij}-y_{kj})^2}$$
It is easy to show, for a single variable $j$, that:
$$\sum\limits_{\mathbf{y}_k\in A}{(y_{ij}-y_{kj})^2}=n_A(y_{ij})^2+\sum\limits_{\mathbf{y}_k\in A}{(y_{kj})^2}-2 y_{ij}\sum\limits_{\mathbf{y}_k\in A}{y_{kj}}=$$
$$=n_A(y_{ij})^2+\left[SSE_{Aj}+n_A(\bar{y}_{Aj})^2\right]-2n_Ay_{ij}\bar{y}_{Aj}=$$
$$=n_A\left(y_{ij}-\bar{y}_{Aj}\right)^2+SSE_{Aj}.$$ Then, the average distance is:
$$(n_A-1)^{-1}\sum\limits_{\mathbf{y}_k\in A}
{(y_{ij}-y_{kj})^2}=\frac{n_A\left(y_{ij}-\bar{y}_{Aj}\right)^2}{n_A-1}+\frac{SSE_{Aj}}{n_A-1}$$
\subsubsection*{The average Euclidean distance of a point from all the other points of a set where it is not contained}
Let consider that $\mathbf{y}_i\in A$ and we want to compute the average distance of $\mathbf{y}_i$ with respect all the other vectors in B. The average squared Euclidean distance between $\mathbf{y}_i$ and all the other vectors in B, for each variable, is given by:
$$(n_B)^{-1}\sum\limits_{\mathbf{y}_k\in B}\sum\limits_{j=1}^P{(y_{ij}-y_{kj})^2}$$
for a single variable $j$, it is easy to show that:
$$\sum\limits_{\mathbf{y}_k\in B}{(y_{ij}-y_{kj})^2}=n_B(y_{ij})^2+\sum\limits_{\mathbf{y}_k\in B}{(y_{kj})^2}-2 y_{ij}\sum\limits_{\mathbf{y}_k\in B}{y_{kj}}=$$
$$=n_B(y_{ij})^2+\left[SSE_{Bj}+n_B(\bar{y}_{Bj})^2\right]-2n_By_{ij}\bar{y}_{Bj}=$$
$$=n_B\left(y_{ij}-\bar{y}_{Bj}\right)^2+SSE_{Bj}.$$ Then, the average distance is that:
$$(n_B)^{-1}\sum\limits_{\mathbf{y}_k\in B}
{(y_{ij}-y_{kj})^2}=\left(y_{ij}-\bar{y}_{Bj}\right)^2+\frac{SSE_{Bj}}{n_B}$$
\subsection*{The Silhouette index}
As stated above, the Silhouette index for a single $\mathbf{y}_i$ is given by:
$$ s(i)=\frac{b(i)-a(i)}{max[a(i),b(i)]}$$
where, in the case of two groups A and B, if $i\in A$:
$$a(i)=(n_A-1)^{-1}\sum\limits_{\mathbf{y}_k\in A}\sum\limits_{j=1}^P{(y_{ij}-y_{kj})^2}$$
and
$$b(i)=(n_B)^{-1}\sum\limits_{\mathbf{y}_k\in B}\sum\limits_{j=1}^P{(y_{ij}-y_{kj})^2}.$$
If we consider the original formulation, for computing $N$ silhouette indexes the computational complexity is in the order of $\mathcal{O}(N^2)$, if $N$ is sufficiently larger than $P$.
If we use the formulas suggested here, once computed the averages and the SSE, the computational cost is approximatively of $\mathcal{O}(N)$. In fact, it is possible to compute $a(i)$ and $b(i)$ as follows:
$$a(i)=\sum\limits_{j=1}^p\left[\frac{n_A\left(y_{ij}-\bar{y}_{Aj}\right)^2}{n_A-1}+\frac{SSE_{Aj}}{n_A-1}\right]$$
and
$$b(i)=\sum\limits_{j=1}^p\left[ \left(y_{ij}-\bar{y}_{Bj}\right)^2+\frac{SSE_{Bj}}{n_B}\right].$$
Considering that the squared L2 Wasserstein distance is an Euclidean distance between quantile functions, that the SSE computed for distribution functions is defined as a sum of squared distance an that the average distributions are Fr\'{e}chet means with respect to the squared $L_2$ Wasserstein distance. The same simplification can be applied for computing the Silhouette Coefficient when data are distributions and they are compared using the squared $L_2$ Wasserstein distance.
\section*{References}
\end{document} |
\begin{document}
\begin{frontmatter}
\title{Dictionary-based Monitoring of Premature Ventricular Contractions: \\ An Ultra-Low-Cost Point-of-Care Service}
\author[iitee]{Bollepalli~S.~Chandra\corref{cor1}} \ead{bschandra@iith.ac.in} \author[iitma]{Challa~S.~Sastry} \ead{csastry@iith.ac.in} \author[mc]{Laxminarayana Anumandla} \ead{laxmin56@gmail.com} \author[iitee]{Soumya~Jana} \ead{jana@iith.ac.in} \address[iitee]{Dept. of Electrical Engineering, Indian Institute of Technology Hyderabad, India - 502285.} \address[iitma]{Dept. of Mathematics, Indian Institute of Technology Hyderabad, India - 502285.} \address[mc]{Dept. of Cardiology, Maxcare Hospital, Warangal, India - 506001} \cortext[cor1]{Corresponding author}
\begin{abstract} While cardiovascular diseases (CVDs) are prevalent across economic strata, the economically disadvantaged population is disproportionately affected due to the high cost of traditional CVD management, involving consultations, testing and monitoring at medical facilities. Accordingly, developing an ultra-low-cost alternative, affordable even to groups at the bottom of the economic pyramid, has emerged as a societal imperative. Against this backdrop, we propose an inexpensive yet accurate home-based electrocardiogram (ECG) monitoring service. Specifically, we seek to provide point-of-care monitoring of premature ventricular contractions (PVCs), high frequency of which could indicate the onset of potentially fatal arrhythmia. Note that a traditional telecardiology system acquires the ECG, transmits it to a professional diagnostic center without processing, and nearly achieves the diagnostic accuracy of a bedside setup, albeit at high bandwidth cost. In this context, we aim at reducing cost without significantly sacrificing reliability. To this end, we develop a dictionary-based algorithm that detects with high sensitivity the anomalous beats only which are then transmitted. We further compress those transmitted beats using class-specific dictionaries subject to suitable reconstruction/diagnostic fidelity. Such a scheme would not only reduce the overall bandwidth requirement, but also localizing anomalous beats, thereby reducing physicians' burden. Finally, using Monte Carlo cross validation on MIT/BIH arrhythmia database, we evaluate the performance of the proposed system. In particular, with a sensitivity target of at most one undetected PVC in one hundred beats, and a percentage root mean squared difference less than 9\% (a clinically acceptable level of fidelity), we achieved about 99.15\% reduction in bandwidth cost, equivalent to 118-fold savings over traditional telecardiology. In the process, our algorithm outperforms known algorithms under various measures in the telecardiological context.
\end{abstract}
\begin{keyword}
Affordable telecardiology \sep Point-of-care service \sep Premature ventricular contractions \sep Dictionary learning \sep High-sensitivity detection \sep High-fidelity compression. \end{keyword}
\end{frontmatter}
\section{Introduction} \label{sect1}
Cardiovascular diseases (CVDs) are a leading cause of death across economic strata \cite{WHO}. Hence a crucial healthcare objective consists in managing those diseases. In this regard, electrocardiogram (ECG) signals acquired from subjects often play a vital role. Specifically, continuous ECG monitoring is central to early diagnosis and improved clinical outcome in certain scenarios. However, such monitoring at a professional facility is often unaffordable to economically disadvantaged individuals due to high cost, low availability and other barriers. Against this backdrop, home-based point-of-care (POC) monitoring assumes significance. In this paper, we propose a POC monitoring service that is highly affordable.
Symptoms indicating CVDs often manifest sporadically. Consequently, to detect deviations from the normal sinus rhythm, subjects should ideally be monitored continuously. Especially, for patients who have suffered myocardial infarction (MI), or developed left ventricular dysfunction (LVD), continuous monitoring has proven essential in promptly detecting sudden deterioration in cardiac functions, and hence preventing mortality \cite{podrid}. The aforementioned as well as various related conditions are associated with premature ventricular contractions (PVCs) that briefly interrupt the normal rhythm of the heart \cite{gertsch}. Although PVCs occur in healthy individuals as well, high frequency of PVCs is known to foretell serious arrhythmic conditions \cite{hinkle}, and significantly correlate with events of mortality \cite{ng}. In short, accurate detection of PVCs assumes clinical significance in stratifying high risk patients, and predicting medical emergencies. In this context, we propose a novel personalized service to monitor the PVC burden.
In particular, we seek to develop a POC service that would appeal to the economically disadvantaged. Worldwide, about 1.2 billion individuals live on less than US\$ 1.25 per day and have little discretionary income \cite{poverty}. To such individuals, the cost of professional monitoring could often be prohibitive. Further barriers to quality care could include travel and hospital expenses. Fortunately, high penetration of mobile phones even in remote communities has mitigated such barriers in certain scenarios \cite{west}. In the present case, can the mobile network be leveraged to provide reliable PVC monitoring at an attractive cost to the communities living at the bottom of the economic pyramid \cite{BOP}?
In response, we take a frugal engineering approach \cite{frugal}, and propose an ultra-low-cost POC service. As depicted in Figure \ref{arch}a, a conventional telecardiology system simply records user ECG and transmits it to a diagnostic center staffed by medical professionals, where anomalies are manually detected and medical intervention is initiated, when necessary. Traditionally, ECG signals are transmitted unaltered, resulting in perfect accuracy (subject only to human error), albeit with the attendant high bandwidth cost and without localizing potentially anomalous beats. To reduce cost, we propose a new telecardiology paradigm, depicted in Figure~\ref{arch}b, where each user is equipped with a heartbeat classifier that detects anomalous beats, and then compresses and transmits only those anomalous beats and delimiting neighbors (forming beat-trios) along with timestamps. Such a system not only reduces the bandwidth requirement but also presents to medical professionals only those beats that warrant closer inspection, thereby potentially improving the responsiveness of the diagnostic center.
In this framework, system design involves a tradeoff among three quantities: (i) classifier sensitivity (the fraction of PVC beats correctly identified), which we take as the reliability criterion, (ii) the fidelity of reconstructed signal at the diagnostic center, which determines the ability of experts to authenticate algorithmic classification, and (iii) the transmission bandwidth, which dictates the operating cost. To ensure accurate clinical outcome, one desires both high reliability (classifier sensitivity) and high fidelity. At the same time, one also seeks low transmission bandwidth in order to operate at a low cost. The main difficulty arises due to the complex three-way tradeoff among the above quantities. In particular, the bandwidth usage increases with sensitivity and decreases with specificity, while sensitivity and specificity themselves exhibit a nonlinear inverse inter-relationship dependent on reconstruction fidelity. The above quantities are further affected by signal compression. In this paper, we study the said tradeoff, and propose a natural design framework for telecardiology systems.
\input{Fig_Arch}
None of the individual tasks, namely, classification and compression, is new in the field of ECG signal processing. In fact, numerous algorithms have been reported specifically for PVC detection. Examples include machine learning algorithms, such as mixture of experts \cite{palreddy}, linear and quadratic discriminant analyses \cite{chazal, Llamedo}, support vector machine \cite{melgani}, and artificial neural networks \cite{jiang, ince, zzhang, kiranyaz, bortolan}. However, such algorithms have not been designed to achieve compression as well, and are not optimized in the high-sensitivity regime. On the other hand, reported ECG compression algorithms are based on techniques, ranging from the classical time and transform domain methods \cite{tai, zigel1}, to the recent overcomplete dictionary learning \cite{fira1, adamo}. Yet, those algorithms too were not designed to achieve the desired high-sensitivity classification. Against this backdrop, we propose a dictionary-based method that attempts to retain the best of both worlds, and achieve the desired classification and the compression goals simultaneously. The proposed approach, however, is different from the (symmetric) joint classification/reconstruction framework \cite{jana}, where a combination of classification and (class-oblivious) reconstruction indices is minimized subject to a rate constraint. Instead, our setup is inherently class-asymmetric as signals detected as normal (excepting delimiting beats) are discarded, i.e., compressed to zero bits, while signals detected as PVCs are compressed at a certain rate so as to meet a target diagnostic fidelity criterion.
To this end, we propose to train separate overcomplete dictionaries for the respective classes of normal and PVC beats using labeled data. Specifically, each test beat is approximated as a linear combination of the columns of each dictionary. Intuitively, a signal should admit sparse representation only in the dictionary of the matching class. In accordance, the ratio of sparsity of representation in each dictionary is computed, and a suitable class is assigned by comparing that ratio to a threshold. The sensitivity level is then tuned to a desired level by varying such threshold. Here, the sparsity of representation is dictated by the desired fidelity of reconstruction, and in turn determines the degree of compression. Although low reconstruction fidelity would result in high sparsity (hence high compression), the resulting representation would also tend to miss the information necessary for accurate classification. Interestingly, high-fidelity regime may not guarantee accurate classification either. As the signal approximation becomes increasingly accurate, the representations based on rival dictionaries would decrease in sparsity, which in turn leads to poor classification. Consequently, our task involves choosing suitable level of fidelity so that high sensitivity and high compression are both achieved.
The efficacy of the proposed scheme is demonstrated on the standard MIT/BIH arrhythmia database using Monte Carlo cross validation (MCCV). Presently, we confine to PVC beat detection, however, the same framework can be extended to detection of other as well as multiple classes of anomalies. Specifically, at a high-sensitivity target of 99\% (i.e., no more than one undetected PVC beat in one hundred), using only classification and only compression, we respectively reduced the bandwidth requirement to 42.4\% and 2.0\% compared to the original. Using both classification and compression, we required a bandwidth of only 0.85\% of the original, which translates to 118-fold savings in the operating cost, and an ultra-low-cost solution. Finally, we compared results obtained by our technique with those obtained using existing algorithms, and demonstrated the criticality of the proposed high-sensitivity approach in realizing practical ultra-low-cost telecardiology.
Our key contributions are as follows. We \begin{enumerate} \item developed a dictionary-based algorithm that achieves high-sensitivity classification and high-fidelity compression; \item demonstrated an affordable POC service based on such algorithm, and evaluated its efficacy using MCCV on the standard MIT/BIH arrhythmia database; \item achieved 118-fold cost reduction over classical telecardiology, which improves upon the cost reduction due to known algorithms. \end{enumerate} The rest of the paper is organized as the following. Sec. \ref{sec:motivation} details our motivation, and identifies the key medical and social goals. In Sec. \ref{sec:sigpro}, the associated signal processing problems are formalized with necessary mathematical treatment. Dictionary-based solutions are developed in Sec. \ref{sec:soln}. Performance evaluation strategy, experimental setup and simulation results are presented in Secs. \ref{sec:perf}, \ref{sec:expt} and \ref{sec:results}, respectively. Finally, Sec. \ref{sec:disc} concludes the paper with a discussion.
\section{Motivation and Envisaged System}
\label{sec:motivation}
We begin by placing the present problem in medical and social contexts. \subsection{Clinical Imperative}
Cardiac anomalies could be caused by various conditions that overwork and/or damage heart muscles. Continuous monitoring has often proven effective in timely detection of such anomalies. In particular, monitoring PVCs, which are an early depolarization of the myocardium originating in the ventricle \cite{gertsch}, assumes significance, even though such beats are found in subjects with as well as without structural heart diseases \cite{hinkle}. In healthy individuals, a PVC prevalence of less than $1\%$ is common, which carries no prognostic significance. In contrast, more frequent PVCs might indicate (or, lead to) structural heart diseases. Specifically, 90\% of patients experience PVCs after acute MI \cite{vlodaver}, and the risk of sudden death in such patients is related to the complexity and frequency of the PVCs. Recent studies also indicate the role of PVCs in inducing cardiomyopathy \cite{cha}.
More generally, continuous monitoring of PVCs has proven effective in stratifying clinical risk. However, there is no clear demarcation between high and low frequencies of PVCs. Recommended lower threshold for the high-risk subjects, such as those with a history of MI or LVD varies between 10,000 and 20,000 in a 24-hour window \cite{niwano}. Another recommendation sets 10\% as the threshold PVC burden \cite{baman}. Besides frequency of PVCs, run of two or more PVCs and their complexity could also indicate an adverse heart condition \cite{ng}. Accordingly, in the present work, we propose a PVC monitoring system that detects PVC beats with high sensitivity and communicates those with high fidelity to the diagnostic center, when suitable high-risk criteria are met.
\input{Fig_BeatTrio}
\subsection{Technological Imperative}
A conventional telecardiology system, depicted in Figure \ref{arch}a, acquires and transmits entire user ECG to the diagnostic center. Such a system not only utilizes the available bandwidth in an inefficient manner but also burdens the medical professional with processing the entire record to identify anomalies. In this framework, telephone based ECG transmission and associated clinical experience were investigated decades ago \cite{scheidt}. With growing ubiquity of mobile phones in recent years, cellular network based as well as ZigBee based wireless systems have been developed \cite{varshney, huang,leehj}. Yet, despite technological progress, the inefficient telecardiology architecture has largely avoided scrutiny. In this backdrop, we propose a novel architecture that makes judicious use of bandwidth while assisting medical professionals by localizing the potential anomalies, without compromising on the quality of care.
In this context, note that efforts have already been made to deliver telecardiology services in the remote and rural communities with rickety networks. In particular, a method to encode ECG signals into ASCII characters to enable communication via SMS (short message service) has been reported \cite{Mukhopadhyay}. In contrast, we assume a reliable network, which is expected to reflect the ground reality better and better with the passage of time in view of the phenomenal advancement in communication technology \cite{ICT}.
\subsection{Social Imperative: Representative Scenario}
As alluded earlier, we seek to provide a low-cost telecardiology solution for individuals with average daily income of about US\$ 1.25. Consider an individual living at the economic threshold of this target population segment, who suffered myocardial infarction in the recent past, and was successfully treated (see \cite{steg} for various treatment options). Post treatment, monitoring PVCs over long intervals has now become a clinical priority as mentioned earlier. In this context, we shall investigate the cost associated with such PVC monitoring.
\subsubsection{Cost incurred in traditional telecardiology} \label{traditionalCost} Let us first estimate the cost incurred in traditional telecardiology. Here, we assume that diagnostic services are rendered free of cost. Such an assumption is realistic in various developing and underdeveloped countries, where free healthcare is dispensed from government-run facilities \cite{purohit}. This welfare paradigm is currently being extended even to the broader context of telemedicine \cite{jaroslawski}. So the cost incurred would only constitute the data transmission cost. Considering a sampling rate of 360Hz and word length of 11 bits (used in MIT/BIH arrhythmia database \cite{physionet}), one would generate about 1.78MB of data per hour. As we plan to use existing mobile networks, communicating entire data to the diagnostic center would cost about US\textcentoldstyle~27 per hour at the rate of US\textcentoldstyle~1.5 per 100KB of data usage\footnote{We use the Indian mobile data tariff of Indian rupee (INR) 1 per 100KB as representative, and an exchange rate of US\$ 1 = INR 66.7.}. At this rate, the cost of ten-hour monitoring of single channel ECG would amount to US\$ 2.7.
\subsubsection{Affordability as necessity}
In general, healthcare expenses exceeding 10\% of household spending is considered catastrophic \cite{affordability}. In the aforementioned scenario, assuming a household size of four excluding the subject (which approximates the average family size in India \cite{household}), the household income amounts to about US\$ 5 a day. Assuming zero savings, the 10-hour PVC monitoring cost of US\$ 2.7, calculated in Sec. \ref{traditionalCost}, amounts to 54\% of the household spending, and would clearly be unaffordable. In this situation, as a catastrophic health condition is expected to be detected rarely, the household could be tempted to view the monitoring expenditure as non-essential. In reality, however, timely detection of a life-threatening condition saves life with high probability, and hence periodic monitoring remains crucial for long-term survival. Hence, it becomes imperative that the monitoring cost be drastically reduced to such an affordable level that even an economically disadvantaged person would find little incentive to forego it.
\subsection{Outline of Envisaged System}
To meet the aforementioned imperative, we seek to reduce the volume of data communicated to the diagnostic center. As a means, it appears natural to compress the data before transmission. In fact, to make the system even more efficient, we propose to detect anomalous beats, and communicate a compressed version of only those beats. More precisely, we shall form beat-trios, each consisting of a PVC beat, and normal beats preceding and following it (See Figure \ref{signal}a for an illustrative example). A representative beat vector for the normal beat and the PVC beat are shown in Figure \ref{signal}b. If a PVC beat is not isolated, but a run of PVCs (two or more) occur, the normal beats preceding and succeeding the run are used as delimiters. Such beat-trios (and delimited PVC runs) will then be communicated to the diagnostic center along with the timing information. Although this scheme adds a worst-case overhead of two beats for each anomalous beat, it preserves the timing and morphological information of neighboring beats, which are known to facilitate professional diagnosis \cite{kiranyaz}. As mentioned earlier, additional bandwidth savings is achieved by transmitting the compressed version of those beats. The original and the reconstructed beat-trio signals along with reconstruction errors for various compression factors are presented in Figure \ref{signal}c. Specific details on the proposed classification and compression techniques are provided later. In summary, we envisage a low-cost system that makes efficient use of bandwidth by suitably classifying and compressing heart beats.
\section{Classification, Compression and Dictionary Learning} \label{sec:sigpro}
As alluded earlier, signal processing in the present work involves classification and compression of ECG signals. In this section, we pose the associated engineering problems, and provide necessary mathematical preliminaries.
\subsection{ECG classification}
A desired classifier specifies two mutually exclusive and exhaustive subsets $\Gamma_1$ and $\Gamma_2$ of set $\Gamma$ of possible ECG beat $x$ as follows. Any beat $x\in\Gamma_1$ is declared normal, while any beat $ x\in \Gamma_2$ is declared a PVC. Presently, we wish to find $\Gamma_2$ (and hence $\Gamma_1$) such that for a given sensitivity $Se$, i.e., fraction of PVC beats correctly detected as PVC beats, the specificity $Sp$ i.e., fraction of normal beats correctly detected as normal beats is maximized \cite{NP}.
Next we examine the bandwidth requirement of the aforementioned classifier, assuming that only beats detected as anomalous (PVC) are transmitted to the diagnostic center, which possesses adequate resources to validate and correct, if necessary, the class of each beat it receives. In other words, one fails to detect a PVC beat only if that beat is originally classified as normal and never transmitted. Thus the fraction of undetected PVCs, $1-Se$, inversely relates to the reliability of the overall system including the diagnostic center. Perfect reliability is achieved when $Se=1$. Denoting by $\rho$ the prevalence rate of PVCs, and taking the bandwidth requirement without classification as the reference, the fraction of actual PVC beats that are classified as PVCs equals $Se \times \rho$, and the fraction of normal beats that are mistakenly classified as PVCs is given by $ (1-Sp) \times (1-\rho)$. Thus the overall fraction of beats declared as PVC equals $(Se \times \rho + (1-Sp) (1-\rho))$.
In the envisaged beat-trio system, assuming a worst-case scenario that each PVC beat is preceded and followed by normal beats, the (conservatively estimated) fraction $B_{cl}$ of bandwidth usage with only classification and no compression is given by \begin{equation} B_{cl} = 3(Se \times \rho + (1-Sp) (1-\rho)). \label{SeSp} \end{equation} Employing an ideal classifier ($Se=1$, $Sp=1$), one would require a bandwidth $B= 3\rho$, amounting to a substantial bandwidth savings (when $\rho <<\frac{1}{3}$), while ensuring perfect reliability. Unfortunately, such an ideal classifier is unrealizable. In practice, we seek to significantly save bandwidth, while still achieving high reliability.
\subsection{ECG compression}
In the same vein, assuming the signal set as composed of only normal and PVC beats with a compression ratio of $\beta_N~(\ge 1)$ and $\beta_V~(\ge 1)$, respectively, for normal and PVC beats, bandwidth usage is a function of prevalence and given by \begin{equation} B_{co} = \rho \times \frac{1}{\beta_V} + (1-\rho) \times \frac{1}{\beta_N}. \label{SeSp} \end{equation} Further bandwidth savings can be achieved by employing a hybrid scheme, where beat-trios are formed around detected PVC beats, which are then compressed and communicated to the diagnostic center. Employing such a scheme, bandwidth usage diminishes to at most \begin{equation} B_{tr} = (Se \times \rho + (1-Sp) (1-\rho))(\frac{1}{\beta_V} + \frac{2}{\beta_N}). \label{overall_cost} \end{equation}
In general, the reconstruction fidelity varies inversely with compression ratio, and the tradeoff is beat-type specific. We shall measure reconstruction fidelity using the percentage root mean squared difference ($PRD$), widely used in the context of ECG:
\begin{equation} \label{eq:error}
PRD = \frac{\|x-\hat x\|_2}{\|x\|_2}, \end{equation} where $x$ and $\hat x$ stand respectively for the original and the reconstructed signals \cite{sornmo}. Further, from a diagnostic perspective, a $PRD$ of no more than 9\% has been found to be ``good" (Table \ref{PRD}) \cite{wdd}. Accordingly, we set the above fidelity constraint in subsequent analysis.
\input{prdTable}
\subsection{Dictionary-based Technique} \label{sec:dict} So far, we have envisaged a system with certain target classification accuracy and reconstruction fidelity. Now we require an enabling technology to achieve those targets. In this regard, we propose a dictionary-based solution. First we need mathematical preliminaries of compressive sampling and dictionary learning.
\subsubsection{Compressive sampling paradigm} \label{sec:cs} Compressive sampling (CS) recovers a high dimensional sparse vector $\alpha \in \mathcal{R}^n$ from a few of its measurements $x =\Phi \alpha, x \in \mathcal{R}^m $, $m <n$, where $\Phi$ denotes the measurement matrix \cite{elad}. Formally, we seek to solve \begin{equation}\label{eq:l_0}
\min_{\alpha} \| \alpha \|_0 \quad subject \quad to \quad \Phi \alpha = x, \end{equation}
where $\| \cdot \|_0$ indicates the $l_0$ (counting) norm. In general, (\ref{eq:l_0}) is intractable. Fortunately, under certain technical conditions, solution to (\ref{eq:l_0}) remains unaltered if $\| \cdot \|_0$ is replaced by the $l_1$ norm $\| \cdot \|_1$. As $l_1$ solver, we shall use orthogonal matching pursuit (OMP) in view of its simplicity, empirical effectiveness (despite its being greedy) \cite{elad}, and relatively low computational complexity of O($m^2n$) \cite{rubinstein}.
\subsubsection{Dictionary learning}
The method of dictionary learning identifies a tunable selection of basis vectors providing sparse representation. Given a set of signals $\{ {x}_{i}\}_{i=1}^M$, $\ K$-SVD obtains the dictionary $D$ that provides the sparsest representation for each example in this set \cite{ksvd}. It involves a two-step procedure. In the first step, for a given dictionary $D$, we obtain matrix $\Psi$ with sparse columns by solving the following optimization problem: \begin{equation} \label{eq:spudate} \Psi = \operatorname{\arg\min}_{\Theta} \sum_{l} \parallel \Theta_l \parallel_{1} \; subject ~ to \; X = D \Theta, \end{equation} where $\Theta_l$ is the $l$-{th} column of $\Theta$, and $ X$ is the matrix whose columns are $x_i$'s. Using the above $\Psi$, the pair ($D, \Psi)$ is then updated as \begin{multline} \label{eq:dicupdate}
({\hat{D}},\hat{\Psi}) = \arg\min_{{D},\Psi} \|{X}-{D}\Psi\|_{F}^{2} ~ subject ~ to \\
\|\Psi_{i}\|_{0} \leq T_{0}~ \forall i, \end{multline}
where $\Psi_{i}$ denotes the $i$-{th} column of $\Psi$, $T_{0}$ the sparsity parameter, and $\|\cdot \|_{F}$ indicates the Frobenius norm. The $\ K$-SVD algorithm alternates between sparse coding (\ref{eq:spudate}), solved by an $l_1$ solver such as OMP (CS theory), and dictionary update (\ref{eq:dicupdate}) based on iterative soft-thresholding, till convergence. The complexity of learning an $m\times n$ dictionary based on $M$ training data (signals) is O($m^2nM$) \cite{rubinstein}. However, as such learning is generally performed offline, complexity of projecting a signal vector on a dictionary and finding dictionary coefficients is a more important consideration. Fortunately, that complexity is O($m^2n$), i.e., the same order as that of OMP. Consequently, the runtime complexity of both dictionary-based classification and compression algorithms is also O($m^2n$).
\section{Proposed Dictionary-based Solution} \label{sec:soln} At this point, we are ready to propose a dictionary-based solution to achieve the desired classification and compression targets.
\subsection{Dictionary-based classification} \label{sec:soln_class}
\input{Fig_Classifier}
Consider labeled dataset $\{ \{ {x}_{il}\}_{i=1}^{M_l} \}_{l=1}^{K}$. Here $l$ indicates the class label: $l=$``$N$'' indicates normal, and $l=$``$V$'' indicates PVC in a two class problem ($K=2$). Further, $i$ indicates beat index, taking values up to $M_l$, the number of beats present in class $l$. Based on such labeled dataset, we learn the dictionary ${D}_l \in \mathbb{R}^{m \times n}$ for class $l$. When a test beat $x$ is presented, to achieve beat classification, we first find the sparsest representation $\alpha_l$ of $x$ using each dictionary $D_l$, $l\in\{N,V\}$, by solving \begin{equation}\label{eq:Rl_1}
\hat{\alpha}_l = \min \| \alpha_l \|_1 \; \text{subject to} \; \|x - D_l \alpha_l\|_2 <\epsilon, \end{equation} where $\epsilon >0$ denotes the representation accuracy and is proportional to PRD, the normalized reconstruction fidelity.
Here, we denote by $PRD^{class}$ the target reconstruction fidelity corresponding to the classification subsystem. Operating at $PRD^{class}$, as depicted in Figure \ref{classifier}a, $x$ is marked as PVC if the ratio of $l_0$ norm (number of non-zero entries) of $\alpha_V^{class}$ to that of $\alpha_N^{class}$ is less than a suitable threshold $\tau$, and as normal otherwise. When $PRD^{class}$ is low, the signal representation tends to be non-sparse and hence our sparsity-based classification could be less accurate. Further, at high levels of $PRD^{class}$, both the dictionaries are expected to represent the signal with only a few coefficients, so that a sparser alternative is harder to pick, thereby also decreasing classification accuracy. Accordingly, we choose to operate at a suitable fidelity level that maximizes classification accuracy. Note that, for a given $PRD^{class}$, classification accuracy depends only on the ratio of sparsity of representation in rival classes, and does not require the signal to actually be reconstructed. Consequently, $PRD^{class}$ has no influence on the signal reconstruction fidelity of the overall system, and remains an internal parameter of the classification subsystem. Further, classifier performance is dictated by the choice of the threshold $\tau$. We plot receiver operating characteristic (ROC) curves for our classifier by varying $\tau$ and pick suitable operating points.
\subsection{Dictionary-based compression} \label{sec:soln_compr}
Recall that each beat marked as anomalous, as well as each delimiting normal beat, is compressed and communicated to the diagnostic center. We intend to maximize compression ratio for a given reconstruction fidelity target $PRD^{compr}$ using a dictionary based method as shown in Figure \ref{classifier}b. Specifically, we first project the test beat on the class-specific dictionary subject to an intermediate PRD constraint $PRD^{int}$ ($\le PRD^{compr}$), and compute the corresponding dictionary coefficients, only a subset of which are expected to be non-zero. Those non-zero coefficients are subsequently quantized such that the PRD degrades enough to meet the overall constraint $PRD^{compr}$. Here, $PRD^{int}$ remains internal to compression subsystem and if we set $PRD^{int}$ to be significantly smaller than $PRD^{compr}$, the number of non-zero coefficients would be large, which would then require coarse quantization so as to increase the overall PRD sufficiently. On the other hand, if $PRD^{int}$ is set too close to $PRD^{compr}$, only a few coefficients are expected to be non-zero, which can only be quantized rather fine because of relatively small room for $PRD$ degradation. In general, $PRD^{int}$ governs the interplay between the number of non-zero coefficients and the coarseness of their quantization; however, it is not straightforward how to optimally set $PRD^{int}$ to obtain the highest compression ratio subject to $PRD^{compr}$. So, we perform a search as follows. In particular, we plot overall PRD versus compression ratio for various choices of $PRD^{int}$, and take the envelop as the plot of $PRD^{compr}$ versus compression ratio. As discussed earlier, we shall operate at $PRD^{compr}=9\%$ for each of PVC and normal classes so as to maximize signal compression while preserving the diagnostic integrity of the ECG signal.
Next we detail our quantization scheme as well as our encoding scheme for the quantized coefficients, which in turn determines compression ratio. We first generate a quantization table for each dictionary coefficient. Specifically, we rank the non-zero coefficients in the descending order of absolute magnitude. At rank $i$, we find the maximum and minimum (signed) values $W_{max}^i$ and $W_{min}^i$, respectively, and adopt uniform quantization with step size $\Delta$. Specifically, $x$ is quantized to \begin{equation} Q^i (x; \Delta) = \begin{cases}
W^i_{min} - \frac{\Delta}{2}, & x < W^i_{min}, \\
k\Delta - \frac{\Delta}{2}, & x \in [(k-1)\Delta, k\Delta), \\
W^i_{max} + \frac{\Delta}{2}, & x \ge W^i_{max}. \end{cases} \end{equation} Note that the quantizer range depends on rank, but not the quantizer step size. Further, as the step size $\Delta$ increases, so does compression ratio as well as PRD. Finally, quantized coefficients, coefficient locations, and differential timestamps are encoded using Huffman coding algorithm based on empirical probabilities \cite{haykin}.
Finally, after quantization and encoding of dictionary coefficients, we compute the class-specific beat compression ratio $\beta$ as follows: \begin{equation} \label{compRatio} \beta_l = \frac{ \text{Number of bits representing} ~ x}{\text{Number of bits representing} ~ C_l + B^{time}_l +1}, \end{equation} where $x$ represents a beat vector, $C_l$ encodes quantized amplitude as well as location of the non-zero elements of sparse dictionary coefficients $\alpha_l$, $l\in {V, N}$, and $B^{time}_l$ represents the number of bits required to encode beat-specific timestamp. Further, one additional bit is used to encode label $l$.
\subsection{End-to-end System} \label{sec:system}
\input{Fig_FlowChart}
At this point, we turn to completing an end-to-end system that utilizes the classification and compression subsystems discussed so far. A flowchart of the proposed system is depicted in Figure \ref{fc}, and consists of the following modules.
\underline{\it Data reading:} We begin by acquiring ECG samples from the subject, and store those in a buffer. Simultaneously, we read stored samples from the buffer to form a beat vector $B_n$. Time $T_n$ of occurrence of corresponding beat is recorded and the communication status flag $S_n$ is set to zero, which would later indicate whether to transmit a specific beat to the diagnostic center.
\underline{\it Classification and compression:} First, to detect anomaly, each beat vector is projected on the pre-learned dictionaries of normal and PVC classes to obtain respective sparse representations, $\alpha_N^{class}$ and $\alpha_V^{class}$, subject to $PRD \le PRD^{class}$. By comparing the ratio of sparsity of representation in either class to a threshold $\tau$, each beat is assigned a class label $L_n$. Internally, we use 0 and 1 to indicate N and V, respectively. Later, beat compression is achieved by projecting the beat vector $B_n$ on the dictionary of the chosen class, subject to $PRD \le PRD^{int}$, the sparse coefficient vector $\alpha_l^{int}$, $l$ = $N$ or $V$. Finally, we encode to $C_n$ only the signed magnitude of non-zero elements and corresponding indices (locations) of $\alpha_l^{int}$ subject to $PRD \le PRD^{compr}$. Here $PRD^{class}$, $\tau$, $PRD^{int}$ and $PRD^{compr}$ are design parameters.
\underline{\it Command flags:} In order to communicate only the beats detected as PVCs and delimiting normal beats, we make use of certain command flags as follows. At instance $n-1$, if a PVC beat is encountered, i.e., $L_{n-1} = V$, the communication status flags for the past, the present and the next beats are all set to 1, i.e., $S_{n-2} = S_{n-1} = S_{n} = 1$. This is repeated after incrementing the counter n. The status flag generation logic is illustrated in Table \ref{statusflags} for three representative label sequences. As the beat label of the current beat could impact the label of the previous beat, our system would incur a delay of two beats.
\input{Table_flags}
\underline{\it Transmission:} Finally, we communicate only the marked beats to the diagnostic center based on certain clinical considerations. Common clinical conditions requiring expert attention include: (i) Average PVC burden exceeds certain threshold over a specified interval; (ii) A run of two or more PVCs is detected. We illustrate the proposed service in Figure \ref{fc} using condition (i). Specifically, we maintain an accumulator flag $Acc$, which is incremented when an anomalous beat is detected. Once the frequency of the anomalous detections exceed the specified threshold $Th$, the user is notified and the data (with a delay of two beat intervals) are communicated to the diagnostic center. Alternatively, if the total number of beats reaches the maximum beat count $N_{th}$, monitoring stops with a notification to the user. Recall that the specific transmission logic described above is presented only as an illustration. In general, one should adopt suitable logic that embodies the desired condition.
\section{Framework for Performance Evaluation} \label{sec:perf}
We now turn to performance evaluation of the proposed classification subsystem, the compression subsystem and the complete end-to-end system. As customary, we evaluate the classification and the compression subsystems based on the tradeoff between sensitivity (reliability) versus specificity, and compression ratio versus reconstruction fidelity, respectively. Further, we evaluate the end-to-end system in terms of bandwidth cost savings subject to clinically motivated reliability and fidelity constraints. For evaluation of various performance indices, we made use of MIT/BIH Arrhythmia database available from the PhysioBank archives \cite{physionet}. In particular, we first partition the database into training and test sets, and train a common dictionary underlying both the classification and compression subsystems. Later, we test the performance of these subsystems and the end-to-end system, and compare with the performance of reported algorithms in the telecardiological context. Clearly, the said partitioning can be carried out in large number of ways. We intend to adopt a partitioning principle that is appropriate for the underlying practical problem.
\subsection{Patient-specific Partitioning}
Traditionally, partitioning of database into training and test sets is performed either in a class-oriented or in a subject-oriented manner \cite{ye}. In the former, partitioning is based only on the heart-beat label, which allows significant amounts of data from the same patient to be represented in both training and test sets, resulting in overly optimistic performance estimates \cite{bortolan, krasteva}. In contrast, the latter seeks to account for inter-subject variability, and constitutes training and test sets with beats from distinct subsets of records, leading to an overly conservative estimate of performance \cite{zzhang}. More recently, a hybrid scheme called patient-specific training has been proposed \cite{palreddy, chazal, jiang, ince, Llamedo, kiranyaz}, in which a subject-oriented approach is taken with the following modification. A few patient-specific beats (generally, segmented from the first 5 minutes of each record) are added to the training set. Such patient-specific approach often provides a reasonable performance estimate, which is less optimistic than the performance estimated using purely class-oriented partitioning, and less conservative than that using purely subject-oriented approach.
Accordingly, we adopt a patient-specific paradigm complying with ANSI/AAMI EC57:1998 recommendation \cite{aami}, and compared with known compliant algorithms. It is worth noting that the said recommendation excludes subjects with paced beats (records 102, 104, 107 and 217 of the MIT/BIH Arrhythmia database) and partitions the remaining 44 records into training and test sets.
\begin{table*}[] \renewcommand{1.2}{1.2} \centering \resizebox{160mm}{!}{ \begin{tabular}{cC{6.5cm}C{6.5cm}C{2.5cm}C{2.5cm}} \hline \multirow{2}{*}{\begin{tabular}[c]{@{}c@{}} Partition \\ index \end{tabular}} & \multicolumn{2}{c}{Subject ID in MIT/BIH arrhythmia database} & Number of normal beats & Number of PVC beats \\ \cline{2-3}
& training set & test set & \begin{tabular}[c]{@{}c@{}}training (\%)\\ testing(\%)\end{tabular} & \begin{tabular}[c]{@{}c@{}}training (\%)\\ testing(\%)\end{tabular} \\ \hline \hline
Partition-1 & 100, 105, 106, 108, 109, 111, 114, 116, 118, 119, 121, 123 and 124 & 200, 201, 202, 203, 205, 207, 208, 209, 210, 213, 124, 215, 219, 221, 223, 228, 230, 231, 233 and 234 & \begin{tabular}[c]{@{}c@{}}25360 (37\%) \\ 43130 (63\%)\end{tabular} & \begin{tabular}[c]{@{}c@{}}1281 (18.3\%) \\ 5727(81.7\%)\end{tabular} \\
Partition-2 & 100, 101, 103, 105, 106, 108, 109, 111, 112, 113, 114, 115, 116, 118, 119, 121, 122, 123 and 124 & 200, 201, 202, 203, 205, 207, 208, 209, 210, 212, 213, 214, 215, 219, 220, 221, 222, 223, 228, 230, 231, 232, 233 and 234 & \begin{tabular}[c]{@{}c@{}}39582 (43.9\%) \\ 50499 (56.1\%) \end{tabular} & \begin{tabular}[c]{@{}c@{}}1281 (18.3\%) \\ 5727 (81.7\%) \end{tabular} \\
Partition-3 & 101, 106, 108, 109, 112, 114, 115, 116, 118, 119, 122, 124, 201, 203, 205, 207, 208, 209, 215, 220, 223 and 230 & 100, 103, 105, 111, 113, 117, 121, 123, 200, 202, 210, 212, 213, 214, 219, 221, 222, 228, 231, 232, 233 and 234 & \begin{tabular}[c]{@{}c@{}} 45798 (50.8\%) \\ 44283 (49.2\%) \end{tabular} & \begin{tabular}[c]{@{}c@{}}3788 (54\%) \\ 3220 (46\%)\end{tabular} \\
Partition-4 & 105, 106, 108, 109, 111, 116, 118, 124, 200, 201, 202, 203, 205, 207, 209, 210, 212, 214, 215, 223, 228 and 232 & 100, 101, 103, 112, 113, 114, 115, 117, 119, 121, 122, 123, 208, 213, 219, 220, 221, 222, 230, 231, 233 and 234 & \begin{tabular}[c]{@{}c@{}}45575 (50.6\%) \\ 44506 (49.4\%) \end{tabular} & \begin{tabular}[c]{@{}c@{}} 4008 (57.2\%) \\ 3000 (42.8\%) \end{tabular} \\ \hline
\end{tabular}} \caption{Various dataset partitions.} \label{data_partition} \end{table*}
\subsection{Hand-picked Partitioning} Even patient-specific partitions are numerous. Traditionally, one research group would hand-pick one such partition based on subjective criteria. Examples include Partition-1, Partition-2 and Partition-3 given in Table \ref{data_partition}. In each case, the total (and relative) numbers of normal and PVC beats are mentioned for each of the training and the test sets. In particular, Partition-1 considers only those records that contain at least one PVC beat. Here, the training and the test sets consist of 13 records with indices in the range 100--124 and 20 records with indices in the range 200--234, respectively \cite{palreddy}. Such partition has relatively a small faction (18.3\%) of PVC beats for training. Subsequently, Partition-2 generalizes Partition-1 by considering all records with indices in the range 100--124 for training, and those with indices in the range 200--234 for testing, without paying attention to occurence of PVC beats \cite{jiang}. Though such partitioning improved numerical balance between training and test sets of normal class, PVC class still remain biased, as in the Partition-1. Finally, Partition-3 possesses the property that training and test sets enjoy approximately equal representation from the rival classes of beats \cite{chazal}. Generally, proposers of a specific algorithm tend to pick a partition that maximizes the algorithmic performance. In this vein, we observed high performance of our classification algorithm for a certain evenly split Partition-4, while adhering to the desired numerical balance (Table \ref{data_partition}). We dub as Proposal-1 the proposal to use our algorithm on Partition-4. However, comparison among algorithms in terms of peak performance observed for a hand-picked partition enjoys limited fairness and may not correlate well with user experience.
\subsection{Randomized Partitioning with Even Split} In response, we advance another Proposal-2, wherein performance is averaged over admissible evenly split partitions. In particular, 22 of 44 subjects' data are randomly chosen for training, and the remaining subjects' data for testing. Further, admissible partitions maintain the numerical balance that 45\%-55\% of the total PVC beats belong to the test set. Here, as the total number of admissible partitions is extremely large ($>10^{11}$), we adopted Monte Carlo cross validation (MCCV) approach, wherein the performance is averaged over multiple (100, in our case) random partitions. Compared to Proposal-1, the randomization in Proposal-2 more satisfactorily accounts for the unseen patient data encountered in practice.
\subsection{Randomized Partitioning with Training Set Larger than Test Set} In our home-based PVC monitoring context, one possesses voluminous historical data, and a few potential subjects to cater. Consequently, partitioning the present database such that the training set is larger than the test set appears more realistic compared to the even split seen in Proposal-1 and Proposal-2. Accordingly, we modify Proposal-2 such that each partition under consideration has 40 subjects for training and 4 for testing, and call the new Proposal-3. In Proposal-3, we also update the numerical balance between PVC and normal beats so that only those partitions, where the test set accounts for 10\% -- 20\% of the total number of PVC beats, are considered. As admissible partitions still number a large 42,294, we adopt the MCCV approach over 100 randomly chosen partitions as earlier. Of course, historical (training) data should in practice be given even more weightage over test data in view of the overwhelming preponderance of the former. However, we settle for the above split in view of the limited size of the dataset at hand.
\subsection{Recommendation}
In summary, Proposal-1 represents the peak performance, which is overly optimistic and should not be used for practical guidance. Proposal-2 provides average performance, which is more satisfactory than the peak performance in certain sense, and helps highlight the significant gap between the two. Yet, the even split in Proposal-2 does not reflect the preponderance of historical data, and hence is too conservative to guide practical design. We recommend performance figures corresponding to Proposal-3, incorporating both a more realistic split and randomization, as a (slightly conservative) design guide.
\section{Experimental Setup} \label{sec:expt}
At this point, we conduct simulation experiments to demonstrate the efficacy of the proposed system. First we describe the experimental setup.
\subsection{Preprocessing}
Recall that the adopted MIT/BIH Arrhythmia database consists of 30-minute excerpts of two channel ambulatory ECG recordings of 48 subjects \cite{physionet}. Each channel collects 360 samples per second with a dynamic range of 10 mV peak-to-peak, and digitized to 11-bit words. Further, each beat is annotated per accepted clinical practice. For our experiments, we used the modified limb lead II (MLII) channel only.
On each record, we performed the following steps. First, the baseline wander was removed using two median filters of respective window sizes 200ms and 600ms in a sequential manner \cite{chazal}. Next the annotated R-peak location in each beat was noted, and 150 samples before, 150 samples after and the R-peak sample were collected in a vector of length 301 \cite{fira1}. Such a signal vector included most of the information contained in one heart cycle. Currently, we considered only PVC and normal beats (Figure \ref{signal}b). Although signal vectors chosen in this manner sometimes overlapped, individual beats still preserved morphological information essential for clinical diagnosis. These signal vectors were used for training dictionaries, and will be called beats from now on for the sake of simplicity.
\input{Fig_DictAtoms}
\subsection{Dictionary Size}
In the proposed dictionary-based classification/compression approach, we trained an overcomplete dictionary for each of normal and PVC classes using K-SVD algorithm. In this regard, the dictionary size assumed importance, as (i) smaller size required less computation, and (ii) larger column size led to sparser representation, both of which properties are desirable. However, sparsity saturates with increasing column size, and has negligible effect on classification performance beyond certain threshold \cite{HC14}. Accordingly, we seek to choose the smallest dictionary that provides acceptable level of sparsity. Empirically, ``good" overcomplete dictionaries have been shown to possess a ratio of column to row size between approximately 2 and 5 \cite{ksvd}. Fortunately, we achieved satisfactory classification and compression performance, even while operating at the lower limit of the said ratio range. In particular, recalling that the row size equals the signal vector length of 301, we made use of dictionaries of size 301$\times$600.
\input{Fig_ROC_Proposed}
\section{Experimental Results} \label{sec:results}
In this section, we present experimental results, and performance analysis for the classification subsystem and the compression subsystem separately, as well as for the overall system. To this end, we made use of MIT/BIH Arrhythmia database \cite{physionet}, adopted patient specific partitioning, evaluated the performance of our dictionary based method according to Proposal-1, Proposal-2 and Proposal-3, and compared with the performance of known algorithms, when relevant. For normal and PVC classes, separate dictionaries were obtained based on the training set, while the performance was evaluated using the test set. See Figure \ref{atoms} for a typical normal beat, a typical PVC beat as well as the three most frequently used atoms of each dictionary. Notice the differing beat morphologies, and how those are captured by the depicted dictionary atoms. For simulations, we used MATLAB v.2014b on a desktop computer with an Intel core i7 3.4 GHz 64-bit processor with 16 GB of memory, and required approximately 0.6 milliseconds to complete both classification and compression, which is several orders faster than real time, and indicates suitability of our system for practical deployment. Now we turn to reporting our results, beginning with classification performance.
\subsection{Classification Performance}
As mentioned earlier, classification performance depends on the reconstruction fidelity target $PRD^{class}$, internal to classification subsystem, which we choose first.
\subsubsection{Key tradeoffs and choice of $PRD^{class}$}
To this end, we studied the relationship among sensitivity, specificity and $PRD^{class}$ of the classifier for Proposal-3. Specifically, we plotted in Figure \ref{classifier_perf}a the tradeoff between specificity and $PRD^{class}$ at various sensitivity levels. At the sensitivity (reliability) target of 99\%, we observed specificity to be maximized at around $PRD^{class} = 9\%$. To appreciate the phenomenon from a different perspective, we plotted in Figure \ref{classifier_perf}b various ROC ($Se$ versus $1-Sp$) curves. There we found that the optimal ROC curve, obtained by varying $PRD^{class}$, is well approximated by the ROC curve at the fixed value $PRD^{class} = 9\%$. Accordingly, we aimed at achieving a target $PRD^{class}$ of $9\%$.
As anticipated in Sec. \ref{sec:soln_class}, we observed in Figure \ref{classifier_perf}a that specificity indeed exhibits steep rise, near constancy (plateau) and steep fall as $PRD^{class}$ increases while keeping sensitivity levels fixed. Equivalently, in Figure \ref{classifier_perf}b, significantly lower (5\%) and higher (30\%) values of $PRD^{class}$ compared to the target $9\%$ lead to poor approximation of the optimal ROC. As an interesting aside, we noticed in Figure \ref{classifier_perf}a that the plateau region shrinks with increasing sensitivity. Further, recall that various levels of sensitivity $Se \in [0,1]$, and hence various points on the ROC, are obtained by varying a threshold $\tau$ on the sparsity ratio (see Sec. \ref{sec:soln_class}). Plotting $\tau$ versus $Se$ in Figure \ref{classifier_perf}c, we noticed that the range of $\tau$ shrinks, as the $PRD^{class}$ increases.
\input{Table_comp}
\subsubsection{Performance statistics}
Now, operating at $PRD^{class}$ = 9\%, we compared the performance of the proposed classifier with various algorithms that adopted patient specific evaluation scheme. Specifically, we report in Table \ref{comparison} sensitivity and specificity of existing classifiers and the proposed classifier along with specific information on training data. Recall that we set a sensitivity target of 99\% for our proposals. Now, comparing the peak performance, our Proposal-1 performs better than most of the reported algorithms. However, Proposal-1 represents an overly optimistic performance specific to Partition-4, and may not capture the performance variation due to randomly chosen partitioning. As a remedy, we reported the performance of our Proposal-2 and Proposal-3, where uncertainty is handled more realistically. Specifically, operating at the target sensitivity of 99\%, we reported the mean and standard deviation of specificity over 100 randomly chosen training and test sets. Subject to evenly split training and testing sets, the mean specificity obtained in Proposal-2 improves upon the peak specificity achieved in Proposal-1 in terms of fairness, although the former is significantly lower as expected. However, the notion of even split diverges from the practical situation, where significantly more data are available for training than testing. Accordingly, we recommend Proposal-3 that incorporates a realistic division with larger proportion of training data, as well as randomization. In this case, desirably, the mean specificity is higher, and the standard deviation is lower.
\input{Fig_ROC_AAMI}
So far, we furnished in Table \ref{comparison} pairs of sensitivity and specificity at the operating point of various algorithms. However, this information does not allow us to compare between those algorithms. Consequently, we used ROC curves to indicate the performance of our classifier across admissible sensitivity and specificity values (Figure \ref{roc_all}). In particular, we plotted ROC curves for Proposal-1, Proposal-2 and Proposal-3 (recommended). Alongside, ROC curves of our classifiers evaluated on Partition-1, Partition-2, Partition-3 and Partition-4 (Proposal-1) are also plotted. As ROCs of existing algorithms remain unavailable, we could only locate their operating points on the same plot. Encouragingly, Proposal-3 offers significant improvement over Proposal-2 as well as the peak performances of majority of reported results evaluated on hand-picked sets.
\subsection{Compression Performance}
Recall from Sec \ref{sec:soln_compr} that compression performance is determined by the number of non-zero elements in the dictionary coefficients, which is in turn dictated by the intermediate reconstruction fidelity $PRD^{int}$. We now choose $PRD^{int}$ that maximizes compression ratio while maintaining desired $PRD^{compr}$.
\input{Fig_Compression_New}
\subsubsection{Key tradeoffs and choice of $PRD^{int}$}
To this end, we first considered Proposal-3, and plotted $PRD^{compr}$ versus compression ratio for various $PRD^{int}$ values for PVC beats (Figure \ref{cr}a). As mentioned earlier, for a small $PRD^{int}$, the number of non-zero elements of dictionary coefficients is large. In this setting, for a small increase in quantization step size $\Delta$, which produces a small increment in compression ratio $\beta_V$, quantization error from all those coefficients accumulate to result in a steep increment in $PRD^{compr}$. In contrast, for a large $PRD^{int}$, non-zero coefficients are few, and hence a similar increase in $\Delta$, producing a similarly small increment in $\beta_V$, now allows accumulation of quantization error from relatively few coefficients, resulting in only a gradual increment in $PRD^{compr}$. So, to plot the optimal curve of $PRD^{compr}$ versus compression ratio, we took the envelop of $PRD^{compr}$ versus compression ratio curves for various $PRD^{int}$ values. At this point, recall from Table \ref{PRD} that to preserve the diagnostic integrity of the ECG signal we should operate at at least $PRD^{compr}= 9\%$, which is indicated by the dotted line. For the choice $PRD^{compr} = 9\%$, we observed that $PRD^{int} \approx 8.8\%$ maximizes the compression ratio. The above steps were then repeated for normal beats, and optimal $PRD^{int} \approx 8.8\%$ was again observed. At this point, optimal $PRD^{compr}$ versus compression ratio curves for both PVC and normal beats were presented on the same plots in Figure \ref{cr}b. Those curves are similar with PVC beats allowing slightly higher compression ratio for any $PRD^{compr}$.
\input{compTable}
\subsubsection{Performance statistics}
Compression performance too depends on the partitioning between training and test data. To remove such dependency, we again adopted MCCV approach to evaluate our compression algorithm and reported the performance statistics for Proposal-1, Proposal-2 and Proposal-3 (Table \ref{table:CR}). Specifically, operating at $PRD^{compr}= 9\%$, we reported the mean and standard deviation of $\beta_N$ and $\beta_V$, compression ratios corresponding to normal beats and PVC beats, respectively. Not unexpectedly, with larger training data (Proposal-3), mean compression performance increased. Interestingly, standard deviation of compression performance also increased.
Here, unlike in the case of classification, the proposed compression technique cannot fairly be compared with state of the art algorithms. To appreciate this, note that certain algorithms achieve high compression ratio for a signal consisting of several beats by stacking such beats before compressing \cite{tai}. In contrast, we avoid stacking to prevent delays. Further, specific fixed partitioning is sometimes chosen so as to maximize reported compression ratio \cite{fira1, adamo}. As mentioned earlier, such unrandomized results cannot be used as guides for practical system design.
\subsection{System Performance}
We now present the overall system performance in terms of savings in bandwidth cost. Operating at the target reliability of $Se = 99\%$ and reconstruction fidelity of $PRD^{compr} = 9\%$, the proposed beat-trio communication system would achieve about 57.6\% savings of original bandwidth if classification alone was used. Using compression alone, bandwidth savings is increased to 97.99\%, while ignoring the communication overheads. Using both classification and compression, the proposed method achieved 99.15\% saving in bandwidth usage, which translates to a proportionate savings in the operating cost.
\input{Fig_Comparison}
Now, let us revisit the representative scenario presented in Section \ref{sec:motivation}, and recall that conventional telecardiology costs about US\$ 2.7 for 10-hour ECG monitoring. Against this reference cost, in Figure \ref{comp} we graphically depicted the performance of our system as well as other reported algorithms in the telecardiological context. Specifically, we located various systems in a reliability versus bandwidth/cost plane. In $y$-axis, we plot the complement of reliability ($1-Se$), i.e., the number of PVCs undetected per one hundred beats, and in $x$-axis, the bandwidth usage (bottom) as well as the cost (top) for ten-hour monitoring. Now, adopting beat-trio transmission, and assuming a PVC prevalence rate $\rho$ = 10\%, we plotted reliability versus the bandwidth cost for various rival algorithms, and our Proposal-1, Proposal-2 and Proposal-3, without as well as with compression. Employing only classification, our recommended proposal required only 42.4\% of bandwidth. Notice that a number of reported classifiers did not perform close to the reliability target of $Se=99\%$, i.e., one undetected PVC in one hundred as indicated by horizontal dashed line. The nearest in this respect, the classifier proposed by Chazal {\em et al.} \cite{chazal}, requires 36.4\% of the reference bandwidth, while missing about six PVCs in one hundred beats, i.e., operating at a rate six-fold higher than the target. Using only compression, our proposal reduced the bandwidth requirement to only 2\% of the original bandwidth. However, such a scheme would burden the medical professional with processing entire record for diagnosis. In comparison, the proposed classifier employing both classification and compression would not only reduce the bandwidth requirement but also assist medical professionals by localizing potential anomalies. Specifically, our system would use only 0.85\% of the original bandwidth, achieving additional 98\% and 57.5\% savings over the bandwidth required for classification alone and compression alone, respectively. This would bring down the operating cost to US\textcentoldstyle~2.3. At this rate, the healthcare expenses of the household, mentioned in Section \ref{sec:motivation}, would be reduced to an affordable 0.46\% of the household income from the original 54\%. We believe that a drastic cost reduction of this scale should enable the targeted BOP communities to opt for continuous monitoring service without severe economic burden.
\section{Discussion} \label{sec:disc}
We conclude by summarizing our contributions, remarking on the anticipated user experience, and reflecting on broader impact of our work.
\subsection{Summary}
In this paper, we presented an ultra-low-cost POC service for PVC monitoring that ensures high accuracy. In particular, we proposed a dictionary-based technique that achieves high-sensitivity classification and high-fidelity compression. We demonstrated the efficacy of our method using Monte Carlo cross validation on the MIT/BIH arrhythmia database \cite{MCCV, physionet}. In particular, the three-way tradeoff between bandwidth, reliability and reconstruction fidelity was characterized. With a reliability target of at most one undetected PVC in one hundred beats, and a reconstruction fidelity of 9\% level of $PRD$, we achieved about forty-fold savings in bandwidth and the associated cost. Our service would cost only US\textcentoldstyle~2.3 for ten-hour monitoring, which, we believe, should be attractive to the economically marginalized.
\subsection{User Experience}
While using our service, the experience of users (both subjects and medical professionals) is anticipated to remain essentially the same as that associated with conventional telecardiology. Specifically, at the subject end, the same transducers are still used to collect the ECG signals from the patient. From the medical professionals' perspective, the inference has to be made from the electronic records at essentially the same quality ($PRD \le 9\%$, from Table \ref{PRD} \cite{wdd}) as the gold (quality) standard of unprocessed signals. In fact, the time and effort required of the medical professional are anticipated to be less than that in the traditional situation, as the proposed method automatically identifies PVCs and presents only delimited anomalous beats. In a nutshell, subjects familiar with convectional telecardiology would require no additional training, whereas medical professionals would only need to focus on the presented beats (beat-trios), and ignore blank spaces, which would just indicate normal (uninformative) beats.
\subsection{Broader Impact}
Monitoring of PVCs is clinically significant in broader scenarios than considered so far. Specifically, high PVC burden could presage adverse heart conditions even in individuals without prior structural heart disease \cite{ng}. In such contexts, our technique with slight modifications could facilitate preventive care. Further, apart from PVCs, the proposed dictionary-based method could be extended to other anomalous indicators such as supraventricular arrhythmias and atrial fibrillation \cite{page}. In addition, incorporating medical professionals' feedback and adaptively learning personalized dictionaries could potentially improve both classification and compression performance levels \cite{mairal}.
\begin{table*}[] \renewcommand{1.2}{1.2} \centering \resizebox{160mm}{!}{ \begin{tabular}{cC{6.5cm}C{6.5cm}C{2.5cm}C{2.5cm}} \hline \multirow{2}{*}{\begin{tabular}[c]{@{}c@{}} Partition \\ index \end{tabular}} & \multicolumn{2}{c}{Subject ID in MIT/BIH arrhythmia database} & Number of normal beats & Number of PVC beats \\ \cline{2-3}
& training set & test set & \begin{tabular}[c]{@{}c@{}}training (\%)\\ testing(\%)\end{tabular} & \begin{tabular}[c]{@{}c@{}}training (\%)\\ testing(\%)\end{tabular} \\ \hline \hline
Partition-1 & 100, 105, 106, 108, 109, 111, 114, 116, 118, 119, 121, 123 and 124 & 200, 201, 202, 203, 205, 207, 208, 209, 210, 213, 124, 215, 219, 221, 223, 228, 230, 231, 233 and 234 & \begin{tabular}[c]{@{}c@{}}25360 (37\%) \\ 43130 (63\%)\end{tabular} & \begin{tabular}[c]{@{}c@{}}1281 (18.3\%) \\ 5727(81.7\%)\end{tabular} \\
Partition-2 & 100, 101, 103, 105, 106, 108, 109, 111, 112, 113, 114, 115, 116, 118, 119, 121, 122, 123 and 124 & 200, 201, 202, 203, 205, 207, 208, 209, 210, 212, 213, 214, 215, 219, 220, 221, 222, 223, 228, 230, 231, 232, 233 and 234 & \begin{tabular}[c]{@{}c@{}}39582 (43.9\%) \\ 50499 (56.1\%) \end{tabular} & \begin{tabular}[c]{@{}c@{}}1281 (18.3\%) \\ 5727 (81.7\%) \end{tabular} \\
Partition-3 & 101, 106, 108, 109, 112, 114, 115, 116, 118, 119, 122, 124, 201, 203, 205, 207, 208, 209, 215, 220, 223 and 230 & 100, 103, 105, 111, 113, 117, 121, 123, 200, 202, 210, 212, 213, 214, 219, 221, 222, 228, 231, 232, 233 and 234 & \begin{tabular}[c]{@{}c@{}} 45798 (50.8\%) \\ 44283 (49.2\%) \end{tabular} & \begin{tabular}[c]{@{}c@{}}3788 (54\%) \\ 3220 (46\%)\end{tabular} \\
Partition-4 & 105, 106, 108, 109, 111, 116, 118, 124, 200, 201, 202, 203, 205, 207, 209, 210, 212, 214, 215, 223, 228 and 232 & 100, 101, 103, 112, 113, 114, 115, 117, 119, 121, 122, 123, 208, 213, 219, 220, 221, 222, 230, 231, 233 and 234 & \begin{tabular}[c]{@{}c@{}}45575 (50.6\%) \\ 44506 (49.4\%) \end{tabular} & \begin{tabular}[c]{@{}c@{}} 4008 (57.2\%) \\ 3000 (42.8\%) \end{tabular} \\ \hline
\end{tabular}} \caption{Various dataset partitions.} \label{data_partition} \end{table*}
\section*{Acknowledgment}
This work was partially supported by the Department of Electronics and Information Technology (DeitY), Govt. of India, under the Cyber Physical Systems Innovation Project: 13(6)/2010- CC\&BT.
\end{document} |
\begin{document}
\title{Bounded Homotopy Path Approach to Find the Solution of Linear Complementarity Problems} \author{ A. Dutta$^{a, 1}$, A. K. Das$^{b, 2}$, R. Jana$^{b, 3}$\\ \emph{\small $^{a}$Department of Mathematics, Jadavpur University, Kolkata, 700 032, India}\\ \emph{\small $^{b}$SQC \& OR Unit, Indian Statistical Institute, Kolkata, 700 108, India}\\ \emph{\small $^{1}$Email: aritradutta001@gmail.com}\\ \emph{\small $^{2}$Email: akdas@isical.ac.in}\\ \emph{\small $^{3}$Email: rwitamjanaju@gmail.com} \\
} \date{} \maketitle
\date{} \maketitle \begin{abstract}
\noindent In this article, we introduce a new homotopy function to trace the trajectory by applying modified homotopy continuation method for finding the solution of the linear complementarity problem. Earlier several authors attempted to propose homotopy functions based on original problems. We propose the homotopy function based on the Karush-Kuhn-Tucker condition of the corresponding quadratic programming problem. The proposed approach extends the processability of the larger class of linear complementarity problem and overcomes the limitations of other existing homotopy approaches. We show that the homotopy path approaching the solution is smooth and bounded with positive tangent direction of the homotopy path. Various classes of numerical examples are illustrated to show the effectiveness of the proposed algorithm and the superiority of the algorithm among other existing iterative methods.\\
\noindent{\bf Keywords:} Linear complementarity problem, homotopy method, interior point method, strictly feasible point. \\
\noindent{\bf AMS subject classifications:} 90C33, 15A39, 15B99, 14F35. \end{abstract}
\footnotetext[1] {Corresponding author} \footnotetext[2] {The author R.Jana presently working in an integrated steel plant of India}
\section{Introduction} Eaves and Saigal \cite{eaves1972homotopies} formed an important class of globally convergent methods for solving systems of non-linear equations, which is known as homotopy method. Such methods have been used to constructively prove the existence of solutions to many economic and engineering problems. Let $X,Y$ be two topologocal spaces and $f, g:X \to Y$ be continuous maps. A homotopy from $f$ to $g$ is a continuous function $H:X \times [0,1] $$ \to Y$ satisfying $H(x,0) = f(x),$ $H(x,1) = g(x) \ \forall x \in X.$ If such a homotopy exists, then $f$ is homotopic to $g $ and it is denoted by $f \simeq g.$ Let $f,g:R\to R$ any two continuous, real functions, then $f \simeq g.$ Now we define a function $H:R\times [0,1] \to R $ by $H(x,t)=(1-t)f(x)+tg(x).$ Clearly $H$ is continuous and $H(x,0)=f(x),$ $H(x,1)=g(x).$ Thus $H$ is a homotopy between $f$ and $g.$ Let $X,Y$ be two topological spaces and Map$(X,Y)$ be the set of all continuous maps from $X$ to $Y.$ Homotopy is an equivalence relation on Map$(X,Y).$ \vskip 0.5em
The fundamental idea of the homotopy continuation method is to solve a problem by tracing a certain continuous path that leads to a solution to the problem. Thus, defining a homotopy mapping that yields a finite continuation path plays an essential role in a homotopy continuation method. The homotopy method \cite{watson1989globally} is itself an important class of globally convergent methods. Many homotopy methods are proposed for constructive proof of the existence of solutions to systems of nonlinear equations, nonlinear optimization problems, Brouwer fixed point problems, nonlinear programming, game problem and complementarity problems \cite{watson1989modern}. Chen et al. \cite{chen2016computing} proposed a homotopy algorithm for computing complex eigenpairs of a tensor in a tensor complementarity problem. Han \cite{han2017homotopy} proposed a homotopy method for finding the unique positive solution to a multilinear system with a nonsingular $M$-tensor and a positive right side vector. \vskip 0.5em The linear complementarity problem is well studied in the literature on mathematical programming and arises in a number of applications in operations research, control theory, mathematical economics, geometry and engineering. For recent works on this problem and applications see \cite{das2017finiteness}, \cite{article12}, \cite{article11} and \cite{article03} and references therein. In complementarity theory several matrix classes are considered due to the study of theoretical properties, applications and its solution methods. For details see \cite{jana2019hidden}, \cite{jana2021more}, \cite{article1}, \cite{mohan2001more}, \cite{neogy2013weak} and \cite{neogy2005almost} and references cited therein. The problem of computing the value vector and optimal stationary strategies for structured stochastic games is formulated as a linear complementary problem for discounted and undiscounded zero-sum games. For details see \cite{mondal2016discounted}, \cite{neogy2008mixture} and \cite{neogy2005linear}. The complementarity problem establishes an important connections with multiobjective programming problem for KKT point and the solution point \cite{article78}. The complementarity problems are considered with respect to principal pivot transforms and pivotal method to its solution point of view. For details see \cite{das2016properties}, \cite{neogy2012generalized} and \cite{neogy2005principal}.
We are interested in solving the complementarity problem, mainly the linear complementarity problem. The linear complementarity problem is identified as an important mathematical programming problem and provides a unifying framework for several optimization problems like linear programming, linear fractional programming, convex quadratic programming and the bimatrix game problem. The linear complementarity problem arising from a free boundary problem can be
reformulated as a fixed-point equation. Zhang \cite{zhang2021modified} presented a modified modulus-based multigrid method to solve this fixed-point equation. The concept of complementarity is synonymous with the notion of system equilibrium. Among the many facets of research in linear complementarity problems, the area that has received thorough attention in recent years is the development of robust and efficient algorithms for solving various kinds of linear complementarity problems. Kojima et al. showed that the interior point method for linear programming problem was a kind of path-following method. This polynomial time-bound method is widely used to solve LCP$(q, A)$, but some matrices are not processable by this method as well as by Lemke's algorithm. For details see \cite{jana2018processability} Modulus based algorithm is one of the proposed iterative method to solve linear complementarity problem. Van Bokhoven proved that the modulus algorithm works when the matrix involved is a symmetric P-matrix. Kappel et al.\cite{kappel1986iterative} extended van Bokhoven's results by showing that the modulus algorithm can be applied to a class of non-symmetric P-matrices. Schafer\cite{schafer2004modulus} showed the convergency of the modulus algorithm for three subclasses of $P$-matrices. Hadjidimos et al. \cite{hadjidimos2009nonstationary}, \cite{hadjidimos2012iterative} proposed a new method, the scaled extrapolated block modulus algorithm, as well as an improved version of the very recently introduced modulus-based matrix splitting modified AOR iteration method to find the solution of thelinear complementarity problem with $H_+$-matrix. Zheng et al. \cite{zheng2013accelerated},\cite{zheng2014convergence}, \cite{zheng2017relaxation} showed that for the large sparse linear complementarity problem, established a relaxation modulus-based matrix splitting iteration method, a class of accelerated modulus-based matrix splitting iteration methods by reformulating it as a general implicit fixed-point equation, which covers the known modulus-based matrix splitting iteration methods and presented the convergence conditions when the matrix involved is either a positive definite matrix or an $H_+$-matrix. Dai et al.\cite{dai2019preconditioned} proposed a preconditioned two-step modulus-based matrix splitting iteration method for linear complementarity problems associated with an $M$-matrix. For further details see \cite{bai1999convergence}, \cite{cui2021relaxation}, \cite{dong2009modified}, \cite{liu2016general}, \cite{chen2016computing}, \cite{article3} and \cite{jana2018semimonotone}.
In the literature it was proved that the homotopy method converges globally to the solution of LCP$(q, A),$ where $A$ is a positive semidefinite matrix \cite{yu2006combined}, a $P$-matrix \cite{xuuu}, an $N$-matrix \cite{N} or a $P_*$-matrix \cite{Wang} with respect to different type of homotopy functions. Han\cite{han2017homotopy}, \cite{han2019continuation} introduced a Kojima–Megiddo–Mizuno type continuation method for solving tensor complementarity problems. He showed that there exists a bounded continuation trajectory when the tensor is strictly semi-positive and any limit point tracing the trajectory gives a solution of the tensor complementarity problem. Moreover, when the tensor is strong strictly semi-positive, tracing the trajectory will converge to the unique solution. In this paper, we attempt to introduce another homotopy function and condition for global convergence of the homotopy method to solve LCP$(q, A),$ where $A$ belongs to various matrix classes. \vskip 0.5em The paper is organized as follows. Section 2 presents some basic notations and results. In section 3, we propose a new homotopy function to find the solution of LCP$(q, A)$. We construct a smooth and bounded homotopy path under some conditions to find the solution of the linear complementarity problem as the homotopy parameter $\lambda$ tends to $0$. We prove an if and only if condition to get the solution of LCP$(q, A)$ from the solution of the homotopy equation. We also find the sign of the positive tangent direction of the homotopy path. We use a modified interior-point bounded homotopy path algorithm for solving the linear complementarity problem in section 4. Finally, in section 4, we consider various matrix classes namely, PSD, $N$, almost $C_0,$ singular $Q_0$, $Q$, ${E_0}^s$, almost $\bar{N}$-matrix, $N_0$-matrix of exact order $2$ and $\bar{N}$-matrix of exact order $2.$ Many of these classes are not processable by Lemke's algorithm, existing homotopy methods and modulus based method. We consider these classes to show the effectiveness of the homotopy function.
\section{Preliminaries} \noindent We denote the $n$ dimensional real space by $R^n$ where $R^n_+$ and $R^{n}_{++}$ denote the nonnegative and positive orthant of $R^n.$ We consider vectors and matrices with real entries. Any vector $x\in R^{n}$ is a column vector and $x^{t}$ denotes the row transpose of $x.$ $e$ denotes the vector of all $1.$ If $A$ is a matrix of order $n,$ $\alpha \subseteq \{1, 2, \cdots, n\}$ and $\bar{\alpha} \subseteq \{1, 2, \cdots, n\} \setminus \alpha$ then $A_{\alpha \bar{\alpha}}$ denotes the submatrix of $A$ consisting of only the rows and columns of $A$ whose indices are in $\alpha$ and $\bar{\alpha}$ respectively. $A_{\alpha \alpha}$ is called a principal submatrix of A and det$(A_{\alpha \alpha})$ is called a principal minor of $A.$ We define $\mathcal{F}=\{x\in R^n:x>0,Ax+q>0\}, \ \mathcal{\bar{F}}=\{x \in R^n:x\geq 0, Ax+q \geq 0\}, \mathcal{F}_1=\mathcal{F} \times R_{++}^n \times R_{++}^n$ and $\mathcal{\bar{F}}_1=\mathcal{\bar{F}} \times R_{+}^n \times R_{+}^n.$ $\partial{\mathcal{F}_1}$ denotes the boundary of $\bar{\mathcal{F}_1}.$
The linear complementarity problem \cite{neogy2005principal} is defined as follows:
Given square matrix $A\in R^{n\times n}$ and a vector $\,q\,\in\,R^{n},\,$ the linear complementarity problem is to find $w \in R^n$ and $x \in R^n$ such that
\begin{equation}\label{1} w - Ax = q, w \geq 0, \, x \geq 0, \end{equation} \begin{equation} \label{2} x^tw = 0. \end{equation}
This problem is denoted as LCP$(q, A).$ Several applications of linear complementarity problems are reported in operations research \cite{pang1995complementarity}, multiple objective programming problems \cite{kostreva1993linear}, mathematical economics and engineering. For details see \cite{ferris1997engineering}, \cite{mohan2001more}, \cite{neogy2006some}, \cite{jana2019hidden} and \cite{jana2021more}.
A matrix $A\in R^{n\times n}$ is said to be a/an \\
\noindent $-$ {\it positive semidefinite} (PSD) matrix if $x^{t}Ax\geq 0,\;\forall\;x\in R^{n}.$ \\ \noindent $-$ {\it $P_0(P)$}-matrix if all its principal minors are nonnegative(positive).\\ \noindent $-$ {\it $N$}-matrix if all its principal minors are negative.\\ \noindent $-$ {\it $P_*$}-matrix if $\exists$ a constant $\tau > 0$ such that for any $x \in R^n,$ $$(1 + \tau)\sum_{i \in I_+(x)} x_i(Mx)_i + \sum_{i \in I_{-}(x)} x_i(Mx)_i \geq 0$$ where $I_{+}(x) = \{i \in N: x_i(Mx)_i > 0\}$ and $I_{-}(x) = \{i \in N: x_i(Mx)_i \leq 0\}.$ \\ $-$ $Z$-matrix if off-diagonal elements are all non-positive and $K\,(K_0)$-matrix if it is a $Z$-matrix as well as $P\,(P_0)$-matrix. ($K$-matrix is also known as $M$-matrix).\\ \noindent $-$ {\it copositive} $(C_{0})$ matrix if $x^{t}Ax\geq 0,\;\forall\;x\geq 0.$ \\
\noindent $-$ {\it almost $C_0$}-matrix if it is copositive of up to order $n-1$ but not of order $n.$ \\ \noindent $-$ {\it $N_0$}-matrix if $\det A_{\alpha \alpha} \leq 0 (< 0) \ \forall\ \alpha \subseteq \{1,2,\cdots,n\}.$\\ \noindent $-$ {\it almost $N0(N)$}-matrix if $\det A_{\alpha \alpha} \leq 0 (< 0) \ \forall\ \alpha \subset \{1,2,\cdots,n\}$ and $\det A > 0$.\\ \noindent $-$ {\it $N_0$-matrix of exact order} $k \, (1 \leq k \leq n)$ if every principal submatrix of order $(n-k)$ is an $N_0$-matrix and every principal minor of order $r,$ $(n-k) < r \leq n$ is positive.\\ \noindent $-$ {\it $\bar{N}$}-matrix \cite{mohan1992} if there exists a sequence $\{A^{(k)}\}$ where $A^{(k)} = [a_{ij}^{(k)}]$ are $N$-matrices such that $a_{ij}^{(k)} \rightarrow a_{ij}$ for all $i, j \in \{1, 2, \cdots n\}.$\\ \noindent $-$ {\it $Q$}-matrix if for every $q\in R^{n},$ LCP$(q,A)$ has a solution. \\ \noindent $-$ {\it $Q_{0}$}-matrix if for any $q\in R^{n},$ (1.1) has a solution implies that LCP$(q, A)$ has a solution. \\ \noindent $-$ {\it ${E_{0}}^s$}-matrix if $x^TAx=0, Ax \geq 0, x\geq 0 \implies A^Tx \leq 0.$\\ \noindent $-$ {\it nondegenerate} matrix if all principal minors of the matrix $A$ are nonzero.\\ For further details about matrix classes see \cite{mohan1992}, \cite{neogy2005principal}, \cite{neogy2005almost}, \cite{dutta2021column}, \cite{dutta2021some}, \cite{neogy2005linear}, \cite{doi:10.1137/040613585}. \vskip 0.5em The basic idea of homotopy methods can be explained as to construct a homotopy from the auxiliary mapping $g$ to the object mapping $p.$ The original problem can be solved by following the homotopy path from the zero set of the auxiliary mapping $g$ to the zero set of the object mapping $p.$ The difficulty of finding a strictly feasible initial point for the interior point algorithm can be avoided by combining the interior point with the homotopy method. Furthermore, the global convergence of the homotopy methods can guarantee the global convergence for the combined homotopy interior point methods. Suppose the given problem is to find a root of the non-linear equation $p(x) = 0$ and suppose $g(x) = 0$ is auxiliary function with an unique solution $x_0.$ Then the homotopy equation can be written as $H(x, \lambda)= \lambda g(x) + (1-\lambda)p(x), \, 0 \leq \lambda \leq 1.$ Then we consider $H(x, \lambda) = 0.$ The value of $\lambda$ will start from $1$ and goes to $0.$ In this way one can find the solution of the given equation $p(x) = 0$ from the solution of $g(x) = 0.$ \vskip 0.5em The key idea to solve LCP$(q,A)$ by the homotopy method is to solve a system of equations of the form $H(x,\lambda)=0,$ where $H:R^n\times [0,1] \to R^n, x\in R^n, \lambda \in [0,1]$ is called homotopy parameter. The homotopy method aims to trace out entire path of equilibria in $H^{-1}=\{(x,\lambda): H(x,\lambda)=0\}$ by varrying both $x$ and $\lambda.$ Now we define a parametric path as a set of functions $(x(s),\lambda(s))\in H^{-1}.$ When we move along the homotopy path, the auxiliary variable $s$ either decreases or increases monotonically. Differentiating $H(x(s),\lambda(s))=0$ with respect to $s$ we get $\frac{\partial H}{\partial x}x'(s)+\frac{\partial H}{\partial \lambda}\lambda'(s)=0,$ where $\frac{\partial H}{\partial x}$ and $\frac{\partial H}{\partial \lambda} $ are $n\times n$ jacobian matrix of $H$ and $n\times 1$ column vector respectively. So this is a system of $n$ differential equations in $n+1$ unknowns ${x_i}'(s) \ \forall \ i$ and $\lambda'(s).$ this system of differential equations has many solutions, which differ by monotone transformation of the auxiliary variable $s.$ \vskip 0.5em Now we state some results which will be required in the next section. \begin{lemma}\cite{cottle2009linear} \label{p01} Let $M$ be a $P_0$-matrix. Then for each vector $z\neq 0$, there exists an index $i$ such that $z_i\neq 0$ and $z_i(Mz)_i \geq 0.$ \end{lemma} \begin{lemma}\cite{cottle2009linear} \label{p02} If $M$ is a $P_0$ matrix, then $M^t$ is also $P_0$. \end{lemma} \begin{lemma} \cite{chow1978finding} \label{main} Let $U \subset R^n$ be an open set and $f :R^n \to R^p$ be smooth. We say $y \in R^p$ is a regular value for $f$ if $\text{Range} \, Df(x) = R^p $ $\forall x \in f^{-1}(y),$ where $Df(x)$ denotes the $n \times p$ matrix of partial derivatives of $f(x).$ \end{lemma} \begin{lemma}\label{par} \cite{Wang} Let $V \subset R^n, U \subset R^m$ be open sets, and let $\phi:V\times U \to R^k$ be a $C^\alpha$ mapping, where $\alpha >\text{max}\{0,m-k\}.$ If $0\in R^k$ is a regular value of $\phi,$ then for almost all $a \in V, 0$ is a regular value of $\phi _ a=\phi(a,.).$ \end{lemma} \begin{lemma}\label{inv} \cite{Wang}
Let $\phi : U \subset R^n \to R^p$ be $C^\alpha$ mapping, where $\alpha >\text{max}\{0,n-p\}.$ Then $\phi^{-1}(0)$ consists of some $(n-p)$ dimensional $C^\alpha$ manifolds. \end{lemma} \begin{lemma}\label{cl} \cite{N} One-dimensional smooth manifold is diffeomorphic to a unit circle or a unit interval. \end{lemma}
\section{Main results} We first discuss some existing homotopy functions.Watson \cite{watson1974variational} illustrated an outline of homotopy approach for complementarity problem. Chow et al. \cite{chow1978finding} developed sufficiently powerful theoretical tools for homotopy methods. In 2006, Yu et al. \cite{yu2006combined} proposed the following homotopy function to solve the LCP$(q, A)$ where $A$ is a \textit{positive semidefinite matrix,} \begin{equation}\label{psdyu} H(w,w^{(0)}, \lambda)=\left[\begin{array}{c} (1-\lambda)[Ax+q-y] + \lambda(x-x^{(0)}) \\ XYe-\lambda e\\ \end{array}\right]=0. \end{equation}
Zhao et al. \cite{N} proposed the following homotopy function in 2010 to solve LCP$(q, A)$ where $A$ is an $N$-matrix, \begin{equation}\label{zhaon} H(w,w^{(0)}, \lambda)=\left[\begin{array}{c} (1-\lambda)[y-Ax-q] + \lambda(x-x^{(0)}) \\ Xy - \lambda X^{(0)}y^{(0)}\\ \end{array}\right]=0. \end{equation}
Later Xu et al. \cite{xuuu} developed another homotopy function for finding the solution of LCP$(q, A)$ where $A$ is a $P$-matrix, \begin{equation}\label{xup} H(w,w^{(0)}, \lambda)=\left[\begin{array}{c} (1 - \lambda)[y-Ax-q] - \lambda(x-x^{(0)}) \\ Xy - \lambda X^{(0)}y^{(0)}\\ \end{array}\right]=0. \end{equation}
Wang et al. \cite{5609642} showed that linear complementarity problem with $P_{*}$-matrix can be solved using the homotopy function \begin{equation}\label{wangp} H(w,w^{(0)}, \lambda)=\left[\begin{array}{c} (1 - \lambda)[Ax+q]- y + \lambda y^{(0)} \\ Xy - \lambda X^{(0)}y^{(0)}\\ \end{array}\right]=0. \end{equation}
We propose a new homotopy function to solve LCP$(q, A)$ based on the KKT condition.\\ \begin{equation} \label{homf} H(y,y^{(0)},\lambda)=\left[\begin{array}{c}
(1-\lambda)[(A+A^t)x+q-z_1-A^tz_2]+\lambda(x-x^{(0)}) \\
Z_1x-\lambda Z_1^{(0)}x^{(0)}\\
Z_2(Ax+q)-\lambda Z_2^{(0)}(Ax^{(0)} + q)\\ \end{array}\right]=0 \end{equation}\\ where $Z_1=\text{diag}(z_1),$ $Z_2=\text{diag}(z_2),$ $Z_1^{(0)}=\text{diag}(z_1^{(0)}),$ $Z_2^{(0)}=\text{diag}(z_2^{(0)}),$ $y=(x,z_1,z_2) \in R_+^n \times R_+^n \times R_+^n,$ $y^{(0)}=(x^{(0)},{z_1}^{(0)},{z_2}^{(0)})\in \mathcal{F}_1,$ and $\lambda \in (0,1].$ We denote $\Gamma_y^{(0)}=\{(y,\lambda)\in R^{3n}\times (0,1]: H(y,y^{(0)},\lambda)=0\} \subset \mathcal{F}_1 \times (0,1]\}.$\\
Here $\lambda$ varies from $1$ to $0,$ and starting from $\lambda =1$ to $\lambda \to 0$ if we get a smooth bounded curve, then we will get a finite solution of the homotopy equation \ref{homf} at $\lambda \to 0.$ At $\lambda \to 1,$ the homotopy equation \ref{homf} gives the solution $(y^{(0)},1),$ and at $\lambda \to 0,$ the homotopy equation \ref{homf} gives the solution of the system of following equations: \begin{center}
$(A+A^t)x+q-z_1-A^tz_2=0$\\
$Z_1x=0$\\
$Z_2(Ax+q)=0$\\ \end{center} where $Z_1=\text{diag}(z_1)$ and $Z_2=\text{diag}(z_2).$
Let $z_{1{I_1}} = 0$ and $x_{I_2}=0,$ where $I_1 \cup I_2=\{n\}.$ Let $z_{2J_1}=0$ and $ (Ax+q)_{J_2}=0,$ where $J_1 \cup J_2=\{n\}.$ If the solution of the homotopy function \ref{homf}, $y=(x,z_1,z_2)$ gives the solution of LCP$(q,A)$ which is $x,$ then $x_{{I_2}^c}\neq0$ $\implies$ ${(Ax+q)_{{I_2}^c}}=0.$ This implies that ${{I_2}^c}\subseteq J_2$ and ${(Ax+q)_{{J_2}^c}}\neq0 \implies x_{{J_2}^c}=0,$ which implies that ${{J_2}^c}\subseteq I_2.$ ${{I_2}^c}=J_2,$ ${{J_2}^c}=I_2$ give the nondegenerate solution of LCP$(q,A)$ and ${{I_2}^c}\subset J_2,$ ${{J_2}^c}\subset I_2$ give the degenerate solution of LCP$(q,A).$ When $I_1 \cap I_2 = \emptyset$ and $ J_1 \cap J_2 = \emptyset,$ it implies $I_1={{I_2}^c}=J_2$ and $J_1={{J_2}^c}=I_2,$ then $x=z_2$ and $Ax+q=z_1$ will give the solution of LCP$(q, A),$ otherwise we get nontrivial solution of LCP$(q, A)$ which is not same as $z_1.$ Therefore the homotopy solution $y$ can not give the LCP solution $x$ when ${{I_2}^c}\nsubseteq J_2$ and ${{J_2}^c}\nsubseteq I_2,$ that is ${{I_2}^c}\subseteq J_1$ and ${{J_2}^c}\subseteq I_1.$
First we show that the smooth curve exists for the homotopy function\ref{homf}.
\begin{theorem}\label{reg}
Let initial point $y^{(0)} \in \mathcal{F}_1.$ Then $0$ is a regular value of the homotopy function $H:R^{3n} \times (0,1] \to R^{3n}$ and the zero point set $H_{y^{(0)}}^{-1}(0)=\{(y,\lambda)\in \mathcal{F}_1:H_{y^{(0)}}(y,\lambda)=0\}$ contains a smooth curve $\Gamma_y^{(0)}$ starting from $(y^{(0)}, 1).$ \end{theorem}
\begin{proof}
The Jacobian matrix of the above homotopy function $H(y, y^{(0)}, \lambda)$ is denoted by $DH(y,y^{(0)},\lambda)$ and we have $DH(y,y^{(0)}, \lambda)=$$\left[\begin{array}{ccc}
\frac{\partial{H(y,y^{(0)},\lambda)}}{\partial{y}} & \frac{\partial{H(y,y^{(0)},\lambda)}}{\partial{y^{(0)}}} & \frac{\partial{H(y,y^{(0)},\lambda)}}{\partial{\lambda}}\\
\end{array}\right].$ For all $y^{(0)} \in \mathcal{F}_1$ and $\lambda \in (0,1],$ we have $\frac{\partial{H(y,y^{(0)},\lambda)}}{\partial{y^{(0)}}}=$$\left[\begin{array}{ccc} -\lambda I & 0 & 0\\
-\lambda Z_1^{(0)} & -\lambda X^{(0)} & 0\\
-\lambda Z_2^{(0)}A & 0 & -\lambda W^{(0)}\\
\end{array}\right],$ where $W^{(0)}=\text{diag}(Ax^{(0)}+q), X^{(0)}=\text{diag}(x^{(0)}),$ $w^{(0)}=Ax^{(0)}+q$ and
$\det(\frac{\partial{H}}{ \partial{y^{(0)}}})$$=(-1)^{3n}\lambda^{3n}\prod_{i=1}^{n} x_i^{(0)}w_i^{(0)}$ $\neq 0$ for $\lambda \in (0,1].$
Thus $DH(y,y^{(0)},\lambda)$ is of full row rank. Therefore, $0$ is a regular value of $H(y,y^{(0)},\lambda)$ by the Lemma \ref{main}. By Lemmata \ref{par} and \ref{inv}, for almost all $y^{(0)} \in \mathcal{F}_1,$ $0$ is a regular value of $H_{y^{(0)}}(y,\lambda)$ and $H_{y^{(0)}}^{-1}(0)$ consists of some smooth curves and $H_{y^{(0)}}(y^{(0)},1)=0.$ Hence there must be a smooth curve $\Gamma_y^{(0)}$ starting from $(y^{(0)},1).$ \end{proof} Hence by implicit function theorem for every $\lambda$ sufficiently close to $1$, the homotopy function has a unique solution $(y,1) $ of \ref{homf}, which is smooth in the parameter $\lambda$, in a neighbourhood of $(y^{(0)},1) $.\\
Now we show that the smooth curve $\Gamma_y^{(0)}$ for the homotopy function \ref{homf} is bounded and converges and establish conditions for global convergence of the homotopy method with the homotopy function \ref{homf}. We show that if the $x$ and $z_2$-components of the point $(x,z_1,z_2,\lambda)$ are bounded, then the homotopy curve $\Gamma_y^{(0)}$ is bounded.
\begin{theorem}\label{bnd}
Let $\mathcal{F}$ be a non-empty set and $A \in R^{n\times n}.$ Assume that there exists a sequence of points $\{u^k\} \subset \Gamma_y^{(0)} \subset \mathcal{F}_1 \times (0,1],$ where $u^k=(x^k,z_1^k,z_2^k, \lambda^k)$ such that $\|x^k\|< \infty \ \text{as} \ k \to \infty$ and $\|z_2^k\|< \infty \ \text{as} \ k \to \infty$ and for a given $y^{(0)} \in \mathcal{F}_1,$ $0$ is a regular value of $H(y,y^{(0)},\lambda).$ Then $\Gamma_y^{(0)}$ is a bounded curve in $\mathcal{F}_1 \times (0,1].$ \end{theorem}
\begin{proof}
Note that $0$ is a regular value of $H(y,y^{(0)},\lambda)$ by Theorem \ref{reg}. Now we assume that $\Gamma_y^{(0)} \subset \mathcal{F}_1 \times (0,1]$ is an unbounded curve. Then there exists a sequence of points $\{u^k\},$ where $u^k=(y^k, \lambda^k) \subset \Gamma_y^{(0)}$ such that $\|(y^k, \lambda^k)\| \to \infty.$ As $(0,1]$ is a bounded set and $x$ component and $z_2$ component of $\Gamma_y^{(0)}$ is bounded, there exists a subsequence of points $\{u^k\}=\{(y^k, \lambda^k)\}=\{x^k,z_1^k,z_2^k, \lambda^k\}$ such that $x^k \to \bar{x},\ {z_2}^k \to \bar{z_2}, \ \lambda^k \to \bar{\lambda} \in [0,1] \ \text{and} \ \|z^k\| \to \infty \ \text{as} \ k \to \infty, \ \text{where} \ z^k=\left[\begin{array}{c} z_1^k\\ z_2^k\\\end{array}\right].$ Since $\Gamma_y^{(0)} \subset H_{y^{(0)}}^{-1}(0),$ we have \begin{equation}\label{zzq} (1-\lambda^k)[(A+A^t)x^k+q-z_1^k-A^tz_2^k]+\lambda^k(x^k-x^{(0)})=0 \end{equation} \begin{equation}\label{yyq} Z_1^kx^k-\lambda^k Z_1^{(0)}x^{(0)}=0 \end{equation} \begin{equation}\label{wwwq} Z_2^k(Ax^k+q)-\lambda^k Z_2^{(0)}(Ax^{(0)}+q)=0 \end{equation} where $Z_1^k=\text{diag}(z_1^k)$ and $Z_2^k=\text{diag}(z_2^k).$ Let
$\bar{\lambda} \in [0,1], \|z_1^k\|=\infty$ and $\|z_2^k\|<\infty$ as $k \to \infty.$ Then $\exists \ i \in \{1,2,\cdots, n\}$ such that $z_{1i}^k \to \infty$ as $k \to \infty.$ Let $I_{1z}=\{i\in\{1,2,\cdots n\} : \lim\limits_{k\to \infty}z_{1i}^k = \infty\}.$ When $\bar{\lambda} \in [0,1),$ for $i\in I_{1z}$ we can get from Equation \ref{zzq}, $(1-\lambda^k)[((A+A^t)x^k)_i+q_i-z_{1i}^k-(A^tz_{2}^k)_i] + \lambda^k(x_i^k-x_i^{(0)})=0$
$\implies (1-\lambda^k)z_{1i}^k=(1-\lambda^k)[((A+A^t)x^k)_i+q_i-(A^tz_{2}^k)_i]+\lambda^k(x_i^k-x_i^{(0)}) \implies z_{1i}^k=[((A+A^t)x^k)_i+q_i-(A^tz_{2}^k)_i]+\frac{\lambda^k}{(1-\lambda^k)}(x_i^k-x_i^{(0)}).$ As $k \to \infty$ right hand side is bounded, but left hand side is unbounded. It contradicts that $\|z_1^k\|=\infty.$ When $\bar{\lambda}=1,$ then from Equation \ref{yyq}, we get, $x_i^k=\frac{\lambda^k z_{1i}^{(0)}x_i^{(0)}}{z_{1i}^k}$ for $i \in I_{1z}.$ As $k \to \infty, x_i^k \to 0.$ Again from Equation \ref{zzq}, we obtain $x_i^{(0)}=\frac{(1-\lambda^k)}{\lambda^k}[((A+A^t)x^k)_i+q_i-z_{1i}^k-(A^tz_2^k)_i]+x_i^k$ for $i \in I_{1z}.$ As $k \to \infty,$ we have $x_i^{(0)}=-\lim\limits_{k\to \infty}\frac{(1-\lambda^k)}{\lambda^k}z_{1i}^k \leq 0.$ It contradicts that $\|z_1^k\|=\infty.$\\ So $\Gamma_y^{(0)}$ is a bounded curve in $\mathcal{F}_1 \times (0,1].$ \end{proof} Now we show the condition to get bounded curve for nonsingular matrix $A$. \begin{corol}\label{222}
Let $\mathcal{F}$ be a non-empty set and $A \in R^{n\times n}$ be a nonsingular matrix. Assume that there exists a sequence of points $\{u^k\} \subset \Gamma_y^{(0)} \subset \mathcal{F}_1 \times (0,1],$ where $u^k=(x^k,z_1^k,z_2^k, \lambda^k)$ such that $\|x^k\|< \infty \ \text{as} \ k \to \infty.$ Further suppose for $\lambda^k \to 1,$ $\|z_2^k\|< \infty \ \text{as} \ k \to \infty.$ Suppose that for a given $y^{(0)} \in \mathcal{F}_1,$ $0$ is a regular value of $H(y,y^{(0)},\lambda).$ Then $\Gamma_y^{(0)}$ is a bounded curve in $\mathcal{F}_1 \times (0,1].$ \end{corol}
\begin{proof}
By theorem \ref{reg}, $0$ is a regular value of $H(y,y^{(0)},\lambda)$. Now we assume that $\Gamma_y^{(0)} \subset \mathcal{F}_1 \times (0,1]$ is an unbounded curve. Then there exists a sequence of points $\{u^k\},$ where $u^k=(y^k, \lambda^k) \subset \Gamma_y^{(0)}$ such that $\|(y^k, \lambda^k)\| \to \infty.$ $(0,1]$ is a bounded set and $x$ component of $\Gamma_y^{(0)}$ is bounded. There exists a subsequence of points $\{u^k\}=\{(y^k, \lambda^k)\}=\{x^k,z_1^k,z_2^k, \lambda^k\}$ such that $x^k \to \bar{x},\ $ and suppose for $\lambda^k \to 1,$ $\|z_2^k\|< \infty \ \text{as} \ k \to \infty.$ Then two cases will arise. \vskip 0.5em
\noindent \textbf{Case 1:} $\bar{\lambda} \in [0,1], \|z_1^k\|<\infty,$ $\|z_2^k\|=\infty.$\\
Let $\|z_2^k\|=\infty.$ Then $\exists \ j \in \{1,2,\cdots n\}$ such that $z_{2j}^k \to \infty$ as $k \to \infty.$ Let $I_{2z}=\{j\in\{1,2,\cdots n\} : \lim\limits_{k\to \infty}z_{2j}^k = \infty\}.$ When $\bar{\lambda} \in [0,1),$ for $j\in I_{2z}$ we can get from Equation \ref{zzq},
$z_{2j}^k=(A^{-t}(A+A^t)x^k)_j+(A^{-t}q)_j-(A^{-t}z_{1}^k)_j+\frac{\lambda^k}{1-\lambda^k}(x_j^k-x_j^{(0)}).$ As $k \to \infty,$ right hand side is bounded, but left hand side is not. This also contradicts that $\|z_2^k\|=\infty.$ So with our assumption for $\lambda^k \to 1,$ $\|z_2^k\|< \infty \ \text{as} \ k \to \infty,$ $\bar{\lambda} \in [0,1],$ the homotopy curve is bounded. \vskip 0.5em
\noindent \textbf{Case 2:} $\bar{\lambda} \in [0,1], \|z_1^k\|=\infty,$ $\|z_2^k\|=\infty.$\\
Let $\|z_1^k\|=\infty, \|z_2^k\|=\infty. $ Then either $\exists \ i \in \{1,2,\cdots n\}$ such that $z_{1i}^k \to \infty$, $z_{2i}^k \to \infty$ as $k \to \infty$ or $\exists \ i,j \in \{1,2,\cdots n\}, i\neq j$ such that $z_{1i}^k \to \infty$ and $z_{2j}^k \to \infty$ as $k \to \infty.$ When $z_{1i}^k \to \infty$, $z_{2i}^k \to \infty$ as $k \to \infty$ and $\bar{\lambda} \in [0,1),$ we have,
$z_{1i}^k+(A^tz_{2}^k)_i=((A+A^t)x^k)_i+q_i+\frac{\lambda^k}{(1-\lambda^k)}(x_i^k-x_i^{(0)}).$ Now as $k \to \infty,$ right hand side is bounded, but left hand side is not, which is impossible. When $\bar{\lambda}=1,$ then our assumption $\|z_2^k\|< \infty \ \text{as} \ k \to \infty$ and the argument of the previous theorem \ref{bnd} contradicts that $z_{1i}^k \to \infty,$ $z_{2i}^k \to \infty$ as $k \to \infty.$ As $k \to \infty,$ when $z_{1i}^k \to \infty,$ $z_{2j}^k \to \infty$ for $i \neq j$ as $k \to \infty$ then considering the $i$th and $j$th component and using same argument similar to the previous theorem \ref{bnd} and case 1, we will get a contradiction.
Thus $\Gamma_y^{(0)}$ is a bounded curve in $\mathcal{F}_1 \times (0,1].$ \end{proof}
Now we show the necessary condition of the homotopy curve $\Gamma_y^{(0)}$ to be bounded. \begin{theorem}\label{001}
Suppose the solution set $\Gamma_y^{(0)}$ of the homotopy function $H(y,y^{(0)},\lambda)=0$ is unbounded. Then there exists $(\xi, \eta, \zeta) \in R_+^{3n}$ such that $e^t \xi =1,$ $\xi^tA\xi \leq 0.$ \end{theorem} \begin{proof}
Assume that the solution set $\Gamma_y^{(0)}$ is unbounded. Then there exists a sequence of points $\{u^k\} \subset \Gamma_y^{(0)} \subset \mathcal{F}_1 \times (0,1],$ where $u^k=(x^k,z_1^k,z_2^k, \lambda^k)$ such that $\lim_{k\to \infty}\lambda^k=\bar{\lambda}$ and either $\|z_2^k\|<\infty$ as $k \to \infty$ with two cases (i) $\lim_{k\to \infty}e^tx^k=\infty$ and (ii) $\lim_{k\to \infty}(1-\lambda^k)e^tx^k= \infty$ or $\lim_{k\to \infty}e^tz_2^k=\infty$ with two cases (i) $\lim_{k\to \infty}e^tx^k=\infty$ and (ii) $\lim_{k\to \infty}(1-\lambda^k)e^tx^k= \infty.$ \\
First we consider that $\|z_2^k\|<\infty$ as $k \to \infty.$\\
Case (i) Let $\lim_{k\to \infty}\frac{x^k}{e^tx^k}=\xi \geq 0 $ and $\lim_{k\to \infty}\frac{z_1^k}{e^tx^k}=\eta \geq 0. $ So it is clear that $e^t\xi=1.$ Then dividing by $e^tx^k$ and taking limit $k \to \infty $ from equations \ref{zzq},\ref{yyq},\ref{wwwq} we get \begin{eqnarray}
(1-\bar{\lambda})[(A+A^t)\xi - \eta]+\bar{\lambda}\xi=0\label{n1}\\
\xi_i \eta_i=0 \ \forall \ i \label{n2}
\end{eqnarray}
From equations \ref{n1} and \ref{n2} we get $\eta=(A+A^t)\xi+\frac{\bar{\lambda}}{(1-\bar{\lambda})}\xi \implies 0=\xi^t\eta=\xi^t[(A+A^t)\xi+\frac{\bar{\lambda}}{(1-\bar{\lambda})}\xi]$ for $\bar{\lambda} \in [0,1).$ This implies that $\xi^t(A+A^t)\xi=-\frac{\bar{\lambda}}{(1-\bar{\lambda})}\xi \leq 0$ i.e. $\xi^tA\xi\leq 0.$ Specifically for $\bar{\lambda}=0,$ $\xi^tA\xi= 0$ and for $\bar{\lambda}\in (0,1),$ $\xi^tA\xi<0.$ For $\bar{\lambda}=1$ $\xi=0,$ contradicts that $e^t\xi=1.$ \\
Case (ii) Let $\lim_{k\to \infty}\frac{(1-\lambda^k)x^k}{(1-\lambda^k)e^tx^k}=\xi'\geq 0.$ Then $e^t\xi'=1.$ Let $\lim_{k\to \infty}\frac{z_1^k}{(1-\lambda^k)e^tx^k}=\eta' \geq 0.$ Then multiplying the equation \ref{zzq} with $(1-\lambda^k)$ and dividing by $(1-\lambda^k)e^tx^k$, multiplying the equation \ref{yyq} with $(1-\lambda^k)$ and dividing by $((1-\lambda^k)e^tx^k)^2$ and multiplying the equation \ref{wwwq} with $(1-\lambda^k)$ dividing by $((1-\lambda^k)e^tx^k)^2$ and taking limit $k \to \infty$, we get \\
\begin{eqnarray}
(1-\bar{\lambda})[(A+A^t)\xi' - (1-\bar{\lambda})\eta']+\bar{\lambda}\xi'=0\label{n11}\\
\xi'_i \eta'_i=0 \ \forall \ i \label{n22}
\end{eqnarray}
Multiplying $(\xi')^t $ in both sides of equation \ref{n11}, we get $(\xi')^tA\xi'\leq 0$ for $\bar{\lambda} \in [0,1).$ Specifically for $\bar{\lambda}=0,$ $(\xi')^tA\xi'= 0$ and for $\bar{\lambda}\in (0,1),$ $(\xi')^tA\xi'<0.$ For $\bar{\lambda}=1,$ $\xi'=0,$ contradicts that $e^t\xi'=1.$ \\
Later we consider that $\lim_{k\to \infty}e^tz_2^k=\infty.$\\
Case (i) Let $\lim_{k\to \infty}\frac{x^k}{e^tx^k}=\xi \geq 0, $ $\lim_{k\to \infty}\frac{z_1^k}{e^tx^k}=\eta \geq 0 $ and $\lim_{k\to \infty}\frac{z_2^k}{e^tx^k}=\zeta \geq 0.$ It is clear that $e^t\xi=1.$ Then dividing by $e^tx^k$ and taking limit $k \to \infty $ from equation \ref{zzq}, dividing by $(e^tx^k)^2$ and taking limit $k \to \infty $ from equation \ref{yyq}, \ref{wwwq}, we get
\begin{eqnarray}
(1-\bar{\lambda})[(A+A^t)\xi-\eta-A^t\zeta]+\bar{\lambda}\xi=0 \label{nn1}\\
\xi_i\eta_i=0 \ \forall \ i \label{nn2}\\
\zeta_i(A\xi)_i=0 \ \forall \ i \label{nn3}
\end{eqnarray}
From equation \ref{nn1} we get $\eta+A^t\zeta=(A+A^t)\xi+\frac{\bar{\lambda}}{1-\bar{\lambda}}\xi$ for $\bar{\lambda} \in [0,1).$ Now multiplying $\xi^t$ in both sides we get $\xi^t(A+A^t)\xi+\frac{\bar{\lambda}}{1-\bar{\lambda}}\xi^t\xi=0.$ Hence $\xi^t(A+A^t)\xi=-\frac{\bar{\lambda}}{1-\bar{\lambda}}\xi^t\xi \leq 0$ for $\bar{\lambda} \in [0,1).$ Specifically for $\bar{\lambda}=0,$ $\xi^tA\xi= 0$ and for $\bar{\lambda}\in (0,1),$ $\xi^tA\xi<0.$ For $\bar{\lambda}=1,$ $\xi=0,$ contradicts that $e^t\xi=1.$\\
Case(ii) Let $\lim_{k\to \infty}\frac{(1-\lambda^k)x^k}{(1-\lambda^k)e^tx^k}=\xi'\geq 0.$ Then $e^t\xi'=1.$ Let $\lim_{k\to \infty}\frac{z_1^k}{(1-\lambda^k)e^tx^k}=\eta' \geq 0$ and $\lim_{k\to \infty}\frac{z_2^k}{(1-\lambda^k)e^tx^k}=\zeta' \geq 0$ Then multiplying the equation \ref{zzq} with $(1-\lambda^k)$ and dividing by $(1-\lambda^k)e^tx^k$, multiplying the equation \ref{yyq} with $(1-\lambda^k)$ and dividing by $((1-\lambda^k)e^tx^k)^2$ and multiplying the equation \ref{wwwq} with $(1-\lambda^k)$ dividing by $((1-\lambda^k)e^tx^k)^2$ and taking limit $k \to \infty,$ we get
\begin{eqnarray}
(1-\bar{\lambda})(A+A^t)\xi'- (1-\bar{\lambda})^2\eta'- (1-\bar{\lambda})^2A^t\zeta'+\bar{\lambda}\xi'=0 \label{nnn1}\\
\xi'_i\eta'_i=0 \ \forall \ i \label{nnn2}\\
\zeta'_i(A\xi')_i=0 \ \forall \ i \label{nnn3}
\end{eqnarray}
Multiplying $(\xi')^t$ in both side of equation \ref{nnn1} we get $(\xi')^t(A+A^t)\xi'- (1-\bar{\lambda})(\xi')^t\eta'- (1-\bar{\lambda})(\xi')^tA^t\zeta'=-\frac{\bar{\lambda}}{(1-\bar{\lambda})}(\xi')^t\xi'\leq 0$ for $\bar{\lambda} \in [0,1).$ Specifically for $\bar{\lambda}=0,$ $(\xi')^tA\xi'= 0$ and for $\bar{\lambda}\in (0,1),$ $(\xi')^tA\xi'<0.$ For $\bar{\lambda}=1, \xi'=0,$ contradicts that $e^t\xi'=1.$
\end{proof}
\begin{remk}\label{00}
Therefore in the neighbouhood of $\bar{\lambda}=1$ the homotopy curve is bounded and for the parameter $\lambda=0,$ $(\xi)^tA\xi= 0$ and for $\lambda\in (0,1),$ $(\xi)^tA\xi<0,$ where $\xi \geq 0,$ $e^t\xi=1.$
\end{remk} \begin{corol}
Suppose $A \in R^{n\times n}$ is a nonsingular matrix and assume that there exists a sequence of points $\{u^k\} \subset \Gamma_y^{(0)} \subset \mathcal{F}_1 \times (0,1],$ where $u^k=(x^k,z_1^k,z_2^k, \lambda^k)$ and $\|x^k\|< \infty \ \text{as} \ k \to \infty.$ For a given $y^{(0)} \in \mathcal{F}_1,$ $0$ is a regular value of $H(y,y^{(0)},\lambda).$ Then $\Gamma_y^{(0)}$ is a bounded curve in $\mathcal{F}_1 \times (0,1].$ \end{corol}
\begin{theorem}\label{01001}
Let $A\in R^{n\times n}$ and the set $\mathcal{F}_1$ be nonempty. For a given $y^{(0)} \in \mathcal{F}_1,$ $0$ is a regular value of $H(y,y^{(0)},\lambda).$ Then the homotopy path $\Gamma_y^{(0)} \subset \mathcal{F}_1 \times (0,1]$ is bounded. \end{theorem} \begin{proof} Suppose $A\in R^{n\times n}$ is a matrix and there exists a sequence of points $\{u^k\} \subset \Gamma_y^{(0)} \subset \mathcal{F}_1 \times (0,1],$ where $u^k=(x^k,z_1^k,z_2^k, \lambda^k).$ Hence by the definition of $\mathcal{F}_1$ $x^k,z_1^k,z_2^k,\\Ax^k+q>0.$ From remark \ref{00} the homotopy curve is bounded in the neighbourhood of $\lambda=1.$ Assume that the homotopy curve $\Gamma_y^{(0)} \subset \mathcal{F}_1 \times (0,1)$ is unbounded. Then from theorem \ref{001} , $(\xi)^tA\xi<0$ for $\lambda\in (0,1).$ But $Ax^k+q>0$ implies that $A\xi \geq 0,$ where $\xi=$$\lim_{k\to \infty}\frac{x^k}{e^tx^k} \geq 0,$ when $\lim_{k\to \infty}{e^tx^k}=\infty$ or $\xi=$$\lim_{k\to \infty}\frac{(1-\lambda^k)x^k}{(1-\lambda^k)e^tx^k}\geq 0,$ when $\lim_{k\to \infty}{(1-\lambda^k)e^tx^k}=\infty.$ Hence $\xi, A\xi \geq 0$ imply that $\xi^tA\xi\geq 0$ for $\lambda\in(0,1)$, which contradicts that the homotopy path is unbounded for $\lambda\in(0,1).$ Therefore the homotopy curve $\Gamma_y^{(0)} \subset \mathcal{F}_1 \times (0,1]$ is bounded. \end{proof} Hence it is proved that the homotopy curve $\Gamma_y^{(0)}$ is bounded for any matrix $A$. \begin{theorem}
For $y^{(0)}=(x^{(0)},z_1^{(0)},z_2^{(0)})\in \mathcal{F}_1,$ the homotopy equation finds a bounded smooth curve $\Gamma_y^{(0)} \subset \mathcal{F}_1 \times (0,1]$ which starts from $(y^{(0)},1)$ and approaches the hyperplane at $\lambda =0.$ As $\lambda \to 0,$ the limit set $L \times \{0\} \subset \bar{\mathcal{F}}_1 \times \{0\}$ of $\Gamma_y^{(0)}$ is nonempty and every point in $L$ is a solution of the following system:
\begin{equation}\label{sys}
\begin{split}
(A+A^t)x+q-z_1-A^tz_2=0 \\
Z_1x=0 \\
Z_2(Ax+q)=0. \\
\end{split}
\end{equation} \end{theorem} \begin{proof}
Note that $\Gamma_y^{(0)}$ is diffeomorphic to a unit circle or a unit interval $(0,1]$ in view of Lemma \ref{cl}. As $\frac{\partial{H(y,y^{(0)},1)}}{\partial{y^{(0)}}}$ is nonsingular, $\Gamma_y^{(0)}$ is diffeomorphic to a unit interval $(0,1].$ Again $\Gamma_y^{(0)}$ is a bounded smooth curve by the Theorem \ref{01001}. Let $(\bar{y},\bar{\lambda})$ be a limit point of $\Gamma_y^{(0)}.$ We consider four cases:
\begin{description}
\item[Case 1:] $(\bar{y},\bar{\lambda})\in \mathcal{F}_1 \times \{1\}.$
\item[Case 2:] $(\bar{y},\bar{\lambda})\in \partial{\mathcal{F}_1} \times \{1\}.$
\item[Case 3:] $(\bar{y},\bar{\lambda})\in \partial{\mathcal{F}_1} \times (0,1).$
\item[Case 4:] $(\bar{y},\bar{\lambda})\in \bar{\mathcal{F}}_1 \times \{0\}.$
\end{description}
As the equation $H_{y^{(0)}}(y,1)=0$ has only one solution $y^{(0)}\in \mathcal{F}_1, $ the case $1$ is impossible. In case $2$ and $3,$ there exists a subsequence of $(y^k, \lambda^k) \in \Gamma_y^{(0)}$ such that $x_i^k \to 0$ or $(Ax^k+q)_i \to 0$ for $i \subseteq \{1,2,\cdots n\}.$ From the last two equalities of the homotopy function \ref{homf}, we have $z_1^k \to \infty$ or $z_2^k \to \infty.$ Hence it contradicts the boundedness of the homotopy path by the Theorem \ref{01001}. Therefore case $4$ is the only possible option. Hence $\bar{y}=(\bar{x},\bar{z_1},\bar{z_2})$ is a solution of the system $(A+A^t)x+q-z_1-A^tz_2=0, \ Z_1x=0, \ Z_2(Ax+q)=0.$ \end{proof}
\begin{remk}\label{me}
From the homotopy function \ref{homf}, we obtain
$\bar{z}_{1i}\bar{x}_i = 0$ and $\bar{z}_{2i}(A\bar{x}+q)_i=0 \ \forall i\in\{1,2, \cdots n\}.$ Now $\bar{z}_1$ and $\bar{z}_2$ can be decomposed as $\bar{z}_1= \bar{w}-\Delta\bar{w} \geq 0$ and $\bar{z}_2= \bar{x}-\Delta\bar{x} \geq 0,$ where $\bar{w}=A\bar{x}+q.$ It is clear that $\bar{w}_i \bar{x}_i=\Delta\bar{w}_i \bar{x}_i=\Delta\bar{x}_i\bar{w}_i \ \forall i \in \{1,2, \cdots n\} .$ \end{remk}
We demonstrate the condition under which the homotopy functions will give the solution of LCP$(q, A).$ \begin{theorem}
The component $\bar{x}$ of $(\bar{x},\bar{z}_1,\bar{z}_2,0) \in L\times \{0\}$ gives the solution of LCP$(q, A)$ if and only if $\Delta\bar{x}_i \Delta\bar{w}_i=0 $ or $\bar{z}_{1i}+\bar{z}_{2i}>0 \ \forall i.$ \end{theorem}
\begin{proof} Suppose $\bar{x} \geq 0$ and $\bar{w}=A\bar{x}+q \geq 0$ are the solution of LCP$(q, A).$ Then $\bar{x}_i\bar{w}_i=0$ \ $ \forall i.$ This implies that $\bar{x}_i=0$ or $\bar{w}_i=0$ \ $ \forall i.$ We consider the following three cases: \vskip 0.5em \noindent \textbf{Case 1:} For at least one $i \in \{1,2,\cdots n\},$ let $\bar{w}_{i}>0, \bar{x}_{i}=0.$ In view of Remark \ref{me}, this implies that $\Delta\bar{x}_i=0 \implies \Delta\bar{x}_i \Delta\bar{w}_i=0.$ \vskip 0.5em \noindent \textbf{Case 2:} For at least one $i \in \{1,2,\cdots n\},$ let $\bar{x}_{i}>0, \bar{w}_{i}=0.$ In view of \ref{me}, this implies that $\Delta\bar{w}_i=0 \implies \Delta\bar{x}_i \Delta\bar{w}_i=0.$ \vskip 0.5em \noindent \textbf{Case 3:} For at least one $i \in \{1,2,\cdots n\},$ let $\bar{w}_{i}=0, \bar{x}_{i}=0.$ This implies that either $\Delta\bar{w}_i\Delta\bar{x}_i=0$ or $\bar{z}_{1i}+\bar{z}_{2i}>0.$ \vskip 0.5em For the converse part, consider $\Delta\bar{x}_i \Delta\bar{w}_i=0 $ or $\bar{z}_{1i}+\bar{z}_{2i}>0 \ \forall i \in \{1,2,\cdots n\}.$ Let $\forall i \in \{1,2,\cdots n\}, \ \Delta\bar{x}_i \Delta\bar{w}_i=0 $ implies either $\Delta\bar{x}_i=0$ or $\Delta\bar{w}_i=0.$ This implies that $\bar{w}_i \bar{x}_i=0 \ \forall i \in \{1,2,\cdots n\}.$ Therefore $\bar{w}$ and $\bar{x}$ are the solution of the LCP$(q, A).$ Consider $\bar{z}_{1i}+\bar{z}_{2i}>0 \ \forall i \in \{1,2,\cdots n\}.$ Then following three cases will arise. \vskip 0.5em \noindent \textbf{Case 1:} Let $\bar{z}_{1i}>0, \bar{z}_{2i}=0$ for at least one $i \in \{1,2,\cdots n\}.$ This implies that $\bar{x}_i=0$ and $\bar{w}_i \geq 0.$ \vskip 0.5em \noindent \textbf{Case 2:} Let $\bar{z}_{1i}=0, \bar{z}_{2i}>0$ for at least one $i \in \{1,2,\cdots n\}.$ This implies that $\bar{x}_i \geq 0$ and $\bar{w}_i=0.$ \vskip 0.5em \noindent \textbf{Case 3:} Let $\bar{z}_{1i}>0, \bar{z}_{2i}>0$ for at least one $i \in \{1,2,\cdots n\}.$ This implies that $\bar{x}_i=0$ and $\bar{w}_i=0.$ \vskip 0.5em Considering the above three cases $\bar{x}, \bar{w}$ solve the LCP$(q, A).$ \end{proof} \begin{theorem}
If $A$ is a $P_0$ matrix, then the component $\bar{x}$ of $(\bar{x},\bar{z}_1,\bar{z}_2,0) \in L\times \{0\}$ gives the solution of LCP$(q, A)$. \end{theorem} \begin{proof} Let $A$ be a $P_0$ matrix. Assume that the component $\bar{x}$ of $(\bar{x},\bar{z}_1,\bar{z}_2,0) \in L\times \{0\}$ does not give the solution of LCP$(q, A)$. Hence $\Delta\bar{x}_i \Delta\bar{w}_i\neq0 $ and $\bar{z}_{1i}+\bar{z}_{2i}=0$ for atleast one $i$. Then $\Delta\bar{x}_i \neq 0, \Delta\bar{w}_i\neq0 , \bar{z}_{1i}=0, \bar{z}_{2i}=0.$ Now $\bar{z}_{1i}=\bar{w}_i-\Delta\bar{w}_i=0$ and $\Delta\bar{x}_i \Delta\bar{w}_i\neq0$ $\implies \bar{w}_i=\Delta\bar{w}_i>0$. In similar way $\bar{z}_{2i}=\bar{x}_i-\Delta\bar{x}_i=0$ and $\Delta\bar{x}_i \Delta\bar{w}_i\neq0$ $\implies \bar{x}_i=\Delta\bar{x}_i>0$. From Equation \ref{sys}, $\Delta\bar{w}_i + (A^t\Delta\bar{x})_i=0$. This implies that $ (A^t\Delta\bar{x})_i<0$ and also $(\bar{x})_i (A^t\Delta\bar{x})_i<0.$ This contradicts that $A$ is a $P_0$-matrix. Therefore the component $\bar{x}$ of $(\bar{x},\bar{z}_1,\bar{z}_2,0) \in L\times \{0\}$ gives the solution of LCP$(q, A)$. \end{proof} \begin{theorem}\label{22222} Suppose the matrix $(\bar{W}+\bar{X}A^t)$ is nonsingular, where $\bar{W}=\text{diag}(\bar{w}),$ $\bar{X}=\text{diag}(\bar{x}).$ Then $\bar{x}$ solves the LCP$(q, A).$ \end{theorem} \begin{proof} Let the matrix $(\bar{W}+\bar{X}A^t)$ be nonsingular. By Equation \ref{sys}, $\Delta\bar{w} + A^t\Delta\bar{x}=0$ and $\bar{X}\Delta\bar{w}=\bar{W}\Delta\bar{x},$ where $\bar{W}=$diag$(\bar{w})=$diag$(A\bar{x}+q).$ Now $\bar{X}\Delta\bar{w} + \bar{X}A^t\Delta\bar{x}=0$ implies that $\bar{W}\Delta\bar{x} + \bar{X}A^t\Delta\bar{x}=0.$ It implies that $\Delta\bar{x}=0.$ Then $\bar{x}$ solves the LCP$(q,A).$ \end{proof} Now we establish a sufficient condition of homotopy method for finding the solution of LCP$(q,A).$ \begin{theorem}
If the matrix $A$ is nondegenerate, then the component $\bar{x}$ of $(\bar{x},\bar{z}_1,\bar{z}_2,0) \in L\times \{0\}$ solves LCP$(q, A)$. \end{theorem} \begin{proof}
Consider that the matrix $A$ associated with LCP$(q,A)$ is nondegenerate. Therefore every principal minor of $A$ is nonzero. By theorem \ref{22222}, if the matrix $(\bar{W}+\bar{X}A^t)$ is nonsingular, then $\bar{x}$ solves the LCP$(q, A)$, where $\bar{W}=\text{diag}(\bar{w}),$ $\bar{X}=\text{diag}(\bar{x}).$ Let $\tilde{\mathcal{A}}=\left[\begin{array}{cc} \bar{W} & \bar{X}\\ -A^t & I\\ \end{array}\right]$. Then $\det(\tilde{\mathcal{A}})= \det(\bar{W}+\bar{X}A^t)$. Assume that the component $\bar{x}$ of $(\bar{x},\bar{z}_1,\bar{z}_2,0) \in L\times \{0\}$ is not the solution of LCP$(q, A)$. Then there exists atleast one $i$, such that $\bar{x}_i\bar{w}_i>0$. Without loss of generality $\bar{w}$ and $\bar{x}$ can be represented as $\bar{w}=\left[\begin{array}{c} \bar{w}_p \\ \bar{w}_q\\ \bar{o}_r\\ \end{array}\right]$, $\bar{x}=\left[\begin{array}{c} \bar{o}_p \\ \bar{x}_q\\ \bar{x}_r\\ \end{array}\right]$, where $\bar{w}_p\in {R^p}_{++}, \ \bar{w}_q, \bar{x}_q \in {R^q}_{++}, \ \bar{x}_r\in {R^r}_{++}$, \ $\bar{o}_r\in R^r, \ \bar{o}_p \in R^p$ and $\bar{o}_r=0, \ \bar{o}_p=0. $ Here $(\bar{w}_q)_i(\bar{x}_q)_i>0$ and $\bar{W}=\text{diag}(\bar{w}), \bar{X}=\text{diag}(\bar{x})$. Now we can rewrite $\left[\begin{array}{cc} \bar{W} & \bar{X}\\ -A^t & I\\ \end{array}\right]=\left[\begin{array}{cccccc} \bar{W}_p & \bar{O}_q & \bar{O}_r & \bar{O}_p & \bar{O}_q & \bar{O}_r \\ \bar{O}_p & \bar{W}_q & \bar{O}_r & \bar{O}_p & \bar{X}_q & \bar{O}_r\\ \bar{O}_p & \bar{O}_q & \bar{O}_r & \bar{O}_p & \bar{O}_q & \bar{X}_r\\ M & B & C & I_p & \bar{O}_q & \bar{O}_r\\ D & E & F & \bar{O}_p & I_q & \bar{O}_r\\ G & H & K & \bar{O}_p & \bar{O}_q & I_r\\ \end{array}\right]$, where $-A^t=\left[\begin{array}{ccc} M & B & C\\ D & E & F \\ G & H & K \\ \end{array}\right]$, \ $\bar{W}=\left[\begin{array}{ccc} \bar{W}_p & \bar{O}_q & \bar{O}_r \\ \bar{O}_p & \bar{W}_q & \bar{O}_r \\ \bar{O}_p & \bar{O}_q & \bar{O}_r \\ \end{array}\right]$, \ $\bar{X}=\left[\begin{array}{ccc}
\bar{O}_p & \bar{O}_q & \bar{O}_r \\
\bar{O}_p & \bar{X}_q & \bar{O}_r\\
\bar{O}_p & \bar{O}_q & \bar{X}_r\\ \end{array}\right]$, \ $\bar{X}_q=\text{diag}(\bar{x}_q)$, \ $\bar{X}_r=\text{diag}(\bar{x}_r)$, \ $\bar{W}_q=\text{diag}(\bar{w}_q)$, $\bar{W}_p=\text{diag}(\bar{w}_p)$, \ $\bar{O}_p=\text{diag}(\bar{o}_p)$, \ $\bar{O}_q=\text{diag}(\bar{o}_q)$ \ $\bar{O}_r=\text{diag}(\bar{o}_r)$, \ $M,D,G, I_p\in R^{p \times p}$, \ $B,E,H, I_q \in R^{q \times q}$, \ $C,F,K, I_r \in R^{r \times r}$ and $I_p,I_q,I_r$ are identity matrices. By elementary row operations we can get \vskip 0.5em $\tilde{\mathcal{B}}=\left[\begin{array}{cccccc} I & \bar{O}_q & \bar{O}_r & \bar{O}_p & \bar{O}_q & \bar{O}_r \\ \bar{O}_p & I & \bar{O}_r & \bar{O}_p & \bar{X}_q{\bar{W}_q}^{-1} & \bar{O}_r\\ \bar{O}_p & \bar{O}_q & \bar{O}_r & \bar{O}_p & \bar{O}_q & I\\ M & B & C & I & \bar{O}_q & \bar{O}_r\\ D & E & F & \bar{O}_p & I & \bar{O}_r\\ G & H & K & \bar{O}_p & \bar{O}_q & I\\ \end{array}\right]$. \vskip 0.5em By interchanging rows this matrix reduces to \\ $\tilde{\mathcal{C}}=$ $\left[\begin{array}{cccccc} I & \bar{O}_q & \bar{O}_r & \bar{O}_p & \bar{O}_q & \bar{O}_r \\ \bar{O}_p & I & \bar{O}_r & \bar{O}_p & \bar{X}_q{\bar{W}_q}^{-1} & \bar{O}_r\\ -G & -H & -K & \bar{O}_p & \bar{O}_q & \bar{O}_r\\ M & B & C & I & \bar{O}_q & \bar{O}_r\\ D & E & F & \bar{O}_p & I & \bar{O}_r\\ G & H & K & \bar{O}_p & \bar{O}_q & I\\ \end{array}\right]$.\\ Hence $\det(\tilde{\mathcal{A}})= \det(\tilde{\mathcal{C}})=(-1)^r\det(K)\neq 0$. Therefore by theorem \ref{22222}, \ $\bar{x}$ solves LCP$(q, A)$. This contradicts the assumption. Hence the component $\bar{x}$ of $(\bar{x},\bar{z}_1,\bar{z}_2,0) \in L\times \{0\}$ is the solution of LCP$(q, A)$. \end{proof} Hence for the $P_0$ and nondegenerate matrix classes the homotopy function \ref{homf} gives the solution of LCP$(q,A)$. \begin{remk}
We trace the homotopy path $\Gamma_y^{(0)} \subset \mathcal{F}_1 \times (0,1]$ from the initial point $(y^{(0)},1)$ as $\lambda \to 0.$ To find the solution of the given LCP$(q, A)$ we consider homotopy path along with other assumptions. Let $s$ denote the arc length of $\Gamma_y^{(0)}.$ We parameterize the homotopy path $\Gamma_y^{(0)}$ with respect to $s$ in the following form\\
\begin{equation}\label{ss}
H_{y^{(0)}} (y(s),\lambda (s))=0, \
y(0)=y^{(0)}, \ \lambda(0)=1.
\end{equation}
Differentiating \ref{ss} with respect to $s,$ we obtain the following system of ordinary differential equations with given initial values
\begin{equation}
H'_{y^{(0)}} (y(s),\lambda (s))\left[\begin{array}{c}
\frac{dy}{ds}\\
\frac{d\lambda}{ds}\\
\end{array}\right]=0, \
\|( \frac{dy}{ds},\frac{d\lambda}{ds})\|=1, \
y(0)=y^{(0)}, \ \lambda(0)=1, \ \frac{d\lambda}{ds}(0)<0,
\end{equation}
and the $y$-component of $(y(\bar{s}),\lambda (\bar{s}))$ gives the solution of LCP$(q,A)$ for $\lambda (\bar{s})=0.$ For details, see \cite{fan}.
Note that the parameter $\lambda$ is updated from the Moore-Penrose inverse of the Jacobian matrix for tracing the homotopy path. However, this approach does not ensure that the updated value of the parameter $\lambda$ is in $(0, 1].$ Value of $\lambda$ beyond $(0, 1]$ leads to a non-homotopy path. To eliminate deviation, we propose a modification by introducing a method called \textit{ensuring feasibility by changing step length.} In this method it is necessary to check whether $0 < (\tilde{\lambda}_i - \hat{\lambda}_i) < 1$ and $(\tilde{y}^{i} - \hat{y}^{i}) \in \bar{\cal{F}}_1$ holds or not. If any of the above-mentioned criteria fails, then the step length will be changed appropriately using geometric series to trace the homotopy path $\Gamma_y^{(0)}.$ This guarantees a homotopy continuation trajectory.
\end{remk}
\subsection{Algorithm} \noindent \textbf{Step 0:} Initialize $(y^{(0)},\lambda_0).$ Set $l_0 \in (0, 1).$ Choose $\epsilon_2 >> \epsilon_3 >> \epsilon_1 > 0$ which are very small positive quantity. \vskip 0.5em \noindent \textbf{Step 1:} $\tau^{(0)}= \xi^{(0)}=(\frac{1}{n})\left[\begin{array}{c} s\\ -1\\
\end{array}\right]$ for $i=0,$ where $n=\|\left[\begin{array}{c} s\\ -1\\
\end{array}\right]\|$ and $s= (\frac{\partial H}{\partial y}(y^{(0)},\lambda_0))^{-1}(\frac{\partial H}{\partial \lambda}(y^{(0)},\lambda_0)).$ If $\det (\frac{\partial H}{\partial y}(y^{(i)},\lambda_i))>0,$ $\tau^{(i)}= \xi^{(i)}$ else $\tau^{(i)}= -\xi^{(i)},$ $i \geq 1.$ Set $l=0.$ \vskip 0.5em
\noindent \textbf{Step 2:} (Predictor point calculation) $(\tilde{y}^{(i)},\tilde{\lambda}_i)=(y^{(i)},\lambda_i)+a\tau^{(i)},$ where $a={l_0}^l.$ Compute $(\hat{y}^{(i)},\hat{\lambda}_{i})=H'_{y^{(0)}}(\tilde{y}^{(i)},\tilde{\lambda}_i)^+ H(\tilde{y}^{(i)}, \tilde{\lambda}_i).$ If $0<(\tilde{\lambda}_i - \hat{\lambda}_{i})<1, $ go to Step 3. Otherwise if $m = \min(a,\|(\tilde{y}^{(i)},\tilde{\lambda}_i)-(\hat{y}^{(i)},\hat{\lambda}_{i})-(y^{(i)},\lambda_i)\|)>a_0,$ update $l$ by $l+1,$ and recompute $(\tilde{\lambda}_i, \hat{\lambda}_{i})$ else go to Step 4. \vskip 0.5em
\noindent \textbf{Step 3:} (Corrector point calculation) $(y^{(i+1)},\lambda_{i+1})=(\tilde{y}^{(i)},\tilde{\lambda}_i)-(\hat{y}^{(i)},\hat{\lambda}_{i}).$ Determine the norm $r=\|H(y^{(i+1)},\lambda_{i+1})\|.$ If $r \leq 1$ and $y^{(i+1)}>0$ go to Step 5, otherwise if $a > \epsilon_3,$ update $l$ by $l+1$ and go to Step 2 else go to Step 4. \vskip 0.5em
\noindent \textbf{Step 4:} If $|\lambda_{i+1} - \lambda_i| < \epsilon_2,$ then if $|\lambda_{i+1}| < \epsilon_2,$ then stop with the solution $(y^{(i+1)},\lambda_{i+1}),$ else terminate (unable to find solution) else $i=i+1$ and go to Step 1. \vskip 0.5em
\noindent \textbf{Step 5:} If $|\lambda_{i+1}| \leq \epsilon_1,$ then stop with solution $(y^{(i+1)},\lambda_{i+1}),$ else $i=i+1$ and go to Step 1. \vskip 0.5em Note that in Step 2, $H'_{y^{(0)}}(y,\lambda)^+ = H'_{y^{(0)}}(y,{\lambda})^{t}(H'_{y^{(0)}}(y,{\lambda})H'_{y^{(0)}}(y,\lambda)^{t})^{-1}$ is the Moore-Penrose inverse of $H'_{y^{(0)}}(y,\lambda).$ We prove the following result to obtain the positive direction of the proposed algorithm. \begin{theorem} \label{direction} If the homotopy curve $\Gamma_y^{(0)}$ is smooth, then the positive predictor direction $\tau^{(0)}$ at the initial point $y^{(0)}$ satisfies $\det \left[\begin{array}{c} \frac{\partial H}{\partial y \partial \lambda}(y^{(0)},1)\\ \tau ^{(0)^t}\\ \end{array}\right]$$ < 0.$ \end{theorem}
\begin{proof}
From the Equation \ref{homf}, we consider the following homotopy function \\
\vskip 0.5em
$H(y,y^{(0)},\lambda)=$ $\left[\begin{array}{c}
(1-\lambda)[(A+A^t)x+q-z_1-A^tz_2]+\lambda(x-x^{(0)}) \\
Z_1x-\lambda Z_1^{(0)}x^{(0)}\\
Z_2(Ax+q)-\lambda Z_2^{(0)}(Ax^{(0)}+q)\\
\end{array}\right]=0.$ Now, \\
\vskip 0.5em
$\frac{\partial H}{\partial y \partial \lambda}(y,\lambda)=\left[\begin{array}{cccc}
(1-\lambda)(A + A^t) + \lambda I & -(1-\lambda)I & -(1-\lambda)A^t & P\\
Z_1 & X & 0 & -Z_1^{(0)}x^{(0)}\\
Z_2A & 0 & W & -Z_2^{(0)}(Ax^{(0)}+q)\\ \
\end{array}\right],$ where $P = (x-x^{(0)})-[(A+A^t)x+q-z_1-A^tz_2]$ and $W=\text{diag}(Ax+q).$ At the initial point $(y^{(0)},1)$\\
$\frac{\partial H}{\partial y \partial \lambda}(y^{(0)},1)=\left[\begin{array}{cccc}
I & 0 & 0 & -[(A+A^t)x^{(0)}+q-z^{(0)}_1-A^tz^{(0)}_2]\\
Z^{(0)}_1 & X^{(0)} & 0 & -Z^{(0)}_1x^{(0)}\\
Z^{(0)}_2A & 0 & W^{(0)} & -Z^{(0)}_2(Ax^{(0)}+q)\\
\end{array}\right].$\\
\vskip 0.5em
\noindent Let positive predictor direction be $\tau^{(0)}=\left[\begin{array}{c}
\kappa \\ -1
\end{array}\right] = \left[\begin{array}{c}
(R^{(0)}_1)^{(-1)}R_2^{(0)} \\ -1
\end{array} \right],$ where
\vskip 0.5em
$R^{(0)}_1=\left[\begin{array}{ccc}
I & 0 & 0 \\
Z^{(0)}_1 & X^{(0)} & 0 \\
Z^{(0)}_2A & 0 & W^{(0)} \\
\end{array}\right],$ $R^{(0)}_2=\left[\begin{array}{c}
-[(A+A^t)x^{(0)}+q-z^{(0)}_1-A^tz^{(0)}_2]\\
-Z^{(0)}_1x^{(0)} \\
-Z^{(0)}_2(Ax^{(0)}+q) \\
\end{array}\right]$ and $\kappa$ is a $n \times 1$ column vector.\\
Hence,
$\det\left[\begin{array}{c}
\frac{\partial H}{\partial y \partial \lambda}(y^{(0)},1)\\
\tau ^{(0)^t}\\
\end{array}\right]$\\
$=\det\left[\begin{array}{cc}
R^{(0)}_1 & R^{(0)}_2\\
(R^{(0)}_2)^t(R^{(0)}_1)^{(-t)} & -1\\
\end{array}\right]$ \\ $= \det\left[\begin{array}{cc}
R^{(0)}_1 & R^{(0)}_2\\
0 & -1-(R^{(0)}_2)^t(R^{(0)}_1)^{(-t)}(R^{(0)}_1)^{(-1)}R_2^{(0)} \\ \end{array}\right] \\$
\vskip 0.5em
\noindent $=\det(R^{(0)}_1) \det(-1-(R^{(0)}_2)^t(R^{(0)}_1)^{(-t)}(R^{(0)}_1)^{(-1)}R_2^{(0)})$ \\
\vskip 0.5em
\noindent $=-\det(R^{(0)}_1) \det(1+(R^{(0)}_2)^t(R^{(0)}_1)^{(-t)}(R^{(0)}_1)^{(-1)}R_2^{(0)})$ \\
\vskip 0.5em
\noindent $=-\prod_{i=1}^{n}x^{(0)}_i y^{(0)}_i \det(1+(R^{(0)}_2)^t(R^{(0)}_1)^{(-t)}(R^{(0)}_1)^{(-1)}R_2^{(0)}) <0. $ \\
So the positive predictor direction $\tau ^{(0)}$ at the initial point $y^{(0)}$ satisfies\\ $\det\left[\begin{array}{c}
\frac{\partial H}{\partial y \partial \lambda}(y^{(0)},1)\\
\tau ^{(0)^t}\\
\end{array}\right]<0.$ \end{proof} \vskip 0.5em \begin{remk}
We conclude from the Theorem \ref{direction} that the positive tangent direction $\tau$ of the homotopy path $\Gamma_y^{(0)}$ at any point $(y,\lambda)$ be negative and it depends on det$(R_1),$ where $R_1=\left[\begin{array}{ccc}
(1-\lambda)(A+A^t)+\lambda I & -(1-\lambda)I & -(1-\lambda)A^t \\
Z_1 & X & 0 \\
Z_2A & 0 & W\\
\end{array}\right].$ \end{remk}
\section{Numerical Examples} In this section we consider some examples of LCP$(q, A)$ based on $P_0$ and nondegenerate matrices to demonstrate the effectiveness of our proposed algorithm. Note that Example \ref{matrix2} - \ref{mtx5} are not processable by the algorithms given in Yu et al. \cite{yu2006combined}, Xu et al. \cite{xuuu}, Zhao et al. \cite{N}. Even these examples are not processable by Lemke's algorithm \cite{das2017finiteness} except example \ref{matrix1} and \ref{matrix2}. Example \ref{matx1} - \ref{mtx5} are also not processable by modulus based algorithm \cite{schafer2004modulus}. We show that the proposed algorithm can process these examples to find the solution. \begin{examp} \label{matrix0}
Consider $A=\left[\begin{array}{cc}
-1 & 2\\
3 & -1\\
\end{array}\right]$ and $q=\left[\begin{array}{c}
1\\
-0.5\\
\end{array}\right].$ Note that $A$ is an $N$-matrix. It is solvable by the homotopy method with the homotopy function \ref{zhaon}, proposed by Zhao et al. \cite{N}. Now we show that the homotopy function \ref{homf} also solves the linear complementarity problem with $N$-matrix. Now choose the initial point $x^{(0)}=\left[\begin{array}{c}
0.4\\
0.1\\
\end{array}\right], \ {z_1}^{(0)}=\left[\begin{array}{c}
1\\
1\\
\end{array}\right]$ and ${z_2}^{(0)}=\left[\begin{array}{c}
1\\
1\\
\end{array}\right].$ Using the proposed algorithm we get the optimal solution of the homotopy function \ref{homf} after 20 iterations and the solution is given by $(\bar{y}, \bar{\lambda})=(1,0,0,2.5,1,0,0).$ Therefore $\bar{x}=\left[\begin{array}{c}
1\\
0\\
\end{array}\right]$ solves LCP$(q, A).$ The homotopy path shown in Figure \ref{fig0} illustrates the convergence with respect to the solution vector $x$ and $\lambda$.
\end{examp}
\begin{examp}\label{matrix1}
Let $A=\left[\begin{array}{cc}
1 & -1 \\
-1 & 1\\
\end{array}\right]$ and $q=\left[\begin{array}{c}
-0.5\\
2\\
\end{array}\right].$ It is easy to show that $A$ is a $PSD$-matrix. It is solvable by the homotopy method with the homotopy function \ref{psdyu}, proposed by Yu et al.\cite{yu2006combined} . Now we show that the homotopy function \ref{homf} also solves the linear complementarity problem with $PSD$-matrix. Now choose the initial point $x^{(0)}=\left[\begin{array}{c}
2\\
1\\
\end{array}\right],$ ${z_1}^{(0)}=\left[\begin{array}{c}
1\\
1\\
\end{array}\right]$ and ${z_2}^{(0)}=\left[\begin{array}{c}
1\\
1\\
\end{array}\right].$ Using the proposed algorithm we obtain $(\bar{y},\bar{\lambda})=(0.5,0,0,1.499,0.499,0,0,0)$ after 22 iterations. Note that $\bar{x}=\left[\begin{array}{c}
0.5\\
0\\
\end{array}\right]$ is the solution of LCP$(q, A).$ The homotopy path shown in Figure \ref{fig1} illustrates the convergence with respect to the solution vector $x$ and $\lambda$. \end{examp}
Now we show that the homotopy function \ref{homf} can solve LCP$(q,A)$ with singular matrix $A$ satisfying some conditions. \begin{examp} \label{matrix2}
Consider $A=\left[\begin{array}{cc}
1 & 1\\
0 & 0\\
\end{array}\right]$ and $q=\left[\begin{array}{c}
-1\\
1\\
\end{array}\right].$ Note that $A$ is a singular $Q_0$-matrix. Now choose the initial point $x^{(0)}=\left[\begin{array}{c}
1\\
0.2\\
\end{array}\right], \ {z_1}^{(0)}=\left[\begin{array}{c}
1\\
1\\
\end{array}\right]$ and ${z_2}^{(0)}=\left[\begin{array}{c}
1\\
1\\
\end{array}\right].$ Using the proposed algorithm we get the optimal solution of the homotopy function \ref{homf} after 15 iterations and the solution is given by $(\bar{y}, \bar{\lambda})=(1,0,0,1,1,0,0).$ Therefore $\bar{x}=\left[\begin{array}{c}
1\\
0\\
\end{array}\right]$ solves LCP$(q, A).$ The homotopy path shown in Figure \ref{fig2} illustrates the convergence with respect to the solution vector $x$ and $\lambda$. \end{examp}
\begin{examp} \label{matx1}
Let $A=\left[\begin{array}{ccc}
0 & 1 & 1\\
2 & 0 & 1\\
-4 & -5 & 0\\
\end{array}\right]$ and $q=\left[\begin{array}{c}
-4\\
-7\\
10\\
\end{array}\right].$ It is easy to show that $A$ is an ${E_0}^s$-matrix. This is not processable by modulus based method. Now choose the initial point $x^{(0)}=\left[\begin{array}{c}
1\\
1\\
6\\
\end{array}\right],$ ${z_1}^{(0)}=\left[\begin{array}{c}
1\\
1\\
1\\
\end{array}\right]$ and ${z_2}^{(0)}=\left[\begin{array}{c}
1\\
1\\
1\\
\end{array}\right].$ Using the proposed algorithm we obtain $(\bar{y},\bar{\lambda})=(0,2,7,5,0,0,0,2,7,0)$ after 14 iterations. Note that $\bar{x}=\left[\begin{array}{c}
0\\
2\\
7\\
\end{array}\right]$ is the solution of LCP$(q, A).$ The convergence of the homotopy function is shown in the Figure \ref{nwfg1}. The first, second and third component of $x$ is represented by data1, data2 and data3 respectively.\\ \end{examp} \begin{examp} \label{matrix3} Let $A=\left[\begin{array}{ccc} -1 & 2 & 1\\ 1 & -0.50 & -0.25\\ -0.50 & -1 & -1\\ \end{array}\right]$ and $q=\left[\begin{array}{c} -0.25\\ -0.10\\ 3\\ \end{array}\right].$ It is easy to show that $A$ is not an $N$-matrix. This matrix is not processable by using existing homotopy functions as well as lemke's algorithm. Now choose the initial point $x^{(0)}=\left[\begin{array}{c} 2.3\\ 1\\ 0.7\\ \end{array}\right],$ ${z_1}^{(0)}=\left[\begin{array}{c} 1\\ 1\\ 1\\ \end{array}\right]$ and ${z_2}^{(0)}=\left[\begin{array}{c} 1\\ 1\\ 1\\ \end{array}\right].$ Using the proposed algorithm we obtain $(\bar{y},\bar{\lambda})=(1.8333,0,2.0833,0,1,2125,0,1.8333,0,2.0833,0)$ after 17 iterations. Note that $\bar{x}=\left[\begin{array}{c} 1.8333\\ 0\\ 2.0833\\ \end{array}\right]$ is the solution of LCP$(q, A).$ The convergence of the homotopy function is shown in the Figure \ref{fig3}. The first, second and third component of $x$ is represented by data1, data2 and data3 respectively. \end{examp}
\begin{examp} \label{matrix4}
Let $A=\left[\begin{array}{ccc}
1 & -2 & 0\\
0 & 1 & -2\\
-2 & 0 & 1\\
\end{array}\right]$ and $q=\left[\begin{array}{c}
-1\\
1\\
7\\
\end{array}\right].$ It is easy to show that $A$ is an almost $C_0$ matrix. This matrix is not processable by lemke's algorithm as well as modulus based algorithm. This matrix is also not processable by existing homotopy methods. Now choose the initial point $x^{(0)}=\left[\begin{array}{c}
3\\
0.5\\
0.5\\
\end{array}\right], \ {z_1}^{(0)}=\left[\begin{array}{c}
1\\
1\\
1\\
\end{array}\right]$ and ${z_2}^{(0)}=\left[\begin{array}{c}
1\\
1\\
1\\
\end{array}\right].$ Using the proposed algorithm we obtain is $(\bar{y}, \bar{\lambda})=(1,0,0,0,1,5,1,0,0,0)$ after $24$ iterations. Note that $\bar{x}=\left[\begin{array}{c}
1\\
0\\
0\\
\end{array}\right]$ solves LCP$(q, A),$ which is a degenerate solution. The convergence of the homotopy function is shown in the Figure \ref{fig4}. The first, second and third component of $x$ is represented by data1, data2 and data3 respectively. \end{examp}
\begin{examp} \label{matrix5}
Let $A=\left[\begin{array}{cccc}
-1 & 1 & 1 & 1\\
1 & 0 & 0 & 0\\
1 & 0 & 0 & -1\\
1 & 0 & -1 & 0\\
\end{array}\right]$ and $q=\left[\begin{array}{c}
-1\\
1\\
-1\\
1\\
\end{array}\right].$ $A$ is a $Q$-matrix by \cite{neogy2005almost} and also almost $\bar{N}$-matrix. This matrix is not processable by lemke's algorithm. Now choose the initial point $x^{(0)}=\left[\begin{array}{c}
4\\
4\\
1\\
1\\
\end{array}\right], \ {z_1}^{(0)}=\left[\begin{array}{c}
1\\
1\\
1\\
1\\
\end{array}\right]$ and ${z_2}^{(0)}=\left[\begin{array}{c}
1\\
1\\
1\\
1\\
\end{array}\right].$ We apply our proposed algorithm to this LCP$(q, A)$ and after 17 iterations we get the approximate optimal solution of the homotopy function \ref{homf}, which is $(\bar{y}, \bar{\lambda})=(1,0,2,0,0,2,0,0,1,0,2,0,0).$ Note that $\bar{x}=\left[\begin{array}{c}
1\\
0\\
2\\
0\\
\end{array}\right]$ solves LCP$(q, A),$ which gives a degenerate solution. The convergence of the homotopy function is shown in the Figure \ref{fig5}. Data1, data2, data3 and data4 represent the first, second, third and fourth component of $x$ respectively. \end{examp} \begin{examp} \label{maat4}
Let $A=\left[\begin{array}{cccc}
-2 & -2 & -2 & 2\\
-2 & -1 & -3 & 3\\
-2 & -3 & -1 & 3\\
2 & 3 & 3 & 0\\
\end{array}\right]$ and $q=\left[\begin{array}{c}
-1001\\
-500\\
-500\\
-500\\
\end{array}\right].$ $A$ is a almost $N_0$-matrix by \cite{neogy2005almost} but not $Q$-matrix. This matrix is not processable by lemke's algorithm as well as modulus based algorithm. This matrix is also not processable by existing homotopy methods. Now choose the initial point $x^{(0)}=\left[\begin{array}{c}
100\\
100\\
200\\
1000\\
\end{array}\right], \ {z_1}^{(0)}=\left[\begin{array}{c}
1\\
1\\
1\\
1\\
\end{array}\right]$ and ${z_2}^{(0)}=\left[\begin{array}{c}
1\\
1\\
1\\
1\\
\end{array}\right].$ We apply our proposed algorithm to this LCP$(q, A)$ and after 17 iterations we get the approximate optimal solution of the homotopy function \ref{homf}, which is $(\bar{y}, \bar{\lambda})=(250,0,0,750.50,0,1251.50,1251.50,0,250,0,0,750.50,0).$ Note that $\bar{x}=\left[\begin{array}{c}
250\\
0\\
0\\
750.50\\
\end{array}\right]$ solves LCP$(q, A),$ which gives a degenerate solution. The convergence of the homotopy function is shown in the Figure \ref{fiiig4}. Data1, data2, data3 and data4 represent the first, second, third and fourth component of $x$ respectively. \end{examp}
\begin{examp} \label{matrix6}
Consider $A=\left[\begin{array}{ccccc}
0 & 0 & 0 & 1 & 2\\
0 & 0 & -1 & -1 & 2\\
0 & -1 & 0 & -1 & 1\\
1 & -1 & -1 & 0 & 0\\
2 & 1 & 0 & 0 & 0\\
\end{array}\right]$ and $q=\left[\begin{array}{c}
-2\\
-1\\
7\\
2\\
-1\\
\end{array}\right].$ $A$ is an $N_0$-matrix of exact order $2.$ This matrix is not processable by lemke's algorithm as well as modulus based algorithm. This matrix is also not processable by existing homotopy methods. Now choose the initial point $x^{(0)}=\left[\begin{array}{c}
3\\
1\\
1\\
1\\
3\\
\end{array}\right],$ ${z_1}^{(0)}=\left[\begin{array}{c}
1\\
1\\
1\\
1\\
1\\
\end{array}\right]$ and ${z_2}^{(0)}=\left[\begin{array}{c}
1\\
1\\
1\\
1\\
1\\
\end{array}\right].$ Using the proposed algorithm, we obtain the approximate optimal solution of the homotopy function \ref{homf}, $(\bar{y}, \bar{\lambda}) = (0.5, 0, 0, 0, 1, 0, 1, 8, 2.5, 0, 0.5, 0, 0, 0, 1, 0)$ after 27 iterations. Note that $\bar{x}=\left[\begin{array}{c}
0.5\\
0\\
0\\
0\\
1\\
\end{array}\right]$ solves LCP$(q, A).$ The convergence of the homotopy function is shown in the Figure \ref{fig6}. Data1, data2, data3, data4 and data5 represent the first, second, third, fourth and fifth component of $x$ respectively.
0,-90,-80,-70,0,-90,-2,-2,-2,2,-70,-2,-1,-3,3,-50,-2,-3,-0.8,3,0,2,3,3,0 \end{examp}
\begin{examp}\label{mtx5}
Consider $A=\left[\begin{array}{ccccc} 0 & -90 & -80 & -70 & 0\\ -90 & -2 & -2 & -2 & 2\\ -70& -2 & -1 & -3 & 3\\ -50 & -2 & -3 & -0.8 & 3\\ 0 & 2 & 3 & 3 & 0\\ \end{array}\right]$ and $q=\left[\begin{array}{c} 400\\ 50\\ 30\\ 20\\ -10\\ \end{array}\right].$ $A$ is an $\bar{N}$-matrix of exact order $2.$ This matrix is not processable by lemke's algorithm as well as modulus based algorithm. This matrix is also not processable by existing homotopy methods. Now choose the initial point $x^{(0)}=\left[\begin{array}{c} 0.1\\ 0.1\\ 0.1\\ 5\\ 100\\ \end{array}\right],$ ${z_1}^{(0)}=\left[\begin{array}{c} 1\\ 1\\ 1\\ 1\\ 1\\ \end{array}\right]$ and ${z_2}^{(0)}=\left[\begin{array}{c} 1\\ 1\\ 1\\ 1\\ 1\\ \end{array}\right].$ Using the proposed algorithm, we obtain the approximate optimal solution of the homotopy function \ref{homf}, $(\bar{y}, \bar{\lambda}) = (0.2403846, 0, 1.634615, 3.846154, 0,$
$0, 17.40385, 0, 0, 6.442308, 0.2403846, 0, 1.634615, 3.846154, 0, 0)$ after 1925 iterations. Note that $\bar{x}=\left[\begin{array}{c} 0.2403846\\ 0\\ 1.634615\\ 3.846154\\ 0\\ \end{array}\right]$ solves LCP$(q, A).$ The convergence of the homotopy function is shown in the Figure \ref{fg5}. Data1, data2, data3, data4 and data5 represent the first, second, third, fourth and fifth component of $x$ respectively. \end{examp}
\begin{figure}
\caption{Example \ref{matrix0}}
\label{fig0}
\caption{Example \ref{matrix1}}
\label{fig1}
\caption{Example \ref{matrix2}}
\label{fig2}
\caption{Example \ref{matx1}}
\label{nwfg1}
\caption{Example \ref{matrix3}}
\label{fig3}
\caption{Example \ref{matrix4}}
\label{fig4}
\caption{Example \ref{matrix5}}
\label{fig5}
\caption{Example \ref{maat4}}
\label{fiiig4}
\caption{Example \ref{matrix6}}
\label{fig6}
\caption{Example \ref{mtx5}}
\label{fg5}
\caption{Homotopy path for the LCP$(q, A)$ to show the convergence}
\label{fig8}
\end{figure}
\section{Conclusion} In this study, we consider an interior point homotopy path to solve linear complementarity problem. We prove a necessary and sufficient condition for the solution of LCP$(q,A)$ based on newly introduced homotopy function. To ensure a homotopy continuation trajectory we introduce a new scheme of choosing step length. Mathematically we find the positive tangent direction of the homotopy path. We show that the smooth curve for the homotopy function is bounded and convergent. Several numerical examples are presented to demonstrate the processability of larger classes of LCP$(q, A)$ based on $P_0$ and nondegenerate matrices namely, $Q$-matrix, almost $\bar{N}$-matrix, $Q_0$-matrix, almost $N_0$-matrix, almost $C_0$-matrix, $N_0$-matrix of exact order $2$ and $\bar{N}$-matrix of exact order $2$. Many of them are not processable by lemke's algoritm, existing homotopy method and modulus based method. However, the proposed method is able to process all the cases to find solution. \section{Acknowledgment} The author A. Dutta is thankful to the Department of Science and Technology, Govt. of India, INSPIRE Fellowship Scheme for financial support. We acknowledge Mr. Abhirup Ganguly(M.Tech 2017-2019, ISI Kolkata) for his contribution. \vskip 0.5em
\end{document} |
\begin{document}
\maketitle
\begin{abstract} A $W^{1,p}$-metric on an $n$-dimensional closed Riemannian manifold naturally induces a distance function, provided $p$ is sufficiently close to $n$. If a sequence of metrics $g_k$ converges in $W^{1,p}$ to a limit metric $g$, then the corresponding distance functions $d_{g_k}$ subconverge to a limit distance function $d$, which satisfies $d\le d_g$.
As an application, we show that the above convergence result applies to a sequence of conformal metrics with $L^{n/2}$-bounded scalar curvatures, under certain geometric assumptions. In particular, in this special setting, the limit distance function $d$ actually coincides with $d_{g}$.
\end{abstract}
\section{Introduction}
In this paper, we are interested in the convergence of a sequence of $W^{1,p}$-metrics on a Riemannian manifold, which is motivated by the study of conformal metrics with $L^{\frac{n}{2}}$-bounded scalar curvatures.
Let $(M, g_0)$ be a smooth closed $n$-dimensional Riemannian manifold and $p<n$ be sufficiently close to $n$. We first observe that, given a $W^{1,p}$-metric $g$ with respect to the background metric $g_0$, there is a well-defined distance function $d_g$ associated to $g$. Actually, using an idea similar to the Trace Embedding Theorem, we can show that $g$ is well-defined almost everywhere except a possible singular set of Hausdorff dimension at most $n-p$, which enables us to define
$$ d_g(x,y):=\inf\left\{ \int_\gamma \sqrt{g(\dot{\gamma},\dot{\gamma})}: \mbox{piecewise smooth $\gamma$ from $x$ to $y$} \right\}, \forall x,y\in M.$$
Now suppose $\{g_k\}$ is a sequence of smooth metrics on $M$ such that $g_k$ and $g_k^{-1}$ converges to $g$ and $g^{-1}$ in $W^{1,p}(M,g_0)$, respectively. Then the limit $W^{1,p}$-metric $g$ induces a distance function $d_g$. On the other hand, the distance function $d_{g_k}$ associated to $g_k$ converges uniformly to a distance function $d$, as the metric space $(M,d_{g_k})$ converge to $(M,d)$ in sense of the Gromov-Hausdorff distance. Then it is very natural to ask what is the relation between $d_g$ and $d$?
Our first main result is
\begin{theorem}\label{main1} Let $(M, g_0)$ be an $n$-dimensional closed Riemannian manifold and $p\in(\frac{2n(n-1)}{2n-1},n]$. Suppose $\{g_k\}$ is a sequence of smooth Riemannian metrics such that $\{g_k\} $ and $\{g^{-1}_k\}$ converge to $g$ and $g^{-1}$ in $W^{1,p}(M,g_0)$, respectively. Then, up to a subsequence, $\{d_{g_k}\}$ converges uniformly to a distance function $d$ with $d\leq d_g$. In particular, if $g$ and $g^{-1}$ are continuous, then $d=d_g$. \end{theorem}
\begin{rem}
Although here we only get the identity $d=d_g$ when $g$ is continuous, we believe it is still true without the assumption that $g$ and $g^{-1}$ are continuous. \end{rem}
As an application, we next study the compactness of a sequence of conformal metrics $\{g_k=u_k^{\frac{4}{n-2}}g_0\}$ with $L^\frac{n}{2}$-bounded scalar curvature $\|R(g_k)\|_{L^{n/2}}\leq C$ and $\mathrm{Vol}(M,g_k)=1$. Since $W^{2,\frac{n}{2}}$-space fails to be embedded into $C^0$, in general one can not expect point-wise compactness without additional assumptions, as shown by the counterexamples in \cite{Brendle,Brendle-Marques,Chang-Gursky-Wolff}. In fact, there is no compactness even if $R(g_k)$ is bounded $L^\infty$, see \cite{Brendle, Brendle-Marques}.
Here we assume in addition to the $L^\frac{n}{2}$-norm of the scalar curvature being bounded, its limiting measure is locally small as specified in the statement of the following theorem. We focus on the convergence of measures and distance functions.
\begin{theorem}\label{main2}
Let $(M,g_0)$ be an $n$-dimensional closed Riemannian manifold with $n\geq 3$. Suppose $\{g_k=u_k^{\frac{4}{n-2}}g_0\}$ is a sequence of conformal metrics such that $\mathrm{Vol}(M,g_k)=1$ and $|R(g_k)|^{\frac{n}{2}}dV_{g_k}$ converges weakly to a measure $\mu$ with $\mu(M)<\Lambda$. Then, for any $q\in (1,\frac{n}{2})$, there exists $\varepsilon_0>0$, which only depends on $(M,g_0)$, $\Lambda$ and $q$, such that if $$ \mu(\{x\})<\varepsilon_0,\forall x\in M, $$ then, after passing to a subsequence, we have
1) $\{u_k\}$, $\{\frac{1}{u_k}\}$ and $\{\log u_k\}$ converge to $u$ , $\frac{1}{u}$ and $\log u$ weakly in $W_{loc}^{2,q}(M)$ respectively.
2)$d_{g_k}$ converges to $d_{g}$ in $C_{loc}^0(M\times M)$, where $d_g$ is the distance function associated to the limit metric $g:=u^\frac{4}{n-2}g_0.$ \end{theorem}
\begin{rem}
Note that the limit metric $g\in W^{2,q}(M), q<\frac{n}{2}$. $W^{2,q}$-space fails to be embedded into $C^0$. Hence in Theorem \ref{main2}, we can not get $g$ is continuous, but we can use the conformal condition to get the same conclusion as in Theorem \ref{main1}. \end{rem} \begin{rem} Note that the $L^{\frac{n}{2}}$-norm of scalar curvature is rescaling invariant, i.e. $$
\int_M|R(\lambda g_k)|^\frac{n}{2}dV_{\lambda g_k}=\int_M|R(g_k)|^\frac{n}{2}dV_{g_k}. $$ So for a general sequence of collapsing metrics $\{g_k\}$, we can normalize the metric and set $g_k'=(c_ku_k)^\frac{4}{n-2}g_0$ such that $\mathrm{Vol}(M,g_k')=1$. Then after passing to a subsequence, the distance functions $d_{g_k'}$ associated to $g_k'$ converges uniformly to a distance function $d_g$, which is defined by $g=v^{\frac{4}{n-2}}g_0$, with $v$ being the $W^{2,q}$-weak limit of $c_ku_k$. \end{rem}
In a recent paper \cite{Aldana-Carron-Tapie}, C. Aldana, G. Carron and S. Tapie obtained the above Gromov-Hausdoff convergence result in a similar setting, see \cite[Theorem 5.1]{Aldana-Carron-Tapie}). But they used a different method from ours, and the equality of $d$ and $d_g$ was left open.
In \cite{Li-Zhou}, the second author studied the bubble tree convergence of $\{g_k\}$ under a stronger assumption that $\|R(g_k)\|_{L^p}<C$ with $p>\frac{n}{2}$. The compactness of conformal metrics with uniformly $L^p$-bounded sectional curvature was discussed in \cite{Chang-Yang1,Chang-Yang2,Gursky}.
The rest of the paper is organized as follows. In Section 2, we study the basic properties of the average limit of a $W^{1,p}$ function with $p<n$.
In Section 3, we discuss the convergence of a sequence of distance functions associated to $W^{1,p}$-metrics and prove Theorem~\ref{main1}. At last, we provide a key $\varepsilon$-regularity theorem and finish the proof of Theorem~\ref{main2} in Section~4.
\textbf{Acknowledgment.}
Part of this work was done while the third author was visiting S.-Y. A. Chang at Princeton University. She would like to thank S.-Y. A. Chang for helpful discussions. The authors would like to thank the referees for their valuable suggestions on the revision.
\section{Traces of $W^{1,p}$-functions} Recall that the \emph{Trace Embedding Theorem} states that the restriction of a $W^{1,p}$-function, which is defined on a domain in $\mathbb{R}^n$, on a $k$-dimensional subset is a $L^q$-function, where $p<n$, $n-p<k\leq n$ and $p\leq q\leq\frac{np}{n-p}$, see \cite[Theorem 4.12]{Adams-Fournier}. There is also the so-called \emph{trace inequality} in Sobolev Spaces, see \cite[Chapter 1]{V.G.-T.O.}. Using similar ideas from those references, here we will show that the \emph{centered average limit} of a given $W^{1,p}$-function is well-defined almost everywhere, except on a subset of Hausdorff dimension smaller than $n-p$ (c.f. \cite{Evans-Gariepy}).
For $r>0$, we denote the $n$-ball centered at $x\in \mathbb{R}^n$ with radius $r$ by $B_r(x)$ and $B_r:=B_r(0)$. Given a function $u$ defined on a domain $U\subset \mathbb{R}^n$, the \emph{$r$-average} of $u$ at $x\in U$ is $$
u_{x,r}:=\frac{1}{|B_r(x)\cap U|}\int_{B_r(x)\cap U}u(y)dy. $$
Now for $u\in W^{1,p}(B_2)$, define the singular set $$ A(u)=\{x\in B_1:\lim_{\tau\rightarrow 0}osc_{r\in(0,\tau]} u_{x,r}>0\}. $$ By Federer and Ziemer's theorem, we know $\dim A(u)\leq n-p$ (see \cite[Theorem 2.1.2]{Lin-Yang} or \cite[p.160]{Evans-Gariepy}). Hence, we can define the \emph{centered average limit} \begin{defi} Given a function $u\in W^{1,p}(B_2), p<n$. For any $x\in B_1\setminus A(u)$, there exists $\hat{u}(x)$, such that $$
\lim_{r\to 0}\frac{1}{B_r(x)}\int_{B_r(x)}|u(y)-\hat{u}(x)|dy=0, $$ $\hat{u}(x)$ is called the \emph{centered average limit} of $u$ at $x$. \end{defi} Thus $\hat{u}$ is well-defined for $\mathcal{H}^s$-a.e. $x\in B_1$, with $s\in (n-p,n)$.
The following estimate will play an essential role in the next section. \begin{lem}\label{measure.estimate} Let $u\in W^{1,p}(B_2)$ and $$
\mathcal{M}(u,t):=\{x\in B_1\setminus A(u): |\hat{u}|(x)>t\}. $$
Assume $\|u\|_{L^1(B_2)}\leq \frac{t\omega_n}{4}$ and $s\in(n-p,n)$. Then $$ \mathcal{H}_{\infty}^s(\mathcal{M}(u,t))\leq\frac{\Lambda}{t^p}\int_{B_2}
|\nabla u|^p, $$ where $\Lambda=\Lambda(n,s,p)$. Moreover, there exists a cover $\{\overline{B_{r_i}(x_i)}\}$ of $\mathcal{M}(u,t)$, such that $$ x_i\in\mathcal{M}(u,t),\,\,\,\, and\,\,\,\, \omega_s\sum_i r_i^s\leq\frac{\Lambda}{t^p}\int_{B_2}
|\nabla u|^p. $$ \end{lem} \proof Fix $x\in\mathcal{M}(u,t)$. By the definition of $\hat{u}$,
$\frac{1}{|B_{r_0}|}|\int_{B_{r_0}(x)}u|>t$ for sufficiently small ${r_0}$. We claim that there exists a small $r_0$ such that \begin{equation}\label{lemma1-claim}
\frac{1}{{r_0}^s}\int_{B_{r_0}(x)}|\nabla u|^p\geq t_1:=(\frac{t}{\Lambda'})^p, \end{equation} where $\Lambda'=\Lambda'(n,p,s)$ is a constant to be determined later.
Assume for contradiction that the claim is not true. By the proof of Poincar\'e inequality (see \cite[pp.275-276]{Evans}), we have $$
\frac{1}{|B_r|}\int_{B_r(x)}|u-u_{x,\frac{r}{2}}|^p\leq
\Lambda_1r^{p-n}\int_{B_r(x)}|\nabla u|^p, $$ where $\Lambda_1$ only depends on $n$. It follows \begin{eqnarray*}
|u_{x,r}-u_{x,\frac{r}{2}}|&=&\frac{1}{|B_r|}\left|\int_{B_r(x)}(u-u_{x,\frac{r}{2}})\right|\nonumber\\\nonumber
&\leq&\frac{1}{|B_r|}\left(\int_{B_r(x)}|u-u_{x,\frac{r}{2}}|^p\right)^\frac{1}{p}|B_r|^{1-\frac{1}{p}}\\
&=&\left(\frac{1}{|B_r|}\int_{B_r(x)}|u-u_{x,\frac{r}{2}}|^p\right)^\frac{1}{p}\nonumber\\
&\leq&\left(\Lambda_1r^{p-n}\int_{B_r(x)}|\nabla u|^p\right)^\frac{1}{p}\nonumber\\
&\leq&\Lambda_2r^\theta t_1^\frac{1}{p}, \end{eqnarray*} where $\theta=\frac{p-n+s}{p}$ and $\Lambda_2=\Lambda_1^\frac{1}{p}$.
For $r_0\in[2^{-k},2^{-k+1})$, we have $$
|u_{x,1}-u_{x,2^{-k}}|\leq \Lambda_2(\sum_{i=0}^{k-1} (2^{-i})^\theta)t_1^\frac{1}{p} \leq \Lambda_3 t_1^\frac{1}{p}, $$ and \begin{eqnarray*}
|u_{x,2^{-k}}-u_{x,r_0}|&=&\frac{1}{|B_{r_0}|}\left|\int_{B_{r_0}(x)}(u-u_{x,2^{-k}})\right|\\
&\leq&\frac{|B_{2^{-k+1}}|}{|B_{r_0}|}\frac{1}{|B_{2^{-k+1}}|}\int_{B_{2^{-k+1}(x)}}\left|u-u_{x,2^{-k}}\right|\\
&\leq&2^n\Lambda_2(2^{1-k})^\theta t_1^\frac{1}{p}, \end{eqnarray*} where $\Lambda_3=2\Lambda_2$. Then \begin{eqnarray}\label{decay}
|u_{x,1}-u_{x,r_0}|\leq\Lambda_3t_1^\frac{1}{p}= \frac{\Lambda_3}{\Lambda'}t. \end{eqnarray}
Note that $|u_{x,1}|\leq\frac{t}{4}$, so we get a contradiction if we set $\Lambda'>2\Lambda_3$. This proves our claim~(\ref{lemma1-claim}).
To complete the proof of the lemma, note that by Vitali Covering Theorem, there exist pairwise disjoint closed balls $\{\overline{B_{r_i}(x_i)}\}_{i=1}^\infty$ such that $$
\frac{1}{r_i^s}\int_{B_{r_i}(x_i)}|\nabla u|^p\geq t_1,\,\,\,\, \mathcal{M}(u,t)\subset\bigcup_i\overline{B_{5r_i}(x_i)}. $$ Therefore, we get \begin{eqnarray*} \mathcal{H}_{\infty}^s(\mathcal{M}(u,t))&\leq& \sum_i\omega_s(5r_i)^s=5^s\omega_s\sum_i r_i^s\\ &\leq&
\frac{1}{t_1}5^s\omega_s\int_{\cup B_{r_i}(x_i)}|\nabla u|^p\\
&\leq&\frac{1}{t_1}5^s\omega_s\int_{B_2}|\nabla u|^p. \end{eqnarray*} $
\Box$\\
As an application of Lemma~\ref{measure.estimate}, we show that $W^{1,p}$-convergence implies $\mathcal{H}^s$-a.e. convergence for each $s>n-p$. \begin{lem}\label{equal}
Assume $u_k, u\in W^{1,p}(B_2)$ and $\|u_k-u\|^p_{W^{1,p}(B_2)}<\frac{1}{2^k}$, then for any $s>n-p$, $\hat{u}_k$ converges to $\hat{u}$ for $\mathcal{H}^{s}$-a.e. $x\in B_1$. \end{lem} \proof Set $$
A=(\bigcup_{i=1}^\infty A(u_k))\bigcup A(u),\,\,\,\, E_{km}=\{x\in B_1\setminus A:|\hat{u}_k-\hat{u}|<\frac{1}{m}\}, $$ and $$ E=\bigcap_{m=1}^\infty\bigcup_{i=1}^\infty\bigcap_{k=i}^\infty E_{km}. $$ It is easy to check that for any $x\in E$, $\hat{u}_k(x)$ converges to $\hat{u}(x)$.
Let $$ F=E^c\cap B_1\setminus A=\bigcup_{m=1}^\infty\bigcap_{i=1}^\infty\bigcup_{k=i}^\infty F_{km}, $$ where $$
F_{km}=\{x\in B_1\setminus A:|\hat{u}_k-\hat{u}|\geq\frac{1}{m}\}. $$
Since $\hat{u}_k-\hat u=\widehat{u_k-u}$ and $\frac{1}{|B_1(x)|}
\int_{B_1(x)}|u_k-u|dx\rightarrow 0$, by Lemma \ref{measure.estimate}, $\mathcal{H}^s_{\infty}(F_{km})\leq Cm2^{-k}$ when $k$ is sufficiently large. It follows that $$ \mathcal{H}_{\infty}^s(\bigcap_{i=1}^\infty\bigcup_{k=i}^\infty F_{km})=0, $$ which implies $\mathcal{H}^s(F)=0$. Since $B_1\setminus E \subset A\cup F$, we get $\mathcal{H}^s(B_1\setminus E)=0$. $
\Box$\\
\begin{rem}\label{second} Lemma \ref{equal} provides another approach to define the value of $u$ at a point. Select a sequence of smooth functions
$u_k$ satisfying $\|u_k-u\|_{W^{1,p}}<2^{-k}$. Since $\hat{u}_k=u_k$, by Lemma \ref{equal}, $u_k$ converges to $\hat{u}$ for $\mathcal{H}^{s}$-a.e. $x$ whenever $s>n-p$. Therefore, $\hat{u}$ is in fact an $\mathcal{H}^s$-a.e. limit of $u_k$. Using this point of view, one can easily check the following:
1) when $f\in C^1$, $\widehat{f u}=f\widehat{u}$ for $\mathcal{H}^s$-a.e. $x$.
2) when $s>n-\frac{p}{q}$, $\widehat{u^q}=\hat{u}^q$ for $\mathcal{H}^s$-a.e. $x$. \end{rem}
Let $p>n-m$ and $\Sigma$ be a compact $m$-dimensional submanifold of $B_1$. We can establish a trace embedding inequality on $\Sigma$. Applying Theorem 1.1.2 in \cite{V.G.-T.O.} to $\mu=\mathcal{H}^m\lfloor\Sigma$, we have $$
\|u\|_{L^1(\Sigma)}\leq C(\Sigma)\|u\|_{W^{1,m}(\mathbb{R}^n)}, $$ where $u\in C_0^\infty(\mathbb{R}^n)$. Given a function $u\in W^{1,p}(B_1)$, after extending it to a function $u'\in W^{1,p}_{0}(B_2)$ with $$
\|u'\|_{W^{1,p}(B_2)}\leq C(n) \|u\|_{W^{1,p}(B_1)}, $$
we can find $u_k\in C^\infty_0(B_2)$ such that $\|u_k-u'\|_{W^{1,p}(\mathbb{R}^n)}\rightarrow 0$. Then $\{u_k\}$ is a Cauchy sequence in $L^1(\Sigma)$. By Lemma \ref{equal}, we may assume $u_k$ converges to $\hat{u}$ for $\mathcal{H}^m-$a.e. $x\in\Sigma$. Therefore, we obtain \begin{equation}\label{trace.inequality}
\int_\Sigma|\hat{u}|d\mathcal{H}^m\lfloor\Sigma\leq C(\Sigma)
\|u\|_{W^{1,p}(B_1)}. \\ \end{equation}
Now we consider the case when $(M,g)$ is a smooth Riemannian manifold and $u\in W^{1,p}(M)$. For $x\in M$, in a local coordinate chart $(x^1,\cdots,x^n)$ centered at $x$, we can define $\hat{u}(x)$ to be the limit of $\frac{1}{\omega_n r^n} \int_{B_r}udx$ as $r\rightarrow 0$, where $B_r$ is the Euclidean ball as before. In view of Remark \ref{second}, one checks that the value of $\hat{u}(x)$ is independent of the choice of coordinate chart for $\mathcal{H}^s-$ a.e. $x\in M$, where $s>n-p$.
\begin{lem}\label{mf} There exists a subset $E\subset M$ with dimension smaller than $(n-p)$ such that for any $x\notin E$, there holds \begin{equation}\label{mfd} \hat{u}(x)=\lim_{r\to 0}\frac{1}{\mathrm{Vol}(B^{g}_r(x))}\int_{B^{g}_r(x)} udV_g, \end{equation} where $$ B_r^g(p)=\{x\in M : d_g(x,p)<r\}. $$ \end{lem} \proof Locally in a coordinate $(x^1,\cdots,x^n)$, we set $$
\Lambda_s=\{x:\varlimsup_{r\to 0} \frac{1}{r^s}\int_{B_r^n(x)}|\nabla u|^pdx>0\},\,\,\,\, and \,\,\,\, \Lambda=\bigcap_{s\in(n-p,n)}\Lambda_s. $$ It is a standard result that $H^s(\Lambda_s)=0$ when $s\in(n-p,n)$ (c.f. \cite[Lemma 2.1.1]{Lin-Yang}). Since $\Lambda_{s'}\subset\Lambda_{s}$ for any $s'<s$, we have $\dim \Lambda<n-p$. We will show that \eqref{mfd} holds for any $x\notin \Lambda$. Obviously, we only need to prove \eqref{mfd} holds for any $x\notin \Lambda_s$ and $s\in (n-p,n)$.
Fix an $x_0\notin \Lambda$. As in \eqref{decay}, we have $$
|u_{x,r}-u_{x,r'}|\leq \Lambda_3\left(\frac{1}{r^s}\int_{B_r(x)}|\nabla u|^pdx\right)^\frac{1}{p},\,\,\,\, whenever\,\,\,\, r'<r. $$ Thus $u_{x,r}$ converges as $r\rightarrow 0$ for any $x\notin \Lambda_s$. Denoting
$u_r(x)=u(x_0+rx)$ and applying the Poincar\'e inequality for a ball with any fixed radius $R>0$, we get $$
\int_{B_R}\left|u_r(x)-\frac{1}{|B_1|}\int_{B_1}u_rdx\right|dx\leq
\int_{B_R}|\nabla u_r|^pdx=R^sr^{s+p-n}\frac{1}{(Rr)^s}\int_{B_{Rr}(x_0)}|\nabla u|^pdx \rightarrow 0. $$
Since $\frac{1}{|B_1|}\int_{B_1}u_rdx=u_{x_0,r}$ converges to $\hat{u}(x_0)$, we have $$
\lim_{r\rightarrow 0}\int_{B_R}\left|u_r(x)-\hat{u}(x_0)\right|dx=0. $$ Note that, when $r$ is sufficiently small, we may assume $B_1^{g(x_0+rx)/r^2}(0)\subset B_R$. Then we conclude \begin{eqnarray*}
& \ &\lim_{r\to0}\frac{1}{\mathrm{Vol}(B_r^{{g}}(x_0))}\int_{B_{r}(x_0)} |u-\hat{u}(x_0)|dV_g\\
&=&\lim_{r\to0}\frac{1}{\mathrm{Vol}(B_1^{{g(x_0+rx)}/r^2}(0))}\int_{B_1^{{g(x_0+rx)}/r^2}(0))} |u_r-\hat{u}(x_0)|dV_{g(x_0+rx)/r^2}\\
&\leq& C(M)\cdot\lim_{r\to0}\int_{B_R}|u_r-\hat{u}(x_0)|dx\\ &=&0. \end{eqnarray*} $
\Box$\\
\section{$W^{1,p}$-metrics} Suppose $(M,g_0)$ is a smooth closed $n$-manifold. Let $g$ be a symmetric tensor of type $(0,2)$, which is positive almost everywhere. Let $g^{-1}$ be the corresponding inverse tensor of type $(2,0)$. We say that $g$ is a $W^{1,p}$-metric if both $g$ and $g^{-1}\in W^{1,p}_{loc}(M,g_0)$. The goal of this section is to define the distance functions induced by $W^{1,p}$-metrics and study the compactness of such metrics. In a local coordinate chart, we can write $$ g=g_{ij}dx^i\otimes dx^j,\,\,\,\, and\,\,\,\, g^{-1}=g^{ij}\frac{\partial}{\partial x^i}\otimes \frac{\partial}{\partial x^j}. $$ Then the functions $g_{ij}$, $g^{ij}$ belong to $W^{1,p}_{loc}$, and $(g^{ij})(g_{ij})=I$ as matrices. Now define the centered average limit of the metric by $$
\hat{g}_{ij}(x)= \lim_{r\to 0}\frac{1}{|B_r(x)|}\int_{B_r(x)} g_{ij}(y)dy, $$ and the corresponding tensor by $$ \hat{g}(x)(V(x),V(x))=\sum_{i,j}\hat{g}_{ij}(x)V_i(x)V_j(x), \quad \forall V\in \Gamma(TM). $$ By Remark \ref{second}, when $s>n-p$, $\hat{g}$ and $\widehat{g^{-1}}$ are well-defined on $T_xM$ and $(T_xM)^*$ for $\mathcal{H}^s$-a.e. $x$. Moreover, by Lemma \ref{mf} \begin{align*} \hat{g}(x)(V(x),V(x))&=\lim_{r\to 0}\frac{1}{\mathrm{Vol}(B_r^{g_0}(x))}\int_{B_r^{g_0}(x)}g(y)(V(y),V(y))dV_{g_0}
\\ &= \sum_{i,j} \left(\lim_{r\to 0}\frac{1}{|B_r(x)|}\int_{B_r(x)} g_{ij}(y)dy\right)V_i(x)V_j(x) \end{align*}
Next, we define the associated distance function by $$ d_g(x,y)=\inf\left\{ \int_\gamma \sqrt{g(\dot{\gamma},\dot{\gamma})}: \mbox{piecewise smooth $\gamma$ from $x$ to $y$} \right\}.$$ First of all, we need to show that $d_g$ is indeed a distance function. \begin{lem} When $p\in (n-1,n)$, $d_g$ is a distance function and it's continuous on $M\times M$. \end{lem}
\proof We first show that $d_g(x,y)<+\infty$ for any $x, y\in M$.
Let $\varphi_x$ be the exponential map from $T_xM$ to $M$. Since $(M,g_0)$ is compact, there exists a number $\tau=\tau(M,g_0)>0$, such that for each $x\in M$, $\varphi_x$ induces normal coordinates $({x'}^1,\cdots,{x'}^n)$ on $B_\tau^{g_0}(x)$ with $$
|g_{0,ij}(x')-\delta_{ij}|<\frac{1}{2}. $$ It follows that the metric $$ g=g_{ij}(x')d{x'}^i\otimes d{x'}^j, $$ satisfies \begin{equation}\label{wip.chart}
\frac{1}{C}\|g\|_{W^{1,p}(B_\tau^{g_0}(x),g_0)}\leq\|(g_{ij})\|_{W^{1,p}(B_\tau^n(0))}\leq C\|g\|_{W^{1,p}(B_\tau^n(0))}, \end{equation} and \begin{equation}\label{wip.chart2}
\frac{1}{C}\|g^{-1}\|_{W^{1,p}(B_\tau^{g_0}(x),g_0)}\leq\|(g^{ij})\|_{W^{1,p}(B_\tau^n(0))}\leq C\|(g^{-1})\|_{W^{1,p}(B_\tau^n(0))}, \end{equation}
where $C$ is a constant independent of $x$. Following \cite[p.178]{Petersen}, we call a curve $\gamma_{pq}$ joining $p$ and $q$ a \emph{segment}in $(M,g_0)$ if $Length(\gamma_{pq})=d_{g_0}(p,q)$ and $|\dot\gamma_{pq}|$ is constant. Let $\gamma:[0,l]\rightarrow M$ be the segment in $(M,g_0)$ from $x$ to $y$. Then we can find $x_1=\gamma(t_1)$, $x_2=\gamma(t_2)$, $\cdots$, $x_m=\gamma(t_m)$, such that $$ m<\frac{2l}{\tau}, \,\,\,\, 0\leq t_{i+1}-t_i\leq \frac{\tau}{2}. $$ For convenience, we set $x_0=\gamma(0)=x$ and $x_{m+1}=\gamma(t_{m+1})=y$.
It suffices to prove $d_{g}(x_i,x_{i+1})<+\infty$. Without loss of generality, we assume the coordinate of $x_{i+1}$ in chart $(B_\tau^{g_0}(x_i),\varphi_{x_i}^{-1})$ is $(\delta_i,0,\cdots,0)$, where $\delta_i=t_{i+1}-t_i$. Obviously, $$ d_g(x_i,x_{i+1})\leq\int_0^{\delta_i}\sqrt{\hat{g}_{11}(t,0,\cdots,0)}dt\leq \delta_i^\frac{1}{2}\sqrt{\int_0^{\delta_i} \hat{g}_{11}(t,0,\cdots,0)dt}. $$ By \eqref{wip.chart}, \eqref{wip.chart2} and \eqref{trace.inequality}, $\hat{g}_{11}(t,0,\cdots,0)$ is integrable on $[0,\delta_i]$. It follows that $d_g(x,y)<+\infty$ and $d_g(x,y)\rightarrow 0$ when $l\to 0$.
Next, we prove that $d_g(x,y)>0$ for any $x\neq y$. In fact, we can prove a stronger result here: for any $\delta>0$, there exists $\delta'>0$, which depends on $g_0$, $\delta$ and
$||g^{-1}||_{W^{1,p}(M,g_0)}$, such that if $d_{g_0}(x,y)\geq\delta$, then \begin{equation}\label{positive.distance} d_g(x,y)\geq \delta'. \end{equation}
Assume $\gamma:[0,l]\rightarrow M$ is an arbitrary piecewise smooth curve from $x$ to $y$ in $M$. We set $$ l'=\sup\{t\in[0,l]:\gamma([0,t])\subset B_\frac{\tau}{2}^{g_0}(x)\}. $$ Obviously, $$ d_{g_0}(x,\gamma(l'))=\min\{\tau/2,d_{g_0}(x,y)\}. $$ In the coordinate defined by $\varphi_x$, we set
$\|g_{ij}\|=\sqrt{\sum (g_{ij})^2}$. It is well-known that
$\|(g_{ij})\|^2$ is the quadratic sum of the eigenvalues of $(g_{ij})$. Let $\lambda$ be the smallest eigenvalue of $(\hat{g}_{ij})$. Since $\frac{1}{\lambda}$ is an eigenvalue of $(\hat{g}^{ij})$, we have $$
E_a:=\{x'\in B_\frac{\tau}{2}^n(0):\lambda(x')<a\}\subset\{x'\in B_\frac{\tau}{2}^n(0):\|(\hat{g}^{ij})\|(x')>\frac{1}{a}\}. $$
By the inequality $\|(\hat{g}^{ij})\|\leq c(n)\sum_{ij} |\hat{g}^{ij}|$, together with Lemma \ref{measure.estimate}, we can find a sufficiently small $a$, which depends on $||g^{-1}||_{W^{1,p}(M)}$, $\delta$ and $\tau$, such that $$
\mathcal{H}_{\infty}^1(\varphi^{-1}_x(\gamma|_{[0,l']})\cap E_a)\leq\mathcal{H}_\infty^1(\{x'\in B_\frac{\tau}{2}^n(0):\|\hat{g}^{ij}\|>\frac{1}{a}\})\leq \sum_{ij}a^{p}\Lambda'\|{g}^{ij}\|^p_{W^{1,p}(B_\frac{\tau}{2}^n(0))}< \frac{d_{g_0}(x,\gamma(l'))}{4}. $$
Note that using balls to cover $\varphi^{-1}_x(\gamma|_{[0,l']})$ might increase $\mathcal{H}_{\infty}^1(\varphi^{-1}_x(\gamma|_{[0,l']}))$ by at most a factor of $2$ (see \cite{Han-Lin}). It follows that $$
d_{\mathbb{R}^n}(0,\varphi_x^{-1}(\gamma(l')))=d_{g_0}(x,\gamma(l')) \leq 2\mathcal{H}_{\infty}^1(\varphi^{-1}_x(\gamma|_{[0,l']})), $$ which implies \begin{eqnarray*}
\mathcal{H}^1(\varphi^{-1}_x(\gamma|_{[0,l']})\setminus E_a)&\geq&\mathcal{H}_{\infty}^1(\varphi^{-1}_x(\gamma|_{[0,l']})\setminus E_a) \geq\mathcal{H}^1_{\infty}(\varphi^{-1}_x(\gamma|_{[0,l']}))-\mathcal{H}^1_{\infty}(\varphi^{-1}_x(\gamma|_{[0,l']})\cap E_a)\\ &\geq&\frac{d_{g_0}(x,\gamma(l'))}{2} -\frac{d_{g_0}(x,\gamma(l'))}{4}=\frac{1}{4}d_{g_0}(x,\gamma(l')). \end{eqnarray*} Since $\gamma$ is locally Lipschitz continuous, $$
\int_{\varphi_x^{-1}(\gamma|_{[0,l']}) \setminus E_a} |\dot\gamma|\geq \mathcal{H}^1(\varphi^{-1}_x(\gamma|_{[0,l']})\setminus E_a). $$ we have $$ \int_\gamma\sqrt{\hat{g}(\gamma)(\dot\gamma,\dot\gamma)}
\geq\int_{\varphi_x^{-1}(\gamma|_{[0,l']})\setminus E_a}\sqrt{\hat{g}_{ij}(\gamma)\dot\gamma^i\dot\gamma^j}
\geq\int_{\varphi_x^{-1}(\gamma|_{[0,l']}) \setminus E_a}\lambda|\dot\gamma|\geq\frac{a}{4}d_{g_0}(x,\gamma(l')). $$ This completes the proof, by letting $\delta'=\frac{a}{4}\min\{\tau/2,\delta\}$.
$
\Box$\\
{\it The proof of Theorem \ref{main1}:}
Since $|\nabla_{g_k,x} d_{g_k}(x,y)|=1$, in local coordinates we have $$
\lambda(x)|\nabla_{x} d_{g_k}(x,y)|\leq 1, $$ where $\lambda$ is the smallest eigenvalue of $(g_{k,ij})$. Since $\frac{1}{\lambda}$ is an eigenvalue of $g_k^{ij}$, we see $$
|\nabla_x d_{g_k}(x,y)|\leq\frac{1}{\lambda}\leq c(n)\sum_{ij} |g_k^{ij}(x)|. $$
Similarly, we have $|\nabla_y d_{g_k}(x,y)|<c(n)\sum_{ij}|g_k^{ij}(y)|$, hence $d_{g_k}$ is bounded in $W^{1,\frac{np}{n-p}}(M\times M,g_0)$. Therefore we may assume $d_{g_k}$ converges to a function $d$ in $C^0(M\times M)$. By \eqref{positive.distance}, we can assume further \begin{equation}\label{positive.distance2} d(x,y)\geq \tau,\,\,\,\, \mbox{whenever} \,\,\,\, d_{g_0}(x,y)>\delta. \end{equation}
Next, we prove that $d\leq d_{g}$. Given two points $x,y \in M$, take a piecewise smooth curve $\gamma$ from $x$ to $y$. By \eqref{trace.inequality}, after passing to a subsequence, $\sqrt{g_{k}(\dot{\gamma},\dot{\gamma}})$ converges to $\sqrt{g(\dot{\gamma},\dot{\gamma})}$ for $\mathcal{H}^1$-a.e. $x\in \gamma$. Moreover, we have $$ \int_\gamma\left(\sqrt{g_k(\dot{\gamma},\dot{\gamma})}\right)^2<C, $$ which implies $$ \lim_{k\rightarrow+\infty}\int_\gamma \sqrt{g_k(\dot{\gamma},\dot{\gamma})}=\int_\gamma\sqrt{g(\dot{\gamma},\dot{\gamma})}. $$ Thus, we arrive at $d(x,y)\leq d_g(x,y)$.
Finally, we show that $d=d_{g}$ in the case when $g$ and $g^{-1}$ are continuous. For any $\varepsilon>0$ fixed, let $$ E_k=\{x:g_k>(1-\varepsilon)g\}=\{x:g-g_k<\varepsilon g\},\,\,\,\, and\,\,\,\, F_k=E_k^c. $$ We claim that $$ \lim_{k\rightarrow+\infty}\mathcal{H}^1_\infty(F_k)=0. $$ Since $M$ is compact, we only need to prove the claim in a local coordinate chart $\varphi: U\rightarrow \mathbb{R}^n$. That is, we only need to check that for any $B_R \subset \varphi(U)$, $$ \mathcal{H}^1_\infty(B_R\cap F_k)\rightarrow 0. $$
For simplicity, we denote the maximum eigenvalue and the minimum eigenvalue of a matrix $A$ by $\Lambda(A)$ and $\lambda(A)$ respectively. Since $g$ and $g^{-1}$ are continuous, we assume for any $x\in B_R$, $$
\frac{\lambda(g_{ij}(x))}{\|(g_{ij}(x))\|}\geq\varepsilon_1 $$ for some $\varepsilon_1>0$. Note that \begin{eqnarray*} F_k\cap B_R&\subset&\{x\in B_R:\Lambda(g_{ij}-g_{k,ij})\geq\varepsilon\lambda(g_{ij})\}\\
&\subset&\{x\in B_R:\|g_{ij}-g_{k,ij}\|\geq\varepsilon\lambda(g_{ij})\}\\
&\subset&\{x\in B_R:\|g_{ij}\|\cdot\|I-g_{k,ij}g^{ij}\|\geq\varepsilon\lambda(g_{ij})\}\\
&\subset&\{x\in B_R:\|I-g_{k,ij}g^{ij}\|\geq\varepsilon_1\epsilon\}. \end{eqnarray*}
From the identity $$ \nabla(g_{k,ij})(g^{ij})=\nabla (g_{k,ij})(g^{ij})+ (g_{k,ij})(\nabla g^{ij}), $$ we see $$
\|I-(g_{k,ij})(g^{ij})\|_{W^{1,q}}\rightarrow 0, $$ for any $q<\frac{np}{2n-p}$. Since $p>2n\frac{n-1}{2n-1}$, we can choose $q$ such that $q>n-1$. By Lemma \ref{measure.estimate}, after passing to a subsequence, we get $$
\lim_{k\rightarrow+\infty}\mathcal{H}^1_\infty(\{x\in B_R:\|I-(g_{k,ij})(g^{ij})\|\geq\varepsilon_1\varepsilon\})=0. $$ Thus the claim follows.
The above claim implies that, given $\varepsilon'>0$, for any $k$ sufficiently large, we can cover $F_k$, which is a compact subset, by finitely many balls $\overline{B_{r_1}}(x_1)$, $\cdots$, $\overline{B_{r_m}}(x_m)$, such that $$ \sum r_i<\varepsilon'. $$ Let $C_1$, $\cdots$, $C_{m'}$ be the connected components of $B=\bigcup \overline{B_{r_i}(x_i)}$ and set $t_1=\inf\{t:\gamma(t)\in B\}$. Without loss of generality, we assume $\gamma(t_1)\in C_1$. Put $t_2=\sup\{t:\gamma(t)\in C_1\}$, and replace
$\gamma|_{[t_1,t_2]}$ with the segment $\overline{\gamma(t_1)\gamma(t_2)}$. In the same manner, we may choose $t_3=\inf\{t:\gamma(t)\in B\setminus C_1\}$ and by induction, we can find $$ 0\leq t_1<t_2<t_3<\cdots<t_{m'}\leq 1, $$ such that $$ \sum_id_{g_0}(\gamma(t_{2i}),\gamma(t_{2i-1}))\leq\sum_idiam(C_i)\leq\sum r_i<\varepsilon'. $$ This give rise to a new curve $\gamma'$ in place of $\gamma$, such that Then \begin{align*} \int_{\gamma} \sqrt{g_k(\dot\gamma,\dot\gamma)} &\geq \int_{\gamma\cap\gamma'} \sqrt{g_k(\dot\gamma,\dot\gamma)}\\ &\geq (1-\varepsilon)^\frac{1}{2}( \int_{\gamma'}\sqrt{g({\dot{\gamma}}',{\dot{\gamma}}')}-\sum_i\int_{\gamma(t_{2i-1})}^{\gamma(t_{2i})}\sqrt{g(\dot{\gamma'},\dot{\gamma'}})) \\ &\geq
(1-\varepsilon)^\frac{1}{2}(d_g(x,y)-\varepsilon'\|\sqrt{g}\|_{C^0}). \end{align*} Therefore, $$
d_{g_k}(x,y)\geq (1-\varepsilon)^\frac{1}{2}(d_g(x,y)-\varepsilon'\|\sqrt{g}\|_{C^0}). $$ Now let $k\rightarrow+\infty$, then $\varepsilon'\rightarrow 0$, and finally $\varepsilon\rightarrow 0$, we get $$ d(x,y)\geq d_g(x,y). $$ $
\Box$\\
Although we only consider a compact manifold in Theorem \ref{main1}, a similar result actually holds for certain complete manifolds. For example, we have \begin{pro}\label{complete.case} Let $\{g_k\}$ be a sequence of metrics defined on $\mathbb{R}^n$ and assume $g_k$ and $g_k^{-1}$ converge to $g_{\mathbb{R}^n}$ and $g_{\mathbb{R}^n}^{-1}$ respectively in $W_{loc}^{1,p}(\mathbb{R}^n)$ for some $p>2n\frac{n-1}{2n-1}$. Then, after passing to a subsequence, $d_{g_k}(x,y)$ converges to
$|x-y|$. \end{pro}
\proof Let $R=|x-y|$. We only need to prove this proposition on $\overline{B_{2R}}$. We omit the details since the proof is almost the same with the one of Theorem \ref{main1}. $
\Box$\\
\section{Conformal metrics with $L^\frac{n}{2}$-bounded scalar curvature}\label{conformal.metrics} First we recall some notations in conformal geometry. Let $(M,g)$ be a closed Riemannian manifold. Denote the scalar curvature by $R(g)$ (or $R_g$). Let $g=u^{\frac{4}{n-2}}g_0$ be a conformal metric, then $u$ satisfies the following equation $$ -\frac{4(n-1)}{n-2}\Delta u+R(g_0)u=R(g)u^\frac{n+2}{n-2}. $$
\subsection{$\varepsilon$-regularity} Again we denote by $B_r$ a ball in $\mathbb{R}^n$ with radius $r$, centered at $0$. Let $u$ be a weak solution of \begin{equation}\label{equation.epsilon} -div(a^{ij}u_{j})=fu, \end{equation} where \begin{equation}\label{aij}
0<\lambda_1 I\leq (a^{ij}),\,\,\,\, \|a^{ij}\|_{C^0(B_2)}+\|\nabla a^{ij}\|_{C^0(B_2)} <\lambda_2. \end{equation}
\begin{lem}\label{Lalpha} Suppose $u\in W^{1,2}(B_2)$ is a positive weak solution of equations \eqref{equation.epsilon} and \eqref{aij}. Assume $$
\int_{B_2}|f|^\frac{n}{2}\leq \Lambda, $$ then $$
r^{2-n}\int_{B_r(x)}|\nabla\log u|^2<C,\,\,\,\, \forall B_r(x)\subset B_1. $$ Moreover, there exist constants $\alpha$ and $C$, which depend on $\Lambda$, $\lambda_1$, $\lambda_2$, such that $$ \int_{B_1}(cu)^{\alpha}+\int_{B_1}(cu)^{-\alpha}<C, $$ where $-\log c$ is the mean value of $\log u$ on $B_1$. \end{lem} \proof
For a ball $B_{2r}(x)\subset B_2(0)$, take $\phi=\eta^2u^{-1}$ as a test function, with $\eta\equiv 1$ on $B_r(x)$, $\eta\in C^\infty_0(B_{2r}(x))$ and $|\nabla \eta|\leq \frac{C}{r}$. Multiplying \eqref{equation.epsilon} by $\phi$ and integrating, we get $$
\int_{B_{2r}(x)}\eta^2u^{-2}|\nabla u|^2\leq C\left(\int_{B_{2r}(x)} |\nabla\eta|^2+(\int_{B_{2r}(x)} f^{\frac{n}{2}})^{\frac{2}{n}}(\int_{B_{2r}(x)} \eta^{\frac{2n}{n-2}})^{\frac{n-2}{n}}\right), $$ which implies \begin{align*}
\int_{B_{r}(x)}|\nabla \log u|^2&\leq Kr^{n-2}. \end{align*} By the Sobolev Embedding Theorem and the John-Nirenberg Lemma \cite[Theorem 3.5]{Han-Lin}, for $\alpha=\frac{C(n)}{K}$, we have \begin{equation}\label{JN}
\|u\|_{L^{\alpha}(B_{1})}\|u^{-1}\|_{L^{\alpha}(B_1)}\leq C. \end{equation} Let $v=\log cu$, where $c$ is chosen such that $$ \int_{B_1}v=0. $$ By the Poincar\'e inequality, we can assume $$
\int_{B_1}|v|\leq \beta_0, $$ where $\beta_0$ only depends on $\Lambda$, $\lambda_1$ and $\lambda_2$. Denote the Lebesgue measure over $\mathbb{R}^n$ by $L^n$. Let $$ E=\{x:v\leq \frac{2\beta_0}{L^n(B_1)}\}. $$ Then $$
L^n(B_1\setminus E)\leq \frac{L^n(B_1)}{2\beta_0}\int_{B_1}|v|\leq\frac{L^n(B_1)}{2}, $$ and $L^n(E)\geq\frac{1}{2}L^n(B_1)$. By \eqref{JN}, we get $$ C\geq \int_{B_1}(cu)^\alpha\int_{B_1}(cu)^{-\alpha}\geq\int_{B_1}(cu)^{\alpha}\int_{E}(cu)^{-\alpha}\geq \frac{1}{2}L^n(B_1) e^{-\frac{2\alpha \beta_0}{L^n(B_1)}}\int_{B_1}(cu)^{\alpha}. $$
In the same way, we can get the estimate of $\int_{B_1}(cu)^{-\alpha}$. $
\Box$\\
\begin{lem}\label{regularity} Suppose $u\in W^{1,2}(B_2)$ is a positive solution of \eqref{equation.epsilon}, \eqref{aij}, and $\log u\in W^{1,2}(B_2)$. Then for any $q\in (0,\frac{n}{2})$, there exists $\varepsilon_0 =\varepsilon(q,\lambda_1,\lambda_2)>0$, such that if $$
\int_{B_2}|f|^\frac{n}{2}<\varepsilon_0, $$ then $$
\|\nabla\log u\|_{W^{1,q}(B_{\frac{1}{2}})}\leq C(\lambda_1,\lambda_2,\epsilon_0). $$ and $$
e^{-\frac{1}{|B_\frac{1}{2}|}\int_{B_\frac{1}{2}}\log u}\|u\|_{W^{2,q}(B_\frac{1}{2})}+e^{\frac{1}{|B_\frac{1}{2}|}\int_{B_\frac{1}{2}}\log u}\|u^{-1}\|_{W^{2,q}(B_\frac{1}{2})} \leq C(\lambda_1,\lambda_2,\epsilon_0). $$ \end{lem} \proof Let $v=\log u$. In order to apply Lemma \ref{Lalpha}, we firstly assume $\int_{B_1}v=0$.
Let $\eta$ be a smooth cutoff function and $\phi=\eta^2u^{\beta}$ be a test function, where $\eta$ and $\beta\neq -1$ or $0$ will be defined later. Multiplying both sides of (\ref{equation.epsilon}) by $\phi$ and integrating, we obtain $$
\int_{B_1} 2\eta\nabla\eta u^{\beta}\nabla u+\int_{B_1} \eta^2\beta u^{\beta-1}|\nabla u|^2=\int f\eta^2u^{\beta+1}. $$ By Young inequality and H\"older inequality: \begin{align}\label{ie1}
|\beta|\int_{B_1} \eta^2 u^{\beta-1} |\nabla u|^2 \leq \frac{C}{|\beta|}\int_{B_1} |\nabla \eta|^2 u^{\beta+1}+(\int_{B_1} |f|^\frac{n}{2})^\frac{2}{n}(\int_{B_1} (\eta^2 u^{\beta+1})^\frac{n}{n-2})^\frac{n-2}{n}. \end{align} Applying the Sobolev inequality and Poincar\'e inequality to $\eta u^\frac{\beta+1}{2}$ , we get \begin{align*}
(\int_{B_1} (\eta u^\frac{\beta+1}{2})^\frac{2n}{n-2})^{\frac{n-2}{n}}&\leq \alpha_n\int_{B_1}|\nabla(\eta u^{\frac{\beta+1}{2}})|^2
\\&\leq 2\alpha_n \int_{B_1}(\nabla \eta)^2 u^{\beta+1}+2\alpha_n \int_{B_1}(\eta)^2|\nabla u^\frac{\beta+1}{2}|^2, \end{align*} which together with (\ref{ie1}) gives \begin{align}\label{ie2}
\frac{4|\beta|}{(\beta+1)^2}\int_{B_1} \eta^2|\nabla u^\frac{\beta+1}{2}|^2 \leq (\frac{C}{|\beta|}+C\varepsilon_0)\int_{B_1} |\nabla\eta|^2 u^{\beta+1}+C\varepsilon_0\int_{B_1} \eta^2|\nabla u^\frac{\beta+1}{2}|. \end{align} When $$
C\varepsilon_0\leq \frac{2|\beta|}{(\beta+1)^2}, $$ we have $$
\frac{2|\beta|}{(\beta+1)^2}\int_{B_1} \eta^2|\nabla u^\frac{\beta+1}{2}|^2 \leq (\frac{C}{|\beta|}+\frac{2|\beta|}{(\beta+1)^2})\int_{B_1} |\nabla\eta|^2 u^{\beta+1}, $$ and $$
\frac{2|\beta|}{(\beta+1)^2}\int_{B_1} |\nabla\eta u^\frac{\beta+1}{2}|^2 \leq (\frac{C}{|\beta|}+\frac{6|\beta|}{(\beta+1)^2})\int_{B_1} |\nabla\eta|^2 u^{\beta+1}. $$
Take $\frac{1}{2}\leq r_1< r_2\leq 1$. Let $\eta\in C^\infty_0(B_{r_2})$, $\eta\equiv 1$ on $B_{r_1}$ and $|\nabla \eta|\leq\frac{C}{|r_2-r_1|}$. By Poincar\'e inequality and Sobolev inequality, we get $$
(\int_{B_{r_1}} |u^{\frac{\beta+1}{2}}|^{2^*})^\frac{1}{2^*}\leq C\left(\frac{(\beta+1)^2}{\beta^2}+1\right)\frac{1}{|r_2-r_1|}(\int_{B_{r_2}} (u^\frac{1+\beta}{2})^2)^\frac{1}{2}, $$ where $2^*=2\frac{n}{n-2}$.
Next we deduce an uniform bound for $\|u\|_{L^p}$.
Let $\frac{\beta+1}{2}=\alpha$. We can choose $\varepsilon_0$ such that
$\|u\|_{L^{2^*\alpha}}<C$. Then by setting $\frac{\beta+1}{2}=2^*\alpha$ we can get $\|u\|_{L^{2^*\cdot 2^* \alpha}}<C$. After several iterations, we get an estimate of $\|u\|_{L^\frac{n}{n-2}}$. So without loss of generality, we assume $\|u\|_{L^\frac{n}{n-2}}<C$.
Denote $\alpha=\frac{n}{n-2}$ and take $$ \frac{n}{n-2}\geq p_0>1. $$ Then $$
(\int_{B_{r_1}} |u^{p_0\frac{\alpha(\beta+1)}{p_0}}|)^{\frac{p_0}{\alpha(\beta+1)}\frac{\beta+1}{2p_0}}\leq C\left(\frac{(\beta+1)^2}{\beta^2}+1\right)\frac{1}{|r_2-r_1|}(\int_{B_{r_2}} u^{p_0\frac{1+\beta}{p_0}})^{\frac{p_0}{\beta+1}\frac{\beta+1}{2p_0}}, $$ i.e. \begin{equation}\label{moser1}
(\int_{B_{r_1}} |u^{p_0\frac{\alpha(\beta+1)}{p_0}}|)^{\frac{p_0}{\alpha(\beta+1)}}\leq \left(C(\frac{(\beta+1)^2}{\beta^2}+1)\frac{1}{|r_2-r_1|}\right)^\frac{2p_0}{\beta+1}(\int_{B_{r_2}} u^{p_0\frac{1+\beta}{p_0}})^{\frac{p_0}{\beta+1}}. \end{equation} Take $\beta+1=\alpha^mp_0$, $r_1=\frac{1}{2}+\frac{1}{2^{m+2}}$ and $r_2=\frac{1}{2}+\frac{1}{2^{m+1}}$, where $m=0,1,2,\cdots$, $m_0$, and $$ m_0=\max\{m:C\varepsilon_0\leq \frac{2(\alpha^mp_0-1)}{(\alpha^mp_0)^2}\}. $$ Rewrite \eqref{moser1} as follows: $$
\|u^{p_0}\|_{L^{\alpha^{m+1}}(B_{r_1})}\leq C^\frac{2m}{\alpha^m}\|u^{p_0}\|_{L^{\alpha^m}(B_{r_2})}, $$ which implies $$
\|u^{p_0}\|_{L^{\alpha^{m_0+1}(B_{\frac{1}{2}})}}\leq C^{\sum\limits_{i=0}^{+\infty} i\alpha^{-i}}\|u^{p_0}\|_{L^{1}(B_{1})}. $$ Given $p\geq p_0$, select $m_0$ such that $p<p_0\alpha^{m_0+1}$ and choose $\varepsilon_0$ under additional assumption: $$ C\varepsilon_0\leq \min\{\frac{2(\alpha^mp_0-1)}{(\alpha^mp_0)^2}:m=0,1,\cdots,m_0\}. $$ It follows $$
\|u\|_{L^p(B_{\frac{1}{2}})}\leq C\|u\|_{L^{p_0\alpha^{m+1}}(B_{\frac{1}{2}})}\leq C\|u\|_{L^{p_0}(B_1)}\leq C. $$
Now we return to the elliptic equation \eqref{equation.epsilon}. For any $q<\frac{n}{2}$, we have $$ (\int_{B_\frac{1}{2}} (fu)^q)^{\frac{1}{q}}\leq (\int_{B_\frac{1}{2}} f^\frac{n}{2})^{\frac{2}{n}}(\int_{B_\frac{1}{2}} u^\frac{n}{n-2q})^{\frac{n-2q}{n}}. $$ Thus, if $p>\frac{n}{n-2q}$, we get $$
\|u\|_{W^{2,q}(B_\frac{1}{4})}<C. $$
Finally, we derive the estimate of $\|u^{-1}\|_{W^{2,q}}$. Similar to the above arguments, one can get $\|u^{-1}\|_{L^p(B_\frac{1}{4})}<C$. The estimate of $\|u^{-1}\|_{W^{2,q}}$ follows from the following: $$ \nabla u^{-1}=\frac{\nabla u}{u^2},\,\,\,\, \nabla^2u^{-1}=
\frac{\nabla^2u}{u^2}-2\frac{|\nabla u|^2}{u^3}. $$ Since $$ \nabla\log u=\frac{\nabla u}{u},\,\,\,\, \nabla^2\log u=
\frac{\nabla^2u}{u}-\frac{|\nabla u|^2}{u^2}, $$ we get the estimate of $\log u$.
Notice that for any positive constant $c$, $cu$ still satisfies the equation. So we can get the estimate of $\|\log u\|_{W^{2,p}}$ without the assumption that $\int_{B_1}\log u=0$. $
\Box$\\
\subsection{Proof of Theorem \ref{main2}}The main goal of this subsection is to prove Theorem \ref{main2}. For any $x\in M$, on a small ball $B_r(x)\subset M$, $g_0|_{B_r(x)}$ can be regarded as a metric over $B_r\subset \mathbb{R}^n$ and we have the following equation $$ -\frac{4(n-1)}{n-2}\Delta_{g_0} u_k=(-R(g_0)+R(g)u_k^\frac{4}{n-2})u_k. $$
By Lemma \ref{Lalpha}, $\|\nabla\log u_k\|_{L^2(M)}<C$. Let $-\log c_k$ be the mean value of $\log u_k$ over $M$. By the Poincar\'e inequality,
$\|\log c_ku_k\|_{L^1(M)}<C$. Note $c_ku_k$ also satisfies $$ -\frac{4(n-1)}{n-2}\Delta_{g_0}c_k u_k=(-R(g_0)+R(g)u_k^\frac{4}{n-2})c_ku_k. $$ Cover $M$ by finitely many balls $B_{r_1}(x_1)$, $\cdots$, $B_{r_m}(x_m)$. Assume for each $B_{r_i}(x_i)$, $$
\int_{B_{2r_i}(x_i)} |R_k|^2dV_{g_k}<\varepsilon_0. $$ Applying Lemma \ref{regularity} to $c_ku_k$, we get
$\|c_ku_k\|_{W^{2,q}(M)}+\|(c_ku_k)^{-1}\|_{W^{2,q}(M)}<C$.
Since $$ 1=\int_Mu_k^\frac{2n}{n-2}=\frac{1}{c_k^\frac{2n}{n-2}}\int_M(c_ku_k)^\frac{2n}{n-2}<C\frac{1}{c_k^\frac{2n}{n-2}}, $$ and \begin{eqnarray*} \mathrm{Vol}^2(M,g_0)&\leq\int_Mu_k^\frac{2n}{n-2} dV_{g_0}\int_Mu_k^{-\frac{2n}{n-2}} dV_{g_0}=\int_Mu_k^{-\frac{2n}{n-2}} dV_{g_0}\\ &=c_k^\frac{2n}{n-2}\int_M(c_ku_k)^{-\frac{2n}{n-2}} dV_{g_0} \leq Cc_k^\frac{2n}{n-2}, \end{eqnarray*} we get the bound of $c_k$. This proves the first part of Theorem \ref{main2}.
By Theorem \ref{main1}, we may assume the sequence $\{d_{g_k}\}$ converges to a distance function $d$ with $d\leq d_g$. To finish the proof of Theorem \ref{main2}, we need to show $d\geq d_g$. The key observation is the following:
\begin{lem}\label{du0/d0} For any $\varepsilon$, we can find $\beta$ and $\tau$, which only depend on $\varepsilon$, such that if $$ \mu(B_{2\delta}(0))<\tau, \,\,\,\, \delta<\delta_0, $$ then $$ \frac{d_g(x,y)}{d(x,y)}\leq1+\varepsilon,\,\,\,\, \forall x, y\in B_{\beta\delta}^{g_0}(0). $$ \end{lem}
\proof We argue by contradiction and assume the lemma is not true. Then we can find $\delta_m\to0$ and $y_m, x_m\in B_{\delta_m}(0)$, such that
$\frac{|x_m-y_m|}{\delta_m}\rightarrow 0$ and $$
\limsup_{k\rightarrow+\infty}\int_{B_{\delta(0)}}|R(g_k)|^\frac{n}{2}dV_{g_k}= 0,\,\,\,\, \frac{d(y_m,x_m)}{d_g(y_m,x_m)}\rightarrow a<1. $$
For any fixed $m$, by the Sobolev Embedding Theorem and Lemma \ref{regularity}, and taking $q$ sufficiently closed to $\frac{n}{2}$, we can find $k_m$ such that $\{u_{k_m}\}$ and $\{\frac{1}{u_{k_m}}\}$ converges to $u$ and $\frac{1}{u}$ respectively in $W^{1,p_0}$, with $p_0<\frac{nq}{n-q}$. Moreover, using the H\"older's Inequality, we can get that, after passing to a subsequence, $\{\frac{u_{k_m}}{u}\}$ converges to $1$ in $W^{1,p}_{loc}(B_{\delta_0}(0))$, for some $p\in (n-1,p_0)$. So we have $$
\left|\frac{d_{g_{k_m}}(y_m,x_m)}{d_g(y_m,x_m)}-\frac{d(y_m,x_m)}{d_g(y_m,x_{m})}\right|<\frac{1}{m}, $$ and \begin{equation}\label{L2}
\int_{B_{\delta_m}(x_m)}\left|\nabla\frac{u_{k_m}(y)}{u(y)}\right|^pdy
+\int_{B_{\delta_m}(x_m)}\left|\frac{u_{k_m}(y)}{u(y)}-1\right|^pdy<\frac{1}{m}, \end{equation}
where $r_m=|y_m-x_m|$ and $B_{\delta_m}(x_m)\subset B_{\delta_0}(0)$. By (\ref{L2}), we have \begin{equation}\label{L1}
r_m^{n-p}\int_{B_{\frac{\delta_m}{r_m}}(0)}\left|\nabla\frac{u_{k_m}(r_mx+x_m)}{u(r_mx+x_m)}\right|^pdx
+r_m^{n}\int_{B_{\frac{\delta_m}{r_m}}(0)}\left|\frac{u_{k_m}(r_mx+x_m)}{u(r_mx+x_m)}-1\right|^pdx<\frac{1}{m}, \end{equation}
For simplicity, we set $y_m=x_m+r_m(1,0,\cdots,0)$ in local coordinates. More precisely, we can take a segment $\gamma_m$ joining $x_m$ and $y_m$, such that $$ \gamma_m(t)=x_m+r_m\gamma(t),\,\,\,\, where\,\,\,\, \gamma(t)=(t,0,\cdots,0),\,\,\,\, t\in[0,1]. $$ Let $u_m'=c_mu_{k_m}(x_m+r_mx)$, where $c_m$ is chosen such that $$ 0=\int_{B_\frac{1}{2}}\log u_m'. $$ In the local coordinate, set $h_m(x)=(g_0)_{ij}(x_m+r_mx)dx^i\otimes dx^j$, which converges to $g_{\mathbb{R}^n}$ smoothly. Let $g_m'(x)=(u_m'(x))^\frac{4}{n-2}h_m(x)=\frac{c_m^\frac{4}{n-2}}{r_m^2}g_{k_m}(x_m+r_mx)$. Since for any $R>0$ $$
\int_{B_R(0)}|R(g_m')|^\frac{n}{2}dV_{g_m'}\leq\int_{B_{\delta_0(0)}}|R(g_k)|^\frac{n}{2} dV_{g_k}\rightarrow 0. $$ By Lemma \ref{regularity}, we may assume $u_m'$ converges to a positive harmonic function $u'$ weakly in $W^{2,q}_{loc}(\mathbb{R}^n)$ with $\int_{B_\frac{1}{2}}\log u'=0$, for some $q<\frac{n}{2}$. By Liouville's theorem, $u'$ is a constant. Since $\int_{B_\frac{1}{2}}\log u'=0$, $u'=1$. Hence, for a fixed $R>1$, $u_m'$ converges to $1$ in $W^{1,p}(B_R(0))$. By \eqref{L1}, $\frac{u(x_m+r_mx)}{u_{k_m}(x_m+r_mx)}$ converges to 1 in $W^{1,p}(B_R(0))$. According to the H\"older's Inequality, $u_m'(x)\frac{u(x_m+r_mx)}{u_{k_m}(x_m+r_{m}x)}$ converges to 1 in $W^{1,p'}(B_R(0))$, for some $p'\in (n-1,p)$. Then we have \begin{align*} \lim_{m\to \infty}c_m^\frac{2}{n-2}d_g(x_m,y_m)&\leq \lim_{m\to \infty}\int_{\gamma_m}(c_mu(\gamma_m(t)))^\frac{2}{n-2} \\ &= \lim_{m\to \infty}\int_0^1(c_mu(\gamma_m(t)))^\frac{2}{n-2} \sqrt{(g_0)_{ij}(\gamma_m(t))\dot{\gamma}_m^i(t)\dot{\gamma}_m^j(t)}dt \\ &=\lim_{m\to \infty} \int_0^1 \left(c_mu_{k_m}(x_m+r_{m}\gamma(t))\frac{u(x_m+r_m\gamma(t))}{u_{k_m}(x_m+r_{m}\gamma(t))}\right)^\frac{2}{n-2}dt \\ &=\lim_{m\to \infty} \int_0^1\left(u_m'(\gamma(t))\frac{u(x_m+r_m\gamma(t))}{u_{k_m}(x_m+r_{m}\gamma(t))}\right)^\frac{2}{n-2}dt. \end{align*} Therefore, we get $$ \lim_{m\rightarrow+\infty} c_m^\frac{2}{n-2}d_g(x_m,y_m)\leq 1. $$ By Proposition \ref{complete.case}, $$ c_m^\frac{2}{n-2}d_{g_{k_m}}(x_m,y_m)=d_{g_m'}(0,(1,0\cdots,0))\rightarrow 1. $$ Hence, $$ \frac{d(x_m,y_m)}{d_g(x_m,y_m)}\geq1 $$ when $m$ is sufficiently large.
$
\Box$\\
{\it The proof of Theorem \ref{main2}:} It remains to show that $d_g\leq d$. Let $\varepsilon$, $\tau$ and $\beta$ be in Lemma \ref{du0/d0} and set $A_\tau=\{x:\mu(\{x\})>\tau\}$. Obviously, $A_\tau$ is a finite set. Then, for any $\delta>0$, we have $$
\int_{B_\delta(x)}|R(g_k)|^\frac{n}{2}dV_{g_k}<\tau,\,\,\,\, whenever\,\,\,\, B_{2\delta}(x) \cap A_\tau=\emptyset, $$ when $k$ is sufficiently large. It follows that $$ \frac{d_g(x,y)}{d(x,y)}<1+\varepsilon, $$ whenever $d_{g_0}(x,y)<\beta\delta$ and $x\notin B_\delta(A_\tau)$.
We say a metric space is a length space if, for each $x$ and $y$ in this space, there exists a minimal geodesic joining them (see \cite[p.148]{Fukaya}). Since $(M,d)$ is also the Gromov-Haudorff limit of $(M,d_{g_k})$, $(M,d)$ is a length space according to \cite [Proposition 1.10]{Fukaya}. Let $\gamma$ be the minimal geodesic defined in $(M,d)$ joining $x_1$ and $x_2$, i.e. $\gamma:[0,a]\rightarrow (M,d)$ is a continuous map which satisfies $$
\gamma(0)=x_1,\,\,\,\, \gamma(a)=x_2,\,\,\,\, and\,\,\,\, d(\gamma(s),\gamma(s'))=|s-s'|,\,\,\,\,\forall s,s'\in[0,a]. $$ We claim that $\gamma$ is also continuous in $(M,g_0)$. For otherwise, we can find $t_k\rightarrow t$ and $a>0$, such that $d_{g_0}(\gamma(t_k),\gamma(t))>a$. By \eqref{positive.distance2}, there exists $a'>0$, such that $$
|t_k-t|\geq d(\gamma(t_k),\gamma(t))>a', $$ which is impossible.
Now we consider two cases. The first case is when $\gamma\cap A_\tau=\emptyset$, say $d(A_\tau,\gamma[0,a])>0$. Since $\gamma$ is continuous, we may assume $$ d_{g_0}(A_\tau,\gamma[0,a])>\delta>0. $$ Then there exists $$ s_0=0<s_1<\cdots<s_m=a, $$ such that $$ d_{g_0}(\gamma(s_{i+1}),\gamma(s_i))<\beta\delta. $$ It follows that \begin{eqnarray}\label{d0.d0'}\nonumber d(x_1,x_2)&=&\sum_{i=0}^{m-1} d(\gamma(s_i),\gamma(s_{i+1}))\\ &\geq& (1+\varepsilon)^{-1}\sum_{i=0}^{m-1}d_g(\gamma(s_i), \gamma(s_{i+1}))\\\nonumber &\geq& (1+\varepsilon)^{-1}d_g(x_1,x_2) \end{eqnarray}
The remaining case is when $\gamma\cap A_\tau\neq \emptyset$. Let $$ \gamma\cap A_\tau=\{\gamma(a_1), \cdots, \gamma(a_i)\}. $$ The distance function $d$ is bounded by \begin{eqnarray*} d(x_1,x_2)&\geq& d(x_1,\gamma(a_1-\varepsilon')) +d(\gamma(a_1+\varepsilon'),\gamma(a_2-\varepsilon'))+ \cdots+d(\gamma(a_i+\varepsilon',x_2))\\ &\geq& (1+\varepsilon)^{-1}(d_g(x_1,\gamma(a_1-\varepsilon')) + \cdots+d_g(\gamma(a_i+\varepsilon',x_2))). \end{eqnarray*} When $\varepsilon'\rightarrow 0$, we get \eqref{d0.d0'} again.
Finally, by letting $\varepsilon\rightarrow 0$, we get the desired inequality. $
\Box$\\ {\small
\end{document} |
\begin{document}
\title{ Weak Necessary and Sufficient Stochastic Maximum Principle for Markovian Regime-Switching Diffusion Models}
\begin{abstract} In this paper we prove a weak necessary and sufficient maximum principle for Markovian regime switching stochastic optimal control problems. Instead of insisting on the maximum condition of the Hamiltonian, we show that $ 0 $ belongs to the sum of Clarke's generalized gradient of the Hamiltonian and Clarke's normal cone of the control constraint set at the optimal control. Under a joint concavity condition on the Hamiltonian and a convexity condition on the terminal objective function, the necessary condition becomes sufficient. We give four examples to demonstrate the weak stochastic maximum principle. \end{abstract}
\noindent\textbf{Keywords}: regime switching stochastic optimal control, weak stochastic maximum principle, necessary and sufficient conditions, Clarke's generalized gradient, Clarke's normal cone, measurable selection.
\noindent\textbf{AMS MSC2010}: 93E20, 49J52.
\section{Introduction} There has been extensive research in the stochastic control theory. Two principal and most commonly used methods in solving stochastic optimal control problems are the dynamic programming principle and the stochastic maximum principle (SMP). The books by Fleming-Rishel \cite{flemingrishel:detestochcontrol}, Fleming-Soner \cite{flemingsoner:controledmarkprocess}, and Yong-Zhou \cite{yong.zhou:stochasticcontrols} provide excellent expositions and rigorous treatment of the subject of the dynamic programming principle in the optimal deterministic and stochastic control theory.
Many people have made great contributions in the research of the SMP. Kushner \cite{kushner:fixedtimecontrol, kushner:nececond} is the first to study the necessary SMP. Haussmann \cite{haussmann:SMPdiffusion}, Bensoussan \cite{bensoussan:lecturestochcontrol} and Bismut \cite{bismut:conjugateconvex, bismut:linearquarcontrol, bismut:dualityoptcontrol} extend Kushner's SMP to more general
stochastic control problems with control-free diffusion coefficients. Peng \cite{peng:SMP} applies the second order spike variation technique to derive the necessary SMP to stochastic control problems with controlled diffusion coefficients. Zhou \cite{zhou:unifiedtreatment} simplifies Peng's proof. Cadenillas-Karatzas \cite{karatzas:SMPrandomcoeff} extends Peng's SMP to systems with random coefficients and Tang-Li \cite{tangli:SMPjump} with jump diffusions. Bismut \cite{bismut:dualityoptcontrol} is the first to investigate the sufficient SMP.
Zhou \cite{zhou:sufficientSMPwithcontroldiffus} proves that Peng's SMP is also sufficient in the presence of certain convexity condition. Framstad-{\O}ksendal-Sulem \cite{oksendal:SSMP} extends the sufficient SMP to systems with jump diffusion, Donnelly \cite{donnelly:SMPregimeswitching} with Markovian regime-switching diffusion and, most recently, Zhang-Elliott-Siu \cite{zhangelliottsiu:SMPregimeswitchingjump} with Markovian regime-switching jump diffusion.
Briefly speaking, the necessary SMP states that any optimal control along with the optimal state trajectory must solve a system of forward-backward SDEs (stochastic differential equations) plus a maximum condition of the optimal control on the Hamiltonian. The necessary condition together with certain concavity conditions on the Hamiltonian give the sufficient condition of optimality. The major difficulty of generalizing the classical Pontryagin's maximum principle to a stochastic control problem with controlled diffusion term is that, in some cases, the Hamiltonian is a convex function of the control variable and achieves the minimum at the optimal control (see \cite[Example 3.3.1]{yong.zhou:stochasticcontrols}). One of the major contributions of Peng's SMP is the introduction of the generalized Hamiltonian and the second order adjoint stochastic processes. In those cases where the Hamiltonian is convex, it is the second order term that turns the generalized Hamiltonian to a concave function which achieves the maximum at the optimal control. The generalized Hamiltonian and the second order adjoint equation are introduced to preserve the maximum condition of Pontryagin's maximum principle.
However, the second order terms also pose problems. Firstly, one has to assume that all functions involved are twice continuously differentiable in the state variable in order to use the second order variation, which limits the scope of problems applicable to the theorem. Secondly, one has to solve the associated second order adjoint backward stochastic differential equation (BSDE) with the dimensionality equal to the square of that of its first order counterpart, which makes the problem more difficult to solve, at least numerically. Lastly, one can not get the sufficient condition by enhancing the necessary condition with some joint concavity condition to the generalized Hamiltonian and instead one has to add some joint concavity condition to the Hamiltonian (compare \cite[Theorem 3.3.2]{yong.zhou:stochasticcontrols} and \cite[Theorem 3.5.2]{yong.zhou:stochasticcontrols}), which illustrates that the necessary SMP is not completely compatible with the sufficient SMP. This motivates us to relax the requirement of the maximality of the Hamiltonian at the optimal control and to seek a weak but compatible necessary and sufficient SMP.
In this paper we assume that the control constraint set is a closed convex set. The second order adjoint processes can also be dropped in \cite{peng:SMP}, see \cite{yong.zhou:stochasticcontrols}, under the differentiability conditions for state and control variables. However, the philosophy of this paper is different from that of \cite{peng:SMP} in the sense that we do not try to preseve Pontryagin's maximum principle but instead try to find all stationary points of the Hamiltonian, which may open the way for new results when the control constraint set is nonconvex.
The main contribution of this paper is that we prove a weak version of the necessary and sufficient SMP for Markovian regime switching diffusion stochastic optimal control problems. Instead of insisting on the Hamiltonian to achieve the maximum at the optimal control, which is in general impossible, we relax the necessary condition by only requiring the optimal control to be a stationary point of the Hamiltonian. Specifically, we prove that 0 belongs to the sum of Clarke's generalized gradient of the Hamiltonian and Clarke's normal cone of the control constraint set at the optimal control almost surely almost everywhere. Under the joint concavity condition on the Hamiltonian and the convexity condition on the terminal objective function, the necessary condition becomes the sufficient condition.
The advantage of the weak SMP is the following. Firstly, the second order differentiability of the coefficients and the objective functions in the state variable is not required as the weak SMP does not have any second order terms.
Secondly, the differentiability of the coefficients and the objective functions in the control variable is not required as the weak SMP uses Clarke's generalized gradients to describe the optimal control. Thirdly, the dimensionality of the BSDE is much reduced as the second order adjoint process is not involved. Lastly, the necessary condition and the sufficient condition are compatible with each other in the sense that the necessary condition provides a stationary point while the sufficient condition confirms its optimality, which is in the same spirit as the necessary and sufficient conditions in the finite dimensional optimization.
The rest of the paper is organized as follows. Section 2 introduces the notations,
the formulation of the regime switching stochastic control problem and the basic assumptions. Section 3 states the main theorems of the paper, the weak necessary SMP (Theorem \ref{WNSMP}) and the weak sufficient SMP (Theorem \ref{WSSMP}). Section 4 gives four examples to demonstrate the usefulness of the weak SMP in solving regime switching stochastic control problems, including nonsmooth noncave case and regime-switching noncave case. Section 5 establishes some useful preliminary results on Clarke's generalized gradient and normal cone, Markovian regime switching SDE and BSDE, moment estimates, Lipschitz property, Taylor expansion and duality analysis. Section 6 proves the main theorems. Section 7 concludes. The appendix gives the proof of Theorem \ref{RSBSDEtheorem} (existence and uniqueness of the solution to a regime switching BSDE) for completeness.
\section{Problem Formulation} In this section, we formulate the stochastic control problem in a regime switching diffusion model and introduce some assumptions. Here we adopt the model in \cite{donnelly:SMPregimeswitching}
Let $ \left(\Omega,\mathcal{F},\mathbb{P}\right) $ be a complete probability space with a $ \mathbb{P} $ complete right continuous filtration. Let the previsible $\sigma$-algebra on $ \Omega\times [0,T] $ associated with the filtration $ \left\{ \mathcal{F}_t:t\in[0,T] \right\} $, denoted by $ \mathcal{P}^{\star} $, be the smallest $ \sigma $-algebra on $ \Omega\times[0,T] $ such that every $ \left\{ \mathcal{F}_t \right\} $-adapted stochastic process which is left continuous with right limit is $ \mathcal{P}^\star $ measurable. A stochastic process $ X $ is previsible, written as $ X\in\mathcal{P}^\star $, provided it is $ \mathcal{P}^\star $ measurable.
Let $ W(\cdot) $ be an $ m $-dimensional standard Brownian motion and $ \alpha(\cdot) $ a continuous time finite state observable Markov chain, which are independent of each other. $ \left\{ \mathcal{F}_t \right\} $ is the natural filtration generated by $ W $ and $ \alpha $, completed with all $ \mathbb{P} $-null sets, denoted by \begin{align*} \mathcal{F}_t=\sigma\left[W(s):0 \leq s \leq t\right]\bigvee\sigma\left[\alpha(s):0\leq s\leq t\right]\bigvee\mathcal{N}, \end{align*} where $ \mathcal{N} $ denotes the totality of $ \mathbb{P} $-null sets.
Let the Markov chain take values in the state space $ I=\left\{ 1,2,\cdots,d-1,d \right\} $ and start from initial state $ i_0\in I $ with a $ d\times d $ generator matrix $ \mathcal{Q}=\left\{ q_{ij} \right\}_{i,j=1}^d $. For each pair of distinct states $ \left( i,j \right) $, define the counting process $ \left[ Q_{ij} \right] :\Omega\times[0,T]\rightarrow\mathbb{N} $ by \begin{align*} [Q_{ij}](\omega,t):=\sum_{0 < s\leq t}\mathcal{X}\left[ \alpha(s-)=i \right](\omega)\mathcal{X}[\alpha(s)=j](\omega),\forall t\in[0,T], \end{align*} and the compensator process $ \langle Q_{ij} \rangle:\Omega\times[0,T]\rightarrow[0,+\infty )$ by \begin{align*} \langle Q_{ij}\rangle(\omega,t):=q_{ij}\int_0^t\mathcal{X}\left[ \alpha(s-)=i \right](\omega)ds,\forall t\in[0,T], \end{align*} where $ \mathcal{X} $ is an indicator function. The processes $$ Q_{ij}(\omega,t):=[Q_{ij}](\omega,t)-\langle Q_{ij}\rangle(\omega,t)$$
is a purely discontinuous square-integrable martingale with initial value zero (\cite[Lemma IV.21.12]{rw:diffusions}).
Consider a stochastic control model where the state of the system is governed by a controlled Markovian regime-switching SDE: \begin{equation}\label{stateeqn} \left\{ \begin{array}{cl} dx(t)= & b(t,x(t),u(t),\alpha(t-))dt+\sigma(t,x(t),u(t),\alpha(t-))dW(t)\\ x(0)= & x_0\in\mathbb{R}^n, \alpha(0)=i_0\in I, \end{array} \right. \end{equation} where $ u(\cdot)$ is a $ \mathbb{R}^k $ valued previsible process, $ T>0 $ is a fixed finite time horizon, $ b:[0,T]\times\mathbb{R}^n\times\mathbb{R}^k\times I \rightarrow \mathbb{R}^n$ and $\sigma:[0,T]\times\mathbb{R}^n\times\mathbb{R}^k\times I \rightarrow \mathbb{R}^{n\times m}$ are given continuous functions satisfying the following assumptions: \begin{enumerate} \item[\textbf{(A1)}] The maps $ b$ and $ \sigma $ are measurable, and there exist constant $ K>0 $ such that for $ \varphi=b \text{ and } \sigma $, we have \begin{align*} \left\{ \begin{array}{ll} \vert \varphi(t,x,u,i)-\varphi(t,\hat{x},\hat{u},i) \vert \leq K\left( \vert x-\hat{x} \vert + \vert u-\hat{u} \vert\right)\\ \forall t\in[0,T]; i \in I ; x,\hat{x}\in\mathbb{R}^n ; u,\hat{u}\in\mathbb{R}^k,\\ \vert \varphi(t,0,0,i) \vert <K, \ \forall t\in[0,T], \forall i\in I. \end{array} \right. \end{align*}
\item[\textbf{(A2)}] The maps $ b $ and $\sigma $ are $ C^1 $ in $ x $ and there exist a constant $ L>0 $ and a modulus of continuity $ \bar{\omega} : [0,+\infty)\rightarrow [0,+\infty)$ such that \begin{align*} \left\{ \begin{array}{ll} \vert \varphi_x(t,x,u,i)-\varphi_x(t,\hat{x},\hat{u},i) \vert \leq L\vert x-\hat{x} \vert + \bar{\omega}(d(u,\bar{u}))\\
\forall t\in[0,T]; i \in I ; x,\hat{x}\in\mathbb{R}^n ; u,\hat{u}\in\mathbb{R}^k,
\end{array}
\right. \end{align*} where $\varphi_x(t,x,u,i)$ is the partial derivative of $\varphi$ with respect to $x$ at the point $(t,x,u,i)$. \end{enumerate}
Consider the cost functional \begin{equation}\label{costfun} J(u)= E\left[ \int_0^T f(t,x(t),u(t),\alpha(t))dt + h(x(T),\alpha(T)) \right], \end{equation} where $ f:[0,T]\times\mathbb{R}^n\times\mathbb{R}^k\times I \rightarrow \mathbb{R} $ and $ h:\mathbb{R}^n\times I \rightarrow \mathbb{R} $ are given functions satisfying the following assumptions: \begin{enumerate} \item[\textbf{(A3)}] The maps $ f $ and $ h $ are measurable and there exist constants $ K_1, K_2 \geq 0 $ such that \begin{align*} \left\{ \begin{array}{ll} \vert f(t,x,u,i)-f(t,x,\hat{u},i) \vert \leq \left[ K_1+K_2(\vert x \vert + \vert u \vert + \vert \hat{u} \vert) \right]\vert u-\hat{u} \vert,\\ \vert f(t,0,0,i) \vert+\vert h(0,i) \vert <K_1, \ \forall t\in[0,T], \forall i\in I. \end{array} \right. \end{align*} \item[\textbf{(A4)}] The maps $ f $ and $ h $ are $ C^1 $ in $ x $ and there exist a constant $ L>0 $ and a modulus of continuity $ \bar{\omega}:[0,+\infty)\rightarrow[0,+\infty) $ such that for $ \varphi=f \text{ and } h $, we have \begin{align*} \left\{ \begin{array}{ll} \vert \varphi_x(t,x,u,i)-\varphi_x(t,\hat{x},\hat{u},i) \vert \leq L\vert x-\hat{x} \vert + \bar{\omega}(d(u,\bar{u})),\\ \forall t\in[0,T]; i \in I ; x,\hat{x}\in\mathbb{R}^n ; u,\hat{u}\in\mathbb{R}^k,\\ \vert \varphi_x(t,0,0,i) \vert \leq L, \forall t\in[0,T], i\in I. \end{array} \right. \end{align*} \end{enumerate} \begin{remark}\rm Assumptions \textbf{(A3)} and \textbf{(A4)} together cover many cases, including all quadratic functions in $ x $ and $ u $. For instance, if $ f $ is Lipschitz in $ u $, then $ K_2=0 $. On the other hand, if $ f $ is differentiable with respect to $ u $ and $ f_u $ satisfies a linear growth condition in $ u $, then $ K_2 $ is a positive constant. \end{remark}
Consider a measure space $ (S, \mathcal{P}^\star, \mu) $, where $ S=\Omega \times [0,T] $ and $ \mu = \mathbb{P}\times Leb $. Define $ L^p(S;\mathbb{R}^q) \text{ for } p,q\in\mathbb{N}^+ $ to be the Banach space of $ \mathbb{R}^q $ valued $ \mathcal{P}^\star $ measurable functions $ f:\Omega\times [0,T]\rightarrow \mathbb{R}^q $ such that \begin{equation}
\|f\|:=\left( \int_0^T E\vert f(t) \vert^p dt \right)^\frac{1}{p} < \infty. \end{equation} Similarly, define $ L^p_{\mathcal{F}}(S;\mathbb{R}^q) \text{ for } p,q\in\mathbb{N}^+ $ to be the space of $ \mathbb{R}^q $ valued $ \mathcal{F}_t $ progressively measurable $ p $th order integrable processes.
According to Theorem \ref{exisuniqSDE}, under assumption \textbf{(A1)}, for any $ u\in L^4(S;\mathbb{R}^k) $, the state equation \eqref{stateeqn} admits a unique solution and the cost functional \eqref{costfun} is well defined. A control is called admissible if it is valued in $ U $, a non-empty closed convex subset of $ \mathbb{R}^k $ and $ u\in L^4(S;\mathbb{R}^k) $. Denoted by $ \mathcal{U}_{ad} $ the set of admissible controls. In the case that $ x $ is a solution of \eqref{stateeqn} corresponding to an admissible control $ u\in \mathcal{U}_{ad} $, we call $ (x,u) $ an admissible pair and $ x $ an admissible state process.
Our optimal control problem can be stated as follows\\ \\ \textbf{Problem (S)} Minimize \eqref{costfun} over $ \mathcal{U}_{ad} $.\\ \\ Any $ \bar{u}\in\mathcal{U}_{ad} $ satisfying \begin{align*} J(\bar{u})=\inf_{u\in\mathcal{U}_{ad}}J(u) \end{align*} is called an \textit{optimal control}. The corresponding $ \bar{x} $ and $ (\bar{x},\bar{u}) $ are called an \textit{optimal state process} and \textit{optimal pair}, respectively.
\section{Weak Stochastic Maximum Principle} In this section we state the weak necessary and sufficient stochastic maximum principle in the regime-switching diffusion model.
The Hamiltonian $ H:[0,T]\times\mathbb{R}^n\times\mathbb{R}^k\times I\times\mathbb{R}^n\times\mathbb{R}^{n\times m} \rightarrow \mathbb{R}$ for the stochastic control problem \eqref{stateeqn} and \eqref{costfun} is defined by: \begin{align}\label{Hamiltonian} \begin{split} H(t,x,u,i,p,q):=&-f(t,x,u,i)+b^\intercal(t,x,u,i)p+tr(\sigma^\intercal(t,x,u,i)q). \end{split} \end{align} Given an admissible pair $ (x,u) $, the adjoint equation in the unknown adapted processes $ p(t)\in\mathbb{R}^n, q(t)\in\mathbb{R}^{n\times m} $ and $ s(t)=(s^{(1)}(t),\cdots,s^{(n)}(t)) $, where $ s^{(l)}(t)\in\mathbb{R}^{d\times d} $ for $ l=1,\cdots,n $, is the following regime-switching BSDE: \begin{equation}\label{ajointeqn} \left\{ \begin{array}{ll} dp(t)=&-H_x(t,x(t),u(t),\alpha(t-),p(t),q(t))dt+q(t)dW(t)+s(t)\bullet dQ(t)\\ p(T)=&-h_x(x(T),\alpha(T)), \end{array} \right. \end{equation} where \begin{align*} s(t)\bullet dQ(t)\equiv\left( \sum_{j\neq i}s^{(1)}_{ij}(t)dQ_{ij}(t),\cdots , \sum_{j\neq i}s^{(n)}_{ij}(t)dQ_{ij}(t)\right)^\intercal. \end{align*}
By Theorem \ref{RSBSDEtheorem}, we claim that under assumptions \textbf{(A1)-(A4)}, for any $ (x,u)\in L^2_{\mathcal{F}}(S;\mathbb{R}^n)\times L^4(S;\mathbb{R}^k) $, \eqref{ajointeqn} admits a unique solution $ \{ (p(t),q(t),s(t))\vert t\in[0,T]\} $ in the sense of Definition \ref{RSBSDEdefn}. If $ (\bar{x},\bar{u}) $ is an optimal (resp. admissible) pair and $ (\bar{p},\bar{q},\bar{s}) $ is the adapted solution of \eqref{ajointeqn}, then $ (\bar{x},\bar{u}, \bar{p},\bar{q},\bar{s}) $ is called an optimal (resp. admissible) 5-tuple.
We can now state the main results of the paper.
\begin{theorem}\label{WNSMP} (Weak Necessary SMP with Regime-Switching) Let assumptions \textbf{(A1)-(A4)} hold. Let $ (\bar{x},\bar{u}) $ be an optimal pair of \textbf{Problem (S)}. Then there exists stochastic process $ (\bar{p},\bar{q},\bar{s}) $ which is an adapted solution to \eqref{ajointeqn}, such that \begin{equation}\label{neceSMPcond} 0\in\partial_u (-H)(t,\bar{x}(t),\bar{u}(t),\alpha(t-),\bar{p}(t),\bar{q}(t))+N_U(\bar{u}(t)), \mbox{a.e. }t\in[0,T], \mathbb{P}\mbox{-a.s.}, \end{equation} where $ \partial_u (-H)(t,\bar{x}(t),\bar{u}(t),\alpha(t-),\bar{p}(t),\bar{q}(t))$ is Clarke's generalized gradient of $-H$ with respect to variable $u$ at point $(t,\bar{x}(t),\bar{u}(t),\alpha(t-),\bar{p}(t),\bar{q}(t))$ and $N_U(\bar{u}(t))$ is Clarke's normal cone of $U$ at point $\bar{u}(t)$ (see Subsection \ref{ClarkeIntro} for details). \end{theorem} \begin{theorem}\label{WSSMP} (Weak Sufficient SMP with Regime-Switching) Let assumptions \textbf{(A1)-(A4)} hold and let $ (\bar{x},\bar{u},\bar{p},\bar{q},\bar{s}) $ be an admissible 5-tuple satisfying \eqref{neceSMPcond}. Suppose further that $ h(\cdot,\alpha(T)) $ is convex and the Hamiltonian $ H(t,\cdot,\cdot,\alpha(t-),\bar{p}(t),\bar{q}(t)) $ is concave for all $ t\in[0,T] $ a.s. Then $ (\bar{x},\bar{u}) $ is an optimal pair for \textbf{Problem (S)}. \end{theorem} \begin{remark}\rm In the special case where $ \mathcal{F}_t=\sigma[W(s):0\leq s\leq t]\bigvee \mathcal{N} $, i.e., the randomness of the system is generated only by the Brownian motion, the Hamiltonian (\ref{Hamiltonian}) and all other functions are free of index $i$ or Markov chain processs value $\alpha(t-)$. The adjoint equation (\ref{ajointeqn}) is a pure Brownian BSDE (no $s(t)\bullet dQ(t)$ term). The weak SMP remains the same as Theorem \ref{WNSMP} and \ref{WSSMP}, but only involves the $ 4 $-tuple $ (\bar{x},\bar{u},\bar{p},\bar{q}) $. \end{remark}
\section{Examples} In this section, we present four examples to demonstrate our main theorems. \subsection{Examples: Weak SMP without Regime-Switching} In this subsection, we consider two examples from \cite{yong.zhou:stochasticcontrols} and derive the same results as those in \cite{yong.zhou:stochasticcontrols} using Theorem \ref{WNSMP} and Theorem \ref{WSSMP}. A key property to use in our approach is the adaptedness of the adjoint process.
\begin{example}(Concave Hamiltonian)\label{example1} {\rm Consider the following stochastic control problem \cite[Example 3.5.3]{yong.zhou:stochasticcontrols}: \begin{equation}\label{example1stateqn} \left\{ \begin{array}{cl} & dx(t)=u(t)dW(t), t\in[0,1]\\ & x(0)=0 \end{array}\right. \end{equation} with the control constraint set $ U=[0,1] $ and the cost functional \begin{align*} J(u) = E\left\{ -\int_0^1 u(t)dt +\frac{1}{2}x(1)^2 \right\}. \end{align*} Suppose $ (\bar{x},\bar{u}) $ is an optimal pair, then the corresponding adjoint equation is \begin{equation}\label{example1adjointqn} \left\{ \begin{array}{cl} & d\bar{p}(t)=\bar{q}(t)dW(t), t\in[0,1]\\ & \bar{p}(1)=-\bar{x}(1). \end{array}\right. \end{equation} Using \eqref{example1stateqn} and \eqref{example1adjointqn} and via a simple calculation we obtain \begin{align*} \bar{p}(t)=-\int_0^t\bar{u}(s)dW(s)-\int_t^1\left( \bar{u}(s)+\bar{q}(s) \right)dW(s). \end{align*} Since the adjoint process $ \bar{p}(t) $ is adapted to the filtration $ \mathcal{F}_t $ , we must have \begin{equation}\label{adapted1} \bar{u}(t)+\bar{q}(t)=0 \textit{ for all } t\in[0,1], \mathbb{P}\textit{-a.s.} \end{equation} The corresponding Hamiltonian is \begin{align*} H(t,x,u,\bar{p}(t),\bar{q}(t)) = \bar{q}(t)u + u. \end{align*} Since the problem satisfies \textbf{(A1)}-\textbf{(A4)}, by Theorem \ref{WNSMP} and (\ref{neceSMPcond}), we have \begin{align*} 0 &\in -(\bar{q}(t)+1)+N_{[0,1]}(\bar{u}(t)) \textit{ for all } t\in[0,1], \mathbb{P}\textit{-a.s}. \end{align*} Consequently, on any nonzero measurable set $ E\in S=\Omega \times [0,1] $, we can only have the following three cases: \begin{description} \item[Case 1]: $ 0<\bar{u}(t)<1 \Longrightarrow N_{[0,1]}\left( \bar{u}(t) \right)=\{0\} \textit{ and } \bar{q}(t)=-1 $. \item[Case 2]: $ \bar{u}(t)=0 \Longrightarrow N_{[0,1]}\left( \bar{u}(t) \right)= (-\infty,0] \textit{ and } \bar{q}(t)+1 \leq 0 $. \item[Case 3]: $ \bar{u}(t)=1 \Longrightarrow N_{[0,1]}\left( \bar{u}(t) \right)= [0,+\infty) \textit{ and } \bar{q}(t)+1 \geq 0 $. \end{description} Suppose Case 1 or Case 2 is true, then $ \bar{u}(t)+\bar{q}(t)\leq \bar{u}(t)-1< 0 $ for some nonzero measurable set $ E\in S $, contradiction to (\ref{adapted1}). Hence, we have $ \bar{u}(t)=1 $ for every $t\in[0,1], \mathbb{P}$-a.s. and $ \bar{x}(t)=W(t) $ and $ (\bar{p}(t),\bar{q(t)})=(-W(t),-1) $ for $ t\in [0,1] $. Since $(x,u)\mapsto H(t,x,u, \bar{p}(t),\bar{q}(t)) = -u+u=0 $ is concave and $x\mapsto h(x)=\frac{1}{2}x^2 $ is convex, we conclude that $ \bar{u}(t)=1 $ is the optimal control using Theorem \ref{WSSMP}. }\end{example}
\begin{example}(Nonconcave nonsmooth Hamiltonian)\label{nonsmoothexample} {\rm Consider the following stochastic control problem \begin{equation}\label{nonsmoothstateqn} \left\{ \begin{array}{cl} & dx(t)=\dfrac{1}{2}\vert u(t)\vert dW(t), t\in[0,1]\\ & x(0)=0 \end{array}\right. \end{equation} with the control constraint set $ U=[-1,1] $ and the cost functional \begin{align*} J(u) = E\left\{\int_0^1 [x(t)^2-\frac{1}{2}u(t)^2]dt + x(1)^2\right\}. \end{align*} Suppose $ (\bar{x},\bar{u}) $ is an optimal pair, then the corresponding adjoint equation is \begin{equation}\label{nonsmoothadjointqn} \left\{ \begin{array}{cl} & d\bar{p}(t)=2\bar{x}(t)dt+\bar{q}(t)dW(t), t\in[0,1]\\ & \bar{p}(1)=-2\bar{x}(1). \end{array}\right. \end{equation} Using \eqref{nonsmoothstateqn}, \eqref{nonsmoothadjointqn} and via a simple calculation, we obtain \begin{align*} \bar{p}(t)=-\int_0^t(2-t)\vert\bar{u}(s)\vert dW(s)-\int_t^1((2-s)\vert\bar{u}(s)\vert +\bar{q}(s))dW(s). \end{align*} Since the adjoint process $ \bar{p}(t) $ is adapted to the filtration $ \mathcal{F}_t $, we must have \begin{equation} \label{4.9} (2-t)\vert\bar{u}(t)\vert +\bar{q}(t)=0 \textit{ for all } t\in[0,1], \mathbb{P}\textit{-a.s.} \end{equation} The corresponding Hamiltonian is \begin{align*} H(t,x,u,\bar{p}(t),\bar{q}(t))=\dfrac{1}{2}\bar{q}(t)\vert u\vert -x^2+\frac{1}{2}u^2. \end{align*} Since the problem satisfies assumptions \textbf{(A1)}-\textbf{(A4)}, by Theorem \ref{WNSMP} and (\ref{neceSMPcond}), we have \begin{align}\label{nonsmoothcontra} 0 & \in \partial_u\left( x(t)^2-\frac{1}{2}q(t)\vert u(t) \vert -\frac{1}{2}u(t)^2 \right) + N_{[-1,1]}(\bar{u}(t))\textit{ for all } t\in[0,1], \mathbb{P}\textit{-a.s}. \end{align} Consequently, on any nonzero measurable set $ E\in S $, we can only have the following five cases: \begin{description} \item[Case 1] $\bar{u}(t)=1 \Longrightarrow 0\in \left\{-\dfrac{1}{2}q(t)-1\right\}+[0,+\infty)$ which is compatible with the adaptedness condition (\ref{4.9}) $ \bar{q}(t)=t-2 $. \item[Case 2] $\bar{u}(t)=-1 \Longrightarrow 0\in \left\{\dfrac{1}{2}q(t)+1\right\}+(-\infty,0]$ which is compatible with (\ref{4.9}) $ \bar{q}(t)=t-2 $. \item[Case 3] $\bar{u}(t)=0 \Longrightarrow 0\in \left[\dfrac{1}{2}q(t),-\dfrac{1}{2}q(t)\right]+\{0\}$ which is compatible with (\ref{4.9}) $ \bar{q}(t)=0 $. \item[Case 4] $\bar{u}(t)\in(0,1) \Longrightarrow 0\in \left\{-\dfrac{1}{2}q(t)-\bar{u}(t)\right\}+\{0\}$ which gives $q(t)=-2\bar{u}(t)<0$, a contradiction to (\ref{4.9}) $ \bar{q}(t)=(t-2)\bar{u}(t)>0$. \item[Case 5] $\bar{u}(t)\in(-1,0) \Longrightarrow 0\in \left\{\dfrac{1}{2}q(t)-\bar{u}(t)\right\}+\{0\}$ which gives $q(t)=2\bar{u}(t)$, a contradiction to (\ref{4.9}) $ \bar{q}(t)=(2-t)\bar{u}(t) $. \end{description} Hence, the set of optimal candidates from Weak Necessary SMP consists of all the progressively measurable processes valued in the set $ \{-1,0,1\} $. However, since the Hamiltonian is not concave, Theorem \ref{WSSMP} cannot be applied. Substituting $ x(t)=\int_0^t \frac{1}{2}\vert u(s)\vert dW(s) $ into the cost functional and by simple calculations, we obtain \begin{align*} J(u) = -\dfrac{1}{4}E\int_0^1 t \vert u(t)\vert ^2 dt. \end{align*}
Hence $J(u)$ reaches the minimum at $|\bar u(t)|=1$ a.s. for all $t$, which implies there are infinitely many optimal controls with any measurable combination of $ 1 $ and $ -1 $. The optimal state process is $\bar x(t)={1\over 2}W(t)$ and the adjoint processes are $\bar p(t)=(t-2)W(t)$ and $\bar q(t)=t-2$ for all $t\in [0,1]$. }\end{example}
\begin{remark} {\rm Example \ref{nonsmoothexample} shows that the weak necessary SMP can find not only the optimal control for minimization problem (any progressively measurable process taking values $-1$ or 1) but also the optimal control for maximization problem (the unique progressively measurable process taking value 0), which is in the same spirit of the necessary condition for finite dimensional optimization. The Hamiltonian in Example \ref{nonsmoothexample} is nonsmooth in control variable $u$, which is beyond any known literature on SMP. } \end{remark}
\begin{remark} {\rm When $dx(t)=u(t)dW(t)$ and $U=[0,1]$ and everything else is kept the same as that in Example \ref{nonsmoothexample}, the problem is the same as that of \cite[Example 3.3.1]{yong.zhou:stochasticcontrols}. Theorem \ref{WNSMP} can again be applied to find the optimal control candidate $\bar u(t)=0$. (We leave this to the reader to check.) The Hamiltonian is a convex function of $u$ and $\bar u(t)=0$ is a minimum point. This is the reason that \cite{peng:SMP} introduces the generalized Hamiltonian ${\cal H}$ which makes $\bar u(t)=0$ a maximum point. } \end{remark}
\subsection{Examples: Weak SMP with Regime-Switching} \begin{example}(Quadratic Loss Minimization) {\rm Here we adopt the setting in \cite[Section 6]{donnelly:SMPregimeswitching}. Let $ (\Omega,\mathcal{F},\left\{\mathcal{F}_t\right\}_{0\leq t\leq T},\mathbb{P}) $ be a complete probability space on which defined a 1-dimensional standard Brownian motion $ W $ and a continuous time Markov chain $ \alpha $ valued in a finite state space $ I=\{1,\cdots d\} $ with generator matrix $ Q=\left[ q_{ij} \right]_{i,j\in I} $ and initial mode $ \alpha(0)=i_0 $. Assume that $ W $ and $ \alpha $ are independent of each other and the filtration is generated jointly by $ W $ and $ \alpha $. Consider a market consisting of one risk-free bank account $ S_0=\left\{ S_0(t),t\in[0,T] \right\} $ and one risky stock $ S_1=\left\{ S_1(t), t\in[0,T] \right\} $. The risk-free asset's price process satisfies the following equation: \begin{align*} \left\{ \begin{array}{l} dS_0(t)=r(t,\alpha(t-))S_0(t)dt \ t\in [0,T]\\ S_0(0)=1, \end{array} \right. \end{align*} where the risk-free rate of return $ r(t,i) $ is a bounded deterministic function for $ i\in I $. The price process of the risky stock is given by \begin{align*} \left\{ \begin{array}{l} dS_1(t)=S_1(t)\left\{ b(t,\alpha(t-))dt+\sigma(t,\alpha(t-))dW(t) \right\} \ t\in[0,T]\\ S_1(0)=S_1>0, \end{array} \right. \end{align*} where the mean rate of return $ b(t,i) $ and the volatility $ \sigma(t,i) $ are bounded non-zero deterministic functions for $ i\in I $. Define the market price of risk $ \theta(t,i)\equiv\sigma^{-1}(t,i)(b(t,i)-r(t,i)) $.
Consider an agent with an initial wealth $ x_0>0 $. Let the $ \mathcal{F}_t $ previsible real valued process $ u(t) $ be the amount allocated to the stock at time $ t $. Then the wealth process $ x $ can be written as \begin{equation}\label{qlmweatheqn} \left\{ \begin{array}{l} dx(t)=\left[ r(t,\alpha(t-))x(t)+u(t)\sigma(t,\alpha(t-))\theta(t,\alpha(t-)) \right]dt+u(t)\sigma(t,\alpha(t-))dW(t)\\ x(0)=x_0. \end{array} \right. \end{equation}
A portfolio $ u(\cdot) $ is said to be admissible, written as $ u(\cdot)\in \mathcal{U}_{ad} $ if it is $ \mathcal{F}_t $-previsible, square integrable and such that the regime switching SDE \eqref{qlmweatheqn} has a unique solution $ x(\cdot) $ corresponding to $ u(\cdot) $. In this case, we refer to $ (x(\cdot),u(\cdot)) $ as an admissible pair. The agent's objective is to find an admissible pair $ (\bar{x}(\cdot),\bar{u}(\cdot)) $ such that \begin{equation*} E\left( \bar{x}(T)-d \right)^2=\inf_{u\in\mathcal{U}_{ad}}E(x(T)-d)^2 \end{equation*} for some fixed constant $ d\in \mathbb{R} $.
To solve this problem, first we find potential optimal candidate using Theorem \ref{WNSMP}. Suppose that $ (\bar{x}(\cdot),\bar{u}(\cdot)) $ is an optimal pair. Then the corresponding adjoint equation is \begin{equation}\label{qlmadjoint} \left\{ \begin{array}{l} dp(t)=-r(t,\alpha(t-))p(t)dt+q(t)dW(t)+s(t)\bullet dQ(t) \ t\in[0,T)\\ -p(T)=2\bar{x}(T)-2d. \end{array} \right. \end{equation}
To find a solution $ (\bar{p},\bar{q},\bar{s}) $ to \eqref{qlmadjoint}, we try a process \begin{equation}\label{qlmpbar} \bar{p}(t)=\phi(t,\alpha(t))\bar{x}(t)+\psi(t,\alpha(t)), \end{equation} where $ \phi(t,i) $ and $ \psi(t,i) $ are deterministic smooth functions with terminal conditions \begin{align*} \phi(T,i)=2 \text{ and } \psi(T,i)=-2d \text{ for } \forall i\in I. \end{align*} Applying Ito's formula to \eqref{qlmpbar} and comparing coefficients with \eqref{qlmadjoint} leads to \begin{align}\label{qlmfirst} &\begin{array}{c} -r(t,\alpha(t-))\bar{p}(t)=\sum_{i=1}^d\mathcal{X}[\alpha(t-)=i]\bigg\{ \bar{x}(t)\left( \phi(t,i)r(t,i)+\Delta\phi(t,i)\right)\\ +\phi(t,i)\bar{u}(t)\sigma(t,i)\theta(t,i)+\Delta\psi(t,i)\bigg\}, \end{array} \\\label{qlmsecond} &\bar{q}(t)=\phi(t,\alpha(t-))\sigma(t,\alpha(t-))\bar{u}(t),\\ &\bar{s}_{ij}(t)=\bar{x}(t)(\phi(t,j)-\phi(t,i))+(\psi(t,j)-\psi(t,i)), \end{align} where for $ \varphi=\phi \text{ and }\psi $, denote by \begin{equation*} \Delta\varphi(t,i)\triangleq\varphi_t(t,i)+\sum_{j=1}^d q_{ij}(\varphi(t,j)-\varphi(t,i)). \end{equation*} The Hamiltonian is given by \begin{equation}\label{qlmHamiltonian} \begin{array}{r} H(t,x,u,\alpha ,p,q)=r(t,\alpha)xp+u\sigma(t,\alpha)q+u\sigma(t,\alpha)\theta(t,\alpha)p. \end{array} \end{equation} By Theorem \ref{WNSMP}, we have \begin{equation*} 0\in\partial_u(-H)(t,\bar{x}(t),\bar{u}(t),\alpha(t-),\bar{p}(t),\bar{q}(t)). \end{equation*} Since $ H $ is a linear function of $ \bar{u} $, we must have \begin{equation}\label{qlmqbar} \bar{q}(t)=-\theta(t,\alpha(t-))\bar{p}(t). \end{equation} Substituting \eqref{qlmqbar} and \eqref{qlmpbar} into \eqref{qlmsecond} we obtain \begin{equation}\label{qlmubar} \bar{u}(t)=-\sigma^{-1}(t,\alpha(t-))\theta(t,\alpha(t-))(\bar{x}(t)+\phi^{-1}(t,\alpha(t-))\psi(t,\alpha(t-))). \end{equation} Substituting \eqref{qlmpbar} and \eqref{qlmubar} into \eqref{qlmfirst} leads to the following two differential equations \begin{align} \phi(t,i)(2r(t,i)-\vert \theta(t,i) \vert^2)+\Delta\phi(t,i)=0,\\ \psi(t,i)(r(t,i)-\vert \theta(t,i) \vert^2)+\Delta\psi(t,i)=0, \end{align} with terminal conditions \begin{equation*} \phi(T,i)=2 \text{ and } \psi(T,i)=-2d \text{ for } \forall i\in I. \end{equation*} It can be showed that the solutions are \begin{align}\label{qlmphi}
\phi(t,i)=2E\bigg\{ \exp\bigg[ \int_t^T \left( 2r(s,\alpha(s))-\vert \theta(s,\alpha(s)) \vert^2\right) ds \bigg]\bigg| \alpha(t)=i \bigg\},\\\label{qlmpsi}
\psi(t,i)=-2dE\bigg\{ \exp\bigg[ \int_t^T \left( r(s,\alpha(s))-\vert \theta(s,\alpha(s)) \vert^2\right) ds \bigg]\bigg| \alpha(t)=i \bigg\}. \end{align} Detailed proofs can be found in \cite[Section 6]{donnelly:SMPregimeswitching} and \cite[Section 5]{zhangelliottsiu:SMPregimeswitchingjump}. Substituting \eqref{qlmphi} and \eqref{qlmpsi} back into \eqref{qlmubar} gives the potential optimal portfolio $ \bar{u} $ and the corresponding potential optimal wealth process $ \bar{x} $.
To verify the optimality of our candidate solution, we apply Theorem \ref{WSSMP}. Since \textbf{(A1)}-\textbf{(A4)} are satisfied, $ h(x(T),\alpha(T))\equiv (x(T)-d)^2 $ is convex and the Hamiltonian \eqref{qlmHamiltonian} is concave, we conclude that $ (\bar{x}(\cdot),\bar{u}(\cdot)) $ is indeed the optimal pair. } \end{example} \begin{remark} {\rm Notice that in this case $ h $ is a convex function and the Hamiltonian is concave. Therefore, one can skip the necessary conditions and use a sufficient stochastic maximum principle of Pontryagin's type directly to find the optimal portfolio process. Detailed steps can be found in \cite[Section 6]{donnelly:SMPregimeswitching} and \cite[Section 5]{zhangelliottsiu:SMPregimeswitchingjump}. However, we follow a different approach here. Instead of using the sufficient SMP directly, we first find all admissible portfolios satisfying the necessary conditions stated in Theorem \ref{WNSMP}. Combining that with the adjoint equations, we then construct candidate optimal portfolio $ \bar{u} $. Finally, an application of Theorem \ref{WSSMP} confirms that $ \bar{u} $ is indeed the optimal portfolio. This approach is particularly useful when the conditions for sufficient SMP are not satisfied, e.g. nonconcave Hamiltonian. } \end{remark}
\begin{example}(Nonconcave Hamiltonian) {\rm Let $ (\Omega,\mathcal{F},\lbrace\mathcal{F}_t\rbrace_{0\leq t\leq 1},\mathbb{P}) $ be a complete probability space. Consider a one-dimensional Brownian motion $ W $ and a continuous time finite state Markov chain $ \lbrace\alpha(t)\vert t\in[0,1]\rbrace $ with state space $ I:=\lbrace 1,2 \rbrace $ and generator matrix $ Q:=[q_{ij}]_{i,j=1,2} $. Assume $ q_{12}+q_{21}\geq 2 $. Consider the following Markovian regime-switching control system \begin{equation*} \left\{ \begin{array}{ll} dx(t)=u(t)dW(t), \ t\in[0,1]\\ x(0)=0 \end{array} \right. \end{equation*} with the control domain $ U=[0,1] $ and the cost functional \begin{align*} J(u(\cdot))=E\bigg[ \int_0^1 \left( A(\alpha(t))u(t)+B(\alpha(t))u^2(t)+C(\alpha(t))x^2(t)\right) dt+D(\alpha(1))x^2(1) \bigg], \end{align*} where functions $ A,B,C,D: I\rightarrow\mathbb{R} $ satisfy \begin{align*} \left\{ \begin{array}{ll} A(1)=-1\\ A(2)=0 \end{array}, \right. \left\{ \begin{array}{ll} B(1)=0\\ B(2)=-\frac{1}{2} \end{array}, \right. \left\{ \begin{array}{ll} C(1)=0\\ C(2)=1 \end{array}, \right. \left\{ \begin{array}{ll} D(1)=\frac{1}{2}\\ D(2)=1 \end{array}. \right. \end{align*}
To solve this problem, first we find potential optimal solutions using Theorem \ref{WNSMP}. Suppose $ (\bar{x}(\cdot),\bar{u}(\cdot)) $ is an optimal pair. Then the corresponding adjoint equation is \begin{equation}\label{egadj} \left\{ \begin{array}{ll} d\bar{p}(t)=2C(\alpha(t))\bar{x}(t)dt+\bar{q}(t)dW(t)+\bar{s}(t)\bullet dQ(t)\\ \bar{p}(1)=-2D(\alpha(1))\bar{x}(1) \end{array}. \right. \end{equation} To find a solution $ (\bar{p},\bar{q},\bar{s}) $ to \eqref{egadj}, we try a process $ \bar{p}(t)=\phi(t,\alpha(t))\bar{x}(t) $, where $ \phi(t,i),\ i=1,2 $ are deterministic functions satisfying the terminal condition $ \phi(1,i)=-2D(i),\ i=1,2 $. Applying Ito's formula \begin{equation}\label{egito} \begin{array}{c} \displaystyle d\bar{p}(t)=\sum_{i=1}^2 \mathcal{X}[\alpha(t-)=i]\bigg\lbrace \bar{x}(t)\bigg( \phi_t(t,i)+\sum_{j=1}^2q_{ij}\left( \phi(t,j)-\phi(t,i) \bigg) \right) \bigg\rbrace dt \\ \displaystyle+\phi(t,\alpha(t))\bar{u}(t)dW(t)+\sum_{i\neq j}\bar{x}(t)\left( \phi(t,j)-\phi(t,i) \right)dQ_{ij}. \end{array} \end{equation} Comparing the coefficients of \eqref{egadj} and \eqref{egito} leads to \begin{align}\label{eglineareqn} 2C(\alpha(t))\bar{x}(t)=&\sum_{i=1}^2\mathcal{X}[\alpha(t-)=i]\bigg\lbrace \bar{x}(t)\bigg( \phi_t(t,i)+\sum_{j=1}^2q_{ij}(\phi(t,j)-\phi(t,i)) \bigg) \bigg\rbrace \\\label{egqcond} \bar{q}(t)=&\phi(t,\alpha(t))\bar{u}(t)\\ \bar{s}_{ij}(t)=&\bar{x}(t)(\phi(t,j)-\phi(t,i)) \end{align} As \eqref{eglineareqn} is a linear equation of $ \bar{x}(t) $, we guess that the coefficient of $ \bar{x}(t) $ vanishes at optimality and obtain the following two equations \begin{align}\label{egode} \left\{ \begin{array}{c} \displaystyle-\phi_t(t,1)-q_{12}(\phi(t,2)-\phi(t,1))=0,\\ 2-\phi_t(t,2)-q_{21}(\phi(t,1)-\phi(t,2))=0, \end{array} \right. \end{align} with terminal conditions \begin{equation}\label{egterminalcond} \phi(1,1)=-1 \text{ and } \phi(1,2)=-2. \end{equation} Solving the system of ordinary differential equations \eqref{egode} with terminal conditions \eqref{egterminalcond} gives \begin{align*} \left\{ \begin{array}{c} \phi(t,1)=\dfrac{q_{12}(q_{12}+q_{21}-2)}{(q_{12}+q_{21})^2}\left( e^{(q_{12}+q_{21}-2)(t-1)}-1 \right)+\dfrac{2q_{12}}{q_{12}+q_{21}}(t-1)-1\\ \phi(t,2)=\dfrac{q_{21}(q_{12}+q_{21}-2)}{(q_{12}+q_{21})^2}\left( 1-e^{(q_{12}+q_{21}-2)(t-1)} \right)+\dfrac{2q_{12}}{q_{12}+q_{21}}(t-1)-2 \end{array} \right. \end{align*} Moreover, since $ q_{12}+q_{21}\geq 2 $ and $ \frac{q_{21}(q_{12}+q_{21}-2)}{(q_{12}+q_{21})^2}<1 $, we obtain that $\phi(t,i)<-1, \ \forall t\in[0,1),i\in I $. Consider the Hamiltonian \begin{align}\label{egHam} \left\{ \begin{array}{l} H(t,x,u,1,p,q)=u+qu\\ H(t,x,u,2,p,q)=\frac{1}{2}u^2-x^2+uq. \end{array} \right. \end{align} By Theorem \ref{WNSMP}, we have \begin{align*} 0\in\partial_u(-H)(t,\bar{x}(t),\bar{u}(t),\alpha(t-),\bar{p}(t),\bar{q}(t))+N_U(\bar{u}(t)) \ \forall t\in[0,1], \ \mathbb{P}-\text{a.s.} \end{align*} Consequently on any nonzero measurable set $ E\in S=\Omega\times[0,1) $ such that $ \alpha(t-)=1 $, we can only have three cases: \begin{description} \item[Case 1]: $\bar{u}(t)=0\Rightarrow N_{[0,1]}(\bar{u}(t))=(-\infty,0] \text{ and } \bar{q}(t)+1\leq 0.$ \\According to \eqref{egqcond}, $ \phi(t,1)\bar{u}(t)\leq-1, \bar{u}(t)\geq-\frac{1}{\phi(t,1)}>0 $, contradiction. \item[Case 2]: $ \bar{u}(t)=1\Rightarrow N_{[0,1]}(\bar{u}(t))=[0,+\infty) \text{ and } \bar{q}(t)+1\geq 0. $\\According to \eqref{egqcond}, $ \phi(t,1)\bar{u}(t)\geq-1, \bar{u}(t)\leq-\frac{1}{\phi(t,1)}<1 $, contradiction. \item[Case 3]: $ 0<\bar{u}(t)<1 \Rightarrow N_{[0,1]}(\bar{u}(t))=\{0\} \text{ and } \bar{q}(t)=-1. $\\According to \eqref{egqcond}, $ \bar{u}(t)=-\frac{1}{\phi(t,1)}\in(0,1). $ \end{description} Hence we conclude that $ \bar{u}(t)=-\frac{1}{\phi(t,1)} $ provided $ \alpha(t-)=1 $. Similarly on any non-zero measurable set $ E\in S=\Omega\times[0,1) $ such that $ \alpha(t-)=2 $, we can only have three cases: \begin{description} \item[Case 1]: $\bar{u}(t)=1\Rightarrow N_{[0,1]}(\bar{u}(t))=[0,+\infty) \text{ and } \bar{q}(t)+\bar{u}(t)\geq 0.$ \\According to \eqref{egqcond}, $ (\phi(t,2)+1)\bar{u}(t)\geq 0, \bar{u}(t)\leq\frac{1}{\phi(t,2)+1}<0 $, contradiction. \item[Case 2]: $ \bar{u}(t)\in(0,1)\Rightarrow N_{[0,1]}(\bar{u}(t))=\{0\} \text{ and } \bar{q}(t)+\bar{u}(t)=0. $\\According to \eqref{egqcond}, $ (\phi(t,2)+1)\bar{u}(t)=0, \bar{u}(t)=0 $, contradiction. \item[Case 3]: $ \bar{u}(t)=0 \Rightarrow N_{[0,1]}(\bar{u}(t))=(-\infty,0] \text{ and } \bar{q}(t)+\bar{u}(t)\leq 0. $\\According to \eqref{egqcond}, $ (\phi(t,2)+1)\bar{u}(t)\leq 0, \bar{u}(t)=0. $ \end{description} Hence we must have $ \bar{u}(t)=0 $ provided $ \alpha(t-)=2 $.\\ In conclusion, the potential optimal control can be written as \begin{equation}\label{egoptimalcontrol} \bar{u}(t)=-\frac{1}{\phi(t,1)}\mathcal{X}[\alpha(t-)=1]. \end{equation}
Let us now show that $ (\bar{x}(\cdot),\bar{u}(\cdot)) $ is indeed an optimal pair. Notice that the Hamiltonian \eqref{egHam} is not concave function of $ u $, and therefore Theorem \ref{WSSMP} cannot be applied. We have to use other methods to check the optimality of $ \bar{u} $. Given any admissible pair $ (x(\cdot),u(\cdot)) $, apply Ito's formula on $ \phi(t,\alpha(t))x^2(t) $ and write it in integral form, \begin{equation}\label{egitointegral} E\left[ \phi(1,\alpha(1))x^2(1) \right]=E\bigg[ \int_0^1 x^2(t)\bigg( \phi_t(t,\alpha(t))+\sum_{j=1}^2q_{ij}(\phi(t,j)-\phi(t,\alpha(t))) \bigg)+\phi(t,\alpha(t))u^2(t)dt \bigg]. \end{equation} Substituting \eqref{egitointegral} into the cost functional and according to \eqref{eglineareqn}, \begin{align*} J(u(\cdot))&=E\bigg[ \int_0^1 \left( A(\alpha(t))u(t)+B(\alpha(t))u^2(t)-\frac{1}{2}\phi(t,\alpha(t))u^2(t)\right) dt \bigg]\\
&=E\bigg[ \int_{S_1}\left( -u(t)-\frac{1}{2}\phi(t,1)u^2(t)\right)dt+\int_{S_2}-\frac{1}{2}(1+\phi(t,2))u^2(t) dt \bigg]\\
&=E\bigg[ \int_{S_1}\left(-\frac{1}{2}\phi(t,1)\left( u(t)+\frac{1}{\phi(t,1)}\right)^2+\frac{1}{2\phi(t,1)}\right) dt+\int_{S_2}-\frac{1}{2}(1+\phi(t,2))u^2(t)dt \bigg], \end{align*} where $ S_1\equiv\left\{ t\vert t\in[0,1] \text{ such that } \alpha(t-)=1 \right\} $ and $ S_2\equiv\left\{ t\vert t\in[0,1] \text{ such that } \alpha(t-)=2 \right\}=[0,1]\backslash S_1 $. Since $ \phi(t,1) \leq 1 $ and $ \phi(t,2)<1 \forall t\in[0,1] $, the minimum value of the cost functional is achieved at $ \bar{u} $ defined in \eqref{egoptimalcontrol}. } \end{example}
\section{Preliminary Results} In this section, we introduce some preliminary results, which will be useful in the sequel. Hereafter, $ K $ represents a generic constant. \subsection{Clarke's Generalized Gradient and Normal Cone}\label{ClarkeIntro} In this subsection we recall some basic concepts and properties in nonsmooth analysis and optimization, which are needed in the statement and proof of the main results (Theorems \ref{WNSMP} and \ref{WSSMP}). Clarke's generalized gradient is first introduced to the finite dimensional space in \cite{clarke:gengradapp} and then extended to the infinite dimensional space in \cite{clarke:shadowprices, clarke:gengradfunctional} and \cite{aubin:setvaluedanalysis}. Interested readers may refer to \cite{clarke:optimization} for a detailed and complete treatment of the topic.
\begin{definition}(Generalized directional derivative) Let $ C $ be an open subset of a Banach space $ X $, and let a function $ f:C\longrightarrow \mathbb{R} $ be given. We suppose that $ f $ is Lipschitz on $ C $. The generalized directional derivative of $ f $ at $ x $ in the direction $ v $, denoted $ f^o(x;v) $, is given by \begin{equation*} f^o(x;v)=\limsup_{y\rightarrow_C x, \lambda \downarrow 0} \dfrac{f(y+\lambda v)-f(y)}{\lambda}. \end{equation*} \end{definition}
\begin{definition}(Clarke's generalized gradient) Let $ X^* $ denote the dual of $ X $ and $ \langle \cdot , \cdot \rangle $ be the duality pairing between $ X $ and $ X^* $. The generalized gradient of $ f $ at $ x $, denoted $ \partial{f}(x) $, is the set of all $ \zeta $ in $ X^* $ satisfying \begin{equation*} f^o(x;v)\geq \langle v, \zeta\rangle \textit{ for } \forall v\in X. \end{equation*} \end{definition}
\begin{theorem}\label{minmaxgradient} If $ f $ attains a local minimum or maximum at x, then $ 0\in \partial{f}(x) $. \end{theorem}
Theorem \ref{minmaxgradient} is only valid in the case where $ C $ is open. When the function is defined on a general non-empty subset of $ X $, we need to introduce the so-called distance function and the concept of Clarke's tangent cone and normal cone.
\begin{definition}(Distance function) Let $ X $ be a Banach space and $ C $ be a non-empty subset of $ X $. The distance function $ d_C:X\rightarrow \mathbb{R} $ is defined as \begin{equation*}
d_C(x)=\inf\{\| x-c \|: c\in C\}. \end{equation*} \end{definition}
\begin{theorem} The function $ d_C $ satisfies the following global Lipschitz condition on $ X $ \begin{equation*}
\vert d_C(x)-d_C(y) \vert \leq \| x-y \|. \end{equation*} \end{theorem}
\begin{definition}(Adjacent cone)\label{ajacentcone} Let $\bar{C}$ be the closure of C and $ x\in \bar{C} $. The adjacent cone to C at x, denoted as $ T_C^b(x) $, is defined by \begin{equation*} T^b_C(x):=\{ v\vert \lim_{h\rightarrow 0^+} d_C(x+hv)/h = 0 \}. \end{equation*} \end{definition}
\begin{definition}(Tangent cone) Suppose $ x\in C $. A vector $ v $ in $ X $ is a tangent to $ C $ at x provided $ d_C^o(x;v)=0 $. The tangent cone to C at x, denoted as $ T_C(x) $, is the set of all tangents to C at x. \end{definition}
In addition, when the set C is convex, it can be proved that the adjacent and tangent cones coincide, see \cite[Proposition 4.2.1]{aubin:setvaluedanalysis}.
\begin{theorem}\label{AdjacentTangent} Assume that C is convex. Then $T_C(x) = T_C^b(x)$. \end{theorem}
\begin{definition}(Normal cone) Let $ x\in C $. The normal cone to C at x is defined by the polarity with $ T_C(x) $: \begin{equation*} N_C(x)=\{ \xi \in X^*: \langle \xi,v \rangle \leq 0 \textit{ for all } v\in T_C(x)\}. \end{equation*} \end{definition}
The following necessary optimality condition is proved in \cite[page 52 Corollary]{clarke:optimization}.
\begin{theorem} \label{NeceOptCond} Assume that f is Lipschitz near x and attains a minimum over C at x. Then $ 0 \in \partial f(x)+N_C(x) $. \end{theorem}
\subsection{Markovian Regime-Switching SDE and BSDE} In this subsection, we establish the existence and uniqueness theorem of solutions to regime switching SDEs of the form \eqref{stateeqn}. First, we give the definition of the solution. \begin{definition}\cite[Definition 3.11]{maoyuan:SDEswitching} An $ \mathbb{R}^n $ valued stochastic process $ \{ x(t) \}_{0\leq t\leq T} $ is called a solution of equation \eqref{stateeqn} if it has the following properties: \begin{enumerate} \item $ \{ x(t) \} $ is continuous and $ \mathcal{F}_t $-adapted; \item $ \{ b(t,x(t),u(t),\alpha(t-)) \}\in L^1_{\mathcal{F}}(S;\mathbb{R}^n) $ and $ \{ \sigma(t,x(t),u(t),\alpha(t-))\in L^2_{\mathcal{F}}(S;\mathbb{R}^{n\times m}) \} $; \item for any $ t\in[0,T] $, equation \begin{align*} x(t)=x_0+\int_0^t b(s,x(s),u(s),\alpha(s-))ds+\int_0^t \sigma(t,x(s),u(s),\alpha(s-))dW(s) \end{align*} holds with probability 1. \end{enumerate} A solution $ \{ x(t) \} $ is said to be unique if any other solution $ \{ \tilde{x}(t) \} $ is indistinguishable from $ \{ x(t) \} $, that is \begin{align*} \mathbb{P}\{ x(t)=\tilde{x}(t) \mbox{ for all }0\leq t\leq T \}=1 \end{align*} \end{definition} Using the same method as in \cite[Chapter 3, Theorem 3.13]{maoyuan:SDEswitching}, the existence and uniqueness of solutions to regime-switching SDE of type \eqref{stateeqn} can be proved. \begin{theorem}\label{exisuniqSDE} Under assumption \textbf{(A1)}, given control $ u\in L^4(S;\mathbb{R}^k) $, there exists a unique solution $ x(t) $ to equation \eqref{stateeqn} and moreover, \begin{equation}\label{SDEEstimate} E\left( \sup_{0\leq t\leq T}\vert x(t) \vert^2 \right)\leq K\left( 1+\vert x \vert^2 + \int_0^TE\vert u(t) \vert^2dt \right) \end{equation} for some constant $ K\geq 0 $. \end{theorem}
We now develop results for existence and uniqueness of adapted solutions to regime switching BSDEs of type \eqref{ajointeqn}. Here we use the method of contraction mapping as in \cite[Chapter 6, Section 3]{yong.zhou:stochasticcontrols} and \cite[Chapter 6, Section 2]{pham:contimuoustimeSC} with the help of a martingale representation theorem for the joint filtration of a vector Brownian motion and a finite state Markov chain. Here we introduce the Dol\'{e}ans measure $ v_{[Q_{ij}]} $ on the measure space $ \left(\Omega\times[0,T],\mathcal{P}^\star\right) $: \begin{align*} v_{[Q_{ij}]}[A]:=E\int_0^T\mathcal{X}_A(\omega,t)d[Q_{ij}](t), \forall A\in\mathcal{P}^\star, \forall i,j\in I, i\neq j. \end{align*} By $ G=H \ v_{[Q]}$-a.e. for $ \mathbb{R}^{d\times d} $ mappings $ G $ and $ H $ on the set $ \Omega\times[0,T] $, we mean that \begin{align*} G_{ij}&=H_{ij} \ v_{[Q_{ij}]}\text{-a.e. } \forall i,j\in I, \ i\neq j\\ \text{and } G_{ii}&=H_{ii} \ \left( \mathbb{P}\otimes\text{Leb} \right)\text{-a.e. } \forall i\in I. \end{align*} We start by defining the following spaces for stochastic processes. \begin{align*} \mathbb{S}^2(\left[0,T\right]):=&\bigg\{Y:\Omega\times[0,T]\rightarrow\mathbb{R}^n\vert Y \text{ is } \mathcal{F}_t \text{ progressively measurable }\\ &\text{and } E\left(\sup_{0\leq t \leq T}\vert Y(t) \vert^2\right) < \infty\bigg\},\\
L^2\left(W,[0,T]\right)&:=\left\{\Lambda:\Omega\times[0,T]\rightarrow\mathbb{R}^{n\times m}\vert \Lambda\in\mathcal{P}^\star \text{ and } E\int_0^T\|\Lambda(t)\|^2 dt<\infty \right\},\\
L^2\left(Q,[0,T]\right)&:=\bigg\{ \Gamma=\left\{\left(\Gamma_{ij}^{(1)}\right)_{i,j=1}^d,\cdots ,\left(\Gamma_{ij}^{(n)}\right)_{i,j=1}^d\bigg\} \middle| \Gamma^{(l)}_{ii}=0 \ \mathbb{P}\otimes\text{Leb}-a.e. \forall i\in I, \right.\\
&\Gamma^{(l)}_{ij}\in \mathcal{P}^\star \text{ and } \sum_{l=1}^n\sum_{i,j=1}^dE\int_0^T\| \Gamma^{(l)}_{ij}(t) \|^2d\left[ Q_{ij} \right](t)<\infty \ \forall i,j\in I, i\neq j \bigg\}. \end{align*} It can be proved that $ L^2(W,[0,T]) $ and $ L^2(Q,[0,T]) $ are Hilbert spaces (see \cite[Lemma A.2.5]{DonnellyPhdThesis}). Next we present a martingale representation theorem for square integrable martingales with joint filtration generated by a Brownian motion and a finite state Markov chain. The proof can be found in \cite[Theorem B.4.6]{DonnellyPhdThesis} and \cite[Proposition 3.9]{donheu:regimeswitching}. \begin{theorem}\label{MRT} Suppose the $ \mathbb{R}^n $-valued process $ \left\{Y(t),t\in[0,T]\right\} $ is a square-integrable $ \{\mathcal{F}_t\} $-martingale and null at the origin. Then there exists processes $ \Lambda\in L^2(W,[0,T]) $ and $ \Gamma\in L^2(Q,[0,T]) $ such that $ Y $ has the stochastic integral representation \begin{equation}\label{martingalerep} Y(t)=Y(0)+\sum_{j=1}^m\int_0^t\Lambda_j(s)dW^j(s)+\int_0^t\Gamma(s)\bullet dQ(s) \ \text{a.s. }\forall t\in[0,T] \end{equation} with the square-bracket quadratic variation process of $ Y $ given by \begin{align*} \left[Y\right](t):=\sum_{i=1}^n\sum_{j=1}^m\int_0^t\Lambda_{ij}^2(s)ds+\sum_{l=1}^n\sum_{i,j=1}^d\int_0^t\left(\Gamma_{ij}^{\left(l\right)}(s)\right)^2d\left[Q_{ij}\right](t)\text{ a.s. }\forall t\in[0,T]. \end{align*} Moreover, $ \Lambda $ and $ \Gamma $ are unique in the sense that if $ \tilde{\Lambda}\in L^2(W,[0,T]) $ and $ \tilde{\Gamma}\in L^2(Q,[0,T]) $ are such that \eqref{martingalerep} holds, then $ \Lambda=\tilde{\Lambda} \ \mathbb{P}\otimes\text{Leb}-a.e. $ and $ \Gamma=\tilde{\Gamma} \ v_{[Q]}-a.e.$ \end{theorem}
Suppose we are given a pair $ (\xi,f) $ called the terminal and generator satisfying the following conditions: \begin{enumerate} \item[(a)] $E\vert \xi \vert^2 <\infty $, \item[(b)] $f: \Omega\times[0,T]\times\mathbb{R}^n\times\mathbb{R}^{n\times m}\rightarrow\mathbb{R}^n \text{ such that}$ \begin{enumerate} \item[(i)] $ f(t,y,z) $ is $ \mathcal{F}_t $-progressively measurable for all $y,z $. \item[(ii)] $ f(t,0,0)\in L_{\mathcal{F}}^2(S;\mathbb{R}^n) $, \item[(iii)] $ f $ satisfies uniform Lipschitz condition in $ \left( y,z \right) $, i.e $\exists C_f>0 \text{ such that } $ \begin{align*} \vert f(t,y_1,z_1)-f(t,y_2,z_2) \vert \leq C_f\left( \vert y_1-y_2 \vert+\vert z_1-z_2 \vert \right) \end{align*} $\forall y_1,y_2\in \mathbb{R}^n,z_1,z_2\in\mathbb{R}^{n\times m} \ \mathbb{P}\otimes\text{Leb} \ a.e. $ \end{enumerate} \end{enumerate} Consider the regime switching BSDE \begin{equation}\label{BSDE} -dY(t)=f(t,Y(t),Z(t))dt-Z(t)dW(t)-S(t)\bullet dQ(t), \ Y(T)=\xi. \end{equation} \begin{definition}\label{RSBSDEdefn} A solution to the regime switching BSDE \eqref{BSDE} is a set $ (Y,Z,S)\in\mathbb{S}^2([0,T])\times L^2(W,[0,T])\times L^2(Q,[0,T]) $ satisfying \begin{align*} Y(t)=\xi+\int_t^T f(s,Y(s),Z(s)))ds-\int_t^T Z(s)dW(s)-\int_t^T S(s)\bullet dQ(t). \end{align*} \end{definition} Now we prove the existence and uniqueness of a solution to the regime switching BSDE of type \eqref{BSDE}. \begin{theorem}\label{RSBSDEtheorem} Given a pair $ (\xi,f) $ satisfying $ (a) $ and $ (b) $, there exists a unique solution $ (Y,Z,S) $ to the regime switching BSDE \eqref{BSDE}. \end{theorem} The proof follows a contraction mapping argument similar to that in \cite[Chapter 6, Section 2]{pham:contimuoustimeSC}. For completeness, we give details in Appendix.
\subsection{A Moment Estimation} In this subsection, we prove a moment estimation result. A simplified version of the moment estimate can be found in \cite[Chapter 3 Lemma 4.2 ]{yong.zhou:stochasticcontrols}. \begin{lemma}\label{momest} Let $ Y(t)\in L^2_{\mathcal{F}}(S;\mathbb{R}^n) $ be the solution of the following regime switching SDE \begin{equation} \left\{ \begin{array}{l} dY(t)=[A(t)Y(t)+\beta(t)]dt+\displaystyle\sum_{j=1}^m\left[ B^j(t)Y(t)+\gamma^j(t) \right]dW^j(t)\\ Y(0)=y_0 \end{array} \right. \end{equation} where $ A,B^j : \Omega\times[0,T]\rightarrow\mathbb{R}^{n\times n}$ and $ \beta,\gamma^j:\Omega\times[0,T]\rightarrow\mathbb{R}^n$ are $ \{\mathcal{F}_t\}_{t\geq 0} $-adapted and \begin{equation} \left\{ \begin{array}{ll} \vert A(t)\vert, \vert B^j(t)\vert \leq K \mbox{ a.e.} t\in[0,T], \mathbb{P}\mbox{-a.s.}\\ \displaystyle\int_0^TE\vert \beta(s) \vert^{2k}ds + \int_0^TE\vert \gamma^j(s) \vert^{2k}ds <\infty \mbox{ for some } k\geq 1. \end{array} \right. \end{equation} Then \begin{equation}\label{momesteqn} \begin{array}{cc} \sup\limits_{t\in[0,T]}E\vert Y(t) \vert^{2k} \leq K\left\{ E\vert y_0 \vert^{2k}+ \displaystyle\int_0^TE\vert \beta(s) \vert^{2k}ds+ \sum_{j=1}^m\int^T_0E\vert \gamma^j(s) \vert^{2k}ds \right\} \end{array} \end{equation} \end{lemma} \begin{proof} For notation simplicity, we prove only the case $ m=n=1 $, leaving the case $ m,n>1 $ to the interested reader. We first assume that $ \beta,\gamma$ are bounded. Let $ \epsilon>0 $ and define \begin{equation}\label{approxabsoluteY} \langle Y \rangle_\epsilon\triangleq\sqrt{\vert Y\vert^2+\epsilon^2}, \forall Y\in\mathbb{R}^L. \end{equation} Note that for any $ \epsilon>0 $, the map $ Y\rightarrow\langle Y \rangle_\epsilon $ is smooth and $ \langle Y \rangle_\epsilon\rightarrow\vert Y \vert $ as $ \epsilon\rightarrow 0 $. Applying Ito's formula to $ \langle Y(t) \rangle_\epsilon^{2k} $, we have \begin{align*} &d\langle Y(t) \rangle_\epsilon^{2k}=2k\langle Y(t) \rangle_\epsilon^{2k-1}\left.\dfrac{\vert Y(t) \vert}{\langle Y(t) \rangle_\epsilon}\right\{ \left[A(t)Y(t)+\beta(t)\right]dt+\left[B(t)Y(t)+\gamma(t)\right]dW(t)\bigg\} \\ &+\left[ k(2k-1)\langle Y(t) \rangle_\epsilon^{2k-2}\dfrac{\vert Y(t) \vert^2}{\langle Y(t) \rangle^2_\epsilon} +k\langle Y(t) \rangle_\epsilon^{2k-1}\dfrac{\epsilon^2}{\langle Y(t) \rangle^3_\epsilon} \right]\left[B(t)Y(t)+\gamma(t)\right]^2dt. \end{align*} Writing it in integral form and taking expectation. Since $ \langle Y(t) \rangle_\epsilon>\vert Y(t) \vert $ and $ 2k-1\geq1 $, we obtain
\begin{align*} E\langle Y(t) \rangle^{2k}_\epsilon\leq & E\langle Y(0) \rangle^{2k}_{\epsilon}+2kE\int_0^t\langle Y(s) \rangle_{\epsilon}^{2k-1}\left\{ \vert A(s)\vert\langle Y(s) \rangle_\epsilon + \vert \beta(s) \vert \right\}ds\\ &+k(2k-1)E\int_0^t\langle Y(s) \rangle_\epsilon^{2k-2}\left[ \vert B(s)\vert\langle Y(s) \rangle_\epsilon +\vert\gamma(s)\vert \right]^2ds\\ \leq & E\langle Y(0) \rangle^{2k}_\epsilon+\left. KE\int_0^t\right\{\langle Y(s) \rangle_\epsilon^{2k}+\vert \beta(s) \vert\langle Y(s) \rangle^{2k-1}_\epsilon +\vert \gamma(s) \vert^2\langle Y(s) \rangle^{2k-2}_\epsilon\bigg\} ds, \end{align*} where $ K $ is a constant independent of $ t $. Applying \textit{Young's inequality}, we get \begin{align*} E\langle Y(t) \rangle_\epsilon^{2k}\leq & E\langle Y(0) \rangle^{2k}_\epsilon +KE\int_0^t\bigg\{\langle Y(s) \rangle_\epsilon^{2k}+\vert \beta(s) \vert^{2k}+\vert \gamma(s) \vert^{2k}\bigg\}ds. \end{align*} Finally, \textit{Gronwall's inequality} yields \begin{equation}\label{momestlasteqn} \begin{array}{ll} \sup\limits_{t\in[0,T]}E\langle Y(t) \rangle^{2k}_\epsilon \leq & K\bigg\{ E\langle Y(0) \rangle^{2k}_\epsilon +E\displaystyle\int_0^T \left[ \vert \beta(s) \vert^{2k} + \vert \gamma(s) \vert^{2k}\right]ds\bigg\}, \end{array} \end{equation} for some constant $ K$. Letting $ \epsilon\rightarrow 0 $ in \eqref{approxabsoluteY}, then \eqref{momestlasteqn} becomes \eqref{momesteqn}. \end{proof}
\subsection{Lipschitz Property} \begin{lemma}\label{Liplemma1} Let $ u_1, u_2 \in L^4(S;\mathbb{R}^k)$ and $ x_1, x_2 $ be the associated state processes satisfying \eqref{stateeqn}. The we have the following inequality: \begin{align*}
\sup_{t\in[0,T]}E\vert x_1(t)-x_2(t) \vert^4 \leq K\|u_1-u_2\|^4 \end{align*} \end{lemma} \begin{proof} Let $\xi(t) \triangleq x_1(t)-x_2(t)$. Then we have \begin{align*} d\xi(t)=&\left[ b(t,x_1(t),u_1(t),\alpha(t-))-b(t,x_2(t),u_2(t),\alpha(t-)) \right]dt\\
&+\left[ \sigma(t,x_1(t),u_1(t),\alpha(t-))-\sigma(t,x_2(t),u_2(t),\alpha(t-)) \right]dW(t) \end{align*} For $ \varphi=b $ and $ \sigma $, let \begin{equation}\label{Lip1} \tilde{\varphi}_x(t)=\int_0^1\varphi_x\left(t,x_2(t)+\theta(x_1(t)-x_2(t)),u_1(t),\alpha(t-)\right)d\theta. \end{equation} Substitute \eqref{Lip1}, we obtain \begin{align*} d\xi(t)=&\left[ \tilde{b}_x(t)\xi(t)+b(t,x_2(t),u_1(t),\alpha(t-))-b(t,x_2(t),u_2(t),\alpha(t-)) \right]dt\\
+&\left[ \tilde{\sigma}_x(t)\xi(t)+ \sigma(t,x_2(t),u_1(t),\alpha(t-))-\sigma(t,x_2(t),u_2(t),\alpha(t-))\right]dW(t). \end{align*} By Lemma \ref{momest}, we obtain \begin{align*} \sup_{t\in[0,T]}E\vert \xi(t) \vert^4 \leq & K\left\{ \int_0^TE\vert b(t,x_2(t),u_2(t),\alpha(t-))-b(t,x_2(t),u_1(t),\alpha(t-)) \vert^4 dt \right.\\
& + \int_0^TE\vert \sigma(t,x_2(t),u_2(t),\alpha(t-))-\sigma(t,x_2(t),u_1(t),\alpha(t-)) \vert^4 dt\\
\leq & K\left\{ \int_0^TE\vert u_1(t)-u_2(t) \vert^4dt \right\} \end{align*} \end{proof} \begin{lemma}\label{lemmalocalLip} The cost functional $ J:L^4(S;\mathbb{R}^k)\rightarrow\mathbb{R} $ is locally Lipschitz, i.e. for all $ \hat{u}\in L^4(S;\mathbb{R}^k) $, there exists a small ball $ B_{\hat{u}}^M $ with radius $ M>0 $ containing $ \hat{u} $ on which, we have \begin{equation}\label{localLip}
\vert J(u_1)-J(u_2) \vert \leq K_{M,\hat{u}}\| u_1-u_2 \|, \end{equation} for $ \forall u_1,u_2\in B_{\hat{u}}^M $, where $ K_{M,\hat{u}} $ is a constant dependent on $ MF\text{ and } \hat{u}$. \end{lemma} \begin{proof} Given $ \hat{u}\in L^4(S;\mathbb{R}^k) $ and $ M>0 $, define \begin{align*}
B_{\hat{u}}^M \triangleq \left\{ u \in L^4(S;\mathbb{R}^k):\| u-\hat{u} \|<M \right\}. \end{align*} For any $ u_1,u_2 \in B^M_{\hat{u}} $ with associated state processes $ x_1,x_2 $, according to \textbf{(A4)}, we have \begin{align*} &E\vert h(x_1(T),\alpha(T))-h(x_2(T),\alpha(T)) \vert\\ &\leq E\int_0^1\vert \langle h_x(x_1(T)+\theta(x_2(T)-x_1(T)),\alpha(T)),x_2(T)-x_1(T) \rangle \vert d\theta\\ &\leq K\left\{ E\left( 1+\vert x_1(T) \vert^2+\vert x_2(T) \vert^2 \right) \right\}^{\frac{1}{2}} \left\{ E\vert x_2(T)-x_1(T) \vert^2 \right\}^{\frac{1}{2}}\\ &\leq K\left\{ E(1+\vert \hat{x}(T) \vert^2+\vert \hat{x}(T)-x_1(T) \vert^2+\vert \hat{x}(T)-x_2(T) \vert^2 ) \right\}^{\frac{1}{2}} \left\{ E\vert x_2(T)-x_1(T) \vert^2 \right\}^{\frac{1}{2}} \end{align*} by H\"{o}lder's inequality and Minkowski's inequality. According to Theorem \ref{exisuniqSDE}, Jensen's inequality and Lemma \ref{Liplemma1}, \begin{align*} E\vert h(x_1(T),\alpha(T))-h(x_2(T),\alpha(T)) \vert \leq &K_M\left\{ 1+\left( \int_0^TE\vert \hat{u}(t) \vert^2 dt \right)^{\frac{1}{2}} \right\}\left\{ E\vert x_2(T)-x_1(T) \vert^2 \right\}^{\frac{1}{2}}\\ \leq & K_M\left\{ 1+\left( \int_0^TE\vert \hat{u}(t) \vert^4 dt \right)^{\frac{1}{4}} \right\}\left\{ E\vert x_2(T)-x_1(T) \vert^4 \right\}^{\frac{1}{4}}\\
\leq & K_{M,\hat{u}}\| u_1-u_2 \|. \end{align*} On the other hand, \begin{align*} E&\int_0^T\vert f(t,x_1(t),u_1(t),\alpha(t)) - f(t,x_2(t),u_2(t),\alpha(t)) \vert dt\\ \leq & E\int_0^T\left(\vert f(t,x_1(t),u_1(t),\alpha(t)) - f(t,x_2(t),u_1(t),\alpha(t)) \vert\right. \\ &+ \left.\vert f(t,x_2(t),u_1(t),\alpha(t)) - f(t,x_2(t),u_2(t),\alpha(t)) \vert\right) dt. \end{align*} Following similar arguments, we have \begin{align*}
E\int_0^T\vert f(t,x_1(t),u_1(t),\alpha(t)) - f(t,x_2(t),u_1(t),\alpha(t)) \vert dt \leq K_{M,\hat{u}}\| u_1-u_2 \|. \end{align*} For the second term, by \textbf{(A3)} we have \begin{align*} &E\int_0^T\vert f(t,x_2(t),u_1(t),\alpha(t)) - f(t,x_2(t),u_2(t),\alpha(t)) \vert dt\\ &\leq E\int_0^T\left\{ K_1+K_2(\vert x_2(t) \vert+\vert u_1(t) \vert+\vert u_2(t) \vert)\right\}\vert u_1(t)-u_2(t) \vert dt\\
&\leq K\left\{ E\int_0^T( 1+\vert x_2(t) \vert^2+\vert u_1(t) \vert^2+\vert u_2(t) \vert^2 )dt \right\}^{\frac{1}{2}}\| u_1-u_2 \|\\
&\leq K_M\left( 1+\left\{ E\int_0^T\vert \hat{u}(t) \vert^4 dt \right\}^{\frac{1}{4}} \right)\| u_1-u_2 \|\\
&\leq K_{M,\hat{u}}\| u_1-u_2 \|, \end{align*} and \eqref{localLip} follows by combining the above inequalities. \end{proof}
\subsection{Taylor Expansions} Let $ (x,u) $ be an admissible pair. Let $ v\in\L^4(S;\mathbb{R}^k) $ and $ \epsilon > 0 $. Define $ u^\epsilon(t) \triangleq u(t)+\epsilon v(t) $ for all $ t\in[0,T] $. Let $ (x^\epsilon,u^\epsilon) $ satisfy the following stochastic control system: \begin{equation*} \left\{ \begin{array}{cl}
dx^\epsilon(t) =& b(t,x^\epsilon(t),u^\epsilon(t),\alpha(t-))dt + \sigma(t,x^\epsilon(t),u^\epsilon(t),\alpha(t-))dW(t), \ t\in[0,T],\\
x^\epsilon(0) =& x_0\in\mathbb{R}^L, \alpha(0)=i_0\in I. \end{array}\right. \end{equation*} Next, for $ \varphi=b,\ \sigma^{j}(1\leq j \leq M) \textit{ and } f $, we define \begin{align*} \left\{ \begin{array}{ll}
&\varphi_x(t) \triangleq \varphi_x(t,x(t),u(t),\alpha(t-)), \\
&\delta\varphi(t) \triangleq \varphi(t,x(t),u^\epsilon(t),\alpha(t-))-\varphi(t,x(t),u(t),\alpha(t-)). \end{array}\right. \end{align*} Let $ y^\epsilon $ be the solution of the following regime-switching SDE: \begin{equation}\label{yepsilon} \left\{ \begin{array}{cl}
dy^\epsilon(t) =& \left\lbrace b_x(t)y^\epsilon(t)+\delta b(t)\right\rbrace dt+ \sum\limits_{j=1}^m \left\lbrace \sigma_x^j(t)y^\epsilon(t)+\delta\sigma^j(t) \right\rbrace dW^j(t), \
t\in [0,T], \\
y^\epsilon(0) =& 0, \alpha(0)=i_0\in I. \end{array}\right. \end{equation}
\begin{remark} {\rm The variation in our proof is different from the so-called spike variation technique in the proof of Peng's maximum principle in \cite{peng:SMP} and \cite{yong.zhou:stochasticcontrols}. In their proof, where $ u^\epsilon(t)=u(t)+1_{[\tau,\tau+\epsilon]}v(t) $, one first perturbs an optimal control on a small set of size $ \epsilon $ and then let $ \epsilon \rightarrow 0 $. Whereas, in our proof we perturbs an optimal control over the whole space. Then reason behind this is that in the definition of Clarke's generalized directional derivative, $ v(t) $ represents a directional vector in $ L^4(S;\mathbb{R}^k) $ and must be fixed. One perturbs the control through multiplication of a scalar $ \epsilon $ and letting $ \epsilon \rightarrow 0 $. } \end{remark}
The following lemma gives the Taylor expansion result of the state process and cost functional. \begin{lemma}\label{TaylorExp} Let assumptions \textbf{(A1)}-\textbf{(A4)} hold. Then, we have \begin{align}\label{firstTaylor}
\sup_{t\in[0,T]} E\left| x^\epsilon(t)-x(t) \right|^2 = O(\epsilon^2),\\\label{secondTaylor}
\sup_{t\in[0,T]} E\left| y^\epsilon(t) \right|^2 = O(\epsilon^2),\\\label{thirdTaylor}
\sup_{t\in[0,T]} E\left| x^\epsilon(t)-x(t)-y^\epsilon(t) \right|^2 = o(\epsilon^2). \end{align} Moreover, the following expansion holds for the cost functional: \begin{equation}\label{fourthTaylor} \begin{array}{cl} J(u^\epsilon)= J(u)+E\langle h_x(x(T),\alpha(T)),y^\epsilon(t) \rangle + E\displaystyle\int_0^T \left\lbrace \langle f_x(t) ,y^\epsilon(t)\rangle + \delta f(t)\right\rbrace dt + o(\epsilon). \end{array} \end{equation} \end{lemma} \begin{proof} For simplicity, we carry out the proof only for the case $ n=m=1 $. \\
\noindent \textit{Proof of \eqref{firstTaylor}}. Let $ \xi^\epsilon(t)\triangleq x^\epsilon(t)-x(t) $. The we have \begin{equation}\label{xiepsilon} \left\{ \begin{array}{cl}
d\xi^\epsilon(t) &= \left\{ \tilde{b}^\epsilon_x(t)\xi^\epsilon(t)+\delta b(t) \right\} dt + \left\{ \tilde{\sigma}_x^\epsilon(t)\xi^\epsilon(t)+\delta\sigma(t) \right\} dW(t)\\
\xi(0) &= 0, \alpha(0)=i_0. \end{array}\right. \end{equation} where for $\phi=b$ and $\sigma$, \begin{equation}\label{bsigmatilda} \begin{array}{cl}
\tilde{\phi}^\epsilon_x(t) \triangleq \displaystyle\int^1_0 \phi_x(t,x(t)+\theta(x^\epsilon(t)-x(t)),u^\epsilon(t),\alpha(t-))d\theta. \end{array} \end{equation} By Lemma \ref{momest}, since $ \tilde{b}^\epsilon_x(t) $, and $ \tilde{\sigma}^\epsilon_x(t) $ are bounded according to assumption \textbf{(A1)}, we obtain \begin{align*} \sup_{t\in[0,T]} E\vert \xi^\epsilon(t) \vert^2 & \leq K\int_0^T E \bigg\{ \vert \delta b(s) \vert^2 + \vert \delta \sigma(s) \vert^2 \bigg\} ds\\ & \leq K \epsilon^2 \int^T_0 E \vert v(s)\vert^2 ds \\ & \leq K \epsilon^2 . \end{align*} This proves \eqref{firstTaylor}. \\ \noindent \textit{Proof of \eqref{secondTaylor}}. Similarly, $ b_x(t) $ and $ \sigma_x(t) $ are bounded according to assumption \textbf{(A1)}. Applying Lemma \ref{momest} to \eqref{yepsilon}, we obtain \begin{align*} \sup_{t\in[0,T]} E\vert y^\epsilon(t) \vert^2 & \leq K\int_0^T E \bigg\{ \vert \delta b(s) \vert^2 + \vert \delta \sigma(s) \vert^2 \bigg\} ds\leq K \epsilon^2. \end{align*} This proves \eqref{secondTaylor}. \\
\noindent \textit{Proof of \eqref{thirdTaylor}}. Let $ \zeta^\epsilon(t)\triangleq x^\epsilon(t)-x(t)-y^\epsilon(t) \equiv \xi^\epsilon(t)-y^\epsilon(t) $. Then, by \eqref{xiepsilon} and \eqref{yepsilon} we have \begin{align*} d\zeta^\epsilon(t) = & d\xi^\epsilon(t)-dy^\epsilon(t)\\
= & \left\{ \tilde{b}_x^\epsilon(t)\xi^\epsilon(t)-b_x(t)y^\epsilon(t) \right\}dt+ \left\{ \tilde{\sigma}_x^\epsilon(t)\xi^\epsilon(t)-\sigma_x(t)y^\epsilon(t) \right\}dW(t)\\
= & \left\{ \tilde{b}_x^\epsilon(t)\zeta^\epsilon(t)+\left[ \tilde{b}_x^\epsilon(t)-b_x(t) \right]y^\epsilon(t) \right\}dt+ \left\{ \tilde{\sigma}_x^\epsilon(t)\zeta^\epsilon(t)+\left[ \tilde{\sigma}_x^\epsilon(t)-\sigma_x(t) \right]y^\epsilon(t) \right\}dW(t) \end{align*} Since $ \tilde{b}_x^\epsilon(t) $ and $ \tilde{\sigma}_x^\epsilon(t) $ are bounded by assumption \textbf{(A1)}, applying Lemma \ref{momest} we obtain \begin{equation}\label{thirdTaylortosub} \begin{array}{cl}
\sup_{t\in[0,T]} E\vert \zeta^\epsilon(t) \vert^2 \leq K\displaystyle\int_0^T E\bigg\{ \left| \left[ \tilde{b}_x^\epsilon(t)-b_x(t) \right]y^\epsilon(t) \right|^2 + \left| \left[ \tilde{\sigma}_x^\epsilon(t)-\sigma_x(t) \right]y^\epsilon(t) \right|^2 \bigg\} dt. \end{array} \end{equation} Recall that $ \bar{\omega} $ appearing in \textbf{(A4)} is a modulus of continuity. Thus for any $ \rho>0 $, there exists a constant $ K_{\rho}>0 $ such that \begin{equation}\label{modulusofcont} \bar{\omega}(r)\leq \rho + rK_{\rho}, \ \forall r\geq 0. \end{equation}
By H\"older's inequality, \eqref{bsigmatilda}, \eqref{secondTaylor}, \eqref{firstTaylor} and \eqref{modulusofcont}, we have \begin{align*}
&\int_0^T E\left| \left[ \tilde{b}^\epsilon_x(t)-b_x(t) \right]y^\epsilon(t) \right|^2 dt \\
& \leq \int_0^T \left( E\left| \tilde{b}_x^\epsilon(t)-b_x(t) \right|^4 \right)^{\frac{1}{2}}\left(E\left| y^\epsilon(t) \right|^4\right)^\frac{1}{2} dt\\
& \leq K\int_0^T \left\{E\int_0^1 \left| b_x\left(t,x(t)+\theta\xi^\epsilon(t),u^\epsilon(t),\alpha(t-)\right)-b_x(t)\right|^4 d\theta\right\}^{\frac{1}{2}}\epsilon^2 dt\\ & \leq K\int_0^T \left\{ E\left(\xi^\epsilon(t)^4+ \bar{\omega}(\epsilon v(t))^4\right) \right\}^{\frac{1}{2}}\epsilon^2 dt \\
&\leq K \int_0^T\left\{ \epsilon^4+E[\rho+K_\rho\epsilon |v(t)|]^4 \right\}^\frac{1}{2}dt\epsilon^2. \end{align*} Hence the first term in (\ref{thirdTaylortosub}) is $o(\epsilon^2)$. Similarly the second and third terms are also $o(\epsilon^2)$, which gives \eqref{thirdTaylor}. \\ \noindent \textit{Proof of \eqref{fourthTaylor}}. By definition of the cost functional \eqref{costfun}, we have \begin{equation} \label{temp1} \begin{array}{cl} &J(u^\epsilon)-J(u)\nonumber \\ &= E\left\{ h(x^\epsilon(T),\alpha(T))-h(x(T),\alpha(T)) \right\} \\ &+ \displaystyle E\int_0^T \left\{f(t,x^\epsilon(t),u^\epsilon(t),\alpha(t))-f(t,x(t),u(t),\alpha(t))\right\}dt \end{array} \end{equation}
For the first term on the right side of (\ref{temp1}) we have \begin{align*} &E\left\{ h(x^\epsilon(T),\alpha(T))-h(x(T),\alpha(T)) \right\} \\ &= E\int^1_0 \langle h_x(x(T)+\theta\xi^\epsilon(T),\alpha(T)),\xi^\epsilon(T) \rangle d\theta\\
&= E\langle h_x(x(T),\alpha(T)),y^\epsilon(T) \rangle + E\langle h_x(x(T),\alpha(T)),\zeta^\epsilon(T) \rangle\\
&+ E\int^1_0 \langle h_x(x(T)+\theta\xi^\epsilon(T),\alpha(T))-h_x(x(T),\alpha(T)),\xi^\epsilon(T)\rangle d\theta. \end{align*} Then, by \eqref{firstTaylor}, \eqref{thirdTaylor}, \textbf{(A4)} and applying H\"{o}lder's inequality, we have \begin{equation}\label{fourthTaylor1} E\left\{ h(x^\epsilon(T),\alpha(T))-h(x(T),\alpha(T)) \right\} = E\langle h_x(x(T),\alpha(T)),y^\epsilon(T) \rangle + o(\epsilon). \end{equation} For the second term on the right side of (\ref{temp1}) we have \begin{align*} E& \int_0^T\left\{ f(t,x^\epsilon(t),u^\epsilon(t),\alpha(t))-f(t,x(t),u(t),\alpha(t)) \right\}dt\\ =& E\int^T_0 \left\{ \int_0^1 \langle f_x(t,x(t)+\theta\xi^\epsilon(t),u^\epsilon(t),\alpha(t)),\xi^\epsilon(t) \rangle d\theta \right\}\\ &+ \left\{ f(t,x(t),u^\epsilon(t),\alpha(t))-f(t,x(t),u(t),\alpha(t)) \right\} dt\\ =& E\int_0^T \left\{ \langle f_x(t),y^\epsilon(t) \rangle + \delta f(t)\right\}\\ &+ \left\{\int_0^1\langle f_x(t,x(t)+\theta\xi^\epsilon(t),u^\epsilon(t),\alpha(t))-f_x(t), y^\epsilon(t) \rangle d\theta\right\}\\ &+ \left\{\int_0^1\langle f_x(t,x(t)+\theta\xi^\epsilon(t),u^\epsilon(t),\alpha(t)),\zeta^\epsilon(t) \rangle d\theta \right\}dt \end{align*} Then, using \textbf{(A4)} and by a similar argument as in the proof of \eqref{thirdTaylor}, we have \begin{equation}\label{fourthTaylor2} \begin{array}{cl} &E\displaystyle\int_0^T\left\{ f(t,x^\epsilon(t),u^\epsilon(t),\alpha(t))-f(t,x(t),u(t),\alpha(t)) \right\}dt \\ &=E\displaystyle\int_0^T \left\{ \langle f_x(t),y^\epsilon(t) \rangle + \delta f(t)\right\}+ o(\epsilon). \end{array} \end{equation} \eqref{fourthTaylor} follows from \eqref{fourthTaylor1} and \eqref{fourthTaylor2}. \end{proof}
\subsection{Duality Analysis} \begin{lemma} Let assumptions \textbf{(A1)}-\textbf{(A4)} hold. Let $ y^\epsilon $ be the solution of \eqref{yepsilon} and $ (p,q,s) $ be the adapted solution of \eqref{ajointeqn}. Then \begin{equation}\label{duality} \begin{array}{cc} E\langle p(T),y^\epsilon(T) \rangle = E\left. \displaystyle\int_0^T \right\{ \langle p(t),\delta b(t) \rangle + \langle f_x(t),y^\epsilon(t) \rangle+ tr\left( q(t)^\intercal\delta\sigma(t) \right)\bigg\} dt \end{array} \end{equation} \end{lemma}
\begin{proof} Applying Ito's lemma and taking expectation immediately lead to \eqref{duality}. \end{proof} Now we are able to give the following lemma, which is of great importance. \begin{lemma}\label{importlemma} Let assumptions \textbf{(A1)}-\textbf{(A4)} hold. For any $ \varepsilon > 0 $ and $ v \in L^4(S;\mathbb{R}^K) $, define \begin{equation*} u^\epsilon(t) \triangleq u(t)+\epsilon v(t) \textit{ for } \forall t\in [0,T]. \end{equation*} Then we have \begin{equation*} \begin{array}{cl} J(u^\epsilon)-J(u) \\ = E\displaystyle\int_0^T (-H(t,x(t),u^\epsilon(t),\alpha(t-),p(t),q(t))) -(-H(t,x(t),u(t),\alpha(t-),p(t),q(t))) dt +o(\epsilon) \end{array} \end{equation*} \end{lemma}
\begin{proof} According to Lemma \ref{TaylorExp}, we have \begin{align*} &J(u^\epsilon)-J(u) \\ &= E\langle h_x(x(T),\alpha(T)),y^\epsilon(T) \rangle
+ E\displaystyle\int_0^T \left\{\langle f_x(t),y^\epsilon(t)\rangle + \delta f(t)\right\} dt + o(\epsilon)\\
&= E\langle -p(T),y^\epsilon(T) \rangle
+ E\displaystyle\int_0^T \left\{\langle f_x(t),y^\epsilon(t)\rangle + \delta f(t)\right\} dt + o(\epsilon). \end{align*} Applying \eqref{duality}, we obtain \begin{align*} &J(u^\epsilon)-J(u)= E\displaystyle\int_0^T -\bigg\{ \langle p(t),\delta b(t) \rangle + tr\left(q(t)^\intercal\delta\sigma(t)\right) -\delta f(t) \bigg\} dt+o(\epsilon)\\ &= E\displaystyle\int_0^T (-H(t,x(t),u^\epsilon(t),\alpha(t-),p(t),q(t)))-(-H(t,x(t),u(t),\alpha(t-),p(t),q(t))) dt +o(\epsilon) \end{align*} \end{proof}
\section{Proof of the Main Theorems} \subsection{Proof of Theorem \ref{WNSMP} } We follow the technique developed in \cite{clarke:shadowprices}. Given an optimal 5-tuple $ (\bar{x},\bar{u},\bar{p},\bar{q},\bar{s}) $, define a functional $ \mathcal{H}^{\bar{u}}: L^4(S;\mathbb{R}^k)\rightarrow \mathbb{R} $ as following \begin{equation*} \mathcal{H}^{\bar{u}}(u) = E\int_0^T -H(t,\bar{x}(t),u(t),\alpha(t-),\bar{p}(t),\bar{q}(t))dt. \end{equation*} By a similar argument as in Lemma \ref{lemmalocalLip}, it can be proved that the functional $ \mathcal{H}^{\bar{u}} $ is also locally Lipschitz on $ L^4(S;\mathbb{R}^k) $. Next, we define Clarke's generalized gradient of the functionals $ J $ and $ \mathcal{H}^{\bar{u}} $ at $ \bar{u} $ and explore their properties. \begin{definition}\label{defJHtangent} Let $L^{\frac{4}{3}}(S;\mathbb{R}^k)$ denote the dual space of $ L^4(S;\mathbb{R}^k) $ and $ \langle\cdot ,\cdot\rangle $ denote the duality pairing between $L^4(S;\mathbb{R}^k) $ and $L^{\frac{4}{3}}(S;\mathbb{R}^k)$. Given an admissible control $ \bar{u}\in L^4(S;\mathbb{R}^k) $, Clarke's generalized gradient of $ J $ at $ \bar{u} $, denoted by $ \partial J(\bar{u}) $, is the set of all $ \zeta\in L^{\frac{4}{3}}(S;\mathbb{R}^k) $ satisfying \begin{equation}\label{ClarkeJ} J^o(\bar{u};v)=\limsup_{u\rightarrow\bar{u},\epsilon\rightarrow 0}\dfrac{J(u+\epsilon v)-J(u)}{\epsilon} \geq \langle v,\zeta \rangle, \end{equation} for all $v\in L^4(S;\mathbb{R}^k)$.
Clarke's generalized gradient of $ \mathcal{H}^{\bar{u}} $ at $ \bar{u} $ is defined similarly. \end{definition}
Then, according to Lemma \ref{importlemma}, given $ u\in L^4(S;\mathbb{R}^k) $, for any $ \epsilon >0 $ and $ v\in L^4(S;\mathbb{R}^k) $ such that $ u+\epsilon v\in L^4(S;\mathbb{R}^k) $, we have \begin{equation*} J(u+\epsilon v)-J(u) = \mathcal{H}^{\bar{u}}(u+\epsilon v) - \mathcal{H}^{\bar{u}}(u) +o(\epsilon). \end{equation*} Hence, we have \begin{equation*} J^o(\bar{u};v) = (\mathcal{H}^{\bar{u}})^o(\bar{u};v), \ \textit{for } \forall v\in L^4(S;\mathbb{R}^k). \end{equation*} Therefore, by Definition \ref{defJHtangent}, we conclude \begin{equation*} \partial J(\bar{u}) = \partial \mathcal{H}^{\bar{u}}(\bar{u}). \end{equation*} Since $ \bar{u} $ is an optimal control on $ \mathcal{U}_{ad} $, according to Theorem \ref{NeceOptCond}, \begin{equation}\label{optcondnormalcone} 0 \in \partial J(\bar{u})+N_{\mathcal{U}_{ad}}(\bar{u}) = \partial\mathcal{H}^{\bar{u}}(\bar{u})+N_{\mathcal{U}_{ad}}(\bar{u}). \end{equation}
To characterize Clarke's tangent cone in the $ L^4(S;\mathbb{R}^k) $ space, we recall \cite[Theorem 8.5.1]{aubin:setvaluedanalysis}. Let $ (\Omega,S,\mu) $ be a complete $ \sigma $-finite measure space and X be a separable Banach space. Consider a measurable set-valued map $K: \Omega\leadsto X $. We associate with it the subset $ \mathcal{K} \subset L^p(\Omega;X,\mu) $ of selections defined by \begin{align*} \mathcal{K}:=\{ x\in L^p(\Omega;X,\mu) \vert \textit{ for almost all }\omega\in\Omega, x(\omega)\in K(\omega) \}. \end{align*} \begin{theorem}\label{TangLeb} Assume that the set-valued map K is measurable and has closed images. Then for every $ x\in\mathcal{K} $, the set valued map $\omega \rightarrow T_{K(\omega)}^b(x(\omega))$ is measurable. Furthermore \begin{equation*} \{ v\in L^p(\Omega;X,\mu)\vert \textit{ for almost all } \omega, v(\omega)\in T_{K(\omega)}^b(x(\omega)) \} \subset T_{\mathcal{K}}^b(x). \end{equation*} \end{theorem} Returning to our proof, since $ U $ is convex, by definition, $ \mathcal{U}_{ad} $ is also a convex subset of $ L^4(S;\mathbb{R}^k) $. Therefore, by Theorem \ref{AdjacentTangent} and Theorem \ref{TangLeb}, we obtain \begin{equation}\label{AccTangLeb} T_{\mathcal{U}_{ad}}(\bar{u}) \supset \{ v\in L^4(S;\mathbb{R}^k) \vert v(\omega,t)\in T_{U}(\bar{u}(\omega,t)) \ \mu\textit{-almost surely}\}. \end{equation} The optimality condition \eqref{optcondnormalcone} together with \eqref{AccTangLeb} implies that $ \exists \zeta \in L^{\frac{4}{3}}(S;\mathbb{R}^k) $ such that \begin{equation}\label{twoinequalities} \left\{ \begin{array}{cl} &E\displaystyle\int_0^T\langle \zeta(t),v(t) \rangle dt \leq 0 \textit{ for } \forall v\in L^4(S;\mathbb{R}^k)\textit{ such that } \\ &v(t)\in T_{U}(\bar{u}(t)) \ \textit{ for every } t\in[0,T], \mathbb{P}\textit{-almost surely} \\ &(\mathcal{H}^{\bar{u}})^o(\bar{u};v) + E\displaystyle\int_0^T \langle\zeta(t),v(t) \rangle dt \geq 0 \textit{ for } \forall v \in L^4(S;\mathbb{R}^k). \end{array} \right. \end{equation}
Now, we recall a version of the measurable selection theorem in \cite{aubin:setvaluedanalysis}. \begin{definition}\cite[Definition 8.1.2]{aubin:setvaluedanalysis} Let $ (\Omega,\mathcal{A}) $ be a measurable space and $ X $ be a complete separable metric space. Consider a set-valued map $ F: \Omega\leadsto X $. A measurable map $ f:\Omega\mapsto X $ satisfying \begin{align*} \forall \omega \in \Omega, f(\omega)\in F(\omega) \end{align*} is called a measurable selection of $ F $. \end{definition} \begin{theorem}\cite[Theorem 8.1.3]{aubin:setvaluedanalysis}\label{measelectionthm} Let $ X $ be a complete separable metric space, $ (\Omega, \mathcal{A}) $ a measurable space, $ F $ a measurable set-valued map from $ \Omega $ to closed nonempty subsets of $ X $. Then there exists a measurable selection of $ F $. \end{theorem}
Return to our problem. Fix $ u\in\mathcal{U}_{ad} $ and $ (\omega,t)\in S $. Let $ \mathbb{Q}_+ $ denote the set of all strictly positive rationals. Following the argument in \cite[Page 325]{aubin:setvaluedanalysis} , we have \begin{align*} T_U(u(\omega,t)) &= T_U^b(u(\omega,t))
=\bigcap_{n>0} cl\left( \bigcup_{\alpha\in\mathbb{Q}_+} \bigcap_{h\in[0,\alpha]\cap \mathbb{Q}_+}\dfrac{U-u(\omega,t)}{h}+\frac{1}{n}B \right), \end{align*} where $ B $ denotes the unit ball centred at 0. By \cite[Theorem 8.2.4]{aubin:setvaluedanalysis}, we conclude that the set-valued function $ T_{U}(\bar{u}) $ is measurable.
For the first inequality in \eqref{twoinequalities}, let $ M > 0 $ and define $ \bar{B}_M \triangleq \{ v\in \mathbb{R}^k : \|v\|\leq M \} $. For any positive integer $ n $, define a set-valued function $ \Pi_n^M $ as follows \begin{equation*} \Pi_n^M (\omega,t) = \left\{ \begin{array}{cl}
&\{0\}, \ \textit{ if } \langle \zeta(\omega,t),v \rangle < \dfrac{1}{n}, \ \forall v\in \bar{B}_M \cap T_U(\bar{u}(\omega,t)) \\
&\{ v\in \bar{B}_M\cap T_U(\bar{u}(\omega,t)): \langle \zeta(\omega,t),v \rangle \geq \dfrac{1}{n} \}, \textit{ otherwise}.
\end{array} \right. \end{equation*}
The map $ (\omega,t,v)\rightarrow \langle \zeta(\omega,t),v \rangle $ is continuous in $ v $. Moreover, since $ \mathbb{R}^k $ is separable, the map can be expressed as the upper limit of a countable family of measurable functions and therefore is measurable. Therefore $ \Pi_n^M $ is measurable since countable intersection of measurable set-valued functions is still measurable. Hence, by Theorem \ref{measelectionthm}, $ \Pi_n^M $ admits a measurable selection $ v_n^M \in L^4(S;\mathbb{R}^k) $. Note that \eqref{twoinequalities} implies that the set \begin{equation*} \{ (\omega,t):\Pi_n^M(\omega,t)\neq \{0\} \} \end{equation*} must have $ \mu $ measure 0. Hence, we conclude that there exists a set, denoted as $ S_n^M $, where \begin{equation*} S_n^M = \{(\omega,t): \Pi^M_n(\omega,t)=\{0\}\} \end{equation*} and $ \mu(S_n^M)=1 $. Consequently, we have \begin{equation}\label{measelecineqfirstineq} \langle \zeta(\omega,t),v \rangle < \dfrac{1}{n} \ \ \forall v\in \bar{B}_M\cap T_{U}(\bar{u}(\omega,t)) \textit{ on } S^M_n. \end{equation} Define $ S^M=\bigcap_{n=1}^\infty S^M_n $ with $ \mu(S^M)=1 $ since $ \mu(S_n^M)=1 \ \forall n\in\mathbb{N}$. Moreover, since \eqref{measelecineqfirstineq} holds for all $ n $, we have \begin{equation}\label{measelecineq2firstineq} \langle \zeta(\omega,t),v \rangle \leq 0 \ \forall v\in\bar{B}_M\cap T_{U}(\bar{u}(\omega,t)) \textit{ on } S^M. \end{equation} Since \eqref{measelecineq2firstineq} holds for arbitrary $ M $, we obtain that \begin{equation}\label{necetocomb1} \langle \zeta(\omega,t),v \rangle \leq 0 \textit{ for } \forall v\in T_{U}(\bar{u}(\omega,t)) \ \textit{$\mu$-almost surely}. \end{equation}
Next, we consider the second inequality in \eqref{twoinequalities}. Define the partial generalized directional derivative of the Hamiltonian $ H $ at $ \bar{u}(t) $ in the direction $ v(t) $ as \begin{equation*} \begin{array}{cl} -H_u^o(\bar{u}(t);v(t))= \limsup\limits_{u\rightarrow\bar{u},\epsilon\rightarrow 0}\dfrac{1}{\epsilon}\bigg\{-H(t,\bar{x}(t), u(t)+\epsilon v(t),\alpha(t-),\bar{p}(t),\bar{q}(t),s(t))\\ +H(t,\bar{x}(t), u(t),\alpha(t-),\bar{p}(t),\bar{q}(t),s(t))\bigg\}. \end{array} \end{equation*} Using Fatou's Lemma on the second inequality in \eqref{twoinequalities}, we have \begin{align} & E\left\{ \int_0^T -H_u^o(\bar{u}(t);v(t))+\langle \zeta(t),v(t) \rangle dt\right\}
\geq (\mathcal{H}^{\bar{u}})^o(\bar{u};v)+E\int_0^T\langle \zeta(t),v(t) \rangle dt \geq 0\label{afterfatou} \end{align} \nocite{*}
Let $ M > 0 $ and define $ \bar{B}_M \triangleq \{ v\in \mathbb{R}^k : \|v\|\leq M \} $. For any $ n \in \mathbb{N}$, define a set-valued function $ \Gamma_n^M $ as follows \begin{equation*} \Gamma_n^M (\omega,t) = \left\{ \begin{array}{cl}
&\{0\}, \ \textit{ if } -H_u^o(\bar{u}(\omega,t);v)+\langle \zeta(\omega,t),v \rangle > -\dfrac{1}{n} \ \forall v\in \bar{B}_M \\
&\{ v\in \bar{B}_M: -H_u^o(\bar{u}(\omega,t);v) +\langle \zeta(\omega,t),v \rangle \leq -\dfrac{1}{n} \}, \textit{ otherwise}.
\end{array} \right. \end{equation*} Using a similar argument as above, with the help of Theorem \ref{measelectionthm} and \eqref{afterfatou}, we can show that the set $\{ (\omega,t):\Gamma_n^M(\omega,t)\neq \{0\} \}$ must have $ \mu $ measure 0, which implies that \begin{equation}\label{necetocomb2} -H_u^o(\bar{u}(\omega,t);v))+\langle \zeta(\omega,t),v \rangle \geq 0 \ \textit{$\mu$-almost surely}. \end{equation} Combining \eqref{necetocomb1} and \eqref{necetocomb2}, we conclude \begin{equation*} 0\in \partial_u(-H)(t,\bar{x}(t),\bar{u}(t),\alpha(t-),\bar{p}(t),\bar{q}(t)) + N_U(\bar{u}(t)),\ a.e. t\in[0,T], \ \textit{$\mathbb{P}$-a.s}. \end{equation*}
\subsection{Proof of Theorem \ref{WSSMP} } Given admissible pair $ (x,u) $, define \begin{equation*} H(t,x(t),u(t)) \triangleq H(t,x(t),u(t),\alpha(t-),\bar{p}(t),\bar{q}(t)) \textit{ for } \forall t\in[0,T],\ \mathbb{P}\textit{-a.s.} \end{equation*} Under the convexity condition, Clarke's generalized gradient and normal cone coincide with the subdifferential and normal cone in the sense of convex analysis. Moreover, combining \eqref{neceSMPcond} and the concavity of $ H(t,\bar{x}(t),\cdot) $ for all $ t\in [0,T] $ a.s, we conclude that \begin{equation*} H(t,\bar{x}(t),\bar{u}(t)) = \max_{u\in U}H(t,\bar{x}(t),u),\ \textit{a.e. } t\in[0,T],\ \mathbb{P}\textit{-a.s.} \end{equation*} Define $ \xi(t)\triangleq x(t)-\bar{x}(t) $ satisfying \begin{equation*} \left\{ \begin{array}{cl}
d\xi(t) &= \left\{b(t,x(t),u(t),\alpha(t-))-b(t,\bar{x}(t),\bar{u}(t),\alpha(t-))\right\}dt \\
&+ \sum\limits_{j=1}^m\left\{\sigma^j(t,x(t),u(t),\alpha(t-))-\sigma^j(t,\bar{x}(t),\bar{u}(t),\alpha(t-))\right\}dW^j(t),\ t\in[0,T],\\
\xi(0) &= 0, \alpha(0)=i_0. \end{array}\right. \end{equation*} Following a standard separating hyperplane argument in convex analysis (see \cite[Chapter 5]{rockafeller:convexanalysis}), we obtain \begin{equation}\label{sufficientclaim} \int_0^T \left\{ H(t,x(t),u(t))-H(t,\bar{x}(t),\bar{u}(t)) \right\} \leq \int_0^T\langle H_x(t,\bar{x}(t),\bar{u}(t)),\xi(t) \rangle dt \end{equation} for any admissible pair $ (x,u) $. Detailed proof of \eqref{sufficientclaim} can be found in \cite{oksendal:SSMP}.
Applying Ito's formula to $\langle \bar{p}(t),\xi(t) \rangle$, noting the convexity of $ h $, the inequality (\ref{sufficientclaim}) and the definition of the Hamilitonian \eqref{Hamiltonian}, we have \begin{align*} &E\{h(x(T),\alpha(T))-h(\bar{x}(T),\alpha(T))\}\\ \geq &E\langle h_x(\bar x(T),\alpha(T)),\xi(T\rangle)\rangle\\
=& -E\langle \bar{p}(T),\xi(T) \rangle \\ = & E \int_0^T \bigg\{ \langle H_x(t,\bar{x}(t),\bar{u}(t)),\xi(t) \rangle \\ &- \langle \bar{p}(t),b(t,x(t),u(t),\alpha(t-))-b(t,\bar{x}(t),\bar{u}(t),\alpha(t-)) \rangle\\ &-\sum_{j=1}^m\langle \bar{q}^j(t),\sigma^j(t,x(t),u(t),\alpha(t-))-\sigma^j(t,\bar{x}(t),\bar{u}(t),\alpha(t-))\rangle \bigg\} dt \\ \geq & -E\int_0^T\{ f(t,x(t),u(t),\alpha(t-))-f(t,\bar{x}(t),\bar{u}(t),\alpha(t-)) \}dt. \end{align*} Therefore $J(\bar{u}) \leq J(u)$ for all $ u\in \mathcal{U}_{ad} $.
\section{Conclusion} We have proved in the paper a weak version of the necessary and sufficient stochastic maximum principle in a regime-switching diffusion model. Instead of insisting on the maximum condition of the Hamiltonian, we showed that $ 0 $ belongs to the sum of Clarke's generalized gradient of $ -H $ and Clarke's normal cone at the optimal control $ \bar{u} $, which also removes the requirement of the differentiability of the functions in the control variable. Under certain concavity conditions on the Hamiltonian, the necessary condition becomes sufficient. The theorem does not involve any second order terms, hence the second order differentiability of the functions in the state variable is not required. Moreover, the absence of the second order adjoint equation considerably simplifies the SMP. Futher research on this topic includes the extension of the weak SMP to more general stochastic control systems such as nonconvex control constraints and locally Lipschitz coefficients. We are currently working on these problems.
{\noindent\bf Acknowledgment}. The authors are grateful to Professor Nicole El Karoui for the useful discussions on the paper, especially on the contents of the measurability of stochastic processes.
\appendix \section{Appendix} \subsection{Proof of Theorem \ref{RSBSDEtheorem}}
\begin{proof} Consider the function $ \Phi $ on $ \mathbb{S}^2([0,T])\times L^2(W,[0,T])\times L^2(Q,[0,T]) $ mapping $ (Y,Z,S)\in \mathbb{S}^2([0,T])\times L^2(W,[0,T])\times L^2(Q,[0,T]) $ to $ \left(\hat{Y},\hat{Z},\hat{S}\right)=\Phi(Y,Z,S) $ defined by \begin{equation*} \hat{Y}(t)=\xi+\int_t^Tf(s,Y(s),Z(s))ds-\int_t^TZ(s)dW(s)-\int_t^TS(s)\bullet dQ(s). \end{equation*} Consider the square-integrable martingale \begin{equation*}
M(t)=E\left[\xi+\int_0^Tf(s,Y(s),Z(s))ds\middle|\mathcal{F}_t\right]. \end{equation*} According to Theorem \ref{MRT}, there exists unique $ \left( \hat{Z},\hat{S} \right)\in L^2(W,[0,T])\times L^2(Q,[0,T]) $ such that \begin{equation*} M(t)=M(0)+\int_0^t\hat{Z}(s)dW(s)+\int_0^t\hat{S}(s)\bullet dQ(s). \end{equation*} We then define the process $ \hat{Y}(t) $ by \begin{align*}
\hat{Y}(t)&=E\left[ \xi+\int_t^Tf(s,Y(s),Z(s))ds\middle|\mathcal{F}_t \right]\\
&=M(t)-\int_0^tf(s,\alpha(s),Y(s),Z(s))ds\\
&=M(0)+\int_0^t\hat{Z}(s)dW(s)+\int_0^t\hat{S}(s)\bullet dQ(s)-\int_0^tf(s,Y(s),Z(s))ds\\
&=\xi+\int_t^Tf(s,Y(s),Z(s))ds-\int_t^T\hat{Z}(s)dW(s)-\int_t^T\hat{S}(s)\bullet dQ(s). \end{align*} By Doob's $ L^2 $ inequality, we have \begin{align*}
&E\left[\sup_{0\leq t\leq T}\middle|\int_t^T\hat{Z}(s)dW(s)\middle|\right]\leq 4E\left[ \int_0^T|\hat{Z}(s)|^2ds \right]<\infty,\\
&E\left[\sup_{0\leq t\leq T}\middle|\int_t^T\hat{S}(s)\bullet dQ(s)\middle|\right]\leq 4E\left[ \sum_{l=1}^n\sum_{i,j=1}^d\int_0^T|\hat{S}_{ij}^{(l)}(s)|^2d[Q_{ij}](s) \right]<\infty. \end{align*} Under the assumptions on $ (\xi,f) $, we conclude that $ \hat{Y}\in S^2([0,T]) $. Hence $ \Phi $ is a well defined function from $ S^2([0,T])\times L^2(W,[0,T])\times L^2(Q,[0,T]) $ into itself. Next, we show that $ (\hat{Y},\hat{Z},\hat{S}) $ is a solution to the regime switching BSDE \eqref{BSDE} if and only if it is a fixed point of $ \Phi $.
Let $ (U,V,\Gamma) $, $ (U',V',\Gamma') \in S^2([0,T])\times L^2(W,[0,T])\times L^2(Q,[0,T])$. Apply function $ \Phi $ and obtain $ (Y,Z,S)=\Phi(U,V,\Gamma),\ (Y',Z',S')=\Phi(U',V',\Gamma') $. Set $ (\bar{U},\bar{V},\bar{\Gamma}) =(U-U',V-V',\Gamma-\Gamma')$, $ (\bar{Y},\bar{Z},\bar{S})=(Y-Y',Z-Z',S-S') $ and $ \bar{f}(t)=f(t,U(t),V(t))-f(t,U'(t),V'(t)) $. Take $ \beta>0 $ to be chosen later and apply Ito's formula to $ e^{\beta s}\vert \bar{Y} \vert^2 $ on $ [0,T] $, \begin{equation}\label{BSDEito} \begin{array}{cl} \vert \bar{Y}(0) \vert^2 =& -\displaystyle\int_0^Te^{\beta t}\left( \beta\vert\bar{Y}(t)\vert^2-2\bar{Y}(t)^\intercal\bar{f}(t) \right)dt-\int_0^Te^{\beta t}\vert \bar{Z}(t) \vert^2dt\\ &-\displaystyle\int_0^Te^{\beta t}\sum_{l=1}^n\sum_{i,j=1}^d\vert\bar{S}^{(l)}_{ij} \vert^2d\left[Q_{ij}\right](t)-2\int_0^T e^{\beta t}\bar{Y}(t)^\intercal\bar{Z}(t)dW(t)\\ &-2\displaystyle\int_0^Te^{\beta t}\sum_{l=1}^n\sum_{i,j=1}^d\bar{Y}^{(l)}(t)\bar{S}_{ij}^{(l)}(t)dQ_{ij}(t). \end{array} \end{equation}
Observe that, according to Young's inequality \begin{align*} &E\left[\left( \int_0^T e^{2\beta t}\vert \bar{Y}(t) \vert^2\vert \bar{Z}(t) \vert^2dt \right)^{\frac{1}{2}}\right]\leq \dfrac{e^{\beta T}}{2}E\left[ \sup_{0\leq t\leq T}\vert \bar{Y}(t) \vert^2+\int_0^T\vert \bar{Z}(t) \vert dt \right]<\infty,\\ &E\left[\left( \int_0^T e^{2\beta t}\vert \bar{Y}^{(l)}(t) \vert^2\vert \bar{S}_{ij}^{(l)}(t) \vert^2 d\left[Q_{ij}\right](t) \right)^{\frac{1}{2}}\right]\\ &\leq \dfrac{e^{\beta T}}{2}E\left[ \sup_{0\leq t\leq T}\vert \bar{Y}^{(l)}(t) \vert^2+\int_0^T\vert \bar{S}^{(l)}_{ij}(t) \vert^2d\left[Q_{ij}\right](t) \right]<\infty. \end{align*} Hence $ \int_0^t e^{\beta s}\bar{Y}(s)^\intercal\bar{Z}(s)dW(s) $ and $ \int_0^te^{\beta s}\sum_{l=1}^n\sum_{i,j=1}^d\bar{Y}^{(l)}(s)\bar{S}_{ij}^{(l)}(s)dQ_{ij}(s) $ are true martingales by the Burkholder-Davis-Gundy inequality. Taking expectation in \eqref{BSDEito}, we get \begin{equation}\label{BSDEtosub} \begin{array}{ll} E\vert \bar{Y}(0) \vert^2 + E\bigg\lbrace\displaystyle\int_0^T e^{\beta t}\bigg[ \left(\beta\vert \bar{Y}(t) \vert^2+\vert \bar{Z}(t) \vert^2\right)dt+\sum_{l=1}^n\sum_{i,j=1}^d\vert \bar{S}^{(l)}_{ij}(t) \vert^2d\left[Q_{ij}\right](t) \bigg]\bigg\rbrace\\ = \displaystyle2E\left[\int_0^Te^{\beta t}\bar{Y}(t)^\intercal\bar{f}(t)dt \right] \leq 2C_fE\left[ \int_0^Te^{\beta t}\vert \bar{Y}(t) \vert\left( \vert \bar{U}(t)\vert+\vert\bar{V}(t) \vert \right)dt \right]\\ \leq \displaystyle 4C_f^2E\bigg[\int_0^T e^{\beta t}\vert \bar{Y}(t) \vert^2dt \bigg]+\dfrac{1}{2}E\bigg[ \int_0^T e^{\beta t}\left( \vert \bar{U}(t) \vert^2+\vert \bar{V}(t) \vert^2 \right) dt \bigg]. \end{array} \end{equation} Take $ \beta=1+4C_f^2 $ and substitute into \eqref{BSDEtosub}, we have \begin{align*} &E\bigg[ \int_0^T e^{\beta t}\left( \vert \bar{Y}(t) \vert^2+\vert \bar{Z}(t) \vert^2 \right)dt+\int_0^Te^{\beta t}\sum_{l=1}^n\sum_{i,j=1}^d\vert \bar{S}^{(l)}_{ij}(t)\vert^2d[Q_{ij}](t) \bigg]\\ &\leq \dfrac{1}{2}E\bigg[ \int_0^T e^{\beta t}\left( \vert \bar{U}(t) \vert^2+\vert \bar{V}(t) \vert^2 \right)dt \bigg] \\ &\leq \dfrac{1}{2}E\bigg[ \int_0^T e^{\beta t}\left( \vert \bar{U}(t) \vert^2+\vert \bar{V}(t) \vert^2 \right)dt \bigg]+\dfrac{1}{2}E\bigg[ \int_0^Te^{\beta t}\sum_{l=1}^n\sum_{i,j=1}^d\vert \bar{\Gamma}^{(l)}_{ij}(t) \vert^2d[Q_{ij}](t) \bigg]. \end{align*} Notice that $ L^2(W,[0,T]) $ and $ L^2(Q,[0,T]) $ are Hilbert spaces and therefore the space $ \mathbb{S}^2([0,T])\times L^2(W,[0,T])\times L^2(Q,[0,T]) $ endowed with the norm \begin{align*}
\| (Y,Z,S) \|_{\beta}=\left\lbrace E\bigg[ \int_0^T e^{\beta t}\left( \vert \bar{Y}(t) \vert^2+\vert \bar{Z}(t) \vert^2 \right)dt+\int_0^Te^{\beta t}\sum_{l=1}^n\sum_{i,j=1}^d\vert\bar{S}^{(l)}_{ij}(t)\vert^2 d[Q_{ij}](t) \bigg] \right\rbrace^{\frac{1}{2}} \end{align*} is a Banach space. We conclude that $ \Phi $ admits a unique fixed point which is the solution to the BSDE \eqref{BSDE}. \end{proof}
\end{document} |
\begin{document}
\begin{Large} \centerline{On a problem related to "second" best approximations to a real number} \vskip+0.5cm \centerline{Pavel Semenyuk} \end{Large}
\vskip+1cm \section{Introduction}
\vskip+0.3cm For a given irrational number $\alpha$ we consider its continued fraction expansion \begin{equation}\label{q}
\alpha = [a_0, a_1, a_2, ...] = a_0 + \dfrac{1}{a_1+\dfrac{1}{a_2+...}}, ~a_0\in\mathbb{Z}, ~a_i\in\mathbb{Z}_{+}, ~i = 1, 2, ... \end{equation}
and its convergents $\dfrac{p_n}{q_n} = [a_0, a_1, ..., a_n]$. We call two irrational numbers $\alpha = [a_0, a_1, a_2, ...]$ and $\beta = [b_0, b_1, b_2, ...]$ equivalent and write $\alpha\sim\beta$, if there exist positive integers $m$ and $n$ such that $a_{n+k} = b_{m+k}, k=1,2,3,...$, that is the tails of continued fraction expansions for $\alpha$ and $\beta$ coincide. Let $$
\psi_{\alpha}(t) = \min\limits_{1\leqslant q \leqslant t, q\in\mathbb{Z}} ||q\alpha|| ,\,\,\,\,\,\text{where}\,\,\,\,\, ||x|| = \min\limits_{n\in\mathbb{Z}}|x-n| $$ be the irrationality measure function for $\alpha$. By Lagrange's theorem on best approximations (see \cite{khinchin1964}) it is a piecewise constant function, namely
$$ \psi_{\alpha}(t) = |q_{\nu}\alpha - p_{\nu}| \,\,\,\,\,\text{ for} \,\,\,\,\, q_\nu\leqslant t\leqslant q_{\nu+1}. $$
Many classical result concerning rational approximations to a real number $\alpha$ can be formulated in terms of $\psi_{\alpha}(t)$.
For example, in terms of irrationality measure function $\psi_\alpha$ one can define the Lagrange constant
\begin{equation}\label{l}
\lambda(\alpha) =
\liminf_{t\to \infty} t\cdot \psi_\alpha (t).
\end{equation}
The set of all possible values of $\lambda(\alpha)$ is known as
Lagrange spectrum $$
\mathbb{L} = \{\lambda ~|~\exists\alpha\in\mathbb{R\setminus Q} \colon \lambda = \lambda(\alpha) \}. $$ It is very well studied (see for example \cite{cusick1989markoff}).
\vskip+0.3cm
In \cite{moshchevitin2017} Moshchevitin considered an irrationality measure function $$
\psi_{\alpha}^{[2]}(t) = \min\limits_{\substack{(q, p)\colon q, p \in\mathbb{Z}, 1\leqslant q\leqslant t, \\ (p, q) \neq (p_n, q_n) ~\forall n\in\mathbb{Z_{+}}}} |q\alpha -p|, $$ related to so-called "second best" approximations, the corresponding Diophantine constant \begin{equation}\label{k}
\mathfrak{k}(\alpha) = \liminf\limits_{t\to\infty} t \cdot\psi_{\alpha}^{[2]}(t) \end{equation} and the corresponding spectrum $$
\mathbb{L}_2 = \{ \lambda ~| ~\exists\alpha\in\mathbb{R\setminus Q} \colon \lambda = \mathfrak{k}(\alpha) \}. $$ In particular, he studied different properties of the function $\psi_{\alpha}^{[2]}(t)$ and calculated two largest elements of the spectrum $\mathbb{L}_2$. In the present paper we calculate the value for the third element of the spectrum $\mathbb{L}_2$.
\vskip+0.3cm
\section{Results on the spectrum $\mathbb{L}_2$}
\vskip+0.3cm
The following structural result about $\mathbb{L}_2$ was proven in \cite{moshchevitin2017}.
\vskip+0.3cm
\textbf{Theorem 1.} \begin{itemize}
\item[1.] The largest element of $\mathbb{L}_2$ is $\dfrac{4}{\sqrt{5}}$. Moreover, $\mathfrak{k}(\alpha) = \dfrac{4}{\sqrt{5}}$ if and only if $\alpha\sim\dfrac{1+\sqrt{5}}{2} = [1; \overline{1}]$; \item[2.] If $\alpha\not\sim\dfrac{1+\sqrt{5}}{2}$, then $\mathfrak{k}(\alpha)\leqslant\dfrac{4}{\sqrt{17}}$; \item[3.] The equality $\mathfrak{k}(\alpha) = \dfrac{4}{\sqrt{17}}$ holds if and only if $\alpha\sim\dfrac{1+\sqrt{17}}{2} = [2; \overline{1,1,3}]$; \item[4.] The point $\dfrac{4}{\sqrt{17}}$ is an isolated point of $\mathbb{L}_2$. \item[5.] The whole segment $\left[ 0, \frac{12}{21+\sqrt{15}}\right]$ belongs to $\mathbb{L}_2$.
\end{itemize} \vskip+0.3cm
The main result of the present paper is as follows.
\vskip+0.3cm
\textbf{Theorem 2.} \begin{enumerate} \item If $\alpha$ is irrational and not equivalent neither to $\frac{1+\sqrt{5}}{2}$ nor to $\frac{1+\sqrt{17}}{2}$, then $\mathfrak{k}(\alpha) \leqslant \frac{164}{13\sqrt{173}}$. \item Let $\alpha_0 = \frac{13\sqrt{173} + 39}{82} = [2; \overline{1, 1, 3, 1, 1, 1, 1, 3}]$. Then $\mathfrak{k}(\alpha) =\mathfrak{k}(\alpha_0) = \frac{164}{13\sqrt{173}}$ if and only if $\alpha \sim \alpha_0$. \item The point $\frac{164}{13\sqrt{173}}$ is an isolated point of $\mathbb{L}_2$. \end{enumerate}
\vskip+0.3cm
\section{Auxiliary results and notation}
\vskip+0.3cm
Consider the tails
$$
\alpha_n = [a_n; a_{n+1}, a_{n+2}, a_{n+3}, ...],\,\,\,\,\,
\alpha^{*}_n = [0; a_n, a_{n-1}, a_{n-2}, ..., a_1]
$$
of continued fraction (\ref{q}). In terms of these quantities it is natural to give a formula
\begin{equation}\label{pe}
\left| \alpha - \frac{p_n}{q_n}\right| = \frac{1}{q_n^2 (\alpha_{n+1}+\alpha_n^*)}
\end{equation}
for the approximation to $\alpha$ by its convergent fraction.
Relation (\ref{pe}) is known as Perron's formula. By means of (\ref{pe}) one can express the value of $\lambda(\alpha)$
defined in (\ref{l}) as
\begin{equation}\label{pe1}
\lambda(\alpha) = \liminf_{n\to \infty} \frac{1}{{\alpha_{n+1}+\alpha_n^*}}.
\end{equation}
An analog of the formula (\ref{pe1}) for the value of $\frak{k}(\alpha)$ was obtained in \cite{moshchevitin2017} .
It is as follows.
Consider the quantities $$ \varkappa^1_n(\alpha) = \dfrac{(1 + \alpha^{*}_{n-1})(\alpha_n - 1)}{\alpha_n + \alpha^{*}_{n-1}},\,\,\,\,\,\, \varkappa^2_n(\alpha) = \dfrac{(1 - \alpha^{*}_n)(\alpha_{n+1} + 1)}{\alpha_{n+1} + \alpha^{*}_n} , \,\,\,\,\,\, \varkappa^4_n(\alpha) = \dfrac{4}{\alpha_n + \alpha^{*}_{n-1}}. $$ (we follow the notation from \cite{moshchevitin2017}). Then
if $\alpha \not\sim \frac{1+\sqrt{5}}{2}$ one has \begin{equation}\label{one} \mathfrak{k}(\alpha) = \liminf\limits_{n\to\infty\colon a_n\geqslant 2}\min(\varkappa^1_n, \varkappa^2_n, \varkappa^4_n) \end{equation} (see Hilfssatz 13 from \cite{moshchevitin2017}). We should note that it is clear that for $\alpha \sim \beta$ one has $ \mathfrak{k}(\alpha) = \mathfrak{k}(\beta) $.
\vskip+0.3cm
Besides formula (\ref{one}) we need some other auxiliary statements from \cite{moshchevitin2017}. We formulate them as the following
\vskip+0.3cm
\textbf{Proposition 1.} Let $\alpha$ be an irrational number not equivalent to $\dfrac{1+\sqrt{5}}{2} = [1; \overline{1}]$ and $\dfrac{1+\sqrt{17}}{2} = [2; \overline{1,1,3}]$. \begin{itemize}
\item[1.] If for infinitely many $n$ in continued fraction expansion (\ref{q}) one has $a_n \geqslant 5$, then $\mathfrak{k}(\alpha) \leqslant \frac{4}{5}$; \item[2.] If for all $n$ sufficiently large in continued fraction expansion (\ref{q}) one has $a_n \leqslant 4$ and for infinitely many $n$ the equality $a_n = 4$ occurs, then $\mathfrak{k}(\alpha) \leqslant \frac{4}{3 + \sqrt{2}}$; \item[3.] If for all $n$ sufficiently large $a_n \leqslant 4$ and for infinitely many $n$ the equality $a_n = 2$ occurs, then $\mathfrak{k}(\alpha) \leqslant \sqrt{2} - \frac{1}{2}$; \item[4.] If for infinitely many $n$ one has $a_n \in \{1, 3\} $ and \item[4.1.] for infinitely many $n$ one has $a_{n-1} = a_n = 3$, then $\mathfrak{k}(\alpha) \leqslant \frac{39}{43}$; \item[4.2.] for infinitely many $n$ one has $a_{n-1} = 3, a_n = 1, a_{n+1} = 3$, then $\mathfrak{k}(\alpha) \leqslant \frac{136}{145}$; \item[4.3.] for infinitely many $n$ one has $a_{n-2} = a_{n-1} = 1, a_n = 3, a_{n+1} = a_{n+2} = a_{n+3} = 1$, then $\mathfrak{k}(\alpha) \leqslant \frac{180}{187}$ \end{itemize}
\vskip+0.3cm
{\bf Remark.} Although the main result on the structure of $\mathbb{L}_2$ from the paper \cite{moshchevitin2017} is correct, its proof from \cite{moshchevitin2017} contains
a mistake: in the cases 4.2 and 4.3 in \cite{moshchevitin2017} instead of upper bounds $\frac{136}{145}$ and $ \frac{180}{187}$ correspondingly, different incorrect bounds were given.\\ \vskip+0.3cm
The bounds for $\mathfrak{k}(\alpha)$ from Proposition 1 can be written as the following table. \vskip+0.3cm
\begin{center}
\begin{tabular}{ | c | c | c | c | } \hline Case & Conditions for partial quotients & Upper bound & Numerical value\\ \hline 1 & $\text{inf. many}\,\, a_n \geqslant 5$ & $4/5$ & 0.8 \\ \hline 2 & $\text{almost all}\,\, a_n\le 4, \,\,\text{inf. many}\,\, a_n = 4$ & $4/(3+\sqrt{2})$ & $0.906163^+$ \\ \hline 3 & $\text{almost all}\,\, a_n\le 4, \,\,\text{inf. many}\,\, a_n = 2$ & $\sqrt{2} - 1/2$ & $0.914213^+$ \\ \hline 4.1 & $\text{inf. many patterns}\,\,$ 3 3 & 39/43 & $0.906976^+$ \\ \hline 4.2 &$\text{inf. many patterns}\,\,$ 3 1 3 & 136/145 & $0.937931^+$ \\ \hline 4.3 & $\text{inf. many patterns}\,\,$ 1 1 3 1 1 1 & 180/187 & $0.962566^+$ \\ \hline \end{tabular} \end{center}
\vskip+0.3cm
\section{The value of $\mathfrak{k} (\alpha_0)$}
\vskip+0.3cm
In this section we prove the following
\vskip+0.3cm
\textbf{Lemma 1.} $\mathfrak{k}(\alpha_0) = \liminf\limits_{n\to\infty\colon a_n\geqslant 2}\varkappa^4_n = \dfrac{164}{13\sqrt{173}}$.
\vskip+0.3cm
\textbf{Proof.} Consider blocks of digits $ {A} = \{1,1,3\}$ and ${ B} = \{1,1,1,1,3\}$. Now $\alpha_0 = [2; \overline{1, 1, 3, 1, 1, 1, 1, 3}]$ can be written as $\alpha_0 =[2;\overline{A B}]$. In view of (\ref{q}), it suffices to prove that for all large enough $n$ such that $a_n = 3$, the value of $\varkappa^4_n$ is less than both $\varkappa^1_n$ and $\varkappa^2_n$. Let us consider separately those values of $n$ which are either $\equiv 0 \pmod{8}$ (if we take a 3 after a block of four 1's) or $\equiv 3 \pmod{8}$ (if we take a 3 after a block of two 1's). Thus, we need to consider two possible limits for each $\varkappa^1_n, \varkappa^2_n, \varkappa^4_n$.
Let us calculate all six of them: $$ \lim\limits_{k\to\infty} \varkappa^1_{8k-5} = \dfrac{(1+[0;\overline{AB}])([3;\overline{BA}]-1)}{[0;\overline{AB}]+[3;\overline{BA}]} = \dfrac{167}{13\sqrt{173}} = 0.976674^+, $$ $$ \lim\limits_{k\to\infty} \varkappa^1_{8k} = \dfrac{(1+[0;\overline{BA}])([3;\overline{AB}]-1)}{[0;\overline{BA}] + [3;\overline{AB}]} = \dfrac{169}{13\sqrt{173}} = 0.988371^+, $$ $$ \lim\limits_{k\to\infty} \varkappa^2_{8k-5} = \dfrac{(1-[0;3,\overline{AB}])([1;1,1,1,3,\overline{AB}]+1)}{[0;3,\overline{AB}]+[1;1,1,1,3,\overline{AB}]} = \dfrac{169}{13\sqrt{173}} = 0.988371^+, $$ $$ \lim\limits_{k\to\infty} \varkappa^2_{8k} = \dfrac{(1-[0;3,\overline{BA}])([1;1,3,\overline{BA}]+1)}{[0;3,\overline{BA}]+[1;1,3,\overline{BA}]} = \dfrac{167}{13\sqrt{173}} = 0.976674^+, $$ $$ \lim\limits_{k\to\infty} \varkappa^4_{8k-5} = \lim\limits_{k\to\infty} \varkappa^4_{8k} = \dfrac{4}{[0;\overline{AB}]+[0;\overline{BA}]+3} = \dfrac{164}{13\sqrt{173}} = 0.959129^+. $$ We see that the last one is the smallest one. This proves Lemma 1.
\vskip+0.3cm
\section{Extremality of $\mathfrak{k} (\alpha_0)$}
\vskip+0.3cm
In this section we finalise the proof of Theorem 2.
\vskip+0.3cm
In cases 1-3, 4.1 and 4.2 of Proposition 1 we have $\mathfrak{k}(\alpha) \leqslant \frac{136}{145} < \mathfrak{k}(\alpha_0) = \frac{164}{13\sqrt{173}}= 0.959129^+$, meanwhile the bound of the case 4.3 satisfies the inequality $ \frac{180}{187} >\mathfrak{k}(\alpha_0)$. So we should consider in more details just the case 4.3. First of all we consider the following four subcases in the case 4.3:
\vskip+0.3cm
\begin{itemize} \item[4.3.1] One has infinitely many patterns $\{1, 1, \boldsymbol{3}, 1, 1, 1, 1, 1\}$ (here and in other cases the bold \textbf{3} represents the $n$th element of the continued fraction). In that case \begin{multline*} \mathfrak{k}(\alpha) \leqslant \liminf\limits_{n\to\infty\colon a_n\geqslant 2}\varkappa^4_n(\alpha) \leqslant \dfrac{4}{[0;1,1,\overline{3,1}] + [3;1,1,1,1,1,\overline{1,3}]} =\\= \dfrac{4}{\frac{\sqrt{21}+1}{10} + \frac{\sqrt{21}+943}{262}} = 0.958087^+. \end{multline*} \item[4.3.2] One has infinitely many patterns $\{1, 1, \boldsymbol{3}, 1, 1, 1, 3\}$. In that case \begin{multline*} \mathfrak{k}(\alpha) \leqslant \liminf\limits_{n\to\infty\colon a_n\geqslant 2}\varkappa^4_n(\alpha) \leqslant \dfrac{4}{[0;1,1,\overline{3,1}] + [3;1,1,1,3,1,1,\overline{3,1}]} =\\= \dfrac{4}{\frac{\sqrt{21}+1}{10} + \frac{\sqrt{21}+4575}{1258}} = 0.952692^+. \end{multline*} \item[4.3.3] One has infinitely many patterns $\{ABB\}$ (or, in other words, infinitely many patterns \\$\{1, 1, 3, 1, 1, 1, 1, \boldsymbol{3}, 1, 1, 1, 1, 3, 1, 1\}$. In that case \begin{multline*} \mathfrak{k}(\alpha) \leqslant \liminf\limits_{n\to\infty\colon a_n\geqslant 2}\varkappa^4_n(\alpha) \leqslant \dfrac{4}{[0;1,1,1,1,3,1,1,\overline{1,3}]+[3;1,1,1,1,3,1,1,\overline{1,3}]} =\\= \dfrac{4}{\frac{3329+\sqrt{21}}{5470} + \frac{19739+\sqrt{21}}{5470}} = 0.948123^+. \end{multline*} \item[4.3.4] One has infinitely many patterns $\{AAB\}$ (or, in other words, infinitely many patterns \\$\{1, 1, 3, 1, 1, 1, 1, \boldsymbol{3}, 1, 1, 3, 1, 1, 3, 1, 1\}$. In that case \begin{multline*} \mathfrak{k}(\alpha) \leqslant \liminf\limits_{n\to\infty\colon a_n\geqslant 2}\varkappa^4_n(\alpha) \leqslant \dfrac{4}{[0;1,1,1,1,3,1,1,\overline{1,3}]+[3;1,1,3,1,1,3,1,1,\overline{3,1}]} =\\= \dfrac{4}{\frac{3329+\sqrt{21}}{5470} + \frac{120383+\sqrt{21}}{33802}}= 0.959006^+. \end{multline*}
\end{itemize}
We see that in all the subcases 4.3.1, 4.3.2, 4.3.3, 4.3.4 the upper bound for $\mathfrak{k}(\alpha) $ obtained is strictly less than $\mathfrak{k}(\alpha_0) = \frac{164}{13\sqrt{173}}$.
Let us describe all the cases considered. Cases 1-3 cover all the continued fractions with infinitely many terms different from 1 and 3. Cases 4.1, 4.2, 4.3.1 and 4.3.2 cover all the fractions with infinitely many patterns of the form $[..., 3, \underbrace{1, ..., 1}_{k}, 3, ...]$, where $k\not\in\{2, 4\}$. Therefore, the only fractions which we did not consider in the cases above, eventually consist of blocks of the form $ {A} $ and ${ B}$. Case 4.3.3 covers all the fractions with infinitely many patterns $ABB$, and case 4.3.4 covers fractions with infinitely many patterns $AAB$. Thus, the only fractions left in consideration are eventually periodic with the period $\overline{AB}$, or, in other words, are equivalent to $\alpha_0=\frac{13\sqrt{173}+39}{82}$. Together with Lemma 1 this proves Theorem 2.
\end{document} |
\begin{document}
\title[On Averages of Order Statistics]{Uniform Estimates for Averages of Order Statistics of Matrices}
\author{Richard Lechner} \address{Institute of Analysis, Johannes Kepler University Linz, Altenberger Stra\ss e 69, 4040 Linz, Austria} \email{richard.lechner@jku.at}
\author{Markus Passenbrunner} \address{Institute of Analysis, Johannes Kepler University Linz, Altenberger Stra\ss e 69, 4040 Linz, Austria} \email{markus.passenbrunner@jku.at}
\author{Joscha Prochno} \address{Institute of Analysis, Johannes Kepler University Linz, Altenberger Stra\ss e 69, 4040 Linz, Austria} \email{joscha.prochno@jku.at}
\date{\today}
\begin{abstract} We prove uniform estimates for the expected value of averages of order statistics of matrices in terms of their largest entries. As an application, we obtain similar probabilistic estimates for $\ell_p$ norms via real interpolation. \end{abstract}
\maketitle
\section{Introduction and Main Results} \label{sec:introduction}
Combinatorial and probabilistic inequalities play an important role in a variety of areas of mathematics, especially in Banach space theory. In \cite{KS1} and \cite{KS2}, S.~Kwapie\'n and C.~Sch\"utt studied combinatorial expressions involving matrices and obtained inequalities in terms of the average of the largest entries of the matrix. To be more precise, they showed that \begin{equation}\label{eq:Kwapien Schuett estimate max-norm}
\frac{1}{n!} \sum_{\pi \in\mathfrak{S}_n} \max_{1\leq i \leq n} |a_{i\pi(i)}| \simeq \frac{1}{n}\sum_{k=1}^n s(k), \end{equation} where $s(k)$ is the $k$-th largest entry of the matrix $a$ and $\mathfrak S_n$ the symmetric group. This estimate seems crucial if one wants to compute the projection constant of symmetric Banach spaces and related invariants. Among other things, the authors obtained estimates for the positive projection constant of finite dimensional Orlicz spaces and estimated the order of the projection constant of the Lorentz spaces $\ell_{2,1}^n$. Also, the symmetric sublattices of $\ell_1(c_0)$ as well as the finite dimensional symmetric subspaces of $\ell_1$ were characterized. Further applications and extensions of \eqref{eq:Kwapien Schuett estimate max-norm} can be found in \cite{KS2,S1,S2,MSS,PS}, just to mention a few.
The main result of this paper is a generalization of \eqref{eq:Kwapien Schuett estimate max-norm} in the sense that we study the expected value of averages of higher order statistics of a matrix in a more general setting described below. Our method of proof is purely probabilistic in nature, whereas the proof of \eqref{eq:Kwapien Schuett estimate max-norm} in \cite{KS1} uses non-trivial combinatorial arguments.
In what follows, given a finite set $G$, we denote the normalized counting measure on $G$ by $\mathbb P$, i.e., \[
\mathbb P(E) = \frac{|E|}{|G|},\qquad E \subseteq G, \]
where $|\cdot|$ denotes the cardinality. $\mathbb E$ will always denote the expectation with respect to the normalized counting measure. Moreover, for a vector $x\in\mathbb R^n$ with non-negative entries, we denote its $k$-th largest entry by \[ \kmax\limits_{1\leq i \leq n} x_i. \] In particular, $\onemax_{1\leq i \leq n} x_i$ is the maximal value, $\nmax_{1\leq i \leq n} x_i$ the minimal value of $x$. Our main result is the following:
\begin{thm}\label{thm:main} Let $n,N\in\mathbb N$ and $a\in \mathbb R^{n\times N}$. Let $G$ be a collection of maps from $I=\{1,\dots,n\}$ to $J=\{1,\dots,N\}$ and $C_G>0$ be a constant only depending on $G$. Assume that for all $i\in I$, $j\in J$ and all different pairs $(i_1,j_1),(i_2,j_2)\in I\times J$ \begin{enumerate}[(i)] \item\label{eq:condition 1} $\mathbb P(\{g\in G: g(i)=j\})=1/N$, \item\label{eq:condition 2} $\mathbb P(\{g\in G: g(i_1)=j_1, g(i_2)=j_2 \}) \leq C_G/N^2$. \end{enumerate} Then, for any $\ell \leq n$, \begin{equation}\label{eq:main result firs part}
\frac{c}{N} \sum_{j=1}^{\ell N}s(j) \leq \int_{G} \sum_{k=1}^{\ell}\kmax\limits_{1\leq i \leq n} |a_{ig(i)}| \,\mathrm{d} \mathbb P(g) \leq \frac{2}{N} \sum_{j=1}^{\ell N}s(j), \end{equation} where $c = 2^{-5}(1+2C_G)^{-2}$. \end{thm}
Observe that estimate \eqref{eq:Kwapien Schuett estimate max-norm} \cite[Theorem 1.1]{KS1} is a special case of our result with the choice $\ell=1$ and $G=\mathfrak S_n$, and that for $\ell=1$ and $G=\{1,\dots,n\}^{\{1,\dots,n\}}$ we directly obtain \cite[Lemma 7]{GLSW1}. Note that in this general setting $\mathbb E \max_{1\leq i \leq n} |a_{ig(i)}|$ was already studied in \cite{KS2}. In a slightly different setting, order statistics were considered also in \cite{GLSW1,GLSW2,GLSW3,GLSW4,GLSW5}.
We will now present two natural choices for the set $G$ that appear frequently in the literature (cf. \cite{KS1,KS2,S1,S2,S3,PS,GLSW1,G,JMST,P}).
\begin{example} If $N=n$ and $G=\mathfrak S_n$ is the group of permutations of the numbers $\{1,\dots,n\}$, then \[ \mathbb P(\pi(i)=j) = \frac{1}{n}, \qquad 1\leq i,j \leq n, \] and for $(i_1,j_1)\neq (i_2,j_2)$ \[ \mathbb P(\pi(i_1)=j_1,\pi(i_2)=j_2) \leq \frac{1}{n(n-1)} \leq \frac{2}{n^2}. \] This means that $C_{G}\leq 2$. Hence, Theorem \ref{thm:main} implies \[
\frac{1}{800} \frac{1}{n}\sum_{j=1}^{\ell n}s(j) \leq \frac{1}{n!} \sum_{\pi\in\mathfrak S_n} \sum_{k=1}^\ell \kmax_{1\leq i \leq n} |a_{i\pi(i)}| \leq 2 \frac{1}{n}\sum_{j=1}^{\ell n}s(j). \] \end{example}
\begin{example} If $N=n$ and $G$ is the set of all mappings from $\{1,\dots,n\}$ into $\{1,\dots,n\}$, then \[ \mathbb P(g(i)=j) = \frac{1}{n}, \qquad 1\leq i,j \leq n, \] and for $(i_1,j_1)\neq (i_2,j_2)$ \[ \mathbb P(g(i_1)=j_1,g(i_2)=j_2) \le \frac{1}{n^2}. \] This means that $C_{G}=1$. Hence, Theorem \ref{thm:main} implies \[
\frac{1}{288} \frac{1}{n}\sum_{j=1}^{\ell n}s(j) \leq \frac{1}{n^n} \sum_{g\in G} \sum_{k=1}^\ell \kmax_{1\leq i \leq n} |a_{ig(i)}| \leq 2 \frac{1}{n}\sum_{j=1}^{\ell n}s(j). \] \end{example}
Another combinatorial inequality that was obtained in \cite[Theorem 1.2]{KS1} and which turned out to be crucial to study and characterize symmetric subspaces of $L_1$ (cf. \cite{S1,S2,PS}) states that for all $1\leq p \leq \infty$ \begin{equation}\label{eq:Kwapien Schuett estimate p-norm}
\frac{1}{n!}\sum_{\pi\in\mathfrak S_n} \Big( \sum_{i=1}^n |a_{i\pi(i)}|^p \Big)^{1/p} \simeq \frac{1}{n}\sum_{k=1}^n s(k) + \Big(\frac{1}{n} \sum_{k=n+1}^{n^2} s(k)^p \Big)^{1/p}. \end{equation} In Section \ref{sec:applications}, we will use Theorem \ref{thm:main} to generalize this result and show that the lower bound in \eqref{eq:Kwapien Schuett estimate p-norm} can be naturally derived via real interpolation. The upper bound is quite easily obtained and we just follow \cite{KS1}. Please note that averages of order statistics of matrices naturally appear, as they are strongly related to the $K$-functional of the interpolation couple $(\ell_1,\ell_\infty)$. Again, two typical choices for the set of maps $G$ are $\mathfrak S_n$ and $\{1,\dots,n\}^{\{1,\dots,n\}}$. We will prove the following result:
\begin{thm}\label{thm:application} Let $n,N\in\mathbb N$, $a\in \mathbb R^{n\times N}$, and $1\leq p < \infty$. Let $G$ be a collection of maps from $I=\{1,\dots,n\}$ to $J=\{1,\dots,N\}$ and $C_G>0$ be a constant only depending on $G$. Assume that for all $i\in I$, $j\in J$ and all different pairs $(i_1,j_1),(i_2,j_2)\in I\times J$ \begin{enumerate}[(i)] \item $\mathbb P(\{g\in G: g(i)=j\})=1/N$, \item $\mathbb P(\{g\in G: g(i_1)=j_1, g(i_2)=j_2 \}) \leq C_G/N^2$. \end{enumerate} Then \begin{align*}
C\bigg[ \frac{1}{N} \sum_{k=1}^N s(k) + \Big(\frac{1}{N}\sum_{k=N+1}^{nN} s(k)^p\Big)^{1/p}\bigg] & \leq \mathbb E \Big( \sum_{i=1}^n |a_{ig(i)}|^p \Big)^{1/p} \\ & \leq \frac{1}{N} \sum_{k=1}^N s(k) + \Big(\frac{1}{N}\sum_{k=N+1}^{nN} s(k)^p\Big)^{1/p}, \end{align*} where $C>0$ is a constant only depending on $C_G$. \end{thm}
The organization of the paper is as follows. In Section \ref{sec: lower bound}, we will prove the lower estimate in \eqref{eq:main result firs part}. This is done by reducing the problem to the case of matrices only taking values in $\{0,1\}$ and showing the estimate for this subclass of matrices. In Section \ref{subsec: upper bound G}, we establish the upper bound in \eqref{eq:main result firs part} by passing from averages of order statistics to equivalent Orlicz norms and using an extreme point argument. Section \ref{sec:applications} contains the proof of Theorem \ref{thm:application}.
\section{Notation and Preliminaries}
Throughout this paper we will use $|E|$ to denote the cardinality of a finite set $E$. By $\mathfrak S_n$ we denote the symmetric group on the set $\{1,\dots,n\}$. We will denote by $\lfloor x \rfloor$ and $\lceil x\rceil$ the largest integer $m\leq x$ and the smallest integer $m\geq x$, respectively.
For an arbitrary matrix $a=(a_{ij})_{i,j=1}^{n,N}$, we denote by $(s(k))_{k=1}^{nN}$ the decreasing rearrangement of $(|a_{ij}|)_{i,j=1}^{n,N}$. To avoid confusion, in certain cases we write $(s_a(k))_{k=1}^{nN}$ to emphasize the underlying matrix $a$.
Please also recall that the Paley-Zygmund inequality for non-negative random variables $Z$ and $0<\theta<1$ states that \begin{equation}\label{ine:paley zygmund}
\mathbb P(Z\geq \theta \cdot\mathbb E Z)\geq (1-\theta)^2\frac{(\mathbb E Z)^2}{\mathbb E Z^2}. \end{equation}
A convex function $M:[0,\infty)\rightarrow[0,\infty)$ is called an Orlicz function if $M(0)=0$ and if $M$ is not constant. Given an Orlicz function $M$, the Orlicz sequence space $\ell_M^n$ is $\mathbb R^n$ equipped with the Luxemburg norm $$
\|x\|_M=\inf\bigg\{\lambda >0 : \sum_{i=1}^n M\left(\frac{|x_i|}{\lambda}\right)\leq 1\bigg\}. $$ For example, the classical $\ell_p$ spaces are Orlicz spaces with $M(t)=p^{-1}t^p$. The closed unit ball of the space $\ell_M^n$ will be denoted by $B_M^n$. We write $\mathrm{ext} (B_M^n)$ for the set of extreme points of $B_M^n$ and $\sconv (M)$ shall denote the set of points of strict convexity of $M$. We will make use of the following characterization of extreme points of $B_M^n$: \begin{lemma}[\cite{W}, Lemma 1]\label{lem:extreme points of orlicz balls} Let $M$ be an Orlicz function. Then $x\in \mathrm{ext} (B_M^n)$ if and only if \begin{enumerate}[(i)]
\item $\sum_{i=1}^nM(|x_i|)=1$, \item there exists at most one index $i_0\in\mathbb N$, $1\leq i_0 \leq n$ such that $x_{i_0}\not\in \pm\sconv (M)$. \end{enumerate} \end{lemma}
For a detailed and thorough introduction to Orlicz spaces we refer the reader to \cite{RR} or \cite{LT77}.
\section{The lower bound} \label{sec: lower bound}
In this section we will prove the lower bound in \eqref{eq:main result firs part}. We begin by recalling some notation and assumptions given in Theorem \ref{thm:main}. Let $a\in\mathbb R^{n\times N}$, $I=\{1,\ldots,n\}$, $J=\{1,\ldots,N\}$, and $G$ be a collection of maps from $I$ to $J$. The matrix $a$ will be fixed throughout the entire section. By $\mathbb P$ we denote the normalized counting measure on $G$, i.e., $\mathbb P(E) = |E|/|G|$ for $E\subset G$. We assume a uniform distribution of the random variable $g \mapsto g(i)$ for each $i\in I$, i.e., \begin{equation*} \mathbb P(g(i)=j)=\frac{1}{N},\qquad i\in I,j\in J. \end{equation*} We assume for all different pairs $(i_1,j_1),(i_2,j_2)\in I\times J$ that \begin{equation*} \mathbb P(g(i_1)=j_1, g(i_2)=j_2 ) \leq \frac{C_G}{N^2}, \end{equation*} with a constant $C_G\geq 1$ that depends on $G$, but not on $n$ or $N$.
Without loss of generality, we will assume that $a$ has only non-negative entries. It is enough to show the lower estimate in \eqref{eq:main result firs part} for matrices $a$ that consist of only the $\ell N$ largest entries, while all others are equal to zero. This is because if we change any entry $a_{i_0j_0}\leq s(\ell N+1)$ by setting $a_{i_0j_0}=0$, the left hand side in \eqref{eq:main result firs part} remains the same, while $\kmax\limits_{1\leq i \leq n} |a_{ig(i)}|$ does not increase for any $g\in G$.
\subsection{The key ingredients}
We will now introduce a bijective function $h$ that determines the ordering of the values of $a$. The crucial point is that this function does not depend on the actual values of the matrix, but merely on their relative size. So let $h:\{1,\dots,n\cdot N\}\rightarrow I\times J$ be a bijective function satisfying \begin{equation}\label{eq:condition on b} \begin{aligned} a(h(j)) & \geq a(h(j+1)),& 1 &\leq j \leq \ell N,\\ a(h(j))& =0, & \ell N +1& \leq j \leq nN. \end{aligned} \end{equation} Observe that there is possibly more than one choice for $h$, since some of the entries of the matrix $a$ might have the same value.
For all $j\in\mathbb N$, $1\leq j \leq n\cdot N$, define the random variable
\[
Y_j:G\to \{0,1\},\qquad
Y_j(g)= \begin{cases}
1, & \text{if }h(j)\in g, \\
0, & \text{if }h(j)\notin g,
\end{cases}
\] and given $m\in\mathbb N$, $1\leq m \leq n\cdot N$, let \[ X_m:G \to \{0,1,\dots, n\}, \qquad
X_m(g):= \sum_{j=1}^mY_j(g)=|h(\{1,\dots,m\})\cap g|, \] where we identify $g$ with its graph $\{(i,g(i)):i\in I \}$. $X_m$ counts the number of elements in the path $\{(i,g(i)):i\in I \}$ that intersect with the positions of the $m$ largest entries of $a$. As we will see in Subsection \ref{subsec:reduction}, the random variables $X_m$ are strongly related to order statistics.
In Lemma \ref{lem:keq1}, Lemma \ref{lem:Xm}, and Lemma \ref{lem:XmXln}, we investigate crucial properties of the distribution function of $X_m$.
\begin{lemma}\label{lem:keq1}
For all $m\in\mathbb N$, $1\leq m \leq n\cdot N$, we have
\begin{equation}\label{eq:Pinduction}
\mathbb P(X_m\geq 1)\geq\frac{m}{N}\Big(1-C_G\frac{m-1}{2N}\Big).
\end{equation}
In particular,
\[
\mathbb P(X_{\lceil N/C_G \rceil}\geq 1)\geq
\frac{1}{2C_G}.
\] \end{lemma} \begin{proof} By using the inclusion-exclusion principle, we obtain \begin{align*} \mathbb P(X_m \geq 1) & = \mathbb P \Big( \bigcup_{j=1}^m \{g\in G: Y_j(g)=1\}\Big) \\ & \geq \sum_{j=1}^m \mathbb P(Y_j=1) - \sum_{i<j}\mathbb P(Y_i=1,Y_j=1) \\ & \geq \frac{m}{N} - \frac{m(m-1)C_G}{2N^2} \\ & = \frac{m}{N} \Big( 1- C_G\frac{m-1}{2N}\Big), \end{align*} where the latter inequality is a direct consequence of conditions \eqref{eq:condition 1} and \eqref{eq:condition 2} in Theorem \ref{thm:main}. \end{proof}
\begin{lemma} \label{lem:Xm}
For all $m\in\mathbb N$, $1\leq m \leq n\cdot N$, and all $\theta\in(0,1)$, we have
\begin{equation}\label{eq:paley estimate}
\mathbb P\Big(X_m\geq \theta\cdot \frac{m}{N}\Big) \geq (1-\theta)^2\frac{m}{N+m\cdot C_G}.
\end{equation}
\end{lemma} \begin{proof}
The result follows as a consequence of Paley-Zygmund's inequality (cf. \eqref{ine:paley zygmund}). Therefore, we need to compute $\mathbb E X_m$ and $\mathbb E X_m^2$. Note that $\mathbb E Y_j=\mathbb P(Y_j=1)=1/N$ and thus $\mathbb E X_m=\sum_{j=1}^m Y_j=m/N$.
Moreover, since $Y_j=Y_j^2$, we have
\begin{align*}
\mathbb E X_m^2 &=\sum_{i,j=1}^m \mathbb E Y_i Y_j = \sum_{j=1}^m \mathbb E Y_j + \sum_{i\neq j} \mathbb E Y_i Y_j \\
& \leq \frac{m}{N}+ C_G\frac{m(m-1)}{N^2},
\end{align*}
where the latter inequality is a direct consequence of conditions \eqref{eq:condition 1} and \eqref{eq:condition 2} in Theorem \ref{thm:main}.
Inserting those estimates in \eqref{ine:paley zygmund}, we obtain the result. \end{proof}
\begin{lemma}\label{lem:XmXln}
For all $m\in\mathbb N$, $1\leq m \leq n\cdot N$, we have
\[
\mathbb P(X_m\geq 1)\geq \min\left\{\frac{m}{2N},\frac{1}{2C_G}\right\}\mathbb P(X_{\ell N}\geq 1),
\]
and for all $k,m\in\mathbb N$ with $2kN\leq m\leq n N$
\begin{equation}\label{eq:Xmk}
\mathbb P(X_m\geq k)\geq \frac{\mathbb P(X_{\ell N}\geq k)}{2+4 C_G}.
\end{equation} \end{lemma} \begin{proof}
Let $1\leq m\leq n\cdot N$. If $m\leq N/C_G$, Lemma \ref{lem:keq1} implies
\[
\mathbb P(X_m\geq 1)\geq \frac{m}{2N}\geq \frac{m}{2N}\cdot\mathbb P(X_{\ell N}\geq 1)=\min\left\{\frac{m}{2N},\frac{1}{2C_G}\right\}\mathbb P(X_{\ell N}\geq 1).
\]
On the other hand, if $m\geq N/C_G$, Lemma \ref{lem:keq1} implies
\begin{align*}
\mathbb P(X_m\geq 1) &\geq \mathbb P(X_{\lceil N/C_G\rceil}\geq 1)
\geq \frac{1}{2C_G} \\
& \geq \frac{1}{2C_G}\mathbb P(X_{\ell N}\geq 1) = \min\left\{\frac{m}{2N},\frac{1}{2C_G}\right\}\mathbb P(X_{\ell N}\geq 1).
\end{align*}
Now we prove \eqref{eq:Xmk}. Let $k\leq n/2$ and $m$ such that $2kN\leq m\leq n\cdot N$. Then Lemma~\ref{lem:Xm} with $\theta=1/2$ implies
\[
\mathbb P(X_m\geq k)\geq \mathbb P(X_{2kN}\geq k)\geq \frac{1}{2+4C_G}\geq \frac{1}{2+4C_G}\mathbb P(X_{\ell N}\geq k). \qedhere
\] \end{proof}
\subsection{Reduction to two valued matrices}\label{subsec:reduction}
We will now reduce the problem of estimating the expected value of averages of order statistics of general matrices to matrices only taking one value different from zero. To do so, we need some more definitions.
Let $\mathcal A_h$ be the collection of all non-negative real $n\times N$ matrices $b$ that satisfy \begin{equation} \begin{aligned} b(h(j)) & \geq b(h(j+1)),& 1 &\leq j \leq \ell N,\\ b(h(j))& =0, & \ell N +1& \leq j \leq nN. \end{aligned} \end{equation} For every $b\in \mathcal A_h$, we set \[ \widetilde{b}(h(j)):=\Big(\frac{1}{\ell N}\sum_{i=1}^{\ell N} b(h(i))\Big) \cdot \mathbbm 1_{\{1,\dots,\ell N\}} (j),\qquad 1\leq j\leq n\cdot N, \] as the matrix that contains the averaged entries of $b$. Note that $\widetilde b \in\mathcal A_h$. Moreover, we define \[ a_m(h(k)):=\mathbbm 1_{\{1,\dots, m\}}(k), \qquad 1\leq m \leq n\cdot N. \] Observe that $a_m\in \mathcal A_h$ for all $1\leq m \leq n\cdot N$. For $b\in\mathcal A_h$ and $g\in G$ we put \[ S_k(b)(g):= \kmax_{1\leq i \leq n} b_{ig(i)}\qquad\text{and}\qquad S(b)(g):=\sum_{k=1}^\ell S_k(b)(g). \]
\begin{lemma}\label{thm:am} Let $m\in\mathbb N$, $1\leq m \leq \ell N$. Then we have \[ \mathbb E S(\widetilde{a_m})\leq (8+16C_G)\cdot \mathbb E S(a_m). \] \end{lemma} \begin{proof} Observe that for every integer $k$ with $1\leq k\leq \ell$, \begin{equation*} \begin{aligned} \mathbb E S_k(a_m)&=\mathbb P(S_k(a_m)=1)=\mathbb P(X_m\geq k),\\ \mathbb E S_k(\widetilde{a_m})&=\frac{m}{\ell N}\cdot\mathbb P\Big(S_k(\widetilde{ a_m})=\frac{m}{\ell N}\Big)= \frac{m}{\ell N}\cdot\mathbb P(X_{\ell N}\geq k). \end{aligned} \end{equation*} As a consequence, \begin{equation*} \begin{aligned} \mathbb E S(\widetilde{a_m})=\frac{m}{\ell N}\sum_{k=1}^\ell \mathbb P(X_{\ell N}\geq k) \qquad\text{and}\qquad \mathbb E S(a_m)=\sum_{k=1}^\ell \mathbb P(X_m\geq k). \end{aligned} \end{equation*} Thus, in order to prove the lemma, it is enough to show that \[ \frac{m}{\ell N}\sum_{k=1}^\ell \mathbb P(X_{\ell N}\geq k) \leq (8+16C_G) \cdot \sum_{k=1}^\ell \mathbb P(X_m\geq k) \] for $1\leq m\leq \ell N$. First, we assume $m\leq 2N$. Then, Lemma \ref{lem:XmXln} implies \begin{align*} \frac{m}{\ell N}\sum_{k=1}^{\ell}\mathbb P(X_{\ell N}\geq k) & \leq \frac{m}{N}\cdot \mathbb P(X_{\ell N}\geq 1) \\ & \leq \frac{2}{N}\max\{N,m\cdot C_G\} \mathbb P(X_m\geq 1) \\ &\leq 4C_G\cdot \sum_{k=1}^\ell \mathbb P(X_m\geq k) \end{align*} i.e., the assertion of the lemma for $m\leq 2N$.
Now, let $m\geq 2N+1$ and choose the integer $t\geq 1$ such that $2tN+1\leq m\leq 2(t+1)N$. The sequence $k\mapsto\mathbb P(X_{\ell N} \geq k)$ is decreasing, hence, noting that $t\leq \ell$, \[ \frac{m}{\ell N}\sum_{k=1}^\ell \mathbb P(X_{\ell N}\geq k)\leq \frac{m}{tN}\sum_{k=1}^t \mathbb P(X_{\ell N}\geq k). \] Then, estimate \eqref{eq:Xmk} of Lemma \ref{lem:XmXln} implies \begin{align*} \frac{m}{\ell N}\sum_{k=1}^\ell \mathbb P(X_{\ell N}\geq k) & \leq \frac{m}{tN}\sum_{k=1}^t (2+4C_G)\cdot\mathbb P(X_m\geq k) \\ &\leq \frac{(4+8C_G)(t+1)}{t}\sum_{k=1}^t \mathbb P(X_m\geq k) \\ &\leq (8+16C_G)\cdot \sum_{k=1}^\ell \mathbb P(X_m\geq k) \end{align*} and the result follows. \end{proof}
\begin{lemma}\label{thm: estimate a and tilde a general case}
We have
\[ \mathbb E S(\widetilde a)\leq (8+16 C_G)\cdot \mathbb E S(a). \] \end{lemma} \begin{proof}
Recall that $X_j(g) = |h(\{1,\dots,j\})\cap g|$. Hence, for all $b\in\mathcal A_h$, \begin{align*} \mathbb E S(b) & = \sum_{k=1}^{\ell}\sum_{j=1}^{\ell N} b(h(j))\cdot \mathbb P(\{g:X_{j-1}(g)=k-1,h(j)\in g \}). \end{align*} Defining \[ f(j):= \sum_{k=1}^{\ell} \mathbb P(\{g:X_{j-1}(g)=k-1,h(j)\in g\}), \qquad 1\leq j \leq \ell N, \] we can write \[ \mathbb E S(b) = \sum_{j=1}^{\ell N}f(j)b(h(j)). \] Since $a,\widetilde a\in\mathcal A_h$, $a(h(j))=s_a(j)$ and $\widetilde a(h(j)) = (\ell N)^{-1}\sum_{i=1}^{\ell N}s_a(i)$ for all $j \leq \ell N$, we obtain \begin{equation*}\label{eq:expectation S with f and g} \mathbb E S(a)=\sum_{j=1}^{\ell N} f(j) s_a(j) \qquad \text{and}\qquad \mathbb E S(\widetilde{a})= \sum_{j=1}^{\ell N } \widetilde f(j)s_a(j), \end{equation*} where for all $1\leq j \leq \ell N$ \[ \widetilde f(j) = \frac{1}{\ell N} \sum_{i=1}^{\ell N} f(i). \] Note that the functions $f$ and $\widetilde f$ only depend on $h$, i.e., only on the positions of the entries in the matrix and not on their values. Since $a_m(h(j)) = 1$ for $j \leq m$ (zero otherwise) and $a_m,\widetilde{a_m}\in\mathcal A_h$, we have \[ \mathbb E S(a_m)=\sum_{j=1}^m f(j) \qquad\text{and}\qquad \mathbb E S(\widetilde{a_m})=\sum_{j=1}^m \widetilde f(j). \] Now we conclude with $C=8+16C_G$ that \begin{align*}
C\mathbb E S(a)- \mathbb E S(\widetilde a)&= s_a(1)[Cf(1)-\widetilde f(1)]+\sum_{j=2}^{\ell N} [Cf(j)-\widetilde f(j)] s_a(j) \\
&\geq s_a(2) \sum_{j=1}^2 [C f(j)-\widetilde f(j)]+ \sum_{j=3}^{\ell N}[Cf(j)-\widetilde f(j)] s_a(j) \end{align*} where we used Lemma \ref{thm:am} for $m=1$. Continuing in this fashion and using Lemma \ref{thm:am} for $m=2,\ldots,\ell N$, we obtain \begin{equation*}
C\mathbb E S(a)- \mathbb E S(\widetilde a) \geq 0.
\qedhere \end{equation*} \end{proof}
\subsection{Conclusion}\label{subsec:conclusion}
As we have seen, we can reduce the case of general $a$ to multiples of matrices only taking values zero and one. Before we finally prove the lower bound in the main theorem, we will need another simple lemma.
\begin{lemma}\label{prop:lower estimate k-max}
Let $b\in \mathcal A_h$ be an $(n\times N)$-matrix consisting of $\ell N$ ones and $(n-\ell)N$ zeros. Then, for all $1\leq k\leq \ell/2$,
$$
\mathbb E \kmax_{1\leq i \leq n} b_{ig(i)} \geq \frac{1}{2+4C_G}.
$$ \end{lemma} \begin{proof}
Let $k\leq \ell/2$. Using Lemma \ref{lem:Xm} with $\theta=1/2$, we obtain
\begin{align*}
\mathbb E \kmax_{1\leq i \leq n} b_{i g(i)}
&\geq \int_{\{g: X_{2kN}(g)\geq k\}} \kmax_{1\leq i \leq n} b_{ig(i)} \,\mathrm{d}\mathbb P(g)\\
& = \mathbb P(X_{2kN}\geq k)
\geq \frac{k}{2(1+2kC_G)}\\
& \geq \frac{1}{2+4C_G}. \qedhere
\end{align*} \end{proof}
\subsection*{Proof of the lower estimate in Theorem \ref{thm:main}}
By Theorem \ref{thm: estimate a and tilde a general case} we obtain \begin{align*} \mathbb E \sum_{k=1}^\ell \kmax_{1\leq i \leq n} a_{ig(i)} & \geq \frac{1}{8(1+2C_G)} \mathbb E \sum_{k=1}^\ell \kmax_{1\leq i \leq n} \widetilde a_{ig(i)}. \end{align*} Now take $b\in\mathcal A_h$ consisting of $\ell N$ ones and $(n-\ell)N$ zeros such that \[ \left(\frac{1}{\ell N}\sum_{i=1}^{\ell N}s_a(i)\right) \cdot b=\widetilde a. \] Then, by Lemma \ref{prop:lower estimate k-max} \begin{align*}
\mathbb E \sum_{k=1}^\ell \kmax_{1\leq i \leq n} \widetilde a_{ig(i)} & = \left[\mathbb E \sum_{k=1}^\ell \kmax_{1\leq i \leq n} b_{ig(i)}\right] \frac{1}{\ell N} \sum_{i=1}^{\ell N}s_a(i) \\ & \geq \left[\mathbb E \sum_{k=1}^{\ell/2} \kmax_{1\leq i \leq n} b_{ig(i)}\right] \frac{1}{\ell N} \sum_{i=1}^{\ell N}s_a(i) \\ & \geq \frac{1}{4+8C_G}\frac{1}{N} \sum_{j=1}^{\ell N}s(j). \end{align*} Combining the above estimates yields \begin{equation}\label{eq:lower_estimate}
\mathbb E \sum_{k=1}^\ell \kmax_{1\leq i \leq n} a_{ig(i)}
\geq \frac{1}{32(1+2C_G)^2} \frac{1}{N} \sum_{j=1}^{\ell N}s(j), \end{equation} which is the lower estimate in Theorem \ref{thm:main}.
\section{The upper bound}\label{subsec: upper bound G}
We will now prove the upper bound of Theorem \ref{thm:main} via an extreme point argument. To do so, we first use the fact that the average of the $j\leq n\cdot N$ largest entries of a matrix $a\in\mathbb R^{n\times N}$ is equivalent to an Orlicz norm $\|a\|_{M_j}$ (cf. Lemma \ref{lem:approximation by orlicz norm}). Then, since the expected value of the average of order statistics defines a norm on $\mathbb R^{n \times N}$ as well, it is enough to prove the upper bound in Theorem \ref{thm:main} for the extreme points of $B^n_{M_j}$.
Recall that, for a vector $(x_i)_{i=1}^n\in\mathbb R^n$, we denote the decreasing rearrangement of $(|x_i|)_{i=1}^n$ by $(x_i^*)_{i=1}^n$. We start with the approximation of sums of decreasing rearrangements of vectors $x\in\mathbb R^n$ by equivalent Orlicz norms.
The following result is due to C. Sch\"utt (private communication).
With his permission we include it here.
\begin{lemma}\label{lem:approximation by orlicz norm} Let $j\in\mathbb N$, $1\leq j \leq n$. Then, for all $x\in\mathbb R^n$, we have \[
\frac{1}{2} \sum_{i=1}^j x_i^* \leq \|x\|_{M_j} \leq \sum_{i=1}^j x_i^*, \] where \begin{equation}\label{eq:definition of Mk} M_j(t) := \begin{cases}
0, & \quad 0\leq t \leq 1/j,\\
t-1/j , & \quad 1/j<t.
\end{cases} \end{equation} \end{lemma} \begin{proof} Let $x\in\mathbb R^n$. We start with the right hand side inequality. Of course, \[ \frac{1}{j}\sum_{i=1}^jx_i^* \geq x_k^*,\qquad\forall j \leq k \leq n. \] Hence, for all $k\geq j$, \begin{equation*} M_j\left(\frac{x_k^*}{\sum_{i=1}^jx_i^*} \right) \leq M_j\left(\frac{1}{j} \right) = 0. \end{equation*} Therefore, we obtain \[
\sum_{k=1}^n M_j\left( \frac{|x_k|}{\sum_{i=1}^jx_i^*}\right) = \sum_{k=1}^{j-1} M_j\left( \frac{x_k^*}{\sum_{i=1}^jx_i^*}\right) \leq \sum_{i=1}^{j-1} \frac{x_k^*}{\sum_{i=1}^jx_i^*} \leq 1. \]
Now the other inequality. Take $\gamma< 1/2$. Then, since $M_j(t) \geq t-1/j$ for all $t\geq 0$, we have \begin{align*}
\sum_{k=1}^n M_j\left(\frac{|x_k|}{\gamma \sum_{i=1}^jx_i^*} \right) & \geq \sum_{k=1}^j M_j\left(\frac{x_k^*}{\gamma \sum_{i=1}^jx_i^*} \right) \\ & \geq \sum_{k=1}^j \left( \frac{x_k^*}{\gamma \sum_{j=1}^ks(j)} - \frac{1}{k} \right) \\ & = \frac{1}{\gamma}-1 >1. \end{align*} Therefore, we have for all $\alpha<1/2$ \[
\|x\|_{M_j} \geq \alpha \sum_{i=1}^jx_i^*. \qedhere \] \end{proof}
We are now able to prove the upper bound of Theorem \ref{thm:main}.
\begin{proposition}\label{prop:upper bound average} Let $a\in\mathbb R^{n\times N}$. Then, for all $\ell \leq n$, \begin{equation}\label{eq:estimate Mk norm}
\mathbb E \sum_{k=1}^\ell \kmax_{1\leq i \leq n} a_{ig(i)} \leq \frac{2}{N} \|a\|_{M_{\ell N}}. \end{equation} \end{proposition} \begin{proof} It is sufficient to show \eqref{eq:estimate Mk norm} for all $a\in\mathrm{ext}{(B^{nN}_{M_\ell})}$. Therefore, by Lemma \ref{lem:extreme points of orlicz balls} (2), we only need to consider matrices $a\in\mathbb R^{n \times N}$ that are of the form \begin{equation}\label{eq:extreme points of B_Mk} a_{ij} := \begin{cases}
\frac{1}{\ell N}, & (i,j)\neq (i_0,j_0),\\
1+\frac{1}{\ell N} , & (i,j)= (i_0,j_0) \end{cases} \end{equation} for some index pair $(i_0,j_0)\in I\times J$ or satisfy $a_{ij}=\frac{1}{\ell N}$ for all $i=1,\dots,n$, $j=1,\dots, N$. However, the latter choice of $a$ does not satisfy condition (1) in Lemma \ref{lem:extreme points of orlicz balls}, since in that case $\sum_{i=1}^{nN}M_{\ell N}(s_a(i))=0$. So the extreme points of $B_{M_{\ell N}}^{nN}$ with positive entries are given by \eqref{eq:extreme points of B_Mk}. Now, let $a$ be such a point in $B^{nN}_{M_{\ell N}}$. Then \begin{align*} & \mathbb E \sum_{k=1}^\ell \kmax_{1\leq i \leq n} a_{ig(i)} \\ & = \int_{\{g:g(i_0)=j_0 \}} \sum_{k=1}^\ell \kmax_{1\leq i \leq n} a_{ig(i)} \,\mathrm{d} \mathbb P(g) + \int_{\{g:g(i_0)\neq j_0 \}} \sum_{k=1}^\ell \kmax_{1\leq i \leq n} a_{ig(i)} \,\mathrm{d} \mathbb P(g) \\ & = \frac{2}{N}. \end{align*} On the other hand, we also have \[ \frac{1}{N}\sum_{j=1}^{\ell N} s(j) = \frac{2}{N}. \] Therefore, \[ \mathbb E \sum_{k=1}^\ell \kmax_{1\leq i \leq n} a_{ig(i)} = \frac{1}{N}\sum_{j=1}^{\ell N} s(j), \] Since by Lemma \ref{lem:approximation by orlicz norm} \[
\sum_{j=1}^{\ell N} s(j) \leq 2 \|a\|_{M_{\ell N}}, \] the result follows. \end{proof}
\subsection*{Conclusion of the proof of Theorem~\ref{thm:main}}
Combining Lemma~\ref{lem:approximation by orlicz norm} and Proposition~\ref{prop:upper bound average}, we get \begin{equation}\label{eq:upper_estimate}
\mathbb E \sum_{k=1}^\ell \kmax_{1\leq i \leq n} a_{ig(i)} \leq \frac{2}{N} \sum_{i=1}^{\ell N}s(i), \end{equation} which is the upper estimate in Theorem \ref{thm:main}. Inequalities~\eqref{eq:lower_estimate} and~\eqref{eq:upper_estimate} together complete the proof.
\section{An application of Theorem~\ref{thm:main}}\label{sec:applications}
We now present an application and use Theorem \ref{thm:main} to prove Theorem \ref{thm:application}. The proof uses real interpolation and is, what we find, a natural approach to combinatorial inequalities such as \eqref{eq:Kwapien Schuett estimate p-norm} that were obtained in \cite{KS1}. Please notice that \cite[Theorem 1.2]{KS1} is a special case of Theorem \ref{thm:application} when $G=\mathfrak S_n$.
Let us first recall some basic notions from interpolation theory. A pair $(X_0,X_1)$ of Banach spaces is called a compatible couple if there is some Hausdorff topological space $\mathcal H$, in which each of $X_0$ and $X_1$ is continuously embedded. For example, $(L_1,L_\infty)$ is a compatible couple, since $L_1$ and $L_\infty$ are continuously embedded into the space of measurable functions that are finite almost everywhere. Of course, any pair $(X,Y)$ for which one of the spaces is continuously embedded in the other is a compatible couple.
For a compatible couple $(X_0,X_1)$ (with corresponding Hausdorff space $\mathcal H$), we equip $X_0+X_1$ with the norm \begin{equation}\label{eq:norm on sum of spaces}
\| x \|_{X_0+X_1} := \inf_{x=x_0+x_1}\big(\|x_0\|_{X_0} + \|x_1\|_{X_1}\big), \end{equation} under which this space becomes a Banach space. This definition is independent of the particular space $\mathcal H$.
The $K$-functional is constructed from the expression \eqref{eq:norm on sum of spaces} by introducing a positive weighting factor $t>0$, as follows:
Let $(X_0,X_1)$ be a compatible couple. The $K$-functional is defined for each $f\in X_0+X_1$ and $t>0$ by \[
K(f,t) = K(f,t;X_0,X_1) := \inf_{f=f_0+f_1}(\|f_0\|_{X_0} + t \|f_1\|_{X_1}), \] where the infimum extends over all representations $f=f_0+f_1$ of $f$ with $f_0 \in X_0$ and $f_1\in X_1$.
Now, let $(X_0,X_1)$ be a compatible couple and suppose $0<\theta < 1$, $1\leq q < \infty$ or $0\leq \theta \leq 1$ and $q=\infty$. The space $(X_0,X_1)_{\theta,q}$ consists of all $f\in X_0+X_1$ for which the functional \begin{equation*}\label{eqn:norm on theta q spaces}
\|f\|_{\theta,q} :=
\begin{cases}
\left( \int_{0}^\infty \big[ t^{-\theta}K(f,t;X_0,X_1)\big]^q \frac{\,\mathrm{d} t}{t}\right)^{1/q}, & \quad 0<\theta<1,~1\leq q < \infty ,\\
\sup_{t>0}t^{-\theta}K(f,t;X_0,X_1) , & \quad 0 \leq \theta \leq 1, ~q=\infty,
\end{cases}
\end{equation*} is finite.
\subsection*{Proof of Theorem \ref{thm:application}} To show the upper bound we use the same argument as in \cite{KS1}. For the sake of completeness we include it here. Let $a\in\mathbb R^{n\times N}$ and write $a=a'+a''$, where $a'$ contains the $N$ largest entries of $a$ and zeros elsewhere, and $a''$ contains $s(N+1)\dots,s(nN)$ and zeros elsewhere. Then, using triangle and Jensen's inequality, we obtain \begin{align*}
\mathbb E \left(\sum_{i=1}^n |a_{ig(i)}|^p\right)^{1/p} & \leq \mathbb E \left(\sum_{i=1}^n |a'_{ig(i)}|^p\right)^{1/p} + \mathbb E \left(\sum_{i=1}^n |a''_{ig(i)}|^p\right)^{1/p} \\
& \leq \mathbb E \sum_{i=1}^n |a'_{ig(i)}| + \mathbb E\left(\sum_{i=1}^n |a''_{ig(i)}|^p\right)^{1/p} \\ & \leq \sum_{k=1}^{N}\sum_{i=1}^n \frac{1}{N}a_{ik}' + \left( \sum_{k=1}^N \sum_{i=1}^n\frac{1}{N}(a_{ik}'')^p \right)^{1/p} \\ & = \frac{1}{N} \sum_{k=1}^N s(k) + \left(\frac{1}{N}\sum_{k=N+1}^{nN}s(k)^p \right)^{1/p}. \end{align*}
We will now prove the lower bound. Let $1\leq p < \infty$ and $\theta=1-1/p$. First, recall that \[
\|a\|_{\theta,p} = \left( \int_{0}^\infty
\Big[t^{-\theta}K\big(a,t;L_1^{|G|}(\ell_1^n),L_1^{|G|}(\ell_\infty^n)\big)\Big]^p
\frac{\,\mathrm{d} t}{t} \right)^{1/p}. \] Second, observe that \begin{align*}
K\big(a,t;L_1^{|G|}(\ell_1^n),L_1^{|G|}(\ell_\infty^n)\big)
& = \inf_{a=b+c} \|b\|_{L_1^{|G|}(\ell_1^n)} + t\, \|c\|_{L_1^{|G|}(\ell_\infty^n)}\\
& = \inf_{a=b+c} \int_G \|b(g)\|_{\ell_1^n} + t\, \|c(g)\|_{\ell_\infty^n} \,\mathrm{d}\mathbb P(g)\\
& = \int_G \inf_{a(g)=b(g)+c(g)} \|b(g)\|_{\ell_1^n} + t\, \|c(g)\|_{\ell_\infty^n} \,\mathrm{d}\mathbb P(g). \end{align*} Hence, we have \begin{equation}\label{eq:representation k functional}
K\big(a,t;L_1^{|G|}(\ell_1^n),L_1^{|G|}(\ell_\infty^n)\big)
= \int_G K\big(a(g),t;\ell_1^n,\ell_\infty^n\big)\,\mathrm{d}\mathbb P(g). \end{equation} Third, the triangle inequality for integrals yields \begin{align*}
\|a\|_{\theta,p}
& = \bigg( \int_0^\infty [
t^{-\theta} K\big(a,t;L_1^{|G|}(\ell_1^n),L_1^{|G|}(\ell_\infty^n)\big)
]^p
\frac{\,\mathrm{d} t}{t} \bigg)^{1/p}\\
& = \bigg( \int_0^\infty [
\int_G t^{-\theta} K\big(a(g),t;\ell_1^n,\ell_\infty^n)\big) \,\mathrm{d}\mathbb P(g)
]^p
\frac{\,\mathrm{d} t}{t} \bigg)^{1/p}\\
& \leq \int_G \bigg( \int_0^\infty
\Big[t^{-\theta} K\big(a(g),t;\ell_1^n,\ell_\infty^n)\big)\Big]^p
\frac{\,\mathrm{d} t}{t} \bigg)^{1/p}
\,\mathrm{d}{\mathbb P(g)}\\
& = \int_G \|a(g)\|_{\ell_p^n} \,\mathrm{d}{\mathbb P(g)}. \end{align*} Therefore, we have \[
\|a\|_{L_1^{|G|}(\ell_p^n)} \geq \|a\|_{\theta,p}, \]
for all $a:G\to\mathbb R^n$, $a(g)(i)=a_{ig(i)}$. Now we compute the $K$-functional of $(L_1^{|G|}(\ell_1^n),L_1^{|G|}(\ell_\infty^n))$ using \eqref{eq:representation k functional}. First, observe that \begin{align*} K(a(g),t;\ell_1^n,\ell_\infty^n) & = \int_0^{\lfloor t\rfloor}(a(g))^*(s) \,\mathrm{d} s + \int_{\lfloor t\rfloor}^t(a(g))^*(s) \,\mathrm{d} s \\
& = \sum_{k=1}^{\lfloor t\rfloor}\kmax |a(g)| + (t-\lfloor t\rfloor)\cdot \tmax |a(g)|. \end{align*} Then, using \eqref{eq:representation k functional}, we obtain \begin{align*}
\|a\|_{1-\frac{1}{p},p}^p & = \int_{0}^\infty t^{-p}K(a,t;L_1^{|G|}(\ell_1^n),L_1^{|G|}(\ell_\infty^n))^p \,\mathrm{d} t \\ & = \int_{0}^\infty t^{-p}\Big( \int_G K(a(g),t;\ell_1^n,\ell_\infty^n) \,\mathrm{d}\mathbb P(g) \Big)^p \,\mathrm{d} t \\
& = \int_{0}^\infty t^{-p}\Big( \int_G \sum_{k=1}^{\lfloor t\rfloor}\kmax |a(g)| + (t-\lfloor t\rfloor)\cdot \tmax |a(g)|) \,\mathrm{d}\mathbb P(g) \Big)^p \,\mathrm{d} t \\ & \geq \int_0^1 \big(\mathbb E a(g)^*(1) \big)^p \,\mathrm{d} t + \int_{1}^{n+1} t^{-p} \Big(\mathbb E \sum_{k=1}^{\lfloor t\rfloor} a(g)^*(k) \Big)^p \,\mathrm{d} t \\ & \geq c_1 \bigg[ \big(\mathbb E a(g)^*(1) \big)^p + \sum_{\ell=1}^{n} \Big(\mathbb E \frac{1}{\ell}\sum_{k=1}^{\ell} a(g)^*(k) \Big)^p\bigg], \end{align*} where $c_1$ is a positive absolute constant. By Theorem \ref{thm:main}, we get \begin{align*}
\Big[\mathbb E a(g)^*(1) \Big]^p + \sum_{\ell=1}^{n} \Big(\mathbb E \frac{1}{\ell}\sum_{k=1}^{\ell} a(g)^*(k) \Big)^p
& \geq c_2\bigg[ \Big(\frac{1}{N}\sum_{j=1}^Ns(j) \Big)^p + \sum_{\ell=1}^n \Big(\frac{1}{\ell N}\sum_{j=1}^{\ell N} s(j) \Big)^p \bigg]\\
& \geq c_2\bigg[ \Big(\frac{1}{N}\sum_{j=1}^Ns(j) \Big)^p + \sum_{\ell=1}^n \frac{1}{N} \sum_{j=\ell N+1}^{(\ell+1)N}s(j)^p \bigg] \\
& = c_2 \bigg[\Big(\frac{1}{N}\sum_{j=1}^Ns(j) \Big)^p + \frac{1}{N} \sum_{\ell=N+1}^{nN} s(j)^p\bigg], \end{align*} where $c_2$ is a positive constant only depending on $C_G$. Taking the $p$-th root concludes the proof.
\subsection*{Acknowledgments} We would like to thank our colleague Erhard Aichinger for helpful discussions.
The first named author is supported by the Austrian Science Fund, FWF P23987 and FWF P22549. The second named author is supported by the Austrian Science Fund, FWF P23987. The third named author is supported by the Austrian Science Fund, FWFM 1628000.
\end{document} |
\begin{document}
\title{Subsumption Algorithms for Three-Valued Geometric Resolution} \author{Hans de Nivelle} \address{School of Science and Technology, Nazarbayev University,
53 Qabanbay Batyr, Astana 010000, Kazakhstan}
\begin{abstract}
In our implementation of geometric resolution, the
most costly operation is subsumption testing (or matching):
One has to decide for a three-valued, geometric formula,
if this formula is false in a given interpretation.
The formula contains only atoms with variables, equality, and existential
quantifiers. The interpretation contains only atoms with constants.
Because the atoms have no term structure,
matching for geometric resolution is hard.
We translate the matching problem into a generalized constraint
satisfaction problem, and discuss several approaches for solving
it efficiently,
one direct algorithm and two translations to propositional SAT.
After that, we study filtering techniques based on local
consistency checking.
Such filtering techniques can a priori refute a large percentage
of generalized constraint satisfaction problems.
Finally, we adapt the matching algorithms in
such a way that they find solutions that use a minimal subset of
the interpretation. The adaptation can be combined with
every matching algorithm.
The techniques presented in this paper may have applications
in constraint solving independent of geometric resolution. \end{abstract}
\maketitle \section{Introduction} \label{Sect_introduction}
Main topic of this paper is \emph{the generalized matching problem}, for example how to match $ p(X,Y), \ q(Y,Z) $ into $ p(0,1), \ p(0,2), \ q(1,3), \ q(2,4), \ r(0,3) $ without matching $ r(X,Z). $ This problem arose in the implemention of geometric resolution. Geometric logic as a theorem proving strategy was introduced in \cite{BezemCoquand2005}. (The authors use the name \emph{coherent logic}.) Bezem and Coquand were motivated mostly by the desire to obtain a theorem proving strategy with a simple normal form transformation, which makes that many natural problems need no transformation at all, others have a much simpler transformation, and which makes that in all cases Skolemization can be avoided. This results in more readable proofs, and proofs that can be backtranslated more easily.
Our motivation for using geometric resolution is different, more engineering-oriented: We hope that three-valued, geometric resolution can be made sufficiently efficient, so that it can be used as a generic reasoning core, into which different kinds of two- or three-valued decision problems (e.g. problems representing type correctness, two-valued decision problems, or simply typed classical problems) can be translated. Because we want the geometric reasoning core to be generic, we are willing to accept transformations that do not preserve much of the structure of the original formula. Subformulas are freely renamed, and functional expressions are flattened and replaced by relations. For details of the calculus, its motivation, and related work, we refer to \cite{deNivelle2014b}.
In the current paper we give only a short introduction, which is aimed at explaining how matching is used in geometric resolution, and how matching instances in geometric resolution are translated into generalized constraint satisfaction problems. If one is interested only in the methods for constraint satisfaction, one can ignore the technical part of this section and continue reading at the overview at the end of this section.
We continue this section by giving a definition of three-valued, geometric formulas. The definition that we give here is slightly too general, but easier to understand than the correct definition in \cite{deNivelle2014b}, which contains some additional, technical restrictions which are not relevant for matching.
\begin{defi}
\label{Def_geometric_atom}
A \emph{geometric literal} has one of the following four forms:
\begin{enumerate}
\item
A simple atom of form $ p_{\lambda}( x_1, \ldots, x_n ), $
where $ x_1, \ldots, x_n $ are variables (with repetitions allowed)
and $ \lambda \in \{ {\bf f}, \ {\bf e}, \ {\bf t} \}. $
(denoting \emph{false}, \emph{error} and \emph{true}.)
\item
An equality atom of form $ x_1 \approx x_2, $ with $ x_1,x_2 $
distinct variables.
\item
A domain atom $ \#_{\bf f} \, x, $ with $x$ a variable.
\item
An \emph{existential atom} of form
$ \exists y \ p_{\lambda}( x_1, \ldots, x_n, y ) $ with
$ \lambda \in \{ {\bf f}, \ {\bf e}, \ {\bf t} \}, $ and
such that $ y $ occurs at least once in the atom, not necessarily
on the last place.
\end{enumerate}
\noindent
A \emph{geometric formula} has form
$ A_1, \ldots, A_p \ | \ B_1, \ldots, B_q, $ where
the $ A_i $ are simple or domain atoms, and the
$ B_j $ are atoms of arbitrary type.
We require that geometric formulas are \emph{range restricted},
which means that every variable that occurs free in a $ B_j $
must occur in an $ A_i $ as well.
The intuitive meaning of $ A_1, \ldots, A_p \ | \ B_1, \ldots, B_q $ is
$ \forall \overline{x} \ A_1 \vee \cdots \vee A_p \vee
B_1 \vee \cdots \vee B_q, $
where $ \overline{x} $ are all the free variables.
The vertical bar $ (|) $ has no logical meaning. Its only purpose is
to separate the two types of atoms. \end{defi}
\noindent A geometric formula that is not range restricted, can always be made range restricted by inserting suitable $ \#_{\bf f} $ atoms into the left hand side. This is the only purpose of the $ \# $-predicate. Interpretations contain predicates of form $ \#_{\bf t} \ c, $ for every domain element $ c. $ Atoms in geometric formulas are variable-only, and are labeled with truth-values, as in \cite{MurrayRosenthal1993}. It is shown in \cite{deNivelle2014a} and \cite{deNivelle2014b} that formulas in classical logic with partial functions (\cite{deNivelle2011a}) can be translated into sets of geometric formulas.
\begin{defi}
\label{Def_interpretation}
We define an \emph{interpretation} $ I $ as a finite set of atoms
of forms $ \#_{\bf t} \, c $ with $ c $ a constant, or
form $ p_{\lambda}( x_1, \ldots, x_n ), $ where $ x_1, \ldots, x_n $
are constants (repetitions allowed).
Interpretations must be \emph{range restricted} as well.
This means that every constant $ x $ occurring in the
interpretation must occur in an atom of form $ \#_{\bf t} \, x. $ \end{defi}
\noindent Matching searches for false formulas. These are formulas whose premises $ A_1, \ldots, A_p $ clash with $ I, $ while none of the $ B_j $ is true in $ I. $
\begin{defi}
\label{Def_conflict_truth}
Let $ I $ be an interpretation. Let $ A $ be a geometric literal.
Let $ \Theta $ be a substitution that assigns constants to variables,
and that is defined on the variables in $ A. $
We say that $ A \Theta $ \emph{conflicts}(or \emph{is in conflict with})
$ I $ if {\bf (1)}
$ A $ has form $ p_{\lambda}( x_1, \ldots, x_n ), $ and there is an
atom of form $ p_{\mu}( x_1 \Theta, \ldots, x_n \Theta ) \in I $
with $ \lambda \not = \mu, $ \
{\bf (2)}
$ A $ has form $ x_1 \approx x_2 $ and $ x_1 \Theta \not = x_2 \Theta, $ or
{\bf (3)}
$ A $ has form
$ \#_{\bf f} \, x $ and $ ( \#_ {\bf t} \, x \Theta ) \in I. $
\noindent
We say that $ A \Theta $ \emph{is true} in $ I $ if \\
{\bf (1)} $ A $ has form $ p_{\lambda}( x_1, \ldots, x_n ) $ and
$ p_{\lambda}( x_1 \Theta, \ldots, x_n \Theta ) \in I, $ \
{\bf (2)} $ A $ has form $ x_1 \approx x_2 $ and
$ x_1 \Theta = x_2 \Theta, $ \
{\bf (3)} $ A $ has form $ \#_{\bf t} \, x $ and
$ ( \#_{\bf t} \, x \Theta ) \in I, $
or \ {\bf (4)}
$ A $ has form $ \exists y \ B_{\lambda}( x_1, \ldots, x_n, y ) $
and there exists a constant $ c, $ s.t.
$ B_{\lambda}( x_1 \Theta, \ldots, x_n \Theta, c ) \in I. $ \end{defi} In the definitions of truth and conflict, $ \# $ is treated as a usual predicate.
\begin{defi}
Let $ I $ be an interpretation. Let $ B $ be a geometric atom.
Let $ \Theta $ be a substitution that instantiates all free variables of
$ B, $
and for which $ B \Theta $ is not true in $ I. $
We define the \emph{extension set} $ E( B, \Theta ) $ as follows:
\begin{itemize}
\item
If $ B $ has form $ p_{\lambda}( x_1, \ldots, x_n) $ or
$ \#_{\bf t} \ x, $ then
$ E( B, \Theta ) = \{ B \Theta \}. $
\item
If $ B $ has form $ x_1 \approx x_2, $ then
$ E( B, \Theta ) = \emptyset. $
\item
If $ B $ has form $ \exists y \ p_{\lambda}( x_1, \ldots, x_n, y ), $ then
\[ E( B, \Theta ) =
\{ \ p_{\lambda}(x_1 \Theta, \ldots,x_n \Theta ,c) \ \} \ | \
c \in I \ \} \cup
\{ \ p_{\lambda}(x_1 \Theta,\ldots,x_n \Theta, \hat{c}) \ \} \ \}. \]
By $ c \in I $ we mean: $ c $ is a constant occurring in
an atom of $ I. $
We assume that $ \hat{c} $ is a fresh constant
for which $ \hat{c} \not \in I. $
\end{itemize} \end{defi} \noindent Intuitively, if for a geometric formula
$ \phi = \ A_1, \ldots, A_p \ | \ B_1, \ldots, B_q $ and a substitution $ \Theta, $ the $ A_i \Theta $ are in conflict with $ I, $ while none of the $ B_j \Theta $ is true in $ I, $ then $ \phi \Theta $ is false in $ I. $ If there exist a $ B_j $ and an atom $ C \in E( B_j, \Theta ) $ that is not in conflict with $ I, $ then $ \phi \Theta $ can be made true by adding $ C. $ If no such $ C $ exists, a conflict was found. If more than one $ C $ exists, the search algorithm has to backtrack through all possibilities. The search algorithm tries to extend an initial interpretation $ I $ into an interpretation $ I' \supset I $ that makes all formulas true. At each stage of the search, it looks for a formula and a substitution that make the formula false. If no formula and substitution can be found, the current interpretation is a model. Otherwise, search continues either by extending $ I, $ or by backtracking. Details of the procedure are described in \cite{deNivelleMeng2006a} for the two-valued case, and in \cite{deNivelle2014b} for the three-valued case. Experiments with the current three-valued version (available from \cite{CASC2016}), and the previous two-valued version (\cite{deNivelleMeng2007f}) show that the search for false formulas consumes nearly all of the resources of the prover.
\begin{defi}
\label{Def_matching}
An instance of \emph{the matching problem} consists of
an interpretation $ I $ and a geometric formula
$ A_1, \ldots, A_p\ | \ B_1, \ldots, B_q. $
Determine if there exists a substitution $ \Theta $ that brings
all $ A_i $ in conflict with $ I, $ and makes none of the
$ B_j $ true in $ \Theta. $
If yes, then return such substitution. \end{defi}
\begin{exas}
\label{Ex_matchings}
Consider an interpretation $ I $ consisting of atoms
\[ P_{\bf t}( x_0, x_0 ), \ P_{\bf e}( x_0, x_1 ), \
P_{\bf t}( x_1, x_1 ), \ P_{\bf e}( x_1, x_2 ), \
Q_{\bf t}( x_2, x_0 ). \]
\noindent
The formula
$ \phi_1 = \ P_{\bf f}(X,Y), \ P_{\bf f}(Y,Z) \ | \
Q_{\bf t}(Z,X) $ can be matched in five ways:
\[ \begin{array}{l}
\Theta_1 = \{ \ X := x_0, \ Y := x_0, \ Z := x_0 \ \} \\
\Theta_2 = \{ \ X := x_0, \ Y := x_0, \ Z := x_1 \ \} \\
\Theta_3 = \{ \ X := x_0, \ Y := x_1, \ Z := x_1 \ \} \\
\Theta_4 = \{ \ X := x_1, \ Y := x_1, \ Z := x_1 \ \} \\
\Theta_5 = \{ \ X := x_1, \ Y := x_1, \ Z := x_2 \ \} \\
\end{array}
\]
The substitution
$ \Theta_6 = \{ \ X := x_0, \ Y := x_1, \ Z := x_2 \ \} $ would make
the conclusion $ Q_{\bf t}(Z,X) $ true.
Next consider the formula
$ \phi_2 = \ P_{\bf f}( X,Y ), \ P_{\bf t}(Y,Z) \ | \
X \approx Y. $ \\
The substitution $ \Theta = \{ \ X := x_0, \ Y := x_1, \ Z := x_2 \ \} $
is the only matching of $ \phi_2 $ into $ I. $
Finally, the formula
$ \phi_3 = \ P_{\bf t}(X,Y) \ | \ \exists Z \ Q_{\bf t}(Y,Z) $
can be matched with $ \Theta = \{ \ X := x_0, \ Y := x_1 \ \}, $
and in no other way. \end{exas}
\noindent The first formula $ \phi_1 $ in example~\ref{Ex_matchings} has five matchings. In case there exists more than one matching, it matters for the geometric prover which matching is returned. This is because the prover analyses which ground atoms in the interpretation $ I $ contributed to the matching, and will consider only those in backtracking. In general, the set of conflicting atoms in $ I $ should be as small as possible, and should depend on as few as possible decisions. (Decisions in the sense of propositional reasoning, see \cite{Handbook_sat:CDCL}.) The simplest solution for finding the best matching would be to enumerate all matchings, and use some preference relation $ \preceq $ to keep the best one. Unfortunately, this approach is not practical because the number of matchings can be extremely high. We will address this problem in Section~\ref{Sect_optimal}.
Even if one is interested in the decision problem only, matching is still intractable because the decision problem is already NP-complete. (See problem {\bf LO18} in \cite{GareyJohnson79}.) In this paper, we introduce several algorithms for efficiently solving the matching problem. The algorithms evolved out of predecessors that have been implemented before in the two-valued version of {\bf Geo} (\cite{deNivelleMeng2007f}), and in the three-valued version of {\bf Geo} that took part in CASC~J8 (see \cite{CASC2016}). The matching algorithm of the three-valued version is discussed in detail in \cite{deNivelle2016a}. Unfortunately, after comparison with other methods, in particular the algorithms in the current paper, and translation to SAT, the approach of \cite{deNivelle2016a} turned out not competitive, and we have abandoned it. The algorithm in this paper, and translation to SAT are on average 500-1000 times faster than the algorithm of \cite{deNivelle2016a}.
The paper is organized as follows: In Section~\ref{Sect_GCSP}, we translate the matching problem into a structure called \emph{generalized constraint satisfaction problem} (GCSP). The generalization consists of the fact that it contains additional constraints, that a solution must not make true. These constraints correspond to the conclusions of the geometric formula that one is trying to match.
After that, we present in Section~\ref{Sect_unary_algo} a backtracking algorithm for solving GCSP, which is based on backtracking combined with a form of propagation. It relies on a data structure that we call \emph{refinement stack}. Refinements stacks were introduced in \cite{deNivelle2016a}. The matching algorithm of \cite{deNivelle2016a} turned out non-competitive, but its data structure is still useful. In Section~\ref{Sect_conflict_learning} we add conflict learning to our matching algorithm. In Section~\ref{Sect_IJCAR2016}, we briefly discuss the algorithm of \cite{deNivelle2016a}. In Section~\ref{Sect_SAT_trans}, we give two translations from GCSP to SAT. The translations are straightforward, and efficiently solved by MiniSat (\cite{Minisat2004}). In order to make it possible to run our matching algorithm independent of geometric logic, possibly opening the way for other applications, we define an input format for matching problems in Section~\ref{Sect_input_format}. The format is derived from the DIMACS format for SAT. We released the sources in \cite{deNivelle2018a}. Section~\ref{Sect_experiments} contains experimental results. The main conclusions are that the algorithm of \cite{deNivelle2016a} is not competitive, and that our own algorithm is comparable to translation to SAT combined with MiniSat. In Section~\ref{Sect_optimal}, we explain how every algorithm that is able to find some solution, can be transformed into an algorithm that finds an optimal solution. This transformation is essential for the application in geometric resolution. In Section~\ref{Sect_local_consistency_checking}, we present a priori filtering techniques, that are able to reject a large percentage of matching instances a priori.
\section{Translation into Generalized Constraint Satisfaction Problem} \label{Sect_GCSP}
We introduce the generalized constraint satisfaction problem, and show how instances of the matching problem can be translated. It is `generalized' because there are additional, negative constraints (called \emph{blockings}), which a solution is not allowed to satisfy. The blockings originate from translations of the $ B_1, \ldots, B_q. $
\begin{defi}
\label{Def_substlet}
A \emph{substlet} $ s $ is a (small) substitution.
We usually write $ s $ in the form $ \overline{v} / \overline{x}, $
where $ \overline{v} $ is a sequence of variables without repetitions,
and $ \overline{x} $ is a sequence of constants of same length as
$ \overline{v}. $
We say that two substlets $ \overline{v}_1/\overline{x}_1 $ and
$ \overline{v}_2/\overline{x}_2 $ are \emph{in conflict}
if there exist $ i,j $ s.t. $ v_{1,i} = v_{2,j} $ and
$ x_{1,i} \not = x_{2,j}. $
If $ \overline{v}_1/\overline{x}_1, \ldots, \overline{v}_n/\overline{x}_n $
is a sequence of substlets not containing a conflicting pair, then
one can merge them into a substitution as follows:
$ \bigcup \{ \overline{v}_1/\overline{x}_1, \ldots,
\overline{v}_n/\overline{x}_n \} =
\{
v_{i,j} := x_{i,j} \ | \ 1 \leq i \leq n, \
1 \leq j \leq \| \overline{v}_i \| \}. $
If $ \Theta $ is a substitution and $ s = \overline{v} / \overline{x} $
is a substlet, we say that $ \Theta $ \mbox{makes} $ s $ \emph{true} if
every $ v_i := x_i $ is present in $ \Theta. $
We say that $ \Theta $ and $ s $ \mbox{are in conflict} if
there is a $ v_i/x_i $ with $ 1 \leq i \leq \| v \|, $ s.t.
$ v_i \Theta $ is defined and distinct from $ x_i. $
A \emph{clause} $ c $ is a finite set of substlets
with the same domain.
We say that a substitution $ \Theta $ \emph{makes} $ c $ \emph{true}
(notation $ \Theta \models c $)
if $ \Theta $ makes a substlet
$ ( \overline{v} / \overline{x} ) \in c $ true.
We say that $ \Theta $ \emph{makes} $ c $ \emph{false}
(notation $ \Theta \models \neg c $) if
every substlet $ ( \overline{v} / \overline{x} ) \in c $ is in conflict
with $ \Theta. $
In the remaining case, we call $ c $ \emph{undecided by} $ \Theta. $ \end{defi}
\begin{defi}
\label{Def_gcsp}
A \emph{generalized constraint satisfaction problem} (GCSP)
is a pair of form $ ( \Sigma^{+}, \Sigma^{-} ) $ in which
$ \Sigma^{+} $ is a finite set of clauses,
and $ \Sigma^{-} $ is a finite set of substlets.
A substitution $ \Theta $ is a \emph{solution} of
$ ( \Sigma^{+}, \Sigma^{-} ), $ if
every clause in $ \Sigma^{+} $ is true in $ \Theta, $ and
there is no $ \sigma \in \Sigma^{-}, $ s.t. $ \Theta $ makes
$ \sigma $ true. \end{defi}
\begin{defi}
Let $ ( \Sigma^{+}, \Sigma^{-} ) $ a GCSP.
We call $ ( \Sigma^{+}, \Sigma^{-} ) $
\emph{range restricted} if for every variable $ v $ that occurs in
a substlet $ \sigma \in \Sigma^{-}, $ there exists a clause
$ c \in \Sigma^{+} $ s.t. every substlet $ s \in c $ has
$ v $ in its domain. \end{defi}
\noindent We now explain how a matching instance is translated into a generalized constraint satisfaction problem.
\begin{defi}
\label{Def_trans_gcsp}
Assume that $ I $ and $ \phi = \ A_1, \ldots, A_p \ | \ B_1, \ldots, B_q $
together form an instance of the matching problem.
The \emph{translation} $ ( \Sigma^{+}, \Sigma^{-} ) $
of $ ( I, \phi ) $
\emph{into} GCSP is obtained as follows:
\begin{itemize}
\item
For every $ A_i, $ let $ \overline{v}_i $ denote the
variables of $ A_i. $
Then $ \Sigma^{+} $ contains the clause
\[ \{ \ \overline{v}_i / \overline{v}_i \Theta \ | \
A_i \Theta \mbox{ is in conflict with } I \ \}. \]
\item
For every $ B_j, $ let $ \overline{w}_j $ denote the variables
of $ B_j. $
For every $ \Theta $ that makes $ B_j \Theta $
true in $ I, $ \ \ $ \Sigma^{-} $ contains the substlet
$ \overline{w}_j / ( \overline{w}_j \Theta ). $
\end{itemize} \end{defi}
\begin{thm}
A matching instance $ ( I, \phi ) $ has a matching iff its corresponding
GCSP has a solution. \end{thm}
\noindent In theory, the set of blockings $ \Sigma^{-} $ can be removed, because a blocking $ \sigma $ can always be replaced by a clause as follows: Let $ \sigma $ be a blocking, let $ \overline{v} $ be its variables. Define $ \sigma_1 = \sigma, $ and let $ \sigma_2, \ldots, \sigma_n \in \Sigma^{-} $ be the blockings whose domain is also $ \overline{v}. $ One can replace $ \sigma_1, \ldots, \sigma_n $ by
the clause $ \{ \ \overline{v}/\overline{c} \ | \
\overline{v} / \overline{c} \mbox{ conflicts all }
\sigma_i \ ( 1 \leq i \leq n ) \ \}. $
We prefer to keep $ \Sigma^{-}, $ because in the worst case,
the resulting clause has size $m^{ \| \overline{v} \| }, $ where $ m $ is the size of the domain. For example, if $ \sigma_1, \ldots, \sigma_n $ result from an equality $ X \approx Y, $ then $ \sigma_i $ has form $ ( X,Y ) / ( x_i, x_i ). $
The resulting clause $ c = \{ (X,Y)/(x_i, x_j) \ | \ i \not = j \} $ has size $ n(n-1) \approx n^2. $
Clauses resulting from a matching problem have the following trivial, but essential property: \begin{lem}
\label{Lemma_conflict_in_clause}
Let $ ( \Sigma^{+}, \Sigma^{-} ) $ be obtained
by the translation in Definition~\ref{Def_trans_gcsp}.
Let $ s_1,s_2 \in c \in \Sigma^{+}. $
Then either $ s_1 = s_2, $ or $ s_1 $ and $ s_2 $ are
in conflict with each other. \end{lem} Lemma~\ref{Lemma_conflict_in_clause} holds because $ s_1 $ and $ s_2 $ have the same domain.
\begin{exas}
\label{Ex_translations}
In example~\ref{Ex_matchings}, the matching
problem $ ( I, \phi_1 ) $ can be translated into
the GCSP below. The clauses are above the horizontal
line, and the blockings are below it. Because substlets
in the same clause always have the same variables, we write
the variables of a clause only once.
\[ \begin{array}{l}
(X,Y) \ / \ ( x_0,x_0 ) \ | \ ( x_0,x_1 ) \ | \
( x_1,x_1 ) \ | \ ( x_1,x_2 ) \\
(Y,Z) \ / \ ( x_0,x_0 ) \ | \ ( x_0,x_1 ) \ | \
( x_1,x_1 ) \ | \ ( x_1,x_2 ) \\
\hline
(X,Z) \ / \ ( x_0,x_2 ) \\
\end{array}
\]
Translating $ (I,\phi_2) $ results in:
\[ \begin{array}{l}
(X,Y) \ / \ ( x_0,x_0 ) \ | \ ( x_0,x_1 ) \ | \
( x_1,x_1 ) \ | \ ( x_1,x_2 ) \\
(Y,Z) \ / \ ( x_0,x_1 ) \ | \ ( x_1,x_2 ) \\
\hline
(X,Y) \ / \ ( x_0, x_0 ) \\
(X,Y) \ / \ ( x_1, x_1 ) \\
(X,Y) \ / \ ( x_2, x_2 ) \\
\end{array}
\]
Translation of $ (I,\phi_3) $ results in:
\[ \begin{array}{l}
(X,Y) \ / \ ( x_0,x_1 ) \ | \ ( x_1,x_2 ) \\
\hline
(Y) \ / \ ( x_2 ) \\
\end{array}
\] \end{exas}
\noindent Before one runs any algorithms on a GCSP, it is useful to do some simplifications. If the GCSP contains a propositional clause (a clause whose domain contains no variables), this clause either has form $ ( \ ) \ / \ $ (no assignments), or $ ( \ ) \ / \ ( \ ) $ (one assignment). In the first case, the problem is trivially unsolvable. In the second case, the clause can be removed.
Similarly, if $ \Sigma^{-} $ contains a propositional blocking, then $ ( \Sigma^{+}, \Sigma^{-} ) $ is trivially unsolvable. Such blockings originate from a $ B_j $ that is purely propositional, or that has form $ \exists y \ P_{\lambda}( y ). $
A third important preprocessing step is \emph{removal of unit blockings}. Let $ \sigma \in \Sigma^{-} $ be a blocking whose domain is included in the domain of some clause $ c \in \Sigma^{+}. $ In that case, one can remove every substlet $ \overline{v}/\overline{c} $ from $ c, $ that has $ \bigcup \{ \overline{v}/\overline{c} \} \models \sigma. $ If this results in $ c $ being empty, then $ ( \Sigma^{+}, \Sigma^{-} ) $ trivially has no solution. If no $ \overline{v}/\overline{c} $ in any clause $ c \in \Sigma^{+} $ implies $ \sigma, $ then $ \sigma $ can be removed from $ \Sigma^{-}, $ because of Lemma~\ref{Lemma_conflict_in_clause}.
Applying removal of unit blockings to the translation of $ (I,\phi_2) $ above results in \[ \begin{array}{l}
(X,Y) \ / \ ( x_0,x_1 ) \ | \ ( x_1,x_2 ) \\
(Y,Z) \ / \ ( x_0,x_1 ) \ | \ ( x_1,x_2 ) \\
\hline
\end{array} \] It is worth noting that removal of propositional blockings can be viewed as a special case of removal of unit blockings.
A GCSP can be solved by backtracking, similar to SAT solving. A backtracking algorithm for GCSP can be either variable or clause based. A variable based algorithm maintains a substitution $ \Theta, $ which it tries to extend into a solution. It backtracks by picking a variable $ v $ and trying to assign it in all possible ways. It backtracks when $ \Theta $ makes a clause $ c \in \Sigma^{+} $ false, or a blocking $ \sigma \in \Sigma^{-} $ true.
A clause based algorithm maintains a consistent set $ S $ of substlets (whose union defines a substitution). It backtracks by picking an undecided clause $ c \in \Sigma^{+}, $ and consecutively inserting all substlets that are consistent with $ S $ into $ S. $ It backtracks when there is a clause $ c $ all of whose atoms are in conflict with $ S, $ or when $ \bigcup S $ makes a blocking true.
Our experiments suggest that there is no significant difference in performance, nor in programming effort, between the two variants. We will stick with clause based algorithms, because it seems that they can be more easily combined with local consistency checking.
\section{Matching Using Refinement Stacks} \label{Sect_unary_algo}
We first present the algorithm without learning, and add learning in the next section. The algorithm that we present here is a simplification of the algorithm in \cite{deNivelle2016a}, which unfortunately could not be made competitive. The previous algorithm was based on a combination of local consistency checking and lemma learning from conflicts. Local consistency checking will be discussed in detail in Section~\ref{Sect_local_consistency_checking}, because there is still a probability that it can be used as priori check.
Local consistency checking means that one generates all subsets of clauses up to some size $ S+1 $ and checks which substlets can occur in solutions. Substlets that do not occur in any solution of some subset, certainly do not occur in a solution of the complete GCSP. In most instances, filtering with a small $ S, $ e.g. $ 1 $ or $ 2 $ results in an empty clause. The algorithm of \cite{deNivelle2016a} was based on a combination of local consistency checking and decision. It is discussed in more detail in Section~\ref{Sect_IJCAR2016}.
The algorithm that we discuss in this section evolved from \cite{deNivelle2016a}. The main differences are: Clauses are not checked against each other anymore. Instead, clauses are checked only against the substitution in combination with blockings. Secondly, learnt lemmas are flat, i.e. finite disjunctions of single assignments to variables. In \cite{deNivelle2016a}, lemmas were finite disjunctions of substlets. It turns out that this simplification improves performance by a factor between 100 and 1000.
In order to implement matching algorithms and local consistency checking, one needs to be able to remove substlets from clauses, and reintroduce them during backtracking. We call the process of removing substlets from a clause \emph{refinement}. Whenever a clause has been refined, it may trigger other refinements. In the earlier algorithm, refinement of a clause could directly trigger more refinements of other clauses. In the current algorithm, refinement of a clause can only trigger possible extension of the substitution, but extension of the substitution may still trigger other clause refinements. As a consequence, one needs to maintain a queue of recent refinements and use this queue to check which more clauses can be refined. We introduce a data structure, called \emph{refinement stack} which supports refinement of clauses, restoring during backtracking, and keeping track of unchecked refinements.
\begin{defi}
\label{Def_refinement_stack}
A \emph{refinement} has form $ c \Rightarrow d, $
where both $ c $ and $ d $ are clauses, and $ d $ is a
subclause of $ c. $
\noindent
A \emph{refinement stack} $ \overline{C} $ is a finite sequence
of refinements $ c_i \Rightarrow d_i. $
If there exists a $ j $ with $ i < j $ and $ c_i = c_j, $ then
$ d_j $ must be a strict subclause of $ d_i. $
\noindent
For a clause $ c, $ if $ c_i \Rightarrow d_i $ is the last
refinement with $ c = c_i $ occurring in $ \overline{C}, $
we call $ d_i $ \emph{the current refinement of} $ c. $
\noindent
We define a predicate $ \alpha_i( \overline{C} ) $ that is true
if $ c_i \Rightarrow d_i $ is the current refinement of $ c_i $
in $ \overline{C}. $ This means that there is no $ j > i $ with
$ c_j = c_i. $
A refinement stack supports gradual refinement of clauses.
If $ \alpha_i( \overline{C} ) $ is true, then clause
$ d_i $ can be refined into $ d' $ by appending
$ c_i \Rightarrow d' $ to $ \overline{C}. $
In the new refinement stack
$ \overline{C}' = \overline{C} + ( c_{i} \Rightarrow d' ), $
we have $ c_i = c_{n+1}, $ \ \
$ \alpha_i( \overline{C}' ) $ is false, and
$ \alpha_{n+1}( \overline{C}' ) $ is true.
The \emph{size} $ \| \overline{C} \| $ of a refinement stack
$ \overline{C} $ is defined as the total number of refinements
that occur in it,
independent of the values of $ \alpha_i( \overline{C} ). $ \end{defi}
\noindent The refinement stack is initialized with the refinements $ c \Rightarrow c, $ for each initial clause $ c. $ Refinement stacks can be efficiently implemented without need to copy clauses by maintaining a stack of intervals of active substlets in the initial clauses. A substlet can be disabled by swapping it with the last active substlet in the interval, and decreasing the size of the interval by one. When the substlet is made active again, it is sufficient to restore the interval, because the order of active substlets in a clause does not matter. Refinement stacks support change driven inspection as well as backtracking.
Change driven inspection of clauses can be implemented by starting at position $ k=1. $
As long as $ k \leq \| \overline{C} \|, $ one first checks $ \alpha_k( \overline{C} ). $ If it is false, then $ d_k $ is not the current version of $ c_k, $ and one can increase $ k. $ If $ \alpha_k( \overline{C} ) $ is current, one can check if $ d_k $ triggers refinement of other clauses. If yes, the results are inserted at the end, so that
they will be inspected at later time. When one reaches $ k > \| \overline{C} \|, $ one has reached a stable state.
When some change involving a variable $ v $ takes place, one needs to check which clauses may be affected by the change, so that they can be refined. These are obviously the clauses that contain $ v, $ but also the clauses that contain a variable occuring in a blocking that contains $ v, $ since the algorithm takes blockings into account, when refining. This gives rise to the following definition:
\begin{defi}
\label{Def_connected_variable}
Let $ v,w $ be two variables. We call $ v $ and $ w $
\emph{connected} if $ v $ and $ w $ occur together in a blocking
$ \sigma \in \Sigma^{-}. $ \end{defi}
\noindent We define the search algorithm. We assume that propositional clauses and unit blockings have been removed from $ ( \Sigma^{+}, \Sigma^{-} ). $ We assume that the substitution $ \Theta $ is an ordered sequence (stack) of assignments $ ( v_1/x_1, \ldots, v_s/x_s ). $
\begin{algo}
\label{Algo_unary}
We want to find a solution for $ ( \Sigma^{+}, \Sigma^{-} ). $
Initially, set
$ \Theta := \emptyset $ and $ \overline{C} := \emptyset. $
After that, for each $ k \ ( 1 \leq k \leq \| \Sigma^{+} \| ), $
do the following:
\begin{description}
\item[PREPROC]
Let $ c_k $ be the $ k $-th clause in $ \Sigma^{+}. $
Append $ (c_k \Rightarrow c_k ) $ to $ \overline{C}. $
For every variable $ v $ occurring in $ c_k, $ for which all
substlets in $ c_k $ agree on the value of $ v, $
let $ x $ be the agreed value.
\begin{itemize}
\item
If $ v \Theta $ is defined, and $ v \Theta \not = x, $
then return $ \bot. $
\item
If $ v \Theta $ is undefined and there is a blocking
$ \sigma $ containing $ v, $ s.t.
$ \Theta \cup \{ v/x \} \models \sigma, $ then
return $ \bot. $ Otherwise, append $ v/x $ to $ \Theta. $
\end{itemize}
\end{description}
\noindent
After that, we call the main search algorithm
$ {\bf findmatch}( \overline{C}, \Theta, s, \Sigma^{-} ) $
with $ s = 1. $
It either returns $ \bot, $ or
it extends $ \Theta $ into a solution of
$ ( \overline{C}, \Sigma^{-} ). $
$ {\bf findmatch}( \overline{C}, \Theta, s, \Sigma^{-} ) $
is defined as follows:
\begin{description}
\item[FORW]
As long as $ s \leq \| \Theta \|, $ let
$ v/x $ be the $ s $-th assignment of $ \Theta. $
\begin{enumerate}
\item
For every $ ( c_i \Rightarrow d_i ) \in \overline{C} $
which has $ \alpha_i( \overline{C} ) $ true, and which
either contains $ v $ itself, or a variable $ w $ that
is connected to $ v, $
let
\[ d' = \{ s \in d_i \ | \ s
\mbox{ is not in conflict with } \Theta \}. \]
If $ d' = \emptyset, $ then return $ \bot. $
Otherwise, let
\[ d'' = \{ s \in d' \ | \
\mbox{there is no } \sigma \in \Sigma^{-}, \mbox{s.t. }
\Theta \cup \{ s \} \models \sigma \}. \]
If $ d'' = \emptyset, $ then return $ \bot. $
Otherwise, if $ d'' \subset d_i, $ then
\begin{enumerate}
\item
append $ ( c_i \Rightarrow d'' ) $ to $ \overline{C}. $
\item
For every variable $ v' $ occurring in $ d'', $ that
is unassigned in $ \Theta, $ for which all substlets in
$ d'' $ agree on the assigned value, let
$ x' $ be the agreed value. Append $ v'/x' $ to $ \Theta. $
\end{enumerate}
\item
Set $ s = s + 1. $
\end{enumerate}
\item[PICK]
Find an $ i $ with $ \alpha_i( \overline{C} ) $ true
and $ \| d_i \| > 1. $
If no such $ i $ exists, then $ \Theta $ is a solution.
\noindent
Otherwise, for every substlet $ \overline{v}_j/\overline{x}_j $
in $ d_i, $ do the following:
\begin{enumerate}
\item
Append
$ c_i \Rightarrow ( \overline{v}_j/\overline{x}_j ) $
to $ \overline{C}, $ and
extend $ \Theta $ with the unassigned
variables in $ \overline{v}_j / \overline{x}_j. $
\item
Recursively call $ {\bf findmatch}( \ \overline{C},
\Theta, s, \ \Sigma^{-} \ ). $
If
$ \Theta $ was extended into a solution, then return $ \Theta. $
\item
Otherwise, restore $ \Theta $ and
$ \overline{C} $ to the sizes that they
had before (1).
\end{enumerate}
At this point, each of the recursive calls has returned
$ \bot. $ Return $ \bot. $
\end{description}
\noindent \end{algo}
\noindent At {\bf FORW}, the algorithm attempts deterministic reasoning. For every new assignment in $ \Theta, $ it is checked if it conflicts with some substlets in some clause. Two types of conflicts are considered, either the substlet contains an assignment that directly conflicts with $ \Theta, $ or it contains an assignment that, together with $ \Theta, $ implies a blocking. As long as conflicts are found, the corresponding clauses are refined. Refinement of a clause may result in $ \Theta $ being extended ({\bf FORW}~b), if the remaining substlets agree on an assignment. Extension of $ \Theta $ may result in further refinements of clauses.
\noindent If {\bf FORW} failed to solve the problem, then at {\bf PICK} a non-unit clause is picked, and non-deterministically refined into a unit clause. This step requires backtracking. It is important (for performance) to pick a clause of minimal length.
Main purpose of {\bf PREPROC} is to initialize the refinement stack $ \overline{C} $ with $ \Sigma^{+}. $ After that, $ \Theta $ is initialized by looking for assignments that are common to all substlets in some clause. If this results in a conflict (either directly, or with a blocking), the problem is rejected.
\noindent Algorithm~\ref{Algo_unary} is similar to DPLL in that it tries to postpone backtracking as long as possible by giving preference to deterministic extension. At {\bf FORW}, blockings are taken into account. It is possible to implement {\bf FORW} without considering blockings. In that case, it has to be checked, whenever the substitution is extended (at {\bf PICK~2} and at {\bf FORW~1b}) that the extended substitution does not imply a blocking. The given version performs better in experiments.
In order to show that Algorithm~\ref{Algo_unary} is correct, i.e. does not report false solutions, we have to show that all necessary checks are made.
\begin{lem}
\label{Lem_not_alone}\leavevmode
\begin{enumerate}
\item
At points {\bf FORW} and {\bf PICK} of Algorithm~\ref{Algo_unary},
there is no $ \sigma \in \Sigma^{-}, $ s.t.
$ \Theta \models \sigma. $
\item
At point {\bf PICK}, no refined clause $ d_i $
contains a substlet that is in conflict with $ \Theta. $
\end{enumerate} \end{lem} Initially, the preprocessor ensures that there is no $ \sigma \in \Sigma^{-}, $ s.t. $ \Theta \models \sigma. $ When $ \sigma $ is extended in {\bf FORW~1b}, it has been checked before that $ \Theta \cup \{ s \} $ does not imply a blocking, for each of the substlets in $ d''. $ At point {\bf PICK}, \ {\bf findmatch} passed through {\bf FORW} which refined away all substlets that conflict with $ \Theta. $
In the next section, we will extend Algorithm~\ref{Algo_unary} with learning. This will prove completeness, because whenever Algorithm~\ref{Algo_unary} does not find a solution, it will construct a lemma that proves that no lemma exists.
\section{Conflict Learning} \label{Sect_conflict_learning}
\noindent It is known from propositional SAT solving that conflict learning dramatically improves the performance of SAT solvers (\cite{Handbook_sat:CDCL}). The matching algorithm in the two-valued version of {\bf Geo} (\cite{deNivelleMeng2007f}) was already equipped with a primitive form of conflict learning. Before releasing {\bf Geo}, we had experimented with naive matching, the algorithm in \cite{GottlobLeitsch85}, and many ad hoc methods. Matching with conflict learning is the only approach that results in acceptable performance. Despite this, matching was still a critical operation in the last two-valued version of {\bf Geo}. In the two-valued version of {\bf Geo}, lemmas had form $ v_1/x_1, \dots, v_n/x_n \rightarrow \bot, $ i.e. they had form $ ( \overline{v} / \overline{x} ) \rightarrow \bot $ for a single substlet.
In \cite{deNivelle2016a} we proposed to replace the lemmas of {\bf Geo}~2007 by arbitrary sets of substlets. It is quite easy to see, that in general such a lemma can be in conflict with more substitutions than a lemma of the previous form. For example, if we assume that the domain is $ \{ X,Y,Z \} $ and the range $ \{ 0,1,2 \}, $ then $ (X,Y,Z)/(0,1,2) \rightarrow \bot $ rejects a single substitution, while $ (X,Y,Z)/(0,1,2), \ (X,Y,Z) / (2,1,0) $ rejects 25 substitutions. Since in case of a conflict, one can always obtain a lemma of the second form, it seemed that lemmas of the second form should be preferred over lemmas of the first form.
The latest version of {\bf Geo} see (\cite{CASC2016}) used the algorithm of \cite{deNivelle2016a} with lemmas of the unrestricted form above. Although this matching algorithm performs better than matching in {\bf Geo}~2007, recent experiments have shown that it performs significantly worse than some other approaches, in particular translation to SAT and Algorithm~\ref{Algo_unary} in combination with flat lemmas. Flat lemmas are lemmas of form $ v_1 \in V_1 \vee \cdots \vee v_n \in V_n. $ Surprisingly, Algorithm~\ref{Algo_unary} with unrestricted lemmas performs several orders worse than Algorithm~\ref{Algo_unary} with flat lemmas. This is surprising, because every general lemma can be flattened into a lemma of the second form by picking a single assignment from each substlet. The resulting lemma is obviously less general than its original, non-flattened version. This loss of generality also applies to the reasoning rules that we use on lemmas. If two substlets in two general lemmas are in conflict, then their flattenings are not necessarily in conflict. Conversely, whenever two flattened substlets are in conflict, their original counterparts are. This means that by using flattened lemmas, one looses conflicts with substitutions, and also resolution derivations involving lemma resolution. Despite this clever reasoning, the first columns of Figure~\ref{Fig_unary_sat} of Section~\ref{Sect_experiments} show that Algorithm~\ref{Algo_unary} with flat lemmas performs approximately 200-400 times worse than Algorithm~\ref{Algo_unary} with unrestricted lemmas. One could assume that this is caused by the fact that handling of unrestricted lemmas is more costly, and that their theoretical advantage is compensated by the increased cost of their maintenance. This assumption is rejected by Figure~\ref{Fig_unary_sat}, because Algorithm~\ref{Algo_unary} with flattened lemmas is not only faster, but it also uses less lemmas, typically by a factor 2-3. The only point where Algorithm~\ref{Algo_unary} with and without flattening can diverge, is when a conflict lemma rejects a substitution $ \Theta, $ and there exists more than one conflict lemma. Since both versions will prefer the shortest lemma, it must be due to the fact that flattening changes the relative sizes of the lemmas.
The outcomes of the experiments make it probable that the best approach to matching will be either Algorithm~\ref{Algo_unary} with flat lemmas, or translation to SAT in combination with a SAT-solver, which we will describe in Section~\ref{Sect_SAT_trans}.
From the practical point of view, the fact that the refining algorithm in \cite{deNivelle2016a} turned out not competitive, is not a serious loss. Despite being elegant on paper, it was hard to implement. Implementation of Algorithm~\ref{Algo_unary} was much easier, and in the long term, it is better that the easier algorithm has the better performance. Moreover, it is clear from Figure~\ref{Fig_unary_sat} that matching in future versions of {\bf Geo} can be approximately $ 1000 $ times faster than it was at {\bf Geo}~2016c (\cite{CASC2016}).
We will now introduce the flat lemmas, and prove that Algorithm~\ref{Algo_unary} can always generate a flat conflict lemma.
\begin{defi}
\label{Def_lemma}
A \emph{lemma} is an object of form $ \{ v_1/V_1, \ldots, v_n/V_n \} $
with $ n \geq 0. $ The $ v_i $ are variables, and the $ V_i $
are finite sets of constants.
\noindent
It is convenient to treat lemmas as total functions from variables
to sets of constants.
For a variable $ v $ and $ \lambda = \{ v_1/V_1, \ldots, v_n/V_n \}, $ \
$ \lambda(v) $ is defined as $ \bigcup \{ V_i \ | \ v_i = v \}. $
\noindent
Let $ \Theta $ be a substitution. We say that $ \Theta $
\emph{makes} $ \lambda $ \emph{true} if
there exists a variable $ v $ in the domain of $ \Theta, $ for which
$ v \Theta \in \lambda(v). $
\noindent
We say that $ \Theta $ makes $ \lambda $ false if all variables
$ v $ for which $ \lambda(v) $ is nonempty, are in the domain
of $ \Theta, $ and $ v \Theta \not \in \lambda(v). $
In that case, we write $ \Theta \models \neg \lambda. $ \end{defi}
\begin{defi}
\label{Def_valid_and_conflict}
Let $ ( \Sigma^{+}, \Sigma^{-} ) $ be
a GCSP.
Let $ \lambda $ be a lemma.
We say that $ \lambda $ is \emph{valid in}
$ ( \Sigma^{+}, \Sigma^{-} ) $
if every solution $ \Theta $ of $ ( \Sigma^{+}, \Sigma^{-} ) $
makes $ \lambda $ true.
\noindent
For a given substitution $ \Theta, $ we call $ \lambda $
a \emph{conflict lemma} if $ \lambda $ is valid and
$ \Theta $ makes $ \lambda $ false. \end{defi}
\noindent If $ \Theta $ is a substitution, and there exists a valid lemma that is false in $ \Theta, $ then it is not possible to extend $ \Theta $ into a solution of $ ( \Sigma^{+}, \Sigma^{-} ). $
In order to derive the conflict lemma, the following rules will be used:
\begin{defi}
\label{Def_reso_proj_sigma}
Given a GCSP $ ( \Sigma^{+}, \Sigma^{-} ), $ we define
the following derivation rules:
\begin{description}
\item[RESOLUTION]
Let $ \lambda_1, \ldots, \lambda_m $ be a sequence of lemmas.
Let $ v $ be a variable.
Let $ V $ be the set of variables $ v, $
for which one of the $ \lambda_j $ has $ \lambda(v) \not = \emptyset. $
We define the $ v $-\emph{resolvent of}
$ \lambda_1, \ldots, \lambda_m $ as
\[ \{ v/\bigcap_{1 \leq j \leq m} \lambda_j(v) \} \cup
\{ v'/\bigcup_{ 1 \leq j \leq m} \lambda_j(v') \ | \
v' \in V \wedge v' \not = v \}. \]
\item[PROJECTION]
Let $ c \in \Sigma^{+} $ be a clause, let $ \lambda $
be a lemma. We call $ \lambda $ a \emph{projection} of $ c, $
if every substlet $ (\overline{v}/\overline{x}) \in c $
contains an assignment $ v/x, $ s.t. $ x \in \lambda(v). $
\item[$\sigma$-RESOLUTION]
Let $ \sigma \in \Sigma^{-} $ be a blocking.
Write $ \sigma = \{ v_1/x_1, \ldots, v_n/x_n \} \ ( n > 0 ). $
Let $ c_1, \ldots, c_n \in \Sigma^{+} $ be clauses,
chosen in such a way that
every variable $ v_i $ occurs in $ c_i. $
For every $ c_i, $ let
\[ V_i = \{ \ x \ | \ c_i \mbox{ contains a substlet }
\overline{w}/\overline{y} \mbox{ which contains }
v_i / x \mbox{ and } x \not = x_i \ \}. \]
Then the lemma
\[ \{ v_1/V_1, \ldots, v_n/V_n \} \]
is called a $ \sigma $-\emph{resolvent of} $ c_1, \ldots, c_n. $
\end{description} \end{defi}
\noindent The lemmas $ \{ \ x / \{ 1, 2, 3 \}, \ y / \{ 2,3 \} \ \} $ and $ \{ \ x / \{ 3, 4 \}, \ y / \{ 3,4 \}, \ z / \{ 2 \} \ \} $ can resolve into $ \{ \ x / \{ 3 \}, \ y / \{ 2,3,4 \}, \ z / \{ 2 \} \ \}. $ Given clauses $ c_1 = \{ \ (x,y)/(1,2), \ (x,y)/(1,1), \ (x,y)/(3,3) \ \} $ and $ c_2 = \{ \ (y,z)/(1,2), \ (y,z)/(2,1) \ \}, $ and a blocking $ (x,z)/(1,2), $ one can obtain the $ \sigma $-resolvent $ \{ \ x / \{ 3 \}, \ z / \{ 1 \} \ \}. $ The lemma $ \lambda'_1 = \{ x / \{ 1,3 \} \} $ is a projection of $ c_1. $ $ \lambda''_1 = \{ \ x / \{ 3 \}, \ y / \{ 1,2 \} \ \} $ is also a projection of $ c_1. $ It is easy to see that the reasoning rules are valid, which implies that every lemma that has been obtained by repeated application from the original clauses in $ \Sigma^{+} $ and blockings in $ \Sigma^{-}, $ is valid.
\begin{lem}
\label{Lem_sigma_resolvent_false}
Let $ ( \Sigma^{+}, \Sigma^{-} ) $ be a GCSP.
Let $ \Theta $ be an interpretation. Let $ \sigma \in \Sigma^{-} $
be a blocking for which $ \Theta \models \sigma. $
Let $ \lambda $ be a $ \sigma $-resolvent of $ \sigma. $
Then $ \Theta $ makes $ \lambda $ false. \end{lem} \begin{proof}
Write $ \sigma = \{ v_1/x_1, \ldots, v_n/x_n \}. $
Let $ c_1, \ldots, c_n \in \Sigma^{+} $ be the clauses
that were used in the construction of $ \lambda. $
Because $ \Theta \models \sigma, $ we know that
for every $ i \ ( 1 \leq i \leq n ), $ we have $ v_i \Theta = x_i. $
From the construction of the $ V_i, $ it follows that $ x_i \not \in V_i. $
Because the variables $ v_i $ are pairwise distinct, we
have $ V_i = \lambda(x_i). $ It follows that $ x_i \not \in \lambda(v_i). $
For all other variables $ v $ that do not occur in $ \sigma, $
we have $ \lambda(v) = \emptyset. $
We can conclude that if
$ \lambda(v) $ is non-empty, then $ v $ equals one of the $ v_i, $
and we have $ v_i \Theta \not \in \lambda(v_i). $ \end{proof} The following lemma states that substlets that are switched off, were switched off because they conflict $ \Theta, $ possibly with help of a blocking.
\begin{lem}
\label{Lemma_weak_conflict}
At every moment during Algorithm~\ref{Algo_unary},
for every refinement $ (c_i \Rightarrow d_i) \in \overline{C}, $ the
following holds:
If $ s \in ( d_i \backslash c_i ), $ then
either
$ s $ is in conflict with $ \Theta, $ or
$ \Theta \cup \{ s \} \models \sigma, $ for a $ \sigma \in \Sigma^{-}. $ \end{lem} \begin{proof}
There are two points at which refinement can take place,
{\bf PICK~1} and
{\bf FORW~1a}. At {\bf PICK~1}, clause $ c_i $ is refined into
$ \overline{v}_j / \overline{x}_j, $ after which $ \Theta $ is
extended with $ \overline{v}_j / \overline{x}_j. $
If some substlet
$ s $ occurs in $ c_i \backslash \{ \overline{v}_j / \overline{x}_j \}, $
then either $ s \in c_i \backslash d_i, $ or
$ s \in d_i \backslash \{ \overline{v}_j / \overline{x}_j \}. $
In the first case, the desired property is inherited from the
previous state, because it is an invariant.
In the second case, because $ \Theta $
is extended by $ \overline{v}_j / \overline{x}_j $ at the same time,
we can apply Lemma~\ref{Lemma_conflict_in_clause}.
At {\bf FORW~1a}, if $ s \in d'' \backslash c_i, $ then either
$ s \in d_i \backslash c_i, \ \
s \in d' \backslash d_i, $ or
$ s \in d'' \backslash d'. $
In the first case,
the desired property is inherited from the previous state.
In the second case, it follows from the construction of $ d', $ that
$ s $ was in conflict with $ \Theta. $
In the third case, it follows from the construction of $ d'', $ that
there is a $ \sigma \in \Sigma^{-}, $ for which
$ \Theta \cup \{s \} \models \sigma. $ \end{proof}
\noindent The following property is the essential property, for proving that Algorithm~\ref{Algo_unary} can always return a conflict lemma. \begin{lem}
\label{Lem_unary_derive}
Let $ ( \Sigma^{+}, \Sigma^{-} ) $ be a GCSP.
Let $ c \in \Sigma^{+} $ be a clause.
Let $ \Theta $ be a substitution. Let $ \Lambda $ be a set
of lemmas. Assume that there is no $ \sigma \in \Sigma^{-}, $ s.t.
$ \Theta \models \sigma, $ and no $ \lambda \in \Lambda, $ s.t.
$ \Theta $ makes $ \lambda $ false.
Assume that for every substlet $ s \in c, $ either
\begin{enumerate}
\item
$ s $ is in conflict with $ \Theta, $
\item
$ \Theta \cup \{ s \} \models \sigma, $
for a $ \sigma \in \Sigma^{-}, $ or
\item
$ \Theta \cup \{ s \} $ makes a $ \lambda \in \Lambda $ false.
\end{enumerate}
Then it is possible to derive a conflict lemma for $ \Theta $
from $ \Sigma^{+} $ and $ \Lambda, $ by applying the rules in
Definition~\ref{Def_reso_proj_sigma}. \end{lem} \begin{proof}
We first remove {\bf (2)} by means of $ \sigma $-resolution.
We will add the resulting $ \sigma $-resolvents to $ \Lambda.$
For every $ s \in c, $ for which $ {\bf (1),(3)} $ do not apply,
$ {\bf (2)} $ must apply.
Write $ \sigma = \{ v_1 /x_1, \ldots, v_n/x_n \}. $
Since $ ( \Sigma^{+}, \Sigma^{-} ) $ is range restricted,
we can find clauses
$ c_1, \ldots, c_n \in \Sigma^{+}, $ s.t.
each $ v_i $ occurs in $ c_i. $
We now can construct the $ \sigma $-resolvent. Write
$ \lambda $ for the resulting lemma.
It follows from Lemma~\ref{Lem_sigma_resolvent_false} that
$ \lambda $ is false in $ \Theta. $ We can add $ \lambda $ to $ \Lambda. $
At this point, we have for every $ s \in c, $ either
$ {\bf (1)} $ or $ {\bf (3)}. $
The rest of the proof is Lemma~\ref{Lem_unary_derive_recurse}. \end{proof}
\begin{lem}
\label{Lem_unary_derive_recurse}
Let $ ( \Sigma^{+}, \Sigma^{-} ) $ be a GCSP.
Let $ c \in \Sigma^{+} $ be a clause.
Let $ \Theta $ be a substitution. Let $ \Lambda $ be a set
of lemmas. Assume that there is no $ \lambda \in \Lambda, $ s.t.
$ \Theta $ makes $ \lambda $ false.
Assume that for every substlet $ s \in c, $ either
\begin{enumerate}
\item
$ s $ is in conflict with $ \Theta, $ or
\item
$ \Theta \cup \{ s \} $ makes a $ \lambda \in \Lambda $ false.
\end{enumerate}
Then it is possible, using the rules in
Definition~\ref{Def_reso_proj_sigma}, to obtain a conflict lemma for
$ \Theta $ from $ c $ and $ \Lambda. $ \end{lem} \begin{proof}
We prove the lemma by induction on the number of unassigned variables
in $ c. $
Let $ c_1 $ be the part of $ c $ to which {\bf (1)} applies,
and let $ c_2 = c \backslash c_1. $
Since each $ s \in c_1 $ is in conflict with $ \Theta, $
one can obtain a projection $ \mu_1 $ of $ c_1 $ by picking
from each $ s \in c_1 $ an assignment $v/x $ for which
$ v \Theta $ is defined and $ v \Theta \not = x. $
By construction, $ \mu_1 $ will be false in $ \Theta. $
If there are no unassigned variables in $ c, $ then
$ c_2 $ must be empty. This means that $ \mu_1 $ is a projection
of $ c $ and false in $ \Theta, $ so we are done.
Otherwise, select a $ v $ in $ c $ that is unassigned by
$ \Theta. $
Let $ V $ be set of values that are assigned to $ v $
by the substlets in $ c_2. $
Define $ \mu_2 = \{ v / V \}. $ Clearly, $ \mu_2 $ is a
projection of $ c_2, $ and $ \mu = \mu_1 \cup \mu_2 $ is a projection
of $ c. $
For each value $ x \in V, $ define $ \Theta_x = \Theta \cup \{ v/x \}. $
If there is no $ \lambda \in \Lambda, $
that is false in $ \Theta_x, $ then $ c, \ \Theta_{x}, \ \Lambda $
still satisfy the conditions of Lemma~\ref{Lem_unary_derive_recurse}.
Moreover, since $ \Theta_x $ contains an assignment to $ v, $
the number of unassigned variables in $ c $ has decreased by one.
This means that we can assume, by induction, that we can derive
a lemma $ \lambda_x $ that is false in $ \Theta_{x}. $
If $ \lambda_{x} $ is also a conflict lemma of $ \Theta, $
we have completed the proof.
Otherwise, we can assume that $ \lambda_x $ is added to $ \Lambda. $
At this point, $ \Lambda $ contains a conflict lemma
$ \lambda_x $ for every
$ \Theta \cup \{ v/x \} $ with $ x \in V. $
Let $ \lambda $ be the $ v $-resolvent of the projection
$ \mu $ constructed above, and the $ \lambda_x, $ i.e.
\[ \lambda =
\{ \ v / ( \ \mu(v) \cap \bigcap_{x \in V} \lambda_x(v) \ ) \ \} \cup
\{ \ v' / ( \ \mu(v') \cup \bigcup_{x \in V} \lambda_x(v') \ )
\ | \ v' \not = v \ \}. \]
In order to show that $ \lambda $ is false in $ \Theta, $
we have to show that for every variable $ v', $ for which
$ \lambda(v') \not = \emptyset, \ \ v' \Theta $ is defined,
and $ v' \Theta \not \in \lambda(v'). $
\begin{itemize}
\item
For $ v, $ we just show that $ \lambda(v) = \emptyset. $
We have $ \mu(v) = \mu_2(v), $ because $ \mu_1(v) = \emptyset. $
It follows from the fact that $ v $ is undefined in $ \Theta, $
and $ \mu_1 $ is false in $ \Theta. $
For each $ x \in \mu_2(v), $ we know that $ \lambda_{x} $ is
false in $ \Theta \cup \{ v/x \}, $ which implies that
$ x \not \in \lambda_x(v). $ This implies that $ x $ is not
in the intersection of all $ \lambda_x(v), $
which in turn implies $ \mu(v) $ and $ \bigcap_{x \in V} \lambda_x(v) $
have no elements in common.
\item
If $ v' \not = v $ and $ \lambda(v') \not = \emptyset, $
then either $ \lambda_x(v') \not = \emptyset, $ for an $ x \in V, $
or $ \mu(v') \not = \emptyset. $ In the first case, it follows
from the fact that $ \lambda_x $ is false in $ \Theta \cup \{ v/x \} $
and $ v' \not = v, $
that $ v' \Theta $ is defined. In the second case,
we know that $ \mu_2(v') $ only assigns to $ v, $ so that
$ \mu_1(v') \not = \emptyset. $ Since we know that $ \mu_1 $ is false
in $ \Theta, $ we know $ v' \Theta $ is defined.
At this point, we are certain that $ v' \Theta $ is defined,
so that we can start showing that
$ v' \Theta \not \in \mu(v') \cup \bigcup_{x \in V} \lambda_x(v'). $
If $ v' \Theta \in \mu(v'), $ then, because $ \mu_2 $
only assigns to $ v, $ we have $ v' \Theta \in \mu_1(v') $
This is impossible because $ \mu_1 $ is false in $ \Theta. $
We can also not have $ v' \Theta \in \lambda_x(v'), $
for any $ x \in V, $ because this would imply that
$ v' ( \Theta \cup \{ v/x \} ) \in \lambda_x(v'), $ which
contradicts the fact that $ \lambda_x $ is false in
$ \Theta \cup \{ v/x \}. $ \qedhere
\end{itemize} \end{proof}
\noindent At this point, it is straightforward to prove that Algorithm~\ref{Algo_unary} can always derive a conflict lemma. There are two points in Algorithm~\ref{Algo_unary} where the substitution is extended. We show for both points that it is possible to obtain a conflict lemma when the substitution is restored.
\begin{description} \item[FORW 1b]
The substitution $ \Theta $ is extended by the common
assignments in $ d''. $
Since the extension of $ \Theta $ had a conflict lemma,
we know that for each $ s \in d'', \ \ \Theta \cup \{ s \} $
has a conflict lemma.
It follows from Lemma~\ref{Lemma_weak_conflict} that for every
substlet $ s $ in $ d'' \backslash \{ c_i \}, $ either $ s$ is in conflict
with $ \Theta, $ or $ \Theta \cup \{ s \} $ implies $ \sigma, $
for a blocking $ \sigma \in \Sigma^{-}. $
From Lemma~\ref{Lem_not_alone}, we know that
there is no $ \sigma \in \Sigma^{-}, $ s.t. $ \Theta \models \sigma. $
It follows that we can apply Lemma~\ref{Lem_unary_derive} with
$ \Lambda = \{ \lambda \} $ to obtain a conflict lemma
for $ \Theta. $ \item[PICK]
Let $ c_i \Rightarrow d_i $ be the refinement that was selected
by PICK.
Let $ \Lambda $ be the set of conflict lemmas that were returned
by the recursive calls of {\bf findmatch}.
If there is a $ \lambda \in \Lambda $ that is false in $ \Theta, $
we can return $ \lambda. $
Otherwise, we know that no $ \lambda \in \Lambda $ is false
in $ \Theta. $
From Lemma~\ref{Lem_not_alone}, we know that
there is no $ \sigma \in \Sigma^{-}, $ s.t. $ \Theta \models \sigma. $
By Lemma~\ref{Lemma_weak_conflict}, every substlet
$ s \in ( c_i \backslash d_i ), $
is either in conflict with $ \Theta, $ or there exists a
$ \sigma \in \Sigma^{-}, $ s.t. $ \Theta \cup \{ s \} \models \sigma. $
This implies that we can apply Lemma~\ref{Lem_unary_derive} to obtain a
conflict lemma of $ \Theta. $ \end{description}
In an implementation of Algorithm~\ref{Algo_unary}, there is no need to follow the rules of Definition~\ref{Def_reso_proj_sigma} carefully, because the conflict lemma can be constructed immediately from the premisses of Lemma~\ref{Lem_unary_derive}.
In order to make Algorithm~\ref{Algo_unary} reuse conflict lemmas, one has to add before {\bf FORW~1}: If there is a $ \lambda \in \Lambda $ containing variable $v, $ s.t. $ \Theta $ makes $ \lambda $ false, then return $ \lambda. $
Integrating lemmas into the refining step of {\bf FORW}~1 seems difficult, because the notion of connection (Definition~\ref{Def_connected_variable}) must be extended to include `$ v $ and $ w $ occur together in a lemma $ \lambda \in \Lambda. $' Currently we don't know how to efficiently enumerate variables that are connected through a lemma.
\section{Matching Based on Local Consistency Checking} \label{Sect_IJCAR2016}
We will discuss the matching algorithm of \cite{deNivelle2016a}. Its performance turned out not competitive, so we will omit most of the details, in particular the completeness proofs for learning. The algorithm is based on the fact that local consistency checking rejects a large percentage of GCSPs without backtracking.
Local consistency checking is the following procedure: For every clause $ c = \{ s_1, \ldots, s_n \} \in \Sigma^{+}, $ check, for all sets of clauses $ C $ with size $ S \geq 1, $ if $ \{ s_i \} \cup C $ has a solution. If not, then remove $ s_i $ from $ c. $ Keep on doing this, until no further changes are possible or a clause has become empty. The procedure is described in detail in Section~\ref{Sect_local_consistency_checking}. Local consistency checking with small $ S $ rejects a large percentage of instances without backtracking. It therefore seemed reasonable to combine local consistency checking with backtracking in the following way: \begin{description} \item[FILTER]
Apply local consistency checking. If this results in an empty
clause, then backtrack to the last decision. If there are no decisions
left, then report failure. \item[DECIDE]
If every clause has become unit, then report a solution.
Otherwise, pick a non-unit clause, and replace it by a singleton
consisting of one of its substlets.
Continue at {\bf FILTER}.
If this results in an empty clause, then backtrack through the
remaining substlets of the clause. \end{description} The assumption was that local consistency checking could play the same role as unit propagation in DPLL, and that local consistency checking would be equally effective on the subproblems obtained during backtracking, as on the initial problem. This assumption turned out false. In \cite{deNivelle2016a}, the algorithm is described for $ S = 1, $ but we have implemented it for arbitrary $ S \geq 1. $
Note that a size of $ S $ means that $ \| C \| = S, $ so that $ \{ c_i \} \cup C $ has size $ S+1. $ Performance results are presented in Figure~\ref{Fig_tyebye_z_maslem} in Section~\ref{Sect_experiments}. It can be seen that $ S > 1 $ does not perform better than $ S = 1. $ It rarely creates less lemmas, and it usually costs more time.
The main observation to be made is that the algorithm is not close to being competitive against Algorithm~\ref{Algo_unary} with flat lemmas, or translation to SAT. In addition to that, it turned out rather unpleasant to implement, much harder than Algorithm~\ref{Algo_unary}. Especially $ S > 1 $ is difficult to handle, because the resolution rules for obtaining lemmas become rather complicated. This does not only apply to the implementation, but also to the theoretical description.
We define the lemmas that were used by the matching algorithm, and the reasoning rules that it uses. A clause can be viewed as a special form of lemma in which the substlets have the same domain.
\begin{defi}
\label{Def_full_lemma}
A \emph{lemma} is a finite set of substlets, possibly
with different domains.
If $ \lambda $ is a lemma, and $ \Theta $ a substitution,
then $ \Theta $ makes $ \lambda $ true if there
is a substlet $ ( \overline{v}/\overline{x} ) \in \lambda, $
s.t. $ \Theta $ makes $ \lambda $ true.
$ \Theta $ makes $ \lambda $ false if every
substlet $ ( \overline{v}/\overline{x} ) \in \lambda $
is in conflict with $ \Theta. $
We say that $ \lambda $ is \emph{valid} relative to
$ ( \Sigma^{+}, \Sigma^{-} ), $ if
$ \Theta $ is true in every solution $ \Theta $ of
$ ( \Sigma^{+}, \Sigma^{-} ). $
We call $ \lambda $ a \emph{conflict lemma} if $ \lambda $ is
false in the current $ \Theta $ and valid
$ ( \Sigma^{+}, \Sigma^{-} ). $ \end{defi} Learning was based on the following resolution rules:
\begin{defi}
\label{Def_full_conflict_resolution}
Let $ \lambda_1 $ and $ \lambda_2 $ be lemmas.
Let $ \mu_1 \subseteq \lambda_1, $
and let $ \mu_2 \subseteq \lambda_2. $
Assume that every $ s_1 \in \mu_1 $ is in conflict with
every $ s_2 \in \mu_2. $
Then $ ( \lambda_1 \backslash \mu_1 ) \cup
( \lambda_2 \backslash \mu_2 ) $
is a resolvent of $ \lambda_1 $ and $ \lambda_2. $ \end{defi}
One can resolve $ \lambda_1 = \{ \ (x,y)/(1,2), \ (x,y)/(1,1), \ (x,y)/(3,3) \ \} $ with $ \lambda_2 = \{ \ (y,z)/(1,2),$ $ \ (y,z)/(2,1) \ \} $ based on $ \mu_1 = \{ \ (x,y)/(1,2), \ (x,y)/(3,3) \ \}, $ and $ \mu_2 = \{ \ (y,z)/(1,2) \ \}. $ The resolvent is $ \{ \ (x,y)/(1,1), \ (y,z)/(2,1) \ \}. $
\begin{defi}
\label{Def_full_sigma_resolution}
Let $ \sigma \in \Sigma^{-} $ be a blocking.
Let $ c_1, \ldots, c_n \in \Sigma^{+} $ be a sequence
of clauses containing all variables of $ \sigma. $
For each $ c_i, $ let
$ \rho_i = \{ \ s \in c_i \ | \ s \mbox{ is in conflict with } \sigma \ \}. $
Then $ \rho_1 \cup \cdots \cup \rho_n $ is a \emph{$ \sigma $-resolvent}
of $ c_1, \ldots, c_n. $ \end{defi}
Using $ \lambda_1, \lambda_2 $ given above, and blocking $ (x,z)/(1,2), $ one can obtain the $ \sigma $-resolvent $\{ \ (x,y) / (3,3), \ (y,z)/(2,1) \ \}.$
It is easy to see that both conflict resolution and $ \sigma $-resolution are valid reasoning rules, which implies that every lemma that was derived by repeated application of resolution from the original clauses in $ \Sigma^{+}, $ is valid.
In \cite{deNivelle2016a}, it was shown that a matching algorithm using $ S = 1 $ can always obtain a conflict lemma using resolution and $ \sigma $-resolution. For $ S > 1, $ an additional rule, called \emph{product resolution}, is required. Results are listed in Figure~\ref{Fig_tyebye_z_maslem}.
After observing that Algorithm~\ref{Algo_unary} improves by a factor $ 500 $ when lemmas are flattened, we tried the same with the refining algorithm. Whenever a new lemma is derived, the assignments that do not contribute to conflicts are removed from the substlets. Different from Algorithm~\ref{Algo_unary}, this does not necessarily lead to a lemma consisting only of single-assignment substlets, but in most cases it does. Surprisingly, this has a strong, negative impact on the performance.
\section{Translation to SAT} \label{Sect_SAT_trans}
\noindent Translating an instance of the matching problem to SAT is easy, and modern SAT solvers have become very effective. As a consequence, translation to SAT should be attempted. In this section, we give two methods of translating GCSP into SAT. The translations are not complicated, and MiniSat \cite{Minisat2004} performs rather well on the results of the translations. Results are listed in the last two columns of Figure~\ref{Fig_unary_sat} and in Figure~\ref{Fig_unary_sat2} in Section~\ref{Sect_experiments}. The results suggest that translation to SAT has a performance that is comparable with Algorithm~\ref{Algo_unary}.
In our first translation only substlets are translated. We assign propositional variables to the substlets, specify that at least one substlet from each clause has to be selected, and list the conflicts between the substlets.
\begin{defi}
\label{Def_trans_general}
We assume a general mapping $ [ \ ] $ that transforms mathematical
objects into distinct propositional variables. \end{defi}
\begin{defi}
\label{Def_trans_SAT1}
Let $ ( \Sigma^{+}, \Sigma^{-} ) $ be GCSP.
The translation into propositional logic has form
$ ( A, P ), $
where $ A $ is a set of atoms, and $ P $ is a set of
clauses over $ A. $
Assume that the GCSP has form $ ( \Sigma^{+}, \Sigma^{-} ), $
assume that $ \Sigma^{+} $ contains $ n $ clauses,
and write $ \{ s_{i,1}, \ldots, s_{i,k_i} \} $ for the $i$-th clause
of $ \Sigma^{+}. $
The set of atoms is defined as
$ A = \{ [ s_{i,j} ] \ | \ 1 \leq i \leq n, \
1 \leq j \leq k_n \}. $
The clause set $ P $ is defined as follows:
\begin{enumerate}
\item
For every $ c_i = \{ s_{i,1}, \ldots, s_{i,k_i} \} \in \Sigma^{+} \
( 1 \leq i \leq n ), $
the propositional clause set $ P $ contains the propositional clause
$ \{ \ [s_{i,1}], \ldots, [s_{i,k_i}] \ \}, $
and for every $ j_1,j_2 \ ( 1 \leq j_1 < j_2 \leq k_i ) $ the
clause $ \{ \ \neg [ s_{i,j_1} ], \ \neg [ s_{i,j_2} ] \ \}. $
\item
For every pair of distinct clauses
$ c_{i_1}, c_{i_2} \in \Sigma^{+} $
that share a variable, for every substlet
$ s \in c_{i_1}, \ \ P $ contains the clause
\[ \{ \ \neg [s] \ \} \cup \{ \ [ s' ] \in c_{i_2} \ | \
s' \in c_{i_2}, \mbox{ and }
s' \mbox{ is not in conflict with } s \ \}. \]
\item
For every blocking $ \sigma \in \Sigma^{-}, $
we assume that there is a way of selecting a
most suitable subset $ C_{\sigma} $ of $ \Sigma^{+} $ that
contains all variables of $ \sigma. $
Then $ P $ contains the clause
\[ \{ \ [s] \ | \ \exists c \in C_{\sigma}, \mbox{ s.t. }
s \in c
\mbox{ and } s
\mbox{ is in conflict with } \sigma \ \}. \]
\end{enumerate} \end{defi} The first part specifies that exactly one substlet must be selected from each $ c \in \Sigma^{+}. $ The second part specifies that if one selects a substlet $ s $ from $ c_{i_1}, $ one has to select a substlet $ s' $ from $ c_{i_2} $ that is not in conflict with $ s. $ The third part of Definition~\ref{Def_trans_SAT1} can be viewed as an application of $ \sigma $-RESOLUTION (Definition~\ref{Def_full_sigma_resolution}).
The second translation differs from the first translation in the fact that it does not only translate substlets, but also variable assignments. In addition to the substlets, it assigns propositional variables to variable assignments $ v/x. $ It specifies the dependencies between substlets and variable assignments. Instead of relying on $ \sigma $-RESOLUTION, blockings can be specified directly in terms of the forbidden variable assignments.
\begin{defi}
\label{Def_trans_SAT2}
Let $ ( \Sigma^{+}, \Sigma^{-} ) $ be a GCSP.
Write $ \Sigma^{+} = \{ c_1, \ldots, c_n \}. $
Write each $ c_i $ in the form $ \{ s_{i,1}, \ldots, s_{i,k_i} \}. $
The translation to propositional logic has form
$ ( A, P ), $ where
$ A $ is the set of atoms used in the translation,
and $ P $ is the set of clauses.
The set of atoms $ A $ is defined as
\[ \{ \ [s_{i,j}] \ | \ 1 \leq i \leq n, \
1 \leq j \leq k_i \ \} \cup
\{ \ [v/x] \ | \
(v/x) \mbox{ occurs in a substlet in } \Sigma^{+} \ \}. \]
The set of propositional clauses $ P $ is obtained as follows:
\begin{enumerate}
\item
For every clause $ c_i, \ (1 \leq i \leq n), $
clause set $ P $ contains the clause
$ \{ \ [ s_{i,1} ], \ldots, [ s_{i,k_i} ] \ \}. $
\item
For every substlet
$ s_{i,j} $ with $ 1 \leq i \leq n, \ 1 \leq j \leq k_{i}, $
for every assignment $ v/x $ that occurs in $ s_{i,j}, $
clause set $ P $ contains the clause
$ \{ \ \neg [ s_{i,j} ], \ [ v/x ] \ \}. $
\item
For every variable $ v $ that occurs in $ \Sigma^{+}, $
for every two distinct values $ x_1, x_2, $ s.t.
$ v/x_1 $ and $ v/x_2$ occur somewhere in substlets in $ \Sigma^{+}, $
clause set $ P $ contains the clause
$ \{ \ \neg [ v/x_1], \ \neg [ v/x_2 ] \ \}. $
\item
For every blocking $ \sigma \in \Sigma^{-}, $
if every $ (v/x) \in \sigma $ occurs somewhere in a clause
in $ \Sigma^{+}, $ then
clause set $ P $ contains the clause
$ \{ \ \neg [ v/x ] \ | \ (v/x) \in \sigma \ \}. $
If some $ (v/x) \in \sigma $ does not occur in $ \Sigma^{+}, $ then
$ \sigma $ is impossible, and there is no need to generate a clause
for it.
\end{enumerate} \end{defi}
\noindent We show correctness of Definition~\ref{Def_trans_SAT2}. If $ ( \Sigma^{+}, \Sigma^{-} ) $ has a solution $ \Theta, $ one can define a satisfying interpretation $ I $ for $ (A,P) $ as follows: \begin{itemize} \item
For $ 1 \leq i \leq n, \ 1 \leq j \leq k_i, $
set $ I( \ [ s_{i,j} ] \ ) = {\bf t} $ iff $ \Theta \models s_{i,j}. $ \item
For every assignment $ v/x $ occurring in a substlet $ s $
occurring in a clause $ c_i, $ set
$ I( \ [v/x] \ ) = {\bf t} $ iff $ v \Theta = x. $ \end{itemize} It is easily checked that $ I $ makes all clauses in Definition~\ref{Def_trans_SAT2} true.
For the other direction, assume that $ (A,P) $ has a satisfying interpretation $ I. $
Define $ \Theta = \{ \ (v/x) \ | \ I( \ [v/x] \ ) = {\bf t} \ \}. $ By part~4, $ \Theta $ does not contain conflicting assignments. By part~1 and part~2, $ \Theta $ contains an assignment for every variable occurring in $ \Sigma^{+}. $ Because of part~3, $ \Theta $ does not imply a blocking $ \sigma \in \Sigma^{-}. $ By part~1 and part~2, every $ c_i \in \Sigma^{+} $ contains one substlet that agrees with $ \Theta. $
\noindent We end the section with an example of both translations:
\begin{exa}
\label{Example_SAT1_SAT2}
We will translate the following GCSP. As usual, $ \Sigma^{+} $
and $ \Sigma^{-} $ are separated by a horizontal bar.
\[
\begin{array}{l}
(X,Y) \ / \ ( 0,1 ) \ | \ ( 1,0 ) \\
(Y,Z) \ / \ ( 0,0 ) \ | \ ( 0,1 ) \ | \ ( 1,0 ) \\
\hline
(X,Z) \ / \ ( 0,0 ) \\
(X,Z) \ / \ ( 1,1 ) \\
\end{array}
\]
$ \Sigma^{+} $ alone has three solutions:
\[ \begin{array}{l}
\Theta_1 = \{ \ X := 0, \ Y := 1, \ Z := 0 \ \}, \\
\Theta_2 = \{ \ X := 1, \ Y := 0, \ Z := 0 \ \}, \\
\Theta_3 = \{ \ X := 1, \ Y := 0, \ Z := 1 \ \}. \\
\end{array}
\]
The first solution is blocked by $ (X,Z)/(0,0), $ the
third solution is blocked by $ (X,Z)/(1,1), $ so that
only $ \Theta_2 $ is a solution of the complete GCSP.
Assume that
\[ [ (X,Y)/(0,1) ] = 1, \ \
[ (X,Y)/(1,0) ] = 2, \ \
[ (Y,Z)/(0,0) ] = 3, \ \
[ (Y,Z)/(0,1) ] = 4, \ \
[ (Y,Z)/(1,0) ] = 5. \]
Definition~\ref{Def_trans_SAT1} constructs the following translation:
\[
{\rm Part}~1:
\left (
\begin{array}{ccc}
1 & 2 \\
3 & 4 & 5 \\
-1 & -2 \\
-3 & -4 \\
-3 & -5 \\
-4 & -5 \\
\end{array}
\right )
\ \ {\rm Part}~2:
\left (
\begin{array}{ccc}
-1 & 5 \\
-2 & 3 & 4 \\
-3 & 2 \\
-4 & 2 \\
-5 & 1 \\
\end{array}
\right )
\ \
{\rm Part}~3:
\left (
\begin{array}{ccc}
1 & 4 \\
2 & 3 & 5 \\
\end{array}
\right )
\]
The only satisfying interpretation is $ \{ -1, 2, 3, -4, -5 \}, $
which corresponds to $ \Theta_2. $
In order to apply the second translation, assume that
\[ [X/0] = 6, \ \
[X/1] = 7, \ \
[Y/0] = 8, \ \
[Y/1] = 9, \ \
[Z/0] = 10, \ \
[Z/1] = 11. \]
The second translation constructs
\[
{\rm Part}~1:
\left (
\begin{array}{ccc}
1 & 2 \\
3 & 4 & 5 \\
\end{array}
\right )
\ \ {\rm Part}~2:
\left (
\begin{array}{ccc}
-1 & 6 \\
-1 & 9 \\
-2 & 7 \\
-2 & 8 \\
-3 & 8 \\
-3 & 10 \\
-4 & 8 \\
-4 & 11 \\
-5 & 9 \\
-5 & 10 \\
\end{array}
\right )
\ \
{\rm Part}~3:
\left (
\begin{array}{ccc}
-6 & -7 \\
-8 & -9 \\
-10 & -11 \\
\end{array}
\right )
\ \ {\rm Part}~4:
\left (
\begin{array}{ccc}
-6 & -10 \\
-7 & -11 \\
\end{array}
\right )
\]
Its only satisfying interpretation is
$ \{ -1, 2, 3, -4, -5, -6, 7, 8, -9, 10, -11 \}, $ which
again corresponds to $ \Theta_2. $ \end{exa}
\section{An Input Format for GCSP} \label{Sect_input_format}
\noindent Since we are claiming that GCSPs are fundamental enough to study on their own, and may have applications outside of geometric resolution, we made our implementation publicly available (\cite{deNivelle2018a}). In this section, we define the input format for GCSP, which is used by our implementation. The format is similar to the DIMACS format for satisfiability (\cite{DimacsSAT1993}). Similar to the DIMACS format, variables and constants are represented by integers. Since GCSP has no polarity (there is no negation), all integers are non-negative.
\begin{defi}
\label{Def_DIMACS}
We define a representation for GCSP.
Input is represented in plain ASCII. The format never distinguishes
between upper and lower case.
\begin{itemize}
\item
Input starts with whitespace, possibly mixed with
comment lines.
A comment line is a line whose first non-whitespace character is
a \verb+`c'+ or a \verb+`C'+. The initial comment lines are ignored.
\item
After that comes a line of form
\begin{verbatim}
p gcsp nrvars nrconsts nrclauses nrblockings \end{verbatim}
\verb+nrvars+ is the number of variables in the problem,
\verb+nrconsts+ is the number of constants in the problem.
Both need not be exact, but must be upperbounds. More
precisely, both variables and constants are represented by
non-negative integers, and \verb+nrvars,nrconsts+ must be bigger
than any variable or constant that appears in the problem.
\verb+nrclauses+ must be the exact number of clauses,
and \verb+nrblockings+ must be the exact number of blockings.
\item
A clause has form \verb+V var1 ... varV S subst1 ... substS+.
Here \verb+V+ is the exact number of variables in the clause,
and \verb+var1 ... varV+ are the variables, represented
by non-negative integers. Each variable must be less than
\verb+nrvars+.
\verb+S+ is the exact number of substlets in the clause.
Each substlet is represented by a sequence of non-negative
integers of length \verb+V+, that specifies the values
assigned to the variables, in the same order as the variables.
Each value must be less than \verb+nrconsts+.
There must be exactly \verb+nrclauses+ clauses.
\item
Blockings are represented in the same way as clauses.
Although in Definition~\ref{Def_gcsp}, blockings are single substlets,
it is convenient to merge blockings with identical domain
into clauses, so that they can be represented more
compactly.
There must be exactly \verb+nrblockings+ (merged) blockings.
Note that it is not obligatory to merge blockings with
same domain.
\item
Everything after the blockings is ignored, so there is room
for more comments.
\item
Solutions are presented in the format
\begin{verbatim}
A V1 C1 ... VA CA \end{verbatim}
Here \verb+A+ is the number of assignments in the substitution,
and each \verb+Vi+ $ \Rightarrow $ \verb+Ci+ is an assignment.
The assigments can be listed in arbitrary order.
\end{itemize} \end{defi}
\begin{exa}
We represent the GCSP of Example~\ref{Example_SAT1_SAT2}.
There are three variables $ X,Y,Z, $ which we will
represent by $ 0,1,2. $ This means that $ 3 $ is an upperbound.
There are two constants $ 0,1, $ so that $ 2 $ is an upperbound.
This is a representation:
\begin{verbatim}
c we did not merge the blockings
p gcsp 3 2 2 2
2 0 1 2 0 1 1 0 2 1 2 3 0 0 0 1 1 0
2 0 2 1 0 0 2 0 2 1 1 1
\end{verbatim}
Since the two blockings in Example~\ref{Example_SAT1_SAT2} have the same
domain, the GCSP can be alternatively represented as follows:
\begin{verbatim} c this time we merged the two blockings c there is no difference in meaning
P GCSP 3 2 2 1
2 0 1 2 0 1 1 0 2 1 2 3 0 0 0 1 1 0
2 0 2 2 0 0 1 1
\end{verbatim}
\noindent
The solution $ \Theta_2 $ can be output as
\verb+ 3 0 1 1 0 2 0+.
Since the order is arbitrary, it can also be
output as \verb+ 3 1 0 2 0 0 1+. \end{exa}
\section{Experiments} \label{Sect_experiments}
\noindent We present measurements on two benchmark sets. Both sets were obtained by running {\bf Geo} on a few input problems, and collecting hard matching instances. The first set consists of problems that took more than an hour to solve with a naive matching algorithm. This set was used in Figures~\ref{Fig_unary_sat} and \ref{Fig_tyebye_z_maslem}.
\begin{figure}
\caption{Comparing Direct Matching with SAT Translations}
\label{Fig_unary_sat}
\end{figure}
\noindent Entries in Figure~\ref{Fig_unary_sat} have form $ t(\lambda), $ where $ t $ is the time used in seconds, and $ \lambda $ the number of lemmas generated. For the 3d and 4th column, the times are the CPU-times reported by MiniSat (Version~2.0~beta) (\cite{Minisat2004}). Since MiniSat is not integrated into {\bf Geo}, it is difficult to measure the total time (conversion+solving). For hard problems, the conversion times are probably negligible, but for trivial problems, they may be significant (in the same order of magnitude as the solving times) because translation is quadratic. Due to the way the benchmarks were collected, they contain no trivial problems. It can be seen from Figure~\ref{Fig_unary_sat} that translation to propositional SAT is comparable to Algorithm~\ref{Algo_unary} with flat lemmas. We will dicuss this more in the context of Figure~\ref{Fig_unary_sat2}. We were not sure how to determine the number of lemmas generated during a run of MiniSat, due to the fact that it performs restarts. Currently, we simply added the numbers reported by the different restarts. Since MiniSat probably reuses lemmas between different restarts, this means that the indicated numbers are likely too high. It can be seen that nearly always, Definition~\ref{Def_trans_SAT2} performs better than Definition~\ref{Def_trans_SAT1}.
\begin{figure}
\caption{Results for Matching Using Local Consistency}
\label{Fig_tyebye_z_maslem}
\end{figure}
Figure~\ref{Fig_tyebye_z_maslem} shows results for the refining algorithm of \cite{deNivelle2016a}, discussed in Section~\ref{Sect_IJCAR2016}. It can be seen that using $ S > 1 $ is hardly worth the effort, and that the refining algorithm performs somewhat worse than Algorithm~\ref{Algo_unary} with learning of unrestricted lemmas. Since flattening of lemmas improves the performance of Algorithm~\ref{Algo_unary} dramatically, we tried the same with the refining algorithm. Unfortunately, the last column of Figure~\ref{Fig_tyebye_z_maslem} shows that flattening has a big, negative impact on the refining algorithm. That means that the refining algorithm can be ruled out as a candidate for being optimal.
\begin{figure}
\caption{Algorithm~\ref{Algo_unary} (flat lemmas)
against {\rm Def}~\ref{Def_trans_SAT2}}
\label{Fig_unary_sat2}
\end{figure}
\noindent Figure~\ref{Fig_unary_sat2} presents another benchmark test that was obtained by running {\bf Geo} on several input problems using Algorithm~\ref{Algo_unary} with flat lemmas. We started by setting a short initial time $ t = 10^{-4}. $ Whenever a matching took more than $ t $ seconds to solve, we added it to the benchmark set, and doubled $ t. $ From several runs, we kept the last three instances generated in this way. Figure~\ref{Fig_unary_sat2} shows that most of the problems obtained in this way, are also hard for MiniSat. We believe Figure~\ref{Fig_unary_sat2} shows the potential of direct matching algorithms, but there are several caveats: Runs of {\bf geo} do not generate really hard matching instances, all instances are solved within seconds, most much faster. The way of collecting problems that are hard for Algorithm~\ref{Algo_unary} puts it at a disadvantage. For example, there may be problems that are hard for MiniSat, which will not enter the benchmark set. On the other hand, MiniSat is not state of the art anymore, and modern SAT solvers probably will perform better. We carefully conclude that there is a chance that in the long term, at least for some applications, our approach of directly implementing matching, may be the optimal approach.
\section{Finding Optimal Matchings} \label{Sect_optimal}
\noindent In this section we address the problem of finding optimal matchings. For the effectiveness of geometric resolution, it is important that a minimal matching is returned, in case more than one exists. A minimal matching is a matching that uses the smallest possible set of assumptions. In terminology of DPLL, assumptions represent decision levels. The assumptions contributing to a conflict represent choice options, which will be replaced by other options during backtracking. In addition to being as few as possible, assumptions at a lower decision level should always be preferred over assumptions at a higher decision level. The reason for this is the fact that in other branches of the search tree, there is a risk that more assumptions will be used, and when assumptions are at a lower level, there is less room for this.
\begin{defi}
Let $ I $ be an interpretation. A weight function $ \alpha $
is a function that assigns finite subsets of natural numbers to
the atoms of $ I. $
Let $ A $ be a geometric literal. Let $ \Theta $ be a substitution
such that $ A \Theta $ is in conflict with $ I. $
Referring to definition~\ref{Def_conflict_truth}, we
define $ \alpha( \ p_{\lambda}( x_1, \ldots, x_n) \Theta, I ) =
\alpha( \ p_{\mu}( x_1 \Theta, \ldots, x_n \Theta ) \ ), \ \
\alpha( \ (x_1 \approx x_2) \Theta, I ) = \{ \}, $ and
$ \alpha( \ ( \#_{\bf f} x ) \Theta, I ) =
\alpha( \ ( \#_{\bf t} x \Theta ) \ ). $ \end{defi}
\begin{defi}
\label{Def_matching_weight}
Let $ I $ and
$ \phi = \ A_1, \ldots, A_p \ | \ B_1, \ldots, B_q $ together form
an instance of the matching problem (Definition~\ref{Def_matching}).
Assume that $ \Theta $ is a solution.
The \emph{weight of} $ \Theta, $ for which we write
$ \alpha( I, \phi, \Theta ), $ is defined as
\[ \bigcup \left \{
\begin{array}{l}
\{ \ \alpha( A_i \Theta, I ) \ | \ 1 \leq i \leq p \} \\
\{ \ \alpha( C, I ) \ | \
1 \leq j \leq q, \ \ C \in E( B_j, \Theta ), \mbox{ and }
C \mbox{ conflicts } I \ \} \\
\end{array}
\right .
\]
Solving optimal matching means:
First establish if $ (I, \phi ) $ has a solution.
If it has, then find a solution $ \Theta $ for which
$ \alpha(I, \phi, \Theta ) $ is multiset minimal. \end{defi} One could try to impose further selection criteria that are harder to explain and whose advantage is less evident.
Solving the minimal matching problem is non-trivial, because the number of possible solutions can be very large. The straightforward solution is to use some efficient algorithm (e.g. the one in this paper) that enumerates all solutions, and keeps the best solution. Unfortunately, this approach is completely impractical because some instances have a very high number of solutions. One frequently encounters instances with $ > 10^{9} $ solutions.
In order to find a minimal solution without enumerating all solutions, one can use any algorithm that stops on the first solution in the following way: The first call is used to find out whether a solution exists. If not, then we are done. Otherwise, the algorithm is called again with its input restricted in such a way that it has to find a better solution than the previous. One can continue doing this, until all possibilities to improve the solution have been exhausted. It can be shown that the number of calls needed to obtain an optimal solution is linear in the size of the assumption set of solution. In this way, it can be avoided that all solutions have to be enumerated.
\begin{defi}
\label{Def_restricted_translation}
Let $ I $ be an interpretation that is equipped
with a weight function $ \alpha. $
Let $ \phi = \ A_1, \ldots, A_p \ | \ B_1, \ldots, B_q $ be a
geometric formula.
Let $ \alpha $ be a fixed set of natural numbers.
We define the $ \alpha $-restricted translation
$ ( \Sigma^{+}, \Sigma^{-} ) $ of
$ ( I, \phi ) $ as follows:
\begin{itemize}
\item
For every $ A_i, $ let $ \overline{v}_i $ be the variables
of $ A_i. $
Then $ \Sigma^{+} $ contains the clause
\[ \{ \overline{v}_i / \overline{v}_i \Theta \ | \
A_i \Theta \mbox{ is in conflict with } I \mbox{ and }
\alpha( A_i \Theta, I ) \subseteq \alpha \}. \]
\item
For each $ B_j, $ let $ \overline{w}_j $ denote the variables of
$ B_j. $ For every $ \Theta $ that makes $ B_j \Theta $ true
in $ I, $ \ \
$ \Sigma^{-} $ contains the substlet
$ \overline{w}_j / ( \overline{w}_j \Theta ). $
In addition, if there exists a $ C \in E( B_j, \Theta ) $
that is in conflict with $ I $ and for which
$ \alpha( C, \Theta ) \not \subseteq \alpha, $ then
$ \Sigma^{-} $ contains the substlet
$ \overline{w}_j / ( \overline{w}_j \Theta ). $
\end{itemize}
\end{defi}
\noindent The $ \alpha $-restricted translation ensures that only conflicts involving atoms $ C $ with $ \alpha(C) \subseteq \alpha $ are considered, and (independently of $ \alpha $), that no $ B_j $ is made true. The translation of Definition~\ref{Def_trans_gcsp} can be viewed as a special case of $ \alpha $-restricted translation with $ \alpha = \mathbb{N}. $
\begin{thm}
\label{Thm_alpha_restricted}
Let $ ( \Sigma^{+}, \Sigma^{-} ) $ be obtained by $ \alpha $-restricted
translation of $ (I, \phi). $
For every substitution $ \Theta, $ \ \
$ \Theta $ is a solution of $ ( \Sigma^{+}, \Sigma^{-} ) $
iff
$ \Theta $ is a solution of $ ( I, \phi ), $ and it
has $ \alpha( I, \phi, \Theta ) \subseteq \alpha. $ \end{thm}
\noindent Using $ \alpha $-restricted translation, we can define the {\bf optimal} matching algorithm:
\begin{algo}
\label{Alg_find_optimal_match}
Let $ {\bf solve}( \Sigma^{+}, \Sigma^{-} ) $ be a function that returns
some solution of $ ( \Sigma^{+}, \Sigma^{-}) $ if it has a solution,
and $ \bot $ otherwise.
We define the algorithm
$ {\bf optimal}( \ I, \phi \ ) $
that returns an optimal solution of $ (I,\phi) $ if one exists and
$ \bot $ otherwise.
\begin{enumerate}
\item
Let $ ( \Sigma^{+}, \Sigma^{-} ) $ be the GCSP obtained by
the translation of Definition~\ref{Def_trans_gcsp}.
If $ \Sigma^{+} $ contains an empty clause, then return $ \bot. $
If $ \Sigma^{-} $ contains a propositional blocking, then return
$ \bot. $
Otherwise, remove unit blockings from $ ( \Sigma^{+}, \Sigma^{-} ). $
If this results in $ \Sigma^{+} $ containing an empty clause, then
return $ \bot. $
\item
Let $ \Theta = {\bf solve}( \Sigma^{+}, \Sigma^{-} ). $
If $ \Theta = \bot, $ then {\bf return} $ \bot. $
\item
Let $ \alpha = \alpha(I, \phi, \Theta), $ and
let $ k := \sup( \alpha ). $
\item
As long as $ k \not = 0, $ do the following:
\begin{itemize}
\item
Set $ k = k - 1. $
If $ k \in \alpha, $ then do
\begin{itemize}
\item
Let $ \alpha' =
( \alpha \backslash \{ k \} ) \cup
\{ 0, 1, 2, \ldots, k - 1 \}. $
\item
Let $ ( \Sigma^{+},
\Sigma^{-} ) $ be the
$ \alpha' $-restricted translation
of $ (I, \phi). $
\item
If $ \Sigma^{+} $ contains an empty clause
or $ \Sigma^{-} $ contains a propositional blocking,
then skip the rest of the loop.
Otherwise, remove the unit blockings from
$ ( \Sigma^{+}, \Sigma^{-} ). $
If this results in $ \Sigma^{+} $ containing the empty clause,
then skip the rest of the loop.
\item
Let $ \Theta' = {\bf solve}( \Sigma^{+}, \Sigma^{-} ). $
If $ \Theta' \not = \bot, $ then set $ \Theta = \Theta' $
and $ \alpha = \alpha( I, \phi, \Theta ). $
\end{itemize}
\end{itemize}
\item
Now $ \Theta $ is an optimal solution,
so we can {\bf return} $ \Theta. $
\end{enumerate} \end{algo} Algorithm $ {\bf optimal} $ first solves $ (I,\phi) $ without restriction. If this results in a solution $ \Theta, $ it checks for each $ k \in \alpha( I, \phi, \Theta ) $ if $ k $ can be removed. The invariant of the main loop is: There exists no $ k' \geq k $ that occurs in $ \alpha( I, \phi, \Theta ), $ and no $ \Theta' $ that is a solution of $ (I, \phi) $ with $ k' \not \in \alpha( I, \phi, \Theta' ). $ In addition, the invariant $ \alpha = \alpha( I,\phi,\Theta ) $ is maintained.
\begin{exa}
Assume that in example~\ref{Ex_matchings}, the
atoms have weights as follows:
\[ \begin{array}{ll}
\alpha( \ P_{\bf t}( c_0, c_0 ) \ ) = \{ 1 \}, \ \
\alpha( \ P_{\bf e}( c_0, c_1 ) \ ) = \{ 2 \}, \ \
\alpha( \ P_{\bf t}( c_1, c_1 ) \ ) = \{ 3 \}, \\
\alpha( \ P_{\bf e}( c_1, c_2 ) \ ) = \{ 4 \}, \ \
\alpha( \ Q_{\bf t}( c_2, c_0 ) \ ) = \{ 5 \}. \\
\end{array}
\]
We have $ \alpha( I, \phi_1, \Theta_1 ) = \{ 1 \}, $ \ \
$ \alpha( I, \phi_1, \Theta_2 ) = \{ 1,2 \}, $
and $ \alpha( I, \phi_1, \Theta_3 ) = \{ 2,3 \}. $
If $ \Theta_3 $ is the first solution generated,
$ {\bf solve} $ will construct the
$ \{ 1,2 \} $-restricted translation of $ (I,\phi_1), $
which equals
\[ \begin{array}{l}
(X,Y) \ / \ ( c_0,c_0 ) \ | \ ( c_0,c_1 ) \\
(Y,Z) \ / \ ( c_0,c_0 ) \ | \ ( c_0,c_1 ) \\
\hline
(X,Z) \ / \ ( c_0,c_2 ) \\
\end{array}
\]
If the next solution found is $ \Theta_2, $ then
$ {\bf solve} $ will construct the $ \{ 1 \} $-restricted translation
\[ \begin{array}{l}
(X,Y) \ / \ ( c_0,c_0 ) \\
(Y,Z) \ / \ ( c_0,c_0 ) \\
\hline
(X,Z) \ / \ ( c_0,c_2 ) \\
\end{array}
\]
whose only solution is $ \Theta_1. $ \end{exa}
\section{Filtering by Local Consistency Checking} \label{Sect_local_consistency_checking}
\noindent Filtering is any procedure that simplifies or possibly rejects a GCSP before the main algorithm is called. In {\bf Geo}, we have used filtering based on local consistency checking. In earlier versions, this was effective because very often, filtering rejects a GCSP without calling the main algorithm. Since the algorithms that we present in this paper, are much more efficient, this is not certain anymore. We still present the local consistency checking procedure, because it is easy to implement using refinement stacks, and it may be still an effective tool for filtering out easy instances.
Local consistency checking (see \cite{Dechter2003,MaloSebag2004,SchefHerbWys94}) is a pre-check that comes in many variations. Local consistency checking is the following procedure: For every clause $ c = \{ s_1, \ldots, s_n \} \in \Sigma^{+}, $ check, for all sets of clauses $ C $ with size $ S \geq 1, $ if $ \{ s_i \} \cup C $ has a solution. If not, then remove $ s_i $ from $ c. $ Keep on doing this, until no further changes are possible or a clause has become empty. Local consistency checking rejects a large percentage of GCSP instances a priori, and usually decreases the size of the clauses involved by a factor two or three.
In \cite{Dechter2003} (Chapter~3), local consistency checking is defined using subsets of variables (instead of clauses). Using subsets of two variables is called \emph{arc consistency checking}, while considering subsets of three variables is called \emph{path consistency checking}. In general, using bigger subsets is a more effective precheck, but also more costly because it gets closer to the original problem.
As discussed in Section~\ref{Sect_IJCAR2016}, we had assumed in \cite{deNivelle2016a} that filtering is so effective, that one can base the complete search algorithm on it. Although this is possible in theory, the resulting algorithm turned out not competitive.
Since the local consistency checks the substlets in a single clause $ c $ against sets of clauses $ C \subseteq \Sigma^{+}, $
we define the size $ S $ of a local consistency check as $ S = \| C \|. $ When performing a local consistency check up to size $ S, $
one has to generate subsets up to size $ S+1, $ and generate their solutions. If $ \| \Sigma^{+} \| = n, $ the total number of such subsets equals $ \left ( \begin{array}{l} n \\ S+1 \\ \end{array} \right ), $ which grows very quickly for realistic $ n. $ The problem can be decreased by not generating all subsets, but only generate subsets whose clauses share variables, or have variables that co-occur in a blocking. \begin{defi}
\label{Def_related}
Let $ c,c' $ be clauses. We write $ c \sim c' $ if either
$ c $ and $ c' $ share a variable, or there exist connected
(Definition~\ref{Def_connected_variable}) variables $ v $ and $ v', $
s.t. $ v $ occurs in $ c $ and $ v' $ occurs in~$c'. $ \end{defi}
\noindent It is sufficient to generate subsets that are connected, because consideration of subsets that are not connected will not lead to the removal of more substlets. We always assume that solutions are non-redundant, i.e. do not contain irrelevant assignments.
\begin{lem}
\label{Lemma_disconnected}
Let $ ( \Sigma^{+}, \Sigma^{-} ) $ be a GCSP.
Let $ C \subseteq \Sigma^{+}. $ If $ C $ can be written as
$ C_1 \cup C_2, $ s.t. there exist no $ c_1 \in C_1 $
and no $ c_2 \in C_2 $ with $ c_1 \sim c_2, $ then
for every two substitutions $ \Theta_1, \Theta_2, $ s.t.
$ \Theta_1 $ is a solution of $ ( C_1, \Sigma^{-} ) $ and
$ \Theta_2 $ is a solution of $ ( C_2, \Sigma^{-} ), $
\ \ $ \Theta_1 \cup \Theta_2 $ is a solution of
$ ( C_1 \cup C_2, \Sigma^{-} ). $ \end{lem} Lemma~\ref{Lemma_disconnected} guarantees that it is not needed to attempt to remove substlets from clauses in $ C_1 \cup C_2, $ after $ C_1 $ and $ C_2 $ have been checked. If some substlet $ s $ in $ C_1 $ occur some solution of $ C_1, $ and $ C_2 $ has a solution, then $ s $ will occur in the combined solution.
We will now show that instead of ignoring disconnected subsets, one can also ignore subsets that are connected only through a single clause:
\begin{lem}
\label{Lemma_pivot}
Let $ ( \Sigma^{+}, \Sigma^{-} ) $ be a GCSP.
Assume that $ C_1, C_2 \subseteq \Sigma^{+}, $ and
$ c \in \Sigma^{+}. $
Assume that for every pair of variables
$ v_1 $ occurring in a clause of $ C_1, $ and
$ v_2 $ occurring in a clause of $ C_2, $ if
either $ v_1 = v_2 $ or $ v_1 $ and $ v_2 $ are connected,
then $ v_1, v_2 $ occur in $ c. $
Then the following holds:
If $ \Theta_1 $ is a solution of
$ ( C_1 \cup \{ c \}, \Sigma^{-} ) $
and $ \Theta_2 $ is a solution of
$ ( C_2 \cup \{ c \}, \Sigma^{-} ), $
s.t. $ \Theta_1, \Theta_2 $ agree on the variables occurring in $ c, $
then $ \Theta_1 \cup \Theta_2 $ is a solution
of $ ( C_1 \cup C_2 \cup \{ c \}, \Sigma^{-} ). $ \end{lem} \begin{proof}
Assume that $ \Theta_1, \Theta_2, C_1,C_2,c $ fulfil the
conditions of the lemma.
By non-redundancy, $ \Theta_1 $ does not contain assignments to
variables not occurring in $ c $ or $ C_1. $
Similarly, $ \Theta_2 $ does not contain assignments to
variables not occurring in $ c $ or $ C_2. $
If $ \Theta_1, \Theta_2 $ share a variable $ v, $ then
this variable must occur in $ c, $
which implies that $ v \Theta_1 = v \Theta_2. $
As a consequence, $ \Theta_1 $ and $ \Theta_2 $ can be merged into
a single substitution $ \Theta = \Theta_1 \cup \Theta_2, $
which has $ \Theta \models C_1 \cup C_2 \cup \{ c \}. $
If there would be a blocking $ \sigma \in \Sigma^{-}, $ s.t.
$ \Theta \models \sigma, $ then we still have
$ \Theta_1 \not \models \sigma $ and $ \Theta_2 \not \models \sigma. $
This implies that there are variables $ v_1 $ in
$ C_1 \backslash \{ c \} $ and
$ v_2 $ occurring in $ C_1 \backslash \{ c \}, $ which
occur together in $ \sigma. $ But this
contradicts the fact that $ v_1 $ and $ v_2 $ cannot be
connected. \end{proof}
As above, if some substlet $ s \in C_1 $ is used in a solution of $ C_1 \cup \{ c \}, $ and $ \{ c \} \cup C_2 $ has a solution, then the solutions can be combined into a single solution that uses $ s. $ If some substlet $ s $ of $ c $ occurs in a solution of $ C_1 \cup \{ c \} $ and in a solution of $ \{ c \} \cup C_2, $ then the solutions can be combined into a single solution that still uses $ s. $
This implies that, if one uses a local consistency checker that gives preference to small subsets, one can ignore subsets that do not contain 'cycles'. If there exist $ c_1,c_2,c_3 \in C, $ s.t. $ c_1,c_3 $ are not connected, and every path from $ c_1 $ to $ c_3 $ has to pass through $ c_2, $ then $ C $ can be ignored. This gives rise to the following definition:
\begin{defi}
\label{Def_circle}
Let $ ( c_1, \ldots, c_{S+1} ) $ with $ S \geq 1 $ be a sequence
of clauses. We call $ ( c_1, \ldots, c_{S+1} ) $ \emph{a circle} if
for every $ i \ ( i \leq S ), $ we have $ c_i \sim c_{i+1}, $
and in addition we have $ c_{S+1} \sim c_1. $
If $ \overline{C} $ is a refinement stack, we call a sequence
of indices $ ( i_1, \ldots, i_{S+1} ) $ \emph{a circle} if
each $ \alpha_{i_j}( \overline{C} ) $ is true, and
$ ( d_{i_1}, \ldots, d_{i_{S+1}} ) $ is a circle. \end{defi}
The local consistency checker checks only circles. Generation of circles in $ \Sigma^{+} $ is easier to implement than generation of all connected subsets, especially if one wants to avoid generating the same subset in different ways. In addition, it is more efficient because there are less circles than connected subsets. The discussion above suggests that generating circles is sufficient to obtain a complete check. We have believed for some time that this is true in general, but we will show below that it is false.
\begin{algo}
\label{Algo_local_consistency_checking}
Let $ S \geq 1 $ be a natural number.
Let $ \Theta $ be a substitution.
Let $ \overline{C} $ be a refinement stack.
A call to $ {\bf local}( s, \Theta,
( k_1, \ldots, k_{S+1} ), \overline{C} ) $
constructs a refinement of $ \overline{C} $ by removing the substlets
that do not occur in any solution of the subset of $ \Sigma^{+} $
of size $ S + 1. $
It returns $ \bot $ if it establishes that $ \Theta $ cannot
be extended into a solution of $ \overline{C}. $
Initially $ s = k_1 = \cdots = k_{S+1} = 1. $
\begin{description}
\item[SUBST]
As long as $ s \leq \| \Theta \|, $ let $ v/x $ be the
$ s $-th assignment in $ \Theta. $
\begin{enumerate}
\item
For every blocking $ \sigma \in \Sigma^{-} $ involving
$ v, $ check if
$ \Theta \models \sigma. $ If yes, then return $ \bot. $
\item
For every $ ( c_i \Rightarrow d_i ) \in \overline{C} $ which
has $ \alpha_i( \overline{C} ) $ true and which contains
$ v, $ let $ d' $ be the set of substlets in $ d_i $ that
are consistent with $ \Theta. $
If $ d' = \emptyset, $ then return $ \bot. $
Otherwise, if $ \emptyset \subset d' \subset d, $ append
$ ( c_i \Rightarrow d' ) $ to $ \overline{C}. $
\end{enumerate}
\item[CLAUSES1]
As long as $ k_1 < \| \overline{C} \| $ do the following:
\begin{enumerate}
\item
If $ \alpha_{k_1}( \overline{C} ) $ is true, and
the $ k_1 $-refinement $ ( c_{k_1} \Rightarrow d_{k_1} ) $
contains a variable $ v,$
s.t. all substlets $ (\overline{v}/\overline{x}) \in d_{k_1} $
agree on the assignment to $ v, $ then let $ x $ be the agreed
value. Append $ v/x $ to $ \Theta. $
\item
Set $ k_1 := k_1 + 1. $
\end{enumerate}
If $ s \leq \| \Theta \|, $ then restart at {\bf SUBST}.
(This means that $ \Theta $ was extended in the previous step.)
\item[CLAUSESN]
As long as there is an $ i $ with $ 2 \leq i \leq S+1, $
s.t. $ k_i \leq \| \overline{C} \|, $
pick the smallest such $ i. $
If $ \alpha_{k_i}( \overline{C} ) $ holds, then
\begin{enumerate}
\item
Enumerate all circles $ ( \lambda_1, \ldots, \lambda_{i} ) $
of size $ i $ starting at $ \lambda_1 = k_i. $
For each such circle $ ( \lambda_1, \ldots, \lambda_{i} ), $
let $ I = \{ d_{\lambda_1}, \ldots, d_{\lambda_{i}} \}. $
Call $ {\bf refine}(I, \Theta, \overline{C} ). $
If the result is $ \bot, $ then return $ \bot. $
If after the call, we have
$ \| \overline{C} \| > k_1, $ then restart at
{\bf CLAUSES1}.
\end{enumerate}
\end{description}
\noindent
Let $ \overline{C} $ be a refinement stack. Let
$ k = \| \overline{C} \|. $
Let $ I $ be a subset of $ \{ 1, \ldots, k \}, $ s.t.
for every $ i \in I, \ \ \alpha_i( \overline{C} ) $ holds.
Algorithm
$ {\bf refine}( I, \Theta, \overline{C} ) $ is defined as follows:
\begin{enumerate}
\item
Initialize a map $ U $ with domain $ I $
by setting $ U(i) = \emptyset, $ for each
$ i \in I. $
Eventually, $ U $ will map
each $ i \in I $ to the set of substlets in $ d_i, $ that can occur
in a solution $ \Theta' $ of
$ \{ d_i \ | \ i \in I \} $ extending $ \Theta. $
\item
Enumerate all maps $ S $ with domain $ I $
that map each $ i \in I $ to
a substlet $ S(i) $ in $ d_i, $ and that have the following properties:
No $ S(i) $ conflicts $ \Theta, $
no $ S(i), S(i') $ are in conflict with each other.
$ \Theta \cup \{ S(i) \ | \ i \in I \} $ does not imply a blocking
$ \sigma \in \Sigma^{-}. $
\noindent
For each of the generated mappings $ S, $ for each $ i \in I, $ set
$ U(i) = U(i) \cup \{ S(i) \}. $
\item
For every $ i \in I, $ for which $ U(i) \not = d_i, $
add the refinement $ ( \ c_i \Rightarrow U(i) \ ) $ to $ \overline{C}. $
\end{enumerate} \end{algo}
The local consistency checker gives priority to checking against the substitution. After checking for conflicts against the substitution, Algorithm~\ref{Algo_local_consistency_checking} generates circles of size up to $ S+1, $ and checks for each of the substlets occurring in the clauses of such a circle, whether it can occur in a solution. Substlets that do not occur in a solution are refined away. Preference is given to small circles. This means that circles of size $ i+1 $ will be checked only after all circles up to size $ i $ have been checked.
We will discuss (and disprove) the conjecture mentioned above, that it is sufficient to check circles, when preference is given to smaller subsets. More precisely: If for a given subset $ C \subseteq \Sigma^{+}, $ all its subcircles have been checked, then $ C $ needs to be checked only if it is a circle by itself. We formally define what 'has been checked' means:
\begin{defi}
\label{Def_filtered}
Let $ ( \Sigma^{+}, \Sigma^{-} ) $ be a GCSP.
Let $ \Theta $ be a substitution.
Let $ C $ be a subset of clauses of $ \Sigma^{+}. $
We write $ \Phi(C) $ for the following property:
For every clause $ c \in C, $ for every substlet $ s \in c, $
there is a solution $ \Theta $ of $ ( C, \Sigma^{-} ), $
s.t. $ \Theta \models s. $ \end{defi}
Algorithm~\ref{Algo_local_consistency_checking} tries to establish $ \Phi(C) $ for every subset $ C $ of size $ i $ up to $ S+1. $ It assumes that
when $ \Phi(C) $ holds for circles with size smaller than $ \| C \|, $ and $C$ is not a circle, then $ \Phi(C) $ automatically holds. We have believed for some time that this assumption is true, because Lemma~\ref{Lemma_disconnected} and Lemma~\ref{Lemma_pivot} provide evidence for it, and it simplifies Algorithm~\ref{Algo_local_consistency_checking}. Unfortunately, the property fails at $ S = 4, $ when circles have size $ 5. $
\begin{conj}
\label{Conj_circles}
Let $ ( \Sigma^{+}, \Sigma^{-} ) $ be a GCSP.
Assume that every
strict subset $ C' \subset \{ c_1, \ldots, c_{S+1} \} $ that
can be arranged into a circle $ c'_1, \ldots, c'_{S'+1} $
has property $ \Phi(C'). $
Then if $ C $ cannot be arranged into a circle, $ C $
has the property $ \Phi(C). $ \end{conj} We prove Conjecture~\ref{Conj_circles} for $ S < 4, $ and provide a counter example for $ S = 4. $ \begin{proof}
\begin{itemize}
\item
$ S = 1 $ follows from Lemma~\ref{Lemma_disconnected}.
\item
In order to prove $ S = 2, $ assume that
$ c_1, c_2, c_3 $ are clauses that do not form a circle.
Without loss of generality, we may assume that
$ c_1 \not \sim c_3. $
If we also have $ c_1 \not \sim c_2, $ then
$ \{ c_1, c_2, c_3 \} $ can be partitioned into
$ \{ c_1 \}, \{ c_2, c_3 \}, $ so that Lemma~\ref{Lemma_disconnected}
can be applied.
If we have $ c_1 \sim c_2, $ we can apply Lemma~\ref{Lemma_pivot}
with $ C_1 = \{ c_1 \}, \ c = c_2, \ C_2 = \{ c_3 \}. $
\item
We prove $ S = 3. $ We use the fact
that Conjecture~\ref{Conj_circles} holds for $ S < 3. $
Let $ c_1, c_2, c_3, c_4 \in \Sigma^{+}. $
If $ \{ c_1, c_2, c_3, c_4 \} $ can be partitioned into
two disjoints sets, we can apply Lemma~\ref{Lemma_disconnected},
and we are done.
Otherwise, if $ \{ c_1, c_2, c_3, c_4 \} $ cannot be
partitioned into disconnected sets, there are two possibilities:
\begin{itemize}
\item
The clauses form a line
$ c_1 \sim c_2 \sim c_3 \sim c_4. $
If $ c_1 \sim c_4, $ then $ ( c_1, c_2, c_3, c_4 ) $
is a circle, so that Conjecture~\ref{Conj_circles} holds trivially.
Otherwise, we can still have $ c_1 \sim c_3 $ or
$ c_2 \sim c_4. $ If we have both, then
$ ( c_1, c_3, c_4, c_2 ) $ is a circle,
so that Conjecture~\ref{Conj_circles} again holds trivially.
If $ c_1 \not \sim c_3, $ we can apply
Lemma~\ref{Lemma_pivot} with
$ C_1 = \{ c_1 \}, \ c = c_2, $ and $ C_2 = \{ c_3, c_4 \}. $
Similarly, if $ c_2 \not \sim c_4, $ we can apply
Lemma~\ref{Lemma_pivot} with
$ C_1 = \{ c_1, c_2 \}, \ c = c_3, $ and $ C_2 = \{ c_4 \}. $
\item
The clauses form a kind of star with $ c_1 $ in the center:
$ c_1 \sim c_2, \ c_1 \sim c_3, \ c_1 \sim c_4. $
For $ c_2, $ if $ c_2 \sim c_3, $ nor $ c_2 \sim c_4, $
we can apply Lemma~\ref{Lemma_pivot} with
$ C_1 = \{ c_2 \}, \ c = c_1, \ C_2 = \{ c_3, c_4 \}. $
If we have both of $ c_2 \sim c_3 $ and $ c_2 \sim c_4, $ then
$ ( c_2,c_3,c_1,c_4 ) $ is a circle.
In the remaining case, we may assume without loss of generality
that $ c_2 \sim c_3, $ but also $ c_2 \not \sim c_4. $
This means that we have $ c_2 \sim c_3, c_2 \not \sim c_4. $
If $ c_4 \sim c_3, $ then $ ( c_1,c_2,c_3,c_4) $ is a circle.
If $ c_4 \not \sim c_3, $ then we can apply Lemma~\ref{Lemma_pivot}
with $ C_1 = \{ c_2,c_3 \}, \ c = c_1, \ C_2 = \{ c_4 \}. $
\end{itemize}
\end{itemize} \end{proof} We give a counter example for $ S = 4. $ \begin{exa}
\label{Exa_failure_conj_circles}
Consider the following GCSP, which has no blockings,
and the following clauses:
\[ \begin{array}{ll}
(c_1) & (X_1,X_2,X_3) \ / \ (0,0,0) \ | \ (0,1,1) \ | \
(1,1,0) \ | \ (1,0,1) \\
(c_2) & (X_1,Y_1) \ / \ (0,0) \ | \ (1,1) \\
(c_3) & (X_2,Y_2) \ / \ (0,0) \ | \ (1,1) \\
(c_4) & (X_3,Y_3) \ / \ (0,0) \ | \ (1,1) \\
(c_5) & (Y_1,Y_2,Y_3) \ / \ (1,0,0) \ | \ (0,1,0) \ | \
(0,0,1) \ | \ (1,1,1) \\
\end{array}
\]
We have $ c_1 \sim c_2, \ c_1 \sim c_3, \ c_1 \sim c_4, $ and
$ c_2 \sim c_5, \ c_3 \sim c_5, \ c_4 \sim c_5. $ There are
no other connections.
The example can be understood as follows: Clause $ c_2 $
requires that $ X_1 \Theta = Y_1 \Theta. $
Similarly, $ c_3 $ requires that $ X_2 \Theta = Y_2 \Theta, $
and $ c_4 $ requires that $ X_3 \Theta = Y_3 \Theta. $
Clause $ c_1 $ requires that $ X_1 \Theta + X_2 \Theta + X_3 \Theta $
is even, while $ c_5 $ requires that
$ Y_1 \Theta + Y_2 \Theta + Y_3 \Theta $
is odd. Since the sums must be equal, and cannot be odd and even at
the same time, $ ( \{ c_1,c_2,c_3,c_4,c_5 \}, \{ \ \} ) $ has no
solution.
Ignoring direction and starting point, there are three
circles of size $ 4: $
\[ ( c_1,c_2,c_5,c_3 ), \ \
( c_1,c_2,c_5,c_4 ), \ \
( c_1,c_3,c_5,c_4 ). \]
Since the circles are symmetric, we show that
every substlet occurring in $ \{ c_1,c_2,c_5,c_3 \} $ can occur
in a solution. One can pick the instance of $ c_1 $ and
$ c_5 $ in such a way that they agree on $ X_1/Y_1 $ and
$ X_2/Y_2. $ They will disagree on $ X_3/Z_3, $ but because
$ c_4 $ is not considered, this is no problem.
After that, the instances to $ c_2 $ and $ c_3 $
are fixed.
It is easily checked that $ c_1,c_2,c_3,c_4,c_5 $ cannot be arranged
into a circle. \end{exa} Example~\ref{Exa_failure_conj_circles} contains a GCSP that would not be refined by Algorithm~\ref{Algo_local_consistency_checking} with $ S = 4, $ despite the fact that it has no solution. We will refrain from trying to make Algorithm~\ref{Algo_local_consistency_checking} complete, because we believe that it is not worth the effort. Experiments suggest that using Algorithm~\ref{Algo_local_consistency_checking} becomes too costly already at $ S \geq 3. $ Implementing a more elaborate check at $ S \geq 3, $ would make Algorithm~\ref{Algo_local_consistency_checking} even more costly, and harder to implement, without much hope for improvement.
It is important to observe that even when Algorithm~\ref{Algo_local_consistency_checking} is used as a precheck, it still needs to be restorable, because it may be called by Algorithm~\ref{Alg_find_optimal_match}, which will turn on and off different substlets, based on $ \alpha. $ Only the first call need not be restorable.
\section{Conclusions}
The problem of matching a geometric formula into an interpretation used to be the bottleneck of our implementation of geometric resolution. In order to improve this situation, we gave a translation of the matching problem into GCSP, and provided efficient approaches for solving GCSP. One approach is to solve the GCSP directly by a combination of refinement, backtracking and learning. The other approach is to translate the problem into SAT. Our experiments suggest that both approaches have comparable performance. Both approaches still have room for improvement. For translation to SAT, one could develop a dedicated SAT solver. For the direct approach using GCSP, one could add heuristics that control when learnt lemmas are forgotten, and probably it will be possible to add deep backtracking.
Independent of the relative performance of the two approaches, one can conclude that the speed of {\bf Geo} can be improved by a very large factor, and that matching is no longer the bottleneck that hinders further development of the geometric resolution calculus.
Since GCSP may have applications outside of geometric logic, we defined an input format for GCSP, similar to DIMACS format for SAT, that can be used for independent applications. We also made the sources of our matching algorithm available.
The fact that the clause refining algorithm based on local consistency checking turned out not competitive, shows that search algorithms that appear good in theory, are not necessarily good in practice. In general, it is difficult to predict what will be the effect of a modification of a search algorithm. A seemingly small change may have a large impact on performance.
As for geometric resolution, one might argue that a calculus that uses an NP-complete problem as its basic operation is not viable, but there is room for interpretation: The complexity of the matching problem is caused by the fact that as result of flattening terms, geometric formulas and interpretations have DAG-structure instead of tree-structure. This increased expressiveness means that a geometric formula possibly represents exponentially many formulas with tree-structure. This may very well result in shorter proofs. Only experiments can determine which of the two effects will be stronger.
\end{document} |
\begin{document}
\title[The Research on Rotational Surfaces in pseudo Euclidean 4-Space with index 2]{The Research on Rotational Surfaces in pseudo Euclidean 4-Space with index 2} \author{Fatma ALMAZ$^{\ast}$} \address{Department of Mathematics, Firat university, 23119 Elazi\u{g}/T\"{U}RK\.{I}YE} \email{fb\_fat\_almaz@hotmail.com} \author{M\.{I}hr\.{I}ban ALYAMA\c{C} K\"{U}LAHCI} \address{Department of Mathematics, Firat university, 23119 Elazi\u{g}/T\"{U}RK\.{I}YE} \email{mihribankulahci@gmail.com } \thanks{This paper is in final form and no version of it will be submitted for publication elsewhere.} \subjclass[2000]{53B30, 53B50, 53C80} \keywords{Pseudo Euclidean 4-space, surfaces of rotation, killing vector field.}
\begin{abstract} In this study, we define a brief description of the hyperbolic and elliptic rotational surfaces using a curve and matrices in 4-dimensional semi-Euclidean space with index 2. That is, we provide different types of rotational matrices, which are the subgroups of $M$ by rotating a selected axis in $E^{4}$. Also, we choose two parameter matrices groups of rotations and we give the matrices of rotation corresponding to the appropriate subgroup in 4-dimensional semi-Euclidean space. Therefore, we generate surfaces of rotation using Killing vector fields in $E_{2}^{4}$ and we give the Gaussian curvature and the mean curvature of the surfaces of rotation. \end{abstract}
\maketitle
\section{Introduction}
From the past to the present many studies have been done that deal with rotational surfaces from algebraic and geometric aspects. The rotational surfaces are parametrized with the help of the Killing vector field. Therefore, the different types of matrices of rotations which are the subgroups of a manifold corresponding to rotation about a chosen axis in the arbitrary 4D-space are expressed. Hence, the two parameter matrices groups of rotations can be chosen and the matrices of rotation corresponding to the appropriate subgroup of an arbitrary 4D-space are expressed. To mention briefly for the publications taken as reference related to the subject studied. In \cite{1}, the geometric quantities associated with the concept of surfaces and the indicatrix of a surface are discussed in four-dimensional Galilean space by the authors. In \cite{2}, the brief description of rotational surfaces are given using a curve and matrices in 4-dimensional (4D) Galilean space. Also, choosing two parameter matrices groups of rotations, the matrices of rotation corresponding to the appropriate subgroup in Galilean 4-space, the rotated surfaces are expressed by the authors. In \cite{3,4}, the authors gave magnetic rotated surfaces in lightlike cone $Q^{2}\subset E_{1}^{3}$. Furthermore, the conditions being geodesic on rotational surface generated by magnetic curve are expressed with the help of Clairaut's theorem. In \cite{5}, the representation formulas of non-null curves are expressed in semi-Euclidean 4-space $ E_{2}^{4}$ and some certain results of describing the nun-null normal curve are presented in $E_{2}^{4}$. In \cite{7,8}, the rotational surfaces are studied by different authors in Minkowski 4-space. In \cite{9}, the some issues of displaying two-dimensional surfaces in 4D space are examined by authors. In \cite{14}, the translation surface in the case being harmonic surface are mainly studied, the necessary and sufficient conditions of being semi-parallel surfaces by considering semi-parallel condition given by the authors. In \cite{15}, the surfaces of revolution are characterized in the three dimensional pseudo-Galilean space.
\section{Preliminaries}
Let $E_{2}^{4}$ denote the $4-$dimensional pseudo-Euclidean space with signature $(2,4)$, that is, the real vector space $
\mathbb{R}
^{4}$ endowed with the metric $\left\langle ,\right\rangle _{E_{2}^{4}}$ which is defined by \begin{equation} \left\langle ,\right\rangle _{E_{2}^{4}}=g=-dx_{1}^{2}-dx_{2}^{2}+dx_{3}^{2}+dx_{4}^{2}, \tag{2.1} \end{equation} where $(x_{1},x_{2},x_{3},x_{4})$ is a standard rectangular coordinate system in $E_{2}^{4}$.
Recall that an arbitrary vector $v\in E_{2}^{4}\backslash \{0\}$ can have one of three characters: it can be space-like if $g(v,v)>0$ or $v=0,$ time-like if $g(v,v)<0$ and null if $g(v,v)=0$ and $v\neq 0.$
The norm of a vector $v$ is given by $\parallel v\parallel =\sqrt{g(v,v)}$ and two vectors $v$ and $w$ are said to be orthogonal if $g(v,w)=0$. An arbitrary curve $x(s)$ in $E_{2}^{4}$ can locally be space-like, time-like or null.
A space-like or time-like curve $x(s)$ has unit speed, if $g(x^{\prime },x^{\prime })=\pm 1.$
Let $ (x_{1},x_{2},x_{3},x_{4}),(y_{1},y_{2},y_{3},y_{4}),(z_{1},z_{2},z_{3},z_{4}) $ be any three vectors in $E_{2}^{4}$. The pseudo Euclidean cross product is given as \begin{equation} x\wedge y\wedge z= \begin{pmatrix} -i_{1} & -i_{2} & i_{3} & i_{4} \\ x_{1} & x_{2} & x_{3} & x_{4} \\ y_{1} & y_{2} & y_{3} & y_{4} \\ z_{1} & z_{2} & z_{3} & z_{4} \end{pmatrix} , \tag{2.2} \end{equation} where $i_{1}=\left( 1,0,0,0\right) ,i_{2}=\left( 0,1,0,0\right) ,i_{3}=\left( 0,0,1,0\right) ,i_{4}=\left( 0,0,0,1\right) $, \cite{11,12,13}.
The pseudo-Riemannian sphere $S_{2}^{3}\left( m,r\right) $ centered at $m\in E_{2}^{4}$ with radius $r>0$ of $E_{2}^{4}$ is defined by \begin{equation*} S_{2}^{3}\left( m,r\right) =\left\{ x\in E_{2}^{4}:\left\langle x-m,x-m\right\rangle =r^{2}\right\} . \end{equation*}
The pseudo-hyperbolic space $H_{1}^{3}\left( m,r\right) $ centered at $m\in E_{2}^{4}$ with radius $r>0$ of $E_{2}^{4}$ is defined by \begin{equation*} H_{1}^{3}\left( m,r\right) =\left\{ x\in E_{2}^{4}:\left\langle x-m,x-m\right\rangle =-r^{2}\right\} . \end{equation*}
The pseudo-Riemannian sphere $S_{2}^{3}\left( m,r\right) $ is diffeomorfic to $
\mathbb{R}
^{2}\times S$ and the pseudo-hyperbolic space $H_{1}^{3}\left( m,r\right) $ is diffeomorfic to $S^{1}\times
\mathbb{R}
^{2}$. The hyperbolic space $H^{3}\left( m,r\right) $ is given by \begin{equation*} H^{3}\left( m,r\right) =\left\{ x\in E_{2}^{4}:\left\langle x-m,x-m\right\rangle =-r^{2},x_{1}>0\right\} . \end{equation*}
Let $\Psi :M\rightarrow E_{2}^{4}$ be an isometric immersion of oriented pseudo-Riemannian submanifold $M$ into $E_{2}^{4}$. Henceforth, a submanifold in $E_{2}^{4}$ always means pseudo-Riemannian. Let $\overset{-}{ \nabla }$ be the Levi-Civita connection of $E_{2}^{4}$ and $\nabla $ be the induced connection on $M$. Also, for any vector fields $X,Y$ tangent to $M$, we get the Gaussian formula \begin{equation} \overset{-}{\nabla }_{X}^{{}}Y=\nabla _{X}^{{}}Y+h(X,Y), \tag{2.3} \end{equation} where $h$ is the second fundamental form which is symmetric in $X$ and $Y$. For a unit normal vector field $\xi $, the Weingarten formula is defined by \begin{equation} \overset{-}{\nabla }_{X}^{{}}\xi =-A_{\xi }X+D_{\xi}X, \tag{2.4} \end{equation} where $A_{\xi }$ is the Weingarten map or the shape operator with respect to $\xi $ and $D $ is the normal connection. The Weingarten map $A_{\xi }$ is a self-adjoint endomorphism of $TM$ which cannot be diagonalized generally. It is known that $h$ and $A_{\xi }$ are related by \begin{equation} \left\langle h(X,Y),\xi \right\rangle =\left\langle A_{\xi }X,Y\right\rangle . \tag{2.5} \end{equation}
The covariant derivative $\overset{\sim }{\nabla }h$ of the second fundamental form $h$ is given by \begin{equation} \overset{\sim }{\nabla }_{X}h\left( Y,Z\right) =\nabla _{X}^{\bot }h\left( Y,Z\right) -h\left( \nabla _{X}^{{}}Y,Z\right) -h\left( Y,\nabla _{X}^{{}}Z\right) , \tag{2.6} \end{equation} where $\nabla ^{\bot }$ indicates the linear connection induced on the normal bundle $T^{\bot }M$. Also, Codazzi equation is given by \begin{equation} \overset{\sim }{\nabla }_{X}h\left( Y,Z\right) =\overset{\sim }{\nabla } _{Y}h\left( X,Z\right) . \tag{2.7} \end{equation}
Let $e_{1},e_{2},...,e_{m}$ be a local orthonormal frame field in $E_{s}^{m}$ such that $e_{1},e_{2},...,e_{n}$ are tangent to $M^{n}$ and $\left\{ e_{n+1},...,e_{m}\right\} $ are normal to $M^{n}$. Let $ w_{1},w_{2},...,w_{m} $ be the coframe of $e_{1},e_{2},...,e_{m}$. We'll make use of the following convention on the ranges of indices $1\leq i,j,...\leq n,n+1\leq s,t,...\leq 4,1\leq A,B,...\leq 4$. Also, $w_{A}\left( e_{B}\right) =\delta _{AB}$ and the pseudo-Riemannian metric on $E_{s}^{m}$ is given by \begin{equation} ds^{2}=\underset{i}{\overset{n}{\sum }}\varepsilon _{A}w_{A}^{2};\varepsilon _{A}=\left\langle e_{A},e_{A}\right\rangle =\pm 1. \tag{2.8} \end{equation}
Let $w_{A}$ be the dual 1-form of $e_{A}$ defined by $w_{A}X=\left\langle e_{A},X\right\rangle $. Also, the connection forms $w_{AB}$ are defined by \begin{equation} de_{A}=\sum \varepsilon _{B}w_{AB}e_{B};w_{AB}+w_{BA}=0. \tag{2.9} \end{equation}
After, the structure equations of $E_{2}^{4}$ are written as follows \begin{equation} dw_{A}=\sum_{B}\varepsilon _{B}w_{AB}\wedge w_{B};dw_{A}=\sum_{C}\varepsilon _{C}w_{AC}\wedge w_{CB.} \tag{2.10} \end{equation}
The canonical forms $\left\{ w_{A}\right\} $ and the connection forms $ \left\{ w_{AB}\right\} $ restricted to $M^{n}$ are also indicated by the same symbols. Also, we get \begin{equation*} w_{s}=0,s=n+1,...,4 \end{equation*} and since $w_{s}$ are zero forms on $M^{n}$, there are symmetric tensor $ h_{ij}^{s}$ by Cartan's lemma such \begin{equation} w_{is}=\sum_{j}\varepsilon _{j}h_{ij}^{s}w_{j};h_{ij}^{s}=h_{ji}^{s}. \tag{2.11} \end{equation}
The mean curvature vector $H$ of $M^{n}$ in $E_{s}^{m}$ is given by \begin{equation} H=\frac{1}{2}\overset{m}{\underset{s=n+1}{\sum }}\underset{i=1}{\overset{n}{ \sum }}\varepsilon _{j}\varepsilon _{s}h_{ij}^{s}e_{s}. \tag{2.12} \end{equation}
Also, the covariant differentiation of $e_{i}$ is given by \begin{equation*} de_{i}=\sum_{A}\varepsilon _{A}w_{iA}e_{A}\text{ or }\overset{-}{\nabla } _{e_{i}}^{{}}e_{j}=\sum_{B}\varepsilon _{B}w_{jB}\left( e_{i}\right) e_{B} \text{,} \end{equation*} \cite{6,10,11}.
\begin{definition} \cite{10}, A one-parameter group of Diffeomorphisms of a manifold $M$ is a regular map $\psi:M\times
\mathbb{R}
\rightarrow M$, such that $\psi_{t}(x)=\psi(x,t),$ where
\begin{enumerate} \item $\psi_{t}:M\rightarrow M$ is a Diffeomorphism
\item $\psi_{0}=id$
\item $\psi_{s+t}=\psi_{s}o\psi_{t}.$ \end{enumerate} \end{definition}
This group is attached with a vector field $W$ given by $\frac{d}{dt}\psi _{t}(x)=W(x),$ and the group of Diffeomorphism is said to be as the flow of $ W$.
\begin{definition} If a one-parameter group of isometries is generated by a vector field $W$, then this vector field is called as a Killing vector field, \cite{10}. \end{definition}
\begin{definition} Let $W$ be a vector field on a smooth manifold $M$ and $\psi_{t}$ be the local flow generated by $W$. For each $t\in
\mathbb{R}
,$ the map $\psi_{t}$ is Diffeomorphism of $M$ and given a function $f$ on $ M $, we consider the Pull-back $\psi_{t}f$. We define the Lie derivative of the function $f$ as to $W$ by \begin{equation*} L_{_{W}}f=\underset{t\longrightarrow0}{\lim}\underset{}{\left( \frac{\psi _{t}f-f}{t}\right) =\frac{d\psi_{t}f}{dt}_{t=0}}. \end{equation*}
Let $g_{xy}$ be any pseudo-Riemannian metric, then the derivative is given as \begin{equation*} L_{_{W}}g_{xy}=g_{xy,z}W^{z}+g_{xz}W_{,y}^{z}+g_{zy}W_{,x}^{z}. \end{equation*}
In Cartesian coordinates in Euclidean spaces where $g_{xy,z}=0,$ and the Lie derivative is given by \begin{equation*} L_{_{W}}g_{xy}=g_{xz}W_{,y}^{z}+g_{zy}W_{,x}^{z}. \end{equation*}
In \cite{10}, the vector $W$ generates a Killing field if and if only \begin{equation*} L_{_{W}}g=0. \end{equation*} \end{definition}
\section{The surfaces of rotation in $E_{2}^{4}$}
In this chapter, we provides a description of surfaces of rotation in $ E_{2}^{4}$. Here, we have used the metric (2.1). Therefore, we will provide different types of matrices of rotations, which are the subgroups of $M$ by rotated a selected axis in $E^{4}$. Hence, we will choose two parameter matrices groups of rotations. In particular, we have defined a brief description of rotational surfaces in four dimensional $E_{2}^{4}$ and we give the rotational matrices corresponding to the appropriate subgroup in $ E_{2}^{4}$. Hence, we generate the rotational surfaces.
The rotation matrices are replaced by Lorentz transformation as follows \begin{equation} M^{T}gM=g, \tag{3.1} \end{equation} where $M^{T}$ is the transpoze, $g$ is the metric matrix of $E_{2}^{4}$ and for the metric (2.1).
Let's obtain the set of all $4\times 4$ type matrices satisfying (3.1). The Lorentz group is a subgroup of the Diffeomorphisms group in $E_{2}^{4}.$
\begin{theorem} Let the pseudo-Euclidean group be a subgroup of the Diffeomorphisms group in $E_{2}^{4}$ and let $W$ be vector field which generate the isometries. Then, the killing vector field associated with the metric $g$ is given as \begin{equation*} W(\xi ,\varrho ,\vartheta ,\eta )=a\left( \eta \partial \xi +\xi \partial \eta \right) +b\left( \vartheta \partial \varrho +\varrho \partial \vartheta \right) +c\left( \vartheta \partial \xi +\xi \partial \vartheta \right) \end{equation*} \begin{equation} +d(\eta \partial \varrho +\varrho \partial \eta )+e(\vartheta \partial \eta -\eta \partial \vartheta )+f\left( \xi \partial \varrho -\varrho \partial \xi \right) , \tag{3.2} \end{equation} where $a,b,c,d,e,f\in
\mathbb{R}
_{0}^{+}.$ \end{theorem}
\begin{proof} Let $W$ be the vector which generate the isometries in $E_{2}^{4}$. We can write as the following the general vector field; \begin{equation} W(\xi ,\varrho ,\vartheta ,\eta )=W^{1}(\xi ,\varrho ,\vartheta ,\eta )\partial \xi +W^{2}(\xi ,\varrho ,\vartheta ,\eta )\partial \varrho +W^{3}(\xi ,\varrho ,\vartheta ,\eta )\partial \vartheta +W^{4}(\xi ,\varrho ,\vartheta ,\eta )\partial \eta , \tag{3.3} \end{equation} where $W^{j}\ $are real functions $($for $j=1,2,3,4)$. Also, by using definition 2 and definition 3, the expression of the (3.3) is \begin{equation} W_{\xi }^{1}=W_{\varrho }^{2}=W_{\vartheta }^{3}=W_{\eta }^{4}=0, \tag{3.4} \end{equation} \begin{equation} W_{\varrho }^{1}+W_{\xi }^{2}=0;W_{\vartheta }^{1}-W_{\xi }^{3}=0;W_{\eta }^{1}-W_{\xi }^{4}=0 \tag{3.5} \end{equation} \begin{equation} W_{\vartheta }^{2}-W_{\varrho }^{3}=0,W_{\eta }^{2}-W_{\varrho }^{4}=0,W_{\eta }^{3}+W_{\vartheta }^{4}=0, \tag{3.6} \end{equation} first, we will obtain the function $W^{1}$, then from (3.4) and (3.5) we write \begin{equation} W_{\varrho }^{1}+W_{\xi }^{2}=0. \tag{3.7} \end{equation} then differentiating with respect to $\varrho $ in\ the previous equation (3.7), we have \begin{equation} W_{\varrho \varrho }^{1}+W_{\xi \varrho }^{2}=0 \tag{3.8} \end{equation} and then differentiating with respect to $\vartheta $ in the equations $ W_{\vartheta }^{1}-W_{\xi }^{3}=0$, we obtain \begin{equation} W_{\vartheta \vartheta }^{1}-W_{\xi \vartheta }^{3}=0, \tag{3.9a} \end{equation} and then differentiating with respect to $\eta $ in the equations $W_{\eta }^{1}-W_{\xi }^{4}=0$, we obtain \begin{equation} W_{\eta \eta }^{1}-W_{\xi \eta }^{4}=0 \tag{3.9b} \end{equation} and from (3.4) we get $W_{\xi \varrho }^{2},W_{\xi \eta }^{4},W_{\xi \vartheta }^{3}=0.$ From (3.8) and (3.9a), (3.9b) which gives $W_{\vartheta \vartheta }^{1},W_{\eta \eta }^{1},W_{\varrho \varrho }^{1}=0.$ Therefore, the function $W^{1}$ can be written as follows \begin{align} W^{1}(\varrho ,\vartheta ,\eta )& =f_{1}^{1}(\varrho ,\eta )\vartheta +g_{1}^{1}(\varrho ,\eta ), \tag{3.10} \\ W^{1}(\varrho ,\vartheta ,\eta )& =f_{1}^{2}(\varrho ,\vartheta )\eta +g_{1}^{2}(\varrho ,\vartheta ), \tag{3.11} \\ W^{1}(\varrho ,\vartheta ,\eta )& =f_{1}^{3}(\eta ,\vartheta )\varrho +g_{1}^{3}(\eta ,\vartheta ), \notag \end{align} where $f_{1}^{i},g_{1}^{i}\in C^{\infty },i=\{i=1,2,3\}.$ From (3.10) and since $W_{\eta \eta }^{1}=0$ we get \begin{equation*} W_{\eta \eta }^{1}(\varrho ,\vartheta ,\eta )=f_{1\eta \eta }^{1}(\varrho ,\eta )\vartheta +g_{1\eta \eta }^{1}(\varrho ,\eta )=0, \end{equation*} this means $f_{1\eta \eta }^{1}(\varrho ,\eta )=g_{1\eta \eta }^{1}(\varrho ,\eta )=$ $0.$ Thus, we can write the equations $f_{1}^{1}(\varrho ,\eta )$ and $g_{1}^{1}(\varrho ,\eta )$ as follows \begin{align*} f_{1}^{1}(\varrho ,\eta )& =h_{1}(\varrho )\eta +m_{1}(\varrho ), \\ g_{1}^{1}(\varrho ,\eta )& =h_{1}^{\ast }(\varrho )\eta +m_{2}(\varrho ). \end{align*}
Furthermore, since $W_{\varrho \varrho }^{1}=0$ we can choose the functions $ h_{1},h_{1}^{\ast },m_{1},m_{2}$ as \begin{align} h_{1}(\varrho )& =a_{1}\varrho +b_{1},h_{1}^{\ast }(\varrho )=a_{1}^{\ast }\varrho +b_{2}, \notag \\ m_{1}(\varrho )& =c_{1}\varrho +d_{1},m_{2}(\varrho )=c_{2}\varrho +d_{2}. \tag{3.12} \end{align}
Furthermore, substituting this equation into (3.10), we have \begin{equation} W^{1}(\varrho ,\vartheta ,\eta )=\left( \left( a_{1}\varrho +b_{1}\right) \eta +c_{1}\varrho +d_{1}\right) \vartheta +\left( a_{1}^{\ast }\varrho +b_{2}\right) \eta +c_{2}\varrho +d_{2}. \tag{3.13} \end{equation}
Similarly, by making the necessary algebraic operations, the following component equations are obtained, respectively. \begin{eqnarray*} W^{2}(\xi ,\vartheta ,\eta ) &=&\left( \left( a_{2}\vartheta +b\right) \eta +x_{2}^{1}\vartheta +y_{2}^{1}\right) \xi +\left( a_{2}^{\ast }\vartheta +b^{\ast }\right) \eta +x_{2}^{2}\vartheta +y_{2}^{2}; \\ W^{3}(\varrho ,\xi ,\eta ) &=&\left( \left( a_{3}\eta +d\right) \xi +x_{3}^{1}\eta +y_{3}^{1}\right) \varrho +\left( a_{3}^{\ast }\eta +d^{\ast }\right) \xi +x_{3}^{2}\eta +y_{3}^{2}; \\ W^{4}(\xi ,\vartheta ,\varrho ) &=&\left( \left( a_{4}\varrho +e\right) \vartheta +x_{4}^{1}\varrho +y_{4}^{1}\right) \xi +\left( a_{4}^{\ast }\varrho +e^{\ast }\right) \vartheta +x_{4}^{2}\varrho +y_{4}^{2}, \end{eqnarray*} where $a_{i},a_{i}^{\ast },x_{i}^{j},y_{i}^{j},b,d,e,b^{\ast },d^{\ast },e^{\ast }\in
\mathbb{R}
;i,j\in I.$
If we assume arbitrary constants as \begin{equation*} a_{i}=x_{i}^{1}=c_{1}=b_{1}=b=d=e=a_{i}^{\ast }=y_{i}^{2}=d_{2}=0;i\in \{1,2,3,4\} \end{equation*} then we obtain \begin{align*} W^{1}(\varrho ,\vartheta ,\eta )& =d_{1}\vartheta +b_{2}\eta +c_{2}\varrho .;W^{2}(\xi ,\vartheta ,\eta )=y_{2}^{1}\xi +b^{\ast }\eta +x_{2}^{2}\vartheta; \\ W^{3}(\varrho ,\xi ,\eta )& =y_{3}^{1}\varrho +d^{\ast }\xi +x_{3}^{2}\eta ;W^{4}(\xi ,\vartheta ,\varrho )=y_{4}^{1}\xi +e^{\ast }\vartheta +x_{4}^{2}\varrho . \end{align*}
Furthermore, by using the equations (3.4), (3.5) and (3.6), we write \begin{eqnarray*} y_{2}^{1} &=&-c_{2}=f;d_{1}=d^{\ast }=c;b_{2}=y_{4}^{1}=a; \\ x_{2}^{2} &=&y_{3}^{1}=b;b^{\ast }=x_{4}^{2}=d;e^{\ast }=-x_{3}^{2}=e. \end{eqnarray*}
Hence, the vector fields $W^{1},W^{2},W^{3},W^{4}$ are given by \begin{align} W^{1}(\varrho ,\vartheta ,\eta )& =c\vartheta +a\eta -f\varrho .;W^{2}(\xi ,\vartheta ,\eta )=f\xi +d\eta +b\vartheta ; \tag{3.14} \\ W^{3}(\varrho ,\xi ,\eta )& =b\varrho +c\xi -e\eta ;W^{4}(\xi ,\vartheta ,\varrho )=a\xi +e\vartheta +d\varrho . \notag \end{align}
By using the equation (3.14) into the equation (3.3), we have \begin{equation*} W(\xi ,\varrho ,\vartheta ,\eta )=\left( c\vartheta +a\eta -f\varrho \right) \partial \xi +\left( f\xi +d\eta +b\vartheta \right) \partial \varrho +\left( b\varrho +c\xi -e\eta \right) \partial \vartheta \end{equation*} \begin{equation*} +\left( a\xi +e\vartheta +d\varrho \right) \partial \eta ; \end{equation*} \begin{equation*} W(\xi ,\varrho ,\vartheta ,\eta )=a\left( \eta \partial \xi +\xi \partial \eta \right) +b\left( \vartheta \partial \varrho +\varrho \partial \vartheta \right) +c\left( \vartheta \partial \xi +\xi \partial \vartheta \right) \end{equation*} \begin{equation*} +d(\eta \partial \varrho +\varrho \partial \eta )+e(\vartheta \partial \eta -\eta \partial \vartheta )+f\left( \xi \partial \varrho -\varrho \partial \xi \right) , \end{equation*} where $a,b,c,d,e,f\in
\mathbb{R}
_{0}^{+}.$ \end{proof}
\begin{theorem} Let $W(\xi ,\varrho ,\vartheta ,\eta )$ be the killing vector field and let $ \gamma=(f_{1},f_{2},f_{3},f_{4})$ be a curve in $E_{2}^{4}$, then the surfaces of rotation are given as follows
\begin{enumerate} \item For the rotations $\Omega _{1}=\vartheta \partial \xi +\xi \partial \vartheta $ and $\Omega _{4}=\eta \partial \varrho +\varrho \partial \eta ,$ the hyperbolic surface of rotation is given as \begin{equation*} S_{14}(x,\alpha ,s)=\left( \begin{array}{c} f_{1}\cosh x+f_{3}\sinh x,f_{2}\cosh \alpha +f_{4}\sinh \alpha , \\ f_{1}\sinh x+f_{3}\cosh x,f_{2}\sinh \alpha +f_{4}\cosh \alpha \end{array} \right) \end{equation*} and for the curve $\gamma (s)=(f_{1}(s),0,0,f_{4}(s))$ the Gaussian curvature $K$ and the mean curvature vector $H$ of the rotational surface $ S_{14}(x(t),\alpha (t),s)=\left( f_{1}\cosh x,f_{4}\sinh \alpha ,f_{1}\sinh x,f_{4}\cosh \alpha \right) $ are given as \begin{equation*} K=\frac{\left( f_{1}^{\prime }f_{4}^{{}}-f_{1}f_{4}^{\prime }\right) ^{2}\left( \overset{.}{x}\overset{.}{\alpha }\right) ^{2}}{f_{4}^{2}\overset{ .}{\alpha }^{2}-f_{1}^{2}\overset{.}{x}^{2}}+\frac{\left( f_{1}^{\prime }f_{4}\overset{.}{\alpha }^{2}-f_{4}^{\prime }f_{1}\overset{.}{x}^{2}\right) \left( f_{1}^{\prime }f_{4}^{\prime \prime }-f_{1}^{\prime \prime }f_{4}^{\prime }\right) }{-f_{1}^{\prime 2}+f_{4}^{\prime 2}}, \end{equation*} \begin{equation*} H=\{\frac{f_{1}f_{4}\left( \overset{..}{x}\overset{.}{\alpha }+\overset{.}{x} \overset{..}{\alpha }\right) }{2\sqrt{f_{4}^{2}\overset{.}{\alpha } ^{2}-f_{1}^{2}\overset{.}{x}^{2}}}+\frac{f_{4}^{\prime }f_{1}\overset{.}{x} ^{2}-f_{1}^{\prime }f_{4}\overset{.}{\alpha }^{2}}{2\sqrt{-f_{1}^{\prime 2}+f_{4}^{\prime 2}}}\}e_{3}+\frac{\left( f_{1}^{\prime }f_{4}^{\prime \prime }-f_{1}^{\prime \prime }f_{4}^{\prime }\right) }{2\sqrt{ -f_{1}^{\prime 2}+f_{4}^{\prime 2}}}e_{4} \end{equation*} where
$e_{3}=\frac{\left( f_{4}\overset{.}{\alpha }\sinh x,f_{1}\overset{.}{x }\cosh \alpha ,f_{4}\overset{.}{\alpha }\cosh x,f_{1}\overset{.}{x}\sinh \alpha \right) }{\sqrt{f_{4}^{2}\overset{.}{\alpha }^{2}-f_{1}^{2}\overset{.} {x}^{2}}},e_{4}=\frac{\left( f_{4}^{\prime }\cosh x,f_{1}^{\prime }\sinh \alpha ,f_{4}^{\prime }\sinh x,f_{1}^{\prime }\cosh \alpha \right) }{\sqrt{ -f_{1}^{\prime 2}+f_{4}^{\prime 2}}}.$
\item For the rotations $\Omega _{2}=\eta \partial \xi +\xi \partial \eta $ and $\Omega _{3}=\vartheta \partial \varrho +\varrho \partial \vartheta ,$ the hyperbolic surface of rotation is given as \begin{equation*} S_{23}(y,z,s)=\left( \begin{array}{c} f_{1}\cosh y+f_{4}\sinh y,f_{2}\cosh z+f_{3}\sinh z, \\ f_{2}\sinh z+f_{3}\cosh z,f_{1}\sinh y+f_{4}\cosh y \end{array} \right) . \end{equation*} and for the curve $\gamma (s)=(f_{1}(s),f_{2}(s),0,0)$ the Gaussian curvature $K$ and the mean curvature vector $H$ of the rotational surface $ S_{23}(y(t),z(t),s)=\left( f_{1}\cosh y,f_{2}\cosh z,f_{2}\sinh z,f_{1}\sinh y\right) $ are given as \begin{equation*} K=-\left( \begin{array}{c} \frac{\left( f_{1}f_{2}^{\prime }+f_{1}^{\prime }f_{2}^{{}}\right) ^{2}\left( \overset{.}{y}\overset{.}{z}\right) ^{2}}{f_{2}^{2}\overset{.}{z} +f_{1}^{2}\overset{.}{y}}+ \\ \frac{\left( f_{1}^{{}}f_{2}^{\prime }\overset{.}{y}^{2}+f_{1}^{\prime }f_{2} \overset{.}{z}^{2}\right) \left( f_{1}^{\prime \prime }f_{2}^{\prime }+f_{1}^{\prime }f_{2}^{\prime \prime }\right) }{f_{1}^{\prime 2}+f_{2}^{\prime 2}} \end{array} \right) ;H=\left( \begin{array}{c} \frac{f_{1}^{{}}f_{2}(\overset{.}{y}\overset{..}{z}+\overset{..}{y}\overset{. }{z})}{2\sqrt{f_{2}^{2}\overset{.}{z}+f_{1}^{2}\overset{.}{y}}}e_{3} \\ +\frac{f_{1}^{{}}f_{2}^{\prime }\overset{.}{y}^{2}+f_{1}^{\prime }f_{2} \overset{.}{z}^{2}-f_{1}^{\prime \prime }f_{2}^{\prime }-f_{1}^{\prime }f_{2}^{\prime \prime }}{2\sqrt{f_{1}^{\prime 2}+f_{2}^{\prime 2}}}e_{4} \end{array} \right) , \end{equation*} where
$e_{3}=\frac{\left( f_{2}\overset{.}{z}\sinh y,f_{1}\overset{.}{y} \sinh z,f_{1}\overset{.}{y}\cosh z,f_{2}\overset{.}{z}\cosh y\right) }{\sqrt{ f_{2}^{2}\overset{.}{z}+f_{1}^{2}\overset{.}{y}}}$; $e_{4}=\frac{\left( f_{2}^{\prime }\cosh y,f_{1}^{\prime }\cosh z,f_{1}^{\prime }\sinh z,f_{2}^{\prime }\sinh y\right) }{\sqrt{f_{1}^{\prime 2}+f_{2}^{\prime 2}}}$
\item For the rotations $\Omega _{5}=\xi \partial \varrho -\varrho \partial \xi $ and $\Omega _{6}=\vartheta \partial \eta -\eta \partial \vartheta ,$ the elliptic surface of rotation is given as \begin{equation*} S_{56}(\beta ,\theta ,s)=\left( \begin{array}{c} f_{1}\cos \beta +f_{2}\sin \beta ,-f_{1}\sin \beta +f_{2}\cos \beta , \\ f_{3}\cos \theta +f_{4}\sin \theta ,-f_{3}\sin \theta +f_{4}\cos \theta \end{array} \right) , \end{equation*} and for the curve $\gamma (s)=(0,f_{2}(s),0,f_{4}(s))$ the Gaussian curvature $K$ and the mean curvature vector $H$ of the rotational surface $ S_{56}(\beta \left( t\right) ,\theta \left( t\right) ,s)=\left( f_{2}\sin \beta ,f_{2}\cos \beta ,f_{4}\sin \theta ,f_{4}\cos \theta \right) $ are given as \begin{equation*} K=-\left( \begin{array}{c} \frac{\left( f_{2}^{\prime }f_{4}-f_{2}f_{4}^{\prime }\right) ^{2}\left( \overset{.}{\beta }\overset{.}{\theta }\right) ^{2}}{-f_{2}^{2}\overset{.}{ \beta }^{2}+f_{4}^{2}\overset{.}{\theta }}+ \\ \frac{\left( -f_{2}^{\prime \prime }f_{4}^{\prime }+f_{2}^{\prime }f_{4}^{\prime \prime }\right) (f_{4}^{\prime }f_{2}\overset{.}{\beta } ^{2}-f_{2}^{\prime }f_{4}\overset{.}{\theta }^{2})^{2}}{-f_{2}^{\prime 2}+f_{4}^{\prime 2}} \end{array} \right) ; \end{equation*} \begin{equation*} H=\frac{f_{4}f_{2}\left( \overset{.}{\beta }\overset{..}{\theta }-\overset{.} {\theta }\overset{..}{\beta }\right) }{2\sqrt{f_{4}^{2}\overset{.}{\theta } -f_{2}^{2}\overset{.}{\beta }^{2}}}e_{3}+\frac{\left( f_{4}^{\prime }f_{2} \overset{.}{\beta }^{2}-f_{2}^{\prime }f_{4}\overset{.}{\theta } ^{2}+f_{2}^{\prime \prime }f_{4}^{\prime }-f_{2}^{\prime }f_{4}^{\prime \prime }\right) }{2\sqrt{f_{4}^{\prime 2}-f_{2}^{\prime 2}}}e_{4} \end{equation*} where $e_{3}=\frac{\left( -f_{4}\overset{.}{\theta }\cos \beta ,f_{4}\overset {.}{\theta }\sin \beta ,-f_{2}\overset{.}{\beta }\cos \theta ,f_{2}\overset{. }{\beta }\sin \theta \right) }{\sqrt{-f_{2}^{2}\overset{.}{\beta } ^{2}+f_{4}^{2}\overset{.}{\theta }}},e_{4}=\frac{\left( f_{4}^{\prime }\sin \beta ,f_{4}^{\prime }\cos \beta ,f_{2}^{\prime }\sin \theta ,f_{2}^{\prime }\cos \theta \right) }{\sqrt{-f_{2}^{\prime 2}+f_{4}^{\prime 2}}}$; $ -\infty <x,y,z,\alpha ,\beta ,\theta <\infty ,s\in I$ and $f_{i}\in C^{\infty }.$ \end{enumerate} \end{theorem}
\begin{proof} Let $W(\xi ,\varrho ,\vartheta ,\eta )=a\Omega _{2}+b\Omega _{3}+c\Omega _{1}+d\Omega _{4}+e$ $\Omega _{6}+f\Omega _{5}$ be the killing vector field. Hence, we can give vector fields generating the rotations as follows \begin{equation} \Omega _{1}=\vartheta \partial \xi +\xi \partial \vartheta ;\text{ }\Omega _{2}=\eta \partial \xi +\xi \partial \eta ;\text{ }\Omega _{3}=\vartheta \partial \varrho +\varrho \partial \vartheta ; \tag{3.15a} \end{equation} \begin{equation} \Omega _{4}=\eta \partial \varrho +\varrho \partial \eta ;\text{ }\Omega _{5}=\xi \partial \varrho -\varrho \partial \xi ;\text{ }\Omega _{6}=\vartheta \partial \eta -\eta \partial \vartheta , \tag{3.15b} \end{equation} by using the equations (3.15), we will find $4\times 4$ matrices of hyperbolic and elliptic by rotating $\Omega _{i}$, $i\in I$.
a) Hyperbolic matrices: we give some one-parameter hyperbolic matrices groups of rotation $\Omega _{i},i=1,2,3,4.$
$1)$ For $\Omega _{1}=\vartheta \partial \xi +\xi \partial \vartheta ,$ we write the vector field \begin{equation} \Lambda _{\Omega _{1}}= \begin{bmatrix} \vartheta & 0 & \xi & 0 \end{bmatrix} ^{\intercal }, \tag{3.16} \end{equation} then, the previous equation can be given as follows \begin{equation} \Delta _{\Lambda _{\Omega _{1}}}= \begin{bmatrix} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{bmatrix} , \tag{3.17} \end{equation} from definition 1, by using the differential equation $\frac{d}{dw}\psi _{w}(x)=W(x)$ we have \begin{equation*} \Pi _{w}(x)=e^{\Delta _{\Lambda _{1}}x}(x)=I_{4\times 4}+\Delta _{\Lambda _{\Omega _{1}}}x+\frac{\left( \Delta _{\Lambda _{\Omega _{1}}}x\right) ^{2}}{ 2!}+... \end{equation*} \begin{equation} \Pi _{\Omega _{1}}(x)= \begin{bmatrix} \cosh x & 0 & \sinh x & 0 \\ 0 & 1 & 0 & 0 \\ \sinh x & 0 & \cosh x & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} . \tag{3.18} \end{equation} $2)$ For $\Omega _{2}=\eta \partial \xi +\xi \partial \eta ,$ we write the vector field \begin{equation} \Lambda _{\Omega _{2}}= \begin{bmatrix} \eta & 0 & 0 & \xi \end{bmatrix} ^{\intercal }, \tag{3.19} \end{equation} then, the previous equation can be given as follows \begin{equation} \Delta _{\Lambda _{\Omega _{2}}}= \begin{bmatrix} 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 \end{bmatrix} , \tag{3.20} \end{equation} from definition 1, by using the differential equation $\frac{d}{du}\psi _{u}(y)=W(y),$ we have \begin{equation*} \Pi _{u}(y)=e^{\Delta _{\Lambda _{\Omega _{2}}}u}(y)=I_{4\times 4}+\Delta _{\Lambda _{\Omega _{2}}}y+\frac{\left( \Delta _{\Lambda _{\Omega _{2}}}y\right) ^{2}}{2!}+... \end{equation*} \begin{equation} \Pi _{\Omega _{2}}(y)= \begin{bmatrix} \cosh y & 0 & 0 & \sinh y \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ \sinh y & 0 & 0 & \cosh y \end{bmatrix} . \tag{3.21} \end{equation}
3) For $\Omega _{3}=\vartheta \partial \varrho +\varrho \partial \vartheta ,$ we write the vector field given as \begin{equation} \Lambda _{\Omega _{3}}= \begin{bmatrix} 0 & \vartheta & \varrho & 0 \end{bmatrix} ^{\intercal }, \tag{3.22} \end{equation} then, the previous equation can be given as follows \begin{equation*} \Delta _{\Lambda _{\Omega _{3}}}= \begin{bmatrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{bmatrix} . \end{equation*}
Now, from definition 1 we can say that the one-parameter group of homomorphism $\psi _{z}(\xi ,\varrho ,\vartheta ,\varsigma )$ is expressed by $\psi _{z}^{\prime }(\xi )=\psi ^{\xi }\psi _{z}(\xi )$. So, we find $ \psi _{z}(\xi )=e^{v\psi _{z}}\xi $ and calculating the matrix exponential, we have \begin{equation*} \Delta _{v}(z)=e^{\Delta _{\Lambda _{\Omega _{3}}}z}(z)=I_{4\times 4}+\Delta _{\Lambda _{\Omega _{3}}}z+\frac{\left( \Delta _{\Lambda _{\Omega _{3}}}z\right) ^{2}}{2!}+... \end{equation*} \begin{equation} \Pi _{\Omega _{3}}(z)= \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & \cosh z & \sinh z & 0 \\ 0 & \sinh z & \cosh z & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} . \tag{3.23} \end{equation}
Similarly for $\Omega _{4}=\eta \partial \varrho +\varrho \partial \eta $, we get \begin{equation*} \Pi _{\Omega _{4}}(\alpha )= \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & \cosh \alpha & 0 & \sinh \alpha \\ 0 & 0 & 1 & 0 \\ 0 & \sinh \alpha & 0 & \cosh \alpha \end{bmatrix} , \end{equation*} and for $\Omega _{5}=\xi \partial \varrho -\varrho \partial \xi $ and $ \Omega _{6}=\vartheta \partial \eta -\eta \partial \vartheta $, we obtain two one-parameter matrix group of rotation.
b) Elliptic matrices: we give some one-parameter elliptic matrices groups of rotation $\Omega _{5}$ and $\Omega _{6}$ \begin{equation*} \Pi _{\Omega _{5}}(\beta )= \begin{bmatrix} \cos \beta & \sin \beta & 0 & 0 \\ -\sin \beta & \cos \beta & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} ;\Pi _{\Omega _{6}}(\theta )= \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & \cos \theta & \sin \theta \\ 0 & 0 & -\sin \theta & \cos \theta \end{bmatrix} . \end{equation*}
Now if we want to express surfaces of rotation generated by two hyperbolic and elliptic subgroups, the sub-algebra of the lie algebra of the Lorentz group can be obtained, then we can write the closed subgroups of Lorentz group. Hence, two parameter subgroups of $SO(4,2)$ are obtain, and two parameter subgroups that fix some axis of rotation can be expressed. Therefore, we can write 2D sub-algebras, and therefore we need to obtain two vectors. In this context, by using Poisson bracket of two vectors $X=\overset {n}{\underset{i=1}{\sum }}X^{i}\partial _{i},Y=\overset{n}{\underset{i=1}{ \sum }}Y^{i}\partial _{i}$ defined by \begin{equation*} \left[ X,Y\right] =\overset{n}{\underset{i=1}{\sum }}\overset{n}{\underset{ j=1}{\sum }}(X^{j}\partial _{j}Y^{i}-Y^{j}\partial _{j}X^{i})\partial _{i}, \end{equation*} we can write the following expressions \begin{eqnarray*} \left[ \Omega _{1},\Omega _{2}\right] &=&\Omega _{6};\left[ \Omega _{1},\Omega _{3}\right] =\Omega _{5};\left[ \Omega _{1},\Omega _{5}\right] =\Omega _{3};\left[ \Omega _{1},\Omega _{6}\right] =\Omega _{2}; \\ \left[ \Omega _{2},\Omega _{4}\right] &=&\Omega _{5};\left[ \Omega _{2},\Omega _{5}\right] =\Omega _{4};\left[ \Omega _{6},\Omega _{2}\right] =\Omega _{1};\left[ \Omega _{3},\Omega _{4}\right] =\Omega _{6}; \\ \left[ \Omega _{5},\Omega _{3}\right] &=&\Omega _{1};\left[ \Omega _{3},\Omega _{6}\right] =\Omega _{4};\left[ \Omega _{5},\Omega _{4}\right] =\Omega _{2};\left[ \Omega _{6},\Omega _{4}\right] =\Omega _{3} \end{eqnarray*} then these Poisson brackets are not in $Sp\{\Omega _{i},\Omega _{j}\}$ excluding $Sp\{\Omega _{1},\Omega _{4}\},Sp\{\Omega _{2},\Omega _{3}\}$ and $ Sp\{\Omega _{5},\Omega _{6}\}.$ Therefore, these are not closed sub-algebra. Also, \begin{equation*} \left[ \Omega _{1},\Omega _{4}\right] =\left[ \Omega _{2},\Omega _{3}\right] =\left[ \Omega _{5},\Omega _{6}\right] =0, \end{equation*} $\{\Omega _{1},\Omega _{4}\},\left\{ \Omega _{2},\Omega _{3}\right\} ,\left\{ \Omega _{5},\Omega _{6}\right\} $ are the closed sub-algebra and we can think $\{\Omega _{1},\Omega _{4}\},$ $\left\{ \Omega _{2},\Omega _{3}\right\} ,$ $\left\{ \Omega _{5},\Omega _{6}\right\} $ as basis. Thus, abelian subgroups of $SO(2,2)$ can be expressed. Then, $\Omega _{1},\Omega _{4}$ and $\Omega _{2},\Omega _{3}$ generate abelian sub-algebras being hyperbolic. Therefore, we can write matrices $\Pi _{\Omega _{1}}(x)\Pi _{\Omega _{4}}(\alpha )$ and $\Pi _{\Omega _{2}}(y)\Pi _{\Omega _{3}}(z)$ being the rotational groups of matrices. Hence, these subgroups don't fix any axis and so it is not a rotation about any axis. First, for the rotations $\Omega _{1}$ and $\Omega _{4},$ the matrices of rotations of this surface can be written as $\Pi _{\Omega _{1}}(x)\Pi _{\Omega _{4}}(\alpha )$ . We are interested in taking a planar curve $\gamma $ with $s$ parameter as follows \begin{equation} \gamma (s)=(f_{1}(s),f_{2}(s),f_{3}(s),f_{4}(s)),s\in I \tag{3.24} \end{equation} and rotating it with 2D subgroup of isometry. Hence, the surface of revolution $S_{14}$ around $\Pi _{\Omega _{1}}(x)$ and $\Pi _{\Omega _{4}}(\alpha )$ can be parametrized as follows \begin{equation} S_{14}(x,\alpha ,s)=\Pi _{\Omega _{1}}(x).\Pi _{\Omega _{4}}(\alpha ). \begin{bmatrix} f_{1}(s) \\ f_{2}(s) \\ f_{3}(s) \\ f_{4}(s) \end{bmatrix} =\left( \begin{array}{c} f_{1}\cosh x+f_{3}\sinh x, \\ f_{2}\cosh \alpha +f_{4}\sinh \alpha , \\ f_{1}\sinh x+f_{3}\cosh x, \\ f_{2}\sinh \alpha +f_{4}\cosh \alpha \end{array} \right) , \tag{3.25a} \end{equation} where for $i\in \{1,2,3,4\},$ $f_{i}$ are smooth functions and $-\infty <x,\alpha <\infty ,s\in I.$ Now, we consider the following rotational surface \begin{equation} S^{_{^{14}}}(x(t),\alpha (t),s)=\left( f_{1}\cosh x(t),f_{4}\sinh \alpha (t),f_{1}\sinh x(t),f_{4}\cosh \alpha (t)\right) , \tag{3.25b} \end{equation} where $f_{1}$ and $f_{4}$ are nonzero smooth functions and the curve $\gamma (s)=(f_{1}(s),0,f_{4}(s))$ lies on the $\xi \eta -$plane. For the rotational surface (3.25b) we have the parametrizations \begin{eqnarray*} S_{s}^{_{^{14}}}(x,\alpha ,s) &=&\left( f_{1}^{\prime }\cosh x,f_{4}^{\prime }\sinh \alpha ,f_{1}^{\prime }\sinh x,f_{4}^{\prime }\cosh \alpha \right) ; \\ S_{ss}^{_{^{14}}}(x,\alpha ,s) &=&\left( f_{1}^{\prime \prime }\cosh x,f_{4}^{\prime \prime }\sinh \alpha ,f_{1}^{\prime \prime }\sinh x,f_{4}^{\prime \prime }\cosh \alpha \right) ; \\ S_{t}^{_{^{14}}}(x,\alpha ,s) &=&\left( f_{1}\overset{.}{x}\sinh x,f_{4} \overset{.}{\alpha }\cosh \alpha ,f_{1}\overset{.}{x}\cosh x,f_{4}\overset{.} {\alpha }\sinh \alpha \right) ; \\ S_{tt}^{_{^{14}}}(x,\alpha ,s) &=&\left( \begin{array}{c} f_{1}(\overset{..}{x}\sinh x+\overset{.}{x}^{2}\cosh x),f_{4}(\overset{..}{ \alpha }\cosh \alpha +\overset{.}{\alpha }^{2}\sinh \alpha ), \\ f_{1}(\overset{..}{x}\cosh x+\overset{.}{x}^{2}\sinh x),f_{4}(\overset{..}{ \alpha }\sinh \alpha +\overset{.}{\alpha }^{2}\cosh \alpha ) \end{array} \right) \\ S_{ts}^{_{^{14}}}(x,\alpha ,s) &=&\left( f_{1}^{\prime }\overset{.}{x}\sinh x(t),f_{4}^{\prime }\overset{.}{\alpha }\cosh \alpha (t),f_{1}^{\prime } \overset{.}{x}\cosh x(t),f_{4}^{\prime }\overset{.}{\alpha }\sinh \alpha \right) \end{eqnarray*} and \begin{equation*} \left\langle S_{s}^{_{^{14}}},S_{s}^{_{^{14}}}\right\rangle =-f_{1}^{\prime 2}+f_{4}^{\prime 2}>0;\left\langle S_{t}^{_{^{14}}},S_{t}^{_{^{14}}}\right\rangle =f_{1}^{2}\overset{.}{x} ^{2}-f_{4}^{2}\overset{.}{\alpha }^{2}<0. \end{equation*} Therefore, we choose the following moving frame $e_{1},e_{2},e_{3},e_{4}$, such that $e_{1},e_{2}$ are tangent to $S^{_{^{14}}}$ and $e_{3},e_{4}$ are normal to $S^{_{^{14}}}.$ Also, we write as \begin{eqnarray*} e_{1} &=&\frac{ \begin{pmatrix} f_{1}\overset{.}{x}\sinh x, \\ f_{4}\overset{.}{\alpha }\cosh \alpha , \\ f_{1}\overset{.}{x}\cosh x, \\ f_{4}\overset{.}{\alpha }\sinh \alpha \end{pmatrix} }{\sqrt{f_{4}^{2}\overset{.}{\alpha }^{2}-f_{1}^{2}\overset{.}{x}^{2}}} ;e_{2}=\frac{ \begin{pmatrix} f_{1}^{\prime }\cosh x, \\ f_{4}^{\prime }\sinh \alpha , \\ f_{1}^{\prime }\sinh x, \\ f_{4}^{\prime }\cosh \alpha \end{pmatrix} }{\sqrt{-f_{1}^{\prime 2}+f_{4}^{\prime 2}}};\frac{ \begin{pmatrix} f_{4}\overset{.}{\alpha }\sinh x, \\ f_{1}\overset{.}{x}\cosh \alpha , \\ f_{4}\overset{.}{\alpha }\cosh x, \\ f_{1}\overset{.}{x}\sinh \alpha \end{pmatrix} }{\sqrt{f_{4}^{2}\overset{.}{\alpha }^{2}-f_{1}^{2}\overset{.}{x}^{2}}} \\ e_{4} &=&\frac{\left( f_{4}^{\prime }\cosh x,f_{1}^{\prime }\sinh \alpha ,f_{4}^{\prime }\sinh x,f_{1}^{\prime }\cosh \alpha \right) }{\sqrt{ -f_{1}^{\prime 2}+f_{4}^{\prime 2}}}. \end{eqnarray*} Then, we can easily get \begin{equation*} \varepsilon _{1}=\left\langle e_{1},e_{1}\right\rangle =-1;\varepsilon _{2}=\left\langle e_{2},e_{2}\right\rangle =1;\varepsilon _{3}=\left\langle e_{3},e_{3}\right\rangle =-1;\varepsilon _{4}=\left\langle e_{4},e_{4}\right\rangle =1. \end{equation*}
By using (2.9), (2.10), (2.11), (2.12), we obtain the following coefficients of the second fundamental form $h$ and the connection forms \begin{eqnarray*} h_{11}^{3} &=&\frac{f_{1}f_{4}\left( \overset{..}{x}\overset{.}{\alpha }+ \overset{.}{x}\overset{..}{\alpha }\right) }{\sqrt{f_{4}^{2}\overset{.}{ \alpha }^{2}-f_{1}^{2}\overset{.}{x}^{2}}};h_{12}^{3}=\frac{\left( f_{1}^{\prime }f_{4}^{{}}-f_{1}f_{4}^{\prime }\right) \overset{.}{x}\overset{ .}{\alpha }}{\sqrt{f_{4}^{2}\overset{.}{\alpha }^{2}-f_{1}^{2}\overset{.}{x} ^{2}}};h_{22}^{3}=0; \\ h_{11}^{4} &=&\frac{f_{1}^{\prime }f_{4}\overset{.}{\alpha } ^{2}-f_{4}^{\prime }f_{1}\overset{.}{x}^{2}}{\sqrt{-f_{1}^{\prime 2}+f_{4}^{\prime 2}}};h_{22}^{4}=\frac{\left( f_{1}^{\prime }f_{4}^{\prime \prime }-f_{1}^{\prime \prime }f_{4}^{\prime }\right) }{\sqrt{-f_{1}^{\prime 2}+f_{4}^{\prime 2}}};h_{12}^{4}=0; \end{eqnarray*} and from (2.12) the mean curvature vector $H$ of the rotational surface $ S^{_{^{14}}}$ is \begin{equation*} H=\left\{ \begin{array}{c} \frac{f_{1}f_{4}\left( \overset{..}{x}\overset{.}{\alpha }+\overset{.}{x} \overset{..}{\alpha }\right) }{2\sqrt{f_{4}^{2}\overset{.}{\alpha } ^{2}-f_{1}^{2}\overset{.}{x}^{2}}} \\ +\frac{f_{4}^{\prime }f_{1}\overset{.}{x}^{2}-f_{1}^{\prime }f_{4}\overset{.} {\alpha }^{2}}{2\sqrt{-f_{1}^{\prime 2}+f_{4}^{\prime 2}}} \end{array} \right\} e_{3}+\left\{ \frac{\left( f_{1}^{\prime }f_{4}^{\prime \prime }-f_{1}^{\prime \prime }f_{4}^{\prime }\right) }{2\sqrt{-f_{1}^{\prime 2}+f_{4}^{\prime 2}}}\right\} e_{4}. \end{equation*}
The Gaussian curvature $K$ of the rotational surface $S^{_{^{14}}}$ is obtained as \begin{equation*} K=\underset{s=3}{\overset{4}{\sum }}\varepsilon _{s}\left[ h_{ij}^{s}\right] =\frac{\left( \begin{array}{c} f_{1}^{\prime }f_{4}^{{}} \\ -f_{1}f_{4}^{\prime } \end{array} \right) ^{2}\left( \overset{.}{x}\overset{.}{\alpha }\right) ^{2}}{f_{4}^{2} \overset{.}{\alpha }^{2}-f_{1}^{2}\overset{.}{x}^{2}}+\frac{\left( \begin{array}{c} f_{1}^{\prime }f_{4}\overset{.}{\alpha }^{2} \\ -f_{4}^{\prime }f_{1}\overset{.}{x}^{2} \end{array} \right) \left( \begin{array}{c} f_{1}^{\prime }f_{4}^{\prime \prime } \\ -f_{1}^{\prime \prime }f_{4}^{\prime } \end{array} \right) }{-f_{1}^{\prime 2}+f_{4}^{\prime 2}}. \end{equation*}
Secondly, for the rotations $\Omega _{2}$ and $\Omega _{3}$, by using the curve $\gamma (s)$, the surface of rotation $S_{23}$ around $\Pi _{\Omega _{2}}(y).\Pi _{\Omega _{3}}(z)$ is given as follows \begin{equation} S_{23}(y,z,s)=\Pi _{\Omega _{2}}(y).\Pi _{\Omega _{3}}(z). \begin{bmatrix} f_{1}(s) \\ f_{2}(s) \\ f_{3}(s) \\ f_{4}(s) \end{bmatrix} =\left( \begin{array}{c} f_{1}\cosh y+f_{4}\sinh y, \\ f_{2}\cosh z+f_{3}\sinh z, \\ f_{2}\sinh z+f_{3}\cosh z, \\ f_{1}\sinh y+f_{4}\cosh y \end{array} \right) , \tag{3.26a} \end{equation} where $-\infty <z,y<\infty ,s\in I.$ Now, we consider the following the surface of rotation \begin{equation} S^{_{^{23}}}(y,z,s)=\left( f_{1}\cosh y,f_{2}\cosh z,f_{2}\sinh z,f_{1}\sinh y\right) , \tag{3.26b} \end{equation} where $f_{1}$ and $f_{2}$ are non-zero smooth functions and the curve $ \gamma (s)=(f_{1}(s),f_{2}(s),0,0)$ lies on the $\xi \rho -$plane. For (3.26b) we have the parametrizations \begin{eqnarray*} S_{t}^{_{^{23}}}(y,z,s) &=&\left( f_{1}\overset{.}{y}\sinh y,f_{2}\overset{.} {z}\sinh z,f_{2}\overset{.}{z}\cosh z,f_{1}\overset{.}{y}\cosh y\right) ; \\ S_{tt}^{_{^{23}}}(y,z,s) &=&\left( \begin{array}{c} f_{1}(\overset{..}{y}\sinh y+\overset{.}{y}^{2}\cosh y),f_{2}(\overset{..}{z} \sinh z+\overset{.}{z}^{2}\cosh z), \\ f_{2}(\overset{..}{z}\cosh z+\overset{.}{z}^{2}\sinh z),f_{1}(\overset{..}{y} \cosh y+\overset{.}{y}^{2}\sinh y) \end{array} \right) ; \\ S_{s}^{_{^{23}}}(y,z,s) &=&\left( f_{1}^{\prime }\cosh y,f_{2}^{\prime }\cosh z,f_{2}^{\prime }\sinh z,f_{1}^{\prime }\sinh y\right) \\ S_{ss}^{^{23}}(y,z,s) &=&\left( f_{1}^{\prime \prime }\cosh y,f_{2}^{\prime \prime }\cosh z,f_{2}^{\prime \prime }\sinh z,f_{1}^{\prime \prime }\sinh y\right) ; \\ S_{st}^{_{^{23}}}(y,z,s) &=&\left( f_{1}^{\prime }\overset{.}{y}\sinh y,f_{2}^{\prime }\overset{.}{z}\sinh z,f_{2}^{\prime }\overset{.}{z}\cosh z,f_{1}^{\prime }\overset{.}{y}\cosh y\right) \end{eqnarray*} and \begin{equation*} \left\langle S_{s}^{^{23}},S_{s}^{_{^{23}}}\right\rangle =-f_{1}^{\prime 2}-f_{2}^{\prime 2}<0;\left\langle S_{t}^{_{^{23}}},S_{t}^{_{^{23}}}\right\rangle =f_{2}^{2}\overset{.}{z} +f_{1}^{2}\overset{.}{y}>0. \end{equation*} Hence, the following moving frame $e_{1},e_{2},e_{3},e_{4}$ can be chosen, such that $e_{1},e_{2}$ are tangent to $S^{_{^{23}}}$ and $e_{3},$ $e_{4}$ are normal to $S^{_{^{23}}}$, we obtain as follows \begin{eqnarray*} e_{1} &=&\frac{\left( f_{1}\overset{.}{y}\sinh y,f_{2}\overset{.}{z}\sinh z,f_{2}\overset{.}{z}\cosh z,f_{1}\overset{.}{y}\cosh y\right) }{\sqrt{ f_{2}^{2}\overset{.}{z}+f_{1}^{2}\overset{.}{y}}}; \\ e_{2} &=&\frac{\left( f_{1}^{\prime }\cosh y,f_{2}^{\prime }\cosh z,f_{2}^{\prime }\sinh z,f_{1}^{\prime }\sinh y\right) }{\sqrt{f_{1}^{\prime 2}+f_{2}^{\prime 2}}}; \\ e_{3} &=&\frac{\left( f_{2}\overset{.}{z}\sinh y,f_{1}\overset{.}{y}\sinh z,f_{1}\overset{.}{y}\cosh z,f_{2}\overset{.}{z}\cosh y\right) }{\sqrt{ f_{2}^{2}\overset{.}{z}+f_{1}^{2}\overset{.}{y}}}; \\ e_{4} &=&\frac{\left( f_{2}^{\prime }\cosh y,f_{1}^{\prime }\cosh z,f_{1}^{\prime }\sinh z,f_{2}^{\prime }\sinh y\right) }{\sqrt{f_{1}^{\prime 2}+f_{2}^{\prime 2}}} \end{eqnarray*} Also, we have $\varepsilon _{1,3}=1;\varepsilon _{2,4}=-1.$ For the equations (2.9), (2.10), (2.11), (2.12), the following coefficients of the second fundamental form $h$ and the connection forms are obtained as \begin{eqnarray*} h_{11}^{3} &=&\frac{f_{1}^{{}}f_{2}(\overset{.}{y}\overset{..}{z}+\overset{.. }{y}\overset{.}{z})}{\sqrt{f_{2}^{2}\overset{.}{z}+f_{1}^{2}\overset{.}{y}}} ;h_{12}^{3}=\frac{\left( f_{1}f_{2}^{\prime }+f_{1}^{\prime }f_{2}^{{}}\right) \overset{.}{y}\overset{.}{z}}{\sqrt{f_{2}^{2}\overset{.}{z }+f_{1}^{2}\overset{.}{y}}};h_{22}^{3}=0; \\ h_{11}^{4} &=&\frac{-f_{1}^{{}}f_{2}^{\prime }\overset{.}{y} ^{2}-f_{1}^{\prime }f_{2}\overset{.}{z}^{2}}{\sqrt{f_{1}^{\prime 2}+f_{2}^{\prime 2}}};h_{22}^{4}=\frac{-f_{1}^{\prime \prime }f_{2}^{\prime }-f_{1}^{\prime }f_{2}^{\prime \prime }}{\sqrt{f_{1}^{\prime 2}+f_{2}^{\prime 2}}};h_{12}^{4}=0; \end{eqnarray*} and from (2.12) the Gaussian curvature $K$ and the mean curvature vector $H$ of the rotational surface $S^{_{^{23}}}$ are obtained as follows \begin{eqnarray*} H &=&\frac{f_{1}^{{}}f_{2}(\overset{.}{y}\overset{..}{z}+\overset{..}{y} \overset{.}{z})}{2\sqrt{f_{2}^{2}\overset{.}{z}+f_{1}^{2}\overset{.}{y}}} e_{3}+\frac{f_{1}^{{}}f_{2}^{\prime }\overset{.}{y}^{2}+f_{1}^{\prime }f_{2} \overset{.}{z}^{2}-f_{1}^{\prime \prime }f_{2}^{\prime }-f_{1}^{\prime }f_{2}^{\prime \prime }}{2\sqrt{f_{1}^{\prime 2}+f_{2}^{\prime 2}}}e_{4} \\ K &=&-\frac{\left( f_{1}f_{2}^{\prime }+f_{1}^{\prime }f_{2}^{{}}\right) ^{2}\left( \overset{.}{y}\overset{.}{z}\right) ^{2}}{f_{2}^{2}\overset{.}{z} +f_{1}^{2}\overset{.}{y}}-\frac{\left( f_{1}^{{}}f_{2}^{\prime }\overset{.}{y }^{2}+f_{1}^{\prime }f_{2}\overset{.}{z}^{2}\right) \left( f_{1}^{\prime \prime }f_{2}^{\prime }+f_{1}^{\prime }f_{2}^{\prime \prime }\right) }{ f_{1}^{\prime 2}+f_{2}^{\prime 2}}. \end{eqnarray*}
Also, $\Omega _{5}$ and $\Omega _{6}$ generate abelian sub-algebra being elliptic. Therefore, we can write matrix $\Pi _{\Omega _{5}}(\beta )\Pi _{\Omega _{6}}(\theta )$ being the rotational group of matrices. This subgroup doesn't fix any axis and so it is not a rotation about any axis. For the rotations $\Omega _{5}$ and $\Omega _{6},$ the matrices of rotations of this surface can be written as $\Pi _{\Omega _{5}}(\beta )\Pi _{\Omega _{6}}(\theta )$, by using a planar curve $\gamma $ with $s$ parameter the surface of rotation $S_{56}$ around $\Pi _{\Omega _{5}}(\beta ).\Pi _{\Omega _{6}}(\theta )$ can be parametrized as follows \begin{equation} S_{56}(\beta ,\theta ,s)=\Pi _{\Omega _{5}}(\beta ).\Pi _{\Omega _{6}}(\theta ). \begin{bmatrix} f_{1}(s) \\ f_{2}(s) \\ f_{3}(s) \\ f_{4}(s) \end{bmatrix} =\left( \begin{array}{c} f_{1}\cos \beta +f_{2}\sin \beta , \\ -f_{1}\sin \beta +f_{2}\cos \beta , \\ f_{3}\cos \theta +f_{4}\sin \theta , \\ -f_{3}\sin \theta +f_{4}\cos \theta \end{array} \right) , \tag{3.27a} \end{equation} where $-\infty <\beta ,\theta <\infty ,s\in I.$ Now, we consider the following rotational surface \begin{equation} S^{_{^{56}}}(\beta \left( t\right) ,\theta \left( t\right) ,s)=\left( f_{2}\sin \beta ,f_{2}\cos \beta ,f_{4}\sin \theta ,f_{4}\cos \theta \right) , \tag{3.27b} \end{equation} where $f_{2}$ and $f_{4}$ are non-zero smooth functions and the curve $ \gamma (s)=(0,f_{2}(s),0,f_{4}(s))$ lies on the $\rho \eta -$plane. From (3.27b) we have the parametrizations \begin{eqnarray*} S_{s}^{_{^{56}}}(\beta ,\theta ,s) &=&\left( f_{2}^{\prime }\sin \beta ,f_{2}^{\prime }\cos \beta ,f_{4}^{\prime }\sin \theta ,f_{4}^{\prime }\cos \theta \right) ; \\ S_{ss}^{_{^{56}}}(\beta ,\theta ,s) &=&\left( f_{2}^{\prime \prime }\sin \beta ,f_{2}^{\prime \prime }\cos \beta ,f_{4}^{\prime \prime }\sin \theta ,f_{4}^{\prime \prime }\cos \theta \right) ; \\ S_{st}^{_{^{56}}}(\beta ,\theta ,s) &=&\left( f_{2}^{\prime }\overset{.}{ \beta }\cos \beta ,-f_{2}^{\prime }\overset{.}{\beta }\sin \beta ,f_{4}^{\prime }\overset{.}{\theta }\cos \theta ,-f_{4}^{\prime }\overset{.}{ \theta }\sin \theta \right) ; \\ S_{t}^{_{^{56}}}(\beta ,\theta ,s) &=&\left( f_{2}\overset{.}{\beta }\cos \beta ,-f_{2}\overset{.}{\beta }\sin \beta ,f_{4}\overset{.}{\theta }\cos \theta ,-f_{4}\overset{.}{\theta }\sin \theta \right) ; \\ S_{tt}^{_{^{56}}}(\beta ,\theta ,s) &=&\left( \begin{array}{c} f_{2}(\overset{..}{\beta }\cos \beta -\overset{.}{\beta }^{2}\sin \beta ),-f_{2}(\overset{..}{\beta }\sin \beta +\overset{.}{\beta }^{2}\cos \beta ), \\ f_{4}(\overset{..}{\theta }\cos \theta -\overset{.}{\theta }^{2}\sin \theta ),-f_{4}(\overset{..}{\theta }\sin \theta +\overset{.}{\theta }^{2}\cos \theta ) \end{array} \right) \end{eqnarray*} and \begin{equation*} \left\langle S_{s}^{_{^{56}}},S_{s}^{_{^{56}}}\right\rangle =-f_{2}^{\prime 2}+f_{4}^{\prime 2}>0;\left\langle S_{t}^{^{56}},S_{t}^{^{56}}\right\rangle =-f_{2}^{2}\overset{.}{\beta }^{2}+f_{4}^{2}\overset{.}{\theta }<0. \end{equation*}
For the following moving frame $e_{1},e_{2},e_{3},e_{4}$, we say that $ e_{1},e_{2}$ are tangent to $S^{_{^{56}}}$ and $e_{3},e_{4}$ are normal to $ S^{_{^{56}}}.$ Therefore, we get \begin{eqnarray*} e_{1} &=&\frac{\left( f_{2}\overset{.}{\beta }\cos \beta ,-f_{2}\overset{.}{ \beta }\sin \beta ,f_{4}\overset{.}{\theta }\cos \theta ,-f_{4}\overset{.}{ \theta }\sin \theta \right) }{\sqrt{-f_{2}^{2}\overset{.}{\beta } ^{2}+f_{4}^{2}\overset{.}{\theta }}}; \\ e_{2} &=&\frac{\left( f_{2}^{\prime }\sin \beta ,f_{2}^{\prime }\cos \beta ,f_{4}^{\prime }\sin \theta ,f_{4}^{\prime }\cos \theta \right) }{\sqrt{ -f_{2}^{\prime 2}+f_{4}^{\prime 2}}} \\ e_{3} &=&\frac{\left( -f_{4}\overset{.}{\theta }\cos \beta ,f_{4}\overset{.}{ \theta }\sin \beta ,-f_{2}\overset{.}{\beta }\cos \theta ,f_{2}\overset{.}{ \beta }\sin \theta \right) }{\sqrt{-f_{2}^{2}\overset{.}{\beta } ^{2}+f_{4}^{2}\overset{.}{\theta }}}; \\ e_{4} &=&\frac{\left( f_{4}^{\prime }\sin \beta ,f_{4}^{\prime }\cos \beta ,f_{2}^{\prime }\sin \theta ,f_{2}^{\prime }\cos \theta \right) }{\sqrt{ -f_{2}^{\prime 2}+f_{4}^{\prime 2}}}, \end{eqnarray*} and we also get $\varepsilon _{1,4}=-1;\varepsilon _{2,3}=1.$ By considering (2.9), (2.10), (2.11), (2.12), we obtain the following coefficients of the second fundamental form $h$ and the connection forms \begin{eqnarray*} h_{11}^{3} &=&\frac{f_{4}f_{2}\left( \overset{.}{\theta }\overset{..}{\beta } -\overset{.}{\beta }\overset{..}{\theta }\right) }{\sqrt{-f_{2}^{2}\overset{. }{\beta }^{2}+f_{4}^{2}\overset{.}{\theta }}};h_{12}^{3}=\frac{\left( f_{2}^{\prime }f_{4}-f_{2}f_{4}^{\prime }\right) \overset{.}{\beta }\overset{ .}{\theta }}{\sqrt{-f_{2}^{2}\overset{.}{\beta }^{2}+f_{4}^{2}\overset{.}{ \theta }}};h_{22}^{3}=0; \\ h_{11}^{4} &=&\frac{f_{4}^{\prime }f_{2}\overset{.}{\beta } ^{2}-f_{2}^{\prime }f_{4}\overset{.}{\theta }^{2}}{\sqrt{-f_{2}^{\prime 2}+f_{4}^{\prime 2}}};h_{22}^{3}=\frac{\left( -f_{2}^{\prime \prime }f_{4}^{\prime }+f_{2}^{\prime }f_{4}^{\prime \prime }\right) }{\sqrt{ -f_{2}^{\prime 2}+f_{4}^{\prime 2}}};h_{12}^{4}=0; \end{eqnarray*} and from (2.12) the mean curvature vector $H$ of the rotation surface $ S^{_{^{56}}}$ is \begin{equation*} H=-\frac{f_{4}f_{2}\left( \overset{.}{\theta }\overset{..}{\beta }-\overset{. }{\beta }\overset{..}{\theta }\right) }{2\sqrt{-f_{2}^{2}\overset{.}{\beta } ^{2}+f_{4}^{2}\overset{.}{\theta }}}e_{3}+\frac{\left( f_{4}^{\prime }f_{2} \overset{.}{\beta }^{2}-f_{2}^{\prime }f_{4}\overset{.}{\theta } ^{2}+f_{2}^{\prime \prime }f_{4}^{\prime }-f_{2}^{\prime }f_{4}^{\prime \prime }\right) }{2\sqrt{-f_{2}^{\prime 2}+f_{4}^{\prime 2}}}e_{4} \end{equation*} and the Gaussian curvature $K$ of the rotation surface $S^{^{_{56}}}$ is obtained as \begin{equation*} K=-\frac{\left( f_{2}^{\prime }f_{4}-f_{2}f_{4}^{\prime }\right) ^{2}\left( \overset{.}{\beta }\overset{.}{\theta }\right) ^{2}}{-f_{2}^{2}\overset{.}{ \beta }^{2}+f_{4}^{2}\overset{.}{\theta }}-\frac{\left( -f_{2}^{\prime \prime }f_{4}^{\prime }+f_{2}^{\prime }f_{4}^{\prime \prime }\right) (f_{4}^{\prime }f_{2}\overset{.}{\beta }^{2}-f_{2}^{\prime }f_{4}\overset{.}{ \theta }^{2})^{2}}{-f_{2}^{\prime 2}+f_{4}^{\prime 2}}. \end{equation*} \end{proof}
\begin{example} We consider the surfaces of rotation given as follows
\begin{enumerate} \item For the curve $\gamma (s)=(s+\sinh s,0,0,s+\cosh s)$ the hyperbolic surface of rotation is given as \begin{equation*} S_{1}(x,\alpha ,s)=\left( \begin{array}{c} (s+\sinh s)\cosh x,(s+\cosh s)\sinh \alpha , \\ (s+\sinh s)\sinh x,(s+\cosh s)\cosh \alpha \end{array} \right) . \end{equation*}
\item For the curve $\gamma (s)=(s\cosh s,s\sinh s,0,0)$ the hyperbolic surface of rotation is given as \begin{equation*} S_{2}(y,z,s)=\left( s\cosh s\cosh y,s\sinh s\cosh z,s\sinh s\sinh z,s\cosh s\sinh y\right) . \end{equation*}
\item For the curve $\gamma (s)=(0,ax^{2}\sin s,0,ax^{2}\cos s)$ the elliptic surface of rotation is given as \begin{equation*} S_{3}(\beta ,\theta ,s)=\left( ax^{2}\sin s\sin \beta ,ax^{2}\sin s\cos \beta ,ax^{2}\cos s\sin \theta ,ax^{2}\cos s\cos \theta \right) ;a,c\in
\mathbb{R}
. \end{equation*} \end{enumerate} \end{example}
\section{Conclusion}
In this paper, we gave different types of matrices of rotation which are the subgroups of the manifold $M$ corresponding to rotation about a chosen axis in $E^{4}$. Hence, we used two parameter matrices groups of rotations and we gave the matrices of rotation corresponding to the appropriate subgroup of the $E_{2}^{4}$ and we defined a brief description of rotational surfaces using a curve and matrices in $E_{2}^{4}$. Furthermore, we examined the special rotated surfaces generated by these matrices of rotation in $ E_{2}^{4}$ and we expressed some certain results of describing the surface obtaining Killing vector field in $E_{2}^{4}$ in detail. Also, we gave the Gaussian curvature and the mean curvature of the surfaces of rotation.
The authors are currently working on the properties of these rotated surfaces with a view to devising suitable metric in $E_{2}^{4}$ by adapting the type of conservation laws considered in the paper. In our future studies, we will study geodesics on the rotational surface obtained in $ E_{2}^{4}$. Also, the physical terms such as specific energy and specific angular momentum will be examined with the help of the conditions obtained by using the Clairaut's theorem on these special surfaces.
\section{Funding}
Not applicable
\section{Conflicts of interest statement}
The authors have NO affiliations with or involvement in any organization or entity with any financial interest or non-financial interest in the subject matter or materials discussed in this manuscript.
\section{Declarations}
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
\end{document} |
\begin{document}
\author{Maren H. Ring\footnote{University of Rostock, maren.ring@uni-rostock.de}, Robert Schüler\footnote{University of Rostock, robert.schueler1989@gmail.com}}
\title{f-vectors of 3-polytopes symmetric under rotations and rotary reflections} \date{\today}
\maketitle \defRoot{$\mathit{f}$}
\begin{abstract} The $f$-vector of a polytope consists of the numbers of its $i$-dimensional faces. An open field of study is the characterization of all possible $f$-vectors. It has been solved in three dimensions by Steinitz in the early 19th century. We state a related question, i.e. to characterize $f$-vectors of three dimensional polytopes respecting a symmetry, given by a finite group of matrices. We give a full answer for all three dimensional polytopes that are symmetric with respect to a finite rotation or rotary reflection group. We solve these cases constructively by developing tools that generalize Steinitz's approach. \end{abstract}
\section{Introduction}
There are many studies of $f$-vectors in higher dimensions, for example see \cite{grunbaum2003convex}, \cite{mcmullen1971minimum_vertices}, \cite{barnette1972inequalities}, \cite{barnette1973proof_of_lower_bound_conjecture}, \cite{bayer1984Counting_faces_and_chains}, \cite{stanley80simplicial}, \cite{stanley1985simplicial} or \cite{bayer1985Generalized_Dehn_Sommerville}. The set of $f$-vectors of four dimensional polytopes has been studied in \cite{barnette1973projections_four_polytopes}, \cite{barnette1974projection}, \cite{altshuler1985complete_enumeration}, \cite{bayer1987extended_f_vectors}, \cite{eppstein2003fat_and_fatter}, \cite{ziegler2002face_numbers} and \cite{brinkmann2018small_f_vectors}. Some insights about $f$-vectors of centrally symmetric polytopes are given in \cite{Barany1982Borsuks_theorem}, \cite{campo-neuen2006on_toric_h_vectors}, \cite{stanley1987centrally_symmetric_simplicial} and \cite{freij2013face_numbers_of_csp}. It is still an open question, even in three dimensions, what the $f$-vectors of \emph{symmetric polytopes} are. This question will be partially answered in this paper.
In particular, given a finite $3\times 3$ matrix group $G$, we ask to determine the set $\mathit{F}(G)$ of vectors $(\mathit{f}_0,\mathit{f}_2)\in \N\times\N$ such that there is a polytope $P$ symmetric under $G$ (i.e. $A\cdot P = P$ for all $A\in G$) with $\mathit{f}_0$ vertices and $\mathit{f}_2$ facets (we omit the number of edges by the Euler-equation). In this paper we give an answer for all groups that don't contain a reflection summarized in the following theorem. For a detailed explanation of the mentioned groups see Theorem \ref{thm:finite_orthogonal_groups}.
\begin{theorem}[main theorem]\label{thm:main} Let $\mathit{F}$ be the set of $\mathit{f}$-vectors of three dimensional polytopes (ommiting the number of edges) and for $M\subset\N\times\N$ we use $M^\diamond \coloneqq \{(y,x) \ : \ (x,y)\in M\}$ as well as $\equiv$ to denote component wise congruence. The $f$-vectors of symmetric polytopes under rotation groups can be classified as follows: \begin{align*} \mathit{F}(\C_n) &= \{\mathit{f}\in\mathit{F} \ : \ \mathit{f}\equiv (1,1) \mod n\} ^\diamond\\ &\cup \{\mathit{f} = (\mathit{f}_0,\mathit{f}_2) \in \mathit{F} \ : \ \mathit{f}\equiv (0,2), 2\mathit{f}_0 - \mathit{f}_2 \geq 2n-2 \mod n\}^\diamond \textnormal{ for } n > 2,\\ \mathit{F}(\C_2) &= \mathit{F},\\ \mathit{F}(\Dih_d) &= \{\mathit{f}\in\mathit{F} \ : \ \mathit{f}\equiv (0,2),(2,d) \mod 2d\}^\diamond \\ &\cup \{\mathit{f} = (\mathit{f}_0,\mathit{f}_2)\in\mathit{F} \ : \mathit{f}\equiv (0,d+2), (d,d+2) \mod 2d , 2\mathit{f}_0 - \mathit{f}_2 \geq 3d-2\}^\diamond\\ &\textnormal{ for } d>2\\ \mathit{F}(\Dih_2) &= \{\mathit{f}\in\mathit{F} \ : \ \mathit{f}\equiv (0,0),(0,2),(2,2) \mod 4\}^\diamond \setminus\{(6,6)\},\\
\mathit{F}(\T) &= \{\mathit{f}\in\mathit{F} \ : \ \mathit{f}\equiv (0,2),(0,8),(4,4),(4,10),(6,8) \mod 12\}^\diamond,\\ \mathit{F}(\Oc) &= \{\mathit{f}\in\mathit{F} \ : \ \mathit{f}\equiv (0,2),(0,14),(6,8),(6,20),(8,18),(12,14) \mod 24\}^\diamond,\\ \mathit{F}(\I) &= \{\mathit{f}\in\mathit{F} \ : \ \mathit{f}\equiv (0,2),(0,32),(12,20), (12,50), \\
& \hspace{19em} (20,42),(30,32) \mod 60\}^\diamond . \end{align*} For rotary reflection groups, the $f$-vectors can be classified as: \begin{align*} \mathit{F}(\G_d) &= \{\mathit{f}\in\mathit{F} \ : \ \mathit{f}\equiv (0,2) \mod 2d\}^\diamond \textnormal{ for } d > 2,
\\ \mathit{F}(\G_2) &= \{\mathit{f}\in\mathit{F} \ : \ \mathit{f}\equiv (0,0),(0,2) \mod 4\}^\diamond.\\ \mathit{F}(\G_1) &= \{\mathit{f}\in\mathit{F} \ : \ \mathit{f}\equiv (0,0) \mod 2\}^\diamond\setminus \{(4,4),(6,6)\} \end{align*}
In this paper we generalize the elementary approach of Steinitz (see \cite{steinitz1906polyederrelationen}). We start by introducing some fundamental terms and concepts relevant to this work in Section \ref{sec:preliminaries}. The coarser structure of $F(G)$ is due to the composition of orbits that the group $\G$ admits. This can be described in general and will be shown Section \ref{sec:conditions}, especially in Lemma~\ref{lem:f_vectors_mod_n}. The extra restrictions arise from certain structures of facets and vertices, e.g. a facet on a 6-fold rotation axis must have at least 6 vertices, which forces the polytope to be 'further away' from being simplicial.
The main difficulty in characterizing $F(G)$ is the construction of $G$-symmetric polytopes with a given $f$-vector. In Section \ref{sec:base} we introduce so called \emph{base polytopes}, symmetric polytopes that can be used to generate an infinite class of $f$-vectors. Since the operations on base polytopes produce $f$-vectors in the same congruence class, we divide the set of possible $f$-vectors in $F(G)$ into several coarser integer cones. To certify the existence of all $f$-vectors in one of these coarser integer cones
we introduce four types of certificates in Section \ref{sec:certificates}. In Corollary \ref{cor:certificates_work} we describe for which $f$-vectors certificates are needed to obtain all $f$-vectors conjectured to be in $F(G)$. To give these certificates we need to find symmetric polytopes with 'small' $f$-vector. To this end, in Section \ref{sec:constructions}, we introduce useful constructions on polytopes that change the $f$-vectors, but preserve the symmetry. As starting points, we then give a list of some well known polytopes taken from the Platonic, the Archimedean and their duals, the Catalan solids. In Section \ref{sec:characterization} we are finally able to connect the theory with explicit constructions of polytopes to prove Theorem \ref{thm:main}.
Lastly, we conclude the paper with some open questions and conjectures in Section \ref{sec:open}. \end{theorem}
\section{Preliminaries}\label{sec:preliminaries}
We start by introducing some fundamental terms and concepts relevant to this work. A polytope is the convex hull of finitely many points in $\R^n$. A face of a polytope $P$ is the intersection of $P$ with a hyperplane that contains no points of the relative interior of $P$. The polytope $P$ itself and the empty set are often considered as non-proper faces of $P$ as well, but are irrelevant for the study of $f$-vectors, since there is always exactly one of each. The dimension of a face is the dimension of its affine hull.
Let $P\subset\R^3$ be a 3-dimensional polytope. For $i\in \{0, 1,2\}$, we denote by $f_i(P)$ the number of $i$-dimensional faces of $P$. We call a facet \emph{simplicial} if it is triangular and a vertex \emph{simple} if it has degree three. A polytope is called simplicial if all of its facets are simplicial and it is called simple if all of its vertices are simple. Furthermore, a $(i,j)$-flag of $P$ for $i\leq j$ consists of an $i$-dimensional face $F_i$ and a $j$-dimensional face $F_j$ of $P$ such that $F_i\subseteq F_j$. The number of all $(i,j)$-flags of $P$ is denoted by $f_{i,j}(P)$. The following result follows directly from the Dehn-Sommerville Equations (see \cite[Section 9.2]{grunbaum2003convex}).
It establishes certain dependencies among the numbers of $i$-dimensional faces of a 3-dimensional polytope:
\begin{theorem}\label{thm:Euler_Steinitz} For any 3-dimensional polytope $P$ we have: \begin{enumerate}[label=(\arabic*)] \item $f_0(P) - f_1(P) + f_2(P) = 2$, \label{fml:Euler} \item $2f_0(P) - f_2(P) \geq 4$,\label{it:simplicial} \item $2f_2(P) - f_0(P) \geq 4$.\label{it:simple} \end{enumerate} We have equality in Equation \ref{it:simplicial} (Equation \ref{it:simple}) if and only if $P$ is simplicial (simple). \end{theorem}
It thus suffices to know two of the three numbers $f_0(P)$, $f_1(P)$ and $f_2(P)$; the missing one can be computed using Theorem \ref{thm:Euler_Steinitz}, Equation \ref{fml:Euler}. That leads us to the following definition of an $f$-vector that differs slightly from the definition for higher dimensional polytopes, where all numbers $f_i(P)$ occur in the $f$-vector. \begin{definition} Let $P$ be a 3-dimensional polytope. We define the \emph{$f$-vector} of $P$ to be $f(P) = (f_0(P), f_2(P))$. \end{definition}
Recall, that we denote by $F$ the set of all possible $f$-vectors of 3-dimensional polytopes, i.e. \begin{equation*} F=\{f\in \Z^2 \ : \ \exists \textnormal{ polytope } P \textnormal{ with } \dim(P)=3 \textnormal{ and } f(P)=f \}. \end{equation*}
It turns out that the conditions of Theorem \ref{thm:Euler_Steinitz} are sufficient for a characterization of $F$.
\begin{theorem}[{\cite[Section 10.3]{grunbaum2003convex}}] We have \begin{align*} \mathit{F} = \{(f_0,f_2)\in \Z^2 \ : \ 2f_0 - f_2 \geq 4 \textnormal{ and } 2f_2 - f_0 \geq 4 \}, \end{align*} i.e. the set of all possible $f$-vectors of 3-dimensional polytopes is a translated integer cone as shown in Figure \ref{fig:f_all}. \end{theorem} \begin{figure}
\caption{The set $F$ of all $f$-vectors of 3-dimensional polytopes.}
\label{fig:f_all}
\end{figure}
Let $G$ be a matrix group. We say that a polytope $P$ is \emph{$G$-symmetric} if $G$ acts on $P$, i.e. $G\cdot P = P$. The set of all possible $f$-vectors of $G$-symmetric polytopes is denoted by $$\mathit{F}(G) = \{f\in\Z^2 \ : \ \exists \ G\text{-symmetric polytope } P \text{ s.t. } \ f(P) = f\}.$$
By the following well known fact, it is sufficient to study orthogonal groups. \begin{lemma} If $G$ is a finite subgroup of $GL_3(\R)$, then there is an inner product $\linspan{\cdot,\cdot}_G$ such that $G$ is an orthogonal group with respect to $\linspan{\cdot,\cdot}_G$. \end{lemma}
Hence, for any finite matrix group $G$ there is an orthogonal group $G'$ with $\mathit{F}(G) = \mathit{F}(G')$ and we can reduce our study to finite orthogonal groups. Luckily, the finite orthogonal groups are well known. In fact, there are only finitely many families. A full list can be found in \cite{benson1985finite_reflection_groups}. Here, we will state the characterization of all finite rotation and rotary reflection groups, since these are our main subject of interest. The remaining finite orthogonal groups are the ones that contain reflections and are discussed in Section \ref{sec:open}.
\begin{theorem}[{\cite[Theorem 2.5.2]{benson1985finite_reflection_groups}}]\label{thm:finite_orthogonal_groups} If $G$ is a finite orthogonal subgroup of $GL_3(\R)$ consisting only of rotations or rotary reflections, then it is isomorphic to one of the following: \begin{enumerate} \item the axis-rotation group $C_n$ \item the dihedral rotation group $\Dih_d$ \item the rotation group of the regular tetrahedron $\T$ \item the rotation group of the regular octahedron $\O$ \item the rotation group of the regular icosahedron $\I$ \item the rotary reflection group $\G_d$ of order $2d$ (generated by a product of a rotation and a reflection) \label{it:rotary} \end{enumerate} \end{theorem}
\begin{remark} Details of the groups will be explained in Section \ref{sec:characterization}. In the notation of Grove and Benson \ref{thm:finite_orthogonal_groups}.\ref{it:rotary} corresponds to $C_3^{2d}]C_3^{d}$ for even $d$ and to $(C_3^d)^\ast$ for odd $d$ and is generated by a 'rotary reflection', i.e. the conjunction of a rotation and a reflection on a hyperplane perpendicular to the rotation axis. \end{remark}
Figure \ref{fig:hasse_diagram} shows a Hasse-diagram of some subgroup relations of these groups. \begin{figure}
\caption{A Hasse diagram of the icosahedral and octahedral rotation group (up to conjugation by orthogonal matrices)}
\label{fig:hasse_diagram}
\end{figure} A detailed description of each group is given in Section \ref{sec:certificates}.
We proceed to discuss certain regularities of the sets $F(G)$. For any set $M\subset\Z^2$ we denote the symmetric set obtained from $M$ as \begin{equation*} M^\diamond = M \cup \{(y,x) \ : \ (x,y)\in M\}. \end{equation*} Considering the $f$-vector of the dual polytope $f(P^\vee) = (f_2(P), f_0(P))$ shows that $\mathit{F}$ is invariant under the $\diamond$ operation. The set $F(G)$ is also invariant under the $\diamond$ operation: Let $P$ be a $G$-symmetric polytope and let $b$ be the barycenter of~$P$. The polytope $P^* = \{x\in\R^3 \ : \ \linspan{x,y} \leq 1 \text{ for all } y\in P-b\}$ is a $G$-symmetric polytope with $f(P^*) = (f_2(P),f_0(P))$. We denote by $P^\vee$ any dual polytope of $P$ that is additionally symmetric under the same symmetry group as $P$ (for example $P^*$). Furthermore, note that for $G_1 \geq G_2$ we have $F(G_1)\subseteq F(G_2)$.
Now we have everything to completely understand our main theorem (Theorem \ref{thm:main}).
If we visualize the set $F$ in a two dimensional coordinate system, it looks like a cone translated by $(4,4)$ (cf. Figure~\ref{fig:f_all}).
To show that all integer points in the cone exist as $f$-vectors of 3-polytopes, Steinitz starts with pyramids over $n$-gons, whose $f$-vectors are $(n+1,n+1)$ for $n\in \Z_{>2}$.
He then shows that by stacking a point on a simplicial facet and by cutting a simple vertex, the $f$-vector of the polytope changes by $+(1,2)$ and $+(2,1)$, respectively. The resulting polytope again has simple vertices and simplicial facets, which means that the construction can be repeated. With this method, it can be shown that for all points $f$ in the set $\{(f_0,f_2)\in \Z^2 \colon 2f_0 - f_2 \geq 4 \textnormal{ and } 2f_2 - f_0 \geq 4 \}$ we can construct a 3-polytope $P$ with $f(P)=f$. Here, the pyramids over $n$-gons in a way act as generators. In fact, the pyramids over a $3$-, a $4$- and a $5$-gon would already suffice to generate all $f$-vectors by stacking on simplicial facets and cutting simple vertices.
Clearly, the sets $F(G)$ for finite symmetry groups $G$ are subsets of $F$. Theorem \ref{thm:main} states that for most groups, $F(G)$ is coarser, since only some values modulo $n$ are allowed. For some groups of $\C_n$ and $\Dih_d$ there are also inequalities restricting some residue classes. The third alteration we observe is that one or several 'small' points are left out as it happens for $\Dih_2$ and $\G_1$. These alterations are illustrated in Figure~\ref{fig:f_divers}. \begin{figure}
\caption{The sets $F(\T)$, $F(\Dih_3)$ and $F(\Dih_6)$ respectively. Different colours indicate different residue classes, crosses indicate points that are not contained in the set due to additional inequalities.}
\label{fig:f_divers}
\end{figure}
\section{Conditions on $\mathit{F}(G)$}\label{sec:conditions}
Let $G$ be a finite orthogonal subgroup of $GL_3(\R)$. In this section we deduce conditions on the sets $\mathit{F}(G)$ in dependence of the group $G$. These conditions mostly depend on the structures of orbits under the action of $G$ on $\R^3$.
We start with some notation. A \emph{ray} in $\R^3$ is a set of the form $$\ray{x}=\{\lambda \cdot x \ : \ \lambda>0\} $$ for some $x\in \R^3\backslash\{0\}$. We say that $\ray{x}$ is the \emph{ray generated by $x$}. If we consider two points in the same ray, say $v \in V\backslash \{0\}$ and $\lambda v$ with $\lambda > 0$, we notice that the orbit polytopes $\conv\{G\cdot v\}$ and $\conv\{G\cdot \lambda v\}=\lambda\cdot \conv\{G\cdot v\}$ are the same up to dilation. It is thus useful to consider different types of orbits of rays as in the following definition:
\begin{definition}
A \emph{ray-orbit} is a set of rays $R = \{\ray{x} \ : x\in G\cdot v\}$ for some $v\in\R^3\setminus\{0\}$. A ray-orbit is called \emph{regular}, if $G$ acts regularly on $R$, i.e. $G$ acts transitively with trivial stabilizer on $R$. In this case $|R|=|G|$. If $G$ is not regular on $R$, then $|R|<|G|$ and we call $R$ \emph{non-regular}.
We further call an
orbit $R$ a \emph{flip orbit}, if the stabilizer of $R$ consists of exactly the identity and a rotation of order $2$. The respective axis is called a \emph{flip-axis}. \end{definition}
The following lemma can be used to define a set $\mathit{F}'$ such that $\mathit{F}(G)\subset\mathit{F}'$. It turns out (cf. Section \ref{sec:characterization}) that this outer approximation is not far from being exact and for certain groups these conditions are already sufficient to describe $F(G)$. Here, for a set $M\subset \Z^2$ and an integer $n$, we define \begin{equation*} (M \mod n) =\{((x\mod n),(y\mod n)) \mid (x,y)\in M \}\subset (\Z / n\Z)^2. \end{equation*}
\begin{lemma}\label{lem:f_vectors_mod_n}
Let $O$ denote the set of all non-regular ray-orbits of $G$. Furthermore, let $O_2$ be the set of flip orbits and define $O':=O\backslash O_2$. For an arbitrary $G$-symmetric polytope $P$ we have
$$(f(P) \mod n) \in \left( \sum_{X\in O'} \{(|X|,0)\}^\diamond + \sum_{X\in O_2} \{(0,0),(|X|,0)\}^\diamond\mod n\right), $$ where the sum denotes the Minkowski sum of sets. Here we set the sum over the empty set to be $ \{(0,0)\}$. \end{lemma}
\begin{example} Consider the group $G=\D_3$, the dihedral group of order 6. It has one 3-fold rotation axis and three 2-fold rotation axes (flip-axes) perpendicular to the 3-fold rotation axis. The orbit of a ray in general position, meaning on none of the rotation axes, has trivial stabilizer and is thus a regular orbit with six elements. Let $r$ be a ray on the $3$-fold rotation axis. All rotations around that axis fix $r$ and any of the three flips map $r$ to $-r$. Hence $\{r,-r\}\subset O'$. For a ray $s$ on one of the flip axes, the stabilizer is generated by the rotation of order 2. The orbit thus is in $O_2$ and has 3 elements. There are two orbits of this kind, namely $D_3 \cdot s$ and $\D_3\cdot (-s)$.
If we now consider a $D_3$- symmetric polytope $P$ and its $f$-vector $f(P)$ modulo $6$, all facets and vertices in general position form orbits of 6 and are thus irrelevant modulo 6. $P$ can either have a vertex on both $r$ and $-r$ or it can have a facet on both. The facet would then have to be perpendicular to the axis and be invariant under the 3-fold rotation with all its vertices in general position. Independently, $P$ can either have a vertex, an edge or a facet on all three elements in the orbit of $s$ and, again independently, a vertex, an edge or a facet on the orbit of $-s$. Accounting for all independent possibilities leads to the calculation given in Lemma \ref{lem:f_vectors_mod_n}: \begin{align*}
(F(P) \mod 6) &\in && ( \{(|\{ r, -r \}|,0) \}^\diamond \\
& &&+ \{(0,0),(|\D_3\cdot s|,0)\}^\diamond + \{(0,0),(|\D_3\cdot (-s)|,0)\}^\diamond \mod 6 ) \\
& = && ( \{ (2,0), (0,2) \} + \{ (0,0), (3,0), (0,3) \} + \{ (0,0), (0,3), (3,0) \} \\
& &&\mod 6)\\
&= && (\{ (0,2), (0,5), (2,3), (3,5) \}^\diamond \mod 6) \end{align*} An $f$-vector $f \equiv (2,3)=(2,0)+(0,3)+(0,0) \mod 6$, for instance, yields that there is a vertex on both sides of the 3-fold axis, a facet on $D_3\cdot s$ and an edge on $D_3\cdot (-s)$ \end{example}
\begin{proof}[Proof of Lemma \ref{lem:f_vectors_mod_n}] Let $\ver(P)$ be the set of vertices of $P$ and $\normals(P)$ the set of outer normal vectors of $P$ (thus representing the facets of $P$).
For a $G$ symmetric set $M\subset\R^3$ we use the notation $$M/G = \{\{\ray{x} \ : \ x\in G\cdot y\} \ : \ y\in M\},$$ the set of all ray-orbits of $M$ under $G$. By partitioning the vertices (resp. facets) into orbits and summing over the cardinality of these orbits, we get
$$\mathit{f}(P) = (\sum_{X\in \ver(P)/G} |X|, \sum_{X\in \normals(P)/G} |X|).$$
If $X$ is a regular ray-orbit, then $|X| = n$ and can hence be omitted modulo~$n$:
$$\mathit{f}(P) \equiv (\sum_{X\in (\ver(P)/G)\cap O} |X|, \sum_{X\in (\normals(P)/G)\cap O} |X|) \mod n,$$ where $O$ is the set of all non-regular ray-orbits of $G$. Now observe, that each non-regular orbit $X$ intersects either vertices, edges or facets of $P$. If $X\in O'$, then the induced symmetry prevents $X$ from containing edges. Therefore, $O'\cap (\ver(P)/G)$ and $O'\cap (\normals(P)/G)$ is in fact a partition of $O'$. Analogously, $O_2\cap (\ver(P)/G)$ and $O_2\cap (\normals(P)/G)$ are disjoint subsets of $O_2$. Altogether, that yields \begin{align*}
f(P)&\equiv (\sum_{X\in (\ver(P)/G) \cap O'} |X|, \sum_{X\in (\normals(P)/G)\cap O'} |X|)\\
&+(\sum_{x\in(\ver(P)/G)\cap O_2} |X|, \sum_{X\in (\normals(P)/G)\cap O_2)} |X| )\\
&\in \sum_{X\in O'} \{(0,|X|),(|X|,0)\} + \sum_{X\in O_2} \{(0,0),(0,|X|),(|X|,0)\}\mod n \end{align*} which is equivalent to the assertion. \end{proof}
\section{Base polytopes}\label{sec:base}
The characterization of $f$-vectors for a given group $G$ mainly consists of two parts. First, we need to find conditions on $\mathit{F}(G)$ to show that $\mathit{F}(G)\subset \mathit{F}'$ for a given set $\mathit{F}'$ as the one in Lemma \ref{lem:f_vectors_mod_n} with a few adjustments. Then we need to construct explicit $G$-symmetric polytopes for each $f\in\mathit{F}'$ to show that $\mathit{F}'\subset \mathit{F}(G)$. To do so, we use polytopes with certain properties to construct infinite families of $G$-symmetric polytopes. These so-called \emph{base polytopes}, introduced in this section, form the foundations of our constructions.
\begin{definition} A \emph{base polytope} w.r.t. $G$ is a $G$-symmetric polytope $P$ with the following properties: \begin{enumerate} \item $P$ contains a simplicial facet with trivial stabilizer, \item $P$ contains a simple vertex with trivial stabilizer. \end{enumerate} \end{definition}
A general approach to construct a $G$-symmetric polytope from a given symmetric polytope $P$ (for example from a base polytope) is to take a some vectors $v_1,\dots,v_k$ and consider the convex hull $\conv(P\cup G\cdot\{v_1,\dots,v_k\})$. In order to keep track of how the number of vertices and faces change due to the construction with respect to the faces and vertices of $P$, the following definition is useful: We say $X$ \emph{sees} $Y$ with respect to $P$, if any of the line segments $\conv(x,y)$ with $x\in X$ and $y\in Y$ does not intersect the interior of $P$.
The next lemma is a technical result ensuring that many operations known for general polytopes can also be used for symmetric polytopes (by adding whole orbits instead of points) without getting unexpected edges and facets.
\begin{lemma} \label{lem:small_disc} Let $F$ be a face of $P$ with stabilizer $H$ and supporting hyperplane $S=\{x\ : \ a^tx=b\}$ such that $P\subset \{x\ : \ a^tx\leq b\}$. Then there exists an $H$-symmetric disc $D$ contained in a hyperplane $S'=\{x\ : \ a^tx=b'\}$ with $b'>b$, such that $D$ does not see the set $((G\cdot D)\backslash D)\cup (P\backslash F)$. Moreover, the center of $D$ is fixed by $H$. \end{lemma}
\begin{proof}
First, note that $D$ sees the set $((G\cdot D)\backslash D)\cup (P\backslash F)$ if and only if $D$ sees the set $((G\cdot D)\backslash D)\cup (\ver(P)\backslash \ver(Q))$:
By definition, $D$ cannot see any interior points of $P$. If we take a point $y$ on the boundary of $P$ that is not a vertex, that is, $y$ lies in the relative interior of a face $F'$ of $P$, then a point $x\in D$ sees $y$ if and only if $x$ sees all vertices of $F'$.
The center point
$$c = \frac{1}{|\ver(F)|}\sum_{x\in\ver(F)} x \quad \in \relint (F)$$ of the vertices as well as the outer normal vector $a$ of $F$ are fixed by $H$. Therefore, $c+\delta a$ is also fixed by $H$ for any choice of $\delta\geq 0$. Then, any disc with normal vector $a$ and center on $c+\R_+ a $ is $H$-symmetric.
For $\eps, \delta \geq 0$ define $D(\delta,\eps)$ to be the $H$-symmetric disc with radius $\eps$ parallel to $F$ with distance $\delta$ by \begin{equation*}
D(\delta,\eps)\coloneqq \{x \ : \ \| x-p_\delta\| \leq \eps, \ a^tx = a^t p_\delta \}, \end{equation*} where $p_\delta:=c+\delta a$. Note that for any vertex $v\in \ver(P)\setminus \ver(F)$ the function $\phi_v$ that sends $(\delta,\eps)$ to the distance between $F\cap \conv(D(\delta, \eps), v)$ and the relative boundary of $F$ is continuous and $\phi_v(0,0)>0$. Analogously, for any $A\in G\backslash H$ the function $\phi_A$ that sends $(\delta, \eps)$ to the distance between $$F\cap \conv(D(\delta,\eps), A\cdot D(\delta ,\eps))$$ and the relative boundary of $F$ is continuous and also $\phi_A(0,0)>0$. Hence, we find a small $\delta_0>0$ and a small $\eps_0>0$, such that $\phi_v(\delta_0, \eps_0)>0$ and $\phi_A(\delta_0,\eps_0)>0$ for all $v\in\ver(P)\backslash\ver(Q)$ and $A\in G\backslash H$. Therefore, for $D\coloneqq D(\delta_0,\eps_0)$, all line segments $\conv(x,y)$ with $x\in D$, $y\in (G\cdot D)\backslash D\cup P\backslash Q$ intersect the interior of $P$. By definition, that means that $D$ does not see the set $((G\cdot D)\backslash D)\cup (P\backslash Q)$. \end{proof}
The most important technique to generate new $G$-symmetric polytopes is by stacking vertices on facets. This is a generalization of the constructions of Steinitz~\cite{steinitz1906polyederrelationen}.
\begin{lemma}\label{lem:stacking} Let $P$ be a $G$-symmetric polytope and let $F$ be a face of $P$ of degree $k$. Let $H\leq G$ be the stabilizer of $F$. There is a $G$-symmetric polytope $P^\prime$ with
$$\mathit{f}(P^\prime) = \mathit{f}(P) + \frac{|G|}{|H|}(1,k-1).$$ Furthermore, $P^\prime$ has simplicial facets with trivial stabilizer and a vertex with stabilizer $H$ of degree $k$.
We denote $P'=\CS_{k,|G|/|H|}(P)$ and call the operation \emph{careful stacking on~$F$}. \end{lemma}
\begin{proof} We choose $w$ to be the center point of a disc as in Lemma \ref{lem:small_disc}. Set $$P^\prime = \conv(P\cup Gw).$$
Clearly $P^\prime$ is $G$-symmetric. By the orbit stabilizer theorem, we know that \mbox{$|Gw| = \frac{|G|}{|H|}$}. Furthermore, we know that all edges incident to $w$ are those between $w$ and the vertices of $F$. Therefore, we can think of $P'$ as the polytope $P$ with pyramids over $F$ and the facets in the orbit of $F$, resulting in $\frac{|G|}{|H|}$ new vertices ($Gw$) and $k\cdot \frac{|G|}{|H|}$ new simplicial facets, while the $\frac{|G|}{|H|}$ facets $GF$ are lost. This yields
$$\mathit{f}(P^\prime) = \mathit{f}(P) + \frac{|G|}{|H|}(1,k-1).$$
The facets of the obtained pyramids are simplicial with trivial stabilizer and $w$ has stabilizer $H$ by definition. \end{proof}
For any operation $\delta$ on polytopes we can define the dual operation that sends $P$ to $(\delta(P^\vee))^\vee)$. Applying that to Lemma \ref{lem:stacking}, we get
\begin{remark}
Let $P$ be a $G$ symmetric polytope and $v$ a vertex of degree $k$ with stabilizer $|H|$. Then there is a $G$ symmetric polytope $P'$ such that
$$f(P') = f(P) + \frac{|G|}{|H|}(k-1,1)$$
obtained by the dual operation of $\CS_{k,|G|/|H|}$ called \emph{careful cutting} $\CC_{k, |G|/|H|}$. More precisely
$$P' = \CC_{k,|G|/|H|}(P) = (\CS_{k,|G|/|H|}(P^\vee))^\vee.$$ The polytope $P'$ has simple vertices with trivial stabilizer and a facet of degree $k$ with stabilizer $H$. \end{remark}
Applying the operations $\CS$ and $\CC$ successively to base polytopes, we can generate infinite families of $G$-symmetric polytopes. That shows the existence of a whole integer cone of $f$-vectors. Using the notation $$\Cf~=~(2,1)\N~+~(1,2)\N,$$ we have:
\begin{corollary}\label{cor:base_cone} Let $P$ be a base polytope with respect to $G$. Then $f(P) + n\Cf\subset \mathit{F}(G)$. \end{corollary} \begin{proof} By Lemma \ref{lem:stacking} we know that there is a polytope $P' = \CS_{3,n}(P)$ with $f(P') = f(P) + n(2,1)$. Furthermore, $P'$ has a simple vertex and a simplicial facet with trivial stabilizer. Thus $P'$ is a base polytope. The same is true for $P'' = \CC_{3,n}(P)$. We can thus apply the operations $\CS_{3,n}$ and $\CC_{3,n}$ successively to get $$f(\CS_{3,n}^a \circ \CC_{3,n}^b (P)) = f(P) + a\cdot (n,2n) + b\cdot (2n,n)$$ for any integers $a,b\geq 0$. Hence, $f(P) + n\Cf\subset\mathit{F}(G)$. \end{proof}
This is the main tool for the construction of polytopes with a given $f$-vector.
\section{Certificates}\label{sec:certificates}
Since it is often impossible to construct symmetric base polytopes with small $f$-vector entries, we show that it is possible to replace one base polytope by several polytopes with certain weaker properties and still get an integer cone of $f$-vectors as in Corollary \ref{cor:base_cone}. We thus introduce the concept of certificates, a collection of one or more polytopes with certain properties that ensure (certify) the existence of an integer cone of f-vectors of the form $v+nf$ for some vector $v\in \N^2$ and $n=|G|$. The weaker properties on polytopes we introduce are called left and right type, where the name is only due to our choice of writing down the certificate.
\begin{definition}\label{def:left_right_type} Let $f\in \Z^2$. A \emph{right type} (\emph{left type}) polytope w.r.t. $G$ and $f$ is a $G$-symmetric polytope $P$ with $f(P) = f$ which has simple vertices (simplicial facets) with trivial stabilizer. If it is understood in the context, we omit the group $G$. \end{definition}
Note that a {left type} polytope $P$ can be used to construct a base polytope $P'$ with $f(P') = f(P) + (n,2n)$ by a single $\CS_{3,n}$ operation. On the other hand, a {right type} polytope $P$ can be used to construct a base polytope $P'$ with $f(P') = f(P) + (2n,n)$ by a single $\CC_{3,n}$ operation. The {left type} and {right type} polytopes can thus be interpreted as 'half base'.
\begin{definition} An \emph{RL-certificate} for a vector $\mathit{f}$ consists of a {right type} polytope $P_R$ w.r.t. $\mathit{f}$ and a {left type} polytope $P_L$ w.r.t. $\mathit{f} + (n,2n)$. It can be visualized as: \begin{center} \defLR {$P_L$} {$P_R$} \certRL \end{center} An \emph{LR-certificate} for a vector $\mathit{f}$ consists of a {left type} polytope $P_L$ w.r.t. $\mathit{f}$ and a {right type} polytope $P_R$ w.r.t. $\mathit{f} + (2n,n)$. It can be visualized as: \begin{center}\defLR {$P_L$} {$P_R$} \certLR\end{center} A \emph{triangle-certificate} for a vector $\mathit{f}$ consists of four polytopes $P_L, P_R, P, Q$ such that \begin{enumerate} \item $P_L$ is a {left type} polytope w.r.t. $\mathit{f} + (n,2n)$, \item $P_R$ is a {right type} polytope w.r.t. $\mathit{f} + (2n,n)$, \item $P$ is some $G$-symmetric polytope with $\mathit{f}(P) = \mathit{f}$ and \item $Q$ is some $G$-symmetric polytope with $\mathit{f}(Q) = \mathit{f} + (3n, 3n)$. \end{enumerate} It can be visualized as: \begin{center}\defLRTX {$P_L$} {$P_R$} {$P$} {$Q$} \certTri\end{center} A \emph{B-certificate} for a vector $\mathit{f}$ consists of a base type polytope $P_B$ with $\mathit{f}(P_B) = \mathit{f}$ or two polytopes $P_L, P_R$ such that $P_L$ is a {left type} polytope and $P_R$ is a {right type} polytope, both with respect to $\mathit{f}$. It can be visualized as: \begin{center}\defT {$P_B$} \begin{adjustbox}{valign=c}\certBtikz\end{adjustbox} \quad or \defT {$P_L,P_R$} \begin{adjustbox}{valign=c}\certBtikz\end{adjustbox}.\end{center} We say that we \emph{have a certificate} for a vector $\mathit{f}$ if there is either an LR-certificate, an RL-certificate, a triangle-certificate or a base-certificate w.r.t. $\mathit{f}$. \end{definition}
The benefit of certificates is the following theorem:
\begin{theorem}\label{thm:certificates} Let $\mathit{f}\in\N^2$. If we have a certificate for $\mathit{f}$ then $\mathit{f} + n\Cf \subset \mathit{F}(G)$. \end{theorem} \begin{proof} No matter the type of the certificate, we can use Corollary \ref{cor:base_cone} and Definition \ref{def:left_right_type} to construct $G$-symmetric polytopes with $\mathit{f}$-vectors $\mathit{f}, \mathit{f} + (n,2n), \mathit{f} + (2n,n)$ and $\mathit{f} + (3n,3n)$ as well as two $G$-symmetric base polytopes $P$ and $Q$ with $\mathit{f}(P) = \mathit{f} + (4n,2n)$ and $\mathit{f}(Q) = \mathit{f} + (2n,4n)$. Consider any vector $v = \mathit{f} + a\cdot (n,2n) + b\cdot (2n,n) \in \mathit{f} + n\Cf$. If $a+b\leq 2$ then $v$ is an $\mathit{f}$-vector of one of the polytopes given above. If $a+b \geq 3$ then either $a$ or $b$ are bigger than two. Thus, if $a\geq 2$, we have $$ v = \mathit{f} + (2n,4n) + (a-2)\cdot (n,2n) + b\cdot (2n,n)$$ and a $G$-symmetric polytope can be constructed via Corollary \ref{cor:base_cone} using $a-2$ careful stacking operations and $b$ careful cutting operations on $Q$. When $b\geq 2$ we can use an analog argument on $P$. \end{proof}
The following Lemma shows, for which vectors $f\in \N^2$ we need a certificate to obtain all $f$-vectors given by a certain congruence relation.
\begin{lemma}\label{lem:finite_cone_distribution}
Let $p,q,n$ be integers such that $0\leq p\leq q < n$. Furthermore, let $m$ be the smallest integer such that $m\geq 4+p-2q$ and $n|m$. Moreover, let $$X(p,q) = \{\mathit{f}\in\mathit{F} \ : \ \mathit{f}\equiv (p,q)\mod n\}.$$ Then $$X(p,q) = (p,q) + \{v_1,v_2,v_3\} + n\Cf.$$ Where \begin{align*} v_1(p,q) =& \begin{cases}(m,m) \text{ if } m+2p-q\geq 4\\ (m+2n, m+n) \text{ otherwise}\end{cases}\\ v_2(p,q) =& \begin{cases}(m+n, m+n) \text{ if } m + n + 2p-q \geq 4\\ (m+3n, m+2n) \text{ otherwise}\end{cases}\\ v_3(p,q) =& \begin{cases}(m+2n, m+2n) \text{ if } m+2n+2p-q \geq 4 \\ (m+4n, m+3n) \text{ otherwise}\end{cases}. \end{align*} \end{lemma}
In other words, Lemma \ref{lem:finite_cone_distribution} shows that if we have certificates for $f=(p,q)+v_i$, $i\in {1,2,3}$, we have shown that $X(p,q)\subset F(G)$. \begin{proof} It is obvious that $\mathit{F}$ is a union of translates of the fundamental domain of the lattice $\Lambda$ generated by $n\cdot(2,1)$ and $n\cdot(1,2)$ starting with $X \coloneqq \{(a,b) : 4\leq 2a-b, 2b-a < 4 + 3n\}$ (the translate in $(4,4)$). There are exactly three points of $X(p,q)$ in $X$ since $\Lambda$ has determinant $3n^2$ while the lattice $n\Z^2$ has determinant $n^2$. It is easy to check, that these are exactly the points $v_1, v_2, v_3$ given above. \end{proof}
With Theorem \ref{thm:certificates} and Lemma \ref{lem:finite_cone_distribution} as well as the fact that $\mathit{F}(G)^\diamond = \mathit{F}(G)$, it easily follows:
\begin{corollary} \label{cor:certificates_work} Let $0\leq p_i\leq q_i <n$ for $i=1,\dots,r$ and $$\mathit{F}' = \{f\in\mathit{F} \ : \ f\equiv (p_i,q_i) \text{ for some } i=1,\dots,r\}^\diamond.$$ If, for every $i=1,\dots,r$ and every $k=1, 2 ,3$ we have a certificate with respect to the group $G$ and the vector $(p_i,q_i) + v_k(p_i,q_i)$ (as in Lemma \ref{lem:finite_cone_distribution}), then $\mathit{F}'\subset\mathit{F}(G)$. \end{corollary}
Consequently, to show that $\mathit{F}(G)$ contains a set $\mathit{F}'$ of the form as in Lemma \ref{lem:f_vectors_mod_n} it suffices to state a list of certificates.
\section{Constructions of symmetric polytopes}\label{sec:constructions}
It is easy to construct polytopes symmetric under $G$ by taking the convex hull of several orbits. However, in most cases it is hard to control the $f$-vector of the resulting polytope. In this section, we describe some special operations that can be applied to $G$-symmetric polytopes to obtain other $G$-symmetric polytopes. These operations will be used in Section \ref{sec:characterization} to construct symmetric polytopes needed for certificates. We will emphasize the implications for the $f$-vector and the types of the polytope. The first operation we discuss is to stack smaller copies of a facet of a polytope on that facet to get a kind of narrowed prism. \begin{figure}
\caption{Projection of the regular prism operation over a $3$-gon and a $4$-gon}
\label{fig:RP}
\end{figure}
\begin{lemma}[$k$-gon prism]\label{lem:RP} Let $F$ be a face of $P$ with degree $k$. Let $H$ be the stabilizer of $F$. Then there exists a $G$-symmetric polytope $P'$ with \begin{equation*}
f(P') = f(P) + \frac{|G|}{|H|}(k,k). \end{equation*}
Furthermore, $P'$ is {right type} and has a facet of degree $k$ with stabilizer $H$. If $P$ is {left type} then $P'$ is base. We denote $P' = \RP_{k,|G|/|H|}(P)$ and call it \emph{regular prisms} or \emph{prisms} over $F$. \end{lemma} \begin{proof}
We choose $F'$ to be a small copy of $F$ contained in a small disc over $F$ chosen as in Lemma \ref{lem:small_disc}. Define $P' = \conv(P\cup G\cdot F')$. $P'$ has $k$ additional vertices over $F$, namely the vertices of $F'$. Furthermore, $P'$ contains $k+1$ additional facets over $F$, one quadrilateral facet between any edge of $F$ and its counterpart in $F'$ as well as $F'$ itself, while $F$ is no face of $P'$. Since the same argument holds for all the $\frac{|G|}{|H|}$ elements in the orbit of $F$, we get
$$f(P') = f(P) + \frac{|G|}{|H|}(k,k)$$ as desired. The vertices of $F'$ are simple with trivial stabilizer, thus $P'$ is {right type}. If $P$ has a simplicial facet with trivial stabilizer which is not contained in the orbit of $F$, then this facet is also a facet of $P'$. If, on the other hand, $F$ is a simplicial facet with trivial stabilizer, then so is $F'$. In any case, if $P$ is {left type} then $P'$ is also {left type} and thus base. \end{proof}
By Lemma \ref{lem:small_disc} we can apply any operation over a two dimensional polytope to a full orbit of facets without unexpected edges and facets. In the following we state a few such operations. The missing proofs are all analougusly to the proof of Lemma \ref{lem:RP}.
For example, we can also stack a smaller rotated copy of a facet over itself to get the following:
\begin{figure}
\caption{Projection of the twisted prism operation over a $3$-gon and a $4$-gon}
\label{fig:TP}
\end{figure}
\begin{lemma}[Twisted prism] Let $\mathit{F}$ be a face of $P$ with $k\geq 3$ vertices and let $H$ be the stabilizer of $F$. Then there exists a $G$-symmetric polytope $P'$ with symmetry group $G$ and \begin{equation*}
f(P')=f(P)+\frac{|G|}{|H|}(k,2k). \end{equation*}
Additionally, $P'$ has simplicial facets with trivial stabilizer and is thus {left type}. We denote $P'=\TP_{k,|G|/|H|}(P)$ and call it \emph{twisted prisms over $F$}. \end{lemma}
Instead of a small copy of itself, one can also stack other facets over a given facet. It can yield interesting transitions of the $f$-vector when the appended faces are not in a general position, lets say some edges are parallel to the edges of the given face. The following Lemma illustrates the example of stacking a $2k$-gon regularly on a $k$-gon.
\begin{figure}
\caption{Orthogonal projection of the big prism operation over a $3$-gon and a $4$-gon}
\label{fig:BP}
\end{figure}
\begin{lemma}[2k-gon on k-gon] Let $F$ be a face of $P$ of degree $k$ and $H$ be the stabilizer of $F$. Then there exists a $G$-symmetric polytope $P'$ with
$$f(P') = f(P) + \frac{|G|}{|H|}(2k,2k).$$
Furthermore, $P'$ is base and has a facet of degree $2k$ with stabilizer $H$. We denote $P' = \BP_{2k, |G|/|H|}(P)$ and call it \emph{big prisms} over $F$. \end{lemma}
Similar to the above construction we may also stack a $k$-gon on a $2k$-gon in a regular manner if the stabilizer allows to do so.
\begin{figure}
\caption{Projection of the half prism operation over a $6$-gon and an $8$-gon}
\label{fig:HP}
\end{figure}
\begin{lemma}[$k$-gon on $2k$-gon] \label{lem:HP}
Let $F$ be a face of $P$ with $2k$ vertices for some $k\in \Z_{\geq3}$. Let $H$ be the stabilizer of $F$ and suppose that $|H|$ divides $k$ and $H$ does not contain any reflections. Then there exists a $G$-symmetric polytope $P'$ with \begin{equation*}
f(P')=f(P)+\frac{|G|}{|H|}(k,2k). \end{equation*} Furthermore, $P'$ is {left type}.
We denote $P'=\HP_{2k,|G|/|H|}(P)$ and call it \emph{half prisms over $F$}. \end{lemma}
The constructions seen before share a strong 'regularity'. The stabilizer of a facet implies a symmetry for all facets we stack over it. So it is a reasonable question if we are able to stack something without such 'regularity' (Say, there are no parallel edges). At least if the stabilizer only contains rotations, this is the case. This is illustrated by the following result.
\begin{figure}
\caption{Projection of the twisted half prism operation over a $6$-gon and an $8$-gon}
\label{fig:THP}
\end{figure} \begin{remark} Under the conditions of Lemma~\ref{lem:HP}, there exists a $G$-symmetric polytope $P'$ with \begin{align*}
f(P')=f(P)+\frac{|G|}{|H|}(k,3k). \end{align*} Furthermore, $P'$ is {left type}.
We denote $P'=\THP_{2k,|G|/|H|}(P)$ and call it \emph{twisted half prisms over $F$}. This can be achieved by slightly rotating $F''$ in the proof of Lemma~\ref{lem:HP}, such that the edges of $F''$ are not parallel to the ones of $F$. \end{remark}
We summarize the constructions given in this section in the following list:
\begin{corollary}\label{cor:operations} For a $G$ symmetric polytope $P$, we have the following operations on $P$ ($P'$ denotes the polytope obtained by the operation):
\hspace*{-16pt} \begin{tabular}{llll} name & conditions on $P$ & type & $f(P')-f(P)$\\
$\CS_{k,m}$ & $\exists$ facet $F$, $\deg(F) =k$, $|G_F| = n/m$ & {left type} & $m(1,k-1)$\\
$\CC_{k,m}$ & $\exists$ vertex $v$, $\deg(v) = k$, $|G_v| = n/m$ & {right type} & $m(k-1,1)$\\ $\RP_{k,m}$ & $\exists$ facet $F$, $\deg(F) = k$, $G_F = n/m$ & {right type} & $m(k,k)$\\ $\RP^\vee_{k,m}$ & $\exists$ vertex $v$, $\deg(v) = k$, $G_v = n/m$ & {left type} & $m(k,k)$\\
$\TP_{k,m}$ & $\exists$ facet $F$, $\deg(F) = k$, $|G_F| = n/m$ & {left type} & $m(k,2k)$\\
$\TP^\vee_{k,m}$ & $\exists$ vertex $v$, $\deg(v) = k$, $|G_v| = n/m$& {right type} & $m(2k,k)$\\
$\HP_{2k,m}$ & $\exists$ facet $F$, $\deg(F) = 2k$, $|G_F| = n/m$ & {left type} & $m(k,2k)$\\
$\THP_{2k,m}$ & $\exists$ facet $F$, $\deg(F) = 2k$, $|G_F| = n/m$ & {left type} & $m(k,3k)$\\
$\BP_{k,m}$ & $\exists$ facet $F$, $\deg(F) = k/2$, $|G_F| = n/m$ & {base} & $m(k,k)$\\
\end{tabular} \end{corollary} \begin{proof} This Corollary is basically a summary of this section.
Note that for every operation $\phi$ of the above with certain conditions on $P$, there is a dual operation defined by $\phi^\vee(P) = (\phi(P^\vee))^\vee$. Clearly, the conditions on the argument and the properties of the image are dual to the conditions and properties according to $\phi$. \end{proof}
For the construction of polytopes via the above described methods, we need explicit examples of polytopes to start with. These polytopes need to have 'small' $f$-vectors, since the constructions can only increase the number of vertices and facets. In the following, we give a list of well known polytopes that are used in this work.
\begin{notation}\label{not:special_polytopes} The following polytopes are used in this paper without further discussion. They are all part of the Platonic, the Archimedean and the Catalan solids and thus well studied in literature. Note that this is not a complete list.
\hspace*{-16pt} \begin{longtable}{llcl}
\textit{abbr.} & \textit{name} & \textit{rotat. symm.} & \textit{f-vector} \\
$Tet$ & tetrahedron & $ \T$ & $(4,4)$\\ $TrTet$ & truncated tetrahedron & $ \T$ & $(12,8)$\\ $Oc$ & octahedron & $ \O$ & $(6,8)$\\
$Cub$ & cube & $ \O$ & $(8,6)$\\ $CubOc$ & cuboctahedron & $ \O$ & $ (12,14)$ \\ $RDo$ & rhombic dodecahedron & $ \O$ & $ (14,12)$ \\ $TrCub$ & truncated cube & $ \O$ & $ (24,14)$ \\
$RCubOc$ & rhombicuboctahedron & $\O$ & $ (24, 26)$ \\
$SnCub$ & snub cube & $ \O$ & $ (24,38)$\\
$TrCubOc$ & truncated cuboctahedron & $ \O$ & $ (48,26)$ \\
$Ico$ & icosahedron & $ \I$ & $ (12,20)$\\
$ID$ & icosidodecahedron & $ \I$ & $ (30,32)$\\
$TrI$ & truncated icosahedron & $ \I$ & $ (60,32)$\\
$RID$ & rhombicosidodecahedron & $ \I$ & $ (60,62)$\\
$SnDo$ & snub dodecahedron & $ \I$ & $ (60,92)$\\
$TrID$ & truncated icosidodecahedron & $ \I$ & $ (120,62)$\\
\end{longtable}
\end{notation}
\section{Characterization of $f$-vectors}\label{sec:characterization}
In this section, we go through all finite orthogonal rotation and rotary reflection groups, as described in Theorem \ref{thm:finite_orthogonal_groups}, and characterize their $f$-vectors using the tools developed in the previous sections.
We start with the group $C_n$, the cyclic group of order $n$, which is generated by a rotation with rotation-angle $2\pi/n$ around a given axis. Thus, $C_n$ has two non-regular orbits of size $1$, namely the two rays of the rotation axis.
These are flip-orbits if and only if $n=2$.
\begin{theorem} \label{thm:C_n} For $n>2$ we have \begin{align*}\mathit{F}(\C_n) &= \{\mathit{f}\in\mathit{F} \ : \ \mathit{f}\equiv (1,1) \mod n\}^\diamond\\ &\cup \{\mathit{f} = (\mathit{f}_0,\mathit{f}_2) \in \mathit{F} \ : \ \mathit{f}\equiv (0,2) \mod n, 2\mathit{f}_0 - \mathit{f}_2 \geq 2n-2\}^\diamond. \end{align*} \end{theorem}
\begin{figure}
\caption{The $\C_6$-symmetric polytopes $Pyr_6$, $Pri_6$ and $TPri_6$. This figure as well as following similiar figures are created using Sage \cite{SageMath} and Polymake \cite{polymake:2000}}
\label{fig:special_cyc}
\end{figure}
In the proof, we need the following special $\C_n$-symmetric polytopes, see Figure~\ref{fig:special_cyc} for images: \begin{enumerate}
\item $Pyr_k$ is a pyramid over a regular $k$-gon. $f(Pyr_k) = (k+1,k+1)$.
\item $Pri_k$ is a prism over a regular $k$-gon. $f(Pri_k) = (2k, k+2)$.
\item $TPri_k$ is a twisted prism over a regular $k$-gon. $f(TPri_k) = (2k, 2k+2)$. \end{enumerate}
\begin{proof} Denote by $\mathit{F}'$ the right hand side of the assertion.
By Lemma \ref{lem:f_vectors_mod_n}, we have for any $\C_n$-symmetric polytope $P$ $$\mathit{F}(P) \subset \{(1,0)\}^\diamond + \{(1,0)\}^\diamond \mod n\equiv \{(0,2),(1,1)\}^\diamond \mod n.$$ The case $f(P)\equiv (0,2) \mod n$ yields that $P$ has one facet with at least $n$ vertices on each non-regular orbit. Counting $\{12\}$ -flags we therefore have $$2f_1(P) = f_{12}(P) \geq 3\cdot (f_2(P) - 2) + 2\cdot n.$$ This is, by the Euler Equation \eqref{thm:Euler_Steinitz} \eqref{fml:Euler}, equivalent to $2\mathit{f}_0(P) - \mathit{f}_2(P) \geq 2n-2.$ Since $\mathit{F}(\C_n)$ is invariant under $\diamond$, we thus know that $\mathit{F}(\C_n)\subset\mathit{F}'$. Consider Corollary \ref{cor:certificates_work} and the following table to see that $F'\subset \mathit{F}(\C_n)$:
\tcs
\begin{tabular}{cc|ccc} $(p,q)$ & \ \hspace{0.7cm} & $v_1(p,q)$ & $v_2(p,q)$ &$v_3(p,q)$\\ \hline $(1,1)$& & \defRoot{$(n+1,n+1)$}\defT {$Pyr_n$} \certB & \defRoot{$(2n+1,2n+1)$}\defT {$Pyr_{2n}$} \certB & \defRoot{$(3n+1,3n+1)$}\defT {$Pyr_{3n}$} \certB\\ \hline $(0,2)$& & \defRoot{$(2n,n+2)$}\defLR {$\TP_{n,1}(*)$} {$Pri_n$}\certRL & \defRoot{$(3n,2n+2)$}\defLR {$\TP_{n,1}(*)$} {$\RP_{n,1}(Pri_n)$}\certRL & \defRoot{$(2n,2n+2)$}\defLR {$TPri_n$} {$\RP_{n,1}^2(Pri_n)$}\certLR \end{tabular}
\end{proof}
Unlike $C_n$ with $n>2$, the group $\C_2$ has two flip-orbits. Thus, any $\C_2$-symmetric polytope may have edges with non-trivial stabilizer. This gives us tremendously more freedom in the construction of $\C_2$ symmetric polytopes.
We consider the group $\C_2$ as rotations around the z-axis. The next result shows that, in fact, any $f$-vector can be realized by a $\C_2$ symmetric polytope.
\begin{theorem} We have \begin{align*} \mathit{F}(\C_2) = \mathit{F}. \end{align*} \end{theorem}
\begin{figure}
\caption{The $\C_2$-symmetric polytopes $ST$, $DT$, $RT_6$ and $TT_6$}
\label{fig:special_cyc_two}
\end{figure}
In the proof we will need the following special $\C_2$ symmetric polytopes with small $f$-vectors. See Figure~\ref{fig:special_cyc_two} for images.
\begin{itemize}
\item $ST := \conv \left(\C_2\cdot\{(-1,0,1),(1,1,0),(1,-2,-1)\}\right)$ is a polytope that looks like a 'scattered tent'. $f(ST) = (6,6)$.
\item $DT := \conv \left(\C_2\cdot\{(1,0,1), (2,2,0), (2,-2,0), (1,0,-1)\}\right)$ is a polytope that looks like a tent above and below a $4$-gon. $\mathit{f}(DT) = (8,8)$.
\item $RT_{2k}$ (regular tent) is constructed by taking a regular $2k$-gon and stack a small edge above it, such that the new edge is paralell to two edges of the $2k$-gon. $f(RT_k) = (k+2,k+1)$.
\item $TT_k$ (twisted tent over $k$-gon) can be constructed by taking a regular $k$-gon and stack a small edge above it such that the new edge is not paralell to any edge of the $k$-gon. $f(TT_k) = (k+2,k+3)$.
\end{itemize}
\begin{proof} Of course $\mathit{F}(\C_2)\subset \mathit{F}$. To see that $$\mathit{F} = \{f\in \mathit{F} \ : \ f\equiv (0,0),(0,1),(1,1) \mod 2 \}^\diamond\subset F(\C_2)$$ consider Corollary \ref{cor:certificates_work} and the following table:
\begin{longtable}{cc|ccc} $(p,q)$ & \hspace{0.0cm} & $v_1(p,q)$ & $v_2(p,q)$ & $v_3(p,q)$\\ \hline $(0,0)$& & \defRoot{$(4,4)$}\defT {$Tet$} \certB & \defRoot{$(6,6)$}\defT {$ST$} \certB & \defRoot{$(8,8)$}\defT {$DT$} \certB\\ \hline $(0,1)$ & & \defRoot{$(6,5)$}\defT {$RT_4$} \certB & \defRoot{$(8,7)$}\defT {$RT_6$} \certB & \defRoot{$(6,7)$}\defT {$TT_4$} \certB\\ \hline $(1,1)$ & & \defRoot{$(5,5)$}\defT {$Pyr_4$} \certB & \defRoot{$(7,7)$}\defT {$Pyr_6$} \certB & \defRoot{$(9,9)$}\defT {$Pyr_8$} \certB\\ \end{longtable} \end{proof}
Next, we characterize the $f$-vectors for the group $\Dih_d$. This group consists of a $d$-fold rotation around a given axis $v$ and $2d$ two-fold rotations around axes that are orthogonal to $v$. So we have one non-regular orbit consisting of the two rays belonging to $v$ which is a flip-orbit if and only if $d=2$.
Furthermore, $\Dih_d$ has two flip orbits of size $d$, whose rays each make up half of the two-fold rotation axes.
\begin{figure}
\caption{The $D_5$-symmetric polytopes $DPri_6$, $Dia_3$, $EB_{6,3}$, $B_{4,3}$ and $B_{6,3}$}
\label{fig:special_dih}
\end{figure}
These observations are sufficient to characterize $f$-vectors for the group $\Dih_d$:
\begin{theorem} For $d > 2$ and $n=2d$ we have \begin{align*} \mathit{F}(\Dih_d) &= \{\mathit{f}\in\mathit{F} \ : \ \mathit{f}\equiv (0,2),(2,d) \mod n\}^\diamond\\ &\cup \{\mathit{f} = (\mathit{f}_0,\mathit{f}_2)\in\mathit{F} \ : \mathit{f}\equiv (0,d+2), (d,d+2) \mod n , 2\mathit{f}_0 - \mathit{f}_2 \geq 3d-2\}^\diamond\\ \end{align*} \end{theorem}
For the construction of small $f$-vectors we need the following polytopes, which are $\Dih_d$ symmetric for certain parameters (see also Figure~\ref{fig:special_dih}): \begin{itemize} \item $DPri_k$ (double prism on a $k$-gon), a regular $k$-gon with a smaller copy above and below. $f(DPri_k)= (3k, 2k+2)$. This is $\Dih_d$ symmetric when $d$ divides $k$.
\item $Dia_k$ (diamond of order $k$). This is the dual of the following polytope: A regular $k$-gon with twisted smaller copies above and below. \\ $f(Dia_k)=(4k+2,3k)$. This is $\Dih_d$ symmetric when $d$ divides $k$.
\item $EB_{2k,l}$ (edge belt): the convex hull of $l$ regular $2k$-gons arranged along an $l$-gon, such that two neighboring $k$-gons intersect in an edge. \\
$f(EB_{k,l})=(l\cdot 2(k-1), 2 \cdot \left\lfloor \frac{k-1}{2} \right\rfloor\cdot l +l+2)$. This is $\Dih_d$ symmetric when $d$ divides $l$.
\item $B_{2k,l}$ (belt): the convex hull of $l$ regular $2k$-gons arranged along an $l$-gon, such that two neighboring $2k$-gons intersect in a vertex. \\ $f(B_{2k,l}) = ((2k-1)\cdot l, 2(1+\lfloor \frac k 2 \rfloor)\cdot l +2)$. This is $\Dih_d$ symmetric, when $d$ divides $l$.
\end{itemize}
\begin{proof} Denote by $\mathit{F}'$ the right hand side of the assertion.
By Lemma \ref{lem:f_vectors_mod_n} we know that \begin{align*} \mathit{F}(\Dih_d) &\subset \{f\in \mathit{F} \ : \ (f\mod n)\in \{(2,0)\}^\diamond + \{(0,0),(0,d)\}^\diamond + \{(0,0),(0,d)\}^\diamond\}\\ &= \{f\in \mathit{F} \ : \ f\equiv (0,2),(0,d+2),(2,d),(d,d+2)\mod n\}^\diamond. \end{align*} If $\mathit{f}_2(P) \equiv d+2 \mod n$ then $P$ has facets on the rotation axis containing at least $d$ vertices. Furthermore, $P$ has also facets on exactly one of the flip-orbits, containing at least $4$ vertices. By counting $\{1,2\}$-flags we have $$2f_1(P) = f_{12}(P) \geq 3\cdot(f_2-d-2) + d\cdot 2 + 4\cdot d.$$
By applying Euler's equation \ref{thm:Euler_Steinitz} \eqref{fml:Euler}, we have $2\mathit{f}_0(P) - \mathit{f}_2(P) \geq 3d-2$. Since $\mathit{F}(\Dih_d)$ is invariant under $\diamond$, we know that $\mathit{F}(\Dih_d)\subset \mathit{F}'$. To see that $\mathit{F}'\subset \mathit{F}(\Dih_d)$ consider Corollary \ref{cor:certificates_work} and the following table:
\tcs
\begin{longtable}{cc|ccc} $(p,q)$ & \ \hspace{0.7cm} & $v_1(p,q)$ & $v_2(p,q)$ & $v_3(p,q)$\\ \hline $(0,2)$ & & \defRoot{$(2n,n+2)$}\defLR {$\BP_{d,2}(TPri_d)$} {$Pri_n$} \certRL & \defRoot{$(n,n+2)$}\defLR {$TPri_d$} {$DPri_n$} \certLR & \defRoot{$(2n,2n+2)$}\defT {$\RP_{d,2}(TPri_d)$} \certB\\ \hline $(2,d)$& & \hspace{-14pt} \defRoot{$(2n+2,n+d)$}\defLR {$\CS_{6,2}\circ\CC_{3,n}(Pri_d)$} {$Dia_d$} \certRL & \defRoot{$(n+2, n+d)$}\defLR {$\CS_{d,2}(Pri_d)$} {$\RP^\vee_{d, 2}(Dia_d)$} \certLR & \hspace{-14pt} \defRoot{$(2n+2, 2n+d)$}\defLR {$\CS_{d,2}\circ\RP_{d,2}(Pri_d)$} {$(\RP^{\vee}_{d, 2})^2(Dia_d)$} \certLR\\ \hline $(0,d+2)$& & \defRoot{$(n,d+2)$}\defLR {$\TP_{d,2}(*)$} {$Pri_d$} \certRL & \defRoot{$(2n,n+d+2)$}\defT {$EB_{6,d}$} \certB & \defRoot{$(3n,2n+d+2)$}\defT {$\BP_{d,2}(Pri_d)$} \certB\\ \hline $(d,d+2)$& & \hspace{-14pt} \defRoot{$(2n+d,n+d+2)$}\defT {$B_{6,d}$} \certB & \defRoot{$(n+d,n+d+2)$}\defLR {$B_{4,d}$} {$B_{8,d}$} \certLR & \defRoot{$(2n+d,2n+d+2)$}\defT {$\RP_{d,2}(B_{4,d})$} \certB\\ \end{longtable} \end{proof}
Next, we consider $\Dih_2$. This group can be interpreted as the group of all flips around coordinate axes. Thus, $\Dih_2$ has exactly three flip-orbits of size $2$, each consisting of the rays of one flip-axis.
Interestingly, the respective characterization of $f$-vectors contains the exceptional case $f = (6,6)$, which can not be realized by a $\Dih_2$ symmetric polytope.
\begin{theorem} We have
\begin{align*} \mathit{F}(\Dih_2) &= \{\mathit{f}\in\mathit{F} \ : \ \mathit{f}\equiv (0,0),(0,2),(2,2) \mod 4\}^\diamond\setminus\{(6,6)\}. \end{align*} \end{theorem}
For small $f$-vectors we consider the following special $\Dih_2$-symmetric polytopes (see Figure~\ref{fig:special_dih_two}): \begin{itemize}
\item $Dih_2(10,10) = \conv(\Dih_2\cdot \{(6,0,0),(1,1,1),(2,1,-2)\}$
\item $Dih_2(10,14) = \conv(\Dih_2\cdot \{(2,0,0),(1,1,1),(\frac 34, 1, \frac 32)\})$
\item $Dih_2(12,12) = \conv \{\Dih_2\cdot (2,1,1), (-2,1,1), (1,0,2)\}$
\item $Dih_2(14,14) = \conv \{\Dih_2\cdot (4,0,0),(1,3,0),(2,0,1),(0,2,1)\}$
\item $Dih_2(18,18) = \conv \{\Dih_2\cdot (4,0,0),(1,3,0),(2,0,1),(0,2,1),(\frac 18, \frac{23}{8}, 1)\}$ \end{itemize}
\begin{figure}
\caption{The $\Dih_2$-symmetric polytopes $Dih_2(10,10)$, $Dih_2(10,14)$, $Dih_2(12,12)$, $Dih_2(14,14)$ and $Dih_2(18,18)$}
\label{fig:special_dih_two}
\end{figure}
\begin{proof} Denote by $\mathit{F}'$ the right hand side of the assertion. By Lemma \ref{lem:f_vectors_mod_n} we have \begin{align*} \mathit{F}(\Dih_2)&\subset \{f\in\mathit{F} \ : \\
& \hspace{2em}\ (f \mod 4)\in \{(0,0),(0,2)\}^\diamond + \{(0,0),(0,2)\}^\diamond + \{(0,0),(0,2)\}^\diamond\}\\ &=\{f\in\mathit{F} \ : \ f\equiv (0,0),(0,2),(2,2)\mod 4\}^\diamond. \end{align*} Next we show that $f(P)\neq (6,6)$. Suppose $f(P)\equiv (2,2) \mod 4$. Then each flip orbit intersects two vertices, two edges and two facets of $P$, respectively. These vertices and facets are incident with at least $4$ edges due to the induced symmetry. Furthermore, due to the symmetry no edge intersects more than one flip-axis. Thus $f_0(P) + f_2(P) -2 = f_1(P)\geq 4\cdot 4 + 2= 18$ and therefore $f(P) \neq (6,6)$.
Altogether, this shows $F(\Dih_2)\subset F'$. To see that $F'\subset F(\Dih_2)$ consider Corollary \ref{cor:certificates_work} and the following table (note that the triangle certificate for $f = (6,6)$ is missing the top entry, which is not relevant for the existence of $f$-vectors other than $(6,6)$). \tcs
\begin{longtable}{cc|ccc} \hspace{1em}$(p,q)$ & \ \hspace{0.7cm} & $v_1(p,q)$ & $v_2(p,q)$ & $v_3(p,q)$\\ \hline \hspace{1em}$(0,0)$ & & \defRoot{$(4,4)$}\defT {$Tet$} \certB & \defRoot{$(8,8)$}\defT {$DT$} \certB & \defRoot{$(12,12)$}\defT {$Dih_2(12,12)$} \certB\\ \hline \hspace{1em}$(0,2)$ & & \defRoot{$(8,6)$}\defLR {$CubOc$} {$Cub$} \certRL & \defRoot{$(12,10)$}\defLR {$\RP_{4,2}(TPri_4)$} {$DPri_4$} \certRL & \defRoot{$(8,10)$}\defLR {$TPri_4$} {$EB_{6,4}$} \certLR\\ \hline \hspace{1em}$(2,2)$ & & \defRoot{$(6,6)$}\defLRTX {$Dih_2(10,14)$\hspace{0.5cm}} {\hspace{0.5cm}$Dih_2(10,14)^\vee$} {$\varnothing$} {$Dih_2(18,18)$}\certTri
& \defRoot{$(10,10)$}\defT {$Dih_2(10,10)$} \certB & \defRoot{$(14,14)$}\defT {$Dih_2(14,14)$} \certB\\ \hline \end{longtable} \end{proof}
Now we consider the tetrahedral rotation group $\T$. Take a given regular tetrahedron with barycenter $0$. The tetrahedral rotation group $\T$ contains four order three rotations around axis which go through a vertex and the respectively opposing facet. Furthermore it contains flips around axes through the midpoints of two opposing edges.
This group has two non-regular orbits of size $4$ which are not flip-orbits and a flip-orbit of size $6$.
We do not need any further polytopes other than the polytopes in Notation~\ref{not:special_polytopes} to proof the following characterization:
\begin{theorem} We have \begin{align*} \mathit{F}(\T) = \{\mathit{f}\in\mathit{F} \ : \ \mathit{f}\equiv (0,2),(0,8),(4,4),(4,10) \mod 12\}^\diamond. \end{align*} \end{theorem}\begin{proof} Denote by $F'$ the right hand side of the assertion.
By Lemma \ref{lem:f_vectors_mod_n} we have $$\mathit{F}(\T) \subset \{\mathit{f}\in\mathit{F} \ : \ (\mathit{f}\mod 12)\in \{(0,8), (8,0), (4,4)\} + \{(0,0),(0,6),(6,0)\} $$ which is equivalent to $\mathit{F}(T)\subset F'$. To see that $F'\subset \mathit{F}(T)$ consider Table \ref{tab:tetra} in the appendix. \end{proof}
Next, we consider the octahedral rotation group $\Oc$. Take a given regular cube with barycenter $0$. The group $\Oc$ contains three four-fold rotations around axis through opposing facets. Furthermore it contains four three-fold rotations around axis through two opposing vertices and flips around axis through the midpoints of the edges.
Thus, the group $\Oc$ has one non-regular orbit of size $6$ consisting of the rays of the threefold rotation axes. Furthermore it has a non-regular orbit of size $8$ consisting of all rays of the fourfold rotation axes. Lastly, it has a flip orbit of size $12$ consisting of all rays belonging to the flip axes.
\begin{theorem} We have \begin{align*} \mathit{F}(\Oc) = \{\mathit{f}\in\mathit{F} \ : \ \mathit{f}\equiv (0,2),(0,14),(6,8),(6,20),(8,18),(12,14) \mod 24\}^\diamond. \end{align*} \end{theorem}
Without loss of generality we may consider $\Oc$ as the group generated by
\begin{equation*} \begin{pmatrix} 0 & -1 & 0 \\ 1& 0 & 0 \\ 0& 0& 1 \end{pmatrix}, \begin{pmatrix} 1 & 0 & 0 \\ 0 & 0 & -1 \\ 0& 1& 0 \end{pmatrix} \end{equation*}
and state the following $\O$ symmetric polytopes in an explicit way (see also Figure~\ref{fig:special_oct}):
\begin{itemize}
\item $Oct(72,50)=\conv(\Oc\cdot \{(1,2,3) , (3,2,1), (1,0,4)\})$
\item $Oct(30,44)=\conv(\Oc\cdot \{(4,0,0) , (-1,2,2)\})$
\item $Oct(32,42)=\conv(\Oc\cdot \{(2,2,2) , (0,1,3)\})$
\item $Oct(60,38)=\conv(\Oc\cdot \{(8,8,0) , (1,7,5), (1,7,-5)\})$
\item $Oct(54,32) = (\CS_{3,8}(SnCub))^\vee$
\end{itemize}
\begin{figure}
\caption{The $\Oc$-symmetric polytopes $Oct(72,50)$, $Oct(30,44)$, $Oct(32,42)$, $Ot(60,38)$ and $Oct(54,32)$}
\label{fig:special_oct}
\end{figure}
\begin{proof} Denote by $\mathit{F}'$ the right hand side of the assertion. By Lemma \ref{lem:f_vectors_mod_n} we have $$\mathit{F}(\Oc)\subset \{ f\in\mathit{F} \ : \ (f \mod 12)\in \{(0,6)\}^\diamond + \{(0,8)\}^\diamond + \{(0,0),(0,12)\}^\diamond\}$$ which is equivalent to $\mathit{F}(\Oc)\subset \mathit{F}'$. To see that $\mathit{F}'\subset\mathit{F}(\Oc)$ consider Corollary \ref{cor:certificates_work} and Table \ref{tab:octa} in the appendix.
\end{proof}
Next, we consider the icosahedral rotation group $\I$. Take a given icosahedron with barycenter $0$. The group $\I$ contains five-fold rotations around axes through opposing vertices. Furthermore, it contains three-fold rotations around axes through opposing facets and flips around axes through the midpoints of edges.
Therefore, the group $\I$ has three different non-regular orbits. One non-flip of size $20$ consisting of the rays belonging to the threefold rotation axes. Furthermore, there is a non-generate orbit of size $12$ consisting of the rays belonging to the five-fold rotations. Lastly, there is a flip-orbit consisting of the $30$ rays belonging to the flip axes.
\begin{theorem} We have \begin{align*} \mathit{F}(\I) = \{\mathit{f}\in\mathit{F} \ : \ \mathit{f}\equiv (0,2),(0,32),(12,20), (12,50), (20,42),(30,32)\mod 60\}^\diamond. \end{align*} \end{theorem}
Let $ \Phi= \frac{1+\sqrt{5}}{2} $ be the golden ratio. Without loss of generality we consider $\I$ as the matrix group generated by
\begin{align*} \begin{pmatrix} -1 & 0 & 0 \\ 0 & -1 & 0\\ 0 & 0 & 1 \end{pmatrix}, \begin{pmatrix} 0 & 0 & 1 \\ 1 & 0 & 0\\ 0 & 1 & 0 \end{pmatrix}, \begin{pmatrix} \frac 1 2 & \frac 1 2 \Phi - \frac 1 2 & - \frac 1 2 \Phi \\
-\frac 1 2 \Phi + \frac 1 2 & - \frac 1 2 \Phi & -\frac 1 2\\ - \frac 1 2 \Phi & \frac 1 2 & -\frac 1 2 \Phi + \frac 1 2 \end{pmatrix} \end{align*}
and state the following $\I$ symmetric polytopes in an explicit way (see also Figure~\ref{fig:special_ico}:
\begin{itemize}
\item $Ico(72, 50)=\conv(\I\cdot \{(1,0,1 ), (-\frac 1 2 \Phi -\frac 1 2, 0, - \frac 1 2 \Phi )\})$
\item $Ico(72, 110)=\conv(\I\cdot \{(0,1,\Phi ), (-\frac 1 4, \frac 3 4 \Phi + \frac 1 4 , \frac 3 4 \Phi - \frac 1 2 )\})$
\item $Ico(80, 42)=\conv(\I\cdot \{(1,0,1 ), (-\frac 1 2, 0, - \frac 1 2 \Phi - \frac 1 2 )\})$
\item $Ico(150, 92)=\conv(\I\cdot \{(0,0,1 ), (\frac 1 6, \frac 2 3 \Phi - \frac 2 3, \frac 1 6 \Phi + \frac 2 3 ), (\frac 1 {12}, \frac 7 {12} \Phi - \frac 7 {12}, \frac 1 {12} \Phi + \frac 5 6)\})$
\item $Ico(180,122) =\conv(\I\cdot \{(\frac{1}{2} \Phi + \frac 1 4, -\frac 1 4 \Phi +\frac 1 4, \frac 1 4 \Phi + \frac 1 2 ), (-\frac 1 6 \Phi +1, - \frac 1 6,\frac 1 6 \Phi + \frac 5 6 ),$\\ $(-\frac 1 3 \Phi +1 , -\frac 1 3, \frac 1 3 \Phi + \frac 2 3 )\})$ \end{itemize}
\begin{figure}
\caption{The $\I$-symmetric polytopes $Ico(72,50)$, $Ico(72,110)$, $Ico(80,42)$, $Ico(150,92)$ and $Ico(180,122)$}
\label{fig:special_ico}
\end{figure}
\begin{proof}
Denote by $\mathit{F}'$ the right hand side of the assertion. By Lemma \ref{lem:f_vectors_mod_n} we therefore have $$\mathit{F}(\I)\subset \{ f\in\mathit{F} \ : \ (f\mod 60) \in \{(0,20)\}^\diamond + \{(0,12)\}^\diamond + \{(0,0),(0,30)\}^\diamond\},$$ which is equivalent to $\mathit{F}(\I)\subset \mathit{F}'$. To see that $\mathit{F}'\subset\mathit{F}(\I)$ consider Corollary \ref{cor:certificates_work} and Table \ref{tab:ico} in the appendix.
\end{proof}
Now, consider the rotary reflection group $G = \G_d$. It is generated by the product $\Theta\cdot \sigma$ where $\Theta$ is a $2d$-fold rotation and $\sigma$ is a reflection orthogonal to the rotation axis of $\Theta$. The order of $\G_d$ is $n=2d$. The group has a non-regular orbit of size two on the rotation axis. All other orbits are regular. We need no further polytopes to derive the following result:
\begin{theorem} We have, $$\mathit{F}(\G_d) = \{\mathit{f}\in\mathit{F} \ : \ \mathit{f}\equiv (0,2) \mod n\}^\diamond \textnormal{ for } d > 2.$$ \end{theorem} \begin{proof} Denote by $F'$ the right hand side of the assertion. By Lemma \ref{lem:f_vectors_mod_n} we have $F(\G_d)\subset F'$. To see that $F' \subset F(\G_d)$ consider the following table:
\tcs
\begin{tabular}{cc|ccc} \hspace{1em}$(p,q)$ & \ \hspace{0.7cm} & $v_1(p,q)$ & $v_2(p,q)$ & $v_3(p,q)$\\ \hline \hspace{1em}$(0,2)$ & & \defRoot{$(2n,n+2)$}\defLR {$\RP_{d,2}^2$} {$Pri_{2d}$} \certRL & \defRoot{$(n,n+2)$}\defLR {$TPri_d$} {$DP_{2d}$} \certLR & \defRoot{$(2n,2n+2)$}\defT {$\RP_{d,2}(TPri_d)$} \certB \hspace{1em}\\ \end{tabular} \end{proof}
Next, we consider the group $G_2$ where the rotation axis provides a flip-orbit of size two. As a special $G_2$-symmetric polytope we consider $G_d(16,18)$ the square orthobi cupola also known as Johnson solid $28$.
\begin{theorem} We have $$F(\G_2) = \{f\in \mathit{F} \ : \ f\equiv (0,0),(0,2) \mod 4\}.$$ \end{theorem} \begin{proof} Denote by $F'$ the right hand side of the assertion. By Lemma \ref{lem:f_vectors_mod_n} we have $F(\G_2)\subset F'$. To see that $F'\subset F(\G_2)$ consider the following table:
\tcs
\begin{tabular}{cc|ccc} \hspace{1em} $(p,q)$ & \ \hspace{0.7cm} & $v_1(p,q)$ & $v_2(p,q)$ & $v_3(p,q)$\\ \hline \hspace{1em} $(0,0)$ & & \defRoot{$(4,4)$} \defT {$Tet$} \certB & \defRoot{$(8,8)$} \defT {$DT$} \certB & \defRoot{$(12,12)$} \defT {$Dih_2(12,12)$} \certB \\ \hline \hspace{1em} $(0,2)$ & & \defRoot{$(8,6)$} \defLR {$CubOc$} {$Cub$} \certRL & \defRoot{$(12,10)$} \defLR {$G_d(16,18)$} {$DPri_4$} \certRL & \defRoot{$(8,10)$} \defLR {$TPri_4$} {$\RP_{4,2}(Cub)$} \certLR\hspace{1em}\\ \end{tabular}
\end{proof}
The group $\G_1$ is the point reflection at the origin. As a matrix group it is generated by the negative identity matrix.
\begin{theorem} We have $$F(\G_1) = \{\mathit{f}\in\mathit{F} \ : \ \mathit{f}\equiv (0,0) \mod 2\}^\diamond\setminus \{(4,4),(6,6)\}.$$ \end{theorem}
For the proof, we give the following $\G_1$-symmetric polytopes in an explicit way (see also Figure~\ref{fig:special_point_reflection}):
\begin{itemize}
\item $PRefl(8,10)=\conv(\G_1\cdot \{(5,0,0) , (0,5,0), (0,0,5),(3, -\frac 1{2},\frac 52)\})$
\item $PRefl(10,10)=\conv(\G_1\cdot \{(4,0,0), (0,4,0) ,(-1, 1,4), (1, -1,4) ,(2, -3,2)\})$
\end{itemize}
\begin{figure}
\caption{The $G_1$-symmetric polytopes $PRefl(8,10)$ and $PRefl(10,10)$}
\label{fig:special_point_reflection}
\end{figure}
\begin{proof} Denote by $F'$ the right hand side of the assertion. First we show that $ (4,4), (6,6)\notin F(\G_1)$. Note that any facet and its reflection at the origin do not intersect. From that we can first conclude that a $\G_1$-symmetric polytope has at least 6 vertices. Secondly, a $G_1$-symmetric polytope with 6 vertices has to be simplicial. By Theorem \ref{thm:Euler_Steinitz}, that means $2f_0-f_2=4$ which is not satisfied for $f=(6,6)$. Together with Lemma \ref{lem:f_vectors_mod_n} we thus have $F(\G_1)\subset F'$.
To see that $F'\subset F(\G_1)$ consider Corollary \ref{cor:certificates_work} and the following table: \tcs
\begin{tabular}{cc|ccc} \hspace{1em} $(p,q)$ & \ \hspace{0.7cm} & $v_1(p,q)$ & $v_2(p,q)$ & $v_3(p,q)$\\ \hline \hspace{1em} $(0,0)$ & & \defRoot{$(4,4)$}\defLRTX {$Cub$} {$Oc$} {$\varnothing$}{$PRefl(10,10)$}\certTri & \defRoot{$(6,6)$}\defLRTX {$PRefl(8,10)$} {\hspace{12pt}$PRefl(8,10)^\vee$} {$\varnothing$}{$\CC_{3,2}(PRefl(8,10))$}\certTri & \defRoot{$(8,8)$} \defT {$DT$} \certB \hspace{1em} \\ \end{tabular}
\end{proof}
This finishes the proof of Theorem \ref{thm:main}.
\section{Open questions}\label{sec:open}
Up to this point only reflection free symmetries have been discussed. To prove the main theorem, Theorem \ref{thm:main}, we mostly followed the characterization \ref{thm:finite_orthogonal_groups} from Grove and Benson \cite[Theorem 2.5.2]{benson1985finite_reflection_groups}. In order to characterize the $f$-vectors of the remaining symmetry groups, it is important to know about the contained reflections and their arrangement. Therefore, we propose the following characterization of symmetry groups containing reflections instead:
Denote a rotation around the $z$-axis by an angle of $2\pi/d$ by $\Theta_d$ and a reflection in a plane $H$ by $\sigma_H$. Let further $z^\perp$ and $x^\perp$ be the planes through the origin orthogonal to the $z$-axis and the $x$-axis, respectively. Then we have the following:
\begin{theorem}\label{thm:alternative_characterization_reflection_groups} Let $G$ be a finite orthogonal subgroup of $GL_3(\R)$. If $G$ contains a reflection then it is isomorphic to one of the following: \begin{enumerate} \item The cyclic rotation group with an additional reflection orthogonal to the rotation axis $C_d^{\perp} \simeq\left <\Theta_d, \sigma_{z^\perp} \right>$, $d\geq 1$, \item the cyclic rotation group with an additional reflection that contains the rotation axis $C_d^{\subset} \simeq \left <\Theta_d, \sigma_{x^\perp} \right>$, $d\geq 2$,
\item the cyclic rotation group containing both additional reflections \\ $C_d^{\perp,\subset}~\simeq~\left <\Theta_d, \sigma_{z^\perp}, \sigma_{x^\perp} \right>$, $d\geq 2$, \item the rotary reflection group of order $2d$ with an additional reflection containing the rotation axis $G_d^{\subset} = \left <-\Theta, \sigma_{x^\perp}\right>$ where $\Theta = \Theta_{2d}$ if $d$ is even and $\Theta = \Theta_d$ if $d$ is odd, $d\geq 2$ \item the full tetrahedral, octahedral and icosahedral symmetry group $\hat\T$, $\hat\O$, $\hat\I$ respectively,
\item the group $\T^\ast = \T\cup \{-X \ : \ X\in\T\}$. \end{enumerate} \end{theorem}
\begin{proof}
In \cite[Theorem 2.5.2]{benson1985finite_reflection_groups} it is shown that any finite orthogonal group consists of rotations and negatives of rotations. Observe that $\Theta$ is a flip around an axis $v$ if and only if $-\Theta$ is a reflection on $v^\perp$. Furthermore, $\left <-\Theta_d \right>$ contains a reflection if and only if $d\equiv 2 \mod 4$. If $d\not\equiv 2 \mod 4$ this group has order $d$ for even $d$ and order $n = 2d$ for odd $d$. Using this, we can compare the above list with the characterization to observe that these characterizations are equivalent. In particular (in the notation of Benson) \begin{enumerate} \item corresponds to $(C_3^{d})^\ast$ for even $d$ and to $C_3^{2d} ] C_3^d$ for odd $d$, \item corresponds to $\mathcal H_3^d ] C_3^d$,
\item corresponds to $(\mathcal H_3^{d})^\ast$ for even $d$ and $\mathcal H_3^{2d} ] \mathcal H_3^d$ for odd $d$, \item corresponds to $\mathcal H_3^{2d} ] \mathcal H_3^{d}$ for even $d$ and to $(\mathcal H_3^{d})^\ast$ for odd $d$, \item corresponds to $\mathcal W ] \mathcal T$, $\mathcal W ^\ast$ and $\mathcal I^\ast$, \item corresponds to $\mathcal T^\ast$,
\end{enumerate} \end{proof}
By applying Lemma \ref{lem:f_vectors_mod_n} we already have the following : \begin{enumerate} \item $F(C_d^\perp) \subset \{f\in F \ : \ f\equiv (0,2) \mod d\}^\diamond$ for $d>2$, \\
$F(C_2^\perp)\subset \{f\in F \ : \ f\equiv (0,0) \mod 2\}^\diamond$,\\
$F(C_1^\perp) \subset F$ , \item $F(C_d^\subset) \subset \{f\in F \ : \ f\equiv (0,2),(1,1) \mod d\}^\diamond$ for $d>2$, \\
$ F(C_2^\subset) \subset F$, \item $F(C_d^{\perp,\subset}) \subset \{f\in F \ : \ f\equiv (0,2) \mod d\}^\diamond$, $d>2$,\\
$F(C_2^{\perp,\subset}) \subset \{f\in F \ : \ f\equiv (0,0) \mod 2\}^\diamond$, \item $F(\G_d^\subset) \subset \{f\in F \ : \ f\equiv (0,2) \mod 2d\}^\diamond$, $d>2$,\\
$F(\G_2^\subset) \subset \{f\in F \ : \ f\equiv (0,0), (0,2) \mod 4\}^\diamond$
\item $F(\hat\T) \subset F(\T)$, $F(\hat\O) \subset F(\O)$, $F(\hat\I) \subset F(\I)$, \item $F(\T^\ast) \subset \{f\in F \ : \ f\equiv (0,2),(0,8),(6,8) \mod 12\}^\diamond$. \end{enumerate}
Next, we consider another related problem. For any three dimensional polytope $P$ we define its linear symmetry group by $\Symm(P) = \{ A\in \R^{3\times 3} \ : \ A\cdot P = P\}$. Furthermore, we denote $$\overline{F(G)} = \{ f \in F \ : \ \text{there is a polytope $P$ with} f(P) = f, \Symm(P) = G\}.$$ Then $\overline{F(G)}\subset F(G)$. We conjecture that every 'large enough' $f$-vector in $F(G)$ is also contained in $\overline{F(G)}$ while there are finitely many $f$-vectors in $F(G)\backslash\overline{F(G)}$. This conjecture is indicated by \cite{Isaacs1977linear_groups_as_stabilizers_of_sets}.
Another interesting problem is the problem in higher dimensions, even the traditional $f$-vector problem is very hard in four dimensions. Nevertheless, the constructions in Corollary \ref{cor:operations} and Lemma \ref{lem:f_vectors_mod_n} can be generalized for any dimension yielding an inner and outer approximation of $f$-vectors of symmetric polytopes. It is expectable that these approximations differ a lot.
To conclude this paper we summarize open problems which deserve further investigation: \begin{enumerate} \item what is $F(G)$ if $G$ is one of the groups described in Theorem \ref{thm:alternative_characterization_reflection_groups}? \item What is $\overline{F(G)}$? \item Is it possible to find good inner and outer approximations of $F(G)$ in dimension~$4$? \item What do we know in arbitrary dimensions? \item What kind of flag-vectors are possible for symmetric 3-polytopes? \end{enumerate}
\section*{Appendix}
\tcs
\begin{longtable}{cc|ccc} \caption{Certificates for the tetrahedral rotation group $\T$.}\\
\label{tab:tetra} $(p,q)$ & \ \hspace{0.7cm} & $v_1(p,q)$ & $v_2(p,q)$ & $v_3(p,q)$\\ \hline $(0,2)$ & & \defRoot{$(24,14)$}\defLR {$\TP_{3,4}(*)$} {$TrCub$} \certRL & \defRoot{$(12,14)$}\defLRTX {$\TP_{3,4}(*)$} {$\RP_{3,4}(TrCub)$} {$CubOc$} {$\TP_{3,4}\circ\RP_{3,4}(TrCub)$}\certTri & \defRoot{$(24,26)$}\defLRTX {$\TP_{3,4}(\ast)$\hspace{0.5cm}} {\hspace{0.5cm}$\RP^2_{3,4}(TrCub)$} {$RCubOc$} {$\RP_{3,8}(\ast)$}\certTri\\ \hline $(0,8)$ & & \defRoot{$(12,8)$}\defLR {$\TP_{3,4}(*)$} {$TrTet$} \certRL & \defRoot{$(24,20)$}\defLR {$\TP_{3,4}(\ast)$} {$\RP_{3,4}(TrTet)$} \certRL & \defRoot{$(12,20)$}\defLRTX {$\THP_{6,4}(TrTet)$} {\hspace{1cm}$\RP^2_{3,4}(TrTet)$} {$Ico$} {$\RP^2_{3,4}\circ\TP_{3,4}(TrTet)$}\certTri\\ \hline $(4,4)$ & & \defRoot{$(4,4)$}\defLRTX {$\TP_{3,4}(*)$} {$\TP^\vee_{3,4}(*)$} {$Tet$} {$\RP_{3,4}^3(*)$}\certTri & \defRoot{$(16,16)$}\defT {$\RP_{3,4}^\vee(Tet) \ \RP_{3,4}(Tet) $} \certB & \defRoot{$(28,28)$}\defT {$(\RP_{3,4}^\vee)^2 (Tet) \ , \ \RP_{3,4}^2(Tet) $} \certB\\ \hline $(4,10)$ & & \defRoot{$(16,10)$}\defLR {$\TP_{3,4}(*)$} {$\CC_{3,4}(Cub)$} \certRL & \defRoot{$(28,22)$}\defLR {$\TP_{3,4}(*)$} {$\RP_{3,4}\circ \CC_{3,4}(Cub)$} \certRL & \defRoot{$(16,22)$}\defLR {$\CS_{3,4}(CubOc)$} {$\RP^2_{3,4}\circ \CC_{3,4}(Cub)$} \certLR\\ \hline $(6,8)$ & & \defRoot{$(6,8)$} \defLRTX {$\TP_{3,4}(\ast)$} {$\CC_{3,4}^2(RDo)$} {$Oc$} {$\RP^3_{3,4}(\ast)$} \certTri & \defRoot{$(18,20)$} \defLR {$\TP_{3,4}(\ast)$} {$\RP_{3,4}(Oc)$} \certRL & \defRoot{$(30,32)$} \defLR {$\TP_{3,4}(\ast)$} {$\RP_{3,4}^2(Oc)$} \certRL\\ \hline
\end{longtable}
\tcs
\begin{longtable}{cc|ccc} \caption{Certificates for the octahedral rotation group $\O$.}\\ \label{tab:octa} $(p,q)$ & \ \hspace{0.7cm} & $v_1(p,q)$ & $v_2(p,q)$ & $v_3(p,q)$\\ \hline $(0,2)$ & & \defRoot{$(48,26)$}\defLR {$\HP_{6,8}(*)$} {$TrCubOc$} \certRL & \defRoot{$(24,26)$}\defLRTX {$\TP_{3,8}(*)$} {$Oct(72,50)$} {$RCubOc$} {$\HP_{6,8}(Oct(72,50))$}\certTri & \defRoot{$(48,50)$}\defLR {$\TP{3,8}(*)$} {$\RP_{3,8}(RCubOc)$} \certRL\\ \hline $(0,14)$ & & \defRoot{$(24,14)$}\defLR {$\TP_{3,8}(*)$} {$TrCub$} \certRL & \defRoot{$(48,38)$}\defLR {$\TP_{3,8}(*)$} {$\RP_{3,8}(TrCub)$} \certRL & \defRoot{$(24,38)$}\defLR {$SnCub$} {$\RP^2_{3,8}(TrCub)$} \certLR\\ \hline $(6,8)$ & & \defRoot{$(6,8)$}\defLRTX {$\TP_{3,8}(*)$} {$Oct(54,32)$} {$Oc$} {$\RP_{3,8}^3(\ast)$}\certTri & \defRoot{$(30,32)$}\defT {$\RP_{3,8}^\vee(Oc) \ \RP_{3,8}(Oc) $} \certB & \defRoot{$(54,56)$}\defT {$(\RP_{3,8}^\vee)^2(Oc) \ \RP_{3,8}^2(Oc) $} \certB\\ \hline $(6,20)$ & & \defRoot{$(30,20)$}\defLR {$\TP_{3,8}(*)$} {$\CC_{3,8}(RDo)$} \certRL & \defRoot{$(54,44)$}\defLR {$\TP_{3,8}(*)$} {$\RP_{3,8}\circ\CC_{3,8}(RDo)$} \certRL & \defRoot{$(30,44)$}\defLR {$Oct(30,44)$} {$\RP^2_{3,8}\circ\CC_{3,8}(RDo)$} \certLR\\ \hline $(8,18)$ & & \defRoot{$(32,18)$}\defLR {$\TP_{4,6}(*)$} {$\CC_{4,6}(RDo)$} \certRL & \defRoot{$(56,42)$}\defLR {$\TP_{4,6}(*)$} {$\RP_{4,6}\circ\CC_{4,6}(RDo)$} \certRL & \defRoot{$(32,42)$}\defLR {$Oct(32,42)$} {$\RP^2_{4,6}\circ\CC_{4,6}(RDo)$} \certLR\\ \hline $(12,14)$ & & \defRoot{$(12,14)$}\defLRTX {$\TP_{3,8}(*)$} {$Oct(60,38)$} {$CubOc$} {$\RP_{3,8}^3(\ast)$}\certTri & \defRoot{$(36,38)$}\defLR {$\TP_{3,8}(*)$} {$\RP_{3,8}(CubOc)$} \certRL & \defRoot{$(60,62)$}\defLR {$\TP_{3,8}(*)$} {$\RP^2_{3,8}(CubOc)$} \certRL\\ \hline
\end{longtable}
\tcs
\begin{longtable}{cc|ccc} \caption{Certificates for the icosahedral rotation group $\I$.}\\ \label{tab:ico} $(p,q)$ & \ \hspace{0.7cm} & $v_1(p,q)$ & $v_2(p,q)$ & $v_3(p,q)$\\ \hline $(0,2)$ & & \defRoot{$(120,62)$}\defLR {$\HP_{6,20}(*)$} {TrID} \certRL & \defRoot{$(60,62)$}\defLRTX {$\TP_{3,20}(*)$} {Ico(180,122)} {$RID$} {$\RP^3_{5,12}(*)$}\certTri & \defRoot{$(120,122)$}\defLR {$\TP_{3,20}(*)$} {$\RP_{5,12}(RID)$} \certRL\\ \hline $(0,32)$ & & \defRoot{$(60,32)$}\defLR {$\HP_{6,20}(*)$} {$TrI$} \certRL & \defRoot{$(120,92)$} \defLR {$\HP_{6,20}(*)$} {$\RP_{5, 12}(TrI)$} \certRL & \defRoot{$(60,92)$}\defLR {$SnDo$} {$\RP^2_{5,12}(TrI)$} \certLR\\ \hline $(12.20)$ & & \defRoot{$(12,20)$}\defLRTX {$\TP_{3,20}(*)$} {$\TP_{5,12}^\vee(*)$} {$Ico$}{$\RP^3_{3,20}(*)$}\certTri & \defRoot{$(72,80)$}\defLR {$\TP_{3,20}(*)$} {$\RP_{3,20}(Ico)$} \certRL & \defRoot{$(132,140)$}\defLR {$\TP_{3,20}(*)$} {$\RP^2_{3,20}(Ico)$} \certRL\\ &\\ \hline $(12,50)$ & & \defRoot{$(72,50)$}\defLR {$\TP_{3,20}(*)$} {Ico(72,50)} \certRL & \defRoot{$(132,110)$}\defLR {$\TP_{3,20}(*)$} {$\RP_{3,20}(Ico(72,50))$} \certRL & \defRoot{$(72,110)$}\defLR {Ico(72,110)} {$\RP_{3,20}^2(Ico(72,50))$} \certLR\\ \hline $(20,42)$ & & \defRoot{$(80,42)$}\defLR {$\TP_{5,12}(\ast)$} {Ico(80,42)}\certRL & \defRoot{$(140,102)$}\defLR {$\TP_{5,12}(*)$} {$\RP_{5,12}(Ico(80,42))$}\certRL & \defRoot{$(80,102)$}\defLR {$\CS_{3,20}(RID)$} {$\TP^\vee_{3,20}(*)$}\certLR\\ \hline $(30,32)$ & & \defRoot{$(30,32)$}\defLRTX {$\TP_{3,20}(*)$} {Ico(150,92)} {$ID$} {$\CS_{3,60}(Ico(150,92))$}\certTri & \defRoot{$(90,92)$}\defLR {$\TP_{5,12}(*)$} {$\RP_{5,12}(ID)$}\certRL & \defRoot{$(150,152)$}\defLR {$\TP_{5,12}(*)$} {$\RP^2_{5,12}(ID)$}\certRL \end{longtable}
\end{document} |
\begin{document} ~
\title{Real-Time Approximate Routing for Smart Transit Systems}
\author{Siddhartha Banerjee}
\email{sbanerjee@cornell.edu}
\affiliation{
\institution{Cornell University}}
\author{Chamsi Hssaine}
\email{ch822@cornell.edu}
\affiliation{
\institution{Cornell University}}
\author{No\'emie P\'erivier}
\email{np2708@columbia.edu}
\affiliation{
\institution{Columbia University}}
\author{Samitha Samaranayake}
\email{samitha@cornell.edu}
\affiliation{
\institution{Cornell University}}
\begin{abstract}
We study real-time routing policies in smart transit systems, where the platform has a combination of cars and high-capacity vehicles (e.g., buses or shuttles) and seeks to serve a set of incoming trip requests. The platform can use its fleet of cars as a feeder to connect passengers to its high-capacity fleet, which operates on fixed routes. Our goal is to find the optimal set of (bus) routes and corresponding frequencies to maximize the social welfare of the system in a given time window. This generalizes the {\it Line Planning Problem}, a widely studied topic in the transportation literature, for which existing solutions are either heuristic (with no performance guarantees), or require extensive computation time (and hence are impractical for real-time use). To this end, we develop a $1-\frac1e-\varepsilon$ approximation algorithm for the {\it Real-Time Line Planning Problem}, using ideas from randomized rounding and the Generalized Assignment Problem. Our guarantee holds under two assumptions: $(i)$ no inter-bus transfers and $(ii)$ access to a pre-specified set of feasible bus lines. We moreover show that these two assumptions are crucial by proving that, if either assumption is relaxed, the Real-Time Line Planning Problem{} does not admit any constant-factor approximation. Finally, we demonstrate the practicality of our algorithm via numerical experiments on real-world and synthetic datasets, in which we show that, given a fixed time budget, our algorithm outperforms Integer Linear Programming-based exact methods. \end{abstract} \maketitle
\section{Introduction}
{In the past decade, the advent of ride-hailing platforms such as Lyft and Uber has revolutionized urban mobility. While commuter transit needs in cities were traditionally satisfied by personal vehicles or mass transit systems, ride-hailing platforms have grown immensely in popularity and gained a seemingly permanent footing in the landscape of mobility solutions. However, despite the increasingly important role played by \emph{Mobility-on-Demand (MoD)} services in today's society, the intermingling of various modes of transportation has yet to make its way into the status quo: by and large, if not for using their personal vehicles, commuters either choose to complete their trips in a low-capacity ride-hailing vehicle, or opt for public mass transit options, each of these options equipped with their respective benefits and disadvantages. On the one hand, ride-hailing services have been lauded for their convenience, competitive pricing, and the creation of flexible, gig economy jobs. On the other, these services have been associated with negative environmental impacts, chief of which are increased emissions due to higher volumes of traffic congestion and vehicle-miles traveled. Moreover, despite the fact that these options are often less expensive than taxi services, they remain out of reach for lower-income populations, for whom mass transit such as bus and subway services remains the most accessible option. And, while these public transit systems are more affordable and environmentally sustainable, they fail to adequately serve areas that are not as densely populated. Further, due to their inability to dynamically adapt to passenger demand, public transit vehicles are often overly packed during rush hour and significantly underfilled in off-peak hours~\citep{nyc_packed_subways}, an inefficiency from which ride-hailing options do not suffer.}
In light of this, it should be clear that there exist potentially massive gains from integrating the on-demand capabilities of ride-hailing services with mass transit options to create a smarter transportation system. The benefits of such a synergy have been uncovered in both the academic literature~\citep{benefit_integration}, as well as in the wild, with ride-hailing platforms such as Lyft experimenting with mass transit-like options in recent years~\citep{hawkins2017lyft}. Indeed, the need for such integration has become all the more stark throughout 2020, when cities have turned to microtransit as a means of addressing reduced public transit services due to the coronavirus pandemic~\citep{pandemic_microtransit}. The value of real-time, adaptive hybrid transportation options that retain both the convenience of ride-hailing and the sustainability of mass transit, is perhaps best evidenced by New York City's months-long overnight, for-hire vehicle program for essential workers, discontinued in August 2020 due to high costs~\citep{nyc_pandemic_buses}. {The extremes of the mobility spectrum to which the Metropolitan Transit Authority (MTA) turned as a stopgap in this relatively short period of time typifies the potential perils of relying on an unintegrated system: the free, late-night for-hire vehicle program was a boon to essential workers who had been deprived of a means to get to their shifts, but the city could not sustain this as a long-term solution; mass transit solutions, though sustainable, were not flexible enough to appropriately serve workers living in communities historically underserved by these services~\citep{nyc_underserved_comms}. As an alternative to these two extremes, the city recently turned to the creation of overnight bus routes that mirror workers' most popular trips~\citep{nyc_pandemic_buses}. In doing so, the MTA is faced with a number of fundamental questions upon which the success of such a system hinges: {given these essential workers' origins and destinations, {\it which} routes should the transit agency operate? {\it How frequently} should it operate each route? How can {\it short, for-hire vehicle trips} help to connect passengers to these routes?}} This paper aims to answer these questions in order to effectively operate such an integrated system.
Just as cities have yet to successfully operate integrated mobility services, the operations research and transportation communities have by and large studied ride-hailing and mass transit systems separately. On the one hand, there exists an active line of work on approximate-optimal policies for dispatching drivers to ride requests, and rebalancing empty vehicles~\citep{banerjee2016pricing,braverman2019empty,banerjee2018state,kanoria2019near}. On the other, the problem of designing the optimal bus routes to serve passenger demand
dates back to the mid-1970s~\citep{magnanti1984network}. And, though the question of integrating mass transit and single-occupancy vehicle solutions has attracted increasing attention in recent years, operational questions have largely been restricted to using ride-hailing services to connect to {\it pre-existing} transit networks~\citep{Ma,MA2019417}. The joint problem of adaptively designing bus routes in near real-time, and connecting passengers to these routes via ride-hailing services has to our knowledge yet to be explored.
The key obstacle in designing real-time algorithms with provable guarantees for transit-network design is the size of the decision space: the number of possible routes is exponential in the number of nodes of the road network. As such, approaches have either been heuristic~\citep{CEDER,Pape,Borndorfer} (lacking any guarantees), or exact~\citep{nachtigall2008simultaneous} (requiring extensive computation time); the former may lead to severe losses in efficiency, while the latter are more properly suited for designing {\it long-term} bus routes, rather than routes that adapt to changing demand patterns.
In this paper, we show that it is possible to design efficient algorithms for line planning that both provide passengers with the experience of near-real-time booking and service {\it and} have theoretical guarantees. However, this is only true up to a point: as the designer expands her solution space of feasible transit options, one runs into fundamental limits in terms of how good an approximation one can hope to achieve via efficient algorithms. Overall, our work provides \emph{theoretically sound and practically meaningful algorithms for real-time line planning, and also exposes the computational limits of line planning.}
\subsection{Summary of our contributions}
We consider a model in which a Mobility-on-Demand provider (henceforth \emph{platform}) has control of a vehicle fleet comprising both single-occupancy and high-capacity vehicles (henceforth \emph{cars} and \emph{buses} respectively). The platform is faced with a number of trip requests to fill during a window of time (e.g., one hour), and has full knowledge of passenger demands (source and destination locations, and constraints on start and end times) prior to the beginning of the time window. This assumption is practically motivated by scheduling services now offered by ride-hailing apps like Lyft and Uber, and/or the use of accurate demand forecasting models. The platform{} can service these trip requests via different \emph{trip options}: it can send a car to transport the passenger from her source to her destination; it can use a car for the first and last legs of the passenger's trip, and have her travel by bus in between; or it can use more complicated trips comprising of multiple car and bus legs.
Each passenger matched to a trip option leads to an associated value (or \emph{reward}), which can reflect both the passenger's utility for the trip-time, comfort, transfers, etc., as well as platform costs in terms of car-miles; in addition, the platform{} also incurs a cost for operating each bus route at a given frequency. We define the combination of a route and a frequency to be a {\it line}. The goal of the platform is to determine the optimal set of lines to operate (given a fixed budget $B$ for opening lines), as well as the assignment of passengers to trip options utilizing these lines, in order to maximize the total reward. We refer to this problem as the {\it Real-Time Line Planning Problem} (\textsc{Rlpp}).
As discussed earlier, though there exist exact methods for solving the Line Planning Problem that can be adapted to the \textsc{Rlpp}\ setting (e.g., by formulating and solving an associated integer linear program), the extensive computation time required to obtain the optimal set of lines runs counter to our goal of computing short-term lines that adapt to demand patterns throughout the day. This motivates studying the task of finding good approximate solutions to \textsc{Rlpp}. In this context, we make two contributions: \begin{itemize}[leftmargin=*] \item[1.] We first demonstrate the computational limits of \textsc{Rlpp}\ by showing that no constant-factor approximation is possible if we relax any one of two assumptions: $(i)$ access to a pre-specified set of feasible bus lines, and $(ii)$ no inter-line (i.e., bus-to-bus) transfers. \item[2.] Under both above assumptions, we design an efficient algorithm for \textsc{Rlpp}\, that respects budget constraints with high probability, while guaranteeing a welfare that is within a $\left(1-\frac1e-\varepsilon\right)$-factor of the optimal (where $\varepsilon$ trades-off the quality of approximation and probability of exceeding the budget). \end{itemize} While assumptions $(i)$ and $(ii)$ are commonly made both in practice and in the academic literature, our work provides \emph{strong theoretical justifications} for these assumptions in that if either fails to hold, there is no hope of obtaining a constant-factor approximation. Assumption $(i)$ forms the basis of all {exact} ILP-based methods; it is also practically relevant due to both constraints imposed by cities on bus routes, as well as expert knowledge of transit designers as to which routes are useful. Assumption $(ii)$ reflects a practical constraint that, given a passenger may already incur car-bus transfers in the first/last legs of her trip, additional bus-bus transfers could be deemed excessive. Even when both hold, however, we show that the problem is still far from trivial: in particular, it does not inherit the attractive combinatorial property of submodularity, and so one cannot employ standard techniques to get the classical $1-\frac1e$ approximation guarantee~\citep{Wolsey}. Moreover, we also show that the natural linear programming (LP) relaxation has a worst-case integrality gap of at least $\frac12$.
In spite of this, in our main technical contribution, we provide a $\left(1-\frac1e-\varepsilon\right)$-factor approximation for Real-Time Line Planning Problem{}. More specifically, our algorithm uses a novel LP relaxation followed by a randomized rounding procedure, that can be tuned to guarantee that the budget constraint is met with any desired high-probability bound, while losing an $\varepsilon$-factor in the welfare guarantee. Our key technical insight is that the Real-Time Line Planning Problem{} can be relaxed and re-formulated as an exponential-size \emph{configuration LP}, and that this formulation then allows us to use ideas from randomized rounding for the Separable Assignment Problem~\citep{SAP}. We then leverage the additional structure in \textsc{Rlpp}{} to show that the rounding step is the {\it only} source of loss in our algorithm. Our results hold under an assumption which we term {\it trip optimality} (i.e., of all the ways in which a passenger can join a given line via car, she must be assigned to the best such option). However, we later show how this assumption can be relaxed, and, with slight modification to our algorithm, we lose at most a constant factor.
Finally, we investigate the practical efficacy of our approach via numerical experiments on real-world and synthetic datasets. We note that, although our algorithm does not guarantee a solution that is always within budget, in practice it is easy to run multiple replications (which are cheap, and can be run in parallel) and choose the best realization satisfying budget constraints. Our numerical experiments simulate this procedure, and we observe that given a time budget on computation (as would be necessary for real-time line planning), our algorithm outperforms state-of-the-art ILP solvers for large problem instances, thereby demonstrating its practicality for the problem of designing integrated and flexible transit networks at scale.
\noindent\textbf{Structure of the paper}. In Section~\ref{sec:related}, we survey relevant literature. We present our model and define the Real-Time Line Planning Problem{} in Section~\ref{sec:preliminaries}. In Section~\ref{sec:hardness}, we characterize fundamental computational limits of \textsc{Rlpp}, establishing the need for a candidate set of lines and precluding bus transfers; we also show that standard techniques are inadequate for our setting. We present our main algorithm and guarantees in Section~\ref{sec:main-result}, and back this up with numerical results in Section~\ref{sec:numerical-experiments} and Appendix~\ref{app:synthetic_experiments}.
Extensions to our main results can be found in Appendix~\ref{ssec:extensions}.
\section{Related work} \label{sec:related}
\noindent\textbf{Line planning in public transportation:} {Our work falls under the large umbrella of transportation network design; see~\citet{magnanti1984network,guihaire2008transit,farahani2013review} for excellent expositions.} Much of this work has historically involved heuristics, including greedy approaches based on simpler network primitives such as shortest-paths and minimum spanning trees~\citep{Dubois,gattermann2017line}, and metaheuristics~\citep{zhao2006simulated,zhao2004transit}.
The largest-scale use of heuristic methods is, to our knowledge, the work of~\citet{Borndorfer}, who rely on column generation and {greedy heuristics}; more importantly, the formulation requires allowing for \emph{arbitrarily many bus transfers}. In practice, it is desirable to enforce a maximum number of allowable transfers (something which we explicitly model in our work); enforcing this however severely impacts computational performance. In a followup work,~\citet{borndorfer2012direct} incorporate transfer penalties (a type of ``soft'' constraint), but the resulting algorithms require on the order of 10 hours of computation time, which for our setting is infeasible. More recently, exact methods based on ILP formulations have gained in popularity~\citep{wan2003mixed,barra2007solving,marin2009urban,nachtigall2008simultaneous}, though these only scale to small networks.
\noindent\textbf{Ride-pooling:} Our problem is also closely related to {\it ride-pooling}, where the goal is to combine multiple trips to improve the efficiency of ride-sharing platforms. To model trade-offs between passenger inconvenience and sharing rides,~\citet{Santi} introduced the abstraction of a {\it shareability network}, and showed via simulations that pairing up to two requests per vehicle could lead to significant savings in cumulative driver miles. Their methods, however, accommodate at most three passengers per vehicle (with heuristics). \citet{Alonso-Mora} develop algorithms which perform well (in simulations) for up to 10 passengers per vehicle. Their method is based on clique decompositions of the shareablity network, which again scales poorly with increasing vehicle capacity; it also imposes strict quality of service constraints leading to fewer feasible trip configurations, which may greatly reduce efficiency in the setting we consider.
\noindent\textbf{Multi-modal solutions to the first-mile/last-mile problem:} From a practical perspective, the transportation community has explored public-private partnerships to exploit both the high capacity of public transit buses and the flexibility of MoD fleets~\citep{benefit_integration,MA2019417}. These works, however, focus not on designing the transit network, but rather on dynamic vehicle dispatching and routing between origin or destination and transit hubs.
\noindent\textbf{Stochastic control for ride-sharing:} A more recent line of work has developed stochastic models for ride-sharing with trip requests arriving via a random process. This has enabled the use of techniques from stochastic control for scheduling and routing~\citep{banerjee2016pricing,braverman2019empty,banerjee2018state,kanoria2019near}, as well as the study of system-level questions such as the effect of competing platforms~\cite{sejourne2018price}. The algorithms developed in these papers largely rely on assuming that under appropriate scaling (in particular, in the `large-market' scaling, where the number of cars scales with the demand), the system is well approximated via a steady-state problem. This is practically meaningful in ride-sharing systems, which can be thought of as being near-stationary over sufficiently small time-scales; such an assumption, however, critically depends on the impact of a single car being ``small'' relative to the rest of the system. In a setting with high-capacity vehicles, however, this ceases to be true, and it is unclear if a stochastic model of our system would exhibit the rapid mixing property with which low-capacity ride-sharing models are endowed, and which allows for these attractive guarantees.
\noindent\textbf{Randomized rounding for resource allocation problems:} Our methodological approach is inspired by the use of {\it configuration programs} for improved approximations for a number of combinatorial optimization problems~\citep{MBA,LP_conf_scheduling,SAP}. At a high level, the approximation algorithms proposed in this line of work reformulate the resource allocation problem as an exponential-size integer program that optimizes over all feasible sets of resources; the LP relaxation of this program can be (approximately) solved in polynomial time, and used to produce approximately optimal solutions to the original problem via rounding. Our main result relies on the randomized rounding scheme proposed by~\citet{SAP} for the Separable Assignment Problem, which comprises a set of bins and items, with a separate packing constraint on each bin, and rewards for each item-bin pair. The objective is to pack items into bins such that the aggregate value of all packed items is maximized. The analogy to the Real-Time Line Planning Problem{} is natural: items correspond to passengers, bins correspond to lines, and the packing constraints correspond to capacity constraints for each bus. The key difference between these two problems is that, in the case of \textsc{Sap}, \emph{bins are provided in advance}, with no associated cost for using a bin. In contrast, the main difficulty in \textsc{Rlpp}{} is in determining which lines to open, given costs for opening each line, and a budget constraint which further couples all lines (bins) together.
\section{Preliminaries} \label{sec:preliminaries}
\subsection{System Model}
We model the transit network as an undirected weighted graph $G=(V,E)$, with $|V|=n$ potential origin/destination nodes, edges representing roads between these nodes, and edge weights $(\tau_e)_{e\in E}$ representing the \emph{cost} (for example, travel time) required to traverse an edge. We assume that $\tau_e \geq \tau_{\min}$ for some constant $\tau_{\min} > 0$.
The network is operated by a single Mobility-on-Demand provider (henceforth \emph{platform}), which employs a fleet comprising two types of vehicles: single-occupancy vehicles (\emph{cars}), and high-capacity vehicles (\emph{buses}). The platform makes all scheduling and routing decisions in a centralized manner. These decisions are made over a fixed time-window, wherein prior to the beginning of the window, the platform receives a set of trip-requests (henceforth \emph{passengers}), and then must decide on a set of bus routes, and match passengers to these routes, using cars to cover `first-last mile' travel. The final trip option presented to each passenger must satisfy her travel needs, which we abstract via the notion of \emph{feasible trip options} for each passenger. The aim of the platform is to maximize some appropriate notion of \emph{system welfare}, which incorporates both utilities of passengers, and costs and constraints of the platform.
\begin{figure}
\caption{Example transit network with a single bus route (marked in red) and a single passenger traveling from source node $s$ to destination node $d$ (marked in green). The passenger has $2$ trip options: she can travel directly by car from $s$ to $d$ (blue arrow), or use a hybrid trip option comprising the dashed portion of the bus route, completing the rest of the trip by car (solid black arrow).}
\label{fig:passenger-options}
\end{figure}
\noindent\textbf{Vehicle Fleet Model.} {As mentioned above, the platform controls both a fleet of cars (which can serve a single passenger) and buses (which are high-capacity). Since in most ride-hailing systems, the former fleet is much larger, and has a high density throughout the city, we primarily focus on the routing/scheduling decisions for buses, incorporating the constraints and costs of the car fleet in the value function of passengers. }
Buses have a fixed capacity $C \in \mathbb{N}$, corresponding to the maximum number of passengers a bus can simultaneously accommodate. We define a {\it route} $r$ to be a fixed sequence of consecutive edges of $G$, and let $\mathcal{R}$ denote the set of all routes of cost at most $D \in (0,T]$, where $D$ is a constant determined by the platform (for example, the duration of the longest bus ride such that the trip is completed within the time window). Moreover, the platform is said to serve route $r \in \mathcal{R}$ at frequency $f \in \mathbb{N}$ if $f$ buses traverse $r$ during the time window. A key abstraction in this paper is that of a {\it line}, which we formally define below. \begin{definition}[Line] The platform is said to operate a {\it line} $\ell = (r_{\ell},f_{\ell})$ if it runs high-capacity vehicles on route $r_{\ell}$ at frequency $f_{\ell}$. \end{definition}
We use $\mathcal{L} = \left\{(r, f) \mid (r,f) \in \mathcal{R}\times\mathbb{N}\right\}$ to denote the set of all \emph{feasible} lines the platform can operate, and let $L=|\mathcal{L}|$. Note that a line can accommodate at most $C\times f_{\ell}$ passengers for each edge $e \in r_{\ell}$, and as such it is without loss of generality to assume that $f_{\ell} \in \{1,\ldots,\lceil N/C \rceil\} \, \forall \, \ell \in \mathcal{L}$, where $N$ is the total number of trip requests during the time window.
The platform has a \emph{budget} $B \in \mathbb{R}_+$ with which to open a set of lines. Let $c_{\ell}$ denote the cost of operating line $\ell$. We assume that line costs are {{\it strictly increasing}} and {\it subadditive} in the frequencies. That is, suppose lines $\ell_1$ and $\ell_2$ use the same route $r$ and have frequencies $f_1, f_2$, respectively. Then: \begin{enumerate}[$(i)$] \item strictly increasing: $f_1 < f_2 \implies c_{\ell_1} < c_{\ell_2}$ \item subadditive: $c_{\ell_1} + c_{\ell_2} \leq c_{\ell_3}$, where $\ell_3 = (r, f_1 + f_2)$. \end{enumerate}
\noindent\textbf{Passenger Model.}
We use $\mathcal{P}$ to denote the set of all passengers requesting a trip during the time window, and $N = |\mathcal{P}|$ the total number of all such passengers. Each passenger $p\in \mathcal{P}$ is associated with fixed source and destination nodes $(s_p,d_p)$. To travel between these nodes, she can use a combination of cars and buses: in particular, she can travel directly from $s_p$ to $d_p$ exclusively by car; alternatively, she can travel by bus for the `middle leg' of her journey, and use cars for the first and last legs (if source/destination is not on the bus route). Figure~\ref{fig:passenger-options} illustrates these possibilities.
In principle, a more complex trip option could also involve multiple bus segments. In this work, however, we restrict passengers to take one of the above two trip options.
\textbf{Assumption 1 (No inter-bus transfers).} A trip can only comprise of a {\it single} bus leg; i.e., the platform cannot assign any passenger to multiple lines.
From a practical perspective, this is a reasonable assumption, given that a passenger may already incur two transfers for the first and last miles of her trip. More importantly, in Section~\ref{sec:hardness} we show that if we relax this assumption by allowing the platform to use trip options involving even just two inter-line transfers, then we can not hope to achieve any constant-factor approximation.
Given line $\ell$, let $\Omega_{\ell p}$ denote the set of all trip options matching passenger $p$ to line $\ell$ that are \emph{feasible}, i.e., where the passenger completes her journey within the time window. Formally, \begin{align*}
\Omega_{\ell p} = \Big\{(s_p,i,j,d_p)|\; i,j \in r_{\ell}, p \text{ travels } s_p \rightarrow i \text{ and } j \rightarrow d_p \text{ by car, and } i \rightarrow j \text{ by bus line } \ell\Big\} \end{align*} Let {$\Omega_p = \{(\omega, \ell):\omega\in\Omega_{\ell p}, \ell \in \mathcal{L}\}$.} For each passenger $p$, there is an associated reward (or value) function {$v_p: \Omega_{p} \mapsto \mathbb{R}_+$}, representing the quality (from either the platform or the passenger's perspective) of a trip option using line $\ell$ (including potential costs incurred by the platform for the passenger's short car trip). We assume that $v_{p}(\cdot)$ is {\it non-decreasing} in the frequency of a line. Formally, suppose lines $\ell_1$ and $\ell_2$ use the same route $r$ and have frequencies $f_1$ and $f_2$, respectively. Since $\ell_1$ and $\ell_2$ share the same route $r$, we have $\Omega_{\ell_1 p} = \Omega_{\ell_2 p}$ for all $p \in \mathcal{P}$. Then, $f_1 \leq f_2 \implies v_p(\omega, \ell_1) \leq v_p(\omega, \ell_2)$ for all $\omega \in \Omega_{\ell_1 p}$.
The above formalism naturally covers trip options that {do not involve a bus segment}; in particular, we use $\omega = \varnothing$ to denote the option which consists of a passenger traveling directly from source to destination by car (the {\it no-line} option). With slight abuse of notation, we assume that $v_p(\varnothing) = 0$ for all $p \in \mathcal{P}$. Hence, one can think of the value associated with assigning a passenger to a trip option as being {relative to} the status quo ride-hailing service.
For any passenger $p$ and line $\ell$, we define the \emph{value} associated with matching the two as follows: \begin{definition}[Passenger-line value] We define $\omega_{\ell p}$ and $v_{\ell p}$ to respectively be the optimal trip option, and its corresponding value, over all feasible trip options matching passenger $p$ to line $\ell$, i.e., \begin{align*}
v_{\ell p} = \max\left\{v_p(\omega,\ell)|\omega \in \Omega_{\ell p}\right\}, \qquad \omega_{\ell p} = \arg\max\left\{v_p(\omega,\ell)|\omega \in \Omega_{\ell p}\right\} \end{align*} \end{definition} If $v_{\ell p} > 0$, we say that line $\ell$ {\it covers} passenger $p$. Let $r_{\ell p}$ denote the sub-route of $r_\ell$ used by passenger $p$ for this option. If $e \in r_{\ell p}$, we say that the passenger {\it uses} edge $e$. Note that computing $v_{\ell p}$ can be done in polynomial time. This follows from the fact that, if $r_{\ell}$ consists of $n_{\ell}$ edges, there are $O(n_{\ell}^2)$ possible trip options to consider for passenger $p$. Since the maximum cost (duration) of a route $D$ is constant, and $\tau_e$ is lower bounded by a constant for all $e \in E$, then $n_\ell$ is polynomial in $n$.
Using the above notation, we assume throughout that if passenger $p$ is matched to line $\ell$, she uses trip option $\omega_{\ell p}$. This assumption is primarily for the sake of simplifying the presentation; in Appendix~\ref{ssec:relaxing-trip-optimality} we discuss how our algorithm can be modified to consider all possible trip options for each line-passenger pair, and show that this only leads to an additional constant factor loss in the approximation guarantee.
\noindent\textbf{Platform Objective.} The following example illustrates a natural value function for a platform seeking to design such an integrated mobility service. \begin{example}\label{ex:value-function} We abuse notation and assume that, for this example, a trip option can be parametrized by the total duration of the trip $T$ and the duration of the portion of the trip completed by car, denoted $t^{\text{car}}$. Consider the following piecewise linear function, representing the reduction in time traveled by car as compared to a direct trip by car: \begin{equation} v_p(T,t^{\text{car}}) = \begin{cases} \beta t_{s_pd_p}^\star-t^{\text{car}} \quad &\text{if } T < (1+\alpha)t_{s_pd_p}^\star, \, t^{\text{car}} < \beta t_{s_pd_p}^\star \\ 0 &\text{otherwise} \end{cases} \end{equation} where $t_{s_pd_p}^\star$ represents the time required to travel from $s_p$ to $d_p$ directly by car, $\alpha \in \mathbb{R}_+$ represents passengers' tolerance for the duration of a trip relative to the most direct route, and $\beta \in (0,1]$ controls the gains in efficiency of a trip option.
For this value function, the trip optimality assumption implies that the passenger must be picked up and dropped off at the bus stops that are closest to $s_p$ and $d_p$, respectively. \end{example}
{Finally, in line with the motivating application of the platform receiving trip requests in advance via a scheduling service, we assume that the platform sees batch demand, and that passengers are willing to wait for the entirety of the time window. As such, we abstract away the notions of travel and clock times.} {In Appendix~\ref{ssec:travel-times} we show that such an assumption is without loss of generality, and that all results hold for a more realistic model in which there are travel times, passengers are associated with the time at which they made the request, and as a result should only be matched to lines whose schedule lines up with the time at which they are traveling.}
\subsection{The Real-Time Line Planning Problem{} (\textsc{Rlpp})}\label{ssec:rlpp}
Let $S \subseteq \mathcal{L}$ denote a subset of lines to be created, and $\mathbf{x} \in \{0,1\}^{N\times L}$ denote an assignment of passengers to the chosen subset of lines. We first define the {\it system welfare} induced by $S$ and $\mathbf{x}$. {\begin{definition}[Welfare] Given $S$ and $\mathbf{x}$, the {\it welfare} $W$ of the system is the sum of all passenger-line values for the lines created under this assignment. Formally: \begin{align*}
W = \sum_{p \in \mathcal{P}}\sum_{\ell \in S} v_{\ell p} x_{\ell p} \end{align*} \end{definition}}
We now define the Real-Time Line Planning Problem{}.
\begin{definition}[Real-Time Line Planning Problem{}] The Real-Time Line Planning Problem{} is defined by a graph $G$, a set of passengers $\mathcal{P}$, costs $\{c_{\ell}\}_{\ell \in \mathcal{L}}$ for opening lines, passenger valuations $\{v_{\ell p}\}_{\ell \in \mathcal{L},p\in\mathcal{P}}$ for using each line, an overall budget $B$, and a bus capacity $C$. The goal is to find a subset of lines to open and an assignment of passengers to lines that maximize the welfare of the system, such that: \begin{enumerate}[$(i)$]
\item the total cost of creating all lines in this subset does not exceed the platform's budget;
\item the number of passengers assigned to line $\ell$ and whose trip uses edge $e \in r_{\ell}$ does not exceed the capacity $C\times f_\ell$ of the buses, for all $e \in r_{\ell}$;
\item a passenger is assigned to at most one line (which implies no inter-bus transfers). \end{enumerate} \end{definition}
We allow for a passenger to not be assigned to any line. In this case, we assume that the passenger's trip is completed exclusively by car, and yields a value of zero.
Formally, the platform's optimization problem is given by: \begin{align}
{(P)} \qquad \max_{\mathbf{y}, \mathbf{x}} \qquad &\sum_{p \in \mathcal{P}}\sum_{\ell \in \mathcal{L}} v_{\ell p} x_{\ell p} \notag \\
\text{s.t.} \qquad & \sum_{\ell \in \mathcal{L}} c_\ell y_\ell \leq B \label{eq:budget-constraint}\\
& \sum_{\substack{p \in P:\\ e \in r_{\ell p}}} x_{\ell p} \leq C \, f_\ell \, y_\ell \quad \forall \, \ell \in \mathcal{L}, e \in r_\ell \label{eq:capacity-constraint}\\
&\sum_{\ell \in \mathcal{L}} x_{\ell p} \leq 1 \quad \forall \, p \in \mathcal{P} \label{eq:transfer-constraint} \\
&x_{\ell p} \in \{0,1\} \quad \forall \, p \in \mathcal{P}, \ell \in \mathcal{L} \quad \notag \\
&y_{\ell} \in \{0,1\} \quad \forall\, \ell \in \mathcal{L} \notag \end{align}
Let $OPT$ denote the optimal value of this optimization problem. In this formulation, the decision variables $\mathbf{y} \in \{0,1\}^{L}$ represent the set of lines to be opened. Recall, $\mathbf{x} \in \{0,1\}^{N\times L}$ corresponds to the assignment of passengers to lines. Constraints~\eqref{eq:budget-constraint},~\eqref{eq:capacity-constraint},~\eqref{eq:transfer-constraint} respectively encode the budget, capacity, and assignment to at most one line.
For any passenger $p \in \mathcal{P}$, in the worst case there are exponentially many routes between $s_p$ and $d_p$, and as a result $(P)$ has exponentially many variables and constraints. For our main result, we make the following assumption regarding the set of routes input to \textsc{Rlpp}.
\textbf{Assumption 2 (Candidate set of routes).} The platform has access to a pre-specified set of feasible routes that is polynomial in the size of the network.
We let $L$ denote the size of the set of lines $\mathcal{L}$ induced by the candidate set of routes and all possible frequencies. Note that the candidate set of routes assumption implies that $L$ is polynomial in $n$.
The assumption of such a candidate set is practically rooted in the reality of transportation systems, in which experts typically have knowledge of a priori ``acceptable'' bus routes and can develop good heuristics. Moreover, such an assumption is in line with the approach adopted in prior work on line planning, which typically generates the candidate set of routes via such heuristics~\citep{CEDER, Chakr, Fan}. In Section~\ref{sec:hardness}, we show that one cannot hope to obtain a constant-factor approximation to the Real-Time Line Planning Problem{} unless the platform has access to such a candidate set.
{We note that the above integer linear programming (ILP) formulation problem is the most natural formulation of the platform's optimization problem, as well as the formulation upon which existing exact methods are based~\citep{wan2003mixed,barra2007solving,marin2009urban,nachtigall2008simultaneous}. In Section~\ref{sec:main-result}, we present an equivalent, less-immediate formulation of the platform's optimization problem upon which our algorithm relies. We nonetheless present this natural formulation, as we will benchmark our algorithm's performance against it in Section~\ref{sec:numerical-experiments}.} Table~\ref{table:notation} summarizes the most frequently-used notation in the paper.
\begin{table*}[h!]
\begin{tabular}{l|l}
\textbf{Symbol} & \textbf{Definition} \\ \hline
$G(V,E)$ & Transit network with $|V|=n$ nodes\\
$\mathcal{L}$ & Pre-specified set of lines, with $L = |\mathcal{L}|$\\
$\mathcal{P}$ & Set of passengers, with $N = |\mathcal{P}|$ \\ $\Omega_{\ell p}$ & Set of feasible trip options for passenger $p$ traveling via line $\ell$ \\ $C$ & Bus capacity \\ $B$ & Platform budget for opening lines \\ $v_{p}(\omega, \ell)$ & Value of trip option $\omega \in \Omega_{\ell p}$ for passenger $p$ traveling via line $\ell$\\ $v_{\ell p}$ & Value of optimal trip option for passenger $p$ on line $\ell$\\ $c_{\ell}$ & Cost of opening line $\ell$ \\ $f_{\ell}$ & Frequency of line $\ell$ \\
\hline \end{tabular} \caption{List of frequently-used notations} \label{table:notation} \end{table*}
\section{Fundamental limits of real-time routing} \label{sec:hardness}
The model in Section~\ref{sec:preliminaries} is endowed with two assumptions: $(i)$ the existence of a pre-specified \emph{candidate set} of feasible lines $\mathcal{L}$ that is polynomial in the number of nodes $n$, and $(ii)$ that trip options can involve at most a single bus segment.
In this section, we show that these assumptions are not just practically relevant, but also have strong theoretical justifications: if either assumption fails to hold, a constant-factor approximation is out of reach. We moreover show that, even in the setting where these two assumptions hold, standard approximation techniques that leverage naive LP relaxations and rely on submodularity are inadequate, emphasizing the non-triviality of the task of designing provably good approximations for fast, real-time routing.
In the remainder of this section, we provide the main ideas of our reductions, and defer proofs of all auxiliary propositions to Appendix~\ref{app:hardness-proofs}.
\subsection{Necessity of a candidate set of lines}
Suppose first that the platform{} does not have access to a candidate set of lines, and thus, for each passenger $p \in \mathcal{P}$, must consider all possible walks of bounded cost between source $s_p$ and destination $d_p$. We show that this problem is hard to approximate even in a particularly simple instance of \textsc{Rlpp}\ with only a single allowed line, which we term the Single Line Problem (\textsc{Slp}).
\begin{definition}[Single Line Problem] In the Single Line Problem, the feasible routes are the walks in the graph of cost at most $D$. Suppose $c_{\ell} = cf_{\ell}$ for all $\ell \in \mathcal{L}$, for some constant $c > 0$. Moreover, suppose $B = c$. That is, only a single line of frequency $f_\ell = 1$ can be opened. The goal is to find the line that maximizes the social welfare of the system. \end{definition}
Using this, we get our first hardness result for \textsc{Rlpp}.
\begin{theorem}\label{thm:necessity-of-candidate-set} Unless NP has polynomial Las Vegas algorithms, the Single Line Problem is hard to approximate to a ratio better than $\Omega(\log^{1-\varepsilon} n)$. \end{theorem}
To establish this inapproximability result, we give a reduction from the Orienteering group TSP problem (\textsc{OgTSP}), for which the approximation lower bound is $\Omega\left(\log^{1-\varepsilon} n\right)$~\citep{recursive_greedy}.
\begin{definition}[Orienteering group TSP] Given an undirected graph $G=(V,E)$, with edge costs $w: E \mapsto \mathbb{R}_+$, $k$ sets (or groups) of vertices $S_{1},\ldots, S_{k} \subseteq V$, a root vertex $r$ and a budget $D>0$, the goal is to find a walk of cost no more than $D$ which spans the maximum number of groups.\footnote{We assume without loss of generality that the root does not belong to any of the groups.} \end{definition}
\begin{proof}[Proof of Theorem~\ref{thm:necessity-of-candidate-set}] Consider an instance of \textsc{OgTSP}. Recall, we've assumed that there exists a constant $\tau_{\min} > 0$ such that $\tau_e > \tau_{\min} \, \forall \, e \in E$. Define $\varepsilon \in (0, \tau_{\min}]$. We use $diam(G)$ to denote the diameter of the graph, and let $t \in \mathbb{R}$ be such that $t > \max\{ diam(G)+\varepsilon, D + \varepsilon\}$.
We construct an instance of \textsc{Slp} as follows. For each group $S_i$, we add a node $g_i$ to $G$, an edge $(r,g_i)$ of cost $t$ and an edge $(j,g_i)$ of cost $t-\varepsilon$ for each node $j \in S_i$. Let $G' = (V',E')$ denote this augmented graph, and let $D$ be the maximum cost of any feasible route on $G'$. For each $i \in [k]$, create a passenger $p_i$ with $s_{p_i} = r$ and $d_{p_i} = g_i$.
For line-passenger pair $(\ell,p_i)$, suppose trip option $\omega$ is such that passenger $p_i$ travels by car from $r$ to $j_1(\omega)$, and from $j_2(\omega)$ to $g_i$, where $j_1(\omega), j_2(\omega) \in V.$ We use $t^{\text{car}}(\omega)$ to denote the total cost of the min-cost paths from $r$ to $j_1(\omega)$ and from $j_2(\omega)$ to $g_i$, and let $t^\star_{p_i}$ denote the min-cost path from $r$ to $g_i$. If $p_i$ travels directly from $r$ to $g_i$ via edge $(r,g_i)$, then $t^{\text{car}}(\omega) = t$.
We define the value function as follows: \begin{align*}
v_{p_i}(\omega,\ell) = \begin{cases}
1 &\quad \text{ if } t^{\text{car}}(\omega) \leq (1-\frac{\varepsilon}{t})t_{p_i}^\star \\
0 &\quad \text{ otherwise.}
\end{cases} \end{align*}
\begin{figure}
\caption{Construction of graph $G'$ from an instance of \textsc{OgTSP} with two groups $S_1$ and $S_2$. The dashed lines represent the edges of the original graph $G$.}
\label{fig:two-lines}
\end{figure}
Propositions~\ref{prop:helper} and~\ref{prop:hardness-helper} characterize the ways in which $p_i$ can feasibly travel from $r$ to $g_i$. {\begin{proposition}\label{prop:helper} For all $\omega$ such that $t^{\text{car}}(\omega) > t-\varepsilon$, $v_{p_i}(\omega,\ell) = 0$. \end{proposition}}
\begin{proposition}\label{prop:hardness-helper} Passenger $p_i$ can travel from $r$ to $g_i$ in one of two ways: \begin{enumerate}[(i)]
\item via edge $(r, g_i)$, in which case this must be by car.
\item by bus from $r$ to $j \in S_i$, and by car via edge $(j,g_i)$. \end{enumerate} \end{proposition}
Let $\ell^\star$ denote the optimal solution to \textsc{Slp} for this instance. \begin{proposition}\label{prop:collect-pos-val} To collect strictly positive value from passenger $p_i$, $\ell^\star$ {\it must} traverse a node $j \in S_i$. \end{proposition}
Finally, observe that $\ell^\star$ necessarily only uses edges from $E$. This follows from the fact that all edges in $E'\setminus E$ have cost greater than $D$ by construction, and thus any route using at least one such edge is infeasible.
Putting these facts together, if line $\ell^\star$ collects value $k' \leq k$ then this implies the existence of a walk of $G$ of cost at most $D$ that has visited $k'$ groups. Thus any $\alpha$-approximation algorithm for the Single Line Problem gives an $\alpha$-approximation for the \textsc{OgTSP}, hence the $\Omega(\log^{1-\varepsilon}(n))$ lower bound for the Single Line Problem.
\end{proof}
\subsection{Hardness of multiple transfers}
Suppose now that the platform{} has access to a candidate set of lines, but allows itself to assign passengers to {at most} {\it two} lines. More specifically, a passenger $p$ can feasibly be assigned to the following trip options: \begin{enumerate}[$(i)$]
\item Travel directly from $s_p$ to $d_p$ by car;
\item Use a single bus line $\ell \in \mathcal{L}$: for some $v_1 \in r_{\ell}, v_2 \in r_{\ell}$, travel from $s_p$ to $v_1$ by car; join line $\ell$ at $v_1$ and travel to $v_2$ by bus; travel from $v_2$ to $d_p$ by car;
\item Use two intersecting bus lines $(\ell_1,\ell_2) \in \mathcal{L}\times\mathcal{L}$: for some $v_1 \in r_{\ell_1}, v_2 \in r_{\ell_1}\bigcap r_{\ell_2}, v_3 \in r_{\ell_2}$, travel from $s_p$ to $v_1$ by car; join line $\ell_1$ at $v_1$ and travel to $v_2$ by bus; join line $\ell_2$ at $v_2$ and travel to $v_3$ by bus; travel from $v_3$ to $d_p$ by car. Figure~\ref{fig:two-lines} illustrates such a trip. {We use $\Omega_{(\ell_1,\ell_2),p}$ to denote the set of all such trips.} \end{enumerate}
Let $v_{(\ell_1,\ell_2),p}$ denote the maximum value passenger $p$ has for all feasible trips using lines $\ell_1$ and $\ell_2$, where $r_{\ell_1}$ and $r_{\ell_2}$ intersect. That is, $v_{(\ell_1,\ell_2),p}= \max\limits_{\substack{\omega \in \Omega_{(\ell_1,\ell_2),p}}} v_{p}(\omega)$. If $v_{(\ell_1,\ell_2),p} > 0$, we say that passenger $p$ is {\it covered} by $\ell_1$ and $\ell_2$.\
\begin{figure}
\caption{Assignment of a passenger to a pair of lines. The passenger travels by car from $s$ to $v_1$. Between $v_1$ and $v_2$, she travels by bus via line $\ell_1$. At $v_2$ she travels via line $\ell_2$ until being dropped off at $v_3$. She completes her trip by car between $v_3$ and $d$.}
\label{fig:two-lines}
\end{figure}
We refer to the problem of matching passengers to at most two bus lines as the {\it Two-Transfer Problem} (\textsc{Ttp}), which we formally define below. \begin{definition}[Two-Transfer Problem] Given a budget $B$ and costs $\{c_\ell\}$, the goal is to find a subset $S \subseteq \mathcal{L}$ of budget-respecting lines to open and a feasible assignment of passengers to $S$ which maximizes the social welfare of the system, given by: \begin{equation*}
\sum_{p\in \mathcal{P}}\left( \sum_{\ell\in S} v_{\ell p} x_{\ell p} + \sum_{(\ell_1,\ell_2)\in S\times S} v_{(\ell_1,\ell_2),p} \, x_{(\ell_1,\ell_2),p}\right). \end{equation*} As before, $\mathbf{x}$ is an indicator variable representing the assignment of passengers to lines. \end{definition}
Our next hardness result shows that allowing even two inter-bus transfers banishes any hope of obtaining a constant-factor approximation for \textsc{Rlpp}.
\begin{theorem}\label{thm:hardness-many-transfers} Under the exponential time hypothesis, the Two-Transfer Problem is hard to approximate to a ratio better than $\Omega\left(n^{{1}/{(\log\log(n))^c}}\right)$, where $c > 0$ is a universal constant. \end{theorem}
To prove the theorem, we give a reduction from the densest $k$-subgraph problem, {which admits an approximation lower bound of $\Omega(n^{1/(\log\log n)^c})$ under the exponential time hypothesis~\citep{manurangsi2017almost}.} Given a graph $G=(V,E)$ and a subgraph $G_s=(V_s,E_s)$ of $G$, the density of any subgraph $G_s$ is the ratio of number of edges to the number of nodes in $G_s$ (i.e.$\frac{|E_s|}{|V_s|}$). Now, the densest $k$-subgraph problem is as follows: \begin{definition}[Densest $k$-subgraph]
Given a graph $G=(V,E)$ with $n = |V|$ and $k\in [n]$, the objective is to find a subgraph $G_s$ of $G$ containing {exactly} $k$ vertices with maximum density. \end{definition}
{Note that, for fixed $k$, finding the subgraph of maximum density is equivalent to finding a subgraph of size $k$ with the maximum number of edges.}
\begin{proof}[Proof of Theorem~\ref{thm:hardness-many-transfers}]
Given an instance of densest $k$-subgraph, we build an instance of \textsc{Ttp} as follows. For each node $i \in V$, construct a line $\ell_i$, with $c_{\ell_i} = 1$ and frequency $f_{\ell_i}$ large enough to cover all passengers. For every edge $(i,j) \in E$, define a passenger $p_{ij}$, and suppose that $p_{ij}$ can only be covered by the pair of lines $(\ell_i, \ell_j)$, with $v_{(\ell_i,\ell_j),p_{ij}} = 1$. That is, $p_{ij}$ {\it has no value associated with a single bus line.} Finally, let $B = k$.
We first claim that, for any \textsc{Ttp} feasible solution of value $k'$ which opens $k'' < k$ lines, one can construct a feasible solution which opens {\it exactly} $k$ lines and has value at least $k'$. This simply follows from non-negativity of the value function and the fact that $c_{\ell_i} = 1$ for all $i$. Thus, the platform can always open $k-k''$ more lines until hitting its budget constraint and not decrease the objective, and it is without loss of generality to only consider feasible solutions that open exactly $k$ lines.
We complete the proof by noting that a feasible solution of value $k'$ corresponds exactly to a subgraph of $G$ containing $k'$ edges (passengers) and $k$ nodes (lines). Thus, if we had a constant-factor approximation algorithm for \textsc{Ttp}, then we would also be able to approximate densest $k$-subgraph within a constant factor. \end{proof}
Henceforth, we operate under the no inter-bus transfers and candidate set of lines assumptions.
\subsection{Inefficacy of standard approximation techniques}
Observe that the ILP formulation of the Real-Time Line Planning Problem{} bears a strong resemblance to the Capacitated Facility Location Problem (\textsc{Cflp}), for which~\citet{Wolsey} provides a $1-\frac1e$ approximation algorithm, a guarantee relying on the underlying {\it submodular} structure of \textsc{Cflp}. Our problem crucially differs from this latter problem, however, in the way capacity is accounted for. Whereas the number of clients assigned to a location cannot exceed its capacity in \textsc{Cflp}, in the Real-Time Line Planning Problem{} the number of passengers assigned to a bus {\it can exceed} its capacity, as passengers may require non-overlapping subpaths of a bus route. In this section, we show that this simple fact fundamentally alters the structure of our problem, and as such precludes the use of standard techniques for submodular function maximization.
Let $w: \{0,1\}^{L} \mapsto \mathbb{R}$ denote the social welfare induced by the optimal assignment of passengers to lines, for a given subset of open lines, represented by $\mathbf{y}$. Formally: \begin{align*}
w(\mathbf{y}) = \max_{\mathbf{x}} \qquad &\sum_{p \in \mathcal{P}}\sum_{\ell \in \mathcal{L}} v_{\ell p} x_{\ell p} \notag \\
\text{s.t.} \qquad
& \sum_{\substack{p \in P:\\ e \in r_{\ell p}}} x_{\ell p} \leq C \, f_\ell \, y_\ell \quad \forall \, \ell \in \mathcal{L}, e \in r_{\ell} \notag \\
&\sum_{\ell \in \mathcal{L}} x_{\ell p} \leq 1 \quad \forall \, p \in \mathcal{P} \notag \\
&x_{\ell p} \in \{0,1\} \quad \forall \, p \in \mathcal{P}, \ell \in \mathcal{L} \notag \end{align*}
Then, we have: \begin{align*}
OPT = \max_{\mathbf{y}} \qquad &w(\mathbf{y}) \\
\text{s.t.} \qquad & \sum_{\ell \in \mathcal{L}} c_\ell y_\ell \leq B\\
&y_{\ell} \in \{0,1\} \quad \forall\, \ell \in \mathcal{L} \notag \end{align*}
\begin{proposition}\label{prop:not-submodular} $w$ is not submodular. \end{proposition}
Another common approach is to develop an approximation algorithm based on an LP relaxation of the ILP. Proposition~\ref{prop:integrality-gap} however shows that such an approach can give strictly worse bounds than the $1-\frac1e$ benchmark.
\begin{proposition}\label{prop:integrality-gap} The worst-case integrality gap for $(P)$ is no better than $\frac12$. \end{proposition}
\section{Main result} \label{sec:main-result}
In this section, we design an approximation algorithm for the Real-Time Line Planning Problem{} that achieves at least $1-\frac1e-\varepsilon$ fraction of the optimal solution in expectation, and produces a solution whose cost is budget-respecting with high probability, as the platform's budget grows large.
{Our high-level approach is as follows. We first formulate the Real-Time Line Planning Problem{} as a configuration ILP, and solve a conservative LP relaxation of this latter program, in the sense that it has a stricter budget than the platform's true budget $B$. We then use a variant of the rounding scheme developed by~\citet{SAP} to produce an approximately feasible integer solution. The key difficulty in such an approach is approximating the exponential-size configuration LP without incurring too much of a loss. Our main contribution in this respect is to show that the structure of \textsc{Rlpp}{} allows us to solve it {\it exactly} in polynomial-time by leveraging the additional structure of our problem in the dual space. Throughout the rest of the section, we defer the proofs of auxiliary facts to Appendix~\ref{app:main-result-proofs}.}
\subsection{An exponential-size configuration ILP}
Consider line $\ell$, and let $\mathcal{I}_{\ell}$ denote the family of all feasible assignments of passengers to $\ell$, where a feasible assignment is such that, for all $e \in r_{\ell}$ the total number of passengers using $e$ does not exceed the capacity of the line. We use $S$ to denote any such assignment in $\mathcal{I}_{\ell}$. $X_{\ell S}$ is the indicator variable representing whether or not the set of passengers $S$ is chosen for line $\ell$. Formally, $S \in \mathcal{I}_{\ell}$ satisfies $\sum\limits_{\substack{p \in S:\\e \in r_{\ell p}}} X_{\ell S} \leq C f_{\ell}$ for all $e \in E$. {Example~\ref{ex:SAP} illustrates this notation.
\begin{example}\label{ex:SAP} Consider lines $\ell_1, \ell_2$ and passengers $p_1, p_2$, with $p_1$ and $p_2$ using the same edges of each line. If $C = 2$, then $\mathcal{I}_{\ell_i} = \left\{\{p_1\}, \{p_2\}, \{p_1, p_2\}\right\}$ for $i \in \{1,2\}$. If $C = 1$, then $\mathcal{I}_{\ell_i} = \left\{\{p_1\}, \{p_2\}\right\}$ for $i \in \{1,2\}$. \end{example}
}
We can now represent {\textsc{Rlpp}} as the following exponential-size integer program: \begin{align} {\widehat{P}:=} \qquad \max_{\left\{X_{\ell S}\right\}} \qquad & \sum_{p \in \mathcal{P}} \; \sum_{\ell\in \mathcal{L}} \sum_{\substack{S \in \mathcal{I}_{\ell}:\\p\in S}}v_{\ell p} X_{\ell S} \notag\\
\text{s.t.}\qquad & \sum_{\ell\in \mathcal{L}} c_\ell\left( \sum_{S\in \mathcal{I}_\ell} X_{\ell S}\right) \leq B \label{budget-config-lp-1}\\
& \sum_{S\in \mathcal{I}_\ell} X_{\ell S} \leq 1\qquad\qquad \forall \, \ell \in \mathcal{L} \label{one-set-per-line-1} \\
&\sum_{\ell\in \mathcal{L}} \sum_{\substack{S \in \mathcal{I}_{\ell}:\\p\in S}} X_{\ell S} \leq 1 \qquad \forall \, p \in \mathcal{P} \label{cust-to-one-set-1} \\ & X_{\ell S} \in \{0,1\} \qquad\qquad \forall \, \ell\in \mathcal{L}, S \in \mathcal{I}_{\ell}\notag \end{align}
Constraint~\eqref{one-set-per-line-1} requires that only one set of passengers be chosen for each line, and Constraint~\eqref{cust-to-one-set-1} ensures that each passenger is only assigned to one line. If a set of passengers is assigned to line $\ell$, that is, if $\sum_{S \in \mathcal{I}_{\ell}} X_{\ell S} > 0$, then $\ell$ is opened and the platform{} incurs cost $c_{\ell}$; else, $\ell$ is not created and no cost is incurred. Let $OPT$ denote the optimal value of $\widehat{P}$.
\subsection{Approximating the exponential-size ILP}
{For a given constant $\varepsilon \in (0,\frac12)$, Algorithm~\ref{alg:rounding} makes use of the following auxiliary configuration LP, which we denote $\widehat{P}^{(\varepsilon)}$.}
{\begin{align} {\,\widehat{P}^{(\varepsilon)}\,:=} \qquad \max_{\left\{X_{\ell S}\right\}} \qquad & \sum_{p \in \mathcal{P}} \; \sum_{\ell\in \mathcal{L}} \sum_{\substack{S \in \mathcal{I}_{\ell}:\\p\in S}}v_{\ell p} X_{\ell S} \notag\\
\text{s.t.}\qquad & \sum_{\ell\in \mathcal{L}} c_\ell\left( \sum_{S\in \mathcal{I}_\ell} X_{\ell S}\right) \leq B(1-\varepsilon) \label{budget-constraint-relaxation}\\
& \sum_{S\in \mathcal{I}_\ell} X_{\ell S} \leq 1\qquad\qquad \forall \, \ell \in \mathcal{L} \label{one-set-per-line-relaxation} \\
&\sum_{\ell\in \mathcal{L}} \sum_{\substack{S \in \mathcal{I}_{\ell}:\\p\in S}} X_{\ell S} \leq 1 \qquad \forall \, p \in \mathcal{P} \label{one-passenger-per-set-relaxation} \\ & X_{\ell S} \in [0,1] \qquad\qquad \forall \, \ell\in \mathcal{L}, S \in \mathcal{I}_{\ell}\notag \end{align}}
Let ${OPT}^{(\varepsilon)}$ denote the optimal value of $\widehat{P}^{(\varepsilon)}$, and $\left\{X_{\ell S}^{(\varepsilon)}\right\}$ its optimal solution. Algorithm~\ref{alg:rounding} presents a high-level description of our algorithm.
\begin{algorithm} \begin{algorithmic} \Require {$G = (V,E), \mathcal{P},\mathcal{L},\left\{\mathcal{I}_{\ell}\right\}_{\ell\in\mathcal{L}}, \varepsilon \in (0,\frac12)$} \Ensure {set of lines to open, passenger assignment to each line} \State Compute $v_{\ell p}$ for all $\ell \in \mathcal{L}, p \in \mathcal{P}$. \State Solve $\widehat{P}^{(\varepsilon)}$.
\State \textbf{Rounding:} For all $\ell \in \mathcal{L}, S \in \mathcal{I}_{\ell}$ such that $X_{\ell S}^{(\varepsilon)} > 0$, {open $\ell$ and}, independently for each line $\ell$, assign $S$ to $\ell$ with probability $X_{\ell S}^{(\varepsilon)}$. \State \textbf{Re-assignment:} If passenger $p$ is assigned to multiple lines, choose the line maximizing $v_{\ell p}$. Close all lines for which no passengers are any longer assigned. \State \textbf{Aggregation:} {If there exist open lines $\ell_1, \ell_2$ such that $r_{\ell_1} = r_{\ell_2} = r$ and $f_{\ell_1} \neq f_{\ell_2}$, close $\ell_1$ and $\ell_2$ and open $\ell^\prime = (r, f_{\ell_1} + f_{\ell_2}).$ Assign all passengers formerly using $\ell_1$ or $\ell_2$ to $\ell^{\prime}$.} \end{algorithmic}
\caption{Randomized rounding for \textsc{Rlpp}\label{alg:rounding}} \end{algorithm}
Let $ALG$ denote the expected value of the solution returned by Algorithm~\ref{alg:rounding}. Theorem~\ref{thm:main-thm} establishes our main result.
\begin{theorem}\label{thm:main-thm} Algorithm~\ref{alg:rounding} respects the budget in expectation, and is of cost no more than $B$ with probability at least $1 -e^{-\varepsilon^2B/3c_{\max}}$, where $c_{\max} = \max_{\ell \in \mathcal{L}} c_\ell$. Moreover, $$ALG \geq \left(1-\frac1e-\varepsilon\right)OPT.$$ \end{theorem}
{Note that the choice of $\varepsilon$ trades off between quality of approximation and feasibility of the rounded solution: as $\varepsilon$ increases, the solution is exponentially more likely to be budget-respecting; on the other hand, we lose $\varepsilon$-fraction of the optimum in terms of the approximation guarantee.}
{To prove Theorem~\ref{thm:main-thm}, we establish the following facts, which characterize the loss incurred in each step of the algorithm: \begin{enumerate}[$(i)$]
\item\label{f5} $OPT^{(\varepsilon)} \geq (1-\varepsilon)OPT$ (Proposition~\ref{prop:loss-from-epsilon}).
\item $\widehat{P}^{(\varepsilon)}$ can be solved in polynomial time (Theorem~\ref{thm:hardness-many-transfers});
\item\label{f4} the loss from rounding and re-assignment is at most $\frac1e$ fraction of the optimal value of $\widehat{P}^{(\varepsilon)}$ (Proposition~\ref{thm:rounding_value});
\item\label{f1} the aggregation step maintains a feasible assignment of passengers to lines, and neither increases the cost of the solution nor decreases the objective (Proposition~\ref{prop:final-step-doesnt-matter});
\item\label{f2} the cost of the final solution respects the platform's budget with high probability (Corollary~\ref{cor:cost}); \end{enumerate}}
We first show that the loss incurred from solving the auxiliary LP is not too large. \begin{proposition}\label{prop:loss-from-epsilon} For all $\varepsilon \in [0,1]$, $$OPT^{(\varepsilon)} \geq (1-\varepsilon)OPT.$$ \end{proposition}
\begin{proof} Let $\{X_{\ell S}^{(0)}\}$ denote the optimal solution to $\widehat{P}^{(0)}$. Observe that $\{(1-\varepsilon)X_{\ell S}^{(0)}\}$ is feasible for the problem $\widehat{P}^{(\varepsilon)}$, and that the objective of $\widehat{P}^{(\varepsilon)}$ evaluated at this feasible solution is: $$(1-\varepsilon)\sum_{p \in \mathcal{P}}\sum_{\ell\in\mathcal{L}}\sum_{\substack{S \in \mathcal{I}_{\ell}:\\ p \in S}} v_{\ell p}X_{\ell S}^{(0)} = (1-\varepsilon) OPT^{(0)}$$
Observe moreover that $\widehat{P}^{(0)}$ corresponds to the LP relaxation of $\widehat{P}$, and thus $OPT^{(0)} \geq OPT$. Chaining these two inequalities together we obtain the fact. \end{proof}
We next observe that Algorithm~\ref{alg:rounding} is underdetermined as defined. In particular, it is a priori unclear how, if at all, one can efficiently solve $\widehat{P}^{(\varepsilon)}$ in polynomial time, or if the best we can hope for is an approximation. Our key contribution is showing that this can in fact efficiently be done, and as a result the only losses potentially incurred by the algorithm come from the rounding, re-assignment, and aggregation steps. \begin{theorem}\label{thm:poly-time-separation} $\widehat{P}^{(\varepsilon)}$ can be solved in polynomial time. \end{theorem}
\begin{proof} Since $\widehat{P}^{(\varepsilon)}$ has an exponential number of variables but only a polynomial number of constraints (in the number of passengers and lines, and hence in $n$), its dual has polynomially many variables, and as such can be solved in polynomial time via the ellipsoid method, {\it assuming access to a polynomial-time separation oracle}~{\citep{bland1981ellipsoid}}. {Given this, one can obtain an optimal primal solution by solving the primal problem with only the variables corresponding to the dual constraints present when the ellipsoid method has terminated (of which there are polynomially many, since the ellipsoid method only makes a polynomial number of calls to the separation oracle)~\citep{carr2000randomized}. Thus, it suffices to design a separation oracle which runs in polynomial time.}
Let $\widehat{D}^{(\varepsilon)}$ denote the dual of $\widehat{P}^{(\varepsilon)}$, with $\alpha, \{q_{\ell}\}, \{\lambda_p\}$ the dual variables corresponding{ to constraints~\eqref{budget-constraint-relaxation}, \eqref{one-set-per-line-relaxation} and \eqref{one-passenger-per-set-relaxation}, respectively.} The dual is given by: \begin{align*} \widehat{D}^{(\varepsilon)} := \qquad \min_{\substack{\{q_{\ell}\}, \{\lambda_p\}, \alpha}}
\qquad & \sum_{\ell \in \mathcal{L}} q_\ell + \sum_{p \in \mathcal{P}} \lambda_p +B(1-\varepsilon)\alpha\\
\text{s.t.} \qquad & q_\ell + \alpha c_\ell \geq \sum_{p\in S} \left(v_{\ell p} - \lambda_p\right) \qquad \forall \, \ell \in \mathcal{L}, S\in \mathcal{I}_\ell \\
& q_\ell \geq 0 \quad \forall \, \ell \in \mathcal{L},\quad \lambda_p \geq 0 \quad \forall \, p \in \mathcal{P}, \quad \alpha \geq 0 \end{align*}
For all $\ell\in\mathcal{L}$, let $\mathcal{F}_{\ell}$ denote the polytope defined by the set of constraints: \begin{equation*}
q_\ell + \alpha c_\ell \geq \sum_{p\in S} (v_{\ell p}-\lambda_p) \qquad \forall \, S\in \mathcal{I}_\ell \end{equation*} It suffices to show that we can design a polynomial time separation algorithm for the polytope $\mathcal{F}_{\ell}$. That is, given $q_{\ell}, \alpha,$ and $\left\{\lambda_p\right\}$, the separation algorithm must be able to find a violated constraint for $\mathcal{F}_\ell$ or certify that all constraints in $\mathcal{F}_\ell$ are satisfied.
Algorithm~\ref{alg:separation} formally describes our separation oracle.
\begin{algorithm} \begin{algorithmic} \Require {$q_{\ell}, \alpha, \{\lambda_p\}, \mathcal{F}_{\ell}$} \Ensure {violated constraint for $\mathcal{F}_{\ell}$, or a certification that all constraints in $\mathcal{F}_{\ell}$ are satisfied} \State Solve the following LP: \begin{align}
\max_{\{x_p\}} \qquad & \sum_{p \in \mathcal{P}} (v_{\ell p}-\lambda_p) x_p\notag\\\
\text{s.t.} \qquad & \sum_{\substack{p \in \mathcal{P}: \\ e \in r_{\ell p}}} x_p \leq Cf_{\ell} \qquad \, \forall \, e \in r_{\ell} \label{single_line_subproblem}\\ & 0 \leq x_p \leq 1 \qquad \forall \, p \in \mathcal{P}.\notag \end{align} Let LP-SEP denote its optimal value, and $\{x_p^\star\}$ an optimal solution to this problem. \State If LP-SEP $\leq q_{\ell} + \alpha c_{\ell}$, then return that all constraints in $\mathcal{F}_{\ell}$ are satisfied. Else, return $S^\star = \{p:x_{p}^\star > 0\}$. \end{algorithmic}
\caption{Separation Algorithm for the Ellipsoid Method\label{alg:separation}} \end{algorithm}
Our separation algorithm solves an LP with polynomially many variables and constraints, and as such runs in polynomial time.{\footnote{We note that, given a dual solution, one can efficiently find a primal solution, as observed by~\citet{carr2000randomized}.}} However, correctness of the algorithm is not immediate: the LP is a relaxation of the set problem we are interested in, and as such $\sum_{p} \left(v_{\ell p} - \lambda_p\right) x_p^\star \geq \max_{S \in \mathcal{I}_{\ell}} \sum_{p \in S}\left(v_{\ell p} - \lambda_p\right)$. If this inequality was strict, the separation algorithm would incorrectly return that a constraint has been violated, when in fact all have been satisfied. {Observe that this would only occur if $\{x^\star_p\}$ were fractional; the separation algorithm we propose, however, is a capacitated variant of the assignment problem, for which the linear programming relaxation is known to admit an integral solution~\citep{bertsekas1991linear}.} Lemma~\ref{lem:correctness} formalizes this high-level intuition, and thus establishes that this inequality is in fact always tight.
This then concludes the proof of the fact that $\widehat{P}^{(\varepsilon)}$ is poly-time solvable. \begin{lemma}\label{lem:correctness} $\{x_p^\star\}$ is integral. Thus, $$\sum_{p} \left(v_{\ell p} - \lambda_p\right)x_p^\star = \max_{S \in \mathcal{I}_{\ell}} \sum_{p \in S}\left(v_{\ell p} - \lambda_p\right).$$ \end{lemma}
\end{proof}
Proposition~\ref{thm:rounding_value} establishes the loss incurred from the rounding step, and follows from~\cite{SAP}. For the sake of completeness, we include the proof in Appendix~\ref{app:main-result-proofs}.
\begin{proposition}\label{thm:rounding_value} Let $\widetilde{ALG}$ denote the value of the solution immediately after the re-assignment step. Then, $\widetilde{ALG} \geq (1-\frac1e)OPT^{(\varepsilon)}$. \end{proposition}
We next show that no additional loss is incurred in the aggregation step of our algorithm. \begin{proposition}\label{prop:final-step-doesnt-matter} The aggregation step maintains a feasible assignment of passengers to lines. Moreover, let $\widetilde{ALG}$ denote the value of the solution {\it before} the final aggregation step, and let $\{\widetilde{Y}_{\ell}\}$ and $\{Y_{\ell}\}$ respectively denote the indicator variables corresponding to whether or not a line was opened, before and after the aggregation step; let $c(\widetilde{Y})$ and $c(Y)$ denote the costs of these respective solutions. Then, ${ALG} \geq \widetilde{ALG}$, and $c(Y) \leq c(\widetilde{Y})$. \end{proposition}
\begin{proof} The fact that the objective weakly increases after the aggregation step follows from the fact that $\ell_1$ and $\ell_2$ share the same route, and $v_p(\cdot)$ is non-decreasing in the line frequency for all $p \in \mathcal{P}$. Moreover, $c(Y) \leq c(\widetilde{Y})$ follows from subadditivity of the cost function.
We now argue that a feasible assignment of passengers to lines is maintained after the aggregation step, i.e., that the bus capacity constraint is not violated for line $\ell' = (r, f_{\ell_1} + f_{\ell_2})$. Let $\left\{X_{\ell p}\right\}$ and $\{\widetilde{X}_{\ell p}\}$ be the indicator variables respectively denoting the assignment of passengers to lines, after and before the aggregation step. For all $e \in r$, we have: \begin{align*}
\sum_{p: e \in r_{\ell' p}}X_{\ell' p} \stackrel{(a)}{=} \sum_{p:e \in r_{\ell_1 p}} \widetilde{X}_{\ell_1 p} + \sum_{p:e \in r_{\ell_2 p}} \widetilde{X}_{\ell_2 p} \stackrel{(b)}{\leq} C(f_{\ell_1} + f_{\ell_2}), \end{align*} where~$(a)$ follows from the aggregation construction and~$(b)$ follows from the fact that the assignment of passengers to lines {\it before} the aggregation step was feasible by construction, for both $\ell_1$ and $\ell_2$. \end{proof}
To complete the proof of the theorem, we characterize the cost of the solution returned by Algorithm~\ref{alg:rounding}. We defer the proof of Proposition~\ref{prop:cost-characterization} to Appendix~\ref{app:main-result-proofs}.
\begin{proposition}\label{prop:cost-characterization} The solution returned by Algorithm~\ref{alg:rounding} satisfies the budget constraint in expectation. Moreover, for all $\delta \in (0,1]$, the cost of the solution returned by Algorithm~\ref{alg:rounding} is at most $B(1-\varepsilon)(1+\delta)$ with probability at least $1 - e^{-\delta^2(1-\varepsilon)B/3c_{\max}}$. \end{proposition}
The probabilistic budget guarantee follows from taking $\delta = \frac{\varepsilon}{1-\varepsilon}$.
\begin{corollary}\label{cor:cost} The cost of the solution returned by Algorithm~\ref{alg:rounding} satisfies the budget constraint with probability at least $1-e^{-\varepsilon^2 B/3c_{\max}}$. \end{corollary}
We complete the proof of Theorem~\ref{thm:main-thm} by putting together the facts established above.
\begin{proof}[Proof of Theorem~\ref{thm:main-thm}.] Corollary~\ref{cor:cost} establishes the cost characterization.
For the approximation guarantee, putting together Theorem~\ref{thm:poly-time-separation} with Propositions~\ref{thm:rounding_value},~\ref{prop:final-step-doesnt-matter} and~\ref{prop:loss-from-epsilon}, we obtain that $$ALG \geq \left(1-\frac1e\right)OPT^{(\varepsilon)} \geq \left(1-\frac1e\right)(1-\varepsilon)OPT \geq \left(1-\frac1e-\varepsilon\right)OPT.$$
\end{proof}
\section{Numerical Experiments} \label{sec:numerical-experiments}
{Finally, we complement our theoretical results by demonstrating the practical efficacy of our algorithm on: $(i)$ the Manhattan network, with real passenger data from for-hire vehicle ride requests, and $(ii)$ on a synthetic dataset based on a random network, designed to minimize any structural advantages. We present the former here, and defer our synthetic experiments to~\cref{app:synthetic_experiments}.}
We compare the solution returned by our algorithm to that of a state-of-the-art ILP solver, run on problem $(P)$ in Section~\ref{ssec:rlpp}. Note that the ILP solver cannot directly solve the configuration LP $\widehat{P}$, due to its exponential size, which is why instead feed it the natural formulation of the problem $(P)$. To emulate the real-time constraints on such a policy in practice, we run both our algorithm and the ILP solver under a strict time budget.
\subsection{Practical Implementation} \label{procedure}
Although the theoretical analysis of our algorithm relies on using the ellipsoid method for solving the configuration LP, in practice, column generation is known to be more efficient (despite lacking poly-time guarantees){~\citep{desaulniers2006column}}. Thus, in our experiments we opt for column generation, where the generation of the new columns is done using our separation algorithm (Algorithm~\ref{alg:separation}).
Given an instance $I$ of \textsc{Rlpp}, and parameters $\varepsilon \in (0,\frac12)$, $m \in \mathbb{N}$, we proceed as follows:
\begin{enumerate}
\item Solve the configuration LP $\widehat{P}^{(\varepsilon)}$ in Algorithm~\ref{alg:rounding} via column generation. Return the current LP solution once the time budget has been exceeded.
\item\label{step-2} Simulate the rounding through re-aggregation steps of Algorithm~\ref{alg:rounding} $m$ times.
\item Let $\mathcal{S}_B(I)$ denote the set of all budget-respecting solutions of the $m$ realized solutions;
return the solution of maximum value in $\mathcal{S}_B(I)$.
\end{enumerate}
We note that this procedure retains our polynomial-time guarantees. Moreover, it benefits from the fact that Step~\ref{step-2} is easily parallelizable. In our experiments, we use $\varepsilon = 0.05$ and $m = 10^4$.
\subsection{Experimental setup and results}
To test the performance of our algorithm in a realistic setting, we develop a new dataset for modeling Mobility-on-Demand platforms, based on the Manhattan road network. We obtain the network from the publicly available OpenStreetMap (OSM) geographical data~\citep{boeing2017osmnx}.
\noindent\textbf{Line inputs.} We set the size of the candidate set of lines to be $L = 1,000$, and generate the candidate set based on the skeleton method proposed by~\citet{SILMAN1974201}, by iteratively choosing four nodes in the graph, uniformly at random, and connecting them via shortest path. We also set {$c_{\ell}$ to be proportional to the total travel time between the start and end nodes of line $\ell$.}
{We set the bus capacity $C = 30$, and assume that all bus routes operate at frequency 1. {Note that increasing the frequency of a line is equivalent to duplicating a route of frequency 1 in our algorithm. In our synthetic experiments (\cref{app:synthetic_experiments}) we observe that our algorithm's performance improves relative to the ILP solver as the size of the candidate set of lines increases. Thus, assuming frequency 1 lines only serves as a lower bound on our algorithm's performance on the real-world dataset.}
\noindent\textbf{Passenger inputs.} We use records of for-hire vehicle trips in Manhattan using the New York City Open Data platform, considering an hour's worth of trip requests between 5pm and 6pm on the first Tuesday of February, March and April 2018. Our time windows have $9983$, $13851$, and $12301$ trip requests respectively. {We note that the more commonly-used taxicab and rideshare datasets are unsuitable for our setting, as these datasets are heavily biased towards short trips (indeed, running our algorithm on this data results in most trips using the car-only option). In contrast, the for-hire trips are longer, and hence lead to significant savings from multi-modal trips.}
{ For each trip, instead of exact pickup and drop-off coordinates, the dataset provides only origin and destination `areas' (the over 4,000 nodes in the Manhattan network are divided into 69 areas). Given the area of an origin or destination, we sample a node in the area from the network uniformly at random. For each passenger $p \in \mathcal{P}$ and line $\ell \in \mathcal{L}$, we define the passenger-line value to be the difference between the time travelled by car when using $\ell$ and the duration of the direct car trip. Thus, our objective function is proportional to the total reduction in miles travelled by car in the system.}
{We moreover impose the constraint that a passenger-line value is only positive if the travel time induced for the passenger is no more than $\beta$ times the time of a direct trip by car, and set this detour factor $\beta = 3$.}
\begin{figure}
\caption{An example of line plan generated by our algorithm for the Manhattan network. We consider here the trip requests made on April 3, 2018 from 5pm to 6pm, with $B=3\cdot10^4$, $L=10^3$ and $\beta =3$.}
\label{fig:manhattan_plan}
\end{figure}
{We run the procedure for each of the three sets of requests, averaging the solutions returned by the procedure over these three instances. Let $ALG$ denote the corresponding empirical average. We also report $n_{\text{ILP}}$ and $n_{\text{ALG}}$, the number of lines respectively opened in the solutions returned by our algorithm and the ILP, and $\alpha$, the fraction of the outputs of the rounding process which were budget-respecting (out of the $m=10^4$ solutions of the rounding process)}. {Finally, we compute the {empirical average of the} multiplicative gap between the solution returned by our procedure and the value of the configuration LP $\widehat{P}^{(\varepsilon)}$ at the end of the allotted time. We use $\eta$ to denote this gap and note that, in cases where the configuration LP is not solved to optimality before rounding, $\eta$ may exceed 1.}
We report the results of our experiments in Table~\ref{tab:results_manhattan}. Our findings illustrate the practicality of our algorithm and relative inadequacy of the ILP for the task of real-time routing at scale.
\begin{table}[ht] \centering
\begin{tabular}{|c|cc|cccc|}
\hline
$B$ & ILP & $ALG$ & $n_{\text{ILP}}$ & $n_{\text{ALG}}$ & $\alpha$ & $\eta$ \\
\hline \hline
$10^4$ & \textbf{289,139} & $279,364$
& $15$ & $16$ & $0.62$ & $0.87$ \\
$2\cdot 10^4$ & $356,621$ & \textbf{509,586}
& $25$ & $27$ & $0.64$ & $0.94$ \\
$3\cdot 10^4$ & --- & \textbf{704,800} & --- & $36$ & $0.65$ & $0.9$\\
$5\cdot 10^4$ & --- & \textbf{917,683} & --- & $60$ & $0.68$ & $0.89$\\
$10^5$ & --- & \textbf{1,140,700} & --- & $106$ & $1$ & $1.12$ \\
$2\cdot 10^5$ & \textbf{2,859,276} & $1,132,616$ & $242$ & $101$ & $1$ & $1.11$ \\
\hline
\end{tabular} \caption{\emph{Numerical results for different budget values}: We set $L=10^3$, $\beta=3$. Bold values indicate the better solution for the corresponding value of $B$. While the ILP outperforms our algorithm for the smallest and largest budgets, our algorithm consistently outperforms the ILP solver for more realistic intermediary budgets, where the ILP solver is often unable to return a solution within the allotted time.\\ The gap $\eta$ between the solution produced by our procedure and the value of the configuration LP at the end of the allotted time, is consistently above $87\%$, which is a significant improvement on the $0.95 \cdot \left(1-1/e\right)$ (i.e., $60\%$) theoretical guarantee. For larger budgets (i.e., between $10^5$ and $2\cdot 10^5$), the performance of our algorithm plateaus, as the column generation process requires more iterations to optimally solve the configuration LP. For the largest budget of $2\cdot 10^5$, the ILP is again able to get a solution by opening 242 lines (approximately a quarter of the candidate set). We conjecture that, with such a large budget, any set of lines is good enough, while more refined search is necessary to find the optimal lines for a more restricted budget. } \label{tab:results_manhattan} \end{table}
\section{Conclusion}\label{sec:conclusion}
The integration of ride-hailing platforms' flexible demand-responsive services with the sustainability of mass transit systems is the next frontier in urban mobility. As ride-hailing platforms such as Uber and Lyft expand their range of services and look to adding high-capacity vehicles such as buses and shuttles to their fleets, they are faced with the following operational question: {\it Given a set of dynamically changing trip requests and a fleet of high-capacity vehicles, what is the optimal set of bus routes and corresponding frequencies with which to operate them?} In this work we provided a partial characterization of the hardness landscape of the Real-Time Line Planning Problem{} by proving that, unless the platform{} has access to an existing candidate set of lines and passengers can only travel via one bus line (but are nevertheless allowed to transfer between bus and car services), the problem is hard to approximate within a constant factor. Under these assumptions, however, we developed a $1-\frac1e-\varepsilon$ approximation algorithm. We moreover demonstrated its efficacy in numerical experiments by showing that, when the platform{} is constrained to short computation times (which is precisely the case if it wishes to be demand-responsive), then our algorithm outperforms exact methods on state-of-the-art ILP solvers.
This paper lends itself to a number of natural directions for future work. From a theory perspective, though we showed that our algorithm can be modified with at most a constant-factor loss when the trip optimality assumption is relaxed, {existing approximation bounds for the interval scheduling problem are quite weak. An important area of investigation is whether we can leverage the additional structure of the Real-Time Line Planning Problem{} to strengthen the bounds of existing interval scheduling techniques.}
\begin{acks} This material is based upon work partially supported by the National Science Foundation under Grant No. CNS-1952011. \end{acks}
\appendix
\section{Extensions}\label{ssec:extensions}
\subsection{Relaxing trip optimality.}\label{ssec:relaxing-trip-optimality} {In this section, we describe how our algorithm and analysis can be modified if the trip-optimality assumption (Assumption 2) is relaxed. Specifically, we no longer assume that passengers must use the trip option which maximizes their value along that line; the platform{} must now consider all possible ways in which a passenger can join each line. We refer to this variant of the problem as the Generalized Real-Time Line Planning Problem{} (\textsc{GRlpp}).
Given line $\ell \in \mathcal{L}$, we define a {\it sub-route} of $r_{\ell}$ to be any set of consecutive edges of $r_{\ell}$. Let $n_{\ell}$ be the size of the set of all sub-routes of $r_{\ell}$. We index sub-routes of $r_{\ell}$ as $r_{\ell}^{(i)}$ for $i \in [n_{\ell}]$. Let $v_{\ell p}^{(i)}$ denote the value associated with passenger $p$ traveling along sub-route $r_{\ell}^{(i)}$ of $r_{\ell}$. Passenger $p$ can be assigned to any sub-route $r_{\ell}^{(i)}$ for which $v_{\ell p}^{(i)} > 0$.
We first define the notion of {\it trip-optimality gap}.
\begin{definition}[Trip-optimality gap] The {\it trip-optimality gap} $\gamma$ characterizes the worst-case multiplicative gap between the optimal values of \textsc{Rlpp}{} and \textsc{GRlpp}. Formally, let $\mathcal{I}$ denote the set of all instances for the Generalized Real-Time Line Planning Problem. For $I \in \mathcal{I}$, $OPT(I)$ and $\widehat{OPT}(I)$ respectively denote the value of the optimal solution to \textsc{Rlpp}{} and \textsc{GLpp}. $$ \gamma = \sup_{I \in \mathcal{I}} \frac{\widehat{OPT}(I)}{OPT(I)}.$$ \end{definition}
\begin{proposition}\label{prop:trip-optimality-gap} The Real-Time Line Planning Problem{} has unbounded trip-optimality gap. \end{proposition}
\begin{proof}
Consider the setting where $|E| = |\mathcal{P}| = n-1$, $C = 1$, and $B$ is such that only one line $\ell$ at frequency 1 can be opened. Let $r_{\ell}^{(n)}$ denote the sub-route which uses all $n-1$ edges of $G$, and suppose $v_{\ell p}^{(n)} = 1$ for all $p \in \mathcal{P}$. Let $r_{\ell}^{(e)}$ denote the sub-route of $r_{\ell}$ which uses a single edge $e$, and suppose $v_{\ell p}^{(e)} = 1/2$ for all $e \in E, p \in \mathcal{P}$. Then, under the trip optimality assumption, the Real-Time Line Planning Problem{} has optimal value 1 (since all passengers must be served on {$r_{\ell}^{(n)}$} but $C = 1$). When this assumption is relaxed, however, the optimal value is {\it at least} $\frac{n-1}{2}$, achieved by having each passenger travel along a different edge. \end{proof}
An unbounded trip-optimality gap would lead one to think that the more general, relaxed problem would require a fundamentally different approach from that of our algorithm. We however demonstrate the flexibility of our approach by proving that our algorithm can easily be modified for this setting, with {\it at most} a constant-factor loss.
We first introduce the following notation. Let ${S}_i$ denote a feasible assignment of passengers to {\it sub-route} $r_{\ell}^{(i)}$ of line $\ell$, for $i \in [n_{\ell}]$. Now, ${S} = ({S}_1,\ldots,{S}_{n_{\ell}})$ denotes a feasible assignment of passengers to {\it line} $\ell$. For ${S}$ to be feasible, $\{{S}_i\}$ must be disjoint subsets of $\mathcal{P}$ (i.e., a passenger can only be matched to one trip option), and the number of passengers using edge $e$ of $r_{\ell}$ must not exceed the capacity of the line. {Let ${\mathcal{I}_{\ell}}$ denote the set of feasible assignments of passengers to $\ell$.} For ease of notation, we use $p\in{S}$ if there exists $i \in [n_{\ell}]$ such that $p \in {S_i}$.
We can still define an exponential-size configuration ILP for the Generalized Real-Time Line Planning Problem{}:
\begin{align} {\widehat{P}:=} \qquad \max_{\left\{X_{\ell {S}}\right\}} \qquad & \sum_{\ell\in \mathcal{L}} \sum_{\substack{{S}\in {\mathcal{I}_{\ell}}}} \sum_{i \in [n_{\ell}]} \sum_{p \in {S}_i} v_{\ell p}^{(i)} X_{\ell {S}} \notag\\
\text{s.t.}\qquad & \sum_{\ell\in \mathcal{L}} c_\ell\left( \sum_{{S}\in \mathcal{I}_\ell} X_{\ell {S}}\right) \leq B \label{budget-config-lp}\\
& \sum_{{S}\in \mathcal{I}_\ell} X_{\ell{S}} \leq 1\qquad\qquad \forall \, \ell \in \mathcal{L} \label{one-set-per-line} \\
&\sum_{\ell\in \mathcal{L}} \sum_{\substack{{S} \in {\mathcal{I}_{\ell}}:\\p\in {S}}} X_{\ell {S}} \leq 1 \qquad \forall \, p \in \mathcal{P} \label{cust-to-one-set} \\ & X_{\ell {S}} \in \{0,1\} \qquad\qquad \forall \, \ell\in \mathcal{L}, {S} \in {\mathcal{I}_{\ell}}\notag \end{align}
We can apply Algorithm~\ref{alg:rounding} to this problem. As before, however, we require a subroutine which (approximately) solves $\widehat{P}^{(\varepsilon)}$, the auxiliary configuration LP.
In Section~\ref{sec:main-result} we showed that we can solve $\widehat{P}^{(\varepsilon)}$ by applying the ellipsoid method to the dual problem $\widehat{D}^{(\varepsilon)}$, { assuming access to a polynomial-time separation oracle}. To this end, we showed that an exact polynomial-time separation algorithm was within reach due to additional structure induced by trip optimality.
We adopt a similar approach for the Generalized Real-Time Line Planning Problem. Consider the dual corresponding to $\widehat{P}^{(\varepsilon)}$, which we denote as before $\widehat{D}^{(\varepsilon)}$:
\begin{align*} \widehat{D}^{(\varepsilon)} := \qquad \min_{\substack{\{q_{\ell}\}, \{\lambda_p\}, \alpha}}
\qquad & \sum_{\ell \in \mathcal{L}} q_\ell + \sum_{p \in \mathcal{P}} \lambda_p +B(1-\varepsilon)\alpha\\
\text{s.t.} \qquad & q_\ell + \alpha c_\ell \geq \sum_{i \in [n_{\ell}]} \sum_{p \in {S}_i} \left(v_{\ell p}^{(i)} - \lambda_p\right) \qquad \forall \, \ell \in \mathcal{L}, {S} \in {\mathcal{I}_{\ell}} \\
& q_\ell \geq 0 \quad \forall \, \ell \in \mathcal{L},\quad \lambda_p \geq 0 \quad \forall \, p \in \mathcal{P}, \quad \alpha \geq 0 \end{align*}
Recall, for fixed $\ell$, given $q_{\ell}, c_{\ell}, \{\lambda_{p}\}$, a separation algorithm for $\widehat{D}^{(\varepsilon)}$ either certifies that $q_\ell + \alpha c_\ell \geq \sum_{i \in [n_{\ell}]} \sum_{p \in {S}_i} \left(v_{\ell p}^{(i)} - \lambda_p\right) \, \forall {S} \in {\mathcal{I}_{\ell}}$, or returns ${S}$ such that this constraint is violated. {This can be done by solving the} following combinatorial optimization problem:
$$\max_{{S} \in {\mathcal{I}_{\ell}}}\sum_{i \in [n_{\ell}]}\sum_{p \in {S}_i} \left(v_{\ell p}^{(i)} - \lambda_p\right).$$
The following lemma follows from~\citet{SAP}.
\begin{lemma}[\citep{SAP}]\label{prop:beta-approx} A $\beta$-approximate separation algorithm for $\widehat{D}^{(\varepsilon)}$ implies a $\beta$-approximation for $\widehat{D}^{(\varepsilon)}$. \end{lemma}
Thus, given a constant-factor approximation for the separation algorithm, a constant-factor approximation for the Generalized Real-Time Line Planning Problem{} follows.
\begin{corollary}\label{prop:glpp} Let $\mathcal{A}$ be a $\beta$-approximate separation algorithm for $\widehat{D}^{(\varepsilon)}$. Then, using $\mathcal{A}$ as a sub-routine to Algorithm~\ref{alg:rounding} guarantees a $\left((1-\frac1e)\beta-\varepsilon\right)$-approximation for the Generalized Real-Time Line Planning Problem{} that is budget-respecting with probability at least $1-e^{-\varepsilon^2 B/3c_{\max}}$. \end{corollary}
It suffices to show that such a constant-factor approximation exists. To see this, we show that the problem of finding a separation algorithm for $\widehat{D}^{(\varepsilon)}$ reduces to an instance of the Weighted Job Interval Selection problem (\textsc{Wjis}), for which a $\tfrac18$-approximation exists~\citep{Erlebach2003}. Establishing this analogy then completes the argument that we can use our algorithm to obtain a constant-factor approximation for the Generalized Real-Time Line Planning Problem{}.
\begin{definition}[Weighted Job Interval Selection Problem] Consider a set of $n$ jobs, $m$ machines, and a set of intervals $\mathcal{I}$ of the real line. Each job $j$ is defined by a set of feasible intervals $I_j \in \mathcal{I}$ in which it can be processed, as well as associated weights $\{w_{ij}\}_{i\in I_j}$. The goal is to select a subset of the intervals of maximum weight such that: $(i)$ at most one interval is selected for each job, and $(ii)$ at any point on the real line, no more than $m$ jobs can be scheduled. \end{definition}
The analogy between the Generalized Real-Time Line Planning Problem{} and the Weighted Job Interval Selection Problem is as follows. Each line $\ell \in \mathcal{L}$ corresponds to the real line, and sub-route $r_{\ell}^{(i)}$ corresponds to an interval $i$ of the real line. Each passenger $p$ corresponds to a job $j$, and $v_{\ell p}^{(i)}-\lambda_p$ corresponds to the weight of processing job $j$ on interval $i$. The bus capacity $C$ is the number of machines. Thus, from any feasible solution to \textsc{Wjis} we can construct an assignment of passengers to sub-routes $\{r_{\ell}^{(i)}\}_{i \in [n_{\ell}]}$ such that each passenger is only assigned to one sub-route and the capacity $C$ of a bus on $r_{\ell}$ is nowhere exceeded. Such an assignment is thus feasible for line $\ell$, and any $\beta$-approximation for \textsc{Wjis} also gives us a $\beta$-approximate separation oracle for $\widehat{D}^{(\varepsilon)}$.}
We briefly note that {$n_{\ell}$ is polynomial in $n$ since we've assumed that the maximum duration (weight) of a route is upper bounded by $D$, and the edge travel times are bounded below by a constant $\tau_{\min} > 0$.}
Thus, since Algorithm~\ref{alg:rounding} runs in polynomial time for the Real-Time Line Planning Problem{}, it also runs in polynomial time for the Generalized Real-Time Line Planning Problem.
\subsection{Travel times.}\label{ssec:travel-times} We now show that abstracting away notions of travel and clock times is indeed without loss of generality, and that all results continue to hold for a more realistic, time-centric model.
Let $T$ denote the length of the discrete time window during which the platform{} must serve the trip requests. A passenger is now defined by her source and destination nodes $s_p$ and $d_p$, as well as the time of her trip request $t_p$. Let $\mathcal{P}_T$ denote the set of all passengers. Clearly, $|\mathcal{P}_T| = |\mathcal{P}|$. In the same vein, a line is now defined by a route, a frequency, and a start time. Formally, the set of all possible lines the platform{} can operate is $\mathcal{L}_T = \left\{(r,f,t) \mid (r,f,t) \in \mathcal{R}\times\mathbb{N}\times [T]\right\}$. In this case, we have $|\mathcal{L}_T| = T|\mathcal{L}|$. Given the set of travel times $\{\tau_{ij}\}$, the platform{} can pre-compute the bus schedule induced by each line (e.g., if $(i,j) \in r_{\ell}$, and the bus leaves node $i$ at time $t$, then it reaches node $j$ at time $t + \tau_{ij}$). With slight abuse of notation, let $t_{\ell i}$ denote the time at which line $\ell$ reaches node $i$. Then, the only feasible trip options for passenger $p$ via line $\ell$ are $\omega = (s_p, i, j, d_p)$ such that $t_p + \tau_{s_p,i}^\star \leq t_{\ell i}$, where $\tau_{s_p,i}^\star$ is the car travel time from $s_p$ to $i$ (i.e., the duration of the shortest path between the two nodes). Given the bus schedule $\{t_{\ell i}\}$ and the passenger set $\mathcal{P}_T$, the platform{} can then pre-compute the passenger-line values $\{v_{\ell p}\}$. The size of each input to Algorithm~\ref{alg:rounding} has increased at most by a constant factor $T$. Hence, our algorithm still runs in polynomial time under this time-sensitive construction.
\section{Omitted proofs}
\subsection{Limits of approximation for the Real-Time Line Planning Problem}\label{app:hardness-proofs}
\subsubsection{Necessity of a candidate set of lines}
\begin{proof}[Proof of Proposition~\ref{prop:helper}]
Since there is an edge of cost $t$ between $r$ and $g_i$, by definition of a min-cost path, $t_{p_i}^\star \leq t$. Thus, $(1-\frac{\varepsilon}{t})t_{p_i}^\star \leq (1-\frac{\varepsilon}{t})t = t-\varepsilon$. This then implies that $v_{p_i}(\omega_{\ell, p_i}) = 0$ for $\omega_{\ell, p_i}$ such that $t^{\text{car}}(\omega_{\ell, p_i}) > t-\varepsilon$.
\end{proof}
\begin{proof}[Proof of Proposition~\ref{prop:hardness-helper}] Consider passenger $p_i$. {We first claim that it is without loss of generality to assume that direct travel by car is completed via edge $(r,g_i)$. This is due to the fact that the cost of each edge of $G$ is lower bounded by $\varepsilon > 0$, and the cost of $(j,g_i)$ is $t-\varepsilon$ for all $j \in S_i$. Thus, traveling from $r$ to $g_i$ via $j \in S_i$ costs {\it at least} $t$, which is exactly the cost of the trip which uses edge $(r,g_i)$.}
The fact that a bus line cannot be routed via edge $(r,g_i)$ follows from the fact that the cost of $(r,g_i)$ is $t > D$, and as such is infeasible {by bus since the maximum cost of a bus line is $D$}.
{The fact that a bus line cannot be routed via edge $(j,g_i)$ follows from the fact that the cost of $(j,g_i)$ is $t-\varepsilon > D$ for all $j \in S_i$. Now, suppose that the passenger travels via line $\ell$, and let $j_{-i}$ denote the vertex at which $p_i$ leaves the line and begins her journey by car. If $j_{-i} \notin S_i$, then reaching $g_i$ by car must incur a cost of at least $t$ (a cost of at least $\varepsilon$ to reach a node $j'\in S_i$, then a cost $t-\varepsilon$ to reach $g_i$ from $j'$). Thus, the value for this trip option is 0 by Proposition~\ref{prop:helper}, and the passenger would opt for a direct travel by car via edge $(r,g_i)$.}
\end{proof}
\begin{proof}[Proof of Proposition~\ref{prop:collect-pos-val}]
{The proposition follows immediately from Proposition~\ref{prop:hardness-helper}. The only feasible options for passenger $p_i$ which use a bus line and collect strictly positive value are those for which a node in $S_i$ can be reached by bus.} \end{proof}
\subsubsection{Inefficacy of standard approximation techniques}
\begin{proof}[Proof of Proposition~\ref{prop:not-submodular}]
Let $S$ denote the set of lines opened under $\mathbf{y}$. With mild abuse of notation, we use $w(S)$ to denote the welfare induced by this set of lines.
Consider the setting with three passengers $p_1, p_2, p_3$, $\mathcal{L} = \{\ell_1, \ell_2, \ell_3\}$, $f_{\ell} = 1 \, \forall \, \ell \in \mathcal{L}$ and $C = 1$. The value functions associated with each passenger are as follows: \begin{align*}
v_{\ell 1} = \begin{cases}
1 &\quad \text{if } \ell = \ell_1 \\
1 &\quad \text{if } \ell = \ell_2 \\
0 &\quad \text{if } \ell = \ell_3
\end{cases}, \qquad v_{\ell 2} = \begin{cases}
1 &\quad \text{if } \ell = \ell_1 \\
0 &\quad \text{if } \ell = \ell_2 \\
1 &\quad \text{if } \ell = \ell_3
\end{cases}, \qquad v_{\ell 3} = \begin{cases}
1 &\quad \text{if } \ell = \ell_1 \\
0 &\quad \text{if } \ell = \ell_2 \\
0 &\quad \text{if } \ell = \ell_3.
\end{cases} \end{align*} Passengers $p_1$ and $p_2$ use disjoint edges of $r_{\ell_1}$. Passenger $p_3$, on the other hand, uses the same edges of $r_{\ell_1}$ as $p_1$ and $p_2$. Thus, any feasible assignment of $p_3$ to $\ell_1$ requires $p_3$ to be its sole passenger.
Let $S_1=\{\ell_1\}$. Then, $w(S_1) = 2$, achieved by assigning $p_1$ and $p_2$ to $\ell_1$. Moreover, $w(S_1\cup \{\ell_3\}) = 2$, by assigning $p_1$ and $p_2$ to $\ell_1$, or $p_1$ to $\ell_1$ and $p_2$ to $\ell_3$. Now, let $S_2 = \{\ell_1, \ell_2\}$. Again, by assigning $p_1$ and $p_2$ to $\ell_1$, we obtain $w(S_2)=2$. Moreover, $w(S_2\cup \{\ell_3\}) = 3$, obtained by assigning $p_1$ to $\ell_2$, $p_2$ to $\ell_3$ and $p_3$ to $\ell_1$.
Since $S_1 \subset S_2$ and $w(S_1 \cup \{\ell_3\}) - w(S_1) < w(S_2 \cup \{\ell_3\}) - w(S_2)$, $w$ is not submodular. \end{proof}
\begin{proof}[Proof of Proposition~\ref{prop:integrality-gap}] Consider passengers $p_1, p_2$ and lines $\ell_1, \ell_2$ such that $$v_{\ell_1,p_1} = v_{\ell_2,p_2} = 1 \quad , \quad v_{\ell_2,p_1} = v_{\ell_1,p_2} = 0$$ with $r_{\ell_1}$ and $r_{\ell_2}$ non-overlapping. Suppose moreover that $c_{\ell_1} = c_{\ell_2} = 1$, and $B = 2-\varepsilon$, for some $\varepsilon \in (0,1)$.
Since the ILP can only open a single line, its optimal value is $OPT = 1$. An optimal solution to the LP relaxation of the ILP, on the other hand, is such that $y_{\ell_1}^\star = 1, y_{\ell_2}^\star = 1-\varepsilon$, and thus its optimal value is $\widehat{OPT} = 2-\varepsilon$. Taking $\varepsilon \to 0$ proves the claim.
\end{proof}
\subsection{Main result: a $1-\frac1e-\varepsilon$ approximation algorithm}\label{app:main-result-proofs}
\begin{proof}[Proof of Lemma~\ref{lem:correctness}]
Let $A \in \mathbb{R}^{|r_{\ell}| \times |\mathcal{P}|}$ denote the constraint matrix corresponding to~\eqref{single_line_subproblem}. $A$ is such that $A_{e, p} = 1$ if passenger $p$ uses edge $e$, and 0 otherwise. Since a passenger exclusively uses {\it consecutive} edges of $r_{\ell}$, the columns of $A$ have the consecutive-ones property. Thus, $A$ is totally unimodular. Since $C$ and $f_{\ell}$ are integral by assumption, these two facts together imply that LP-SEP is integral. \end{proof}
\begin{proof}[Proof of Proposition~\ref{thm:rounding_value}~\citep{SAP}] Let $ALG(p)$ denote passenger $p$'s expected contribution to the objective in the solution returned by our algorithm. To prove the approximation guarantee, it suffices to show the following:
$$ALG(p) \geq \left(1-\frac1e\right)\sum_{\ell \in \mathcal{L}}\sum_{S \in \mathcal{I}_{\ell}:p\in S} {v_{\ell p}}X_{\ell S}^{(\varepsilon)} \quad \forall \, p \in \mathcal{P}$$ where $\left\{X_{\ell S}^{(\varepsilon)}\right\}$ is the solution to $\widehat{P}^{(\varepsilon)}$. Summing over all $p$ and using $\sum_p \sum_{\ell \in \mathcal{L}}\sum_{S \in \mathcal{I}_{\ell}:p\in S} {v_{\ell p}}X_{\ell S}^{(\varepsilon)} = OPT^{(\varepsilon)}$ completes the proof of the result.
For each passenger $p\in \mathcal{P}$, let $Y_{\ell p} = \sum_{S \in \mathcal{I}_{\ell}: p \in S}X_{\ell S}^{(\varepsilon)}$. Sort the lines for which $Y_{\ell p} > 0$ in decreasing order of $v_{\ell p}$. Let $\{\ell_1,\ell_2,\ldots,\ell_k\}$ denote these lines, with $v_{\ell_1, p} \geq v_{\ell_2, p} \geq \ldots v_{\ell_k, p}$.
After rounding and re-assignment (R\&R), passenger $p$ is assigned to line $\ell_1$ if any set containing $p$ is assigned to $\ell_1$. Thus, $p$ is assigned to $\ell_1$ with probability $Y_{\ell_1, p}$. If no set containing passenger $p$ is assigned to $\ell_1$ after R\&R, then we look to $\ell_2$. The probability that a set containing $p$ is assigned to $\ell_2$ after R\&R is $Y_{\ell_2, p}$. Thus, $p$ is assigned to $\ell_2$ with probability $(1-Y_{\ell_1, p})Y_{\ell_2,p}$. It follows that, for all $k' \leq k$, passenger $p$ is assigned to $k'$ with probability $\prod_{i=1}^{k'-1}(1-{Y_{\ell_i, p}})Y_{k'}$. Hence, we have $$ALG(p) = \sum_{k'=1}^k v_{\ell_{k'} p} Y_{k'} \left( \prod_{i=1}^{k'-1}(1-Y_{\ell_i, p})\right).$$
Lemma~\ref{lem:main-techical-lemma} relates $ALG(p)$ to the contribution of passenger $p$ {\it before} rounding and re-assignment, $\sum_{k'=1}^k v_{\ell_{k'},p} Y_{\ell_{k'},p}$.
\begin{lemma}[\citep{SAP}]\label{lem:main-techical-lemma} Suppose $Y_{\ell p} \geq 0$ for all $\ell \in \mathcal{L}$, $\sum_{\ell \in \mathcal{L}}Y_{\ell p} \leq 1$, and $v_{\ell_1,p}\geq v_{\ell_2, p} \geq \ldots \geq v_{\ell_k, p} \geq 0$. Then $$\sum_{k'=1}^k {v_{\ell_{k'} p}Y_{k'}}\left(\prod_{i=1}^{k'-1}(1-Y_{\ell_i, p})\right) \geq \left(1-(1-\frac1L)^L \right)\sum_{k'=1}^k v_{\ell_{k'},p} Y_{\ell_{k'},p}.$$
\end{lemma}
Using the fact that $(1-(1-\frac{1}{L})^L)\geq 1-\frac1e$ for all $L \geq 1$ completes the proof of Proposition~\ref{thm:rounding_value}. \end{proof}
\begin{proof}[Proof of Proposition~\ref{prop:cost-characterization}] Let $\widetilde{Y}_{\ell}$ be the indicator variable denoting the event that line $\ell$ was opened {\it before} the re-assignment step, and let $Y_{\ell}$ denote the final line status, after the aggregation step. Let $c(Y)$ and $c(\widetilde{Y})$ denote the total costs associated with $Y$ and $\widetilde{Y}$, respectively. Between the re-assignment step and the aggregation step, the cost of the solution could only have decreased, since lines were potentially closed. Similarly, by Proposition~\ref{prop:final-step-doesnt-matter}, the cost of the solution could only have decreased after the aggregation step. Thus, we have $c(Y) \leq c(\widetilde{Y})$, and\begin{align*} \mathbb{E}\left[c(Y)\right] \leq \mathbb{E}\left[c(\widetilde{Y})\right] = \sum_{\ell \in \mathcal{L}} c_{\ell}\mathbb{P}\left[\widetilde{Y}_{\ell} = 1\right] = \sum_{\ell \in \mathcal{L}} c_{\ell}\left(\sum_{S \in \mathcal{I}_{\ell}}X_{\ell S}^{(\varepsilon)}\right) \leq B(1-\varepsilon), \end{align*} where the second inequality is by feasibility of $X_{\ell S}^{(\varepsilon)}$. Thus, the budget constraint is satisfied in expectation.
We now prove the second part of the claim: \begin{align}
\mathbb{P}\left(\sum_{\ell\in \mathcal{L}} c_\ell \, Y_{\ell}\geq (1+\delta)(1-\varepsilon)B\right) &{\leq} \mathbb{P}\left(\sum_{\ell\in \mathcal{L}} c_\ell \, \widetilde{Y}_{\ell}\geq (1+\delta)(1-\varepsilon)B\right) \notag \\
&= \mathbb{P}\left(\sum_{\ell\in \mathcal{L}} \frac{c_\ell}{c_{\max}} \, \widetilde{Y}_{\ell}\geq (1+\delta)\frac{(1-\varepsilon)B}{c_{\max}}\right) \notag
\\&{\leq} e^{-\delta^2(1-\varepsilon)B/3c_{\max}} \label{eq:hoeff} \end{align} where~\eqref{eq:hoeff} follows from an application of the Chernoff bound to the independent random variables $\left\{\frac{c_{\ell}}{c_{\max}}\widetilde{Y}_{\ell}\right\}_{\ell \in \mathcal{L}}$, and uses the fact that $\mathbb{E}\left[\sum_{\ell}\frac{c_{\ell}}{c_{\max}}\widetilde{Y}_{\ell}\right] \leq \frac{(1-\varepsilon)B}{c_{\max}}$ by feasibility of $\left\{X_{\ell S}^{(\varepsilon)}\right\}$.
\end{proof}
\section{Additional numerical experiments on synthetic data} \label{app:synthetic_experiments} {To complement our real-world data experiments, we consider a synthetic dataset and show how the performance of our algorithm depends on the number of requests and the cardinality of the candidate set of lines, using the ILP as a benchmark}.
Observe that our algorithm relies on the underlying road network solely through the candidate set of lines $\mathcal{L}$, the line costs $\{c_{\ell}\}$, and the passenger-line values $\{v_{\ell p}\}$. Thus, it suffices to directly generate these latter sets of inputs, rather than inheriting them from an underlying structured network. We note that generating inputs in this manner, rather than running our algorithm on a synthetic network (e.g., a grid network), further underscores the strength and generalizability of our scheme, as its success is not tied to the geometry of any underlying graph.
\noindent\textbf{Line inputs.} We generate the candidate set of lines as follows. For each $\ell \in \mathcal{L}$, we associate $D_{\ell}$ edges, where $D_{\ell}\sim Unif\{5,50\}$. Moreover, let $c_{\ell} = 1 \, \forall \, \ell \in \mathcal{L}$. {This implies that a platform{} with budget $B$ can open {at most} $B$ lines.} Let $\mathcal{F}$ denote the set of possible frequencies with which to operate each bus route. In our first set of experiments, we let $\mathcal{F} = \{1\}$. Doing so is without loss of generality since, by definition, bus routes operated at different frequencies are considered to be different lines. Thus, considering a larger set of frequencies is computationally equivalent to increasing the size of candidate set of lines (e.g., considering 1,000 lines with 2 different frequencies is equivalent to considering 2,000 lines with a single frequency). We set the bus capacity $C = 30$.
\noindent\textbf{Passenger inputs.} For each passenger $p \in \mathcal{P}$ and line $\ell \in \mathcal{L}$, we let $r_{\ell p}$ be a random subset of contiguous edges of $r_{\ell}$. To model the fact that, in a realistic network, passengers would not be covered by all lines, we define random variable $Z_{\ell p}~\sim Ber(0.1)$ representing whether or not passenger $p$ is covered by line $\ell$. Given $Z_{\ell p}$, we define the passenger-line value as follows:
$$v_{\ell p} = \begin{cases}
Unif[0,1] &\quad \text{ if } Z_{\ell p} = 1 \\
0 &\quad \text { otherwise.}
\end{cases}
$$
\noindent\textbf{Performance metrics.} We investigate the performance of this practical procedure along three dimensions: $(i)$ the number of passengers $N$, $(ii)$ the size of the candidate set of lines $L$, and $(iii)$ the platform's budget $B$. {For both the ILP and our algorithm, we set a strict time limit of 20 minutes, and compare the solutions returned by the two schemes at the end of the allotted time.}
Given an instance of line and passenger inputs, we run the procedure described in Section~\ref{procedure} $M = 500$ times for each combination of parameters $(L,N,B)$ (i.e., we find the maximum of the $m=10^4$ realized solutions $M=500$ times).
We run the procedure for 5 randomly-generated instances of line and passenger inputs. Let $ALG$ denote the empirical average of the solution returned by the procedure.
As before, we compute $\eta$ the {empirical average of the} multiplicative gap between the solution returned by our procedure and the value of the configuration LP $\widehat{P}^{(\varepsilon)}$ at the end of the allotted time.
We report the results of our experiments in Table~\ref{tab:results}.
Whereas in theory the ILP solver provides an upper bound on $ALG$, this does not necessarily hold in our numerical results. This is due to the fact that, for large-scale problems and under reduced time budgets (i.e., the real-time application we are interested in), the ILP solver cannot solve the problem to optimality, and as such the objective it achieves is not necessarily an upper bound on $ALG$ in practice.
\begin{table}[ht] \centering
\begin{tabular}{|ccc|ccc|}
\hline
$L$ & $N$ & $B$ & ILP & $ALG$ & $\eta$ \\
\hline \hline
$1,000$ & $5,000$ & 20 & {2363} & 2173 & 0.81 \\
$5,000$ & $5,000$ & 20 & 2098 & \textbf{2171} & 0.81 \\
$7,000$ & $5,000$ & 20 & 807 & \textbf{2173} & 0.80\\
$10,000$ & $5,000$ & 20 & --- & \textbf{2174} & 0.81 \\
\hline
$5,000$ & $5,000$ & 20 & 2098 & \textbf{2171} & 0.81 \\
$5,000$ & $10,000$ & 20 & --- &\textbf{3498} & 0.84 \\
$5,000$ & $15,000$ & 20 & --- & \textbf{4445} & 0.88 \\
\hline
\end{tabular}
\qquad
\begin{tabular}{|ccc|ccc|}
\hline
$L$ & $N$ & $B$ & ILP & $ALG$ & $\eta$ \\
\hline \hline
$1,000$ & $5,000$ & 40 & 3671 & 2744 & 0.88 \\
$5,000$ & $5,000$ & 40 & 3686 & 2743 & 0.88\\
$7,000$ & $5,000$ & 40 & 2750 & \textbf{2754} & 0.88\\
$10,000$ & $5,000$ & 40 & --- & \textbf{2748} & 0.88\\
\hline
$5,000$ & $5,000$ & 40 & 3686 & 2743 & 0.88 \\
$5,000$ & $10,000$ & 40 & --- & \textbf{4949} & 0.85 \\
$5,000$ & $15,000$ & 40 & --- & \textbf{6691} & 0.82 \\
\hline \end{tabular} \caption{Numerical results for budgets $B \in \{20,40\}$. Bolded values of $ALG$ indicate that our procedure outperforms the ILP benchmark for the corresponding $L, N, B$. While the ILP outperforms our algorithm {on smaller instances}, for larger values of $L$ and $N$, our algorithm consistently outperforms the ILP. {As the budget increases from 20 to 40, the ILP outperforms our algorithm for a larger set of values of $L$ and $N$; however, there still exists a threshold past which our algorithm outperforms the ILP. This difference is especially stark when $L$ and $N$ are both very large (we note that it is reasonable to expect $L$ and $N$ to grow with $B$): for these large-scale settings, the ILP is incapable of returning any feasible solution in the allotted time.} {Observe moreover that $\eta$, the gap between the solution produced by our procedure and the value of the configuration LP, is consistently above $80\%$, which is a significant improvement upon the $0.95 \cdot (1-\frac1e)$ (i.e., $60\%$) theoretical guarantee.}}\label{tab:results} \end{table}
\end{document} |
\begin{document}
\title{On the time to absorption in $\Lambda$-coalescents}
\author{G\"otz Kersting\thanks{Institut f\"ur Mathematik, Goethe Universit\"at, Frankfurt am Main, Germany \newline kersting@math.uni-frankfurt.de, wakolbinger@math.uni-frankfurt.de \newline Work partially supported by the DFG Priority Programme SPP 1590 ``Probabilistic Structures in Evolution''}$\ $ and Anton Wakolbinger$^*$} \date{} \maketitle \begin{abstract} We present a law of large numbers and a central limit theorem for the time to absorption of $\Lambda$-coalescents, started from $n$ blocks, as $n \to \infty$. The proofs rely on an approximation of the logarithm of the block-counting process of $\Lambda$-coalescents with a dust component by means of a drifted subordinator.\\
{\em AMS 2010 subject classification:} 60J75 (primary), 60J27, 60F05 (secondary)$^{\color{white} \big|}$\\ {\em Keywords:} coalescents, time to absorption, law of large numbers, central limit theorem, subordinator with drift \end{abstract}
\section{Introduction and main results} How long does it take for the ancestral lineages of a large sample of individuals back to its common ancestor? For population of constant size this turns into a question on the absorption time of a coalescents, which describes the genealogical tree of $n$ individuals by means of merging partitions. Here we consider coalescent with multiple mergers, also known as $\Lambda$-coalescents, which were introduced in 1999 by Pitman \cite{Pi} and Sagitov \cite{Sa}. If $\Lambda$ is a finite, non-zero measure on $[0,1]$, then the $\Lambda$-coalescent started with $n$ blocks is a continuous-time Markov chain $(\Pi_n(t), t \geq 0)$ taking its values in the set of partitions of $\{1, \dots, n\}$. It has the property that whenever there are $b$ blocks, each possible transition that involves merging $k \geq 2$ of the blocks into a single block happens at rate \[ \lambda_{b,k} = \int_{[0,1]} p^{k} (1-p)^{b-k} \: \frac{\Lambda(dp)}{p^2}\ , \] and these are the only possible transitions. Let $N_n(t)$ be the number of blocks in the partition $\Pi_n(t)$, $t \ge 0$. Then \[\tau_n := \inf\{t\ge 0: N_n(t) = 1\}\] is the time of the last merger, also called the {\em absorption time} of the coalescent started in $n$ blocks. We will investigate the asymptotic distribution of $\tau_n$ as $n \to \infty$.
Our first result is a law of large numbers for the times $\tau_n$. Let \[ \mu:=\int_{[0,1]} \log \frac 1{1-p} \, \frac {\Lambda(dp)}{p^2} \ , \] in particular $\mu=\infty$ in case of $\Lambda(\{0\})>0$ or $\Lambda(\{1\})>0$.
\begin{Theo}\label{LLN} For any $\Lambda$-coalescent, \begin{align} \frac{\tau_n}{\log n} \to \frac 1\mu \ . \label{lln} \end{align} in probability as $n \to \infty$. \end{Theo}
This theorem says that in a $\Lambda$-coalescent the number of blocks decays at least at an exponential rate. If $\mu=\infty$, then the right-hand limit is 0, and the coalescent decreases even super-exponentially fast. The case $\mu<\infty$ is equivalently captured by the simultaneous validity of the conditions \[ \int_{[0,1]}\frac{\Lambda(dp)}p < \infty \quad \text{ and }\quad \int_{[0,1]} \log \frac 1{1-p} \, \Lambda (dp) < \infty \ . \] The first one is a requirement on $\Lambda$ in the vicinity of 0, it prohibits a swarm of small mergers (as they occur in coalescents coming down from infinity, meaning that the $\tau_n$ are bounded in probability uniformly in $n$). The second is a condition on $\Lambda$ in the vicinity of 1. It rules out the possibility of mergers which, although appearing only every now and then, are so vast that they make the coalescent collapse. -- A counterpart to Theorem \ref{LLN}, with $\tau_n$ in \eqref{lln} replaced by its expectation, was obtained by Herriger and M\"ohle \cite{HeMoe}.
Our second result is a central limit theorem. Here we confine ourselves to coalescents with $\mu<\infty$. Then the function \begin{equation} \label{ourf} f(y):= \int_{[0,1]} \frac{1-(1-p)^{e^y} }{e^y} \, \frac{\Lambda(dp)}{p^2} \ , \ y \in \mathbb R \end{equation} is everywhere finite. Also $f$ is a positive, monotone decreasing, continuous function with the property $f(y)\to 0$ for $y \to \infty$. Let \[ b_n := \int_\kappa^{\log n} \frac{dy}{\mu- f(y)} \ , \] where we choose $\kappa \ge 0$ such that \[ f(y) \le \frac \mu 2 \text{ for all } y \ge \kappa \ . \]
\begin{Theo}\label{CLT} Assume that $\mu< \infty$ and moreover \[ \sigma^2 := \int_{[0,1]} \Big(\log \frac 1{1-p}\Big)^2\, \frac{\Lambda(dp)}{p^2} < \infty \ . \] Then \begin{align}\label{celith}
\frac{\tau_n - b_n}{\sqrt{\log n}}\ \stackrel d \to\ N(0, \sigma^2/\mu^3) \end{align}
as $n \to \infty$. \end{Theo}
Under the additional condition \begin{align}\label{integral} \int_{[0,1]} \log \frac 1p \, \frac{\Lambda(dp)}{p} < \infty \ . \end{align} the CLT \eqref{celith} has been obtained by Gnedin, Iksanov and Marynych \cite{Gne}, with $b_n $ replaced by $\log n/\mu$. (Their condition (9) is equivalent to the above condition \eqref{integral}, see Remark 13 in \cite{Ke}). Thus the question arises, whether the simplified centering by $\log n/\mu$ is always feasible. The next proposition shows that this can be done under a condition that is weaker than \eqref{integral}, but not in any case.
\begin{Prop} \label{Prop2} Let $0\le c < \infty$. Then \begin{align} \label{condi2} b_n= \frac{\log n}{\mu}+\frac {2c} {\mu^2}\sqrt {\log n} + o(\sqrt {\log n}) \end{align} as $n \to \infty$, if and only if \begin{align}\label{condi} \sqrt{\log \tfrac 1r}\int_{[0,r]} \frac{\Lambda(dp)}p \to c \end{align} as $r\to 0$. \end{Prop}
\paragraph{Example.} We consider for $\gamma \in \mathbb R$ the finite measures \[ \Lambda(dp) = \big(1+\log \tfrac 1p\big)^{-\gamma} \, dp\ , \ 0\le p \le 1 \ . \] For $\gamma=0$ this gives the Bolthausen-Sznitman coalescent. For $\gamma >1$ it leads to coalescents with $\mu, \sigma^2<\infty$. Note that \eqref{integral} is satisfied iff $\gamma >2$, and \eqref{condi} is fulfilled iff $\gamma >3/2$. Thus within the range $1<\gamma \le 3/2$ one has to come back to the constants $b_n$ in the central limit theorem.
The law of large numbers from Theorem 1 holds for all $\gamma >1$. For the regime $\gamma \le 1$, Theorem~\ref{LLN} just tells us that $\tau_n= o_P(\log n)$. For $\gamma=0$, the Bolthausen-Sznitman coalescent, it is known that $\tau_n$ is already down to the order $\log\log n$ \cite{Go}. For $\gamma <0$, applying Schweinsberg's criterion \cite{Schw}, it can be shown that the coalescents come down from infinity. There remains the gap $0< \gamma \le 1$. It is tempting to conjecture that $\tau_n$ is of order $(\log n)^\gamma$ for $0<\gamma < 1$. \qed
\mbox{}\\ If equation \eqref{condi} is violated then the subsequent approximation to $b_n$ may be practical. Starting from the identity \[ \frac{1}{\mu-f(y)} = \frac 1\mu +\frac {f(y)}{\mu^2} + \frac{f^2(y)}{\mu^3} +\cdots + \frac{f^k(y)}{\mu^{k+1}}+ \frac {f^{k+1}(y)}{\mu^{k+1}(\mu-f(y))}\] we obtain the expansion \begin{align*} b_n = \frac {\log n}\mu + \frac {1}{\mu^2} \int_0^{\log n} f(y)\, dy + \cdots + \frac 1{\mu^{k+1}}\int^{\log n}_0 f^k(y)\, dy + O\Big( \int_0^{\log n} f^{k+1}(y)\, dy\Big) \ . \end{align*}
Let us now explain the method of proving Theorems \ref{LLN} and \ref{CLT}. We are mainly dealing with $\Lambda$-coalescents having a {\em dust component}. Shortly speaking these are the coalescents for which the rate, at which a single lineage merges with some others from the sample, stays bounded as the sample size tends to infinity. As is well-known this property is characterized by the condition \begin{align} \label{dustcond} \int_{[0,1]} \frac{\Lambda(dp)}{p} < \infty \ . \end{align} An established tool for the analysis of a $\Lambda$-coalescent with dust is the subordinator $S=(S_t)_{t \ge 0}$, which is used to approximate the logarithm of its block-counting process $N_n=(N_n(t))_{t \ge 0}$ (see e.g. Pitman \cite{Pi}, M\"ohle \cite{Moe}, and the above mentioned paper by Gnedin et al \cite{Gne}). We will recall this subordinator in Sec.~3. Indeed, analogues of Theorems 1 and 2 are well-known for first-passage times of subordinators with finite first resp. second moment. However, this approximation neglects the subtlety that a coalescent of $b$ lineages results in a downward jump of size $b-1$ (and not $b$) for the process $N_n$. This effect becomes significant when many small jumps accumulate over time, as it happens close to the dustless case (and as it becomes visible in Proposition \ref{Prop2} and in the above example). Then the appropriate approximation is provided by a {\em drifted} subordinator $Y_n=(Y_n(t))_{t \ge0}$, given by the SDE \[ Y_n(t) = \log n -S_t + \int_0^t f(Y_n(s))\, ds \ , \ t \ge 0\ ,\] with initial value $Y_n(0)=\log n$. The drift compensates the just mentioned difference between $b$ and $b-1$. In Kersting et al \cite{Ke} it is shown that
\[ \sup_{t<\tau_n}\big|Y_n(t)- \log N_n(t)\big|=O_P(1)\] as $n \to \infty$, that is, these random variables are bounded in probability. In Sec.~3 we suitably strengthen this result. In Sec.~2 we provide the required limit theorems for passage times for a more general class of drifted subordinators. The above results are then proved in Sec.~4.
It turns out that the regime considered by Gnedin et al \cite{Gne} is the one in which the random variables $\int_0^{\tau_n} f(Y_n(s))\, ds$ are bounded in probability uniformly in $n$. This can be seen to be equivalent to the requirement $\int_0^\infty f(y)\, dy<\infty$, which likewise is equivalent to \eqref{integral} (see the proof of Corollary 12 in \cite{Ke}). Under this assumption Gnedin et al \cite{Gne} proved their central limit theorem also with non-normal (stable or Mittag-Leffler) limiting distributions of $\tau_n$. A similar generalization of Theorem \ref{CLT} is feasible in the general dust case, without the requirement \eqref{integral}.
\section{Limit theorems for a drifted subordinator}
Let $S=(S_t)_{t \ge 0}$ be a pure jump subordinator with L\'evy measure $\lambda$ on $(0,\infty)$. Recall that this requires \[ \int_0^\infty (y\wedge 1) \, \lambda(dy) < \infty \ .\] With regard to the mentioned properties of the function in \eqref{ourf}, let $f:\mathbb R \to \mathbb R$ be an arbitrary positive, non-increasing, continuous function with \[ \lim_{y\to \infty} f(y)=0\ . \] Let the process $Y^z=(Y^z_t)_{t \ge 0}$ denote the unique solution of the SDE \begin{align} \label{SDE} Y^z_t = z-S_t + \int_0^t f(Y^z_s)\, ds \end{align} with initial value $z> 0$. We will investigate the asymptotic behaviour of its passage times across $x \in \mathbb R$, \[ T^z_x := \inf\{ t \ge 0: Y^z_t < x \}\ ,\]
in the limit $z \to \infty$.
The first result provides a law of large numbers. Denote \begin{align} \mu := \int_{(0,\infty)} y\, \lambda(dy) \ . \label{defmu} \end{align} \begin{Prop} \label{Prop3} Assume that $\mu<\infty$. Then for any $x \in \mathbb R$ \[ \frac{1}z \, T^z_x \to \frac 1\mu \] in probability as $z\to \infty$. \end{Prop}
\begin{proof} Let $z >x$. Then \[ \{T^z_x \ge t\} = \{ Y^z_s \ge x \text{ for all }s \le t \} = \Big\{S_s \le z-x + \int_0^sf(Y^z_u)\, du \text{ for all } s \le t\Big\} \ . \] By positivity of the function $f$ it follows $ \mathbf P( T^z_x \ge t) \ge \mathbf P( S_t \le z-x) $, thus for any $\varepsilon >0$ \begin{equation}\label{oneplusepsilon} \mathbf P\Big(T_x^{ z}\ge (1-\varepsilon)\frac z{\mu} \Big) \ge \mathbf P\big( S_{(1-\varepsilon)z/\mu} \le z -x\big) \ . \end{equation} Now $\mu=\mathbf E[S_1]$, thus by the law of large numbers \[\frac{S_t}t \to \mu \] a.s., hence the right-hand term in \eqref{oneplusepsilon} converges to 1 for $z \to \infty$ and also \[\mathbf P\Big(T_x^{ z}\ge (1-\varepsilon)\frac z{\mu} \Big) \to 1 \ . \]
On the other hand, \begin{align*}\{ T^z_x \ge t\} = \{ Y^z_s \ge x \text{ for all } s \le t\}=\Big\{ Y^z_s \ge x \text{ for all }s \le t\, ,\, S_t \le z -x + \int_0^t f(Y_s^z)\,ds \Big\} \ . \end{align*} Monotonicity of $f$ implies $ \mathbf P( T^z_x \ge t ) \le \mathbf P \big(S_t \le z -x + t f(x) \} \big) $. Therefore, since $f(x)\to 0$ as $x \to \infty$, \begin{align*} \mathbf P\Big(T_x^{ z}\ge (1+\varepsilon)\frac z{\mu} \Big) &\le \mathbf P\Big( S_{(1+\varepsilon)z/\mu } \le z -x + (1+\varepsilon)\frac z{\mu}f(x) \Big)\\&\le \mathbf P\big( S_{(1+\varepsilon)z/\mu } \le z(1+\varepsilon/2) -x\big) \end{align*} if only $x$ is sufficiently large. Now the right-hand term converges to 0, thus it follows that \[ \mathbf P\Big(T_x^{ z}\ge (1+\varepsilon)\frac z{\mu} \Big) \to 0\ . \] Note that we proved this result only for $x$ sufficiently large, depending on $\varepsilon$. However, this restriction may be skipped, since for fixed $x_1<x_2$ the random variables $T_{x_1}^z-T_{x_2}^z$ are bounded in probability uniformly in $z$. Thus altogether we have for any $x$ \[ \mathbf P\Big( (1-\varepsilon) \frac z{\mu} \le T^z_x < (1+\varepsilon) \frac z{\mu} \Big)\to 1\] as $z \to \infty$, which (since $\varepsilon > 0$ was arbitrary) is our assertion. \end{proof}
\mbox{}\\ Now we turn to a central limit theorem for passage times of the processes $Y^z$. Let the function $\beta_z$, $z\ge \kappa$, be given by \begin{equation}\label{defbeta}
\beta_z:= \int_\kappa^z \frac{dy}{\mu-f(y)} \ ,
\end{equation} where we choose $\kappa\ge 0$ so large that \[ \sup_{y \ge \kappa} f(y) \le \frac \mu 2 \ . \]
\begin{Prop}\label{Prop4} Suppose that \begin{align}\label{sigma} \sigma^2 := \int_{(0,\infty)} y^2 \, \lambda (dy) < \infty\ . \end{align}
Then \[ \frac{ T_x^z- \beta_z}{\sqrt z}\ \stackrel d \to\ N(0, \sigma^2/\mu^3) \] as $z \to \infty$. \end{Prop}
\begin{proof} (i) Note again that for $x_1 < x_2$ the random variables $T^z_{x_1}-T^z_{x_2}$ are bounded in probability uniformly in $z$. Thus it suffices to prove our theorem for all $x \ge x_0$ for some $x_0\in \mathbb R$. Therefore we may change $f(x)$ for all $x<x_0$. We do it in such a way that $f(x)\le \mu/2$ for all $x\in \mathbb R$, without touching the other properties of $f$. Thus we assume from now that \begin{align} \label{fmu} f(y) \le \frac \mu 2 \quad \text{ for all } y \in \mathbb R\ \end{align} and set $\kappa=0\ $ in \eqref{defbeta}. Consequently, \begin{equation}\label{betabounds} \frac z\mu \le \beta_z\le \frac{2z}\mu\ , \quad z >0\ . \end{equation}
For any $z >0$ we define the function $\rho^z(t)=\rho^z_t$, $0 \le t \le \beta_z$, such that \[ \beta_{\rho^z(t)}=\beta_z-t\ , 0\le t \le \beta_z\ , \] in particular $\rho^z(0)=z$ and $\rho^z(\beta_z)=0$. This means that $\rho^z$ arises by first inverting the function $\beta$ (restricted to the interval $[0, z]$), and then reversing the time parameter on its domain $[0, \beta_z]$. By differentiation we obtain \[ \dot\rho^z_t= f(\rho^z_t)-\mu\ , \]
consequently $\dot \rho_t \le - \mu/2$ and \[ \rho^z_t = z - \mu t + \int_0^t f(\rho^z_s)\, ds\ . \]
(ii) A glimpse on \eqref{SDE} suggests that $\rho^z$ will make a good approximation for the process $Y^z$. In order to estimate their difference observe that \[ Y_t^z- \rho^z_t = -(S_t-\mu t) + \int_0^t (f(Y^z_s)-f(\rho^z_s))\, ds \ .\] For given $t>0$ define \begin{align*} u_t =\begin{cases} \sup \{ s<t: Y^z_s \le \rho^z_s\} \mbox{ on the event } Y_t^z > \rho^z_t \\ \sup \{ s<t: Y^z_s \ge \rho^z_s\} \mbox{ on the event } Y_t^z < \rho^z_t \end{cases} \end{align*} and $u_t:=t$ on the event $Y_t^z = \rho^z_t$. We have $0 \le u_t \le t$, since $Y^z_0=z= \rho_0^z$. Because $f$ is a decreasing function, the event $Y_t^z > \rho^z_t$ implies that \begin{align*}Y_t^z- \rho^z_t &\le Y_t^z- \rho^z_t - \int_{u_t}^t (f(Y^z_s)-f(\rho^z_s))\, ds - (Y^z_{u_t-}- \rho^z_{u_t-})\\ &= -(S_t- \mu t) + (S_{u_t-}-\mu u_t) \ . \end{align*} On the event $Y_t^z < \rho^z_t$ there is an analogous estimate from below, altogether \begin{align*}
|Y_t^z- \rho^z_t| \le 2 M_t \quad \text{ with } M_t:= \sup_{u \le t} |S_u-\mu u| \ .
\end{align*} Consequently, $Y^z_s \ge \rho^z_s-2M_s \ge \rho^z_s-2M_t$ for $s \le t$ and by means of the monotonicity of $f$ \begin{align*} \int_0^t f(Y^z_s)\, ds -\int_0^tf(\rho^z_s)\, ds \le \int_0^t f(\rho^z_s-2M_t)\, ds -\int_0^tf(\rho^z_s)\, ds \le2M_t f(\rho^z_t-2M_t)\ . \end{align*} An analoguous estimate is valid from below and we obtain \begin{align}
\Big|\int_0^t f(Y^z_s)\, ds -\int_0^tf(\rho^z_s)\, ds \Big| \le 2M_t f(\rho^z_t-2M_t)\ . \label{Mrho} \end{align}
At this point we recall that under the above assumptions on the subordinator $S$ by Donsker's invariance principle we have \[ M_t = O_P( \sqrt t) \] as $t \to \infty$.
(iii) Now we derive some upper estimates of probabilities. Given $a,x\in \mathbb R$, we have for any $c>0$ \begin{align*} \mathbf P&( T^z_x \ge \beta_z + a \sqrt z) = \mathbf P(Y^z_t \ge x \text{ for all } t \le \beta_z+a\sqrt z)\\ &= \mathbf P\Big(S_{\beta_z + a\sqrt z} \le z-x + \int_0^{\beta_z + a\sqrt z} f(Y^z_s) \, ds\, , \,Y^z_t \ge x \text{ for all } t \le \beta_z+a\sqrt z\Big)\\
&\le \mathbf P\Big(S_{\beta_z + a\sqrt z} \le z-x + f(x)(c+|a|)\sqrt z+ \int_0^{\beta_z -c\sqrt z} f(Y^z_s) \, ds\Big) \end{align*} We now bring \eqref{Mrho} into play. From the definition of $\rho^z$ we have, writing $\beta(y) = \beta_y$, that \[\beta(\rho^z(\beta_z-c\sqrt z))= c\sqrt z,\] thus because of \eqref{betabounds} \[\rho^z(\beta_z-c\sqrt z) \ge \frac{c\sqrt z}{2\mu}.\] Then on the event $M_{\beta_z} \le {c\sqrt z} /{(8\mu)}$ we have \[\rho^z(\beta_z-\sqrt z) - 2M_{\beta_z-c\sqrt z}\ge \frac{c\sqrt z}{2\mu}- \frac{c\sqrt z}{4\mu} = \frac{c\sqrt z}{4\mu}. \] Consequently, by means of \eqref{Mrho} and since $\beta_z \le 2z/\mu$ \begin{align} \mathbf P( T^z_x &\ge \beta_z + a \sqrt z) \le \mathbf P\Big(M_{2z/\mu} > \frac{c\sqrt z}{8\mu}\Big) \notag\\ & \mbox{}+
\mathbf P\Big(S_{\beta_z + a\sqrt z} \le z-x + f(x)(c+|a|)\sqrt z+ \int_0^{\beta_z} f(\rho^z_s) \, ds+ \frac{c\sqrt z}{4\mu} f\Big(\frac{c\sqrt z}{4\mu}\Big) \Big) \ . \label{Ungl1} \end{align} Moreover, by definition of $\rho^z$, \[ z+\int_0^{\beta_z} f(\rho^z_s) \, ds =\rho^z(\beta_z)+\mu \beta_z = \mu \beta_z . \]
Therefore, if we fix $\varepsilon>0$, let $c$ be so large that the first right-hand probability in \eqref{Ungl1} is smaller than $\varepsilon$, then choose $z$ so large that $(c/4\mu) f(\frac{c\sqrt z}{ 4\mu}) \le \varepsilon $, and also choose $x >0$ and so large that $cf(x)(c+|a|) \le \varepsilon $, then we end up with \[ \mathbf P( T^z_x \ge \beta_z + a \sqrt z) \le \varepsilon + \mathbf P\Big(S_{\beta_z + a\sqrt z} \le \mu\beta_z +2 \varepsilon\sqrt z \Big)\ . \] Also by the law of large numbers \[ S_{\beta_z + a\sqrt z}- S_{\beta_z} \sim \mu a\sqrt z \] in probability. Therefore \[\mathbf P( T^z_x \ge \beta_z + a \sqrt z) \le 2\varepsilon+ \mathbf P\big(S_{\beta_z} \le \mu\beta_z + (-\mu a+3\varepsilon)\sqrt z \big)\ .\] Moreover $\mu \beta_z \sim z $, hence \[\mathbf P( T^z_x \ge \beta_z + a \sqrt z) \le 2 \varepsilon+ \mathbf P\big(S_{\beta_z} \le \mu\beta_z + (-\mu a+4\varepsilon)\mu^{1/2} \sqrt{\beta_z} \big)\] for large $z$. Now from assumption \eqref{sigma} and the central limit theorem there follows \[ \frac{S_t-\mu t} {\sqrt {\sigma^2 t}} \ \stackrel d\to\ L \ , \] where $L$ denotes a standard normal random variable. Thus \[ \limsup_{z \to \infty} \mathbf P( T^z_x \ge \beta_z + a \sqrt z)\le 2\varepsilon + \mathbf P (L \le (-\mu a+4\varepsilon)\mu^{1/2} \sigma^{-1})\ . \] Note that the choice of $x$ depends on $\varepsilon$ in our proof. However, since again the differences $T^z_{x_1}-T^z_{x_2}$ are bounded in probability uniformly in $z$, this estimate generalizes to all $x$. Now letting $\varepsilon \to 0$ we obtain \[\limsup_{z \to \infty} \mathbf P \Big( \frac{T^z_x- \beta_z}{\sqrt z} \ge a\Big) \le \mathbf P(L \le -\mu^{3/2} \sigma^{-1}a) \ . \] This is the first part of our claim.
(iv) For the lower estimates we first introduce the random variable \[ R_{z,x} := \sup\{ t\ge 0: Y^z_t \ge x\} - \inf \{ t \ge 0: Y^z_t < x\} \] which is the length of the time interval where $Y^z_t-x$ is changing from positive sign to ultimately negative sign (note that the paths of $Y^z$ are {\em not} monotone). We claim that these random variables are bounded in probability, uniformly in $z$ and $x$. Indeed, with \[ \eta_{z,x}:= \inf \{t \ge 0: Y^z_t < x\} \] we have for $t>\eta=\eta_{z,x} $ because of $Y^z_\eta \le x$ and \eqref{fmu} \begin{align*}Y_t^z &= Y_\eta^z -(S_t-S_\eta) +\int_\eta^tf(Y^z_s)\, ds\le x -(S_t-S_\eta) + \frac \mu 2 (t-\eta) \ . \end{align*} Thus $R_{z,x}$ is bounded from above by \[ R_{z,x}' := \sup \{u \ge 0: (S_{\eta_{z,x}+u}-S_{\eta_{z,x}}) - \mu u/2\le 0 \} \] These random variables are a.s. finite. Moreover, they are identically distributed, since $\eta_{z,x}$ are stopping times. This proves that the $R_{z,x}$ are uniformly bounded in probability.
Now for the lower bounds we have for $a,b \in \mathbb R$ \begin{align*} \mathbf P(T^z_x \ge \beta_z + a \sqrt z) &\ge \mathbf P( Y^z_t \ge x \text{ for all } t \le \beta_z+a\sqrt z\, ,\, R_{z,x} \le b)\\ &= \mathbf P( Y^z_t \ge x \text{ for all } \beta_z +a\sqrt z- b\le t \le \beta_z+a\sqrt z\, ,\, R_{z,x} \le b)\ . \end{align*} For these $t$ we have \[Y_t^z = z- S_t+ \int_0^t f(Y^z_s)\, ds \ge z-S_{\beta_z+a\sqrt z}+ \int_0^{\beta_z +a\sqrt z- b} f(Y^z_s)\, ds \ , \] therefore \begin{align*} \mathbf P(T^z_x \ge \beta_z + a \sqrt z) &\ge \mathbf P\Big( S_{\beta_z+a\sqrt z} \le z- x+ \int_0^{\beta_z+a\sqrt z- b}f(Y^z_s)\, ds \, , \, R_{z,x} \le b\Big)\\ &\ge \mathbf P\Big( S_{\beta_z+a\sqrt z} \le z- x+ \int_0^{\beta_z-c\sqrt z}f(Y^z_s)\, ds\Big) - \mathbf P(R_{z,x}>b) \end{align*} for $c$ sufficiently large.
We now bring, as in part (iii), \eqref{Mrho} into play. Proceeding analogously we obtain instead of \eqref{Ungl1} the estimate \begin{align*} \mathbf P( T^z_x \ge \beta_z + a \sqrt z) \ge &- \mathbf P(R_{z,x}>b)-\mathbf P\Big(M_{2z/\mu} > \frac{c\sqrt z}{8\mu}\Big) \\ & \mbox{}+ \mathbf P\Big(S_{\beta_z + a\sqrt z} \le z-x + \int_0^{\beta_z-c\sqrt z} f(\rho^z_s) \, ds- \frac{c\sqrt z}{4\mu} f\Big(\frac{c\sqrt z}{4\mu}\Big) \Big) \ . \end{align*} Also, since $\rho^z_{\beta_z}=0$ and $\dot \rho^z_t \le -\mu/2$, \[ \int_{\beta_z-c\sqrt z}^{\beta_z} f(\rho^z_s) \, ds \le \int_0^{c\sqrt z} f(\mu s/2) \, ds = o(\sqrt z) \ .\] Hence, for given $\varepsilon>0$ and $z$ sufficiently large \begin{align*} \mathbf P( T^z_x \ge \beta_z + a \sqrt z) \ge &- \mathbf P(R_{z,x}>b)-\mathbf P\Big(M_{2z/\mu} > \frac{c\sqrt z}{8\mu}\Big) \\ & \mbox{}+ \mathbf P\Big(S_{\beta_z + a\sqrt z} \le z-\varepsilon \sqrt z + \int_0^{\beta_z} f(\rho^z_s) \, ds- \frac{c\sqrt z}{4\mu} f\Big(\frac{c\sqrt z}{4\mu}\Big) \Big) \ . \end{align*} Returning to the arguments of part (iii) we choose $b$, $c$ and then $z$ so large that we arrive at \[ \mathbf P( T^z_x \ge \beta_z + a \sqrt z) \ge - 2\varepsilon + \mathbf P\Big(S_{\beta_z + a\sqrt z} \le \mu \beta_z- 2\varepsilon \sqrt z \Big) \] and further at \[ \liminf_{z \to \infty} \mathbf P( T^z_x \ge \beta_z + a \sqrt z)\ge -3\varepsilon + \mathbf P (L \le (-\mu a-3\varepsilon)\mu^{1/2}\sigma^{-1} )\ . \] The limit $\varepsilon \to 0$ leads to the desired lower estimate. \end{proof}
\section{Approximating the block counting process} In this section we derive a strengthening of a result in Kersting, Schweinsberg and Wakolbinger ~\cite{Ke} on the approximation to the logarithm of the block counting processes in the dust case. To this end, let us quickly recall the Poisson point process construction of the $\Lambda$-coalescent given in \cite{Ke}, which is a slight variation of the construction provided by Pitman in \cite{Pi}.
This construction requires $\Lambda(\{0\}) = 0$, which is fulfilled for coalescents with dust. Consider a Poisson point process $\Psi$ on $(0, \infty) \times (0, 1] \times [0, 1]^n$ with intensity $$dt \times p^{-2} \Lambda(dp) \times du_1 \times \dots \times du_n\ ,$$ and let $\Pi_n(0) = \{\{1\}, \dots, \{n\}\}$ be the partition of the integers $1, \dots, n$ into singletons. Suppose $(t, p, u_1, \dots, u_n)$ is a point of $\Psi$, and $\Pi_n(t-)$ consists of the blocks $B_1, \dots, B_b$, ranked in order by their smallest element. Then $\Pi_n(t)$ is obtained from $\Pi_n(t-)$ by merging together all of the blocks $B_i$ for which $u_i \leq p$ into a single block. These are the only times that mergers occur. This construction is well-defined because almost surely for any fixed $t' < \infty$, there are only finitely many points $(t, p, u_1, \dots, u_n)$ of $\Psi$ for which $t \leq t'$ and at least two of $u_1, \dots, u_n$ are less than or equal to $p$. The resulting process $\Pi_n = (\Pi_n(t), t \geq 0)$ is the $\Lambda$-coalescent. When $(t, p, u_1, \dots, u_n)$ is a point of $\Psi$, we say that a $p$-merger occurs at time $t$.
Condition \eqref{dustcond} allows us to approximate the number of blocks in the $\Lambda$-coalescent by a subordinator. Let $\phi: (0, \infty) \times (0, 1] \times [0, 1]^n \rightarrow (0, \infty) \times (0, \infty]$ be the function defined by $$\phi(t, p, u_1, \dots, u_n) = (t, -\log(1-p)).$$ Now $\phi(\Psi)$ is a Poisson point process, and we can define a pure jump subordinator $(S(t), t \geq 0)$ having the property that $S(0) = 0$ and, if $(t, x)$ is a point of $\phi(\Psi)$, then $S(t) = S(t-) + x$. With $\lambda$ the L\'evy measure of $S$, the formulas \eqref{defmu} and \eqref{sigma} now read \[ \mu=\int_{[0,1]} \log \frac 1{1-p} \, \frac {\Lambda(dp)}{p^2} \ \text{ and }\ \sigma^2 = \int_{[0,1]} \Big(\log \frac 1{1-p}\Big)^2\, \frac{\Lambda(dp)}{p^2} \ .\]
This subordinator first appeared in the work of Pitman \cite{Pi} and was used to approximate the block-counting process by Gnedin et al. \cite{Gne} and M\"ohle \cite{Moe}; the benefits of a refined approximation by a {\em drifted} subordinator were discovered in \cite{Ke}. We recall that the drift appears because a merging of $b$ out of $N_n(t)$ lines results in a decrease by $b-1$ and not by $b$ lines, see equation (23) in \cite{Ke} for an explanation of the form of the drift.
The next result provides a refinement of Theorem 10 in \cite{Ke}.
\begin{Prop}\label{Prop5} Let \[ \int_{[0,1]} \frac{\Lambda(dp)}p < \infty \ , \] let $f$ be as in \eqref{ourf}, and let $Y_n$ be the solution of \eqref{SDE} with $z:= \log n$. Then for any $\varepsilon >0$ there is an $\ell <\infty$ such that
\[ \mathbf P\big( \sup_{t<\tau_n}| \log N_n(t)-Y_n(t)| \le \ell \, , \, Y_n(\tau_n) < \ell \big) \ge 1-\varepsilon \ . \] \end{Prop}
\begin{proof} From \cite{Ke} we know that for given $\varepsilon >0$ there is an $r <\infty$ such that \begin{align*}
\mathbf P\big( \sup_{t<\tau_n}| \log N_n(t)-Y_n(t)| \le r \big) \ge 1-\varepsilon/2 \ . \end{align*} Now we consider the size $\Delta_n$ of the last jump. Letting $(u_i,p_i)$, $i \ge 1$, be the points of the underlying Poisson point process with intensity measure $dt\, \Lambda(dp)/p^2$, the associated subordinator $S$ has jumps of size $v_i=-\log(1-p_i)$ at times $t_i$. Thus for any $c>0$ we have \begin{align*} \{ \Delta_n \le \log N_n(\tau_n-)-c\} &= \{ \tau_n=t_i \text{ and } -\log(1-p_i) \le \log N_n(t_i-)-c \text{ for some } i\ge 1\}\\ &= \Big\{ \tau_n=t_i \text{ and } p_i \le 1- \frac{e^c}{N_n(t_i-)} \text{ for some } i\ge 1\Big\} \end{align*} Given $N_n(t-)$ this event appears at time $t$ with rate \[ \nu_{n,t}= \int_{[0,1-e^c/N_n(t-)]} p^{N_n(t-)} \frac{\Lambda(dp)}{p^2} \ .\] Using the inequalities $p^b=(1-(1-p))^b\le e^{-(1-p)b} \le 1/((1-p)b)$ we get \[ \nu_{n,t} \le \int_{[0,1-e^c/N_n(t-)]} e^{-(1-p)(N_n(t-)-2)} \, \Lambda(dp)\le \int_{[0,1-e^c/N_n(t-)]}\frac {e^2}{(1-p)N_n(t-)} \, \Lambda(dp) \ . \] It follows \begin{align*} \mathbf E\Big[ \int_0^\infty \nu_{n,t} \, dt \Big] \le \mathbf E \Big[ \int_{[0,1]} \int_0^\infty \frac {e^2}{(1-p)N_n(t-)} I_{\{N_n(t-) \ge \lceil e^c/(1-p)\rceil \}} \, dt \, \Lambda(dp) \Big] \end{align*}
Lemma 14 of \cite{Ke} yields the estimate \[\mathbf E \Big[ \int_0^\infty \frac {1}{N_n(t-)} I_{\{N_n(t-) \ge \lceil e^c/(1-p)\rceil \}} \, dt\Big] \le c_1 \lceil e^c/(1-p)\rceil ^{-1}\le c_1 \frac{1-p}{e^c}\] with some $c_1>0$, hence \[ \mathbf E\Big[ \int_0^\infty \nu_{n,t} \, dt \Big] \le c_1 e^{2-c} \Lambda ([0,1])\ . \] Therefore for $c$ sufficiently large \[ \mathbf E\Big[ \int_0^\infty \nu_{n,t} \, dt \Big] \le \varepsilon/2 \ , \] which implies \[ \mathbf P\big(\Delta_n \le \log N_n(\tau_n-)-c\big) = 1-\exp \Big(- \mathbf E\Big[ \int_0^\infty \nu_{n,t} \, dt \Big]\Big) \le \varepsilon/2 \ .\]
Altogether we obtain
\[ \mathbf P\big(\sup_{t<\tau_n}| \log N_n(t)-Y_n(t)| \le r \, , \, \Delta_n > \log N_n(\tau_n-)-c\big) \ge 1-\varepsilon \ . \] The event in the previous formula implies \[ Y_n(\tau_n) = Y_n(\tau_n-)-\Delta_n < \log N_n(\tau_n-)+r -(\log N_n(\tau_n-) -c) = r+c \ ,\] and the claim of the theorem follows with $\ell=r+c$. \end{proof}
\section{Proof of the main results}
\begin{proof}[Proof of Theorem \ref{LLN}] Let us first assume that $\mu < \infty$. Then we have a coalescent with dust, and we may apply Proposition \ref{Prop5}. Fix $\eta >0$. Note that on the event that $Y_n(\tau_n)<\ell$ the event $\tau_n < (1- \eta)\log n/\mu$ implies the inequality $T_\ell^{\log n} < (1-\eta) \log n/\mu$. Thus in view of Proposition~\ref{Prop5} there exists for any $\varepsilon >0$ an $\ell$ such that \begin{align*} \mathbf P( \tau_n < (1- \eta)\log n/\mu) \le \mathbf P( T_\ell^{\log n} < (1-\eta) \log n/\mu)+ \varepsilon \ . \end{align*} Proposition \ref{Prop3} implies that the right-hand probability converges to 0 as $n \to \infty$. Letting $\varepsilon \to 0$ we obtain \[ \lim_{n \to \infty} \mathbf P( \tau_n < (1- \eta)\log n/\mu) =0\ . \]
Also on the event $\sup_{t<\tau_n}| \log N_n(t)-Y_n(t)| \le \ell$, the event $\tau_n > (1+\eta)\log n/\mu $ implies $Y_n(t) \ge -\ell$ for all $t\le (1+\eta)\log n/\mu$, and consequently \[ \mathbf P(\tau_n > (1+\eta)\log n/\mu) \le \mathbf P( T_{-\ell}^{\log n} > (1+\eta) \log n/\mu)+ \varepsilon \ . \] Again the right-hand probability converges to zero in view of Proposition \ref{Prop3}, and we obtain \[\lim_{n \to \infty} \mathbf P(\tau_n > (1+\eta)\log n/\mu) =0 \ . \] Altogether our claim follows in the case $\mu < \infty$.
Now assume $\mu = \infty$. If $\Lambda(\{0\})>0$, then the coalescent comes down from infinity and $\tau_n$ stays bounded in probability. The same is true if $\Lambda(\{1\})>0$, thus we may assume that $\Lambda(\{0,1\})=0$.
For given $\varepsilon >0$ define the measure $\Lambda^\varepsilon$ by $\Lambda^\varepsilon(B):= \Lambda(B\cap [\varepsilon,1-\varepsilon])$. Obviously \[ \mu^\varepsilon:=\int_0^1 \log \frac 1{1-p} \, \frac{\Lambda^\varepsilon(dp)}{p^2} < \infty\ . \] Thus for the absorption times $\tau_n^\varepsilon$ of the $\Lambda^\varepsilon$-coalescent we have \[ \frac {\tau^\varepsilon_n}{\log n} \to \frac 1{\mu^\varepsilon}\] in probability as $n \to \infty$. Now we may couple the $\Lambda^\varepsilon$-coalescent in an obvious manner to the $\Lambda$-coalescent in such a way that $N_n(t)\le N_n^\varepsilon(t)$ a.s. for all $t \ge 0$, in particular $\tau_n \le \tau_n^\varepsilon$. Hence it follows that \[ \mathbf P(\tau_n/\log n > 2 /\mu^\varepsilon) \to 0 \ . \] Because of $\Lambda(\{0,1\})=0$ we have $\mu^\varepsilon \to \mu=\infty $ with $\varepsilon \to 0$, consequently \[ \mathbf P(\tau_n/\log n > \eta) \to 0\] for all $\eta >0$. This is our claim. \end{proof}
\begin{proof}[Proof of Theorem \ref{CLT}] Because of the condition $\mu<\infty$ we again may apply Proposition \ref{Prop5}. We follow the same line as in the previous proof: For $\varepsilon >0$ there exists an $\ell$ such that for all $a\in \mathbb R$ \begin{align*} \mathbf P( \tau_n < b_n+ a \sqrt n) \le \mathbf P( T_\ell^{\log n} < b_n+ a \sqrt n)+ \varepsilon \end{align*} and \begin{align*} \mathbf P(\tau_n > b_n+a\sqrt n) \le \mathbf P( T_{-\ell}^{\log n} > b_n+a\sqrt n)+ \varepsilon \end{align*} Now apply Proposition \ref{Prop4} and let $\varepsilon \to 0$. \end{proof}
\begin{proof}[Proof of Proposition \ref{Prop2}] (i) Let us first assume \eqref{condi}. Because of $1-(1-p)^{1/r} \le \min (p/r,1)$ for $0<r<1$ we have for $\alpha > 0$ \begin{align} f\big(\log \tfrac 1r\big) \le \int_0^{r^\alpha} \frac{\Lambda (dp)}p + r\int_{r^\alpha}^1 \frac{ \Lambda(dp)}{p^2} \le \int_0^{r^\alpha} \frac{\Lambda (dp)}p + r^{1-\alpha }\int_0^1 \frac{\Lambda(dp)}{p} \ . \label{estimate1} \end{align} Also, because of $1-(1-p)^{1/r} \ge 1- e^{-p/r} \ge e^{-p/r}p/r$, it follows for $\beta >0$ that \begin{align} f\big(\log \tfrac 1r\big) \ge e^{-r^{\beta -1}}\int_0^{r^\beta} \frac{\Lambda (dp)}p\ . \label{estimate2} \end{align} Together with \eqref{condi} these two estimates yield for $\alpha < 1 < \beta$ \[ c \beta^{-1/2} \le \liminf_{r \to 0} f\big(\log \tfrac 1r\big) \sqrt{\log \tfrac 1r} \le \limsup_{r \to 0} f\big(\log \tfrac 1r\big) \sqrt{\log \tfrac 1r} \le c \alpha^{-1/2} \ .\] Letting $\alpha, \beta \to 1$ we arrive at $f(y)= (c+o(1))/\sqrt y$ as $y \to \infty$ and consequently \[ \int_0^{\log n} f(y) \, dy = (c+o(1))2\sqrt {\log n} \] as $n \to \infty$.
Now, because of \begin{align*}
\frac{1}{\mu-f(y)}= \frac 1\mu + \frac {f(y)}{\mu(\mu-f(y))} \end{align*} and $f(y)=o(1)$ as $y\to \infty$, we have \begin{align} \label{formula} \int_\kappa^z \frac{dy}{\mu-f(y)} = \frac z\mu + \frac{1+o(1)}{\mu^2} \int_0^z f(y)\, dy +O(1) \end{align} as $z\to \infty$, and consequently, as claimed, \begin{align*} b_n = \frac{\log n}\mu + \frac{2c+o(1)}{\mu^2} \, \sqrt{\log n} \ . \end{align*}
(ii) Now suppose that \eqref{condi2} is satisfied. Then in view of \eqref{formula} with $z=\log n$ it follows that \[ \int_0^{\log n} f(y)\, dy = (2c+o(1)) \sqrt {\log n} \] as $n \to \infty$, or equivalently \[ \int_0^{z} f(y)\, dy = (2c+o(1)) \sqrt z \] for $z \to \infty$. This implies that $f(z)= (c+o(1))/ \sqrt z$ as $z\to \infty$. For $c=0$ this claim follows because $f$ is decreasing, which entails \[ zf(z) \le \int_0^z f(y)\, dy = o(\sqrt z) \ . \] For $c>0$ we use the estimate \[ \frac 1{\eta \sqrt z}\int_z^{(1+\eta) z} f(y)\, dy \le \sqrt z f(z) \le \frac 1{\eta \sqrt z} \int_{(1-\eta)z}^z f(y) \, dy \]
with $\eta >0$. Taking the limit $z\to \infty$ and then $\eta \to 0$ yields $f(z)= (c+o(1))/ \sqrt z$. Now, similar as in part (i) we get from \eqref{estimate1} and \eqref{estimate2} \[c\sqrt \alpha \le \liminf_{r \to 0}\sqrt {\log \tfrac 1r}\int_{[0,r]} \frac{\Lambda(dp)}p\le \limsup_{r \to 0}\sqrt {\log \tfrac 1r}\int_{[0,r]} \frac{\Lambda(dp)}p \le c\sqrt \beta\ .\] With $\alpha, \beta \to 1$ we arrive at \eqref{condi}. \end{proof}
\paragraph{Acknowledgement.} It is our pleasure to dedicate this work to Peter Jagers.
\end{document} |
\begin{document}
\title{The valuation difference rank of a quasi-ordered difference field} \subjclass{ Primary 03C60, 06A05, 12J15: Secondary 12L12, 26A12} \author{Salma Kuhlmann} \address{Universit\"at Konstanz\\ FB Mathematik und Statistik\\ 78457 Konstanz, Germany} \email{salma.kuhlmann@uni-konstanz.de} \author{Micka\"{e}l Matusinski} \address{IMB, Universit\'e Bordeaux 1, 33405 Talence, France} \email{mmatusin@math.u-bordeaux1.fr} \author{Fran\c coise Point} \address{Institut de math{\'e}matique, Le Pentagone\\ Universit{\'e} de Mons\\ B-7000 Mons, Belgium} \email{Francoise.Point@umons.ac.be} \thanks{Supported by a
Research in Paris grant from Institut Henri Poincar\'e, Konstanz University, Bordeaux 1 University and the Fonds de la Recherche Scientifique FNRS-FRS} \maketitle
\begin{abstract} There are several equivalent characterizations of the valuation rank of an ordered or valued field. In this paper, we extend the theory to the case of an ordered or valued {\it difference} field (that is, ordered or valued field endowed with a compatible field automorphism). We introduce the notion of {\it difference rank}. To treat simultaneously the cases of ordered and valued fields, we consider quasi-ordered fields. We characterize the difference rank as the quotient modulo the equivalence relation naturally induced by the automorphism (which encodes its growth rate). In analogy to the theory of convex valuations, we prove that any linearly ordered set can be realized as the difference rank of a maximally valued quasi-ordered difference field. As an application, we show that for every regular uncountable cardinal $\kappa$ such that $\kappa= \kappa^{< \kappa}$, there are $2^{\kappa}$ pairwise non-isomorphic quasi-ordered difference fields of cardinality $\kappa$, but all isomorphic as quasi-ordered fields. \end{abstract} \section{Introduction}
The theory of convex valuations and coarsenings of valuations is a special chapter in classical valuation theory. It is a basic tool in algebraic and real algebraic geometry. Surveys can be found in \salmacite{[LAM2]}, \salmacite{[LAN]} and \salmacite{[PC]}. This special chapter is in turn closely related to ordered algebraic structures, see \cite{[Fu]}. In particular, an important isomorphism invariant of an ordered or valued field is its rank as a valued field, which has several equivalent characterizations: via the ideals of the valuation ring, the value group, or the residue field, see \cite{[ZS]}. \par
\noindent This can be extended to ordered and valued fields with extra structure, giving a characterization completely analogous to the above, but taking into account the corresponding induced structure on the ideals, value group, or residue field. For example, in \cite[Chapter 3]{[K]} the notion of the exponential rank of an ordered exponential field is introduced and analysed in light of the above classical tools. The exponential rank measures the growth rate of the given exponential function, and is thus closely related to asymptotic analysis in the sense of G. H. Hardy \cite{[H]}. \par
\noindent In this paper, we push this analogy to the case of an ordered or valued difference field. We work with quasi-ordered fields, see \cite{[F]}. In Section \ref{sectprelconv} we review classical notions and results on ordered or valued fields. We thereby present a uniform approach via quasi-orders, treating simultaneously the cases of ordered and valued fields. Theorem \ref{wconv} gives a characterization of the rank of a quasi-ordered field in terms of coarsenings of its natural valuation. Descending down to the value group of the quasi-ordered field, and yet further down to the value set $\Gamma$ of the value group, the rank and principal rank are finally characterized by the chain $\Gamma$, see Theorems \ref{theorem1OSMT} and \ref{theorem2OSMT}. In Section \ref{partIIOSMT} we start by a key remark regarding equivalence relations defined by monotone maps on chains. We describe in Theorem \ref{theorem3OSMT} the rank of a quasi-ordered field via the equivalence relations induced by addition and multiplication on the field. This approach allows us to develop in Section \ref{diffanal} the notion of difference compatible valuations and introduce the difference rank. We characterize in Theorem \ref{wconvsigma} the difference rank, in analogy to Theorem \ref{wconv}. \cite[Lemma 1]{[S]} is a special case of our Corollary \ref{weakiso} on weak isometries. Corollary \ref{intersection} describes the set of fixed points of an automorphism $\sigma$ in terms of its difference rank, whereas Corollary \ref{omega} examines the special case of $\omega$-increasing or $\omega$-contracting automorphisms. In the last Section \ref{principalsigmarank} we describe the principal difference rank, see Theorem \ref{theorem3OSMTsigma} and its Corollaries \ref{theorem1OSMTsigma}, \ref{theorem2OSMTsigma} and \ref{arbitrary}. In Theorem \ref{last} we construct large families of quasi-ordered difference fields with distinct difference ranks. \par
\noindent Some closing comments are in place. The theory of well-quasi orders \cite{[Kru]} is currently a highly developed part of combinatorics with surprising applications in logic, mathematics and computer science. Quasi-ordered algebraic structures are interesting in their own right, and we will continue our investigations of these fascinating objects. Quasi-orders \cite{[Bir]} appear in the literature also as {\it preorders}, see e.g. \cite[p.1]{[Fu]}. However we will not use this terminology, in order to avoid confusion with the notion of preorders appearing in real algebraic geometry (e.g. in \cite{[Kr]}). The theory of quasi-ordered abelian groups is closely related to that of C-groups \cite{[H-M]} and has already found interesting applications in \cite{[L]} to the study of the asymptotic couple associated to a valued differential field. Throughout the paper, Hahn groups and Hahn fields play a fundamental role. The group of automorphisms of Hahn structures have been extensively studied, see \cite{[B]}, \cite{[DG]}, \cite{[Ho]} and \cite{[S]}. In future work, we will analyse the behaviour of the difference rank as function defined on these automorphism groups.
\section{ The rank of a quasi-ordered field} \label{sectprelconv}
\par
\noindent A {\bf quasi-order (q.o.)} on a set $S$ is a binary relation $\preceq$ which is reflexive and transitive. Throughout this paper, we will deal only with {\bf total quasi-order}, i.e. either $a\preceq b$ or $b\preceq a$, for any $a,\;b\in S$. We will omit henceforth `total'. Note that an order is a q.o which is in addition anti-symmetric. In the latter case, we say that $S$ is an ordered set or a {\bf chain}. The {\bf induced equivalence relation} is defined by $a\asymp b$ if and only if ($a\preceq b$ and $b\preceq a$). We shall write $a \prec b $ if $a\preceq b$ but $b\asymp a$ fails. Note that $\preceq$ induces canonically a total order on $S/\asymp$. Conversely if $\asymp$ is an equivalence relation on a set $S$ such that $S/\asymp$ is a total order, then $\asymp$ induces canonically a q.o. on $S$. A subset $E$ of $S$ is {\bf $\preceq$-convex} if for all $a, b, c$ in $S$, if $a\preceq c\preceq b$ and $a,\;b\in E$, then $c\in E$. We shall write convex instead of $\preceq$-convex if the context is clear. \par
\noindent A {\bf quasi-ordered field} $(K,\preceq)$ is a field $K$ endowed with a quasi-order $\preceq$ which satisfies the following compatibility conditions, for any $a, b, c \in K$. \begin{description} \item[qo1] If $a\asymp 0$, then $a=0$. \item[qo2] If $0\preceq c$ and $a\preceq b$, then $ac\preceq bc$. \item[qo3] If $a\preceq b$ and $b\not\asymp c$ , then $a+c\preceq b+c$. \end{description} From {\bf qo2} one deduces that if $a\preceq b$ and $0\preceq c\preceq d$, then $ac\preceq bc\preceq bd$, so $ac\preceq bd$.\par
\noindent
Given a valuation $w$ on $K$ we denote the {\bf valuation ring} by $K_w\,$, its {\bf group of units} $K_w^{\times}$ by $\mathcal{U}_w$, its {\bf valuation ideal} (i.e. its unique maximal ideal) by $I_w\,$, its {\bf value group} by $w(K^{\times})$ and {\bf residue field} $K_w/I_w$ by $Kw \>.$ \par
\noindent An ordered field $(K, \leq)$ is a q.o. field. The valuation on a valued field $(K,w)$ induces a quasi-order: $a\preceq_{w} b$ if and only if $w(b)\leq w(a)$, i.e. if and only if $ab^{-1}\in K_w$.
S. Fakhruddin \salmacite{[F]} showed that if $\preceq$ is a q.o. on a field $K$, then $\preceq$ is either an order or there is a (unique up to equivalence of valuations) valuation $v$ on $K$ such that $\preceq\> = \preceq_{v}$. The dichotomy is achieved by considering the equivalence class $E_1$ of $1$ with respect to $\asymp \>$. In the order case, $E_1 = \{1\}$ and $\asymp \>$ is just equality. The quasi-order is said to be a {\bf proper quasi-order (p.q.o.)} if $E_1 \not= \{1\}$. In this case, $E_1 \not= \{1\}$ is a non-trivial subgroup of $K^{\times}$ and $K^{\times}/E_1$ is an ordered abelian group. Then $\mathcal{U}_v$ is just $E_1$ and $v(K^{\times})\>$ is $K^{\times}/E_1$. In the p.q.o case $a\succeq 0$ for all $a\in K\>$.
\par
\noindent Given two valuations $v$ and $w$ on $K$, recall that $w$ is said to be a \slind{coarsening} of $v$ ($w$ is coarser than $v$) or that $v$ a \slind{refinement} of $w$ ($v$ is finer than $w$) if $K_v \subseteq K_w$. In case the inclusion of the valuation rings is strict, we add the predicate strict in the terminology coarser and finer. Note that $w$ is coarser than $v$ if and only if $a\preceq_{v} b$ implies $a\preceq_{w} b\>$. If $\sim _1$ and $\sim _2$ are two equivalence relations defined on the same set, then $\sim _1$ is said to be {\bf coarser} than $\sim _2$ (or $\sim_{2}$ {\bf finer} than $\sim_{1}$) if $\sim _2$-equivalence implies $\sim _1$-equivalence.
\par
\noindent Let us now fix a q.o. $\preceq$ on $K$ . A valuation $w$ on $K$ is called \slind{convex} with respect to $\preceq$ if its valuation ring $K_w$ is convex. It is called \slind{compatible} with $\preceq$ (or $\preceq$ is compatible with $w$ or $w$ and $\preceq$ are compatible) if
for all $a,b\in K\>:$ $$ 0\preceq b \preceq a \;\;\;\Longrightarrow\;\;\;w(a) \leq w(b)\>.$$ Equivalently, $w$ is compatible with $\preceq$
if and only if for all $a,b\in K\>:$ $$0\preceq b \preceq a \;\;\;\Longrightarrow\;\;\; b \preceq_w a\>.$$ \begin{remark} \label{OSRGA1} \par
\noindent
(i) If $\preceq$ is an order, then this is the usual notion of compatibility for orders and valuations, see e.g. \salmacite{[LAM1]}, \salmacite{[LAM2]}, \salmacite{[PC]}, or \salmacite{[PR1]}. \par
\noindent
(ii) If $\preceq = \preceq _v$ is a p.q.o. then $w$ compatible with $\preceq_v$ just means that for all $a,b\in K\>, v(a) \leq v(b) \;\;\;\Longrightarrow\;\;\;w(a) \leq w(b)\>.$
This in turn just means that $K_v \subseteq K_w$ or $w$ is a coarsening of $v$, equivalently $\asymp_{w}$ is coarser than $\asymp_{v}$. \end{remark} \par
\noindent The following gives the characterization of valuations compatible with a quasi-order. Theorem \ref{wconv} is in complete analogy to the characterization of valuations compatible with an order. So for $\preceq$ an order, we omit the proof and refer the reader to \cite [Proposition~5.1]{ [LAM1]} , or \cite[Theorem~2.3 and Proposition~2.9]{ [LAM2]} , or \cite[Lemma~3.2.1]{ [PC]}, or \cite[ Lemma~7.2]{ [PR1]} or \cite[ Proposition 2.2.4]{ [EP]}. \begin{theorem} \label{wconv} Let $(K, \preceq)$ be a q.o. field and $w$ a valuation on $K$. The following assertions are equivalent:\par
\noindent 1)\ \ $w$ is compatible with $\preceq$,\par
\noindent 2)\ \ $w$ is convex, \par
\noindent 3)\ \ $I_w$ is convex,\par
\noindent 4)\ \ $I_w\prec 1\,$, \par
\noindent 5)\ \ the quasi-order $\preceq$ induces canonically via the residue map $a \mapsto aw$ a quasi-order on the residue field $Kw\>.$ \end{theorem} \begin{proof} Assume $\preceq = \preceq _v$ is a p.q.o. Compatible valuations are clearly convex, this follows from the definitions. Conversely if $w$ is convex and $0 =v(1) \leq v(a)\>,$ i.e. $a \preceq 1\>,$ then $a\in K_w$ by convexity. So $w$ is a coarsening of $v$. This establishes the equivalence of 1) and 2).\par
\noindent If $w$ is convex, $a \preceq b$ with $b \in I_w\>,$ then $0 < w(b) \leq w(a)$ by compatibility, so $a\in I_w$. Conversely assume $I_w$ convex, and let $a \preceq b$ with $b \in K_w\setminus I_w$. If $a\notin K_w$ then $a^{-1} \in I_w$. Now $b^{-1} \preceq a^{-1}$, so $b^{-1} \in I_w\>,$ a contradiction. This establishes the equivalence of 2) and 3). \par
\noindent If $I_w$ is convex, then $w$ is a coarsening of $v$, so $I_w \subseteq I_v\prec 1$. Conversely, assume $I_w \prec 1$ and let $a \preceq b$ with $b \in K_w\>$. If $a\notin K_w\>,$ then $a^{-1} \in I_w\>.$ So $a^{-1}b \in I_w$ whence $a^{-1}b\prec 1$. Multiplying by $a$ gives $b \prec a$, a contradiction. This establishes the equivalence of 3) and 4). \par
\noindent Now let $w$ be a coarsening of $v\>$. Then $v$ induces canonically a valuation $v/w$ on the residue field $Kw$, defined by $v/w(aw) := \infty$ if $aw = 0$ and $v/w(aw):=v(a)$ otherwise (\salmacite{ [EP]} p. 44) . The p.q.o. $\preceq_{v/w}$ is precisely the induced well defined quasi-order in 5), i.e. $aw \preceq _{v/w} bw$ if and only if $a \preceq _v b$ holds. Conversely, let $\preceq _{v/w}$
be a p.q.o. on $Kw$ induced by the residue map. This means that $aw \preceq _{v/w} bw$ if and only if $a \preceq _v b$ holds. Then $w$ is a coarsening of $v$ (see \cite[ p. 45]{ [EP]}). This establishes the equivalence of 1) and 5). \end{proof} \begin{remark} \label{OSRGA2}
If $\preceq$ is an order then the induced quasi-order in 5) is also an order, if $\preceq$ is a p.q.o then the induced quasi-order in 5) is also a p.q.o. \end{remark} Let $(K,\preceq)$ be a q.o. field. We define its {\bf natural valuation}, denoted by $v$, to be the finest $\preceq$- convex valuation of $K$. If $(K, \leq)$ is ordered, then the natural valuation is the valuation $v$ whose valuation ring $K_v$ is the convex hull of $\lv Q$ in $K$. In this case, the natural valuation on $K$ satisfies $v(x+y) = \min\{v(x), v(y)\}$ if sign($x$) = sign($y$) and for all $a,b\in K\,: a\geq b>0\;\;\;\Longrightarrow\;\;\;v(a)\leq v(b)\>.$ It is characterized by the fact that the induced order on its residue field $Kv$ is archimedean, i.e. the only equivalence classes for the archimean equivalence relation (see definition below following Lemma \ref{wconvvg}) are those of 0 and 1. If $w$ is a coarsening of a convex valuation,
then $w$ also is convex. Conversely, a convex subring containing $1$ is a valuation ring, see \cite[Section 2.2.2]{[EP]}. The set ${\mathcal R}$ of all valuation rings $K_w$ of convex valuations $w\ne v$ (i. e. all strict corsenings of $v$) is totally ordered by inclusion. Its order type is called the {\bf rank of the ordered field} $K$. For convenience, we will identify it with ${\mathcal R}$. For example, the rank of an archimedean ordered field is empty since its natural valuation is trivial (i.e. its valuation ring is the field itself). The rank of the rational function field $K=\lv R(t)$ with any order is a singleton: ${\mathcal R}= \{K\}$. Theorem \ref{wconv} is a characterization of the elements of the rank of the ordered field $(K, \leq)$. Note that the rank of $(K, \leq)$ is invariant under isomorphisms of ordered fields. \par
\noindent If $(K,\preceq)$ is p.q.o. then the unique valuation $v$ such that $\preceq= \preceq _v$ is the natural valuation. A compatible valuation $w$ is a coarsening of $v$. We define the \slind{rank of the valued field} $(K, v)$ to be the (order type of the) totally ordered set ${\mathcal R}$ of all strict corsenings of $v$. Thus, Theorem \ref{wconv} is a characterization of the elements of the rank of $(K,v)$. Note that the rank of $(K, v)$ is invariant under isomorphisms of valued fields. As we recalled in the proof of Theorem \ref{wconv}, the natural valuation $v$ induces canonically a valuation $v/w$ on the residue field $Kw$ and $v$ is the \slind{compositum} of $w$ and $v/w$ (see \cite[ pp. 44-45]{ [EP]}) . The p.q.o. $\preceq_{v/w}$ is precisely the induced quasi-order in Theorem \ref{wconv} 5). If $w = v\>,$ then $v/w$ is trivial. Thus $v$ is characterized by the fact that the induced p.q.o on its residue field $Kv$ is \slind{trivial}, i.e. the only equivalence classes of $\asymp$ are those of $0$ and $1$. \begin{remark} \label{prime} The maximal ideals $I_w$ appearing in Theorem \ref{wconv} 4) are prime ideals of the valuation ring $K_v$. The strict coarsenings $K_w$ of $K_v$ are the localizations of $K_v$ at the prime ideals $\{0\} \subseteq I \subset I_v$, \cite[Lemma 2.3.1 p. 43]{[EP]}, \cite[Theorem 15, p. 40]{[ZS]}. Thus the rank is also isomorphic to the totally ordered (by reverse inclusion) set of prime ideals of $K_v$ which are strictly contained in the maximal ideal $I_v$. \end{remark} We now want to characterize the rank by going down to the value group. Let $v$ be the natural valuation on the q.o. field $(K,\preceq)$. We set $G = v(K^{\times})$.
Recall that the set of all convex subgroups $G_w \ne \{0\}$ of the value group $G$ is totally ordered by inclusion. Its order type is called the {\bf rank} of~$G$, it is an isomorphism invariant, see \cite {[Fu]} or \cite{ [PC]}. To every convex valuation ring $K_w$, we associate a convex subgroup $G_w\>:=\>\{v(a)\mid a\in K\,\wedge\ w(a)=0\}\>=\> v({\mathcal U}_w)\;.$ We call $G_w$ the {\bf convex subgroup associated to $w$}. Note that $G_v=\{0\}$. Conversely, given a convex subgroup $G_w$ of $v(K^{\times})$, we define $w: K \rightarrow v(K^{\times})/G_w$ by $w(a)=v(a)+G_w$. Then $w$ is a convex valuation with $v({\mathcal U}_w)=G_w$ (and $v$ is strictly finer than $w$ if and only if $G_w \not= \{0\}$). We call $w$ the {\bf convex valuation associated to $G_w$}.
We summarize the above discussion in the following lemma, for more details see \cite{[EP]}, or \cite {[Fu]} or \cite{ [PC]}. \begin{lemma}\label{wconvvg} The correspondence $K_w \mapsto G_w$ is an order preserving bijection, thus ${\mathcal R}$ is (isomorphic to) the rank of $G$. \end{lemma} We now want to characterize the rank by going further down to the value set of the value group. Recall that on the negative cone $G^{< 0}$ of an ordered abelian group $G$, the {\bf archimedean equivalence} relation $\sim$ is defined by: $a\sim b$ if and only if there is $n\in \lv N$ such that $a\geq nb$ and $b\geq na$. Let $v_G$ be the map $a\mapsto [a]_{\sim}\,$, where $[a]_{\sim}$ denotes the equivalence class of $a$. The order on $\Gamma:=G^{<0}/\sim$ is the one induced by the order of $G^{<0}$. We call $v_G(G^{<0}):=\Gamma$ the {\bf value set of $G$}. By convention we also write $v_G(G):=\Gamma\cup\{\infty\}$ extending the archimedean equivalence relation to the positive cone of $G$ by setting $v_G(g) := v_G(-g)$ and $v_{G}(0)=\infty>\Gamma$. The map $v_G$ on $G$ satisfies the ultrametric triangle inequality, and in particular we have: $v_G(x+y)
= \min\{v_G(x), v_G(y)\}$ if sign($x$) = sign($y$). We call $v_G$ the {\bf natural valuation on $G$}.
We now recall the relation between the rank of $G$ and the value set $\Gamma$ of $G$. To $G_w \ne \{0\}$ a convex subgroup, we associate $\Gamma_w := v_G(G_w ^{<0})$ a non-empty final segment of $\Gamma$. Conversely, if $\Gamma _w$ is a non-empty final segment of $\Gamma$, then $G_w = \{g\mid g\in G, v_G(g) \in \Gamma _w \} \cup \{0\}$ is a convex subgroup, with $\Gamma _w = v_G(G_w)$. Let us denote by $\Gamma ^{\rm fs}$ the set of non-empty final segments of $\Gamma $, totally ordered by inclusion. We summarize the above discussion in the following lemma, for more details see \cite{[EP]}, or \cite {[Fu]} or \cite{ [PC]}. \begin{lemma} \label{wconvvs} The correspondence $G_w \mapsto \Gamma _w$ is an order preserving bijection, thus the rank of $G$ is (isomorphic to) $\Gamma ^{\rm fs}$. \end{lemma} Combining Lemmas \ref{wconvvg} and \ref{wconvvs} we obtain the following result. Note that Theorem \ref{theorem1OSMT} will also follow, by a different argument, from Theorem \ref {theorem3OSMT} in the next section. \begin{theorem} \label{theorem1OSMT} The correspondence $K_w \mapsto \Gamma _w$ is an order preserving bijection, thus ${\mathcal R}$ is (isomorphic to) $\Gamma ^{\rm fs}$. \end{theorem} \par
\noindent A final segment which has a least element is a \slind{principal final segment}. It is of the form $\{\gamma'\mid\gamma'\in\Gamma,\gamma'\geq\gamma\}$, for some $\gamma \in \Gamma$. Let $\Gamma ^*$ denote the set $\Gamma$ with its reversed ordering. The proof of the following Lemma is now routine. \begin{lemma} \label{vsr} The map from $\Gamma$ to $\Gamma ^{\rm fs}$ defined by $\gamma\mapsto\{\gamma'\mid\gamma'\in\Gamma,\gamma'\geq\gamma\}$ is an order reversing embedding. Its image is the set of principal final segments. Thus $\Gamma ^*$ is (isomorphic to) the totally ordered set of principal final segments. \end{lemma} \par
\noindent For the notions and results in this last paragraph of the section, we refer the reader to \cite{[Fu]} or \cite{[PC]} for more details. Recall that a convex subgroup $G_w$ of $G$ is called {\bf principal generated by $g$}, $g\in G$, if $G_w$ is the minimal convex subgroup containing $g$. The {\bf principal rank} of $G$ is the subset of the rank of $G$ consisting of all principal $G_w \ne \{0\}$ . \begin{lemma} \label{cpcs} Let $G_w \ne \{0\}$ be a convex subgroup. Then $G_w$ is principal if and only if $v_G(G_w) = \Gamma _w$ is a principal final segment. \end{lemma} \begin{lemma} \label{star} The map $G_w \mapsto \min \Gamma_w$ is an order reversing bijection from the principal rank of $G$ onto $\Gamma$ . Thus the principal rank of $G$ is (isomorphic to) $\Gamma ^*$. \end{lemma} \par
\noindent We set: $\mbox{\bf P}_K:=K^{\succeq 0}\setminus K_v$, where $K^{\succeq 0}: = \{a\in K \>; a\succeq 0\}$. A $K_w \in \mathcal{R}$ is \slind{principal generated by $a$} for $a \in \mbox{\bf P} _K$ if $K_w$ is the smallest (convex) subring containing $a$. We observe: \par
\noindent \begin{lemma}\label{principal} Let $K_w\in\mathcal{R}$. Then, $K_{w}$ is principal generated by $a$ if and only if $K_{w}=\{b\in K:\;\exists n\in \lv N_{0} \> s.t.\;\;b\preceq_{v} a^n\}$. \end{lemma} \begin{proof} It is enough to verify that $\{b\in K:\;\exists n\in \lv N_{0}\;\;b\preceq_{v} a^n\}$ is a subring of $K$. Let $b_{1}\preceq_{v} a^{n_{1}}$ and $b_{2}\preceq_{v} a^{n_{2}}$. Then $b_{1}b_{2}\preceq_{v} a^{n_{1}+n_{2}}$ and $b_{1}+b_{2}\preceq_{v} a^{max\{n_{1},n_{2}\}}$. Clearly, this ring contains $K_v$ and $a$. \end{proof} \par
\noindent The {\bf principal rank} of $K$ is the subset \gloss{${\mathcal R}^{\rm pr}$} of ${\mathcal R}$ consisting of all principal $K_w\in {\mathcal R}$. Combining the last three lemmas we obtain: \begin{theorem} \label{theorem2OSMT} The correspondence $K_w \mapsto \Gamma _w$ is an order preserving bijection between ${\mathcal R}^{\rm pr}$ and the principal rank of $G$, thus ${\mathcal R}^{\rm pr}$ is (isomorphic to) $\Gamma ^*$. \end{theorem} Note that Theorem \ref{theorem2OSMT} will also follow, by a different argument, from Theorem \ref {theorem3OSMT} in the next section. \begin{remark}\label{completion} It is straightforward to verify that an order preserving isomorphism $\psi:\Gamma_{1}\rightarrow \Gamma_{2}$ induces an order preserving isomorphism $\psi^{\rm fs}:\Gamma_{1}^{\rm fs}\rightarrow\Gamma_{2}^{\rm fs}$ (\cite[p.19]{[R]}). Thus $\Gamma$ determines $\Gamma^{\rm fs}$ up to isomorphism. It follows from Theorems \ref{theorem1OSMT} and \ref{theorem2OSMT} that if two q.o. fields have isomorphic principal ranks, then they have isomorphic ranks. In the next section we shall hence focus our attention on the principal rank. \end{remark}
\section{The principal rank via equivalence relations} \label{partIIOSMT} \par
\noindent We begin by the following key observation: \par
\noindent
\begin{remark} \label{mm}
Let $\varphi$ be a map from a q.o. ordered set $(S, \preceq)$ into itself, and assume that $\varphi$ is q.o. preserving,\ i.\ e. \ $a\preceq a'$ implies $\varphi(a)\preceq \varphi(a')$, for all $a,\> a' \in S$. Assume that $\varphi$ has an orientation or is {\bf oriented}, \ i.\ e. \ $\varphi(a)\succeq a$ for all $a\in S\>$ ($\varphi$ is a \slind{right shift}) or $\varphi(a)\preceq a$ for all $a\in S\>$ ($\varphi$ is a \slind{left shift}). We set $\varphi^0(a):=a$ and $\varphi^{n+1}(a):=\varphi(\varphi^n(a))$ for $n\in \lv N _{0}:=\lv N \cup \{0\}$.
It is then straightforward that the following defines an equivalence relation on $S$: \par
\noindent
(i) If $\varphi$ is a right shift, set $a\sim_{\varphi} a'$ if and only if there is some $n\in\lv N_{0}$ such that $\varphi^n (a)\succeq a' \> \mbox{ and } \> \varphi^n (a')\succeq a$ (equivalently for some $n,\;m\in \lv N_{0}$, $\varphi^n (a)\succeq a' \> \mbox{ and } \> \varphi^m(a')\succeq a\;$),
\par
\noindent (ii) If $\varphi$ is a left shift, set $a\sim_{\varphi} a'$ if and only if there is some $n\in\lv N_{0}$ such that $\varphi^n (a)\preceq a' \> \mbox{ and } \> \varphi^n (a')\preceq a$ (equivalently for some $n,\;m\in \lv N_{0}$, $\varphi^n (a)\preceq a' \> \mbox{ and } \> \varphi^m(a')\preceq a\;$). \par
\noindent (iii) The equivalence classes $[a]_{\varphi}$ of $\sim_{\varphi}$ are $\preceq$-convex and closed under application of $\varphi$. By the $\preceq$-convexity, the quasi-order of $S$ induces an order on $S/{\sim}_{\varphi}$ such that $[a]_{\varphi}\prec [b]_{\varphi}$ if and only if $a'\prec b'$ for all $a'\in [a]_{\varphi}$ and $b'\in [b]_{\varphi}\,$. \par
\noindent
Note that if $\varphi$ is the identity map $\mathbb{I}$, then the equivalence relation $\sim_{\mathbb{I}}$ is just $\asymp$ associated to the q.o., and is the finest one such that $S/\sim_{\mathbb{I}}$ is an ordered set.
\end{remark}
\par
\noindent We exploit Remark \ref{mm} to give an interpretation of the rank and principal rank as quotients via an appropriate equivalence relation, thereby providing - as promised in the previous section- alternative proofs for Theorem \ref{theorem1OSMT} and Theorem \ref{theorem2OSMT}. It is precisely this approach that we will generalize to the difference rank in Section \ref{principalsigmarank}. Let $v$ be the natural valuation on the q.o. field $(K,\preceq)$. Recall that $\mbox{\bf P}_K$ denotes $K^{\succeq 0}\setminus K_v$. Consider the following commutative diagram: \par\noindent \parbox[c]{.4\textwidth}{ \begin{center} \setlength{\unitlength}{0.002\textwidth} \begin{picture}(200,250)(0,20) \put(50,250){\bbox{${\bf P} _K$}} \put(50,150){\bbox{$G ^{<0}$}} \put(50,50){\bbox{$v_G (G)$}} \put(150,250){\bbox{${\bf P} _K$}} \put(150,150){\bbox{$G ^{<0}$}} \put(150,50){\bbox{$v_G (G)$}} \put(80,50){\vector(1,0){40}} \put(80,150){\vector(1,0){40}} \put(80,250){\vector(1,0){40}} \put(50,230){\vector(0,-1){60}} \put(150,230){\vector(0,-1){60}} \put(50,130){\vector(0,-1){60}} \put(150,130){\vector(0,-1){60}} \put(60,200){\bbox{$v$}} \put(160,200){\bbox{$v$}} \put(60,100){\bbox{$v_G$}} \put(160,100){\bbox{$v_G$}} \put(100,260){\bbox{$\varphi$}} \put(100,160){\bbox{$\varphi_{G}$}} \put(100,60){\bbox{$\varphi_{\Gamma}$}} \put(100,200){\bbox{\tiny\rm ///}} \put(100,100){\bbox{\tiny\rm ///}} \end{picture} \end{center}}
\parbox[c]{.58\textwidth}{with
$\varphi(a)\>:=\> a^2$ for all $a\in {\bf P}_K\>,$ $\varphi$ is a right shift, \par
\noindent\par
\noindent $\varphi_{G}(v(a))\>:=\>v(\varphi(a))$ for all $a\in {\bf P}_K\>,$\par
\noindent that is $\varphi_{G}(g)= 2g$ for all $g\in G^{<0}\>,$ $\varphi_{G}$ is a left shift and \par
\noindent\par
\noindent $\varphi_{\Gamma}(v_G(g))\>:=\> v_G (\varphi_{G} (g))$ for all $g\in G^{<0}\>,$ \par
\noindent that is $\varphi_{\Gamma}(\gamma)=\gamma$ for all $\gamma\in\Gamma\>$, so that $\varphi_{\Gamma}$ is just the identity map.} \par
\noindent By Remark \ref{mm}, we can work with the equivalence relations associated to the following oriented maps: the q.o. preserving map $\varphi$ and the order preserving maps $\varphi_G$ and $\varphi_{\Gamma}$ (as defined on the right hand side of the above diagram). Note that $\>\sim_{\varphi_G}\>$ is just archimedean equivalence on $G$ and $\sim_{\varphi_{\Gamma}}$ is just equality on $\Gamma$. The following straightforward observation will be useful for the proof of Theorem \ref{theorem3OSMT} below: \begin{lemma} \label{corsavarphi} The equivalence classes of $\sim_{\varphi}$ are closed under multiplication. \end{lemma} \begin{proof}
The proof is similar to that of Lemma \ref{principal}. Let $a,\;b \in {\bf P}_K$, and without loss of generality assume that $a\preceq b$ and $a\sim_{\varphi} b$. We show that $ab\sim_{\varphi} a$. Let $n\in \lv N_{0}$, such that $b\preceq a^{2^n}$. By axiom qo$2$, $ab\preceq b^2$. Thus $b^2\preceq a^{2^n}b$ and $ab\preceq a^{2^n}b$. So, $ab\preceq a^{2^{n+1}}$. Since $1\preceq b$, by axiom qo$2$, we get that $a\preceq ab$.
Therefore, $ab\sim_{\varphi} a$. \end{proof} \begin{remark}\label{mmc} We note that \begin{equation} \varphi^n _G (v(a))\> = \> v(\varphi ^n (a)) \mbox{ and } \varphi^n _{\Gamma} (v_G(g))\> = \> v_G(\varphi^n _G (g)) \end{equation} thus \begin{equation} a\sim_{\varphi} a' \mbox{ if and only if } v(a)\sim_{\varphi_{G}}v(a') \mbox{ if and only if } v_G (v(a))\sim_{\varphi_{\Gamma}} v_G (v(a'))\> \end{equation} Thus we have an order reversing bijection from ${\bf P}_K/\sim_{\varphi}$ onto $\Gamma/\sim_{\varphi_{\Gamma}}=\Gamma$. Thus the chain $[{\bf P}_K/\sim_{\varphi}]^{\rm is}$ of non-empty initial segments of ${\bf P}_K/\sim_{\varphi}$ ordered by inclusion is isomorphic to $\Gamma^{\rm fs}$. In particular, initial segments which have a last element are in bijective correspondence to principal final segments. Thus the subchain of $[{\bf P}_K/\sim_{\varphi}]^{\rm is}$ of initial segments which have a last element is isomorphic to $\Gamma ^*$ \footnote{Note that the subchain of $[{\bf P}_K/\sim_{\varphi}]^{\rm is}$ of initial segments which have a last element is isomorphic to $[{\bf P}_K/\sim_{\varphi}]$ itself.} Therefore, as promised in the previous section, Theorems \ref{theorem1OSMT} and \ref{theorem2OSMT} will now follow from the following result: \end{remark} \begin{theorem}\label{theorem3OSMT} The rank ${\mathcal R}$ is isomorphic to the chain $[{\bf P}_K/\sim_{\varphi}]^{\rm is}$ and the principal rank ${\mathcal R}^{\rm pr}$ is isomorphic to the subchain of $[{\bf P}_K/\sim_{\varphi}]^{\rm is}$ of initial segments which have a last element. \end{theorem} \begin{proof} First we note that if $K_w$ is a convex valuation ring, then clearly $K_w^{\succ 0}\setminus K_v^{\succ 0}$ is an initial segment of ${\bf P}_K$. Moreover by Lemma \ref{principal} if $K_w$ is principal generated by $a$, then $[a]_{\sim _{\varphi}}$ is the last class. Furthermore, if $K_w$ intersects an equivalence class $[a]_{\sim_{\varphi}}$ then it must contain it, since the sequence $a^n; n\in \lv N_0$ is cofinal in $[a]_{\sim_{\varphi}}$ and $K_w$ is a convex subring. We conclude that $(K_w^{\succ 0}\setminus K_v^{\succ 0})/{\sim}_{\varphi}$ is an initial segment of ${\bf P}_K/{\sim}_{\varphi}$.
Conversely set ${\mathcal I}_w\>=\> \{[a]_{\varphi}\mid a\in K_w^{\succ 0}\setminus K_v^{\succ 0}\}\>.$ Given ${\mathcal I}\in [{\bf P}_K/ {\sim}_{\varphi}]^{\rm is}$, we show that there is a convex valuation ring $K_w$ such that ${\mathcal I}_w = {\mathcal I}$. Given ${\mathcal I}$, let $(\bigcup {\mathcal I})$ denote the set theoretic union of the elements of ${\mathcal I}$ and $-(\bigcup {\mathcal I})$ the set of additive inverses. Set $K_w=-\left(\bigcup {\mathcal I}\right)\cup K_v \cup \left(\bigcup {\mathcal I}\right)\>.$ We claim that $K_w$ is the required ring. Clearly, ${\mathcal I}_w = {\mathcal I}$. Further $K_w$ is convex (by its construction), and strictly contains $K_v$. We leave it to the reader, using Lemma \ref{corsavarphi} and Lemma \ref{principal}, to verify that $K_w$ is a ring, and that $K_w$ is principal generated by $a$ if $[a]_{\sim {\varphi}}$ is the last element of ${\mathcal I}$. \end{proof}
\section{The difference analogue of the rank} \label{diffanal} In this section, we develop a difference analogue of what has been reviewed above. That is, we develop a theory of difference compatible valuations, in analogy to the theory of convex valuations. The automorphism will play the role that multiplication plays in the previous case. \par
\noindent Let $(K, \preceq)$ be a q.o. field and $\sigma$ be a {\bf q.o. preserving} field automorphism of $K$, that is, $a\preceq a'$ if and only if $\sigma(a)\preceq \sigma(a')$, for all $a,\> a' \in K$. We say that $(K, \preceq, \sigma)$ is a {\bf q.o. difference field.} \begin{remark} \label{ordervsnatval} Let $(K, \leq, \sigma)$ be an ordered difference field. Recall that the natural valuation $v$ on $K$ is defined by archimedean equivalence. Since archimedean equivalence is preserved under order preserving automorphisms, we see that $\sigma$ is also $\preceq_v$ preserving (so that $(K, \preceq_v, \sigma)$ is a q.o. difference field). The converse fails: Consider the field of real Laurent series $K:=\lv R((t))$ endowed with the lexicographic order and the corresponding natural valuation $v_{\min}$ (see definitions following Corollary \ref{omega} below). The map $t\mapsto (-t)$ defines a field automorphisme $\sigma$ on $K$ which clearly preserves $v_{\min}$ but not the lexicographic order on $K$. \end{remark} \par
\noindent Now let $(K, \preceq, \sigma)$ be a non-trivial (i.e. $\sigma\not=$ identity) q.o. difference field and $v$ its natural valuation. By definition,
$\sigma$ satisfies for all $a,b\in K\>: v(a)\leq v(b)\;\mbox{ if and only if }v(\sigma(a))\leq v(\sigma(b))$ and thus induces an order preserving automorphism $\sigma_G$ and $\sigma_{\Gamma}$ such that the following diagram commutes: \par\noindent \parbox[c]{.4\textwidth}{ \begin{center} \setlength{\unitlength}{0.002\textwidth} \begin{picture}(200,250)(0,20) \put(50,250){\bbox{${\bf P}_K$}} \put(50,150){\bbox{$G ^{<0}$}} \put(50,50){\bbox{$v_G (G)$}} \put(150,250){\bbox{${\bf P}_K$}} \put(150,150){\bbox{$G ^{<0}$}} \put(150,50){\bbox{$v_G (G)$}} \put(80,50){\vector(1,0){40}} \put(80,150){\vector(1,0){40}} \put(80,250){\vector(1,0){40}} \put(50,230){\vector(0,-1){60}} \put(150,230){\vector(0,-1){60}} \put(50,130){\vector(0,-1){60}} \put(150,130){\vector(0,-1){60}} \put(60,200){\bbox{$v$}} \put(160,200){\bbox{$v$}} \put(60,100){\bbox{$v_G$}} \put(160,100){\bbox{$v_G$}} \put(100,260){\bbox{$\sigma$}} \put(100,160){\bbox{$\sigma_{G}$}} \put(100,60){\bbox{$\sigma_{\Gamma}$}} \put(100,200){\bbox{\tiny\rm ///}} \put(100,100){\bbox{\tiny\rm ///}} \end{picture} \end{center}}
\parbox[c]{.58\textwidth}{with
$\sigma_{G}(v(a))\>:=\>v(\sigma(a))$ for all $a\in {\bf P}_K\>,$ \par
\noindent\par
\noindent and \par
\noindent\par
\noindent $\sigma_{\Gamma}(v_G(g))\>:=\> v_G (\sigma_{G} (g))$ for all $g\in G^{<0}\>.$} \par
\noindent Now let $w$ be a convex valuation on $K$. Say $w$ is {\bf $\sigma$-compatible} if for all $a,b\in K\>: w(a)\leq w(b)\;\mbox{ if and only if }w(\sigma(a))\leq w(\sigma(b))\>.$ Thus $w$ is $\sigma$-compatible if and only if $\sigma$ preserves the q.o. $\preceq _w$. \par
\noindent The subset ${\mathcal R _{\sigma}}:=\{\> K_w \in {\mathcal R}\>;\> w \mbox{ is } \sigma$-\mbox{ compatible }\} is the {\bf $\sigma$-rank} of $(K,\preceq,\sigma)$. Similarly, the subset of all convex subgroups $G_w \ne \{0\}$ such that $\sigma_G(G_w)=G_w$, i.e $G_w$ is {\bf $\sigma_G$- invariant}, is the {\bf $\sigma$-rank} of~$G$. Finally, we denote by $\sigma_{\Gamma}$-$\Gamma ^{\rm fs}$ the subset of non-empty final segments $\Gamma _w$ such that $\sigma_{\Gamma}(\Gamma_w)=\Gamma_w$, i.e. $\Gamma _w$ is {\bf $\sigma_{\Gamma}$- invariant}. \par
\noindent The following Theorem \ref{wconvsigma}, Lemmas \ref{wconvvgsigma} and \ref{wconvvssigma} are analogues of Theorem \ref{wconv}, Lemma \ref{wconvvg} and Lemma \ref{wconvvs} respectively. They are verified by straightforward computations, using basic properties of valuations rings on the one hand and of automorphisms on the other (e.g. $\sigma(A\setminus B) =\sigma(A)\setminus \sigma(B)$, $\sigma(A) \subseteq B$ if and only if $A \subseteq \sigma^{-1}(B)$ and $\sigma(A) \subseteq B$ if and only if $\sigma(-A) \subseteq -B$). The equivalence of 1) and 7) in Theorem \ref{wconvsigma} follows from the compatibility of $\sigma$ with $w$ on the one hand, and from the definition of the induced q.o. on $Kw$ on the other. We call $K_w$ $\sigma$-compatible if any of the equivalent conditions below holds. \begin{theorem} \label{wconvsigma} The following assertions are equivalent for a convex valuation $w\>$:\par\noindent 1)\ \ $w$ is $\sigma$--compatible\par\noindent 2)\ \ $w$ is $\sigma^{-1}$--compatible\par\noindent 3)\ \ $\sigma(K_w)\>=\>K_w$\par\noindent 4)\ \ $\sigma(I_w)\>=\>I_w$ \par\noindent 5)\ \ $\sigma({\mathcal U}_w)\>=\>{\mathcal U}_w$ \par\noindent 6)\ \ $\sigma(K_w^{\succ 0}\setminus K_v^{\succ 0})\>=\>K_w^{\succ 0}\setminus K_v^{\succ 0}$ \par\noindent 7)\ \ the map $\sigma w: Kw\rightarrow Kw$ defined by $aw\mapsto \sigma(a)w$ is well-defined and is a q.o. (with respect to the induced q.o. on $Kw$ ) preserving field automorphism of $Kw$ . \end{theorem} \begin{remark}\label{sixandsixprime} Let $(K, \leq, \sigma)$ be an ordered field with natural valuation $v$. In this case, condition 7) on $\sigma w$ in Theorem \ref{wconvsigma} is referring to the induced order on the residue field $Kw$. Consider instead the following condition:\par\noindent 8)\ \ the map $\sigma w: Kw\rightarrow Kw$ defined by $aw\mapsto \sigma(a)w$ is well-defined and is a q.o. (with respect to the
q.o. $\preceq _{v/w}$ on $Kw$ ) preserving field automorphism of $Kw$ .\par
\noindent We observe that 7) implies 8). Indeed, $ \sigma w$ is assumed to be order preserving on $Kw$ by 7). Now $(Kw)(v/w) = Kv$ (see \cite[ Lemma 2.1]{[K-K]}). Therefore $v/w$ has archimedean residue field and is thus the natural valuation on the ordered field $Kw$. By Remark
\ref{ordervsnatval} we obtain the assertion. \end{remark} \begin{remark} \label{sigmaprime} The maximal ideals $I_w$ appearing in Theorem \ref{wconvsigma} 4) are $\sigma$- invariant prime ideals (also called transformally prime ideals in \cite{[C]}) of the valuation ring $K_v$ and the coarsenings $K_w$ are just the localizations of $K_v$ at those $\sigma$- invariant prime ideals, see \cite[ Lemma 2.3.1 p. 43]{[EP]}. Thus the $\sigma$- rank is also characterized by the chain of $\sigma$- invariant prime ideals of $K_v$. \end{remark} \begin{lemma}\label{wconvvgsigma} The correspondence $K_w \mapsto G_w$ is an order preserving bijection from ${\mathcal R}_{\sigma}$ onto the $\sigma_G$-rank of $G$. \end{lemma} \begin{lemma} \label{wconvvssigma} The correspondence $G_w \mapsto \Gamma _w$ is an order preserving bijection from the $\sigma_G$-rank of $G$ onto $\sigma_{\Gamma}$-$\Gamma ^{\rm fs}$. \end{lemma} \par
\noindent We deduce from Lemma \ref{wconvvgsigma} and Lemma \ref{wconvvssigma} that the $\sigma$-rank is the order type of $\sigma_{\Gamma}$-$\Gamma ^{\rm fs}$: \begin{theorem} \label{theorem1OSMTsigma} The correspondence $K_w \mapsto \Gamma _w$ is an order preserving bijection from ${\mathcal R}_{\sigma}$ onto $\sigma_{\Gamma}$-$\Gamma ^{\rm fs}$. \end{theorem} \par
\noindent We now exploit this observation. An automorphism $\sigma$ is an {\bf isometry} if $v(\sigma(a))=v(a)$ for all $a\in K$, equivalently $\sigma_G$ is the identity automorphism, and a {\bf weak isometry} if $\sigma_{\Gamma}$ is the identity automorphism. Every isometry is a weak isometry. Note that if $\Gamma$ is a rigid chain (i.e the only order preserving automorphism is the identity map), then $\sigma$ is necessarily a weak isometry. If $\sigma$ is a weak isometry, then $\sigma_{\Gamma}(v_G(g))\>=\>v_G(\sigma_{G}(g))\>=\>v_G(g)$, thus $g$ is archimedean equivalent to $\sigma_{G}(g)$ for all $g$, and so every convex subgroup is $\sigma_G$-invariant. \begin{corollary} \label{weakiso}
If $\sigma$ is a weak isometry, then ${\mathcal R}_{\sigma}\>=\> {\mathcal R}$. \end{corollary} \begin{corollary}\label{intersection} The correspondence $K_w \mapsto \min \Gamma_w$ is an order (reversing) isomorphism from ${\mathcal R}_{\sigma}\cap {\mathcal R}^{\rm pr}$ onto the chain $\{\gamma\>;\> \sigma_{\Gamma}(\gamma) = \gamma\}$ of fixed points of $\sigma_{\Gamma}$. \end{corollary} \begin{proof} By Lemma \ref{cpcs}, set $\mbox{ min } \Gamma_w:= \gamma_0$. By Lemma \ref{wconvvgsigma} and Lemma \ref{wconvvssigma}, $\Gamma_w$ in invariant under $\sigma_{\Gamma}$. Since $\sigma_{\Gamma}$ is order preserving, we must have $\sigma_{\Gamma}(\gamma_0) = \gamma_0$ \end{proof} \par
\noindent At the other extreme $\sigma$ is said to be {\bf $\omega$-increasing} if $a^n \prec \sigma(a)$ for all $n\in\lv N_0$ and all $a\in {\bf P}_K$, and {\bf $\omega$-contracting} if $\sigma^{-1}$ is $\omega$-increasing. \begin{remark}\label{strict left shift} Note that $\sigma$ is $\omega$-increasing (respectively, $\omega$-contracting) if and only if $\sigma_{\Gamma}$ is a {\bf strict left shift}, that is, $\sigma_{\Gamma}(\gamma) < \gamma$ for all $\gamma \in \Gamma\>$
(respectively, a {\bf strict right shift}, i.e. $\sigma_{\Gamma}(\gamma)> \gamma$ for all $\gamma \in \Gamma\>$). Thus if $\sigma$ $\omega$-increasing or $\omega$-contracting, then $\sigma_{\Gamma}$ has no fixed points. \end{remark} \begin{corollary}\label{omega} If $\sigma$ is $\omega$-increasing or $\omega$-contracting, then ${\mathcal R}_{\sigma}\cap {\mathcal R}^{\rm pr}$ is empty. \end{corollary}
\par
\noindent Recall that the {\bf Hahn group} \cite{[Ha]} over the chain $\Gamma$ and components $\lv R$, denoted ${\bf H}_{\Gamma} \lv R$, is the totally ordered abelian group whose elements are formal sums $\>g:= \sum g_{\gamma} 1_{\gamma}\>$, with well-ordered $\mbox{ support } g: =\{\gamma\>;\> g_{\gamma}\not= 0\}\>.$ Here $g_{\gamma}\in \lv R$ and $1_{\gamma}$ denotes the characteristic function on the singleton $\{\gamma\}$.
Addition is pointwise and the order lexicographic. Similarly, given a field $F$,
the field of {\bf generalized power series} over the ordered abelian group $G$ (or {\bf Hahn field} over $G$) with coefficients in $F$,
denoted $\mathbb{F}:=F((G))\>$, is the field whose elements are formal series $\>s:= \sum s_{g} t^g\>$, with well-ordered $\mbox{ support } s: =\{g\>;\> s_{g}\not= 0\}\>.$ Addition is pointwise, multiplication is given by the usual convolution formula. The field $\mathbb{F}$ has the same characteristic as that of $F$. The canonical valuation $v_{\min}$ on $\mathbb{F}$ is defined by $v_{\min}(s) : = \min \mbox{ support } s$ for $s \not= 0$. Its value group is $G$ and its residue field is $F$. Thus ($\mathbb{F}, \preceq _{v_{\min}}$ ) is a q.o. field.
If $F$ is an ordered field, its order extends to the lexicographic order on $\mathbb{F}$: a series $s$ is positive if and only if the coefficient of $t^{v_{\min}(s)}$ is positive in $F$. Thus, in that case $(\mathbb{F}, \leq)$ is an ordered field. Hahn fields are maximally valued: they admit no proper immediate extension, that is, no proper valued field extension preserving the value group and the residue field. They were extensively studied e.g. by Hahn \cite{[Ha]} and in the seminal paper of Kaplansky \cite{[KA]}.
\begin{lemma}\label{lifting} Any order preserving automorphism $\sigma_{\Gamma}$ of the chain $\Gamma$ lifts to an order preserving automorphism $\sigma_G$ of the Hahn group $G$ over $\Gamma$, and $\sigma_G$ lifts in turn to a q.o. preserving automorphism $\sigma$ of the Hahn field over $G$. \end{lemma} \begin{proof} Set $\>\sigma_G(\sum g_{\gamma} 1_{\gamma}):= \sum g_{\gamma} 1_{\sigma_{\Gamma}(\gamma)}\>$. It is straightforward to verify that the thus defined $\>\sigma_G$ induces the given automorphism $\sigma_{\Gamma}$ on $\Gamma$. Thus $\>\sigma_G$ is a lifting of $\sigma_{\Gamma}$. Now set $\>\sigma(\sum s_{g} t^g):= \sum s_{g} t^{\sigma_{G}(g)}\>.$ Again, it is clear that $\sigma$ induces $\>\sigma_G$ on $G$. Thus $\sigma$ is a lifting of $\>\sigma_G$ as asserted. \end{proof} \begin{corollary} \label{firstconstruction} Given any order type $\tau$ there exists an
ordered difference field $(K, \leq, \sigma)$, and also a p.q.o. difference field $(K, \preceq, \sigma)$ such that the order type of ${\mathcal R}_{\sigma}\cap {\mathcal R}^{\rm pr}$ is $\tau$. \end{corollary} \begin{proof} Set $\mu: = \tau^*$, and consider e.g. the linear ordering $\Gamma:=\sum _\mu \lv Q^{\geq 0}$, that is, the concatenation of $\mu$ copies of the non-negative rationals. Fix a non-trivial order automorphism $\eta$ of $\lv Q^{>0}$. Define $\sigma _{\Gamma}$ to be the uniquely defined order automorphism of $\Gamma$ fixing every $0\in \lv Q^{\geq 0}$ in every copy and extending $\eta$ on every copy. It is clear that the order type of the chain of fixed points (the zeros in every copy) of $\sigma _{\Gamma}$ is $\mu$. Set e.g. $G:={\bf H}_{\Gamma} \lv R$. By Lemma \ref{lifting}, $\sigma_{\Gamma}$ lifts canonically to $\sigma_G$ on $G$. Now consider e.g. the ordered field $\mathbb{F}:= \lv R((G))$. Again by Lemma \ref{lifting}, $\sigma_G$ lifts canonically to an order automorphism $\sigma$ of $\mathbb{F}$. This is our required $\sigma$, by Corollary \ref{intersection}. To obtain a p.q.o difference field, take $F$ any field and the corresponding $(\mathbb{F}, \preceq _{v_{\min}}, \sigma)$. \end{proof} \par
\noindent In the next section, we will exploit appropriate equivalence relations to define the principal difference rank and construct difference fields of arbitrary principal difference rank.
\section{The $\sigma$-rank and principal $\sigma$-rank via equivalence relations} \label{principalsigmarank} Let $(K, \preceq, \sigma)$ be a q.o. difference field. As promised in Section \ref{partIIOSMT}, we now exploit Remark \ref{mm} to give an interpretation of the $\sigma$- rank and define the principal $\sigma$-rank as quotients via appropriate equivalence relations. Our aim is to state and prove the analogues to Theorems \ref{theorem3OSMT}, \ref{theorem1OSMT} and \ref{theorem2OSMT}. We recall that the q.o. preserving maps considered in Remark \ref{mm} are assumed to be oriented. Moreover, scrutinizing the proof of Theorem \ref{theorem3OSMT} we quickly realize that we need Lemma \ref{corsasigma} below, an analogue of Lemma \ref{corsavarphi}. Thus we need further assumptions on $\sigma$, to ensure that $\sigma$ satisfies Lemma \ref{corsasigma}. For simplicity from now on we will assume that $\sigma$ or $\sigma^{-1}$ satisfy $\sigma(a) \succeq a^2$ for all $a\in {\bf P}_K$. Note that this implies that $\sigma(a) \succ a$, so $\sigma$ is an oriented strict right-shift. Note that our condition on $\sigma$ is fulfilled for $\omega$-increasing or $\omega$-contracting automorphisms. \par
\noindent A convex subring $K_w \ne K_v$ is \slind{$\sigma$-principal generated by $a$} for $a \in \mbox{\bf P} _K$ if $K_w$ is the smallest convex $\sigma$-compatible subring containing $a$. The {\bf $\sigma$-principal rank} of $K$ is the subset \gloss{${\mathcal R}_{\sigma}^{\rm pr}$} of ${\mathcal R}_{\sigma}$ consisting of all $\sigma$-principal $K_w\in {\mathcal R}$. We will use the analogue of Remark \ref{mmc}: \begin{remark} \label{mmcsigma} The maps $\sigma$, $\sigma_G$ and $\sigma_{\Gamma}$ are q.o. preserving and we can define the corresponding equivalence relations $\sim_{\sigma}$, $\sim_{\sigma_G}$ and $\sim_{\sigma_{\Gamma}}$. As before we have \begin{equation} \label{reduction} a\sim_{\sigma} a' \mbox{ if and only if } v(a)\sim_{\sigma_{G}}v(a') \mbox{ if and only if } v_G (v(a))\sim_{\sigma_{\Gamma}} v_G (v(a')) \end{equation} Thus we have an order reversing bijection from ${\bf P}_K/\sim_{\sigma}$ onto $\Gamma/\sim_{\sigma_{\Gamma}}$. Thus the chain $[{\bf P}_K/\sim_{\sigma}]^{\rm is}$ of initial segments of ${\bf P}_K/\sim_{\sigma}$ ordered by inclusion is isomorphic to $(\Gamma/\sim_{\sigma_{\Gamma}}) ^{\rm fs}$. As before, the subchain of initial segments which have a last element is isomorphic to $(\Gamma/\sim_{\sigma_{\Gamma}}) ^*$. \end{remark} \begin{lemma} \label{corsasigma} The equivalence classes of $\sim_{\sigma}$ are closed under $\sigma$ and under multiplication. \end{lemma} \begin{proof} The condition on $\sigma$ implies by induction that $\sigma^n(a) \succeq a^{2^n}$. Thus given $n\in\lv N_0\>,$ there exists $l\in \lv N_0$ such that $\sigma^l(a) \succeq a^n$. Thus $a\sim_{\sigma}\sigma (a)$. So the
equivalence classes of $\sigma$ are closed under $\sigma$ .
Recall that the natural valuation $v_G$ on $G$ satisfies $v_G(x+y)
= \min\{v_G(x), v_G(y)\}$ if sign($x$) = sign($y$). Again one easily deduces from this fact and the equivalences (\ref{reduction}) above that the
equivalence classes of $\sigma$ are closed under multiplication. Indeed assume that $a\sim_{\sigma} b$ and $a\sim_{\sigma} c$. We want to show that $a\sim_{\sigma} bc$. Set $x: = v(b)$, $y: = v(c)$ and $z: = v(a) \in G ^{<0}\>$. By the first equivalence in (\ref{reduction}), it is enough to show that $v(a)\sim_{\sigma_{G}}v(bc)$ i.e. that $x + y \sim_{\sigma_{G}} z$. By the second equivalence in (\ref{reduction}), it is enough to show that $ v_G (x + y)\sim_{\sigma_{\Gamma}} v_G (z)\>$. Without loss of generality $ v_G (x + y) = v_G(x)$. But since $a\sim_{\sigma} b$ it follows by (\ref{reduction}) that $ v_G (x)\sim_{\sigma_{\Gamma}} v_G (z)$ as required. \end{proof} We can now prove the analogue of Theorem \ref{theorem3OSMT}: \begin{theorem}\label{theorem3OSMTsigma} The $\sigma$-rank ${\mathcal R_{\sigma}}$ is isomorphic to $[{\bf P}_K/\sim_{\sigma}]^{\rm is}$ and the principal $\sigma$-rank ${\mathcal R}_{\sigma}^{\rm pr}$ is isomorphic to the subset of $[{\bf P}_K/\sim_{\sigma}]^{\rm is}$ of initial segments which have a last element.\footnote{Note that the subchain of $[{\bf P}_K/\sim_{\sigma}]^{\rm is}$ of initial segments which have a last element is isomorphic to $[{\bf P}_K/\sim_{\sigma}]$ itself.} \end{theorem} \begin{proof} First we note that if $K_w$ is a convex $\sigma$-compatible valuation ring, then clearly $K_w^{\succ 0}\setminus K_v^{\succ 0}$ is an initial segment of ${\bf P}_K$. Furthermore, if $K_w$ intersects a $\sigma$- equivalence class $[a]_{\sim_{\sigma}}$ then it must contain it, since the sequence $\sigma(a)^n; n\in \lv N_0$ is cofinal in $[a]_{\sim_{\sigma}}$ and $K_w$ is a convex subring. We conclude that $(K_w^{\succ 0}\setminus K_v^{\succ 0})/{\sim}_{\sigma}$ is an initial segment of ${\bf P}_K/{\sim}_{\sigma}$ and moreover $[a]_{\sim _{\sigma}}$ is the last class in case $K_w$ is $\sigma$- principal generated by $a$. Conversely set ${\mathcal I}_w\>=\> \{[a]_{\sigma}\mid a\in K_w^{\succ 0}\setminus K_v^{\succ 0}\}\>.$ Given ${\mathcal I}\in [{\bf P}_K/ {\sim}_{\sigma}]^{\rm is}$, we show that there is a $\sigma$-compatible convex valuation ring $K_w$ such that ${\mathcal I}_w = {\mathcal I}$. Given ${\mathcal I}$, let $(\bigcup {\mathcal I})$ denote the set theoretic union of the elements of ${\mathcal I}$ and $-(\bigcup {\mathcal I})$ the set of additive inverses. Set $K_w=-\left(\bigcup {\mathcal I}\right)\cup K_v \cup \left(\bigcup {\mathcal I}\right)\>.$ We claim that $K_w$ is the required ring. Clearly, ${\mathcal I}_w = {\mathcal I}$. Further $K_w$ is convex (by its construction), and strictly contains $K_v$. We leave it to the reader, using Lemma \ref{corsasigma}, to verify that $K_w$ is a $\sigma$-compatible subring, and that $K_w$ is $\sigma$-principal generated by $a$ if $[a]_{\sim {\sigma}}$ is the last element of ${\mathcal I}$. \end{proof} We now deduce from this theorem combined with Remark \ref{mmcsigma} the promised analogues of Theorems \ref{theorem1OSMT} and \ref{theorem2OSMT} respectively: \begin{corollary} \label{theorem2OSMTsigma} ${\mathcal R}_{\sigma}$ is (isomorphic to) $(\Gamma/\sim _{\sigma_{\Gamma}}) ^{\rm fs}$. \end{corollary} \begin{corollary} \label{theorem1OSMTsigma} ${\mathcal R_{\sigma} ^{\rm pr}}$ is (isomorphic to) $(\Gamma/\sim _{\sigma_{\Gamma}}) ^*$. \end{corollary} \par
\noindent We call the order type of $(\Gamma/\sim _{\sigma_{\Gamma}})$ the {\bf rank} of the automorphism $\sigma_{\Gamma}$ . We now can construct $\omega$-increasing automorphisms of arbitrary principal difference rank. Corollary \ref{arbitrary} below, compared to Corollary \ref{omega} demonstrates the discrepancy between the chains ${\mathcal R_{\sigma} ^{\rm pr}}$ and ${\mathcal R}_{\sigma}\cap {\mathcal R}^{\rm pr}$. \begin{corollary}\label{arbitrary} Given any order type $\tau$ there exists a maximally valued ordered field endowed with an $\omega$-increasing automorphism of principal difference rank $\tau$. \end{corollary} \begin{proof} Set $\mu: = \tau^*$, and consider e.g. the linear ordering $\Gamma:=\sum _\mu \lv Q$, that is, the concatenation of $\mu$ copies of the non-negative rationals. Let $\ell$ be e.g. translation by $-1$ on $\lv Q$. Define $\sigma _{\Gamma}$ to be the uniquely defined order automorphism of $\Gamma$ extending $\ell$ on every copy. It is clearly a strict left shift of rank $\mu$. Set e.g. $G:={\bf H}_{\Gamma} \lv R$. Then by Lemma \ref{lifting} $\sigma_{\Gamma}$ lifts canonically to $\sigma_G$ on $G$. Now set e.g. $K:= \lv R((G))$. By Lemma \ref{lifting}, Remark \ref{strict left shift} and Corollary \ref{theorem1OSMTsigma}, $\sigma_G$ lifts canonically to an $\omega$-increasing automorphism of $K$ of principal difference rank $\mu^*=\tau$. \end{proof} \begin{example} \label{k-s}
Consider the chain $\Gamma= \lv Z \times \lv Z$ (the lexicographic product of two copies of $\lv Z$ ). We endow $\Gamma$ with the automorphisms $\tau((x, y)):=(x-1, y)$ and $\sigma((x, y)):=(x, y-1)$. The rank of $\tau$ is one and that of $\sigma$ is $\lv Z$. Both are strict left shifts. Lifting those automorphisms to $G:={\bf H}_{\Gamma} \lv R$ and then to $K:= \lv R((G))$ as in the proof of Corollary \ref{arbitrary}, we obtain $\omega$-increasing automorphisms of $K$ of distinct principal difference ranks. \end{example}
For a regular uncountable cardinal ${\kappa}$, let us denote by $G_{\kappa}$ the $\kappa$-bounded Hahn group, that is, the subgroup of $G={\bf H}_{\Gamma} \lv R$ consisting of elements with support of cardinality $< {\kappa}$. Similarly, we denote by $\lv R((G))_{\kappa}$ the $\kappa$-bounded Hahn field, i.e. the subfield of $K= \lv R((G))$ consisting of series with support of cardinality $< {\kappa}$. If $\kappa = \kappa^{< \kappa}$ then $\lv R((G_{\kappa}))_{\kappa}$ has cardinality $\kappa$, see \cite{[A-K]}.
We now generalize Example \ref{k-s}. In \cite[ Corollary 14]{[K-S]}, we construct for every infinite cardinal $\kappa$ a chain $\Gamma$ of cardinality $\kappa$ which admits of family of $2^{\kappa}$ strict left shift automorphisms, of pairwise distinct ranks. Lifting those automorphisms to $\lv R((G_{\kappa}))_{\kappa}$, we conclude as in \cite[ Theorem 9]{[K-S]}: \begin{theorem} \label{last}
Let $\kappa = \kappa^{< \kappa}$ be a regular uncountable cardinal and $\Gamma$ be any chain of cardinality $\kappa$ which admits a family of $2^{\kappa}$ strict left shift automorphisms of pairwise distinct ranks. Then the corresponding $\kappa$-bounded Hahn field $\lv R((G_{\kappa}))_{\kappa}$ of cardinality $\kappa$ admits a family of $2^{\kappa}$ $\omega$-increasing automorphisms of distinct principal difference ranks. \end{theorem}
\end{document} |
\begin{document}
\allowdisplaybreaks \title{Littlewood--Paley--Stein Estimates for Non-local Dirichlet Forms}
\author{Huaiqian Li\qquad\quad Jian Wang} \thanks{\emph{H.\ Li:} Center for Applied Mathematics, Tianjin University, Tianjin 300072, P. R. China \texttt{huaiqianlee@gmail.com}}
\thanks{\emph{J.\ Wang:}
College of Mathematics and Informatics \& Fujian Key Laboratory of Mathematical Analysis and Applications (FJKLMAA), Fujian Normal University, 350007 Fuzhou, P. R. China. \texttt{jianwang@fjnu.edu.cn}}
\maketitle \begin{abstract} We obtain the boundedness in $L^p$ spaces for all $1<p<\infty$ of the so-called vertical Littlewood--Paley functions for non-local Dirichlet forms in the metric measure space under some mild assumptions. For $1<p\leqslant 2$, the pseudo-gradient is introduced to overcome the difficulty that chain rules are not available for non-local operators, and then the Mosco convergence is used to pave the way from the finite jumping kernel case to the general case, while for $2\leqslant p<\infty$, the Burkholder--Davis--Gundy inequality is effectively applied. The former method is analytic and the latter one is probabilistic. The results extend those ones for pure jump symmetric L\'evy processes in Euclidean spaces.
\noindent\textbf{Keywords:} Littlewood--Paley--Stein estimate; non-local Dirichlet form; pseudo-gradient; Mosco convergence; Burkholder--Davis--Gundy inequality.
\noindent \textbf{MSC 2010:} 60G51; 60G52; 60J25; 60J75. \end{abstract} \allowdisplaybreaks
\allowdisplaybreaks
\section{Introduction}\label{section1} Let $(M,d)$ be a locally compact and separable metric space, and $\mu$ be a positive Radon measure on $M$ with full support. We will refer to such triple $(M,d,\mu)$ as a metric measure space. As usual, the real $L^p$ space is denoted by $L^p(M,\mu)$ with the norm
$$\|f\|_p:= \left(\int_M |f(x)|^p\, \mu(d x)\right)^{1/p},\quad1\leqslant p <\infty,$$ and
$$\|f\|_\infty := \textup{ess}\sup_{x\in M} |f(x)|,$$ where $\textup{ess}\sup$ is the essential supremum. The inner product of functions $f,g\in L^2(M,\mu)$ is denoted by $\langle f, g\rangle$.
Consider a Dirichlet form $(D,\mathscr{F})$ in $L^2(M,\mu)$, which is a closed, symmetric, non-negative definite, bilinear form $D: \mathscr{F}\times \mathscr{F}\rightarrow\mathds R$ defined on a dense subspace $\mathscr{F}$ of $L^2(M,\mu)$, satisfying in addition the Markov property. The closedness means that $\mathscr{F}$ is a Hilbert space with respect to the $D_1^{1/2}$-inner product defined by $$D_1(f,g)=D(f,g)+\langle f, g\rangle.$$ The Markov property means that if $f\in\mathscr{F}$ then the function $\hat{f}:=\max\{0,\min\{1,f\}\}$ belongs to $\mathscr{F}$ and $D(\hat{f})\leqslant D(f)$. Here and in the sequel, we write $D(f)$ instead of $D(f,f)$ for short.
Let $L$ be the non-negative definite $L^2$-generator of the Dirichlet form $(D,\mathscr{F})$, which is a self-adjoint operator on $L^2(M,\mu)$ with domain $\mathscr{D}(L)$ such that $$D(f,g)=\langle Lf,g\rangle,$$ for all $f\in\mathscr{D}(L)$ and $g\in\mathscr{F}$. The generator $L$ give rises to the semigroup $(P_t)_{t\geq0}$ with $P_t=e^{-tL}$ for all $t\geq0$ in the sense of functional calculus. It turns out that $(P_t)_{t\geq0}$ is a strongly continuous, contractive, symmetric semigroup in $L^2(M,\mu)$, and satisfies the Markov property which means that $0\leqslant P_tf\leq1$ for every $t>0$ provided $0\leqslant f\leq1$.
Let $C_c(M)$ be the space of all continuous functions on $M$ with compact support. Recall that the Dirichlet form $(D,\mathscr{F})$ is called regular if $\mathscr{F}\cap C_c(M)$ is dense both in $\mathscr{F}$ (with respect to the $D_1^{1/2}$-norm) and in $C_c(M)$ (with respect to the supremum norm). It follows that if $(D,\mathscr{F})$ is regular, then every function $f\in\mathscr{F}$ admits a quasi-continuous version $\tilde{f}$ (see e.g. \cite[Theorem 2.1.3]{FOT}). Throughout this paper, we abuse the notation and represent $f\in \mathscr{F}$ by its quasi-continuous version without writing $\tilde f$.
In order to introduce the so-called vertical Littlewood--Paley square function, the ``module of gradient'' is necessary. The suitable candidate in this general setting should be the \emph{carr\'{e} du champ} operator. It is a non-negative, symmetric and continuous bilinear form $\Gamma: \mathscr{F}\times\mathscr{F}\rightarrow L^1(M,\mu)$ such that $$D(f,g)=\int_M \Gamma(f,g)\,d\mu\quad\mbox{for every } f,g\in \mathscr{F},$$ which is uniquely characterized in the algebra $L^\infty(X,\mu)\cap \mathscr{F}$ by $$\int_M\Gamma(f,g)h\,d\mu=D(f,gh)+D(g,fh)-D(fg,h),$$ for every $f,g,h\in L^\infty(X,\mu)\cap \mathscr{F}$. See \cite{BH1991} for more details. In the sequel, we use the notation $\Gamma(f):=\Gamma(f,f)$ for convenience.
\ \
In this paper, we are concerned with non-local Dirichlet forms. Let $(D,\mathscr{F})$ be a regular Dirichlet form of pure jump type in $L^2(M,\mu)$ defined as \begin{equation}\label{nondi} D(f,g)=\frac{1}{2}\iint_{M\times M \backslash {\rm diag}} (f(x)-f(y))(g(x)-g(y)) \,J(x,dy)\,\mu(dx),\quad f,g\in\mathscr{F},\end{equation} where ${\rm diag}$ denotes the diagonal set $\{(x,x):x \in M\}$ and $J(x,dy)$ is a non-negative kernel satisfying the symmetry property
$$J(x,dy)\,\mu(dx)=J(y,dx)\,\mu(dy).$$ $J(x,dy)$ is called jumping kernel associated with the Dirichlet form $(D,\mathscr{F})$ in the literature. Then the\emph{ carr\'e du champ} operator $\Gamma$ is defined as follows $$\Gamma(f,g)(x)=\frac{1}{2}\int_M (f(x)-f(y))(g(x)-g(y))\,J(x,dy),\quad f,g\in \mathscr{F} \text{ and } x\in M.$$ Clearly, $$D(f)=\displaystyle\int_M \Gamma(f)(x)\,\mu(dx)\quad\mbox{for every }f\in\mathscr{F}.$$ This motivates us to define the gradient (more precisely, the module of gradient) of a function $f\in \mathscr{F}$ by \begin{equation}\label{g-1}
|\nabla f|(x)= \sqrt{ \Gamma(f)}(x)= \left(\frac{1}{2}\int_M (f(x)-f(y))^2\,J(x,dy)\right)^{1/2},\quad x\in M.\end{equation} Note that, due to the symmetry of $J(x,dy)\,\mu(dx)$, for every $f\in\mathscr{F}$, $$D(f)= \iint_{\{(x,y)\in M\times M: f(x)\geqslant f(y)\}} (f(x)-f(y))^2 \,J(x,dy)\,\mu(dx).$$ Then, we can also well define the following (module of) modified gradient for every $f\in\mathscr{F}$, \begin{equation}\label{g-2}
|\widetilde\nabla f|_*(x):= \left(\int_{\{y\in M:\,f(x)\geqslant f(y)\}} (f(x)-f(y))^2\,J(x,dy)\right)^{1/2},\quad x\in M.\end{equation} From \eqref{g-1} and \eqref{g-2}, it is easy to know that, for every $f\in\mathscr{F}$,
$$0\leqslant |\widetilde\nabla f|_*\leqslant \sqrt{2}|\nabla f|\quad\mbox{and}\quad\||\nabla f|\|_2^2= \||\widetilde\nabla f|_*\|_2^2=D(f,f).$$
Actually, motivated by \cite{BBL}, we need a further modification of the gradient (and this is a crucial point; see some remarks at the end of Section \ref{section2}). For any $f\in\mathscr{F}$, we define \begin{equation}\label{g-21}
|\widetilde\nabla f|(x)=
\left(\int_{\{y\in M:\,|f|(x)\geqslant |f|(y)\}} (f(x)-f(y))^2\,J(x,dy)\right)^{1/2},\quad x\in M.\end{equation} It is easy to see that
$|\widetilde\nabla f|=|\widetilde\nabla f|_*$ for any $0\leqslant f\in \mathscr{F}$; however, for general $f\in \mathscr{F}$, they are not comparable to each other. We also note that, similar to the standard module of gradient,
$|\widetilde\nabla f|= |\widetilde\nabla (-f)|$ for any $f\in \mathscr{F}$; however, such property is not satisfied for $|\widetilde\nabla \cdot|_*$. This in some sense indicates that the definition of the modified gradient $|\widetilde\nabla \cdot|$ above is more reasonable than that of $|\widetilde\nabla \cdot|_*$.
For every $f\in L^1(M,\mu)\cap L^\infty(M,\mu)$, we now define the vertical Littlewood--Paley $\mathscr{H}$-functions $\mathscr{H}_\nabla(f)$ and $\mathscr{H}_{\widetilde\nabla}(f)$ corresponding to the non-local Dirichlet form $(D,\mathscr{F})$ in \eqref{nondi} as
$$\mathscr{H}_\nabla(f)(x)=\left(\int_0^\infty |\nabla P_t f|^2(x)\,dt\right)^{1/2},$$ and
\begin{equation}\label{eeefff}\mathscr{H}_{\widetilde\nabla}(f)(x)=\left(\int_0^\infty |\widetilde\nabla P_tf|^2(x)\,dt\right)^{1/2},\end{equation} for every $x\in M$.
The purpose of this paper is to establish Littlewood--Paley--Stein estimates in $L^p(M,\mu)$ for non-local Dirichlet form $(D,\mathscr{F})$ and for all $1<p<\infty$. The main result is the following theorem (see Theorems \ref{th1} and \ref{thp} below for precise expressions). \begin{theorem}\label{main} Let $(M,d,\mu)$ be a metric measure space. Consider the non-local Dirichlet form $(D,\mathscr{F})$ defined in \eqref{nondi}. Under some mild assumptions, for $p\in (1,2]$ the vertical Littlewood--Paley operator $\mathscr{H}_{\widetilde\nabla}$ is bounded in $L^p(M,\mu)$; for $p\in [2,\infty)$ the vertical Littlewood--Paley operator $\mathscr{H}_\nabla$ is bounded in $L^p(M,\mu)$. \end{theorem}
The prototype of Littlewood--Paley--Stein estimates is the $L^p$ boundedness of the Littlewood--Paley $g$-function in the Euclidean space for all $1<p<\infty$; see \cite[Chapter IV, Theorem 1]{St1970}. There are a lot of extensions on this result in various directions, and we only recall some of them. We are interested in the vertical (i.e., derivative with respect to the spatial variable) Littlewood--Paley--Stein estimates for heat or Poisson semigroups. Let $M$ be a complete and connected (smooth) Riemannian manifold with Riemannian volume measure $dx$, the non-negative Laplace--Beltrami operator $\Delta$, the corresponding heat semigroup $(e^{-t\Delta})_{t\ge0}$ and Poisson semigroup $(e^{-t\sqrt{\Delta}})_{t\ge0}$, as well as the gradient operator $\nabla$. For every $f\in C_c^\infty(M)$, the vertical Littlewood--Paley $\mathscr{H}$- and $\mathscr{G}$-functions are given by \begin{equation}\label{cla}
\mathscr{H}(f)(x)=\left(\int_0^\infty |\nabla e^{-t\Delta} f|^2(x)\,dt\right)^{1/2}, \end{equation}
and \begin{equation}\label{cla-G}
\mathscr{G}(f)(x)=\left(\int_0^\infty t|\nabla e^{-t\sqrt{\Delta}} f|^2(x)\,dt\right)^{1/2}, \end{equation}
for every $x\in M$, where $|\cdot|$ is the length induced by the Riemannian distance in the tangent space. The operator $\mathscr{H}$ is called bounded in $L^p(M,dx)$ (or the Littlewood--Paley--Stein estimate holds for $\mathscr{H}$) for any $p\in (1,\infty)$, if there exists a constant $c_p>0$ such that
$$\|\mathscr{H}(f)\|_p\leqslant c_p\|f\|_p,\quad f\in C_c^\infty(M).$$
(The same for $\mathscr{G}$.) On the aspect of analytic approaches, Stein \cite[Chapter II]{Stein} proved the $L^p$ boundedness of $\mathscr{G}$ for all $p\in (1,\infty)$ on compact Lie groups. Lohou\'{e} \cite{Lou1987} investigated the $L^p$ boundedness of the Littlewood--Paley $\mathscr{H}_a$- and $\mathscr{G}_a$-functions, defined as $$\mathscr{H}_a(f)(x)=\Big(\int_0^\infty e^{at}|\nabla e^{-t\Delta}f(x)|^2\, dt\Big)^{1/2}$$
and $$\mathscr{G}_a(f)(x)=\Big(\int_0^\infty te^{at}|\nabla e^{-t\sqrt{\Delta}}f(x)|^2\, dt\Big)^{1/2}$$ in the Cartan--Hadamard manifold, where $a$ is a real number to be determined. In fact, no additional assumptions on $M$ are needed for the boundedness of $\mathscr{H}$ and $\mathscr{G}$ in $L^p(M,dx)$ for $1<p\leq2$ (see e.g. \cite{CDD}), while, for the case when $2<p<\infty$, much stronger assumptions are need (see e.g.\ \cite[Proposition 3.1]{CD2003}). On the aspect of probabilistic approaches, we should mention that Meyer \cite{Mey,Mey1981} studied the $L^p$ boundedness for all $1<p<\infty$ on the Littlewood--Paley $\mathscr{G}_*$-function, defined as
$$\mathscr{G}_*(f)(x)=\Big(\int_0^\infty te^{-2t\sqrt{\Delta}}|\nabla e^{-t\sqrt{\Delta}}f(x)|^2\, dt\Big)^{1/2}.$$ Bakry established a slightly different Littlewood--Paley--Stein estimate for diffusion processes under the condition that the Bakry--Emery $\Gamma_2$ is non-negative in \cite{Bakry1985}, and then proved it under the condition that $\Gamma_2$ is lower bounded on complete Riemannian manifolds in \cite{Bakry1987}. See also \cite{ShYo}, where strong assumptions are needed to guarantee a nice algebra and to run the $\Gamma$ calculus for diffusion processes. Li \cite{Li2006} established the Littlewood--Paley--Stein estimate for $\mathscr{G}$ on complete Riemannian manifolds, as well as the $L^p$ boundedness for $p\in(1,2]$ of Littlewood--Paley square functions for Poisson semigroups generated by the Hodge--Laplacian. We do not mention many studies on Wiener spaces here.
For non-local Dirichlet forms, to the authors' knowledge, the study on the $L^p$ boundedness of the vertical Littlewood--Paley operator $\mathscr{H}$ is not too much. Dungey \cite{Nick} obtained the $L^p$ boundedness of the vertical Littlewood--Paley operator with $1<p\leq2$ for random walks on graphs and groups. Ba\~{n}uelos, Bogdan and Luks \cite{BBL} studied Littlewood--Paley--Stein estimates for symmetric L\'evy processes in the Euclidean space recently (see \cite{BK} for more recent extension on non-symmetric L\'evy processes). In the aforementioned papers on the L\'evy process case, the Euclidean structure is nice to apply the Hardy--Stein identity and it is used in a crucial way. However, the approach to prove our main result
(Theorem \ref{main} above), which is presented in the metric measure space setting, is different. Indeed, when $p\in (1,2]$, we prove the boundedness of $\mathscr{H}_{\widetilde\nabla}$ by using the pseudo-gradient to overcome the difficulty that chain rules are not available for non-local operators, and then by applying the Mosco convergence from finite jumping kernel case to general case (see Theorem \ref{th1-23} below); when $p\in [2,\infty)$, we verify the boundedness of $\mathscr{H}_\nabla$, by following the idea of \cite{BBL} to express the square function as a conditional expectation of the quadratic variation of a suitable martingale and then applying the Burkholder--Davis--Gundy inequality (see Theorem \ref{thp} below). We would like to mention that, though $|\nabla f|$
seems more natural than $|\widetilde\nabla f|$, since $|\widetilde\nabla f|$ is of a certain asymmetry, \cite[Example 2]{BBL} indicates that in some settings $|\nabla f|$ may be too large to yield the boundedness of $\mathscr{H}_\nabla$ in $L^p(M,\mu)$ for $1<p< 2$.
To indicate clearly the contribution of our paper, we present two examples, for which Theorem \ref{main} (see also Theorems \ref{th1} and \ref{thp} below) is applicable.
\begin{example}\it Let $(D,\mathscr{F})$ be a regular Dirichlet form on $L^2(M,\mu)$ as follows \begin{align*}D(f,f)=\frac{1}{2}\iint_{M\times M \backslash {\rm diag}}(f(y)-f(x))^2J(x,y)\,\mu(dx)\,\mu(dy),\quad f\in \mathscr{F} \end{align*} where $J(x,y)$ is a non-negative measurable symmetric function on $M\times M\backslash {\rm diag}$ such that $$\int_{\{y\in M: d(x,y)>r\}} J(x,y)\,\mu(dy)<\infty,\quad x\in M, r>0.$$ Then, for any $p\in (1,2]$, the vertical Littlewood--Paley operator $\mathscr{H}_{\widetilde\nabla}$ is bounded in $L^p(M,\mu)$. \end{example}
The next example includes symmetric stable-like processes on $\mathds R^d$ of variable orders.
\begin{example}\label{exm2}\it Let $s:\mathds R^d\to [s_1,s_2]\subset (0,2)$ such that
$$|s(x)-s(y)|\leqslant \frac{c}{\log(2/|x-y|)},\quad |x-y|\leqslant 1$$ holds for some constant $c>0$. Let $(D,\mathscr{F})$ be a regular Dirichlet form on $L^2(\mathds R^d,dx)$ as follows \begin{align*}D(f,f)=&\frac{1}{2}\iint(f(y)-f(x))^2j(x,y)\,dx\,dy,\\ \mathscr{F}=&\overline{ C_c^1(\mathds R^d)}^{D^{1/2}_1}, \end{align*}
where $D_1(f,f)=D(f,f)+\|f\|_2^2$, $j(x,y)$ is a non-negative symmetric measurable function on $\mathds R^d\times \mathds R^d$ such that
$$\sup_{x\in \mathds R^d} \int_{\{|y-x|>1\}}j(x,y)\,dy<\infty,$$ and for some constants $c_1,c_2>0$
\begin{equation}\label{ed4}\frac{c_1}{|x-y|^{d+s(x)\wedge s(y)}}\leqslant j(x,y)\leqslant \frac{c_2}{|x-y|^{d+s(x)\vee s(y)}},\quad |x-y|\leqslant 1.\end{equation} Then, for $p\in [2,\infty)$, the vertical Littlewood--Paley operator $\mathscr{H}_{\nabla}$ is bounded in $L^p(\mathds R^d,dx)$. \end{example}
\ \
The next two sections are devoted to proving the main result (Theorem \ref{main} above), which is treated separately according to $p\in (1,2]$ and $p\in [2,\infty)$.
\section{Littlewood--Paley--Stein estimates for $1<p\leqslant 2$}\label{section2}
\subsection{Pseudo-gradient} In order to show the motivation, for the moment, let $M$ be a (smooth) Riemannian manifold, $\Delta$ be the Laplace--Beltrami operator, and $\nabla$ be the Riemannian gradient, and denote by $|\cdot|$ the length induced by the Riemannian distance in the tangent space. A beautiful way to prove the $L^p$ boundedness of $\mathscr{H}$ and $\mathscr{G}$, defined by \eqref{cla-G} and \eqref{cla} respectively, is based on the following chain rule: \begin{equation}\label{e:rie}
\Delta (f^p)=pf^{p-1} \Delta f-p(p-1)f^{p-2}|\nabla f|^2, \end{equation} which is valid for all $1<p<\infty$ and for all
smooth functions $f$ on the Riemannian manifold $M$; see \cite[Chapter IV, Lemma 1, p.\ 86]{Stein} or the proof of \cite[Lemma 2.1]{CDD} for example.
However, this so-called chain rule no longer holds for non-local Dirichlet forms. For this, following the idea of \cite{Nick}, we may make use of the following pseudo-gradient, which is defined by \begin{equation}\label{e:ger} \widetilde{\Gamma}_p(f)=pf(Lf)-f^{2-p}L(f^p),\end{equation}
for $p\in (1,\infty)$ and suitable non-negative functions $f$, where $L$ is the generator of the regular non-local Dirichlet form $(D,\mathscr{F})$ given in \eqref{nondi}. From \eqref{e:rie}, when $L$ is the Laplace--Beltrami operator $\Delta$ on the Riemannian manifold $M$, it is clear that the right hand side of \eqref{e:ger} is just $p(p-1)|\nabla f|^2$. Due to this, one can reasonably imagine that, in the general setting, $\widetilde{\Gamma}_p(f)$ should play the same role as $|\nabla f|^2$ in the Riemannian manifold setting. This is the reason why we call $\widetilde{\Gamma}_p(f)$ the pseudo-gradient of $f$. For more details on the pseudo-gradient, refer to its origination \cite{Nick}.
For our purpose, we need to define the pseudo-gradient $\Gamma_p$ for suitable function $f$ (which is not necessarily non-negative) as follows: for $p\in (1,2]$,
\begin{equation}\label{e:ger0}\Gamma_p(f)=pf(Lf)-|f|^{2-p}L(|f|^p).\end{equation} In particular, when $p=2$, $\Gamma_2(f)=2fLf-L(f^2)$. (In fact, for $p\in (2,\infty)$, we can also define $\Gamma_p(f)$ by the right hand side of \eqref{e:ger0} for suitable function $f$; however, we will not use it in this work.) We emphasis that, to extend the definition of $\Gamma_p$ for signed function is one of the crucial points in our argument. This is a key difference between the discrete setting as in \cite{Nick} and the present setting for general metric measure spaces.
Recall that $(M,d,\mu)$ is a metric measure space and $(D,\mathscr{F})$ is a non-local regular Dirichlet form of pure jump type defined in \eqref{nondi}. In the present setting, for a suitable function $f$ on $M$, $|\nabla f|$ and $|\widetilde\nabla f|$ are defined in \eqref{g-1} and \eqref{g-21}, respectively. In order to compare $\Gamma_p(f)$ with $|\nabla f|$ and $|\widetilde\nabla f|$, we need the closed expression of the generator $L$, which is difficult to seek in general (see e.g.\ \cite{SW2014}). However, if \begin{equation}\label{e:bound} \int_{M\backslash\{x\} } J(x,dy)<\infty,\quad x\in M,\end{equation} then it is easy to see that, for all $f\in \mathscr{D}(L)$,
\begin{equation}\label{a12}Lf(x)=\int_M (f(x)-f(y))\,J(x,dy).\end{equation} Note also that under \eqref{e:bound}, $Lf$ is pointwise $\mu$-a.e.\ well defined by \eqref{a12} for every $f\in L^\infty(M,\mu)$. We call that the jumping kernel $J(x,dy)$ is finite, if \eqref{e:bound} is satisfied.
\subsection{Boundedness of Littlewood--Paley functions for $1<p\leqslant 2$: the finite case} Throughout this subsection, we always suppose that the jumping kernel $J(x,dy)$ is finite, i.e., \eqref{e:bound} holds. The next lemma provides an explicit formula for $\Gamma_p(f)$ when $p\in (1,2]$ and $f\ge0$ (see \cite[Lemma 3.2]{Nick} for the case on graphs). \begin{lemma}\label{Gamma_p}Under \eqref{e:bound}, for any $p\in (1,2]$ and $0\leqslant f\in \mathscr{D}(L)\cap L^\infty(M,\mu)$, it holds $$\Gamma_p(f)(x)=p(p-1)\int_{\{y\in M:f(y)\neq f(x)\}}(f(x) -f(y))^2 \, I(f(x),f(y);p)\,J(x,dy),$$ for any $x\in M$, where $$I(f(x),f(y);p)= \int_0^1 \frac{(1-u)f(x)^{2-p}}{((1-u)f(x)+uf(y))^{2-p}}\,du,$$ and $0^0:=1$. \end{lemma} \begin{remark}\label{remark333} On the one hand, Lemma \ref{Gamma_p} holds trivially when $p=2$; indeed, it is well known that, for every $f\in \mathscr{D}(L)\cap L^\infty(M,\mu)$ and every $x\in M$, $$\Gamma_2(f)(x)=\big[2f(Lf)-L(f^2)\big](x)= 2 \Gamma(f)(x)=\int_M (f(x)-f(y))^2\,J(x,dy).$$ On the other hand, we note that for $p\in (1,2)$ and $0\leqslant f\in \mathscr{D}(L)\cap L^\infty(M,\mu)$ such that $f(x)\neq f(y)$ for $x,y\in M$, $$\int_0^1 \frac{(1-u)f(x)^{2-p}}{((1-u)f(x)+uf(y))^{2-p}}\,du \leqslant \int_0^1 \frac{1}{(1-u)^{1-p}}\,du=\frac{1}{p},$$ and hence, $\Gamma_p(f)\leqslant (p-1) \Gamma_2(f)<\infty$. \end{remark} \begin{proof}[Proof of Lemma $\ref{Gamma_p}$] By the remark above, we only need to prove the case when $p\in (1,2)$. According to \eqref{a12}, for $0\leqslant f\in \mathscr{D}(L)\cap L^\infty(M,\mu)$, we have $$\Gamma_p(f)(x)=\int_M\big(pf(x)(f(x)-f(y))-f(x)^{2-p}(f(x)^p-f(y)^p)\big)\,J(x,dy).$$ Note that, to further calculate the right hand side of the equality above, we only need to consider the case when $f(y)\neq f(x)$ inside the integral.
By the Taylor expansion of the function $t\mapsto t^p$ on $[0,\infty)$, we have \begin{align*}t^p-s^p=&ps^{p-1}(t-s)+p(p-1)\int_s^t v^{p-2}(t-v)\,dv\\ =&ps^{p-1}(t-s)+p(p-1)(t-s)^2\int_0^1 \frac{1-u}{((1-u)s+ut)^{2-p}}\,du\end{align*} for any $s,t\ge0$ with $s\neq t$, where the second equality follows by a change of variable, i.e., $v=(1-u)s+ut$. When $s=0$, the condition $p>1$ ensures that the integral exists. If $f(y)\neq f(x)$, then, setting $s=f(x)$ and $t=f(y)$, we obtain that \begin{align*}&pf(x)(f(x)-f(y))-f(x)^{2-p}(f(x)^p-f(y)^p)\\ &=f(x)^{2-p}[(f(y)^p-f(x)^p)-pf(x)^{p-1} (f(y)-f(x))]\\ &=p(p-1)(f(x)-f(y))^2\int_0^1\frac{(1-u)f(x)^{2-p}}{((1-u)f(x)+uf(y))^{2-p}}\,du\\ &=p(p-1)(f(x)-f(y))^2 I(f(x),f(y);p).\end{align*} This yields \begin{equation*} \begin{split}\Gamma_p(f)(x)=&p(p-1)\int_{\{y\in M:f(y)\neq f(x)\}} (f(x)-f(y))^2 I(f(x),f(y);p)\,J(x,dy).\end{split}\end{equation*} We prove the desired assertion. \end{proof}
Now we can immediately compare $\Gamma_p(f)$ with $|\nabla f|^2$ and $|\widetilde{\nabla} f|^2$ for suitable non-negative $f$. \begin{corollary}\label{comp} Under \eqref{e:bound}, for $p\in (1,2]$ and $0\leqslant f\in \mathscr{D}(L)\cap L^\infty(M,\mu)$,
$$0\leqslant \Gamma_p(f)(x)\leqslant 2(p-1)|\nabla f|^2(x)$$ and
\begin{equation}\label{e:ffcc}|\widetilde \nabla f|^2(x)\leqslant 2/(p(p-1)) \Gamma_p(f)(x),\end{equation}
for any $x\in M$, where $|\nabla f|$ and $|\widetilde\nabla f|$ are defined by \eqref{g-1} and \eqref{g-21}, respectively.
\end{corollary} \begin{proof} Let $p\in (1,2)$ and $0\leqslant f\in \mathscr{D}(L)\cap L^\infty(M,\mu)$. We assume that \eqref{e:bound} holds.
(i) It is clear from Lemma \ref{Gamma_p} that $\Gamma_p(f)(x)\ge0$. Observing that $$(1-u)^{2-p}f(x)^{2-p}\leqslant ((1-u)f(x)+uf(y))^{2-p},$$ for any $0\leqslant u\leq1$ and $f\ge0$, we obtain that \begin{align*} \Gamma_p(f)(x)&\leqslant p(p-1)\int_0^1(1-u)^{p-1} \, du\int_M (f(x)-f(y))^2 J(x,y)\,\mu(dy)\\
&= 2(p-1)|\nabla f|^2(x). \end{align*}
(ii) Observing that, for $0\leqslant f(y)< f(x)$, one has $(1-u)f(x)+uf(y)\leqslant f(x)$ for any $0\leqslant u\leq1$. Hence
$$I(f(x),f(y);p)=\int_0^1 \frac{(1-u)f(x)^{2-p}}{((1-u)f(x)+uf(y))^{2-p}}\,du\geqslant \int_0^1 (1-u)\,du =1/2.$$ This along with the definition of $|\widetilde \nabla f|^2$ yields the desired assertion. \end{proof}
\begin{remark} It is easy to see that, in general, $\Gamma_p(f)(x)< 2(p-1)|\nabla f|^2(x)$ holds for any $0\leqslant f\in \mathscr{D}(L)\cap L^\infty(M,\mu)$ under the assumption \eqref{e:bound}. For example,
for any function $f$ with $f\neq 0$ and $f(x)=0$ for some $x\in M$, we have $|\nabla f|^2(x)>0$ and $\Gamma_p(f)(x)=0$. Therefore, for $p\in (1,2]$, in general situations, one can use the bound on $\Gamma_p(f)$ to control $|\widetilde \nabla f|^2$ but not $|\nabla f|^2$. \end{remark}
The following statement shows that \eqref{e:ffcc} indeed holds for all $f\in \mathscr{D}(L)\cap L^\infty(M,\mu)$, which is one of the key ingredients in our proof.
\begin{proposition}\label{P:comp} Under \eqref{e:bound}, for $p\in (1,2]$ and $f\in \mathscr{D}(L)\cap L^\infty(M,\mu)$,
$$0\leqslant |\widetilde \nabla f|^2(x)\leqslant 2/(p(p-1)) \Gamma_p(f)(x),$$
for any $x\in M$, where $|\widetilde\nabla f|$ is defined by \eqref{g-21}.
\end{proposition}
\begin{proof} By Remark \ref{remark333} again, without loss of generality we may and can
assume that $p\in (1,2)$. The proof is a little bit delicate and is based on Corollary \ref{comp}.
For any $f\in \mathscr{D}(L)\cap L^\infty(M,\mu)$, by \eqref{e:ger0}, \begin{align*}
\Gamma_p(f)&=pfLf-p|f|(L|f|)+p|f|L|f|-|f|^{2-p}L(|f|^p)\\
&=pfLf-p|f|L|f|+\Gamma_p(|f|). \end{align*} According to \eqref{e:ffcc} in Corollary \ref{comp}, it holds that
\begin{align*}\Gamma_p(|f|)(x)\geqslant &\frac{p(p-1)}{2}\big|\widetilde \nabla |f|\big|^2(x)\\
\geqslant &\frac{p(p-1)}{2}\int_{\{y\in M:|f|(x)\geqslant |f|(y)\}}(|f|(x)-|f|(y))^2\,J(x,dy). \end{align*}
On the other hand,
\begin{align*}&pf(x)(Lf)(x)-p|f|(x)(L|f|)(x)\\
&=p\left(\int_M f(x)(f(x)-f(y))\,J(x,dy)-\int_M |f|(x)(|f|(x)-|f|(y))\,J(x,dy)\right)\\
&=p\int_M (|f|(x)|f|(y)-f(x)f(y))\,J(x,dy)\\
&\geqslant p\int_{\{y\in M:|f|(x)\geqslant |f|(y)\}}(|f|(x)|f|(y)-f(x)f(y))\,J(x,dy), \end{align*} where in the last inequality we used the fact that
$|f|(x)|f|(y)-f(x)f(y)\ge0$ for all $x,y\in M$.
Furthermore, we deduce that
\begin{align*}&\frac{p-1}{2}(|f|(x)-|f|(y))^2+|f|(x)|f|(y)-f(x)f(y)\\
&=\frac{p-1}{2}(f^2(x)+f^2(y))+(2-p)|f|(x)|f|(y)-f(x)f(y)\\
&\geqslant \frac{p-1}{2}(f^2(x)+f^2(y))+(1-p)|f|(x)|f|(y)\\ &\geqslant \frac{p-1}{2}(f^2(x)+f^2(y))-(p-1)f(x)f(y)\\ &=\frac{p-1}{2}(f(x)-f(y))^2,\end{align*} where both inequalities follow from the fact that
$|f|(x)|f|(y)-f(x)f(y)\ge0$ for all $x,y\in M$ again.
Combining with all the inequalities above, we arrive at the desired assertion.
\end{proof}
Recall that $(P_t)_{t\geq0}$ with $P_t=e^{-tL}$ is the semigroup corresponding to the Dirichlet form $(D,\mathscr{F})$. \begin{proposition}\label{pre}\,\ Let $(M,d,\mu)$ be a metric measure space, and $(D,\mathscr{F})$ be the Dirichlet form defined in \eqref{nondi}. Suppose that \eqref{e:bound} holds. Then, for any $p\in (1,2]$, the following assertions hold. \begin{itemize} \item[{\rm(i)}] There exists a constant $c_p>0$ such that, for all $t>0$ and $f\in L^1(M,\mu)\cap L^\infty(M,\mu)$,
$$\|\Gamma_p^{1/2}(P_tf)\|_p\leqslant c_p t^{-1/2} \|f\|_p.$$
\item[{\rm(ii)}] For any $f\in L^1(M,\mu)\cap L^\infty(M,\mu)$, define $$(\mathscr{H}_pf)(x)=\left(\int_0^\infty\Gamma_p (P_tf)(x)\,dt\right)^{1/2}\quad\mbox{for every } x\in M.$$ Then there is a constant $c_p'>0$ such that, for all $f\in L^1(M,\mu)\cap L^\infty(M,\mu)$,
$$\|\mathscr{H}_pf\|_p\leqslant c'_p\|f\|_p.$$ \end{itemize} \end{proposition}
\begin{proof}
Noticing that $(P_t)_{t\ge0}$ is a strongly continuous Markovian semigroup defined on $L^2(M,\mu)$, the operator $P_t$ can be extended to $L^\infty(M,\mu)$ such that $\|P_tf\|_{\infty}\leqslant \|f\|_{\infty}$ (see \cite[Page\ 56]{FOT}). On the other hand, we can also extend $P_t$ on $L^1(M,\mu)\cap L^2(M,\mu)$ to $L^1(M,\mu)$ uniquely. Since $(P_t)_{t\ge0}$ is a symmetric Markovian semigroup, it holds that $\|P_tf\|_{1}\leqslant \|f\|_{1}$. By the Riesz--Thorin interpolation theorem, for all $p\in(1,\infty)$, we have $$\|P_t\|_{p\to p}:=\sup_{f\in L^p(M,\mu)\setminus\{0\}}\frac{\|P_tf\|_{p}}{\|f\|_{p}}\leqslant 1.$$
(i) In the following, let $p\in (1,2]$, and consider any non-zero function $f\in L^1(M,\mu)\cap L^\infty(M,\mu)$. Set $u_t=P_tf$ for all $t\geqslant 0$. Then $u_t\in \mathscr{D}(L)\cap L^1(M,\mu)\cap L^\infty(M,\mu)$ for every $t>0$. In what follows, let $\partial_t$ denote the differentiation with respect to $t$. The fundamental idea of the proof below is due to \cite{Stein}; however, we may not reduce the problem to take non-negative function $f$ as in the aforementioned reference, since $|\widetilde\nabla \cdot|$ does not enjoy the sublinear property.
Instead, we should take into account $|u_t|$ not $u_t$ itself, which also explains the reason why we need to define the pseudo-gradient $\Gamma_p$ for all suitable signed functions by \eqref{e:ger0}.
By the definition of $\Gamma_p$ and the fact that \begin{equation}\label{eq-t}\partial_t(|u_t|^p)=pu_t|u_t|^{p-2}\partial_tu_t=-pu_t|u_t|^{p-2}Lu_t,\end{equation} we have
\begin{align*}|u_t|^{p-2}\Gamma_p(u_t)&=|u_t|^{p-2}\left(pu_t Lu_t-|u_t|^{2-p}L|u_t|^p\right)\\
&=pu_t |u_t|^{p-2}(Lu_t)-L(|u_t|^p)\\
&=-(\partial_t+L)(|u_t|^p).\end{align*} It follows that
$$\Gamma_p(u_t)=-|u_t|^{2-p}(\partial_t+L)(|u_t|^p).$$ Set
$$J_t:=-(\partial_t+L)(|u_t|^p).$$ Then $J_t\geq0$ since $\Gamma_p(u_t)\geq0$ by Lemma \ref{Gamma_p}. Using the H\"{o}lder inequality, we have
\begin{align*} \|\Gamma_p^{1/2}(u_t)\|_p^p&=\int_M \Gamma_p^{p/2}(u_t)(x)\,\mu(dx)=\int_M |u_t|^{p(2-p)/2}(x) J_t^{p/2}(x)\,\mu(dx)\\
&\leqslant \left(\int_M J_t(x)\,\mu(dx)\right)^{p/2}\left(\int_M |u_t|^p(x)\,\mu(dx)\right)^{(2-p)/2}.\end{align*}
Note that, by the contraction property of the semigroup $(P_t)_{t\ge0}$ on $L^p(M,\mu)$,
$$\left(\int_M |u_t|^p(x)\,\mu(dx)\right)^{(2-p)/2}\leqslant \|f\|_p^{p(2-p)/2}.$$ On the other hand, by H\"{o}lder's inequality and the contraction property again,
\begin{align*}\int_M |\partial _t|u_t|^p|(x)\,\mu(dx)=&p\int_M |u_t|^{p-1}| |\partial_tu_t|\,d\mu \leqslant p\|\partial_tu\|_p\||u_t|^{p-1}\|_{p/(p-1)}\\
\leqslant& p\|\partial_tu_t\|_p\|f\|_p^{p-1}. \end{align*}
Recall that $L$ is self-adjoint in $L^2(M,\mu)$ and $P_t$ is continuous as a map from $L^p(M,\mu)$ to itself for every $t\ge0$ and for every $p\in[1,\infty]$. By the classical theory developed by Stein (see e.g. \cite[Chapter III, Theorem 1]{Stein}), we derive that $(P_t)_{t>0}$ is an analytic semigroup in $L^p(M,\mu)$ for every $p\in(1,\infty)$; more precisely, the map $t\mapsto P_t$ has an analytic extension in the sense that it extends to an analytic $L^p(M, \mu)$-operator-valued function $t + is\mapsto P_{t+is}=e^{-(t+is)L}$, which is defined in the sector of the complex plane
$$|\arg(t+is)|<\frac{\pi}{2}\Big(1-\Big|\frac{2}{p}-1\Big|\Big).$$
Hence, we find that $$\|\partial_tu_t\|_p=\|Lu_t\|_p\leqslant c_p t^{-1}\|f\|_p.$$ Thus, together with \eqref{eq-t},
\begin{equation}\label{eeff} \int_M |\partial _t|u_t|^p|(x)\,\mu(dx)\leqslant pc_p t^{-1}\|f\|_p^{p}.\end{equation} In particular, due to $J_t\ge0$ and
$$L|u_t|^p=-\partial_t|u_t|^p-J_t,$$ we get that
\begin{equation}\label{eeff0} \mu((L|u_t|^p)^+)<\infty.\end{equation}
Thus, \begin{equation}\label{eeff1}\int_M L|u_t|^p(x)\,\mu(dx)\ge0.\end{equation} Indeed, let $\{K_n\}_{n\ge1}$ be a sequence of increasing compact sets such that $\cup_{n=1}^\infty K_n=M$, and let $\{\varphi_n\}_{n\ge1}$ be a sequence of bounded measurable functions such that $\varphi_n=1$ on $K_n$, and $0\leqslant \varphi_n\leqslant 1$ on $K_n^c$. Using \eqref{eeff0} and the extension of Fatou's lemma (see \cite[Theorem 3.2.6 (2), p.\ 52]{YJA}), we get \begin{equation*}\begin{split}
\int_M L|u_t|^p(x)\,\mu(dx)&\geqslant\limsup_{n\to \infty} \int_M \varphi_n(x) (L|u_t|^p)(x)\,\mu(dx)\\
&= \limsup_{n\to \infty} \int_M |u_t|^p(x) (L \varphi_n)(x)\,\mu(dx), \end{split} \end{equation*}
where in the equality above we used the symmetry property of $J(x,dy)\,\mu(dx)$ and the facts that $Lf$ is pointwise $\mu$-a.e. well defined for any bounded measurable function $f$ and $|u_t|^p\in L^1(M,\mu)$. On the other hand, since $(D,\mathscr{F})$ is a regular Dirichlet form on $L^2(M,\mu)$, for any relatively compact open sets $U$ and $V$ with $\bar{U}\subset V$, there is a function $\psi\in \mathscr{F}\cap C_c(M)$ such that $\psi=1$ on $U$ and $\psi=0$ on $V^c$. Consequently, \begin{equation}\label{e:ffee}\begin{split}\iint_{U\times V^c} J(x,dy)\,\mu(dx)=&\iint_{U\times V^c}(\psi(x)-\psi(y))^2J(x,dy)\,\mu(dx)\\ \leqslant &D(\psi,\psi)<\infty.\end{split}\end{equation} For any fixed $x\in M$ and any $\varepsilon>0$, by \eqref{e:bound}, we can choose $R:=R(x,\varepsilon)>0$ large enough such that $$\displaystyle\int_{\{y\in M: d(x,y)\geqslant R\}}\,J(x,dy)<\varepsilon.$$
Fix this $R$. Then, for $n\geqslant 1$ large enough, $\varphi_n(y)=1$ for all $y\in M$ with $d(x,y)< R$. Thus, for $n\geqslant 1$ large enough,
$$|L\varphi_n(x)|=\bigg|\int_M\left(\varphi_n(x)-\varphi_n(y)\right)\, J(x,d y) \bigg|\leqslant \int_{\{y\in M: d(x,y)\geqslant R\}}\,J(x,dy)<\varepsilon,$$
which means that $\lim_{n\to\infty }L\varphi_n(x)=0$ for all $x\in M$. This, along with \eqref{e:ffee}, the fact that $|u_t|^p\in L^1(M,\mu)\cap L^\infty(M,\mu)$ and the dominated convergence theorem, gives
$$\limsup_{n\to \infty} \int_M |u_t|^p(x) (L \varphi_n)(x)\,\mu(dx)=0.$$ So, \eqref{eeff1} holds true.
\eqref{eeff1} together with \eqref{eeff} yields that
\begin{align*}\int_M J_t(x)\,\mu(dx)&=-\int_M\partial _t|u_t|^p(x)\,\mu(dx)-\int_M L|u_t|^p(x)\,\mu(dx)\leqslant pc_p t^{-1}\|f\|_p^{p}.\end{align*} Hence,
$$ \left(\int_M J_t(x)\,\mu(dx)\right)^{p/2}\leqslant c'_p t^{-p/2}\|f\|_p^{p^2/2}.$$
Combining with all the conclusions above, we prove the first assertion.
(ii) Note that \begin{align*}(\mathscr{H}_pf)^2(x)&=\int_0^\infty \Gamma_p(u_t)(x)\,dt\\
&=-\int_0^\infty |u_t|(x)^{2-p}(\partial_t+L)(|u_t|^p)(x)\,dt\\ &\leqslant f^*(x)^{2-p} J(x), \end{align*} where $f^*$ is the semigroup maximal function defined by \eqref{maxf} below, and
$$J(x)=-\int_0^\infty (\partial_t+L)(|u_t|^p)(x)\,dt,$$ which is non-negative. Thus, by using the H\"{o}lder inequality, \begin{align*}\int_M( \mathscr{H}_p f)^p(x)\,\mu(dx)\leqslant&\int_M f^*(x)^{p(2-p)/2}J(x)^{p/2}\,\mu(dx)\\ \leqslant &\left(\int_M f^*(x)^p\,\mu(dx)\right)^{(2-p)/2}\left(\int_M J(x)\,\mu(dx)\right)^{p/2}.\end{align*}
Lemma \ref{max} below further yields that
$$\left(\int_M f^*(x)^p\,\mu(dx)\right)^{(2-p)/2}\leqslant c_p'\|f\|_p^{p(2-p)/2}.$$
On the other hand, by \eqref{eeff1},
\begin{align*}\int_M J(x)\,\mu(dx) =&-\int_0^\infty dt \int_M (\partial_t+L) |u_t|^p(x)\,\mu(dx)\\
\leqslant&-\int_0^\infty dt \int_M \partial_t |u_t|^p(x)\,\mu(dx)\\
\leqslant& \int_M |f|^p(x)\,\mu(dx)=\|f\|_p^p.\end{align*}
Combining all the inequalities above, we obtain that
$$\int_M( \mathscr{H}_p f)^p(x)\,\mu(dx)\leqslant c_p'\|f\|_p^p,$$
which is the second assertion.
Therefore, the proof is complete.
\end{proof}
For any $f\in L^1(M,\mu)\cap L^\infty(M,\mu)$, define the semigroup maximal function $f^*$ by
\begin{equation}\label{maxf}f^*(x)=\sup_{t>0} |P_tf(x)|,\quad x\in M.\end{equation} Since $(P_t)_{t\ge0}$ is a symmetric sub-Markovian semigroup, we have the following \begin{lemma}\label{max}$($\cite[Section III. 3, p.\ 73]{Stein}$)$ For all $p\in (1,\infty]$, there exists a constant $c_p>0$ such that for all $f\in L^p(M,\mu)$,
$$\|f^*\|_p\leqslant c_p\|f\|_p,$$
where, for $p=\infty$, the right hand side is just $\|f\|_{\infty}$ $($i.e., the constant $c_\infty=1)$. \end{lemma}
Furthermore, according to Proposition \ref{P:comp} and Proposition \ref{pre}, we immediately have the following \begin{theorem}\label{th1} {\bf (Finite jumping kernel case)}\,\, Under the same assumption of Proposition $\ref{pre}$, for any $p\in (1,2]$,
$\mathscr{H}_{\widetilde \nabla}$ is bounded in $L^p(M,\mu)$, i.e., there exists a constant $c_p>0$ such that, for every $f\in L^p(M,\mu)$,
$$\|\mathscr{H}_{\widetilde \nabla}f\|_{p}\leqslant c_p\|f\|_{p}.$$ Moreover, there exists a constant $\tilde{c}_p>0$ such that, for every $f\in L^p(M,\mu)$,
$$\||\widetilde \nabla P_t f|\|_{p}\leqslant \tilde{c}_p t^{-1/2}\|f\|_{p}.$$ \end{theorem} \begin{proof} For any $f\in L^1(M, \mu)\cap L^\infty(M,\mu)$, $P_tf$ belongs to $\mathscr{D}(L)\cap L^1(M,\mu)\cap L^\infty(M,\mu)$ for every $t>0$. By Proposition \ref{P:comp}, we deduce that \begin{align*}
\mathscr{H}_{\widetilde \nabla} f(x)&=\left(\int_0^\infty |\widetilde \nabla P_tf|^2(x)\,dt \right)^{1/2}\\
&\leqslant\left(\int_0^\infty\Big|\frac{2}{p(p-1)}\Gamma_p(P_tf)(x)\Big|\,d t\right)^{1/2}\\ &=\Big(\frac{2}{p(p-1)}\Big)^{1/2}(\mathscr{H}_pf)(x). \end{align*} By Proposition \ref{pre}(ii), we have
$$\|\mathscr{H}_{\widetilde \nabla} f\|_p\leqslant c_p'\Big(\frac{2}{p(p-1)}\Big)^{1/2}\|f\|_p=:c_p\|f\|_p.$$ Now the general case when $f\in L^p(M,\mu)$ follows from the fact that $L^1(M,\mu)\cap L^\infty(M,\mu)$ is dense in $L^p(M,\mu)$ and the application of Fatou's lemma.
The last assertion is also an immediate application of Proposition \ref{P:comp} and Proposition \ref{pre}(i). \end{proof} \subsection{Boundedness of Littlewood--Paley functions for $1<p\leqslant 2$: the general case} In this part, we consider the case that \eqref{e:bound} is not necessarily satisfied. In this general setting, it is not clear that $Lf^p$ and so $\Gamma_p(f)$ given by \eqref{e:ger} are well defined for $p\in(1,2]$ and $0\leqslant f\in \mathscr{D}(L)\cap L^\infty(M,\mu)$ (even in the pointwise sense). (Note that, according to \cite[Theorem 2]{Mey}, if $f\in \mathscr{D}(L)\cap L^\infty(M,\mu)$, then $f^p\in \mathscr{D}(L)\cap L^\infty(M,\mu)$ for all $p\in[2,\infty)$.) To overcome this difficulty, we will make use of the Mosco convergence of non-local Dirichlet forms, and impose the absolute continuity and the local finiteness assumptions on the jumping kernel $J(x,dy)$, i.e., there is a non-negative measurable function $J(x,y)$ on $M\times M \setminus {\rm diag}$ such that, for every $x,y\in M$, \begin{equation}\label{e:ack} J(x,dy)=J(x,y)\,\mu(dy), \end{equation} and \begin{equation} \label{e:ack1}\int_{\{y\in M: d(x,y)>r\}}J(x,y)\,\mu(dy)<\infty,\quad x\in M,\, r>0. \end{equation}
\ \
Recall that, a sequence of Dirichlet forms $\{(D^n,\mathscr{F}^n)\}_{n\ge1}$ on $L^2(M,\mu)$ is said to be convergent to a Dirichlet form $(D,\mathscr{F})$ in $L^2(M,\mu)$ in the sense of Mosco if \begin{itemize} \item[(a)] for every sequence $\{f_n\}_{n\ge1}$ in $L^2(M,\mu)$ converging weakly to $f$ in $L^2(M,\mu)$, $$\liminf_{n\to \infty} D^n(f_n,f_n)\geqslant D(f,f);$$ \item[(b)] for every $f\in L^2(M,\mu)$, there is a sequence $\{f_n\}_{n\ge1}$ in $L^2(M,\mu)$ converging strongly to $f$ in $L^2(M,\mu)$ such that
$$\limsup_{n\to \infty} D^n(f_n,f_n)\leqslant D(f,f).$$ \end{itemize}
\begin{remark}\label{r:mos} We make the following two comments on Mosco convergence. \begin{itemize} \item[(1)] Condition (b) in the definition of Mosco convergence is implied by the following condition: \begin{itemize} \item[(b)'] There is a common core $\mathscr{C}$ for the Dirichlet forms $\{(D^n,\mathscr{F}^n)\}_{n\ge1}$ and $(D,\mathscr{F})$ such that $$\lim_{n\to\infty} D^n(f,f)=D(f,f),\quad\mbox{for every } f\in \mathscr{C}.$$ See the proof of \cite[Thoorem 2.3]{BBCK}. \end{itemize} \item[(2)] Let $\{(D^n,\mathscr{F}^n)\}_{n\ge1}$ and $(D,\mathscr{F})$ be Dirichlet forms on $L^2(M,\mu)$. Then, the sequence $\{(D^n,\mathscr{F}^n)\}_{n\ge1}$ converges to $(D,\mathscr{F})$ in the sense of Mosco, if and only if, for every $t>0$ and $f\in L^2(M,\mu)$, $P^{(n)}_tf$ converges to $P_tf$ in $L^2(M,\mu)$, where $(P_t)_{t\ge0}$ and $(P^{(n)}_t)_{t\ge0}$ are the semigroups corresponding to $(D,\mathscr{F})$ and $(D^n,\mathscr{F}^n)$, respectively. See \cite[Corollary 2.6.1]{Mo1994}. \end{itemize} \end{remark}
\ \
Now, we consider the regular non-local Dirichlet form $(D,\mathscr{F})$ given by \eqref{nondi}, and suppose that \eqref{e:ack} is satisfied. For any $n\geqslant 1$ and $x,y\in M$, define $$J_n(x,y)=J(x,y)\mathds 1_{\{d(x,y)>1/n\}}.$$ Note that \eqref{e:ffee} holds for general regular Dirichlet form $(D,\mathscr{F})$.
Then, by the definition of $J_n(x,y)$ and \eqref{e:ffee}, the sequence $\{J_n(x,y)\}_{n\ge1}$ converges to $J(x,y)$ locally in $L^1(M\times M\backslash {\rm diag}, \mu\times \mu)$.
Let $\mathscr{C}:=\mathscr{F}\cap C_c(M)$ be a core of $(D,\mathscr{F})$. For any $n\ge1$, we define the regular Dirichlet form $(D^n,\mathscr{F}^n)$ as follows \begin{align*} D^n(f,f)=&\iint (f(x)-f(y))^2 J_n(x,y)\,\mu(dx)\,\mu(dy),\\
\mathscr{F}^n=& \overline{\mathscr{C}}^{\sqrt{D^{n,1}}},\end{align*} where ${D^{n,1}}(f,f)=D^n(f,f)+\|f\|_2^2$ and $\overline{\mathscr{C}}^{\sqrt{D^{n,1}}}$ denotes the closure of $\mathscr{C}$ with respect to the metric $\sqrt{D^{n,1}}$. Note that $J_n(x,y)\leqslant J(x,y)$, $\mathscr{F}\subset \mathscr{F}^n$ for all $n\geqslant 1$. In particular, $\mathscr{C}$ is a common core for all $(D^n,\mathscr{F}^n)$, $n\geq1$. Furthermore, we have the following statement.
\begin{proposition}\label{P:mos} Under \eqref{e:ack}, the sequence of Dirichlet forms $\{(D^n,\mathscr{F}^n)\}_{n\ge1}$ above converges to $(D,\mathscr{F})$ in the sense of Mosco. \end{proposition} \begin{proof} We split the proof into two parts.
(1) In this part, the argument is inspired by the proof of \cite[Theorem 1.4]{SU}. Suppose that $u_n$ is weakly convergent to $u$ in $L^2(M,\mu)$ as $n\rightarrow\infty$, and $$\liminf_{n\to \infty} \iint (u_n(x)-u_n(y))^2 J_n(x,y)\,\mu(dx)\,\mu(dy)<\infty.$$ We may assume that $$\lim_{n\to \infty} \iint (u_n(x)-u_n(y))^2 J_n(x,y)\,\mu(dx)\,\mu(dy)<\infty.$$ For $(x,y)\in M\times M \backslash {\rm diag}$ and $n\geqslant 1$, define $$\tilde u_n(x,y)=(u_n(x)-u_n(y))J_n(x,y)^{1/2}.$$ Then $\{\tilde u_n\}_{n\ge1}$ is a bounded sequence in $L^2(M\times M \backslash {\rm diag}, \mu\times \mu)$, and hence there exists a subsequence $\{\tilde u_{n_k}\}_{k\ge1}$, which converges to some element $\tilde u$ weakly in $L^2 (M\times M \backslash {\rm diag}, \mu\times \mu)$. We now claim that \begin{equation}\label{e:ff0}\tilde u(x,y)= (u(x)-u(y))J(x,y)^{1/2},\quad \mu\times \mu\mbox{-}a.e.\,\ (x,y) \text{ with }x\neq y.\end{equation}
To simplify the notation, without confusion in double integrals below we will omit the integral domain $M\times M\backslash {\rm diag}$. For any non-negative function $v\in C_c(M\times M \backslash{\rm diag})$ and for any $n_k$, we have \begin{equation*}\begin{split}
&\bigg|\iint \big[\tilde u(x,y)-(u(x)-u(y)) J(x,y)^{1/2} \big] v(x,y)\,\mu(dx)\,\mu(dy)\bigg|\\
&\leqslant \bigg|\iint\big[\tilde u(x,y)-\big(u_{n_k}(x)-u_{n_k}(y)\big) J_{n_k}(x,y)^{1/2}\big] v(x,y)\,\mu(dx)\,\mu(dy)\bigg|\\
&\quad + \bigg|\iint \big(u_{n_k}(x)-u_{n_k}(y)\big)\big(J_{n_k}(x,y)^{1/2}-J(x,y)^{1/2}\big)v(x,y)\,\mu(dx)\,\mu(dy)\bigg|\\
&\quad + \bigg|\iint \big[\big(u_{n_k}(x)-u_{n_k}(y)\big)-\big(u(x)-u(y)\big)\big]J(x,y)^{1/2}v(x,y)\,\mu(dx)\,\mu(dy)\bigg|\\ &=:I_{1,n_k}+I_{2,n_k}+I_{3,n_k}.\end{split} \end{equation*}
Firstly, since $\tilde u_n$ converges to $\tilde u$ weakly in $L^2(M\times M \backslash {\rm diag}, \mu\times \mu)$, we see that $\lim_{k\to \infty}I_{1,n_k}=0.$ Secondly, by using the Cauchy--Schwarz inequality and the fact that $\{u_{n_k}\}_{k\ge1} $ is a bounded sequence in $L^2(M,\mu)$, we derive that \begin{align*}I_{2,n_k}\leqslant &\left( \iint\big(u_{n_k}(x)-u_{n_k}(y)\big)^2 v(x,y)\,\mu(dx)\,\mu(dy)\right)^{1/2}\\ &\times \left(\iint\big(J_{n_k}(x,y)^{1/2}-J(x,y)^{1/2}\big)^2 v(x,y)\,\mu(dx)\,\mu(dy)\right)^{1/2}\\
\leqslant& \sqrt{2} \|u_{n_k}\|_2 \|v\|_\infty \\ &\times
\left(\sup_{x\in M} \int_{\{y\in M: (x,y)\in {\rm supp} v\}} \,\mu(dy)+\sup_{y\in M} \int_{\{x\in M: (x,y)\in {\rm supp} v\}} \,\mu(dx)\right)^{1/2}\\
&\times \left(\iint_{{\rm supp}v} |J_{n_k}(x,y)-J(x,y)|\,\mu(dx)\,\mu(dy)\right)^{1/2}, \end{align*}
where the right hand side of the above inequality converges to 0 as $k\to \infty$. Here, in the second inequality above, we used the elementary inequalities $(a-b)^2\leqslant 2(a^2+b^2)$ for all $a,b\in \mathds R$, and $|\sqrt{a}-\sqrt{b}|\leqslant \sqrt{|a-b|}$ for all $a,b\geqslant 0$, and in the last inequality, we used the fact that
$$\sup_{x\in M} \int_{\{y\in M: (x,y)\in {\rm supp} v\}} \,\mu(dy)+\sup_{y\in M} \int_{\{x\in M: (x,y)\in {\rm supp} v\}} \,\mu(dx)<\infty,$$ due to $v\in C_c(M\times M \backslash{\rm diag})$. Thirdly, for $I_{3,n_k}$, note that, by the Cauchy--Schwarz inequality and \eqref{e:ffee}, both
$$\phi(x):=\int_M J(x,y)^{1/2} v(x,y)\,\mu(dy),\quad x\in M$$
and
$$\psi(y):=\int_M J(x,y)^{1/2} v(x,y)\,\mu(dx),\quad y\in M$$
are in $L^2(M,\mu)$.
Hence, we see
\begin{align*}I_{3,n_k}\leqslant & \bigg| \int_M \big(u_{n_k}(x)-u(x)\big)\phi(x)\,\mu(dx)\bigg|+ \bigg| \int_M \big(u_{n_k}(y)-u(y)\big)\psi(y)\,\mu(dy)\bigg|\end{align*} goes to 0 as $k\to \infty$. Thus, we conclude that \eqref{e:ff0} holds.
Choose a sequence $\{v_k\}_{k\geq1}$ from $C_c(M\times M \backslash{\rm diag})$ such that $0\leqslant v_k\uparrow 1$ as $k\to \infty$. Letting $n\rightarrow\infty$, by Fatou's lemma, we deduce that, for any $k\geqslant 1$, \begin{align*} \liminf_{n\to \infty} D^n(u_n,u_n)\geqslant &\liminf_{n\to \infty} \iint \big(u_n(x)-u_n(y)\big)^2 J_n(x,y) v_k(x,y)\,\mu(dx)\,\mu(dy)\\ \geqslant &\iint \big(u(x)-u(y)\big)^2 J(x,y) v_k(x,y)\,\mu(dx)\,\mu(dy). \end{align*} Taking $k\to \infty$, by the monotone convergence theorem, we arrive at $$ \liminf_{n\to \infty} D^n(u_n,u_n)\geqslant D(u,u).$$
(2) Note that $\mathscr{C}$ is the common core of $(D^n,\mathscr{F}^n)$ for all $n\geqslant 1$ and $(D,\mathscr{F})$. Hence, by the monotone convergence theorem, $$\lim_{n\to \infty} D^n(u,u)= D(u,u),\quad u\in \mathscr{C}.$$ This proves that the condition $(b)'$ in Remark \ref{r:mos} (1) holds.
Combining both conclusions from (1) and (2), we prove the desired assertion. \end{proof}
Now, we are in a position to prove the main result in this section. \begin{theorem}\label{th1-23} {\bf (General jumping kernel case)}\,\, Let $(M,d,\mu)$ be a metric measure space, and $(D,\mathscr{F})$ be the non-local regular Dirichlet form defined in \eqref{nondi}. Suppose that \eqref{e:ack} and \eqref{e:ack1} hold. Then, for any $p\in (1,2]$, $\mathscr{H}_{\widetilde \nabla}$ is bounded in $L^p(M,\mu)$, i.e., there exists a constant $c_p>0$ such that, for every $f\in L^p(M,\mu)$,
$$\|\mathscr{H}_{\widetilde \nabla}f\|_{p}\leqslant c_p\|f\|_{p}.$$
\end{theorem}
\begin{proof} For each $n\geqslant 1$, denote by $(L_n, \mathscr{D}(L_n))$ and $(P_t^n)_{t\geq0}$ the generator and the semigroup associated with the Dirichlet form $(D^n,\mathscr{F}^n)$, respectively. According to \eqref{e:ack1} and Theorem \ref{th1}, for any $p\in (1,2]$, we can find a constant $c_p>0$ such that for all $n\ge1$ and $f\in L^p(M,\mu)$,
$$\|\mathscr{H}_{\widetilde \nabla, n}f\|_p\leqslant c_p\|f\|_p,$$ where $\mathscr{H}_{\widetilde \nabla, n}$ is defined by \eqref{eeefff} with $L_n$ in place of $L$ and \eqref{g-21} with the jumping kernel $J_n(x,dy)=J_n(x,y)\,\mu(dy)$; more precisely,
$$\mathscr{H}_{\widetilde \nabla, n}(f)(x)=\Big(\int_0^\infty\!\!\int_{\{y\in M: |P_t^nf|(x)\geqslant |P_t^nf|(y)\}}\big(P_t^nf(x)-P_t^nf(y)\big)^2J_n(x,dy)\,dt\Big)^{1/2}.$$ Note that, from the argument of Theorem \ref{th1}, the constant $c_p$ here is independent of $n\ge1$.
On the other hand, by Proposition \ref{P:mos} and Remark \ref{r:mos} (2), we know that for every $t>0$ and $f\in C_c(M)$, $P_t^nf$ converges to $P_tf$ in $L^2(M,\mu)$ as $n\rightarrow\infty$. Hence, there is a subsequence $\{P_t^{n_k}f\}_{k\geq1}$ which converges to $P_tf$ $\mu$-a.e. as $k\rightarrow\infty$. By the definition of $\mathscr{H}_{\widetilde \nabla,n}(f)$ and the assumption \eqref{e:ack1}, we obtain that $\mathscr{H}_{\widetilde \nabla,n_k}(f)\rightarrow \mathscr{H}_{\widetilde \nabla}(f)$ $\mu$-a.e. as $k\rightarrow\infty$. By using the Fatou lemma twice,
\begin{align*}&c_p\|f\|_p\\ &\geqslant \liminf_{n\to\infty} \int_M \mathscr{H}_{\widetilde \nabla, n}(f)^p(x)\,\mu(dx)\\
&=\liminf_{n\to\infty} \int_M \!\Big(\int_0^\infty\!\!\int_{\{y\in M: |P_t^nf|(x)\geqslant |P_t^nf|(y)\}}\big(P_t^nf(x)-P_t^nf(y)\big)^2J_n(x,dy)\,dt\Big)^{p/2}\,\mu(dx)\\
&\geqslant \int_M \!\Big(\liminf_{n\to\infty}\int_0^\infty\!\!\int_{\{y\in M: |P_t^nf|(x)\geqslant |P_t^nf|(y)\}}\big(P_t^nf(x)-P_t^nf(y)\big)^2J_n(x,dy)\,dt\Big)^{p/2}\,\mu(dx)\\
&\geqslant \int_M\Big(\int_{\{y\in M: |P_tf|(x)\geqslant |P_tf|(y)\}}\big(P_tf(x)-P_tf(y)\big)^2J(x,dy)\,dt\Big)^{p/2}\,\mu(dx) .\end{align*} Hence, there is a constant $c_p>0$ such that
$$\|\mathscr{H}_{\widetilde \nabla}(f)\|_p\leqslant c_p\|f\|_p\quad\mbox{for any } f\in C_c(M).$$
The general case for $f\in L^p(M,\mu)$ is accomplished by approximation since $C_c(M)$ is dense in $L^p(M,\mu)$ for all $1\leqslant p<\infty$ and by Fatou's lemma.
\end{proof}
\ \
It is easy to know that
\begin{equation}\label{bu2}\|\mathscr{H}_{\widetilde \nabla}f\|_{2}\leqslant\frac{\sqrt{2}}{2}\|f\|_{2}\quad\mbox{for all }0\leqslant f\in L^2(M,\mu).\end{equation} Indeed, letting $\{E_\lambda: 0\leqslant\lambda<\infty\}$ be the spectral representation of $L$, for any $h\in L^2(M,\mu)$, we have $$D(h,h)=\int_0^\infty \lambda \,d\langle E_\lambda h,E_\lambda h\rangle$$ and $$D(P_th, P_t h)=\int_{[0,\infty)} \lambda e^{-2\lambda t}\,d\langle E_\lambda h,E_\lambda h\rangle.$$
Then, for all $0\leqslant f\in L^2(M,\mu)$, \begin{align*}\|\mathscr{H}_{\widetilde \nabla}f\|_{2}^2&=\int_{[0,\infty)} D(P_tf, P_t f)\,dt=\int_{[0,\infty)} \lambda e^{-2\lambda t}\,dt \int_{[0,\infty)} \,d \langle E_\lambda f, E_\lambda f\rangle\\
&=\frac{1}{2}\int_{(0,\infty)} \,d \langle E_\lambda f, E_\lambda f\rangle\leqslant\frac{1}{2}\|f\|_{2}^2.\end{align*} However, for general $f\in L^2(M,\mu)$, we can not derive \eqref{bu2} as above. This shows that even $L^2$ boundedness of $\mathscr{H}_{\widetilde \nabla}$ seems to be non-trivial, which differs from the classic Littlewood--Paley theory in the case $p=2$.
In harmonic analysis, we are also interested in the so-called vertical Littlewood--Paley $\mathscr{G}$-function of the following form: for $f\in L^1(M,\mu)\cap L^\infty(M,\mu)$,
$$\mathscr{G}_{\widetilde \nabla}f(x)= \left(\int_0^\infty t |\widetilde \nabla e^{-t\sqrt{L}}f|^2(x)\,dt \right)^{1/2},\quad x\in M.$$ Inspired by the argument of \cite[Remark 1.3(ii)]{CDD}, we know that the function $\mathscr{G}_{\widetilde \nabla}f$ is dominated pointwise by $\mathscr{H}_{\widetilde \nabla}f$. Indeed, by using the fact $$\int_0^\infty e^{-u} u^{1/2}\,du=\frac{\sqrt{\pi}}{2},$$ and applying the formula $$e^{-t \sqrt{L}}=\frac{1}{\sqrt{\pi}}\int_0^\infty e^{-t^2L/(4u)} e^{-u} u^{-1/2}\,du,\quad t\geq0,$$ we deduce from Jensen's inequality, Fubini's theorem and the change-of-variables formula that
\begin{align*}(\mathscr{G}_{\widetilde \nabla}f)^2(x)&=\int_0^\infty t|\widetilde \nabla e^{-t\sqrt{ L}} f|^2(x)\,dt\\
&=\frac{1}{\pi}\int_0^\infty t\Big(\int_0^\infty |\widetilde \nabla e^{-t^2L/(4u)}f|(x)e^{-u}u^{-1/2}\,du\Big)^2\,dt\\
&\leqslant \frac{1}{\sqrt{\pi}}\int_0^\infty t \Big( \int_0^\infty |\widetilde \nabla e^{-t^2L/(4u)}f|^2(x)\,e^{-u}u^{-1/2}\,du\Big) \, dt\\
&=\frac{1}{\sqrt{\pi}}\int_0^\infty \Big(\int_0^\infty t|\widetilde \nabla e^{-t^2L/(4u)}f|^2(x)\,dt\Big)e^{-u}u^{-1/2}\,du\\
&=\frac{2}{\sqrt{\pi}} \int_0^\infty e^{-u}u^{1/2}\,du\int_0^\infty |\widetilde \nabla e^{-sL}f|^2(x)\,ds\\ &=(\mathscr{H}_{\widetilde \nabla}f)^2(x);
\end{align*} hence,
$$(\mathscr{G}_{\widetilde \nabla}f)(x)\leqslant (\mathscr{H}_{\widetilde \nabla}f)(x).$$ In particular, the $L^p$ boundedness of $\mathscr{H}_{\widetilde \nabla}$ implies the $L^p$ boundedness of $\mathscr{G}_{\widetilde \nabla}$.
\ \
At the end of this section, we make some comments on Theorem \ref{th1-23} and its proof. In $\mathds R^d$, the boundedness of $\mathscr{H}_{\widetilde \nabla}$ in $L^p(\mathds R^d,dx)$, $1<p\leqslant 2$, for L\'evy process $X$ has been proved in \cite[Theorem 4.1 and Lemma 4.5]{BBL}, when the process $X$ satisfies the Hartman--Wintner condition. (Note that such condition implies that the process $X$ has a transition density function $p_t(x)$ with respect to the Lebesgue measure such that $\lim_{t\to\infty}p_t(0)=0$.) The aforementioned approach differs from ours, and it is based on the Hardy--Stein identity (see \cite[Theorem 3.2 and (3.5)]{BBL}). It seems that such identity depends heavily on the characterization of L\'evy processes, and may not hold for general jump processes. The authors mentioned in the introduction section of their paper \cite{BBL} that --- \emph{The results should hold in a much more general setting, but the scope of the extension is unclear at this moment}.
\ \
As mentioned in Section \ref{section1}, it seems more natural to study the boundedness of the vertical Littlewood--Paley $\mathscr{H}$-function defined in terms of $\nabla$, that is, the $L^p$ boundedness of the operator
$$\mathscr{H}_{\nabla} f(x)=\left(\int_0^\infty |\nabla e^{-tL}f|^2(x)\,dt \right)^{1/2},\quad x\in M.$$ By the same argument for \eqref{bu2}, it holds true that $\|\mathscr{H}_{\nabla}f\|_{2}\leqslant\|f\|_{2}/\sqrt{2}$ for all $0\leqslant f\in L^2(M,\mu).$ However, \cite[Example 2]{BBL}, which is inspired by \cite[Page 165-166]{Ben}, shows that in general settings the operator $\mathscr{H}_{\nabla}$
may fail to be bounded on $L^p(M,\mu)$ if $1<p<2$. Thus, $\mathscr{H}_{\nabla}$ and $\mathscr{H}_{\widetilde\nabla}$ differ considerably. Another point is that, if we define $\mathscr{H}_{\widetilde\nabla_*}$ as $\mathscr{H}_{\widetilde\nabla}$ by using the modified gradient $|\widetilde\nabla f|_*$ defined by \eqref{g-2} instead of $|\widetilde\nabla f|$ defined by \eqref{g-21}, one may wonder whether $\mathscr{H}_{\widetilde\nabla_*}$ is bounded on $L^p(M,\mu)$ for $p\in (1,2]$ or not. The answer is no! Indeed, suppose that $\mathscr{H}_{\widetilde\nabla_*}$ is bounded on $L^p(M,\mu)$ for $p\in (1,2]$. Then, a simple change of $f$ into $-f$ will give us the boundedness of the operator $\mathscr{H}_{\nabla}-\mathscr{H}_{\widetilde\nabla_*}$, which forces that $\mathscr{H}_{\nabla}$ is bounded too. However, this is a contradiction.
In comparison with the discrete setting in \cite{Nick}, one key point of Theorem \ref{th1-23} is the boundedness of $\mathscr{H}_{\widetilde\nabla}$ in $L^p(M,\mu)$ for $p\in (1,2]$, not only in the particular cone $L^p_+(M,\mu):=\{f\geq0: f\in L^p(M,\mu)\}$. The reason is that the operator $\mathscr{H}_{\widetilde\nabla}$ does not enjoy the sublinear property. Also due to this, we need to define the pseudo-gradient $\Gamma_p$ for suitable signed function $f$; see \eqref{e:ger0}. By some simple calculation, we can deduce that, for any measurable function $f$ on $M$, \begin{align*}\label{absolute-control}
\mathds 1_{\{z\in M:|f(x)|\geqslant|f(z)|\}}(y)\big(f(x)-f(y)\big)^2 \leqslant & 4\mathds 1_{\{z\in M:f^+(x)\geqslant f^+(z)\}}(y)\big(f^+(x)-f^+(y)\big)^2\\ & + 4\mathds 1_{\{z\in M:f^-(x)\geqslant f^-(z)\}}(y)\big(f^-(x)-f^-(y)\big)^2 \end{align*}holds for all $x,y\in M$. So, it should be a feasible way to prove the boundedness of $\mathscr{H}_{\widetilde\nabla}$ in $L^p(M,\mu)$ from its boundedness in $L^p_+(M,\mu)$.
\section{Littlewood--Paley--Stein estimates for $2\leqslant p<\infty$} Recall that $(M,d,\mu)$ is a metric measure space, $(D,\mathscr{F})$ given by \eqref{nondi} is a regular Dirichlet form, $(L,\mathscr{D}(L))$ and $(P_t)_{t\ge0}:=(e^{-tL})_{t\ge0}$ are the corresponding $L^2$-generator and $L^2$-semigroup, respectively. Associated with the regular Dirichlet form $(D,\mathscr{F})$ on $L^2(M,\mu)$ there is a symmetric Hunt process $X=\{X_t,t\ge0, \mathds P^x, x\in M\backslash\mathscr{N}\}$. Here $\mathscr{N}$ is a properly exceptional set for $(D,\mathscr{F})$ in the sense that $\mu(\mathscr{N})=0$ and $\mathds P^x(X_t\in \mathscr{N}\textrm{ for some }t>0)=0$ for all $x\in M\backslash\mathscr{N}.$ See \cite{CF, FOT} for more details.
\ \
Throughout this section, we make the following assumptions.
\begin{itemize} \item[$(A1)$] For any $ f\in C_c(M)$, the function $(t,x)\mapsto P_tf(x)$ is continuous on $(0,\infty)\times M$.
\item[$(A2)$] The process $X$ has a transition density function $p_t(x,y)$ with respect to the reference measure $\mu$, i.e., for any $t>0$, $x\in M\backslash\mathscr{N}$ and any Borel set $B\subset M$, $$\mathds P^x(X_t\in B)=\int_B p_t(x,y)\,\mu(dy).$$
\item[$(A3)$] The process $X$ is conservative, i.e., for any $t>0$ and $x\in M\backslash\mathscr{N}$, $$\int_M p_t(x,y)\,\mu(dy)=1.$$
\item[$(A4)$] There exist a $\sigma$-finite measure space $(U,\mathscr{U},\nu)$ and a function $k:$ $M\times U\rightarrow M$ such that for any Borel set $B\subset M$ and $\mu$-a.e. $x\in M$, \begin{equation}\label{ss3}\nu\left\{z\in U: k(x,z)\in B\right\}=J(x,B).\end{equation} \end{itemize}
We make some comments on assumptions above. Firstly, according to \cite[Chapter 1, Lemma 1.4, p.\ 5]{BSW}, assumption $(A1)$ holds if the semigroup $(P_t)_{t\ge0}$ enjoys the $C_\infty$-Feller property; that is, for any $t>0$ and $f\in C_\infty(M)$, $P_tf\in C_\infty(M)$, and $\lim_{t\to 0}\|P_tf-f\|_\infty =0$, where $C_\infty(M)$ denotes the set of continuous functions which varnish at infinity. Secondly, there are already a few works on the conservativeness of processes generated by non-local Dirichlet forms on metric measure spaces; see e.g. \cite{MUW} and the references therein. Thirdly, assumption $(A4)$ is our technique condition. When $M=\mathds R^d$, one can take $U=\mathds R^n$ and $\nu(dz)= |z|^{-n-1}\,dz$ with $n\ge2$, and find a measurable function $k:\mathds R^d\times \mathds R^n\to \mathds R^d$ such that \eqref{ss3} is satisfied; see \cite[Chapeter 3, Theorem 3.2.5]{Sto}. Hence, assumption $(A4)$ always holds in the Euclidean space. As a general result on construction of the coefficient $k(x,z)$ in \eqref{ss3}, we refer to El Karoui and Lepeltier \cite{KL}, where they constructed $k(x,z)$ under the condition that $U$ is a Lusin space and $\nu$ is a $\sigma$-finite diffusive measure on $U$ with infinite total mass.
\ \
For fixed $f\in C_c(M)$ and $T>0$, let $$H_t=P_{T-t}f(X_t)-P_Tf(X_0),\quad 0\leqslant t\leqslant T.$$ Denote by $(\mathscr{F}_t)_{t\ge0}$ the natural filtration of the process $X$. Then, we have \begin{lemma}\label{lemm31} Under the assumption $(A1)$, $\{H_t, \mathscr{F}_t\}_{0\leqslant t\leqslant T}$ defined above is a martingale starting at $0$, and for any $0\leqslant t\leqslant T$, \begin{equation} \label{eek01}[H]_t= \int_0^t\int_M\left( P_{T-s} f(y)- P_{T-s}f(X_{s-})\right)^2\,J(X_{s-},dy)\,ds, \end{equation} where $[H]_t$ is the quadratic variation of $H_t$. \end{lemma}
The statement above for symmetric L\'evy processes can be obtained directly via the It\^{o} formula; see \cite[Section 4]{BBL}. However, since the It\^{o} formula is not available in the present setting, we will use a different approach.
\begin{proof}[Proof of Lemma $\ref{lemm31}$] For any $0\leqslant s\leqslant t\leqslant T$, by the Markov property, $$P_{T-t}f(X_t)=\mathds E^{X_t} f(X_{T-t})=\mathds E (f(X_T)|\mathscr{F}_t),$$ and so
\begin{align*}\mathds E(H_t|\mathscr{F}_s)=&\mathds E(P_{T-t}f(X_t)-P_Tf(X_0)|\mathscr{F}_s)=\mathds E(P_{T-t}f(X_t)|\mathscr{F}_s)-P_Tf(X_0)\\
=&\mathds E(\mathds E (f(X_T)|\mathscr{F}_t)|\mathscr{F}_s)-P_Tf(X_0)=\mathds E(f(X_T)|\mathscr{F}_s)-P_Tf(X_0)\\ =&P_{T-s} f(X_s)-P_T f(X_0)=H_s.\end{align*} This proves that $\{H_t, \mathscr{F}_t\}_{0\leqslant t\leqslant T}$ is a martingale.
For any $x\in M$ and $0\leqslant t\leqslant T$, we have \begin{align*}\mathds E^x(H_t^2)=& \mathds E^x\big(P_{T-t}f(X_t)-P_Tf(X_0)\big)^2\\ =&\mathds E^x(P_{T-t}f)^2(X_t)-2P_Tf(x)\mathds E^xP_{T-t}f(X_t)+(P_Tf)^2(x)\\ =& P_t(P_{T-t}f)^2(x)- (P_Tf)^2(x).\end{align*} Then, for any $x\in M$ and $0\leqslant s\leqslant t\leqslant T$, \begin{equation}\label{e:fffee}\begin{split}
\mathds E^x(H_t^2-H_s^2)=& P_t(P_{T-t}f)^2(x)- P_s(P_{T-s}f)^2(x)\\ =& \int_s^t \frac{ d(P_u(P_{T-u} f)^2)(x)}{du}\,du\\ =&\int_s^t\Big(-LP_u (P_{T-u}f)^2(x)+P_u(2P_{T-u}f \cdot L P_{T-u}f)(x)\Big)\,du \\
=& \int_s^t P_u\Big(-L(P_{T-u} f)^2+2P_{T-u} f\cdot LP_{T-u}f\Big)(x)\,du\\
=&2\int_s^t P_u\Gamma (P_{T-u}f )(x)\,du\\
=&2\mathds E^x \Big( \int_s^t \Gamma (P_{T-u}f)(X_u)\,du\Big),\end{split}\end{equation}
where in the penultimate equality above we used the fact that
$$\Gamma(f)= \frac{1}{2}\left(2fLf - L(f^2)\right).$$ See e.g. \cite[Theorem (3.7)]{CKS}. \eqref{e:fffee} together with the Markov property in turn yields that \begin{equation}\label{eek00}\Big\{H_t^2 - 2\int_0^t \Gamma (P_{T-u}f)(X_u)\,du, \mathscr{F}_t\Big\}_{\ 0\leqslant t\leqslant T}\end{equation} is a martingale. Denote by $\langle H\rangle_t$ and $[H]_t$ the predicable quadratic variation and the quadratic variation of $H_t$, respectively.
Thus, according to \eqref{eek00} and \cite[Chapter 4, Theorem 4.2, p.\ 38]{JS},
$$\langle H\rangle _t=2\int_0^t \Gamma (P_{T-u}f)(X_u)\,du, \quad 0\leqslant t\leqslant T.$$
Furthermore, under assumption $(A1)$ and by the fact that $t\mapsto X_t$ is quasi-left-continuous (since $X$ is a Hunt process enjoying the strong Markov property), $\{H_t,\mathscr{F}_t\}_{0\leqslant t\leqslant T}$ is a martingale which has a continuous version, see again \cite[Chapter 4, Theorem 4.2, p.\ 38]{JS}. Hence, by \cite[Chapter 2, Definition 2.25, p.\ 22; Chapter 4, Theorem 4.52, p.\ 55]{JS}, for any $0\leqslant t\leqslant T$, \begin{align*}[H]_t=&\langle H\rangle _t=2\int_0^t \Gamma (P_{T-u}f)(X_u)\,du\\ =&\int_0^t\int_M\big( P_{T-u} f(y)- P_{T-u}f(X_{u-})\big)^2\,J(X_{u-},dy)\,du.\end{align*} The proof is complete. \end{proof}
Next, we will make use of the space-time parabolic martingale $\{H_t, \mathscr{F}_t\}_{0\leqslant t\leqslant T}$ defined above with $T\in(0,\infty]$ to prove the boundedness of Littlewood--Paley functions in $L^p(M,\mu)$ for $2\leqslant p<\infty$. We mainly follow the approach of \cite[Section 4]{BBL} with necessary modifications.
First, we note that, under assumption $(A3)$, one can rewrite \eqref{eek01} for the quadratic variation $[H]_t$ of the martingale $H_t$ as \begin{equation}\label{eek02} [H]_t= \int_0^t\int_M\big( P_{T-s} f(k(X_{s-},y))- P_{T-s}f(X_{s-})\big)^2\,\nu(dy)\,ds,\quad 0\leqslant t\leqslant T. \end{equation}
Second, we need to define the Littlewood--Paley function $G$, which can be regard as the conditional expectation of the quadratic variation $[H]_t$. For $f\in L^1(M,\mu)\cap L^\infty(M,\mu)$, define
$$Gf(x)=\left(\int_0^\infty\!\! \int_M\!\int_M |P_t f(k(z,y))-P_tf(z)|^2p_t(x,z)\,\mu(dz)\,\nu(dy)\,dt\right)^{1/2},$$ and
$$G_{T}f(x)=\left(\int_0^T \!\!\int_M\!\int_M |P_t f(k(z,y))-P_tf(z)|^2p_t(x,z)\,\mu(dz)\,\nu(dy)\,dt \right)^{1/2}.$$ Clearly, $\lim_{T\to \infty}G_{T} f(x)=Gf(x)$ for all $x\in M$. \begin{lemma}\label{lem-1} Under assumptions $(A1)$, $(A2)$, $(A3)$ and $(A4)$, for any $f\in C_c(M)$, $x\in M$ and $T>0$,
$$(G_{T} f)^2(x)=\int_M \mathds E^z ([H]_T|X_T=x)p_T(z,x)\,\mu(dz).$$ \end{lemma}
\begin{proof}The proof is almost the same as that of \cite[(4.5)]{BBL}, and we present it here for the sake of completeness. By \eqref{eek02}, we have
\begin{align*}
&\int_M\mathds E^z ([H]_T|X_T=x)p_T(z,x)\,\mu(dz)\\
&=\int_M\mathds E^z\left( \int_0^T\int_M\big( P_{T-s} f(k(X_{s-},y))- P_{T-s}f(X_{s-})\big)^2\,\nu(dy)\,ds\Big| X_T=x\right)\\ &\qquad\qquad\qquad\times p_T(z,x)\,\mu(dz)\\ &=\int_M \bigg(\int_0^T\int_M\frac{p_s(z,w)p_{T-s}(w,x)}{p_T(z,x)} \\
&\qquad\qquad\qquad\times \int_M |P_{T-s} f(k(w, y))-P_{T-s}f(w)|^2\,\nu(dy)\,\mu(dw)\,ds\bigg)p_T(z,x)\,\mu(dz)\\
&=\int_0^T\int_M p_{T-s}(w,x) \int_M|P_{T-s} f(k(w, y))-P_{T-s}f(w)|^2\,\nu(dy)\,\mu(dw)\,ds\\
&=\int_0^T \int_M \int_M|P_{T-s} f(k(w, y))-P_{T-s}f(w)|^2p_{T-s}(x,w)\,\mu(dw)\,\nu(dy)\,ds\\ &=(G_{T}f)^2(x), \end{align*} where in the third equality we used the fact that $$\int_M p_s(z,w)\,\mu(dz)= \int_M p_s(w,z)\,\mu(dz)=1\quad\mbox{for any }w\in M\backslash\mathscr{N}, $$ due to the symmetry of $p_t(x,y)$ and the conservativeness of the process $X$, and in the fourth equality we used symmetry again. \end{proof}
\begin{lemma}\label{lem-2} Under assumptions $(A1)$, $(A2)$ and $(A4)$, for any $f\in C_c(M)$ and $x\in M$, $$(\mathscr{H}_\nabla f)(x)\leqslant Gf(x).$$ \end{lemma} \begin{proof} For any $f\in C_c(M)$ and $x\in M$, using the Cauchy--Schwarz inequality, the property of the semigroup $(P_t)_{t\ge0}$ and \eqref{ss3}, we get
\begin{align*}(G f)^2(x)=&\int_0^\infty \!\!\int_M\! \int_M|P_{t}f(k(z,y))-P_{t}f(z)|^2p_{t}(x,z)\,\mu(dz)\,\nu(dy)\,dt\\
\geqslant&\int_0^\infty \int_M\! \left(\int_M |P_{t}f(k(z,y))-P_{t}f(z)|p_{t}(x,z)\,\mu(dz)\right)^2\,\nu(dy)\,dt\\
=&\int_0^\infty \int_M\! \Big( P_t |P_{t}f(k(\cdot,y))-P_{t}f(\cdot)| (x)\Big)^2\,\nu(dy)\,dt\\
\geqslant &\int_0^\infty \int_M\! \big(|P_{2t}f(k(\cdot,y))-P_{2t}f(\cdot)| (x)\big)^2\,\nu(dy)\,dt\\
=&\int_0^\infty \left(\int_M\! \big(|P_{2t}f(z)-P_{2t}f(\cdot)|\big)^2\,J(\cdot, dz)\right) (x)\,dt\\ =&2\int_0^\infty \Gamma(P_{2t}f ) (x)\,dt=\int_0^\infty \Gamma(P_{t}f ) (x)\,dt\\
= &\int_0^\infty|\nabla P_t f|^2(x)\,dt=(\mathscr{H}_\nabla f)^2(x). \end{align*} This proves the desired assertion. \end{proof}
Now, we are in a position to present the main result in this section. \begin{theorem}\label{thp} Let $(M,d,\mu)$ be a metric measure space, and $(D,\mathscr{F})$ be the non-local regular Dirichlet form defined in \eqref{nondi}. Suppose that $(A1)$, $(A2)$, $(A3)$ and $(A4)$ hold. Then, for any $p\in [2,\infty)$,
$\mathscr{H}_\nabla$ is bounded in $L^p(M,\mu)$, i.e., there exists a constant $C_p>0$ such that, for every $f\in L^p(M,\mu)$,
$${\|\mathscr{H}_\nabla f\|_{p}\leqslant C_p\|f\|_{p}.}$$
\end{theorem}
\begin{proof}
Let $p\geqslant 2$. For any $f\in C_c(M)$,
by Lemma \ref{lem-1} and
by applying
Jensen's inequality twice, we get
\begin{align*}\int_{M} (G_T f)^p(x)\,\mu(dx)&\leqslant \int_{M}\!\!\int_{M} \left(\mathds E^z([H]_T|X_T=x)\right)^{p/2} p_T(z,x)\,\mu(dz)\,\mu(dx)\\
&\leqslant \int_{M}\!\int_{M} \mathds E^z([H]_T^{p/2}|X_T=x)p_T(z,x)\,\mu(dz)\,\mu(dx)\\ &=\int_{M} \mathds E^z([H]_T^{p/2})\,\mu(dz). \end{align*}
By the Burkholder--Davis--Gundy inequality (see e.g. \cite[Page 234]{Ap}), we have
\begin{align*}
\mathds E^z([H]_T^{p/2})&\leqslant C_p'\mathds E^z|H_T|^p=C_p'\mathds E^z\big(|f(X_T)-P_Tf(X_0)|^p\big)\\
&\leqslant 2^pC_p' P_T|f|^p(z),
\end{align*}
where in the last inequality we have used the elementary fact that
$$(a+b)^p\leqslant 2^{p-1}(a^p+b^p),$$
for all $a,b\ge0$ and $p\geq1$. Thus, by the contraction property of the semigroup $(P_t)_{t\ge0}$ on $L^p(M,\mu)$, we arrive at
$$\int_{M} (G_T f)^p(x)\,\mu(dx)\leqslant 2^pC_p'\int_{M} P_T|f|^p(x)\,\mu(dx)\leqslant 2^pC_p'\|f\|_{p}^p,$$ which along with the monotone convergence theorem yields that
$$ \int_{M} (Gf)^p(x)\,\mu(dx)\leqslant 2^pC_p'\|f\|_{p}^p.$$ Combining this with Lemma \ref{lem-2}, we obtain that
$$\|\mathscr{H}_\nabla f\|_p\leqslant C_p\|f\|_p\quad\mbox{for every }f\in C_c(M).$$ For every $f\in L^p(M,\mu)$, since $C_c(M)$ is dense in $L^p(M,\mu)$ for all $1\leqslant p<\infty$, we may choose a sequence $\{f_n\}_{n\geq1}$ from $C_c(M)$ such that $f_n$ converges to $f$ in $L^p(M,\mu)$ as $n\rightarrow\infty$. Then, applying Fatou's lemma, we obtain the desired conclusion.
\end{proof}
Finally, we present the proof of Example \ref{exm2}. \begin{proof}[Proof of Example $\ref{exm2}$] According to comments after assumptions in the beginning of this section, we only need to verify assumptions $(A1)$---$(A3)$. According to \cite[Proposition 3.3]{BKK}, the associated resolvent of $(D,\mathscr{F})$ is H\"{o}lder continuous, which entails that the $L^2$-semigroup $(P_t)_{t\ge0}$ generated by the Dirichlet form $(D,\mathscr{F})$ is a $C_\infty$-Feller semigroup; see \cite[Proposition 4.3 and Corollary 6.4]{RU}. Thus, $(A1)$ holds. From \eqref{ed4}, one can easily deduce that the Nash type inequality holds for $(D,\mathscr{F})$, which implies that the associated process $X$ enjoys the transition density function (see \cite{BKK} for more details); that is, $(A2)$ is also satisfied. $(A3)$ immediately follows from \cite[Theorem 1.1]{MUW}. Therefore, we can obtain the desired assertion from Theorem \ref{thp}.\end{proof}
\subsection*{Acknowledgment} \hskip\parindent\!\!\! The second author would like to thank Professor Kazuhiro Kuwae for very helpful comments on the Mosco convergence of Dirichlet forms and the proof of Lemma \ref{lemm31}. The research of Huaiqian Li is supported by National Natural Science Foundation of China (Nos.\ 11401403, 11571347). The research of Jian Wang is supported by National Natural Science Foundation of China (No.\ 11522106), the Fok Ying Tung Education Foundation (No.\ 151002) and the Program for Probability and Statistics: Theory and Application (No.\ IRTL1704).
\end{document} |
\begin{document}
\title{\textsc{Heterofusion: fusing genomics data of different measurement scales}}
\author{A.K. Smilde*$^{1}$, Y. Song$^1$, J.A. Westerhuis$^1$, H.A.L. Kiers$^2$, \\N. Aben$^3$, L.F.A. Wessels$^3$}
\maketitle
\begin{tabular}{l} $^1$Biosystems Data Analysis, Swammerdam Institute for Life Sciences, \\ University of Amsterdam, Amsterdam, The Netherlands.\\ $^2$Heymans Institute, University of Groningen, Groningen, The Netherlands.\\ $^3$Oncode Institute, Netherlands Cancer Institute, Amsterdam, The Netherlands.\\
$^*$Corresponding author (a.k.smilde@uva.nl)\\ \end{tabular}
\section{Abstract}
In systems biology, it is becoming increasingly common to measure biochemical entities at different levels of the same biological system. Hence, data fusion problems are abundant in the life sciences. With the availability of a multitude of measuring techniques, one of the central problems is the heterogeneity of the data. In this paper, we discuss a specific form of heterogeneity, namely that of measurements obtained at different measurement scales, such as binary, ordinal, interval and ratio-scaled variables. Three generic fusion approaches are presented of which two are new to the systems biology community. The methods are presented, put in context and illustrated with a real-life genomics example.\\
\noindent Keywords: data fusion, data integration, low-level fusion, measurement scales, heterogeneous data
\section{Introduction}
\subsection{General}
With the availability of comprehensive measurements collected in multiple related data sets in the life sciences, the need for a simultaneous analysis of such data to arrive at a global view on the system under study is of increasing importance. There are many ways to perform such a simultaneous analysis and these go also under very different names in different areas of data analysis: data fusion, data integration, global analysis, multi-set or multi-block analysis to name a few. We will use the term \emph{data fusion} in this paper.\\
\noindent Data fusion plays an especially important role in the life sciences, e.g., in genomics it is not uncommon to measure gene-expression (array data or RNA-sequencing (RNAseq) data), methylation of DNA and copy number variation. Sometimes, also proteomics and metabolomics measurements are available. All these examples serve to show that having methods in place to integrate these data is not a luxury anymore.
\subsection{Types of data fusion}
Without trying to build a rigorous taxonomy of data fusion it is worthwhile to distinguish several distinctions in data fusion. The first distinction is between model-based and exploratory data fusion. The former uses background knowledge in the form of models to fuse the data; one example being genome-scale models in biotechnology \citep{Zimmermann2017}. The latter does not rely on models, since these are not available or poorly known, and thus uses empirical modeling to explore the data. In this paper, we will focus on exploratory data fusion.\\
\noindent The next distinction is between low-, medium-, and high-level fusion \cite{Steinmetz1999}. In low-level fusion, the data sets are combined at the lowest level, that is, at the level of the (preprocessed) measurements. In medium-level fusion, each separate data set is first summarized, e.g., by using a dimension reduction method or through variable selection. The reduced data sets are subsequently subjected to the fusion. In high-level fusion, each data set is used for prediction or classification of an outcome and the prediction or classification results are then combined, e.g, by using majority voting \citep{Doeswijk2011}. In machine learning this is known as early, intermediate and late integration. All these types of fusions have advantage and disadvantages which are beyond the scope of this paper. In this paper, we will focus on low- and medium-level fusion.\\
\noindent The final characteristic of data fusion relevant for this paper is heterogeneity of the data sets to be fused. Different types of heterogeneity can be distinguished. The first one is the type of \emph{data}, such as metabolomics, proteomics and RNAseq data in genomics. Clearly, these data relate to different parts of the biological system. The second one is the type of \emph{measurement-scale} in which the data are measured that are hoing to be fused. In genomics, an example is a data set where gene-expressions are available and mutation data in the processed form of Single Nucleotide Variants (SNVs). The latter are binary data and gene-expression is quantitative data. They are clearly measured at a different scale. Ideally, data fusion methods should consider the scale of such measurements and this will be the topic of this paper.
\subsection{Types of measurement scales}
The history of measurement scales goes back a long time. A seminal paper drawing attention to this issue appeared in the 40-ties \citep{Stevens1946}. Since then numerous papers, reports and books have appeared \citep{Suppes1962,Krantz1971,Narens1981,Narens1986,Luce1987,Hand1996}. The measuring process assigns numbers to aspects of objects (an \textit{empirical system}), e.g, weights of persons. Hence, measurements can be regarded as a mapping from the empirical system to numbers, and scales are properties of these mappings.\\
\noindent In measurement theory, there are two fundamental theorems \citep{Krantz1971}: the representation theorem and the uniqueness theorem. The \textit{representation theorem} asserts the axioms to be imposed on an empirical system to allow for a homomorphism of that system to a set of numerical values. Such a homomorphism into the set of real numbers is called a scale and thus represents the empirical system. A scale possesses \textit{uniqueness} properties: we can measure the weight of persons in kilograms or in grams, but if one person weighs twice as much as another person, this ratio holds true regardless the measurement unit. Hence, weight is a so-called ratio-scaled variable and this ratio is unique. The transformation of measuring in grams instead of kilograms is called a \textit{permissible} transformation since it does not change the ratio of two weights. For a ratio-scaled variable, only similarity transformations are permissible; i.e. $\widetilde{x}=\alpha x; \alpha>0$ where $x$ is the variable on the original scale and $\widetilde{x}$ is the variable on the transformed scale. This is because \begin{equation}\label{eRatio}
\frac{\widetilde{x_i}}{\widetilde{x_j}}=\frac{\alpha x_i}{\alpha x_j}=\frac{x_i}{x_j}. \end{equation} Note that this coincides with the intuition that the unit of measurement is immaterial.\\
\noindent The next level of scale is the interval-scaled measurement. The typical example of such a scale is degrees Celsius and the permissible transformation is affine; i.e. $\widetilde{x}=\alpha x +\beta; \alpha>0$. In that case, the ratio of two intervals is unique because \begin{equation}\label{eInterval}
\frac{\widetilde{x_i}-\widetilde{x_j}}{\widetilde{x_k}-\widetilde{x_l}}=\frac{(\alpha x_i + \beta)-(\alpha x_j + \beta)}{(\alpha x_k + \beta)-(\alpha x_l + \beta)}=\frac{\alpha (x_i-x_j)}{\alpha (x_k-x_l)}=\frac{x_i-x_j}{x_k-x_l}. \end{equation} Stated differently, the zero point and the unit are arbitrary on this scale.\\
\noindent Ordinal-scaled variables represent the next level of measurements. Typical examples are scales of agreement in surveys: strongly disagree, disagree, neutral, agree and strongly agree. There is a rank-order in these answers, but no relationship in terms of ratios or intervals. The permissible transformation of an ordinal-scale is a monotonic increasing transformation since such transformations keep the order of the original scale intact.\\
\noindent Nominal-scaled variables are next on the list. These variables are used to encode categories and are sometimes also called categorical. Typical example are gender, race, brands of cars and the like. The only permissible transformation for a nominal-scaled variable is the one-to-one mapping. A special case of a nominal-scaled variable is the binary (0/1) scale. Binary data can have different meanings; they can be used as categories (e.g. gender) and are then nominal-scale variables. They can also be two points on a higher-level scale, such as absence and presence (e.g. for methylation data).\\
\noindent The above four scales are the most used ones but others exists \citep{Suppes1962,Krantz1971}. Counts, e.g., have a fixed unit and are therefore sometimes called absolute-scaled variables \citep{Narens1986}. Another scale is the one for which the power transformation is permissible; i.e. $\widetilde{x}=\alpha x^\beta; \alpha, \beta>0$ which is called a log-interval scale because a logarithmic transformation of such a scale results in an interval-scale. An example is density \citep{Krantz1971}. Sometimes the scales are lumped in quantitative (i.e. ratio and interval) and qualitative (ordinal and nominal) data.\\
\noindent An interesting aspect of measurement scales is to what extent meaningful statistics can be derived from such scales (see Table 1 in \citep{Stevens1946}). A prototypical example is using a mean of a sample of nominal-scaled variables which is generally regarded as being meaningless. This has also provoked a lot of discussion \citep{Adams1965,Hand1996} and there are nice counter-examples of apparently meaningless statistics that still convey information about the empirical system \citep{Michell1986}. As always, the world is not black or white.
\subsection{Motivating example}
Examples of fusing data of different measurement scales are abundant in modern life science research. We will first give a short description of modern measurements in genomics that will illustrate this. In a sample extracted from biological systems (e.g. cells) it is possible to measure the mRNA molecules. This is done nowadays with RNAseq techniques and in essence the mRNA are counts per volume, hence, a concentration. Epigenetics concerns, amongst other, the methylation of some of the sites of a DNA molecule and is in essence a binary variable (yes/no methylated at a given location of the DNA). Another feature in genetics is whether a location of the DNA is mutated, a phenomenon called SNVs (single nucleotide variants), which is also binary. Lastly, there are Copy Number Variations (CNVs) of genes on the genome which is in essence a (limited) number of counts and sometimes expressed as Copy Number Abberations (CNA) with a binary coding (no: normal number of copies, yes: aberrant number of copies). If we move to the field of metabolomics and proteomics, then most of the measurements are relative intensities and in some cases - when calibration lines have been made - concentrations which are ratio-scaled.\\
\noindent The above exposition clearly shows that if we want to fuse different types of genomics data, or fuse genomics data with metabolomics and/or proteomics then there is a problem of different measurement scales. This problem is aggravated by the fact that some of this data is very high-dimensional. SNP and methylation data can contain 100.000 features or variables, RNAseq data has usually around 20.000 genes. Shotgun proteomics data (based on LC-MS or LC-MS/MS) can also easily contain 100.000 features. Hence, in many cases dimension reduction has to take place, asking for methods to deal properly with the corresponding measurement scale. For some of the methods to be discussed in this paper there are already examples in the literature. There are examples of the use of the parametric approach using latent variables \citep{Shen2009,Mo2013} and also of the optimal scaling approach \citep{Wietmarschen2011,Wietmarschen2012}. For the third approach to be discussed, we have not found examples yet in the life sciences. We will come back to these examples in Section \ref{Discussion}.
\subsection{Goal of the paper}
In this paper, we describe low- and mid-level fusion ideas of data of different measurement scales. We will restrict ourselves to data sharing the object mode. Mid-level fusion first selects variables and then is subjected to the methods described below. These methods can be applied in different fields of science, but we will illustrate them by using a genomics example.\\
\noindent We think this paper is needed since the different methods originate from different fields of data analysis, psychometrics and bioinformatics with limited cross-talk between those fields; we will try to fill this gap. Moreover, there are relationships between the methods and this might help in selecting the proper method for a particular application. Hence, we will also discuss the properties of the different methods.\\
\noindent We will select and discuss methods that provide coordinates of the objects that can be used for plotting and visualizing the relationships between the objects. Moreover, we think it is also worthwhile to consider methods that generate importance values for the variables in the different blocks since this will facilitate interpretation of the results in substantive terms.
\section{Theory}
\subsection{Three basic ideas}
We will describe three basic ideas that can be used for fusion of data of different measurement scales on a conceptual level. A more detailed explanation is given in following Sections. One of these methods is parametric and thus depends on distributions \citep{Mo2013}. The other two methods are non-parametric and based on concepts of representation matrices \citep{Zegers1986c,Kiers1989} and optimal scaling \citep{Gifi1990}.\\
\noindent The first idea is illustrated in Figure \ref{FigureRM} \citep{Kiers1989}. Suppose we have three blocks of data, the first block ($\mathbf{X}_1$) contains ratio-scaled data, the second block ($\mathbf{X}_2$) binary data and the third block ($\mathbf{X}_3$) categorical data with each of the $J_3$ variables having four categories (labeled A, B, C and D). Each variable in each block is represented by an $I \times I$ representation matrix (to be explained later). Then these representation matrices can be stacked and the resulting three-way array can be analyzed by a suitable three-way method using $R$ components giving coordinates for the objects and weights for the variables.\\
\begin{figure}
\caption{\footnotesize Heterofusion using representation matrices (see text).}
\label{FigureRM}
\end{figure}
\noindent The second idea is illustrated in Figure \ref{FigureOS} \citep{Gifi1990,Michailidis1998}. The original matrices are subjected to optimal scaling and the fusion problem is solved as one global optimization problem (to be explained later). The idea of optimal scaling goes back already to R. Fisher and nice introductions are available \citep{Young1981}. For the first block, the variables remain the same but for the second and third block these variables are (optimally) transformed. Using optimal scaling, the three blocks are made comparable and are analyzed simultaneously by a multiblock method (e.g. Simultaneous Component Analysis or Consensus PCA) giving $R$ coordinates for the objects (the $I \times R$ matrix) and loadings (the ($J_1 \times R$), ($J_2 \times R$) and ($J_3 \times R$) matrices) for the transformed variables.\\
\begin{figure}
\caption{\footnotesize Heterofusion using optimal scaling (see text).}
\label{FigureOS}
\end{figure}
\noindent The third idea relies on the explicit use of the $R$ latent variables collected in $\mathbf{Z}$ (see Figure \ref{FigurePM}) \citep{Mo2013}. These latent variables are then thought to generate the manifest variables in the different blocks using different distributions. For the ratio-scaled block, a regression model is assumed based on the normal distribution and with parameters $\boldsymbol{\alpha_{j1}}$ and $\boldsymbol{\beta_{j1}}$. For the binary block, a logit or probit model is assumed with parameters $\boldsymbol{\alpha_{j2}}$ and $\boldsymbol{\beta_{j2}}$. The final - categorical - block is modeled by a multilogit model with parameters $\boldsymbol{\alpha_{j3c}}$ and $\boldsymbol{\beta_{j3c}}$ where $c=A,B,C,D$.\\
\begin{figure}
\caption{\footnotesize Heterofusion using parametric models (see text).}
\label{FigurePM}
\end{figure}
\noindent We will use the following conventions for notations. A vector $\mathbf{x}$ is a bold lowercase and a matrix ($\mathbf{X}$) a bold uppercase. Running indices will be used for samples ($i$=1,\ldots ,$I$) with $I$ is the number of samples; we will use likewise the indices $k$=1,\ldots ,$K$ for the data blocks; variables within a data block are indexed by $j_k$=1,\ldots ,$J_k$ and we will use $r$=1,\ldots ,$R$ as an index for latent variables or components.\\
\subsection{Representation matrices approaches}
\subsubsection{Representation matrices}
\noindent \textsc{Idea of representation matrices.}\\
\noindent Suppose we have a data matrix $\mathbf{X} (I \times J)$ with columns $\mathbf{x}_j$ containing the scores of the objects on variable $j$. Such a score can be a ratio-scaled value, but can also be a binary value, a categorical value or an ordinal-scaled value. A representation operator works on this vector and produces a representation matrix which serves as a building block to calculate associations between variables and to analyze several variables simultaneously \citep{Zegers1986c, Kiers1989}. Such a representation matrix can be a vector ($I \times 1$), a rectangular matrix ($I \times R; R<I$) or a square matrix ($I \times I$). Let $\mathbf{S}_j$ and $\mathbf{S}_k$ be the representation matrices for variables $j$ and $k$, respectively, then a general equation of the association between variables $j$ and $k$ is \begin{equation}\label{eAss1}
q_{jk}=\frac{2tr(\mathbf{S}_j^T\mathbf{S}_k)}{tr(\mathbf{S}_j^T\mathbf{S}_j)+tr(\mathbf{S}_k^T\mathbf{S}_k)} \end{equation} where the symbol 'tr' is used to indicate the trace of a matrix. In most cases that follow below the representation matrices are standardized (centered and scaled to length one\footnote{An alternative is scaling to variance one, but this only differs with the same constant for each variable.}) and in these cases Eqn. \ref{eAss1} simplifies to \begin{equation}\label{eAss2}
\tilde{q}_{jk}=tr(\mathbf{S}_j^T\mathbf{S}_k) \end{equation} since both $tr(\mathbf{S}_j^T\mathbf{S}_j)$ and $tr(\mathbf{S}_k^T\mathbf{S}_k)$ are one. As will be shown in the following, Eqn. \ref{eAss2} can generate the familiar associations such as the Pearson correlation or the Spearman correlation. An extensive description of all kinds of representation matrices is beyond the scope of this paper; we will discuss the most relevant ones for the problem of heterofusion. The idea of representation matrices\footnote{Their original name was quantification matrices but that name has also been used differently. Hence our choice to rename such matrices.} goes back to the work of \citet{Janson1982} and \citet{Zegers1986c}. Examples of different representation matrices are given in Section \ref{Appendix}.\\
\noindent \textsc{Representation matrices for ratio- and interval-scaled values.}\\
\noindent For ratio- and interval-scaled values, two types of representation matrices can be defined: vectors and square matrices. If $\mathbf{x}_j$ represents the raw scores of the objects on variable $j$ then the vector quantification can be this vector itself (i.e. $\mathbf{s}_j=\mathbf{x}_j$) or a standardized version of it. When the latter is used in Eqn. \ref{eAss2}, Pearson's $R$-value is obtained. In standard multivariate analysis this is by far the most used representation matrix.\\
\noindent There is also another possibility for ratio- and interval-scaled values, namely square representation matrices. Two examples are the following. Define \begin{equation}\label{eRM1}
\widetilde{\mathbf{S}}_j=(\mathbf{x}_j\textbf{1}^T-\textbf{1}\mathbf{x}_j^T) \end{equation} where $\textbf{1}$ is an $I \times 1$ column of ones. This $\widetilde{\mathbf{S}}_j$ generates a skew-symmetric matrix enumerating all differences between the object-scores of variable $j$ (for an example, see \ref{Examples ratio scaled}). Hence, distances between objects are obtained per variable and these distance matrices can be subjected to an INDSCAL model \citep{Kiers1989}. Upon standardizing $\widetilde{\mathbf{S}}_j$ by $\mathbf{S}_j=(tr\widetilde{\mathbf{S}}_j^T\widetilde{\mathbf{S}}_j)^{-1/2}\widetilde{\mathbf{S}}_j$ and using this $\mathbf{S}_j$ (and a similarly defined $\mathbf{S}_k$) in Eqn. \ref{eAss2} gives again Pearson's $R$-value. Another example is using $\mathbf{S}_j=\mathbf{s}_j\mathbf{s}_j^T$ where $\mathbf{s}_j$ is the standardized version of $\mathbf{x}_j$. Using this $\mathbf{S}_j$ (and a similarly defined $\mathbf{S}_k$) in Eqn. \ref{eAss2} gives Pearson's $R^2$ value. Such representation matrices correspond to the blue-squared matrices in Figure \ref{FigureRM} and are the basis of Kernel and Multidimensional Scaling methods (\textbf{check!}).\\
\noindent \textsc{Representation matrices for ordinal-scaled values.}\\
\noindent When the data are ordinal-scaled, then the vector of readings can be encoded in terms of rank-orders $\mathbf{r}_j (I \times 1)$. For the earlier example of strongly disagree, disagree, neutral, agree, strongly agree such a ranking may be encoded as 1 (strongly disagree) to 5 (strongly agree). Then again - as in the ratio-scaled variables - representation can be done using the vectors $\mathbf{r}_j$ or their standardized version. In the latter case, applying Eqn. \ref{eAss2} to this version gives the Spearman's rank-order correlation coefficient. Another representation is by using (the raw-)$\mathbf{r}_j$ in Eqn. \ref{eRM1} instead of $\mathbf{x}_j$ and this generates Spearman's rank-order correlation coefficient after using Eqn. \ref{eAss1}.\\
\noindent \textsc{Representation matrices for nominal-scaled values.}\\
\noindent We will discuss the representation matrices for nominal-scaled variables separately for binary data and categorical data. We first discuss representation matrices for categorical data. We have to distinguish two situations: one in which all categorical variables have the same number of categories and the situation that this is not the case. Since the latter is more general and encountered more often, we will restrict ourselves to this case. Then only square representation matrices are available. These are based on indicator matrices \citep{Zegers1986c,Kiers1989,Gifi1990}. If variable $\mathbf{x}_j$ has four categories (A,B,C,D), then this can be encoded in the rectangular matrix $\mathbf{G}_j (I \times 4)$ where each column $\mathbf{g}_{jk}$ in $\mathbf{G}_j$ represents a category and each row an object. This matrix has only zeros or ones; where $g_{ijk}$ is one, if and only if object $i$ belongs to the category represented by $k$. The representation matrix can now be built using the products $\mathbf{G}_j \mathbf{G}_j^T (I \times I)$. There are very many versions of such square representation matrices based on indicator matrices and some of them give rise to a known correlation, e.g., \begin{equation}\label{eRMNom1}
\mathbf{J} \mathbf{G}_j \mathbf{D}_j^{-1} \mathbf{G}_j^T \mathbf{J} \end{equation} where $\mathbf{J} (I \times I)$ is the centering operator and $\mathbf{D}_j (C_j \times C_j)$ is a diagonal matrix containing the marginal frequencies of categories $1,..,C_j$ for variable $j$. The corresponding correlation coefficient is the so-called $T^2$ coefficient \citep{Tschuprow1939}. These representation matrices correspond to the red-square matrices in Figure \ref{FigureRM}. Examples are given in Section \ref{Examples nominal data}.\\
\noindent For binary data (if all variables are binary) then rectangular representation matrices are possible. This comes down to the same idea as above, namely, to consider the binary variables as representing two categories. This results then in representation matrices $\mathbf{G}_j$ of sizes $(I \times 2)$ . When fusing with other types of variables is the goal, then a squared type of representation is needed such as in Eqn. \ref{eRMNom1} and visualized in Figure \ref{FigureRM} (green matrices). Examples are given in Section \ref{Examples binary data}.\\
\subsubsection{Data fusion using representation matrices} \label{Data fusion using representation matrices}
To illustrate how to use representation matrices we will work with four data matrices, each on a different measurement scale and sharing the same set of $I$ samples. The first matrix $\mathbf{X}_1 (I \times J_1)$ contains ratio- or interval-scaled data; the second matrix $\mathbf{X}_2 (I \times J_2)$ contains ordinal-scaled data; the third $\mathbf{X}_3 (I \times J_3)$ contains nominal data and the last matrix $\mathbf{X}_4 (I \times J_4)$ contains binary data.\\
\noindent The representation matrices $\mathbf{S}_j$ can now be used in a three-way model for symmetric data. The basic model for a single data block is the INDSCAL (INdividual Differences SCALing) model: \begin{equation}\label{eINDSCAL}
\min_{\mathbf{Z},\mathbf{A}_j} \sum_{j=1}^J||\mathbf{S}_j-\mathbf{Z} \mathbf{A}_j \mathbf{Z}^T||^2 \end{equation} where $\mathbf{A}_j$ is the diagonal matrix with the $j^{th}$ row of the loadings $\mathbf{A} (J \times R)$ on its diagonal and the matrix $\mathbf{Z}(I \times R)$ contains the object scores. The loadings $\mathbf{A} (J \times R)$ are nonnegative to ensure the fitted part of the model ($\mathbf{Z} \mathbf{A}_j \mathbf{Z}^T$) to be positive (semi-) definite. If the additional constraint that $\mathbf{Z}^T\mathbf{Z}=\mathbf{I}$ is used, then the model is called INDORT (INDscal with ORThogonal constraints) \citep{Kiers1989}.\\
\noindent The INDORT method can now be generalized to analyze simultaneously all blocks by simply stacking all similarity matrices on top of each other (see Figure \ref{FigureRM}): \begin{equation}\label{eINDOMIX}
\min_{\mathbf{Z},\mathbf{A}_{j_k}} \sum_{k=1}^4 \sum_{j_{k}=1}^{J_{k}}||\mathbf{S}_{j_k}-\mathbf{Z} \mathbf{A}_{j_k} \mathbf{Z}^T||^2 \end{equation} where $\mathbf{A}_{j_k}$ is the diagonal matrix with the $j_k^{th}$ row of the loadings $\mathbf{A}_k (J \times R)$ on its diagonal and the matrix $\mathbf{Z}(I \times R)$ contains the object scores. This model is called IDIOMIX for obvious reasons \citep{Kiers1989}.
\subsection{Optimal scaling approaches}
There are many ways to explain optimal scaling; we will follow the exposition given by \cite{Michailidis1998}. Suppose that the matrix $\mathbf{X} (I \times J)$ contains $J$ categorical variables not necessarily with the same number of categories. Each variable $\mathbf{x}_j$ can now be encoded with indicator matrix $\mathbf{G}_j (I \times L_j)$ where $L_j$ is the number of categories for variable $j$ as discussed before. The idea of optimal scaling is to find objects scores $\mathbf{Z} (I \times R)$ and category quantification matrices $\mathbf{Y}_j (L_j \times R; j=1,...,J)$ such that the following problem is solved \citep{Michailidis1998}: \begin{equation}\label{eOS1}
\min_{\mathbf{Z},\mathbf{Y}_j} \sum_{j=1}^J||\mathbf{Z}-\mathbf{G}_j \mathbf{Y}_j||^2 \end{equation} under the constraints that $(1/I) \mathbf{Z}^T \mathbf{Z} = \mathbf{I}$ and these scores are centered around zero (to avoid trivial solutions of Eqn. \ref{eOS1}). This method - including the alternating optimization method to solve Eqn. \ref{eOS1} - is called homogeneity analysis or HOMALS for short \citep{Gifi1990}. The rows of $\mathbf{Z}$ give a low dimensional representation of the objects and the matrices $\mathbf{Y}_j (j=1,...,J)$ give the optimal quantifications of the categorical variables. Note that these matrices $\mathbf{Y}_j (j=1,...,J)$ are not loadings; they give quantifications for the categorical variables which are different for the $R$ components, namely $\mathbf{y}_{jr} (L_j \times 1; r=1,...,R)$ where $\mathbf{y}_{jr}$ is the $r-th$ column of $\mathbf{Y}_j$.\\
\noindent Upon restricting the rank of $\mathbf{Y}_j (j=1,...,J)$ to be one, we arrive at non-linear PCA (PRINCALS) \citep{Gifi1990,Michailidis1998}. Then Eqn. \ref{eOS1} can be rewritten as \begin{equation}\label{eOS2}
\min_{\mathbf{Z},\mathbf{y}_j,\mathbf{a}_j} \sum_{j=1}^J||\mathbf{Z}-\mathbf{G}_j \mathbf{y}_j {\mathbf{a}^T}\!_j||^2 \end{equation} with the same constraints on $\mathbf{Z}$ as before (i.e. $(1/I)\mathbf{Z}^T\mathbf{Z}=\mathbf{I}$). As an identification constraint for $\mathbf{y}_j$ and $\mathbf{a}_j$ we impose $\mathbf{y}_j^T\mathbf{G}_j^T\mathbf{G}_j\mathbf{y}_j=I$. Now, the vectors $\mathbf{a}_j (R \times 1)$ are the loadings and $\mathbf{y}_j (L_j \times 1)$ contain the quantifications which are the same for all $R$ dimensions of the solution. The relationship between (linear) PCA and non-linear PCA becomes clear when rewriting Eqn. \ref{eOS2} (following \citep{Gifi1990}, p.167-168) as
\begin{gather}\label{eOS3}
\min_{\mathbf{Z},\mathbf{y}_j,\mathbf{a}_j} \sum_{j=1}^J||\mathbf{Z}-\mathbf{G}_j \mathbf{y}_j \mathbf{a}_j^T||^2= \\ \nonumber
\min_{\mathbf{Z},\mathbf{y}_j,\mathbf{a}_j} \sum_j tr(\mathbf{Z}^T\mathbf{Z})-2 \sum_j tr(\mathbf{Z}^T\mathbf{G}_j\mathbf{y}_j\mathbf{a}_j^T)+\sum_jtr(\mathbf{a}_j\mathbf{y}_j^T\mathbf{G}_j^T\mathbf{G}_j\mathbf{y}_j\mathbf{a}_j^T)= \\ \nonumber
\min_{\mathbf{Z},\mathbf{y}_j,\mathbf{a}_j} IJtr\mathbf{I}-2\sum_j tr(\mathbf{Z}^T\mathbf{G}_j\mathbf{y}_j\mathbf{a}_j^T)+I \sum_j tr(\mathbf{a}_j \mathbf{a}_j^T) \nonumber \end{gather}
using the constraints on $\mathbf{Z}$ and $\mathbf{y}_j$. The function in Eqn. \ref{eOS3} differs only a constant from the function
\begin{equation}\label{eOS4}
g(\mathbf{Z},\mathbf{y}_j,\mathbf{a}_j)=\sum_j\|\mathbf{G}_j\mathbf{y}_j-\mathbf{Z}\mathbf{a}_j\|^2, \end{equation}
as follows from rewriting $g(\mathbf{Z},\mathbf{y}_j,\mathbf{a}_j)$ using the constraints on $\mathbf{Z}$ and $\mathbf{y}_j$:
\begin{gather}\label{eOS5}
g(\mathbf{Z},\mathbf{y}_j,\mathbf{a}_j)= \\ \nonumber
\sum_j \mathbf{y}_j^T \mathbf{G}_j^T \mathbf{G}_j \mathbf{y}_j-2 \sum_j tr(\mathbf{a}_j^T \mathbf{Z}^T \mathbf{G}_j \mathbf{y}_j)+\sum_j tr(\mathbf{a}_j^T \mathbf{Z}^T \mathbf{Z} \mathbf{a}_j)= \\ \nonumber
IJ-2\sum_j tr(\mathbf{Z}^T \mathbf{G}_j \mathbf{y}_j \mathbf{a}_j^T)+I\sum_j tr(\mathbf{a}_j^T \mathbf{a}_j). \nonumber \end{gather}
Thus, it has been shown that problem Eqn. \ref{eOS2} subject to the constraints $(1/I)\mathbf{Z}^T\mathbf{Z}=\mathbf{I}$ and $\mathbf{y}_j^T\mathbf{G}_j^T\mathbf{G}_j\mathbf{y}_j=I$ is equivalent to the problem
\begin{gather}\label{eOS6}
\min_{\mathbf{Z},\mathbf{y}_j,\mathbf{a}_j} \sum_{j=1}^J\|\mathbf{G}_j \mathbf{y}_j - \mathbf{Z}_j \mathbf{a}_j\|^2= \\ \nonumber
\min_{\mathbf{Z},\mathbf{y}_j,\mathbf{a}_j} \|[\mathbf{G}_1 \mathbf{y}_1|...|\mathbf{G}_J \mathbf{y}_J]-\mathbf{Z} \mathbf{A}^T\|^2= \\ \nonumber
\min_{\mathbf{Z},\mathbf{y}_j,\mathbf{a}_j} \|\mathbf{X}^*-\mathbf{Z} \mathbf{A}^T\|^2 \nonumber \end{gather}
where $\mathbf{A}$ has rows $\mathbf{a}_j^T$ and $[\mathbf{G}_1 \mathbf{y}_1|...|\mathbf{G}_J \mathbf{y}_J]$ is written as $\mathbf{X}^*$ where the superscript '*' represents the optimal scaled data, and this is seen to be the (non-linear) analog of ordinary PCA \citep{Gifi1990}.\\
\noindent The nature of the measurement scale can now be incorporated by allowing the quantifications to be free for nominal-scale data and monotonic for ordinal-scaled data. The latter quantification ensures the order in the ordinal-scaled data. Framed in terms of Eqn. \ref{eOS6} this becomes: \begin{equation}\label{eOS7}
{x^*}\!_{ij}>{x^*}\!_{kj}\;\; if \;\; x_{ij}>x_{kj} \end{equation} where ${x^*}\!_{ij}$ and ${x^*}\!_{kj}$ are elements of $\mathbf{X}^*$; $x_{ij}$ and $x_{kj}$ are elements of $\mathbf{X}$. Ties in the original data can be treated in different ways depending on whether the underlying measurements can be considered continuous or discrete \citep{DeLeeuw1976,Takane1977,Young1978} but this is beyond the scope of this paper.\\
\noindent There are close similarities between optimal scaling and multiple correspondence analysis \citep{Kiers1989,Michailidis1998}. Binary data represents a special case. When considered as categorical data, non-linear PCA using optimal scaling is the same as performing a (linear) PCA on the standardized binary data, for a proof, see Appendix \citep{DeLeeuw1973,Kiers1989}.
\subsubsection{Data fusion using optimal scaling matrices}
There are different ways to use optimal scaling for fusing data. One method generalizes (generalized) canonical correlation analysis (OVERALS \citep{VanderBurg1988}) and the other method generalizes simultaneous component analysis (SCA) (MORALS \citep{Young1981}). Experiences with generalized canonical correlations show that this method tends to overfit for high-dimensional data. An attempt to overcome this problem is by introducing sparsity constraints \citep{Waaijenborg2008}, but it is not trivial to combine this with optimal scaling. Hence, we chose to use the extension of SCA. Note that SCA was originally developed for analyzing multiple data sets sharing the same set of variables \citep{TenBerge1992}, but it can likewise be formulated for multiple data sets having the sampling mode in common \citep{VanDEun2009}. Using the latter interpretation of SCA leads to the following approach.
We take the same data matrices as in Section \ref{Data fusion using representation matrices} and upon writing $\mathbf{X}^*=[{\mathbf{X}^*}\!_1|{\mathbf{X}^*}\!_2|{\mathbf{X}^*}\!_3|{\mathbf{X}^*}\!_4]$ the problem becomes \begin{equation}\label{eOS8}
\min_{Par}||\mathbf{X}^*-\mathbf{Z} \mathbf{A}^T||^2=\min_{Par}||[{\mathbf{X}^*}\!_1|{\mathbf{X}^*}\!_2|{\mathbf{X}^*}\!_3|{\mathbf{X}^*}\!_4]-\mathbf{Z} [{\mathbf{A}^T}\!_1|{\mathbf{A}^T}\!_2|{\mathbf{A}^T}\!_3|{\mathbf{A}^T}\!_4]||^2 \end{equation} with an obvious partition of the loading matrix $\mathbf{A}$ and where the term 'Par' stands for all parameters. Apart from the scores $\mathbf{Z}$ and loadings $\mathbf{A}$ these are the following. For the ratio- interval-scaled block there are no extra parameters since the original scale is used (i.e. ${\mathbf{X}^*}\!_1=\mathbf{X}_1$. The second - ordinal-scaled - block puts restrictions on ${\mathbf{X}^*}$ following the restrictions of Eqn. \ref{eOS7}. The third (nominal-) block has underlying indicator matrices $\mathbf{G}$ and associated quantifications $\mathbf{y}$ and loadings $\mathbf{A}_3$ obey the rules of Eqn. \ref{eOS2}. Finally, the binary block ${\mathbf{X}^*}\!_4$ is simply the standardized version of $\mathbf{X}_4$ and this ensures an optimal scaling as mentioned above. Note that this way of fusing data assumes an identity link function \citep{VanMechelen2010} and is thus an extension of methods like Consensus PCA and SCA. We will call this method OS-SCA in the sequel. There is no differentiation between common and distinct components \citep{Smilde2017}\\
\subsection{Parametric approaches}
A different class of methods has its roots in factor analysis and can be summarized as follows (see Figure \ref{FigurePM}). The basic idea is that a set of (shared) latent variables is responsible for the variation in all the blocks \citep{Shen2009,Curtis2012,Mo2013} and, subsequently, models are built for the individual blocks based on those shared latent variables. We will describe the Generalized Simultaneous Component Analysis (GSCA) method \citep{Song2018} in more detail since that is the method used in this paper. If $\mathbf{X}_1$ is the binary data matrix, then we assume that there is a low-dimensional deterministic structure $\boldsymbol{\Theta_1} (I \times J_1)$ underlying $\mathbf{X}_1$ and the elements of $\mathbf{X}_1$ follow a Bernoulli distribution with parameters $\phi(\theta_{1ij})$, thus $x_{1ij} \sim B(\phi(\theta_{1ij}))$. The function $\phi(.)$ can be taken as the logit link $\phi(\theta)=(1+exp(-\theta))^{-1}$ and $x_{1ij}$, $\theta_{1ij}$ are the $ij^{th}$ elements of $\mathbf{X}_1$ and $\boldsymbol{\Theta_1}$, respectively. The $\boldsymbol{\Theta_1}$ is now assumed to be equal to $\boldsymbol{1} \boldsymbol{\mu}_1^T + \mathbf{Z} \mathbf{A}_1$ where $\boldsymbol{\mu}_1$ represent the off-set term, $\mathbf{Z}$ the common scores and $\mathbf{A}_1$ the loadings for the binary data.\\
\noindent The quantitative measurements $\mathbf{X}_2$ are assumed to follow the model $\mathbf{X}_2=\boldsymbol{1} \boldsymbol{\mu}_2^T + \mathbf{Z} \mathbf{A}_2 + \mathbf{E}$ where the elements $e_{ij}$ of $\mathbf{E}$ are normally distributed with mean 0 and variance $\sigma^2$. The matrix $\mathbf{A}_2$ contains the loadings of the quantitative data set; $\mathbf{Z}$ are again the common scores and the constraints $\mathbf{Z}^T\mathbf{Z}=I \mathbf{I}_R$ and $\boldsymbol{1}^T\mathbf{Z}=0$ are imposed for identifiability. The shared information between $\mathbf{X}_1$ and $\mathbf{X}_2$ is assumed to be represented fully by the common latent variables $\mathbf{Z}$. Thus $\mathbf{X}_1$ and $\mathbf{X}_2$ are stochastically independent given these latent variables and the negative log-likelihoods of both parts can be summed:
\begin{gather}
f_1(\boldsymbol{\Theta_1}) = -\sum_i^I \sum_j^{J_1}[x_{1ij}log(\phi(\theta_{1ij}))+(1-x_{1ij})log(1-\phi(\theta_{1ij})] \\ \nonumber
f_2(\boldsymbol{\Theta_2},\sigma^2) = \frac{1}{2\sigma^2}\|\mathbf{X}_2-\boldsymbol{\Theta_2}\|_F^2 + \frac{1}{2}log(2\pi\sigma^2) \\ \nonumber
f(\boldsymbol{\Theta_1},\boldsymbol{\Theta_2},\sigma^2) = f_1(\boldsymbol{\Theta_1})+f_2(\boldsymbol{\Theta_2},\sigma^2) \end{gather}
and minimized simultaneously. This requires some extra constraints; details are given elsewhere \citep{Song2018}.\\
\section{Practical issues and examples}
\subsection{Genomics example}
The genomics example is from the field of cancer research and the data are obtained from the Genomics in Drugs Sensitivity in Cancer from the Sanger Institute (http://www.cancerrxgene.org/). Briefly, this repository consists of measurements performed on cell lines pertaining to different types of cancer. We used the copy number aberration (CNA) and gene-expression data of the cell lines related to breast cancer (BRCA), lung cancer (LUAD) and skin cancer (SKCM). After selecting the samples which had values for all these types of cancer we filtered the gene-expression data by selecting the 1000 variables with the highest variance across the samples. The CNA data contains amplifications and losses of DNA-regions as compared to the average copy numbers in the population. Both amplifications and losses are encoded as ones indicating deviances. The zeros in the CNA data indicate a normal diploid copy number. This provides us with $I=160$ samples; $J_1=410$ binary values for the CNA data and $J_2=1000$ variables for the gene-expression data.\\
\noindent For the representation approach we built a three-way array of size $160 \times 160 \times (410+1000)$ and performed an IDIOMIX analysis. For the binary part, this array contains the slabs $\mathbf{S}_j$ according to Eqn. \ref{eRMNom1} and for the gene-expression part, the slabs $\mathbf{S}_j$ are defined by the outer products of the samples in the gene-expression data after auto-scaling the columns of that data. The optimal scaling result are obtained by auto-scaling both raw data sets and subsequently perform an (OS-)SCA on the concatenated data $\mathbf{X}_{sc}=[\mathbf{X}_{1sc} | \mathbf{X}_{2sc}]$. The final way of fusing the two data sets is by using the GSCA model.\\
\noindent The amounts of explained variations are shown in Table \ref{Table1} which contains a lot of information and should be interpreted with care.
\begin{table}[h!] \footnotesize \centering
\begin{tabular}{|c|ccc|ccc|ccc|c|}
\hline
Method & IDIOMIX & & & OS-SCA & & & GSCA & & & PCA \\ \hline
Data type & Binary & Quant & Total & Binary & Quant & Total & Binary & Quant & Total & Quant \\ \hline
SC1 & 0.06 & 9.32 & 6.60 & 13.65 & 22.15 & 16.14 & 63.58 & 22.11 & 23.64 & 22.15 \\
SC2 & 0.03 & 2.38 & 1.69 & 5.66 & 10.17 & 7.48 & 15.72 & 10.19 & 10.39 & 10.17 \\
SC3 & 3.73 & 0.01 & 1.10 & 4.99 & 4.52 & 4.35 & 6.17 & 4.48 & 4.54 & 4.52 \\ \hline
Cum & 3.82 & 11.70 & 9.39 & 24.29 & 36.84 & 27.96 & 85.47 & 36.77 & 38.58 & 36.84 \\ \hline \end{tabular} \caption{\label{Table1}\footnotesize Variances explained by the various methods. SC is the abbreviation of simultaneous component. For more explanation, see main text.} \label{Table1} \end{table}
First, for IDIOMAX, OS-SCA and the quantitative part of the GSCA model the explained variation is calculated using sums-of-squares. This is not the case for the binary part of GSCA (for details, see \cite{Song2018}). Second, IDIOMAX on the one hand and OS-SCA, GSCA on the other hand are very different types of models, i.e., they use the data directly (OS-SCA, GSCA) or indirectly (IDIOMAX) so a simple comparison of explained sums-of-squares between these types of models is difficult. The final column of the table reports the amounts of explained variation of a regular PCA on the (autoscaled) gene-expression data.\\
\noindent The first observation to make regarding the values in Table \ref{Table1} is that the amounts of explained variations of the PCA model of the gene-expression data is closely followed by the amounts of explained variations in the gene-expression simultaneous components for OS-SCA and GSCA. This means that the data fusion is dominated by the gene-expression block. This is confirmed by plotting the scores of PC1 and PC2 of the PCA on gene-expression against the SC-scores 1 and 2 of OS-SCA and GSCA: these are almost on a straight line (plot not shown). Although the explained variances for IDIOMAX are much lower, the same observation is valid for IDIOMAX: also for this method the first two SC-scores resembles the ones of a PCA on the gene-expression almost perfectly. This dominance of the gene-expression block in the data fusion as reflected in the first two components cannot completely be explained by the differences in block sizes (1000 variables for gene-expression and 410 variables for the CNA block) but is also due to dominant intrinsic systematic patterns in the gene-expression data.\\
\noindent To get a feeling for what is represented in the first two SCs (that are virtually identical across the three methods), we show the scores for the GSCA method on SC1 and SC2 in Figure \ref{FigurePMPC1PC2}.
\begin{figure}
\caption{\footnotesize Scores on SC1 and SC2 for the GSCA model (see text).}
\label{FigurePMPC1PC2}
\end{figure}
The scores show a clear separation in cancer types with specific sub-clusters for hormone-positive breast cancer (within the BRCA-group) and MITF-high melanoma (in the SKCM group) (for a more elaborate interpretation see \cite{Song2018}).\\
\noindent Whereas the three approaches give similar results for the first two simultaneous components, qualitative differences can be seen in SC3. This is especially apparent in Table \ref{Table1} where the third component for IDIOMIX is now dominated by the CNA data. This is visualized in Figure \ref{FigureGenPC1PC3} which shows the score plots of SC1 versus SC3 for all methods which are clearly different.
\begin{figure}
\caption{\footnotesize Score plots of SC1 versus SC3 of in the genomics example using all models (see text). Legend: left: IDIOMIX; middle: OS-SCA; right: GSCA.}
\label{FigureGenPC1PC3}
\end{figure}
To further confirm this, the scores of the different methods for the three different components were plotted against each other (see Figure \ref{PC1PC2PC3}) and this confirms that indeed the first two SCs are very similar for all methods, but that SC3 shows differences where GSCA is the most deviating.
\begin{figure}
\caption{\footnotesize Score plots of SC1-SC3 for all fusion methods. Optimal scaling is OS-SCA; Representation is IDIOMIX.}
\label{PC1PC2PC3}
\end{figure}
To shed some light on this deviating behavior, we plotted the scores of a PCA on the gene-expression data against the SC-scores of the fusion methods for the third component, see Figure \ref{PCACNAFreq}.
\begin{figure}
\caption{\footnotesize Left panels: score plots of PC3 of PCA on gene-expression data (x-axis) compared to the SC3 from the fusion results (y-axis); optimal scaling is OS-SCA, representation is IDIOMAX. Right panels: scores of SC3 of all methods compared to CNA frequency.}
\label{PCACNAFreq}
\end{figure}
The left panels in this figure show that SC3 from GSCA is very similar to the PC3 of a PCA on the gene-expression data alone (see also again Table \ref{Table1}). The same does not hold true for the other methods. The CNA values are available for each sample and thus the scores on the fusion SC3 can be plotted against the frequency at which such an aberration occurs (number of ones divided by the total). From the right panels of Figure \ref{PCACNAFreq}, it then becomes clear that SC3 of IDIOMAX and OS-SCA are mostly picking up the differences in frequencies, contrary to the GSCA-SC3 scores.\\
\noindent A similar comparison can be made for the loadings, see Figure \ref{PC3GainsLoss}. The left panels show the PC3 loadings from gene-expression using PCA and the fusion methods. In the right panels the fusion loadings are plotted against CNA frequencies (now across DNA-positions) and those show no correlation. As explained earlier, the aberrations can either be amplifications or losses and those are clearly picked up by the loadings of IDIOMAX and OS-SCA.\\
\begin{figure}
\caption{\footnotesize Left panels: PC3 loadings of PCA on gene-expression data (x-axis) compared to the SC3 loadings from the fusion results (y-axis); optimal scaling is OS-SCA, representation is IDIOMIX. Right panels: loadings of SC3 of all fusion methods Amplifications (black) and Losses (red).}
\label{PC3GainsLoss}
\end{figure}
\noindent To interpret the GSCA-loadings, these loadings were subjected to a Gene Set Enrichment Analysis (GSEA). This resulted in a highly significant enrichment for epithelial-mesenchymal transition (EMT), a process undergone by tumor cells frequently associated with invasion of surrounding tissues and subsequent metastases. The largest positive loading on GSCA-PC3 for the gene-expression is ZEB1, a transcription factor associated with EMT. A plot of the loadings of the CNA data is shown in Figure \ref{PC3EMT} and one of the loadings identifies SMAD4 loss as an important factor. SMAD4 is required for TGF-$\beta$ driven EMT which confirms the finding that the GSCA gene-expression loadings are strongly enriched for EMT \citep{Tian2009}.\\
\begin{figure}
\caption{\footnotesize SC3 loadings of GSCA. In red: SMAD4loss (see text).}
\label{PC3EMT}
\end{figure}
\noindent Summarizing, IDIOMAX and OS-SCA are very similar for the whole analysis. For the first two SCs, also the GSCA resembles the other approaches. The difference of GSCA is in the third SC. It seems that GSCA is focussing more on the gene-expression data; whereas IDIOMAX and OS-SCA pick up specific aspects of the CNA data in this third SC. The results of the GSCA-SC3 are biologically relevant; this is less the case for SC3 of the other approaches. It may be that GSCA is focussing more on the common variation between the two data sets and is less influenced by the distinctive parts \citep{Smilde2017}. This needs further exploration in a follow-up paper.
\section{Discussion} \label{Discussion}
In this paper we have described and compared three methods of fusing data of different measurement scales. We used the example of quantitative and binary data, but all methods can also deal with ordinal data. For the example, it appears that IDIOMAX and OS-SCA give very similar results whereas GSCA is different. One of the reasons may be that the methods deal differently with common and distinct parts of the data.\\
\noindent All methods have meta-parameters, that is, prior choices have to be made. For IDIOMAX, this is the type of representation to select; for OS-SCA it is the type of restrictions to apply; for GSCA it is the distribution to assume for the separate data sets. All methods also require selecting the complexity of the models, i.e., the number of components. The selection of all these meta-parameters will, in practice, be made based on a mixture of domain knowledge and validation, such as cross-validation or scree-tests for selecting model complexity.\\
\noindent We hesitate in giving recommendations regarding which method to use for a particular application. First, the example of this paper concerns an exploratory study for which it is always difficult to judge the relative merits of the methods. Secondly, the cultural background of the investigator plays a role. In data analysis and chemometrics, the culture is to avoid distributional assumptions and have a more data analytic approach, thus resulting in a preference for IDIOMAX or OS-SCA. In statistics and, to some extent, in bioinformatics there is more a tendency to go for parametric modes, hence, GSCA in our context. Thirdly, these methods have not yet been used to a large extent by researchers, hence, experience on their behavior upon which a recommendation can be based is lacking.\\
\noindent In terms of ease of use, we have a slight preference for IDIOMAX. Once the representation matrices are built, standard three-way analysis software can be used to fit the models. There is also software available for OS-SCA and GSCA, but this software is more difficult to implement.\\
\noindent There remain open issues to be investigated. Some of the more prominent ones is to understand the behavior of the methods regarding common, distinct and local components in fusing data sets. Little has been done in this field regarding data of different measurement scales.
\section{Appendix} \label{Appendix}
\subsection{Optimal scaling of binary data equals analyzing standardized data}
The fact that optimal scaling of binary data equals the analysis of standardized data can be shown as follows. Suppose that a binary vector has $n_0$ values of zero, $n_1$ values of one and $n$ values in total, and $a$ is the optimal scaled value for the zeros and $b$ for the ones. In optimal scaling, the optimal scaled variables need to get some kind of normalization. A common set of choices (see \citep{Gifi1990}) is to make sure that the scaled values have mean zero and variance one. This leads to the following two equations:
\begin{gather}\label{eProof1}
n_0a+n_1b=0 \\ \nonumber
n_0a^2+n_1b^2=n \nonumber \end{gather} and these equations can be solved for $a$ and $b$ since $n_0, n_1$ and $n$ are known. This gives two values for $a$; one positive and one negative. The values of $b$ follow automatically with the opposite sign. Hence, both solutions are practically equal.
\subsection{Examples of representation matrices for ratio- and interval-scaled data} \label{Examples ratio scaled}
We will illustrate some ideas of representation matrices using a small example of an $(4 \times 2)$ matrix $\mathbf{X}=[\mathbf{x}_1|\mathbf{x}_2]$: \begin{eqnarray}
\mathbf{X}
=
\left(
\begin{array}{cc}
2 & 9 \\
4 & 9 \\
6 & 10 \\
8 & 12 \\
\end{array}
\right) \label{esmallX} \end{eqnarray} and the standardized version of this is \begin{eqnarray}
\mathbf{X}_s
=
\left(
\begin{array}{cc}
-0.671 & -0.408 \\
-0.224 & -0.408 \\
0.224 & 0 \\
0.671 & 0.816 \\
\end{array}
\right) \label{esmallXs} \end{eqnarray} where indeed ${\mathbf{x}^T}\!_{s1}{\mathbf{x}}_{s1}=1$, ${\mathbf{x}^T}\!_{s2}\mathbf{x}_{s2}=1$ and ${\mathbf{x}^T}\!_{s1}\mathbf{x}_{s2}=0.913$ the latter being the correlation between $\mathbf{x}_1$ and $\mathbf{x}_2$. The square representation using Eqn. \ref{eRM1} on $\mathbf{x}_1$ gives \begin{eqnarray}
\widetilde{\mathbf{S}}_1
=
\left(
\begin{array}{cccc}
0 & -2 & -4 & -6\\
2 & 0 & -2 & -4\\
4 & 2 & 0 & -2\\
6 & 4 & 2 & 0\\
\end{array}
\right) \label{esmallS1} \end{eqnarray} which is skew-symmetric (${\widetilde{\mathbf{S}}^T}_1=-\widetilde{\mathbf{S}}_1$) and contains all the differences between the elements of $\mathbf{x}_1$. The standardized version of $\widetilde{\mathbf{S}}_1$ is \begin{eqnarray}
\mathbf{S}_1
=
\left(
\begin{array}{cccc}
0 & -0.158 & -0.316 & -0.474\\
0.158 & 0 & -0.158 & -0.316\\
0.316 & 0.158 & 0 & -0.158\\
0.474 & 0.316 & 0.158 & 0\\
\end{array}
\right) \label{esmallS1s} \end{eqnarray} and a similar matrix can be made for $\mathbf{x}_2$. Then using Eqn. \ref{eAss2} on the pairs $(\mathbf{S}_1,\mathbf{S}_1)$ and $(\mathbf{S}_2,\mathbf{S}_2)$ gives a value of one; and on the pair $(\mathbf{S}_1,\mathbf{S}_2)$ gives 0.913, which is the Pearson's correlation again.\\
\noindent Alternative square representations of $\mathbf{x}_{s1}$ and $\mathbf{x}_{s2}$ are \begin{eqnarray}
\mathbf{S}_{A1}
=\mathbf{x}_{s1} {\mathbf{x}^T}\!_{s1}=
\left(
\begin{array}{cccc}
0.45 & 0.15 & -0.15 & -0.45\\
0.15 & 0.05 & -0.05 & -0.15\\
-0.15 & -0.05 & 0.05 & 0.15\\
-0.45 & -0.15 & 0.15 & 0.45\\
\end{array}
\right) \label{esmallSA1} \end{eqnarray} and \begin{eqnarray}
\mathbf{S}_{A2}
=\mathbf{x}_{s2} {\mathbf{x}^T}\!_{s2}=
\left(
\begin{array}{cccc}
0.167 & 0.167 & 0 & -0.333\\
0.167 & 0.167 & 0 & -0.333\\
0 & 0 & 0 & 0\\
-0.333 & -0.333 & 0 & 0.667\\
\end{array}
\right) \label{esmallSA2} \end{eqnarray} and using Eqn. \ref{eAss2} on $\mathbf{S}_{A1}$ and $\mathbf{S}_{A2}$ gives 0.833 which is the squared Pearson's correlation between the original variables.
\subsection{Examples of representation matrices for nominal data} \label{Examples nominal data}
We will illustrate some ideas on representing nominal data using two categorical variables $\mathbf{x}_1$ and $\mathbf{x}_2$. The first variable contains four categories encoded as A,B,C,D and reads $\mathbf{x}_1=(A,B,A,C,D,C,B,D)^T$; the second variable has three categories encoded as I,II,III and reads $\mathbf{x}_2=(I,II,II,I,III,III,I,II)^T$ where the roman capitals are used to show that the two variables encode different types of categories. The indicator matrices are now \begin{eqnarray}
\mathbf{G}_1
=
\left(
\begin{array}{cccc}
1 & 0 & 0 & 0\\
0 & 1 & 0 & 0\\
1 & 0 & 0 & 0\\
0 & 0 & 1 & 0\\
0 & 0 & 0 & 1\\
0 & 0 & 1 & 0\\
0 & 1 & 0 & 0\\
0 & 0 & 0 & 1\\
\end{array}
\right) \label{eG1nom} \end{eqnarray} and \begin{eqnarray}
\mathbf{G}_2
=
\left(
\begin{array}{ccc}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 1 & 0 \\
1 & 0 & 0 \\
0 & 0 & 1 \\
0 & 0 & 1 \\
1 & 0 & 0 \\
0 & 1 & 0 \\
\end{array}
\right) \label{eG2nom} \end{eqnarray} and a special feature of this kind of data becomes present namely that some objects have exactly the same rows in $\mathbf{G}_1$ (and similarly in $\mathbf{G}_2$). Moreover, the matrices show closure (${\mathbf{G}^T}\!_1 \mathbf{1}=\mathbf{1},{\mathbf{G}^T}\!_2 \mathbf{1}=\mathbf{1}$). The marginal frequencies are collected in \begin{eqnarray}
\mathbf{D}_1
=
\left(
\begin{array}{cccc}
2 & 0 & 0 & 0\\
0 & 2 & 0 & 0\\
0 & 0 & 2 & 0\\
0 & 0 & 0 & 2\\
\end{array}
\right)
=\mathbf{G}^T_1\mathbf{G}_1 \label{eD1nom} \end{eqnarray} and \begin{eqnarray}
\mathbf{D}_2
=
\left(
\begin{array}{ccc}
3 & 0 & 0 \\
0 & 3 & 0 \\
0 & 0 & 2 \\
\end{array}
\right)
= \mathbf{G}^T_2\mathbf{G}_2 \label{eD2nom} \end{eqnarray} with obvious properties.\\
\noindent Simple representations of these variables are now \begin{eqnarray}
\mathbf{S}_{1s}
=
\left(
\begin{array}{cccccccc}
1 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\
0 & 1 & 0 & 0 & 0 & 0 & 1 & 0\\
1 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\
0 & 0 & 0 & 1 & 0 & 1 & 0 & 0\\
1 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\
0 & 0 & 0 & 0 & 1 & 0 & 0 & 1\\
0 & 0 & 0 & 1 & 0 & 1 & 0 & 0\\
0 & 0 & 0 & 0 & 1 & 0 & 0 & 1\\
\end{array}
\right)
=\mathbf{G}_1\mathbf{G}^T_1 \label{eS1simple} \end{eqnarray} and \begin{eqnarray}
\mathbf{S}_{2s}
=
\left(
\begin{array}{cccccccc}
1 & 0 & 0 & 1 & 0 & 0 & 1 & 0\\
0 & 1 & 1 & 0 & 0 & 0 & 0 & 1\\
0 & 1 & 1 & 0 & 0 & 0 & 0 & 1\\
1 & 0 & 0 & 1 & 0 & 0 & 1 & 0\\
0 & 0 & 0 & 0 & 1 & 1 & 0 & 0\\
0 & 0 & 0 & 0 & 1 & 1 & 0 & 0\\
1 & 0 & 0 & 1 & 0 & 0 & 1 & 0\\
0 & 1 & 1 & 0 & 0 & 0 & 0 & 1\\
\end{array}
\right)
=\mathbf{G}_2\mathbf{G}^T_2 \label{eS2simple} \end{eqnarray} and these representations are encoding which objects have equal categories in the variables. The more complex representations (according to Eqn \ref{eRMNom1}) are now \begin{eqnarray}
\mathbf{S}_{1c}
=
\left(
\begin{array}{cccccccc}
0.375 & -0.125 & 0.375 & -0.125 & -0.125 & -0.125 & -0.125 & -0.125\\
-0.125 & 0.375 & -0.125 & -0.125 & -0.125 & -0.125 & 0.375 & -0.125\\
0.375 & -0.125 & 0.375 & -0.125 & -0.125 & -0.125 & -0.125 & -0.125\\
-0.125 & -0.125 & -0.125 & 0.375 & -0.125 & 0.375 & -0.125 & -0.125\\
-0.125 & -0.125 & -0.125 & -0.125 & 0.375 & -0.125 & -0.125 & 0.375\\
-0.125 & -0.125 & -0.125 & 0.375 & -0.125 & 0.375 & -0.125 & -0.125\\
-0.125 & 0.375 & -0.125 & -0.125 & -0.125 & -0.125 & 0.375 & -0.125\\
-0.125 & -0.125 & -0.125 & -0.125 & 0.375 & -0.125 & -0.125 & 0.375\\
\end{array}
\right) \label{eS1complex} \end{eqnarray} and \begin{eqnarray}
\mathbf{S}_{2c}
=
\left(
\begin{array}{cccccccc}
0.208 & -0.125 & -0.125 & 0.208 & -0.125 & -0.125 & 0.208 & -0.125\\
-0.125 & 0.208 & 0.208 & -0.125 & -0.125 & -0.125 & -0.125 & 0.208\\
-0.125 & 0.208 & 0.208 & -0.125 & -0.125 & -0.125 & -0.125 & 0.208\\
0.208 & -0.125 & -0.125 & 0.208 & -0.125 & -0.125 & 0.208 & -0.125\\
-0.125 & -0.125 & -0.125 & -0.125 & 0.375 & 0.375 & -0.125 & -0.125\\
-0.125 & -0.125 & -0.125 & -0.125 & 0.375 & 0.375 & -0.125 & -0.125\\
0.208 & -0.125 & -0.125 & 0.208 & -0.125 & -0.125 & 0.208 & -0.125\\
-0.125 & 0.208 & 0.208 & -0.125 & -0.125 & -0.125 & -0.125 & 0.208\\
\end{array}
\right) \label{eS2complex} \end{eqnarray} which are indeed double centered and standardized. Using Eqn. \ref{eAss2} on the matrices $\mathbf{S}_{1c}$ and $\mathbf{S}_{2c}$ gives the correlation coefficient $T^2=0.5$.
\subsection{Examples of representation matrices for binary data} \label{Examples binary data}
As an example for binary data we will use a simple data set consisting of two binary variables $\mathbf{x}_1$ and $\mathbf{x}_2$ which are columns of \begin{eqnarray}
\mathbf{X}
=
\left(
\begin{array}{ccc}
0 & 1 \\
0 & 1 \\
1 & 0 \\
1 & 1 \\
0 & 1 \\
1 & 0 \\
0 & 1 \\
0 & 0 \\
\end{array}
\right) \label{eG1bin} \end{eqnarray} with indicator matrices \begin{eqnarray}
\mathbf{G}_1
=
\left(
\begin{array}{ccc}
1 & 0 \\
1 & 0 \\
0 & 1 \\
0 & 1 \\
1 & 0 \\
0 & 1 \\
1 & 0 \\
1 & 0 \\
\end{array}
\right) \label{eG1bin} \end{eqnarray} and \begin{eqnarray}
\mathbf{G}_2
=
\left(
\begin{array}{ccc}
0 & 1 \\
0 & 1 \\
1 & 0 \\
0 & 1 \\
0 & 1 \\
1 & 0 \\
0 & 1 \\
1 & 0 \\
\end{array}
\right). \label{eG2bin} \end{eqnarray} A correlation measure between binary variables is the $\phi$-coefficient which is defined as \begin{equation}\label{ePhi1}
\frac{n_{11}n_{00}-n_{10}n_{01}}{\sqrt{n_{1.}n_{0.}n_{.0}n_{.1}}} \end{equation} where the values $n$ are shown in Table \ref{table2}. For the example, this $\phi$-coefficient equals $-0.4667$ which is also equivalent to the Pearson correlation between $\mathbf{x}_1$ and $\mathbf{x}_2$.
\begin{table}[h!] \centering
\begin{tabular}{|c|cc|c|}
\hline
& $x_2=0$ & $x_2=1$ & total \\ \hline
$x_1=1$ & $n_{11}(1)$ & $n_{10}(2)$ & $n_{1.}(3)$ \\
$x_1=0$ & $n_{01}(4)$ & $n_{00}(1)$ & $n_{0.}(5)$ \\ \hline
total & $n_{.1}(5)$ & $n_{.0}(3)$ & $n(8)$ \\ \hline \end{tabular} \caption{\label{Table2}\footnotesize Calculation of the $\phi$-coefficient (between brackets the values of the example).} \label{table2} \end{table}
\noindent There are two alternative square representations of $\mathbf{x}_1$ and $\mathbf{x}_2$. The first uses Eqn. \ref{eRMNom1} based on the indicator matrices and the results are \begin{eqnarray}
\mathbf{S}_{1c}
=
\left(
\begin{array}{cccccccc}
0.075 & 0.075 & -0.125 & -0.125 & 0.075 & -0.125 & 0.075 & 0.075 \\
0.075 & 0.075 & -0.125 & -0.125 & 0.075 & -0.125 & 0.075 & 0.075 \\
-0.125 & -0.125 & 0.208 & 0.208 & -0.125 & 0.208 & -0.125 & -0.125 \\
-0.125 & -0.125 & 0.208 & 0.208 & -0.125 & 0.208 & -0.125 & -0.125 \\
0.075 & 0.075 & -0.125 & -0.125 & 0.075 & -0.125 & 0.075 & 0.075 \\
-0.125 & -0.125 & 0.208 & 0.208 & -0.125 & 0.208 & -0.125 & -0.125 \\
0.075 & 0.075 & -0.125 & -0.125 & 0.075 & -0.125 & 0.075 & 0.075 \\
0.075 & 0.075 & -0.125 & -0.125 & 0.075 & -0.125 & 0.075 & 0.075 \\
\end{array}
\right) \label{eS1Nomcomplex} \end{eqnarray} and \begin{eqnarray}
\mathbf{S}_{2c}
=
\left(
\begin{array}{cccccccc} 0.075 & 0.075 & -0.125 & 0.075 & 0.075 & -0.125 & 0.075 & -0.125 \\ 0.075 & 0.075 & -0.125 & 0.075 & 0.075 & -0.125 & 0.075 & -0.125 \\ -0.125 & -0.125 & 0.208 & -0.125 & -0.125 & 0.208 & -0.125 & 0.208 \\
0.075 & 0.075 & -0.125 & 0.075 & 0.075 & -0.125 & 0.075 & -0.125 \\
0.075 & 0.075 & -0.125 & 0.075 & 0.075 & -0.125 & 0.075 & -0.125 \\
-0.125 & -0.125 & 0.208 & -0.125 & -0.125 & 0.208 & -0.125 & 0.208 \\
0.075 & 0.075 & -0.125 & 0.075 & 0.075 & -0.125 & 0.075 & -0.125 \\
-0.125 & -0.125 & 0.208 & -0.125 & -0.125 & 0.208 & -0.125 & 0.208\\
\end{array}
\right) \label{eS2Nomcomplex} \end{eqnarray} and when these are used in Eqn. \ref{eAss2} the result is $0.2178$ which is the square of the $\phi$-coefficient.\\
\noindent The other representations are based on the standardized $x$-variables $\mathbf{z}_1$ and $\mathbf{z}_2$ (with ${\mathbf{z}^T}\!_1 \mathbf{z}_2=-0.4667$). It now holds that $\mathbf{J} \mathbf{G}_j \mathbf{D}_j^{-1} \mathbf{G}_j^T \mathbf{J}=\mathbf{z}_j {\mathbf{z}^T}\!_j$ and, hence, both representations coincide.
\end{document} |
\begin{document}
\title[Population models at stochastic times]{Population models at stochastic times} \author{Enzo Orsingher} \email{enzo.orsingher@uniroma1.it} \author{Costantino Ricciuti} \email{costantino.ricciuti@uniroma1.it} \author{Bruno Toaldo}
\email{bruno.toaldo@uniroma1.it} \address{Department of Statistical Sciences, Sapienza - University of Rome} \keywords{Non-linear birth processes, sublinear and linear death processes, sojourn times, fractional birth processes, random time} \date{\today} \subjclass[2010]{60G22; 60G55}
\begin{abstract} In this article, we consider time-changed models of population evolution $\mathcal{X}^f(t)=\mathcal{X}(H^f(t))$, where $\mathcal{X}$ is a counting process and $H^f$ is a subordinator with Laplace exponent $f$. In the case $\mathcal{X}$ is a pure birth process, we study the form of the distribution, the intertimes between successive jumps and the condition of explosion (also in the case of killed subordinators). We also investigate the case where $\mathcal{X}$ represents a death process (linear or sublinear) and study the extinction probabilities as a function of the initial population size $n_0$. Finally, the subordinated linear birth-death process is considered. A special attention is devoted to the case where birth and death rates coincide; the sojourn times are also analysed. \end{abstract}
\maketitle
\section{Introduction} Birth and death processes can be applied in modelling many dynamical systems, such as cosmic showers, fragmentation processes, queueing systems, epidemics, population growth and aftershocks in earthquakes. The time-changed version of such processes has also been analysed since it is useful to describe the dynamics of various systems when the underlying environmental conditions randomly change. For example, the fractional birth and death processes, studied in \citet{orspolber3, orspolber4, orspolber1, orspolber2}, are time-changed processes where the distribution of the time is related to the fractional diffusion equations. On this point consult \citet{cahoy, cahoypol} for some applications and simulations.
In this paper, we consider the case where the random time is a subordinator. Actually, subordinated Markov processes have been extensively studied since the Fifties. The case of birth and death processes merits however a further investigation and this is the role of the present paper. We consider here compositions of point processes $\mathcal{X}(t)$, $t>0$, with an arbitrary subordinator $H^f(t)$ related to the Bern\v{s}tein functions $f$. We denote such processes as $\mathcal{X}^f(t)= \mathcal{X}(H^f(t))$. The general form of $f$ is as follows \begin{align} f(x)= \alpha+\beta x+\int _0^\infty (1-e^{-xs}) \nu(ds) \qquad \alpha \geq 0, \beta \geq 0, \label{Bernstein} \end{align} where $\nu$ is the L\'evy measure satisfying \begin{align} \int_0^\infty (s \wedge 1 ) \nu(ds) < \infty. \label{misura di Levy} \end{align} In this paper we refer to the case $\alpha=\beta=0$, unless explicitly stated. The structure of the paper is as follows: section 2 treats the subordinated non-linear birth process; section 3 deals with the subordinated linear and sublinear death processes; section 4 analyses the linear birth-death process, with particular attention to the case where birth and death rates coincide. In all three cases, we compute directly the state probabilities by means of the composition formula \begin{align} \Pr \ll \mathcal{X}^f(t)=k \rr = \int_0^{\infty} \Pr \ll \mathcal{X}(s)=k \rr \Pr \ll H^f(t) \in ds \rr. \end{align} Despite most of the subordinators do not possess an explicit form for the probability density function, the distribution of $\mathcal{X}(H^f(t))$ always presents a closed form in terms of the Laplace exponent $f$. We also study the transition probabilities, both for finite and infinitesimal time intervals. We emphasize that the subordinated point processes have a fundamental difference with respect to the classical ones, in that they perform upward or downward jumps of arbitrary size. For infinitesimal time intervals, we provide a direct and simple proof of the following fact: \begin{align}
\Pr \ll \mathcal{X}^f(t+dt)=k | \mathcal{X}^f(t)=r \rr = dt \int_0^{\infty} \Pr \ll \mathcal{X}(s)=k | \mathcal{X}(0)=r \rr \nu(ds), \label{14b} \end{align} which is related to Bochner subordination (see \cite{phillips}).\\ The first case taken into account is that of a non-linear birth process with birth rates $\lambda_k$, $k\geq 1$, which is denoted by $\mathcal{N}(t)$. The subordinated process $\mathcal{N}^f(t)$ does not explode if and only if the following condition is fullfilled \begin{align} \sum_{j=1}^\infty \frac{1}{\lambda_j} \, = \, \infty. \end{align} This is the same condition of non-explosion holding for the classical case. Such a condition ceases to be true if we consider a L\'evy exponent with $\alpha \neq 0$, which is related to the so-called killed subordinator. In this case, indeed, the process $\mathcal{N}^f(t)$ can explode in a finite time, even if $\mathcal{N}(t)$ does not; more precisely \begin{align} \Pr \ll \mathcal{N}^f(t)= \infty \rr = 1-e^{-\alpha t}. \end{align} We note that $\mathcal{N}^f(t)$ can be regarded as a process where upward jumps are separated by exponentially distribuited time intervals $Y_k$ such that \begin{align}
\Pr \ll Y_k>t | \mathcal{N}^f(T_{k-1})=r \rr= e^{-f(\lambda_r) t} \end{align} where $T_{k-1}$ is the instant of the ($k-1$)-th jump.
In section 3 we study the subordinated linear and sublinear death processes, that we respectively denote by $M^f(t)$ and $\mathbb{M}^f(t)$, with an initial number of components $n_0$. We emphasize that in the sublinear case the annihilation is initially slower, then accelerates when few survivors remain. So, despite $M^f(t)$ and $\mathbb{M}^f(t)$ present different state probabilities, we observe that the extinction probabilities coincide and we prove that they decrease for increasing values of $n_0$.
In section 4, the subordinated linear birth-death process $L^f(t)$ is considered. If the birth and death rates coincide and $H^f$ is a stable subordinator, we compute the mean sojourn time in each state and find, in some particular cases, the distribution of the intertimes between successive jumps. We finally study the probability density of the sojourn times, by giving a sketch of the derivation of their Laplace transforms.
\section{Subordinated non-linear birth process} We consider in this section the process $ \mathcal{N}^f(t)= \mathcal{N}(H^f(t))$, where $\mathcal{N}$ is a non-linear birth process with one progenitor and rates $\lambda _k$, $k \geq 1 $, and $H^f(t)$ is a subordinator independent from $\mathcal{N}(t)$. It is well known that the state probabilities of $\mathcal{N}(t)$ read \begin{align}
\Pr \ll\mathcal{N}(t)=k| \mathcal{N}(0)=1 \rr = \, & \begin{cases} \prod_{j=1}^{k-1} \lambda _ j\sum_{m=1}^k \frac{e^{-\lambda_m t }}{ \prod_{l=1 , l \neq m}^k (\lambda_l-\lambda_m )}, \qquad & k>1, \\ e^{-t f(\lambda_1) }, & k=1. \end{cases} \end{align} The subordinated process $\mathcal{N}^f(t)$ thus possesses the following distribution: \begin{align}
\Pr \ll \mathcal{N}^f(t)=k|\mathcal{N}^f(0)=1 \rr \, = \, & \int_0^\infty \Pr \ll \mathcal{N}(s)=k | \mathcal{N}(0)=1\rr \Pr \ll H^f(t) \in ds \rr \notag \\
\, = \, & \begin{cases} \prod_{j=1}^{k-1} \lambda _ j\sum_{m=1}^k \frac{e^{-t \, f(\lambda _m )}}{ \prod_{l=1 , l \neq m}^k (\lambda_l-\lambda_m )}, \qquad & k>1, \\ e^{-t f(\lambda_1) }, & k=1. \end{cases} \label{non linear birth 1 progenitor} \end{align} The distribution \eqref{non linear birth 1 progenitor} can be easily generalised to the case of $r$ progenitors and reads \begin{align}
\Pr \ll \mathcal{N}^f(t)=r+k | \mathcal{N}^f(0)=r \rr \, = \,\begin{cases} \prod_{j=r}^{r+k-1} \lambda _ j\sum_{m=r}^{r+k} \frac{e^{-tf(\lambda _m) }}{ \prod_{l=r , l \neq m}^{r+k} (\lambda_l-\lambda_m )}, \quad & k>0, \\ e^{-tf(\lambda_r) }, & k=0. \end{cases} \label{23} \end{align} The subordinated process $\mathcal{N}^f(t)$ is time-homogeneous and Markovian. So, the last formula permits us to write \begin{align}
&\Pr \ll \mathcal{N}^f(t+dt)=r+k | \mathcal{N}^f(t)=r \rr \notag \\
= \, & \begin{cases} \prod_{j=r}^{r+k-1} \lambda _ j\sum_{m=r}^{r+k} \frac{1-dt f(\lambda_m )}{ \prod_{l=r , l \neq m}^{r+k} (\lambda_l-\lambda_m )}, \quad & k>0, \\
1-dt f(\lambda_r), & k=0. \label{rprog} \end{cases} \end{align} To find an alternative expression for the transition probabilities we need the following \begin{lem} For any sequence of $k+1$ distinct positive numbers $\lambda_r, \lambda_{r+1} \cdots \lambda _{r+k}$ the following relationship holds: \begin{align} c_{r,k}=\sum_{m=r}^{r+k} \frac{1}{ \prod_{l=r , l \neq m}^{r+k} (\lambda_l-\lambda_m )}=0. \label{Vandermonde} \end{align} \end{lem} \begin{proof} It is a consequence of \eqref{23} by letting $t \to 0$. An alternative proof can be obtained by suitably adapting the calculation in Theorem 2.1 of \cite{orspolber4}. \end{proof} We are now able to state the following theorem. \begin{te} For $k>r$ the transition probability takes the form \begin{align}
\Pr \ll \mathcal{N}^f(t+dt)=k| \mathcal{N}^f(t)=r \rr = dt\, \int _0 ^{\infty} \Pr \ll \mathcal{N}(s)=k|\mathcal{N}(0)=r \rr \nu (ds) \end{align} \end{te} \begin{proof} By repeatedly using both \eqref{Vandermonde} and the representation \eqref{Bernstein} of the Bern\v{s}tein functions $f$, we have that \begin{align}
\Pr \ll \, \mathcal{N}^f(t+dt)=k | \mathcal{N}^f(t)=r \rr \, &= \, \prod_{j=r}^{r+k-1} \lambda_ j\sum_{m=r}^{r+k} \frac{1-dt f(\lambda _m )}{ \prod_{l=r , l \neq m}^{r+k} (\lambda_l-\lambda_m )} \notag \\ &= \, -dt\prod_{j=r}^{r+k-1} \lambda_ j\sum_{m=r}^{r+k} \frac{f(\lambda _m )}{ \prod_{l=r , l \neq m}^{r+k} (\lambda_l-\lambda_m )}\notag \\ & =- dt \int _0^\infty \prod_{j=r}^{r+k-1} \lambda _ j\sum_{m=r}^{r+k} \frac{1-e^{-\lambda_m s }}{ \prod_{l=r , l \neq m}^{r+k} (\lambda_l-\lambda_m )} \nu (ds) \notag \\
&= dt \int _0^\infty \prod_{j=r}^{r+k-1} \lambda _ j\sum_{m=r}^{r+k} \frac{e^{-\lambda_m s }}{ \prod_{l=r , l \neq m}^{r+k} (\lambda_l-\lambda_m )} \nu (ds) \label{ccc}. \end{align} In light of \eqref{Vandermonde}, the integrand in \eqref{ccc} is $ \mathcal{O}( s)$ for $s \to 0$. Reminding \eqref{misura di Levy}, this ensures the convergence of \eqref{ccc}, and the proof is thus complete. \end{proof} \begin{os} For the sake of completeness, we observe that in the case $k=0$ we have \begin{align}
\Pr \ll \, \mathcal{N}^f(t+dt)=r | \mathcal{N}^f(t)=r \rr \, = \, & 1-dt f(\lambda _r) \notag \\ = \, & 1-dt \int _0^\infty (1-e^{-\lambda _r s}) \nu(ds)\\
= \, & 1-dt \int _0^\infty (1-\Pr \ll \mathcal{N}(s)=r | \mathcal{N}(0)=r \rr) \nu(ds). \end{align} \end{os} \begin{os}
The subordinated non-linear birth process performs jumps of arbitrary height as the subordinated Poisson process (see, for example, \citet{orstoa}). Thus, in view of markovianity, we can write the governing equations for the state probabilities $p_k^f(t)= \Pr \ll \mathcal{N}^f(t)=k | \mathcal{N}^f(0) =1 \rr$. For $k>1$ we have that \begin{align} &\frac{d}{dt} p_k^f(t)\, = \, -f(\lambda_k) p_k^f(t)+ \sum_{r=1}^{k-1} p_r^f(t) \int _0^\infty \prod_{j=r}^{k-1} \lambda_j \sum_{m=r}^{k} \frac{e^{-\lambda_m s }}{ \prod_{l=r , l \neq m}^{k} (\lambda_l-\lambda_m )} \nu (ds), \end{align} while for $k=1$ \begin{align} &\frac{d}{dt} p_1^f(t) \, = \, - f(\lambda_1) p_1^f(t). \end{align} \end{os} \begin{os} The process $\mathcal{N}(H^f(t))$ presents positive and integer-valued jumps occurring at random times $T_1, T_2, \cdots T_n $. The inter-arrival times $Y_1, Y_2, \cdots Y_n $ are defined as \begin{align} Y_k= T_k-T_{k-1}. \end{align} It is easy to prove that \begin{align}
\Pr \ll Y_k >t |\mathcal{N}^f(T_{k-1})=r \rr= e^{-f(\lambda_r)t}. \end{align} This can be justified by considering that in the time intervals $[T_{k-1}, T_{k-1}+t]$, no new offspring appears in the population and thus, by \eqref{rprog}, we have \begin{align}
\Pr \ll Y_k >t |\mathcal{N}^f(T_{k-1})=r \rr= \Pr \ll \mathcal{N}^f(t+T_{k-1})=r|\mathcal{N}^f(T_{k-1})=r)\rr= e^{-f(\lambda_r)t}. \end{align} \end{os}
\subsection{Condition of explosion for the subordinated non-linear birth process} We note that the explosion of the process $\mathcal{N}^f(t)$, $t>0$, in a finite time is avoided if and only if \begin{align} T_{\infty}= Y_1+Y_2 \cdots Y_{\infty}= \infty \end{align} where $Y_j$ , $j \geq 1$, are the intertimes between successive jumps (see \cite{grimmet}, p. 252). For the non-linear classical process we have that \begin{align} \mathbb{E} e^{-T_{\infty}}\, = \, & \mathbb{E}e^{- \sum _{j=1}^{\infty}Y_j}= \lim_{n \to \infty} \prod_{j=1}^{n} \mathbb{E} e^{-Y_j}= \lim_{n \to \infty} \prod _{j=1}^n \frac{\lambda _j}{1+\lambda _j} \notag \\
= \, & \prod _{j=1}^{\infty} \frac{1}{1+\frac{1}{\lambda _j}}= \frac{1}{1+ \sum _{j=1}^{\infty}\frac{1}{\lambda _j}+ \cdots}. \end{align}
So, if $\sum_{j=1}^\infty \frac{1}{\lambda_j}= \infty$ we have $e^{-T_{\infty}}=0$ a.s., that is $T_{\infty}=\infty$. Therefore, for the subordinated non-linear birth process we have that \begin{align} \Pr \ll \mathcal{N}^f(t) < \infty \rr \, = \, & \int _0 ^{\infty} \sum _{k=1}^{\infty} \Pr \ll \mathcal{N}(s)=k \rr \Pr\ll {H^f(t) \in ds}\rr \notag \\ = \, & \int_0^{\infty} \Pr\ll {H^f(t) \in ds}\rr =1, \qquad \forall t>0. \end{align} Instead, if $\sum _{j=1}^{\infty} \frac{1}{\lambda _j} <\infty$, we get $\sum _{k=1}^{\infty} \Pr \ll \mathcal{N}(s)=k \rr < \infty $, and this implies that $\Pr \ll \mathcal{N}^f(t)<\infty \rr <1$.
We can now consider the case of killed subordinators $\mathcal{H}^g(t)$, defined as \begin{align} \mathcal{H}^g(t)= \begin{cases} H^f(t), &\qquad t< T,\\ \infty, &\qquad t \geq T, \end{cases} \end{align} where $T\sim Exp(\alpha)$ and $H^f(t)$ is an ordinary subordinator related to the function $f(x)= \int _0^{\infty}(1-e^{-sx}) \nu(ds)$. It is well-known that $\mathcal{H}^g(t)$ is related to a Bern\v{s}tein function \begin{align} g(x)= \alpha+f(x). \end{align} In this case, even if $\sum_{j=1}^\infty \frac{1}{\lambda_j}= \infty$ , the probability of explosion for $\mathcal{N}^f(t)$ is positive and equal to \begin{align} \Pr \ll \mathcal{N}^f(t)= \infty \rr = 1- e^{-t \alpha}. \end{align} This can be proven by observing that \begin{align} \Pr \ll \mathcal{N}^f(t) < \infty \rr \, = \, & \int _0 ^{\infty} \sum _{k=1}^{\infty} \Pr \ll \mathcal{N}(s)=k \rr \Pr\ll {H^f(t) \in ds}\rr \notag \\
= \, & \int _0^{\infty} \Pr\ll {H^f(t) \in ds}\rr = \int _0^{\infty} e^{- \mu s} \Pr\ll {H^f(t) \in ds}\rr\bigg|_{\mu =0} \notag \\
= \, & e^{- \alpha t- f(\mu) t}\bigg|_{\mu =0}= e^{-\alpha t}. \end{align} If, instead, $\sum_{j=1}^\infty \frac{1}{\lambda_j}< \infty$, we have $\sum _{k=1}^{\infty} \Pr \ll \mathcal{N}(s)=k \rr <1$ and, a fortiori, $\Pr \ll \mathcal{N}^f(t)<\infty \rr < e ^{-\alpha t}$.
\subsection{Subordinated linear birth process} The subordinated Yule-Furry process $N^f(t)$ with one initial progenitor possesses the following distribution \begin{align} p_k^f(t) \, = \, & \int _0 ^ {\infty} e^{- \lambda s } (1-e^{- \lambda s })^{k-1} \Pr \lbrace H^ {f}(t) \in ds \rbrace \notag \\ = \, & \int _0 ^ {\infty} e^{- \lambda s }\sum _{J=0}^{k-1} \binom{k-1}{j}(-1)^j e^{- \lambda sj }\Pr \lbrace H^ {f}(t) \in ds \rbrace\notag \\ = \, & \sum _{j=0}^{k-1}\binom{k-1}{j} (-1)^j \int _0 ^{\infty} e^{- s(\lambda + \lambda j) } \Pr \lbrace H^ {f}(t) \in ds \rbrace \notag \\ = \, & \sum _{j=0}^{k-1}\binom{k-1}{j} (-1)^j e^{-t\, f(\lambda (j+1))}. \end{align} Of course, this is obtainable from the distribution $\mathcal{N}^f(t)$ by assuming that $\lambda _j=\lambda j$ We now compute the factorial moments of the subordinated linear birth process. The probability generating function is \begin{align} G^f(u,t)= \sum_{k=1}^\infty u^k \int_0^\infty e^{-\lambda s} (1-e^{-\lambda s})^{k-1} \Pr (H^f(t) \in ds). \end{align} The $r$-th order factorial moments are \begin{align}
& \frac{\partial^r}{\partial u^r} G^f(u,t)\bigg|_{u=1} \notag \\ = \, & \sum_{k=r}^\infty k(k-1)\cdots(k-r+1) \int_0^\infty e^{-\lambda s} (1-e^{-\lambda s})^{k-1} \Pr \ll H^f(t) \in ds\rr \notag \\ = \, & \sum_{k=r}^\infty k(k-1)\cdots(k-r+1) \int_0^\infty e^{-\lambda s} (1-e^{-\lambda s})^{k-r} (1-e^{-\lambda s})^{r-1}\Pr \ll H^f(t) \in ds \rr \end{align} and since \begin{align} \sum_{k=r}^\infty k(k-1)...(k-r+1)(1-p)^{k-r} = (-1)^r\frac{d^r}{dp^r} \sum_{k=0}^\infty (1-p)^k = (-1)^r\frac{d^r}{dp^r} \frac{1}{p}= \frac{r!}{p^{r+1}} \end{align} we have that \begin{align}
\frac{\partial^r}{\partial u^r} G(u,t)\bigg|_{u=1} \, = \, & r!\int_0^\infty e^{\lambda r s }(1-e^{-\lambda s})^{r-1} \Pr \ll H^f(t) \in ds \rr \notag \\ = \, & r! \sum_{m=0} ^{r-1} \begin{pmatrix} r-1 \\ m \end{pmatrix}(-1)^m \int_0^\infty e^{- \lambda s(m-r)} \Pr \ll H^f(t) \in ds\rr \\
= \, & r! \sum_{m=0} ^{r-1} \begin{pmatrix} r-1 \\ m \end{pmatrix}(-1)^m e^{-t f(\lambda (m-r))}. \end{align} By $f(-x)$, $x>0$ we mean the extended Bern\v{s}tein function, having representation \begin{align} f(-x)=\int_0^{\infty} (1-e^{sx}) \nu(ds), \qquad x>0, \label{extended bernstein} \end{align} provided that the integral in \eqref{extended bernstein} is convergent. In particular, we infer that \begin{align} \mathbb{E}(\mathcal{N}^f(t))=e^{-tf(-\lambda)} \end{align} and \begin{align} \textrm{Var} ( \mathcal{N}^f(t) )= 2e^{-tf(-2\lambda)}- e^{-tf(-\lambda)}-e^{-2tf(-\lambda)}. \end{align} For a stable subordinator, that is with L\'evy measure $\nu(ds)= \frac{\alpha s^{-\alpha -1}}{\Gamma(1-\alpha)}ds$ , $\alpha \in (0,1)$, all the factorial moments are infinite. Instead, for a tempered stable subordinator, where $\nu(ds)= \frac{\alpha e^{-\theta s} s^{-\alpha -1}}{\Gamma(1-\alpha)}ds$, $\alpha \in (0,1)$ and $\theta>0$, only the factorial moments of order $r$ such that $r< \frac{\theta}{\lambda}$ are finite. If we then consider the Gamma subordinator, with $\nu(ds)= \frac{e^{-\alpha s }}{s}ds$, only the factorial moments of order $r$ such that $r< \frac{\alpha}{\lambda}$ are finite.
\subsection{Fractional subordinated non-linear birth process} The fractional non-linear birth process has state probabilities $ p_k^{\nu}(t)$ solving the fractional differential equation \begin{align} \frac{d^{\nu}p_k^{\nu}(t)}{dt^{\nu}}= -\lambda _k p_k^{\nu}(t) +\lambda _{k-1}p_{k-1}^{\nu}(t) \qquad \nu \in (0,1), k \geq 1 \end{align} with initial condition \begin{align} p_k^{\nu}(0)= \begin{cases} 1, \qquad &k=1, \\ 0, & k>1. \end{cases} \end{align} The state probabilities read (see Orsingher and Polito \cite{orspolber1}) \begin{align}
p_k^{\nu}(t)= \Pr \ll \mathcal{N}^{\nu}(t)=k | \mathcal{N}^{\nu}(0)=1 \rr= \prod_{j=1}^{k-1} \lambda_j \sum_{m=1}^{k} \frac{E_{\nu,1}(-\lambda_m t^{\nu} )}{ \prod_{l=1 , l \neq m}^{k} (\lambda_l-\lambda_m )} \qquad \nu \in (0,1), \end{align} where \begin{align} E_{\nu,1}(-\eta t^{\nu})= \frac{\sin (\nu \pi) }{\pi} \int _0 ^{\infty} \frac{r^{\nu -1 } e^{-r \eta ^{\frac{1}{\nu}}t}}{r^{2 \nu}+2r^{\nu}\cos(\nu \pi)+1}dr \end{align} is the Mittag-Leffler function (see formula (7.3) in \citet{saxena}). So, the subordinated non-linear fractional birth process has distribution \begin{align}
& \Pr \ll \mathcal{N}^{\nu}(H^f(t))=k | \mathcal{N}^{\nu}(0)=1 \rr \notag \\ = \, & \prod_{j=1}^{k-1} \lambda_j \sum_{m=1}^{k} \frac{1}{ \prod_{l=1 , l \neq m}^{k} (\lambda_l-\lambda_m )}\frac{\sin (\nu \pi) }{\pi} \int _0 ^{\infty} \frac{r^{\nu -1 } e^{-tf(r \lambda_m ^{\frac{1}{\nu}})}}{r^{2 \nu}+2r^{\nu}\cos(\nu \pi)+1}dr. \end{align}
\section{Subordinated death processes} We now consider the process $M^f(t)= M(H^f(t))$, where $M$ is a linear death process with $n_0$ progenitors. The state probabilities read \begin{align}
&\Pr \ll M^f(t)=k | M^f(0) = n_0 \rr = \int _0^{\infty} \binom{n_0}{k} e^{-\mu ks}(1-e^{-\mu s}) ^{n_0-k} \Pr \ll H^f(t) \in ds \rr \notag \\
= \, & \begin{pmatrix} n_0 \\ k \end{pmatrix} \sum _ {j=0} ^ {n_0-k}\begin{pmatrix} n_0-k \\ j \end{pmatrix}(-1)^j \int _0 ^ {\infty}e^{- (\mu k + \mu j)s }\Pr \ll H^f(t) \in ds \rr \notag \\ = \, & \begin{pmatrix} n_0 \\ k \end{pmatrix} \sum _ {j=0} ^ {n_0-k}\begin{pmatrix} n_0-k \\ j \end{pmatrix}(-1)^j e^ {-tf(\mu k + \mu j)}. \end{align} In particular, the extinction probability is \begin{align}
\Pr \ll M^f(t)=0 | M^f(0) = n_0 \rr \,= & \sum _ {j=0} ^ {n_0}\begin{pmatrix} n_0 \\ j \end{pmatrix}(-1)^j e^ {-tf(\mu j)}\notag \\ = \, & 1+ \sum _ {j=1} ^ {n_0}\begin{pmatrix} n_0 \\ j \end{pmatrix}(-1)^j e^ {-tf(\mu j)} \end{align} and converges to $1$ exponentially fast with rate $f(\mu)$. \begin{os} We observe that the extinction probability is a decreasing function of $n_0$ for any choice of the subordinator $H^f(t)$. This can be shown by observing that \begin{align}
&\Pr \ll M^f(t)=0 |M^f(0)=n_0 \rr -\Pr \ll M^f(t)=0 |M^f(0)=n_0-1 \rr \notag \\ = \, & \sum _ {j=1} ^ {n_0}\begin{pmatrix} n_0 \\ j \end{pmatrix}(-1)^j e^ {-tf(\mu j)} -\sum _ {j=1} ^ {n_0-1}\begin{pmatrix} n_0-1 \\ j \end{pmatrix}(-1)^j e^ {-tf(\mu j)} \notag \\ = \, & \sum _ {j=1} ^ {n_0-1}\begin{pmatrix} n_0-1 \\ j-1 \end{pmatrix}(-1)^j e^ {-tf(\mu j)}+(-1)^{n_0}e^{-tf(\mu n_0)}\notag \\ = \, & \sum _ {j=1} ^ {n_0}\begin{pmatrix} n_0-1 \\ j-1 \end{pmatrix}(-1)^j e^ {-tf(\mu j)} \notag \\ = \, & -\sum _ {j=0} ^ {n_0-1}\begin{pmatrix} n_0-1 \\ j \end{pmatrix}(-1)^j e^ {-tf(\mu (j+1)} \notag \\ = \, &-\int_0 ^{\infty} \sum _ {j=0} ^ {n_0-1}\begin{pmatrix} n_0-1 \\ j \end{pmatrix}(-1)^j e^ {-s\mu (j+1)} \Pr \ll H^f(t) \in ds \rr \notag \\ = \, &-\int_0^\infty e^{-\mu s}(1-e^{-\mu s})^{n_0-1}\Pr \ll H^f(t) \in ds \rr < 0. \end{align} This permits us also to establish the following upper bound which is valid for all values of $n_0$. \begin{align}
\Pr \ll M^f(t)=0 |M^f(0)=n_0 \rr < \Pr \ll M^f(t)=0 |M^f(0)=1 \rr = 1-e^{-tf(\mu)}. \end{align} We also infer that \begin{align*}
& \Pr \ll M^f(t)= k|M^f(0)=n_0 \rr = \\ & \Pr \ll M^f(t)=k|M^f(0)=n_0 -1 \rr - \frac{1}{n_0} \Pr \ll M^f(t)=1|M^f(0)=n_0 \rr \qquad \forall k< n_0 \end{align*} \end{os} \begin{os} The probability generating function of the subordinated linear death process is \begin{align} G(u,t)= \int_0^\infty (ue^{- \mu s}+1-e^{- \mu s})^{n_0} \Pr \ll H^f(t) \in ds \rr. \end{align} We now compute the factorial moments of order $r$ for the process $M^f(t)$: \begin{align} &\mathbb{E} \bigl ( M^f(t) (M^f(t)-1)(M^f(t)-2) \cdots (M^f(t)-r+1) \bigr ) \notag \\
= \, &\int_0^\infty \frac{\partial^r}{\partial u^r} (ue^{- \mu s}+1-e^{- \mu s})^{n_0}|_{u=1} \Pr \ll H^f(t) \in ds \rr \notag \\
= \, & n_0(n_0-1)(n_0-2)...(n_0-r+1) \int_0^\infty e^ {-\mu r s } \Pr \ll H^f(t) \in ds \rr \notag \\
= \, & n_0(n_0-1)(n_0-2)...(n_0-r+1) e^{-tf(\mu r )} \notag \\ = \, & r! \binom{n_0}{r} e^{-tf(\mu r )} \qquad \qquad \textrm{for } r \leq n_0. \end{align} In particular, we extract the expressions \begin{align} \mathbb{E} \, M^f(t)= n_0 e^ {-t \, f(\mu)} \end{align} and \begin{align} \textrm{Var} \,M^f(t)= n_0 e^{-tf(\mu)}-n_0 e^{-tf(2 \mu)} + n_0^2 e^{-t f(2 \mu)} -n_0 ^2 e^{-2t f(\mu)}. \end{align} The variance can be also be obtained as \begin{align}
\textrm{Var} \, M^f(t)= \, & \mathbb{E} \ll \textrm{Var} \, (M(H^f(t))|H^f(t)) \rr + \textrm{Var} \, \ll \mathbb{E}(M(H^f(t))|H^f(t)) \rr \notag \\ = \, & \mathbb{E}\bigl ( n_0 e^{-\mu H^f(t)}(1-e^{-\mu H^f(t)}) \bigr )+ \textrm{Var}\, ( n_0 e ^{-\mu H^f(t)})\notag \\ = \, & n_0 e^{-tf(\mu)}-n_0 e^{-tf(2 \mu)} + n_0^2 e^{-t f(2 \mu)} -n_0 ^2 e^{-2t f(\mu)}. \end{align} \end{os} \begin{os} The transition probabilities \begin{align}
\Pr \ll M^f(t_0+t)=k|M^f(t_0)=r \rr = \binom{r}{k} \sum_{j=0}^{r-k}\binom{r-k}{j} (-1)^j e^{-tf(\mu k + \mu j)} \end{align} permit us to write, for a small time interval $[t,t+dt)$, \begin{align}
& \Pr \ll M^f(t_0+dt)=k | M^f(t_0)=r \rr \notag \\ = \, & \binom{r}{k} \sum_{j=0}^{r-k}\binom{r-k}{j}(-1)^j (1-dt \,f(\mu k + \mu j) ) \notag \\ = \, & -dt \binom{r}{k} \sum_{j=0}^{r-k}\binom{r-k}{j}(-1)^j \, f(\mu k + \mu j) \notag \\ = \, & -dt \binom{r}{k} \sum_{j=0}^{r-k}\binom{r-k}{j}(-1)^j \, \int _0 ^\infty (1-e^{-(\mu k + \mu j)s}) \nu (ds) \notag \\
= \, & dt \binom{r}{k} \int _0 ^\infty \sum _ {j=0} ^ {r-k}\binom{r-k}{j}(-1)^j \, e^{- \mu js} e^{-\mu k s} \nu (ds) \notag \\ = \, & dt \int _0 ^\infty \binom{r}{k} (1-e^{- \mu s})^{r-k} e^{-\mu k s} \nu (ds)\notag \\
= \, & dt \int_0^\infty \Pr \ll M(s) = k | M(0) = r \rr \nu(ds) \label{trans prob death process} \qquad 0 \leq k <r \leq n_0 \end{align} It follows that the subordinated death process decreases with downwards jumps of arbitrary size. Formula \eqref{trans prob death process} is a special case of \eqref{14b} for the linear death process. \end{os} \begin{os} If $M^f(t_0)=r$, the probability that the number of individuals does not change during a time interval of length $t$ is \begin{align}
\Pr \ll M^f(t_0+t)=r| M^f(t_0)=r \rr \, = e^{-tf(r \mu)}. \label{312} \end{align} As a consequence, the random time between two successive jumps has exponential distribution with rate $f(\mu r)$, i.e. \begin{align} T_r \sim \textrm{Exp}(f(\mu r)). \end{align} From \eqref{312} we have also that \begin{align}
\Pr \ll M^f(t+dt) = r | M^f(t) = r \rr \, = \, 1-dtf(\mu r). \end{align} \end{os} \begin{os}
In view of \eqref{trans prob death process} we can write the governing equations for the transition probabilities $p_k^f(t)= \Pr \ll M^f(t)=k|M^f(0)=n_0 \rr$, for $0 \leq k \leq n_0$ \begin{align} \frac{d}{dt} p_k^f(t)= -p_k^f(t)f(\mu k)+ \sum_{j=k+1}^{n_0} p_j^f(t) \int _0 ^\infty \binom{j}{k} (1-e^{- \mu s})^{j-k} e^{-\mu k s} \nu (ds) . \end{align} \end{os} \subsection{The subordinated sublinear death process} In the sublinear death process we have that, for $0 \leq k \leq n_0$, \begin{align}
\Pr \ll \mathbb{M}(t+dt)=k-1| \mathbb{M}(t)=k, \mathbb{M}(0)=n_{0}\rr= \mu (n_0-k+1)dt+o(dt) \end{align} so that the probability that a particle disappears in $[t,t+dt)$ is proportional to the number of deaths occurred in $[0,t)$. It is well-known that \begin{align}
\Pr \ll \mathbb{M}(t)=k| \mathbb{M}(0)=n_0\rr \, = \, \begin{cases} e^{-\mu t}(1-e^{-\mu t })^{n_0-k}, \qquad &k=1,2, \dots, n_0, \\ (1-e^{-\mu t})^{n_0} , & k=0. \end{cases} \end{align} So, the probability law of the subordinated process immediately follows \begin{align}
&\Pr \ll \mathbb{M}^f(t)=k| \mathbb{M}^f(0)=n_0\rr \notag \\
= \, & \begin{cases} \sum_{j=0}^{n_0-k} \begin{pmatrix} n_0-k \\ j \end{pmatrix} (-1)^j e^{- tf(\mu (j+1))}, \qquad &k=0,1, \dots, n_0, \\ \sum _{k=0}^{n_0} \begin{pmatrix} n_0 \\k \end{pmatrix} (-1)^k e^{-t f( \mu k) }, &k=0 \end{cases} \end{align} The extinction probability is a decreasing function of $n_0$ as in the sublinear death process. Furthermore we observe that the extinction probabilities for the subordinated linear and sublinear death process coincide.
\section{Subordinated linear birth-death processes} In this section we consider the linear birth and death process $L(t)$ with one progenitor at the time $H^f(t)$. We recall that, for $k\geq 1$ (see \citet{bailey}, page 90), \begin{align}
\Pr \ll L(t)=k | L(0) =1 \rr = \begin{cases} \frac{(\lambda-\mu)^2e^{-(\lambda-\mu)t}(\lambda(1-e^{-(\lambda-\mu)t}))^{k-1}}{(\lambda-\mu e^{-(\lambda-\mu)t})^{k+1}}, \qquad & \lambda > \mu, \\ \frac{(\mu-\lambda)^2 e^{-(\mu-\lambda)t}(\lambda(1-e^{-(\mu-\lambda)t}))^{k-1}}{(\mu-\lambda e^{-(\mu-\lambda)t})^{k+1}}, & \lambda < \mu, \\ \frac{(\lambda t)^{k-1}}{(1+\lambda t)^{k+1}}, & \lambda = \mu. \end{cases} \end{align} while the extinction probabilities have the form \begin{align}
\Pr \ll L(t)=0 |L(0) = 1 \rr = \begin{cases} \frac{\mu-\mu e^{-t(\lambda-\mu)}}{\lambda-\mu e ^{-t(\lambda- \mu)}}, \qquad & \lambda > \mu,\\ \frac{\mu-\mu e^{-t(\mu- \lambda)}}{\lambda-\mu e ^{-t(\mu- \lambda)}}, & \mu > \lambda, \\ \frac{\lambda t}{1+\lambda t}, & \lambda= \mu. \end{cases} \end{align} We now study the subordinated process $L^f(t)=L(H^f(t))$. When $\lambda \neq \mu$, after a series expansion we easily obtain that \begin{align}
&\Pr \ll L^f(t)=k |L^f(0) =1 \rr \notag \\ = \, & \begin{cases}
\l \frac{\lambda - \mu }{\lambda} \r^2 \sum_{l=0}^\infty \binom{l+k}{l} \l \frac{\mu }{\lambda} \r^l \sum_{r=0}^{k-1} (-1)^r \binom{k-1}{r} e^{-t f \l \l \lambda - \mu \r \l l+r+1 \r \r} , \qquad &\lambda >\mu, \\
\l \frac{\mu -\lambda}{\mu} \r^2 \l \frac{\lambda}{\mu} \r^{k-1} \sum_{l=0}^\infty \binom{l+k}{l} \l \frac{\lambda }{\mu} \r^l \sum_{r=0}^{k-1} (-1)^r \binom{k-1}{r} e^{-t f\l \l \mu - \lambda \r \l l+r+1 \r \r}, & \lambda <\mu, \end{cases} \end{align} provided that $k \geq 1$. Moreover, the extinction probabilities have the following form \begin{align} \Pr \ll L^f(t)=0 \rr \, = \, \begin{cases} \frac{\mu - \lambda}{\lambda} \l \sum_{m=1}^\infty \l \frac{\mu}{\lambda} \r^m e^{-tf\l (\lambda - \mu)m \r} \r+\frac{\mu}{\lambda}, \qquad &\lambda>\mu, \\
1- \l \frac{\mu -\lambda}{\lambda} \r \sum_{m=1}^\infty \l \frac{\lambda}{\mu} \r^m e^{-tf \l \l \mu - \lambda \r m \r}, &\lambda <\mu. \end{cases} \end{align} Similarly to the classical process, we have \begin{align} \lim_{t \to \infty} \Pr \ll L^f(t)=0 \rr = \begin{cases} \frac{\mu}{\lambda}, \qquad &\lambda > \mu, \\ 1, & \lambda < \mu. \end{cases} \end{align} \subsection{Processes with equal birth and death rates} We concentrate ourselves on the case $\lambda= \mu$, which leads to some interesting results. The extinction probability reads
\begin{align}
\Pr \ll L^f(t) = 0 | L^f(0) =1 \rr \, = \, & \int_0^\infty \frac{\lambda s}{1+\lambda s} \Pr \ll H^f(t) \in ds \rr \notag \\ = \, &1- \int_0^\infty \frac{1}{1+\lambda s} \Pr \ll H^f(t) \in ds \rr \notag \\ = \, & 1-\int_0^\infty \Pr \ll H^f(t) \in ds \rr \int_0^\infty dw \, e^{-w\lambda s} \, e^{-w} \notag \\ = \, &1-\int_0^\infty dw \, e^{-w} e^{-tf(\lambda w)} \label{exctinction probability}. \end{align} We note that \begin{align}
\lim_{t \to \infty} \Pr \ll L^f(t)=0 |L^f(0) = 1 \rr = 1 \end{align} as in the classical case. From \eqref{exctinction probability} we infer that the distribution of the extinction time $T_0^f = \inf \ll t \geq 0 : L^f(t) = 0 \rr$, has the following form \begin{align} \Pr \ll T_0^f \in dt \rr / dt \, = \, \int_0^\infty e^{-w} f(\lambda w) e^{-tf(\lambda w)} dw. \end{align} We now observe that all the state probabilities of the process $L(t)$ depend on the extinction probability (see \cite{orspolber1}) \begin{align}
\Pr \ll L(t) = k |L(0) =1 \rr \, = \, & \frac{(\lambda t)^{k-1}}{\l 1+\lambda t \r^{k+1}} \qquad \qquad \qquad \qquad \qquad \qquad \qquad k\geq 1 \notag \\ = \, & \frac{(-1)^{k-1} \lambda^{k-1}}{k!} \frac{d^k}{d\lambda^k} \l \frac{\lambda}{1+\lambda t} \r \notag \\ = \, & \frac{(-1)^{k-1} \lambda^{k-1}}{k!} \frac{d^k}{d\lambda^k} \l \lambda \l 1-\Pr \ll L(t)=0\rr \r \r. \label{state prob for L(t)} \end{align} Hence, the state probabilities of $L^f(t)$ can be written, for $k \geq 1$, as \begin{align}
&\Pr \ll L^f(t) = k |L^f(0) =1 \rr \, \notag \\ = \, &\frac{(-1)^{k-1} \lambda^{k-1}}{k!} \frac{d^k}{d\lambda^k}\left[ \lambda \int_0^\infty \l 1-\Pr \ll L(s)=0 \rr \r \Pr \ll H^f(t) \in ds \rr \right] \notag \\ = \, & \frac{(-1)^{k-1} \lambda^{k-1}}{k!} \frac{d^k}{d\lambda^k} \left[ \lambda \l 1- \Pr \ll L^f(t) = 0 \rr \r \right] \notag \\ =\, &\frac{(-1)^{k-1} \lambda^{k-1}}{k!} \frac{d^k}{d\lambda^k} \left[ \lambda \int_0^\infty dw \, e^{-w} e^{-tf(\lambda w)} \right]. \label{state probabilities L^f(t)} \end{align}
\subsection{Transition probabilities} To compute the transition probabilities of $L^f(t)$, we recall that the linear birth-death process with $r$ progenitors has the following probability law (see \cite{bailey}, page 94, formula 8.47): \begin{align}
\Pr \ll L(t)= n| L(0)=r \rr = \sum _{j=0}^{min(r,n)} \binom{r}{j} \binom{r+n-j-1}{r-1} \alpha ^{r-j} \beta ^{n-j}(1-\alpha-\beta)^{j}, \label{transition prob birth death} \end{align} where $n \geq 0$ and \begin{align} \alpha \,= \, \frac{\mu(e^{(\lambda-\mu)t}-1)}{\lambda e^{(\lambda-\mu)t}-\mu } \qquad \textrm{and} \qquad \beta \,= \, \frac{\lambda(e^{(\lambda-\mu)t}-1)}{\lambda e^{(\lambda-\mu)t}-\mu }. \end{align} In the case $\lambda= \mu$ we have \begin{align} \lim_{\mu \to \lambda} \alpha \, = \, \lim_{\mu \to \lambda} \beta = \frac{\lambda t}{1+\lambda t} \end{align} so that \begin{align}
&\Pr \ll L(t)=n|L(0)=r \rr \notag \\
= \, & \sum _{j=0}^{min(r,n)} \binom{r}{j} \binom{r+n-j-1}{r-1} \biggl ( \frac{\lambda t}{1+\lambda t}\biggr )^{r+n-2j}\biggl (1-2 \frac{\lambda t}{1+ \lambda t}\biggr)^{j} \notag \\ = \, & \sum _{j=0}^{min(r,n)} \sum_{k=0}^j \binom{r}{j} \binom{r+n-j-1}{r-1} \binom{j}{k}(-2)^k \biggl ( \frac{\lambda t}{1+\lambda t}\biggr )^{r+n-2j+k}. \label{mi serve per phillips nascita e morte 2} \end{align} One can check that for $r=1$ the last formula reduces to \begin{align}
\Pr \ll L(t)=n |L(0) =1 \rr= \frac{(\lambda t)^{n-1}}{(1+\lambda t)^{n+1}}. \end{align} The transition probabilities related to the subordinated process $L^f(t)$ can be written in an elegant form, as shown in the following theorem. \begin{te} In the subordinated linear birth-death process $L^f(t)$, when $\lambda=\mu$, $n\geq 0$, $r\geq 1$, $n \neq r$, we have that \begin{align}
&\Pr \ll L^f(t+t_0)=n |L^f(t_0)=r \rr \notag \\= \, &\sum _{j=0}^{min(r,n)} \sum_{k=0}^j \binom{r}{j} \binom{r+n-j-1}{r-1} \binom{j}{k}2^k \frac{ (-1)^{r+n-1} \lambda ^ {r+n+k-2j}}{(r+n-2j+k-1)!} \notag \\ & \times \frac{d^{r+n-2j+k-1}}{d \lambda ^{r+n-2j+k-1}} \biggl [ \frac{1}{\lambda}- \frac{1}{\lambda}\int_0^\infty dw \, e^{-w} e^{-tf(\lambda w)} \biggr ] \label{probabilità di transizione processo nascita morte subordinato} \end{align} \end{te} \begin{proof} By subordination we have \begin{align}
\Pr \ll L^f(t)=n | L^f(0)=r \rr \, = \, & \int _0^{\infty} \Pr \ll L(s)=n |L(0)=r \rr \Pr \ll H^f(t) \in ds \rr \notag \\
= \, & \sum _{j=0}^{min(r,n)} \sum_{k=0}^j \binom{r}{j} \binom{r+n-j-1}{r-1} \binom{j}{k}(-2)^k \notag \\
& \times \int_0 ^ {\infty} \Pr \ll H(t) \in ds \rr \biggl ( \frac{\lambda s}{1+\lambda s}\biggr )^{r+n-2j+k}. \end{align} To compute the last integral, we preliminarly observe that \begin{align} \frac{d^m}{d \lambda ^m} \frac{1}{1+\lambda s}= (-1)^m m!\, s^m \frac{1}{(1+\lambda s)^{m+1}} \end{align} and consequently \begin{align} \biggl ( \frac{\lambda s}{1+ \lambda s } \biggr )^m = \frac{(-1)^{m-1} s \, \lambda ^m}{(m-1)!}\frac{d^{m-1}}{d \lambda ^{m-1}} \frac{1}{1+\lambda s}. \label{mi serve per phillips in nascita e morte 1} \end{align} So, we have \begin{align}
&\Pr \ll L^f(t)=n|L^f(0)=r \rr \notag \\
= \, & \sum _{j=0}^{min(r,n)} \sum_{k=0}^j \binom{r}{j} \binom{r+n-j-1}{r-1} \binom{j}{k}2^k \frac{ (-1)^{r+n-1} \lambda^{r+n-2j+k}}{(r+n-2j+k-1)!} \notag \\ & \times \frac{d^{r+n-2j+k-1}}{d \lambda ^{r+n-2j+k-1}} \int _0 ^{\infty} \frac{s}{1+\lambda s} \Pr \ll H^f(t) \in ds \rr \end{align} where, by using \eqref{exctinction probability}, we write \begin{align}
\int _0 ^{\infty} \frac{s}{1+\lambda s} \Pr \ll H^f(t) \in ds \rr &= \frac{1}{\lambda} \int _0 ^{\infty} \frac{\lambda s}{1+\lambda s} \Pr \ll H^f(t) \in ds \rr \notag \\
&= \frac{1}{\lambda} \biggl [1-\int_0^\infty dw \, e^{-w} e^{-tf(\lambda w)} \biggr ] \end{align} and the desired result immediately follows. \end{proof} \begin{os} For a small time interval $dt$, the quantity in square brackets in (\ref{probabilità di transizione processo nascita morte subordinato}) can be written as \begin{align*} & \frac{1}{\lambda}- \frac{1}{\lambda} \int_0^{\infty} dw \, e^{-w}(1-dt f(\lambda w)) \\ & =dt \, \frac{1}{\lambda} \int_0^{\infty} dw \, e^{-w} \int_0^{\infty} \nu(ds) (1-e^{-\lambda w s})\\ & = dt \int_0^{\infty} \nu(ds) \frac{s}{1+\lambda s} \end{align*} Then, by using (\ref{mi serve per phillips in nascita e morte 1}) e (\ref{mi serve per phillips nascita e morte 2}), formula (\ref{probabilità di transizione processo nascita morte subordinato}) reduces to \begin{align*}
\Pr \ll L^f(t_0+dt)=n| L^f(t_0)=k \rr = dt\int_0^{\infty} \nu(ds) \Pr \ll L(s)=n | L(0)=k \rr \end{align*} thus proving relation (\ref{14b}) for subordinated birth-death processes. \end{os} \begin{os} If $L^f(0)=1$, from (\ref{state probabilities L^f(t)}) we have that the probability that the number of individuals does not change during a time interval of length $dt$ is \begin{align*}
\Pr \ll L^f(dt)=1|L^f(0)=1 \rr= 1-dt\, \frac{d}{d \lambda} \bigl (\lambda\int _0^{\infty}dw \, e^{-w}f(\lambda w) \bigr ) \end{align*} Thus the waiting time for the first jump, i.e. \begin{align*} T_1= inf \ll t>0: L^f(t) \neq 1 \rr , \end{align*} has the following distribution \begin{align} \Pr \ll T_1>t \rr = e^{-t \frac{d}{d \lambda} (\lambda\int _0^{\infty}dw \, e^{-w}f(\lambda w)}). \end{align} For example, in the case $H^f(t)$ is a stable subordinator with index $\alpha \in (0,1)$, $T_1$ has an exponential distribution with parameter $ \lambda ^{\alpha} \Gamma (\alpha +2)$. \end{os}
\subsection{Mean sojourn times} Let $V_k(t)$, $k\geq 1$ the total amount of time that the process $L(t)$ spends in the state $k$ up to time $t$, i.e. \begin{align} V_k(t)= \int_0 ^t I_k(L(s))\, ds, \end{align} where $I_k(.)$ is the indicator function of the state $k$. The mean sojourn time up to time $t$ is given by \begin{align}
\mathbb{E}V_k(t)=\int_0 ^t \Pr \ll L(s)=k |L(0) =1 \rr ds. \end{align} By means of \eqref{state prob for L(t)} we have that \begin{align}
\mathbb{E}V_k(t) \, = \, & \int_0^t \Pr \ll L(s)=k |L(0)=1 \rr ds \notag \\ = \, & \frac{(-1)^{k-1} \lambda^{k-1}}{k!} \frac{d^k}{d\lambda^k} \l \lambda \l t-\int_0^t \Pr \ll L(s)=0\rr ds \r \r \notag \\ = \, & \frac{(-1)^{k-1} \lambda^{k-1}}{k!} \frac{d^k}{d\lambda^k} \l \lambda \l t-\int_0^t \frac{\lambda s}{1+\lambda s}ds \r \r \notag \\ = \, & \frac{(-1)^{k-1} \lambda^{k-1}}{k!} \frac{d^k}{d\lambda^k} \log (1+\lambda t) \notag \\ = \, & \frac{1}{\lambda k} \l \frac{\lambda t}{1+\lambda t} \r^k \end{align} and the mean asymptotic sojourn time is therefore given by \begin{align} \mathbb{E}V_k(\infty)=\frac{1}{\lambda k}. \label{mean classic sojourn time} \end{align}
In view of \eqref{state probabilities L^f(t)}, for the sojourn time $V_k^f(t)$ of the subordinated process $L^f(t)$ we have that \begin{align}
\mathbb{E}V_k^f(t) \, = \, &\int_0^t \Pr \ll L^f(s)=k |L^f(0) =1 \rr ds \notag \\ = \, & \frac{(-1)^{k-1} \lambda^{k-1}}{k!} \frac{d^k}{d\lambda^k} \left[ \lambda \int_0^\infty dw \, e^{-w} \frac{1}{f(\lambda w)} \l 1- e^{-tf(\lambda w)} \r \right] \end{align} and the mean asymptotic sojourn time is given by \begin{align} \mathbb{E}V_k^f(\infty)= \frac{(-1)^{k-1} \lambda^{k-1}}{k!} \frac{d^k}{d\lambda^k} \left[ \lambda \int_0^\infty dw \, e^{-w} \frac{1}{f(\lambda w)} \right]. \end{align} It is possible to obtain an explicit expression for $ \mathbb{E}V_k^f(\infty)$ in the case of a stable subordinator, when $f(x)=x^{\alpha}$, $\alpha \in (0,1)$, i.e. \begin{align} \mathbb{E}V_k^f(\infty) \, = \, & \frac{(-1)^{k-1} \lambda^{k-1}}{k!} \frac{d^k}{d\lambda^k} \left[ \lambda \int_0^\infty dw \, e^{-w} \frac{1}{\lambda ^{\alpha}w^{\alpha}} \right] \notag \\ = \, & \frac{(-1)^{k-1} \lambda^{k-1}\Gamma (1-\alpha)}{k!} \frac{d^k}{d \lambda ^k} \lambda ^{1-\alpha} \notag\\ = \, & \frac{(-1)^{k-1} \lambda^{k-1}\Gamma (1-\alpha)}{k!} (1-\alpha)(-\alpha)(-\alpha-1)\cdots (-\alpha -k+1) \lambda ^{-\alpha -k+1} \notag\\
= \, & \frac{\Gamma(1-\alpha)\Gamma(\alpha+k)}{k!\Gamma(\alpha)\lambda^{\alpha}}\notag \\
= \, & \frac{B(1-\alpha, k+ \alpha)}{\Gamma(\alpha) \lambda^{\alpha}}, \qquad \textrm{for } k \geq 1. \label{tempo medio di soggiorno stabile} \end{align} In the case $\alpha= \frac{1}{2}$, by using the duplication formula for the Gamma function and the Stirling formula, the quantity in \eqref{tempo medio di soggiorno stabile} can be estimated, for large values of $k$, in the following way: \begin{align} \mathbb{E} V_k^f(\infty)=\frac{\Gamma(\frac{1}{2}+k)}{k! \sqrt{\lambda}} = \frac{\Gamma(\frac{1}{2})2^{1-2k}\Gamma(2k)}{k! \sqrt{\lambda}\Gamma(k)} \simeq \frac{1}{\sqrt{\lambda k}} \end{align} which is somehow related to \eqref{mean classic sojourn time}. We finally note that \begin{align} \frac{1}{(\alpha + k) \Gamma(\alpha) \lambda^\alpha} < \mathbb{E} V_k^f(\infty) < \frac{1}{(1-\alpha)\Gamma(\alpha)\lambda^{\alpha}}, \qquad \forall k \geq 1, \end{align} since \begin{align} \frac{1}{(\alpha +k)} < B(1-\alpha, k+\alpha)< \frac{1}{1-\alpha}. \end{align}
\subsection{On the distribution of the sojourn times} Let $L^f_k(t)$ be a linear birth-death process with $k$ progenitors. We now study the distribution of the sojourn time \begin{align} V_{k}(t) \, = \, \int_0^t I_k \l L^f_k(s) \r ds \end{align} which represents the total amount of time that the process spends in the state $k$ up to time $t$. We now define the Laplace transform \begin{align} r_k(\mu) \, = \, \int_0^\infty e^{-\mu t} \Pr \ll L^f_k(t) = k \rr dt. \end{align} The hitting time \begin{align} V_k^{-1}(t) \, = \, \inf \ll w > 0 : V_k(w) > t \rr \end{align} is such that \begin{align} \mathbb{E} \int_0^\infty e^{-\mu V_k^{-1}(t)} dt \, = \, & \mathbb{E} \int_0^\infty e^{-\mu t} dV_k(t) \notag \\ = \, & \mathbb{E} \int_0^\infty e^{-\mu t} I_k\l L_k^f(t) \r dt \notag \\ = \, &r_k(\mu). \label{14} \end{align} By Proposition 3.17, chapter V, of \cite{getoor} we have \begin{align} \mathbb{E} e^{-\mu V_k^{-1}(t)} \, = \, e^{-t\frac{1}{r_k(\mu)}}.\label{sss} \end{align} Now we resort to the fact that \begin{align} \Pr \ll V_k(t) > x \rr \, = \, \Pr \ll V_k^{-1}(x) < t \rr \end{align} and thus we can write \begin{align} \Pr \ll V_k(t) \in dx \rr /dx\, = \, -\frac{\partial}{\partial x} \int_0^t \Pr \ll V_k^{-1}(x) \in dw \rr. \label{inverso} \end{align} We therefore have that \begin{align} \frac{1}{dx}\int_0^\infty e^{-\mu t}\Pr \ll V_k(t) \in dx \rr dt \, & = -\frac{d}{dx} \int_0^{\infty} dt \, e^{-\mu t} \int_0^{t} \Pr \ll V_k^{-1}(x) \in dw\rr \notag \\ &= - \frac{d}{dx} \int_0^{\infty} dw \int _w^{\infty} dt \, e^{-\mu t} \Pr \ll V_k^{-1}(x) \in dw\rr \notag \\ &= -\frac{1}{\mu} \frac{d}{dx} \int_0^{\infty} dw\, e^{- \mu w} \Pr \ll V_k^{-1}(x) \in dw\rr \notag \\
& =-\frac{1}{\mu} \frac{d}{d x} e^{-x \frac{1}{r_k(\mu)}} \notag \\ &= \frac{1}{\mu \, r_k(\mu)}e^{-x \frac{1}{r_k(\mu)}}. \end{align} If $r_k(0)<\infty$, from \eqref{sss} it emerges that $\Pr \ll V_k^{-1}(t) < \infty \rr<1$; so the sample paths of $V_k(t)$ become constant after a random time with positive probability. This is related to the fact that the subordinated birth and death process extinguishes with probability one in a finite time when $\lambda = \mu$.
We finally observe that in the case $k=1$ by \eqref{state probabilities L^f(t)} we have \begin{align} r_1(\mu) \, = \, \int_0^{\infty} e^{-\mu t}\Pr \ll L^f(t)=k \rr dt \, = \, \frac{d}{d\lambda}\left[ \lambda \int_0^{\infty}dw \, e^{-w}\frac{1}{\mu+f(\lambda w)}\right], \end{align} provided that the Fubini Theorem holds true.
\end{document} |
\begin{document}
\title{
A Constructive Logic with Classical Proofs and Refutations\\
(Extended Version) }
\author{\IEEEauthorblockN{Pablo Barenbaum} \IEEEauthorblockA{ Departamento de Computación, \\ Facultad de Ciencias Exactas y Naturales, \\ Universidad de Buenos Aires, Argentina.\\ Universidad Nacional de Quilmes, Argentina.\\ Email: pbarenbaum@dc.uba.ar} \and \IEEEauthorblockN{Teodoro Freund} \IEEEauthorblockA{ Departamento de Computación, \\ Facultad de Ciencias Exactas y Naturales, \\ Universidad de Buenos Aires, Argentina.\\ Email: tfreund95@gmail.com} }
\maketitle
\begin{abstract} We study a conservative extension of classical propositional logic distinguishing between four modes of statement: a proposition may be affirmed or denied, and it may be strong or classical. Proofs of strong propositions must be constructive in some sense, whereas proofs of classical propositions proceed by contradiction. The system, in natural deduction style, is shown to be sound and complete with respect to a Kripke semantics. We develop the system from the perspective of the propositions-as-types correspondence by deriving a term assignment system with confluent reduction. The proof of strong normalization relies on a translation to System F with Mendler-style recursion. \end{abstract}
\IEEEpeerreviewmaketitle
\section{Introduction}
Intuitionistic logic was born out of Brouwer's remark that the {\em law of excluded middle} ($A \lor \negA$) allows one to prove propositions in a seemingly non-constructive way. But what constitutes a {\em constructive} proof, exactly? A possible answer to this question may be found in the {\em realizability} interpretation, also known as the Brouwer--Heyting--Kolmogorov interpretation, which establishes what kinds of mathematical constructions can be regarded as a {\em realizer} or {\em canonical proof} of a proposition. For example, a canonical proof of a conjunction $(A\landB)$ is given by a pair $\langle{p,q}\rangle$, where $p$ and $q$ are in turn canonical proofs of $A$ and $B$ respectively. From the works of Gentzen~\cite{gentzen1935untersuchungen} and Prawitz~\cite{prawitz1965natural} we know that, in intuitionistic natural deduction, an arbitrary proof of a proposition $A$ can always be {\em normalized} to a canonical proof of~$A$.
These ideas culminate in the {\em propositions-as-types} correspondence, the realization that a proposition $A$ may be understood as a {\em type} that expresses the specification of a program. A proof $p$ of $A$ may be understood as a program fulfilling the specification $A$. Running the program corresponds to applying a computational procedure that normalizes the proof $p$ to obtain a canonical proof $p'$ of~$A$. Under this paradigm, proofs in intuitionistic natural deduction can be identified with programs in the {\em simply typed $\lambda$-calculus}. This correspondence has been extended to encompass many logical systems, including first-order~\cite{debruijn1970mathematical,martinlof1971theory} and second-order intuitionistic logic~\cite{thesisgirard,reynolds1974towards}, linear logic~\cite{girard1987linear}, classical logic~\cite{griffin1989formulae,Curien00theduality,symmetric-Barbanera-berardi,lambdamu-parigot}, and modal logic~\cite{bierman2000intuitionistic,DBLP:journals/jacm/DaviesP01}. These developments unveil the deep connection between logic and computer science, and they have practical applications in the development of programming languages and proof assistants based in type theory such as \textsc{Coq} and \textsc{Agda}.
In this paper, we define a logical system $\textsc{prk}$, presented in natural deduction style, that distinguishes between four ``modes'' of stating a proposition $A$, which are written $A{}^+$ (strong affirmation), $A{}^-$ (strong denial), $A{}^\oplus$ (classical affirmation), and $A{}^\ominus$ (classical denial). As the name implies, strong affirmation is stronger than classical affirmation, {\em i.e.}\xspace from $A{}^+$ one may deduce $A{}^\oplus$, and likewise from $A{}^-$ one may deduce $A{}^\ominus$. Affirmation and denial are contradictory, {\em i.e.}\xspace from $A{}^+$ and $A{}^-$ one may derive any conclusion, and similarly for $A{}^\oplus$ and $A{}^\ominus$. This logic turns out to be a {\em conservative extension} of classical propositional logic, in the sense that a proposition $A$ is classically valid if and only if $A{}^\oplus$ is valid in $\textsc{prk}$.
System $\textsc{prk}$ is then shown to be {\em sound} and {\em complete} with respect to a Kripke-style semantics. This helps to elucidate the difference between the four modes of statement. In particular, strong affirmation and denial have a constructive ``flavor''---for example, the law of excluded middle holds classically, {\em i.e.}\xspace $(A\lor\negA){}^\oplus$ is valid, but it does not hold strongly, {\em i.e.}\xspace $(A\lor\negA){}^+$ is not valid.
Furthermore, following the propositions-as-types paradigm, we derive an associated calculus $\lambda^{\PRK}$ and we show that it enjoys the expected meta-theoretical properties: {\em confluence}, {\em subject reduction} and {\em strong normalization}. Besides, we characterize the set of {\em normal forms}. This sheds a new light on the structure of classical proofs, and it may form the basis of the type systems for future programming languages and proof assistants.
{\bf Classical Proofs and Refutations.} It is well-known that intuitionistic propositional logic enjoys the {\em disjunctive property}, that is, a canonical proof of a disjunction $(A \lor B)$ is given by either a canonical proof of $A$ or a canonical proof of $B$. In particular, the proof {\em contains one bit of information}, indicating whether it encloses a proof of $A$ or of $B$. In contrast, the intuitionistic notion of {\em refutation} (proof of a negation) is not dual to the notion of proof. For example, the set of refutations of a conjunction $(A \land B)$ is {\em not} the disjoint union of the set of refutations of $A$ and the set of refutations of $B$. This is related to the fact that one of De~Morgan's laws, namely $\neg(A \land B) \to (\negA \lor \negB)$, which is classically valid, does not hold intuitionistically. The reason is that the proof of a negation in intuitionistic logic proceeds by contradiction, {\em i.e.}\xspace the equivalence $\negA \equiv (A \to \bot)$ holds. As a matter of fact, a refutation in intuitionistic logic {\em contains no information}\footnote{As attested for example by the fact that, in {\em homotopy type theory}, a type of the form $\negA$ can always be shown to be a mere proposition, {\em i.e.}\xspace if it is inhabited, it is equivalent to the unit type; see for instance~\cite[Section~3.6]{hottbook}.}.
The attempt to recover the symmetry between the notions of proof and refutation in a constructive setting lead Nelson to study logical systems with {\em strong negation}~\cite{nelson1949constructible}. One way to formulate Nelson's system is to distinguish between two modes to state a proposition $A$, that we may call affirmation~($A{}^+$) and denial~($A{}^-$), whose witnesses are called {\em proofs} and {\em refutations} of $A$ respectively\footnote{These are called ``P-realizers'' and ``N-realizers'' by Nelson.}. The following (informal) equations suggest a realizability interpretation for affirmations and denials of conjunction, disjunction and negation, found in Nelson's system: \[ {\small
\begin{array}{rcl@{\hspace{.5cm}}rcl}
(A\landB){}^+ & \approx & A{}^+ \times B{}^+
&
(A\landB){}^- & \approx & A{}^- \uplus B{}^-
\\
(A\lorB){}^+ & \approx & A{}^+ \uplus B{}^+
&
(A\lorB){}^- & \approx & A{}^- \times B{}^-
\\
(\negA){}^+ & \approx & A{}^-
&
(\negA){}^- & \approx & A{}^+
\end{array} } \] These equations state, for example, that the set of proofs of a conjunction $(A\landB)$ is the cartesian product of the set of proofs of $A$ and the set of proofs of $B$, while the set of refutations of a conjunction $(A\landB)$ is the disjoint union of the set of refutations of $A$ and the set of refutations of $B$.
This paper was conceived with the goal in mind of providing a {\bf realizability interpretation for classical logic} based on this strong notion of negation. Long-established embeddings of classical logic into intuitionistic logic, such as G\"odel's, are based on {\em double-negation translations}. These translations rely on the equivalence $A \equiv \neg\negA$, which is classically, but not intuitionistically, valid.
Our starting point is a different equivalence, namely $A \equiv (\negA \to A)$, which is again classically, but not intuitionistically, valid. To formulate the interpretation, we introduce a further distinction, according to which a proposition~$A$ may be qualified as {\em strong} or {\em classical}, resulting in four possible modes: \[
\begin{array}{l|ll}
& \text{affirmation} & \text{denial} \\
\hline
\text{strong} & A{}^+ & A{}^- \\
\text{classical} & A{}^\oplus & A{}^\ominus \\
\end{array} \] As before, the witness of an affirmation (resp. denial) is called a proof (resp. refutation). Our first intuition is that a classical proof of a proposition $A$ should be given by a construction that transforms a {\em strong} refutation of $A$ into a strong proof of $A$. Hence, informally speaking, the realizability interpretation should include an equation like $A{}^\oplus \approx (A{}^- \to A{}^+)$, where $(X \to Y)$ is expected to denote the set of ``transformations'', from $X$ to $Y$ in some suitable sense.
In a preliminary version of this work, we explored a realizability interpretation based on such an equation, and its dual equation, $A{}^\ominus \approx (A{}^+ \to A{}^-)$. But, unfortunately, we were not able to formulate a well-behaved system from the computational point of view\footnote{The difficulty is that it is not obvious how to normalize a proof of falsity derived from $A{}^\oplus$ and $A{}^\ominus$, {\em i.e.}\xspace a contradiction obtained from combining a classical proof and a classical refutation of a proposition $A$.}. The study of proof normalization suggests that the ``right'' equations should instead be $A{}^\oplus \approx (A{}^\ominus \to A{}^+)$ and its dual, $A{}^\ominus \approx (A{}^\oplus \to A{}^-)$. This means that a classical proof of a proposition $A$ should be given by a transformation that takes a {\em classical} refutation of $A$ as an input and produces a strong proof of $A$ as an output. This is indeed the path that we follow.
The complete set of equations that suggest the realizability interpretation that we study in this paper is: \[ {\small
\begin{array}{rcl@{\hspace{.5cm}}rcl}
(A\landB){}^+ & \approx & A{}^\oplus \times B{}^\oplus
&
(A\landB){}^- & \approx & A{}^\ominus \uplus B{}^\ominus
\\
(A\lorB){}^+ & \approx & A{}^\oplus \uplus B{}^\oplus
&
(A\lorB){}^- & \approx & A{}^\ominus \times B{}^\ominus
\\
(\negA){}^+ & \approx & A{}^\ominus
&
(\negA){}^- & \approx & A{}^\oplus
\\
A{}^\oplus & \approx & A{}^\ominus \to A{}^+
&
A{}^\ominus & \approx & A{}^\oplus \to A{}^-
\\
\end{array} } \] Observe that a strong proof of a conjunction is given by a pair of {\em classical} (and not strong) proofs. Similarly for the other connectives, {\em e.g.}\xspace a strong refutation of $\negA$ is given by a {\em classical} (and not a strong) proof of $A$. For the sake of brevity, in this paper we will only consider three logical connectives: conjunction, disjunction, and negation. Extending our results and techniques to incorporate other propositional connectives, such as implication, and truth and falsity constants, should not present major obstacles.
One technical difficulty that we confront is the fact that the last two equations are mutually recursive. This means, in particular, that these equations cannot be understood as a translation from formulae of $\textsc{prk}$ to formulae of other systems (such as the simply typed $\lambda$-calculus), at least not in the naive sense. However, as we shall see, these recursive equations do fulfill Mendler's {\em positivity requirement}~\cite{mendler1991inductive}, which allows us to give a translation from $\lambda^{\PRK}$ to System~F extended with (non-strictly positive) recursive type constraints.
{\bf Structure of This Paper.} The remainder of this paper is organized as follows. In \rsec{prk_natural_deduction} ({\bf Natural Deduction}) we present the proof system $\textsc{prk}$ in natural deduction style, and we study some basic facts, such as weakening and substitution. In \rsec{prk_kripke_semantics} ({\bf Kripke Semantics}) we define an ad~hoc notion of Kripke model, and we show that $\textsc{prk}$ is sound and complete with respect to this Kripke semantics, {\em i.e.}\xspace, a sequent $\Gamma \vdash A$ is provable in $\textsc{prk}$ if and only if it holds in every Kripke model. In \rsec{prk_lambdaC} ({\bf The $\lambda^{\PRK}$-calculus}) we derive a term assignment for $\textsc{prk}$, and we endow it with a small-step reduction semantics. We show that the system is confluent and that it enjoys subject reduction. To show that $\lambda^{\PRK}$ is strongly normalizing, we rely on the aforementioned translation to System~F extended with recursive type constraints. We also provide an inductive characterization of the set of normal forms, and we show that an extensionality rule akin to $\eta$-reduction may be incorporated to the system. In \rsec{prk_classical_logic} ({\bf Relation with Classical Logic}) we show that $\textsc{prk}$ is a conservative extension of classical logic. We show how this provides a new computational interpretation for classical logic. In \rsec{prk_conclusion} ({\bf Conclusion}) we conclude, and we discuss related and future work.
\section{The Natural Deduction System $\textsc{prk}$} \lsec{prk_natural_deduction}
In this section, we define the logical system $\textsc{prk}$, formulated in natural deduction style~(\rdef{system_prk}). We then prove that some typical reasoning principles, namely weakening, cut, and substitution, as well as some principles specific to this system, are admissible in $\textsc{prk}$~(\rlem{admissible_rules_logic}). An important result in this section is the projection lemma~(\rlem{projection_lemma}). We also formulate an explicit duality principle~(\rlem{duality_principle}).
We suppose given a denumerable set of {\em propositional variables} $\alpha,\beta,\gamma,\hdots$. The set of {\em pure propositions} is given by the abstract syntax: \[ \begin{array}{rrll}
A,B,C,\hdots
& ::= & \alpha
& \text{propositional variable}
\\
& \mid & A \land B
& \text{conjunction}
\\
& \mid & A \lor B
& \text{disjunction}
\\
& \mid & \negA
& \text{negation}
\\ \end{array} \] The set of {\em moded propositions} (or just {\em propositions}) is given by the abstract syntax: \[ \begin{array}{rrll}
P,Q,R,\hdots ::=
& ::= & A{}^+ & \text{strong affirmation}
\\
& \mid & A{}^- & \text{strong denial}
\\
& \mid & A{}^\oplus & \text{classical afirmation}
\\
& \mid & A{}^\ominus & \text{classical denial} \end{array} \] As mentioned in the introduction, propositions are classified into four modes, which arise from discriminating two dimensions. The first dimension (called {\em sign}) distinguishes between {\em affirmations} ($A{}^+$ and $A{}^\oplus$) and {\em denials} ($A{}^-$ and $A{}^\ominus$), sometimes also called {\em positive} and {\em negative} propositions. The second dimension (called {\em strength}) distinguishes between {\em strong propositions} ($A{}^+$ and $A{}^-$) and {\em classical propositions} ($A{}^\oplus$ and $A{}^\ominus$). Note that modes cannot be nested, {\em e.g.}\xspace $(A{}^+ \land B{}^+){}^-$ is not a well-formed proposition.
The {\em opposite proposition} $P{}^{\sim}$ of a given proposition $P$ is defined by flipping the sign, but preserving the strength: \[
\begin{array}{rcl@{\hspace{1cm}}rcl}
(A{}^+){}^{\sim} & \eqdef & A{}^-
&
(A{}^-){}^{\sim} & \eqdef & A{}^+
\\
(A{}^\oplus){}^{\sim} & \eqdef & A{}^\ominus
&
(A{}^\ominus){}^{\sim} & \eqdef & A{}^\oplus
\end{array} \] The {\em classical projection} of a given proposition $P$ is written $\trunc{P}$ and defined by preserving the sign and making the strength classical: \[
\begin{array}{rcl@{\hspace{.5cm}}rcl}
\trunc{(A{}^+)} & \eqdef & A{}^\oplus &
\trunc{(A{}^-)} & \eqdef & A{}^\ominus \\
\trunc{(A{}^\oplus)} & \eqdef & A{}^\oplus &
\trunc{(A{}^\ominus)} & \eqdef & A{}^\ominus \\
\end{array} \] Note that $P{}^{\sim}{}^{\sim} = P$, $\trunc{\trunc{P}} = \trunc{P}$, and $\trunc{(P{}^{\sim})} = (\trunc{P}){}^{\sim}$.
\begin{definition}[System $\textsc{prk}$] \ldef{system_prk} Judgments in $\textsc{prk}$ are of the form $\Gamma \vdash P$, where $\Gamma$ is a finite {\em set} of moded propositions, {\em i.e.}\xspace we work implicitly up to structural rules of contraction and exchange. Derivability of judgments is defined inductively by the following inference schemes.
Except for the first two rules, the system is defined following the realizability interpretation of propositions discussed in the introduction. For instance, rules $\rulename{I$\land\pp$}$ and $\Eandp$ embody the equation for the strong affirmation of a conjunction, $(A\landB){}^+ \approx (A{}^\oplus \times B{}^\oplus)$.
\[ {\small \indrule{\rulename{Ax}}{ }{
\Gamma,P \vdash P } \indrule{\rulename{Abs}}{
\Gamma \vdash P
\hspace{.5cm}
\Gamma \vdash P{}^{\sim}
\hspace{.5cm}
\text{$P$ strong} }{
\Gamma \vdashQ } } \] \[ {\small \indrule{\rulename{I$\land\pp$}}{
\Gamma \vdash A{}^\oplus
\hspace{.5cm}
\Gamma \vdash B{}^\oplus }{
\Gamma \vdash (A \land B){}^+ } \indrule{\rulename{I$\lor\nn$}}{
\Gamma \vdash A{}^\ominus
\hspace{.5cm}
\Gamma \vdash B{}^\ominus }{
\Gamma \vdash (A \lor B){}^- } } \] \[ {\small \indrule{\Eandp}{
\Gamma \vdash (A_1 \land A_2){}^+
\hspace{.5cm} i \in \set{1, 2} }{
\Gamma \vdash A_i{}^\oplus } } \] \[ {\small \indrule{\Eorn}{
\Gamma \vdash (A_1 \lor A_2){}^-
\hspace{.5cm} i \in \set{1, 2} }{
\Gamma \vdash A_i{}^\ominus } } \] \[ {\small \indrule{\Iorp}{
\Gamma \vdash A_i{}^\oplus
\hspace{.5cm} i \in \set{1, 2} }{
\Gamma \vdash (A_1 \lor A_2){}^+ } \indrule{\Iandn}{
\Gamma \vdash A_i{}^\ominus
\hspace{.5cm} i \in \set{1, 2} }{
\Gamma \vdash (A_1 \land A_2){}^- } } \] \[ {\small \indrule{\rulename{E$\lor\pp$}}{
\Gamma \vdash (A \lor B){}^+
\hspace{.5cm}
\Gamma, A{}^\oplus \vdash P
\hspace{.5cm}
\Gamma, B{}^\oplus \vdash P }{
\Gamma \vdash P } }
\]
\[ {\small \indrule{\rulename{E$\land\nn$}}{
\Gamma \vdash (A \land B){}^-
\hspace{.5cm}
\Gamma, A{}^\ominus \vdash P
\hspace{.5cm}
\Gamma, B{}^\ominus \vdash P }{
\Gamma \vdash P } } \] \[ {\small \indrule{\rulename{I$\lnot\pp$}}{
\Gamma \vdash A{}^\ominus }{
\Gamma \vdash (\negA){}^+ } \indrule{\rulename{I$\lnot\nn$}}{
\Gamma \vdash A{}^\oplus }{
\Gamma \vdash (\negA){}^- } } \] \[ {\small \indrule{\rulename{E$\lnot\pp$}}{
\Gamma \vdash (\negA){}^+ }{
\Gamma \vdash A{}^\ominus } \indrule{\rulename{E$\lnot\nn$}}{
\Gamma \vdash (\negA){}^- }{
\Gamma \vdash A{}^\oplus } } \] \[ {\small \indrule{\rulename{IC$\pp$}}{
\Gamma, A{}^\ominus \vdash A{}^+ }{
\Gamma \vdash A{}^\oplus } \indrule{\rulename{IC$\nn$}}{
\Gamma, A{}^\oplus \vdash A{}^- }{
\Gamma \vdash A{}^\ominus } } \] \[ {\small \indrule{\rulename{EC$\pp$}}{
\Gamma \vdash A{}^\oplus
\hspace{.5cm}
\Gamma \vdash A{}^\ominus }{
\Gamma \vdash A{}^+ } \indrule{\rulename{EC$\nn$}}{
\Gamma \vdash A{}^\ominus
\hspace{.5cm}
\Gamma \vdash A{}^\oplus }{
\Gamma \vdash A{}^- } } \] \end{definition}
Rule $\rulename{Ax}$ is the standard axiom rule. Rule $\rulename{Abs}$ is the {\em absurdity} rule, which allows one to derive any proposition $Q$ from a strong proposition $P$ and its opposite $P{}^{\sim}$. Rules $\rulename{I$\land\pp$}$ and $\Eandp$ are introduction and elimination rules for the strong affirmation of a conjunction. Rules $\Iorp$ and $\rulename{E$\lor\pp$}$ are introduction and elimination rules for the strong affirmation of a disjunction; note that $\rulename{E$\lor\pp$}$ allows one to conclude {\em any} proposition. Rules $\rulename{I$\lnot\pp$}$ and $\rulename{E$\lnot\pp$}$ are introduction and elimination rules for the strong affirmation of a negation. Rules $\rulename{IC$\pp$}$ and $\rulename{EC$\pp$}$ are introduction and elimination rules for classical affirmation (resembling introduction and elimination rules for implication: $\rulename{IC$\pp$}$ resembles the deduction theorem, while $\rulename{EC$\pp$}$ resembles {\em modus ponens}). The negative rules are dual to the positive ones, {\em i.e.}\xspace the rules for an affirmation of a given connective have the same structure as the rules for the denial of the dual connective. Note that conjunction is dual to disjunction and negation and classical proposition are dual to themselves.
In the rest of this paper, we frequently use the following lemma without explicit mention. It establishes a number of basic reasoning principles that are valid in $\textsc{prk}$.
\begin{lemma} \llem{admissible_rules_logic} \llem{projection_of_conclusions} \llem{classical_strengthening} The following inference schemes are admissible in $\textsc{prk}$: \begin{enumerate} \item {\bf Weakening} ($\rulename{W}$):
if $\Gamma \vdash P$ then $\Gamma,Q \vdash P$. \item {\bf Cut} ($\rulename{Cut}$):
if $\Gamma,P \vdash Q$ and $\Gamma \vdash P$
then $\Gamma \vdash Q$. \item {\bf Substitution}:
if $\Gamma \vdash Q$
then $\Gamma\sub{\alpha}{A} \vdash Q\sub{\alpha}{A}$,
where $-\sub{\alpha}{A}$ denotes the substitution of the
propositional variable $\alpha$ for the pure proposition~$A$. \item {\bf Generalized absurdity} ($\rulename{Abs}'$):
if $\Gamma \vdash P$ and $\Gamma \vdash P{}^{\sim}$,
where $P$ is not necessarily strong,
then $\Gamma \vdash Q$. \item {\bf Projection of conclusions} ($\rulename{PC}$):
if $\Gamma \vdash P$ then $\Gamma \vdash \trunc{P}$. \item {\bf Contraposition} ($\rulename{Contra}$):
if $P$ is classical and
$\Gamma,P \vdash Q$
then $\Gamma,Q{}^{\sim} \vdash P{}^{\sim}$. \item {\bf Classical strengthening} ($\rulename{CS}$):
if $P$ is classical and
$\Gamma,P{}^{\sim} \vdash P$ then $\Gamma \vdash P$. \end{enumerate} \end{lemma} \begin{proof} {\bf Weakening}, {\bf cut}, and {\bf substitution} are routine proofs by induction on the derivation of the first judgment.
For {\bf generalized absurdity}, suppose that $\Gamma \vdash P$ and $\Gamma \vdash P{}^{\sim}$. If $P$ is strong, applying the $\rulename{Abs}$ rule we may conclude $\Gamma \vdash Q$. If $P$ is classical, there are two cases, depending on whether $P$ is positive or negative. If $P$ is positive, {\em i.e.}\xspace $P = A{}^\oplus$ then: \[
\indruleN{\rulename{Abs}}{
\indruleN{\rulename{EC$\pp$}}{
\Gamma \vdash A{}^\oplus
\hspace{.5cm}
\Gamma \vdash A{}^\ominus
}{
\Gamma \vdash A{}^+
}
\hspace{.5cm}
\indruleN{\rulename{EC$\nn$}}{
\Gamma \vdash A{}^\ominus
\hspace{.5cm}
\Gamma \vdash A{}^\oplus
}{
\Gamma \vdash A{}^-
}
}{
\Gamma \vdash Q
} \] If $P$ is negative, {\em i.e.}\xspace $P = A{}^\ominus$, the proof is symmetric.
For {\bf projection of conclusions}, if $P$ is classical, {\em i.e.}\xspace of the form $A{}^\oplus$ or $A{}^\ominus$, we are done. If $P$ is strong, {\em i.e.}\xspace of the form $A{}^+$ or $A{}^-$, we conclude by applying the $\rulename{IC$\pp$}$ or the $\rulename{IC$\nn$}$ rule respectively. For example, if $P = A{}^+$: \[
\indruleN{\rulename{IC$\pp$}}{
\indruleN{\rulename{W}}{
\Gamma \vdash A{}^+
}{
\Gamma,A{}^\ominus \vdash A{}^+
}
}{
\Gamma \vdash A{}^\oplus
} \]
For {\bf contraposition} we only study the case when $P$ is positive, {\em i.e.}\xspace $P = A{}^\oplus$; the negative case is symmetric. So let $\Gamma,A{}^\oplus \vdash Q$. Then: \[
\indruleN{\rulename{IC$\nn$}}{
\indruleN{\rulename{Abs}'}{
\indruleN{\rulename{W}}{
\Gamma,A{}^\oplus \vdash Q
}{
\Gamma,Q{}^{\sim},A{}^\oplus \vdash Q
}
\hspace{.5cm}
\indruleN{\rulename{Ax}}{
}{
\Gamma,Q{}^{\sim},A{}^\oplus \vdash Q{}^{\sim}
}
}{
\Gamma,Q{}^{\sim},A{}^\oplus \vdash A{}^-
}
}{
\Gamma,Q{}^{\sim} \vdash A{}^\ominus
} \]
For {\bf classical strengthening} we only study the case when $P$ is positive, {\em i.e.}\xspace $P = A{}^\oplus$; the negative case is symmetric. So let $\Gamma,A{}^\ominus \vdash A{}^\oplus$. Then: \[
\indruleN{\rulename{IC$\pp$}}{
\indruleN{\rulename{EC$\pp$}}{
\Gamma,A{}^\ominus \vdash A{}^\oplus
\hspace{.5cm}
\indruleN{\rulename{Ax}}{}{\Gamma,A{}^\ominus \vdash A{}^\ominus}
}{
\Gamma,A{}^\ominus \vdash A{}^+
}
}{
\Gamma \vdash A{}^\oplus
}\qedhere \] \end{proof}
\begin{example}[Law of excluded middle] \lexample{lem_and_noncontr} The law of excluded middle holds classically in $\textsc{prk}$, that is, $\vdash (A\lor\negA){}^\oplus$. Indeed, let $\Gamma = \set{(A\lor\negA){}^\ominus, (\negA){}^\ominus}$, and let $\pi$ be the following derivation: \[
\begin{array}{c}
\indruleN{\rulename{IC$\pp$}}{
\indruleN{\Iorp[1]}{
\indruleN{\rulename{IC$\pp$}}{
\indruleN{\rulename{Abs}'}{
\indruleN{\rulename{Ax}}{
}{
\Gamma, A{}^\ominus \vdash (\negA){}^\ominus
}
\hspace{.5cm}
\indruleN{\rulename{IC$\pp$}}{
\indruleN{\rulename{I$\lnot\pp$}}{
\indruleN{\rulename{Ax}}{
}{
\Gamma, A{}^\ominus, (\negA){}^\ominus \vdash A{}^\ominus
}
}{
\Gamma, A{}^\ominus, (\negA){}^\ominus \vdash (\negA){}^+
}
}{
\Gamma, A{}^\ominus \vdash (\negA){}^\oplus
}
}{
\Gamma, A{}^\ominus \vdash A{}^+
}
}{
\Gamma \vdash A{}^\oplus
}
}{
\Gamma \vdash (A\lor\negA){}^+
}
}{
\Gamma \vdash (A\lor\negA){}^\oplus
}
\end{array} \] Then we have that: \[
\indruleN{\rulename{IC$\pp$}}{
\indruleN{\Iorp[2]}{
\indruleN{\rulename{IC$\pp$}}{
\indruleN{\rulename{I$\lnot\pp$}}{
\indruleN{\Eorn}{
\indruleN{\rulename{EC$\nn$}}{
\indruleN{\rulename{Ax}}{
}{
(A\lor\negA){}^\ominus, (\negA){}^\ominus \vdash (A\lor\negA){}^\ominus
}
\hspace{.5cm}\hspace{.5cm}
\derivdots{\pi}
\hspace{.5cm}
}{
(A\lor\negA){}^\ominus, (\negA){}^\ominus \vdash (A\lor\negA){}^-
}
}{
(A\lor\negA){}^\ominus, (\negA){}^\ominus \vdash A{}^\ominus
}
}{
(A\lor\negA){}^\ominus, (\negA){}^\ominus \vdash (\negA){}^+
}
}{
(A\lor\negA){}^\ominus \vdash (\negA){}^\oplus
}
}{
(A\lor\negA){}^\ominus \vdash (A\lor\negA){}^+
}
}{
\vdash (A\lor\negA){}^\oplus
} \] Dually, the law of non-contradiction holds classically in $\textsc{prk}$, that is, $\vdash (A\land\negA){}^\ominus$ holds. Results from the following section will entail that the strong law of excluded middle, $\vdash (A\lor\negA){}^+$, does not hold in $\textsc{prk}$ (see~\rexample{counter_model_lem}). The reader may attempt to derive this judgment to convince herself that it does not hold. \end{example}
{\bf Projection Lemma.} The proof of the following lemma is subtle. It will be a key tool in order to prove completeness of $\textsc{prk}$ with respect to the Kripke semantics: \begin{lemma} \llem{projection_lemma} If $\Gamma,P \vdash Q$ then $\Gamma,\trunc{P} \vdash \trunc{Q}$. \end{lemma} \begin{proof} By induction on the derivation of $\Gamma,P \vdash Q$. The difficult cases are conjunction and disjunction elimination. \SeeAppendix{See~\rsec{appendix:projection_lemma} in the appendix for the proof.} \end{proof} A corollary obtained from iterating the projection lemma is that if $P_1,\hdots,P_n \vdash Q$ then $\trunc{P_1},\hdots,\trunc{P_n} \vdash \trunc{Q}$.
{\bf Duality Principle.} The {\em dual} of a pure proposition $A$ is written $A^\bot$ and defined as: \[
\begin{array}{rcl@{\ }rcl}
\alpha^\bot & \eqdef & \alpha
&
(A\landB)^\bot & \eqdef & A^\bot \lor B^\bot
\\
(A\lorB)^\bot & \eqdef & A^\bot \land B^\bot
&
(\negA)^\bot & \eqdef & \neg(A^\bot)
\end{array} \] The dual of a proposition $P$ is written $P^\bot$ and defined as: \[
\begin{array}{rcl@{\hspace{.5cm}}rcl}
(A{}^+)^\bot & \eqdef & (A^\bot){}^-
&
(A{}^-)^\bot & \eqdef & (A^\bot){}^+
\\
(A{}^\oplus)^\bot & \eqdef & (A^\bot){}^\ominus
&
(A{}^\ominus)^\bot & \eqdef & (A^\bot){}^\oplus
\end{array} \] The following duality principle is then straightforward to prove by induction on the derivation of the judgment:
\begin{lemma} \llem{duality_principle} If $P_1,\hdots,P_n \vdash Q$ then $P_1^\bot,\hdots,P_n^\bot \vdash Q^\bot$. \end{lemma}
\section{Kripke Semantics for $\textsc{prk}$} \lsec{prk_kripke_semantics}
In this section, we define a Kripke semantics~(\rdef{kripke_model}, \rdef{kripke_forcing}), for which system $\textsc{prk}$ turns out to be sound~(\rprop{kripke_soundness}) and complete~(\rthm{kripke_completeness}). Recall that a Kripke model $\mathcal{M}$ in intuitionistic logic\footnote{See for instance~\cite[Section~5.3]{DBLP:books/daglib/0080654}.} is given by a set $\mathcal{W}$ of elements called {\em worlds}, a partial order $\leq$ on $\mathcal{W}$ called the {\em accessibility relation}, and for each world $w \in \mathcal{W}$ a set $\wlint{w}$ of propositional variables verifying a {\em monotonicity} property, namely, $w \leq w'$ implies $\wlint{w} \subseteq \wlint{w'}$. A relation of forcing $\kripforce{w}{A}$ is defined for each proposition $A$ by structural recursion on $A$. In the base case, $\kripforce{w}{\alpha}$ is declared to hold for a propositional variable $\alpha$ whenever $\alpha \in \wlint{w}$.
This standard notion of Kripke model is adapted for $\textsc{prk}$ by replacing the set $\wlint{w}$ with two sets $\wlpos{w}$ and $\wlneg{w}$ and by imposing an additional condition we call {\em stabilization}, stating that a propositional variable must eventually belong to the union $\wlpos{w} \cup \wlneg{w}$, but never to the intersection $\wlpos{w} \cap \wlneg{w}$. The relation of forcing $\kripforce{w}{P}$ is then defined in such a way that $\kripforce{w}{\alpha{}^+}$ is declared to hold if $\alpha \in \wlpos{w}$. Similarly, $\alpha{}^-$ is declared to hold if $\alpha \in \wlneg{w}$. One difficulty that we find is how to define the forcing relation for a classical proposition like $A{}^\oplus$. The forcing relation for $A{}^\oplus$ should behave, informally speaking, like an intuitionistic implication ``$A{}^\ominus \to A{}^+$''. However this does not provide a {\em bona fide} definition, because the interpretation of $A{}^\oplus$ depends on $A{}^\ominus$, and the interpretation of $A{}^\ominus$ depends in turn on $A{}^\oplus$. What we do is define the interpretations of $A{}^\oplus$ and $A{}^\ominus$ without referring to each other. A key lemma~(\rlem{rule_of_classical_forcing}) then ensures that $A{}^\oplus$ is given the same semantics as an intuitionistic implication of the form ``$A{}^\ominus \to A{}^+$''.
\begin{definition} \ldef{kripke_model} A {\em Kripke model} (for $\textsc{prk}$) is a structure $\mathcal{M} = (\mathcal{W},\mathrel{\leq},\wlpos{},\wlneg{})$ where $\mathcal{W} = \set{w,w',\hdots}$ is a set of worlds, $\mathrel{\leq}$ is a partial order on $\mathcal{W}$, and for each world $w \in \mathcal{W}$ there are sets $\wlpos{w}$ and $\wlneg{w}$ of propositional variables, such that the following conditions hold: \begin{enumerate} \item {\bf Monotonicity.}
If $w \mathrel{\leq} w'$
then $\wlpos{w} \subseteq \wlpos{w'}$
and $\wlneg{w} \subseteq \wlneg{w'}$. \item {\bf Stabilization.}
For all $w \in \mathcal{W}$ and all $\alpha$,
there exists $w' \mathrel{\geq} w$
such that $\alpha \in \wlpos{w'} \triangle \wlneg{w'}$. \end{enumerate} Note that we write $w' \mathrel{\geq} w$ for $w \mathrel{\leq} w'$, and $\triangle$ denotes the symmetric difference on sets, that is, $X \triangle Y =
(X \setminus Y)
\cup (Y \setminus X)$. \end{definition}
The definition of the forcing relation is given by induction on the following notion of {\em measure} $\#(-)$ of a proposition $P$: \[
\begin{array}{rcl@{\hspace{.5cm}}rcl}
\#(A{}^+) & \eqdef & 2 |A| &
\#(A{}^-) & \eqdef & 2 |A| \\
\#(A{}^\oplus) & \eqdef & 2 |A| + 1 &
\#(A{}^\ominus) & \eqdef & 2 |A| + 1
\end{array} \]
where $|A|$ denotes the {\em size}, {\em i.e.}\xspace the number of symbols, in the formula $A$. Note in particular that $\#(A{}^\oplus) = \#(A{}^\ominus) > \#(A{}^+) = \#(A{}^-)$, that $\#((A\starB){}^+) = \#((A\starB){}^-) > \#(A{}^\oplus) = \#(A{}^\ominus)$ for $\star \in \set{\land,\lor}$, and that $\#((\negA){}^+) = \#((\negA){}^-) > \#(A{}^\oplus) = \#(A{}^\ominus)$.
\begin{definition}[Forcing] \ldef{kripke_forcing} Given a Kripke model, we define the {\em forcing} relation, written $\kripforce{w}{P}$ for each world $w \in \mathcal{W}$ and each proposition $P$, as follows, by induction on the {\em measure} $\#(P)$: \begin{center} {\small \begin{tabular}{l@{\ }l@{\ }l@{\ }l@{\ }l}
$\kripforce{w}{\alpha{}^+}$
& $\iff$ &
$\alpha \in \wlpos{w}$ \\
$\kripforce{w}{\alpha{}^-}$
& $\iff$ &
$\alpha \in \wlneg{w}$ \\
$\kripforce{w}{(A\landB){}^+}$
& $\iff$ &
$\kripforce{w}{A{}^\oplus}$
& \text{and} &
$\kripforce{w}{B{}^\oplus}$ \\
$\kripforce{w}{(A\landB){}^-}$
& $\iff$ &
$\kripforce{w}{A{}^\ominus}$
& \text{or} &
$\kripforce{w}{B{}^\ominus}$ \\
$\kripforce{w}{(A\lorB){}^+}$
& $\iff$ &
$\kripforce{w}{A{}^\oplus}$
& \text{or} &
$\kripforce{w}{B{}^\oplus}$ \\
$\kripforce{w}{(A\lorB){}^-}$
& $\iff$ &
$\kripforce{w}{A{}^\ominus}$
& \text{and} &
$\kripforce{w}{B{}^\ominus}$ \\
$\kripforce{w}{(\negA){}^+}$
& $\iff$ &
$\kripforce{w}{A{}^\ominus}$ \\
$\kripforce{w}{(\negA){}^-}$
& $\iff$ &
$\kripforce{w}{A{}^\oplus}$ \\
$\kripforce{w}{A{}^\oplus}$
& $\iff$ &
$\kripnotforce{w'}{A{}^-}$
& \text{for all} & $w' \mathrel{\geq} w$ \\
$\kripforce{w}{A{}^\ominus}$
& $\iff$ &
$\kripnotforce{w'}{A{}^+}$
& \text{for all} & $w' \mathrel{\geq} w$ \end{tabular} } \end{center} Furthermore, if $\Gamma$ is a (possibly infinite) set of propositions, we write: \[ {\small \begin{array}{l@{\ }l}
\kripforcefull{w}{\Gamma}
&
\iff \text{$\kripforce{w}{P}$ for every $P \in \Gamma$} \\
\kripforcefull{\Gamma}{P}
&
\iff
\text{$\kripforcefull{w}{\Gamma}$ implies $\kripforce{w}{P}$
for every $w$} \\
\kripentails{\Gamma}{P}
&
\iff
\text{$\kripforcefull{\Gamma}{P}$ for every Kripke model $\mathcal{M}$} \end{array}} \] \end{definition}
Note that most cases in the definition of forcing do not mention the accessibility relation, other than for classical propositions.
\begin{example}[Counter-model for the strong excluded middle] \lexample{counter_model_lem} There is a Kripke model $\mathcal{M}$ with a world $w_0$ such that $\kripnotforce{w_0}{(\alpha\lor\neg\alpha){}^+}$. Indeed, let $\mathcal{P}$ be the set of all propositional variables, and let $\mathcal{M}$ be the Kripke model such that $\mathcal{W} = \set{w_0,w_1,w_2}$ with $w_0 \leq w_1$ and $w_0 \leq w_2$, where $\wlpos{}$ and $\wlneg{}$ are defined as follows: \[
\begin{array}{r|l|l}
& \wlpos{} & \wlneg{} \\
\hline
w_0 & \varnothing & \varnothing \\
w_1 & \mathcal{P} & \varnothing \\
w_2 & \varnothing & \mathcal{P} \\
\end{array} \] It is easy to verify that $\mathcal{M}$ is a Kripke model and that $\kripnotforce{w_0}{(\alpha\lor\neg\alpha){}^+}$. Note, on the other hand, that the classical excluded middle holds, {\em i.e.}\xspace $\kripforce{w_0}{(\alpha\lor\neg\alpha){}^\oplus}$. \end{example}
Before going on, we introduce typical nomenclature. If $\Gamma$ is a possibly infinite set of propositions, we say that $\Gamma \vdash Q$ holds whenever the judgment $\Delta \vdash Q$ is derivable in $\textsc{prk}$ for some finite subset $\Delta \subseteq \Gamma$. A set $\Gamma$ of propositions is {\em consistent} if there is a proposition $P$ such that $\Gamma \nvdash P$. Otherwise, $\Gamma$ is {\em inconsistent}.
In the remainder of this section we shall prove that $\textsc{prk}$ is sound and complete with respect to this notion of Kripke model. {\em i.e.}\xspace that $\Gamma \vdash P$ holds if and only if $\kripentails{\Gamma}{P}$ holds. We begin by establishing some basic properties of the forcing relation. \SeeAppendix{See~\rsec{appendix:properties_forcing} in the appendix for the proofs.} \begin{lemma}[Properties of Forcing] \llem{properties_forcing} \llem{monotonicity_forcing} \llem{stabilization_forcing} \llem{non_contradiction_forcing} \quad \begin{enumerate} \item {\bf Monotonicity.}
If $\kripforce{w}{P}$ and $w \leq w'$
then $\kripforce{w'}{P}$. \item {\bf Stabilization.}
For every world $w$ and every proposition $P$,
there is a world $w' \mathrel{\geq} w$
such that
either $\kripforce{w'}{P}$
or $\kripforce{w'}{P{}^{\sim}}$ hold,
but not both. \item {\bf Non-contradiction.}
If $\kripforce{w}{P}$ then $\kripnotforce{w}{P{}^{\sim}}$. \end{enumerate} \end{lemma}
To prove {\bf soundness}, we first need an auxiliary lemma that gives necessary and sufficient conditions for a classical proposition to hold. \SeeAppendix{See~\rsec{appendix:kripke_soundness} in the appendix for the full proof of soundness.} \begin{lemma}[Rule of Classical Forcing] \llem{rule_of_classical_forcing} \quad \begin{enumerate} \item $(\kripforce{w}{A{}^\oplus})$
if and only if \\
$(\forall w' \mathrel{\geq} w)
((\kripforce{w'}{A{}^\ominus}) \implies (\kripforce{w'}{A{}^+}))$. \item $(\kripforce{w}{A{}^\ominus})$
if and only if \\
$(\forall w' \mathrel{\geq} w)
((\kripforce{w'}{A{}^\oplus}) \implies (\kripforce{w'}{A{}^-}))$. \end{enumerate} \end{lemma}
\noindent With these tools at our disposal, it is immediate to prove soundness: \begin{proposition}[Soundness] \lprop{kripke_soundness} If $\Gamma \vdash P$ then $\kripentails{\Gamma}{P}$. \end{proposition} \begin{proof} By induction on the derivation of $\Gamma \vdash P$. The interesting cases are the $\rulename{IC$\pp$}$ and $\rulename{IC$\nn$}$ rules, which follow from the rule of classical forcing~(\rlem{rule_of_classical_forcing}), and the $\rulename{Abs}$, $\rulename{EC$\pp$}$, and $\rulename{EC$\nn$}$ rules, which follow from the property of non-contradiction~(\rlem{non_contradiction_forcing}). \end{proof}
To prove {\bf completeness}, the methodology that we follow is the standard one, which proceeds by contraposition assuming that $\Gamma \nvdash P$ and building a counter-model. The counter-model is given by a Kripke model $\mathcal{M}_0$ and a world $w$ such that $\kripforcefull[\mathcal{M}_0]{w}{\Gamma}$ but $\kripnotforce[\mathcal{M}_0]{w}{P}$. In fact, the choice of the Kripke model $\mathcal{M}_0$ does not depend on $\Gamma$ nor $P$. Rather, $\mathcal{M}_0$ is always chosen to be the {\em canonical} Kripke model whose worlds are ``saturated'' sets of propositions ({\em prime theories}, sometimes called {\em disjunctive theories}). Completeness is obtained by taking $\Gamma$ and {\em saturating} it it to a prime theory $\Gamma'$ which then verifies $\kripforcefull[\mathcal{M}_0]{\Gamma'}{\Gamma}$ but $\kripnotforce[\mathcal{M}_0]{\Gamma'}{P}$. \SeeAppendix{In the remainder of this section, the proofs of the technical lemmas are only sketched; see~\rsec{appendix:kripke_completeness} in the appendix for the full proofs.}
\begin{definition}[Prime theory] A {\em prime theory} is a set of propositions $\Gamma$ such that the following hold: \begin{enumerate} \item {\bf Closure by deduction.}
If $\Gamma \vdash P$ then $P \in \Gamma$. \item {\bf Consistency.}
$\Gamma$ is consistent. \item {\bf Disjunctive property.} \\
$\bullet$ If $(A \lor B){}^+ \in \Gamma$ then either $A{}^\oplus \in \Gamma$ or $B{}^\oplus \in \Gamma$. \\
$\bullet$ If $(A \land B){}^- \in \Gamma$ then either $A{}^\ominus \in \Gamma$ or $B{}^\ominus \in \Gamma$. \end{enumerate} \end{definition}
\begin{lemma}[Saturation] \llem{kripke_saturation} Let $\Gamma$ be a consistent set of propositions, and let $Q$ be a proposition such that $\Gamma \nvdash Q$. Then there exists a prime theory $\Gamma' \supseteq \Gamma$ such that $\Gamma' \nvdash Q$. \end{lemma} \begin{proof} \SeeAppendix{\rlem{appendix:kripke_saturation} in the appendix.} \end{proof}
\begin{definition}[Canonical model] The {\em canonical model} is the structure $\mathcal{M}_0 = (\mathcal{W}_0,\subseteq,\wlpos{},\wlneg{})$, where: \begin{enumerate} \item $\mathcal{W}_0$ is the set of all prime theories. \item $\subseteq$ denotes the set-theoretic inclusion between prime theories. \item $\wlpos{\Gamma} = \set{\alpha \ |\ \alpha{}^+ \in \Gamma}$
and
$\wlneg{\Gamma} = \set{\alpha \ |\ \alpha{}^- \in \Gamma}$. \end{enumerate} \end{definition}
\begin{lemma} The canonical model is a Kripke model. \end{lemma} \begin{proof} \SeeAppendix{\rlem{appendix:canonical_model_is_kripke} in the appendix.} The difficult part is proving the stabilization property, which relies on the fact that if $\Gamma$ is a consistent set and $P$ is a proposition, then $\Gamma \cup \set{P}$ and $\Gamma \cup \set{P{}^{\sim}}$ are not both inconsistent. \end{proof}
\begin{lemma}[Main Semantic Lemma] \llem{kripke_main_semantic_lemma} Let $\Gamma$ be a prime theory. Then $\kripforce[\mathcal{M}_0]{\Gamma}{P}$ holds in the canonical model if and only if $P \in \Gamma$. \end{lemma} \begin{proof} \SeeAppendix{\rlem{appendix:kripke_main_semantic_lemma} in the appendix.} By induction on the measure $\#(P)$. The difficult case is when $P$ is a classical proposition, which requires resorting to the Saturation lemma~(\rlem{kripke_saturation}). \end{proof}
\begin{theorem}[Completeness] \lthm{kripke_completeness} If $\kripentails{\Gamma}{P}$ then $\Gamma \vdash P$. \end{theorem} \begin{proof} The proof is by contraposition, {\em i.e.}\xspace let $\Gamma \nvdash P$ and let us show that there is a Kripke model $\mathcal{M}$ and a world $w$ such that $\kripforcefull{w}{\Gamma}$ but $\kripnotforce{w}{P}$. Note that $\Gamma$ is consistent, so by Saturation~(\rlem{kripke_saturation}) there exists a prime theory $\Gamma' \supseteq \Gamma$ such that $\Gamma' \nvdash P$. Note that $\kripforcefull[\mathcal{M}_0]{\Gamma'}{\Gamma}$ because, by the Main~Semantic~Lemma~(\rlem{kripke_main_semantic_lemma}), we have that $\kripforce[\mathcal{M}_0]{\Gamma'}{Q}$ for every $Q \in \Gamma \subseteq \Gamma'$. Moreover, also by the Main~Semantic~Lemma~(\rlem{kripke_main_semantic_lemma}), we have that $\kripnotforce[\mathcal{M}_0]{\Gamma'}{P}$ because $P \notin \Gamma'$. \end{proof}
\section{Propositions as types: the $\lambda^{\PRK}$-Calculus} \lsec{prk_lambdaC}
In this section, we formulate a typed $\lambda$-calculus, called $\lambda^{\PRK}$, by deriving a system of term assignment for derivations in $\textsc{prk}$, and furnishing it with reduction rules. Besides the basic results of confluence~(\rprop{lambdaC_confluent}) and subject reduction~(\rprop{subject_reduction}) the central result in this section is a translation~(\rdef{semF_translation_types}, \rdef{semF_translation}) from $\textsc{prk}$ to System~F extended with recursive type constraints, following Mendler~\cite{mendler1991inductive}. The translation maps each type $P$ of $\textsc{prk}$ to a type $\semF{P}$, and each term $t$ of type $P$ to a term $\semF{t}$ of type $\semF{P}$. Recursion is needed in order to be able to translate classical propositions, which are characterized by the recursive equations discussed in the introduction, $A{}^\oplus \approx (A{}^\ominus \to A{}^+)$ and its dual $A{}^\ominus \approx (A{}^\oplus \to A{}^-)$. Moreover the translation is such that each reduction step $t \to s$ in $\lambda^{\PRK}$ is simulated in one or more steps $\semF{t} \to^+ \semF{s}$ in the extended System~F. This translation provides a {\em syntactical model} for $\lambda^{\PRK}$ in the sense of~\cite{DBLP:conf/cpp/BoulierPT17}, and one of its consequences is that $\lambda^{\PRK}$ is strongly normalizing~(\rthm{lambdaC_canonical}). This allows us to prove {\em canonicity}~(\rthm{canonicity}), for which we study an inductive characterization of the set of normal forms. Finally, we consider an extension $\lambda^{\PRK}_\eta$ of the system that incorporates an extensionality rule for classical proofs~(\rdef{lambdaCeta_calculus}, \rthm{lambdaCeta_canonical}).
Propositions $P,Q,\hdots$ are sometimes also called {\em types} in this section. We assume given a denumerable set of variables $x,y,z,\hdots$. The set of {\em typing contexts} is defined by the grammar $\Gamma ::= \varnothing \mid \Gamma,x:P$, where each variable is assumed to occur at most once in a typing context. Typing contexts are considered implicitly up to reordering\footnote{Remark that the type system $\lambda^{\PRK}$ is a {\em refinement} of the logical system $\textsc{prk}$ because contexts are {\em multisets}, rather than {\em sets}, of assumptions: there is no structural rule of contraction in $\lambda^{\PRK}$. This means, for example, that in $\lambda^{\PRK}$ there are two different derivations of the sequent $P,P \vdash P$ using the $\rulename{Ax}$ rule, depending on which one of the two assumptions is used, whereas in $\textsc{prk}$ there is only one such proof. This is a typical situation in a propositions-as-types setting.}
The set of terms is given by the following abstract syntax. The letter $i$ ranges over the set $\set{1,2}$. Some terms are decorated with a plus or a minus sign. In the grammar we write ``${}^{\pm}$'' to stand for either ``${}^+$'' or ``${}^-$''. \[ {\small \begin{array}{rrll}
t,s,u,\hdots
& ::= & x
& \text{variable}
\\
& \mid & \strongabs{P}{t}{s}
& \text{absurdity}
\\
& \mid & \pairpn{t}{s}
& \text{$\land{}^+$ / $\lor{}^-$ introduction}
\\
& \mid & \projipn{t}
& \text{$\land{}^+$ / $\lor{}^-$ elimination}
\\
& \mid & \inipn{t}
& \text{$\lor{}^+$ / $\land{}^-$ introduction}
\\
& \mid & \casepn{t}{x:P}{s}{y:Q}{u}
& \text{$\lor{}^+$ / $\land{}^-$ elimination}
\\
& \mid & \negipn{t}
& \text{$\neg{}^+$ / $\neg{}^-$ introduction}
\\
& \mid & \negepn{t}
& \text{$\neg{}^+$ / $\neg{}^-$ elimination}
\\
& \mid & \claslampn{(x:P)}{t}
& \text{classical introduction}
\\
& \mid & \clasappn{t}{s}
& \text{classical elimination} \end{array} } \] The notions of free and bound occurrences of variables are defined as expected considering that $\casepn{t}{x:P}{s}{y:Q}{u}$ binds occurrences of $x$ in $s$ and occurrences of $y$ in $u$, whereas $\claslampn{x:P}{t}$ binds occurrences of $x$ in $t$. We implicitly work modulo $\alpha$-renaming of bound variables. We write $\fv{t}$ for the set of free variables of $t$, and $t\sub{x}{s}$ for the capture-avoiding substitution of $x$ by $s$ in $t$. Sometimes we omit type decorations if they are irrelevant or clear from the context, for example, we may write $\claslamp{x}{t}$ rather than $\claslamp{(x:A{}^\ominus)}{t}$, and $\strongabs{}{t}{s}$ rather than $\strongabs{P}{t}{s}$. Sometimes we also omit the name of unused bound variables, writing ``$\underline{\,\,\,}$'' instead; {\em e.g.}\xspace if $x \not\in \fv{t}$ we may write $\claslamp{\underline{\,\,\,}}{t}$ rather than $\claslamp{(x:A{}^\ominus)}{t}$.
\begin{definition}[The $\lambda^{\PRK}$ type system] \ldef{lambdaC_type_system} Typing judgments are of the form $\Gamma \vdash t : P$. Derivability of judgments is defined inductively by the following typing rules: \[ {\small \indrule{\rulename{Ax}}{ }{
\Gamma,x:P \vdash x:P } \indrule{\rulename{Abs}}{
\Gamma \vdash t : P
\hspace{.3cm}
\Gamma \vdash s : P{}^{\sim}
\hspace{.3cm}
\text{$P$ strong} }{
\Gamma \vdash \strongabs{Q}{t}{s} : Q } } \] \[ {\small \indrule{\rulename{I$\land\pp$}}{
\Gamma \vdash t : A{}^\oplus
\hspace{.3cm}
\Gamma \vdash s : B{}^\oplus }{
\Gamma \vdash \pairp{t}{s} : (A \land B){}^+ } \indrule{\rulename{I$\lor\nn$}}{
\Gamma \vdash t : A{}^\ominus
\hspace{.3cm}
\Gamma \vdash s : B{}^\ominus }{
\Gamma \vdash \pairn{t}{s} : (A \lor B){}^- } } \] \[ {\small \indrule{\Eandp}{
\Gamma \vdash t : (A_1 \land A_2){}^+
\hspace{.3cm} i \in \set{1, 2} }{
\Gamma \vdash \projip{t} : A_i{}^\oplus } }
\]
\[ {\small \indrule{\Eorn}{
\Gamma \vdash t : (A_1 \lor A_2){}^-
\hspace{.3cm} i \in \set{1, 2} }{
\Gamma \vdash \projin{t} : A_i{}^\ominus } } \] \[ {\small \indrule{\Iorp}{
\Gamma \vdash t : A_i{}^\oplus
\hspace{.3cm} i \in \set{1, 2} }{
\Gamma \vdash \inip{t} : (A_1 \lor A_2){}^+ } \indrule{\Iandn}{
\Gamma \vdash t : A_i{}^\ominus
\hspace{.3cm} i \in \set{1, 2} }{
\Gamma \vdash \inin{t} : (A_1 \land A_2){}^- } } \] \[ {\small \indrule{\rulename{E$\lor\pp$}}{
\Gamma \vdash t : (A \lor B){}^+
\hspace{.3cm}
\Gamma, x:A{}^\oplus \vdash s : P
\hspace{.3cm}
\Gamma, y:B{}^\oplus \vdash u : P }{
\Gamma \vdash \casep{t}{x:A{}^\oplus}{s}{y:B{}^\oplus}{u}
: P } }
\]
\[ {\small \indrule{\rulename{E$\land\nn$}}{
\Gamma \vdash t : (A \land B){}^-
\hspace{.3cm}
\Gamma, x:A{}^\ominus \vdash s : P
\hspace{.3cm}
\Gamma, y:B{}^\ominus \vdash u : P }{
\Gamma \vdash \casen{t}{x:A{}^\ominus}{s}{y:B{}^\ominus}{u}
: P } } \] \[ {\small \indrule{\rulename{I$\lnot\pp$}}{
\Gamma \vdash t : A{}^\ominus }{
\Gamma \vdash \negip{t} : (\negA){}^+ } \indrule{\rulename{I$\lnot\nn$}}{
\Gamma \vdash t : A{}^\oplus }{
\Gamma \vdash \negin{t} : (\negA){}^- } } \] \[ {\small \indrule{\rulename{E$\lnot\pp$}}{
\Gamma \vdash t : (\negA){}^+ }{
\Gamma \vdash \negep{t} : A{}^\ominus } \indrule{\rulename{E$\lnot\nn$}}{
\Gamma \vdash t : (\negA){}^- }{
\Gamma \vdash \negen{t} : A{}^\oplus } } \] \[ {\small \indrule{\rulename{IC$\pp$}}{
\Gamma, x : A{}^\ominus \vdash t : A{}^+ }{
\Gamma \vdash \claslamp{(x:A{}^\ominus)}{t} : A{}^\oplus } \indrule{\rulename{IC$\nn$}}{
\Gamma, x : A{}^\oplus \vdash t : A{}^- }{
\Gamma \vdash \claslamp{(x:A{}^\oplus)}{t} : A{}^\ominus } } \] \[ {\small \indrule{\rulename{EC$\pp$}}{
\Gamma \vdash t : A{}^\oplus
\hspace{.3cm}
\Gamma \vdash s : A{}^\ominus }{
\Gamma \vdash \clasapp{t}{s} : A{}^+ } \indrule{\rulename{EC$\nn$}}{
\Gamma \vdash t : A{}^\ominus
\hspace{.3cm}
\Gamma \vdash s : A{}^\oplus }{
\Gamma \vdash \clasapn{t}{s} : A{}^- } } \] \end{definition}
\begin{remark} Each typing rule in $\lambda^{\PRK}$~(\rdef{lambdaC_type_system}) corresponds exactly to the rule of the same name in $\textsc{prk}$~(\rdef{system_prk}). It is immediate to show that $P_1,\hdots,P_n \vdash Q$ is derivable in $\textsc{prk}$ if and only if $x_1:P_1,\hdots,x_n:P_n \vdash t : Q$ is derivable in $\lambda^{\PRK}$ for some term $t$. \end{remark}
We begin by studying properties of $\lambda^{\PRK}$ from the {\em logical} point of view, as a type system. In particular, the following lemma adapts some of the results in~\rlem{admissible_rules_logic} and~\rexample{lem_and_noncontr} to $\lambda^{\PRK}$, providing explicit proof terms for derivations.
\begin{lemma} \llem{admissible_rules} \llem{lem_and_noncontr} \label{lemma:admissible_rules} The following rules are admissible in $\lambda^{\PRK}$: \begin{enumerate} \item {\bf Weakening} ($\rulename{W}$):
If $\Gamma \vdash t : P$
and $x \not\in \fv{t}$
then $\Gamma, x:Q \vdash t : P$. \item {\bf Cut} ($\rulename{Cut}$):
\label{lemma:admissible_rules:subst}
if $\Gamma,x:P \vdash t : Q$
and $\Gamma \vdash s : P$
then $\Gamma \vdash t\sub{x}{s} : Q$. \item {\bf Generalized absurdity} ($\rulename{Abs}'$):
if $\Gamma \vdash t : P$
and $\Gamma \vdash s : P{}^{\sim}$,
where $P$ is not necessarily strong,
there is a term $\abs{Q}{t}{s}$
such that $\Gamma \vdash \abs{Q}{t}{s} : Q$. \item {\bf Contraposition} ($\rulename{Contra}$):
if $P$ is classical and
$\Gamma, x : P \vdash t : Q$,
there is a term $\contrapose{x}{y}{t}$
such that
$\Gamma, y : Q{}^{\sim} \vdash \contrapose{x}{y}{t} : P{}^{\sim}$. \item {\bf Excluded middle}:
there is a term $\lemP{A}$
such that $\Gamma \vdash \lemP{A} : (A \lor \negA){}^\oplus$. \item {\bf Non-contradiction}:
there is a term $\lemN{A}$
such that $\Gamma \vdash \lemN{A} : (A \land \negA){}^\ominus$. \end{enumerate} \end{lemma} \begin{proof} {\bf Weakening} and {\bf cut} are routine by induction on the derivation of the first premise of the rule. For {\bf generalized absurdity}, it suffices to take: \[
\abs{Q}{t}{s} \eqdef
\begin{cases}
\strongabs{Q}{t}{s}
& \text{if $P$ is strong} \\
\strongabs{Q}{(\clasapp{t}{s})}{(\clasapn{s}{t})}
& \text{if $P = A{}^\oplus$} \\
\strongabs{Q}{(\clasapn{t}{s})}{(\clasapp{s}{t})}
& \text{if $P = A{}^\ominus$} \\
\end{cases} \] For {\bf contraposition}, it suffices to take: \[
\contrapose{x}{y}{t} \eqdef
\begin{cases}
\claslamn{(x:A{}^\oplus)}{
(\abs{A{}^-}{
t
}{
y
})
}
& \text{if $P = A{}^\oplus$}
\\
\claslamn{(x:A{}^\ominus)}{
(\abs{A{}^+}{
t
}{
y
})
}
& \text{if $P = A{}^\ominus$}
\end{cases} \] For {\bf excluded middle}, it suffices to take:
\[
\!\!
{\small
\begin{array}{r@{\,}c@{\,}l}
\lemP{A} & \eqdef &
\claslamp{(x:(A\lor\negA){}^\ominus)}{
\inip[2]{
\claslamp{(y:\negA{}^\ominus)}{
\negip{
\projin[1]{
\clasapn{
x
}{
\lemPinner{y}{A}
}
}
}
}
}
}
\\
\lemPinner{y}{A} & \eqdef &
\claslamp{(\underline{\,\,\,}:(A\lor\negA){}^\ominus)}{
\inip[1]{
\claslamp{(z:A{}^\ominus)}{
(\abs{
A{}^+
}{
y
}{
\claslamp{(\underline{\,\,\,}:\negA{}^\ominus)}{
\negip{
z
}
}
})
}
}
}
\end{array}
}
\] Dually, for {\bf non-contradiction}:
\[
\!\!
{\small
\begin{array}{r@{\,}c@{\,}l}
\lemN{A} & \eqdef &
\claslamn{(x:(A \land \negA){}^\oplus)}{
\inin[2]{
\claslamn{(y:\negA{}^\oplus)}{
\negin{
\projip[1]{
\clasapp{
x
}{
\lemNinner{y}{A}
}
}
}
}
}
}
\\
\lemNinner{y}{A} & \eqdef &
\claslamn{(\underline{\,\,\,}:(A \land \negA){}^\oplus)}{
\inin[1]{
\claslamn{(z:A{}^\oplus)}{
(\abs{
A{}^-
}{
y
}{
\claslamn{(\underline{\,\,\,}:\negA{}^\oplus)}{
\negin{
z
}
}
})
}
}
}
\end{array}
}
\] \end{proof}
We now turn to studying the {\em computational} properties of $\lambda^{\PRK}$, provided with the following notion of reduction:
\begin{definition}[The $\lambda^{\PRK}$-calculus] \ldef{the_lambdaC_calculus} Typable terms of $\lambda^{\PRK}$ are endowed with the following rewriting rules, closed under arbitrary contexts. \[
{\small
\begin{array}{rcl@{\hspace{.5cm}}l}
\projipn{\pairpn{t_1}{t_2}}
& \toa{\rewritingRuleName{proj}} &
t_i
& \text{if $i \in \set{1,2}$}
\\
\casepn{(\inipn{t})}{x}{s_1}{x}{s_2}
& \toa{\rewritingRuleName{case}} &
s_i\sub{x}{t}
& \text{if $i \in \set{1,2}$}
\\
\negepn{(\negipn{t})}
& \toa{\rewritingRuleName{neg}} &
t
\\
\clasappn{(\claslampn{x}{t})}{s}
& \toa{\rewritingRuleName{beta}} &
t\sub{x}{s}
\\
\strongabs{}{\pairpn{t_1}{t_2}}{\ininp{s}}
& \toa{\rewritingRuleName{absPairInj}} &
\abs{}{t_i}{s}
& \text{if $i \in \set{1,2}$}
\\
\strongabs{}{\inipn{t}}{\pairnp{s_1}{s_2}}
& \toa{\rewritingRuleName{absInjPair}} &
\abs{}{t}{s_i}
& \text{if $i \in \set{1,2}$}
\\
\strongabs{}{(\negipn{t})}{(\neginp{s})}
& \toa{\rewritingRuleName{absNeg}} &
\abs{}{t}{s}
\end{array}
} \] If many occurrences of ``$\pm$'' appear in the same expression, they are all supposed to stand for the same sign (either $+$ or $-$), and $\mp$ is supposed to stand for the opposite sign. \end{definition}
\begin{example} If $x:A{}^\ominus \vdash t : A{}^+$ and $y:A{}^\oplus \vdash s : A{}^-$ then: \[ {\small
\begin{array}{rrl}
&&
\strongabs{}{(\negin{(\claslamp{x}{t})})}{(\negip{(\claslamn{y}{s})})}
\\
& \longrightarrow &
\abs{}{(\claslamp{x}{t})}{(\claslamn{y}{s})}
\\
& = &
\strongabs{}{
(\clasapp{(\claslamp{x}{t})}{(\claslamn{y}{s})})
}{
(\clasapn{(\claslamn{y}{s})}{(\claslamp{x}{t})})
}
\\
& \longrightarrow &
\strongabs{}{
t\sub{x}{(\claslamn{y}{s})}
}{
(\clasapn{(\claslamn{y}{s})}{(\claslamp{x}{t})})
}
\\
& \longrightarrow &
\strongabs{}{
t\sub{x}{(\claslamn{y}{s})}
}{
s\sub{y}{(\claslamp{x}{t})}
}
\end{array} } \] \end{example}
A first observation is that $\textsc{prk}$'s duality principle~(\rlem{duality_principle}) can be strengthened to obtain a {\bf computational duality principle} for $\lambda^{\PRK}$. The proof is immediate given that all typing and reduction rules are symmetric:
\begin{lemma} If $t^\bot$ is the term that results from flipping all the signs in $t$, then $\Gamma \vdash t : P$ if and only if $\Gamma^\bot \vdash t^\bot : P^\bot$, and $t \toa{} s$ if and only if $t^\bot \toa{} s^\bot$. \end{lemma}
The second computational property that we study is {\bf subject reduction}, also known as {\em type preservation}. This fundamental property ensures that reduction is well-defined over the set of typable terms. More precisely:
\begin{proposition} \lprop{subject_reduction} If $\Gamma \vdash t : P$ and $t \toa{} s$, then $\Gamma \vdash s : P$. \end{proposition} \begin{proof} The core of the proof consists in checking that each rewriting rule preserves the type of the term. \SeeAppendix{See~\rsec{appendix:subject_reduction} in the appendix for the proof.} \end{proof}
Third, the $\lambda^{\PRK}$-calculus enjoys {\bf confluence}, the basic property of a rewriting system stating that given reduction sequences $t_0 \to^* t_1$ and $t_0 \to^* t_2$ there must exist a term $t_3$ such that $t_1 \to^* t_3$ and $t_2 \to^* t_3$.
\begin{proposition} \lprop{lambdaC_confluent} The $\lambda^{\PRK}$-calculus is confluent. \end{proposition} \begin{proof} The rewriting system $\lambda^{\PRK}$ can be modeled as a higher-order rewriting system (HRS) in the sense of Nipkow\footnote{It suffices to model it with a single sort $\iota$, with constants such as $\pi{}^+_i : \iota \to \iota$, $\mathsf{IC}{}^- : (\iota \to \iota) \to \iota$, etc., and rules such as $\delta{}^+(\mathsf{in}^+_1 x)\,f\,g \to f\,x$. Strictly speaking, two constants for $\strongabs{}{}{}$ are needed, depending on the signs.}. This HRS is {\em orthogonal}, {\em i.e.}\xspace left-linear without critical pairs, which entails that it is confluent~\cite{nipkow1991higher}. \end{proof}
Our next goal is to prove that $\lambda^{\PRK}$ enjoys {\bf strong normalization}, that is, that there are no infinite reduction sequences $t_1 \to t_2 \to t_3 \to \hdots$. To do so, we give a translation to System~F extended with {\em recursive type constraints}.
Type constraints are a way to define types as solutions to recursive equations. For instance, the type $\mathsf{T}$ of binary trees is given by $\mathsf{T} \equiv 1 + (\mathsf{T} \times \mathsf{T})$. In our case, the idea is to define $A{}^\oplus$ and $A{}^\ominus$ as solutions to the mutually recursive equations $A{}^\oplus \equiv (A{}^\ominus \to A{}^+)$ and $A{}^\ominus \equiv (A{}^\oplus \to A{}^-)$.
We begin by recalling the extended System~F and its relevant properties.
{\bf System~F Extended with Recursive Type Constraints.} In this subsection we recall the definition of System~F$\extwith{\mathcal{C}}$, {\em i.e.}\xspace System~F parameterized by an {\em arbitrary} set of recursive type constraints $\mathcal{C}$, as formulated by Mendler~\cite{mendler1991inductive}.
The set of {\em types} in System~F$\extwith{\mathcal{C}}$ is given by $A ::= \alpha \mid A \to A \mid \forall\alpha.A$ where $\alpha,\beta,\hdots$ are called {\em base types}. The set of {\em terms} is given by $
t ::= x^A
\mid \lam{x^A}{t} \mid t\,t
\mid \lam{\alpha}{t} \mid t\,A $, where $\lam{\alpha}{t}$ is type abstraction and $t\,A$ is type application. A {\em type constraint} is an equation of the form $\alpha \equiv A$. System~F$\extwith{\mathcal{C}}$ is parameterized by a set $\mathcal{C}$ of type constraints. Each set $\mathcal{C}$ of type constraints induces a notion of equivalence between types, written $A \equiv B$ and defined as the congruence generated by $\mathcal{C}$. Typing rules are those of the usual System~F~\cite[Section~11.3]{girard1989proofs} extended with a {\em conversion} rule: \[ {\small
\indrule{Conv}{
\Gamma \vdash t : A
\hspace{.5cm}
A \equiv B
}{
\Gamma \vdash t : B
} } \] Variables occurring {\em positively} (resp. {\em negatively}) in a type $A$ are written $\posvars{A}$ (resp. $\negvars{A}$) and defined as usual: \[
{\small
\begin{array}{r@{\ }c@{\ }l@{\hspace{.5cm}}r@{\ }c@{\ }l}
\posvars{\alpha} & \eqdef & \set{\alpha}
&
\negvars{\alpha} & \eqdef & \varnothing
\\
\posvars{A \to B} & \eqdef & \negvars{A} \cup \posvars{B}
&
\negvars{A \to B} & \eqdef & \posvars{A} \cup \negvars{B}
\\
\posvars{\forall\alpha.A} & \eqdef & \posvars{A} \setminus \set{\alpha}
&
\negvars{\forall\alpha.A} & \eqdef & \negvars{A} \setminus \set{\alpha}
\end{array}
} \] A set of type constraints $\mathcal{C}$ verifies the {\em positivity condition} if for every type constraint $(\alpha \equiv A) \in \mathcal{C}$ and every type $B$ such that $\alpha \equiv B$ one has that $\alpha \not\in \negvars{B}$. Mendler's main result~\cite[Theorem~13]{mendler1991inductive} is: \begin{theorem}[Mendler, 1991] \lthm{systemF_SN_Mendler} If $\mathcal{C}$ verifies the positivity condition, then System~F$\extwith{\mathcal{C}}$ is strongly normalizing. \end{theorem} \noindent We define the empty ($\mathbf{0}$), unit ($\mathbf{1}$), product ($A \times B$), and sum types ($A + B$) via their usual encodings in System~F (see for instance~\cite[Section~11.3]{girard1989proofs}). For example, the product type is defined as $(A \times B) \eqdef
\forall\alpha.((A \to B \to \alpha) \to \alpha)$. with a constructor $\pairF{t}{s}$ and an eliminator $\projiF{t}$. \SeeAppendix{See~\rsec{appendix:system_f} for a more detailed description of the Extended System~F.}
{\bf System~F Extended with $\mathcal{C}_{\mathbf{pn}}$ Constraints.} In this subsection, we describe System~F$\extwith{\mathcal{C}_{\mathbf{pn}}}$, an extension of System~F with a {\em specific} set of recursive type constraints called $\mathcal{C}_{\mathbf{pn}}$. Given that the set of base types is countably infinite, we may assume without loss of generality that, for any two types $A,B$ in System~F there are two type variables, called $\Pos{A}{B}$ and $\Neg{A}{B}$. More precisely, the set of type variables can be partitioned as $\mathbf{V} \uplus \mathbf{P} \uplus \mathbf{N}$ in such a way that the propositional variables of $\lambda^{\PRK}$ are identified with type variables of $\mathbf{V}$, and there are bijective mappings $(A,B) \mapsto \Pos{A}{B} \in \mathbf{P}$ and $(A,B) \mapsto \Neg{A}{B} \in \mathbf{N}$. Note that we do not forbid $A$ and $B$ to have occurrences of type variables in $\mathbf{P}$ and $\mathbf{N}$.\footnote{An alternative, perhaps cleaner, presentation would be to define types inductively as $A,B,\hdots ::= \alpha \mid \Pos{A}{B} \mid \Neg{A}{B} \mid A \to B \mid \forall\alpha.A$.}
System~F$\extwith{\mathcal{C}_{\mathbf{pn}}}$ is given by extending System~F with the set of recursive type constraints $\mathcal{C}_{\mathbf{pn}}$, including the following equations for all types $A,B$: \[
\Pos{A}{B}
\equiv
(\Neg{A}{B} \to A)
\hspace{.5cm}
\Neg{A}{B}
\equiv
(\Pos{A}{B} \to B) \] This extension is in fact {\bf strongly normalizing}: \begin{corollary} \lcoro{systemF_SN_posneg} System~F$\extwith{\mathcal{C}_{\mathbf{pn}}}$ is strongly normalizing. \end{corollary} \begin{proof} A corollary of the previous theorem. It suffices to show that the recursive type constraints $\mathcal{C}_{\mathbf{pn}}$ verify Mendler's positivity condition. The proof of this fact is slightly technical. \SeeAppendix{See~\rsec{appendix:positivity_condition} in the appendix for the proof.} \end{proof}
{\bf Translating $\lambda^{\PRK}$ to System~F$\extwith{\mathcal{C}_{\mathbf{pn}}}$.} We are now in a position to define the translation from $\lambda^{\PRK}$ to System~F$\extwith{\mathcal{C}_{\mathbf{pn}}}$. \begin{definition}[Translation of Propositions] \ldef{semF_translation_types} A proposition $P$ of $\lambda^{\PRK}$ is translated into a type $\semF{P}$ of System~F$\extwith{\mathcal{C}_{\mathbf{pn}}}$, according to the following definition, given by induction on the {\em measure} $\#(P)$ (defined in \rsec{prk_kripke_semantics}): \[ {\small \!\!
\begin{array}{r@{\ }c@{\ }l}
\semF{\alpha{}^+}
& \eqdef &
\alpha
\\
\semF{(A \land B){}^+}
& \eqdef &
\semF{A{}^\oplus} \times \semF{B{}^\oplus}
\\
\semF{(A \lor B){}^+}
& \eqdef &
\semF{A{}^\oplus} + \semF{B{}^\oplus}
\\
\semF{(\negA){}^+}
& \eqdef &
\mathbf{1} \to \semF{A{}^\ominus}
\\
\semF{A{}^\oplus}
& \eqdef &
\Pos{\semF{A{}^+}}{\semF{A{}^-}}
\end{array}
\begin{array}{r@{\ }c@{\ }l}
\semF{\alpha{}^-}
& \eqdef &
\alpha \to \mathbf{0}
\\
\semF{(A \land B){}^-}
& \eqdef &
\semF{A{}^\ominus} + \semF{B{}^\ominus}
\\
\semF{(A \lor B){}^-}
& \eqdef &
\semF{A{}^\ominus} \times \semF{B{}^\ominus}
\\
\semF{(\negA){}^-}
& \eqdef &
\mathbf{1} \to \semF{A{}^\oplus}
\\
\semF{A{}^\ominus}
& \eqdef &
\Neg{\semF{A{}^+}}{\semF{A{}^-}}
\end{array} } \] Moreover, a typing context $\Gamma = (x_1:P_1,\hdots,x_n:P_n)$ is translated as $\semF{\Gamma} \eqdef (x_1:\semF{P_1},\hdots,x_n:\semF{P_n})$. \end{definition} Note that the translation of propositions mimicks the equations for the realizability interpretation discussed in the introduction. In fact, the translation of $\semF{A{}^\oplus}$ is $\Pos{\semF{A{}^+}}{\semF{A{}^-}}$, which is equivalent to $\Neg{\semF{A{}^+}}{\semF{A{}^-}} \to \semF{A{}^+}$ according to the recursive type constraints in $\mathcal{C}_{\mathbf{pn}}$, and this in turn equals $\semF{A{}^\ominus} \to \semF{A{}^+}$, just as required. Similarly for the translation of $A{}^\ominus$. The translation of $(\negA){}^+$ is $(\mathbf{1} \to \semF{A{}^\ominus})$ rather than just $\semF{A{}^\ominus}$ for a technical reason, in order to ensure that each reduction step in $\lambda^{\PRK}$ is simulated by {\em at least one} step in System~F$\extwith{\mathcal{C}_{\mathbf{pn}}}$.
\begin{definition}[Translation of Terms] \ldef{semF_translation} First, we define a family of terms $\vdash \funabsF{P}{Q} : \semF{P} \to \semF{P{}^{\sim}} \to \semF{Q}$ in System~F$\extwith{\mathcal{C}_{\mathbf{pn}}}$ as follows, by induction on the measure $\#(P)$: \[ {\small
\begin{array}{r@{\ }c@{\ }l}
\funabsF{\alpha{}^+}{Q}
& \eqdef &
\lam{x\,y}{
\abortF{\semF{Q}}{y\,x}
}
\\
\funabsF{\alpha{}^-}{Q}
& \eqdef &
\lam{x\,y}{
\abortF{\semF{Q}}{x\,y}
}
\\
\funabsF{(A\landB){}^+}{Q}
& \eqdef &
\lam{x\,y}{
\caseF{y}{
z
}{
\funabsF{A{}^\oplus}{Q}\,\projiF[1]{x}\,z
}{
z
}{
\funabsF{B{}^\oplus}{Q}\,\projiF[2]{x}\,z
}
}
\\
\funabsF{(A\landB){}^-}{Q}
& \eqdef &
\lam{x\,y}{
\caseF{x}{
z
}{
\funabsF{A{}^\ominus}{Q}\,z\,\projiF[1]{y}
}{
z
}{
\funabsF{B{}^\ominus}{Q}\,z\,\projiF[2]{x}
}
}
\\
\funabsF{(A\lorB){}^+}{Q}
& \eqdef &
\lam{x\,y}{
\caseF{x}{
z
}{
\funabsF{A{}^\oplus}{Q}\,x\,\projiF[1]{y}
}{
z
}{
\funabsF{B{}^\oplus}{Q}\,x\,\projiF[2]{y}
}
}
\\
\funabsF{(A\lorB){}^-}{Q}
& \eqdef &
\lam{x\,y}{
\caseF{y}{
z
}{
\funabsF{A{}^\ominus}{Q}\,\projiF[1]{x}\,z
}{
z
}{
\funabsF{B{}^\oplus}{Q}\,\projiF[2]{x}\,z
}
}
\\
\funabsF{(\negA){}^+}{Q}
& \eqdef &
\lam{x\,y}{
\funabsF{A{}^\ominus}{Q}\,(x\,\trivsym)\,(y\,\trivsym)
}
\\
\funabsF{(\negA){}^-}{Q}
& \eqdef &
\lam{x\,y}{
\funabsF{A{}^\oplus}{Q}\,(x\,\trivsym)\,(y\,\trivsym)
}
\\
\funabsF{A{}^\oplus}{Q}
& \eqdef &
\lam{x\,y}{
\funabsF{A{}^+}{Q}
\,(x\,y)
\,(y\,x)
}
\\
\funabsF{A{}^\ominus}{Q}
& \eqdef &
\lam{x\,y}{
\funabsF{A{}^-}{Q}
\,(x\,y)
\,(y\,x)
}
\end{array} } \] where: $\abort{A}{t}$ denotes an inhabitant of $A$ whenever $t$ is an inhabitant of the empty type; $\caseF{t}{x}{s}{x}{u}$ is the eliminator of the sum type; $\projiF{t}$ is the eliminator of the product type; and $\trivsym$ denotes the trivial inhabitant of the unit type. Now each typable term $\Gamma \vdash t : P$ in $\lambda^{\PRK}$ can be translated into a term $\semF{\Gamma} \vdash \semF{t} : \semF{P}$ of System~F$\extwith{\mathcal{C}_{\mathbf{pn}}}$ as follows: \[ {\small
\begin{array}{rcll}
\semF{x}
& \eqdef &
x
\\
\semF{\strongabs{Q}{t}{s}}
& \eqdef &
\funabsF{P}{Q}\,\semF{t}\,\semF{s}
\\
&& \hspace{.5cm}\text{if $\Gamma \vdash t : P$ and $\Gamma \vdash s : P{}^{\sim}$}
\\
\semF{\pairpn{t}{s}}
& \eqdef &
\pairF{\semF{t}}{\semF{s}}
\\
\semF{\projipn{t}}
& \eqdef &
\projiF{\semF{t}}
\\
\semF{\inipn{t}}
& \eqdef &
\iniF{\semF{t}}
\\
\semF{\casepn{t}{(x:P)}{s}{(y:Q)}{u}}
& \eqdef &
\caseF{\semF{t}}{
(x:\semF{P})
}{
\semF{s}
}{
(y:\semF{Q})
}{
\semF{u}
}
\\
\semF{\negipn{t}}
& \eqdef &
\lam{x^{\mathbf{1}}}{\semF{t}}
\hspace{.5cm}\text{where $x \not\in \fv{t}$}
\\
\semF{\negepn{t}}
& \eqdef &
\semF{t}\,\trivsym
\\
\semF{\claslampn{(x:P)}{t}}
& \eqdef &
\lam{x^{\semF{P}}}{\semF{t}}
\\
\semF{\clasappn{t}{s}}
& \eqdef &
\semF{t}\,\semF{s}
\\
\end{array} } \] \end{definition} It is easy to check that $\semF{\Gamma} \vdash \semF{t} : \semF{P}$ holds in System~F$\extwith{\mathcal{C}_{\mathbf{pn}}}$ by induction on the derivation of the judgment $\Gamma \vdash t : P$ in $\lambda^{\PRK}$. Two straightforward properties of the translation are:
\begin{lemma} \llem{semF_properties} \textnormal{1.} $\fv{\semF{t}} = \fv{t}$; \textnormal{2.} $\semF{t\sub{x}{s}} = \semF{t}\sub{x}{\semF{s}}$. \end{lemma}
The key result is the following {\bf simulation} lemma from which strong normalization follows: \begin{lemma} \llem{semF_simulation} If $t \toa{} s$ in $\lambda^{\PRK}$ then $\semF{t} \toa{}^+ \semF{s}$ in System~F$\extwith{\mathcal{C}_{\mathbf{pn}}}$. \end{lemma} \begin{proof} \SeeAppendix{\rlem{appendix:semF_simulation} in the appendix.} By case analysis on the rewriting rule used to derive the step $t \toa{} s$. The interesting case is when the rule is applied at the root of the term. As an illustrative example, consider an instance of the $\rewritingRuleName{absPairInj}$ rule, with $\Gamma \vdash t_1 : A_1{}^\oplus$, and $\Gamma \vdash t_2 : A_2{}^\oplus$, and $\Gamma \vdash s : A_i{}^\ominus$ for some $i \in \set{1,2}$. Then: \[ {\small
\begin{array}[b]{ll}
&
\semF{\strongabs{P}{\pairp{t_1}{t_2}}{\inin{s}}}
\\
= &
\funabsF{(A_1 \land A_2){}^+}{P}
\,\pairF{\semF{t_1}}{\semF{t_2}}
\,\iniF{\semF{s}}
\\
\toa{}^+ &
\!\!\!
\caseFtablex{\iniF{\semF{s}}}{
(z:\semF{A_1{}^\ominus})
}{
\funabsF{A_1{}^\oplus}{P}\,\projiF[1]{ \pairF{\semF{t_1}}{\semF{t_2}} }\,z
}{
(z:\semF{A_2{}^\ominus})
}{
\funabsF{A_2{}^\oplus}{P}\,\projiF[2]{ \pairF{\semF{t_1}}{\semF{t_2}} }\,z
}
\\
\toa{} &
\funabsF{A_i{}^\oplus}{P}\,\projiF{ \pairF{\semF{t_1}}{\semF{t_2}} }\,\semF{s}
\\
\toa{} &
\funabsF{A_i{}^\oplus}{P}\,\semF{t_i}\,\semF{s}
\\
\toa{}^+ &
\funabsF{A_i{}^+}{P}\,(\semF{t_i}\,\semF{s})
(\semF{s}\,\semF{t_i})
\\
= &
\semF{\strongabs{P}{(\clasapp{t_i}{s})}{(\clasapn{t_i}{s})}}
\\
= &
\semF{\abs{P}{t_i}{s}}
\end{array} \qedhere } \] \end{proof}
\begin{theorem} \lthm{lambdaC_canonical} The $\lambda^{\PRK}$-calculus is strongly normalizing. \end{theorem} \begin{proof} An easy consequence of~\rlem{semF_simulation} given that the extended System~F is strongly normalizing (\rcoro{systemF_SN_posneg}). \end{proof}
{\bf Canonicity.} In the previous subsections we have shown that the $\lambda^{\PRK}$-calculus enjoys subject reduction and strong normalization. This implies that each typable term $t$ reduces to a normal form $t'$ of the same type. In this subsection, these results are refined to prove a {\em canonicity} theorem, stating that each closed, typable term $t$ reduces to a {\em canonical} term $t'$ of the same type. For example, canonical terms of type $(A\lorB){}^+$ are of the form $\inip{t}$. From the logical point of view, this means that given a strong proof of $(A\lorB)$, in a context without assumptions, one can always recover either a classical proof of $A$ or a classical proof of $B$. This shows that $\textsc{prk}$ has a form of disjunctive property.
First we provide an inductive characterization of the set of {\bf normal forms} of $\lambda^{\PRK}$.
\begin{definition}[Normal terms] \ldef{normal_terms} The sets of {\em normal terms} ($N,\hdots$) and {\em neutral terms} ($S,\hdots$) are defined mutually inductively by: \[ \begin{array}{rrllllllllll}
N
& ::= & S
& \mid & \pairpn{N}{N}
& \mid & \inipn{N}
\\
& \mid & \negipn{N}
& \mid & \claslampn{x : P}{N} \\ \\
S
& ::= & x
& \mid & \projipn{S}
& \mid & \casepn{S}{x}{N}{x}{N}
\\
& \mid & \negepn{S}
& \mid & \clasappn{S}{N}
\\
& \mid & \strongabs{P}{S}{N}
& \mid & \strongabs{P}{N}{S} \end{array} \] \end{definition}
\begin{proposition} \lprop{characterization_of_normal_terms} A term is in the grammar of normal terms if and only if it is a normal form, {\em i.e.}\xspace it does not reduce in $\lambda^{\PRK}$. \end{proposition} \begin{proof} Straightforward by induction. \SeeAppendix{See~\rsec{appendix:characterization_of_normal_forms} in the appendix for a detailed proof.} \end{proof}
In order to state a canonicity theorem succintly, we introduce some nomenclature. A term is {\em canonical} if it has any of the following shapes: \[
\pairpn{t_1}{t_2}
\hspace{.5cm}
\inipn{t}
\hspace{.5cm}
\negipn{t}
\hspace{.5cm}
\claslampn{x}{t} \] A typing context is {\em classical} if all the assumptions are classical, {\em i.e.}\xspace of the form $A{}^\oplus$ or $A{}^\ominus$. Recall that a context is a term $\mathtt{C}$ with a single free occurrence of a hole $\Box$, and that $\gctxof{t}$ denotes the capturing substitution of the term $t$ into the hole of $\mathtt{C}$. A {\em case-context} is a context of the form $
\mathtt{K} ::= \Box
\mid \casepn{\mathtt{K}}{x}{t}{y}{s} $. An {\em eliminative context} is a context of the form $
\mathtt{E} ::= \Box
\mid \projipn{\mathtt{E}}
\mid \casepn{\mathtt{E}}{x}{t}{y}{s}
\mid \negepn{\mathtt{E}} $. Note that $\clasappn{\Box}{t}$ is not eliminative and that all case-contexts are eliminative. An {\em explosion} is a term of the form $\strongabs{P}{t}{s}$ or of the form $\clasappn{t}{s}$. A term is {\em closed} if it has no free variables. A term is {\em open} if it not closed, {\em i.e.}\xspace it has at least one free variable.
The following theorem has three parts; the first one provides guarantees for {\em closed} terms, whereas the two other ones provide weaker guarantees for terms typable under an arbitrary classical context. \begin{theorem}[Canonicity] \lthm{canonicity} \quad \begin{enumerate} \item
Let $\vdash t : P$.
Then $t$ reduces to a canonical term. \item
Let $\Gamma \vdash t : P$
where $\Gamma$ is classical and $P$ is strong.
Then either $t \toa{}^* t'$ where $t'$ is canonical
or $t \toa{}^* \casectxof{t'}$
where $\mathtt{K}$ is a case-context
and $t'$ is an open explosion. \item
Let $\Gamma \vdash t : P$
where $\Gamma$ and $P$ are classical.
Then either $t \toa{}^* \claslampn{x}{t'}$
or $t \toa{}^* \elctxof{t'}$,
where $\mathtt{E}$ is an eliminative context
and $t'$ is a variable or an open explosion. \end{enumerate} \end{theorem} \begin{proof} By subject reduction~(\rprop{subject_reduction}) and strong normalization~(\rthm{lambdaC_canonical}) the term $t$ reduces to a normal form $t'$. Moreover, by~\rprop{characterization_of_normal_terms}, we have that $t'$ is generated by the grammar of~\rdef{normal_terms}. The proof then proceeds by induction on the derivation of $t'$ in the grammar of normal terms. \SeeAppendix{See~\rsec{appendix:canonicity} in the appendix for the proof.} \end{proof}
{\bf Extensionality for Classical Proofs.} To conclude the syntactic study of $\lambda^{\PRK}$, we discuss that an extensionality rule, akin to $\eta$-reduction in the $\lambda$-calculus, may be incorporated to $\lambda^{\PRK}$, obtaining a calculus $\lambda^{\PRK}_\eta$.
\begin{definition} \ldef{lambdaCeta_calculus} The $\lambda^{\PRK}_\eta$-calculus is defined by extending the $\lambda^{\PRK}$ calculus with the following reduction rule: \[
\claslampn{x}{(\clasappn{t}{x})}
\ \toa{\rewritingRuleName{eta}}\
t
\hspace{.5cm}\text{if $x \notin \fv{t}$} \] \end{definition}
\begin{theorem} \lthm{lambdaCeta_canonical} The $\lambda^{\PRK}_\eta$-calculus enjoys subject reduction, and it is strongly normalizing and confluent. \end{theorem} \begin{proof} Subject reduction is straightforward, extending~\rprop{subject_reduction} with an easy case for the $\rewritingRuleName{eta}$ rule. Local confluence is also straightforward by examining the critical pairs. The key lemma to prove strong normalization is that $\rewritingRuleName{eta}$ reduction steps can be postponed after steps of other kinds. \noindent\SeeAppendix{See~\rsec{appendix:extensionality} for the details.} \end{proof}
\section{Embedding Classical Logic into $\textsc{prk}$} \lsec{prk_classical_logic}
Intuitionistic logic {\em refines} classical logic: each intuitionistically valid formula $A$ is also classically valid, but there may be many classically equivalent ``readings'' of a classical formula which are not intuitionistically equivalent, such as $\neg(A \land \negB)$ and $\negA \lor B$. System $\textsc{prk}$ refines classical logic in a similar sense. For example, the classical sequent $\alpha \vdash \alpha$ may be ``read'' in $\textsc{prk}$ in various different ways, such as $\alpha{}^+ \vdash \alpha{}^\oplus$ and $\alpha{}^\oplus \vdash \alpha{}^+$, of which the former holds but the latter does not. In this section we show that $\textsc{prk}$ is {\em conservative}~(\rprop{prk_conservative}) with respect to classical logic, and that classical logic may be {\em embedded}~(\rthm{prk_embedding}) in $\textsc{prk}$. We also describe the computational behavior of the terms resulting from this embedding~(\rlem{classical_embedding_computation}).
First, we claim that $\textsc{prk}$ is a {\bf conservative extension} of classical logic, {\em i.e.}\xspace if $A{}^\oplus_1,\hdots,A{}^\oplus_n \vdash B{}^\oplus$ holds in $\textsc{prk}$ then the sequent $A_1,\hdots,A_n \vdash B$ holds in classical logic. In general: \begin{proposition} \lprop{prk_conservative} Define $\classem{P}$ as follows: \[ {\small
\begin{array}{r@{\ }c@{\ }l@{\hspace{.5cm}}r@{\ }c@{\ }l}
\classem{A{}^\oplus} & \eqdef & A
&
\classem{A{}^\ominus} & \eqdef & \negA
\\
\classem{A{}^+} & \eqdef & A
&
\classem{A{}^-} & \eqdef & \negA \\
\end{array} } \] If the sequent $P_1,\hdots,P_n \vdash Q$ holds in $\textsc{prk}$ then the sequent $\classem{P_1},\hdots,\classem{P_n} \vdash \classem{Q}$ holds in classical propositional logic. \end{proposition} \begin{proof} By induction on the derivation of the judgment, observing that all the inference rules in $\textsc{prk}$ are mapped to classically valid inferences. For example, for the $\rulename{E$\land\nn$}$ rule, note that if $\Gamma \vdash \neg(A \land B)$ and $\Gamma, \negA \vdash C$ and $\Gamma, \negB \vdash C$ hold in classical propositional logic then $\Gamma \vdash C$. \end{proof}
Second, we claim that classical logic may be {\bf embedded} in $\textsc{prk}$, that is: \begin{theorem} \lthm{prk_embedding} If $A_1,\hdots,A_n \vdash B$ holds in classical logic then $A{}^\oplus_1,\hdots,A{}^\oplus_n \vdash B{}^\oplus$ holds in $\textsc{prk}$. \end{theorem} \begin{proof} The proof is by induction on the proof of the sequent $A_1,\hdots,A_n \vdash B$ in Gentzen's system of natural deduction for classical logic \textsf{NK}, including introduction and elimination rules for conjunction, disjunction, and negation (encoding falsity as the pure proposition $\bot \eqdef (\alpha_0 \land \neg\alpha_0)$ for some fixed propositional variable $\alpha_0$), the explosion principle, and the law of excluded middle. We build the corresponding proof terms in $\lambda^{\PRK}$:
\noindent{1.} {\bf Conjunction introduction.}
Let $\Gamma \vdash t : A{}^\oplus$ and $\Gamma \vdash s : B{}^\oplus$.
Then $\Gamma \vdash \pairc{t}{s} : (A \land B){}^\oplus$
where:
\[
\pairc{t}{s} \eqdef
\claslamp{(\underline{\,\,\,}:(A\landB){}^\ominus)}{
\pairp{t}{s}
}
\]
\noindent{2.} {\bf Conjunction elimination.}
Let $\Gamma \vdash t : (A_1 \land A_2){}^\oplus$.
Then $\Gamma \vdash \projic{t} : A_i{}^\oplus$ where:
\[
\projic{t} \eqdef
\claslamp{(x:A_i{}^\ominus)}{
\clasapp{
\projip{
\clasapp{
t
}{
\claslamn{(\underline{\,\,\,}:(A_1 \land A_2){}^\oplus)}{\inin{x}}
}
}
}{
x
}
}
\]
\noindent{3.} {\bf Disjunction introduction.}
Let $\Gamma \vdash t : A_i{}^\oplus$.
Then $\Gamma \vdash \inic{t} : (A_1 \lor A_2){}^\oplus$ where:
\[
\inic{t} \eqdef
\claslamp{(\underline{\,\,\,}:(A_1\lorA_2){}^\ominus)}{
\inip{t}
}
\]
\noindent{4.} {\bf Disjunction elimination.}
Let
$\Gamma \vdash t : (A \lor B){}^\oplus$
and $\Gamma, x : A{}^\oplus \vdash s : C{}^\oplus$
and $\Gamma, x : B{}^\oplus \vdash u : C{}^\oplus$.
Then $\Gamma \vdash
\casec{t}{(x:A{}^\oplus)}{s}{(x:B{}^\oplus)}{u}
: C{}^\oplus$,
where:
\[
\claslamp{(y:C{}^\ominus)}{
\caseptablex{
(\clasapp{
t
}{
\claslamn{(\underline{\,\,\,}:(A\lorB){}^\oplus)}{
\pairn{
\contrapose{x}{y}{
s
}
}{
\contrapose{x}{y}{
u
}
}
}
})
}{
(x : A{}^\oplus)
}{
\clasapp{
s
}{
y
}
}{
(x : B{}^\oplus)
}{
\clasapp{
u
}{
y
}
}
}
\]
Recall that $\contrapose{x}{y}{t}$
stands for the witness of contraposition~(\rlem{admissible_rules}).
\noindent{5.} {\bf Negation introduction.}
By \rlem{lem_and_noncontr} we have that
$\Gamma \vdash \lemN{\alpha_0} : (\alpha_0 \land \neg\alpha_0){}^\ominus$,
that is $\Gamma \vdash \lemN{\alpha_0} : \bot{}^\ominus$.
Moreover, suppose that $\Gamma, x:A{}^\oplus \vdash t : \bot{}^\oplus$.
Then $\Gamma \vdash \neglamc{(x:A{}^\oplus)}{t} : (\negA){}^\oplus$,
where:
\[
\neglamc{(x:A{}^\oplus)}{t} \eqdef
\claslamp{(\underline{\,\,\,}:(\negA){}^\ominus)}{
\negip{
\claslamn{(x:A{}^\oplus)}{
(\abs{
A{}^-
}{
t
}{
\lemN{\alpha_0}
})
}
}
}
\]
\noindent{6.} {\bf Negation elimination.}
Let $\Gamma \vdash t : (\negA){}^\oplus$
and $\Gamma \vdash s : A{}^\oplus$.
Then $\Gamma \vdash \negapc{t}{s} : \bot{}^\oplus$,
where:
\[
\negapc{t}{s} \eqdef
\abs{
\bot{}^\oplus
}{
t
}{
\claslamn{(\underline{\,\,\,}:(\negA){}^\oplus)}{
\negin{
s
}
}
}
\]
\noindent{7.} {\bf Explosion.}
Let $\Gamma \vdash t : \bot{}^\oplus$.
Then $\Gamma \vdash (\abs{Q}{t}{\lemN{\alpha_0}}) : Q$.
\noindent{8.} {\bf Excluded middle.}
It suffices to take $\lemC{A} \eqdef \lemP{A}$.
Then by \rlem{lem_and_noncontr},
$\Gamma \vdash \lemC{A} : (A \lor \negA){}^\oplus$. \end{proof}
Finally, this embedding may be understood as providing a {\bf computational interpretation} for classical logic. In fact, besides the introduction and elimination rules that have been proved above, implication may be defined as an abbreviation, $(A \Rightarrow B) \eqdef (\negA \lor B)$, and witnesses for its introduction rule $\lamc{x:A}{t}$ and its elimination rule $\appc{t}{s}$ may be defined as follows. If $\Gamma,x:A{}^\oplus \vdash t : B{}^\oplus$ then $\Gamma \vdash \lamc{(x:A)}{t} : (A \Rightarrow B){}^\oplus$ where: \[ {\small \begin{array}{rcl}
\lamc{x}{t} & \eqdef &
\claslamp{(y:(A\IMPB){}^\ominus)}{
\inip[2]{
t\sub{x}{\mathbf{X}_{y}}
}
} \\ \mathbf{X}_y & \eqdef &
\claslamp{(z:A{}^\ominus)}{
\clasapp{
(\negen{(
\clasapn{
\mathbf{X'}_{y,z}
}{
\claslamp{(\underline{\,\,\,}:(\negA){}^\ominus)}{
\negip{
z
}
}
}
)})
}{
z
}
} \\ \mathbf{X'}_{y,z} & \eqdef &
\projip[1]{
\clasapn{
y
}{
\claslamp{(\underline{\,\,\,}:(A\IMPB){}^\ominus)}{
\inip[1]{
\claslamp{(\underline{\,\,\,}:(\negA){}^\ominus)}{
\negip{
z
}
}
}
}
}
} \end{array} } \] If $\Gamma \vdash t : (A \Rightarrow B){}^\oplus$ and $\Gamma \vdash s : A{}^\oplus$, then $\Gamma \vdash \appc{t}{s} : B{}^\oplus$, where: \[ {\small \begin{array}{r@{}c@{}l}
\appc{t}{s}
& \eqdef &
\claslamptable{(x:B{}^\ominus)}{
\caseptablex{
(\clasapp{
t
}{
\claslamn{(\underline{\,\,\,}:(A\RightarrowB){}^\oplus)}{
\pairn{
(\claslamn{(\underline{\,\,\,}:(\negA){}^\oplus)}{
\negin{
s
}
})
}{
x
}
}
})
}{
(y:(\negA){}^\oplus)
}{
\abs{B{}^+}{s}{
\negen{
(\clasapp{
y
}{
\claslamn{(\underline{\,\,\,}:(\negA){}^\oplus)}{
\negin{
x
}
}
})
}
}
}{
(z:B{}^\oplus)
}{
\clasapp{z}{x}
}
} \end{array} } \]
\begin{lemma} \llem{classical_embedding_computation} The following hold in $\lambda^{\PRK}_\eta$ (with $\rewritingRuleName{eta}$ reduction): \[ \begin{array}{rcl}
\projic{\pairc{t_1}{t_2}}
& \to^* & t_i \\
\casec{\inic{t}}{x}{s_1}{x}{s_2}
& \to^* &
s_i\sub{x}{t} \\
\appc{(\lamc{x}{t})}{s}
& \to^* &
t\sub{x}{s} \\
\casec{\lemC{A}}{x}{s_1}{x}{s_2}
& \to^* &
\claslamp{y}{
(\clasapp{
s_2\sub{x}{s^*_1}
}{
y
})
} \end{array} \]
where $
s^*_1 :=
\claslamp{\underline{\,\,\,}}{
\negip{
(\claslamn{x}{
\abs{}{
s_1
}{
y
}
})
}
}$. \end{lemma} \begin{proof} By calculation. The last rule describes the behaviour of the law of excluded middle. \SeeAppendix{See~\rsec{classical_simulation} in the appendix.} \end{proof}
\section{Conclusion} \lsec{prk_conclusion}
This work explores a logical system $\textsc{prk}$, formulated in natural deduction style~(\rdef{system_prk}), based on a, to the best of our knowledge, new realizability interpretation for classical logic. The key idea is that a classical proof of a proposition can be understood as a transformation from a classical refutation to a strong proof of the proposition. We summarize our contributions: system $\textsc{prk}$ has been shown to be {\bf sound}~(\rprop{kripke_soundness}) and {\bf complete}~(\rthm{kripke_completeness}) with respect to a Kripke semantics. A calculus $\lambda^{\PRK}$~(\rdef{lambdaC_type_system}, \rdef{the_lambdaC_calculus}) based on $\textsc{prk}$ via the propositions-as-types correspondence, has been defined. The calculus enjoys good properties, the most relevant ones being {\bf confluence}~(\rprop{lambdaC_confluent}), {\bf subject reduction}~(\rprop{subject_reduction}), {\bf strong normalization}~(\rthm{lambdaC_canonical}), and {\bf canonicity}~(\rthm{canonicity}). Finally, we have shown that $\textsc{prk}$ is a {\bf conservative extension}~(\rprop{prk_conservative}) of classical logic, and classical logic may be {\bf embedded}~(\rthm{prk_embedding}) in $\textsc{prk}$. This provides a {\bf computational interpretation}~(\rlem{classical_embedding_computation}) for classical logic.
{\bf Future Work.} It is a natural question whether $\lambda^{\PRK}$ can be extended to second-order logic. In fact, formulating such a system is straightforward by extending the realizability interpretation described in the introduction with equations: \[ {\small
\begin{array}{rcl@{\hspace{.5cm}}rcl}
(\forall\alpha.A){}^+ & \approx & \forall\alpha.(A{}^\oplus)
&
(\forall\alpha.A){}^- & \approx & \exists\alpha.(A{}^\ominus)
\\
(\exists\alpha.A){}^+ & \approx & \exists\alpha.(A{}^\oplus)
&
(\exists\alpha.A){}^- & \approx & \forall\alpha.(A{}^\ominus)
\end{array} } \] For instance, introduction and elimination rules for positive universal quantification in second-order $\lambda^{\PRK}$ would be: \[ {\small
\indrule{\rulename{I$\forall\pp$}}{
\Gamma \vdash t : A{}^\oplus
\hspace{.5cm}
\alpha \not\in \fv{\Gamma}
}{
\Gamma \vdash \lamtp{\alpha}{t} : (\all{\alpha}{A}){}^+
}
\indrule{\rulename{E$\forall\pp$}}{
\Gamma \vdash t : (\all{\alpha}{B}){}^+
}{
\Gamma \vdash \apptp{t}{A} : B\sub{\alpha}{A}{}^\oplus
} } \] From the logical point of view, the system turns out to be a conservative extension of second-order classical logic, and from the computational point of view it still enjoys confluence and subject reduction. However, the techniques described in this paper do not suffice to prove strong normalization. A different normalization proof, possibly based on Tait--Girard's technique of {\em reducibility candidates}, should be explored.
We have not addressed decision problems, such as determining the validity of a formula in $\textsc{prk}$, corresponding to the type inhabitation problem for $\lambda^{\PRK}$. Unfortunately, $\lambda^{\PRK}$ does not enjoy the {\em subformula property}. For a counter\-example consider $\strongabs{\alpha{}^+}{(\strongabs{\beta{}^+}{x}{y})}{(\strongabs{\beta{}^-}{x}{y})}$, which is a normal term of type $\alpha{}^+$ under the context $x:\alpha{}^+,y:\alpha{}^-$ such that the unrelated type $\beta{}^+$ appears in the derivation.
We have not stated explicitly a computational rule for negation, {\em i.e.}\xspace for $\negapc{(\neglamc{x}{t})}{s}$ in \rlem{classical_embedding_computation} \SeeAppendix{but see~\rsec{appendix:simulation_of_negation} in the appendix}. Intriguingly, it does not reduce to $t\sub{x}{s}$ in general, {\em i.e.}\xspace the inference schemes for classical negation that we have constructed are not consistent with a definition of negation as $\negA \equiv (A \to \bot)$ in intuitionistic logic. We believe this to be not just an artifact of a faulty construction, but due to a deeper reason.
{\bf Related Work.} That classical logic may be embedded in intuitionistic logic has been known as early as Glivenko's proof of his theorem in the late 1920s. For a long time, however, the generalized belief seemed to be that classical proofs had no computational content. In the late 1980s, Griffin~\cite{griffin1989formulae} remarked that the type of Felleisen's $\mathcal{C}$ operator (similar to \texttt{call/cc}) corresponds to Peirce's law $(((A \to B) \to A) \to A)$. This sparked research on calculi for classical logic. Many of these works are based on classical axioms that behave as {\em control operators}, {\em i.e.}\xspace operators that can manipulate their computational context. The literature is abundant on this topic---we limit ourselves to pointing out some influential works.
Parigot~\cite{lambdamu-parigot} proposes a calculus $\lambda\mu$ based on cut elimination in natural deduction with multiple conclusions for second-order classical logic. Its control operator $\mu$ is related with the rule we call contraposition. The study of $\lambda\mu$ is mature: topics such as separability~\cite{david2001lambdamu,saurin2005separation,DBLP:conf/csl/Saurin08}, abstract machines~\cite{DBLP:journals/mscs/Groote98}, call-by-need~\cite{pedrot2016classical}, intersection types~\cite{DBLP:journals/lmcs/KesnerV19}, and encodings into linear logic~\cite{DBLP:journals/tcs/Laurent03,DBLP:conf/csl/KesnerBV20} have been developed.
Barbanera and Berardi~\cite{symmetric-Barbanera-berardi} propose a symmetric $\lambda$-calculus based on a system of natural deduction including a ``symmetric application'' operator $(t \bigstar s)$, closely related to our witness of absurdity ($\strongabs{}{t}{s}$). The system of~\cite{symmetric-Barbanera-berardi} is sound and complete with respect to second-order classical logic, and it is strongly normalizing, but not confluent.
Curien and Herbelin~\cite{Curien00theduality} derive a calculus $\bar{\lambda}\mu\tilde{\mu}$ from Gentzen's classical sequent calculus. This exposes the symmetry between a program yielding an output and a continuation consuming an input. The interaction between a program and a continuation, written $\langle{t\mids}\rangle$ is also reminiscent to our witness of absurdity ($\strongabs{}{t}{s}$). Many variants of this system have been studied; for example, recently, Miquey~\cite{DBLP:journals/toplas/Miquey19} has extended $\bar{\lambda}\mu\tilde{\mu}$ to incorporate dependent types.
Classical calculi such as $\lambda\mu$ and $\bar{\lambda}\mu\tilde{\mu}$ are typically translated into the $\lambda$-calculus by means of continuation-passing style (CPS) translations, whereas our translation from $\lambda^{\PRK}$ to the extended System~F is simpler. Part of the complexity appears to be factored into the proof that classical inference schemes hold in $\lambda^{\PRK}$~({\em i.e.}\xspace \rlem{classical_embedding_computation}).
The works of Andreoli~\cite{andreoli1992logic} and Girard~\cite{girard1993unity} in linear logic introduced the notions of {\em focusing} and {\em polarity}, which allow to formulate linear, intuitionistic, and classical logic as fragments of a single system (Unified~Logic). Our notions of positive and negative formulae, which express affirmation and denial, should not be confused with the subtler notions of positive and negative formulae in the sense of polarity.
Krivine~\cite{krivine2009} defines a realizability interpretation for classical logic using an abstract machine $\lambda_c$ that extends Krivine's abstract machine with further instructions. This approach is based on the idea that adding logical axioms corresponds to adding instructions to the machine, and it has been adapted to provide computational meaning to reasoning principles such as the axiom of dependent choice~\cite{DBLP:journals/tcs/Krivine03,herbelin2012constructive}.
Ilik, Lee, and Herbelin~\cite{DBLP:journals/apal/IlikLH10} study a Kripke semantics for classical logic. Note that our work in~\rsec{prk_kripke_semantics} provides a different Kripke semantics for $\textsc{prk}$, and hence for classical logic. The semantics given in~\cite{DBLP:journals/apal/IlikLH10} and our own have some similarities, but the relation between them is not obvious. For example, in~\cite{DBLP:journals/apal/IlikLH10} a Kripke model involves a relation of ``exploding'' world, which has no counterpart in our system.
\ifCLASSOPTIONcompsoc
\section*{Acknowledgments} \else
\section*{Acknowledgment} \fi This work was partially supported by project grant ECOS~Sud~A17C01. The authors would like to thank Eduardo\- Bo\-ne\-lli and the anonymous reviewers for feedback on an early draft.
\begin{alphasection}
\section{Technical Appendix}
\subsection{Proof of the Projection Lemma~(\rlem{projection_lemma})} \lsec{appendix:projection_lemma}
\begin{lemma}[Projection] If $\Gamma,P \vdash Q$ then $\Gamma,\trunc{P} \vdash \trunc{Q}$. \end{lemma} \begin{proof} We call $P$ the {\em target assumption}. The proof proceeds by induction on the derivation of $\Gamma,P \vdash Q$. We only study the cases with positive signs, the negative cases are symmetric.
\Case{$\rulename{Ax}$}:
let $\Gamma,Q \vdash Q$.
There are two cases, depending on whether the target assumption is
in $\Gamma$ or not.
\begin{enumerate}
\item
{\em If the target assumption is in $\Gamma$, {\em i.e.}\xspace $\Gamma = \Gamma',P$.}
Note that we have $\Gamma',\trunc{P},Q \vdash Q$
by the $\rulename{Ax}$ rule.
By truncating the conclusion (\rlem{projection_of_conclusions})
we conclude that $\Gamma',\trunc{P},Q \vdash \trunc{Q}$,
as required.
\item
{\em If the target assumption is $Q$.}
Then we have that $\Gamma,\trunc{Q} \vdash \trunc{Q}$
by the $\rulename{Ax}$ rule.
\end{enumerate}
\Case{$\rulename{Abs}$}:
let $\Gamma,P \vdash Q$ be derived
from $\Gamma,P \vdash R$ and $\Gamma,P \vdash R{}^{\sim}$
for some strong proposition $R$.
By IH\xspace we have that
$\Gamma,\trunc{P} \vdash \trunc{R}$ and $\Gamma,\trunc{P} \vdash \trunc{R{}^{\sim}}$
so by the generalized absurdity rule ($\rulename{Abs}'$) we have that
$\Gamma,\trunc{P} \vdash \trunc{Q}$.
\Case{\rulename{I$\land\pp$}}:
let $\Gamma,P \vdash (A\landB){}^+$
be derived from $\Gamma,P \vdash A{}^\oplus$ and $\Gamma,P \vdash A{}^\oplus$.
By IH\xspace, $\Gamma,\trunc{P} \vdash A{}^\oplus$ and $\Gamma,\trunc{P} \vdash A{}^\oplus$.
By the $\rulename{I$\land\pp$}$ rule,
$\Gamma,\trunc{P} \vdash (A\landB){}^+$.
Projecting the conclusion (\rlem{projection_of_conclusions}),
$\Gamma,\trunc{P} \vdash (A\landB){}^\oplus$
as required.
\Case{$\Eandp$}:
let $\Gamma,P \vdash A_i{}^\oplus$ be derived from
$\Gamma,P \vdash (A_1\landA_2){}^+$.
Then the proof is of the form:
\[
\indruleN{\rulename{CS}}{
\indruleN{\Eandp}{
\indruleN{\rulename{EC$\pp$}}{
\derivdots{\pi}
\hspace{.5cm}
\derivdots{\xi}
}{
\Gamma,\trunc{P},A_i{}^\ominus \vdash (A_1\landA_2){}^+
}
}{
\Gamma,\trunc{P},A_i{}^\ominus \vdash A_i{}^\oplus
}
}{
\Gamma,\trunc{P} \vdash A_i{}^\oplus
}
\]
where:
\[
\begin{array}{rcl}
\pi & \eqdef &
\indruleNParen{\rulename{W}}{
\indruleN{}{\text{IH\xspace}}{\Gamma,\trunc{P} \vdash (A_1\landA_2){}^\oplus}
}{
\Gamma,\trunc{P},A_i{}^\ominus \vdash (A_1\landA_2){}^\oplus
}
\\
\xi & \eqdef &
\indruleNParen{\rulename{PC}}{
\indruleN{\Iandn}{
\indruleN{\rulename{Ax}}{}{\Gamma,\trunc{P},A_i{}^\ominus \vdash A_i{}^\ominus}
}{
\Gamma,\trunc{P},A_i{}^\ominus \vdash (A_1\landA_2){}^-
}
}{
\Gamma,\trunc{P},A_i{}^\ominus \vdash (A_1\landA_2){}^\ominus
}
\end{array}
\]
\Case{$\Iorp$}:
let $\Gamma,P \vdash (A_1 \lor A_2){}^+$
be derived from $\Gamma,P \vdash A_i{}^\oplus$.
By IH\xspace, $\Gamma,\trunc{P} \vdash A_i{}^\oplus$.
By the $\Iorp$ rule, $\Gamma,\trunc{P} \vdash (A_1 \lor A_2){}^+$.
Projecting the conclusion~(\rlem{projection_of_conclusions}),
$\Gamma,\trunc{P} \vdash (A_1 \lor A_2){}^\oplus$.
\Case{$\rulename{E$\lor\pp$}$}:
let $\Gamma,P \vdash Q$
be derived from
$\Gamma,P \vdash (A_1\lorA_2){}^+$
and
$\Gamma,P,A_i{}^\oplus \vdash Q$ for each $i \in \set{1,2}$.
By IH\xspace,
$\Gamma,\trunc{P} \vdash (A_1\lorA_2){}^\oplus$
and
$\Gamma,\trunc{P},A_i{}^\oplus \vdash \trunc{Q}$
for each $i \in \set{1,2}$. Then the proof is of the form:
\[
\indruleN{\rulename{CS}}{
\indruleN{\rulename{E$\lor\pp$}}{
\indruleN{\rulename{EC$\pp$}}{
\derivdots{\rho}
\hspace{.5cm}\hspace{.5cm}
\derivdots{\xi}
\hspace{.5cm}
}{
\Gamma,\trunc{P},\trunc{Q{}^{\sim}} \vdash (A_1\lorB_2){}^+
}
\hspace{.5cm}
\derivdots{\pi_1} \hspace{.5cm} \derivdots{\pi_2}
\hspace{.5cm}
}{
\Gamma,\trunc{P},\trunc{Q{}^{\sim}} \vdash \trunc{Q}
}
}{
\Gamma,\trunc{P} \vdash \trunc{Q}
}
\]
where:
\[
\begin{array}{rcl}
\rho & \eqdef &
\indruleNParen{\rulename{W}}{
\indruleN{}{\text{IH\xspace}}{\Gamma,\trunc{P} \vdash (A_1\lorA_2){}^\oplus}
}{
\Gamma,\trunc{P},\trunc{Q{}^{\sim}} \vdash (A_1\lorA_2){}^\oplus
}
\\
\\
\xi & \eqdef &
\indruleNParen{\rulename{I$\lor\nn$}}{
\derivdots{\xi_1} \hspace{.5cm} \derivdots{\xi_2}
}{
\Gamma,\trunc{P},\trunc{Q{}^{\sim}} \vdash (A_1\lorA_2){}^\ominus
}
\end{array}
\]
and for each $i \in \set{1,2}$ the derivations
$\pi_i$ and $\xi_i$ are given by:
\[
\begin{array}{rcl}
\pi_i & \eqdef &
\indruleNParen{\rulename{W}}{
\indruleN{}{\text{IH\xspace}}{\Gamma,\trunc{P},A_i{}^\oplus \vdash \trunc{Q}}
}{
\Gamma,\trunc{P},\trunc{Q{}^{\sim}},A_i{}^\oplus \vdash \trunc{Q}
}
\\
\\
\xi_i & \eqdef &
\indruleNParen{\rulename{Contra}}{
\indruleN{}{\text{IH\xspace}}{\Gamma,\trunc{P},A_i{}^\oplus \vdash \trunc{Q}}
}{
\Gamma,\trunc{P},\trunc{Q{}^{\sim}} \vdash A_i{}^\ominus
}
\end{array}
\]
\Case{$\rulename{I$\lnot\pp$}$}:
let $\Gamma,P \vdash (\negA){}^+$ be derived from $\Gamma,P \vdash A{}^\ominus$.
By IH\xspace we have that $\Gamma,\trunc{P} \vdash A{}^\ominus$.
By the $\rulename{I$\lnot\pp$}$ rule,
$\Gamma,\trunc{P} \vdash (\negA){}^+$.
Projecting the conclusion~(\rlem{projection_of_conclusions}),
$\Gamma,\trunc{P} \vdash (\negA){}^\oplus$.
\Case{$\rulename{E$\lnot\pp$}$}:
let $\Gamma,P \vdash A{}^\ominus$ be derived from $\Gamma,P \vdash (\negA){}^+$.
Then the proof is of the form:
\[
\hspace{-.5cm}
\indruleN{\rulename{CS}}{
\indruleN{\rulename{E$\lnot\pp$}}{
\indruleN{\rulename{EC$\pp$}}{
\derivdots{\pi}
\hspace{.5cm}
\derivdots{\xi}
}{
\Gamma,\trunc{P},A{}^\oplus \vdash (\negA){}^+
}
}{
\Gamma,\trunc{P},A{}^\oplus \vdash A{}^\ominus
}
}{
\Gamma,\trunc{P} \vdash A{}^\ominus
}
\]
where:
\[
\begin{array}{rcl}
\pi & \eqdef &
\indruleNParen{\rulename{W}}{
\indruleN{}{\text{IH\xspace}}{\Gamma,\trunc{P} \vdash (\negA){}^\oplus}
}{
\Gamma,\trunc{P},A{}^\oplus \vdash (\negA){}^\oplus
}
\\
\\
\xi & \eqdef &
\indruleNParen{\rulename{IC$\nn$}}{
\indruleN{\rulename{I$\lnot\nn$}}{
\indruleN{\rulename{Ax}}{
}{
\Gamma,\trunc{P},A{}^\oplus,(\negA){}^\oplus \vdash A{}^\oplus
}
}{
\Gamma,\trunc{P},A{}^\oplus,(\negA){}^\oplus \vdash (\negA){}^-
}
}{
\Gamma,\trunc{P},A{}^\oplus \vdash (\negA){}^\ominus
}
\end{array}
\]
\Case{$\rulename{IC$\pp$}$}:
let $\Gamma,P \vdash A{}^\oplus$
be derived from $\Gamma,P,A{}^\ominus \vdash A{}^+$.
By IH\xspace, $\Gamma,\trunc{P},A{}^\ominus \vdash A{}^\oplus$,
so by \rlem{classical_strengthening} we have that
$\Gamma,\trunc{P} \vdash A{}^\oplus$.
\Case{$\rulename{EC$\pp$}$}:
let $\Gamma,P \vdash A{}^+$
be derived from $\Gamma,P \vdash A{}^\oplus$
and $\Gamma,P \vdash A{}^\ominus$.
Then, in particular, by IH\xspace on the first premise,
we have $\Gamma,\trunc{P} \vdash A{}^\oplus$,
as required. \end{proof}
\subsection{Proof of Properties of Forcing~(\rlem{properties_forcing})} \lsec{appendix:properties_forcing}
\begin{lemma}[Monotonicity of forcing] \llem{appendix:monotonicity_forcing} If $\kripforce{w}{P}$ and $w \mathrel{\leq} w'$ then $\kripforce{w'}{P}$. \end{lemma} \begin{proof} By induction on the measure $\#(P)$. We only check the positive propositions; the negative cases are dual---{\em e.g.}\xspace the proof for $(A\landB){}^-$ is symmetric to the proof for $(A\lorB){}^+$: \begin{enumerate} \item
{\em Propositional variable, $P = \alpha{}^+$.}
Let $\kripforce{w}{\alpha{}^+}$,
that is $\alpha \in \wlpos{w}$.
Then by the monotonicity property we have that $\alpha \in \wlpos{w'}$,
so $\kripforce{w'}{\alpha{}^+}$. \item
{\em Conjunction, $P = (A\landB){}^+$.}
Let $\kripforce{w}{(A\landB){}^+}$,
that is
$\kripforce{w}{A{}^\oplus}$ and $\kripforce{w}{B{}^\oplus}$.
Then by IH\xspace
$\kripforce{w'}{A{}^\oplus}$ and $\kripforce{w'}{B{}^\oplus}$
so $\kripforce{w'}{(A\landB){}^+}$. \item
{\em Disjunction, $P = (A\lorB){}^+$.}
Let $\kripforce{w}{(A\lorB){}^+}$,
that is
$\kripforce{w}{A{}^\oplus}$ or $\kripforce{w}{B{}^\oplus}$.
We consider the two possibilities.
On one hand, if $\kripforce{w}{A{}^\oplus}$
then by IH\xspace $\kripforce{w'}{A{}^\oplus}$
so $\kripforce{w'}{(A\lorB){}^+}$.
On the other hand, if $\kripforce{w}{B{}^\oplus}$
then by IH\xspace $\kripforce{w'}{B{}^\oplus}$
so $\kripforce{w'}{(A\lorB){}^+}$. \item
{\em Negation, $P = (\negA){}^+$.}
Let $\kripforce{w}{(\negA){}^+}$,
that is $\kripforce{w}{A{}^\ominus}$.
Then by IH\xspace $\kripforce{w'}{A{}^\ominus}$
so $\kripforce{w'}{(\negA){}^+}$. \item
{\em Classical proposition, $P = A{}^\oplus$.}
Let $\kripforce{w}{A{}^\oplus}$,
that is, for every $w'' \mathrel{\geq} w$
we have that $\kripnotforce{w''}{A{}^-}$.
Our goal is to prove that $\kripforce{w'}{A{}^\oplus}$,
so let $w'' \mathrel{\geq} w'$
and let us check that $\kripnotforce{w''}{A{}^-}$.
Indeed, given that $w'' \mathrel{\geq} w' \mathrel{\geq} w$
we have that $\kripnotforce{w''}{A{}^-}$. \end{enumerate} \end{proof}
\begin{lemma}[Stabilization of forcing] \llem{appendix:stabilization_forcing} For every world $w$ and every proposition $P$, there is a world $w' \mathrel{\geq} w$ such that either $\kripforce{w'}{P}$ or $\kripforce{w'}{P{}^{\sim}}$, but not both. \end{lemma} \begin{proof} By induction on the measure $\#(P)$. We only check the positive propositions; the negative cases are dual. \begin{enumerate} \item
{\em Propositional variable, $P = \alpha{}^+$ and $P{}^{\sim} = \alpha{}^-$.}
By the stabilization property,
there exists $w' \mathrel{\geq} w$
such that $\alpha \in \wlpos{w'} \triangle \wlneg{w'}$,
{\em i.e.}\xspace $\alpha \in \wlpos{w'}$ or $\alpha \in \wlneg{w'}$ but not both,
so we consider two cases:
\begin{enumerate}
\item
If $\alpha \in \wlpos{w'} \setminus \wlneg{w'}$
then $\kripforce{w'}{\alpha{}^+}$ and $\kripnotforce{w'}{\alpha{}^-}$.
\item
If $\alpha \in \wlneg{w'} \setminus \wlpos{w'}$
then $\kripforce{w'}{\alpha{}^-}$ and $\kripnotforce{w'}{\alpha{}^+}$.
\end{enumerate} \item
{\em Conjunction,
$P = (A\landB){}^+$ and $P{}^{\sim} = (A\landB){}^-$.}
By IH\xspace there is a world $w_1 \mathrel{\geq} w$
such that either $\kripforce{w_1}{A{}^\oplus}$
or $\kripforce{w_1}{A{}^\ominus}$ but not both,
so we consider two subcases:
\begin{enumerate}
\item
If $\kripforce{w_1}{A{}^\oplus}$ and $\kripnotforce{w_1}{A{}^\ominus}$,
then by IH\xspace there is a world $w_2 \mathrel{\geq} w_1$
such that either $\kripforce{w_2}{B{}^\oplus}$
or $\kripforce{w_2}{B{}^\ominus}$ but not both,
so we consider two further subcases:
\begin{enumerate}
\item
If $\kripforce{w_2}{B{}^\oplus}$ and $\kripnotforce{w_2}{B{}^\ominus}$,
then we take $w' := w_2$.
By monotonicity~(\rlem{appendix:monotonicity_forcing})
we have that $\kripforce{w_2}{A{}^\oplus}$
so indeed $\kripforce{w_2}{(A\landB){}^+}$.
We are left to show that $\kripnotforce{w_2}{(A\landB){}^-}$.
We already know that $\kripnotforce{w_2}{B{}^\ominus}$,
so to conclude it suffices to show that $\kripnotforce{w_2}{A{}^\ominus}$.
Indeed, suppose that $\kripforce{w_2}{A{}^\ominus}$ holds.
By IH\xspace there exists $w_3 \mathrel{\geq} w_2$ such that
either $\kripforce{w_3}{A{}^\oplus}$ or $\kripforce{w_3}{A{}^\ominus}$
but {\em not both}.
However, by monotonicity~(\rlem{appendix:monotonicity_forcing})
---given that both
$\kripforce{w_2}{A{}^\oplus}$ and $\kripforce{w_2}{A{}^\ominus}$ hold---
we know that
both $\kripforce{w_3}{A{}^\oplus}$ and $\kripforce{w_3}{A{}^\ominus}$
hold,
a contradiction.
\item
If $\kripforce{w_2}{B{}^\ominus}$ and $\kripnotforce{w_2}{B{}^\oplus}$,
then we take $w' := w_2$,
and we have that $\kripforce{w_2}{(A\landB){}^-}$
and $\kripnotforce{w_2}{(A\landB){}^+}$.
\end{enumerate}
\item
If $\kripforce{w_1}{A{}^\ominus}$ and $\kripnotforce{w_1}{A{}^\oplus}$,
then we take $w' := w_1$,
and we have that $\kripforce{w_1}{(A\landB){}^-}$
and $\kripnotforce{w_1}{(A\landB){}^+}$.
\end{enumerate} \item
{\em Disjunction, $P = (A\lorB){}^+$ and $P{}^{\sim} = (A\lorB){}^-$.}
By IH\xspace there is a world $w_1 \mathrel{\geq} w$ such that
either $\kripforce{w_1}{A{}^\oplus}$ or $\kripforce{w_1}{A{}^\ominus}$
but not both, so we consider two subcases:
\begin{enumerate}
\item
If $\kripforce{w_1}{A{}^\oplus}$ and $\kripnotforce{w_1}{A{}^\ominus}$,
then we take $w' := w_1$,
and we have that $\kripforce{w_1}{(A\lorB){}^+}$
and $\kripnotforce{w_1}{(A\lorB){}^-}$.
\item
If $\kripforce{w_1}{A{}^\ominus}$ and $\kripnotforce{w_1}{A{}^\oplus}$,
then by IH\xspace there is a world $w_2 \mathrel{\geq} w_1$ such that
either $\kripforce{w_2}{B{}^\oplus}$ or $\kripforce{w_2}{B{}^\ominus}$
but not both, so we consider two further subcases:
\begin{enumerate}
\item
If $\kripforce{w_2}{B{}^\oplus}$ and $\kripnotforce{w_2}{B{}^\ominus}$,
then we take $w' := w_2$, and we have that
$\kripforce{w_2}{(A\lorB){}^+}$
and
$\kripnotforce{w_2}{(A\lorB){}^-}$.
\item
If $\kripforce{w_2}{B{}^\ominus}$ and $\kripnotforce{w_2}{B{}^\oplus}$,
then we take $w' := w_2$.
By monotonicity~(\rlem{appendix:monotonicity_forcing})
we have that $\kripforce{w_2}{A{}^\ominus}$
so indeed $\kripforce{w_2}{(A\lorB){}^-}$.
We are left to show that $\kripnotforce{w_2}{(A\lorB){}^+}$.
We already know that $\kripnotforce{w_2}{B{}^\oplus}$,
so we are left to show that $\kripnotforce{w_2}{A{}^\oplus}$.
Indeed, suppose that $\kripforce{w_2}{A{}^\oplus}$ holds.
By IH\xspace there exists $w_3 \mathrel{\geq} w_2$ such that
either $\kripforce{w_3}{A{}^\oplus}$ or $\kripforce{w_3}{A{}^\ominus}$
holds but {\em not both}.
However, by monotonicity~(\rlem{appendix:monotonicity_forcing})
---given that both
$\kripforce{w_2}{A{}^\oplus}$ and $\kripforce{w_2}{A{}^\ominus}$ hold---
we know that both
$\kripforce{w_3}{A{}^\oplus}$ and $\kripforce{w_3}{A{}^\ominus}$ hold,
a contradiction.
\end{enumerate}
\end{enumerate} \item
{\em Negation, $P = (\negA){}^+$ and $P{}^{\sim} = (\negA){}^-$.}
By IH\xspace there is a world $w' \mathrel{\geq} w$
such that either $\kripforce{w'}{A{}^\oplus}$ or $\kripforce{w'}{A{}^\ominus}$
hold but not both, so we consider two cases:
\begin{enumerate}
\item
If $\kripforce{w'}{A{}^\oplus}$ and $\kripnotforce{w'}{A{}^\ominus}$,
then $\kripforce{w'}{(\negA){}^-}$
and $\kripnotforce{w'}{(\negA){}^+}$.
\item
If $\kripforce{w'}{A{}^\ominus}$ and $\kripnotforce{w'}{A{}^\oplus}$,
then $\kripforce{w'}{(\negA){}^+}$
and $\kripnotforce{w'}{(\negA){}^-}$.
\end{enumerate} \item
{\em Classical proposition, $P = A{}^\oplus$ and $P{}^{\sim} = A{}^\ominus$.}
By IH\xspace there is a world $w' \mathrel{\geq} w$ such that
either $\kripforce{w'}{A{}^+}$ or $\kripforce{w'}{A{}^-}$
but not both.
We consider two subcases:
\begin{enumerate}
\item
If $\kripforce{w'}{A{}^+}$ and $\kripnotforce{w'}{A{}^-}$,
then we claim that
$\kripforce{w'}{A{}^\oplus}$ and $\kripnotforce{w'}{A{}^\ominus}$.
Indeed, let us prove each condition:
\begin{enumerate}
\item
In order to show that $\kripforce{w'}{A{}^\oplus}$,
it suffices to check that given
$w'' \mathrel{\geq} w'$ we have that $\kripnotforce{w''}{A{}^-}$.
Indeed, suppose that $\kripforce{w''}{A{}^-}$.
Then by IH\xspace there exists $w''' \mathrel{\geq} w''$
such that either $\kripforce{w'''}{A{}^+}$
or $\kripforce{w'''}{A{}^-}$ but {\em not both}.
However, by monotonicity~(\rlem{appendix:monotonicity_forcing})
---given that both $\kripforce{w'}{A{}^+}$
and $\kripforce{w''}{A{}^-}$ hold,
and $w' \mathrel{\leq} w'' \mathrel{\leq} w'''$---
we know that both
$\kripforce{w'''}{A{}^+}$
and
$\kripforce{w'''}{A{}^-}$
hold, a contradiction.
\item
In order to show that $\kripnotforce{w'}{A{}^\ominus}$,
it suffices to note that $\kripforce{w'}{A{}^+}$,
which contradicts the definition of $\kripforce{w'}{A{}^\ominus}$,
given that accessibility is reflexive, {\em i.e.}\xspace $w' \mathrel{\leq} w'$.
\end{enumerate}
\item
If $\kripforce{w'}{A{}^-}$ and $\kripnotforce{w'}{A{}^+}$,
then we claim that
$\kripforce{w'}{A{}^\ominus}$ and $\kripnotforce{w'}{A{}^\oplus}$.
Indeed, let us prove each condition:
\begin{enumerate}
\item
In order to show that $\kripforce{w'}{A{}^\ominus}$,
it suffices to check that given $w'' \mathrel{\geq} w'$
we have that $\kripnotforce{w''}{A{}^+}$.
Indeed, suppose that $\kripforce{w''}{A{}^+}$.
Then by IH\xspace there exists $w''' \mathrel{\geq} w''$
such that either $\kripforce{w'''}{A{}^+}$ and
$\kripforce{w'''}{A{}^-}$ but {\em not both}.
However, by monotonicity~(\rlem{appendix:monotonicity_forcing})
---given that both
$\kripforce{w''}{A{}^+}$ and $\kripforce{w'}{A{}^-}$ hold,
and $w' \mathrel{\leq} w'' \mathrel{\leq} w'''$---
we know that both $\kripforce{w'''}{A{}^+}$
and $\kripforce{w'''}{A{}^-}$ hold, a contradiction.
\item
In order to show that $\kripnotforce{w'}{A{}^\oplus}$
it suffices to note that $\kripforce{w'}{A{}^-}$,
which contradicts the definition of $\kripforce{w'}{A{}^\oplus}$,
given that accessibility is reflexive, {\em i.e.}\xspace $w' \mathrel{\leq} w'$.
\end{enumerate}
\end{enumerate} \end{enumerate} \end{proof}
\begin{lemma}[Non-contradiction of forcing] \llem{appendix:non_contradiction_forcing} If $\kripforce{w}{P}$ then $\kripnotforce{w}{P{}^{\sim}}$. \end{lemma} \begin{proof} Suppose that both $\kripforce{w}{P}$ and $\kripforce{w}{P{}^{\sim}}$ hold. By stabilization~(\rlem{appendix:stabilization_forcing}) there is a world $w' \mathrel{\geq} w$ such that either $\kripforce{w'}{P}$ or $\kripforce{w'}{P{}^{\sim}}$ but {\em not both}. However, by monotonicity~(\rlem{appendix:monotonicity_forcing}) we know that both $\kripforce{w'}{P}$ and $\kripforce{w'}{P{}^{\sim}}$ must hold, a contradiction. \end{proof}
\subsection{Proof of Soundness of $\textsc{prk}$ with respect to the Kripke semantics} \lsec{appendix:kripke_soundness}
\begin{lemma}[Rule of classical forcing] \llem{appendix:rule_of_classical_forcing} \quad \begin{enumerate} \item $(\kripforce{w}{A{}^\oplus})$
if and only if,
for all $w' \mathrel{\geq} w$,
$(\kripforce{w'}{A{}^\ominus})$ implies
$(\kripforce{w'}{A{}^+})$. \item $(\kripforce{w}{A{}^\ominus})$
if and only if,
for all $w' \mathrel{\geq} w$,
$(\kripforce{w'}{A{}^\oplus})$
implies
$(\kripforce{w'}{A{}^-})$. \end{enumerate} \end{lemma} \begin{proof} We only prove the first item. The second one is symmetric, flipping all the signs. \begin{itemize} \item[$(\Rightarrow)$]
Suppose that $\kripforce{w}{A{}^\oplus}$,
let $w' \mathrel{\geq} w$,
and let us show that the implication
$(\kripforce{w'}{A{}^\ominus}) \implies (\kripforce{w'}{A{}^+})$
holds.
In fact, the implication holds vacuously,
given that
$\kripforce{w'}{A{}^\oplus}$ by monotonicity~(\rlem{monotonicity_forcing}),
and therefore
$\kripnotforce{w'}{A{}^\ominus}$ by non-contradiction~(\rlem{non_contradiction_forcing}). \item[$(\Leftarrow)$]
Suppose that for every $w' \mathrel{\geq} w$
the implication
$(\kripforce{w'}{A{}^\ominus}) \implies (\kripforce{w'}{A{}^+})$
holds.
Let us show that $\kripforce{w}{A{}^\oplus}$ holds,
{\em i.e.}\xspace that for every $w' \mathrel{\geq} w$
we have that $\kripnotforce{w'}{A{}^-}$.
Let $w'$ be a world such that $w' \mathrel{\geq} w$
and, by contradiction, suppose that $\kripforce{w'}{A{}^-}$.
Then by non-contradiction~(\rlem{non_contradiction_forcing}) we
have that $\kripnotforce{w'}{A{}^+}$.
Hence, to obtain a contradiction, using the implication of the hypothesis,
it suffices to show that $\kripforce{w'}{A{}^\ominus}$,
that is, that for every $w'' \mathrel{\geq} w'$ we have that
$\kripnotforce{w''}{A{}^+}$.
Indeed, let $w'' \mathrel{\geq} w'$.
By monotonicity~(\rlem{monotonicity_forcing}) $\kripforce{w''}{A{}^-}$,
so by non-contradiction~(\rlem{non_contradiction_forcing}) $\kripnotforce{w''}{A{}^+}$,
as required. \end{itemize} \end{proof}
\begin{proposition}[Soundness] If $\Gamma \vdash P$ is provable in $\textsc{prk}$, then $\kripentails{\Gamma}{P}$. \end{proposition} \begin{proof} By induction on the derivation of $\Gamma \vdash P$. The axiom rule, and the introduction and elimination rules for conjunction, disjunction, and negation are straightforward using the definition of Kripke model. The interesting cases are the following rules: \begin{enumerate} \item
\indrulename{\rulename{Abs}}:
let $\Gamma \vdash Q$
be derived from $\Gamma \vdash P$ and $\Gamma \vdash P{}^{\sim}$
for some strong proposition $P$.
Suppose that $\kripforcefull{w}{\Gamma}$ holds in an arbitrary world $w$
under an arbitrary Kripke model $\mathcal{M}$,
and let us show that $\kripforce{w}{Q}$.
Note that by IH\xspace we have that
$\kripforce{w}{P}$ and $\kripforce{w}{P{}^{\sim}}$.
But this is impossible by non-contradiction~(\rlem{non_contradiction_forcing}).
Hence $\kripforce{w}{Q}$. \item
\indrulename{\rulename{IC$\pp$}}:
let $\Gamma \vdash A{}^\oplus$
be derived from $\Gamma,A{}^\ominus \vdash A{}^+$.
Suppose that $\kripforcefull{w}{\Gamma}$ holds
in an arbitrary world $w$
under an arbitrary Kripke model $\mathcal{M}$,
and let us show that $\kripforce{w}{A{}^\oplus}$.
We claim that for every $w' \mathrel{\geq} w$ the implication
$(\kripforce{w'}{A{}^\ominus}) \implies (\kripforce{w'}{A{}^+})$
holds.
Indeed, suppose that $\kripforce{w'}{A{}^\ominus}$.
Moreover, by monotonicity~(\rlem{monotonicity_forcing}),
we have that $\kripforcefull{w'}{\Gamma}$.
So $\kripforcefull{w'}{\Gamma,A{}^\ominus}$ holds.
Hence by IH\xspace we have that $\kripforce{w'}{A{}^+}$.
Given that the implication
$(\kripforce{w'}{A{}^\ominus}) \implies (\kripforce{w'}{A{}^+})$
holds for all $w' \mathrel{\geq} w$,
using the rule of classical forcing (\rlem{rule_of_classical_forcing})
we conclude that $\kripforce{w}{A{}^\oplus}$,
as required. \item
\indrulename{\rulename{IC$\nn$}}:
similar to the \indrulename{\rulename{IC$\pp$}} case. \item
\indrulename{\rulename{EC$\pp$}}, \indrulename{\rulename{EC$\nn$}}:
similar to the \indrulename{\rulename{Abs}} case. \end{enumerate} \end{proof}
\subsection{Auxiliary lemmas to prove Completeness of $\textsc{prk}$ with respect to the Kripke semantics} \lsec{appendix:kripke_completeness}
In the following proof we use an encoding of falsity with the pure proposition $\bot \eqdef (\alpha_0 \land \neg\alpha_0)$ for some fixed propositional variable $\alpha_0$. Remark that $\Gamma \vdash \bot{}^\ominus$ is provable, being an instance of the law of non-contradiction~(\rexample{lem_and_noncontr}).
\begin{lemma}[Consistent extension] \llem{appendix:consistent_extension} Let $\Gamma$ be a consistent set, and let $P$ be a proposition. Then $\Gamma \cup \set{P}$ and $\Gamma \cup \set{P{}^{\sim}}$ are not both inconsistent. \end{lemma} \begin{proof} Suppose that $\Gamma \cup \set{P}$ and $\Gamma \cup \set{P{}^{\sim}}$ are both inconsistent. In particular we have that $\Gamma,P \vdash \bot{}^\oplus$ and $\Gamma,P{}^{\sim} \vdash \bot{}^\oplus$. By the projection lemma~(\rlem{projection_lemma}) we have that $\Gamma,\trunc{P} \vdash \bot{}^\oplus$ and $\Gamma,\trunc{P{}^{\sim}} \vdash \bot{}^\oplus$. Moreover, by contraposition~(\rlem{admissible_rules_logic}) we have that $\Gamma,\bot{}^\ominus \vdash \trunc{P{}^{\sim}}$ and $\Gamma,\bot{}^\ominus \vdash \trunc{P}$. Since $\bot{}^\ominus$ is provable~(\rexample{lem_and_noncontr}), applying the cut rule~(\rlem{admissible_rules_logic}) we have that $\Gamma \vdash \trunc{P{}^{\sim}}$ and $\Gamma \vdash \trunc{P}$. The generalized absurdity rule allows us to derive $\Gamma \vdash Q$ for any $Q$ from these two sequents, so $\Gamma$ is inconsistent. This contradicts the hypothesis that $\Gamma$ is consistent. \end{proof}
\begin{lemma}[Saturation] \llem{appendix:kripke_saturation} Let $\Gamma$ be a consistent set of propositions, and let $Q$ be a proposition such that $\Gamma \nvdash Q$. Then there exists a prime theory $\Gamma' \supseteq \Gamma$ such that $\Gamma' \nvdash Q$. \end{lemma} \begin{proof} Consider an enumeration of all propositions $(P_1,P_2,\hdots)$. We build a sequence of sets $\Gamma = \Gamma_0 \subseteq \Gamma_1 \subseteq \Gamma_2 \subseteq \hdots$, with the invariant that $\Gamma_n \nvdash Q$ for all $n \geq 0$, according to the following construction.
In the $n$-th step, suppose that $\Gamma_1,\hdots,\Gamma_n$ have already been constructed, and consider the first proposition $P$ in the enumeration such that $\Gamma_n \vdash P$ but the disjunctive property fails for $P$, that is, either $P$ is of the form $(A \lor B){}^+$ with $A{}^\oplus,B{}^\oplus \notin \Gamma_n$ or $P$ is of the form $(A \land B){}^-$ with $A{}^\ominus,B{}^\ominus \notin \Gamma_n$. There are two subcases: \begin{enumerate} \item
If $P = (A \lor B){}^+$ with $A{}^\oplus,B{}^\oplus \notin \Gamma_n$,
note that $\Gamma_n,A{}^\oplus \vdash Q$ and $\Gamma_n,B{}^\oplus \vdash Q$
cannot both hold simultaneously.
Indeed, if both $\Gamma_n,A{}^\oplus \vdash Q$ and $\Gamma_n,B{}^\oplus \vdash Q$ hold,
given that also $\Gamma_n \vdash (A\lorB){}^+$,
applying $\rulename{E$\lor\pp$}$ we would have $\Gamma_n \vdash Q$, contradicting the hypothesis.
Hence we may define $\Gamma_{n+1}$ as follows:
\[
\Gamma_{n+1} \eqdef
\begin{cases}
\Gamma_n \cup \set{A{}^\oplus} & \text{if $\Gamma_n,A{}^\oplus \nvdash Q$} \\
\Gamma_n \cup \set{B{}^\oplus} & \text{otherwise} \\
\end{cases}
\]
Note that,
in the second case, $\Gamma_n,B{}^\oplus \nvdash Q$ holds. \item
If $P = (A \land B){}^-$ with $A{}^\ominus,B{}^\ominus \notin \Gamma_n$,
the construction is similar,
defining $\Gamma_{n+1}$ as
either $\Gamma_n \cup \set{A{}^\ominus}$
or $\Gamma_n \cup \set{B{}^\ominus}$. \end{enumerate} Now we define $\Gamma_\omega$ and $\Gamma'$ as follows: \[
\begin{array}{rcl}
\Gamma_\omega & \eqdef & \bigcup_{n \in \mathbb{N}} \Gamma_n \\
\Gamma' & \eqdef & \Gamma_\omega
\cup \set{A^{\pm} \ |\ \Gamma_\omega \vdash A^{\pm}}
\end{array} \] Note that $\Gamma \subseteq \Gamma_\omega \subseteq \Gamma'$. Moreover, we claim that $\Gamma'$ is a prime theory:
{\bf Closure by deduction.}
Let $\Gamma' \vdash P$, and let us show that $P \in \Gamma'$.
Since all assumptions in $\Gamma'$ of the form $A^{\pm}$ are provable
from $\Gamma_\omega$, this means that $\Gamma_\omega \vdash P$
by the cut rule~(\rlem{admissible_rules}).
We consider four subcases,
depending on whether $P$ is a strong/classical proof/refutation.
We only study the positive cases; the negative cases are symmetric:
\begin{enumerate}
\item {\em Strong proof, {\em i.e.}\xspace $P = A{}^+$.}
Then $\Gamma_\omega \vdash A{}^+$
so $A{}^+ \in \Gamma'$ by definition of $\Gamma'$.
\item {\em Classical proof, {\em i.e.}\xspace $P = A{}^\oplus$.}
Then $\Gamma_\omega \vdash A{}^\oplus$
so in particular $\Gamma_\omega \vdash (A \lor A){}^+$
applying the $\Iorp[1]$ rule.
Then there is an $n_0$ such that
$\Gamma_n \vdash (A \lor A){}^+$ for all $n \geq n_0$.
Then it cannot be the case that $A{}^\oplus \notin \Gamma_n$ for all $n \geq n_0$,
because the proposition $(A \lor A){}^+$
must be eventually treated by
the construction of $(\Gamma_n)_{n \in \mathbb{N}}$ above.
This means that there is an $n \geq n_0$
such that $A{}^\oplus \in \Gamma_n$,
and therefore $A{}^\oplus \in \Gamma_\omega \subseteq \Gamma'$,
as required.
\end{enumerate}
{\bf Consistency.}
It suffices to note that $\Gamma' \nvdash Q$.
Indeed, suppose that $\Gamma' \vdash Q$.
Then $\Gamma_\omega \vdash Q$
by the cut rule~(\rlem{admissible_rules}),
so there exists an $n_0$ such that $\Gamma_n \vdash Q$ for all $n \geq n_0$.
This contradicts the invariant of
the construction of $(\Gamma_n)_{n \in \mathbb{N}}$ above.
{\bf Disjunctive property.}
We consider only the positive case. The negative case is symmetric.
Suppose that $\Gamma' \vdash (A \lor B){}^+$.
Then $\Gamma_\omega \vdash (A \lor B){}^+$
by the cut rule~(\rlem{admissible_rules}),
so there exists an $n_0$ such that $\Gamma_n \vdash (A \lor B){}^+$ for all $n \geq n_0$.
Then it cannot be the case that $A{}^\oplus,B{}^\oplus \notin \Gamma_n$ for all $n \geq n_0$,
because the proposition $(A \lor B){}^+$
must be eventually treated by
the construction of $(\Gamma_n)_{n \in \mathbb{N}}$ above.
This means that there is an $n \geq n_0$
such that either $A{}^\oplus \in \Gamma_n$ or $B{}^\oplus \in \Gamma_n$,
and therefore we have that
either $A{}^\oplus \in \Gamma_\omega \subseteq \Gamma'$,
or $B{}^\oplus \in \Gamma_\omega \subseteq \Gamma'$, as required.
Finally, note that $\Gamma' \nvdash Q$, as has already been shown in the proof of consistency above. \end{proof}
\begin{definition}[Canonical model] The {\em canonical model} is the structure $\mathcal{M}_0 = (\mathcal{W}_0,\subseteq,\wlpos{},\wlneg{})$: \begin{enumerate} \item $\mathcal{W}_0$ is the set of all prime theories,
{\em i.e.}\xspace $\mathcal{W}_0 \eqdef \set{\Gamma \ |\ \text{$\Gamma$ is prime}}$. \item $\subseteq$ is the set-theoretic inclusion between prime theories. \item $\wlpos{\Gamma} = \set{\alpha \ |\ \alpha{}^+ \in \Gamma}$
and
$\wlneg{\Gamma} = \set{\alpha \ |\ \alpha{}^- \in \Gamma}$. \end{enumerate} \end{definition}
\begin{lemma} \llem{appendix:canonical_model_is_kripke} The canonical model is a Kripke model. \end{lemma} \begin{proof} Let us check the two required properties. {\bf Monotonicity} is immediate, since if $\Gamma \subseteq \Gamma'$ then $\alpha^{\pm} \in \Gamma$ implies $\alpha^{\pm} \in \Gamma'$. For {\bf stabilization}, let $\Gamma$ be a prime theory and let $\alpha$ be a propositional variable. First note that $\Gamma \cup \set{\alpha{}^+}$ and $\Gamma \cup \set{\alpha{}^-}$ cannot both be inconsistent, by the consistent extension lemma~(\rlem{appendix:consistent_extension}). We consider two subcases, depending on whether $\Gamma \cup \set{\alpha{}^+}$ is consistent: \begin{enumerate} \item {\em If $\Gamma \cup \set{\alpha{}^+}$ is consistent.}
Then $\Gamma,\alpha{}^+ \nvdash \alpha{}^-$
because $\Gamma,\alpha{}^+ \vdash \alpha{}^-$ would make
the set $\Gamma\cup\set{\alpha{}^+}$ inconsistent.
Then by saturation~(\rlem{appendix:kripke_saturation})
there is a prime theory $\Gamma' \supseteq \Gamma \cup \set{\alpha{}^+}$
such that $\Gamma' \nvdash \alpha{}^-$.
Hence we have that $\Gamma' \supseteq \Gamma$
with $\alpha \in \wlpos{\Gamma'} \setminus \wlneg{\Gamma'}$. \item {\em Otherwise, so $\Gamma \cup \set{\alpha{}^-}$ is consistent.}
Similarly as in the previous case,
we have that $\Gamma,\alpha{}^- \nvdash \alpha{}^+$,
so by saturation~(\rlem{appendix:kripke_saturation})
there is a prime theory $\Gamma' \supseteq \Gamma \cup \set{\alpha{}^-}$
such that $\Gamma' \nvdash \alpha{}^+$,
and this implies that $\alpha \in \wlneg{\Gamma'} \setminus \wlpos{\Gamma'}$. \end{enumerate} \end{proof}
\begin{lemma}[Main Semantic Lemma] \llem{appendix:kripke_main_semantic_lemma} Let $\Gamma$ be a prime theory. Then $\kripforce[\mathcal{M}_0]{\Gamma}{P}$ holds in the canonical model if and only if $P \in \Gamma$. \end{lemma} \begin{proof} We proceed by induction on the measure $\#(P)$. We only study the positive cases, the negative cases are symmetric.
{\bf Propositional variable, $P = \alpha{}^+$.}
\[
\kripforce[\mathcal{M}_0]{\Gamma}{\alpha{}^+}
\iff
\alpha \in \wlpos{\Gamma}
\iff
\alpha{}^+ \in \Gamma
\]
{\bf Strong conjunction, $P = (A \land B){}^+$.}
\[
\begin{array}{rcll}
&&
\kripforce[\mathcal{M}_0]{\Gamma}{(A \land B){}^+}
\\
& \iff &
\kripforce[\mathcal{M}_0]{\Gamma}{A{}^\oplus} \text{ and }
\kripforce[\mathcal{M}_0]{\Gamma}{B{}^\oplus}
\\
& \iff &
A{}^\oplus \in \Gamma \text{ and } B{}^\oplus \in \Gamma
& \text{by IH\xspace}
\\
& \iff &
(A \land B){}^+ \in \Gamma
\end{array}
\]
The last equivalence uses the
fact that $\Gamma$ is closed by deduction,
using rule $\rulename{I$\land\pp$}$ for the ``only if'' direction
and rules $\Eandp[1],\Eandp[2]$ for the ``if'' direction.
{\bf Strong disjunction, $P = (A \lor B){}^+$.}
\[
\begin{array}{rcll}
&&
\kripforce[\mathcal{M}_0]{\Gamma}{(A \lor B){}^+}
\\
& \iff &
\kripforce[\mathcal{M}_0]{\Gamma}{A{}^\oplus}
\text{ or }
\kripforce[\mathcal{M}_0]{\Gamma}{B{}^\oplus}
\\
& \iff &
A{}^\oplus \in \Gamma
\text{ or }
B{}^\oplus \in \Gamma
& \text{by IH\xspace}
\\
& \iff &
(A \lor B){}^+ \in \Gamma
\end{array}
\]
The last equivalence uses the fact that
$\Gamma$ is a prime theory,
using rules $\Iorp[1]$ and $\Iorp[2]$
for the ``only if'' direction, and the fact that $\Gamma$ is disjunctive
for the ``if'' direction.
{\bf Strong negation, $P = (\negA){}^+$.}
\[
\begin{array}{rcll}
\kripforce[\mathcal{M}_0]{\Gamma}{(\negA){}^+}
& \iff &
\kripforce[\mathcal{M}_0]{\Gamma}{A{}^\ominus}
\\
& \iff &
A{}^\ominus \in \Gamma
& \text{by IH\xspace}
\\
& \iff &
(\negA){}^+ \in \Gamma
\end{array}
\]
The last equivalence uses the fact that
$\Gamma$ is closed by deduction,
using rule $\rulename{I$\lnot\pp$}$ for the ``only if'' direction
and rule $\rulename{E$\lnot\pp$}$ for the ``if'' direction.
{\bf Classical proposition, $P = A{}^\oplus$.}
\[
\begin{array}{rcll}
\kripforce[\mathcal{M}_0]{\Gamma}{A{}^\oplus}
& \iff &
\forall \Gamma' \supseteq \Gamma,\
\kripnotforce[\mathcal{M}_0]{\Gamma'}{A{}^-}
\\
& \iff &
\forall \Gamma' \supseteq \Gamma,\
A{}^- \notin \Gamma'
& \text{by IH\xspace}
\\
& \iff &
A{}^\oplus \in \Gamma
\end{array}
\]
Note that $\Gamma'$ does not vary over arbitrary sets of propositions,
but only over prime theories.
To justify the last equivalence, we prove each implication separately:
\begin{itemize}
\item[$(\Rightarrow)$]
We show the contrapositive.
Let $A{}^\oplus \notin \Gamma$
and let us show that there is a prime theory $\Gamma' \supseteq \Gamma$
such that $A{}^- \in \Gamma'$.
First we claim that $\Gamma\cup\set{A{}^-}$ is consistent.
\begin{itemize}
\item[] {\em Proof of the claim.}
Suppose by contradiction that $\Gamma\cup\set{A{}^-}$ is inconsistent.
Then in particular $\Gamma,A{}^- \vdash \bot{}^\oplus$.
(Recall that we encode falsity
as $\bot \eqdef (\alpha_0 \land \neg\alpha_0)$).
By the projection lemma~(\rlem{projection_lemma})
we have that $\Gamma,A{}^\ominus \vdash \bot{}^\oplus$.
By contraposition~(\rlem{admissible_rules_logic})
$\Gamma,\bot{}^\ominus \vdash A{}^\oplus$.
Since $\bot{}^\ominus$ is provable~(\rexample{lem_and_noncontr}),
by the cut rule~(\rlem{admissible_rules_logic})
we have that $\Gamma \vdash A{}^\oplus$.
But $\Gamma$ is closed by deduction, so $A{}^\oplus \in \Gamma$.
This contradicts the fact that $A{}^\oplus \notin \Gamma$
and concludes the proof of the claim.
\end{itemize}
Now since $\Gamma\cup\set{A{}^-}$ is consistent,
by saturation~(\rlem{appendix:kripke_saturation}),
we may extend it
to a prime theory $\Gamma' \supseteq \Gamma\cup\set{A{}^-}$.
This concludes this case.
\item[$(\Leftarrow)$]
Suppose that $A{}^\oplus \in \Gamma$, and let $\Gamma' \supseteq \Gamma$
such that $A{}^- \in \Gamma'$.
Then since $\Gamma'$ is closed by deduction, using the $\rulename{IC$\pp$}$ rule
we have that $A{}^\ominus \in \Gamma'$.
Since $\Gamma'$ contains both $A{}^\oplus$ and $A{}^\ominus$,
using the generalized absurdity rule we may
derive an arbitrary proposition from $\Gamma'$,
which means that $\Gamma'$ is inconsistent,
contradicting the fact that $\Gamma'$ is a prime theory.
\end{itemize} \end{proof}
\subsection{Proof of Subject~Reduction of $\lambda^{\PRK}$~(\rprop{subject_reduction})} \lsec{appendix:subject_reduction}
\begin{proposition}[Subject reduction] \lprop{appendix:subject_reduction} If $\Gamma \vdash t : P$ and $t \toa{} s$, then $\Gamma \vdash s : P$. \end{proposition} \begin{proof} Since reduction is closed under arbitrary contexts, the term on the left hand side is of the form $\gctxof{t_0}$ and it reduces to $\gctxof{t_1}$ contracting the redex $t_0$. We proceed by induction on the context $\mathtt{C}$ under which the rewriting step takes place. The interesting case is when the context is empty. All other cases are easy by resorting to the IH\xspace. We proceed by case analysis on each of the reduction rules. Note that each rule actually stands for two rules, depending on the instantiations of the signs. We write only one of these cases; if the signs are flipped the proof is symmetric. We use the admissible typing rules $\rulename{Cut}$ and $\rulename{Abs}'$~(\rlem{admissible_rules}).
\noindent$\bullet$ $\rewritingRuleName{proj}$: let $i \in \set{1,2}$. We have:
\[
{\small
\indrule{\Eandp}{
\indrule{\rulename{I$\land\pp$}}{
\indrule{}{
\pi_1
}{
\Gamma \vdash t_1 : A_1{}^\oplus
}
\hspace{.5cm}
\indrule{}{
\pi_2
}{
\Gamma \vdash t_2 : A_2{}^\oplus
}
}{
\Gamma \vdash \pairp{t_1}{t_2} : (A_1 \land A_2){}^+
}
}{
\Gamma \vdash t_i : A_i{}^\oplus
}
}
\] Then:
\[
{\small
\prooftree
\pi_i
\justifies \Gamma \vdash t_i : A{}^\oplus
\thickness=0.05em
\endprooftree
}
\]
\noindent$\bullet$ $\rewritingRuleName{case}$:
let $i \in \set{1,2}$. We have:
\[
{\small
\indrule{\rulename{E$\lor\pp$}}{
\indrule{\Iorp}{
\indrule{}{
\pi
}{
\Gamma \vdash t : A_i{}^\oplus
}
}{
\Gamma \vdash \inip{t} : (A_1 \lor A_2){}^+
}
\hspace{.5cm}
\pi_1
\hspace{.5cm}
\pi_2
}{
\Gamma \vdash \casep{(\inip{t})}{x}{s_1}{x}{s_2} : P
}
}
\]
where for each $j \in \set{1,2}$,
the derivation $\pi_j$ is:
\[
\indrule{}{
\pi'_j
}{
\Gamma, x : A_j{}^\oplus \vdash s_j : P
}
\]
Then:
\[
{\small
\prooftree
\[
\pi'_i
\justifies \Gamma, x : A_i{}^\oplus \vdash s_i : P
\]
\[
\pi
\justifies \Gamma \vdash t : A_i{}^\oplus
\]
\justifies \Gamma \vdash s_i\sub{x}{t} : P
\using{\indrulename{\rulename{Cut}}}
\thickness=0.05em
\endprooftree
}
\]
\noindent$\bullet$ $\rewritingRuleName{neg}$:
We have:
\[
{\small
\prooftree
\[
\[
\pi
\justifies \Gamma \vdash t : A{}^\ominus
\]
\justifies \Gamma \vdash \negip{t} : (\lnotA){}^+
\using{\rulename{I$\lnot\pp$}}
\thickness=0.05em
\]
\justifies \Gamma \vdash \negep{(\negip{t})} : A{}^\ominus
\thickness=0.05em
\using{\indrulename{\rulename{E$\lnot\pp$}}}
\endprooftree
}
\]
Then:
\[
{\small
\prooftree
\pi
\justifies \Gamma \vdash t : A{}^\ominus
\thickness=0.05em
\endprooftree
}
\]
\noindent$\bullet$ $\rewritingRuleName{beta}$:
we have:
\[
{\small
\prooftree
\[
\[
\pi
\justifies \Gamma, x : A{}^\ominus \vdash t : A{}^+
\]
\justifies \Gamma \vdash \claslamp{x}{t} : A{}^\oplus
\using{\rulename{IC$\pp$}}
\thickness=0.05em
\]
\[
\pi'
\justifies \Gamma \vdash s : A{}^\ominus
\]
\justifies \Gamma \vdash \clasapp{(\claslamp{x}{t})}{s} : A{}^+
\thickness=0.05em
\using{\indrulename{\rulename{EC$\pp$}}}
\endprooftree
}
\]
Then:
\[
{\small
\prooftree
\[
\pi
\justifies \Gamma, x : A{}^\ominus \vdash t : A{}^+
\]
\[
\pi'
\justifies \Gamma \vdash s : A{}^\ominus
\]
\justifies \Gamma \vdash t\sub{x}{s} : A{}^+
\using{\indrulename{\rulename{Cut}}}
\thickness=0.05em
\endprooftree
}
\] \noindent$\bullet$ $\rewritingRuleName{absPairInj}$:
we have:
\[
{\footnotesize
\prooftree
\[
\[
\pi_1
\justifies \Gamma \vdash t_1 : A_1{}^\oplus
\]
\[
\pi_2
\justifies \Gamma \vdash t_2 : A_2{}^\oplus
\]
\justifies \Gamma \vdash \pairp{t_1}{t_2} : (A_1 \land A_2){}^+
\using{\rulename{I$\land\pp$}}
\thickness=0.05em
\]
\[
\[
\pi'
\justifies \Gamma \vdash s : A_i{}^\ominus
\]
\justifies \Gamma \vdash \inin{s} : (A_1 \land A_2){}^-
\using{\indrulename{\Iandn}}
\]
\justifies \Gamma \vdash \strongabs{P}{\pairp{t_1}{t_2}}{\inin{s}} : P
\thickness=0.05em
\using{\indrulename{\rulename{Abs}}}
\endprooftree
}
\]
Then:
\[
{\small
\prooftree
\[
\pi_i
\justifies \Gamma \vdash t_i : A_i{}^\oplus
\]
\[
\pi'
\justifies \Gamma \vdash s : A_i{}^\ominus
\]
\justifies \Gamma \vdash \abs{P}{t_i}{s} : P
\using{\indrulename{\rulename{Abs}'}}
\thickness=0.05em
\endprooftree
}
\]
\noindent$\bullet$ $\rewritingRuleName{absInjPair}$:
similar to the previous case.
\noindent$\bullet$ $\rewritingRuleName{absNeg}$:
\[
{\small
\prooftree
\[
\[
\pi
\justifies \Gamma \vdash t : A{}^\ominus
\]
\justifies \Gamma \vdash \negip{t} : (\lnotA){}^+
\using{\indrulename{\rulename{I$\lnot\pp$}}}
\]
\[
\[
\pi'
\justifies \Gamma \vdash s : A{}^\oplus
\]
\justifies \Gamma \vdash \negin{s} : (\lnotA){}^-
\using{\indrulename{\rulename{I$\lnot\nn$}}}
\]
\justifies \Gamma \vdash \strongabs{P}{(\negip{t})}{(\negin{s})} : P
\thickness=0.05em
\using{\indrulename{\rulename{Abs}}}
\endprooftree
}
\]
Then:
\[
{\small
\prooftree
\[
\pi
\justifies \Gamma \vdash t : A{}^\ominus
\]
\[
\pi'
\justifies \Gamma \vdash s : A{}^\oplus
\]
\justifies \Gamma \vdash \abs{P}{t}{s} : P
\thickness=0.05em
\using{\indrulename{\rulename{Abs}'}}
\endprooftree
}
\] \end{proof}
\subsection{Proof of the Positivity Condition for~\rcoro{systemF_SN_posneg}} \lsec{appendix:positivity_condition}
\begin{definition} Recall that the set of type constraints $\mathcal{C}_{\mathbf{pn}}$ is given by all equations of the following form, for all types $A,B$ of System~F: \[
\Pos{A}{B}
\equiv
(\Neg{A}{B} \to A)
\hspace{.5cm}
\Neg{A}{B}
\equiv
(\Pos{A}{B} \to B) \] \end{definition}
\begin{proposition} The set of type constraints $\mathcal{C}_{\mathbf{pn}}$ verifies Mendler's positivity condition (stated in the body of the paper, and also in the appendix in \rdef{positivity}). \end{proposition} \begin{proof} Define the {\em complexity} of a type as follows: \[
\begin{array}{rcl@{\hspace{1cm}}rcl}
\compl{\alpha} & \eqdef & 1 \text{\hspace{.5cm} if $\alpha \in \mathbf{V}$}
\\
\compl{\Pos{A}{B}}
= \compl{\Neg{A}{B}}
= \compl{A \to B}
& \eqdef & 1 + \compl{A} + \compl{B}
\\
\compl{\forall\alpha.A} & \eqdef & 1 + \compl{A}
\end{array} \] Recall that $\posvars{A}$ (resp. $\negvars{A}$) stand for the set of type variables occurring positively (resp. negatively) in a given type $A$. Moreover, the set of type variables occurring {\em weakly positively} (resp. {\em weakly negatively}) in $A$ are written $\wposvars{A}$ (resp. $\wnegvars{A}$) and defined as follows: \[
\begin{array}{rcl}
\wposvars{\alpha} & \eqdef & \set{\alpha}
\text{\hspace{.5cm} if $\alpha \in \mathbf{V}$}
\\
\wposvars{\Pos{A}{B}} & \eqdef &
\set{\Pos{A}{B}}
\cup \wposvars{A} \cup \wnegvars{B}
\\
\wposvars{\Neg{A}{B}} & \eqdef &
\set{\Neg{A}{B}}
\cup \wnegvars{A} \cup \wposvars{B}
\\
\wposvars{A \to B} & \eqdef & \wnegvars{A} \cup \wposvars{B}
\\
\wposvars{\forall\alpha.A} & \eqdef & \wposvars{A} \setminus \set{\alpha}
\end{array} \] \[
\begin{array}{rcl}
\\
\wnegvars{\alpha} & \eqdef & \varnothing
\text{\hspace{.5cm} if $\alpha \in \mathbf{V}$}
\\
\wnegvars{\Pos{A}{B}} & \eqdef &
\wnegvars{A} \cup \wposvars{B}
\\
\wnegvars{\Neg{A}{B}} & \eqdef &
\wposvars{A} \cup \wnegvars{B}
\\
\wnegvars{A \to B} & \eqdef & \wposvars{A} \cup \wnegvars{B}
\\
\wnegvars{\forall\alpha.A} & \eqdef & \wnegvars{A} \setminus \set{\alpha}
\end{array} \] It is easy to check that $\posvars{A} \subseteq \wposvars{A}$ and $\negvars{A} \subseteq \wnegvars{A}$ by simultaneous induction on $A$. It is also easy to check that if $\alpha \in \wposvars{A} \cup \wnegvars{A}$ then $\compl{\alpha} \leq \compl{A}$, by induction on $A$. Moreover, let $X, Y$ be types. A type $A$ is said to be {\em $(X,Y)$-positive} if $\Pos{X}{Y} \in \wposvars{A}$ or $\Neg{X}{Y} \in \wnegvars{A}$. Symmetrically, a type $A$ is said to be {\em $(X,Y)$-negative} if $\Pos{X}{Y} \in \wnegvars{A}$ or $\Neg{X}{Y} \in \wposvars{A}$. It is straightforward to prove the following {\bf invariant} for the equivalence $A \equiv B$ between types induced by the recursive type constraints, by induction on the derivation of $A \equiv B$. \begin{enumerate} \item
If $A \equiv B$, then
$A$ is $(X,Y)$-positive if and only if $B$ is $(X,Y)$-positive. \item
If $A \equiv B$, then
$A$ is $(X,Y)$-negative if and only if $B$ is $(X,Y)$-negative. \end{enumerate} To prove Mendler's positivity condition, we must check that given any type variable $\alpha$ of the form $\Pos{A}{B}$ or of the form $\Neg{A}{B}$, then whenever $\alpha \equiv C$ we have that $\alpha$ does not occur negatively in $C$. We consider two cases, depending on whether $\alpha = \Pos{A}{B}$ or $\alpha = \Neg{A}{B}$: \begin{enumerate} \item
Let $\Pos{A}{B} \equiv C$
and suppose that $\Pos{A}{B} \in \negvars{C}$.
Then we have that $\Pos{A}{B} \in \wnegvars{C}$,
so $C$ is $(A,B)$-negative.
By the invariant, $\Pos{A}{B}$ is also $(A,B)$-negative,
so either $\Pos{A}{B} \in \wnegvars{\Pos{A}{B}}$
or $\Neg{A}{B} \in \wposvars{\Pos{A}{B}}$.
Both conditions are impossible, indeed:
\begin{enumerate}
\item
Suppose that $\Pos{A}{B} \in \wnegvars{\Pos{A}{B}}$.
Then, given that $\Pos{A}{B}$ does not occur weakly negatively
at the root of $\Pos{A}{B}$, so it
must occur either inside $A$ or inside $B$,
so $\compl{\Pos{A}{B}} < \compl{\Pos{A}{B}}$,
which is a contradiction.
\item
Suppose that $\Neg{A}{B} \in \wnegvars{\Pos{A}{B}}$.
Then, again, $\Neg{A}{B}$
must occur either inside $A$ or inside $B$,
so $\compl{\Neg{A}{B}} < \compl{\Pos{A}{B}}$,
which is a contradiction.
\end{enumerate} \item
If $\Neg{A}{B} \equiv C$ then, symmetrically as above,
we have that $\Neg{A}{B} \notin \negvars{C}$. \end{enumerate} \end{proof}
\subsection{Proof of the Simulation Lemma for the Translation from $\textsc{prk}$
to the Extended System~F}
\begin{lemma} \llem{appendix:semF_simulation} If $t \toa{} s$ in $\lambda^{\PRK}$ then $\semF{t} \toa{}^+ \semF{s}$ in System~F extended with $\mathcal{C}_{\mathbf{pn}}$. \end{lemma} \begin{proof} By case analysis on the rewriting rule used to derive the step $t \toa{} s$. Note that showing contextual closure is immediate, so we only study the cases in which the rewriting rule is applied at the root:
\noindent$\bullet$ $\rewritingRuleName{proj}$:
$
\semF{\projipn{\pairpn{t_1}{t_2}}}
=
\projiF{\pairF{\semF{t_1}}{\semF{t_2}}}
\toa{}
\semF{t_i}
$.
\noindent$\bullet$ $\rewritingRuleName{case}$:
\[
\begin{array}{rcll}
&&
\semF{\casepn{\inipn{t}}{(x:P)}{s_1}{(x:Q)}{s_2}}
\\
& = &
\caseF{\iniF{\semF{t}}}{(x:\semF{P})}{\semF{s_1}}{(x:\semF{Q})}{\semF{s_2}}
\\
& \to &
\semF{s_i}\sub{x}{\semF{t}}
\\
& = &
\semF{s_i\sub{x}{\semF{t}}}
\\
&& \hspace{.5cm}\text{by \rlem{semF_properties}}
\end{array}
\]
\noindent$\bullet$ $\rewritingRuleName{neg}$:
$
\semF{\negepn{\negipn{t}}}
=
(\lam{x^\mathbf{1}}{\semF{t}})\,\trivsym
\toa{}
\semF{t}\sub{x}{\trivsym}
=
\semF{t}
$
by \rlem{semF_properties}, since $x \not\in \fv{t}$
by definition of $\semF{\negipn{t}}$.
\noindent$\bullet$ $\rewritingRuleName{beta}$:
$
\semF{\clasappn{(\claslampn{(x:P)}{t})}{s}}
=
(\lam{x^{\semF{P}}}{\semF{t}})\,\semF{s}
\to
\semF{t}\sub{x}{\semF{s}}
=
\semF{t\sub{x}{s}}
$
by \rlem{semF_properties}.
\noindent$\bullet$ $\rewritingRuleName{absPairInj}$:
we consider two subcases, depending on the signs:
\begin{enumerate}
\item
Let $\vdash t_1 : A_1{}^\oplus$,
$\vdash t_2 : A_2{}^\oplus$,
and $\vdash s : A_i{}^\ominus$ for some $i \in \set{1,2}$.
Then:
\[
\begin{array}{ll}
&
\semF{\strongabs{P}{\pairp{t_1}{t_2}}{\inin{s}}}
\\
= &
\funabsF{(A_1 \land A_2){}^+}{P}
\,\pairF{\semF{t_1}}{\semF{t_2}}
\,\iniF{\semF{s}}
\\
\toa{}^+ &
\caseFtable{\iniF{\semF{s}}}{
(z:\semF{A_1{}^\ominus})
}{
\funabsF{A_1{}^\oplus}{P}\,\projiF[1]{ \pairF{\semF{t_1}}{\semF{t_2}} }\,z
}{
(z:\semF{A_2{}^\ominus})
}{
\funabsF{A_2{}^\oplus}{P}\,\projiF[2]{ \pairF{\semF{t_1}}{\semF{t_2}} }\,z
}
\\
& \hspace{.5cm}\text{by definition of $\funabsF{(A_1 \land A_2){}^+}{P}$}
\\
\toa{} &
\funabsF{A_i{}^\oplus}{P}\,\projiF{ \pairF{\semF{t_1}}{\semF{t_2}} }\,\semF{s}
\\
\toa{} &
\funabsF{A_i{}^\oplus}{P}\,\semF{t_i}\,\semF{s}
\\
\toa{}^+ &
\funabsF{A_i{}^+}{P}\,(\semF{t_i}\,\semF{s})
(\semF{s}\,\semF{t_i})
\\&\hspace{.5cm}\text{by definition of $\funabsF{A_i{}^\oplus}{P}$}
\\
= &
\semF{\strongabs{P}{(\clasapp{t_i}{s})}{(\clasapn{t_i}{s})}}
\\
= &
\semF{\abs{P}{t_i}{s}}
\end{array}
\]
\item
Let $\Gamma \vdash t_1 : A_1{}^\ominus$,
$\Gamma \vdash t_2 : A_2{}^\ominus$,
and $\Gamma \vdash s : A_i{}^\oplus$ for some $i \in \set{1,2}$.
Then, symmetrically as for the previous case,
$
\semF{\strongabs{P}{\pairn{t_1}{t_2}}{\inip{s}}}
\toa{}^+
\semF{\abs{P}{t_i}{s}}
$.
\end{enumerate}
\noindent$\bullet$ $\rewritingRuleName{absInjPair}$:
symmetric to the previous case.
\noindent$\bullet$ $\rewritingRuleName{absNeg}$:
we consider two subcases, depending on the signs:
\begin{enumerate}
\item
Let $\Gamma \vdash t : A{}^\ominus$ and $\Gamma \vdash s : A{}^\oplus$.
Then:
\[
\begin{array}{rcll}
&&
\semF{\strongabs{P}{(\negip{t})}{(\negin{s})}}
\\
& = &
\funabsF{(\negA){}^+}{P}
(\lam{x^{\mathbf{1}}}{\semF{t}})
(\lam{y^{\mathbf{1}}}{\semF{s}})
\\&& \hspace{.5cm}\text{where $x \not\in \fv{t}$,
$y \not\in \fv{s}$}
\\
& \toa{}^+ &
\funabsF{A{}^\ominus}{P}
((\lam{x^{\mathbf{1}}}{\semF{t}})\,\trivsym)
((\lam{y^{\mathbf{1}}}{\semF{s}})\,\trivsym)
\\&& \hspace{.5cm}\text{by definition of $\funabsF{(\negA){}^+}{P}$}
\\
& \toa{}^+ &
\funabsF{A{}^\ominus}{P}\,\semF{t}\,\semF{s}
\\
& \toa{}^+ &
\funabsF{A{}^-}{P}\,(\semF{t}\,\semF{s})\,(\semF{s}\,\semF{t})
\\&& \hspace{.5cm}\text{by definition of $\funabsF{A{}^\ominus}{P}$}
\\
& = &
\semF{\strongabs{P}{(\clasapn{t}{s})}{(\clasapp{s}{t})}}
\\
& = &
\semF{\abs{P}{t}{s}}
\end{array}
\]
\item
Let $\Gamma \vdash t : A{}^\oplus$ and $\Gamma \vdash s : A{}^\ominus$.
Then, symmetrically as for
the previous case:
$
\semF{\strongabs{P}{(\negin{t})}{(\negip{s})}}
\toa{}^+
\semF{\abs{P}{t}{s}}
$.
\end{enumerate} \end{proof}
\subsection{Proof of Characterization of Normal Forms~(\rprop{characterization_of_normal_terms})} \lsec{appendix:characterization_of_normal_forms}
\begin{proposition} \lprop{appendix:characterization_of_normal_terms} A term is normal if and only if it does not reduce in $\lambda^{\PRK}$. \end{proposition} \begin{proof} $(\Rightarrow)$ Let $t$ be a normal term, and let us check that it is a $\toa{}$-normal form. We proceed by induction on the derivation that $t$ is a normal term.
The cases corresponding to introduction rules are straightforward by IH\xspace. For example, if $t = \pairpn{N_1}{N_2}$, then by IH\xspace $N_1$ and $N_2$ have no $\toa{}$-redexes. Moreover, there are no rules involving a pair $\pairpn{-}{-}$ at the root, so $\pairpn{N_1}{N_2}$ is in $\toa{}$-normal form.
The cases corresponding to elimination rules and the absurdity rule are also straightforward by IH\xspace, observing that there cannot be a redex at the root. For example, if $t = \projipn{S}$, then by IH\xspace $S$ has no $\toa{}$-redexes. Moreover, the only rule involving a projection $\projipn{-}$ at the root is $\rewritingRuleName{proj}$, which would require that $S = \pairpn{t_1}{t_2}$. But this is impossible ---as can be checked by exhaustive case analysis on $S$---, so $t$ is in $\toa{}$-normal form.
\noindent $(\Leftarrow)$
Let $t$ be a $\toa{}$-normal form,
let us check that it is a normal term.
We proceed by induction on the structure of the term $t$:
\begin{enumerate}
\item {\em Variable, $x$:} it is a neutral term.
\item {\em Absurdity, $\strongabs{P}{t}{s}$:}
by IH\xspace, $t$ and $s$ are normal terms.
If either $t$ or $s$ is a neutral term, we are done.
We are left to analyze the case in which they are not neutral terms,
{\em i.e.}\xspace both $t$ and $s$ are built using introduction rules.
Note that the types of $t$ and $s$ are $Q$ and $Q{}^{\sim}$
respectively, for some strong type $Q$.
We proceed by case analysis on the form of the proposition $Q$.
There are four cases:
\begin{enumerate}
\item {\em Proof/refutation of a propositional variable, $Q = \alpha^\pm$.}
This case is impossible, since $t$ only may be of one of the following forms:
$\pairpn{N}{N}$, $\inipn{N}$, $\claslampn{x : P}{N}$, or
$\negipn{N}$, none of which are of type $\alpha^\pm$.
\item {\em Proof of a conjunction, $Q = (A\landB){}^+$
or refutation of a disjunction $Q = (A\lorB){}^-$.}
Then $t$ is of the form $\pairpn{t_1}{t_2}$
and $s$ is of the form $\ininp{s'}$ for some $i \in \set{1,2}$,
so the rule $\rewritingRuleName{absPairInj}$ may be applied at the root, contradicting
the hypothesis that the term is $\to{}$-normal.
\item {\em Disjunction, $Q = (A\landB)^\pm$.}
Then $t$ is of the form $\inipn{s'}$ for some $i \in \set{1,2}$
and $s$ is of the form $\pairnp{t_1}{t_2}$,
so the rule $\rewritingRuleName{absInjPair}$ may be applied at the root, contradicting
the hypothesis that the term is $\to{}$-normal.
\item {\em Negation, $Q = (\negA)^\pm$.}
Then $t$ is of the form $\negipn{t'}$
and $s$ is of the form $\neginp{s'}$,
so the rule $\rewritingRuleName{absNeg}$ may be applied at the root, contradicting
the hypothesis that the term is $\to{}$-normal.
\end{enumerate}
\item {\em Pair, $\pairpn{t}{s}$:}
by IH\xspace, $t$ and $s$ are normal terms, so $\pairpn{t}{s}$
is also a normal term.
\item {\em Projection, $\projipn{t}$:}
by IH\xspace, $t$ is a normal term.
It suffices to show that $t$ is neutral.
Indeed, if $t$ is a normal but not neutral term,
then since the type of $t$ may be
either of the form $(A\landB){}^+$ or of the form $(A\lorB){}^-$,
we have that $t$ is of the form $\pairpn{s}{u}$.
Then the rule $\rewritingRuleName{proj}$ may be applied at the root, contradicting
the hypothesis that the term is $\to{}$-normal.
\item {\em Injection, $\inipn{t}$:}
by IH\xspace, $t$ is a normal term, so $\inipn{t}$ is also normal.
\item {\em Case, $\casepn{t}{x}{s}{x}{u}$:}
by IH\xspace $t$, $s$ and $u$ are normal terms.
It suffices to show that $t$ is neutral.
Indeed, if $t$ is a normal but not neutral term,
then since the type of $t$ may be
either of the form $(A\lorB){}^+$ or of the form $(A\landB){}^-$,
we have that $t$ is of the form $\inipn{t'}$ for some $i \in \set{1,2}$.
Then the rule $\rewritingRuleName{case}$ may be applied at the root, contradicting
the hypothesis that the term is $\to{}$-normal.
\item {\em Negation introduction, $\negipn{t}$:}
by IH\xspace, $t$ is a normal term. Then $\negipn{t}$ is also normal.
\item {\em Negation elimination, $\negepn{t}$:}
by IH\xspace, $t$ is a normal term.
It suffices to show that $t$ is neutral.
Indeed, if $t$ is a normal but not neutral term,
then since the type of $t$ is of the form $(\negA)^\pm$,
then $t$ is of the form $\negipn{t'}$.
Then the rule $\rewritingRuleName{neg}$ may be applied at the root, contradicting
the hypothesis that the term is $\to{}$-normal.
\item {\em Classical introduction, $\claslampn{x : P}{t}$:}
by IH\xspace, $t$ is a normal term, so $\claslampn{x : P}{t}$
is also normal.
\item {\em Classical elimination, $\clasappn{t}{s}$:}
by IH\xspace, $t$ and $s$ are normal terms.
It suffices to show that $t$ is neutral.
Indeed, if $t$ is a normal but not neutral term,
then since the type of $t$ may be
either of the form $A{}^\oplus$ or of the form $A{}^\ominus$,
we have that $t$ is of the form $\claslampn{x}{t'}$.
Then the rule $\rewritingRuleName{beta}$ may be applied at the root, contradicting
the hypothesis that the term is $\to{}$-normal.
\end{enumerate} \end{proof}
\subsection{Proof of Canonicity~(\rthm{canonicity})} \lsec{appendix:canonicity}
We give a slightly different statement of Canonicity, adding the additional hypothesis that $t$ is already a normal form. This addition comes at no loss of generality, given that $\lambda^{\PRK}$ enjoys subject reduction~(\rprop{subject_reduction}) and strong normalization~(\rthm{lambdaC_canonical}).
\begin{theorem}[Canonicity] \lthm{appendix:canonicity} \quad \begin{enumerate} \item
Let $\vdash t : P$ where $t$ is a normal form.
Then $t$ is canonical. \item
Let $\Gamma \vdash t : A^\pm$
where $\Gamma$ is classical and $t$ is a normal form.
Then either $t$ is canonical
or $t$ is of the form $\casectxof{t'}$
where $\mathtt{K}$ is a case-context
and $t'$ is an open explosion. \item
Let $\Gamma \vdash t : A{}^\oplus$
or $\Gamma \vdash t : A{}^\ominus$,
where $\Gamma$ is classical and $t$ is a normal form.
Then either $t = \claslampn{x}{t'}$
or $t = \elctxof{t'}$,
where $\mathtt{E}$ is an eliminative context
and $t'$ is a variable or an open explosion. \end{enumerate} \end{theorem} \begin{proof} \quad \begin{enumerate} \item
Let $\vdash t : P$ where $t$ is a normal form.
Note,
by induction on the formation rules for neutral terms~(\rdef{normal_terms})
that a neutral term must have at least one free variable.
But $t$ is typed in the empty typing context, so it must be closed.
Hence $t$ is not a neutral term, so by
\rprop{characterization_of_normal_terms}, it must be canonical. \item
Let $\Gamma \vdash t : P$ where $\Gamma$ is classical
and $t$ is a normal form.
By \rprop{characterization_of_normal_terms} either $t$ is canonical
or it is a neutral term. If $t$ is canonical we are done.
If $t$ is a neutral term it suffices to show the following claim,
namely that if $\Gamma \vdash t : B^\pm$ is a derivable
judgment such that $\Gamma$ is classical
and $t$ is a neutral term,
then $t$ is of the form $t = \casectxof{t'}$,
where $\mathtt{K}$ is a case-context
and $t'$ is an open explosion.
We proceed by induction on the formation rules
for neutral terms~(\rdef{normal_terms}):
\begin{enumerate}
\item {\em Variable, $t = x$.}
this case is impossible, given that $\Gamma$ is assumed to be classical,
so $\Gamma \vdash x : P$
where $P$ must be of the form $C{}^\oplus$ or $C{}^\ominus$,
hence $P$ cannot be of the form $B^\pm$.
\item {\em Projection, $\projipn{S}$:}
this case is impossible, as $\Gamma \vdash \projipn{S} : P$
where $P$ must be of the form $C{}^\oplus$ or $C{}^\ominus$,
hence $P$ cannot be of the form $B^\pm$.
\item {\em Case, $\casepn{S}{x}{N_1}{x}{N_2}$:}
by inversion of the typing rules we have that
either $\Gamma \vdash S : (A \lor B){}^+$
or $\Gamma \vdash S : (A \land B){}^-$.
In both cases we may apply the IH\xspace to conclude that
$S$ is of the form
$S = \casectxof{t'}$ where $\mathtt{K}$ is a case-context
and $t'$ is an open explosion.
Therefore $t = \casepn{(\casectxof{t'})}{x}{N_1}{x}{N_2}$
where now
$\casepn{(\mathtt{K})}{x}{N_1}{x}{N_2}$
is a case-context.
\item {\em Classical elimination, $\clasappn{S}{N}$:}
then $t$ is an explosion
under the empty case-context.
Moreover, $S$ must have at least one free variable
so $t$ is indeed an open explosion.
\item {\em Negation elimination, $\negepn{S}$:}
this case is impossible, as $\Gamma \vdash \negepn{S} : P$
where $P$ must be of the form $C{}^\oplus$ or $C{}^\ominus$,
hence $P$ cannot be of the form $B^\pm$.
\item {\em Absurdity, $\strongabs{}{S}{N}$ or $\strongabs{}{N}{S}$:}
then $t$ is an explosion
under the empty case-context.
Moreover, $S$ must have at least one free variable
so $t$ is indeed an open explosion.
\end{enumerate} \item
Let $\Gamma \vdash t : A{}^\oplus$
or $\Gamma \vdash t : A{}^\ominus$,
where $\Gamma$ is classical and $t$ is a normal form.
By \rprop{characterization_of_normal_terms} either $t$ is canonical
or it is a neutral term.
If $t$ is canonical, then by the constraints on its type it
must be of the form $t = \claslampn{x}{t'}$, so we are done.
If $t$ is neutral, it suffices to show the following claim
namely that if $\Gamma \vdash t : P$
is a derivable judgment, with $P \in \set{B{}^\oplus,B{}^\ominus}$,
such that $\Gamma$ is classical
and $t$ is a neutral term,
then $t$ is of the form $t = \elctxof{t'}$,
where $\mathtt{E}$ is an eliminative context
and $t'$ is a variable or an open explosion.
We proceed by induction on the formation rules
for neutral terms~(\rdef{normal_terms}):
\begin{enumerate}
\item {\em Variable, $t = x$.}
immediate, as $t$ is a variable under the empty eliminative context.
\item {\em Projection, $\projipn{S}$:}
by inversion of the typing rules, we have that
either $\Gamma \vdash S : (A \land B){}^+$
or $\Gamma \vdash S : (A \lor B){}^-$.
In both cases we may apply the second item of this lemma
to conclude that $S$ is of the form $S = \casectxof{t'}$
where $\mathtt{K}$ is a case-context and $t'$ is an open explosion.
Therefore $t = \projipn{\casectxof{t'}}$,
where now $\projipn{\mathtt{K}}$ is an eliminative context.
\item {\em Case, $\casepn{S}{x}{N_1}{x}{N_2}$:}
by inversion of the typing rules, we have that
either $\Gamma \vdash S : (A \lor B){}^+$
or $\Gamma \vdash S : (A \land B){}^-$.
In both cases we may apply the second item of
this lemma to conclude that
$S$ is of the form $S = \casectxof{t'}$
where $\mathtt{K}$ is an eliminative context
and $t'$ is an open explosion.
Therefore
$t = \casepn{(\casectxof{t'})}{x}{N_1}{x}{N_2}$,
where now $\casepn{(\mathtt{K})}{x}{N_1}{x}{N_2}$
is an eliminative context.
\item {\em Classical elimination, $\clasappn{S}{N}$:}
then $t$ is an explosion
under the empty eliminative context.
Moreover, $S$ must have at least one free variable
so $t$ is indeed an open explosion.
\item {\em Negation elimination, $\negepn{S}$:}
by inversion of the typing rules, we have that
$\Gamma \vdash S : (\negA)^\pm$.
By the second item of this lemma,
$S$ is of the form $S = \casectxof{t'}$
where $\mathtt{K}$ is a case-context and $t'$ is an open explosion.
Therefore $t = \negepn{\casectxof{t'}}$,
where now $\negepn{\mathtt{K}}$ is an eliminative context.
\item {\em Absurdity, $\strongabs{}{S}{N}$ or $\strongabs{}{N}{S}$:}
then $t$ is an explosion
under the empty eliminative context.
Moreover, $S$ must have at least one free variable
so $t$ is indeed an open explosion.
\end{enumerate} \end{enumerate} \end{proof}
\subsection{Proof that $\lambda^{\PRK}_\eta$ is Strongly Normalizing and Confluent~(\rthm{lambdaCeta_canonical})} \lsec{appendix:extensionality}
\begin{lemma}[Local confluence] \llem{lambdaCeta_local_confluence} The $\lambda^{\PRK}_\eta$-calculus has the weak Church--Rosser property. \end{lemma} \begin{proof} Let $t_0 \toa{} t_1$ and $t_0 \toa{} t_2$, and let us show that the diagram can be closed, {\em i.e.}\xspace that there is a term $t_3$ such that $t_1 \mathrel{\rightarrow^*} t_3$ and $t_2 \mathrel{\rightarrow^*} t_3$. The proof is by induction on $t_0$ and by case analysis on the relative positions of the steps $t_0 \toa{} t_1$ and $t_0 \toa{} t_2$. Most cases are straightforward by resorting to the IH\xspace. We study only the interesting cases, when the patterns of the redexes overlap. There are two such cases: \begin{enumerate} \item $\rewritingRuleName{beta}$/$\rewritingRuleName{eta}$:
Let $x \notin \fv{t}$.
The overlap involves a step
$\clasappn{(\claslampn{x}{\clasappn{t}{x}})}{s}
\toa{\rewritingRuleName{beta}}
\clasappn{t}{s}$
and a step
$\clasappn{(\claslampn{x}{\clasappn{t}{x}})}{s}
\toa{\rewritingRuleName{eta}}
\clasappn{t}{s}$,
so the diagram is trivially closed in zero rewriting steps. \item $\rewritingRuleName{eta}$/$\rewritingRuleName{beta}$:
Let $x \notin \fv{t}$.
The overlap involves a step
$\claslampn{x}{\clasappn{(\claslampn{y}{t})}{x}}
\toa{\rewritingRuleName{eta}} \claslampn{y}{t}$
and a step
$\claslampn{x}{\clasappn{(\claslampn{y}{t})}{x}}
\toa{\rewritingRuleName{beta}}
\claslampn{x}{t\sub{y}{x}}$.
Note that the targets of the steps are $\alpha$-equivalent,
so the diagram is trivially closed in zero rewriting steps. \end{enumerate} \end{proof}
\begin{lemma}[Properties of reduction in $\lambda^{\PRK}_\eta$] \llem{lambdaCeta_properties} \quad \begin{enumerate} \item
{\em Reduction does not create free variables.}
If $t \to t'$ then $\fv{t} \supseteq \fv{t'}$. \item
{\em Substitution (I).}
Let $\Gamma,x:A \vdash t : B$
and $\Gamma \vdash s : A$.
If $t \to t'$
then $t\sub{x}{s} \to t'\sub{x}{s}$. \item
{\em Substitution (II).}
Let $\Gamma,x:A \vdash t : B$
and $\Gamma \vdash s : A$.
If $s \to s'$
then $t\sub{x}{s} \mathrel{\rightarrow^*} t\sub{x}{s'}$. \item
{\em Substitution (III).}
Let $\Gamma,x:A \vdash t : B$
and $\Gamma \vdash s : A$.
If $t \mathrel{\rightarrow^*} t'$ and $s \mathrel{\rightarrow^*} s'$
then $t\sub{x}{s} \mathrel{\rightarrow^*} t'\sub{x}{s'}$. \end{enumerate} \end{lemma} \begin{proof} Items 1., 2., and 3. are by induction on $t$. Item 4. is by induction on the sum of the lengths of the sequences $t \mathrel{\rightarrow^*} t'$ and $s \mathrel{\rightarrow^*} s'$, resorting to the two previous items. \end{proof}
\begin{lemma}[Postponement of $\rewritingRuleName{eta}$ steps] \llem{lambdaCeta_postponement} Let $t \toa{\rewritingRuleName{eta}} s \toa{\rewritingRuleName{r}} u$ where $\rewritingRuleName{r}$ is a rewriting rule other than $\rewritingRuleName{eta}$. Then there exists a term $s'$ such that $t \ptoa{\rewritingRuleName{r}} s' \rtoa{\rewritingRuleName{eta}} u$. \end{lemma} \begin{proof} By induction on $t$. If the $\rewritingRuleName{eta}$ step and the $\rewritingRuleName{r}$ step are not reduction steps at the root, it is immediate to conclude, resorting to the IH\xspace when appropriate.
If the $\rewritingRuleName{eta}$ step is at the root, then the first step is of the form $t = \claslampn{x}{(\clasappn{s}{x})}
\toa{\rewritingRuleName{eta}} s$, where $x \notin \fv{s}$. Taking $s' := \claslampn{x}{(\clasappn{u}{x})}$ we have that $t = \claslampn{x}{(\clasappn{s}{x})}
\toa{\rewritingRuleName{r}} \claslampn{x}{(\clasappn{u}{x})}
\toa{\rewritingRuleName{eta}} u$, so we are done. For the last reduction step, we use the fact that reduction does not create free variables~(\rlem{lambdaCeta_properties}).
Otherwise, we have that the $\rewritingRuleName{eta}$ step is {\em not} at the root and the $\rewritingRuleName{r}$ step is at the root. Then we proceed by case analysis, depending on the kind of rule applied. We only study the positive cases (the negative cases are symmetric): \begin{enumerate} \item $\rewritingRuleName{proj}$:
then we have that
$t \toa{\rewritingRuleName{eta}} s = \projip{\pairp{s_1}{s_2}}
\toa{\rewritingRuleName{proj}} s_i$.
Recall that the $\rewritingRuleName{eta}$ step is not at the root of $t$.
Moreover, it cannot be the case that $t = \projip{t'}$
and the $\rewritingRuleName{eta}$ step is at the root of $t'$,
because the type of $t'$ must be of the form $(A \land B){}^+$
but the $\rewritingRuleName{eta}$ rule can only be applied on a term constructed with a
$\claslampn{-}{-}$, whose type is classical.
This means that $t$ must be of the form $\projip{\pairp{t_1}{t_2}}$
and that the $\rewritingRuleName{eta}$ step is
either internal to $t_1$
or internal to $t_2$,
which
implies that $t_1 \rtoa{\rewritingRuleName{eta}} s_1$
and $t_2 \rtoa{\rewritingRuleName{eta}} s_2$.
Taking $s' := t_i$
we have that
$t = \projip{\pairp{t_1}{t_2}}
\toa{\rewritingRuleName{proj}} t_i
\rtoa{\rewritingRuleName{eta}} s_i$,
as required. \item $\rewritingRuleName{case}$:
then we have that
$t
\toa{\rewritingRuleName{eta}} s
= \casep{\inip{s_0}}{y}{s_1}{y}{s_2}
\toa{\rewritingRuleName{case}} s_i\sub{y}{s_0}$.
Recall that the $\rewritingRuleName{eta}$ step is not at the root of $t$.
Moreover, it cannot be the case that
$t = \casep{t'}{y}{s_1}{y}{s_2}$
and the $\rewritingRuleName{eta}$ step is at the root of $t'$,
because the type of $t'$ must be of the form $(A \lor B){}^+$,
but the $\rewritingRuleName{eta}$ rule can only be applied on a term constructed with a
$\claslampn{-}{-}$, whose type is classical.
This means that $t$ must be of the form
$\casep{\inip{t_0}}{y}{t_1}{y}{t_2}$
and that the $\rewritingRuleName{eta}$-step is
either internal to $t_0$, or internal to $t_1$, or internal to $t_2$,
which implies that
$t_0 \rtoa{\rewritingRuleName{eta}} s_0$
and $t_1 \rtoa{\rewritingRuleName{eta}} s_1$
and $t_2 \rtoa{\rewritingRuleName{eta}} s_2$.
Taking $s' := t_i\sub{y}{t_0}$
we have that
$t = \casep{\inip{t_0}}{y}{t_1}{y}{t_2}
\toa{\rewritingRuleName{case}} t_i\sub{y}{t_0}
\rtoa{\rewritingRuleName{eta}} s_i\sub{y}{s_0}$
resorting to~\rlem{lambdaCeta_properties} for the last step. \item $\rewritingRuleName{neg}$:
then we have that
$t \toa{\rewritingRuleName{eta}} \negep{(\negip{s_1})} \toa{\rewritingRuleName{neg}} s_1$.
Recall that the $\rewritingRuleName{eta}$-reduction step is not at the root of $t$.
Moreover, it cannot be the case that $t = \negep{t'}$
and the $\rewritingRuleName{eta}$-reduction step is at the root of $t'$,
because the type of $t'$ must be of the form $(\negA){}^+$
but the $\rewritingRuleName{eta}$ rule can only be applied on a term constructed with a
$\claslampn{-}{-}$, whose type is classical.
This means that $t$ must be of the form
$\negep{(\negip{t_1})}$
and that the $\rewritingRuleName{eta}$ step is internal to $t_1$,
{\em i.e.}\xspace $t_1 \toa{\rewritingRuleName{eta}} s_1$.
Then taking $s' := t_1$
we have that
$t = \negep{(\negip{t_1})} \toa{\rewritingRuleName{neg}} t_1 \toa{\rewritingRuleName{eta}} s_1$
as required. \item $\rewritingRuleName{beta}$:
then we have that
$t
\toa{\rewritingRuleName{eta}} \clasapp{(\claslamp{y}{s_1})}{s_2}
\toa{\rewritingRuleName{beta}} s_1\sub{y}{s_2}$.
Recall that the $\rewritingRuleName{eta}$ step is not at the root of $t$.
There are three cases, depending on the position of the $\rewritingRuleName{eta}$-step:
\begin{enumerate}
\item
{\em Immediately to the left of the application.}
That is, $t = \clasapp{t'}{s_2}$
and the $\rewritingRuleName{eta}$ step is at the root of $t'$,
{\em i.e.}\xspace $t' \toa{\rewritingRuleName{eta}} \claslamp{y}{s_1}$
is a reduction step at the root.
Then
$t' = \claslamp{x}{(\clasapp{(\claslamp{y}{s_1})}{x})}$.
Hence taking $s' := s_1\sub{y}{s_2}$
we have that
\[
\begin{array}{cl}
&
t = \clasapp{(\claslamp{x}{(\clasapp{(\claslamp{y}{s_1})}{x})})}{s_2}
\\
\toa{\rewritingRuleName{beta}} &
\clasapp{(\claslamp{y}{s_1})}{s_2}
\\
\toa{\rewritingRuleName{beta}} &
s_1\sub{y}{s_2}
\end{array}
\]
using two $\rewritingRuleName{beta}$ steps and no $\rewritingRuleName{eta}$ steps.
\item
{\em Inside the abstraction.}
That is, $t = \clasapp{(\claslamp{y}{t_1})}{s_2}$
with $t_1 \toa{\rewritingRuleName{eta}} s_1$.
Then taking $s' := t_1\sub{y}{s_2}$
we have that
$t = \clasapp{(\claslamp{y}{t_1})}{s_2}
\toa{\rewritingRuleName{beta}} t_1\sub{y}{s_2}
\toa{\rewritingRuleName{eta}} s_1\sub{y}{s_2}$
resorting to~\rlem{lambdaCeta_properties} for the last step.
\item
{\em To the right of the application.}
That is, $t = \clasapp{(\claslamp{y}{s_1})}{t_2}$
with $t_2 \toa{\rewritingRuleName{eta}} s_2$.
Then taking $s' := s_1\sub{y}{t_2}$
we have that
$t = \clasapp{(\claslamp{y}{s_1})}{t_2}
\toa{\rewritingRuleName{beta}} s_1\sub{y}{t_2}
\rtoa{\rewritingRuleName{eta}} s_1\sub{y}{s_2}$
resorting to~\rlem{lambdaCeta_properties} for the last step.
\end{enumerate} \item $\rewritingRuleName{absPairInj}$:
then we have that
$t \toa{\rewritingRuleName{eta}} \strongabs{}{\pairp{s_1}{s_2}}{\inin{s_3}}
\toa{\rewritingRuleName{absPairInj}} \abs{}{s_i}{s_3}$.
Recall that the $\rewritingRuleName{eta}$ step is not at the root of $t$.
Moreover, it cannot be the case that
$t = \strongabs{}{t'}{\inin{s_3}}$
and the $\rewritingRuleName{eta}$ step is at the root of $t'$,
because the type of $t'$ must be of the form $(A \land B){}^+$,
but the $\rewritingRuleName{eta}$ rule can only be applied on a term constructed with a
$\claslampn{-}{-}$, whose type is classical.
For similar reasons,
it cannot be the case that $t = \strongabs{}{\pairp{s_1}{s_2}}{t'}$
with the $\rewritingRuleName{eta}$ step is at the root of $t'$,
because then the type of $t'$ must be of the form $(A \land B){}^-$.
This means that $t$ must be of the form
$\strongabs{}{\pairp{t_1}{t_2}}{\inin{t_3}}$
and that the $\rewritingRuleName{eta}$ step is
either internal to $t_1$, or internal to $t_2$, or internal to $t_3$.
This implies that
$t_1 \rtoa{\rewritingRuleName{eta}} s_1$
and $t_2 \rtoa{\rewritingRuleName{eta}} s_2$
and $t_3 \rtoa{\rewritingRuleName{eta}} s_3$.
Taking $s' := \abs{}{t_i}{t_3}$
we have that
$t = \strongabs{}{\pairp{t_1}{t_2}}{\inin{t_3}}
\toa{\rewritingRuleName{absPairInj}} \abs{}{t_i}{t_3}
= \strongabs{}{(\clasapp{t_i}{t_3})}{(\clasapn{t_3}{t_i})}
\rtoa{\rewritingRuleName{eta}}
\strongabs{}{(\clasapp{s_i}{s_3})}{(\clasapn{s_3}{s_i})}
= \abs{}{s_i}{s_3}$. \item $\rewritingRuleName{absInjPair}$:
Symmetric to the previous case. \item $\rewritingRuleName{absNeg}$:
then we have that
$t
\toa{\rewritingRuleName{eta}} \strongabs{}{(\negip{s_1})}{(\negin{s_2})}
\toa{\rewritingRuleName{absNeg}} \abs{}{s_1}{s_2}$.
Recall that the $\rewritingRuleName{eta}$ step is not at the root of $t$.
Moreover, it cannot be the case that
$t = \strongabs{}{t'}{(\negin{s_2})}$
and the $\rewritingRuleName{eta}$ step is at the root of $t'$,
because the type of $t'$ must be of the form $(\negA){}^+$,
but the $\rewritingRuleName{eta}$ rule can only be applied on a term constructed with a
$\claslampn{-}{-}$, whose type is classical.
For similar reasons,
it cannot be the case that
$t = \strongabs{}{\negip{s_1}}{t'}$
with the $\rewritingRuleName{eta}$ step is at the root of $t'$,
because then the type of $t'$ must be of the form $(\negA){}^-$.
This means that $t$ must be of the form
$\strongabs{}{(\negip{t_1})}{(\negin{t_2})}$
and that the $\rewritingRuleName{eta}$ step is either internal to $t_1$
or internal to $t_2$.
This implies that $t_1 \rtoa{\rewritingRuleName{eta}} s_1$
and $t_2 \rtoa{\rewritingRuleName{eta}} s_2$.
Taking $s' := \abs{}{t_1}{t_2}$
we have that
$t = \strongabs{}{(\negip{t_1})}{(\negin{t_2})}
\toa{\rewritingRuleName{absNeg}} \abs{}{t_1}{t_2}
= \strongabs{}{(\clasapp{t_1}{t_2})}{(\clasapp{t_2}{t_1})}
\rtoa{\rewritingRuleName{eta}}
\strongabs{}{(\clasapp{s_1}{s_2})}{(\clasapp{s_2}{s_1})}
= \abs{}{s_1}{s_2}$. \end{enumerate} \end{proof}
\begin{theorem} \lthm{appendix:lambdaCeta_canonical} The $\lambda^{\PRK}_\eta$-calculus is strongly normalizing and confluent. \end{theorem} \begin{proof} Strong normalization follows from postponement of the $\rewritingRuleName{eta}$ rule~(\rlem{lambdaCeta_postponement}) and strong normalization of the calculus without $\rewritingRuleName{eta}$~(\rthm{lambdaC_canonical}) by the usual rewriting techniques.
More precisely, let us write $\toa{\neg\rewritingRuleName{eta}}$ for reduction not using $\rewritingRuleName{eta}$, that is, $\toa{\neg\rewritingRuleName{eta}} \eqdef (\toa{\,} \setminus \toa{\rewritingRuleName{eta}})$. Suppose there is an infinite reduction sequence $t_1 \to t_2 \to t_3 \hdots$ in $\lambda^{\PRK}_\eta$. Let $t_1 \rtoa{\neg\rewritingRuleName{eta}} t_i$ be the longest prefix of the sequence whose steps are not $\rewritingRuleName{eta}$ steps. This prefix cannot be infinite given that $\lambda^{\PRK}$ is strongly normalizing. Let $t_i \rtoa{\rewritingRuleName{eta}} t_{i+n}$ be the longest sequence of $\rewritingRuleName{eta}$ steps starting on $t_i$. This sequence cannot be infinite given that an $\rewritingRuleName{eta}$ step decreases the size of the term. Now there must be a step $t_{i+n} \toa{\neg\rewritingRuleName{eta}} t_{i+n+1}$. Applying the postponement lemma~(\rlem{lambdaCeta_postponement}) $n$ times, we obtain an infinite sequence of the form $t_1 \rtoa{\neg\rewritingRuleName{eta}} t_i \toa{\neg\rewritingRuleName{eta}} t'_{i+1} \hdots$. By repeatedly applying this argument, we may build an infinite sequence of $\toa{\neg\rewritingRuleName{eta}}$ steps, contradicting the fact that $\lambda^{\PRK}$ is strongly normalizing.
Confluence of $\lambda^{\PRK}_\eta$ follows from the fact that it is strongly normalizing and locally confluent~(\rlem{lambdaCeta_local_confluence}), resorting to Newman's Lemma~\cite[Theorem~1.2.1]{terese}. \end{proof}
\subsection{Computation Rules for the Embedding of Classical Logic into $\textsc{prk}$} \lsec{classical_simulation}
The statements of all of the following lemmas are in $\lambda^{\PRK}_\eta$ (with $\rewritingRuleName{eta}$ reduction).
\subsubsection{Simulation of conjunction}
\begin{definition}[Conjunction introduction] Let $\Gamma \vdash t : A{}^\oplus$ and $\Gamma \vdash s : B{}^\oplus$. Then $\Gamma \vdash \pairc{t}{s} : (A \land B){}^\oplus$ where: \[
\pairc{t}{s} \eqdef
\claslamp{(\underline{\,\,\,}:(A\landB){}^\ominus)}{
\pairp{t}{s}
} \] \end{definition}
\begin{definition}[Conjunction elimination] Let $\Gamma \vdash t : (A_1 \land A_2){}^\oplus$. Then $\Gamma \vdash \projic{t} : A_i{}^\oplus$ where: \[
\projic{t} \eqdef
\claslamp{(x:A_i{}^\ominus)}{
\clasapp{
\projip{
\clasapp{
t
}{
\claslamn{(\underline{\,\,\,}:(A_1 \land A_2){}^\oplus)}{\inin{x}}
}
}
}{
x
}
} \] \end{definition}
\begin{lemma}
$\projic{\pairc{t_1}{t_2}} \mathrel{\rightarrow^*} t_i$ \end{lemma} \begin{proof}
\[
{\small
\begin{array}{rcll}
&&
\projic{\pairc{t_1}{t_2}}
\\
& = &
\claslamp{x:A_i{}^\ominus}{
\clasapp{
\projip{
\clasapp{
(\claslamp{\underline{\,\,\,}}{\pairp{t_1}{t_2}})
}{
\claslamn{\underline{\,\,\,}}{\inin{x}}
}
}
}{
x
}
}
\\
& \toa{\rewritingRuleName{beta}} &
\claslamp{x:A_i{}^\ominus}{
\clasapp{
\projip{\pairp{t_1}{t_2}}
}{
x
}
}
\\
& \toa{\rewritingRuleName{proj}} &
\claslamp{x:A_i{}^\ominus}{
\clasapp{
t_i
}{
x
}
}
\\
& \toa{\rewritingRuleName{eta}} &
t_i
\end{array}
}
\] \end{proof}
\subsubsection{Simulation of disjunction}
\begin{definition}[Disjunction introduction] Let $\Gamma \vdash t : A_i{}^\oplus$. Then $\Gamma \vdash \inic{t} : (A_1 \lor A_2){}^\oplus$ where: \[
\inic{t} \eqdef
\claslamp{(\underline{\,\,\,}:(A_1\lorA_2){}^\ominus)}{
\inip{t}
} \] \end{definition}
\begin{definition}[Disjunction elimination] Let $\Gamma \vdash t : (A \lor B){}^\oplus$ and $\Gamma, x : A{}^\oplus \vdash s : C{}^\oplus$ and $\Gamma, x : B{}^\oplus \vdash u : C{}^\oplus$. Then $\Gamma \vdash
\casec{t}{(x:A{}^\oplus)}{s}{(x:B{}^\oplus)}{u}
: C{}^\oplus$, where: \[
\claslamp{(y:C{}^\ominus)}{
\caseptablex{
(\clasapp{
t
}{
\claslamn{(\underline{\,\,\,}:(A\lorB){}^\oplus)}{
\pairn{
\contrapose{x}{y}{
s
}
}{
\contrapose{x}{y}{
u
}
}
}
})
}{
(x : A{}^\oplus)
}{
\clasapp{
s
}{
y
}
}{
(x : B{}^\oplus)
}{
\clasapp{
u
}{
y
}
}
} \] \end{definition}
\begin{lemma}
$\casec{\inic{t_i}}{x}{s_1}{x}{s_2}
\mathrel{\rightarrow^*} s_i\sub{x}{t}$ \end{lemma} \begin{proof}
\[
{\small
\begin{array}{rcll}
&&
\casec{\inic{t_i}}{x}{s_1}{x}{s_2}
\\
& = &
\claslamptable{(y:C{}^\ominus)}{
\caseptablex{
\clasapptable{
\claslamp{(\underline{\,\,\,}:(A_1\lorA_2){}^\ominus)}{
\inip{t}
}
}{
\claslamn{(\underline{\,\,\,}:(A\lorB){}^\oplus)}{
\pairn{
\contrapose{x}{y}{
s_1
}
}{
\contrapose{x}{y}{
s_2
}
}
}
}
}{
(x : A{}^\oplus)
}{
\clasapp{
s_1
}{
y
}
}{
(x : B{}^\oplus)
}{
\clasapp{
s_2
}{
y
}
}
}
\\
& \toa{\rewritingRuleName{beta}} &
\claslamp{(y:C{}^\ominus)}{
\casep{
\inip{t}
}{
(x : A{}^\oplus)
}{
\clasapp{
s_1
}{
y
}
}{
(x : B{}^\oplus)
}{
\clasapp{
s_2
}{
y
}
}
}
\\
& \toa{\rewritingRuleName{case}} &
\claslamp{(y:C{}^\ominus)}{
\clasapp{s_i\sub{x}{t}}{y}
}
\\
& \toa{\rewritingRuleName{eta}} &
s_i\sub{x}{t}
\end{array}
}
\] \end{proof}
\subsubsection{Simulation of negation} \lsec{appendix:simulation_of_negation}
\begin{definition}[Negation introduction] By \rlem{lem_and_noncontr} we have that $\Gamma \vdash \lemN{\alpha_0} : (\alpha_0 \land \neg\alpha_0){}^\ominus$, that is $\Gamma \vdash \lemN{\alpha_0} : \bot{}^\ominus$. Moreover, suppose that $\Gamma, x:A{}^\oplus \vdash t : \bot{}^\oplus$. Then $\Gamma \vdash \neglamc{(x:A{}^\oplus)}{t} : (\negA){}^\oplus$, where: \[
\neglamc{(x:A{}^\oplus)}{t} \eqdef
\claslamp{(\underline{\,\,\,}:(\negA){}^\ominus)}{
\negip{
\claslamn{(x:A{}^\oplus)}{
(\abs{
A{}^-
}{
t
}{
\lemN{\alpha_0}
})
}
}
} \] \end{definition}
\begin{definition}[Negation elimination] Let $\Gamma \vdash t : (\negA){}^\oplus$ and $\Gamma \vdash s : A{}^\oplus$. Then $\Gamma \vdash \negapc{t}{s} : \bot{}^\oplus$, where: \[
\negapc{t}{s} \eqdef
\abs{
\bot{}^\oplus
}{
t
}{
\claslamn{(\underline{\,\,\,}:(\negA){}^\oplus)}{
\negin{
s
}
}
} \] \end{definition}
\begin{lemma}
$\negapc{(\neglamc{x}{t})}{s} \mathrel{\rightarrow^*}
\strongabs{}{
(\abs{}{
t\sub{x}{s}
}{
\lemN{\alpha_0}
})
}{
(\clasapn{
s
}{
(\claslamn{x}{
(\abs{}{
t
}{
\lemN{\alpha_0}
})
})
})
}$ \end{lemma} \begin{proof} \[ {\small \begin{array}{rl}
& \negapc{(\neglamc{x}{t})}{s} \\ \\ = & \abs{
\bot{}^\oplus
}{
(\claslamp{\underline{\,\,\,}}{
\negip{
\claslamn{x}{
(\abs{}{
t
}{
\lemN{\alpha_0}
})
}
}
})
}{
\claslamn{\underline{\,\,\,}}{
\negin{
s
}
}
} \\ \\ = & \strongabstable{}{
(\clasapp{
(\claslamp{\underline{\,\,\,}}{
\negip{
\claslamn{x}{
(\abs{}{
t
}{
\lemN{\alpha_0}
})
}
}
})
}{
\claslamn{\underline{\,\,\,}}{
\negin{
s
}
}
})
}{
(\clasapn{
\claslamn{\underline{\,\,\,}}{
\negin{
s
}
}
}{
(\claslamp{\underline{\,\,\,}}{
\negip{
\claslamn{x}{
(\abs{}{
t
}{
\lemN{\alpha_0}
})
}
}
})
})
} \\ \\ \toa{\rewritingRuleName{beta}}(2) &
\strongabs{}{
(\negip{
\claslamn{x}{
(\abs{}{
t
}{
\lemN{\alpha_0}
})
}
})
}{
(\negin{s})
} \\ \\ \toa{\rewritingRuleName{absNeg}} &
\strongabs{}{
(\clasapp{
(\claslamn{x}{
(\abs{}{
t
}{
\lemN{\alpha_0}
})
})
}{
s
})
}{
(\clasapn{
s
}{
(\claslamn{x}{
(\abs{}{
t
}{
\lemN{\alpha_0}
})
})
})
} \\ \\ \toa{\rewritingRuleName{beta}} &
\strongabs{}{
(\abs{}{
t\sub{x}{s}
}{
\lemN{\alpha_0}
})
}{
(\clasapn{
s
}{
(\claslamn{x}{
(\abs{}{
t
}{
\lemN{\alpha_0}
})
})
})
} \end{array} } \] \end{proof}
\subsubsection{Simulation of implication}
Define implication $A \Rightarrow B$ as an abbreviation of $\negA \lor B$.
\begin{definition}[Implication introduction] If $\Gamma,x:A{}^\oplus \vdash t : B{}^\oplus$ then $\Gamma \vdash \lamc{(x:A)}{t} : (A \Rightarrow B){}^\oplus$ where: \[ {\small \begin{array}{rcl}
\lamc{x}{t} & \eqdef &
\claslamp{(y:(A\IMPB){}^\ominus)}{
\inip[2]{
t\sub{x}{\mathbf{X}_{y}}
}
} \\ \mathbf{X}_y & \eqdef &
\claslamp{(z:A{}^\ominus)}{
\clasapp{
(\negen{(
\clasapn{
\mathbf{X'}_{y,z}
}{
\claslamp{(\underline{\,\,\,}:(\negA){}^\ominus)}{
\negip{
z
}
}
}
)})
}{
z
}
} \\ \mathbf{X'}_{y,z} & \eqdef &
\projip[1]{
\clasapn{
y
}{
\claslamp{(\underline{\,\,\,}:(A\IMPB){}^\ominus)}{
\inip[1]{
\claslamp{(\underline{\,\,\,}:(\negA){}^\ominus)}{
\negip{
z
}
}
}
}
}
} \end{array} } \] \end{definition}
\begin{definition}[Implication elimination] If $\Gamma \vdash t : (A \Rightarrow B){}^\oplus$ and $\Gamma \vdash s : A{}^\oplus$, then $\Gamma \vdash \appc{t}{s} : B{}^\oplus$, where: \[ {\small \begin{array}{r@{}c@{}l}
\appc{t}{s}
& \eqdef &
\claslamptable{(x:B{}^\ominus)}{
\caseptablex{
(\clasapp{
t
}{
\claslamn{(\underline{\,\,\,}:(A\RightarrowB){}^\oplus)}{
\pairn{
(\claslamn{(\underline{\,\,\,}:(\negA){}^\oplus)}{
\negin{
s
}
})
}{
x
}
}
})
}{
(y:(\negA){}^\oplus)
}{
\abs{B{}^+}{s}{
\negen{
(\clasapp{
y
}{
\claslamn{(\underline{\,\,\,}:(\negA){}^\oplus)}{
\negin{
x
}
}
})
}
}
}{
(z:B{}^\oplus)
}{
\clasapp{z}{x}
}
} \end{array} } \] \end{definition}
The following lemma is the computational rule for implication:
\begin{lemma} $\appc{(\lamc{x}{t})}{s} \mathrel{\rightarrow^*} t\sub{x}{s}$ \end{lemma} \begin{proof}
First let $u =
\claslamn{\underline{\,\,\,}}{
\pairn{
(\claslamn{\underline{\,\,\,}}{
\negin{
s
}
})
}{
x'
}
}$ and note that: \[ {\small
\begin{array}{r@{\ }c@{\ }l}
&&
\mathbf{X'}_{u,z}
\\
& = &
\projip[1]{
\clasapn{
u
}{
\claslamp{(\underline{\,\,\,}:(A\IMPB){}^\ominus)}{
\inip[1]{
\claslamp{(\underline{\,\,\,}:(\negA){}^\ominus)}{
\negip{
z
}
}
}
}
}
}
\\
& \toa{\rewritingRuleName{beta}} &
\projip[1]{
\pairn{
(\claslamn{\underline{\,\,\,}}{
\negin{
s
}
})
}{
x'
}
}
\\
& \toa{\rewritingRuleName{proj}} &
\claslamn{\underline{\,\,\,}}{
\negin{
s
}
}
\end{array} } \] Hence: \[ {\small
\begin{array}{r@{\ }c@{\ }l}
&&
\mathbf{X}_{u}
\\
& = &
\claslamp{(z:A{}^\ominus)}{
\clasapp{
(\negen{(
\clasapn{
\mathbf{X'}_{u,z}
}{
\claslamp{(\underline{\,\,\,}:(\negA){}^\ominus)}{
\negip{
z
}
}
}
)})
}{
z
}
}
\\
& \mathrel{\rightarrow^*} &
\claslamp{(z:A{}^\ominus)}{
\clasapp{
(\negen{(
\clasapn{
(\claslamn{\underline{\,\,\,}}{
\negin{
s
}
})
}{
\claslamp{(\underline{\,\,\,}:(\negA){}^\ominus)}{
\negip{
z
}
}
}
)})
}{
z
}
}
\\
& \toa{\rewritingRuleName{beta}} &
\claslamp{(z:A{}^\ominus)}{
\clasapp{
(\negen{(
\negin{
s
}
)})
}{
z
}
}
\\
& \toa{\rewritingRuleName{neg}} &
\claslamp{(z:A{}^\ominus)}{
\clasapp{
s
}{
z
}
}
\\
& \toa{\rewritingRuleName{eta}} &
s
\end{array} } \] Hence: \[ {\small
\begin{array}{r@{\ }c@{\ }l}
& &
\appc{(\lamc{x}{t})}{s}
\\
& = &
\claslamptable{x'}{
\caseptablex{
\clasapptable{
(\claslamp{y}{
\inip[2]{
t\sub{x}{\mathbf{X}_{y}}
}
})
}{
\claslamn{\underline{\,\,\,}}{
\pairn{
(\claslamn{\underline{\,\,\,}}{
\negin{
s
}
})
}{
x'
}
}
}
}{
y'
}{
\abs{B{}^+}{s}{
\negen{
(\clasapp{
y'
}{
\claslamn{\underline{\,\,\,}}{
\negin{
x'
}
}
})
}
}
}{
z'
}{
\clasapp{z'}{x'}
}
}
\\
& \toa{\rewritingRuleName{beta}} &
\claslamptable{x'}{
\caseptablex{
\inip[2]{
t\sub{x}{\mathbf{X}_{u}}
}
}{
y'
}{
\abs{B{}^+}{s}{
\negen{
(\clasapp{
y'
}{
\claslamn{\underline{\,\,\,}}{
\negin{
x'
}
}
})
}
}
}{
z'
}{
\clasapp{z'}{x'}
}
}
\\
& \toa{\rewritingRuleName{case}} &
\claslamp{x'}{
\clasapp{
t\sub{x}{\mathbf{X}_{u}}
}{x'}
}
\\
& \mathrel{\rightarrow^*} &
\claslamp{x'}{
\clasapp{
t\sub{x}{s}
}{x'}
}
\\
& \toa{\rewritingRuleName{eta}} &
t\sub{x}{s}
\end{array} } \] \end{proof}
\subsubsection{Computational content of the law of excluded middle}
\begin{lemma} \quad \[ {\small
\begin{array}{ll}
&
\casec{\lemC{A}}{x}{s_1}{x}{s_2}
\\
\mathrel{\rightarrow^*}
&
\claslamp{y}{
\clasapp{
s_2\sub{x}{
\claslamp{\underline{\,\,\,}}{
\negip{
(\claslamn{x}{
\abs{}{
s_1
}{
y
}
})
}
}
}
}{
y
}
}
\end{array} } \] \end{lemma} \begin{proof} Recall that $\lemC{A} = \lemP{A}$, where: \[
{\footnotesize
\begin{array}{r@{\,}c@{\,}l}
\lemP{A} & \eqdef &
\claslamp{(x:(A\lor\negA){}^\ominus)}{
\inip[2]{
\claslamp{(y:\negA{}^\ominus)}{
\negip{
\projin[1]{
\clasapn{
x
}{
\lemPinner{y}{A}
}
}
}
}
}
}
\\
\lemPinner{y}{A} & \eqdef &
\claslamp{(\underline{\,\,\,}:(A\lor\negA){}^\ominus)}{
\inip[1]{
\claslamp{(z:A{}^\ominus)}{
(\abs{
A{}^+
}{
y
}{
\claslamp{(\underline{\,\,\,}:\negA{}^\ominus)}{
\negip{
z
}
}
})
}
}
}
\end{array}
} \] Let $u = \claslamn{\underline{\,\,\,}}{
\pairn{
\contrapose{x'}{y'}{
s_1
}
}{
\contrapose{x'}{y'}{
s_2
}
}
}$. Then:
\[
{\small
\begin{array}{r@{\ }c@{\ }l}
&&
\casec{\lemC{A}}{x'}{s_1}{x'}{s_2}
\\
& = &
\claslamptable{y'}{
\caseptablex{
\clasapptable{
\claslamp{x}{
\inip[2]{
\claslamp{y}{
\negip{
\projin[1]{
\clasapn{
x
}{
\lemPinner{y}{A}
}
}
}
}
}
}
}{
u
}
}{
x'
}{
\clasapp{
s_1
}{
y'
}
}{
x'
}{
\clasapp{
s_2
}{
y'
}
}
}
\\
& \toa{\rewritingRuleName{beta}} &
\claslamptable{y'}{
\caseptablex{
\inip[2]{
\claslamp{y}{
\negip{
\projin[1]{
\clasapn{
u
}{
\lemPinner{y}{A}
}
}
}
}
}
}{
x'
}{
\clasapp{
s_1
}{
y'
}
}{
x'
}{
\clasapp{
s_2
}{
y'
}
}
}
\\
& \toa{\rewritingRuleName{case}} &
\claslamp{y'}{
\clasapp{
s_2\sub{x'}{
\claslamp{y}{
\negip{
\projin[1]{
\clasapn{
u
}{
\lemPinner{y}{A}
}
}
}
}
}
}{
y'
}
}
\\
& \toa{\rewritingRuleName{beta}} &
\claslamp{y'}{
\clasapp{
s_2\sub{x'}{
\claslamp{\underline{\,\,\,}}{
\negip{
\projin[1]{
\pairn{
\contrapose{x'}{y'}{
s_1
}
}{
\contrapose{x'}{y'}{
s_2
}
}
}
}
}
}
}{
y'
}
}
\\
& \toa{\rewritingRuleName{proj}} &
\claslamp{y'}{
\clasapp{
s_2\sub{x'}{
\claslamp{\underline{\,\,\,}}{
\negip{
\contrapose{x'}{y'}{
s_1
}
}
}
}
}{
y'
}
}
\\
& = &
\claslamp{y'}{
\clasapp{
s_2\sub{x'}{
\claslamp{\underline{\,\,\,}}{
\negip{
(\claslamn{x'}{
\abs{}{
s_1
}{
y'
}
})
}
}
}
}{
y'
}
}
\\
\end{array}
}
\] \end{proof}
\section{Formal Systems}
\subsection{System~F Extended with Recursive Type Constraints} \lsec{appendix:system_f}
\begin{definition}[System~F$\extwith{\mathcal{C}}$] The set of {\em types} is given by: \[
A,B,\hdots ::= \alpha \mid A \to B \mid \forall\alpha.A \] The set of {\em terms} is given by: \[
t,s,\hdots
::= x
\mid \lam{x^A}{t} \mid t\,s
\mid \lam{\alpha}{t} \mid t\,A \] we omit type annotations over variables when clear from the context. A {\em type constraint} is an equation of the form $\alpha \equiv A$. Each set $\mathcal{C}$ of type constraints induces a notion of equivalence between types, written $A \equiv B$ and defined as the congruence generated by $\mathcal{C}$. More precisely: \[ {\small
\indrule{constr}{
(A \equiv B) \in \mathcal{C}
}{
A \equiv B
}
\indrule{refl}{}{
A \equiv A
}
\indrule{sym}{
A \equiv B
}{
B \equiv A
} } \] \[ {\small
\indrule{trans}{
A \equiv B
\hspace{.5cm}
B \equiv C
}{
A \equiv C
}
\indrule{cong}{
A \equiv B
}{
C\sub{\alpha}{A} \equiv C\sub{\alpha}{B}
} } \] We suppose that $\mathcal{C}$ is fixed. Typing judgments are of the form $\Gamma \vdash t : A$. \[ {\small
\indrule{\rulename{Ax}}{}{
\Gamma,x:A \vdash x : A
}
\indrule{Conv}{
\Gamma \vdash t : A
\hspace{.5cm}
A \equiv B
}{
\Gamma \vdash t : B
} } \] \[ {\small
\indrule{\rulename{I$\rightarrow$}}{
\Gamma,x:A \vdash t : B
}{
\Gamma \vdash \lam{x^A}{t} : A \to B
}
\indrule{E$\rightarrow$}{
\Gamma \vdash t : A \to B
\hspace{.5cm}
\Gamma \vdash s : A
}{
\Gamma \vdash t\,s : B
} } \] \[ {\small
\indrule{I$\forall$}{
\Gamma \vdash t : A
\hspace{.5cm}
\alpha \notin \fv{\Gamma}
}{
\Gamma \vdash \lam{\alpha}{t} : \forall\alpha.A
}
\indrule{E$\forall$}{
\Gamma \vdash t : \forall\alpha.A
}{
\Gamma \vdash t\,B : A\sub{\alpha}{B}
} } \] Reduction is defined as the closure by arbitrary contexts of the following rewriting rules: \[
\begin{array}{rcl}
(\lam{x}{t})\,s & \to & t\sub{x}{s} \\
(\lam{\alpha}{t})\,A & \to & t\sub{\alpha}{A} \\
\end{array} \] \end{definition}
\begin{definition}[Positive/negative occurrences] The set of type variables occurring positively (resp. negatively) in a type $A$ are written $\posvars{A}$ (resp. $\negvars{A}$) and defined by: \[ {\small
\begin{array}{r@{\ }c@{\ }l@{\hspace{.5cm}}r@{\ }c@{\ }l}
\posvars{\alpha} & \eqdef & \set{\alpha}
&
\negvars{\alpha} & \eqdef & \varnothing
\\
\posvars{A \to B} & \eqdef & \negvars{A} \cup \posvars{B}
&
\negvars{A \to B} & \eqdef & \posvars{A} \cup \negvars{B}
\\
\posvars{\forall\alpha.A} & \eqdef & \posvars{A} \setminus \set{\alpha}
&
\negvars{\forall\alpha.A} & \eqdef & \negvars{A} \setminus \set{\alpha}
\end{array} } \] \end{definition}
\begin{definition}[Positivity condition] \ldef{positivity} A set of type constraints $\mathcal{C}$ verifies the {\em positivity condition} if for every type constraint $(\alpha \equiv A) \in \mathcal{C}$ and every type $B$ such that $\alpha \equiv B$ one has that $\alpha \not\in \negvars{B}$. \end{definition}
\begin{theorem}[Mendler] \lthm{appendix:systemF_SN_Mendler} If $\mathcal{C}$ verifies the positivity condition, then System~F$\extwith{\mathcal{C}}$ is strongly normalizing. \end{theorem} \begin{proof} See~\cite[Theorem~13]{mendler1991inductive}. \end{proof}
{\bf Abbreviations.} We define the following standard abbreviations for types: \[
\begin{array}{rcl}
\mathbf{1} & \eqdef & \forall\alpha.(\alpha \to \alpha) \\
\mathbf{0} & \eqdef & \forall\alpha.\alpha \\
\negA & \eqdef & A \to \mathbf{0} \\
A \times B & \eqdef &
\forall\alpha.((A \to B \to \alpha) \to \alpha) \\
A + B & \eqdef &
\forall\alpha.((A \to \alpha) \to (B \to \alpha) \to \alpha) \\
\end{array} \] And the following terms. We omit the typing contexts for succintness: \[ {\small
\begin{array}{rcllll}
\trivsym
& \eqdef
& \lam{\alpha}{\lam{x^\alpha}{x}}
\\
& : & \mathbf{1}
\\
\\
\abortF{A}{t}
& \eqdef
& t\,A
\\
& :
& A
\\&&\text{if $t : \mathbf{0}$}
\\
\\
\pairF{t}{s}
& \eqdef
& \lam{\alpha}{\lam{f^{A \to B \to \alpha}}{f\,t\,s}}
\\
& :
& A \times B
\\&&\text{if $t : A$ and $s : B$}
\\
\\
\projiF{t}
& \eqdef
& t\,A_i\,(\lam{x_1^{A_1}}{\lam{x_2^{A_2}}{x_i}})
\\
& :
& A_i
\\&&\text{if $t : A_1 \times A_2$}
\\
\\
\iniF{t}
& \eqdef
& \lam{\alpha}{
\lam{f_1^{A_1 \to \alpha}}{\lam{f_2^{A_2 \to \alpha}}{f_i\,t}}
}
\\
& :
& A_1 + A_2
\\&&\text{if $t : A_i$ and $i \in \set{1,2}$}
\\
\\
\caseF{t}{x:A_1}{s_1}{x:A_2}{s_2}
& \eqdef
& t\,B
\,(\lam{x^{A_1}}{s_1})
\,(\lam{x^{A_2}}{s_2})
\\
& :
& B
\\
&&
\text{if $t : A_1 + A_2$ and
$s_i : B$}
\\
&&
\text{for each $i \in \set{1,2}$}\hspace{-6cm}
\end{array} } \]
\end{alphasection}
\end{document} |
\begin{document}
\pagestyle{empty}
\begin{titlepage}
\begin{center}\large
\quad\includegraphics[height=17mm]{kit_logo_de.pdf}
\includegraphics[height=20mm]{grouplogo-algo-blue.pdf}\quad\null
Bachelor Thesis
\vspace*{2cm}
\textbf{\huge Engineering Faster Sorters} \\
\textbf{\huge for Small Sets of Items}
Jasper Anton Marianczuk
\vspace*{15mm}
Date: May 09, 2019
\vspace*{45mm}
\begin{tabular}{rl}
Supervisors: & Prof. Dr. Peter Sanders \\
& Dr. Timo Bingmann \\
\end{tabular}
\vspace*{10mm}
Institute of Theoretical Informatics, Algorithmics \\
Department of Informatics \\
Karlsruhe Institute of Technology
\vspace*{12mm}
\end{center}
\end{titlepage}
\vspace*{0pt}
\hrule
Hiermit versichere ich, dass ich diese Arbeit selbständig verfasst und keine anderen, als die angegebenen Quellen und Hilfsmittel benutzt, die wörtlich oder inhaltlich übernommenen Stellen als solche kenntlich gemacht und die Satzung des Karlsruher Instituts für Technologie zur Sicherung guter wissenschaftlicher Praxis in der jeweils gültigen Fassung beachtet habe.
\noindent Karlsruhe, den 09.05.2019
\vspace*{5cm}
\vspace*{0pt}
\selectlanguage{english} \begin{abstract} {\centering \textbf{Abstract}
}
Sorting a set of items is a task that can be useful by itself or as a building block for more complex operations. That is why a lot of effort has been put into finding sorting algorithms that sort large sets as efficiently as possible. But the more sophisticated and fast the algorithms become asymptotically, the less efficient they are for small sets of items due to large constant factors.
A relatively simple sorting algorithm that is often used as a base case sorter is insertion sort, because it has small code size and small constant factors influencing its execution time.
This thesis aims to determine if there is a faster way to sort these small sets of items to provide an efficient base case sorter. We looked at sorting networks, at how they can improve the speed of sorting few elements, and how to implement them in an efficient manner by using conditional moves. Since sorting networks need to be implemented explicitly for each set size, providing networks for larger sizes becomes less efficient due to increased code sizes. To also enable the sorting of slightly larger base cases, we modified Super Scalar Sample Sort and created Register Sample Sort, to break down those larger sets into sizes that can in turn be sorted by sorting networks.
From our experiments we found that when sorting only small sets, the sorting networks outperform insertion sort by at least 25\% for any array size between 2 and 16. When integrating sorting networks as a base case sorter into quicksort, we achieved far less performance improvements over using insertion sort, which is due to the networks having a larger code size and cluttering the L1 instruction cache. The same effect occurs when including Register Sample Sort as a base case sorter for IPS${}^4$o. But for computers that have a larger L1 instruction cache of 64 KiB or more, we obtained speed-ups of 6.4\% when using sorting networks as a base case sorter in quicksort, and of 9.2\% when integrating Register Sample Sort as a base case sorter into IPS${}^4$o, each in comparison to using insertion sort as the base case sorter.
In conclusion, the desired improvement in speed could only be achieved under special circumstances, but the results clearly show the potential of using conditional moves in the field of sorting algorithms.
\end{abstract}
\selectlanguage{german} \begin{abstract} {\centering \textbf{Zusammenfassung}
}
Das Sortieren einer Menge von Elementen ist ein Prozess der f\"ur sich alleine n\"utzlich sein kann oder als Baustein f\"ur komplexere Operationen dient. Deswegen wurde in den Entwurf von Sortieralgorithmen, die eine gro\ss{}e Menge an Elementen effizient sortieren, bereits gro\ss{}er Aufwand investiert. Doch je ausgefeilter und schneller die Algorithmen asymptotisch sind, desto ineffizienter werden sie beim Sortieren kleinerer Mengen aufgrund hoher konstanter Faktoren.
Ein relativ einfacher Sortieralgorithmus, der oft als Basisfall Sortierer genutzt wird, ist Insertion Sort, weil dessen Code kurz ist und er kleine konstante Faktoren hat.
Diese Bachelorarbeit hat das Ziel herauszufinden ob es einen schnelleren Algorithmus gibt um solche wenigen Elemente zu sortieren, damit dieser als effizienter Basisfall Sortierer genutzt werden kann. Wir haben uns dazu Sortiernetzwerke angeschaut, wie man durch sie das Sortieren kleiner Listen beschleunigen kann und wie man sie effizient implementiert: Durch das Ausnutzen von konditionellen \texttt{move}-Befehlen. Weil Sortiernetzwerke f\"ur jede Listengr\"o\ss{}e explizit implementiert werden m\"ussen, nimmt die Effizienz des Sortierens mittels Sortiernetwerken wegen erh\"ohter Codegr\"o\ss{}e ab je gr\"o\ss{}er die Listen sind, die sortiert werden sollen. Um auch das Sortieren etwas gr\"o\ss{}erer Basisf\"alle zu erm\"oglichen haben wir Super Scalar Sample Sort modifiziert und Register Sample Sort entworfen, welcher eine gr\"o\ss{}ere Liste in mehrere kleine Listen zerteilt, die dann von den Sortiernetzwerke sortiert werden k\"onnen.
In unseren Experimenten sind wir zu dem Ergebnis gekommen, dass, wenn nur kleine Mengen sortiert werden, die Sortiernetzwerke um mindestens 25\% schneller sind als Insertion Sort, f\"ur alle Listen, die zwischen 2 und 16 Elementen enthalten. Beim Integrieren der Sortiernetzwerke als Basisfall Sortierer in Quicksort haben wir weit weniger Geschwindigkeitszuwachs gegen\"uber der Benutzung von Insertion Sort erhalten, was daran liegt, dass der Code der Netzwerke mehr Platz ben\"otigt und den Code f\"ur Quicksort aus dem L1 Instruktionscache verdr\"angt. Derselbe Effekt tritt auch beim Benutzen von Register Sample Sort as Basisfall Sortierer f\"ur IPS${}^4$o\, auf. Allerdings konnten wir uns bei Rechnern, die \"uber einen gr\"o\ss{}eren L1 Instruktionscache von 64 KiB oder mehr verf\"ugen, mit Sortiernetzwerken bei Quicksort um 6,4\% und mit Register Sample Sort bei IPS${}^4$o\, um 9,2\% gegen\"uber Insertion Sort als Basisfall Sortierer verbessern.
Zusammenfassend haben wir die angestrebte Verbesserung nur unter besonderen Bedingungen erreicht, aber die Ergebnisse weisen deutlich darauf hin, dass die konditionellen \texttt{move}-Befehle Potential im Anwendungsbereich von Sortieralgorithmen haben.
\end{abstract} \selectlanguage{english}
\pagestyle{normal}
\renewcommand\sectionmark[1]{\markboth{\thesection\quad\MakeUppercase{#1}}{\thesection\quad\MakeUppercase{#1}}} \renewcommand\subsectionmark[1]{\markright{\thesubsection\quad\MakeUppercase{#1}}}
\tableofcontents
\listoffigures \listoftables \listofalgorithms
\section{Introduction}
\subsection{Motivation}
Sorting, that is rearranging the elements in a set to be in a specific order, is one of the basic algorithmic problems. In school and university, basic sorting algorithms like bubble sort, insertion sort, and merge sort, as well as a simple variant of quicksort are taught at first. These algorithms are rated by the number of comparisons they require to sort a set of items. This amount of comparisons is put into relation to the input size and looked at on an asymptotic level. Only later one realizes that what looks good on paper does not have to work well in practice, so factors like average cases, cache effects, hardware setups, and constant factors need to be taken into consideration, too. A sophisticated choice on which sorting algorithm to use (for a particular use case) should be influenced by all of these factors. \\
Complex sorting algorithms aim to sort a large number of items quickly, and a lot of them follow the divide-and-conquer idea of designing an algorithm. However, sorting small sets of items, e.g. with 16 elements or less, is usually fast enough that investing a lot of effort into optimizing sorting algorithms for those cases results in very small gains, looking at the absolute amount of time saved. \\
The complex sorters do not perform as well when sorting small sets of items, having good asymptotic properties but larger constant factors that become more important for the small sizes. Because of that the \emph{base case} of sorting small enough subsets is performed using a simpler algorithm, which is often insertion sort. It has a worst-case run-time of $\mathcal{O}(n^2)$, but small constant factors that make it suitable to use for small $n$. If this sorter is executed many times as base case of a larger sorter though, the times do sum up to contribute to a substantial part of the sorting time. \\
The guiding question of this thesis is:
\begin{center}
Is there a faster way to sort sets of up to 16 elements than insertion sort?
\end{center}
When sorting a set of uniformly distributed random numbers, the chance of any number being greater than another is on average 50\%. Therefore, whenever a conditional branch is influenced by one element's relation to another, one in two of those branches will be mispredicted, which leads to an overall performance penalty.\\
This is a problem that has already been looked at by Michael Codish, Lu\'is Cruz-Filipe, Markus Nebel and Peter Schneider-Kamp in \enquote{Optimizing sorting algorithms by using sorting networks} \cite{DBLP:journals/fac/CodishCNS17} in 2017, and this thesis has taken a great deal of inspiration from it.
\subsection{Overview of the Thesis}
We will first look at sorting networks in section \ref{section:networks}. Section \ref{section:preliminaries:networks} gives a basis of sorting networks and assembly code. After that, we look at different ways of implementing sorting networks efficiently in C++ in section \ref{section:implementation-networks}. For that we focused on elements that consist of a key and an additional reference value. This enables the sorting of complex items, not being limited to integers. \\
In section \ref{section:samplesort} we will take a small detour to look at Super Scalar Sample Sort and develop an efficient modified version for sets with 256 elements or less by holding the splitters in general purpose registers instead of an array. After that section \ref{section:results} discusses the results and improvements of using sorting networks we achieved in our experiments, measuring the performance of the sorting networks and sample sort individually, and also including them as base cases into quicksort and IPS${}^4$o~ \cite{DBLP:conf/esa/AxtmannWF017}. After that we conclude the results of this thesis in section \ref{section:conclusion}.
\section{Sorting Networks} \label{section:networks}
\subsection{Preliminaries} \label{section:preliminaries:networks}
Sorting algorithms can generally be classified into two groups: Those of which the behaviour depends on the input, e.g. quicksort where the sorting speed depends on how well the chosen pivot partitions the set into equally-sized halves, and those of which the behaviour is not influenced by the configuration of the input. The latter are also called \emph{data-oblivious}.
One example of a data-oblivious sorting algorithm is the sorting network. A sorting network of size $n$ consists of a number of $n$ so-called channels numbered $1$ to $n$, each representing one of the inputs, and connections between the channels, called comparators. Where two channels are connected by a comparator it means that the values are to be compared, and if the channel with the lower number currently holds a value that is greater than the value of the channel with the higher number, the values are to be exchanged between the channels. The comparators are given in a fixed order that determines the sequence of executing these \emph{conditional swaps}, so that in the end
\begin{enumerate}
\item the channels contain a permutation of the original input, and
\item the values held by the channels are in nondecreasing order.
\end{enumerate}
Sorting networks are data-oblivious because all the comparisons are always performed, and in the same order, no matter which permutation of an input is given.
For any sorting network, two metrics can be used to quantify it: the \emph{length} and the \emph{depth}. A network's length refers to the number of comparators it contains, and a network's depth describes the minimal amount of levels a network can be divided into. \\
Where two comparators are ordered one after the other, and no channel is used by both comparators, they can be combined into a level. In other words: the result of the second comparator does not depend upon the result of the first. Inductively, any comparator can be merged into a level that executes right before or after it, if its channels are not already used by any comparator in the level. Since now all the comparators in a level are independent from one another, they can be executed in parallel.
\subsubsection{Networks in Practice}
\begin{itemize}
\item \textbf{Best known networks:} For networks of up to size 16 there exist proven optimal lengths and proven optimal depths. For example, the network for 10 elements with optimal length 29 has depth 9, the one with optimal depth 7 has length 31 \cite{DBLP:books/lib/Knuth98a, DBLP:conf/ictai/CodishCFS14}. For networks of greater size there only exist currently known lowest numbers of length or depth. Those best networks are acquired through optimizations that were initially done by hand and nowadays are realized e.g.~with the help of computers and evolutionary algorithms \cite{SorterHunter}.
\item \textbf{Recursive networks:} For creating sorting networks there also exist algorithms that work in a recursive divide-and-conquer way: split the input into two parts, sort each part recursively, and merge the two parts together in the end. Representatives for this kind of approach are the constructions of R.J. Nelson and B.C. Bose \cite{DBLP:journals/jacm/BoseN62} and the algorithm by K.E. Batcher \cite{DBLP:conf/afips/Batcher68}. Bose and Nelson split the input sequence into first and second half, while Batcher partitions into elements with an even index and elements with an odd index. The advantage of those recursive networks over the specially optimized ones is that they can easily be created even for large network sizes. While the generated networks may have more comparators than the best known networks, the number of comparators in a network acquired from either Bose-Nelson or Batcher of size $n$ has an upper bound of $\mathcal{O}(n\, (\log n)^2)$ \cite{DBLP:books/lib/Knuth98a}.
\end{itemize}
\begin{figure}
\caption{Sorting network by Bose and Nelson for 6 elements}
\label{network:bosenelson:6}
\end{figure}
Sorting networks are usually depicted by using horizontal lines for the channels, and vertical connections between these lines for the comparators. A network by Bose and Nelson for 6 elements displayed like that can be seen in figure \ref{network:bosenelson:6}. \\
\subsubsection{Improving the Speed of Sorting through Sorting Networks}
An important question to ask is how sorting networks can improve the sorting speed on a set of elements (on average), if they can not take any shortcuts for \enquote{good} inputs, like an insertion sort that would leverage an already sorted input and do one comparison per element. The answer to this question is \emph{branching}. Because the compiler knows in advance which comparisons are going to be executed in which order, the control flow does not contain conditional branches, in particular getting rid of expensive branch mispredictions. On uniformly distributed random inputs, the chances that any number is smaller than another is 50\% on average, making branches unpredictable. In the case of insertion sort that means not knowing in advance with how many elements the next one has to be compared until it is inserted into the right place. \\
Even though with sorting networks the compiler knows in advance when to execute which comparator, implementing the compare-and-swap operation in a naive way (as seen in \ref{section:preliminaries:compare-and-swap}) the compiler might still generate branches. In that case, the sorting networks are no faster than insertion sort, or even slower.
\subsubsection{Compare-and-Swap} \label{section:preliminaries:compare-and-swap}
For sorting networks, the basic operation used is to compare two values against each other. If they are in the wrong order (the \enquote{smaller} element occurs after the \enquote{bigger} one in the sequence), they are swapped. Intuitively, one might implement the operation in C++ like this:
\begin{verbatim}
void ConditionalSwap(TValueType& left, TValueType& right)
{
if (left > right) { std::swap(left, right); }
}
\end{verbatim}
Here \verb|TValueType| is a template typename and can be instantiated with any type that implements the \verb|>| operator. \\
As suggested in \cite{DBLP:journals/fac/CodishCNS17}, the same piece of code can be rewritten like this:
\begin{verbatim}
void ConditionalSwap2(TValueType& left, TValueType& right)
{
TValueType temp = left;
if (temp > right) { left = right; }
if (temp > right) { right = temp; }
}
\end{verbatim}
At first glance it looks like we now have two branches that can be taken. But the code executed if the condition is true now only consists of a single assignment each, which can be expressed in x86-Architecture through a \emph{conditional move} instruction. In AT\&T syntax (see section \ref{section:preliminaries:asm}), a conditional move (\verb|cmov a,b|) will write a value in register \verb|a| into register \verb|b|, if a condition is met. If the condition is not met, no operation takes place (still taking the same number of CPU cycles as the move operation would have). Since the address of the next instruction no longer depends upon the previously evaluated condition, the control flow now does not contain branches. The only downside of the conditional move is that it can take longer than a normal move instruction on certain architectures, and can only be executed when the comparison has performed and its result is available. \\
When the elements to be sorted are only integers, some compilers do generate code with conditional moves for those operations. When the elements are more general (in this thesis we will look at pairs of an unsigned 64 bit integer key and an unsigned 64 bit reference value, which could be a pointer or an address in an array), gcc 7.3.0, the compiler used for the experiments, does not generate conditional moves. To force the usage of conditional moves, a feature of gcc was used that allows the programmer to specify small amounts of assembly code to be inserted into the regular machine code generated by gcc, called inline assembly \cite{GccInlineAssembly}. This mechanic and the notation is further explained in section \ref{section:preliminaries:asm}.
\subsubsection{Assembly Code} \label{section:preliminaries:asm}
Assembly code represents the machine instructions executed by the CPU. It can be given as the actual opt-codes or as human-readable text. There are two different conventions for the textual representation, the Intel syntax or MASM syntax and the AT\&T syntax. The main differences are:
\\
\begin{tabular}{c | p{6.2cm} | p{6.2cm}}
& \multicolumn{1}{c |}{Intel} & \multicolumn{1}{c}{AT\&T} \\ \hline
Operand size & The size of the operand does not have to be specified & The size of the operand is appended to the instruction: \verb|b| (byte = 8 bit), \verb|l| (long = 32 bit), \verb|q| (quad-word = 64 bit) \\ \hline
Parameter order & The destination is written first, then the source of the value: \verb|mov dest,src| & The source is written first, then the destination: \verb|movq src,dest| \\
\end{tabular}
\\
In this thesis only the AT\&T syntax will be used. \\
The gcc C++ compiler has a feature that allows the programmer to write assembly instructions in between regular C++ code, called \enquote{inline assembly} (\verb|asm|) \cite{GccInlineAssembly}. A set of assembly instructions to be executed must be given, followed by a definition for input and output variables and a list of clobbered registers. This extra information is there to communicate to the compiler what is happening inside the \verb|asm| block. Gcc itself does not parse or optimize the given assembly statements, they are only after compilation added into the generated assembly code by the GNU Assembler.
A variable being in the output list means that the value will be modified, a clobbered register is one where gcc cannot assume that the value it held before the \verb|asm| block will be the same as after the block. In this thesis, the clobbered registers will almost always be the conditional-codes registers (\verb|cc|), which include the carry-flag, zero-flag and the signed-flag, which are modified during a compare-instruction. This way of specifying the input, output and clobbered registers is also called \emph{extended asm}. \\
Taking the code from \ref{section:preliminaries:compare-and-swap}, and assuming \verb|TValueType = uint64_t|, the statement
\begin{verbatim}
if (temp > right) { left = right; }
\end{verbatim}
can now be written as
\begin{verbatim}
__asm__(
"cmpq
"cmovbq
: [left] "=&r"(left) //output
: "0"(left), [right] "r"(right), [temp] "r"(temp) //input
: "cc" //clobber
);
\end{verbatim}
In extended \verb|asm|, one can define C++ variables as input or output operands, and gcc will assign a register for that value (if it has the \verb|"r"| modifier), and also write the value in an output register back to the given variable after the \verb|asm| block. Note that the names in square brackets are symbolic names only valid in the context of the assembly instructions and independent from the names in the C++ code before. The link between the C++ names and the symbolic names happens in the input and output declarations. \\
With the conditional moves it is important to properly declare the input and output variables, because they perform a task that is a bit unusual: an output variable may be overwritten, and also may not. For the output register for \verb|left|, two things must apply:
\begin{enumerate}
\item if the condition is false, it must hold the value of \verb|left|, and \label{cmov:condition:1}
\item if the condition is true, it must hold the value of \verb|right|.
\end{enumerate}
For optimizations purposes, the compiler might reduce the number of registers used by placing the output of one operation into a register that previously held the input for some other operation. To prevent this, the declaration for the output \verb|[left] "=&r"(left)| has the \verb|"&"| modifier added to it, meaning it is an \enquote{early clobber} register and that no other input can be placed in that register. In combination with \verb|"0"(left)| in the input line, it is also tied to an input, so that the previous value of \verb|left| is loaded into the register beforehand, to comply with constraint \ref{cmov:condition:1}. Because we already declared it as output, instead of giving it a new symbolic name we tie it to the output by referencing its index in the output list, which since it is the first output variable is \verb|"0"|. The \verb|"="| in the output declaration solely means that this register will be written to. Any output needs to have the \verb|"="| modifier. \\
We see that each assembly instruction is postfixed with \verb|\n\t|. That is because the instruction strings are appended into a single instruction string during compilation and \verb|\n\t| tells the GNU assembler where one instruction ends and the next begins. \\
The \verb|cmov| instruction is postfixed with a \verb|b| in this example, which stands for \enquote{below}. So the \verb|cmov| will be executed if \verb|right| is below \verb|temp| (unsigned comparison \verb|right < temp|). Apart from below we will also see not equal (\verb|ne|) and carry (\verb|c|) as a postfix. \\
In addition to that, both the \verb|cmp| and the \verb|cmovb| are postfixed with a \verb|q| (quad-word) to indicate that the operands are 64-bit values. \\
When a subtraction $\mathtt{minuend} - \mathtt{subtrahend}$ is performed and \texttt{subtrahend} is larger than \texttt{minuend} (interpreted as unsigned numbers), the operation causes an underflow which results in the carry flag being set to 1. The check for that carry flag being 1 can be used as a condition by itself, and the carry flag influences other condition checks like \emph{below}. This property of the comparison setting the carry flag will be used in section \ref{section:samplesort:impl}.
\subsection{Implementation of Sorting Networks} \label{section:implementation-networks}
\subsubsection{Providing the Network Frame}
\begin{figure}
\caption{Best network with optimal length for 16 elements}
\label{network:best:16}
\end{figure}
\begin{figure}
\caption{Bose Nelson network for 16 elements optimizing locality}
\label{network:bosenelson:16}
\end{figure}
\begin{figure}
\caption{Bose Nelson network for 16 elements optimizing parallelism}
\label{network:bosenelson:parl:16}
\end{figure}
The best networks for sizes of up to 16 elements were taken from John Gamble's Website \cite{JgambleNetworks} and are length-optimal. \\
The Bose Nelson networks have been generated using the instructions from their paper \cite{DBLP:journals/jacm/BoseN62}. \\
For sizes of 8 and below the best and generated networks have the same amount of comparators and levels. For sizes larger than 8 the generated networks are at a disadvantage because they have more comparators and/or levels. As a trade-off their recursive structure makes it possible to leverage a different trait: locality. Instead of optimizing them to sort as parallel as possible, we can first sort the first half of the set, then the second half, and then apply the merger. This way, chances are higher that all $\frac{n}{2}$ elements of the first half might fit into the processor's general purpose registers. During this part of the sorting routine, no accesses to memory or cache are required. To determine if there is a visible speed-up, the networks were generated optimizing (a) locality and (b) parallelism. \\
As an extra idea, the Bose Nelson networks were generated in a way that one can pass the elements as separate parameters instead of as an array. That way one can sort elements that are not contiguously placed in memory. Because the networks were implemented as method calls to the smaller sorters and merge methods, there would be a large overhead in placing many elements onto the call stack for each method call. While we hoped this would make a difference by reducing code size, the overhead for the method call was too large. That is why all the methods are declared \verb|inline| which results in the same flat sequence of swaps for each size the networks optimizing locality have. \\
Examples of networks for 16 elements can be seen in figures \ref{network:best:16}, \ref{network:bosenelson:16} and \ref{network:bosenelson:parl:16}.
All networks are implemented so that they have an entry method that takes a pointer to an array \verb|A| and an array size \verb|n| as input and delegates the call to the specific method for that number of elements, which in turn executes all the comparators. To measure different implementations for the conditional swaps, the network methods and the swap are templated, so that when calling the network with an array of a specific type the respective specialized conditional-swap implementation will be used. \\
Our approach differs from the work in \cite{DBLP:journals/fac/CodishCNS17} in the type of elements that were sorted. While they measured the sorting of \verb|int|s, which are usually 32-bit sized integers, we made the decision to sort elements that consist of a 64-bit integer key and a 64-bit integer reference value, enabling not only the sorting of numbers but also the sorting of complex elements, when giving a pointer or an array index as the reference value to the original set. This was implemented by creating \verb|struct|s that contain a key and reference value each, having the following structure:
\begin{verbatim}
struct SortableRef
{
uint64_t key, reference;
}
\end{verbatim}
They also define the operators~~\verb|>, >=, ==, <, <=|~~and~~\verb|!=|~~for reasons of usability, and templated methods \verb|uint64_t GetKey(TSortable)| and \verb|uint64_T GetReference(TSortable)| are available.
\subsubsection{Implementing the Conditional Swap} \label{section:implementation-conditionalswap}
The \texttt{ConditionalSwap} is implemented as a templated method like this:
\begin{verbatim}
template <typename TValueType>
inline
void ConditionalSwap(TValueType& left, TValueType& right)
{
//body
}
\end{verbatim}
The following variants will represent the body of one specialization of the template function for a specific struct. Each of them was given a three letter abbreviation to name them in the results. We implemented the following approaches:
\newcommand{using std::swap (\texttt{Def})}{using std::swap (\texttt{Def})}
\newcommand{using inline if statements (\texttt{QMa})}{using inline if statements (\texttt{QMa})}
\newcommand{using std::tie and std::tuple (\texttt{Tie})}{using std::tie and std::tuple (\texttt{Tie})}
\newcommand{using jmp and xchg (\texttt{JXc})}{using jmp and xchg (\texttt{JXc})}
\newcommand{using four cmovs and temp variables (\texttt{4Cm})}{using four cmovs and temp variables (\texttt{4Cm})}
\newcommand{using four cmovs split from one another and temp variables (\texttt{4CS})}{using four cmovs split from one another and temp variables (\texttt{4CS})}
\newcommand{using six cmovs and temp variables (\texttt{6Cm})}{using six cmovs and temp variables (\texttt{6Cm})}
\newcommand{moving pointers with cmov instead of values (\texttt{Cla})}{moving pointers with cmov instead of values (\texttt{Cla})}
\newcommand{moving pointers and supporting a predicate (\texttt{CPr})}{moving pointers and supporting a predicate (\texttt{CPr})}
\begin{itemize}
\item using std::swap (\texttt{Def})
\item using inline if statements (\texttt{QMa})
\item using std::tie and std::tuple (\texttt{Tie})
\item using jmp and xchg (\texttt{JXc})
\item using four cmovs and temp variables (\texttt{4Cm})
\item using four cmovs split from one another and temp variables (\texttt{4CS})
\item using six cmovs and temp variables (\texttt{6Cm})
\item moving pointers with cmov instead of values (\texttt{Cla})
\item moving pointers and supporting a predicate (\texttt{CPr})
\end{itemize}
The details of implementation can be seen in the following paragraphs.
\paragraph*{using std::swap (\texttt{Def})} The default implementation for the template makes use of the defined \verb|<| operator:
\begin{verbatim}
if (right < left)
std::swap(left, right);
\end{verbatim}
This is the intuitive way of writing the conditional swap we already saw in section \ref{section:preliminaries:compare-and-swap}, without any inline assembly.
\paragraph*{using inline if statements (\texttt{QMa})}
\begin{verbatim}
bool r = (left > right);
auto temp = left;
left = r ? right : left;
right = r ? temp : right;
\end{verbatim}
Here it was attempted to convince the compiler to generate conditional moves by using the \emph{inline if}-statements with trivial values in the else part.
\paragraph*{using std::tie and std::tuple (\texttt{Tie})}
\begin{verbatim}
std::tie(left, right) =
(right < left) ? std::make_tuple(right, left) : std::make_tuple(left, right);
\end{verbatim}
This approach uses assignable tuples (tie).
\paragraph*{using jmp and xchg (\texttt{JXc})}
\begin{verbatim}
__asm__(
"cmpq
"jae
"xchg
"xchg
"
: [left_key] "=&r"(left.key), [right_key] "=&r"(right.key),
[left_reference] "=&r"(left.reference),
[right_reference] "=&r"(right.reference)
: "0"(left.key), "1"(right.key), "2"(left.reference), "3"(right.reference)
: "cc"
);
\end{verbatim}
The \verb|
\paragraph*{using four cmovs and temp variables (\texttt{4Cm})}
\begin{verbatim}
uint64_t tmp = left.key;
uint64_t tmpRef = left.reference;
__asm__(
"cmpq
"cmovbq
"cmovbq
"cmovbq
"cmovbq
: [left_key] "=&r"(left.key), [right_key] "=&r"(right.key),
[left_reference] "=&r"(left.reference),
[right_reference] "=&r"(right.reference)
: "0"(left.key), "1"(right.key), "2"(left.reference), "3"(right.reference),
[tmp] "r"(tmp), [tmp_ref] "r"(tmpRef)
: "cc"
);
\end{verbatim}
\paragraph*{using four cmovs split from one another and temp variables (\texttt{4CS})}
\begin{verbatim}
uint64_t tmp = left.key;
uint64_t tmpRef = left.reference;
__asm__ volatile (
"cmpq
:
: [left_key] "r"(left.key), [right_key] "r"(right.key)
: "cc"
);
__asm__ volatile (
"cmovbq
: [left_key] "=&r"(left.key)
: "0"(left.key), [right_key] "r"(right.key)
:
);
__asm__ volatile (
"cmovbq
: [left_reference] "=&r"(left.reference)
: "0"(left.reference), [right_reference] "r"(right.reference)
:
);
__asm__ volatile (
"cmovbq
: [right_key] "=&r"(right.key)
: "0"(right.key), [tmp] "r"(tmp)
:
);
__asm__ volatile (
"cmovbq
: [right_reference] "=&r"(right.reference)
: "0"(right.reference), [tmp_ref] "r"(tmpRef)
:
);
\end{verbatim}
Because we split the \verb|asm| blocks, they have to be declared \verb|volatile| so that the optimizer does not move them around or out of order. Without declaring them \verb|volatile|, some of the networks were not sorting correctly. The blocks were split because we hoped the compiler would be able to insert operations that do not affect the conditional codes and are unrelated to the current conditional swap between the \verb|cmp|-instruction and the conditional moves, to reduce the amount of wait cycles that have to be performed. This was successful as can be seen in the experimental results in section \ref{section:experiments:normal}. \\
\paragraph*{using six cmovs and temp variables (\texttt{6Cm})}
\begin{verbatim}
uint64_t tmp;
uint64_t tmpRef;
__asm__ (
"cmpq
"cmovbq
"cmovbq
"cmovbq
"cmovbq
"cmovbq
"cmovbq
: [left_key] "=&r"(left.key), [right_key] "=&r"(right.key),
[left_reference] "=&r"(left.reference),
[right_reference] "=&r"(right.reference),
[tmp] "=&r"(tmp), [tmp_ref] "=&r"(tmpRef)
: "0"(left.key), "1"(right.key), "2"(left.reference), "3"(right.reference),
"4"(tmp), "5"(tmpRef)
: "cc"
);
\end{verbatim}
\paragraph*{moving pointers with cmov instead of values (\texttt{Cla})} This idea came from a result created by the clang compiler from the special code as seen in the ConditionalSwap2 method in \ref{section:preliminaries:compare-and-swap}. For the transformation to gcc, we took only the minimal necessary instructions concerning the conditional move into the \verb|asm| block:
\begin{verbatim}
SortableRef_ClangVersion* leftPointer = &left;
SortableRef_ClangVersion* rightPointer = &right;
uint64_t rightKey = right.key;
SortableRef_ClangVersion tmp = left;
__asm__ volatile(
"cmpq
"cmovbq
: [left_pointer] "=&r"(leftPointer)
: "0"(leftPointer), [right_pointer] "r"(rightPointer),
[tmp_key] "m"(tmp.key), [right_key] "r"(rightKey)
: "cc"
);
left = *leftPointer;
leftPointer = &tmp;
__asm__ volatile(
"cmovbq
: [right_pointer] "=&r"(rightPointer)
: "0"(rightPointer), [left_pointer] "r"(leftPointer)
:
);
right = *rightPointer;
\end{verbatim}
\paragraph*{moving pointers and supporting a predicate (\texttt{CPr})} Instead of performing the comparison inside the \texttt{asm} block, which requires knowledge of the datatype of the key, it can also be done over a predicate, using the result of that comparison inside the inline assembly:
\begin{verbatim}
SortableRef_ClangPredicate* leftPointer = &left;
SortableRef_ClangPredicate* rightPointer = &right;
SortableRef_ClangPredicate temp = left;
int predicateResult = (int) (right < temp);
__asm__ volatile(
"cmp $0,
"cmovneq
: [left_pointer] "=&r"(leftPointer)
: "0"(leftPointer), [right_pointer] "r"(rightPointer),
[predResult] "r"(predicateResult)
: "cc"
);
left = *leftPointer;
leftPointer = &temp;
__asm__ volatile(
"cmovneq
: [right_pointer] "=&r"(rightPointer)
: "0"(rightPointer), [left_pointer] "r"(leftPointer)
:
);
right = *rightPointer;
\end{verbatim}
For the \texttt{Cla} implementation the \verb|b| in \verb|cmovb| was used to execute the conditional move if \verb|right_key| was smaller than \verb|temp_key|. If that is the case, the predicate will return true, or as an int a value not equal to zero. When comparing this result to 0, the \verb|cmov| is to be executed if the result was any value other than zero, so the postfix here is \verb|ne| (not equal). \\
Note that while the knowledge of how to compare the elements is still present by doing the comparison directly (\verb|right < temp|), the compiler now needs to take the result from the comparison, and put it into an integer that is then used in the \verb|asm| block. The only addition to make it completely independent from the sorted elements would be to pass a predicate to do the comparison, which would also involve modifying the network frame to take and pass the predicate. To measure on the same network frame we took this shortcut of doing the comparison using the \verb|<| operator.
\section{Register Sample Sort} \label{section:samplesort}
\subsection{Preliminaries} \label{section:samplesort:preliminaries}
Sample sort is a sorting algorithm that follows the divide-and-conquer principle. The input is separated into $k$ subsets, that each contain elements within an interval of the total ordering, with the intervals being distinct from one another. That is done by first choosing a subset $S$ of $a \cdot k$ elements and sorting $S$. Afterwards the splitters $\{s_0, s_1, \ldots, s_{k-1}, s_k\} = \{-\infty, S_a, S_{2a}, \ldots, S_{(k-1)a}, \infty\}$ are taken from $S$. The parameter $a$ denotes the oversampling factor. Oversampling is used to get a better sample of splitters to achieve more evenly-sized partitions, trading for the time that is required to sort the larger sample. \\
With the splitters the elements $e_i$ are then \emph{classified}, placing them into buckets $b_j$, where $j \in \{1, \ldots, k\}$ and $s_{j-1} < e_i \leq s_j$. For $k$ being a power of 2, this placement can be achieved by viewing the splitters as a binary tree, with $s_{k/2}$ being the root, all $s_l$ with $l < k/2$ representing the left subtree and those with $l > k/2$ the right one. To place an element, one must only traverse this binary tree, resulting in a binary search instead of a linear one \cite{DBLP:conf/esa/SandersW04}.\\
Quicksort is therefore a specialization of sample sort with fixed parameter $k = 2$, having only one splitter, the pivot, and splitting the input into two partitions.
\subsection{Implementing Sample Sort for Medium-Sized Sets} \label{section:samplesort:impl}
The motivation to look at sample sort was that we wanted to see how well the sorting networks perform when using them as a base case for the \textit{In-Place Parallel Super Scalar Samplesort} (IPS${}^4$o) by Michael Axtmann, Sascha Witt, Daniel Ferizovic and Peter Sanders \cite{DBLP:conf/esa/AxtmannWF017}. The problem that occured is that IPS${}^4$o~can go into the base case with sizes larger than 16, while the networks we looked at only sort sets of up to 16 elements. \\
To close that gap, we created a sequential version of Super Scalar Sample Sort \cite{DBLP:conf/esa/SandersW04} that can reduce base case sizes of up to 256 down to blocks of 16 or less in an efficient manner.
Since the total size was expected to not be much greater than 256, not much effort was made to keep the algorithm in-place. The central idea was to place the splitters not into an array, as described in \cite{DBLP:conf/esa/SandersW04}, but to hold them in general purpose registers for the whole duration of the element classification.
The question now arose as to which splitter an element needs to be compared to after the first comparison with the middle splitter. When the splitters are organized in a binary heap in an array, that can be done by using array indices, the children of splitter $j$ being at positions $2j$ and $2j + 1$. If an element is smaller than $s_j$, it would afterwards be compared to $s_{2j}$, otherwise to $s_{2j+1}$. But this way of accessing the splitters does not work when they are placed in registers. The solution was to create a copy of the left subtree, and to conditionally overwrite that with the right subtree, should the element be greater than the root node. The next comparison is then made against the root of the temporary tree that now contains the correct splitters to compare that element against. For 3 splitters that requires 1 conditional move, and for 7 splitters would require 3 conditional moves after the first comparison and 1 more after the second comparison, per element.
After finding the correct splitters to compare to, we are left with one more problem: How to know in which bucket the element is to be placed into at the end. In \cite{DBLP:conf/esa/SandersW04} this was done by making use of the calculated index determining the next splitter to compare to. We chose an approach similar to creating this index, using the correlation between binary numbers and the tree-like structure of the splitters. We will be viewing the splitters not as a binary heap but just as a list where the middle of the list represents the root node of the tree, its children being the middle element of the left and the middle element of the right list. \\
If an element $e_i$ is larger than the first splitter $s_{k/2}$ (with $k-1$ being the number of splitters), it must be placed in a bucket $b_j$ with $j \geq \frac{k}{2}$ (assuming 0-based indexing for $b$). That also means that the index of that bucket, represented as a binary number, must have its bit at position $l := \log \frac{k}{2}$ set to 1. That way, the result of the comparison ($e_i > s_{k/2}$) can be interpreted as an integer (1 for \verb|true|, 0 for \verb|false|) and added to j. If that was not the last comparison, $j$ is then multiplied by 2 (meaning its bits are shifted left by one position). This means the bit from the first comparison makes its way \enquote{left} in the binary representation while the comparison traverses down the tree, and so forth with the other comparisons. After traversing the splitter tree to the end, $e_i$ will have been compared to the correct splitters and $j$ will hold the index of the bucket that $e_i$ belongs into. These operations can be implemented without branches by making use of the way comparisons are done: \\
At the end of section \ref{section:preliminaries:asm} we explained that when comparing (unsigned) numbers (which is nothing but a subtraction), and the \texttt{subtrahend} being greater than the \texttt{minuend}, the operation causes an underflow and the carry flag is set. We also notice that when converting the result of the predicate ($e_i > s_{k/2}$) to an integer value, the integer will be 1 for \verb|true| and 0 for \verb|false|. So in assembly code, we can compare the result from evaluating the predicate to the value 0: \verb|cmp
As an addition to the efficient classification, while looping over the elements we allow to place multiple elements into buckets per loop, allowing for all the registers in the machine to be used. This additional parameter is called \emph{blockSize}.
There is one downside to this approach: The keys of the splitters (since we only need a splitter's key for classifying an element) must be small enough to fit into a general purpose register. Needing more than one register per key would mean either running out of registers or spending extra time to conditionally move the splitter keys around. For three splitters the needed number of registers for block sizes 1 to 5 are as seen in table \ref{table:samplesort:registerusage}. We can see that the trade-off for classifying multiple elements at the same time is the amount of registers needed. \\
If we were to use 7 splitters instead of three, the number of registers required for classifying just 1 element at a time would go up to 15. Also, with 8 buckets, if we get recursive subproblems with sizes just over 16, classifying into 8 buckets again would be greatly inefficient, resulting in many buckets containing very few. This is why we decided to only use three splitters for this particular sorter. \\
Pseudocode to implement the classification can be seen as an example for an array of integers and \texttt{blockSize = 1} in algorithm \ref{algo:samplesort:cversion}. $j$ is here called \texttt{state}, and the temporary subtree consists of one splitter which we gave the name \texttt{splitterx}. For the branchless implementation we used the \verb|cmovc| for line \ref{algo:samplesort:cmov} and the \verb|rcl| instruction for line \ref{algo:samplesort:rcl}. At the last level of classification no more moving of splitters is required, so instead of doing another comparison against the predicate result and using \verb|rcl|, we can just shift \texttt{state} left by one position and add the predicate's result to it (line \ref{algo:samplesort:laststate}). Alternatively we could use a bitwise \verb|OR| or \verb|XOR| after the shift, which would have the same result. But we decided that adding the predicate result was more readable. \\
For sorting the splitter sample, the same sorting method can be used as for the base case. \\
\begin{table}[!h]
\begin{center}
\begin{tabular}{ r | c c c c c | c c c c c}
& \multicolumn{5}{c |}{3 splitters} & \multicolumn{5}{c}{7 splitters} \\ \hline
& \multicolumn{5}{c |}{block size} & \multicolumn{5}{c}{block size} \\
& 1 & 2 & 3 & 4 & 5 & 1 & 2 & 3 & 4 & 5 \\ \hline
splitters & 3 & 3 & 3 & 3 & 3 & 7 & 7 & 7 & 7 & 7 \\
buckets pointer & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\
current element index & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\
element count & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\ \hline
state & 1 & 2 & 3 & 4 & 5 & 1 & 2 & 3 & 4 & 5 \\
predicate result & 1 & 2 & 3 & 4 & 5 & 1 & 2 & 3 & 4 & 5 \\
splitterx & 1 & 2 & 3 & 4 & 5 & 3 & 6 & 9 & 12 & 15 \\ \hline
sum & 9 & 12 & 15 & 18 & 21 & 15 & 20 & 25 & 30 & 35
\end{tabular}
\end{center}
\caption{Registers required by Register Sample Sort with three or seven splitters} \label{table:samplesort:registerusage}
\end{table}
\begin{algorithm}[!h]
\caption{Register Sample Sort Classification(\texttt{array}, \texttt{elementCount}, \texttt{predicate})} \label{algo:samplesort:cversion}
int splitter0, splitter1, splitter2 $\leftarrow$ determineSplitters() \\
int state, predicateResult, splitterx \\
int* b${}_0$, b${}_1$, b${}_2$, b${}_3$ $\leftarrow$ allocateBuckets(elementCount) \\
\For{$1 \leq i \leq \mathtt{elementCount}$}{
state $\leftarrow$ 0\\
predicateResult $\leftarrow$ (int) predicate(splitter1 < array[i]) \\
splitterx $\leftarrow$ splitter0 \\
\If{predicateResult > 0}{
splitterx $\leftarrow$ splitter2 \label{algo:samplesort:cmov} \\
state $\leftarrow$ (state \texttt{<<} 1) + 1 \label{algo:samplesort:rcl}
}
predicateResult $\leftarrow$ (int) predicate(splitterx < array[i]) \\
state $\leftarrow$ (state \texttt{<<} 1) + predicateResult \label{algo:samplesort:laststate}\\
place array[i] in buckets $b_{state}$
}
\end{algorithm}
\section{Experimental Results} \label{section:results}
In the tests we ran, different sorting algorithms and conditional-swap implementations were compared. For the details about the different sorters and swaps refer to section \ref{section:implementation-networks}. \\
The names of the sorters are built in an abbrevatory way that matches the following format:
{ \setlength\parskip{
amount}
\begin{enumerate}
\item It starts with an \verb|I| or an \verb|N|, indicating if the used algorithm is insertion sort or a sorting network. \label{enumeration:experimentnaming:algtype}
\begin{itemize}
\item In case of sorting networks, if it is a \verb|Best| network or a Bose Nelson network (\verb|BoNe|).
\begin{itemize}
\item For a Bose Nelson network whether it was optimized for Locality (\verb|L|), Parallelism (\verb|P|) or generated to take the items as single parameters \verb|M| (see section \ref{section:implementation-networks})
\end{itemize}
\end{itemize}
\item Then follows the type of benchmark, \verb|-N| for sorting one set of items (\enquote{normal sort}, section \ref{section:experiments:normal}), \verb|-I| for sorting many continuous sets of items (\enquote{inrow sort}, section \ref{section:experiments:inrow}), \verb|-S| for sorting with Sample Sort (section \ref{section:experiments:samplesort}), \verb|-Q| for sorting with quicksort (section \ref{section:experiments:quicksort}) and \verb|-4| for sorting with IPS${}^4$o~ (section \ref{section:experiments:ipso}).
\begin{itemize}
\item In case of Sample Sort, the parameters \texttt{numberOfSplitters}, \texttt{oversamplingFactor} and \texttt{blockSize} are appended as numbers
\end{itemize}
\item Lastly, the name of the \verb|struct| used for the template specialization is appended (see section \ref{section:preliminaries:compare-and-swap} for the abbreviations for conditional swaps) as well as a single \verb|K| for elements that have only a key and \verb|KR| for those that have a key and a reference value.
\end{enumerate}
}
Where for comparison \verb|std::sort| was run, the name in step \ref{enumeration:experimentnaming:algtype} is \verb|StdSort|.
For example, when measuring sample sort with parameters 332 and a Bose Nelson network optimizing parallelism as the base case with conditional swap \verb|4CS|, the sorter name would be \verb|N BoNeP -S332 KR 4CS| .
\subsection{Environment}
\begin{table}[!h]
\begin{tabular}{r | r | r | r}
Machine Name & \multicolumn{1}{c|}{A} & \multicolumn{1}{c|}{B} & \multicolumn{1}{c}{C} \\ \hline
\multirow{2}{*}{CPU} & 2 x Intel Xeon 8-core & 2 x Intel Xeon 12-core & AMD Ryzen 8-core \\
& E5-2650 v2 2.6 GHz & E5-2670 v3 2.3 GHz & 1800X 3.6 GHz \\
RAM & 128 GiB DDR3 & 128 GiB DDR4 & 32GB DDR4 \\
L1 Cache (per Core) & 32 KiB I + 32 KiB D & 32 KiB I + 32 KiB D & 64 KiB I + 32 KiB D \\
L2 Cache (per Core) & 256 KiB & 256 KiB & 512 KiB \\
L3 Cache (total) & 20 MiB & 30 MiB & 16 MiB [8 MiB]
\end{tabular}
\caption{Hardware properties of the machines used} \label{table:machines}
\end{table}
As compiler the gcc C++ compiler in version 7.3.0 was used with the \verb|-O3| flag. \\
The measurements were done with only essential processes running on the machine apart from the measurement. To prevent the process from being swapped to another core during execution it was run with \verb|taskset 0x1|. \\
In total, three different machines were used to do the measurements. Their hardware properties can be seen in table \ref{table:machines}. \enquote{I} and \enquote{D} refer to dedicated Instruction and Data caches. Also note that while the AMD Ryzen's L3 cache has a total size of 16 MiB, it is divided into two 8 MiB caches that are exclusive to 4 cores each. Since all measurements were done on a single core, the L3 cache size in brackets is the one available to the program. The operating system on all machine was Ubuntu 18.04.
\subsection{Generating Plots}
Due to the high number of dimensions in the measurements (machine the measurement is run on, type of network, conditional swap implementation, array size) the results could not always be plotted two-dimensionally. We used box-plots where applicable to show more than just an average value for a measurement. The box incloses all values between the first quartile (\texttt{1Q}) and third quartile (\texttt{3Q}). The line in the middle shows the median. Further the inter-quartile-range (\texttt{IQR}) is calculated as the distance between first and third quartile. The lines (called whiskers) left and right of the boxes go until the smallest value greater than $\mathtt{1Q} - 1.5 \cdot \mathtt{IQR}$ and the greatest value smaller than $\mathtt{3Q} + 1.5 \cdot \mathtt{IQR}$ respectively. Values below these ranges are called outliers and shown as individual dots.
\subsection{Conducting the Measurements} \label{section:measurements}
\paragraph*{Random Numbers} In order to measure the time needed to sort some data, one has to have data first. For these measurements, the data consisted of pairs of a 64-bit unsigned integer key and a 64-bit unsigned integer reference value. Those were generated as uniformly distributed random numbers by a lightweight implementation of the std::minstd\_rand generator from the C++ <random> library that works as follows: \\
First a \texttt{seed} is set, taken e.g. from the current time. When a new random number is requested, the generator calculates $\mathtt{seed} = \mathtt{seed}~ \cdot~ 48271~ \%~ 2147483647$ and returns the current \texttt{seed}. \\
The numbers generated like that do not use all 64 bits available, which is only for practicality with the permutation check as will be seen below. \\
For each measurement $i$, a new $\mathtt{seed}_i$ is taken from the current time. The same $\mathtt{seed}_i$ is then set before the execution of each sorter, to provide all sorters with the same random inputs.
\paragraph*{Measuring} The actual measuring was done via linux's PERF\_EVENT interface that allows to do fine-grained measurements. Here, the number of cpu cycles spent on sorting was the unit of measurement. That also means that the results do not depend on clock speeds (e.g. when overclocking), but only on the CPU's architecture.
\paragraph*{Compilation} When we started this project, it was only a single source file (.cpp) with an increasing amount of headers that were all included in that single file. That is also due to the fact that templated methods cannot be placed in source files because they need to be visible to all including files at compile time. The increasing amount of code and the many different templates brought the compiler to a point where it took over a minute to compile the project. The problem we encountered was that the compiler only gives itself a limited amount of time for compiling a single source file. In order to stay within the time boundaries for a single file, the optimization became poor. We saw measurements being slower for no apparent reason. To solve that problem, we used code generation to create source files that contain an acceptable amount of methods that initiate part of a measurement in a wrapper method. This way, from the main source file we only need to call the correct wrapper methods to perform the measurements, and this way we achieved results that were more stable and reproducible. \\
For compilation, the flag \verb|-O3| was used to achieve high optimization and speed. That also means that, without using the sorted data in some way, the compiler would deem the result unimportant and skip the sorting altogether. That is why after each sort, to generate a side-effect, the set is checked for two properties: That it is sorted, and that it is a permutation of the previously generated set. The first can easily be done by checking for each value that it is not greater than the value before it.
\paragraph*{Permutation Check} The permutation check is done probabilistically: At design time, a (preferably large) prime number $p$ is chosen. \\
Before sorting, $v = \prod_{i = 1}^{n} (z - a_i) \mod p$ is calculated for a number $z$ and values $a = \{a_1, \ldots, a_n\}$. \\
To check the permutation after sorting and obtaining $a' = \{a'_1, \ldots, a'_n\}$, $w = \prod_{i = 1}^{n} (z - a'_i) \mod p$ is calculated. If $v \neq w$, $a'$ cannot be a permutation of $a$. If $v = w$, we claim that $a'$ is a permutation of $a$. \\
To minimize the chances of $a'$ not being a permutation of $a$, but $v$ being equal to $w$, $v = 0$ was disallowed in the first step. If $v$ is zero, $z$ is incremented by one and the product calculated again, until $v \neq 0$.
\paragraph*{Benchmarks} The benchmark seen in algorithm \ref{algo:normal} was used for most of the measurements. \\
To reduce the chance of cache misses at the beginning of the measurement, one warmup run of random generation, sorting and sorted checking is done beforehand (lines \ref{algo:normal:warmup:start} to \ref{algo:normal:warmup:end}). The array is then sorted \texttt{numberOfIterations} times and checked for the sorted and permutation properties. After that only the generation of the random numbers and the sorted and permutation checking is measured, to later subtract the time from the previously measured one, resulting in the time needed for the sorting alone. Since this is not deterministic in time, and both measurements are subjects to their own deviation, it can occasionally happen that the second measurement takes longer than the first, even though less work has been done. We get those negative times more often for the sorters with small array sizes, where the sorting itself takes relatively little time compared to the random generation and sorted checking. The negative times show up as outliers in the results. \\
The function \texttt{simulateCheckSorted} checks the permutation like \texttt{checkSorted}, but since randomly generated arrays are rarely ordered, instead of checking for each element if it is smaller than its predecessor, it checks for equality. That should never happen with the random number generator used, and thus run for the same amount of cycles. \\
The function MeasureSorting is called a total of \texttt{numberOfMeasures} times for each \texttt{arraySize} that is sorted. \\
For the measurements shown in section \ref{section:experiments:inrow} the benchmark was slightly modified as can be seen in algorithm \ref{algo:inrow}. Here the goal was to look at cache- and memory-effects by creating an array that does not fit into the CPU's L3-cache, and then filling the cache with something else, in this case the reference array. We then split the original array into many blocks of size \texttt{arraySize} and sort each independently. Because we have to create the whole array at the beginning, we can generate the numbers before and check for correct sorting after measuring, so there is no need to do a second measurement like in the first benchmark (lines \ref{algo:normal:random:start} to \ref{algo:normal:random:end} in algorithm \ref{algo:normal}). \\
Here, instead of giving a \texttt{numberOfIterations} parameter to indicate how often the sorting is to be executed, we provide a \texttt{numberOfArrays} value that says how many arrays of size \texttt{arraySize} are to be created contiguously. This parameter is chosen for each \texttt{arraySize} in a way that $\mathtt{numberOfArrays} ~\times~ \mathtt{arraySize}$ does not fit into the L3 cache of the machine the measurement is performed on.
\begin{algorithm}[!p]
\caption{MeasureSorting(\texttt{arraySize}, \texttt{numberOfIterations}, \texttt{seed})} \label{algo:normal}
\ForEach{sorter}{
setSeed(seed) \\
arr $\leftarrow$ makeArray(arraySize) \\
numberOfBadSorts $\leftarrow$ 0 \\
arr $\leftarrow$ generateRandomArray() \label{algo:normal:warmup:start} \\
sorter(arr) \\
checkSorted(arr) // create side-effect \label{algo:normal:warmup:end}
startMeasuring() \\
\For{i $\leftarrow$ 0 \KwTo numberOfIterations}{
arr $\leftarrow$ generateRandomArray() \\
sorter(arr) \\
checkSorted(arr) // create side-effect
}
stopMeasuring() \\
outputResult() \\
setSeed(seed) \label{algo:normal:random:start} \\
startMeasuring() \\
\For{i $\leftarrow$ 0 \KwTo numberOfIterations}{
arr $\leftarrow$ generateRandomArray() \\
simulateCheckSorted(arr) // create side-effect
}
stopMeasuring() \\
outputResult() \label{algo:normal:random:end} \\
}
\end{algorithm}
\begin{algorithm}[!p]
\caption{MeasureSortingInRow(\texttt{arraySize}, \texttt{numberOfArrays}, \texttt{seed})} \label{algo:inrow}
\ForEach{sorter}{
SetSeed(seed) \\
arr $\leftarrow$ makeArray(arraySize $\times$ numberOfArrays) \\
arr $\leftarrow$ GenerateRandomArray() \\
compareArr $\leftarrow$ makeArray(arraySize $\times$ numberOfArrays) \\
compareArr $\leftarrow$ CopyArray(arr) \\
\ForEach{currentArr \KwForIn compareArr \KwOfSize arraySize}{
sort(currentArray, arraySize) //sort reference array
}
//warmup on single array of size \textit{arraySize} like in algorithm \ref{algo:normal}, lines \ref{algo:normal:warmup:start} to \ref{algo:normal:warmup:end}
StartMeasuring() \\
\ForEach{currentArr \KwForIn arr \KwOfSize arraySize}{
sorter(currentArray, arraySize)
}
StopMeasuring() \\
CheckArraysForEquality(arr, compareArr) //check correct sorting, create side-effect
OutputResult() \\
}
\end{algorithm}
\subsection{Sorting One Set of 2-16 items} \label{section:experiments:normal}
The benchmark from algorithm \ref{algo:normal} was used with parameters
\begin{itemize}
\item $\mathtt{numberOfIterations} = 100$
\item $\mathtt{numberOfMeasures} = 500$
\item $\mathtt{arraySize} \in \{2, \ldots, 16\}$.
\end{itemize}
The results seen in tables \ref{table:normalsort:avg:A}, \ref{table:normalsort:avg:B} and \ref{table:normalsort:avg:C} contain the name of the sorter and the average number of cycles per iteration, over the total of all measurements, for machines A, B and C. The algorithm that performed best in a column is marked in bold font, and for each column the value relative to the best in that column was calculated. For each row the geometric mean is calculated over the relative values and from that the rank is determined. \\
Table \ref{table:normalsort:avg:all} contains the geometric mean and rank taking the results from all three machines into consideration. \\
Here it becomes visible that the implementations that have conditional branches and those that do not are clearly separated by rank, the former occupy the lower share of the ranks, while the latter get all the higher ranks. We see that the claim from section \ref{section:implementation-conditionalswap} for the \verb|4CS| conditional swap is true for machines A and B, but not for machine C. We also see in table \ref{table:normalsort:avg:all} that the first three ranks have the same geometric mean, so the Bose Nelson networks can compete with the optimized networks that have fewer comparators due to their locality. \\
The boxplots for array size 8 are given for each machine in figures \ref{plot:normal:8:A}, \ref{plot:normal:8:B} and \ref{plot:normal:8:C}, showing that these higher-ranked implementations are not only faster on average, but that their distribution is almost entirely faster than any of the insertion sort implementations, together with a lower variance. To improve readability, the variants \verb|JXc|, \verb|6Cm| and \verb|QMa| are omitted. Also one outlier was removed from dataset of machine B for the \texttt{'N BoNeL -N KR Cla'} sorter with value $-42.6$ so that the plot has a scale similar to those of the other two machines, to improve comparability. The result set for machine A contains a lot of outliers that we did not want to exclude. To be able to compare it easily with the other two plots we added an additional axis at the top that shows the CPU cycles per iteration as percentages where the average of the best insertion sort is 100\%. \\
To see a trend in increasing array size, we chose a few Conditional Swap implementations that do best for more than one network and array size on all machines. Their average sorting times can be seen in figures \ref{plot:normal:lineplot:A}, \ref{plot:normal:lineplot:B} and \ref{plot:normal:lineplot:C}. For visibility reasons, we omitted the Bose Nelson Parameter networks in these plot. What we already saw from the tables is here visible as well, the \verb|4Cm| and \verb|4CS| implementations have good performance and are almost always faster on average than insertion sort (apart from \verb|arraySize = 2| on machine A). \\
These results indicate that there is potential in using sorting networks, showing an improvement of 32\% of the best network over the best insertion sort, on average, for any array size. Problems with this way of measurement are that the same space in memory is sorted over and over again, which is rarely a use case when sorting a base case. Because of this, the measurements probably reflect unrealistic conditions regarding cache accesses and cache misses. To get a bit closer to actual base case sorting, the next section has a different approach to not sort the same space in memory twice.
\begin{sidewaystable}[!p]
\begin{scriptsize}
\begin{tabular}{l | r @{~~} r | r@{~~}r@{~~}r@{~~}r@{~~}r@{~~}r@{~~}r@{~~}r@{~~}r@{~~}r@{~~}r@{~~}r@{~~}r@{~~}r@{~~}r@{~~}r|}
& \multicolumn{2}{c|}{Overall} & \multicolumn{15}{c}{Array Size} \\
& Rank & GeoM & 2&3&4&5&6&7&8&9&10&11&12&13&14&15&16\\ \hline \verb+I -N KR POp+ & 22 & 1.85 & 15.21&37.82&80.52&124.17&166.14&204.37&250.08&282.97&323.87&369.31&417.57&437.93&509.37&520.20&579.44\\ \verb+I -N KR STL+ & 26 & 1.92 & 13.82&39.99&83.75&128.44&178.50&213.03&257.62&287.15&346.35&382.08&434.47&455.29&532.36&554.80&614.56\\ \verb+I -N KR Def+ & 33 & 2.07 & 17.23&40.53&84.80&132.12&178.09&220.69&277.29&311.27&378.66&412.51&475.48&499.27&595.68&614.74&693.90\\ \verb+I -N KR AIF+ & 36 & 2.21 & 16.55&50.76&90.56&148.37&202.86&252.22&307.29&342.66&400.98&442.48&485.63&518.62&593.07&609.29&672.98
\\ \verb+N Best -N KR 4CS+ & 1 & 1.07 & 11.59&\textbf{24.11}&\textbf{34.35}&66.58&82.54&96.95&125.56&134.92&\textbf{183.73}&\textbf{218.14}&\textbf{254.95}&278.53&356.75&353.24&395.58\\ \verb+N Best -N KR 4Cm+ & 5 & 1.12 & 16.33&24.21&38.05&54.80&74.74&\textbf{85.23}&127.40&141.99&201.85&238.39&279.15&301.47&375.39&399.06&450.54\\ \verb+N Best -N KR Cla+ & 8 & 1.23 & 8.91&31.22&40.41&86.73&110.04&144.05&163.77&188.67&220.54&240.00&298.76&285.42&347.36&\textbf{349.35}&400.98\\ \verb+N Best -N KR CPr+ & 10 & 1.27 & \textbf{8.13}&32.20&46.88&87.99&112.46&146.10&164.91&190.61&208.00&256.03&296.58&301.14&382.17&381.71&438.18\\ \verb+N Best -N KR 6Cm+ & 13 & 1.37 & 17.20&25.57&46.86&64.68&96.36&107.56&146.34&176.34&259.56&289.24&339.02&392.64&490.05&502.11&585.95\\ \verb+N Best -N KR Def+ & 21 & 1.84 & 20.09&37.08&73.94&113.14&144.92&179.09&248.33&268.15&302.44&338.76&417.69&429.27&555.59&552.84&712.22\\ \verb+N Best -N KR Tie+ & 25 & 1.90 & 20.47&38.06&63.58&98.92&139.21&182.82&238.59&271.77&316.34&369.96&477.94&519.14&597.69&639.07&753.42\\ \verb+N Best -N KR JXc+ & 32 & 2.04 & 18.44&36.50&68.67&113.06&167.99&207.12&264.82&293.56&347.16&409.67&506.11&522.21&680.63&711.20&791.41\\ \verb+N Best -N KR QMa+ & 37 & 2.60 & 17.72&44.69&96.02&149.03&207.73&252.95&341.19&397.81&438.98&573.64&681.13&700.09&832.27&910.77&1057.45
\\ \verb+N BoNeL -N KR 4CS+ & 2 & 1.08 & 11.45&24.99&35.82&67.94&82.41&98.05&128.16&\textbf{132.68}&186.46&224.66&262.69&\textbf{275.88}&\textbf{344.76}&352.46&\textbf{387.59}\\ \verb+N BoNeL -N KR 4Cm+ & 3 & 1.11 & 13.53&25.18&38.33&55.62&76.06&86.06&132.02&142.06&193.99&232.70&284.86&302.12&383.21&386.96&423.71\\ \verb+N BoNeL -N KR 6Cm+ & 15 & 1.42 & 15.96&28.18&45.90&73.18&90.18&115.24&148.93&214.27&278.32&298.28&365.05&414.27&493.26&508.85&560.16\\ \verb+N BoNeL -N KR Cla+ & 16 & 1.42 & 8.72&31.04&40.71&82.99&112.46&143.75&163.76&239.80&270.36&325.30&354.56&403.83&452.67&493.90&550.58\\ \verb+N BoNeL -N KR CPr+ & 17 & 1.44 & 9.03&33.01&47.21&88.37&113.62&147.21&166.12&238.67&265.87&321.12&347.10&401.81&446.13&482.28&536.51\\ \verb+N BoNeL -N KR Tie+ & 27 & 1.93 & 20.87&40.57&64.35&99.65&137.68&173.18&231.53&265.85&343.47&383.88&472.81&513.56&636.85&676.79&782.22\\ \verb+N BoNeL -N KR Def+ & 28 & 1.94 & 20.11&40.52&78.60&102.58&139.42&170.73&237.18&265.27&366.93&372.58&478.09&481.87&637.88&636.35&764.29\\ \verb+N BoNeL -N KR JXc+ & 31 & 2.04 & 18.95&36.18&68.86&108.92&160.18&196.61&256.54&314.83&368.23&427.12&504.82&570.41&642.64&662.73&789.99\\ \verb+N BoNeL -N KR QMa+ & 38 & 2.67 & 18.16&45.95&93.38&143.03&196.39&241.82&326.07&406.05&514.55&578.58&685.08&776.92&912.15&998.34&1163.99
\\ \verb+N BoNeM -N KR 4Cm+ & 7 & 1.22 & 16.08&27.11&38.84&54.86&82.51&94.36&119.90&214.28&252.78&251.85&284.48&318.74&415.51&394.31&500.68\\ \verb+N BoNeM -N KR 4CS+ & 11 & 1.28 & 11.79&24.70&43.96&73.23&83.76&130.14&\textbf{115.48}&204.19&273.27&286.72&281.43&314.56&487.85&464.53&518.94\\ \verb+N BoNeM -N KR 6Cm+ & 18 & 1.51 & 15.65&27.98&52.71&93.07&85.76&113.21&134.35&223.11&340.69&340.96&393.85&434.99&575.98&499.30&630.78\\ \verb+N BoNeM -N KR Cla+ & 19 & 1.63 & 13.05&32.46&54.07&90.32&116.27&149.86&175.80&278.17&314.13&348.62&395.85&448.91&559.24&566.87&639.09\\ \verb+N BoNeM -N KR CPr+ & 20 & 1.67 & 15.10&33.91&46.42&103.89&120.13&153.62&203.55&287.35&322.22&347.58&374.67&423.75&521.46&565.28&726.59\\ \verb+N BoNeM -N KR Def+ & 29 & 1.94 & 18.38&39.34&75.91&113.68&157.87&199.64&237.50&259.60&352.19&369.31&455.58&479.48&596.34&633.54&748.67\\ \verb+N BoNeM -N KR Tie+ & 30 & 2.01 & 21.34&38.64&62.66&96.05&135.19&172.29&229.37&265.31&368.96&452.05&554.50&544.95&769.84&767.68&788.78\\ \verb+N BoNeM -N KR JXc+ & 35 & 2.18 & 19.47&38.29&70.01&108.73&147.29&184.76&252.61&358.55&454.97&478.92&594.31&589.14&769.05&751.62&924.53\\ \verb+N BoNeM -N KR QMa+ & 39 & 2.71 & 24.28&54.14&100.79&136.42&204.57&251.56&325.24&403.05&480.82&547.65&651.81&739.13&864.35&966.34&1092.30
\\ \verb+N BoNeP -N KR 4CS+ & 4 & 1.11 & 11.41&24.83&35.67&61.12&84.51&96.59&130.30&159.81&199.35&230.24&271.39&302.50&363.86&388.34&422.56\\ \verb+N BoNeP -N KR 4Cm+ & 6 & 1.14 & 13.00&25.08&38.14&\textbf{53.95}&\textbf{74.04}&94.47&119.49&151.09&209.73&237.70&291.55&334.29&398.82&426.06&468.23\\ \verb+N BoNeP -N KR Cla+ & 9 & 1.25 & 8.81&31.30&41.47&80.30&100.54&130.65&147.00&211.21&233.02&265.58&286.03&320.45&363.05&385.07&438.35\\ \verb+N BoNeP -N KR CPr+ & 12 & 1.28 & 9.68&33.04&47.46&82.36&94.76&116.33&147.20&212.97&223.40&267.75&289.20&337.12&401.57&417.11&468.35\\ \verb+N BoNeP -N KR 6Cm+ & 14 & 1.41 & 15.17&27.86&45.69&66.43&86.38&102.07&151.31&207.81&275.71&317.69&375.96&418.47&505.54&545.92&617.54\\ \verb+N BoNeP -N KR Tie+ & 23 & 1.88 & 20.56&37.00&64.49&97.05&131.03&171.37&223.33&270.21&324.68&395.20&462.10&534.33&590.53&642.31&744.88\\ \verb+N BoNeP -N KR Def+ & 24 & 1.88 & 19.80&41.86&74.79&111.41&144.15&174.45&237.91&265.89&325.01&350.58&420.83&473.65&563.51&599.15&727.96\\ \verb+N BoNeP -N KR JXc+ & 34 & 2.08 & 20.22&36.90&69.50&109.37&150.28&188.40&251.74&293.71&379.19&439.07&525.51&585.25&719.42&744.81&844.24\\ \verb+N BoNeP -N KR QMa+ & 40 & 2.79 & 24.23&52.09&99.01&148.85&192.75&263.79&338.27&395.25&517.71&584.97&677.76&794.20&938.78&1006.31&1166.80\\ \end{tabular}
\end{scriptsize}
\caption{Average number of CPU cycles per iteration of single array sorting on machine A} \label{table:normalsort:avg:A}
\end{sidewaystable}
\begin{sidewaystable}[!p]
\begin{scriptsize}
\begin{tabular}{l | r @{~~} r | r@{~~}r@{~~}r@{~~}r@{~~}r@{~~}r@{~~}r@{~~}r@{~~}r@{~~}r@{~~}r@{~~}r@{~~}r@{~~}r@{~~}r@{~~}r|}
& \multicolumn{2}{c|}{Overall} & \multicolumn{15}{c}{Array Size} \\
& Rank & GeoM & 2&3&4&5&6&7&8&9&10&11&12&13&14&15&16\\ \hline \verb+I -N KR POp+ & 25 & 1.84 & 12.56&36.78&73.37&111.91&151.52&183.05&221.86&263.07&302.35&353.36&399.83&439.54&475.60&508.11&550.21\\ \verb+I -N KR STL+ & 29 & 1.93 & 10.97&37.26&78.04&122.75&161.05&201.27&247.30&280.78&321.94&376.15&420.50&461.52&493.07&519.96&559.69\\ \verb+I -N KR Def+ & 32 & 1.98 & 14.03&40.14&76.93&117.68&157.59&198.50&242.60&280.73&322.72&375.54&422.67&465.00&511.51&558.47&599.95\\ \verb+I -N KR AIF+ & 36 & 2.32 & 14.98&54.78&92.20&135.59&195.79&245.14&292.73&327.82&380.85&438.38&489.94&538.38&573.31&612.57&656.83
\\ \verb+N Best -N KR 4CS+ & 1 & 1.06 & 8.00&22.10&35.70&60.36&71.61&93.40&112.30&144.17&\textbf{171.34}&\textbf{214.03}&238.42&285.88&316.71&344.10&364.40\\ \verb+N Best -N KR 4Cm+ & 3 & 1.10 & 8.75&\textbf{20.34}&\textbf{34.47}&\textbf{54.70}&70.88&90.27&115.48&\textbf{137.92}&189.67&222.14&262.03&307.09&353.77&391.72&432.70\\ \verb+N Best -N KR CPr+ & 8 & 1.19 & \textbf{5.23}&25.15&38.71&71.13&92.24&124.12&145.15&188.31&194.53&234.72&266.15&307.63&343.99&372.53&409.52\\ \verb+N Best -N KR Cla+ & 9 & 1.20 & 6.86&28.30&38.70&75.69&92.84&129.24&146.96&190.31&196.30&233.05&261.56&\textbf{278.43}&\textbf{306.94}&\textbf{335.57}&363.08\\ \verb+N Best -N KR 6Cm+ & 14 & 1.36 & 9.52&24.23&39.97&69.73&88.59&111.52&136.40&174.35&228.98&283.96&321.57&402.13&458.39&499.02&553.25\\ \verb+N Best -N KR Def+ & 21 & 1.78 & 16.38&33.75&64.57&95.65&125.05&164.26&204.86&246.37&265.34&336.21&399.28&428.57&508.38&550.37&631.51\\ \verb+N Best -N KR Tie+ & 22 & 1.82 & 16.04&33.74&57.37&88.86&114.24&165.85&203.09&250.32&280.61&352.47&435.29&508.75&539.74&598.76&685.05\\ \verb+N Best -N KR JXc+ & 33 & 2.00 & 16.52&33.44&63.44&98.88&138.36&179.19&218.44&282.01&309.11&385.21&480.67&541.52&626.83&665.20&766.23\\ \verb+N Best -N KR QMa+ & 37 & 2.53 & 14.98&42.43&86.31&140.41&184.73&235.35&301.62&364.06&383.06&539.44&623.15&659.71&744.50&846.43&932.15
\\ \verb+N BoNeL -N KR 4CS+ & 2 & 1.08 & 8.79&23.49&37.04&62.21&72.40&93.37&112.70&142.31&173.74&218.67&\textbf{237.75}&281.41&309.13&342.61&\textbf{349.19}\\ \verb+N BoNeL -N KR 4Cm+ & 4 & 1.11 & 9.06&21.77&35.45&55.14&72.10&90.98&116.37&142.51&181.85&223.36&263.63&319.19&355.23&380.37&404.78\\ \verb+N BoNeL -N KR CPr+ & 13 & 1.34 & 6.54&27.47&39.79&72.94&93.51&125.24&145.99&215.92&243.90&285.57&315.19&374.44&404.65&434.14&454.70\\ \verb+N BoNeL -N KR Cla+ & 15 & 1.39 & 7.30&29.84&39.35&76.71&92.91&130.63&147.23&226.02&247.07&293.28&324.55&393.59&415.47&460.03&477.18\\ \verb+N BoNeL -N KR 6Cm+ & 16 & 1.43 & 10.65&25.90&40.56&70.90&87.99&113.19&137.92&217.54&260.44&292.33&353.98&427.97&475.74&503.83&529.54\\ \verb+N BoNeL -N KR Tie+ & 26 & 1.87 & 17.59&33.75&57.62&87.79&115.16&157.70&198.00&245.89&307.31&367.01&444.25&506.00&579.13&658.81&722.07\\ \verb+N BoNeL -N KR Def+ & 27 & 1.89 & 16.64&36.37&67.23&90.58&120.61&158.18&200.27&250.83&320.55&355.67&453.90&487.58&575.63&625.45&688.18\\ \verb+N BoNeL -N KR JXc+ & 31 & 1.97 & 16.28&33.15&62.43&91.86&133.58&171.44&214.22&279.84&338.71&398.70&456.86&536.44&615.27&660.74&748.77\\ \verb+N BoNeL -N KR QMa+ & 38 & 2.63 & 14.51&43.23&82.70&134.53&178.88&225.65&290.45&391.15&462.34&547.55&646.34&742.97&835.85&940.49&1057.89
\\ \verb+N BoNeM -N KR 4Cm+ & 7 & 1.17 & 9.86&21.49&34.89&55.43&72.68&\textbf{89.56}&\textbf{106.83}&212.18&224.25&233.20&260.33&325.45&378.21&405.97&438.89\\ \verb+N BoNeM -N KR 4CS+ & 12 & 1.28 & 8.90&23.66&41.74&72.84&72.88&129.91&108.65&206.26&248.00&262.31&262.53&321.93&431.00&445.25&452.85\\ \verb+N BoNeM -N KR 6Cm+ & 18 & 1.56 & 11.72&25.59&48.66&95.36&84.11&110.15&133.02&234.05&318.24&343.21&382.43&447.82&527.23&528.53&599.74\\ \verb+N BoNeM -N KR CPr+ & 19 & 1.58 & 11.92&29.69&39.66&93.65&94.49&144.63&168.80&258.49&301.06&331.26&341.15&432.34&444.36&531.04&566.47\\ \verb+N BoNeM -N KR Cla+ & 20 & 1.60 & 12.11&33.08&47.86&88.18&94.52&149.64&147.27&249.49&280.22&330.82&350.09&443.00&490.15&546.82&537.15\\ \verb+N BoNeM -N KR Def+ & 28 & 1.92 & 15.74&35.40&67.25&105.51&137.72&179.21&211.66&250.63&320.06&350.38&443.83&481.71&549.25&621.84&689.81\\ \verb+N BoNeM -N KR Tie+ & 30 & 1.96 & 16.95&33.72&56.67&86.72&115.31&154.85&198.75&245.12&339.09&443.65&526.52&553.74&695.92&741.39&723.53\\ \verb+N BoNeM -N KR JXc+ & 35 & 2.14 & 16.46&35.71&60.93&95.72&123.50&167.92&218.13&359.78&433.33&449.88&539.73&577.80&689.96&729.62&862.97\\ \verb+N BoNeM -N KR QMa+ & 39 & 2.66 & 20.64&48.64&90.61&126.40&185.62&229.86&300.87&373.85&431.10&518.01&615.57&735.18&780.40&895.37&951.65
\\ \verb+N BoNeP -N KR 4CS+ & 5 & 1.12 & 8.46&23.34&37.09&57.80&73.68&92.70&113.69&159.60&190.06&219.94&252.55&307.79&328.41&377.27&392.42\\ \verb+N BoNeP -N KR 4Cm+ & 6 & 1.14 & 8.95&22.69&35.57&55.35&\textbf{69.60}&90.41&111.13&149.56&197.46&225.24&271.98&335.76&362.40&414.60&447.03\\ \verb+N BoNeP -N KR CPr+ & 10 & 1.22 & 5.97&26.99&39.44&67.44&80.12&111.68&129.78&190.36&211.08&251.35&278.01&336.65&370.57&405.37&436.74\\ \verb+N BoNeP -N KR Cla+ & 11 & 1.23 & 7.10&29.97&39.96&73.81&84.93&116.40&131.89&200.48&213.50&243.83&265.25&317.23&337.10&377.03&402.82\\ \verb+N BoNeP -N KR 6Cm+ & 17 & 1.44 & 10.32&25.62&40.47&69.19&84.75&110.24&134.78&210.52&252.52&313.76&368.79&434.89&481.18&543.99&580.31\\ \verb+N BoNeP -N KR Def+ & 23 & 1.83 & 16.91&38.39&65.18&97.07&120.63&156.82&195.89&242.43&288.94&336.14&403.45&458.41&522.17&589.12&656.89\\ \verb+N BoNeP -N KR Tie+ & 24 & 1.84 & 17.37&32.93&57.10&86.35&114.55&156.12&193.53&248.25&299.72&369.58&429.28&509.05&566.42&628.05&716.46\\ \verb+N BoNeP -N KR JXc+ & 34 & 2.04 & 16.51&32.47&60.17&95.97&137.22&171.44&221.18&283.43&339.45&422.44&492.87&576.73&655.46&723.94&784.93\\ \verb+N BoNeP -N KR QMa+ & 40 & 2.75 & 20.44&48.92&87.15&142.84&174.98&248.01&298.49&385.09&462.66&544.00&626.85&763.33&847.00&940.79&1027.04\\ \end{tabular}
\end{scriptsize}
\caption{Average number of CPU cycles per iteration of single array sorting on machine B} \label{table:normalsort:avg:B}
\end{sidewaystable}
\begin{sidewaystable}[!p]
\begin{scriptsize}
\begin{tabular}{l | r @{~~} r | r@{~~}r@{~~}r@{~~}r@{~~}r@{~~}r@{~~}r@{~~}r@{~~}r@{~~}r@{~~}r@{~~}r@{~~}r@{~~}r@{~~}r@{~~}r|}
& \multicolumn{2}{c|}{Overall} & \multicolumn{15}{c}{Array Size} \\
& Rank & GeoM & 2&3&4&5&6&7&8&9&10&11&12&13&14&15&16\\ \hline \verb+I -N KR POp+ & 21 & 2.45 & 11.39&36.05&77.28&128.96&181.15&227.59&265.00&298.11&335.56&370.76&404.10&450.91&488.88&529.10&582.69\\ \verb+I -N KR Def+ & 33 & 2.90 & 15.33&48.24&95.20&148.27&195.18&249.58&305.80&347.41&395.45&434.98&479.17&524.59&575.01&628.20&688.63\\ \verb+I -N KR STL+ & 35 & 2.97 & 16.64&52.56&102.10&154.79&194.80&248.41&306.96&346.42&393.33&436.37&486.46&536.80&580.03&634.94&702.84\\ \verb+I -N KR AIF+ & 36 & 3.23 & 16.77&56.45&109.56&160.66&215.97&268.93&332.20&379.01&432.62&479.89&540.53&601.36&643.93&708.10&776.50
\\ \verb+N Best -N KR 4Cm+ & 2 & 1.11 & 7.31&\textbf{16.27}&29.45&47.03&70.03&82.22&\textbf{96.48}&126.39&\textbf{143.46}&169.87&176.58&233.61&268.71&310.79&345.07\\ \verb+N Best -N KR 4CS+ & 5 & 1.19 & 7.07&18.10&34.24&55.89&67.86&93.54&110.54&137.94&146.50&175.16&195.19&239.20&274.19&315.00&343.90\\ \verb+N Best -N KR 6Cm+ & 8 & 1.25 & 6.85&17.84&33.82&62.44&79.27&97.94&116.81&142.67&165.77&184.17&206.94&256.17&295.75&316.26&352.90\\ \verb+N Best -N KR CPr+ & 11 & 1.43 & \textbf{3.09}&30.86&36.06&73.57&89.71&124.63&140.12&190.47&199.37&251.14&265.94&292.43&335.35&364.68&389.62\\ \verb+N Best -N KR Cla+ & 12 & 1.45 & 4.73&26.60&37.62&75.06&87.90&116.55&142.33&194.22&196.00&251.06&260.73&294.26&326.31&356.21&384.08\\ \verb+N Best -N KR Tie+ & 26 & 2.75 & 15.76&38.05&73.01&109.62&144.16&211.29&266.32&314.66&362.63&424.85&500.30&595.58&687.34&733.87&897.51\\ \verb+N Best -N KR JXc+ & 27 & 2.79 & 15.56&42.27&75.44&114.30&147.68&205.71&254.78&309.67&375.45&433.90&491.17&605.14&703.28&774.28&901.10\\ \verb+N Best -N KR Def+ & 30 & 2.85 & 15.40&37.48&80.60&126.48&158.41&222.60&294.91&346.24&358.13&456.70&514.14&561.78&694.25&758.85&846.88\\ \verb+N Best -N KR QMa+ & 38 & 3.79 & 11.05&54.53&110.76&180.78&239.95&310.40&389.88&468.14&494.45&665.40&741.91&783.56&920.78&1000.01&1110.08
\\ \verb+N BoNeL -N KR 4Cm+ & 1 & 1.07 & 6.05&16.77&29.70&47.32&70.14&83.57&96.87&\textbf{125.01}&149.19&164.25&\textbf{174.84}&\textbf{216.16}&\textbf{244.09}&\textbf{266.22}&\textbf{288.18}\\ \verb+N BoNeL -N KR 4CS+ & 4 & 1.15 & 6.51&18.51&33.64&57.84&69.08&90.70&112.78&134.28&153.70&182.76&185.91&234.77&256.08&272.71&299.99\\ \verb+N BoNeL -N KR 6Cm+ & 9 & 1.31 & 8.37&20.04&35.32&67.72&77.64&99.97&120.89&156.16&177.52&199.67&217.96&260.26&282.87&316.31&348.27\\ \verb+N BoNeL -N KR CPr+ & 17 & 1.58 & 3.27&32.08&37.08&73.85&91.57&124.20&141.78&205.08&251.50&290.99&312.06&369.32&405.85&437.23&462.54\\ \verb+N BoNeL -N KR Cla+ & 18 & 1.59 & 4.48&27.21&37.60&76.08&89.28&117.83&141.78&200.79&240.48&289.48&306.41&371.91&403.27&443.11&457.93\\ \verb+N BoNeL -N KR JXc+ & 22 & 2.56 & 15.24&35.21&66.96&100.40&131.09&189.59&235.21&304.23&364.99&418.31&472.06&536.86&655.17&704.77&777.83\\ \verb+N BoNeL -N KR Tie+ & 24 & 2.69 & 15.15&37.39&64.10&112.33&139.39&191.19&240.18&314.21&375.78&435.35&500.68&610.78&682.65&765.66&882.64\\ \verb+N BoNeL -N KR Def+ & 31 & 2.89 & 14.65&39.25&78.65&115.49&145.62&206.33&273.29&347.40&409.92&460.74&560.31&611.72&725.22&827.99&932.90\\ \verb+N BoNeL -N KR QMa+ & 39 & 3.81 & 10.33&50.08&97.42&168.02&223.17&286.64&359.03&484.01&578.52&663.41&766.77&906.75&1002.35&1112.73&1223.32
\\ \verb+N BoNeM -N KR 4Cm+ & 7 & 1.23 & 7.29&17.72&\textbf{29.35}&\textbf{45.03}&\textbf{61.88}&82.52&98.43&202.36&212.83&200.65&243.93&238.92&308.91&320.33&364.00\\ \verb+N BoNeM -N KR 6Cm+ & 15 & 1.50 & 7.00&21.07&40.85&84.36&80.01&99.31&117.10&197.20&266.33&235.02&258.16&323.18&415.37&352.34&399.50\\ \verb+N BoNeM -N KR 4CS+ & 16 & 1.51 & 6.96&17.60&40.66&80.80&70.93&145.19&112.51&208.82&246.55&256.60&238.25&282.13&411.16&416.42&433.64\\ \verb+N BoNeM -N KR CPr+ & 19 & 1.95 & 8.86&33.28&38.15&94.93&98.62&147.79&186.70&251.71&298.36&323.52&341.95&417.32&448.73&548.35&592.07\\ \verb+N BoNeM -N KR Cla+ & 20 & 1.96 & 13.01&32.55&47.51&90.72&93.87&154.40&146.24&232.55&276.11&316.92&340.40&421.70&482.01&533.05&533.90\\ \verb+N BoNeM -N KR JXc+ & 28 & 2.82 & 16.39&37.79&66.83&110.69&134.03&194.48&243.91&385.89&444.32&496.26&538.42&586.56&705.07&767.01&877.98\\ \verb+N BoNeM -N KR Tie+ & 29 & 2.84 & 17.52&41.29&65.86&105.65&145.14&198.25&259.15&322.99&398.23&468.78&547.83&627.10&796.34&798.66&842.88\\ \verb+N BoNeM -N KR Def+ & 34 & 2.95 & 11.71&37.08&84.63&136.06&176.03&231.43&289.16&351.31&413.91&456.64&542.57&641.53&720.54&826.37&908.10\\ \verb+N BoNeM -N KR QMa+ & 37 & 3.77 & 17.63&58.32&107.62&142.40&237.64&264.96&384.33&435.37&500.00&574.65&726.09&825.43&913.20&980.12&1129.61
\\ \verb+N BoNeP -N KR 4Cm+ & 3 & 1.15 & 6.86&17.92&29.84&49.01&61.90&\textbf{79.90}&106.49&129.82&156.20&\textbf{163.84}&201.19&251.63&287.49&319.13&345.12\\ \verb+N BoNeP -N KR 4CS+ & 6 & 1.21 & 6.63&20.99&33.47&50.87&69.91&89.23&114.03&132.59&155.05&197.48&204.60&262.52&285.92&321.02&342.46\\ \verb+N BoNeP -N KR 6Cm+ & 10 & 1.31 & 7.80&20.58&34.62&60.07&83.27&101.94&126.02&143.48&174.07&207.35&228.25&262.75&287.39&344.92&355.76\\ \verb+N BoNeP -N KR CPr+ & 13 & 1.48 & 3.24&31.77&37.47&74.46&84.18&123.89&146.83&196.26&213.12&254.61&274.11&323.22&347.05&384.41&414.97\\ \verb+N BoNeP -N KR Cla+ & 14 & 1.48 & 4.77&27.88&38.17&69.83&75.02&116.28&141.25&191.49&221.49&250.44&274.70&325.92&350.47&394.49&424.23\\ \verb+N BoNeP -N KR JXc+ & 23 & 2.69 & 13.90&35.16&65.39&105.11&139.71&197.48&255.87&294.85&364.97&461.81&505.71&605.46&686.42&814.11&919.46\\ \verb+N BoNeP -N KR Tie+ & 25 & 2.72 & 15.69&34.72&68.55&112.03&142.03&209.70&257.04&312.52&367.93&434.85&509.64&578.63&702.64&752.57&892.05\\ \verb+N BoNeP -N KR Def+ & 32 & 2.90 & 14.81&39.78&78.61&127.92&181.05&231.43&274.24&346.11&379.57&453.44&491.92&610.98&708.39&773.08&859.50\\ \verb+N BoNeP -N KR QMa+ & 40 & 3.99 & 19.09&56.49&99.26&180.09&222.01&298.15&371.18&475.00&549.86&660.00&745.95&881.24&994.07&1082.76&1196.13\\ \end{tabular}
\end{scriptsize}
\caption{Average number of CPU cycles per iteration of single array sorting on machine C} \label{table:normalsort:avg:C}
\end{sidewaystable}
\begin{sidewaystable}[!p]
\begin{scriptsize}
\begin{tabular}{l | r @{~~} r | r@{~~}r@{~~}r@{~~}r@{~~}r@{~~}r@{~~}r@{~~}r@{~~}r@{~~}r@{~~}r@{~~}r@{~~}r@{~~}r@{~~}r@{~~}r|}
& \multicolumn{2}{c|}{Overall} & \multicolumn{15}{c}{Array Size} \\
& Rank & GeoM & 2&3&4&5&6&7&8&9&10&11&12&13&14&15&16\\ \hline \verb+I -N KR POp+ & 21 & 1.97 & 13.31&36.84&77.52&121.73&166.39&204.93&245.35&281.53&320.20&364.03&407.29&443.03&491.19&519.29&570.38\\ \verb+I -N KR STL+ & 29 & 2.18 & 13.99&44.13&88.72&136.05&178.19&221.54&271.65&306.46&354.48&399.11&448.19&485.94&536.15&570.51&626.51\\ \verb+I -N KR Def+ & 34 & 2.22 & 15.49&43.53&85.92&133.04&177.05&222.94&275.77&313.38&366.00&407.47&459.20&496.26&561.57&600.20&659.81\\ \verb+I -N KR AIF+ & 36 & 2.48 & 16.19&53.96&98.14&148.52&205.27&255.57&311.14&350.39&404.94&454.16&505.94&553.38&604.04&645.66&703.20
\\ \verb+N Best -N KR 4CS+ & 1 & 1.08 & 9.49&21.14&34.61&61.39&74.12&93.88&116.89&138.90&\textbf{166.86}&\textbf{201.46}&\textbf{229.80}&267.67&318.35&338.59&367.90\\ \verb+N Best -N KR 4Cm+ & 4 & 1.10 & 11.80&\textbf{20.24}&\textbf{33.80}&\textbf{51.24}&71.67&\textbf{85.75}&113.85&\textbf{135.38}&178.33&208.90&238.10&278.08&333.61&367.57&413.45\\ \verb+N Best -N KR Cla+ & 8 & 1.26 & 7.07&28.71&38.97&80.04&96.97&129.58&150.98&191.32&204.88&242.45&273.71&286.36&326.97&347.76&382.99\\ \verb+N Best -N KR CPr+ & 9 & 1.27 & \textbf{5.62}&29.14&40.79&78.30&98.99&133.63&150.52&189.73&200.78&247.67&278.47&300.13&353.26&373.06&412.66\\ \verb+N Best -N KR 6Cm+ & 12 & 1.31 & 11.86&21.84&40.18&65.61&87.83&105.09&133.64&163.41&217.18&252.83&286.92&353.76&417.93&429.41&497.07\\ \verb+N Best -N KR Tie+ & 23 & 2.07 & 17.79&36.53&64.86&99.31&132.75&186.83&236.91&278.75&320.47&382.78&471.00&542.17&608.62&658.26&779.34\\ \verb+N Best -N KR Def+ & 24 & 2.07 & 17.67&36.25&73.23&111.95&143.00&189.50&250.12&287.76&308.47&378.63&443.90&474.71&587.42&626.30&731.54\\ \verb+N Best -N KR JXc+ & 33 & 2.19 & 17.11&37.58&69.36&108.46&151.93&197.17&245.69&295.11&343.64&409.43&492.56&556.92&670.48&716.70&818.76\\ \verb+N Best -N KR QMa+ & 37 & 2.86 & 14.49&47.71&97.71&157.50&211.46&266.79&344.29&410.69&439.47&592.67&682.56&714.70&833.13&919.82&1034.29
\\ \verb+N BoNeL -N KR 4Cm+ & 2 & 1.08 & 9.98&21.36&34.70&51.99&73.39&87.23&116.70&135.64&174.36&205.56&240.54&275.67&330.05&346.83&375.29\\ \verb+N BoNeL -N KR 4CS+ & 3 & 1.08 & 9.11&22.19&35.65&62.92&75.16&94.38&119.00&136.72&173.22&211.53&232.70&\textbf{265.93}&\textbf{304.89}&\textbf{326.46}&\textbf{348.16}\\ \verb+N BoNeL -N KR 6Cm+ & 14 & 1.37 & 12.10&24.61&41.19&70.46&84.83&108.23&135.38&196.49&245.98&259.64&310.46&366.85&429.46&437.78&485.76\\ \verb+N BoNeL -N KR Cla+ & 16 & 1.43 & 6.67&29.23&39.13&79.28&99.19&130.78&150.76&222.19&252.77&303.20&328.71&389.41&422.62&465.09&493.86\\ \verb+N BoNeL -N KR CPr+ & 17 & 1.43 & 6.24&30.74&41.61&79.86&100.76&134.10&151.81&219.75&254.20&299.55&327.56&382.07&421.80&452.26&485.46\\ \verb+N BoNeL -N KR Tie+ & 25 & 2.08 & 18.14&37.39&61.98&100.26&130.55&174.16&223.09&275.50&342.71&395.67&472.43&544.67&633.90&701.33&794.77\\ \verb+N BoNeL -N KR JXc+ & 27 & 2.12 & 17.03&34.89&65.91&100.48&142.81&186.18&235.30&299.63&356.66&414.47&478.47&548.58&637.53&677.42&771.92\\ \verb+N BoNeL -N KR Def+ & 28 & 2.15 & 17.48&38.62&74.44&103.11&135.11&178.95&237.15&288.96&365.87&397.72&498.02&529.04&646.10&699.94&796.40\\ \verb+N BoNeL -N KR QMa+ & 38 & 2.92 & 14.43&46.63&90.93&149.35&199.66&251.64&325.05&427.89&518.88&596.66&699.04&810.17&917.14&1017.55&1148.28
\\ \verb+N BoNeM -N KR 4Cm+ & 7 & 1.19 & 11.67&22.36&34.35&51.89&72.19&89.01&\textbf{109.41}&208.81&231.50&227.94&263.31&294.41&367.71&374.03&431.20\\ \verb+N BoNeM -N KR 4CS+ & 13 & 1.32 & 9.54&21.89&42.09&76.18&77.20&136.41&112.08&206.68&256.74&270.60&260.43&306.41&441.37&442.21&467.06\\ \verb+N BoNeM -N KR 6Cm+ & 18 & 1.49 & 11.67&24.90&47.78&90.53&83.76&107.69&128.11&219.37&311.29&297.59&345.98&401.83&503.23&457.17&541.58\\ \verb+N BoNeM -N KR Cla+ & 19 & 1.68 & 12.87&32.68&50.31&89.81&103.25&151.29&157.12&253.04&290.36&332.44&361.42&437.62&510.11&548.59&572.66\\ \verb+N BoNeM -N KR CPr+ & 20 & 1.69 & 12.25&32.26&42.04&98.50&105.12&149.09&186.85&265.49&308.33&334.27&355.31&424.85&476.11&548.73&628.61\\ \verb+N BoNeM -N KR Def+ & 30 & 2.18 & 15.33&37.26&76.07&119.07&157.49&203.52&246.69&289.45&362.59&392.89&482.09&536.33&621.77&697.33&783.48\\ \verb+N BoNeM -N KR Tie+ & 31 & 2.18 & 18.97&38.05&61.85&96.35&131.68&175.20&229.28&278.30&368.86&455.01&542.54&576.28&753.78&769.33&784.82\\ \verb+N BoNeM -N KR JXc+ & 35 & 2.30 & 17.73&37.10&65.93&104.73&135.12&182.40&237.82&368.48&444.19&474.94&558.74&584.25&722.61&749.45&889.37\\ \verb+N BoNeM -N KR QMa+ & 39 & 2.92 & 21.00&53.83&99.47&135.04&209.36&248.78&337.02&403.97&470.19&546.77&664.62&768.21&853.46&946.93&1058.06
\\ \verb+N BoNeP -N KR 4Cm+ & 5 & 1.12 & 10.10&21.80&34.53&52.94&\textbf{69.20}&88.78&112.80&143.67&188.17&210.52&252.70&303.77&355.74&392.22&419.03\\ \verb+N BoNeP -N KR 4CS+ & 6 & 1.13 & 9.31&23.11&35.51&56.29&77.11&92.99&120.16&151.19&179.63&217.68&244.75&292.50&328.03&362.46&387.20\\ \verb+N BoNeP -N KR Cla+ & 10 & 1.28 & 6.86&29.69&39.90&74.93&87.15&122.21&139.99&201.27&222.60&253.53&275.37&321.67&350.16&385.47&421.30\\ \verb+N BoNeP -N KR CPr+ & 11 & 1.30 & 6.55&30.51&42.04&75.27&86.68&117.72&140.35&199.82&216.17&258.35&280.93&331.44&373.93&402.71&440.35\\ \verb+N BoNeP -N KR 6Cm+ & 15 & 1.38 & 11.51&24.77&40.86&65.58&84.98&105.34&137.82&187.49&235.82&276.33&324.97&372.66&435.94&470.59&525.82\\ \verb+N BoNeP -N KR Tie+ & 22 & 2.06 & 18.15&34.93&63.42&98.77&129.21&179.81&224.98&277.16&331.25&399.71&466.61&540.55&620.63&675.01&785.68\\ \verb+N BoNeP -N KR Def+ & 26 & 2.11 & 17.48&39.97&72.50&112.59&149.29&187.89&236.08&284.89&332.48&380.62&439.68&515.16&600.16&656.77&749.71\\ \verb+N BoNeP -N KR JXc+ & 32 & 2.19 & 17.01&34.81&65.09&103.38&143.31&185.32&242.27&290.60&361.36&441.20&508.06&589.21&687.21&762.41&849.86\\ \verb+N BoNeP -N KR QMa+ & 40 & 3.05 & 21.39&52.56&94.71&158.02&196.77&270.19&335.88&419.22&509.74&596.41&684.62&813.48&926.26&1010.03&1129.40\\ \end{tabular}
\end{scriptsize}
\caption{Average number of CPU cycles per iteration of single array sorting across all machines} \label{table:normalsort:avg:all}
\end{sidewaystable}
\begin{figure}
\caption{Single sort for array size = 8 on machine A}
\label{plot:normal:8:A}
\end{figure}
\begin{figure}
\caption{Single sort for array size = 8 on machine B}
\label{plot:normal:8:B}
\end{figure}
\begin{figure}
\caption{Single sort for array size = 8 on machine C}
\label{plot:normal:8:C}
\end{figure}
\begin{figure}
\caption{Single sort of array sizes 2 to 16 on machine A}
\label{plot:normal:lineplot:A}
\end{figure}
\begin{figure}
\caption{Single sort of array sizes 2 to 16 on machine B}
\label{plot:normal:lineplot:B}
\end{figure}
\begin{figure}
\caption{Single sort of array sizes 2 to 16 on machine C}
\label{plot:normal:lineplot:C}
\end{figure}
\subsection{Sorting Many Continuous Sets of 2-16 Items} \label{section:experiments:inrow}
Here the benchmark shown in algorithm \ref{algo:inrow} was used. Instead of sorting a single array multiple times, multiple arrays are created adjacent to each other and sorted in series. \\
The number of arrays used is chosen in a way that all of them do not fit into the CPU's L3 cache. Since the reference array is sorted before the measurement, the original array should not be present in the cache, causing a cache miss on every access.
The results are similar to the previous ones. A difference we can see when comparing figures \ref{plot:inrow:lineplot:A}, \ref{plot:inrow:lineplot:B} and \ref{plot:inrow:lineplot:C} to figures \ref{plot:normal:lineplot:A}, \ref{plot:normal:lineplot:B} and \ref{plot:normal:lineplot:C} from the single sort measurement is that the \verb|CPr| swap that operates on pointers and moves values around in memory became worse compared to the \verb|4Cm| and \verb|4CS| implementations for array sizes greater than 2. Here the values can probably get pre-loaded for the next conditional swap while the current one is finishing, while \verb|CPr| accesses the element's reference value only when the destination address is calculated, which results in less pre-loading that can be done. \\
The complete overview over the average values of each sorter across all three machines can be seen in table \ref{table:inrowsort:avg:all}. We see speed-ups for using the sorting networks from 25\% at array size 2 all the way up to 59\% at array size 15.
\begin{figure}
\caption{Continuous sorting of array sizes 2 to 16 on machine A}
\label{plot:inrow:lineplot:A}
\end{figure}
\begin{figure}
\caption{Continuous sorting of array sizes 2 to 16 on machine B}
\label{plot:inrow:lineplot:B}
\end{figure}
\begin{figure}
\caption{Continuous sorting of array sizes 2 to 16 on machine C}
\label{plot:inrow:lineplot:C}
\end{figure}
\begin{sidewaystable}[!tbp]
\begin{scriptsize}
\begin{tabular}{l | r @{~~} r | r@{~~}r@{~~}r@{~~}r@{~~}r@{~~}r@{~~}r@{~~}r@{~~}r@{~~}r@{~~}r@{~~}r@{~~}r@{~~}r@{~~}r@{~~}r|}
& \multicolumn{2}{c|}{Overall} & \multicolumn{15}{c}{Array Size} \\
& Rank & GeoM & 2&3&4&5&6&7&8&9&10&11&12&13&14&15&16\\ \hline \verb+I -I KR POp+ & 26 & 2.39 & 20.67&44.67&72.23&104.33&142.53&182.47&222.07&262.90&302.13&342.53&382.87&425.57&468.73&513.80&556.07\\ \verb+I -I KR Def+ & 27 & 2.42 & 22.17&47.30&74.60&106.70&143.40&183.47&223.37&263.23&302.53&342.47&383.63&425.73&467.77&511.97&554.67\\ \verb+I -I KR STL+ & 34 & 2.57 & 27.07&53.47&82.40&114.07&150.63&191.53&232.87&273.00&313.67&355.40&398.73&441.80&484.63&529.97&573.47\\ \verb+I -I KR AIF+ & 35 & 2.57 & 23.67&50.00&79.53&111.97&149.87&193.77&237.77&280.87&324.37&367.43&412.23&456.13&499.43&544.10&589.47
\\ \verb+N Best -I KR 4CS+ & 2 & 1.05 & 15.97&25.00&35.67&48.73&57.60&70.90&79.30&96.50&\textbf{111.23}&133.43&148.70&175.07&222.03&226.00&261.80\\ \verb+N Best -I KR 4Cm+ & 5 & 1.08 & 15.97&\textbf{23.67}&34.87&47.20&56.30&70.20&82.20&\textbf{95.07}&114.20&\textbf{131.30}&\textbf{146.00}&205.50&236.97&271.73&299.80\\ \verb+N Best -I KR 6Cm+ & 8 & 1.29 & 15.97&25.70&37.03&55.13&66.80&85.50&94.37&117.63&139.07&160.90&181.33&270.23&301.70&347.80&379.53\\ \verb+N Best -I KR Cla+ & 11 & 1.40 & 14.00&25.30&39.90&70.83&87.67&114.43&128.97&169.67&172.10&201.00&218.60&231.20&254.33&278.03&295.50\\ \verb+N Best -I KR CPr+ & 13 & 1.43 & \textbf{13.90}&28.37&41.07&74.03&91.33&117.27&130.77&167.40&165.93&201.17&220.90&233.23&259.73&288.53&307.23\\ \verb+N Best -I KR Def+ & 22 & 2.33 & 23.67&44.33&72.00&96.07&129.27&167.40&205.97&245.73&262.57&328.70&373.57&408.03&478.33&531.47&605.10\\ \verb+N Best -I KR Tie+ & 23 & 2.34 & 25.97&43.77&69.00&92.00&119.37&166.30&200.83&235.50&274.13&324.33&380.43&435.33&486.90&555.63&639.40\\ \verb+N Best -I KR JXc+ & 32 & 2.52 & 24.60&45.30&70.63&99.83&137.83&173.73&213.70&253.73&295.23&363.17&418.60&479.23&560.80&638.80&696.60\\ \verb+N Best -I KR QMa+ & 37 & 3.10 & 24.67&46.00&83.13&130.07&176.83&223.30&279.90&333.30&357.03&500.63&575.63&580.67&684.17&773.97&845.93
\\ \verb+N BoNeL -I KR 4CS+ & 1 & 1.04 & 16.00&25.00&35.63&48.50&57.10&69.30&78.83&100.57&115.27&136.13&150.63&\textbf{174.33}&200.53&\textbf{207.23}&\textbf{247.57}\\ \verb+N BoNeL -I KR 4Cm+ & 3 & 1.06 & 15.97&24.00&34.97&47.70&57.00&70.50&79.10&100.80&117.30&136.77&148.63&180.77&\textbf{198.23}&245.17&268.67\\ \verb+N BoNeL -I KR 6Cm+ & 9 & 1.32 & 16.00&26.00&37.60&55.70&66.93&85.33&94.57&124.93&145.57&173.90&194.77&269.73&328.20&350.90&355.43\\ \verb+N BoNeL -I KR Cla+ & 17 & 1.65 & 13.93&25.13&40.00&70.97&87.63&114.33&129.00&200.47&223.80&268.67&286.87&344.30&365.33&400.97&416.60\\ \verb+N BoNeL -I KR CPr+ & 18 & 1.67 & 14.60&28.40&41.10&74.00&91.40&117.00&130.73&197.03&229.17&265.43&282.93&335.90&362.13&389.33&407.03\\ \verb+N BoNeL -I KR Tie+ & 25 & 2.36 & 25.97&43.63&68.33&92.87&120.40&166.40&200.63&237.40&282.20&331.43&384.17&444.20&494.67&562.53&639.57\\ \verb+N BoNeL -I KR Def+ & 28 & 2.42 & 24.10&45.40&74.37&95.90&128.67&166.37&205.67&248.33&306.17&330.20&414.60&438.80&525.63&575.17&630.03\\ \verb+N BoNeL -I KR JXc+ & 31 & 2.47 & 24.33&45.33&70.67&99.67&137.83&173.80&213.67&260.80&303.70&347.07&402.80&459.87&533.17&576.37&640.67\\ \verb+N BoNeL -I KR QMa+ & 39 & 3.27 & 25.00&46.50&83.20&130.23&177.13&223.37&280.17&371.97&434.07&505.10&582.43&688.13&756.13&833.53&945.70
\\ \verb+N BoNeM -I KR 4Cm+ & 7 & 1.19 & 16.00&23.93&\textbf{34.33}&47.33&55.90&66.60&\textbf{76.57}&170.50&177.00&167.90&194.00&194.70&253.57&262.33&307.97\\ \verb+N BoNeM -I KR 4CS+ & 12 & 1.41 & 16.00&24.63&41.20&63.03&55.77&111.00&77.50&165.30&203.97&204.63&200.43&294.33&321.33&349.40&382.30\\ \verb+N BoNeM -I KR 6Cm+ & 16 & 1.52 & 16.00&25.93&36.90&53.23&65.30&125.07&92.47&193.63&244.90&239.67&236.83&344.47&348.43&383.13&363.77\\ \verb+N BoNeM -I KR Cla+ & 19 & 1.86 & 16.67&29.33&41.20&83.67&89.00&135.63&128.30&220.20&259.97&297.53&311.73&402.10&427.67&484.07&477.33\\ \verb+N BoNeM -I KR CPr+ & 20 & 1.93 & 16.27&29.73&41.90&92.67&91.70&137.23&157.13&240.17&275.77&304.70&309.10&399.97&404.70&479.83&531.63\\ \verb+N BoNeM -I KR Tie+ & 29 & 2.45 & 26.00&44.33&69.03&92.70&121.00&162.47&197.57&239.17&305.27&373.97&427.80&516.20&569.87&566.87&648.47\\ \verb+N BoNeM -I KR Def+ & 30 & 2.46 & 23.33&45.40&73.73&105.90&144.73&182.37&212.07&246.03&305.07&329.70&410.60&432.07&501.03&575.37&633.07\\ \verb+N BoNeM -I KR JXc+ & 36 & 2.64 & 24.40&42.67&70.63&123.73&130.33&168.57&212.20&281.13&333.73&432.47&458.37&492.10&583.77&632.10&741.07\\ \verb+N BoNeM -I KR QMa+ & 38 & 3.20 & 29.00&52.00&89.77&126.97&177.13&225.23&278.20&346.27&400.33&467.37&533.50&611.23&702.33&774.00&878.87
\\ \verb+N BoNeP -I KR 4CS+ & 4 & 1.08 & 16.00&24.97&35.70&\textbf{45.70}&\textbf{54.77}&66.70&78.73&99.60&122.90&145.37&156.00&190.20&210.63&263.43&279.30\\ \verb+N BoNeP -I KR 4Cm+ & 6 & 1.09 & 16.00&23.80&36.73&46.93&54.97&\textbf{65.80}&78.73&101.50&119.97&138.13&155.37&218.03&211.63&285.87&305.37\\ \verb+N BoNeP -I KR 6Cm+ & 10 & 1.35 & 16.33&26.00&37.30&52.43&63.17&79.57&93.27&124.70&151.13&174.90&241.00&287.13&329.40&385.30&411.77\\ \verb+N BoNeP -I KR Cla+ & 14 & 1.45 & 14.63&25.03&40.03&66.00&81.47&108.33&116.67&178.70&188.40&215.53&227.33&269.33&285.97&316.60&333.97\\ \verb+N BoNeP -I KR CPr+ & 15 & 1.47 & 13.90&28.33&41.13&70.27&83.43&108.47&116.93&177.00&185.57&210.00&228.57&263.30&295.20&318.77&341.63\\ \verb+N BoNeP -I KR Def+ & 21 & 2.32 & 23.67&46.10&71.00&96.90&128.83&160.97&196.50&241.30&278.13&321.10&364.47&419.27&478.67&544.53&597.17\\ \verb+N BoNeP -I KR Tie+ & 24 & 2.35 & 25.50&44.67&69.63&90.33&124.73&160.10&193.67&236.70&284.33&327.73&383.80&440.30&500.37&557.47&633.60\\ \verb+N BoNeP -I KR JXc+ & 33 & 2.54 & 24.67&43.33&70.60&100.33&133.17&169.03&217.23&258.40&318.80&377.17&428.93&489.80&564.90&625.70&713.63\\ \verb+N BoNeP -I KR QMa+ & 40 & 3.29 & 26.33&49.67&87.57&130.00&165.33&224.03&274.07&367.53&428.20&509.07&570.77&684.67&776.17&849.17&927.27\\ \end{tabular}
\end{scriptsize}
\caption{Average number of CPU cycles per array of continuous sorting across all machines} \label{table:inrowsort:avg:all}
\end{sidewaystable}
\subsection{Sorting a Large Set of Items with Quicksort} \label{section:experiments:quicksort}
After seeing the first two results, we wanted to know how the base case sorters perform when used inside a scalable sorting algorithm. For that we modified introsort, a quicksort implementation from the STL library, as follows: Introsort calls insertion sort only once, right at the end. Since that is not possible with the sorting networks, they had to be called directly when the partitioning resulted in a partition of 16 elements or less. Also we determined the pivot using the 3-element Bose Nelson parameter network instead of using \verb|if-else| and \verb|std::swap|. \\
The sorters were measured using benchmark \ref{algo:normal} with parameters
\begin{itemize}
\item $\mathtt{numberOfIterations} = 50$
\item $\mathtt{numberOfMeasures} = 200$
\item $\mathtt{arraySize} = 1024 \times 16 = 16384 = 2^{14}$.
\end{itemize}
To have a basis of comparison we also measured sorting with \verb|std::sort|. These times can be taken from figures \ref{plot:quicksort:A}, \ref{plot:quicksort:B} and \ref{plot:quicksort:C}. \\
The \verb|QSort -Q KR Def| sorter is just a direct copy of the STL sort doing a final insertion sort at the end. That was measured to see that our code copy does as well as \verb|std::sort| before doing the modifications.
\begin{figure}
\caption{Sorting times of quicksort with different base cases on machine A}
\label{plot:quicksort:A}
\end{figure}
\begin{figure}
\caption{Sorting times of quicksort with different base cases on machine B}
\label{plot:quicksort:B}
\end{figure}
\begin{figure}
\caption{Sorting times of quicksort with different base cases on machine C}
\label{plot:quicksort:C}
\end{figure}
\begin{table}[!h]
\begin{center}
\begin{small}
\begin{tabular}{ c | c | c | c }
& A:~~ \verb|N Best -Q KR Cla| & B:~~ \verb|N Best -Q KR Cla| & C:~~ \verb|N Best -Q KR Cla| \\ \hline
\verb|I -Q KR Def| & 1.76\% & 2.1\% & 8.76\% \\
\verb|I -Q KR POp| & 3.99\% & 2.58\% & 6.47\%\\
\verb|StdSort -Q| & 12.3\% & 10.6\% & 14\% \\
\end{tabular}
\end{small}
\end{center}
\caption{Average speed-ups of the fastest sorting network over the fastest insertion sort as base case in quicksort and unmodified std::sort} \label{table:completesort:speedups}
\end{table}
Speed-ups of including sorting networks into a sorting algorithm like quicksort can be seen in table \ref{table:completesort:speedups}. \\
What is notable is that the variants with insertion sort at the base are faster than the one with the final insertion sort, which should come from the fact that they are already specialized for the item they sort and do not require a predicate for the sorting. Also, the base case is called right after the partitioning is at a low enough level, which means that the elements are still present in the first- or second-level cache. That also explains why the \verb|Cla| conditional swap performs the best with quicksort, while we saw in the last section that this is not necessarily the case when we have a cache miss.
Recalling the results from the previous sections, we appeared to be achieving great improvements in reducing the time needed for sorting sets of 2-16 items. By measuring only the sorting of the small sets we have exploited the networks' strength: not containing conditional branches. The results from the measurements with quicksort highlight the networks' weakness: The larger code size. \\
When integrating the sorting networks into quicksort for sorting the base cases, every time a partition results in one part having 16 elements or less, we switch from the code for quicksort to the code for the sorting network. Thus, the code for quicksort is partly removed from the L1 instruction cache and replaced with the code for the sorting network. Because the network's code is just a flat sequence of conditional swaps, each line of code is accessed exactly once per sort. That means it caused a lot of quicksort's code to be removed from the instruction cache without gaining a speed-up because its code is now in the cache, and will be removed again when quicksort is handed back the flow of control and loads its code back into the instruction cache. \\
We can see that effect especially for machines A and B which have 32 KiB of L1 instruction cache, where the speed-up is hardly over 2\% for the best network base case over the best insertion sort base case. Where we got a much more improvement is on machine C, which has double the space in its L1 instruction cache. Here we achieved a speed-up of almost 6.5\% when making use of the best networks. \\
It is no surprise that we do not see improvements similar to those in section \ref{section:experiments:normal} or \ref{section:experiments:inrow} because the partitioning that quicksort performs takes the same amount of time no matter which base case sorter is used, representing a part of the algorithm that is not optimizable through using sorting networks.
\begin{table}[!h]
\begin{center}
\begin{small}
\begin{tabular}{ c | c | c | c }
& A:~~ \verb|N BoNeL -s332 4CS| & B:~~ \verb|N BoNeL -s332 4CS| & C:~~ \verb|N BoNeL -s332 4CS| \\ \hline
\verb|I -s332 Def| & 17.4\% & 17.5\% & 29.2\% \\
\verb|StdSort -s| & 43.6\% & 43.49\% & 51\% \\
\end{tabular}
\end{small}
\end{center}
\caption{Average speed-ups of the fastest sorting network over the fastest insertion sort as base case in sample sort and unmodified std::sort} \label{table:samplesort:speedups}
\end{table}
\subsection{Sorting a Medium-Sized Set of Items with Sample Sort} \label{section:experiments:samplesort}
Sample sort was measured using benchmark \ref{algo:normal} with parameters:
\begin{itemize}
\item $\mathtt{numberOfIterations} = 50$
\item $\mathtt{numberOfMeasures} = 200$
\item $\mathtt{arraySize} = 256$.
\end{itemize}
The measurements were done with two different goals in mind:
The first was to see which parameters work best for the machines used and the array size set. This can be seen in figures \ref{plot:samplesort:bonel:A}, \ref{plot:samplesort:bonel:B} and \ref{plot:samplesort:bonel:C} for the Bose Nelson networks optimizing locality. To be able to compare the results on the different machines, the configurations were ordered based on the times from machine A, and are in the same order in the other two plots. An oversampling factor of 3 and block size of 2 performed best on machine A and B. That configuration also performs best when using the other networks or insertion sort as a base case. \\
On machine C block sizes larger than 2 performed better (on average) along with an oversampling factors of 3 or greater. We measured larger variances and got a lot more outliers, so here choosing a \enquote{best} configuration was not so easy. When looking at the other networks and insertion sort as base case, consistently well performing parameters are an oversampling factor of 3 and a block size of 4, but with very little lead over other configurations. That is interesting to see because all three machines run x86 assembly instructions and have the same number of general purpose registers available. What comes into play here is the size of the instruction cache: Machine C has double the amount of L1 instruction cache of what machines A and B have. We can only assume that the instructions for classifying three elements need more space than the smaller 32 KiB instruction caches can provide, while the 64 KiB instruction cache that machine C has fits the instructions for classifying four and / or almost five elements at once, considering that block size 5 also performs well. \\
The second goal was to see if the results from section \ref{section:experiments:normal}, \ref{section:experiments:inrow} and \ref{section:experiments:quicksort} would relate to the results from using sample sort with the sorting networks as base cases. These results can be seen in figures \ref{plot:samplesort:s332:A}, \ref{plot:samplesort:s332:B} and \ref{plot:samplesort:s332:C} for the 332 configuration. All measurements were made with a base case limit of 16. Here, too, a single outlier was excluded from the dataset for scaling purposes: A value of $40177$ measured on machine B for the \texttt{'N BoNeP -s332 KR 4Cm'} sorter. \\
The achieved speed-ups of using the sorting networks are given in table \ref{table:samplesort:speedups}. On the left we see sample sort with insertion sort as base case and std::sort that was also measured sorting 256 elements. On the top we see the best performing network \texttt{'N BoNeL -s332 4CS'} as a base case for sample sort on all three machines. The number indicates the speed-up of sample sort with the network over sample sort with insertion sort and over std::sort. \\
Again we see that due to machine C having a larger L1 instruction cache the performance gain is almost double that for the other machines. Unlike in the previous section though we got much greater speed-ups as a result of using the sorting networks as a base case. That comes from the fact that sample sort has no unpredictable branches classifying the elements, as opposed to quicksort having to deal with conditional branches during the partitioning, while both need to invest the same time to sort all the base cases. So with sample sort, the base case sorting takes up a larger time slot of the whole execution than it does with quicksort. We also see that with very few conditional branches we can get up to 50\% faster than std::sort (for sets of up to 256 items at least).
\begin{figure}\label{plot:samplesort:bonel:A}
\end{figure}
\begin{figure}\label{plot:samplesort:bonel:B}
\end{figure}
\begin{figure}\label{plot:samplesort:bonel:C}
\end{figure}
\begin{figure}
\caption{Sample sort 332 with different base cases on machine A}
\label{plot:samplesort:s332:A}
\end{figure}
\begin{figure}
\caption{Sample sort 332 with different base cases on machine B}
\label{plot:samplesort:s332:B}
\end{figure}
\begin{figure}
\caption{Sample sort 332 with different base cases on machine C}
\label{plot:samplesort:s332:C}
\end{figure}
\subsection{Sorting a Large Set of Items with IPS${}^4$o} \label{section:experiments:ipso}
\newcommand{\texttt{BaseCaseSize}${}_4$}{\texttt{BaseCaseSize}${}_4$}
With the efficient implementation of sample sort for medium-sized sets we can now include the new base case sorters into a complex sorting algorithm. The \textit{In-Place Parallel Super Scalar Samplesort} (IPS${}^4$o) \cite{DBLP:conf/esa/AxtmannWF017} was executed without introducing parallelism. The algorithm has many parameters that can be adjusted. The important parameter for us was the \texttt{BaseCaseSize}${}_4$\footnote{we will use the ${}_4$ to distinguish IPS${}^4$o's \texttt{BaseCaseSize} from Register Sample Sort's base case size}: it tells IPS${}^4$o~ to aim for base case sizes that are smaller or equal to \texttt{BaseCaseSize}${}_4$. Even though that is the goal, for a large-scale sorter like IPS${}^4$o~ it would be far less efficient to partition e.g. 32 elements into many buckets, that might end up not containing many elements each, than just using the base case sorter for these situations, even though the number of items is larger than the specified \texttt{BaseCaseSize}${}_4$. \\
That was the reason to develop Register Sample Sort that can break those medium-sized sets down into sizes that can be sorted using the sorting networks. \\
We started the measuring using the best combination of sample sort from section \ref{section:experiments:samplesort} as a base case for IPS${}^4$o, together with using the default \texttt{BaseCaseSize}${}_4$ = 16, but that turned out to perform worse than just insertion sort. \\
The distribution of the base case array sizes can be seen in figure \ref{plot:distr:16} for \texttt{BaseCaseSize}${}_4$\, = 16 and figure \ref{plot:distr:32} for \texttt{BaseCaseSize}${}_4$\, = 32. From that it was evident that in most of the instances with parameter \texttt{BaseCaseSize}${}_4$\, = 16 the base case sorter was being invoked on sets smaller than even 32 elements. That also meant that sample sort had to deal with a larger overhead than insertion sort, not justified by a larger amount of items. \\
In addition to that the size of the instruction cache that had already had a great influence on the measurements of quicksort seemed to be another factor for the bad performance of Register Sample Sort as a base case. \\
That is why we decided to measure the following setups:
\begin{itemize}
\item Pure insertion sort as base case (\verb|I|) with
\begin{itemize}
\item \texttt{BaseCaseSize}${}_4$ = 16 and 32
\end{itemize}
\item Register sample sort as base case (\verb|S+N|) with
\begin{itemize}
\item \texttt{BaseCaseSize}${}_4$ = 16, 32, and 64,
\item Configurations 331 and 332, and
\item Best networks and Bose Nelson networks (optimizing locality) as base case for Register Sample Sort, with the \verb|4CS| conditional swap and base case size 16
\end{itemize}
\item A combination of the sorting networks and insertion sort (\verb|I+N|): \\
Since the base case sizes were often smaller than 16, we wanted to make use of that by using the sorting networks, while not having to rely on Register Sample Sort with its larger overhead for the slightly larger base cases. The solution was to use the Bose Nelson networks (optimizing locality) if the set had 16 elements or less, and insertion sort otherwise.
\end{itemize}
\begin{figure}
\caption{Distribution of the size of the array passed to the base case sorter when executing IPS${}^4$o\, with parameter \texttt{BaseCaseSize}${}_4$\, = 16}
\label{plot:distr:16}
\end{figure}
\begin{figure}
\caption{Distribution of the size of the array passed to the base case sorter when executing IPS${}^4$o\, with parameter \texttt{BaseCaseSize}${}_4$\, = 32}
\label{plot:distr:32}
\end{figure}
Figures \ref{plot:ipso:A}, \ref{plot:ipso:B} and \ref{plot:ipso:C} display the results from the measurements with the above variants. The \texttt{BaseCaseSize}${}_4$~was appended after the \verb|-4|, along with an underscore followed by the Register Sample Sort configuration. \\
The the benchmark from algorithm \ref{algo:normal} was used with parameters
\begin{itemize}
\item $\mathtt{numberOfIterations} = 50$
\item $\mathtt{numberOfMeasures} = 200$
\item $\mathtt{arraySize} = 1024 \times 32 = 32768 = 2^{15}$.
\end{itemize}
As already seen in \cite{DBLP:conf/esa/AxtmannWF017}, we get a speed-up of over 59\% over \verb|std::sort| with unchanged IPS${}^4$o\, on all machines. On machine A, none of the variants we tried led to an improvement in sorting speed over the default use of insertion sort at \texttt{BaseCaseSize}${}_4$~ 16. For machine B, interestingly, using Register Sample Sort did not lead to an improvement, but the combination of insertion sort and Bose Nelson networks did manage to reduce the sorting time by 4.3\%. For machine C we see the impact of the large L1 instruction cache in the visible improvement of 9.2\% for having Register Sample Sort as a base case instead of insertion sort, though the combinations of insertion sort and the sorting network also performed well. It is notable to see that, while Register Sample Sort by itself did well with blockSizes of 4 or 5, here it is beneficial to use blockSize = 1, having a smaller impact on the instruction cache.
\begin{table}[!h]
\begin{center}
\begin{small}
\begin{tabular}{ c | c | c | c }
& A:~~ \verb|S+N BoNeL 16_331 4CS| & B:~~ \verb|I+N 16| & C:~~ \verb|S+N BoNeL 16_331 4CS| \\ \hline
\verb|I 16 Def| & \red{-3.4\%} & 4.3\% & 9.2\% \\
\verb|StdSort -s| & 59.1\% & 61,7\% & 65\% \\
\end{tabular}
\end{small}
\end{center}
\caption{Average speed-ups of the fastest sorting network over the fastest insertion sort as base case in IPS${}^4$o\, and unmodified std::sort} \label{table:ipso:speedups}
\end{table}
\begin{figure}
\caption{Sorting times for IPS${}^4$o\, on machine A with different base cases and base case sizes}
\label{plot:ipso:A}
\end{figure}
\begin{figure}
\caption{Sorting times for IPS${}^4$o\, on machine B with different base cases and base case sizes}
\label{plot:ipso:B}
\end{figure}
\begin{figure}
\caption{Sorting times for IPS${}^4$o\, on machine C with different base cases and base case sizes}
\label{plot:ipso:C}
\end{figure}
\section{Conclusion} \label{section:conclusion}
\subsection{Results and Assessment}
In this thesis we have seen that for sorting sets of up to 16 elements it can be viable to use sorting algorithms other than insertion sort. We looked at sorting networks in particular, paying special attention to the implementation of the conditional swap and giving multiple alternative ways of realizing that implementation. \\
After seeing that the sorting networks outperform insertion sort each on their own for a specific array size in section \ref{section:experiments:normal} and \ref{section:experiments:inrow}, we saw in section \ref{section:experiments:quicksort} that this improvement does not necessarily transfer to sorting networks being used as base case sorter in quicksort. Because the networks have a larger code size, the code for quicksort is removed from the instruction cache and the advantage of not having conditional branches is impaired by that larger code size. But we also saw that for machines with larger instruction caches using sorting networks with quicksort can lead to visible improvements of about 6.4\%. \\
After that we integrated the sorting networks into a very advanced sorter like IPS${}^4$o, which was possible by adding an intermediate sorter into the procedure. For that we created Register Sample Sort, which is an implementation of Super Scalar Sample Sort that holds the splitters in general-purpose registers instead of an array. When measuring IPS${}^4$o\, with Register Sample Sort as a base case, we found that the instruction cache makes even more of a difference, because we now add the code size for Register Sample Sort on top of the code size for the sorting networks. \\
We proposed an additional alternative to Register Sample Sort, using a combination of insertion sort and sorting networks: For base cases of 16 elements or less, we used the sorting network, for any size above that insertion sort. \\
On one of the machines with a smaller instruction cache of 32 KiB we could not achieve a speed-up with any of the variants, on the other the combination of insertion sort and sorting networks led to an improvement in sorting time of 4.3\%. The only substantial improvement we achieved with IPS${}^4$o\, was on the machine with 64 KiB of L1 instruction cache, where using Register Sample Sort led to an improvement of 9.2\% over plain insertion sort.
In closing, we want to mention that this particular implementation only compiles when using the gcc C++ compiler due to compiler-dependent inline-assembly statements. This also means that the code is probably not as fast as it could be due to the inline-assembly not being optimized by the compiler. The complete project is available on github at\\ \url{https://github.com/JMarianczuk/SmallSorters}.
\subsection{Experiences and Hurdles}
The greatest hurdle we encountered during this project was, as mentioned in section \ref{section:measurements}, the fact that the compiler reduces its optimizations with increasing compilation effort, when compiling only a single source file. That can lead to performance variations that happen for no \enquote{apparent} reason, and is especially tricky when dealing with templated methods that can not be moved from header files into source files. The solution was to use code generation and to include all logically coherent method invocations in one wrapper method that is then placed in its own source file, to not have different parts of the program influencing each other over the decision which one gets to be optimized and which one not.
\subsection{Possible Additions}
In addition to the work in this thesis, we would like to explore further possibilities to implement the conditional swap for the sorting networks, as well as seeing which of the C++ compilers generate conditional moves when using portable C++ code instead of compiler-dependent inline-assembly. That also includes looking at conditional swaps for elements that differ from the 64-bit key and reference value pair that we looked at in this thesis. \\
Furthermore we would like to take a look at implementing sorting networks in a way that they take up less code space, and what the trade-off for that decreased code size would be. \\
Apart from the sorting networks we would also like to take another look at Register Sample Sort to find out if using seven splitters instead of three can be more practical when increasing the input size to sizes larger than 256.
\end{document} |
\begin{document}
\title[]{On the HJY Gap Conjecture in CR geometry vs. the SOS Conjecture for polynomials} \author{Peter Ebenfelt} \address{Department of Mathematics, University of California at San Diego, La Jolla, CA 92093-0112} \email{pebenfel@math.ucsd.edu}
\thanks{The author was supported in part by the NSF grant DMS-1301282.} \begin{abstract} We show that the Huang-Ji-Yin (HJY) Gap Conjecture concerning CR mappings between spheres follows from a conjecture regarding Sums of Squares (SOS) of polynomials. The connection between the two problems is made by the CR Gauss equation and the fact that the former conjecture follows from the latter follows from a recent result, due to the author, on partial rigidity of CR mappings of strictly pseudoconvex hypersurfaces into spheres. \end{abstract}
\thanks{2000 {\em Mathematics Subject Classification}. 32H02, 32V30}
\maketitle
\section{Introduction}
The purpose of this note is to explain how the Huang-Ji-Yin (HJY) Gap Conjecture concerning CR mappings between spheres \cite{HuangJiYin09} follows from a conjecture regarding Sums of Squares (SOS) of polynomials. The connection between the two problems is made by the CR Gauss equation (a well known fact) and the implication follows from a recent result, due to the author \cite{E12}, on partial rigidity ("flatness") of CR mappings of strictly pseudoconvex hypersurfaces into spheres.
The HJY Gap Conjecture concerns CR mappings $f$ of an open piece of the unit sphere $\mathbb S^n\subset\mathbb C^{n+1}$ into the unit sphere $\mathbb S^N\subset \mathbb C^{N+1}$ when the codimension $N-n$ lies in the integral interval $[0,D_n]$, where $D_n$ is a specific integer that depends on $n$ (with $D_n\sim \sqrt{2}n^{3/2}$, see below); here, we use the non-standard convention that the superscript $m$ on a real hypersurface $M^m\subset \mathbb C^{m+1}$ refers to the CR dimension, and not the real dimension (which is $2m+1$). The mappings $f$ are assumed to be (sufficiently) smooth and, by results in \cite{Forstneric89} and \cite{CS90}, they therefore extend as rational maps without poles on $\overline{\mathbb B_{n+1}}$, where $\mathbb B_{n+1}\subset \mathbb C^{n+1}$ denotes the unit ball. In particular, there is no loss of generality in considering globally defined CR mappings $f\colon \mathbb S^{n}\to \mathbb S^N$. The conjecture asserts that there is a collection of finitely many disjoint integral subintervals $I_1,\ldots, I_{\kappa_0}\subset [0,D_n]$ with the property that if the codimension $N-n$ belongs to one of these subintervals, $N-n\in I_\kappa=[a_\kappa,b_\kappa]$, then \begin{equation}\label{f=TLf0} f=T\circ L\circ f_0,
\end{equation} where $f_0$ is a CR mapping $S^n\to S^{N_0}$ for some $N_0$ with codimension $N_0-n<a_\kappa\leq N-n$ (in particular, then $N_0<N$), and where $L\colon S^{N_0}\to S^N$ is the standard linear embedding in which the last $N-N_0$ coordinates are zero and $T\colon S^N\to S^N$ is an automorphism of the target sphere $S^N$. It is well known and easy to see that the representation \eqref{f=TLf0} is equivalent to the statement that the image $f(\mathbb S^n)$ is contained in an affine complex subspace $A^{N_0+1}$ of dimension $N_0+1$.
Before formulating the HJY Gap Conjecture more precisely, we must introduce the integral intervals $I_\kappa$. For $n\geq 2$, we define \begin{equation}\label{Ik} I_\kappa:=\left \{j\in \mathbb N\colon (\kappa-1)n+\kappa\leq j\leq \sum_{i=0}^{\kappa-1}(n-i)-1=n+(n-1)+\ldots+(n-\kappa+1)-1\right\}, \end{equation} for $\kappa=1,\ldots, \kappa_0$, where $\kappa_0=\kappa_0(n)$ is the largest integer $\kappa$ such that the integral interval $I_\kappa$ is non-trivial, i.e., \begin{equation}\label{k0} (\kappa-1)n+\kappa\leq \sum_{i=0}^{\kappa-1}(n-i)-1. \end{equation} A simple calculation shows that $\kappa_0=\kappa_0(n)$ is increasing in $n$ (clearly, with $\kappa_0<n$) and grows like $\sqrt{2n}$. We have, e.g., $\kappa_0(2)=1$, $\kappa_0(4)=2$, and for $\kappa_0(n)\geq 3$, we need $n\geq 7$. For the integer $D_n$ referenced above, we can then take $$ D_n=\kappa_0n-\frac{\kappa_0(\kappa_0-1)}{2}-1=\sqrt{2}n^{3/2}-n-\sqrt{2n}+O(1). $$ Now, the conjecture made by X. Huang, S. Ji, and W. Yin in \cite{HuangJiYin09} can be formulated as follows:
\begin{conjecture}[HJY Gap Conjecture]\label{HJYConj} For $n\geq 2$, let $\kappa_0$ and $I_1,\ldots I_{\kappa_0}$ be as above and assume that $f\colon \mathbb S^n\to \mathbb S^{N}$ is a sufficiently smooth CR mapping. If the codimension $N-n\in I_\kappa$, then there exists an integer $n\leq N_0<N$ with \begin{equation}\label{No-nest} N_0-n\leq (\kappa-1)n-\kappa-1
\end{equation} and an affine complex subspace $A^{N_0+1}$ of dimension $N_0+1$ such that $f(\mathbb S^n)\subset S^N\cap A^{N_0+1}$. \end{conjecture}
The $\kappa$th integral interval $I_\kappa$ with the property described in the conjecture above is referred to as the $\kappa$th {\it gap}. We note that the existence of the first gap is the statement that if $f\colon \mathbb S^n\to \mathbb S^{N}$ is a sufficiently smooth CR mapping and $1\leq N-n\leq n-1$, then $f(\mathbb S^n)\subset \mathbb S^N\cap A^{n+1}$. Since $\mathbb S^N\cap A^{n+1}$ is a sphere in the $(n+1)$-dimensional complex space $A^{n+1}$ and, thus, CR equivalent to $S^n\subset \mathbb C^{n+1}$, we can write $f=T\circ L\circ f_0$, where $T$ and $L$ are as in \eqref{f=TLf0} and $f_0$ is a map of $S^n$ to itself. By work of Poincar\'e \cite{Poincare07}, Alexander \cite{Alexander74}, and Pinchuk \cite{Pincuk74}, $f_0$ is in fact an automorphism of $\mathbb S^n$ (unless it is constant, of course) and by an appropriate choice of $T$, we can in fact make $f_0$ linear. The existence of the first gap, under the assumption that $f$ is real-analytic, was established by Faran in \cite{Faran86}; the smoothness required for this was subsequently lowered to $C^{N-n}$ by Forstneric \cite{Forstneric89} and then to $C^2$ by X. Huang in \cite{Huang99}. The existence of the second gap (when $n\geq 4$) and the third gap (when $n\geq 7$) was established under the assumption of $C^3$-smoothness of $f$ in \cite{HuangJiXu06} and \cite{HuangJiYin12}, respectively. The existence of the $\kappa$th gap for $3< \kappa\leq \kappa_0$ is an open problem at this time. It is, however, known \cite{JPDLebl09} that when the codimension $N-n$ is sufficiently large, then there are no more gaps (in the sense of Conjecture \ref{HJYConj}).
For the first three gaps, one can also classify the possible maps $f_0$ that appear in \eqref{f=TLf0}, as in the (very simple) Poincar\'e-Alexander-Pinchuk classification corresponding to the first gap described above; see \cite{HuangJi01}, \cite{Hamada05}, \cite{HuangJiYin12}. For the gaps beyond these, such a classification is most likely beyond what one can hope for at this time, at least for large $\kappa$. To the best of the author's knowledge, there is no conjecture as to what such "model" maps would be for general $\kappa$.
For a CR mapping $f\colon \mathbb S^n\to \mathbb S^N$, there is a notion of the CR second fundamental form of $f$ and its covariant derivatives, and if we form the corresponding sectional curvatures (defined more precisely in the next section), then we obtain a collection of polynomials $\Omega^1(z), \ldots,\Omega^{N-n}(z)$ in the variables $z=(z^1,\ldots,z^n)\in\mathbb C^n$, whose coefficients consist of components of the second fundamental form and its covariant derivatives up to some finite order (bounded from above by the codimension $N-n$); we shall refer to the polynomial mapping $\Omega=(\Omega^1,\ldots\Omega^{N-n})$ as the total second fundamental polynomial. These polynomials satisfy a Sums Of Squares (SOS) identity as a consequence of a CR version of the Gauss equation. The SOS identity has the following form \begin{equation}\label{SOSid}
\sum_{j=1}^{N-n}|\Omega^j(z)|^2 = A(z,\bar z)\sum_{i=1}^n|z^i|^2, \end{equation} where $A(z,\bar z)$ is a Hermitian (real-valued) polynomial in $z$ and $\bar z$. To simplify the notation, for a polynomial mapping $P(z)=(P^1(z),\ldots,P^q(z))$ we shall write $|\!| P(z)|\!|^2$ for the SOS of moduli of the components, i.e., \begin{equation}
|\!| P(z)|\!|^2:=\sum_{k=1}^q|P^k(z)|^2. \end{equation} The number $q$ of terms in the norm will differ depending on the mapping in question, but will be clear from the context. Using this notation, the identity \eqref{SOSid} can be written in the following way: \begin{equation}\label{SOSid'} |\!|\Omega(z)|\!|^2=A(z,\bar z)|\!| z|\!|^2. \end{equation} The polynomial $A(z,\bar z)$ is in principle computable from $f$, but useful properties of $A$ seem difficult to extract directly in this way, and often it suffices to know that $\Omega$ satisfies an SOS identity of this form, for some Hermitian polynomial $A$. SOS identities of the form \eqref{SOSid'} appear in many different contexts, and there is an abundance of literature considering various aspects of such identities. We mention here only a few, and only ones with a connection to CR geometry and complex analysis: e.g., \cite{Quillen68}, \cite{CatlinDangelo96}, \cite{CatlinJPD99}, \cite{EHZ05}, \cite{JPDLebl09}, \cite{JPD11}, \cite{HuangY13}, \cite{GrHa13}, \cite{GruLV14}, \cite{E15}, and refer the reader to these papers for further connections and references to the literature. The reader is especially referred to the paper \cite{JPD11} by D'Angelo, which contains an excellent discussion of SOS identities and positivity conditions.
We shall here be concerned with a very specific property of polynomial maps $\Omega$ that satisfy \eqref{SOSid'}, namely the possible linear ranks that can occur. For a polynomial mapping $P(z)=(P^1(z),\ldots,P^q(z))$, we define its {\it linear rank} to be the dimension of the complex vector space $V_P$ spanned by its components, in the polynomial ring $\mathbb C[z]$. The main result in this note is that the HJY Gap Conjecture will follow from the following conjecture regarding the possible linear ranks of polynomial mappings $P(z)$ that satisfy an SOS identity:
\begin{conjecture}[SOS Conjecture]\label{SOSConj} Let $P(z)=(P^1(z),\ldots,P^q(z))$ be a polynomial mapping in $z=(z^1,\ldots, z^n)\in \mathbb C^n$, and assume that there exists a Hermitian polynomial $A(z,\bar z)$ such that the SOS identity \begin{equation}\label{SOSidP} |\!| P(z)|\!|^2=A(z,\bar z)|\!| z|\!|^2 \end{equation} holds. If $r$ denotes the linear rank of $P(z)$, then either \begin{equation}\label{rmax} r\geq (\kappa_0+1)n-\frac{\kappa_0(\kappa_0+1)}{2}-1, \end{equation} where $\kappa_0$ is the largest integer $\kappa$ such that \eqref{k0} holds, or there exists a integer $1\leq \kappa\leq \kappa_0<n$ such that \begin{equation}\label{e:ASOSbound} \sum_{i=0}^{\kappa-1} (n-i)=n\kappa-\frac{\kappa(\kappa-1)}{2}\leq r\leq \kappa n. \end{equation} \end{conjecture}
\begin{remark} {\rm The integer $\kappa_0$ is also the integer for which the integral intervals in $\kappa$, defined by \eqref{e:ASOSbound} start overlapping for $\kappa=\kappa_0+1$. } \end{remark}
The main result in this note is that this SOS Conjecture implies the HJY Gap Conjecture:
\begin{theorem}\label{MainThm} If the SOS Conjecture $\ref{SOSConj}$ holds, then the HJY Gap Conjecture $\ref{HJYConj}$ holds. \end{theorem}
The connection between the two conjectures is explained in Section \ref{GaussSec}. The conclusion of Theorem \ref{MainThm} will then be derived, in Section \ref{Proof}, as a consequence of Theorem 1.1 in \cite{E12}, reproduced here in a special case as Theorem \ref{Thm1.1}.
\subsection{Results on the SOS Conjecture; reduction to an alternative SOS Conjecture} While the literature on SOS of polynomials is vast, as mentioned above, there are very few results that have a direct impact on the SOS Conjecture \ref{SOSConj}. To the best of the author's knowledge, the only general result on this conjecture is what is now known as Huang's Lemma, which first appeared in \cite{Huang99}, and which establishes the first gap in the SOS Conjecture: If $r<n$, then $A\equiv 0$, and, hence $r=0$. Huang used this result in \cite{Huang99} to give a new proof of Faran's result regarding existence of the first gap in the Gap Conjecture \ref{HJYConj}, and to show that it suffices to assume that the mappings are merely $C^2$-smooth.
In another recent paper \cite{GrHa13} by Grundmeier and Halfpap, the SOS Conjecture \ref{SOSConj} was established in the special case where $A(z,\bar z)$ is itself an SOS, i.e., \begin{equation}\label{ASOS} A(z,\bar z)=|\!| F(z)|\!|^2, \end{equation} for some polynomial mapping $F(z)$. The integer $\kappa$ in the conjecture in this case is the linear rank of the polynomial mapping $F(z)$; it is assumed in \cite{GrHa13} that the components of $P(z)$ are homogeneous polynomials, but a simple homogenization argument can remove this assumption (cf. \cite{E15}). The Grundmeier-Halfpap result by itself does not seem to have any direct implications for the Gap Conjecture \ref{HJYConj}, as the needed information regarding the Hermitian polynomial $A(z,\bar z)$ seems difficult to glean from the mapping $f$, but it offers the opportunity to formulate an alternative, arguably simplified version of the SOS conjecture, which would imply Conjecture \ref{SOSConj} as a consequence of the Grundmeier-Halfpap result. We shall formulate this alternative SOS Conjecture in what follows.
We observe that, by standard linear algebra arguments, any Hermitian polynomial $A(z,\bar z)$ can be expressed as a difference of squared norms of polynomial mappings, \begin{equation}\label{DOS} A(z,\bar z)=|\!| F(z)|\!|^2-|\!| G(z)|\!|^2, \end{equation} where $F=(F^1,\ldots,F^{q_+})$ and $G=(G^1,\ldots, G^{q_-})$ are mappings whose components are polynomials in $z$. We may further assume that the complex vector spaces $V_F$, $V_G$ spanned by their respective components have dimensions $q_+$, $q_-$, respectively (i.e., the components of $F$ and $G$ are linearly independent, so their linear ranks are $q_+$, $q_-$, respectively), and that $V_F\cap V_G=\{0\}$. The Grundmeier-Halfpap result proves Conjecture \ref{SOSConj} in the special case where $G=0$. Thus, it suffices to prove the conjecture in the case where $G\neq 0$. In this case, the product $A(z,\bar z)|\!| z|\!|^2$ need of course not be an SOS, so this must be assumed. An optimistic view of the situation in the conjecture would be to hope that the "gaps" in linear ranks that are predicted in \eqref{e:ASOSbound} can only occur when $G=0$, and when $G\neq 0$, but $A(z,\bar z)|\!| z|\!|^2$ is still an SOS, the lower bound \eqref{rmax} always holds. The author has reasons to believe that this optimistic view is indeed what happens, though at this point the reasons are too vague to try to explain in this note. In any case, the following "weak", or alternative form of the SOS Conjecture, if true, then implies the SOS Conjecture \ref{SOSConj}, in view of the Grundmeier-Halfpap result.
\begin{conjecture}[Weak (Alternative) SOS Conjecture]\label{SOSConj2} Let $P(z)=(P^1(z),\ldots,P^q(z))$ be a polynomial mapping in $z=(z^1,\ldots, z^n)\in \mathbb C^n$, and assume that there exists a Hermitian polynomial $A(z,\bar z)$ of the form \eqref{DOS} such that the SOS identity \eqref{SOSid} holds. If $r$ denotes the linear rank of $P(z)$ and if the polynomial mapping $G$ in \eqref{DOS} is not identically zero, then \eqref{rmax} holds. \end{conjecture}
One of the main difficulties in Conjecture \ref{SOSConj2} when $G\neq 0$ comes from the fact that it seems hard to characterize when $A(z,\bar z)|\!| z|\!|^2$ is in fact an SOS of the form \eqref{SOSid}. The reader is referred to, e.g., \cite{JPDVar04}, \cite{JPD11} for discussions related to this difficulty. We can mention here that a necessary condition for an SOS identity \eqref{e:ASOSbound} to hold is that $V_{G\otimes z}\subset V_{F\otimes z}$, where the tensor product of two mappings $F\otimes H$ is defined as the mapping whose components comprise all the products of components $F^jH^k$. From this one can easily see that the linear rank $r=\dim_\mathbb C V_P$ in Conjecture \ref{SOSConj2} must satisfy \begin{equation}\label{specrk} \dim_\mathbb C V_{F\otimes z}/V_{G\otimes z}\leq r\leq \dim_\mathbb C V_{F\otimes z}. \end{equation} The lower bound can only be realized if a maximum number of "cancellations" occur. If we consider the 1-parameter family of Hermitian polynomials $$ A_t(z,\bar z):=|\!| F(z)|\!|^2-t|\!| G(z)|\!|^2 $$ for $0\leq t\leq 1$, where $A(z,\bar z)=A_1(z,\bar z)$ satisfies an SOS identity \eqref{e:ASOSbound}, then clearly $A_t(z,\bar z)|\!| z|\!|^2$ is an SOS for each $0\leq t\leq 1$ (since $A_t(z,\bar z)=A_1(z,\bar z)+(1-t)|\!| G(z)|\!|^2$). One can show that "cancellations" causing strict inequality in the upper bound in \eqref{specrk} do not occur for general $t$ in this range, and the linear rank of $A_t(z,\bar z)|\!| z|\!|^2$ for such $t$ is then $r=\dim_\mathbb C V_{F\otimes z}$. Nevertheless, for the given $A(z,\bar z)=A_1(z,\bar z)$, all we can say seems to be that the estimate \eqref{specrk} holds.
\section{The second fundamental form and the Gauss equation}\label{GaussSec}
We shall utilize E. Cartan's differential systems ("moving frames") approach to CR geometry, as well as S. Webster's theory of psuedohermitian structures. We will follow the set-up and notational conventions introduced in \cite{BEH08} (see also \cite{E15} and \cite{EHZ04}). We shall summarize the notation very briefly here, but refer the reader to \cite{BEH08} (which, on occasion, refers to \cite{EHZ04}) for all details. We shall also from the beginning specialize the general set-up to the special case of CR mappings between spheres, which simplifies matters significantly due to the vanishing of the CR curvature tensor of the sphere. Thus, let $f\colon \mathbb S^n\to \mathbb S^N$ be a smooth CR mapping with $2\leq n\leq N$. For a point $p_0\in \mathbb S^n$, we may choose local adapted (to $f$), admissible (in the sense of Webster \cite{Webster78}) CR coframes $(\theta,\theta^\alpha,\theta^{\bar\alpha})$ on $\mathbb S^n$ near $p_0$ and $(\hat \theta,\hat \theta^A,\hat \theta^{\bar A})$ on $\mathbb S^N$ near $\hat p_0:=f(p_0)$, where the convention in \cite{BEH08} dictates that Greek indices, $\alpha$, etc., range over $\{1,\ldots, n\}$, capital Latin letters, $A$, etc., range over $\{1,\ldots N\}$, and where barring an index on a
previously defined object corresponds to complex conjugation, e.g., $\theta^{\bar\alpha}:=\overline{\theta^\alpha}$. Being adapted means that \begin{equation} f^*\hat\theta=\theta,\quad f^*\hat\theta^\alpha=\theta^\alpha,\quad f^*\hat\theta^a=0, \end{equation} where we have used the further convention that lower case Latin letters $a$, etc., run over the indices $\{N-n+1,\ldots, N\}$. Thus, in particular, $f$ is a (local) pseudohermitian mapping between the (local) pseudohermitian structures obtained on $\mathbb S^n$ and $\mathbb S^N$ by fixing the contact forms $\theta$ and $\hat \theta$ near $p_0$ and $\hat p_0$, respectively. We denote by $g_{\alpha\bar\beta}$, $\hat g_{A\bar B}$ the respective Levi forms (which can, and later will be both assumed to be the identity), and by $\omega_\alpha{}^\beta$, $\hat \omega_A{}^B$ the Tanaka-Webster connection forms. We shall pull all forms and tensors back to $\mathbb S^n$ by $f$, and for convenience of notation, we shall simply denote by $\hat\omega_A{}^B$ the pulled back form $f^*\hat\omega_A{}^B$, etc. Moreover, the fact that the two coframes are adapted implies that we can drop the $\hat{}$ on the pullbacks to $\mathbb S^n$ without any risk of confusion; in other words, we have, e.g., $\omega_\alpha{}^\beta=\hat\omega_\alpha{}^\beta$ and $g_{\alpha\bar\beta}=\hat g_{\alpha\bar\beta}$ (we repeat here that we refer to \cite{BEH08} and \cite{EHZ04} for the details), and of course $\omega_\alpha{}^a$, e.g., can have only one meaning.
The collection of 1-forms $(\omega_{\alpha}^{\:\:\:a})$ on $\mathbb S^n$ defines the \emph{second fundamental form} of the mapping $f$, denoted $\Pi_f\colon T^{1,0}\mathbb S^n\times T^{1,0}\mathbb S^n\to T^{1,0}\mathbb S^N/f_*T^{1,0}\mathbb S^n$, as described in \cite{BEH08}. We recall from there that \begin{equation}\label{SFF1} \omega_{\alpha}^{\:\:\:a} = \omega_{\alpha \:\:\: \beta}^{\:\:\:a}\theta^{\beta}, \qquad \omega_{\alpha \:\:\: \beta}^{\:\:\:a} = \omega_{\beta \:\:\: \alpha}^{\:\:\:a}. \end{equation} If we identify the CR-normal space $T_{f(p)}^{1,0}\mathbb S^N/f_*T_{p}^{1,0}\mathbb S^n$, also denoted by $N_{p}^{1,0}{\mathbb S^n}$, with $\mathbb{C}^{N-n}$, then we may identify $\Pi_f$ with the $\mathbb C^{N-n}$-valued, symmetric $n\times n$ matrix $(\omega_{\alpha}{}^a{}_ \beta)_{a=n+1}^{N}$. We shall not be so concerned with the matrix structure of this object, and consider $\Pi_f$ as the collection, indexed by $\alpha, \beta$, of its component vectors $(\omega_{\alpha}{}^a{}_ \beta)_{a=n+1}^{N}$ in $\mathbb{C}^{N-n}$. By viewing the second fundamental form as a section over $\mathbb S^n$ of the bundle $(T^*)^{1,0}\mathbb S^n\otimes N^{1,0}{\mathbb S^n} \otimes (T^*)^{1,0}\mathbb S^n$, we may use the pseudohermitian connections on $\mathbb S^n$ and $\mathbb S^N$ to define the covariant differential \begin{equation*} \nabla \omega_{\alpha\:\:\beta}^{\:\:a} = d\omega_{\alpha\:\:\beta}^{\:\:a} - \omega_{\mu\:\:\beta}^{\:\:a}\omega_{\alpha}^{\:\:\mu} + \omega_{\alpha\:\:\beta}^{\:\:b}\omega_{b}^{\:\:a} - \omega_{\alpha\:\:\mu}^{\:\:a}\omega_{\beta}^{\:\:\mu}. \end{equation*} We write $\omega_{\alpha\:\:\beta ; \gamma}^{\:\:a}$ to denote the component in the direction $\theta^{\gamma}$ and define higher order derivatives inductively as: \begin{equation*} \nabla \omega_{\gamma_{1}\:\:\gamma_{2};\gamma_{3}\ldots\gamma_{j}}^{\:\:a} = d\omega_{\gamma_{1}\:\:\gamma_{2};\gamma_{3}\ldots\gamma_{j}}^{\:\:a} + \omega_{\gamma_{1}\:\:\gamma_{2};\gamma_{3}\ldots\gamma_{j}}^{\:\:b}\omega_{b}^{\:\:a} - \sum_{l=1}^{j}\omega_{\gamma_{1}\:\:\gamma_{2};\gamma_{3}\ldots\gamma_{l-1}\mu \gamma_{l+1}\ldots\gamma_{j}}^{\:\:a}\omega_{\gamma_{l}}^{\:\:\mu}. \end{equation*} A tensor $T_{\alpha_1\ldots\alpha_r\bar\beta_1\ldots\bar\beta_s}{}^{a_1\ldots a_t\bar b_1\ldots\bar b_q}$, with $r,s\geq1$, is called {\em conformally flat} if it is a linear combination of $g_{\alpha_i\bar\beta_j}$ for $i=1,\ldots,r$, $j=1,\ldots,s$, i.e. \begin{equation} T_{\alpha_1\ldots\alpha_r\bar\beta_1\ldots\bar\beta_s}{}^{a_1\ldots a_t\bar b_1\ldots\bar b_q}=\sum_{i=1}^r\sum_{j=1}^s g_{\alpha_i\bar\beta_j} (T_{ij})_{\alpha_1\ldots\widehat{\alpha_i}\ldots \alpha_r\bar\beta_1\ldots\widehat{\bar\beta_j}\ldots\ldots\bar\beta_s}{}^{a_1\ldots a_t\bar b_1\ldots\bar b_q}, \end{equation} where e.g.\ $\widehat{\alpha}$ means omission of that factor. (A similar definition can be made for tensors with different orderings of indices.) The following observation gives a motivation for this definition. Let $T_{\alpha_1\ldots\alpha_r\bar\beta_1\ldots\bar\beta_s}{}^{a_1\ldots a_t\bar b_1\ldots\bar b_q}$ be a tensor, symmetric in $\alpha_1,\ldots,\alpha_r$ as well as in $\beta_1,\ldots,\beta_s$, and form the homogeneous vector-valued polynomial of bi-degree $(r,s)$ whose components are given by $$T^{a_1\ldots a_t\bar b_1\ldots\bar b_q}(z,\bar z):= T_{\alpha_1\ldots\alpha_r\bar\beta_1\ldots\bar\beta_s}{}^{a_1\ldots a_t\bar b_1\ldots\bar b_q}z^{\alpha_1}\ldots z^{\alpha_r}\overline{z^{\beta_1}}\ldots\overline{z^{\beta_s}}, $$ where $z=(z^1,\ldots,z^n)$ and the usual summation convention is used. Then, the reader can check that the tensor is conformally flat if and only if all the polynomials $T^{a_1\ldots a_t\bar b_1\ldots\bar b_q}(z,\bar z)$ are divisible by the Hermitian form $g(z,\bar z):=g_{\alpha\bar\beta}z^\alpha\overline{z^\beta}$. Moreover, and importantly, a conformally flat tensor has the property that its covariant derivatives are again conformally flat, since one of the defining properties of the pseudohermitian connection is that $\nabla g_{\alpha\bar\beta}=0$. We shall use the terminology that $T_{\alpha_1\ldots\alpha_r\bar\beta_1\ldots\bar\beta_s}{}^{a_1\ldots a_t\bar b_1\ldots\bar b_q}\equiv 0 \mod \CFT$ if the tensor is conformally flat.
Now, the Gauss equation for the second fundamental form of a CR mapping $f\colon \mathbb S^n\to \mathbb S^N$ takes the following simple form (since the CR curvature tensors of $\mathbb S^n$ and $\mathbb S^N$ vanish): \begin{equation}\label{Gauss} g_{a\bar b}\omega_{\alpha}{}^a{}_{\nu}\omega_{\bar\beta}{}^{\bar b}{}_{\bar\mu}\equiv 0\mod\CFT. \end{equation} We proceed as in the proof of Theorem 5.1 in \cite{BEH08} and take repeated covariant derivatives in $\theta^{\gamma_r}$ and $\theta^{\bar \lambda_s}$ in the Gauss equation. By using the fact that $\omega_\alpha{}^a{}_{\beta;\bar\mu}$ is conformally flat (Lemma 4.1 in \cite{BEH08}) and the commutation formula in Lemma 4.2 in \cite{BEH08}, we obtain the full family of Gauss equations, for any $r,s\geq 2$: \begin{equation}\label{GaussFull} g_{a\bar b}\omega_{\gamma_1}{}^a{}_{\gamma_2;\ldots\gamma_r}\omega_{\bar\lambda_1}{}^{\bar b}{}_{\bar\lambda_2;\ldots\bar\lambda_s}\equiv 0\mod\CFT. \end{equation} We now consider also the component vectors of higher order derivatives of $\Pi_f$ as elements of $\mathbb{C}^{N-n}\cong N_p^{1,0}S^n$ and define an increasing sequence of vector spaces \begin{equation*} E_{2}(p) \subseteq \ldots \subseteq E_{l}(p) \subseteq \ldots \subseteq \mathbb{C}^{N-n}\cong N_p^{1,0}\mathbb S^n \end{equation*} by letting $E_{l}(p)$ be the span of the vectors \begin{equation}\label{Eldef} (\omega_{\gamma_{1}\:\:\gamma_{2};\gamma_{3}\ldots\gamma_{j}}^{\:\:a})_{a=n+1}^{N}, \qquad \forall\, 2 \leq j \leq l, \gamma_{j}\in \{1,\ldots,n\}, \end{equation} evaluated at $p \in \mathbb S^n$. We let $d_l(p)$ be the dimension of $E_l(p)$, and for convenience we set $d_1(p)=0$. As is mentioned in \cite{E12}, it is shown in \cite{EHZ04} that $d_l(p)$ defined in this way coincides with the $d_l(p)$ defined by (1.3) in \cite{E12}. By moving to a nearby point $p_0$ if necessary, we may assume that all $d_l=d_l(p)$ are locally constant near $p_0$ and \begin{equation}\label{dimstab} 0=d_1<d_2<\ldots<d_{l_0}=d_{l_0+1}=\ldots\leq N-n \end{equation} for some $1\leq l_0\leq N-n+1$ (with $l_0=1$ if $d_2=0$ near such generic $p_0$). The mapping $f$ is said to be constantly $l_0$-degenerate of rank $d:=d_{l_0}\leq N-n$ at $p_0$; the codimension $N-n-d$ is called the degeneracy and if the degeneracy is $0$, then the mapping is also said to be $l_0$-nondegenerate.
For each integer $l\geq 2$, we form the $\mathbb C^{N-n}$-valued, homogeneous polynomial $\Omega_{(l)}=(\Omega^1_{(l)},\ldots,\Omega^{N-n}_{(l)})$ in $z=(z^1,\ldots,z^n)\in \mathbb C^n$ as follows: \begin{equation}\label{Omega(l)} \Omega^j_{(l)}(z):= \omega_{\gamma_{1}\:\:\gamma_{2};\gamma_{3}\ldots\gamma_{l}}^{\:\:a}z^{\gamma_1}\ldots z^{\gamma_l}, \quad a=n+j, \end{equation} and we define the {\it total second fundamental polynomial} $\Omega=(\Omega^1,\ldots,\Omega^{N-n})$ of $f$ near $p_0$ as follows: \begin{equation} \Omega^j(z):=\sum_{l=2}^{l_0}\Omega^j_{(l)}(z), \end{equation} where $l_0$ is the integer, defined above, where the dimensions $d_l$ stabilize. The following proposition is easily proved by using the fact that the rank of a matrix equals that of its transpose; the details are left to the reader.
\begin{proposition}\label{Omegarank} The rank $d=d_{l_0}$ of the $l_0$-degeneracy is also the linear rank of the polynomial mapping $\Omega(z)$, i.e., the dimension of the vector space in $\mathbb C[z]$ spanned by the polynomials $\Omega^1(z),\ldots, \Omega^{N-n}(z)$. \end{proposition}
We now recall, as mentioned above, that we may choose the adapted, admissible CR coframes (near $p_0$ and $\hat p_0=f(p_0)$) in such a way that the Levi forms of $\mathbb S^n$ and $\mathbb S^N$ both equal the identity matrix. Let us now insist on such a choice of coframes. We then notice that the full family of Gauss equations in \eqref{GaussFull} for $r,s\leq l_0$ can be summarized in the following Sum-Of-Squares identity for the total second fundamental polynomial.
\begin{lemma}[Total polynomial Gauss equation]\label{GaussLemma} There exists a Hermitian polynomial $A(z,\bar z)$ such that \begin{equation}\label{GaussLemmaId} |\!|\Omega(z)|\!|^2=A(z,\bar z)|\!| z|\!|^2, \end{equation}
where the notation $|\!|\Omega(z)|\!|^2:=\sum_{j=1}^{N-n} |\Omega^j(z)|^2$ introduced in the introduction has been used. \end{lemma}
\begin{proof} The proof consists of multiplying the identities \eqref{GaussFull} by $z^{\gamma_1}\ldots z^{\gamma_r}\overline{z^{\lambda_1}\ldots z^{\lambda_s}}$ and summing according to the summation convention. The conformally flat tensors on the right hand sides all contain a factor of $|\!| z|\!|^2$. The proof is then completed by comparing the polynomial identities obtained in this way to the result of expanding the left hand side of \eqref{GaussLemmaId} and collecting terms of a fixed bidegree $(r,s)$. The details are left to the reader. \end{proof}
\section{Proof of Theorem \ref{MainThm}}\label{Proof}
We shall prove Conjecture \ref{HJYConj} under the assumption that the conclusion of Conjecture \ref{SOSConj} holds. We quote first Theorem 1.1 in \cite{E12}, in the special case of CR mappings $f\colon \mathbb S^n\to \mathbb S^N$:
\begin{theorem}[\cite{E12}]\label{Thm1.1} Let $f\colon \mathbb S^n \to \mathbb S^N$ be a smooth CR mapping and the dimensions $d_l(p)$ be as defined in Section $\ref{GaussSec}$. Let $U$ be an open subset of $\mathbb S^n$ on which $f$ is constantly $l_0$-degenerate, and such that $d_l=d_l(p)$, for $2\leq l\leq l_0$, are constant on $U$ and \eqref{dimstab} holds. Assume that there are integers $0\leq k_2, k_3,\ldots,k_{l_0}\leq n-1$, such that: \begin{equation} \begin{aligned}\label{conds0} d_l-d_{l-1} <&\sum_{j=0}^{k_l}(n-j),\quad l=2,\ldots, l_0,\quad (d_1=0)\\ k:=&\sum_{l=2}^{l_0}k_l<n. \end{aligned} \end{equation} Then $f(\mathbb S^n)$ is contained in a complex affine subspace $A^{n+d+k+1}$ of dimension $n+d+k+1$, where $k$ is defined in \eqref{conds0} and $d:=d_{l_0}$ is the rank of the $l_0$-degeneracy. \end{theorem}
\begin{remark} {\rm The integers $k_2,\ldots, k_{l_0}$ become invariants of the mapping $f$ if we require them to be minimal in an obvious way. The invariant $k_2$ was introduced in \cite{Huang03} and called there the geometric rank of $f$. This geometric rank plays an important role in \cite{Huang03}, \cite{HuangJiXu06}, and \cite{HuangJiYin12}. } \end{remark}
\begin{proof}[Proof of Theorem $\ref{MainThm}$] We assume now that there is a mapping $f\colon \mathbb S^n \to \mathbb S^N$ with codimension $N-n\in I_\kappa$ for some $\kappa\leq \kappa_0<n$. Thus, we have $$ N-n\leq \sum_{i=0}^{\kappa-1}(n-i)-1. $$ We consider an open subset $U\subset \mathbb S^n$ as in Theorem \ref{Thm1.1}. Since the rank of the $l_0$-degeneracy satisfies $d\leq N-n$, we then have \begin{equation}\label{dest} d\leq \sum_{i=0}^{\kappa-1}(n-i)-1, \end{equation} in Theorem \ref{Thm1.1}. By Proposition \ref{Omegarank}, $d$ is also the linear rank of the total second fundamental polynomial $\Omega(z)$, and by Lemma \ref{GaussLemma}, an SOS identity of the form \eqref{GaussLemmaId} holds. If we now assume that the SOS Conjecture \ref{SOSConj} holds, then \eqref{dest} implies that in fact \begin{equation}\label{SOScons} d\leq (\kappa-1)n. \end{equation} It is also clear from \eqref{dest} that there exist integers $0\leq k_l\leq \kappa-1$ such that the first identity in \eqref{conds0} hold. We shall choose the $k_j$ minimal, so that in addition we have \begin{equation}\label{klmin} d_l-d_{l-1}\geq \sum_{j=0}^{k_l-1}(n-j), \end{equation} where the right hand side is understood to be $0$ if $k_l=0$. We claim that \begin{equation}\label{kclaim} k:=\sum_{l=2}^{l_0}k_l \leq \kappa-1. \end{equation} If we can prove this claim, then it follows from Theorem \ref{Thm1.1}, since $\kappa\leq \kappa_0<n$, that $f(\mathbb S^n)$ is contained in a complex affine subspace $A^{N_0+1}$ of dimension $N_0=n+d+k$, and the codimension satisfies, by \eqref{SOScons} and \eqref{kclaim}, $$ N_0-n=d+k\leq (\kappa-1)n+\kappa-1, $$ which is precisely the desired conclusion in the Gap Conjecture \ref{HJYConj}. Thus, we proceed to prove \eqref{kclaim}. Let us denote by $g(j)$ the non-increasing function \begin{equation} g(j)=\left\{ \begin{aligned} n-j,&\quad 0\leq j< n\\ 0,&\quad j\geq n. \end{aligned} \right. \end{equation} Using the fact that we have set $d_1=0$, we can telescope $d$ as follows \begin{equation} d=(d_{l_0}-d_{l_0-1})+\ldots (d_2-d_1)=\sum_{l=2}^{l_0}(d_l-d_{l-1}), \end{equation} and deduce from \eqref{klmin} that \begin{equation}\label{dest2} d\geq \sum_{l=2}^{l_0}\sum_{j=0}^{k_l-1}(n-j)=\sum_{l=2}^{l_0}\sum_{j=0}^{k_l-1}g(j). \end{equation} Since $g(j)$ is non-increasing, we can estimate \begin{equation}\label{shift} \sum_{l=2}^{l_0}\sum_{j=0}^{k_l-1}g(j)\geq \sum_{l=2}^{l_0}\sum_{j=0}^{k_l-1}g\left(j+m_l\right), \end{equation} where we have set $m_2=0$ and, for $3\leq l\leq l_0$, \begin{equation} m_l:=\sum_{i=2}^{l-1}k_{i}. \end{equation} Substituting $i=j+m_l$ in \eqref{shift}, we deduce from \eqref{dest2} \begin{equation} d\geq \sum_{l=2}^{l_0}\sum_{i=m_l}^{m_l+k_l-1}g(i)=\sum_{l=2}^{l_0}\sum_{i=m_l}^{m_{l+1}-1}g(i)= \sum_{i=0}^{m_{l_0+1}-1}g(i). \end{equation} Since $m_{l_0+1}=k$, we conclude that \begin{equation} d\geq \sum_{i=0}^{k-1}g(i), \end{equation} and since $k<n$, we also have $g(i)=n-i$ for $i=1,\ldots, k-1$, and therefore we can write \begin{equation} \sum_{i=0}^{k-1}(n-i)\leq d. \end{equation} By comparing this with \eqref{dest}, we conclude that $k-1<\kappa-1$, which establishes the claim \eqref{kclaim}. This completes the proof of Theorem \ref{MainThm}.
\end{proof}
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\title[Involutions of second kind on Shimura surfaces]{Involutions of second kind on Shimura surfaces and surfaces of general type with $q=p_g=0$}
\author{Amir D\v{z}ambi\'{c}, Xavier Roulleau}
\maketitle
\begin{abstract} Quaternionic Shimura surfaces are quotient of the bidisc by an irreducible cocompact arithmetic group. In the present paper we are interested in (smooth) quaternionic Shimura surfaces admitting an automorphism with one dimensional fixed locus; such automorphisms are involutions. We propose a new construction of surfaces of general type with $q=p_{g}=0$ as quotients of quaternionic Shimura surfaces by such involutions. These quotients have finite fundamental group. \end{abstract}
\section{Introduction}
Among smooth minimal surfaces of general type, the ones with vanishing geometric genus $p_{g}$ are of main interest (see e.g. \cite{bauercatanesepignatelli2}). For such surfaces, the Chern number $c_{1}^{2}=K^{2}$ belongs to the set $\{1,\dots,9\}$ and $c_{2}=12-c_{1}^{2}$. We are far away from a complete classification, although great advances have been done recently, e.g. for surfaces with $c_{1}^{2}=9$, the fake projective planes, which have been completely classified, see \cite{Prasad}, \cite{Steger}. In the other cases, a major task is to construct new examples of such surfaces.
In this paper we give an uniform construction of surfaces with $q=p_{g}=0$ and $c_{1}^{2}=1,\dots,7$. These surfaces are obtained as quotients of smooth quaternionic Shimura surfaces $X$ by a special kind of involution. Recall that a smooth Shimura surface $X=X_{\Gamma}$ is the quotient of $\mathbb{H}\times\mathbb{H}$, the product of two copies of the complex upper half plane, by a discrete cocompact torsion free group $\Gamma\subset{\rm Aut}(\mathbb{H})\times{\rm Aut}(\mathbb{H})$ of holomorphic automorphisms defined by certain quaternion algebra. The invariants of $X$ are $c_{1}^{2}(X)=2c_{2}(X)=8(1+p_{g}(X))$ and $q(X)=0$.
In the first step we prove that an automorphism of a Shimura surface is quite special: \begin{thm} Let $\sigma$ be an automorphism of a smooth Shimura surface. Then either $\sigma$ has a finite number of fixed points or the fixed point set of $\sigma$ is a divisor and $\sigma^{2}=1$. \end{thm} Under certain conditions on the group $\Gamma$, the involution $\mu\in{\rm Aut}(\mathbb{H}\times\mathbb{H})$ exchanging the two factors induces an involution $\sigma$ on the surface $X$. We call such an automorphism of $X$ an involution of second kind. We obtain \begin{thm} Let $X$ be a smooth Shimura surface admitting an involution of second kind $\sigma.$ The fixed point set $C$ of $\sigma$ is a a union of disjoint smooth Shimura curves. The arithmetic genus $g$ of $C$ satisfies $2\leq g\leq p_{g}(X)$ and the quotient surface $Z=X/\sigma$ is smooth with finite fundamental group. If moreover $g=p_{g}(X)\leq8$, $Z$ is a surface of general type with invariants: \[ c_{1}(Z)^{2}=9-p_{g}(X),\, c_{2}(Z)=3+p_{g}(X),\, p_{g}(Z)=q(Z)=0. \]
\end{thm} If $p_{g}=2$ or $3$, one can prove that $g=p_{g}$ and therefore $c_{1}^{2}=7$ and $6$ respectively.
We then concentrate on the construction of such Shimura surfaces with low $p_{g}$ and admitting an involution of second kind. It turns out that such surfaces are rather exceptional. For instance, if we restrict our consideration to totally real fields of degree 2, there are at most 14 isomorphism classes of quaternion algebras leading to smooth Shimura surfaces of geometric genus $2\leq p_{g}\leq8$ admitting an involution of second kind (see Theorem \ref{list_real_quadratic_candidates}). On the other hand we consider Shimura surfaces corresponding to congruence subgroups and we are able to show: \begin{thm} For $k=5,6$ there exists a smooth Shimura surface $X$ with an involution of second kind $\sigma$ and $p_{g}(X)=k$. In the case $k=5$, the curve $C$ fixed by $\sigma$ is irreducible of genus $g(C)=5=p_{g}(X)$. \end{thm} In the light of some open questions concerning fundamental groups of surfaces with geometric genus zero, see \cite{bauercatanesepignatelli2}, an example of a smooth Shimura surface with $p_{g}=2$ admitting an involution of second kind would be highly interesting. However, finding such a surface turns out to be very difficult (see Remark \ref{remark_deg_2}).
This paper mixes two fields: Theory of Shimura surfaces and classical algebraic geometry of surfaces. Shimura surfaces are closely related to Hilbert modular surfaces (which were first systematically studied by Hirzebruch, see for instance \cite{VanderGeer}, but are less known and studied. It was also one of our aim to develop that theory of Shimura surfaces.
The paper is organized as follows: In Section \ref{sec:Involutions and group gamma} we discuss the conditions on $\Gamma$ under which the Shimura surface $X_{\Gamma}$ has an involution of second kind. In Section \ref{sec:Quotient and fundamental group} we study the quotient surface of a smooth Shimura surface by the action of an involution of second kind and we prove that its fundamental group is finite. In Section \ref{sec:Examples}, we investigate examples of Shimura surfaces with low geometric genus admitting an involution of second kind. In particular we develop new tools to create smooth quaternionic Shimura surfaces and we make a systematic study of Shimura surfaces defined over quadratic fields with low geometric genus. We give examples of surfaces with $p_{g}=5,6$ admitting an involution of second kind. Finally, in section \ref{sec:determination-of-the-curve} we present the method to identify the Shimura curve fixed by an involution of second kind acting on a Shimura surface and we furthermore examine the example with $p_{g}=5$.
\section{Involutions of second kind acting on Shimura surfaces}
\label{sec:Involutions and group gamma}
\subsection{Quaternionic Shimura surfaces.\label{sub:Notations,-quaternionic-Shimura}}
Let us recall the construction of quaternionic Shimura surfaces.
Let $k$ be a totally real number field of degree $n=[k:\mathbb{Q}]$. The places of $k$ are the equivalence classes of valuations on $k$, and the infinite places of $k$ correspond to embeddings $\sigma_{i}\in{\rm Hom}_{\mathbb{Q}}(k,\mathbb{R})$, $i=1,\ldots,n$.
Let $A$ be a division quaternion algebra whose center is $k$. For every place $v$ of $k$, we denote by $k_{v}$ the completion of $k$ with respect to $v$ and define $A_{v}=A\otimes_{k}k_{v}$. The algebra $A$ is \emph{ramified} at $v$ if $A_{v}$ is a division algebra over $k_{v}$ and \emph{unramified} otherwise, that is, if $A_{v}\cong M_{2}(k_{v})$. By the classical theorem of Hasse and the product formula for Hilbert symbols, the isomorphism class of $A$ is uniquely determined by the set $Ram(A)$ of ramified places of $A$. Assume that $A$ is unramified at the first, say $m\leq n$, infinite places $\sigma_{1},\ldots,\sigma_{m}$ and ramified at the remaining $n-m$ infinite places. Equivalently we assume that we have an isomorphism \[ A\otimes_{\mathbb{Q}}\mathbb{R}\to M_{2}(\mathbb{R})^{m}\times\mathbf{H}^{n-m} \]
where $\mathbf{H}$ denotes the skew field of Hamiltonian quaternions. Then, $A$ is uniquely determined up to an isomorphism by the tuple \[ (k,\sigma_{1},\ldots,\sigma_{m},\mathfrak{p}_{1},\dots,\mathfrak{p}_{r}), \]
where $\mathfrak{p}_{1},\dots,\mathfrak{p}_{r}$ are the non-archimedean places of $k$ where $A$ is ramified, and $\sigma_{1},\ldots,\sigma_{m}$ are the infinite places of $k$ such that $A\otimes_{\sigma_{i}(k)}\mathbb{R}\cong M_{2}(\mathbb{R})$. Since we are interested in algebraic surfaces, we suppose that there are exactly two infinite places $\sigma_{1},\sigma_{2}$ such that $A$ is unramified at the $\sigma_{i}$. With this fixed ramification behaviour at the infinite places, the isomorphism class of $A$ depends only on the tuple $(\mathfrak{p}_{1},\dots,\mathfrak{p}_{r})$. Let us write \[ A=A(k,\mathfrak{p}_{1},\dots,\mathfrak{p}_{r}), \] (omitting the explicit indication of two unramified infinite places) for such a quaternion algebra in the following. The subgroup $A^{+}$ of $A$ consisting of the units of $A$ having totally positive reduced norm can be identified via the isomorphism $A\otimes_{\mathbb{Q}}\mathbb{R}\stackrel{f}{\rightarrow}M_{2}(\mathbb{R})^{2}\times\mathbf{H}^{n-2}$ with a subgroup of $GL_{2}^{+}(\mathbb{R})^{2}\times\mathbf{H}^{*n-2}$, and projecting to the first two factors gives an injection of $A^{+}$ into $GL_{2}^{+}(\mathbb{R})^{2}$. We denote by \[ \rho=(\rho_{1},\rho_{2}):A\rightarrow M_{2}(\mathbb{R})^{2} \] this representation of $A$. Note that these $\rho_{i},\, i=1,2$ are extensions of two morphisms $k\rightarrow\mathbb{R}$ (where $\mathbb{R}\subset M_{2}(\mathbb{R})$ is identified with diagonal matrices) corresponding to the places $\sigma_{1},\sigma_{2}$.
Let us denote by $\mathcal{O}$ and $\mathcal{O}_{k}$ a maximal order of $A$ and the ring of integers of $k$. Let $\mathcal{O}^{*}$ denote the group of units of $\mathcal{O}$, $\mathcal{O}^{+}$ the group of units in $\mathcal{O}$ with totally positive reduced norm and $\mathcal{O}^{1}\subset\mathcal{O}$ the group of units of reduced norm $1$. The group $\mathcal{O}^{1}$ is via $\rho$ a discrete subgroup of $SL_{2}(\mathbb{R})^{2}$, whereas $\mathcal{O}^{+}$ is embedded as a discrete subgroup in $GL_{2}^{+}(\mathbb{R})^{2}$.
The group $A^{+}$ and any subgroup $G\subset A^{+}$ acts on the product $\mathbb{H}\times\mathbb{H}$ of two copies of the upper half plane as follows: if $(z,w)\in\mathbb{H}\times\mathbb{H}$ is a point and $g\in G$, then $\rho(g)$ is represented by two matrices $g_{i}=\rho_{i}(g)=\left(\begin{array}{cc} a_{i} & b_{i}\\ c_{i} & d_{i} \end{array}\right)$ ($i=1,2$) and: \[ g(z,w)=(g_{1}z,g_{2}w):=(\frac{a_{1}z+b_{1}}{c_{1}z+d_{1}},\frac{a_{2}w+b_{2}}{c_{2}w+d_{2}}), \]
where $a_{1},a_{2}$ (resp. $(b_{1},b_{2})\ldots$) are conjugates with respect to the places $\sigma_{1},\sigma_{2}$ over the Galois closure of $k$ in $\mathbb{R}$. The action of $G$ is not effective; the center $Z(G)$ acts trivially on $\mathbb{H}\times\mathbb{H}$ and therefore we will consider subgroups $\Gamma=G/Z(G)\subset A^{+}/k^{\ast}$ rather than subgroups $G\subset A^{+}$. Let us write $\Gamma_{\mathcal{O}}(1)$ or sometimes simply $\Gamma(1)$ for the group $\mathcal{O}^{1}/\{\pm1\}$. The group $\Gamma(1)$ and also any subgroup $\Gamma$ of $A^{+}/k^{\ast}$ commensurable with $\Gamma(1)$ acts properly discontinuously on $\mathbb{H}\times\mathbb{H}$.
The quotient $X_{\Gamma}=\mathbb{H}\times\mathbb{H}/\Gamma$ is a compact algebraic surface, which will be called \emph{quaternionic Shimura surface} (corresponding to $\Gamma$) in the sequel.
\subsection{The involution exchanging factors and involutions of the second kind.}
Let, as above, $k$ be a totally real number field and consider the quaternion algebra $A=A(k,\mathfrak{p}_{1},\dots,\mathfrak{p}_{r})$ over $k$ unramified at two infinite places of $k$ and ramified at all the other infinite places of $k$ and at the finite places $\mathfrak{p}_{1},\ldots,\mathfrak{p}_{r}$. \begin{defn} An \emph{involution of second kind} on $A$ is a map $\tau:A\to A$ such that $\tau^{2}(a)=a$, $\tau(a+b)=\tau(a)+\tau(b)$, $\tau(ab)=\tau(a)\tau(b)$ for all $a,b\in A$ and such that the restriction of $\tau$ to $k$ is a non-trivial automorphism of $k$. \end{defn} Let $\ell=k^{\tau}$ be the fixed field of $\tau$. Then $k/\ell$ is a quadratic extension and in this case we will say that $\tau$ is a \emph{$k/\ell$-involution}. \begin{rem} (See also \cite[Lemma 4.2]{Granath}). Let $\tau$ and $\sigma$ be two $k/\ell$-involutions of second kind on $A$. Then there exists $m\in A^{*}$ such that $\sigma(a)=m^{-1}\tau(a)m$ and $\tau(m)^{*}=m$, where $a\to a^{*}$ is the canonical anti-involution on $A$, that is, the uniquely determined anti-involution $\ast:A\to A$ such that the reduced norm is $Nrd(x)=xx^{*}$ and the reduced trace is $Trd(x)=x+x^{*}$.
Following \cite{Granath} we choose to work with involutions on $A$ which are, by above definition, particularly ring homomorphisms of $A$. In the literature more often one works with anti-involutions, that is, with maps $\rho:A\to A$ satisfying $\rho(ab)=\rho(b)\rho(a)$. These two kinds of maps are linked by the canonical anti-involution. First observe that every involution of second kind commutes with the canonical anti-involution, that is, for every $a\in A$ we have $\tau(a)^{*}=\tau(a^{*})$, see \cite{Granath}. From this it follows that there is one-to-one correspondence between involutions and anti-involutions on $A$ given by the rule \[ \tau\mapsto\tau^{*}\ \text{and}\ \rho\mapsto\rho^{*} \] where $\tau^{*}$, resp.~$\rho^{*}$, is defined as $\tau^{*}(a)=\tau(a)^{*}$, resp.~$\rho^{*}(a)=\rho(a)^{*}$, for every $a\in A$. In this way, involutions of second kind correspond to classically studied anti-involutions of second kind. \end{rem} A criterion for the existence of anti-involutions of second kind is well-known and goes back to work of Albert and Landherr. \begin{prop} \label{existence_type_2}(See also \cite[Lemma 4.3]{Granath} and more generally \cite[Theorem 3]{Landherr}). Let $k/\ell$ be a quadratic extension of totally real fields. Let $\alpha$ be the non-trivial $\ell$-automorphism of the extension $k/\ell$ and let $A=A(k,\mathfrak{p}_{1},\dots,\mathfrak{p}_{r})$. Then, there exists a $k/\ell$-involution $\tau$ of second kind on $A$ (i.~e.~$\tau(xa)=x^{\alpha}\tau(a)\ \text{for all}\ x\in k,a\in A$) if and only if \begin{enumerate} \item $r$ is even and after a suitable renumbering of the $\mathfrak{p}_{i}$, we have $\mathfrak{p}_{2i-1}=\mathfrak{p}_{2i}^{\alpha}$ and $\mathfrak{p}_{i}\neq\mathfrak{p}_{i}^{\alpha}$ ($i=1,\ldots,r/2$). \item With the notations and hypothesis of section \ref{sub:Notations,-quaternionic-Shimura}, it holds that $\sigma_{2}=\sigma_{1}\circ\alpha$. \end{enumerate} \end{prop} \begin{proof} Let us first show that there exists an $k/\ell$-involution $\tau$ on $A$ if and only if there exists a quaternion subalgebra $A'\subset A$ whose center is $\ell$.\\
Namely, if such $A'$ exists then $A'\otimes_{\ell}k=A$ and $a\otimes x\stackrel{\tau}{\mapsto}a\otimes x^{\alpha}$ is a $k/\ell$-involution. Conversely, let $A'=\{a\in A\mid\tau(a)=a\}$. This is a $\ell$-subalgebra of $A$. Let $\theta\in k$ be such that $k=\ell(\theta)$ and $\tau(\theta)=-\theta$. We claim that $A=A'\oplus\theta A'$. To see this we write an element $a\in A$ as $a=\frac{{1}}{2}(a+\tau(a))+\frac{1}{2}\theta\frac{a-\tau(a)}{\theta}$. It follows that $A=A'+\theta A'$. Clearly, an element $a\in A'\cap\theta A'$ satisfies $\tau(a)=a=-a,$ hence $a=0$ and it follows that $kA'\cong k\otimes_{\ell}A'=A$ and therefore $A'$ is a quaternion algebra over $\ell$ since $A=A'\otimes_{\ell}k$ is a quaternion algebra.
Now we compare the sets of ramification of $A$ and $A'$. If $v$ is a place of $\ell$ such that $A'\otimes_{\ell}\ell_{v}\cong M_{2}(\ell_{v})$, then for any place $w$ of $k$ lying over $v$ we have $A\otimes_{k}k_{w}\cong M_{2}(k_{w})$, so if $A'$ is unramified at a place $v$ of $\ell$, then $A$ is unramified at every place $w$ of $k$ lying over $v$. Particularly, since $k$ and $\ell$ are totally real, if $v$ is an infinite place of $\ell$, that is, an embedding $v:\ell\hookrightarrow\mathbb{R}$ then there are exactly two places of $k$ lying over $v$, that is, two embeddings $w_{1},w_{2}:k\hookrightarrow\mathbb{R}$ extending $v$ and satisfying $w_{2}=w_{1}\circ\alpha$.
Assume now that $v$ is a place of $\ell$ where $A'$ is ramified. If $v$ is an infinite place of $\ell$, then, again since $k$ and $\ell$ are both totally real, $A$ is ramified at every place $w$ of $k$ over $v$. Assume that $v$ is a finite place of $\ell$ corresponding to a prime ideal $\mathcal{P}$. There are two different possibilities: $\mathcal{P}$ is split in $k$, that is, $\mathcal{P}\mathcal{O}_{k}=\mathfrak{p}\mathfrak{p}^{\alpha}$, with two prime ideals $\mathfrak{p}\neq\mathfrak{p}^{\alpha}$ of $k$. As $k_{\mathfrak{p}}=\ell_{\mathcal{P}}$, $A'$ is ramified at $\mathfrak{\mathcal{P}}$ if and only if $A$ is ramified at $\mathfrak{p}$. If $\mathcal{P}$ is non-split in $k$, i.~e., if only one prime $\mathfrak{p}$ is lying over $\mathcal{P}$, then by the theorem of Hasse, the field $k_{\mathfrak{p}}$ can be embedded into $A'\otimes_{\ell}\ell_{\mathcal{P}}$ and therefore $A'\otimes_{\ell}\ell_{\mathcal{P}}\cong M_{2}(k_{\mathfrak{p}})$. In this case $A$ is unramified at $\mathfrak{p}$. It follows, that the existence of a $k/\ell$-involution implies the conditions (1) and (2).
Conversely, choosing the set of ramified places according to (1) and (2), we can construct a quaternion algebra $A'$ over $\ell$ such that $A'\otimes_{\ell}k$ and $A$ are ramified exactly at the same set of primes. This implies an isomorphism $A\cong A'\otimes_{\ell}k$ and thus the existence of an $k/\ell$-involution of second kind \end{proof} \begin{rem} Let us note that with notations of section \ref{sub:Notations,-quaternionic-Shimura} the above proposition implies that in the case of the existence of a $k/\ell$-involution $\tau$ on $A$ we particularly have the relation $\rho_{2}=\rho_{1}\circ\tau$ on $A$. Note also that the condition (2) is superfluous in the case where $k$ is a real quadratic field, since there is only one choice of $\sigma_{2}=\sigma_{1}\circ\alpha$. \end{rem} Let $\tau$ be an involution of second kind on $A$. Let $\Gamma\subset A^{+}/k^{\ast}$ be a subgroup stable by $\tau$, that is, for all $\gamma\in\Gamma$ we have $\tau(\gamma)\in\Gamma$. Since $\rho_{2}=\rho_{1}\circ\tau$, the images of $\Gamma$ in $PSL_{2}(\mathbb{R})$ by $\rho_{1}$ and $\rho_{2}$ are the same, and since this image is isomorphic to $\Gamma$ we denote it also by $\Gamma$. In order to avoid more long-winded notations, let us write for an involution of second kind $\tau$ shortly \[ \tau(a)=\overline{a}\ \text{for all}\ a\in A. \] In particular, the non-trivial $\ell$-automorphism of $k$ is denoted by $x\mapsto\overline{x}$ (for $x\in k$). Identifying $k$ with $\sigma_{1}(k)$, say, the action of any element $\gamma\in A^{+}/k^{\ast}$ on $\mathbb{H}\times\mathbb{H}$ given by \[ \gamma(z_{1},z_{2})=(\rho_{1}(\gamma)z_{1},\rho_{2}(\gamma)z_{2}) \]
is written as \[ \gamma(z_{1},z_{2})=(\gamma z_{1},\bar{\gamma}z_{2}). \]
Let $\mu:\mathbb{H}\times\mathbb{H}\rightarrow\mathbb{H}\times\mathbb{H}$ be the involution that exchanges the two factors. The group $Aut(\mathbb{H}\times\mathbb{H})$ is the semi direct product of $Aut(\mathbb{H})\times Aut(\mathbb{H})$ and the group $\mathbb{Z}/2\mathbb{Z}$ generated by $\mu$. \begin{prop} \label{pro:involution second kind implies involution} Let $\Gamma$ be a subgroup of $A^{+}$ commensurable with $\mathcal{O}^{1}$ for some maximal order $\mathcal{O}$ in $A$. Suppose that there exists an involution $\tau$ of second kind on $A$ preserving $\Gamma$. Then, the automorphism $\mu$ of $\mathbb{H}\times\mathbb{H}$ induces an involution $\sigma$ on the surface $X{}_{\Gamma}=\mathbb{H}\times\mathbb{H}/\Gamma$. \end{prop} \begin{proof} We need to prove that $\mu$ sends an orbit under the action of $\Gamma$ to another $\Gamma$-orbit. We have: \[
\Gamma(z_{1},z_{2})=\{(\gamma z_{1},\bar{\gamma}z_{2})|\,\gamma\in\Gamma\}, \]
thus \[
\mu(\Gamma(z_{1},z_{2}))=\{(\bar{\gamma}z_{2},\gamma z_{1})|\,\gamma\in\Gamma\}. \]
Since $\tau:a\to\bar{a}$ preserves $\Gamma$, we get: \[
\mu(\Gamma(z_{1},z_{2}))=\{(\gamma z_{2},\bar{\gamma}z_{1})|\,\gamma\in\Gamma\}=\Gamma(z_{2},z_{1}), \]
therefore $\mu(\Gamma(z_{1},z_{2}))=\Gamma(z_{2},z_{1})$ is an orbit under the action of $\Gamma$ on $\mathbb{H}\times\mathbb{H}$ and $\mu$ acts on the orbit space $X_{\Gamma}$. \end{proof}
\subsection{Automorphisms of Shimura surfaces.}
Let $\mu\in{\rm Aut}\mathbb{H}\times\mathbb{H}$ be the involution that exchanges the two factors. Let $X_{\Gamma}=\mathbb{H}\times\mathbb{H}/\Gamma$ be a smooth quaternionic Shimura surface. By \cite[Theorem 3.12]{DzambicRoulleau} and its proof, we get: \begin{prop} \label{pro:fixed point of dimension 1}Suppose that the fixed locus $C$ of an automorphism $\sigma$ of $X=X_{\Gamma}$ contains an one-dimensional component. Then $\sigma$ is an involution that lifts on the universal cover to (a conjugate of) $\mu$. \end{prop} Let us consider an involution $a\mapsto\bar{a}$ on $A$ of second kind and a torsion-free group $\Gamma$ commensurable with a group $\Gamma(1)$, and as above stable under $a\mapsto\overline{a}$. Recall that for $\gamma\in\Gamma$ and $t=(z_{1},z_{2})\in\mathbb{H}\times\mathbb{H}$ the action of $\gamma$ on $\mathbb{H}\times\mathbb{H}$ is given by \[ \gamma t=(\gamma z_{1},\bar{\gamma}z_{2}). \]
The image of $t$ on $X_{\Gamma}=\mathbb{H}\times\mathbb{H}/\Gamma$ is a fixed point of the involution $\sigma$ induced by $\mu$ if and only if \[ \Gamma(z_{1},z_{2})=\Gamma(z_{2},z_{1}), \]
which is the case if and only if there exists $\gamma$ in $\Gamma$ such that \[ (z_{1},z_{2})=\gamma(z_{2},z_{1})=(\gamma z_{2},\bar{\gamma}z_{1}), \]
that is, if and only if $z_{1}=\gamma z_{2}$ and $z_{2}=\bar{\gamma}\gamma z_{2}$. Since $\Gamma$ is torsion-free (because $X$ is smooth), the image of $t$ in $X$ is a fixed point of $\sigma$ if and only if $\bar{\gamma}\gamma=1$.
\textcolor{black}{As in \cite{Granath}, for any $\beta\in A\setminus\{0\}$, let $\Delta_{\beta}$ be the disk $\Delta_{\beta}=\{(z,\beta z)/z\in\mathbb{H}\}$}. \textcolor{black}{We have $\lambda\Delta_{\beta}=\Delta_{\bar{\lambda}\beta\lambda^{-1}}$ for any $\lambda\in A\setminus\{0\}$.} We denote by $F_{\beta}$ the image of $\Delta_{\beta}$ in $X_{\Gamma}$.
Let $\gamma\in\Gamma$ be such that $\bar{\gamma}=\gamma^{-1}$. Since $\bar{\gamma}\gamma=1$, we have $(\gamma z,z)=\gamma(z,\gamma z)$, therefore for any point $t$ of $\Delta_{\gamma}$, we obtain \[ \mu\Gamma t=\Gamma\mu t=\Gamma\gamma t=\Gamma t, \]
and the image $F_{\gamma}$ of $\Delta_{\gamma}$ on $X_{\Gamma}$ is fixed by the involution $\sigma$. That implies that $F_{\gamma}$ is a smooth irreducible algebraic curve, more precisely a Shimura curve, and that there is only a finite number of such curves. We obtain: \begin{cor} The fixed point set of $\sigma$ is the union of smooth disjoint Shimura curves $F_{\gamma}$ with $\gamma\in\Gamma$ such that $\gamma\bar{\gamma}=1$. \end{cor} \begin{rem} Since the irreducible components of the fixed locus $C$ are smooth disjoint Shimura curves on the Shimura surface $X$, we get by the Hirzebruch Proportionality Theorem \[ 2C^{2}=-K_{X}C=4(1-g) \] where $g>1$ is the arithmetic genus of $C$. If the irreducible componants are $C_{j},\, j=1\cdots,k$ of genus $g_{j}$, then $C_{j}^{2}=2(1-g_{j})$ and $C^{2}=\sum C_{j}^{2}=\sum2(1-g_{j})$. Moreover \[ K_{X}C=\sum4(g_{j}-1) \] thus $C^{2}+K_{X}C=\sum2g_{i}-2=2g-2$ and $C^{2}=2-2g$. \end{rem} Recall that \textcolor{black}{$\lambda C_{\beta}=C_{\bar{\lambda}\beta\lambda^{-1}}$ for any $\lambda\in A\setminus\{0\}$}, and in particular for every $\lambda\in\Gamma$ we have: \[ F_{1}=F_{\bar{\lambda}\lambda^{-1}}. \] Of course, for $\alpha=\bar{\lambda}\lambda^{-1}$ we have $\alpha\bar{\alpha}=1$. We conjecture that for a smooth surface $X$ the fixed locus of such $\sigma$ is irreducible. As we see immediatly, this is equivalent to the following: \begin{conjecture} \label{conjecture1}Let be $\lambda\in\Gamma$ such that $\lambda\bar{\lambda}=1$. There exists $\gamma\in\Gamma$ such that $\lambda=\bar{\gamma}\gamma^{-1}$. \end{conjecture}
\section{Quotient of a quaternionic Shimura surface by an involution of second kind.}
\label{sec:Quotient and fundamental group}
\subsection{Invariants of the quotient}
Let $\Gamma$ be a lattice preserved by an involution of the second kind and let $\sigma$ be the corresponding involution acting on the Shimura surface $X=X_{\Gamma}$. Let $C$ be the smooth curve of arithmetic genus $g$ fixed by the involution. \begin{prop} \label{invariants_of_quotient} The quotient surface $Z=X/\sigma$ is smooth and has invariants: \[ K_{Z}^{2}=e(X)+5(1-g),\, c_{2}=\frac{1}{2}e(X)+1-g,\, p_{g}=\frac{e(X)-4-4g}{8},\, q=0, \]
where $e(X)=c_{2}(X)$ is the topological Euler number of $X$. \\
If $(K_{X}-C)^{2}>0$, then $Z$ has general type; this condition on the positivity is satisfied if $e(X)\le36$.\\
Suppose $e(X)=12$, then $C$ is irreducible of genus $g=2$ and the quotient surface $X/\sigma$ has invariants: \[ K_{X}^{2}=7,\, c_{2}=5,\, p_{g}=0,\, q=0. \]
Suppose $e(X)=16$, then $C$ is irreducible of genus $g=3$ and the quotient surface $Z$ has invariants: \[ K_{X}^{2}=6,\, c_{2}=6,\, p_{g}=0,\, q=0. \] \end{prop} \begin{proof} Let $\pi:X\to Z=X/\sigma$ be the quotient map. Since $K_{X}^{2}=2e$ (for $e=c_{2}(X)$), $K_{X}=\pi^{*}K_{Z}+C$ and $C^{2}=2(1-g),\, K_{X}C=4(g-1)$ (here we use that $C$ is a disjoint union of smooth Shimura curves) we get \[ K_{Z}^{2}=\frac{1}{2}(K_{X}-C)^{2}=\frac{1}{2}(K_{X}^{2}-2K_{X}C+C{}^{2})=e-5(g-1). \]
Moreover by general formulas on quotient surfaces (see e.g. \cite{DzambicRoulleau}) \[ e(Z)=\frac{1}{2}(e(X)+e(C))=\frac{1}{2}e(X)+1-g. \]
As $q(X)=0$, we get $q(Z)=0$ and \[ p_{g}(Z)=\chi(\mathcal{O}_{Z})-1=\frac{e(X)-4(g+1)}{8}. \]
As $g\geq2$ because $X$ is hyperbolic and $p_{g}(Z)\geq0$, we get that \[ 2\leq g\leq\frac{e(X)-4}{4}, \]
thus \[ e(Z)\geq\frac{e(X)}{4}+2,\,\chi(\mathcal{O}_{Z})\geq1 \]
and $K_{Z}^{2}\geq10-\frac{e}{4}.$ Let us prove that $Z$ has general type if $K_{Z}^{2}>0$. Since $\pi^{*}K_{Z}=K_{X}-C$, it is enough to prove that powers $\mathcal{L}^{m}$ of $\mathcal{L}=\mathcal{O}_{X}(K_{X}-C)$ have sections growing in $c\cdot m^{2}$, where $c>0$ is a constant. Suppose that $K_{Z}^{2}=\frac{1}{2}(K_{X}-C)^{2}>0$ (note that by the preceding discussion, this condition is always satisfied for a surface $X$ with $e(X)<40$). By the Riemann-Roch Theorem, we have \[ \chi(\mathcal{L}^{m})=\frac{m^{2}}{2}(K_{X}-C)^{2}-\frac{m}{2}K_{X}(K_{X}-C)+\chi(\mathcal{O}_{X}). \]
Serre duality gives \[ \chi(\mathcal{L}^{m})=H^{0}(X,\mathcal{L}^{m})-H^{1}(X,\mathcal{L}^{m})+H^{0}(X,mC-(m-1)K_{X}). \]
Suppose that $D=mC-(m-1)K_{X}$ is effective. As $K_{X}$ is ample $K_{X}D>0$. But \[ K_{X}D=m(4g-4-2e)+2e \]
and as $g\leq\frac{e(X)-4}{4}$, we get $4g-4\leq e-8$ and \[ K_{X}D\leq m(-8-e)+2e<0 \]
for $m\geq3$. Therefore $H^{0}(X,mC-(m-1)K_{X})=0$ and $Z$ has general type. \end{proof} The possibilities for values $12,16,\dots,36$ of $e(X)$ and for the genus $g$ of $C$ are listed in the above tables:
\begin{tabular}{|c|c|c|c|c|} \hline $e(X)$ & $g$ & $K_{Z}^{2}$ & $c_{2}(Z)$ & $p_{g}(Z)$\tabularnewline \hline $e=12$ & $2$ & $7$ & $5$ & $0$\tabularnewline \hline $e=16$ & $3$ & $6$ & $6$ & $0$\tabularnewline \hline $e=20$ & $2$ & $15$ & $9$ & $1$\tabularnewline \cline{2-5}
& $4$ & $5$ & $7$ & $0$\tabularnewline \hline $e=24$ & $3$ & $14$ & $10$ & $1$\tabularnewline \cline{2-5}
& $5$ & $4$ & $8$ & $0$\tabularnewline \hline $e=28$ & $2$ & $23$ & $13$ & $2$\tabularnewline \cline{2-5}
& $4$ & $13$ & $11$ & $1$\tabularnewline \hline \end{tabular}\hspace{0.1in}
\begin{tabular}{|c|c|c|c|c|} \hline $e(X)$ & $g$ & $K_{Z}^{2}$ & $c_{2}(Z)$ & $p_{g}(Z)$\tabularnewline \hline $e=28$ & $6$ & $3$ & $9$ & $0$\tabularnewline \hline
& $3$ & $22$ & $14$ & $2$\tabularnewline \cline{2-5} $e=32$ & $5$ & $12$ & $12$ & $1$\tabularnewline \cline{2-5}
& $7$ & $2$ & $10$ & $0$\tabularnewline \hline
& $2$ & $31$ & $17$ & $3$\tabularnewline \cline{2-5}
$e=36$ & $4$ & $21$ & $15$ & $2$\tabularnewline \cline{2-5}
& $6$ & $11$ & $13$ & $1$\tabularnewline \cline{2-5}
& $8$ & $1$ & $11$ & $0$\tabularnewline \hline \end{tabular}
\subsection{The fundamental group of the quotient}
Let us recall some results about fundamental groups. Let $G$ be a discontinuous group of homeomorphisms of a path connected, simply connected, locally compact metric space $M$, and let $G_{tor}$ be the normal subgroup of $G$ generated by those elements which have fixed points, or equivalently, the torsion elements. Then \begin{thm} \cite{Armstrong}. The fundamental group of the orbit space $M/G$ is isomorphic to the factor group $G/G_{tor}$. \end{thm} Let $X=X_{\Gamma}$ be our Shimura surface with fundamental group $\Gamma$ such that the involution $\mu$ switching the two factors of the bi-disk $\mathbb{H}\times\mathbb{H}$ acts on $X$ by an involution denoted by $\sigma$. The fundamental group of the quotient surface $X/\sigma$ is isomorphic to $\Gamma'/\Gamma'_{tor}$, where $\Gamma'$ is the group generated by $\Gamma$ and $\mu$.
Recall that a group $G$ acting on the space $M$ is discontinuous if: \\
(1) the stabilizer of each point of $M$ is finite, and\\
(2) each point $x\in M$ has a neighborhood $U$ such that any element of $G$ not in the stabilizer of $x$ maps $U$ outside itself (i.e. if for $g\in G$, $gx\not=x$ then $U\cap gU$ is empty). \begin{lem} The group $\Gamma'$ is discontinuous. \end{lem} \begin{proof} Since $M=\mathbb{H}\times\mathbb{H}$ is a locally compact Hausdorff space, $\Gamma'$ is discontinuous if and only if it is discrete subgroup of $Aut(\mathbb{H}\times\mathbb{H})$. The latter assertion follows from the fact that $\Gamma'$ is an index-2 extension of the discrete group $\Gamma$. \end{proof} For any $\gamma\in\Gamma$, we have $\mu\gamma=\bar{\gamma}\mu.$ Let $g\in\Gamma'_{tor}$ be a non-trivial torsion element. Since $\Gamma$ is torsion free, $g\not\in\Gamma$ and there exists $\lambda\in\Gamma$ such that $g=\lambda\mu$. The order of $g$ is then divisible by $2$ and we have $g^{2n}=(\lambda\bar{\lambda})^{n}$. Since $\Gamma$ is torsion free, $g^{2n}=1$ if and only if $\lambda\bar{\lambda}=1$. Therefore a torsion element $g$ of $\Gamma'$ has order $2$ and there exists $\lambda\in\Gamma$ such that $g=\lambda\mu$ and $\lambda\bar{\lambda}=1$.
As an immediate consequence we obtain: \begin{lem} The fundamental group $\Gamma'/\Gamma'_{tor}$ of $X/\sigma$ is isomorphic to the group $\Gamma/N$ where $N$ is the normal group generated by the $\lambda\in\Gamma$ such that $\lambda\bar{\lambda}=1$. \end{lem} Since for any $\gamma\in\Gamma$, the group $\Gamma'_{tor}$ contains $\bar{\gamma}\mu\bar{\gamma}^{-1}=\bar{\gamma}\gamma^{-1}\mu$, we see that $N$ contains $\bar{\gamma}\gamma^{-1}$ for every $\gamma\in\Gamma$, therefore the quotient $\Gamma/N$ forces the relation $\gamma=\bar{\gamma}$ for any $\gamma\in\Gamma$.
Let us denote by $H$ the normal subgroup of $\Gamma$ generated by the elements $\bar{\gamma}\gamma^{-1}$, $\gamma\in\Gamma$. Note that under Conjecture \ref{conjecture1}, the group $H$ is equal to $N$. The group $\pi_{1}(X/\sigma)=\Gamma'/\Gamma'_{tor}\simeq\Gamma/N$ is a quotient of $\Gamma/H$. \begin{thm} Suppose that $\Gamma$ is a subgroup of $\Gamma(1)$. Then $\Gamma/H$ is a finite group and the fundamental group of $X/\sigma$ is finite.\end{thm} \begin{proof} Let $A'$ be a quaternion algebra over the field $\ell$ as above such that $A=A'\otimes k$ and the involution of second kind on $A$ is given by $a'\otimes u\to a'\otimes\bar{u}$. Let $k'$ be a degree $2$ extension of $\ell$ such that $A'\otimes_{\ell}k'=M_{2}(k')$ and let $K$ be the compositum of $k,k'$ : $K=k\otimes_{\ell}k'$. Then the algebra $A\otimes_{k}K$ is $A'\otimes_{\ell}K=M_{2}(k')\otimes_{\ell}k=M_{2}(K)$. The involution of second kind $a\to\bar{a}$ extends to $M_{2}(K)$ and acts on each entries fixing $M_{2}(k')\subset M_{2}(K)$. The embedding \[ j:A\hookrightarrow M_{2}(K) \] is equivariant for the action of the involution: $\forall a\in A$, $j(\bar{a})=\overline{j(a)}$, where the action on the left hand side is the conjugation on each entries of the matrix.
The group $j(\Gamma(1))$ is a subgroup of $PSL_{2}(\mathcal{O}_{K})$. Let $I\subset\mathcal{O}_{K}$ be the (non-trivial) ideal generated by the elements $\bar{a}-a,\, a\in\mathcal{O}_{K}$. The ring $O_{K}/I$ is a finite ring therefore the subgroup $\Gamma/H$ of $PSL_{2}(\mathcal{O}_{K}/I)$ is finite.
The fundamental group of $X/\sigma$ is (isomorphic to) $\Gamma/N$ which is a quotient of the finite group $\Gamma/H$, therefore $\pi_{1}(X/\sigma)$ is finite. \end{proof}
\section{Examples}
\label{sec:Examples}
\subsection{Aim and terminology}
Our goal is to find examples of smooth quaternionic Shimura surfaces $X_{\Gamma}$ together with an involution $\sigma$ on $X_{\Gamma}$ having one-dimensional fixed locus. So, we consider an indefinite quaternion algebra $A$ over a totally real field $k$ of degree $n=[k:\mathbb{Q}]$, unramified exactly at two infinite places of $k$ and consider groups $\Gamma$ commensurable with $\Gamma_{\mathcal{O}}(1)$, the group of norm-1 elements of a maximal order $\mathcal{O}\subset A$ (modulo center). \begin{defn} Let $k,A,\mathcal{O}$ be as above. We say that a discrete group $\Gamma$ in the commensurability class of $\Gamma_{\mathcal{O}}(1)$ is \emph{admissible of type $e$} if: \begin{enumerate} \item $\Gamma$ is torsion-free. \item $e(X_{\Gamma})=e$ where $e(X_{\Gamma})$ is denoting the (orbifold-) Euler number. \item On $A$ there exists an involution $\tau$ of second kind such that $\Gamma$ is invariant under $\tau$. \end{enumerate} \end{defn} Let us remark that according to Proposition \ref{invariants_of_quotient}, the quotient $X_{\Gamma}/\sigma$ will be of general type if $e(X_{\Gamma})\leq36$. Hence, we will focus on admissible groups of type $e=12,16,20,...,36$. Also, the proposition \ref{existence_type_2} gives us a condition which guarantees the existence of an involution of second kind on $A$.
\subsection{Smoothness and the Euler number\label{sub:Smoothness-and-the-Euler}}
Let $A=A(k,\mathfrak{p}_{1},...,\mathfrak{p}_{2m})$ be as above and assume that there exists a $k/\ell$-involution on $A$ with respect to a subfield $\ell\subset k$. According to Proposition \ref{existence_type_2} we particularly assume that the primes $\mathfrak{p}_{i},\, i=1,\dots,2m$ in $\mathcal{O}_{k}$ come in pairs: there exist primes $\mathcal{P}_{1}$,...,$\mathcal{P}_{m}$ of $\mathcal{O}_{\ell}$ such that $\mathfrak{p}_{1}\mathfrak{p}_{2}=(\mathcal{P}_{1})$,...,$\mathfrak{p}_{2m-1}\mathfrak{p}_{2m}=(\mathcal{P}_{m})$.
If $\Gamma$ is commensurable with $\Gamma_{\mathcal{O}}(1)$, we have the following general formula for the orbifold Euler number of the Shimura surface $X_{\Gamma}$ \textcolor{red}{} \begin{prop} (see \cite{shimizu}, \cite{vign}) \label{shimizu} Let $k$ and $A=A(k,\mathfrak{p}_{1},...,\mathfrak{p}_{2m})$ be as above. Assume that there exists a $k/\ell$-involution on $A$. Let $n=[k:\mathbb{Q}]$, $\zeta_{k}(\ )$ be the Dedekind zeta function of $k$ and $d_{k}$ denote the discriminant of $k$. Then the orbifold Euler number of $X_{\Gamma}$ equals \[ e(X_{\Gamma})=[\Gamma_{\mathcal{O}}(1):\Gamma]\cdot\frac{d_{k}^{3/2}\zeta_{k}(2)}{2^{2n-3}\pi^{2n}}\prod_{i=1}^{m}(N\mathcal{P}_{i}-1)^{2}, \]
where $N\mathcal{P}=|\mathcal{\mathcal{O}}_{\ell}/\mathcal{P}|$ denotes the norm of $\mathcal{P}$ and where \[ [\Gamma_{\mathcal{O}}(1):\Gamma]=\frac{[\Gamma_{\mathcal{O}}(1):\Gamma\cap\Gamma_{\mathcal{O}}(1)]}{[\Gamma:\Gamma\cap\Gamma_{\mathcal{O}}(1)]} \]
is the generalized index of two commensurable groups. \end{prop} When $k=\mathbb{Q}(\sqrt{d})$ is a real quadratic field, we have a particularly handable formula $\zeta_{k}(2)=\frac{\pi^{4}B_{2,k}}{6d_{k}^{3/2}}$ for the value $\zeta_{k}(2)$ in terms of the second generalized Bernoulli number. This implies \begin{cor} (see \cite{Shavel78}) Let $k=\mathbb{Q}(\sqrt{d})$ be a real quadratic field. We denote by $B_{2,k}$ the second generalized Bernoulli number associated with the quadratic Dirichlet character $\chi_{k}(p)=\left(\frac{d_{k}}{p}\right)$ of $k$. Then, \[ e(X_{\Gamma})=[\Gamma_{\mathcal{O}}(1):\Gamma]\cdot\frac{B_{2,k}}{12}\prod_{i=1}^{m}(p_{i}-1)^{2}, \]
where $p_{1},\ldots,p_{m}$ are rational primes such that $\mathfrak{p}_{1}\mathfrak{p}_{2}=(p_{1})$,...,$\mathfrak{p}_{2m-1}\mathfrak{p}_{2m}=(p_{m})$. \end{cor} As next, we would like to discuss the question about the smoothness of $X_{\Gamma}$. Note that $X_{\Gamma}$ is smooth if and only if $\Gamma$ is torsion-free. Here, we will concentrate on subgroups $\Gamma\subset\Gamma_{\mathcal{O}}(1)$. It is worth to recall that the torsions in $\Gamma_{\mathcal{O}}(1)$ correspond to ring embeddings $\mathcal{O}_{k}[\xi_{n}]\longrightarrow\mathcal{O}$ of the roots of unity into the maximal order $\mathcal{O}$. The general criterion for the existence of torsions in $\Gamma_{\mathcal{O}}(1)$ is as follows. \begin{lem} (see \cite{Shavel78}) \label{torsions} Let $\xi_{n}$ be a primitive $n$-th root of unity. There exists an element of order $n$ in $\Gamma_{\mathcal{O}}(1)$ if and only if: \begin{enumerate} \item $\xi_{n}+\xi_{n}^{-1}\in k$ \item every ramified prime in $A$ is non-split in $k(\xi_{n})$ \end{enumerate} \end{lem} The above lemma already gives us a bound for the order of possible torsion elements. Namely, for any $a\in A\setminus k$, the algebra $k(a)$ is commutative subfield of $A$. Since $\dim_{k}A=4$, and $a\notin k$, $\dim_{k}k(a)=2$ and $k(a)$ is a quadratic extension of $k$. Assume now that $\xi$ is a primitive $n$-th root of unity embedded in $\mathcal{O}$, then as $L=k(\xi)\subset A$ is a quadratic extension of $k$, we have $\varphi(n)\leq2[k:\mathbb{Q}]$, since $\varphi(n)=[\mathbb{Q}(\xi):\mathbb{Q}]\leq[L:\mathbb{Q}]\leq2[k:\mathbb{Q}]$.\\
Above results provide us with criteria to test the conditions in the definition of admissible groups. Let us state a classification result in the case of a real quadratic field $k$. \begin{thm} \label{list_real_quadratic_candidates} Let $k=\mathbb{Q}(\sqrt{d})$ be a real quadratic field and consider the totally indefinite quaternion algebra $A=A(k,\mathfrak{p}_{1},\overline{\mathfrak{p}}_{1}\ldots,\mathfrak{p}_{m},\overline{\mathfrak{p}}_{m})$ over $k$ ramified at the prime ideals dividing rational primes $p_{1},\ldots,p_{m}$ which are split in $k$. If $\Gamma\subset\Gamma_{\mathcal{O}}(1)$ is an admissible group of type $e$, then the possibilities are as follows: \end{thm} \begin{center}
\begin{tabular}{|c|c|c|c|} \hline type $e$ & $k$ & Ram$(A)$ & $[\Gamma_{\mathcal{O}}(1):\Gamma]$\tabularnewline \hline $12$ & $\mathbb{Q}(\sqrt{17})$ & $\mathfrak{p}_{2},\overline{\mathfrak{p}}_{2}$ & $18$\tabularnewline \hline $16$ & $\mathbb{Q}(\sqrt{13})$ & $\mathfrak{p}_{3},\overline{\mathfrak{p}}_{3}$ & $12$\tabularnewline \hline $16$ & $\mathbb{Q}(\sqrt{17})$ & $\mathfrak{p}_{2},\overline{\mathfrak{p}}_{2}$ & $24$\tabularnewline \hline $20$ & $\mathbb{Q}(\sqrt{17})$ & $\mathfrak{p}_{2},\overline{\mathfrak{p}}_{2}$ & $30$\tabularnewline \hline $24$ & $\mathbb{Q}(\sqrt{13})$ & $\mathfrak{p}_{3},\overline{\mathfrak{p}}_{3}$ & $18$\tabularnewline \hline $24$ & $\mathbb{Q}(\sqrt{17})$ & $\mathfrak{p}_{2},\overline{\mathfrak{p}}_{2}$ & $36$\tabularnewline \hline $24$ & $\mathbb{Q}(\sqrt{2})$ & $\mathfrak{p}_{7},\overline{\mathfrak{p}}_{7}$ & $4$\tabularnewline \hline $24$ & $\mathbb{Q}(\sqrt{33})$ & $\mathfrak{p}_{2},\overline{\mathfrak{p}}_{2}$ & $12$\tabularnewline \hline $28$ & $\mathbb{Q}(\sqrt{17})$ & $\mathfrak{p}_{2},\overline{\mathfrak{p}}_{2}$ & $42$\tabularnewline \hline $32$ & $\mathbb{Q}(\sqrt{13})$ & $\mathfrak{p}_{3},\overline{\mathfrak{p}}_{3}$ & $24$\tabularnewline \hline $32$ & $\mathbb{Q}(\sqrt{17})$ & $\mathfrak{p}_{2},\overline{\mathfrak{p}}_{2}$ & $48$\tabularnewline \hline $32$ & $\mathbb{Q}(\sqrt{28})$ & $\mathfrak{p}_{3},\overline{\mathfrak{p}}_{3}$ & $6$\tabularnewline \hline $36$ & $\mathbb{Q}(\sqrt{17})$ & $\mathfrak{p}_{2},\overline{\mathfrak{p}}_{2}$ & $54$\tabularnewline \hline $36$ & $\mathbb{Q}(\sqrt{33})$ & $\mathfrak{p}_{2},\overline{\mathfrak{p}}_{2}$ & $18$\tabularnewline \hline \end{tabular} \par\end{center}
In order to prove this theorem we will need the following elementary lemma. \begin{lem} \label{TdividesG:H}Let $G$ be an arbitrary group and $H\subset G$
a torsion-free subgroup of finite index. If $T\subset G$ is a finite subgroup, then $|T|$ divides $[G:H]$. \end{lem} \begin{proof} Let $G/H=\{g_{1}H,\ldots,g_{I}H\}$ be a set of left cosets of $H$ in $G$. The group $T$ acts by left multiplication on this set. And moreover this action is free. Otherwise, we would have $t\cdot g_{i}H=g_{i}H$ with some non-trivial $t\in T$ and consequently, $g_{i}^{-1}tg_{i}\in H$, which is not possible, since $H$ is torsion-free and $t$ as well as $g_{i}^{-1}tg_{i}$, is of finite order. By elementary group theory, the length of any $T$-orbit on $G/H$ is the same as the order of
$|T|$. And since $G/H$ is the union of different $T$-orbits, $|G/H|$
is divisible by $|T|$. \end{proof} \emph{Proof of Theorem \ref{list_real_quadratic_candidates}.} Recall that we restrict the type $e$ of an admissible group to values $e=12+4k\leq36$. If $\Gamma\subset\Gamma_{\mathcal{O}}(1)$ is admissible of type $e$, then \[ 36\geq[\Gamma_{\mathcal{O}}(1):\Gamma]\frac{B_{2,k}}{12}\prod_{i=1}^{m}(p_{i}-1)^{2}. \]
Since $[\Gamma_{\mathcal{O}}(1):\Gamma]$ and $(p_{i}-1)^{2}$ are positive we have the condition $36\geq\frac{B_{2,k}}{12}$. By \cite{Shavel78}, Proposition 3.2, the Bernoulli number $B_{2,k}$ is bounded below by $3d_{k}^{3/2}/50$ and this implies the upper bound $d_{k}<373$ for the discriminant of $k$. Using the formula for the second generalized Bernoulli number given in \cite{Shavel78}, p.~228, we can compute all the values $B_{2,k}$ for $d_{k}<373$ easily with the help of a computer. With the list of all these values we check the necessary conditions: \begin{itemize} \item $36\geq B_{2,k}/12$ \item $B_{2,k}\mid12\cdot e=144,192,240,\ldots432$ for integral $B_{2,k}$ ( $\Leftrightarrow$ $d_{k}\neq5$) and with obvious modification for $d_{k}=5$. \item The square part of $12e/B_{2,k}$ is divisible by the product $\prod_{p}(p-1)^{2}$ where $p$ runs over subsets of rational primes which are split in $k$ (note that $p=2$, if split in $k$, contributes the factor $1$ to the product). \end{itemize} We obtain the following list of tuples satisfying all the conditions
\begin{center}
\begin{tabular}{|c|c|c|c|} \hline $d_{k}$ & $B_{2,k}$ & $e$ & $12e/B_{2,k}=I\cdot\prod(p-1)^{2}$\tabularnewline \hline $137$ & $192$ & $16$ & $1=(2-1)^{2}$\tabularnewline \hline $113$ & $144$ & $12$ & $1=(2-1)^{2}$\tabularnewline \hline $109$ & $108$ & $36$ & $4=(3-1)^{2}$\tabularnewline \hline $105$ & $144$ & $12$ & $1=(2-1)^{2}$\tabularnewline \hline $85$ & $72$ & $24$ & $1=(3-1)^{2}$\tabularnewline \hline $40$ & $28$ & $28$ & $12=3\cdot(3-1)^{2}$\tabularnewline \hline $37$ & $20$ & $20$ & $12=3\cdot(3-1)^{2}$\tabularnewline \hline $33$ & $24$ & $24$ & $12\cdot(2-1)^{2}$\tabularnewline \hline $29$ & $12$ & $16$ & $16=(5-1)^{2}$\tabularnewline \hline $29$ & 12 & $32$ & $32=2\cdot(5-1)^{2}$\tabularnewline \hline $29$ & 12 & $36$ & $36=(7-1)^{2}$\tabularnewline \hline $28$ & $16$ & $16$ & $12=3\cdot(3-1)^{2}$\tabularnewline \hline $28$ & 16 & $32$ & $24=6(3-1)^{2}$\tabularnewline \hline $24$ & $12$ & $16$ & $16=(5-1)^{2}$\tabularnewline \hline $24$ & 12 & $32$ & $32=2\cdot(5-1)^{2}$\tabularnewline \hline $17$ & $8$ & $12+4k$, $k=0,\ldots,6$ & $e\cdot(2-1)^{2}$\tabularnewline \hline $13$ & $4$ & $12+4k$, $k=0,\ldots,6$ & $(9+3k)\cdot(3-1)^{2}$\tabularnewline \hline $8$ & $2$ & $12,24,36$ & $2\cdot(7-1)^{2}$,$4\cdot(7-1)^{2}$, $6\cdot(7-1)^{2}$\tabularnewline \hline $5$ & $\frac{4}{5}$ & $20$ & $3\cdot(11-1)^{2}$\tabularnewline \hline \end{tabular} \par\end{center}
We observe (keeping also in mind the splitting behavior of $2$ in $k$) that the set of ramified places in $A$ is determined by the value $12e/B_{2,k}$ in the table.
As next we identify those subgroups $\Gamma\subset\Gamma_{\mathcal{O}}(1)$ which cannot be torsion-free. For this we use the two Lemmas \ref{torsions} and \ref{TdividesG:H}. Namely, note first that $\Gamma_{\mathcal{O}}(1)$ contains at most torsions of order $2,3$ and $6$ for $k\neq\mathbb{Q}(\sqrt{5}),\mathbb{Q}(\sqrt{2})$ and additionally elements of order $5$ for $k=\mathbb{Q}(\sqrt{5})$ and of order $4$ for $k=\mathbb{Q}(\sqrt{2})$. Case by case analysis leads to the final statement; to check the splitting behavior in $k(\xi_{n})$ one can use the criterion of Shavel (see \cite{Shavel78}, Theorem 4.8). We double-checked the the conditions of Lemma \ref{torsions} explicitly with PARI/GP.
\subsection{Admissible groups defined by congruences}
Let $k$ a totally real number field and $A(k,\mathfrak{p}_{1},\overline{\mathfrak{p}}_{1},\ldots,\mathfrak{p}_{m}\overline{\mathfrak{p}}_{m})$ an indefinite quaternion algebra over $k$, $\mathcal{O}$ a maximal order in $A$, $\mathcal{O}$ and $\Gamma_{\mathcal{O}}(1)$ as in the previous sections. If $\mathfrak{a}$ is a two-sided $\mathcal{O}$ ideal in $\mathcal{O}$, the \emph{principal congruence subgroup} in $\mathcal{O}(1)$ associated with $\mathfrak{a}$ is defined as \[ \mathcal{O}(\mathfrak{a})=\{g\in\mathcal{O}\mid Nrd(g)=1,g-1\in\mathfrak{a}\} \]
Additionally we define $\Gamma_{\mathcal{O}}(\mathfrak{a})=\mathcal{O}(\mathfrak{a})/Z$ where $Z$ denotes the center of $\mathcal{O}(\mathfrak{a})$. A \emph{congruence subgroup} in $\mathcal{O}(1)$, resp.~$\Gamma_{\mathcal{O}}(1)$, is a subgroup $G\subset\mathcal{O}(1)$, resp.~$\Gamma\subset\Gamma_{\mathcal{O}}(1)$, which contains some $\mathcal{O}(\mathfrak{a})$, resp.~$\Gamma_{\mathcal{O}}(\mathfrak{a})$. The group $\mathcal{O}(\mathfrak{a})$ is a normal subgroup of finite index in $\mathcal{O}(1)$ and we have $\mathcal{O}(1)/\mathcal{O}(\mathfrak{a})\cong\Gamma_{\mathcal{O}}(1)/\pm\Gamma_{\mathcal{O}}(\mathfrak{a})$ if $2\notin\mathfrak{a}$ and $\mathcal{O}(1)/\mathcal{O}(\mathfrak{a})\cong\Gamma_{\mathcal{O}}(1)/\Gamma_{\mathcal{O}}(\mathfrak{a})$
if $2\in\mathfrak{a}$. The size $|\mathcal{O}(1)/\mathcal{O}(\mathfrak{a})|$ is computed as follows \begin{itemize} \item Any two-sided ideal $\mathfrak{a}$ in $\mathcal{O}$ has a unique decomposition $\mathfrak{a}=\mathfrak{Q}_{1}^{e_{1}}\cdot\ldots\cdot\mathfrak{Q}_{r}^{e_{r}}$ as a product of prime ideal powers. Then, $\mathcal{O}(1)/\mathcal{O}(\mathfrak{a})$ is a direct product \[ \mathcal{O}(1)/\mathcal{O}(\mathfrak{a})=\mathcal{O}(1)/\mathcal{O}(\mathfrak{Q}_{1}^{e_{1}})\times\ldots\times\mathcal{O}(1)/\mathcal{O}(\mathfrak{Q}_{r}^{e_{r}}). \]
\item Let $\mathfrak{Q}$ be a prime ideal in $\mathcal{O}$. The $\mathcal{O}_{k}$-ideal $\mathfrak{q}=Nrd(\mathfrak{Q})$ generated by the reduced norms of elements in $\mathfrak{Q}$, which is also the intersection $\mathfrak{q}=\mathfrak{Q\cap\mathcal{O}}_{k}$, is a prime ideal and there are two possible cases:
\begin{itemize} \item $\mathfrak{q}\notin Ram(A)$. Then, $\mathfrak{Q}=\mathfrak{q}\mathcal{O}$ and $\mathcal{O}(1)/\mathcal{O}(\mathfrak{Q}_{1}^{e})\cong SL_{2}(\mathcal{O}_{k}/\mathfrak{q}^{e})$. \item $\mathfrak{q}\in Ram(A)$. Then, $\mathfrak{Q}^{2}=\mathfrak{q}\mathcal{O}$ and $\mathcal{O}(1)/\mathcal{O}(\mathfrak{Q}^{e})\cong(\mathcal{O}/\mathfrak{Q}^{e})_{1}=\ker\left((\mathcal{O}/\mathfrak{Q}^{e})^{\ast}\stackrel{Nr}{\rightarrow}(\mathcal{O}_{k}/\mathfrak{q}^{e})^{\ast}\right)$, where $Nr$ is the norm map induced by the reduced norm $Nrd:\mathcal{O}\rightarrow\mathcal{O}_{k}$. \end{itemize} \end{itemize} \begin{rem} If we want to search for admissible groups among the principal congruence subgroups, then we must note the following: for $g\in\mathcal{O}(\mathfrak{Q})$ we have $\overline{g}\in\mathcal{O}(\overline{\mathfrak{Q}})$, so, $\Gamma_{\mathcal{O}}(\mathfrak{Q})$ will be admissible if and only if $\mathfrak{Q}=\overline{\mathfrak{Q}}$ (for more precise statement see Theorem \ref{invariantthm} and Lemma \ref{invarianceunderinvolution} below). This is already a strong condition: If $k$ is a quadratic field, the prime $\mathfrak{q}$ under $\mathfrak{Q}$ must be inert or ramified over $\ell$. For instance, this in combination with list of possible candidates from Theorem \ref{list_real_quadratic_candidates} shows there are no admissible principal congruence subgroups of any type $e$ defined over a real quadratic field. \end{rem} In the following we will make use of the following well-known fact. \begin{lem} \label{congruencetorsion} Let $\mathfrak{q}$ be a prime ideal in $\mathcal{O}_{k}$, unramified in a $k$-central quaternion algebra $A$ and $\mathfrak{Q=\mathfrak{q}\mathcal{O}}$ the corresponding $\mathcal{O}$-ideal . Let $q\mathbb{Z}=\mathfrak{q}\cap\mathbb{Z}$ be the rational prime divisible by $\mathfrak{q}$ and finally, let $x\in\mathcal{O}^{1}(\mathfrak{Q})$ be an element of order $p$, where $p$ is a prime. Then $p=q$. \end{lem} \begin{proof} We have $x\in\mathcal{O}^{1}(\mathfrak{Q})\Leftrightarrow Nrd(x-1)\in\mathfrak{q}$. We can assume that $x$ is a primitive $p$-th root of unity contained in $A$. Since $k(x)\subset A$, we have $Nrd(x-1)=N_{k(x)/k}(x-1)$. Taking $N_{k/\mathbb{Q}}(\cdot)$ on both sides we obtain \[ N_{k/\mathbb{Q}}(Nrd(x-1))=N_{k(x)/\mathbb{Q}}(x-1)\in N_{k/\mathbb{Q}}(\mathfrak{q})=q^{f}\mathbb{Z}. \]
where $f>0$ is the inertia degree of $\mathfrak{q}.$ On the other hand $x-1\in\mathbb{Q}(x)$, and therefore $N_{k(x)/\mathbb{Q}}(x-1)=N_{\mathbb{Q}(x)/\mathbb{Q}}(x-1)^{d}$, where $d=[k(x):\mathbb{Q}(x)]$. Altogether, we obtain the relation \[ N_{\mathbb{Q}(x)/\mathbb{Q}}(x-1)^{d}\in q^{f}\mathbb{Z}. \]
Finally, it is well-known that $N_{\mathbb{Q}(x)/\mathbb{Q}}(x-1)=\pm p$ and from this the claim follows. \end{proof} Assume that the prime ideal $\mathfrak{q}\subset\mathcal{O}_{k}$ is unramified in $A$, then $\mathfrak{q}\mathcal{O}$ is a prime ideal in $\mathcal{O}\subset A$. Since in this case $\mathfrak{Q=\mathfrak{q}\mathcal{O}}$ we will write $\Gamma_{\mathcal{O}}(\mathfrak{\mathfrak{q}})=\Gamma_{\mathcal{O}}(\mathfrak{Q}).$
Let $s=q^{f}$ be the absolute norm $N_{k/\mathbb{Q}}(\mathfrak{q})=|\mathcal{O}_{k}/\mathfrak{q}|$ of $\mathfrak{q}$. Then $\Gamma_{\mathcal{O}}(1)/\Gamma_{\mathcal{O}}(\mathfrak{q})\cong PSL_{2}(\mathcal{O}_{k}/\mathfrak{q})\cong PSL_{2}(s)=PSL_{2}(\mathbb{F}_{s})$. By the classification theorem of Dickson, we know all the subgroups of $PSL_{2}(s)$. Let us mention two particular subgroups which we will use later on: \begin{enumerate} \item Borel subgroup $B\subset PSL_{2}(s)$ consisting of all upper triangular matrices in $PSL_{2}(s)$. The group $B$ is a maximal subgroup of $PSL_{2}(s)$ of index $s+1$ and order $s(s-1)/t$, with $t=gcd(s-1,2)$. \item Unipotent subgroup $U\subset PSL_{2}(s)$ consisting of all elements in $B$ of the form $\left(\begin{smallmatrix}1 & \ast\\ 0 & 1 \end{smallmatrix}\right)$. The group $U$ is a subgroup of index $(s^{2}-1)/t$ and order $s$. \end{enumerate} With above notations let $\pi:\Gamma_{\mathcal{O}}(1)\longrightarrow PSL_{2}(s)$ denote the epimorphism induced by the canonical projection $\Gamma_{\mathcal{O}}(1)\longrightarrow Q=\Gamma_{\mathcal{O}}(1)/\Gamma_{\mathcal{O}}(\mathfrak{q})$. Let $\Gamma^{B}(\mathfrak{q})=\pi^{-1}(B)$ and $\Gamma^{U}(\mathfrak{q})=\pi^{-1}(U)$ be the inverse images of $B$ and $U$ respectively. These are subgroups of $\Gamma_{\mathcal{O}}(1)$ of index equal to the index of its image in $PSL_{2}(s)$ under $\pi$. It is important to mention that $\Gamma_{\mathcal{O}}^{B}(\mathfrak{q})$ is also constructed as the group of the norm-1 elements (modulo center) in an appropriate Eichler order $\mathcal{E}$. The construction is as follows: Let $\mathcal{O}$ be a maximal order and denote $\mathcal{O}_{v}=\mathcal{O}\otimes_{\mathcal{O}_{k}}R_{v}$ the localizations of $\mathcal{O}$ at finite places $v$ of $k$. There $R_{v}$ denotes the valuation ring in the localization $k_{v}$ of $k$ at $v$. Note that $\mathcal{O}=\bigcap_{v}\mathcal{O}_{v}$. Let $\mathcal{O}'$ be another maximal order with the property that for all finite places $v\neq\mathfrak{q}$ we have $\mathcal{O}_{v}=\mathcal{O}'_{v}$ and additionally $\mathcal{O}_{\mathfrak{q}}\cap\mathcal{O}'_{\mathfrak{q}}$ has index $N(\mathfrak{q})=\#R_{\mathfrak{q}}/\mathfrak{q} R_{\mathfrak{q}}$ in both, $\mathcal{O}_{\mathfrak{q}}$ and $\mathcal{O}'_{\mathfrak{q}}$. Put $\mathcal{E=\mathcal{O}\cap\mathcal{O}}'$. Then, by definition, $\mathcal{E}$ is an Eichler order of level $\mathfrak{q}$. If $\mathfrak{q}$ is ramified in $A$, we have $\mathcal{E=\mathcal{O}}$ and in the case where $\mathfrak{q}$ is unramified, after a possible conjugation, we can assume that $\mathcal{O}_{\mathfrak{q}}=M_{2}(R_{\mathfrak{q}})$ and we can choose $\mathcal{O}'_{\mathfrak{q}}=PM_{2}(R_{\mathfrak{\mathfrak{q}}})P^{-1}$ with $P=diag(1,\varpi)$, where $\varpi$ is a generator of the valuation ideal $\mathfrak{q}R_{\mathfrak{q}}$. The reduction modulo $\mathfrak{q}$ maps $\mathcal{E_{\mathfrak{q}}=}\mathcal{O}_{\mathfrak{q}}\cap\mathcal{O}'_{\mathfrak{q}}$ surjectively to the subalgebra of upper triangular matrices in $M_{2}(R_{\mathfrak{q}}/\mathfrak{q}R_{\mathfrak{q}})$. Therefore the group $\mathcal{E}^{1}$ of norm-1 elements in $\mathcal{E}$ corresponds to exactly those elements in $\mathcal{O}^{1}$ which modulo $\mathfrak{q}\mathcal{O}$ are upper triangular.
In general, a $k/\ell$-involution on a quaternion algebra $A$ does not preserve a maximal order. But under certain conditions on $A$ we can ensure the existence of such an order: \begin{thm} (Scharlau, \cite[Theorem 4.6]{Sharlau}) \label{invariantthm} Let $A$ be a quaternion algebra over $k$ admitting a $k/\ell$-involution $\tau$. Then there exists a maximal order $\mathcal{O}$ invariant under $\tau$ unless the following exceptional situation is given: \begin{itemize} \item the extension $k/\ell$ is unramified and \item the number of places $v\in Ram(A)$ is $\equiv2\bmod4$. \end{itemize} \end{thm} \begin{cor} \label{invarianceunderinvolution} Assume that $A$ admits a $k/\ell$-involution $\tau$ which preserves a maximal order $\mathcal{O}$ and let $\mathfrak{q}\subset\mathcal{O}_{k}$ be a prime ideal which is unramified in $A$ and $\mathfrak{q}\mathcal{O}$ the corresponding prime ideal in $\mathcal{O}$. Assume that the non-trivial $k/\ell$-automorphism $x\mapsto\overline{x}$ maps $\mathfrak{q}$ to itself, that is, $\overline{\mathfrak{q}}=\mathfrak{q}$. Then, for each of the groups $\Gamma=\Gamma_{\mathcal{O}}(1)$, $\Gamma_{\mathcal{O}}^{B}(\mathfrak{q})$, $\Gamma_{\mathcal{O}}^{U}(\mathfrak{q})$ and $\Gamma_{\mathcal{O}}(\mathfrak{q})$ there exists a $k/\ell$-involution on $A$ leaving $\Gamma$ invariant.\end{cor} \begin{proof} The group $\Gamma_{\mathcal{O}}(1)$ is $\tau$-invariant since $\mathcal{O}$ is $\tau$-invariant and for any $a\in A$ we have $Nrd(\overline{a})=\overline{Nrd(a)}$. Also, $\Gamma(\mathfrak{q})$ is invariant, since for every $x\in\Gamma_{\mathcal{O}}(\mathfrak{q})$ there is a representative $x'\in\mathcal{O}$ of the class $x$ satisfying $x-1\in\mathfrak{q}\mathcal{O}$, thus $\overline{(x'-1)}\in\overline{\mathfrak{q}\mathcal{O}}=\overline{\mathfrak{q}}\mathcal{O}=\mathfrak{q}\mathcal{O}$, by assumption, hence $\overline{x'}\equiv1\bmod\mathfrak{q}\mathcal{O}$.
In order to prove the invariance of other groups, we first localize at $\mathfrak{q}$; since $\mathfrak{q\notin}Ram(A)$, the local algebra $A_{\mathfrak{q}}=A\otimes_{k}k_{\mathfrak{q}}$ is isomorphic to $M_{2}(k_{\mathfrak{q}})$ and since $\mathcal{O}$ is maximal $\mathcal{O_{\mathfrak{q}}=\mathcal{O}\otimes}_{\mathcal{O}_{k}}R_{\mathfrak{q}}$ is isomorphic to $M_{2}(R_{\mathfrak{q}})$ where $R_{\mathfrak{q}}$ is the valuation ring in $k_{\mathfrak{q}}$. Choosing the appropriate isomorphism $A_{\mathfrak{q}}\cong M_{2}(k_{\mathfrak{q}})$ we can assume that $\mathcal{O_{\mathfrak{q}}}=M_{2}(R_{\mathfrak{q}})$. Consider the order $\mathcal{\mathcal{E_{\mathfrak{q}}}=}M_{2}(R_{\mathfrak{\mathfrak{q}}})\cap PM_{2}(R_{\mathfrak{\mathfrak{q}}})P^{-1}$ with $P=diag(1,\varpi)$, where $\varpi$ is a generator of the valuation ideal $\mathfrak{q}R_{\mathfrak{q}}$. This is the localization of the global Eichler order $\mathcal{E}$ corresponding to a group $\Gamma_{\mathcal{O}}^{B}(\mathfrak{q})$. The involution $\tau$ which leaves $\mathcal{O}$ invariant extends to an involution $\hat{\tau}$ on $\mathcal{O}_{\mathfrak{q}}=\mathcal{O}\otimes_{\mathcal{O}_{k}}R_{\mathfrak{q}}$ in an obvious way by defining $\hat{\tau}(x\otimes r)=\tau(x)\otimes\bar{r}$ where $r\mapsto\overline{r}$ is the generator of $Gal(k_{\mathfrak{q}}/\ell_{\mathcal{Q}})$ and $\mathcal{Q=\mathfrak{q}\cap\ell}$ is the prime of $\ell$ lying under $\mathfrak{q}$ and $\ell_{\mathcal{Q}}$ its localization. By this the involution $\hat{\tau}$ maps the matrix $P$ to $\pm P$ depending on whether $k_{\mathfrak{q}}/\ell_{\mathcal{Q}}$ is unramified or ramified but in any case $\hat{\tau}$ preserves $\mathcal{E_{\mathfrak{q}}}.$ From the construction of $\hat{\tau}$ we see that $\hat{\tau}$ also preserves $\mathcal{O}\cap\mathcal{E_{\mathfrak{q}}=\mathcal{E}}$ and particularly the norm-1 group $\mathcal{E}^{1}$ whose quotient by the center is $\Gamma_{\mathcal{O}}^{B}(\mathfrak{q})$. The group $\Gamma_{\mathcal{O}}^{U}(\mathfrak{q})$ consists of those elements in $\Gamma_{\mathcal{O}}^{B}(\mathfrak{q})$ which reduce modulo $\mathfrak{q}$ to upper triangular matrices with only 1 on the diagonal. The preimage of such matrices in $\mathcal{E_{\mathfrak{q}}}$ is preserved by $\hat{\tau}$ and hence $\tau$ preserves the preimage of these matrices in $\mathcal{O}\cap\mathcal{E_{\mathfrak{q}}}$. This implies that with $\Gamma_{\mathcal{O}}^{B}(\text{\ensuremath{\mathfrak{q}}})$ also $\Gamma_{\mathcal{O}}^{U}(\mathfrak{q})$ is preserved. \end{proof}
\subsection{Construction with the Borel subgroup}
Let $A(k,\mathfrak{p}_{1},\overline{\mathfrak{p}}_{1},\ldots,\mathfrak{p}_{m},\overline{\mathfrak{p}}_{m})$ be as before. Let $\mathfrak{q}$ be a prime ideal of $k$ such that $\mathfrak{q}\neq\mathfrak{p}_{i},\overline{\mathfrak{p}}_{i}$ for $i=1,\ldots,m$ and consider the group $\Gamma_{\mathcal{O}}^{B}(\mathfrak{q})$, the inverse image $\pi^{-1}(B)$ of a Borel subgroup $B\subset\Gamma_{\mathcal{O}}(1)/\Gamma_{\mathcal{O}}(\mathfrak{q})\cong PSL_{2}(\mathcal{O}_{k}/\mathfrak{q})$. The group $\Gamma_{\mathcal{O}}^{B}(\mathfrak{q})$ is a subgroup of index $N_{k/\mathbb{Q}}(\mathfrak{q})+1$ in $\Gamma_{\mathcal{O}}(1)$. In order to discuss the torsions in $\Gamma_{\mathcal{O}}^{B}(\mathfrak{q})$ it will again be useful to interpret $\Gamma_{\mathcal{O}}^{B}(\mathfrak{q})$ as the norm-1 group of an Eichler order as explained in previous section. Let us give conditions under which $\Gamma_{\mathcal{O}}^{B}(\mathfrak{q})$ is torsion-free. \begin{lem} \label{eichlercrit} Let $k$, $A$, $\mathcal{O}$ and $\mathfrak{q}$ be as above. Then, $\Gamma_{\mathcal{O}}^{B}(\mathfrak{q})$ contains a torsion if and only if a primitive $n$-th root of unity $\xi$ can be embedded in $\mathcal{E}(\mathfrak{q})$. This happens if and only if every prime $\mathfrak{p}\in Ram(A)$ is either ramified or inert in $k(\xi)$ and $\mathfrak{q}$ is split in $k(\xi)$. \end{lem} \begin{proof} Let $\gamma$ be a torsion in $\Gamma_{\mathcal{O}}^{B}(\mathfrak{q})$. Then there is a minimal $N$ such that $\gamma^{N}=\pm1$, which implies that $\gamma$ is an $N$-th (or $2N$-nth) root of unity contained in $\mathcal{E}(\mathfrak{q})$. Conversely, let $\xi$ be a root of unity, let $L=k(\xi)$ and assume that there exists an embedding $\sigma:L\hookrightarrow A$ such that $\sigma(L)\cap\mathcal{E}(\mathfrak{q})=\mathcal{O}_{k}(\xi)$ (that is, $\mathcal{O}_{k}(\xi)$ is embedded in $\mathcal{E}(\mathfrak{q})$). Then $\sigma(\xi)$ is a torsion in $\Gamma_{\mathcal{O}}^{B}(\mathfrak{q})$. By a theorem of Eichler (see \cite[Satz 6]{eichler}) such an embedding is possible if and only if the splitting condition mentioned in the statement of Lemma is satisfied. \end{proof}
\subsection{A Shimura surface with an involution of second kind and $p_{g}=5$.\label{sub:pg5}}
Let $k=\mathbb{Q}(\sqrt{33})$ and $A=A(k,\mathfrak{p}_{2}\overline{\mathfrak{p}}_{2})$ the indefinite quaternion algebra ramified exactly at the two places over $2$ (note that $2$ is split in $k$, since $33\equiv1\bmod8$). By Theorem \ref{existence_type_2}, $A$ admits a $k/\mathbb{Q}$-involution and since $k/\mathbb{Q}$ is not totally ramified, Theorem \ref{invariantthm} ensures the existence of an involution invariant order $\mathcal{O}$. Let $\mathfrak{q}=\mathfrak{q}_{11}$ be the prime over $11$. Since $11$ is ramified in $k$, we have $N_{k/\mathbb{Q}}(\mathfrak{q})=11$ and $\Gamma_{\mathcal{O}}^{B}(\mathfrak{q}_{11})$ is of index $12$ in $\Gamma_{\mathcal{O}}$. By the volume formula from Theorem \ref{shimizu}, we have $e(\Gamma_{\mathcal{O}}(1))=2$, hence $e(\Gamma_{\mathcal{O}}^{B}(\mathfrak{q}))=24$. Let us show that $\Gamma_{\mathcal{O}}^{B}(\mathfrak{q})$ is torsion-free. For this we need to exclude the existence of elements of order $2$ and $3$ only, since these are the only primes for which an embedding of $\xi_{p}$ in $A$ is possible. Elements of order $2$ come from embeddings of $k(\xi_{4})=k(\sqrt{-1})$ in $A$ and those of order $3$ from embeddings of $k(\xi_{6})=k(\sqrt{-3})$. We use Lemma \ref{eichlercrit}: $k(\xi_{4})\cong\mathbb{Q}[x]/\langle x^{4}-64x^{2}+1156\rangle$ and we find that $11\mathcal{O}_{k(\zeta_{4})}=\mathfrak{Q}^{2}$ with $(\mathcal{O}_{k(\xi_{4})}/\mathfrak{Q}:\mathbb{F}_{11})=2$. It follows that $\mathfrak{q}_{11}$ is inert in $k(\xi_{4})$ and by Lemma \ref{eichlercrit}, $\Gamma_{\mathcal{O}}^{B}(\mathfrak{q})$ contains no elements of order $2$. Similar argument excludes the existence of elements of order $3$. Namely, $k(\xi_{6})\cong\mathbb{Q}[x]/\langle x^{4}-60x^{2}+1296\rangle$ and in $k(\xi_{6})$ we again have $11\mathcal{O}_{k(\zeta_{6})}=\mathfrak{Q}^{2}$ with a prime ideal $\mathfrak{Q}$ in $\mathcal{O}_{k(\zeta_{6})}$ whose inertia degree is $(\mathcal{O}_{k(\xi_{6})}/\mathfrak{Q}:\mathbb{F}_{11})=2$. Again this implies that $\mathfrak{q}_{11}$ is inert in $\mathcal{O}_{k(\xi_{6})}$ and by Lemma \ref{eichlercrit}, there are no elements of order 3 in $\Gamma_{\mathcal{M}}^{B}(\mathfrak{q}_{11})$. Finally by Corollary \ref{invarianceunderinvolution}, $\Gamma_{\mathcal{O}}^{B}(\mathfrak{q})$ is invariant under the involution on $\mathcal{O}$ and we get: \begin{thm} The group $\Gamma_{\mathcal{O}}^{B}(\mathfrak{q}_{11})$ is admissible of type $24$. \end{thm} \begin{rem} \label{remark_deg_2}Unfortunately, one promising candidate for a admissible group of type $12$ ($p_{g}=2$) fails to be torsion-free. Namely, let $A=A(\mathbb{Q}(\sqrt{17}),\mathfrak{p}_{2}\overline{\mathfrak{p}}_{2})$ and take $\mathfrak{q}=\mathfrak{q}_{17}$, the ideal over $17$. Then the index of $\Gamma_{\mathcal{O}}^{B}(\mathfrak{p}_{17})$ in $\Gamma_{\mathcal{O}}(1)$ is $18$ and as $e(\Gamma_{\mathcal{O}}(1))=2/3$, we get $e(\Gamma_{\mathcal{O}}^{B}(\mathfrak{p}_{17}))=12$. The invariance under the involution of second kind is guaranteed by the condition $\mathfrak{q}_{17}=\overline{\mathfrak{q}}_{17}$. But in $L=k(\xi_{4})\cong\mathbb{Q}[x]/\langle x^{4}-32x^{2}+324\rangle$, both primes $\mathfrak{p}_{2}$ and $\overline{\mathfrak{p}}_{2}$ are non-split and $\mathfrak{p}_{17}$ is split. This implies that there are 2-torsions in $\Gamma_{\mathcal{O}}^{B}(\mathfrak{q}_{17})$. There are more examples of non-smooth Shimura surfaces with ``good'' invariants. Consider for instance $k=\mathbb{Q}(\sqrt{28})$ and $A=A(k,\mathfrak{p}_{3}\overline{\mathfrak{p}}_{3})$ the indefinite quaternion algebra ramified at the two places over $3$. Theorems \ref{existence_type_2} and \ref{invarianceunderinvolution} ensure that $A$ has an involution of second kind and that there is an order $\mathcal{O}$ invariant under the involution. From Theorem \ref{shimizu} we know that $e(\Gamma_{\mathcal{O}}(1))=16/3$. The rational prime $2$ is ramified in $k$. Let $\mathfrak{q}=\mathfrak{q}_{2}$ be the prime ideal of $k$ with $\mathfrak{q}_{2}^{2}=2\mathcal{O}_{k}$ and consider the principal congruence subgroup $\Gamma_{\mathcal{O}}(\mathfrak{q})$. It is a subgroup in $\Gamma_{\text{\ensuremath{\mathcal{O}}}}(1)$ of index $6$. There are no elements of order $3$ in $\Gamma_{\mathcal{O}}(\mathfrak{q})$ by Lemma \ref{congruencetorsion} but there are elements of order 2 coming from embeddings of $\sqrt{-1}$ in $\mathcal{O}$. \\
Looking at the list in Theorem \ref{list_real_quadratic_candidates}, we can also prove that no other admissible groups of type $e=12,\ldots,36$ can be obtained from $k=\mathbb{Q}(\sqrt{d})$ and $\Gamma=\Gamma_{\mathcal{O}}^{B}(\mathfrak{q})$ or $\Gamma_{\mathcal{O}}^{U}(\mathfrak{q})$. \end{rem} Instead we can consider totally real fields of higher degree:
\subsection{A Shimura surface with an involution of second kind and $p_{g}=6$.\label{sub:pg6}}
In this example we consider the unique totally real number field $k$ of degree $4$ and discriminant $d_{k}=725$. Its defining polynomial is $x^{4}-x^{3}-3x^{2}+x+1$ and $k$ contains $\ell=\mathbb{Q}(\sqrt{5})$ as a subfield of degree $2$. Let us consider the $k$-central quaternion algebra $A(k,\emptyset)$ ramified exactly at two infinite places $v_{1}$ and $v_{2}$ of $k$ such that $v_{2}=v_{1}\circ\sigma$, where $\langle\sigma\rangle=Gal(k/\ell)$. We remark that $k$ is not a Galois extension of $\mathbb{Q}$. The algebra $A$ admits an involution of second kind $\tau$ and by Theorem \ref{invariantthm}, there is a maximal order $\mathcal{O}$ invariant under $\tau$. Consider now the prime $q=29$. In $\ell=\mathbb{Q}(\sqrt{5})$, $29\mathcal{O}_{\ell}=\mathcal{Q}_{29}\mathcal{Q}_{29}'$ is a product of two primes. On the other hand, a computation with PARI/GP shows that the ideal $29\mathcal{O}_{k}=\mathfrak{q}_{29}^{2}\mathfrak{q}_{29}'$ is also a product of two prime ideals $\mathfrak{q}_{29}$ (with multiplicity 2) and $\mathfrak{q}_{29}'$, hence neither $\mathcal{Q}_{29}$ nor $\mathcal{Q}_{29}'$ is split in $k$. Moreover we deduce that $\mathfrak{q}_{29}^{2}=\mathcal{Q}_{29}\mathcal{O}_{k}$ and $\mathfrak{q}_{29}'=\mathcal{Q}'_{29}\mathcal{O}_{k}$ as well as $\mathcal{O}_{k}/\mathfrak{q}_{29}\cong\mathbb{F}_{29}$ and $\mathcal{O}_{k}/\mathfrak{q}'_{29}\cong\mathbb{F}_{29^{2}}$. By Theorem \ref{shimizu} we have $e(X_{\Gamma_{\mathcal{O}}(1)})=1/15$ (we compute $\zeta_{k}(2)$ with PARI/GP command \verb zetak). Consider the congruence subgroup $\Gamma_{\mathcal{O}}^{U}(\mathfrak{q}_{29})$. We obtain $[\Gamma_{\mathcal{O}}(1):\Gamma_{\mathcal{O}}^{U}(\mathfrak{q}_{29})]=420$ and $c_{2}(X_{\Gamma_{\mathcal{O}}^{U}(\mathfrak{q}_{29})})=28$. By corollary \ref{invarianceunderinvolution}, $\Gamma_{\mathcal{O}}^{U}(\mathfrak{q}_{29})$ is stable under $\tau$. Also $\Gamma_{\mathcal{O}}^{U}(\mathfrak{q}_{29})$ is torsion-free. Namely, as the order of $U$ is $s$, any non-trivial torsion element in $\Gamma_{\mathcal{O}}^{U}(\mathfrak{q}_{29})$ has order $29$ (which is impossible by lemma \ref{torsions}) or lies already in $\Gamma_{\mathcal{O}}(\mathfrak{q}_{29})$. But this latter group is torsion-free by Lemma \ref{congruencetorsion}. \begin{rem} \label{remark_in_degree_4} Of course, the strategy of Section \ref{sub:Smoothness-and-the-Euler} leading to Theorem \ref{list_real_quadratic_candidates} could be applied also in the case of quaternion algebras over totally real fields $k$ of degree $>2$ but becomes very soon computationally involved. Restricting ourselves to totally real quartic fields of discriminant $\leq10^{4}$ and groups of type $\Gamma_{\mathcal{O}}^{B}(\mathfrak{q})$ or $\Gamma_{\mathcal{O}}^{U}(\mathfrak{q})$ we find that example \ref{sub:pg6} is the only example of an admissible group (of any type $e=12+4k$, $0\leq k\leq6$). But similarly to Remark \ref{remark_deg_2} we find some interesting non-smooth examples: Let $k_{4,D}$ denote a totally real field of degree $4$ and discriminant $D$ (in the examples below, there will be only one such field up to isomorphism) containing a real quadratic field $\ell$. Let $A(k_{4,D},\emptyset)$ denote the quaternion algebra over $k_{4,D}$ ramified exactly at the two infinite places of $k_{4,D}$ which are conjugate under the non-trivial $\ell$-automorphism of $k_{4,D}$. Such quaternion algebra admits a $k_{4,D}/\ell$ -involution. Let $\mathcal{O}$ be a maximal order in $A(k_{4,D},\emptyset)$ and $\Gamma_{\mathcal{O}}(1)$ the corresponding projectivized modular group. We get several \emph{singular} Shimura surfaces $X_{\Gamma}$ admitting an involution of second kind with $\Gamma\subset\Gamma_{\mathcal{O}}(1)$ given in the table below: \end{rem}
\begin{tabular}{|c|c|c|c|c|} \hline $k_{n,D}$ & $\ell$ & $e(\Gamma(1))$ & $\Gamma$ & $e(\Gamma)$\tabularnewline \hline \hline $k_{4,2624}$ & $\mathbb{Q}(\sqrt{5})$ & $1/2$ & $\Gamma^{B}(\mathfrak{q}_{7,1})\cap\Gamma^{B}(\text{\ensuremath{\mathfrak{q}}}_{7,2})$ & 32\tabularnewline \hline $k_{4,2000}$ & $\mathbb{Q}(\sqrt{5})$ & $1/3$ & $\Gamma(\mathfrak{q}_{5})$ & 20\tabularnewline \hline $k_{4,2000}$ & $\mathbb{Q}(\sqrt{5})$ & $1/3$ & $\Gamma(\mathfrak{q}_{2})$ & 20\tabularnewline \hline $k_{4,2525}$ & $\mathbb{Q}(\sqrt{5})$ & $2/3$ & $\Gamma^{B}(\mathfrak{q}_{5})\cap\Gamma^{B}(\overline{\mathfrak{q}}_{5})$ & 24\tabularnewline \hline $k_{4,3600}$ & $\mathbb{Q}(\sqrt{3})$, & $4/5$ & $\Gamma^{U}(\mathfrak{q}_{2})$ & 12\tabularnewline \hline \end{tabular}
\section{determination of the fixed curve.\label{sec:determination-of-the-curve}}
Let $X_{\Gamma}=\mathbb{H}_{\mathbb{C}}^{2}/\Gamma$ be a smooth Shimura surface such that the involution $\mu$ on $\mathbb{H}_{\mathbb{C}}^{2}$ exchanging the factors descends to an involution $\sigma$ on the quotient $X_{\Gamma}$. The image of the diagonal $\Delta\subset\mathbb{H}\times\mathbb{H}$ is a smooth Shimura curve $C_{\Gamma}$ fixed by $\sigma$. The aim of this section is to determine that curve in examples we investigated in Sections \ref{sub:pg5} and \ref{sub:pg6}.
The analogous problem for Hilbert modular surfaces is well-known, see for instance \cite{HirzebruchModular} or \cite{Hausmann}. The quaternion algebra $A$ over $k$ defining the group $\Gamma$ has a non trivial involution $\tau$ of second kind. That involution leaves invariant a subfield $\ell$. Recall that $A=A(k\mathfrak{,p}_{1},\dots,\mathfrak{p}_{2r})$ with $\mathfrak{p}_{2i-1}=\mathfrak{p}_{2i}^{\alpha}$ as in Prop. \ref{existence_type_2} and $\alpha:k\to k$ the involution of the extension $k/\ell$ with $\sigma_{1}=\sigma_{2}\circ\alpha$ for $\sigma_{i}:k\to\mathbb{R}$ the unramified infinite places in $A$. Also note that the involution $\sigma:X_{\Gamma}\longrightarrow X_{\Gamma}$ is determined by the involution of second kind $\tau$ on $A$. The fixed point set of $\sigma$ is associated with the invariant $\ell$-subalgebra $A'=\left\{ a\in A\ \mid\ \tau(a)=a\right\} $ of $A$. From the proof of Proposition \ref{existence_type_2} we know that $A'$ is a quaternion algebra over $\ell$ with the property $A'\otimes k\cong A$. Moreover, $A'$ is ramified at every prime $\mathcal{P}_{i}$ of $\ell$ such that $\mathcal{P}_{i}\mathcal{O}_{k}=\mathfrak{p}_{2i-1}\mathfrak{p}_{2i}$ for $i=1,\ldots,r$. \begin{lem} Let $A=A(k,\mathfrak{p}_{1},\ldots,\mathfrak{p}_{2r})$ be a quaternion algebra admitting an involution of second kind $\tau$ and $A'$ the elementwise $\tau$-invariant subalgebra. Let $\mathcal{O}$ be a order in $A$, then $\mathcal{O}'=\mathcal{O}\cap A'$ is an order in $A'$. Conversely, assume that $A'=A'(\ell,\mathcal{P}_{1},\ldots,\mathcal{P}_{s})$ is a quaternion algebra over $\ell$ and $\mathcal{O}'$ an order in $A'$ then $\mathcal{O}=\mathcal{O}'\otimes_{\mathcal{O}_{\ell}}\mathcal{O}_{k}$ is an order in $A.$ Assume that $\mathcal{O}'$ is maximal and each $\mathcal{P}_{i}$ is split in $k$ then $\mathcal{O}'\otimes_{\mathcal{O}_{\ell}}\mathcal{O}_{k}$ is a maximal order in $A=A'\otimes_{\ell}k$. $ $\end{lem} \begin{proof} The first part of the Lemma concerning the correspondence between orders $\mathcal{O}'$ in $A'$ and orders in $A$ is obvious and we shall therefore prove only the second part. Assume that $\mathcal{O}'\subset A'$ is a maximal order and let $\mathcal{O}=\mathcal{O}'\otimes_{\mathcal{O}_{\ell}}\mathcal{O}_{k}$. The order $\mathcal{O}$ is maximal if and only if each of its localizations $\mathcal{O}_{\mathfrak{p}}$ is maximal in the local algebra $A_{\mathfrak{p}}$. Here, $\mathcal{O}_{\mathfrak{p}}=\mathcal{O}'_{\mathcal{P}}\otimes\mathcal{O}_{k_{\mathfrak{p}}}$ arises from the local maximal order $\mathcal{O}'_{\mathcal{P}}$ corresponding to a finite place $\mathcal{P}$ of $\ell$ lying under $\mathfrak{p}$. Assume now that $\mathcal{P\neq\mathcal{P}}_{i}$ is a finite place at which $B$ is unramified. Then $\mathcal{O}'_{\mathcal{P}}\cong M_{2}(\mathcal{O}_{\ell_{\mathcal{P}}})$ and clearly $\mathcal{O}_{\mathfrak{p}}\cong M_{2}(\mathcal{O}_{k_{\mathfrak{p}}})$ is also maximal. If $\mathcal{P=\mathcal{P}}_{i}$ is a place such that $A'_{\mathcal{P}_{i}}$ is a division algebra, as by assumption $k_{\mathfrak{p}_{i}}=k_{\bar{\mathfrak{p}_{i}}}=\ell_{\mathcal{P}_{i}}$, $\mathcal{O}_{\mathfrak{p}_{i}}$ and $\mathcal{O}_{\overline{\mathfrak{p}}_{i}}$ are maximal.\end{proof} \begin{rem} We shall note that in the case where $A'=A'(\ell,\mathcal{P}_{1},\ldots,\mathcal{P}_{1},\mathcal{Q})$ ramifies also at some prime $\mathcal{Q}$ that is non-split in $k$ the order $\mathcal{O}=\mathcal{O}'\otimes\mathcal{O}_{k}$ is not maximal even if $\mathcal{O}'$ is maximal. \end{rem} \begin{example} Let $A'=\left(\frac{2,5}{\mathbb{Q}}\right)$ be the quaternion algebra over $\mathbb{Q}$ generated by elements $1,i,j,ij$ such that $i^{2}=2$, $j^{2}=5$ (and $ij=-ji$). The algebra $A'$ is ramified exactly at the primes $2$ and $5$. Let $\mathcal{O}'$ be a maximal order in $A'$. The group $\Gamma_{\mathcal{O}'}(1)$ is a Fuchsian group. Let $\Gamma_{\mathcal{O}'}^{B}(11)$ be the subgroup corresponding to the Borel subgroup of $PSL_{2}(\mathbb{F}_{11})$. This subgroup can be interpreted as the group of elements of reduced norm 1 of an Eichler order $\mathcal{E}(11)$ of level $11$. The group $\Gamma_{\mathcal{O}'}(1)$ is of index $12$ in $\Gamma_{\mathcal{O}'}(1)$ and is torsion-free by Lemma \ref{eichlercrit}, as $5$ is split in $\mathbb{Q}(i)$ and $11$ is non-split in $\mathbb{Q}(\sqrt{3})$. The genus of the curve $C=\Gamma_{\mathcal{O}'}(11)\backslash\mathbb{H}$ can be easily computed with the already used general volume formula from \cite{shimizu} (see also \cite[III, Prop. 2.10]{vign2}) by which $2-2g(C)=-8$. Let $k=\mathbb{Q}(\sqrt{33})$. Then $A=A'\otimes k=\left(\frac{2,5}{k}\right)$ is a quaternion algebra over $k$ and is ramified exactly at the two primes lying over $2$. The Eichler order $\mathcal{E}(11)$ in $A'$ is naturally contained in the order $\mathcal{E}(11)\otimes_{\mathbb{Z}}\mathcal{O}_{k}$ of $A$ and the latter one is contained in $\mathcal{E}(\mathfrak{q}_{11})$ the Eichler order of $A$ corresponding to the prime $\mathfrak{q}_{11}$ of $k$ lying over $11$, since the elements $\mathcal{E}(11)\otimes_{\mathbb{Z}}\mathcal{O}_{k}$ become upper triangular modulo $11$, hence modulo $\mathfrak{q}_{11}$. This gives an embedding of $\Gamma_{\mathcal{O}'}^{B}(11)$ in $\Gamma_{\mathcal{O}}^{B}(\mathfrak{q}_{11})$ and hence an embedding of a Shimura curve of genus $5$ into the Shimura surface $X=\Gamma_{\mathcal{O}}^{B}(\mathfrak{q}_{11})\backslash\mathbb{H}\times\mathbb{H}$ (see Section \ref{sub:pg5}) which by construction must be fixed by the involution of second kind on $X$. This gives a precise characterization of the Shimura surface $Z=X/\sigma$, the quotient of $X$ by the involution on $X$ induced by the involution of second kind:\end{example} \begin{prop} The surface $Z$ is a smooth surface of general type with $p_{g}=0$, $K_{Z}^{2}=4$ and $e(Z)=8$. \end{prop}
{\large{\setlength{\parindent}{0.2in} }}{\large \par}
{\large{Amir D\v{z}ambi\'{c},}}{\large \par}
{\large{Johann Wolfgang Goethe Universität, Institut für Mathematik,}}{\large \par}
{\large{Robert-Mayer-Str. 6-8, 60325 Frankfurt am Main,}}{\large \par}
{\large{Germany}}{\large \par}
\texttt{\large{dzambic@math.uni-frankfurt.de}}{\large \par}
{\large{
\setlength{\parindent}{0.5in}}}{\large \par}
{\large{Xavier Roulleau,}}{\large \par}
{\large{Laboratoire de Mathématiques et Applications, Université de Poitiers,}}{\large \par}
{\large{Téléport 2 - BP 30179 - 86962 Futuroscope Chasseneuil }}{\large \par}
{\large{France}}{\large \par}
\texttt{\large{roulleau@math.univ-poitiers.fr{} }}{\large{{} }}
\end{document} |
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\title[Pre-anti-flexible bialgebrask]{Pre-anti-flexible bialgebras} \author[Mafoya Landry Dassoundo]{Mafoya Landry Dassoundo} \address[]{Chern Institute of Mathematics \& LPMC, Nankai University, Tianjin 300071, China} \email{dassoundo@yahoo.com}
\begin{abstract} In this paper, we derive pre-anti-flexible algebras structures in term of zero weight's Rota-Baxter operators defined on anti-flexible algebras, view pre-anti-flexible algebras as a splitting of anti-flexible algebras, introduce the notion of pre-anti-flexible bialgebras and establish equivalences among matched pair of anti-flexible algebras, matched pair of pre-anti-flexible algebras and pre-anti-flexible bialgebras. Investigation on special class of pre-anti-flexible bialgebras leads to the establishment of the pre-anti-flexible Yang-Baxter equation. Both dual bimodules of pre-anti-flexible algebras and dendriform algebras have the same shape and this induces that both pre-anti-flexible Yang-Baxter equation and $\mathcal{D}$-equation are identical. Symmetric solution of pre-anti-flexible Yang-Baxter equation gives a pre-anti-flexible bialgebra. Finally, we recall and link $\mathcal{O}$-operators of anti-flexible algebras to bimodules of pre-anti-flexible algebras and built symmetric solutions of anti-flexible Yang-Baxter equation. \end{abstract}
\subjclass[2010]{17A20, 17D25, 16T10, 16T15, 17B38, 16T25}
\keywords{(pre-)anti-flexible algebra, dendriform algebras,
(pre-)anti-flexible bialgebra, Yang-Baxter equation, Rota-Baxter operator}
\maketitle
\tableofcontents
\numberwithin{equation}{section}
\tableofcontents
\numberwithin{equation}{section}
\allowdisplaybreaks
\section{Introduction and Preliminaries}
The notion of pre-anti-flexible algebras are introduced in \cite{DBH3} to derive
the $\mathcal{O}$-operators of anti-flexible algebras which allow to built the
skew-symmetric solutions of anti-flexible Yang-Baxter equation.
Pre-anti-flexible algebras are closed dendriform algebras
which are introduced by J.-L. Loday (\cite{Loday}).
Besides, pre-anti-flexible algebras can be considered as a
generalization of dendriform algebras and
as very well known and widespread in the literature, dendriform algebras
are also induced by the well known notion of Rota-Baxter operators of weight zero
(\cite{Aguiar1}) which are introduced around 1960’s by
G. Baxter (\cite{Baxter}) and G.-C. Rota (\cite{Rota}). Recently,
significant advances contributions on Rota-Baxter operators and related
applications are summarized in \cite{Guo} and the references therein.
Since dendriform algebras are closed to associative algebras,
pre-anti-flexible algebras are strongly linked to anti-flexible algebras
(also known as center-symmetric algebras)
and themselves associated with Lie algebras (\cite{Hounkonnou_D_CSA}) and other
similar deduction are derived and readable
in the following diagram which summarizes underlying relations
among pre-anti-flexible algebras (PAFA),
anti-flexible algebras (AFA),
Lie algebras (LA), associative algebras (AA), dendriform algebras (DA),
and finally with what we call
derived Lie-admissible algebra of a given pre-anti-flexible algebra (DLAd-PAFA)
\begin{eqnarray*}
\xymatrix{
&{\bf PAFA\;} (A, \prec, \succ) \ar[dl]_-{\mbox{C2}}\ar[d]^-{\mbox{C6}}\ar[rr]^-{\mbox{C1}}
&&{\bf AFA\;}(A, \cdot) \ar[d]^-{\mbox{C5}}\\
{\bf DA\;} (A, \prec_{_1}, \succ_{_1})\ar[dr]_-{\mbox{C3}}
& {\bf{DLAd-PAFA}\;}(A, \circ )\ar[rr]^-{\mbox{C7}}&&{\bf LA\;} (A, [,]), \\
&{\bf AA\;}(A, \cdot_{_1} )\ar[rru]_-{\mbox{C4}}
}
\end{eqnarray*}
where, for any $x,y, z\in A$, the condition C1 translates
$x\cdot y=x\prec y+x\succ y$,
C2 means $\prec_{_1}:=\prec; \succ_{_1}:=\succ$ and
$(x,y,z)_m=0, (x,y,z)_l=0, (x,y,z)_r=0$ which are given by
Eqs.~\eqref{eq:biasso} (trivial
pre-anti-flexible algebras are dendriform algebras), C3 describes
$x\cdot_{_1} y= x\prec_{_1} y+x\succ_{_1}y$, C4 expresses the commutator
$[x,y]=x\cdot_{_1}y-y\cdot_{_1} x$, C5 translates
$[x,y]=x\cdot y-y\cdot x$, C6 describes $x\circ y=x\succ y-y\prec x$ and finally
C7 expresses $[x,y]=x\circ y-y\circ x$. It is also useful to recall that any
associative algebra is a trivial anti-flexible algebra.
Thus, anti-flexible algebras generalize associative algebras.
Notice here that, although the goal of this paper
is not to construct a cohomology theory for
anti-flexible algebras, cohomology of associative and Lie algebras,
and other algebras are well known. Unfortunately, despite their links to associative algebras and to Lie algebras described above,
anti-flexible algebras and pre-anti-flexible algebras lack a suitable cohomology theory
which can justify certain shortcomings
on anti-flexible algebras such as for instance among many other,
coboundary anti-flexible algebras and those of pre-anti-flexible algebras
which are well known on associative and Lie algebras.
That said, all is not lost for avoid talking about some notions on the cohomology
of anti-flexible and pre-anti-flexible algebras.
As proof, the analogue to the classical Yang-Baxter
equation on Lie algebras derived
by Drinfeld (\cite{Drinfeld}), that of associative Yang-Baxter
on associative algebras (\cite{Aguiar, Bai_Double})
as well as $\mathcal{D}$-equation of dendriform algebras (\cite{Bai_Double}),
anti-flexible Yang-Baxter equation recovered in special consideration of
anti-flexible algebras (\cite{DBH3}) as well as
pre-anti-flexible Yang-Baxter equation on special class of
pre-anti-flexible algebras.
Furthermore, alternative $\mathcal{D}$-bialgebras are also
provided and described on Cayley-Dickson matrix algebras (\cite{Gon}).
Besides, by keeping the spirit that dendriform algebras are viewed as
splitted associative algebras (\cite{Bai_spit}),
pre-anti-flexible algebras are regarded as splitted anti-flexible algebras and
more generally, operadic definition for the notion of
splitting algebra structures are introduced
and provided some equivalence with Manin products of operads in
quadratic operads (\cite{Pei_Bai_Guo}).
Before straight although to the goal of this paper, recall some
fundamentals which will be necessary throughout our
concern on pre-anti-flexible bialgebras.
In this paper, all considered vector spaces are finite-dimensional
over a base field $\mathbb{F}$ whose characteristic is $0$.
Many derived results still hold regardless the dimension of vector spaces
on which they are stated. For this purpose, we mean by anti-flexible algebra (\cite{Hounkonnou_D_CSA}),
a couple $(A, \ast)$ where $A$ is a vector space equipped with a linear product
"$\ast$" such that for any $x,y, z\in A$, $(x,y,z)=(z,y,x)$, where the triple is defined as
$(x,y,z):=(x\ast y)\ast z-x\ast (y\ast z)$.
If in addition $A$ is equipped with two linear maps $l,r:A\rightarrow \End(V)$, where $V$
is a vector space, such that for any $x,y\in A$
\begin{subequations}
\begin{eqnarray}\label{eqbimodule1}
l{(x\ast y)}-l(x)l(y)=r(x)r(y)-r({y \ast x}),
\end{eqnarray}
\begin{eqnarray}\label{eqbimodule2}
\left[l(x), r(y)\right]= \left[l(y), r(x)\right],
\end{eqnarray}
\end{subequations}
then the triple $(l,r,V)$ is called bimodule of $(A, \ast)$.
\begin{thm}\label{thm_2} \cite{Hounkonnou_D_CSA}
Let $( A, \ast)$ and $(B, \circ)$ be two anti-flexible algebras.
Suppose that $(l_{ A}, r_{ A}, B)$ and $(l_{B}, r_{B}, A)$
are bimodules of $(A, \ast)$ and $(B, \circ)$, respectively, where
$l_{ A}, r_{ A}:A\rightarrow \End(B)$ and
$l_{B}, r_{B}:B\rightarrow \End(A)$ are four linear maps and
obeying the relations,
for any $x, y \in A$ and for any $a, b \in B,$
\begin{subequations}
\begin{eqnarray}\label{eqq1}
l_{B}(a)(x\ast y) +r_{B}(a)(y\ast x)-r_{B}(l_{ A}(x)a)y-
y\ast(r_{B}(a)x) -l_{B}(r_{ A}(x)a)y - (l_{B}(a)x)\ast y = 0,
\end{eqnarray}
\begin{eqnarray}\label{eqq2}
l_{ A}(x)(a\circ b) +r_{ A}(x)(b\circ a)-r_{ A}(l_{B}(a)x)b-
b\circ (r_{ A}(x)a)+l_{ A}(r_{B}(a)x)b - (l_{ A}(x)a)\circ b=0,
\end{eqnarray}
\begin{eqnarray}\label{eqq3}
\begin{array}{lll}
y\ast (l_{B}(a)x)+(r_{B}(a)x)\ast y - (r_{B}(a)y)\ast x-l_{B}(l_{ A}(y)a)x+ \cr
r_{B}(r_{ A}(x)a)y+l_{B}(l_{ A}(x)a)y -x\ast (l_{B}(a)y)-r_{B}(r_{ A}(y)a)x=0,
\end{array}
\end{eqnarray}
\begin{eqnarray}\label{eqq4}
\begin{array}{lll}
b \circ (l_{ A}(x)a)+(r_{ A}(x)a)\circ b -(r_{ A}(x)b)\circ a-
l_{ A}(l_{B}(b)x)a+\cr
r_{ A}(r_{B}(a)x)b+l_{ A}(l_{B}(a)x)b -
a\circ (l_{ A}(x)b) -r_{ A}(r_{B}(b)x)a=0.
\end{array}
\end{eqnarray}
\end{subequations}
Then, there is an anti-flexible algebra structure on $ A \oplus B$
given by for any $x,y\in A$ and any $a, b\in B$
\begin{eqnarray*}
(x+a)\star (y+b)= (x \ast y + l_{B}(a)y+r_{B}(b)x)+ (a \circ b + l_{A}(x)b+r_{A}(y)a).
\end{eqnarray*}
\end{thm}
\begin{defi}\cite{DBH3}
A pre-anti-flexible algebra is a vector space $ A$ equipped with two bilinear products
$\prec, \succ: A\otimes A \rightarrow A$
satisfying the following relations
\begin{subequations}
\begin{eqnarray}\label{eq_pre_antiflexible_1}
(x,y,z)_{_m}=(z,y,x)_{_m}, \; \; \; \forall x,y,z\in A,
\end{eqnarray}
\begin{eqnarray}\label{eq_pre_antiflexible_2}
(x,y,z)_{_l}=(z,y,x)_{_r}, \; \; \; \forall x,y,z\in A,
\end{eqnarray}
\end{subequations}
where for any $x,y, z\in A$,
\begin{subequations}\label{eq:biasso}
\begin{eqnarray}\label{eq_biasso_m}
(x,y,z)_{_m}:=(x \succ y) \prec z-x \succ (y \prec z),
\end{eqnarray}
\begin{eqnarray}\label{eq_biasso_l}
(x,y,z)_{_l}:=(x\cdot y)\succ z-x\succ (y\succ z),
\end{eqnarray}
\begin{eqnarray}\label{eq_biasso_r}
(x,y,z)_{_r}:=(x\prec y)\prec z-x\prec (y\cdot z),
\end{eqnarray}
\end{subequations}
with $x\cdot y=x\prec y+x\succ y$.
Equivalently, a pre-anti-flexible algebra is a triple
$(A, \prec, \succ)$ such that $A$ is a vector space and
$\prec, \succ: A\times A \rightarrow A$ are
two linear maps satisfying the relations for any
$x,y,z\in A$
\begin{subequations}
\begin{eqnarray}\label{eq:pre-antiflexible1}
(x\succ y) \prec z-x\succ (y\prec z)=(z\succ y)\prec x-
z\succ(y\prec x),
\end{eqnarray}
\begin{eqnarray}\label{eq:pre-antiflexible2}
(x\prec y +x\succ y )\succ z-x\succ (y\succ z)=
(z\prec y)\prec x-z\prec (y\prec x+y\succ x).
\end{eqnarray}
\end{subequations}
\end{defi}
\begin{ex}
For a given associative $(A, \ast)$, setting for any $x,y\in A$,
$x\succ y=x\ast y$ (or $x\succ y=y\ast x$) and
$x\prec y=0$, then $(A, \prec, \succ)$ is
a pre-anti-flexible algebra. Similarly, $(A, \prec, \succ)$ is a
pre-anti-flexible algebra by setting for any $x,y\in A$,
$x\prec y=x\ast y$ (or $x\prec y=y\ast x$) and $x\succ y=0$.
\end{ex}
\begin{rmk}\label{rmk_1}
Let $( A, \prec, \succ)$ be a pre-anti-flexible algebra.
\begin{enumerate}
\item\label{rmk_flex}
It is well known that the couple $( A, \cdot)$ is an
anti-flexible algebra (\cite{DBH3}), where
for any $x,y\in A$, $x\cdot y=x\prec y+x\succ y$, i.e.
$(x,y,z)=(z,y,x)$,
and we will denote this anti-flexible algebra by $aF(A)$
call it underlying anti-flexible algebra of the pre-anti-flexible algebra
$(A, \prec, \succ)$.
\item
As we can see, if both sides of equality in the
Eqs.~\eqref{eq:pre-antiflexible1} and \eqref{eq:pre-antiflexible2} are zero
i.e. for any $ x,y,z\in A, (x,y,z)_{_m}=0$, $(x,y,z)_{_l}=0$
and $(x,y,z)_{_r}=0$, then $(A, \prec, \succ)$
is a dendriform algebra i.e. the couple $(A, \prec, \succ)$ such that
for any $x,y,z\in A,$
\begin{eqnarray*}
&&(x\succ y) \prec z-x\succ (y\prec z)=0,\cr
&&(x\prec y +x\succ y )\succ z-x\succ (y\succ z)=0,\cr
&&(x\prec y)\prec z-x\prec (y\prec z+y\succ z)=0,
\end{eqnarray*}
introduced by J.-L.~Loday (\cite{Loday}).
Clearly, dendriform algebra is a pre-anti-flexible algebra
and then pre-anti-flexible algebras can viewed as a generalization
of dendriform algebras.
\end{enumerate}
\end{rmk}
Throughout this paper, if there is no other consideration, for any
$x,y\in A$, $x\ast y$ or $x\cdot y$ will simply written by $xy$.
Furthermore, underlying anti-flexible algebra structure
of a given pre-anti-flexible algebra will generally denote by "$\cdot$" as
in Remark~\ref{rmk_1}~\eqref{rmk_flex}.
Moreover, for a given pre-anti-flexible algebra
$(A, \prec, \succ)$, we denote by
$L_{\prec}, R_{\prec}: A\rightarrow \End(A)$ the left
and right multiplication operators, respectively, on $(A, \prec)$
and similarly by $L_{\succ}, R_{\succ}: A\rightarrow \End(A)$ these on
$(A, \succ)$ which are defined as
$\forall x,y \in A,$
\begin{equation*}
L_{_\prec}(x)y=x\prec y,\;\; R_{_\prec}(x)y=y\prec x,\;\;
L_{_\succ}(x)y=x\succ y,\;\; R_{_\succ}(x)y=y\succ x.
\end{equation*}
\begin{thm}\cite{DBH3}\label{Theo_existance_pre_anti_flexible}
Let $(A, \ast)$ be an anti-flexible algebra equipped with a
non-degenerate bilinear form $\omega: A\otimes A\rightarrow \mathbb{F}$ satisfying
\begin{equation}\label{eq:simplectic_form}
\omega(x\ast y, z)+\omega(y\ast z, x)+\omega(z\ast x, y)=0,
\quad \forall x,y\in A.
\end{equation}
Then there is a pre-anti-flexible algebra structure
"$\prec, \succ$" defined on $ A$
satisfying the following relation, for any $x,y, z\in A$,
\begin{eqnarray}\label{eq_useful1}
\omega(x\prec y,z)=\omega(x, y\ast z), \quad
\omega(x\succ y, z)=\omega(y, z\ast x).
\end{eqnarray}
\end{thm}
To make this introductory section devoted to
fundamentals necessary for addressed main issues
as short as possible, we end it by outlined the content
of this article as follows.
In section~\ref{section1}, we prove and generalize that Rota-Baxter
operators on an anti-flexible algebras
induce pre-anti-flexible algebras.
In Section~\ref{section2},
we study bimodules and matched pairs of pre-anti-flexible algebras.
Precisely, we derive the dual bimodules of
bimodules of an pre-anti-flexible algebras.
In Section~\ref{section3},
we establish the equivalences among matched pair of the underlying
anti-flexible algebras of pre-anti-flexible algebras,
matched pair of pre-anti-flexibles algebras,
and to pre-anti-flexible bialgebras.
In Section~\ref{section4},
we rule on a special class of pre-anti-flexible bialgebras
which lead to the introduction
of the pre-anti-flexible Yang-Baxter equation.
A symmetric solution of the pre-anti-flexible Yang-Baxter
equation gives a such pre-anti-flexible bialgebra.
Finally in Section~\ref{section5},
we recall the notions of $\mathcal{O}$-operators of anti-flexible algebras
and intertwine this notion to that of bimodules of pre-anti-flexible algebras
and use the relationships among
them to provide the
symmetric solutions of pre-anti-flexible Yang-Baxter equation in
pre-anti-flexible bialgebras.
\section{Rota-Baxter operators and pre-anti-flexible algebras}\label{section1}
In this section, we are going to express pre-anti-flexible algebras
in terms of Rota-Baxter operator of weight zero defined on anti-flexible algebras.
\begin{defi}
Let $(A,\ast)$ be an anti-flexible algebra.
A Rota-Baxter operator ($\mathrm{R_B}$) of weight zero on $A$
is a linear operator $\mathrm{R_B}:A\rightarrow A$ satisfying
\begin{eqnarray}\label{eq:Rota-Baxter}
\mathrm{R_B}(x)\ast \mathrm{R_B}(y)=
\mathrm{R_B}(x\ast \mathrm{R_B}(y)+\mathrm{R_B}(x)\ast y), \;\forall x,y\in A.
\end{eqnarray}
\end{defi}
\begin{thm}
Let $(A, \ast)$ be an anti-flexible algebra equipped with a linear map
$\alpha:A\rightarrow A$. Consider the bilinear products "$\prec, \succ$"
given by for any $x,y\in A,$
\begin{eqnarray}\label{eq:pre-anti-flexible-Rota-Baxter}
x\succ y=\alpha(x)\ast y,\;\quad
x\prec y=x\ast \alpha(y).
\end{eqnarray}
Then the triple $(A, \prec, \succ)$ is a pre-anti-flexible algebra
if and only if for any $x,y,z\in A,$
\begin{eqnarray}\label{eq:identity-RB-pre-anti-flexible}
(\alpha(x)\ast \alpha(y)-\alpha(x\ast \alpha(y)+\alpha(x)\ast y))\ast z+
z\ast (\alpha(y)\ast \alpha(x)-\alpha(y\ast \alpha(x)+\alpha(y)\ast x))=0.
\end{eqnarray}
\end{thm}
\begin{proof}
For any $x,y,z\in A$, we have
\begin{eqnarray*}
(x,y,z)_m=(\alpha(z), y, \alpha(x))=(\alpha(x), y, \alpha(z))=(z, y, x)_m.
\end{eqnarray*}
Thus the bilinear products given by Eq.~\eqref{eq:pre-anti-flexible-Rota-Baxter} satisfy Eq.~\eqref{eq_pre_antiflexible_1}.
Besides, we have
\begin{eqnarray*}
(x,y,z)_l=-(\alpha(x)\ast \alpha(y)-\alpha(x\ast \alpha(y)+\alpha(x)\ast y))\ast z+(z, \alpha(y), \alpha(x))
\end{eqnarray*}
and
\begin{eqnarray*}
(z, y, x)_r=z\ast (\alpha(y)\ast \alpha(x)-
\alpha(y\ast \alpha(x)+\alpha(y)\ast x))+(\alpha(x), \alpha(y), z).
\end{eqnarray*}
Therefore, Eq.~\eqref{eq_pre_antiflexible_2} is equivalent to
Eq.~\eqref{eq:identity-RB-pre-anti-flexible}.
\end{proof}
It is obvious to remark that the previous theorem generalizes
the following corollary which links Rota-Baxter operators to
pre-anti-flexible algebras.
\begin{cor}
If $\alpha:A\rightarrow A$ is a Rota-Baxter operator
defined on an anti-flexible algebra $(A, \ast)$
i.e. $\alpha$ is a linear map satisfies Eq.~\eqref{eq:Rota-Baxter}, then there is
a pre-anti-flexible algebra structure "$\prec, \succ$" on $A$ given by
Eq.~\eqref{eq:pre-anti-flexible-Rota-Baxter}.
\end{cor}
\section{Bimodules and matched pair of pre-anti-flexible algebras}\label{section2}
In this section, we provide bimodules and dual bimodules of
pre-anti-flexible algebras. We also
introduce matched pair of pre-anti-flexible algebras and
equivalently link them to a matched pair of their
underlying anti-flexible algebras. Finally, we define anti-flexible bialgebras
and establish related identities.
\begin{defi}
Let $( A, \prec, \succ)$ be a pre-anti-flexible algebra and $V$ be a vector space.
Consider the four linear maps
$l_{_\succ}, l_{_\prec},r_{_\succ}, r_{_\prec}: A\rightarrow \End(V) $.
The quintuple $(l_{_\succ},r_{_\succ}, l_{_\prec}, r_{_\prec}, V)$
is called a bimodule of $( A, \prec, \succ)$ if for any $x,y\in A$,
\begin{subequations}
\begin{eqnarray}\label{eq_bimodule_pre_anti_flexible1}
[r_{_\prec}(x), l_{_\succ}(y)]=[r_{\prec}(y), l_{_\succ}(x)],
\end{eqnarray}
\begin{eqnarray}\label{eq_bimodule_pre_anti_flexible2}
l_{_\prec}(x\succ y)-l_{_\succ}(x)l_{_\prec}(y)=
r_{_\prec}(x)r_{_\succ}(y)-r_{_\succ}(y\prec x),
\end{eqnarray}
\begin{eqnarray}\label{eq_bimodule_pre_anti_flexible3}
l_{_\succ}(x\cdot y)-l_{_\succ}(x)l_{_\succ}(y)=
r_{_\prec}(x)r_{_\prec}(y)-r_{_\prec}(y\cdot x),
\end{eqnarray}
\begin{eqnarray}\label{eq_bimodule_pre_anti_flexible4}
r_{_\succ}(x)l_{\cdot}(y)-l_{_\succ}(y)r_{_\succ}(x)=
r_{_\prec}(y)l_{_\prec}(x)-l_{_\prec}(x)r_{\cdot}(y),
\end{eqnarray}
\begin{eqnarray}\label{eq_bimodule_pre_anti_flexible5}
r_{_\succ}(x)r_{\cdot}(y)-r_{_\succ}(y\succ x)=
l_{_\prec}(x\prec y)-l_{_\prec}(x)l_{\cdot}(y),
\end{eqnarray}
\end{subequations}
where, $x\cdot y=x\prec y+x\succ y, l_{_\cdot}=l_{_\prec}+l_{_\succ}$ and
$r_{_\cdot}=r_{_\prec}+r_{_\succ}$.
\end{defi}
\begin{pro}
Let $( A, \prec, \succ)$ be a pre-anti-flexible algebra and $V$ be a vector space.
Consider the four linear maps
$l_{_\succ}, l_{_\prec},r_{_\succ}, r_{_\prec}: A\rightarrow \End(V) $.
The quintuple $(l_{_\succ},r_{_\succ}, l_{_\prec}, r_{_\prec}, V)$
is called a bimodule of $( A, \prec, \succ)$ if and only if there is a
pre-anti-flexible algebra defined on $ A\oplus V$ by for any
$x, y\in A$, $u, v\in V$,
\begin{eqnarray}\label{eq_pre_anti_flexible_bimodule}
\begin{array}{ccc}
(x+u)\prec'(y+v)=x\prec y+l_{_\prec}(x)v+r_{_\prec}(y)u,\cr
(x+u)\succ'(y+v)=x\succ y+l_{_\succ}(x)v+r_{_\succ}(y)u,
\end{array}
\end{eqnarray}
and
$
(x+u)\cdot'(y+v)=(x+u)\prec'(y+v)+(x+u)\succ'(y+v).
$
\end{pro}
\begin{proof}
According to Eq.~\eqref{eq_pre_anti_flexible_bimodule}
we have for any $x,y,z\in A$ and for any $u,v,w\in V$,
\begin{eqnarray*}
(x+u, y+v, z+w)_{_m}&=&((x+u)\succ'(y+v) )\prec' (z+w)-(x+u)\succ'((y+v)\prec'(z+w))\cr
&=&(x,y,z)_{_m}+\{l_{_\prec}(x\succ y)-l_{_\succ}(x)(l_{_\prec}(y))\}w\cr
&+&\{r_{_\prec}(z)l_{_\succ}(x)-l_{_\succ}(x)r_{_\prec}(z)\}v+
\{r_{_\prec}(z)r_{_\succ}(y)-r_{_\prec}(y\prec z)\}u.
\end{eqnarray*}
\begin{eqnarray*}
(x+u, y+v, z+w)_{_l}&=&((x+u)\cdot'(y+v) )\succ' (z+w)-(x+u)\succ'((y+v)\succ'(z+w))\cr
&=&(x,y,z)_{_l}+\{ l_{\cdot}(x\cdot y)-l_{_\succ}(x)l_{_\succ}(y) \}w\cr
&+&\{r_{_\succ}(z)l_{\cdot}(x)-l_{_\succ}(x)r_{_\succ}(z)\}v+
\{r_{_\succ}(z)r_{\cdot}(y)-r_{_\succ}(y\succ z)\}u.
\end{eqnarray*}
\begin{eqnarray*}
(x+u, y+v, z+w)_{_r}&=&((x+u)\prec'(y+v) )\prec' (z+w)-(x+u)\prec'((y+v)\cdot'(z+w))\cr
&=&(x,y,z)_{_r}+\{l_{_\prec}(x\prec y)-l_{_\prec}(x)l_{\cdot}(y) \}w\cr
&+&\{r_{_\prec}(z)l_{_\prec}(x)-l_{_\prec}(x)r_{\cdot}(z) \}v+
\{r_{_\prec}(z)r_{_\prec}(y)-r_{_\prec}(y\cdot z)\}u.
\end{eqnarray*}
If in addition the four linear maps $l_{_\succ}, l_{_\prec},r_{_\succ}, r_{_\prec}$
satisfy Eqs.~\eqref{eq_bimodule_pre_anti_flexible1}~-~
\eqref{eq_bimodule_pre_anti_flexible5},
then for any $x,y,z\in A$, and for any
$u, v,w\in V$ the following conditions
are satisfied,
\begin{eqnarray*}
(x+u, y+v, z+w)_{_m}=(z+w, y+v, x+u)_{_m},\;\;
(x+u, y+v, z+w)_{_l}=(z+w, y+v, x+u)_{_r}.
\end{eqnarray*}
Therefore, holds the equivalence.
\end{proof}
In the following of this paper, for a given pre-anti-flexible algebra
$(A, \prec, \succ)$, a vector space $V$ and a linear map
$\varphi : A\rightarrow \End(V)$,
its dual linear map is defined as $\varphi^* : A\rightarrow \End(V^*)$ by
\begin{eqnarray}\label{eq_dual_map}
\langle \varphi^*(x)u^*, v\rangle=
\langle u^*, \varphi(x)v\rangle , \quad \forall x\in A, \; v\in V, u^*\in V^*,
\end{eqnarray}
where $\langle , \rangle $ is the usual pairing between $V$ and $V^*$.
In addition, we also denote by $\sigma$ a linear map from
$V\otimes V$ into itself or from $V^*\otimes V^*$ into itself define by for any
$u,v\in V$, $u^*, v^*\in V^*$,
$\sigma(u\otimes v)=v\otimes u$ and $\sigma(u^*\otimes v^*)=v^*\otimes u^*$.
\begin{pro}\label{prop_operation_bimodule_pre_anti_flexible}
Let $(l_{_\succ},r_{_\succ}, l_{_\prec}, r_{_\prec}, V)$ be a bimodule of a
pre-anti-flexible algebra $( A, \prec, \succ)$, where
$V$ is a vector space and $l_{_\succ}, l_{_\prec},r_{_\succ}, r_{_\prec}:
A\rightarrow \End(V) $
are four linear maps. We have:
\begin{enumerate}
\item $(l_{_\succ},0, 0, r_{_\prec}, V)$ ,
$(r^*_{_\prec}, l^*_{_\prec} , l^*_{_\succ}, r^*_{_\succ} , V^*)$and
$(r^*_{_\prec}, 0 , 0,l^*_{_\succ}, V^*)$
are bimodules of the pre-anti-flexible algebra $( A, \prec, \succ)$,
\item\label{eq:one} $(l_{\cdot}, r_{\cdot}, V)$, $(l_{_\succ}, r_{_\prec}, V)$,
$(r^*_{\cdot} , l^*_{\cdot} , V^*)$ and
$(r^*_{_\prec} , l^*_{_\succ} , V^*)$ are bimodules of
the underlying anti-flexible algebra $aF(A)$ of $( A, \prec, \succ)$,
\end{enumerate}
where, $l_{_\succ}+l_{_\prec}=l_{\cdot}, r_{_\succ}+r_{_\prec}=r_{\cdot}$.
\end{pro}
\begin{proof}
It is well known that
$aF(A)$ is an anti-flexible algebra.
\begin{enumerate}
\item
Since $(l_{_\succ},r_{_\succ}, l_{_\prec}, r_{_\prec}, V)$ is a bimodule of the
pre-anti-flexible algebra $( A , \prec, \succ)$, then
the four linear maps $l_{_\succ}, l_{_\prec},r_{_\succ}, r_{_\prec}:
A \rightarrow \End(V)$ satisfy
Eqs.~\eqref{eq_bimodule_pre_anti_flexible1}~-~\eqref{eq_bimodule_pre_anti_flexible5}
which still satisfied by setting
$ l_{_\prec}=0$ and $r_{_\succ}=0$. Thus $(l_{_\succ},0, 0, r_{_\prec}, V)$ is a
bimodule of the pre-anti-flexible algebra $( A , \prec, \succ)$.
Furthermore, using Eq.~\eqref{eq_dual_map}, we deduce that
$(r^*_{_\prec}, l^*_{_\prec} , l^*_{_\succ}, r^*_{_\succ} , V^*)$ and
$(r^*_{_\prec}, 0 , 0,l^*_{_\succ}, V^*)$ are also
bimodules of the pre-anti-flexible algebra $( A , \prec, \succ)$.
\item
Since the four linear maps $l_{_\succ}, l_{_\prec},r_{_\succ}, r_{_\prec}: A \rightarrow \End(V)$
satisfy Eqs.~\eqref{eq_bimodule_pre_anti_flexible1}~-~\eqref{eq_bimodule_pre_anti_flexible5},
then both linear maps $l_{_\succ}$ and $r_{_\prec}$ satisfy
Eqs.~\eqref{eq_bimodule_pre_anti_flexible1} and \eqref{eq_bimodule_pre_anti_flexible3}
which is exactly Eqs.~\eqref{eqbimodule2} and \eqref{eqbimodule1}, respectively.
Thus $(l_{_\succ}, r_{_\prec},V)$ is a bimodule of
$aF(A)$. In view Eqs.~\eqref{eq_bimodule_pre_anti_flexible1}
and \eqref{eq_bimodule_pre_anti_flexible4},
we have for any $x,y\in A $,
\begin{eqnarray*}
[l_{_\cdot}(x), r_{_\cdot}(y)]- [l_{_\cdot}(y), r_{_\cdot}(x)]&=&
\{l_{_\prec}(x)r_{_\cdot}(y)+r_{_\succ}(x)l_{_\cdot}(y)-
l_{_\succ}(y)r_{_\succ}(x)-r_{_\prec}(y)l_{_\prec}(x) \}\cr
&-&\{ l_{_\prec}(y)r_{_\cdot}(x)+r_{_\succ}(y)l_{_\cdot}(x)-
l_{_\succ}(x)r_{_\succ}(y)-r_{_\prec}(x)l_{_\prec}(y) \}\cr
&+&\{[l_{_\succ}(x), r_{_\prec}(y)]-[l_{_\succ}(y), r_{_\prec}(x)] \}=0.
\end{eqnarray*}
In addition, considering Eqs.~\eqref{eq_bimodule_pre_anti_flexible2},
\eqref{eq_bimodule_pre_anti_flexible3} and \eqref{eq_bimodule_pre_anti_flexible5},
we have for any $x,y\in A $,
\begin{eqnarray*}
l_{_\cdot}(x\cdot y)-l_{_\cdot}(x)l_{_\cdot}(y)-r_{_\cdot}(x)r_{_\cdot}(y)+r_{_\cdot}(y\cdot x)
&=&\{l_{_\succ}(x\cdot y) -l_{_\succ}(x)l_{_\succ}(y)-r_{_\prec}(x)r_{_\prec}(y)\cr
&+&r_{_\prec}(y\cdot x)\}+ \{l_{_\prec}(x\succ y)-l_{_\succ}(x)l_{_\prec}(y) \cr
&+&r_{_\succ}(y\prec x)-r_{_\prec}(x)r_{_\succ}(y) \}+\{l_{_\prec}(x\prec y) \cr
&+& r_{_\prec}(y\succ x)-l_{_\prec}(x)l_{_\cdot}(y)-r_{_\succ}(x)r_{_\cdot}(y)\}
=0.
\end{eqnarray*}
Therefore both $( l_{_\cdot}, r_{_\cdot}, V)$ and $(l_{_\succ}, r_{_\prec}, V)$
are bimodules of $aF(A)$.
According to Eq.~\eqref{eq_dual_map}, both
$(r^*_{_\cdot} , l^*_{_\cdot} , V^*)$ and
$(r^*_{_\prec} , l^*_{_\succ} , V^*)$ are bimodules of $aF(A)$.
\end{enumerate}
\end{proof}
\begin{ex}
Consider a pre-anti-flexible algebra $(A, \prec, \succ)$.
We have $( L_{_\prec}, R_{_\prec} , L_{_\succ}, R_{_\succ} , A)$ and
$(L_{_\succ},0, 0, R_{_\prec}, A)$ are bimodules of
$(A, \prec, \succ)$. Besides,
$(R^*_{_\prec}, L^*_{_\prec} , L^*_{_\succ}, R^*_{_\succ} , A^*)$ and
$(R^*_{_\prec}, 0 , 0, L^*_{_\succ}, A^*)$
are also bimodules of the pre-anti-flexible algebra $( A, \prec, \succ)$.
\end{ex}
\begin{rmk}\label{rmk_useful}
For a given bimodule $(l_{_\succ},r_{_\succ}, l_{_\prec}, r_{_\prec}, V)$
of a pre-anti-flexible algebra $(A, \prec, \succ)$ we have
\begin{enumerate}
\item
If both side of
Eqs.~\eqref{eq_bimodule_pre_anti_flexible1}~-~\eqref{eq_bimodule_pre_anti_flexible5}
are zero i.e.
the linear maps $l_{_\succ}, l_{_\prec},r_{_\succ}, r_{_\prec}$ satisfy
\begin{eqnarray*}
r_{_\prec}(x)l_{_\succ}(y)=l_{_\succ}(y)r_{_\prec}(x),\;
l_{_\prec}(x\succ y)=l_{_\succ}(x)l_{_\prec}(y),\;
r_{_\prec}(x)r_{_\succ}(y)=r_{_\succ}(y\prec x),\\
l_{_\succ}(x\cdot y)=l_{_\succ}(x)l_{_\succ}(y),\;
r_{_\prec}(x)r_{_\prec}(y)=r_{_\prec}(y\cdot x),\;
r_{_\succ}(x)l_{\cdot}(y)=l_{_\succ}(y)r_{_\succ}(x),\\
r_{_\prec}(y)l_{_\prec}(x)=l_{_\prec}(x)r_{\cdot}(y),\;
r_{_\succ}(x)r_{\cdot}(y)=r_{_\succ}(y\succ x),\;
l_{_\prec}(x\prec y)=l_{_\prec}(x)l_{\cdot}(y),
\end{eqnarray*}
with $l_{\cdot}=l_{_\succ}+l_{_\prec}$ and $r_{\cdot}=r_{_\succ}+r_{_\prec}$.
\item
For any $x,y \in A$ we have
\begin{eqnarray}\label{eq:useful}
L_{\cdot}(x)L_{\cdot}(y)+R_{\cdot}(x) R_{\cdot}(y)&=&
(L_{\prec}(x)+L_{\succ}(x))(L_{\prec}(y)+L_{\succ}(y))+
(R_{\prec}(x)+R_{\succ}(x))(R_{\prec}(y)+R_{\succ}(y))\cr
&=&(L_{\succ}(x)L_{\succ}(y)+R_{\prec}(x)R_{\prec}(y))+
(L_{\succ}(x)L_{\prec}(y)+L_{\prec}(x)R_{\succ}(y))\cr
&+&(L_{\prec}(x)L_{\succ}(y)+L_{\prec}(x)L_{\prec}(y)+
(R_{\succ}(x)R_{\succ}(y)+R_{\succ}(x)L_{\prec}(y) )\cr
L_{\cdot}(x)L_{\cdot}(y)+R_{\cdot}(x) R_{\cdot}(y)&=&L_{\cdot}(x\cdot y)+R_{\cdot}(y\cdot x)
\end{eqnarray}
\item Besides, for any $x,y\in A$
\begin{eqnarray}\label{eq:useful1}
[L_{\cdot}(x), R_{\cdot}(y)]-[L_{\cdot}(y), R_{\cdot}(x)]&=&
\{L_{_\prec}(x)R_{_\cdot}(y)+R_{_\succ}(x)L_{_\cdot}(y)-L_{_\succ}(y)R_{_\succ}(x)-R_{_\prec}(y)L_{_\prec}(x) \}\cr
&-&\{ L_{_\prec}(y)R_{_\cdot}(x)+R_{_\succ}(y)L_{_\cdot}(x)-L_{_\succ}(x)R_{_\succ}(y)-R_{_\prec}(x)L_{_\prec}(y) \}\cr
&+&\{[L_{_\succ}(x), R_{_\prec}(y)]-[L_{_\succ}(y), R_{_\prec}(x)] \}=0
\end{eqnarray}
\item\label{dual-bimodule}
Both dendriform and pre-anti-flexible algebras have the same shape of dual bimodules.
This fact induces some consequences which we will derive and explain in the following of this paper.
\end{enumerate}
\end{rmk}
\begin{thm}\label{Theo_pre_Sum}
Let $( A , \prec_{_A }, \succ_{_A })$ be a pre-anti-flexible algebra. Suppose there is a pre-anti-flexible algebra structure
"$ \prec_{_{ A^*}}, \succ_{_{A^*}}$" on $ A^*$. The following statements are equivalent:
\begin{enumerate}
\item\label{1} $(R^*_{\prec_{_ A}},L^*_{\succ_{_ A}}, R^*_{\prec_{_{ A^*}}}, L^*_{\succ_{_{ A^*}}}, A, A^*)$ is a matched pair of anti-flexible algebras $aF(A)$ and $aF(A^*)$.
\item\label{2} There exists an anti-flexible algebra structure on $ A\oplus A^*$ given by for any $x,y\in A$ and for any $a,b\in A^*$,
\begin{eqnarray}\label{eq_anti_flexible_sum}
(x+a)\star(y+b)=
(x\cdot y+R^*_{\prec_{_ A}}(a)y+L^*_{\succ_{_ A}}(b)x)+
(a\circ b+R^*_{\prec_{_{ A^*}}}(x)b+L^*_{\succ_{_{ A^*}}}(y)a),
\end{eqnarray} where $x\cdot y= x\prec_{_ A}y+ x\succ_{_ A} y$, $a\circ b= a\prec_{_{ A^*}}b+a\succ_{_{ A^*}}b$,
and a non-degenerate closed skew-symmetric bilinear form $\omega$ defined on $ A\oplus A^*$ given by
for any $x,y\in A$ and for any $a,b \in A^*$,
\begin{eqnarray}\label{eq_skew_symmetric_form}
\omega(x+a, y+b)=\langle x,b\rangle -\langle y,a\rangle.
\end{eqnarray}
\end{enumerate}
\end{thm}
\begin{proof}
Let $( A , \prec_{_A }, \succ_{_A })$ be a pre-anti-flexible algebra.
Suppose there is a pre-anti-flexible algebra structure
"$ \prec_{_{ A^*}}, \succ_{_{A^*}}$" on $ A^*$.
\begin{itemize}
\item[$\eqref{1}\Longrightarrow \eqref{2}$]
Suppose that $(R^*_{\prec_{_ A}},L^*_{\succ_{_ A}}, R^*_{\prec_{_{ A^*}}},L^*_{\succ_{_{ A^*}}}, A, A^*)$
is a matched pair of anti-flexible algebras $aF(A)$ and $aF(A^*)$.
Then, there is an anti-flexible algebra structure "$\star$" on $ A\oplus A^*$
given by for any $x,y\in A$ and for any $a,b, \in A^*$,
\begin{eqnarray*}
(x+a)\star(y+b)=
(x\cdot y+R^*_{\prec_{_ A}}(a)y+L^*_{\succ_{_ A}}(b)x)+
(a\circ b+R^*_{\prec_{_{ A^*}}}(x)b+L^*_{\succ_{_{ A^*}}}(y)a),
\end{eqnarray*}
where, $\forall x,y\in A$, $\forall a,b\in A^*$,
\begin{eqnarray*}
x\cdot y= x\prec_{_ A}y+ x\succ_{_ A} y, a\circ b= a\prec_{_{ A^*}}b+a\succ_{_{ A^*}}b.
\end{eqnarray*}
Besides, considering the skew-symmetric bilinear form $\omega$ defined on $ A\oplus A^*$ by
Eq.~\eqref{eq_skew_symmetric_form}, we have for any $x,y\in A$ and for any $a,b\in A^*$,
\begin{eqnarray*}
&&\omega((x+a)\star(y+b), (z+c))+\omega((y+b)\star(z+c), (x+a))+\omega((z+c)\star(x+a), (y+b))\cr
&&=
\langle x\cdot y, c\rangle +\langle y, c\prec_{_{ A^*}} a\rangle +\langle x, b\succ_{_{ A^*}} c\rangle
-\langle z, a\circ b\rangle -\langle z\prec_{_ A} x, b\rangle -\langle y\succ_{_{ A}} z, a\rangle \cr&&+
\langle y\cdot z, a\rangle +\langle z, a\prec_{_{ A^*}} b\rangle +\langle y, c\succ_{_{ A^*}} a\rangle
-\langle x, b\circ c\rangle -\langle x\prec_{_ A} y, c\rangle -\langle z\succ_{_{ A}} x, b\rangle \cr&&
+\langle z\cdot y-x, b\rangle +\langle x, b\prec_{_{ A^*}} c\rangle +\langle z, a\succ_{_{ A^*}} b\rangle
-\langle y, c\circ a\rangle -\langle y\prec_{_ A} z, a\rangle -\langle x\succ_{_{ A}} y, c\rangle =0.
\end{eqnarray*}
Clearly, $\omega$ is closed and $\omega( A, A)=0=\omega( A^*, A^*)$, then $( A, \cdot)$
and $( A^*, \circ)$ are Lagrangian anti-flexible subalgebras
of the anti-flexible algebra $( A\oplus A^*, \star)$.
\item[$\eqref{2}\Longrightarrow \eqref{1}$]
Suppose that there exists an anti-flexible algebra structure "$\star$" on $ A\oplus A^*$ given by Eq.~\eqref{eq_anti_flexible_sum}
and a non-degenerate closed skew-symmetric bilinear form on $ A\oplus A^*$ given by Eq.~\eqref{eq_skew_symmetric_form}.
According to Proposition~\ref{prop_operation_bimodule_pre_anti_flexible} the triple
$(R^*_{\prec_{_ A}},L^*_{\succ_{_ A}}, A^*)$ is a bimodule
of $aF(A)$ and $(R^*_{\prec_{_{ A^*}}},L^*_{\succ_{_{ A^*}}}, A)$ is a bimodule of $aF(A^*)$, thus "$\star$" defines
an anti-flexible algebra structure on
$ A\oplus A^*$ if $(R^*_{\prec_{_ A}},L^*_{\succ_{_ A}},R^*_{\prec_{_{ A^*}}},L^*_{\succ_{_{ A^*}}}, A, A^*)$ is a
matched pair of the anti-flexible algebras $aF(A)$ and $aF(A^*)$.
\end{itemize}
Therefore, holds the conclusion.
\end{proof}
\begin{thm}\label{thm_matchedpair}
Let $( A, \prec_{_ A}, \succ_{_ A})$ and $(B, \prec_{_B}, \succ_{_B})$ be two pre-anti-flexible algebras.
Suppose that there are four linear maps
$ l_{_{\succ_ A}}, r_{_{\succ_ A}}, l_{_{\prec_ A}}, r_{_{\prec_ A}}: A\rightarrow \End(B)$
such that $(l_{_{\succ_ A}}, r_{_{\succ_ A}}, l_{_{\prec_ A}}, r_{_{\prec_ A}}, B)$ is a bimodule of
$( A, \prec_{_ A}, \succ_{_ A})$ and another four linear maps
$ l_{_{\succ_B}}, r_{_{\succ_B}}, l_{_{\prec_B}}, r_{_{\prec_B}}:B\rightarrow \End( A)$
such that $(l_{_{\succ_B}}, r_{_{\succ_B}}, l_{_{\prec_B}}, r_{_{\prec_B}}, A)$ is a bimodule of
$(B, \prec_{_B}, \succ_{_B})$. If in addition the eight linear maps
$l_{_{\succ_ A}}, r_{_{\succ_ A}}, l_{_{\prec_ A}},$ $r_{_{\prec_ A}},
l_{_{\succ_B}}, r_{_{\succ_B}}, l_{_{\prec_B}}$, and $r_{_{\prec_B}}$
satisfying the relations, for any $ x,y\in A$, and for any $a,b\in B$,
\begin{subequations}
\begin{eqnarray}\label{eq_matched_pre_1}
(l_{_\succ{_{_B}}}(a)x)\prec_{_ A}y+ l_{_{\succ_{_B}}}(r_{_\succ{_ A}}(x)a)y-
l_{_{\succ{_B}}}(a)(x\prec_{_ A} y)=\cr r_{_{\prec_{_B}}}(a)(y\succ_{_ A}x)-
y\succ_{_ A}(r_{_{\prec_{_B}}}(a)x)-r_{_{\succ_{_B}}}(l_{_{\prec_{_ A}}}(x)a)y,
\end{eqnarray}
\begin{eqnarray}\label{eq_matched_pre_2}
(l_{_{\succ_{_ A}}}(x)b)\prec_{_B}a+l_{_{\prec_{_ A}}}(r_{_{\succ_{_B}}}(b)x)a-
l_{_{\succ_{_ A}}}(x)(b\prec_{_B}a)=\cr
r_{_{\prec_{_ A}}}(x)(a\succ_{_B} b)-a\succ_{_B}(r_{_{\prec_{_ A}}}(x)b)-
r_{_{\succ_{_ A}}}(l_{_{\prec_{_B}}}(b)x)a,
\end{eqnarray}
\begin{eqnarray}\label{eq_matched_pre_3}
(l_{_{\cdot_B}}(a)x )\succ_{_ A}y+l_{_{\succ_{_B}}}(r_{_{ \cdot_ A}}(x)a)y-
l_{_{\succ_{B}}}(a)(x\succ_{_ A}y) =\cr
r_{_{\prec_{_B}}}(a)(y\prec_{_ A} x)-y\prec_{_ A}(r_{\cdot_{_B}}(a)x)-
r_{_{\prec_{_B}}}(l_{{\cdot_{_ A}}}(x)a)y,
\end{eqnarray}
\begin{eqnarray}\label{eq_matched_pre_4}
r_{_{\succ_{B}}}(a)(x\cdot_{_ A} y)-x\succ{_{_ A}}(r_{_\succ{_B}}(a)y)-
r_{_{_\succ{_B}}}(l_{_{\succ{_ A}}}(y)a)x =\cr
(l_{_{\prec_{_B}}}(a)y)\prec_{_ A}x+l_{_{\prec_{_B}}}(r_{_{\prec_{_ A}}}(y)a)x-
l_{_{\prec_{_B}}}(a)(y\cdot_{_ A} x),
\end{eqnarray}
\begin{eqnarray}\label{eq_matched_pre_5}
(l_{_{\cdot_ A}}(x)b )\succ_{_B}a+l_{_{\succ_{_ A}}}(r_{_{\cdot_B}}(b)x)a-
l_{_{\succ_{ A}}}(x)(b\succ_{_B}a) =\cr
r_{_{\prec_{_ A}}}(x)(a\prec_{_B} b)-a\prec_{_B}(r_{\cdot_{_ A}}(x)b)-
r_{_{\prec_{_ A}}}(l_{{\cdot_{_B}}}(b)x)a,
\end{eqnarray}
\begin{eqnarray}\label{eq_matched_pre_6}
r_{_{\succ_{ A}}}(x)(a\cdot_{_B} b)-a\succ{_{_B}}(r_{_\succ{_ A}}(x)b)-
r_{_{_\succ{_ A}}}(l_{_{\succ{_B}}}(b)x)a=\cr
(l_{_{\prec_{_ A}}}(x)b)\prec_{_ A}a+l_{_{\prec_{_ A}}}(r_{_{\prec_{_B}}}(b)x)a-
l_{_{\prec_{_ A}}}(x)(b\cdot_{_B} a),
\end{eqnarray}
\begin{eqnarray}\label{eq_matched_pre_7}
(r_{_{\succ_B}}(a)x)\prec_{_ A} y+l_{_{\prec_B}}( l_{_{\succ_ A}}(x)a)y
-x\succ_{_ A}(l_{_{\prec_{_B}}}(a)y)-r_{_{\succ_{_B}}}(r_{_{\prec_{_ A}}}(y)a )x=\cr
(r_{_{\succ_B}}(a)y)\prec_{_ A} x+l_{_{\prec_B}}( l_{_{\succ_ A}}(y)a)x
-y\succ_{_ A}(l_{_{\prec_{_B}}}(a)x)-r_{_{\succ_{_B}}}(r_{_{\prec_{_ A}}}(x)a )y,
\end{eqnarray}
\begin{eqnarray}\label{eq_matched_pre_8}
(r_{_{\succ_ A}}(x)a)\prec_{_B} b+l_{_{\prec_ A}}( l_{_{\succ_B}}(a)x)b
-a\succ_{_B}(l_{_{\prec_{_ A}}}(x)b)-r_{_{\succ_{_ A}}}(r_{_{\prec_{_B}}}(b)x )a=\cr
(r_{_{\succ_ A}}(x)b)\prec_{_B} a+l_{_{\prec_ A}}( l_{_{\succ_B}}(b)x)a
-b\succ_{_B}(l_{_{\prec_{_ A}}}(x)a)-r_{_{\succ_{_ A}}}(r_{_{\prec_{_B}}}(a)x )b,
\end{eqnarray}
\begin{eqnarray}\label{eq_matched_pre_9}
(r_{ \cdot_{_B}}(a)x)\succ_{_ A} y +l_{_{_\succ{_B}}}(l_{_{\cdot_{_ A}}}(x)a)y
-x\succ_{_ A}(l_{_{\succ_{_B}}}(a)y)-r_{_{\succ_{_B}}}(r_{_{\succ_{_ A}}}(y)a)x=\cr
(r_{_{\prec{_B}}}(a)y)\prec_{_ A} x+l_{_{\prec_{_B}}}(l_{_{\prec_{_ A}}}(y) a)x
-y\prec_{_ A}(l_{\cdot_{_B}} (a)x)-r_{_{\prec_{_B}}}(r_{\cdot_{_ A}}(x)a)y ,
\end{eqnarray}
\begin{eqnarray}\label{eq_matched_pre_10}
(r_{\cdot_{_ A}}(x)a)\succ_{_B} b+l_{_{_\succ{_ A}}}(l_{_{\cdot_{_B}}}(a)x)b
-a\succ_{_B}(l_{_{\succ_{_ A}}}(x)b)-r_{_{\succ_{_ A}}}(r_{_{\succ_{_B}}}(b)x)a=\cr
(r_{_{\prec{_ A}}}(x)b)\prec_{_B} a+ l_{_{\prec_{_ A}}}(l_{_{\prec_{_B}}}(b) x)a
-b\prec_{_B}(l_{\cdot_{_ A}} (x)a)-r_{_{\prec_{_ A}}}(r_{\cdot_{_B}}(a)x)b,
\end{eqnarray}
\end{subequations}
there is a pre-anti-flexible product on $ A\oplus B$ given by, for any $x,y\in A$, and for any $a,b\in B$,
\begin{eqnarray}\label{eq_pre_anti_matched}
\begin{array}{cccc}
(x+a)\prec (y+b)= \{x\prec_{_ A} y+l_{_{\prec_{_B}}}(a)y+r_{_{\prec_{_B}}}(b)x\}+
\{a\prec_{_B} b+l_{_{\prec_{_ A}}}(x)b+r_{_{\prec_{_ A}}}(y)a\}, \cr
(x+a)\succ (y+b)=
\{ x\succ_{_ A} y+l_{_{\succ_{_B}}}(a)y+r_{_{\succ_{_B}}}(b)x\}+
\{a\succ_{_B} b+l_{_{\succ_{_ A}}}(x)b+r_{_{\succ_{_ A}}}(y)a\},
\end{array}
\end{eqnarray}
where for any $x,y, \in A$ and any $a,b\in B$,
\begin{eqnarray*}
x\cdot_{_ A} y=x\prec_{_ A} y+x\succ_{_ A} y,\quad l_{ \cdot_{_ A}}
=l_{_\prec{_{_ A}}}+l_{_\succ{_{_ A}}}, \quad r_{\cdot_{_ A}}
=r_{_\prec{_{_ A}}}+r_{_\succ{_{_ A}}},\cr
a\cdot_{_B} b=a\prec_{_B} b+a\succ_{_B} b,\quad l_{\cdot_{_B}}
=l_{_\prec{_{_B}}}+l_{_\succ{_{_B}}}, \quad r_{\cdot{_B}}
=r_{_\prec{_{_B}}}+r_{_\succ{_{_B}}}.
\end{eqnarray*}
\end{thm}
\begin{proof}
Let $( A, \prec_{_ A}, \succ_{_ A})$ and $(B, \prec_{_B}, \succ_{_B})$ be two pre-anti-flexible algebras.
Suppose in addition that $(l_{_{\succ_ A}}, r_{_{\succ_ A}}, l_{_{\prec_ A}}, r_{_{\prec_ A}}, B)$
is a bimodule of $( A, \prec_{_ A}, \succ_{_ A})$ and $(l_{_{\succ_B}}, r_{_{\succ_B}}, l_{_{\prec_B}}, r_{_{\prec_B}}, A)$
is a bimodule of $(B, \prec_{_B}, \succ_{_B})$, where
$ l_{_{\succ_B}}, r_{_{\succ_B}}, l_{_{\prec_B}}, r_{_{\prec_B}}:B\rightarrow \End( A)$ and
$ l_{_{\succ_ A}}, r_{_{\succ_ A}}, l_{_{\prec_ A}}, r_{_{\prec_ A}}: A\rightarrow \End(B)$
are eight linear maps. Considering the product given in Eq.~\eqref{eq_pre_anti_matched}, we have
for any $x,y, z\in A$ and any $a,b,c\in B$,
\begin{eqnarray*}
(x+a, y+b, z+c)_{_m}&=&(x,y,z)_{_m}+(a,b,c)_{_m}+
\{l_{_{\prec_B}}(a\succ_{_B}b)z -l_{_\succ{_B}}(a)( l_{_{\prec_B}}(b)z) \}\cr
&+&\{(l_{_\succ{_{_B}}}(a)y)\prec_{_ A}z+ l_{_\succ{_B}}(r_{_\succ{_ A}}(y)a)z-
l_{_{\succ{_B}}}(a)(y\prec_{_ A} z) \}\cr
&+&\{r_{_{\prec_{_B}}}(c)(x\succ_{_ A} y)-x\succ_{_ A}(r_{_{\prec_{_B}}}(c)y)-
r_{_{\succ_{_B}}}(l_{_{\prec_{_ A}}}(y)c)x \}\cr
&+&\{ r_{_{\prec_{_ A}}}(z)(a\succ_{_B} b)-a\succ_{_B}(r_{_{\prec_{_ A}}}(z)b)-
r_{_{\succ_{_ A}}}(l_{_{\prec_{_B}}}(b)z)a \}\cr
&+&\{ (l_{_{\succ_{_ A}}}(x)b)\prec_{_B}c+l_{_{\prec_{_ A}}}(r_{_{\succ_{_B}}}(b)x)c-
l_{_{\succ_{_ A}}}(x)(b\prec_{_B}c) \}\cr
&+&\{r_{_{\prec_{_B}}}(c)(l_{_{\succ_{_B}}}(a)y)-
l_{_{\succ_{_B}}}(a)(r_{_{\prec_{_B}}}(c)y) \}+\{ (r_{_{\succ_B}}(b)x)\prec_{_ A} z
\cr &+&l_{_{\prec_B}}( l_{_{\succ_ A}}(x)b)z
-x\succ_{_ A}(l_{_{\prec_{_B}}}(b)z)
-r_{_{\succ_{_B}}}(r_{_{\prec_{_ A}}}(z)b )x \}\cr
&+&\{ r_{_{\prec_{_ A}}}(z)(r_{_{\succ_{_ A}}}(y)a)-
r_{_{\succ_{_ A}}}(y\prec_{_ A}z)a \}+
\{ (r_{_{\succ_ A}}(y)a)\prec_{_B} c
\cr&+&l_{_{\prec_ A}}( l_{_{\succ_B}}(a)y)c
-a\succ_{_B}(l_{_{\prec_{_ A}}}(y)c)
-r_{_{\succ_{_ A}}}(r_{_{\prec_{_B}}}(c)y )a \}\cr
&+&\{r_{_{\prec_{_B}}}(c)(r_{_{\succ_{_B}}}(b)x)-
r_{_{\succ_{_B}}}(b\prec_{_B}c)x \}\cr
&+&\{ l_{_{\prec_ A}}(x\succ_{_ A}y)c -
l_{_\succ{_ A}}(x)( l_{_{\prec_ A}}(y)c) \}+
\{ r_{_{\prec_{_ A}}}(z)(l_{_{\succ_{_ A}}}(x)b)-
l_{_{\succ_{_ A}}}(x)(r_{_{\prec_{_ A}}}(z)b) \}
\end{eqnarray*}
\begin{eqnarray*}
(x+a, y+b, z+c)_{_l}&=&(x,y,z)_{_l}+(a,b,c)_{_l}+
\{r_{_{\succ_{_A}}}(z)(r_{\cdot_{A}}(y)a)-r_{_{\succ_{_A}}}(y\succ_{_A} y)a \}\cr
&+&\{(l_{_{\cdot_B}}(a)y )\succ_{_A}z+l_{_{\succ_{_B}}}(r_{_{\cdot_A}}(y)a)z-
l_{_{\succ_{B}}}(a)(y\succ_{_A}z) \}\cr
&+&\{r_{_{\succ_{B}}}(c)(x\ast_{_A} y)-x\succ{_{_A}}(r_{_\succ{_B}}(c)y)-
r_{_{_\succ{_B}}}(l_{_{\succ{_A}}}(y)c)x \}\cr
&+&\{(l_{_{\cdot_A}}(x)b )\succ_{_B}c+l_{_{\succ_{_A}}}(r_{_{\cdot_B}}(b)x)c-
l_{_{\succ_{A}}}(x)(b\succ_{_B}c) \}\cr
&+&\{ r_{_{\succ_{A}}}(z)(a\cdot_{_B} b)-a\succ{_{_B}}(r_{_\succ{_A}}(z)b)-
r_{_{_\succ{_A}}}(l_{_{\succ{_B}}}(b)z)a \}\cr
&+&\{ l_{_\succ{_B}}(a\cdot_{_B} b)z-l_{_{\succ_{_B}}}(a)(l_{_{\succ_{_B}}}(b)z)\}+
\{(r_{\cdot_{_B}}(b)x)\succ_{_A} z \cr
&+&l_{_{_\succ{_B}}}(l_{_{\cdot_{_A}}}(x)b)z-x\succ_{_A}(l_{_{\succ_{_B}}}(b)z)-
r_{_{\succ_{_B}}}(r_{_{\succ_{_A}}}(z)b)x \}\cr
&+&\{ l_{_\succ{_A}}(x\cdot_{_A} y)c-l_{_{\succ_{_A}}}(x)(l_{_{\succ_{_A}}}(y)c) \}+
\{ (r_{\cdot_{_A}}(y)a)\succ_{_B} c \cr
&+&l_{_{_\succ{_A}}}(l_{_{\cdot_{_B}}}(a)y)c-a\succ_{_B}(l_{_{\succ_{_A}}}(y)c)-
r_{_{\succ_{_A}}}(r_{_{\succ_{_B}}}(c)y)a \}\cr
&+&\{ r_{_{\succ_{_B}}}(c)(l_{_\cdot{_B}}(a)y)-l_{_{_{\succ{_B}}}}(a)(r_{_{\succ{_B}}}(c)y) \}\cr
&+&\{ r_{_{\succ_{_B}}}(c)(r_{_\cdot{_B}}(b)x)-r_{_{\succ_{_B}}}(b\succ_{_B} c)x \}+
\{ r_{_{\succ_{_A}}}(z)(l_{_\cdot{_A}}(x)b)-l_{_{_{\succ{_A}}}}(x)(r_{_{\succ{_A}}}(z)b) \}
\end{eqnarray*}
\begin{eqnarray*}
(z+c, y+b, x+a)_{_r}&=&(z,y,x)_{_r}+(c,b,a)_{_r}
+\{r_{_{\prec_{_B}}}(a)(r_{_{\prec_{_B}}}(b)z)-r_{_{\prec_{_B}}}(b\ast_{_B} a)z \}\cr
&+&\{(l_{_{\prec_{_B}}}(c)y)\prec_{_A}x+l_{_{\prec_{_B}}}(r_{_{\prec_{_A}}}(y)c)x-
l_{_{\prec_{_B}}}(c)(y\cdot_{_A} x) \}\cr
&+&\{r_{_{\prec_{_B}}}(a)(z\prec_{_A} y)-z\prec_{_A}(r_{\cdot_{_B}}(a)y)-
r_{_{\prec_{_B}}}(l_{{\cdot_{_A}}}(y)a)z \}\cr
&+&\{ (l_{_{\prec_{_A}}}(z)b)\prec_{_A}a+l_{_{\prec_{_A}}}(r_{_{\prec_{_B}}}(b)z)a-
l_{_{\prec_{_A}}}(z)(bAst_{_B} a) \}\cr
&+&\{ r_{_{\prec_{_A}}}(x)(c\prec_{_B} b)-c\prec_{_B}(r_{\cdot_{_A}}(x)b)-
r_{_{\prec_{_A}}}(l_{{\cdot_{_B}}}(b)x)c \}\cr
&+&\{l_{_{\prec_{_B}}}(c\prec_{_B} b)x-l_{_{\prec_{_B}}}(c)(l_{_{\cdot_{_B}}}(b)c) \}+
\{(r_{_{\prec{_B}}}(b)z)\prec_{_A} x\cr
&+&l_{_{\prec_{_B}}}(l_{_{\prec_{_A}}}(z) b)x-z\prec_{_A}(l_{\cdot_{_B}} (b)x)-
r_{_{\prec_{_B}}}(r_{\cdot_{_A}}(x)b)z \}\cr
&+&\{r_{_{\prec{_B}}}(a)(l_{_{_\prec{_B}}}(c)y)-l_{_{\prec{_B}}}(c)(r_{_{\cdot{_B}}}(a)y) \}+
\{ (r_{_{\prec{_A}}}(y)c)\prec_{_B} a \cr
&+& l_{_{\prec_{_A}}}(l_{_{\prec_{_B}}}(c) y)a-c\prec_{_B}(l_{\cdot_{_A}} (y)a)-
r_{_{\prec_{_A}}}(r_{\cdot_{_B}}(a)y)c \}\cr
&+&\{r_{_{\prec_{_A}}}(x)(r_{_{\prec_{_A}}}(y)c)-r_{_{\prec_{_A}}}(y\cdot_{_A} x)c \}\cr
&+&\{l_{_{\prec_{_A}}}(z\prec_{_A} y)a-l_{_{\prec_{_A}}}(z)(l_{_{\cdot_{_A}}}(y)z)\}
+\{r_{_{\prec{_A}}}(x)(l_{_{_\prec{_A}}}(z)b)-l_{_{\prec{_A}}}(z)(r_{_{\cdot{_A}}}(x)b) \}
\end{eqnarray*}
Besides, for any $x,y, z\in A$ and for any $a,b,c\in B$,
$
(x+a,y+b,z+c)_{_m}=(z+c,y+b,x+a)_{_m}
$
and
$
(x+a,y+b,z+c)_{_l}=(z+c,y+b,x+a)_{_r}
$ are equivalent to
$(l_{_{\succ_A}}, r_{_{\succ_A}}, l_{_{\prec_A}}, r_{_{\prec_A}}, B)$ is a bimodule of
$(A, \prec_{_A}, \succ_{_A})$, $(l_{_{\succ_B}}, r_{_{\succ_B}}, l_{_{\prec_B}}, r_{_{\prec_B}}, A)$
is a bimodule of $(B, \prec_{_B}, \succ_{_B})$ and
Eqs.~\eqref{eq_matched_pre_1}~-~\eqref{eq_matched_pre_10} are satisfied.
\end{proof}
\begin{defi}
Let $(A, \prec_{_A}, \succ_{_A})$ and $(B, \prec_{_B}, \succ_{_B})$ be two
pre-anti-flexible algebras. Suppose that there are four linear maps
$ l_{_{\succ_A}}, r_{_{\succ_A}}, l_{_{\prec_A}}, r_{_{\prec_A}}:A\rightarrow \End(B)$ such that
$(l_{_{\succ_A}}, r_{_{\succ_A}}, l_{_{\prec_A}}, r_{_{\prec_A}}, B)$ is a bimodule of
$(A, \prec_{_A}, \succ_{_A} )$ and another four linear maps
$l_{_{\succ_B}}, r_{_{\succ_B}}, l_{_{\prec_B}}, r_{_{\prec_B}}:B\rightarrow \End(A)$
such that
$(l_{_{\succ_B}}, r_{_{\succ_B}}, l_{_{\prec_B}}, r_{_{\prec_B}}, A)$
is a bimodule of $(B, \prec_{_B}, \succ_{_B} )$ and
Eqs.~\eqref{eq_matched_pre_1}~-~\eqref{eq_matched_pre_10} hold.
Then we call the ten-tuple $(A, B,l_{_{\succ_A}}, r_{_{\succ_A}}, l_{_{\prec_A}},
r_{_{\prec_A}},l_{_{\succ_B}}, r_{_{\succ_B}}, l_{_{\prec_B}}, r_{_{\prec_B}} )$ a
{\bf matched pair of the pre-anti-flexible} algebras $(A, \prec_{_A}, \succ_{_A})$
and $(B, \prec_{_B}, \succ_{_B})$.
We also denote the pre-anti-flexible algebra defined by Eq.~\eqref{eq_pre_anti_matched} by
$A\bowtie^{l_{_{\succ_A}}, r_{_{\succ_A}}, l_{_{\prec_A}},
r_{_{\prec_A}}}_{l_{_{\succ_B}}, r_{_{\succ_B}}, l_{_{\prec_B}}, r_{_{\prec_B}}} B$ or simply by
$A \bowtie B$.
\end{defi}
\begin{cor}\label{corollary_matched_pair_pre}
If
$(A, B, l_{_{\succ_A}}, r_{_{\succ_A}}, l_{_{\prec_A}}, r_{_{\prec_A}},
l_{_{\succ_B}}, r_{_{\succ_B}}, l_{_{\prec_B}}, r_{_{\prec_B}} )$
is a matched pair of the pre-anti-flexible algebras
$(A, \prec_{_A}, \succ_{_A})$ and $(B, \prec_{_B}, \succ_{_B})$ then
$(l_{_{\cdot_{_A}}}, r_{_{\cdot_{_A}}}, l_{_{\cdot_{_B}}}, r_{_{\cdot_{_B}}}, A, B )$
is a matched pair of anti-flexible algebras $aF(A)$ and $aF(B)$.
\end{cor}
\begin{proof}
Suppose that $(A, B, l_{_{\succ_A}}, r_{_{\succ_A}}, l_{_{\prec_A}},
r_{_{\prec_A}}, l_{_{\succ_B}}, r_{_{\succ_B}}, l_{_{\prec_B}}, r_{_{\prec_B}} )$
is a matched pair of the pre-anti-flexible algebras $(A, \prec_{_A}, \succ_{_A})$
and $(B, \prec_{_B}, \succ_{_B})$, where
$(l_{_{\succ_A}}, r_{_{\succ_A}}, l_{_{\prec_A}}, r_{_{\prec_A}}, B)$
is a bimodule of the pre-anti-flexible algebra $(A, \prec_{_A}, \succ_{_A})$ and
$(l_{_{\succ_B}}, r_{_{\succ_B}}, l_{_{\prec_B}}, r_{_{\prec_B}}, A)$
is a bimodule of the pre-anti-flexible algebra $(B, \prec_{_B}, \succ_{_B})$.
According to Proposition~\ref{prop_operation_bimodule_pre_anti_flexible},
$(l_{_{\cdot_{_A}}}, r_{_{\cdot_{_A}}}, B)$
is a bimodule of the anti-flexible algebra $aF(A)$ and
$(l_{_{\cdot_{_B}}}, r_{_{\cdot_{_B}}}, A)$ is a bimodule of the anti-flexible algebra $aF(B)$.
In addition, underlying anti-flexible product defined on
$A\oplus B$ in Eq.~\eqref{eq_pre_anti_matched} is exactly that obtained from the matched pair
$(l_{_{\cdot_{_A}}}, r_{_{\cdot_{_A}}},l_{_{\cdot_{_B}}}, r_{_{\cdot_{_B}}}, A, B)$
of anti-flexible algebra $aF(A)$ and $aF(B)$.
\end{proof}
\begin{thm}
Let $(A, \prec_{_A}, \succ_{_A})$ be a pre-anti-flexible algebra.
Suppose there is a pre-anti-flexible algebra structure
"$ \prec_{_{A^*}}, \succ_{_{A^*}}$" on its dual space $A^*$.
The following statements are equivalent:
\begin{enumerate}
\item\label{un}
$(A, A^*, R^*_{\prec_{_A}},L^*_{\succ_{_A}}, R^*_{\prec_{_{A^*}}}, L^*_{\succ_{_{A^*}}})$
is a matched pair of anti-flexible algebras $aF(A)$ and $aF(A^*)$.
\item\label{deux}
$(A, A^*, -R^*_{\succ_{_A}},-L^*_{\prec_{_A}} ,R^*_{\cdot_{_A}},L^*_{\cdot_{_A}} ,
-R^*_{\succ_{_{A^*}}},-L^*_{\prec_{_{A^*}}},
R^*_{\circ_{_{A^*}}}, L^*_{\circ_{_{A^*}}})$
is a matched pair of the pre-anti-flexible algebras $(A, \prec_{_A}, \succ_{_A})$ and
$(A^*, \prec_{_{A^*}}, \succ_{_{A^*}})$.
\end{enumerate}
\end{thm}
\begin{proof}
Let $(A, \prec_{_A}, \succ_{_A})$ be a pre-anti-flexible algebra.
Suppose there is a pre-anti-flexible algebra structure
"$ \prec_{_{A^*}}, \succ_{_{A^*}}$" on its dual space $A^*$.
According to Corollary~\ref{corollary_matched_pair_pre},
we have $\eqref{deux} \Longrightarrow \eqref{un}$.
If $(A, A^*, R^*_{\prec_{_A}},L^*_{\succ_{_A}}, R^*_{\prec_{_{A^*}}}, L^*_{\succ_{_{A^*}}})$
is a matched pair of anti-flexible algebras $aF(A)$ and $aF(A^*)$,
According to Theorem~\ref{Theo_pre_Sum},
there exists an anti-flexible algebra structure on $A\oplus A^*$ given by Eq.~\eqref{eq_anti_flexible_sum}
and a non-degenerate closed skew-symmetric bilinear form on $A\oplus A^*$
given by Eq.~\eqref{eq_skew_symmetric_form}. In view of Theorem~\ref{Theo_existance_pre_anti_flexible},
there exists a pre-anti-flexible algebra structure "$\prec, \succ$" defined on
$A\oplus A^*$ and satisfying Eq.~\eqref{eq_useful1}
i.e. for any $x,y, z\in A$ and for any $a, b, c\in A^*$,
\begin{eqnarray*}
\omega((x+a)\prec (y+b), (z+c))=\omega((x+a), (y+b)\star(z+c) ),\cr
\omega((x+a)\succ (y+b), (z+c))=\omega((y+b), (z+c)\star(x+a)),
\end{eqnarray*}
where "$\star$" is given by Eq.~\eqref{eq_anti_flexible_sum}.
More precisely, we have for any $x,y,z\in A$ and for any $a, b, c\in A^*$,
\begin{eqnarray*}
\omega(x+a, (y+b)\star(z+c) )&=&
\omega(x+a, (y\cdot z+R^*_{\prec_{_A}}(b)z+L^*_{\succ_{_A}}(c)y)\cr&+&
(b\circ c+R^*_{\prec_{_{A^*}}}(y)c+L^*_{\succ_{_{A^*}}}(z)b))\cr &=&
\langle x,b\circ c+R^*_{\prec_{_{A^*}}}(y)c+L^*_{\succ_{_{A^*}}}(z)b) \rangle
\cr&-&\langle y\cdot z+R^*_{\prec_{_A}}(b)z+L^*_{\succ_{_A}}(c)y, a\rangle
\cr &=&\langle x, b\circ c\rangle+
\langle x\prec_{_{A}}y, c\rangle+
\langle z\succ_{_{A}}x, b\rangle
-\langle y\cdot z, a\rangle\cr
&-&\langle z, a\prec_{_{_{A^*}}}b\rangle
-\langle y, c\succ_{_{_{A^*}}}a\rangle\cr
&=&\langle x\prec_{_{A}}y+L_{\circ}^*(b)x-R_{_{_{\succ_{A^*}}}}^*(a)y, c\rangle
\cr&-&
\langle z, a\prec_{_{_{A^*}}}b+L_{\cdot}^*(y)a-R_{_{\succ_{A}}}^*(x)b\rangle \cr
&=&\omega((x\prec_{_{A}}y-R_{_{_{\succ_{A^*}}}}^*(a)y+L_{\circ}^*(b)x)\cr&+&
( a\prec_{_{_{A^*}}}b-R_{_{\succ_{A}}}^*(x)b+L_{\cdot}^*(y)a), z+c).
\end{eqnarray*}
Thus
\begin{eqnarray*}
(x+a)\prec (y+b)
=(x\prec_{_{A}}y-R_{_{_{\succ_{A^*}}}}^*(a)y+L_{\circ}^*(b)x)
+( a\prec_{_{_{A^*}}}b-R_{_{\succ_{A}}}^*(x)b+L_{\cdot}^*(y)a).
\end{eqnarray*}
Similarly, we have
\begin{eqnarray*}
(x+a)\succ (y+b)=(x\prec{_{_A}}y+R_{\circ}^*(a)y-L_{_{\prec_{A^*}}}^*(b)x)+
(a\prec{_{_{A^*}}}b+R_{\cdot}^*(x)b-L_{_{\prec_{A}}}^*(y)a)
\end{eqnarray*}
Therefore, $( A, A^*, -R_{\succ_{_A}}^*,-L^*_{\prec_{_A}} ,R^*_{\cdot},L^*_{\cdot} ,
-R^*_{\succ_{_{A^*}}}, -L^*_{\prec_{_{A^*}}},
R^*_{\circ}, L^*_{\circ})$
is a matched pair of the pre-anti-flexible algebras $(A, \prec_{_A}, \succ_{_A})$ and
$(A^*, \prec_{_{A^*}}, \succ_{_{A^*}})$. Hence \eqref{un} $\Longrightarrow$ \eqref{deux}
\end{proof}
\section{Pre-anti-flexible bialgebras}\label{section3}
In this section, we are going to provide the definition of a pre-anti-flexible bialgebra
and provide they equivalent notions previously announced.
To achieve this goal, we have
\begin{thm}
Let $(A, \prec_{_A}, \succ_{_A})$ be a pre-anti-flexible algebra whose products are given
by two linear maps
$\beta_{_{\succ}}^*, \beta_{_{\prec}}^*: A\otimes A\rightarrow A$.
Suppose in addition that there is a pre-anti-flexible algebra structure
"$ \prec_{_{A^*}}, \succ_{_{A^*}}$" on $A^*$ given by:
$\Delta_{_{\succ}}^*, \Delta_{_{\prec}}^*: A^*\otimes A^*\rightarrow A^*$.
Then the following relations are equivalent:
\begin{enumerate}
\item $(A, A^*, R^*_{\prec_{_A}},L^*_{\succ_{_A}}, R^*_{\prec_{_{A^*}}}, L^*_{\succ_{_{A^*}}})$
is matched pair of anti-flexible algebras $aF(A)$ and $aF(A^*)$.
\item
The fourth linear maps $\beta_{_{\succ}}, \beta_{_{\prec}}:A^*\rightarrow A^*\otimes A^*$ and
$\Delta_{_{\succ}}, \Delta_{_{\prec}}:A\rightarrow A\otimes A$ satisfying
for any $x,y\in A$ and for any $a,b\in A^*$,
\begin{subequations}
\begin{eqnarray}\label{eq_pre_matched_1}
\begin{array}{llll}
\Delta_{_{\succ}}(x\cdot y)-(R_{_{\prec_A}}(y)\otimes \id)\Delta_{_{\succ}}(x)-
(\id\otimes L_{\cdot}(x))\Delta_{_{\succ}}(y)=\cr
\sigma(\id \otimes L_{_{\succ_{A}}}(y))\Delta_{_{\prec}}(x)+
\sigma(R_{\cdot}(x)\otimes \id)\Delta_{_{\prec}}(y)-\sigma \Delta_{_{\prec}}(y\cdot x),
\end{array}
\end{eqnarray}
\begin{eqnarray}\label{eq_pre_matched_3}
\begin{array}{llll}
&&(\sigma(L_{\cdot}(y)\otimes \id-\id\otimes R_{_{\prec_{_A}}}(y)))\Delta_{_{\prec}}(x)
+(L_{_{\succ_{A}}}(x)\otimes\id-\id\otimes R_{\cdot}(x))\Delta_{_{\succ}}(y)=\cr&&
(\sigma(L_{\cdot}(x)\otimes\id -\id\otimes R_{_{\prec_{_A}}}(x) ) )\Delta_{_{\prec}}(y)+
(L_{_{\succ_{_{A}}}}(y)\otimes\id -\id\otimes R_{{\cdot}}(y))\Delta_{_{\succ}}(x),
\end{array}
\end{eqnarray}
\begin{eqnarray}\label{eq_pre_matched_2}
\begin{array}{llll}
\beta_{_{\succ}}(a\circ b)-
(R_{_{{\prec_{A^*}}}}(b)\otimes \id)\beta_{_{\succ}}(a)-
(\id\otimes L_{\circ}(a))\beta_{_{\succ}}(b)=\cr
\sigma(\id \otimes L_{_{\succ_{A^*}}}(b))\beta_{_{\prec}}(a)+
\sigma(R_{\circ}(a)\otimes \id)\beta_{_{\prec}}(b)-\sigma \beta_{_{\prec}}(b\circ a),
\end{array}
\end{eqnarray}
\begin{eqnarray}\label{eq_pre_matched_4}
\begin{array}{llll}
&&(\sigma(L_{\circ}(b)\otimes \id-\id\otimes R_{_{\prec_{A^*}}}(b))\beta_{_{\prec}}(a)+
(L_{_{\succ_{A^*}}}(a)\otimes\id-\id\otimes R_{\circ}(a))\beta_{_{\succ}}(b)=\cr &&
(\sigma(L_{\circ}(a)\otimes\id -\id\otimes R_{_{\prec_{A^*}}}(a) ) )\beta_{_{\prec}}(b)+
(L_{_{\succ_{A^*}}}(b)\otimes\id-\id\otimes R_{{\circ}}(b) )\beta_{_{\succ}}(a),
\end{array}
\end{eqnarray}
\end{subequations}
where
$R_{\cdot}=R_{_{\succ_{A}}}+R_{_{\prec_{A}}}$,
$L_{\cdot}=L_{_{\succ_{A}}}+L_{_{\prec_{A}}}$,
$R_{\circ}=R_{_{\succ_{A^*}}}+R_{_{\prec_{A^*}}}$ and
$L_{\circ}=L_{_{\succ_{A^*}}}+L_{_{\prec_{A^*}}}.$
\end{enumerate}
\end{thm}
\begin{proof}
According to Remark~\ref{rmk_1}, $(R^*_{\prec_{_A}},L^*_{\succ_{_A}}, A^*)$ and
$(R^*_{\prec_{_{A^*}}}, L^*_{\succ_{_{A^*}}},A)$ are bimodules of $aF(A)$ and $aF(A^*)$, respectively.
Taking into account the following, for any $x,y\in A$ and any $a,b\in A^*$
\begin{eqnarray*}
&&\langle \sigma\circ \Delta_{_{\prec}}(x\cdot y), a\otimes b \rangle=
\langle \Delta_{_{\prec}}(x\cdot y), b\otimes a \rangle=
\langle x\cdot y, R_{_{\prec_{A^*}}}(a)b \rangle=
\langle R_{_{\prec_{A^*}}}^*(a)(x\cdot y), b \rangle, \cr
&& \langle \Delta_{_{\succ}}(y\cdot x), a\otimes b \rangle=
\langle y\cdot x, a\succ_{_{A^*}} b \rangle=
\langle y\cdot x, L_{_{_{\succ_{A^*}}}}(a)b \rangle=
\langle L_{_{{\succ_{A^*}}}}^*(a)(y\cdot x), b \rangle, \cr
&&\langle (R_{_{\prec_A}}(x)\otimes \id)\Delta_{_{\succ}}(y), a\otimes b \rangle=
\langle \Delta_{_{\succ}}(y),R_{_{\prec_A}}^*(x)a\otimes b \rangle=
\langle L_{_{\succ_{A^*}}}^*(R_{_{\prec_A}}^*(x)a)y, b \rangle, \cr
&&\langle (\id\otimes L_{\cdot}(y))\Delta_{_{\succ}}(x), a\otimes b \rangle=
\langle x, a\succ_{_{A^*}}(L_{\cdot}^*(y)b) \rangle=
\langle L_{_{_{\succ_{A^*}}}}^*(a)x ,L_{\cdot}^*(y)b \rangle=
\langle y\cdot (L_{_{{\succ_{A^*}}}}^*(a)x) , b \rangle, \cr
&&\langle \sigma(\id \otimes L_{_{\succ_{A}}}(x))\Delta_{_{\prec}}(y), a\otimes b \rangle=
\langle y, b\prec_{_{A^*}} (L_{_{\succ_{_{A}}}}^*(x)a) \rangle=
\langle R_{_{\prec_{_{A^*}}}}^*(L_{_{\succ_{_{A}}}}^*(x)a)y, b \rangle, \cr
&&\langle \sigma(R_{\cdot}(y)\otimes \id)\Delta_{_{\prec}}(x), a\otimes b \rangle=
\langle x, (R_{\cdot}^*(y)b)\prec_{_{A^*}} a \rangle=
\langle R_{_{{\prec_{_{A^*}}}}}^*(a)x ,R_{\cdot}^*(y)b \rangle=
\langle (R_{_{{\prec_{_{A^*}}}}}^*(a)x)\cdot y ,b \rangle,
\end{eqnarray*}
we deduce that Eq.~\eqref{eq_pre_matched_1} is equivalent to Eq.~\eqref{eqq1}.
Similarly, we get equivalence between Eqs.~\eqref{eq_pre_matched_2} and \eqref{eqq2}.
Besides, taking into account the following,
\begin{eqnarray*}
&&\langle (\sigma(L_{\cdot}(y)\otimes \id)\Delta_{_{\prec}}(x), a\otimes b\rangle=
\langle x, (L_{\cdot}^*(y)b) \prec_{_{A^*}} a \rangle=
\langle R_{\prec_{_{A^*}}}^*(a)x, L_{\cdot}^*(y)b \rangle=
\langle y\cdot (R_{\prec_{_{A^*}}}^*(a)x) ,b\rangle, \cr
&&\langle (\sigma(\id\otimes R_{_{\prec_{_A}}}(y)))\Delta_{_{\prec}}(x), a\otimes b\rangle=
\langle x, b\prec_{_{A^*}} (R_{_{\prec_{_A}}}^*(y)a)\rangle=
\langle R_{\prec_{_{A^*}}}^*(R_{_{\prec_{_A}}}^*(y)a)x,b\rangle, \cr
&&\langle (L_{_{\succ_{_{A}}}}(y)\otimes\id) \Delta_{_{\succ}}(x), a\otimes b\rangle=
\langle x, (L_{_{\succ_{_{A}}}}^*(y)a)\succ_{_{ A^*}} b \rangle=
\langle L_{\succ_{_{ A^*}}}^*(L_{_{\succ_{_{A}}}}^*(y)a)x,b\rangle, \cr
&&\langle (\id\otimes R_{{\cdot}}(y))\Delta_{_{\succ}}(x), a\otimes b\rangle=
\langle x, a \succ_{_{ A^*}} (R_{{\cdot}}^*(y)b) \rangle=
\langle L_{\succ_{_{ A^*}}}^*(a)x, R_{{\cdot}}^*(y)b \rangle=
\langle (L_{\succ_{_{ A^*}}}^*(a)x)\cdot y,b\rangle,
\end{eqnarray*}
we deduce equivalence between Eqs.~\eqref{eq_pre_matched_3} and \eqref{eqq3}.
Similarly, we get equivalence between Eqs.~\eqref{eq_pre_matched_4} and \eqref{eqq4}.
\end{proof}
\begin{rmk}\label{rmk:identities}
Let $x,y\in A$ and $a, b\in A^*$. Setting by
$\Delta=\Delta_{_\prec}+\Delta_{_\succ}$ we have the following
\begin{eqnarray*}
\langle \beta_{_\succ}(a\circ b), x\otimes y \rangle=
\langle a\otimes b,\Delta (x\succ y) \rangle, \quad
\langle \sigma\beta_{_\prec}(b\circ a), x\otimes y \rangle=
\langle a\otimes b, \sigma\Delta(y\prec x)\rangle,
\end{eqnarray*}
\begin{eqnarray*}
\langle( R_{\prec{_{A^*}}}(b)\otimes \id )\beta_{_\succ}(a), x\otimes y \rangle=
\langle a\otimes b, (R_{_{\succ}}(y)\otimes \id)\Delta_{_{\prec}}(x) \rangle,
\end{eqnarray*}
\begin{eqnarray*}
\langle (\id \otimes L_{\circ}(a))\beta_{{_\succ}}(b), x\otimes y\rangle=
\langle a\otimes b, (\id\otimes L_{{_\succ}}(x))\Delta(y)\rangle,
\end{eqnarray*}
\begin{eqnarray*}
\langle \sigma(\id \otimes L_{{_\succ}}(b))\beta_{{_\prec}}(a), x\otimes y\rangle=
\langle a\otimes b,(L_{_{\prec}}(y)\otimes \id)\sigma\Delta_{_{\succ}}(x) \rangle,
\end{eqnarray*}
\begin{eqnarray*}
\langle \sigma(R_{\circ}(a)\otimes \id )\beta_{{_\prec}}(b), x\otimes y\rangle=
\langle a\otimes b, (\id \otimes R_{{_\prec}}(x))\sigma\Delta(y)\rangle,
\end{eqnarray*}
\begin{eqnarray*}
\langle (L_{{_\succ}}(a)\otimes \id )\beta_{{_\succ}}(b), x\otimes y\rangle=
\langle a\otimes b, (\id \otimes R_{_{\succ}}(y))\Delta_{_{\succ}}(x)\rangle,
\end{eqnarray*}
\begin{eqnarray*}
\langle (\id\otimes R_{\circ}(a))\beta_{_{\succ}}(b), x\otimes y\rangle=
\langle a\otimes b, (\id\otimes L_{{_\succ}}(x))\sigma\Delta(y) \rangle,
\end{eqnarray*}
\begin{eqnarray*}
\langle \sigma(L_{\circ}(b)\otimes \id)\beta_{_{\prec}} (a), x\otimes y\rangle=
\langle a\otimes b, (R_{_{\prec}}(x)\otimes \id )\sigma\Delta(y)\rangle,
\end{eqnarray*}
\begin{eqnarray*}
\langle \sigma(\id \otimes R_{_{\prec}})\beta_{_{\prec}}(a), x\otimes y\rangle=
\langle a\otimes b, (L_{_{\prec}}(y)\otimes \id )\Delta_{_{\prec}}(x)\rangle.
\end{eqnarray*}
\end{rmk}
Considering above identities, we derive and can prove the following lemma and theorem
\begin{lem}
Let $(A, \prec_{_A}, \succ_{_A})$ be a pre-anti-flexible algebra whose
products are given by the linear maps
$\beta_{_{\succ}}^*, \beta_{_{\prec}}^*: A\otimes A\rightarrow A$.
Suppose in addition that there is a pre-anti-flexible algebra structure
"$ \prec_{_{A^*}}, \succ_{_{A^*}}$" on its dual space $A^*$ given by:
$\Delta_{_{\succ}}^*, \Delta_{_{\prec}}^*: A^*\otimes A^*\rightarrow A^*$.
Let $x,y\in A$. We have
\begin{subequations}
\mbox{ Eq.~\eqref{eq_pre_matched_2} is equivalent to}
\begin{eqnarray}\label{eq_pre_matched_2'}
\Delta (x\succ y) -(R_{_{\succ}}(y)\otimes \id)\Delta_{_{\prec}}(x) -
(\id\otimes L_{{_\succ}}(x))\Delta(y)=\cr
(L_{_{\prec}}(y)\otimes \id)\sigma\Delta_{_{\succ}}(x) +
(\id \otimes R_{{_\prec}}(x))\sigma\Delta(y)
-\sigma\Delta(y\prec x).
\end{eqnarray}
\mbox{ Eq.~\eqref{eq_pre_matched_4} is equivalent to}
\begin{eqnarray}\label{eq_pre_matched_4'}
(\id \otimes R_{_{\succ}}(y))\Delta_{_{\succ}}(x)-
(L_{_{\prec}}(y)\otimes \id )\Delta_{_{\prec}}(x)
+(R_{_{\prec}}(x)\otimes \id-\id\otimes L_{{_\succ}}(x))\sigma\Delta(y)=\cr
(R_{_{\succ}}(y) \otimes \id)\sigma\Delta_{_{\succ}}(x)-
(\id\otimes L_{_{\prec}}(y) )\sigma\Delta_{_{\prec}}(x)
+(\id \otimes R_{_{\prec}}(x)- L_{{_\succ}}(x)\otimes \id)\Delta(y).
\end{eqnarray}
\end{subequations}
\end{lem}
\begin{thm}
Let $(A, \prec_{_A}, \succ_{_A})$ be a pre-anti-flexible algebra.
Suppose in addition that there is a pre-anti-flexible algebra structure
"$ \prec_{_{A^*}}, \succ_{_{A^*}}$" on $A^*$ given by:
$\Delta_{_{\succ}}^*, \Delta_{_{\prec}}^*: A^*\otimes A^*\rightarrow A^*$.
Then the following relations are equivalent:
\begin{enumerate}
\item $(A, A^*, R^*_{\prec_{_A}},L^*_{\succ_{_A}}, R^*_{\prec_{_{A^*}}}, L^*_{\succ_{_{A^*}}})$
is matched pair of anti-flexible algebras $aF(A)$ and $aF(A^*)$.
\item The two linear maps
$\Delta_{_{\succ}}, \Delta_{_{\prec}}:A\rightarrow A\otimes A$ satisfying
Eqs.~\eqref{eq_pre_matched_1}, \eqref{eq_pre_matched_3}, \eqref{eq_pre_matched_2'}
and \eqref{eq_pre_matched_4'}.
\end{enumerate}
\end{thm}
In addition we have
\begin{defi}
A pre-anti-flexible bialgebra structure on a vector space $A$ is given by the four linear maps
$\Delta_{_{\prec}}, \Delta_{_{\succ}}: A\rightarrow A\otimes A$
and $\beta_{_{\prec}}, \beta_{_{\succ}}:A^*\rightarrow A^*\otimes A^*$,
where $A^*$ is the dual of $A$, such that:
\begin{enumerate}
\item the dual maps $\Delta_{_{\prec}}^*, \Delta_{_{\succ}}^*: A^*\otimes A^*\rightarrow A^*$
induce a pre-anti-flexible algebra structure $\prec_{_{A^*}}, \succ_{_{A^*}}$ on $A^*$,
\item the dual maps $\beta_{_{\prec}}^*, \beta_{_{\succ}}^*:A\otimes A \rightarrow A$
induce a pre-anti-flexible algebra structure $\prec_{_{A}}, \succ_{_{A}}$ on $A$,
\item the linear maps $\Delta_{_{\prec}}, \Delta_{_{\succ}}$
satisfy Eqs.~\eqref{eq_pre_matched_1}, \eqref{eq_pre_matched_3}, \eqref{eq_pre_matched_2'}
and \eqref{eq_pre_matched_4'}.
\end{enumerate}
\end{defi}
\begin{thm}
Let $(A, \prec_{_A}, \succ_{_A})$ and $(A^*, \prec_{_{A^*}}, \succ_{_{A^*}})$
be two pre-anti-flexible algebras. The following relations are equivalent:
\begin{enumerate}
\item there is an anti-flexible algebra structure on $aF(A)\oplus aF(A^*)$ and a
nondegenerate skew-symmetric bilinear form $\omega$
is given by Eq.~\eqref{eq_skew_symmetric_form} and satisfying Eq.~\eqref{eq:simplectic_form},
\item $(-R^*_{\succ_{_A}},-L^*_{\prec_{_A}} ,R^*_{\cdot_{_A}},L^*_{\cdot_{_A}} ,
-R^*_{\succ_{_{A^*}}},-L^*_{\prec_{_{A^*}}},
R^*_{\circ_{_{A^*}}}, L^*_{\circ_{_{A^*}}}, A, A^*)$
is a matched pair of the pre-anti-flexible algebras $(A, \prec_{_A}, \succ_{_A})$ and
$(A^*, \prec_{_{A^*}}, \succ_{_{A^*}})$,
\item $(R^*_{\prec_{_A}},L^*_{\succ_{_A}}, R^*_{\prec_{_{A^*}}},L^*_{\succ_{_{A^*}}},A, A^*)$
is a matched pair of the underlying anti-flexible algebras $aF(A)$ and $aF(A^*)$,
\item $(A, A^*)$ is a pre-anti-flexible bialgebra.
\end{enumerate}
\end{thm}
\begin{defi}
A homomorphism of pre-anti-flexible bialgebras
$(A, A^*, \Delta_{_{\prec_{A}}}, \Delta_{_{\succ_{A}}},\beta_{_{\prec_{A^*}}}, \beta_{_{\succ_{A^*}}})$
and\\
$(B,B^*,\Delta_{_{\prec_{B}}},\Delta_{_{\succ_{B}}},\beta_{_{\prec_{B^*}}},\beta_{_{\succ_{B^*}}})$
is a homomorphism of pre-anti-flexible algebras
$\psi:A\rightarrow B$ such that its dual $\psi^*:B^*\rightarrow A^*$
is also a homomorphism of pre-anti-flexible algebras i.e. for any $x\in A$,
$a\in B^*,$
\begin{eqnarray*}
(\psi\otimes \psi)\Delta_{_{\prec_{A}}}(x)=\Delta_{_{\prec_{B}}}(\psi(x)),\;\;
(\psi\otimes\psi) \Delta_{_{\succ_{A}}}(x)= \Delta_{_{\succ_{B}}}(\psi(x)),
\end{eqnarray*}
\begin{eqnarray*}
(\psi^*\otimes \psi^*)\beta_{_{\prec_{B^*}}}(a)=\beta_{_{\prec_{A^*}}}(\psi^*(a)),\;\;
(\psi^*\otimes \psi^*)\beta_{_{\succ_{B^*}}}(a)=\beta_{_{\succ_{A^*}}}(\psi^*(a)).
\end{eqnarray*}
An invertible homomorphism of pre-anti-flexible bialgebras is an isomorphism of
pre-anti-flexible algebras.
\end{defi}
\begin{rmk}\label{rmk:dual_preantiflexible}
According to Remark~\ref{rmk:identities}, if
$(A, A^*, \Delta_{\succ}, \Delta_{\prec}, \beta_{{_\succ}}, \beta_{{_\prec}})$
be a pre-anti-flexible
bialgebra, then its associated dual
$(A^*, A, \beta_{{_\succ}}, \beta_{{_\prec}}, \Delta_{\succ}, \Delta_{\prec})$ is also
a pre-anti-flexible bialgebra.
\end{rmk}
\section{Special pre-anti-flexible bialgebras and pre-anti-flexible Yang-Baxter equation}\label{section4}
We deal here with a special class of pre-anti-flexible bialgebras
provided by the linear maps $\Delta_{_{\prec}}, \Delta_{_{\succ}}:A\rightarrow A\otimes A$
defined by for any $x\in A$
\begin{subequations}\label{eq:coboundary}
\begin{equation}\label{eq:coboundary_a}
\Delta_{\succ}(x)=(\id \otimes L_{\cdot}(x))\mathrm{r}_{_\succ}+
(R_{\prec}(x)\otimes \id)\sigma \mathrm{r}_{_\prec},
\end{equation}
\begin{equation}\label{eq:coboundary_b}
\Delta_{\prec}(x)=(\id\otimes L_{\succ}(x))\mathrm{r}_{_\prec}+
(R_{\cdot}(x)\otimes \id)\sigma \mathrm{r}_{_\succ},
\end{equation}
\end{subequations}
which generate the an analogue equation of $\mathcal{D}$-equation of dendriform algebras
called pre-anti-flexible Yang-Baxter equation.
In the following of this paper, we will refer to both Eqs.~\eqref{eq:coboundary_a} and
\eqref{eq:coboundary_b} by Eq.~\eqref{eq:coboundary}. Other similar referring are
scattered throughout this paper.
\begin{lem}\label{lem:sigma}
Let $(A, \prec, \succ)$ be a pre-anti-flexible algebra and
$\mathrm{r}_{_\prec}, \mathrm{r}_{_\succ}\in A\otimes A$. Consider
$\Delta_{\prec}, \Delta_{\succ}:A\rightarrow A\otimes A$
two linear maps defined by Eq.~\eqref{eq:coboundary}. Then for any $x\in A,$
\begin{subequations}
\begin{eqnarray}\label{eq:sigma_succ}
\sigma\Delta_{\succ}(x)=(L_{\cdot}(x)\otimes \id )\sigma \mathrm{r}_{_\succ}+
(\id\otimes R_{\prec}(x)) \mathrm{r}_{_\prec},
\end{eqnarray}
\begin{eqnarray}\label{eq:sigma_prec}
\sigma\Delta_{\prec}(x)=(L_{\succ}(x)\otimes \id )\sigma \mathrm{r}_{_\prec}+
(\id\otimes R_{\cdot}(x)) \mathrm{r}_{_\succ},
\end{eqnarray}
\begin{eqnarray}\label{eq:delta}
\Delta(x)=(\id \otimes L_{\cdot}(x))\mathrm{r}_{_\succ}+
(R_{\prec}(x)\otimes \id)\sigma \mathrm{r}_{_\prec}+
(\id\otimes L_{\succ}(x))\mathrm{r}_{_\prec}+
(R_{\cdot}(x)\otimes \id)\sigma \mathrm{r}_{_\succ},
\end{eqnarray}
\begin{eqnarray}\label{eq:sigma_delta}
\sigma\Delta(x)=
(L_{\cdot}(x)\otimes \id )\sigma \mathrm{r}_{_\succ}+
(\id\otimes R_{\prec}(x)) \mathrm{r}_{_\prec}+
(L_{\succ}(x)\otimes \id )\sigma \mathrm{r}_{_\prec}+
(\id\otimes R_{\cdot}(x)) \mathrm{r}_{_\succ}.
\end{eqnarray}
\end{subequations}
\end{lem}
As consequences following the definition given by
Eq.~\eqref{eq:coboundary} we have
\begin{itemize}
\item For any $x,y \in A$. By Eq.~\eqref{eq:sigma_prec} we have
\begin{eqnarray*}
\Delta_{\succ}(x\cdot y)+\sigma\Delta_{\prec}(y\cdot x)=
(\id \otimes(L_{\cdot}(x\cdot y)+R_{\cdot}(y\cdot x)))\mathrm{r}_{_\succ}+
((R_{\prec}(x\cdot y)+L_{\succ}(y\cdot x))\otimes\id)\sigma \mathrm{r}_{_\prec}.
\end{eqnarray*}
According to Eq.~\eqref{eq:useful} we have
\begin{eqnarray*}
\Delta_{\succ}(x\cdot y)+\sigma\Delta_{\prec}(y\cdot x)&=&
(\id \otimes(L_{\cdot}(x)L_{\cdot}(y)+R_{\cdot}(x)R_{\cdot}(y)))\mathrm{r}_{_\succ}\cr&+&
((R_{\prec}(y)R_{\prec}(x)+L_{\succ}(y)L_{\succ}( x))\otimes\id)\sigma \mathrm{r}_{_\prec}.
\end{eqnarray*}
In addition
\begin{eqnarray*}
(R_{\prec}(y)\otimes\id)\Delta_{\succ}(x)&=&
(R_{\prec}(y)\otimes L_{\cdot}(x))\mathrm{r}_{_\succ}+
(R_{\prec}(y)R_{\prec}(x)\otimes \id)\sigma \mathrm{r}_{_\prec}\cr
(\id \otimes L_{\cdot}(x))\Delta_{\succ}(y)&=&
(\id\otimes L_{\cdot}(x)L_{\cdot}(y))\mathrm{r}_{_\succ}+
(R_{\prec}(y)\otimes L_{\cdot}(x))\sigma \mathrm{r}_{_\prec}\cr
\sigma(\id \otimes L_{\succ}(y))\Delta_{\prec}(x)&=&
(L_{\succ}(y)L_{\succ}(x)\otimes \id )\sigma \mathrm{r}_{_\prec}+
(L_{\succ}(y)\otimes R_{\cdot}(x) ) \mathrm{r}_{_\succ}\cr
\sigma(R_{\cdot}(x)\otimes \id )\Delta_{\prec}(y)&=&
(L_{\succ}(y)\otimes R_{\cdot} (x))\sigma \mathrm{r}_{_\prec}+
(\id \otimes R_{\cdot}(x)R_{\cdot}(y))\mathrm{r}_{_\succ}.
\end{eqnarray*}
Thus
\begin{eqnarray*}
&&(R_{\prec}(y)\otimes\id)\Delta_{\succ}(x)+
(\id \otimes L_{\cdot}(x))\Delta_{\succ}(y)-
\Delta_{\succ}(x\cdot y)+
\sigma(\id \otimes L_{\succ}(y))\Delta_{\prec}(x)-
\sigma\Delta_{\prec}(y\cdot x)\cr&&+
\sigma(R_{\cdot}(x)\otimes \id )\Delta_{\prec}(y)=
(R_{\prec}(y) \otimes L_{\cdot}(x) +
L_{\succ}(y) \otimes R_{\cdot}(x) )(\mathrm{r}_{_\succ}+\sigma \mathrm{r}_{_\prec}).
\end{eqnarray*}
Therefore
Eq.~\eqref{eq_pre_matched_1} is equivalent to the following
\begin{eqnarray}{\label{eq:coboundary1}}
(R_{_\prec}(y)\otimes L_{\cdot}(x)+L_{\succ}(y)\otimes R_{\cdot}(x))(\mathrm{r}_{_\succ}+
\sigma \mathrm{r}_{_\prec})=0, \;\; \forall x,y\in A.
\end{eqnarray}
\item Besides, for any $x, y\in A$ we have
\begin{eqnarray*}
(L_{\succ}(x)\otimes \id -\id \otimes R_{\cdot}(x))\Delta_{_\succ}(y)&=&
(L_{\succ}(x)\otimes L_{\cdot}(y))\mathrm{r}_{_\succ}+
(L_{\succ}(x)R_{\prec}(y)\otimes\id )\sigma \mathrm{r}_{_\prec}\cr
&-&(\id\otimes R_{\cdot}(x)L_{\cdot}(y) )\mathrm{r}_{_\succ}-
(R_{\prec}(y)\otimes R_{\cdot}(x))\sigma \mathrm{r}_{_\prec}\cr
\sigma(L_{\cdot}(y)\otimes \id -\id \otimes R_{\prec}(y) )\Delta_{\prec}(x)&=&
(L_{\succ}(x)\otimes L_{\cdot}(y))\sigma \mathrm{r}_{_\prec}+
(\id\otimes L_{\cdot}(y)R_{\cdot}(x))\mathrm{r}_{_\succ}\cr
&-&(R_{\prec}(y)L_{\succ}(x)\otimes\id)\sigma \mathrm{r}_{_\prec}-
(R_{\prec}(y)\otimes R_{\cdot}(x))\mathrm{r}_{_\succ}
\end{eqnarray*}
Then
\begin{eqnarray*}
&&(L_{\succ}(x)\otimes \id -\id \otimes R_{\cdot}(x))\Delta_{_\succ}(y)+
\sigma(L_{\cdot}(y)\otimes \id -\id \otimes R_{\prec}(y) )\Delta_{\prec}(x)\cr
&&-(L_{\succ}(y)\otimes \id -\id \otimes R_{\cdot}(y))\Delta_{_\succ}(x)-
\sigma(L_{\cdot}(x)\otimes \id -\id \otimes R_{\prec}(x) )\Delta_{\prec}(y)\cr&&=
(L_{\succ}(x)\otimes L_{\cdot}(y)-R_{\prec}(y)\otimes R_{\cdot}(x)-
L_{\succ}(y)\otimes L_{\cdot}(x)+
R_{\prec}(x)\otimes R_{\cdot}(y))(\mathrm{r}_{_\succ}+\sigma \mathrm{r}_{_\prec})\cr
&&+ (([L_{\succ}(x), R_{\prec}(y)]-
[L_{\succ}(y), R_{\prec}(x)])\otimes\id)\sigma \mathrm{r}_{_\prec}+
(\id\otimes ([L_{\cdot}(y), R_{\cdot}(x)]-
[L_{\cdot}(x), R_{\cdot}(y)]))\mathrm{r}_{_\succ}\cr
&&=(L_{\succ}(x)\otimes L_{\cdot}(y)-R_{\prec}(y)\otimes R_{\cdot}(x)-
L_{\succ}(y)\otimes L_{\cdot}(x)+
R_{\prec}(x)\otimes R_{\cdot}(y))(\mathrm{r}_{_\succ}+\sigma \mathrm{r}_{_\prec})
\end{eqnarray*}
Note that the last equal sign in above equation is due to
Eq.~\eqref{eq_bimodule_pre_anti_flexible1} and Eq.~\eqref{eq:useful1}.
Therefore, Eq.~\eqref{eq_pre_matched_3} is equivalent to the following
\begin{eqnarray}{\label{eq:coboundary2}}
(L_{\succ}(x)\otimes L_{\cdot}(y)-R_{\prec}(y)\otimes R_{\cdot}(x)-
L_{\succ}(y)\otimes L_{\cdot}(x)+R_{\prec}(x)\otimes R_{\cdot}(y))(\mathrm{r}_{_\succ}+
\sigma \mathrm{r}_{_\prec})=0,\; \forall x,y\in A.
\end{eqnarray}
\item Furthermore, by Eqs.~\eqref{eq:delta} and
\eqref{eq:sigma_delta} we have for any $x,y\in A$
\begin{eqnarray*}
\Delta(x\succ y)+\sigma\Delta(y\prec x)&=&
(\id \otimes (L_{\cdot}(x\succ y)+R_{\cdot}(y\prec x)))\mathrm{r}_{_\succ}+
(\id \otimes(R_{_{\prec}}(y\prec x)+L_{_{\succ}}(x\succ y)))\mathrm{r}_{_\prec}\cr&+&
((R_{_{\prec}}(x\succ y)+L_{_{\succ}}(y\prec x))\otimes \id)\sigma \mathrm{r}_{_\prec}+
((R_{\cdot}(x\succ y)+L_{\cdot}(y\prec x))\otimes \id)\sigma \mathrm{r}_{_\succ}
\end{eqnarray*}
Taking into account to Eqs.~\eqref{eq_bimodule_pre_anti_flexible2} and
\eqref{eq_bimodule_pre_anti_flexible5} we have
\begin{eqnarray*}
\Delta(x\succ y)+\sigma\Delta(y\prec x)&=&
(\id\otimes (L_{_{\succ}}(x\succ y)+
R_{_{\prec}}(y\prec x)))(\mathrm{r}_{_\succ}+\mathrm{r}_{_\prec})\cr
&+&
((L_{_{\succ}}(y\prec x)+R_{_{\prec}}(x\succ y))\otimes \id)(\sigma \mathrm{r}_{_\succ}+
\sigma \mathrm{r}_{_\prec})\cr
&+&
(\id \otimes (L_{_{\succ}}(x)L_{_{\prec}}(y)+
R_{_{\prec}}(x)R_{_{\succ}}(y)))\mathrm{r}_{_\succ}+\cr
&+&
((L_{_{\prec}}(y)L_{\cdot}(x)+R_{_{\succ}}(y)R_{\cdot}(x))\otimes \id )\sigma \mathrm{r}_{_\succ}.
\end{eqnarray*}
Using Eqs.~ \eqref{eq:sigma_succ} , \eqref{eq:delta} and
\eqref{eq:sigma_delta} we have for any $x,y\in A$
\begin{eqnarray*}
(R_{_{\succ}}(y)\otimes \id)\Delta_{_{\prec}}(x)&=&
(R_{_{\succ}}(y)\otimes L_{_{\succ}}(x))\mathrm{r}_{_{\prec}}+
(R_{_{\succ}}(y)R_{\cdot}(x)\otimes\id)\sigma \mathrm{r}_{_{\succ}}\cr
(L_{_{\prec}}(y)\otimes \id )\sigma\Delta_{_{\succ}}(x)&=&
(L_{_{\prec}}(y)L_{\cdot}(x)\otimes\id)\sigma \mathrm{r}_{_{\succ}}+
(L_{_{\prec}}(y)\otimes R_{_{\prec}}(x))\mathrm{r}_{_{\prec}}\cr
(\id \otimes L_{_{\succ}} (x))\Delta(y)&=&
(\id\otimes L_{_{\succ}}(x)L_{\cdot}(y))\mathrm{r}_{_{\succ}}+
(R_{_{\prec}}(y)\otimes L_{_{\succ}}(x)) \sigma \mathrm{r}_{_{\prec}}
\cr&+&
(\id\otimes L_{_{\succ}}(x)L_{_{\succ}}(y))\mathrm{r}_{_{\prec}}+
(R_{\cdot}(y)\otimes L_{_{\succ}}(x))\sigma \mathrm{r}_{_{\succ}}\cr
(\id \otimes R_{_{\prec}}(x) )\sigma\Delta(y)&=&
(L_{\cdot}(y)\otimes R{_{\prec}}(x))\sigma \mathrm{r}_{_{\succ}}+
(\id\otimes R_{_{\prec}}(x)R_{_{\prec}}(y))\mathrm{r}_{_{\prec}}\cr
&+&
(L_{_{\succ}}(y)\otimes R_{_{\prec}}(x)) \sigma \mathrm{r}_{_{\prec}}+
(\id\otimes R_{_{\prec}}(x)R_{\cdot}(y)) \mathrm{r}_{_{\succ}}
\end{eqnarray*}
Thus
\begin{eqnarray*}
&&(R_{_{\succ}}(y)\otimes \id)\Delta_{_{\prec}}(x)+
(L_{_{\prec}}(y)\otimes \id )\sigma\Delta_{_{\succ}}(x)+
(\id \otimes L_{_{\succ}} (x))\Delta(y)+
(\id \otimes R_{_{\prec}}(x) )\sigma\Delta(y)
\cr&&=
(\id\otimes(L_{_{\succ}}(x)L_{_{\succ}}(y)+
R_{_{\prec}}(x)R_{_{\prec}}(y)) )(\mathrm{r}_{_{\succ}}+\mathrm{r}_{_{\prec}})+
((R_{_{\succ}}(y)R_{\cdot}(x)+
L_{_{\prec}}(y)L_{\cdot}(x))\otimes\id )\sigma \mathrm{r}_{_{\succ}}\cr&&+
(R_{_{\prec}}(y)\otimes L_{_{\succ}}(x) +
L_{_{\succ}}(y)\otimes R_{_{\prec}}(x) )(\sigma \mathrm{r}_{_{\succ}}+
\sigma \mathrm{r}_{_{\prec}})+
(\id \otimes(L_{_{\succ}}(x)L_{_{\prec}}(y)+
R_{_{\prec}}(x)R_{_{\succ}}(y)) )\mathrm{r}_{_{\succ}}\cr&&+
(R_{_{\succ}}(y)\otimes L_{_{\succ}}(x) +
L_{_{\prec}}(y)\otimes R_{_{\prec}}(x) )(\mathrm{r}_{_{\prec}}+
\sigma \mathrm{r}_{_{\succ}})
\end{eqnarray*}
Therefore, using Eq.~\eqref{eq_bimodule_pre_anti_flexible3} we deduce that
Eq.~\eqref{eq_pre_matched_2'} is equivalent to the following
\begin{eqnarray}{\label{eq:coboundary3}}
\begin{array}{llll}
&&
(R_{_{\succ}}(y)\otimes L_{_{\succ}}(x) +
L_{_{\prec}}(y) \otimes R_{_{\prec}}(x) )(\mathrm{r}_{_{\prec}}+
\sigma \mathrm{r}_{_{\succ}})\cr&&+
(R_{_{\prec}}(y)\otimes L_{_{\succ}}(x) +
L_{_{\succ}}(y)\otimes R_{_{\prec}}(x) )(\sigma \mathrm{r}_{_{\succ}}+
\sigma \mathrm{r}_{_{\prec}})\cr&&+
(\id\otimes(L_{_{\succ}}(x\prec y)+R_{_{\prec}}(y\succ x)))(\mathrm{r}_{_{\succ}}+\mathrm{r}_{_{\prec}})\cr&&-
((L_{_{\succ}}(y\prec x)+
R_{_{\prec}}(x\succ y))\otimes \id)(\sigma \mathrm{r}_{_{\succ}}+\sigma \mathrm{r}_{_{\prec}})=0.
\end{array}
\end{eqnarray}
\item Finally, in view of Lemma~\ref{lem:sigma} we have for any $x,y\in A$
\begin{eqnarray*}
(\id \otimes R_{_{\succ}}(y))\Delta_{_{\succ}}(x)&=&
(\id\otimes R_{_{\succ}}(y)L_{\cdot}(x))\mathrm{r}_{_{\succ}}+
(R_{_{\prec}}(x)\otimes R_{_{\succ}}(y))\sigma \mathrm{r}_{_{\prec}}
\cr
(L_{_{\prec}}(y)\otimes \id)\Delta_{_{\prec}}(x)&=&
(L_{_{\prec}}(y)\otimes L_{_{\succ}}(x))\mathrm{r}_{_{\prec}}+
(L_{_{\prec}}(y)R_{\cdot}(x)\otimes \id)\sigma \mathrm{r}_{_{\succ}}
\cr
(R_{_{\prec}}(x)\otimes \id)\sigma \Delta(y)&=&
(R_{_{\prec}}(x)L_{\cdot}(y)\otimes\id )\sigma \mathrm{r}_{_{\succ}}+
(R_{_{\prec}}(x)\otimes R_{_{\prec}}(y))\mathrm{r}_{_{\prec}}\cr&+&
(R{_{\prec}}(x)L_{_{\succ}}(y)\otimes \id)\sigma \mathrm{r}_{_{\prec}}+
(R_{_{\prec}}(x)\otimes R_{\cdot}(y) )\mathrm{r}_{_{\succ}}
\cr
(\id \otimes L_{_{\succ}}(x))\sigma \Delta(y)&=&
(L_{\cdot}(y)\otimes L_{_{\succ}}(x) )\sigma \mathrm{r}_{_{\succ}}+
(\id\otimes L_{_{\succ}}(x)R_{_{\prec}}(y))\mathrm{r}_{_{\prec}}\cr&+&
(L_{_{\succ}}(y)\otimes L_{_{\succ}}(x) )\sigma \mathrm{r}_{_{\prec}}+
(\id\otimes L_{_{\succ}}(x)R_{\cdot}(y) )\mathrm{r}_{_{\succ}}
\cr
(R_{_{\succ}}(y)\otimes \id)\sigma\Delta_{_{\succ}}(x)&=&
(R_{_{\succ}}(y)L_{\cdot}(x)\otimes\id)\sigma \mathrm{r}_{_{\succ}}+
(R_{_{\succ}}(y)\otimes R_{_{\prec}}(x))\mathrm{r}_{_{\prec}}\cr
(\id \otimes L_{_{\prec}}(y))\sigma\Delta_{_{\prec}}(x)&=&
(L_{_{\succ}}(x)\otimes L_{_{\prec}}(y))\sigma \mathrm{r}_{_{\prec}}+
(\id\otimes L_{_{\prec}}(y)R_{\cdot}(x))\mathrm{r}_{_{\succ}}\cr
(\id \otimes R_{_{\prec}}(x))\Delta(y)&=&
(\id\otimes R_{_{\prec}}(x)L_{\cdot}(y))\mathrm{r}_{_{\succ}}+
(R_{_{\prec}}(y)\otimes R_{_{\prec}}(x))\sigma \mathrm{r}_{_{\prec}}\cr
&+&
(\id\otimes R_{_{\prec}}(x)L_{_{\succ}}(y))\mathrm{r}_{_{\prec}}+
(R_{\cdot}(y)\otimes R_{_{\prec}}(x))\sigma \mathrm{r}_{_{\succ}}\cr
(L_{_{\succ}}(x)\otimes \id)\Delta(y)&=&
(L_{_{\succ}}(x)\otimes L_{\cdot}(y))\mathrm{r}_{_{\succ}}+
(L_{_{\succ}}(x)R_{_{\prec}}(y)\otimes \id)\sigma \mathrm{r}_{_{\prec}}\cr
&+&
(L_{_{\succ}}(x)\otimes L_{_{\succ}}(y))\mathrm{r}_{_{\prec}}+
(L_{_{\succ}}(x)R_{\cdot}(y)\otimes\id )\sigma \mathrm{r}_{_{\succ}}
\end{eqnarray*}
According to Eq.~\eqref{eq_bimodule_pre_anti_flexible4} we have for any $x,y\in A$
\begin{eqnarray*}
&&(\id \otimes R_{_{\succ}}(y))\Delta_{_{\succ}}(x)-
(L_{_{\prec}}(y)\otimes \id )\Delta_{_{\prec}}(x)
+(R_{_{\prec}}(x)\otimes \id-\id\otimes L_{{_\succ}}(x))\sigma\Delta(y)\cr
&&-(R_{_{\succ}}(y) \otimes \id)\sigma\Delta_{_{\succ}}(x)+
(\id\otimes L_{_{\prec}}(y) )\sigma\Delta_{_{\prec}}(x)
-(\id \otimes R_{_{\prec}}(x)- L_{{_\succ}}(x)\otimes \id)\Delta(y)\cr
&=&((R_{_{\prec}}(x)\otimes R_{_{\prec}}(x)+ L_{_{\succ}}(x)\otimes L_{_{\succ}}(y))
-(\id\otimes (L_{_{\succ}}(x)R_{_{\prec}}(y)+
R_{_{\prec}}(x)L_{_{\succ}}(y))))(\mathrm{r}_{_{\succ}}+\mathrm{r}_{_{\prec}})\cr
&+&
((R_{_{\prec}}(x)L_{_{\succ}}(y)+L_{_{\succ}}(x)R_{_{\prec}}(y))\otimes \id-
(L_{_{\succ}}(y)\otimes L_{_{\succ}}(x) +
R_{_{\prec}}(y)\otimes R_{_{\prec}}(x)) )(\sigma \mathrm{r}_{_{\prec}}+
\sigma \mathrm{r}_{_{\succ}})\cr
&+&
(R_{_{\prec}}(x)\otimes R_{_{\succ}}(y)+L_{_{\succ}}(x)\otimes
L_{_{\prec}}(y))(\mathrm{r}_{_{\succ}}+\sigma \mathrm{r}_{_{\prec}})-
(L_{_{\prec}}(y)\otimes L_{_{\succ}}(x)+R_{_{\succ}}(y)\otimes
R_{_{\prec}}(x))(\mathrm{r}_{_{\prec}}+\sigma \mathrm{r}_{_{\succ}})
\end{eqnarray*}
Therefore, Eq.~\eqref{eq_pre_matched_4'} is equivalent to the following
\begin{eqnarray}{\label{eq:coboundary4}}
&&0=((R_{_{\prec}}(x)\otimes R_{_{\prec}}(x)+
L_{_{\succ}}(x)\otimes L_{_{\succ}}(y))
-(\id\otimes(L_{_{\succ}}(x)R_{_{\prec}}(y)+
R_{_{\prec}}(x)L_{_{\succ}}(y))))(\mathrm{r}_{_{\succ}}+\mathrm{r}_{_{\prec}})\cr
&&+
((R_{_{\prec}}(x)L_{_{\succ}}(y)+L_{_{\succ}}(x)R_{_{\prec}}(y))\otimes \id-
(L_{_{\succ}}(y)\otimes L_{_{\succ}}(x) +
R_{_{\prec}}(y)\otimes R_{_{\prec}}(x)) )(\sigma \mathrm{r}_{_{\prec}}+\sigma \mathrm{r}_{_{\succ}})\\
&&+
(R_{_{\prec}}(x)\otimes R_{_{\succ}}(y)+
L_{_{\succ}}(x)\otimes L_{_{\prec}}(y))(\mathrm{r}_{_{\succ}}+\sigma \mathrm{r}_{_{\prec}})-
(L_{_{\prec}}(y)\otimes L_{_{\succ}}(x)+
R_{_{\succ}}(y)\otimes R_{_{\prec}}(x))(\mathrm{r}_{_{\prec}}+\sigma \mathrm{r}_{_{\succ}})\nonumber
\end{eqnarray}
\end{itemize}
Clearly, we have provided the proof of the following theorem
\begin{thm}
Let $(A, \prec, \succ)$ be a pre-anti-flexible algebra and
$\mathrm{r}_{_\prec}, \mathrm{r}_{_\succ}\in A\otimes A$. Consider
$\Delta_{_\prec}, \Delta_{_\succ}:A\rightarrow A\otimes A$
two linear maps defined by Eq.~\eqref{eq:coboundary} such that their dual maps
$\Delta_{\prec}^*, \Delta_{\succ}^*:A^*\otimes A^*\rightarrow A^*$
define a pre-anti-flexible algebra on $A^*$.
Then $(A, A^*)$ is a pre-anti-flexible bialgebra if and only if
$\Delta_{\prec}, \Delta_{\succ}$ satisfying
Eqs.~\eqref{eq:coboundary1}~-~\eqref{eq:coboundary4}.
\end{thm}
\begin{lem}
Let $A$ be a vector space and let
$\Delta_{_\prec}, \Delta_{\succ} :A\rightarrow A\otimes A$ be two linear maps.
Then $\Delta_{_{\prec}}^*, \Delta_{_{\succ}}^*:A^*\otimes A^*\rightarrow A^*$
define a pre-anti-flexible algebra structure
on $A^*$ if and only if the following conditions are satisfied
\begin{subequations}
\begin{eqnarray}\label{eq:rmatrix1}
(\Delta_{_\succ}\otimes\id )\Delta_{_{\prec}}-
(\id\otimes\Delta_{_\prec})\Delta_{_{\succ}}=
(\id\otimes\sigma\Delta_{_\succ})\sigma\Delta_{_{\prec}}-
(\sigma\Delta_{_{\prec}}\otimes\id)\sigma\Delta_{_{\succ}},
\end{eqnarray}
\begin{eqnarray}\label{eq:rmatrix2}
((\Delta_{_{\prec}}+\Delta_{_{\succ}})\otimes\id)\Delta_{\succ}-
(\id\otimes\Delta_{\succ})\Delta_{\succ}=
(\id\otimes \sigma\Delta_{_{\prec}})\sigma\Delta_{_\prec}-
(\sigma(\Delta_{_{\prec}}+\Delta_{_{\succ}})\otimes\id)\sigma\Delta_{_\prec}.
\end{eqnarray}
\end{subequations}
\end{lem}
\begin{proof}
Denote by "$\prec_{_{A^*}}, \succ_{_{A^*}}$" the bilinear products on
$A^*$ defined respectively by
$\Delta_{_{\prec}}, \Delta_{_{\succ}}$, i.e.
for any $x\in A$ and for any $a, b\in A^*$
\begin{equation*}
\langle a\prec_{_{A^*}} b, x \rangle=\langle \Delta_{_{\prec}}^*(a\otimes b), x \rangle
=\langle a\otimes b,\Delta_{_{\prec}}(x)\rangle;\;\;
\langle a\succ_{_{A^*}} b, x\rangle=\langle \Delta_{_{\succ}}^*(a\otimes b), x \rangle=
\langle a\otimes b, \Delta_{_{\succ}}(x)\rangle.
\end{equation*}
Furthermore, according to Eq.~\eqref{eq:biasso} for any $a,b, c\in A^*$
and any $x\in A$, we have
\begin{eqnarray*}
\langle (a,b,c)_{_m}, x \rangle
&=&\langle (a\succ_{_{A^*}} b)\prec_{_{A^*}} c-a\succ_{_{A^*}}
(b\prec_{_{A^*}} c), x\rangle \cr&=&
\langle(\Delta_{_{\prec}}^*(\Delta_{_\succ}^*\otimes\id )-
\Delta_{_{\succ}}^*(\id\otimes\Delta_{_\prec}^*))(a\otimes b\otimes c), x\rangle\cr
\langle (a,b,c)_{_m}, x \rangle&=&\langle a\otimes b\otimes c,
( (\Delta_{_\succ}\otimes\id )\Delta_{_{\prec}}-
(\id\otimes\Delta_{_\prec})\Delta_{_{\succ}})(x)\rangle,\cr
\langle (c,b,a)_{_m}, x \rangle&=&\langle (c\succ_{_{A^*}} b)\prec_{_{A^*}} a-
c\succ_{_{A^*}} (b\prec_{_{A^*}} a), x\rangle \cr&=&
\langle( \Delta_{_{\prec}}^*\sigma(\id\otimes\Delta_{_\succ}^*\sigma)-
\Delta_{_{\succ}}^*\sigma(\Delta_{_{\prec}}^*\sigma\otimes\id))
(a\otimes b\otimes c), x\rangle\cr
\langle (c,b,a)_{_m}, x \rangle&=&\langle a\otimes b\otimes c,
((\id\otimes\sigma\Delta_{_\succ})\sigma\Delta_{_{\prec}}-
(\sigma\Delta_{_{\prec}}\otimes\id)\sigma\Delta_{_{\succ}})(x)\rangle,\cr
\langle (a,b,c)_{_l}, x \rangle&=&\langle (a \prec_{_{A^*}}b+
a\succ_{_{A^*}} b) \succ_{_{A^*}} c- a\succ_{_{A^*}} (b \succ_{_{A^*}} c), x\rangle \cr
&=&\langle(\Delta_{\succ}^*((\Delta_{_{\prec}}^*+\Delta_{_{\succ}}^*)\otimes\id)-
\Delta_{\succ}^*(\id\otimes\Delta_{\succ}^*))(a\otimes b\otimes c) ,
x\rangle\cr&=&\langle a\otimes b\otimes c,
(((\Delta_{_{\prec}}+\Delta_{_{\succ}})\otimes\id)\Delta_{\succ}-
(\id\otimes\Delta_{\succ})\Delta_{\succ})(x) \rangle,\cr
\langle (c,b,a)_{_r}, x \rangle&=& \langle (c\prec_{_{A^*}} b) \prec_{_{A^*}}a-
c\prec_{_{A^*}}(b\prec_{_{A^*}} a+b\succ_{_{A^*}} a), x\rangle\cr
&= &\langle ((\id\otimes \Delta_{_{\prec}}^*\sigma)\Delta_{_\prec}^*\sigma-
((\Delta_{_{\prec}}^*+\Delta_{_{\succ}}^*)\sigma\otimes\id)
\Delta_{_\prec}^*\sigma)(a\otimes b\otimes c),
x\rangle\cr
&=&\langle a\otimes b\otimes c, ((\id\otimes \sigma\Delta_{_{\prec}})\sigma\Delta_{_\prec}-
(\sigma(\Delta_{_{\prec}}+\Delta_{_{\succ}})\otimes\id)\sigma\Delta_{_\prec})(x)\rangle.
\end{eqnarray*}
Therefore, $(A^*, \prec_{_{A^*}}, \succ_{_{A^*}})$ is a
pre-anti-anti-flexible algebra if and only if
Eqs.~\eqref{eq:rmatrix1} and \eqref{eq:rmatrix2} are satisfied.
\end{proof}
For a given pre-anti-flexible algebra $(A,\prec, \succ)$ and
two elements $\mathrm{r}_{_\prec}, \mathrm{r}_{_\succ}$ in $A\otimes A$ given by
$\displaystyle \mathrm{r}_{_\succ}=\sum_i{a_i\otimes b_i}$ and
$\displaystyle \mathrm{r}_{_\prec}=\sum_i{c_i\otimes d_i},$
for any $a_i, b_i, c_i$ and $d_i$ in $A$, we designate by
\begin{equation*}
\mathrm{r}_{_{\succ, 12}}=\sum_ia_i\otimes b_i\otimes 1,\quad
\mathrm{r}_{_{\succ, 13}}=\sum_{i}a_i\otimes 1\otimes b_i,
\quad \mathrm{r}_{_{\succ, 23}}=\sum_i1\otimes a_i\otimes b_i,
\mbox{ etc} \cdots,
\end{equation*}
\begin{equation*}
\mathrm{r}_{_{\prec, 12}}=\sum_i c_i\otimes d_i\otimes 1,\quad
\mathrm{r}_{_{\prec, 13}}=\sum_{i}c_i\otimes 1\otimes d_i,
\quad \mathrm{r}_{_{\prec, 23}}=\sum_i1\otimes c_i\otimes d_i,
\mbox{ etc} \cdots,
\end{equation*}
where $1$ is the unit element if $(A,\prec, \succ)$ unitary, otherwise is a
symbol playing a similar role as that of the unit element on $A$. Then operations
between two $\mathrm{r}_{_{\prec, ..}}, \mathrm{r}_{_{\succ, ..}}$ are in an obvious way. For instance,
\begin{equation*}
\mathrm{r}_{_{\succ,12}}\cdot \mathrm{r}_{_{\succ, 13}}=
\sum_{i,j}a_i\cdot a_j\otimes b_i\otimes b_j,
\mathrm{r}_{_{\succ, 13}}\prec \mathrm{r}_{_{\succ, 23}}=
\sum_{i,j}a_i\otimes a_j\otimes b_i\prec b_j,
\mathrm{r}_{_{\prec, 23}}\succ \mathrm{r}_{_{\prec, 12}}=
\sum_{i,j}c_j\otimes c_i\succ d_j\otimes d_i,
\mbox{ etc}
\end{equation*}
and similarly
\begin{equation*}
\mathrm{r}_{_{\prec, 12}}\succ \mathrm{r}_{_{\succ, 13}}=
\sum_{i,j}c_i\succ a_j\otimes d_i\otimes b_j,
\mathrm{r}_{_{\succ,13}}\prec \mathrm{r}_{_{\prec, 23}}=
\sum_{i,j}a_i\otimes c_j\otimes b_i\prec d_j,
\mathrm{r}_{_{\prec, 23}}\cdot \mathrm{r}_{_{\succ,12}}=
\sum_{i,j}a_j\otimes c_i\cdot b_j\otimes d_i,
\end{equation*}
and so on.
\begin{pro}
Let $(A, \prec, \succ)$ be a pre-anti-flexible algebra and
$\mathrm{r}_{_\prec}, \mathrm{r}_{_\succ} \in A\otimes A$. Define
$\Delta_{_{\prec}}, \Delta_{_{\succ}}:A\rightarrow A\otimes A$ by
Eqs.~\eqref{eq:coboundary1} and \eqref{eq:coboundary2}.
Then $\Delta_{_\prec}^*, \Delta_{_{\succ}}^*: A^*\otimes A^*\rightarrow A^*$
define a pre-anti-flexible algebra structure on $A^*$
if and only if the following equations are satisfied for any $x\in A$
\begin{subequations}
\begin{eqnarray}\label{eq:ybe1}
&&(\id\otimes\id\otimes L_{\succ}(x))M(\mathrm{r})-
(R_{\prec}(x)\otimes \id\otimes\id)N(\mathrm{r})\cr
&&+(\id\otimes \id \otimes R_{\prec}(x))P(\mathrm{r})-
(L_{\succ}(x)\otimes\id \otimes \id)Q(\mathrm{r})=0,
\end{eqnarray}
\begin{eqnarray}\label{eq:ybe2}
\begin{array}{llll}
&&(\id \otimes\id\otimes L_{\cdot}(x))M'(\mathrm{r})
+(\id \otimes \id \otimes R_{\cdot}(x))N'(\mathrm{r})+R'(x)
\cr&&-(R_{\prec}(x)\otimes\id \otimes \id)P'(\mathrm{r})
-(L_{\succ}(x)\otimes \id \otimes \id)Q'(\mathrm{r})=0,
\end{array}
\end{eqnarray}
\end{subequations}
where
\begin{eqnarray*}
M(\mathrm{r})=\mathrm{r}_{_{\prec, 23}}\cdot \mathrm{r}_{_{\succ, 12}}+
\mathrm{r}_{_{\prec, 21}}\prec \mathrm{r}_{_{\prec, 13}}-
\mathrm{r}_{_{\succ, 13}} \succ \mathrm{r}_{_{\prec, 23}},\;
P(\mathrm{r})=\mathrm{r}_{_{\succ, 12}}\cdot \mathrm{r}_{_{\prec, 23}}+
\mathrm{r}_{_{\prec, 13}}\succ \mathrm{r}_{_{\prec, 21}}-
\mathrm{r}_{_{\prec, 23}} \prec \mathrm{r}_{_{\succ, 13}},\cr
N(\mathrm{r})=
\mathrm{r}_{_{\succ, 32}}\cdot \mathrm{r}_{_{\prec, 21}}+
\mathrm{r}_{_{\prec, 31}}\succ \mathrm{r}_{_{\prec, 23}}-
\mathrm{r}_{_{\prec, 21}} \prec \mathrm{r}_{_{\succ,31}},\;
Q(\mathrm{r})=
\mathrm{r}_{_{\prec , 21}}\cdot \mathrm{r}_{_{\succ, 32}}+
\mathrm{r}_{_{\prec, 23}}\prec \mathrm{r}_{_{\prec, 31}}-
\mathrm{r}_{_{\succ, 31}} \succ \mathrm{r}_{_{\prec, 21}},
\end{eqnarray*}
\begin{eqnarray*}
M'(\mathrm{r})&=&
\mathrm{r}_{_{\succ, 23}}\prec \mathrm{r}_{_{\succ, 12}}+
\mathrm{r}_{_{\succ, 21}}\succ \mathrm{r}_{_{\succ, 13}}-
\mathrm{r}_{_{\succ, 13}}\cdot \mathrm{r}_{_{\succ,23}}+
\mathrm{r}_{_{\succ, 23}}\succ (\mathrm{r}_{_{\prec, 12}}+\mathrm{r}_{_{\succ, 12}})+
(\mathrm{r}_{_{\prec, 21}}+\mathrm{r}_{_{\succ, 21}})\prec \mathrm{r}_{_{\succ, 13}},\cr
N'(\mathrm{r})&=&
\mathrm{r}_{_{\succ, 12}}\succ \mathrm{r}_{_{\succ, 23}}+
\mathrm{r}_{_{\succ, 13}}\prec \mathrm{r}_{_{\succ, 21}}-
\mathrm{r}_{_{\succ,23}}\cdot \mathrm{r}_{_{\succ, 13}}+
(\mathrm{r}_{_{\prec, 12}}+\mathrm{r}_{_{\succ, 12}})\prec \mathrm{r}_{_{\succ, 23}}+
\mathrm{r}_{_{\succ, 13}}\succ (\mathrm{r}_{_{\prec, 21}}+\mathrm{r}_{_{\succ, 21}}),\cr
P'(\mathrm{r})&=&
\mathrm{r}_{_{\prec, 32}}\prec \mathrm{r}_{_{\prec, 21}}+
\mathrm{r}_{_{\prec, 31}}\cdot \mathrm{r}_{_{\succ, 23}}-
\mathrm{r}_{_{\succ, 21}}\succ \mathrm{r}_{_{\prec , 31}}-
(\mathrm{r}_{_{\prec, 21}}+\mathrm{r}_{_{\succ, 21}})\prec \mathrm{r}_{_{\prec, 31}},\cr
Q'(\mathrm{r})&=&
\mathrm{r}_{_{\prec, 21}}\succ \mathrm{r}_{_{\prec, 32}}+
\mathrm{r}_{_{\succ, 23}}\cdot \mathrm{r}_{_{\prec, 31}}
-\mathrm{r}_{_{\prec,31}}\prec \mathrm{r}_{_{\succ, 21}}-
\mathrm{r}_{_{\prec,31}}\succ (\mathrm{r}_{_{\prec, 21}}+\mathrm{r}_{_{\succ, 21}}),
\end{eqnarray*}
and
\begin{eqnarray*}
R'(x)&=&[(\id\otimes (R_{_{\prec}}(x)+L_{_{\succ}}(x))\otimes \id)\mathrm{r}_{_{\prec, 32}}]
\succ(\mathrm{r}_{_{\prec, 12}}+\mathrm{r}_{_{\succ, 12}})\cr
&-&[((L_{_{\prec}}(x)+R_{_{\succ}}(x))\otimes\id \otimes \id)\mathrm{r}_{_{\prec, 31}}]
\succ(\mathrm{r}_{_{\prec, 21}}+\mathrm{r}_{_{\succ, 21}}).
\end{eqnarray*}
\end{pro}
\begin{proof}
Let $x\in A$.
Setting $\displaystyle \mathrm{r}_{_\succ}=\sum_i a_i\otimes b_i, \;\;
\mathrm{r}_{_\prec}=\sum_i c_i\otimes d_i$ we have
\begin{eqnarray*}
(\Delta_{_\succ}\otimes\id )\Delta_{_{\prec}}(x)&=&
\sum_i\{a_{j}\otimes(c_i\cdot b_j)\otimes (x\succ d_{i})+
(d_j\prec c_i)\otimes c_j\otimes (x\succ d_i)\cr&+&
a_j\otimes ((b_i\cdot x)\cdot b_j)\otimes a_i+
(d_j\prec (b_i\cdot x))\otimes c_j\otimes a_i\}\cr
&=&
(\id\otimes\id\otimes L_{\succ}(x))(\mathrm{r}_{_{\prec, 23}}\cdot \mathrm{r}_{_{\succ, 12}}+
\mathrm{r}_{_{\prec, 21}}\prec \mathrm{r}_{_{\prec, 13}})\cr
&+&\sum_i\{a_j\otimes ((b_i\cdot x)\cdot b_j)\otimes a_i+
(d_j\prec (b_i\cdot x))\otimes c_j\otimes a_i\}
\cr
(\id\otimes\Delta_{_\prec})\Delta_{_{\succ}}(x)&=&
\sum_i\{a_i\otimes c_j\otimes ((x\cdot b_i)\prec d_j)+
a_i\otimes (b_j\cdot (x\cdot b_i))\otimes a_j\cr&+&
(d_i\prec x)\otimes c_j\otimes (c_i\succ d_j)+
(d_i\prec x)\otimes (b_j\cdot c_i)\otimes a_j\}\cr
&=&
(R_{\prec}(x)\otimes \id\otimes\id)(\mathrm{r}_{_{\prec, 31}}\succ \mathrm{r}_{_{\prec, 23}}+
\mathrm{r}_{_{\succ, 32}}\cdot \mathrm{r}_{_{\prec, 21}})\cr
&+&\sum_i \{a_i\otimes c_j\otimes ((x\cdot b_i)\prec d_j)+
a_i\otimes (b_j\cdot (x\cdot b_i))\otimes a_j\}\cr
(\id\otimes\sigma\Delta_{_\succ})\sigma\Delta_{_{\prec}}(x)&=&
\sum_i\{(x\succ d_i)\otimes (c_i\cdot b_j)\otimes a_j+
(x\succ d_i)\otimes c_j\otimes (d_j\prec c_i)\cr&+&
a_i\otimes ((b_i\cdot x)\cdot b_j)\otimes a_j+
a_i\otimes c_j\otimes (d_j\prec (b_i\cdot x))\}\cr&=&
(L_{\succ}(x)\otimes\id \otimes \id)(\mathrm{r}_{_{\prec , 21}}\cdot \mathrm{r}_{_{\succ, 32}}+
\mathrm{r}_{_{\prec, 23}}\prec \mathrm{r}_{_{\prec, 31}})\cr
&+&\sum_i\{a_i\otimes ((b_i\cdot x)\cdot b_j)\otimes a_j+
a_i\otimes c_j\otimes (d_j\prec (b_i\cdot x))\}\cr
(\sigma\Delta_{_{\prec}}\otimes\id)\sigma\Delta_{_{\succ}}(x)&=&
\sum_i\{((x\cdot b_i)\succ d_j)\otimes c_j\otimes a_i+
a_j\otimes (b_j\cdot (x\cdot b_i))\otimes a_i\cr
&+&(c_i\succ d_j)\otimes c_j\otimes (d_i \prec x)+
a_j\otimes (b_j\cdot c_i)\otimes (b_i\prec x)\}\cr
&=&
(\id\otimes \id \otimes R_{\prec}(x))(\mathrm{r}_{_{\prec, 13}}\succ \mathrm{r}_{_{\prec, 21}}+
\mathrm{r}_{_{\succ, 12}}\cdot \mathrm{r}_{_{\prec, 23}})\cr
&+&\sum_i\{((x\cdot b_i)\succ d_j)\otimes c_j\otimes a_i+
a_j\otimes (b_j\cdot (x\cdot b_i))\otimes a_i\}
\end{eqnarray*}
\begin{eqnarray*}
((\Delta_{_{\prec}}+\Delta_{_{\succ}})\otimes\id)\Delta_{\succ}(x)&=&
\sum_i\{c_j\otimes (a_i\succ d_j)\otimes (x\cdot b_i)+
(b_j\cdot a_i)\otimes a_j\otimes (x\cdot b_i)\cr&+&
a_j\otimes (a_i\cdot b_j)\otimes (x\cdot b_i)
+(d_j\prec a_i)\otimes c_j\otimes(x\cdot b_i)\cr&+&
c_j\otimes ((d_i\prec x)\succ d_j)\otimes c_i+
(b_j\cdot(d_i\prec x))\otimes a_j\otimes c_i\cr&+&
a_j\otimes ((d_i\prec x)\cdot b_j)\otimes c_i+
(d_j\prec (d_i\prec x))\otimes c_j\otimes c_i\}\cr
&=&
(\id \otimes\id\otimes L_{\cdot}(x))(\mathrm{r}_{_{\succ, 23}}\succ \mathrm{r}_{_{\prec, 12}}+
\mathrm{r}_{_{\succ, 21}}\cdot \mathrm{r}_{_{\succ, 13}}\cr&+&
\mathrm{r}_{_{\succ, 23}}\cdot \mathrm{r}_{_{\succ, 12}}+
\mathrm{r}_{_{\prec, 21}}\prec \mathrm{r}_{_{\succ, 13}})\cr
&+&\sum_i\{c_j\otimes ((d_i\prec x)\succ d_j)\otimes c_i+
(b_j\cdot(d_i\prec x))\otimes a_j\otimes c_i\cr&+&
a_j\otimes ((d_i\prec x)\cdot b_j)\otimes c_i+
(d_j\prec (d_i\prec x))\otimes c_j\otimes c_i\}\cr
(\id\otimes\Delta_{\succ})\Delta_{\succ}(x)&=&
\sum_i\{a_i\otimes a_j\otimes ((x\cdot b_i)\cdot b_j)
+a_i\otimes (d_j\prec (x\cdot b_i))\otimes c_j\cr
&+&(d_i\prec x)\otimes a_j\otimes (c_i\cdot b_j)+
(d_i\prec x)\otimes (d_j\prec c_i)\otimes c_j\}\cr
&=&
(R_{\prec}(x)\otimes\id \otimes \id)(\mathrm{r}_{_{\prec, 31}}\cdot \mathrm{r}_{_{\succ, 23}}+
\mathrm{r}_{_{\prec, 32}}\prec \mathrm{r}_{_{\prec, 21}})\cr
&+&\sum_i\{a_i\otimes a_j\otimes ((x\cdot b_i)\cdot b_j)
+a_i\otimes (d_j\prec (x\cdot b_i))\otimes c_j\}\cr
(\id\otimes \sigma\Delta_{_{\prec}})\sigma\Delta_{_\prec}(x)&=&
\sum_i\{(x\succ d_i)\otimes (c_i\succ d_j)\otimes c_j+
(x\succ d_i)\otimes a_j\otimes (b_j\cdot c_i)\cr
&+&a_i\otimes ((b_i\cdot x)\succ d_j)\otimes c_j+
a_i\otimes a_j\otimes (b_j\cdot(b_i\cdot x))\}\cr
&=&(L_{\succ}(x)\otimes \id \otimes \id)(\mathrm{r}_{_{\prec, 21}}\succ \mathrm{r}_{_{\prec, 32}}+
\mathrm{r}_{_{\succ, 23}}\cdot \mathrm{r}_{_{\prec, 31}})\cr
&+& \sum_i\{a_i\otimes ((b_i\cdot x)\succ d_j)\otimes c_j+
a_i\otimes a_j\otimes (b_j\cdot(b_i\cdot x))\}\cr
(\sigma(\Delta_{_{\prec}}+\Delta_{_{\succ}})\otimes\id)\sigma\Delta_{_\prec}(x)&=&
\sum_i\{((x\succ d_i)\succ d_j)\otimes c_j\otimes c_i+
a_j\otimes (b_j\cdot(x\succ d_i))\otimes c_i\cr
&+&
((x\succ d_i)\cdot b_j)\otimes a_j\otimes c_i+
c_j\otimes (d_j\prec(x\succ d_i))\otimes c_i\cr
&+&
(a_i\succ d_j)\otimes c_j\otimes (b_i\cdot x)+
a_j\otimes (b_j\cdot a_i)\otimes (b_i\cdot x)
\cr&+&
(a_i\cdot b_j)\otimes a_j\otimes (b_i\cdot x)+
c_j\otimes (d_j\prec a_i)\otimes (b_i\cdot x)\}\cr
&=&
(\id \otimes \id \otimes R_{\cdot}(x))(\mathrm{r}_{_{\succ, 13}}\succ \mathrm{r}_{_{\prec, 21}}+
\mathrm{r}_{_{\succ, 12}}\cdot \mathrm{r}_{_{\succ, 23}}\cr&+&
\mathrm{r}_{_{\succ, 13}}\cdot \mathrm{r}_{_{\succ, 21}}+
\mathrm{r}_{_{\prec, 12}}\prec \mathrm{r}_{_{\succ, 23}})\cr
&+&
\sum_i\{((x\succ d_i)\succ d_j)\otimes c_j\otimes c_i+
a_j\otimes (b_j\cdot(x\succ d_i))\otimes c_i\cr
&+&
((x\succ d_i)\cdot b_j)\otimes a_j\otimes c_i+
c_j\otimes (d_j\prec(x\succ d_i))\otimes c_i\}
\end{eqnarray*}
Then
\begin{eqnarray*}
&&(\Delta_{_\succ}\otimes\id )\Delta_{_{\prec}}(x)-
(\id\otimes\Delta_{_\prec})\Delta_{_{\succ}}(x)-
(\id\otimes\sigma\Delta_{_\succ})\sigma\Delta_{_{\prec}}(x)+
(\sigma\Delta_{_{\prec}}\otimes\id)\sigma\Delta_{_{\succ}}(x)=A1(x)+A2(x)\cr&&+
(\id\otimes\id\otimes L_{\succ}(x))(\mathrm{r}_{_{\prec, 23}}\cdot \mathrm{r}_{_{\succ, 12}}+
\mathrm{r}_{_{\prec, 21}}\prec \mathrm{r}_{_{\prec, 13}})
-(R_{\prec}(x)\otimes \id\otimes\id)(\mathrm{r}_{_{\prec, 31}}\succ \mathrm{r}_{_{\prec, 23}}+
\mathrm{r}_{_{\succ, 32}}\cdot \mathrm{r}_{_{\prec, 21}})\cr &&
+(\id\otimes \id \otimes R_{\prec}(x))(\mathrm{r}_{_{\prec, 13}}\succ \mathrm{r}_{_{\prec, 21}}+
\mathrm{r}_{_{\succ, 12}}\cdot \mathrm{r}_{_{\prec, 23}})
-(L_{\succ}(x)\otimes\id \otimes \id)(\mathrm{r}_{_{\prec , 21}}\cdot \mathrm{r}_{_{\succ, 32}}+
\mathrm{r}_{_{\prec, 23}}\prec \mathrm{r}_{_{\prec, 31}})
\end{eqnarray*}
where
\begin{eqnarray*}
A1(x)&=&\sum_i\{ a_j\otimes ((b_i\cdot x)\cdot b_j+b_j\cdot (x\cdot b_i))\otimes a_i-
a_i\otimes (b_j\cdot (x\cdot b_i)+(b_i\cdot x)\cdot b_j)\otimes a_j\}\\
A2(x)&=&\sum_i\{(d_j\prec (b_i\cdot x)+(x\cdot b_i)\succ d_j)\otimes c_j\otimes a_i -
a_i\otimes c_j\otimes ((x\cdot b_i)\prec d_j+d_j\prec (b_i\cdot x)) \}\\
\end{eqnarray*}
By exchanging $i$ and $j$, and using Remark~\ref{rmk_1}~\eqref{rmk_flex},
we have $A1(x)=0$.
Using Eqs.~\eqref{eq:pre-antiflexible2} we have
\begin{eqnarray*}
A2(x)&=&(L_{\succ}(x)\otimes\id \otimes\id)
(\mathrm{r}_{_{\succ, 31}} \succ \mathrm{r}_{_{\prec, 21}})
-(\id \otimes \id \otimes L_{\succ}(x))
(\mathrm{r}_{_{\succ, 13}} \succ \mathrm{r}_{_{\prec, 23}})
\cr&+&
(R_{\prec}(x)\otimes\id \otimes\id) (\mathrm{r}_{_{\prec, 21}} \prec \mathrm{r}_{_{\succ,31}})-
(\id \otimes \id \otimes R_{\prec}(x))(\mathrm{r}_{_{\prec, 23}} \prec \mathrm{r}_{_{\succ, 13}}).
\end{eqnarray*}
Besides,
\begin{eqnarray*}
&&((\Delta_{_{\prec}}+\Delta_{_{\succ}})\otimes\id)\Delta_{\succ}(x)-
(\id\otimes\Delta_{\succ})\Delta_{\succ}(x)
-(\id\otimes \sigma\Delta_{_{\prec}})\sigma\Delta_{_\prec}(x)
+(\sigma(\Delta_{_{\prec}}+
\Delta_{_{\succ}})\otimes\id)\sigma\Delta_{_\prec}(x)\cr
&=&(\id \otimes\id\otimes L_{\cdot}(x))(\mathrm{r}_{_{\succ, 23}}\succ \mathrm{r}_{_{\prec, 12}}
+\mathrm{r}_{_{\succ, 21}}\cdot \mathrm{r}_{_{\succ, 13}}+
\mathrm{r}_{_{\succ, 23}}\cdot \mathrm{r}_{_{\succ, 12}}+
\mathrm{r}_{_{\prec, 21}}\prec \mathrm{r}_{_{\succ, 13}})+B(x)
\cr &+&(\id \otimes \id \otimes R_{\cdot}(x))(\mathrm{r}_{_{\succ, 13}}\succ \mathrm{r}_{_{\prec, 21}
}+\mathrm{r}_{_{\succ, 12}}\cdot \mathrm{r}_{_{\succ, 23}}+
\mathrm{r}_{_{\succ, 13}}\cdot \mathrm{r}_{_{\succ, 21}}+
\mathrm{r}_{_{\prec, 12}}\prec \mathrm{r}_{_{\succ, 23}})+
\cr&-&(R_{\prec}(x)\otimes\id \otimes \id)(\mathrm{r}_{_{\prec, 31}}\cdot \mathrm{r}_{_{\succ, 23}
}+\mathrm{r}_{_{\prec, 32}}\prec \mathrm{r}_{_{\prec, 21}})
-(L_{\succ}(x)\otimes \id \otimes \id)(\mathrm{r}_{_{\prec, 21}}\succ \mathrm{r}_{_{\prec, 32}}+
\mathrm{r}_{_{\succ, 23}}\cdot \mathrm{r}_{_{\prec, 31}}),
\end{eqnarray*}
where $B(x)=B1(x)+B2(x)+B3(x)+B4(x)+B5(x)$ with
$$
B1(x)=-\sum_i\{a_i\otimes a_j\otimes ((x\cdot b_i)\cdot b_j+
b_j\cdot(b_i\cdot x))\},\;\;
B2(x)=\sum_i\{c_j\otimes ((d_i\prec x)\succ d_j+
d_j\prec(x\succ d_i))\otimes c_i\},
$$
$$
B3(x)=\sum_i\{((x\succ d_i)\succ d_j+
d_j\prec(d_i\prec x))\otimes c_j\otimes c_i\},\;\;
B4(x)=\sum_i\{ (b_j\cdot(d_i\prec x)+(x\succ d_i)\cdot b_j)\otimes a_j\otimes c_i\}
$$
\begin{eqnarray*}
B5(x)=\sum_i\{a_j\otimes (b_j\cdot(x\succ d_i)+(d_i\prec x)\cdot b_j)\otimes c_i-
a_i\otimes ((b_i\cdot x)\succ d_j+d_j\prec (x\cdot b_i))\otimes c_j\}.
\end{eqnarray*}
Considering Remark~\ref{rmk_1}~\eqref{rmk_flex} we have
\begin{eqnarray*}
B1(x)=-(\id \otimes \id \otimes L_{\cdot}(x))
(\mathrm{r}_{_{\succ, 13}}\cdot \mathrm{r}_{_{\succ,23}})-
(\id \otimes \id \otimes R_{\cdot}(x))(\mathrm{r}_{_{\succ,23}}\cdot \mathrm{r}_{_{\succ, 13}}).
\end{eqnarray*}
Using Eq.~\eqref{eq_bimodule_pre_anti_flexible5} we have
\begin{eqnarray*}
B3(x)&=&(x\succ (d_i\succ d_j)+(d_j\prec d_i)\prec x)\otimes c_j\otimes c_i-
((x\prec d_i)\succ d_j+d_j\prec (d_i\succ x))\otimes c_j\otimes c_i\cr
&=&(L_{_{\succ}}(x)\otimes\id \otimes\id)(\mathrm{r}_{_{\prec,31}}\succ \mathrm{r}_{_{\prec, 21}})+
(R_{_{\prec}}(x)\otimes\id \otimes\id)(\mathrm{r}_{_{\prec, 21}}\prec \mathrm{r}_{_{\prec, 31}})
\cr&-&((x\prec d_i)\succ d_j+d_j\prec (d_i\succ x))\otimes c_j\otimes c_i.
\end{eqnarray*}
By Eqs.~\eqref{eq_bimodule_pre_anti_flexible2} and \eqref{eq_bimodule_pre_anti_flexible5} we have
\begin{eqnarray*}
B4(x)&=& (b_j\succ(d_i\prec x)+(x\succ d_i)\prec b_j ) \otimes a_j\otimes c_i+
(b_j\prec(d_i\prec x)+(x\succ d_i)\succ b_j ) \otimes a_j\otimes c_i\cr
&=&(L_{_{\succ}}(x)\otimes\id \otimes\id)(\mathrm{r}_{_{\prec, 31}}\prec \mathrm{r}_{_{\succ, 21}}+
\mathrm{r}_{_{\prec, 31}}\succ \mathrm{r}_{_{\succ, 21}})\cr
&+&
(R_{_{\prec}}(x)\otimes\id \otimes\id)(\mathrm{r}_{_{\succ, 21}}\succ \mathrm{r}_{_{\prec , 31}}+
\mathrm{r}_{_{\succ, 21}}\prec \mathrm{r}_{_{\prec, 31}})\cr
&-&(b_j\prec(d_i\succ x)+(x\prec d_i)\succ b_j ) \otimes a_j\otimes c_i.
\end{eqnarray*}
Furthermore, we have
\begin{eqnarray*}
B5(x)&=&\sum_i\{a_j\otimes (b_j\cdot(x\succ d_i)+(d_i\prec x)\cdot b_j)\otimes c_i-
a_i\otimes ((b_i\cdot x)\succ d_j+d_j\prec (x\cdot b_i))\otimes c_j\}\cr
&=&\sum_i\{a_j\otimes (b_j\prec (x\succ d_i)+b_j\succ (x\succ d_i))\otimes c_i+
a_j\otimes ((d_i\prec x)\prec b_j+(d_i\prec x)\succ b_j) \otimes c_i\cr
&-&a_i\otimes ((b_i\cdot x)\succ d_j+d_j\prec (x\cdot b_i))\otimes c_j\}
=\sum_i\{a_j\otimes (b_j\succ (x\succ d_i)+(d_i\prec x)\prec b_j)\otimes c_i\cr
&-&a_i\otimes ((b_i\cdot x)\succ d_j+d_j\prec (x\cdot b_i))\otimes c_j\}+
\sum_i\{a_j\otimes (b_j\prec (x\succ d_i)+(d_i\prec x)\succ b_j)\otimes c_i\}\cr
B5(x)&=&\sum_i\{a_j\otimes (b_j\prec (x\succ d_i)+(d_i\prec x)\succ b_j)\otimes c_i\}
\end{eqnarray*}
The last equal sign in above equation is due to
Eq.~\eqref{eq:pre-antiflexible2} and changing indices
$i$ to $j$ in the last term of the first summation.
Finally, we have
\begin{eqnarray*}
R'(x)&=&\sum_i\{c_j\otimes ((d_i\prec x)\succ d_j+d_j\prec(x\succ d_i))\otimes c_i-
((x\prec d_i)\succ d_j+d_j\prec (d_i\succ x))\otimes c_j\otimes c_i\cr
&-&(b_j\prec(d_i\succ x)+(x\prec d_i)\succ b_j ) \otimes a_j\otimes c_i+
a_j\otimes (b_j\prec (x\succ d_i)+(d_i\prec x)\succ b_j)\otimes c_i\}\cr
R'(x)&=&
[(\id\otimes R_{_{\prec}}(x)\otimes \id)\mathrm{r}_{_{\prec, 32}}]
\succ(\mathrm{r}_{_{\prec, 12}}+\mathrm{r}_{_{\succ, 12}})
+
[(\id\otimes L_{_{\succ}}(x)\otimes \id)\mathrm{r}_{_{\prec, 32}}]
\prec(\mathrm{r}_{_{\prec, 12}}+\mathrm{r}_{_{\succ, 12}})\cr
&-&
[(L_{_{\prec}}(x)\otimes\id \otimes \id)\mathrm{r}_{_{\prec, 31}}]
\succ(\mathrm{r}_{_{\prec, 21}}+\mathrm{r}_{_{\succ, 21}})
-
[(R_{_{\succ}}(x)\otimes\id \otimes \id)\mathrm{r}_{_{\prec, 31}}]
\prec (\mathrm{r}_{_{\prec, 21}}+\mathrm{r}_{_{\succ, 21}}).
\end{eqnarray*}
Therefore, hold the equivalences.
\end{proof}
\begin{rmk}
Considering the flipping map "$flp$" defined on $A\otimes A$ such that for any elements
$\mathrm{r}, \mathrm{r}'\in A\otimes A$,
$flp(\mathrm{r}\prec \mathrm{r}' )=
\mathrm{r}\succ \mathrm{r}'$,
$flp( \mathrm{r}\succ \mathrm{r}'=
\mathrm{r}\prec \mathrm{r}'$, we establish finally
the following relations
$
P(\mathrm{r})=flp(M(\mathrm{r})),\;
N(\mathrm{r})=\sigma_{13}(flp(M(\mathrm{r}))),\;
Q(\mathrm{r})=flp( \sigma_{13} (flp(M(\mathrm{r})))),
$
with $\sigma_{13}(x\otimes y\otimes z)=z\otimes y\otimes x$
for any $x,y,z \in A$. Besides,
we also have $N'(\mathrm{r})=flp(M'(\mathrm{r}))$ and
$Q'(\mathrm{r})=flp(P'(\mathrm{r}))$.
\end{rmk}
\begin{pro}
Let $(A, \prec, \succ)$ be a pre-anti-flexible algebra and
$\mathrm{r}_{_\prec}, \mathrm{r}_{_\succ} \in A\otimes A$. Define
the linear maps
$\Delta_{_{\prec}}, \Delta_{_{\succ}}:A\rightarrow A\otimes A$ by
Eq.~\eqref{eq:coboundary}.
Then $\Delta_{_\prec}^*, \Delta_{_{\succ}}^*: A^*\otimes A^*\rightarrow A^*$
define a pre-anti-flexible algebra structure on $A^*$
if and only if the following equations are satisfied for any $x\in A$
\begin{subequations}
\begin{eqnarray}\label{eq:ybe1'}
&&((\id\otimes\id\otimes L_{\succ}(x))-
(R_{\prec}(x)\otimes \id\otimes\id)\sigma_{13}\circ flp
+(\id\otimes \id \otimes R_{\prec}(x))flp
\cr&&-(L_{\succ}(x)\otimes\id \otimes \id)flp\circ
\sigma_{13} \circ flp)M(\mathrm{r})=0,
\end{eqnarray}
\begin{eqnarray}\label{eq:ybe2'}
\begin{array}{llll}
&&((\id \otimes\id\otimes L_{\cdot}(x))
+(\id \otimes \id \otimes R_{\cdot}(x))flp)M'(\mathrm{r})
\cr&&-((R_{\prec}(x)\otimes\id \otimes \id)
+(L_{\succ}(x)\otimes \id \otimes \id)flp)P'(\mathrm{r})+R'(x)=0,
\end{array}
\end{eqnarray}
\end{subequations}
where
$M(\mathrm{r})=\mathrm{r}_{_{\prec, 23}}\cdot \mathrm{r}_{_{\succ, 12}}+
\mathrm{r}_{_{\prec, 21}}\prec \mathrm{r}_{_{\prec, 13}}-
\mathrm{r}_{_{\succ, 13}} \succ \mathrm{r}_{_{\prec, 23}}$,
\begin{eqnarray*}
M'(\mathrm{r})&=&
\mathrm{r}_{_{\succ, 23}}\prec \mathrm{r}_{_{\succ, 12}}+
\mathrm{r}_{_{\succ, 21}}\succ \mathrm{r}_{_{\succ, 13}}-
\mathrm{r}_{_{\succ, 13}}\cdot \mathrm{r}_{_{\succ,23}}+
\mathrm{r}_{_{\succ, 23}}\succ (\mathrm{r}_{_{\prec, 12}}+
\mathrm{r}_{_{\succ, 12}})+(\mathrm{r}_{_{\prec, 21}}+
\mathrm{r}_{_{\succ, 21}})\prec \mathrm{r}_{_{\succ, 13}},\cr
P'(\mathrm{r})&=&
\mathrm{r}_{_{\prec, 32}}\prec \mathrm{r}_{_{\prec, 21}}+
\mathrm{r}_{_{\prec, 31}}\cdot \mathrm{r}_{_{\succ, 23}}-
\mathrm{r}_{_{\succ, 21}}\succ \mathrm{r}_{_{\prec , 31}}-
(\mathrm{r}_{_{\prec, 21}}+\mathrm{r}_{_{\succ, 21}})\prec \mathrm{r}_{_{\prec, 31}},\cr
R'(x)&=&
[(\id\otimes (R_{_{\prec}}(x)+L_{_{\succ}}(x))\otimes \id)
\mathrm{r}_{_{\prec, 32}}]\succ(\mathrm{r}_{_{\prec, 12}}+\mathrm{r}_{_{\succ, 12}})\cr
&-&
[((L_{_{\prec}}(x)+R_{_{\succ}}(x))\otimes\id \otimes \id)
\mathrm{r}_{_{\prec, 31}}]\succ(\mathrm{r}_{_{\prec, 21}}+\mathrm{r}_{_{\succ, 21}}).
\end{eqnarray*}
\end{pro}
\begin{thm}\label{thm_coboundary}
Let $(A, \prec, \succ)$ be a pre-anti-flexible algebra and
$\mathrm{r}_{_\prec}, \mathrm{r}_{_\succ} \in A\otimes A$. Define the linear maps
$\Delta_{_{\prec}}, \Delta_{_{\succ}}:A\rightarrow A\otimes A$ by
Eq.~\eqref{eq:coboundary}.
Then $(A, A^*)$ is a pre-anti-flexible bialgebra
if and only if $\mathrm{r}_{_\prec}, \mathrm{r}_{_\succ}$ satisfy
Eqs.~\eqref{eq:coboundary1}~-~\eqref{eq:coboundary4},
Eqs.~\eqref{eq:ybe1'} and \eqref{eq:ybe2'}.
\end{thm}
In view of symmetries brought out by equations characterizing
pre-anti-flexible bialgebras provided by the linear maps given
by Eq.~\eqref{eq:coboundary}, i.e.
Eqs.~\eqref{eq:coboundary1}~-~\eqref{eq:coboundary4},
Eqs.~\eqref{eq:ybe1'} and \eqref{eq:ybe2'}, we will consider pre-anti-flexible balgebras
generated by $\mathrm{r}\in A\otimes A$ in the following cases.
\begin{itemize}
\item[Case1]
\begin{eqnarray}\label{eq:particular1-r}
\mathrm{r}_{_{\prec}}=\mathrm{r},\;\; \mathrm{r}_{_{\succ}}=
-\sigma \mathrm{r}, \quad \mathrm{r}\in A\otimes A.
\end{eqnarray}
\begin{cor}
Let $(A, \prec, \succ)$ be a pre-anti-flexible algebra and
$\mathrm{r}\in A\otimes A$. Then the maps $\Delta_{_{\prec}}, \Delta_{_{\succ}}$
defined by Eq.~\eqref{eq:coboundary} with $\mathrm{r}_{_{\succ}}, \mathrm{r}_{_{\prec}}$ given by
Eq.~\eqref{eq:particular1-r} induce a pre-anti-flexible algebra structure on $A^*$
such that $(A, A^*)$ is a pre-anti-flexible bialgebra if and only if
$\mathrm{r}$ satisfies the following equations
\begin{subequations}
\begin{eqnarray}\label{eq:A}
\begin{array}{lll}
&&(\id\otimes(L_{_{\succ}}(x\prec y)+R_{_{\prec}}(y\succ x))+
((L_{_{\succ}}(y\prec x)+R_{_{\prec}}(x\succ y))\otimes \id)
(\mathrm{r}-\sigma \mathrm{r})\cr
&&-
(R_{_{\prec}}(y)\otimes L_{_{\succ}}(x) +
L_{_{\succ}}(y)\otimes R_{_{\prec}}(x) )
(\mathrm{r}-\sigma \mathrm{r})=0.
\end{array}
\end{eqnarray}
\begin{eqnarray}\label{eq:B}
&&0=((R_{_{\prec}}(x)\otimes R_{_{\prec}}(x)+
L_{_{\succ}}(x)\otimes L_{_{\succ}}(y))+(L_{_{\succ}}(y)\otimes L_{_{\succ}}(x) +
R_{_{\prec}}(y)\otimes R_{_{\prec}}(x))
)(\mathrm{r}-\sigma \mathrm{r})\cr
&&+
((R_{_{\prec}}(x)L_{_{\succ}}(y)+L_{_{\succ}}(x)R_{_{\prec}}(y))\otimes \id+
(\id\otimes(L_{_{\succ}}(x)R_{_{\prec}}(y)+
R_{_{\prec}}(x)L_{_{\succ}}(y))))(\sigma \mathrm{r}- \mathrm{r})
\end{eqnarray}
\begin{eqnarray}\label{eq:C}
&&((\id\otimes\id\otimes L_{\succ}(x))-
(R_{\prec}(x)\otimes \id\otimes\id)\sigma_{13}\circ flp
+(\id\otimes \id \otimes R_{\prec}(x))flp\cr&&-
(L_{\succ}(x)\otimes\id \otimes \id)flp\circ \sigma_{13} \circ flp)M_1(\mathrm{r})=0,
\end{eqnarray}
\begin{eqnarray}\label{eq:D}
\begin{array}{llll}
&&((\id \otimes\id\otimes L_{\cdot}(x))
+(\id \otimes \id \otimes R_{\cdot}(x))flp)M_1'(\mathrm{r})+R_1'(x)
\cr&&-((R_{\prec}(x)\otimes\id \otimes \id)
+(L_{\succ}(x)\otimes \id \otimes \id)flp)P_1'(\mathrm{r})=0,
\end{array}
\end{eqnarray}
\end{subequations}
where $x,y\in A$,
\begin{eqnarray*}
M_1(\mathrm{r})&=&-\mathrm{r}_{_{23}}\cdot \mathrm{r}_{_{21}}+
\mathrm{r}_{_{21}}\prec \mathrm{r}_{_{13}}+
\mathrm{r}_{_{31}} \succ \mathrm{r}_{_{23}}\cr
M_1'(\mathrm{r})&=&
\mathrm{r}_{_{32}}\prec \mathrm{r}_{_{21}}+
\mathrm{r}_{_{12}}\succ \mathrm{r}_{_{31}}-
\mathrm{r}_{_{31}}\cdot \mathrm{r}_{_{32}}-
\mathrm{r}_{_{32}}\succ (\mathrm{r}_{_{12}}- \mathrm{r}_{_{21}})-
(\mathrm{r}_{_{21}}- \mathrm{r}_{_{12}})\prec \mathrm{r}_{_{31}},\cr
P_1'(\mathrm{r})&=&\mathrm{r}_{_{32}}\prec \mathrm{r}_{_{21}}-
\mathrm{r}_{_{31}}\cdot \mathrm{r}_{_{32}}+
\mathrm{r}_{_{12}}\succ \mathrm{r}_{_{31}}-
(\mathrm{r}_{_{21}}- \mathrm{r}_{_{12}})\prec \mathrm{r}_{_{31}}, \; \; \mbox{ and }\cr
R_1'(x)&=&[(\id\otimes (R_{_{\prec}}(x)+
L_{_{\succ}}(x))\otimes \id)\mathrm{r}_{_{\prec, 32}}]
\succ(\mathrm{r}_{_{12}}-\mathrm{r}_{_{21}})\cr&-&
[((L_{_{\prec}}(x)+R_{_{\succ}}(x))\otimes\id \otimes \id)\mathrm{r}_{_{31}}]
\succ(\mathrm{r}_{_{21}}-\mathrm{r}_{_{12}}).
\end{eqnarray*}
\end{cor}
\begin{rmk}
It is straight to identify $M_1'(\mathrm{r})=
P_1'(\mathrm{r})-\mathrm{r}_{_{32}}\succ (\mathrm{r}_{_{12}}- \mathrm{r}_{_{21}})$ and
setting in addition
$\sigma_{_{123}}: A\otimes A\otimes A \rightarrow A\otimes A\otimes A$, by
$\sigma_{_{123}}(x\otimes y\otimes z)=z\otimes y\otimes x$, we have
\begin{eqnarray*}
M_1'(\mathrm{r})=\sigma_{_{123}}(M_1(\mathrm{r}))-
\mathrm{r}_{_{32}}\succ (\mathrm{r}_{_{12}}- \mathrm{r}_{_{21}})-
(\mathrm{r}_{_{21}}- \mathrm{r}_{_{12}})\prec \mathrm{r}_{_{31}}.
\end{eqnarray*}
Besides, if in addition $\mathrm{r}$ commutes then $R_1'(x)=0$ and
Eqs.~\eqref{eq:A} and \eqref{eq:B}
are satisfied and finally $\mathrm{r}$ satisfies the following equation
\begin{eqnarray}\label{eq:AFPYBE}
\mathrm{r}_{_{23}}\cdot \mathrm{r}_{_{12}}=
\mathrm{r}_{_{12}}\prec \mathrm{r}_{_{13}}+
\mathrm{r}_{_{13}} \succ \mathrm{r}_{_{23}}.
\end{eqnarray}
\end{rmk}
\item[Case2]
\begin{eqnarray}\label{eq:particular2-r}
\mathrm{r}_{_{\prec}}+\mathrm{r}_{_{\succ}}=0,\;
\mathrm{r}_{_{\succ}}=\mathrm{r}, \quad \mathrm{r}\in A\otimes A.
\end{eqnarray}
\begin{cor}
Let $(A, \prec, \succ)$ be a pre-anti-flexible algebra and
$\mathrm{r}\in A\otimes A$. Then the maps $\Delta_{_{\prec}}, \Delta_{_{\succ}}$
defined by Eq.~\eqref{eq:coboundary} with
$\mathrm{r}_{_{\succ}}, \mathrm{r}_{_{\prec}}$ given by
Eq.~\eqref{eq:particular2-r} induce a pre-anti-flexible algebra structure on $A^*$
such that $(A, A^*)$ is a pre-anti-flexible bialgebra if and only if
$\mathrm{r}$ satisfies the following equations for any $x,y\in A$
\begin{subequations}
\begin{eqnarray}\label{eq:A'}
(R_{_\prec}(y)\otimes L_{\cdot}(x)+L_{\succ}(y)\otimes R_{\cdot}(x))(\mathrm{r}-
\sigma \mathrm{r})=0,
\end{eqnarray}
\begin{eqnarray}\label{eq:B'}
(L_{\succ}(x)\otimes L_{\cdot}(y)-R_{\prec}(y)\otimes R_{\cdot}(x)-
L_{\succ}(y)\otimes L_{\cdot}(x)+R_{\prec}(x)\otimes R_{\cdot}(y))(\mathrm{r}-
\sigma \mathrm{r})=0,
\end{eqnarray}
\begin{eqnarray}\label{eq:C'}
(R_{_{\succ}}(y)\otimes L_{_{\succ}}(x) +
L_{_{\prec}}(y) \otimes R_{_{\prec}}(x) )(\mathrm{r}-\sigma \mathrm{r}),
\end{eqnarray}
\begin{eqnarray}\label{eq:D'}
(R_{_{\prec}}(x)\otimes R_{_{\succ}}(y)+
L_{_{\succ}}(x)\otimes L_{_{\prec}}(y)+
L_{_{\prec}}(y)\otimes L_{_{\succ}}(x)+
R_{_{\succ}}(y)\otimes R_{_{\prec}}(x))(\mathrm{r}-\sigma \mathrm{r})=0,
\end{eqnarray}
\begin{eqnarray}\label{eq:E'}
&&((\id\otimes\id\otimes L_{\succ}(x))-
(R_{\prec}(x)\otimes \id\otimes\id)\sigma_{13}\circ flp
+(\id\otimes \id \otimes R_{\prec}(x))flp
\cr&&-(L_{\succ}(x)\otimes\id \otimes \id)flp\circ
\sigma_{13} \circ flp)M_2(\mathrm{r})=0,
\end{eqnarray}
\begin{eqnarray}\label{eq:F'}
\begin{array}{llll}
&&((\id \otimes\id\otimes L_{\cdot}(x))
+(\id \otimes \id \otimes R_{\cdot}(x))flp)M_2'(\mathrm{r})
\cr&&-((R_{\prec}(x)\otimes\id \otimes \id)
+(L_{\succ}(x)\otimes \id \otimes \id)flp)P_2'(\mathrm{r})=0,
\end{array}
\end{eqnarray}
\end{subequations}
where
\begin{eqnarray*}
M_2(\mathrm{r})&=&-\mathrm{r}_{_{23}}\cdot \mathrm{r}_{_{12}}+
\mathrm{r}_{_{21}}\prec \mathrm{r}_{_{13}}+
\mathrm{r}_{_{13}} \succ \mathrm{r}_{_{23}},\cr
M_2'(\mathrm{r})&=&
-\mathrm{r}_{_{13}}\cdot \mathrm{r}_{_{23}}+
\mathrm{r}_{_{23}}\prec \mathrm{r}_{_{12}}+
\mathrm{r}_{_{21}}\succ \mathrm{r}_{_{13}},\cr
P_2'(\mathrm{r})&=&
-\mathrm{r}_{_{31}}\cdot \mathrm{r}_{_{23}}+
\mathrm{r}_{_{32}}\prec \mathrm{r}_{_{21}}+
\mathrm{r}_{_{21}}\succ \mathrm{r}_{_{31}}.
\end{eqnarray*}
\end{cor}
\begin{rmk}
Clearly, we have $P_2'(\mathrm{r})=\sigma_{_{123}}(M_2(\mathrm{r}))$
and if $\mathrm{r}$ commutes then
$P_2'(\mathrm{r})=M_2'(\mathrm{r})$ and Eqs.~\eqref{eq:A'}~-~\eqref{eq:D'}
are satisfied and finally
$\mathrm{r}$ satisfied Eq.~\eqref{eq:AFPYBE}.
\end{rmk}
\end{itemize}
We finally deduct the following pre-anti-flexible bialgebras
provided by a given $\mathrm{r}\in A\otimes A$ possessing
some internal symmetries while browsing $A\otimes A$.
\begin{cor}
Let $(A, \prec, \succ)$ be a pre-anti-flexible algebra and
consider symmetric element $\mathrm{r}\in A\otimes A$
satisfying Eq.~\eqref{eq:AFPYBE}. Then the linear maps
$\Delta_{_{\prec}}, \Delta_{_{\succ}}$
defined by Eq.~\eqref{eq:coboundary} with $\mathrm{r}_{_{\succ}}=\mathrm{r}$
and $\mathrm{r}_{_{\prec}}=-\mathrm{r}$ induce a
pre-anti-flexible algebra structure on $A^*$ such that
$(A, A^*)$ is a pre-anti-flexible bialgebra.
\end{cor}
\begin{defi}
Let $(A, \prec, \succ)$ be a pre-anti-flexible algebra and $\mathrm{r} \in A \otimes A$.
The Eq.~\eqref{eq:AFPYBE} is called the
\textbf{pre-anti-flexible Yang-Baxter equation} (PAFYBE) in $(A, \prec, \succ)$.
\end{defi}
\begin{rmk}
We due the notion of pre-anti-flexible Yang-Baxter equation in pre-anti-flexible algebras
as an analogue of the anti-flexible Yang-Baxter
equation in anti-flexible algebras (\cite{DBH3}) or
classical Yang-Baxter equation in Lie algebras (\cite{Drinfeld}) or
the associative Yang-Baxter equation in associative algebras (\cite{Aguiar, Bai_Double})
and $\mathcal{D}$-equation in dendriform algebras (\cite{Bai_Double}).
For no other specific reason than which showing that
both dendriform and pre-anti-flexible algebras possessing the
same shape of dual bimodules (see Remark~\ref{rmk_useful}~\eqref{dual-bimodule}),
to our amazement, $\mathcal{D}$-equation
in dendriform algebras and PAFYBE in pre-anti-flexible algebras
own the same form translated by Eq.~\eqref{eq:AFPYBE}.
This could making parallel with associative Yang-Baxter equation in associative algebras
and anti-flexible Yang-Baxter equation in anti-flexible algebras (\cite{DBH3}).
\end{rmk}
\section{Solutions of the pre-anti-flexible Yang-Baxter equation}\label{section5}
Let $A$ be a vector space. For any $\mathrm{r} \in A \otimes A$, $\mathrm{r}$
can be regarded as a linear map $\mathrm{r}:A^*\rightarrow A$
in the following way:
\begin{equation*}
\langle \mathrm{r}, u^*\otimes v^*\rangle=
\langle \mathrm{r}(u^*), v^* \rangle, \;\; \forall u^*, v^*\in A^*.
\end{equation*}
As PAFYBE in pre-anti-flexible algebras have the same form of the
$\mathcal{D}$-equation in dendriform algebra, we omitted proofs (too similar
to the case of dendriform algebra and related $\mathcal{D}$-equation) of the following
in which $(A, \prec, \succ)$ is a pre-anti-flexible algebra.
\begin{pro}\label{pro_pre_anti_flexible dual}
For a given $\mathrm{r}\in A\otimes A$, $r$ is a symmetric solution of
the PAFYBE in $A$ if and only if for any $x\in A$ and any $a, b\in A^*$
\begin{eqnarray}\label{eq:pre-anti-flexible-dual}
\begin{array}{llllllllll}
a\prec b&=&-R_{_{\succ}}^*(\mathrm{r}(a))b+L_{\cdot}^*(\mathrm{r}(b))a,\;
a\succ b= R^*_{\cdot}(\mathrm{r}(a))b-L_{_{\prec}}^*(\mathrm{r}(b))a,\;\cr
a\cdot b&=& a\prec b+a\succ b =
R^*_{_{\prec}}(\mathrm{r}(a))b+L_{_{\succ}}^*(\mathrm{r}(b))a, \;\cr
x\prec a&=& x\prec \mathrm{r}(a)+\mathrm{r}(R^*_{_{\succ}}(x)a)-R^*_{_{\succ}}(x)a,\;
x\succ a=x\succ \mathrm{r}(a)-\mathrm{r}(R^*_{\cdot}(x)a)+R_{\cdot}^*(x)a, \cr
x\cdot a&=&x\cdot \mathrm{r}(a)-R^*_{_{\prec}}(x)a+R^*_{_{\prec}}(x)a,\;
a\cdot x= \mathrm{r}(a)\cdot x-\mathrm{r}(L^*_{_{\succ}}(x)a)+L^*_{_{\succ}}(x)a, \cr
a\prec x&=&\mathrm{r}(a)\prec x- \mathrm{r}(L^*_{\cdot}(x)a)+L^*_{\cdot}(x)a, \;
a\succ x=\mathrm{r}(a)\succ x+ \mathrm{r}(L^*_{_{\prec}}(x)a)-L^*_{_{\prec}}(x)a.
\end{array}
\end{eqnarray}
\end{pro}
\begin{thm}
Consider a symmetric and non-degenerate element $\mathrm{r}\in A\otimes A$.
Then $\mathrm{r}$ is a solution of the PAFYBE in $A$ if and only if
the inverse homomorhpism $A^*\rightarrow A$ induced by $\mathrm{r}$
regarded as a bilinear form $\mathfrak{B}$ on $A$
(i.e. $\mathfrak{B}(x, y)= \langle \mathrm{r}^{-1}(x), y\rangle$,
for any $x,y\in A$) and satisfies
\begin{equation}\label{eq:2-cocycle}
\mathfrak{B}(x\cdot y, z)=\mathfrak{B}(y, z\prec x)+
\mathfrak{B}(x, y\succ z), \mbox{ for any } x,y,z\in A.
\end{equation}
\end{thm}
\begin{cor}
Let $\mathrm{r}\in A\otimes A$ be a symmetric solution of PAFYBE in $A$.
Suppose in addition
by "$\prec_{_{ A^*}}, \succ_{_{ A^*}}$" the pre-anti-flexible algebra structure on $A^*$
induced by $r$ via Proposition~\ref{pro_pre_anti_flexible dual}. Then we have for
any $a,b\in A^*$
\begin{equation*}
a\prec_{_{ A^*}} b=\mathrm{r}^{-1}(\mathrm{r}(a)\prec_{_ A} \mathrm{r}(b)), \;
a\succ_{_{ A^*}} b=\mathrm{r}^{-1}(\mathrm{r}(a)\succ_{_ A} \mathrm{r}(b)).
\end{equation*}
Therefore, $\mathrm{r}:A^*\rightarrow A$ is an isomorphism of pre-anti-flexible algebras.
\end{cor}
\begin{thm}
Let $(A, \prec, \succ)$ be a pre-anti-flexible algebra and
$\mathrm{r}\in A\otimes A$ symmetric.
Then, $\mathrm{r}$ is a solution of {PAFYBE} if and only if its satisfies
\begin{equation*}
\mathrm{r}(a)\cdot\mathrm{r}(b)=\mathrm{r}(R^*_{_{\prec}}(\mathrm{r}(a))b+
L^*_{_{\succ}}(\mathrm{r}(b))a ), \;\forall a,b \in A^*.
\end{equation*}
\end{thm}
Recall that a $\mathcal{O}$-operator related to the bimodule $(l, r, V )$
of an anti-flexible algebra $(A, \cdot)$ is a linear map $T :V\rightarrow A$ satisfies
\begin{equation*}
T (u) \cdot T (v) = T (l(T (u))v + r(T (v))u), \; \forall u, v \in V.
\end{equation*}
In addition, for a given pre-anti-flexible algebra $(A, \prec, \succ)$, according
to Proposition~\ref{prop_operation_bimodule_pre_anti_flexible}~\eqref{eq:one},
$(L_{_{\succ}},R_{_{\prec}}, A)$ is a bimodule of
its underlying anti-flexible algebra $aF(A)$. Furthermore, for any $x,y\in A$
\begin{eqnarray}\label{eq:o-operator}
\id(x)\cdot \id(y)= \id (L_{_{\succ}}(\id(x))y+R_{_{\prec}}(\id(y))x),
\end{eqnarray}
then $\id:A\rightarrow A$ is an $\mathcal{O}$-operator of $aF(A)$ associated to
the bimodule $(L_{_{\succ}}, R_{_{\prec}}, A)$.
\begin{cor}
Consider a symmetric element $\mathrm{r}\in A\otimes A$.
Then $\mathrm{r}$ is a solution
of PAFYBE in $A$ if and only if it is an $\mathcal{O}$-operator
of the underlying anti-flexible $aF(A)$ associated to the bimodule
$(R^*_{_{\prec}}, L^*_{_{\succ}}, A^*)$. Furthermore, there is a pre-anti-flexible
algebra structure on $A^*$ given by
\begin{eqnarray*}
a\prec b=L^*_{_{\succ}}(\mathrm{r}(b))a;\;\;
a\succ b=R^*_{_{\prec}}(\mathrm{r}(a))b;\; \forall a,b\in A^*,
\end{eqnarray*}
which is the same of that associated to the pre-anti-flexible bialgebra derived on $A^*$
by Eq~\eqref{eq:pre-anti-flexible-dual}. If in addition $\mathrm{r}$ is
non degenerate, then there is a new compatible pre-anti-flexible algebraic
structure given on $A$ by
\begin{eqnarray*}
x\prec'y=\mathrm{r}(L^*_{_{\succ}}(y)\mathrm{r}^{-1}(x)),\;
x\succ'y=\mathrm{r}(R^*_{_{\prec}}(x)\mathrm{r}^{-1}(y)), \;\forall x,y\in A,
\end{eqnarray*}
which is the pre-anti-flexible algebra structure given by
\begin{eqnarray*}
\mathfrak{B}(x\prec'y, z)=\mathfrak{B}(x, y\ast z), \;
\mathfrak{B}(x\succ'y, z)=\mathfrak{B}(y, z\cdot x), \;\forall x,y,z\in A,
\end{eqnarray*}
where $\mathfrak{B}$ is given by $
\mathfrak{B}(x,y)=\langle \mathrm{r}^{-1}(x), y\rangle$ for
any $x,y\in A$ and satisfies Eq.~\eqref{eq:2-cocycle}.
\end{cor}
Taking into account \cite[Proposition 2.7.]{DBH3} we have
\begin{thm}
Let $(A, \cdot)$ be an anti-flexible algebra, $(l,r, V)$ a bimodule
of $(A, \cdot)$ and $T:V\rightarrow A$ an $\mathcal{O}$-operator associated
to $(l,r, V)$. Then $\mathrm{r}=T+\sigma T$ is a symmetric solution of
the PAFYBE in $T(V)\ltimes_{r^*, 0,0, l^*} V^*$, where
$T(V)\subset A$ is endowed with a pre-anti-flexible given by
for any $u,v\in V, $
\begin{eqnarray*}
T(v)\prec T(v)=T(r(T(v))u), \;
T(u)\succ T(v)=T(l(T(u))v),
\end{eqnarray*}
such that $(r^*, 0,0, l^*, T(V)^*)$ is its associated bimodule
and underlying anti-flexible algebra is a sub-algebra of $A$, and finally
$T$ can be identified with an element in
$T(V)\otimes V^*\subset(T(V)\ltimes_{r^*, 0,0, l^*} V^*)\otimes
T(V)\ltimes_{r^*, 0,0, l^*} V^*$.
\end{thm}
Considering the above theorem,
Proposition~\ref{prop_operation_bimodule_pre_anti_flexible}~\eqref{eq:one} and
Eq.~\eqref{eq:o-operator}, we have
\begin{cor}
Let $(A, \prec, \succ)$ be a $n$-dimensional pre-anti-flexible algebra. Then the element
\begin{eqnarray*}
\mathrm{r}=\sum_{i}^{n} (e_i\otimes e_i^*+e_i^*\otimes e_i)
\end{eqnarray*}
is a symmetric solution of PAFYBE in
$A\ltimes_{R_{_{\prec}}^*, 0,0, L_{_{\succ}}^*} A^*$,
where $\{e_1, \cdots , e_n\}$ it a basis of $A$ and
$\{e^*_1, \cdots , e^*_n\}$ its associated dual basis.
Furthermore, $\mathrm{r}$ is non degenerate and it
induced bilinear form $\mathfrak{B}$ on
$A\ltimes_{R_{_{\prec}}^*, 0,0, L_{_{\succ}}^*} A^*$
is given by
\begin{eqnarray*}
\mathfrak{B}(x+a, y+b)=\langle x, b\rangle+\langle y, a\rangle, \;
\forall x,y\in A, a, b\in A^*.
\end{eqnarray*}
\end{cor}
\noindent
{\bf Acknowledgments.}
The author thanks Professor C. Bai for helpful discussions and
his encouragement, and Nankai ZhiDe Foundation.
\end{document} |
\begin{document}
\title{The Series Product for Gaussian Quantum Input Processes } \author{John E. Gough \\
Aberystwyth University, SY23 3BZ, Wales, United Kingdom \\ e-mail: jug@aber.ac.uk \\ [2ex] Matthew R. James \\ Australian National University, Canberra, ACT 0200, Australia \\ e-mail: Matthew.James@anu.edu.au} \maketitle
\begin{abstract} We present a theory for connecting quantum Markov components into a network with quantum input processes in a Gaussian state (including thermal and squeezed). One would expect on physical grounds that the connection rules should be independent of the state of the input to the network. To compute statistical properties, we use a version of Wicks' Theorem involving fictitious vacuum fields (Fock space based representation of the fields) and while this aids computation, and gives a rigorous formulation, the various representations need not be unitarily equivalent. In particular, a naive application of the connection rules would lead to the wrong answer. We establish the correct interconnection rules, and show that while the quantum stochastic differential equations of motion display explicitly the covariances (thermal and squeezing parameters) of the Gaussian input fields We introduce the Wick-Stratonovich form which leads to a way of writing these equations that does not depend on these covariances and so corresponds to the universal equations written in terms of formal quantum input processes. We show that a wholly consistent theory of quantum open systems in series can be developed in this way, and as required physically, is universal and in particular representation-free. \end{abstract}
\noindent \textbf{Keywords:} Gaussian Wick Theorem, Wick-Stratonovich Form, Quantum Gaussian Feedback Networks.
\section{Introduction}
The quantum input-output theory has had an immense impact on quantum optics, and in recent years has extended to opto-mechanical systems and beyond. The prospect of routing the inputs through a network, or indeed using feedback has lead to a burgeoning field of quantum feedback control \cite{I_12}-\cite {JAM_14}. The development of a systems engineering approach to quantum technology has benefited from having a systematic framework in which traditional open quantum systems models can be combined according to physical connection architectures.
The initial work on how to cascade two quantum input-output systems can be traced back to Gardiner \cite{Gardiner_cascade} and Carmichael \cite {Carmichael_cascade}. More generally, the authors have introduced the theory of \textit{Quantum Feedback Networks} (QFN) which generalizes this to include cascading, feedback, beam-splitting and general scattering of inputs, etc., \cite{GJ-QFN}, \cite{GJ-Series}. One of the basic constructs is the series product which gives the instantaneous feedforward limit of two components connected in series via quantum input processes: in fact, the systems need not necessarily be distinct and the series product generalizes cascading by allowing for feedback. The original work was done for input processes where the input fields where in the Fock vacuum field state. A generalization to squeezed fields and squeezing components has been given \cite{GJN_squeeze}, however this was restricted to the case of linear coupling and dynamics: there it was shown that the resulting transform analysis could be applied in a completely consistent manner. More recent work has shown that non-classical states for the input fields, such as shaped single-photon or multi-photon states, or cat states of coherent fields, may in principle be generated from signal models \cite{GJNC}, \cite {GZ_filter} - that is, where a field in the Fock vacuum state was passed through an ancillary dynamical system (the signal generator) which is then to be cascaded to the desired system. Quantum feedback network (QFN) theory concerns the interconnection of open quantum systems. The interconnections are mediated by quantum fields in the input-output theory, \cite {GarZol00,GJ-QFN,GJ-Series}. The idea is that an output from one node is fed back in as input to another (not necessarily distinct) node, the simplest case being the cascade connection (e.g., light exiting one cavity being directed into another). The components are specified by Markovian models determined by SLH parameters which describe the self-energy of the system and how the system interacts with the fields (via idealized Jaynes-Cummings type interactions and scattering).
Here we turn to the problem of the general class of Gaussian states for quantum fields. This includes thermal fields, and of course squeezed fields. In principle, these may be approximated as the output of a degenerate parametric amplifier (DPA) driven by vacuum input, see \cite{GarZol00}. In a sense, we have that a singular DPA may serve is the appropriate signal generator to modify a vacuum field into a squeezed field before passing into a given network. We will exploit this in the paper, however, we will have to pay attention to the operator ordering problem when inserting these approximations into quantum dynamical equations of motion and input-output relations.
The programme turns out to be rather more involved than one might expect at first glance. It is always possible to represent a collection of $d$ Gaussian fields using $2d$ vacuum fields (a Bogoliubov transformation!) and one might hope that the corresponding connection rules applied to the representation in terms of vacuum fields would agree with the intuitive rules one would desire. This turns out not to be the case, and the various feedback constraints cannot be naively applied to the representing fields: the reason is that the representations are a linear combination of creation and annihilation operators for the representing vacuum fields, and we have broken the Wick ordered form of the original equations.
If applied naively, the series product would predict a contribution to the global network model that depended on the covariance parameters of the state. From the physical point of view, this ought to be spurious. In comparison with classical analog linear electronics, we see that the components (e.g. resistors, capacitors, inductors) are described by impedances. When components are interconnected to form a network, the network may be described by an equivalent impedance, derived through an application of Kirchhoff laws. Impedances do not depend on the applied currents or voltages, and are therefore intrinsic to the device or network. Similarly the rules for connecting a quantum feedback network should be intrinsic, and not depend on the state of the noise fields.
\section{Background and Problem statement}
Let us begin in the concrete setting of the quantum stochastic calculus of Hudson and Parthasarathy \cite{HP} with a fixed initial space $\mathfrak{h} _{0}$ and a noise space that is the (Bose) Fock space over $\mathbb{C}^{d}$ -valued $L^{2}$-functions on the time interval $[0,\infty )$. In the language of Hudson and Parthasarathy, we have a multiplicity space of dimension $d$ and we select an orthonormal basis which determines $d$ channels. We denote by $A_{k}\left( t\right) $, $A_{k}\left( t\right) ^{\ast }$, and $\Lambda _{jk}\left( t\right) $ the processes of annihilation, creation (for channel $j$) and scattering (from channel $k$ to channel $j$). In the following, we shall introduce an Einstein summation convention for repeated channel indices. We will deal with the class of quantum stochastic integrals processes satisfying the appropriate conditions of local integrability, square-integrability \cite{HP} without explicit reference. We have for instance the QSDE \begin{equation} dX\left( t\right) =x_{jk}\left( t\right) d\Lambda _{jk}\left( t\right) +x_{j0}\left( t\right) dA_{j}\left( t\right) ^{\ast }+x_{0k}dA_{k}\left( t\right) +x_{00}\left( t\right) dt \end{equation} where the coefficients are adapted and the increments are (quantum) It\={o}. We have the quantum It\={o} product formula \begin{equation} d\left( X\left( t\right) Y\left( t\right) \right) =dX\left( t\right) \,Y\left( t\right) +X\left( t\right) \,dY\left( t\right) +dX\left( t\right) \,dY\left( t\right) \end{equation} where the It\={o} correction comes from the quantum It\={o} table \cite{HP} \begin{eqnarray} d\Lambda _{jk}\left( t\right) \,d\Lambda _{lm}\left( t\right) &=&\delta _{kl}d\Lambda _{jm}\left( t\right) ,\quad d\Lambda _{jk}\left( t\right) \,dA_{l}\left( t\right) ^{\ast }=\delta _{kl}dA_{j}\left( t\right) ^{\ast }, \nonumber \\ dA_{k}\left( t\right) \,d\Lambda _{lm}\left( t\right) &=&\delta _{kl}dA_{m}\left( t\right) ,\quad dA_{j}\left( t\right) \,dA_{k}=\delta _{jk}dt, \end{eqnarray} with all other products of the fundamental increments vanishing.
\begin{definition}{Definition} The Stratonovich integral is defined algebraically via \begin{eqnarray} X\left( t\right) \circ dY\left( t\right) &=&X\left( t\right) dY\left( t\right) +\frac{1}{2}dX\left( t\right) \,dY\left( t\right) \\ dX\left( t\right) \circ Y\left( t\right) &=&dX\left( t\right) \,Y\left( t\right) +\frac{1}{2}dX\left( t\right) \,dY\left( t\right) . \end{eqnarray} \end{definition}
This turns out to be equivalent to a mid-point rule \cite{Chebotarev}.
If we consider the QSDE $dU\left( t\right) =-idE\left( t\right) \circ U\left( t\right) $, with $U\left( 0\right) $ the identity and $E\left( t\right) =E_{jk}\Lambda _{jk}\left( t\right) +E_{j0}B_{j}\left( t\right) ^{\ast }+E_{0k}B_{k}\left( t\right) +E_{00}$ a self-adjoint quantum stochastic integral process, then we may convert to the It\={o} form to get \begin{equation} dU\left( t\right) =\bigg\{\left( S_{jk}-\delta _{jk}\right) d\Lambda _{jk}\left( t\right) +L_{j}dA_{j}^{\ast }\left( t\right) -L_{j}^{\ast }S_{jk}dA_{k}\left( t\right) +Kdt\bigg\}\,U\left( t\right) , \end{equation} where (setting $E_{\ell \ell }$ to be the $d\times d$ matrix with entries $ E_{jk\text{)}}$ \begin{equation} S=\left[ \begin{array}{ccc} S_{11} & \cdots & S_{1d} \\ \vdots & \ddots & \vdots \\ S_{d1} & \cdots & S_{dd} \end{array} \right] =\frac{I-\frac{i}{2}E_{\ell \ell }}{I+\frac{i}{2}E_{\ell \ell }} \end{equation} is called the matrix of scattering coefficients unitary (that is, $ S_{jk}^{\ast }S_{jl}=\delta _{kl}=S_{lj}S_{kj}^{\ast }$), \begin{equation} L=\left[ \begin{array}{c} L_{1} \\ \vdots \\ L_{d} \end{array} \right] =\frac{i}{I+\frac{i}{2}E_{\ell \ell }}\left[ \begin{array}{c} E_{10} \\ \vdots \\ E_{d0} \end{array} \right] \end{equation} which is the column vector of coupling operators, and \begin{equation} K=-\frac{1}{2}L_{k}^{\ast }L_{k}-iH, \label{eq:K_Fock} \end{equation} where $H$ is the Hamiltonian ($H^{\ast }=H=E_{00}+\frac{1}{2}E_{0j}\left[ \text{Im}\left\{ \frac{1}{I+\frac{i}{2}E_{\ell \ell }}\right\} \right] _{jk}E_{k0}$). For simplicity we will assume that the terms $S_{jk},L_{j}$ and $H$ are bounded operators on the system Hilbert space $\mathfrak{h}_{0}$.
We generally refer to the triple $\mathbf{G}\sim \left( S,L,H\right) $ as the Hudson-Parthasarathy parameters, or informally the ``SLH'' parameters specifying the model. The unitary process they generate may be denoted as $ U^{\mathbf{G}}\left( t\right) $ if we wish to emphasize the dependence on these parameters.
For $X$ an operator of the initial space, we introduce $j_{t}\left( X\right) =U\left( t\right) ^{\ast }X\,U\left( t\right) $ and from the quantum It\={o} rule obtain the Heisenberg-Langevin equation \begin{equation} dj_{t}(X)=j_{t}\left( \mathcal{L}_{jk}X\right) \,d\Lambda _{jk}+j_{t}( \mathcal{L}_{j0}X)\,dA_{j}^{\ast }+j_{t}(\mathcal{L}_{0k}X)\,dA_{k}+j_{t}( \mathcal{L}_{00}X)dt \end{equation} where \begin{equation} \mathcal{L}_{jk}X=S_{lj}^{\ast }XS_{lk}-\delta _{jk}X,\quad \mathcal{L} _{j0}X=S_{lj}^{\ast }\left[ X,L_{l}\right] ,\quad \mathcal{L}_{0k}X=\left[ L_{l}^{\ast },X\right] S_{lk} \end{equation} and the Lindblad generator $\mathcal{L}_{00} \equiv \mathcal{L}$ is \begin{equation} \mathcal{L} X=\frac{1}{2}L_{k}^{\ast }\left[ X,L_{k}\right] +\frac{1}{2} \left[ L_{k}^{\ast },X\right] L_{k}-i\left[ X,H\right] . \label{eq:Linblad_Fock} \end{equation} The maps $\mathcal{L}_{\alpha \beta }$ are known as the \textit{Evans-Hudson super-operators}. We shall occasionally write $j_{t}^{\mathbf{G}}\left( X\right) $ for the dynamical flow of $X$ when we wish to emphasis the dependence on the SLH parameters $\mathbf{G}$.
Let us now write the input processes as $A_{\mathrm{in},j}\left( t\right) =A_{j}\left( t\right) $ and introduce the output processes as $A_{\mathrm{out },j}\left( t\right) =U\left( t\right) ^{\ast }A_{\mathrm{in},j}\left( t\right) U\left( t\right) $ then from the quantum It\={o} rule we see that \begin{equation} dA_{\mathrm{out},j}\left( t\right) =j_{t}\left( S_{jk}\right) \,dA_{\mathrm{ in},k}\left( t\right) +j_{t}\left( L_{l}\right) \,dt. \end{equation}
\subsection{Thermal Fields}
Considering the single channel $\left( d=1\right) $ case for the moment, we may introduce non-Fock quantum stochastic processes as follows \cite{HL}. For $n>0$, we set \begin{equation} B\left( t\right) =\sqrt{n+1}A_{+}\left( t\right) +\sqrt{n}A_{-}\left( t\right) ^{\ast },\quad \tilde{B}\left( t\right) =\sqrt{n}A_{+}\left( t\right) +\sqrt{n+1}A_{-}\left( t\right) ^{\ast } \end{equation} which are canonical fields on the Fock space with a pair of channels labeled as $k=\pm $. In fact, the map $\left( A_{+},A_{-}\right) \mapsto \left( B,\tilde{B}\right) $ is a Bogoliubov transformation with inverse \begin{equation} \left[ \begin{array}{c} A_{+} \\ A_{-} \end{array} \right] =\left[ \begin{array}{cc} \sqrt{\left( n+1\right) } & -\sqrt{n} \\ -\sqrt{n} & \sqrt{\left( n+1\right) } \end{array} \right] \left[ \begin{array}{c} B \\ \tilde{B} \end{array} \right] . \end{equation}
This is of course based on an Araki-Woods representation of the fiels \cite{AW63}. As is well known, these transformation cannot be implemented unitarily. However, from a quantum optics point of view, devices transforming or even squeezing fields in this manner are frequently considered, and it is useful to imagine a hypothetical device - a Bogoliubov box - performing such a canonical transformation on our idealized fields.
Ignoring the second process $\tilde{B}$, we obtain the non-Fock quantum It\={o} table \begin{equation} dB\left( t\right) dB\left( t\right) ^{\ast }=\left( n+1\right) dt,\quad dB\left( t\right) ^{\ast }dB\left( t\right) =ndt. \end{equation} It problematic (read impossible) to incorporate a scattering process $ \Lambda $ into this table. We refer to $B$ as non-Fock quantum noise.
We need to drop the scattering term from the unitary evolution equation, i.e. set $S\equiv I$, and with \begin{equation} L=\left[ \begin{array}{c} L_{+} \\ L_{-} \end{array} \right] =\left[ \begin{array}{c} \sqrt{n+1}L \\ -\sqrt{n}L^{\ast } \end{array} \right] \end{equation} we have \begin{eqnarray} dU\left( t\right) &=&\bigg\{LdB\left( t\right) ^{\ast }-L^{\ast }dB\left( t\right) +K^{\text{th}}dt\bigg\}\,U\left( t\right) \nonumber \\ &=&\bigg\{L_{j}dA_{j}^{\ast }\left( t\right) -L_{j}^{\ast }dA_{j}\left( t\right) +Kdt\bigg\}\,U\left( t\right) , \end{eqnarray} where \begin{equation} K^{\text{th}}=-\frac{1}{2}L_{+}^{\ast }L_{+}-\frac{1}{2}L_{-}^{\ast }L_{-}-iH=-\frac{n+1}{2}L^{\ast }L-\frac{n}{2}LL^{\ast }-iH. \end{equation} For the flow, we need that the Hudson-Evans super-operator associated with the scattering terms are trivial. This is the case when the entries of the scattering matrix $S$ are (e.g. scalars) commuting with operators of the initial space, but we can get away without assuming that $\left[ \begin{array}{cc} S_{++} & S_{+-} \\ S_{-+} & S_{--} \end{array} \right] $ is the identity. By inspection we find that flow equation will take the form \begin{equation} dj_{t}\left( X\right) =j_{t}\left( \left[ X,L\right] \right) S^{\ast }dB\left( t\right) ^{\ast }+j_{t}\left( \left[ L,X\right] \right) SdB\left( t\right) +j_{t}\left( \mathcal{L}^{\text{th}}X\right) dt \end{equation} if and only if we take $\left[ \begin{array}{cc} S_{++} & S_{+-} \\ S_{-+} & S_{--} \end{array} \right] \equiv \left[ \begin{array}{cc} S & 0 \\ 0 & S^{\ast } \end{array} \right] $ - otherwise we obtain the other noise $\tilde{B}$ - and in which case the Lindbladian is \begin{eqnarray} \mathcal{L}^{\text{th}}X &=&\frac{1}{2}\left[ L_{+}^{\ast },X\right] L_{+}+ \frac{1}{2}L_{+}^{\ast }\left[ X,L_{+}\right] +\frac{1}{2}\left[ L_{-}^{\ast },X\right] L_{-}+\frac{1}{2}L_{-}^{\ast }\left[ X,L_{-}\right] -i[X,H] \nonumber \\ &=&\frac{n+1}{2}\left\{ \left[ L^{\ast },X\right] L+L^{\ast }\left[ X,L \right] \right\} +\frac{n}{2}\left\{ \left[ L^{\ast },X\right] L+L^{\ast } \left[ X,L\right] \right\} -i[X,H]. \end{eqnarray}
\subsection{The Series Product - Vacuum Inputs}
In \cite{GJ-Series} the authors introduce a rule for combining SLH models in series. For instance, we have the output of the $\mathbf{G}_{\mathscr{A}}\sim \left( S_{\mathscr{A}},L_{\mathscr{A} },H_{\mathscr{A}}\right) $ fed instantaneously as input to $\mathbf{G}_{ \mathscr{B}}\sim \left( S_{\mathscr{B}},L_{\mathscr{B}},H_{\mathscr{B} }\right) $ and it is shown that this is equivalent to the model generated by \begin{eqnarray} \mathbf{G}_{\mathscr{B}}\vartriangleleft \mathbf{G}_{\mathscr{A}} &\sim &\left( S_{\mathscr{A}},L_{\mathscr{A}},H_{\mathscr{A}}\right) \vartriangleleft \left( S_{\mathscr{B}},L_{\mathscr{B}},H_{\mathscr{B} }\right) \notag \\ &=&\bigg(S_{\mathscr{B}}S_{\mathscr{A}},L_{\mathscr{B}}+S_{\mathscr{B}}L_{ \mathscr{A}},H_{\mathscr{A}}+H_{\mathscr{B}}+\mathrm{Im}\left\{ L_{ \mathscr{B}}^{\ast }S_{\mathscr{B}}L_{\mathscr{A}}\right\} \bigg). \label{eq:series_prod} \end{eqnarray} Here $\mathrm{Im}\left\{ C\right\} $ means $\frac{1}{2i}\left( C-C^{\ast }\right) $.We note that every model may be written as a purely scattering component and a non-scattering component in series, since we have the law $ (S,L,H)=(I,L,H)\vartriangleleft (S,0,0)$.
We should remark that it is not necessary to view the two systems $ \mathscr{A}$ and $\mathscr{B}$ as separate systems - specifically, in the derivation of the series product\cite{GJ-Series} it is not assumed that the $ \mathscr{A}$ and $\mathscr{B}$ operators need commute!
\subsection{Statement of the Problem}
If we wish to have a pair of systems $\mathscr{A}$ and $\mathscr{B}$ (both accepting $d$ inputs) in series, then we obtain an equivalent Markov model in the limit where the intervening connection is instantaneous. Let $L_{ \mathscr{A}}$ be the column of the $d$ operators $L_{\mathscr{A},k}$, $ k=1,\cdots , d$, and similar for system $\mathscr{B}$. The series product says that the equivalent model has coupling $L_{\mathscr{A}}+L_{\mathscr{B}}$ and Hamiltonian \begin{eqnarray} H_{\mathscr{A}}+H_{\mathscr{B}}+\mathrm{Im}\left\{ L_{ \mathscr{B}}^{\ast }L_{\mathscr{A}}\right\} . \end{eqnarray}
Suppose we were to apply the series product to two systems with the same single thermal input $B$, and try and describe this as a series connection using the two vacuum inputs $A_{+}$ and $A_{-}$. Naively applying the series product to the construction in the $A_{\pm }$ format leads to the correct rule $L_{\mathscr{A}}+L_{\mathscr{B}}$ for the coupling terms, but \begin{equation} H_{\mathscr{A}}+H_{\mathscr{B}}+\mathrm{Im}\left\{ L_{\mathscr{B}}^{\ast }L_{ \mathscr{A}}\right\} +n\mathrm{Im}\left[ L_{\mathscr{B}}^{\ast },L_{ \mathscr{A}}\right] . \end{equation} We have picked up an $n$-dependent term. For pure cascading, the systems $ \mathscr{A}$ and $\mathscr{B}$ are distinct and so $\left[ L_{\mathscr{B} }^{\ast },L_{\mathscr{A}}\right] =0$. However, the series product should also apply to the situation where the systems share degrees of freedom. In such cases the additional term is physically unreasonable as it depends on the state of the noise.
It is not immediately obvious what is wrong with the construction. Going to the double Fock vacuum representations and then using the vacuum version of the series product would seem a reasonable thing to do. However, a fully quantum description would involve the $\tilde{B}$ fields as well, and at a schematic level this would involve one or more Bogoliubov boxes - something conspicuously. We will give the correct procedure in this paper.
\section{Multi-Dimensional Gaussian Processes}
\subsection{Notation}
We will use the symbol $\triangleq $ to signify a defining equation. We will denote the operations of complex conjugation, hermitean conjugation, and more generally adjoint by *. For $X=[x_{ij}]$ an $n\times m$ array with complex-valued entries, or more generally operator-valued entries, we write $ X^{\ast }$ for the $m\times n$ array obtained by transposition of the array and conjugation of the entries: that is the $ij$ entry is $x_{ji}^{\ast }$. The transpose alone will be denotes as $X^{\top }$, that is the $m\times n$ array with $ij$ entry $x_{ji}$. We will also use the notation $ X^{\#}=(X^{\top })^{\ast }$ which is the $n\times m$ array with $ij$ entry $ x_{ij}^{\ast }$.
\subsection{Finite Dimensional Gaussian States}
\label{sec:FD_Gaussian}Let $a_{1},\cdots ,a_{d}$ be the annihilation operators for $d$ independent oscillators. We consider a mean zero Gaussian state with second moments
\begin{equation} n_{ij}=\langle a_{i}^{\ast }a_{j}\rangle ,\quad m_{ij}=\langle a_{i}a_{j}\rangle , \end{equation} which we assemble into a hermitean $d\times d$ matrix, $N$, with entries $ n_{ji}^{\ast }=n_{ij}$, and a symmetric matrix $M$ is the $d\times d$ matrix with entries $m_{ij}=m_{ji}$. The \emph{covariance matrix} is \begin{equation} F=\left[ \begin{array}{cc} I+N^{\top } & M \\ M^{\ast } & N \end{array} \right] . \end{equation} In order to yield mathematically correct variances, we must have both $F$ and $N$ positive. The vacuum state is characterized by having $N=M=0$, that is \begin{equation} F_{\mathrm{vac}}\equiv \left[ \begin{array}{cc} I & 0 \\ 0 & 0 \end{array} \right] . \label{eq:cov} \end{equation}
The covariance matrix $F$ defined by (\ref{eq:cov}) must be positive semi-definite, as will be the matrices $N$ and $I+N^{\top }$. We must also have ran$\left( M\right) \subseteq $ran$\left( I+N^{\top }\right) $ and $ MN^{-}M^{\ast }\leq I+N$, where $N^{-}$ is the Moore-Penrose inverse of $N$.
A linear transformation of the form \begin{eqnarray} \tilde{a}=Ua+Va^{\#}, \end{eqnarray} that is $\tilde{a}_{j}=\sum_{k}\left( U_{jk}a_{k}+V_{jk}a_{k}^{\ast }\right) $, is called a \textit{Bogoliubov transformation} if we have again the canonical commutation relations for the primed operators.
The transformation $\tilde{a}=Ua+Va^{\#}$ is Bogoliubov if and only if the following identities hold $UU^{\ast }=I+VV^{\ast },\quad UV^{\top }=VU^{\top }.$
This is easily established by inspection, as are the following.
\begin{lemma}{Lemma} Let $\tilde{a}=Ua+Va^{\#}$ be a Bogoliubov transformation, then the covariance matrix for $\tilde{a}$ is \begin{eqnarray} \tilde{F}=WFW^{\dag } \end{eqnarray} where $W=\Delta \left( U,V\right) $. In particular, the new matrices are \begin{eqnarray} N^{\prime } &=&V^{\#}V^{\top }+V^{\#}N^{\top }V^{\top }+U^{\#}M^{\ast }V^{\top } +V^{\#}MU^{\top }+U^{\#}NU^{\top }, \nonumber\\ M^{\prime } &=&UV^{\top }+UN^{\top }V^{\top }+VM^{\ast }V^{\top } +UM^{\ast }U^{\top }+VNU^{\top }. \end{eqnarray} \end{lemma}
\begin{lemma}{Lemma} \label{Prop:W_vac}
Given $a_{\mathrm{vac}}$ with the choice of the vacuum state, the Bogoliubov transformation $a=Ua_{\mathrm{vac}}+Va_{\mathrm{vac}}^{\#}$ leads to operators with the covariance matrix \begin{eqnarray} F=WF_{\mathrm{vac}}W^{\ast }=\left[ \begin{array}{cc} I+N^{\top } & M \\ M^{\ast } & N \end{array} \right] \end{eqnarray} where $W=\Delta \left( U,V\right) $ and \begin{eqnarray} N=V^{\#}V^{\top },\quad M=UV^{\top }. \end{eqnarray} \end{lemma}
We note that the determinant of the covariance matrix is preserved under Bogoliubov transformations. In particular, if we have $F=WF_{\mathrm{vac} }W^{\ast }$, as in the last Proposition, then $F$ must also be singular. This means that if we wish to obtain a given covariance matrix $F$ for $d$ modes by a Bogoliubov transformation of vacuum modes, we will typically need a larger number $D$ of these modes with $F$ being a sub-block of a transformed matrix $WF_{\mathrm{vac}}W^{\ast }$. The example in the Theorem shows that in order to obtain the $d=1$ covariance \begin{eqnarray} F=\left[ \begin{array}{cc} 1+n & 0 \\ 0 & n \end{array} \right] \end{eqnarray} we need a Bogoliubov transformation of $D=2$ modes. We remark that we may obtain the covariance \begin{eqnarray} F=\left[ \begin{array}{cc} 1+n & m \\ m^{\ast } & n \end{array} \right] , \end{eqnarray}
with the constraint $\left| m\right| ^{2}\leq n\left( n+1\right) $ ensuring positivity, from 2 vacuum modes via \cite{HHKKR02,G_QWN_ME} \begin{eqnarray}
\tilde{a}=\sqrt{n+1-\frac{1}{n}\left| m\right| ^{2}}a_{1}+\sqrt{n} a_{2}^{\ast }+\frac{m}{\sqrt{n}}a_{2}. \label{eq:bog_m} \end{eqnarray}
The maximal case $\left| m\right| ^{2}=n\left( n+1\right) $ may be obtained from a \textit{single} mode $a_{1}$ via $a=\sqrt{n+1}a_{1}+e^{i\theta }\sqrt{ n}a_{1}^{\ast }$ where $m\equiv \sqrt{n\left( n+1\right) }e^{i\theta }$.
\subsection{Quantum Ito Calculus: Gaussian Noise}
One would like to extend this to non-vacuum inputs, in particular, those with general flat power Gaussian states for the noise. (We restrict to a single noise channel for transparency but the generalization is straightforward enough.) It is possible to construct noises having the following quantum It\={o} table \begin{eqnarray} dB_{i}dB_{j}^{\ast } &=&\left( n_{ji}+\delta _{ij}\right) dt,\quad dB_{i}^{\ast }dB_{j}=n_{ij}dt, \notag \\ dB_{i}dB_{j} &=&m_{ij}dt,\quad dB_{i}^{\ast }dB_{j}^{\ast }=m_{ji}^{\ast }dt, \label{eq:table_non_Fock} \end{eqnarray} where $N=[n_{ij}]$ and $M=[m_{ij}]$ have the same properties and constraints as introduced above.
In reality, we are assuming that the fields $B_{j}\left( t\right) $ correspond to a representation on a double Fock space, say, \begin{equation} B(t)=U\left[ \begin{array}{c} A_{+}\left( t\right) \otimes I \\ I\otimes A_{-}(t) \end{array} \right] +V\left[ \begin{array}{c} A_{+}\left( t\right) ^{\#}\otimes I \\ I\otimes A_{-}(t)^{\#} \end{array} \right] \end{equation} where $A_{k}\left( t\right) =\left[ \begin{array}{c} A_{k,1}\left( t\right) \\ \vdots \\ A_{k.d}\left( t\right) \end{array} \right] $ are copies of the Fock fields encountered above, and where $N=V^{\#}V,M=UV^{\top }$ as in Proposition \ref{Prop:W_vac}.
The underlying mathematical problem is that we are trying to implement a canonical transformation that is not inner \cite{partha,Shale,DG}- specifically the various representations for different pairs $\left( N,M\right) $ are not unitarily equivalent.
Instead we must restrict to QSDE models in the general Gaussian case which are driven by $B$ and $B^{\ast }$ only. We in fact find the class of QSDEs \begin{equation} dU\left( t\right) =\left\{ L_{k}dB_{k}^{\ast }\left( t\right) -L_{k}^{\ast }dB_{k}\left( t\right) +K^{\left( N,M\right) }dt\right\} \,U\left( t\right) \label{eq:QSDE_non_Fock1} \end{equation} generating unitaries and we now require that \begin{equation} K^{\left( N,M\right) }=-\frac{1}{2}(\delta _{ij}+n_{ji})L_{i}^{\ast }L_{j}- \frac{1}{2}n_{ij}L_{i}L_{j}^{\ast }+\frac{1}{2}m_{ij}L_{i}^{\ast }L_{j}^{\ast }+\frac{1}{2}m_{ji}^{\ast }L_{i}L_{j}-iH, \end{equation} with $H$ again self-adjoint.
Let us denote the conditional expectation from the algebra of operators on the system-tensor-Fock Hilbert space down to the system operators (i.e., the partial trace over the Gaussian state) as $\mathbb{E}_{\left(
N,M\right) }\left[ \cdot |\mathrm{sys}\right] $. As the differentials $dB_{k}\left( t\right) $ and $ dB_{k}\left( t\right) ^{\ast }$ are It\={o} (future pointing) their products with adapted operators will have conditional expectation zero. Therefore \begin{equation}
\mathbb{E}_{\left( N,M\right) }\left[ dU_{t}|\mathrm{sys}\right] =K^{\left(
N,M\right) }\,\mathbb{E}_{\left( N,M\right) }\left[ U_{t}|\mathrm{sys}\right] \,dt \end{equation} and we deduce that \begin{equation}
\mathbb{E}_{\left( N,M\right) }\left[ U_{t}|\mathrm{sys}\right] =e^{tK^{\left( N,M\right) }}. \end{equation}
The corresponding Heisenberg-Langevin equations are of the form \begin{equation} dj_{t}(X)=j_{t}(\left[ X,L_{k}\right] )dB_{k}^{\ast }+j_{t}(\left[ L_{k}^{\ast },X\right] )dB_{k}+j_{t}(\mathcal{L}^{\left( N,M\right) }X)dt \end{equation} where the new Lindbladian is \begin{eqnarray} \mathcal{L}^{\left( N,M\right) }X &=&\frac{1}{2}(\delta _{ij}+n_{ji})\big\{ L_{i}^{\ast }\left[ X,L_{j}\right] +\left[ L_{i}^{\ast },X\right] L_{j}\big\} \notag \\ &&+\frac{1}{2}n_{ij}\big\{L_{i}\left[ X,L_{j}^{\ast }\right] +\left[ L_{i},X \right] L_{j}^{\ast }\big\} \notag \\ &&-\frac{1}{2}m_{ij}\big\{L_{i}^{\ast }\left[ X,L_{j}^{\ast }\right] +\left[ L_{i}^{\ast },X\right] L_{j}^{\ast }\big\} \notag \\ &&-\frac{1}{2}m_{ji}^{\ast }\big\{L_{i}\left[ X,L_{j}\right] +\left[ L_{i},X \right] L_{j}\big\}-i\left[ X,H\right] . \notag \\ && \label{eq:Lindblad_non_Fock} \end{eqnarray} Likewise, we find that \begin{equation}
\mathbb{E}_{\left( N,M\right) }\left[ j_{t}\left( X\right) |\mathrm{sys} \right] =e^{t\mathcal{L}^{\left( N,M\right) }}X. \end{equation}
A little algebra allows us to relate these to the vacuum expressions: \begin{eqnarray} K^{\left( N,M\right) } =K-\frac{1}{2}n_{ji}L_{i}^{\ast }L_{j}-\frac{1}{2} n_{ij}L_{i}L_{j}^{\ast } +\frac{1}{2}m_{ij}L_{i}^{\ast }L_{j}^{\ast }+\frac{1 }{2}m_{ji}^{\ast }L_{i}L_{j}, \label{eq:K_form} \end{eqnarray} \begin{eqnarray} \mathcal{L}^{\left( N,M\right) }X &=&\mathcal{L}X+\frac{1}{2}n_{ji}\big\{ L_{i}^{\ast }\left[ X,L_{j}\right] +\left[ L_{i}^{\ast },X\right] L_{j}\big\} \notag \\ &&+\frac{1}{2}n_{ij}\big\{L_{i}\left[ X,L_{j}^{\ast }\right] +\left[ L_{i},X \right] L_{j}^{\ast }\big\} \notag \\ &&-\frac{1}{2}m_{ij}\big\{L_{i}^{\ast }\left[ X,L_{j}^{\ast }\right] +\left[ L_{i}^{\ast },X\right] L_{j}^{\ast }\big\} \notag \\ &&-\frac{1}{2}m_{ji}^{\ast }\big\{L_{i}\left[ X,L_{j}\right] +\left[ L_{i},X \right] L_{j}\big\} \notag \\ &\equiv &\mathcal{L}X+\frac{1}{2}n_{ji}\big\{\left[ L_{i}^{\ast },\left[ X,L_{j}\right] \right] +\left[ \left[ L_{i}^{\ast },X\right] ,L_{j}\right] \big\} \notag \\ &&+\frac{1}{2}m_{ij}\left[ L_{j}^{\ast }\left[ L_{i}^{\ast },X\right] \right] +\frac{1}{2}m_{ij}^{\ast }\left[ \left[ X,L_{i}\right] ,L_{j}\right] . \notag \\ && \label{eq:Lind_form} \end{eqnarray}
\section{Representation-Free Form}
\label{Sec:Rep_Free} Returning to the problem stated in the Introduction, we have that \textit{all} the $U_{t}^{\left( N,M\right) }$ arise from the \textit{same} physical dynamical evolution $U_{t}$, and the dynamics show not depend on the state! The $U_{t}^{\left( N,M\right) }$ unfortunately belong to representations that are \textit{not} generally unitarily equivalent! There should be some sense in which the QSDEs for the various $U_{t}^{\left( N,M\right) }$ should in some sense be equivalent. These QSDEs will depend explicitly on the state parameters $\left( N,M\right) $ of the input field, but what we would like to do is to show that there is nevertheless a representation-free version of each of these QSDEs in each fixed representation.
We now show that there is a way of presenting the unitary (\ref {eq:QSDE_non_Fock1}) and Heisenberg (\ref{eq:Lindblad_non_Fock}) QSDEs so as to be independent of the state parameters $(N,M)$.
\begin{theorem}{Theorem} \textbf{(Representation-Free Form)} The non-Fock QSDEs (\ref{eq:QSDE_non_Fock1}) and (\ref{eq:Lindblad_non_Fock} ) may be written in the equivalent Stratonovich forms \begin{eqnarray} dU &=&dA_{k}^{\ast }\circ L_{k}U-L_{k}^{\ast }U\circ d A_{k}+KU\left( t\right) \circ dt, \label{eq:Strat_QSDE} \\ dj_{t}(X) &=& d A_{k}^{\ast }\circ j_{t}(\left[ X,L_{k}\right] )+j_{t}( \left[ L_{k}^{\ast },X\right] )\circ d A_{k} +j_{t}(\mathcal{L}X)\circ dt, \label{eq:Strat_Heis} \end{eqnarray} respectively, where $K$ and $\mathcal{L}$ are the Fock representation expressions (\ref{eq:K_Fock}) and (\ref{eq:Linblad_Fock}). \end{theorem}
\begin{proof} We first observe that \begin{equation} dB_{k}^{\ast }\circ L_{k}U=dB_{k}^{\ast }L_{k}U+\frac{1}{2}dB_{k}^{\ast }L_{k}dU \end{equation} and substituting the QSDE (\ref{eq:QSDE_non_Fock1}) for $dU$ and using the quantum It\={o} table (\ref{eq:table_non_Fock}) gives \begin{equation} dB_{k}^{\ast }\circ L_{k}U=L_{k}UdB_{k}^{\ast }+\frac{1}{2}L_{k}\left( m_{kj}^{\ast }L_{j}-n_{kj}L_{j}^{\ast }\right) Udt, \end{equation} and similarly \begin{equation} -L_{k}^{\ast }U\circ dB_{k}=-L_{k}^{\ast }UdB_{k}-\frac{1}{2}L_{k}^{\ast }dUdB_{k}=-L_{k}^{\ast }UdB_{k}-\frac{1}{2}L_{k}^{\ast }\left( n_{jk}L_{j}-m_{ki}L_{j}^{\ast }\right) dt. \end{equation} Combining these terms and using the identity (\ref{eq:K_form}) shows that ( \ref{eq:Strat_QSDE}) is equivalent to (\ref{eq:QSDE_non_Fock1}).
For the Heisenberg equation, we first note that \begin{eqnarray} dB_{k}^{\ast }\circ j_{t}(\left[ X,L_{k}\right] ) &=&dB_{k}^{\ast }j_{t}( \left[ X,L_{k}\right] )+\frac{1}{2}dB_{k}^{\ast }dj_{t}(\left[ X,L_{k}\right] ) \nonumber\\ &=&j_{t}(\left[ X,L_{k}\right] )dB_{k}^{\ast } \nonumber\\ &&+\frac{1}{2}dB_{k}^{\ast } \bigg\{j_{t}(\left[ \left[ X,L_{k}\right] ,L_{j}\right] )dB_{j}^{\ast }+j_{t}\left( \left[ L_{j}^{\ast },\left[ X,L_{k}\right] \right] \right) dB_{j}\bigg\} \nonumber\\ &=&j_{t}(\left[ X,L_{k}\right] )dB_{k}^{\ast }+j_{t}\big(\frac{1}{2} m_{kj}^{\ast }\big[\left[ X,L_{k}\right] ,L_{j}\big]+\frac{1}{2}n_{kj}\left[ L_{j}^{\ast },\left[ X,L_{k}\right] \right] \big)dt, \nonumber \\ \quad \end{eqnarray} and similarly \begin{equation} j_{t}(\left[ L_{k}^{\ast },X\right] )\circ dB_{k}=j_{t}(\left[ L_{k}^{\ast },X\right] )dB_{k}+j_{t}\bigg(\frac{1}{2}n_{jk}\big[\left[ L_{k}^{\ast },X \right] ,L_{j}\big]+\frac{1}{2}m_{jk}\left[ L_{j}^{\ast },\big[L_{k}^{\ast },X\right] \big]\bigg)dt. \end{equation} Combining these terms and using the identity (\ref{eq:Lind_form}) shows that (\ref{eq:Strat_Heis}) is equivalent to (\ref{eq:Lindblad_non_Fock}). \end{proof}
Note that in both equations (\ref{eq:Strat_QSDE}) and (\ref{eq:Strat_Heis}) the Stratonovich differentials occur in Wick order relative to the integrand terms. What is remarkable about these relations is that they are structurally the same as the Fock vacuum form of the QSDEs with $S=I$. We say that the equations (\ref{eq:Strat_QSDE}) and (\ref{eq:Strat_Heis}) are \textit{representation-free} in the sense that they do not depend on the parameters $N$ and $M$ determining the state of the noise.
\section{White Noise Description}
We now present a more formal, but insightful account of quantum stochastic processes. Consider a collection of quantum noise input processes $ \{b_{k}\left( t\right) :t\in \mathbb{R},k=1,\cdots ,d\}$ obeying the commutation relations \begin{equation} \left[ b_{j}\left( t\right) ,b_{k}^{\ast }\left( s\right) \right] =\delta \left( t-s\right) ,\qquad \left[ b_{j}^{\ast }\left( t\right) ,b_{k}^{\ast }\left( s\right) \right] =\left[ b_{j}\left( t\right) ,b_{k}\left( s\right) \right] =0. \end{equation} We wish to model the interaction of a quantum mechanical system driven by these processes, and to this end introduce a unitary dynamics given by \begin{equation} U\left( t\right) =\vec{\mathbf{T}}\exp \left\{ -i\int_{0}^{t}\Upsilon _{s}ds\right\} \end{equation} where (with an implied summation convention with range 1,$\cdots ,d$) \begin{equation} -i\Upsilon _{t}=L_{k}\otimes b_{k}^{\ast }\left( t\right) -L_{k}^{\ast }\otimes b_{k}\left( t\right) -iH\otimes I. \end{equation} Here $L_{k}$ and $H=H^{\ast }$ are system operators. The time ordering $\vec{ \mathbf{T}}$ is understood in the usual sense of a Dyson series expansion. From this we may arrive at \begin{equation} \dot{U}\left( t\right) =L_{k}b_{k}^{\ast }\left( t\right) U\left( t\right) -L_{k}^{\ast }b_{k}\left( t\right) U\left( t\right) -iHU\left( t\right) . \label{eq:SCHROd} \end{equation}
We claim that $U\left( t\right) $ should correspond to the evolution operator for $\mathbf{G}\sim \left( S=I,L,H\right) $ without due reference to a particular state for the noise. If we fix the state, say the vacuum, then we use Wick ordering to compute the partial expectations with respect to that state.
To see how to proceed, let us consider a general quantum stochastic integral $X\left( t\right) $ described by a formal equation \begin{equation} \dot{X}\left( t\right) =b_{j}\left( t\right) ^{\ast }x_{jk}\left( t\right) b_{k}\left( t\right) +b_{j}\left( t\right) ^{\ast }x_{j0}\left( t\right) +x_{0k}\left( t\right) b_{k}\left( t\right) +x_{00}\left( t\right) . \label{eq:wn_qsi} \end{equation} where the terms $x_{\alpha \beta }\left( t\right) $ are ``adapted'' in the formal sense that they do not depend on the noises $b_{k}\left( s\right) $
for $s>t$. As we are talking about the vacuum representation for the time being, we can bootstrap from the vacuum $|\Omega \rangle $ to construct the Fock space as the completion of the span of all vectors of the type $\int f_{k\left( 1\right) }\left( t_{1}\right) b_{k\left( 1\right) }\left( t_{1}\right) ^{\ast }\cdots f_{k\left( n\right) }\left( t_{n}\right)
b_{k\left( n\right) }\left( t_{m}\right) ^{\ast }|\Omega \rangle $, and moreover we can build up the domain of exponential vectors. We quickly see that (\ref{eq:wn_qsi}), with Wick ordered right hand side, corresponds to the QSDE \begin{equation} dX\left( t\right) =x_{jk}\left( t\right) d\Lambda _{lk}\left( t\right) +x_{j0}\left( t\right) dB_{j}\left( t\right) ^{\ast }+x_{0k}\left( t\right) dB_{k}\left( t\right) +x_{00}\left( t\right) dt. \end{equation} Our issue however is how do we put to Wick order a given expression, for instance, the right hand side of (\ref{eq:SCHROd}).
\begin{proposition}{Proposition} For the process $X\left( t\right) $ described by (\ref{eq:wn_qsi}), we have \begin{eqnarray} b_{k}\left( t\right) X\left( t\right) &=&X\left( t\right) b_{k}\left( t\right) +\frac{1}{2}x_{kl}\left( t\right) b_{l}\left( t\right) +\frac{1}{2} x_{k0}\left( t\right) , \notag \\ X\left( t\right) b_{k}\left( t\right) ^{\ast } &=&b_{k}\left( t\right) ^{\ast }X\left( t\right) +\frac{1}{2}b_{j}\left( t\right) ^{\ast }x_{j0}\left( t\right) +\frac{1}{2}x_{0k}\left( t\right) . \label{eq:wn_Strat} \end{eqnarray} \end{proposition}
We may justify this as follows: \begin{eqnarray} \left[ b_{k}\left( t\right) ,X\left( t\right) \right] &=&\int_{0}^{t}\left[ b_{k}\left( t\right) ,\dot{X}\left( s\right) \right] ds=\int_{0}^{t}\delta \left( t-s\right) \left\{ x_{kl}\left( s\right) b_{l}\left( s\right) +x_{k0}\left( s\right) \right\} \nonumber \\ &=&\frac{1}{2}x_{kl}\left( t\right) b_{l}\left( t\right) +\frac{1}{2} x_{k0}\left( t\right) \end{eqnarray} with the factor of $\frac{1}{2}$ coming from the half-contribution of the $ \delta $-function. Evidently what the equations in (\ref{eq:wn_Strat}) correspond to is our definition of a Stratonovich differential - at least for the Fock vacuum representation. While we can make a connection between ( \ref{eq:wn_qsi}) and the rigorously defined Hudson-Parthasarathy processes, it should be appreciated at the very least that (\ref{eq:wn_Strat}) is the correct mnemonic for doing the Wick ordering - an attempt to convert into a Dyson-type series expansion and Wick ordering under the iterated integral signs to get a Maassen-Meyer kernel expansion shows this. At work here is an old principle that ``It\^{o}’s formula is the chain rule with Wick ordering'' \cite{HS}. Let us now examine (\ref{eq:SCHROd}) and put it to Wick ordered form. By a similar argument, we have \begin{equation} \left[ b_{k}\left( t\right) ,U\left( t\right) \right] =\int_{0}^{t}\left[ b_{k}\left( t\right) ,\Upsilon \left( s\right) \right] U\left( s\right) ds\equiv \frac{1}{2}L_{k}U\left( t\right) , \end{equation} or $b_{k}\left( t\right) U\left( t\right) =U\left( t\right) b_{k}\left( t\right) +\frac{1}{2}L_{k}U\left( t\right) $. By means of this we may place ( \ref{eq:SCHROd}) into the Wick-ordered form \begin{equation} U\left( t\right) =L_{k}b_{k}^{\ast }\left( t\right) U\left( t\right) -L_{k}^{\ast }U\left( t\right) b_{k}\left( t\right) -(\frac{1}{2}L_{k}^{\ast }L_{k}+iH)U\left( t\right) , \end{equation} and picking up the correct vacuum damping (\ref{eq:K_Fock}), $K$, as a result.
Setting $X_{t}=U\left( t\right) (X\otimes I)U\left( t\right) $, the same Wick ordering rule can be applied to the Heisenberg equations to obtain \begin{equation} \dot{X}_{t}=\left\{ b_{k}^{\ast }\left( t\right) +\frac{1}{2}L_{k,t}^{\ast }\right\} \left[ X,L_{k}\right] _{t}+\left[ L_{k}^{\ast },X\right] _{t}\left\{ b_{k}\left( t\right) +\frac{1}{2}L_{k,t}\right\} +\frac{1}{i} U\left( t\right) \left[ X,H\right] U\left( t\right) . \end{equation} Here we use the notation $L_{k,t}=U\left( t\right) (L_{k}\otimes I)U\left( t\right) $, etc.
We also remark that we may define the corresponding \textit{output fields} by \begin{eqnarray} b^{\mathrm{out}}_k (t) \triangleq U^\ast_T \, b(t) \, U_T, \end{eqnarray} where $T>t$. One may show that the input-output relations are \begin{eqnarray} b^{\mathrm{out}}_k (t) \equiv b_k (t) + L_{k,t}. \label{eq:i-o} \end{eqnarray}
If, on the other hand, we want the state of the noise to be a mean-zero Gaussian with correlations, say \begin{equation} \left\langle b_{j}\left( t\right) ^{\ast }b_{k}\left( s\right) \right\rangle =n_{jk}\,\delta \left( t-s\right) ,\quad \left\langle b_{j}\left( t\right) b_{k}\left( s\right) \right\rangle =m_{jk}\,\delta \left( t-s\right) , \label{eq:cov_flat} \end{equation} then we represent the noise as \begin{equation} b_{k}\left( t\right) =U_{jk}a_{+,k}\left( t\right) +V_{jk}a_{-,k}\left( t\right) ^{\ast } \label{eq:wn_bog} \end{equation} employing a suitable Bogoliubov transformation. Here we now have double the number of quantum white noises $a_{+,k}$ and $a_{-,k}$ but these are represented as Fock processes.
If we now substitute (\ref{eq:wn_bog}) into (\ref{eq:SCHROd}) we see explicitly that the $a_{\pm ,k}$ are out Wick order, but this can be rectified by the same sort of manipulation as above. Once the $a_{\pm ,k}\left( t\right) $ are Wick ordered, we have a equation which we can interpret as the It\={o} non-Fock QSDE, and this leads to the correct expressions $K^{\left( N,M\right) }$ and $\mathcal{L}^{\left( N,M\right) }$ in the unitary and flow equations respectively.
Given a Gaussian state $\left\langle \cdot \right\rangle $ on the noise, we may introduce a conditional expectation according to $\mathbb{E}\left[ \cdot
|\mathrm{sys}\right] :A\otimes B\mapsto \left\langle B\right\rangle \,A$. For instance, $\mathbb{E}\left[ U\left( t\right) |\mathrm{sys}\right] $ then defines a contraction on the system Hilbert space and we have \begin{equation}
\mathbb{E}\left[ U\left( t\right) |\mathrm{sys}\right] =I_{\mathrm{sys} }+\sum_{n\geq 1}\left( -i\right) ^{n}\int_{\Delta _{n}\left( t\right) }
\mathbb{E}\left[ \Upsilon _{s_{n}}\cdots \Upsilon _{s_{1}}|\mathrm{sys} \right] . \end{equation}
Now the expression $\mathbb{E}\left[ \Upsilon _{s_{n}}\cdots \Upsilon _{s_{1}}|\mathrm{sys}\right] $ will be a sum of products of the operators $ L,-L^{\ast }$ and $H$ times a $n$-point function in the fields. Similarly, we obtain a reduced Heisenberg equation. To compute these averages we need to be able to calculate $n$-point functions of chronologically ordered Gaussian fields - this is the realm of Wick's Theorem, so what we have presented may be interpreted as a Gaussian Wick's Theorem \cite{Evans_Steer}. We of course recover the partial traces appearing in the previous section.
\section{Approximate Signal Generator for Thermal States}
In this section we show how to go from a general SLH model driven by the output of a Degenerate Parametric Amplifier (DPA) to the limit where the same SLH model is driven by a thermal white noise. We start with the single channel for simplicity.
\subsection{The Thermal White Noise as Idealization of the Output of a Degenerate Parametric Amplifier}
We now show that in the strong coupling limit the output of a degenerate parametric amplifier approximates a thermal white noise. the model consists of a system of two cavities modes $c_{+}$ and $c_{-}$ coupled to input processes $A_{+}\left( t\right) $ and $A_{-}\left( t\right) $ respectively. Both inputs are taken to be in the vacuum state and the Schr\"{o}dinger equation is \begin{equation} \dot{U}_{t}=\sum_{i=+,-}L_{i}U\left( t\right) dA_{i}\left( t\right) ^{\ast }-\sum_{i=+,-}L_{i}^{\ast }U\left( t\right) dA_{i}\left( t\right) -iH_{ \mathrm{amp}}U_{t}, \end{equation} with initial condition $U_{0}=I$ and \begin{equation} L_{+}=\sqrt{2\kappa k}c_{+},\quad L_{-}=\sqrt{2\kappa k}c_{-}\text{ and }H_{ \mathrm{amp}}=\frac{\varepsilon k}{i}\left( c_{+}c_{-}-c_{+}c_{-}\right) . \end{equation} Here $\varepsilon >\kappa $ and $k>0$ is a scaling parameter which we eventually model to be large. It is more convenient to work with the white noises $a_{\pm }\left( t\right) $.
The model is linear and we obtain the input-output relations in the Laplace domain to be \cite{GJN_squeeze} \begin{equation} \left[ \begin{array}{c} b\left[ s\right] \\ \tilde{b}\left[ s\right] \end{array} \right] =\Xi _{-}^{\left( k\right) }\left( s\right) \left[ \begin{array}{c} a_{+}\left[ s\right] \\ a_{-}\left[ s\right] \end{array} \right] +\Xi _{+}^{\left( k\right) }\left( s\right) \left[ \begin{array}{c} a_{+}\left[ s\right] \\ a_{-}\left[ s\right] \end{array} \right] \end{equation} where $\Xi _{-}^{\left( k\right) }\left( s\right) =\left[ \begin{array}{cc} u\left( s/k\right) & 0 \\ 0 & u\left( s/k\right) \end{array} \right] ,\quad \Xi _{+}^{\left( k\right) }\left( s\right) =\left[ \begin{array}{cc} 0 & v\left( s/k\right) \\ v\left( s/k\right) & 0 \end{array} \right] $ with the functions $u\left( s\right) =\frac{s^{2}-\kappa ^{2}-\varepsilon ^{2}}{s^{2}+2s\kappa +\kappa ^{2}-\varepsilon ^{2}},\quad v\left( s\right) =\frac{2\kappa \varepsilon }{s^{2}+2\kappa +\kappa ^{2}-\varepsilon ^{2}}$.
In the limit $k\rightarrow \infty $ we find the static ($s$-independent) coefficients \begin{equation} \lim_{k\rightarrow \infty }\Xi _{-}^{\left( k\right) }\left( s\right) =\frac{ \varepsilon ^{2}+\kappa ^{2}}{\varepsilon ^{2}-\kappa ^{2}}\left[ \begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array} \right] ,\quad \lim_{k\rightarrow \infty }\Xi _{+}^{\left( k\right) }\left( s\right) =\frac{2\varepsilon \kappa }{\varepsilon ^{2}-\kappa ^{2}}\left[ \begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array} \right] . \end{equation} and returning to the time domain, the limit output fields are just a Bogoliubov transform of the inputs \begin{equation} b\left( t\right) =\sqrt{n+1}a_{+}\left( t\right) +\sqrt{n}a_{-}\left( t\right) ,\quad \tilde{b}\left( t\right) =\sqrt{n}a_{+}\left( t\right) + \sqrt{n+1}a_{-}\left( t\right) , \end{equation} Here the parameter $n$ corresponds is $n=\left( \frac{2\varepsilon \kappa }{ \varepsilon ^{2}-\kappa ^{2}}\right) ^{2}.$
It is instructive to look closely at the finite $k$ equations. We have the Heisenberg equations \begin{eqnarray} \dot{c}_{+}\left( t\right) &=&-k\kappa c_{+}\left( t\right) +k\varepsilon c_{-}\left( t\right) -\sqrt{2\kappa k}a_{+}\left( t\right) , \nonumber \\ \dot{c}_{-}\left( t\right) &=&-k\kappa c_{-}\left( t\right) +k\varepsilon c_{+}\left( t\right) -\sqrt{2\kappa k}a_{-}\left( t\right) , \end{eqnarray} and for $k$ large we may ignore the $\dot{c}_{+}\left( t\right) $ and $ \dot{c}_{-}\left( t\right) $ terms leaving a pair of simultaneous equations which we may solve to get \begin{eqnarray} \sqrt{k}c_{+}\left( t\right) &\simeq &\frac{\sqrt{2\kappa }}{\varepsilon ^{2}-\kappa ^{2}}\left[ \kappa a_{+}\left( t\right) +\varepsilon a_{-}\left( t\right) ^{\ast }\right] ,\nonumber \\
\sqrt{k}c_{-}\left( t\right) &\simeq &\frac{ \sqrt{2\kappa }}{\varepsilon ^{2}-\kappa ^{2}}\left[ \kappa a_{-}\left( t\right) +\varepsilon a_{+}\left( t\right) ^{\ast }\right] . \label{eq:approx_eq} \end{eqnarray} The output is then \begin{eqnarray} b\left( t\right) &=&a_{+}\left( t\right) +\sqrt{2\kappa k}c_{+}\left( t\right) \simeq a_{+}\left( t\right) +\frac{2\kappa }{\varepsilon ^{2}-\kappa ^{2}}\left[ \kappa a_{+}\left( t\right) +\varepsilon a_{-}\left( t\right) ^{\ast }\right] \nonumber \\ &\equiv &\sqrt{n+1}a_{+}\left( t\right) +\sqrt{n}a_{-}\left( t\right) , \end{eqnarray} and likewise \begin{eqnarray} \tilde{b}\left( t\right) &=&a_{-}\left( t\right) +\sqrt{2\kappa k} c_{-}\left( t\right) \simeq a_{-}\left( t\right) +\frac{2\kappa }{ \varepsilon ^{2}-\kappa ^{2}}\left[ \kappa a_{-}\left( t\right) +\varepsilon a_{+}\left( t\right) ^{\ast }\right] \nonumber \\ &\equiv &\sqrt{n}a_{+}\left( t\right) +\sqrt{n+1}a_{-}\left( t\right) . \end{eqnarray}
It is relatively straightforward to find a multi-dimensional version of this for a general Bogoliubov transformation \begin{equation} \left[ \begin{array}{c} b\left( t\right) \\ \tilde{b}\left( t\right) \end{array} \right] =U\left[ \begin{array}{c} a_{+}\left( t\right) \\ a_{-}\left( t\right) \end{array} \right] +V\left[ \begin{array}{c} a_{+}\left( t\right) \\ a_{-}\left( t\right) \end{array} \right] . \end{equation}
\subsection{Cascade Approximation}
The DPA which is described by \begin{equation} \mathbf{G}_{DPA}\sim \left( \left[ \begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array} \right] ,\left[ \begin{array}{c} \sqrt{2\kappa k}c_{+} \\ \sqrt{2\kappa k}c_{-} \end{array} \right] ,H_{\mathrm{amp}}\right) \end{equation} driven by the (vacuum) input pair $\left[ \begin{array}{c} a_{+}\left( t\right) \\ a_{-}\left( t\right) \end{array} \right] $. It is then put in series with \begin{equation} \mathbf{G}\sim \left( S,L,H\right) \boxplus \left( 1,0,0\right) =\left( \left[ \begin{array}{cc} S & 0 \\ 0 & 1 \end{array} \right] ,\left[ \begin{array}{c} L \\ 0 \end{array} \right] ,H\right) \end{equation} which means that the output $a_{+}\left( t\right) $ is fed in as input to the system $\mathbf{G}\sim \left( S,L,H\right) $ and $a_{-}\left( t\right) $ is left to go away unhindered, $\mathbf{G}_{\mathrm{trivial}}\sim \left( 1,0,0\right) $. According to the series product rule, we get DPA and system in series is described by, \begin{equation} \mathbf{G}\vartriangleleft \mathbf{G}_{DPA}\sim \bigg(\left[ \begin{array}{cc} S & 0 \\ 0 & 1 \end{array} \right] ,\left[ \begin{array}{c} L+S\sqrt{2\kappa k}c_{+} \\ \sqrt{2\kappa k}c_{-} \end{array} \right] ,H+H_{\mathrm{amp}}+\frac{\sqrt{\kappa k}}{\sqrt{2}i}\left( L^{\ast }Sc_{+}-c_{+}^{\ast }S^{\ast }L\right) \bigg). \end{equation}
From this we obtain the Heisenberg equations \begin{eqnarray} \dot{X}_{t} &=&a_{+}\left( t\right) ^{\ast }\left( S^{\ast }XS-X\right) _{t}a_{+}\left( t\right) +a_{+}\left( t\right) ^{\ast }S_{t}^{\ast }\left[ X,L\right] _{t}+\left[ L^{\ast },X\right] _{t}S_{t}a_{+}\left( t\right) \nonumber \\ &&+\frac{1}{2}\left[ L^{\ast },X\right] _{t}\left( L+S\sqrt{2\kappa k} c_{+}\right) _{t}+\frac{1}{2}\left( L+S\sqrt{2\kappa k}c_{+}\right) _{t}^{\ast }\left[ X,L\right] _{t} \nonumber \\ &&-i\left[ X,H+\frac{\sqrt{2\kappa k}}{2i}\left( L^{\ast }Sc_{+}-c_{+}^{\ast }S^{\ast }L\right) \right] _{t}. \end{eqnarray}
We now make the approximation $\sqrt{k}c_{+}\left( t\right) \simeq \frac{ \sqrt{2\kappa }}{\varepsilon ^{2}-\kappa ^{2}}\left[ \kappa a_{+}\left( t\right) +\varepsilon a_{-}\left( t\right) ^{\ast }\right] $ which leads to
\begin{eqnarray} \dot{X}_{t} &\simeq &a_{+}\left( t\right) ^{\ast }\left( S^{\ast }XS-X\right) _{t}a_{+}\left( t\right) +a_{+}\left( t\right) ^{\ast }S_{t}^{\ast }\left[ X,L\right] _{t}+\left[ L^{\ast },X\right] _{t}S_{t}a_{+}\left( t\right) +\mathcal{L}\left( X\right) _{t} \nonumber \\ &&+\left\{ \left[ L^{\ast },X\right] _{t}S_{t}+\frac{1}{2}L_{t}^{\ast }\left[ S,X\right] _{t}\right\} \left[ \left( \sqrt{n+1}-1\right) a_{+}\left( t\right) +\sqrt{n}a_{-}\left( t\right) ^{\ast }\right] \nonumber \\ &&+\left[ \left( \sqrt{n+1}-1\right) a_{+}\left( t\right) ^{\ast }+\sqrt{n} a_{-}\left( t\right) \right] \left\{ S_{t}^{\ast }\left[ X,L\right] _{t}+ \frac{1}{2}\left[ X,S^{\ast }\right] _{t}L_{t}\right\} . \end{eqnarray} Here we have $n=\left( \frac{2\varepsilon \kappa }{\varepsilon ^{2}-\kappa ^{2}}\right) ^{2}$, as before.
We now make a key assumption: \textbf{the scattering term} $S$ \textbf{ corresponds to a static element}. In this case $S\equiv e^{i\theta }$ for some real $\theta $. The limit Heisenberg equation therefore simplifies to \begin{eqnarray} \dot{X}_{t} &=&a_{+}\left( t\right) ^{\ast }S^{\ast }\left[ X,L\right] _{t}+ \left[ L^{\ast },X\right] _{t}Sa_{+}\left( t\right) +\mathcal{L}\left( X\right) _{t} \nonumber \\ &&+\left[ L^{\ast },X\right] _{t}S\left[ \left( \sqrt{n+1}-1\right) a_{+}\left( t\right) +\sqrt{n}a_{-}\left( t\right) ^{\ast }\right] \nonumber \\ &&+\left[ \left( \sqrt{n+1}-1\right) a_{+}\left( t\right) ^{\ast }+\sqrt{n}a_{-}\left( t\right) \right] S^{\ast }\left[ X,L\right] _{t} \nonumber \\ &=&\sqrt{n+1}a_{+}\left( t\right) ^{\ast }S^{\ast }\left[ X,L\right] _{t}+ \sqrt{n+1}\left[ L^{\ast },X\right] _{t}Sa_{+}\left( t\right) \nonumber \\ &&+\sqrt{n}\left[ L^{\ast },X\right] _{t}Sa_{-}\left( t\right) ^{\ast }+ \sqrt{n}a_{-}\left( t\right) S^{\ast }\left[ X,L\right] _{t}+\mathcal{L} \left( X\right) _{t}. \end{eqnarray}
We are not quite finished as the operators $a_{-}\left( t\right) $ and $ a_{-}\left( t\right) $ are out of Wick order. However, this is easily remedied. For instance, we easily deduce that \begin{eqnarray} \left[ Y_{t},a_{-}\left( t\right) ^{\ast }\right] &=&\int_{0}^{t}\left[ \dot{Y}_{s},a_{-}\left( t\right) ^{\ast }\right] ds \nonumber \\ &=&\int_{0}^{t}\left[ \sqrt{n}a_{-}\left( s\right) S^{\ast }\left[ Y,L\right] _{s},a_{-}\left( t\right) ^{\ast }\right] ds \nonumber \\ &=&\frac{1}{2}\sqrt{n}S^{\ast }\left[ Y,L\right] _{t} \end{eqnarray} so that we arrive at \begin{equation} \left[ L^{\ast },X\right] _{t}Sa_{-}\left( t\right) ^{\ast }=a_{-}\left( t\right) ^{\ast }\left[ L^{\ast },X\right] _{t}S+\frac{1}{2}\sqrt{n}\left[ \left[ L^{\ast },X\right] ,L\right] _{t}. \end{equation}
Similarly $\left[ a_{-}\left( t\right) ,Y_{t}\right] =\frac{1}{2}\sqrt{n} \left[ L^{\ast },Y\right] _{t}$ and therefore we get the Wick re-ordering \begin{eqnarray} a_{-}\left( t\right) S^{\ast }\left[ X,L\right] _{t}=S^{\ast }\left[ X,L \right] _{t}a_{-}\left( t\right) +\frac{1}{2}\sqrt{n}\left[ L^{\ast },\left[ X,L\right] \right] _{t}. \end{eqnarray}
This leads to the form of the quantum white noise equation with both $a_{+}$ and $a_{-}$ Wick ordered as
\begin{eqnarray} \dot{X}_{t} &=&\sqrt{n+1}a_{+}\left( t\right) ^{\ast }S^{\ast }\left[ X,L \right] _{t}+\sqrt{n+1}\left[ L^{\ast },X\right] _{t}Sa_{+}\left( t\right) \notag \\ &&+\sqrt{n}a_{-}\left( t\right) ^{\ast }\left[ L^{\ast },X\right] _{t}S+ \sqrt{n}S^{\ast }\left[ X,L\right] _{t}a_{-}\left( t\right) \notag \\ &&+\mathcal{L}\left( X\right) _{t}+\frac{1}{2}n\left[ \left[ L^{\ast },X \right] ,L\right] _{t}+\frac{1}{2}n\left[ L^{\ast },\left[ X,L\right] \right] _{t}. \label{eq:approx_Heis} \end{eqnarray}
At this stage we recognize (\ref{eq:approx_Heis}) as the equivalent form of the Heisenberg quantum stochastic differential equation for thermal noise.
We also remark that the output process determined by systems in series is $ B^{\mathrm{out}}\left( t\right) =U_{t}^{\ast }A_{+}\left( t\right) U_{t}$, and from the quantum stochastic calculus we have \begin{equation} dB^{\mathrm{out}}\left( t\right) =dA_{+}\left( t\right) +\left( L+S\sqrt{ 2\kappa k}c_{+}\right) _{t}dt. \end{equation} Using (\ref{eq:approx_eq}) we approximate this as \begin{equation} dB^{\mathrm{out}}\left( t\right) \simeq dA_{+}\left( t\right) +L_{t}dt+S \frac{2\kappa }{\varepsilon ^{2}-\kappa ^{2}}\left[ \kappa dA_{+}\left( t\right) +\varepsilon dA_{-}\left( t\right) ^{\ast }\right] \equiv SdB^{ \mathrm{in}}\left( t\right) +L_{t}dt, \end{equation} that is, the thermal input $B^{\mathrm{in}}\left( t\right) =\sqrt{n+1} A_{+}\left( t\right) +\sqrt{n}A_{-}\left( t\right) ^{\ast }$ produces the output $B^{\mathrm{out}}\left( t\right) $ according to the usual rules one would expect of a quantum Markov component with the parameters $\mathbf{G} \sim \left( S,L,H\right) $.
Therefore the description of a component with the parameters $\mathbf{G}\sim \left( S,L,H\right) $, at least in the case where $S$ is a static beam-splitter matrix, with Gaussian input processes may be considered as the same component cascaded with a degenerate parametric amplifier with vacuum inputs in the singular coupling limit of the DPA.
\section{The General Series Product}
\subsection{\label{Sec:NS_SP}Without Scattering}
Let us now consider the situation where a Gaussian input $B_{\mathrm{in}}=B_{ \mathrm{in}}^{\left( \mathscr{A}\right) }$ is driving a system with SLH parameters $\left( I,L_{\mathscr{A}},H_{\mathscr{A}}\right) $ and that its output $B_{\mathrm{out}}^{\left( \mathscr{A}\right) }$ acts as input $B_{ \mathrm{in}}^{\left( \mathscr{B}\right) }$ to a second system $\left( I,L_{ \mathscr{B}},H_{\mathscr{B}}\right) $. (We do not assume that any of the various SLH operators commute!)
\textbf{(Components in Series: The no scattering case)} The Heisenberg QSDE for the systems $\left( I,L_{\mathscr{A}},H_{\mathscr{A}}\right) $ and $ \left( I,L_{\mathscr{B}},H_{\mathscr{B}}\right) $ given by \begin{equation} dj_{t}(X)=\sum_{\mathscr{S}=\mathscr{A},\mathscr{B}}\bigg\{dB_{\mathrm{in} }^{\left( \mathscr{S}\right) \ast }\circ j_{t}(\left[ X,L_{\mathscr{S}} \right] )+j_{t}(\left[ L_{\mathscr{S}}^{\ast },X\right] )\circ dB_{\mathrm{in }}^{\left( \mathscr{S}\right) }+j_{t}(\mathcal{L}_{\mathscr{S}}X)\circ dt \bigg\}, \end{equation} where \begin{equation} \mathcal{L}_{\mathscr{S}}X=\frac{1}{2}L_{\mathscr{S}}^{\ast }\left[ X,L_{ \mathscr{S}}\right] +\frac{1}{2}\left[ L_{\mathscr{S}}^{\ast },X\right] L_{ \mathscr{S}}-i\left[ X,H_{\mathscr{S}}\right] . \end{equation} and we have the constraints $B_{\mathrm{in}}^{\left( \mathscr{A}\right) }=B_{ \mathrm{in}}$ and $dB_{\mathrm{in}}^{\left( \mathscr{B}\right) }=dB_{\mathrm{ in}}^{\left( \mathscr{A}\right) }+j_{t}\left( L_{\mathscr{A}}\right) dt$, consistent with $B_{\mathrm{in}}$ driving system $\mathscr{A}$ which in turn drives $\mathscr{B}$, corresponds to the dynamics given by the intrinsic series product (\ref{eq:series_prod}).
\begin{proof} We have to show consistency of the quantum stochastic Heisenberg evolution $ j_{t}(\cdot )$. To this end we take the open loop equations and impose the constraint $dB_{\mathrm{in}}^{\left( \mathscr{B}\right) }=dB_{\mathrm{in} }^{\left( \mathscr{A}\right) }+j_{t}\left( L_{\mathscr{A}}\right) dt$ giving \begin{eqnarray} dj_{t}\left( X\right) &=&dB_{\mathrm{in}}^{\ast }\circ j_{t}(\left[ X,L_{ \mathscr{A}}\right] )+j_{t}(\left[ L_{\mathscr{A}}^{\ast },X\right] )\circ dB_{\mathrm{in}} \notag \\ &+&\left( dB_{\mathrm{in}}+j_{t}(L_{\mathscr{A}})\right) ^{\ast }\circ j_{t}( \left[ X,L_{\mathscr{B}}\right] ) \notag \\ &+&j_{t}(\left[ L_{\mathscr{B}}^{\ast },X\right] )\circ \left( dB_{\mathrm{in }}+j_{t}(L_{\mathscr{A}})dt\right) \notag \\ &+&j_{t}(\mathcal{L}_{\mathscr{A}}X)\circ dt+j_{t}(\mathcal{L}_{\mathscr{B} }X)\circ dt, \end{eqnarray} which we may rearrange as \begin{eqnarray} dj_{t}(X) &=&dB_{\mathrm{in}}^{\ast }\circ j_{t}(\left[ X,L_{\mathscr{A}}+L_{ \mathscr{B}}\right] )+j_{t}(\left[ L_{\mathscr{A}}^{\ast }+L_{\mathscr{B} }^{\ast },X\right] )\circ dB_{\mathrm{in}} \nonumber \\ &&+j_{t}\bigg(\mathcal{L}_{\mathscr{A}}X+\mathcal{L}_{\mathscr{B}}X+L_{ \mathscr{A}}^{\ast }\left[ X,L_{\mathscr{B}}\right] +\left[ L_{\mathscr{B} }^{\ast },X\right] L_{\mathscr{A}}\bigg)\circ dt. \end{eqnarray} However, the $dt$ term can be recast using the identity \begin{eqnarray} &&\mathcal{L}_{\mathscr{A}}X+\mathcal{L}_{\mathscr{B}}X+L_{\mathscr{A} }^{\ast }\left[ X,L_{\mathscr{B}}\right] +\left[ L_{\mathscr{B}}^{\ast },X \right] L_{\mathscr{A}} \nonumber \\ &=&\frac{1}{2}\left( L_{\mathscr{A}}+L_{\mathscr{B}}\right) ^{\ast }\left[ X,L_{\mathscr{A}}+L_{\mathscr{B}}\right] +\frac{1}{2}\left[ L_{\mathscr{A} }^{\ast }+L_{\mathscr{B}}^{\ast },X\right] \left( L_{\mathscr{A}}+L_{ \mathscr{B}}\right) \nonumber \\ &&-i\left[ X,H_{\mathscr{A}}+H_{\mathscr{B}}+\frac{1}{2i}\left( L_{ \mathscr{B}}^{\ast }L_{\mathscr{A}}-L_{\mathscr{A}}^{\ast }L_{\mathscr{B} }\right) \right] . \end{eqnarray} The resulting Heisenberg dynamics is therefore the same as for the model $ (I,L,H)$ with $L=L_{\mathscr{A}}+L_{\mathscr{B}}$, and $H=H_{\mathscr{A}}+H_{ \mathscr{B}}+\mathrm{Im}\{L_{\mathscr{B}}^{\ast }L_{\mathscr{A}}\}$. This is, of course, the form predicted by the series product in the Fock case ( \ref{eq:series_prod}). \end{proof}
\subsection{Including Scattering}
As mentioned above, it is not possible to construct a well defined scattering processes $\Lambda _{jk}$ in the non-Fock theory. Nevertheless, the effects of static beam-splitter scattering $S$ may be included in a straightforward manner without directly considering unitary QSDE models involving the scattering processes. A clue on how to proceed is given by our earlier observation that if the scattering matrix $S$ entries commute with systems operators - physically, a static beam-splitter - the scattering processes disappears.
In the Fock representation, we could always take the input field $A_{\mathrm{ in}}$ and apply a unitary rotation $A=SA_{\mathrm{in}}$ before passing it though as drive for component. As we have seen, this will require a compensating rotation of the coupling operators, but no change to the Lindbladian. There is also a rotation of the output, however, anticipating this we make the following definition.
\begin{definition}{Definition} Let $\mathbf{G}$ and $\mathbf{\tilde{G}}$ be SLH model parameters which, for given input noise $A_{\mathrm{in}}=\tilde{A}_{\mathrm{in}}$ lead to output noises $A_{\mathrm{out}}$ and $\tilde{A}_{\mathrm{out}}$ respectively. We say that the models' input-output relations are \textbf{related by a static beam-splitter matrix} $S$ if we have \begin{eqnarray} A_{\mathrm{out}}=S\,\tilde{A}_{\mathrm{out}}. \end{eqnarray} \end{definition}
The following result shows that for the Fock representation, if the scattering is just a static beam-splitter, then we can produce a related model which avoids the use of the scattering processes.
\begin{theorem}{Theorem} \label{Thm:S} Let $S$ be a static beam-splitter matrix and set $\mathbf{G} \sim \left( S,L,H\right) $ and $\mathbf{\tilde{G}}\sim \left( I,S^{\ast }L,H\right) $. Then the model parameters $\mathbf{G}$ and $\mathbf{\tilde{G}} $ generate the same Heisenberg dynamics. Moreover, their input-output relations are related by the static beam-splitter matrix $S$. \end{theorem}
\begin{proof} The Heisenberg dynamics generated by $\mathbf{G}$ is (the scattering terms vanish for a static beam-splitter) \begin{equation} dj_{t}^{\mathbf{G}}(X)=\sum_{j}j_{t}(\mathcal{L}_{j0}^{\mathbf{G} }X)\,dA_{j}^{\ast }+\sum_{k}j_{t}(\mathcal{L}_{0k}^{\mathbf{G} }X)\,dA_{k}+j_{t}(\mathcal{L}^{\mathbf{G}}X)dt \end{equation} where \begin{equation} \mathcal{L}_{j0}^{\mathbf{G}}X=S_{lj}^{\ast }\left[ X,L_{l}\right] ,\quad \mathcal{L}_{0k}^{\mathbf{G}}X=\left[ L_{l}^{\ast },X\right] S_{lk} \end{equation} and the Lindblad generator is $\mathcal{L}^{\mathbf{G}}X=\frac{1}{2} L_{k}^{\ast }\left[ X,L_{k}\right] +\frac{1}{2}\left[ L_{k}^{\ast },X\right] L_{k}-i\left[ X,H\right] $. The Heisenberg dynamics for $\mathbf{\tilde{G}}$ similarly has no scattering terms in its QSDE, and we see that \begin{equation} \mathcal{L}_{j0}^{\mathbf{G}}X=[X,S_{lj}^{\ast }L_{l}]\equiv \mathcal{L} _{j0}^{\mathbf{\tilde{G}}}X,\quad \mathcal{L}_{0k}^{\mathbf{G} }X=[L_{l}^{\ast }S_{lk},X]\equiv \mathcal{L}_{0k}^{\mathbf{\tilde{G}}}X. \end{equation} From the unitarity and scalar nature of $S$ we have that \begin{eqnarray} \mathcal{L}^{\mathbf{\tilde{G}}}X &=&\frac{1}{2}L_{k}^{\ast }S_{kl}\left[ X,S_{jl}^{\ast }L_{j}\right] +\frac{1}{2}\left[ L_{k}^{\ast }S_{kl},X\right] S_{jl}^{\ast }L_{j}-i\left[ X,H\right] \nonumber\\ &=&\frac{1}{2}L_{k}^{\ast }\left[ X,L_{k}\right] +\frac{1}{2}\left[ L_{k}^{\ast },X\right] L_{k}-i\left[ X,H\right] \nonumber \\ &\equiv &\mathcal{L}^{\mathbf{G}}X. \end{eqnarray} Therefore the QSDEs corresponding to the Heisenberg dynamics for $\mathbf{G}$ and $\mathbf{\tilde{G}}$ are identical.
The input-output relations for $\mathbf{G}$ are \begin{equation} dA_{\mathrm{out},j}\left( t\right) =S_{jk}\,dA_{\mathrm{in},k}+j_{t}\left( L_{j}\right) \,dt \end{equation} while for $\mathbf{\tilde{G}}$ we have \begin{equation} dB_{\mathrm{out},j}\left( t\right) =dB_{\mathrm{in},j}+S_{jk}\,j_{t}\left( L_{k}\right) \,dt. \end{equation} If we require the inputs to be the same ($A_{\mathrm{in}}=B_{\mathrm{in}}$) then we have $A_{\mathrm{out}}=S\,B_{\mathrm{out}}$. \end{proof}
Our strategy for introducing static beam-splitter scattering into the situation where we have non-Fock noise input fields is to say that the initial input $A_{\mathrm{in}}$ be replaced by the rotated input $SA_{ \mathrm{in}}$, and exploit the fact that the Heisenberg dynamics no longer involves the scattering processes $\Lambda _{jk}$ explicitly.
\begin{lemma}{Lemma} \textbf{(The Universal Heisenberg QSDE Description)} The Heisenberg dynamics for a general $\left( S,L,H\right) $ model with a static beam-splitter matrix $S$ are given by the QSDE \begin{eqnarray} dj_{t}(X)&=&dA_{\mathrm{in}}^{\ast }\circ S^{\ast }j_{t}(\left[ X,L\right] )+j_{t}(\left[ L^{\ast },X\right] )S\circ dA_{\mathrm{in}} +j_{t}(\mathcal{L}X)\circ dt \end{eqnarray} for all mean-zero Gaussian input fields $A_{\mathrm{in}}$. \end{lemma}
This is of course just the equation (\ref{eq:approx_Heis}) written in the Wick-Stratonovich form so as to be representation free!
Now let us try and repeat or analysis from Section \ref{Sec:NS_SP}. Let us now consider the situation where a Gaussian input $A_{\mathrm{in}}=A_{ \mathrm{in}}^{\left( 1\right) }$ is driving a system with SLH parameters $ \left( S_{\mathscr{A}},L_{\mathscr{A}},H_{\mathscr{A}}\right) $ and that its output $A_{\mathrm{out}}^{\left( 1\right) } $ acts as input for a second system $\left( S_{\mathscr{B}},L_{\mathscr{B}},H_{\mathscr{B}}\right) $.
\begin{lemma}{Lemma} \label{prop:fig} \textbf{(Components in series: With a static beam-splitter scattering)} The Heisenberg QSDE for a pair of systems $\left( S_{\mathscr{A} },L_{\mathscr{A}},H_{\mathscr{A}}\right) $ and $\left( S_{\mathscr{B}},L_{ \mathscr{B}},H_{\mathscr{B}}\right) $ in series is \begin{eqnarray} dj_{t}(X)=\sum_{\mathscr{S}=\mathscr{A},\mathscr{B}}\bigg\{dA_{\mathrm{in} }^{\left( \mathscr{S}\right) \ast }\circ j_{t}(\left[ X,L_{\mathscr{S}} \right] ) +j_{t}(\left[ L_{\mathscr{S}}^{\ast },X\right] )\circ dA_{\mathrm{in} }^{\left( \mathscr{S}\right) }+j_{t}(\mathcal{L}_{\mathscr{S}}X)\circ dt \bigg\} , \end{eqnarray} where $A_{\mathrm{in}}^{\left( \mathscr{A}\right) }=S_{\mathscr{A}}A_{\mathrm{in} }$ and $A_{\mathrm{in}}^{\left( \mathscr{B}\right) }=S_{\mathscr{B}}A_{\mathrm{ out}}^{\left( \mathscr{A}\right) }$ where $dA_{\mathrm{out}}^{\left( \mathscr{A}\right) }=S_{\mathscr{A}}dA_{\mathrm{in}}^{\left( \mathscr{A} \right) }+j_{t}\left( L_{\mathscr{A}}\right) dt$, and the Lindbladians $\mathcal{L}_{\mathscr{S}}$ are as before. \end{lemma}
\begin{proof} Substituting the processes into the QSDEs yields \begin{eqnarray} dj_{t}(X) &=&\left( S_{\mathscr{A}}dA_{\mathrm{in}}\right) ^{\ast }\circ j_{t}(\left[ X,L_{\mathscr{A}}\right] ) +j_{t}(\left[ L_{\mathscr{A}}^{\ast },X\right] )\circ S_{\mathscr{A}}dA_{\mathrm{in}} \nonumber\\ &&+\left( S_{\mathscr{B}}S_{\mathscr{A}}dA_{\mathrm{in}}+S_{\mathscr{B}}L_{ \mathscr{A}}dt\right) ^{\ast }\circ j_{t}(\left[ X,L_{\mathscr{B}}\right] ) \nonumber \\ && +j_{t}(\left[ L_{\mathscr{B}}^{\ast },X\right] )\circ \left( S_{ \mathscr{B}}S_{\mathscr{A}}dA_{\mathrm{in}}+S_{\mathscr{B}}L_{\mathscr{A} }dt\right) \nonumber \\ &&+j_{t}(\mathcal{L}_{\mathscr{A}}X)\circ dt+j_{t}(\mathcal{L}_{\mathscr{B} }X)\circ dt, \nonumber \\ &=&\left( dA_{\mathrm{in}}\right) ^{\ast }\circ j_{t}(\left[ X,S_{\mathscr{A} }^{\ast }L_{\mathscr{A}}+S_{\mathscr{A}}^{\ast }S_{\mathscr{B}}^{\ast }L_{ \mathscr{B}}\right] ) +j_{t}(\left[ L_{\mathscr{A}}^{\ast }S_{\mathscr{A} }+L_{\mathscr{B}}^{\ast }S_{\mathscr{B}}S_{\mathscr{A}},X\right] )\circ dA_{ \mathrm{in}} \nonumber \\ &&+j_{t}(\mathcal{L}_{\mathscr{A}}X+\mathcal{L}_{\mathscr{B}}X +L_{ \mathscr{A}}^{\ast }S_{\mathscr{B}}^{\ast }\left[ X,L_{\mathscr{B}}\right] + \left[ L_{\mathscr{B}}^{\ast },X\right] S_{\mathscr{B}}L_{\mathscr{A}})\circ dt. \end{eqnarray}
A similar calculation to before shows that \begin{eqnarray} &&\mathcal{L}_{\mathscr{A}}X+\mathcal{L}_{\mathscr{B}}X+L_{\mathscr{A} }^{\ast }S_{\mathscr{B}}^{\ast }\left[ X,L_{\mathscr{B}}\right] +\left[ L_{ \mathscr{B}}^{\ast },X\right] S_{\mathscr{B}}L_{\mathscr{A}} \nonumber \\ &=&\frac{1}{2}\left( S_{\mathscr{B}}L_{\mathscr{A}}+L_{\mathscr{B}}\right) ^{\ast }\left[ X,S_{\mathscr{B}}L_{\mathscr{A}}+L_{\mathscr{B}}\right] + \frac{1}{2}\left[ L_{\mathscr{A}}^{\ast }S_{\mathscr{B}}^{\ast }+L_{ \mathscr{B}}^{\ast },X\right] \left( S_{\mathscr{B}}L_{\mathscr{A}}+L_{ \mathscr{B}}\right) \nonumber \\ &&-[iX,H_{\mathscr{A}}+H_{\mathscr{B}}+\frac{1}{2i}\left( L_{\mathscr{B} }^{\ast }S_{\mathscr{B}}L_{\mathscr{A}}-L_{\mathscr{A}}^{\ast }S_{\mathscr{B} }^{\ast }L_{\mathscr{B}}\right) ]. \end{eqnarray} The resulting Heisenberg dynamics is therefore same as for the model $ \mathbf{\tilde{G}}\sim (I,\tilde{L},H)$ with coupling operators $\tilde{L} =S_{\mathscr{A}}^{\ast }L_{\mathscr{A}}+S_{\mathscr{A}}^{\ast }S_{\mathscr{B} }^{\ast }L_{\mathscr{B}}\equiv S_{\mathscr{A}}^{\ast }S_{\mathscr{B}}^{\ast }\left( S_{\mathscr{B}}L_{\mathscr{A}}+L_{\mathscr{B}}\right) $, and Hamiltonian $H=H_{\mathscr{A}}+H_{\mathscr{B}}+\mathrm{Im}\{L_{\mathscr{B} }^{\ast }S_{\mathscr{B}}L_{\mathscr{A}}\}$.
The output is then $B_{\mathrm{out}}$ where \begin{equation} dB_{\mathrm{out}}\left( t\right) =dA_{\mathrm{in}}\left( t\right) +j_{t}\left( S_{\mathscr{A}}^{\ast }L_{\mathscr{A}}+S_{\mathscr{A}}^{\ast }S_{\mathscr{B}}^{\ast }L_{\mathscr{B}}\right) dt. \end{equation} The correct output for this should however be $A_{\mathrm{out}}=S_{ \mathscr{B}}S_{\mathscr{A}}B_{\mathrm{out}}$ so that
\begin{eqnarray} dA_{\mathrm{out}}\left( t\right) =S_{\mathscr{B}}S_{\mathscr{A}}dA_{\mathrm{ in}}\left( t\right) +j_{t}\left( S_{\mathscr{B}}L_{\mathscr{A}}+L_{ \mathscr{B}}\right) dt \end{eqnarray} and we have the desired matrix $S_{\mathscr{B}}S_{\mathscr{A}}$ multiply the inputs corresponding to scattering first by matrix $S_{\mathscr{A}}$ and then by $S_{\mathscr{B}}$. The model $\mathbf{G}$ obtained from postulate Ia is then the one related to $\mathbf{\tilde{G}}$ by the static beam-splitter matrix $S_{\mathscr{B}}S_{\mathscr{A}}$, that is (from Theorem \ref{Thm:S} with $S=S_{\mathscr{B}}S_{\mathscr{A}}$ and $\tilde{L}=S^\ast L$ \begin{eqnarray} \mathbf{G} & \sim & \left( S,L,H\right) =\left( S_{\mathscr{B}}S_{\mathscr{A} },S_{\mathscr{B}}S_{\mathscr{A}}\tilde{L},H\right) \notag \\ &=& \left( S_{\mathscr{B}}S_{\mathscr{A}},S_{\mathscr{B}}L_{\mathscr{A}}+L_{ \mathscr{B}},H_{\mathscr{A}}+H_{\mathscr{B}}+\mathrm{Im}\{L_{\mathscr{B} }^{\ast }S_{\mathscr{B}}L_{\mathscr{A}}\}\right) , \notag \end{eqnarray} and again we have the same form as the series product in the Fock case (\ref {eq:series_prod}). \end{proof}
\section{Conclusions}
We have shown that there is a consistent theory for quantum input-output models in series when the driving input processes are in general Gaussian states with a flat power spectrum. This emerges fairly explicitly at the level of the singular input processes $b_k (t)$ themselves, but to have a working theory we need to make the connection to the Hudson-Parthasarathy quantum stochastic calculus. This involves quantum stochastic differential equations on the Fock spaces used to represent the noise (which are a mathematical convenience and not physical objects) with the result that the associated dynamical equations appear to depend on the choice of Gaussian state of the noise. In reality this is a mathematical artifact and we show that even here there is a way of expressing the quantum stochastic differential equations (the Wick-Stratonovich form introduced in this paper) which removes these terms. In effect, it is the Wick-Stratonovich form that translates in the physically relevant dynamical equations written in terms of the quantum input processes $b_k (t)$.
The connection rules are then shown to be genuinely independent of the choice of state. We were also able to include the effects of a static beam-splitter component. At first sight this would seem problematic as the scattering terms $\Lambda _{jk}(t)$ are not well-defined for non-vacuum states, however, it is possible to ignore them from the model: in fact we need to work at the level of the Heisenberg flow and the input-output relations, neither of which involve the scattering terms. The result is that we may account for static scattering and we find that the series product of \cite{GJ-Series} again gives the correct rule. In this way we extend the series product to deal with quantum feedback networks driven by general Gaussian input processes.
We have restricted our analysis to Bose systems, however, there is an Araki-Woods type double Fock space representation for Fermi fields with quasi-free states as well, and is applicable to Fermi stochastic processes \cite{HP_Fermi}, \cite{BSW}. The network rules for Fermi stochastic processes can be similarly derived and one would naturally expect these to again be state-independent.
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{\large UNIVERSIT\'E PIERRE ET MARIE CURIE}
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\'Ecole Doctorale de Sciences Math\'ematiques de Paris Centre
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\textsc{\textbf{{\LARGE Th\`ese de doctorat}}} \end{center}
\begin{center} {\large Discipline: Math\'ematiques} \end{center}
\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}} \begin{center} {\Large \textbf{Periods of the motivic fundamental groupoid of $\boldsymbol{\mathbb{P}^{1} \diagdown \lbrace 0, \mu_{N}, \infty \rbrace}$.}} \noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}
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\begin{center} Pr\'esent\'ee par \\
\textbf{{\Large Claire GLANOIS}} \end{center}
\begin{center} Dirig\'ee par {\Large Francis BROWN.} \end{center}
\vspace*{3cm} \begin{center} Pr\'{e}sent\'{e}e et soutenue publiquement le 6 janvier 2016 devant le jury compos\'{e} de : \end{center} \begin{center}
\begin{tabular}{lll} Francis {\sc Brown} & Directeur de th\`{e}se & Oxford University\\ Don {\sc Zagier} & Rapporteur & Max Planck Institute, Bonn\\ Jianqiang {\sc Zhao} & Rapporteur & ICMAT, Madrid\\ Yves {\sc Andr\'{e}} & Examinateur & UPMC (IMJ), Paris\\ Pierre {\sc Cartier} & Examinateur & IH\'{E}S, Paris-Saclay\\ Herbert {\sc Gangl} & Examinateur & Durham University\\ \end{tabular} \end{center}
\vspace*{3cm} \begin{flushright} \textit{Aux inconnues},\\ \textit{Aux variables},\\ \textit{A} Elle, \end{flushright} \epigraph{$\mlq$\textit{Prairie de tous mes instants, ils ne peuvent me fouler.}\\ \textit{Leur voyage est mon voyage et je reste obscurit\'e.}$\mrq$}{R. Char}
\begin{center} {\Huge \texttt{Abstract}} \end{center} In this thesis, following F. Brown's point of view, we look at the Hopf algebra structure of motivic cyclotomic multiple zeta values, which are motivic periods of the fundamental groupoid of $\mathbb{P}^{1} \diagdown \lbrace 0, \mu_{N}, \infty \rbrace $. By application of a surjective \textit{period} map (which, under Grothendieck's period conjecture, is an isomorphism), we deduce results (such as generating families, identities, etc.) on cyclotomic multiple zeta values, which are complex numbers. The coaction of this Hopf algebra (explicitly given by a combinatorial formula from A. Goncharov and F. Brown's works) is the dual of the action of a so-called \textit{motivic} Galois group on these specific motivic periods. This entire study was actually motivated by the hope of a Galois theory for periods, which should extend the usual Galois theory for algebraic numbers.
In the first part, we focus on the case of motivic multiple zeta values ($N = 1$) and Euler sums ($N = 2$). In particular, we present new bases for motivic multiple zeta values: one via motivic Euler sums, and another (depending on an analytic conjecture) which is known in the literature as the Hoffman $\star$ basis; under a general motivic identity that we conjecture, these bases are identical.
In the second part, we apply some Galois descents ideas to the study of these periods, and examine how periods of the fundamental groupoid of $\mathbb{P}^{1} \diagdown \lbrace 0, \mu_{N'}, \infty \rbrace $ are embedded into periods of $\pi_{1}(\mathbb{P}^{1} \diagdown \lbrace 0, \mu_{N}, \infty \rbrace )$, when $N'\mid N$. After giving some general criteria for any $N$, we focus on the cases $N=2,3,4,\mlq 6 \mrq, 8$, for which the motivic fundamental group generates the category of mixed Tate motives on $\mathcal{O}_{N}[\frac{1}{N}]$ (unramified if $N=6$). For those $N$, we are able to construct Galois descents explicitly, and extend P. Deligne's results.\\ \\ \texttt{Key words}:\textit{ Periods, Polylogarithms, multiple zeta values, Mixed Tate Motives, cyclotomic field, Hopf algebra, Motivic fundamental group, Galois Descent.}
\begin{center} {\Huge \texttt{R\'{e}sum\'{e}.}} \end{center} A travers ce manuscrit, en s'inspirant du point de vue adopt\'{e} par F. Brown, nous examinons la structure d'alg\`{e}bre de Hopf des multiz\^{e}tas motiviques cyclotomiques, qui sont des p\'{e}riodes motiviques du groupo\"{i}de fondamental de $\mathbb{P}^{1} \diagdown \lbrace 0, \mu_{N}, \infty \rbrace $. Par application d'un morphisme \textit{p\'{e}riode} surjectif (isomorphisme sous la conjecture de Grothendieck), nous pouvons d\'{e}duire des r\'{e}sultats (tels des familles g\'{e}n\'{e}ratrices, des identit\'{e}s, etc.) sur ces nombres complexes que sont les multiz\^{e}tas cyclotomiques. La coaction de cette alg\`{e}bre de Hopf (explicite par une formule combinatoire due aux travaux de A.B. Goncharov et F. Brown) est duale \`{a} l'action d'un d\'{e}nomm\'{e} \textit{groupe de Galois motivique} sur ces p\'{e}riodes motiviques. Ces recherches sont ainsi motiv\'{e}es par l'espoir d'une th\'{e}orie de Galois pour les p\'{e}riodes, \'{e}tendant la th\'{e}orie de Galois usuelle pour les nombres alg\'{e}briques.
Dans un premier temps, nous nous concentrons sur les multiz\^{e}tas ($N=1$) et les sommes d'Euler ($N=2$) motiviques. En particulier, de nouvelles bases pour les multizetas motiviques sont pr\'{e}sent\'{e}es: une via les sommes d'Euler motiviques, et une seconde (sous une conjecture analytique) qui est connue sous le nom de \textit{Hoffman} $\star$; soulignons que sous une identit\'{e} motivique g\'{e}n\'{e}rale que nous conjecturons \'{e}galement, ces bases sont identiques.
Dans un second temps, nous appliquons des id\'{e}es de descentes galoisiennes \`{a} l'\'{e}tude de ces p\'{e}riodes, en regardant notamment comment les p\'{e}riodes du groupo\"{i}de fondamental de $\mathbb{P}^{1} \diagdown \lbrace 0, \mu_{N'}, \infty \rbrace $ se plongent dans les p\'{e}riodes de $\pi_{1}(\mathbb{P}^{1} \diagdown \lbrace 0, \mu_{N}, \infty \rbrace )$, lorsque $N'\mid N$. Apr\`{e}s avoir fourni des crit\`{e}res g\'{e}n\'{e}raux (quel que soit $N$), nous nous tournons vers les cas $N=2,3,4,\mlq 6 \mrq, 8$, pour lesquels le groupo\"{i}de fondamental motivique engendre la cat\'{e}gorie des motifs de Tate mixtes sur $\mathcal{O}_{N}[\frac{1}{N}]$ (non ramifi\'{e} si $N=6$). Pour ces valeurs, nous sommes en mesure d'expliciter les descentes galoisiennes, et d'\'{e}tendre les r\'{e}sultats de P. Deligne.\\ \\ \texttt{Mots cl\'{e}s}: \textit{P\'{e}riodes, Polylogarithmes, multiz\^{e}tas, corps cyclotomiques, Motifs de Tate Mixtes, alg\`{e}bre de Hopf, groupe motivique fondamental, descente galoisienne.}
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\chapter{Introduction}
\epigraph{$\mlq$\textit{Qui est-ce ? Ah, tr\`{e}s bien, faites entrer l'infini.}$\mrq$}{Aragon}
\section{Motivation}
\subsection{Periods} A \textbf{\textit{period}} \footnote{For an enlightening survey, see the reference article $\cite{KZ}$.} denotes a complex number that can be expressed as an integral of an algebraic function over an algebraic domain.\footnote{We can equivalently restrict to integral of rational functions over a domain in $\mathbb{R}^{n}$ given by polynomial inequalities with rational coefficients, by introducing more variables.} They form the algebra of periods $\mathcal{P}$\nomenclature{$\mathcal{P}$}{the algebra of periods, resp. $\widehat{\mathcal{P}}$ of extended periods}, fundamental class of numbers between algebraic numbers $\overline{\mathbb{Q}}$ and complex numbers.\\ The study of these integrals is behind a large part of algebraic geometry, and its connection with number theory, notably via L-functions \footnote{One can associate a L$-$ function to many arithmetic objects such as a number field, a modular form, an elliptic curve, or a Galois representation. It encodes its properties, and has wonderful (often conjectural) meromorphic continuation, functional equations, special values, and non-trivial zeros (Riemann hypothesis).}; and many of the constants which arise in mathematics, transcendental number theory or in physics turn out to be periods, which motivates the study of these particular numbers.\\ \\ \texttt{Examples:} \begin{itemize} \item[$\cdot$] The following numbers are periods: $$\sqrt{2}= \int_{2x^{2} \leq 1} dx \text{ , } \quad \pi= \int_{x^{2}+y^{2} \leq 1} dx dy \quad\text{ and } \quad \log(z)=\int_{1}^{z} \frac{dx}{x}, z>1, z\in \overline{\mathbb{Q}}.$$ \item[$\cdot$] Famous -alleged transcendental- numbers which conjecturally are not periods: $$e =\lim_{n \rightarrow \infty} \left( 1+ \frac{1}{n} \right)^{n} \text{ , } \quad \gamma=\lim_{n \rightarrow \infty} \left( -ln(n) + \sum_{k=1}^{n} \frac{1}{k} \right) \text{ or } \quad \frac{1}{\pi}.$$ It can be more useful to consider the ring of extended periods, by inverting $\pi$: $$\widehat{\mathcal{P}}\mathrel{\mathop:}=\mathcal{P} \left[ \frac{1}{\pi} \right].$$ \item[$\cdot$] Multiple polylogarithms at algebraic arguments (in particular cyclotomic multiple zeta values), by their representation as iterated integral given below, are periods. Similarly, special values of Dedekind zeta function $\zeta_{F}(s)$ of a number field, of L-functions, of hypergeometric series, modular forms, etc. are (conjecturally at least) periods or extended periods. \item[$\cdot$] Periods also appear as Feynman integrals: Feynman amplitudes $I(D)$ can be written as a product of Gamma functions and meromorphic functions whose coefficients of its Laurent series expansion at any integer $D$ are periods (cf. $\cite{BB}$), where D is the dimension of spacetime. \end{itemize} Although most periods are transcendental, they are constructible; hence, the algebra $\mathcal{P}$ is countable, and any period contains only a finite amount of information. Conjecturally (by Grothendieck's conjecture), the only relations between periods comes from the following rules of elementary integral calculus\footnote{However, finding an algorithm to determine if a real number is a period, or if two periods are equal seems currently out of reach; whereas checking if a number is algebraic, or if two algebraic numbers are equal is rather \say{easy} (with \say{LLL}-type reduction algorithm, resp. by calculating the g.c.d of two vanishing polynomials associated to each).}: \begin{itemize} \item[$(i)$] Additivity (of the integrand and of the integration domain) \item[$(ii)$] Invertible changes of variables \item[$(iii)$] Stokes's formula.\\ \end{itemize}
Another way of viewing a period $ \boldsymbol{\int_{\gamma} \omega }$ is via a comparison between two cohomology theories: the algebraic \textit{De Rham} cohomology, and the singular (\textit{Betti}) cohomology. More precisely, let $X$ a smooth algebraic variety defined over $\mathbb{Q}$ and $Y$ a closed subvariety over $\mathbb{Q}$. \begin{itemize} \item[$\cdot$] On the one hand, the \textit{algebraic} De Rham cohomology $H^{\bullet}_{dR}(X)$\nomenclature{$H^{\bullet}_{dR}$}{the \textit{algebraic} De Rham cohomology} is the hypercohomology of the sheaf of algebraic (K{\"a}hler) differentials on $X$. If $X$ is affine, it is defined from the de Rham complex $\Omega^{\bullet}(X)$ which is the cochain complex of global algebraic (K{\"a}hler) differential forms on X, with the exterior derivative as differential. Recall that the \textit{classical} $k^{\text{th}}$ de Rham cohomology group is the quotient of smooth closed $k$-forms on the manifold X$_{\diagup \mathbb{C}}$ modulo the exact $k$-forms on $X$.\\ Given $\omega$ a closed algebraic $n$-form on $X$ whose restriction on $Y$ is zero, it defines an equivalence class $[\omega]$ in the relative de Rham cohomology groups $H^{n}_{dR}(X,Y)$, which are finite-dimensional $\mathbb{Q}-$ vector space. \item[$\cdot$] On the other hand, the Betti homology $H_{\bullet}^{B}(X)$\nomenclature{$H_{\bullet}^{B}(X)$}{the Betti homology} is the homology of the chain complex induced by the boundary operation of singular chains on the manifold $X(\mathbb{C})$; Betti cohomology groups $H_{B}^{n}(X,Y)= H^{B}_{n}(X,Y)^{\vee}$ are the dual $\mathbb{Q}$ vector spaces (taking here coefficients in $\mathbb{Q}$, not $\mathbb{Z}$).\\ Given $\gamma$ a singular $n$ chain on $X(\mathbb{C}) $ with boundary in $ Y(\mathbb{C})$, it defines an equivalence class $[\gamma]$ in the relative Betti homology groups $ H_{n}^{B}(X,Y)=H^{n}_{B}(X,Y)^{\vee}$.\footnote{Relative homology can be calculated using the following long exact sequence:
$$ \cdots \rightarrow H_{n}(Y) \rightarrow H_{n}(X) \rightarrow H_{n}(X,Y) \rightarrow H_{n-1} (Y) \rightarrow \cdots.$$
} \end{itemize} Furthermore, there is a comparison isomorphism\nomenclature{$\text{comp}_{B,dR}$}{the comparison isomorphism between de Rham and Betti cohomology} between relative de Rham and relative Betti cohomology (due to Grothendieck, coming from the integration of algebraic differential forms on singular chains): $$\text{comp}_{B,dR}: H^{\bullet}_{dR} (X,Y) \otimes_{\mathbb{Q}} \mathbb{C} \rightarrow H^{\bullet}_{B} (X,Y) \otimes_{\mathbb{Q}} \mathbb{C}.$$ By pairing a basis of Betti homology to a basis of de Rham cohomology, we obtain the \textit{matrix of periods}, which is a square matrix with entries in $\mathcal{P}$ and determinant in $\sqrt{\mathbb{Q}^{\ast}}(2i\pi)^{\mathbb{N}^{\ast}}$\nomenclature{$\mathbb{N}^{\ast}$}{the set of positive integers, $\mathbb{N}:=\mathbb{N}^{\ast}\cup\lbrace 0\rbrace$ }; i.e. its inverse matrix has its coefficients in $\widehat{\mathcal{P}}$. Then, up to the choice of these two basis: \begin{framed} The period $ \int_{\gamma} \omega $ is the coefficient of this pairing $\langle [\gamma], \text{comp}_{B,dR}([\omega]) \rangle$. \end{framed} \noindent \texttt{Example}: Let $X=\mathbb{P}^{1}\diagdown \lbrace 0, \infty \rbrace$, $Y=\emptyset$ and $\gamma_{0}$ the counterclockwise loop around $0$:
$$H^{B}_{i} (X)= \left\lbrace \begin{array}{ll} \mathbb{Q} & \text{ if } i=0 \\ \mathbb{Q}\left[\gamma_{0}\right] & \text{ if } i=1 \\ 0 & \text{ else }. \end{array} \right. \quad \text{ and } \quad H^{i}_{dR} (X)= \left\lbrace \begin{array}{ll} \mathbb{Q} & \text{ if } i=0 \\ \mathbb{Q}\left[ \frac{d}{dx}\right] & \text{ if } i=1 \\ 0 & \text{ else }. \end{array} \right. $$ Since $\int_{\gamma_{0}} \frac{dx}{x}= 2i\pi$, $2i\pi$ is a period; as we will see below, it is a period of the Lefschetz motive $\mathbb{L}\mathrel{\mathop:}=\mathbb{Q}(-1)$.\\ \\
Viewing periods from this cohomological point of view naturally leads to the definition of \textbf{\textit{motivic periods}} given below \footnote{The definition of a motivic period is given in $\S 2.4$ in the context of a category of Mixed Tate Motives. In general, one can do with Hodge theory to define $ \mathcal{P}^{\mathfrak{m}}$, which is not strictly speaking \textit{motivic}, once we specify that the mixed Hodge structures considered come from the cohomology of algebraic varieties.}, which form an algebra $\mathcal{P}^{\mathfrak{m}}$, equipped with a period homomorphism: $$\text{per}: \mathcal{P}^{\mathfrak{m}} \rightarrow \mathcal{P}.$$ A variant of Grothendieck's conjecture, which is a presently inaccessible conjecture in transcendental number theory, predicts that it is an isomorphism.\\ There is an action of a so-called motivic Galois group $\mathcal{G}$ on these motivic periods as we will see below in $\S 2.1$. If Grothendieck's period conjecture holds, this would hence extend the usual Galois theory for algebraic numbers to periods (cf. $\cite{An2}$). \\
In this thesis, we will focus on motivic (cyclotomic) multiple zeta values, defined in $\S 2.3$, which are motivic periods of the motivic (cyclotomic) fundamental group, defined in $\S 2.2$. Their images under this period morphism are the (cyclotomic) multiple zeta values; these are fascinating examples of periods, which are introduced in the next section (see also $\cite{An3}$). $$ \quad $$
\subsection{Multiple zeta values}
The Zeta function is known at least since Euler, and finds itself nowadays, in its various generalized forms (multiple zeta values, Polylogarithms, Dedekind zeta function, L-functions, etc), at the crossroad of many different fields as algebraic geometry (with periods and motives), number theory (notably with modular forms), topology, perturbative quantum field theory (with Feynman diagrams, cf. $\cite{Kr}$), string theory, etc. Zeta values at even integers are known since Euler to be rational multiples of even powers of $\pi$:
\begin{lemme} $$\text{For } n \geq 1, \quad \zeta(2n)=\frac{\mid B_{2n}\mid (2\pi)^{2n}}{2(2n)!}, \text{ where } B_{2n} \text{ is the } 2n^{\text{th}} \text { Bernoulli number.} $$ \end{lemme} However, the zeta values at odd integers already turn out to be quite interesting periods: \begin{conje} $\pi, \zeta(3), \zeta(5), \zeta(7),\cdots $ are algebraically independent. \end{conje} This conjecture raises difficult transcendental questions, rather out of reach; currently we only know $\zeta(3)\notin \mathbb{Q}$ (Ap\'{e}ry), infinitely many odd zeta values are irrational (Rivoal), or other quite partial results (Zudilin, Rivoal, etc.); recently, F. Brown paved the way for a pursuit of these results, in $\cite{Br5}$.\\ \\ \textbf{Multiple zeta values relative to the $\boldsymbol{N^{\text{th}}}$ roots of unity} $\mu_{N}$, \nomenclature{$\mu_{N}$}{$N^{\text{th}}$ roots of unity}which we shall denote by \textbf{MZV}$\boldsymbol{_{\mu_{N}}}$\nomenclature{\textbf{MZV}$\boldsymbol{_{\mu_{N}}}$}{Multiple zeta values relative to the $\boldsymbol{N^{\text{th}}}$ roots of unity, denoted $\zeta()$} are defined by: \footnote{Beware, there is no consensus on the order for the arguments of these MZV: sometimes the summation order is reversed.} \begin{framed} \begin{equation}\label{eq:mzv} \text{ } \zeta\left(n_{1}, \ldots , n_{p} \atop \epsilon_{1} , \ldots ,\epsilon_{p} \right)\mathrel{\mathop:}= \sum_{0<k_{1}<k_{2} \cdots <k_{p}} \frac{\epsilon_{1}^{k_{1}} \cdots \epsilon_{p}^{k_{p}}}{k_{1}^{n_{1}} \cdots k_{p}^{n_{p}}} \text{, } \epsilon_{i}\in \mu_{N} \text{, } n_{i}\in\mathbb{N}^{\ast}\text{, } (n_{p},\epsilon_{p})\neq (1,1). \end{equation} \end{framed} The \textit{weight}, often denoted $w$ below, is defined as $\sum n_{i}$, the \textit{depth} is the length $p$, whereas the \textit{height}, usually denoted $h$, is the number of $n_{i}$ greater than $1$. The weight is conjecturally a grading, whereas the depth is only a filtration. Denote also by $\boldsymbol{\mathcal{Z}^{N}}$\nomenclature{$\mathcal{Z}^{N}$}{the $\mathbb{Q}$-vector space spanned by the multiple zeta values relative to $\mu_{N}$} the $\mathbb{Q}$-vector space spanned by these multiple zeta values relative to $\mu_{N}$.\\ These MZV$_{\mu_{N}}$ satisfy both \textit{shuffle} $\shuffle$ relation (coming from the integral representation below) and \textit{stuffle} $\ast$ relation (coming from this sum expression), which turns $\mathcal{Z}^{N}$ into an algebra. These relations, for $N=1$, are conjectured to generate all the relations between MZV if we add the so-called Hoffman (regularized double shuffle) relation; cf. \cite{Ca}, \cite{Wa} for a good introduction to this aspect. However, the literature is full of other relations among these (cyclotomic) multiple zeta values: cf. $\cite{AO},\cite{EF}, \cite{OW}, \cite{OZa}, \cite{O1}, \cite{Zh2}, \cite{Zh3}$. Among these, we shall require the so-called pentagon resp. \textit{hexagon} relations (for $N=1$, cf. $\cite{Fu}$), coming from the geometry of moduli space of genus $0$ curves with $5$ ordered marked points $X=\mathcal{M}_{0,5}$ resp. with $4$ marked points $X=\mathcal{M}_{0,4}=\mathbb{P}^{1}\diagdown\lbrace 0,1, \infty\rbrace$ and corresponding to a contractible path in X; hexagon relation (cf. Figure $\ref{fig:hexagon}$) is turned into an \textit{octagon} relation (cf. Figure $\ref{fig:octagon}$) for $N>1$ (cf. $\cite{EF}$) and is used below in $\S 4.2$.\\ \\
One crucial point about multiple zeta values, is their \textit{integral representation}\footnote{Obtained by differentiating, considering there variables $z_{i}\in\mathbb{C}$, since: $$\frac{d}{dz_{p}} \zeta \left( n_{1}, \ldots, n_{p} \atop z_{1}, \ldots, z_{p-1}, z_{p}\right) = \left\lbrace \begin{array}{ll} \frac{1}{z_{p}} \zeta \left( n_{1}, \ldots, n_{p}-1 \atop z_{1}, \ldots, z_{p-1}, z_{p}\right) & \text{ if } n_{p}\neq 1\\ \frac{1}{1-z_{p}} \zeta \left( n_{1}, \ldots, n_{p-1} \atop z_{1}, \ldots, z_{p-1} z_{p}\right) & \text{ if } n_{p}=1. \end{array} \right. $$}, which makes them clearly \textit{periods} in the sense of Kontsevich-Zagier. Let us define first the following iterated integrals and differential forms, with $a_{i}\in \lbrace 0, \mu_{N} \rbrace$:\nomenclature{$I(0; a_{1}, \ldots , a_{n} ;1)$}{particular iterated integrals, with $a_{i}\in \lbrace 0, \mu_{N} \rbrace$} \begin{equation}\label{eq:iterinteg} \boldsymbol{I(0; a_{1}, \ldots , a_{n} ;1)}\mathrel{\mathop:}= \int_{0< t_{1} < \cdots < t_{n} < 1} \frac{dt_{1} \cdots dt_{n}}{ (t_{1}-a_{1}) \cdots (t_{n}-a_{n}) }=\int_{0}^{1} \omega_{a_{1}} \ldots \omega_{a_{n}} \text{, with } \omega_{a}\mathrel{\mathop:}=\frac{dt}{t-a}. \end{equation}\nomenclature{$\omega_{a}$}{$\mathrel{\mathop:}=\frac{dt}{t-a}$, differential form.} In this setting, with $\eta_{i}\mathrel{\mathop:}= (\epsilon_{i}\ldots \epsilon_{p})^{-1}\in\mu_{N}$, $n_{i}\in\mathbb{N}^{\ast}$\footnote{The use of bold in the iterated integral writing indicates a repetition of the corresponding number, as $0$ here.}: \begin{framed} \begin{equation}\label{eq:reprinteg} \zeta \left({ n_{1}, \ldots , n_{p} \atop \epsilon_{1} , \ldots ,\epsilon_{p} }\right) = (-1)^{p} I(0; \eta_{1}, \boldsymbol{0}^{n_{1}-1}, \eta_{2},\boldsymbol{0}^{n_{2}-1}, \ldots, \eta_{p}, \boldsymbol{0}^{n_{p}-1} ;1). \end{equation} \end{framed} \nomenclature{$\epsilon_{i}$, $\eta_{i}$}{\textit{(Usually)} The roots of unity corresponding to the MZV resp. to the iterated integral, i.e. $\eta_{i}\mathrel{\mathop:}= (\epsilon_{i}\cdots \epsilon_{p})^{-1}$.} \textsc{Remarks}: \begin{itemize} \item[$\cdot$] Multiple zeta values can be seen as special values of generalized multiple polylogarithms, when $\epsilon_{i}$ are considered in $\mathbb{C}$\footnote{The series is absolutely convergent for $ \mid \epsilon_{i}\mid <1$, converges also for $\mid \epsilon_{i}\mid =1$ if $n_{p} >1$. Cf. \cite{Os} for an introduction.}. First, notice that in weight $2$, $Li_{1}(z)\mathrel{\mathop:}=\zeta\left( 1\atop z\right) $ is the logarithm $-\log (1-z)$. Already the dilogarithm, in weight $2$, $Li_{2}(z)\mathrel{\mathop:}=\zeta \left( 2\atop z\right) =\sum_{k>0} \frac{z^{k}}{k^{2}}$, satisfies nice functional equations\footnote{As the functional equations with $Li_{2}\left( \frac{1}{z}\right) $ or $Li_{2}\left( 1-z \right) $ or the famous five terms relation, for its sibling, the Bloch Wigner function $D(z)\mathrel{\mathop:}= Im\left( Li_{2}(z) +\log(\mid z\mid) \log(1-z) \right)$: $$D(x)+D(y)+ D\left( \frac{1-x}{1-xy}\right) + D\left( 1-xy\right)+ D\left( \frac{1-y}{1-xy}\right)=0. $$} and arises in many places such as in the Dedekind zeta value $\zeta_{F}(2)$ for F an imaginary quadratic field, in the Borel regulator in algebraic K-theory, in the volume of hyperbolic manifolds, etc.; cf. $\cite{GZ}$; some of these connections can be generalized to higher weights. \item[$\cdot$] Recall that an iterated integral of closed (real or complex) differential $1-$forms $\omega_{i}$ along a path $\gamma$ on a 1-dimensional (real or complex) differential manifold $M$ is homotopy invariant, cf. $\cite{Ch}$. If $M=\mathbb{C}\diagdown \lbrace a_{1}, \ldots , a_{N}\rbrace$ \footnote{As for cyclotomic MZV, with $a_{i}\in \mu_{N}\cup\lbrace 0\rbrace$; such an $I=\int_{\gamma}\omega_{1}\ldots \omega_{n}$ is a multivalued function on $M$.} and $\omega_{i}$ are meromorphic closed $1-$forms, with at most simple poles in $a_{i}$, and $\gamma(0)=a_{1}$, the iterated integral $I=\int_{\gamma}\omega_{1}\cdots \omega_{n}$ is divergent. The divergence being polynomial in log $\epsilon$ ($\epsilon \ll 1$) \footnote{More precisely, we can prove that $\int_{\epsilon}^{1} \gamma^{\ast} (\omega_{1}) \cdots \gamma^{\ast} (\omega_{n}) = \sum_{i=0} \alpha_{i} (\epsilon) \log^{i} (\epsilon)$, with $\alpha_{i} (\epsilon)$ holomorphic in $\epsilon=0$; $\alpha_{0} (\epsilon) $ depends only on $\gamma'(0)$.}, we define the iterated integral I as the constant term, which only depends on $\gamma'(0)$. This process is called \textit{regularization}, we need to choose the tangential base points to properly define the integral. Later, we will consider the straight path $dch$ from $0$ to $1$, with tangential base point $\overrightarrow{1}$ at $0$ and $\overrightarrow{-1}$ at $1$, denoted also $\overrightarrow{1}_{0}, \overrightarrow{-1}_{1}$ or simply $\overrightarrow{01}$ for both. \end{itemize} \texttt{Notations}: In the case of \textit{multiple zeta values} (i.e. $N=1$) resp. of \textit{Euler sums} (i.e. $N=2$), since $\epsilon_{i}\in \left\{\pm 1\right\}$, the notation is simplified, using $ z_{i}\in \mathbb{Z}^{\ast}$:\nomenclature{ES}{Euler sums, i.e. multiple zeta values associated to $\mu_{2}=\lbrace\pm 1\rbrace$}
\begin{equation}\label{eq:notation2} \zeta\left(z_{1}, \ldots, z_{p} \right) \mathrel{\mathop:}= \zeta\left(n_{1}, \ldots , n_{p} \atop \epsilon_{1} , \ldots ,\epsilon_{p} \right)\text{ with } \left( n_{i} \atop \epsilon_{i} \right)\mathrel{\mathop:}=\left( \mid z_{i} \mid \atop sign(z_{i} ) \right) .
\end{equation} Another common notation in the literature is the use of \textit{overlines} instead of negative arguments, i.e.: $ z_{i}\mathrel{\mathop:}=\left\lbrace \begin{array}{ll} n_{i} &\text{ if } \epsilon_{i}=1\\ \overline{n_{i}} &\text{ if } \epsilon_{i}=-1 \end{array} \right. .$\nomenclature{$\overline{n}$}{another notation to denote a negative argument in the Euler sums: when the corresponding root of unity is $\epsilon=-1$.}
\section{Contents}
In this thesis, we mainly consider the \textit{\textbf{motivic}} versions of these multiple zeta values, denoted $\boldsymbol{\zeta^{\mathfrak{m}}}(\cdot)$ and shortened \textbf{MMZV}$\boldsymbol{_{\mu_{N}}}$ and defined in $\S 2.3$\nomenclature{\textbf{MMZV}$\boldsymbol{_{\mu_{N}}}$}{the motivic multiple zeta values relative to $\mu_{N}$, denoted $\zeta^{\mathfrak{m}}(\ldots)$, defined in $\S 2.3$}. They span a $\mathbb{Q}$-vector space $\boldsymbol{\mathcal{H}^{N}}$ of motivic multiple zetas relative to $\mu_{N}$\nomenclature{$\mathcal{H}^{N}$}{the $\mathbb{Q}$-vector space $\boldsymbol{\mathcal{H}^{N}}$ of motivic multiple zetas relative to $\mu_{N}$}. There is a surjective homomorphism, called the \textit{period map}, which is conjectured to be an isomorphism (this is a special case of the period conjecture)\nomenclature{per}{the surjective period map, conjectured to be an isomorphism.}: \begin{equation}\label{eq:per} \textbf{per} : w:\quad \mathcal{H}^{N} \rightarrow \mathcal{Z}^{N} \text{ , } \quad \zeta^{\mathfrak{m}} (\cdot) \mapsto \zeta ( \cdot ). \end{equation}
Working on the motivic side, besides being conjecturally identical to the complex numbers side, turns out to be somehow simpler, since motivic theory provides a Hopf Algebra structure as we will see throughout this thesis. Notably, each identity between motivic MZV$_{\mu_{N}}$ implies an identity for their periods; a motivic basis for MMZV$\mu_{N}$ is hence a generating family (conjecturally basis) for MZV$\mu_{N}$.
Indeed, on the side of motivic multiple zeta values, there is an action of a \textit{motivic Galois group} $\boldsymbol{\mathcal{G}}$\footnote{Later, we will define a category of Mixed Tate Motives, which will be a tannakian category: consequently equivalent to a category of representation of a group $\mathcal{G}$; cf. $\S 2.1$. }, which, passing to the dual, factorizes through a \textit{coaction} $\boldsymbol{\Delta}$ as we will see in $\S 2.4$. This coaction, which is given by an explicit combinatorial formula (Theorem $(\ref{eq:coaction})$, [Goncharov, Brown]), is the keystone of this PhD. In particular, it enables us to prove linear independence of MMZV, as in the theorem stated below (instead of adding yet another identity to the existing zoo of relations between MZV), and to study Galois descents. From this, we deduce results about numbers by applying the period map. \\ \\ This thesis is structured as follows: \begin{description} \item[Chapter $2$] sketches the background necessary to understand this work, from Mixed Tate Motives to the Hopf algebra of motivic multiple zeta values at $\mu_{N}$, with some specifications according the values of $N$, and results used throughout the rest of this work. The combinatorial expression of the coaction (or of the weight graded \textit{derivation operators} $D_{r}$ extracted from it, $(\ref{eq:Der})$) is the cornerstone of this work. We shall also bear in mind Theorem $2.4.4$ stating which elements are in the kernel of these derivations), which sometimes allows to lift identities from MZV to motivic MZV, up to rational coefficients, as we will see throughout this work.\\ \texttt{Nota Bene}: A \textit{motivic relation} is indeed stronger; it may hence require several relations between MZV in order to lift an identity to motivic MZV. An example of such a behaviour occurs with some Hoffman $\star$ elements, in Lemma $\ref{lemmcoeff}$.\\ \item[Chapter $3$] explains the main results of this PhD, ending with a wider perspective and possible future paths. \item[Chapter $4$] focuses on the cases $N=1$, i.e. multiple zeta values and $N=2$, i.e. Euler sums, providing some new bases: \begin{itemize} \item[$(i)$] First, we introduce \textit{Euler $\sharp$ sums}, variants of Euler sums, defined in $\S 2.3$ as in $(\ref{eq:reprinteg})$, replacing each $\omega_{\pm 1}$ by $\omega_{\pm \sharp}\mathrel{\mathop:}=2 \omega_{\pm 1}-\omega_{0}$, except for the first one and prove: \begin{theom} Motivic Euler $\sharp$ sums with only positive $odd$ and negative $even$ integers as arguments are unramified: i.e. motivic multiple zeta values. \end{theom} By application of the period map above: \begin{corol} Each Euler $\sharp$ sums with only positive $odd$ and negative $even$ integers as arguments is unramified, i.e. $\mathbb{Q}$ linear combination of multiple zeta values. \end{corol} Moreover, we can extract a basis from this family: \begin{theom} $ \lbrace \zeta^{\sharp, \mathfrak{m}} \left( 2a_{0}+1, 2a_{1}+3, \ldots, 2a_{p-1}+3 , -(2 a_{p}+2)\right) , a_{i} \geq 0 \rbrace$ is a graded basis of the space of motivic multiple zeta values. \end{theom} By application of the period map: \begin{corol} Each multiple zeta value is a $\mathbb{Q}$ linear combination of elements of the same weight in $\lbrace \zeta^{ \sharp} \left( 2a_{0}+1, 2a_{1}+3, \ldots, 2a_{p-1}+3 , -(2 a_{p}+2)\right) , a_{i} \geq 0 \rbrace$. \end{corol} \item[$(ii)$] We also prove the following, where Euler $\star$ sums are defined (cf. $\S 2.3$) as in $(\ref{eq:reprinteg})$, replacing each $\omega_{\pm 1}$ by $\omega_{\pm \star}\mathrel{\mathop:}=\omega_{\pm 1}-\omega_{0}$, except the first: \begin{theom} If the analytic conjecture ($\ref{conjcoeff}$) holds, then the motivic \textit{Hoffman} $\star$ family $\lbrace \zeta^{\star,\mathfrak{m}} (\lbrace 2,3 \rbrace^{\times})\rbrace$ is a basis of $\mathcal{H}^{1}$, the space of MMZV. \end{theom} \item[$(iii)$] Conjecturally, the two previous basis, namely the Hoffman $^{\star}$ family and the Euler$^{\sharp}$ family, are the same. Indeed, we conjecture a generalized motivic Linebarger-Zhao equality (Conjecture $\ref{lzg}$) which expresses each motivic multiple zeta $\star$ value as a motivic Euler $\sharp$ sum. It extends the Two One formula [Ohno-Zudilin], the Three One Formula [Zagier], and Linebarger Zhao formula, and applies to motivic MZV. If this conjecture holds, then $(i)$ implies that the Hoffman$^{\star}$ family is a basis. \end{itemize} Such results on linear independence of a family of motivic MZV are proved recursively, once we have found the \textit{appropriate level} filtration on the elements; ideally, the family considered is stable under the derivations \footnote{If the family is not \textit{a priori} \textit{stable} under the coaction, we need to incorporate in the recursion an hypothesis on the coefficients which appear when we express the right side with the elements of the family.}; the filtration, as we will see below, should correspond to the \textit{motivic depth} defined in $\S 2.4.3$, and decrease under the derivations \footnote{In the case of Hoffman basis ($\cite{Br2}$), or Hoffman $\star$ basis (Theorem $4.4.1$) it is the number of $3$, whereas in the case of Euler $\sharp$ sums basis (Theorems $4.3.2$), it is the depth minus one; for the \textit{Deligne} basis given in Chapter $5$ for $N=2,3,4, \mlq 6 \mrq ,8$, it is the usual depth. The filtration by the level has to be stable under the coaction, and more precisely, the derivations $D_{r}$ decrease the level on the elements of the conjectured basis, which allows a recursion.}; if the derivations, modulo some spaces, act as a deconcatenation on these elements, linear independence follows naturally from this recursion. Nevertheless, to start this procedure, we need an analytic identity\footnote{Where F. Brown in $\cite{Br2}$, for the Hoffman basis, used the analytic identity proved by Zagier in $\cite{Za}$, or $\cite{Li}$.}, which is left here as a conjecture in the case of the Hoffman $\star $ basis. This conjecture is of an entirely different nature from the techniques developed in this thesis. We expect that it could be proved using analytic methods along the lines of $\cite{Za}, \cite{Li}$.\\ \item[Chapter $5$] applies ideas of Galois descents on the motivic side. Originally, the notion of Galois descent was inspired by the question: which linear combinations of Euler sums are \textit{unramified}, i.e. multiple zeta values?\footnote{This was already a established question, studied by Broadhurst (which uses the terminology \textit{honorary}) among others. Notice that this issue surfaces also for motivic Euler sums in some results in Chapter $3$ and $5$.} More generally, looking at the motivic side, one can ask which linear combinations of MMZV$_{\mu_{N}}$ lie in MMZV$_{\mu_{N'}}$ for $N'$ dividing $N$. This is what we call \textit{descent} (the first level of a descent) and can be answered by exploiting the motivic Galois group action. General descent criteria are given; in the particular case of $N=2,3,4,\mlq 6 \mrq,8$\footnote{\texttt{Nota Bene}: $N=\mlq 6 \mrq$ is a special case; the quotation marks indicate here that we restrict to \textit{unramified} MMZV cf. $\S 2.1.1$.}, Galois descents are made explicit and our results lead to new bases of MMZV relative to $\mu_{N'}$ in terms of a basis of MMZV relative to $\mu_{N}$, and in particular, a new proof of P. Deligne's results $\cite{De}$.\\ Going further, we define ramification spaces which constitute a tower of intermediate spaces between the elements in MMZV$_{\mu_{N}}$ and the whole space of MMZV$_{\mu_{N'}}$. This is summed up in $\S 3.2$ and studied in detail Chapter $5$ or article $\cite{Gl1}$.\\ Moreover, as we will see below, these methods enable us to construct the motivic periods of categories of mixed Tate motives which cannot be reached by standard methods: i.e. are not simply generated by a motivic fundamental group. \item[Chapter $6$] gathers some applications of the coaction, from maximal depth terms, to motivic identities, via unramified motivic Euler sums; other potential applications of these Galois ideas to the study of these periods are still waiting to be further investigated. \\ \\ \\ \end{description} \texttt{\textbf{Consistency:}}\\ Chapter $2$ is fundamental to understand the tools and the proofs of both Chapter $4$, $5$ and $6$ (which are independent between them), but could be skimmed through before the reading of the main results in Chapter $3$. The proofs of Chapter $4$ are based on the results of Annexe $A.1$, but could be read independently.
\chapter{Background}
\section{Motives and Periods}
Here we sketch the motivic background where the motivic iterated integrals (and hence this work) mainly take place; although most of it can be taken as a black box. Nevertheless, some of the results coming from this rich theory are fundamental to our proofs.
\subsection{Mixed Tate Motives}
\paragraph{Motives in a nutshell.} Motives are supposed to play the role of a universal (and algebraic) cohomology theory (see $\cite{An}$). This hope is partly nourished by the fact that, between all the classical cohomology theories (de Rham, Betti, $l$-adique, crystalline), we have comparison isomorphisms in characteristic $0$ \footnote{Even in positive characteristic, $\dim H^{i}(X)$ does not depend on the cohomology chosen among these.}. More precisely, the hope is that there should exist a tannakian (in particular abelian, tensor) category of motives $\mathcal{M}(k)$, and a functor $\text{Var}_{k} \xrightarrow{h} \mathcal{M}(k) $ such that:\\ \textit{For each Weil cohomology}\footnote{This functor should verify some properties, such as Kunneth formula, Poincare duality, etc. as the classic cohomology theories. \\ If we restrict to smooth projective varieties, $\text{SmProj}_{k}$, we can construct such a category, the category of pure motives $ \mathcal{M}^{pure}(k)$ starting from the category of correspondence of degree $0$. For more details, cf. $\cite{Ka}$.}: $\text{Var}_{k} \xrightarrow{H} \text{Vec}_{k}$, \textit{there exists a realization map $w_{H}$ such that the following commutes}:\nomenclature{$\text{Var}_{k}$}{the category of varieties over k}\nomenclature{$\text{Vec}_{k}$}{the category of $k$-vector space of finite dimension}\nomenclature{$\text{SmProj}_{k}$}{the category of smooth projective varieties over k.} $$\xymatrix{ \text{Var}_{k} \ar[d]^{\forall H} \ar[r]^{ h} & \mathcal{M}(k) \ar[dl]^{\exists w_{H}}\\ \text{Vec}_{K} & },$$ \textit{where $h$ satisfy properties such as }$h(X\times Y)=h(X)\oplus h(Y)$, $h(X \coprod Y)= h(X)\otimes h(Y)$. The realizations functors are conjectured to be full and faithful (conjecture of periods of Grothendieck, Hodge conjecture, Tate conjecture, etc.)\footnote{In the case of Mixed Tate Motives over number fields as seen below, Goncharov proved it for Hodge and l-adique Tate realizations, from results of Borel and Soule.}.\\ To this end, Voedvosky (cf. $\cite{Vo}$) constructed a triangulated category of Mixed Motives $DM^{\text{eff}}(k)_{\mathbb{Q}}$, with rational coefficients, equipped with tensor product and a functor: \begin{center} $M_{gm}: \text{Sch}_{\diagup k} \rightarrow DM^{\text{eff}}$ satisfying some properties such as: \end{center} \begin{description} \item[Kunneth] $M_{gm}(X \times Y)=M_{gm}(X)\otimes M_{gm}(Y)$. \item[$\mathbb{A}^{1}$-invariance] $M_{gm}(X \times \mathbb{A}^{1})= M_{gm}(X)$. \item[Mayer Vietoris] $M_{gm}(U\cap V)\rightarrow M_{gm}(U) \otimes M_{gm}(V) \rightarrow M_{gm}(U\cup V)\rightarrow M_{gm}(U\cap V)[1] $, $U,V$ open, is a distinguished triangle.\footnote{Distinguished triangles in $DMT^{\text{eff}}(k)$, i.e. of type Tate, become exact sequences in $\mathcal{MT}(k)$.} \item[Gysin] $M_{gm}(X\diagdown Z)\rightarrow M_{gm}(X) \rightarrow M_{gm}(Z)(c)[2c]\rightarrow M_{gm}(X\diagdown Z)[1] $, $X$ smooth, $Z$ smooth, closed, of codimension $c$, is a distinguished triangle. \end{description} We would like to extract from the triangulated category $DM^{\text{eff}}(k)_{\mathbb{Q}}$ an abelian category of Mixed Motives over k\footnote{ A way would be to define a $t$ structure on this category, and the heart of the t-structure, by Bernstein, Beilinson, Deligne theorem is a full admissible abelian sub-category.}. However, we still are not able to do it in the general case, but it is possible for some triangulated tensor subcategory of type Tate, generated by $\mathbb{Q}(n)$ with some properties.\\ \\ \textsc{Remark}: $\mathbb{L}\mathrel{\mathop:}=\mathbb{Q}(-1)= H^{1}(\mathbb{G}_{\mathfrak{m}})=H^{1}(\mathbb{P}^{1} \diagdown \lbrace 0, \infty\rbrace)$\nomenclature{$\mathbb{G}_{\mathfrak{m}}$}{the multiplicative group} which is referred to as the \textit{Lefschetz motive}, is a pure motive, and has period $(2i\pi)$. Its dual is the so-called \textit{Tate motive} $\mathbb{T}\mathrel{\mathop:}=\mathbb{Q}(1)=\mathbb{L}^{\vee}$. More generally, let us define $\mathbb{Q}(-n)\mathrel{\mathop:}= \mathbb{Q}(-1)^{\otimes n}$ resp. $\mathbb{Q}(n)\mathrel{\mathop:}= \mathbb{Q}(1)^{\otimes n}$ whose periods are in $(2i\pi)^{n} \mathbb{Q}$ resp. $(\frac{1}{2i\pi})^{n} \mathbb{Q}$, hence extended periods in $\widehat{P}$; we have the decomposition of the motive of the projective line: $h(\mathbb{P}^{n})= \oplus _{k=0}^{n}\mathbb{Q}(-k)$.
\paragraph{Mixed Tate Motives over a number field.} Let's first define, for $k$ a number field, the category $\mathcal{DM}(k)_{\mathbb{Q}}$ from $\mathcal{DM}^{\text{eff}}(k)_{\mathbb{Q}}$ by formally \say{inverting} the Tate motive $\mathbb{Q}(1)$, and then $\mathcal{DMT}(k)_{\mathbb{Q}}$ as the smallest triangulated full subcategory of $\mathcal{DM}(k)_{\mathbb{Q}}$ containing $\mathbb{Q}(n), n\in\mathbb{Z}$ and stable by extension.\\ By the vanishing theorem of Beilinson-Soule, and results from Levine (cf. $\cite{Le}$), there exists:\footnote{A \textit{tannakian} category is abelian, $k$-linear, tensor rigid (autoduality), has an exact faithful fiber functor, compatible with $\otimes$ structures, etc. Cf. $\cite{DM}$ about Tannakian categories.} \begin{framed} A tannakian \textit{category of Mixed Tate motives} over k with rational coefficients, $\mathcal{MT}(k)_{\mathbb{Q}}$\nomenclature{$\mathcal{MT}(k)$}{category of Mixed Tate Motives over $k$} and equipped with a weight filtration $W_{r}$ indexed by even integers such that $gr_{-2r}^{W}(M)$ is a sum of copies of $\mathbb{Q}(r)$ for $M\in \mathcal{MT}(k)$, i.e.,\\
Every object $M\in \mathcal{MT}(k)_{\mathbb{Q}}$ is an iterated extension of Tate motives $\mathbb{Q}(n), n\in\mathbb{Z}$. \end{framed} such that (by the works of Voedvodsky, Levine $\cite{Le}$, Bloch, Borel (and K-theory), cf. $\cite{DG}$): $$\begin{array}{ll} \text{Ext}^{1}_{\mathcal{MT}(k)}(\mathbb{Q}(0),\mathbb{Q}(n) )\cong K_{2n-1}(k)_{\mathbb{Q}} \otimes \mathbb{Q} \cong & \left\lbrace \begin{array}{ll} k^{\ast}\otimes_{\mathbb{Z}} \mathbb{Q} & \text{ if } n=1 .\\
\mathbb{Q}^{r_{1}+r_{2}} & \text{ if } n>1 \text{ odd }\\ \mathbb{Q}^{r_{2}} & \text{ if } n>1 \text{ even } \end{array} \right. . \\ \text{Ext}^{i}_{\mathcal{MT}(k)}(\mathbb{Q}(0),\mathbb{Q}(n) )\cong 0 & \quad \text{ if } i>1 \text{ or } n\leq 0.\\ \end{array}$$ Here, $r_{1}$ resp $r_{2}$ stand for the number of real resp. complex (and non real, up to conjugate) embeddings from $k$ to $\mathbb{C}$.\\ In particular, the weight defines a canonical fiber functor\nomenclature{$\omega$}{the canonical fiber functor}: $$\begin{array}{lll} \omega: & \mathcal{MT}(k) \rightarrow \text{Vec}_{\mathbb{Q}} & \\
&M \mapsto \oplus \omega_{r}(M) & \quad \quad \text{ with } \left\lbrace \begin{array}{l}
\omega_{r}(M)\mathrel{\mathop:}= \text{Hom}_{\mathcal{MT}(k)}(\mathbb{Q}(r), gr_{-2r}^{W}(M))\\
\text{ i.e. } \quad gr_{-2r}^{W}(M)= \mathbb{Q}(r)\otimes \omega_{r}(M). \end{array} \right. \end{array} $$ \\ The category of Mixed Tate Motives over $k$, since tannakian, is equivalent to the category of representations of the so-called \textit{\textbf{motivic Galois group}} $\boldsymbol{\mathcal{G}^{\mathcal{MT}}}$\nomenclature{$\mathcal{G}^{\mathcal{M}}$}{the motivic Galois group of the category of Mixed Tate Motives $\mathcal{M}$ } of $\mathcal{MT}(k)$ \footnote{With the equivalence of category between $A$ Comodules and Representations of the affine group scheme $\text{Spec}(A)$, for $A$ a Hopf algebra. Note that $\text{Rep}(\mathbb{G}_{m})$ is the category of $k$-vector space $\mathbb{Z}$-graded of finite dimension.}:\nomenclature{$\text{Rep}(\cdot)$}{the category of finite representations} \begin{framed} \begin{equation}\label{eq:catrep} \mathcal{MT}(k)_{\mathbb{Q}}\cong \text{Rep}_{k} \mathcal{G}^{\mathcal{MT}} \cong \text{Comod } (\mathcal{O}(\mathcal{G}^{\mathcal{MT}})) \quad \text{ where } \mathcal{G}^{\mathcal{MT}}\mathrel{\mathop:}=\text{Aut}^{\otimes } \omega . \end{equation} \end{framed} The motivic Galois group $\mathcal{G}^{\mathcal{MT}}$ decomposes as, since $\omega$ is graded: \begin{center} $\mathcal{G}^{\mathcal{MT}}= \mathbb{G}_{m} \ltimes \mathcal{U}^{\mathcal{MT}}$, $\quad \text{i.e. } \quad 1 \rightarrow \mathcal{U}^{\mathcal{MT}} \rightarrow \mathcal{G}^{\mathcal{MT}} \leftrightarrows \mathbb{G}_{m} \rightarrow 1 \quad \textit{ is an exact sequence, }$ \end{center} \begin{center} \textit{where} $\mathcal{U}^{\mathcal{MT}}$\nomenclature{$\mathcal{U}^{\mathcal{M}}$}{the pro-unipotent part of the motivic Galois group $\mathcal{G}^{\mathcal{M}}$.} \textit{is a pro-unipotent group scheme defined over }$\mathbb{Q}$. \end{center} The action of $\mathbb{G}_{m}$ is a grading, and $\mathcal{U}^{\mathcal{MT}}$ acts trivially on the graded pieces $\omega(\mathbb{Q}(n))$.\\ \\ Let $\mathfrak{u}$ denote the completion of the pro-nilpotent graded Lie algebra of $\mathcal{U}^{\mathcal{MT}}$ (defined by a limit); $\mathfrak{u}$ is free and graded with negative degrees from the $\mathbb{G}_{m}$-action. Furthermore\footnote{Since $\text{Ext}^{2}_{\mathcal{MT}} (\mathbb{Q}(0), \mathbb{Q}(n))=0$, which implies $\forall M$, $H^{2}(\mathfrak{u},M)=0$, hence $\mathfrak{u}$ free. Moreover, $(\mathfrak{u}^{ab})= \left( \mathfrak{u} \diagup [\mathfrak{u}, \mathfrak{u}]\right)= H_{1}(\mathfrak{u}; \mathbb{Q})$, then, for $\mathcal{U}$ unipotent: $$\left( \mathfrak{u}^{ab}\right)^{\vee}_{m-n} \cong \text{Ext}^{1}_{\text{Rep} _{\mathbb{Q}}} (\mathbb{Q}(n), \mathbb{Q}(m)).$$}: \begin{equation} \label{eq:uab} \mathfrak{u}^{ab} \cong \bigoplus \text{Ext}^{1}_{\mathcal{MT}} (\mathbb{Q}(0), \mathbb{Q}(n))^{\vee} \text{ in degree } n. \end{equation} Hence the \textit{fundamental Hopf algebra} is\nomenclature{$\mathcal{A}^{\mathcal{M}}$}{the fundamental Hopf algebra of $\mathcal{M}$} \footnote{Recall the anti-equivalence of Category, between Hopf Algebra and Affine Group Schemes: $$\xymatrix@R-1pc{ k-Alg^{op} \ar[r]^{\sim} & k-\text{AffSch } \\ k-\text{HopfAlg}^{op} \ar@{^{(}->}[u] \ar[r]^{\sim} & k-\text{ AffGpSch } \ar@{^{(}->}[u]\\
A \ar@{|->}[r] & \text{ Spec } A \\
\mathcal{O}(G) & G: R \mapsto \text{Hom}_{k}(\mathcal{O}(G),R) \ar@{|->}[l] }. $$ It comes from the fully faithful Yoneda functor $C^{op} \rightarrow \text{Fonct}(C, \text{Set})$, leading to an equivalence of Category if we restrict to Representable Functors: $k-\text{AffGpSch } \cong \text{RepFonct } (\text{Alg }^{op},Gp)$. Properties for Hopf algebra are obtained from Affine Group Scheme properties by 'reversing the arrows' in each diagram.\\ Remark that $G$ is unipotent if and only if $A$ is commutative, finite type, connected and filtered.}: \begin{equation} \label{eq:Amt} \mathcal{A}^{\mathcal{MT}}\mathrel{\mathop:}=\mathcal{O}(\mathcal{U}^{\mathcal{MT}})\cong (U^{\wedge} (\mathfrak{u}))^{\vee} \cong T(\oplus_{n\geq 1} \text{Ext}^{1}_{\mathcal{MT}_{N}} (\mathbb{Q}(0), \mathbb{Q}(n))^{\vee} ). \end{equation} Hence, by the Tannakian dictionary $(\ref{eq:catrep})$: $\mathcal{MT}(k)_{\mathbb{Q}}\cong \text{Rep}_{k}^{gr} \mathcal{U}^{\mathcal{MT}} \cong \text{Comod}^{gr} \mathcal{A}^{\mathcal{MT}} .$ \\ \\ Once an embedding $\sigma: k \hookrightarrow \mathbb{C}$ is fixed, Betti cohomology leads to a functor \textit{Betti realization}:\nomenclature{$\omega_{B_{\sigma}}$}{Betti realization functor} $$\omega_{B_{\sigma}}: \mathcal{MT}(k) \rightarrow \text{Vec}_{\mathbb{Q}} ,\quad M \mapsto M_{\sigma}.$$ De Rham cohomology leads similarly to the functor \textit{de Rham realization}\nomenclature{$\omega_{dR}$}{de Rham realization functor}: $$\omega_{dR}: \mathcal{MT}(k) \rightarrow \text{Vec}_{k} , \quad M \mapsto M_{dR} \text{ , } \quad M_{dR} \text{ weight graded}.$$ Beware, the de Rham functor $\omega_{dR}$ here is not defined over $\mathbb{Q}$ but over $k$ and $\omega_{dR}=\omega \otimes_{\mathbb{Q}} k$, so the de Rham realization of an object $M$ is $M_{dR}=\omega(M)\otimes_{\mathbb{Q}} k$.\\ Between all these realizations, we have comparison isomorphisms, such as: $$ M_{\sigma}\otimes_{\mathbb{Q}} \mathbb{C} \xrightarrow[\sim]{\text{comp}_{dR, \sigma}} M_{dR} \otimes_{k,\sigma} \mathbb{C} \text{ with its inverse } \text{comp}_{\sigma,dR}.$$ $$ M_{\sigma}\otimes_{\mathbb{Q}} \mathbb{C} \xrightarrow[\sim]{\text{comp}_{\omega, B_{\sigma}}} M_{\omega} \otimes_{\mathbb{Q}} \mathbb{C} \text{ with its inverse } \text{comp}_{B_{\sigma}, \omega}.$$ Define also, looking at tensor-preserving isomorphisms: $$\begin{array}{lll} \mathcal{G}_{B}\mathrel{\mathop:}=\text{Aut}^{\otimes}(\omega_{B}), & \text{resp. } \mathcal{G}_{dR}\mathrel{\mathop:}=\text{Aut}^{\otimes}(\omega_{dR}) & \\ P_{\omega, B}\mathrel{\mathop:}=\text{Isom}^{\otimes}(\omega_{B},\omega), & \text{resp. } P_{B,\omega}\mathrel{\mathop:}=\text{Isom}^{\otimes}(\omega,\omega_{B}), & (\mathcal{G}^{\mathcal{MT}}, \mathcal{G}_{B}) \text{ resp. } (\mathcal{G}_{B}, \mathcal{G}^{\mathcal{MT}}) \text{ bitorsors }. \end{array}$$ Comparison isomorphisms above define $\mathbb{C}$ points of these schemes: $\text{comp}_{\omega,B}\in P_{B,\omega} (\mathbb{C})$.\\ \\ \textsc{Remarks}: By $(\ref{eq:catrep})$:\footnote{The different cohomologies should be viewed as interchangeable realizations. Etale chomology, with the action of the absolute Galois group $\text{Gal}(\overline{\mathbb{Q}}\diagup\mathbb{Q})$ (cf $\cite{An3}$) is related to the number $N_{p}$ of points of reduction modulo $p$. For Mixed Tate Motives (and conjecturally only for those) $N_{p}$ are polynomials modulo $p$, which is quite restrictive.} \begin{framed} A Mixed Tate motive over a number field is uniquely defined by its de Rham realization, a vector space $M_{dR}$, with an action of the motivic Galois group $\mathcal{G}^{\mathcal{MT}}$. \end{framed}
\texttt{Example:} For instance $\mathbb{Q}(n)$, as a Tate motive, can be seen as the vector space $\mathbb{Q}$ with the action $\lambda\cdot x\mathrel{\mathop:}= \lambda^{n} x$, for $\lambda\in\mathbb{Q}^{\ast}=\text{Aut}(\mathbb{Q})=\mathbb{G}_{m}(\mathbb{Q})$.
\paragraph{Mixed Tate Motives over $\mathcal{O}_{S}$.}
Before, let's recall for $k$ a number field and $\mathcal{O}$\nomenclature{$\mathcal{O}_{k}$}{ring of integers of $k$} its ring of integers, \textit{archimedian values} of $k$ are associated to an embedding $k \xhookrightarrow{\sigma} \mathbb{C}$, such that: $$\mid x \mid \mathrel{\mathop:}= \mid\sigma(x)\mid_{\infty}\text{ , where } \mid\cdot \mid_{\infty} \text{ is the usual absolute value},$$ and \textit{non archimedian values} are associated to non-zero prime ideals of $\mathcal{O}$\footnote{$ \mathcal{O}$ is a Dedekind domain, $ \mathcal{O}_{\mathfrak{p}}$ a discrete valuation ring whose prime ideals are prime ideals of $\mathcal{O}$ which are included in $(\mathfrak{p})\mathcal{O}_{\mathfrak{p}}$.}: $$v_{\mathfrak{p}}: k^{\times} \rightarrow \mathbb{Z}, \quad v_{\mathfrak{p}}(x) \text{ is the integer such that } x \mathcal{O}_{\mathfrak{p}} = \mathfrak{p}^{v_{\mathfrak{p}}(x)} \mathcal{O}_{\mathfrak{p}} \text{ for } x\in k^{\times}.$$ For $S$ a finite set of absolute values in $k$ containing all archimedian values, the \textit{ring of S-integers}\nomenclature{$\mathcal{O}_{S}$}{ring of $S$-integers}: $$\mathcal{O}_{S}\mathrel{\mathop:}= \left\lbrace x\in k \mid v(x)\geq 0 \text{ for all valuations } v\notin S \right\rbrace. $$ Dirichlet unit's theorem generalizes for $\mathcal{O}^{\times}_{S}$, abelian group of type finite:\nomenclature{$\mu(K)$}{the finite cyclic group of roots of unity in $K$} \footnote{ It will be used below, for dimensions, in $\ref{dimensionk}$. Here, $\text{ card } (S)= r_{1}+r_{2}+\text{ card }(\text{non-archimedian places}) $; as usual, $r_{1}, r_{2}$ standing for the number of real resp. complex (and non real, and up to conjugate) embeddings from $k$ to $\mathbb{C}$; $\mu(K) $ is the finite cyclic group of roots of unity in $K$.} $$\mathcal{O}_{S}^{\times} \cong \mu(K) \times \mathbb{Z}^{\text{ card }(S)-1}.$$ \texttt{Examples}: \begin{itemize} \item[$\cdot$] Taking S as the set of the archimedian values leads to the usual ring of integers $\mathcal{O}$, and would lead to the unramified category of motives $\mathcal{MT}(\mathcal{O})$ below. \item[$\cdot$] For $k=\mathbb{Q}$, $p$ prime, with $S=\lbrace v_{p}, \mid \cdot \mid_{\infty}\rbrace$, we obtain $\mathbb{Z}\left[ \frac{1}{p} \right] $. Note that the definition does not allow to choose $S= \lbrace \cup_{q \text{prime} \atop q\neq p } v_{q}, \mid \cdot \mid_{\infty}\rbrace$, which would lead to the localization $\mathbb{Z}_{(p)}\mathrel{\mathop:}=\lbrace x\in\mathbb{Q} \mid v_{p}(x) \geq 0\rbrace$. \end{itemize} Now, let us define the categories of Mixed Tate Motives which interest us here:\nomenclature{$\mathcal{MT}_{\Gamma}$}{a tannakian category associated to $\Gamma$ a sub-vector space of $\text{Ext}^{1}_{\mathcal{MT}(k)}( \mathbb{Q}(0), \mathbb{Q}(1))$} \begin{defin}\label{defimtcat} \begin{description} \item[$\boldsymbol{\mathcal{MT}_{\Gamma}}$:] For $\Gamma$ a sub-vector space of $\text{Ext}^{1}_{\mathcal{MT}(k)}( \mathbb{Q}(0), \mathbb{Q}(1)) \cong k^{\ast}\otimes \mathbb{Q}$: \begin{center} $\mathcal{MT}_{\Gamma}:$ the tannakian subcategory formed by objects $M$ such that each subquotient $E$ of $M$: $$0 \rightarrow \mathbb{Q}(n+1) \rightarrow E \rightarrow \mathbb{Q}(n) \rightarrow 0 \quad \Rightarrow \quad [E]\in \Gamma \subset \text{Ext}^{1}_{\mathcal{MT}(k)}( \mathbb{Q}(0), \mathbb{Q}(1)) \footnote{$\text{Ext}^{1}_{\mathcal{MT}(k)}( \mathbb{Q}(0), \mathbb{Q}(1)) \cong \text{Ext}^{1}_{\mathcal{MT}(k)}( \mathbb{Q}(n), \mathbb{Q}(n+1)).$}.$$ \end{center}
\item[$\boldsymbol{\mathcal{MT}(\mathcal{O}_{S})}$:] The category of mixed Tate motives unramified in each finite place $v\notin S$: \begin{center} $\mathcal{MT}(\mathcal{O}_{S})\mathrel{\mathop:}=\mathcal{MT}_{\Gamma}, \quad $ for $\Gamma=\mathcal{O}_{S}^{\ast}\otimes \mathbb{Q}$. \end{center} \end{description} \end{defin} Extension groups for these categories are then identical to those of $\mathcal{MT}(k)$ except:
\begin{equation} \label{eq:extension} \text{Ext}^{1}_{\mathcal{MT}_{\Gamma}}( \mathbb{Q}(0), \mathbb{Q}(1))= \Gamma, \quad \text{ resp. } \quad \text{Ext}^{1}_{\mathcal{MT}(\mathcal{O}_{S})}( \mathbb{Q}(0), \mathbb{Q}(1))= K_{1}(\mathcal{O}_{S})\otimes \mathbb{Q}=\mathcal{O}^{\ast}_{S}\otimes \mathbb{Q}. \end{equation}
\paragraph{Cyclotomic Mixed Tate Motives.} In this thesis, we focus on the cyclotomic case and consider the following categories, and sub-categories, for $k_{N}$\nomenclature{$k_{N}$}{the $N^{\text{th}}$ cyclotomic field} the $N^{\text{th}}$ cyclotomic field, $\mathcal{O}_{N}\mathrel{\mathop:}= \mathbb{Z}[\xi_{N}]$\nomenclature{$\mathcal{O}_{N}$}{the ring of integers of $k_{N}$ i.e. $ \mathbb{Z}[\xi_{N}]$ } its ring of integers, with $\xi_{N}$\nomenclature{$\xi_{N}$}{a primitive $N^{\text{th}}$ root of unity} a primitive $N^{\text{th}}$ root of unity:\nomenclature{$\mathcal{MT}_{N,M}$}{the tannakian Mixed Tate subcategory of $\mathcal{MT}(k_{N})$ ramified in $M$} \begin{framed} $$\begin{array}{ll} \boldsymbol{\mathcal{MT}_{N,M}} & \mathrel{\mathop:}= \mathcal{MT} \left( \mathcal{O}_{N} \left[ \frac{1}{M}\right] \right).\\ \boldsymbol{\mathcal{MT}_{\Gamma_{N}}}, & \text{ with $\Gamma_{N}$ the $\mathbb{Q}$-sub vector space of } \left( \mathcal{O}\left[ \frac{1}{N} \right] \right) ^{\ast} \otimes \mathbb{Q}\\ & \text{ generated by $\lbrace 1-\xi^{a}_{N}\rbrace_{0< a < N}$ (modulo torsion). } \end{array}$$ \end{framed} \noindent Hence: $$ \mathcal{MT} \left( \mathcal{O}_{N} \right) \subsetneq \mathcal{MT}_{\Gamma_{N}} \subset \mathcal{MT} \left( \mathcal{O}_{N} \left[ \frac{1}{N}\right] \right)$$ The second inclusion is an equality if and only if $N$ has all its prime factors inert\footnote{I.e. each prime $p$ dividing $N$, generates $(\mathbb{Z} /m \mathbb{Z})^{\ast}$, for $m$ such as $N=p^{v_{p}(N)} m$.\nomenclature{$v_{p}(\cdot)$}{$p$-adic valuation} It could occur only in the following cases: $N= p^{s}, 2p^{s}, 4p^{s}, p^{s}q^{k}$, with extra conditions in most of these cases such as: $2$ is a primitive root modulo $p^{s}$ etc.}, since: \begin{equation}\label{eq:gamma} \Gamma_{N}= \left\lbrace \begin{array}{ll} \left( \mathcal{O}\left[ \frac{1}{p} \right] \right) ^{\ast} \otimes \mathbb{Q} & \text{ if } N = p^{r} \\ (\mathcal{O} ^{\ast} \otimes \mathbb{Q} )\oplus \left( \oplus_{ p \text{ prime } \atop p\mid N} \langle p \rangle\otimes \mathbb{Q} \right) &\text{ else .} \end{array} \right. . \end{equation} The motivic cyclotomic MZVs lie in the subcategory $\mathcal{MT}_{\Gamma_{N}}$, as we will see more precisely in $\S 2.3$.\\ \\ \texttt{Notations:} We may sometimes drop the $M$ (or even $N$), to lighten the notations:\footnote{For instance, $\mathcal{MT}_{3}$ is the category $\mathcal{MT} \left( \mathcal{O}_{3} \left[ \frac{1}{3} \right] \right) $.}:\nomenclature{$\mathcal{MT}_{N}$}{a tannakian Mixed Tate subcategory of $\mathcal{MT}(k_{N})$}\\ $$\mathcal{MT}_{N}\mathrel{\mathop:}= \left\lbrace \begin{array}{ll} \mathcal{MT}_{N,N} & \text{ if } N=2,3,4,8\\ \mathcal{MT}_{6,1} & \text{ if } N=\mlq 6 \mrq. \\ \end{array} \right. $$
\subsection{Motivic periods} Let $\mathcal{M}$ a tannakian category of mixed Tate motives. Its \textit{algebra of motivic periods} is defined as (cf. $\cite{D1}$, $\cite{Br6}$, and $\cite{Br4}$, $\S 2$):\nomenclature{$\mathcal{P}_{\mathcal{M}}^{\mathfrak{m}}$}{the algebra of motivic period of a tannakian category of MTM $\mathcal{M}$} $$\boldsymbol{\mathcal{P}_{\mathcal{M}}^{\mathfrak{m}}}\mathrel{\mathop:}=\mathcal{O}(\text{Isom}^{\otimes}_{\mathcal{M}}(\omega, \omega_{B}))=\mathcal{O}(P_{B,\omega}).$$ \begin{framed} A \textbf{\textit{motivic period}} denoted as a triplet $\boldsymbol{\left[M,v,\sigma \right]^{\mathfrak{m}}}$,\nomenclature{$\left[M,v,\sigma \right]^{\mathfrak{m}}$}{a motivic period} element of $\mathcal{P}_{\mathcal{M}}^{\mathfrak{m}}$, is constructed from a motive $M\in \text{ Ind } (\mathcal{M})$, and classes $v\in\omega(M)$, $\sigma\in\omega_{B}(M)^{\vee}$. It is a function $P_{B,\omega} \rightarrow \mathbb{A}^{1}$, which, on its rational points, is given by: \begin{equation}\label{eq:mper} \quad P_{B,\omega} (\mathbb{Q}) \rightarrow \mathbb{Q}\text{ , } \quad \alpha \mapsto \langle \alpha(v), \sigma\rangle . \end{equation} Its \textit{period} is obtained by the evaluation on the complex point $\text{comp}_{B, dR}$: \begin{equation}\label{eq:perm} \begin{array}{lll} \mathcal{P}_{\mathcal{M}}^{\mathfrak{m}} & \rightarrow & \mathbb{C} \\ \left[M,v,\sigma \right]^{\mathfrak{m}} & \mapsto & \langle \text{comp}_{B,dR} (v\otimes 1), \sigma \rangle . \end{array} \end{equation} \end{framed} \noindent \texttt{Example}: The first example is the \textit{Lefschetz motivic period}: $\mathbb{L}^{\mathfrak{m}}\mathrel{\mathop:}=[H^{1}(\mathbb{G}_{m}), [\frac{dx}{x}], [\gamma_{0}]]^{\mathfrak{m}}$, period of the Lefschetz motive $\mathbb{L}$; it can be seen as the \textit{motivic} $(2 i\pi)^{\mathfrak{m}}$; this notation appears below.\nomenclature{$\mathbb{L}$}{Lefschetz motive, $\mathbb{L}^{\mathfrak{m}}$ the Lefschetz motivic period}\\ \\ This construction can be generalized for any pair of fiber functors $\omega_{1}$, $\omega_{2}$ leading to: \begin{center} \textit{Motivic periods of type} $(\omega_{1},\omega_{2})$, which are in the following algebra of motivic periods: $$\boldsymbol{\mathcal{P}_{\mathcal{M}}^{\omega_{1},\omega_{2}}}\mathrel{\mathop:}= \mathcal{O}\left( P_{\omega_{1},\omega_{2}}\right) = \mathcal{O}\left( \text{Isom}^{\otimes}(\omega_{2}, \omega_{1})\right).$$ \end{center} \textsc{Remarks}: \begin{itemize} \item[$\cdot$] The groupoid structure (composition) on the isomorphisms of fiber functors on $\mathcal{M}$, by dualizing, leads to a coalgebroid structure on the spaces of motivic periods: $$ \mathcal{P}_{\mathcal{M}}^{\omega_{1},\omega_{3}} \rightarrow \mathcal{P}_{\mathcal{M}}^{\omega_{1},\omega_{2}} \otimes \mathcal{P}_{\mathcal{M}}^{\omega_{2},\omega_{3}}.$$ \item[$\cdot$] Any structure carried by these fiber functors (weight grading on $\omega_{dR}$, complex conjugation on $\omega_{B}$, etc.) is transmitted to the corresponding ring of periods. \end{itemize} \texttt{Examples}: \begin{itemize} \item[$\cdot$ ] For $(\omega,\omega_{B})$, it comes down to (our main interest) $\mathcal{P}_{\mathcal{M}}^{\mathfrak{m}}$ as defined in $(\ref{eq:mper})$. By the last remark, $\mathcal{P}_{\mathcal{M}}^{\mathfrak{m}}$ inherits a weight grading and we can define (cf. $\cite{Br4}$, $\S 2.6$):\nomenclature{$\mathcal{P}_{\mathcal{M}}^{\mathfrak{m},+}$}{the ring of geometric motivic periods of $\mathcal{M}$} \begin{framed} \begin{center} $\boldsymbol{\mathcal{P}_{\mathcal{M}}^{\mathfrak{m},+}} \subset \mathcal{P}_{\mathcal{M}}^{\mathfrak{m}}$, the ring of \textit{geometric periods}, is generated by periods of motives with non-negative weights: $\left\lbrace \left[M,v,\sigma \right]^{\mathfrak{m}}\in \mathcal{P}_{\mathcal{M}}^{\mathfrak{m}} \mid W_{-1} M=0 \right\rbrace $. \end{center} \end{framed} \item[$\cdot$] The ring of periods of type $(\omega,\omega)$ is $\mathcal{P}_{\mathcal{M}}^{\omega}\mathrel{\mathop:}= \mathcal{O} \left( \text{Aut}^{\otimes}(\omega)\right)= \mathcal{O} \left(\mathcal{G}^{\mathcal{MT}}\right)$.\footnote{ In the case of a mixed Tate category over $\mathbb{Q}$, as $\mathcal{MT}(\mathbb{Z})$, this is equivalent to the \textit{De Rham periods} in $\mathcal{P}_{\mathcal{M}}^{\mathfrak{dR}}\mathrel{\mathop:}= \mathcal{O} \left( \text{Aut}^{\otimes}(\omega_{dR})\right)$, defined in $\cite{Br4}$; however, for other cyclotomic fields $k$ considered later ($N>2$), we have to consider the canonical fiber functor, since it is defined over $k$.} \\ \textit{Unipotent variants} of these periods are defined when restricting to the unipotent part $\mathcal{U}^{\mathcal{MT}}$ of $\mathcal{G}^{\mathcal{MT}}$, and appear below (in $\ref{eq:intitdr}$):\nomenclature{$\mathcal{P}_{\mathcal{M}}^{\mathfrak{a}}$}{the ring of unipotent periods} $$\boldsymbol{\mathcal{P}_{\mathcal{M}}^{\mathfrak{a}}}\mathrel{\mathop:}=\mathcal{O} \left( \mathcal{U}^{\mathcal{MT}}\right)= \mathcal{A}^{\mathcal{MT}}, \quad \text{ the fundamental Hopf algebra}.$$ They correspond to the notion of framed objects in mixed Tate categories, cf. $\cite{Go1}$. By restriction, there is a map: $$ \mathcal{P}_{\mathcal{M}}^{\omega} \rightarrow \mathcal{P}_{\mathcal{M}}^{\mathfrak{a}}.$$
\end{itemize}
By the remark above, there is a coaction: $$\boldsymbol{\Delta^{\mathfrak{m}, \omega}}:\mathcal{P}_{\mathcal{M}}^{\mathfrak{m}} \rightarrow \mathcal{P}_{\mathcal{M}}^{\omega} \otimes \mathcal{P}_{\mathcal{M}}^{\mathfrak{m}}.$$ Moreover, composing this coaction by the augmentation map $\epsilon: \mathcal{P}_{\mathcal{M}}^{\mathfrak{m},+} \rightarrow (\mathcal{P}_{\mathcal{M}}^{\mathfrak{m},+})_{0} \cong \mathbb{Q}$, leads to the morphism (details in $\cite{Br4}$, $\S 2.6$): \begin{equation}\label{eq:projpiam} \boldsymbol{\pi_{\mathfrak{a},\mathfrak{m}}}: \quad \mathcal{P}_{\mathcal{M}}^{\mathfrak{m},+} \rightarrow \mathcal{P}_{\mathcal{M}}^{\mathfrak{a}}, \end{equation} which is, on periods of a motive $M$ such that $W_{-1} M=0$: $\quad \left[M,v,\sigma \right]^{\mathfrak{m}} \rightarrow \left[M,v,^{t}c(\sigma) \right]^{\mathfrak{a}}, $ $$ \text{ where } c \text{ is defined as the composition}: \quad M_{\omega} \twoheadrightarrow gr^{W}_{0}M_{\omega}= W_{0} M_{\omega} \xrightarrow{\text{comp}_{B,\omega}} W_{0} M_{B} \hookrightarrow M_{B} .$$ Bear in mind also the non-canonical isomorphisms, compatible with weight and coaction ($\cite{Br4}$, Corollary $2.11$) between those $\mathbb{Q}$ algebras: \begin{equation} \label{eq:periodgeom}
\mathcal{P}_{\mathcal{M}}^{\mathfrak{m}} \cong \mathcal{P}_{\mathcal{M}}^{\mathfrak{a}} \otimes_{\mathbb{Q}} \mathbb{Q} \left[ (\mathbb{L}^{\mathfrak{m}} )^{-1} ,\mathbb{L}^{\mathfrak{m}} \right], \quad \text{and} \quad \mathcal{P}_{\mathcal{M}}^{\mathfrak{m},+} \cong \mathcal{P}_{\mathcal{M}}^{\mathfrak{a}} \otimes_{\mathbb{Q}} \mathbb{Q} \left[ \mathbb{L}^{\mathfrak{m}} \right]. \end{equation} In particular, $\pi_{\mathfrak{a},\mathfrak{m}}$ is obtained by sending $\mathbb{L}^{\mathfrak{m}}$ to $0$.\\ \\ In the case of a category of mixed Tate motive $\mathcal{M}$ defined over $\mathbb{Q}$, \footnote{As, in our concerns, $\mathcal{MT}_{N}$ above with $N=1,2$; in these exceptional (real) cases, we want to keep track of only even Tate twists.} the complex conjugation defines the \textit{real Frobenius} $\mathcal{F}_{\infty}: M_{B} \rightarrow M_{B}$, and induces an involution on motivic periods $\mathcal{F}_{\infty}: \mathcal{P}_{\mathcal{M}}^{\mathfrak{m}} \rightarrow\mathcal{P}_{\mathcal{M}}^{\mathfrak{m}}$. Furthermore, $\mathbb{L}^{\mathfrak{m}}$ is anti invariant by $\mathcal{F}_{\infty}$ (i.e. $\mathcal{F}_{\infty} (\mathbb{L}^{\mathfrak{m}})=-\mathbb{L}^{\mathfrak{m}}$). Then, let us define:\nomenclature{$\mathcal{F}_{\infty}$}{real Frobenius} \begin{framed} \begin{center} $\boldsymbol{\mathcal{P}_{\mathcal{M}, \mathbb{R} }^{\mathfrak{m},+}}$ the subset of $\mathcal{P}_{\mathcal{M}}^{\mathfrak{m},+}$ invariant under the real Frobenius $\mathcal{F}_{\infty}$, \text{ which, by } $(\ref{eq:periodgeom})$ \text{ satisfies }:\nomenclature{$\mathcal{P}_{\mathcal{M}, \mathbb{R} }^{\mathfrak{m},+}$}{the ring of the Frobenius-invariant geometric periods} \end{center} \begin{equation}\label{eq:periodgeomr} \mathcal{P}_{\mathcal{M}}^{\mathfrak{m},+}\cong \mathcal{P}_{\mathcal{M}, \mathbb{R}}^{\mathfrak{m},+} \oplus \mathcal{P}_{\mathcal{M}, \mathbb{R}}^{\mathfrak{m},+}. \mathbb{L}^{\mathfrak{m}} \quad \text{and} \quad \mathcal{P}_{\mathcal{M}, \mathbb{R}}^{\mathfrak{m},+}\cong \mathcal{P}_{\mathcal{M}}^{\mathfrak{a}} \otimes_{\mathbb{Q}} \mathbb{Q}\left[ (\mathbb{L}^{\mathfrak{m}})^{2} \right] . \end{equation} \end{framed}
\paragraph{Motivic Galois theory.} The ring of motivic periods $\mathcal{P}_{\mathcal{M}}^{\mathfrak{m}} $ is a bitorsor under Tannaka groups $(\mathcal{G}^{\mathcal{MT}}, \mathcal{G}_{B})$. If Grothendieck conjecture holds, via the period isomorphism, there is therefore a (left) action of the motivic Galois group $\mathcal{G}^{\mathcal{MT}}$ on periods. \\ More precisely, for each period $p$ there would exist: \begin{itemize} \item[$(i)$] well defined conjugates: elements in the orbit of $\mathcal{G}^{\mathcal{MT}}(\mathbb{Q})$. \item[$(ii)$] an algebraic group over $\mathbb{Q}$, $\mathcal{G}_{p}= \mathcal{G}^{\mathcal{MT}} \diagup Stab(p)$, where $Stab(p)$ is the stabilizer of $p$; $\mathcal{G}_{p}$, the Galois group of $p$, transitively permutes the conjugates. \end{itemize} \texttt{Examples}: \begin{itemize} \item[$\cdot$] For $\pi$ for instance, the Galois group corresponds to $\mathbb{G}_{m}$. Conjugates of $\pi$ are in fact $\mathbb{Q}^{\ast} \pi$, and the associated motive would be the Lefschetz motive $\mathbb{L}$, motive of $\mathbb{G}_{m}=\mathbb{P}^{1}\diagdown \lbrace0,\infty\rbrace$, as seen above. \item[$\cdot$] For $\log t$, $t>0$, $t\in\mathbb{Q} \diagdown \lbrace -1, 0, 1\rbrace$, this is a period of the Kummer motive in degree $1$:\footnote{Remark the short exact sequence: $ 0 \rightarrow \mathbb{Q}(1) \rightarrow H_{1}(X, \lbrace 1,t \rbrace) \rightarrow \mathbb{Q}(0) \rightarrow 0 .$}
$$K_{t}\mathrel{\mathop:}=M_{gm}(X, \lbrace 1,t \rbrace)\in \text{Ext}^{1}_{\mathcal{MT}(\mathbb{Q})}(\mathbb{Q}(0),\mathbb{Q}(1)) \text{ , where } X\mathrel{\mathop:}=\mathbb{P}^{1}\diagdown \lbrace 0, \infty \rbrace.$$
Since a basis of $H^{B}_{1}(X, \lbrace 1,t \rbrace)$ is $[\gamma_{0}]$, $[\gamma_{1,t}]$ with $\gamma_{1,t}$ the straight path from $1$ to $t$, and a basis of $H^{1}_{dR}(X, \lbrace 1,t \rbrace) $ is $[dx], \left[ \frac{dx}{x} \right] $, the period matrix is:
$$ \left( \begin{array}{ll} \mathbb{Q} & 0\\ \mathbb{Q} \log(t) & 2i\pi \mathbb{Q} \\
\end{array} \right). $$ The conjugates of $\log t$ are $\mathbb{Q}^{\ast}\log t+\mathbb{Q}$, and its Galois group is $\mathbb{Q}^{\ast} \ltimes \mathbb{Q}$. \item[$\cdot$] Similarly for zeta values $\zeta(n)$, $n$ odd in $\mathbb{N}^{\ast}\diagdown\lbrace 1 \rbrace$ which are periods of a mixed Tate motive over $\mathbb{Z}$ (cf. below): its conjugates are $\mathbb{Q}^{\ast}\zeta(n)+\mathbb{Q}$, and its Galois group is $\mathbb{Q}^{\ast} \ltimes \mathbb{Q}$. Grothendieck's conjecture implies that $\pi,\zeta(3), \zeta(5), \ldots$ are algebraically independent.\\ More precisely, $\zeta(n)$ is a period of $E_{n}\in \mathcal{MT}(\mathbb{Q})$, where: $$ 0\rightarrow \mathbb{Q}(n) \rightarrow E_{n} \rightarrow \mathbb{Q}(0) \rightarrow 0.$$ Notice that for even $n$, by Borel's result, $\text{Ext}_{\mathcal{MT}(\mathbb{Q})}^{1}(\mathbb{Q}(0),\mathbb{Q}(n))=0$, which implies $E_{n}= \mathbb{Q}(0)\oplus \mathbb{Q}(n)$, and hence $\zeta(n)\in (2i\pi)^{n}\mathbb{Q}$. \item[$\cdot$] More generally, multiple zeta values at roots of unity $\mu_{N}$ occur as periods of mixed Tate motives over $\mathbb{Z}[\xi_{N}]\left[ \frac{1}{N}\right] $, $\xi_{N}$ primitive $N^{\text{th}}$ root of unity. The motivic Galois group associated to the algebra $\mathcal{H}^{N}$ generated by MMZV$_{\mu_{N}}$ is conjectured to be a quotient of the motivic Galois group $\mathcal{G}^{\mathcal{MT}_{N}}$, equal for some values of $N$: $N=1,2,3,4,8$ for instance, as seen below. We expect MZV to be simple examples in the conjectural Galois theory for transcendental numbers. \end{itemize} \textsc{Remark}: By K-theory results above, non-zero Ext groups for $\mathcal{MT}(\mathbb{Q})$ are: $$\text{Ext}^{1}_{\mathcal{MT}(\mathbb{Q})} (\mathbb{Q}(0), \mathbb{Q}(n))\cong \left\lbrace \begin{array}{ll} \mathbb{Q}^{\ast}\otimes_{\mathbb{Z}} \mathbb{Q} \cong \oplus_{p \text{ prime} } \mathbb{Q} & \text{ if } n=1\\ \mathbb{Q} & \text{ if } n \text{ odd} >1.\\ \end{array}\right. $$ Generators of these extension groups correspond exactly to periods $\log(p)$, $p$ prime in degree 1 and $\zeta(odd)$ in degree odd $>1$, which are periods of $\mathcal{MT}(\mathbb{Q})$.
\section{Motivic fundamental group}
\paragraph{Prounipotent completion.}
Let $\Pi$ be the group freely generated by $\gamma_{0}, \ldots, \gamma_{N}$. The completed Hopf algebra $\widehat{\Pi}$ is defined by: $$\widehat{\Pi}\mathrel{\mathop:}= \varprojlim \mathbb{Q}[\Pi] \diagup I^{n} , \quad \text{ where }I\mathrel{\mathop:}= \langle \gamma-1 , \gamma\in \Pi \rangle \text{ is the augmentation ideal} .$$ Equipped with the completed coproduct $\Delta$ such that the elements of $\Pi$ are primitive, it is isomorphic to the Hopf algebra of non commutative formal series:\footnote{Well defined inverse since the log converges in $\widehat{\Pi}$; $exp(e_{i})$ are then group-like for $\Delta$. Notice that the Lie Algebra of the group of group-like elements is formed by the primitive elements and conversely; besides, the universal enveloping algebra of primitive elements is the whole Hopf algebra.} $$\widehat{\Pi} \xrightarrow[\gamma_{i}\mapsto \exp(e_{i}) ]{\sim} \mathbb{Q} \langle\langle e_{0}, \ldots, e_{N}\rangle\rangle. $$ \\ The \textit{prounipotent completion} of $\Pi$ is an affine group scheme $\Pi^{un}$:\nomenclature{$\Pi^{un}$}{prounipotent completion of $\Pi$} \begin{equation}\label{eq:prounipcompletion} \boldsymbol{\Pi^{un}}(R)=\lbrace x\in \widehat{\Pi} \widehat{\otimes} R \mid \Delta x=x\otimes x\rbrace \cong \lbrace S\in R\langle\langle e_{0}, \ldots, e_{N} \rangle\rangle^{\times}\mid \Delta S=S\otimes S, \epsilon(S)=1\rbrace , \end{equation} i.e. the set of non-commutative formal series with $N+1$ generators which are group-like for the completed coproduct for which $e_{i}$ are primitive. \\ It is dual to the shuffle $\shuffle$ relation between the coefficients of the series $S$\footnote{It is a straightforward verification that the relation $\Delta S= S\otimes S$ implies the shuffle $\shuffle$ relation between the coefficients of S.}. Its affine ring of regular function is the Hopf algebra (filtered, connected) for the shuffle product, and deconcatenation coproduct: \begin{equation} \boldsymbol{\mathcal{O}(\Pi^{un})}= \varinjlim \left( \mathbb{Q}[\Pi] \diagup I^{n+1} \right) ^{\vee} \cong \mathbb{Q} \left\langle e^{0}, \ldots, e^{N} \right\rangle . \end{equation} $$\boldsymbol{\mathcal{O}(\Pi^{\mathfrak{m}}(X_{N},x,y))}\in\mathcal{MT}(k_{N}).$$
\paragraph{Motivic Fundamental pro-unipotent groupoid.}\footnote{ \say{\textit{Esquisse d'un programme}}$\cite{Gr}$, by Grothendieck, vaguely suggests to study the action of the absolute Galois group of the rational numbers $ \text{Gal}(\overline{\mathbb{Q}} \diagup \mathbb{Q} ) $ on the \'{e}tale fundamental group $\pi_{1}^{et}(\mathcal{M}_{g,n})$, where $\mathcal{M}_{g,n}$ is the moduli space of curves of genus $g$ and $n$ ordered marked points. In the case of $\mathcal{M}_{0,4}= \mathbb{P}^{1}\diagdown \lbrace 0, 1, \infty \rbrace$, Deligne proposed to look instead (analogous) at the pro-unipotent fundamental group $\pi_{1}^{un}(\mathbb{P}^{1}\diagdown \lbrace 0, 1, \infty \rbrace)$. This motivates also the study of multiple zeta values, which arose as periods of this fundamental group. } The previous construction can be applied to $\pi_{1}(X,x)$, resp. $\pi_{1}(X,x,y)$, if assumed free, the fundamental group resp. groupoid of $X$ with base point $x$, resp. $x,y$, rational points of $X$, an algebraic variety over $\mathbb{Q}$; the groupoid $\pi_{1}(X,x,y)$, is a bitorsor formed by the homotopy classes of path from $x$ to $y$. \\
From now, let's turn to the case $X_{N}\mathrel{\mathop:}=\mathbb{P}^{1}\diagdown \lbrace 0, \infty, \mu_{N} \rbrace$. There, the group $\pi_{1}(X_{N}, x)$ is freely generated by $\gamma_{0}$ and $(\gamma_{\eta})_{\eta\in\mu_{N}}$, the loops around $0$ resp. $\eta\in\mu_{N}$.\footnote{Beware, since $\pi_{1}(X,x,y)$ is not a group, we have to pass first to the dual in the previous construction: $$ \pi^{un}_{1}(X,x,y)\mathrel{\mathop:}= \text{Spec} \left( \varinjlim \left( \mathbb{Q}[\pi_{1}] \diagup I^{n+1} \right) ^{\vee} \right) .$$}\\ Chen's theorem implies here that we have a perfect pairing: \begin{equation}\label{eq:chenpairing} \mathbb{C}[\pi_{1} (X_{N},x,y)] \diagup I^{n+1} \otimes \mathbb{C}\langle \omega_{0}, (\omega_{\eta})_{\eta\in\mu_{N}} \rangle_{\leq n} \rightarrow \mathbb{C} . \end{equation} In order to define the motivic $\pi^{un}_{1}$, let us introduce (cf. $\cite{Go2}$, Theorem $4.1$): \begin{equation}\label{eq:y(n)} Y^{(n)}\mathrel{\mathop:}=\cup_{i} Y_{i}, \text{ where } \quad \begin{array}{ll} Y_{0}\mathrel{\mathop:}= \lbrace x\rbrace \times X^{n-1}& \\ Y_{i}\mathrel{\mathop:}= X^{i-1}\times \Delta \times X^{n-i-1}, & \Delta \subset X \times X \text{ the diagonal} \\
Y_{n}\mathrel{\mathop:}= X^{n-1} \times \lbrace y\rbrace & \end{array}. \end{equation} Then, by Beilinson theorem ($\cite{Go2}$, Theorem $4.1$), coming from $\gamma \mapsto [\gamma(\Delta_{n})]$: $$H_{k}(X^{n},Y^{(n)}) \cong \left\lbrace \begin{array}{ll}
\mathbb{Q}[\pi_{1}(X,x,y)] \diagup I^{n+1}& \text{ for } k=n \\ 0 & \text{ for } k<n \end{array} \right. .$$ The left side defines a mixed Tate motive and: \begin{equation} \label{eq:opiunvarinjlim}
\mathcal{O}(\pi_{1}^{un}(X,x,y)) \xrightarrow{\sim} \varinjlim_{n} H^{n}(X^{n}, Y^{(n)}). \end{equation} By $(\ref{eq:opiunvarinjlim})$, $\mathcal{O}\left( \pi_{1}^{un}(X,x,y)\right)$ defines an Ind object \footnote{Ind objects of a category $\mathcal{C}$ are inductive filtered limit of objects in $\mathcal{C}$.} in the category of Mixed Tate Motives over $k$, since $Y_{I}^{(n)}\mathrel{\mathop:}=\cap Y_{i}^{(n)}$ is the complement of hyperplanes, hence of type Tate: \begin{framed} \begin{equation}\label{eq:pi1unTate0} \mathcal{O}\left( \pi_{1}^{un}(\mathbb{P}^{1}\diagdown \lbrace 0, \infty, \mu_{N} \rbrace,x,y)\right) \in \text{Ind } \mathcal{MT}(k). \end{equation} \end{framed} We denote it $\boldsymbol{\mathcal{O}\left( \pi_{1}^{\mathfrak{m}}(X,x,y)\right) }$, and $\mathcal{O}\left( \pi_{1}^{\omega}(X,x,y)\right)$, $\mathcal{O}\left( \pi_{1}^{dR}(X,x,y)\right)$, $\mathcal{O}\left( \pi_{1}^{B}(X,x,y)\right)$ its realizations, resp. $\boldsymbol{\pi_{1}^{\mathfrak{m}}(X)}$ for the corresponding $\mathcal{MT}(k)$-groupoid scheme, called the \textit{\textbf{motivic fundamental groupoid}}, with the composition of path. \\ \\ \textsc{Remark: } The pairing $(\ref{eq:chenpairing})$ can be thought in terms of a perfect pairing between homology and de Rham cohomology, since (Wojtkowiak $\cite{Wo2}$): $$H_{dR}^{n}(X^{n},Y^{(n)}) \cong k_{N}\langle \omega_{0}, \ldots, \omega_{N} \rangle_{\leq n}.$$ \\ The construction of the prounipotent completion and then the motivic fundamental groupoid would still work for the case of \textit{tangential base points }, cf. $\cite{DG}$, $\S 3$\footnote{I.e. here non-zero tangent vectors in a point of $\lbrace 0, \mu_{N}, \infty\rbrace $ are seen as \say{base points at infinite}. Deligne explained how to replace ordinary base points with tangential base points.}. Let us denote $\lambda_{N}$ the straight path between $0$ and $\xi_{N}$, a primitive root of unity. In the following, we will particularly consider the tangential base points $\overrightarrow{0\xi_{N}}\mathrel{\mathop:}=(\overrightarrow{1}_{0}, \overrightarrow{-1}_{\xi_{N}})$, defined as $(\lambda_{N}'(0), -\lambda_{N}'(1))$; but similarly for each $x,y\in \mu_{N}\cup \lbrace 0, \infty\rbrace $, such that $_{x}\lambda_{y}$ the straight path between $x,y$ in in $\mathbb{P}^{1} (\mathbb{C} \diagdown \lbrace 0, \mu_{N}, \infty \rbrace)$, we associate the tangential base points $\overrightarrow{xy}\mathrel{\mathop:}= (_{x}\lambda_{y}'(0), - _{x}\lambda_{y}'(1))$\footnote{In order that the path does not pass by $0$, we have to exclude the case where $x=-y$ if $N$ even.}. Since the motivic torsor of path associated to such tangential basepoints depends only on $x,y$ (cf. $\cite{DG}$, $\S 5$) we will denote it $_{x}\Pi^{\mathfrak{m}}_{y}$. This leads to a groupoid structure via $_{x}\Pi^{\mathfrak{m}}_{y} \times _{y}\Pi^{\mathfrak{m}}_{z} \rightarrow _{x}\Pi^{\mathfrak{m}}_{z}$: cf. Figure $\ref{fig:Pi}$ and $\cite{DG}$.\\ In fact, by Goncharov's theorem, in case of these tangential base points, the motivic torsor of path corresponding has good reduction outside N and (cf. $\cite{DG}, \S 4.11$): \begin{equation}\label{eq:pi1unTate} \mathcal{O}\left( _{x}\Pi^{\mathfrak{m}}_{y} \right) \in \text{ Ind } \mathcal{MT}_{\Gamma_{N}} \subset \text{ Ind } \mathcal{MT}\left( \mathcal{O}_{N}\left[ \frac{1}{N} \right] \right) . \end{equation} The case of ordinary base points, lying in $\text{ Ind } \mathcal{MT}(k)$, has no such good reduction.\\ In summary, from now, we consider, for $x,y\in \mu_{N}\cup\lbrace 0\rbrace$\footnote{$_{x}\Pi^{\mathfrak{m}}_{y}$ is a bitorsor under $(_{x}\Pi^{\mathfrak{m}}_{x}, _{y}\Pi^{\mathfrak{m}}_{y})$.}:\nomenclature{$_{x}\Pi^{\mathfrak{m}}_{y}$}{motivic bitorsor of path, and $_{x}\Pi_{y}$, $_{x}\Pi_{y}^{dR}$, $_{x}\Pi_{y}^{B}$, its $\omega$, resp. de Rham resp. Betti realizations} \begin{framed} \textit{The motivic bitorsors of path} $_{x}\Pi^{\mathfrak{m}}_{y}\mathrel{\mathop:}=\pi_{1}^{\mathfrak{m}} (X_{N}, \overrightarrow{xy})$ on $X_{N}\mathrel{\mathop:}=\mathbb{P}^{1} -\left\{0,\mu_{N},\infty\right\}$ with tangential basepoints given by $\overrightarrow{xy}\mathrel{\mathop:}= (\lambda'(0), -\lambda'(1))$ where $\lambda$ is the straight path from $x$ to $y$, $x\neq -y$.\\ \end{framed} Let us denote $_{x}\Pi_{y}\mathrel{\mathop:}=_{x}\Pi^{\omega}_{y}$, resp. $_{x}\Pi_{y}^{dR}$, $_{x}\Pi_{y}^{B}$ its $\omega$, resp. de Rham resp. Betti realizations. In particular, Chen's theorem implies that we have an isomorphism: $$_{0}\Pi_{1}^{B}\otimes\mathbb{C}\xrightarrow{\sim} {} _{0}\Pi_{1}\otimes \mathbb{C}.$$\\ Therefore, the motivic fundamental group above boils down to: \begin{itemize} \item[$(i)$] The affine group schemes $_{x}\Pi_{y}^{B}$, $x,y\in\mu_{N}\cup \lbrace0, \infty\rbrace $, with a groupoid structure. The Betti fundamental groupoid is the pro-unipotent completion of the ordinary topological fundamental groupoid, i.e. corresponds to $\pi_{1}^{un}(X,x,y)$ above. \item[$(ii)$] $\Pi(X)=\pi^{\omega}_{1}(X)$, the affine group scheme over $\mathbb{Q}$. It does not depend on $x,y$ since the existence of a canonical de Rham path between x and y implies a canonical isomorphism $\Pi(X)\cong _{x}\Pi(X)_{y}$; however, the action of the motivic Galois group $\mathcal{G}$ is sensitive to the tangential base points $x,y$. \item[$(iii)$] a canonical comparison isomorphism of schemes over $\mathbb{C}$, $\text{comp}_{B,\omega}$. \end{itemize}
\begin{figure}
\caption{Part of the Fundamental groupoid $\Pi$.\\ This picture however does not represent accurately the tangential base points.}
\label{fig:Pi}
\end{figure}
Moreover, the dihedral group\footnote{Symmetry group of a regular polygon with $N$ sides.} $Di_{N}= \mathbb{Z}\diagup 2 \mathbb{Z} \ltimes \mu_{N}$\nomenclature{$Di_{N}$}{dihedral group of order $2n$} acts on $X_{N}=\mathbb{P}^{1}\diagdown \lbrace 0, \mu_{N},\infty\rbrace$\nomenclature{$X_{N}$}{defined as $\mathbb{P}^{1}\diagdown \lbrace 0, \mu_{N},\infty\rbrace$}: the group with two elements corresponding to the action $x \mapsto x^{-1}$ and the cyclic group $\mu_{N}$ acting by $x\mapsto \eta x$. Notice that for $N=1,2,4$, the group of projective transformations $X_{N}\rightarrow X_{N}$ is larger than $Di_{N}$, because of special symmetries, and detailed in $A.3$. \footnote{Each homography $\phi$ defines isomorphisms: $$\begin{array}{lll}
_{a}\Pi_{b} & \xrightarrow[\sim]{\phi}& _{\phi(a)}\Pi_{\phi(b)} \\ f(e_{0}, e_{1}, \ldots, e_{n}) &\mapsto &f(e_{\phi(0)}, e_{\phi(1)}, \ldots, e_{\phi(n)}) \end{array} \text{ and, passing to the dual } \mathcal{O}(_{\phi(a)}\Pi_{\phi(b)}) \xrightarrow[\sim]{\phi^{\vee}} \mathcal{O}( _{a}\Pi_{b}) .$$} \\ The dihedral group $Di_{N}$ acts then on the motivic fundamental groupoid $\pi^{\mathfrak{m}}_{1}(X,x,y)$, $x,y \in \lbrace 0 \rbrace \cup \mu_{N}$ by permuting the tangential base points (and its action is respected by the motivic Galois group): $$\text{For } \quad \sigma\in Di_{N}, \quad _{x}\Pi_{y} \rightarrow _{\sigma.x}\Pi_{\sigma.y} $$
The group scheme $\mathcal{V}$ of automorphisms on these groupoids $_{x}\Pi_{y}$, respecting their structure, i.e.: \begin{itemize} \item[$\cdot$] groupoid structure, i.e. the compositions $_{x}\Pi_{y}\times _{y}\Pi_{z} \rightarrow _{x}\Pi_{z}$, \item[$\cdot$] $\mu_{N}$-equivariance as above, \item[$\cdot$] inertia: the action fixes $\exp(e_{x})\in _{x}\Pi_{x}(\mathbb{Q})$, \end{itemize} is isomorphic to (cf. $\cite{DG}$, $\S 5$ for the detailed version): \begin{equation}\label{eq:gpaut} \begin{array}{ll} \mathcal{V}\cong _{0}\Pi_{x} \\ a\mapsto a\cdot _{0}1_{x} \end{array}. \end{equation} In particular, the \textit{Ihara action} defined in $(\ref{eq:iharaaction})$ corresponds via this identification to the composition law for these automorphisms, and then can be computed explicitly. Its dual would be the combinatorial coaction $\Delta$ used through all this work.\\ \\ In consequence of these equivariances, we can restrict our attention to: \begin{framed} $$_{0}\Pi^{\mathfrak{m}}_{ \xi_{N}}\mathrel{\mathop:}=\pi_{1}^{\mathfrak{m}}(X_{N}, \overrightarrow{0\xi_{N}} ) \text{ or equivalently at } _{0}\Pi^{\mathfrak{m}}_{1}.$$ \end{framed} \noindent Keep in mind, for the following, that $_{0}\Pi_{1}$ is the functor:\nomenclature{$R\langle X \rangle$ resp. $R\langle\langle X \rangle\rangle$}{the ring of non commutative polynomials, resp. of non commutative formal series in elements of X} \begin{framed}
\begin{equation}\label{eq:pi}_{0}\Pi_{1}: R \text{ a } \mathbb{Q}-\text{algebra } \mapsto \left\{S\in R\langle\langle e_{0}, (e_{\eta})_{\eta\in\mu_{N}}\rangle\rangle^{\times} | \Delta S= S\otimes S \text{ and } \epsilon(S)= 1 \right\} ,\end{equation} whose affine ring of regular functions is the graded (Hopf) algebra for the shuffle product: \begin{equation}\label{eq:opi} \mathcal{O}(_{0}\Pi_{1})\cong \mathbb{Q} \left\langle e^{0}, (e^{\eta})_{\eta\in\mu_{N}} \right\rangle. \end{equation} \end{framed} \noindent The Lie algebra of $_{0}\Pi_{1}(R)$ would naturally be the primitive series ($\Delta S= 1 \otimes S+ S\otimes 1$). \\ \\ Let us denote $dch_{0,1}^{B}=_{0}1^{B}_{1}$,\nomenclature{$dch_{0,1}^{B}$}{the image of the straight path} the image of the straight path (\textit{droit chemin}) in $_{0}\Pi_{1}^{B}(\mathbb{Q})$, and $dch_{0,1}^{dR}$ or $\Phi_{KZ_{N}}$ the corresponding element in $_{0}\Pi_{1}(\mathbb{C})$ via the Betti-De Rham comparison isomorphism: \begin{equation}\label{eq:kz}
\boldsymbol{\Phi_{KZ_{N}}}\mathrel{\mathop:}= dch_{0,1}^{dR}\mathrel{\mathop:}= \text{comp}_{dR,B}(_{0}1^{B}_{1})= \sum_{W\in \lbrace e_{0}, (e_{\eta})_{\eta\in\mu_{N}} \rbrace^{\times}} \zeta_{\shuffle}(w) w \quad \in \mathbb{C} \langle\langle e_{0}, (e_{\eta})_{\eta\in\mu_{N}} \rangle\rangle , \end{equation} where the correspondence between MZV and words in $e_{0},e_{\eta}$ is similar to the iterated integral representation $(\ref{eq:reprinteg})$, with $\eta_{i}$. It is known as the \textit{Drinfeld associator} and arises also from the monodromy of the famous Knizhnik$-$Zamolodchikov differential equation.\footnote{Indeed, for $N=1$, Drinfeld associator is equal to $G_{1}^{-1}G_{0}$, where $G_{0},G_{1}$ are solutions, with certain asymptotic behavior at $0$ and $1$ of the Knizhnik$-$Zamolodchikov differential equation:$$ \frac{d}{dz}G(z)= \left(\frac{e_{0}}{z}+ \frac{e_{1}}{1-z} \right) G(z) .$$}
\paragraph{Category generated by $\boldsymbol{\pi_{1}^{\mathfrak{m}}}$. } Denote by:\nomenclature{$\mathcal{MT}'_{N}$}{the full Tannakian subcategory of $\mathcal{MT}_{N}$ generated by the fundamental groupoid} \begin{framed}
$\boldsymbol{\mathcal{MT}'_{N}}$ the full Tannakian subcategory of $\mathcal{MT}_{N}$ generated by the fundamental groupoid,
\end{framed} (i.e. generated by $\mathcal{O}(\pi_{1}^{\mathfrak{m}} (X_{N},\overrightarrow{01}))$ by sub-objects, quotients, $\otimes$, $\oplus$, duals) and let:\nomenclature{$\mathcal{G}^{N}$}{the motivic Galois group of $\boldsymbol{\mathcal{MT}'_{N}}$ }\nomenclature{$\mathcal{A}^{N}$}{the fundamental Hopf algebra of $\boldsymbol{\mathcal{MT}'_{N}}$ }\nomenclature{$\mathcal{L}^{N}$}{the motivic coalgebra associated to $\boldsymbol{\mathcal{MT}'_{N}}$ } \begin{itemize} \item[$\cdot$] $\mathcal{G}^{N}=\mathbb{G}_{m} \ltimes \mathcal{U}^{N} $ its motivic \textit{Galois group} defined over $\mathbb{Q}$, \item[$\cdot$] $\mathcal{A}^{N}=\mathcal{O}(\mathcal{U}^{N})$ its \textit{fundamental Hopf algebra}, \item[$\cdot$] $\mathcal{L}^{N}\mathrel{\mathop:}= \mathcal{A}^{N}_{>0} / \mathcal{A}^{N}_{>0} \cdot\mathcal{A}^{N}_{>0}$ the Lie \textit{coalgebra of indecomposable elements}. \end{itemize} \texttt{Nota Bene}: $\mathcal{U}^{N}$ is the quotient of $\mathcal{U}^{\mathcal{MT}} $ by the kernel of the action on $_{0}\Pi_{1}$: i.e. $\mathcal{U}^{N}$ acts faithfully on $_{0}\Pi_{1}$.\nomenclature{$\mathcal{U}^{N}$}{the motivic prounipotent group associated to $\boldsymbol{\mathcal{MT}'_{N}}$ }\\ \\ \textsc{Remark:} In the case of $N=1$ (by F. Brown in \cite{Br2}), or $N=2,3,4,\mlq 6 \mrq,8 $ (by P. Deligne, in \cite{De}, proven in a dual point of view in Chapter $5$), these categories $\mathcal{MT}'_{N}$ and $\mathcal{MT}(\mathcal{O}_{N}\left[ \frac{1}{N} \right] )$ are equal. More precisely, for $\xi_{N}\in\mu_{N}$ a fixed primitive root, the following motivic torsors of path are sufficient to generate the category: \begin{description} \item[$\boldsymbol{N=2,3,4}$:] $\Pi^{\mathfrak{m}} (\mathbb{P}^{1} \diagdown \lbrace 0, 1, \infty \rbrace, \overrightarrow{0 \xi_{N}})$ generates $\mathcal{MT}(\mathcal{O}_{N}\left[ \frac{1}{N} \right] )$. \item[$\boldsymbol{N=\mlq 6\mrq}$:]\footnote{The quotation marks around $6$ underlines that we consider the unramified category in this case.} $\Pi^{\mathfrak{m}} (\mathbb{P}^{1} \diagdown \lbrace 0, 1, \infty \rbrace, \overrightarrow{0 \xi_{6}})$ generates $\mathcal{MT}(\mathcal{O}_{6})$. \item[$\boldsymbol{N=8}$:] $\Pi^{\mathfrak{m}} (\mathbb{P}^{1} \diagdown \lbrace 0, \pm 1, \infty \rbrace, \overrightarrow{0 \xi_{8}})$ generates $\mathcal{MT}(\mathcal{O}_{8}\left[ \frac{1}{2}\right] )$.\\ \end{description} However, if $N$ has a prime factor which is non inert, the motivic fundamental group is in the proper subcategory $\mathcal{MT}_{\Gamma_{N}}$ and hence can not generate $\mathcal{MT}(\mathcal{O}_{N}\left[ \frac{1}{N}\right] )$.
\section{Motivic Iterated Integrals}
Taking from now $\mathcal{M}=\mathcal{MT}'_{N}$, $M=\mathcal{O}(\pi^{\mathfrak{m}}_{1}(\mathbb{P}^{1}-\lbrace 0,\mu_{N},\infty\rbrace ,\overrightarrow{xy} ))$, the definition of motivic periods $(\ref{eq:mper})$ leads to motivic iterated integrals relative to $\mu_{N}$. Indeed:\nomenclature{$I^{\mathfrak{m}}(x;w;y)$}{motivic iterated integral} \begin{framed}\label{mii} A \textit{\textbf{motivic iterated integral}} is the triplet $I^{\mathfrak{m}}(x;w;y)\mathrel{\mathop:}= \left[\mathcal{O} \left( \Pi^{\mathfrak{m}} \left( X_{N}, \overrightarrow{xy}\right) \right) ,w,_{x}dch_{y}^{B}\right]^{\mathfrak{m}}$ where $w\in \omega(M)$, $_{x}dch_{y}^{B}$ is the image of the straight path from $x$ to $y$ in $\omega_{B}(M)^{\vee}$ and whose period is: \begin{equation}\label{eq:peri} \text{per}(I^{\mathfrak{m}}(x;w;y))= I(x;w;y) =\int_{x}^{y}w= \langle \text{comp}_{B,dR}(w\otimes 1),_{x}dch_{y}^{B} \rangle \in\mathbb{C}. \end{equation} \end{framed} \noindent \\ \\ \textsc{Remarks: } \begin{itemize} \item[$\cdot$] There, $w\in \omega(\mathcal{O}(_{x}\Pi^{\mathfrak{m}}_{y}))\cong \mathbb{Q} \left\langle \omega_{0}, (\omega_{\eta})_{\eta\in\mu_{N}} \right\rangle $ where $\omega_{\eta}\mathrel{\mathop:}= \frac{dt}{t-\eta}$. Similarly to $\ref{eq:iterinteg}$, let: \begin{equation}\label{eq:iterintegw} I^{\mathfrak{m}} (a_{0}; a_{1}, \ldots, a_{n}; a_{n+1})\mathrel{\mathop:}= I^{\mathfrak{m}} (a_{0}; \omega_{\boldsymbol{a}}; a_{n+1}), \quad \text{ where } \omega_{\boldsymbol{a}}\mathrel{\mathop:}=\omega_{a_{1}} \cdots \omega_{a_{n}}, \text{ for } a_{i}\in \lbrace 0\rbrace \cup \mu_{N} \end{equation} \item[$\cdot$] The Betti realization functor $\omega_{B}$ depends on the embedding $\sigma: k \hookrightarrow \mathbb{C}$. Here, by choosing a root of unity, we fixed the embedding $\sigma$. \end{itemize} For $\mathcal{M}$ a category of Mixed Tate Motives among $\mathcal{MT}_{N}, \mathcal{MT}_{\Gamma_{N}}$ resp. $\mathcal{MT}'_{N}$, let introduce the graded $\mathcal{A}^{\mathcal{M}}$-comodule, with trivial coaction on $\mathbb{L}^{\mathfrak{m}}$ (degree $1$):
\begin{equation}\label{eq:hn}
\boldsymbol{\mathcal{H}^{\mathcal{M}}} \mathrel{\mathop:}= \mathcal{A}^{\mathcal{M}} \otimes \left\{ \begin{array}{ll} \mathbb{Q}\left[ (\mathbb{L}^{\mathfrak{m}})^{2} \right] & \text{ for } N=1,2 \\ \mathbb{Q}\left[ \mathbb{L}^{\mathfrak{m}} \right] & \text{ for } N>2 . \end{array} \right. \subset \mathcal{O}(\mathcal{G}^{\mathcal{M}})= \mathcal{A}^{\mathcal{M}}\otimes \mathbb{Q}[\mathbb{L}^{\mathfrak{m}}, (\mathbb{L}^{\mathfrak{m}})^{-1}]. \end{equation} \texttt{Nota Bene}: For $N>2$, it corresponds to the geometric motivic periods, $\mathcal{P}_{\mathcal{M}}^{\mathfrak{m},+}$ whereas for $N=1,2$, it is the subset $\mathcal{P}_{\mathcal{M},\mathbb{R}}^{\mathfrak{m},+}$ invariant by the real Frobenius; cf. $(\ref{eq:periodgeom}), (\ref{eq:periodgeomr})$.\\ For $\mathcal{M}=\mathcal{MT}'_{N}$, we will simply denote it $\mathcal{H}^{N}\mathrel{\mathop:}=\mathcal{H}^{\mathcal{MT}'_{N}}$. Moreover: $$\mathcal{H}^{N}\subset \mathcal{H}^{\mathcal{MT}_{\Gamma_{N}}} \subset \mathcal{H}^{\mathcal{MT}_{N}} .$$ \\ Cyclotomic iterated integrals of weight $n$ are periods of $\pi^{un}_{1}$ (of $X^{n}$ relative to $Y^{(n)}$): \footnote{Notations of $(\ref{eq:y(n)})$. Cf. also $(\ref{eq:pi1unTate})$. The case of tangential base points requires blowing up to get rid of singularities. Most interesting periods are often those whose integration domain meets the singularities of the differential form.} \begin{framed}
Any motivic iterated integral $I^{\mathfrak{m}}$ relative to $\mu_{N}$ is an element of $\mathcal{H}^{N}$, which is the graded $\mathcal{A}^{N}-$ comodule generated by these motivic iterated integrals relative to $\mu_{N}$. \end{framed}
In a similar vein, define:\nomenclature{$ I^{\mathfrak{a}}$ resp. $I^{\mathfrak{l}}$}{versions of motivic iterated integrals in $\mathcal{A}$ resp. in $\mathcal{L}$} \begin{itemize} \item[$\cdot \boldsymbol{I^{\omega}}$: ] A motivic period of type $(\omega,\omega)$, in $\mathcal{O}(\mathcal{G})$: \begin{equation} \label{eq:intitdr} I^{\omega}(x;w;y)=\left[\mathcal{O} \left( _{x}\Pi^{\mathfrak{m}}_{y}\right) ,w,_{x}1^{\omega}_{y} \right]^{\omega}, \quad \text{ where } \left\lbrace \begin{array}{l} w\in\omega(\mathcal{O} \left( _{x}\Pi^{\mathfrak{m}}_{y}\right) )\\ _{x}1^{\omega}_{y}\in \omega(M)^{\vee}=\mathcal{O}\left( _{x}\Pi_{y}\right)^{\vee} \end{array}\right. . \end{equation} where $_{x}1^{\omega}_{y}\in \mathcal{O}\left( _{x}\Pi_{y}\right)^{\vee}$ is defined by the augmentation map $\epsilon:\mathbb{Q}\langle e^{0}, (e^{\eta})_{\eta\in\mu_{N}}\rangle \rightarrow \mathbb{Q}$, corresponding to the unit element in $_{x}\Pi_{y}$. This defines a function on $\mathcal{G}=\text{Aut}^{\otimes}(\omega)$, given on the rational points by $g\in\mathcal{G}(\mathbb{Q}) \mapsto \langle g\omega, \epsilon\rangle \in \mathbb{Q}$. \item[$\cdot \boldsymbol{I^{\mathfrak{a}}}$: ] the image of $I^{\omega}$ in $\mathcal{A}= \mathcal{O}(\mathcal{U})$, by the projection $\mathcal{O}(\mathcal{G})\twoheadrightarrow \mathcal{O}(\mathcal{U})$. These \textit{unipotent} motivic periods are the objects studied by Goncharov, which he called motivic iterated integrals; for instance, $\zeta^{\mathfrak{a}}(2)=0$. \item[$\cdot \boldsymbol{I^{\mathfrak{l}}}$: ] the image of $I^{\mathfrak{a}}$ in the coalgebra of indecomposables $\mathcal{L}\mathrel{\mathop:}=\mathcal{A}_{>0} \diagup \mathcal{A}_{>0}. \mathcal{A}_{>0}$.\footnote{Well defined since $\mathcal{A}= \mathcal{O} (\mathcal{U})$ is graded with positive degrees.} \end{itemize}
\textsc{Remark:} It is similar (cf. $\cite{Br2}$) to define $\mathcal{H}^{N}$, as $\mathcal{O}(_{0}\Pi_{1}) \diagup J \quad$, with:
\begin{itemize}
\item[$\cdot$] $J\subset \mathcal{O}(_{0}\Pi_{1})$ is the biggest graded ideal $\subset \ker per$ closed by the coaction $\Delta^{c}$, corresponding to the ideal of motivic relations, i.e.: $$\Delta^{c}(J)\subset \mathcal{A}\otimes J + J\mathcal{A} \otimes \mathcal{O}(_{0}\Pi_{1}).$$ \item[$\cdot$] the $\shuffle$-homomorphism: $\text{per}: \mathcal{O}(_{0}\Pi_{1}) \rightarrow \mathbb{C} \text{ , } e^{a_{1}} \cdots e^{a_{n}} \mapsto I(0; a_{1}, \ldots, a_{n} ; 1)\mathrel{\mathop:}= \int_{dch} \omega.$ \\
\end{itemize} Once the motivic iterated integrals are defined, motivic cyclotomic multiple zeta values follow, as usual (cf. $\ref{eq:iterinteg}$): \begin{center} \textit{Motivic multiple zeta values} relative to $\mu_{N}$ are defined by, for $\epsilon_{i}\in\mu_{N}, k\geq 0, n_{i}>0$ \begin{equation}\label{mmzv} \boldsymbol{\zeta_{k}^{\mathfrak{m}} \left({ n_{1}, \ldots , n_{p} \atop \epsilon_{1} , \ldots ,\epsilon_{p} }\right) }\mathrel{\mathop:}= (-1)^{p} I^{\mathfrak{m}} \left(0;\boldsymbol{0}^{k}, (\epsilon_{1}\cdots \epsilon_{p})^{-1}, \boldsymbol{0}^{n_{1}-1} ,\cdots, (\epsilon_{i}\cdots \epsilon_{p})^{-1}, \boldsymbol{0}^{n_{i}-1} ,\cdots, \epsilon_{p}^{-1}, \boldsymbol{0}^{n_{p}-1} ;1 \right) \end{equation} \end{center}
An \textit{admissible} (motivic) MZV is such that $\left( n_{p}, \epsilon_{p}\right) \neq \left( 1, 1 \right) $; otherwise, they are defined by shuffle regularization, cf. ($\ref{eq:shufflereg}$) below; the versions $\boldsymbol{\zeta_{k}^{\mathfrak{a}}} (\cdots)$, or $\boldsymbol{\zeta_{k}^{\mathfrak{l}}} (\cdots)$ are defined similarly, from $I^{\mathfrak{a}}$ resp. $I^{\mathfrak{l}}$ above. The roots of unity in the iterated integral will often be denoted by $\eta_{i}\mathrel{\mathop:}= (\epsilon_{i}\cdots \epsilon_{p})^{-1}$\\ \\ From $(\ref{eq:projpiam})$, there is a surjective homomorphism called the \textbf{\textit{period map}}, conjectured to be isomorphism:
\begin{equation}\label{eq:period}\text{per}:\mathcal{H} \rightarrow \mathcal{Z} \text{ , } \zeta^{\mathfrak{m}} \left(n_{1}, \ldots , n_{p} \atop \epsilon_{1} , \ldots ,\epsilon_{p} \right)\mapsto \zeta\left(n_{1}, \ldots , n_{p} \atop \epsilon_{1} , \ldots ,\epsilon_{p} \right).
\end{equation} \texttt{Nota Bene:} Each identity between motivic cyclotomic multiple zeta values is then true for cyclotomic multiple zeta values and in particular each result about a basis with motivic MZV implies the corresponding result about a generating family of MZV by application of the period map.\\ Conversely, we can sometimes \textit{lift} an identity between MZV to an identity between motivic MZV, via the coaction (as in $\cite{Br2}$, Theorem $3.3$); this is discussed below, and illustrated throughout this work in different examples or counterexamples, as in Lemma $\ref{lemmcoeff}$. It is similar in the case of motivic Euler sums ($N=2$). We will see (Theorem $2.4.4$) that for other roots of unity there are several rational coefficients which appear at each step (of the coaction calculus) and prevent us from concluding by identification.\\
\paragraph{Properties.}\label{propii} Motivic iterated integrals satisfy the following properties:\nomenclature{$\mathfrak{S}_{p}$}{set of permutations of $\lbrace 1, \ldots, p\rbrace$.} \begin{itemize}
\item[(i)] $I^{\mathfrak{m}}(a_{0}; a_{1})=1$.
\item[(ii)] $I^{\mathfrak{m}}(a_{0}; a_{1}, \cdots a_{n}; a_{n+1})=0$ if $a_{0}=a_{n+1}$.
\item[(iii)] Shuffle product:\footnote{Product rule for iterated integral in general is: $$ \int_{\gamma} \phi_{1} \cdots \phi_{r} \cdot \int_{\gamma} \phi_{r+1} \cdots \phi_{r+s} = \sum_{\sigma\in Sh_{r,s}} \int_{\gamma} \phi_{\sigma^{-1}(1)} \cdots \phi_{\sigma^{-1}(r+s)} , $$
where $Sh_{r,s}\subset \mathfrak{S}_{r+s}$ is the subset of permutations which respect the order of $\lbrace 1 , \ldots, r\rbrace $ and $\lbrace r+1 , \ldots, r+s\rbrace$. Here, to define the non convergent case, $(iii)$ is sufficient, paired with the other rules.}
\begin{multline}\label{eq:shufflereg} \zeta_{k}^{\mathfrak{m}} \left( {n_{1}, \ldots , n_{p} \atop \epsilon_{1}, \ldots ,\epsilon_{p} }\right)= \\ (-1)^{k}\sum_{i_{1}+ \cdots + i_{p}=k} \binom {n_{1}+i_{1}-1} {i_{1}} \cdots \binom {n_{p}+i_{p}-1} {i_{p}} \zeta^{\mathfrak{m}} \left( {n_{1}+i_{1}, \ldots , n_{p}+i_{p} \atop \epsilon_{1}, \ldots ,\epsilon_{p} }\right).
\end{multline}
\item[(iv)] Path composition:
$$ \forall x\in \mu_{N} \cup \left\{0\right\} , I^{\mathfrak{m}}(a_{0}; a_{1}, \ldots, a_{n}; a_{n+1})=\sum_{i=1}^{n} I^{\mathfrak{m}}(a_{0}; a_{1}, \ldots, a_{i}; x) I^{\mathfrak{m}}(x; a_{i+1}, \ldots, a_{n}; a_{n+1}) .$$
\item[(v)] Path reversal: $I^{\mathfrak{m}}(a_{0}; a_{1}, \ldots, a_{n}; a_{n+1})= (-1)^n I^{\mathfrak{m}}(a_{n+1}; a_{n}, \ldots, a_{1}; a_{0}).$
\item[(vi)] Homothety: $\forall \alpha \in \mu_{N}, I^{\mathfrak{m}}(0; \alpha a_{1}, \ldots, \alpha a_{n}; \alpha a_{n+1}) = I^{\mathfrak{m}}(0; a_{1}, \ldots, a_{n}; a_{n+1})$. \end{itemize}
\textsc{Remark}: These relations, for the multiple zeta values relative to $\mu_{N}$, and for the iterated integrals $I(a_{0}; a_{1}, \cdots ,a_{n}; a_{n+1})$ ($\ref{eq:reprinteg}$), are obviously all easily checked.\\ \\ It has been proven that motivic iterated integrals verify stuffle $\ast$ relations, but also pentagon, and hexagon (resp. octagon for $N>1$) ones, as iterated integral at $\mu_{N}$. In depth $1$, by Deligne and Goncharov, the only relations satisfied by the motivic iterated integrals are distributions and conjugation relations, stated in $\S 2.4.3$.
\paragraph{Motivic Euler $\star$, $\boldsymbol{\sharp}$ sums.} Here, assume that $N=1$ or $2$.\footnote{Detailed definitions of these $\star$ and $\sharp$ versions are given in $\S 4.1$.} In the motivic iterated integrals above, $I^{\mathfrak{m}}(\cdots, a_{i}, \cdots)$, $a_{i}$ were in $\lbrace 0, \pm 1 \rbrace$. We can extend by linearity to $a_{i}\in \lbrace \pm \star, \pm \sharp\rbrace$, which corresponds to a $\omega_{\pm \star}$, resp. $\omega_{ \pm\sharp}$ in the iterated integral, with the differential forms:\nomenclature{$\omega_{\pm\star}$, $\omega_{\pm\sharp}$}{specific differential forms} $$\boldsymbol{\omega_{\pm\star}}\mathrel{\mathop:}= \omega_{\pm 1}- \omega_{0}=\frac{dt}{t(\pm t -1)} \quad \text{ and } \quad \boldsymbol{\omega_{\pm\sharp}}\mathrel{\mathop:}=2 \omega_{\pm 1}-\omega_{0}=\frac{(t \pm 1)dt}{t(t\mp 1)}.$$ It means that, by linearity, for $A,B$ sequences in $\lbrace 0, \pm 1, \pm \star, \pm \sharp \rbrace$: \begin{equation} \label{eq:miistarsharp}
I^{\mathfrak{m}}(A, \pm \star, B)= I^{\mathfrak{m}}(A, \pm 1, B) - I^{\mathfrak{m}}(A, 0, B), \text{ and } I^{\mathfrak{m}}(A, \pm \sharp, B)= 2 I^{\mathfrak{m}}(A, \pm 1, B) - I^{\mathfrak{m}}(A, 0, B). \end{equation} \nomenclature{$\zeta^{\star, \mathfrak{m}}$, resp. $\zeta^{\sharp, \mathfrak{m}}$}{Motivic Euler $\star$ Sums, resp. Motivic Euler $\sharp$ Sums} \begin{itemize} \item[$\boldsymbol{\zeta^{\star, \mathfrak{m}}}$: ]\textit{Motivic Euler $\star$ Sums} are defined by a similar integral representation as MES ($\ref{eq:reprinteg}$), with $\omega_{\pm \star}$ replacing the $\omega_{\pm 1}$, except the first one, which stays a $\omega_{\pm 1}$. \\ Their periods, Euler $\star$ sums, which are already common in the literature, can be written as a summation similar than for Euler sums replacing strict inequalities by large ones: $$ \zeta^{\star}\left(n_{1}, \ldots , n_{p} \right) = \sum_{0 < k_{1}\leq k_{2} \leq \cdots \leq k_{p}} \frac{\epsilon_{1}^{k_{1}} \cdots \epsilon_{p}^{k_{p}}}{k_{1}^{\mid n_{1}\mid} \cdots k_{p}^{\mid n_{p}\mid}}, \quad \epsilon_{i}\mathrel{\mathop:}=sign(n_{i}), \quad n_{i}\in\mathbb{Z}^{\ast}, n_{p}\neq 1. $$ \item[$\boldsymbol{\zeta^{\sharp, \mathfrak{m}}}$: ] \textit{Motivic Euler $\sharp$ Sums} are defined by a similar integral representation as MES ($\ref{eq:reprinteg}$), with $\omega_{\pm \sharp}$ replacing the $\omega_{\pm 1}$, except the first one, which stays a $\omega_{\pm 1}$. \end{itemize} They are both $\mathbb{Q}$-linear combinations of multiple Euler sums, and appear in Chapter $4$, via new bases for motivic MZV (Hoffman $\star$, or with Euler $\sharp$ sums) and in the Conjecture $\ref{lzg}$.\\
\paragraph{Dimensions.} Algebraic $K$-theory provides an \textit{upper bound} for the dimensions of motivic cyclotomic iterated integrals, since: \begin{equation} \begin{array}{ll}
\text{Ext}_{\mathcal{MT}_{N,M}}^{1} (\mathbb{Q}(0), \mathbb{Q}(1)) = (\mathcal{O}_{k_{N}}[\frac{1}{M}])^{\ast} \otimes \mathbb{Q}& \\
\text{Ext}_{\mathcal{MT}_{\Gamma_{N}}}^{1} (\mathbb{Q}(0), \mathbb{Q}(1)) = \Gamma_{N} & \\
\text{Ext}_{\mathcal{MT}_{N,M}}^{1} (\mathbb{Q}(0), \mathbb{Q}(n)) = \text{Ext}_{\mathcal{MT}_{\Gamma}}^{1} (\mathbb{Q}(0), \mathbb{Q}(n)) = K_{2n-1}(k_{N}) \otimes \mathbb{Q} & \text{ for } n >1 .\\
\text{Ext}_{\mathcal{MT}_{N,M}}^{i} (\mathbb{Q}(0), \mathbb{Q}(n))= \text{Ext}_{\mathcal{MT}_{\Gamma}}^{i} (\mathbb{Q}(0), \mathbb{Q}(n)) =0 & \text{ for } i>1 \text{ or } n\leq 0 .
\end{array} \end{equation}
Let $ n_{\mathfrak{p}_{M}}$\nomenclature{$ n_{\mathfrak{p}_{M}}$}{ the number of different prime ideals above the primes dividing $M$} denote the number of different prime ideals above the primes dividing $M$, $\nu_{N}$\nomenclature{$\nu_{N}$}{the number of primes dividing $N$} the number of primes dividing $N$ and $\varphi$ Euler's indicator function\nomenclature{$\varphi$}{Euler's indicator function}. For $M|N$ (cf. $\cite{Bo}$), using Dirichlet $S$-unit theorem when $n=1$: \begin{equation}\label{dimensionk}\dim K_{2n-1}(\mathcal{O}_{k_{N}} [1/M]) \otimes \mathbb{Q} = \left\{ \begin{array}{ll}
1 & \text{ if } N =1 \text{ or } 2 , \text{ and } n \text{ odd }, (n,N) \neq (1,1) .\\
0 & \text{ if } N =1 \text{ or } 2 , \text{ and } n \text{ even } .\\
\frac{\varphi(N)}{2}+ n_{\mathfrak{p}_{M}}-1& \text{ if } N >2, n=1 . \\
\frac{\varphi(N)}{2} & \text{ if } N >2 , n>1 . \end{array} \right. \end{equation} The numbers of generators in each degree, corresponding to the categories $\mathcal{MT}_{N,M}$ resp. $\mathcal{MT}_{\Gamma_{N}}$, differ only in degree $1$: \begin{equation}\label{eq:agamma} \begin{array}{ll}
\text{In degree } >1 : & b_{N}\mathrel{\mathop:}=b_{N,M}= b_{\Gamma_{N}}= \frac{\varphi(N)}{2} \\
\text{In degree } 1 : & a_{N,M}\mathrel{\mathop:}=\frac{\varphi(N)}{2}+ n_{\mathfrak{p}_{M}}-1 \quad \text{ whereas } \quad a_{\Gamma_{N}}\mathrel{\mathop:}= \frac{\varphi(N)}{2}+\nu(N)-1. \end{array} \end{equation} \texttt{Nota Bene}: The following formulas in this paragraph can be applied for the categories $\mathcal{MT}_{N,M}$ resp. $\mathcal{MT}_{\Gamma_{N}}$, replacing $a_{N}$ by $a_{N,M}$ resp. $a_{\Gamma_{N}}$.\\ \\ In degree $1$, for $\mathcal{MT}_{M,N}$, only the units modulo torsion matter whereas for the category $\mathcal{MT}_{\Gamma_{N}}$, only the cyclotomic units modulo torsion matter in degree $1$, cf. $\S 2.4.3$. Recall that cyclotomic units form a subgroup of finite index in the group of units, and generating families for cyclotomic units modulo torsion are (cf. $\cite{Ba}$)\footnote{If we consider cyclotomic units in $\mathbb{Z}[\xi_{N}]\left[ \frac{1}{M}\right] $, with $M=\prod r_{i}$, $r_{i}$ prime power, we have to add $\lbrace 1- \xi_{r_{i}}\rbrace$.}:\nomenclature{ $a\wedge b$}{$gcd(a,b)$} $$\begin{array}{ll} \text{ For } N=p^{s} : &\left\lbrace \frac{1-\xi_{N}^{a}}{1-\xi_{N}} , a\wedge p=1 \right\rbrace, \quad \text{ where } a\wedge b\mathrel{\mathop:}= gcd(a,b).\\ \text{ For } N=\prod_{i} p_{i}^{s_{i}}= \prod q_{i} : &\left\lbrace \frac{1-\xi_{q_{i}}^{a}}{1-\xi_{q_{i}}} , a\wedge p_{i}=1 \right\rbrace \cup \left\lbrace 1-\xi_{d}^{a}, \quad a\wedge d=1, d\mid N, d\neq q_{i} \right\rbrace \\ \end{array}. $$ Results on cyclotomic units determine depth $1$ weight $1$ results for MMZV$_{\mu_{N}}$ (cf. $\S. 2.4.3$).\\ \\ \\ Knowing dimensions, we lift $(\ref{eq:uab})$ to a non-canonical isomorphism with the free Lie algebra: \begin{equation} \label{eq:LieAlg} \mathfrak{u}^{\mathcal{MT} } \underrel{n.c}{\cong} L\mathrel{\mathop:}= \mathbb{L}_{\mathbb{Q}} \left\langle \left( \sigma^{j}_{1}\right)_{1 \leq j \leq a_{N}}, \left( \sigma^{j}_{i}\right)_{1 \leq j \leq b_{N}}, i>1 \right\rangle \quad \sigma_{i} \text{ in degree } -i. \end{equation} The generators $\sigma^{j}_{i}$\nomenclature{$\sigma^{j}_{i}$}{generators of the graded Lie algebra $\mathfrak{u}$} of the graded Lie algebra $\mathfrak{u}$ are indeed non-canonical, only their classes in the abelianization are.\footnote{In other terms, this means: \begin{equation} H_{1}(\mathfrak{u}^{\mathcal{MT}}; \mathbb{Q}) \cong \bigoplus_{i,j \text{ as above }} [\sigma^{j}_{i}]\mathbb{Q} , \quad H^{B}_{i}(\mathfrak{u}^{\mathcal{MT} }; \mathbb{Q}) =0 \text{ for } i>1. \end{equation}} For the fundamental Hopf algebra, with $f^{j}_{i}=(\sigma^{j}_{i})^{\vee}$\nomenclature{$f^{j}_{i}$}{are defined as $(\sigma^{j}_{i})^{\vee}$} in degree $j$: \begin{equation} \label{HopfAlg} \mathcal{A}^{\mathcal{MT}} \underrel{n.c}{\cong} A\mathrel{\mathop:}= \mathbb{Q} \left\langle \left( f^{j}_{1}\right)_{1 \leq j \leq a_{N}}, \left( f^{j}_{i}\right)_{1 \leq j \leq b_{N}}, i>1 \right\rangle . \end{equation} \begin{framed} $\mathcal{A}^{\mathcal{MT}}$ is a cofree commutative graded Hopf algebra cogenerated by $a_{N}$ elements $f^{\bullet}_{1}$ in degree 1, and $b_{N}$ elements $f^{\bullet}_{r}$ in degree $r>1$. \end{framed}
The comodule $\mathcal{H}^{N}\subseteq \mathcal{O}(_{0}\Pi_{1})$ embeds, non-canonically\footnote{We can fix the image of algebraically independent elements with trivial coaction.\\ For instance, for $N=3$, we can choose to send: $ \zeta^{\mathfrak{m}}\left( r \atop j \right) \xmapsto{\phi} f_{r} , \quad \text{ and } \quad \left( 2i \pi \right)^{\mathfrak{m}} \xmapsto{\phi} g_{1}$.}, into $\mathcal{H}^{\mathcal{MT}_{N}}$ and hence:\nomenclature{$\phi^{N}$}{the Hopf algebra morphism $\mathcal{H}^{N} \hookrightarrow H^{N}$}\nomenclature{$H^{N}$}{the Hopf algebra $\mathbb{Q} \left\langle \left(f^{j}_{1}\right) _{1\leq j \leq a_{N}}, \left( f^{j}_{r}\right)_{r>1\atop 1\leq j \leq b_{N}} \right\rangle \otimes \mathbb{Q}\left[ g_{1} \right]$}
\begin{framed} \begin{equation}\label{eq:phih} \mathcal{H}^{N} \xhookrightarrow[n.c.]{\quad\phi^{N}\quad} H^{N}\mathrel{\mathop:}= \mathbb{Q} \left\langle \left(f^{j}_{1}\right) _{1\leq j \leq a_{N}}, \left( f^{j}_{r}\right)_{r>1\atop 1\leq j \leq b_{N}} \right\rangle \otimes \mathbb{Q}\left[ g_{1} \right]. \end{equation}
\end{framed}
\noindent
\texttt{Nota Bene:} This comodule embedding is an isomorphism for $N=1,2,3,4,\mlq 6\mrq,8$ (by F. Brown $\cite{Br2}$ for $N=1$, by Deligne $\cite{De}$ for the other cases; new proof in Chapter $5$), since the categories $\mathcal{MT}'_{N}$, $\mathcal{MT}(\mathcal{O}\left[ \frac{1}{N}\right] )$ and $\mathcal{MT}_{\Gamma_{N}}$ are equivalent. However, for some other $N$, such as $N$ prime greater than $5$, it is not an isomorphism.\\ \noindent Looking at the dimensions $d^{N}_{n}\mathrel{\mathop:}= \dim \mathcal{H}^{\mathcal{MT}_{N}}_{n}$:\nomenclature{$d^{N}_{n}$}{the dimension of the $\mathbb{Q}$vector space $\mathcal{H}^{\mathcal{MT}_{N}}_{n}$} \begin{lemm} For $N>2$, $d^{N}_{n}$ satisfies two (equivalent) recursive formulas\footnote{Those two recursive formulas, although equivalent, leads to two different perspective for counting dimensions.}: $$\begin{array}{lll} d^{N}_{n} = & 1 + a_{N} d_{n-1}+ b_{N}\sum_{i=2}^{n} d_{n -i} & \\ d^{N}_{n} = & (a_{N}+1)d_{n-1}+ (b_{N}-a_{N})d_{n -2} & \text{ with } \left\lbrace \begin{array}{l}
d_{0}=1\\ d_{1}=a_{N}+1 \end{array}\right. \end{array} .$$ Hence the Hilbert series for the dimensions of $\mathcal{H}^{\mathcal{MT}}$ is: $$h_{N}(t)\mathrel{\mathop:}=\sum_{k} d_{k}^{N} t^{k}=\frac{1}{1-(a_{N}+1)t+ (a_{N}-b_{N})t^{2}}. $$ \end{lemm} In particular, these dimensions (for $\mathcal{H}^{\mathcal{MT}_{\Gamma_{N}}}$) are an upper bound for the dimensions of motivic MZV$_{\mu_{N}}$ (i.e. of $\mathcal{H}^{N}$), and hence of MZV$_{\mu_{N}}$ by the period map. In the case $N=p^{r}$, $p\geq 5$ prime, this upper bound is conjectured to be not reached; for other $N$ however, this bound is still conjectured to be sharp (cf. $\S 3.4$). \\ \\ \texttt{Examples:} \begin{itemize} \item[$\cdot$] For the unramified category $\mathcal{MT}(\mathcal{O}_{N})$: $$d_{n}= \frac{\varphi(N)}{2}d_{n-1}+ d_{n-2}.$$ \item[$\cdot$] For $M \mid N$ such that all primes dividing $M$ are inert, $ n_{\mathfrak{p}_{M}}=\nu(N)$. In particular, it is the case if $N=p^{r}$: $$\text{ For } \mathcal{MT}\left( \mathcal{O}_{p^{r}}\left[ \frac{1}{p} \right] \right) \text{ , } \quad d_{n}= \left( \frac{\varphi(N)}{2}+1\right) ^{n}.$$ Let us detail the cases $N=2,3,4,\mlq 6\mrq,8$ considered in Chapter $5$:\\ \end{itemize}
\begin{tabular}{|c|c|c|c|}
\hline
& & & \\
$N \backslash$ $d_{n}^{N}$& $A$ & Dimension relation $d_{n}^{N}$ & Hilbert series \\
\hline
$N=1$\footnotemark[2] & \twolines{$1$ generator in each odd degree $>1$\\
$\mathbb{Q} \langle f_{3}, f_{5}, f_{7}, \cdots \rangle$ } &\twolines{$d_{n}=d_{n-3} +d_{n-2}$,\\ $d_{2}=1$, $d_{1}=0$} & $ \frac{1}{1-t^{2}-t^{3}}$ \\
\hline
$N=2$\footnotemark[3] & \twolines{$1$ generator in each odd degree $\geq 1$\\
$\mathbb{Q} \langle f_{1}, f_{3}, f_{5}, \cdots \rangle$ } & \twolines{$d_{n}=d_{n-1} +d_{n-2}$\\ $d_{0}=d_{1}=1$} & $ \frac{1}{1-t-t^{2}}$ \\
\hline
$N=3,4$ & \twolines{$1$ generator in each degree $\geq 1$\\
$\mathbb{Q} \langle f_{1}, f_{2}, f_{3}, \cdots \rangle$ } & $d_{k}=2d_{k-1} = 2^{k}$ & $\frac{1}{1-2t}$ \\
\hline
$N=8$ & \twolines{$2$ generators in each degree $\geq 1$ \\ $\mathbb{Q} \langle f^{1}_{1}, f^{2}_{1}, f^{1}_{2}, f^{2}_{2}, \cdots \rangle$ } & $d_{k}= 3 d_{k -1}=3^{k}$ & $\frac{1}{1-3t}$ \\
\hline
\twolines{$N=6$ \\ $\mathcal{MT}(\mathcal{O}_{6}\left[\frac{1}{6}\right])$} & \twolines{$1$ in each degree $> 1$, $2$ in degree $1$\\
$\mathbb{Q} \langle f^{1}_{1}, f^{2}_{1}, f_{2}, f_{3}, \cdots \rangle$ } & \twolines{$d_{k}= 3 d_{k -1} -d_{k-2}$, \\$d_{1}=3$} & $\frac{1}{1-3t+t^{2}}$ \\
\hline
\twolines{$N=6$ \\ $\mathcal{MT}(\mathcal{O}_{6})$} & \twolines{$1$ generator in each degree $>1$\\
$\mathbb{Q} \langle f_{2}, f_{3}, f_{4}, \cdots \rangle$} & \twolines{$d_{k}= 1+ \sum_{i\geq 2} d_{k-i}$\\$=d_{k -1} +d_{k-2}$} & $ \frac{1}{1-t-t^{2}}$ \\
\hline \end{tabular} \footnotetext[2]{For $N=1$, Broadhurst and Kreimer made a more precise conjecture for dimensions of multiple zeta values graded by the depth, which transposes to motivic ones: \begin{equation}\label{eq:bkdepth}
\sum \dim (gr^{\mathfrak{D}}_{d} \mathcal{H}^{1}_{n})s^{n}t^{d} = \frac{1+\mathbb{E}(s)t}{1- \mathbb{O}(s)t+\mathbb{S}(s)t^{2}-\mathbb{S}(s) t^{4}} , \quad \text{ where } \begin{array}{l}
\mathbb{E}(s)\mathrel{\mathop:}= \frac{s^{2}}{1-s^{2}}\\
\mathbb{O}(s)\mathrel{\mathop:}= \frac{s^{3}}{1-s^{2}}\\
\mathbb{S}(s)\mathrel{\mathop:}= \frac{s^{12}}{(1-s^{4})(1-s^{6})}
\end{array} \end{equation} where $ \mathbb{E}(s)$, resp. $ \mathbb{O}(s)$, resp. $ \mathbb{S}(s)$ are the generating series of even resp. odd simple zeta values resp. of the space of cusp forms for the full modular group $PSL_{2}(\mathbb{Z})$. The coefficient $\mathbb{S}(s)$ of $t^{2}$ can be understood via the relation between double zetas and cusp forms in $\cite{GKZ}$; The coefficient $\mathbb{S}(s)$ of $t^{4}$, underlying exceptional generators in depth $4$, is now also understood by the recent work of F. Brown $\cite{Br3}$, who gave an interpretation of this conjecture via the homology of an explicit Lie algebra.} \footnotetext[3]{For $N=2$, the dimensions are Fibonacci numbers.}
\section{Motivic Hopf algebra}
\subsection{Motivic Lie algebra.}
Let $\mathfrak{g}$\nomenclature{$\mathfrak{g}$}{ the free graded Lie algebra generated by $e_{0},(e_{\eta})_{\eta\in\mu_{N}}$ in degree $-1$} the free graded Lie algebra generated by $e_{0},(e_{\eta})_{\eta\in\mu_{N}}$ in degree $-1$. Then, the completed Lie algebra $\mathfrak{g}^{\wedge}$ is the Lie algebra of $_{0}\Pi_{1}(\mathbb{Q})$ and the universal enveloping algebra $ U\mathfrak{g}$ is the cocommutative Hopf algebra which is the graded dual of $O(_{0}\Pi_{1})$: \begin{equation} \label{eq:ug} (U\mathfrak{g})_{n}=\left( \mathbb{Q}e_{0} \oplus \left( \oplus_{\eta\in\mu_{N}} \mathbb{Q}e_{\eta}\right) \right) ^{\otimes n}= (O(_{0}\Pi_{1})^{\vee})_{n}. \end{equation} The product is the concatenation, and the coproduct is such that $e_{0},e_{\eta}$ are primitive.\\ \\ Considering the motivic version of the Drinfeld associator:\nomenclature{$\Phi^{\mathfrak{m}}$}{the motivic Drinfeld associator} \begin{equation} \label{eq:associator} \Phi^{\mathfrak{m}}\mathrel{\mathop:}= \sum_{w} \zeta^{\mathfrak{m}} (w) w \in \mathcal{H}\left\langle \left\langle e_{0},e_{\eta} \right\rangle \right\rangle \text{, where :} \end{equation} $$ \zeta^{\mathfrak{m}} (e_{0}^{n}e_{\eta_{1}}e_{0}^{n_{1}-1}\cdots e_{\eta_{p}}e_{0}^{n_{p}-1}) =\zeta^{\mathfrak{m}}_{n}\left( n_{1}, \ldots, n_{p} \atop \epsilon_{1} , \ldots, \epsilon_{p}\right) \text{ with } \begin{array}{l} \epsilon_{p}\mathrel{\mathop:}=\eta_{p}^{-1}\\
\epsilon_{i}\mathrel{\mathop:}=\eta_{i}^{-1}\eta_{i+1} \end{array}.$$ \texttt{Nota Bene:} This motivic Drinfeld associator satisfies the double shuffle relations, and, for $N=1$, the associator equations defined by Drinfeld (pentagon and hexagon), replacing $2\pi i$ by the Lefschetz motivic period $\mathbb{L}^{\mathfrak{m}}$; for $N>1$, an octagon relation generalizes this hexagon relation, as we will see in $\S 4.2.2$.\\ Moreover, it defines a map: $$\oplus \mathcal{H}_{n}^{\vee} \rightarrow U \mathfrak{g} \quad \text{ which induces a map: } \oplus \mathcal{L}_{n}^{\vee} \rightarrow U \mathfrak{g}.$$ Define $\boldsymbol{\mathfrak{g}^{\mathfrak{m}}}$, the \textit{Lie algebra of motivic elements} as the image of $\oplus \mathcal{L}_{n}^{\vee}$ in $U \mathfrak{g}$:\footnote{The action of the Galois group $\mathcal{U}^{\mathcal{MT}}$ turns $\mathcal{L}$ into a coalgebra, and hence $\mathfrak{g}^{\mathfrak{m}}$ into a Lie algebra.} \begin{equation} \label{eq:motivicliealgebra} \oplus \mathcal{L}_{n}^{\vee} \xrightarrow{\sim} \mathfrak{g}^{\mathfrak{m}} \hookrightarrow U \mathfrak{g}. \end{equation} The Lie algebra $\mathfrak{g}^{\mathfrak{m}}$ is equipped with the Ihara bracket given precisely below. Notice that for the cases $N=1,2,3,4,\mlq 6\mrq,8$, $\mathfrak{g}^{\mathfrak{m}}$ is non-canonically isomorphic to the free Lie algebra $L$ defined in $(\ref{eq:LieAlg})$, generated by $(\sigma_{i})'s$.
\paragraph{Ihara action.} As said above, the group scheme $\mathcal{V}$ of automorphisms of $_{x}\Pi_{y}, x,y\in\lbrace 0, \mu_{N} \rbrace$ is isomorphic to $_{0}\Pi_{1}$ ($\ref{eq:gpaut}$), and the group law of automorphisms leads to the Ihara action. More precisely, for $a\in _{0}\Pi_{1}$ (cf. $\cite{DG}$): \begin{equation} \label{eq:actionpi01} \begin{array}{lllll} \text{ The action on } _{0}\Pi_{0} : \quad \quad &\langle a\rangle_{0} : & _{0}\Pi_{0} & \rightarrow &_{0} \Pi_{0} \\ && \exp(e_{0}) &\mapsto & \exp(e_{0}) \\ &&\exp(e_{\eta}) &\mapsto &([\eta]\cdot a) \exp(e_{\eta}) ([\eta]\cdot a)^{-1} \\ \text{ Then, the action on } _{0}\Pi_{1} :\quad \quad & \langle a\rangle : & _{0}\Pi_{1} &\rightarrow &_{0}\Pi_{1} \\ & & b &\mapsto & \langle a\rangle _{0} (b)\cdot a \end{array} \end{equation} This action is called the \textbf{\textit{Ihara action}}:\nomenclature{$\circ$}{Ihara action} \begin{equation} \label{eq:iharaaction} \begin{array}{llll} \circ : & _{0}\Pi_{1} \times _{0}\Pi_{1} & \rightarrow & _{0}\Pi_{1} \\ & (a,b) & \mapsto & a\circ b \mathrel{\mathop:}= \langle a\rangle _{0} (b)\cdot a. \end{array} \end{equation} At the Lie algebra level, it defines the \textit{Ihara bracket} on $Lie(_{0}\Pi_{1})$: \begin{equation} \lbrace a, b\rbrace \mathrel{\mathop:}= a \circ b - b\circ a. \end{equation}
\texttt{Nota Bene:} The dual point of view leads to a combinatorial coaction $\Delta^{c}$, which is the keystone of this work.
\subsection{Coaction}
The motivic Galois group $\mathcal{G}^{\mathcal{MT}_{N}}$ and hence $\mathcal{U}^{\mathcal{MT}}$ acts on the de Rham realization $_{0}\Pi_{1}$ of the motivic fundamental groupoid (cf. $\cite{DG}, \S 4.12$). It is fundamental, since the action of $\mathcal{U}^{\mathcal{MT}}$ is compatible with the structure of $_{x}\Pi_{y}$ (groupoid, $\mu_{N}$ equivariance and inertia), that this action factorizes through the Ihara action, using the isomorphism $\mathcal{V}\cong _{0}\Pi_{1}$ ($\ref{eq:gpaut}$): $$ \xymatrix{ \mathcal{U}^{\mathcal{MT}}\times _{0}\Pi_{1} \ar[r] \ar[d] &_{0}\Pi_{1} \ar[d]^{\sim}\\ _{0}\Pi_{1} \times _{0}\Pi_{1} \ar[r]^{\circ} & _{0}\Pi_{1} \\ }$$ Since $\mathcal{A}^{\mathcal{MT}}= \mathcal{O}(\mathcal{U}^{\mathcal{MT}})$, this action gives rise by duality to a coaction: $\Delta^{\mathcal{MT}}$, compatible with the grading, represented below. By the previous diagram, the combinatorial coaction $\Delta^{c}$ (on words on $0, \eta\in\mu_{N}$), which is explicit (the formula being given below), factorizes through $\Delta^{\mathcal{MT}}$. Remark that $\Delta^{\mathcal{MT}}$ factorizes through $\mathcal{A}$, since $\mathcal{U}$ is the quotient of $\mathcal{U}^{\mathcal{MT}}$ by the kernel of its action on $_{0}\Pi_{1}$. By passing to the quotient, it induces a coaction $\Delta$ on $\mathcal{H}$:
$$ \label{Coaction} \xymatrix{ \mathcal{O}(_{0}\Pi_{1}) \ar[r]^{\Delta^{c}} \ar[d]^{\sim} & \mathcal{A} \otimes_{\mathbb{Q}} \mathcal{O} (_{0}\Pi_{1}) \ar[d] \\ \mathcal{O}(_{0}\Pi_{1}) \ar[d]\ar[r]^{\Delta^{\mathcal{MT}}} & \mathcal{A}^{\mathcal{MT}} \otimes_{\mathbb{Q}} \mathcal{O} (_{0}\Pi_{1}) \ar[d]\\ \mathcal{H} \ar[r]^{\Delta} & \mathcal{A} \otimes \mathcal{H}. \\ }$$ \\ The coaction for motivic iterated integrals is given by the following formula, due to A. B. Goncharov (cf. $\cite{Go1}$) for $\mathcal{A}$ and extended by F. Brown to $\mathcal{H}$ (cf. $\cite{Br2}$):\nomenclature{$\Delta$}{Goncharov coaction} \begin{theom} \label{eq:coaction} The coaction $\Delta: \mathcal{H} \rightarrow \mathcal{A} \otimes_{\mathbb{Q}} \mathcal{H}$ is given by the combinatorial coaction $\Delta^{c}$: $$\Delta^{c} I^{\mathfrak{m}}(a_{0}; a_{1}, \cdots a_{n}; a_{n+1}) =$$ $$\sum_{k ;i_{0}= 0<i_{1}< \cdots < i_{k}<i_{k+1}=n+1} \left( \prod_{p=0}^{k} I^{\mathfrak{a}}(a_{i_{p}}; a_{i_{p}+1}, \cdots a_{i_{p+1}-1}; a_{i_{p+1}}) \right) \otimes I^{\mathfrak{m}}(a_{0}; a_{i_{1}}, \cdots a_{i_{k}}; a_{n+1}) .$$ \end{theom} \noindent \textsc{Remark:} It has a nice geometric formulation, considering the $a_{i}$ as vertices on a half-circle: $$\Delta^{c} I^{\mathfrak{m}}(a_{0}; a_{1}, \cdots a_{n}; a_{n+1})=\sum_{\text{ polygons on circle } \atop \text{ with vertices } (a_{i_{p}})} \prod_{p} I^{\mathfrak{a}}\left( \text{ arc between consecutive vertices } \atop \text{ from } a_{i_{p}} \text{ to } a_{i_{p+1}} \right) \otimes I^{\mathfrak{m}}(\text{ vertices } ).$$ \texttt{Example}: In the reduced coaction\footnote{$\Delta'(x):=\Delta(x)-1\otimes x-x\otimes 1$} of $\zeta^{\mathfrak{m}}(-1,3)=I^{\mathfrak{m}}(0; -1,1,0,0;1)$, there are $3$ non zero cuts: \includegraphics[]{dep1.pdf}. Hence: \begin{multline}\nonumber \Delta'(I^{\mathfrak{m}}(0; -1,1,0,0;1))\\ = I^{\mathfrak{a}}(0; -1;1) \otimes I^{\mathfrak{m}}(0; 1,0,0;1)+ I^{\mathfrak{a}}(-1; 1;0) \otimes I^{\mathfrak{m}}(0; -1,0,0;1)+ I^{\mathfrak{a}}(-1; 1,0,0;1) \otimes I^{\mathfrak{m}}(0; -1;1) \end{multline} I.e, in terms of motivic Euler sums, using the properties of motivic iterated integrals ($\S \ref{propii}$): $$\Delta'(\zeta^{\mathfrak{m}}(-1,3))= \zeta^{\mathfrak{a}}(-1)\otimes \zeta^{\mathfrak{m}}(3)-\zeta^{\mathfrak{a}}(-1)\otimes \zeta^{\mathfrak{m}}(-3)+ (\zeta^{\mathfrak{a}}(3)-\zeta^{\mathfrak{a}}(-3) )\otimes \zeta^{\mathfrak{m}}(-1).$$ \\ Define for $r\geq 1$, the \textit{derivation operators}: \begin{equation}\label{eq:dr} \boldsymbol{D_{r}}: \mathcal{H} \rightarrow \mathcal{L}_{r} \otimes_{\mathbb{Q}} \mathcal{H}, \end{equation}
composite of $\Delta'= \Delta^{c}- 1\otimes id$ with $\pi_{r} \otimes id$, where $\pi_{r}$ is the projection $\mathcal{A} \rightarrow \mathcal{L} \rightarrow \mathcal{L}_{r}$.\\
\\ \texttt{Nota Bene:} It is sufficient to consider these weight-graded derivation operators to keep track of all the information of the coaction.\\ \\ According to the previous theorem, the action of $D_{r}$ on $I^{\mathfrak{m}}(a_{0}; a_{1}, \cdots a_{n}; a_{n+1})$ is:\nomenclature{$D_{r}$}{the $r$-weight-graded part of the coaction $\Delta$ } \begin{framed} \begin{equation} \label{eq:Der} D_{r}I^{\mathfrak{m}}(a_{0}; a_{1}, \cdots, a_{n}; a_{n+1})= \end{equation} $$\sum_{p=0}^{n-1} I^{\mathfrak{l}}(a_{p}; a_{p+1}, \cdots, a_{p+r}; a_{p+r+1}) \otimes I^{\mathfrak{m}}(a_{0}; a_{1}, \cdots, a_{p}, a_{p+r+1}, \cdots, a_{n}; a_{n+1}) .$$ \end{framed} \textsc{Remarks} \begin{itemize} \item[$\cdot$] Geometrically, it is equivalent to keep in the previous coaction only the polygons corresponding to a unique cut of (interior) length $r$ between two elements of the iterated integral. \item[$\cdot$] These maps $D_{r}$ are derivations: $$D_{r} (XY)= (1\otimes X) D_{r}(Y) + (1\otimes Y) D_{r}(X).$$ \item[$\cdot$] This formula is linked with the differential equation satisfied by the iterated integral $I(a_{0}; \cdots; a_{n+1})$ when the $a_{i} 's$ vary (cf. \cite{Go1})\footnote{Since $I(a_{i-1};a_{i};a_{i+1})= \log(a_{i+1}-a_{i})-\log(a_{i-1}-a_{i})$.}: $$dI(a_{0}; \cdots; a_{n+1})= \sum dI(a_{i-1};a_{i};a_{i+1}) I(a_{0}; \cdots, \widehat{a_{i}}, \cdots; a_{n+1}).$$ \end{itemize} \texttt{Example}: By the previous example:\\ $$D_{3}(\zeta^{\mathfrak{m}}(-1,3))=(\zeta^{\mathfrak{a}}(3)-\zeta^{\mathfrak{a}}(-3) )\otimes \zeta^{\mathfrak{m}}(-1)$$ $$D_{1}(\zeta^{\mathfrak{m}}(-1,3))= \zeta^{\mathfrak{a}}(-1)\otimes ( \zeta^{\mathfrak{m}}(3)- \zeta^{\mathfrak{m}}(-3)) $$
\subsection{Depth filtration}
The inclusion of $\mathbb{P}^{1}\diagdown \lbrace 0, \mu_{N},\infty\rbrace \subset \mathbb{P}^{1}\diagdown \lbrace 0,\infty\rbrace$ implies the surjection for the de Rham realizations of fundamental groupoid: \begin{equation} \label{eq:drsurj}
_{0}\Pi_{1} \rightarrow \pi_{1}^{dR}(\mathbb{G}_{m}, \overrightarrow{01}).
\end{equation} Looking at the dual, it corresponds to the inclusion of: \begin{equation} \label{eq:drsurjdual}
\mathcal{O} \left( \pi_{1}^{dR}(\mathbb{G}_{m}, \overrightarrow{01} ) \right) \cong \mathbb{Q} \left\langle e^{0} \right\rangle \xhookrightarrow[\quad \quad]{} \mathcal{O} \left( _{0}\Pi_{1} \right) \cong \mathbb{Q} \left\langle e^{0}, (e^{\eta})_{\eta} \right\rangle .
\end{equation} This leads to the definition of an increasing \textit{depth filtration} $\mathcal{F}^{\mathfrak{D}}$ on $\mathcal{O}(_{0}\Pi_{1})$\footnote{It is the filtration dual to the filtration given by the descending central series of the kernel of the map $\ref{eq:drsurj}$; it can be defined also from the cokernel of $\ref{eq:drsurjdual}$, via the decontatenation coproduct.} such that:\nomenclature{$\mathcal{F}_{\bullet}^{\mathfrak{D}}$}{the depth filtration}
\begin{equation}\label{eq:filtprofw} \boldsymbol{\mathcal{F}_{p}^{\mathfrak{D}}\mathcal{O}(_{0}\Pi_{1})} \mathrel{\mathop:}= \left\langle \text{ words } w \text{ in }e^{0},e^{\eta}, \eta\in\mu_{N} \text{ such that } \sum_{\eta\in\mu_{N}} deg _{e^{\eta}}w \leq p \right\rangle _{\mathbb{Q}}.
\end{equation} This filtration is preserved by the coaction and thus descends to $\mathcal{H}$ (cf. $\cite{Br3}$), on which:
\begin{equation}\label{eq:filtprofh} \mathcal{F}_{p}^{\mathfrak{D}}\mathcal{H}\mathrel{\mathop:}= \left\langle \zeta^{\mathfrak{m}}\left( n_{1}, \ldots, n_{r} \atop \epsilon_{1}, \ldots, \epsilon_{r} \right) , r\leq p \right\rangle _{\mathbb{Q}}.
\end{equation} In the same way, we define $ \mathcal{F}_{p}^{\mathfrak{D}}\mathcal{A}$ and $\mathcal{F}_{p}^{\mathfrak{D}}\mathcal{L}$. Beware, the corresponding grading on $\mathcal{O}(_{0}\Pi_{1})$ is not motivic and the depth is not a grading on $\mathcal{H}$\footnote{ For instance: $\zeta^{\mathfrak{m}}(3)=\zeta^{\mathfrak{m}}(1,2)$. }. The graded spaces $gr^{\mathfrak{D}}_{p}$ are defined as the quotient $\mathcal{F}_{p}^{\mathfrak{D}}/\mathcal{F}_{p-1}^{\mathfrak{D}}$.\\ Similarly, there is an increasing depth filtration on $U\mathfrak{g}$, considering the degree in $\lbrace e_{\eta}\rbrace_{\eta\in\mu_{N}}$, which passes to the motivic Lie algebra $\mathfrak{g}^{\mathfrak{m}}$($\ref{eq:motivicliealgebra}$) such that the graded pieces $gr^{r}_{\mathfrak{D}} \mathfrak{g}^{\mathfrak{m}}$ are dual to $gr^{\mathfrak{D}}_{r} \mathcal{L}$.\\ In depth $1$, there are canonical elements:\footnote{For $N=1$, there are only the $\overline{\sigma}_{2i+1}\mathrel{\mathop:}=(\text{ad} e_{0})^{2i} (e_{1}) \in gr^{1}_{\mathfrak{D}} \mathfrak{g}^{\mathfrak{m}}$, $i>0$ and the subLie algebra generated by them is not free, which means also there are other \say{exceptional} generators in higher depth, cf. \cite{Br2}.\\ For $N=2,3,4,\mlq 6\mrq,8$, when keeping $\eta_{i}$ as in Lemma $5.2.1$, $(\overline{\sigma}^{(\eta_{i})}_{i})$ then generate a free Lie algebra in $gr_{\mathfrak{D}} \mathfrak{g}$.} \begin{equation}\label{eq:oversigma} \overline{\sigma}^{(\eta)}_{i}\mathrel{\mathop:}=(\text{ad } e_{0})^{i-1} (e_{\eta}) \in gr^{1}_{\mathfrak{D}} \mathfrak{g}^{\mathfrak{m}}. \end{equation} They satisfy the distribution and conjugation relations stated below.\\
\paragraph{Depth $\boldsymbol{1}$.} In depth $1$, it is known for $\mathcal{A}$ (cf. $\cite{DG}$ Theorem $6.8$): \begin{lemm}[Deligne, Goncharov] The elements $\zeta^{\mathfrak{a}} \left( r; \eta \right)$ are subject only to the following relations in $\mathcal{A}$: \begin{description} \item[Distribution]
$$\forall d|N \text{ , } \forall \eta\in\mu_{\frac{N}{d}} \text{ , } (\eta,r)\neq(1,1)\text{ , } \zeta^{\mathfrak{a}} \left({r \atop \eta}\right)= d^{r-1} \sum_{\epsilon^{d}=\eta} \zeta^{\mathfrak{a}} \left({r \atop \epsilon}\right).$$ \item[Conjugation] $$\zeta^{\mathfrak{a}} \left({r \atop \eta}\right)= (-1)^{r-1} \zeta^{\mathfrak{a}} \left({r \atop \eta^{-1}}\right).$$ \end{description} \end{lemm} \textsc{Remark}: More generally, distribution relations for MZV relative to $\mu_{N}$ are:
$$\forall d| N, \quad \forall \epsilon_{i}\in\mu_{\frac{N}{d}} \text{ , } \quad \zeta\left( { n_{1}, \ldots , n_{p} \atop \epsilon_{1} , \ldots ,\epsilon_{p} } \right) = d^{\sum n_{i} - p} \sum_{\eta_{1}^{d}=\epsilon_{1}} \cdots \sum_{\eta_{p}^{d}=\epsilon_{p}} \zeta \left( {n_{1}, \ldots , n_{p} \atop \eta_{1} , \ldots ,\eta_{p} } \right) .$$ They are deduced from the following identity:
$$\text{ For } d|N , \epsilon\in\mu_{\frac{N}{d}} \text { , } \sum_{\eta^{d}=\epsilon} \eta^{n}= \left\{ \begin{array}{ll}
d \epsilon ^{\frac{n}{d}}& \text{ if } d|n \\
0 & \text{ else }.\\ \end{array} \right. $$ These relations are obviously analogous of those satisfied by the cyclotomic units modulo torsion. \\ \\ In weight $r>1$, a basis for $gr_{1}^{\mathfrak{D}} \mathcal{A}$ is formed by depth $1$ MMZV at primitive roots up to conjugation. However, MMZV$_{\mu_{N}}$ of weight $1$, $\zeta^{\mathfrak{m}} \left( 1 \atop \xi^{a}_{N}\right) = -\log(1-\xi^{a}_{N})$, are more subtle. For instance (already in $\cite{CZ})$: \begin{lemme} A $\mathbb{Z}$-basis for $\mathcal{A}_{1}$ is hence: \begin{description} \item[$\boldsymbol{N=p^{r}}$:] $ \quad \quad \left\lbrace \zeta^{\mathfrak{a}}\left( 1 \atop \xi^{k}\right) \quad a\wedge p=1 \quad 1 \leq a \leq \frac{p-1}{2} \right\rbrace$. \item[$\boldsymbol{N=pq}$:] With $p<q$ primes: $$ \quad\left\lbrace \left\lbrace \zeta^{\mathfrak{a}}\left( 1 \atop \xi^{k}\right) \quad a\wedge p=1 \quad 1 \leq a \leq \frac{p-1}{2} \right\rbrace \bigcup_{a\in (\mathbb{Z}/q\mathbb{Z})^{\ast}\diagup \langle -1, p\rangle } \left\lbrace \zeta^{\mathfrak{a}}\left( 1 \atop \xi^{ap}\right)\right\rbrace \diagdown \left\lbrace\zeta^{\mathfrak{a}}\left( 1 \atop \xi^{a}\right) \right\rbrace \right. $$ $$\left. \bigcup_{a\in (\mathbb{Z}/p\mathbb{Z})^{\ast}\diagup \langle -1, q\rangle}\left\lbrace \zeta^{\mathfrak{a}}\left( 1 \atop \xi^{aq}\right)\right\rbrace \diagdown \left\lbrace\zeta^{\mathfrak{a}}\left( 1 \atop \xi^{a}\right) \right\rbrace \right\rbrace $$ \end{description} \end{lemme} \textsc{Remarks}: \begin{itemize} \item[$\cdot$] Indeed, for $N=pq$, a phenomenon of loops occurs: orbits via the action of $p$ and $-1$ on $(\mathbb{Z}/q \mathbb{Z})^{\ast}$, resp. of $q$ and $-1$ on $(\mathbb{Z}/p \mathbb{Z})^{\ast}$. Consequently, for each loop we have to remove a primitive root $\zeta\left( 1 \atop \xi^{a}\right) $ and add the non primitive $\zeta\left( 1 \atop \xi^{ap}\right) $ to the basis.\footnote{Cardinal of an orbit $ \lbrace \pm ap^{i} \mod N\rbrace$ is either the order of $p$ modulo $q$, if odd, or half of the order of $p$ modulo $q$, if even.} The situation for $N$ a product of primes would be analogous, considering different orbits associated to each prime; we just have to pay more attention when orbits intersect, for the choice of the representatives $a$: avoid to withdraw or add an element already chosen for previous orbits. \item[$\cdot$] Depth $1$ results also highlight a nice behavior in the cases $N=2,3,4,\mlq 6\mrq,8$: primitive roots of unity modulo conjugation form a basis (as in the case of prime powers) and if we restrict (for dimension reasons) for non primitive roots to $1$ (or $\pm 1$ for $N=8$), it is annihilated in weight $1$ and in weight $>1$ modulo $p$. \item[$\cdot$] In weight $1$, there always exists a $\mathbb{Z}$- basis.\footnote{Conrad and Zhao conjectured (\cite{CZ}) there exists a basis of MZV$_{\mu_{N}}$ for the $\mathbb{Z}$-module spanned by MZV$_{\mu_{N}}$ for each $N$ and fixed weight $w$, except $N=1$, $w=6,7$.} \end{itemize} \texttt{Example}: For $N=34$, relations in depth $1$, weight $1$ lead to two orbits, with $(a)\mathrel{\mathop:}=\zeta^{\mathfrak{a}} \left( 1 \atop \xi^{a}_{N}\right) $: $$\begin{array}{ll} (2)= (16) +(1) & \quad \quad (6)=(3)+(14) \\ (16)=(8) +(9) & \quad \quad (14)=(7)+(10) \\ (8)= (4) +(13) & \quad \quad (10)=(5)+(12) \\ (4)= (2) +(15) & \quad \quad (12)=(11)+(6) \\ \end{array},$$ Hence a basis could be chosen as: $$\left\lbrace \zeta^{\mathfrak{a}}\left( 1 \atop \xi_{34}^{k}\right), k\in \lbrace 5,7,9,11,13,15,\boldsymbol{2,6} \rbrace \right\rbrace .$$
\paragraph{Motivic depth.} The \textit{motivic depth} of an element in $\mathcal{H}^{\mathcal{MT}_{N}}$ is defined, via the correspondence ($\ref{eq:phih}$), as the degree of the polynomial in the $(f_{i})$. \footnote{Beware, $\phi$ is non-canonical, but the degree is well defined.} It can also be defined recursively as, for $\mathfrak{Z}\in \mathcal{H}^{N}$: $$\begin{array}{lll} \mathfrak{Z} \text{ of motivic depth } 1 & \text{ if and only if } & \mathfrak{Z}\in \mathcal{F}_{1}^{\mathfrak{D}}\mathcal{H}^{N}.\\ \mathfrak{Z} \text{ of motivic depth } \leq p & \text{ if and only if } & \left( \forall r< n, D_{r}(\mathfrak{Z}) \text{ of motivic depth } \leq p-1 \right). \end{array}.$$ For $\mathfrak{Z}= \zeta^{\mathfrak{m}} \left( n_{1}, \ldots, n_{p} \atop \epsilon_{1}, \ldots, \epsilon_{p} \right) \in \mathcal{H}^{N}$ of motivic depth $p_{\mathfrak{m}}$, we clearly have the inequalities: $$ \text {depth } p \geq p_{c} \geq p_{\mathfrak{m}} \text{ motivic depth}, \quad \text{ where $ p_{c}$ is the smallest $i$ such that $\mathfrak{Z}\in \mathcal{F}_{i}^{\mathfrak{D}}\mathcal{H}^{N}$}. $$ \texttt{Nota Bene:} For $N=2,3,4, \mlq 6\mrq, 8$, $p_{\mathfrak{m}}$ always coincides with $p_{c}$, whereas for $N=1$, they may differ.
\subsection{Derivation space}
Translating ($\ref{eq:dr}$) for cyclotomic MZV: \begin{lemm} \label{drz} $$D_{r}: \mathcal{H}_{n} \rightarrow \mathcal{L}_{r} \otimes \mathcal{H}_{n-r} $$ \begin{multline}
D_{r} \left(\zeta^{\mathfrak{m}} \left({n_{1}, \ldots , n_{p} \atop \epsilon_{1}, \ldots ,\epsilon_{p}} \right)\right) = \delta_{r=n_{1}+ \cdots +n_{i}} \zeta^{\mathfrak{l}} \left({n_{1}, \cdots, n_{i} \atop \epsilon_{1}, \ldots, \epsilon_{i}}\right) \otimes \zeta^{\mathfrak{m}} \left( { n_{i+1},\cdots, n_{p} \atop \epsilon_{i+1}, \ldots, \epsilon_{p} }\right) \\ + \sum_{1 \leq i<j\leq p \atop \lbrace r \leq \sum_{k=i}^{j} n_{k} -1\rbrace} \left[ \delta_{ \sum_{k=i+1}^{j} n_{k} \leq r } \zeta^{\mathfrak{l}}_{r- \sum_{k=i+1}^{j}n_{k}} \left({ n_{i+1}, \ldots , n_{j} \atop \epsilon_{i+1}, \ldots, \epsilon_{j}}\right) +(-1)^{r} \delta_{ \sum_{k=i}^{j-1} n_{k} \leq r} \zeta^{\mathfrak{l}}_{r- \sum_{k=i}^{j-1}n_{k}} \left({ n_{j-1}, \cdots, n_{i} \atop \epsilon_{j-1}^{-1}, \ldots, \epsilon_{i}^{-1}}\right) \right] \\
\otimes \zeta^{\mathfrak{m}} \left( {\cdots, \sum_{k=i}^{j} n_{k}-r,\cdots \atop \cdots , \prod_{k=i}^{j}\epsilon_{k}, \cdots}\right) \end{multline} \end{lemm} \begin{proof} Straightforward from $(\ref{eq:dr})$, passing to MZV$_{\mu_{N}}$ notation. \end{proof} A key point is that the Galois action and hence the coaction respects the weight grading and the depth filtration\footnote{Notice that $(\mathcal{F}_{0}^{\mathfrak{D}} \mathcal{L}=0$.}: $$D_{r} (\mathcal{H}_{n}) \subset \mathcal{L}_{r} \otimes_{\mathbb{Q}} \mathcal{H}_{n-r}.$$ $$ D_{r} (\mathcal{F}_{p}^{\mathfrak{D}} \mathcal{H}_{n}) \subset \mathcal{L}_{r} \otimes_{\mathbb{Q}} \mathcal{F}_{p-1}^{\mathfrak{D}} \mathcal{H}_{n-r}.$$ Indeed, the depth filtration is motivic, i.e.: $$\Delta (\mathcal{F}^{\mathfrak{D}}_{n}\mathcal{H}) \subset \sum_{p+q=n} \mathcal{F}^{\mathfrak{D}}_{p}\mathcal{A} \otimes \mathcal{F}^{\mathfrak{D}}_{q}\mathcal{H}.$$ Furthermore, $\mathcal{F}^{\mathfrak{D}}_{0}\mathcal{A}=\mathcal{F}^{\mathfrak{D}}_{0}\mathcal{L}=0$. Therefore, the right side of $\Delta(\bullet)$ is in $\mathcal{F}^{\mathfrak{D}}_{q}\mathcal{H}$, with $q<n$. This feature of the derivations $D_{r}$ (decreasing the depth) will enable us to do some recursion on depth through this work.\\ \\ Passing to the depth-graded, define:
$$gr^{\mathfrak{D}}_{p} D_{r}: gr_{p}^{\mathfrak{D}} \mathcal{H} \rightarrow \mathcal{L}_{r} \otimes gr_{p-1}^{\mathfrak{D}} \mathcal{H} \text{, as the composition } (id\otimes gr_{p-1}^{\mathfrak{D}}) \circ D_{r |gr_{p}^{\mathfrak{D}}\mathcal{H}}.$$ By Lemma $\ref{drz}$, all the terms appearing in the left side of $gr^{\mathfrak{D}}_{p} D_{2r+1}$ have depth $1$. Hence, let's consider from now the derivations $D_{r,p}$:\nomenclature{$D_{r,p}$}{depth graded derivations} \begin{lemm} \label{Drp} $$\boldsymbol{D_{r,p}}: gr_{p}^{\mathfrak{D}} \mathcal{H} \rightarrow gr_{1}^{\mathfrak{D}} \mathcal{L}_{r} \otimes gr_{p-1}^{\mathfrak{D}} \mathcal{H} $$ $$ D_{r,p} \left(\zeta^{\mathfrak{m}} \left({n_{1}, \ldots , n_{p} \atop \epsilon_{1}, \ldots ,\epsilon_{p}} \right)\right) = \textsc{(a0) } \delta_{r=n_{1}}\ \zeta^{\mathfrak{l}} \left({r \atop \epsilon_{1}}\right) \otimes \zeta^{\mathfrak{m}} \left( { n_{2},\cdots \atop \epsilon_{2}, \cdots }\right) $$ $$\textsc{(a) } + \sum_{i=2}^{p-1} \delta_{n_{i}\leq r < n_{i}+ n_{i-1}-1} (-1)^{r-n_{i}} \binom {r-1}{r-n_{i}} \zeta^{\mathfrak{l}} \left({ r \atop \epsilon_{i}}\right) \otimes \zeta^{\mathfrak{m}} \left( {\cdots, n_{i}+n_{i-1}-r,\cdots \atop \cdots , \epsilon_{i-1}\epsilon_{i}, \cdots}\right) $$ $$ \textsc{(b) } -\sum_{i=1}^{p-1} \delta_{n_{i}\leq r < n_{i}+ n_{i+1}-1} (-1)^{n_{i}} \binom{r-1}{r-n_{i}} \zeta^{\mathfrak{l}} \left( {r \atop \epsilon_{i}^{-1}}\right) \otimes \zeta^{\mathfrak{m}} \left( {\cdots, n_{i}+n_{i+1}-r, \cdots \atop \cdots , \epsilon_{i+1}\epsilon_{i}, \cdots}\right) $$ $$\textsc{(c) } +\sum_{i=2}^{p-1} \delta_{ r = n_{i}+ n_{i-1}-1 \atop \epsilon_{i-1}\epsilon_{i}\neq 1} \left( (-1)^{n_{i}} \binom{r-1}{n_{i}-1} \zeta^{\mathfrak{l}} \left( {r \atop \epsilon_{i-1}^{-1}} \right) + (-1)^{n_{i-1}-1} \binom{r-1}{n_{i-1}-1} \zeta^{\mathfrak{l}} \left( {r \atop \epsilon_{i}} \right) \right)$$ $$\otimes \zeta^{\mathfrak{m}} \left( {\cdots, 1, \cdots \atop \cdots, \epsilon_{i-1} \epsilon_{i}, \cdots}\right) $$
$$ \textsc{(d) } +\delta_{ n_{p} \leq r < n_{p}+ n_{p-1}-1} (-1)^{r-n_{p}} \binom{r-1}{r-n_{p}} \zeta^{\mathfrak{l}} \left({r \atop \epsilon_{p}} \right) \otimes \zeta^{\mathfrak{m}} \left( {\cdots, n_{p-1}+n_{p}-r\atop \cdots, \epsilon_{p-1}\epsilon_{p}}\right) $$
$$\textsc{(d') } +\delta_{ r = n_{p}+ n_{p-1}-1 \atop \epsilon_{p-1}\epsilon_{p}\neq 1} (-1)^{n_{p-1}}\left( \binom{r-1}{n_{p}-1} \zeta^{\mathfrak{l}} \left( {r \atop \epsilon_{p-1}^{-1}} \right) - \binom{r-1}{n_{p-1}-1} \zeta^{\mathfrak{l}} \left( {r \atop \epsilon_{p}} \right) \right) \otimes \zeta^{\mathfrak{m}} \left( { \cdots, 1 \atop \cdots \epsilon_{p-1}\epsilon_{p}}\right) .$$ \end{lemm} \textsc{Remarks}: \begin{itemize} \item[$\cdot$] The terms of type \textsc{(d, d')}, corresponding to a \textit{deconcatenation}, play a particular role since modulo some congruences (using depth $1$ result for the left side of the coaction), we will get rid of the other terms in the cases $N=2,3,4,\mlq 6\mrq,8$ for the elements in the basis. In the dual point of view of Lie algebra, like in Deligne's article \cite{De} or Wojtkowiak \cite{Wo}, this corresponds to showing that the Ihara bracket $\lbrace,\rbrace$ on these elements modulo some vector space reduces to the usual bracket $[,]$. More generally, for other bases, like Hoffman's one for $N=1$, the idea is still to find an appropriate filtration on the conjectural basis, such that the coaction in the graded space acts on this family, modulo some space, as the deconcatenation, as for the $f_{i}$ alphabet. Indeed, on $H$ ($\ref{eq:phih}$), the weight graded part of the coaction, $D_{r}$ is defined by: \begin{equation}\label{eq:derf} D_{r} : \quad H_{n} \quad \longrightarrow \quad L_{r} \otimes H_{n-r} \quad \quad\text{ such that :} \end{equation} $$ f^{j_{1}}_{i_{1}} \cdots f^{j_{k}}_{i_{k}}\longmapsto\left\{ \begin{array}{ll}
f^{j_{1}}_{i_{1}} \otimes f^{j_{2}}_{i_{2}} \ldots f^{j_{k}}_{i_{k}} & \text{ if } i_{1}=r .\\
0 & \text{ else }.\\ \end{array} \right.$$
\item[$\cdot$] One fundamental feature for a family of motivic multiple zeta values (which makes it \say{natural} and simple) is the \textit{stability} under the coaction. For instance, if we look at the following family which appears in Chapter $5$:
$$\zeta^{\mathfrak{m}}\left(n_{1}, \cdots, n_{p-1}, n_{p} \atop \epsilon_{1}, \ldots , \epsilon_{p-1},\epsilon_{p}\right) \quad \text{ with } \epsilon_{p}\in\mu_{N} \quad \text{primitive} \quad \text{ and } (\epsilon_{i})_{i<p} \quad \text{non primitive}.$$
If N is a power of a prime, this family is stable via the coaction. \footnote{Since in this case, $(\text{non primitive}) \cdot (\text{ non primitive})=$ non primitive and non primitive $\cdot$ primitive $=$ primitive root. Note also, for dimensions reasons, if we are looking for a basis in this form, we should have $N-\varphi(N)\geq \frac{\varphi(N)}{2}$, which comes down here to the case where $N$ is a power of $2$ or $3$.} It is also stable via the Galois action if we only need to take $1$ as a non primitive ($1$-dimensional case), as for $\mathcal{MT} (\mathcal{O}_{6})$.\\
\end{itemize}
\begin{proof} Straightforward from $\ref{drz}$, using the properties of motivic iterated integrals previously listed ($\S \ref{propii}$). Terms of type \textsc{(a)} correspond to cuts from a $0$ (possibly the very first one) to a root of unity, $\textsc{(b)}$ terms from a root of unity to a $0$, $\textsc{(c)}$ terms between two roots of unity and $\textsc{(d,d')}$ terms are the cuts ending in the last $1$, called \textit{deconcatenation terms}. \end{proof}
\paragraph{Derivation space.} By Lemma $2.4.1$ (depth $1$ results), once we have chosen a basis for $gr_{1}^{\mathfrak{D}} \mathcal{L}_{r}$, composed by some $\zeta^{\mathfrak{a}}(r_{i};\eta_{i})$, we can well define: \footnote{Without passing to the depth-graded, we could also define $D^{\eta}_{r}$ as $D_{r}: \mathcal{H}\rightarrow gr^{\mathfrak{D}}_{1}\mathcal{L}_{r} \otimes \mathcal{H}$ followed by $\pi^{\eta}_{r}\otimes id$ where $\pi^{\eta}:gr^{\mathfrak{D}}_{1}\mathcal{L}_{r} \rightarrow \mathbb{Q}$ is the projection on $\zeta^{\mathfrak{m}}\left( r \atop \eta \right) $, once we have fixed a basis for $gr^{\mathfrak{D}}_{1}\mathcal{L}_{r}$; and define as above $\mathscr{D}_{r}$ as the set of the $D^{\eta}_{r,p}$, for $\zeta^{\mathfrak{m}}(r,\eta)$ in the basis of $gr_{1}^{\mathfrak{D}} \mathcal{A}_{r}$.} \begin{itemize} \item[$(i)$] For each $(r_{i}, \eta_{i})$:\nomenclature{$D^{\eta}_{r,p}$}{defined from $D_{r,p}$ followed by a projection} \begin{equation} \label{eq:derivnp} \boldsymbol{D^{\eta_{i}}_{r_{i},p}}: gr_{p}^{\mathfrak{D}}\mathcal{H} \rightarrow gr_{p-1}^{\mathfrak{D}} \mathcal{H}, \end{equation}
as the composition of $D_{r_{i},p}$ followed by the projection: $$\pi^{\eta}: gr_{1}^{\mathfrak{D}} \mathcal{L}_{r}\otimes gr_{p-1}^{\mathfrak{D}} \mathcal{H}\rightarrow gr_{p-1}^{\mathfrak{D}} \mathcal{H}, \quad \quad\zeta^{\mathfrak{m}}(r; \epsilon) \otimes X \mapsto c_{\eta, \epsilon, r} X , $$ with $c_{\eta, \epsilon, r}\in \mathbb{Q}$ the coefficient of $\zeta^{\mathfrak{m}}(r; \eta)$ in the decomposition of $\zeta^{\mathfrak{m}}(r; \epsilon)$ in the basis. \item[$(ii)$] \begin{equation}\label{eq:setdrp}
\boldsymbol{\mathscr{D}_{r,p}} \text{ as the set of } D^{\eta_{i}}_{r_{i},p} \text{ for } \zeta^{\mathfrak{m}}(r_{i},\eta_{i}) \text{ in the chosen basis of } gr_{1}^{\mathfrak{D}} \mathcal{A}_{r}.
\end{equation} \item[$(iii)$] The \textit{derivation set} $\boldsymbol{\mathscr{D}}$ as the (disjoint) union: $\boldsymbol{\mathscr{D}} \mathrel{\mathop:}= \sqcup_{r>0} \left\lbrace \mathscr{D}_{r} \right\rbrace $. \end{itemize}\nomenclature{$\mathscr{D}$}{the derivation set}
\textsc{Remarks}: \begin{itemize} \item[$\cdot$] In the case $N=2,3,4,\mlq 6\mrq$, the cardinal of $\mathscr{D}_{r,p}$ is one (or $0$ if $r$ even and $N=2$, or if $(r,N)=(1,6)$), whereas for $N=8$ the space generated by these derivations is $2$-dimensional, generated by $D^{\xi_{8}}_{r}$ and $D^{-\xi_{8}}_{r}$ for instance. \item[$\cdot$] Following the same procedure for the non-canonical Hopf comodule $H$ defined in $(\ref{eq:phih})$, isomorphic to $\mathcal{H}^{\mathcal{MT}_{N}}$, since the coproduct on $H$ is the deconcatenation $(\ref{eq:derf})$, leads to the following derivations operators: $$\begin{array}{llll} D^{j}_{r} : & H_{n} & \rightarrow & H_{n-r} \\ & f^{j_{1}}_{i_{1}} \cdots f^{j_{k}}_{i_{k}} & \mapsto & \left\{ \begin{array}{ll}
f^{j_{2}}_{i_{2}} \ldots f^{j_{k}}_{i_{k}} & \text{ if } j_{1}=j \text{ and } i_{1}=r .\\
0 & \text{ else }.\\ \end{array} \right. \end{array}.$$ \end{itemize} Now, consider the following application, depth graded version of the derivations above, fundamental for several linear independence results in $\S 4.3$ and Chapter $5$:\nomenclature{$\partial _{n,p}$}{a depth graded version of the infinitesimal coactions} \begin{equation} \label{eq:pderivnp} \boldsymbol{\partial _{n,p}} \mathrel{\mathop:}=\oplus_{r<n\atop D\in \mathscr{D}_{r,p}} D : gr_{p}^{\mathfrak{D}}\mathcal{H}_{n} \rightarrow \oplus_{r<n } \left( gr_{p-1}^{\mathfrak{D}}\mathcal{H}_{n-r}\right) ^{\oplus \text{ card } \mathscr{D}_{r,p}} \end{equation}
\paragraph{Kernel of $\boldsymbol{D_{<n}}$. }
A key point for the use of these derivations is the ability to prove some relations (and possibly lift some from MZV to motivic MZV) up to rational coefficients. This comes from the following theorem, looking at primitive elements:\nomenclature{$D_{<n}$ }{is defined as $\oplus_{r<n} D_{r}$} \begin{theo} Let $D_{<n}\mathrel{\mathop:}= \oplus_{r<n} D_{r}$, and fix a basis $\lbrace \zeta^{\mathfrak{a}}\left( n \atop \eta_{j} \right) \rbrace$ of $gr_{1}^{\mathfrak{D}} \mathcal{A}_{n}$. Then: $$\ker D_{<n}\cap \mathcal{H}^{N}_{n} = \left\lbrace \begin{array}{ll} \mathbb{Q}\zeta^{\mathfrak{m}}\left( n \atop 1 \right) & \text{ for } N=1,2 \text{ and } n\neq 1.\\ \oplus \mathbb{Q} \pi^{\mathfrak{m}} \bigoplus_{1 \leq j \leq a_{N}} \mathbb{Q} \zeta^{\mathfrak{m}}\left( 1 \atop \eta_{j} \right). & \text{ for } N>2, n=1.\\ \oplus \mathbb{Q} (\pi^{\mathfrak{m}})^{n} \bigoplus_{1 \leq j \leq b_{N}} \mathbb{Q} \zeta^{\mathfrak{m}}\left( n \atop \eta_{j} \right). & \text{ for } N>2, n>1. \end{array}\right. .$$ \end{theo} \begin{proof} It comes from the injective morphism of graded Hopf comodules $(\ref{eq:phih})$, which is an isomorphism for $N=1,2,3,4,\mlq 6\mrq,8$: $$\phi: \mathcal{H}^{N} \xrightarrow[\sim]{n.c} H^{N} \mathrel{\mathop:}= \mathbb{Q}\left\langle \left( f^{j}_{i}\right) \right\rangle \otimes_{\mathbb{Q}} \mathbb{Q} \left[ g_{1}\right] .$$ Indeed, for $H^{N}$, the analogue statement is obviously true, for $\Delta'=1\otimes \Delta+ \Delta\otimes 1$: $$\ker \Delta' \cap H_{n} = \oplus_{j} f^{j}_{n} \oplus g_{1}^{n} .$$ \end{proof} \begin{coro}\label{kerdn} Let $D_{<n}\mathrel{\mathop:}= \oplus_{r<n} D_{r}$.\footnote{For $N=1$, we restrict to $r$ odd $>1$; for $N=2$ we restrict to r odd; for $N=\mlq 6\mrq$ we restrict to $r>1$.} Then: $$\ker D_{<n}\cap \mathcal{H}^{N}_{n} = \left\lbrace \begin{array}{ll} \mathbb{Q}\zeta^{\mathfrak{m}}\left( n \atop 1 \right) & \text{ for } N=1,2.\\ \mathbb{Q} (\pi^{\mathfrak{m}})^{n} \oplus \mathbb{Q} \zeta^{\mathfrak{m}}\left( n \atop \xi_{N} \right) & \text{ for } N=3,4,\mlq 6\mrq.\\ \mathbb{Q} (\pi^{\mathfrak{m}})^{n} \oplus \mathbb{Q} \zeta^{\mathfrak{m}}\left( n \atop \xi_{8} \right) \oplus \mathbb{Q} \zeta^{\mathfrak{m}}\left( n \atop -\xi_{8} \right) & \text{ for } N=8.\\ \end{array}\right. .$$ \end{coro} In particular, by this result (for $N=1,2$), proving an identity between motivic MZV (resp. motivic Euler sums), amounts to: \begin{enumerate} \item Prove that the coaction is identical on both sides, computing $D_{r}$ for $r>0$ smaller than the weight. If the families are not stable under the coaction, this step would require other identities. \item Use the corresponding analytic result for MZV (resp. Euler sums) to deduce the remaining rational coefficient; if the analytic equivalent is unknown, we can at least evaluate numerically this rational coefficient. \end{enumerate} Some examples are given in $\S 6.3 $ and $\S 4.4.3$.\\ Another important use of this corollary, is the decomposition of (motivic) multiple zeta values into a conjectured basis, which has been explained by F. Brown in \cite{Br1}.\footnote{He gave an exact numerical algorithm for this decomposition, where, at each step, a rational coefficient has to be evaluated; hence, for other roots of unity, the generalization, albeit easily stated, is harder for numerical experiments.}\\ However, for greater $N$, several rational coefficients appear at each step, and we would need linear independence results before concluding.\\
\chapter{Results} \section{Euler $\star,\sharp$ sums \textit{[Chapter 4]}}
In Chapter $4$, we focus on motivic Euler sums ($N=2$), shortened ES, and motivic multiple zeta values ($N=1$), with in particular some new bases for the vector space of MMZV: one with Euler $\sharp$ sums and, under an analytic conjecture, the so-called \textit{Hoffman $\star$ family}. These two variants of Euler sums are (cf. Definition $4.1.1$): \begin{description} \item[Euler $\star$ sums] corresponds to the analogue multiple sums of ES with $\leq$ instead of strict inequalities. It verifies: \begin{equation} \label{eq:esstar}\zeta ^{\star}(n_{1}, \ldots, n_{p})= \sum_{\circ=\mlq + \mrq \text{ or } ,} \zeta (n_{1}\circ \cdots \circ n_{p}). \end{equation} \texttt{Notation:} This $\mlq + \mrq$ operation on $n_{i}\in\mathbb{Z}$, is a summation of absolute values, while signs are multiplied.\\ These have already been studied in many papers: $\cite{BBB}, \cite{IKOO}, \cite{KST}, \cite{LZ}, \cite{OZ}, \cite{Zh3}$. \item[Euler $\sharp$ sums] are, similarly, linear combinations of MZV but with $2$-power coefficients: \begin{equation} \label{eq:essharp} \zeta^{\sharp}(n_{1}, \ldots, n_{p})= \sum_{\circ=\mlq + \mrq \text{ or } ,} 2^{p-n_{+}} \zeta(n_{1}\circ \cdots \circ n_{p}), \quad \text{ with } n_{+} \text{ the number of } +. \end{equation} \end{description} We also pave the way for a motivic version of a generalization of a Linebarger and Zhao's equality (Conjecture $\ref{lzg}$) which expresses each motivic multiple zeta $\star$ as a motivic Euler $\sharp$ sums; under this conjecture, Hoffman $\star$ family is a basis, identical to the one presented with Euler sums $\sharp$.\\ \\ The first (naive) idea, when looking for a basis for the space of multiple zeta values, is to choose: $$\lbrace \zeta\left( 2n_{1}+1,2n_{2}+1, \ldots, 2n_{p}+1 \right) (2 i \pi)^{2s}, n_{i}\in\mathbb{N}^{\ast}, s\in \mathbb{N} \rbrace .$$ However, considering Broadhurst-Kreimer conjecture $(\ref{eq:bkdepth})$, the depth filtration clearly does \textit{not} behave so nicely in the case of MZV \footnote{Remark, as we will see in Chapter $5$, or as we can see in $\cite{De}$ that for $N=2,3,4,\mlq 6\mrq,8$, the depth filtration is dual of the descending central series of $\mathcal{U}$, and, in that sense, does \textit{behave well}. For instance, the following family is indeed a basis of motivic Euler sums: $$\lbrace \zeta^{\mathfrak{m}}\left( 2n_{1}+1,2n_{2}+1, \ldots, 2n_{p-1}+1,-(2n_{p}+1) \right) (\mathbb{L}^{\mathfrak{m}})^{2s}, n_{i}\in\mathbb{N}, s\in \mathbb{N} \rbrace .$$ } and already in weight $12$, they are not linearly independent: $$28\zeta(9,3)+150\zeta(7,5)+168\zeta(5,7) = \frac{5197}{691}\zeta(12).$$ Consequently, in order to find a basis of motivic MZV, we have to: \begin{itemize} \item[\texttt{Either}: ] Allow \textit{higher} depths, as the Hoffman basis (proved by F Brown in $\cite{Br2}$), or the $\star$ analogue version: $$\texttt{ Hoffman } \star \quad : \left\lbrace \zeta^{\star, \mathfrak{m}} \left( \boldsymbol{2}^{a_{0}},3, \boldsymbol{2}^{a_{1}}, \ldots, 3, \boldsymbol{2}^{a_{p}} \right), a_{i}\geq 0 \right\rbrace .$$ The analogous real family Hoffman $\star$ was also conjectured (in $\cite{IKOO}$, Conjecture $1$) to be a basis of the space of MZV. Up to an analytic conjecture ($\ref{conjcoeff}$), we prove (in $\S 4.4$) that the motivic Hoffman $\star$ family is a basis of $\mathcal{H}^{1}$, the space of motivic MZV\footnote{Up to this analytic statement, $\ref{conjcoeff}$, the Hoffman $\star$ family is then a generating family for MZV.}. In this case, the notion of \textit{motivic depth} (explained in $\S 2.4.3$) is the number of $3$, and is here in general much smaller than the depth. \item[\texttt{Or}: ] Pass by motivic Euler sums, as the Euler $\sharp$ basis given below; it is also another illustration of the descent idea of Chapter $5$: roughly, it enables to reach motivic periods in $\mathcal{H}^{N'}$ coming from above, i.e. via motivic periods in $\mathcal{H}^{N}$, for $N' \mid N$. \end{itemize} More precisely, let look at the following motivic Euler $\sharp$ sums: \begin{theom} The motivic Euler sums $\zeta^{\sharp, \mathfrak{m}} \left( \lbrace \overline{\text{even }}, \text{odd } \rbrace^{\times} \right) $ are motivic geometric periods of $\mathcal{MT}(\mathbb{Z})$. Hence, they are $\mathbb{Q}$ linear combinations of motivic multiple zeta values.\footnote{Since, by $\cite{Br2}$, we know that Frobenius invariant geometric motivic periods of $\mathcal{MT}(\mathbb{Z})$ are $\mathbb{Q}$ linear combinations of motivic multiple zeta values.} \end{theom} \texttt{Notations}: Recall that an overline $\overline{x}$ corresponds to a negative sign, i.e. $-x$ in the argument. Here, the family considered is a family of Euler $\sharp$ sums with only positive odd and negative even integers for arguments.\\ This motivic family is even a generating family of motivic MZV from which we are able to extract a basis: \begin{theom} A basis of $\boldsymbol{\mathcal{P}_{\mathcal{MT}(\mathbb{Z}), \mathbb{R}}^{\mathfrak{m},+}}=\mathcal{H}^{1}$, the space of motivic multiple zeta values is: $$\lbrace\zeta^{\sharp,\mathfrak{m}} \left( 2a_{0}+1,2a_{1}+3,\cdots, 2 a_{p-1}+3, \overline{2a_{p}+2}\right) \text{ , } a_{i}\geq 0 \rbrace .$$ \end{theom} The proof is based on the good behaviour of this family with respect to the coaction and the depth filtration; the suitable filtration corresponding to the \textit{motivic depth} for this family is the usual depth minus $1$.\\ By application of the period map, combining these results: \begin{corol} Each Euler sum $\zeta^{\sharp} \left( \lbrace \overline{\text{even }}, \text{odd } \rbrace^{\times} \right) $ (i.e. with positive odd and negative even integers for arguments) is a $\mathbb{Q}$ linear combination of multiple zeta values of the same weight.\\ Conversely, each multiple zeta value of depth $<d$ is a $\mathbb{Q}$ linear combination of elements $\zeta^{\sharp} \left( 2a_{0}+1,2a_{1}+3,\cdots, 2 a_{p-1}+3, \overline{2a_{p}+2}\right) $, of the same weight with $a_{i}\geq 0$, $p\leq d$. \end{corol}
\textsc{Remarks}: \begin{itemize} \item[$\cdot$] Finding a \textit{good} basis for the space of motivic multiple zeta values is a fundamental question. Hoffman basis may be unsatisfactory for various reasons, while this basis with Euler sums (linear combinations with $2$ power coefficients) may appear slightly more natural, in particular since the motivic depth is here the depth minus $1$. However, both of those two baess are not bases of the $\mathbb{Z}$ module and the primes appearing in the determinant of the \textit{passage matrix}\footnote{The inverse of the matrix expressing the considered basis in term of a $\mathbb{Z}$ basis.} are growing rather fast.\footnote{Don Zagier has checked this for small weights with high precision; he suggested that the primes involved in the case of this basis could have some predictable features, such as being divisor of $2^{n}-1$.} \item[$\cdot$] Looking at how periods of $\mathcal{MT}(\mathbb{Z})$ embed into periods of $\mathcal{MT}(\mathbb{Z}[\frac{1}{2}])$, is a fragment of the Galois descent ideas of Chapter $5$.\\ Euler sums which belong to the $\mathbb{Q}$-vector space of multiple zeta values, sometimes called \textit{honorary}, have been studied notably by D. Broadhurst (cf. $\cite{BBB1}$) among others. We define then \textit{unramified} motivic Euler sums as motivic ES which are $\mathbb{Q}$-linear combinations of motivic MZVs, i.e. in $\mathcal{H}^{1}$. Being unramified for a motivic period implies that its period is unramified, i.e. honorary; some examples of unramified motivic ES are given in $\S 6.2$, or with the family above. In Chapter 5, we give a criterion for motivic Euler sums to be unramified \ref{criterehonoraire}, which generalizes for some other roots of unity; by the period map, this criterion also applies to Euler sums. \item[$\cdot$] For these two theorems, in order to simplify the coaction, we crucially need a motivic identity in the coalgebra $\mathcal{L}$, proved in $\S 4.2$, coming from the octagon relation pictured in Figure $\ref{fig:octagon2}$. More precisely, we need to consider the linearized version of the anti-invariant part by the Frobenius at infinity of this relation, in order to prove this hybrid relation (Theorem $\ref{hybrid}$), for $n_{i}\in\mathbb{N}^{\ast}$, $\epsilon_{i}\in\pm 1$: $$\zeta^{\mathfrak{l}}_{k}\left(n_{0},\cdots, n_{p} \atop \epsilon_{0} , \ldots, \epsilon_{p} \right) + \zeta^{\mathfrak{l}}_{n_{0}+k}\left( n_{1}, \ldots, n_{p} \atop \epsilon_{1} , \ldots, \epsilon_{p} \right) \equiv (-1)^{w+1}\left( \zeta^{\mathfrak{l}}_{k}\left( n_{p}, \ldots, n_{0} \atop \epsilon_{p} , \ldots, \epsilon_{0}\right) + \zeta^{\mathfrak{l}}_{k+n_{p}}\left( n_{p-1}, \ldots,n_{0} \atop \epsilon_{p-1}, \ldots, \epsilon_{0}\right) \right).$$ Thanks to this hybrid relation, and the antipodal relations presented in $\S 4.2.1$, the coaction expression is considerably simplified in Appendix $A.1$. \end{itemize}
\begin{theom} If the analytic conjecture ($\ref{conjcoeff}$) holds, then the motivic \textit{Hoffman} $\star$ family $\lbrace \zeta^{\star,\mathfrak{m}} (\lbrace 2,3 \rbrace^{\times})\rbrace$ is a basis of $\mathcal{H}^{1}$, the space of MMZV. \end{theom} \texttt{Nota Bene:} A MMZV $\star$, in the depth graded, is obviously equal to the corresponding MMZV. However, the motivic Hoffman (i.e. with only $2$ and $3$) multiple zeta $(\star)$ values are almost all zero in the depth graded (the \textit{motivic depth} there being the number of $3$). Hence, the analogous result for the non $\star$ case\footnote{I.e. that the motivic Hoffman family is a basis of the space of MMZV, cf $\cite{Br1}$.}, proved by F. Brown, does not make the result in the $\star$ case anyhow simpler.\\ \\ Denote by $\mathcal{H}^{2,3}$\nomenclature{$\mathcal{H}^{2,3}$}{the $\mathbb{Q}$-vector space spanned by the motivic Hoffman $\star$ family} the $\mathbb{Q}$-vector space spanned by the motivic Hoffman $\star$ family. The idea of the proof is similar as in the non-star case done by Francis Brown. We define an increasing filtration $\mathcal{F}^{L}_{\bullet}$ on $\mathcal{H}^{2,3}$, called the \textit{level}, such that:\footnote{Beware, this notion of level is different than the level associated to a descent in Chapter $5$. It is similar as the level notion for the Hoffman basis, in F. Brown paper's $\cite{Br2}$. It corresponds to the motivic depth, as we will see through the proof.}\nomenclature{$\mathcal{F}^{L}_{l}$}{level filtration on $\mathcal{H}^{2,3}$} \begin{center} $\mathcal{F}^{L}_{l}\mathcal{H}^{2,3}$ is spanned by $\zeta^{\star,\mathfrak{m}} (\boldsymbol{2}^{a_{0}},3,\cdots,3, \boldsymbol{2}^{a_{l}}) $, with less than \say{l} $3$. \end{center} One key feature is that the vector space $\mathcal{F}^{L}_{l}\mathcal{H}^{2,3}$ is stable under the action of $\mathcal{G}$.\\ The linear independence is then proved thanks to a recursion on the level and on the weight, using the injectivity of a map $\partial$ where $\partial$ came out of the level and weight-graded part of the coaction $\Delta$ (cf. $\S 4.4.1$). The injectivity is proved via $2$-adic properties of some coefficients with Conjecture $\ref{conjcoeff}$.\\ One noteworthy difference is that, when computing the coaction on the motivic MZV$^{\star}$, some motivic MZV$^{\star\star}$ arise, which are a non convergent analogue of MZV$^{\star}$ and have to be renormalized. Therefore, where F. Brown in the non-star case needed an analytic formula proven by Don Zagier ($\cite{Za}$), we need some slightly more complicated identities (in Lemma $\ref{lemmcoeff}$) because the elements involved, such as $\zeta^{\star \star,\mathfrak{m}} (\boldsymbol{2}^{a},3, \boldsymbol{2}^{b}) $ for instance, are not of depth $1$ but are linear combinations of products of depth $1$ motivic MZV times a power of $\pi$.\\ \\ \\ These two bases for motivic multiple zeta values turn to be identical, when considering this conjectural motivic identity, more generally: \begin{conje} For $a_{i},c_{i} \in \mathbb{N}^{\ast}$, $c_{i}\neq 2$, \begin{equation} \zeta^{\star, \mathfrak{m}} \left( \boldsymbol{2}^{a_{0}},c_{1},\cdots,c_{p}, \boldsymbol{2}^{a_{p}}\right) = \end{equation} $$(-1)^{1+\delta_{c_{1}}} \zeta^{\sharp, \mathfrak{m}} \left( \pm (2a_{0}+1-\delta_{c_{1}}),\boldsymbol{1}^{ c_{1}-3},\cdots,\boldsymbol{1}^{ c_{i}-3 },\pm(2a_{i}+3-\delta_{c_{i}}-\delta_{c_{i+1}}), \ldots, \pm ( 2 a_{p}+2-\delta_{c_{p}}) \right) . $$ where the sign $\pm$ is always $-$ for an even argument, $+$ for an odd one, $\delta_{c}=\delta_{c=1}$, Kronecker symbol, and $\boldsymbol{1}^{n}:=\boldsymbol{1}^{min(0,n)}$ is a sequence of $n$ 1 if $n\in\mathbb{N}$, an empty sequence else. \end{conje} This conjecture expresses each motivic MZV$^{\star}$ as a linear combination of motivic Euler sums, which gives another illustration of the Galois descent between the Hopf algebra of motivic MZV and the Hopf algebra of motivic Euler sums.\\ \\ \texttt{Nota Bene}: Such a \textit{motivic relation} between MMZV$_{\mu_{N}}$ is stronger than its analogue between MZV$_{\mu_{N}}$ since it contains more information; it implies many other relations because of its Galois conjugates. This explain why its is not always simple to lift an identity from MZV to MMZV from the Theorem $\ref{kerdn}$. If the family concerned is not stable via the coaction, such as $(iv)$ in Lemma $\ref{lemmcoeff}$, we may need other analytic equalities before concluding.\\ \\ This conjecture implies in particular the following motivic identities, whose analogue for real Euler sums are proved as indicated in the brackets\footnote{Beware, only the identity for real Euler sums is proved; the motivic analogue stays a conjecture.}: \begin{description} \item[Two-One] [For $c_{i}=1$, Ohno Zudilin: $\cite{OZ}$]: \begin{equation} \zeta^{\star, \mathfrak{m}} (\boldsymbol{2}^{a_{0}},1,\cdots,1, \boldsymbol{2}^{a_{p}})= - \zeta^{\sharp, \mathfrak{m}} \left( \overline{2a_{0}}, 2a_{1}+1, \ldots, 2a_{p-1}+1, 2 a_{p}+1\right) . \end{equation} \item[Three-One] [For $c_{i}$ alternatively $1$ and $3$, Zagier conjecture, proved in $\cite{BBB}$] \begin{equation} \zeta^{\star, \mathfrak{m}} (\boldsymbol{2}^{a_{0}},1,\boldsymbol{2}^{a_{1}},3 \cdots,1, \boldsymbol{2}^{a_{p-1}}, 3, \boldsymbol{2}^{a_{p}}) = -\zeta^{\sharp, \mathfrak{m}} \left( \overline{2a_{0}}, \overline{2a_{1}+2}, \ldots, \overline{2a_{p-1}+2}, \overline{2 a_{p}+2} \right) . \end{equation} \item[Linebarger-Zhao $\star$] [With $c_{i}\geq 3$, Linebarger Zhao in $\cite{LZ}$]: \begin{equation} \zeta^{\star, \mathfrak{m}} \left( \boldsymbol{2}^{a_{0}},c_{1},\cdots,c_{p}, \boldsymbol{2}^{a_{p}}\right) = -\zeta^{\sharp, \mathfrak{m}} \left( 2a_{0}+1,\boldsymbol{1}^{ c_{1}-3 },\cdots,\boldsymbol{1}^{ c_{i}-3 },2a_{i}+3, \ldots, \overline{ 2 a_{p}+2} \right) \end{equation} In particular, restricting to all $c_{i}=3$: \begin{equation}\label{eq:LZhoffman} \zeta^{\star, \mathfrak{m}} \left( \boldsymbol{2}^{a_{0}},3,\cdots,3,\boldsymbol{ 2}^{a_{p}}\right) = - \zeta^{\sharp, \mathfrak{m}} \left( 2a_{0}+1, 2a_{1}+3, \ldots, 2a_{p-1}+3, \overline{2 a_{p}+2}\right) . \end{equation} \end{description} \texttt{Nota Bene}: Hence the previous conjecture $(\ref{eq:LZhoffman})$ implies that the motivic Hoffman $\star$ is a basis, since we proved the right side of $(\ref{eq:LZhoffman})$ is a basis: $$ \text{ Conjecture } \ref{lzg} \quad \Longrightarrow \quad \text{ Hoffman } \star \text{ is a basis of MMZV} . $$ \\ \texttt{Examples:} The previous conjecture would give such relations:\\ $$\begin{array}{ll} \zeta^{\star, \mathfrak{m}} (2,2,3,3,2) =-\zeta^{\sharp, \mathfrak{m}} (5,3,-4) &\zeta^{\star, \mathfrak{m}} (5,6,2) =-\zeta^{\sharp, \mathfrak{m}} (1,1,1,3,1,1,1,-4) \\ \zeta^{\star, \mathfrak{m}} (1,6) =\zeta^{\sharp, \mathfrak{m}} (-2,1,1,1,-2) &\zeta^{\star, \mathfrak{m}} (2,4, 1, 2,2,3) =-\zeta^{\sharp, \mathfrak{m}} (3,1, -2, -6,-2) . \end{array}$$
\section{Galois Descents \textit{[Chapter 5]}}
There, we study Galois descents for categories of mixed Tate motives $\mathcal{MT}_{\Gamma_{N}}$, and how periods of $\pi_{1}^{un}(X_{N'})$ are embedded into periods of $\pi_{1}^{un}(X_{N})$ for $N'\mid N$. Indeed, for each $N, N'$ with $N'|N$ there are the motivic Galois group $\mathcal{G}^{\mathcal{MT}_{N}}$ acting on $\mathcal{H}^{\mathcal{MT}_{N}} $ and a Galois descent between $\mathcal{H}^{\mathcal{MT}_{N'}}$ and $\mathcal{H}^{\mathcal{MT}_{N}}$, such that: $$(\mathcal{H}^{\mathcal{MT}_{N}})^{\mathcal{G}^{N/N'}}=\mathcal{H}^{\mathcal{MT}_{N'}}.$$ Since for $N=2,3,4,\mlq 6\mrq,8$, the categories $\mathcal{MT}_{N}$ and $\mathcal{MT}'_{N}$ are equal, this Galois descent has a parallel for the motivic fundamental group side; we will mostly neglect the difference in this chapter: \begin{figure}
\caption{Galois descents, $N=2,3,4,\mlq 6\mrq,8$ (level $0$).\protect\footnotemark }
\label{fig:paralleldescent}
\end{figure} \footnotetext{The (non-canonical) horizontal isomorphisms have to be chosen in a compatible way.} \texttt{Nota Bene:} For $N'=1$ or $2$, $i\pi^{\mathfrak{m}}$ has to be replaced by $\zeta^{\mathfrak{m}}(2)$ or $(\pi^{\mathfrak{m}})^{2}$, since we consider, in $\mathcal{H}^{N'}$ only periods invariant by the Frobenius $\mathcal{F}_{\infty}$. In the descent between $\mathcal{H}^{N}$ and $\mathcal{H}^{N'}$, we require hence invariance by the Frobenius in order to keep only those periods; this condition get rid of odd powers of $i\pi^{\mathfrak{m}}$.\\
The first section of Chapter $5$ gives an overview for the Galois descents valid for any $N$: a criterion for the descent between MMZV$_{\mu_{N'}}$ and MMZV$_{\mu_{N}}$ (Theorem $5.1.1$), a criterion for being unramified (Theorem $5.1.2$), and their corollaries. The conditions are expressed in terms of the derivations $D_{r}$, since they reflect the Galois action. Indeed, looking at the descent between $\mathcal{MT}_{N,M}$ and $\mathcal{MT}_{N',M'}$, sometimes denoted $(\mathcal{d})=(k_{N}/k_{N'}, M/M')$, it has possibly two components:\nomenclature{$\mathcal{d}$}{a specific Galois descent between $\mathcal{MT}_{N,M}$ and $\mathcal{MT}_{N',M'}$} \begin{itemize} \item[$\cdot$] The change of cyclotomic fields $k_{N}/k_{N'}$; there, the criterion has to be formulated in the depth graded. \item[$\cdot$] The change of ramification $M/M'$, which is measured by the $1$ graded part of the coaction i.e. $D_{1}$ with the notations of $\S 2.4$.\\ \end{itemize}
The second section specifies the descents for $N\in \left\{2, 3, 4, \mlq 6\mrq, 8\right\}$ \footnote{As above, the quotation marks underline that we consider the unramified category for $N=\mlq 6\mrq$.} represented in Figure $\ref{fig:d248}$, and $\ref{fig:d36}$. In particular, this gives a basis of motivic multiple zeta values relative to $\mu_{N'}$ via motivic multiple zeta values relative to $\mu_{N}$, for these descents considered, $N'\mid N$. It also gives a new proof of Deligne's results ($\cite{De}$): the category of mixed Tate motives over $\mathcal{O}_{k_{N}}[1/N]$, for $N\in \left\{2, 3, 4,\mlq 6\mrq, 8\right\}$ is spanned by the motivic fundamental groupoid of $\mathbb{P}^{1}\setminus\left\{0,\mu_{N},\infty \right\}$ with an explicit basis; as claimed in $\S 2.2$, we can even restrict to a smaller fundamental groupoid.\\ Let us present our results further and fix a descent $(\mathcal{d})=(k_{N}/k_{N'}, M/M')$ among these considered (in Figures $\ref{fig:d248}$, $\ref{fig:d36}$), between the category of mixed Tate motives of $\mathcal{O}_{N}[1/M]$ and $\mathcal{O}_{N'}[1/M']$.\footnote{Usually, the indication of the descent (in the exponent) is omitted when we look at a specific descent.} Each descent $(\mathcal{d})$ is associated to a subset $\boldsymbol{\mathscr{D}^{\mathcal{d}}} \subset \mathscr{D}$ of derivations, which represents the action of the Galois group $\mathcal{G}^{N/N'}$. It defines, recursively on $i$, an increasing motivic filtration $\mathcal{F}^{\mathcal{d}}_{i}$ on $\mathcal{H}^{N}$ called \textit{motivic level}, stable under the action of $\mathcal{G}^{\mathcal{MT}_{N}}$:\nomenclature{$\mathcal{F}^{\mathcal{d}}_{\bullet}$}{the increasing filtration by the motivic level associated to $\mathcal{d}$} $$\texttt{Motivic level:} \left\lbrace \begin{array}{l}
\mathcal{F}^{\mathcal{d}} _{-1} \mathcal{H}^{N} \mathrel{\mathop:}=0\\ \boldsymbol{\mathcal{F}^{\mathcal{d}}_{i}} \text{ the largest submodule of } \mathcal{H}^{N} \text{ such that } \mathcal{F}^{\mathcal{d}}_{i}\mathcal{H}^{N}/\mathcal{F}^{\mathcal{d}} _{i-1}\mathcal{H}^{N} \text{ is killed by } \mathscr{D}^{\mathcal{d}}. \end{array} \right. . $$ The $0^{\text{th}}$ level $\mathcal{F}^{\mathcal{d}}_{0}\mathcal{H}^{N}$, corresponds to invariants under the group $\mathcal{G}^{N/N'}$ while the $i^{\text{th}}$ level $\mathcal{F}^{\mathcal{d}}_{i}$, can be seen as the $i^{\text{th}}$ \textit{ramification space} in generalized Galois descents. Indeed, they correspond to a decreasing filtration of $i^{\text{th}}$ ramification Galois groups $\mathcal{G}_{i}$, which are the subgroups of $\mathcal{G}^{N/N'}$ which acts trivially on $\mathcal{F}^{i}\mathcal{H}^{N}$.\footnote{\textit{On \textbf{ramification groups} in usual Galois theory}: let $L/K$ a Galois extension of local fields. By Hensel's lemma, $\mathcal{O}_{L}=\mathcal{O}_{K}[\alpha]$ and the i$^{\text{th}}$ ramification group is defined as: \begin{equation}\label{eq:ramifgroupi} G_{i}\mathrel{\mathop:}=\left\lbrace g\in \text{Gal}(L/K) \mid v(g(\alpha)-\alpha) >i \right\rbrace , \quad \text{ where } \left\lbrace \begin{array}{l}
v \text{ is the valuation on } L \\
\mathfrak{p}= \lbrace x\in L \mid v(x) >0 \rbrace \text{ maximal ideal for } L \end{array}\right. . \end{equation} Equivalently, this condition means $g$ acts trivially on $\mathcal{O}_{L}\diagup \mathfrak{p}^{i+1}$, i.e. $g(x)\equiv x \pmod{\mathfrak{p}^{i+1}}$. This decreasing filtration of normal subgroups corresponds, by the Galois fundamental theorem, to an increasing filtration of Galois extensions: $$G_{0}=\text{ Gal}(L/K) \supset G_{1} \supset G_{2} \supset \cdots \supset G_{i} \cdots$$ $$K=K_{0} \subset K_{1} \subset K_{2} \subset \cdots \subset K_{i} \cdots$$ $G_{1}$, the inertia subgroup, corresponds to the subextension of minimal ramification.}
\begin{figure}
\caption{Representation of a Galois descent.}
\label{eq:descent}
\label{fig:descent}
\end{figure} \noindent Those ramification spaces constitute a tower of intermediate spaces between the elements in MMZV$_{\mu_{N}}$ and the whole space of MMZV$_{\mu_{N'}}$.\\ \\ Let define the quotients associated to the motivic level: $$\boldsymbol{\mathcal{H}^{\geq i}} \mathrel{\mathop:}= \mathcal{H}/ \mathcal{F}_{i-1}\mathcal{H}\text{ , } \quad\mathcal{H}^{\geq 0}=\mathcal{H}.$$
\noindent The descents considered are illustrated by the following diagrams:\\
\begin{figure}
\caption{\textsc{The cases $N=1,2,4,8$}. }
\label{fig:d248}
\end{figure}
\begin{figure}
\caption{\textsc{The cases $N=1,3,\mlq 6\mrq$}. }
\label{fig:d36}
\end{figure}
\textsc{Remarks:} \begin{enumerate} \item[$\cdot$] The vertical arrows represent the change of field and the horizontal arrows the change of ramification. The full arrows are the descents made explicit in Chapter $5$.\\ More precisely, for each arrow $A \stackrel{\mathcal{F}_{0}}{\leftarrow}B$ in the above diagrams, we give a basis $\mathcal{B}^{A}_{n}$ of $\mathcal{H}_{n}^{A}$, and a basis of $\mathcal{H}_{n}^{B}= \mathcal{F}_{0} \mathcal{H}_{n}^{A}$ in terms of the elements of $\mathcal{B}_{n}^{A}$; similarly for the higher level of these filtrations.
\item[$\cdot$] The framed spaces $\mathcal{H}^{\cdots}$ appearing in these diagrams are not known to be associated to a fundamental group and there is presently no other known way to reach these (motivic) periods. For instance, we obtain by descent, a basis for $\mathcal{H}_{n}^{\mathcal{MT}(\mathbb{Z}\left[ \frac{1}{3}\right] )}$ in terms of the basis of $\mathcal{H}_{n}^{\mathcal{MT}\left( \mathcal{O}_{3}\left[ \frac{1}{3}\right] \right) }$.\\ \end{enumerate} \texttt{Example: Descent between Euler sums and MZV.} The comodule $\mathcal{H}^{1}$ embeds, non-canonically, into $\mathcal{H}^{2}$. Let first point out that:\footnote{Since all the motivic iterated integrals with only $0,1$ of length $1$ are zero by properties stated in $\S \ref{propii}$, hence the left side of $D_{1}$, defined in $(\ref{eq:Der})$, would always cancel.} $D_{1}(\mathcal{H}^{1})=0$; the Galois descent between $\mathcal{H}^{2}$ and $\mathcal{H}^{1}$ is precisely measured by $D_{1}$: \begin{theom} Let $\mathfrak{Z}\in\mathcal{H}^{2}$, a motivic Euler sum. Then: $$\mathfrak{Z}\in\mathcal{H}^{1}, \text{ i.e. is a motivic MZV } \Longleftrightarrow D_{1}(\mathfrak{Z})=0 \textrm{ and } D_{2r+1}(\mathfrak{Z})\in\mathcal{H}^{1}.$$ \end{theom} This is a useful recursive criterion to determine if a (motivic) Euler sum is in fact a (motivic) multiple zeta value. It can be generalized for other roots of unity, as we state more precisely in $\S 5.1$. These unramified motivic Euler sums are the $0^{\text{th}}$-level of the filtration by the motivic level here defined as: \begin{center} $\mathcal{F}_{i}\mathcal{H}^{2}$ is the largest sub-module such that $\mathcal{F}_{i}/ \mathcal{F}_{i-1}$ is killed by $D_{1}$. \end{center}
\paragraph{Results.} More precisely, for $N\in \left\{2, 3, 4, \mlq 6\mrq, 8\right\}$, we define a particular family $\mathcal{B}^{N}$ of motivic multiple zeta values relative to $\mu_{N}$ with different notions of \textbf{\textit{level}} on the basis elements, one for each Galois descent considered above:\nomenclature{$\mathcal{B}^{N}$}{basis of $\mathcal{H}^{N}$} \begin{equation}\label{eq:base} \mathcal{B}^{N}\mathrel{\mathop:}=\left\{ \zeta^{\mathfrak{m}}\left(x_{1}, \cdots x_{p-1}, x_{p} \atop \epsilon_{1}, \ldots, \epsilon_{p-1}, \epsilon_{p}\xi_{N}\right) (2\pi i)^{s ,\mathfrak{m}} \text{ , } x_{i}\in\mathbb{N}^{\ast} , s\in\mathbb{N}^{\ast} , \left\{ \begin{array}{ll}
x_{i} \text{ odd, } s \text{ even}, \epsilon_{i}=1 &\text{if } N=2 \\
\epsilon_{i}=1 &\text{if } N=3,4\\
x_{i} >1 \text{, } \epsilon_{i}=1 &\text{if } N=6\\
\epsilon_{i}\in\lbrace\pm 1\rbrace &\text{if } N=8
\end{array} \right. \right\} \end{equation}
Denote by $\mathcal{B}_{n,p,i}$ the subset of elements with weight $n$, depth $p$ and level $i$. \\
\\
\texttt{Examples}: \begin{description}
\item[$\cdot N=2$: ] The basis for motivic Euler sums: $\mathcal{B}^{2}\mathrel{\mathop:}=\left\{ \zeta^{\mathfrak{m}}\left(2y_{1}+1, \ldots , 2 y_{p}+1 \atop 1, 1, \ldots, 1, -1\right) \zeta^{\mathfrak{m}} (2)^{s}, y_{i} \geq 0, s\geq 0 \right\}$ .
The level for the descent from $\mathcal{H}^{2}$ to $\mathcal{H}^{1}$ is defined as the number of $y_{i}'s$ equal to $0$.
\item[$\cdot N=4$: ] The basis is: $\mathcal{B}^{4}\mathrel{\mathop:}= \left\{ \zeta^{\mathfrak{m}}\left(x_{1}, \ldots , x_{p} \atop 1,1, \ldots, 1, \sqrt{-1}\right) (2\pi i)^{s ,\mathfrak{m}}, s\geq 0, x_{i} >0 \right\} $. $$\text{The level is:} \begin{array}{l} \cdot \text{ the number of even $x_{i}'s$ for the descent from $\mathcal{H}^{4}$ to $\mathcal{H}^{2}$ }\\ \cdot \text{ the number of even $x_{i}'s + $ number of $x_{i}'s$ equal to 1 for the descent from $\mathcal{H}^{4}$ to $\mathcal{H}^{1}$ } \end{array}$$
\item[$\cdot N=8$: ] the level includes the number of $\epsilon_{i}'s$ equal to $-1$, etc.\\
\end{description} The quotients $\mathcal{H}^{\geq i}$, respectively filtrations $\mathcal{F}_{i}$ associated to the descent $\mathcal{d}$, will match with the sub-families (level restricted) $\mathcal{B}_{n,p, \geq i}$, respectively $\mathcal{B}_{n,p, \leq i}$. Indeed, we prove: \footnote{Cf. Theorem $5.2.4$ slightly more precise.}\nomenclature{$\mathbb{Z}_{1[P]}$}{subring of $\mathbb{Z}$} \begin{theom} With $ \mathbb{Z}_{1[P]} \mathrel{\mathop:}=\left\{ \frac{a}{1+b P}, a,b\in\mathbb{Z} \right\}$ where $ P \mathrel{\mathop:}= \left\lbrace \begin{array}{ll} 2 & \text{ for } N=2,4,8 \\ 3 & \text{ for } N=3,\mlq 6\mrq \end{array} \right. $. \begin{enumerate} \item[$\cdot$] $\mathcal{B}_{n,\leq p, \geq i}$ is a basis of $\mathcal{F}_{p}^{\mathfrak{D}} \mathcal{H}_{n}^{\geq i}$ and $\mathcal{B}_{n,\cdot, \geq i} $ a basis of $\mathcal{H}^{\geq i}_{n}$. \item[$\cdot$] $\mathcal{B}_{n, p, \geq i}$ is a basis of $gr_{p}^{\mathfrak{D}} \mathcal{H}_{n}^{\geq i}$ on which it defines a $\mathbb{Z}_{1[P]}$-structure: \begin{center} Each $\zeta^{\mathfrak{m}}\left( z_{1}, \ldots , z_{p} \atop \epsilon_{1}, \ldots, \epsilon_{p}\right)$ decomposes in $gr_{p}^{\mathfrak{D}} \mathcal{H}_{n}^{\geq i}$ as a $\mathbb{Z}_{1[P]}$-linear combination of $\mathcal{B}_{n, p, \geq i}$ elements. \end{center}
\item[$\cdot$] We have the two split exact sequences in bijection: $$ 0\longrightarrow \mathcal{F}_{i}\mathcal{H}_{n} \longrightarrow \mathcal{H}_{n} \stackrel{\pi_{0,i+1}} {\rightarrow}\mathcal{H}_{n}^{\geq i+1} \longrightarrow 0$$ $$ 0 \rightarrow \langle \mathcal{B}_{n, \cdot, \leq i} \rangle_{\mathbb{Q}} \rightarrow \langle\mathcal{B}_{n} \rangle_{\mathbb{Q}} \rightarrow \langle \mathcal{B}_{n, \cdot, \geq i+1} \rangle_{\mathbb{Q}} \rightarrow 0 .$$
\item[$\cdot$] A basis for the filtration spaces $\mathcal{F}_{i} \mathcal{H}_{n}$ is: $$\cup_{p} \left\{ \mathfrak{Z}+ cl_{n, \leq p, \geq i+1}(\mathfrak{Z}), \mathfrak{Z}\in \mathcal{B}_{n, p, \leq i} \right\},$$
$$\text{ where } cl_{n,\leq p,\geq i}: \langle\mathcal{B}_{n, p, \leq i-1}\rangle_{\mathbb{Q}} \rightarrow \langle\mathcal{B}_{n, \leq p, \geq i}\rangle_{\mathbb{Q}} \text{ such that } \mathfrak{Z}+cl_{n,\leq p,\geq i}(\mathfrak{Z})\in \mathcal{F}_{i-1}\mathcal{H}_{n}.$$ \item[$\cdot$] A basis for the graded space $gr_{i} \mathcal{H}_{n}$: $$\cup_{p} \left\{ \mathfrak{Z}+ cl_{n, \leq p, \geq i+1}(\mathfrak{Z}), \mathfrak{Z}\in \mathcal{B}_{n, p, i} \right\}.$$ \end{enumerate} \end{theom} \noindent \texttt{Nota Bene}: The morphism $cl_{n, \leq p, \geq i+1}$ satisfying those conditions is unique.\\ \\ The linear independence is obtained first in the depth graded, and the proof relies on the bijectivity of the following map $\partial^{i, \mathcal{d}}_{n,p}$ by an argument involving $2$ or $3$ adic properties:\footnote{The first $ c ^{\mathcal{d}}_{r}$ components of $\partial^{i, \mathcal{d}}_{n,p}$ correspond to the derivations in $\mathscr{D}^{\mathcal{d}}$ associated to the descent, which hence decrease the motivic level.} \begin{equation}\label{eq:derivintro} \partial^{i, \mathcal{d}}_{n,p}: gr_{p}^{\mathfrak{D}} \mathcal{H}_{n}^{\geq i} \rightarrow \oplus_{r<n} \left( gr_{p-1}^{\mathfrak{D}} \mathcal{H}_{n-r}^{\geq i-1}\right) ^{\oplus c ^{\mathcal{d}}_{r}} \oplus_{r<n} \left( gr_{p-1}^{\mathfrak{D}} \mathcal{H}_{n-r}^{\geq i}\right) ^{\oplus c^{\backslash\mathcal{d}}_{r}} \text{, } c ^{\mathcal{d}}_{r}, c^{\backslash\mathcal{d}}_{r}\in\mathbb{N}, \end{equation} which is obtained from the depth and weight graded part of the coaction, followed by a projection for the left side (by depth $1$ results), and by passing to the level quotients ($(\ref{eq:pderivinp})$). Once the freeness obtained, the generating property is obtained from counting dimensions, since K-theory gives an upper bound for the dimensions.\\ \\ This main theorem generalizes in particular a result of P. Deligne ($\cite{De}$), which we could formulate by different ways: \footnote{The basis $\mathcal{B}$, in the cases where $N\in \left\{3, 4,8\right\}$ is identical to P. Deligne's in $\cite{De}$. For $N=2$ (resp. $N=\mlq 6\mrq$ unramified) it is a linear basis analogous to his algebraic basis which is formed by Lyndon words in the odd (resp. $\geq 2$) positive integers (with $\ldots 5 \leq 3 \leq 1$); a Lyndon word being strictly smaller in lexicographic order than all of the words formed by permutation of its letters. Deligne's method is roughly dual to this point of view, working in Lie algebras, showing the action is faithful and that the descending central series of $\mathcal{U}$ is dual to the depth filtration.} \begin{corol} \begin{itemize} \item[$\cdot$] The map $\mathcal{G}^{\mathcal{MT}} \rightarrow \mathcal{G}^{\mathcal{MT}'}$ is an isomorphism. \item[$\cdot$] The motivic fundamental group $\pi_{1}^{\mathfrak{m}} \left( \mathbb{P}^{1}\diagdown \lbrace 0, \mu_{N}, \infty \rbrace, \overline{0 \xi_{N}} \right)$ generates the category of mixed Tate motives $\mathcal{MT}_{N}$. \item[$\cdot$] $\mathcal{B}_{n}$ is a basis of $ \mathcal{H}^{N}_{n}$, the space of motivic MZV relative to $\mu_{N}$. \item[$\cdot$] The geometric (and Frobenius invariant if $N=2$) motivic periods of $\mathcal{MT}_{N}$ are $\mathbb{Q}$-linear combinations of motivic MZV relative to $\mu_{N}$ (unramified for $N=\mlq 6\mrq$). \end{itemize} \end{corol} \textsc{Remarks: } \begin{itemize} \item[$\cdot$] For $N=\mlq 6\mrq$ the result remains true if we restrict to iterated integrals relative not to all $6^{\text{th}}$ roots of unity but only to these relative to primitive roots. \item[$\cdot$] We could even restrict to: $\begin{array}{ll} \pi_{1}^{\mathfrak{m}} \left( \mathbb{P}^{1}\diagdown \lbrace 0, 1, \infty \rbrace, \overline{0 \xi_{N}} \right) & \text{ for $N=2,3,4,\mlq 6\mrq$} \\ \pi_{1}^{\mathfrak{m}} \left( \mathbb{P}^{1}\diagdown \lbrace 0, \pm 1, \infty \rbrace, \overline{0 \xi_{N}} \right) & \text{ for } N=8 \end{array}$ . \end{itemize}
The previous theorem also provides the Galois descent from $\mathcal{H}^{\mathcal{MT}_{N}}$ to $\mathcal{H}^{\mathcal{MT}_{N'}}$: \begin{corol} A basis for MMZV$_{\boldsymbol{\mu_{N'}}}$ is formed by MMZV$_{\boldsymbol{\mu_{N}}}$ $\in \mathcal{B}^{N}$ of level $0$ each corrected by a $\mathbb{Q}$-linear combination of MMZV $_{\boldsymbol{\mu_{N}}}$ of level greater than or equal to $1$: $$\text{ Basis of } \mathcal{H}^{N'}_{n} : \quad \left\{ \mathfrak{Z}+ cl_{n,\cdot, \geq 1}(\mathfrak{Z}), \mathfrak{Z}\in \mathcal{B}^{N}_{n, \cdot, 0} \right\}.$$ \end{corol} \textsc{Remark}: Descent can be calculated explicitly in small depth, less than or equal to $3$, as we explain in the Appendix $A.2$. In the general case, we could make the part of maximal depth of $cl(\mathfrak{Z})$ explicit (by inverting a matrix with binomial coefficients) but motivic methods do not enable us to describe the other coefficients for terms of lower depth.\\ \\ \texttt{Example, $N=2$:} A basis for motivic multiple zeta values is formed by: $$ \left\lbrace \zeta^{\mathfrak{m}}\left( 2x_{1}+1, \ldots, \overline{2x_{p}+1}\right) \zeta^{\mathfrak{m}}(2)^{s} + \sum_{\exists i, y_{i}=0 \atop q\leq p} \alpha_{\textbf{y}}^{\textbf{x}} \zeta^{\mathfrak{m}}(2y_{1}+1, \ldots, \overline{2y_{q}+1})\zeta^{\mathfrak{m}}(2)^{s} \text{ , } x_{i}>0, \alpha^{\textbf{x}}_{\textbf{y}}\in\mathbb{Q} \right\rbrace .$$ Starting from a motivic Euler sum with odd numbers greater than $1$, we add some \textit{correction terms}, in order to get an element in $\mathcal{H}^{1}$, the space of MMZV. At this level, correction terms are motivic Euler sums with odds, and at least one $1$ in the arguments; i.e. they are of level $\geq 1$ with the previous terminology. For instance, the following linear combination is a motivic MZV: $$\zeta^{\mathfrak{m}}(3,3,\overline{3})+ \frac{774}{191} \zeta^{\mathfrak{m}}(1,5, \overline{3}) - \frac{804}{191} \zeta^{\mathfrak{m}}(1,3, \overline{5}) -6 \zeta^{\mathfrak{m}}(3,1,\overline{5}) +\frac{450}{191}\zeta^{\mathfrak{m}}(1,1, \overline{7}).$$
\section{Miscellaneous Results \textit{[Chapter 6]}}
Chapter $6$ is devoted on the Hopf algebra structure of motivic multiple zeta values relative to $\mu_{N}$, particularly for $N=1,2$, presenting various uses of the coaction, and divided into sections as follows:\\ \begin{enumerate} \item An important use of the coaction, is the decomposition of (motivic) multiple zeta values into a conjectured basis, as explained in $\cite{Br1}$. It is noteworthy to point out that the coaction always enables us to determine the coefficients of the maximal depth terms. We consider in $\S 6.1$ two simple cases, in which the space $gr^{\mathfrak{D}}_{max}\mathcal{H}_{n}$ is $1$ dimensional: \begin{itemize} \item[$(i)$] For $N=1$, when the weight is a multiple of $3$ ($w=3d$), such that the depth $p>d$:\footnote{This was a question asked for by D. Broadhurst: an algorithm, or a formula for the coefficient of $\zeta(3)^{d}$ of such a MZV, when decomposed in Deligne basis.} $$gr^{\mathfrak{D}}_{p}\mathcal{H}_{3d} =\mathbb{Q} \zeta^{\mathfrak{m}}(3)^{d}.$$ \item[$(ii)$] For $N=2,3,4$, when weight equals depth: $$gr^{\mathfrak{D}}_{p}\mathcal{H}_{p} =\mathbb{Q} \zeta^{\mathfrak{m}}\left( 1 \atop \xi_{N}\right) ^{p}.$$ The corresponding Lie algebra, called the \textit{diagonal Lie algebra}, has been studied by Goncharov in $ \cite{Go2}, \cite{Go3} $. \end{itemize} In these cases, we are able to determine the projection: $$\vartheta : gr_{max}^{\mathfrak{D}} \mathcal{H}_{n}^{N} \rightarrow \mathbb{Q},$$ either via the linearized Ihara action $\underline{\circ}$, or via the dual point of view of infinitesimal derivations $D_{r}$. For instance, for $(i)$ ($N=1$, $w=3d$), it boils down to look at: $$\frac{D_{3}^{\circ d }}{d! } \quad \text{ or } \quad \exp_{\circ} ( \overline{\sigma}_{3}), \text{ where } \overline{\sigma}_{2i+1}= (-1)^{i}(\text{ad} e_{0} )^{2i}(e_{1}) \footnote{ These $\overline{\sigma}_{2i+1}$ are the generators of $gr_{1}^{\mathfrak{D}} \mathfrak{g}^{\mathfrak{m}}$, the depth $1$ graded part of the motivic Lie algebra; cf. $(\ref{eq:oversigma})$.}.$$ In general, the space $gr^{\mathfrak{D}}_{max}\mathcal{H}^{N}_{n}$ is more than $1$-dimensional; nevertheless, these methods could be generalized. \item Using criterion $\ref{criterehonoraire}$, we provide in the second section infinite families of honorary motivic multiple zeta values up to depth 5, with specified alternating odd or even integers. It was inspired by some isolated examples of honorary multiple zeta values found by D. Broadhurst\footnote{ Those emerged when looking at the \textit{depth drop phenomena}, cf. $\cite{Bro2}$.}, such as $\zeta( \overline{8}, 5, \overline{8}), \zeta( \overline{8}, 3, \overline{10}), \zeta(3, \overline{6}, 3, \overline{6}, 3)$, where we already could observe some patterns of even and odd. Investigating this trail in a systematic way, looking for any general families of unramified (motivic) Euler sums (without linear combinations first), we arrive at the families presented in $\S 6.2$, which unfortunately, stop in depth $5$. However, this investigation does not cover the unramified $\mathbb{Q}$-linear combinations of motivic Euler sums, such as those presented in Chapter $4$, Theorem $\ref{ESsharphonorary}$ (motivic Euler $\sharp$ sums with positive odds and negative even integers). \item By Corollary $\ref{kerdn}$, we can lift some identities between MZV to motivic MZV (as in $\cite{Br2}$, Theorem $4.4$), and similarly in the case of Euler sums. Remark that, as we will see for depth $1$ Hoffman $\star$ elements (Lemma $\ref{lemmcoeff}$), the lifting may not be straightforward, if the family is not stable under the coaction. In this section $\S 6.3$, we list some identities that we are able to lift to motivic versions, in particular some \textit{Galois trivial} elements\footnote{Galois trivial here means that the unipotent part of the Galois group acts trivially, not $\mathbb{G}_{m}$; hence not strictly speaking Galois trivial.} or product of simple zetas, and sum identities. \end{enumerate}
\textsc{Remark}: The stability of a family on the coaction is a precious feature that allows to prove easily (by recursion) properties such as linear independence\footnote{If we find an appropriate filtration respected by the coaction, and such as the $0$ level elements are Galois-trivial, it corresponds then to the motivic depth filtration; for the Hoffman ($\star$) basis it is the number of $3$; for the Euler $\sharp$ sums basis, it is the number of odds, also equal to the depth minus $1$; for Deligne basis relative to $\mu_{N}$, $N=2,3,4,\mlq 6\mrq,8$, it is the usual depth.}, Galois descent features (unramified for instance), identities ($\S 6.3$), etc. $$\quad $$
\section{And Beyond?}
For most values of $N$, the situation concerning the periods of $\mathcal{MT}_{\Gamma_{N}} \subset \mathcal{MT} ( \mathcal{O}_{N}[\frac{1}{N}])$ is still hazy, although it has been studied in several articles, notably by Goncharov (\cite{Go2},\cite{Go3}, \cite{Go4}\footnote{Goncharov studied the structure of the fundamental group of $\mathbb{G}_{m} \diagdown \mu_{N}$ and made some parallels with the topology of some modular variety for $GL_{m, \diagup\mathbb{Q}}$, $m>1$ notably. He also proved, for $N=p\geq 5$, that the following morphism, given by the Ihara bracket, is not injective: $$\beta: \bigwedge^{2} gr^{\mathfrak{D}}_{1} \mathfrak{g}^{\mathfrak{m}}_{1} \rightarrow gr^{\mathfrak{D}}_{2} \mathfrak{g}^{\mathfrak{m}}_{2} \quad \text{ and } \quad \begin{array}{ll} \dim \bigwedge^{2} gr^{\mathfrak{D}}_{1} \mathfrak{g}^{\mathfrak{m}}_{1} & = \frac{(p-1)(p-3)}{8}\\ \dim \ker \beta & = \frac{p^{2}-1}{24}\\ \dim Im \beta & = \dim gr^{\mathfrak{D}} \mathfrak{g}_{2}^{\mathfrak{m}}= \frac{(p-1)(p-5)}{12} \\ \dim gr^{\mathfrak{D}} \mathfrak{g}_{3}^{\mathfrak{m}}& \geq \frac{(p-5)(p^{2}-2p-11)}{48}. \end{array}$$ Note that $gr^{\mathfrak{D}}_{2} \mathfrak{g}^{\mathfrak{m}}$ corresponds to the space generated by $\zeta^{\mathfrak{m}}\left(1,1 \atop \epsilon_{1}, \epsilon_{2}\right)$ quotiented by dilogarithms $\zeta^{\mathfrak{m}}\left(2 \atop \epsilon \right)$, modulo torsion.}) and Zhao: some bounds on dimensions, tables in small weight, and other results and thoughts on cyclotomic MZV can be seen in $\cite{Zh2}$, $\cite{Zh1}$, $\cite{CZ}$. \\ \\ \texttt{Nota Bene}: As already pointed out, as soon as $N$ has a non inert prime factor $p$\footnote{In particular, as soon as $N\neq p^{s}, 2p^{s}, 4p^{s}, p^{s}q^{k}$ for $p,q$ odd prime since $(\mathbb{Z} \diagup m\mathbb{Z})^{\ast}$ is cyclic $\Leftrightarrow m=2,4,p^{k}, 2p^{k}$.}, $ \mathcal{MT}_{\Gamma_{N}} \subsetneq \mathcal{MT}(\mathcal{O}_{N}\left[ \frac{1}{N} \right] )$. Hence, some motivic periods of $\mathcal{MT}(\mathcal{O}_{N}\left[ \frac{1}{N}\right] )$ are not motivic iterated integrals on $\mathcal{P}^{1}\diagdown \lbrace 0, \mu_{N}, \infty\rbrace$ as considered above; already in weight $1$, there are more generators than the logarithms of cyclotomic units $\log^{\mathfrak{m}} (1-\xi_{N}^{a})$.\\ \\ Nevertheless, we can \textit{a priori} split the situation (of $\mathcal{MT}_{\Gamma_{N}}$) into two main schemes: \begin{itemize} \item[$(i)$] As soon as $N$ has two distinct prime factors, or $N$ power of $2$ or $3$, it is commonly believed that the motivic fundamental group $\pi_{1}^{\mathfrak{m}}(\mathbb{P}^{1}\diagdown \lbrace 0, \infty, \mu_{N}\rbrace, \overrightarrow{01})$ generates $\mathcal{MT}_{\Gamma_{N}}$, even though no suitable basis has been found. Also, in these cases, Zhao conjectured there were non standard relations\footnote{Non standard relations are these which do not come from distribution, conjugation, and regularised double shuffle relation, cf. $\cite{Zh1}$}. Nevertheless, in the case of $N$ power of $2$ or power of $3$, there seems to be a candidate for a basis ($\ref{eq:firstidea}$) and some linearly independent families were exhibited: \begin{equation}\label{eq:firstidea} \zeta^{\mathfrak{m}}\left(n_{1}, \cdots n_{p-1}, n_{p} \atop \epsilon_{1}, \ldots , \epsilon_{p-1},\epsilon_{p}\right) \quad \text{ with } \epsilon_{p}\in\mu_{N} \quad \text{primitive} \quad \text{ and } (\epsilon_{i})_{i<p} \quad \text{non primitive}. \end{equation} Indeed, when $N$ is a power of $2$ or $3$, linearly independent subfamilies of $\ref{eq:firstidea}$, keeping $\frac{3}{4}$ resp. $\frac{2}{3}$ generators in degree $1$, and all generators in degree $r>1$ are presented in $\cite{Wo}$ (in a dual point of view of the one developed here).\\ \\ \texttt{Nota Bene}: Some subfamilies of $\ref{eq:firstidea}$, restricting to $\lbrace \epsilon_{i}=1, x_{i} \geq 2\rbrace$ (here $\epsilon_{p}$ still as above) can be easily proven (via the coaction, by recursion on depth) to be linearly independent for any $N$; if N is a prime power, we can widen to $x_{i}\geq 1$, and for $N$ even to $\epsilon_{i}\in \lbrace \pm 1 \rbrace$; nevertheless, these families are considerably \textit{small}.\\
\item[$(ii)$] For $N=p^{s}$, $p$ prime greater than $5$, there are \textit{missing periods}: i.e. it is conjectured that the motivic fundamental group $\pi_{1}^{\mathfrak{m}}(\mathbb{P}^{1}\diagdown \lbrace 0, \infty, \mu_{N}\rbrace, \overrightarrow{01})$ does not generate $\mathcal{MT}_{\Gamma_{N}}$. For $N=p \geq 5$, it can already be seen in weight $2$, depth $2$. More precisely, (taking the dual point of view of Goncharov in $\cite{Go3}$), the following map is not surjective: \begin{equation}\label{eq:d1prof2} \begin{array}{llll} D_{1}: & gr^{\mathfrak{D}}_{2} \mathcal{A}_{2} & \rightarrow & \mathcal{A}_{1} \otimes \mathcal{A}_{1}\\
&\zeta^{\mathfrak{a}} \left( 1,1 \atop \xi^{a}, \xi^{b}\right)& \mapsto &(a) \otimes (b) + \delta_{a+b \neq 0} ((b)-(a))\otimes (a+b) \end{array}, \text{ where } \boldsymbol{(a)}\mathrel{\mathop:}= \zeta^{\mathfrak{a}} \left( 1 \atop \xi^{a}\right). \end{equation} These missing periods were a motivation for instance to introduce Aomoto polylogarithms (in \cite{Du})\footnote{Aomoto polylogarithms generalize the previous iterated integrals, with notably differential forms such as $\frac{dt_{i}}{t_{i}-t_{i+1}-a_{i}}$; there is also a coaction acting on them.}.\\ Another idea, in order to reach these missing periods would be to use Galois descents: coming from a category above, in order to arrive at the category underneath, in the manner of Chapter $5$. For instance, missing periods for $N=p$ prime $>5$, could be reached via a Galois descent from the category $\mathcal{MT}_{\Gamma_{2p}}$ \footnote{This category is equal to $\mathcal{MT}(\mathcal{O}_{2p} ([\frac{1}{2p}]))$ iff $2$ is a primitive root modulo $p$. Some conditions on $p$ necessary or sufficient are known: this implies that $p\equiv 3,5 ± \mod 8$; besides, if $p\equiv 3,11 \mod 16$, it is true, etc.}. First, let point out that this category has the same dimensions than $\mathcal{MT}_{p}$ in degree $>1$, and has one more generator in degree $1$, corresponding to $\zeta^{\mathfrak{a}} \left( 1 \atop \xi^{p} \right) $. Furthermore, for $p$ prime, the descent between $\mathcal{H}^{2p}$ and $\mathcal{H}^{p}$ is measured by $D_{1}^{p}$, the component of $D_{1}$ associated to $\zeta^{\mathfrak{a}} \left( 1 \atop \xi^{p}\right) $:\\ $$\text{Let } \mathfrak{Z} \in \mathcal{H}^{2p}, \text{ then } \mathfrak{Z} \in \mathcal{H}^{p} \Leftrightarrow \left\lbrace \begin{array}{l} D^{p}_{1}(\mathfrak{Z})=0\\ D_{r} (\mathfrak{Z}) \in \mathcal{H}^{p} \end{array}\right.$$ The situation is pictured by: \begin{equation}\label{eq:descent2p} \xymatrix{ \mathcal{H}^{2p}:=\mathcal{H}^{\Gamma_{2p}} \ar@{^{(}->}[r] & \mathcal{H}^{\mathcal{MT}\left( \mathcal{O}_{2p}\left[ \frac{1}{2p}\right]\right) }\\ \mathcal{H}^{p}:= \mathcal{H}^{\Gamma_{p}}=\mathcal{H}^{\mathcal{MT}\left( \mathcal{O}_{p}\left[ \frac{1}{p}\right]\right)} \ar[u]^{D^{p}_{1}} \ar@{=}[r] & \mathcal{H}^{\mathcal{MT}\left( \mathcal{O}_{2p}\left[ \frac{1}{p}\right]\right) } \\ \mathcal{H}^{\mathcal{MT}\left( \mathcal{O}_{p}\right) }\ar[u]^{\lbrace D^{2a}_{1}-D^{a}_{1}\rbrace_{2 \leq a \leq \frac{p-1}{2}}} \ar@{=}[r] & \mathcal{H}^{\mathcal{MT}\left( \mathcal{O}_{2p}\right) } }. \end{equation} \texttt{Example, for N=5}:\nomenclature{$\text{Vec}_{k} \left\langle X \right\rangle$}{the $k$ vector space generated by elements in $X$ } A basis of $gr^{\mathfrak{D}}_{p}\mathcal{A}_{1}$ corresponds to the logarithms of the roots of unity $\xi^{1}, \xi^{2}$; here, $\xi=\xi_{5}$ is a primitive fifth root of unity. Moreover, the image of $D_{1}: \mathcal{A}_{2} \rightarrow \mathcal{A}_{1} \otimes \mathcal{A}_{1}$ on $\zeta^{\mathfrak{a}} \left( 1,1 \atop \xi_{5}^{a}, \xi_{5}^{b}\right) $ is (cf. $\ref{eq:d1prof2}$): $$\text{Vec}_{\mathbb{Q}} \left\langle (1)\otimes (1), (2)\otimes (2),(1)\otimes (2)+ (2)\otimes (1) \right\rangle .$$ We notice that one dimension is missing ($3$ instead of $4$). Allowing the use of tenth roots of unity, adding for instance here in depth $2$, $\zeta^{\mathfrak{a}} \left( 1,1 \atop \xi_{10}^{1}, \xi_{10}^{2}\right)$ recovers the surjection for $D_{1}$. Since we have at our disposal criterion to determine if a MMZV$_{\mu_{10}}$ is in $\mathcal{H}^{5}$, we could imagine constructing a base of $\mathcal{H}^{5}$ from tenth roots of unity. \\ \texttt{Nota Bene}: More precisely, we have the following spaces, descents and dimensions: \begin{equation}\label{eq:descent10} \xymatrix{ \mathcal{H}^{\mathcal{MT}\left( \mathcal{O}_{10}\left[ \frac{1}{10}\right]\right) }= \mathcal{H}^{\Gamma_{10}} & \\ \mathcal{H}^{\mathcal{MT}\left( \mathcal{O}_{10}\left[ \frac{1}{5}\right]\right) }=\mathcal{H}^{\mathcal{MT}\left( \mathcal{O}_{5}\left[ \frac{1}{5}\right] \right) }= \mathcal{H}^{\Gamma_{5}} \ar[u]^{D^{5}_{1}} & d_{n}= 2d_{n-1}+3d_{n-2}= 3d_{n-1}\\ \mathcal{H}^{\mathcal{MT}\left( \mathcal{O}_{5}\right) }=\mathcal{H}^{\mathcal{MT}\left( \mathcal{O}_{10}\right) } \ar[u]^{D^{4}_{1}+ D^{2}_{1}} & d'_{n}= 2d'_{n-1}+d'_{n-2} }\\ \end{equation}
\end{itemize} \noindent \textsc{Remarks}: \begin{itemize} \item[$\cdot$] Recently (in $\cite{Bro3}$), Broadhurst made some conjectures about \textit{multiple Landen values}, i.e. periods associated to the ring of integers of the real subfield of $\mathbb{Q}(\xi_{5})$, i.e. $\mathbb{Z} \left[\rho \right]$, with $\rho\mathrel{\mathop:}= \frac{1+\sqrt{5}}{2} $, the golden ratio\footnote{He also looked at the case of the real subfield of $\mathbb{Q}(\xi_{7})$ in his latest article: $\cite{Bro4}$}. Methods presented through this thesis could be transposed in such context.\\ \item[$\cdot$] It also worth noticing that, for $N=p>5$, modular forms obstruct the freeness of the Lie algebra $gr_{\mathfrak{D}} \mathfrak{g}^{\mathfrak{m}}$\footnote{Goncharov proved that the subspace of cuspidal forms of weight 2 on the modular curve $X_{1}(p)$ (associated to $\Gamma_{1}(p)$), of dimension $\frac{(p-5)(p-7)}{24}$ embeds into $\ker \beta$, for $N=p \geq 11$ which leaves another part of dimension $\frac{p-3}{2}$.}, as in the case of $N=1$ (cf. $\cite{Br3}$). Indeed, for $N=1$ one can associate, to each cuspidal form of weight $n$, a relation between weight $n$ double and simple multiple zeta values, cf. $\cite{GKZ}$. Notice that, on the contrary, for $N=2,3,4,8$, $gr_{\mathfrak{D}} \mathfrak{g}^{\mathfrak{m}}$ is free. This fascinating connection with modular forms still waits to be explored for cyclotomic MZV. \footnote{We could hope also for an interpretation, in these cyclotomic cases, of exceptional generators and relations in the Lie algebra, in the way of $\cite{Br3}$ for $N=1$.}\\ \item[$\cdot$] In these cases where $gr_{\mathfrak{D}} \mathfrak{g}^{\mathfrak{m}}$ is not free, since we have to turn towards other basis (than $\ref{eq:firstidea}$), we may remember the Hoffman basis (of $\mathcal{H}^{1}_{n}$, cf $\cite{Br2}$): $\left\lbrace \zeta^{\mathfrak{m}} \left( \lbrace 2, 3\rbrace^{\times}\right)\right\rbrace_{\text{ weight } n}$, whose dimensions verify $d_{n}=d_{n-2}+d_{n-3}$. Looking at dimensions in Lemma $2.3.1$, two cases bring to mind a basis in the \textit{\textbf{Hoffman's way}}: \begin{itemize} \item[$(i)$] For $\mathcal{MT}(\mathcal{O}_{N})$, since $d_{n}= \frac{\varphi(N)}{2}d_{n-1}+ d_{n-2}$, this suggests to look for a basis with $\boldsymbol{1}$ (with $\frac{\varphi(N)}{2}$ choices of $N^{\text{th}}$ roots of unity) and $\boldsymbol{2}$ (1 choice of $N^{\text{th}}$ roots of unity). \item[$(ii)$] For $\mathcal{MT}\left( \mathcal{O}_{p^{r}}\left[ \frac{1}{p} \right] \right)$, where $p \mid N$ and $p$ inert, since $ d_{n}= \left( \frac{\varphi(N)}{2}+1\right)^{n},$ this suggests a basis with only $1$ above, and $( \frac{\varphi(N)}{2}+1)$ choices of $N^{\text{th}}$ roots of unity; in particular if $N=p^{k}$.\\ \end{itemize} \texttt{Example}: For $N=2$, the recursion relation for dimensions $d_{n}=d_{n-1}+d_{n-2}$ of $\mathcal{H}_{n}^{2}$ suggests, \textit{in the Hoffman's way}, a basis composed of motivic Euler sums with only $1$ and $2$. For instance, the following are candidates conjectured to be a basis, supported by numerical computations: $$\left\lbrace \zeta^{\mathfrak{m}} \left( n_{1}, \ldots, n_{p-1}, n_{p} \atop 1, \ldots, 1, -1 \right) , n_{i}\in \lbrace 2, 1\rbrace \right\rbrace, \textsc{ or } \left\lbrace \zeta^{\mathfrak{m}} \left( 1, \cdots 1, \atop \boldsymbol{s}, -1 \right)\zeta^{\mathfrak{m}} (2)^{\bullet} ,\boldsymbol{s}\in \left\lbrace \lbrace 1\rbrace, \lbrace -1,-1\rbrace\right\rbrace ^{\ast }\right\rbrace.$$ However, there is not a nice \textit{suitable} filtration\footnote{In the second case, it appears that we could proceed as follows to show the linear independence of these elements, where $p$ equals $1+$ the number of $1$ in the $E_{n}$ element: Prove that, for $x\in E_{n,p}$ there exists a linear combination $cl(x)\in E_{n,>p}$ such that $x+cl(x)\in\mathcal{F}^{\mathfrak{D}}_{p} \mathcal{H}_{n}$, and then that $\lbrace x+cl(x), x\in E_{n,p} \rbrace$ is precisely a basis for $gr^{\mathfrak{D}}_{p} \mathcal{H}_{n}$, considering, for $2r \leq n-p$: $$D_{2r+1}: gr^{\mathfrak{D}}_{p} \mathcal{H}_{n} \rightarrow gr^{\mathfrak{D}}_{p-1} \mathcal{H}_{n-2r-1}.$$} corresponding to the motivic depth which would allow a recursive proof \footnote{A suitable filtration, whose level $0$ would be the power of $\pi$, level $1$ would be linear combinations of $\zeta(odd)\cdot\zeta(2)^{\bullet}$, etc.; as in proofs in $\S 4.5.1$.}. \end{itemize}
\chapter{MZV $\star$ and Euler $\sharp$ sums}
\paragraph{\texttt{Contents}:} After introducing motivic Euler $\star$, and $\sharp$ sums, with some useful motivic relations (antipodal and hybrid), the third section focuses on some specific Euler $\sharp$ sums, starting by a broad subfamily of \textit{unramified} elements (i.e. which are motivic MZV) and extracting from it a new basis for $\mathcal{H}^{1}$. The fourth section deals with the Hoffman star family, proving it is a basis of $\mathcal{H}^{1}$, up to an analytic conjecture ($\ref{conjcoeff}$). In Appendix $\S 4.7$, some missing coefficients in Lemma $\ref{lemmcoeff}$, although not needed for the proof of the Hoffman $\star$ Theorem $\ref{Hoffstar}$, are discussed. The last section presents a conjectured motivic equality ($\ref{lzg}$) which turns each motivic MZV $\star$ into a motivic Euler $\sharp$ sums of the previous honorary family; in particular, under this conjecture, the two previous bases are identical. The proofs here are partly based on results of Annexe $\S A.1$, which themselves use relations presented in $\S 4.2$.
\section{Star, Sharp versions}
Here are the different variants of motivic Euler sums (MES) used in this chapter, where a $ \pm \star$ resp. $\pm \sharp$ in the notation below $I(\cdots)$ stands for a $\omega_{\pm \star} $ resp. $\omega_{\pm\sharp}$ in the iterated integral:\footnote{Possibly regularized with $(\ref{eq:shufflereg})$.} \begin{defi} Using the expression in terms of motivic iterated integrals ($\ref{eq:reprinteg}$), motivic Euler sums are, with $n_{i}\in\mathbb{Z}^{\ast}$, $\epsilon_{i}\mathrel{\mathop:}=sign(n_{i})$: \begin{equation}\label{eq:mes} \zeta^{\mathfrak{m}}_{k} \left(n_{1}, \ldots , n_{p} \right) \mathrel{\mathop:}= (-1)^{p}I^{\mathfrak{m}} \left(0; 0^{k}, \epsilon_{1}\cdots \epsilon_{p}, 0^{\mid n_{1}\mid -1} ,\ldots, \epsilon_{i}\cdots \epsilon_{p}, 0^{\mid n_{i}\mid -1} ,\ldots, \epsilon_{p}, 0^{\mid n_{p}\mid-1} ;1 \right). \end{equation} $$\text{ With the differentials: } \omega_{\pm\star}\mathrel{\mathop:}= \omega_{\pm 1}- \omega_{0}=\frac{dt}{t(\pm t-1)}, \quad \quad \omega_{\pm\sharp}\mathrel{\mathop:}=2 \omega_{\pm 1}-\omega_{0}=\frac{(t \pm 1)dt}{t(t\mp 1)},$$ \begin{description} \item[MES ${\star}$] are defined similarly than $(\ref{eq:mes})$ with $\omega_{\pm \star}$ (instead of $\omega_{\pm 1}$), $\omega_{0}$ and a $\omega_{\pm 1}$ at the beginning: $$\zeta_{k}^{\star,\mathfrak{m}} \left(n_{1}, \ldots , n_{p} \right) \mathrel{\mathop:}= (-1)^{p} I^{\mathfrak{m}} \left(0; 0^{k}, \epsilon_{1}\cdots \epsilon_{p}, 0^{\mid n_{1}\mid-1}, \epsilon_{2}\cdots \epsilon_{p}\star, 0^{\mid n_{2}\mid -1}, \ldots, \epsilon_{p}\star, 0^{\mid n_{p}\mid-1} ;1 \right).$$ \item[MES ${\star\star}$] similarly with only $\omega_{\pm \star}, \omega_{0}$ (including the first):\nomenclature{MES ${\star\star}$, $\zeta^{\star\star,\mathfrak{m}}$}{Motivic Euler sums $\star\star$} $$\zeta_{k}^{\star\star,\mathfrak{m}} \left(n_{1}, \ldots , n_{p} \right) \mathrel{\mathop:}= (-1)^{p} I^{\mathfrak{m}} \left(0; 0^{k}, \epsilon_{1}\cdots \epsilon_{p}\star, 0^{\mid n_{1}\mid-1}, \epsilon_{2}\cdots \epsilon_{p}\star, 0^{\mid n_{2}\mid-1}, \ldots, \epsilon_{p}\star, 0^{\mid n_{p}\mid-1} ;1 \right).$$ \item[MES ${\sharp}$] with $\omega_{\pm \sharp},\omega_{0} $ and a $\omega_{\pm 1}$ at the beginning: $$\zeta_{k}^{\sharp,\mathfrak{m}} \left(n_{1}, \ldots , n_{p} \right) \mathrel{\mathop:}= 2 (-1)^{p} I^{\mathfrak{m}} \left(0; 0^{k}, \epsilon_{1}\cdots \epsilon_{p}, 0^{\mid n_{1}\mid-1}, \epsilon_{2}\cdots \epsilon_{p}\sharp, 0^{\mid n_{2}\mid -1}, \ldots, \epsilon_{p}\sharp, 0^{\mid n_{p}\mid-1} ;1 \right).$$ \item[MES $\sharp\sharp$] similarly with only $\omega_{\pm \sharp}, \omega_{0}$ (including the first):\nomenclature{MES ${\sharp\sharp}$, $\zeta^{\sharp\sharp,\mathfrak{m}}$}{Motivic Euler sums $\sharp\sharp$} $$\zeta_{k}^{\sharp\sharp,\mathfrak{m}} \left(n_{1}, \ldots , n_{p} \right) \mathrel{\mathop:}= (-1)^{p} I^{\mathfrak{m}} \left(0; 0^{k}, \epsilon_{1}\cdots \epsilon_{p}\sharp, 0^{\mid n_{1}\mid-1}, \epsilon_{2}\cdots \epsilon_{p}\sharp, 0^{\mid n_{2}\mid-1}, \ldots, \epsilon_{p}\sharp, 0^{\mid n_{p}\mid-1} ;1 \right).$$ \end{description} \end{defi} \textsc{Remarks}: \begin{itemize} \item[$\cdot$] The Lie algebra of the fundamental group $\pi_{1}^{dR}(\mathbb{P}^{1}\diagdown \lbrace 0, 1, \infty\rbrace)=\pi_{1}^{dR}(\mathcal{M}_{0,4})$ is generated by $e_{0}, e_{1},e_{\infty}$ with the only condition than $e_{0}+e_{1}+e_{\infty}=0$\footnote{ For the case of motivic Euler sums, it is the Lie algebra generated by $e_{0}, e_{1}, e_{-1}, e_{\infty}$ with the only condition than $e_{0}+e_{1}+e_{-1}+e_{\infty}=0$; similarly for other roots of unity with $e_{\eta}$. Note that $e_{i}$ corresponds to the class of the residue around $i$ in $H_{dR}^{1}(\mathbb{P}^{1} \diagdown \lbrace 0, \mu_{N}, \infty \rbrace)^{\vee}$. }. If we keep $e_{0}$ and $e_{\infty}$ as generators, instead of the usual $e_{0},e_{1}$, it leads towards MMZV $^{\star\star}$ up to a sign, instead of MMZV since $-\omega_{0}+\omega_{1}- \omega_{\star}=0$. We could also choose $e_{1}$ and $e_{\infty}$ as generators, which leads to another version of MMZV that has not been much studied yet. These versions are equivalent since each one can be expressed as $\mathbb{Q}$ linear combination of another one. \item[$\cdot$] By linearity and $\shuffle$-regularisation $(\ref{eq:shufflereg})$, all these versions ($\star$, $\star\star$, $\sharp$ or $\sharp\sharp$) are $\mathbb{Q}$-linear combination of motivic Euler sums. Indeed, with $n_{+}$ the number of $+$ among $\circ$: $$\begin{array}{llll} \zeta ^{\star,\mathfrak{m}}(n_{1}, \ldots, n_{p}) &=& \sum_{\circ=\mlq + \mrq \text{ or } ,} & \zeta ^{\mathfrak{m}}(n_{1}\circ \cdots \circ n_{p}) \\ \\ \zeta ^{\mathfrak{m}}(n_{1}, \ldots, n_{p}) &=& \sum_{\circ=\mlq + \mrq \text{ or } ,} (-1)^{n_{+}} & \zeta ^{\star,\mathfrak{m}}(n_{1}\circ \cdots \circ n_{p}) \\ \\
\zeta ^{ \sharp,\mathfrak{m}}(n_{1}, \ldots, n_{p}) &=& \sum_{\circ=\mlq + \mrq \text{ or } ,} 2^{p-n_{+}} & \zeta ^{\mathfrak{m}}(n_{1}\circ \cdots \circ n_{p}) \\
\\
\zeta ^{\mathfrak{m}}(n_{1}, \ldots, n_{p}) &=& \sum_{\circ=\mlq + \mrq \text{ or } ,} (-1)^{n_{+}} 2^{-p} & \zeta ^{ \sharp,\mathfrak{m}}(n_{1}\circ \cdots \circ n_{p}) \\
\\
\zeta ^{\star\star,\mathfrak{m}}(n_{1}, \ldots, n_{p}) &=& \sum_{i=0}^{p-1} & \zeta ^{\star,\mathfrak{m}}_{\mid n_{1}\mid+\cdots+\mid n_{i}\mid}(n_{i+1}, \cdots , n_{p}) \\
& = & \sum_{\circ=\mlq + \mrq \text{ or } ,\atop i=0}^{p-1} & \zeta^{\mathfrak{m}}_{\mid n_{1}\mid+\cdots+\mid n_{i}\mid}(n_{i+1}\circ \cdots \circ n_{p})\\
\\
\zeta ^{ \sharp\sharp,\mathfrak{m}}(n_{1}, \ldots, n_{p}) &=& \sum_{ i=0}^{p-1} & \zeta ^{\sharp,\mathfrak{m}}_{\mid n_{1}\mid+\cdots+\mid n_{i}\mid}(n_{i+1}, \cdots , n_{p}) \\
& = & \sum_{\circ=\mlq + \mrq \text{ or } ,\atop i=0}^{p-1} 2^{p-i-n_{+}} & \zeta^{\mathfrak{m}}_{\mid n_{1}\mid+\cdots+\mid n_{i}\mid}(n_{i+1}\circ \cdots \circ n_{p}) \\
\\
\zeta ^{\star,\mathfrak{m}}(n_{1}, \ldots, n_{p}) &=& \zeta ^{\star\star,\mathfrak{m}}(n_{1}, \ldots, n_{p})& -\zeta ^{\star\star,\mathfrak{m}}_{\mid n_{1}\mid}(n_{2}, \ldots, n_{p})
\\ \\ \zeta ^{\sharp,\mathfrak{m}}(n_{1}, \ldots, n_{p}) &=& \zeta ^{\sharp\sharp}(n_{1}, \ldots, n_{p}) &-\zeta ^{\sharp\sharp,\mathfrak{m}}_{\mid n_{1}\mid}(n_{2}, \ldots, n_{p}) \\
\end{array}$$ \texttt{Notation:} Beware, the $\mlq + \mrq$ here is on $n_{i}\in\mathbb{Z}^{\ast}$ is a summation of absolute values while signs are multiplied: $$n_{1} \mlq + \mrq \cdots \mlq + \mrq n_{i} \rightarrow sign(n_{1}\cdots n_{i})( \vert n_{1}\vert +\cdots + \vert n_{i} \vert).$$ \end{itemize} \texttt{Examples:} Expressing them as $\mathbb{Q}$ linear combinations of motivic Euler sums\footnote{To get rid of the $0$ in front of the MZV, as in the last example, we use the shuffle regularisation $\ref{eq:shufflereg}$.}: $$\begin{array}{lll} \zeta^{\star,\mathfrak{m}}(2,\overline{1},3) & = & -I^{\mathfrak{m}}(0;-1,0,-\star, \star,0,0; 1) \\ & = & \zeta^{\mathfrak{m}}(2,\overline{1},3)+ \zeta^{\mathfrak{m}}(\overline{3},3)+ \zeta^{\mathfrak{m}}(2,\overline{4})+\zeta^{\mathfrak{m}}(\overline{6}) \\ \zeta^{\sharp,\mathfrak{m}}(2,\overline{1},3) &=& - 2 I^{\mathfrak{m}}(0;-1,0,-\sharp, \sharp,0,0; 1) \\
&=& 8\zeta^{\mathfrak{m}}(2,\overline{1},3)+ 4\zeta^{\mathfrak{m}}(\overline{3},3)+ 4\zeta^{\mathfrak{m}}(2,\overline{4})+2\zeta^{\mathfrak{m}}(\overline{6})\\
\zeta^{\star\star,\mathfrak{m}}(2,\overline{1},3) &=& -I^{\mathfrak{m}}(0;-\star,0,-\star, \star,0,0; 1) \\
&=& \zeta^{\star,\mathfrak{m}}(2,\overline{1},3)+ \zeta^{\star,\mathfrak{m}}_{2}(\overline{1},3)+\zeta^{\star,\mathfrak{m}}_{3}(3) \\
&=& \zeta^{\star,\mathfrak{m}}(2, \overline{1}, 3)+\zeta^{\star,\mathfrak{m}}(\overline{3},3)+3 \zeta^{\star,\mathfrak{m}}(\overline{2},4)+6\zeta^{\star,\mathfrak{m}}(1, 5)-10\zeta^{\star,\mathfrak{m}}(6)\\
&=& 11\zeta^{\mathfrak{m}}(\overline{6})+2\zeta^{\mathfrak{m}}(\overline{3}, 3)+\zeta^{\mathfrak{m}}(2, \overline{4})+\zeta^{\mathfrak{m}}(2,\overline{1}, 3)+3\zeta^{\mathfrak{m}}(\overline{2}, 4)+6\zeta^{\mathfrak{m}}(\overline{1}, 5)-10\zeta^{\mathfrak{m}}(6)\\ \end{array}$$
\paragraph{Stuffle.}
One of the most famous relations between cyclotomic MZV, the\textit{ stuffle} relation, coming from the multiplication of series, has been proven to be \textit{motivic} i.e. true for cyclotomic MMZV, which was a priori non obvious. \footnote{The stuffle for these motivic iterated integrals can be deduced from works by Goncharov on mixed Hodge structures, but was also proved in a direct way by G. Racinet, in his thesis, or I. Souderes in $\cite{So}$ via blow-ups. Remark that shuffle relation, coming from the iterated integral representation is clearly \textit{motivic}.} In particular: \begin{lemm} $$\zeta^{\mathfrak{m}}\left( a_{1}, \ldots, a_{r} \atop \alpha_{1}, \ldots, \alpha_{r}\right) \zeta^{\mathfrak{m}}\left( b_{1}, \ldots, b_{s} \atop \beta_{1}, \ldots, \beta_{s}\right)=\sum_{ \left( c_{j} \atop \gamma_{j}\right) = \left( a_{i} \atop \alpha_{i} \right) ,\left( b_{i'} \atop \beta_{i'}\right) \text{ or }\left( a_{i}+b_{i'} \atop \alpha_{i}\beta_{i'}\right) \atop \text{order } (a_{i}), (b_{i}) \text{ preserved} }\zeta^{\mathfrak{m}}\left( c_{1}, \ldots, c_{m} \atop \gamma_{1}, \ldots, \gamma_{m} \right) .$$ $$\zeta^{\star,\mathfrak{m}}\left( a_{1}, \ldots, a_{r} \atop \alpha_{1}, \ldots, \alpha_{r}\right) \zeta^{\star, \mathfrak{m}}\left( b_{1}, \ldots, b_{s} \atop \beta_{1}, \ldots, \beta_{s}\right)=\sum_{ \left( c_{j} \atop \gamma_{j}\right) = \left( a_{i} \atop \alpha_{i} \right) ,\left( b_{i} \atop \beta_{i}\right) \text{ or }\left( a_{i}+b_{i'} \atop \alpha_{i}\beta_{i'}\right) \atop \text{order } (a_{i}), (b_{i}) \text{ preserved} }(-1)^{r+s+m}\zeta^{\star, \mathfrak{m}}\left( c_{1}, \ldots, c_{m} \atop \gamma_{1}, \ldots, \gamma_{m} \right) .$$ $$\zeta^{\sharp , \mathfrak{m}}\left( \textbf{a} \atop \boldsymbol{\alpha} \right) \zeta^{\sharp, \mathfrak{m}}\left( \textbf{ b } \atop \boldsymbol{\beta} \right)=\sum_{ \left( c_{j} \atop \gamma_{j}\right) = \left( a_{i}+\sum_{l=1}^{k} a_{i+l} +b_{i'+l} \atop \alpha_{i}\prod_{l=1}^{k}\alpha_{i+l}\beta_{i'+l}\right) \text{ or } \left( b_{i'}+\sum_{l=1}^{k} a_{i+l} +b_{i'+l} \atop \beta_{i'}\prod_{l=1}^{k}\alpha_{i+l}\beta_{i'+l}\right) \atop k\geq 0, \text{ order } (a_{i}), (b_{i}) \text{ preserved}}(-1)^{\frac{r+s-m}{2}}\zeta^{\sharp, \mathfrak{m}}\left( c_{1}, \ldots, c_{m} \atop \gamma_{1}, \ldots, \gamma_{m} \right) .$$ \end{lemm} \noindent\textsc{Remarks:} \begin{itemize} \item[$\cdot$] In the depth graded, stuffle corresponds to shuffle the sequences $\left( \boldsymbol{a} \atop \boldsymbol{\alpha} \right) $ and $\left( \boldsymbol{b} \atop \boldsymbol{\beta} \right) $. \item[$\cdot$] Other identities mixing the two versions could also be stated, such as $$\zeta^{\star, \mathfrak{m}}\left( a_{1}, \ldots, a_{r} \atop \alpha_{1}, \ldots, \alpha_{r}\right) \zeta^{\mathfrak{m}}\left( b_{1}, \ldots, b_{s} \atop \beta_{1}, \ldots, \beta_{s}\right)=\sum_{ \left( c_{j} \atop \gamma_{j}\right) = \left( a_{i} \atop \alpha_{i} \right) ,\left( b_{i'} \atop \beta_{i'}\right) \text{ or }\left( (\sum_{l=1}^{k} a_{i+l})+b_{i'} \atop (\prod_{l=1}^{k}\alpha_{i+l})\beta_{i'}\right) \atop k \geq 1, \text{order } (a_{i}), (b_{i}) \text{ preserved} }\zeta^{\mathfrak{m}}\left( c_{1}, \ldots, c_{m} \atop \gamma_{1}, \ldots, \gamma_{m} \right) .$$ \end{itemize}
\section{Relations in $\mathcal{L}$}
\subsection{Antipode relation}
In this part, we are interested in some Antipodal relations for motivic Euler sums in the coalgebra $\mathcal{L}$, i.e. modulo products. To explain quickly where they come from, let's go back to two combinatorial Hopf algebra structures.\\ \\ First recall that if $A$ is a graded connected bialgebra, there exists an unique antipode S (leading to a Hopf algebra structure)\footnote{It comes from the usual required relation for the antipode in a Hopf algebra, but because it is graded and connected, we can apply the formula recursively to construct it, in an unique way. }, which is the graded map defined by: \begin{equation} \label{eq:Antipode} S(x)= -x-\sum S(x_{(1)}) \cdot x_{(2)}, \end{equation} where $\cdot$ is the product and using Sweedler notations for the coaction: $$\Delta (x)= 1\otimes x+ x\otimes 1+ \sum x_{(1)}\otimes x_{(2)}= \Delta'(x)+ 1\otimes x+ x\otimes 1 .$$ Hence, in the quotient $A/ A_{>0}\cdot A_{>0} $: $$S(x) \equiv -x . $$
\subsubsection{The $\shuffle$ Hopf algebra}
Let $X=\lbrace a_{1},\cdots, a_{n} \rbrace$ an alphabet and $A_{\shuffle}\mathrel{\mathop:}=\mathbb{Q} \langle X^{\times} \rangle$ the $\mathbb{Q}$-vector space generated by words on X, i.e. non commutative polynomials in $a_{i}$. It is easy to see that $A_{\shuffle}$ is a Hopf algebra with the $\shuffle$ shuffle product, the deconcatenation coproduct $\Delta_{D}$ and antipode $S_{\shuffle}$:\nomenclature{$\Delta_{D}$}{the deconcatenation coproduct} \begin{equation} \label{eq:shufflecoproduct} \Delta_{D}(a_{i_{1}}\cdots a_{i_{n}})= \sum_{k=0}^{n} a_{i_{1}}\cdots a_{i_{k}} \otimes a_{i_{k+1}} \cdots a_{i_{n}}. \end{equation} \begin{equation} \label{eq:shuffleantipode} S_{\shuffle} (a_{i_{1}} \cdots a_{i_{n}})= (-1)^{n} a_{i_{n}} \cdots a_{i_{1}}. \end{equation} $A_{\shuffle}$ is even a connected graded Hopf algebra, called the \textit{ shuffle Hopf algebra}; the grading coming from the degree of polynomial. By the equivalence of category between $\mathbb{Q}$-Hopf algebra and $\mathbb{Q}$-Affine Group Scheme, it corresponds to: \begin{equation} \label{eq:gpschshuffle} G=\text{Spec} A_{\shuffle} : R \rightarrow \text{Hom}(\mathbb{Q} \langle X \rangle, R)=\lbrace S\in R\langle\langle a_{i} \rangle\rangle \mid \Delta_{\shuffle} S= S\widehat{\otimes} S, \epsilon(S)=1 \rbrace, \end{equation} where $\Delta_{\shuffle}$ is the coproduct dual to the product $\shuffle$:\nomenclature{$\Delta_{\shuffle}$}{the $\shuffle$ coproduct} $$\Delta_{\shuffle}(a_{i_{1}}\cdots a_{i_{n}})= \left( 1\otimes a_{i_{1}}+ a_{i_{1}}\otimes 1\right) \cdots \left( 1\otimes a_{i_{n}}+ a_{i_{n}}\otimes 1\right) .$$ Let restrict now to $X=\lbrace 0,\mu_{N}\rbrace$; our main interest in this Chapter is $N=2$, but it can be extended to other roots of unity. The shuffle relation for motivic iterated integral relative to $\mu_{N}$: \begin{equation}\label{eq:shuffleim} I^{\mathfrak{m}}(0; \cdot ; 1) \text{ is a morphism of Hopf algebra from } A_{\shuffle} \text{ to } (\mathbb{R},\times): \end{equation} $$I^{\mathfrak{m}}(0; w ; 1) I^{\mathfrak{m}}(0; w' ; 1)= I^{\mathfrak{m}}(0; w\shuffle w' ;1) \text{ with } w,w' \text{ words in } X.$$
\begin{lemm}[\textbf{Antipode $\shuffle$}] In the coalgebra $\mathcal{L}$, with $w$ the weight, $\bullet$ standing for MMZV$_{\mu_{N}}$, or $\star\star$ ($N=2$) resp. $\sharp\sharp$-version ($N=2$): $$\zeta^{\bullet,\mathfrak{l}}_{n-1}\left( n_{1}, \ldots, n_{p} \atop \epsilon_{1}, \ldots, \epsilon_{p} \right) \equiv (-1)^{w+1}\zeta^{\bullet,\mathfrak{l}}_{n_{p}-1}\left( n_{p-1}, \ldots, n_{1},n \atop \epsilon_{p-1}^{-1}, \ldots, \epsilon_{1}^{-1}, \epsilon \right) \text{ where } \epsilon\mathrel{\mathop:}=\epsilon_{1}\cdot\ldots\cdot\epsilon_{p}.$$ \end{lemm} \noindent This formula stated for any $N$ is slightly simpler in the case $N=1,2$ since $n_{i}\in\mathbb{Z}^{\ast}$: \begin{framed} \begin{equation}\label{eq:antipodeshuffle2} \textsc{Antipode } \shuffle \quad : \begin{array}{l}
\zeta^{\bullet,\mathfrak{l}}_{n-1}\left( n_{1}, \ldots, n_{p} \right) \equiv(-1)^{w+1}\zeta^{\bullet,\mathfrak{l}}_{\mid n_{p}\mid -1}\left( n_{p-1}, \ldots, n_{1},sign(n_{1}\cdots n_{p}) n \right)\\
\text{ } \\
I^{\mathfrak{l}}(0;X;\epsilon)\equiv (-1)^{w} I^{\mathfrak{l}}(\epsilon;\widetilde{X};0) \equiv (-1)^{w+1} I^{\mathfrak{l}}(0; \widetilde{X}; \epsilon) \end{array}. \end{equation} \end{framed} Here $X$ is any word in $0,\pm 1$ or $0, \pm \star$ or $0, \pm\sharp$, and $\widetilde{X}$ denotes the \textit{reversed} word.
\begin{proof} For motivic iterated integrals, as said above: $$ S_{\shuffle} (I^{\mathfrak{m}}(0; a_{1}, \ldots, a_{n}; 1))= (-1)^{n}I^{\mathfrak{m}}(0; a_{n}, \ldots, a_{1}; 1),$$ which, in terms of the MMZV$_{\mu_{N}}$ notation is: $$S_{\shuffle}\left( \zeta^{\bullet,\mathfrak{l}}_{n-1}\left( n_{1}, \ldots, n_{p} \atop \epsilon_{1}, \ldots, \epsilon_{p} \right) \right) \equiv (-1)^{w}\zeta^{\bullet,\mathfrak{l}}_{n_{p}-1}\left( n_{p-1}, \ldots, n_{1},n \atop \epsilon_{p-1}^{-1}, \ldots, \epsilon_{1}^{-1}, \epsilon \right) \text{ where } \epsilon\mathrel{\mathop:}=\epsilon_{1}\cdot\ldots\cdot\epsilon_{p}.$$ Then, if we look at the antipode recursive formula $\eqref{eq:Antipode}$ in the coalgebra $\mathcal{L}$, for $a_{i}\in \lbrace 0, \mu_{N} \rbrace$: $$ S_{\shuffle} (I^{\mathfrak{l}}(0; a_{1}, \ldots, a_{n}; 1))\equiv - I^{\mathfrak{l}}(0; a_{1}, \ldots, a_{n}; 1).$$ This leads to the lemma above. The $\shuffle$-antipode relation can also be seen at the level of iterated integrals as the path composition modulo products followed by a reverse of path. \end{proof}
\subsubsection{The $\ast$ Hopf algebra} Let $Y=\lbrace \cdots, y_{-n}, \ldots, y_{-1}, y_{1},\cdots, y_{n}, \cdots \rbrace$ an infinite alphabet and $A_{\ast}\mathrel{\mathop:}=\mathbb{Q} \langle Y^{\times} \rangle$ the non commutative polynomials in $y_{i}$ with rational coefficients, with $y_{0}=1$ the empty word. Similarly, it is a graded connected Hopf algebra called the\textit{ stuffle Hopf algebra}, with the stuffle $\ast$ product and the following coproduct:\footnote{For the $\shuffle$ algebra, we had to use the notation in terms of iterated integrals, with $0,\pm 1$, but for the $\ast$ stuffle relation, it is more natural with the Euler sums notation, which corresponds to $y_{n_{i}}, n_{i}\in\mathbb{Z}$.} \begin{equation} \label{eq:stufflecoproduct} \Delta_{D\ast}(y_{n_{1}} \cdots y_{n_{p}})= \sum_{} y_{n_{1}} \cdots y_{n_{i}}\otimes y_{n_{i+1}}, \ldots, y_{n_{p}}, \quad n_{i}\in \mathbb{Z}^{\ast}. \end{equation} \texttt{Nota Bene}: Remark that here we restricted to Euler sums, $N=2$, but it could be extended for other roots of unity, for which stuffle relation has been stated in $\S 4.1$.\\ The completed dual is the Hopf algebra of series $\mathbb{Q}\left\langle \left\langle Y \right\rangle \right\rangle $ with the coproduct: $$\Delta_{\ast}(y_{n})= \sum_{ k =0 \atop sgn(n)=\epsilon_{1}\epsilon_{2}}^{\mid n \mid} y_{\epsilon_{1} k} \otimes y_{\epsilon_{2}( n-k)}.$$ Now, let introduce the notations:\footnote{Here $\star$ resp. $\sharp$ refers naturally to the Euler $\star$ resp. $\sharp$, sums, as we see in the next lemma. Beware, it is not a $\ast$ homomorphism.} $$(y_{n_{1}} \cdots y_{n_{p}})^{\star} \mathrel{\mathop:}= \sum_{1=i_{0}< i_{1} < \cdots < i_{k-1}\leq i_{k+1}=p \atop k\geq 0} y_{n_{i_{0}}\mlq + \mrq\cdots \mlq + \mrq n_{i_{1}-1}} \cdots y_{n_{i_{j}}\mlq + \mrq\cdots \mlq + \mrq n_{i_{j+1}-1}} \cdots y_{n_{i_{k}}\mlq + \mrq \cdots \mlq + \mrq n_{i_{k+1}}}.$$ $$(y_{n_{1}} \cdots y_{n_{p}})^{\sharp} \mathrel{\mathop:}= \sum_{1=i_{0}< i_{1} < \cdots < i_{k-1}\leq i_{k+1}=p \atop k\geq 0} 2^{k+1} y_{n_{i_{0}}\mlq + \mrq\cdots \mlq + \mrq n_{i_{1}-1}} \cdots y_{n_{i_{j}} \mlq + \mrq \cdots \mlq + \mrq n_{i_{j+1}-1}} \cdots y_{n_{i_{k}}\mlq + \mrq \cdots \mlq + \mrq n_{i_{k+1}}},$$ where $n_{i}\in\mathbb{Z}^{\ast}$ and the operation $\mlq + \mrq$ indicates that signs are multiplied whereas absolute values are summed. It is straightforward to check that: \begin{equation} \Delta_{D\ast}(w^{\star})=(\Delta_{D\ast}(w))^{\star} , \quad \text{ and } \quad \Delta_{D\ast}(w^{\sharp})=(\Delta_{D\ast}(w))^{\sharp}. \end{equation} As said above, the relation stuffle is motivic: \begin{center} $\zeta^{\mathfrak{m}}(\cdot)$ is a morphism of Hopf algebra from $A_{\ast}$ to $(\mathbb{R},\times)$. \end{center}
\begin{lemm}[\textbf{Antipode $\ast$}] In the coalgebra $\mathcal{L}$, with $n_{i}\in\mathbb{Z}^{\ast}$ $$\zeta^{\mathfrak{l}}_{n-1}(n_{1}, \ldots, n_{p}) \equiv (-1)^{p+1}\zeta^{\star,\mathfrak{l}}_{n-1}(n_{p}, \ldots, n_{1}).$$ $$\zeta^{\sharp,\mathfrak{l}}_{n-1}(n_{1}, \ldots, n_{p})\equiv (-1)^{p+1}\zeta^{\sharp,\mathfrak{l}}_{n-1}(n_{p}, \ldots, n_{1}).$$ \end{lemm} \begin{proof} By recursion, using the formula $\eqref{eq:Antipode}$, and the following identity (left to the reader): $$\sum_{i=0}^{p-1} (-1)^{i}(y_{n_{i}} \cdots y_{n_{1}})^{\star} \ast (y_{n_{i+1}} \cdots y_{n_{p}})= -(-1)^{p} (y_{n_{p}} \cdots y_{n_{1}})^{\star}, $$ we deduce the antipode $S_{\ast}$: $$S_{\ast} (y_{n_{1}} \cdots y_{n_{p}})= (-1)^{p} (y_{n_{p}} \cdots y_{n_{1}})^{\star} .$$ Similarly: $$S_{\ast}((y_{n_{1}} \cdots y_{n_{p}})^{\sharp})=-\sum_{i=0}^{n-1} S_{\ast}((y_{n_{1}} \cdots y_{n_{i}})^{\sharp}) \ast (y_{n_{i+1}} \cdots y_{n_{p}})^{\sharp}$$ $$=-\sum_{i=0}^{n-1} (-1)^{i}(y_{n_{i}} \cdots y_{n_{1}})^{\sharp} \ast (y_{n_{i+1}} \cdots y_{n_{p}})^{\sharp}=(-1)^{p}(y_{n_{p}} \cdots y_{n_{1}})^{\sharp}.$$ Then, we deduce the lemma, since $\zeta^{\mathfrak{m}}(\cdot)$ is a morphism of Hopf algebra. Moreover, the formula $\eqref{eq:Antipode}$ in the coalgebra $\mathcal{L}$ gives that: $$S(\zeta^{\mathfrak{l}}(\textbf{s}))\equiv -\zeta^{\mathfrak{l}}(\textbf{s}).$$ \end{proof}
\subsection{Hybrid relation in $\mathcal{L}$}
In this part, we look at a new relation called \textit{hybrid relation} between motivic Euler sums in the coalgebra $\mathcal{L}$, i.e. modulo products, which comes from the motivic version of the octagon relation (for $N>1$, cf. $\cite{EF}$) \footnote{\begin{figure}\label{fig:hexagon}
\end{figure}}
\begin{figure}\label{fig:octagon}
\end{figure} \noindent This relation is motivic, and hence valid for the motivic Drinfeld associator $\Phi^{\mathfrak{m}}$ ($\ref{eq:associator}$), replacing $2 i \pi$ by the Lefschetz motivic period $\mathbb{L}^{\mathfrak{m}}$. \\
Let focus on the case $N=2$ and recall that the space of motivic periods of $\mathcal{MT}\left( \mathbb{Z}[\frac{1}{2}]\right)$ decomposes as (cf. $\ref{eq:periodgeomr}$): \begin{equation}\label{eq:perioddecomp2} \mathcal{P}_{\mathcal{MT}\left( \mathbb{Z}[\frac{1}{2}]\right)}^{\mathfrak{m}}= \mathcal{H}^{2} \oplus \mathcal{H}^{2}. \mathbb{L}^{\mathfrak{m}}, \quad \text{ where } \begin{array}{l} \mathcal{H}^{2} \text{ is } \mathcal{F}_{\infty} \text{ invariant} \\ \mathcal{H}^{2}. \mathbb{L}^{\mathfrak{m}} \text{ is } \mathcal{F}_{\infty} \text{ anti-invariant} \end{array}. \end{equation} For the motivic Drinfeld associator, seeing the path in the Riemann sphere, it becomes: \begin{figure}\label{fig:octagon2}
\end{figure}
Let $X= \mathbb{P}^{1}\diagdown \left\lbrace 0, \pm 1, \infty \right\rbrace $. The action of the \textit{real Frobenius} $\boldsymbol{\mathcal{F}_{\infty}}$ on $X(\mathbb{C})$ is induced by complex conjugation. The real Frobenius acts on the Betti realization $\pi^{B}(X (\mathbb{C}))$\footnote{ It is compatible with the groupoid structure of $\pi^{B}$, and the local monodromy. }, and induces an involution on motivic periods, compatible with the Galois action: $$\mathcal{F}_{\infty}: \mathcal{P}_{\mathcal{MT}(\mathbb{Z}[\frac{1}{2}])}^{\mathfrak{m}} \rightarrow\mathcal{P}_{\mathcal{MT}(\mathbb{Z}[\frac{1}{2}])}^{\mathfrak{m}}.$$ The Lefschetz motivic period $\mathbb{L}^{\mathfrak{m}}$ is anti-invariant by $\mathcal{F}_{\infty}$: $$\mathcal{F}_{\infty} \mathbb{L}^{\mathfrak{m}}= -\mathbb{L}^{\mathfrak{m}},$$ whereas terms corresponding to real paths in Figure $\ref{fig:octagon2}$, such as Drinfeld associator terms, are obviously invariant by $\mathcal{F}_{\infty}$.\\ \\ The linearized $\mathcal{F}_{\infty}$-anti-invariant part of this octagon relation leads to the following hybrid relation.
\begin{theo}\label{hybrid} In the coalgebra $\mathcal{L}^{2}$, with $n_{i}\in \mathbb{Z}^{\ast}$, $w$ the weight: $$\zeta^{\mathfrak{l}}_{k}\left( n_{0}, n_{1},\ldots, n_{p} \right) + \zeta^{\mathfrak{l}}_{\mid n_{0} \mid +k}\left( n_{1}, \ldots, n_{p} \right) \equiv (-1)^{w+1}\left( \zeta^{\mathfrak{l}}_{k}\left( n_{p}, \ldots, n_{1}, n_{0} \right) + \zeta^{\mathfrak{l}}_{k+\mid n_{p}\mid}\left( n_{p-1}, \ldots, n_{1},n_{0} \right)\right)$$ Equivalently, in terms of motivic iterated integrals, for $X$ any word in $\lbrace 0, \pm 1 \rbrace$, with $\widetilde{X}$ the reversed word, we obtain both: $$I^{\mathfrak{l}} (0; 0^{k}, \star, X; 1)\equiv I^{\mathfrak{l}} (0; X, \star, 0^{k}; 1)\equiv (-1)^{w+1} I^{\mathfrak{l}} (0; 0^{k}, \star, \widetilde{X}; 1), $$ $$I^{\mathfrak{l}} (0; 0^{k}, -\star, X; 1)\equiv I^{\mathfrak{l}} (0;- X, -\star, 0^{k}; 1)\equiv (-1)^{w+1} I^{\mathfrak{l}} (0; 0^{k}, -\star, -\widetilde{X}; 1) $$ \end{theo} The proof is given below, firstly for $k=0$, using octagon relation (Figure $\ref{fig:octagon2}$). The generalization for any $k >0$ is deduced directly from the shuffle regularization $(\ref{eq:shufflereg})$.\\ \\ \textsc{Remarks}: \begin{itemize} \item[$\cdot$] This theorem implies notably the famous \textit{depth-drop phenomena} when weight and depth have not the same parity (cf. Corollary $\ref{hybridc}$). \item[$\cdot$] Equivalently, this statement is true for $X$ any word in $\lbrace 0, \pm \star \rbrace$. Recall that ($\ref{eq:miistarsharp}$), by linearity: $$ I^{\mathfrak{m}}(\ldots, \pm \star, \ldots)\mathrel{\mathop:}= I^{\mathfrak{m}}(\ldots, \pm 1, \ldots) - I^{\mathfrak{m}}(\ldots, 0, \ldots).$$ \item[$\cdot$] The point of view adopted by Francis Brown in $\cite{Br3}$, and its use of commutative polynomials (also seen in Ecalle work) can be applied in the coalgebra $\mathcal{L}$ and leads to a new proof of Theorem $\ref{hybrid}$ in the case of MMZV, i.e. $N=1$, sketched in Appendix $A.4$; it uses the stuffle relation and the antipode shuffle. Unfortunately, generalization for motivic Euler sums of this proof is not clear, because of this commutative polynomial setting. \end{itemize} Since Antipode $\ast$ relation expresses $\zeta^{\mathfrak{l}}_{n-1}(n_{1}, \ldots, n_{p})+(-1)^{p} \zeta^{\mathfrak{l}}_{n-1}(n_{p}, \ldots, n_{1})$ in terms of smaller depth (cf. Lemma $4.2.2$), when weight and depth have not the same parity, it turns out that a (motivic) Euler sum can be expressed by smaller depth:\footnote{Erik Panzer recently found a new proof of this depth drop result for MZV at roots of unity, which appear as a special case of some functional equations of polylogarithms in several variables. } \begin{coro}\label{hybridc} If $w+p$ odd, a motivic Euler sum in $\mathcal{L}$ is reducible in smaller depth: $$2\zeta^{\mathfrak{l}}_{n-1}(n_{1}, \ldots, n_{p}) \equiv$$ $$-\zeta^{\mathfrak{l}}_{n+\mid n_{1}\mid -1}(n_{2}, \ldots, n_{p})+(-1)^{p} \zeta^{\mathfrak{l}}_{n+\mid n_{p}\mid -1}(n_{p-1}, \ldots, n_{1})+\sum_{\circ=+ \text{ or } ,\atop \text{at least one } +} (-1)^{p+1} \zeta^{\mathfrak{l}}_{n-1}(n_{p}\circ \cdots \circ n_{1}).$$ \end{coro}
\paragraph{Proof of Theorem $\ref{hybrid}$} First, the octagon relation (Figure $\ref{fig:octagon2}$) is equivalent to: \begin{lemm} In $\mathcal{P}_{\mathcal{MT}\left( \mathbb{Z}[\frac{1}{2}]\right)}^{\mathfrak{m}}\left\langle \left\langle e_{0}, e_{1}, e_{-1}\right\rangle \right\rangle $, with $e_{0} + e_{1} + e_{-1} +e_{\infty} =0$: \begin{equation}\label{eq:octagon21} \Phi^{\mathfrak{m}}(e_{0}, e_{1},e_{-1}) e^{\frac{\mathbb{L}^{\mathfrak{m}} e_{0}}{2}} \Phi^{\mathfrak{m}}(e_{-1}, e_{0},e_{\infty}) e^{\frac{\mathbb{L}^{\mathfrak{m}} e_{-1}}{2}} \Phi^{\mathfrak{m}}(e_{\infty}, e_{-1},e_{1}) e^{\frac{\mathbb{L}^{\mathfrak{m}} e_{\infty}}{2}} \Phi^{\mathfrak{m}}(e_{1}, e_{\infty},e_{0}) e^{\frac{\mathbb{L}^{\mathfrak{m}} e_{1}}{2}} =1, \end{equation} Hence, the linearized octagon relation is: \begin{multline}\label{eq:octagonlin}
- e_{0} \Phi^{\mathfrak{l}}(e_{-1}, e_{0},e_{\infty})+ \Phi^{\mathfrak{l}}(e_{-1}, e_{0},e_{\infty})e_{0} +(e_{0}+e_{-1}) \Phi^{\mathfrak{l}}(e_{\infty}, e_{-1},e_{1}) - \Phi^{\mathfrak{l}}(e_{\infty}, e_{-1},e_{1}) (e_{0}+e_{-1})\\
- e_{1} \Phi^{\mathfrak{l}}(e_{1}, e_{\infty},e_{0}) + \Phi^{\mathfrak{l}}(e_{1}, e_{\infty},e_{0}) e_{1} \equiv 0. \end{multline} \end{lemm} \begin{proof} \begin{itemize} \item[$\cdot$] Let's first remark that: $$ \Phi^{\mathfrak{m}}(e_{0}, e_{1},e_{-1})= \Phi^{\mathfrak{m}}(e_{1}, e_{0},e_{\infty})^{-1} .$$ Indeed, the coefficient in the series $\Phi^{\mathfrak{m}}(e_{1}, e_{0},e_{\infty})$ of a word $e_{0}^{a_{0}} e_{\eta_{1}} e_{0}^{a_{1}} \cdots e_{\eta_{r}} e_{0}^{a_{r}}$, where $\eta_{i}\in \lbrace\pm 1 \rbrace$ is (cf. $\S 4.6$):
$$ I^{\mathfrak{m}} \left(0; (\omega_{1}-\omega_{-1})^{a_{0}} (-\omega_{\mu_{1}}) (\omega_{1}-\omega_{-1})^{a_{1}} \cdots (-\omega_{\mu_{r}})(\omega_{1}-\omega_{-1})^{a_{r}} ;1 \right) \texttt{ with } \mu_{i}\mathrel{\mathop:}= \left\lbrace \begin{array}{ll} -\star& \texttt{if } \eta_{i}=1\\ -1 & \texttt{if } \eta_{i}=-1 \end{array} \right. .$$ Let introduce the following homography $\phi_{\tau\sigma}$ (cf. Annexe $(\ref{homography2})$): $$\phi_{\tau\sigma}= \phi_{\tau\sigma}^{-1}: t \mapsto \frac{1-t}{1+t} :\left\lbrace \begin{array}{l} -\omega_{\star}\mapsto \omega_{\star} \\ -\omega_{1}\mapsto \omega_{-\star}\\ \omega_{-1}-\omega_{1} \mapsto -\omega_{0}\\ \omega_{-1} \mapsto -\omega_{-1}\\ \omega_{-\star} \mapsto -\omega_{1} \end{array} \right..$$ If we apply $\phi_{\tau\sigma}$ to the motivic iterated integral above, it gives: $ I^{\mathfrak{m}} \left(1; \omega_{0}^{a_{0}} \omega_{\eta_{1}} \omega_{0}^{a_{1}} \cdots \omega_{\eta_{r}} \omega_{0}^{a_{r}} ;0 \right)$. Hence, summing over words $w$ in $e_{0},e_{1},e_{-1}$: $$ \Phi^{\mathfrak{m}}(e_{1}, e_{0},e_{\infty})= \sum I^{\mathfrak{m}}(1; w; 0) w$$ Therefore: $$\Phi^{\mathfrak{m}}(e_{0}, e_{1},e_{-1})\Phi^{\mathfrak{m}}(e_{1}, e_{0},e_{\infty})= \sum_{w, w=uv} I^{\mathfrak{m}}(0; u; 1) I^{\mathfrak{m}}(1; v; 0) w= 1.$$ We used the composition formula for iterated integral to conclude, since for $w$ non empty, $\sum_{w=uv} I^{\mathfrak{m}}(0; u; 1) I^{\mathfrak{m}}(1; v; 0)= I^{\mathfrak{m}}(0; w; 0) =0$.\\ Similarly: $$\Phi^{\mathfrak{m}}(e_{0}, e_{-1},e_{1})= \Phi^{\mathfrak{m}}(e_{-1}, e_{0},e_{\infty})^{-1} , \quad \text{ and } \quad \Phi^{\mathfrak{m}}(e_{\infty}, e_{1},e_{-1})= \Phi^{\mathfrak{m}}(e_{1}, e_{\infty},e_{0})^{-1}.$$ The identity $\ref{eq:octagon21}$ follows from $\ref{fig:octagon2}$.\\
\item[$\cdot$] Let consider both paths on the Riemann sphere $\gamma$ and $\overline{\gamma}$, its conjugate: \footnote{Path $\gamma$ corresponds to the cycle $\sigma$, $1 \mapsto \infty \mapsto -1 \mapsto 0 \mapsto 1$ (cf. in Annexe $\ref{homography2}$). Beware, in the figure, the position of both path is not completely accurate in order to distinguish them.} \\ \\ \includegraphics[]{octagon3.pdf}\\ Applying $(id-\mathcal{F}_{\infty})$ to the octagon identity $\ref{eq:octagon21}$ \footnote{The identity $\ref{eq:octagon21}$ corresponds to the path $\gamma$ whereas applying $\mathcal{F}_{\infty}$ to the path $\gamma$ corresponds to the path $\overline{\gamma}$ represented.} leads to: \begin{small} \begin{multline}\label{eq:octagon22} \Phi^{\mathfrak{m}}(e_{0}, e_{1},e_{-1}) e^{\frac{\mathbb{L}^{\mathfrak{m}} e_{0}}{2}} \Phi^{\mathfrak{m}}(e_{-1}, e_{0},e_{\infty}) e^{\frac{\mathbb{L}^{\mathfrak{m}} e_{-1}}{2}} \Phi^{\mathfrak{m}}(e_{\infty}, e_{-1},e_{1}) e^{\frac{\mathbb{L}^{\mathfrak{m}} e_{\infty}}{2}} \Phi^{\mathfrak{m}}(e_{1}, e_{\infty},e_{0}) e^{\frac{\mathbb{L}^{\mathfrak{m}} e_{1}}{2}}\\ -\Phi^{\mathfrak{m}}(e_{0}, e_{1},e_{-1}) e^{-\frac{\mathbb{L}^{\mathfrak{m}} e_{0}}{2}} \Phi^{\mathfrak{m}}(e_{-1}, e_{0},e_{\infty}) e^{-\frac{\mathbb{L}^{\mathfrak{m}} e_{-1}}{2}} \Phi^{\mathfrak{m}}(e_{\infty}, e_{-1},e_{1}) e^{-\frac{\mathbb{L}^{\mathfrak{m}} e_{\infty}}{2}} \Phi^{\mathfrak{m}}(e_{1}, e_{\infty},e_{0}) e^{-\frac{\mathbb{L}^{\mathfrak{m}} e_{1}}{2}}=0. \end{multline} \end{small} By $(\ref{eq:perioddecomp2})$, the left side of $(\ref{eq:octagon22})$, being anti-invariant by $\mathcal{F}_{\infty}$, lies in $ \mathcal{H}^{2}\cdot \mathbb{L}^{\mathfrak{m}} \left\langle \left\langle e_{0}, e_{1}, e_{-1} \right\rangle \right\rangle $. Consequently, we can divide it by $\mathbb{L}^{\mathfrak{m}}$ and consider its projection $\pi^{\mathcal{L}}$ in the coalgebra $\mathcal{L} \left\langle \left\langle e_{0}, e_{1}, e_{-1} \right\rangle \right\rangle $, which gives firstly: \begin{small} \begin{multline}\label{eq:octagon23}\hspace*{-0.5cm} 0=\Phi^{\mathfrak{l}}(e_{0}, e_{1},e_{-1}) \pi^{\mathcal{L}} \left( (\mathbb{L}^{\mathfrak{m}})^{-1} \left[ e^{\frac{\mathbb{L}^{\mathfrak{m}} e_{0}}{2}} e^{\frac{\mathbb{L}^{\mathfrak{m}} e_{-1}}{2}}e^{\frac{\mathbb{L}^{\mathfrak{m}} e_{\infty}}{2}} e^{\frac{\mathbb{L}^{\mathfrak{m}} e_{1}}{2}} - e^{-\frac{\mathbb{L}^{\mathfrak{m}} e_{0}}{2}} e^{-\frac{\mathbb{L}^{\mathfrak{m}} e_{-1}}{2}}e^{-\frac{\mathbb{L}^{\mathfrak{m}} e_{\infty}}{2}} e^{-\frac{\mathbb{L}^{\mathfrak{m}} e_{1}}{2}} \right] \right) \\ \hspace*{-0.5cm} +\pi^{\mathcal{L}} \left( (\mathbb{L}^{\mathfrak{m}})^{-1} \left[ e^{\frac{\mathbb{L}^{\mathfrak{m}} e_{0}}{2}} \Phi^{\mathfrak{l}}(e_{-1}, e_{0},e_{\infty}) e^{\frac{\mathbb{L}^{\mathfrak{m}} e_{-1}}{2}} e^{\frac{\mathbb{L}^{\mathfrak{m}} e_{\infty}}{2}} e^{\frac{\mathbb{L}^{\mathfrak{m}} e_{1}}{2}}- e^{-\frac{\mathbb{L}^{\mathfrak{m}} e_{0}}{2}} \Phi^{\mathfrak{l}}(e_{-1}, e_{0},e_{\infty}) e^{-\frac{\mathbb{L}^{\mathfrak{m}} e_{-1}}{2}} e^{-\frac{\mathbb{L}^{\mathfrak{m}} e_{\infty}}{2}} e^{-\frac{\mathbb{L}^{\mathfrak{m}} e_{1}}{2}} \right] \right) \\
\hspace*{-0.5cm} + \pi^{\mathcal{L}} \left( (\mathbb{L}^{\mathfrak{m}})^{-1} \left[ e^{\frac{\mathbb{L}^{\mathfrak{m}} e_{0}}{2}} e^{\frac{\mathbb{L}^{\mathfrak{m}} e_{-1}}{2}} \Phi^{\mathfrak{l}}(e_{\infty}, e_{-1},e_{1}) e^{\frac{\mathbb{L}^{\mathfrak{m}} e_{\infty}}{2}} e^{\frac{\mathbb{L}^{\mathfrak{m}} e_{1}}{2}} - e^{-\frac{\mathbb{L}^{\mathfrak{m}} e_{0}}{2}} e^{-\frac{\mathbb{L}^{\mathfrak{m}} e_{-1}}{2}} \Phi^{\mathfrak{l}}(e_{\infty}, e_{-1},e_{1}) e^{-\frac{\mathbb{L}^{\mathfrak{m}} e_{\infty}}{2}} e^{-\frac{\mathbb{L}^{\mathfrak{m}} e_{1}}{2}} \right] \right) \\ \hspace*{-0.5cm} +\pi^{\mathcal{L}} \left( (\mathbb{L}^{\mathfrak{m}})^{-1} \left[ e^{\frac{\mathbb{L}^{\mathfrak{m}} e_{0}}{2}} e^{\frac{\mathbb{L}^{\mathfrak{m}} e_{-1}}{2}} e^{\frac{\mathbb{L}^{\mathfrak{m}} e_{\infty}}{2}} \Phi^{\mathfrak{l}}(e_{1}, e_{\infty},e_{0}) e^{\frac{\mathbb{L}^{\mathfrak{m}} e_{1}}{2}} - e^{-\frac{\mathbb{L}^{\mathfrak{m}} e_{0}}{2}} e^{-\frac{\mathbb{L}^{\mathfrak{m}} e_{-1}}{2}} e^{-\frac{\mathbb{L}^{\mathfrak{m}} e_{\infty}}{2}} \Phi^{\mathfrak{l}}(e_{1}, e_{\infty},e_{0}) e^{-\frac{\mathbb{L}^{\mathfrak{m}} e_{1}}{2}} \right] \right) \end{multline} \end{small} The first line is zero (since $e_{0}+e_{1}+ e_{-1}+e_{\infty}=0$) whereas each other line will contribute by two terms, in order to give $(\ref{eq:octagonlin})$. Indeed, the projection $\pi^{\mathcal{L}}(x)$, when seeing $x$ as a polynomial (with only even powers) in $\mathbb{L}^{\mathfrak{m}}$, only keep the constant term; hence, for each term, only one of the exponentials above $e^{x}$ contributes by its linear term i.e. $x$, while the others contribute simply by $1$. For instance, if we examine carefully the second line of $(\ref{eq:octagon23})$, we get: $$\begin{array}{ll} = & e_{0} \Phi^{\mathfrak{l}}(e_{-1}, e_{0},e_{\infty}) + \Phi^{\mathfrak{l}}(e_{-1}, e_{0},e_{\infty}) (e_{-1}+e_{\infty}+e_{1})\\ & - (-e_{0}) \Phi^{\mathfrak{l}}(e_{-1}, e_{0},e_{\infty}) - \Phi^{\mathfrak{l}}(e_{-1}, e_{0},e_{\infty}) ( - e_{-1}- e_{\infty}- e_{1}) \\ = & 2 \left[ e_{0} \Phi^{\mathfrak{l}}(e_{-1}, e_{0},e_{\infty}) - \Phi^{\mathfrak{l}}(e_{-1}, e_{0},e_{\infty}) e_{0}\right] \end{array}.$$ Similarly, the third line of $(\ref{eq:octagon23})$ is equal to $(e_{0}+e_{-1}) \Phi^{\mathfrak{l}}(e_{\infty}, e_{-1},e_{1}) - \Phi^{\mathfrak{l}}(e_{\infty}, e_{-1},e_{1}) (e_{0}+e_{-1})$ and the last line is equal to $ -e_{1} \Phi^{\mathfrak{l}}(e_{1}, e_{\infty},e_{0}) + \Phi^{\mathfrak{l}}(e_{1}, e_{\infty},e_{0}) e_{1}$. Therefore, $(\ref{eq:octagon23})$ is equivalent to $(\ref{eq:octagonlin})$, as claimed. \end{itemize} \end{proof}
This linearized octagon relation $\ref{eq:octagonlin}$, while looking at the coefficient of a specific word in $\lbrace e_{0},e_{1}, e_{-1}\rbrace$, provides an identity between some $ \zeta^{\star\star,\mathfrak{l}} (\bullet)$ and $\zeta^{\mathfrak{l}} (\bullet)$ in the coalgebra $\mathcal{L}$. The different identities obtained in this way are detailed in the $\S 4.6$. In the following proof of Theorem $\ref{hybrid}$, two of those identities are used.
\begin{proof}[Proof of Theorem $\ref{hybrid}$] The identity with MMZV$_{\mu_{2}}$ is equivalent to, in terms of motivic iterated integrals:\footnote{Indeed, if $\prod_{i=0}^{p} \epsilon_{i}=1$, it corresponds to the first case, whereas if $\prod_{i=0}^{p} \epsilon_{i}$, we need the second case.} $$I^{\mathfrak{l}} (0; 0^{k}, \star, X; 1)\equiv I^{\mathfrak{l}} (0; X, \star, 0^{k}; 1) \text{ and } I^{\mathfrak{l}} (0; 0^{k}, -\star, X; 1)\equiv I^{\mathfrak{l}} (0;- X, -\star, 0^{k}; 1).$$ Furthermore, by shuffle regularization formula ($\ref{eq:shufflereg}$), spreading the first $0$ further inside the iterated integrals, the identity $I^{\mathfrak{l}} (0;\boldsymbol{0}^{k}, \star, X; 1)\equiv (-1)^{w+1} I^{\mathfrak{l}} (0;\boldsymbol{0}^{k}, \star, \widetilde{X}; 1)$ boils down to the case $k=0$. \\ The notations are as usual: $\epsilon_{i}=\text{sign} (n_{i})$, $\epsilon_{i}=\eta_{i}\eta_{i+1}$,$\epsilon_{p}= \eta_{p}$, $n_{i}=\epsilon_{i}(a_{i}+1)$. \begin{itemize} \item[$(i)$] In $(\ref{eq:octagonlin})$, if we look at the coefficient of a specific word in $\lbrace e_{0},e_{1}, e_{-1}\rbrace$ ending and beginning with $e_{-1}$ (as in $\S 4.6$), only two terms contribute, i.e.: \begin{equation}\label{eq:octagonlinpart1}
e_{-1}\Phi^{\mathfrak{l}}(e_{\infty}, e_{-1},e_{1})- \Phi^{\mathfrak{l}}(e_{\infty}, e_{-1},e_{1})e_{-1}
\end{equation} The coefficient of $e_{0}^{a_{0}}e_{\eta_{1}} e_{0}^{a_{1}} \cdots e_{\eta_{p}} e_{0}^{a_{p}}$ in $\Phi^{\mathfrak{m}}(e_{\infty}, e_{-1},e_{1})$ is $(-1)^{n+p}\zeta^{\star\star,\mathfrak{m}}_{n_{0}-1} \left( n_{1}, \cdots, n_{p-1}, -n_{p}\right)$.\footnote{The expressions of those associators are more detailed in the proof of Lemma $\ref{lemmlor}$.} Hence, the coefficient in $(\ref{eq:octagonlinpart1})$ (as in $(\ref{eq:octagonlin})$) of the word $e_{-1} e_{0}^{a_{0}} e_{\eta_{1}} \cdots e_{\eta_{p}} e_{0}^{a_{p}} e_{-1}$ is: $$ \zeta^{\star\star, \mathfrak{l}}_{\mid n_{0}\mid -1}(n_{1}, \cdots, - n_{p}, 1) - \zeta^{\star\star, \mathfrak{l}}(n_{0}, n_{1}, \cdots, n_{p-1}, -n_{p})=0, \quad \text{ with} \prod_{i=0}^{p} \epsilon_{i}=1.$$ In terms of iterated integrals, reversing the first one with Antipode $\shuffle$, it is: $$ I^{\mathfrak{l}} \left(0;-X , \star ;1 \right)\equiv I^{\mathfrak{l}} \left(0; \star, -X ;1 \right), \text{ with } X\mathrel{\mathop:}=0^{n_{0}-1} \eta_{1} 0^{n_{1}-1} \cdots \eta_{p} 0^{n_{p}-1}.$$ Therefore, since $X$ can be any word in $\lbrace 0, \pm \star \rbrace$, by linearity this is also true for any word X in $\lbrace 0, \pm 1 \rbrace$: $ I^{\mathfrak{l}} \left(0;X, \star ;1 \right)\equiv I^{\mathfrak{l}} \left(0; \star, X ;1 \right)$. \item[$(ii)$] Now, let look at the coefficient of a specific word in $\lbrace e_{0},e_{1}, e_{-1}\rbrace$ beginning by $e_{1}$, and ending by $e_{-1}$. Only two terms in the left side of $(\ref{eq:octagonlin})$ contribute, i.e.: \begin{equation}\label{eq:octagonlinpart2}
-e_{1}\Phi^{\mathfrak{l}}(e_{1}, e_{\infty},e_{0})- \Phi^{\mathfrak{l}}(e_{\infty}, e_{-1},e_{1})e_{-1}
\end{equation} The coefficient in this expression of the word $e_{1} e_{0}^{a_{0}} e_{\eta_{1}} \cdots e_{\eta_{p}} e_{0}^{a_{p}} e_{-1}$ is: $$ \zeta^{\star\star, \mathfrak{l}}_{\mid n_{0}\mid -1}(n_{1}, \cdots, n_{p}, -1) - \zeta^{\star\star, \mathfrak{l}}(n_{0}, n_{1}, \cdots, n_{p})=0, \quad \text{ with} \prod_{i=0}^{p} \epsilon_{i}=-1.$$ In terms of iterated integrals, reversing the first one with Antipode $\shuffle$, it is: $$ I^{\mathfrak{l}} \left(0; - X , -\star ;1 \right)\equiv I^{\mathfrak{l}} \left(0; -\star, X ;1 \right).$$ Therefore, since $X$ can be any word in $\lbrace 0, \pm \star \rbrace$, by linearity this is also true for any word X in $\lbrace 0, \pm 1 \rbrace$. \end{itemize} \end{proof}
\paragraph{For Euler $\boldsymbol{\star\star}$ sums. }
\begin{coro} In the coalgebra $\mathcal{L}^{2}$, with $n_{i}\in\mathbb{Z}^{\ast}$, $n\geq 1$: \begin{equation}\label{eq:antipodestaresss} \zeta^{\star\star,\mathfrak{l}}_{n-1}(n_{1}, \ldots, n_{p})\equiv (-1)^{w+1}\zeta^{\star\star,\mathfrak{l}}_{n-1}(n_{p}, \ldots, n_{1}). \end{equation} Motivic Euler $\star\star$ sums of depth $p$ in $\mathcal{L}$ form a dihedral group of order $p+1$: $$\textsc{(Shift) } \quad \zeta^{\star\star,\mathfrak{l}}_{\mid n\mid -1}(n_{1}, \ldots, n_{p})\equiv \zeta^{\star\star,\mathfrak{l}}_{\mid n_{1}\mid -1}(n_{2}, \ldots, n_{p},n) \quad \text{ where } sgn(n)\mathrel{\mathop:}= \prod_{i} sgn(n_{i}).$$ \end{coro} \noindent Indeed, these two identities lead to a dihedral group structure of order $p+1$: $(\ref{eq:antipodestaresss})$, respectively $\textsc{Shift}$, correspond to the action of a reflection resp. of a cycle of order $p$ on motivic Euler $\star\star$ sums of depth $p$ in $\mathcal{L}$. \begin{proof} Writing $\zeta^{\star\star,\mathfrak{m}}$ as a sum of Euler sums: \begin{small} $$\zeta^{\star\star,\mathfrak{m}}_{n-1}(n_{1}, \ldots, n_{p})=\sum_{i=1}^{p} \zeta^{\mathfrak{m}}_{n-1+\mid n_{1}\mid+\cdots+ \mid n_{i-1}\mid}(n_{i} \circ \cdots \circ n_{p})=\sum_{r \atop A_{i}} \left( \zeta^{\mathfrak{m}}_{n-1}(A_{1}, \ldots, A_{r})+ \zeta^{\mathfrak{m}}_{n-1+\mid A_{1}\mid}(A_{2}, \ldots, A_{r})\right),$$ \end{small} where the last sum is over $(A_{i})_{i}$ such that each $A_{i}$ is a non empty \say{sum} of consecutive $(n_{j})'s$, preserving the order; the absolute value being summed whereas the sign of the $n_{i}$ involved are multiplied; moreover, $\mid A_{1}\mid \geq \mid n_{1}\mid $ resp. $\mid A_{r} \mid \geq \mid n_{p}\mid $.\\ Using Theorem $(\ref{hybrid})$ in the coalgebra $\mathcal{L}$, the previous equality turns into: $$(-1)^{w+1}\sum_{r \atop A_{i}} \left( \zeta^{\mathfrak{l}}_{n-1}(A_{r}, \ldots, A_{1})+ \zeta^{\mathfrak{l}}_{n-1+\mid A_{r}\mid}(A_{r-1}, \ldots, A_{1})\right) \equiv (-1)^{w+1}\zeta^{\star\star,\mathfrak{m}}_{n-1}(n_{p}, \ldots, n_{1}). $$ The identity $\textsc{Shift}$ is obtained as the composition of Antipode $\shuffle$ $(\ref{eq:antipodeshuffle2})$ and the first identity of the corollary. \end{proof}
\paragraph{For Euler $\boldsymbol{\sharp\sharp}$ sums.}
\begin{coro} In the coalgebra $\mathcal{L}$, for $n\in\mathbb{N}$, $n_{i}\in\mathbb{Z}^{\ast}$, $\epsilon_{i}\mathrel{\mathop:}=sign(n_{i})$:\footnote{Here, $\mlq - \mrq$ denotes the operation where absolute values are subtracted whereas sign multiplied.}\\ \begin{tabular}{lll} \textsc{Reverse} & $\zeta^{\sharp\sharp,\mathfrak{l}}_{n}(n_{1}, \ldots, n_{p})+ (-1)^{w}\zeta^{\sharp\sharp,\mathfrak{l}}_{n}(n_{p}, \ldots, n_{1}) \equiv \left\{ \begin{array}{l} 0 . \\ \zeta^{\sharp,\mathfrak{l}}_{n}(n_{1}, \ldots, n_{p})
\end{array}\right.$ & $ \begin{array}{l} \textrm{ if } w+p \textrm{ even } . \\
\textrm{ if } w+p \textrm{ odd } \end{array} .$\\
&&\\ \textsc{Shift} & $\zeta^{\sharp\sharp,\mathfrak{l}}_{ n -1}(n_{1}, \ldots, n_{p})\equiv \zeta^{\sharp\sharp,\mathfrak{l}}_{\mid n_{1}\mid-1}(n_{2}, \ldots, n_{p},\epsilon_{1}\cdots \epsilon_{p} \cdot n)$ & for $w+p$ even.\\ &&\\ \textsc{Cut} & $\zeta^{\sharp\sharp,\mathfrak{l}}_{n}(n_{1},\cdots, n_{p}) \equiv \zeta^{\sharp\sharp,\mathfrak{l}}_{n+\mid n_{p}\mid}(n_{1},\cdots, n_{p-1}),$ & for $w+p$ odd.\\ &&\\ \textsc{Minus} & $\zeta^{\sharp\sharp,\mathfrak{l}}_{n-i}(n_{1},\cdots, n_{p}) \equiv \zeta^{\sharp\sharp,\mathfrak{l}}_{n}\left( n_{1},\cdots, n_{p-1}, n_{p} \mlq - \mrq i)\right)$, & for $\begin{array}{l} w+p \text{ odd }\\ i \leq \min(n,\mid n_{p}\mid) \end{array}$.\\ &&\\ \textsc{Sign} & $\zeta^{\sharp\sharp,\mathfrak{l}}_{n}(n_{1},\cdots, n_{p-1}, n_{p}) \equiv \zeta^{\sharp\sharp,\mathfrak{l}}_{n}(n_{1},\cdots, n_{p-1},-n_{p})$,& for $w+p$ odd. \end{tabular} $$\quad\Rightarrow \forall W \in \lbrace 0, \pm \sharp\rbrace^{\times} \text{with an odd number of } 0, \quad I^{\mathfrak{l}}(-1; W ; 1) \equiv 0 .$$ \end{coro} \noindent \textsc{Remark}: In the coaction of Euler sums, terms with $\overline{1}$ can appear\footnote{More precisely, using the notations of Lemma $\ref{lemmt}$, a $\overline{1}$ can appear in terms of the type $T_{\epsilon, -\epsilon}$ for a cut between $\epsilon$ and $-\epsilon$.}, which are clearly not motivic multiple zeta values. The left side corresponding to such a term in the coaction part $D_{2r+1}(\cdot)$ is $I^{ \mathfrak{l}}(1;X;-1)$, X odd weight with $0, \pm \sharp$. It is worth underlying that, for the $\sharp$ family with $\lbrace \overline{even}, odd\rbrace$, these terms disappear by $\textsc{Sign}$, since by constraint on parity, X will always be of even depth for such a cut. This $\sharp$ family is then more suitable for an unramified criterion, cf. $\S 4.3$. \begin{proof} These are consequences of the hybrid relation in Theorem $\ref{hybrid}$. \begin{itemize} \item[$\cdot$] \textsc{Reverse:} Writing $\zeta^{\sharp\sharp,\mathfrak{l}}$ as a sum of Euler sums: \begin{flushleft} $\hspace*{-0.5cm}\zeta^{\sharp\sharp,\mathfrak{m}}_{k}(n_{1}, \ldots, n_{p}) +(-1)^{w} \zeta^{\sharp\sharp,\mathfrak{m}}_{k}(n_{p}, \ldots, n_{1})$ \end{flushleft} \begin{small} $$\hspace*{-0.5cm}\begin{array}{l} =\sum_{i=1}^{p} 2^{p-i+1-n_{+}} \zeta^{\mathfrak{m}}_{k+n_{1}+\cdots+ n_{i-1}}(n_{i} \circ \cdots \circ n_{p}) + (-1)^{w}2^{i-n_{+}} \zeta^{\mathfrak{m}}_{k+n_{p}+\cdots+ n_{i+1}}(n_{i} \circ \cdots \circ n_{1})\\
\\ =\sum_{r \atop A_{i}}2^{r-1} \left(\right. 2 \zeta^{\mathfrak{m}}_{k}(A_{1}, \ldots, A_{r}) +2 (-1)^{w} \zeta^{\mathfrak{m}}_{k}(A_{r}, \ldots, A_{1}) + \zeta^{\mathfrak{m}}_{k+A_{1}}(A_{2}, \ldots, A_{r}) +(-1)^{w} \zeta^{\mathfrak{m}}_{k+A_{r}}(A_{r-1}, \ldots, A_{1}) \left. \right) \end{array}$$ \end{small} where the sum is over $(A_{i})$ such that each $A_{i}$ is a non empty \say{sum} of consecutive $(n_{j})'s$, preserving the order; i.e. absolute values of $n_{i}$ are summed whereas signs are multiplied; moreover, $A_{1}$ resp. $A_{r}$ are no less than $n_{1}$ resp. $n_{p}$.\\ By Theorem $\ref{hybrid}$, the previous equality turns into, in $\mathcal{L}$: $$\sum_{r \atop A_{i}}2^{r-1} \left( \zeta^{\mathfrak{l}}_{k}(A_{1}, \ldots, A_{r})+ (-1)^{w} \zeta^{\mathfrak{l}}_{k}(A_{r}, \ldots, A_{1})\right)$$ $$ \equiv 2^{-1} \left( \zeta^{\sharp,\mathfrak{l}}_{k}(n_{1}, \ldots, n_{p})+ (-1)^{w} \zeta^{\sharp,\mathfrak{l}}_{k}(n_{p}, \ldots, n_{1})\right)\equiv 2^{-1} \zeta^{\sharp,\mathfrak{l}}_{k}(n_{1}, \ldots, n_{p}) \left( 1+ (-1)^{w+p+1} \right).$$ By the Antipode $\star$ relation applied to $\zeta^{\sharp,\mathfrak{l}}$, it implies the result stated, splitting the cases $w+p$ even and $w+p$ odd. \item[$\cdot$] \textsc{Shift:} Obtained when combining \textsc{Reverse} and \textsc{Antipode} $\shuffle$, when $w+p$ even. \item[$\cdot$] \textsc{Cut:} Reverse in the case $w+p$ odd implies: $$\zeta^{\sharp\sharp,\mathfrak{l}}_{n+\mid n_{1}\mid }(n_{2}, \ldots, n_{p})+ (-1)^{w}\zeta^{\sharp\sharp,\mathfrak{l}}_{n}(n_{p}, \ldots, n_{1}) \equiv 0,$$ Which, reversing the variables, gives the Cut rule. \item[$\cdot$] \textsc{Minus} follows from \textsc{Cut} since, by \textsc{Cut} both sides are equal to $\zeta^{\sharp\sharp,\mathfrak{l}}_{n-i+ \mid n_{p}\mid}(n_{1},\cdots, n_{p-1})$. \item[$\cdot$] In \textsc{Cut}, the sign of $n_{p}$ does not matter, hence, using \textsc{Cut} in both directions, with different signs leads to \textsc{Sign}: $$\zeta^{\sharp\sharp,\mathfrak{l}}_{n} (n_{1},\ldots,n_{p})\equiv \zeta^{\sharp\sharp,\mathfrak{l}}_{n+ \mid n_p\mid } (n_{1}, \ldots,n_{p-1})\equiv \zeta^{\sharp\sharp,\mathfrak{l}}_{n} ( n_{1},\ldots,-n_{p}).$$ Note that, translating in terms of iterated integrals, it leads to, for $X$ any sequence of $0, \pm \sharp$, with $w+p$ odd: $$ I^{\mathfrak{l}}(0; X ; 1) \equiv I^{\mathfrak{l}}(0; -X; 1), $$ where $-X$ is obtained from $X$ after exchanging $\sharp$ and $-\sharp$. Moreover, $I^{\mathfrak{l}}(0; -X; 1)\equiv I^{\mathfrak{l}}(0; X; -1) \equiv - I^{\mathfrak{l}}(-1; X; 0)$. Hence, we obtain, using the composition rule of iterated integrals modulo product: $$I^{\mathfrak{l}}(0; X ; 1) + I^{\mathfrak{l}}(-1; X; 0)\equiv I^{\mathfrak{l}}(-1; X ; 1) \equiv 0.$$ \end{itemize} \end{proof}
\section{Euler $\sharp$ sums}
Let's consider more precisely the following family, appearing in Conjecture $\ref{lzg}$, ith only positive odd and negative even integers for arguments: $$\zeta^{\sharp, \mathfrak{m}} \left( \lbrace \overline{\text{even }} , \text{odd } \rbrace^{\times} \right) .$$ In the iterated integral, this condition means that we see only the following sequences: \begin{center} $\epsilon 0^{2a} \epsilon$, $\quad$ or $\quad\epsilon 0^{2a+1} -\epsilon$, $\quad$ with $\quad\epsilon\in \lbrace \pm\sharp \rbrace$. \end{center} \begin{theo}\label{ESsharphonorary} The motivic Euler sums $\zeta^{\sharp, \mathfrak{m}} (\lbrace \overline{\text{even }}, \text{odd } \rbrace^{\times} )$ are motivic geometric$^{+}$ periods of $\mathcal{MT}(\mathbb{Z})$.\\ Hence, they are $\mathbb{Q}$ linear combinations of motivic multiple zeta values. \end{theo} The proof, in $\S 4.3.2$, relies mainly upon the stability under the coaction of this family.\\ This motivic family is even a generating family of motivic MZV:\nomenclature{$\mathcal{B}^{\sharp}$}{is a family of (unramified) motivic Euler $\sharp$ sums, basis of MMZV} \begin{theo}\label{ESsharpbasis} The following family is a basis of $\mathcal{H}^{1}$: $$\mathcal{B}^{\sharp}\mathrel{\mathop:}= \left\lbrace \zeta^{ \sharp,\mathfrak{m}} (2a_{0}+1,2a_{1}+3,\cdots, 2 a_{p-1}+3, \overline{2a_{p}+2})\text{ , } a_{i}\geq 0\right\rbrace .$$ \end{theo} First, it is worth noticing that this subfamily is also stable under the coaction.\\ \\ \textsc{Remark}: It is conjecturally the same family as the Hoffman star family $\zeta^{\star} (\boldsymbol{2}^{a_{0}},3,\cdots, 3, \boldsymbol{2}^{a_{p}})$, by Conjecture $(\ref{lzg})$.\\ \\ For that purpose, we use the increasing \textit{depth filtration} $\mathcal{F}^{\mathfrak{D}}$ on $\mathcal{H}^{2}$ such that (cf. $\S 2.4.3$): \begin{center}
$\mathcal{F}_{p}^{\mathfrak{D}} \mathcal{H}^{2}$ is generated by Euler sums of depth smaller than $p$.
\end{center}
Note that it is not a grading, but we define the associated graded as the quotient $gr_{p}^{\mathfrak{D}}\mathrel{\mathop:}=\mathcal{F}_{p}^{\mathfrak{D}} \diagup \mathcal{F}_{p-1}^{\mathfrak{D}}$. The vector space $\mathcal{F}_{p}^{\mathfrak{D}}\mathcal{H}$ is stable under the action of $\mathcal{G}$. The linear independence of this $\sharp$ family is proved below thanks to a recursion on the depth and on the weight, using the injectivity of a map $\partial$ where $\partial$ came out of the depth and weight-graded part of the coaction $\Delta$.
\subsection{Depth graded Coaction}
In Chapter $2$, we defined the depth graded derivations $D_{r,p}$ (cf. $\ref{Drp}$), and $D^{-1}_{r,p}$ ($\ref{eq:derivnp}$) after the projection on the right side, using depth $1$ results: $$gr^{\mathcal{D}}_{1} \mathcal{L}_{2r+1}=\mathbb{Q}\zeta^{\mathfrak{l}}(2r+1).$$ Let look at the following maps, whose injectivity is fundamental to the Theorem $\ref{ESsharpbasis}$: $$ D^{-1}_{2r+1,p} : gr^{\mathfrak{D}}_{p}\mathcal{H}_{n}\rightarrow gr^{\mathfrak{D}}_{p-1}\mathcal{H}_{n-2r-1} .$$ $$\partial_{<n,p} \mathrel{\mathop:}=\oplus_{2r+1<n} D^{-1}_{2r+1,p} .$$ Their explicit expression is:
\begin{lemm}\footnotemark[2]\footnotetext[2]{To be accurate, the term $i=0$ in the first sum has to be understood as: $$ \frac{2^{2r+1}}{1-2^{2r}}\binom{2r}{2a_{1}+2} \zeta^{\sharp,\mathfrak{m}} (2\alpha+3, 2 a_{2}+3,\cdots, \overline{2a_{p}+2}) . $$ Meanwhile the terms $i=1$, resp. $i=p$ in the second sum have to be understood as: $$ \frac{2^{2r+1}}{1-2^{2r}}\binom{2r}{2a_{0}+2} \zeta^{\sharp,\mathfrak{m}} (2\alpha+3, 2 a_{2}+3,\cdots, \overline{2a_{p}+2}) \quad \text{ resp. } \quad \frac{2^{2r+1}}{1-2^{2r}}\binom{2r}{2a_{p-1}+2} \zeta^{\sharp,\mathfrak{m}} (\cdots, 2 a_{p-2}+3, \overline{2\alpha+2}).$$} \begin{multline} \label{eq:dgrderiv} D^{-1}_{2r+1,p} \left( \zeta^{\sharp,\mathfrak{m}} (2a_{0}+1,2a_{1}+3,\cdots, 2 a_{p-1}+3, \overline{2a_{p}+2}) \right) = \\ \delta_{r=a_{0}} \frac{2^{2r+1}}{1-2^{2r}}\binom{2r}{2r+2} \zeta^{\sharp,\mathfrak{m}} (2 a_{1}+3,\cdots, \overline{2a_{p}+2})\\ + \sum_{0 \leq i \leq p-2, \quad \alpha \leq a_{i}\atop r=a_{i+1}+a_{i}+1-\alpha} \frac{2^{2r+1}}{1-2^{2r}}\binom{2r}{2a_{i+1}+2} \zeta^{\sharp,\mathfrak{m}} (\cdots, 2 a_{i-1}+3,2\alpha+3, 2 a_{i+2}+3,\cdots, \overline{2a_{p}+2})\\ + \sum_{1 \leq i \leq p-1, \quad \alpha \leq a_{i} \atop r=a_{i-1}+a_{i}+1-\alpha} \frac{2^{2r+1}}{1-2^{2r}}\binom{2r}{2a_{i-1}+2} \zeta^{\sharp,\mathfrak{m}} (\cdots, 2 a_{i-2}+3,2\alpha+3, 2 a_{i+1}+3,\cdots, \overline{2a_{p}+2})\\ + \textsc{(Deconcatenation)} \sum_{\alpha \leq a_{p} \atop r=a_{p-1}+a_{p}+1-\alpha} 2 \binom{2r}{2a_{p}+1}\zeta^{\sharp,\mathfrak{m}} (\cdots, 2 a_{p-1}+3,\overline{2\alpha+2}). \end{multline} \end{lemm} \begin{proof} Looking at the Annexe $A.1$ expression for $D_{2r+1}$, we obtain for $D_{2r+1,p}$ keeping only the cuts of depth one (removing exactly one non zero element): \begin{multline} \nonumber D_{2r+1,p} \zeta^{\sharp,\mathfrak{m}} (2a_{0}+1,2a_{1}+3,\cdots, 2 a_{p-1}+3, \overline{2a_{p}+2})=\\ \sum_{i, \alpha \leq a_{i}\atop r=a_{i+1}+a_{i}+1-\alpha} 2 \zeta^{\mathfrak{l}} _{2a_{i}-2\alpha}(2a_{i+1}+3) \otimes \zeta^{\sharp,\mathfrak{m}} (\cdots, 2 a_{i-1}+3,2\alpha+3, 2 a_{i+2}+3,\cdots, \overline{2a_{p}+2})\\ +\sum_{i, \alpha \leq a_{i} \atop r=a_{i-1}+a_{i}+1-\alpha} 2 \zeta^{\mathfrak{l}} _{2a_{i}-2\alpha}(2a_{i-1}+3) \otimes \zeta^{\sharp,\mathfrak{m}} (\cdots, 2 a_{i-2}+3,2\alpha+3, 2 a_{i+1}+3,\cdots, \overline{2a_{p}+2})\\ +\sum_{\alpha \leq a_{p} \atop r=a_{p-1}+a_{p}+1-\alpha} 2 \zeta^{\mathfrak{l}} _{2a_{p-1}-2\alpha+1}(\overline{2a_{p}+2}) \otimes \zeta^{\sharp,\mathfrak{m}} (\cdots, 2 a_{p-1}+3,\overline{2\alpha+2}). \end{multline} To lighten the result, some cases at the borders ($i=0$, or $i=p$) have been included in the sum, being fundamentally similar (despite some index problems). These are clarified in the previous footnote\footnotemark[2].\\ In particular, with notations of the Lemma $\ref{lemmt}$, $T_{0,0}$ terms can be neglected as they decrease the depth by at least $2$; same for the $T_{0,\epsilon}$ and $T_{\epsilon,0}$ for cuts between $\epsilon$ and $\pm \epsilon$. To obtain the lemma, it remains to check the coefficient of $\zeta^{\mathfrak{l}}(\overline{2r+1})$ for each term in the left side thanks to the known identities: $$ \zeta^{\mathfrak{l}}(2r+1)= \frac{-2^{2r}}{2^{2r}-1} \zeta^{\mathfrak{l}}(\overline{2r+1})\quad \text{ and } \quad \zeta^{\mathfrak{l}}_{2r+1-a}(a)=(-1)^{a+1}\binom{2r}{a-1} \zeta^{\mathfrak{l}}(2r+1).$$ \end{proof}
\subsection{Proofs of Theorem $4.3.1$ and $4.3.2$}
\begin{proof}[\textbf{Proof of Theorem $4.3.1$}] By Corollary $5.1.2$, we can prove it in two steps: \begin{itemize} \item[$\cdot$] First, checking that $D_{1}(\cdot)=0$ for this family, which is rather obvious by Lemma $\ref{condd1}$ since there is no sequence of the type $\lbrace 0, \epsilon, -\epsilon \rbrace$ or $\lbrace \epsilon, -\epsilon, 0 \rbrace$ in the iterated integral. \item[$\cdot$] Secondly, we can use a recursion on weight to prove that $D_{2r+1}(\cdot)$, for $r> 0$, are unramified. Consequently, using recursion, this follows from the following statement: \begin{center} The family $\zeta^{\sharp, \mathfrak{m}} \left( \lbrace \overline{\text{even }}, + \text{odd } \rbrace^{\times} \right) $ is stable under $D_{2r+1}$. \end{center} This is proved in Lemma $A.1.3$, using the relations of $\S 4.2$ in order to simplify the \textit{unstable cuts}, i.e. the cuts where a sequence of type $\epsilon, 0^{2a+1}, \epsilon$ or $\epsilon, 0^{2a}, -\epsilon$ appears; indeed, these cuts give rise to a $\text{even}$ or to a $\overline{ \text{ odd}}$ in the $\sharp$ Euler sum. \end{itemize} One fundamental observation about this family, used in Lemma $A.1.3$ is: for a subsequence of odd length from the iterated integral, because of these patterns of $\epsilon, \boldsymbol{0}^{2a}, \epsilon$, or $\epsilon, \boldsymbol{0}^{2a+1}, -\epsilon$, we can put in relation the depth $p$, the weight $w$ and $s$ the number of sign changes among the $\pm\sharp$: $$w\equiv p-s \pmod{2}.$$ It means that if we have a cut $\epsilon_{0},\cdots \epsilon_{p+1}$ of odd weight, then: \begin{center} \textsc{Either:} Depth $p$ is odd, $s$ even, $\epsilon_{0}=\epsilon_{p+1}$, \textsc{Or:} Depth $p$ is even, $s$ odd, $\epsilon_{0}=-\epsilon_{p+1}$. \end{center} \end{proof}
\begin{proof}[\textbf{Proof of Theorem $4.3.2$}] By a cardinality argument, it is sufficient to prove the linear independence of the family, which is based on the injectivity of $\partial_{<n,p}$. Let us define: \footnote{Sub-$\mathbb{Q}$ vector space of $\mathcal{H}^{1}$ by previous Theorem.}\nomenclature{$\mathcal{H}^{odd\sharp}$}{$\mathbb{Q}$-vector space generated by $\zeta^{ \sharp,\mathfrak{m}} (2a_{0}+1,2a_{1}+3,\cdots, 2 a_{p-1}+3, \overline{2a_{p}+2})$} \begin{center} $\mathcal{H}^{odd\sharp}$: $\mathbb{Q}$-vector space generated by $\zeta^{ \sharp,\mathfrak{m}} (2a_{0}+1,2a_{1}+3,\cdots, 2 a_{p-1}+3, \overline{2a_{p}+2})$. \end{center} The first thing to remark is that $\mathcal{H}^{odd\sharp}$ is stable under these derivations, by the expression obtained in Lemma $A.1.4$.: $$D_{2r+1} (\mathcal{H}_{n}^{odd\sharp}) \subset \mathcal{L}_{2r+1} \otimes \mathcal{H}_{n-2r-1}^{odd\sharp},$$ Now, let consider the restriction on $\mathcal{H}^{odd\sharp}$ of $\partial_{<n,p}$ and prove: $$\partial_{<n,p}: gr^{\mathfrak{D}}_{p} \mathcal{H}_{n}^{odd\sharp} \rightarrow \oplus_{2r+1<n} gr^{\mathfrak{D}}_{p-1}\mathcal{H}_{n-2r-1}^{odd\sharp} \text{ is bijective. }$$ The formula $\eqref{eq:dgrderiv}$ gives the explicit expression of this map. Let us prove more precisely: \begin{center} $M^{\mathfrak{D}}_{n,p}$ the matrix of $\partial_{<n,p}$ on $\left\lbrace \zeta^{ \sharp ,\mathfrak{m}} (2a_{0}+1,2a_{1}+3,\cdots, 2 a_{p-1}+3, \overline{2a_{p}+2})\right\rbrace $ in terms of $\left\lbrace \zeta^{ \sharp ,\mathfrak{m}} (2b_{0}+1,2b_{1}+3,\cdots, 2 b_{p-2}+3, \overline{2b_{p-1}+2})\right\rbrace $ is invertible. \end{center} \texttt{Nota Bene}: The matrix $M^{\mathfrak{D}}_{n,p}$ is well (uniquely) defined provided that the $\zeta^{ \sharp ,\mathfrak{m}}$ of the second line are linearly independent. So first, we have to consider the formal matrix associated $\mathbb{M}^{\mathfrak{D}}_{n,p}$ defined explicitly (combinatorially) by the formula for the derivations given, and prove $\mathbb{M}^{\mathfrak{D}}_{n,p}$ is invertible. Afterwards, we could state that $M^{\mathfrak{D}}_{n,p}$ is well defined and invertible too since equal to $\mathbb{M}^{\mathfrak{D}}_{n,p}$. \begin{proof} The invertibility comes from the fact that the (strictly) smallest terms $2$-adically in $\eqref{eq:dgrderiv}$ are the deconcatenation ones, which is an injective operation. More precisely, let $\widetilde{M}^{\mathfrak{D}}_{n,p}$ be the matrix $\mathbb{M}_{n,p}$ where we have multiplied each line corresponding to $D_{2r+1}$ by ($2^{-2r}$). Then, order elements on both sides by lexicographical order on ($a_{p}, \ldots, a_{0}$), resp. ($r,b_{p-1}, \ldots, b_{0}$), such that the diagonal corresponds to $r=a_{p}+1$ and $b_{i}=a_{i}$ for $i<p$. The $2$ -adic valuation of all the terms in $(\ref{eq:dgrderiv})$ (once divided by $2^{2r}$) is at least $1$, except for the deconcatenation terms since: $$v_{2}\left( 2^{-2r+1} \binom{2r}{2a_{p}+1} \right) \leq 0 \Longleftrightarrow v_{2}\left( \binom{2r}{2a_{p}+1} \right) \leq 2r-1.$$ Then, modulo $2$, only the deconcatenation terms remain, so the matrix $\widetilde{M}^{\mathfrak{D}}_{n,p}$ is triangular with $1$ on the diagonal. This implies that $\det (\widetilde{M}^{\mathfrak{D}}_{n,p})\equiv 1 \pmod{2}$, and in particular is non zero: the matrix $\widetilde{M}^{\mathfrak{D}}_{n,p}$ is invertible, and so does $\mathbb{M}^{\mathfrak{D}}_{n,p}$. \end{proof} This allows us to complete the proof since it implies: \begin{center} The elements of $\mathcal{B}^{\sharp}$ are linearly independent. \end{center} \begin{proof} First, let prove the linear independence of this family of the same depth and weight, by recursion on $p$. For depth $0$, this is obvious since $\zeta^{\mathfrak{m}}(\overline{2n})$ is a rational multiple of $\pi^{2n}$.\\ Assuming by recursion on the depth that the elements of weight $n$ and depth $p-1$ are linearly independent, since $M^{\mathfrak{D}}_{n,p}$ is invertible, this means both that the $\zeta^{ \sharp,\mathfrak{m}} (2a_{0}+1,2a_{1}+3,\cdots, 2 a_{p-1}+3, \overline{2a_{p}+2})$ of weight $n$ are linearly independent and that $\partial_{<n,p}$ is bijective, as announced before.\\ The last step is just to realize that the bijectivity of $\partial_{<n,l}$ also implies that elements of different depths are also linearly independent. The proof could be done by contradiction: by applying $\partial_{<n,p}$ on a linear combination where $p$ is the maximal depth appearing, we arrive at an equality between same level elements. \end{proof} \end{proof}
\section{Hoffman $\star$}
\begin{theo}\label{Hoffstar} If the analytic conjecture ($\ref{conjcoeff}$) holds, then the motivic \textit{Hoffman} $\star$ family $\lbrace \zeta^{\star,\mathfrak{m}} (\lbrace 2,3 \rbrace^{\times})\rbrace$ is a basis of $\mathcal{H}^{1}$, the space of MMZV. \end{theo}
For that purpose, we define an increasing filtration $\mathcal{F}^{L}_{\bullet}$ on $\mathcal{H}^{2,3}$, called \textbf{level}, such that: \begin{equation}\label{eq:levelf}
\mathcal{F}^{L}_{l}\mathcal{H}^{2,3} \text{ is spanned by } \zeta^{\star,\mathfrak{m}} (\boldsymbol{2}^{a_{0}},3,\cdots,3, \boldsymbol{2}^{a_{p}}) \text{, with less than 'l' } 3. \end{equation} It corresponds to the motivic depth for this family, as we see through the proof below and the coaction calculus.\\ \paragraph{Sketch. } The vector space $\mathcal{F}^{L}_{l}\mathcal{H}^{2,3}$ is stable under the action of $\mathcal{G}$ ($\ref{eq:levelfiltstrable}$). The linear independence of the Hoffman $\star$ family is proved below ($ § 4.4.2$) thanks to a recursion on the level and on the weight, using the injectivity of a map $\partial^{L}$ where $\partial^{L}$ came out of the level and weight-graded part of the coaction $\Delta$ (cf. $4.4.2$). The injectivity is proved via $2$-adic properties of some coefficients conjectured in $\ref{conjcoeff}$.\\ Indeed, when computing the level graded coaction (cf. Lemma $4.4.2$) on the Hoffman $\star$ elements, looking at the left side, some elements appear, such as $\zeta^{\star\star,\mathfrak{m}}(\boldsymbol{2}^{a},3,\boldsymbol{2}^{b})$ but also $\zeta^{\star\star,\mathfrak{m}}(\boldsymbol{2}^{a},3,\boldsymbol{2}^{b})$. These are not always of depth $1$ as we could expect,\footnote{As for the Hoffman non $\star$ case done by Francis Brown, using a result of Don Zagier for level $1$.} but at least are abelians: product of motivic simple zeta values, as proved in Lemma $\ref{lemmcoeff}$.\\ To prove the linear independence of Hoffman $\star$ elements, we will then need to know some coefficients appearing in Lemma $\ref{lemmcoeff}$ (or at least the 2-adic valuation) of $\zeta(weight)$ for each of these terms, conjectured in $\ref{conjcoeff}$, which is the only missing part of the proof, and can be solved at the analytic level.\\
\subsection{Level graded coaction}
Let use the following form for a MMZV$^{\star}$, gathering the $2$: $$\zeta^{\star, \mathfrak{m}} (\boldsymbol{2}^{a_{0}},c_{1},\cdots,c_{p}, \boldsymbol{2}^{a_{p}}), \quad c_{i}\in\mathbb{N}^{\ast}, c_{i}\neq 2.$$ This writing is suitable for the Galois action (and coaction) calculus, since by the antipode relations ($\S 4.2$), many of the cuts from a $2$ to a $2$ get simplified (cf. Annexe $\S A.1$).\\ For the Hoffman family, with only $2$ and $3$, the expression obtained is:\footnote{Cf. Lemma $A.1.2$; where $\delta_{2r+1}$ means here that the left side has to be of weigh $2r+1$.}\\ \begin{flushleft} \hspace*{-0.7cm}$D_{2r+1} \zeta^{\star, \mathfrak{m}} (\boldsymbol{2}^{a_{0}},3,\cdots,3, \boldsymbol{2}^{a_{p}})$ \end{flushleft} \begin{multline} \label{eq:dr3} \hspace*{-1.3cm}= \delta_{2r+1}\sum_{i<j} \left[ \begin{array}{lll}
+ \quad \zeta^{\star\star, \mathfrak{l}}_{1} (\boldsymbol{2}^{a_{i+1}},3,\cdots,3, \boldsymbol{2}^{\leq a_{j}}) & \otimes & \zeta^{\star, \mathfrak{m}} (\cdots,3, \boldsymbol{2}^{1+a_{i}+ \leq a_{j}},3, \cdots)\\
- \quad \zeta^{\star\star, \mathfrak{l}}_{1} (\boldsymbol{2}^{\leq a_{i}},3,\cdots,3, \boldsymbol{2}^{ a_{j-1}}) & \otimes & \zeta^{\star, \mathfrak{m}} (\cdots,3, \boldsymbol{2}^{1+a_{j}+ \leq a_{i}},3, \cdots)\\
+ \left( \zeta^{\star\star, \mathfrak{l}}_{2} (\boldsymbol{2}^{a_{i+1}},3,\cdots, \boldsymbol{2}^{a_{j}},3) + \zeta^{\star\star, \mathfrak{l}}_{1} (\boldsymbol{2}^{<a_{i}},3,\cdots, \boldsymbol{2}^{a_{j}},3) \right) & \otimes& \zeta^{\star, \mathfrak{m}} (\cdots,3, \boldsymbol{2}^{<a_{i}},3,\boldsymbol{2}^{a_{j+1}},3, \cdots)\\
- \left(\zeta^{\star\star, \mathfrak{l}}_{2} (\boldsymbol{2}^{a_{j+1}},3,\cdots,3) + \zeta^{\star\star, \mathfrak{l}}_{1}(\boldsymbol{2}^{<a_{j}},3,\cdots,3) \right)& \otimes & \zeta^{\star, \mathfrak{m}} (\cdots,3, \boldsymbol{2}^{a_{i-1}},3,\boldsymbol{2}^{< a_{j}},3, \cdots) \\ \end{array} \right] \\ \quad \quad \begin{array}{lll} \quad \quad+ \quad\delta_{2r+1} \quad \left( \zeta^{\star, \mathfrak{l}} (\boldsymbol{2}^{a_{0}},3,\cdots,3, \boldsymbol{2}^{\leq a_{i}})- \zeta^{\star\star, \mathfrak{l}} (\boldsymbol{2}^{\leq a_{i}},3,\cdots,3, \boldsymbol{2}^{a_{0}}) \right) & \otimes & \zeta^{\star, \mathfrak{m}} (\boldsymbol{2}^{\leq a_{i}},3, \cdots)\\
\quad\quad +\quad \delta_{2r+1} \quad\zeta^{\star\star, \mathfrak{l}} (\boldsymbol{2}^{\leq a_{j}},3,\cdots,3, \boldsymbol{2}^{ a_{p}}) & \otimes & \zeta^{\star, \mathfrak{m}} (\cdots,3, \boldsymbol{2}^{\leq a_{j}}). \end{array} \end{multline} In particular, the coaction on the Hoffman $\star$ elements is stable. \\ By the previous expression $(\ref{eq:dr3})$, we see that each cut (of odd length) removes at least one $3$. It means that the level filtration is stable under the action of $\mathcal{G}$ and: \begin{equation} \label{eq:levelfiltstrable} D_{2r+1}(\mathcal{F}^{L}_{l}\mathcal{H}^{2,3}) \subset \mathcal{L}_{2r+1} \otimes \mathcal{F}^{L}_{l-1}\mathcal{H}_{n-2r-1}^{2,3} . \end{equation} Then, let consider the level graded derivation: \begin{equation} gr^{L}_{l} D_{2r+1}: gr^{L}_{l}\mathcal{H}_{n}^{2,3} \rightarrow \mathcal{L}_{2r+1} \otimes gr^{L}_{l-1}\mathcal{H}_{n-2r-1}^{2,3}. \end{equation} If we restrict ourselves to the cuts in the coaction that remove exactly one $3$ in the right side, the formula $(\ref{eq:dr3})$ leads to: \begin{flushleft} \hspace*{-0.5cm}$gr^{L}_{l} D_{2r+1} \zeta^{\star, \mathfrak{m}} (\boldsymbol{2}^{a_{0}},3,\cdots,3, \boldsymbol{2}^{a_{p}}) =$ \end{flushleft} \begin{multline}\label{eq:gdr3} \hspace*{-1.5cm}\begin{array}{lll} \quad - \delta_{a_{0} < r \leq a_{0}+a_{1}+2} \quad \zeta^{\star\star, \mathfrak{l}}_{2} (\boldsymbol{2}^{a_{0}}, 3, \boldsymbol{2}^{r-a_{0}-2}) &\otimes & \zeta^{\star, \mathfrak{m}} (\boldsymbol{2}^{ a_{0}+a_{1}+1-r},3, \cdots) \end{array}\\ \hspace*{-1.3cm}\sum_{i<j} \left[ \begin{array}{l} \delta_{r\leq a_{i}} \quad \zeta^{\star\star, \mathfrak{l}}_{1} (\boldsymbol{2}^{r}) \quad \quad \quad \quad \quad \otimes \left( \zeta^{\star, \mathfrak{m}} (\cdots,3, \boldsymbol{2}^{a_{i-1}+ a_{i}-r+1},3, \cdots) - \zeta^{\star, \mathfrak{m}} (\cdots,3, \boldsymbol{2}^{a_{i+1}+ a_{i}-r+1},3, \cdots) \right) \\
+ \left( \delta_{r=a_{i}+2} \zeta^{\star\star, \mathfrak{l}}_{2} (\boldsymbol{2}^{a_{i}},3) + \delta_{r< a_{i}+a_{i-1}+3} \zeta^{\star\star, \mathfrak{l}}_{1} (\boldsymbol{2}^{r-a_{i}-3}, 3, \boldsymbol{2}^{a_{i}},3) \right) \otimes \zeta^{\star, \mathfrak{m}} (\cdots,3, \boldsymbol{2}^{a_{i}+a_{i-1}-r+1},3,\boldsymbol{2}^{a_{i+1}},3, \cdots)\\
- \left( \delta_{r=a_{i}+2} \zeta^{\star\star, \mathfrak{l}}_{2} (\boldsymbol{2}^{a_{i}},3) + \delta_{r< a_{i}+a_{i+1}+3} \zeta^{\star\star, \mathfrak{l}}_{1}(\boldsymbol{2}^{r-a_{i}-3},3, \boldsymbol{2}^{a_{i}}, 3) \right) \otimes \zeta^{\star, \mathfrak{m}} (\cdots,3, \boldsymbol{2}^{a_{i-1}},3,\boldsymbol{2}^{a_{i}+a_{i+1}-r+1},3, \cdots) \end{array} \right] \\ \hspace*{-2cm} \textsc{(D)} \begin{array}{lll}
+\delta_{a_{p}+1 \leq r \leq a_{p}+a_{p-1}+1} \quad \zeta^{\star\star, \mathfrak{l}} (\boldsymbol{2}^{r- a_{p}-1},3, \boldsymbol{2}^{ a_{p}}) &\otimes & \zeta^{\star, \mathfrak{m}} (\cdots,3, \boldsymbol{2}^{a_{p}+ a_{p-1}-r+1}).
\end{array}
\end{multline} By the antipode $\shuffle$ relation (cf. $\ref{eq:antipodeshuffle2}$): $$\zeta^{\star\star, \mathfrak{l}}_{1} (\boldsymbol{2}^{a},3, \boldsymbol{2}^{b},3)= \zeta^{\star\star, \mathfrak{l}}_{2} (\boldsymbol{2}^{b},3, \boldsymbol{2}^{a+1})=\zeta^{\star\star, \mathfrak{l}}(\boldsymbol{2}^{b+1},3, \boldsymbol{2}^{a+1})- \zeta^{\star, \mathfrak{l}}(\boldsymbol{2}^{b+1},3, \boldsymbol{2}^{a+1}).$$ Then, by Lemma $\ref{lemmcoeff}$, all the terms appearing in the left side of $gr^{L}_{l} D_{2r+1}$ are product of simple MZV, which turns into, in the coalgebra $\mathcal{L}$ a rational multiple of $\zeta^{\mathfrak{l}}(2r+1)$: $$gr^{L}_{l} D_{2r+1} (gr^{L}_{l}\mathcal{H}_{n}^{2,3}) \subset \mathbb{Q}\zeta^{\mathfrak{l}}(2r+1)\otimes gr^{L}_{l-1}\mathcal{H}_{n-2r-1}^{2,3}.$$ \\ Sending $\zeta^{\mathfrak{l}}(2r+1)$ to $1$ with the projection $\pi:\mathbb{Q} \zeta^{\mathfrak{l}}(2r+1)\rightarrow\mathbb{Q}$, we can then consider:\nomenclature{$\partial^{L}_{r,l}$ and $\partial^{L}_{<n,l}$}{defined as composition from derivations} \begin{description} \item[$\boldsymbol{\cdot\quad \partial^{L}_{r,l}}$] $ : gr^{L}_{l}\mathcal{H}_{n}^{2,3}\rightarrow gr^{L}_{l-1}\mathcal{H}_{n-2r-1}^{2,3}, \quad \text{ defined as the composition }$ $$\partial^{L}_{r,l}\mathrel{\mathop:}=gr_{l}^{L}\partial_{2r+1}\mathrel{\mathop:}=m\circ(\pi\otimes id)(gr^{L}_{l} D_{r}): \quad gr^{L}_{l}\mathcal{H}_{n}^{2,3} \rightarrow \mathbb{Q}\otimes_{\mathbb{Q}} gr^{L}_{l-1}\mathcal{H}_{n-2r-1}^{2,3} \rightarrow gr^{L}_{l-1}\mathcal{H}_{n-2r-1}^{2,3} .$$ \item[$\boldsymbol{\cdot\quad \partial^{L}_{<n,l}}$] $\mathrel{\mathop:}=\oplus_{2r+1<n}\partial^{L}_{r,l} .$ \\ \end{description} The injectivity of this map is the keystone of the Hoffman$^{\star}$ proof. Its explicit expression is: \begin{lemm} \begin{flushleft} $\partial^{L}_{r,l} (\zeta^{\star, \mathfrak{m}} (\boldsymbol{2}^{a_{0}},3,\cdots,3, \boldsymbol{2}^{a_{p}}))=$ \end{flushleft} $$\begin{array}{l}
\quad - \delta_{a_{0} < r \leq a_{0}+a_{1}+2} \widetilde{B}^{a_{0}+1,r-a_{0}-2} \zeta^{\star, \mathfrak{m}} (\boldsymbol{2}^{ a_{0}+a_{1}+1-r},3, \cdots) \\
\\ + \sum_{i<j} \left[ \begin{array}{l}
\delta_{r\leq a_{i}}C_{r} \left( \zeta^{\star, \mathfrak{m}} (\cdots,3, \boldsymbol{2}^{a_{i-1}+ a_{i}-r+1},3, \cdots) - \zeta^{\star, \mathfrak{m}} (\cdots,3, \boldsymbol{2}^{a_{i+1}+ a_{i}-r+1},3, \cdots) \right) \\
\\ +\delta_{a_{i}+2\leq r \leq a_{i}+a_{i-1}+2} \widetilde{B}^{a_{i}+1,r-a_{i}-2} \zeta^{\star, \mathfrak{m}} (\cdots,3, \boldsymbol{2}^{a_{i}+a_{i-1}-r+1},3,\boldsymbol{2}^{a_{i+1}},3, \cdots) \\
\\ - \delta_{a_{i}+2 \leq r\leq a_{i}+a_{i+1}+2} \widetilde{B}^{a_{i}+1,r-a_{i}-2} \zeta^{\star, \mathfrak{m}} (\cdots,3, \boldsymbol{2}^{a_{i-1}},3,\boldsymbol{2}^{a_{i}+a_{i+1}-r+1},3, \cdots) \\ \end{array} \right] \\
\\
\textsc{(D)} + \delta_{a_{p}+1 \leq r \leq a_{p}+a_{p-1}+1} B^{r-a_{p}-1,a_{p}} \zeta^{\star, \mathfrak{m}} (\cdots,3, \boldsymbol{2}^{a_{p}+ a_{p-1}-r+1}) , \\
\\
\quad \quad \quad \text{ with } \widetilde{B}^{a,b}\mathrel{\mathop:}=B^{a,b}C_{a+b+1}-A^{a,b}. \end{array}$$
\end{lemm} \begin{proof} Using Lemma $\ref{lemmcoeff}$ for the left side of $gr^{L}_{p} D_{2r+1}$, and keeping just the coefficients of $\zeta^{2r+1}$, we obtain easily this formula. In particular: \begin{flushleft} $\zeta^{\star\star, \mathfrak{l}}_{2} (\boldsymbol{2}^{a},3, \boldsymbol{2}^{b})=\zeta^{\star\star, \mathfrak{l}}(\boldsymbol{2}^{a+1},3, \boldsymbol{2}^{b})- \zeta^{\star, \mathfrak{l}}(\boldsymbol{2}^{a+1},3, \boldsymbol{2}^{b}) = \widetilde{B}^{a+1,b} \zeta^{\mathfrak{l}}(\overline{2a+2b+5}).$\\ $\zeta^{\star\star, \mathfrak{l}}_{1} (\boldsymbol{2}^{a},3, \boldsymbol{2}^{b},3)= \zeta^{\star\star, \mathfrak{l}}_{2} (\boldsymbol{2}^{b},3, \boldsymbol{2}^{a+1})= \widetilde{B}^{b+1,a+1}\zeta^{\mathfrak{l}}(\overline{2a+2b+7}).$ \end{flushleft} \end{proof}
\subsection{Proof of Theorem $4.4.1$}
Since the cardinal of the Hoffman $\star$ family in weight $n$ is equal to the dimension of $\mathcal{H}_{n}^{1}$, \footnote{Obviously same recursive relation: $d_{n}=d_{n-2}+d_{n-3}$} it remains to prove that they are linearly independent: \begin{center} \texttt{Claim 1}: The Hoffman $\star$ elements are linearly independent. \end{center} It fundamentally use the injectivity of the map defined above, $\partial^{L}_{<n,l}$, via a recursion on the level. Indeed, let first prove the following statement: \begin{equation} \label{eq:bijective} \texttt{Claim 2}: \quad \partial^{L}_{<n,l}: gr^{L}_{l}\mathcal{H}_{n}^{2,3}\rightarrow \oplus_{2r+1<n} gr^{L}_{l-1}\mathcal{H}_{n-2r-1}^{2,3} \text{ is bijective}. \end{equation} Using the Conjecture $\ref{conjcoeff}$ (assumed for this theorem), regarding the $2$-adic valuation of these coefficients, with $r=a+b+1$:\footnote{The last inequality comes from the fact that $v_{2} (\binom{2r}{2b+1} )<2r $.} \begin{equation}\label{eq:valuations} \hspace*{-0.7cm}\left\lbrace \begin{array}{ll}
C_{r}=\frac{2^{2r+1}}{2r+1} &\Rightarrow v_{2}(C_{r})=2r+1 .\\
\widetilde{B}^{a,b}\mathrel{\mathop:}= B^{a,b}C_{r}-A^{a,b}=2^{2r+1}\left( \frac{1}{2r+1}-\frac{\binom{2r}{2a}}{2^{2r}-1} \right) &\Rightarrow v_{2}(\widetilde{B}^{a,b}) \geq 2r+1.\\ B^{a,b}C_{r}=C_{r}-2\binom{2r}{2b+1} &\Rightarrow v_{2}(B^{0,r-1}C_{r})= 2+ v_{2}(r) \leq v_{2}(B^{a,b}C_{r}) < 2r+1 . \end{array} \right. \end{equation} The deconcatenation terms in $\partial^{L}_{<n,l}$, which correspond to the terms with $B^{a,b}C_{r}$ are then the smallest 2-adically, which is crucial for the injectivity.\\ \\ Now, define a matrix $M_{n,l}$ as the matrix of $\partial^{L}_{<n,l}$ on $\zeta^{\star, \mathfrak{m}} (\boldsymbol{2}^{a_{0}},3,\cdots,3, \boldsymbol{2}^{a_{l}})$ in terms of $\zeta^{\star, \mathfrak{m}} (\boldsymbol{2}^{b_{0}},3,\cdots,3, \boldsymbol{2}^{b_{l-1}})$; even if up to now, we do not know that these families are linearly independent. We order the elements on both sides by lexicographical order on ($a_{l}, \ldots, a_{0}$), resp. ($r,b_{l-1}, \ldots, b_{0}$), such that the diagonal corresponds to $r=a_{l}$ and $b_{i}=a_{i}$ for $i<l$ and claim: \begin{center} \texttt{Claim 3}: The matrix $M_{n,l}$ of $\partial^{L}_{<n,l}$ on the Hoffman $\star$ elements is invertible \end{center} \begin{proof}[\texttt{Proof of Claim 3}] Indeed, let $\widetilde{M}_{n,l}$ be the matrix $M_{n,l}$ where we have multiplied each line corresponding to $D_{2r+1}$ by ($2^{-v_{2}(r)-2}$). Then modulo $2$, because of the previous computations on the $2$-adic valuations of the coefficients, only the deconcatenations terms remain. Hence, with the previous order, the matrix is, modulo $2$, triangular with $1$ on the diagonal; the diagonal being the case where $B^{0,r-1}C_{r}$ appears. This implies that $\det (\widetilde{M}_{n,l})\equiv 1 \pmod{2}$, and in particular is non zero. Consequently, the matrix $\widetilde{M}_{n,l}$ is invertible and so does $M_{n,l}$. \end{proof} Obviously, $\texttt{Claim 3} \Rightarrow \texttt{Claim 2} $, but it will also enables us to complete the proof:
\begin{proof}[\texttt{Proof of Claim 1}] Let first prove it for the Hoffman $\star$ elements of a same level and weight, by recursion on level. Level $0$ is obvious: $\zeta^{\star,\mathfrak{m}}(2)^{n}$ is a rational multiple of $(\pi^{\mathfrak{m}})^{2n}$. Assuming by recursion on the level that the Hoffman $\star$ elements of weight $\leq n$ and level $l-1$ are linearly independent, since $M_{n,l}$ is invertible, this means both that the Hoffman $\star$ elements of weight $n$ and level $l$ are linearly independent.\\ The last step is to realize that the bijectivity of $\partial^{L}_{<n,l}$ also implies that Hoffman $\star$ elements of different levels are linearly independent. Indeed, proof can be done by contradiction: applying $\partial^{L}_{<n,l}$ to a linear combination of Hoffman $\star$ elements, $l$ being the maximal number of $3$, we arrive at an equality between same level elements, and at a contradiction. \end{proof}
\subsection{Analytic conjecture}
Here are the equalities needed for Theorem $4.4.1$, known up to some rational coefficients: \begin{lemm} \label{lemmcoeff} With $w$, $d$ resp. $ht$ denoting the weight, the depth, resp. the height: \begin{itemize} \item[$(o)$] $\begin{array}{llll} \zeta^{\mathfrak{m}}(\overline{r}) & = & (2^{1-r}-1) &\zeta^{\mathfrak{m}}(r).\\ \zeta^{\mathfrak{m}}(2n) & = & \frac{\mid B_{n}\mid 2^{3n-1}3^{n}}{(2n)!} &\zeta^{\mathfrak{m}}(2)^{n}. \end{array}$ \item[$(i)$] $\zeta^{\star,\mathfrak{m}}(\boldsymbol{2}^{n})= -2 \zeta^{\mathfrak{m}}(\overline{2n}) =\frac{(2^{2n}-2)6^{n}}{(2n)!}\vert B_{2n}\vert\zeta^{\mathfrak{m}}(2)^{n}.$ \item[$(ii)$] $\zeta^{\star,\mathfrak{m}}_{1}(\boldsymbol{2}^{n})= -2 \sum_{r=1}^{n} \zeta^{\mathfrak{m}}(2r+1)\zeta^{\star,\mathfrak{m}}(\boldsymbol{2}^{n-r}).$ \item[$(iii)$] \begin{align} \zeta^{\star\star,\mathfrak{m}}(\boldsymbol{2}^{n}) & = \sum_{d \leq n} \sum_{w(\textbf{m})=2n \atop ht(\textbf{m})=d(\textbf{m})=d} 2^{2n-2d}\zeta^{\mathfrak{m}}(\textbf{m}) \\ & =\sum_{2n=\sum s_{k}(2i_{k}+1)+2S \atop i_{k}\neq i_{j}} \left( \prod_{k=1}^{p} \frac{C_{i_{k}}^{s_{k}}} {s_{k}!} \zeta^{\mathfrak{m}}(\overline{2i_{k}+1})^{s_{k}} \right) D_{S} \zeta^{\mathfrak{m}}(2)^{S}. \nonumber\\ \zeta^{\star\star,\mathfrak{m}}_{1}(\boldsymbol{2}^{n}) & =-\sum_{d \leq n} \sum_{w(\textbf{m})=2n+1 \atop ht(\textbf{m})=d(\textbf{m})=d} 2^{2n+1-2d}\zeta^{\mathfrak{m}}(\textbf{m}) \\ &=\sum_{2n+1=\sum s_{k}(2i_{k}+1)+2S \atop i_{k}\neq i_{j}} \left( \prod_{k=1}^{p} \frac{C_{i_{k}}^{s_{k}}} {s_{k}!} \zeta^{\mathfrak{m}}(\overline{2i_{k}+1})^{s_{k}}\right) D_{S} \zeta^{\mathfrak{m}}(2)^{S}\nonumber \end{align}
\item[$(iv)$] $\zeta^{\star,\mathfrak{m}}(\boldsymbol{2}^{a},3,\boldsymbol{2}^{b})= \sum A^{a,b}_{r} \zeta^{\mathfrak{m}}(\overline{2r+1})\zeta^{\star,\mathfrak{m}}(\boldsymbol{2}^{n-r}).$ \item[$(v)$] \begin{align} \zeta^{\star\star,\mathfrak{m}}(\boldsymbol{2}^{a},3,\boldsymbol{2}^{b}) &= \sum_{w=\sum s_{k}(2i_{k}+1)+2S \atop i_{k}\neq i_{j}} B^{a,b}_{i_{1},\cdots, i_{p}\atop s_{1}\cdots s_{p}} \left( \prod_{k=1}^{p} \frac{C_{i_{k}}^{s_{k}}} {s_{k}!} \zeta^{\mathfrak{m}}(\overline{2i_{k}+1})^{s_{k}}\right) D_{S} \zeta^{\mathfrak{m}}(2)^{S}.\\ \zeta^{\star\star,\mathfrak{m}}_{1}(\boldsymbol{2}^{a},3,\boldsymbol{2}^{b}) &=D^{a,b} \zeta^{\mathfrak{m}}(2)^{\frac{w}{2}}+ \sum_{w=\sum s_{k}(2i_{k}+1)+2S \atop i_{k}\neq i_{j}} B^{a,b}_{i_{1},\cdots, i_{p}\atop s_{1}\cdots s_{p}} \left( \prod_{k=1}^{p} \frac{C_{i_{k}}^{s_{k}}} {s_{k}!} \zeta^{\mathfrak{m}}(\overline{2i_{k}+1})^{s_{k}}\right) D_{S}\zeta^{\mathfrak{m}}(2)^{S}. \end{align} \end{itemize} Where: \begin{itemize}
\item[$\cdot$] $C_{r}=\frac{2^{2r+1}}{2r+1}$, $D_{S}$ explicit\footnote{Cf. Proof.} and with the following constraint: \begin{equation} \label{eq:constrainta} A^{a,b}_{r}=A_{r}^{a,r-a-1}+C_{r} \left( B^{r-b-1,b}- B^{r-a-1,a} +\delta_{r\leq b}-\delta_{r\leq a} \right). \end{equation} \item[$\cdot$] The recursive formula for $B$-coefficients, where $B^{x,y}\mathrel{\mathop:}=B^{x,y}_{x+y+1 \atop 1}$ and $r<a+b+1$:
\begin{equation} \label{eq:constraintb} \begin{array}{lll }
B^{a,b}_{r \atop 1} & = & \delta_{r\leq b} - \delta_{r< a}+ B^{r-b-1,b}+\frac{D^{a-r-1,b}}{a+b-r+1}+\delta_{r=a} \frac{2(2^{2b+1}-1)6^{b+1} \mid B_{2b+2} \mid}{(2b+2)! D_{b+1}}.\\
B^{a,b}_{i_{1},\cdots, i_{p}\atop s_{1}\cdots s_{p}} &=& \left\{ \begin{array}{l}
\delta_{i_{1}\leq b } - \delta_{i_{1}< a } + B^{i_{1}-b-1,b} + B^{a-i_{1}-1,b}_{i_{1}, \ldots, i_{p}\atop s_{1}-1, \ldots, s_{p}} \quad \text{ for } \sum s_{k} \text{ odd } \\
\delta_{i_{1}\leq b } - \delta_{i_{1}\leq a } + B^{i_{1}-b-1,b} +B^{a-i_{1},b}_{i_{1}, \ldots, i_{p}\atop s_{1}-1, \ldots, s_{p}} \quad \text{ else }.
\end{array}
\right. \end{array}
\end{equation} \end{itemize}
\end{lemm} \noindent Before giving the proof, here is the (analytic) conjecture remaining on some of these coefficients, sufficient to complete the Hoffman $\star$ basis proof (cf. Theorem $\ref{Hoffstar}$): \begin{conj}\label{conjcoeff} The equalities $(v)$ are satisfied for real MZV, with: $$B^{a,b}=1-\frac{2}{C_{a+b+1}}\binom{2a+2b+2}{2b+1}.$$ \end{conj} \textsc{Remarks:} \begin{itemize} \item[$\cdot$] This conjecture is of an entirely different nature from the techniques developed in this thesis. We can expect that it can proved using analytic methods as the usual techniques of identifying hypergeometric series, as in $\cite{Za}$, or $\cite{Li}$. \item[$\cdot$] The equality $(iv)$ is already proven in the analytic case by Ohno-Zagier (cf.$\cite{IKOO}$, $\cite{Za}$), with the values of the coefficient $A_{r}^{a,b}$ given below. Nevertheless, as we will see through the proofs below, to make the coefficients for the (stronger) motivic identity $(iv)$ explicit, we need to prove the other identities in $(v)$. \item[$\cdot$] We will use below a result of Ohno and Zagier on sums of MZV of fixed weight, depth and height to conclude for the coefficients for $(iii)$. \end{itemize}
\begin{theo} If the analytic conjecture ($\ref{conjcoeff}$) holds, the equalities $(iv)$, $(v)$ are true in the motivic case, with the same values of the coefficients. In particular: $$A_{r}^{a,b}= 2\left( -\delta_{r=a}+ \binom{2r}{2a} \right) \frac{2^{2r}}{2^{2r}-1}-2\binom{2r}{2b+1}.$$ \end{theo} \begin{proof} Remind that if we know a motivic equality up to one unknown coefficient (of $\zeta(weight)$), the analytic result analogue enables us to conclude on its value by Corollary $\ref{kerdn}$.\\ Let assume now, in a recursion on $n$, that we know $\lbrace B^{a,b}, D^{a,b}, B_{i_{1} \cdots i_{p} \atop s_{1} \cdots s_{p} }^{a,b} \rbrace_{a+b+1<n}$ and consider $(a,b)$ such that $a+b+1=n$. Then, by $(\ref{eq:constraintb})$, we are able to compute the $B_{\textbf{i}\atop \textbf{s}}^{a,b}$ with $(s,i)\neq (1,n)$. Using the analytic $(v)$ equality, and Corollary $\ref{kerdn}$, we deduce the only remaining unknown coefficient $B^{a,b}$ resp. $D^{a,b}$ in $(v)$.\\ Lastly, by recursion on $n$ we deduce the $A_{r}^{a,b}$ coefficients: let assume they are known for $a+b+1<n$, and take $(a,b)$ with $a+b+1=n$. By the constraint $(\ref{eq:constrainta})$, since we already know $B$ and $C$ coefficients, we deduce $A_{r}^{a,b}$ for $r<n$. The remaining coefficient, $A_{n}^{a,b}$, is obtained using the analytic $(iv)$ equality and Corollary $\ref{kerdn}$. \end{proof}
\paragraph{\texttt{Proof of} Lemma $\ref{lemmcoeff}$.}: \begin{proof} Computing the coaction on these elements, by a recursive procedure, we are able to prove these identities up to some rational coefficients, with the Corollary $\ref{kerdn}$. When the analytic analogue of the equality is known for MZV, we \textit{may} conclude on the value of the remaining rational coefficient of $\zeta^{\mathfrak{m}}(w)$ by identification (as for $(i),(ii),(iii)$). However, if the family is not stable under the coaction , (as for $(iv)$) knowing the analytic case is not enough.\\ \texttt{Nota Bene:} This proof refers to the expression of $D_{2r+1}$ in Lemma $\ref{lemmt}$: we look at cuts of length $2r+1$ among the sequence of $0, 1, $ or $\star$ (in the iterated integral writing); there are different kind of cuts (according their extremities), and each cut may bring out two terms ($T_{0,0}$ and $T_{0,\star}$ for instance). The simplifications are illustrated by the diagrams, where some arrows (term of a cut) get simplified by rules specified in Annexe $A$.\\ \begin{itemize} \item[$(i)$] The corresponding iterated integral: $$I^{\mathfrak{m}}(0; 1, 0, \star, 0 \cdots, \star, 0; 1).$$ The only possible cuts of odd length are between two $\star$ ($T_{0,\star}$ and $T_{\star,0}$) or $T_{1,0}$ from the first $1$ to a $\star$, or $T_{0,1}$ from a $\star$ to the last $1$. By \textsc{ Shift }(\ref{eq:shift}), these cuts get simplified two by two. Since $D_{2r+1}(\cdot)$, for $2r+1<2n$ are all zero, it belongs to $\mathbb{Q}\zeta^{\mathfrak{m}}(2n)$, by Corollary $\ref{kerdn}$). Using the (known) analytic equality, we can conclude. \item[$(ii)$] It is quite similar to $(i)$: using $\textsc{ Shift }$ $(\ref{eq:shift})$, it remains only the cut:\\ \includegraphics[]{dep2.pdf}\\ $$\text{i.e.}: \quad D_{2r+1} (\zeta^{\star,\mathfrak{m}}_{1}(\boldsymbol{2}^{n}))= \zeta^{\mathfrak{l},\star}_{1}(\boldsymbol{2}^{r})\otimes \zeta^{\star,\mathfrak{m}}(\boldsymbol{2}^{n-r})=-2 \zeta^{ \mathfrak{l}}(\overline{2r+1})\otimes \zeta^{ \star,\mathfrak{m}}(\boldsymbol{2}^{n-r}).$$ The last equality is deduced from the recursive hypothesis (smaller weight). The analytic equality (coming from the Zagier-Ohno formula, and the $\shuffle$ regulation) enables us to conclude on the value of the remaining coefficient of $\zeta^{\mathfrak{m}}(2n+1)$. \item[$(iii)$] Expressing these ES$\star\star$ as a linear combination of ES by $\shuffle$ regularisation: $$\hspace*{-0.5cm}\zeta^{\star\star,\mathfrak{m}}(\boldsymbol{2}^{n})= \sum_{k_{i} \text{ even}} \zeta^{\mathfrak{m}}_{2n-\sum k_{i}}(k_{1},\cdots, k_{p})=\sum_{n_{i}\geq 2} \left( \sum_{k_{i} \text{ even} \atop k_{i} \leq n_{i}} \binom{n_{1}-1}{k_{1}-1} \cdots \binom{n_{d}-1}{k_{d}-1} \right) \zeta^{\mathfrak{m}}(n_{1},\cdots, n_{d}) .$$ Using the multi-binomial formula: $$2^{\sum m_{i}}=\sum_{l_{i} \leq m_{i}} \binom{m_{1}}{l_{1}}(1-(-1))^{l_{1}} \cdots \binom{m_{d}}{l_{d}}(1-(-1))^{l_{d}}= 2^{d} \sum_{l_{i} \leq m_{i}\atop l_{i} \text{ odd }} \binom{m_{1}}{l_{1}} \cdots \binom{m_{d}}{l_{d}} .$$ Thus: $$\zeta^{\star\star,\mathfrak{m}}(\boldsymbol{2}^{n})=\sum_{d \leq n} \sum_{w(\textbf{m})=2n \atop ht(\textbf{m})=d(\textbf{m})=d} 2^{2n-2d}\zeta^{\mathfrak{m}}(\textbf{m}).$$ Similarly for $(4.27)$, since: $$\zeta^{\star\star,\mathfrak{m}}_{1}(\boldsymbol{2}^{n})=\sum_{k_{i} \text{ even}} \zeta^{\mathfrak{m}}_{2n+1-\sum k_{i}}(k_{1},\cdots, k_{p})=\sum_{d \leq n} \sum_{w(\textbf{m})=2n \atop ht(\textbf{m})=d(\textbf{m})=d} 2^{2n-2d}\zeta^{\mathfrak{m}}(\textbf{m}).$$ Now, using still only $\textsc{ Shift }$ $(\ref{eq:shift})$, it remains the following cuts:\\ \includegraphics[]{dep3.pdf}\\ With a recursion on $n$ for both $(4.26)$, $(4.27)$, we deduce: $$D_{2r+1}(\zeta^{\star\star,\mathfrak{m}}(\boldsymbol{2}^{n}))=\zeta^{\star\star,\mathfrak{m}}_{1}(\boldsymbol{2}^{r})\otimes \zeta^{\star\star,\mathfrak{l}}_{1}(\boldsymbol{2}^{n-r})=C_{r} \zeta^{ \mathfrak{l}}(\overline{2r+1})\otimes \zeta^{\star\star,\mathfrak{m}}_{1}(\boldsymbol{2}^{n-r-1}).$$ $$D_{2r+1}(\zeta^{\star\star,\mathfrak{m}}_{1}(\boldsymbol{2}^{n}))=\zeta^{\star\star,\mathfrak{l}}_{1}(\boldsymbol{2}^{r})\otimes \zeta^{\star\star,\mathfrak{m}}(\boldsymbol{2}^{n-r})=C_{r} \zeta^{ \mathfrak{l}}(\overline{2r+1})\otimes \zeta^{\star\star,\mathfrak{m}}(\boldsymbol{2}^{n-r}).$$ To find the remaining coefficients, we need the analytic result corresponding, which is a consequence of the sum relation for MZV of fixed weight, depth and height, by Ohno and Zagier ($\cite{OZa}$, Theorem $1$), via the hypergeometric functions.\\ Using $\cite{OZa}$, the generating series of these sums is, with $\alpha,\beta=\frac{x+y \pm \sqrt{(x+y)^{2}-4z}}{2}$: $$\begin{array}{lll} \phi_{0}(x,y,z)\mathrel{\mathop:} & = & \sum_{s\leq d \atop w\geq d+s} \left( \sum \zeta(\textbf{k}) \right) x^{w-d-s}y^{d-s}z^{s-1} \\ & = & \frac{1}{xy-z} \left( 1- \exp \left( \sum_{m=2}^{\infty} \frac{\zeta(m)}{m}(x^{m}+y^{m}-\alpha^{m}-\beta^{m}) \right) \right) . \end{array}$$ From this, let express the generating series of both $\zeta^{\star\star}(\boldsymbol{2}^{n})$ and $\zeta^{\star\star}_{1}(\boldsymbol{2}^{n})$: $$\phi(x)\mathrel{\mathop:}= \sum_{w} \left( \sum_{ht(\textbf{k})=d(\textbf{k})=d\atop w\geq 2d} 2^{w-2d} \zeta(\textbf{k}) \right) x^{w-2}= \phi_{0}(2x, 0, x^{2}).$$ Using the result of Ohno and Don Zagier: $$\phi(x)= \frac{1}{x^{2}} \left(\exp \left( \sum_{m=2}^{\infty} \frac{2^{m}-2}{m} \zeta(m) x^{m} \right) -1\right).$$ Consequently, both $\zeta^{\star\star}(\boldsymbol{2}^{n})$ and $\zeta^{\star\star}_{1}(\boldsymbol{2}^{n})$ can be written explicitly as polynomials in simple zetas. For $\zeta^{\star\star}(\boldsymbol{2}^{n})$, by taking the coefficient of $x^{2n-2}$ in $\phi(x)$: $$\zeta^{\star\star}(\boldsymbol{2}^{n})= \sum_{\sum m_{i} s_{i}=2n \atop m_{i}\neq m_{j}} \prod_{i=1}^{k} \left( \frac{1}{s_{i} !}\left( \zeta(m_{i}) \frac{2^{m_{i}}-2}{m_{i}}\right)^{s_{i}} \right) .$$ Gathering the zetas at even arguments, it turns into: $$\zeta^{\star\star}(\boldsymbol{2}^{n})= \sum_{\sum (2i_{k}+1) s_{k}+2S=2n \atop i_{k}\neq i_{j}} \prod_{i=1}^{p} \left( \frac{1}{s_{k} !}\left( \zeta(2 i_{k}+1) \frac{2^{2 i_{k}+1}-2}{2i_{k}+1}\right)^{s_{k}} \right) d_{S} \zeta(2)^{S}, $$ \begin{equation}\label{eq:coeffds}
\text{ where } d_{S}\mathrel{\mathop:}=3^{S}\cdot 2^{3S}\sum_{\sum m_{i} s_{i}=S \atop m_{i}\neq m_{j}} \prod_{i=1}^{k} \left( \frac{1}{s_{i}!} \left( \frac{\mid B_{2m_{i}}\mid (2^{2m_{i}-1}-1) } {2m_{i} (2m_{i})!}\right)^{s_{i}} \right). \end{equation} It remains to turn $\zeta(odd)$ into $\zeta(\overline{odd})$ by $(o)$ to fit the expression of the Lemma: $$\zeta^{\star\star}(\boldsymbol{2}^{n})= \sum_{\sum (2i_{k}+1) s_{k}+2S=2n \atop i_{k}\neq i_{j}} \prod_{i=1}^{p} \left( \frac{1}{s_{k} !}\left( c_{i_{k}}\zeta(\overline{2 i_{k}+1}) \right)^{s_{k}} \right) d_{S} \zeta(2)^{S}, \text{ where } c_{r}=\frac{2^{2r+1}}{2r+1}.$$ It is completely similar for $\zeta^{\star\star}_{1}(\boldsymbol{2}^{n})$: by taking the coefficient of $x^{2n-3}$ in $\phi(x)$, we obtained the analytic analogue of $(4.25)$, with the same coefficients $d_{S}$ and $c_{r}$.\\ Now, using these analytic results for $(4.26)$, $(4.27)$, by recursion on the weight, we can identify the coefficient $D_{S}$ and $C_{r}$ with resp. $d_{S}$ and $c_{r}$, since there is one unknown coefficient at each step of the recursion. \item[$(iv)$] After some simplifications by Antipodes rules ($\S A.1$), only the following cuts remain:\\ \includegraphics[]{dep4.pdf}\\ This leads to the formula:\\ $$D_{2r+1} (\zeta^{\star,\mathfrak{m}}(\boldsymbol{2}^{a},3,\boldsymbol{2}^{b}))= \left(\zeta^{\star,\mathfrak{m}}(\boldsymbol{2}^{a},3,\boldsymbol{2}^{r-a-1})+\right.$$ $$ \left.\left( \delta_{r \leq b}-\delta_{r \leq a}\right) \zeta^{\star\star,\mathfrak{m}}_{1}(\boldsymbol{2}^{r}) + \zeta^{\star\star , \mathfrak{m}}(\boldsymbol{2}^{r-b-1},3,\boldsymbol{2}^{b}) -\zeta^{\star\star ,\mathfrak{m}}(\boldsymbol{2}^{r-a-1},3,\boldsymbol{2}^{a})\right) \otimes \zeta^{\star,\mathfrak{m}}(\boldsymbol{2}^{n-r}).$$ In particular, the Hoffman $\star$ family is not stable under the coaction, so we need first to prove $(v)$, and then: $$\hspace*{-0.7cm}D_{2r+1} (\zeta^{\star ,\mathfrak{m}}(\boldsymbol{2}^{a},3,\boldsymbol{2}^{b}))= \left( A_{r}^{a,r-a-1}+C_{r} \left( B^{r-b-1,b}- B^{r-a-1,a} +\delta_{r\leq b}-\delta_{r\leq a} \right)\right) \zeta^{ \mathfrak{l}}(\overline{2r+1})\otimes \zeta^{\star ,\mathfrak{m}}(\boldsymbol{2}^{n-r}). $$ It leads to the constraint $(\ref{eq:constrainta})$ above for coefficients $A$. To make these coefficients explicit, apart from the known analytic Ohno Zagier formula, we need the analytic analogue of $(v)$ identities, as stated in Conjecture $\ref{conjcoeff}$. \item[$(v)$] By Annexe rules, the following cuts get simplified (by colors, above with below):\footnote{The vertical arrows indicates a cut from the $\star$ to a $\star$ of the same group.}\\ \includegraphics[]{dep5.pdf}\\ Indeed, cyan arrows get simplified by \textsc{Antipode} $\shuffle$, $T_{0,0}$ resp. $T_{0, \star}$ above with $T_{0,0}$ resp. $T_{\star,0}$ below; magenta ones by $\textsc{ Shift }$ $(\ref{eq:shift})$, term above with the term below shifted by two on the left. It remains the following cuts for $(4.28)$:\\ \includegraphics[]{dep6.pdf}\\ In a very similar way, the simplifications lead to the following remaining terms:\\ \includegraphics[]{dep7.pdf}\\ Then, the derivations reduce to: $$\hspace*{-0.7cm}D_{2r+1} (\zeta^{\star\star ,\mathfrak{m}}(\boldsymbol{2}^{a},3,\boldsymbol{2}^{b}))= \left( \left( \delta_{r\leq b}-\delta_{r \leq a}\right) \zeta^{\star\star ,\mathfrak{l}}_{1}(\boldsymbol{2}^{r}) +\delta_{r> b}\zeta^{\star\star, \mathfrak{m-l}}(\boldsymbol{2}^{r-b-1},3,\boldsymbol{2}^{b})\right) \otimes \zeta^{\star\star ,\mathfrak{m}}(\boldsymbol{2}^{n-r}) +$$ $$\hspace*{+1cm} +\delta_{r\leq a-1} \zeta^{\star\star ,\mathfrak{l}}_{1}(\boldsymbol{2}^{r}) \otimes \zeta^{\star\star ,\mathfrak{m}}_{1}(\boldsymbol{2}^{a-r-1},3,\boldsymbol{2}^{b})+ \delta_{r=a} \zeta^{\star\star ,\mathfrak{l}}_{1}(\boldsymbol{2}^{a})\otimes \zeta^{\star\star ,\mathfrak{m}}_{2}(\boldsymbol{2}^{b}) .$$ $$\hspace*{-1.4cm}D_{2r+1} (\zeta^{\star\star ,\mathfrak{m}}_{1}(\boldsymbol{2}^{a},3,\boldsymbol{2}^{b}))= \left( \left( \delta_{r\leq b}-\delta_{r \leq a}\right) \zeta^{\star\star ,\mathfrak{l}}_{1}(\boldsymbol{2}^{r}) + \zeta^{\star\star ,\mathfrak{l}}(\boldsymbol{2}^{r-b-1},3,\boldsymbol{2}^{b})\right)\otimes \zeta^{\star\star,\mathfrak{m}}_{1}(\boldsymbol{2}^{n-r}) +$$ $$ + \zeta^{\star\star ,\mathfrak{l}}_{1}(\boldsymbol{2}^{r})\otimes \zeta^{\star\star,\mathfrak{m}}(\boldsymbol{2}^{a-r},3,\boldsymbol{2}^{b}) .$$ \hspace*{-0.5cm}With a recursion on $w$ for both: $$\hspace*{-1.4cm}\begin{array}{ll} D_{2r+1} (\zeta^{\star\star ,\mathfrak{m}}(\boldsymbol{2}^{a},3,\boldsymbol{2}^{b})) & = C_{r} \zeta^{ \mathfrak{l}}(\overline{2r+1})\otimes\\ & \left( \left( \delta_{r\leq b}-\delta_{r < a} + B_{r}^{r-b-1,b}\right) \zeta^{\star\star ,\mathfrak{m}}(\boldsymbol{2}^{n-r}) + \zeta^{\star\star ,\mathfrak{m}}_{1}(\boldsymbol{2^{a-r-1},}3,\boldsymbol{2}^{b})+ \delta_{r=a}\zeta^{\star ,\mathfrak{m}}(\boldsymbol{2}^{b+1}) \right) .\\ & \\ D_{2r+1} (\zeta^{\star\star ,\mathfrak{m}}_{1}(\boldsymbol{2}^{a},3,\boldsymbol{2}^{b}))& = C_{r} \zeta^{ \mathfrak{l}}(\overline{2r+1})\otimes\left( \left( \delta_{r\leq b}-\delta_{r \leq a} + B_{r}^{r-b-1,b}\right) \zeta^{ \star\star ,\mathfrak{m}}_{1}(\boldsymbol{2}^{n-r}) + \zeta^{\star\star ,\mathfrak{m}}(\boldsymbol{2}^{a-r},3,\boldsymbol{2}^{b}) \right). \end{array}$$ This leads to the recursive formula $(\ref{eq:constraintb})$ for $B$. \end{itemize} \end{proof}
\section{Motivic generalized Linebarger Zhao Conjecture}
We conjecture the following motivic identities, which express each motivic MZV $\star$ as a motivic Euler $\sharp$ sum: \begin{conj}\label{lzg} For $a_{i},c_{i} \in \mathbb{N}^{\ast}$, $c_{i}\neq 2$, $$\zeta^{\star, \mathfrak{m}} \left( \boldsymbol{2}^{a_{0}},c_{1},\cdots,c_{p}, \boldsymbol{2}^{a_{p}}\right) =(-1)^{1+\delta_{c_{1}}}\zeta^{\sharp, \mathfrak{m}} \left(B_{0},\boldsymbol{1}^{c_{1}-3 },\cdots,\boldsymbol{1}^{ c_{i}-3 },B_{i}, \ldots, B_{p}\right), $$ where $\left\lbrace \begin{array}{l} B_{0}\mathrel{\mathop:}= \pm (2a_{0}+1-\delta_{c_{1}})\\ B_{i}\mathrel{\mathop:}= \pm(2a_{i}+3-\delta_{c_{i}}-\delta_{c_{i+1}})\\ B_{p}\mathrel{\mathop:}=\pm ( 2 a_{p}+2-\delta_{c_{p}}) \end{array}\right.$, with $\pm\mathrel{\mathop:}=\left\lbrace \begin{array}{l} - \text{ if } \mid B_{i}\mid \text{ even} \\ + \text{ if } \mid B_{i}\mid \text{ odd} \end{array} \right.$, $\begin{array}{l} \delta_{c}\mathrel{\mathop:}=\delta_{c=1},\\ \text{the Kronecker symbol}. \end{array}$ and $\boldsymbol{1}^{n}:=\boldsymbol{1}^{min(0,n)}$ is a sequence of $n$ 1 if $n\in\mathbb{N}$, an empty sequence else. \end{conj} \textsc{Remarks}: \begin{itemize} \item[$\cdot$] Motivic Euler $\sharp$ sums appearing on the right side have already been proven to be unramified in $\S 4.3$, i.e. MMZV. \item[$\cdot$] This conjecture implies that the motivic Hoffman $\star$ family is a basis, since it corresponds here to the motivic Euler $\sharp$ sum family proved to be a basis in Theorem $\ref{ESsharpbasis}$: cf. ($\ref{eq:LZhoffman}$). \item[$\cdot$] The number of sequences of consecutive $1$ in $\zeta^{\star}$, $n_{1}$ is linked with the number of even in $\zeta^{\sharp}$, $n_{e}$, here by the following formula:\\ $$n_{e}=1+2n_{1}-2\delta_{c_{p}} -\delta_{c_{1}}.$$ In particular, when there is no $1$ in the MMZV $\star$, there is only one even (at the end) in the Euler sum $\sharp$. There are always at least one even in the Euler sums. \end{itemize}
Special cases of this conjecture, which are already proven for real Euler sums (references indicated in the braket), but remain conjectures in the motivic case: \begin{description} \item[Two-One] [Ohno Zudilin, $\cite{OZ}$.] \begin{equation}\label{eq:OZ21} \zeta^{\star, \mathfrak{m}} (\boldsymbol{2}^{a_{0}},1,\cdots,1, \boldsymbol{2}^{a_{p}})= - \zeta^{\sharp, \mathfrak{m}} \left( \overline{2a_{0}}, 2a_{1}+1, \ldots, 2a_{p-1}+1, 2 a_{p}+1\right) . \end{equation} \item [Three-One] [Broadhurst et alii, $\cite{BBB}$.] \footnote{The Three-One formula was conjectured for real Euler sums by Zagier, proved by Broadhurst et alii in $\cite{BBB}$.} \begin{equation}\label{eq:Z31} \zeta^{\star, \mathfrak{m}} (\boldsymbol{2}^{a_{0}},1,\boldsymbol{2}^{a_{1}},3 \cdots,1, \boldsymbol{2}^{a_{p-1}}, 3, \boldsymbol{2}^{a_{p}}) = -\zeta^{\sharp, \mathfrak{m}} \left( \overline{2a_{0}}, \overline{2a_{1}+2}, \ldots, \overline{2a_{p-1}+2}, \overline{2 a_{p}+2} \right) . \end{equation} \item[Linebarger-Zhao$\star$] [Linebarger Zhao, $\cite{LZ}$] With $c_{i}\geq 3$: \begin{equation}\label{eq:LZ} \zeta^{\star, \mathfrak{m}} \left( \boldsymbol{2}^{a_{0}},c_{1},\cdots,c_{p}, \boldsymbol{2}^{a_{p}}\right) = -\zeta^{\sharp, \mathfrak{m}} \left( 2a_{0}+1,\boldsymbol{1}^{ c_{1}-3 },\cdots,\boldsymbol{1}^{ c_{i}-3 },2a_{i}+3, \ldots, \overline{ 2 a_{p}+2} \right) \end{equation} In particular, when all $c_{i}=3$: \begin{equation}\label{eq:LZhoffman} \zeta^{\star, \mathfrak{m}} \left( \boldsymbol{2}^{a_{0}},3,\cdots,3, \boldsymbol{2}^{a_{p}}\right) = - \zeta^{\sharp, \mathfrak{m}} \left( 2a_{0}+1, 2a_{1}+3, \ldots, 2a_{p-1}+3, \overline{2 a_{p}+2}\right) . \end{equation} \end{description} \texttt{Examples}: Particular identities implied by the previous conjecture, sometimes known for MZV and which could then be proven for motivic Euler sums directly with the coaction: \begin{itemize} \item[$\cdot$] $ \zeta^{\star, \mathfrak{m}}(1, \left\lbrace 2 \right\rbrace^{n} )=2 \zeta^{ \mathfrak{m}}(2n+1).$ \item[$\cdot$] $ \zeta^{\star, \mathfrak{m}}(1, \left\lbrace 2 \right\rbrace^{a}, 1, \left\lbrace 2 \right\rbrace^{b} )= \zeta^{ \sharp\mathfrak{m} }(2a+1,2b+1)= 4 \zeta^{ \mathfrak{m} }(2a+1,2b+1)+ 2 \zeta^{ \mathfrak{m} }(2a+2b+2). $ \item[$\cdot$] $ \zeta^{ \mathfrak{m}} (n)= - \zeta^{\sharp, \mathfrak{m}} (\lbrace 1\rbrace^{n-2}, -2)= -\sum_{ w(\boldsymbol{k})=n \atop \boldsymbol{k} \text{admissible}} \boldsymbol{2}^{p} \zeta^{\mathfrak{m}}(k_{1}, \ldots, k_{p-1}, -k_{p}).$ \item[$\cdot$] $ \zeta^{ \star, \mathfrak{m}} (\lbrace 2 \rbrace ^{n})= \sum_{\boldsymbol{k} \in \lbrace \text{ even }\rbrace^{\times} \atop w(\boldsymbol{k})= 2n} \zeta^{\mathfrak{m}} (\boldsymbol{k})=- 2 \zeta^{\mathfrak{m}} (-2n) .$ \end{itemize} We paved the way for the proof of Conjecture $\ref{lzg}$, bringing it back to an identity in $\mathcal{L}$: \begin{theo} Let assume: \begin{itemize} \item[$(i)$] The analytic version of $\ref{lzg}$ is true. \item[$(ii)$] In the coalgebra $\mathcal{L}$, i.e. modulo products, for odd weights: \begin{equation}\label{eq:conjid}
\zeta^{\sharp, \mathfrak{l}} _{B_{0}-1}(\boldsymbol{1}^{ \gamma_{1}},\cdots, \boldsymbol{1}^{\gamma_{p} },B_{p})\equiv
\zeta^{\star\star, \mathfrak{l}}_{2} (\boldsymbol{2}^{a_{0}-1},c_{1},\cdots,\boldsymbol{2}^{a_{p}})-\zeta^{\star\star, \mathfrak{l}}_{1} (\boldsymbol{2}^{a_{0}}, c_{1}-1, \ldots, \boldsymbol{2}^{a_{p}}) , \end{equation} \begin{flushright} with $c_{1}\geq 3$, $a_{0}>0$, $\gamma_{i}=c_{i}-3 + 2\delta_{c_{i}}$ and $\left\lbrace \begin{array}{l} B_{0}= 2a_{0}+1-\delta_{c_{1}}\\ B_{i}=2a_{i}+3-\delta_{c_{i}}-\delta_{c_{i+1}}\\ B_{p}=2a_{p}+3-\delta_{c_{p}} \end{array} \right. $. \end{flushright} \end{itemize} Then: \begin{enumerate}[I.] \item Conjecture $\ref{lzg}$ is true, for motivic Euler sums. \item In the coalgebra $\mathcal{L}$, for odd weights, with $c_{1}\geq 3$ and the previous notations: \begin{equation}\label{eq:toolid} \zeta^{\sharp, \mathfrak{l}} (\boldsymbol{1}^{ \gamma_{1}},\cdots, \boldsymbol{1}^{ \gamma_{p} },B_{p})\equiv - \zeta^{\star, \mathfrak{l}}_{1} (c_{1}-1, \boldsymbol{2}^{a_{1}},c_{2},\cdots,c_{p}, \boldsymbol{2}^{a_{p}}). \end{equation} \end{enumerate} \end{theo} \texttt{ADDENDUM:} The hypothesis $(i)$ is proved: J. Zhao deduced it from its Theorem 1.4 in $\cite{Zh3}$.\\ \\ \textsc{Remark:} The $(ii)$ hypothesis should be proven either directly via the various relations in $\mathcal{L}$ proven in $\S 4.2$ (as for $\ref{eq:toolid}$), or using the coaction, which would require the analytic identity corresponding. Beware, $(ii)$ would only be true in $\mathcal{L}^{2}$, not in $\mathcal{H}^{2}$.
\begin{proof} To prove this equality $1.$ at a motivic level by recursion, we would need to proof that the coaction is equal on both side, and use the conjecture analytic version of the same equality. We prove $I$ and $II$ successively, in a same recursion on the weight: \begin{enumerate}[I.] \item Using the formulas of the coactions $D_{r}$ for these families (Lemma $A.1.2$ and $A.1.4$), we can gather terms in both sides according to the right side, which leads to three types: $$ \begin{array}{llll} (a) & \zeta^{\star, \mathfrak{m}} (\cdots,\boldsymbol{2}^{a_{i}}, \alpha, \boldsymbol{2}^{\beta}, c_{j+1}, \cdots) & \longleftrightarrow & \zeta^{\sharp ,\mathfrak{m}}(B_{0} \cdots, B_{i}, \textcolor{magenta}{1^{\gamma}, B}, 1^{\gamma_{j+1}}, \ldots, B_{p}) \\ (b) & \zeta^{\star, \mathfrak{m}} (\cdots,\boldsymbol{2}^{a_{i-1}}, c_{i}, \boldsymbol{2}^{\beta}, c_{j+1}, \cdots) & \longleftrightarrow & \zeta^{\sharp ,\mathfrak{m}}(B_{0} \cdots, B_{i-1}, 1^{\gamma_{i}}, \textcolor{green}{B}, 1^{\gamma_{j+1}}, \ldots, B_{p}) \\ (c) & \zeta^{\star, \mathfrak{m}} (\cdots, c_{i}, \boldsymbol{2}^{\beta}, \alpha, \boldsymbol{2}^{a_{j}}, \cdots) & \longleftrightarrow & \zeta^{\sharp, \mathfrak{m}}(B_{0} \cdots, 1^{\gamma_{i+1}},\textcolor{cyan}{ B, 1^{\gamma}}, B_{j+1}, \ldots, B_{p}) \end{array},$$ with $ \gamma=\alpha-3$ and $B=2\beta +3-\delta_{c_{j+1}}$, or $B=2\beta+3 - \delta_{c_{i}}- \delta_{c_{j+1}}$ for $(b)$.\\ The third case, antisymmetric of the first case, may be omitted below. By recursive hypothesis, these right sides are equal and it remains to compare the left sides associated: \begin{enumerate} \item On the one hand, by lemma $A.1.2$, the left side corresponding: $$ \delta_{3\leq \alpha \leq c_{i+1}-1 \atop 0\leq \beta a_{j}} \zeta^{\star, \mathfrak{l}}_{c_{i+1}-\alpha}- (\boldsymbol{2}^{ a_{j}-\beta}, \ldots, \boldsymbol{2}^{a_{i+1}}).$$ On the other hand (Lemma $A.1.4$), the left side is: $$-\delta_{2 \leq B \leq B_{j} \atop 0\leq\gamma\leq\gamma_{i+1}-1}\zeta^{\sharp,\mathfrak{l}}(B_{j}-B+1, 1^{\gamma_{j}}, \ldots, 1^{\gamma_{i+1}-\gamma-1}).$$ They are both equal, by $\ref{eq:toolid}$, where $c_{i+1}-\alpha+2$ corresponds to $c_{1} $ and is greater than $3$.\\ \item By lemma $A.1.2$, the left side corresponding for $\zeta^{\star}$: $$\hspace*{-1.2cm}\begin{array}{llll} -& \delta_{c_{i}>3} \zeta^{\star\star, \mathfrak{l}}_{2} (\boldsymbol{2}^{a_{i}}, \ldots, \boldsymbol{2}^{ a_{j}-\beta-1}) & + & \delta_{c_{j}>3} \zeta^{\star\star, \mathfrak{l}}_{2} (\boldsymbol{2}^{a_{j}}, \ldots, \boldsymbol{2}^{ a_{i}-\beta-1}) \\
- & \delta_{c_{i}=1} \zeta^{\star\star, \mathfrak{l}} (\boldsymbol{2}^{a_{j}-\beta}, \ldots, \boldsymbol{2}^{ a_{i}}) & +& \delta_{c_{j+1}=1} \zeta^{\star\star, \mathfrak{l}} (\boldsymbol{2}^{a_{i}-\beta}, \ldots, \boldsymbol{2}^{ a_{j}})\\
+ & \delta_{c_{i+1}=1 \atop \beta> a_{i}} \zeta^{\star\star, \mathfrak{l}}_{1} (\boldsymbol{2}^{a_{i}+a_{j}-\beta}, \ldots, \boldsymbol{2}^{ a_{i+1}}) & - & \delta_{c_{j}=1 \atop \beta> a_{j}} \zeta^{\star\star, \mathfrak{l}}_{1} (\boldsymbol{2}^{a_{i}+a_{j}-\beta}, \ldots, \boldsymbol{2}^{ a_{j-1}})\\
- & \delta_{a_{j}< \beta \leq a_{i}+ a_{j}+1} \zeta^{\star\star, \mathfrak{l}}_{c_{j}-2} (\boldsymbol{2}^{a_{j-1}}, \ldots, \boldsymbol{2}^{a_{i}+ a_{j}-\beta+1}) & + & \delta_{a_{i}< \beta \leq a_{j}+a_{i}+1} \zeta^{\star\star, \mathfrak{l}}_{c_{i+1}-2} (\boldsymbol{2}^{a_{i+1}}, \ldots, \boldsymbol{2}^{ a_{i}+ a_{j} -\beta+1}) . \end{array}$$ It should correspond to (using still lemma $A.1.4$), with $B_{k}=2a_{k}+3-\delta_{c_{k}}-\delta_{c_{k+1}}$, $\gamma_{k}=c_{k}-3+2\delta_{c_{k}}$ and $B=2\beta+3 - \delta_{c_{i}}- \delta_{c_{j+1}}$: $$\left( \delta_{B_{i}< B}\zeta^{\sharp\sharp,\mathfrak{l}}_{B_{i}+B_{j}-B}(1^{\gamma_{j}}, \ldots, 1^{\gamma_{i+1}}) - \delta_{B_{j}< B}\zeta^{\sharp\sharp,\mathfrak{l}}_{B_{i}+B_{j}-B}(1^{\gamma_{i+1}}, \ldots, 1^{\gamma_{j}}) \right. $$ $$\left. + \zeta^{\sharp\sharp,\mathfrak{l}}_{B_{i}-B}(1^{\gamma_{i+1}}, \ldots, B_{j}) - \zeta^{\sharp\sharp,\mathfrak{l}}_{B_{j}-B}(1^{\gamma_{j}}, \ldots, B_{i})\right) .$$ The first line has even depth, while the second line has odd depth, as noticed in Lemma $A.1.4$. Let distinguish three cases, and assume $a_{i}<a_{j}$:\footnote{The case $a_{j}<a_{i}$ is anti-symmetric, hence analogue.} \begin{itemize}
\item[$(i)$] When $\beta< a_{i}<a_{j}$, we should have: \begin{equation}\label{eq:ci}
\zeta^{\sharp\sharp,\mathfrak{l}}_{B_{i}-B}(1^{\gamma_{i+1}}, \ldots, B_{j}) - \zeta^{\sharp\sharp,\mathfrak{l}}_{B_{j}-B}(1^{\gamma_{j}}, \ldots, B_{i}) \text{ equal to:} \end{equation} $$\begin{array}{llll} - \delta_{c_{i} >3} & \zeta^{\star\star, \mathfrak{l}}_{2} (\boldsymbol{2}^{a_{j}-\beta-1}, \ldots, \boldsymbol{2}^{ a_{i}}) & - \delta_{c_{i}=1} & \zeta^{\star\star, \mathfrak{l}} (\boldsymbol{2}^{a_{j}-\beta}, \ldots, \boldsymbol{2}^{ a_{i}}) \\ +\delta_{c_{j+1}>3} & \zeta^{\star\star, \mathfrak{l}}_{2} (\boldsymbol{2}^{a_{i}-\beta-1}, \ldots, \boldsymbol{2}^{ a_{j}}) & + \delta_{c_{j+1}=1} & \zeta^{\star\star, \mathfrak{l}} (\boldsymbol{2}^{a_{i}-\beta}, \ldots, \boldsymbol{2}^{ a_{j}}) \end{array}$$ \begin{itemize} \item[$\cdot$] Let first look at the case where $c_{i}>3, c_{j+1}>3$. Renumbering the indices, using $\textsc{Shift}$ for odd depth for the second line, it is equivalent to, with $\alpha=\beta +1$, $B_{p}=2a_{p}+3, B_{0}=2a_{0}+3$: $$\begin{array}{llll} &\zeta^{\star\star, \mathfrak{l}}_{2} (\boldsymbol{2}^{a_{0}-\alpha},c_{1},\cdots,c_{p},\boldsymbol{2}^{a_{p}}) & -& \zeta^{\star\star, \mathfrak{l}}_{2} (\boldsymbol{2}^{a_{0}},c_{1},\cdots,c_{p},\boldsymbol{2}^{a_{p}-\alpha}) \\ \equiv & \zeta^{\sharp\sharp, \mathfrak{l}} _{B_{0}-B}(1^{\gamma_{1}},\cdots, 1^{\gamma_{p}},B_{p}) & - & \zeta^{\sharp\sharp, \mathfrak{l}} _{B_{p}-B}(B_{0}, 1^{\gamma_{1}},\cdots, 1^{\gamma_{p}})\\ \equiv & \zeta^{\sharp\sharp, \mathfrak{l}} _{B_{p}-1}(B_{0}-B+1,1^{\gamma_{1}},\cdots, 1^{\gamma_{p}}) & -& \zeta^{\sharp\sharp,\mathfrak{l}} _{B_{p}-B}(B_{0}, 1^{ \gamma_{1}},\cdots, 1^{\gamma_{p}})\\ \equiv & \zeta^{\sharp, \mathfrak{l}} _{B_{p}-1}(B_{0}-B+1,1^{\gamma_{1}},\cdots, 1^{\gamma_{p}}) & -& \zeta^{\sharp,\mathfrak{l}} _{B_{p}-B}(B_{0}, 1^{ \gamma_{1}},\cdots, 1^{\gamma_{p}}). \end{array}$$ This boils down to $(\ref{eq:conjid})$ applied to each $\zeta^{\star\star}_{2}$, since by \textsc{Shift} $(\ref{eq:shift})$ the two terms of the type $\zeta^{\star\star}_{1}$ get simplified.\\ \item[$\cdot$] Let now look at the case where $c_{i}=1, c_{j+1}>3$ \footnote{The case $c_{j+1}=1, c_{i}>3$ being analogue, by symmetry.}; hence $B_{i}=2a_{i}+2-\delta_{c_{i+1}}$, $B=2\beta+2$. In a first hand, we have to consider: $$ \zeta^{\star\star, \mathfrak{l}}_{2} (\boldsymbol{2}^{a_{i}-\beta-1},c_{i+1},\cdots,c_{j},\boldsymbol{2}^{a_{j}}) - \zeta^{\star\star, \mathfrak{l}} (\boldsymbol{2}^{a_{j}-\beta},c_{j},\cdots,c_{i+1},\boldsymbol{2}^{a_{i}}).$$ By renumbering indices in $\ref{eq:ci}$, the correspondence boils down here to the following $\boldsymbol{\diamond} = \boldsymbol{\Join}$, where $B_{0}=2a_{0}+3-\delta_{c_{1}}$, $B_{i}=2a_{i}+3-\delta_{c_{i}}-\delta_{c_{i+1}}$, $B=2\beta +2$: $$ (\boldsymbol{\diamond}) \quad \zeta^{\star\star, \mathfrak{l}}_{2} (\boldsymbol{2}^{a_{0}-\beta},c_{1},\cdots,c_{p},\boldsymbol{2}^{a_{p}}) - \zeta^{\star\star, \mathfrak{l}} (\boldsymbol{2}^{a_{0}+1},c_{1},\cdots,c_{p},\boldsymbol{2}^{a_{p}-\beta})$$ $$(\boldsymbol{\Join}) \quad \zeta^{\sharp\sharp,\mathfrak{l}}_{B_{0}-B+1}(1^{\gamma_{1}}, \ldots,1^{\gamma_{p}}, B_{p}) - \zeta^{\sharp\sharp,\mathfrak{l}}_{B_{p}-B}(1^{\gamma_{p}}, \ldots,1^{\gamma_{1}}, B_{0}+1).$$ Turning in $(\boldsymbol{\diamond})$ the second term into a $\zeta^{\star, \mathfrak{l}}(2, \cdots)+ \zeta^{\star\star, \mathfrak{l}}_{2} (\cdots)$, and applying the identity $(\ref{eq:conjid})$ for both terms $\zeta^{\star\star, \mathfrak{l}}_{2}(\cdots)$ leads to: $$\hspace*{-0.7cm}(\boldsymbol{\diamond}) \left\lbrace \begin{array}{lll}
+ \zeta^{\star\star, \mathfrak{l}}_{1} (\boldsymbol{2}^{a_{0}-\beta+1},c_{1}-1,\cdots,c_{p},\boldsymbol{2}^{a_{p}}) & - \zeta^{\star\star, \mathfrak{l}}_{1} (\boldsymbol{2}^{a_{0}+1},c_{1}-1,\cdots,c_{p},\boldsymbol{2}^{a_{p}-\beta})& \quad (\boldsymbol{\diamond_{1}}) \\ + \zeta^{\sharp, \mathfrak{l}}_{B_{0}-B+1} (\boldsymbol{1}^{\gamma_{1}},\cdots,\boldsymbol{1}^{\gamma_{p}},B_{p}) &- \zeta^{\sharp, \mathfrak{l}}_{B_{0}-1} (\boldsymbol{1}^{\gamma_{1}},\cdots,\boldsymbol{1}^{\gamma_{p}},B_{p}-B+2)& \quad(\boldsymbol{\diamond_{2}}) \\ - \zeta^{\star, \mathfrak{l}} (\boldsymbol{2}^{a_{0}+1},c_{1},\cdots,c_{p},\boldsymbol{2}^{a_{p}-\beta}) & & \quad (\boldsymbol{\diamond_{3}}) \\ \end{array} \right. $$ The first line, $(\boldsymbol{\diamond}_{1}) $ by $\textsc{Shift}$ is zero. We apply $\textsc{Antipode}$ $\ast$ on the terms of the second line, then turn each into a difference $\zeta^{\sharp\sharp}_{n}(m, \cdots)- \zeta^{\sharp\sharp}_{n+m}(\cdots)$; the terms of the type $\zeta^{\sharp\sharp}_{n+m}(\cdots)$, are identical and get simplified:
$$(\boldsymbol{\diamond_{2}}) \quad \begin{array}{lll} \equiv & \zeta^{\sharp\sharp, \mathfrak{l}}_{B_{0}-B+1} (B_{p},\boldsymbol{1}^{\gamma_{p}},\cdots,\boldsymbol{1}^{\gamma_{1}}) & - \zeta^{\sharp\sharp, \mathfrak{l}}_{B_{0}-B+1+ B_{p}} (\boldsymbol{1}^{\gamma_{p}},\cdots,\boldsymbol{1}^{\gamma_{1}}) \\ & -\zeta^{\sharp\sharp, \mathfrak{l}}_{B_{0}-1} (B_{p}-B+2,\boldsymbol{1}^{\gamma_{p}},\cdots,\boldsymbol{1}^{\gamma_{1}}) & + \zeta^{\sharp\sharp, \mathfrak{l}}_{B_{0}-B+1+ B_{p}} (\boldsymbol{1}^{\gamma_{p}},\cdots,\boldsymbol{1}^{\gamma_{1}}) \\ \equiv & \zeta^{\sharp\sharp, \mathfrak{l}}_{B_{0}-B+1} (B_{p},\boldsymbol{1}^{\gamma_{p}},\cdots,\boldsymbol{1}^{\gamma_{1}}) & - \zeta^{\sharp\sharp, \mathfrak{l}}_{B_{0}-1} (B_{p}-B+2,\boldsymbol{1}^{\gamma_{p}},\cdots,\boldsymbol{1}^{\gamma_{1}}). \end{array}$$ Furthermore, applying the recursion hypothesis (I.), i.e. conjecture $\ref{lzg}$ on $(\boldsymbol{\diamond}_{3})$, and turn it into a difference of $\zeta^{\sharp\sharp}$: $$(\boldsymbol{\diamond_{3}})\quad \begin{array}{ll} & - \zeta^{\star, \mathfrak{l}} (\boldsymbol{2}^{a_{0}+1},c_{1},\cdots,c_{p},\boldsymbol{2}^{a_{p}-\beta})\\ \equiv & - \zeta^{\sharp, \mathfrak{l}} (B_{p}-B+1,\boldsymbol{1}^{\gamma_{p}},\cdots,\boldsymbol{1}^{\gamma_{1}},B_{0})\\ \equiv & - \zeta^{\sharp\sharp, \mathfrak{l}} (B_{p}-B+1,\boldsymbol{1}^{\gamma_{p}},\cdots,\boldsymbol{1}^{\gamma_{1}},B_{0}) + \zeta^{\sharp\sharp, \mathfrak{l}}_{B_{p}-B+1} (\boldsymbol{1}^{\gamma_{p}},\cdots,\boldsymbol{1}^{\gamma_{1}},B_{0}) \end{array}$$ When adding $(\boldsymbol{\diamond_{2}})$ and $(\boldsymbol{\diamond_{3}})$ to get $(\boldsymbol{\diamond})$, the two last terms (odd depth) being simplified by $\textsc{Shift}$, it remains: $$(\boldsymbol{\diamond}) \quad \zeta^{\sharp\sharp, \mathfrak{l}}_{B_{0}-B+1} (B_{p},\boldsymbol{1}^{\gamma_{p}},\cdots,\boldsymbol{1}^{\gamma_{1}}) - \zeta^{\sharp\sharp, \mathfrak{l}} (B_{p}-B+1,\boldsymbol{1}^{\gamma_{p}},\cdots,\boldsymbol{1}^{\gamma_{1}},B_{0}). $$ This, applying \textsc{Antipode} $\ast$ to the first term, $\textsc{Cut}$ and $\textsc{Shift}$ to the second, corresponds to $(\boldsymbol{\Join})$.\\ \end{itemize} \item[$(ii)$] When $\beta >a_{j}>a_{i}$, we should have: $$\begin{array}{lll} & - \zeta^{\star\star, \mathfrak{l}}_{c_{j}-2} (\boldsymbol{2}^{a_{j-1}}, \ldots, \boldsymbol{2}^{a_{i}+ a_{j}-\beta+1}) & + \zeta^{\star\star, \mathfrak{l}}_{c_{i+1}-2} (\boldsymbol{2}^{a_{i+1}}, \ldots, \boldsymbol{2}^{ a_{i}+ a_{j} -\beta+1})\\ \equiv & + \zeta^{\sharp\sharp,\mathfrak{l}}_{B_{i}+B_{j}-B}(1^{\gamma_{j}}, \ldots, 1^{\gamma_{i+1}}) & -\zeta^{\sharp\sharp,\mathfrak{l}}_{B_{i}+B_{j}-B}(1^{\gamma_{i+1}}, \ldots, 1^{\gamma_{j}}). \end{array}$$ Using \textsc{Shift} $(\ref{eq:shift})$ for the first line, and renumbering the indices, it is equivalent to, with $c_{1},c_{p} \geq 3$ and $a_{0}>0$: \begin{equation} \label{eq:corresp3} \zeta^{\star\star, \mathfrak{l}}_{1} (\boldsymbol{2}^{a_{0}},c_{1}-1,\cdots,c_{p})- \zeta^{\star\star, \mathfrak{l}}_{1} (\boldsymbol{2}^{a_{0}},c_{1},\cdots,c_{p}-1) \end{equation} $$ \equiv \zeta^{\sharp\sharp, \mathfrak{l}} _{B_{0}+2}(1^{\gamma_{1}},\cdots, 1^{\gamma_{p}})-\zeta^{\sharp\sharp, \mathfrak{l}} _{B_{0}+2}(1^{\gamma_{p}},\cdots, 1^{\gamma_{1}}) \equiv \zeta^{\sharp, \mathfrak{l}} _{B_{0}+2}(1^{\gamma_{1}}, \ldots, 1^{\gamma_{p}}).$$ The last equality comes from Corollary $4.2.7$, since depth is even. By $(\ref{eq:corresp})$ applied on each term of the first line $$\zeta^{\star\star, \mathfrak{l}}_{1} (\boldsymbol{2}^{a_{0}},c_{1}-1,\cdots,c_{p})- \zeta^{\star\star, \mathfrak{l}}_{1} (\boldsymbol{2}^{a_{0}},c_{1},\cdots,c_{p}-1)$$ $$\hspace*{-1.5cm} \equiv \zeta^{\star\star, \mathfrak{l}}_{2} (\boldsymbol{2}^{a_{0}-1},c_{1},\cdots,c_{p})+ \zeta^{\sharp \mathfrak{l}} _{2a_{0}}(3,1^{\gamma_{p}},\cdots, 1^{\gamma_{1}}) - \zeta^{\star\star, \mathfrak{l}}_{2} (c_{p},\cdots,c_{1},\boldsymbol{2}^{a_{0}-1}) - \zeta^{\sharp\sharp, \mathfrak{l}}_{2}(2a_{0}+1,1^{\gamma_{p}},\cdots, 1^{\gamma_{1}}).$$ By Antipode $\shuffle$, the $\zeta^{\star\star}$ get simplified, and by the definition of $\zeta^{\sharp\sharp}$, the previous equality is equal to: $$\equiv- \zeta^{\sharp \mathfrak{l}} _{2a_{0}+3}(1^{\gamma_{p}},\cdots, 1^{\gamma_{1}}) + \zeta^{\sharp\sharp \mathfrak{l}} _{2a_{0}}(3,1^{\gamma_{p}},\cdots, 1^{\gamma_{1}}) + \zeta^{\sharp\sharp \mathfrak{l}} _{2a_{0}+4}(1^{\gamma_{p}-1},\cdots, 1^{\gamma_{1}}) $$ $$ - \zeta^{\sharp\sharp \mathfrak{l}} _{2}(2a_{0}+1, 1^{\gamma_{p}},\cdots, 1^{\gamma_{1}}) + \zeta^{\sharp\sharp \mathfrak{l}} _{2a_{0}+3}(1^{\gamma_{r}},\cdots, 1^{\gamma_{1}}).$$ Then, by \textsc{Shift} $(\ref{eq:shift})$, the second and fourth term get simplified while the third and fifth term get simplified by \textsc{Cut} $(\ref{eq:cut})$. It remains: $$- \zeta^{\sharp , \mathfrak{l}} _{2a_{0}+3}(1^{\gamma_{p}},\cdots, 1^{\gamma_{1}}), \quad \text{ which leads straight to } \ref{eq:corresp3}.$$ \item[$(iii)$] When $a_{i}< \beta <a_{j}$, we should have: \begin{multline}\nonumber
- \zeta^{\star\star, \mathfrak{l}}_{2} (\boldsymbol{2}^{a_{i}}, \ldots, \boldsymbol{2}^{a_{j}-\beta-1}) + \zeta^{\star\star, \mathfrak{l}}_{c_{i+1}-2} (\boldsymbol{2}^{a_{i+1}}, \ldots, \boldsymbol{2}^{a_{i}+ a_{j} -\beta+1}) \\ \equiv \zeta^{\sharp\sharp,\mathfrak{l}}_{B_{i}+B_{j}-B}(1^{\gamma_{j}}, \ldots, 1^{\gamma_{i+1}})-\zeta^{\sharp\sharp,\mathfrak{l}}_{B_{j}-B}(1^{\gamma_{j}}, \ldots, B_{i}). \end{multline} Using resp. \textsc{Antipode} \textsc{Shift} $(\ref{eq:shift})$ for the first line, and re-ordering the indices, it is equivalent to, with $c_{1}\geq 3$, $B_{0}=2a_{0}+1-\delta c_{1}$ here: \begin{equation} \label{eq:corresp} \zeta^{\star\star, \mathfrak{l}}_{2} (\boldsymbol{2}^{a_{0}-1},c_{1},\cdots,c_{p},\boldsymbol{2}^{a_{p}})- \zeta^{\star\star, \mathfrak{l}}_{1} (\boldsymbol{2}^{a_{0}},c_{1}-1,\cdots,c_{p},\boldsymbol{2}^{a_{p}}) \end{equation} $$\equiv \zeta^{\sharp\sharp, \mathfrak{l}} _{B_{p}-1}(B_{0}, 1^{\gamma_{1}},\cdots, 1^{\gamma_{p}}) - \zeta^{\sharp\sharp, \mathfrak{l}} _{B_{0}+B_{p}-1}(1^{ \gamma_{p}},\cdots, 1^{\gamma_{1}}) \equiv \zeta^{\sharp, \mathfrak{l}} _{B_{0}-1}(1^{\gamma_{1}},\cdots, 1^{\gamma_{p}},B_{p}).$$ This matches with the identity $\ref{eq:conjid}$; the last equality coming from $\textsc{Shift}$ since depth is odd. \end{itemize} \item Antisymmetric of the first case.\\ \end{enumerate}
\item Let us denote the sequences $\textbf{X}=\boldsymbol{2}^{a_{1}}, \ldots , \boldsymbol{2}^{a_{p}}$ and $\textbf{Y}= \boldsymbol{1}^{\gamma_{1}-1}, B_{1},\cdots, \boldsymbol{1}^{\gamma_{p}} $.\\ We want to prove that: \begin{equation} \label{eq:1234567} \zeta^{\sharp,\mathfrak{l}} (1,\textbf{Y},B_{p})\equiv -\zeta^{\star,\mathfrak{l}}_{1} (c_{1}-1,\textbf{X}) \end{equation} Relations used are mostly these stated in $\S 4.2$. Using the definition of $\zeta^{\star\star}$: \begin{equation}\label{eq:12345} \begin{array}{ll} -\zeta^{\star,\mathfrak{l}}_{1} (c_{1}-1,\textbf{X}) & \equiv -\zeta^{\star\star,\mathfrak{l}}_{1} (c_{1}-1,\textbf{X})+ \zeta^{\star\star,\mathfrak{l}}_{c_{1}} (\textbf{X})\\ & \equiv - \zeta^{\star\star,\mathfrak{l}} (1,c_{1}-1,\textbf{X})+ \zeta^{\star,\mathfrak{l}} (1,c_{1}-1,\textbf{X})+ \zeta^{\star\star,\mathfrak{l}}(c_{1},\textbf{X})- \zeta^{\star,\mathfrak{l}} (c_{1},\textbf{X}) \\ & \equiv \zeta^{\star,\mathfrak{l}} (1,c_{1}-1,\textbf{X})- \zeta^{\star,\mathfrak{l}} (c_{1},\textbf{X})- \zeta^{\star,\mathfrak{l}}(c_{1}-1,\textbf{X},1). \end{array} \end{equation} There, the first and third term in the second line, after applying \textsc{Shift}, have given the last $\zeta^{\star}$ in the last line.\\ Using then Conjecture $\ref{lzg}$, in terms of MMZV$^{\sharp}$, then MMZV$^{\sharp\sharp}$, it gives: \begin{multline} \zeta^{\sharp,\mathfrak{l}} (2,\textbf{Y},B_{p}-1)+ \zeta^{\sharp,\mathfrak{l}} (1,1,\textbf{Y},B_{p}-1)+ \zeta^{\sharp,\mathfrak{l}} (1,\textbf{Y},B_{p}-1,1)\\
\equiv \zeta^{\sharp\sharp,\mathfrak{l}} (2,\textbf{Y},B_{p}-1)- \zeta^{\sharp\sharp,\mathfrak{l}} _{2}(\textbf{Y},B_{p}-1)+ \zeta^{\sharp\sharp,\mathfrak{l}} (1,1,\textbf{Y},B_{p}-1)\\ -\zeta^{\sharp\sharp,\mathfrak{l}}_{1} (1,\textbf{Y},B_{p}-1)+ \zeta^{\sharp\sharp,\mathfrak{l}} (1,\textbf{Y},B_{p}-1,1)-\zeta^{\sharp\sharp,\mathfrak{l}}_{1} (\textbf{Y},B_{p}-1,1) \end{multline} First term (odd depth)\footnote{Since weight is odd, we know also depth parity of these terms.} is simplified with the last, by $\textsc{Schift}$. Fifth term (even depth) get simplified by \textsc{Cut} with the fourth term. Hence it remains two terms of even depth: $$\equiv - \zeta^{\sharp\sharp,\mathfrak{l}} _{2}(\textbf{Y},B_{p}-1)+ \zeta^{\sharp\sharp,\mathfrak{l}} (1,1,\textbf{Y},B_{p}-1) \equiv - \zeta^{\sharp\sharp,\mathfrak{l}} _{1}(\textbf{Y},B_{p})+ \zeta^{\sharp\sharp,\mathfrak{l}}_{B_{p}-1} (1,1,\textbf{Y}) , $$ where \textsc{Minus} resp. \textsc{Cut} have been applied. This matches with $(\ref{eq:1234567})$ since, by $\textsc{Shift}:$ $$\equiv - \zeta^{\sharp\sharp,\mathfrak{l}} _{1}(\textbf{Y},B_{p})+ \zeta^{\sharp\sharp,\mathfrak{l}} (1,\textbf{Y},B_{p})\equiv \zeta^{\sharp,\mathfrak{l}}(1,\textbf{Y},B_{p}). $$ The case $c_{1}=3$ slightly differs since $(\ref{eq:12345})$ gives, by recursion hypothesis I.($\ref{lzg}$): $$ -\zeta^{\star,\mathfrak{l}}_{1} (2,\textbf{X})\equiv \zeta^{\sharp,\mathfrak{l}} (B_{1}+1,\textbf{Y}',B_{p}-1)+ \zeta^{\sharp,\mathfrak{l}} (1,B_{1},\textbf{Y}',B_{p}-1)+ \zeta^{\sharp,\mathfrak{l}} (B_{1},\textbf{Y}',B_{p}-1,1),$$ where $\textbf{Y}'= \boldsymbol{1}^{\gamma_{2}},\cdots, \boldsymbol{1}^{\gamma_{p}} $, odd depth. Turning into MES$^{\sharp\sharp}$, and using identities of $\S 4.2$ in the same way than above, leads to the result. Indeed, from: $$\equiv \zeta^{\sharp\sharp,\mathfrak{l}} (B_{1}+1,\textbf{Y}',B_{p}-1)+ \zeta^{\sharp\sharp,\mathfrak{l}} (1,B_{1},\textbf{Y}',B_{p}-1)+ \zeta^{\sharp\sharp,\mathfrak{l}} (B_{1},\textbf{Y}',B_{p}-1,1)$$ $$-\zeta^{\sharp\sharp,\mathfrak{l}}_{B_{1}+1} (\textbf{Y}',B_{p}-1)- \zeta^{\sharp\sharp,\mathfrak{l}}_{1} (B_{1},\textbf{Y}',B_{p}-1)-\zeta^{\sharp\sharp,\mathfrak{l}}_{B_{1}} (\textbf{Y}',B_{p}-1,1)$$ First and last terms get simplified via $\textsc{Shift}$, while third and fifth term get simplified by $\textsc{Cut}$; besides, we apply \textsc{minus} for second term, and \textsc{minus} for the fourth term, which are both of even depth. This leads to $\ref{eq:toolid}$, using again $\textsc{Shift}$ for the first term: $$\begin{array}{l} \equiv\zeta^{\sharp\sharp,\mathfrak{l}}_{B_{p}-1} (1,B_{1},\textbf{Y}')-\zeta^{\sharp\sharp,\mathfrak{l}}_{B_{1}} (\textbf{Y}',B_{p}) \\ \equiv \zeta^{\sharp\sharp,\mathfrak{l}} (B_{1},\textbf{Y}',B_{p})-\zeta^{\sharp\sharp,\mathfrak{l}}_{B_{1}} (\textbf{Y}',B_{p}) \\ \equiv \zeta^{\sharp,\mathfrak{l}} (B_{1},\textbf{Y}',B_{p}). \end{array}$$ \end{enumerate} \end{proof}
\section{Appendix $1$: From the linearized octagon relation}
The identities in the coalgebra $\mathcal{L}$ obtained from the linearized octagon relation $\ref{eq:octagonlin}$:
\begin{lemm}\label{lemmlor} In the coalgebra $\mathcal{L}$, $n_{i}\in\mathbb{Z}^{\ast}$:\footnote{Here, $\mlq + \mrq$ still denotes the operation where absolute values are summed and signs multiplied.} \begin{itemize} \item[$(i)$] $\zeta^{\star\star, \mathfrak{l}}(n_{0},\cdots, n_{p})= (-1)^{w+1} \zeta^{\star\star,\mathfrak{l}}(n_{p},\cdots, n_{0})$. \item[$(ii)$] $\zeta^{\mathfrak{l}}(n_{0},\cdots, n_{p})+(-1)^{w+p} \zeta^{\star\star\mathfrak{l}}(n_{0},\cdots, n_{p})+(-1)^{p} \zeta^{\star\star\mathfrak{l}}_{n_{p}}(n_{p-1},\cdots,n_{1},n_{0})=0$. \item[$(iii)$] $$\hspace*{-1cm}\zeta^{\mathfrak{l}}_{n_{0}-1}(n_{1},\cdots, n_{p})- \zeta^{\mathfrak{l}}_{n_{0}}(n_{1},\cdots,n_{p-1}, n_{p}\mlq + \mrq 1 )=(-1)^{w} \left[ \zeta^{\star\star,\mathfrak{l}}_{n_{0}-1}(n_{1},\cdots, n_{p})- \zeta^{\star\star,\mathfrak{l}}_{n_{0}}(n_{1},\cdots,n_{p-1}, n_{p}\mlq + \mrq 1)\right].$$ \end{itemize} \end{lemm} \begin{proof} The sign of $n_{i}$ is denoted $\epsilon_{i}$ as usual. First, we remark that, with $\eta_{i}=\pm 1$, $n_{i}=\epsilon_{i} (a_{i}+1)$, and $\epsilon_{i}= \eta_{i}\eta_{i+1}$: $$\hspace*{-0.5cm}\begin{array}{ll} \Phi^{\mathfrak{m}}(e_{\infty}, e_{-1},e_{1}) & = \sum I^{\mathfrak{m}} \left(0; (-\omega_{0})^{a_{0}} (-\omega_{-\eta_{1}\star}) (-\omega_{0})^{a_{1}} \cdots (-\omega_{-\eta_{p}\star}) (-\omega_{0})^{a_{p}} ;1 \right) e_{0}^{a_{0}}e_{\eta_{1}} e_{0}^{a_{1}} \cdots e_{\eta_{p}} e_{0}^{a_{p}}\\ & \\ & = \sum (-1)^{n+p}\zeta^{\star\star,\mathfrak{m}}_{n_{0}-1} \left( n_{1}, \cdots, n_{p-1}, -n_{p}\right) e_{0}^{a_{0}}e_{\eta_{1}} e_{0}^{a_{1}} \cdots e_{\eta_{p}} e_{0}^{a_{p}}. \\ \end{array}$$ Similarly, with $ \mu_{i}\mathrel{\mathop:}= \left\lbrace \begin{array}{ll} \star & \texttt{if } \eta_{i}=1\\ 1 & \texttt{if } \eta_{i}=-1 \end{array} \right. $, applying the homography $\phi_{\tau\sigma}$ to get the second line: $$\hspace*{-0.5cm}\begin{array}{ll} \Phi^{\mathfrak{m}}(e_{-1}, e_{0},e_{\infty}) & = \sum I^{\mathfrak{m}} \left(0; (\omega_{1}-\omega_{-1})^{a_{0}} \omega_{\mu_{1}} (\omega_{1}-\omega_{-1})^{a_{1}} \cdots \omega_{\mu_{p}} (\omega_{1}-\omega_{-1})^{a_{p}} ;1 \right) e_{0}^{a_{0}}e_{\eta_{1}} e_{0}^{a_{1}} \cdots e_{\eta_{p}} e_{0}^{a_{p}}\\ & \\ \Phi^{\mathfrak{l}}(e_{-1}, e_{0},e_{\infty}) & = \sum (-1)^{p} I^{\mathfrak{m}} \left(0; 0^{a_{0}} \omega_{-\eta_{1}} 0^{a_{1}} \cdots \omega_{-\eta_{p}} 0^{a_{p}} ;1 \right) e_{0}^{a_{0}}e_{\eta_{1}} e_{0}^{a_{1}} \cdots e_{\eta_{p}} e_{0}^{a_{p}}\\ & \\ & = \sum \zeta^{\mathfrak{m}}_{n_{0}-1} \left( n_{1}, \cdots, n_{p-1}, -n_{p}\right) e_{0}^{a_{0}}e_{\eta_{1}} e_{0}^{a_{1}} \cdots e_{\eta_{p}} e_{0}^{a_{p}}. \\ \end{array}$$ Lastly, still using $\phi_{\tau\sigma}$, with here $\mu_{i}\mathrel{\mathop:}= \left\lbrace \begin{array}{ll} \star & \texttt{if } \eta_{i}=1\\ 1 & \texttt{if } \eta_{i}=-1 \end{array} \right. $: $$\hspace*{-0.5cm}\begin{array}{ll} \Phi^{\mathfrak{m}}(e_{1}, e_{\infty},e_{0}) & = \sum I^{\mathfrak{m}} \left(0; (\omega_{-1}-\omega_{1})^{a_{0}} \omega_{\mu_{1}} (\omega_{-1}-\omega_{1})^{a_{1}} \cdots \omega_{\mu_{p}} (\omega_{-1}-\omega_{1})^{a_{p}} ;1 \right) e_{0}^{a_{0}}e_{\eta_{1}} e_{0}^{a_{1}} \cdots e_{\eta_{p}} e_{0}^{a_{p}}\\ & \\ \Phi^{\mathfrak{l}}(e_{1}, e_{\infty},e_{0}) & = \sum (-1)^{w+1} I^{\mathfrak{m}} \left(0; 0^{a_{0}} \omega_{\eta_{1}\star} 0^{a_{1}} \cdots \omega_{\eta_{p}\star} 0^{a_{p}} ;1 \right) e_{0}^{a_{0}}e_{\eta_{1}} e_{0}^{a_{1}} \cdots e_{\eta_{p}} e_{0}^{a_{p}}\\ & \\ & = \sum (-1)^{n+p+1}\zeta^{\star\star,\mathfrak{m}}_{n_{0}-1} \left( n_{1}, \cdots, n_{p-1}, n_{p}\right) e_{0}^{a_{0}}e_{\eta_{1}} e_{0}^{a_{1}} \cdots e_{\eta_{p}} e_{0}^{a_{p}}. \\ \end{array}$$
\begin{itemize} \item[$(i)$] This case is the one used in Theorem $\ref{hybrid}$. This identity is equivalent to, in terms of iterated integrals, for $X$ any sequence of $\left\lbrace 0, \pm 1 \right\rbrace $ or of $\left\lbrace 0, \pm \star \right\rbrace $: $$\left\lbrace \begin{array}{llll} I^{\mathfrak{l}}(0;0^{k}, \star, X ;1) & = & I^{\mathfrak{l}}(0; X, \star, 0^{k}; 1) & \text{ if } \prod_{i=0}^{p} \epsilon_{i}=1 \Leftrightarrow \eta_{0}=1\\ I^{\mathfrak{l}}(0;0^{k}, -\star, X ;1) & = & I^{\mathfrak{l}}(0; -X, -\star, 0^{k}; 1) & \text{ if } \prod_{i=0}^{p} \epsilon_{i}=-1 \Leftrightarrow \eta_{0}=-1\\ \end{array} \right. $$ The first case is deduced from $\ref{eq:octagonlin}$ when looking at the coefficient of a word beginning and ending by $e_{-1}$ (or beginning and ending by $e_{1}$), whereas the second case is obtained from the coefficient of a word beginning by $e_{-1}$ and ending by $e_{1}$, or beginning by $e_{1}$ and ending by $e_{-1}$.
\item[$(ii)$] Let split into two cases, according to the sign of $\prod \epsilon_{i}$: \begin{itemize} \item[$\cdot$] In $\ref{eq:octagonlin}$, when looking at the coefficient of a word beginning by $e_{1}$ and ending by $e_{0}$, only these three terms contribute: $$ \Phi^{\mathfrak{l}}(e_{-1}, e_{0},e_{\infty})e_{0}- \Phi^{\mathfrak{l}}(e_{\infty}, e_{-1},e_{1})e_{0}- e_{1} \Phi^{\mathfrak{l}}(e_{1}, e_{\infty},e_{0}) .$$ Moreover, the coefficient of $e_{-1} e_{0}^{a_{0}} e_{\eta_{1}} \cdots e_{\eta_{p}} e_{0}^{a_{p}+1}$ is, using the expressions above for $\Phi^{\mathfrak{l}}(\cdot)$: \begin{multline}\nonumber
(-1)^{p} I^{\mathfrak{l}}(0; -1, -X; 1)+ (-1)^{w+1} I^{\mathfrak{l}}(0; -\star, -X_{\star}; 1)+ (-1)^{w}I^{\mathfrak{l}}(0; X_{\star}, 0; 1)=0, \\ \text{where }\begin{array}{l} X:= \omega_{0}^{a_{0}} \omega_{\eta_{1}} \cdots \omega_{\eta_{p}} \omega_{0}^{a_{p}}\\ X_{\star}:= \omega_{0}^{a_{0}} \omega_{\eta_{1}\star} \cdots \omega_{\eta_{p}\star} \omega_{0}^{a_{p}} \end{array}. \end{multline} In terms of motivic Euler sums, it is, with $\prod \epsilon_{i}=1$: $$ \zeta^{\mathfrak{l}} (n_{0},\cdots, -n_{p}) +(-1)^{w+p} \zeta^{\star\star\mathfrak{l}}(n_{0},\cdots, -n_{p})+(-1)^{w+p} \zeta^{\star\star\mathfrak{l}}_{n_{0}-1}(n_{1},\cdots,n_{p-1}, n_{p}\mlq + \mrq 1)=0.$$ Changing $n_{p}$ into $-n_{p}$, and applying Antipode $\shuffle$ to the last term, it gives, with now $\prod \epsilon_{i}=-1$: $$ \zeta^{\mathfrak{l}} (n_{0},\cdots, n_{p}) +(-1)^{w+p} \zeta^{\star\star\mathfrak{l}}(n_{0},\cdots, n_{p})+(-1)^{p} \zeta^{\star\star\mathfrak{l}}_{n_{p}}(n_{p-1},\cdots,n_{1},n_{0})=0.$$ \item[$\cdot$] Similarly, for the coefficient of a word beginning by $e_{-1}$ and ending by $e_{0}$, only these three terms contribute: $$ \Phi^{\mathfrak{l}}(e_{-1}, e_{0},e_{\infty})e_{0}- \Phi^{\mathfrak{l}}(e_{\infty}, e_{-1},e_{1})e_{0}+ e_{-1} \Phi^{\mathfrak{l}}(e_{\infty}, e_{-1},e_{1}) .$$ Similarly than above, it leads to the identity, with $\prod \epsilon_{i}=-1$: $$ \zeta^{\mathfrak{l}} (n_{0},\cdots, -n_{p}) +(-1)^{w+p} \zeta^{\star\star\mathfrak{l}}(n_{0},\cdots, -n_{p})+(-1)^{w+p} \zeta^{\star\star\mathfrak{l}}_{n_{0}-1}(n_{1},\cdots,n_{p-1},-(n_{p}\mlq + \mrq 1))=0.$$ Changing $n_{p}$ into $-n_{p}$, and applying Antipode $\shuffle$ to the last term, it gives, with now $\prod \epsilon_{i}=1$: $$ \zeta^{\mathfrak{l}} (n_{0},\cdots, n_{p}) +(-1)^{w+p+1} \zeta^{\star\star\mathfrak{l}}(n_{0},\cdots, n_{p})+(-1)^{p} \zeta^{\star\star\mathfrak{l}}_{n_{p}}(n_{p-1},\cdots,n_{1},n_{0})=0.$$ \end{itemize}
\item[$(iii)$] When looking at the coefficient of a word beginning by $e_{0}$ and ending by $e_{0}$ in $\ref{eq:octagonlin}$, only these three terms contribute: $$ -e_{0} \Phi^{\mathfrak{l}}(e_{-1}, e_{0},e_{\infty})+ \Phi^{\mathfrak{l}}(e_{-1}, e_{0},e_{\infty})e_{0}+ e_{0} \Phi^{\mathfrak{l}}(e_{\infty}, e_{-1},e_{1})- \Phi^{\mathfrak{l}}(e_{\infty}, e_{-1},e_{1})e_{0}.$$ If we identify the coefficient of the word $ e_{0}^{a_{0}+1} e_{-\eta_{1}} \cdots e_{-\eta_{p}} e_{0}^{a_{p}+1}$, it leads straight to the identity $(iii)$. \end{itemize} \textsc{Remark}: Looking at the coefficient of words beginning by $e_{0}$ and ending by $e_{1}$ or $e_{-1}$ in $\ref{eq:octagonlin}$ would lead to the same identity than the second case. \end{proof}
\section{Appendix $2$: Missing coefficients}
In Lemma $\ref{lemmcoeff}$, the coefficients $D_{a,b}$ appearing (in $(v)$) are the only one which are not conjectured. Albeit these values are not required for the proof of Theorem $4.4.1$, we provide here a table of values in small weights. Let examine the coefficient corresponding to $\zeta^{\star}(\boldsymbol{2}^{n})$ instead of $\zeta^{\star}(2)^{n}$, which is (by $(i)$ in Lemma $\ref{lemmcoeff}$), with $n=a+b+1$: \begin{equation} \widetilde{D}^{a,b}\mathrel{\mathop:}= \frac{(2n)!}{6^{n}\mid B_{2n}\mid (2^{2n}-2)} D^{a,b} \quad \text{ and } \quad \widetilde{D}_{n}\mathrel{\mathop:}= \frac{(2n)!}{6^{n}\mid B_{2n}\mid (2^{2n}-2)}D_{n} . \end{equation} We have an expression $(\ref{eq:coeffds})$ for $D_{n}$, albeit not very elegant, which would give: \begin{equation} \label{eq:coeffdstilde} \widetilde{D}_{n}= \frac{2^{2n} (2n)!}{(2^{2n}-2)\mid B_{2n}\mid }\sum_{\sum m_{i} s_{i}=n \atop m_{i}\neq m_{j}} \prod_{i=1}^{k} \left( \frac{1}{s_{i}!} \left( \frac{\mid B_{2m_{i}}\mid (2^{2m_{i}-1}-1) } {2m_{i} (2m_{i})!}\right)^{s_{i}} \right). \end{equation} Here is a table of values for $\widetilde{D}_{n}$ and $\widetilde{D}^{k,n-k-1}$ in small weights:\\ \\
\begin{tabular}{| l || c | c |c | c |}
\hline
$\cdot \quad \quad \diagdown n$ & $2$ & $3$ & $4$ & $5$ \\ \hline
$\widetilde{D_{n}}$ & $\frac{19}{2^{3}-1}$ & $\frac{275}{2^{5}-1}$& $\frac{11813}{3(2^{7}-1)}$ & $\frac{783}{7}$\\
& & & & \\ \hline
$\widetilde{D}_{k,n-1-k}$ &$\frac{-12}{7}$ & $\frac{-84}{31}, \frac{160}{31}$& $\frac{1064}{127}, \frac{-1680}{127}, \frac{-9584}{381}$ & $\frac{189624}{2555}$,$\frac{-137104}{2555}$,$\frac{-49488}{511}$,$\frac{-17664}{511}$ \\
& & & &\\
\hline
\hline
$\cdot \quad \quad \diagdown n$ & $6$ & $7$ & $8$ & $9$ \\ \hline
$\widetilde{D_{n}} \quad \quad $ & $\frac{581444793}{691(2^{11}-1)}$& $\frac{263101079}{21(2^{13}-1)}$& $\frac{6807311830555}{3617(2^{15}-1)}$& $\frac{124889801445461}{43867(2^{17}-1)}$\\
& & & & \\
\hline
\end{tabular} \\
\\
\\ The denominators of $\widetilde{D_{n}},\widetilde{D}_{k,n-1-k}$ can be written as $(2^{2n-1}-1)$ times the numerator of the Bernoulli number $B_{2n}$. No formula has been found yet for their numerators, that should involve binomial coefficients. These coefficients are related since, by shuffle: $$\begin{array}{lll}
& \zeta^{\star\star, \mathfrak{m}}_{2} (\boldsymbol{2}^{n})+ \sum_{k=0}^{n-1}\zeta^{\star\star, \mathfrak{m}}_{1} (\boldsymbol{2}^{k},3,\boldsymbol{2}^{n-k-1}) & =0\\
& \zeta^{\star\star, \mathfrak{m}}(\boldsymbol{2}^{n+1})-\zeta^{\star, \mathfrak{m}}(\boldsymbol{2}^{n+1}) \sum_{k=0}^{n-1}\zeta^{\star\star, \mathfrak{m}}_{1} (\boldsymbol{2}^{k},3,\boldsymbol{2}^{n-k-1}) & =0. \end{array}$$ Identifying the coefficients of $\zeta^{\star}(\boldsymbol{2}^{n})$ in formulas $(iii),(v)$ in Lemma $\ref{lemmcoeff}$ leads to: \begin{equation}\label{eq:coeffdrel} 1-\widetilde{D_{n}}= \sum_{k=0}^{n-1} \widetilde{D}_{k,n-1-k}. \end{equation}
\chapter{Galois Descents}
\paragraph{\texttt{Contents}: } The first section gives the general picture (for any $N$), sketching the Galois descent ideas. The second section focuses on the cases $N=2,3,4,\mlq 6\mrq, 8$, defining the filtrations by the motivic level associated to each descent, and displays both results and proofs. Some examples in small depth for are given in the Annexe $\S A.2$.\\ \\ \texttt{Notations}: For a fixed $N$, let $k_{N}\mathrel{\mathop:}=\mathbb{Q}(\xi_{N})$, where $\xi_{N}\in\mu_{N}$ is a primitive $N^{\text{th}}$ root of unity, and $\mathcal{O}_{N}$ is the ring of integers of $k_{N}$. The subscript or exponent $N$ will be omitted when it is not ambiguous. For the general case, the decomposition of $N$ is denoted $N=\prod q_{i}= \prod p_{i}^{\alpha}$.\\
\section{Overview}
\paragraph{Change of field. }
As said in Chapter $3$, for each $N, N'$ with $N' | N$, the Galois action on $\mathcal{H}_{N}$ and $\mathcal{H}_{N'}$ is determined by the coaction $\Delta$. More precisely, let consider the following descent\footnote{More generally, there are Galois descents $(\mathcal{d})=(k_{N}/k_{N'}, M/M')$ from $\mathcal{H}^{\mathcal{MT}\left( \mathcal{O}_{k_{N}} \left[ \frac{1}{M}\right] \right) }$, to $\mathcal{H}^{\mathcal{MT}\left( \mathcal{O}_{k_{N'}} \left[ \frac{1}{M'}\right]\right) }$, with $N'\mid N$, $M'\mid M$, with a set of derivations $\mathscr{D}^{\mathcal{d}} \subset \mathscr{D}^{N}$ associated.}, \texttt{assuming $\phi_{N'}$ is an isomorphism of graded Hopf comodules}: \footnote{Conjecturally as soon as $N'\neq p^{r}$, $p\geq 5$. Proven for $N'=1,2,3,4,\mlq 6\mrq,8$.} $$\xymatrixcolsep{5pc}\xymatrix{ \mathcal{H}^{N} \ar@{^{(}->}[r] ^{\phi_{N}}_{n.c} & \mathcal{H}^{\mathcal{MT}_{\Gamma_{N}}} \\ \mathcal{H}^{N'}\ar[u]_{\mathcal{G}^{N/N'}} \ar@{^{(}->}[r] _{n.c}^{\phi_{N'}\atop \sim} &\mathcal{H}^{\mathcal{MT}_{\Gamma_{N'}}} \ar[u]^{\mathcal{G}^{\mathcal{MT}}_{N/N'}} }$$ Let choose a basis for $gr_{1}\mathcal{L}_{r}^{\mathcal{MT}_{N'}}$, and extend it into a basis of $gr_{1}\mathcal{L}_{r}^{\mathcal{MT}_{N}}$: $$ \left\lbrace \zeta^{\mathfrak{m}}(r; \eta'_{i,r}) \right\rbrace_{i} \subset \left\lbrace \zeta^{\mathfrak{m}}(r; \eta'_{i,r}) \right\rbrace \cup \left\lbrace \zeta^{\mathfrak{m}}(r; \eta_{i}) \right\rbrace_{1\leq i \leq c_{r}}, $$ $$\text{where } \quad c_{r} =\left\{ \begin{array}{ll} a_{N}-a_{N'}= \frac{\varphi(N)-\varphi(N')}{2}+p(N)-p(N') & \text{ if } r=1\\ b_{N}-b_{N'}= \frac{\varphi(N)-\varphi(N')}{2} & \text{ if } r>1\\ \end{array} \right. .$$ Then, once this basis fixed, let split the set of derivations $\mathscr{D}^{N}$ into two parts (cf. $\S 2.4.4$), one corresponding to $\mathcal{H}^{N'}$:\nomenclature{$\mathscr{D}^{\backslash \mathcal{d}}$ and $\mathscr{D}^{\mathcal{d}} $}{sets of derivations associated to a descent $\mathcal{d}$} \begin{equation} \label{eq:derivdescent} \mathscr{D}^{N} = \mathscr{D}^{\backslash \mathcal{d}} \uplus \mathscr{D}^{\mathcal{d}} \quad \text{ where }\quad \left\lbrace \begin{array}{l} \mathscr{D}^{\backslash \mathcal{d}} =\mathscr{D}^{N'}\mathrel{\mathop:}= \cup_{r} \left\lbrace D_{r}^{\eta'_{i,r}} \right\rbrace_{1\leq i \leq c_{r}} \\ \mathscr{D}^{\mathcal{d}}\mathrel{\mathop:}= \cup_{r} \left\lbrace D^{\eta_{i,r}}_{r} \right\rbrace_{1\leq i \leq c_{r}} \end{array} \right. . \end{equation}
\texttt{Examples:} \begin{itemize} \item[$\cdot$] For the descent from $\mathcal{MT}_{3}$ to $\mathcal{MT}_{1}$: $\mathscr{D}^{(k_{3}/\mathbb{Q}, 3/1)}=\left\lbrace D^{\xi_{3}}_{1}, D^{\xi_{3}}_{2r}, r>0 \right\rbrace $. \item[$\cdot$] For the descent from $\mathcal{MT}_{8}$ to $\mathcal{MT}_{4}$: $\mathscr{D}^{(k_{8}/k_{4}, 2/2)}=\left\lbrace D^{\xi_{8}}_{r}-D^{-\xi_{8}}_{r}, r>0 \right\rbrace $. \item[$\cdot$] For the descent from $\mathcal{MT}_{9}$ to $\mathcal{MT}_{3}$: $\mathscr{D}^{(k_{9}/k_{3}, 3/3)}=\left\lbrace D^{\xi_{9}}_{r}-D^{-\xi^{4}_{9}}_{r}, D^{\xi_{9}}_{r}-D^{-\xi^{7}_{9}}_{r} r>0 \right\rbrace $.\footnote{By the relations in depth $1$, since: $$\zeta^{\mathfrak{a}} \left( r\atop \xi^{3}_{9}\right)= 3^{r-1} \left( \zeta^{\mathfrak{a}} \left( r\atop \xi^{1}_{9}\right) + \zeta^{\mathfrak{a}} \left( r\atop \xi^{4}_{9}\right)+ \zeta^{\mathfrak{a}} \left( r\atop \xi^{7}_{9}\right) \right) \quad \text{etc.}$$} \end{itemize}
\begin{theo} Let $N'\mid N$ such that $\mathcal{H}^{N'}\cong \mathcal{H}^{\mathcal{MT}_{\Gamma_{N'}}}$.\\ Let $\mathfrak{Z}\in gr^{\mathfrak{D}}_{p}\mathcal{H}_{n}^{N}$, depth graded MMZV relative to $\mu_{N}$.\\ Then $\mathfrak{Z}\in gr^{\mathfrak{D}}_{p}\mathcal{H}^{N'}$, i.e. $\mathfrak{Z}$ is a depth graded MMZV relative to $\mu_{N'}$ modulo smaller depth if and only if: $$ \left( \forall r<n, \forall D_{r,p}\in\mathscr{D}_{r}^{\mathcal{d}},\quad D_{r,p}(\mathfrak{Z})=0\right) \quad \textrm{ and } \quad \left( \forall r<n, \forall D_{r,p} \in\mathscr{D}^{\diagdown\mathcal{d}}, \quad D_{r,p}(\mathfrak{Z})\in gr^{\mathfrak{D}}_{p-1}\mathcal{H}^{N'}\right) .$$ \end{theo} \begin{proof} In the $(f_{i})$ side, the analogue of this theorem is pretty obvious, and the result can be transported via $\phi$, and back since $\phi_{N'}$ isomorphism by assumption. \end{proof} This is a very useful recursive criterion (derivation strictly decreasing weight and depth) to determine if a (motivic) multiple zeta value at $\mu_{N}$ is in fact a (motivic) multiple zeta value at $\mu_{N'}$, modulo smaller depth terms; applying it recursively, it could also take care of smaller depth terms. This criterion applies for motivic MZV$_{\mu_{N}}$, and by period morphism is deduced for MZV$_{\mu_{N}}$.\\
\paragraph{Change of Ramification.} If the descent has just a ramified part, the criterion can be stated in a non depth graded version. Indeed, there, since only weight $1$ matters, to define the derivation space $\mathcal{D}^{\mathcal{d}}$ as above ($\ref{eq:derivdescent}$), we need to choose a basis for $\mathcal{O}_{N}^{\ast}\otimes \mathbb{Q}$, which we complete with $\left\lbrace \xi^{\frac{N}{q_{i}}}_{N}\right\rbrace_{i\in I}$ into a basis for $\Gamma_{N}$. Then, with $N=\prod p_{i}^{\alpha_{i}}=\prod q_{i}$: \begin{theo}\label{ramificationchange} Let $\mathfrak{Z}\in \mathcal{H}_{n}^{N}\subset \mathcal{H}^{\mathcal{MT}_{\Gamma_{N}}}$, MMZV relative to $\mu_{N}$.\\ Then $\mathfrak{Z}\in \mathcal{H}^{\mathcal{MT}(\mathcal{O}_{N})}$ unramified if and only if: $$ \left( \forall i\in I, D^{\xi^{\frac{N}{q_{i}}}}_{1}(\mathfrak{Z})=0\right) \quad \textrm{ and } \quad \left( \forall r<n, \forall D_{r}\in\mathscr{D}^{\diagdown\mathcal{d}}, \quad D_{r}(\mathfrak{Z})\in \mathcal{H}^{\mathcal{MT}(\mathcal{O}_{N})}\right) .$$ \end{theo} \texttt{Nota Bene}: Intermediate descents and change of ramification, keeping part of some of the weight $1$ elements $\left\lbrace \xi^{\frac{N}{q_{i}}}_{N}\right\rbrace$ could also be stated.\\ \\ \texttt{Examples}: \begin{description} \item[$\boldsymbol{N=2}$:] As claimed in the introduction, the descent between $\mathcal{H}^{2}$ and $\mathcal{H}^{1}$ is precisely measured by $D_{1}$:\footnote{$\mathscr{D}^{(\mathbb{Q}/\mathbb{Q}, 2/1)}=\left\lbrace D^{-1}_{1} \right\rbrace $ with the above notations; and $D^{-1}_{1}$ is here simply denoted $D_{1}$ .} \begin{coro}\label{criterehonoraire} Let $\mathfrak{Z}\in\mathcal{H}^{2}=\mathcal{H}^{\mathcal{MT}_{2}}$, a motivic Euler sum.\\ Then $\mathfrak{Z}\in\mathcal{H}^{1}=\mathcal{H}^{\mathcal{MT}_{1}}$, i.e. $\mathfrak{Z}$ is a motivic multiple zeta value if and only if: $$D_{1}(\mathfrak{Z})=0 \quad \textrm{ and } \quad D_{2r+1}(\mathfrak{Z})\in\mathcal{H}^{1}.$$ \end{coro} \item[$\boldsymbol{N=3,4,6}$:] \begin{coro}\label{ramif346} Let $N\in \lbrace 3,4,6\rbrace$ and $\mathfrak{Z}\in\mathcal{H}^{\mathcal{MT}(\mathcal{O}_{N} \left[ \frac{1}{N}\right] )}$, a motivic MZV$_{\mu_{N}}$.\\ Then $\mathfrak{Z}$ is unramified, $\mathfrak{Z}\in\mathcal{H}^{\mathcal{MT} (\mathcal{O}_{N})}$ if and only if: $$D_{1}(\mathfrak{Z})=0 \textrm{ and } \quad D_{r}(\mathfrak{Z})\in\mathcal{H}^{\mathcal{MT} (\mathcal{O}_{N})}.$$ \end{coro} \item[$\boldsymbol{N=p^{r}}$:] A basis for $\mathcal{O}^{N}\otimes \mathbb{Q}$ is formed by: $\left\lbrace \frac{1-\xi^{k}}{1-\xi} \right\rbrace_{k\wedge p=1 \atop 0<k\leq\frac{N}{2}} $, which corresponds to $$\text{ a basis of } \mathcal{A}_{1}^{\mathcal{MT}(\mathcal{O}_{N})} \quad : \left\lbrace \zeta^{\mathfrak{m}}\left( 1 \atop \xi^{k} \right)- \zeta^{\mathfrak{m}}\left( 1 \atop \xi \right) \right\rbrace _{ k\wedge p=1 \atop 0<k\leq\frac{N}{2}} .$$ It can be completed in a basis of $\mathcal{A}_{1}^{N}$ with $\zeta^{\mathfrak{m}}\left( 1 \atop \xi^{1} \right)$. \footnote{With the previous theorem notations, $\mathcal{D}^{\mathcal{d}}=\lbrace D^{\xi}_{1}\rbrace$ whereas $\mathcal{D}^{\diagdown \mathcal{d}}= \lbrace D^{\xi^{k}}_{1}-D^{\xi}_{1} \rbrace_{k\wedge p=1 \atop 1<k\leq\frac{N}{2} } \cup_{r>1} \lbrace D^{\xi^{k}}_{r}\rbrace_{k\wedge p=1 \atop 0<k\leq\frac{N}{2}}$; where $D^{\xi}_{1}$ has to be understood as the projection of the left side over $\zeta^{\mathfrak{a}}\left( 1 \atop \xi \right)$ in respect to the basis above of $\mathcal{H}_{1}^{\mathcal{MT}(\mathcal{O}_{N})}$ more $\zeta^{\mathfrak{a}}\left( 1 \atop \xi \right)$. This leads to a criterion equivalent to $(\ref{ramifpr})$.} However, if we consider the basis of $\mathcal{A}_{1}^{N}$ formed by primitive roots of unity up to conjugates, the criterion for the descent could also be stated as follows: \begin{coro}\label{ramifpr} Let $N=p^{r}$ and $\mathfrak{Z}\in\mathcal{H}^{\mathcal{MT}_{\Gamma_{N}}}=\mathcal{H}^{\mathcal{MT}(\mathcal{O}_{N} \left[ \frac{1}{p}\right] )}$, relative to $\mu_{N}$\footnote{For instance a MMZV relative to $\mu_{N}$. Beware, for $p>5$, there could be other periods.}.\\ Then $\mathfrak{Z}$ is unramified, $\mathfrak{Z}\in\mathcal{H}^{\mathcal{MT} (\mathcal{O}_{N})}$ if and only if: $$\sum_{k\wedge p=1 \atop 0<k\leq\frac{N}{2}} D^{\xi^{k}_{N}}_{1}(\mathfrak{Z})=0 \quad \textrm{ and } \quad \forall \left\lbrace \begin{array}{l}
r>1 \\ 1<k\leq\frac{N}{2}\\ k\wedge p=1 \\ \end{array} \right. , \quad D^{\xi^{k}_{N}}_{r}(\mathfrak{Z})\in\mathcal{H}^{\mathcal{MT} (\mathcal{O}_{N})}.$$ \end{coro}
\end{description}
\section{Descents for $\boldsymbol{N=2,3,4,\mlq 6\mrq,8}$.}
\subsection{Depth $\boldsymbol{1}$}
Let start with depth $1$ results, deduced from Lemma $2.4.1$ (from $\cite{De}$), fundamental to initiate the recursion later. \begin{lemm} The basis for $gr^{\mathfrak{D}}_{1} \mathcal{A}$ is: $$\left\{ \zeta^{\mathfrak{a}}\left(r; \xi \right) \text{ such that } \left\{ \begin{array}{ll} r>1 \text{ odd } & \text{ if }N=1 \\ r \text{ odd } & \text{ if }N=2 \\ r>0 & \text{ if } N=3,4 \\ r>1 & \text{ if } N=6 \\ \end{array} \right. \right\rbrace $$ For $N=8$, the basis for $gr^{\mathfrak{D}}_{1} \mathcal{A}_{r}$ is two dimensional, for all $r>0$: $$\left\{ \zeta^{\mathfrak{a}}\left(r; \xi \right), \zeta^{\mathfrak{a}}\left(r; -\xi \right)\right\rbrace.$$ \end{lemm}
Let make these relations explicit in depth $1$ for $N=2,3,4,\mlq 6\mrq,8$, since we would use some $p$-adic properties of the basis elements in our proof:
\begin{description} \item[\textsc{For $N=2$:}] The distribution relation in depth 1 is: $$\zeta^{\mathfrak{a}}\left( {2 r + 1 \atop 1}\right) = (2^{-2r}-1)\zeta^{\mathfrak{a}}\left( {2r+1 \atop -1}\right) .$$ \item[\textsc{For $N=3$:}] $$ \zeta^{\mathfrak{l}} \left( {2r+1 \atop 1} \right)\left(1-3^{2r}\right)= 2\cdot 3^{2r}\zeta^{\mathfrak{l}}\left({2r+1 \atop \xi}\right) \quad \quad \zeta^{\mathfrak{l}}\left({2r \atop 1}\right)=0 \quad \quad \zeta^{\mathfrak{l}}\left({r \atop \xi}\right) =\left(-1\right)^{r-1} \zeta^{\mathfrak{l}}\left({r \atop \xi^{-1}}\right). $$ \item[\textsc{For $N=4$:}] $$\begin{array}{lllllll} \zeta^{\mathfrak{l}}\left({ r \atop 1} \right) (1-2^{r-1}) & = & 2^{r-1}\cdot \zeta^{\mathfrak{l}}\left( {r\atop -1} \right) \text{ for } r\neq 1 & \quad & \zeta^{\mathfrak{l}}\left({1\atop 1}\right) & = & \zeta^{\mathfrak{l}}\left( {2r\atop -1} \right)=0 \\ \zeta^{\mathfrak{l}}\left({2r+1\atop -1}\right) & = & 2^{2r+1} \zeta^{\mathfrak{l}}\left( {2r+1\atop \xi} \right) & \quad & \zeta^{\mathfrak{l}}\left( {r \atop \xi} \right) & = & \left(-1\right)^{r-1} \zeta^{\mathfrak{l}}\left({r\atop \xi^{-1}}\right). \end{array}$$ \item[\textsc{For $N=6$:}] $$\begin{array}{lllllll} \zeta^{\mathfrak{l}}\left({r\atop 1}\right)\left(1-2^{r-1}\right) & = & 2^{r-1}\zeta^{\mathfrak{l}}\left({ r \atop -1} \right) \text{ for } r\neq 1 & \quad & \zeta^{\mathfrak{l}}\left( {1 \atop 1} \right) & = & \zeta^{\mathfrak{l}}\left({2r\atop -1}\right)=0\\ \zeta^{\mathfrak{l}}\left( {2r+1 \atop -1} \right) & = & \frac{2\cdot 3^{2r}}{1-3^{2r}} \zeta^{\mathfrak{l}}\left( {2r+1 \atop \xi} \right)& \quad & \zeta^{\mathfrak{l}}\left( {r \atop \xi^{2}} \right) & = & \frac{2^{r-1}}{1-(-2)^{r-1}} \zeta^{\mathfrak{l}}\left( {r \atop \xi} \right).\\ \zeta^{\mathfrak{l}}\left({r\atop \xi} \right) &=&\left(-1\right)^{r-1} \zeta^{\mathfrak{l}}\left( {r \atop \xi^{-1}} \right) &\quad & \zeta^{\mathfrak{l}}\left({r \atop -\xi} \right) & = & \left(-1\right)^{r-1} \zeta^{\mathfrak{l}}\left( {r \atop -\xi^{-1}} \right). \end{array}$$ \item[\textsc{For $N=8$:}] $$\begin{array}{lllllll}
\zeta^{\mathfrak{l}}\left({ r \atop 1} \right)& =& \frac{ 2^{r-1}}{\left(1-2^{r-1}\right)}\zeta^{\mathfrak{l}}\left({r\atop -1}\right) \text{ for } r\neq 1 & \quad & \zeta^{\mathfrak{l}}\left( {1 \atop 1} \right) &=&\zeta^{\mathfrak{l}}\left({2r\atop -1}\right)=0 \\
\zeta^{\mathfrak{l}}\left({ r \atop -i }\right) &=& 2^{r-1} \left( \zeta^{\mathfrak{l}}\left({r\atop \xi}\right) + \zeta^{\mathfrak{l}}\left({r\atop -\xi}\right) \right) & \quad & \zeta^{\mathfrak{l}}\left( {2r+1 \atop -1} \right) &=& 2^{2r+1} \zeta^{\mathfrak{l}}\left({2r+1\atop i}\right) \\ \zeta^{\mathfrak{l}}\left({ r\atop \pm \xi} \right) &=&\left(-1\right)^{r-1} \zeta^{\mathfrak{l}}\left( {r \atop \pm \xi^{-1} }\right) & \quad & \zeta^{\mathfrak{l}}\left( {r \atop i} \right) &=&\left(-1\right)^{r-1} \zeta^{\mathfrak{l}}\left( {r \atop -i}\right) \\ \end{array}$$
\end{description}
\subsection{Motivic Level filtration}
Let fix a descent $(\mathcal{d})=(k_{N}/k_{N'}, M/M')$ from $\mathcal{H}^{\mathcal{MT}\left( \mathcal{O}_{k_{N}} \left[ \frac{1}{M}\right] \right) }$, to $\mathcal{H}^{\mathcal{MT}\left( \mathcal{O}_{k_{N'}} \left[ \frac{1}{M'}\right]\right) }$, with $N'\mid N$, $M'\mid M$, among these considered in this section, represented in Figures $\ref{fig:d248}, \ref{fig:d36}$.\\ Let us define a motivic level increasing filtration $\mathcal{F}^{\mathcal{d}}$ associated, from the set of derivations associated to this descent, $\mathscr{D}^{\mathcal{d}} \subset \mathscr{D}^{N}$, defined in $(\ref{eq:derivdescent})$.
\begin{defi} The filtration by the \textbf{motivic level} associated to a descent $(\mathcal{d})$ is defined recursively on $\mathcal{H}^{N}$ by: \begin{itemize} \item[$\cdot$] $\mathcal{F}^{\mathcal{d}} _{-1} \mathcal{H}^{N}=0$. \item[$\cdot$] $\mathcal{F}^{\mathcal{d}} _{i} \mathcal{H}^{N}$ is the largest submodule of $\mathcal{H}^{N}$ such that $\mathcal{F}^{\mathcal{d}}_{i}\mathcal{H}^{N}/\mathcal{F}^{\mathcal{d}} _{i-1}\mathcal{H}^{N}$ is killed by $\mathscr{D}^{\mathcal{d}}$, $\quad$ i.e. is in the kernel of $\oplus_{D\in \mathscr{D}^{\mathcal{d}}} D$. \end{itemize} \end{defi} It's a graded Hopf algebra's filtration: $$\mathcal{F} _{i}\mathcal{H}. \mathcal{F}_{j}\mathcal{H} \subset \mathcal{F}_{i+j}\mathcal{H} \text{ , } \quad \Delta (\mathcal{F}_{n}\mathcal{H})\subset \sum_{i+j=n} \mathcal{F}_{i}\mathcal{A} \otimes \mathcal{F}_{j}\mathcal{H}.$$ The associated graded is denoted: $gr^{\mathcal{d}} _{i}$ and the quotients, coalgebras compatible with $\Delta$: \begin{equation} \label{eq:quotienth} \mathcal{H}^{\geq 0} \mathrel{\mathop:}= \mathcal{H} \text{ , } \boldsymbol{\mathcal{H}^{\geq i}}\mathrel{\mathop:}= \mathcal{H}/ \mathcal{F}_{i-1}\mathcal{H} \text{ with the projections :}\quad \quad \forall j\geq i \text{ , } \pi_{i,j}: \mathcal{H}^{\geq i} \rightarrow \mathcal{H}^{\geq j}. \end{equation} Note that, via the isomorphism $\phi$, the motivic filtration on $\mathcal{H}^{\mathcal{MT}_{N}}$ corresponds to\footnote{In particular, remark that $\dim \mathcal{F}^{\mathcal{d}} _{i} \mathcal{H}_{n}^{\mathcal{MT}_{N}}$ are known.}: \begin{equation} \label{eq:isomfiltration}\mathcal{F}^{\mathcal{d}} _{i} \mathcal{H}^{\mathcal{MT}_{N}} \longleftrightarrow \left\langle x\in H^{N} \mid Deg^{\mathcal{d}} (x) \leq i \right\rangle _{\mathbb{Q}} , \end{equation} where $Deg^{\mathcal{d}}$ is the degree in $\left\lbrace \lbrace f^{j}_{r} \rbrace_{b_{N'}<j\leq b_{N} \atop r>1} , \lbrace f^{j}_{1} \rbrace_{a_{N'}<j\leq a_{N}} \right\rbrace $, which are the images of the complementary part of $ gr_{1}\mathcal{L}^{\mathcal{MT}_{N'}}$ in the basis of $gr_{1}\mathcal{L}^{\mathcal{MT}_{N}}$.\\ \\ \texttt{Example}: For the descent between $\mathcal{H}^{\mathcal{MT}_{2}}$ and $\mathcal{H}^{\mathcal{MT}_{1}}$, since $gr_{1}\mathcal{L}^{\mathcal{MT}_{2}}= \left\langle \zeta^{\mathfrak{m}}(-1), \left\lbrace \zeta^{\mathfrak{m}}(2r+1)\right\rbrace _{r>0}\right\rangle$: $$\mathcal{F} _{i} \mathcal{H}^{\mathcal{MT}_{2}} \quad \xrightarrow[\sim]{\quad \phi}\quad \left\langle x\in \mathbb{Q}\langle f_{1}, f_{3}, \cdots \rangle\otimes \mathbb{Q}[f_{2}] \mid Deg_{f_{1}} (x) \leq i \right\rangle _{\mathbb{Q}} \text{ , where } Deg_{f_{1}}= \text{ degree in } f_{1}.$$ \\ By definition of these filtrations: \begin{equation}D_{r,p}^{\eta} \left( \mathcal{F}_{i}\mathcal{H}_{n} \right) \subset \left\lbrace \begin{array}{ll} \mathcal{F}_{i-1}\mathcal{H}_{n-r} & \text{ if }D_{r,p}^{\eta}\in\mathscr{D}^{\mathcal{d}}_{r} \\ \mathcal{F}_{i}\mathcal{H}_{n-r} & \text{ if } D_{r,p}^{\eta}\in\mathscr{D}^{\backslash\mathcal{d}}_{r} \end{array} \right. . \end{equation} Similarly, looking at $\partial_{n,p}$ (cf. $\ref{eq:pderivnp}$): \begin{equation} \partial_{n,p}(\mathcal{F}_{i-1}\mathcal{H}_{n}) \subset \oplus_{r<n} \left( gr_{p-1}^{\mathfrak{D}} \mathcal{F}_{i-2}\mathcal{H}_{n-r}\right)^{\text{ card } \mathscr{D}^{\mathcal{d}}_{r}} \oplus_{r<n} \left( gr_{p-1}^{\mathfrak{D}} \mathcal{F}_{i-1}\mathcal{H}_{n-r}\right) ^{\text{ card } \mathscr{D}^{\backslash\mathcal{d}}_{r}}. \end{equation} This allows us to pass to quotients, and define $D^{\eta,i,\mathcal{d}}_{n,p}$ and $\partial^{i,\mathcal{d}}_{n,p}$:\nomenclature{$D^{\eta,i,\mathcal{d}}_{n,p}$ and $\partial^{i,\mathcal{d}}_{n,p}$}{quotient maps} \begin{equation} \label{eq:derivinp} \boldsymbol{D^{\eta,i,\mathcal{d}}_{n,p}}: gr_{p}^{\mathfrak{D}} \mathcal{H}_{n}^{\geq i} \rightarrow \left\lbrace \begin{array}{ll}
gr_{p-1}^{\mathfrak{D}} \mathcal{H}_{n-r}^{\geq i-1} & \text{ if }D_{r,p}^{\eta}\in\mathscr{D}^{\mathcal{d}}_{r} \\ gr_{p-1}^{\mathfrak{D}} \mathcal{H}_{n-r}^{\geq i} & \text{ if } D_{r,p}^{\eta}\in\mathscr{D}^{\backslash\mathcal{d}}_{r} \end{array} \right. \end{equation} \begin{framed} \begin{equation} \label{eq:pderivinp} \boldsymbol{\partial^{i,\mathcal{d}}_{n,p}}: gr_{p}^{\mathfrak{D}} \mathcal{H}_{n}^{\geq i} \rightarrow \oplus_{r<n} \left( gr_{p-1}^{\mathfrak{D}} \mathcal{H}_{n-r}^{\geq i-1}\right)^{\text{ card } \mathscr{D}^{\mathcal{d}}_{r}} \oplus_{r<n} \left( gr_{p-1}^{\mathfrak{D}} \mathcal{H}_{n-r}^{\geq i}\right)^{\text{ card } \mathscr{D}^{\backslash\mathcal{d}}_{r}} . \end{equation} \end{framed} The bijectivity of this map is essential to the results stated below.
\subsection{General Results} In the following results, the filtration considered $\mathcal{F}_{i}$ is the filtration by the motivic level associated to the (fixed) descent $\mathcal{d}$ while the index $i$, in $\mathcal{B}_{n, p, i}$ refers to the level notion for elements in $\mathcal{B}$ associated to the descent $\mathcal{d}$.\footnote{Precisely defined, for each descent in $\S 5.2.5 $.}\\ We first obtain the following result on the depth graded quotients, for all $i\geq 0$, with: $$\mathbb{Z}_{1[P]} \mathrel{\mathop:}= \frac{\mathbb{Z}}{1+ P\mathbb{Z}}=\left\{ \frac{a}{1+b P}, a,b\in\mathbb{Z} \right\} \text{ with } \begin{array}{ll} P=2 & \text{ for } N=2,4,8 \\ P=3 & \text{ for } N=3,6 \end{array} .$$
\begin{lemm} \begin{itemize} \item[$\cdot$] $$\mathcal{B}_{n, p, \geq i} \text{ is a linearly free family of } gr_{p}^{\mathfrak{D}} \mathcal{H}_{n}^{\geq i} \text{ and defines a } \mathbb{Z}_{1[P]} \text{ structure :}$$ Each element $\mathfrak{Z}= \zeta^{\mathfrak{m}}\left( z_{1}, \ldots , z_{p} \atop \epsilon_{1}, \ldots, \epsilon_{p}\right)\in \mathcal{B}_{n,p} $ decomposes in a $\mathbb{Z}_{1[P]}$-linear combination of $\mathcal{B}_{n, p, \geq i}$ elements, denoted $cl_{n,p,\geq i}(\mathfrak{Z})$ in $gr_{p}^{\mathfrak{D}} \mathcal{H}_{n}^{\geq i}$, which defines, in an unique way: $$cl_{n,p,\geq i}: \langle\mathcal{B}_{n, p, \leq i-1}\rangle_{\mathbb{Q}} \rightarrow \langle\mathcal{B}_{n, p, \geq i}\rangle_{\mathbb{Q}}.$$ \item[$\cdot$] The following map $\partial^{i,\mathcal{d}}_{n,p}$ is bijective: $$\partial^{i,\mathcal{d}}_{n,p}: gr_{p}^{\mathfrak{D}} \langle \mathcal{B}_{n, \geq i} \rangle_{\mathbb{Q}} \rightarrow \oplus_{r<n} \left( gr_{p-1}^{\mathfrak{D}} \langle \mathcal{B}_{n-1, \geq i-1} \rangle_{\mathbb{Q}} \right) ^{\oplus \text{ card } \mathcal{D}^{\mathcal{d}}_{r}} \oplus_{r<n} \left( gr_{p-1}^{\mathfrak{D}} \langle \mathcal{B}_{n-2r-1, \geq i} \rangle_{\mathbb{Q}} \right) ^{\oplus \text{ card } \mathcal{D}^{\backslash\mathcal{d}}_{r}}.$$ \end{itemize} \end{lemm} \nomenclature{$cl_{n,p,\geq i}$, or $cl_{n,\leq p,\geq i}$}{maps whose existence is proved in $\S 5.2$}Before giving the proof, in the next section, let present its consequences such as bases for the quotient, the filtration and the graded spaces for each descent considered: \begin{theo} \begin{itemize} \item[$(i)$] $\mathcal{B}_{n,\leq p, \geq i}$ is a basis of $\mathcal{F}_{p}^{\mathfrak{D}} \mathcal{H}_{n}^{\geq i}=\mathcal{F}_{p}^{\mathfrak{D}} \mathcal{H}_{n}^{\geq i, \mathcal{MT}}$. \item[$(ii)$] \begin{itemize} \item[$\cdot$] $\mathcal{B}_{n, p, \geq i}$ is a basis of $gr_{p}^{\mathfrak{D}} \mathcal{H}_{n}^{\geq i}=gr_{p}^{\mathfrak{D}} \mathcal{H}_{n}^{\geq i, \mathcal{MT}}$ on which it defines a $\mathbb{Z}_{1[P]}$-structure:\\ Each element $\mathfrak{Z}= \zeta^{\mathfrak{m}}\left( z_{1}, \ldots , z_{p} \atop \epsilon_{1}, \ldots, \epsilon_{p}\right)$ decomposes in a $\mathbb{Z}_{1[P]}$-linear combination of $\mathcal{B}_{n, p, \geq i}$ elements, denoted $cl_{n,p,\geq i}(\mathfrak{Z})$ in $gr_{p}^{\mathfrak{D}} \mathcal{H}_{n}^{\geq i}$, which defines in an unique way: $$cl_{n,p,\geq i}: \langle\mathcal{B}_{n, p, \leq i-1}\rangle_{\mathbb{Q}} \rightarrow \langle\mathcal{B}_{n, p, \geq i}\rangle_{\mathbb{Q}} \text{ such that } \mathfrak{Z}+cl_{n,p,\geq i}(\mathfrak{Z})\in \mathcal{F}_{i-1}\mathcal{H}_{n}+ \mathcal{F}^{\mathfrak{D}}_{p-1}\mathcal{H}_{n}.$$ \item[$\cdot$] The following map is bijective: $$\partial^{i, \mathcal{d}}_{n,p}: gr_{p}^{\mathfrak{D}} \mathcal{H}_{n}^{\geq i} \rightarrow \oplus_{r<n} \left( gr_{p-1}^{\mathfrak{D}} \mathcal{H}_{n-1}^{\geq i-1}\right) ^{\oplus \text{ card } \mathcal{D}^{\mathcal{d}}_{r}} \oplus_{r<n} \left( gr_{p-1}^{\mathfrak{D}} \mathcal{H}_{n-r}^{\geq i}\right) ^{\oplus \text{ card } \mathcal{D}^{\backslash\mathcal{d}}_{r}}. $$ \item[$\cdot$] $\mathcal{B}_{n,\cdot, \geq i} $ is a basis of $\mathcal{H}^{\geq i}_{n} =\mathcal{H}^{\geq i, \mathcal{MT}}_{n}$. \end{itemize}
\item[$(iii)$] We have the two split exact sequences in bijection: $$ 0\longrightarrow \mathcal{F}_{i}\mathcal{H}_{n} \longrightarrow \mathcal{H}_{n} \stackrel{\pi_{0,i+1}} {\rightarrow}\mathcal{H}_{n}^{\geq i+1} \longrightarrow 0$$ $$ 0 \rightarrow \langle \mathcal{B}_{n, \cdot, \leq i} \rangle_{\mathbb{Q}} \rightarrow \langle\mathcal{B}_{n} \rangle_{\mathbb{Q}} \rightarrow \langle \mathcal{B}_{n, \cdot, \geq i+1} \rangle_{\mathbb{Q}} \rightarrow 0 .$$ The following map, defined in an unique way:
$$cl_{n,\leq p,\geq i}: \langle\mathcal{B}_{n, p, \leq i-1}\rangle_{\mathbb{Q}} \rightarrow \langle\mathcal{B}_{n, \leq p, \geq i}\rangle_{\mathbb{Q}} \text{ such that } \mathfrak{Z}+cl_{n,\leq p,\geq i}(\mathfrak{Z})\in \mathcal{F}_{i-1}\mathcal{H}_{n}.$$ \item[$(iv)$] A basis for the filtration spaces $\mathcal{F}_{i} \mathcal{H}^{\mathcal{MT}}_{n}=\mathcal{F}_{i} \mathcal{H}_{n}$: $$\cup_{p} \left\{ \mathfrak{Z}+ cl_{n, \leq p, \geq i+1}(\mathfrak{Z}), \mathfrak{Z}\in \mathcal{B}_{n, p, \leq i} \right\}.$$ \item[$(v)$] A basis for the graded space $gr_{i} \mathcal{H}^{\mathcal{MT}}_{n}=gr_{i} \mathcal{H}_{n}$: $$\cup_{p} \left\{ \mathfrak{Z}+ cl_{n, \leq p, \geq i+1}(\mathfrak{Z}), \mathfrak{Z}\in \mathcal{B}_{n, p, i} \right\}.$$ \end{itemize} \end{theo} The proof is given in $\S 5.2.4$, and the notion of level resp. motivic level, some consequences and specifications for $N=2,3,4,\mlq 6\mrq,8$ individually are provided in $\S 5.2.5$. Some examples in small depth are displayed in Appendice $A.2$.\\ \\ \\ \texttt{{\large Consequences, level $i=0$:}} \begin{itemize} \item[$\cdot$] The level $0$ of the basis elements $\mathcal{B}^{N}$ forms a basis of $\mathcal{H}^{N} = \mathcal{H}^{\mathcal{MT}_{N}}$, for $N=2,3,4,\mlq 6 \mrq, 8$. This gives a new proof (dual) of Deligne's result (in $\cite{De}$).\\ The level $0$ of this filtration is hence isomorphic to the following algebras:\footnote{The equalities of the kind $\mathcal{H}^{\mathcal{MT}_{N}}= \mathcal{H}^{N}$ are consequences of the previous theorem for $N=2,3,4,\mlq 6 \mrq,8$, and by F. Brown for $N=1$ (cf. $\cite{Br2}$). Moreover, we have inclusions of the kind $\mathcal{H}^{\mathcal{MT}_{N'}} \subseteq \mathcal{F}_{0}^{k_{N}/k_{N'},M/M'}\mathcal{H}^{\mathcal{MT}_{N}}$ and we deduce the equality from dimensions at fixed weight.} $$ \mathcal{F}_{0}^{k_{N}/k_{N'},M/M'}\mathcal{H}^{\mathcal{MT}_{N}}=\mathcal{F}_{0}^{k_{N}/k_{N'},M/M'}\mathcal{H}^{N}=\mathcal{H}^{\mathcal{MT}_{N',M'}}="\mathcal{H}^{N',M'}" .$$ Hence the inclusions in the following diagram are here isomorphisms: $$\xymatrix{ \mathcal{F}_{0}^{k_{N}/k_{N'},M/M'}\mathcal{H}^{\mathcal{MT}_{N}} & \mathcal{H}^{\mathcal{MT}_{N'}} \ar@{^{(}->}[l]\\ \mathcal{F}_{0}^{k_{N}/k_{N'},M/M'}\mathcal{H}^{N} \ar@{^{(}->}[u] & \mathcal{H}^{N'} \ar@{^{(}->}[l] \ar@{^{(}->}[u]}.$$ \item[$\cdot$] It gives, considering such a descent $(k_{N}/k_{N'},M/M')$, a basis for $\mathcal{F}^{0}\mathcal{H}^{N}= \mathcal{H}^{\mathcal{MT}_{N',M'}}$ in terms of the basis of $\mathcal{H}^{N}$. For instance, it leads to a new basis for motivic multiple zeta values in terms of motivic Euler sums, or motivic MZV$_{\mu_{3}}$.\\ Some other $0$-level such as $\mathcal{F}_{0}^{k_{N}/k_{N},P/1}$, $N=3,4$ which should reflect the descent from $\mathcal{MT}(\mathcal{O}_{N}\left[ \frac{1}{P}\right] )$ to $\mathcal{MT}(\mathcal{O}_{N})$ are not known to be associated to a fundamental group, but the previous result enables us to reach them. We obtain a basis for: \begin{itemize} \item[$\bullet$] $\boldsymbol{\mathcal{H}^{\mathcal{MT}(\mathbb{Z}\left[\frac{1}{3}\right])}}$ in terms of the basis of $\mathcal{H}^{\mathcal{MT}(\mathcal{O}_{3}[\frac{1}{3}])}$. \item[$\bullet$] $\boldsymbol{\mathcal{H}^{\mathcal{MT}(\mathcal{O}_{3})}}$ in terms of the basis of $\mathcal{H}^{\mathcal{MT}(\mathcal{O}_{3}[\frac{1}{3}])}$. \item[$\bullet$] $\boldsymbol{\mathcal{H}^{\mathcal{MT}(\mathcal{O}_{4})}}$ in terms of the basis of $\mathcal{H}^{\mathcal{MT}(\mathcal{O}_{4}[\frac{1}{4}])}$. \\ \end{itemize} \end{itemize}
\subsection{Proofs}
As proved below, Theorem $5.2.4$ boils down to the Lemma $5.2.3$. Remind the map $\partial^{i,\mathcal{d}}_{n,p}$: $$\partial^{i,\mathcal{d}}_{n,p}: gr_{p}^{\mathfrak{D}} \mathcal{H}_{n}^{\geq i} \rightarrow \oplus_{r<n} \left( gr_{p-1}^{\mathfrak{D}} \mathcal{H}_{n-r}^{\geq i-1}\right)^{\text{ card } \mathscr{D}^{\mathcal{d}}_{r}} \oplus_{r<n} \left( gr_{p-1}^{\mathfrak{D}} \mathcal{H}_{n-r}^{\geq i}\right)^{\text{ card } \mathscr{D}^{\backslash\mathcal{d}}_{r}}.$$ We will look at its image on $\mathcal{B}_{n,p,\geq i} $ and prove both the injectivity of $\partial^{i,\mathcal{d}}_{n,p}$ as considered in Lemma $5.2.3$, and the linear independence of these elements $\mathcal{B}_{n,p,\geq i}$.
\paragraph{\Large { Proof of Lemma $\boldsymbol{5.2.3}$ for $\boldsymbol{N=2}$:}}
The formula $(\ref{Drp})$ for $D^{-1}_{2r+1,p}$ on $\mathcal{B}$ elements:\footnote{Using identity: $\zeta^{\mathfrak{a}}(\overline{2 r + 1 })= (2^{-2r}-1)\zeta^{\mathfrak{a}}(2r+1 )$. Projection on $\zeta^{\mathfrak{l}}(\overline{2r+1})$ for the left side.} \begin{multline} \label{Deriv2} D^{-1}_{2r+1,p} \left(\zeta^{\mathfrak{m}} (2x_{1}+1, \ldots , \overline{2x_{p}+1}) \right) = \\
\frac{2^{2r}}{1-2^{2r}}\delta_{r =x_{1}} \cdot \zeta^{\mathfrak{m}} (2 x_{2}+1, \ldots, \overline{2x_{p}+1}) \\ \frac{2^{2r}}{1-2^{2r}} \left\lbrace \begin{array}{l} \sum_{i=1}^{p-2} \delta_{x_{i+1}\leq r < x_{i}+ x_{i+1} } \binom{2r}{2x_{i+1}} \\ -\sum_{i=1}^{p-1} \delta_{x_{i}\leq r < x_{i}+ x_{i+1}} \binom{2r}{2x_{i}} \end{array} \right. \cdot \zeta^{\mathfrak{m}} \left( \cdots ,2x_{i-1}+1, 2 (x_{i}+x_{i+1}-r) +1, 2 x_{i+2}+1, \cdots \right) \\ \textrm{\textsc{(d) }} +\delta_{x_{p} \leq r \leq x_{p}+ x_{p-1}} \binom{2r}{2x_{p}} \cdot\zeta^{\mathfrak{m}} \left( \cdots ,2x_{p-2}+1, \overline{2 (x_{p-1}+x_{p}-r) +1}\right) \end{multline} Terms of type \textsc{(d)} play a particular role since they correspond to deconcatenation for the coaction, and will be the terms of minimal $2$-adic valuation.\\ $D^{-1}_{1,p}$ acts as a deconcatenation on this family: \begin{equation} \label{Deriv21} D^{-1}_{1,p} \left(\zeta^{\mathfrak{m}} (2x_{1}+1, \ldots , \overline{2x_{p}+1}) \right) = \left\{ \begin{array}{ll}
0 & \text{ if } x_{p}\neq 0 \\
\zeta^{\mathfrak{m}} (2x_{1}+1, \ldots , \overline{2x_{p-1}+1}) & \text{ if } x_{p}=0 .\\ \end{array} \right. \end{equation} For $N=2$, $\partial^{i}_{n,p}$ ($\ref{eq:pderivinp}$) is simply: \begin{equation}\label{eq:pderivinp2} \partial^{i}_{n,p}: gr_{p}^{\mathfrak{D}} \mathcal{H}_{n}^{\geq i} \rightarrow gr_{p-1}^{\mathfrak{D}} \mathcal{H}_{n-1}^{\geq i-1} \oplus_{1<2r+1\leq n-p+1} gr_{p-1}^{\mathfrak{D}} \mathcal{H}_{n-2r-1}^{\geq i}. \end{equation} Let prove all statements of Lemma $5.2.3$, recursively on the weight, and then recursively on depth and on the level, from $i=0$. \begin{proof} By recursion hypothesis, weight being strictly smaller, we assume that:
$$\mathcal{B}_{n-1,p-1,\geq i-1} \oplus_{1<2r+1\leq n-p+1} \mathcal{B}_{n-2r-1,p-1,\geq i} \text{ is a basis of } $$
$$gr_{p-1}^{\mathfrak{D}} \mathcal{H}_{n-1}^{\geq i-1,\mathcal{B}} \oplus_{1<2r+1\leq n-p+1} gr_{p-1}^{\mathfrak{D}} \mathcal{H}_{n-2r-1}^{\geq i,\mathcal{B}}. $$
\begin{center} \textsc{Claim:} The matrix $M^{i}_{n,p}$ of $\left(\partial^{i,\mathcal{d}}_{n,p} (z) \right)_{z\in \mathcal{B}_{n, p, \geq i}}$ on these spaces is invertible.
\end{center} \texttt{Nota Bene:} Here $D^{-1}_{1}(z)$, resp. $D^{-1}_{2r+1,p}(z)$ are expressed in terms of $\mathcal{B}_{n-1,p-1,\geq i-1} $ resp. $\mathcal{B}_{n-2r-1,p-1,\geq i}$.\\ It will prove both the bijectivity of $\partial^{i,\mathcal{d}}_{n,p}$ as considered in the lemma and the linear independence of $\mathcal{B}_{n, p, \geq i}$. Let divide $M^{i}_{n,p}$ into four blocks, with the first column corresponding to elements of $\mathcal{B}_{n, p, \geq i}$ ending by $1$: \begin{center}
\begin{tabular}{ l || c | c ||}
& $x_{p}=0$ & $x_{p}>0$ \\ \hline
$D_{1,p}$ & M$1$ & M$2$ \\
$D_{>1,p}$ & M$3$ & M$4$ \\
\hline
\end{tabular} \end{center} According to ($\ref{Deriv21}$), $D^{-1}_{1,p}$ is zero on the elements not ending by 1, and acts as a deconcatenation on the others. Therefore, M$3=0$, so $M^{i}_{n,p}$ is lower triangular by blocks, and the left-upper-block M$1$ is diagonal invertible. It remains to prove the invertibility of the right-lower-block $\widetilde{M}\mathrel{\mathop:}=M4$, corresponding to $D^{-1}_{>1,p}$ and to the elements of $\mathcal{B}_{n, p, \geq i}$ not ending by 1.\\ \\ Notice that in the formula $(\ref{Deriv2})$ of $D_{2r+1,p}$, applied to an element of $\mathcal{B}_{n, p, \geq i}$, most of terms appearing have a number of $1$ greater than $i$ but there are also terms in $\mathcal{B}_{n-2r-1,p-1,i-1}$, with exactly $(i-1)$ \say{$1$} for type $\textsc{a,b,c}$ only. We will make disappear the latter modulo $2$, since they are $2$-adically greater. \\ More precisely, using recursion hypothesis (in weight strictly smaller), we can replace them in $gr_{p-1} \mathcal{H}^{\geq i}_{n-2r-1}$ by a $\mathbb{Z}_{\text{odd}}$-linear combination of elements in $\mathcal{B}_{n-2r-1, p-1, \geq i}$, which does not lower the $2$-adic valuation. It is worth noticing that the type \textsc{d} elements considered are now always in $\mathcal{B}_{n-2r-1,p-1,\geq i}$, since we removed the case $x_{p}= 0$.\\
Once done, we can construct the matrix $\widetilde{M}$ and examine its entries.\\ Order elements of $\mathcal{B}$ on both sides by lexicographic order of its \say{reversed} elements: \begin{center} $(x_{p},x_{p-1},\cdots, x_{1})$ for the colums, $(r,y_{p-1},\cdots, y_{1})$ for the rows. \end{center} Remark that, with such an order, the diagonal corresponds to the deconcatenation terms: $r=x_{p}$ and $x_{i}=y_{i}$.\\ Referring to $(\ref{Deriv2})$, and by the previous remark, we see that $\widetilde{M}$ has all its entries of 2-adic valuation positive or equal to zero, since the coefficients in $(\ref{Deriv2})$ are in $2^{2r}\mathbb{Z}_{\text{odd}}$ (for types \textsc{a,b,c}) or of the form $\mathbb{Z}_{\text{odd}}$ for types \textsc{d,d'}. If we look only at the terms with $2$-adic valuation zero, (which comes to consider $\widetilde{M}$ modulo $2$), it only remains in $(\ref{Deriv2})$ the terms of type \textsc{(d,d')}, that is: \begin{multline}\nonumber
D_{2r+1,p} (\zeta^{\mathfrak{m}}(2x_{1}+1, \ldots, \overline{2x_{p}+1})) \equiv \delta_{ r = x_{p}+ x_{p-1}} \binom{2r}{2x_{p}} \zeta^{\mathfrak{m}} (2x_{1}+1, \ldots ,2x_{p-2}+1, \overline{1}) \\
+ \delta_{x_{p} \leq r < x_{p}+ x_{p-1}} \binom{2r}{2x_{p}} \zeta^{\mathfrak{m}} (2x_{1}+1, \ldots ,2x_{p-2}+1, \overline{2 (x_{p-1}+x_{p}-r) +1}) \pmod{ 2}.
\end{multline} Therefore, modulo 2, with the order previously defined, it remains only an upper triangular matrix ($\delta_{x_{p}\leq r}$), with 1 on the diagonal ($\delta_{x_{p}= r}$, deconcatenation terms). Thus, $\det\widetilde{M}$ has a 2-adic valuation equal to zero, and in particular can not be zero, that's why $\widetilde{M}$ is invertible.\\
The $\mathbb{Z}_{odd}$ structure is easily deduced from the fact that the determinant of $\widetilde{M}$ is odd, and the observation that if we consider $D_{2r+1,p} (\zeta^{\mathfrak{m}} (z_{1}, \ldots, z_{p}))$, all the coefficients are integers. \end{proof}
\paragraph{{\Large Proof of Lemma $\boldsymbol{5.2.3}$ for other $\boldsymbol{N}$.}}
These cases can be handled in a rather similar way than the case $N=2$, except that the number of generators is different and that several descents are possible, hence there will be several notions of level and filtrations by the motivic level, one for each descent. Let fix a descent $\mathcal{d}$ and underline the differences in the proof:
\begin{proof} In the same way, we prove by recursion on weight, depth and level, that the following map is bijective: $$\partial^{i,\mathcal{d}}_{n,p}: gr_{p}^{\mathfrak{D}} \langle \mathcal{B}_{n, \geq i} \rangle_{\mathbb{Q}} \rightarrow \oplus_{r<n} \left( gr_{p-1}^{\mathfrak{D}} \langle \mathcal{B}_{n-1, \geq i-1} \rangle_{\mathbb{Q}} \right) ^{\oplus \text{ card } \mathscr{D}^{\mathcal{d}}_{r}} \oplus_{r<n} \left( gr_{p-1}^{\mathfrak{D}} \langle \mathcal{B}_{n-2r-1, \geq i} \rangle_{\mathbb{Q}} \right) ^{\oplus \text{ card } \mathscr{D}^{\backslash\mathcal{d}}_{r}}.$$ \begin{center} I.e the matrix $M^{i}_{n,p}$ of $\left(\partial^{i}_{n,p} (z) \right)_{z\in \mathcal{B}_{n, p, \geq i}}$ on $\oplus_{r<n} \mathcal{B}_{n-r,p-1,\geq i-1}^{\text{ card } \mathscr{D}^{\mathcal{d}}_{r}} \oplus_{r<n} \mathcal{B}_{n-r,p-1,\geq i}^{\text{ card } \mathscr{D}^{\backslash\mathcal{d}}_{r}}$ \footnote{Elements in arrival space are linearly independent by recursion hypothesis.} is invertible. \end{center}
As before, by recursive hypothesis, we replace elements of level $\leq i$ appearing in $D^{i}_{r,p}$, $r\geq 1$ by $\mathbb{Z}_{1[P]}$-linear combinations of elements of level $\geq i$ in the quotient $gr_{p-1}^{\mathfrak{D}} \mathcal{H}_{n-r}^{\geq i}$, which does not decrease the $P$-adic valuation.\\ Now looking at the expression for $D_{r,p}$ in Lemma $2.4.3$, we see that on the elements considered, \footnote{i.e. of the form $\zeta^{\mathfrak{m}} \left({x_{1}, \ldots , x_{p} \atop \epsilon_{1}, \ldots ,\epsilon_{p-1}, \epsilon_{p}\xi_{N} }\right)$, with $\epsilon_{i}\in \pm 1$ for $N=8$, $\epsilon_{i}=1$ else.} the left side is: \begin{center} Either $\zeta^{\mathfrak(l)}\left( r\atop 1 \right) $ for type $\textsc{a,b,c} \qquad $ Or $\zeta^{\mathfrak(l)}\left( r\atop \xi \right) $ for Deconcatenation terms. \end{center} Using results in depth $1$ of Deligne and Goncharov (cf. $\S 2.4.3$), the deconcatenation terms are $P$-adically smaller. \\ \texttt{For instance}, for $N=\mlq 6 \mrq$, $r$ odd: $$\zeta^{\mathfrak{l}}\left( r; 1\right) =\frac{2\cdot 6^{r-1}}{(1-2^{r-1})(1-3^{r-1})} \zeta^{\mathfrak{l}}(r; \xi) , \quad \text{ and } \quad v_{3} \left( \frac{2\cdot 6^{r-1}}{(1-2^{r-1})(1-3^{r-1})}\right) >0 .$$ \texttt{Nota Bene:} For $N=8$, $D_{r}$ has two independent components, $D_{r}^{\xi}$ and $D_{r}^{-\xi}$. We have to distinguish them, but the statement remains similar since the terms appearing in the left side are either $\zeta^{\mathfrak(l)}\left( r\atop \pm 1 \right)$, or deconcatenation terms, $\zeta^{\mathfrak(l)}\left( r\atop \pm \xi \right)$, $2$-adically smaller by $\S 4.1$.\\ Thanks to congruences modulo $P$, only the deconcatenation terms remain:\\ $$D_{r,p} \left(\zeta^{\mathfrak{m}} \left({x_{1}, \ldots , x_{p} \atop \epsilon_{1}, \ldots ,\epsilon_{p-1},\epsilon_{p} \xi }\right)\right) = $$ $$ \delta_{ x_{p} \leq r \leq x_{p}+ x_{p-1}-1} (-1)^{r-x_{p}} \binom{r-1}{x_{p}-1} \zeta ^{\mathfrak{l}} \left( r\atop \epsilon_{p}\xi \right) \otimes \zeta^{\mathfrak{m}} \left({ x_{1}, \ldots, x_{p-2}, x_{p-1}+x_{p}-r\atop \epsilon_{1}, \cdots, \epsilon_{p-2}, \epsilon_{p-1}\epsilon_{p}\xi} \right) \pmod{P}.$$ As in the previous case, the matrix being modulo $P$ triangular with $1$ on the diagonal, has a determinant congruent at $1$ modulo $P$, and then, in particular, is invertible. \\ \end{proof}
\paragraph{{\Large \texttt{EXAMPLE for} $\boldsymbol{N=2}$}:} Let us illustrate the previous proof by an example, for weight $n=9$, depth $p=3$, level $i=0$, with the previous notations.\\ Instead of $\mathcal{B}_{9, 3, \geq 0}$, we will restrict to the subfamily (corresponding to $\mathcal{A}$): $$\mathcal{B}_{9, 3, \geq 0}^{0}\mathrel{\mathop:}= \left\{ \zeta^{\mathfrak{m}}(2a+1,2b+1,\overline{2c+1}) \text{ of weight } 9 \right\} \subset$$ $$ \mathcal{B}_{9, 3, \geq 0}\mathrel{\mathop:}= \left\{ \zeta^{\mathfrak{m}}(2a+1,2b+1,\overline{2c+1})\zeta^{\mathfrak{m}}(2)^{s}\text{ of weight } 9 \right\}$$ Note that $\zeta^{\mathfrak{m}}(2)$ being trivial under the coaction, the matrix $M_{9,3}$ is diagonal by blocks following the different values of $s$ and we can prove the invertibility of each block separately; here we restrict to the block $s=0$. The matrix $\widetilde{M}$ considered represents the coefficients of: $$\zeta^{\mathfrak{m}}(\overline{2r+1})\otimes \zeta^{\mathfrak{m}}(2x+1,\overline{2y+1})\quad \text{ in }\quad D_{2r+1,3}(\zeta^{\mathfrak{m}}(2a+1,2b+1,\overline{2c+1})).$$ The chosen order for the columns, resp. for the rows \footnote{I.e. for $\zeta^{\mathfrak{m}}(2a+1,2b+1,2c+1)$ resp. for $(D_{2r+1,3}, \zeta^{\mathfrak{m}}(2x+1,\overline{2y+1}))$.} is the lexicographic order applied to $(c,b,a)$ resp. to $(r,y,x)$. Modulo $2$, it only remains the terms of type \textsc{d,d'}, that is: $$ D_{2r+1,3} (\zeta^{\mathfrak{m}}(2a+1, 2b+1, \overline{2c+1})) \equiv \delta_{c \leq r \leq b+c} \binom{2r}{2c} \zeta^{\mathfrak{m}} (2a+1, \overline{2 (b+c-r) +1}) \text{ } \pmod{ 2}.$$ With the previous order, $\widetilde{M}_{9,3}$ is then, modulo $2$:\footnote{Notice that the first four rows are exact: no need of congruences modulo $2$ for $D_{1}$ because it acts as a deconcatenation on the base.}\\ \\
\begin{tabular}{c|c|c|c|c|c|c|c|c|c|c}
$D_{r}, \zeta\backslash$ $\zeta$& $7,1,\overline{1}$ & $5,3,\overline{1}$ & $3,5,\overline{1}$& $1,7,\overline{1}$& $5,1,\overline{3}$&$3,3,\overline{3}$&$1,5,\overline{3}$&$3,1,\overline{5}$&$1,3,\overline{5}$ & $1,1,\overline{7}$ \\
\hline
$D_{1},\zeta^{\mathfrak{m}}(7,\overline{1})$ & $1$ & $0$ &$0$ &$0$ &$0$ &$0$ &$0$ &$0$ &$0$ &$0$ \\
$D_{1},\zeta^{\mathfrak{m}}(5,\overline{3})$ & $0$ & $1$ &$0$ &$0$ &$0$ &$0$ &$0$ &$0$ &$0$ &$0$ \\
$D_{1},\zeta^{\mathfrak{m}}(3,\overline{5})$ & $0$ & $0$ &$1$ &$0$ &$0$ &$0$ &$0$ &$0$ &$0$ &$0$ \\
$D_{1},\zeta^{\mathfrak{m}}(1,\overline{7})$ & $0$ & $0$ &$0$ &$1$ &$0$ &$0$ &$0$ &$0$ &$0$ &$0$ \\
$D_{3},\zeta^{\mathfrak{m}}(5,\overline{7})$ & $0$ & $0$ &$0$ &$0$ &$1$ &$0$ &$0$ &$0$ &$0$ &$0$ \\
$D_{3},\zeta^{\mathfrak{m}}(3,\overline{3})$ & $0$ & $0$ &$0$ &$0$ &$0$ &$1$ &$0$ &$0$ &$0$ &$0$ \\
$D_{3},\zeta^{\mathfrak{m}}(1,\overline{5})$ & $0$ & $0$ &$0$ &$0$ &$0$ &$0$ &$1$ &$0$ &$0$ &$0$ \\
$D_{5},\zeta^{\mathfrak{m}}(3,\overline{1})$ & $0$ & $0$ &$0$ &$0$ &$0$ &$\binom{4}{2}$ &$0$ &$1$ &$0$ &$0$ \\
$D_{5},\zeta^{\mathfrak{m}}(1,\overline{3})$ & $0$ & $0$ &$0$ &$0$ &$0$ &$0$ &$\binom{4}{2}$ &$0$ &$1$ &$0$ \\
$D_{7},\zeta^{\mathfrak{m}}(1,\overline{1})$ & $0$ & $0$ &$0$ &$0$ &$0$ &$0$ &$\binom{6}{2}$ &$0$ &$\binom{6}{4}$ &$1$ \\
\\ \end{tabular}. As announced, $\widetilde{M}$ modulo $2$ is triangular with $1$ on the diagonal, thus obviously invertible.
\paragraph{ {\Large Proof of the Theorem $\boldsymbol{5.2.4}$}.} \begin{proof} This Theorem comes down to the Lemma $5.2.3$ proving the freeness of $\mathcal{B}_{n, p, \geq i}$ in $gr_{p}^{\mathfrak{D}} \mathcal{H}_{n}^{\geq i}$ defining a $\mathbb{Z}_{odd}$-structure: \begin{itemize}
\item[$(i)$] By this Lemma, $\mathcal{B}_{n, p, \geq i}$ is linearly free in the depth graded, and $\partial^{i,\mathcal{d}}_{n,p}$, which decreases strictly the depth, is bijective on $\mathcal{B}_{n, p, \geq i}$. The family $\mathcal{B}_{n, \leq p, \geq i}$, all depth mixed is then linearly independent on $\mathcal{F}_{p}^{\mathfrak{D}} \mathcal{H}_{n}^{\geq i}\subset \mathcal{F}_{p}^{\mathfrak{D}} \mathcal{H}_{n}^{\geq i, \mathcal{MT}}$: easily proved by application of $\partial^{i,\mathcal{d}}_{n,p}$.\\
By a dimension argument, since $\dim \mathcal{F}_{p}^{\mathfrak{D}} \mathcal{H}_{n}^{\geq i, \mathcal{MT}}= \text{ card } \mathcal{B}_{n, \leq p, \geq i}$, we deduce the generating property.
\item[$(ii)$] By the lemma, this family is linearly independent, and by $(i)$ applied to depth $p-1$,
$$gr_{p}^{\mathfrak{D}} \mathcal{H}_{n}^{\geq i}\subset gr_{p}^{\mathfrak{D}} \mathcal{H}_{n}^{\geq i, \mathcal{MT}}.$$
Then, by a dimension argument, since $\dim gr_{p}^{\mathfrak{D}} \mathcal{H}_{n}^{\geq i, \mathcal{MT}} = \text{ card } \mathcal{B}_{n, p, \geq i}$ we conclude on the generating property. The $\mathbb{Z}_{odd}$ structure has been proven in the previous lemma.\\
By the bijectivity of $\partial_{n,p}^{i,\mathcal{d}}$ (still previous lemma), which decreases the depth, and using the freeness of the elements of a same depth in the depth graded, there is no linear relation between elements of $\mathcal{B}_{n,\cdot, \geq i}$ of different depths in $\mathcal{H}_{n}^{\geq i} \subset \mathcal{H}^{\geq i \mathcal{MT}}_{n}$. The family considered is then linearly independent in $\mathcal{H}_{n}^{\geq i}$. Since $\text{card } \mathcal{B}_{n,\cdot, \geq i} =\dim \mathcal{H}^{\geq i, \mathcal{MT}}_{n}$, we conclude on the equality of the previous inclusions.
\item[$(iii)$] The second exact sequence is obviously split since $ \mathcal{B}_{n, \cdot,\geq i+1}$ is a subset of $\mathcal{B}_{n}$. We already know that $\mathcal{B}_{n}$ is a basis of $\mathcal{H}_{n}$ and $\mathcal{B}_{n, \cdot, \geq i+1}$ is a basis of $\mathcal{H}_{n}^{\geq i+1}$. Therefore, it gives a map $\mathcal{H}_{n} \leftarrow\mathcal{H}_{n}^{\geq i+1}$ and split the first exact sequence. \\
The construction of $cl_{n,\leq p, \geq i}(x)$, obtained from $cl_{n,p, \geq i}(x)$ applied repeatedly, is the following: \begin{center}
$x\in\mathcal{B}_{n, \cdot, \leq i-1} $ is sent on $\bar{x}\in \mathcal{H}_{n}^{\geq i} \cong \langle\mathcal{B}_{n, \leq p, \geq i}\rangle_{\mathbb{Q}} $ by the projection $\pi_{0,i}$ and so $x -\bar{x} \in \mathcal{F}_{i-1}\mathcal{H}$. \end{center} Notice that the problem of making $cl(x)$ explicit boils down to the problem of describing the map $\pi_{0,i}$ in the bases $\mathcal{B}$.
\item[$(iv)$] By the previous statements, these elements are linearly independent in $\mathcal{F}_{i} \mathcal{H}^{MT}_{n}$. Moreover, their cardinal is equal to the dimension of $\mathcal{F}_{i} \mathcal{H}^{MT}_{n}$. It gives the basis announced, composed of elements $x\in \mathcal{B}_{n, \cdot, \leq i}$, each corrected by an element denoted $cl(x)$ of $ \langle\mathcal{B}_{n, \cdot, \geq i+1}\rangle_{\mathbb{Q}}$.
\item[$(v)$] By the previous statements, these elements are linearly independent in $gr_{i} \mathcal{H}_{n}$, and by a dimension argument, we can conclude. \end{itemize} \end{proof}
\subsection{Specified Results}
\subsubsection{\textsc{The case } $N=2$.} Here, since there is only one Galois descent from $\mathcal{H}^{2}$ to $\mathcal{H}^{1}$, the previous exponents for level filtrations can be omitted, as the exponent $2$ for $\mathcal{H}$ the space of motivic Euler sums. Set $\mathbb{Z}_{\text{odd}}= \left\{ \frac{a}{b} \text{ , } a\in\mathbb{Z}, b\in 2 \mathbb{Z}+1 \right\}$, rationals having a $2$-adic valuation positive or infinite. Let us define particular families of motivic Euler sums, a notion of level and of motivic level. \begin{defi} \begin{itemize}
\item[$\cdot$] $\mathcal{B}^{2}\mathrel{\mathop:}=\left\{\zeta^{\mathfrak{m}}(2x_{1}+1, \ldots, 2 x_{p-1}+1,\overline{2 x_{p}+1}) \zeta(2)^{\mathfrak{m},k}, x_{i} \geq 0, k \in \mathbb{N} \right\}.$\\ Here, the level is defined as the number of $x_{i}$ equal to zero.
\item[$\cdot$] The filtration by the motivic ($\mathbb{Q}/\mathbb{Q},2/1$)-level, $$\mathcal{F}_{i}\mathcal{H}\mathrel{\mathop:}=\left\{ \mathfrak{Z} \in \mathcal{H}, \textrm{ such that } D^{-1}_{1}\mathfrak{Z} \in \mathcal{F}_{i-1} \mathcal{H} \text{ , } \forall r>0, D^{1}_{2r+1}\mathfrak{Z} \in \mathcal{F}_{i}\mathcal{H} \right\}.$$ \begin{center} I.e. $\mathcal{F}_{i}$ is the largest submodule such that $\mathcal{F}_{i} / \mathcal{F}_{i-1}$ is killed by $D_{1}$. \end{center} \end{itemize} \end{defi} This level filtration commutes with the increasing depth filtration.\\ \\ \textsc{Remarks}: The increasing or decreasing filtration defined from the number of 1 appearing in the motivic multiple zeta values is not preserved by the coproduct, since the number of 1 can either decrease or increase (by at the most 1) and is therefore not \textit{motivic}.\\ \\ Let list some consequences of the results in $\S 5.2.3$, which generalize in particular a result similar to P. Deligne's one (cf. $\cite{De}$): \begin{coro} The map $\mathcal{G}^{\mathcal{MT}} \rightarrow \mathcal{G}^{\mathcal{MT}'}$ is an isomorphism.\\ The elements of $\mathcal{B}_{n}$, $\zeta^{\mathfrak{m}}(2x_{1}+1, \ldots, \overline{2 x_{p}+1}) \zeta(2)^{k}$ of weight $n$, form a basis of motivic Euler sums of weight $n$, $\mathcal{H}^{2}_{n}=\mathcal{H}^{\mathcal{MT}_{2}}_{n}$, and define a $\mathbb{Z}_{odd}$-structure on the motivic Euler sums. \end{coro} \noindent The period map, $\text{per}: \mathcal{H} \rightarrow \mathbb{C}$, induces the following result for the Euler sums: \begin{center} Each Euler sum is a $\mathbb{Z}_{odd}$-linear combination of Euler sums \\ $\zeta(2x_{1}+1, \ldots, \overline{2 x_{p}+1}) \zeta(2)^{k}, k\geq 0, x_{i} \geq 0$ of the same weight. \end{center}
\noindent Here is the result on the $0^{\text{th}}$ level of the Galois descent from $\mathcal{H}^{1}$ to $\mathcal{H}^{2}$: \begin{coro} $$\mathcal{F}_{0}\mathcal{H}^{\mathcal{MT}_{2}}=\mathcal{F}_{0}\mathcal{H}^{2}=\mathcal{H}^{\mathcal{MT}_{1}}=\mathcal{H}^{1} .$$ A basis of motivic multiple zeta values in weight $n$, is formed by terms of $\mathcal{B}_{n}$ with $0$-level each corrected by linear combinations of elements of $\mathcal{B}_{n}$ of level $1$: \begin{multline}\nonumber \mathcal{B}_{n}^{1}\mathrel{\mathop:}=\left\{ \zeta^{\mathfrak{m}}(2x_{1}+1, \ldots, \overline{2x_{p}+1})\zeta^{\mathfrak{m}}(2)^{s} + \sum_{y_{i} \geq 0 \atop \text{at least one } y_{i} =0} \alpha_{\textbf{x} , \textbf{y}} \zeta^{\mathfrak{m}}(2y_{1}+1, \ldots, \overline{2y_{p}+1})\zeta^{\mathfrak{m}}(2)^{s} + \right.\\ \left. \sum_{\text{lower depth } q<p, z_{i}\geq 0 \atop \text{ at least one } z_{i} =0} \beta_{\textbf{x}, \textbf{z}} \zeta^{\mathfrak{m}}(2 z_{1}+1, \ldots, \overline{2 z_{q}+1})\zeta^{\mathfrak{m}}(2)^{s}, x_{i}>0 , \alpha_{\textbf{x} , \textbf{y}} , \beta_{\textbf{x} , \textbf{z}} \in\mathbb{Q},\right\}_{\sum x_{i}= \sum y_{i}=\sum z_{i}= \frac{n-p}{2} -s}. \end{multline} \end{coro}
\paragraph{Honorary.} About the first condition in $\ref{criterehonoraire}$ to be honorary: \begin{lemm}\label{condd1} Let $\zeta^{\mathfrak{m}}(n_{1},\cdots,n_{p}) \in\mathcal{H}^{2}$, a motivic Euler sum, with $n_{i}\in\mathbb{Z}^{\ast}$, $ n_{p}\neq 1$. Then: $$\forall i \text{ , } n_{i}\neq -1 \Rightarrow D_{1}(\zeta^{\mathfrak{m}}(n_{1},\cdots,n_{p}))=0 $$ \end{lemm} \begin{proof} Looking at all iterated integrals of length $1$ in $\mathcal{L}$, $I^{\mathfrak{l}}(a;b;c)$, $a,b,c\in \lbrace 0,\pm 1\rbrace$: the only non zero ones are these with a consecutive $\lbrace 1,-1\rbrace$ or $\lbrace -1,1\rbrace$ sequence in the iterated integral, with the condition that extremities are different, that is: $$I(0;1;-1), I(0;-1;1), I(1;-1;0), I(-1;+1;0), I(-1;\pm 1;1), I(1;\pm 1;-1).$$ Moreover, they are all equal to $\pm \log^{\mathfrak{a}} (2)$ in the Hopf algebra $\mathcal{A}$. Consequently, if there is no $-1$ in the Euler sums notation, it implies that $D_{1}$ would be zero. \end{proof}
\paragraph{Comparison with Hoffman's basis. } Let compare: \begin{itemize} \item[$(i)$] The Hoffman basis of $\mathcal{H}^{1}$ formed by motivic MZV with only $2$ and $3$ ($\cite{Br2}$) $$\mathcal{B}^{H}\mathrel{\mathop:}= \left\{\zeta^{\mathfrak{m}} (x_{1}, \ldots, x_{k}), \text{ where } x_{i}\in\left\{2,3\right\} \right\}.$$ \item[$(ii)$] $\mathcal{B}^{1}$, the base of $\mathcal{H}^{1}$ previously obtained (Corollary $5.2.7$). \end{itemize}
Beware, the index $p$ for $\mathcal{B}^{H}$ indicates the number of 3 among the $x_{i}$, whereas for $\mathcal{B}^{1}$, it still indicates the depth; in both case, it can be seen as the \textit{motivic depth} (cf. $\S 2.4.3$):
\begin{coro} $\mathcal{B}^{1}_{n,p}$ is a basis of $gr_{p}^{\mathfrak{D}} \langle\mathcal{B}^{H}_{n,p}\rangle_{\mathbb{Q}}$ and defines a $\mathbb{Z}_{\text{odd}}$-structure.\\ I.e. each element of the Hoffman basis of weight $n$ and with $p$ three, $p>0$, decomposes into a $\mathbb{Z}_{\text{odd}}$-linear combination of $\mathcal{B}^{1}_{n,p}$ elements plus terms of depth strictly less than $p$. \end{coro} \begin{proof} Deduced from the previous results, with the $\mathcal{Z}_{odd}$ structure of the basis for Euler sums. \end{proof}
\subsubsection{\textsc{The cases } $N=3,4$.}
For $N=3,4$ there are a generator in each degree $\geq 1$ and two Galois descents. \\ \begin{defi} \begin{itemize}
\item[$\cdot$] \textbf{Family:} $\mathcal{B}\mathrel{\mathop:}=\left\{\zeta^{\mathfrak{m}}\left({x_{1}, \ldots,x_{p}\atop 1, \ldots , 1, \xi }\right) (2i \pi)^{s,\mathfrak{m}}, x_{i} \geq 1, s \geq 0 \right\}$.
\item[$\cdot$] \textbf{Level:} $$\begin{array}{lll} \text{ The $(k_{N}/k_{N},P/1)$-level } & \text{ is defined as } & \text{ the number of $x_{i}$ equal to 1 }\\ \text{ The $(k_{N}/\mathbb{Q},P/P)$-level } & \text{ } & \text{ the number of $x_{i}$ even }\\ \text{ The $(k_{N}/\mathbb{Q},P/1)$-level } & \text{ } & \text{ the number of even $x_{i}$ or equal to $1$ } \end{array} $$
\item[$\cdot$] \textbf{Filtrations by the motivic level:} $\mathcal{F}^{\mathcal{d}} _{-1} \mathcal{H}^{N}=0$ and $\mathcal{F}^{\mathcal{d}} _{i} \mathcal{H}^{N}$ is the largest submodule of $\mathcal{H}^{N}$ such that $\mathcal{F}^{\mathcal{d}}_{i}\mathcal{H}^{N}/\mathcal{F}^{\mathcal{d}} _{i-1}\mathcal{H}^{N}$ is killed by $\mathscr{D}^{\mathcal{d}}$, where
$$\mathscr{D}^{\mathcal{d}} = \begin{array}{ll} \lbrace D^{\xi}_{1} \rbrace & \text{ for } \mathcal{d}=(k_{N}/k_{N},P/1)\\ \lbrace(D^{\xi}_{2r})_{r>0} \rbrace & \text{ for } \mathcal{d}=(k_{N}/\mathbb{Q},P/P)\\ \lbrace D^{\xi}_{1},(D^{\xi}_{2r})_{r>0} \rbrace & \text{ for } \mathcal{d}=(k_{N}/\mathbb{Q},P/1) \\ \end{array}. $$ \end{itemize} \end{defi} \textsc{Remarks}: \begin{itemize} \item[$\cdot$] As before, the increasing, or decreasing, filtration that we could define by the number of 1 (resp. number of even) appearing in the motivic multiple zeta values is not preserved by the coproduct, since the number of 1 can either diminish or increase (at most 1), so is not motivic. \item[$\cdot$] An effective way of seeing those motivic level filtrations, giving a recursive criterion: $$\hspace*{-0.5cm}\mathcal{F}_{i}^{k_{N}/\mathbb{Q},P/P }\mathcal{H}= \left\{ \mathfrak{Z} \in \mathcal{H}, \textrm{ s. t. } \forall r > 0 \text{ , } D^{\xi}_{2r}(\mathfrak{Z}) \in \mathcal{F}_{i-1}^{k_{N}/\mathbb{Q},P/P}\mathcal{H} \text{ , } \forall r \geq 0 \text{ , } D^{\xi}_{2r+1}(\mathfrak{Z}) \in \mathcal{F}_{i}^{ k_{N}/\mathbb{Q},P/P}\mathcal{H} \right\}.$$ \end{itemize} \noindent We deduce from the result in $\S 5.2.3$ a result of P. Deligne ($i=0$, cf. $\cite{De}$): \begin{coro} The elements of $\mathcal{B}^{N}_{n,p, \geq i}$ form a basis of $gr_{p}^{\mathfrak{D}} \mathcal{H}_{n}/ \mathcal{F}_{i-1} \mathcal{H}_{n}$.\\ In particular the map $\mathcal{G}^{\mathcal{MT}_{N}} \rightarrow \mathcal{G}^{\mathcal{MT}_{N}'}$ is an isomorphism. The elements of $\mathcal{B}_{n}^{N}$, form a basis of motivic multiple zeta value relative to $\mu_{N}$, $\mathcal{H}_{n}^{N}$. \end{coro} The level $0$ of the filtrations considered for $N' \vert N\in \left\lbrace 3,4 \right\rbrace $ gives the Galois descents: \begin{coro}
A basis of $\mathcal{H}_{n}^{N'} $ is formed by elements of $\mathcal{B}_{n}^{N}$ of level $0$ each corrected by linear combination of elements $\mathcal{B}_{n}^{N}$ of level $ \geq 1$. In particular, with $\xi$ primitive: \begin{itemize}
\item[$\cdot$] \textbf{Galois descent} from $N'=1$ to $N=3,4$: A basis of motivic multiple zeta values: $$\hspace*{-0.5cm}\mathcal{B}^{1 ; N} \mathrel{\mathop:}= \left\{ \zeta^{\mathfrak{m}}\left({2x_{1}+1, \ldots, 2x_{p}+1\atop 1, \ldots, 1, \xi} \right) \zeta^{\mathfrak{m}}(2)^{s} + \sum_{y_{i} \geq 0 \atop \text{ at least one $y_{i}$ even or } = 1} \alpha_{\textbf{x},\textbf{y}} \zeta^{\mathfrak{m}} \left({y_{1}, \ldots, y_{p}\atop 1, \ldots, 1, \xi } \right)\zeta^{\mathfrak{m}}(2)^{s} \right.$$ $$ \left. + \sum_{\text{ lower depth } q<p, \atop \text{ at least one even or } = 1} \beta_{\textbf{x},\textbf{z}} \zeta^{\mathfrak{m}}\left({z_{1}, \ldots, z_{q}\atop 1, \ldots, 1, \xi } \right)\zeta^{\mathfrak{m}}(2)^{s} \text{ , } x_{i}>0 , \alpha_{\textbf{x},\textbf{y}} , \beta_{\textbf{x},\textbf{z}} \in\mathbb{Q} \right\}. $$
\item[$\cdot$] \textbf{Galois descent} from $N'=2$ to $N=4$: A basis of motivic Euler sums: $$\hspace*{-0.5cm}\mathcal{B}^{2; 4}\mathrel{\mathop:}= \left\{ \zeta^{\mathfrak{m}} \left({2x_{1}+1, \ldots, 2x_{p}+1\atop 1, \ldots, 1, \xi_{4}} \right)\zeta^{\mathfrak{m}}(2)^{s} + \sum_{y_{i}>0 \atop \text{at least one even}} \alpha_{\textbf{x},\textbf{y}} \zeta^{\mathfrak{m}}\left({y_{1}, \ldots, y_{p}\atop 1, \ldots, 1, \xi_{4} } \right)\zeta^{\mathfrak{m}}(2)^{s} \right.$$ $$ \left. +\sum_{\text{lower depth } q<p \atop z_{i}>0, \text{at least one even}} \beta_{\textbf{x},\textbf{z}} \zeta^{\mathfrak{m}}\left( z_{1}, \ldots, z_{q} \atop 1, \ldots, 1, \xi_{4} \right) \zeta^{\mathfrak{m}}(2)^{s} \text{ , } x_{i}\geq 0 , \alpha_{\textbf{x},\textbf{y}}, \beta_{\textbf{x},\textbf{z}}\in\mathbb{Q} \right\} .$$ \item[$\cdot$] Similarly, replacing $\xi_{4}$ by $\xi_{3}$ in $\mathcal{B}^{2; 4}$, this gives a basis of: $$\mathcal{F}^{k_{3}/\mathbb{Q},3/3}_{0} \mathcal{H}_{n}^{3}=\boldsymbol{\mathcal{H}_{n}^{\mathcal{MT}(\mathbb{Z}[\frac{1}{3}])}}.$$ \item[$\cdot$] A basis of $\mathcal{F}^{k_{N}/k_{N},P/1}_{0} \mathcal{H}_{n}^{N}=\boldsymbol{\mathcal{H}_{n}^{\mathcal{MT}(\mathcal{O}_{N})}}$, with $N= 3,4$: $$\hspace*{-0.5cm}\mathcal{B}^{N \text{ unram}}\mathrel{\mathop:}= \left\{ \zeta^{\mathfrak{m}} \left({x_{1}, \ldots, x_{p} \atop 1, \ldots, 1, \xi} \right)\zeta^{\mathfrak{m}}(2)^{s} + \sum_{y_{i}>0 \atop \text{at least one } 1} \alpha_{\textbf{x},\textbf{y}} \zeta^{\mathfrak{m}}\left({y_{1}, \ldots, y_{p}\atop 1, \ldots, 1, \xi} \right)\zeta^{\mathfrak{m}}(2)^{s} \right.$$ $$ \left. +\sum_{\text{lower depth } q<p \atop z_{i}>0, \text{at least one } 1} \beta_{\textbf{x},\textbf{z}} \zeta^{\mathfrak{m}}\left( z_{1}, \ldots, z_{q} \atop 1, \ldots, 1, \xi \right) \zeta^{\mathfrak{m}}(2)^{s} \text{ , } x_{i} > 0 , \alpha_{\textbf{x},\textbf{y}}, \beta_{\textbf{x},\textbf{z}}\in\mathbb{Q} \right\} .$$ \end{itemize} \end{coro} \noindent \texttt{Nota Bene:} Notice that for the last two level $0$ spaces, $\mathcal{H}_{n}^{\mathcal{MT}(\mathcal{O}_{N})}$, $N=3,4$ and $\mathcal{H}_{n}^{\mathcal{MT}(\mathbb{Z}[\frac{1}{3}])}$, we still do not have another way to reach them, since those categories of mixed Tate motives are not simply generated by a motivic fundamental group.
\subsubsection{\textsc{The case } $N=8$.}
For $N=8$ there are two generators in each degree $\geq 1$ and three possible Galois descents: with $\mathcal{H}^{4}$, $\mathcal{H}^{2}$ or $\mathcal{H}^{1}$.\\
\begin{defi} \begin{itemize}
\item[$\cdot$] \textbf{Family:} $\mathcal{B}\mathrel{\mathop:}=\left\{\zeta^{\mathfrak{m}}\left( {x_{1}, \ldots,x_{p}\atop \epsilon_{1}, \ldots , \epsilon_{p-1},\epsilon_{p} \xi }\right)(2i \pi)^{s,\mathfrak{m}}, x_{i} \geq 1, \epsilon_{i}\in \left\{\pm 1\right\} s \geq 0 \right\}$.
\item[$\cdot$] \textbf{Level}, denoted $i$: $$\begin{array}{lll}
\text{ The $(k_{8}/k_{4},2/2)$-level } & \text{ is the number of } & \text{ $\epsilon_{j}$ equal to $-1$ } \\
\text{ The $(k_{8}/\mathbb{Q},2/2)$-level } & \text{ } & \text{ $\epsilon_{j}$ equal to $-1$ $+$ even $x_{j}$ } \\
\text{ The $(k_{8}/\mathbb{Q},2/1)$-level } & \text{ } & \text{ $\epsilon_{j}$ equal to $-1$, $+$ even $x_{j}$ $+$ $x_{j}$ equal to $1$. } \end{array}$$
\item[$\cdot$] \textbf{Filtrations by the motivic level:} $\mathcal{F}^{\mathcal{d}} _{-1} \mathcal{H}^{8}=0$ and $\mathcal{F}^{\mathcal{d}} _{i} \mathcal{H}^{8}$ is the largest submodule of $\mathcal{H}^{8}$ such that $\mathcal{F}^{\mathcal{d}}_{i}\mathcal{H}^{8}/\mathcal{F}^{\mathcal{d}} _{i-1}\mathcal{H}^{8}$ is killed by $\mathscr{D}^{\mathcal{d}}$, where $$\mathscr{D}^{\mathcal{d}} = \begin{array}{ll} \left\lbrace (D^{\xi}_{r}- D^{-\xi}_{r})_{r>0} \right\rbrace & \text{ for } \mathcal{d}=(k_{8}/k_{4},2/2)\\ \left\lbrace (D^{\xi}_{2r+1}- D^{-\xi}_{2r+1})_{r\geq 0}, (D^{\xi}_{2r})_{r>0},( D^{-\xi}_{2r})_{r>0} \right\rbrace & \text{ for } \mathcal{d}=(k_{8}/\mathbb{Q},2/2)\\ \left\lbrace (D^{\xi}_{2r+1}- D^{-\xi}_{2r+1})_{r> 0}, D^{\xi}_{1}, D^{-\xi}_{1}, (D^{\xi}_{2r})_{r>0},( D^{-\xi}_{2r})_{r>0} \right\rbrace & \text{ for } \mathcal{d}=(k_{8}/\mathbb{Q},2/1) \\ \end{array}. $$ \end{itemize} \end{defi}
\begin{coro}
A basis of $\mathcal{H}_{n}^{N'} $ is formed by elements of $\mathcal{B}_{n}^{N}$ of level $0$ each corrected by linear combination of elements $\mathcal{B}_{n}^{N}$ of level $ \geq 1$. In particular, with $\xi$ primitive: \begin{description} \item[$\boldsymbol{8 \rightarrow 1} $:] A basis of MMZV: $$\hspace*{-0.5cm}\mathcal{B}^{1;8}\mathrel{\mathop:}= \left\{ \zeta^{\mathfrak{m}}\left( 2x_{1}+1, \ldots, 2x_{p}+1 \atop 1, \ldots, 1, \xi \right)\zeta^{\mathfrak{m}}(2)^{s} + \sum_{y_{i} \text{at least one even or } =1 \atop { or one } \epsilon_{i}=-1 } \alpha_{\textbf{x},\textbf{y}} \zeta^{\mathfrak{m}}\left( y_{1}, \ldots, y_{p} \atop \epsilon_{1}, \ldots, \epsilon_{p-1}, \epsilon_{p}\xi \right)\zeta^{\mathfrak{m}}(2)^{s} \right.$$ $$\left. + \sum_{q<p \text{ lower depth, level } \geq 1} \beta_{\textbf{x},\textbf{z}} \zeta^{\mathfrak{m}}\left(z_{1}, \ldots, z_{q} \atop \widetilde{\epsilon}_{1}, \ldots, \widetilde{\epsilon}_{q}\xi \right)\zeta^{\mathfrak{m}}(2)^{s} \text{ , }x_{i}>0 , \alpha_{\textbf{x},\textbf{y}}, \beta_{\textbf{x},\textbf{z}}\in\mathbb{Q}\right\}.$$ \item[$\boldsymbol{8 \rightarrow 2 } $:] A basis of motivic Euler sums: $$\hspace*{-0.5cm}\mathcal{B}^{2;8} \mathrel{\mathop:}= \left\{ \zeta^{\mathfrak{m}} \left(2x_{1}+1, \ldots, 2x_{p}+1 \atop 1, \ldots, 1, \xi\right)\zeta^{\mathfrak{m}}(2)^{s} +\sum_{y_{i} \text{ at least one even} \atop \text{or one }\epsilon_{i}=-1} \alpha_{\textbf{ x},\textbf{y}} \zeta^{\mathfrak{m}}\left( y_{1}, \ldots, y_{p} \atop \epsilon_{1}, \ldots, \epsilon_{p-1}, \epsilon_{p}\xi \right)\zeta^{\mathfrak{m}}(2)^{s} \right.$$ $$\left. + \sum_{\text{lower depth} q<p \atop \text{with level} \geq 1 } \beta_{\textbf{x},\textbf{z}} \zeta^{\mathfrak{m}}\left(z_{1}, \ldots, z_{q} \atop \widetilde{\epsilon}_{1}, \ldots, \widetilde{\epsilon}_{q}\xi\right)\zeta^{\mathfrak{m}}(2)^{s} \text{ , }x_{i}\geq 0,\alpha_{\textbf{x},\textbf{y}}, \beta_{\textbf{x},\textbf{y}} \in\mathbb{Q} \right\}.$$
\item[$\boldsymbol{ 8 \rightarrow 4 } $:] A basis of MMZV relative to $\mu_{4}$: $$\hspace*{-0.5cm}\mathcal{B}^{4;8}\mathrel{\mathop:}= \left\{ \zeta^{\mathfrak{m}}\left( x_{1}, \ldots, x_{p} \atop 1, \ldots, 1, \xi \right)(2i\pi)^{s} + \sum_{\text{ at least one }\epsilon_{i}=-1} \alpha_{\textbf{x}, \textbf{y}} \zeta^{\mathfrak{m}}\left(y_{1}, \ldots, y_{p} \atop \epsilon_{1}, \ldots, \epsilon_{p-1},\epsilon_{p} \xi \right)(2 i \pi)^{s} \right.$$ $$ \left. + \sum_{\text{lower depth, level } \geq 1} \beta_{\textbf{x},\textbf{z}} \zeta^{\mathfrak{m}}\left( z_{1}, \ldots,z_{q} \atop \widetilde{\epsilon}_{1}, \ldots, \widetilde{\epsilon}_{q}\xi \right)(2i \pi)^{s} \alpha_{\textbf{x},\textbf{y}}, \beta_{\textbf{x},\textbf{z}}\in\mathbb{Q} \right\}.$$ \end{description} \end{coro}
\subsubsection{\textsc{The case } $N=\mlq 6 \mrq$.}
For the unramified category $\mathcal{MT}(\mathcal{O}_{6})$, there is one generator in each degree $>1$ and one Galois descent with $\mathcal{H}^{1}$.\\ \\ First, let us point out this sufficient condition for a MMZV$_{\mu_{6}}$ to be unramified: \begin{lemm} $$\text{Let } \quad \zeta^{\mathfrak{m}} \left( n_{1},\cdots,n_{p} \atop \epsilon_{1}, \ldots, \epsilon_{p} \right) \in\mathcal{H}^{\mathcal{MT}(\mathcal{O}_{6} \left[ \frac{1}{6}\right] )} \text{ a motivic MZV} _{\mu_{6}}, \quad \text{ such that : \footnote{In the iterated integral notation, the associated roots of unity are $\eta_{i}\mathrel{\mathop:}= (\epsilon_{i}\cdots \epsilon_{p})^{-1}$.}}$$ $$\begin{array}{ll}
& \text{ Each } \eta_{i} \in \lbrace 1, \xi_{6} \rbrace \\ \textsc{ or }& \text{ Each } \eta_{i} \in \lbrace 1, \xi^{-1}_{6} \rbrace \end{array} \quad \quad \text{ Then, } \quad \zeta^{\mathfrak{m}} \left( n_{1},\cdots,n_{p} \atop \epsilon_{1}, \ldots, \epsilon_{p} \right) \in \mathcal{H}^{\mathcal{MT}(\mathcal{O}_{6})}$$ \end{lemm} \begin{proof} Immediate, by Corollary, $\ref{ramif346}$, and with the expression of the derivations $(\ref{drz})$ since these families are stable under the coaction. \end{proof}
\begin{defi} \begin{itemize}
\item[$\cdot$] \textbf{Family}: $\mathcal{B}\mathrel{\mathop:}=\left\{\zeta^{\mathfrak{m}}\left( {x_{1}, \ldots,x_{p}\atop 1, \ldots , 1,\xi) } \right)(2i \pi)^{s,\mathfrak{m}}, x_{i} > 1, s \geq 0 \right\}$.
\item[$\cdot$] \textbf{Level:} The $(k_{6}/\mathbb{Q},1/1)$-level, denoted $i$, is defined as the number of even $x_{j}$.
\item[$\cdot$] \textbf{Filtration by the motivic } $(k_{6}/\mathbb{Q},1/1)$-\textbf{level}: \begin{center}
$\mathcal{F}^{(k_{6}/\mathbb{Q},1/1)} _{-1} \mathcal{H}^{6}=0$ and $\mathcal{F}^{(k_{6}/\mathbb{Q},1/1)} _{i} \mathcal{H}^{6}$ is the largest submodule of $\mathcal{H}^{6}$ such that $\mathcal{F}^{(k_{6}/\mathbb{Q},1/1)}_{i}\mathcal{H}^{6}/\mathcal{F}^{(k_{6}/\mathbb{Q},1/1)} _{i-1}\mathcal{H}^{6}$ is killed by $\mathscr{D}^{(k_{6}/\mathbb{Q},1/1)}=\left\lbrace D^{\xi}_{2r} , r>0 \right\rbrace $.
\end{center} \end{itemize} \end{defi}
\begin{coro} Galois descent from $N'=1$ to $N=\mlq 6 \mrq$ unramified. A basis of MMZV: $$\mathcal{B}^{1;6} \mathrel{\mathop:}= \left\{ \zeta^{\mathfrak{m}}\left( 2x_{1}+1, \ldots, 2x_{p}+1 \atop 1, \ldots, 1, \xi \right)\zeta^{\mathfrak{m}}(2)^{s} + \sum_{y_{i} \text{ at least one even}} \alpha_{\textbf{x},\textbf{y}}\zeta^{\mathfrak{m}}\left( y_{1}, \ldots, y_{p} \atop 1, \ldots, 1, \xi \right)\zeta^{\mathfrak{m}}(2)^{s} \right.$$ $$\left. +\sum_{\text{lower depth, level } \geq 1}\beta_{\textbf{x},\textbf{z}} \zeta^{\mathfrak{m}} \left( z_{1}, \ldots, z_{q} \atop 1, \ldots, 1, \xi \right)\zeta^{\mathfrak{m}}(2)^{s} \text{ , } \alpha_{\textbf{x},\textbf{y}}, \beta_{\textbf{x},\textbf{z}}\in \mathbb{Q}, x_{i}>0 \right\}.$$ \end{coro}
\chapter{Miscellaneous uses of the coaction}
\section{Maximal depth terms, $\boldsymbol{gr^{\mathfrak{D}}_{\max}\mathcal{H}_{n}}$}
The coaction enables us to compute, by a recursive procedure, the coefficients of the terms of \textit{maximal depth}, i.e. the projection on the graded $\boldsymbol{gr^{\mathfrak{D}}_{\max}\mathcal{H}_{n}}$. In particular, let look at: \begin{itemize} \item[$\cdot$] For $N=1$, when weight is a multiple of $3$ ($w=3d$), such as depth $p>d$: $$gr^{\mathfrak{D}}_{p}\mathcal{H}_{3d} =\mathbb{Q} \zeta^{\mathfrak{m}}(3)^{d}.$$ \item[$\cdot$] Another simple case is for $N=2,3,4$, when weight equals depth, which is referred to as the \textit{diagonal comodule}: $$gr^{\mathfrak{D}}_{p}\mathcal{H}_{p} =\mathbb{Q} \zeta^{\mathfrak{m}}\left( 1 \atop \xi_{N}\right) ^{p}.$$ \end{itemize} The space $gr^{\mathfrak{D}}_{\max}\mathcal{H}^{N}_{n}$ is usually more than $1$ dimensional, but the methods presented below could generalize.
\subsection{MMZV, weight $\boldsymbol{3d}$.}
\paragraph{Preliminaries: Linearized Ihara action.} The linearisation of the map $\circ: U\mathfrak{g} \otimes U\mathfrak{g} \rightarrow U\mathfrak{g}$ induced by Ihara action (cf. $\S 2.4$) can be defined recursively on words by, with $\eta\in\mu_{N}$:\nomenclature{$\underline{\circ}$}{linearized Ihara action} \begin{equation}\label{eq:circlinear} \begin{array}{lll}
\underline{\circ}: \quad U\mathfrak{g} \otimes U\mathfrak{g} \rightarrow U\mathfrak{g}: & a \underline{\circ} e_{0}^{n} & = e_{0}^{n} a \\
& a \underline{\circ} e_{0}^{n}e_{\eta} w & = e_{0}^{n} ([\eta].a) e_{\eta}w + e_{0}^{n} e_{\eta } ([\eta].a)^{\ast} w + e_{0}^{n} e_{\eta} (a\underline{\circ} w), \\ \end{array} \end{equation} where ${\ast}$ stands for the following involution: $$(a_{1} \cdots a_{n})^{\ast}\mathrel{\mathop:}=(-1)^{n}a_{n} \cdots a_{1}.$$ For this paragraph, from now, let $N=1$ and let use the \textit{commutative polynomial setting}, introducing the isomorphism:\nomenclature{ $\rho$}{isomorphism used to pass to a commutative polynomial setting} \begin{align}\label{eq:rho}
\rho: U \mathfrak{g} & \longrightarrow \mathbb{Q} \langle Y\rangle\mathrel{\mathop:}=\mathbb{Q} \langle y_{0}, y_{1}, \ldots, y_{n},\cdots \rangle \\
e_{0}^{n_{0}}e_{1} e_{0}^{n_{1}} \cdots e_{1} e_{0}^{n_{p}} & \longmapsto y_{0}^{n_{0}} y_{1}^{n_{1}} \cdots y_{p}^{n_{p}} \nonumber
\end{align} Remind that if $\Phi\in U \mathfrak{g}$ satisfies the linearized $\shuffle$ relation, it means that $\Phi$ is primitive for $\Delta_{\shuffle}$, and equivalently that $\phi_{u \shuffle v}=0$, with $\phi_{w}$ the coefficient of $w$ in $\Phi$. In particular, this is verified for $\Phi$ in the motivic Lie algebra $\mathfrak{g}^{\mathfrak{m}}$.\\ This property implies for $f=\rho(\Phi)$ a translation invariance (cf. $6.2$ in $\cite{Br3}$) \begin{equation} \label{eq:translationinv} f(y_{0},y_{1},\cdots, y_{p})= f(0,y_{1}-y_{0}, \ldots, y_{p}-y_{0}). \end{equation} Let consider the map: \begin{align} \label{eq:fbar} \mathbb{Q} \langle Y\rangle & \longrightarrow\mathbb{Q} \langle X\rangle = \mathbb{Q} \langle x_{1}, \ldots, x_{n},\cdots\rangle , & \\ \quad \quad f & \longmapsto \overline{f} & \text{ where } \overline{f}(x_{1},\cdots, x_{p})\mathrel{\mathop:}=f(0,x_{1},\cdots, x_{p}).\nonumber \end{align} If $f$ is translation invariant, $f(y_{0}, y_{1}, \ldots, y_{p})=\overline{f}(y_{1}-y_{0},\cdots, y_{p}-y_{0})$.\\ The image of $\mathfrak{g}^{\mathfrak{m}}$ under $\rho$ is contained in the subspace of polynomial in $y_{i}$ invariant by translation. Hence we can consider alternatively in the following $\phi\in\mathfrak{g}^{\mathfrak{m}}$, $f=\rho(\phi)$ or $\overline{f}$.\\ \\ Since the linearized action $\underline{\circ}$ respects the $\mathcal{D}$-grading, it defines, via the isomorphism $\rho: gr^{r}_{\mathfrak{D}} U \mathfrak{g} \rightarrow \mathbb{Q}[y_{0}, \ldots, y_{r}]$, graded version of $(\ref{eq:rho})$, a map: $$\underline{\circ}: \mathbb{Q}[y_{0}, \ldots, y_{r}]\otimes \mathbb{Q}[y_{0}, \ldots, y_{s}] \rightarrow \mathbb{Q}[y_{0}, \ldots, y_{r+s}] \text{ , which in the polynomial representation is:}$$ \begin{multline}\label{eq:circpolynom} f\underline{\circ} g (y_{0}, \ldots, y_{r+s})=\sum_{i=0}^{s} f(y_{i}, \ldots, y_{i+r})g(y_{0}, \ldots, y_{i}, y_{i+r+1}, \ldots, y_{r+s}) \\ + (-1)^{\deg f+r}\sum_{i=1}^{s} f(y_{i+r}, \ldots, y_{i})g(y_{0}, \ldots, y_{i-1}, y_{i+r}, \ldots, y_{r+s}). \end{multline} Or via the isomorphism $\overline{\rho}: gr^{r}_{\mathfrak{D}} U \mathfrak{g} \rightarrow \mathbb{Q}[x_{1}, \ldots, x_{r}]$, graded version of $(\ref{eq:fbar})\circ (\ref{eq:rho}) $: \begin{multline}\label{eq:circpolynomx} f\underline{\circ} g (x_{1}, \ldots, x_{r+s})=\sum_{i=0}^{s} f(x_{i+1}-x_{i}, \ldots, x_{i+r}-x_{i})g(y_{1}, \ldots, x_{i}, x_{i+r+1}, \ldots, x_{r+s}) \\ + (-1)^{\deg f+r}\sum_{i=1}^{s} f(x_{i+r-1}-x_{i+r}, \ldots, x_{i}-x_{i+r})g(x_{1}, \ldots, x_{i-1}, x_{i+r}, \ldots, x_{r+s}). \end{multline}
\paragraph{Coefficient of $\boldsymbol{\zeta(3)^{d}}$.} If the weight $w$ is divisible by $3$, for motivic multiple zeta values, it boils down to compute the coefficient of $\zeta^{\mathfrak{m}}(3)^{\frac{w}{3}}$ and a recursive procedure is given in Lemma $6.1.1$.\\ \\ Since $gr_{d}^{\mathfrak{D}} \mathcal{H}_{3d}^{1} $ is one dimensional, generated by $\zeta^{\mathfrak{m}}(3)^{d}$, we can consider the projection: \begin{equation} \vartheta : gr_{d}^{\mathfrak{D}} \mathcal{H}_{3d}^{1} \rightarrow \mathbb{Q}. \end{equation} Giving a motivic multiple zeta value $\zeta^{\mathfrak{m}}(n_{1}, \ldots, n_{d})$, of depth $d$ and weight $w=3d$, there exists a rational $\alpha_{\underline{\textbf{n}}}= \vartheta(\zeta(n_{1}, \ldots, n_{d}))$ such that: \begin{framed} \begin{equation} \zeta^{\mathfrak{m}}(n_{1}, \ldots, n_{d}) = \frac{\alpha_{\underline{\textbf{n}}}} {d!} \zeta^{\mathfrak{m}}(3)^{d} + \text{ terms of depth strictly smaller than } d. \footnote{The terms of depth strictly smaller than $d$ can be expressible in terms of the Deligne basis for instance.} \end{equation} \end{framed} \noindent In the depth graded in depth 1, $\partial \mathfrak{g}^{\mathfrak{m}}_{1}$, the generators are: $$\overline{\sigma}_{2i+1}= (-1)^{i} (\text{ad} e_{0})^{2i} (e_{1}) .$$ We are looking at, in the depth graded: \begin{equation} \label{eq:expcirc3} \exp_{\circ}(\overline{\sigma_{3}})\mathrel{\mathop:}=\sum_{n=0}^{n}\frac{1}{n!} \overline{\sigma_{3}} \circ \cdots \circ \overline{\sigma_{3}}= \sum_{n=0}^{n}\frac{1}{n!} (\text{ad}(e_{0})^{2} (e_{1}))^{\underline{\circ} n}. \end{equation} In the commutative polynomial representation, via $\overline{\rho}$, since $ \overline{\rho}(\overline{\sigma}_{2n+1})= x_{1}^{2n}$, it becomes: $$\sum_{n=0}^{n}\frac{1}{n!} x_{1}^{2} \underline{\circ} (x_{1}^{2} \underline{\circ}( \cdots (x_{1}^{2} \underline{\circ}x_{1}^{2} ) \cdots )).$$
\begin{lemm} The coefficient of $\zeta^{\mathfrak{m}}(3)^{p}$ in $\zeta^{\mathfrak{m}}(n_{1}, \ldots, n_{p})$ of weight $3p$ is given recursively: \begin{multline} \label{coeffzeta3} \alpha_{n_{1}, \ldots, n_{p}}= \delta_{n_{p}=3} \alpha_{n_{1}, \ldots, n_{p-1}}\\ +\sum_{k=1 \atop n_{k}=1 }^{p} \left( \delta_{n_{k-1}\geq 3} \alpha_{n_{1}, \ldots, n_{k-1}-2,n_{k+1}, \ldots, n_{p}} -\delta_{n_{k+1}\geq 3} \alpha_{n_{1}, \ldots, n_{k-1},n_{k+1}-2, \ldots, n_{p}} \right) \\ +2 \sum_{k=1 \atop n_{k}=2 }^{p} \left(-\delta_{n_{k-1}\geq 3} \alpha_{n_{1}, \ldots, n_{k-1}-2,n_{k+1}, \ldots, n_{p}} + \delta_{n_{k+1}\geq 3} \alpha_{n_{1}, \ldots, n_{k-1},n_{k+1}-2, \ldots, n_{p}} \right) . \end{multline} \end{lemm} \noindent \textsc{Remarks}: \begin{itemize} \item[$\cdot$] This is proved for motivic multiple zeta values, and by the period map, it also applies to multiple zeta values. \item[$\cdot$] This lemma (as the next one, more precise) could be generalized for unramified motivic Euler sums. \item[$\cdot$] All the coefficients $\alpha$ are all integers. \end{itemize} \begin{proof} Recursively, let consider: \begin{equation} P_{n+1} (x_{1},\cdots, x_{n+1})\mathrel{\mathop:}=x_{1}^{2} \underline{\circ}P_{n} (x_{1},\cdots, x_{n}). \end{equation} By the definition of the linearized Ihara action $(\ref{eq:circpolynom})$: \begin{multline} \nonumber P_{n+1} (x_{1},\cdots, x_{n+1}) =\sum_{i=0}^{n} (x_{i+1}-x_{i})^{2} P_{n} (x_{1},\cdots, x_{i}, x_{i+2}, \ldots, x_{n+1}) \\ - \sum_{i=1}^{n} (x_{i+1}-x_{i})^{2} P_{n} (x_{1},\cdots, x_{i-1}, x_{i+1}, \ldots, x_{n+1})\\ = (x_{n+1}-x_{n})^{2} P_{n}(x_{1}, \ldots, x_{n})+ \sum_{i=0}^{n-1} (x_{i}-x_{i+2})(x_{i}+x_{i+2}-2x_{i+1}) P_{n} (x_{1},\cdots, x_{i}, x_{i+2}, \ldots, x_{n+1}). \end{multline} Turning now towards the coefficients $c^{\textbf{i}}$ defined by: $$ P_{p} (x_{1},\cdots, x_{p})= \sum c^{\textbf{i}} x_{1}^{i_{1}}\cdots x_{p}^{i_{p}}, \quad \text{ we deduce: } $$ \begin{multline} \nonumber c^{i_{1}, \ldots, i_{p}}= -\delta_{i_{1}=0 \atop i_{2} \geq 2} c^{i_{2}-2,i_{3}, \ldots, i_{p}} + \delta_{i_{p}=2 } c^{i_{1},\cdots, i_{p-1}} + \delta_{i_{n}=0 \atop i_{p-1} \geq 2} c^{i_{1}, \ldots, i_{p-2},i_{p-1}-2} - 2 \delta_{i_{p}=1 \atop i_{p-1} \geq 2} c^{i_{1}, \ldots, i_{p-2}, i_{p-1}-1} \\ + \sum_{k=2 \atop i_{k}=0 }^{p-1} \left( \delta_{i_{k-1}\geq 2} c_{i_{1}, \ldots, i_{k-1}-2,i_{k+1}, \ldots, i_{p}} -\delta_{i_{k+1}\geq 2} c_{i_{1}, \ldots, i_{k-1},i_{k+1}-2, \ldots, i_{p}} \right)\\ + 2 \sum_{k=2 \atop i_{k}=1 }^{p-1} \left( -\delta_{i_{k-1}\geq 1} c_{i_{1}, \ldots, i_{k-1}-2,i_{k+1}, \ldots, i_{p}} + \delta_{i_{k+1}\geq 1} c_{i_{1}, \ldots, i_{k-1},i_{k+1}-2, \ldots, i_{p}} \right) , \end{multline} which gives the recursive formula of the lemma. \end{proof}
\paragraph{Generalization. }
Another proof of the previous lemma is possible using the dual point of view with the depth-graded derivations $D_{3,p}$, looking at cuts of length $3$ and depth $1$.\footnote{The coefficient $\alpha$ indeed emerges when computing $(D_{3,p})^{\circ p}$.}\\ A motivic multiple zeta value of weight $3d$ and of depth $p>d$ could also be expressed as: \begin{equation}\label{eq:zeta3d} \zeta^{\mathfrak{m}}(n_{1}, \ldots, n_{p}) = \frac{\alpha_{\underline{\textbf{n}}}} {d!} \zeta^{\mathfrak{m}}(3)^{d} + \text{ terms of depth strictly smaller than } d. \end{equation} However, to compute this coefficient $\alpha_{\underline{\textbf{n}}}$, we could not work as before in the depth graded; i.e. this time, we have to consider all the possible cuts of length $3$. Then, the coefficient emerges when computing $\boldsymbol{(D_{3})^{\circ d}}$. \begin{lemm} The coefficient of $\zeta^{\mathfrak{m}}(3)^{d}$ in $\zeta^{\mathfrak{m}}(n_{1}, \ldots, n_{p})$ of weight $3d$ such that $p>d$, is given recursively: \begin{multline} \label{coeffzeta3g} \alpha_{n_{1}, \ldots, n_{p}}= \delta_{n_{p}=3} \alpha_{n_{1}, \ldots, n_{p-1}}\\ +\sum_{k=1 \atop n_{k}=1 }^{p} \left( \delta_{n_{k-1}\geq 3 \atop k\neq 1} \alpha_{n_{1}, \ldots, n_{k-1}-2,n_{k+1}, \ldots, n_{p}} -\delta_{n_{k+1}\geq 3 \atop k\neq p} \alpha_{n_{1}, \ldots, n_{k-1},n_{k+1}-2, \ldots, n_{p}} \right) \\ +2 \sum_{k=1 \atop n_{k}=2 }^{p} \left(-\delta_{n_{k-1}\geq 3} \alpha_{n_{1}, \ldots, n_{k-1}-2,n_{k+1}, \ldots, n_{p}} + \delta_{n_{k+1}\geq 3 \atop k\neq p} \alpha_{n_{1}, \ldots, n_{k-1},n_{k+1}-2, \ldots, n_{p}} \right) \\ + \sum_{k=1 \atop n_{k}=1, n_{k+1}=1 }^{p-1} \left(-\delta_{n_{k-1}\geq 3 \atop k\neq 1} \alpha_{n_{1}, \ldots, n_{k-1}-1,n_{k+2}, \ldots, n_{p}} + \delta_{n_{k+2}\geq 3} \alpha_{n_{1}, \ldots, n_{k-1},n_{k+2}-1, \ldots, n_{p}} \right)\\ + \sum_{k=1 \atop n_{k}=1, n_{k+1}=2 }^{p-1} \left( \delta_{n_{k-1}\geq 3 \atop \text{ or } k=1 } \alpha_{n_{1}, \ldots, n_{k-1},n_{k+2}, \ldots, n_{p}} +2 \delta_{n_{k+2}\geq 2 \atop k\neq p-1} \alpha_{n_{1}, \ldots, n_{k-1},n_{k+2}, \ldots, n_{p}} \right)\\ + \sum_{k=1 \atop n_{k}=2, n_{k+1}=1 }^{p-1} \left(-2\delta_{n_{k-1}\geq 2 \atop \text{ or } k=1} \alpha_{n_{1}, \ldots, n_{k-1},n_{k+2}, \ldots, n_{p}} - \delta_{n_{k+2}\geq 3 \atop k\neq p-1} \alpha_{n_{1}, \ldots, n_{k-1},n_{k+2}, \ldots, n_{p}} \right).\\ \\ \end{multline} \end{lemm}
\begin{proof} Let list first all the possible cuts of length $3$ and depth $1$ in a iterated integral with $\lbrace 0,1 \rbrace$:\\ \includegraphics[]{dep8.pdf}\\ The coefficient above the arrow is the coefficient of $\zeta^{\mathfrak{m}}(3)$ in $I^{\mathfrak{m}}(cut)$, using that: $$\zeta^{\mathfrak{m}}_{1}(2)=-2\zeta^{\mathfrak{m}}(3), \quad \zeta^{\mathfrak{m}}(1,2)=\zeta^{\mathfrak{m}}(3), \quad \zeta^{\mathfrak{m}}(2,1)=-2\zeta^{\mathfrak{m}}(3), \quad \zeta^{\mathfrak{m}}_{1}(1,1)=\zeta^{\mathfrak{m}}(3).$$ Therefore, when there is a $1$ followed or preceded by something greater than $4$, the contribution is $\pm 1$, while when there is a $2$ followed or preceded by something greater than $3$, the contribution is $\pm 2$ as claimed in the lemma above. The contributions of a $3$ in the third line when followed of preceded by something greater than $2$ get simplified (except if there is a $3$ at the very end); when a $3$ is followed resp. preceded by a $1$ however, we assimilate it to the contribution of a $1$ preceded resp. followed by a $3$; which leads exactly to the penultimate lemma.\\ Additionally to the cuts listed above:\\ \includegraphics[]{dep9.pdf} This analysis leads to the given formula.\\ \end{proof} In particular, a sequence of the type $\boldsymbol{Y12X}$ resp. $\boldsymbol{X21Y}$ ($X \geq 2, Y \geq 3$) will imply a $(3)$ resp. $(-3)$ times the coefficient of the same sequence without $\boldsymbol{ 12}$, resp. $\boldsymbol{ 21}$.\\ \\ \\ \texttt{{\Large Examples:}} Let list a few families of multiple zeta values for which we have computed explicitly the coefficient $\alpha$ of maximal depth:\\ \\
\begin{tabular}{| c | c | c | } \hline Family & Recursion relation & Coefficient $\alpha$\\ \hline $\zeta^{\mathfrak{m}}(\lbrace 3\rbrace^{p})$ & $\alpha_{\lbrace 3\rbrace^{ p}}=\alpha_{\lbrace 3\rbrace^{ p-1}}$ & $1$\\ $\zeta^{\mathfrak{m}}(\lbrace 1,2\rbrace^{p})$ & $\alpha_{\lbrace 1,2 \rbrace^{ p}}=\alpha_{\lbrace 1,2 \rbrace^{ p-1}}$ & $1$\\ $\zeta^{\mathfrak{m}}(2,4,3\lbrace 3\rbrace^{p})$ & $\alpha_{2,4,\lbrace 3\rbrace^{p}}=\alpha_{2,4,\lbrace 3\rbrace^{p-1}}+2 \alpha_{\lbrace 3\rbrace^{p+1}} $ & $2(p+1)$\\ $\zeta^{\mathfrak{m}}(4,2,\lbrace 3\rbrace^{p})$ & $\alpha_{4,2,\lbrace 3\rbrace^{p}}=3\alpha_{4,2,\lbrace 3\rbrace^{p-1}}-2 \alpha_{\lbrace 3\rbrace^{p+1}} $ & $-3^{p+1}+1$\\ $\zeta^{\mathfrak{m}}(\lbrace 3\rbrace^{p},4,2)$ & $\alpha_{\lbrace 3\rbrace^{p},4,2}=-2\alpha_{\lbrace 3\rbrace^{p+1}} $ & $-2$\\ $\zeta^{\mathfrak{m}}(\lbrace 3\rbrace^{p},2,4)$ & $\alpha_{\lbrace 3\rbrace^{p},2,4}=2\alpha_{\lbrace 3\rbrace^{p-1},2,4}-2 \alpha_{\lbrace 3\rbrace^{p+1}} $ & $(-2)^{p}\frac{4}{3}+\frac{2}{3}$\\ $\zeta^{\mathfrak{m}}(2,\lbrace 3\rbrace^{p},4)$ & $\alpha_{2,\lbrace 3\rbrace^{p},4}=2\alpha_{2,\lbrace 3\rbrace^{p-1},4} $ & $2^{p+1}$\\ $\zeta^{\mathfrak{m}}(4,\lbrace 3\rbrace^{p},2)$ & $\alpha_{4,\lbrace 3\rbrace^{p},2}=-2\alpha_{4,\lbrace 3\rbrace^{p-1},2} $ & $(-2)^{p+1}$\\ $\zeta^{\mathfrak{m}}(1,5,\lbrace 3\rbrace^{p})$ & $\alpha_{1,5,\lbrace 3\rbrace^{p}}=\alpha_{1,5,\lbrace 3\rbrace^{p-1}}-1 $ & $-(p+1)$\\ $\zeta^{\mathfrak{m}}(\lbrace 2\rbrace^{p},\lbrace 4\rbrace^{p})$ & $\alpha_{\lbrace 2\rbrace^{p},\lbrace 4\rbrace^{p}}= 4 \alpha_{\lbrace 2\rbrace^{p-1},\lbrace 4\rbrace^{p-1}}$ & $2^{2p-1}$\\ $\zeta^{\mathfrak{m}}(\lbrace 2\rbrace^{p},\lbrace 3\rbrace^{a} \lbrace 4\rbrace^{p})$ & $\alpha_{\lbrace 2\rbrace^{p},\lbrace 3\rbrace^{a} \lbrace 4\rbrace^{p}}= 2^{a}\alpha_{\lbrace 2\rbrace^{p},\lbrace 4\rbrace^{p}}$ & $2^{a+2p-1}$\\ $\zeta^{\mathfrak{m}}(\lbrace 2\rbrace^{p},p+3)$ & $\alpha _{\lbrace 2\rbrace^{p},p+3}= 2\alpha _{\lbrace 2\rbrace^{p-1},p+1}$ & $2^{p}$\\ $\zeta^{\mathfrak{m}}(2,3,4,\lbrace 3\rbrace^{p})$ & $\alpha_{2,3,4,\lbrace 3\rbrace^{p}}=\alpha_{2,3,4,\lbrace 3\rbrace^{p-1}}+ 2 \alpha_{2,4,\lbrace 3\rbrace^{p}}$ & $2(p+1)(p+2)$\\ $\zeta^{\mathfrak{m}}(2,1,5,4,\lbrace 3\rbrace^{p})$ & $\alpha_{2,1,5,4,\lbrace 3\rbrace^{p}}=\alpha_{2,1,5,4,\lbrace 3\rbrace^{p-1}}- \alpha_{2,3,4,\lbrace 3\rbrace^{p}}$ & $-\frac{2(p+1)(p+2)(p+3)}{3}$\\ $\zeta^{\mathfrak{m}}(\lbrace 2\rbrace^{a}, a+3, \lbrace 3\rbrace^{b})$ & $\alpha_{a; b}=2 \alpha_{a-1;b}+\alpha_{a;b-1 }$ & $2^{a}\binom{a+b}{a} $\\ $\zeta^{\mathfrak{m}}(\lbrace 5,1\rbrace^{n})\text{ with } 3$\footnotemark[1] & $\alpha= \sum_{i=1}^{2p-1} (-1)^{i-1} \alpha_{\lbrace 5,1\rbrace^{p \text{ or } p-1} \text{with } 3} $ \footnotemark[2] & 1\\ \hline \end{tabular}\\ \footnotetext[1]{Any $\zeta^{\mathfrak{m}}(\lbrace 5,1\rbrace^{p})$ where we have inserted some $3$ in the subsequence.} \footnotetext[2]{Either a $3$ has been removed, either a $5,1$ resp. $1,5$ has been converted into a $3$ (with a sign coming from if we consider the elements before or after a $1$). If it ends with $3$, the contribution of a $3$ cancel with the contribution of the last $1$.} \\ \\ \textsc{For instance}, for the coefficient $\alpha_{a;b;c}$ associated to $\zeta^{\mathfrak{m}}(\lbrace 3\rbrace^{a},2, \lbrace 3\rbrace^{b}, 4, \lbrace 3\rbrace^{c} )$, the recursive relation is: \begin{equation} \alpha_{a;b;c}=\alpha_{a;b;c-1}+2\alpha_{a;b-1;c}-2\alpha_{a-1;b;c}, \quad \text{ which leads to the formula:} \end{equation} \begin{multline}\nonumber \alpha_{a;b;c}= (-2)^{a} \sum_{l=0}^{b-1} \sum_{m=0}^{c-1} 2^{l}\frac{(a+l+m-1) !}{(a-1) ! l ! m! } \alpha_{0;b-l;c-m} + 2^{b} \sum_{k=0}^{a-1} \sum_{m=0}^{c-1} (-2)^{k} \frac{(b+k+m-1) !}{k !(b-1) ! m! } \alpha_{a-k;0;c-m}\\
+ \sum_{l=0}^{b-1} \sum_{k=0}^{a-1} (-2)^{k} 2^{l} \frac{(k+l+c-1) !}{k! l ! (c-1)! } \alpha_{a-k;b-l;0}. \end{multline} Besides, we can also obtain very easily: $$\hspace*{-0.5cm}\alpha_{a;0;0}= (-2)^{a}\frac{4}{3}+\frac{2}{3} \text{,} \quad \alpha_{0;b;0}=2^{b+1}, \quad \alpha_{0;0;c}= 2(c+1), \quad \text{ and } \quad \alpha_{0;b;c}=2^{b+1}\binom{b+c+1}{c} $$ Indeed, using $\sum_{k=0}^{n}\binom{a+k}{a}= \binom{n+a+1}{a+1}$, and $\sum_{k=0}^{n}(n-k) \binom{k}{a}= \binom{n+1}{a+2}$, we deduce: $$\begin{array}{lll} \alpha_{0;b;c} &= 2\alpha_{0;b-1;c}+\alpha_{0;b;c-1}& = 2^{b+1} \left( \sum_{k=0}^{c-1} \binom{b+k-1}{b-1}(c-k+1) + \sum_{k=0}^{b-1}\binom{c+k-1}{c-1} \right)\\ & & =2^{b+1} \left( \binom{b+c+1}{b+1}-\binom{b+c-1}{b-1} + \binom{b+c-1}{b-1}\right) = 2^{b+1}\binom{b+c+1}{c}. \end{array}$$ \\ \texttt{Conjectured examples:} \\ \\ \begin{center}
\begin{tabular}{| c | c | } \hline Family & Conjectured coefficient $\alpha$\\ \hline $\zeta^{\mathfrak{m}}(\lbrace 2, 4 \rbrace^{p })$ & $\alpha_{p}$ such that $ 1-\sqrt{cos(2x)}=\sum \frac{\alpha_{p}(-1)^{p+1} x^{2p}}{(2p)!}$ \\ $\zeta^{\mathfrak{m}}(\lbrace 1, 5 \rbrace^{p})$ & Euler numbers: $\frac{1}{cosh(x)}=\sum \frac{\alpha_{p} x^{2p}}{(2p)!}$ \\ $\zeta^{\mathfrak{m}}(\lbrace 1, 5 \rbrace^{p}, 3)$ & $(-1)^{p} \text{Euler ZigZag numbers } E_{2p+1} $ $= 2^{2p+2}(2^{2p+2}-1)\frac{ B_{2p+2}}{2p+2} $ \\ \hline \end{tabular} \end{center}
\subsection{$\boldsymbol{N>1}$, The diagonal algebra.}
For $N=2,3,4$, $gr^{\mathfrak{D}}_{d} \mathcal{H}_{d}$ is $1$ dimensional, generated by $\zeta^{\mathfrak{m}}({ 1 \atop \xi})$, where $\xi\in\mu_{N}$ primitive fixed, which allows us to consider the projection: \begin{equation} \vartheta^{N} : gr_{d}^{\mathfrak{D}} \mathcal{H}_{d}^{1} \rightarrow \mathbb{Q}. \end{equation} Giving a motivic multiple zeta value relative to $\mu_{N}$, of weight $d$, depth $d$, there exists a rational such that: \begin{framed} \begin{equation}\label{eq:zeta1d} \zeta^{\mathfrak{m}}\left(1, \ldots, 1 \atop \epsilon_{1} , \ldots, \epsilon_{d} \right) = \frac{\alpha_{\boldsymbol{\epsilon}}} {n!} \zeta^{\mathfrak{m}}\left( 1 \atop \xi \right) ^{d} + \text{ terms of depth strictly smaller than } d. \end{equation} \end{framed} The coefficient $\alpha$ being calculated recursively, using depth $1$ results: \begin{lemm} $$ \hspace*{-0.5cm}\alpha_{\epsilon_{1}, \ldots, \epsilon_{d}}= \left\lbrace \begin{array}{ll} 1 & \text{if } \boldsymbol{N\in \lbrace 2,3 \rbrace}.\\
\sum_{k=1 \atop \epsilon_{k}\neq 1 }^{d} \beta_{\epsilon_{k}}\left( \delta_{\epsilon_{k-1}\epsilon_{k}\neq 1} \alpha_{\epsilon_{1}, \ldots, \epsilon_{k-1}\epsilon_{k},\epsilon_{k+1}, \ldots, \epsilon_{d}} -\delta_{\epsilon_{k+1}\epsilon_{k}\neq 1 \atop k < d} \alpha_{\epsilon_{1}, \ldots, \epsilon_{k-1},\epsilon_{k+1}\epsilon_{k}, \ldots, \epsilon_{d}} \right) & \text{if } \boldsymbol{N=4}. \end{array} \right. $$ \begin{flushright} with $\beta_{\epsilon_{k}}= \left\lbrace \begin{array}{ll} 2 & \text{ if } \epsilon_{k}=-1\\ 1 & \text{ else} \end{array}\right. $. \end{flushright} \end{lemm} \begin{proof} In regards to redundancy, the proof being in the same spirit than the previous section ($N=1, w=3d$), is left to the reader.\footnote{The cases $N=2,3$ correspond to the case $N=4$ with $\beta$ always equal to $1$.} \end{proof} \textsc{Remarks:} \begin{itemize} \item[$\cdot$] For the following categories, the space $gr^{\mathfrak{D}}_{d} \mathcal{H}_{d}$ is also one dimensional: $$\mathcal{MT}\left( \mathcal{O}_{6}\left[ \frac{1}{2}\right] \right) ,\quad \mathcal{MT}\left( \mathcal{O}_{6}\left[ \frac{1}{3}\right] \right) , \quad \mathcal{MT}\left( \mathcal{O}_{5}\right) , \mathcal{MT}\left( \mathcal{O}_{10}\right) , \quad \mathcal{MT}\left( \mathcal{O}_{12}\right) .$$ The recursive method to compute the coefficient of $\zeta^{\mathfrak{m}}\left( 1 \atop \eta\right) ^{d}$ would be similar, except that we do not know a proper basis for these spaces. \item[$\cdot$] For $N=1$, and $w\equiv 2 \mod 3$ for instance, $gr^{\mathfrak{D}}_{\max}\mathcal{H}_{n}$ is generated by the elements of the Euler $\sharp$ sums basis:\\ $\zeta^{\sharp, \mathfrak{m}} (1, \boldsymbol{s}, \overline{2})$ with $\boldsymbol{s}$ composed of $3$'s and one $5$, $\zeta^{\sharp, \mathfrak{m}} (3, 3 , \ldots, 3, \overline{2})$ and $\zeta^{\sharp, \mathfrak{m}} (1, 3 , \ldots, 3, \overline{4})$. \end{itemize}
\section{Families of unramified Euler sums.}
The proof relies upon the criterion $\ref{criterehonoraire}$,, which enables us to construct infinite families of unramified Euler sums with parity patterns by iteration on the depth, up to depth $5$.\\ \\ \texttt{Notations:} The occurrences of the symbols $E$ or $O$ can denote arbitrary even or odd integers, whereas every repeated occurrence of symbols $E_{i}$ (respectively $O_{i}$) denotes the same positive even (resp. odd) integer. The bracket $\left\{\cdot, \ldots, \cdot \right\}$ means that every permutation of the listed elements is allowed. \begin{theo} The following motivic Euler sums are unramified, i.e. motivic MZV:\footnote{Beware, here, $\overline{O}$ and $\overline{n}$ must be different from $\overline{1}$, whereas $O$ and $n$ may be 1. There is no $\overline{1}$ allowed in these terms if not explicitly written.} \\ \\
\hspace*{-0.5cm} \begin{tabular}{| c | l | l | }
\hline
& \textsc{Even Weight} & \textsc{Odd Weight} \\ \hline \textsc{Depth } 1 & \text{ All } & \text{ All }\footnotemark[1] \\
\hline \textsc{Depth } 2 & $\zeta^{\mathfrak{m}}(\overline{O},\overline{O}), \zeta^{\mathfrak{m}}(\overline{E},\overline{E})$ & \text{ All } \\
\hline \multirow{2}{*}{ \textsc{Depth } 3 } & $\zeta^{\mathfrak{m}}(\left\{E,\overline{O},\overline{O}\right\}), \zeta^{\mathfrak{m}}(O,\overline{E},\overline{O}), \zeta^{\mathfrak{m}}(\overline{O},\overline{E}, O)$ & $\zeta^{\mathfrak{m}}(\left\{\overline{E},\overline{E},O\right\}), \zeta^{\mathfrak{m}}(\overline{E},\overline{O},E), \zeta^{\mathfrak{m}}(E,\overline{O},\overline{E})$ \\
& $ \zeta^{\mathfrak{m}}(\overline{O_{1}}, \overline{E},\overline{O_{1}}), \zeta^{\mathfrak{m}}(O_{1}, \overline{E},O_{1}), \zeta^{\mathfrak{m}}(\overline{E_{1}}, \overline{E},\overline{E_{1}}) .$ & \\
\hline \multirow{2}{*}{ \textsc{Depth } 4 } & $\zeta^{\mathfrak{m}}(E,\overline{O},\overline{O},E),\zeta^{\mathfrak{m}}(O,\overline{E},\overline{O},E), $ & \\ & $\zeta^{\mathfrak{m}}(O,\overline{E},\overline{E},O), \zeta^{\mathfrak{m}}(E,\overline{O},\overline{E},O)$ & \\
\hline
\textsc{Depth } 5 & & $\zeta^{\mathfrak{m}}(O_{1}, \overline{E_{1}},O_{1},\overline{E_{1}}, O_{1}).$ \\
\hline
\end{tabular} Similarly for these linear combinations, in depth $2$ or $3$:
$$\zeta^{\mathfrak{m}}(n_{1},\overline{n_{2}}) + \zeta^{\mathfrak{m}}(\overline{n_{2}},n_{1}) , \zeta^{\mathfrak{m}}(n_{1},\overline{n_{2}}) + \zeta^{\mathfrak{m}}(\overline{n_{1}},n_{2}), \zeta^{\mathfrak{m}}(n_{1},\overline{n_{2}}) - \zeta^{\mathfrak{m}}(n_{2}, \overline{n_{1}}) .$$
$$(2^{n_{1}}-1) \zeta^{\mathfrak{m}}(n_{1},\overline{1}) + (2^{n_{1}-1}-1) \zeta^{\mathfrak{m}}(\overline{1},n_{1}).$$
$$ \zeta^{\mathfrak{m}}(n_{1},n_{2},\overline{n_{3}}) + (-1)^{n_{1}-1} \zeta^{\mathfrak{m}}(\overline{n_{3}},n_{2},n_{1}) \text{ with } n_{2}+n_{3} \text{ odd }.$$ \end{theo} \texttt{Examples}: These motivic Euler sums are motivic multiple zeta values: $$\zeta^{\mathfrak{m}}(\overline{25}, 14,\overline{17}),\zeta^{\mathfrak{m}}(17, \overline{14},17), \zeta^{\mathfrak{m}}(\overline{24}, \overline{14},\overline{24}), \zeta^{\mathfrak{m}}(6, \overline{23}, \overline{17}, 10) , \zeta^{\mathfrak{m}}(13, \overline{24}, 13,\overline{24}, 13).$$ \textsc{Remarks:} \begin{itemize} \item[$\cdot$] This result for motivic ES implies the analogue statement for ES. \item[$\cdot$] Notice that for each honorary MZV above, the reverse sum is honorary too, which was not obvious a priori, since the condition $\textsc{c}1$ below is not symmetric. \end{itemize} \begin{proof} The proof amounts to the straight-forward control that $D_{1}(\cdot)=0$ (immediate) and that all the elements appearing in the right side of $D_{2r+1}$ are unramified, by recursion on depth: here, these elements satisfy the sufficient criteria given below. Let only point out a few things, referring to the expression $(\ref{eq:derhonorary})$: \begin{description} \item[\texttt{Terms} $\textsc{c}$:] The symmetry condition $(\textsc{c}4)$, obviously true for these single elements above, get rid of these terms. For the few linear combinations of MES given, the cuts of type (\textsc{c}) get simplified together. \item[\texttt{Terms} $\textsc{a,b}$:] Checking that the right sides are unramified is straightforward by depth-recursion hypothesis, since only the (previously proven) unramified elements of lower depth emerge. For example, the possible right sides (not canceled by a symmetric cut and up to reversal symmetry) are listed below, for some elements from depth 3. \\
\hspace*{-1cm} \begin{tabular}{| c | l | l | }
\hline
& Terms \textsc{a0} & Terms \textsc{a,b} \\ \hline
$(O,\overline{E},\overline{E}) $ & & $(\overline{E},\overline{E})$ ,$(O,O)$ \\
$(\overline{E},O,\overline{E}) $ & / & $(\overline{E},\overline{E})$ \\
$(E,\overline{O},\overline{E}) $ & / & $(\overline{E},\overline{E}), (E,E)$ \\
\hline
$(E,\overline{O},\overline{O},E) $ & $(\overline{O},E)$ & $(\overline{E},\overline{O},E), (E,O,E),(E,\overline{O},\overline{E}), (E,O),(O,E)$ \\
$(O,\overline{E},\overline{O},E) $ & $(\overline{E},\overline{O},E),(\overline{O},E)$ & $(\overline{E},\overline{O},E),(O,E,E),(O,\overline{E},\overline{E}), (O,E)$ \\
$(O,\overline{E},\overline{E},O) $& $(\overline{E},\overline{E},O) ,(\overline{E},O)$ & $(\overline{E},\overline{E},O), (O,O,O), (O,\overline{E},\overline{E}), (O,E), (E,O)$ \\
$(\overline{E}, O_{1},\overline{E},O_{1}) $& $(\overline{E},O)$ & $(\overline{E},\overline{E},O) , (\overline{E},O,\overline{E}) ,(\overline{E},\overline{O}), (E,O)$ \\ $(\overline{E_{1}}, \overline{E_{2}},\overline{E_{1}}, \overline{E_{2}}) $& / & $(O,\overline{E},\overline{E}) ,(\overline{E},O,\overline{E}) ,(\overline{E},\overline{E},O) ,(\overline{E},\overline{O}),(\overline{O},\overline{E})$ \\
\hline
$(O_{1}, \overline{E_{1}},O_{1},\overline{E_{1}}, O_{1})$ & $(\overline{E_{1}},O_{1},\overline{E_{1}}, O_{1}),$ & $(\overline{E},O_{1},\overline{E}, O_{1}), (O_{1}, \overline{E},\overline{E}, O_{1}), (O_{1}, \overline{E},O_{1},\overline{E}),$ \\
& $(O_{1},\overline{E}, O_{1})$ & $(\overline{O},\overline{E},O),(O,\overline{E},\overline{O}), (O,O)$\\
\hline
\end{tabular} \end{description} It refers to the expression of the derivations $D_{2r+1}$ (from Lemma $\ref{drz}$$\ref{drz}$): \begin{multline}\label{eq:derhonorary}
D_{2r+1} \left(\zeta^{\mathfrak{m}} \left(n_{1}, \ldots , n_{p} \right)\right) = \textsc{(a0) } \delta_{2r+1 = \sum_{k=1}^{i} \mid n_{k} \mid} \zeta^{\mathfrak{l}} (n_{1}, \ldots , n_{i}) \otimes \zeta^{\mathfrak{m}} (n_{i+1},\cdots n_{p}) \\ \textsc{(a,b) } \sum_{1\leq i < j \leq p \atop 2r+1=\sum_{k=i}^{j} \mid n_{k}\mid - y } \left\lbrace \begin{array}{l}
-\delta_{2\leq y \leq \mid n_{j}\mid } \zeta_{\mid n_{j}\mid -y}^{\mathfrak{l}} (n_{j-1}, \ldots ,n_{i+1}, n_{i}) \\
+\delta_{2\leq y \leq \mid n_{i}\mid} \zeta_{\mid n_{i}\mid -y}^{\mathfrak{l}} (n_{i+1}, \cdots ,n_{j-1}, n_{j})
\end{array} \right. \otimes \zeta^{\mathfrak{m}} (n_{1}, \ldots, n_{i-1},\prod_{k=i}^{j}\epsilon_{k} \cdot y,n_{j+1},\cdots n_{p}). \\ \textsc{(c) } + \sum_{1\leq i < j \leq p\atop {2r+2=\sum_{k=i}^{j} \mid n_{k}\mid} } \delta_{ \prod_{k=i}^{j} \epsilon_{k} \neq 1} \left\lbrace \begin{array}{l} + \zeta_{\mid n_{i}\mid -1}^{\mathfrak{l}} (n_{i+1}, \cdots ,n_{j-1}, n_{j}) \\ - \zeta_{\mid n_{j}\mid -1}^{\mathfrak{l}} (n_{j-1}, \cdots ,n_{i+1}, n_{i})
\end{array} \right. \otimes \zeta^{\mathfrak{m}} (n_{1}, \ldots, n_{i-1},\overline{1},n_{j+1},\cdots n_{p}). \end{multline} \end{proof}
\paragraph{Sufficient condition. }\label{sufficientcondition} Let $\mathfrak{Z}= \zeta^{\mathfrak{m}}(n_{1}, \ldots, n_{p})$ a motivic Euler sum. These four conditions are \textit{sufficient} for $\mathfrak{Z}$ to be unramified: \begin{description}
\item [\textsc{c}1]: No $\overline{1}$ in $\mathfrak{Z}$.
\item [\textsc{c}2]: For each $(n_{1}, \ldots, n_{i})$ of odd weight, the MES $\zeta^{\mathfrak{m}}(n_{i+1}, \ldots, n_{p})$ is a MMZV.
\item [\textsc{c}3]: If a cut removes an odd-weight part (such that there is no symmetric cut possible), the remaining MES (right side in terms \textsc{a,b}), is a MMZV.
\item [\textsc{c}4]: Each sub-sequence $(n_{i}, \ldots, n_{j})$ of even weight such that $\prod_{k=i}^{j} \epsilon_{k} \neq 1$ is symmetric. \end{description} \begin{proof} The condition $\textsc{c}1$ implies that $D_{1}(\mathfrak{Z})=0$; conditions $\textsc{c}2$, resp. $\textsc{c}3$ take care that the right side of terms \textsc{a0}, resp. \textsc{a,b} are unramified, while the condition $\textsc{c}4$ cancels the (disturbing) terms \textsc{c}: indeed, a single ES with an $\overline{1}$ can not be unramified.\\ Note that a MES $\mathfrak{Z}$ of depth $2$, weight $n$ is unramified if and only if $ \left\lbrace \begin{array}{l} D_{1}(\mathfrak{Z})=0\\ D_{n-1}(\mathfrak{Z})=0 \end{array}\right.$. \end{proof} \noindent \texttt{Nota Bene:} This criterion is not \textit{necessary}: it does not cover the unramified $\mathbb{Q}$-linear combinations of motivic Euler sums, such as those presented in section $4$, neither some isolated (\textit{symmetric enough}) examples where the unramified terms appearing in $D_{2r+1}$ could cancel between them. However, it embrace general families of single Euler sums which are unramified.\\ \\ Moreover, \begin{framed} \emph{If we \textit{comply with these conditions}, the \textit{only} general families of single MES obtained are the one listed in Theorem $6.2.1$.} \end{framed} \begin{proof}[Sketch of the proof] Notice first that the condition \textsc{c}$4$ implies in particular that there are no consecutive sequences of the type (since it would create type $\textsc{c}$ terms): $$\textsc{ Seq. not allowed : } O\overline{O}, \overline{O}O, \overline{E}E, E\overline{E}.$$ It implies, from depth $3$, that we can't have the sequences (otherwise one of the non allowed depth $2$ sequences above appear in $\textsc{a,b}$ terms): $$\textsc{ Seq. not allowed : } \overline{E}\overline{E}\overline{O}, \overline{E}\overline{E}\overline{E}O, \overline{E}\overline{E}O\overline{E}, E\overline{O}\overline{E},EE\overline{O}, \overline{O}EE, \overline{E}OE, \overline{E}\overline{O}\overline{E}, \overline{O}\overline{O}\overline{O}.$$ Going on in this recursive way, carefully, leads to the previous theorem.\\ \texttt{For instance,} let $\mathfrak{Z}$ a MES of even weight, with no $\overline{1}$, and let detail two cases: \begin{description} \item[\texttt{Depth} $3$:] The right side of $D_{2 r+1}$ has odd weight and depth smaller than $2$, hence is always MMZV if there is no $\overline{1}$ by depth $2$ results. It boils down to the condition $\textsc{c}4$: $\mathfrak{Z}$ must be either symmetric (such as $O_{1}E O_{1}$ or $E_{1}EE_{1}$ with possibly one or three overlines) either have exactly two overlines. Using the analysis above of the allowed sequences in depth $2$ and $3$ for condition $\textsc{c3,4}$ leads to the following: $$(E,\overline{O},\overline{O}),(\overline{O},\overline{O},E), (O,\overline{E},\overline{O}), (\overline{O},\overline{E}, O), (\overline{O},E,\overline{O}), (\overline{O_{1}}, \overline{E},\overline{O_{1}}), (O_{1}, \overline{E},O_{1}), (\overline{E_{1}}, \overline{E},\overline{E_{1}}) .$$ \item[\texttt{Depth} $4$:] Let $\mathfrak{Z}=\zeta^{\mathfrak{m}}\left( n_{1}, \ldots, n_{4}\right) $, $\epsilon_{i}=sign(n_{i})$. To avoid terms of type $ \textsc{c}$ with a right side of depth $1$: if $\epsilon_{1}\epsilon_{2}\epsilon_{3}\neq 1$, either $n_{1}+n_{2}+n_{3}$ is odd, or $n_{1}=n_{3}$ and $\epsilon_{2}=-1$; if $\epsilon_{2}\epsilon_{3}\epsilon_{4}\neq 1$, either $n_{2}+n_{3}+n_{4}$ is odd, or $n_{2}=n_{4}$ and $\epsilon_{3}=-1$. The following sequences are then not allowed: $$ (\overline{E}, O,O,\overline{E}), (\overline{E}, \overline{O},\overline{O},\overline{E}), (\overline{E}, \overline{O},E,O), (\overline{E}, \overline{E},O,O), (O,O,\overline{E}, \overline{E}).$$ \end{description} \end{proof}
\section{Motivic Identities}
As we have seen above, in particular in Lemma $\ref{lemmcoeff}$, the coaction enables us to prove some identities between MMZV or MES, by recursion on the depth, up to one rational coefficient at each step. This coefficient can be deduced then, if we know the analogue identity for MZV, resp. Euler sums. Nevertheless, a \textit{motivic identity} between MMZV (resp. MES) is stronger than the corresponding relation between real MZV (resp. Euler sums); it may hence require several relations between MZV in order to lift an identity to motivic MZV. An example of such a behaviour occurs with some Hoffman $\star$ elements, ($(iv)$ in Lemma $\ref{lemmcoeff}$).\\ \\ In this section, we list a few examples of identities, picked from the zoo of existing identities, that we are able to lift easily from Euler sums to motivic Euler sums: \textit{Galois trivial} elements (action of the unipotent part of the Galois group being trivial), sums identities, etc. \\ \\ \texttt{Nota Bene}: For other cyclotomic MMZV, we could somehow generalize this idea, but there would be several unknown coefficients at each step, as stated in Theorem $2.4.4$. For $N=3$ or $4$, we have to consider all $D_{r}, 0<r<n$, and there would be one resp. two (if weight even) unknown coefficients at each step ; for $N=\mlq 6 \mrq$, if unramified, considering $D_{r}, r>1$, there would be also one or two unknown coefficients at each step. \\ \\ \texttt{Example:} Here is an identity known for Euler sums, proven at the motivic level by recursion on $n$ via the coaction for motivic Euler sums (and using the analytic identity): \begin{equation} \zeta^{\mathfrak{m}}(\lbrace 3 \rbrace^{n})= \zeta^{\mathfrak{m}}(\lbrace 1,2 \rbrace^{n}) = 8^{n} \zeta^{\mathfrak{m}}(\lbrace 1, \overline{2} \rbrace^{n}). \end{equation} \begin{proof} These three families are stable under the coaction: $$\begin{array}{lllll} D_{2r+1} (\zeta^{\mathfrak{m}}(\lbrace 3 \rbrace^{n})) & = & \delta_{2r+1=3s} \zeta^{\mathfrak{a}}(\lbrace 3 \rbrace^{s}) & \otimes & \zeta^{\mathfrak{m}}(\lbrace 3 \rbrace^{n-s}) .\\ D_{2r+1} (\zeta^{\mathfrak{m}}(\lbrace 1,2 \rbrace^{n})) & = & \delta_{2r+1=3s} \zeta^{\mathfrak{a}}(\lbrace 1,2 \rbrace^{s}) & \otimes & \zeta^{\mathfrak{m}}(\lbrace 1,2 \rbrace^{n-s}) .\\ D_{2r+1} (\zeta^{\mathfrak{m}}(\lbrace 1,\overline{2} \rbrace^{n})) & = & \delta_{2r+1=3s} \zeta^{\mathfrak{a}}(\lbrace 1,\overline{2} \rbrace^{s}) & \otimes & \zeta^{\mathfrak{m}}(\lbrace 1,\overline{2} \rbrace^{n-s}) . \end{array}$$ Indeed, in both case, in the diagrams below, cuts $(3)$ and $(4)$ are symmetric and get simplified by reversal, as cuts $(1)$ and $(2)$, except for last cut of type $(1)$ which remains alone:\\ \includegraphics[]{dep10.pdf}\\ Similarly for $\zeta^{\mathfrak{m}}(\lbrace 1,\overline{2} \rbrace^{n})$: cuts of type $(3)$, $(4)$ resp. $(1), (2)$ get simplified together, except the first one, when $\epsilon=\epsilon'$ in the diagram below. The other possible cuts of odd length would be $(5)$ and $(6)$ below, when $\epsilon=-\epsilon'$, but each is null since antisymmetric.\\ \includegraphics[]{dep11.pdf} \end{proof}
\paragraph{Galois trivial.} The Galois action of the unipotent group $\mathcal{U}$ is trivial on $\mathbb{Q}[\mathbb{L}^{\mathfrak{m}, 2n}]= \mathbb{Q}[\zeta^{\mathfrak{m}}(2)^{n}]$. To prove an element of $\mathcal{H}_{2n}$ is a rational multiple of $\mathbb{L}^{\mathfrak{m},2n}$, it is equivalent to check it is in the kernel of the derivations $D_{2r+1}$, for $1\leq 2r+1<2n$, by Corollary $\ref{kerdn}$. We have to use the (known) analogue identities for MZV to conclude on the value of such a rational.\\ \\ \texttt{Example:} \begin{itemize} \item[$\cdot$] Summing on all the possible ways to insert n $\boldsymbol{2}$'s. \begin{equation} \zeta^{\mathfrak{m}}(\left\lbrace 1,3 \right\rbrace^{p} \text{with n } \boldsymbol{2} \text{ inserted } )= \binom{2p+n}{n} \frac{\pi^{4p+2n, \mathfrak{m}}}{(2n+1) (4p+2n+1)!}. \end{equation} \item[$\cdot$] More generally, with fixed $(a_{i})$ such that $\sum a_{i}=n$: \footnote{Both appears also in Charlton's article.$\cite{Cha}$.} \begin{equation} \sum_{\sigma\in\mathfrak{S}_{2p}} \zeta^{\mathfrak{m}}(2^{a_{\sigma(0)}} 1 , 2^{a_{\sigma(1)}}, 3, 2^{a_{\sigma(2)}}, \ldots, 1, 2^{a_{\sigma(2p-1)}}, 3, 2^{a_{\sigma(2p)}} )\in\mathbb{Q} \pi^{4p+2n, \mathfrak{m}}. \end{equation} \end{itemize}
\begin{proof} In order to justify why all the derivations $D_{2r+1}$ cancel, the possible cuts of odd length are, with $X= \lbrace 01 \rbrace^{a_{2i+2}+1} \lbrace 10 \rbrace^{a_{2i+3}+1} \cdots \lbrace 01 \rbrace^{a_{2j-2}} \lbrace 10 \rbrace^{a_{2j-1}} $:\\ \includegraphics[]{dep12.pdf} All the cuts get simplified by \textsc{Antipode} $\shuffle$ $\ref{eq:antipodeshuffle2}$, which proves the result, as follows: \begin{itemize} \item[$\cdot$] Cut $(1)$ for $(a_{0}, \ldots, a_{2p})$ with Cut $(2)$ for $(a_{0}, \ldots, a_{2i-1}, a_{2j+1} \cdots, a_{2i}, a_{2j+2 },\cdots, a_{2p})$. \item[$\cdot$] Similarly between $(3)$ and $(4)$, which get simplified considering the sequence where $(a_{2i+1}, \ldots, a_{2j})$ is reversed. \end{itemize} \end{proof}
\paragraph{Polynomial in simple zetas.} A way to prove that a family of (motivic) MZV are polynomial in simple (motivic) zetas, by recursion on depth: \begin{lemm} Let $\mathfrak{Z}\in\mathcal{H}^{1}_{n}$ a motivic multiple zeta value of depth $p$. \\ If the following conditions hold, $\forall \quad 1<2r+1<n$, $m\mathrel{\mathop:}=\lfloor\frac{n}{2}\rfloor-1$: \begin{itemize} \item[$(i)$] $D_{2r+1,p}(\mathfrak{Z})= P^{\mathfrak{Z}}_{r}(\zeta^{\mathfrak{m}}(3),\zeta^{\mathfrak{m}}(5), \ldots, \zeta^{\mathfrak{m}}(2m+1), \zeta^{\mathfrak{m}}(2)),$ $$\text{with } P^{\mathfrak{Z}}_{r}(X_{1},\cdots, X_{m}, Y )= \sum_{2s+\sum (2k+1)\cdot a_{k}=n-2r-1 } \beta^{r}_{a_{1}, \ldots, a_{m}, s} X_{1}^{a_{1}} \cdots X_{m}^{a_{m}} Y^{s}.$$ \item[$(ii)$] For $ a_{k},a_{r}>0 \text{ : } \frac{ \beta^{r}_{a_{1}, \ldots, a_{r}-1,\cdots, a_{m},s}}{a_{r}+1} =\frac{ \beta^{k}_{a_{1}, \ldots, a_{k}-1, \ldots, a_{m},s}}{a_{k}}.$ \end{itemize} Then, $\mathfrak{Z}$ is a polynomial in depth $1$ MMZV: $$ \mathfrak{Z}= \alpha \zeta^{\mathfrak{m}}(n)+ \sum_{2s+\sum (2k+1)a_{k}=n} \alpha_{a_{1}, \ldots, a_{m},s} \zeta^{\mathfrak{m}}(3)^{a_{1}} \cdots \zeta^{\mathfrak{m}}(2m+1)^{a_{m}} \zeta^{\mathfrak{m}}(2)^{s}. \footnote{In particular, $\alpha_{a_{1}, \ldots, a_{m},s} =\frac{\beta^{r}_{a_{1}, \ldots,a_{r}-1, \ldots, a_{m}, s}}{a_{r}}$ for $a_{r}\neq 0$. }$$ \end{lemm} \begin{proof} Immediate with Corollary $\ref{kerdn}$ since: $$D_{2r+1,p } \left( \zeta^{\mathfrak{m}}(3)^{a_{1}} \cdots \zeta^{\mathfrak{m}}(2m+1)^{a_{m}} \zeta^{\mathfrak{m}}(2)^{s}\right) = a_{r} \zeta^{\mathfrak{m}}(3)^{a_{1}} \cdots \zeta^{\mathfrak{m}}(2r+1)^{a_{r}} \cdots \zeta^{\mathfrak{m}}(2m+1)^{a_{m}} \zeta^{\mathfrak{m}}(2)^{s}. $$ \end{proof} \noindent \texttt{Example}: Some examples were given in the proof of Lemma $\ref{lemmcoeff}$; the following family is polynomial in zetas \footnote{Proof method: with recursion hypothesis on coefficients, using: $$D_{2r+1}(\zeta^{\mathfrak{m}} (\left\lbrace 1 \right\rbrace ^{n}, m))= - \sum_{j=\max(0,2r+2-m)}^{\min(n-1,2r-1)} \zeta^{\mathfrak{l}} (\left\lbrace 1 \right\rbrace ^{j}, 2r+1-j) \otimes \zeta^{\mathfrak{m}} (\left\lbrace 1 \right\rbrace ^{n-j-1}, m-2r+j).$$}: $$\zeta^{\mathfrak{m}} (\left\lbrace 1 \right\rbrace ^{n}, m).$$
\paragraph{Sum formulas.} Here are listed a few examples of the numerous \textit{sum identities} known for Euler sums\footnote{Usually proved considering the generating function, and expressing it as a hypergeometric function.} which we can lift to motivic Euler sums, via the coaction. For these identities, as we see through the proof, the action of the Galois group is trivial; the families being stable under the derivations, we are able to lift the identity to its motivic version via a simple recursion.
\begin{theo} Summations, if not precised are done over the admissible multi-indices, with $w(\cdot)$, resp. $d(\cdot)$, resp. $h(\cdot)$ indicating the weight, resp. the depth, resp. the height: \begin{itemize} \item[(i)] With fixed even (possibly negative) $\left\lbrace a_{i}\right\rbrace _{1 \leq i \leq p}$ of sum $2n$:\footnote{This would be clearly also true for MMZV$^{\star}$.} $$\sum_{\sigma\in\mathfrak{S}_{p}} \zeta^{\mathfrak{m}}(a_{\sigma(1)}, \ldots, a_{\sigma(p)}) \in \mathbb{Q} \pi^{2n, \mathfrak{m}}.$$ In particular:\footnote{The precise coefficient is given in $\cite{BBB1}$, $(48)$ and can then be deduced also for the motivic identity.} $$\zeta^{\mathfrak{m}}(\left\lbrace 2n \right\rbrace^{p} ) , \zeta^{\mathfrak{m}}(\left\lbrace \overline{2n} \right\rbrace^{p} ) \in \mathbb{Q} \pi^{2np, \mathfrak{m}}.$$ More precisely, with Hoffman $\cite{Ho}$ \footnotemark[6] \begin{multline}\nonumber \sum_{\sum n_{i}= 2n} \zeta^{\mathfrak{m}}\left( 2n_{1}, \ldots, 2n_{k}\right) =\\ \frac{1}{2^{2(k-1)}} \binom{2k-1}{k} \zeta^{\mathfrak{m}}(2n) - \sum_{j=1}^{\lfloor\frac{k-1}{2}\rfloor} \frac{1}{2^{2k-3}(2j+1) B_{2j}} \binom{2k-2j-1}{k} \zeta^{\mathfrak{m}}(2j) \zeta^{\mathfrak{m}}(2n-2j) . \end{multline} \item[(ii)] With Granville $\cite{Gra}$, or Zagier $\cite{Za1}$ \footnotemark[6] $$ \sum_{w(\textbf{k})=n, d(\textbf{k})=d } \zeta^{\mathfrak{m} }(\textbf{k})= \zeta^{\mathfrak{m}}(n). $$ \item[(iii)] With Aoki, Ohno $\cite{AO}$\footnotemark[6] \footnotemark[2] \begin{align*} \sum_{w(\textbf{k})=n, d(\textbf{k})=d } \zeta^{\star,\mathfrak{m}}(\textbf{k}) & = \binom{n-1}{d-1} \zeta^{\mathfrak{m}}(n).\\
\sum_{w(\textbf{k})=n, h(\textbf{k})=s } \zeta^{\star,\mathfrak{m}}(\textbf{k})& = 2\binom{n-1}{2s-1} (1-2^{1-n}) \zeta^{\mathfrak{m}}(n). \end{align*} \item[(iv)] With Le, Murakami$\cite{LM}$\footnotemark[6] $$\sum_{w(\textbf{k})=n, h(\textbf{k})=s } (-1)^{d(\textbf{k})}\zeta^{\mathfrak{m}}(\textbf{k})=\left\lbrace \begin{array}{ll} 0 & \text{ if } n \text{ odd} . \\ \frac{(-1)^{\frac{n}{2}} \pi^{\mathfrak{m},n}}{(n+1)!} \sum_{k=0}^{\frac{n}{2}-s}\binom{n+1}{2k} (2-2^{2k})B_{2k} & \text{ if } n \text{ even} .\\ \end{array} \right. $$ \item[(v)] With S. Belcher (?)\footnotemark[6] $$\hspace*{-0.5cm}\begin{array}{llll} \sum_{w(\cdot)=2n \atop d(\cdot)=2p } \zeta^{\mathfrak{m}}(odd, odd>1, odd, \ldots, odd, odd>1)& =& \alpha^{n,p} \zeta^{\mathfrak{m}} (2)^{n}, & \alpha^{n,p} \in \mathbb{Q}\\ \sum_{w(\cdot)=2n+1 \atop d(\cdot)=2p+1} \zeta^{\mathfrak{m}}(odd, odd>1, odd, \ldots, odd>1, odd)&=& \sum_{i=1}^{n} \beta^{n,p}_{i} \zeta^{\mathfrak{m}}(2i+1) \zeta^{\mathfrak{m}}(2)^{n-i} , & \beta^{n,p}_{i}\in\mathbb{Q}\\ \sum_{w(\cdot)=2n+1 \atop d(\cdot)=2p+1} \zeta^{\mathfrak{m}}(odd>1, odd, \ldots, odd, odd>1)&=& \sum_{i=1}^{n} \gamma^{n,p}_{i} \zeta^{\mathfrak{m}}(2i+1) \zeta^{\mathfrak{m}}(2)^{n-i}, & \gamma^{n,p}_{i}\in\mathbb{Q}
\end{array}$$ \end{itemize} \footnotetext[6]{The person(s) at the origin of the analytic equality for MZV, used in the proof for motivic MZV.} \end{theo} \noindent \textsc{Remark}: The permutation identity $(i)$ would in particular imply that all sum of MZV at even arguments times a symmetric function of these same arguments are rational multiple of power of $\mathbb{L}^{\mathfrak{m}}$. \\ Many specific identities, in small depth have been already found (as Machide in $\cite{Ma}$, resp. Zhao, Guo, Lei in $\cite{GLZ}$, etc.), and can be directly deduced for motivic MZV, such as: \begin{align*} \hspace*{-2.5cm}\sum_{k=1}^{n-1} \zeta(2k, 2n-2k) \quad\quad & \left\lbrace \begin{array}{lll} 1 & =& \frac{3}{4} \zeta(2n)\\ 4^{k}+4^{n-k} &=& (n+\frac{4}{3}+\frac{4^{n}}{6})\zeta(2n) \\ (2k-1)(2n-2k-1) &=& \frac{3}{4} (n-3) \zeta(2n) \end{array} \right. \\
\hspace*{-0.3cm}\sum \zeta(2i, 2j, 2n-2i-2j) & \left\lbrace \begin{array}{lll} 1 & =& \frac{5}{8} \zeta(2n)- \frac{1}{4} \zeta(2n-2) \zeta(2)\\ ij +jk+ki &=& \frac{5n}{64} \zeta(2n)+(4n-\frac{9}{10}) \zeta(2n-2) \zeta(2) \\ ijk &=& \frac{n}{128} (n-3) \zeta(2n)-\frac{1}{32} \zeta(2n-2) \zeta(2)+\frac{2n-5}{8} \zeta(2n-4) \zeta(4)\\ \end{array} \right.\\
& \\ \end{align*}
\begin{proof} We refer to the formula of the derivations $D_{r}$ in Lemma $\ref{lemmt}$. For many of these equalities, when summing over all the permutations of a certain subset, most of the cuts will get simplified two by two as followed: \begin{equation}\label{eq:termda} \zeta^{\mathfrak{m}}\left( k_{1}, \ldots, k_{i}, k_{i+1}, \ldots, k_{j}, k_{j+1}, \cdots k_{d}\right) \text{ : } 0; \cdots 1 0^{k_{i}-1} \boldsymbol{1 } 0^{k_{i+1}-1} \cdots 0^{k_{j-1}-1} 1 \boldsymbol{0^{k_{j}-1}} 1 0^{k_{j+1}-1}\cdots ; 1 . \end{equation} \begin{equation}\label{eq:termdb} \zeta^{\mathfrak{m}}(k_{1},\cdots, k_{i}, k_{j}, \cdots, k_{i+1}, k_{j+1}, \ldots, k_{d}) \text{ : } 0; \cdots 1 0^{k_{i}-1} 1 \boldsymbol{0^{k_{j}-1}} \cdots 0^{k_{i+2}-1} 1 0^{k_{i+1}-1} \boldsymbol{1} 0^{k_{j+1}-1}\cdots ; 1. \end{equation} It remains only the first cuts, beginning with the first $0$, such as: \begin{equation}\label{eq:termd1} \delta_{2r+1= \sum_{j=1}^{i} k_{j}} \zeta^{\mathfrak{m}}\left( k_{1}, \ldots, k_{i})\otimes \zeta^{\mathfrak{m}}(k_{i+1}, \ldots, k_{d}\right) , \end{equation} and possibly the cuts from a $k_{i}=1$ to $k_{d}$, if the sum is over admissible MMZV: \footnote{There, beware, the MZV at the left side can end by $1$.} \begin{equation}\label{eq:termdr} -\delta_{2r+1< \sum_{j=i+1}^{d} k_{j}} \zeta^{\mathfrak{m}}\left( k_{i+1}, \ldots, k_{d-1}, 2r+1- \sum_{j=i+1}^{d-1} k_{j}\right) \otimes \zeta^{\mathfrak{m}}\left( k_{1}, \ldots, k_{i-1}, \sum_{j=i+1}^{d} k_{j} -2r\right) . \end{equation} \begin{itemize} \item[(i)] From the terms above in $D_{2r+1}$, $(\ref{eq:termda})$, and $(\ref{eq:termdb})$ get simplified together, and there are no terms $(\ref{eq:termd1})$ since the $a_{i}$ are all even. Therefore, it is in the kernel of $\oplus_{2r+1<2n} D_{2r+1}$ with even weight, hence Galois trivial.\\ For instance, for $\zeta^{\mathfrak{m}}(\left\lbrace \overline{2n} \right\rbrace^{p} ) $, with $\epsilon, \epsilon'\in \lbrace\pm 1\rbrace$:\\ \includegraphics[]{dep13.pdf}\\ Either, $\epsilon=\epsilon'$ and $X$ is symmetric, and by reversal of path (cf. $\S A.1.1$), cuts above get simplified, or $\epsilon=-\epsilon'$ and $X$ is antisymmetric, and the cuts above still get simplified since $I^{\mathfrak{m}}(\epsilon;0^{a+1} X;0)=-I^{\mathfrak{m}}(0;\widetilde{X} 0^{a+1};\epsilon)=-I^{\mathfrak{m}}(0;X 0^{a+1};-\epsilon)$. \item[(ii)] Let us denote this sum $G(n,d)$, and $G_{1}(n,d)$ the corresponding sum where a $1$ at the end is allowed. As explained in the proof's preamble, the remaining cuts being the first ones and the one from a $k_{i}=1$ to the last $k_{d}$: $$\hspace*{-0.5cm}D_{2r+1}(G (n,d))= \sum_{i=0}^{d-1} G^{\mathfrak{l}}_{1}(2r+1,i) \otimes G(n-2r-1,d-i) -\sum_{i=0}^{d-1} G^{\mathfrak{l}}_{1}(2r+1,i) \otimes G(n-2r-1,d-i) =0 .$$
\item[(iii)] This can be proven also computing the coaction, or noticing that it can be deduced from Euler relation above, turning a MZV$^{\star}$ into a sum of MZV of smaller depth, it turns to be: $$\sum_{i=1}^{d} \sum_{w(\boldsymbol{k})= n, d(\boldsymbol{k})=i} \binom{n-i-1}{d-i} \zeta^{\mathfrak{m}} (\boldsymbol{k}).$$ For the Aoki-Ohno identity, using the formula for MZV $\star$, and with recursion hypothesis, we could similarly prove that the coaction is zero on these elements, and conclude with the result for MZV.
\item[(iv)] Let us denote this sum $G_{-}(n,s)$ and $G_{-,(1)}(n,s)$ resp. $G_{-,1}(n,s)$ the analogue sums with possibly a $1$ at the end, resp. with necessarily a $1$ at the end. Looking at the derivations, since we sum over all the permutations of the admissible indices, all the cuts get simplified with its symmetric cut as said above, and it remains only the beginning cut (with the first $0$), and the cut from a $k_{j}=1$ to the last $k_{d}$, which leads to: \begin{multline}\nonumber \hspace*{-1cm}D_{2r+1}(G_{-}(n,s))= \sum_{i=0}^{s-1} \left( G^{\mathfrak{l}}_{-,(1)}(2r+1,i) -G^{\mathfrak{l}}_{-}(2r+1,i+1)- G^{\mathfrak{l}}_{-,1}(2r+1,i)\right) \otimes G_{-}(n-2r-1,s-i) \\ = \sum_{i=0}^{s-1} (G^{\mathfrak{l}}_{-}(2r+1,i) -G^{\mathfrak{l}}_{-}(2r+1,i+1))\otimes G_{-}(n-2r-1,s-i). \end{multline} Using recursion hypothesis, it cancels, and thus, $G_{alt}(n,s)\in\mathbb{Q} \zeta^{\mathfrak{m}}(n)$. Using the analogue analytic equality, we conclude.
\item[(v)] For odd sequences with alternating constraints ($>1$ or $\geq 1$ for instance), cuts between $k_{i}$ and $k_{j}$ will get simplified with some symmetric terms in the sum, except possibly (when odd length), the first (i.e. from the first $1$ to a first $0$) and the last (i.e. from a last $0$ to the very last $1$) one. More precisely, with $O$ any odd integer, possibly all different: \begin{itemize} \item[$\cdot$] \begin{small} \begin{multline}\nonumber \hspace*{-1cm}D_{2r+1} \left( \sum_{w(\cdot)=2n \atop d(\cdot)=2p } \zeta^{\mathfrak{m}}(O, O>1, \cdots, O, O>1) \right) \\
= \sum_{i=0}^{p-1} \left( \sum_{w(\cdot)=2r+1 \atop d(\cdot)=2i+1 } \begin{array}{l} + \zeta^{\mathfrak{l}}(O, O>1, \cdots, O>1, O)\\ -\zeta^{\mathfrak{l}}(O, O >1, \ldots, O>1, O) \end{array} \right) \otimes \sum_{w(\cdot)=2n-2r-1 \atop d(\cdot)=2p-2i-1 } \zeta^{\mathfrak{m}}(O, O>1, \ldots, O, O>1)=0.
\end{multline}
\end{small} \item[$\cdot$] \begin{small} \begin{multline}\nonumber \hspace*{-1cm}D_{2r+1} \left( \sum_{w(\cdot)=2n+1 \atop d(\cdot)=2p+1 } \zeta^{\mathfrak{m}}(O>1, O, \ldots, O, O>1) \right) \\ = \sum_{i=0}^{p-1} \left( \sum_{w(\cdot)=2r+1 \atop d(\cdot)=2i+1 } \zeta^{\mathfrak{l}}(O>1, O, \ldots, O>1) \right) \otimes \sum_{w(\cdot)=2n-2r \atop d(\cdot)=2p-2i-1 } \zeta^{\mathfrak{m}}(O, O>1, \cdots, O, O>1).
\end{multline}
\end{small} By the previous identity, the right side is in $\mathbb{Q}\pi^{2n-2r}$, which proves the result claimed; it gives also the recursion for the coefficients: $\beta^{n,p}_{r}= \sum_{i=0}^{p-1} \beta^{r,i}_{r} \alpha^{n-r,p-i} $. \item[$\cdot$] \begin{small} \begin{multline}\nonumber \hspace*{-1cm}D_{2r+1} \left( \sum_{w(\cdot)=2n+1 \atop d(\cdot)=2p+1 } \zeta^{\mathfrak{m}}(O, O>1, \ldots, O>1, O) \right) =\\
\begin{array}{l} +\sum_{i=0}^{p-1} \left( \sum_{w(\cdot)=2r+1 \atop d(\cdot)=2i+1 } \zeta^{\mathfrak{m}}(O, O>1, \ldots, O) \right) \otimes \sum_{w(\cdot)=2n-2r \atop d(\cdot)=2p-2i-1 } \zeta^{\mathfrak{m}}(O, O>1, \cdots, O>1)\\ +\sum_{i=0}^{p-1} \left( \sum_{w(\cdot)=2r+1, \atop d(\cdot)=2i+1 } \begin{array}{l} + \zeta^{\mathfrak{m}}(O, O>1, \ldots, O)\\
- \zeta^{\mathfrak{m}}(O, O>1, \ldots, O) \end{array} \right) \otimes \sum_{w(\cdot)=2n-2r \atop d(\cdot)=2p-2i-1 } \zeta^{\mathfrak{m}}(O>1, O, \cdots, O>1, O) \end{array}
\end{multline}
\end{small} As above, by recursion hypothesis, the right side of the first sum is in $\mathbb{Q}\pi^{2n-2r}$, which proves the result claimed, the second sum being $0$; the rational coefficients $\gamma$ are given by a recursive relation. \end{itemize} \end{itemize} \end{proof}
\appendix \chapter{} \section{Coaction}
The coaction formula given by Goncharov and extended by Brown for motivic iterated integrals applies to the $\star$, $\star\star$, $\sharp$ or $\sharp\sharp$ version by linearity \footnote{Recall the identities $\ref{eq:miistarsharp}$ to turn a $\star$ (resp. $\sharp$) into a $1$ (resp. two times a $1$) minus a $0$.}. Here is the version obtained for MMZV $\star,\star\star$, $\sharp$ or $\sharp\sharp$:\footnote{\textit{For purpose of stability}: if there is a $\pm 1$ at the beginning, as for $\star$ or $\sharp$ versions, the cut with this first $\pm 1$ will be let as a $T_{\pm 1, 0}$ term (and not converted into a $T_{\epsilon, 0}$ less a $T_{0,0}$), in order to still have a $\pm 1$ at the beginning; whereas, if there is no $\pm 1$ at the beginning, as for $\star\star$ or $\sharp\sharp$ version even the first cut (first line) has to be converted into a $T_{0, \epsilon}$ less a $T_{0,0}$, in order to still have a $\epsilon$ at the beginning.} \begin{lemm}\label{lemmt} $L$ being a sequence in $\lbrace 0, \pm\star\rbrace$ resp. $\lbrace 0,\pm\sharp\rbrace$, with possibly $1$ at the beginning, $\epsilon \in \lbrace \pm\star \rbrace$ resp. $\in \lbrace\pm\sharp\rbrace$, and $s_{\epsilon}\mathrel{\mathop:}=sign(\epsilon)$. $$D_{r} I^{\mathfrak{m}}_{s}\left(0;L;1 \right)=\delta_{ L= A \epsilon B \atop w(A)=r} I^{\mathfrak{l}}_{k}\left(0;A ;s_{\epsilon} \right) \otimes I^{\mathfrak{m}}_{s-k}\left(0;s_{\epsilon}, B ;1 \right) $$ $$+ \sum_{L=A \epsilon B 0 C \atop w(B)=r} I^{\mathfrak{l}}\left(s_{\epsilon};B ;0 \right) \otimes \left( \underbrace{I^{\mathfrak{m}}_{s}\left(0;A, \epsilon, 0, C ;1\right)}_{T_{\epsilon,0}} + \underbrace{I^{\mathfrak{m}}_{s}(0;A,0,0,C ;1)} _{T_{0,0}} \right)$$ $$+ \sum_{L=A 0 B \epsilon C \atop w(B)=r} I^{\mathfrak{l}}\left(0;B, s_{\epsilon}\right) \otimes \left( \underbrace{I^{\mathfrak{m}}_{s}\left(0;A,0, \epsilon, C ;1 \right)}_{T_{0, \epsilon}} + \underbrace{I^{\mathfrak{m}}_{s}(0;A,0,0,C ;1)}_{T_{0,0}} \right)$$ $$+ \sum_{L=A \epsilon B \epsilon C\atop w(B)=r} I^{\mathfrak{l}}\left(0;B, s_{\epsilon}\right) \otimes \left( \underbrace{I^{\mathfrak{m}}_{s}(0;A,\epsilon,0,C ;1)}_{T_{\epsilon,0}} - \underbrace{I^{\mathfrak{m}}_{s}\left(0;A,0, \epsilon, C ;1 \right)}_{T_{0,\epsilon}} \right)$$
\begin{multline}\nonumber + \sum_{L=A \epsilon B -\epsilon C\atop w(B)=r} \left[ I^{\mathfrak{l}}\left(0;B; -s_{\epsilon} \right) \otimes \underbrace{I^{\mathfrak{m}}_{s}(0;A,\epsilon, 0,C ;1)}_{T_{\epsilon,0}} + I^{\mathfrak{l}}\left(s_{\epsilon};B;0\right)\otimes \underbrace{I^{\mathfrak{m}}_{s}(0; A,0,-\epsilon, C ;1)}_{T_{0,\epsilon}} \right. \\ \left. I^{\mathfrak{l}}\left(s_{\epsilon};B; -s_{\epsilon} \right) \otimes \underbrace{I^{\mathfrak{m}}_{s}\left(0;A, \epsilon, - \epsilon, C ;1 \right)}_{T_{\epsilon, -\epsilon}} \right] . \end{multline} \end{lemm}
\noindent \textsc{Remarks}: \begin{itemize} \item[$\cdot$] We will refer to these different terms $T$ for each cut in the whole appendix when using the coaction. \item[$\cdot$] The expression of $D_{r}$ for specific MMZV $\star$ and Euler $\sharp$ sums is simplified below thanks to antipodal and hybrid relations, and is fundamentally used in the proofs of Chapter $4$. \end{itemize} \begin{proof} The proof is straightforward from $\eqref{eq:Der}$, using the linearity (with $\ref{eq:miistarsharp}$) in both directions: \begin{itemize} \item[$(i)$] First, to turn $\epsilon$ into a difference of $\pm 1$ minus $0$ in order to use $\eqref{eq:Der}$. \item[$(ii)$] Then, in the right side, a $\pm 1$ appeared inside the iterated integral when looking at the usual coaction formula which is turned into a sum of a term with $\epsilon$ (denoted $T_{\epsilon,0}$ or $T_{0,\epsilon}$) and a term with $0$ (denoted $T_{0,0}$) by linearity of the iterated integrals and in order to end up only with $0,\epsilon$ in the right side. \end{itemize} Listing now the different cuts leads to the expression of the lemma, since: \begin{itemize} \item[$\cdot$] The first line corresponds to the initial cut (from the $s+1$ first $0$). \item[$\cdot$] The second line corresponds to a cut either from $\pm\epsilon$ to $0$; the $\pm\epsilon$ being $\pm 1$. \item[$\cdot$] The third line corresponds to a cut from $0$ to $\pm\epsilon$. \item[$\cdot$] The fourth line corresponds to cut from $\epsilon$ to $\epsilon$, with two choices: a $\epsilon$ being fixed to $0$, the other one fixed to $1$. Replacing $1$ by $(\epsilon)+(0)$, this leads to a $T_{0,\epsilon}$, a $T_{\epsilon,0}$ and two $T_{0,0}$ terms which get simplified together. \item[$\cdot$] The last lines correspond to cuts from $\epsilon$ to $-\epsilon$, with three possibilities: one being fixed to $0$, the other one fixed to $\pm 1$, or the first being $1$, the second $-1$. This leads to a $T_{\epsilon,0}$, a $T_{0, -\epsilon}$ and a $T_{\epsilon,-\epsilon}$, since the $T_{0,0}$ terms get simplified. \end{itemize} \end{proof}
\subsection{Simplification rules} This section is devoted on the simplification of the coaction, in the case of motivic Euler sums: we gather terms in $D_{2r+1}$ according to their right side, using relations ($§ 4.2$) between motivic iterated integrals $I^{\mathfrak{l}}\in\mathcal{L}$ to simplify the left side.\\ \\ \texttt{Notations:} We use the notation of the iterated integrals inner sequences and represent a term of a cut in $D_{2r+1}$ (referring to Lemma $4.4.2$) by arrows between two elements of this sequence. The weight of the cut (which is the length of the subsequence in the iterated integral) would always be considered odd here.\footnote{Since we are here only interested in motivic Euler sums, the non zero weight graded parts in the coaction are these corresponding to odd weights: $D_{2r+1}$, $r\geq 0$.} The diagrams show which terms get simplified together: i.e. these which have same right side, but opposite left side, by the relation considered in the coalgebra $\mathcal{L}$. \\ \begin{description} \item[\textsc{ Composition }:] The composition rule (cf. $§ 1.6$) in the coalgebra $\mathcal{L}$ boils down to: \begin{equation} I^{\mathfrak{l}}(a; X; b)\equiv -I^{\mathfrak{l}}(b; X; a), \quad \text{ with $X$ any sequence of } 0, \pm 1, \pm \star, \pm \sharp. \end{equation} It allows us to switch the two extremities of the integral if we multiply by $-1$ the integral: this exchange is considerably used below, without mentioning. \item[\textsc{ Antipode } $\shuffle$:] It corresponds to a reversal of path for iterated integrals (cf. $\ref{eq:antipodeshuffle2}$): \begin{center} $I^{\mathfrak{l}}(a; X; b)\equiv(-1)^{w}I^{\mathfrak{l}}(b; \widetilde{X}; a)$ for any X sequence of $0, \pm 1, \pm \star, \pm \sharp$. Hence: \end{center} \begin{itemize} \item[$\cdot$] If X symmetric, i.e. $\widetilde{X}=X$, these two cuts get simplified, \includegraphics[]{dep14.pdf} since $ I^{\mathfrak{l}}(\epsilon; X \epsilon^{i+1}; 0) \equiv - I^{\mathfrak{l}}(0; \epsilon^{i+1}\widetilde{X} ; \epsilon) \equiv - I^{\mathfrak{l}}(0; \epsilon^{i+1} X ; \epsilon)$. \item[$\cdot$] If X antisymmetric, i.e. $\widetilde{X}=-X$,the cut \includegraphics[]{dep15.pdf} is zero since:\\ $ \begin{array}{ll} I^{\mathfrak{l}}(\epsilon; X; -\epsilon) & \equiv I^{\mathfrak{l}}(\epsilon; X; 0)+ I^{\mathfrak{l}}(0; X; -\epsilon)\\ &\equiv I^{\mathfrak{l}}(\epsilon; X; 0)- I^{\mathfrak{l}}(-\epsilon; \widetilde{X}; 0) \\ &\equiv I^{\mathfrak{l}}(\epsilon; X; 0)- I^{\mathfrak{l}}(-\epsilon; -X; 0)\\ &\equiv 0 \end{array}$. \end{itemize}
\item[\textsc{ Shift }] For MES $\star\star$ and, when weight and depth odd for Euler $\sharp\sharp$ sums: \begin{equation}\label{eq:shift} \textsc{(Shift) } \zeta^{\bullet}_{n-1} (n_{1},\cdots, n_{p})= \zeta^{\bullet}_{n_{1}-1} (n_{2},\cdots, n_{p},n) \end{equation} \includegraphics[]{dep16.pdf} A dot belongs to $\lbrace \pm\star,\pm\sharp\rbrace$, and two dots with a same index, $\bullet_{i}$ shall be identical. \item[\textsc{ Cut }] For ES $\sharp\sharp$, with even depth\footnotemark[1], odd weight:\footnotetext[1]{Note that the depth considered here needed to be even is the depth of the bigger cut.}\\ \begin{equation}\label{eq:cut} \includegraphics[]{dep17.pdf} \end{equation} \item[\textsc{ Cut Shifted }]: For ES $\sharp\sharp$, with even depth\footnotemark[1], odd weight, composing Cut with Shift: \begin{equation}\label{eq:cutshifted} \includegraphics[]{dep18.pdf} \end{equation} \item[\textsc{ Minus }] For ES $\sharp\sharp$, with even depth, odd weight: \begin{equation}\label{eq:minus} \includegraphics[]{dep19.pdf} \end{equation} \item[\textsc{ Sign }] For ES $\sharp\sharp$ with even depth, odd weight, i.e. $X\in \lbrace 0, \pm \sharp\rbrace^{\times}$: \begin{equation}\label{eq:sign} \includegraphics[]{dep20.pdf} \end{equation} \textsc{ Sign } hence also means that the $\pm$ sign at one end of a cut does not matter. \end{description}
\subsection{MMZV $\star$}
Let express each MMZV$^{\star}$ as: $$\zeta^{\star, \mathfrak{m}} (2^{a_{0}},c_{1},\cdots,c_{p}, 2^{a_{p}}).$$ As we will see below, this writing is suitable for the coaction expression, since most of the cuts from a $2$ to another $2$ get simplified by the rules above. The iterated integral corresponding:\\ \begin{equation}\label{eq:iistar} I\left( 0; 1, 0, \left(\star, 0\right)^{a_{0}-1}, \star \cdots 0^{c_{i}-1} \left( \star 0 \right)^{a_{i}} \star , \ldots, 0^{c_{j}-1} \left(\star 0 \right)^{a_{j}} \star, \ldots, 0^{c_{p}-1} \left( \star 0 \right)^{a_{p}} ; 1\right) . \end{equation} Considering $D_{2r+1}$ after some simplifications:\footnote{Here $\delta_{r}$ underlines that the left side must have a weight equal to $2r+1$.} \begin{lemm} \begin{equation} \label{eq:DerivStar} D_{2r+1} \left( \zeta^{\star, \mathfrak{m}} (2^{a_{0}},c_{1},\cdots,c_{p}, 2^{a_{p}})\right) = \end{equation} $$\delta_{r} \sum_{i<j} \left[ \delta_{3\leq \alpha \leq c_{i+1}-1 \atop 0\leq \beta \leq a_{j}} \zeta^{\star, \mathfrak{l}}_{c_{i+1}-\alpha} (2^{a_{j}-\beta}, \ldots, 2^{ a_{i+1}}) \otimes \zeta^{\star, \mathfrak{m}} (\cdots,2^{a_{i}}, \alpha, 2^{\beta}, c_{j+1}, \cdots) \right.$$ $$\left( -\delta_{c_{i+1}>3} \zeta^{\star\star, \mathfrak{l}}_{2} (2^{a_{j}-\beta-1}, \ldots, 2^{ a_{i+1}}) + \delta_{c_{j+1}>3} \zeta^{\star\star, \mathfrak{l}}_{2} (2^{a_{j}}, \ldots, 2^{ a_{i+1}-\beta -1}) + \right. $$ $$ - \delta_{c_{i+1}=1} \zeta^{\star\star, \mathfrak{l}} (2^{a_{j}-\beta}, \ldots, 2^{ a_{i+1}}) + \delta_{c_{j+1}=1} \zeta^{\star\star, \mathfrak{l}} (2^{a_{i+1}-\beta}, \ldots, 2^{ a_{j}}) $$ $$+ \delta_{c_{i+2}=1 \atop \beta>a_{i+1}} \zeta^{\star\star, \mathfrak{l}}_{1} (2^{a_{j}+a_{i+1}-\beta}, \ldots, 2^{ a_{i+2}}) - \delta_{c_{j}=1 \atop \beta>a_{j}} \zeta^{\star\star, \mathfrak{l}}_{1} (2^{a_{i+1}+a_{j}-\beta}, \ldots, 2^{ a_{j-1}}) .$$ $$ \left. +\delta_{\beta > a_{i+1}}\zeta^{\star\star, \mathfrak{l}}_{c_{i+2}-2} (2^{a_{i+1}+a_{j}-\beta+1}, \ldots, 2^{ a_{i+2}}) - \delta_{\beta > a_{j}} \zeta^{\star\star, \mathfrak{l}}_{c_{j}-2} (2^{a_{i+1}+a_{j}-\beta+1}, \ldots, 2^{ a_{j-1}}) \right) $$ $$\otimes \zeta^{\star, \mathfrak{m}} (\cdots,2^{a_{i}}, c_{i+1}, 2^{\beta}, c_{j+1}, \cdots)$$ $$\left. - \delta_{3\leq \alpha \leq c_{j}-1 \atop 0\leq \beta \leq a_{i}} \zeta^{\star, \mathfrak{l}}_{c_{j}-\alpha} (2^{a_{i}-\beta}, \ldots, 2^{ a_{j-1}}) \otimes \zeta^{\star, \mathfrak{m}} (\cdots, c_{i}, 2^{\beta}, \alpha, 2^{a_{j}}, \cdots)\right] . $$ \end{lemm}
\begin{proof} We look at cuts of odd interior length between two elements of the sequence inside $\ref{eq:iistar}$. By \textsc{Shift}, the following cuts get simplified:\\ \includegraphics[]{dep21.pdf}\ More precisely, $T_{0,0}$ resp. $T_{0,\star}$ above get simplified with $T_{0,0}$ resp. $T_{\star,0}$ below (shifted by one at the right), by colors, two by two. The dotted arrows mean that in the particular case where $c_{i}=1$ resp. $c_{j}=1$, only $T_{0,0}$ get simplified.\\ The following arrows get simplified by $\textsc{Shift}$ $(\ref{eq:shift})$, still above with below and by colors:\\ \includegraphics[]{dep22.pdf}\\ \includegraphics[]{dep23.pdf}\\
Cyan arrows above resp. below are $T_{0,\star}$ resp. $T_{0,\star}$ terms; magenta ones above resp. below stand for $T_{0,0}$ and $T_{0,\star}$ resp. for $T_{0,0}$ and $T_{\star,0}$ terms. \\ If $c_{i}=1$ (the case $c_{j}=1$, antisymmetric, is omitted), it remains also the following cuts ($T_{\star,0}$ for black ones or $T_{0,\star}$ for cyan ones):\\ \includegraphics[]{dep24.pdf}\\
Gathering the remaining cuts in this diagram, according the right side:\\ \begin{enumerate} \item For $\zeta^{\star, \mathfrak{m}} (\cdots,2^{a_{i}}, \boldsymbol{\alpha, 2^{\beta}}, c_{j+1}, \cdots)$: $$ \left( \textcolor{magenta}{\delta_{3\leq \alpha < c_{i+1} \atop 0\leq \beta < a_{j}}} \zeta^{\star\star, \mathfrak{l}}_{c_{i+1}-\alpha} (2^{a_{i+1}}, \ldots, 2^{ a_{j}-\beta}) - \left( \textcolor{magenta}{\delta_{4\leq \alpha < c_{i+1} \atop 0\leq \beta < a_{j}}} + \textcolor{cyan}{\delta_{\alpha=3 \atop 0\leq \beta < a_{j}}} \right) \zeta^{\star\star, \mathfrak{l}}_{c_{i+1}-\alpha+2} (2^{a_{i+1}}, \ldots, 2^{ a_{j}-\beta-1}) \right).$$ Using \textsc{Shift} $(\ref{eq:shift})$ for the first term and then using the definition of $\zeta^{\star}$ it turns into: $$ \delta_{3\leq \alpha < c_{i+1} \atop 0\leq \beta <a_{j}} \left( \zeta^{\star\star, \mathfrak{l}}_{1} (c_{i+1}-\alpha+1, 2^{a_{i+1}}, \ldots, 2^{ a_{j}-\beta-1}) -\zeta^{\star\star, \mathfrak{l}}_{c_{i+1}-\alpha+2} (2^{a_{i+1}}, \ldots, 2^{ a_{j}-\beta-1}) \right) $$ $$= \delta_{3\leq \alpha < c_{i+1} \atop 0\leq \beta < a_{j}} \zeta^{\star, \mathfrak{l}}_{1} (c_{i+1}-\alpha+1, 2^{a_{i+1}}, \ldots, 2^{ a_{j}-\beta-1}). $$ Applying antipodes $A_{\shuffle} \circ A_{\ast} \circ A_{\shuffle}$: $$ =\delta_{3\leq \alpha < c_{i+1} \atop 0\leq \beta < a_{j}} \zeta^{\star, \mathfrak{l}}_{c_{i+1}-\alpha} (2^{ a_{j}-\beta}, \ldots, 2^{a_{i+1}}),$$ which gives the first line in $\eqref{eq:DerivStar}$.
\item For $\zeta^{\star, \mathfrak{m}} (\cdots,2^{a_{i}}, \boldsymbol{\alpha}, 2^{a_{j}}, c_{j+1}, \cdots)$, the corresponding left sides are: $$ -\left( \textcolor{magenta}{\delta_{c_{j}+2\leq \alpha \leq c_{i+1}}} + \textcolor{cyan}{\delta_{\alpha=c_{j}+1\atop c_{i+1}>c_{j}} } \right) \zeta^{\star\star, \mathfrak{l}}_{c_{i+1}+ c_{j}-\alpha} (2^{a_{i+1}}, \ldots, 2^{a_{j-1}}) $$ $$ +\left( \textcolor{magenta}{ \delta_{c_{i+1}+2 \leq \alpha \leq c_{j}}} + \textcolor{cyan}{ \delta_{\alpha = c_{i+1}+1 \atop c_{j}> c_{i+1}}} \right) \zeta^{\star\star, \mathfrak{l}}_{c_{i+1}+c_{j}-\alpha} (2^{a_{j-1}}, \ldots, 2^{a_{i+1}}) $$ $$+\delta_{3\leq \alpha < c_{i+1}} \zeta^{\star\star, \mathfrak{l}}_{c_{i+1}-\alpha} (2^{a_{i+1}}, \ldots, c_{j}) -\delta_{3\leq \alpha <c_{j}}\zeta^{\star\star, \mathfrak{l}}_{c_{j}-\alpha} (2^{a_{j-1}}, \ldots,c_{i+1})$$ Using Antipode $\ast$ and turning some $\epsilon$ into $'1+0'$: $$=+\delta_{3\leq \alpha < c_{i+1}} \zeta^{\star\star, \mathfrak{l}}_{c_{i+1}-\alpha} (2^{a_{i+1}}, \ldots, c_{j}) -\delta_{3\leq \alpha < c_{j}}\zeta^{\star\star, \mathfrak{l}}_{c_{j}-\alpha} (2^{a_{j-1}}, \ldots,c_{i+1})$$ $$+ (-1)^{{c_{j}<c_{i+1}}} \delta_{\min(c_{j},c_{i+1}) < \alpha \leq \max(c_{j},c_{i+1})} \zeta^{\star\star, \mathfrak{l}}_{c_{i+1}+ c_{j}-\alpha} (2^{a_{i+1}}, \ldots, 2^{a_{j-1}}) $$ $$= \delta_{3\leq \alpha < c_{i+1}} \zeta^{\star, \mathfrak{l}}_{c_{i+1}-\alpha} (c_{j}, \ldots, 2^{ a_{i+1}}) - \delta_{3\leq \alpha < c_{j} } \zeta^{\star, \mathfrak{l}}_{c_{j}-\alpha} (c_{i+1}, \ldots, 2^{ a_{j-1}})$$ This gives exactly the same expression than the first and fourth case for $\beta=a_{i}$ or $a_{j}$, and are integrated to them in $\eqref{eq:DerivStar}$.
\item For $\zeta^{\star, \mathfrak{m}} (\cdots, c_{i+1},\boldsymbol{2^{\beta}}, c_{j+1}, \cdots)$: \footnote{It includes the case $\alpha=2$.} $$-\delta_{c_{i+1}>3 \atop 0 \leq \beta < a_{j}}\zeta^{\star\star, \mathfrak{l}}_{2} (2^{a_{j}-\beta-1}, \ldots, 2^{ a_{i+1}}) + \delta_{c_{j+1}>3 \atop 0 \leq \beta < a_{i+1}} \zeta^{\star\star, \mathfrak{l}}_{2} (2^{a_{j}}, \ldots, 2^{ a_{i+1}-\beta -1}) $$ $$ +\delta_{\beta > a_{i+1} \atop c_{i+1}>3 }\zeta^{\star\star, \mathfrak{l}}_{c_{i+2}-2} (2^{a_{i+1}+a_{j}-\beta+1}, \ldots, 2^{ a_{i+2}}) - \delta_{\beta > a_{j}} \zeta^{\star\star, \mathfrak{l}}_{c_{j}-2} (2^{a_{i+1}+a_{j}-\beta+1}, \ldots, 2^{ a_{j-1}}) $$ $$- \delta_{c_{i+1}=1 \atop 1\leq \beta < a_{j}} \zeta^{\star\star, \mathfrak{l}} (2^{a_{j}-\beta}, \ldots, 2^{ a_{i+1}}) + \delta_{c_{j+1}=1 \atop 1 \leq \beta <a_{i+1}} \zeta^{\star\star, \mathfrak{l}} (2^{a_{i+1}-\beta}, \ldots, 2^{ a_{j}}) $$ $$+ \delta_{c_{i+2}=1 \atop \beta>a_{i+1}} \zeta^{\star\star, \mathfrak{l}}_{1} (2^{a_{j}+a_{i+1}-\beta}, \ldots, 2^{ a_{i+2}}) - \delta_{c_{j}=1 \atop \beta>a_{j}} \zeta^{\star\star, \mathfrak{l}}_{1} (2^{a_{i+1}+a_{j}-\beta}, \ldots, 2^{ a_{j-1}}) .$$ \item For $ \zeta^{\star, \mathfrak{m}} (\cdots, c_{i}, \boldsymbol{2^{\beta}, \alpha}, 2^{a_{j}}, \cdots)$, antisymmetric to 1: $$ \left( - \textcolor{magenta}{\delta_{2\leq \alpha \leq c_{j}-1 \atop 0\leq \beta \leq a_{i}}} \zeta^{\star\star, \mathfrak{l}}_{c_{j}-\alpha} (2^{a_{j-1}}, \ldots, 2^{ a_{i}-\beta}) + \left(\textcolor{magenta}{\delta_{4\leq \alpha \leq c_{j}+1 \atop 0\leq \beta \leq a_{i}-1}} + \textcolor{cyan}{\delta_{\alpha=3 \atop 0\leq \beta \leq a_{i}-1}} \right) \zeta^{\star\star, \mathfrak{l}}_{c_{j}-\alpha+2} (2^{a_{j-1}}, \ldots, 2^{ a_{i}-\beta-1}) \right) $$ \end{enumerate} This leads to the lemma, with the second case incorporated in the first and last line. \end{proof}
\subsection{Euler $\sharp$ sums with $\boldsymbol{\overline{even}}, \boldsymbol{odd}$}
Let us consider the following family: $$\zeta^{\sharp, \mathfrak{m}}\left( \lbrace\boldsymbol{\overline{even}}, \boldsymbol{odd}\rbrace^{\times} \right) , \text{i.e. negative even and positive odd integers}$$ which, in terms of iterated integrals corresponds to, with $\epsilon\in \lbrace\pm \sharp\rbrace$: \begin{equation}\label{eq:iisharp} I^{ \mathfrak{m}} \left( 0; \left\lbrace \begin{array}{l} 1, \boldsymbol{0}^{odd} ,-\sharp \\ 1, \boldsymbol{0}^{even} , \sharp \end{array}\right\rbrace , \cdots, \quad \left\lbrace \begin{array}{l} \epsilon, \boldsymbol{0}^{odd}, -\epsilon \\ \epsilon, \boldsymbol{0}^{even}, \epsilon \end{array}\right\rbrace \quad, \cdots; 1 \right) . \end{equation}
\begin{lemm} The family $\zeta^{\sharp \mathfrak{m}}\left( \lbrace\overline{even}, odd\rbrace^{\times} \right) $ is stable under the coaction. \end{lemm} \begin{proof} Looking at the possible kinds of cuts, and gathering them according the right side: \includegraphics[]{dep25.pdf}\\ These cuts have the same form for the right side in the coaction: $$I^{\mathfrak{m}}(0; \cdots, \boldsymbol{\epsilon_{i} 0^{\alpha} \epsilon_{j}}, \cdots ;1).$$ Notice there would be no term $T_{\epsilon, -\epsilon}$ in a cut from $\epsilon$ to $-\epsilon$ because of \textsc{Sign} $(\ref{eq:sign})$ identity, therefore you have there all the possible cuts pictured.\\ A priori, cuts can create in the right side a sequence $\left\lbrace \epsilon, \boldsymbol{0}^{even}, -\epsilon \right\rbrace $ or $\left\lbrace \epsilon, \boldsymbol{0}^{odd}, \epsilon\right\rbrace $ inside the iterated integral; these cuts are the \textit{unstable} ones, since they are out of the considered family. However, by coupling these cuts two by two, and using the rules listed at the beginning of the Annexe, the unstable cuts would all get simplified.\\ Indeed, let examine each of the terms $(1-6)$\footnote{There is no remaining cuts between $\epsilon$ and $\epsilon$. Notice also that the left sides of the remaining terms have an even depth.}:\\ \\
\begin{tabular}{c | c | l | l} Term & Left side & Unstable if & Simplified with \\ \hline \multirow{4}{*}{$(1)$} & $\zeta^{ \sharp\sharp,\mathfrak{l}}_{a-1-\alpha} (a_{1}, \ldots, a_{n},b)$, & $n$ even & the previous cut: \\
& with $\alpha<a$. & & either $(6)$ by \textsc{Minus} \\
& & & or $(5)$ by \textsc{Cut} \\
& & & or $(3)$ by \textsc{Cut} \footnotemark[1]\\ \hline \multirow{4}{*}{$(2)$} & $\zeta^{ \sharp\sharp,\mathfrak{l}}_{b-1} (a_{n}, \ldots, a_{1})$ & $\epsilon_{i+1}=\epsilon_{j}$ & the previous cut: \\
& with $\alpha=a$. & & either with $(5)$ by \textsc{Shift} \\
& & & or with $(6)$ by \textsc{Cut Shifted} \\
& & & or with $(3)$ by \textsc{Shift}.\\ \hline \multirow{4}{*}{$(3)$} & $-\zeta^{ \sharp\sharp,\mathfrak{l}}_{a+b-\alpha-1} (a_{1}, \ldots, a_{n})$ & $n$ odd & the following cut:\\
& with $\alpha>b$. & & either $(1)$ by \textsc{Cut} \\
& & & or $(2)$ by \textsc{Shift} \\
& & & or $(4)$ by \textsc{Shift}. \\ \hline
\multirow{4}{*}{$(4)$} & $\zeta^{ \sharp\sharp,\mathfrak{l}}_{a+b-\alpha-1}(a_{n}, \ldots, a_{1})$ & $n$ odd & the previous cut:\\
& with $\alpha>a$ .& & either $(6)$ by \textsc{Cut Shifted} \\
& & & or with $(5)$ by \textsc{Shift} \\
& & & or with $(3)$ by \textsc{Shift}. \\ \hline
\multirow{4}{*}{$(5)$}& $-\zeta^{ \sharp\sharp,\mathfrak{l}}_{a-1} (a_{1}, \ldots, a_{n})$ & $\epsilon_{i}=\epsilon_{j-1}$ & the following cut:\\
& with $\alpha=b$. & & either with $(1)$ by \textsc{Cut}, \\
& & & or with $(2)$ by \textsc{Shift}, \\
& & & or with $(4)$ by \textsc{Shift}. \\ \hline
\multirow{4}{*}{$(6)$} & $-\zeta^{ \sharp\sharp,\mathfrak{l}}_{b-1-\alpha} (a_{n}, \cdots a_{1}, a)$ & $n$ even & the following cut:\\
& & & either $(1)$ by \textsc{Minus} \\
& & & or with $(2)$ by \textsc{Cut Shifted} \\
& & & or with $(4)$ by \textsc{Cut Shifted} \\ \hline \end{tabular} \\ \footnotetext[1]{It depends on the sign of $b+1-\alpha$ here for instance.} \end{proof} \noindent \paragraph{Derivations.} Let use the writing of the Conjecture $\ref{conjcoeff}$: \begin{equation}\label{eq:essharpgather}
\zeta^{\sharp,\mathfrak{m}}(B_{0}, 1^{\gamma_{1}}, \ldots, 1^{\gamma_{p}}, B_{p}) \text{ with } B_{i}<0 \text{ if and only if } B_{i} \text{ even }. \end{equation} \texttt{Nota Bene:} Beware, for instance $B_{i}$ may be equal to $1$, which implies that $\gamma_{i}= \gamma_{i+1}=0$. Indeed, we look at the indices corresponding to a sequence $(2^{a_{0}}, c_{1}, \ldots, c_{p}, 2^{a_{p}})$ as in the Conjecture $\ref{conjcoeff}$: $$\begin{array}{l} B_{i}= 2a_{i}+3 - \delta_{c_{i}}- \delta_{c_{i+1}}\\ B_{0}= 2a_{0}+1 - \delta_{c_{1}}\\ B_{p}= 2a_{p}+2 - \delta_{c_{p}}\\ \end{array}, \gamma_{i}\mathrel{\mathop:}= c_{i}-3 +2 \delta_{c_{i}}, \quad \text{ where } \left\lbrace \begin{array}{l} a_{i} \geq 0 \\ c_{i}>0,c_{i}\neq 2 \\ \delta_{c}\mathrel{\mathop:}= \left\lbrace \begin{array}{ll} 1 & \text{ if } c=1\\ 0 & \text{ else }. \end{array}\right. \end{array}. \right. $$
\begin{lemm} \begin{equation} D_{2r+1}\left( \zeta^{\sharp,\mathfrak{m}}(B_{0}, 1^{\gamma_{1}}, \ldots, 1^{\gamma_{p}}, B_{p})\right) =\footnotemark[2] \end{equation} $$\delta_{r} \left[ -\delta_{{2 \leq B \leq B_{j}+1 \atop 0\leq\gamma\leq\gamma_{i+1}-1 }}\zeta^{\sharp,\mathfrak{l}}(B_{j}-B+1, 1^{\gamma_{j}}, \ldots, 1^{\gamma_{i+1}-\gamma-1})\otimes\zeta^{\sharp,\mathfrak{m}}(B_{0} \cdots, B_{i}, \textcolor{magenta}{1^{\gamma}, B}, 1^{\gamma_{j+1}}, \ldots, B_{p}) \right. $$
$$\left[ \begin{array}{l} + \delta_{B_{i+1}< B}\zeta^{\sharp\sharp,\mathfrak{l}}_{B_{i+1}+B_{j}-B}(1^{\gamma_{j}}, \ldots, 1^{\gamma_{i+2}})\\ - \delta_{B_{j}< B}\zeta^{\sharp\sharp,\mathfrak{l}}_{B_{i+1}+B_{j}-B}(1^{\gamma_{i+2}}, \ldots, 1^{\gamma_{j}})\\
+ \zeta^{\sharp\sharp,\mathfrak{l}}_{B_{i+1}-B}(1^{\gamma_{i+2}}, \ldots, B_{j}) - \zeta^{\sharp\sharp,\mathfrak{l}}_{B_{j}-B}(1^{\gamma_{j}}, \ldots, B_{i+1}) \end{array} \right] \otimes\zeta^{\sharp,\mathfrak{m}}(B_{0} \cdots, B_{i}, 1^{\gamma_{i+1}}, \textcolor{green}{B}, 1^{\gamma_{j+1}}, \ldots, B_{p}) $$
$$\left. \delta_{{1 \leq B \leq B_{i+1}+1 \atop 0\leq\gamma\leq\gamma_{j+1}-1}}\zeta^{\sharp,\mathfrak{l}}(B_{i+1}-B+1, 1^{\gamma_{i+2}}, \ldots, 1^{\gamma_{j+1}-\gamma-1})\otimes\zeta^{\sharp,\mathfrak{m}}(B_{0} \cdots, 1^{\gamma_{i+1}},\textcolor{cyan}{ B, 1^{\gamma}}, B_{j+1}, \ldots, B_{p}) \right],$$ where $B$ is positive if odd, negative if even. \end{lemm} \begin{proof} \footnotetext[2]{Here $\delta_{r}$ indicates that left side has to be of weight $2r+1$.} \texttt{Nota Bene}: For the left side, we only look at odd weight $w$, and the parity of the depth $d$ is fundamental since the relations stated above depend on the parity of $w-d$. For instance, for such a sequence $(\boldsymbol{1}^{\gamma_{i}},B_{i}, \ldots,B_{j-1},\boldsymbol{1}^{\gamma_{j}})$ (with the previous notations), $weight- depth$ has the same parity than $\delta_{c_{i}}+\delta_{c_{j}}$. \\ \\ The following cuts get simplified, with \textsc{Shift}, since depth is odd ($B_{i}$ odd if $c_{i},c_{i+1}\neq 1$):\\ \includegraphics[]{dep26.pdf}\\ It remains, where all the unstable cuts are simplified by the Lemma $A.1.4$, cuts that we can gather into four groups, according to the right side of the coaction: \begin{itemize} \item[$(i)$] $\zeta^{\sharp,\mathfrak{m}}(B_{0} \cdots, B_{i}, \textcolor{magenta}{1^{\gamma}, B}, 1^{\gamma_{j+1}}, \ldots, B_{p}) $. \item[$(ii)$] $\zeta^{\sharp,\mathfrak{m}}(B_{0} \cdots, B_{i}, \textcolor{yellow}{1^{\gamma}}, B_{j}, \ldots, B_{p}) $. \item[$(iii)$] $\zeta^{\sharp,\mathfrak{m}}(B_{0} \cdots, B_{i}, 1^{\gamma_{i+1}}, \textcolor{green}{B}, 1^{\gamma_{j+1}}, \ldots, B_{p}) $. \item[$(iv)$] $\zeta^{\sharp,\mathfrak{m}}(B_{0} \cdots, 1^{\gamma_{i+1}},\textcolor{cyan}{ B, 1^{\gamma}}, B_{j+1}, \ldots, B_{p}) $. \end{itemize} It remains, where $(iv)$ terms, antisymmetric of $(i)$ ones, are omitted to lighten the diagrams: \\ \includegraphics[]{dep27.pdf}\\ Now, let list these remaining terms, gathered according to their right side as above: \begin{itemize} \item[$(i)$] Looking at the magenta terms, with $2 \leq B \leq B_{j}-1$ or $B=B_{j}+1$ and $0\leq\gamma\leq\gamma_{i+1}-1$: $$\zeta^{\sharp\sharp,\mathfrak{l}}_{B_{j}-B+1}(1^{\gamma_{j}}, \ldots, 1^{\gamma_{i+1}-\gamma-1})-\zeta^{\sharp\sharp,\mathfrak{l}}_{B_{j}-B}(1^{\gamma_{j}}, \ldots, 1^{\gamma_{i+1}-\gamma}) =-\zeta^{\sharp,\mathfrak{l}}(B_{j}-B+1, 1^{\gamma_{j}}, \ldots, 1^{\gamma_{i+1}-\gamma-1})$$ With $even$ depth for the first term and $odd$ for the second since otherwise the cuts would be unstable and simplified by $\textsc{Cut}$; here also $c_{i+1}\neq 1$. \item[$(ii)$] These match exactly with the left side of $(i)$ for $B=B_{j}$ and $(iv)$ terms for $B=B_{i}$. \item[$(iii)$] The following cuts: $$\delta_{B_{i+1}\geq B}\zeta^{\sharp\sharp,\mathfrak{l}}_{B_{i+1}-B}(1^{\gamma_{i+2}}, \ldots, B_{j}) -\delta_{B_{j}\geq B} \zeta^{\sharp\sharp,\mathfrak{l}}_{B_{j}-B}(1^{\gamma_{j}}, \ldots, B_{i+1})+$$ $$\delta_{B_{i+1}< B}\zeta^{\sharp\sharp,\mathfrak{l}}_{B_{i+1}+B_{j}-B}(1^{\gamma_{j}}, \ldots, 1^{\gamma_{i+2}}) - \delta_{B_{j}< B}\zeta^{\sharp\sharp,\mathfrak{l}}_{B_{i+1}+B_{j}-B}(1^{\gamma_{i+2}}, \ldots, 1^{\gamma_{j}}) .$$ The parity of $weight-depth$ for the first line is equal to the parity of $\delta_{c_{i+1}}+ \delta_{c_{j+1}}+B$. Notice that if this is even, the first line has odd depth whereas the second line has even depth, and by $\textsc{Cut}$ and $\textsc{Antipode} \ast$, all terms got simplified. Hence, we can restrict to $B$ written as $2\beta+3- \delta_{c_{i+1}}- \delta_{c_{j+1}}$, the first line being of even depth, the second line of odd depth.
\item[$(iv)$] Antisymmetric of $(i)$. \end{itemize} \end{proof}
\section{Galois descent in small depths, $N=2,3,4,\mlq 6 \mrq,8$}
\subsection{$\boldsymbol{N=2}$: Depth $\boldsymbol{2,3}$} Here we have to consider only one Galois descent, from $\mathcal{H}^{2}$ to $\mathcal{H}^{1}$. \\ In depth $1$ all the $\zeta^{\mathfrak{m}}(\overline{s})$, $s>1$ are MMZV. Let us detail the case of depth 2 and 3 as an application of the results of Chapter $5$. In depth 2, coefficients are explicit: \begin{lemm} The depth $2$ part of the basis of the motivic multiple zeta values is: $$\left\{ \zeta^{\mathfrak{m}}(2a+1, \overline{2b+1})- \binom {2(a+b)}{2b} \zeta^{\mathfrak{m}}(1,\overline{2(a+b)+1}), a,b> 0 \right\}.$$ \end{lemm} \begin{proof} Indeed, we have if $a,b>0$, $D_{1}(\zeta^{\mathfrak{m}}(2a+1, \overline{2b+1}))=0$ and for $r>0$: \begin{multline}\nonumber D_{2r+1,2}(\zeta^{\mathfrak{m}}(2a+1, \overline{2b+1}))= \zeta^{\mathfrak{l}}(2r+1)\otimes \zeta^{\mathfrak{m}}(\overline{2(a+b-r)+1})\\ \left(-\delta_{a \leq r < a+b} \binom{2r}{2a} + \delta_{r=a} + \delta_{b\leq r < a+b} \binom{2r}{2b}(2^{-2r}-1)+ \delta_{r=a+b}(2^{-2r}-2)\binom{2(a+b)}{2b}\right). \end{multline} There is only the case $r=a+b$ where a term ($\zeta^{\mathfrak{m}}(\overline{1}))$ which does not belong to $\mathcal{F}_{0}\mathcal{H}$ appears: $$D_{2r+1,2}(\zeta^{\mathfrak{m}}(2a+1, \overline{2b+1}))\equiv \delta_{r=a+b}(2^{-2r}-2)\binom{2(a+b)}{2b} \zeta^{\mathfrak{l}}(2r+1)\otimes \zeta^{\mathfrak{m}}(\overline{1}) \text{ in the quotient } \mathcal{H}^{\geq 1}.$$ Referring to the previous results, we can correct $\zeta^{\mathfrak{m}}(2a+1, \overline{2b+1})$ with terms of the same weight, same depth, and with at least one $1$ (not at the end), which here corresponds only to $\zeta^{\mathfrak{m}}(1,\overline{2(a+b)+1})$.\\ Furthermore, the last equality being true in the quotient $\mathcal{H}^{\geq 1}$: \begin{align*} D_{2r+1,2}(\zeta^{\mathfrak{m}}(1, \overline{2(a+b)+1})) & = \zeta^{\mathfrak{l}}(2r+1)\otimes (-\delta_{r < a+b}+ \delta_{r=a+b}(2^{-2r}-2)) \zeta^{\mathfrak{m}}(\overline{2(a+b-r)+1})\\ & \equiv \delta_{r=a+b}(2^{-2r}-2) \zeta^{\mathfrak{l}}(2r+1)\otimes \zeta^{\mathfrak{m}}(\overline{1}) . \end{align*} According to these calculations of infinitesimal coactions: $$\zeta^{\mathfrak{m}}(2a+1, \overline{2b+1})- \binom {2(a+b)}{2b} \zeta^{\mathfrak{m}}(1,\overline{2(a+b)+1}) \text{ belongs to } \mathcal{F}_{0}\mathcal{H} \text{ , i.e. is a MMZV.}$$ \end{proof} \noindent \texttt{Examples:} Here are some motivic multiple zeta values: $$\zeta^{\mathfrak{m}}(3, \overline{3})-6 \zeta^{\mathfrak{m}}(1,\overline{5}) \text{ , } \zeta^{\mathfrak{m}}(3, \overline{5})-15 \zeta^{\mathfrak{m}}(1,\overline{7}) \text{ , }\zeta^{\mathfrak{m}}(5, \overline{3})-15 \zeta^{\mathfrak{m}}(1,\overline{7}) \text{ , } \zeta^{\mathfrak{m}}(5, \overline{7})-210 \zeta^{\mathfrak{m}}(1,\overline{11}) .$$ \\ \textsc{Remarks:} \begin{itemize}
\item[$\cdot$] The corresponding Euler sums $\left\{ \zeta(2a+1, \overline{2b+1})- \binom {2(a+b)}{2b} \zeta(1,\overline{2(a+b)+1}), a,b> 0 \right\}$ are a generating family of MZV in depth $2$.
\item[$\cdot$] Similarly, we can prove that the following elements are (resp. motivic) MZV, if no $\overline{1}$:
$$\begin{array}{l|ll}
\text{ } \zeta(\overline{A}, \overline{B}) & \zeta(A,\overline{B}) +\zeta(\overline{A},B) & \text{ if } A,B \text{ odd } \\
\left. \begin{array}{l}
\zeta(A, \overline{B})\\
\zeta(\overline{A}, B)
\end{array} \right\rbrace \text{ if } A+B \text{ odd } & \zeta(A, \overline{B}) + (-1)^{A} \binom{A+B-2}{A-1} \zeta(1,\overline{A+B-1}) & \text{ if } A+B \text{ even} \\ \begin{array}{l} \zeta(\overline{1},\overline{1}) -\frac{1}{2}\zeta(\overline{1})^{2} \\
\zeta(1,\overline{1}) -\frac{1}{2}\zeta(\overline{1})^{2} \end{array} & \zeta(\overline{A}, B)- (-1)^{A} \binom{A+B-2}{A-1} \zeta(1,\overline{A+B-1}) & \text{ if } \left\lbrace \begin{array}{l}
A+B \text{ even } \\
A,B\neq 1 \end{array}\right.
\end{array}$$ \end{itemize}
\begin{lemm} The depth $2$ part of the basis of $\mathcal{F}_{1}\mathcal{H}$ is: $$\left\{ \zeta^{\mathfrak{m}}(2a+1, \overline{2b+1}) , (a,b)\neq(0, 0) \right\}.$$ \end{lemm} \begin{proof} No need of correction ($\mathcal{B}_{n,2,\geq 2}$ is empty for $n\neq 2$), these elements belong to $\mathcal{F}_{1}\mathcal{H}$. \end{proof}
\begin{lemm} The depth $3$ part of the basis of motivic multiple zeta values is: \begin{multline}
\left\{\zeta^{\mathfrak{m}}(2a+1,2b+1,\overline{2c+1})-\sum_{k=1}^{a+b+c}\alpha_{k}^{a,b,c}\zeta^{\mathfrak{m}}(1,2(a+b+c-k)+1, \overline{2k+1}) \right. \\ \left. -\binom {2(b+c)}{2c}\zeta^{\mathfrak{m}}(2a+1,1,\overline{2(b+c)+1}),a,b,c>0 \right\}.
\end{multline}
where $\alpha_{k}^{a,b,c} \in\mathbb{Z}_{\text{odd}}$ are solutions of $M_{3}X=A^{a,b,c}$. With $A^{a,b,c}$ such that $r^{\text{th}}-$coefficient is: $$\delta_{b \leq r < a+b} \binom{2(n-r)}{2c}\binom{2r}{2b} - \delta_{a < r< a+b} \binom{2(n-r)}{2c}\binom{2r}{2a}-\delta_{b\leq r < b+c}\binom{2(n-r)}{2a}\binom{2r}{2b} $$ $$- \delta_{r\leq a}\binom{2(n-r)}{2(b+c)}\binom{2(b+c)}{2c} + \delta_{r<b+c}\binom{2(n-r)}{2a}\binom{2(b+c)}{2c} + \delta_{c\leq r < b+c}\binom{2r}{2c}\binom{2(n-r)}{2a}(2^{-2r}-1).$$ $M_{3}$ the matrix whose $(r,k)^{\text{th}}$ coefficient is: $$\delta_{r=a+b+c}(2^{-2r}-2)\binom{2n}{2k}+ \delta_{k \leq r < n} \binom{2r}{2k}(2^{-2r}-1) - \delta_{r<n-k} \binom{2(n-r)}{2k} - \delta_{n-k \leq r<n} \binom{2r}{2(n-k)}. $$ \end{lemm} \begin{proof} Let $\zeta^{\mathfrak{m}}(2a+1,2b+1,\overline{2c+1})$, $a,b,c >0$ fixed, and substract elements of the same weight, of depth 3 until it belongs to $gr_{3} \mathcal{F}_{0}\mathcal{H}$.\\ Let calculate infinitesimal coproducts referring to the formula ($\ref{Deriv2}$) in the quotient $\mathcal{H}^{\geq 1}$ and use previous results for depth 2, with $n=a+b+c$: \begin{small} $$D_{2r+1,3}(\zeta^{\mathfrak{m}}(2a+1,2b+1,\overline{2c+1}))\equiv \zeta^{\mathfrak{l}}(2r+1)\otimes \left[ \delta_{r=b+c} \binom{2(b+c)}{2c}(2^{-2r}-2) \zeta^{\mathfrak{m}}(2a+1, \overline{1}) \right.$$ $$\left. + \zeta^{\mathfrak{m}}(1, \overline{2(n-r)+1}) \left( \delta_{a = r} \binom{2(n-r)}{2c}+ \delta_{b \leq r < a+b} \binom{2r}{2b}\binom{2(n-r)}{2c} - \delta_{a\leq r<a+b} \binom{2r}{2a}\binom{2(n-r)}{2c} \right. \right.$$ $$\left. \left. - \delta_{b\leq r<b+c} \binom{2r}{2b}\binom{2(n-r)}{2a} +\delta_{c \leq r <b+c} \binom{2r}{2c} \binom{2(n-r)}{2a}(2^{-2r}-1)\right) \right].$$ \end{small} At first, let substract $\binom {2(b+c)}{2c}\zeta(2a+1,1,\overline{2(b+c)+1})$ such that the $D^{-1}_{1,2} D^{1}_{2r+1,3} $ are equal to zero, which comes to eliminate the term $\zeta^{\mathfrak{m}}(2a+1, \overline{1})$ appearing (case $r=b+c$).\\ So, we are left to substract a linear combination $$\sum_{k=1}^{a+b+c} \alpha_{k}^{a,b,c} \zeta^{\mathfrak{m}}(1,2(a+b+c-k)+1, \overline{2k+1})$$
such that the coefficients $\alpha_{k}^{a,b,c}$ are solutions of the system $M_{3}X=A^{a,b,c}$ where $A^{a,b,c}= (A^{a,b,c}_{r})_{r}$ satisfying in $\mathcal{H}^{\geq 1}$: \begin{small} \begin{multline}\nonumber D_{2r+1,3} \left( \zeta^{\mathfrak{m}}(2a+1,2b+1,\overline{2c+1})- \binom {2(b+c)}{2c}\zeta(2a+1,1,\overline{2(b+c)+1})\right) \equiv\\ A^{a,b,c}_{r}\zeta^{\mathfrak{l}}(2r+1)\otimes \zeta^{\mathfrak{m}}(1, \overline{2(n-r)+1}), \end{multline} \end{small} and $M_{3}= (m_{r,k})_{r,k}$ matrix such that: $$D_{2r+1,3}( \zeta^{\mathfrak{m}}(1,2(a+b+c-k)+1, \overline{2k+1}))= m_{r, k} \zeta^{\mathfrak{l}}(2r+1)\otimes \zeta^{\mathfrak{m}}(1, \overline{2(n-r)+1}).$$ This system has solutions since, according to Chapter $5$ results, the matrix $M_{3}$ is invertible.\footnote{Indeed, modulo 2, $M_{3}$ is an upper triangular matrix with $1$ on diagonal.}\\ Then, the following linear combination will be in $\mathcal{F}_{0}\mathcal{H}$: \begin{small} $$\zeta^{\mathfrak{m}}(2a+1,2b+1,\overline{2c+1})-\sum_{k=1}^{a+b+c} \alpha_{k}^{a,b,c} \zeta^{\mathfrak{m}}(1,2(a+b+c-k)+1, \overline{2k+1})-\binom {2(b+c)}{2c}\zeta(2a+1,1,\overline{2(b+c)+1}).$$ \end{small} The coefficients $\alpha_{k}^{a,b,c}$ belong to $\mathbb{Z}_{\text{odd}}$ since coefficients are integers, and $\det(M_{3})$ is odd. Referring to the calculus of infinitesimal coactions, $A^{a,b,c}$ and $M_{3}$ are as claimed in lemma. \end{proof} \noindent \texttt{Examples:} \begin{itemize} \item[$\cdot$] By applying this lemma, with $a=b=c=1$ we obtain the following MMZV: $$\zeta^{\mathfrak{m}}(3,3,\overline{3})+ \frac{774}{191} \zeta^{\mathfrak{m}}(1,5, \overline{3}) - \frac{804}{191} \zeta^{\mathfrak{m}}(1,3, \overline{5}) + \frac{450}{191}\zeta^{\mathfrak{m}}(1,1, \overline{7}) -6 \zeta^{\mathfrak{m}}(3,1,\overline{5}).$$ Indeed, in this case, with the previous notations: $$M_{3}=\begin{pmatrix} \frac{27}{4}&-1&-1 \\ -\frac{53}{8}&-\frac{111}{16}&-1\\ -\frac{1905}{64}&-\frac{1905}{64}&-\frac{127}{64} \end{pmatrix} \text{ , } \quad A^{1,1,1}=\begin{pmatrix} \frac{51}{2} \\ 0\\ 0 \end{pmatrix} .$$ \item[$\cdot$] Similarly, we obtain the following motivic multiple zeta value: $$\hspace*{-1.2cm}\zeta^{\mathfrak{m}}(3,3,\overline{5})+ \frac{850920}{203117}\zeta^{\mathfrak{m}}(1,7, \overline{3}) +\frac{838338}{203117}\zeta^{\mathfrak{m}}(1,5, \overline{5}) -\frac{3673590}{203117}\zeta^{\mathfrak{m}}(1,3, \overline{7})+ \frac{20351100}{203117} \zeta^{\mathfrak{m}}(1,1, \overline{9}) -15\zeta^{\mathfrak{m}}(3,1,\overline{7}).$$ $$\hspace*{-2cm}\text{ There: } \quad \quad M_{3}=\begin{pmatrix} -\frac{63}{4}& 15 & -1& -1\\ -\frac{93}{8}&-\frac{31}{16}&-6&-1 \\ -\frac{1009}{64}&-\frac{1905}{64}&-\frac{1023}{64}&-1\\ -\frac{3577}{64}&-\frac{17885}{128}&-\frac{3577}{64}&-\frac{511}{256} \end{pmatrix} \text{ , } \quad \quad A^{1,1,2}=\begin{pmatrix} 210\\ \frac{387}{8} \\ 0\\ 0 \end{pmatrix} .$$ \end{itemize}
\begin{lemm} The depth $3$ part of the basis of $\mathcal{F}_{1}\mathcal{H}$ is: $$\left\{\zeta^{\mathfrak{m}}(2a+1,2b+1,\overline{2c+1})-\delta_{a=0 \atop\text{ or } c=0} (-1)^{\delta_{c=0}} \binom{2(a+b+c)}{2b} \zeta^{\mathfrak{m}}(1,1, \overline{2(a+b+c)+1}) \right.$$ $$ \left. - \delta_{c=0} \binom{2(a+b)}{2b} \zeta(1,2(a+b)+1,\overline{1}), \text{ at most one of } a,b,c \text{ equals zero }\right\}.$$ \end{lemm} \begin{proof} Let $\zeta^{\mathfrak{m}}(2a+1,2b+1,\overline{2c+1})$ with at most one $1$.\\ \begin{center}
\emph{Our goal is to annihilate $D^{-1}_{1,3}$ and $\lbrace D^{-1}_{1,3} \circ D^{1}_{2r+1}\rbrace_{r>0}$, in the quotient $\mathcal{H}^{\geq 1}$.}
\end{center} Let first cancel $D^{-1}_{1,3}$: if $c\neq 0$, it is already zero; otherwise, for $c=0$, in $\mathcal{H}^{\geq 1}$, according to the results in depth $2$ for $\mathcal{F}_{0}$, we can substract $\binom{2(a+b)}{2a} \zeta(1,2(a+b)+1,\overline{1})$ since: \begin{small} $$D_{1,3} (\zeta^{\mathfrak{m}}(2a+1,2b+1,\overline{1}))\equiv \binom{2(a+b)}{2a}\zeta^{\mathfrak{m}}(1,\overline{2(a+b)+1})\equiv \binom{2(a+b)}{2a} D_{1,3} (\zeta^{\mathfrak{m}}(1,2(a+b)+1,\overline{1})).$$ \end{small} Furthermore, with $\equiv$ standing for an equality in $\mathcal{H}^{\geq 1}$: \begin{small} \begin{align*} D^{-1}_{1,2} D^{1}_{2r+1,3} (\zeta^{\mathfrak{m}}(2a+1,2b+1,\overline{2c+1})) & = \delta_{r=b+c} \binom{2r}{2c}(2^{-2r}-2)\zeta^{\mathfrak{m}}(\overline{2a+1}) \\
& \equiv \delta_{r=b+c \atop a=0} \binom{2(b+c)}{2c}(2^{-2(b+c)}-2)\zeta^{\mathfrak{m}}(\overline{1}). \\
D^{-1}_{1,2} D^{1}_{2r+1,3} (\zeta^{\mathfrak{m}}(1,1,\overline{2(a+b+c)+1})) & = \delta_{r=a+b+c} (2^{-2(a+b+c)}-2)\zeta^{\mathfrak{m}}(\overline{1}).\\
D^{-1}_{1,2} D^{1}_{2r+1,3} (\zeta^{\mathfrak{m}}(1,2(a+b+c)+1, \overline{1})) & = \delta_{r=a+b+c} (2^{-2(a+b+c)}-2)\zeta^{\mathfrak{m}}(\overline{1}). \end{align*} \end{small} Therefore, to cancel $D^{-1}_{1,2}\circ D^{1}_{2r+1,3}$: \begin{itemize} \item[$\cdot$] If $a=0$ we substract $\binom{2(b+c)}{2c}\zeta^{\mathfrak{m}}(1,1,\overline{2(b+c)+1}) $. \item[$\cdot$] If $c=0$, we add $\binom{2(b+c)}{2c}\zeta^{\mathfrak{m}}(1,1,\overline{2(a+b)+1})$. \end{itemize} \end{proof}
\paragraph{\textsc{Depth }$\boldsymbol{4}$.} The simplest example in depth $4$ of MMZV obtained by this way, with $\alpha_{i}\in\mathbb{Q}$: $$-\zeta^{\mathfrak{m}}(3, 3, 3, \overline{3})-\frac{3678667587000}{4605143289541}\zeta^{\mathfrak{m}}(1, 1, 1, \overline{9})+\frac{9187768536750}{4605143289541}\zeta^{\mathfrak{m}}(1, 1, 3, \overline{7})+\frac{41712466500}{4605143289541}\zeta^{\mathfrak{m}}(1, 1, 5, \overline{5})$$ $$-\frac{9160668717750}{4605143289541} \zeta^{\mathfrak{m}}(1, 1, 7, \overline{3})+\frac{11861255103300}{4605143289541}\zeta^{\mathfrak{m}}(1, 3, 1, \overline{7})+\frac{202283196216}{4605143289541}\zeta^{\mathfrak{m}}(1, 3, 3, \overline{5})$$ $$-\frac{993033536436}{4605143289541}\zeta^{\mathfrak{m}}(1, 3, 5, \overline{3})+\frac{8928106562124}{4605143289541}\zeta^{\mathfrak{m}}(1, 5, 1, \overline{5})-\frac{1488017760354}{4605143289541}\zeta^{\mathfrak{m}}(1, 5, 3, \overline{3})$$ $$-\frac{450}{191}\zeta^{\mathfrak{m}}(3, 1, 1, \overline{7})+\frac{804}{191}\zeta^{\mathfrak{m}}(3, 1, 3, \overline{5})-\frac{774}{191}\zeta^{\mathfrak{m}}(3, 1, 5, \overline{3})+6\zeta^{\mathfrak{m}}(3, 3, 1, \overline{5})$$ $$+ \alpha_{1} \zeta^{\mathfrak{m}}(1,-11)+ \alpha_{2} \zeta^{\mathfrak{m}}(1,-9)\zeta^{\mathfrak{m}}(2)+ \alpha_{3} \zeta^{\mathfrak{m}}(1,-7)\zeta^{\mathfrak{m}}(2)^{2}+ \alpha_{4} \zeta^{\mathfrak{m}}(1,-5)\zeta^{\mathfrak{m}}(2)^{3}+ \alpha_{5} \zeta^{\mathfrak{m}}(1,-3)\zeta^{\mathfrak{m}}(2)^{4}.$$ $$\quad $$
\subsection{ $\boldsymbol{N=3,4}$: Depth $\boldsymbol{2}$}
Let us detail the case of depth 2 as an application of the results in Chapter $5$ and start by defining some coefficients appearing in the next examples: \begin{defi} Set $\alpha^{a,b}_{k}\in\mathbb{Z}$ such that $M(\alpha^{a,b}_{k})_{b+1 \leq k \leq \frac{n}{2}-1 }= A^{a,b}$ with $n=2(a+b+1)$: $$\hspace*{-0.5cm}M\mathrel{\mathop:}= \left( \binom{2r-1}{2k-1} \right)_{b+1 \leq r,k \leq \frac{n}{2}-1}; A^{a,b}\mathrel{\mathop:}=\left(-\binom{2r-1}{2b}\right)_{b+1 \leq r \leq \frac{n}{2}-1}; \beta^{a,b}\mathrel{\mathop:}= \binom{n-2}{2b} + \sum_{k=b+1}^{a+b} \alpha_{k} \binom{n-2}{2k-1}.$$ \end{defi} \noindent \texttt{Nota Bene}: The matrix $M$ having integers as entries and determinant equal to $1$, and $A$ having integer components, the coefficients $\alpha^{a,b}_{k}$ are obviously integers; the matrix $M$ and its inverse are lower triangular with $1$ on the diagonal. Furthermore:\footnote{There $c_{i}\in\mathbb{N} $ does not depend neither on $b$ nor on $a$.}: \begin{multline}\nonumber \alpha^{a,b}_{b+i}= (-1)^{i} \binom{2b+2i-1}{2i-1} c_{i},\\ \alpha^{a,b}_{b+1}=-(2b+1), \quad \alpha^{a,b}_{b+2}=2\binom{2b+3}{3}, \quad \alpha^{a,b}_{b+3}=-16\binom{2b+5}{5}, \quad \alpha^{a,b}_{b+4}=272\binom{2b+7}{7}.\\ \end{multline}
\begin{lemm} The depth $2$ part of the basis of MMZV, for even weight $n=2(a+b+1)$, is: \begin{small} $$\left\{ \zeta^{\mathfrak{m}}\left( 2a+1, 2b+1 \atop 1, \xi \right)- \beta^{a,b} \zeta^{\mathfrak{m}}\left(1,n-1 \atop 1, \xi \right) - \sum_{k=b+1}^{\frac{n}{2}-1} \alpha^{a,b}_{k} \zeta^{\mathfrak{m}}\left( n-2k, 2k \atop 1, \xi \right), a,b> 0 \right\}.$$ \end{small} \end{lemm} \begin{proof}\footnote{We omit the exponent $\xi$ indicating the projection on the second factor of the derivations $D_{r}$, to lighten the notations.} Let $Z=\zeta^{\mathfrak{m}}(2a+1, \overline{2b+1})$ fixed, with $a,b>0$.\\ First we substract a linear combination of $\zeta^{\mathfrak{m}}\left(n-2k, 2k \atop 1, \xi \right)$ in order to cancel $\lbrace D_{2r}\rbrace$. It is possible since in depth 2, because $\zeta^{\mathfrak{l}}\left( 2r \atop 1\right) =0$: \begin{small} $$ D_{2r} (\zeta^{\mathfrak{m}}(x_{1}, \overline{x_{2}}))= \delta_{x_{2} \leq 2r \leq x_{1}+x_{2}-1} (-1)^{x_{2}} \binom{2r-1}{x_{2}-1} \zeta^{\mathfrak{l}}\left( 2 r \atop \xi\right)\otimes \zeta^{\mathfrak{m}}\left( x_{1}+x_{2}-r\atop \xi \right).$$ \end{small} Hence it is sufficient to choose $\alpha_{k}$ such that $M\alpha^{a,b}= A^{a,b}$ as in Definition $A.2.5$.\\ Now, it remains to satisfy $D_{1}\circ D_{2r+1}(\cdot)=0$ (for $r=n-1$ only) in order to have an element of $\mathcal{F}^{k_{N}/\mathbb{Q},P/1}_{0}\mathcal{H}_{n}$. In that purpose, let substract $\beta^{a,b} \zeta^{\mathfrak{m}}(1,n-1 ; 1, \xi)$ with $\beta^{a,b}$ as in Definition $A.2.5$) according to the calculation of $D_{1}\circ D_{2r+1}(\cdot)$, left to the reader.\\ \end{proof} \noindent \texttt{Examples}: The following are motivic multiple zeta values: \begin{itemize} \item[$\cdot$] $\zeta^{\mathfrak{m}}\left(5,3 \atop 1, \xi \right) -75 \zeta^{\mathfrak{m}}\left( 1,7 \atop 1, \xi \right) + 3 \zeta^{\mathfrak{m}}\left(4, 4 \atop 1, \xi \right) - 20 \zeta^{\mathfrak{m}}\left( 2, 6 \atop 1, \xi \right).$ \item[$\cdot$] $\zeta^{\mathfrak{m}}\left(3,5 \atop 1, \xi \right) +15 \zeta^{\mathfrak{m}}\left( 1, 7 \atop 1, \xi \right) + 5 \zeta^{\mathfrak{m}}\left(6, 2 \atop 1, \xi \right) .$ \item[$\cdot$] $\zeta^{\mathfrak{m}}\left(5,5 \atop 1, \xi \right) -350 \zeta^{\mathfrak{m}}\left( 1, 9 \atop 1, \xi \right) + 5 \zeta^{\mathfrak{m}}\left( 4, 6 \atop 1, \xi \right) -70 \zeta^{\mathfrak{m}}\left( 2, 8 \atop 1, \xi \right).$ \item[$\cdot$] $\zeta^{\mathfrak{m}}\left( 7, 5 \atop 1, \xi \right) +12810 \zeta^{\mathfrak{m}}\left( 1, 11 \atop 1, \xi \right) + 5 \zeta^{\mathfrak{m}}\left( 6, 6 \atop 1, \xi \right) -70 \zeta^{\mathfrak{m}}\left( 4,8 \atop 1, \xi \right)+ 2016 \zeta^{\mathfrak{m}}\left( 2, 10 \atop 1, \xi \right).$
\item[$\cdot$]$\zeta^{\mathfrak{m}}\left(9, 5 \atop 1, \xi \right) -685575 \zeta^{\mathfrak{m}}\left( 1, 13 \atop 1, \xi \right) + 5 \zeta^{\mathfrak{m}}\left( 8, 6 \atop 1, \xi \right) -70 \zeta^{\mathfrak{m}}\left( 6, 8 \atop 1, \xi \right)+ 2016 \zeta^{\mathfrak{m}}\left( 4, 10 \atop 1, \xi \right)- 89760 \zeta^{\mathfrak{m}}\left( 2, 12 \atop 1, \xi \right).$ \end{itemize}
\begin{lemm} The depth $2$ part of the basis of $\mathcal{F}^{k_{N}/\mathbb{Q},P/1}_{1}\mathcal{H}_{n}$ is for even $n$: \begin{small} $$\left\{ \zeta^{\mathfrak{m}}\left(2a+1, 2b+1 \atop 1, \xi\right)- \sum_{k=b+1}^{\frac{n}{2}-1} \alpha^{a,b}_{k} \zeta^{\mathfrak{m}}\left(n-2k, 2k \atop 1, \xi \right), a,b\geq 0, (a,b)\neq(0,0) \right\},$$ \end{small} For odd $n$, the part in depth $2$ of the basis of $\mathcal{F}^{k_{N}/\mathbb{Q},P/1}_{1}\mathcal{H}_{n}$ is: \begin{small} $$\left\{ \zeta^{\mathfrak{m}}\left(x_{1}, x_{2} \atop 1, \xi \right)+ (-1)^{x_{2}+1} \binom{n-2}{x_{2}-1} \zeta^{\mathfrak{m}}\left(1, n-1 \atop 1, \xi \right), x_{1},x_{2} >1, \text{ one even, the other odd } \right\}.$$ \end{small} \end{lemm} \begin{proof} \begin{itemize} \item[$\cdot$] For even $n$, we need to cancel $D_{2r}$ (else $D_{2s}\circ D_{2r}(\cdot) \neq 0$), so we substract the same linear combination than in the previous lemma. \item[$\cdot$] For odd $n$, we need to cancel $D_{1}\circ D_{2r}$. Since $D_{1}\circ D_{2r}(Z)= (-1)^{x_{2}} \binom{n-2}{x_{2}-1}$, we substract $(-1)^{x_{2}+1} \binom{n-2}{x_{2}-1} \zeta^{\mathfrak{m}}(1, \overline{n-1})$. \end{itemize} \end{proof}
\begin{lemm} The depth $2$ part of the basis of $\mathcal{F}^{k_{N}/\mathbb{Q},P/P}_{0}\mathcal{H}_{n}$ ($=\mathcal{H}_{n}^{\mathcal{MT}_{2}}$ if $N=4$) is: \begin{small} $$\left\{ \zeta^{\mathfrak{m}}\left(2a+1, 2b+1 \atop 1, \xi \right)- \sum_{k=b+1}^{\frac{n}{2}-1} \alpha^{a,b}_{k} \zeta^{\mathfrak{m}}\left(n-2k, 2k \atop 1, \xi \right), a,b \geq 0 \right\}.$$ \end{small} \end{lemm} \begin{proof} To cancel $D_{2r}$, we substract the same linear combination than above. \end{proof}
\begin{lemm} The depth $2$ part of the basis of $\mathcal{F}^{k_{N}/\mathbb{Q},P/P}_{1}\mathcal{H}_{n}$ is for even $n$: \begin{small} $$\left\{ \zeta^{\mathfrak{m}}\left(2a+1, 2b+1 \atop 1, \xi \right)- \sum_{k=b+1}^{\frac{n}{2}-1} \alpha^{a,b}_{k} \zeta^{\mathfrak{m}}\left(n-2k, 2k \atop 1, \xi \right), a,b\geq 0, \right\},$$ \end{small} And for odd $n$, the part in depth $2$ of the basis of $\mathcal{F}^{k_{N}/\mathbb{Q},P/P}_{1}\mathcal{H}_{n}$ is: \begin{small} $$\left\{ \zeta^{\mathfrak{m}}\left( x_{1}, x_{2} \atop 1, \xi \right), x_{1},x_{2} \geq 1 \text{, one even, the other odd } \right\}.$$ \end{small} \end{lemm} \begin{proof} If $n$ is even, to cancel $\lbrace D_{2r}\rbrace$, we use the same linear combination than above.\\ If $n$ is odd, we already have $\zeta^{\mathfrak{m}}(x_{1}, x_{2}; 1, \xi)\in \mathcal{F}^{k_{N}/\mathbb{Q},P/P}_{1}\mathcal{H}_{n}$.\\ \end{proof}
\subsection{$\boldsymbol{N=8}$: Depth $\boldsymbol{2}$}
Let us illustrate the results for the depth 2; proofs being similar (albeit longer) as in the previous sections are left to the reader; same notations than the previous case.
\begin{lemm} \begin{itemize}
\item[$\cdot$] The depth $2$ part of the basis of MMZV$_{\mu_{4}}$ is: {\small $$\left\{ \zeta^{\mathfrak{m}}\left(x_{1}, x_{2} \atop 1, \xi\right)+ \zeta^{\mathfrak{m}}\left(x_{1},x_{2} \atop -1, -\xi\right)+ \zeta^{\mathfrak{m}}\left(x_{1},x_{2} \atop 1, -\xi\right)+ \zeta^{\mathfrak{m}}\left(x_{1},x_{2} \atop -1, \xi\right), x_{i} \geq 1 \right\}.$$}
\item[$\cdot$] The depth $2$ part of the basis of motivic Euler sums is: \begin{small} $$\left\{ \zeta^{\mathfrak{m}}\left(2a+1, 2b+1 \atop 1, \xi\right)+ \zeta^{\mathfrak{m}}\left(2a+1, 2b+1 \atop -1, -\xi\right) + \zeta^{\mathfrak{m}}\left(2a+1, 2b+1 \atop 1, -\xi\right)+ \zeta^{\mathfrak{m}}\left(2a+1, 2b+1 \atop -1, \xi\right) \right. $$ $$\left. - \sum_{k=b+1}^{\frac{n}{2}-1} \alpha^{a,b}_{k} \left( \zeta^{\mathfrak{m}}\left(n-2k, 2k \atop 1, \xi\right) + \zeta^{\mathfrak{m}}\left(n-2k, 2k \atop -1, -\xi\right) + \zeta^{\mathfrak{m}}\left(n-2k, 2k \atop 1, -\xi\right) + \zeta^{\mathfrak{m}}\left(n-2k, 2k \atop -1, \xi\right)\right)\right\}_{a,b \geq 0}$$ \end{small}
\item[$\cdot$] The depth $2$ part of the basis of MMZV is:
\begin{small} $$\left\{ \zeta^{\mathfrak{m}}\left(2a+1, 2b+1\atop 1, \xi\right) + \zeta^{\mathfrak{m}}\left(2a+1, 2b+1\atop -1, -\xi\right) + \zeta^{\mathfrak{m}}\left(2a+1, 2b+1\atop 1, -\xi\right) + \zeta^{\mathfrak{m}}\left(2a+1, 2b+1\atop -1, \xi\right) \right.$$ $$ \left. - \sum_{k=b+1}^{\frac{n}{2}-1} \alpha^{a,b}_{k} \left( \zeta^{\mathfrak{m}}\left(n-2k, 2k\atop 1, \xi\right)+ \zeta^{\mathfrak{m}}\left(n-2k, 2k\atop -1,-\xi\right) + \zeta^{\mathfrak{m}}\left(n-2k, 2k\atop 1, -\xi\right) + \zeta^{\mathfrak{m}}\left(n-2k, 2k\atop -1, \xi\right) \right) \right.$$ $$\left. - \beta^{a,b} \left( \zeta^{\mathfrak{m}}\left(1,n-1\atop 1, \xi\right)+ \zeta^{\mathfrak{m}}\left(1,n-1\atop -1, \xi\right)+ \zeta^{\mathfrak{m}}\left(1,n-1\atop 1, -\xi\right)+ \zeta^{\mathfrak{m}}\left(1,n-1\atop -1, -\xi\right) \right), a,b> 0 \right\} $$ \end{small} \end{itemize} \end{lemm}
\begin{lemm} \begin{itemize}
\item[$\cdot$] The depth $2$ part of the basis of $\mathcal{F}^{k_{8}/k_{4},2/2}_{1}\mathcal{H}_{n}$ is, for even $n$:
\begin{small} $$\left\{ \zeta^{\mathfrak{m}}\left( x_{1}, x_{2}\atop 1, \xi\right)+ \zeta^{\mathfrak{m}}\left( x_{1},x_{2} \atop -1, -\xi\right), \zeta^{\mathfrak{m}}\left( x_{1},x_{2} \atop 1, -\xi\right)- \zeta^{\mathfrak{m}}\left( x_{1},x_{2} \atop -1, -\xi\right), \zeta^{\mathfrak{m}}\left( x_{1},x_{2} \atop -1, \xi\right)+ \zeta^{\mathfrak{m}}\left( x_{1},x_{2} \atop -1, -\xi\right), x_{i} \geq 1 \right\}.$$ \end{small}
\item[$\cdot$] The depth $2$ part of the basis of $\mathcal{F}^{k_{8}/\mathbb{Q},2/2}_{1}\mathcal{H}_{n}$ is for odd $n$:
\begin{small} $$\left\{ \zeta^{\mathfrak{m}}\left( x_{1}, x_{2}\atop 1, \xi\right)+ \zeta^{\mathfrak{m}}\left( x_{1}, x_{2}\atop -1, -\xi\right) + \zeta^{\mathfrak{m}}\left( x_{1}, x_{2}\atop 1, -\xi\right)+ \zeta^{\mathfrak{m}}\left( x_{1}, x_{2}\atop -1, \xi\right) , \text{ exactly one even } x_{i} \right\}.$$ \end{small} The depth $2$ part of the basis of $\mathcal{F}^{k_{8}/\mathbb{Q},2/2}_{1}\mathcal{H}_{n}$ is for even $n$: \begin{small} \begin{multline}\nonumber \left\{ \zeta^{\mathfrak{m}}\left( 2a+1, 2b+1\atop -1, \xi\right)+ \zeta^{\mathfrak{m}}\left( 2a+1, 2b+1\atop -1, -\xi\right) - \sum_{k=b+1}^{\frac{n}{2}-1} \alpha^{a,b}_{k} \left( \zeta^{\mathfrak{m}}\left( n-2k, 2k\atop -1, \xi\right) + \zeta^{\mathfrak{m}}\left( n-2k, 2k\atop -1, -\xi\right) \right)\right\}_{a,b\geq 0}\\ \cup \left\{ \zeta^{\mathfrak{m}}\left( 2a+1, 2b+1\atop 1, -\xi\right)- \zeta^{\mathfrak{m}}\left( 2a+1, 2b+1\atop -1, -\xi\right)- \sum_{k=b+1}^{\frac{n}{2}-1} \alpha^{a,b}_{k} \left( \zeta^{\mathfrak{m}}\left( n-2k, 2k\atop 1, -\xi\right) - \zeta^{\mathfrak{m}}\left( n-2k, 2k\atop -1, -\xi\right) \right)\right\}_{ a,b\geq 0}. \end{multline} \end{small}
\item[$\cdot$] The depth $2$ part of the basis of $\mathcal{F}^{k_{8}/\mathbb{Q},2/1}_{1}\mathcal{H}_{n}$ is for odd $n$:
\begin{small} \begin{multline}\nonumber
\left\{ \zeta^{\mathfrak{m}}\left( x_{1}, x_{2}\atop 1, \xi\right)+ \zeta^{\mathfrak{m}}\left( x_{1}, x_{2}\atop -1, -\xi\right)+ \zeta^{\mathfrak{m}}\left( x_{1}, x_{2}\atop 1, -\xi\right) + \zeta^{\mathfrak{m}}\left( x_{1}, x_{2}\atop -1, \xi\right) \right. \\
\left. - \gamma^{x_{1},x_{2}} \left( \zeta^{\mathfrak{m}}\left( 1, n-1\atop 1, \xi\right) + \zeta^{\mathfrak{m}}\left( 1, n-1\atop -1, -\xi\right)+ \zeta^{\mathfrak{m}}\left( 1, n-1\atop -1, \xi\right)+ \zeta^{\mathfrak{m}}\left( 1, n-1\atop 1, -\xi\right) \right), \text{ exactly one even } x_{i} \right\}.
\end{multline} \end{small} In even weight $n$, depth $2$ part of the basis of $\mathcal{F}^{k_{8}/\mathbb{Q},2/1}_{1}\mathcal{H}_{n}$ is: \begin{small} \begin{multline}\nonumber \left\{ \zeta^{\mathfrak{m}}\left( 1, n-1\atop 1, \xi\right)+ \zeta^{\mathfrak{m}}\left( 1, n-1\atop -1,-\xi \right)+ \zeta^{\mathfrak{m}}\left( 1, n-1\atop 1,-\xi \right) + \zeta^{\mathfrak{m}}\left( 1, n-1\atop -1,\xi \right) \right\} \\ \cup \left\{ \zeta^{\mathfrak{m}}\left( n-1,1\atop 1, \xi\right)+ \zeta^{\mathfrak{m}}\left( n-1,1\atop -1,-\xi \right)+ \zeta^{\mathfrak{m}}\left( n-1,1\atop 1,-\xi \right) + \zeta^{\mathfrak{m}}\left( n-1,1\atop -1,\xi \right) + \right.\\ \left. -\sum_{k=1}^{\frac{n}{2}-1} \alpha^{0,\frac{n}{2}-1}_{k} \left( \zeta^{\mathfrak{m}}\left( n-2k, 2k\atop 1, \xi\right) + \zeta^{\mathfrak{m}}\left( n-2k, 2k\atop -1, -\xi\right) + \zeta^{\mathfrak{m}}\left( n-2k, 2k\atop 1, -\xi\right)+ \zeta^{\mathfrak{m}}\left( n-2k, 2k\atop -1, \xi\right) \right)\right\}\\ \cup \left\{ \zeta^{\mathfrak{m}}\left( 2a+1, 2b+1\atop \epsilon_{1}, \epsilon_{2}\xi\right)+ \epsilon_{2} \zeta^{\mathfrak{m}}\left( 2a+1,2b+1\atop -1, -\xi\right) \right. - \beta^{a,b} \left( \zeta^{\mathfrak{m}}\left( 1, n-1\atop \epsilon_{1}, \epsilon_{2} \xi\right) + \epsilon_{2} \zeta^{\mathfrak{m}}\left( 1,n-1\atop -1, -\xi\right) \right)\\ \left. -\sum_{k=b+1}^{\frac{n}{2}-1} \alpha^{a,b}_{k} \left( \zeta^{\mathfrak{m}}\left( n-2k, 2k\atop \epsilon_{1}, \epsilon_{2}\xi \right) + \epsilon_{2} \zeta^{\mathfrak{m}}\left( n-2k, 2k\atop -1, -\xi \right) \right), a,b >0 , \epsilon_{i}\in\left\{\pm 1\right\}, \epsilon_{1}=- \epsilon_{2} \right\} . \end{multline} \end{small} Where $\gamma^{x_{1},x_{2}}=(-1)^{x_{2}} \binom{2r-1}{2r-x_{2}}$. \end{itemize} \end{lemm}
\subsection{$\boldsymbol{N=\mlq 6 \mrq}$: Depth $\boldsymbol{2}$}
In depth 2, coefficients are explicit as previously:
\begin{lemm} The depth $2$ part of the basis of MMZV, for even weight $n$ is: $$\left\{ \zeta^{\mathfrak{m}}\left(2a+1, 2b+1 \atop 1, \xi \right)- \sum_{k=b+1}^{\frac{n}{2}-1} \alpha^{a,b}_{k} \zeta^{\mathfrak{m}}\left(n-2k, 2k\atop 1, \xi\right), a,b> 0 \right\},$$ \end{lemm} \begin{proof} Proof being similar than the cases $N=3,4$ is left to the reader.\\ \\ \end{proof}
\section{Homographies of $\boldsymbol{\mathbb{P}^{1}\diagdown \lbrace 0, \mu_{N}, \infty\rbrace}$}
The homographies of the projective line $\mathbb{P}^{1}$ which permutes $\lbrace 0, \mu_{N}, \infty \rbrace$, induce automorphisms $\mathbb{P}^{1}\diagdown \lbrace 0, \mu_{N}, \infty\rbrace \rightarrow \mathbb{P}^{1}\diagdown \lbrace 0, \mu_{N}, \infty\rbrace$. The projective space $\mathbb{P}^{1} \diagdown \lbrace 0, \mu_{N}, \infty \rbrace$ has a dihedral symmetry, the dihedral group $Di_{N}= \mathbb{Z}\diagup 2 \mathbb{Z} \ltimes \mu_{N}$ acting with $x \mapsto x^{-1}$ and $x\mapsto \eta x$. In the special case of $N=1,2,4$, and for these only, the group of homographies is bigger than the dihedral group, due to particular symmetries of the points $\mu_{N}\cup \lbrace 0, \infty\rbrace$ on the Riemann sphere. Let specify these cases: \begin{itemize} \item[For $N=1:$] The homography group is the anharmonic group generated by $z\mapsto\frac{1}{z}$ and $z \mapsto 1-z$, and corresponds to the permutation group $\mathfrak{S}_{3}$. Precisely, projective transformations of $\mathbb{P}^{1}\diagdown \lbrace 0, 1, \infty\rbrace$ are: $$\begin{array}{lll}\label{homography1} \phi_{\tau}: & t \mapsto 1-t : & \left\lbrace \begin{array}{l} (0,1,\infty)\mapsto (1,0,\infty)\\ (\omega_{0},\omega_{1}, \omega_{\star}, \omega_{\sharp}) \mapsto (\omega_{1},\omega_{0}, -\omega_{\star}, \omega_{0}-\omega_{\star}). \end{array} \right. \\ \phi_{c}: & t \mapsto \frac{1}{1-t} :& \left\lbrace \begin{array}{l} 0\mapsto 1 \mapsto \infty \mapsto 0\\ (\omega_{0},\omega_{1}, \omega_{\star}, \omega_{\sharp}) \mapsto (\omega_{\star},-\omega_{0}, -\omega_{1}, -\omega_{0}-\omega_{1}) \end{array} \right. \\ \phi_{\tau c} : & t \mapsto \frac{t}{t-1} :& \left\lbrace \begin{array}{l}
(0,1,\infty)\mapsto (0,\infty,1)\\
(\omega_{0},\omega_{1}, \omega_{\star}) \mapsto (-\omega_{\star},-\omega_{1}, -\omega_{0}) \end{array} \right.\\ \phi_{c\tau}: & t \mapsto \frac{1}{t} : & \left\lbrace \begin{array}{l} (0,1,\infty)\mapsto (\infty,1,0)\\ (\omega_{0},\omega_{1}, \omega_{\star},\omega_{\sharp}) \mapsto (-\omega_{0},\omega_{\star}, \omega_{1}, \omega_{\sharp}) \end{array} \right. \\
\phi_{c^{2}}: & t \mapsto \frac{t-1}{t} : & \left\lbrace \begin{array}{l} 0\mapsto \infty \mapsto 1 \mapsto 0\\ (\omega_{0},\omega_{1}, \omega_{\star}) \mapsto (-\omega_{1},-\omega_{\star}, \omega_{0}) \end{array} \right. \\ \end{array}$$ Remark that hexagon relation ($\ref{fig:hexagon}$) corresponds to a cycle $c$ whereas the reflection relation corresponds to a transposition $\tau$, and : $$\mathfrak{S}_{3}= \langle c, \tau \mid c^{3}=id, \tau^{2},c\tau c =\tau \rangle= \lbrace 1, c, c^{2}, \tau, \tau c, c\tau\rbrace.$$ \item[For $N=2:$] Here, $(0,\infty, 1,-1)$ has a cross ratio $-1$ (harmonic conjugates) and there are $8$ permutations of $(0,\infty, 1,-1)$ preserving its cross ratio. The homography group corresponds indeed to the group of automorphisms of a square with consecutive vertices $(0, 1, \infty, -1)$, i.e. the dihedral group of degree four $Di_{4}$ defined by the presentation $\langle \sigma, \tau \mid \sigma^{4}= \tau^{2}=id, \sigma\tau \sigma= \tau \rangle$: $$\begin{array}{lll}\label{homography2} \phi_{\tau}: & t \mapsto \frac{1}{t} :& \left\lbrace \begin{array}{l} \pm 1\mapsto \pm 1 \quad 0 \leftrightarrow \infty\\ (\omega_{0},\omega_{1}, \omega_{\star},\omega_{-1}, \omega_{-\star}, \omega_{\pm\sharp}) \mapsto (-\omega_{0},\omega_{\star}, \omega_{1},\omega_{-\star},\omega_{-1}, \omega_{\pm\sharp}) \end{array} \right.\\ \\
\phi_{\sigma}: & t \mapsto \frac{1+t}{1-t} : & \left\lbrace \begin{array}{l} -1\mapsto 0\mapsto 1\mapsto \infty\mapsto -1\\ (\omega_{0},\omega_{1},\omega_{\star},\omega_{-1}, \omega_{-\star}) \mapsto (\omega_{-1}- \omega_{1}, -\omega_{-1}, - \omega_{1}, - \omega_{-\star}, - \omega_{\star})\\ (\omega_{\sharp}, \omega_{-\sharp}) \mapsto (-\omega_{1}-\omega_{-1}, -\omega_{\star}-\omega_{-\star}) \end{array} \right. \\ \\
\phi_{\sigma^{2}\tau}: & t \mapsto -t:& \left\lbrace \begin{array}{l} -1 \leftrightarrow 1 \\ (\omega_{0},\omega_{ 1}, \omega_{-1}, \omega_{ \pm \ast}, \omega_{\pm \sharp}) \mapsto (\omega_{0},\omega_{-1}, \omega_{1},\omega_{\mp \ast}, \omega_{\mp \sharp}) \end{array} \right. \\ \\ \phi_{\sigma^{2}}: & t \mapsto \frac{-1}{t} : & \left\lbrace \begin{array}{l} 0 \leftrightarrow \infty \quad -1 \leftrightarrow 1\\ (\omega_{0},\omega_{1},\omega_{\star},\omega_{-1}, \omega_{-\star}, \omega_{\pm \sharp}) \mapsto (-\omega_{0}, \omega_{-\star}, - \omega_{-1}, \omega_{\star}, \omega_{1}, \omega_{\mp\sharp}) \end{array} \right.\\ \\ \phi_{\sigma^{-1}}: & t \mapsto \frac{t-1}{1+t} : & \left\lbrace \begin{array}{l} 0 \mapsto -1 \mapsto \infty \mapsto 1 \mapsto 0 \\ (\omega_{0}, \omega_{1}, \omega_{-1}, \omega_{\star}, \omega_{-\star}) \mapsto (\omega_{-1}-\omega_{1}, - \omega_{\star}, -\omega_{1}, -\omega_{-\star}, -\omega_{-1}) \\ (\omega_{\sharp}, \omega_{-\sharp}) \mapsto ( -\omega_{\star}-\omega_{-\star}, -\omega_{1}-\omega_{-1}) \end{array} \right.\\ \\ \phi_{\tau \sigma}: & t \mapsto \frac{1-t}{1+t} : & \left\lbrace \begin{array}{l} -1 \leftrightarrow \infty \quad 0 \leftrightarrow 1 \\ (\omega_{0},\omega_{1},\omega_{\star},\omega_{-1}, \omega_{-\star}) \mapsto (\omega_{1}-\omega_{-1},-\omega_{-\star},-\omega_{\star},-\omega_{-1}, -\omega_{1}) \\ (\omega_{\sharp}, \omega_{-\sharp}) \mapsto ( -\omega_{\star}-\omega_{-\star}, -\omega_{1}-\omega_{-1}) \end{array} \right.\\ \\ \phi_{\sigma \tau}: & t \mapsto \frac{1+t}{t-1} : & \left\lbrace \begin{array}{l} -1 \leftrightarrow 0 \quad 1 \leftrightarrow \infty\\ (\omega_{0},\omega_{1},\omega_{\star},\omega_{-1}, \omega_{-\star}) \mapsto (\omega_{1}-\omega_{-1},-\omega_{1},-\omega_{-1},-\omega_{\star}, -\omega_{-\star}) \\ (\omega_{\sharp}, \omega_{-\sharp}) \mapsto ( -\omega_{1}-\omega_{-1}, -\omega_{\star}-\omega_{-\star}) \end{array} \right. \end{array} $$ Remark that the octagon relation ($\ref{fig:octagon}$) comes from the cycle $\sigma$ of order $4$; the other permutations above could also leads to relations. \item[For $N=4:$] $\mathbb{P}^{1}\diagdown \lbrace 0, 1,-1,i,-i, \infty\rbrace$ has an octahedral symmetry, and the homography group is the group of automorphisms of this octahedron placed on the Riemann sphere of vertices $(0,1,i,-1, -i,\infty)$.\footnote{Zhao showed this octahedral symmetry allows to reach the \say{non standard} relations which appeared in weight $3$, $4$ for $N=4$; non standard relations are these which do not come from distribution, conjugation, and regularised double shuffle relation, cf. $\cite{Zh1}$.} It is composed by 48 transformations, corresponding to 24 rotational symmetries, and a reflection. \end{itemize} We could also look at other projective transformations: $\mathbb{P}^{1}\diagdown \lbrace 0, \mu_{N}, \infty\rbrace \rightarrow \mathbb{P}^{1}\diagdown \lbrace 0, \mu_{N'}, \infty\rbrace _{N'\mid N}$.\\ \\ \texttt{Examples:} \begin{itemize} \item[$\cdot$] $\mathbb{P}^{1}\diagdown \lbrace 0, -1, \infty\rbrace \rightarrow \mathbb{P}^{1}\diagdown \lbrace 0, +1, \infty\rbrace \text{ , } t\mapsto 1+t $. \item[$\cdot$] $\mathbb{P}^{1}\diagdown \lbrace 0, -1, \infty\rbrace \rightarrow \mathbb{P}^{1}\diagdown \lbrace 0, +1, \infty\rbrace \text{ , } t\mapsto \frac{1}{1+t} $. \item[$\cdot$] $\mathbb{P}^{1}\diagdown \lbrace 0, \pm 1, \infty\rbrace \rightarrow \mathbb{P}^{1}\diagdown \lbrace 0, 1, \infty \rbrace \text{ , } t\mapsto t^{2} $. \end{itemize}
\section{Hybrid relation for MMZV}
The commutative polynomial setting is briefly introduced in $\S 6.1.1$. \\ Let consider the following involution, which represents the Antipode $\shuffle$ as seen in $\S 4.2.1$:\\ \begin{equation} \label{eq:sigma} \boldsymbol{\sigma}: \quad \mathbb{Q} \langle Y\rangle \rightarrow \mathbb{Q} \langle Y\rangle \text{ , } \quad f(y_{0},y_{1},\cdots, y_{p}) \mapsto (-1)^{w} f(y_{p},y_{p-1},\cdots, y_{1}), \end{equation} with $w$ the weight, equal to the degree of $f$ plus $p$. In particular, for $f\in\rho(\mathfrak{g}^{\mathfrak{m}})$: \begin{equation} \label{eq:fantipodesh} \textsc{ Antipode } \shuffle\text{ : } f+\sigma(f)=0. \end{equation} Note that $f^{(p)}$ denotes the part of $f$ involving $y_{0},\cdots, y_{p}$. We can also consider: \begin{equation} \label{eq:tau} \boldsymbol{\tau}: \quad \mathbb{Q} \langle X\rangle \rightarrow \mathbb{Q} \langle X\rangle\text{ , }\quad \overline{f}^{(p)}(x_{1},\cdots, x_{p}) \mapsto (-1)^{p}\overline{f}^{(p)}(x_{p}\cdots, x_{1}). \end{equation} The Antipode stuffle corresponds to $\tau(\overline{f}^{\star})$, where $\overline{f}^{\star}$ is defined by: \begin{equation} \label{eq:fstar} \overline{f}^{\star}(x_{1}, \ldots, x_{p})\mathrel{\mathop:}= \sum_{s \leq p, i_{k} \atop p=\sum i_{k}} f(\lbrace x_{1} \rbrace ^{i_{1}}, \ldots, \lbrace x_{s} \rbrace ^{i_{s}}) (-1)^{d-1} \prod_{k=1}^{s} x^{i_{k}-1}_{k} . \end{equation} It corresponds naturally to the Euler sums $\star$ version. Then, for $\overline{f}\in \overline{\rho} (\mathfrak{g}^{\mathfrak{m}})$: \begin{equation} \label{eq:fantipodest} \textsc{ Antipode } \ast\text{: } \overline{f}+ \tau(\overline{f}^{\star})=0 . \end{equation} \\ The \textit{hybrid relation} (Theorem $\ref{hybrid}$) for motivic multiple zeta values is equivalent to, in this setting of commutative polynomials to the following, already in some notes of F. Brown: \begin{theom}[\textsc{F. Brown}] For $\overline{f}\in \overline{\rho} (\mathfrak{g}^{\mathfrak{m}})$, the 6 terms relation holds: \begin{multline}\nonumber \overline{f}^{(p)} (x_{1},\cdots, x_{p}) + \frac{\overline{f}^{(p-1)} (x_{2}-x_{1}, \ldots, x_{p}-x_{1}) - \overline{f}^{(d-1)} (x_{2},\cdots, x_{p})}{x_{1}} \\ =(-1)^{w+1} \left( \overline{f}^{(p)} (x_{p},\cdots, x_{1}) + \frac{\overline{f}^{(p-1)} (x_{p-1}-x_{p}, \ldots, x_{1}-x_{p}) - \overline{f}^{(p-1)} (x_{p-1},\cdots, x_{1})}{x_{p}} \right). \end{multline} \end{theom} \noindent Before giving the proof, to be convinced these statements are equivalent, let just write $\overline{f}$ as: $$\overline{f}=\sum \alpha_{n_{1}, \ldots, n_{k}} x_{1}^{n_{1}-1} \cdots x_{k}^{n_{k}-1}.$$ Then: \begin{flushleft} $\frac{\overline{f}^{(p-1)} (x_{2}-x_{1}, \ldots, x_{p}-x_{1}) - \overline{f}^{(p-1)} (x_{2},\cdots, x_{p})}{x_{1}} $ \end{flushleft} \begin{align*} \quad \quad = & \sum \alpha_{n_{1}, \ldots, n_{p-1}} \frac{(x_{2}-x_{1})^{n_{1}-1} \cdots (x_{p}-x_{1})^{n_{p}-1} -x_{2}^{n_{1}-1} \cdots x_{p}^{n_{p}-1} }{x_{1}} \\ \quad\quad = & \sum \alpha_{n_{1}, \ldots, n_{p-1}} \sum_{1 \leq k_{i} \leq n_{i} \atop k\mathrel{\mathop:}=\sum n_{i}-k_{i}>0} (-1)^{k}x_{1}^{k-1} \prod_{i=1}^{d-1}\left( \binom{n_{i}-1}{k_{i}-1} x_{i+1}^{k_{i}-1} \right) \\
\quad \quad = & \sum_{\sum i_{j}=k} \alpha_{k_{1}+i_{1}, \ldots, k_{p-1}+i_{p-1}} \binom{k_{1}+i_{1}-1}{k_{1}-1} \cdots \binom{k_{p-1}+i_{p-1}-1}{k_{p-1}-1} (-1)^{k}x_{1}^{k-1} x_{2}^{k_{1}-1} \cdots x_{p}^{k_{p-1}-1} \end{align*} This, according to the shuffle regularization $(\ref{eq:shufflereg})$, matches exactly with the definition of $\zeta^{\mathfrak{m}}_{k}(k_{1}, \ldots, k_{p-1})$.
\begin{proof}[Proof of the previous theorem] The proof combines the shuffle relation (using that $f$ is translation invariant notably), the linearized stuffle relation (giving a relation between depth $p$ and depth $p-1$) and the antipode $\shuffle$. \\ Let's take $f$ in $\rho (\mathfrak{g}^{\mathfrak{m}})$ and let consider the difference \begin{equation} \label{eq:if} I(y_{0},y_{1},\cdots, y_{p})\mathrel{\mathop:}= f^{(p)} (y_{0}, y_{1},\cdots, y_{p})+(-1)^{w} f^{(p)} (y_{0}, y_{p},\cdots, y_{1}) \end{equation} $$= f^{(p)} (y_{0}, y_{1},\cdots, y_{p})- f^{(p)} (y_{1},\cdots, y_{p}, y_{0}). $$ Consider also the relation given by the linearized stuffle relation (in $\mathcal{L})$, between depth $p$ and depth $p-1$, defining $St$: \begin{equation} \label{eq:stf0} St(y_{0}, y_{1} \shuffle y_{2} \cdots y_{p})\mathrel{\mathop:}= f^{(p)} (y_{0}, y_{1} \shuffle y_{2}\cdots y_{p}), \end{equation} Where $St$ can then be expressed by $f^{(p-1)}$ using stuffle: \begin{equation} \label{eq:stf} St(y_{0}, y_{1} \shuffle y_{2} \cdots y_{p})= \sum \frac{1}{y_{i}-y_{1}} \left(f^{(p-1)} (y_{0},y_{2},,\cdots y_{i-1},y_{1},y_{i+1}, \ldots, y_{p}) - f^{(p-1)} (y_{0}, y_{2},\cdots, y_{p})\right). \end{equation} The theorem is then equivalent to the following identity $$ (\Join) \text{ } I(y_{0},y_{1},\cdots, y_{p}) = (-1)^{w+1}St(y_{p}, y_{0} \shuffle y_{p-1} \cdots y_{1})- St(y_{1}, y_{0} \shuffle y_{2} \cdots y_{p}).$$ Indeed, looking at the previous definition ($\ref{eq:stf}$ ), most of the terms of $St$ in the right side of $(\Join)$ get simplified together, and it remains only: $$ (-1)^{w+1} \frac{f^{(p-1)} (y_{p}, \ldots, y_{2}, y_{0}) - f^{(p-1)} (y_{p},\cdots, y_{1})}{y_{1}-y_{0}} - \frac{f^{(p-1)} (y_{1}, \ldots, y_{p-1}, y_{0}) - f^{(p-1)} (y_{1},\cdots, y_{p})}{y_{p}-y_{0}}. $$ Passing to the $x_{i}$ variables, we conclude that $ (\Join)$ is equivalent to the theorem's statement; let now prove $(\Join) $. By definition: $$St(y_{1},y_{0}\shuffle y_{2} \cdots y_{p})= f^{(p)}(y_{1},y_{0}\shuffle y_{2} \cdots y_{p})= f^{(p)}(y_{1},y_{0}\shuffle y_{2} \cdots y_{p-1}, y_{p})+ f^{(p)}(y_{1}, y_{2} , \ldots, y_{p}, y_{0}) .$$ Doing a right shift, using the definition of $I$: \begin{equation} \label{eq:a} St(y_{1},y_{0}\shuffle y_{2} \cdots y_{p}) \end{equation} $$=f^{(p)}(y_{p}, y_{1},y_{0}\shuffle y_{2} \cdots y_{p-1}) - I(y_{p}, y_{1},y_{0}\shuffle y_{2} \cdots y_{p-1}) + f^{(p)}(y_{0},y_{1}, y_{2} \cdots, y_{p}) -I(y_{0},y_{1}, y_{2} \cdots, y_{p}).$$ Since: \begin{align*} f^{(p)}(y_{p}, y_{1},y_{0}\shuffle y_{2} \cdots y_{p-1}) & = St(y_{p}, y_{0}\shuffle y_{1} y_{2} \cdots y_{p-1})- f^{(p)}(y_{p}, y_{0}, y_{1}, y_{2}, \ldots, y_{p-1}) \\ f^{(p)}(y_{0},y_{1}, y_{2} \cdots, y_{p}) & = -I (y_{p}, y_{0}, y_{1}, y_{2}, \ldots, y_{p-1})+ f^{(p)}(y_{p}, y_{0}, y_{1}, y_{2}, \ldots, y_{p-1}). \end{align*} Then, $\eqref{eq:a}$ becomes: \begin{multline}\nonumber St(y_{1},y_{0}\shuffle y_{2} \cdots y_{p})- St(y_{p}, y_{0}\shuffle y_{1} y_{2} \cdots y_{p-1}) \\ =-I (y_{p}, y_{0}, y_{1}, y_{2}, \ldots, y_{p-1}) - I(y_{p}, y_{1},y_{0}\shuffle y_{2} \cdots y_{p-1}) -I(y_{0},y_{1}, y_{2} \cdots, y_{p}). \end{multline} The sum of the first two $I$ is $I(y_{p}, y_{0}\shuffle y_{1} \cdots y_{p-1})$ which gives: \begin{multline} \label{eq:b} I(y_{0},y_{1}, y_{2} \cdots, y_{p})= - St(y_{1},y_{0}\shuffle y_{2} \cdots y_{p})+ St(y_{p}, y_{0}\shuffle y_{1} y_{2} \cdots y_{p-1})-I(y_{p}, y_{0}\shuffle y_{1} \cdots y_{p-1}) \\ =- St(y_{1},y_{0}\shuffle y_{2} \cdots y_{p}) + (-1)^{w+1}St(y_{p}, y_{0} \shuffle y_{p-1} \cdots y_{1}). \end{multline} The identity $(\Join)$ holds, and the identity of the theorem follows. \end{proof}
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\title{Optimal control of semi-Markov processes with a backward stochastic differential equations approach}
\begin{abstract} \noindent In the present work we employ, for the first time, backward stochastic differential equations (BSDEs) to study the optimal control of semi-Markov processes on finite horizon, with general state and action spaces. More precisely, we prove that the value function and the optimal control law can be represented by means of the solution of a class of BSDEs driven by a semi-Markov process or, equivalently, by the associated random measure. The peculiarity of the semi-Markov framework, with respect to the pure jump Markov case, consists in the proof of the relation between BSDE and optimal control problem. This is done, as usual, via the Hamilton-Jacobi-Bellman (HJB) equation, which however in the semi-Markov case is characterized by an additional differential term $\partial_a$. Taking into account the particular structure of semi-Markov processes we rewrite the HJB equation in a suitable integral form which involves a directional derivative operator $D$ related to $\partial_a$. Then, using a formula of It$\hat{\mbox{o}}$ type tailor-made for semi-Markov processes and the operator $D$, we are able to prove that the BSDE provides the unique classical solution to the HJB equation, which is shown to be the value function of our control problem.
\noindent{\small\textbf{Keywords:} Backward stochastic differential equations, optimal control problems, semi-Markov processes, marked point processes.} \end{abstract}
\section{Introduction} The aim of the present paper is to study optimal control problems for a class of semi-Markov processes using a suitable class of backward stochastic differential equations (BSDEs), driven by the random measure associated to the semi-Markov process itself.
Let us briefly describe our framework. Our starting point is a semi-Markov pure jump process $X$ on a general state space $K$. It is constructed starting from a jump rate function $\lambda(x,a)$ and a jump measure $A\mapsto\bar{q}(x,a,A)$ on $K$, depending on $x\in K$ and $a\ge0 $. Our approach is to consider a semi-Markov pure jump process as a two dimensional time-homogeneous and strong Markov process $\{(X_s,a_s), \, s \geq 0\}$ with its natural filtration $\mathcal{F}$ and a family of probabilities $\mathbb P^{x,a}$ for $x \in K$, $a \in [0,\infty)$ such that $\mathbb P^{x,a}(X_0=x,a_0=a)=1$. If the process starts from $(x,a)$ at time $t=0$ then the distribution of its first jump time $T_1$ under $\mathbb P^{x,a}$ is described by the formula \begin{equation}\label{jumptimeintro} \mathbb P^{x,a}(T_1> s)=
\exp\left(-\int_a^{a+s}\lambda(x,r)\,dr\right), \end{equation} and the conditional probability that the process is in $A$ immediately after a jump at time $T_1=s$ is $$
\mathbb P^{x,a}(X_{T_1}\in A\,|\,T_1=s ) =
\bar{q}(x,s,A). $$ $X_s$ is called the state of the process at time $s$, and $a_s$ is the duration period in this state up to moment $s$: \begin{eqnarray*} a_{s}= \left\{ \begin{array}{ll} a + s \qquad \qquad \qquad \qquad \qquad \quad \qquad \text{if}\,X_{p} = X_{s} \quad \forall \, 0 \leqslant p \leqslant s,\,\,p,s \in\mathbb R, \\ s - \sup\{\,p : \, 0\leqslant p \leqslant s,\, X_{p} \neq X_{s}\} \quad \text{otherwise.} \end{array} \right. \end{eqnarray*} We note that $X$ alone is not a Markov process. We limit ourselves to the case of a semi-Markov process $X$ such that the survivor function of $T_1$ under $\mathbb P^{x,0}$ is absolutely continuous and admits a hazard rate function $\lambda$ as in \eqref{jumptimeintro}. The holding times of the process are not necessarily exponentially distributed and can be infinite with positive probability. Our main restriction is that the jump rate function $\lambda$ is uniformly bounded, which implies that the process $X$ is non explosive.
Denoting by $T_n$ the jump times of $X$, we consider the marked point process $(T_n,X_{T_n})$ and the associated random measure $p(dt\,dy)= \sum_{n} \delta_{(T_n,X_{T_n})} $ on $(0,\infty)\times K$, where $\delta$ denotes the Dirac measure. The dual predictable projection $\tilde p$ of $p$ (shortly, the compensator) has the following explicit expression $$ \tilde p(ds\,dy)= \lambda(X_{s-},a_{s-})\,\bar{q}(X_{s-},a_{s-},dy)\,ds. $$
In Section \ref{section semi_markov_HJB} we address an optimal intensity-control problem for the semi-Markov process. This is formulated in a classical way by means of a change of probability measure, see e.g. \cite{ElK}, \cite{E}, \cite{B}. We define a class ${\cal A}$ of admissible control processes $(u_s)_{s \in [0,\,T]}$; for every fixed $t \in [0,\,T]$ and $(x,a)\in K \times [0,\infty)$, the cost to be minimized and the corresponding value function are \begin{eqnarray*} J(t,x,a, u(\cdot)) & = & \sperxaut{ \int_0^{T-t} l(t+ s,X_s,a_s,u_s)\,ds + g(X_{T-t},a_{T-t})}, \\ v(t,x,a)& = & \inf_{u(\cdot)\in{\cal A} }J(t,x,a,u(\cdot)), \end{eqnarray*} where $g,l$ are given real functions. Here $\mathbb E_{u,t}^{x,a}$ denotes the expectation with respect to another probability $\mathbb P_{u,t}^{x,a}$, depending on $t$ and on the control process $u$ and constructed in such a way that the compensator under $\mathbb P_{u,t}^{x,a}$ equals $r(t+s,X_{s-},a_{s-},y,u_s)\, \lambda(X_{s-},a_{s-})\,\bar{q}(X_{s-},a_{s-},dy)\,ds$, for some function $r$ given in advance as another datum of the control problem. Since the process $(X_s,a_s)_{s \geq 0}$ we want to control is time-homogeneous and starts from $(x,a)$ at time $s = 0$, we introduce a temporal translation which allows to define the cost functional for all $t \in [0,T]$. For more details see Remark \ref{rem:controllo 1}.
Our approach to this control problem consists in introducing a family of BSDEs parametrized by $(t,x,a)\in [0,T]\times K \times [0,\infty)$: \begin{equation}\label{intro_BSDE_controllo} Y^{x,a}_{s,t} + \int_{s}^{T-t}\int_{K}Z^{x,a}_{\sigma,t}(y)\,q(d\sigma\,dy) =
g(X_{T-t},a_{T-t}) + \int_{s}^{T-t}\,f\Big(t+\sigma,X_{\sigma},a_{\sigma},Z^{x,a}_{\sigma,t}(\cdot)\Big)\,d\sigma, \quad s\in [0,\,T-t], \end{equation} where the generator is given by the Hamiltonian function $f$ defined for every $s \in [0,\,T]$, $(x,a) \in K\times[0,\,+\infty)$, $z \in L^{2}(K,\mathcal{K},\lambda(x,a)\bar{q}(x,a,dy))$, as \begin{equation}\label{intro_hamilton_function} f(s,x,a,z(\cdot))=
\inf_{u \in U}\Big\{\,l(s,x,a,u) + \int_{K}z(y)(r(s,x,a,y,u)-1)\lambda(x,a)\bar{q}(x,a,dy)\,\Big\}. \end{equation} Under appropriate assumptions we prove that the optimal control problem has a solution and that the value function and the optimal control can be represented by means of the solution to the BSDE \eqref{intro_BSDE_controllo}.
Backward equations driven by random measures have been studied in many papers, within \cite{TaLi}, \cite{BaBuPa}, \cite{Roy}, \cite{KhMaPhZh}, \cite{Xia}, and more recently \cite{Be}, \cite{Cre-Mat}, \cite{KaTa-Po-Zh_1}, \cite{KaTa-Po-Zh_2}, \cite{CoFu-mpp}, \cite{CoFu-m}. In many of them the stochastic equations are driven by a Wiener process and a Poisson process, see, e.g., \cite{TaLi}, \cite{BaBuPa}, \cite{Roy}, \cite{KhMaPhZh}. A more general results on BSDEs driven by random measures is given by \cite{Xia}, but in this case
the generator $f$ depends on the process $Z$ in a specific way and this condition prevents a direct application to optimal control problems. In \cite{Be}, \cite{Cre-Mat}, \cite{KaTa-Po-Zh_1}, \cite{KaTa-Po-Zh_2}, the authors deal with BSDEs with jumps with a random compensator more general than the compensator of a Poisson random measure; here are involved random compensators which are absolutely continuous with respect to a deterministic measure, that can be reduced to a Poisson measure by a Girsanov change of probability. Finally, BSDEs driven by a random measure related to a pure jump process have been recently studied in \cite{CoFu-mpp}, and in \cite{CoFu-m} the pure jump Markov case is considered.
Our backward equation \eqref{intro_BSDE_controllo} is driven by a random measure associated to a two dimensional Markov process $(X,a)$, and his compensator is a stochastic random measure with a non-dominated intensity as in \cite{CoFu-m}. Even if the associated process is not pure jump, the existence, uniqueness and continuous dependence on the data for the BSDE \eqref{intro_BSDE_controllo} can be deduced extending in a straightforward way the results in \cite{CoFu-m}.
Concerning the optimal control of semi-Markov processes, the case of a finite number of states has been studied in \cite{chito}, \cite{Howard}, \cite{Jewell}, \cite{Osaki}, while the case of arbitrary state space is considered in \cite{Ross} and \cite{St1}. As in \cite{chito} and in \cite{St1}, in our formulation we admit control actions that can depend not only on the state process but also on the length of time the process has remained in that state. The approach based on BSDEs is classical in the diffusive context and is also present in the literature in the case of BSDEs with jumps, see as instance \cite{LimQuenez}. However, it seems to us be pursued here for the first time in the case of the semi-Markov processes. It allows to treat in a unified way a large class of control problems, where the state space is general and the running and final cost are not necessarily bounded. We remark that, comparing with \cite{St1}, the controlled processes we deal with have laws absolutely continuous with respect to a given, uncontrolled process; see also a more detailed comment in Remark \ref{confrontocontrollo} below. Moreover, in \cite{St1} optimal control problems for semi-Markov processes are studied in the case of infinite time horizon.
In Section \ref{section semi_markov_Kolmogorov_equation} we solve a nonlinear variant of the Kolmogorov equation for the process $(X,a)$, with the BSDEs approach. The process $(X,a)$ is time-homogeneous and Markov, but is not a pure jump process. In particular it has the integro-differential infinitesimal generator \begin{displaymath} \mathcal{\tilde{L}}\Phi(x,a) := \partial_a \Phi(x,a) + \int_{K}[\Phi(y,0)-\Phi(x,a)]\,\lambda(x,a)\,\bar{q}(x,a,dy), \qquad (x,a)\in K \times [0, \infty). \end{displaymath} The additional differential term $\partial_a $ do not allow to study the associated nonlinear Kolmogorov equation proceeding as in the pure jump Markov processes framework (see \cite{CoFu-m}). On the other hand, the two dimensional Markov process $(X_s,a_s)_{s \geqslant 0}$ belongs to the larger class of piecewise-deterministic Markov processes (PDPs) introduced by M.H.A. Davis in \cite{Da-bo}, and studied in the optimal control framework by several authors, within \cite{Da-Fa}, \cite{Ver}, \cite{Dem}, \cite{LenYam}. Moreover, we deal with a very specific PDP: taking into account the particular structure of semi-Markov processes, we present a reformulation of the Kolmogorov equation which allows us to consider solutions in a classical sense. In particular, we notice that the second component of the process $(X_s,\,a_s)_{s \geqslant 0}$ is linear in $s$. This fact suggests to introduce the formal directional derivative operator \begin{equation*} (Dv)(t,x,a):= \lim_{h \downarrow 0}\frac{v(t+h,x,a+h)-v(t,x,a)}{h}, \end{equation*} and to consider the following nonlinear Kolmogorov equation \begin{eqnarray}\label{Kolmogorov_diff_rif_intro} \left\{ \begin{array}{ll} Dv(t,x,a) +\mathcal{L}v(t,x,a) + f(t,x,a,v(t,x,a),v(t,\cdot,0)-v(t,x,a))=0 ,\\ \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad t \in [0,T], \, x \in K, \, a \in [0, \infty), \\ v(T,x,a)= g(x,a), \end{array} \right. \end{eqnarray} where $$\mathcal{L}\Phi(x,a): = \int_{K}[\Phi(y,0)-\Phi(x,a)]\,\lambda(x,a)\,\bar{q}(x,a,dy), \qquad (x,a)\in K \times [0, \infty).$$ Then we look for a solution $v$ such that the map $t \mapsto v(t,x,t+c)$ is absolutely continuous on $[0,T]$, for all constants $c \in [-T,\,+ \infty)$. The functions $f,g$ in \eqref{Kolmogorov_diff_rif_intro} are given. While it is easy to prove well-posedness of \eqref{Kolmogorov_diff_rif_intro} under boundedness assumptions, we achieve the purpose of finding a unique solution under much weaker conditions related to the distribution of the process $(X,a)$: see Theorem \ref{thm_kolm}. To this end we need to define a formula of It$\hat{\mbox{o}}$ type, involving the directional derivative operator $D$, for the composition of the process $(X_s,\,a_s)_{s \geqslant 0}$ with functions $v$ smooth enough (see Lemma \ref{Ito formula} below).\\ We construct the solution $v$ by means of a family of BSDEs of the form \eqref{intro_BSDE_controllo}. By the results above there exists a unique solution $(Y_{s,t}^{x,a}, Z_{s,t}^{x,a})_{s\in [0,\,T-t]}$ and
the estimates on the BSDEs are used to prove well-posedness of \eqref{Kolmogorov_diff_rif_intro}. As a by-product we also obtain the representation formulae $$ v(t,x,a)=Y_{0,t}^{x,a}, \qquad
Y_{s,t}^{x,a}=v(t+s,X_s,a_s), \qquad Z_{s,t}^{x,a}(y)= v(t+s,y,0)- v(t+s,X_{s-},a_{s-}), $$ which are sometimes called, at least in the diffusive case, non linear Feynman-Kac formulae.\\ Finally we can go back to the original control problem and observe that the associated Hamilton-Jacobi-Bellman equation has the form \eqref{Kolmogorov_diff_rif_intro} where $f$ is the Hamiltonian function \eqref{intro_hamilton_function}. By previous results we are able to identify the HJB solution $v(t,x,a)$, constructed probabilistically via BSDEs, with the value function.
\section{Notation, preliminaries and basic assumptions}\label{section_notation}
\subsection{Semi-Markov jump processes}\label{subsection_construction_SMP} We recall the definition of a semi-Markov process, as given, for instance, in \cite{G-S}. More precisely we will deal with a semi-Markov process with infinite lifetime (i.e. non explosive).
Suppose we are given a measurable space $(K, \mathcal{K})$, a set $\Omega$ and two functions $X: \Omega \times [0,\infty) \rightarrow K$, $a: \Omega \times [0,\infty) \rightarrow [0,\infty)$. For every $t \geq 0$, we denote by $\mathcal{F}_t$ the $\sigma$-algebra $\sigma((X_s,a_s), \, s \in [0,t] )$. We suppose that for every $x \in K$ and $a \in [0,\infty)$, a probability $\mathbb P^{x,a}$ is given on $(\Omega, \mathcal{F}_{[0,\infty)})$ and the following conditions hold.
\begin{enumerate} \item $\mathcal{K}$ contains all one-point sets. $\Delta$ denotes a point not included in $K$.
\item $\mathbb P^{x,a}(X_0=x,a_0=a)=1$ for every $x \in K$, $a \in [0,\infty)$. \item For every $s,\,p \geqslant 0$ and $A\in {\cal K}$ the function $(x,\,a)\mapsto \Bbb{P}^{x,a}(X_{s}\in A,\,a_{s}\leqslant p)$ is $\mathcal{K}\otimes \mathcal{B}^+$-measurable.
\item For every $0\le t\le s$, $p \geqslant 0$, and $A\in {\cal K}$ we have
$\mathbb P^{x,a}(X_s\in A,\,a_{s}\leqslant p\,|\,{\cal F}_{t})=\mathbb P^{X_t,a_t}(X_s\in A,\,a_{s}\leqslant p)$, $\mathbb P^{x,a}$-a.s.
\item All the trajectories of the process $X$ have right limits when $K$ is given the discrete topology (the one where all subsets are open). This is equivalent to require that for every $\omega\in\Omega$ and $t\ge 0$ there exists $\delta>0$ such that $X_s(\omega)=X_t(\omega)$ for $s\in [t,t+\delta]$.
\item All the trajectories of the process $a$ are continuous from the right piecewise linear functions. For every $\omega\in\Omega$, if $[\alpha, \beta)$ is the interval of linearity of $a_{\cdot}(\omega)$ then $a_s(\omega)= a_{\alpha}(\omega) + s -\alpha$ and $X_{\alpha}(\omega)=X_s(\omega)$; if $\beta$ is a discontinuity point of $a_{\cdot}(\omega)$ then $a_{\beta+}(\omega)=0$ and $X_{\beta}(\omega) \neq X_{\beta-}(\omega)$.
\item For every $\omega\in\Omega$ the number of jumps of the trajectory $t \mapsto X_t(\omega)$ is finite on every bounded interval. \end{enumerate} $X_s$ is called the \emph{state} of the process at time $s$, $a_s$ is the \emph{duration period} in this state up to moment $s$. Also we call $X_s$ the \emph{phase} and $a_s$ the \emph{age} or the \emph{time component} of a semi-Markov process. $X$ is
a non explosive process because of condition 7. We note, moreover, that the two-dimensional process $(X,a)$ is a strong Markov process with time-homogeneous transition probabilities because of conditions 2,\,3, and 4. It has right-continuous sample paths because of conditions 1, 5 and 6, and it is not a pure jump Markov process, but only a PDP.
The class of semi-Markov processes we consider in the paper will be described by means of a special form of joint law $Q$ under $\Bbb{P}^{x,a}$ of the first jump time $T_{1}$, and the corresponding position $X_{T_{1}}$. To proceed formally, we fix $X_{0} = x \in K$ and define the first jump time \begin{equation*} T_{1} = \inf\{ p >0:\, X_{p} \neq x \}, \end{equation*} with the convention that $T_{1} = +\infty$ if the indicated set is empty.\\ We introduce $S := K \times [0,\,+\infty) $ an we denote by $\mathcal{S}$ the smallest $\sigma$-algebra containing all sets of $\mathcal{K} \otimes \mathcal{B}([0,\,+\infty))$. (Here and in the following ${\cal B}(\Lambda)$ denotes the Borel $\sigma$-algebra of a topological space $\Lambda$). Take an extra point $\Delta \notin K$ and define $X_{\infty}(\omega)= \Delta$ for all $\omega \in \Omega$, so that $X_{T_{1}}: \Omega \rightarrow K \cup \{ \Delta \}$ is well defined. Then on the extended space $S \cup \{ (\Delta ,\,\infty) \}$ we consider the smallest $\sigma$-algebra, denoted by $\mathcal{S}^{\text{enl}}$, containing $ \{ (\Delta ,\,\infty) \}$ and all sets of $\mathcal{K} \otimes \mathcal{B}([0,\,+\infty))$. Then $(X_{T_{1}},\,T_{1})$ is a random variable with values in $(S\cup \{ (\Delta ,\,\infty) \},\mathcal{S}^{\text{enl}})$. Its law under $\Bbb{P}^{x,a}$ will be denoted by $Q(x,a,\cdot)$.
We will assume that $Q$ is constructed from two given functions denoted by $\lambda$ and $\bar{q}$. More precisely we assume the following. \begin{hypothesis}\label{hp_dati} There exist two functions \begin{equation*}\lambda: S \rightarrow [0,\infty) \mbox{ and } \bar{q}:S \times \mathcal{K}\rightarrow [0,1] \end{equation*} such that \begin{itemize} \item[(i)] $(x, a) \mapsto \lambda(x,a)$ is $\mathcal{S}$-measurable; \item[(ii)] $\sup_{(x,a) \in S}\lambda(x,a) \leqslant C \in \mathbb R^+$; \item[(iii)] $(x, a) \mapsto \bar{q}(x,a,A)$ is $\mathcal{S}$-measurable $\forall A \in \mathcal{K}$; \item[(iv)] $A \mapsto \bar{q}(x,a,A)$ is a probability measure on $\mathcal{K}$ for all $(x,\,a)\in S$. \end{itemize} \end{hypothesis} We define a function $H$ on $ K \times[0,\infty]$ by \begin{equation}\label{def_H} H(x, s):= 1- e^{-\int_{0}^{s}\lambda(x,r)dr}. \end{equation} Given $\lambda$ and $\bar{q}$, we will require that for the semi-Markov process $X$ we have, for every $(x,a)\in S$ and for $A\in {\cal K}$, $0\le c< d\le \infty$,
\begin{eqnarray}\label{jumpkernel}
Q(x,a, A \times (c,d) ) & = & \displaystyle \frac{1}{1- H(x,a)}\int_c^d
\bar{q}(x,s,A) \frac{d }{d\,s} \, H(x,a+s)\,ds\nonumber\\
& = & \displaystyle \int_c^d
\bar{q}(x,s,A)\;\lambda (x,a+s)\;
\exp\left(-\int_a^{a+s}\lambda(x,r)\,dr\right) \,ds, \end{eqnarray} where
$Q$ was described above as the law of $(X_{T_1},T_1)$ under $\mathbb P^{x,a}$.
The existence of a semi-Markov process satisfying \eqref{jumpkernel} is a well known fact, see for instance \cite{St1} Theorem 2.1, where it is proved that $X$ is in addition a strong Markov process. The nonexplosive character of $X$ is made possible by Hypothesis \ref{hp_dati}-(ii).
We note that our data only consist initially in a measurable space $(K,{\cal K})$ ($\mathcal{K}$ contains all singleton subsets of $K$), and in two functions $\lambda$, $\bar{q}$ satisfying Hypothesis \ref{hp_dati}. The semi-Markov process $X$ can be constructed in an arbitrary way provided \eqref{jumpkernel} holds.
\begin{remark}\label{processosemi-markov} \begin{enumerate} \item Note that \eqref{jumpkernel} completely specifies the probability measure $Q(x,a,\cdot)$ on
$(S\,\cup\, \{ (\Delta ,\,\infty) \},\mathcal{S}^{\text{enl}})$: indeed simple computations show that, for $s\ge 0$,
\begin{equation}\label{jumpkerneldue}
\mathbb P^{x,a}(T_1\in (s,\infty]) =1- Q(x,a,K \times (0,s] )
= \exp\left(-\int_a^{a+s}\lambda(x,r)\,dr\right),
\end{equation}
and
we clearly have
\begin{equation*}\label{jumpkerneltre}
\begin{array}{lll}
\mathbb P^{x,a}(T_1=\infty)&=&\displaystyle
Q(x, a,\{(\Delta,\infty)\})=\exp\left(-\int_a^\infty\lambda(x,r)\,dr\right).
\end{array}
\end{equation*}
Moreover, the kernel $Q$ is well defined, because $H(x,a)< 1$ for all $(x,a) \in S$ by assumption \ref{hp_dati}-(ii).
\item
The data $\lambda$ and $\bar{q}$ have themselves a probabilistic interpretation.
In fact if in \eqref{jumpkerneldue} we set $a = 0$ we obtain \begin{equation}
\mathbb P^{x,0}(T_1 > s) = \exp\left(-\int_0^s\lambda(x,r)\,dr\right) = 1 - H(x,s).
\end{equation}
This means that under $\mathbb P^{x,0}$ the law of $T_1$ is described by the distribution function $H$, and
\begin{displaymath}
\lambda(x,a)= \frac{\frac{\partial H}{\partial a}(x,a)}{1- H(x,a)}.
\end{displaymath}
Then $\lambda(x,a)$ is the jump rate of the process $X$ given that it has been in state $x$ for a time $a$.\\
Moreover, the probability
$\bar{q}(x,s,\cdot)$ can be interpreted as the conditional probability that $X_{T_1}$
is in $A\in{\cal K}$ given that $T_1=s$; more precisely,
$$
\mathbb P^{x,a}(X_{T_1}\in A, T_1<\infty\,|\,T_1 ) =
\bar{q}(x,T_1,A)\,1_{T_1<\infty}, \qquad \mathbb P^{x,a}-a.s.
$$
\item
In \cite{G-S} the following observation is made: starting from $T_0=t$ define inductively
$
T_{n+1}=\inf\{s>T_n\,:\, X_s\neq
X_{T_{n}}\},
$
with the convention that $T_{n+1} =\infty$ if the indicated
set is empty; then, under the probability
$\mathbb P^{x,a}$, the sequence of the successive states of the semi-Markov $X$ is a Markov chain, as in the case of Markov processes.
However, while for the latter the duration period in the state depends only on this state and it is necessarily exponentially distributed, in the case of a semi Markov process the duration period depends also on the state into which the process moves and the distribution of the duration period may be arbitrary.
\item In \cite{G-S} is also
proved that the sequence $(X_{T_n},T_n)_{n\ge 0}$ is a discrete-time
Markov process in $(S\cup \{(\Delta,\,\infty)\},$ $\mathcal{S}^{\text{enl}})$ with transition kernel $Q$, provided we extend
the definition of $Q$ making the state $(\Delta,\,\infty)$ absorbing, i.e. we define
$$
Q(\Delta,\,\infty,\, S)=0,\qquad
Q(\Delta,\,\infty, \,\{(\Delta,\,\infty)\})=1.
$$
Note that $(X_{T_n},T_n)_{n\ge 0}$ is time-homogeneous.
This fact allows for a simple description of the process $X$.
Suppose one starts with a discrete-time Markov process $(\tau_n,\xi_n)_{n\ge 0}$
in $S$ with transition probability kernel $Q$ and a given starting
point $(x,a)\in S$ (conceptually, trajectories
of such a process are easy to simulate). One can then define a process $Y$
in $K$ setting $Y_t=\sum_{n=0}^N\xi_n 1_{[\tau_n, \tau_{n+1}) }(t)$,
where $N=\sup\{n\ge 0\,:\, \tau_n\leqslant\infty\}$. Then $Y$ has the same law
as the process $X$ under $\mathbb P^{x,a}$.
\item We stress that \eqref{def_H} limits ourselves to deal with a class of semi-Markov processes for which the survivor function $T_1$ under $\mathbb P^{x,0}$ admits a hazard rate function $\lambda$.
\end{enumerate} \end{remark}
\subsection{BSDEs driven by a semi-Markov process} Let be given a measurable space $(K,\mathcal K)$, a transition measure $\bar{q}$ on $K$ and a given positive function $\lambda$, satisfying Hypothesis \ref{hp_dati}. Let $X$ be the associated semi-Markov process constructed out of them as described in Section \ref{subsection_construction_SMP}. We fix a deterministic terminal time $T>0$ and a pair $(x,a)\in S$, and we look at all processes under the probability $\Bbb{P}^{x,a}$. We denote by $\mathcal{F}$ the natural filtration $(\mathcal{F}_t)_{t \in[0, \infty)}$ of $X$. Conditions 1, 5 and 6 above imply that the filtration $\mathcal{F}$ is right continuous (see \cite{B}, Appendix A2, Theorem T26). The predictable $\sigma$-algebra (respectively, the progressive $\sigma$-algebra) on $\Omega \times [0, \, \infty)$ is denoted by $\mathcal{P}$ (respectively, by $Prog$). The same symbols also denote the restriction to $\Omega \times [0, \, T]$.
We define a sequence $(T_{n})_{n \geqslant 1}$ of random variables with values in $[0, \, \infty]$, setting \begin{equation}\label{Tn_def} T_{0}(\omega) = 0,\quad T_{n+1}(\omega) =\inf\{ s \geqslant T_{n}(\omega): \, X_{s}(\omega) \neq X_{T_{n}}(\omega) \}, \end{equation} with the convention that $T_{n+1}(\omega)= \infty$ if the indicated set is empty. Being $X$ a jump process we have $T_{n}(\omega)\leqslant T_{n+1}(\omega)$ if $T_{n+1}(\omega)< \infty$, while the non explosion of $X$ means that $T_{n+1}(\omega)\rightarrow \infty$. We stress the fact that $(T_{n})_{n \geqslant 1}$ coincide by definition with the time jumps of the two dimensional process $(X,a)$.
For $\omega \in \Omega$ we define a random measure on $( [0, \, \infty)\times K ,\, \mathcal{B}[0, \, \infty)\otimes \mathcal{K})$ setting \begin{equation} p(\omega, C) = \sum_{n \geqslant 1}\ensuremath{\mathonebb{1}}_{\{(T_{n}(\omega), \, X_{T_{n}}(\omega)) \in C \}}, \qquad C \in \mathcal{B}[0, \, \infty)\otimes \mathcal{K}. \end{equation} The random measure $\lambda(X_{s-},a_{s-})\,\bar{q}(X_{s-},a_{s-},dy)\,ds$ is called the compensator, or the dual predictable projection, of $p(ds,dy)$. We are interested in the following family of backward equations driven by the compensated random measure $q(ds\,dy)=p(ds\,dy) -\lambda(X_{s-},a_{s-})\,\bar{q}(X_{s-},a_{s-},dy)\,ds$ and parametrized by $(x,a)$: $\Bbb{P}^{x,a}$-a.s., \begin{equation}\label{BSDE} Y_{s} + \int_{s}^{T}\int_{K}Z_{r}(y)\,q(dr\,dy) =
g(X_{T},a_{T}) + \int_{s}^{T}f\Big(r,X_{r},a_{r},Y_{r},Z_{r}(\cdot)\Big)\,dr, \qquad s\in [0,\,T]. \end{equation} We consider the following assumptions on the data $f$ and $g$. \begin{hypothesis}\label{H_1} \begin{itemize} \item[\emph{(1)}] The final condition $g : S \rightarrow \mathbb R$ is $\mathcal S$-measurable \\
and $\sperxa{\abs{g(X_{T},a_{T})}^{2}} < \infty$. \item[\emph{(2)}] The generator $f$ is such that \begin{itemize} \item[\emph{(i)}] for every $s \in [0,\,T]$, $(x,a)\in S$, $r \in \mathbb R$, $f$ is a mapping \\ $f(s,x,a,r,\cdot) : \mathcal{L}^{2}(K,\mathcal{K},\,\lambda(x,a)\,\bar{q}(x,a,dy))\rightarrow \mathbb R$; \item[\emph{(ii)}] for every bounded and $\mathcal{K}$-measurable $z: K \rightarrow \mathbb R$ the mapping \begin{equation}\label{map} (s,x,a,r)\mapsto f(s,x,a,r,z(\cdot)) \end{equation} is $\mathcal{B}([0,\,T]) \otimes \mathcal S \otimes \mathcal{B}(\mathbb R)$-measurable; \item[\emph{(iii)}] there exist $L \geqslant 0$, $L' \geqslant 0 $ such that for every $s \in [0, \, T]$, $(x,a)\in S$, $r, r' \in \mathbb R, \\
z, z' \in \mathcal{L}^{2}(K,\mathcal{K},\lambda(x,a)\,\bar{q}(x,a,dy))$ we have \begin{equation}\label{f_inequality} \abs{f(s,x,a,r,z(\cdot)) - f(s,x,a,r',z'(\cdot))} \leqslant L'\abs{r - r'} + L\left( \int_{K} \abs{z(y) - z'(y)}^{2} \lambda(x,a)\,\bar{q}(x,a,dy) \right)^{1/2}; \end{equation} \item[\emph{(iv)}]we have \begin{equation}\label{finite_sper} \sperxa{\int_{0}^{T}\abs{f(s,X_{s},a_{s},0,0)}^{2}ds} < \infty. \end{equation} \end{itemize} \end{itemize} \end{hypothesis}
\begin{remark} Assumptions (i), (ii), and (iii) imply the following measurability properties of \\ $f(s,X_{s},a_{s},Y_{s},Z_{s}(\cdot))$: \begin{itemize} \item if $Z\in \mathcal{L}^2(p)$, then the mapping \begin{displaymath} (\omega,s,y) \mapsto f(s,X_{s-}(\omega),a_{s-}(\omega),y,Z_{s}(\omega, \cdot)) \end{displaymath} is $\mathcal{P}\otimes \mathcal{B}(\mathbb R)$-measurable; \item if, in addition, $Y$ is a $Prog$-measurable process, then \begin{displaymath} (\omega,s) \mapsto f(s,X_{s-}(\omega),a_{s-}(\omega),Y_{s}(\omega),Z_{s}(\omega, \cdot)) \end{displaymath} is $Prog$-measurable. \end{itemize} \end{remark} We introduce the space $\Bbb{M}^{x,a}$ of the processes $(Y,Z)$ on $[0,\,T]$ such that $Y$ is real-valued and $Prog$-measurable, $Z: \Omega \times K \rightarrow \mathbb R$ is $\mathcal{P}\otimes \mathcal{K}$-measurable, and \begin{displaymath}
||(Y,Z)||^{2}_{\Bbb{M}^{x,a}} := \sperxa{\int_{0}^{T}\abs{Y_{s}}^{2}ds} + \sperxa{\int_{0}^{T}\int_{K}\abs{Z_{s}(y)}^{2}\lambda(X_{s},a_{s})\,\bar{q}(X_{s},a_{s},dy)\,ds} <\infty. \end{displaymath} The space $\Bbb{M}^{x,a}$ endowed with this norm is a Banach space, provided we identify pairs of processes whose difference has norm zero. \begin{theorem}\label{thm: uniqueness_existence_BSDE} Suppose that Hypothesis \ref{H_1} holds for some $(x,a) \in S$.\\ Then there exists a unique pair $(Y,Z)$ in $\Bbb{M}^{x,a}$ which solves the BSDE \eqref{BSDE}. Let moreover $(Y',Z')$ be another solution in $\Bbb{M}^{x,a}$ to the BSDE \eqref{BSDE} associated with the driver $f'$ and final datum $g'$. Then \begin{align}\label{stima-differenza}
&\sup_{s \in [0,\,T]}\sperxa{|Y_s-Y'_{s}|^2} + \sperxa{\int_{0}^{T}|Y_s-Y'_{s}|^2ds} + \sperxa{\int_{0}^{T}\int_{K}|Z_{s}(y)-Z'_{s}(y)|^2\lambda(X_{s},a_{s})\,\bar{q}(X_{s},a_{s},dy)\,ds} \nonumber\\
& \leqslant C\sperxa{|g(X_{T})-g'(X_{T})|^2 + \int_{0}^{T}|f(s,X_{s},a_{s},Y'_{s},Z'_{s}(\cdot) )-f'(s,X_{s},a_{s},Y'_{s},Z'_{s}(\cdot))|^2ds}, \end{align} where $C$ is a constant depending on $T$, $L$, $L'$. \end{theorem} \begin{remark} The construction of a solution to the BSDE \eqref{BSDE} is based on the integral representation theorem of marked point process martingales (see, e.g., \cite{Da-bo}), and on a fixed-point argument. Similar results of well-posedness for BSDEs driven by random measures can be found in literature, see, in particular, the theorems given in \cite{CoFu-m}, Section 3, and in \cite{Be}. Notice that these results can not be a priori straight applied to our framework: in \cite{Be} are involved random compensators which are absolutely continuous with respect to a deterministic measure, instead in our case the compensator is a stochastic random measure with a non-dominated intensity;
\cite{CoFu-m} apply to BSDEs driven by a random measure associated to a pure jump Markov process, while the two dimensional process $(X,a)$ is Markov but not pure jump. Nevertheless, under Hypothesis \ref{H_1}, Theorem 3.4 and Proposition 3.5 in \cite{CoFu-m} can be extended to our framework without additional difficulties. The proofs turn out to be very similar to those of the mentioned results, and we do not report them here to alleviate the presentation. \end{remark}
\section{Optimal control}\label{section semi_markov_HJB} \subsection{Formulation of the problem} In this section we consider again a measurable space $(K,\mathcal{K})$, a transition measure $\bar{q}$ and a function $\lambda$ satisfying Hypothesis \ref{hp_dati}. The data specifying the optimal control problem we will address to are an action (or decision) space $U$, a running cost function $l$, a terminal cost function $g$, a (deterministic, finite) time horizon $T>0$ and another function $r$ specifying the effect of the control process. We define an admissible control process, or simply a control, as a predictable process $(u_{s})_{s \in [0, \, T]}$ with values in $U$. The set of admissible control processes is denoted by $\mathcal{A}$. We will make the following assumptions: \begin{hypothesis} \label{hyp_control} \begin{itemize} \item[\emph{(1)}] $(U,\mathcal U)$ is a measurable space. \item[\emph{(2)}] The function $r: [0, \, T]\times S \times K \times U \rightarrow \mathbb R $ is $\mathcal{B}([0,\,T]) \otimes \mathcal S \otimes \mathcal K \otimes \mathcal U$-measurable and there exists a constant $C_{r}>1$ such that, \begin{equation}\label{hyp: r_bound} 0\leqslant r(t,x,a,y,u)\leqslant C_{r}, \qquad t\in[0, \, T],\,(x,a)\in S,\, y \in K,\, u \in U. \end{equation} \item[\emph{(3)}] The function $g: S \rightarrow \mathbb R $ is $ \mathcal S$-measurable, and for all fixed $t \in [0,\,T]$, \begin{equation}\label{hyp: g_boundedness} \sperxa{\abs{g(X_{T-t},a_{T-t})}^{2}}<\infty,\qquad \forall (x,a)\in S. \end{equation} \item[\emph{(4)}] The function $l: [0, \, T]\times S\times U \rightarrow \mathbb R $ is $\mathcal{B}([0\,\,T]) \otimes \mathcal S \otimes \mathcal U$-measurable and there exists $\alpha >1$ such that, for every fixed $t \in [0,\,T]$, for every $(x,a) \in S$ and $u(\cdot) \in \mathcal A$, \begin{eqnarray}\label{hyp: l_inf_bound} \begin{array}{ll} \inf_{u \in U}l(t,x,a,u) > \infty;\\ \sperxa{\int_{0}^{T-t}\abs{\inf_{u \in U}l(t+s,X_{s},a_{s},u)}^{2}\,ds}<\infty,\\ \sperxa{\int_{0}^{T-t}\abs{l(t+s,X_{s},a_{s},u_s)}\,ds}^{\alpha}<\infty. \end{array} \end{eqnarray} \end{itemize} \end{hypothesis} To any $(t,x,a) \in [0,\,T] \times S$ and any control $u(\cdot)\in \mathcal{A}$ we associate a probability measure $\Bbb{P}^{x,a}_{u,t}$ by a change of measure of Girsanov type, as we now describe. Recalling the definition of the jump times $T_{n}$ in \eqref{Tn_def}, we define, for every fixed $t \in [0,\,T]$, $$ L^{t}_{s} = \exp\left(\int_{0}^{s}\!\!\int_{K}(1-r(t+\sigma,X_{\sigma},a_{\sigma},y,u_{\sigma}) )\,\lambda(X_{\sigma},a_{\sigma})\,\bar{q}(X_{\sigma},a_{\sigma},dy)\,d\sigma \!\right)\!\!\!\prod_{n\geqslant 1: T_{n} \leqslant s}\!\!\!r(t+T_{n}, X_{T_{n}-},a_{T_{n}-},X_{T_{n}},u_{T_{n}}), $$ for all $s \in [0,\,T-t]$, with the convention that the last product equals $1$ if there are no indices $n \geqslant 1$ satisfying $T_{n}\leqslant s$. As a consequence of the boundedness assumption on $\bar{q}$ and $\lambda$ it can be proved, using for instance Lemma 4.2 in \cite{CoFu-mpp}, or \cite{B} Chapter VIII Theorem T11, that for every fixed $t \in [0,\,T]$ and for every $\gamma >1$ we have \begin{equation}\label{L_martingale} \sperxa{\abs{L^t_{T-t}}^{\gamma}}<\infty,\qquad \sperxa{L^t_{T-t}}=1, \end{equation} and therefore the process $L^t$ is a martingale (relative to $\Bbb{P}^{x,a}$ and $\mathcal{F}$). Defining a probability $\Bbb{P}_{u,t}^{x,a}(d\omega)= L^t_{T-t}(\omega)\,\Bbb{P}^{x,a}(d\omega)$, we introduce the cost functional corresponding to $u(\cdot)\in \mathcal{A}$ as \begin{equation}\label{cost_functional} J(t,x,a,u(\cdot))= \sperxaut{ \int_{0}^{T-t}\,l(t+s,X_{s},a_{s},u_{s})\,ds + g(X_{T-t},a_{T-t}) }, \end{equation} where $\Bbb{E}_{u,t}^{x,a}$ denotes the expectation under $\Bbb{P}_{u,t}^{x,a}$. Taking into account \eqref{hyp: g_boundedness}, \eqref{hyp: l_inf_bound} and \eqref{L_martingale}, and using H\"{o}lder inequality it is easily seen that the cost is finite for every admissible control. The control problem starting at $(x,a)$ at time $s=0$ with terminal time $s=T-t$ consists in minimizing $J(t,x,a,\cdot)$ over $\mathcal{A}$.
We finally introduce the value function \begin{displaymath} v(t,x,a) = \inf_{u(\cdot)\in \mathcal{A}}J(t,x,a,u(\cdot)),\qquad t \in [0,\,T],\,\, (x,a)\in S. \end{displaymath} The previous formulation of the optimal control problem by means of change of probability measure is classical (see e.g. \cite{ElK}, \cite{E}, \cite{B}). Some comments may be useful at this point. \begin{remark}\label{rem:controllo 1} \begin{itemize} \item[1.] The particular form of cost functional \eqref{cost_functional} is due to the fact that the time-homogeneous Markov process ${(X_{s},a_{s})}_{s \geqslant 0}$ satisfies
\begin{equation*}
\mathbb P^{x,a}(X_{0}=x,\,a_{0}=a) = 1;
\end{equation*}
the introduction of the temporal translation in the first component allows us to define $J(t,x,a, u(\cdot))$ for all $t \in [0,\,T]$. \item [2.] We recall (see e.g. \cite{B}, Appendix A2, Theorem T34) that a process $u$ is $\mathcal{F}$-predictable if and only if it admits the representation
\begin{displaymath}
u_{s}(\omega) = \sum_{n \geqslant 0}u_{s}^{(n)}(\omega)\,\ensuremath{\mathonebb{1}}_{(T_{n}(\omega),T_{n+1}(\omega)]}(s)
\end{displaymath}
where for each $(\omega,s)\mapsto u_{s}^{(n)}(\omega)$ is $\mathcal{F}_{[0,\,T_{n}]}\otimes \mathcal{B}(\mathbb R^+)$-measurable, with $\mathcal{F}_{[0,\,T_{n}]}= \sigma(T_{i},X_{T_{i}},\, 0\leqslant i \leqslant n)$ (see e.g. \cite{B}, Appendix A2, Theorem T30).
Thus the fact that controls are predictable processes admits the following interpretation: at each time $T_{n}$ (i.e. immediately after a jump) the controller, having observed the random variables $T_{i},\, X_{T_{i}},\, (0\leqslant i \leqslant n)$, chooses his current action, and updates her/his decisions only at time $T_{n+1}$. \item[3.] It can be proved (see \cite{J} Theorem 4.5) that the compensator of $p(ds \,dy)$ under $\Bbb{P}_{u,t}^{x,a}$ is
\begin{displaymath}
r(t+s,X_{s-},a_{s-},y,u_{s})\,\lambda(X_{s-},a_{s-})\,\bar{q}(X_{s-},a_{s-},dy)\,ds,
\end{displaymath}
whereas the compensator of $p(ds\, dy)$ under $\Bbb{P}^{x,a}$ was $\lambda(X_{s-},a_{s-})\,\bar{q}(X_{s-},a_{s-},dy)\,ds$. This explains that the choice of a given control $u(\cdot)$ affects the stochastic system multiplying its compensator by $r(t+s,x,a,y,u_{s})$. \item[4.]We call control law an arbitrary measurable function $\underbar{u}: [0,\,T]\times S\rightarrow U$. Given a control law one can define an admissible control $u$ setting $u_{s}= \underbar{u}(s,X_{s-},a_{s-})$.\\ Controls of this form are called feedback controls. For a feedback control the compensator of $p(ds \,dy)$ is
$r(t+s,X_{s-},a_{s-},y,\underbar{u}(s,X_{s-},a_{s-}))\,\lambda(X_{s-},a_{s-})\,\bar{q}(X_{s-},a_{s-},dy)\,ds$ under $\Bbb{P}_{u,t}^{x,a}$. Thus, the process $(X,a)$ under the optimal probability is a two-dimensional Markov process corresponding to the transition measure
\begin{displaymath}
r(t+s,x,a,y,\underbar{u}(s,x,a))\,\lambda(x,a)\,\bar{q}(x,a,dy)
\end{displaymath}
instead of $\lambda(x,a)\,\bar{q}(x,a,dy)$.
However, even if the optimal control is in the feedback form, the optimal process is not, in general, time-homogeneous since the control law may depend on time. In this case, according to the definition given in Section \ref{section_notation}, the process $X$ under the optimal probability is not a semi-Markov process. \end{itemize} \end{remark}
\begin{remark}\label{confrontocontrollo} Our formulation of the optimal control should be compared with another approach (see e.g. \cite{St1}). In \cite{St1} is given a family of jump measures on K $\{\bar{q}(x, b ,\cdot),\,b \in B\}$ with $B$ some index set endowed with a topology. In the so called \emph{strong formulation} a control $u$ is an ordered pair of functions $(\lambda',\beta)$ with $\lambda': S \rightarrow \mathbb R^+$, $\beta: S \rightarrow B$ such that \begin{align*} \begin{array}{lll} \lambda'\,\,\text{ and}\,\, \beta\,\, \text{are}\,\, \mathcal{S}-\text{measurable}; \\ \forall x \in K,\, \exists\, t(x)>0:\,\,\int_{0}^{t(x)}\,\lambda'(x,r)\, dr < \infty;\\ \bar{q}(\cdot, \beta ,A)\,\, \text{is}\,\, \mathcal{B}^+\text{-measurable}\,\, \forall A \in \mathcal{K}. \end{array} \end{align*} If $\mathcal{A}$ is the class of controls which satisfies the above conditions, then a control $u = (\lambda', \beta)\in \mathcal{A}$ determines a controlled process $X^{u}$ in the following manner. Let \begin{displaymath} H^{u}(x, s):= 1- e^{-\int_{0}^{s}\,\lambda'(x,r)\,dr}, \quad \forall (x,s)\in S, \end{displaymath} and suppose that $(X^{u}_{0},a_{0}^{u})=(x,a)$. Then at time $0$, the process starts in state $x$ and remains there a random time $S_{1}>0$, such that \begin{equation}\label{tempo_soggiorno_controllato} \probxa{S_{1} \leqslant s} = \frac{H^{u}(x,a+s)-H^{u}(x,a)}{1- H^{u}(x,a)}. \end{equation} At time $ S_{1}$ the process transitions to the state $X^{u}_{ S_{1}}$, where \begin{displaymath}
\probxa{X^{u}_{S_{1}} \in A| S_{1}} = \bar{q}(x,\beta(x,S_{1}),A). \end{displaymath} The process stays in state $X^{u}_{ S_{1}}$ for a random time $S_{2}>0$ such that \begin{displaymath}
\probxa{S_{2} \leqslant s| S_{1}, \,X^{u}_{S_{1}}} = H^{u}(X^{u}_{ S_{1}},s) \end{displaymath} and then at time $ S_{1}+S_{2}$ transitions to $X^{u}_{ S_{1}+S_{2}}$, where \begin{displaymath}
\probxa{X^{u}_{ S_{1}+S_{2}} \in A| S_{1},\, X^{u}_{S_{1}},\,S_{2}} = \bar{q}(X^{u}_{ S_{1}},\beta(X^{u}_{S_{1}},S_{2}),A). \end{displaymath} We remark that the process $X^u$ constructed in this way turns out to be semi-Markov.
We also mention that the class of control problems specified by the initial data $\lambda'$ and $\beta$ is in general larger that the one we address in this paper. This can be seen noticing that in our framework all the controlled processes have laws which are absolutely continuous with respect to a single uncontrolled process (the one corresponding to $r\equiv 1$) whereas this might not be the case for the rate measures $\lambda'(x,a)\,\bar{q}(x,\beta(x,a),A)$ when $u= (\lambda',\,\beta)$ ranges in the set of all possible control laws. \end{remark}
\subsection{BSDEs and the synthesis of the optimal control} We next proceed to solve the optimal control problem formulated above. A basic role is played by the BSDE: for every fixed $t \in [0,\,T]$, $\Bbb{P}^{x,a}$-a.s. \begin{equation}\label{BSDE_controllo} Y^{x,a}_{s,t} + \int_{s}^{T-t}\int_{K}Z^{x,a}_{\sigma,t}(y)q(d\sigma\,dy) =
g(X_{T-t},a_{T-t}) + \int_{s}^{T-t}f\Big(t+\sigma,X_{\sigma},a_{\sigma},Z^{x,a}_{\sigma,t}(\cdot)\Big)d\sigma, \quad \forall s\in [0,\,T-t], \end{equation} with terminal condition given by the terminal cost $g$ and generator given by the Hamiltonian function $f$ defined for every $s \in [0,\,T],\, (x,a) \in S,\, z \in L^{2}(K,\mathcal{K},\,\lambda(x,a)\,\bar{q}(x,a,dy))$, as \begin{equation}\label{hamilton_function} f(s,x,a,z(\cdot))=
\inf_{u \in U}\Big\{\,l(s,x,a,u) + \int_{K}z(y)(r(s,x,a,y,u)-1)\lambda(x,a)\bar{q}(x,a,dy)\,\Big\}. \end{equation} In \eqref{BSDE_controllo} the superscript $(x,a)$ denotes the starting point at time $s=0$ of the process $(X_s,\,a_s)_{s \geqslant 0}$, while the dependence of $Y$ and $Z$ on the parameter $t$ is related to the temporal horizon of the considered optimal control problem. For every $t \in [0\,\,T]$, we look for a process $Y^{x,a}_{s,t}(\omega)$ adapted and càdlàg and a process $Z^{x,a}_{s,t}(\omega,y)$ $\mathcal{P}\otimes \mathcal{K}$-measurable satisfying the integrability conditions $$ \sperxa{\int_{0}^{T-t}\abs{Y^{x,a}_{s,t}}^{2}ds}<\infty,\qquad \sperxa{\int_{0}^{T-t}\int_{K}\abs{Z^{x,a}_{s,t}(y)}^{2}\lambda(X_{s},a_{s})\,\bar{q}(X_{s},a_{s},dy)\,ds}<\infty. $$ One can verify that, under Hypothesis \ref{hyp_control} on the optimal control problem, all the assumptions of Hypothesis \ref{H_1} hold true for the generator $f$ and the terminal condition $g$ in the BSDE \eqref{BSDE_controllo}. The only non trivial verification is the Lipschitz condition \eqref{f_inequality}, which follows from the boundedness assumption \eqref{hyp: r_bound}. Indeed, for every $s\in [0,\,T]$, $(x,a)\in S$, $z,\,z'\in L^{2}(K, \mathcal{K},\lambda(x,a)\,\bar{q}(x,a,dy))$, \begin{align*} &\int_{K}z(y)(r(s,x,a,y,u))-1)\,\lambda(x,a) \, \bar{q}(x,a,dy)\\ & \leqslant \int_{K}\abs{z(y)-z'(y)}\,(r(s,x,a,y,u)-1)\,\lambda(x,a) \, \bar{q}(x,a,dy) + \int_{K}z'(y)(r(s,x,a,y,u)-1)\,\lambda(x,a) \, \bar{q}(x,a,dy)\\ & \leqslant (C_{r} + 1)\,(\lambda(x,a) \, \bar{q}(x,a,K))^{1/2}\,\cdot\left(\int_{K}\abs{z(y)-z'(y)}^2\,\lambda(x,a) \, \bar{q}(x,a,dy)\right)^{1/2}\\ & + \int_{K}z'(y)(r(s,x,a,y,u)-1)\,\lambda(x,a) \, \bar{q}(x,a,dy), \end{align*} so that, adding $l(s,x,a,u)$ on both sides and taking the infimum over $u \in U$, it follows that \begin{equation} f(s,x,a,z)\leqslant L\left(\int_{K}\abs{z(y)-z'(y)}^2\lambda(x,a) \, \bar{q}(x,a,dy)\right)^{1/2}
+ f(s,x,a,z'), \end{equation} where $L:= (C_{r} + 1)\sup_{(x,a) \in S}\,(\lambda(x,a)\,\bar{q}(x,a,K))^{1/2}$; exchanging $z$ and $z'$ roles we obtain \eqref{f_inequality}.
Then by Theorem \ref{thm: uniqueness_existence_BSDE}, for every fixed $t \in [0,\,T]$, for every $(x,a)\in S$, there exists a unique solution of \eqref{BSDE_controllo} $(Y^{x,a}_{s,t},Z^{x,a}_{s,t})_{s \in [0,\,T-t]}$, and $Y_{0,t}^{x,a}$ is deterministic. Moreover, we have the following result: \begin{proposition} Assume that Hypotheses \ref{hyp_control} hold. Then, for every $t \in [0,\,T]$, $(x,a) \in S$, and for every $u(\cdot) \in \mathcal{A}$, \begin{displaymath} Y_{0,t}^{x,a} \leqslant J(t,x,a,u(\cdot)). \end{displaymath} \end{proposition}
\proof We consider the BSDE \eqref{BSDE_controllo} at time $s=0$ and we apply the expected value $\Bbb{E}_{u,t}^{x,a}$ associated to the controlled probability $\Bbb{P}_{u,t}^{x,a}$. Since the $\Bbb{P}_{u,t}^{x,a}$-compensator of $p(ds dy)$ is\\ $r(t+s,X_{s-},a_{s-},y,u_{s})\,\lambda(X_{s-},a_{s-})\,\bar{q}(X_{s-},a_{s-},dy)\,ds$, we have that \begin{align*} \sperxaut{\int_{0}^{T-t}\int_{K}Z_{s,t}^{x,a}(y)\,q(ds dy)} &= \sperxaut{\int_{0}^{T-t}\int_{K}Z_{s,t}^{x,a}(y)\,p(ds dy)} \\ &\quad -\sperxaut{\int_{0}^{T-t}\int_{K}Z_{s,t}^{x,a}(y)\,\lambda(X_{s},a_{s})\,\bar{q}(X_{s},a_{s},dy)\,ds}\\ & = \sperxaut{\int_{0}^{T-t}\!\!\int_{K}Z_{s,t}^{x,a}(y)\,[ r(t+s,X_{s},a_{s},y,u_{s})-1 ]\, \lambda(X_{s},a_{s})\,\bar{q}(X_{s},a_{s},dy)\,ds}\!. \end{align*} Then \begin{align*} Y_{0,t}^{x,a} &= \sperxaut{g(X_{T-t},a_{T-t})} + \sperxaut{\int_{0}^{T-t}f(t+s,X_{s},a_{s},Z_{s,t}^{x,a}(\cdot))\,ds}\\ & - \sperxaut{\int_{0}^{T-t}\int_{K}Z_{s,t}^{x,a}(y)\,[r(t+s,X_{s},a_{s},y,u_{s})-1]\, \lambda(X_{s},a_{s})\,\bar{q}(X_{s},a_{s},dy)\,ds}. \end{align*} Adding and subtracting $\sperxaut{\int_{0}^{T-t}l(t+s, X_{s},a_{s},u_{s})\,ds}$ on the right side we obtain the following relation: \begin{align}\label{BSDE_fundam_rel} &Y_{0,t}^{x,a} = J(t,x,a, u(\cdot))+ \sperxaut{\int_{0}^{T-t} \left[f(t+s,X_{s},a_{s},Z_{s,t}^{x,a}(\cdot)) -l(t+s, X_{s},a_{s},u_{s})\right]\,ds} \nonumber\\ &\qquad -\sperxaut{\int_{0}^{T-t}\int_{K} Z_{s,t}^{x,a}(\cdot)\,[r(t+s,X_{s},a_{s},y,u_{s})-1]\, \lambda(X_{s},a_{s})\,\bar{q}(X_{s},a_{s},dy)\,ds}. \end{align} By the definition of the Hamiltonian function $f$, the two last terms are non positive, and it follows that \[ Y_{0,t}^{x,a} \leqslant J(t,x,a, u(\cdot)),\qquad \forall u(\cdot)\in \mathcal{A}. \] \endproof \noindent We define the following, possibly empty, set: \begin{align}\label{Gamma} &\Gamma(s,x,a,z(\cdot))= \{ \,u \in U: f(s,x,a,z(\cdot))=l(s,x,a,u) + \int_{K}z(y)\,(r(s,x,a,y,u)-1)\,\lambda(x,a)\,\bar{q}(x,a,dy);\nonumber\\ & \qquad \qquad \qquad \qquad \qquad \qquad \qquad \,s \in [0,\,T],\,(x,a)\in S,\,z \in L^{2}(K,\mathcal{K},\lambda(x,a)\,\bar{q}(x,a,dy))\,\}. \end{align} In order to prove the existence of an optimal control we need to require that the infimum in the definition of $f$ is achieved. Namely we assume that \begin{hypothesis}\label{hyp_assumed_min} The sets $\Gamma$ introduced in \eqref{Gamma} are non empty; moreover, for every fixed $t \in [0,\,T]$ and $(x,a)\in S$, one can find an $\mathcal{F}$-predictable process $u^{\ast\,t,x,a}(\cdot)$ with values in $U$ satisfying \begin{equation}\label{u_in_Gamma} u^{\ast\,t,x,a}_{s} \in \Gamma(t+s, X_{s-}, a_{s-},Z^{x,a}_{s,t}(\cdot)), \qquad \Bbb{P}^{x,a}-\text{a.s}. \,\,\forall s \in [0,\,T-t]. \end{equation} \end{hypothesis} \begin{theorem}\label{control_sol_BSDEs} Under Hypothesis \ref{hyp_control} and \ref{hyp_assumed_min} for every fixed $t \in [0,\,T]$ and $(x,a) \in S$, $u^{\ast\,t,x,a}(\cdot) \in \mathcal{A}$ is an optimal control for the control problem starting from $(x,a)$ at time $s=0$ with terminal value $s=T-t$. Moreover, $Y_{0,t}^{x,a}$ coincides with the value function, i.e. $Y_{0,t}^{x,a}=J(t,x,a,u^{*\,t,x,a}(\cdot))$. \end{theorem} \proof It follows immediately from the relation \eqref{BSDE_fundam_rel} and from the definition of the Hamiltonian function $f$. \endproof We recall that general conditions can be formulated for the existence of a process $u^{\ast\,t,x,a}(\cdot)$ satisfying \eqref{u_in_Gamma}, hence of an optimal control; this is done by means of an appropriate selection theorem, see e.g. Proposition 5.9 in \cite{CoFu-m}.
We end this section with an example where the BSDE \eqref{BSDE_controllo} can be explicitly solved and a closed form solution of an optimal control problem can be found. \begin{example} We consider a fixed time interval $[0,\,T]$ and a state space consisting of three states: $K = \{x_1,x_2,x_3,x_4\}$. We introduce $(T_n,\xi_n)_{n \geqslant 0}$ setting $(T_0,\xi_0)=(0,x_1)$, $(T_n,\xi_n)=( + \infty, x_1)$ if $n \geqslant 3$ and on $(T_1,\xi_1)$ and $(T_2,\xi_2)$ we make the following assumptions: $\xi_1$ takes values $x_2$ with probability $1$, $\xi_2$ takes values $x_3,x_4$ with probability $1/2$. This means that the system starts at time zero in a given state $x_1$, jumps into state $x_2$ with probability $1$ at the random time $T_1$ and into state $x_3$ or $x_4$ with equal probability at the random time $T_2$. It has no jumps after. We take $U=[0,\,2]$ and define the function $r$ specifying the effects of the control process as $r(x_1,u)=r(x_2,u)=1$, $r(x_3,u)=u$, $r(x_4,u)=2-u$, $u \in U$.
Moreover, the final cost $g$ assumes the value $1$ in $(x,a)=(x_4,T-T_2)$ and zero otherwise, and the running cost is defined as $l(s,x,a,u)= \frac{\alpha\,u}{2} \,\lambda(x,a)$, where $\alpha >0$ is a fixed parameter. The BSDE we want to solve takes the form: \begin{equation}\label{example} Y_s + \int_{s}^{T}\int_K Z_{\sigma}(y) \,p(d\sigma\,dy) = g(X_T,\,a_T)+\int_{s}^{T}\inf_{u \in [0,\,2]}\left \{ \frac{\alpha \,u }{2} + \int_K Z_{\sigma}(y)\,r(y,u)\,\bar{q}(X_{\sigma},a_{\sigma},dy)\right\} \lambda(X_{\sigma},a_{\sigma})d\sigma \end{equation} that can be written as \begin{align*} Y_s + \sum_{n \geqslant 1} Z_{T_n}(X_{T_n}) \,\ensuremath{\mathonebb{1}}_{\{s < T_n \leqslant T\}} & = g(X_T,\,a_T)+\int_{s}^{T}\inf_{u \in [0,\,2]}\left \{ \frac{\alpha \,u }{2}+ Z_{\sigma}(x_2) \right\}\lambda(x_1,a+\sigma) \ensuremath{\mathonebb{1}}_{\{0 \leqslant \sigma <T_1 \wedge T\}} \,d\sigma\\
& + \int_{s}^{T}\inf_{u \in [0,\,2]}\left \{ \frac{\alpha \,u }{2}+ Z_{\sigma}(x_3)\frac{u}{2} + Z_{\sigma}(x_4)(1-\frac{u}{2}) \right\}\lambda(x_2,\sigma-T_1) \ensuremath{\mathonebb{1}}_{\{T_1 \leqslant\sigma <T_2 \wedge T\}}\,d\sigma. \end{align*} It is known by \cite{CFJ} that BSDEs of this type admit the following explicit solution $(Y_s,Z_s(\cdot))_{s\in[0,\,T]}$: \begin{eqnarray*} Y_s &=& y^0(s)\ensuremath{\mathonebb{1}}_{\{ s <T_1\}}+y^1(s,T_1,\xi_1)\,\ensuremath{\mathonebb{1}}_{\{T_1 \leqslant s <T_2\}}+y^2(s,T_2,\xi_2,T_1,\xi_1)\,\ensuremath{\mathonebb{1}}_{\{ T_2 \leqslant s\}}\\ Z_s(y) &= & z^0(s,y)\,\ensuremath{\mathonebb{1}}_{\{ s \leqslant T_1\}}+z^1(s,y,T_1,\xi_1)\,\ensuremath{\mathonebb{1}}_{\{T_1 < s \leqslant T_2\}},\quad y \in K. \end{eqnarray*} To deduce $y^0$ and $y^1$ we reduce the BSDE to a system of two ordinary differential equation. To this end, it suffices to consider the following cases: \begin{itemize} \item{$\omega \in \Omega \,\,\text{such that}\,\, T < T_1(\omega) < T_2(\omega)$:} \eqref{example} reduces to \begin{align}\label{y0} y^0(s) & = \int_{s}^{T}\inf_{u \in [0,\,2]}\left \{ \frac{\alpha \,u }{2}+ z^0(\sigma,x_2) \right\}\lambda(x_1,a+\sigma) \,d\sigma
= \int_{s}^{T} z^0(\sigma,x_2)\, \lambda(x_1,a+\sigma) \,d\sigma\nonumber\\ & = \int_{s}^{T}(y^1(\sigma,\sigma,x_2)-y^0(\sigma))\,\lambda(x_1,a+\sigma) \,d\sigma; \end{align} \item{$\omega \in \Omega \,\,\text{such that}\,\, T_1(\omega) < T < T_2(\omega)$, \,$s > T_1$:} \eqref{example} reduces to \begin{eqnarray}\label{y1} y^1(s, T_1,\xi_1) & = & \int_{s}^{T}\inf_{u \in [0,\,2]}\left \{ \frac{\alpha \,u }{2}+ z^1(\sigma,x_3,T_1,\xi_1)\frac{u}{2} + z^1(\sigma,x_4,T_1,\xi_1)(1-\frac{u}{2}) \right\}\lambda(\xi_1,\sigma-T_1) \,d\sigma \nonumber\\ & = & \int_{s}^{T}[z^1(\sigma,x_4,T_1,\xi_1)\wedge (\alpha + z^1(\sigma,x_3,T_1,\xi_1))]\,\lambda(\xi_1,\sigma-T_1) \,d\sigma\nonumber\\ & = & \int_{s}^{T}[(1\wedge \alpha)-y^1(\sigma,T_1,\xi_1)]\,\lambda(\xi_1,\sigma-T_1)\, d\sigma. \end{eqnarray} \end{itemize} Solving \eqref{y0} and \eqref{y1} we obtain \begin{align*} &y^0(s) = (1 \wedge \alpha)\left( 1- e^{-\int_s^T\lambda(x_1,a+\sigma)\,d\sigma} \right) - (1 \wedge \alpha)\,e^{- \int_s^T\lambda(x_1,a+\sigma)\,d\sigma}\int_s^T \lambda(x_1,a+\sigma)\, e^{\int_{\sigma}^T\lambda(x_1,a+z)\,dz } e^{-\int_{\sigma}^{T}\lambda(x_2,z - \sigma)\,dz}\,d\sigma\}, \\ &y^1(s,T_1, \xi_1) = (1 \wedge \alpha)\left(1 - e^{-\int_{s}^{T}\lambda(\xi_1,\sigma - T_1)\,d\sigma} \right); \end{align*} moreover, \begin{align*} & y^2(s,T_2, \xi_2, T_1,\xi_1)=\ensuremath{\mathonebb{1}}_{\{\xi_2 = x_4\}},\\ &z^0(s,x_1)=z^0(s,x_3)=z^0(s,x_4)=0,\,\,\, \quad \quad z^0(s,x_2)=y^1(s,s,x_2)-y^0(s),\\ & z^1(s,x_1,T_1,\xi_1)= z^1(s,x_2,T_1,\xi_1)=0, \quad \quad z^1(s,x_3,T_1,\xi_1) = (1 \wedge \alpha)\left(e^{-\int_{s}^{T}\lambda(\xi_1,\sigma-T_1)\,d\sigma}-1 \right),\\ & z^1(s,x_4,T_1,\xi_1)=1+z^1(s,x_3,T_1,\xi_1), \end{align*} where $z^0$ and $z^1$ are obtained respectively from $y^2$, $y^1$ and $y^1$, $y^0$ by subtraction.\\ \noindent The optimal cost is then given by $Y_0= y^0(0)$. The optimal control is obtained during the computation of the Hamiltonian function: it is the process $u_s = 2\,\ensuremath{\mathonebb{1}} _{(T_1, T_2]}(s)$ if $\alpha \leqslant 1$, and the process $u_s=0$ if $\alpha \geqslant 1$ (both are optimal if $\alpha = 1$). \end{example}
\section{Nonlinear variant of Kolmogorov equation}\label{section semi_markov_Kolmogorov_equation} Throughout this section we still assume that a semi-Markov process $X$ is given. It is constructed as in Section \ref{subsection_construction_SMP} by the rate function $\lambda$ and the measure $\bar{q}$ on $K$, and $(X,a)$ is the associated time-homogeneous Markov process. We assume that $\lambda$ and $\bar{q}$ satisfy Hypothesis \ref{hp_dati}.
It is our purpose to present here some nonlinear variants of the classical backward Kolmogorov equation associated to the Markov process $(X,a)$ and to show that their solution can be represented probabilistically by means of an appropriate BSDE of the type considered above.
We will suppose that two functions $f$ and $g$ are given, satisfying Hypothesis \ref{H_1}, and that moreover $g$ verifies, for every fixed $t \in [0,\,T]$, \begin{equation}\label{g_integrability} \sperxa{\abs{g(X_{T-t},a_{T-t})}^2}< \infty. \end{equation} We define the operator \begin{equation}\label{L_operator} \mathcal{L}\psi(x,a):= \int_{K}[\psi(y,0)-\psi(x,a)] \,\lambda(x,a)\,\bar{q}(x,a,dy), \qquad (x,a) \in S, \end{equation} for every measurable function $\psi : S\rightarrow \mathbb R$ for which the integral is well defined.\\ The equation \begin{align}\label{Kolmogorov_int} &v(t,x,a)= g(x,a+T-t) + \int_{t}^{T}\mathcal{L}v(s,x,a+s-t)\,ds \\ & + \int_{t}^{T} f(s,x,a+s-t,v(s,x,a+s-t),v(s,\cdot,0)-v(s,x,a+s-t))\,ds, \quad t \in [0,\,T],\,\,(x,a)\in S, \nonumber \end{align} with unknown function $v:[0,\,T]\times S \rightarrow \mathbb R$ will be called the nonlinear Kolmogorov equation.
Equivalently, one requires that for every $x \in K$ and for all constant $c \in [-T,\,+ \infty)$, \begin{eqnarray} \begin{array}{ll}\label{absolute_continuity} t \mapsto v(t,x,t+c) \,\,\text{is absolutely continuous on $[0,T]$,} \end{array} \end{eqnarray} and \begin{align}\label{Kolmogorov_diff} \left\{ \begin{array}{ll} Dv(t,x,a) +\mathcal{L}v(t,x,a) + f(t,x,a,v(t,x,a),v(t,\cdot,0)-v(t,x,a))=0 \\ v(T,x,a)= g(x,a), \end{array} \right. \end{align} where $D$ denotes the formal directional derivative operator \begin{equation}\label{directional_der} (Dv)(t,x,a):= \lim_{h \downarrow 0}\frac{v(t+h,x,a+h)-v(t,x,a)}{h}. \end{equation} In other words, the presence of the directional derivative operator \eqref{directional_der} allows us to understand the nonlinear Kolmogorov equation \eqref{Kolmogorov_diff} in a classical sense. In particular, the first equality in \eqref{Kolmogorov_diff} is understood to hold almost everywhere on $[0,\,T]$ outside of a $dt$-null set of points which can depend on $(x,a)$.
Under appropriate boundedness assumptions we have the following result: \begin{lemma}\label{lem_existence_uniqueness_Kolmogorov_boundedness} Suppose that $f$ and $g$ verify Hypothesis \ref{H_1} and that \eqref{g_integrability} holds; suppose, in addition, that \begin{equation}\label{bound_f_g} \sup_{t \in [0,\,T],\,(x,a) \in S}\Big( \abs{g(x,a)} + \abs{f(t,x,a,0,0)} \Big) < \infty. \end{equation} Then the nonlinear Kolmogorov equation \eqref{Kolmogorov_int} has a unique solution $v$ in the class of measurable bounded functions. \end{lemma}
\proof The result follows as usual from a fixed-point argument, that we only sketch. Let us define a map $\Gamma$ setting $v=\Gamma(w)$ where \begin{align*}\label{eq_v_w} & v(t,x,a) = g(x,a+T-t) + \int_{t}^{T}\mathcal{L}w(s,x,a+s-t)\,ds \\ &\qquad \qquad + \int_{t}^{T} f(s,x,a+s-t,w(s,x,a+s-t),w(s,\cdot,0)-w(s,x,a+s-t))\,ds. \nonumber \end{align*}
Using the Lipshitz character of $f$ and Hypothesis \ref{hp_dati}-ii), one can show that, for some $\beta >0$ sufficiently large, the above map is a contraction in the space of bounded measurable real functions on $[0,\,T]\times S$ endowed with the supremum norm: \begin{displaymath}
||v||_{\ast}:= \sup_{0\leqslant t\leqslant T}\sup_{(x,a)\in S}e^{-\beta(T-t)}\abs{v(t,x,a)}. \end{displaymath} The unique fixed point of $\Gamma$ gives the required solution. \endproof Our goal is now to remove the boundedness assumption \eqref{bound_f_g}. To this end we need to define a formula of It$\hat{\mbox{o}}$ type for the composition of the process $(X_s,\,a_s)_{s \geqslant 0}$ with functions $v$ smooth enough defined on $[0,\,T]\times S$. Taking into account the particular form of \eqref{Kolmogorov_int}, and the fact that the second component of the process $(X_s,\,a_s)_{s \geqslant 0}$ is linear in $s$, the idea is to use in this formula the directional derivative operator $D$ given by \eqref{directional_der}. \begin{lemma}[A formula of It$\hat{\mbox{o}}$ type]\label{Ito formula} Let consider functions $v: [0,\,T]\times S \rightarrow \mathbb R$ such that \begin{itemize} \item[(i)] $\forall\, x \in K$, $\forall\, c \in [-T,\,+\infty)$, the map $t \mapsto v(t,x,t+c)$ is absolutely continuous on $[0,\,T]$, with directional derivative $D$ given by \eqref{directional_der}; \item[(ii)] for fixed $t \in [0,\,T]$, $\{ v(t+s,y,0)- v(t+s,X_{s-},a_{s-}),\, s \in [0,\,T-t],\ y \in K\}$ belongs to $\mathcal{L}^{1}_{loc}(p)$. \end{itemize} Then $\mathbb P^{x,a}$-a.s., for every $t \in [0,\,T]$, \begin{align}\label{Ito's_formula} v(T,X_{T-t},a_{T-t}) - v(t,x,a) &=\int_{0}^{T-t}Dv(t+s,X_{s},a_{s})\,ds + \int_{0}^{T-t}\mathcal{L}v(t+s,X_{s},a_{s})\,ds \nonumber\\ &+ \int_{0}^{T-t}\int_{K}\left(v(t+s,y,0) - v(t+s,X_{s-},a_{s-})\right)\,q(ds,dy), \end{align} where the stochastic integral is a local martingale. \end{lemma} \proof We proceed by reasoning as in the proof of Theorem 26.14 in \cite{Da-bo}. We consider a function $v: [0,\,T]\times S \rightarrow \mathbb R$ satisfying (i) and (ii), and we denote by $N_t$ the number of jumps in the interval $[0,\,t]$: \[ N_t = \sum_{n \geqslant 1}\ensuremath{\mathonebb{1}}_{\{T_n \leqslant t\}}. \] We have \begin{align*} v(T,X_T,a_T)-v(0,x,a) &= v(T,X_T,a_T) - v(T_{N_T},X_{T_{N_T}}, a_{T_{N_T}})+\, \sum_{n=2}^{N_T} \left\{v(T_n,X_{T_n}, a_{T_n}) - v(T_{n-1},X_{T_{n-1}}, a_{T_{n-1}})\right\}\nonumber\\ & +\, v(T_1,X_{T_1},a_{T_1}) - v(0,x,a). \end{align*} Noticing that $X_{T_{n-}}=X_{T_{n-1}}$ for all $n \in [1,\,N_T]$, $X_{T}= X_{T_{N_T}}$, and that $a_{T_{n}}=0$ for all $n \in [1,\,N_T]$, $a_{T_{1-}}=a+T_1$, and $a_{T_{n-}}=T_n - T_{n-1}$ for all $n \in [2,\,N_T]$, we have $$ v(T,X_T,a_T)-v(0,x,a) = I + II + III, $$ where \begin{align*} I &= (v(T_1,X_{T_1},0) - v(T_1,X_{T_1-},a_{T_1-})) + (v(T_1,x,a + T_1) - v(0,x,a))=: I' + I'',\\ II &= \sum_{n = 2}^{N_T} (v(T_{n},X_{T_{n}},0) - v(T_{n},X_{T_{n}-},a_{T_{n}-}) +
+ \sum_{n = 2}^{N_T}(v(T_{n},X_{T_{n-1}},T_{n}- T_{n-1}) - v(T_{n-1},X_{T_{n-1}},0)))=: II'+II'',\\ III &= v(T,X_{T},T - T_N) - v(T_N,X_{T_N},0). \end{align*} Let $H$ denote the $\mathcal{P}\otimes \mathcal{K}$-measurable process \begin{displaymath} H_s(y)= v(s,y,0)-v(s,X_{s-},a_{s-}), \end{displaymath} with the convention $X_{0-}=X_0$, $a_{0-}=a_0$. We have \begin{align*} I'+II' &= \sum_{n \geqslant 1: T_n \leqslant T} (v(T_n,X_{T_n},0)-v(T_n,X_{T_{n-}},a_{T_{n-}})) = \sum_{n \geqslant 1: T_n \leqslant T} H_{T_n}(X_{T_N})= \int_{0}^{T}\int_{K}H_s(y)\,p(ds,dy). \end{align*} On the other hand, since $v$ satisfies (i) and recalling the definition \ref{directional_der} of the directional derivative operator $D$, \begin{align*} &I''+II''+ III = \int_0^{T_1}\lim_{h \rightarrow 0}\frac{v(0 + h s,x,a + h s) - v(0,x,a)}{h}\,ds\\ &+\sum_{n \geqslant 2: T_n \leqslant T}\int_{T_{n-1}}^{T_{n}}\lim_{h \rightarrow 0}\frac{v(T_{n-1} + h (s-T_{n-1}),X_{T_{n-1}},a_{T_{n-1}} + h (s- T_{n-1})) - v(T_{n-1},X_{T_{n-1}},a_{T_{n-1}})}{h}\,ds\\ &+\int_{T_{N_T}}^{T}\lim_{h \rightarrow 0}\frac{v(T_{N_T} + h (s-T_{N_T}),X_{T_{N_T}},a_{T_{N_T}} + h (s- T_{N_T})) - v(T_{N_T},X_{T_{N_T}},a_{T_{N_T}})}{h}\,ds\\ & = \int_0^{T}Dv(s,X_s,a_s)\,ds. \end{align*} Then $\mathbb P^{x,a}$-a.s., \begin{align*} &v(T,X_{T},a_{T}) - v(0,x,a) = \int_{0}^{T} D v(s, X_s,a_s)\, ds + \int_{0}^{T}\int_{K}\left(v(s,y,0) - v(s,X_{s-},a_{s-})\right)\,p(ds,dy)\\ &\qquad = \int_{0}^{T} D v(s, X_s,a_s)\, ds + \int_{0}^{T}\mathcal{L}v(s,X_{s},a_{s})\,ds + \int_{0}^{T}\int_{K}\left(v(s,y,0) - v(s,X_{s-},a_{s-})\right)\,q(ds,dy), \end{align*} where the second equality is obtained using the identity $q(dt \,dy)= p(dt \,dy)-\lambda(X_{t-},a_{t-})\,\bar{q}(X_{t-},a_{t-},dy)\,dt$ together with the definition \eqref{L_operator} of the operator $\mathcal{L}$.
Finally, applying a shift in time, i.e. considering for every $t \in [0,\,T]$ the differential of the process $v(s+t,X_{s-},a_{s-})$ with respect to $s \in [0,\,T-t]$, the previous formula becomes: $\mathbb P^{x,a}$-a.s., for every $t \in [0,\,T]$, \begin{align*} v(T-t,X_{T},a_{T}) - v(t,x,a) &= \int_{0}^{T-t} D v(s+t, X_s,a_s)\, ds + \int_{0}^{T-t}\mathcal{L}v(s+t,X_{s},a_{s})\,ds \\ & + \int_{0}^{T-t}\int_{K}\left(v(s+t,y,0) - v(s+t,X_{s-},a_{s-})\right)\,q(ds,dy), \end{align*} where the stochastic integral is a local martingale thanks to condition (ii). \endproof We will call \eqref{Ito's_formula} the It$\hat{\mbox{o}}$ formula for $v(t+s, \cdot,\cdot) \circ {(X_{s},a_{s})}_{s \in [0,\,T-t]}$. In differential notation: \begin{align*} dv(t+s,X_{s-},a_{s-}) &= Dv(t+s,X_{s-},a_{s-})\,ds \,+\, \mathcal{L}v(t+s,X_{s-},a_{s-})\,ds \\ &\quad + \int_{K}\left(v(t+s,y,0) - v(t+s,X_{s-},a_{s-})\right)\,q(ds,dy). \end{align*} \begin{remark} With respect to the classical It$\hat{\mbox{o}}$ formula, we underline that in \eqref{Ito's_formula} we have \begin{itemize} \item[-] the directional derivative operator $D$ instead of the usual time derivative; \item[-] the temporal translation in the first component of $v$, i.e. we consider the differential of the process \\$v(t+s,X_{s-},a_{s-})$ with respect to $s \in [0,\,T-t]$.
Indeed, the time-homogeneous Markov process ${(X_{s},a_{s})}_{s \geqslant 0}$ satisfies
\begin{equation*}
\mathbb P^{x,a}(X_{0}=x,\,a_{0}=a) = 1,
\end{equation*}
and the temporal translation in the first component allows us to consider $dv(t,X_t,a_t)$ for all $t \in [0,\,T]$. \end {itemize} \end{remark} We go back to consider the Kolmogorov equation \eqref{Kolmogorov_int} in a more general setting. More precisely, on the functions $f$, $g$ we will only ask that they satisfy Hypothesis \ref{H_1} for every $(x,a)\in S$ and that \eqref{g_integrability} holds. \begin{definition}\label{def_v_sol_Kolmogorov} We say that a measurable function $v: [0,\,T] \times S \rightarrow \mathbb R$ is a solution of the nonlinear Kolmogorov equation \eqref{Kolmogorov_int}, if, for every fixed $t \in [0,\,T]$, $(x,a) \in S$, \begin{itemize} \item[1.] $\sperxa{\int_{0}^{T-t}\int_{K}\abs{v(t+s,y,0)-v(t+s,X_{s},a_{s})}^{2}\lambda(X_{s},a_{s})\,\bar{q}(X_{s},a_{s},dy)\,ds}<\infty$; \item[2.] $\sperxa{\int_{0}^{T-t}\abs{v(t+s,X_{s},a_{s})}^{2}ds}<\infty$; \item[3.] \eqref{Kolmogorov_int} is satisfied. \end{itemize} \end{definition}
\begin{remark} Condition 1. is equivalent to the fact that $v(t+s,y,0) - v(t+s,X_{s-},a_{s-})$ belongs to $\mathcal{L}^{2}(p)$. Conditions 1. and 2. together are equivalent to the fact that the pair \\$\{ v(t+s,X_{s},a_{s}) , \, v(t+s,y,0) - v(t+s,X_{s-},a_{s-});\,s \in [0,\,T-t],\, y \in K \}$ belongs to the space $\Bbb{M}^{x,a}$; in particular they hold true for every measurable bounded function $v$. \end{remark}
\begin{remark} We need to verify the well-posedness of equation \eqref{Kolmogorov_int} for a function $v$ satisfying the condition 1. and 2. above. We start by noticing that, for every $(x,a)\in S$, $\Bbb{P}^{x,a}$-a.s., \begin{displaymath} \int_{0}^{T}\int_{K}\abs{v(s,y,0)-v(s,X_{s},a_{s})}^{2}\lambda(X_{s},a_{s})\,\bar{q}(X_{s},a_{s},dy)\,ds + \int_{0}^{T}\abs{v(s,X_{s},a_{s})}^{2}ds <\infty. \end{displaymath} By the law \eqref{jumpkerneldue} of the first jump it follows that the set $\{\omega \in \Omega:\, T_{1}(\omega) >T\}$ has positive $\Bbb{P}^{x,a}$ probability, and on this set we have $X_{s-}(\omega) = x$, $a_{s-}(\omega) = a+s$. Taking such an $\omega$ we get \begin{displaymath} \int_{0}^{T}\int_{K}\abs{v(s,y,0)-v(s,x,a+s)}^{2}\,\lambda(x,a+s)\,\bar{q}(x,a+s,dy)\,ds + \int_{0}^{T}\abs{v(s,x,a+s)}^{2}ds <\infty,\,\,\forall (x,a) \in S. \end{displaymath} Since $\sup_{(x,a)\in S}\lambda(x,a)\bar{q}(x,a,K) < \infty$ by assumption, H\"{o}lder's inequality implies that \begin{align*} \int_{0}^{T}\abs{\mathcal{L}(v(s,x,a+s))}\,ds &\leqslant \int_{0}^{T}\int_{K}\abs{v(s,y,0)-v(s,x,a+s)}\,\lambda(x,a+s)\,\bar{q}(x,a+s,dy)\,ds \\ &\leqslant c\left(\int_{0}^{T}\int_{K}\abs{v(s,y,0)-v(s,x,a+s)}^{2}\,\lambda(x,a+s)\,\bar{q}(x,a+s,dy)\,ds\right)^{1/2} < \infty \end{align*} for some constant $c$ and for all $(x,a)\in S$. Similarly, since $\sperxa{\int_{0}^{T}\abs{f(s,X_{s},a_{s},0,0)}^2 ds}< \infty$ and arguing again on the jump time $T_{1}$, we deduce that \begin{displaymath} \int_{0}^{T}\abs{f(s,x,a+s,0,0)}^2\, ds< \infty,\,\,\forall (x,a)\in S; \end{displaymath} finally, from the Lipschitz conditions on $f$ we can conclude that \begin{align*} &\int_{0}^{T}\abs{f(s,x,a+s,v(s,x,a+s),v(s,\cdot,0)-v(s,x,a+s))}\,ds \\ & \leqslant c_{1}\left(\int_{0}^{T}\abs{f(s,x,a+s,0,0)}^{2}ds\right)^{1/2} + c_{2}\left (\int_{0}^{T}\abs{v(s,x,a+s)}^{2}ds\right)^{1/2}\\ & + c_{3}\left(\int_{0}^{T}\int_{K}\abs{v(s,y,0)-v(s,x,a+s)}^{2}\,\lambda(x,a+s)\,\bar{q}(x,a+s,dy)\,ds\right)^{1/2} < \infty \end{align*} for some constants $c_{i}$, $i = 1,2,3$, and for all $(x,a) \in S$. Therefore, all terms occurring in equation \eqref{Kolmogorov_int} are well defined. \end{remark}
For every fixed $t \in [0,\,T]$ and $(x,a) \in S$, we consider now a BSDE of the form \begin{equation}\label{BSDE_txa} Y^{x,a}_{s,t} + \int_{s}^{T-t}\int_{K}Z^{x,a}_{r,t} (y)\,q(dr\,dy) = g(X_{T-t},a_{T-t}) + \int_{s}^{T-t}\,f\Big(t+r,X_{r-},a_{r-},Y^{x,a}_{r,t} ,Z^{x,a}_{r,t} (\cdot)\Big)\,dr,\,\,s \in [0,\,T-t]. \end{equation} Then there exists a unique solution $(Y^{x,a}_{s,t} ,Z^{x,a}_{s,t} (\cdot))_{s\in [0,\,T-t]}$, in the sense of Theorem \ref{thm: uniqueness_existence_BSDE}, and $Y^{x,a}_{0,t}$ is deterministic. We are ready to state the main result of this section. \begin{theorem}\label{thm_kolm} Suppose that $f$, $g$ satisfy Hypothesis \ref{H_1} for every $(x,a) \in S$ and that \eqref{g_integrability} holds. Then for every $t \in [0,\,T]$, the nonlinear Kolmogorov equation \eqref{Kolmogorov_int} has a unique solution $v(t,x,a)$ in the sense of Definition \ref{def_v_sol_Kolmogorov}.
Moreover, for every fixed $t \in [0,\,T]$, for every $(x,a) \in S$ and $s \in [0,\,T-t]$ we have \begin{align} Y^{x,a}_{s,t} &= v(t+s,X_{s-},a_{s-}),\label{ident_1}\\ Z^{x,a}_{s,t}(y) &= v(t+s,y,0)-v(t+s,X_{s-},a_{s-})\label{ident_2}, \end{align} so that in particular $v(t,x,a) = Y_{0,t}^{x,a}$. \end{theorem}
\begin{remark}\label{rem_interpret_identities} The equalities \eqref{ident_1} and \eqref{ident_2} are understood as follows. \begin{itemize} \item $\Bbb{P}^{x,a}$-a.s., equality \eqref{ident_1} holds for all $s \in [0,\,T-t]$.
The trajectories of $(X_s)_{s \in [0,\,T-t]}$ are piecewise constant and càdlàg, while the trajectories of $(a_s)_{s \in [0,\,T-t]}$ are piecewise linear in $s$ (with unitary slope) and càdlàg; moreover the processes $(X_s)_{s \in [0,\,T-t]}$ and $(a_s)_{s \in [0,\,T-t]}$ have the same jump times $(T_{n})_{n\geqslant 1}$.
Then the equality \eqref{ident_1} is equivalent to the condition
\begin{displaymath}
\sperxa{\int_{0}^{T-t}\abs{Y^{x,a}_{s,t}-v(t+s,X_{s},a_{s})}^{2}ds}=0.
\end{displaymath} \item The equality \eqref{ident_2} holds for all $(\omega,s,y)$ with respect to the measure \\ $\lambda(X_{s-}(\omega),a_{s-}(\omega))\,\bar{q}(X_{s-}(\omega),a_{s-}(\omega),dy)\,\Bbb{P}^{x,a}(d\omega)ds$, i.e.,
\begin{displaymath}
\sperxa{\int_{0}^{T-t}\int_{K}\abs{Z^{x,a}_{s,t}(y)-v(t+s,y,0)+v(t+s,X_{s},a_{s})}^{2}\lambda(X_{s},a_{s})\,\bar{q}(X_{s},a_{s},dy)\,ds}=0.
\end{displaymath} \end{itemize} \end{remark}
\proof \emph{Uniqueness.} Let $v$ be a solution of the nonlinear Kolmogorov equation \eqref{Kolmogorov_int}. It follows from equality \eqref{Kolmogorov_int} itself that for every $x \in K$ and every $\tau \in [-T ,\, + \infty)$,\,\,$t \mapsto v(t,x,t+\tau)$ is absolutely continuous on $[0,\,T]$. Indeed, applying in \eqref{Kolmogorov_int} the change of variable $\tau:= a-t$, we obtain $\forall t \in [0,\,T]$, $\forall \tau \in [-T ,\, + \infty)$, $$ v(t,x,t+\tau) = g(x,T+\tau) +\int_{t}^{T}\mathcal{L}v(s,x,s+\tau)\,ds+\int_{t}^{T}\, f(s,x,s+\tau,v(s,x,s+\tau),v(s,\cdot,0)-v(s,x,s+\tau))\,ds. \nonumber $$ Then, since by assumption the process $v(t+s,y,0)-v(t+s,X_{s-},a_{s-})$ belongs to $\mathcal{L}^2(p)$, we are in a position to apply the It$\hat{\mbox{o}}$ formula \eqref{Ito's_formula} to the process $v(t+s,X_{s-},a_{s-})$, $s \in [0,\,T-t]$. We get: $\Bbb{P}^{x,a}$-a.s., \begin{align*} v(t+s,X_{s-},a_{s-}) &= v(t,x,a) + \int_{0}^{s}Dv(t+r,X_{r},a_{r})\,dr + \int_{0}^{s}\mathcal{L}v(t+r,X_{r},a_{r})\,dr \nonumber\\ &+ \int_{0}^{s}\int_{K}\left(v(t+r,y,0) - v(t+r,X_{r},a_{r})\right)q(dr,dy),\qquad s \in [0,\,T-t]. \end{align*} We know that $v$ satisfies \eqref{Kolmogorov_diff}; moreover the process $X$ has piecewise constant trajectories, the process $a$ has linear trajectories in $s$, and they have the same time jumps. Then, $\mathbb P^{x,a}$-a.s., $$ Dv(t+s,X_{s-},a_{s-}) + \mathcal{L}v(t+s,X_{s-},a_{s-}) +f(t+s, X_{s-},a_{s-},v(t+s,X_{s-},a_{s-}),v(t+s,\cdot,0) - v(t+s,X_{s-},a_{s-})) = 0, $$ for almost $s \in [0,\,T-t]$. In particular, $\mathbb P^{x,a}$-a.s., \begin{align*} &v(t+s,X_{s-},a_{s-}) = v(t,x,a) + \int_{0}^{s}\int_{K}\left(v(t+r,y,0) - v(t+r,X_{r-},a_{r-})\right)q(dr,dy) \\ &\qquad - \int_{0}^{s}f(t+r, X_{r},a_{r},v(t+s,X_{s},a_{s}),v(t+r,y,0) - v(t+r,X_{r},a_{r}))\,dr, \qquad s \in [0,\,T-t]. \end{align*} Since $v(T,x,a)=g(x,a)$ for all $(x,a)\in S$, by simple computations we can prove that, $\forall s \in [0,\,T-t]$, \begin{align*} &v(t+s,X_{s-},a_{s-}) + \int_{s}^{T-t}\int_{K}\left(v(t+r,y,0) - v(t+r,X_{r-},a_{r-})\right)q(dr,dy)\\ & \qquad = g(X_{T-t},a_{T-t}) + \int_{s}^{T-t}\,f(t+r, X_{r},a_{r},v(t+r,X_{r},a_{r}),v(t+r,y,0) - v(t+r,X_{r},a_{r}))\,dr \nonumber. \end{align*} Since the pairs $(Y^{x,a}_{s,t},Z^{x,a}_{s,t}(\cdot))_{s \in [0,\,T-t]}$ and $(v(t+s,X_{s-},a_{s-}),v(t+s,y,0)-v(t+s,X_{s-},a_{s-}))_{s \in [0,\,T-t]}$ are both solutions to the same BSDE under $\mathbb P^{x,a}$, they coincide as members of the space $\Bbb{M}^{x,a}$. It follows that equalities \eqref{ident_1} and \eqref{ident_2} hold. In particular, $v(t,x,a)= Y_{0,t}^{x,a}$, and this yields the uniqueness of the solution.
\emph{Existence.} We proceed by an approximation argument, following the same lines of the proof of Theorem 4.4 in \cite{CoFu-m}. We recall that, by Theorem \ref{thm: uniqueness_existence_BSDE}, for every fixed $t \in [0,\,T]$, the BSDE \eqref{BSDE_txa} has a unique solution $(Y^{x,a}_{s,t},Z^{x,a}_{s,t}(\cdot))_{s \in [0,\,T-t]}$ for every $(x,a) \in S$; moreover, $Y_{0,t}^{x,a}$ is deterministic, i.e., there exists a real number, denoted by $v(t,x,a)$, such that $\mathbb P^{x,a}(Y_{0,t}^{x,a}= v(t,x,a))=1$. At this point, we set $f^{n} = (f \wedge n)\vee (-n)$ and $g^{n} = (g \wedge n)\vee (-n)$ as the truncations of $f$ and $g$ at level $n$. By Lemma \ref{lem_existence_uniqueness_Kolmogorov_boundedness}, for $t \in [0,\,T]$, $(x,a)\in S$, equation \begin{align}\label{Kolmogorov_truncated}
v^{n}(t,x,a) &= g^{n}(x,a+T-t) + \int_{t}^{T}\mathcal{L}v^{n}(s,x,a+s-t)\,ds \nonumber\\ & \quad + \int_{t}^{T}\, f^{n}(s,x,a+s-t,v^{n}(s,x,a+s-t),v^{n}(s,\cdot,0)-v^{n}(s,x,a+s-t))\,ds. \end{align} admits a unique bounded measurable solution $v^{n}$. In particular, the first part of the proof yield the following identifications: \begin{align*} v^{n}(t,x,a) &= Y^{x,a,n}_{0,t},\\ v^{n}(t+s,X_{s-},a_{s-}) &= Y^{x,a,n}_{s,t},\\ v^{n}(t+s,y,0) - v^{n}(t+s,X_{s-},a_{s-}) &= Z^{x,a,n}_{s,t}(y), \end{align*} in the sense of Remark \ref{rem_interpret_identities}, where $(Y_{s,t}^{x,a,n},Z_{s,t}^{x,a,n}(\cdot))_{s \in [0,\,T-t]}$ is the unique solution to the BSDE $$ Y^{x,a,n}_{s,t} + \int_{s}^{T-t}\int_{K}Z^{x,a,n}_{r,t}(y)\,q(dr\,dy)= g^{n}(X_{T-t},a_{T-t}) + \int_{s}^{T-t}\,f^{n}\left(t+r,X_{r},a_{r},Y^{x,a,n}_{r,t},Z^{x,a,n}_{r,t}(\cdot)\right)\,dr, $$ for all $s\in [0,\,T-t]$. Recalling \eqref{BSDE_txa} and applying Theorem \ref{thm: uniqueness_existence_BSDE}, we deduce that, for some constant $c$, \begin{align}\label{convergence}
&\sup_{s \in [0,\,T-t]}\mathbb E^{x,a}\left[|Y^{x,a}_{s,t}-Y^{x,a,n}_{s,t}|^2\right] +
\mathbb E^{x,a}\left[\int_{0}^{T-t}|Y^{x,a}_{s,t}-Y^{x,a,n}_{s,t}|^2ds\right] \nonumber \\
&\qquad + \mathbb E^{x,a}\left[\int_{0}^{T-t}\int_{K}|Z^{x,a}_{s,t}(y)-Z^{x,a,n}_{s,t}(y)|^2\lambda(X_{s},a_{s})\,\bar{q}(X_{s},a_{s},dy)\,ds\right]\nonumber\\ &
\leqslant c \mathbb E^{x,a}\left[|g(X_{T-t},a_{T-t})-g^{n}(X_{T-t},a_{T-t})|^2\right] \nonumber \\
&\qquad+c\mathbb E^{x,a}\left[\int_{0}^{T-t}|f(t+s,X_{s},a_{s},Y^{x,a}_{s,t},Z^{x,a}_{s,t}(\cdot))-f^{n}(t+s,X_{s},a_{s},Y^{x,a}_{s,t},Z^{x,a}_{s,t}(\cdot))|^2ds\right] \longrightarrow 0, \end{align} where the two final terms tend to zero by monotone convergence. In particular \eqref{convergence} yields \begin{displaymath}
|v(t,x,a)-v^{n}(t,x,a)|^{2} = |Y^{x,a}_{0,t}-Y^{x,a,n}_{0,t}|^2 \leqslant
\sup_{s \in [0,\,T-t]}\mathbb E^{x,a}\left[|Y^{x,a}_{s,t}-Y^{x,a,n}_{s,t}|^2\right]\longrightarrow 0, \end{displaymath} and therefore $v$ is a measurable function. At this point, applying the Fatou Lemma we get \begin{align*} &\sperxa{\int_{0}^{T-t}\abs{Y^{x,a}_{s,t}-v(t+s,X_{s},a_{s})}^{2}\,ds} \\ &+ \sperxa{\int_{0}^{T-t}\int_{K}\abs{Z^{x,a}_{s,t}(y)-v(t+s,y,0)+v(t+s,X_{s},a_{s})}^{2}\,\lambda(X_{s},a_{s})\,\bar{q}(X_{s},a_{s},dy)\,ds}\\ &\leqslant \liminf_{n \rightarrow \infty}\sperxa{\int_{0}^{T-t}\abs{Y^{x,a}_{s,t}-v^{n}(t+s,X_{s},a_{s})}^{2}\,ds}\\ &+ \liminf_{n \rightarrow \infty}\sperxa{\int_{0}^{T-t}\int_{K}\abs{Z^{x,a}_{s,t}(y)-v^n(t+s,y,0)+v^n(t+s,X_{s},a_{s})}^{2}\,\lambda(X_{s},a_{s})\,\bar{q}(X_{s},a_{s},dy)\,ds}\\ & = \liminf_{n \rightarrow \infty}\sperxa{\int_{0}^{T-t}\abs{Y^{x,a}_{s,t}-Y_{s,t}^{x,a,n}}^{2}\,ds} \\ &+ \liminf_{n \rightarrow \infty}\sperxa{\int_{0}^{T-t}\int_{K}\abs{Z^{x,a}_{s,t}(y)-Z_{s,t}^{x,a,n}(y)}^{2}\lambda(X_{s},a_{s})\,\bar{q}(X_{s},a_{s},dy)\,ds}= 0 \end{align*} by \eqref{convergence}. The above calculations show that \eqref{ident_1} and \eqref{ident_2} hold. Moreover, they imply that \begin{align*} &\sperxa{\int_{0}^{T-t}\abs{v(t+s,X_{s},a_{s})}^{2}ds} + \sperxa{\int_{0}^{T-t}\int_{K}\abs{v(t+s,y,0)-v(t+s,X_{s},a_{s})}^{2}\lambda(X_{s},a_{s})\,\bar{q}(X_{s},a_{s},dy)\,ds}\\ & = \sperxa{\int_{0}^{T-t}\abs{Y^{x,a}_{s,t}}^{2}ds}+ \sperxa{\int_{0}^{T-t}\int_{K}\abs{Z^{x,a}_{s,t}(y)}^{2}\lambda(X_{s-},a_{s-})\,\bar{q}(X_{s},a_{s},dy)\,ds}< \infty, \end{align*} that accords to requirement of Definition \ref{def_v_sol_Kolmogorov}.
It remains to show that $v$ satisfies \eqref{Kolmogorov_int}. This would follow from a passage to the limit in \eqref{Kolmogorov_truncated}, provided we show that \begin{equation}\label{convergence_Kolm_trunc_1} \int_{t}^{T}\mathcal{L}v^{n}(s,x,a+s-t)ds \rightarrow \int_{t}^{T}\mathcal{L}v(s,x,a+s-t)ds, \end{equation} and \begin{align}\label{convergence_Kolm_trunc_2} &\int_{t}^{T} f^{n}(s,x,a+s-t,v^{n}(s,x,a+s-t),v^{n}(s,\cdot,0)-v^{n}(s,x,a+s-t))\,ds \nonumber\\ & \qquad \qquad \rightarrow \int_{t}^{T} f(s,x,a+s-t,v(s,x,a+s-t),v(s,\cdot,0)-v(s,x,a+s-t))\,ds. \end{align} To prove \eqref{convergence_Kolm_trunc_1}, we observe that \begin{align*} &\mathbb E^{x,a}\abs{\int_{0}^{T-t}\mathcal{L}v(t+s,X_{s-},a_{s-})\,ds - \int_{0}^{T-t}\mathcal{L}v^{n}(t+s,X_{s-},a_{s-})\,ds} \\ &= \mathbb E^{x,a}\abs{\int_{0}^{T-t}\int_{K}(Z^{x,a}_{s,t}-Z_{s,t}^{x,a,n})\,\lambda(X_{s},a_{s})\,\bar{q}(X_{s},a_{s},dy)\,ds} \\ & \leqslant (T-t)^{1/2}\sup_{x,a}[\lambda(x,a)\,\bar{q}(x,a,K)]^{1/2}\left (\sperxa{\int_{0}^{T-t}\int_{K}\abs{Z^{x,a}_{s,t}-Z_{s,t}^{x,a,n}}\,\lambda(X_{s},a_{s})\,\bar{q}(X_{s},a_{s},dy)\,ds}\right)^{1/2}\rightarrow 0, \end{align*} by \eqref{convergence}. Then, for a subsequence (still denoted $v^n$) we get \[ \int_{0}^{T-t}\mathcal{L}v^n(t+s,X_{s},a_{s})\,ds \rightarrow \int_{0}^{T-t}\mathcal{L}v(t+s,X_{s},a_{s})\,ds, \quad \mathbb P^{x,a}\textup{-a.s}. \] Recalling the law \eqref{jumpkerneldue} of the first jump $T_1$, we see that the set $\{\omega \in \Omega:\, T_{1}(\omega) >T\}$ has positive $\Bbb{P}^{x,a}$ probability, and on this set we have $X_{s-}(\omega) = x$, $a_{s-}(\omega) = a+s$. Choosing such an $\omega$ we have \begin{displaymath} \int_{0}^{T-t}\mathcal{L}v^n(t+s,x,a+s)ds \rightarrow \int_{0}^{T-t}\mathcal{L}v(t+s,x,a+s)ds, \end{displaymath} i.e., by a translation of $t$ in the temporal line, \begin{displaymath} \int_{t}^{T}\mathcal{L}v^n(s,x,a+s-t)ds \rightarrow \int_{t}^{T}\mathcal{L}v(s,x,a+s-t)ds. \end{displaymath}
To show \eqref{convergence_Kolm_trunc_2}, we compute \begin{align*} &\mathbb E^{x,a}\abs{\int_{0}^{T-t}f(t+s,X_{s},a_{s},Y^{x,a}_{s,t},Z^{x,a}_{s,t})-f^n(t+s,X_{s},a_{s},Y_{s,t}^{x,a,n},Z_{s,t}^{x,a,n}))\,ds} \\ &\leqslant \sperxa{\int_{0}^{T-t}\abs{f(t+s,X_{s},a_{s},Y^{x,a}_{s,t},Z^{x,a}_{s,t})-f^n(t+s,X_{s},a_{s},Y^{x,a}_{s,t},Z^{x,a}_{s,t})}\,ds}\\ & + \sperxa{\int_{0}^{T-t}\abs{f^{n}(t+s,X_{s},a_{s},Y^{x,a}_{s,t},Z^{x,a}_{s,t})-f^n(t+s,X_{s},a_{s},Y_{s,t}^{x,a,n},Z_{s,t}^{x,a,n})}\,ds}. \end{align*} The first integral term in the right-hand side tends to zero by monotone convergence. At this point, we notice that $f^n$ is a truncation of $f$, and therefore it satisfies the Lipschitz condition \eqref{f_inequality} with the same constants $L$, $L'$, independent of $n$. This yields the following estimate for the second integral: \begin{align*} & L' \,\sperxa{\int_{0}^{T-t}\abs{Y^{x,a}_{s,t}-Y_{s,t}^{x,a,n}}ds}+ L\,\sperxa{\int_{0}^{T-t}\Big(\int_{K}\abs{Z^{x,a}_{s,t}(y)-Z_{s,t}^{x,a,n}(y)}^2\lambda(X_{s},a_{s})\,\bar{q}(X_{s},a_{s},dy)\Big)^{1/2}ds}\\ & \leqslant L' \, \left( \,(T-t)\,\sperxa{\int_{0}^{T-t}\abs{Y^{x,a}_{s,t}-Y_{s,t}^{x,a,n}}^2\,ds} \right)^{1/2}\\ & + L\,\left(\,(T-t)\,\sperxa{\int_{0}^{T-t}\int_{K}\abs{Z^{x,a}_{s,t}(y)-Z_{s,t}^{x,a,n}(y)}^2\lambda(X_{s},a_{s})\,\bar{q}(X_{s},a_{s},dy)\,ds}\,\right)^{1/2}, \end{align*} which tends to zero, again by \eqref{convergence}. Considering a subsequence (still denoted $v^{n}$) we get, \begin{eqnarray*} &&\int_{0}^{T-t}f^{n}(t+s,X_{s},a_{s},v^{n}(t+s,X_{s},a_{s}),v^{n}(t+s,y,0)-v^{n}(t+s,X_{s},a_{s}))\,ds \\ && \qquad \rightarrow \int_{0}^{T-t}f(t+s,X_{s},a_{s},v(t+s,X_{s},a_{s}),v(t+s,y,0)-v(t+s,X_{s},a_{s}))\,ds,\,\, \mathbb P^{x,a}\mbox{-a.s.} \end{eqnarray*} Choosing also in this case an $\omega$ in the set $\{\omega \in \Omega: \, T_{1}(\omega)> T\}$, we find \begin{eqnarray*} &&\int_{0}^{T-t}f^{n}(t+s,x,a+s,v^{n}(t+s,x,a+s),v^{n}(t+s,y,0)-v^{n}(t+s,x,a+s))\,ds \\ && \qquad \rightarrow \int_{0}^{T-t}f(t+s,x,a+s,v(t+s,x,a+s),v(t+s,y,0)-v(t+s,x,a+s))\,ds, \end{eqnarray*} and a change of temporal variable allows to prove that \eqref{Kolmogorov_int} holds, and to conclude the proof. \endproof
We finally introduce the Hamilton-Jacobi-Bellman (HJB) equation associated to the control problem considered in Section \ref{section semi_markov_HJB}: for every $t \in [0,\,T]$ and $(x,a)\in S$, \begin{equation}\label{HJB} v(t,x,a) = g(x,a+T-t) + \int_{t}^{T}\mathcal{L}v(s,x,a+s-t)\,ds + \int_{t}^{T} f(s,x,a+s-t,v(s,\cdot,0)-v(s,x,a+s-t))\,ds, \end{equation} where $\mathcal{L}$ denotes the operator introduced in \eqref{L_operator}, $f$ is the Hamiltonian function defined by \eqref{hamilton_function} and $g$ is the terminal cost. Since \eqref{HJB} is a nonlinear Kolmogorov equation of the form \eqref{Kolmogorov_int}, we can apply Theorem \ref{thm_kolm} and conclude that the value function and an optimal control law can be represented by means of the HJB solution $v(t,x,a)$. \begin{corollary} Let Hypotheses \ref{hyp_control} and \ref{hyp_assumed_min} hold. For every fixed $t \in [0,\,T]$, for every $(x,a) \in S$ and $s \in [0,\,T-t]$, there exists a unique solution $v$ to the HJB equation \eqref{HJB}, satisfying \begin{eqnarray*} v(t+s,X_{s-},a_{s-}) &=& Y^{x,a}_{s,t},\\ v(t+s,y,0)-v(t+s,X_{s-},a_{s-})&=& Z^{x,a}_{s,t}(y), \end{eqnarray*} where the above equalities are understood as explained in Remark \ref{rem_interpret_identities}.\\ In particular an optimal control is given by the formula $$ u^{*\,t,x,a}_{s} \in \Gamma(t+s, X_{s-}, a_{s-}, v(t+s,\cdot,0)-v(t+s,X_{s-},a_{s-})), $$ while the value function coincides with $v(t,x,a)$, i.e. $J(t,x,a,u^{*\,t,x,a}(\cdot)) = v(t,x,a)= Y_{0,t}^{x,a}$. \end{corollary}
\footnotesize
\end{document} |
\begin{document}
\begin{abstract} We prove an incidence theorem for points and planes in the projective space ${\mathbb P}^3$ over any field $\mathbb F$, whose characteristic $p\neq 2.$ An incidence is viewed as an intersection along a line of a pair of two-planes from two canonical rulings of the Klein quadric. The Klein quadric can be traversed by a generic hyperplane, yielding a line-line incidence problem in a three-quadric, the Klein image of a regular line complex. This hyperplane can be chosen so that at most two lines meet. Hence, one can apply an algebraic theorem of Guth and Katz, with a constraint involving $p$ if $p>0$.
This yields a bound on the number of incidences between $m$ points and $n$ planes in ${\mathbb P}^3$, with $m\geq n$ as $$O\left(m\sqrt{n}+ m k\right),$$ where $k$ is the maximum number of collinear planes, provided that $n=O(p^2)$ if $p>0$. Examples show that this bound cannot be improved without additional assumptions.
This gives one a vehicle to establish geometric incidence estimates when $p>0$. For a non-collinear point set $S\subseteq \mathbb{F}^2$ and a non-degenerate symmetric or skew-symmetric bilinear form $\omega$, the number of distinct values of $\omega$ on pairs of points of $S$ is $\Omega\left[\min\left(|S|^{\frac{2}{3}},p\right)\right]$. This is also the best known bound over ${\mathbb R}$, where it follows from the Szemer\'edi-Trotter theorem. Also, a set $S\subseteq \mathbb F^3$, not supported in a single semi-isotropic plane contains a point, from which $\Omega\left[\min\left(|S|^{\frac{1}{2}},p\right)\right]$ distinct distances to other points of $S$ are attained.
\end{abstract}
\title{On the number of incidences between points and planes in three dimensions}
\section{Introduction}
Let $\mathbb{F}$ be a field of characteristic $p$ and ${\mathbb P}^d$ the $d$-dimensional projective space over $\mathbb{F}$. Our methods do not work for $p=2$, but the results in view of constraints in terms of $p$ hold trivially for $p=O(1)$. As usual, we use the notation $|\cdot|$ for cardinalities of finite sets. The symbols $\ll$, $\gg,$ suppress absolute constants in inequalities, as well as respectively do $O$ and $\Omega$. Besides, $X=\Theta(Y)$ means that $X=O(Y)$ and $X=\Omega(Y)$. The symbols $C$ and $c$ stand for absolute constants, which may sometimes change from line to line, depending on the context. When we turn to sum-products, we use the standard notation $$A+B=\{a+b:\,a\in A,\,b\in B\}$$ for the sumset $A+B$ of $A, B\subseteq \mathbb F$, similarly for the product set $AB$.
The Szemer\'edi-Trotter theorem \cite{ST} on the number of incidences between lines and points in the Euclidean plane has many applications in combinatorics. The theorem is also valid over $\mathbb{C}$, this was first proved by T\'oth \cite{T}. In positive characteristic, however, no universal satisfactory for applications point-line incidence estimate is available. The current ``world record" for partial results in this direction for the prime residue field $\mathbb{F}_p$ is due to Jones \cite{J}.
This paper shows that in three dimensions there is an incidence estimate between a set $P$ of $m$ points and a set $\Pi$ of $n$ planes in ${\mathbb P}^3$, valid for any field of characteristic $p\neq 2$. If $p>0$, there is a constraint that $\min(m,n)=O(p^2).$ Hence, the result is trivial, unless $p$ is regarded as a large parameter. Still, since our geometric set-up in terms of $\alpha$-and $\beta$-planes in the Klein quadric breaks down for $p= 2$, we have chosen to state that $p\neq 2$ explicitly in the formulation of main results. Extending the results to more specific situations, when the constraint in terms of $p$ can be weakened may not be impossible but way beyond the methodology herein. A few more words address this issue in the sequel.
The set of incidences is defined as \begin{equation} I(P,\Pi) :=\{(q,\pi)\in P\times \Pi:\,q\in\pi\}. \label{ins}\end{equation}
Over the reals, the point-plane incidence problem has been studied quite thoroughly throughout the past 25 years and several tight bounds are known. In general one can have all the points and planes involved to be incident to a single line in space, in which case the number of incidences is trivially $mn$. To do better than that, one needs some non-degeneracy assumption regarding collinearity, and the results quoted next differ as to the exact formulation of such an assumption.
In the 1990 paper of Edelsbrunner et al. \cite{EGS} it was proven (modulo slow-growing factors that can be removed, see \cite{AS}) that if no three planes are collinear in $\mathbb R^3$, \begin{equation}\label{egsest}
|I(P,\Pi)| = O\left(m^{\frac{4}{5}}n^{\frac{3}{5}}+m+n\right). \end{equation} This bound was shown to be tight for a wide range of $m$ and $n$, owing to a construction by Brass and Knauer \cite{BK}. A thorough review of the state of the art by the year 2007 can be found in the paper of Apfelbaum and Sharir \cite{AS}.
Elekes and T\'oth \cite{ET} weakened the non-collinearity assumption down to that all planes were ``not-too-degenerate". That is a single line in a plane may support only a constant proportion of incidences in that plane. They proved a bound \begin{equation}\label{etest}
|I(P,\Pi)| = O\left((mn)^{\frac{3}{4}} + m\sqrt{n} + n\right) \end{equation} and presented a construction, showing it to be generally tight. The constructions supporting the tightness of both latter estimates are algebraic and extend beyond the real case.
More recently, research in incidence geometry over ${\mathbb R}$ has intensified after the introduction of the polynomial partitioning technique in a breakthrough paper of Guth and Katz \cite{GK}. E.g., there is now a ``continuous" generalisation of the bound \eqref{egsest} by Basit and Sheffer \cite{BS}:
\begin{equation}\label{bsest}
|I(P,\Pi)| = O^*\left(m^{\frac{4}{5}+\epsilon}n^{\frac{3}{5}}k^{\frac{2}{5}} +mk+n\right), \end{equation} where $k$ is the maximum number of collinear planes. For any $\epsilon>0$, the constant hidden in the $O^*$-symbol depends on $\epsilon$.
The proofs of the above results rely crucially on the order properties of $\mathbb R$. Some of them, say \eqref{etest} extend over $\mathbb C$, for it is based on the Szemer\'edi-Trotter theorem. Technically harder partitioning-based works like \cite{BS} have so far defied generalisation beyond ${\mathbb R}$.
This paper presents a different approach to point-plane incidences in the projective three-space ${\mathbb P}^3$. The approach appears to be robust enough to embrace, in principle, all fields $\mathbb{F}$, but for the apparently special case of characteristic $2$. When we have a specific field $\mathbb{F}$ in mind, we use the notation $\mathbb{FP}$ for the projective line ${\mathbb P}$. The novelty of our approach is on its geometric side: we fetch and use extensively the classical XIX century Pl\"ucker-Klein formalism for line geometry in ${\mathbb P}^3$. This is combined with a recent algebraic incidence theorem for counting line-line intersections in three dimensions by Guth and Katz.
The work of Guth and Katz, see \cite{GK} and the references contained therein for its predecessors, established two important theorems. Both rested on the polynomial Nullstellensatz principle, which was once again demonstrated to be so efficient a tool for discrete geometry problems by Dvir, who used it to resolve the finite field Kakeya conjecture \cite{D}. The proof of the first Guth-Katz theorem, Theorem 2.10 in \cite{GK}, was in essence algebraic, using the Nullstellensatz and basic properties of ruled surfaces, which come into play due to the use of the classical XIX century geometry Monge-Salmon theorem. See \cite{Sa} for the original exposition of the latter theorem, as well as \cite{K} (wholly dedicated to the prominent role this theorem plays in today's incidence geometry) and Appendix in \cite{Ko}.
The second Guth-Katz theorem, Theorem 2.11 in \cite{GK}, introduced the aforementioned method of polynomial partitioning of the real space, based on the Borsuk-Ulam theorem. It is the latter theorem of Guth and Katz that has recently attracted more attention and follow-ups. Since we work over any field, we cannot not use polynomial partitioning.
It is a variant of Theorem 2.10 from \cite{GK} that plays a key role here, and is henceforth referred to as {\em the} Guth-Katz theorem. We share this, at least in part, with a recent work of Koll\'ar \cite{Ko} dedicated to point-line incidences in $3D$, in particular over fields with positive characteristic.
\begin{theorem}[Guth-Katz] \label{gkt} Let $\mathcal L$ be a set of $n$ straight lines in ${\mathbb R}^3$. Suppose, no more then two lines are concurrent. Then the number of pair-wise intersections of lines in $\mathcal L$ is $$ O\left(n^{\frac{3}{2}}+ nk\right), $$ where $k$ is the maximum number of lines, contained in a plane or regulus. \end{theorem}
The proof of Theorem \ref{gkt} goes about over the complex field.\footnote{For a reader not familiar with the proof of Theorem \ref{gkt}, that is Theorem 2.10 in \cite{GK}, we recommend Katz's note \cite{K} for more than an outline of the proof. See also a post {\sf www.terrytao.wordpress.com/2014/03/28/the-cayley-salmon-theorem-via-classical-differential-geometry/} by Tao and the links contained therein.} Moreover, it extends without major changes to any algebraically closed field, under the constraint $n=O(p^2)$ in the positive characteristic case. This was spelt out by Koll\'ar, see \cite{Ko} Corollary 40, with near-optimal values of constants.
To complete the introduction, let us briefly discuss, in slightly more modern terms, the ``continuous'' Monge-Salmon theorem, brought in by Guth and Katz to discrete geometry. Suppose the field $\mathbb F$ is algebraically closed field and $Z$ is a surface in $\mathbb{F}{\mathbb P}^3$, defined as the zero set of a minimal polynomial $Q$ of degree $d$. A point $x\in Z$ is called {\em flechnodal} if there is a line $l$ with at least fourth order contact with $Z$ at $x$, that is apart from $x\in l$, at least three derivatives of $Q$ in the direction of $l$ vanish at $x$. Monge showed that flechnodal points are cut out by a homogeneous polynomial, whose degree Salmon claimed to be equal to $11d-24$ (which is sharp for $d=3$, due to the celebrated Cayley-Salmon theorem). Thus, for an irreducible $Z$, either all points are flechnodal, or flechnodal points lie on a curve of degree $d(11d-24)$. Over the complex field Salmon proved that assuming that all points of $Z$ are flechnodal implies that $Z$ is ruled. In positive characteristic it happens that there exist high degree non-ruled surfaces, where each point is flechnodal. But not for $d<p$. Voloch \cite{V} adapted the Monge-Salmon proof to modern terminology, $p>0$, $d<p$, and also conjectured that counterexamples may take place only if $p$ divides $d(d-1)(d-2)$.
The following statement is implicit in the proof of Proposition 1 in \cite{V}.
\begin{theorem}[Salmon] \label{Salmon} An irreducible algebraic surface in ${\mathbb P}^3$ over an algebraically closed field $\mathbb{F}$, containing more than $d(11d-24)$ lines must be ruled, under the additional constraint that $d<p$ if $\mathbb{F}$ has positive characteristic $p$. \end{theorem}
Once Theorem \ref{Salmon} gets invoked within the proof of Theorem \ref{gkt}, the rest of it uses basic properties of ruled surfaces, for a summary see \cite{Ko}, Section 7. We complement it with some additional background material in Section \ref{ruled}, working with the Grassmannian parameterising the set of lines in ${\mathbb P}^3$, that is the Klein quadric.
\section{Main results} The main geometric idea of this paper is to interpret incidence problems between points and planes in ${\mathbb P}^3$ as line-line incidence problems in a projective three-quadric $\mathcal{G}$. $\mathcal{G}$ is contained in the Klein quadric $\mathcal{K}$ representing the space of lines in the ``physical space'' ${\mathbb P}^3$ in the ``phase space'' ${\mathbb P}^5$. $\mathcal{G}$ is the Klein image of a so-called {\em regular line complex} and has many well-known geometric properties. In comparison to ${\mathbb P}^3$, where the space of lines is four-dimensional, the space of lines in $\mathcal{G}$ is three-dimensional, and this enables one to satisfy the no-multiple-concurrencies hypotheses of Theorem \ref{gkt}. It will also turn out that the parameters denoted as $k$ in both the point-plane incidence estimate (\ref{bsest}) and Theorem \ref{gkt} are closely related.
Our main result is as follows.
\begin{theorem} \label{mish} Let $P, \Pi$ be sets of points and planes, of cardinalities respectively $m$ and $n$ in ${\mathbb P}^3$. Suppose, $m\geq n$ and if $\mathbb{F}$ has positive characteristic $p$, then $p\neq 2$ and $n=O(p^2)$. Let $k$ be the maximum number of collinear planes.
Then \begin{equation}\label{pups}
|I(P,\Pi)|=O\left( m\sqrt{n}+ km\right).\end{equation} \end{theorem} The statement of the theorem can be reversed in an obvious way, using duality in the case when the number of planes is greater than the number of points. Note that the $km$ term may dominate only if $k\geq\sqrt{n}$.
The estimate (\ref{pups}) of Theorem \ref{mish} is a basic universal estimate. It is weaker than the above quoted estimates (\ref{egsest}), as well as (\ref{bsest}) for small values of $k$, and slightly weaker than (\ref{etest}). Later in Section \ref{example}, for completeness sake, we present a construction, not so dissimilar from those in \cite{BK} and \cite{ET}, showing that in the case $n=m$ and $k=m^{\frac{1}{2}}$, the estimate (\ref{pups}) is tight, for any admissible $n$.
\begin{remark}\label{sharps} Let us argue that in positive characteristic and solely under the constraint $\min(m,n)=O(p^2)$ the main term in the estimate \eqref{pups} cannot be improved. This suggests that analogues of stronger Euclidean point-plane incidence bounds like \eqref{egsest} do not extend to positive characteristic without additional assumptions stronger than in Theorem \ref{mish}.
Let $\mathbb{F}=\mathbb{F}_p$, and take the point set $P$ as a smooth cubic surface in $\mathbb{F}^3$, so $|P|=O(p^2)$. Suppose, $|\Pi|>|P|$, so the roles of $m, n$ in Theorem \ref{mish} get reversed. A generic plane intersects $P$ at $\Omega(p)$ points. By the classical Cayley-Salmon theorem (which follows from the statement of Theorem \ref{Salmon} above) $P$ may contain at most $27$ lines. Delete them, still calling $P$ the remaining positive proportion of $P$. Now no more than three points in $P$ are collinear, so $k=3$. However, for a set of generic planes $\Pi$, $|I(P,\Pi)|=\Omega(|\Pi|\sqrt{|P|})$, which matches up to constants the bound \eqref{pups} when $|\Pi|>|P|$. \end{remark}
We also give two applications of Theorem \ref{mish} and show how it yields reasonably strong geometric incidence estimates over fields with positive characteristic. The forthcoming Theorem \ref{spr} claims that any plane set $S\subset \mathbb{F}^2$ of $N$ non-collinear points determines $\Omega\left[\min\left(N^{\frac{2}{3}},p\right)\right]$ distinct pair-wise bilinear -- i.e., wedge or dot -- products, with respect to any origin.
If $S=A\times A$, $A\subseteq \mathbb{F}$ this improves to a sum-product type inequality \begin{equation}
|AA+AA|=\Omega\left[\min\left(|A|^{\frac{3}{2}},p\right)\right]. \label{2spr}\end{equation}
In the special case of $A$ being a multiplicative subgroup of $\mathbb{F}^*_p$, the same bound was proved by Heath-Brown and Konyagin \cite{HBK} and improved by V'jugin and Shkredov \cite{VS} (for suitably small multiplicative subgroups) to $\Omega\left(\frac{|A|^{\frac{5}{3}}} {\log^{\frac{1}{2}}|A|}\right).$ Theorem \ref{mish} becomes a vehicle to extend bounds for multiplicative subgroups to approximate subgroups.
For more applications of Theorem \ref{mish} to questions of sum-product type see \cite{RRS}. Results in the latter paper include a new state of the art sum-product estimate
$$\max(|A+A|,\,|AA|)\gg |A|^{\frac{6}{5}},\qquad\mbox{for }|A|<p^{\frac{5}{8}},$$ obtained from Theorem \ref{mish} in a manner, similar to how the sum-product exponent $\frac{5}{4}$ gets proven over ${\mathbb R}$ using the Szemer\'edi-Trotter theorem in the well-known construction by Elekes \cite{E0}. The previously known best sum-product exponent $\frac{12}{11}-o(1)$ over $\mathbb{F}_p$ was proven by the author \cite{R}, ending a stretch of many authors' incremental contributions based on the so-called {\em additive pivot} technique introduced by Bourgain, Katz and Tao \cite{BKT}.\footnote{In a forthcoming paper with E. Aksoy, B. Murphy, and I. D. Shkredov we present further applications of Theorem \ref{mish} to sum-product type questions in positive characteristic.}
Such reasonably strong bounds in positive characteristic have been available so far only for subsets of finite fields, large enough relative to the size of the field itself: see, e.g., \cite{HI}. Theorem \ref{mish} enables one to extend these bounds to small sets, and the barrier it imposes in terms of $p$ is often exactly where the two types of bounds over $\mathbb{F}_p$ meet. See \cite{RRS} for more discussion along these lines.
The same can be said about our second application of Theorem \ref{mish}, Theorem \ref{erd}. It yields a new result for the Erd\H os distance problem in three dimensions in positive characteristic, which is not too far off what is known over the reals. A set $S$ of $N$ points not supported in a single semi-isotropic plane in $\mathbb{F}^3$, contains a point, from which $\Omega\left[\min\left(\sqrt{N},p\right)\right]$ distances are realised. Semi-isotropic planes are planes spanned by two mutually orthogonal vectors $\boldsymbol e_1,\boldsymbol e_2$, such that $\boldsymbol e_1\cdot \boldsymbol e_1=0$, while $\boldsymbol e_2\cdot \boldsymbol e_2\neq 0$. They always exist in positive characteristic -- see \cite{HI} for explicit constructions in finite fields -- and one can have point sets with very few distinct distances within these planes.
We mention in passing another application of Theorem \ref{mish}, which is Corollary \ref{intersections} appearing midway through the paper, concerning the prime residue field $\mathbb{F}_p$. Given {\em any} family of $\Omega(p^2)$ straight lines in $G={SL}_2(\mathbb{F}_p)$, their union takes up a positive proportion of $G$. In Lie group-theoretical terminology these lines are known as generalised horocycles, that is right cosets of one-dimensional subgroups conjugate to one of the two one-dimensional subgroups of triangular matrices with $1$'s on the main diagonal. (See, e.g., \cite{BM} as a general reference.) A similar claim in $\mathbb{F}_p^3$ is false, for all the lines may lie in a small number of planes. Nonetheless our Corollary \ref{intersections} is not new and follows from a result of Ellenberg and Hablicsek \cite{EH}. They extend to $\mathbb{F}_p^3$ another, earlier relative to Theorem \ref{gkt}, algebraic theorem of Guth and Katz over $\mathbb{C}$, from another breakthrough paper \cite{GKprime}. The assumption required for that in \cite{EH} is that all planes be relatively ``poor''.
\begin{remark} The presence of the characteristic $p$ in the constraint of Theorem \ref{mish} and its applications makes a positive characteristic field $\mathbb{F}$ somewhat morally just $\mathbb{F}_p$, for Theorem \ref{Salmon} is not true otherwise. Replacing this constraint by more elaborate ones in the context of finite extensions of $\mathbb{F}_p$ may be possible for $p>2$ provided that classification of exceptional cases as to Salmon's theorem becomes available. Voloch \cite{V} conjectures that an irreducible flechnodal surface of degree $d$ may be unruled only if $p$ divides $d(d-1)(d-2)$ and gives evidence in this direction. See also \cite{EH} for examples of such {\em flexy} surfaces and discussion from the incidence theory viewpoint.\label{rem} \end{remark}
Let us give an outline of the proof of Theorem \ref{mish} to motivate the forthcoming background material in Section \ref{setup}. First off, Theorem \ref{gkt} needs to be extended to the case of pair-wise intersections between two families of $m$ and $n$ lines, respectively. The only way to do so to meet our purpose, in view of Remark \ref{sharps}, is the cheap one. If $m$ is much bigger than $n$, partition the $m$ lines into $\sim \frac{m}{n}$ groups of $\sim n$ lines each and apply a generalisation to $\mathbb{F}$ of Theorem \ref{gkt} separately to count incidences of each group with the family of $n$ lines.
Let us proceed assuming $m=n$. Let $q\in P,\,\pi\in\Pi$ be a point and a plane in ${\mathbb P}^3,$ and $q\in\pi$. Draw in the plane $\pi$ all lines, incident to the point $q$. In line geometry literature this figure is called a {\em plane pencil} of lines. It is represented by a line in the space of lines, that is the Klein quadric ${\mathcal K}$, a four-dimensional hyperbolic projective quadric in ${\mathbb P}^5$, whose points are in one-to-one correspondence with lines in ${\mathbb P}^3$ via the so-called {\em Klein map}. If the characteristic of $\mathbb{F}$, $p\neq 2$, the line pencil gets represented in $\mathcal{K}$ as follows. The Klein image of the family of all lines incident to $q$ is a copy of ${\mathbb P}^2$ contained in $\mathcal{K}$, a so-called $\alpha$-plane. The family of all lines contained in $\pi$ is also represented by a copy of ${\mathbb P}^2$ contained in $\mathcal{K}$, a so-called $\beta$-plane. A pair of planes of two distinct types in ${\mathcal K}$ typically do not meet. If they do, this happens along a copy of ${\mathbb P}^1,$ a line in ${\mathcal K}$, which is the Klein image of the above line pencil, if and only if $q\in\pi$.
Thus the number of incidences $|I(P,\Pi)|$ equals the number of lines along which the corresponding sets of $\alpha$ and $\beta$-planes meet in ${\mathcal K}$. One can now restrict the arrangement of planes in ${\mathcal K}$ from ${\mathbb P}^5$ to a generic hyperplane ${\mathbb P}^4$ intersecting $\mathcal{K}$ transversely. Its intersection with $\mathcal{K}$ is a three-dimensional sub-quadric $\mathcal{G}$, whose pre-image under the Klein map is called a {\em regular line complex.} There is a lot of freedom in choosing the generic subspace ${\mathbb P}^4$ to cut out $\mathcal{G}$. Or, one can fix the subspace ${\mathbb P}^4$ in the ``phase space'' ${\mathbb P}^5$ and realise this freedom to allow for certain projective transformations of the ``physical space'' ${\mathbb P}^3$ and its dual. For there is a one-to-one correspondence between regular line complexes and so-called {\em null polarities} -- transformations from ${\mathbb P}^3$ to its dual by non-degenerate skew-symmetric matrices. See \cite{PW}, Chapter 3 for general theory.
The benefit of having gone down in dimension from $\mathcal{K}$ to $\mathcal{G}$ is that $\alpha$ and $\beta$-planes restrict to $\mathcal{G}$ as lines, which may generically meet only if they are of different type. This is because two planes of the same type intersect at one and only one point in $\mathcal{K}$. So one can choose the subspace ${\mathbb P}^4$, defining $\mathcal{G}$ in such a way that it contains none of the above finite number of points. If the field $\mathbb{F}$ is finite, the latter finite set may appear to be sizable in comparison with the size of $\mathcal{G}$ itself. However, just like in the proof of Theorem \ref{gkt} one works in the algebraic closure of $\mathbb{F}$, which is infinite. Thus the only place where the characteristic $p$ of $\mathbb{F}$ makes a difference is within the body of the Guth-Katz theorem to ensure the validity of Salmon's Theorem \ref{Salmon}. The corresponding constraint in terms of $p$ is stated explicitly and with constants in \cite{Ko}, Corollary 40.
Having restricted the $\alpha$ and $\beta$-planes as lines in $\mathcal{G}$ we end up with two families of lines there, such that lines of the same type do not meet. The number of incidences $|I(P,\Pi)|$ equals the number of pair-wise intersections of these lines. The lines satisfy the input conditions of Theorem \ref{gkt}, the only difference being that they live in the three-quadric $\mathcal{G}\subset{\mathbb P}^4$, rather than ${\mathbb P}^3$. But one can always project a finite family of lines from higher dimension to ${\mathbb P}^3$ so that skew lines remain skew. Thereupon we find ourselves in ${\mathbb P}^3$, and what's left for the proof of Theorem \ref{mish} has been essentially worked out by Guth-Katz and Koll\'ar. This seems a bit like a waste, for the space of lines in $\mathcal{G}$ is three, rather than four-dimensional. Yet we could not conceive a better theorem for $\mathcal{G}$, but for a chance of slightly better constants. Plus, Remark \ref{sharps} suggests that a stronger theorem about $\mathcal{G}$ must have more restrictive assumptions than Theorem \ref{mish}.
To conclude this section, we briefly summarise the key steps in the beautiful proof of Guth and Katz, which we retell with small modifications as the proof of Theorem \ref{gkt2} in the main body of the paper. We could have almost got away with just citing \cite{Ko}, Sections 3 and 4 but for a few extra details, since we still need to bring the collinearity parameter $k$ into play.
Assuming that there are some $C n^{\frac{3}{2}}$ pair-wise line intersections in ${\mathbb P}^3$ enables one to put all the lines, supporting more than roughly the average number of incidences per line, on a polynomial surface $Z$ of degree $d\sim \frac{ \sqrt{n} }{C},$ so most of the incidences come from within factors of $Z$. One can use induction in $n$ to effectively assume that the number of these lines is $\Omega(n)$. Then Salmon's theorem implies that $Z$ must have a ruled component, containing a vast majority of the latter lines. One should not bother about non-ruled factors by the induction hypothesis. However, a non-cone ruled factor of degree $d>2$ can only support a relatively small number of incidences. Since lines of the same type do not meet, having many incidences within planes or cones is not an option either.
Hence if there are $C n^{\frac{3}{2}}$ incidences, $Z\subset {\mathbb P}^3$ must have a doubly-ruled quadric factor, containing many lines. Once we lift $Z$ back to $\mathcal{G}$, this means having many lines of each type in the intersection of $\mathcal{G}\subset{\mathbb P}^4$ with a ${\mathbb P}^3$. Finally, an easy argument in the forthcoming Lemma \ref{lem} shows that intersections of $\mathcal{G}$ with a ${\mathbb P}^3$ can be put into correspondence with what happens within the original arrangement of points and planes in the ``physical space''. Namely lines of the two types meeting in $\mathcal{G}\cap{\mathbb P}^3$ represent precisely incidences of the original points and planes along some line in the ${\mathbb P}^3$. This brings the collinearity parameter $k$ into the incidence estimate and completes the proof.
\section{Acknowledgment}
The author is grateful to Jon Selig for educating him about the Klein quadric. Special thanks to J\'ozsef Solymosi for being the first one to point out a mistake in the estimate of Theorem \ref{mish} in the original version of the paper and two anonymous Referees for their patience and attention to detail. This research was conceived in part when the author was visiting the Institute for Pure and Applied Mathematics (IPAM), which is supported by the National Science Foundation.
\section{Geometric set-up} \subsection{Background} \label{setup} We begin with a brief introduction of the Klein, alias Klein-Pl\"ucker quadric ${\mathcal K}$. See \cite{PW}, Chapter 2 or \cite{JS}, Chapter 6 for a more thorough treatment.
The space of lines in ${\mathbb P}^3$ is represented as a projective quadric, known as the {\em Klein quadric} $\mathcal K$ in ${\mathbb P}^5$, with projective coordinates $(P_{01}:P_{02}:P_{03}:P_{23}:P_{31}:P_{12})$, known as {\em Pl\"ucker coordinates.} The latter {\em Pl\"ucker vector} yields the {\em Klein image} of a line $l$ defined by a pair of points $q=(q_0:q_1:q_2:q_3)$ and $u=(u_0:u_1:u_2:u_3)$ in ${\mathbb P}^3$ that it contains, under the {\em Klein map}, defined as follows: \begin{equation} P_{ij}=q_iu_j-q_ju_i,\qquad i,j=0,\ldots,3. \label{Pc}\end{equation} It is easy to verify that once $\{P_{ij}\}$ are viewed as homogeneous coordinates, this definition does not depend on the particular choice of the pair of points on the ``physical line'' $l$, and there are $6=4\cdot 3/2$ independent projective Pl\"ucker coordinates $P_{ij}$. We use the capital $L\in {\mathbb P}^5$ for the Pl\"ucker vector, which is the Klein image of the line $l\subset {\mathbb P}^3$.
For an affine line in $\mathbb{F}^3$, obtained by setting $q_0=u_0=1$, the Pl\"ucker coordinates acquire the meaning of a projective pair of three-vectors $(\boldsymbol \omega: \boldsymbol v)$, where $\boldsymbol \omega=(P_{01},P_{02},P_{03})$ is a vector in the direction of the line and for any point $\boldsymbol q=(q_1,q_2,q_3)$ on the line, $\boldsymbol v = (P_{23},P_{31},P_{12}) = \boldsymbol q\times\boldsymbol \omega$ is the line's moment vector\footnote{In this section we use boldface notation for three-vectors. The essentially Euclidean vector product notation is to keep the exposition as elementary as possible: in $\mathbb{F}^3$ the notation $\boldsymbol v = \boldsymbol q\times\boldsymbol \omega$ means only that $\boldsymbol v$ arises from $\boldsymbol \omega$ after multiplication on the left by the skew-symmetric matrix $T=ad({\boldsymbol q})$, with $T_{12}=-q_3,\,T_{13}=q_2,\,T_{23}=-q_1$.\label{ads}} with respect to the fixed origin. Lines in the plane at infinity have $\boldsymbol \omega=\boldsymbol 0$. We use the boldface notation for three-vectors throughout.
Conversely, one can denote $\boldsymbol \omega=(P_{01},P_{02},P_{03}),\; \boldsymbol v=(P_{23},P_{31},P_{12}),$ the Pl\"ucker coordinates then become $(\boldsymbol \omega:\boldsymbol v)$, and treat $\boldsymbol \omega$ and $\boldsymbol v$ as vectors in $\mathbb{F}^3$, bearing in mind that as a pair they are projective quantities. The equation of the Klein quadric ${\mathcal K}$ in ${\mathbb P}^5$ is \begin{equation} P_{01}P_{23}+P_{02}P_{31}+P_{03}P_{12}=0,\;\mbox{ i.e., }\; \boldsymbol \omega\cdot\boldsymbol v=0. \label{Klein}\end{equation} More formally, equation \eqref{Klein} arises after writing out, with the notations \eqref{Pc}, the truism $$ \det\left(\begin{array}{cccccc} q_0&u_0&q_0&u_0\\q_1&u_1&q_1&u_1\\q_2&u_2&q_2&u_2\\q_3&u_3&q_3&u_3\end{array}\right) = 0. $$ Two lines $l,l'$ in ${\mathbb P}^3$, with Klein images $$L=(P_{01}:P_{02}:P_{03}:P_{23}:P_{31}:P_{12}),\qquad L'=(P'_{01}:P'_{02}:P'_{03}:P'_{23}:P'_{31}:P'_{12})$$ meet in ${\mathbb P}^3$ if and only if \begin{equation}\label{intersection} P_{01}P'_{23} + P_{02}P'_{31} + P_{03}P'_{12} + P'_{01}P_{23} + P'_{02}P_{31} + P'_{03}P_{12}\;=\;0.\end{equation}
The left-hand side of \eqref{intersection} is called the {\em reciprocal product} of two Pl\"ucker vectors. If they are viewed as $L=(\boldsymbol \omega:\boldsymbol v)$ and $L'=(\boldsymbol \omega':\boldsymbol v')$, the intersection condition becomes \begin{equation}\label{intersectionv} \boldsymbol \omega\cdot \boldsymbol v'+ \boldsymbol v\cdot \boldsymbol \omega' = 0.\end{equation} Condition \eqref{intersection} can be restated as \begin{equation} L^T\mathcal QL' = 0,\qquad \mathcal Q = \left(\begin{array}{ccc} 0 & I_3\\ I_3 & 0\end{array}\right),\label{qum}\end{equation} where $I_3$ is the $3\times 3$ identity matrix.
It is easy to see by \eqref{intersection}, after taking the gradient in \eqref{Klein} that a hyperplane ${\mathbb P}^4$ in ${\mathbb P}^5$ is tangent to $\mathcal{K}$ at some point $L$ if and only if the covector defining the hyperplane is itself in the Klein quadric in the dual space. Moreover, it follows from (\ref{intersection}) that the intersection of $\mathcal{K}$ with the tangent hyperplane $T_L \mathcal{K}\cap \mathcal{K}$ through $L$ consists of $L'\in \mathcal{K}$, which are the Klein images of all lines $l'$ in ${\mathbb P}^3$, incident to the line $l$, represented by $L$. The union of all these lines $l'$ is called a {\em singular line complex.}
\subsubsection{Two rulings by planes and line complexes} The largest dimension of a projective subspace contained in $\mathcal{K}$ is two. This can be seen as follows. After the coordinate change $\boldsymbol x = \boldsymbol \omega-\boldsymbol v$, $\boldsymbol y = \boldsymbol \omega+\boldsymbol v$, the equation \eqref{Klein} becomes \begin{equation}
\|\boldsymbol x\|^2= \|\boldsymbol y\|^2. \label{xform}\end{equation} This equation cannot be satisfied by a ${\mathbb P}^3$. It can be satisfied by a ${\mathbb P}^2$ if and only if $\boldsymbol y= M\boldsymbol x,$ for some orthogonal matrix $M$. We further assume that ${\rm char}(\mathbb F)\neq 2$, which is crucial. For then there are two cases, corresponding to $\det M=\pm 1$. The two cases correspond to two ``rulings'' of $\mathcal{K}$ by planes, which lie entirely in it, the fibre space of each ruling being ${\mathbb P}^3$.
To characterise the two rulings, called $\alpha$ and $\beta$-planes, corresponding to $\det M=\pm1$, respectively, one returns to the original coordinates $(\boldsymbol \omega:\boldsymbol v)$. After a brief calculation, see \cite{JS}, Section 6.3, it turns out that Pl\"ucker vectors in a single $\alpha$-plane in $\mathcal{K}$ are Klein images of lines in ${\mathbb P}^3$, which are concurrent at some point $(q_0:q_1:q_2:q_3)\in {\mathbb P}^3$. If the concurrency point is $(1:\boldsymbol q)$, which is identified with $\boldsymbol q\in \mathbb F^3$, the $\alpha$-plane is a graph $\boldsymbol v = \boldsymbol q\times \boldsymbol \omega$. Otherwise, an ideal concurrency point $(0:\boldsymbol \omega)$ gets identified with some fixed $\boldsymbol \omega$, viewed as a projective vector. The corresponding $\alpha$-plane is the union of the set of parallel lines in $\mathbb{F}^3$ in the direction of $\boldsymbol \omega$, with Pl\"ucker coordinates $(\boldsymbol \omega:\boldsymbol v)$, so $\boldsymbol v\cdot\boldsymbol \omega=0,$ by \eqref{Klein}, and the set of lines in the plane at infinity incident to the ideal point $(0:\boldsymbol \omega)$. The latter lines have Pl\"ucker coordinates $(\boldsymbol 0:\boldsymbol v),$ with once again $\boldsymbol v\cdot\boldsymbol \omega=0$.
Similarly, Pl\"ucker vectors lying in a $\beta$-plane represent co-planar lines in ${\mathbb P}^3$. A ``generic'' $\beta$-plane is a graph $\boldsymbol \omega = \boldsymbol u\times \boldsymbol v$, for some $\boldsymbol u \in \mathbb F^3$. The case $\boldsymbol u =\boldsymbol 0$ corresponds to the plane at infinity, otherwise the equation of the co-planarity plane in $\mathbb{F}^3$ becomes \begin{equation}\boldsymbol u\cdot \boldsymbol q =-1.\label{refer}\end{equation} If $\boldsymbol u$ gets replaced by a fixed ideal point $(0:\boldsymbol v)$, the corresponding $\beta$-plane comprises lines, coplanar in planes through the origin: $\boldsymbol v\cdot \boldsymbol q = 0$. The $\beta$-plane in the Klein quadric is formed by the set of lines with Pl\"ucker coordinates $(\boldsymbol \omega:\boldsymbol v)$, plus the set of lines through the origin in the co-planarity plane. The latter lines have Pl\"ucker coordinates $(\boldsymbol \omega:\boldsymbol 0)$. In both cases one requires $\boldsymbol \omega\cdot\boldsymbol v = 0$.
Two planes of the same ruling of $\mathcal{K}$ always meet at a point, which is the line defined by the two concurrency points in the case of $\alpha$-planes. An $\alpha$ and a $\beta$-plane typically do not meet. If they do -- this means that the concurrency point $q$, defining the $\alpha$-plane lives in the plane $\pi$, defining the $\beta$-plane. The intersection is then a line, a copy of ${\mathbb P}^1$ in ${\mathcal K}$, representing a {\em plane pencil of lines}. These are the lines in ${\mathbb P}^3$, which are co-planar in $\pi$ and concurrent at $q$. Conversely, each line in ${\mathcal K}$ identifies the pair ($\alpha$-plane, $\beta$-plane), that is the plane pencil of lines uniquely. Moreover points $L,L'\in \mathcal{K}$ can be connected by a straight line in $\mathcal{K}$ if and only if the corresponding lines $l,l'$ in ${\mathbb P}^3$ meet, cf. \eqref{qum}.
From non-degeneracy of the reciprocal product it follows that the reciprocal-orthogonal projective subspace to a $\alpha$ or $\beta$-plane is the plane itself. Hence, a hyperplane in ${\mathbb P}^5$ contains a $\alpha$ or $\beta$-plane if and only if it is a $T_L(\mathcal{K})$ at some point $L$, lying in the plane.
It follows that a singular line complex arises if and only if the equation of the hyperplane intersecting $\mathcal{K}$ is $(\boldsymbol u:\boldsymbol w)^T(\boldsymbol \omega:\boldsymbol v)=0$, with the dual vector $(\boldsymbol u:\boldsymbol w)$ itself such that $\boldsymbol u\cdot\boldsymbol w=0$. Otherwise the Klein pre-image of the intersection of the hyperplane with $\mathcal{K}$ is called a regular line complex.
We remark that a geometric characterisation of a regular line complex is that it is a set of invariant lines of some {\em null polarity}, that is a projective map from ${\mathbb P}^3$ to its dual ${\mathbb P}^{3*}$ defined via a $4\times 4$ non-degenerate skew-symmetric matrix. In particular, a null polarity assigns to each point $q\in {\mathbb P}^3$ a plane $\pi(q)$, such that $q\in \pi$. See \cite{PW}, Chapter 3 for more detail.
A particular example of the Klein image of a regular line complex arises if one sets $\omega_3=v_3$, i.e. $x_3=0$ in coordinates \eqref{xform}. One can identify $(-x_1:x_2:0:y_1:y_2:1)$ with $\mathbb F^4$, getting $$ x_1y_1 - x_2y_2 = 1 $$ for the affine part of $\mathcal{G}$, which can be identified with the group $SL_2(\mathbb F)$.
The following lemma describes the intersection of a regular complex with a singular one. \begin{lemma} \label{lem} Let $l$ be a line in ${\mathbb P}^3$, represented by $L\in\mathcal{K}$. Then $\mathcal{K}\cap T_L\mathcal{K}$ contains $\alpha$ and $\beta$-planes, corresponding, respectively to points on $l$ and planes containing $l$. Given two hyperplanes $S_1$, $S_2$ in ${\mathbb P}^5$, suppose $\mathcal{K}\cap S_1$ is the Klein image of a regular line complex. Consider the intersection $\mathcal{K}\cap S_1\cap S_2$. If the field $\mathbb{F}$ is algebraically closed, $\mathcal{K}\cap S_1\cap S_2 = \mathcal{K}\cap S_1\cap S'_2$, where $S'_2$ is tangent to $\mathcal{K}$ at some point $L$. That is, $\mathcal{K}\cap S_2'$ is the Klein image of the singular line complex of lines in ${\mathbb P}^3$ meeting the Klein pre-image $l$ of $L$. \end{lemma}
\begin{proof} The first statement follows immediately by definitions above. To prove the second statement, suppose $S_2$ is not tangent to $\mathcal{K}$. Let the two line complexes be defined by dual vectors $(\boldsymbol u:\boldsymbol w)$ and $(\boldsymbol u':\boldsymbol w')$. If $\mathbb{F}$ is algebraically closed, the line $t_1(\boldsymbol u:\boldsymbol w) + t_2 (\boldsymbol u':\boldsymbol w')$ in the dual space will then intersect the Klein quadric in the dual space, a point of intersection $L$ defining $S_2'$.
Note, however, that if $S_2$ is itself tangent to $\mathcal{K}$ at $L$, then there is only one solution, $L$ itself, otherwise there are two. \end{proof}
\subsubsection{Reguli} For completeness purposes and since reguli appear in the formulation of Theorem \ref{gkt} we give a brief account in this section. See also the next section on ruled surfaces.
The $\alpha$ and $\beta$-planes represent a degenerate case when a subspace $S= {\mathbb P}^2$ of ${\mathbb P}^5$ is contained in $\mathcal{K}$. Assume that $\mathbb{F}$ is algebraically closed, then any $S$ intersects $\mathcal{K}$. The non-degenerate situation would be $S$ intersecting $\mathcal{K}$ along a irreducible conic curve. This curve in $\mathcal{K}$ is called a {\em regulus}, and the union of lines corresponding to in in the physical space forms a single ruling of a doubly-ruled quadric surface. One uses the term regulus to refer to both the above curve in $\mathcal{K}$ and the family of lines in ${\mathbb P}^3$ this curve represents.
Choose affine coordinates, so that the equations of the two-plane $S$ can be written as $$ A\boldsymbol \omega + B\boldsymbol v = \boldsymbol 0, $$ where $A,B$ are some $3\times3$ matrices. For points in $S\cap\mathcal{K}$, which do not represent lines in the plane at infinity in ${\mathbb P}^3$, we can write $\boldsymbol v =\boldsymbol q\times \boldsymbol \omega$, where $\boldsymbol q$ is some point in $\mathbb{F}^3$, on the line with Pl\"ucker coordinates $(\boldsymbol \omega:\boldsymbol v)$, and $\boldsymbol \omega\neq \boldsymbol 0$. If $T$ denotes the skew-symmetric matrix $ad(\boldsymbol q)$ we obtain $$ (A-BT)\boldsymbol \omega =\boldsymbol 0\qquad \Rightarrow\qquad \det(A-BT)=0. $$ This a quadratic equation in $\boldsymbol q$, since $T$ is a $3\times 3$ skew-symmetric matrix, so $\det T=0$. If the above equation has a linear factor in $\boldsymbol q$, defining a plane $\pi\subset{\mathbb P}^3$, then $S\cap \mathcal{K}$ contains a line, which represents a pencil of lines in $\pi$. If the above quadratic polynomial in $\boldsymbol q$ is irreducible, and $\mathbb{F}$ is algebraically closed, one always gets a quadric irreducible surface in ${\mathbb P}^3$ as the union of lines in the regulus, see Lemma \ref{rs} in the next section.
In the latter case, by Lemma \ref{lem}, the two-plane $S$ in ${\mathbb P}^5$ can be obtained as the intersection of three four-planes, tangent to $\mathcal{K}$ at some three points $L_1,L_2,L_3$, corresponding to three mutually skew lines in ${\mathbb P}^3$. Thus a regulus can be redefined as the set of all lines in ${\mathbb P}^3$, meeting three given mutually skew lines $l_1,l_2,l_3$. Its Klein image is a conic.
Each regulus has a reciprocal one, the Klein image of the union of all lines incident to any three lines, represented in the former regulus. These lines form the second ruling of the same quadric doubly-ruled surface. See \cite{JS}, Section 6.5.1 for coordinate description of reciprocal reguli. \label{rgl}
\subsubsection{Algebraic ruled surfaces} \label{ruled} Differential geometry of ruled surfaces is a rich and classical field of study. From a historical perspective, it was Pl\"ucker who pretty much invented the subject in the two-volume treatise \cite{Pl}, which was completed after his death by Klein.
We give the minimum background on algebraic ruled surfaces in ${\mathbb P}^3$. In this whole section the field $\mathbb{F}$ is assumed to be algebraically closed, of characteristic $p\neq 2$. See \cite{PW}, Chapter 5 for the discussion in the case $p=0$. In positive characteristic the basics of algebraic theory of ruled surfaces are in many respects the same, and for our modest designs we need only these basics.
A ruled surface is defined as a smooth projective surface over an algebraically closed field that is birationally equivalent to a surface ${\mathbb P}\times \mathcal C$ where $\mathcal C$ is a smooth projective curve of genus $g\geq0$. See, e.g. \cite{Ba}, \cite{Li} for general theory of algebraic surfaces. Also Koll\'ar (see \cite{Ko}, Section 7) presents in terms of more formal algebraic geometry a brief account of facts about ruled surfaces, necessary for the proof of Theorem \ref{gkt}. Since he only mentions the Klein quadric implicitly through a citation we review these facts below.
Informally, an algebraic ruled surface is a surface in $\mathbb P^3$ composed of a polynomial family of lines. We assume the viewpoint from Chapter 5 of the book by Pottmann and Wallner \cite{PW}, where an algebraic ruled surface is identified with a polynomial curve $\Gamma$ in the Klein quadric. The union of lines, Klein pre-images of the points of $\Gamma$ draws a surface $Z\subset \mathbb P^3$ called the {\em point set} of $\Gamma$. It is easy to show that $Z$ is then an algebraic surface, that is a projective variety of dimension $2$. A line in $Z$, which is the Klein pre-image of a point of $\Gamma$ is called a {\em generator.} A regular generator $L$, that is a regular point of $\Gamma\subset \mathbb K$ is called {\em torsal} in the special case when the tangent vector to $\Gamma$ at $L$ is also in $\mathcal{K}$. The Klein pre-image of a regular torsal generator necessarily supports a singular surface point, called {\em cuspidal point}. An irreducible component of $\Gamma$ is referred to as a {\em ruling} of $\Gamma$. The same term {\em ruling} is applied to the corresponding family of lines, ruling the surface $Z$.
Here is s basic genericity statement about ruled surfaces. See, e.g., \cite{PW}, Chapter 5.
\begin{lemma}\label{rs} Let $\Gamma$ be an algebraic curve in $\mathcal{K}$, with no irreducible component contained in the intersection of $\mathcal{K}$ with any ${\mathbb P}^2$. Let $Z$ be the point set of $\Gamma$. The subset of $Z$, which is the union of all pair-wise intersections of different rulings of $\Gamma$ and all cuspidal points is a subset of the set of singular points of $Z$. It is contained in an algebraic subvariety of dimension $\leq 1$.
Besides, the curve $\Gamma$ is irreducible if and only if its point set $Z$ is irreducible.
\end{lemma}
We do not give a proof but for a few remarks. The conditions of Lemma \ref{rs} rule out the cases when $Z$ has a plane or smooth quadric component. Clearly, a plane can be the point set for many rulings of lines lying therein, a smooth quadric has two reciprocal reguli, and is therefore an example when the union of the two reguli, not irreducible as a ruled surface has an irreducible point set.
Let $Z$ further denote the point set of a ruling. Suppose, $Z$ contains three lines $l_{1}, l_{2}, l_3$ incident to every line in the ruling. If, say $l_{1}$ and $l_{2}$ meet, then $Z$ is either a plane, and hence the ruling lies in an $\alpha$ or $\beta$-plane, depending on whether or not $l_3$ also meets $l_1$ and $l_2$ at the same point. If the three lines are mutually skew, then the ruling is contained in the intersection of three singular line complexes $T_1,T_2,T_3$, corresponding to the three lines. Their intersection is represented in $\mathcal{K}$ as the latter's transverse section by a ${\mathbb P}^2$ along a conic, that is a regulus. Then $Z$ is a irreducible quadric surface, which has a reciprocal ruling: the set of lines incident to any three lines in the former ruling. See the above discussion of reguli, as well as \cite{JS}, Chapter 6 for more details.
Conversely, if a ruling is contained in a $\alpha$-plane, then $Z$ is a cone: all the generators are incident at the concurrency point defining the $\alpha$-plane. It a ruling lies in a $\beta$-plane, then $Z$ is a plane. If the ruling arises as a result of transverse intersection of a ${\mathbb P}^2$ with $\mathcal{K}$, it is either a pencil of lines or a regulus. In the former case $Z$ is a plane, in the latter case an irreducible doubly-ruled quadric.
An important part of the proof of Theorem \ref{gkt} is the claim that one cannot have too many line-line incidences within a higher degree irreducible ruled surface, which is not a cone. It is essentially the rest of this section that is directly relevant to Theorems \ref{gkt} and \ref{mish}.
\begin{lemma}\label{rss} Let $\Gamma$ be an algebraic ruled surface of degree $d$, whose point set $Z$ has no plane component. Then the degree of $Z$ equals $d$. A generator in a ruled surface of degree $d$, which does not have a cone component, meets at most $d-2$ other generators. \end{lemma}
\begin{proof} By the preceding argument, the theorem is true for $d=2$, so one may assume that conditions of Lemma \ref{rs} are satisfied. Since $\mathbb{F}$ is algebraically closed, a generic line $l$ in ${\mathbb P}^3$ intersects $Z$ exactly $d$ times at points meeting one generator each. It follows that for the Klein image $L$ of $l$, one has $$ L^T \mathcal Q L' = 0, $$ for $d$ distinct $L'\in \Gamma$. Thus the curve $\Gamma$ meets a hyperplane $T_L\mathcal{K}$ in ${\mathbb P}^5$ transversely $d$ times, and hence has degree $d$.
If in the latter equation $L$ no longer represents a generic line in ${\mathbb P}^3$ but a generator of $\Gamma$, and the above equation must still have $d$ solutions, counting multiplicities. Besides $L'=L$ has multiplicity at least $2$, since the intersection of $\Gamma$ with $T_L\mathcal{K}$ at $L$ is not transverse. \end{proof} It also follows that the point set of an irreducible ruled surface $\Gamma$ of degree $d\geq3$ cannot be a smooth projective surface. The point set of $\Gamma$ will necessarily have singular points where two generators meet or a cuspidal points of torsal generators.
It is also well known that the point set of an irreducible ruled surface of degree $d\geq3$ can support at most two non-generator {\em special lines} which intersect each generator. This is because special lines must be skew to each other, or one has a plane. But then if there are three or more special lines, one has a quadric.
\subsection{Point-plane incidences in ${\mathbb P}^3$ are line incidences in a three-quadric in ${\mathbb P}^4$} \label{geom} We can now start moving towards Theorem \ref{mish}. Assume that $\mathbb{F}$ is algebraically closed or pass to the algebraic closure still calling it $\mathbb{F}$. It is crucial for this section that $\mathbb{F}$ not have characteristic $2$. Let $\mathcal{K}\subset{\mathbb P}^5$ be the Klein quadric, $\mathcal{G}=\mathcal{K}\cap S$, for a four-hyperplane $S$ whose defining covector is not in the Klein quadric in ${\mathbb P}^{5*}$. E.g., $\mathcal{G}$ may be defined by the equation $P_{03}=P_{12}$. Since $\mathcal{G}$ contains no planes, each $\alpha$ or $\beta$-plane in ${\mathcal K}$ intersects ${\mathcal G}$ along a line. We therefore have two line families $L_\alpha,L_\beta$ in $\mathcal{G}$. We warn the reader from confusing lines lying in the three-quadric $\mathcal{G}\subset {\mathbb P}^4\subset\mathcal{K}\subset{\mathbb P}^5$ in the ``phase space'' with lines from the regular line complex in the ``physical space'' ${\mathbb P}^3$ that $\mathcal{G}$ is the Klein image of.
The following lemma states that one can assume $L_\alpha\cap L_\beta=\emptyset,$ as well as that the lines within each family do not meet each other.
\begin{lemma} \label{tog} Suppose, $\mathbb{F}$ is algebraically closed and not of characteristic $2$. To every finite point-plane arrangement $(P,\Pi)$ in ${\mathbb P}^3$ one can associate two distinct families of lines $L_\alpha,L_\beta$ contained in some three quadric $\mathcal{G}=\mathcal{K}\cap S$, where the four-hyperplane $S$ is not tangent to $\mathcal{K}$, with the following property. No two lines of the same family meet; $|L_\alpha|=m$, $|L_\beta|=n$, and $|I(P,\Pi)| = |I(L_\alpha,L_\beta)|$, where $I(L_\alpha,L_\beta)$ is the set of pair-wise incidences between the lines in $L_\alpha$ and $L_\beta$.
Alternatively, one can regard $S$ as fixed and find a new point-plane arrangement $(P',\Pi')$ in ${\mathbb P}^3$ with the same $m,n$ and the number of incidences, to which the above claim applies.
Besides, if $k_m,k_n$ are the maximum numbers of, respectively, collinear points and planes in $P,\Pi$, they are now the maximum numbers of lines in the families $L_\alpha,L_\beta$, respectively, contained in the intersection of $\mathcal{G}\subset S$ with a projective three-subspace in $S$. \end{lemma}
\begin{proof} Suppose, we have an incidence $(p,\pi)\in P\times\Pi$. This means that the $\alpha$-plane defined by $q\in P$ and the $\beta$-plane defined by $\pi\in \Pi$ intersect along a line in ${\mathcal K}$. There are at most $m^2+n^2$ points in ${\mathcal{K}}$ where planes of the same type meet and at most $mn$ lines along which the planes of different type may possibly intersect.
We must choose $\mathcal{G}$ that is a hyperplane $S$ in ${\mathbb P}^5$ intersecting $\mathcal{K}$ transversely, so that it supports none of the above lines or points in $\mathcal{K}$. This means avoiding a finite number of linear constraints on the dual vector $U^T\in {\mathbb P}^{5*}$ defining $S$. Since $\mathbb{F}$ is algebraically closed, it is infinite, and such $S$ always exists, for $m,n$ are finite. The covector $U^T$ defining $S$ must (i) not lie in the Klein quadric in ${\mathbb P}^{5*},$ and (ii) be such that $U^T L_i\neq 0$ for at most $m^2+n^2 + mn$ Pl\"ucker vectors $L_i$. There is a nonempty Zariski open set of such covectors in ${\mathbb P}^{5*}$.
To justify the second claim of the lemma we use the fact that there is one-to-one correspondence between so-called null polarities and regular line complexes. A null polarity is a projective transformation from ${\mathbb P}^3$ to its dual, given by a non-degenerate $4\times 4$ skew-symmetric matrix. The six above-diagonal entries of the matrix are in one-to-one correspondence with the covector defining the regular line complex. The fact that the skew-symmetric matrix is non-degenerate is precisely that the covector not lie in the Klein quadric. See \cite{PW}, Chapter 3 for general theory of line complexes.
Hence the following procedure is equivalent to the above-described one of choosing the transverse hyperplane $S$ defining $\mathcal{G}$. Fix $S$ and find a null polarity, whose application to the original arrangement of planes and points in ${\mathbb P}^3$ yields a new point-plane arrangement as follows. The roles of points and planes get reversed, and we now have the set of $m$ planes $\Pi'$ and the set of $n$ points $P'$, with the same number of incidences $|I(P,\Pi)|$. Take a dual arrangement so points become again points and planes are planes. However, no two lines of the same type, arising in $\mathcal{G}\subset S$ after the procedure described in the beginning of this section applied to the arrangement $(P',\Pi')$, will intersect.
The last claim of Lemma \ref{tog} follows from Lemma \ref{lem}. \end{proof}
Fixing the transverse hyperplane $S$ may be interesting for applications, when the affine part of the quadric $\mathcal{G}$ becomes, say the Lie group ${SL}_2(\mathbb{F})$, with its standard embedding in $\mathbb{F}^4$. Suppose there are $n$ lines supported in a fixed $\mathcal{G}$. Each line in $\mathcal{G}$ is a line in $\mathcal{K}$ and therefore corresponds to a unique plane pencil of lines in the ``physical space" ${\mathbb P}^3$, that is a unique pair $\alpha$ and $\beta$-plane intersecting along this line. I.e., there is a unique pair $(q,\pi(q))$, where the point $q$ lies in the plane $\pi(q)$. (Conversely, $\mathcal{G}$ viewed as a null polarity is defined by the linear skew-symmetric linear map $q\to\pi(q),$ see \cite{PW}, Chapter 3.) Hence, given a family of $n$ lines in $\mathcal{G}$, the problem of counting their pair-wise intersections can be expressed as counting the number of incidences in $I(P,\Pi)$, where $P=\{q\}$ and $\Pi=\{\pi(q)\}$. Moreover, $|P|,|\Pi|=n$, for two different planes of the same type will never intersect $\mathcal{G}$ along the same line (that is a null polarity is an isomorphism). Besides, if $k$ was the maximum number of lines in the intersection of $\mathcal{G}\subset {\mathbb P}^4$ with a ${\mathbb P}^3$, then the same $k$ stands for the maximum number of collinear points or planes, by Lemma \ref{lem}.
We have established the following statement.
\begin{lemma}\label{convert} Suppose, $\mathbb{F}$ is algebraically closed and not of characteristic $2$. Let $\mathcal L$ be a family of $n$ lines in $\mathcal{G}.$ Then there is an arrangement $(P,\Pi)$ of $n$ points and $n$ planes in ${\mathbb P}^3$, such that the number of pair-wise intersections of lines in $\mathcal L$ equals $|I(P,\Pi)|-n$. Moreover, there are two disjoint families of $n$ new lines in $\mathcal{G}$ each, such that lines within each family are mutually skew, and the total number of incidences is $|I(P,\Pi)|-n$. \end{lemma}
Note, the $-n$ comes from the fact that each $\pi(q)$ contains $q$. Lemma \ref{convert} and Theorem \ref{mish} have the following corollary. This fact also follows from the results in \cite{EH} after a projection argument. We present the proof along the lines of exposition in this section, for it also gives an application of the formalism here.
\begin{corollary}\label{intersections} The union of any $n=\Omega(p^2)$ straight lines in $G={SL}_2(\mathbb{F}_p)$ has cardinality $\Omega(p^3)$, that is takes up a positive proportion of $G$. \end{corollary} \begin{proof} The statement is trivial for small $p$, so let $p>2$. View lines in $G\subset \mathbb{F}_p^4$ as lines in $\mathcal{G}\subset {\mathbb P}^4$ over the algebraic closure of $\mathbb{F}_p$. Pass to a point-plane incidence problem in ${\mathbb P}^3$ using Lemma \ref{convert} and then by Lemma \ref{tog} back to a line-line incidence problem in $\mathcal{G}.$ We may change $n$ to $cn$ to make Theorem \ref{mish} applicable. The value of the absolute $c$ may be further decreased to justify subsequent steps. By the inclusion-exclusion principle one needs to show that the number of pair-wise intersections of lines is at most a fraction of $pn$. This would follow if one could apply the incidence bound \eqref{pups} with $m=n$ and, say $k=\frac{p}{2}$.
By Lemma \ref{tog} the quantity $k$ is the maximum number of ``new lines'' in the intersection of $\mathcal{G}$ with a projective three-hyperplane. Observe that there are more than $\frac{p}{2}$ of new lines in the intersection of $\mathcal{G}$ with a hyperplane if and only if there was the same number of ``old lines'' in the intersection of $G$ with an affine hyperplane in $\mathbb{F}_p^4$.
Let us throw away from the initial set of lines in $G$ those lines, contained in intersections of $G\subset \mathbb F_p^4$ with affine three-planes $H$, with $H\cap G$ having more than $\frac{p}{2}$ lines. Either we have a positive proportion of lines left, and no more rich hyperplanes $H$, or we have had $\Omega(p)$ quadric surfaces $H\cap G$ in $G$, with at least $\frac{p}{2}$ lines in each. In the former case, if $c$ is small enough, we are done by (\ref{pups}). In the latter case, by the inclusion-exclusion principle applied within each surface, the union of lines contained therein takes up a positive proportion of each $H\cap G$, i.e., has cardinality $\Omega(p^2)$. Since $H\cap H'\cap G$, $H\neq H'$ is at most two lines, by the inclusion-exclusion principle, the union of $\Omega(p)$ of them has cardinality $\Omega(p^3)$. \end{proof}
\section{Proof of Theorem \ref{mish}}
We use Lemma \ref{tog} to pass to the incidence problem between two disjoint line families $L_\alpha,L_\beta$ lying in $\mathcal{G}$, now using $m= |L_\alpha|$, $n= |L_\beta|$. Lines within each family are mutually skew.
All we need on the technical side is to consider the case $m\geq n$ and adapt the strategy of the proof of Theorem \ref{gkt} to the three-quadric $\mathcal{G}$ instead of ${\mathbb P}^3$. The latter is done via a generic projection argument, and the rest of the proof follows the outline in the opening sections. We skip some easy intermediate estimates throughout the proof, since they have been worked out accurately up to constants in \cite{Ko}, Sections 3,4.
The key issue is that any finite line arrangement over an infinite field in higher dimension can be projected into three dimensions with the same number of incidences; this fact is also stated in \cite{Ko}. Our lines lie in ${\mathbb P}^4$, containing the quadric $\mathcal{G}$. A pair of skew lines defines a three-hyperplane $H_i$ in ${\mathbb P}^4$. This hyperplane is projected one-to-one onto a fixed three-hyperplane $H$ if and only if the projective vector $u\in{\mathbb P}^4$ defining $H$ does not lie in $H_i$. Since we are dealing with a finite number of pairs of skew lines and $\mathbb{F}$ is infinite, the set of $u$, such that the projection of the line arrangement on the corresponding three-hyperplane $H$ acts one-to-one on the set of incidences is non-empty and Zariski open.
\begin{theorem}\label{gkt2}
Let $L_\alpha,L_\beta$ be two disjoint sets of respectively $m,n$ lines contained in the quadric $\mathcal{G}=\mathcal{K}\cap S$, where the hyperplane $S$ is not tangent to the Klein quadric $\mathcal{K}$. Suppose, lines within each family are mutually skew. Assume that $m\geq n$, $\mathbb{F}$ is algebraically closed, with characteristic $p\neq 2$. Let $n\leq cp^2,$ for some absolute $c$.
Then \begin{equation}\label{ibou}
|I(L_\alpha,L_\beta)|=O\left( m\sqrt{n}+ km\right), \end{equation} where $k$ is the maximum number of lines in $L_\beta$, contained in the intersection of $\mathcal{G}\subset {\mathbb P}^4$ with a subspace ${\mathbb P}^3$ in ${\mathbb P}^4$. \end{theorem}
\begin{proof}
Following Guth and Katz, it is technically very convenient to use induction in $\min(m,n)$ and a probabilistic argument. The estimate $I=O( m\sqrt{n} )$ is true for all sufficiently small $m,n$, given a sufficiently large $O(1)$ value $C$ of the constant in the $O$-symbol, which we fix. We do not specify how large $C$ should be, however Koll\'ar evaluates it explicitly, see \cite{Ko}. For the induction assumption to work throughout let us reset $n = \min(|L_\alpha|,|L_\beta|)$ and $m = \max(|L_\alpha|,|L_\beta|)$. The induction assumption will be used throughout the proof as the bound for incidences between sub-families of $(m',n')$ lines, with $n'$ sufficiently less than $n$, no matter what $m'$ is. This will enable us to exclude from consideration the incidences that some undesirable subsets of lines in $L_\beta$ account for, as long as they constitute a reasonably small fraction of $L_\beta$ itself.
Suppose, we have the smallest value of $n$, such that for some $m\geq n$ the main term in the right-hand side of \eqref{ibou} fails to do the job, that is \begin{equation}
|I(L_\alpha,L_\beta)| = Cm\sqrt{n},\label{contr}\end{equation} for some large enough constant $C$. We will show that this assumption implies the bound $I=O(km)$, independent of $C$, which will therefore finish the proof.
Note that since the right-hand side of the assumption \eqref{contr} is linear in $m$, it implies, by the pigeonhole principle, that there is a subset $\tilde L_\alpha$ of $L_\alpha$ of some $\tilde m\leq m$ lines, with $\tilde m=O(n)$, such that
$$|I(\tilde L_\alpha, L_\beta)| \geq C\tilde m\sqrt{n}.$$ We reset the notations $\tilde L_\alpha$ to $L_\alpha$ and $\tilde m$ back to $m$, but now $m=O(n)$, which is necessary for the next step.
A large proportion of incidences must be supported on lines in $L_\alpha$, which are intersected not much less than average, say by at least $\frac{1}{4} C \sqrt{n}$ lines from $L_\beta$ each. Let us call this popular set $L'_\alpha$. We now delete lines from $L_\beta$ randomly and independently, with probability $1-\rho$ to be chosen. Let the random surviving subset of $L_\beta$ be denoted as $\tilde L_\beta$. By the law of large numbers, the probability that an individual line in $L'_\alpha$ is met by lines from $\tilde L_\beta$ less than half the expected number of times is exponentially small in $n$, and so is $m$ times this probability, since now $m=O(n)$. Thus there is a realisation of $\tilde L_\beta \subset L_\beta$, of size close to the expected one, i.e., between $\frac{1}{2}\rho n$ and $2\rho n$ such that every line in $L'_\alpha$ meets at least, say \begin{equation}\label{avrg}\frac{1}{8} C \rho \sqrt{n}\end{equation} lines in $\tilde L_\beta$.
Our lines live in $\mathcal{G}\subset{\mathbb P}^4$, with homogeneous coordinates $(x_0:\ldots:x_4)$. By the projection argument, preceding the formulation of Theorem \ref{gkt2}, the coordinates can be chosen in such a way that lines in the union of the two families project one-to-one as lines in the $(x_1:\ldots:x_4)$-space, and skew lines remains skew.
Let $Q$ be a nonzero homogeneous polynomial in $(x_1:\ldots:x_4)$ that vanishes on the projections of the lines in $\tilde L_\beta$ to the $(x_1:\ldots:x_4)$-space, so it will also vanish on the lines in $\tilde L_\beta$.
The degree $d$ of $Q$ can be taken as $O\left((\rho n)^{\frac{1}{2}}\right)$. This fact is well known, see e.g. the survey \cite{D1}. For completeness, we give a quick argument. Choose $t$ points on each of the projected lines from $\tilde L_\beta$, with or without repetitions. Let $X\subset{\mathbb P}^3$ be the corresponding set of at most $t|\tilde L_\beta|$ points. There is a nonzero homogeneous polynomial of degree $d= O[ (t|\tilde L_\beta|)^{1/3} ]$ vanishing on $X$. More precisely, it suffices to satisfy the inequality $\left(\begin{array}{c} d+3\\3\end{array}\right)>|X|$ for the degree of the polynomial. The left-hand side of the latter inequality is the dimension of the vector space of degree $d$ homogeneous polynomials in four variables; if it is bigger than $|X|$, the evaluation map on $X$ has nontrivial kernel, by the rank-nullity theorem.
By construction of the point set $X$, the polynomial $Q$ has $t$ zeroes on each line from $\tilde L_\beta$, so in order to have it vanish identically on the union of these lines one must merely ensure that $t>d$. Hence, the above claim for $d$.
We choose the parameter $\rho$, so that the degree $d$ of $Q$ is smaller than the number of its zeroes on each line in $L_\alpha'$, which is at least \eqref{avrg}. I.e., $$\rho =O\left(\frac{1}{C^2}\right)<1,$$
and thus \begin{equation}\label{d1}d=O(\sqrt{\rho n}) =O\left( \frac{\sqrt{n}}{C}\right). \end{equation}
Reduce $Q$ to the minimal product of irreducible factors. Denote $\bar Z$ the zero set of the polynomial $Q$ in ${\mathbb P}^3$ defined by the $(x_1:\ldots:x_4)$ variables and $\bar L'_\alpha, \bar L'_\beta$ the projections of the corresponding line families. Let also $Z$ denote the zero set of the polynomial $Q$ in $\mathcal{G}\subset{\mathbb P}^4$. Recall that the projection has been chosen so that $|I(\bar L'_\alpha, \bar L'_\beta)|=|I(L'_\alpha, L'_\beta)|$ and lines in the same family still do not meet. In the sequel, when we speak of zero sets of factors of $Q$, we mean point sets in ${\mathbb P}^3$, in the $(x_1:\ldots:x_4)$ variables.
It follows that all the lines in $\bar L'_\alpha$ are contained in $\bar Z$, for each supports more zeroes of $Q$ than the degree $d$. For all lines from $\bar L_\beta$ that do not live in $\bar Z$, every such line will intersect $\bar Z$ at most $d$ times. The number of incidences these lines can create altogether is thus \begin{equation}O\left( C^{-1} n^{\frac{3}{2}} \right) = O\left( C^{-1} m\sqrt{n}\right),\label{transverse}\end{equation} which is too small in comparison with the supposedly large total number of incidences \eqref{contr}. Therefore, we may assume that, say at least $\frac{1}{2} C m\sqrt{n}$ incidences are supported on lines in $\bar L'_\alpha$ and those lines from $\bar L_\beta$ that are also contained in $\bar Z$. Suppose, the number of the latter lines is less than, say $\frac{n}{16}$. This will contradict the induction assumption -- no matter how many lines $m'$ are there in $\bar L'_\alpha$. If $m'\geq n$, then the number of incidences, by the induction assumption, must be at most $Cm'\sqrt{n}/4$; if $m'<\frac{n}{16}$, it is at most $C n \sqrt{m'}/16<C m \sqrt{n}/16.$
Hence, there are at least $\frac{n}{16}$ lines from $\bar L_\beta$ in $\bar Z$, and we call the set of these lines $\bar L'_\beta$. To avoid taking further fractions of $n$, let us proceed assuming that $|\bar L'_\beta|=n$.
We can repeat the transverse intersection incidence counting argument for the zero set of each irreducible factor of $Q$. Suppose, the factor has degree $d'$. Then the number of incidences of lines in the zero set $\bar Z'$ of the factor with those not contained in $\bar Z'$ is at most $d' (m+n)$. Summing over the factors, we can use the right-hand side of \eqref{transverse} as the estimate for the total over all the irreducible factors of $Q$ number of transverse incidences. We therefore proceed assuming that there are $\Omega( C m\sqrt{n})$ of pairs of intersecting lines from the two families, each incidence occurring inside the zero set of some irreducible factor of $Q$.
Invoking Salmon's Theorem \ref{Salmon} we deduce that if $n>11d^2-24d$, and given that $d<p$ if the characteristic $p>0$, the zero set $\bar Z$ of the polynomial $Q$ must have a ruled factor. The latter inequality entails that almost 100\% of lines in the $\beta$-family must lie in ruled factors. Indeed, we have $|\bar L'_\beta|=n$ lines in $Z$, and at most $11d^2=O(n/C^2)$ may lie in the union of non-ruled factors, provided that $d=O(\sqrt{n}/C)< p$, that is the constraint in Theorem \ref{Salmon} has been satisfied. Thus, we may not bother about what happens in non-ruled factors of $\bar Z$ by the induction assumption and proceed, having redefined $n$ slightly one more time, so that now $n$ lines from $\bar L'_\beta$ lie in ruled factors of $\bar Z$. They still have to account for $\Omega(Cm\sqrt{n})$ incidences with the lines from $\bar L'_\alpha$, for all the lines in $\bar L'_\beta$ that have been disregarded so far could only account for a small percentage of the total number of incidences.
A single ruled factor cannot be a cone, for no more than two of our lines meet at a point. However, a ruled factor of degree $d'>2$, which is not a cone, can contribute, by Lemma \ref{rss}, at most $n(d'-2)+2n+(m+n)d'$ incidences. The latter three summands come, respectively, from mutual intersections of generators, intersections of generators with special lines -- see the discussion from Lemma \ref{rss} through the end of Section \ref{ruled} -- and intersections of lines within the factor with lines outside the factor.
Once again, summing over irreducible ruled factors with $d'>2$, we arrive in the right-hand side term in \eqref{transverse} again -- this is too small in comparison with \eqref{contr}. Hence $Q$ must contain one or more irreducible factors $Q'$ of degree at most $2$, that is the zero set of each such $Q'$ is an irreducible doubly-ruled quadric or a plane in ${\mathbb P}^3$. If the union of these low degree factors contains only a small proportion of the lines from $\bar L'_\beta$, we once again invoke the induction assumption and contradict \eqref{contr}.
Let us reset $n$ to its original value. The argument up to now has calmed that if \eqref{contr} is true, we have at least $cn$ lines from $\bar L'_\beta$ lying in the union of the zero sets of low degree -- meaning degree at most two -- factors of $Q$, creating at least $cCm\sqrt{n}$ incidences with lines from $\bar L'_\alpha$ inside these factors. By the pigeonhole principle, there is a low degree factor $Q'$, whose zero set contains at least $c\frac{n}{d}=\Omega(C\sqrt{n})$ lines from $\bar L'_\beta$. Moreover, we can disregard whatever happens inside the union of low degree factors, each containing fewer than some $cC\sqrt{n}$ lines from $\bar L'_\beta$, by the induction assumption.
The contribution of plane factors of $Q$ is negligible, for each plane in ${\mathbb P}^3$ may contain only one line from each (projected) family. Thus there is a rich degree $2$ irreducible factor $Q'$, which defines a doubly ruled quadric surface $\bar Z'$ in the $(x_1:\ldots:x_4)$ variables. $\bar Z'$ supports at least two lines from $\bar L'_\alpha$ in one ruling, for otherwise the total number of incidences within all such rich quadrics would be $O(C^{-1}n^{\frac{3}{2}})$. These two lines are crossed by all lines in the second ruling, that is by $\Omega(C\sqrt{n})$ lines from the family $\bar L'_\beta.$
It remains to bring the parameter $k$ in, the maximum number of lines from $L_\beta$, per intersection of $\mathcal{G}\subset {\mathbb P}^4$ with a three-hyperplane. Let $Z'=\mathcal{G}\cap (\bar Z'\times {\mathbb P}^1)$, that is the intersection of the quadric $\mathcal{G}$ with the quadric, which is the zero set of $Q'$ in ${\mathbb P}^4$. Lifting lines from $\bar Z'$ to $Z'$ preserves incidences, so we arrive at the following figure in $Z'\subset \mathcal{G}$: a pair of skew lines from $L_\alpha$ crossed by $\Omega(C\sqrt{n})$ lines from $L_\beta$. The two lines from $L_\alpha$ determine a three-hyperplane $H$, which also contains all the $\Omega(C\sqrt{n})$ lines in question from $L_\beta$.
By the assumption of the theorem, $H$ may contain at most $k$ lines from $L_\beta$. This means $C=O\left(\frac{k}{\sqrt{n}}\right)$. Substituting this into \eqref{contr} yields the inequality $ |I(L_\alpha,L_\beta)|=O(km).$ This completes the proof of Theorem \ref{gkt2}. \end{proof}
Theorem \ref{gkt2} together with the preceding it discussion in Sections \ref{setup} and \ref{geom} and its outcomes stated as Lemma \ref{lem} and \ref{tog}, result straight into the claim of our main Theorem \ref{mish}.
\section{Applications of Theorem \ref{mish}} This section has three main parts. First, we develop an application of Theorem \ref{mish} to the problem of counting vector products defined by a plane point set, extending to positive characteristic the estimates obtained over ${\mathbb R}$ via the Szemer\'edi-Trotter theorem. Then we use that application in a specific example to show that in a certain parameter regime Theorem \ref{mish} is tight. Finally, we use Theorem \ref{mish} to consider a pinned version of the Erd\H os distance problem on the number of distinct distances determined by a set of $N$ points in $\mathbb{F}^3$, where we also get a new bound in positive characteristic, which is not too far off the best known bound over the reals.
Before we do this, we state a slightly stronger version of Theorem \ref{mish}, which is more tuned for applications. The need for it comes from the fact that sometimes, when questions of geometric and arithmetic combinatorics are reformulated as incidence problems, there are certain geometrically identifiable subsets of the incidence set that should be excluded from the count, for they correspond to some in some sense ``pathological'' scenario. We encountered this in \cite{RR}, where the Guth-Katz approach to the the Erd\H os distance problem was applied to Minkowski distances in the real plane. In order to get the lower bound for the number of distinct Minkowski distances, one claims an upper bound on the number of pairs of congruent, that is equal Minkowski length line segments with endpoints in the given plane point set. However, it is easy to construct an example where the number of pairs of zero Minkowski length segments is forbiddingly large. Hence, the analysis in \cite{RR} considered only nonzero Minkowski distances, and had to elucidate how this fact gets reflected in the corresponding incidence problem for lines in three dimensions. Discounting pairs of line segments of zero Minkowski length was equivalent to discounting pair-wise line intersections within a set of specific two planes in $3D$; these planes could violate the assumption of Theorem \ref{gkt} about the maximum number of coplanar lines.
Such a restricted application of the Guth-Katz approach was further generalised in \cite{RS}, where more $2D$ combinatorial problems have been identified, where the tandem of incidence Theorems 2.10 and 2.11 from \cite{GK} worked ``as a hammer'', if used in the restricted form, that is discounting pairwise line intersections within certain ``bad'' planes, as well as at certain ``bad'' points.
Technically, it is Theorem 2.11 from \cite{GK}, whose restricted version required most of the work in \cite{RR}; adapting Theorem 2.10 took only a few lines of argument, and this is all that is essentially needed here regarding Theorem \ref{mish}, where we wish to discount point-plane incidences supported on a certain set of forbidden lines in ${\mathbb P}^3$.
Suppose, we have a finite set of lines $L^*$ in ${\mathbb P}^3$. Define the restricted set of incidences between a point set $P$ and set of planes $\Pi$ as \begin{equation}\label{inss} I^*(P,\Pi) = \{(q,\pi)\in P\times \Pi: q\in \pi \mbox{ and } \forall l\in L^*, \,q\not \in l \mbox{ or } l \not\subset\pi\}. \end{equation}
\addtocounter{theorem}{-10}
\renewcommand{\arabic{theorem}}{\arabic{theorem}*} \begin{theorem} \label{mishh} Let $P, \Pi$ be sets of points and planes in ${\mathbb P}^3$, of cardinalities respectively $m,n$, with $m\geq n$. If $\mathbb{F}$ has positive characteristic $p$, then $p\neq 2$ and $n=O(p^2)$. For a finite set of lines $L^*$, let $k^*$ be the maximum number of planes, incident to any line not in $L^*$.
Then \begin{equation}\label{pupss}
|I^*(P,\Pi)|=O\left( m\sqrt{n}+ k^*m\right).\end{equation} \end{theorem} \begin{proof} We return to Section \ref{geom} to map the incidence problem between points and planes to one between line families $L_\alpha,L_\beta$ in $\mathcal G\subset {\mathbb P}^4$. By Lemmas \ref{lem}, \ref{tog} the set of lines $L^*$ now displays itself as a set ${\mathcal H}^*$ of three-hyperplanes in ${\mathbb P}^4$. One comes to Theorem \ref{gkt2}, only now aiming to claim \eqref{pupss} as the estimate for the cardinality of the restricted incidence set $I^*(L_\alpha,L_\beta)$, which discounts pair-wise line intersections within the intersections of $\mathcal{G}$ with each $h\in \mathcal H^*$, $k^*$ replacing $k$.
The proof of Theorem \ref{gkt2} is modified as follows. Since the number of bad hyperplanes is finite, one can choose coordinates so that the intersection of each $h\in \mathcal H^*$ with $\mathcal{G}$ is defined by a quadratic polynomial $Q_h$ in $(x_1:\ldots:x_4)$. The arguments of Theorem \ref{gkt2} are copied modulo that one assumes \eqref{contr} about the quantity $|I^*(P,\Pi)|$ and having reduced the problem to counting incidences only inside factors of a polynomial $Q$ of degree satisfying \eqref{d1}, does not take into account incidences in common factors of $Q$ and $\prod_{h\in \mathcal H} Q_h$. As a result, the modified assumption \eqref{contr} forces one to have a rich irreducible degree $2$ factor of $Q$, which is not forbidden. This corresponds, within Theorem \ref{gkt2} to $\Omega(C\sqrt{n})$ lines from the family $L_\beta$ lying inside the intersection of $\mathcal{G}$ with some three-hyperplane $H\not\in \mathcal H^*$. In terms of Theorem \ref{mishh} this means collinearity of $\Omega(C\sqrt{n})$ planes in $\Pi$ along some line not in $L^*$. This establishes the estimate \eqref{pupss}.\end{proof}
\renewcommand{\arabic{theorem}}{\arabic{theorem}} \addtocounter{theorem}{9}
Throughout the rest of the section, $\mathbb{F}$ is a field of odd characteristic $p$.
\subsection{On distinct values of bilinear forms}\label{vpr} Established sum-product type inequalities over fields with positive characteristic have been weaker than over ${\mathbb R}$, where one can take advantage of the order structure and use geometric, rather than additive combinatorics. See, e.g., \cite{E0}, \cite{So}, \cite{KR}, \cite{BJ}, \cite{KS} for some key methods and ``world records''.
The closely related geometric problem discussed in this section is one of lower bounds on the cardinality of the set of values of a non-degenerate bilinear form $\omega$, evaluated on pairs of points from a set $S$ of $N$ non-collinear points in the plane. One may conjecture the bound $\Omega(N)$, possibly modulo factors, growing slower than any power of $N$. This may clearly hold in full generality in positive characteristic only if $N=O(p)$.
The problem was claimed to have been solved over ${\mathbb R}$ up to the factor of $\log N$ in \cite{IRR}, $\omega$ being the cross or dot product. However, the proof was flawed. The error came down to ignoring the presence of nontrivial weights or multiplicities, as they appear below. The best bound over ${\mathbb R},\mathbb{C}$ that the erratum \cite{IRRE} sets is $\Omega(N^{9/13})$, for a skew-symmetric $\omega$. The bound $\Omega(N^{2/3})$ for any non-degenerate form $\omega$ follows just from applying the Szemer\'edi-Trotter theorem to bound the number of realisations of any particular nonzero value of $\omega$.
In this section we prove the following theorem. \begin{theorem}\label{spr} Let $\omega$ be a non-degenerate symmetric or skew-symmetric bilinear form and the set $S\subseteq \mathbb{F}^2$ of $N$ points not be supported on a single line. Then
\begin{equation}|\omega(S) := \{\omega(s,s'):\,s,s'\in S\}| = \Omega\left[\min \left(N^{\frac{2}{3}},p\right)\right].\label{worst}\end{equation}
If $S$ has a subset $S'$ of $N'<p$ points, lying in distinct directions from the origin, then $|\omega(S)|\gg N'.$ \end{theorem}
\begin{proof} From now on we assume that $S$ does not have more than $N^{\frac{2}{3}}$ points on a single line through the origin, for since $S$ also contains a point outside this line, the estimate \eqref{worst} follows. This assumption will be seen not to affect the second claim of the theorem. Suppose also, without loss of generality, that $S$ does not contain the origin, nor does it have points on the two coordinate axes.
We may assume that $\mathbb{F}$ is algebraically closed, in which case one may take a symmetric form $\omega$ as given by the $2\times 2$ identity matrix and a skew-symmetric one by the canonical symplectic matrix. We consider the latter situation only. The former case is similar. One can also replace $S$ with its union with $S^\perp=\{(-q_2,q_1):\,(q_1,q_2)\in S\}$ and repeat the forthcoming argument.
Consider the equation \begin{equation}\label{eng} \omega(s,s')=\omega(t,t')\neq 0, \qquad (s,s',t,t')\in S\times S\times S\times S. \end{equation}
Assuming that $\omega$ represents wedge products, this equation can be viewed as counting the number of incidences between the set of points $P\subset {\mathbb P}^3$ with homogeneous coordinates $(s_1:s_2:t_1:t_2)$ and planes in a set $\Pi$ defined by covectors $(s_2':-s_1':-t_2':t_1')$. However, both points and planes are weighted. Namely, the weight $w(p)$ of a point $p=(s:t)$ is the number of points $(s,t)\in \mathbb{F}^4$, which are projectively equivalent that is lie on the same line through the origin. The same applies to planes. The total weight of both sets of points and planes is $W=N^2$. Like in the case of the Szemer\'edi-Trotter theorem, the weighted variant of estimate of Theorem \ref{mish} gets worse with maximum possible weight.
The number of solutions of \eqref{eng}, plus counting also quadruples yielding zero values of $\omega$ is the number of weighted incidences \begin{equation}\label{weightin} I_w := \sum_{q\in P, \pi\in \Pi} w(q)w(\pi) \delta_{q\pi}, \end{equation} where $\delta_{q\pi}$ is $1$ when $q\in \pi$ and zero otherwise.
Consider two cases: {\em (i) $S$ only has points in $O(N^{2/3})$ distinct directions through the origin; (ii) there exists $S'\subset S$ with exactly one point in $\Omega(N^{2/3})$ distinct directions.}
To deal with (i) we need the following weighted version of Theorem \ref{mish}. \begin{theorem}\label{wmish}
Let $P, \Pi$ be weighted sets of points and planes in ${\mathbb P}^3$, both with total weight $W$. Suppose, maximum weights are bounded by $w_0>1$. Let $k$ be the maximum number of collinear points, counted without weights. Suppose, $\frac{W}{w_0}=O(p^2)$, where $p>2$ is the characteristic of $\mathbb{F}$. Then the number $I_w$ of weighted incidences is bounded as follows: \begin{equation}\label{pupsweight} I_w=O\left( W\sqrt{w_0W}+ k w_0 W\right).\end{equation} The same estimate holds for the quantity $I^*_w$, which discounts weighted incidences along a certain set $L^*$ of lines in ${\mathbb P}^3$, the quantity $k^*$, denoting the maximum number of points incident to a line not in $L^*$ replacing $k$ in estimate \eqref{pupsweight}. \end{theorem}
\begin{proof} It is a simple weight rearrangement argument, the same as, e.g., in \cite{IKRT} apropos of the Szemer\'edi-Trotter theorem. Pick a subset $P'\subseteq P$, containing $n=O\left(\frac{W}{w_0}\right)$ richest points in terms of non-weighted incidences. Assign to each one of the points in $P'$ the weight $w_0$, delete the rest of the points in $P$, so $P'$ now replaces $P$. The number of weighted incidences will thereby not decrease. Now of all planes pick a subset $\Pi'$ of the same number $n$ of the richest ones, in terms of their non-weighted incidences with $P'$. Assign once again the weight $w_0$ to each plane in $\Pi'$. We now replace $P,\Pi$ with $P',\Pi'$ -- the sets of respectively $n$ points and planes, for which we apply Theorem \ref{mish}, counting each incidence $w_0^2$ times. Note that we may still have $k$ collinear points in $P'$ or planes in $\Pi'$. This yields \eqref{pupsweight}.
For the last claim of Theorem \ref{wmish} we use Theorem \ref{mishh} instead of Theorem \ref{mish}.\end{proof}
Returning to the proof of Theorem \ref{spr}, suppose we are in case (i). We will apply the $I_w^*$ estimate of Theorem \ref{wmish} to the weighted arrangement of planes and points in ${\mathbb P}^3$, representing \eqref{eng}. Let us show that the quantity $k^*$ can be bounded as $O(N^{\frac{2}{3}})$, after it becomes clear what the set $L^*$ of forbidden lines is. The quantity $k$ is the maximum number of collinear points in the set $S\times S\in \mathbb{F}^4$, viewed projectively. Suppose, $k\geq N^{\frac{2}{3}}$. This means we have a two-plane through the origin in $\mathbb{F}^4$, which contains points of $S\times S$ in at least $N^{\frac{2}{3}}$ directions in this plane. If this two-plane projects on the first two coordinates in $\mathbb{F}^4$ one-to-one, then $S$ itself has points in $N^{\frac{2}{3}}$ directions. But in case (i) this is not the case.
We now define the finite set $L^*$ of forbidden lines in ${\mathbb P}^3$ as two-planes in $\mathbb{F}^4$, which are Cartesian products of pairs of lines through the origin in $\mathbb{F}^2$, each supporting a point of $S$. Hence $k^*$ is the maximum number of points incident to any other line in ${\mathbb P}^3$. If the two-plane through the origin in $\mathbb{F}^4$ projects on each coordinate two-plane $\mathbb{F}^2$, containing $S$ as a line through the origin, it is a Cartesian product of two lines $l_1$ and $l_2$ through the origin in $\mathbb{F}^2$. Such a plane may contain a point $(s_1,s_2,t_1,t_2)\in \mathbb{F}^4$ or be incident to a three-hyperplane through the origin in $\mathbb{F}^4$, defined by the covector $(s_2', -s_1', -t_2', t_1')=0$ only if the lines $l_1,l_2$ contain points of $S$.
Applying the $I_w^*$-version of estimate \eqref{pupsweight}, we therefore obtain \begin{equation}\label{weste} I^*_w=O\left(N^{\frac{10}{3}} + N^{\frac{10}{3}}\right). \end{equation}
It remains to show that point-plane incidences along the lines in $L^*$ correspond to zero values of the form $\omega$ in \eqref{eng}. By definition, a line in $L^*$ is represented by a pair $(l_1,l_2)$ lines through the origin in $\mathbb{F}^2$. If the $\mathbb{F}^4$-point $(s,t)=(s_1,s_2,t_1,t_2)$ lies in the two-plane, which is the Cartesian product $l_1\times l_2$, this means $s\in l_1$, $t\in l_2$. If a three-hyperplane through the origin in $\mathbb{F}^4$, defined by the covector $(s_2', -s_1', -t_2', t_1')=0$ contains both lines $l_1,l_2$, this means $s'\in l_1$, $t'\in l_2$. Hence $\omega(s,s')=\omega(t,t')=0.$
So, if case (i) takes place, the bound \eqref{worst} follows from \eqref{eng} and \eqref{weste} by the Cauchy-Schwarz inequality. Observe that Theorem \ref{wmish} applies under the constraint $N\leq cp^{\frac{3}{2}}$ for some absolute $c$. In particular, when $N= \lfloor cp^{\frac{3}{2}}\rfloor$, it yields $I_w=O(p^5)$, hence one has $\Omega(N^{\frac{2}{3}})= \Omega(p)$ distinct values of the form $\omega$. For $N\geq cp^{\frac{3}{2}}$ we do no more than retain this estimate.
Finally, if case (ii) takes place, we apply Theorem \ref{mish} to the set $S'$. For now planes and points bear no weights other than $1$, and the above argument about collinear planes and points applies. Namely one can set $k=N'$ and zero values of $\omega$ may no longer be excluded. Then equation \eqref{eng} with variables in $S'$ alone has $O({N'}^3)$ solutions, and the last claim of Theorem \ref{spr} follows by the Cauchy-Schwarz inequality. \end{proof}
It is easy to adapt the proof of Theorem \ref{spr} to the special case when $S=A\times B$ for then one can set $w_0 = \min(|A|,|B|)$. This results in the following corollary. There is also a more economical way of deriving the following statement from Theorem \ref{mish}. See \cite{RRS}, Corollary 4.
\begin{corollary}\label{hbk} Let $A,B\subseteq \mathbb{F}$, with $|A|\geq|B|$.
Then
\begin{equation} |AB\pm AB|=\Omega\left[\min\left(|A|\sqrt{|B|},p\right)\right].\label{mebd}\end{equation} \end{corollary}
\subsection{Tightness of Theorem \ref{mish}} \label{example} We use the considerations of the previous section, looking at the number of distinct dot products of pairs of vectors in the set $$ S=\{(a,b):\,a,b\in[1,\ldots, n]:\; \mbox{gcd}(a,b)=1\}. $$ The set can be thought of lying in ${\mathbb R}^2$ or $\mathbb{F}_p^2$, for $p\gg n^2$. Clearly, $S$ has $N=\Theta(n^2)$ elements.
But now there are no weights in excess of $1$, in the sense of the discussion in the preceding section. So we can apply the argument from case (ii) within the proof of Theorem \ref{spr} and get a $O(N^3)$ bound for the number of solutions $E$ of the equation, with the standard dot product, \begin{equation} s\cdot s' = t\cdot t', \qquad (s,s',t,t')\in S\times S\times S\times S. \label{integers}\end{equation} Note that zero dot products can only contribute $O(N^2)$.
On the other hand, the same, up to constants, bound for $E$ from below follows by the Cauchy-Schwarz inequality. Indeed, $x=s\cdot s' $ in equation \eqref{integers} assumes integer values in $[1\ldots4n^2]$. If $n(x)$ is the number of realisations of $x$, one has $$E=\sum_x n^2(x) \geq \frac{1}{4n^2} \left(\sum_x n(x)\right)^2 \gg n^6\gg N^3.$$
\subsection{On distinct distances in $\mathbb{F}^3$} \newcommand{\boldsymbol s}{\boldsymbol s} \newcommand{\boldsymbol t}{\boldsymbol t} Once again in this section $\mathbb{F}$ is an algebraically closed field of positive characteristic $p>2$.
The Erd\H os distance conjecture is open in ${\mathbb R}^3$, where it claims that a set $S$ of $N$ points determines $\Omega(N^{\frac{2}{3}})$ distinct distances\footnote{The conjecture is often formulated more cautiously, that there are $\Omega^*(N^{\frac{2}{3}})$ distinct distances, the symbol $\Omega^*$ swallowing terms, growing slower than any power of $N$.}. The best known bound in ${\mathbb R}^3$ is $\Omega(N^{.5643})$, due to Solymosi and Vu \cite{SV}.
We prove the bound $\Omega(\sqrt{N})$ for the positive characteristic pinned version of the problem, i.e., for the number of distinct distances, attained from some point $\boldsymbol s\in S$, for $N=O(p^2)$, assuming that $S$ is not contained in a single semi-isotropic plane, as described below.
Define the distance set
$$\Delta(S) = \{\|\boldsymbol s-\boldsymbol t\|^2:\,\boldsymbol s,\boldsymbol t \in S\},$$
with the notation $\boldsymbol s=(s_1,s_2,s_3)$, $\|\boldsymbol s\|^2 = s_1^2+s_2^2+s_3^2.$ Let us call a pair $(\boldsymbol s,\boldsymbol t)$ a {\em null-pair} if $\|\boldsymbol s-\boldsymbol t\|=0$.
In positive characteristic, the space $\mathbb{F}^3$ (even if $\mathbb{F}=\mathbb{F}_p$) always has a cone of {\em isotropic directions} from the origin, that is $\{\boldsymbol \omega\in \mathbb{F}^3:\,\boldsymbol \omega\cdot \boldsymbol \omega=0\}$, with respect to the standard dot product. See \cite{HI}, in particular Theorem 2.7 therein for explicit calculations of isotropic vectors and their orthogonal complements over $\mathbb{F}_p$.
The equation for the isotropic cone through the origin in $\mathbb{F}^3$ is clearly \begin{equation}\label{cone} x^2+y^2+z^2=0. \end{equation} It is a degree two ruled surface, whose ruling is not a regulus, see Section \ref{rgl}.
If $\boldsymbol e_1$ is an isotropic vector through the origin, its orthogonal complement $ \boldsymbol e_1^\perp$ is a plane, containing $\boldsymbol e_1$. Let $\boldsymbol e_2$ be another basis vector in this plane, orthogonal to $\boldsymbol e_2$. Then $\boldsymbol e_2$ is not isotropic, for otherwise the whole plane $\boldsymbol e_1^\perp$ would be isotropic. This is impossible, for equation \eqref{cone} is irreducible. We call the plane $\boldsymbol e_1^\perp$ or its translate {\em semi-isotropic.}
The fact that $\boldsymbol e_2$ is not isotropic implies that there are no {\em nontrivial null triangles} that is triangles with three zero length sides, unless the three vertices lie on an isotropic line. With this terminology, there exist only {\em trivial} null triangles in $\mathbb{F}^3$.
In a semi-isotropic plane one can have $N=kl$ points, with $1\leq k\leq l$, with just $O(k)$ distinct pairwise distances: place $l$ points on each of $k$ parallel lines in the direction of $\boldsymbol e_1$, whose $\boldsymbol e_2$-intersects are in arithmetic progression.
To deal with zero distances we use the following lemma. \begin{lemma} \label{easy} Let $T$ be a set of $K$ points on the level set $$Z_R=\{(x,y,z):\,x^2+y^2+z^2=R\}.$$ For $K\gg1$ sufficiently large, either $\Omega(K)$ points in $T$ are collinear, or a possible proportion of $(\boldsymbol t,\boldsymbol t')\in T\times T$ are not null pairs. \end{lemma} \begin{proof} First note, as an observation, that even if $R\neq 0$, when $Z_R$ is a doubly-ruled quadric it may well be ruled by isotropic lines. Indeed, representing lines in $\mathbb{F}^3$ by Pl\"ucker vectors $(\boldsymbol \omega:\boldsymbol v)$ in the Klein quadric $\mathcal{K}$, defined by the relation \eqref{Klein}, i.e., $\boldsymbol \omega\cdot\boldsymbol v =0$, isotropic vectors are cut out by the quadric $\boldsymbol \omega\cdot\boldsymbol \omega=0$, while a regulus is a conic curve cut out from $\mathcal{K}$ by a two-plane. If the intersection of the three varieties in question is non-degenerate, it is at most four points, that is there are at most four isotropic lines per regulus.
However, take the two-plane as $\boldsymbol v =\lambda \boldsymbol \omega$, for some $\lambda\neq 0$. (The case $\lambda=0$ corresponds to the isotropic cone through the origin.) Write $\boldsymbol v=ad(\boldsymbol q)\,\boldsymbol \omega$, for some point $\boldsymbol q\in \mathbb{F}^3$ lying on the line in question, where $ad(\boldsymbol q)$ is a skew-symmetric matrix, see Footnote \ref{ads}. This yields the eigenvalue equation $\det(ad(\boldsymbol q) - \lambda I) =0,$ which means that $\boldsymbol q$ satisfies $$
\lambda^2+ \|\boldsymbol q^2\|=0, $$ that is $\boldsymbol q\in Z_{-\lambda^2},$ the point set of a regulus of isotropic lines.
Turning to the actual proof of the lemma, consider a simple undirected graph $G$ with the vertex set $T$, where there is an edge connecting distinct vertices $\boldsymbol t$ and $\boldsymbol t'$ if $(\boldsymbol t,\boldsymbol t')$ is a null pair. Suppose $G$ is close to the complete graph, that is $G$ has at least $.99 K(K-1)/2$ edges.
Note that $K'\leq K$ points of $T$, lying on an isotropic line, yield a clique of size $K'$ in $K$. Suppose there is no clique of size, say $K'\geq.01K,$ or we are done.
Then in each clique of size $K'$ one can delete at most $(K'+1)^2/4$ edges, turning it into a bipartite graph, whereupon there are no triangles left within that clique. After that one is left with no triangles in $G$, corresponding to trivial null triangles in $Z_R$.
However if $K'\leq .01K$, the number of remaining edges is still greater than $K^2/4$, clearly the former cliques had no edges in common. By Turan's theorem there is a triangle in what is left of $G$, and it corresponds to a nontrivial null triangle in $Z_R$.
This contradiction finishes the proof of Lemma \ref{easy}
\end{proof}
We are now ready to prove the last theorem in this paper. \begin{theorem}\label{erd} A set $S$ of $N$ points in $\mathbb{F}^3$, such that all points in $S$ do not lie in a single semi-isotropic plane, determines $\Omega[\min(\sqrt{N},p)]$ distinct pinned distances, i.e., distances from some fixed $\boldsymbol s\in S$ to other points of $S$. \end{theorem}
\begin{proof} First off, let us restrict $S$, if necessary, to a subset of at most $cp^2$ points, where $c$ is some small absolute constant, later to enable us to use Theorem \ref{mish}. We keep using the notation $S$ and $N$. Furtermore, we assume that $S$ has at most $\sqrt{N}$ collinear points or there is nothing to prove: even if $\sqrt{N}$ collinear points lie on an isotropic line, $S$ has another point $\boldsymbol s$ outside this line, such that the plane containing $\boldsymbol s$ and the line is not semi-isotropic. It is easy to see that then there are $\Omega(\sqrt{N})$ distinct distances from $\boldsymbol s$ to the points on the line.
Let $E$ be the number of solutions of the equation \begin{equation}\label{energy}
\|\boldsymbol s-\boldsymbol t\|^2 = \|\boldsymbol s-\boldsymbol t'\|^2\neq 0,\qquad (\boldsymbol s,\boldsymbol t,\boldsymbol t')\in S\times S\times S.\end{equation} Let us show that either $S$ contains a line with $\Omega(\sqrt{N})$ points or \begin{equation}\label{clm} E=O(N^{\frac{5}{2}}). \end{equation}
We claim, by the pigeonhole principle and Lemma \ref{easy}, that assuming $E\gg N^{5/2}$ implies that either there is a line with $\Omega(\sqrt{N})$ points, or $E=O(E^*)$, where $E^*$ is the number of solutions of the equation \begin{equation}\label{energystar}
\|\boldsymbol s-\boldsymbol t\|^2 = \|\boldsymbol s-\boldsymbol t'\|^2\neq 0,\qquad (\boldsymbol s,\boldsymbol t,\boldsymbol t')\in S\times S\times S:\;\|\boldsymbol t-\boldsymbol t'\|\neq 0. \end{equation}
Indeed, the quantity $E$ counts the number of equidistant pairs of points from each $\boldsymbol s\in S$ and sums over $\boldsymbol s$. Therefore, a positive proportion of $E$ is contributed by points $\boldsymbol s$ and level sets $Z_R(\boldsymbol s)=\{\boldsymbol t\in \mathbb{F}^3:\, \|\boldsymbol s-\boldsymbol t\|=R\}$, such that $Z_R(\boldsymbol s)$ supports $\Omega(\sqrt{N})$ points of $S$. By Lemma \ref{easy} either there is a line with $\Omega(\sqrt{N})$ points, or a positive proportion of pairs of distinct $\boldsymbol t,\boldsymbol t'\in Z_R(\boldsymbol s)$ is non-null.
This establishes the claim in question.
Now observe that to evaluate the quantity $E^*$, for each pair $(\boldsymbol t,\boldsymbol t')$ we have a plane through the midpoint of the segment $[\boldsymbol t\,\boldsymbol t']$, normal to the vector $\boldsymbol t-\boldsymbol t'$ and need to count points $\boldsymbol s$ incident to this plane. The plane in question does not contain $\boldsymbol t$ or $\boldsymbol t'$.
We arrive at an incidence problem $(S,\Pi)$ between $N$ points and a family of planes, but the planes have weights in the range $[1,\ldots,N]$, for the same plane can bisect up to $N/2$ segments $[\boldsymbol t \,\boldsymbol t']$, {\em provided that} $(\boldsymbol t,\boldsymbol t')$ is not a null pair. That is given the plane, there is at most one $\boldsymbol t'$ for each $\boldsymbol t$, so that the plane may bisect $[\boldsymbol t \,\boldsymbol t']$.
Thus number $m$ of distinct planes is $\Omega(N)$ and at most $N^2$, the maximum weight per plane is $N$, the total weight of the planes $W=N^2$.
It is immediate to adapt the formula (\ref{pups}) to the case of planes with weights. Note that the number of distinct planes is not less than the number of points, so in the formula \eqref{pups}, the notation $m$ will now pertain to planes, $n$ to points, and $k$ to the maximum number of collinear points. Since the estimate (\ref{pups}) is linear in $m$, the case of weighted planes and non-weighted points arises by replacing $m$ with $N^2$, $n$ with $N$, and $k$ with $\sqrt{N}$, for otherwise, once again, there is nothing to prove.
Theorem \ref{mish} now applies for $N=O(p^2)$ and yields the estimate \eqref{clm}. Theorem \ref{erd} follows from \eqref{energy} by the Cauchy-Schwarz inequality. In particular, when $N=cp^2$ for some absolute $c$, we get $\Omega(p)$ distinct pinned distances. If $N\geq cp^2$ we simply retain this estimate.
\end{proof}
\end{document} |
\begin{document}
\title{Poisson's spot and Gouy phase} \author{I. G. da Paz$^{1}$, Rodolfo Soldati $^{4}$, L. A. Cabral$^{2}$, J. G. G. de Oliveira Jr$^{3}$, Marcos Sampaio$^{4}$}
\affiliation{$^1$ Departamento de F\'{\i}sica, Universidade Federal do Piau\'{\i}, Campus Ministro Petr\^{o}nio Portela, CEP 64049-550, Teresina, PI, Brazil}
\affiliation{$^{2}$ Curso de F\'{\i}sica, Universidade Federal do Tocantins, Caixa Postal 132, CEP 77804-970, Aragua\'{\i}na, TO, Brazil}
\affiliation{$^{3}$ Departamento de Ci\^{e}ncias Exatas e Tecnol\'{o}gicas, Universidade Estadual de Santa Cruz, Caixa Postal 45662-900, Ilh\'{e}us, BA, Brazil}
\affiliation{$^{4}$ Departamento de F\'{\i}sica, Instituto de Ci\^{e}ncias Exatas, Universidade Federal de Minas Gerais, Caixa Postal 702, CEP 30161-970, Belo Horizonte, Minas Gerais, Brazil}
\begin{abstract} Recently there have been experimental results on Poisson spot matter wave interferometry followed by theoretical models describing the relative importance of the wave and particle behaviors for the phenomenon. We propose an analytical theoretical model for the Poisson's spot with matter waves based on Babinet principle in which we use the results for a free propagation and single slit diffraction. We take into account effects of loss of coherence and finite detection area using the propagator for a quantum particle interacting with an environment. We observe that the matter wave Gouy phase plays a role in the existence of the central peak and thus corroborates the predominantly wavelike character of the Poisson's spot. Our model shows remarkable agreement with the experimental data for deuterium ($D_{2}$) molecules.
\end{abstract}
\pacs{03.75.-b, 03.65.Vf, 03.75.Be \\ \\ {\it Keywords}: Poisson spot, Gouy phase, partially coherent matter waves}
\maketitle
\section{Introduction} The wave nature of light which explains the Poisson's spot (``Tache de Poisson-Fresnel-Arago") has an interesting history. In the beginning of the $19^{th}$ century, Fresnel submitted a paper on the theory of diffraction supporting the wave nature of light for a contest sponsored by the French Academy. Poisson, a member of the judging committee, used Fresnel's theory to show the odd prediction that a bright spot should appear behind a circular obstacle. Arago, another member of the committee, thus observed the spot experimentally. Fresnel won the competition, and the phenomenon is known in history as Poisson's or Arago's spot \cite{Fresnel}.
Particle interferometry far-field diffraction behind a grating and near-field interference behind an opaque sphere or disk, namely the observation of the Poisson's spot for matter waves provide experimental evidence that matter can exhibit wave-particle duality. Technology has greatly evolved since electron diffraction in the 1920s to interferometry in a grating with macromolecules like fullerene in the 1999s \cite{Zeilinger}. Poisson spot has been demonstrated by means of matter-waves with electrons \cite{Komrska} and deuterium molecules \cite{Thomas1}. Some theoretical models study the feasibility of the Poisson spot setup for fullerene \cite{Thomas2} and gold clusters \cite{Juffmann}. The transverse coherence of the matter wave beam is achieved for a source pinhole sufficiently far away from the obstacle. Thus multi-path interference leads to a bright spot at the center of the shadow region behind the obstacle. From the experimental viewpoint \cite{Juffmann,Thomas3}, it is believed that the diffraction pattern is significantly affected by the dispersive interaction between the matter-waves and the obstacle namely modifying the width and the height of the central Poisson spot, invalidating the Fresnel zone construction and Babinet principle. They argue that the spot could appear in the case of classical particles passing the obstacle following deflected trajectories due to the attractive force towards the obstacle (van-der-Walls). In addition to that, we have the unavoidable edge-corrugation of the disc. In \cite{Thomas3} it was studied the effect of Casimir-Polder/van der Waals (CP/vdW) dispersion forces on Poisson spot diffraction at a dielectric sphere which may obscure the distinction between particle and wave nature. Obviously these effects blemish the distinction between the quantum and the classical description for large enough interaction strengths, such that the appearance of the spot in itself is not exclusively due to wave-like behaviour of the particles.
Recently it was shown that two fundamental but seemingly independent optical phenomena, namely the Poisson spot and the orbital angular momentum (OAM) of light, can be well connected by a phase changing. It was demonstrated that spiral phase modulation can be added to the optical tool to effectively shape the diffraction of light which may have potential applications in the field of optical manipulations \cite{Zhang}.
In 1890 L. G. Gouy observed the effect of a phase shift in light optics that further was named Gouy phase \cite{gouy1,gouy2}. The physical origin of this phase attracted the attention of several researchers as can be seen in the works \cite{Visser1,simon1993,feng2001,yang,boyd,hariharan,feng98, Pang}. As is known today, the Gouy phase shift appears in any kind of wave that is submitted to transverse spatial confinement, either by focusing or by diffraction through small apertures. When a wave is focused \cite{feng2001}, the Gouy phase shift is associated to the propagation from $-\infty$ to $+\infty$ and is equal to $\pi/2$ for cylindrical waves (line focus), and $\pi$ for spherical waves (point focus). In the case of diffraction by a slit it was shown that the Gouy phase shift is $\pi/4$ \cite{Paz4}. The Gouy phase shift has been observed in water waves \cite{chauvat}, acoustic \cite{holme}, surface plasmon-polariton \cite{zhu}, phonon-polariton \cite{feurer} pulses, and recently in matter waves \cite{cond,elec2,elec1}. As some examples of applications of Gouy phase we can mention: the Gouy phase has to be taken into account to determine the resonant frequencies in laser cavities \cite{siegman} or the phase matching in high-order harmonic generation (HHG) \cite{Balcou} and to describe the spatial variation of the carrier envelope phase of ultrashort pulses in a laser focus \cite{Lindner}. Also, it plays important role in the evolution of optical vortex beams \cite{Allen} as well as electron beams which acquire an additional Gouy phase dependent on the absolute value of the orbital angular momentum \cite{elec2}.
In the coherent matter wave context the Gouy phase has been explored in \cite{Paz4,Paz1,Paz2,Ducharme}. Experimental realizations were made in different systems such as Bose-Einstein condensates \cite{cond}, electron vortex beams \cite{elec2} and astigmatic electron matter waves using in-line holography \cite{elec1}. Matter wave Gouy phases have interesting applications, for instance, they serve as mode converters important in quantum information \cite{Paz1}, in the development of singular electron optics \cite{elec1} and in the study of non-classical (looping) paths in interference experiments \cite{Paz3}.
It is the main purpose of this contribution to perform a complete analytical calculation of partially coherent matter-wave Poisson's spot due to an unidimensional obstacle. We also define a generalized expression for the Gouy phase for partially coherent matter waves and study the effect of this phase in the Poisson's spot intensity. The shape of the diffraction pattern on the screen can be computed using Babinet principle \cite{Wolf}: the superposition principle implies that the wave amplitudes behind a slit of certain length, $\psi_{slit}$ and behind the corresponding obstacle of the same length, $\psi_{obst}$, must add up such that $\psi_{slit} + \psi_{obst} = 1$. In order to keep track of all important phases such as Gouy phase and display fully analytical results we consider a gaussian-shaped slit (obstacle). In this way we may compare with experimental results and clearly distinguish wave interference from mutually induced dipoles which give rise to van der Waals-type attracting forces on the particle towards the obstacle as well as imperfections of the blocking object. In order to incorporate loss of coherence in our model, we obtain the reduced density matrix of the particles evolving effectively and autonomously according to a ``Boltzmann-type" master equation. The effect of the environment is summarized by ``a collision term" in the propagator which takes into account the decoherence, that is to say, the damping of off-diagonal terms of the density matrix in position representation just as in \cite{Viale}. The Gouy phase for partially coherent light wave was treated in \cite{Visser} which define a generalized expression for the Gouy phase in terms of the cross-spectral density. For a model of matter waves with loss of coherence we do not have an expression for the Gouy phase. However, since the cross-spectral density and density matrix have analogous meaning, in this contribution we follow the treatment adopted in \cite{Visser} and define the Gouy phase as the phase of the density matrix.
The article is organized as follows: in section II we use the Babinet principle to obtain analytical expression for the Poisson spot with coherent matter waves. In section III we obtain analytical expression for the Poisson's spot with partially coherent matter waves and define a generalized expression for the Gouy phase. These results are used in section IV to analyze the existing experimental data. We draw our concluding remarks in section V.
\section{Babinet principle: Poisson spot and Gouy phase}
In this section we model the Poisson spot problem using the Babinet principle and show that the intensity at the detector depends on the Gouy phase and plays an important role particularly at the central peak.
For the sake of simplicity we will treat with a coherent model in order to demonstrate the action of the Babinet principle as well as the contribution of the Gouy phase for the intensity. We shall obtain simple analytical expression for the Poisson intensity which enables us to distinguish the role played by each phase. A source of particles positioned behind an opaque disc of radius $\beta$ emits particles one-by-one and a detector browses over a screen of detection. It is a good approximation as we shall see to suppose an one-dimensional model in which quantum effects are manifested only in the $x$-direction as depicted in Fig.1 by a red line along a diameter of the disc.
\begin{figure}
\caption{Sketch of the Poisson spot problem. A source of particles positioned behind of an opaque disc of radius $\beta$ send particles one-by-one and a bright spot is observed by a detector in a screen of detection. The red line along one diameter of the disc is used to illustrate the treatment of the disc as a one dimensional problem.}
\label{Figure1}
\end{figure}
The propagation through the obstacle can be obtained by the Babinet principle which enables us write $\psi_{obst}(x,t,\tau)=\psi_{free}(x,t+\tau)-\psi_{slit}(x,t,\tau)$. Here, $\psi_{obst}(x,t,\tau)$ stands for the wave function describing the propagation through the obstacle, $\psi_{free}(x,t+\tau)$ the wave function for free propagation and $\psi_{slit}(x,t,\tau)$ the wave function characterizing propagation through a single slit. To calculate the corresponding wave functions, we consider that a coherent Gaussian wavepacket of initial transverse width $\sigma_{0}$ is produced at the source and propagates during a time $t$ before arriving at a single slit with Gaussian aperture from which the Gaussian wavepacket propagates. After crossing the slit the wavepacket propagates during a time $\tau$ before arriving at detector in the detection screen. The superposition of the wavepackets that propagate free and through the slit gives rise to a interference pattern as a function of the transverse coordinate $x$. Quantum effects are realized only in $x$-direction as we consider that the energy associated with the momentum of the particles in the $z$-direction is very high such that the momentum component $p_{z}$ is sharply defined, i.e., $\Delta p_{z}\ll p_{z}$. Then we can consider a classical movement in this direction with velocity $v_{z}$. Because the propagation is free, the $x$, $y$ and $z$ dimensions decouple for a given longitudinal location and thus we may write $z=v_{z}t$. Because $v_{z}$ is assumed to be a well defined velocity we can neglect statistical fluctuations in the time of flight, i.e., $\Delta t\ll t$. Such approximation leaves the Schr\"{o}dinger equation analogous to the optical paraxial Helmholtz equation \cite{Viale, Berman}.
The wave functions at the screen of detection are given by
\begin{equation} \psi_{free}(x,t+\tau)=\int dx_{0} K_{t}(x,t+\tau; x_{0},0)\psi_{0}(x_{0}), \end{equation}
and
\begin{eqnarray} \psi_{slit}(x,t,\tau)&=&\int \int dx_{j} dx_{0} K_{\tau}(x,t+\tau; x_{j},t)
F(x_{j})\nonumber\\
&\times&K_{t}(x_{j},t; x_{0},0)\psi_{0}(x_{0}), \end{eqnarray}
with \begin{equation} K(x_{j},t_{j};x_{0},t_0)=\sqrt{\frac{m}{2\pi i\hbar (t_{j}-t_{0})}}\exp\left[\frac{im(x_{j}-x_{0})^{2}}{2\hbar (t_{j}-t_0)}\right], \end{equation}
\begin{equation} F(x_{j})=\exp\left[-\frac{(x_{j})^{2}}{2\beta^{2}}\right], \end{equation} and \begin{equation} \psi_{0}(x_{0})=\frac{1}{\sqrt{\sigma_{0}\sqrt{\pi}}}\exp\left(-\frac{x_{0}^{2}}{2\sigma_{0}^{2}}\right). \end{equation} The kernels $K_{t}(x_{j},t;x_{0},0)$ and $K_{\tau}(x,t+\tau;x_{j},t)$ are the free propagators for the particle, the function $F(x_{j})$ describes the slit transmission function which is taken to be Gaussian of width $\beta$; $\sigma_{0}$ is the effective width of the wavepacket emitted from the source, $m$ is the mass of the particle, $t$ ($\tau$) is the time of flight from the source (slit) to the slit (screen).
After some algebraic manipulations, we obtain
\begin{equation} \psi_{free}(x,t+\tau)=\frac{1}{\sqrt{b\sqrt{\pi}}}\exp\left(-\frac{x^{2}}{2b^{2}}\right)\exp\left(\frac{imx^{2}}{2\hbar r}+i\mu_{f}\right), \label{free} \end{equation}
and
\begin{equation} \psi_{slit}(x,t,\tau) = \frac{1}{\sqrt{B\sqrt{\pi}}}\exp \left(-\frac{x^{2}}{2B^{2}}\right)\exp \left(\frac{imx^2}{2\hbar R} + i\mu_{s}\right), \label{slit} \end{equation}
where
\begin{equation} b(t+\tau)=\sigma_{0}\left[1+\left(\frac{t+\tau}{\tau_{0}}\right)^{2}\right]^{\frac{1}{2}}, \end{equation}
\begin{equation} r(t+\tau)=(t+\tau)\left[1+\left(\frac{\tau_{0}}{t+\tau}\right)^{2}\right], \end{equation}
\begin{equation} \mu_{f}(t+\tau)=-\frac{1}{2}\arctan\left(\frac{t+\tau}{\tau_{0}}\right), \end{equation}
\begin{equation} B(t,\tau) =\sqrt{\frac{\left(\frac{1}{\beta^{2}}+\frac{1}{b(t)^{2}}\right)^{2}+\frac{m^{2}}{\hbar^{2}}\left(\frac{1}{\tau}+\frac{1}{r(t)}\right)^{2}} {\left(\frac{m}{\hbar\tau}\right)^{2}\left(\frac{1}{\beta^{2}}+\frac{1}{b(t)^{2}}\right)},} \label{Bt} \end{equation}
\begin{equation} R(t,\tau)=\tau\frac{\left(\frac{1}{\beta^{2}}+\frac{1}{b(t)^{2}}\right)^{2}+\frac{m^{2}}{\hbar^{2}}\left(\frac{1}{\tau}+\frac{1}{r(t)}\right)^{2}} {\left(\frac{1}{\beta^{2}}+\frac{1}{b(t)^{2}}\right)^{2}+\frac{t}{\sigma_{0}^{2}b(t)^{2}}\left(\frac{1}{\tau}+\frac{1}{r(t)}\right)}, \label{Rt} \end{equation}
\begin{equation} \mu_{s}(t,\tau)=-\frac{1}{2}\arctan\left[\frac{t+\tau\left(1+\frac{\sigma_{0}^{2}}{\beta^{2}}\right)}{\tau_{0}\left(1-\frac{t\tau\sigma_{0}^{2}}{\tau_{0}^{2} \beta^{2}}\right)}\right] \label{Gouy1}, \end{equation}
and
\begin{equation} \tau_{0}=\frac{m\sigma_{0}^{2}}{\hbar}. \end{equation}
Here, $b(t+\tau)$, $r(t+\tau)$ and $\mu_{f}(t+\tau)$ are respectively the beam width, the radius of curvature of the wavefronts and Gouy phase for the free propagation during the total time $t+\tau$. Moreover, $B(t,\tau)$, $R(t,\tau)$ and $\mu_{s}(t,\tau)$ are respectively the beam width, the radius of curvature of the wavefronts and Gouy phase for the propagation through a single slit. $B(t,\tau)$ and $R(t,\tau)$ can be written in terms of $b(t)$ and $r(t)$, i.e, the beam width and the radius of curvature of the wavefronts for the free evolution from the source to the slit (or disc). The parameter $\tau_{0}=m\sigma_{0}^{2}/\hbar$ is viewed as a characteristic time for the ``aging" of the initial state \cite{solano}.
According to the Babinet principle the intensity at the screen of detection is given by
\begin{eqnarray}
I(x,t,\tau)&=&|\psi_{obst}(x,t,\tau)|^{2}\nonumber\\ &=&\frac{1}{\sqrt{\pi}b}\exp\left(-\frac{x^{2}}{b^{2}}\right)+\frac{1}{\sqrt{\pi}B}\exp\left(-\frac{x^{2}}{B^{2}}\right)\nonumber\\ &-&\frac{2}{\sqrt{\pi b B}}\exp\left[-\left(\frac{1}{2b^{2}}+\frac{1}{2B^{2}}\right)x^{2}\right]\nonumber\\ &\times&\cos\left[\frac{mx^{2}}{2\hbar}\left(\frac{1}{R}-\frac{1}{r}\right)+\mu(t,\tau)\right], \label{I_Free} \end{eqnarray} where \begin{eqnarray} \mu(t,\tau)&=&\mu_{s}(t,\tau)-\mu_{f}(t+\tau)\nonumber\\ &=& -\frac{1}{2}\arctan\bigg\{\frac{\tau[\tau_{0}^{2}+t(t+\tau)]}{\tau_{0}\tau^{2}+\frac{\beta^{2}}{\sigma_{0}^{2}}\tau[(t+\tau)^{2}+\tau_{0}^{2}]}\bigg\} \label{gouy_c} \end{eqnarray} is the coherent Gouy phase difference. Therefore, from equation (\ref{I_Free}) we clearly observe the Gouy phase effect on the Poisson's spot intensity. We illustrate such an effect in Fig. 2 by plotting the normalized intensity $I$ for the parameters of deuterium molecules of Ref. \cite{Thomas1}, i.e., $m=3.34\times10^{-27}\;\mathrm{kg}$, $\sigma_{0}=50\;\mathrm{\mu m}$ and $\beta=60\;\mathrm{\mu m}$. We consider the propagation times $t=20\;\mathrm{ms}$ and $\tau=40\;\mathrm{ms}$. For solid line we consider and for pointed line we do not consider the Gouy phase effect. A pronounced peak at $x=0$ appears for the case in which we consider the Gouy phase difference.
\begin{figure}
\caption{Gouy phase effect on the Poisson spot for a coherent model. Solid line we consider and for pointed line we do not consider the Gouy phase effect on the normalized intensity $I$.}
\label{Gouy}
\end{figure}
The dependence of the Poisson's spot intensity on the Gouy phase obviously appears corroborates the wave nature of the Poisson's spot since the Gouy phase is a wave property. The simple model treated so far in this section does not take into account some effects that a realist model to Poisson spot have to present. It would be interesting to investigate if a more realistic model for the Poisson spot can still be related to the Gouy phase. It is the purpose of the next section to investigate such extension.
\section{A model with loss of coherence}
The result obtained in equation (\ref{I_Free}) for the Poisson's spot intensity does not take account any loss of coherence. We shall consider that the loss of coherence is produced from the obstacle to the screen and therefore starting from time $t+\epsilon$ (with $\epsilon\rightarrow0$ being the propagation time through the obstacle) until the detection screen. Now, the evolution during the time $\tau$ is given by the propagator for a quantum particle interacting with an environment. In order to include such a loss of coherence we follow the result obtained in Ref. \cite{Viale} and write the Poisson spot intensity as
\begin{eqnarray} I_{\ell}(x,t,\tau)&\equiv &\rho(x=x^{\prime},t,\tau)=N\int\int dx_{0}dx_{0}^{\prime}\nonumber\\ &\times&\exp\left\{\frac{im}{2\hbar \tau}[x_{0}^{2}-x_{0}^{\prime2}+2x(x_{0}-x_{0}^{\prime})]\right\}\nonumber\\ &\times&\exp\left[-\frac{(x_{0}-x_{0}^{\prime})^{2}}{2\ell^{2}(\tau)}\right] \tilde{\rho}(x_{0},x_{0}^{\prime},t), \label{int_l} \end{eqnarray} where
\begin{equation} \tilde{\rho}(x_{0},x_{0}^{\prime},t)=\psi_{obst}(x_{0},t,\epsilon\rightarrow0)\psi_{obst}^{*}(x_{0}^{\prime},t,\epsilon\rightarrow0), \end{equation}
and
\begin{eqnarray} \ell(\tau)\equiv\frac{\ell_{0}}{\sqrt{1+\frac{2\Lambda \tau}{3}}\ell_{0}^{2}}. \end{eqnarray} Here, $N$ is a normalization constant, $\tilde{\rho}(x_{0},x_{0}^{\prime},t)$ is the density matrix in the obstacle, $\tau$ is the propagation time from the obstacle to the screen in which we have loss of coherence, $\ell(\tau)$ is the time dependent coherence length and $\ell_{0}$ is the coherence length in the obstacle which is the same of the source since we consider that the propagation from the source to the obstacle is free, i.e., $\ell_{0}=\ell(t)$. The parameter $\Lambda$ encodes decohering events such as scattering and photon emission and $\ell_{0}$ carries incoherence effects of the source \cite{Viale}.
In order to obtain the density matrix in the obstacle we have to take the limit when $\epsilon\rightarrow0$ in the parameters $B(t,\epsilon)$, $R(t,\epsilon)$ and $\mu_{s}(t,\epsilon)$ of the wavefunction given by equation (\ref{slit}). After performing such limits using the expressions (\ref{Bt}), (\ref{Rt}) and (\ref{Gouy1}), we obtain the following results $\lim_{\epsilon\rightarrow0}\;B(t,\epsilon)=\sqrt{\frac{b^{2}(t)\beta^{2}}{\beta^{2}+b^{2}(t)}}$, $\lim_{\epsilon\rightarrow0}\;R(t,\epsilon)=r(t)$ and $\lim_{\epsilon\rightarrow0}\;\mu_{s}(t,\epsilon)=\mu_{f}(t)$. Notice that only the parameter $B(t,\epsilon\rightarrow0)$ is changed by the slit. Using the results above we obtain the density matrix in the obstacle $\tilde{\rho}(x_{0},x_{0}^{\prime},t)$.
After performing the integration in equation (\ref{int_l}) and some algebraic manipulation we obtain
\begin{eqnarray} I_{\ell}(x,t,\tau)&=&\sqrt{\frac{\pi}{\eta}}\exp\left[-\frac{m^{2}x^{2}}{4\eta\hbar^{2}\tau^{2}}\right]\nonumber\\ &+&\sqrt{\frac{\pi}{\eta^{\prime}}}\exp\left[-\frac{m^{2}x^{2}}{4\eta^{\prime}\hbar^{2}\tau^{2}}\right]\nonumber\\ &-&\frac{2\sqrt{2\pi\hbar \tau}}{\sqrt{\sqrt{C[1+(b(t)/\beta)^{2}]}}}\exp(-\alpha x^{2})\nonumber\\ &\times&\cos(\delta x^{2}+\mu_{\ell}), \label{I_Decoh} \end{eqnarray}
where
\begin{eqnarray} \eta(t,\tau)&=&b^{2}(t)\bigg[\frac{1}{2b^{2}(t)}\left(\frac{1}{2b^{2}(t)}+\frac{1}{\ell^{2}}\right)\nonumber\\ &+&\left(\frac{m}{2\hbar r(t)}+\frac{m}{2\hbar \tau}\right)^{2}\bigg], \end{eqnarray}
\begin{eqnarray} \eta^{\prime}(t,\tau)&=&\left(\frac{\beta^{2}b^{2}(t)}{\beta^{2}+b^{2}(t)}\right)\bigg[\left(\frac{m}{2\hbar r(t)}+\frac{m}{2\hbar \tau}\right)^{2}\nonumber\\ &+&\left(\frac{1}{2b^{2}(t)}+\frac{1}{2\beta^{2}}\right)\nonumber\\ &\times&\left(\frac{1}{2b^{2}(t)}+\frac{1}{2\beta^{2}}+\frac{1}{\ell^{2}}\right)\bigg], \end{eqnarray}
\begin{eqnarray} \alpha(t,\tau)&=&\frac{m^{2}}{C}\left(\frac{1}{b^{2}(t)}+\frac{1}{2\beta^{2}}\right)\nonumber\\ &\times&\bigg[\left(\frac{\beta^{2}+b^{2}(t)}{4\beta^{2}b^{2}(t)}\right)\left(\frac{1}{b^{2}(t)}+\frac{1}{\ell^{2}}\right)\nonumber\\ &+&\frac{1}{4\ell^{2}b^{2}(t)}+\left(\frac{m}{2\hbar r(t)}+\frac{m}{2\hbar \tau}\right)^{2}\bigg], \end{eqnarray}
\begin{equation} \delta(t,\tau)=\frac{m^{3}}{4\hbar \beta^{2}C}\left(\frac{1}{b^{2}(t)}+\frac{1}{2\beta^{2}}\right)\left(\frac{1}{r(t)}+\frac{1}{\tau}\right), \end{equation} and \begin{eqnarray} C(t,\tau)&=&4\hbar^{2}\tau^{2}\bigg\{\frac{m^{2}}{16\hbar^{2} \beta^{4}}\left(\frac{1}{r(t)}+\frac{1}{\tau}\right)^{2}\nonumber\\ &+&\bigg[\left(\frac{\beta^{2}+b^{2}(t)}{4\beta^{2}b^{2}(t)}\right)\left(\frac{1}{b^{2}(t)}+\frac{1}{\ell^{2}}\right)\nonumber\\ &+&\frac{1}{4\ell^{2}b^{2}(t)}+\left(\frac{m}{2\hbar r(t)}+\frac{m}{2\hbar \tau}\right)^{2}\bigg]^{2}\bigg\}. \end{eqnarray} The Poisson spot intensity given by equation (\ref{I_Decoh}) is the main result of this paper. To our knowledge, an analytical expression incorporating such effects for the Poisson's spot has not been obtained. The result of equation (\ref{I_Decoh}) is useful to define the Gouy phase for partially coherent matter waves and explore the role of this phase in the Poisson spot.
\subsection{Generalized Gouy phase for partially coherent matter waves}
In Ref. \cite{Paz1} was shown that the matter waves Gouy phase is related with the off diagonal elements of the covariance matrix which indirectly enabled to extract the Gouy phase from the beam width. It was shown that the experimental data for the diffraction of fullerene molecules is quantitatively consistent with the existence of a Gouy phase. Since the fullerene molecules have to be treated as partially coherent matter waves in that work was conjectured that the Gouy phase can be obtained by integrating the inverse of the squared beam width, as is valid for coherent case. Further, a complete definition for the Gouy phase for partially coherent light waves was given in Ref. \cite{Visser}. In this work we use the definition of Ref. \cite{Visser} to obtain the Gouy phase for partially coherent matter waves as \begin{equation} \mu_{\ell}(t,\tau)=\arg[I_{\ell}(0,t,\tau)]. \end{equation} We have a complete analogy with the generalized definition for the Gouy phase of Ref. \cite{Visser}, since here $I_{\ell}(0,t,\tau)\equiv\rho(0,t,\tau)$ is the density matrix in the propagation axis $z$ which is similar to the cross-spectral density and $(t,\tau)$ can be used to obtain two different positions in the propagation axis since we are substituting the propagation time by $z/v_{z}$. In the case of light waves the mechanism of loss of coherence is attributed to the source incoherence whereas for matter waves such effects can be attributed both to the source incoherence and environment decoherence. We calculate the Gouy phase here and obtain the following result \begin{equation} \mu_{\ell}(t,\tau)=-\frac{1}{2}\arctan\left[\frac{r(t)+\tau}{a_{1}+a_{2}+\frac{r(t)\tau}{\tau_{0}}\left(1+\frac{2\beta^{2}}{b^{2}(t)}\right)\frac{\sigma_{0}^{2}}{\ell^{2}}}\right], \label{gouy_l} \end{equation} where \begin{equation} a_{1}(t,\tau)=\left(\frac{\beta^{2}\tau_{0}}{\sigma_{0}^{2}r(t)\tau}\right)(r(t)+\tau)^{2}, \end{equation} and \begin{equation} a_{2}(t,\tau)=\frac{r(t)\tau}{\tau_{0}}\left(1+\frac{\beta^{2}}{b^{2}(t)}\right)\left(\frac{\sigma_{0}}{b(t)}\right)^{2}. \end{equation} We can observe from equation (\ref{gouy_l}) that the Gouy phase is dependent of the coherence length $\ell$. The same dependence was discussed in Ref. \cite{Visser} for partially coherent light wave. We can easily obtain that the result of equation (\ref{gouy_l}) reduces to that of equation (\ref{gouy_c}) for coherent matter waves in the limit $\ell\rightarrow\infty$. On the other hand, in the limit of completely non-coherent matter waves $\ell\rightarrow0$ we have $\mu_{\ell}\rightarrow0$.
Therefore, just as in the coherent case the Poisson spot intensity is changed by the Gouy phase. This can be clearly seen in the figure below. In Fig. 3(a) we show the Gouy phase $\mu_{\ell}(t,\tau)$ as a function of the propagation time $\tau$ for $t=20\;\mathrm{ms}$ and for the data of the deuterium molecules. Solid line corresponds to $\ell=1.0\;\mathrm{m}$ and pointed line corresponds to $\ell=100\;\mathrm{\mu m}$. In Fig. 3(b) we show the normalized intensity $I_{\ell}$ as a function of $x$ for $t=20\;\mathrm{ms}$ and $\tau=40\;\mathrm{ms}$ for the data of deuterium molecules. Solid line we consider the effect of the phase $\mu_{\ell}$ and pointed line we do not consider such effect. \begin{figure}
\caption{(a) Gouy for partially coherent matter waves as a function of $\tau$ for two different values of coherence length and $t=20\;\mathrm{ms}$. Solid line corresponds to $\ell=1.0\;\mathrm{m}$ and pointed line corresponds to $\ell=100\;\mathrm{\mu m}$. (b) Normalized intensity $I_{\ell}$ as a function of $x$ for $\ell=100\;\mathrm{\mu m}$, $t=20\;\mathrm{ms}$ and $\tau=40\;\mathrm{ms}$. Solid line we consider the effect of the phase $\mu_{\ell}$ and pointed line we do not consider such effect.}
\label{gouy_dec}
\end{figure}
Fixing a set values of parameters we observe that the Gouy phase decreases when the coherence length decreases. This is a novel result in the context of matter waves. The visibility of the Poisson spot tends to decrease as an effect of the loss of coherence. We can observe this by comparing the intensity for the coherent and partially coherent case, Fig. 2 and Fig. 3(b) respectively, which show that the minimum intensity for the partially coherent case is not zero. The partially coherent Gouy phase changes the intensity in a such way that the intensity in the central peak is not observed if one neglected this phase. The presence of the Gouy phase is a signature of the wave behaviour. Thus, the relationship between Poisson spot and Gouy phase for a model of partially coherent matter waves with analytical results as obtained in this section is useful to treat experimental data and to elucidate the wave behaviour of the Poisson spot with matter waves. In order to test our results in the next section we will analyse the experimental data for deuterium molecules of Ref. \cite{Thomas1}.
\section{Analysis of existing experimental data}
In this section we compare our model with the experimental data for deuterium molecules of Ref. \cite{Thomas1}. We take into account loss of coherence and the finite detection area. The loss of coherence was obtained in equation (\ref{I_Decoh}). To include the detector effect we perform a convolution to obtain the effective intensity \begin{equation} I_{eff}(x,t,\tau)=\int^{\infty}_{-\infty} I_{\ell}(x^{\prime},t,\tau)D(x-x^{\prime})dx^{\prime}. \end{equation} Considering a Gaussian profile to the detector aperture as $D(x)=\exp\left(-x^{2}/2\sigma_{D}^{2}\right)$, where $\sigma_{D}$ is the detector width, the integral above is easily done.
In order to compare our model with experimental results previously published in Ref. \cite{Thomas1} we relate our model and the one in \cite{Thomas1} by $I_{eff}(x,l,\sigma_D)=a+b\;I(x,l,\sigma_D)$. The parameters $a$ and $b$ are necessary to convert our results in units ($rate/s$) used in Ref. \cite{Thomas1}. The numerical calculations obtained within these units are summarized in Table 1. The obtained results are in good agreement with the experimental values, as we can observe in Fig. 4.
\begin{table}[t] \caption{\label{table1}Parameters of analytic model and numerical results} \begin{ruledtabular} \begin{tabular}{ll}
Coherence parameter & $\ell=0.3369\;\mathrm{\mu m}$ \\ Detector width & $\sigma_{D}=3.96\;\mathrm{\mu m}$ \\ Gaussian width & $\sigma_0=51\;\mathrm{\mu m}$\\ Disc aperture \footnote{The physical parameters are compatible with Ref. \cite{Thomas1}}& $\beta=60\;\mathrm{\mu m}$\\ Time before disc & $t=1.4\;\mathrm{ms}$\\ Time after disc & $\tau=0.606\;\mathrm{ms}$\\ Partially coherent fit [PC] $^{b}$ & $a=29829.11, \; b=-348.71$\\ Detector convolution fit [DC] \footnote{See the corresponding curve on Fig. 4 and experimental data [EX] from Ref. \cite{Thomas1}} & $a=40465.09,\; b=-466.29$ \\
Gouy Phase [PC] & $\mu_\ell = 0.00060097028\;\mathrm{rad}$ \\ Gouy Phase [DC]& $\mu_\ell = 0.00069360626\;\mathrm{rad}$\\ \end{tabular} \end{ruledtabular} \end{table}
\begin{figure}
\caption{Comparison of experimental data Ref. \cite{Thomas1} with analytical model (table 1). Black points with error bars are the experimental data. Green line is the coherent model. Blue line is the model with loss of coherence and red line is the model with loss of coherence and detector convolution. }
\label{Figure1}
\end{figure}
We observe by red line of Fig. 4 that an analytical model including loss of coherence as well as finite detector area is in full agreement with existing experimental data. On the other hand, the blue line shows that considering only loss of coherence there is an agreement between the model and the experimental data but to obtain a full agreement it is necessary to consider the detector convolution. Green line shows that is not possible to adjust the data by considering a completely coherent model. In the adjustment a given value of partially coherent Gouy phase is necessary. The small value found here is related with the set value of parameters used in the experiment, specially the propagation times. Therefore, different values of partially coherent Gouy phase can be obtained if the experiment is realized with different set value of parameters.
\section{Conclusions} We developed a theoretical model for the Poisson's spot problem by using the Babinet principle. It was possible to include loss of coherence and detector convolution in the observed intensity. Firstly, we treated the coherent model and then we studied the effect of the loss of coherence. Based in the previous definition to the Gouy phase for partially coherent light waves (source incoherence) we obtained an expression to the Gouy phase for partially coherent matter waves (source incoherence + environment decoherence). We observed that this phase influences the Poisson spot intensity. Therefore, we have found a relationship between two old physical problems (Gouy phase and Poisson spot). We obtained full agreement between our results and existent experimental data. We observed that the Gouy phase depends on the set value of parameters used in the Poisson spot experiment. Thus, the Poisson spot experiment can be used to measure the Gouy phase for partially coherent matter waves.
\vskip1.0cm \begin{acknowledgments} The authors would like to thank CNPq-Brazil for financial support. I. G. da Paz thanks support from the program PROPESQ (UFPI/PI) under grant number PROPESQ 23111.011083/2012-27. \end{acknowledgments}
\end{document} |
\begin{document}
\begin{abstract}
Results from a few years ago of Kennedy and Schafhauser characterize simplicity of reduced crossed products $A \rtimes_\lambda G$, where $A$ is a unital C*-algebra and $G$ is a discrete group, under an assumption which they call \emph{vanishing obstruction}. However, this is a strong condition that often fails, even in cases of $A$ being finite-dimensional and $G$ being finite. In this paper, we give the complete, two-way characterization, of when the crossed product is simple, in the case of $G$ being an FC-hypercentral group. This is a large class of amenable groups that, in the finitely-generated setting, is known to coincide with the set of groups which have polynomial growth. With some additional effort, we can learn even more about the ideal structure of $A\rtimes_\lambda G$ for the slightly less general class of FC-groups. Finally, for minimal actions of arbitrary discrete groups on unital C*-algebras, we are able to generalize a result by Hamana for finite groups, and characterize when the crossed product $A \rtimes_\lambda G$ is prime. All of our characterizations are initially given in terms of the dynamics of $G$ on $I(A)$, the injective envelope of $A$. If $A$ is separable, this is shown to be equivalent to an intrinsic condition on the dynamics of $G$ on $A$ itself.
\end{abstract}
\maketitle
\tableofcontents
\renewcommand*{\thetheoremintro}{\Alph{theoremintro}}
\section{Introduction}
The last several years have seen tremendous progress in problems related to C*-algebras generated by groups and their dynamical systems. In particular, the problem of determining when the reduced group C*-algebra $C^*_\lambda(G)$ is simple, which has its origin in Powers' work from 1975 \cite{Powers}, was completely solved by Breuillard, Kalantar, Kennedy, and Ozawa in \cite{kalantar_kennedy_boundaries}, \cite{breuillard_kalantar_kennedy_ozawa_c_simplicity}, and \cite{kennedy_intrinsic} (see also \cite{HaagerupCstarSimplicity}), using new refreshing ideas. This theory was shortly after generalized to the case of reduced crossed products $C(X) \rtimes_\lambda G$ involving commutative C*-algebras, by Kawabe in \cite{kawabe_crossed_products}. Somewhat more recently, the corresponding result for {\'e}tale groupoid C*-algebras was obtained in collaborated work of the second author \cite{kklru_groupoids}. This last result is still in essence a result about dynamics on commutative C*-algebras, with the C*-algebra $C_0(\mathcal{G}^{(0)})$ (where $\mathcal{G}^{(0)} \subseteq \mathcal{G}$ is the unit space) admitting a partial action of the inverse semigroup of open bisections, in the appropriate sense.
The problem of determining when a reduced crossed product $A \rtimes_\lambda G$, where $A$ is now noncommutative, is simple (or more generally, has the intersection property) was addressed already in the classical work of Archbold and Spielberg \cite{archbold_spielberg_amenable_crossed_products}.
They show that $A \rtimes_\lambda G$ is simple under minimality and a (noncommutative) topological freeness assumption on the action. While minimality is necessary for simplicity, topological freeness is known to be only a sufficient condition. However, if $A$ is commutative and the group $G$ is amenable (or acts amenably on $A$), then it is shown in \cite{archbold_spielberg_amenable_crossed_products} that $A \rtimes_\lambda G$ is simple if and only if the action is minimal and topologically free.
This leaves the question of a necessary and sufficient condition for simplicity in the noncommutative setting open and mysterious.
A seemingly weaker condition of noncommutative topological freeness, which nowadays is considered to be ``the right condition'', is called \emph{proper outerness} and it was introduced by Kishimoto in \cite{Kishimoto_freely_acting}. A different version of this property was introduced a few years prior by Elliott in \cite{elliott_properly_outer}, and the two notions are known to coincide when the underlying C*-algebra is separable. The first general result on proper outerness (together with minimality of the action and separability of the base algebra) implying simplicity of a reduced crossed product can be found in a paper by Olesen and Pedersen \cite[Theorem~7.2]{olesen_pedersen_III} (see also \cite[Remark~2.23]{Sierakowski}), generalizing older results of Elliott and Kishimoto in special cases.
Using the powerful tools developed for the theory of C*-simplicity, Kennedy and Schafhauser provide a new proof for these results (which cover the non-separable case as well) \cite[Section~6]{kennedy_schafhauser_noncommutative_crossed_products} (we refer to \cite{Zarikian} for similar results which were achieved independently by Zarikian).
Moreover, under an ``untwisting'' assumption, which they call \emph{vanishing obstruction}, they also manage to obtain some converse results. Under this assumption (which is automatic for commutative C*-dynamical systems), they show that simplicity of $A\rtimes_\lambda G$ is equivalent to proper outerness and minimality of the action, whenever $G$ is an amenable group \cite[Corollary~9.7]{kennedy_schafhauser_noncommutative_crossed_products}.
However, \emph{vanishing obstruction} is a strong condition, which can fail even in the setting of finite abelian groups acting on finite-dimensional C*-algebras. See \cite[Example~5.6]{kennedy_schafhauser_noncommutative_crossed_products} for a counterexample of the form $M_2 \rtimes_\lambda (\mathbb{Z} / 2 \mathbb{Z} \times \mathbb{Z} / 2 \mathbb{Z})$.
Also worth mentioning are the works of Olesen and Pedersen \cite{olesen_pedersen_I} and \cite{olesen_pedersen_II}, where they fully characterize when a crossed product is prime, or has the intersection property, in the case of locally compact abelian groups. Their characterizations are in terms of the Connes spectrum $\Gamma(\alpha)$ of the homomorphism $\alpha\colon G \to \Aut(A)$, which is a certain subgroup of the dual group $\what{G}$. The precise definition of the Connes spectrum, found in \cite[Section~1]{olesen_inner_automorphisms}, is a fairly complicated construction that involves considering all possible $G$-invariant hereditary C*-subalgebras $B \subseteq A$, and the $L^1(G)$-action on each of these algebras.
In addition, there is also the work of Rieffel in \cite{rieffel_finite_groups}, where he considers characterizing simplicity and primality of crossed products in the case of finite groups, using quite similar notions of \emph{partly inner} and \emph{purely outer} automorphisms. However, these results are written in terms of a suitable subalgebra $C \subseteq A \rtimes_\lambda G$ that is in principle easier to understand, but more intrinsic characterizations in terms of the dynamical system $(A,G)$ are not given.
We now begin describing the results in this article, and the main ideas surrounding them. First of all, we would like to mention that, as in all recent results concerning C*-simplicity and ideal structure for crossed products, we will heavily use the theory of injective envelopes developed by Hamana. More precisely, for a C*-dynamical system $(A,G)$, we will frequently consider the two additional induced dynamical systems: $(I(A),G)$ and $(I_G(A),G)$, where $I(A)$, the \emph{injective envelope} of $A$, denotes the minimal injective C*-algebra containing $A$, and $I_G(A)$ denotes its equivariant version. The existence of these objects was proven by Hamana in \cite{hamana79_injective_envelopes_cstaralg, hamana85-injective_envelopes_equivariant}. We refer the reader to Section~\ref{sec:preliminaries:injective_envelopes} for more details.
In Section~\ref{sec:intersection_property_fc}, we consider the case of FC-groups, which are groups where every conjugacy class is finite. Recall that a C*-dynamical system $(A,G)$ is said to have the \emph{intersection property} if any non-zero ideal of $A \rtimes_\lambda G$ has non-zero intersection with $A$. It is clear that for minimal actions, this is equivalent to simplicity of $A \rtimes_\lambda G$. We write down our main results in terms of the negations of these properties, as this makes the statements of the characterizations less cumbersome.
\begin{thmintro}[Theorem~\ref{thm:mainSec4} and Theorem~\ref{thm:equivariant_proper_outer_equivalence}]
\label{thmIntro:A}
Let $G$ be an FC-group acting on a unital C*-algebra $A$. Then the crossed product $A \rtimes_\lambda G$ does not have the intersection property if and only if there exist $r \in G \setminus \set{e}$, a non-zero $r$-invariant central projection $p \in I(A)$, and a unitary $u \in U(I(A)p)$ such that
\begin{enumerate}
\item $r$ acts by $\Ad u$ on $I(A)p$;
\item $s \cdot p = p$ and $s \cdot u = u$ for all $s \in C_G(r) \coloneqq \setbuilder{g \in G}{gr = rg}$.
\end{enumerate}
If $A$ is separable, then this is equivalent to the existence of an $r \in G \setminus \set{e}$, a non-zero $r$-invariant ideal $J \subseteq A$, and a unitary $u$ in the multiplier algebra $M(J)$ such that:
\begin{enumerate}
\item $J \cap s \cdot J$ is essential in both $J$ and $s \cdot J$, for all $s \in C_G(r)$. In particular, $M(J)$ and $M(s \cdot J)$ both canonically embed into $M(J \cap s \cdot J)$;
\item Letting $\varepsilon_1 = \norm{\alpha_r|_J - (\Ad u)|_J}$ and $\varepsilon_2 = \sup_{s \in C_G(r)} \norm{s \cdot u - u}$, we have
\[ 2 \sqrt{2 - \sqrt{4 - \varepsilon_1^2}} + \varepsilon_2 < \sqrt{2}. \]
\end{enumerate}
\end{thmintro}
Similar methods allow us to also characterize when a crossed product $A \rtimes_\lambda G$ is prime - see Theorem~\ref{thm:mainSec4Prime}. This generalizes a characterization by Hamana, which was obtained in the setting of finite groups, see \cite[Theorem~10.1]{hamana85-injective_envelopes_equivariant}.
The first step in our argument is that if the crossed product $A \rtimes_\lambda G$ does not have the intersection property, then through the machinery of Section~\ref{sec:pseudoexpectations}, we are fairly easily able to obtain an element $u\in I_G(A)$ with the above requirements. However, in Theorem~\ref{thmIntro:A} we want $u$ to rather belong to $I(A)$. A general theme in all of the modern C*-simplicity results is to take any results on $I_G(A)$, which is an extremely mysterious and poorly-understood space, and try to push them down onto a more tractable space, such as $I(A)$ or $A$ itself. In the commutative case, this ends up being relatively straightforward, given that we have dual maps between the spectra of all of these spaces. In the noncommutative case, this is not nearly as straightforward, and we end up developing a far more roundabout argument.
Finally, after we have our unitary $u \in U(I(A)p)$ as in the statement of Theorem~\ref{thmIntro:A}, we may instead convert this to a more tractable property involving $A$ instead. Observe that if we removed the statements ``$s \cdot p = p$ and $s \cdot u = u$ for all $s \in C_G(r)$'' from the requirements in the theorem, then the characterization would essentially be asking whether or not the action of $G$ on $I(A)$ is properly outer, which is well-known to coincide with an appropriate analogue on $A$ (with a little bit of work, this follows from \cite[Theorem~7.4]{hamana85-injective_envelopes_equivariant}). In the separable setting at least, the simplest characterization of proper outerness is that of Elliott \cite{elliott_properly_outer}. A brief overview of these classical notions is given in Section~\ref{sec:preliminaries:properly_outer}. The equivariant version of this characterization is developed in Section~\ref{sec:proper_outerness_on_A}. This turns out to be substantially more difficult than one would expect, and is the most involved part of our paper. Another glance at the second half of Theorem~\ref{thmIntro:A} will lead to the observation that there is no $C_G(r)$-invariance on any of the constituent components, but rather an ``almost invariance'' in all of the appropriate senses. The proof of the equivalence is also far from being a straightforward generalization of the classical case, and requires the application of new ideas and techniques along every step of the way.
In Section~\ref{sec:simplicity_fch}, we upgrade these results to the setting of FC-hypercentral groups. A brief review of FC-hypercentral groups can be found in Section~\ref{sec:preliminaries:fc_hypercentral}. This is a much larger class of groups, which, in the finitely generated setting, coincides with the set of groups with polynomial growth. Denoting by $\FC(G)$ the FC-center of a a group $G$ (that is, the subset of elements which have finite conjugacy class), the main theorem reads as follows:
\begin{thmintro}[Theorem~\ref{thm:mainFCHSimpleMinimal} and Theorem~\ref{thm:equivariant_proper_outer_equivalence}]
\label{thmIntro:B}
Let $G$ be an FC-hypercentral group acting minimally on a unital C*-algebra $A$. Then $A \rtimes_\lambda G$ is not simple if and only if there exist $r \in \FC(G) \setminus \set{e}$, a non-zero $r$-invariant central projection $p \in I(A)$, and a unitary $u \in U(I(A)p)$ such that
\begin{enumerate}
\item $r$ acts by $\Ad u$ on $I(A)p$;
\item $s \cdot p = p$ and $s \cdot u = u$ for all $s \in C_G(r)$.
\end{enumerate}
If $A$ is separable, then this is equivalent to the existence of an $r \in \FC(G) \setminus \set{e}$, a non-zero $r$-invariant ideal $J \subseteq A$, and a unitary $u$ in the multiplier algebra $M(J)$ such that:
\begin{enumerate}
\item $J \cap s \cdot J$ is essential in both $J$ and $s \cdot J$, for all $s \in C_G(r)$. In particular, $M(J)$ and $M(s \cdot J)$ both canonically embed into $M(J \cap s \cdot J)$;
\item Letting $\varepsilon_1 = \norm{\alpha_r|_J - (\Ad u)|_J}$ and $\varepsilon_2 = \sup_{s \in C_G(r)} \norm{s \cdot u - u}$, we have
\[ 2 \sqrt{2 - \sqrt{4 - \varepsilon_1^2}} + \varepsilon_2 < \sqrt{2}. \]
\end{enumerate}
\end{thmintro}
The proof of this theorem is inspired by the work of B{\'e}dos and Omland \cite{bedos_omland_fc_hypercentral_simplicity}, where they characterize when twisted reduced group C*-algebras $C^*_\lambda(G,\sigma)$ involving FC-hypercentral groups are simple, or have a unique trace. In their work, as well as in ours, the characterization reduces down to the FC-center of the group $G$, via a technical lemma that allows working up the entire FC-central series. Note that in our proof, we really do need minimality for one very essential part of our argument, and we do not see an easy way to work around it to obtain a characterization of the intersection property (or primality) like in the previous setting of FC-groups.
Before proceeding further, we would like to take a small detour and point out an interesting observation. It is well known that every FC-hypercentral group is amenable. It is moreover known that, for every amenable group $G$, any injective von Neumann algebra is automatically $G$-injective. It is therefore natural to ask if any injective C*-algebra is $G$-injective for any amenable group $G$ (equivalently, $I(A) = I_G(A)$ for any C*-algebra $A$ and any amenable group $G$), which would significantly simplify the arguments mentioned above for both FC-groups and FC-hypercentral groups. In fact, we strongly recommend that the reader keeps this question in mind while reading through the rest of the paper. To our knowledge, this problem is open, and we do not see an easy argument for proving it is true, nor for constructing a counterexample. We give a slightly more detailed discussion on this matter in Section~\ref{sec:preliminaries:injective_envelopes}.
Our final main result stems from the observation that most of the arguments presented in the proofs of the previous two theorems can be modified to prove, in full generality, when a crossed product $A \rtimes_\lambda G$ is prime, as long as the action is minimal (see Section~\ref{sec:primality_minimal}). Again, minimality is very crucial for a specific part of the argument, and we do not see an easy way to reduce it down to just assuming that $A$ is $G$-prime.
\begin{thmintro}[Theorem~\ref{thm:DiscreteMinimal} and Theorem~\ref{thm:equivariant_proper_outer_equivalence}]
\label{thmIntro:C}
\label{thm:primality_minimal}
Let $G$ be any discrete group acting minimally on a unital C*-algebra $A$. Then $A \rtimes_\lambda G$ is not prime if and only if there exist $r \in \FC(G) \setminus \set{e}$, a non-zero $r$-invariant central projection $p \in I(A)$, and a unitary $u \in U(I(A)p)$ such that
\begin{enumerate}
\item $r$ acts by $\Ad u$ on $I(A)p$;
\item $s \cdot p = p$ and $s \cdot u = u$ for all $s \in C_G(r)$.
\end{enumerate}
If $A$ is separable, then this is equivalent to the existence of an $r \in FC(G) \setminus \set{e}$, a non-zero $r$-invariant ideal $J \subseteq A$, and a unitary $u$ in the multiplier algebra $M(J)$ such that:
\begin{enumerate}
\item $J \cap s \cdot J$ is essential in both $J$ and $s \cdot J$, for all $s \in C_G(r)$. In particular, $M(J)$ and $M(s \cdot J)$ both canonically embed into $M(J \cap s \cdot J)$;
\item Letting $\varepsilon_1 = \norm{\alpha_r|_J - (\Ad u)|_J}$ and $\varepsilon_2 = \sup_{s \in C_G(r)} \norm{s \cdot u - u}$, we have
\[ 2 \sqrt{2 - \sqrt{4 - \varepsilon_1^2}} + \varepsilon_2 < \sqrt{2}. \]
\end{enumerate}
\end{thmintro}
If the group $G$ is amenable, then the proof of Theorem~\ref{thmIntro:C} is not difficult. However, the non-amenable setting makes use of a highly non-trivial technical lemma, Lemma~\ref{lem:infinitely_many_distinct_translates_unbounded_sum}, which was not needed for the previous results due to FC-groups and FC-hypercentral groups always being amenable.
We remark that Theorem~\ref{thmIntro:B} follows from Theorem~\ref{thmIntro:C} and the fact that for FC-hypercentral groups acting minimally on a unital C*-algebra, the attached crossed product is simple if and only if it is prime. Nevertheless, we keep the proofs of Theorem~\ref{thmIntro:B} and Theorem~\ref{thmIntro:C} separate, as the FC-hypercentral case is more direct. In addition, it is worth noting that the equivalence between simplicity and primality in the FC-hypercentral setting was originally proven by Echterhoff in much greater generality in \cite[Theorem~3.1]{echterhoff_jot} (see also \cite[Satz~5.3.1]{echterhoff_thesis} in his thesis). This is something that we initially overlooked and gave a different proof of, based on techniques introduced by B{\'e}dos and Omland in \cite{bedos_omland_fc_hypercentral_simplicity}. We still keep the proof in Section~\ref{sec:simplicity_fch} as-is, as we consider the underlying details interesting on their own. See the introductory paragraph in that section for more details.
Finally, we obtain an immediate corollary to Theorem~\ref{thmIntro:C}, one which is perhaps a bit surprising at first. It is reminiscent of a likely well-known but somewhat folklore analogue in the theory of tracial von Neumann algebras. Consider an ICC group $G$ acting on a tracial von Neumann algebra $(M,\tau)$, and assume that both the action on $M$ is trace-preserving and the action on $Z(M)$ is ergodic. In this case, the von Neumann crossed product $M \closure{\rtimes} G$ is a factor. In the C*-algebra setting, we have the following:
\begin{corintro}[Corollary~\ref{cor:ICC}]
\label{cotIntro:D}
If $G$ is a discrete ICC group acting minimally on a unital C*-algebra $A$, then the reduced crossed product $A\rtimes_\lambda G$ is prime.
\end{corintro}
\section{Preliminaries}
\subsection{Monotone complete C*-algebras}
\label{sec:preliminaries:monotone_complete}
As made abundantly clear in the introduction, the injective envelopes $I(A)$ and $I_G(A)$ (reviewed in Section~\ref{sec:preliminaries:injective_envelopes}) played a crucial role in all recent ideal-structure results for $A \rtimes_\lambda G$. One of the most important facts about injective algebras that make them especially convenient to work with is that they are what is known as \emph{monotone complete}. This means that every bounded increasing net of self-adjoint elements admits a supremum.
Every von Neumann algebra is monotone complete, and while the converse is false, such algebras will still in many ways behave as if they were von Neumann algebras. Quite important to us is the fact that polar decompositions still work inside monotone complete C*-algebras (for a more general version, we refer to \cite[Section~21]{Berberian}).
\begin{proposition}[{\cite[Lemma~2.1]{YenPolarDec}} and {\cite[Lemma~4.2]{SasakiPolarDec}}] \label{prop:monotonecompletePolarDec}
Let $A$ be a monotone complete C*-algebra, and let $a \in A$. There exists a unique projection $p\in A$, called the \emph{right projection} of $a$, such that $ap = a$, and $ab = 0$ if and only if $pb = 0$, for every $b\in A$. This projection is denoted by $\RP(a)$. The \emph{left projection} of $a$ is defined similarly, and is denoted by $\LP(a)$.
Moreover, there is a unique partial isometry $u \in A$ with the property that $a = u \abs{a}$ and $u^*u = \RP(a)$.
\end{proposition}
It is important to recognize a subtle point, and that is that the left and right projections $\LP(a)$ and $\RP(a)$ were defined \emph{intrinsically} in terms of the monotone complete C*-algebra $A$, and \emph{not} in terms of any representation $A \subseteq B(H)$. In the setting of von Neumann algebras $M \subseteq B(H)$, the polar decomposition $x = u \abs{x}$ of an element $x \in M$ is typically done with requiring that the support projection of $u$ is the projection onto $\closure{\operatorname{ran}} \abs{x}$. In this setting, this projection lies in $M$ and coincides with $\RP(x)$,
but in the general monotone complete setting, there is no reason to expect them to coincide.
Also worth mentioning is the fact that, just like von Neumann algebras canonically admit a weak*-topology, monotone complete C*-algebras admit notions of order convergence, with the one of interest to us given by Hamana in \cite[Section~1]{hamana82_mc_tensor_products_I}. Note that Hamana defines a notion of convergence for nets, without necessarily showing that it arises out of a topology. While we will not actually need to use this notion of convergence most of the time, it is still worth keeping its existence in mind, as in particular it is a very necessary part of the definition of Hamana's monotone complete crossed product, which we make heavy use of.
\begin{definition}
Let $A$ be a monotone complete C*-algebra. We say that a net $(x_\alpha)_\alpha$ in $A$ \emph{order-converges} to $x \in A$, and write $O-\lim x_\alpha = x$, if for every $k = 0, 1, 2, 3$, there are bounded nets $(y_\alpha^{(k)})_\alpha$ and $(z_\alpha^{(k)})_\alpha$ of self-adjoint elements in $A$, and self-adjoint elements $y^{(k)} \in A$, such that the following hold:
\begin{itemize}
\item $0 \leq y_\alpha^{(k)} - y^{(k)} \leq z_\alpha^{(k)}$;
\item The net $(z_\alpha^{(k)})_\alpha$ is decreasing and has infimum zero;
\item $\sum_{k=0}^3 i^k y_\alpha^{(k)} = x_\alpha$ and $\sum_{k=0}^3 i^k y^{(k)} = x$.
\end{itemize}
\end{definition}
Many times, certain properties of a non-monotone complete C*-algebra can be studied more easily by embedding it into a certain monotone completion. This is done, for example, in \cite{hamana81_regular_embeddings}, where the \emph{regular monotone completion} $\closure{A}$ of $A$ is constructed. This object admits nice abstract properties that describe it uniquely, but is perhaps a bit difficult to get a concrete handle on. If we instead consider a crossed product C*-algebra $A \rtimes_\lambda G$, then, under certain conditions, there is another monotone complete C*-algebra that it embeds into, and is far easier to explicitly write down. In \cite[Section~3]{hamana82_mc_tensor_products_II}, the \emph{monotone complete crossed product} is defined as follows:
Assume $G$ is a discrete group acting on a unital C*-algebra $A \subseteq B(H)$, and recall that a reduced crossed product $A \rtimes_\lambda G$ can be viewed as bounded operators acting on the Hilbert space $H \otimes \ell^2(G)$. One may concretely view every operator $T \in B(H \otimes \ell^2(G))$ as a matrix $T = [T_{r,s}]_{r,s \in G}$ over $B(H)$. With respect to this representation, every finitely supported element $\sum_{t \in G} a_t \lambda_t\in A\rtimes_\lambda G$ embeds as the matrix $[r^{-1} \cdot a_{rs^{-1}}]_{r,s \in G}$.
Using the fact that such symmetry is still present after taking limits, every element $a\in A\rtimes_\lambda G$ can be written as a formal sum $\sum_{t\in G}a_t\lambda_t$, in a unique way.
However, there is nothing stopping us from instead considering the following, perfectly valid, operator subsystem of $B(H \otimes \ell^2(G))$.
\[M(A,G) \coloneqq \setbuilder{\sum_{t \in G} a_t \lambda_t \text{ (formal sum)}}{[r^{-1} \cdot a_{rs^{-1}}]_{r,s \in G}\in B(H \otimes \ell^2(G))}. \]
It is immediate that the coefficients $(a_t)_{t\in G}$ uniquely determine the elements of $M(A,G)$. What is not immediately clear is why $M(A,G)$ is, in any way, closed under multiplication. We summarize below results from
\cite[Section~6]{hamana82_mc_tensor_products_I} and \cite[Section~3]{hamana82_mc_tensor_products_II} (see also \cite[Section~3]{hamana85-injective_envelopes_equivariant}), which show that things mostly still behave how one would expect, as long as we specify the correct multiplication structure to use.
\begin{proposition}
Let $G$ be a discrete group acting on a monotone complete C*-algebra $A$. Then $M(A,G)$ is a C*-algebra, when inheriting the involution and Banach space structure from $B(H \otimes \ell^2(G))$, but equipped with the \emph{new} multiplication
\[ [ x_{r,s} ]_{r,s\in G} \cdot [ y_{r,s} ]_{r,s\in G} \coloneqq \left[ O-\sum_{t \in G} x_{r,t} y_{t,s} \right]_{r,s\in G} \]
where $[x_{r,s}]_{r,s \in G}$ and $[y_{r,s}]_{r,s \in G}$ are the matrix representations with respect to the Hilbert space $H \otimes \ell^2(G)$, and $O-\sum_{t \in G} x_{r,t} y_{t,s}$ denotes the order-limit in $A$ of the finite sums. The multiplication of the formal sums is reflected in the following way:
\[ \left( \sum_{g \in G} a_g \lambda_g \right) \cdot \left( \sum_{g \in G} b_g \lambda_g \right) = \sum_{g \in G} \left( O-\sum_{t \in G} a_{t^{-1}} (t^{-1} \cdot b_{tg}) \right) \lambda_g. \]
Moreover, $M(A,G)$ is monotone complete.
\end{proposition}
\begin{remark}
As remarked in \cite[Section~3]{hamana85-injective_envelopes_equivariant}, the above multiplication does not necessarily coincide with the usual multiplication in $B(H \otimes \ell^2(G))$, due to the fact that we are taking an order-limit in $A$ instead of, say, a strong limit in $B(H)$.
\end{remark}
Most of the time, we will not be multiplying arbitrary elements in $M(A,G)$ together, and thus the reader should not worry too much about order-convergent sums. However, one fact that we absolutely will be making heavy use of is the fact that $M(A,G)$ is itself monotone complete, whenever $A$ is.
\begin{remark}\label{rem:CondExpeMonComp}
Just as for reduced crossed products, there is an equivariant faithful conditional expectation $E \colon M(A,G) \to A$, extending the canonical conditional expectation on $A \rtimes_\lambda G$. Every $x \in M(A,G)$ is given by the formal sum $x=\sum_{g\in G}E(x\lambda_g^*)\lambda_g$.
\end{remark}
Moving back to the more basic theory of monotone complete C*-algebras, they admit fairly canonical monotone complete subalgebras. The following are two results along these lines.
\begin{proposition}
\label{prop:monotone_complete_fixed_point_algebra}
Let $G$ be a discrete group acting on a monotone complete C*-algebra $A$ by *-automorphisms. The fixed point algebra
\[ A^G \coloneqq \setbuilder{a \in A}{g \cdot a = a \text{ for all } g \in G} \]
is itself monotone complete.
\end{proposition}
\begin{proof}
In general, it is clear that if $(x_\lambda)_\lambda$ is a bounded increasing net of self-adjoint elements in $A$, and $\alpha \colon A \to B$ is a *-isomorphism between two monotone complete C*-algebras, then $(\alpha(x_\lambda))_\lambda$ is a bounded increasing net of self-adjoint elements in $B$, and ${\sup\limits_\lambda}^{B} \alpha(x_\lambda) = \alpha({\sup\limits_\lambda}^{A} x_\lambda)$.
If $(x_\lambda)_\lambda$ is such a net in $A^G$, then we know that the supremum in $A$ exists. Using the above observation, given any $g \in G$, we have
\[ g \cdot ({\sup_\lambda}^{A} x_\lambda) = {\sup_\lambda}^{A} (g \cdot x_\lambda) = {\sup_\lambda}^{A} x_\lambda. \]
In other words, ${\sup\limits_\lambda}^{A} x_\lambda \in A^G$. Given, however, that this was a supremum in the \emph{larger} C*-algebra $A$, then it automatically serves as a supremum in the smaller algebra $A^G$.
\end{proof}
\begin{proposition}
\label{prop:monotone_complete_center}
Let $A$ be a monotone complete C*-algebra. Then the center $Z(A)$, is automatically monotone complete.
\end{proposition}
\begin{proof}
This follows from applying Proposition~\ref{prop:monotone_complete_fixed_point_algebra} to $A$ and the group of unitaries $U(A)$ acting on $A$ by conjugation.
\end{proof}
We will often be interested with working more hands-on with the center of a monotone complete C*-algebra. It is a well-known result (we include a more modern citation) that, in the commutative setting, monotone complete C*-algebras are determined by a topological condition on their spectra.
\begin{theorem}[{\cite[Theorem~2.3.7]{SWMonotoneComplete}}]
\label{thm:commutative_monotone_complete_iff_extremally_disconnected}
Let $X$ be a compact Hausdorff space. The C*-algebra $C(X)$ is monotone complete if and only if $X$ is \emph{extremally disconnected}, i.e.\ the closure of any open set $U \subseteq X$ is in fact clopen.
\end{theorem}
\subsection{Injective envelopes}
\label{sec:preliminaries:injective_envelopes}
The theory of injective envelopes of C*-algebras was introduced by Hamana in the 1970s and 1980s, originally in \cite{hamana79_injective_envelopes_cstaralg} and \cite{hamana79_injective_envelopes_opsys}, along with an equivariant version in \cite{hamana85-injective_envelopes_equivariant}.
When dealing with Hamana's theory, the right category to work in is the category of \emph{$G$-operator systems}. That is, the category where objects are operator systems with an action of a discrete group $G$ by unital complete order isomorphisms (these automatically become *-isomorphisms whenever the operator system $G$ is acting upon is a C*-algebra). The non-equivariant version can be obtained by letting $G = \set{e}$. However, the objects we deal with in this paper will always end up being C*-algebras. Therefore, for convenience, we recall some of Hamana's theory here in a way that requires only very basic theory of operator systems (see \cite{paulsen_cb_maps} for an extensive introduction).
\begin{definition}
Let $S$ and $T$ be operator systems. We say that $\phi\colon S\to T$ is a \emph{complete order isomorphism} if $\phi$ is a unital completely positive linear isomorphism with completely positive inverse.
\end{definition}
When two operators systems $S$ and $T$ are equipped with an action of a discrete group $G$ by complete order isomorphisms, we say that $S$ embeds equivariantly into $T$, and write $(S,G)\subseteq (T,G)$, if there is a $G$-equivariant unital completely positive map $\phi: S\to T$ which is a complete order isomorphism onto its range. Such a map is called an \emph{equivariant embedding}.
\begin{theorem}
Let $G$ be a discrete group acting on a unital C*-algebra $A$. Then there exists a unique $G$-C*-algebra, denoted $I_G(A)$, with the following properties:
\begin{enumerate}
\item $(A,G)\subseteq (I_G(A),G)$ (inclusion of C*-algebras).
\item Let $S$ and $T$ be $G$-operator systems such that $(S,G)\subseteq (T,G)$. Then every unital $G$-equivariant completely positive map $\phi\colon S\to I_G(A)$ extends to a unital $G$-equivariant completely positive map $\tilde{\phi}\colon T\to I_G(A)$.
\item Let $S$ be a $G$-operator system and assume that $\phi\colon I_G(A)\to S$ is a $G$-equivariant unital completely positive map which restricts to an embedding on $A$. Then $\phi$ is an embedding.
\end{enumerate}
\end{theorem}
When $G=\set{e}$, one obtains the \emph{injective envelope} of $A$, which is denoted by $I(A)$.
\begin{definition}
\label{def:GinjectiveGrigidetc}
Property (2) expresses the fact that $I_G(A)$ is a \emph{$G$-injective} object in the category of $G$-operator systems. Property (3) is known as \emph{$G$-essentiality} of the inclusion $A\subseteq I_G(A)$. Moreover, the inclusion $A\subseteq I_G(A)$ is \emph{$G$-rigid}. Namely, the identity map on $I_G(A)$ is the only $G$-equivariant unital completely positive map $\phi\colon I_G(A)\to I_G(A)$ which restricts to the identity map on $A$ (see \cite[Lemma~2.4]{hamana85-injective_envelopes_equivariant}). If we set $G = \set{e}$, we obtain the non-equivariant versions of these properties, and we simply call them \emph{injectivity}, \emph{essentiality}, and \emph{rigidity}.
\end{definition}
Very importantly, we may apply all of our theory of monotone complete C*-algebras to the theory of injective envelopes, as the following proposition shows (though well-known, we include a quick argument for convenience).
\begin{proposition}
\label{prop:injective_is_mc_and_ginjective_is_injective}
An injective C*-algebra is automatically monotone complete, and in particular this applies to $I(A)$ for any unital C*-algebra $A$. Moreover, a $G$-injective C*-algebra is automatically injective, and is therefore monotone complete as well. In particular, this applies to $I_G(A)$ for any unital C*-algebra $A$.
\end{proposition}
\begin{proof}
It suffices to show that any $G$-injective C*-algebra $B$ is injective and monotone complete. We claim that there is an \emph{equivariant} embedding of $B$ into some $B(H)$ equipped with a $G$-action. To see this, consider the reduced crossed product $B \rtimes_\lambda G$ and faithfully represent it on some Hilbert space, so that $B \rtimes_\lambda G \subseteq B(H)$. This embedding is equivariant with respect to the action of conjugation by the unitaries $\lambda_g$ in both algebras. By $G$-injectivity, there is now a $G$-equivariant unital completely positive map $\phi \colon B(H) \to B$ that restricts to the identity map on $B$. It is well-known that $B(H)$ is both injective (in the non-equivariant sense) and monotone complete, and moreover that these properties necessarily will pass to $B$. For the specific details, the fact that $B$ is injective can be found in \cite[Proposition~15.1]{paulsen_cb_maps}, and the fact that $B$ is monotone complete can be found in \cite[Section~8.1.1]{SWMonotoneComplete}. Letting $G = \set{e}$ also shows that injective C*-algebras are monotone complete in general.
\end{proof}
As remarked in \cite[Remark~2.3]{hamana85-injective_envelopes_equivariant}, $I_G(A)$ is always an injective C*-algebra. However, as $I(A)$ is an essential extension of $A$, it is not hard to construct the following chain of inclusions, where the embeddings are operator system embeddings:
\[ A\subseteq I(A)\subseteq I_G(A). \]
The next proposition shows that we may in fact obtain such inclusions in a $G$-C*-algebra sense.
\begin{proposition}[{\cite[Section~3]{hamana85-injective_envelopes_equivariant}}]
\label{prop:injective_envelope_inclusions}
Let $G$ be a discrete group acting on a unital C*-algebra $A$. The action of $G$ on $A$ extends uniquely to an action of $G$ on $I(A)$, and there is a $G$-equivariant injective *-homomorphism from $I(A)$ to $I_G(A)$ restricting to the identity map on $A$. In other words, we may view
\[ (A,G) \subseteq (I(A),G) \subseteq (I_G(A),G). \]
\end{proposition}
It is also convenient to have the above inclusions on the respective centers.
\begin{proposition}[{\cite[Corollary~4.3]{hamana79_injective_envelopes_cstaralg}} and {\cite[Lemma~6.2]{hamana85-injective_envelopes_equivariant}}]
\label{prop:injective_envelope_center_inclusions}
Let $G$ be a discrete group acting on a unital C*-algebra $A$. The inclusions $A \subseteq I(A) \subseteq I_G(A)$ from Proposition~\ref{prop:injective_envelope_inclusions} automatically restrict to inclusions on the center of each algebra. That is,
\[ (Z(A),G) \subseteq (Z(I(A)),G) \subseteq (Z(I_G(A)),G). \]
\end{proposition}
A slightly surprising version of Proposition~\ref{prop:injective_envelope_inclusions} for crossed products is the following theorem by Hamana, which will be instrumental in passing from $I_G(A)$ to $I(A)$.
\begin{theorem}[{\cite[Theorem~3.4]{hamana85-injective_envelopes_equivariant}}]
\label{thm:crossed_product_injective_envelope_inclusions}
Let $G$ be a discrete group acting on a unital C*-algebra $A$. We have a *-homomorphic embedding $I_G(A) \rtimes_\lambda G \hookrightarrow I(A \rtimes_\lambda G)$ that restricts to the identity on the copy of $A \rtimes_\lambda G$ in both algebras. Consequently, we may view
\[ A \rtimes_\lambda G \subseteq I(A) \rtimes_\lambda G \subseteq I_G(A) \rtimes_\lambda G \subseteq I(A \rtimes_\lambda G). \]
\end{theorem}
Keeping the above in mind, the following result is a very easy but very important observation, which follows from the uniqueness of the injective envelope. It will let us transfer properties between $A \rtimes_\lambda G$, $I(A) \rtimes_\lambda G$, and $I_G(A) \rtimes_\lambda G$.
\begin{proposition}
\label{prop:intermediate_algebra_shares_injective_envelope}
Let $A$ and $B$ be unital C*-algebras such that $A \subseteq B \subseteq I(A)$. Then there is a *-isomorphism $I(B) \cong I(A)$ which restricts to the identity on $B$.
\end{proposition}
Hamana proves in \cite[Theorem~7.1]{hamana81_regular_embeddings} the equivalence between a C*-algebra $B$ being prime, and the regular monotone completion $\closure{B}$ being a factor. This is also true when considering the injective envelope $I(B)$ instead, which can be proven using the exact same proof, or the fact that $Z(\closure{B}) = Z(I(B))$ (see \cite[Theorem~6.3]{hamana81_regular_embeddings}).
Together with Theorem~\ref{thm:crossed_product_injective_envelope_inclusions} and Proposition~\ref{prop:intermediate_algebra_shares_injective_envelope}, we obtain the following proposition:
\begin{proposition}
\label{proposition:primeifffactor}
Let $B$ be a unital C*-algebra. Then $B$ is prime if and only if $I(B)$ is a factor.
In particular, for a discrete group $G$ acting on a unital C*-algebra $A$, the following are equivalent:
\begin{enumerate}
\item $A\rtimes_\lambda G$ is prime.
\item $I(A)\rtimes_\lambda G$ is prime.
\item $I_G(A)\rtimes_\lambda G$ is prime.
\end{enumerate}
\end{proposition}
Although $I(A)\rtimes_\lambda G$ and $I_G(A)\rtimes_\lambda G$ embed into the monotone complete C*-algebra $I(A\rtimes_\lambda G)$, which is also their common injective envelope, we may alternatively consider the embeddings of $I(A)\rtimes_\lambda G$ and $I_G(A)\rtimes_\lambda G$ into the monotone complete C*-algebras $M(I(A),G)$ and $M(I_G(A),G)$, respectively. This is usually more helpful, because it allows us to work with elements that still admit well-behaved series $x = \sum_{t \in G} x_t \lambda_t$, with $x_t$ inside $I(A)$ or $I_G(A)$.
We will also need the following easy observation.
\begin{proposition}
\label{prop:minimality_transfers_to_injective_envelopes_and_centers}
Let $G$ be a discrete group acting minimally on a unital C*-algebra $A$. The induced actions on the C*-algebras $I(A)$, $I_G(A)$, $Z(A)$, $Z(I(A))$, and $Z(I_G(A))$ are all minimal as well.
\end{proposition}
\begin{proof}
Assume that $J \subseteq I_G(A)$ is a non-trivial $G$-invariant ideal of $I_G(A)$. The quotient map $q \colon I_G(A) \to I_G(A) / J$ is necessarily injective on $A$ by minimality, but is not injective on $I_G(A)$. This contradicts $G$-essentiality of the inclusion $A\subseteq I_G(A)$.
The exact same proof will work for $I(A)$, as long as we can show that the inclusion $A\subseteq I(A)$ is $G$-essential. Assume $\phi \colon I(A) \to B$ is an equivariant unital completely positive map that is an embedding on $A$. By $G$-injectivity, it can be extended to an equivariant unital completely positive map $\wtilde{\phi} \colon I_G(A) \to I_G(B)$. By essentiality of $(A,G)\subseteq (I_G(A),G)$, this extension is necessarily an embedding, so in particular the original map $\phi$ was also an embedding.
Since $A$ was arbitrary, if we show that $(Z(A),G)$ is minimal, this will automatically hold for $I(A)$ and $I_G(A)$ as well. To see this, identify $Z(A)$ with $C(X)$, for some compact space $X$, and assume that $U \subseteq X$ is a non-zero proper invariant open subset for the induced action on $X$.
Choose $x\in X\setminus U$, and extend the evaluation map $\delta_x\colon Z(A)\to\mathbb{C}$ to a state $\rho\colon A\to \mathbb{C}$. Then $\rho$ vanishes on the ideal generated by $C_0(U)$ in $A$, namely $\closure{\operatorname{span}} (C_0(U) \cdot A)$, since $Z(A)$ lies in the multiplicative domain (see \cite[Proposition~1.5.7]{Brown_ozawa} and \cite[Definition~1.5.8]{Brown_ozawa}) of $\rho$. (As a side note, the ideal $\closure{\operatorname{span}} (C_0(U) \cdot A)$ is just $C_0(U) \cdot A$ by the Cohen-Hewitt factorization theorem). In any case, as $\rho$ is non-trivial, this ideal (which is clearly $G$-invariant and non-zero) cannot be all of $A$, contradicting minimality of the action.
\end{proof}
We also briefly mentioned in the introduction of this paper that a large part of the difficulty in characterizing simplicity of $A \rtimes_\lambda G$ is the difficulty in passing from results on $I_G(A)$ to results on $I(A)$. The following proposition, which we prove here for convenience, was observed by Hamana in \cite[Remark~3.8]{hamana85-injective_envelopes_equivariant}.
\begin{proposition}
Let $G$ be a discrete amenable group and let $M$ be an injective von Neumann algebra. Then $M$ is automatically $G$-injective.
\end{proposition}
\begin{proof}
Consider an inclusion $(S,G)\subseteq (T,G)$ of $G$-operator systems, and assume $\phi\colon S \to M$ is a $G$-equivariant unital completely positive map. We know that there is at least one extension $\psi\colon T \to M$ which is not necessarily equivariant, by injectivity. Hence, the following set is nonempty:
\[ \setbuilder{\psi\colon T \to M}{\psi \text{ is a unital completely positive map extending } \phi}. \]
Observe that this set admits a compact Hausdorff topology under the point-weak* topology, due to the fact that $M$ is a von Neumann algebra. It also canonically has a convex structure. Finally, it admits a $G$-action given by $\psi \mapsto g \psi g^{-1}$, and this is an action by affine homeomorphisms. By Day's fixed fixed point theorem, this action admits a fixed point, and in this setting, a fixed point is a $G$-equivariant unital and completely positive map $\psi\colon T \to M$ extending $\phi\colon S \to M$.
\end{proof}
This proof is highly contingent on $M$ being a von Neumann algebra, thus having the unit ball be compact under a nice topology. In general, $I(A)$ is only monotone complete, and thus the closest analogue is a notion of order convergence. See, for example, the discussion near the start of \cite[Section~1]{hamana82_mc_tensor_products_I}. It is unlikely that the unit ball of $I(A)$ is compact under any order topology, and thus this crucial piece of the puzzle is missing in the above proof. In other words, one expects $I_G(A) \neq I(A)$.
With that being said, this does not rule out the existence of some other deep reason for the equality $I_G(A) = I(A)$ to still hold, and such a result would indeed trivialize half of our work. Evidence for it being true is that they are completely indistinguishable when $G$ is amenable and acts minimally on a commutative C*-algebra $C(X)$. In this setting, \cite[Theorem~3.4]{kawabe_crossed_products} says that the crossed product $C(X) \rtimes_\lambda G$ will be simple if and only if the action on the spectrum of $I_G(C(X))$ is free. On the other hand, by \cite[Theorem~2]{archbold_spielberg_amenable_crossed_products}, simplicity will be equivalent to topological freeness of the action on $X$, which in turn is known to be equivalent to freeness of the action on the spectrum of $I(C(X))$.
\subsection{Properly outer automorphisms}
\label{sec:preliminaries:properly_outer}
This subsection serves two purposes. The first is simply for intuition on the statements of the main theorems (Theorem~\ref{thmIntro:A}, Theorem~\ref{thmIntro:B}, and Theorem~\ref{thmIntro:C}), as all of them characterize simplicity (or primality) using an equivariant version of proper outerness, or the lack thereof. However, observe that in each theorem, \emph{two} characterizations are given. One is a more elegant, but perhaps more intractable characterization involving the somewhat mysterious injective envelope $I(A)$. The other is a characterization involving the ideals of $A$ (and their multiplier algebras), and it is more down-to-earth and understandable, even if it is more technically cumbersome. These are both based off of two equivalent characterizations for what is known as \emph{proper outerness} of an action of $G$ on a C*-algebra $A$.
Our story begins with von Neumann algebras. Recall that an automorphism $\alpha$ of a measure space $(X,\mu)$ is said to be \emph{essentially free} if the set of fixed points of $\alpha$ has measure zero, and this notion is very important when studying actions on abelian von Neumann algebras $L^\infty(X,\mu)$. In \cite{kallman_free_actions}, Kallman generalizes this notion to the noncommutative setting, and introduces the notion of a \emph{freely acting} automorphism of a von Neumann algebra. However, it was observed by Hamana in \cite[Proposition~5.1]{hamana82_mc_tensor_products_I} that because monotone complete C*-algebras in many ways behave like von Neumann algebras, in particular with being able to take polar decompositions, all of Kallman's results work exactly the same in this more general setting. The following series of results are those originally observed by Kallman and Hamana:
\begin{definition}\label{def:Prop_outer_monotone_complete}
Let $A$ be a monotone complete C*-algebra. We say that a *-automorphism $\alpha \in \Aut(A)$ is \emph{freely acting} if whenever $x \in A$ satisfies $xy = \alpha(y) x$ for all $y \in A$, we have $x = 0$. We say that $\alpha$ is \emph{properly outer} if there is no non-zero $\alpha$-invariant central projection $p \in A$ with $\alpha|_{Ap}$ inner.
\end{definition}
The following result shows that in the setting of monotone complete C*-algebras, there is no difference between the notions defined above. Actually, there is a bit of a subtle point in the statement of the theorem. Namely, in the definition of proper outerness, one can ask whether $\alpha|_{pAp}$ is ever inner for $p$ not necessarily central, but this does not actually change the definition.
\begin{theorem}[{\cite[Proposition~5.1]{hamana82_mc_tensor_products_I}}]
\label{thm:PropInnerDecoMonCom}
Let $A$ be a monotone complete C*-algebra, and let $\alpha \in \Aut(A)$. There is a largest $\alpha$-invariant projection $p \in A$ such that $\alpha|_{pAp}$ is inner. Moreover, $p$ is central and $\alpha|_{A(1-p)}$ is freely acting. In fact, this decomposition of $A \cong Ap \oplus A(1-p)$ into inner and freely acting parts is unique.
\end{theorem}
For the proof of Theorem~\ref{thm:PropInnerDecoMonCom}, Hamana uses the following theorem, which we explicitly state below as we will make use of it as well. We provide the proof for convenience, but we closely follow the proof given by Kallman in \cite[Theorem~1.1]{kallman_free_actions}, in the setting of von Neumann algebras.
\begin{theorem}
\label{thm:PropOuterCorner}
Let $A$ be a monotone complete C*-algebra, let $\alpha \in \Aut(A)$, and assume that $x \in A$ implements the automorphism in the sense that $xy = \alpha(y) x$ for all $y \in A$. Let $x = u \abs{x}$ be the polar decomposition of the element $x$, and let $p = u^*u$ be the domain projection of the partial isometry $u$. We have:
\begin{enumerate}
\item The projection $p$ is an $\alpha$-invariant central projection in $A$;
\item The partial isometry $u$ is a unitary in $U(Ap)$, and $\alpha|_{Ap} = \Ad u$.
\end{enumerate}
\end{theorem}
\begin{proof}
We claim that $x^*x\in Z(A)$. To see this, first note that, by taking adjoints in the equality $xy = \alpha(y)x$, for all $y \in A$, we have that $x^*\alpha(y) = yx^*$, for all $y \in A$. Consequently,
\[ x^*x y = x^* \alpha(y) x = y x^*x. \]
Therefore, $x^*x \in Z(A)$, and so $\RP(x^*x)\in Z(A)$ (see for example \cite[Chapter~1, Section~3, Corollary~1]{Berberian}).
It is easy to verify that $\RP(x)=\RP(x^*x)$. Thus, the projection $p = \RP(x)$ in the statement of the theorem is indeed central.
Observe that $xx^*\RP(x)=x\RP(x)x^*=xx^*$, using centrality of $\RP(x)$.
Therefore $xx^*(1-\RP(x))=0$, which implies that $\RP(xx^*)(1-\RP(x))=0$, by definition of polar decomposition (see Proposition~\ref{prop:monotonecompletePolarDec}). In other words, we have that $\RP(x^*)=\RP(xx^*)\leq \RP(x)$.
A similar computation shows that $\RP(x)\leq \RP(x^*)$.
Hence, $uu^*=\RP(x^*)=\RP(x)=u^*u$.
We now wish to show that $\alpha(p) = p$.
To this end, note that because $p$ and therefore $\alpha(p)$ is central, we have $x\alpha(p)=\alpha(p)x=xp=x$, which implies that $x(1-\alpha(p))=0$, and so $p(1-\alpha(p))=0$, by definition of polar decomposition. Therefore, $p \leq \alpha(p)$. For the reverse inequality, we have
\[\alpha(x)p=\alpha(x\alpha^{-1}(p))=\alpha(px)=\alpha(xp)=\alpha(x), \]
which implies that $\alpha(p) \leq p$ using the fact that $\alpha(p)=\RP(\alpha(x))$.
Finally, we prove that the automorphism $\alpha$ is implemented by $u$ on the corner $Ap$.
Let $y\in A$ be given. Keeping in mind that $\abs{x}$ is central, we have
\[ \abs{x} uy = u\abs{x}y = xy = \alpha(y)x = \alpha(y) u\abs{x} = \abs{x} \alpha(y) u. \]
Thus, $\abs{x}(uy-\alpha(u)y)=0$, and we conclude that $\RP(\abs{x})(uy-\alpha(y)u)=0$. It is again easy to verify that $\RP(\abs{x})=\RP(x^*x)$, and so we have $p(uy-\alpha(y)u)=0$. As $p\in Z(A)$ and $up=u$, we see that $uy=\alpha(y)u$. Observe that, because $u$ is a unitary in the $\alpha$-invariant corner $Ap$, if $y \in Ap$, then in fact $\alpha(y) = uyu^*$.
\end{proof}
We state the following corollary (which we mentioned earlier) to Theorem~\ref{thm:PropInnerDecoMonCom}.
\begin{corollary}
Given a *-automorphism $\alpha \in \Aut(A)$ on a monotone complete C*-algebra $A$, it is freely acting if and only if it is properly outer.
\end{corollary}
Everything that was done above shows that proper outerness is a very intuitive and well-behaved property in the setting of monotone complete C*-algebras. Our discussion now takes a turn towards the setting of general C*-algebras, given that an equivariant version of proper outerness forms the second half of each of Theorem~\ref{thmIntro:A}, Theorem~\ref{thmIntro:B}, and Theorem~\ref{thmIntro:C}. As mentioned in the introduction, such notions (in their non-equivariant versions) have played an important role in studying the simplicity of reduced crossed products $A \rtimes_\lambda G$. Multiple generalizations of proper outerness to automorphisms on arbitrary C*-algebras were given by various authors, and they generally all coincide when the underlying C*-algebra is separable. In particular, consider the properties listed in Theorem~\ref{thm:properly_outer_equivalence} below. The proofs of these implications are somewhat subtle and trace back through several papers (with citations in the proof of Theorem~\ref{thm:properly_outer_equivalence}). In the interest of the reader's experience, we will note ahead of time that:
\begin{itemize}
\item These are already somewhat well-known results, and the proof is given just in the interest of gathering everything in one convenient place.
\item The Borchers spectrum $\Gamma_B(\cdot)$ is used in one of the equivalences. A precise definition of this spectrum is given below, but we will not make use of it anywhere else in our paper.
\item Derivations are used in the proof. They will not be used anywhere else in this paper.
\item This proof makes brief use of the theory of injective envelopes of non-unital C*-algebras, discussed in Section~\ref{sec:proper_outerness_on_A}. Non-unital C*-algebras and their basic injective envelope theory will not be used anywhere else in the paper, except of course in that specific section.
\end{itemize}
The precise definition of the Borchers spectrum itself can be found, for example, in \cite[Section~8.8]{pedersen_cstar_algebras_and_their_automorphism_groups} (alternatively, see \cite[Section~3]{olesen_pedersen_III} or \cite[Section~1]{Kishimoto_freely_acting}), along with \cite[Section~1]{olesen_inner_automorphisms} for the Arveson spectrum component. We repeat the full definition here, for convenience:
\begin{definition}
\label{def:borchers_spectrum}
Assume $\alpha \colon G \to \Aut(A)$ is an action of a discrete abelian group $G$ on a not necessarily unital C*-algebra $A$. (This definition can be modified to work in the locally compact setting). The Borchers spectrum is defined as
\[ \Gamma_B(\alpha) \coloneqq \bigcap_{B \in H^\alpha_B(A)} \operatorname{Sp}(\alpha|_B), \]
where $H^\alpha_B(A)$ is the set of $\alpha$-invariant hereditary C*-algebras $B \subseteq A$ generating an essential ideal in $A$, and $\operatorname{Sp}(\beta)$ of an automorphism $\beta \in \Aut(B)$ is the Arveson spectrum, given by
\[ \operatorname{Sp}(\beta) = \setbuilder{\gamma \in \what{G}}{\what{f}(\gamma) = 0 \text{ for all } f \in I^\beta_B}, \]
where
\[ I^\beta_B = \setbuilder{f \in \ell^1(G)}{\sum_{g \in G} f(g) \beta_g(x) = 0 \text{ for all } x \in B}, \]
and
\[ \what{f}(\gamma) = \sum_{g \in G} f(g) \gamma(g). \]
The Borchers spectrum of a single automorphism $\alpha \in \Aut(A)$ is defined as the Borchers spectrum for the corresponding $\mathbb{Z}$-action on $A$, where $n \in \mathbb{Z}$ acts by $\alpha^n$.
\end{definition}
\begin{theorem}
\label{thm:properly_outer_equivalence}
Let $A$ be a unital C*-algebra, and let $\alpha \in \Aut(A)$, with its unique extension to $I(A)$ also denoted by $\alpha$. Consider the following conditions:
\begin{enumerate}
\item
\label{thm:properly_outer_equivalence:borchers}
There is a non-zero $\alpha$-invariant ideal $J \subseteq A$ such that $\Gamma_B(\alpha|_J) = \set{e}$.
\item
\label{thm:properly_outer_equivalence:injective_envelope}
There is a non-zero $\alpha$-invariant central projection $p \in I(A)$, and a unitary $u \in U(I(A)p)$ such that $\alpha$ acts by $\Ad u$ on $I(A)p$.
\item
\label{thm:properly_outer_equivalence:elliott}
There is a non-zero $\alpha$-invariant ideal $J \subseteq A$ and a unitary $u \in M(J)$, such that $\norm{\alpha|_J - (\Ad u)|_J} < 2$.
\item
\label{thm:properly_outer_equivalence:elliott_strong}
Given any $\varepsilon\in (0,2)$, there exists a non-zero $\alpha$-invariant ideal $J \subseteq A$ and a unitary $u \in M(J)$ such that $\norm{\alpha|_J - (\Ad u)|_J} < \varepsilon$.
\end{enumerate}
We have that (\ref{thm:properly_outer_equivalence:borchers}) and (\ref{thm:properly_outer_equivalence:injective_envelope}) are equivalent, and both are implied by (\ref{thm:properly_outer_equivalence:elliott}) and (\ref{thm:properly_outer_equivalence:elliott_strong}). If $A$ is separable, all four are equivalent.
\end{theorem}
\begin{proof}
It is clear that (\ref{thm:properly_outer_equivalence:elliott_strong}) $\implies$ (\ref{thm:properly_outer_equivalence:elliott}). Now assume that (\ref{thm:properly_outer_equivalence:elliott}) holds, with $J \subseteq A$ and $u \in M(I)$ so that $\norm{\alpha|_J - (\Ad u)|_J} < 2$. The following argument is based on what is observed in \cite[Remark~2.2]{elliott_properly_outer}. Our inequality is equivalent to $\norm{(\Ad u^* \circ \alpha)|_J - \id|_J} < 2$. However, automorphisms that are ``close'' to the identity have very special structure, and in particular, a result of Kadison and Ringrose (specifically \cite[Theorem~7]{kadison_ringrose}) tells us that such automorphisms lie on a norm-continuous one-parameter subgroup of $\Aut(J)$ (automorphisms lying in the image of a continuous homomorphism from $\mathbb{R}$ to $\Aut(J)$). It is implicitly recognized in the proof of \cite[Lemma~2]{kadison_ringrose} (using an even more classical result) that such automorphisms are of the form $\exp \delta$ for a derivation $\delta \colon J \to J$ satisfying $\delta(x^*) = \delta(x)^*$. Consequently, we have that
\[ \alpha|_J = (\Ad u \circ \exp \delta)|_J. \]
Consider $J$ on its own as a C*-algebra, one that is not necessarily unital, and let $J^+$ be $J$ if it is coincidentally unital (with respect to its own unit, and not the one in $A$), and the unitization if it is not. Observe that the canonical extension of $\alpha|_J$ to $J^+$ is also of the form $\Ad u \circ \exp \delta$, where we are also extending $\delta$ canonically to $J^+$. (Because all of the extensions we are dealing with are canonical or unique, we will avoid re-labeling our maps.) By \cite[Theorem~2.1]{hamana82_okayasu_saito}, there exists a unique extension of $\delta$ to a derivation on the injective envelope $I(J^+)$, and furthermore, it is of the form $\Ad x$ for some $x \in I(J^+)$. The same theorem tells us that, because $\delta$ is self-adjoint on $J^+$ (equivalently, $i \delta$ is skew-adjoint), then its unique extension to $I(J^+)$ is also self-adjoint. It is not hard to check that this implies $\operatorname{Re} x \in Z(I(A))$, and so replacing $x$ by $x - \operatorname{Re} x$, we may assume without loss of generality that $x^* = -x$. Thus, the exponential $\exp \delta$ is also inner and given by $\Ad e^x$, where $e^x$ is unitary. By realizing $M(J)$ as the idealizer of $J$ in $I(J^+)$ (see also \cite[Section~1]{hamana82_centre} or Proposition~\ref{prop:multiplier_algebra_contained_in_injective_envelope}), we have $M(J) \subseteq I(J^+)$, and so the unique extension of $\alpha|_J$ to $I(J^+)$ is inner and of the form $\Ad (u e^x)$.
In summary, (\ref{thm:properly_outer_equivalence:elliott}) implies that $\alpha|_J$ is inner on the injective envelope $I(J^+)$. Applying \cite[Theorem~7.3]{hamana85-injective_envelopes_equivariant} (while keeping in mind that the Borchers spectrum in this case is that of the corresponding $\mathbb{Z}$-action of $\alpha$), this is equivalent to $\Gamma_B(\alpha|_{J^+}) = \set{e}$. It is quite possible that Hamana's result was stated in the context of not necessarily unital C*-algebras, but in the interest of avoiding figuring that out, we just remark that $\Gamma_B(\alpha|_{J^+}) = \Gamma_B(\alpha|_J)$. Thus, (\ref{thm:properly_outer_equivalence:elliott}) $\implies$ (\ref{thm:properly_outer_equivalence:borchers}).
To show that (\ref{thm:properly_outer_equivalence:borchers}) and (\ref{thm:properly_outer_equivalence:injective_envelope}) are equivalent, first assume (\ref{thm:properly_outer_equivalence:borchers}), and again note that $\Gamma_B(\alpha|_{J^+}) = \Gamma_B(\alpha|_J)$. Now, $\Gamma_B(\alpha|_{J^+})$ is a subgroup of the dual group $\what{\mathbb{Z}}$, and so $\Gamma_B(\alpha|_{J^+}) = \set{e}$ if and only if $1 \in \Gamma_B(\alpha|_{J^+})^{\perp}$, which according to \cite[Theorem~7.3]{hamana85-injective_envelopes_equivariant} is equivalent to $\alpha$ acting inner on $I(J^+)$. Because of the correspondence given in \cite[Lemma~1.1]{hamana82_centre} (alternatively, see Proposition~\ref{prop:ideal_injective_envelope}) between ideals $J \subseteq A$ and corners $I(A)p \subseteq A$ generated by central projections in $A$, we get that (\ref{thm:properly_outer_equivalence:borchers}) is equivalent to (\ref{thm:properly_outer_equivalence:injective_envelope}).
Finally, the implication (\ref{thm:properly_outer_equivalence:injective_envelope}) $\implies$ (\ref{thm:properly_outer_equivalence:elliott_strong}) in the separable setting is given in one of the many equivalences listed in \cite[Theorem~6.6]{olesen_pedersen_III}. Specifically, it follows from the equivalence between (viii) and (iii), or more specifically their negations.
\end{proof}
In all of the above properties, the intuition behind them is that the automorphism $\alpha \in \Aut(A)$ is ``almost inner'' on an ideal $J \subseteq A$. Given that proper outerness is supposed to mean ``not inner on any piece of the C*-algebra'', the following definition makes sense.
\begin{definition}
Let $A$ be a unital C*-algebra. A *-automorphism $\alpha \in \Aut(A)$ is said to be \emph{properly outer in the sense of Kishimoto} if it satisfies the \emph{negation} of condition~(\ref{thm:properly_outer_equivalence:borchers}) in Theorem~\ref{thm:equivariant_proper_outer_equivalence} (originally introduced in \cite{Kishimoto_freely_acting} as ``freely acting''), and hence equivalently the negation of condition~(\ref{thm:properly_outer_equivalence:injective_envelope}) in that theorem. It is called \emph{properly outer in the sense of Elliott} if it satisfies the \emph{negation} of condition~(\ref{thm:properly_outer_equivalence:elliott}) in Theorem~\ref{thm:equivariant_proper_outer_equivalence}. Typically, ``properly outer'' on its own means properly outer in the sense of Kishimoto.
An action $\alpha\colon G\to \Aut(A)$ of a discrete group $G$ on the C*-algebra $A$ is said to be properly outer if each $\alpha_t$ corresponding to $t \in G \setminus \set{e}$ is properly outer.
\end{definition}
Recall that $I(A)$ is in many ways small enough to still remember many of the basic properties of $A$.
Thus, given that in the monotone complete setting, it is clear what the ``correct'' definition of proper outerness is (recall Definition~\ref{def:Prop_outer_monotone_complete}), the ``correct'' definition for an automorphism $\alpha\in \Aut(A)$ on a general C*-algebra $A$ should be the one that always coincides with the (unique) extension of $\alpha$ to $I(A)$ being properly outer. Hence, Kishimoto's definition is in some sense preferable to Elliott's definition (which only coincides with being properly outer on $I(A)$ in the separable setting). However, in the separable setting, where all of these definitions coincide, Elliott's definition is likely the more easy of the two to verify. Kishimoto's definition involves the use of the Borchers spectrum, and a quick glance at Definition~\ref{def:borchers_spectrum} will reveal that this spectrum has several quite cumbersome components to deal with, including
\begin{itemize}
\item Considering hereditary C*-subalgebras, as opposed to just ideals.
\item Considering the $\ell^1(\mathbb{Z})$-action of an automorphism, and its kernel.
\end{itemize}
\subsection{FC-hypercentral groups}
\label{sec:preliminaries:fc_hypercentral}
The center of a group $G$ can be phrased as the set of elements having a conjugacy class of size $1$. The FC-center $\FC(G)$ is a slight generalization of this concept, where we instead consider the set of elements admitting conjugacy classes of finite size. It is not hard to check that this is indeed a normal subgroup of $G$. This concept can once again be taken further. Consider now the quotient $G/\FC(G)$. There is no guarantee that there are no non-trivial elements of finite conjugacy class in this new quotient. That is, it may be the case that $\FC(G/\FC(G))$ is again non-trivial, and so we may quotient by it again.
The FC-central series is what is obtained after performing the above process ordinal-many times. Start with $F_1 = \FC(G)$, and define the sets $F_\alpha$ for the ordinal numbers $\alpha$ as follows:
\begin{enumerate}
\item For successor ordinals $\alpha + 1$, define $F_{\alpha + 1}$ as the (necessarily normal) subgroup of $G$ satisfying $F_{\alpha + 1} / F_\alpha = \FC(G / F_\alpha)$, that is, $F_{\alpha + 1}$ is the preimage of $\FC(G / F_\alpha)$ under the quotient map $G \twoheadrightarrow G / F_\alpha$.
\item For limit ordinals $\beta$, define $F_\beta = \bigcup_{\alpha < \beta} F_\alpha$ (again, necessarily normal in $G$).
\end{enumerate}
\begin{definition}
The \emph{FC-hypercenter} of $G$, denoted $\FCH(G)$, is the union of all of the $F_\alpha$ above. We say that $G$ is \emph{FC-hypercentral} if $\FCH(G) = G$. We say that $G$ is an \emph{FC-group} if $\FC(G)=G$.
\end{definition}
Recall that a group $G$ is called \emph{ICC} if every $g\in G\setminus \{e\}$ has infinite conjugacy class. FC-hypercentral groups are precisely the class of groups which have no non-trivial ICC quotients. This class contains all virtually nilpotent groups, and, in the finitely generated setting, coincides with the class of virtually nilpotent groups, and thus, also with the class of polynomial growth groups by Gromov's Theorem (see \cite[Theorem~2]{McLain_remarks_upper_central_series} and \cite[Theorem~2]{DM56_FcNipFcSol}).
In recent breakthrough results, the class of countable (not necessarily finitely generated) FC-hypercentral groups was identified with the class of strongly amenable groups \cite{FrischTamuzPooya} and with the class of groups which satisfy the \emph{Choquet-Deny property}, by \cite[Theorem 4.8]{Jaworski} and \cite{FrischHartmanTamuzPooya}.
We refer the reader to \cite{FrischThesis} for a smooth introduction to these exciting results.
\section{Pseudo-expectations and central elements}
\label{sec:pseudoexpectations}
Recall that there is always a canonical conditional expectation $E\colon A \rtimes_\lambda G \to A$, determined by mapping a finitely supported element $\sum_{t \in G} x_t \lambda_t\in A\rtimes_\lambda G$ to $x_e$. All of the existing ideal structure results for crossed products from the last few years pass through the machinery of what are known as \emph{pseudo-expectations}, which is an analogue obtained by expanding the codomain to the $G$-injective envelope $I_G(A)$. We recall the precise definition below.
\begin{definition}[{\cite[Definition~6.1]{kennedy_schafhauser_noncommutative_crossed_products}}]
\label{def:pseudoexpectation}
An \emph{equivariant pseudo-expectation} for a reduced crossed product $A \rtimes_\lambda G$ is a $G$-equivariant unital and completely positive map $F \colon A \rtimes_\lambda G \to I_G(A)$, such that $F|_A = \id_A$.
\end{definition}
Pseudo-expectations were first introduced in the non-equivariant setting by Pitts in \cite{Pitts}, and even in that setting were found to be helpful for understanding the ideal structure of $A\rtimes_\lambda G$ (see \cite{Zarikian}). However, we will focus on equivariant pseudo-expectations, following the work of Kennedy and Schafhauser \cite{kennedy_schafhauser_noncommutative_crossed_products}.
Before we proceed further, the following was not explicitly mentioned in \cite{kennedy_schafhauser_noncommutative_crossed_products}, but is a very convenient result that is used implicitly. We prove it for convenience, and will certainly make use of it as well.
\begin{proposition}
\label{prop:pseudoexpectations_conditional_expectations_bijection}
Equivariant conditional expectations $F'\colon I_G(A) \rtimes_\lambda G \to I_G(A)$ are in canonical bijection (given by restriction) with equivariant pseudo-expectations $F\colon A \rtimes_\lambda G \to I_G(A)$.
\end{proposition}
\begin{proof}
It is clear that equivariant conditional expectations on $I_G(A)\rtimes_\lambda G$ restrict to pseudo-expectations on $A \rtimes_\lambda G$. The map $F' \mapsto F'|_{A \rtimes_\lambda G}$ is injective, since $I_G(A)$ lies in the multiplicative domain of $F'$, and so $F'$ is uniquely determined by the values it takes on the unitaries $\{\lambda_t\}_{t\in G}$.
The restriction map is also surjective, as every equivariant pseudo-expectation $F\colon A \rtimes_\lambda G \to I_G(A)$ extends to some $G$-equivariant completely positive map $F'\colon I_G(A) \rtimes_\lambda G \to I_G(A)$, thanks to $G$-injectivity of $I_G(A)$. This extension, being the identity on $A$, is necessarily the identity on $I_G(A)$ by $G$-rigidity (see Definition~\ref{def:GinjectiveGrigidetc}).
\end{proof}
There are many reasons why pseudo-expectations are useful for studying simplicity of $A \rtimes_\lambda G$. One evidence is the following proposition, which directly characterizes simplicity, or more generally the intersection property, in terms of pseudo-expectations.
\begin{proposition}[{\cite[Theorem~6.6]{kennedy_schafhauser_noncommutative_crossed_products}}]
\label{prop:intersection_property_iff_faithful}
Let $G$ be a discrete group acting on a unital C*-algebra $A$. Then $A \rtimes_\lambda G$ has the intersection property if and only if every equivariant pseudo-expectation $F\colon A \rtimes_\lambda G \to I_G(A)$ is faithful.
\end{proposition}
This requirement that every equivariant pseudo-expectation be faithful is perhaps a bit mysterious. If $A$ is commutative, it follows from a combination of Proposition~\ref{prop:intersection_property_iff_faithful}, \cite[Theorem~3.4]{kawabe_crossed_products} and \cite[Theorem~6.4]{kennedy_schafhauser_noncommutative_crossed_products} that this is in fact equivalent to having a \emph{unique} equivariant pseudo-expectation, namely the canonical conditional expectation. (See also \cite[Theorem~4.6]{PittsZarikian} for the non-equivariant setting.) In the noncommutative setting, it is not known whether they are equivalent. This is one of the obstructions to obtaining a nice characterization of simplicity in the noncommutative case, as precisely what makes a pseudo-expectation non-faithful is poorly understood, while in contrast we have a better idea of what arbitrary pseudo-expectations look like. Proposition~\ref{prop:pseudoexpectation_coefficients} paints a picture on the latter (see also \cite{Ursu}, where similar ideas are used by the second coauthor in the context of traces on crossed products).
We would also like to remark that in an unpublished work of Zarikian
it was proven that, in the non-equivariant setting, having all pseudo-expectations $A \rtimes_\lambda G\to I(A)$ faithful implies that there is a unique pseudo-expectation.
\begin{proposition}
\label{prop:pseudoexpectation_coefficients}
Let $G$ be a discrete group acting on a unital C*-algebra $A$ and let $F \colon A \rtimes_\lambda G \to I_G(A)$ be an equivariant pseudo-expectation. Then $F$ is uniquely determined by the ``coefficients'' $x_t \coloneqq F(\lambda_t)$, which satisfy the following properties:
\begin{enumerate}
\item $x_e = 1$;
\item $x_t y = (t \cdot y) x_t$, for all $t\in G$ and all $y \in I_G(A)$;
\item $s \cdot x_t = x_{sts^{-1}}$, for all $s,t\in G$;
\item The matrices $[x_{st^{-1}}]_{s,t \in \mathcal{F}}$ are positive for every finite set $\mathcal{F} \subseteq G$.
\end{enumerate}
\end{proposition}
\begin{proof}
By Proposition~\ref{prop:pseudoexpectations_conditional_expectations_bijection}, we may transfer the discussion to equivariant conditional expectations $F\colon I_G(A) \rtimes_\lambda G \to I_G(A)$. These are uniquely determined by the coefficients $(x_t)_{t\in G}$ since $I_G(A)$ lies in the multiplicative domain of $F$.
We now turn to proving each individual property of the coefficients. It is clear that $x_e = 1$. Now let $y \in I_G(A)$ and $t\in G$ be given, and using multiplicative domain again, observe that:
\[ x_t y = F(\lambda_t) y = F(\lambda_t y) = F((t \cdot y) \lambda_t) = (t \cdot y) F(\lambda_t) = (t \cdot y) x_t. \]
Furthermore, $s \cdot x_t = x_{sts^{-1}}$ is due to $G$-equivariance of $F$. Finally, the last positivity condition is due to the fact that
\[ F^{(n)} \left( \begin{bmatrix} \lambda_{s_1} \\ \vdots \\ \lambda_{s_n} \end{bmatrix} \begin{bmatrix} \lambda_{s_1} \\ \vdots \\ \lambda_{s_n} \end{bmatrix}^* \right) = \left[x_{s_i s_j^{-1}}\right]_{i,j = 1, \dots, n}. \]
\end{proof}
\begin{remark}
It is not immediately obvious, but it can actually be shown that every time one has such coefficients $(x_t)_{t \in G}$ as above, this will define a pseudo-expectation $F\colon A \rtimes_\lambda G \to I_G(A)$ through $F(a \lambda_t) \coloneqq a x_t$. This is essentially what is proven, in a very special case, in \cite[Lemma~9.1]{kennedy_schafhauser_noncommutative_crossed_products}. However, we will not make use of this converse, and will not make use of the positive-definiteness requirement either.
\end{remark}
The following highlights another, seemingly unrelated, area in which such coefficients $(x_t)_{t \in G}$ show up. This will be crucial for us when constructing non-trivial central elements.
\begin{proposition}
\label{prop:crossed_product_center_coefficients}
Let $G$ be a discrete group acting on a unital C*-algebra $A$, and consider an element $x\coloneqq \sum_{t \in G} x_t^* \lambda_t\in A\rtimes_\lambda G$. (The stars on $x_t$ are intentional). Then, $x\in Z(A\rtimes_\lambda G)$ if and only if
\begin{enumerate}
\item $x_t y = (t \cdot y) x_t$, for all $t\in G$ and all $y \in A$;
\item $s \cdot x_t = x_{sts^{-1}}$, for all $s,t\in G$.
\end{enumerate}
\end{proposition}
\begin{proof}
Given $a\in A$, it is not hard to verify that
\[ a\cdot x=\sum_{t\in G}a x_t^*\lambda_{t} \quad \text{and} \quad x\cdot a=\sum_{t\in G} x_t^*(t\cdot a)\lambda_{t}. \]
Using the fact that elements in $A\rtimes_\lambda G$ are uniquely determined by their coefficients, we see that $x$ commutes with the copy of $A$ inside $A\rtimes_\lambda G$ if and only if condition~(1) of the proposition is satisfied.
Now given $g\in G$, we also have
\[ \lambda_s\cdot x=\sum_{t\in G}(s \cdot x_{s^{-1}t}^*)\lambda_t \quad \text{and} \quad x\cdot \lambda_s = \sum_{t\in G} x_{ts^{-1}}^*\lambda_t. \]
Comparing coefficients, we see that $x$ commutes with the set of unitaries $\setbuilder{\lambda_s}{s \in G}$ if and only if condition~(2) of the proposition is satisfied.
The result then follows.
\end{proof}
The following closely related proposition was proven by Hamana.
\begin{proposition}[{\cite[Lemma~10.3]{hamana85-injective_envelopes_equivariant}}]
\label{prop:mc_crossed_product_commutant_coefficients}
Let $G$ be a discrete group acting on a monotone complete C*-algebra $A$, and let $x\coloneqq \sum_{t \in G} x_t^* \lambda_t\in M(A,G)$ be given. The following conditions are equivalent.
\begin{enumerate}
\item $x\in Z(M(A,G))$;
\item $x\in (A\rtimes_\lambda G)'\cap M(A,G)$;
\item\begin{enumerate}
\item $x_t y = (t \cdot y) x_t$, for all $t\in G$ and all $y \in A$;
\item $s \cdot x_t = x_{sts^{-1}}$, for all $s,t\in G$.
\end{enumerate}
\end{enumerate}
\end{proposition}
The equivalence between properties (2) and (3) will be the one which is relevant for us, and its proof is similar to the proof of Proposition~\ref{prop:crossed_product_center_coefficients}. The equivalence to condition~(1) will not be used later, but it is interesting nonetheless.
\begin{proposition}
\label{prop:properly_outer_equivariant_polar_decompositions}
Let $G$ be a discrete group acting on a monotone complete C*-algebra $A$. Assume that $(x_t)_{t \in G}$ are elements of $A$ satisfying the following properties
\begin{enumerate}
\item $x_t y = (t \cdot y) x_t$, for all $t\in G$ and all $y \in A$;
\item $s \cdot x_t = x_{sts^{-1}}$, for all $s,t\in G$.
\end{enumerate}
Let $x_t = u_t \abs{x_t}$ be the polar decomposition of $x_t$, with $p_t = u_t^*u_t$ being the domain projection of $u_t$. Then the following relations hold:
\begin{enumerate}
\item The projection $p_t$ is a $t$-invariant central projection in $A$, the partial isometry $u_t$ is a unitary in $U(Ap_t)$, and $t$ acts by $\Ad u_t$ on the corner $Ap_t$;
\item $s \cdot u_t = u_{sts^{-1}}$, for all $s,t\in G$.
\end{enumerate}
\end{proposition}
\begin{proof}
The first condition is an immediate application of Theorem~\ref{thm:PropOuterCorner}. For the second condition, first note that if $\alpha \colon A \to B$ is a *-isomorphism between monotone complete C*-algebras, and $x \in A$ admits polar decomposition $x = u \abs{x}$, then $\alpha(x)$ admits polar decomposition $\alpha(u) \abs{\alpha(x)}$. In particular, applying this to the *-automorphism of $s \in G$ tells us that the element $x_{sts^{-1}} = s \cdot x_t$ admits the polar decomposition $(s \cdot u_t) \abs{s \cdot x_t}$. Uniqueness of polar decomposition then tells us that $s \cdot u_t = u_{sts^{-1}}$.
\end{proof}
For convenience, we will sometimes want to go from a subset of pairs $(p_t,u_t)_{t\in C}$ indexed over a conjugacy class $C\subseteq G$, and satisfying the conditions above, to a single pair $(p,u)$, and vice versa.
\begin{proposition}
\label{prop:equivariant_unitaries_to_single_unitary}
Let $G$ be a discrete group acting on a unital C*-algebra $A$, and let $C \subseteq G$ be the conjugacy class of a fixed element $r \in C$. The following are in canonical bijection with each other:
\begin{enumerate}
\item Sets of pairs $\set{(p_t, u_t)}_{t \in C}$, where $p_t$ is a $t$-invariant central projection in $A$, $u_t \in U(A p_t)$ is a unitary such that $t$ acts by $\Ad u_t$ on the corner $Ap_t$, and moreover $s \cdot u_t = u_{sts^{-1}}$, for all $s \in G$ and $t \in C$.
\item A single $r$-invariant central projection $p$ in $A$ and a unitary $u \in U(Ap)$ such that $r$ acts by $\Ad u$ on the corner $Ap$, and moreover $s \cdot u = u$, for all $s \in C_G(r)$.
\end{enumerate}
The map from the first setting to the second setting is given by $p = p_r$ and $u = u_r$.
\end{proposition}
\begin{proof}
Since we are dealing with a conjugacy class, the pairs $\{(p_t,u_t)\}_{t\in C}$, are uniquely determined by the values of $p_r$ and $u_r$, due to the fact that $p_{srs^{-1}} = s \cdot p_r$ and $u_{srs^{-1}} = s \cdot u_r$. Thus, the map between the two settings mentioned above is injective.
To show that this map is surjective, we need to check that the elements $p_{srs^{-1}} \coloneqq s \cdot p$ and $u_{srs^{-1}} \coloneqq s \cdot u$, for $s\in G$, are well-defined and satisfy the required properties. Assume that $s_1 r s_1^{-1} = s_2 r s_2^{-1}$, for some $s_1,s_2\in G$. This is equivalent to the requirement that $s_2^{-1} s_1$ belongs to $C_G(r)$, and so $s_2^{-1} s_1 \cdot u = u$, or, in other words, $s_1 \cdot u = s_2 \cdot u$. Thus, the aforementioned elements give rise to well-defined pairs $\{(p_t,u_t)\}_{t\in C}$. Observe that we indeed have $p_r = p$ and $u_r = u$.
It is straightforward to verify that each $p_t$ is a $t$-invariant central projection in $A$, that $u_t\in U(Ap_t)$, and that $s\cdot u_t=u_{sts^{-1}}$, for all $s\in G$ and all $t\in C$.
Finally, we check that the action of $t$ is given by $\Ad u_t$ on the corner $Ap_t$, for every $t\in C$. Given $srs^{-1} \in C$ and $x \in A p_{srs^{-1}}$ we have that $s^{-1} x \in A p$, and so
\[ srs^{-1} \cdot x = sr \cdot (s^{-1} \cdot x) = s \cdot (u (s^{-1} \cdot x) u^*) = (s \cdot u) x (s \cdot u)^* = u_{srs^{-1}} x u_{srs^{-1}}^*. \]
\end{proof}
\section{The intersection property for crossed products by FC-groups}
\label{sec:intersection_property_fc}
The goal of this section is to prove the first half of Theorem~\ref{thmIntro:A}, which characterizes the intersection property for the crossed product $A \rtimes_\lambda G$ in terms of the dynamics of $G$ on $I(A)$. Specifically, the lack of the intersection property is equivalent to the existence of a triple $(p,u,r)$, where
\begin{itemize}
\item $r \in G \setminus \set{e}$;
\item $p$ is an $r$-invariant central projection in $I(A)$;
\item $u$ is a unitary in $I(A)p$ such that $r$ acts by $\Ad u$ on $I(A)p$;
\item $u$ is $C_G(r)$-invariant.
\end{itemize}
We would first like to give some brief intuition and sketch the proof of this theorem in the minimal setting. The setting of the intersection property relies on some technical results that make perhaps not as much intuitive sense without the appropriate context.
Assume that the action of $G$ on $A$ is minimal, and assume that $A \rtimes_\lambda G$ is not simple. Through the technology of pseudo-expectations (see Section~\ref{sec:pseudoexpectations}), it is possible to write down a non-trivial element of $Z(I_G(A) \rtimes_\lambda G)$, which in particular implies that $I_G(A) \rtimes_\lambda G$ is not prime, and, by Proposition~\ref{proposition:primeifffactor}, neither is $I(A) \rtimes_\lambda G$. From here, letting $I$ and $J$ be two non-trivial ideals of $I(A) \rtimes_\lambda G$ with $I\cdot J = 0$, we may take a supremum of the positive elements $x\in I$ with $\norm{x}<1$ inside of the monotone complete crossed product $M(I(A),G)$. Denote this supremum by $q$. We will see in Proposition~\ref{prop:ideal_sup_projection} that this element must commute with the copy of $I(A) \rtimes_\lambda G$, and it cannot be a scalar (since it must still be orthogonal to $J$ - see Proposition~\ref{prop:ideal_sup_orthogonality}). By Proposition~\ref{prop:mc_crossed_product_commutant_coefficients}, the coefficients of $q= \sum_{r \in G} x_r \lambda_r\in M(I(A),G)$ satisfy certain properties that we are after, and moreover, there is at least one $r\in G\setminus\{e\}$ for which $x_r\neq 0$ (otherwise, $q$ would be a non-trivial element of $Z(I(A))^G$, and this is impossible due to minimality - see Proposition~\ref{prop:minimality_transfers_to_injective_envelopes_and_centers}). Following the procedure carried out in Proposition~\ref{prop:properly_outer_equivariant_polar_decompositions} and Proposition~\ref{prop:equivariant_unitaries_to_single_unitary}, we deduce that the polar decomposition of $x_r$ inside $I(A)$ gives rise to a triple $(p,u,r)$ that has the desired properties.
In order to deal with the general case, we will at some point need to define a suitable variant of being prime (see Lemma~\ref{lemma:stronglynonprime}), one that is appropriate for dealing with the intersection property when the action is not minimal.
We would now like to also motivate the converse direction of the Theorem, relative to the ideas of Kennedy and Schafhauser in \cite{kennedy_schafhauser_noncommutative_crossed_products} (specifically for the case of amenable groups, which slightly simplifies their proof).
The existence of a triple $(p,u,r)$ as in the theorem is clearly stronger than requiring the action on $I(A)$ (equivalently, on $A$) to be non properly outer. However, in \cite[Theorem~9.5]{kennedy_schafhauser_noncommutative_crossed_products}, it is instead assumed that the action on $A$ is not properly outer \emph{and} that the system $(A,G)$ has \emph{vanishing obstruction}. We now briefly explain what this means and how they use it in order to prove the existence of a non-faithful pseudo-expectation on $A\rtimes_\lambda G$ (which then violates the intersection property for $(A,G)$, by Proposition~\ref{prop:intersection_property_iff_faithful}). Let $u_t \in U(I(A)p_t)$ be the unitaries arising out of the decompositions $I(A) = I(A) p_t \oplus I(A) (1-p_t)$ into inner and properly outer parts, as described in Theorem~\ref{thm:PropInnerDecoMonCom}.
If it were possible to ``untwist'' these unitaries in such a way so that the map $t \mapsto u_t$ behaves essentially like a group homomorphism (or, more precisely, if this map is a \emph{partial representation} of $G$, see \cite[Definition~9.1]{Exel_book} or \cite[Definition~8.2]{kennedy_schafhauser_noncommutative_crossed_products}), then the proofs in the commutative case can be mimicked, and a new non-faithful equivariant pseudo-expectation $F'\colon A \rtimes_\lambda G \to I_G(A)$ can be defined by $F'(a \lambda_t) = a u_t$.
(One can also start with proper outerness on $I_G(A)$, and untwisting the resulting unitaries from the decomposition of $I_G(A)$ instead. In the case of non-amenable groups, doing this becomes mandatory, due to coefficients from $I(A)$ only defining pseudo-expectations on the universal crossed product $A \rtimes G$, and not the reduced $A\rtimes_\lambda G$. See the proof of \cite[Theorem~9.1]{kennedy_schafhauser_noncommutative_crossed_products}.)
This ability to untwist the unitaries (vanishing obstruction) trivially holds when $A$ is commutative, but as mentioned before is a very strong property that frequently fails in the noncommutative case. Likewise, even if these unitaries $u_t$ arise out of the polar decomposition of elements $x_t \in I_G(A)$ satisfying the positive-definiteness condition mentioned in Proposition~\ref{prop:pseudoexpectation_coefficients} (i.e.\ $[x_{st^{-1}}]_{s,t \in \mathcal{F}} \geq 0$ for all finite subsets $\mathcal{F} \subseteq G$), there is no guarantee that the matrices $[u_{st^{-1}}]_{s,t \in \mathcal{F}}$ are positive, a necessary condition for defining the aforementioned pseudo-expectation.
Let us now explain the different route taken in our proof: Starting with a triple $(p,u,r)$ as in the theorem, we obtain a set of pairs $\{(p_t,u_t)\}_{t\in C}$ as in Proposition~\ref{prop:equivariant_unitaries_to_single_unitary}, where $C$ denotes the (finite) conjugacy class of $r$. These do not necessarily allow us to construct a non-faithful pseudo-expectation for $A\rtimes_\lambda G$, as explained above and done in \cite{kennedy_schafhauser_noncommutative_crossed_products}. However, a direct application of Proposition~\ref{prop:crossed_product_center_coefficients} shows that the element $\sum_{t\in C}u_t^*\lambda_t$ lies in the center of $I(A)\rtimes_\lambda G$, but does not belong to the copy of $I(A)$. Using this fact, we then prove that $I(A)\rtimes_\lambda G$ is not prime in a strong way (see Lemma~\ref{lemma:stronglynonprime} for the precise statement), which allows to deduce that $(I(A),G)$ does not have the intersection property, and so neither does $(A,G)$ \cite[Theorem~3.2]{bryder_injective_envelopes}.
We now start building up the technical results that are necessary in order to give a complete proof. The following proposition is very similar to \cite[Lemma~1.1]{hamana82_centre} (see also the proof of \cite[Theorem~7.1]{hamana81_regular_embeddings}), which is done in the context of having $B$ below being either $\closure{A}$, Hamana's regular monotone completion, or $I(A)$. For our purposes, we will usually consider $A \rtimes_\lambda G$ and $M(A,G)$ as the monotone complete algebra, which is in general neither isomorphic to $\closure{A \rtimes_\lambda G}$ nor to $I(A \rtimes_\lambda G)$ (this is easy to see in the case of $A = \mathbb{C}$ and $G$ abelian).
\begin{proposition}
\label{prop:ideal_sup_projection}
Let $A \subseteq B$ be an inclusion of unital C*-algebras, and assume that $B$ is monotone complete. If $I \subseteq A$ is any ideal of $A$, we let $I^+_1 \coloneqq \setbuilder{x \in I}{\norm{x} < 1}$. Recalling that this is an upwards-directed set, we have that $\sup^B I^+_1$ is a projection in $A' \cap B$, which acts on $I$ like a unit.
\end{proposition}
\begin{proof}
For convenience, we will denote $\sup^B I^+_1$ by $p$. First, we show that it commutes with all of $A$. To see this, note that any *-isomorphism $\alpha \colon B \to C$ between monotone complete C*-algebras will preserve supremums. In other words, if $(b_\lambda)_\lambda$ is a bounded increasing net of self-adjoint elements in $B$, then $(\alpha(b_\lambda))_\lambda$ is such a net in $C$, and
\[ \sup_\lambda \alpha(b_\lambda) = \alpha( \sup_\lambda b_\lambda). \]
In particular, if we let $u \in U(A)$ be any unitary element, then the *-automorphism $\alpha = \Ad u$ will also preserve the supremum of $I^+_1$. However, $x \mapsto uxu^*$ is a bijection on this set. Thus,
\[ upu^* = u ( \sup{^B} I^+_1 ) u^* = \sup{^B} ( u I^+_1 u^* ) = \sup{^B} I^+_1 = p. \]
Since $U(A)$ spans $A$, we conclude that $p \in A' \cap B$.
To see that $p$ is a projection, note that for any $x \in I^+_1$, we have that $x^{1/2} \in I^+_1$ as well, and so $p \geq x^{1/2}$. Since $p$ and $x$ commute, we also have $p^2\geq x$.
In other words, $p^2$ is an upper bound to $I^+_1$. But since $p \leq 1$, we have that $p^2 \leq p$. Given that $p$ was the \emph{least} upper bound, it follows that $p^2 = p$.
Finally, we show that $p$ acts on $I$ like a unit. Let $j\in I_+^1$. Since $j\leq p$, we have
\[(1-p)j(1-p)\leq (1-p)p(1-p)=0. \]
This implies $(1-p)j=0$. Since $I^+_1$ spans $I$, we have $pj=j$, for every $j\in I$.
\end{proof}
It is interesting to note that in \cite[Theorem~7.1]{hamana81_regular_embeddings}, Hamana does not take a supremum of $I^+_1$ directly, but rather a supremum of the left projections of the elements in this set. It is possible that he wished for the result to be a projection, but at least in this earlier paper, overlooked the fact that $\sup I^+_1$ is itself always a projection. In \cite[Lemma~1.1]{hamana82_centre}, approximate units are used directly.
\begin{lemma}[{\cite[Lemma~1.9 and Corollary~4.10]{hamana81_regular_embeddings}}]
\label{lem:sup_bFbstar}.
Let $B$ be any C*-algebra, and let $\mathcal{F} \subseteq B$ be a bounded set of self-adjoint elements that admits a supremum. Then, for every $b\in B$, $\sup ( b \mathcal{F} b^* )$ exists, and
\[ b ( \sup \mathcal{F} ) b^* = \sup ( b \mathcal{F} b^* ). \]
\end{lemma}
\begin{proposition}
\label{prop:ideal_sup_orthogonality}
Let $A \subseteq B$ be an inclusion of unital C*-algebras, where $B$ is monotone complete. Let $I,J \subseteq A$ be non-trivial ideals of $A$ with $I\cdot J = 0$. Let $p \coloneqq \sup^B I^+_1$ and $q \coloneqq \sup^B J^+_1$. Then $pq = 0$. Moreover, $p\cdot J=0$ and $q\cdot I=0$.
\end{proposition}
\begin{proof}
Let $j \in J$ be any positive element. Using Lemma~\ref{lem:sup_bFbstar} and the fact that $p$ is a projection that commutes with $A$ (see Proposition~\ref{prop:ideal_sup_projection}), we have that
\[ jp = j^{1/2} p j^{1/2} = j^{1/2} (\sup{^B} I^+_1) j^{1/2} = \sup{^B} (j^{1/2} I^+_1 j^{1/2}) = 0. \]
Since positive elements span a C*-algebra, we have that $p\cdot J=0$. Similarly, $q\cdot I=0$.
Applying Lemma~\ref{lem:sup_bFbstar} once again, we have
\[ pqp = p (\sup{^B} J^+_1) p = \sup{^B} (p J^+_1 p) = 0, \]
and thus $pq=0$.
\end{proof}
One of Hamana's goals with these sorts of results was to prove the equivalence between a C*-algebra $B$ being prime, and the regular monotone completion $\closure{B}$ being a factor, although the same proof directly works with the injective envelope $I(B)$ instead. One direction is clear from the above result. Orthogonal ideals $I$ and $J$ in $B$ give orthogonal projections in $I(B)$ that commute with $B$, and therefore lie in $Z(I(B))$ (see \cite[Corollary~4.3]{hamana79_injective_envelopes_cstaralg}).
We will also need the following concrete computation that shows we may move back and forth between projections in $I(B)$ and ideals in $B$, in a certain sense. This is essentially just part of \cite[Lemma~1.3(i)]{hamana82_centre}, but Hamana phrases it in terms of injective envelopes of non-unital C*-algebras. To avoid referencing this theory just yet (and leave it contained to Section~\ref{sec:proper_outerness_on_A}), we give a short re-proof here.
\begin{proposition}
\label{prop:ConcreteSupProj}
Let $B$ be a unital C*-algebra and let $r\in Z(I(B))$ be a projection. Then, for $J\coloneqq r\cdot I(B)\cap B$, one has $\sup^{I(B)}J_1^+=r$.
\end{proposition}
\begin{proof}
Denote $p=\sup^{I(B)}J_1^+$. By Proposition~\ref{prop:ideal_sup_projection} and \cite[Corollary~4.3]{hamana79_injective_envelopes_cstaralg}, we have that $p\in I(B)\cap B'=Z(I(B))$.
In addition, given that every $x \in J_1^+$ satisfies $x \leq r$, we have $p \leq r$.
On the other hand, since $(r-p)\cdot I(B)\cap B\subseteq J$ and $p$ acts on $J$ like a unit (see Proposition~\ref{prop:ideal_sup_projection}), it follows that $p$ acts on $(r-p)\cdot I(B)\cap B$ as a unit. Since $p\perp (r-p)$, we conclude that $(r-p)\cdot I(B)\cap B=0$, and therefore also $(r-p)\cdot I(B)=0$ by essentiality of the inclusion $B\subseteq I(B)$. This finishes the proof.
\end{proof}
As mentioned in the beginning of this section, in the minimal setting, the proof of the main theorem of this section (Theorem~\ref{thm:mainSec4}) involves showing that $I_G(A) \rtimes_\lambda G$ is not prime and conclude that $I(A) \rtimes_\lambda G$ is not prime, since they share the same injective envelope. In the context of the intersection property, we need a stronger notion of ``not prime'' that plays well relative to the subalgebras $A$, $I(A)$, and $I_G(A)$.
\begin{lemma}
\label{lemma:stronglynonprime}
Let $G$ be a discrete group acting on a unital C*-algebra $A$. Assume that the canonical inclusion $Z(I_G(A))^G\subseteq Z(I_G(A) \rtimes_\lambda G)$ is proper. Then there exist non-trivial ideals $J,K\subseteq I(A) \rtimes_\lambda G$ with the following properties:
\begin{enumerate}
\item $J\cdot K=0$;
\item If $J\cdot a=0$ for some $a\in I(A)$, then $a=0$.
\end{enumerate}
\end{lemma}
\begin{proof}
View the commutative C*-algebras $Z(I_G(A))^G$ and $Z(I_G(A) \rtimes_\lambda G)$ as $C(X)$ and $C(Y)$, respectively. The inclusion $C(X) \subsetneqq C(Y)$ is dual to a surjective, non-injective, continuous map $q : Y \to X$. Choose distinct elements $y_1,y_2$ in $Y$ mapping to the same point $x \in X$ under $q$, i.e.\ $x = q(y_1) = q(y_2)$.
Let $U_1,U_2\subseteq Y$ be open disjoint neighborhoods of $y_1$ and $y_2$, respectively, and let $z_1,z_2\in C(Y)$ be positive contractions such that $z_i$ is supported on $U_i$ and $z_i(y_i)=1$, for $i=1,2$. Clearly, $z_1z_2=0$. Moreover, we claim that $z_i\notin C(X)$, for $i=1,2$. Indeed, the inclusion $C(X)\subseteq C(Y)$ is given by viewing a function $f\in C(X)$ as a function of $C(Y)$ which is constant (and equal to $f(x)$) on the fiber $q^{-1}(\{x\})$, for every $x\in X$. However, the functions $z_i\in C(Y)$, $i=1,2$, are clearly not constant along the fiber $q^{-1}(\{x\})$, and therefore do not belong to the copy of $C(X)$.
Note that $E(z_1)\in C(X)$, since the canonical conditional expectation $E\colon I_G(A) \rtimes_\lambda G\to I_G(A)$ maps $Z(I_G(A) \rtimes_\lambda G)$ onto $Z(I_G(A))^G$. Moreover, $Z(I_G(A))^G$ is a monotone complete C*-algebra, and so $X$ is an extremally disconnected space (see the discussion at the end of Section~\ref{sec:preliminaries:monotone_complete}). This implies that
\[ \closure{\supp (E(z_1))} = \closure{\setbuilder{x \in X}{E(z_1)(x) \neq 0}} \]
is a clopen subset of $X$. Let $p_1\in C(X)$ be the characteristic function of $\overline{\supp(E(z_1))}$ (in other words, $p_1$ is the support projection of $E(z_1)$). Set $w_1\coloneqq (1-p_1)+z_1$ and $w_2\coloneqq z_2p_1$. We will show the following properties:
\begin{enumerate}
\item $w_1, w_2\in Z(I_G(A) \rtimes_\lambda G)$ are non-zero orthogonal positive contractions.
\item If $w_1a=0$ for some $a\in I_G(A)$, then $a=0$.
\end{enumerate}
It is immediate that $w_1, w_2\in Z(I_G(A) \rtimes_\lambda G)$ are positive contractions and that $w_1\neq 0$. We have to show that $w_2\neq 0$. For this, we first observe that $p_1z_1=z_1$. This is due the fact that $E((1-p_1)z_1)=(1-p_1)E(z_1)=0$, and since $E$ is faithful, we get $(1-p_1)z_1=0$. By definition, we have that $z_1(y_1)=1$, and thus we must have that $p_1(y_1)=1$. As $p_1\in C(X)$ and $q(y_1)=q(y_2)$, we see that $p_1(y_1)=p_1(y_2)=1$ under the embedding $C(X) \subseteq C(Y)$. Now, $w_2(y_2)=z_2(y_2)p_1(y_2)=1$, so $w_2\neq 0$.
Orthogonality of $w_1$ and $w_2$ follows from orthogonality of $z_1$ and $z_2$.
Finally, consider the set
\[ I = \setbuilder{a\in I_G(A)}{w_1a=0}. \]
Using that $w_1\in Z(I_G(A) \rtimes_\lambda G)$, it is easy to check that $I$ is a closed two-sided $G$-invariant ideal of $I_G(A)$. Let $q=\sup^{I_G(A)}I_1^+$. By Proposition~\ref{prop:ideal_sup_projection}, we know that $q$ is a projection inside $Z(I_G(A))$, which is also $G$-invariant since
\[ g \cdot q = g \cdot (\sup I_1^+) = \sup(g \cdot I_1^+) = \sup I_1^+ = q. \]
By Lemma~\ref{lem:sup_bFbstar}, we have $w_1qw_1=w_1(\sup I_1^+)w_1=\sup(w_1I_1^+w_1)=0$. Thus $w_1q=0$.
This in turn implies that $E(w_1q)=0$, or, equivalently, $E(w_1)q=0$, which means that $((1-p_1)+E(z_1))q=0$ inside $Z(I_G(A))^G = C(X)$. However, by definition of $p_1$, we see that the function $(1-p_1)+E(z_1)$ is non-vanishing on
\[ \supp(E(z_1)) \cup (X \setminus \closure{\supp(E(z_1))}), \]
which is a dense subset of $X$. This forces $q$ to be the zero element, and consequently, $I=0$ as required.
Next, recalling that $I(I_G(A)\rtimes_\lambda G)=I(A\rtimes_\lambda G)$ (see Theorem~\ref{thm:crossed_product_injective_envelope_inclusions} and Proposition~\ref{prop:intermediate_algebra_shares_injective_envelope}) and that $Z(B)\subseteq Z(I(B))$ for every unital C*-algebra $B$ \cite[Corollary~4.3]{hamana79_injective_envelopes_cstaralg}, we have that $Z(I_G(A)\rtimes_\lambda G) \subseteq Z(I(A\rtimes_\lambda G))$. This allows us to view $w_1,w_2$ as elements of $Z(I(A\rtimes_\lambda G))$, which is again monotone complete. Denote by $r_1$ and $r_2$ the right (or equivalently left) projections of $w_1$ and $w_2$, respectively, inside of this commutative and monotone complete C*-algebra (recall the definition of these projections from Proposition~\ref{prop:monotonecompletePolarDec}). The following properties hold:
\begin{enumerate}
\item $r_1,r_2\in Z(I(A\rtimes_\lambda G))$ are non-zero orthogonal projections.
\item If $r_1a=0$ for some $a\in I(A)\subseteq I(A\rtimes_\lambda G)$, then $a=0$.
\end{enumerate}
The first property follows from the fact that, because $w_1 w_2 = 0$, then $w_1 r_2 = 0$, and so $r_1 r_2 = 0$ as well. The second property is also not hard to see, as if $r_1 a = 0$ for some $a \in I(A)$, then
\[ a^* w_1^* w_1 a \leq a^* r_1 a = 0, \]
and so $w_1 a = 0$, which in turn we know implies $a = 0$.
Now we may finally construct the ideals that we are after. Define
\begin{align*}
&J \coloneqq (r_1 \cdot I(A \rtimes_\lambda G)) \cap (I(A) \rtimes_\lambda G), \\
&K \coloneqq (r_2 \cdot I(A \rtimes_\lambda G)) \cap (I(A) \rtimes_\lambda G).
\end{align*}
By essentiality of the inclusion $I(A)\rtimes_\lambda G\subseteq I(A\rtimes_\lambda G)$, it follows that $J$ and $K$ are non-zero ideals inside $I(A) \rtimes_\lambda G$, and they are clearly orthogonal as well. Let $a \in I(A)$ be given, and assume that $ja=0$ for every $j\in J$. By Proposition~\ref{prop:ConcreteSupProj}, we have that $\sup^{I(A\rtimes_\lambda G)} J_1^+ = r_1$, and so $a^*r_1a=0$ by Lemma~\ref{lem:sup_bFbstar}, and so $r_1 a = 0$. Due to condition~(2) above, we obtain $a=0$. In summary, if $J\cdot a=0$ for some $a\in I(A)$, then $a=0$.
\end{proof}
\begin{remark}
\label{remark:stronglynonprimeinjective}
Consider the statement of Lemma~\ref{lemma:stronglynonprime}, and assume that instead of requiring the inclusion $Z(I_G(A))^G \subseteq Z(I_G(A) \rtimes_\lambda G)$ to be proper, we required $Z(I(A))^G \subseteq Z(I(A) \rtimes_\lambda G)$ to be proper. This would automatically imply that the former inclusion is proper as well. Indeed, assume $x \in Z(I(A) \rtimes_\lambda G) \setminus Z(I(A))^G$. It is clear that $x \notin Z(I_G(A))^G$. Moreover, if we consider the inclusions
\[ A \rtimes_\lambda G \subseteq I(A) \rtimes_\lambda G \subseteq I_G(A) \rtimes_\lambda G \subseteq I(A \rtimes_\lambda G) \]
from Theorem~\ref{thm:crossed_product_injective_envelope_inclusions}, then we see that any element of $Z(I(A) \rtimes_\lambda G)$ in particular commutes with $A \rtimes_\lambda G$, and therefore with all of $I(A \rtimes_\lambda G)$ by \cite[Corollary~4.3]{hamana79_injective_envelopes_cstaralg}. Hence, it must lie in $Z(I_G(A) \rtimes_\lambda G)$.
\end{remark}
\begin{theorem}
\label{thm:mainSec4}
Let $G$ be an FC-group acting on a unital C*-algebra $A$. Then $(A,G)$ does not have the intersection property if and only if there exist $r\in G\setminus\{e\}$, a non-zero $r$-invariant central projection $p\in I(A)$ and a unitary $u\in U(I(A)p)$ such that
\begin{enumerate}
\item $r$ acts by $\Ad u$ on $I(A)p$;
\item $s \cdot p = p$ and $s\cdot u=u$ for all $s\in C_G(r)$.
\end{enumerate}
\end{theorem}
\begin{proof}
First, assume that there exists a triple $(p,u,r)$ satisfying properties $(1)$ and $(2)$.
By Proposition~\ref{prop:equivariant_unitaries_to_single_unitary}, we obtain well-defined pairs of elements $\{(p_t,u_t)\}_{t\in C}$, where $C=\setbuilder{grg^{-1}}{g \in G}$ denotes the conjugacy class of $r$, each $p_t$ is a $t$-invariant central projection in $I(A)$, each $u_t\in I(A)$ is a unitary element in the corner $I(A)p_t$ such that $t$ acts by $\Ad u_t$ on this corner, and moreover $s \cdot u_t=u_{sts^{-1}}$ for all $s\in G$ and $t\in C$.
Letting $u_g=0$ for every $g\in G\setminus C$, we check that the elements $(u_g)_{g\in G}$ satisfy the conditions stated in Proposition~\ref{prop:crossed_product_center_coefficients}. Indeed, for every $g\in G\setminus C$ and every $y\in I(A)$, it is clear that $u_gy=(g\cdot y)u_g$. However, if $g\in C$, we have that
\[ u_gy =u_gp_gy = (u_g(p_gy)u_g^*)u_g = (g \cdot (p_g y)) u_g = p_g (g\cdot y) u_g = (g\cdot y)u_g, \]
and so condition~(1) of Proposition~\ref{prop:crossed_product_center_coefficients} is satisfied, and condition~(2) is immediate by construction.
Therefore,
\[z\coloneqq\sum\limits_{c\in C} u_c^*\lambda_c\in Z(I(A)\rtimes_\lambda G). \]
However, $z\notin Z(I(A))^G$, since $u_r=u$ and $r\neq e$. Hence, $Z(I(A))^G\subseteq Z(I(A)\rtimes_\lambda G)$ is a proper inclusion and we can apply Lemma~\ref{lemma:stronglynonprime} together with Remark~\ref{remark:stronglynonprimeinjective} in order to obtain non-trivial orthogonal ideals $J,K\subseteq I(A)\rtimes_\lambda G$ with the property that $J^\perp\cap I(A)=\{0\}$. In particular, $K$ is a non-trivial ideal of $I(A)\rtimes_\lambda G$ which intersects $I(A)$ trivially. This violates the intersection property for $(I(A),G)$, and thus for $(A,G)$ as well by \cite[Theorem~3.2]{bryder_injective_envelopes}.
Conversely, assume that $(A,G)$ does not have the intersection property, and let $F\colon A\rtimes_\lambda G\to I_G(A)$ be a non-canonical pseudo expectation, which exists by Proposition~\ref{prop:intersection_property_iff_faithful}. The map $F$ gives us non-trivial coefficients $(x_t)_{t \in G}$ in $I_G(A)$ by letting $x_t\coloneqq F(\lambda_t)$, for every $t\in G$. These satisfy the properties listed in Proposition~\ref{prop:pseudoexpectation_coefficients}. Let $r\in G \setminus \set{e}$ be such that $x_r \neq 0$ and denote by $C \subseteq G$ its conjugacy class. We obtain that $\sum_{t \in C} x_t^* \lambda_t$ lies in $Z(I_G(A) \rtimes_\lambda G)$ by Proposition~\ref{prop:crossed_product_center_coefficients}, and so we conclude that the canonical inclusion
\[Z(I_G(A))^G\subseteq Z(I_G(A) \rtimes_\lambda G) \]
is proper.
Applying Lemma~\ref{lemma:stronglynonprime}, we obtain non-trivial orthogonal ideals $J,K\subseteq I(A) \rtimes_\lambda G$ with the property that whenever $J\cdot a=0$, for some $a\in I(A)$, then $a=0$. Denote $q \coloneqq \sup^{M(I(A),G)}K_1^+$. By Proposition~\ref{prop:ideal_sup_projection}, we have that $q$ is a non-zero projection which commutes with the copy of $I(A) \rtimes_\lambda G \subseteq M(I(A),G)$.
Moreover, $J\cdot q=0$, by Proposition~\ref{prop:ideal_sup_orthogonality}, which implies that $q\notin I(A)$.
Let $(q_t)_{t\in G}$ denote the coefficients in $I(A)$ of $q = \sum_{t \in G} q_t \lambda_t \in M(I(A),G)$, and let $z_t \coloneqq q_t^*$, for every $t\in G$. By Proposition~\ref{prop:mc_crossed_product_commutant_coefficients}, we have that $z_t y =(t\cdot y) z_t$, for all $t \in G$ and all $y \in I(A)$, and moreover that $s\cdot z_t=z_{sts^{-1}}$, for all $s,t\in G$.
Given that $q \notin I(A)$, at least one coefficient $z_r$ is non-zero for some $r \in G \setminus \set{e}$. Letting $z_r=u\abs{z_r}$ be the polar decomposition of this element, and $p=u^*u$, we conclude by Proposition~\ref{prop:properly_outer_equivariant_polar_decompositions} and Proposition~\ref{prop:equivariant_unitaries_to_single_unitary} that the triple $(p,u,r)$ satisfies the desired properties.
\end{proof}
Since, under minimality of the action of $G$ on $A$, the intersection property is equivalent to simplicity of the associated crossed product, the theorem above also gives a characterization of simplicity of crossed products by FC-groups. Moreover, in general (not assuming the action is minimal), we almost immediately obtain a characterization for when crossed products by FC-groups are prime, and we state it separately below. This is in slight contrast to what is done in Section~\ref{sec:primality_minimal}, which characterizes exactly when a crossed product $A \rtimes_\lambda G$ is prime \emph{regardless} of what $G$ is, but assuming the action is minimal.
Recall that a C*-algebra $A$ equipped with a $G$-action is called \emph{$G$-prime} if whenever $I,J\subseteq A$ are non-trivial $G$-invariant ideals in $A$, their product $I\cdot J$ is non-trivial.
\begin{theorem}\label{thm:mainSec4Prime}
Let $G$ be an FC-group acting on a unital, $G$-prime C*-algebra $A$. Then $A\rtimes_\lambda G$ is not prime if and only if there exist $r\in G\setminus\{e\}$, a non-zero $r$-invariant central projection $p\in I(A)$ and a unitary $u\in U(I(A)p)$ such that
\begin{enumerate}
\item $r$ acts by $\Ad u$ on $I(A)p$;
\item $s \cdot p = p$ and $s\cdot u=u$, for all $s\in C_G(r)$.
\end{enumerate}
\end{theorem}
\begin{proof}
Assume that there is a triple $(p,u,r)$ satisfying the properties listed in the Theorem. Exactly as in the proof of Theorem~\ref{thm:mainSec4}, the triple gives rise to a central element of $I(A)\rtimes_\lambda G$ which does not lie in $Z(I(A))^G$. In particular, such an element is non-scalar, and so $I(A)\rtimes_\lambda G$ is not prime. By Proposition~\ref{proposition:primeifffactor}, $A\rtimes_\lambda G$ is not prime either.
Conversely, assume that $A\rtimes_\lambda G$ is not prime. We claim that the intersection property for $(A,G)$ cannot hold. Indeed, since $A\rtimes_\lambda G$ is not prime, we can find two non-trivial orthogonal ideals $J,K\subseteq A\rtimes_\lambda G$. As $J\cap A$ and $K\cap A$ are $G$-invariant orthogonal ideals in $A$, and $A$ is $G$-prime, we have that at least one of them must be the zero ideal. In other words, at least one of $J$ or $K$ violates the intersection property for $(A,G)$. By Theorem~\ref{thm:mainSec4}, we obtain a triple $(p,u,r)$ which satisfies the desired properties.
\end{proof}
\begin{remark}
Theorem~\ref{thm:mainSec4Prime} had already been proven in \cite[Theorem~10.1]{hamana85-injective_envelopes_equivariant} in the special case of finite groups.
\end{remark}
\section{Simplicity for crossed products by FC-hypercentral groups}
\label{sec:simplicity_fch}
This section aims to take the ideas from Section~\ref{sec:intersection_property_fc}, and show that at least in the minimal setting, they can be generalized to the case of FC-hypercentral groups.
After the first public release of this preprint, Siegfried Echterhoff kindly pointed out to us that the main theorem of this section (Theorem~\ref{thm:mainFCHSimpleMinimal}) follows from a combination of the main result of his paper \cite{echterhoff_jot} with our main result of Section~\ref{sec:primality_minimal} (we will comment about it further in Remark~\ref{rem:Sec5from6}). In particular, in this section, we reproduce a special case of Echterhoff's result \cite[Theorem~3.1]{echterhoff_jot}, which was also obtained in his thesis \cite[Satz~5.3.1]{echterhoff_thesis}, and implies that a crossed product by an FC-hypercentral group is prime if and only if it is simple (see Lemma~\ref{lem:fc_simplicity_equivalence}). Our proof uses different techniques, motivated by \cite{bedos_omland_fc_hypercentral_simplicity}, which reveal interesting information about the structure of pseudo-expectations of crossed products by FC-hypercentral groups. As such, we still find it worthwhile to keep this alternate proof.
Assume that the crossed product $A \rtimes_\lambda G$ does not have the intersection property. We know from Propositions~\ref{prop:pseudoexpectations_conditional_expectations_bijection} that there exists a non-canonical equivariant pseudo-expectation $F \colon A \rtimes_\lambda G \to I_G(A)$. This was used in Section~\ref{sec:intersection_property_fc} to define a certain non-trivial element of the center of $I_G(A) \rtimes_\lambda G$, taking advantage of the fact that we may sum over a finite conjugacy class.
We claim that the same thing can be done in the setting of FC-hypercentral groups. Specifically, we will show that, at least in the minimal setting, every non-canonical equivariant pseudo-expectation $F\colon A \rtimes_\lambda G \to I_G(A)$ necessarily has $F(\lambda_t)\neq 0$, for some $t\in \FC(G)\setminus \{e\}$. From there, we may proceed as before. An analogous result was observed by B{\'e}dos and Omland in the context of traces on twisted group C*-algebras $C^*_\lambda(G,\sigma)$ associated to FC-hypercentral groups, where they wished to characterize when such C*-algebras are simple, or have unique trace. The following lemma is in particular adapted from \cite[Lemma~2.1]{bedos_omland_fc_hypercentral_simplicity}, and allows us to work our way up the entire FC-central series of $G$.
\begin{lemma}
\label{lem:fc_central_series_expectation_successor_triviality}
Let $G$ be an amenable group acting minimally on a unital C*-algebra $A$. Assume that $N$ and $M$ are two normal subgroups of $G$ with $\FC(G) \leq N \leq M \leq G$, and moreover that $M/N \subseteq \FC(G/N)$. If $F \colon A \rtimes_\lambda G \to I_G(A)$ is an equivariant pseudo-expectation that restricts to the canonical conditional expectation on $A \rtimes_\lambda N$, then it necessarily restricts to the canonical conditional expectation on $A \rtimes_\lambda M$.
\end{lemma}
\begin{proof}
Before we begin, we recall that such a map extends uniquely to a an equivariant conditional expectation $F \colon I_G(A) \rtimes_\lambda G \to I_G(A)$ by Proposition~\ref{prop:pseudoexpectations_conditional_expectations_bijection}. There is no problem with denoting this by the same letter.
Let $h \in M$ by given. We must show that $F(\lambda_h) = 0$. Given that $F$ is canonical on $I_G(A) \rtimes_\lambda N$, and $N$ contains $\FC(G)$, we necessarily have that $h$ has infinitely many $G$-conjugates. However, in the quotient $M/N$, it has only finitely many $G$-conjugates, by assumption. Thus, by the pigeonhole principle, we may choose infinitely many $(g_i)_{i=1}^\infty$ such that $h_i \coloneqq g_i h g_i^{-1}$ are all distinct, but $h_i N$ are all the same coset. Observe that, in particular, if $i \neq j$, then $h_i h_j^{-1}$ lies in $N \setminus \set{e}$.
Now let $a_i \coloneqq F(\lambda_{h_i}) \in I_G(A)$, for every $i\in\mathbb{N}$, and for each $k \in \mathbb{N}$, consider the element
\[ x_k = 1 - \sum_{i=1}^k a_i^* \lambda_{h_i} \in I_G(A) \rtimes_\lambda G \]
We have that
\[ x_k x_k^* = 1 - \sum_{i=1}^k a_i^* \lambda_{h_i} - \sum_{j=1}^k \lambda_{h_j}^* a_j + \sum_{i=1}^k \sum_{j=1}^k a_i^* \lambda_{h_i h_j^{-1}} a_j. \]
However, observe that when computing $F(x_k x_k^*)$, most of the terms in the double sum at the end will disappear, as if $i \neq j$, we have
\[ F(a_i^* \lambda_{h_i h_j^{-1}} a_j) = a_i^* F(\lambda_{h_i h_j^{-1}}) a_j = 0, \]
given that $h_i h_j^{-1} \in N \setminus \set{e}$. Hence, $F(x_k x_k^*)$ simplifies greatly as
\[ F(x_k x_k^*) = 1 - 2\cdot\sum_{i=1}^k a_i^* a_i = 1 - 2\cdot\sum_{i=1}^k g_i \cdot (F(\lambda_h)^*F(\lambda_h)), \]
using equivariance of $F$ in the last equality.
In particular, we must have that
\[ \sum_{i=1}^k g_i \cdot (F(\lambda_h)^*F(\lambda_h)) \leq \frac{1}{2}, \]
no matter what $k \in \mathbb{N}$ was chosen to be.
Observe that $F(\lambda_h)^*F(\lambda_h)$ necessarily lies in $Z(I_G(A))$, as using Proposition~\ref{prop:pseudoexpectation_coefficients}, we get for any $y \in I_G(A)$ that
\[ F(\lambda_h)^*F(\lambda_h) y = F(\lambda_h)^* (h \cdot y) F(\lambda_h) = y F(\lambda_h)^* F(\lambda_h). \]
However, we also know by Proposition~\ref{prop:minimality_transfers_to_injective_envelopes_and_centers} that the induced action on $Z(I_G(A))$ is minimal. Given that $G$ is amenable, it must be the case that $Z(I_G(A))$ admits a $G$-invariant state $\omega$, which is necessarily faithful by minimality. Consequently, for a large enough value of $k$,
\[ \omega \left( \sum_{i=1}^k g_i \cdot (F(\lambda_h)^*F(\lambda_h)) \right) = k \cdot \omega(F(\lambda_h)^*F(\lambda_h)) \]
cannot be bounded above by $\frac{1}{2}$ if $F(\lambda_h) \neq 0$. This forces $F(\lambda_h) = 0$.
\end{proof}
\begin{remark}
It is worth noting that the above lemma needs that $G$ is amenable only to obtain a $G$-invariant state on $Z(I_G(A))$. This can moreover be bypassed altogether and be made to work for arbitrary discrete groups, using Lemma~\ref{lem:infinitely_many_distinct_translates_unbounded_sum}.
\end{remark}
\begin{corollary}
\label{cor:fch_nontrivial_pseudoexpectation_nontrivial_fc_coefficients}
Let $G$ be an FC-hypercentral group acting minimally on a unital C*-algebra $A$. If $F \colon A \rtimes_\lambda G \to I_G(A)$ is a non-canonical equivariant pseudo-expectation, there is necessarily some $t \in FC(G) \setminus \set{e}$ for which $F(\lambda_t) \neq 0$.
\end{corollary}
\begin{proof}
Recall from Section~\ref{sec:preliminaries:fc_hypercentral} that the FC-central series of $G$ is a series of normal subgroups $(F_\alpha)_\alpha$ indexed by the ordinal numbers, which is given by:
\begin{enumerate}
\item $F_1 = \FC(G)$;
\item $F_{\alpha + 1} / F_\alpha = \FC(G / F_\alpha)$;
\item $F_\beta = \cup_{\alpha < \beta} F_\alpha$ for any limit ordinal $\beta$.
\end{enumerate}
We prove the desired result by proving the contrapositive. Assume that $F(\lambda_t) = 0$ for all $t \in \FC(G) \setminus \set{e}$. By multiplicative domain considerations, this means that $F$ is canonical on $A \rtimes_\lambda F_1$. Furthermore, Lemma~\ref{lem:fc_central_series_expectation_successor_triviality} tells us that if $F$ is trivial on $A \rtimes_\lambda F_\alpha$, then it is necessarily trivial on $A \rtimes_\lambda F_{\alpha+1}$. Finally, it is clear that if $\beta$ is a successor ordinal, and $F$ is trivial on $A \rtimes_\lambda F_\alpha$ for all $\alpha < \beta$, then $F$ is trivial on $A \rtimes_\lambda F_\beta$. By transfinite induction, $F$ is trivial on all of $A \rtimes_\lambda G$.
\end{proof}
From here, the proof of the main result proceeds as in Section~\ref{sec:intersection_property_fc}, except much more easily thanks to minimality, due to only needing to construct non-trivial central elements in the appropriate algebra with no additional special properties. We first need the following lemma, which, as explained in the beginning of this section, was proven in much greater generality in \cite{echterhoff_jot}.
\begin{lemma}
\label{lem:fc_simplicity_equivalence}
Let $G$ be an FC-hypercentral group acting minimally on a unital C*-algebra $A$. Then
$A \rtimes_\lambda G$ is simple if and only if $A \rtimes_\lambda G$ is prime.
\end{lemma}
\begin{proof}
We prove the non-trivial direction. Assume that $A \rtimes_\lambda G$ is not simple. Then, by Proposition~\ref{prop:intersection_property_iff_faithful}, there exists a non-canonical equivariant pseudo-expectation $F\colon A\rtimes_\lambda G\to I_G(A)$. Moreover, Corollary~\ref{cor:fch_nontrivial_pseudoexpectation_nontrivial_fc_coefficients} guarantees there exists $s\in \FC(G)\setminus\{e\}$ for which $F(\lambda_s)\neq 0$. Let $C$ denote the conjugacy class of $s$. Using Proposition~\ref{prop:pseudoexpectation_coefficients}, the coefficients $(x_t)_{t\in G}$ given by $x_t\coloneqq F(\lambda_t)$ for $t\in C$, and $x_t=0$ for $t\in G\setminus C$, satisfy the properties listed in Proposition~\ref{prop:crossed_product_center_coefficients}, and so
\[\sum\limits_{c\in C} x_c^*\lambda_c\in Z(I_G(A)\rtimes_\lambda G).\]
This means that $I_G(A)\rtimes_\lambda G$ has non-trivial center, and in particular, it is not prime. Proposition~\ref{proposition:primeifffactor} then implies that $A\rtimes_\lambda G$ is not prime.
\end{proof}
\begin{theorem}\label{thm:mainFCHSimpleMinimal}
Let $G$ be an FC-hypercentral group acting minimally on a unital C*-algebra $A$. Then $A \rtimes_\lambda G$ is not simple if and only if there exist $r \in \FC(G) \setminus \set{e}$, a non-zero $r$-invariant central projection $p \in I(A)$, and a unitary $u \in U(I(A)p)$ such that
\begin{enumerate}
\item $r$ acts by $\Ad u$ on $I(A)p$;
\item $s \cdot p = p$ and $s \cdot u = u$ for all $s \in C_G(r)$.
\end{enumerate}
\end{theorem}
\begin{proof}
The proof of Theorem~\ref{thm:mainSec4} works verbatim for showing that a triple $(p,u,r)$, with properties as listed in the theorem, gives rise to a non-trivial central element of $I(A)\rtimes_\lambda G$. This means that $I(A)\rtimes_\lambda G$ is not prime, and therefore neither is $A \rtimes_\lambda G$ by Proposition~\ref{proposition:primeifffactor}. Clearly, this means that $A \rtimes_\lambda G$ is not simple either.
Conversely, assume that $A \rtimes_\lambda G$ is not simple. By Lemma~\ref{lem:fc_simplicity_equivalence}, $A\rtimes_\lambda G$ is not prime, and hence neither is $I(A) \rtimes_\lambda G$ (by Proposition~\ref{proposition:primeifffactor} again).
Choose non-trivial orthogonal ideals $J,K \subseteq I(A) \rtimes_\lambda G$, and let \[q\coloneqq \sup {}^{M(I(A),G)}K_1^+.\] By Proposition~\ref{prop:ideal_sup_projection} and Proposition~\ref{prop:ideal_sup_orthogonality}, we have that $q$ is a non-zero projection which commutes with the copy of $I(A) \rtimes_\lambda G \subseteq M(I(A),G)$, and satisfies $J\cdot q=0$. Moreover, $q\notin I(A)$, as otherwise $q$ would be an element of $Z(I(A))^G$. This is impossible, since the induced system $(Z(I(A)),G)$ is minimal (see Proposition~\ref{prop:minimality_transfers_to_injective_envelopes_and_centers}), and so $Z(I(A))^G=\mathbb{C}$.
View $q = \sum_{t \in G} q_t \lambda_t$ inside of $M(I(A),G)$.
If we could find $t\in \FC(G)\setminus \{e\}$ such that $q_t\neq 0$, the proof is essentially done, as we may proceed exactly as was done in the proof of Theorem~\ref{thm:mainSec4}. However, so far, we only know that there exists some $t\in G\setminus \{e\}$ for which $q_t\neq 0$.
Let us remark that one can show that the coefficients of $q$ must actually be supported on elements with finite conjugacy class, so that $t$ must belong to $\FC(G)$. This will be shown in the proof of Theorem~\ref{thm:primality_minimal}, using a fairly non-trivial technical lemma (Lemma~\ref{lem:infinitely_many_distinct_translates_unbounded_sum}). However, in the case of FC-hypercentral groups, we can use more direct methods in order to show that there exists at least one $r\in \FC(G) \setminus \{e\}$ for which $q_r\neq 0$, which will be sufficient for our purposes.
Let $E \colon M(I(A),G) \to I(A)$ be the canonical conditional expectation (see Remark~\ref{rem:CondExpeMonComp}), and consider $q_e = E(q)$. Note that $q_e\in Z(I(A))^G$ (it is therefore just a complex number), and that $0\leq q_e \leq 1$, since $E$ is contractive. Consequently, the element
\[ w \coloneqq (1 - q_e) + q \in M(I(A),G) \]
is positive, commutes with $I(A) \rtimes_\lambda G$, and satisfies $E(w)=1$. The non-trivial coefficients of $w = \sum_{t \in G} w_t \lambda_t$ are identical to those of $q$. That is, $w_t = q_t$ for $t \neq e$.
From here, we may define a map $F \colon I(A) \rtimes_\lambda G \to I(A)$, by
\[ F(x) \coloneqq E(wx) = E(w^{1/2}xw^{1/2}), \]
for every $x\in I(A)\rtimes_\lambda G$, where the product $wx = w^{1/2}xw^{1/2}$ takes place in $M(I(A),G)$. It is immediate to verify that $F$ is an equivariant conditional expectation.
Moreover $F(\lambda_t^*) = w_t$ is non-zero for at least one $t \neq e$, and so $F$ is non-canonical.
Viewing the codomain of $F$ as $I_G(A)$ and restricting to $A \rtimes_\lambda G$ in the domain, we obtain that $F$ induces a non-canonical equivariant pseudo-expectation on $A\rtimes_\lambda G$.
Corollary~\ref{cor:fch_nontrivial_pseudoexpectation_nontrivial_fc_coefficients} now tells us that there exists $r\in \FC(G)\setminus\{e\}$ such that $F(\lambda_r^*) \neq 0$, which is exactly the same as saying that $q_r\neq 0$. As explained above, the proof now proceeds exactly as in Theorem~\ref{thm:mainSec4}. More precisely, if $q_r^*=u\abs{q_r^*}$ is the polar decomposition of $q_r^*$, with $p=u^*u$, then the triple $(p,u,r)$ has the desired properties.
\end{proof}
\section{Primality for crossed products of minimal actions by discrete groups}
\label{sec:primality_minimal}
This section is essentially an observation that most of the arguments in the previous sections, which deal with characterizing when a reduced crossed product is simple, actually also apply in the context of characterizing when a reduced crossed product is prime, for all minimal actions of any discrete groups.
First, we require a seemingly arbitrary lemma with quite a convoluted proof, and it will have its usefulness become clear near the end of the proof of the main theorem. The proof of the lemma makes use of the basic properties of the Furstenberg boundary $\partial_F G$. This was originally introduced by Furstenberg several decades ago in \cite{furstenberg_proceedings} (see also \cite{furstenberg_poisson}), as a topological boundary to be used largely in the study of Lie groups. It was more recently studied in the C*-simplicity results \cite{kalantar_kennedy_boundaries} and \cite{breuillard_kalantar_kennedy_ozawa_c_simplicity}, where in particular it was observed that for discrete groups, $I_G(\mathbb{C}) \cong C(\partial_F G)$ as $G$-C*-algebras. If $G$ is a discrete group, the Furstenberg boundary $\partial_F G$ is the universal compact Hausdorff $G$-space with the following properties:
\begin{enumerate}
\item It is \emph{minimal}.
\item It is \emph{strongly proximal}, in the sense that for any probability measure $\nu \in P(\partial_F G)$, there is a net of group elements $(g_\lambda)_\lambda$ such that $\lim_\lambda g_\lambda \nu = \delta_x$, for some $x \in X$.
\end{enumerate}
As noted in Proposition~\ref{prop:injective_is_mc_and_ginjective_is_injective}, $I_G(\mathbb{C})$ is monotone complete, and hence $\partial_F G$ is an extremally disconnected space (see, for example, \cite[Theorem~2.3.7]{SWMonotoneComplete}).
Before we proceed with the proof of the lemma, it might also interest the reader to realize that in the case of amenable groups, we are guaranteed the existence of an invariant state on any unital C*-algebra, and the proof of the lemma becomes extremely easy. The significant difficulty in the non-amenable case is that now, we are only guaranteed the existence of $G$-equivariant unital completely positive maps into $I_G(\mathbb{C})$.
\begin{lemma}
\label{lem:infinitely_many_distinct_translates_unbounded_sum}
Let $G$ be a discrete group acting minimally on a compact Hausdorff space $X$, and let $f \in C(X)$ be a non-zero positive function. Consider the set
\[ \setbuilder{g \cdot f}{g \in G}, \]
without counting repetition among the elements.
If this set is infinite, then the sum of its elements cannot be uniformly bounded from above, in the sense that there cannot exist some scalar $k \geq 0$ such that every sum of finitely many elements is bounded by $k$ from above.
\end{lemma}
\begin{proof}
This proof will, somewhat interestingly, first assume that $G$ is countable, and from there deduce the case of uncountable groups.
For convenience, let $W \subseteq G$ be a choice of elements of $G$ so that
\[ \setbuilder{g \cdot f}{g \in W} = \setbuilder{g \cdot f}{g \in G}, \]
and $g \cdot f$ are distinct for distinct values of $g \in W$. We will do a proof by contradiction, and start with the assumption that $\sum_{g \in W} g \cdot f$ is uniformly bounded.
Since $I_G(\mathbb{C})$ is $G$-injective ans isomorphic to $C(\partial_F G)$, we know that there is at least one $G$-equivariant unital completely positive map $\phi \colon C(X) \to C(\partial_F G)$. We also know that the left-ideal
\[ \setbuilder{h \in C(X)}{\phi(h^*h) = 0} \]
is a two-sided $G$-invariant ideal in $C(X)$. Since $1$ does not lie in this ideal, by minimality, this ideal must necessarily be zero. In other words, the map $\phi$ is faithful.
We know that $\phi(f) \neq 0$, by faithfulness. We now divide the proof into two cases:
If it were the case that $\phi(f) \geq \delta$ for some $\delta > 0$, then the sum $\sum_{g \in W} g \cdot f$ could not be bounded, as neither could $\sum_{g \in W} \phi(g \cdot f)$.
Assume therefore that $\phi(f)$ is not bounded from below. From here, choose some fixed $\delta > 0$ so that $\phi(f)(y) > \delta$ for some $y \in \partial_F G$. Let $U \coloneqq \phi(f)^{-1}(\delta,\infty)$. Given that $\partial_F G$ is extremally disconnected, we have that $\closure{U}$ is a clopen subset, and $\phi(f)(z) \geq \delta$ for every $z\in \closure{U}$, while $\phi(f)(z) \leq \delta$ for every $z\in \partial_FG\setminus \closure{U}$. Note that, by assumption of $\phi(f)$ not being bounded below by any strictly positive scalar, we have $\closure{U} \neq \partial_FG$. For convenience, we will also let $p \in C(\partial_F G)$ be the non-trivial projection coming from this clopen subset, so that $\phi(f) \geq \delta \cdot p$.
Another observation we make is that if $g_1 \cdot p \neq g_2 \cdot p$, this implies that $g_1 U \neq g_2 U$, which are exactly the set of points on which $g_1 \cdot \phi(f)$ and $g_2 \cdot \phi(f)$ are strictly greater than $\delta$, respectively. Consequently, it must be the case that $g_1 \cdot \phi(f) \neq g_2 \cdot \phi(f)$.
In other words, distinct translates of $p$ give rise to distinct translates of $\phi(f)$. Thus, if we show that the sum of the distinct translates of $p$ is unbounded, then, since $\phi(f) \geq \delta \cdot p$, this will automatically imply the same for the distinct translates of $\phi(f)$.
Since $G$ is countable, the Furstenberg boundary $\partial_F G$ is separable. This is a direct consequence of minimality of the space. Let $(y_n)_{n=1}^\infty$ be a dense subset of this space. Observe that there cannot exist any $g \in G$ with the property that $g \cdot y_n \in \closure{U}$ for all $n\in\mathbb{N}$. Otherwise, we would have $y_n \in g^{-1} \closure{U}$ for all $n\in\mathbb{N}$, implying $g^{-1} \closure{U} = \partial_F G$ by density. This is impossible, since we have already shown $\closure{U} \neq \partial_F G$.
Thus, if we construct a probability measure $\omega \in P(\partial_F G)$ by $\omega = \sum_{n=1}^\infty \frac{1}{2^n} \delta_{y_n}$ (which we will also view as a state on $C(\partial_F G)$), the previous paragraph essentially tells us that $(g\cdot \omega)(p)$ is never exactly equal to $1$ for any $g \in G$. However, choosing any $z \in \closure{U}$, we know that since $\partial_F G$ is a strongly proximal, there is a net $(g_\lambda)_\lambda$ in $G$ such that $\lim_\lambda g_\lambda \cdot \omega = \delta_z$. In other words, even though $(g\cdot \omega)(p) \neq 1$ for any $g\in G$, we can still find infinitely many elements $g\in G$ for which $(g\cdot \omega)(p) = \omega(g^{-1} \cdot p)$ is arbitrarily close to $1$ and, say, greater than $\frac{1}{2}$. Thus, we have infinitely many $g\in G$ for which $g^{-1} p$ are distinct, and $\omega(g^{-1} \cdot p) \geq \frac{1}{2}$.
Consequently, the sum of the distinct translates of $p$ cannot be bounded. By what was mentioned earlier, the same must be true for the corresponding distinct translates of $\phi(f)$, and thus the distinct translates of $f \in C(X)$.
The case of uncountable groups is not that hard to deduce from the countable case. Assume $G$ is uncountable, and consider our starting function $f \in C(X)$. Let $U \subseteq X$ be a nonempty open set such that $f(x) > 0$ for all $x \in U$. As $\cup_{g\in G} g\cdot U$ is an open invariant set, it must cover $X$ by minimality. Compactness then tells us that there is a finite collection $g_1 U, \dots, g_n U$ covering $X$. In terms of our function translates, this tells us that $g_1 \cdot f, \dots, g_n \cdot f$ have the property that for any $x \in X$, there is some $i\in \{1,\ldots,n\}$ with $(g_i \cdot f)(x) > 0$.
By assumption, we may also choose countably many distinct group elements $(k_i)_{i=1}^\infty$ such that the translates $k_i \cdot f$ are all distinct. Now let $H$ be the necessarily countable subgroup of $G$ generated by $g_1, \dots, g_n$ and $(k_i)_{i=1}^\infty$. Although the restricted action of $H$ on $X$ is not necessarily minimal, we can pass to a minimal $H$-subsystem $Y\subseteq X$ by Zorn's lemma. Consider the restriction map $\pi \colon C(X) \to C(Y)$. The function $\pi(f)$ is non-zero, by construction.
If the set $\setbuilder{h \cdot \pi(f)}{h \in H}$ is finite, then by the pigeonhole principle, because $\setbuilder{h \cdot f}{h \in H}$ is infinite, there are infinitely many distinct translates $h \cdot f$ mapping to $h_0 \cdot \pi(f)$, for some $h_0\in H$. Thus, the sum of the distinct translates $h \cdot f$ mapping to this value is unbounded, as under the image of $\pi$, the sum of infinitely many copies of $h_0 \cdot \pi(f)$ is certainly unbounded.
If the set $\setbuilder{h \cdot \pi(f)}{h \in H}$ is infinite, then we may simply apply the lemma we have already proven in the countable setting to deduce that the sum of the infinitely many distinct translates $h \cdot \pi(f)$ in $C(Y)$ is unbounded. Thus, the sum of the infinitely many distinct translates in the original space $C(X)$, i.e.\ the sum of the distinct $h \cdot f$, is certainly unbounded as well.
\end{proof}
\begin{theorem}\label{thm:DiscreteMinimal}
Let $G$ be a discrete group acting minimally on a unital C*-algebra $A$. Then $A \rtimes_\lambda G$ is not prime if and only if there exist $r \in \FC(G) \setminus \set{e}$, a non-zero $r$-invariant central projection $p \in I(A)$, and a unitary $u \in U(I(A)p)$ such that
\begin{enumerate}
\item $r$ acts by $\Ad u$ on $I(A)p$;
\item $s \cdot p = p$ and $s \cdot u = u$ for all $s \in C_G(r)$.
\end{enumerate}
\end{theorem}
\begin{proof}
First, if there is a triple $(p,u,r)$ satisfying the properties listed above, then exactly as in the proof of Theorem~\ref{thm:mainSec4Prime} (see also Theorem~\ref{thm:mainSec4}), we conclude that $A\rtimes_\lambda G$ is not prime.
Conversely, assume that $A \rtimes_\lambda G$ is not prime.
Arguing the same as in the proof of Theorem~\ref{thm:mainFCHSimpleMinimal}, leads to a projection $q\in M(I(A),G)$, which commutes with the copy of $I(A)\rtimes_\lambda G$, but does not lie in the copy of $I(A)$.
Letting $(q_t)_{t\in G}$ by the coefficients of $q$ as an element of $M(I(A),G)$, i.e.\ $q = \sum_{t\in G}q_t\lambda_t$, and let $z_t\coloneqq q_t^*$, for every $t\in G$. We know that there exists $r\in G\setminus \{e\}$ so that $q_r\neq 0$. We will show that $r$ must have finite conjugacy class. Note that this is stronger than what is shown in Theorem~\ref{thm:mainFCHSimpleMinimal}, and therefore would again lead to the desired triple $(p,u,r)$.
To see this, observe that $q_e=E(q)=E(qq^*)$ is given by
\[ O-\sum_{t \in G} z_t^* z_t, \]
so that the net of finite sums is order-convergent to a concrete element in $I(A)$. The precise details of order convergence are not important. What is important is that this in particular implies that
\[\sum_{t \in \mathcal{F}} z_t^* z_t\leq E(q), \]
for every finite subset $\mathcal{F}\subseteq G$ (see \cite[Lemma~1.2.(iv)]{hamana82_mc_tensor_products_I}).
We recall from Proposition~\ref{prop:mc_crossed_product_commutant_coefficients} that $z_t y = (t \cdot y) z_t$, for all $y \in I(A)$. Consequently,
\[ z_t^* z_t y = z_t^* (t \cdot y) x_t = y z_t^* z_t, \]
for all $y \in I(A)$, or, in other words, $z_t^*z_t$ lies in $Z(I(A))$. Proposition~\ref{prop:mc_crossed_product_commutant_coefficients} also tells us that $s \cdot z_t = z_{sts^{-1}}$, and so $s \cdot (z_t^*z_t) = z_{sts^{-1}}^*z_{sts^{-1}}$ for all $s,t\in G$.
Let us restrict our attention to an \emph{infinite} conjugacy class $C \subseteq G$, and show that the coefficients $z_c$ must be zero for every $c\in C$. Assume otherwise, so that one, hence all, of these coefficients are non-zero. If $\setbuilder{z_c^*z_c}{c \in C}$ is a finite set, then by the pigeonhole principle, there are infinitely many $c \in C$ for which $z_c^*z_c$ are all the same value. Given that $z_c^*z_c \neq 0$, this clearly contradicts the assumption that the sum $\sum_{t \in G} z_t^*z_t$ is uniformly bounded above.
Hence, we are left to conclude that $\setbuilder{z_c^*z_c}{c \in C}$ is an infinite set. Equivalently, if we choose a fixed $c_0 \in C$, the set
\[ \setbuilder{g \cdot (z_{c_0}^*z_{c_0})}{g \in G} \]
has infinitely many distinct elements in $Z(I(A))$. By Lemma~\ref{lem:infinitely_many_distinct_translates_unbounded_sum}, these once again cannot have a bounded sum, as $Z(I(A))$ is minimal by Proposition~\ref{prop:minimality_transfers_to_injective_envelopes_and_centers}. Therefore $z_c = 0$.
As we had $z_r=q_r^*\neq 0$, we conclude that $r\in\FC(G)$.
\end{proof}
\begin{remark}\label{rem:Sec5from6}
Note that Theorem~\ref{thm:mainFCHSimpleMinimal} follows from a combination of Theorem~\ref{thm:DiscreteMinimal} and of the fact that in the FC-hypercentral case, $A \rtimes_\lambda G$ is simple if and only if it is prime (this follows from \cite[Theorem~3.1]{echterhoff_jot}, or alternatively Lemma~\ref{lem:fc_simplicity_equivalence}). Despite this fact, we keep Section~\ref{sec:simplicity_fch} separate, given that the full proof of Theorem~\ref{thm:DiscreteMinimal} (which in particular relies upon Lemma~\ref{lem:infinitely_many_distinct_translates_unbounded_sum}) is substantially more difficult in the setting of non-amenable groups.
\end{remark}
Finally, we state the following immediate, and somewhat surprising, corollary to Theorem~\ref{thm:DiscreteMinimal}.
\begin{corollary}
\label{cor:ICC}
If $G$ is a discrete ICC group acting minimally on a unital C*-algebra $A$, then the reduced crossed product $A\rtimes_\lambda G$ is prime.
\end{corollary}
\section{From the injective envelope to the original C*-algebra}
\label{sec:proper_outerness_on_A}
The first dynamical characterizations of simplicity and primality that we obtain in our paper are initially written in terms of the dynamics of $G$ on $I(A)$, the injective envelope of $A$. While this gives the most elegant characterizations from a theory perspective, the injective envelope is still a somewhat mysterious object that is not that easy to describe concretely in many cases. Strictly speaking, it was also possible in all of our results to simply use Hamana's \emph{regular monotone completion} $\overline{A}$, which is significantly smaller in many cases, but at the end of the day it suffers from the same problem of still being relatively difficult to write down concretely. It would be desirable to have results which relate back to the dynamics on the original C*-algebra $A$.
It was mentioned in the introduction that the notion of a \emph{properly outer} automorphism plays an important role in the study of simplicity of crossed products, and indeed, our results are an equivariant version of this notion. Recall the classical, non-equivariant notion first, which was discussed in Section~\ref{sec:preliminaries:properly_outer} (especially Theorem~\ref{thm:properly_outer_equivalence}, which we state again for convenience). Let $A$ be a unital C*-algebra, and let $\alpha \in \Aut(A)$, with its unique extension to $I(A)$ also denoted by $\alpha$. Consider the following conditions:
\begin{enumerate}
\item
There is a non-zero $\alpha$-invariant ideal $J \subseteq A$ such that $\Gamma_B(\alpha|_J) = \set{e}$.
\item
There is a non-zero $\alpha$-invariant central projection $p \in I(A)$, and a unitary $u \in U(I(A)p)$ such that $\alpha$ acts by $\Ad u$ on $I(A)p$.
\item
There is a non-zero $\alpha$-invariant ideal $J \subseteq A$ and a unitary $u \in M(J)$, such that $\norm{\alpha|_J - (\Ad u)|_J} < 2$.
\item
Given any $\varepsilon\in (0,2)$, there exists a non-zero $\alpha$-invariant ideal $J \subseteq A$ and a unitary $u \in M(J)$ such that $\norm{\alpha|_J - (\Ad u)|_J} < \varepsilon$.
\end{enumerate}
We have that (\ref{thm:properly_outer_equivalence:borchers}) and (\ref{thm:properly_outer_equivalence:injective_envelope}) are equivalent, and both are implied by (\ref{thm:properly_outer_equivalence:elliott}) and (\ref{thm:properly_outer_equivalence:elliott_strong}). If $A$ is separable, all four are equivalent.
We will ultimately require some notion of invariance when it comes to all of the pieces involved in the above theorem. In particular, our characterization of simplicity, or lack thereof, is a modified version of condition~(\ref{thm:properly_outer_equivalence:injective_envelope}), where we require the unitary $u$ corresponding to some automorphism $\alpha_r$ (where $r \in \FC(G) \setminus \set{e}$) to be $C_G(r)$-invariant. Note that this is in contrast to the usual definition of proper outerness, where the automorphisms are considered individually without any regard to the rest of the group action.
Let us briefly investigate the feasibility and outcome of generalizing each characterization of proper outerness on the original C*-algebra $A$. First, consider (\ref{thm:properly_outer_equivalence:borchers}). The Borchers spectrum is normally defined for actions of abelian groups (and for single automorphisms $\alpha$, it secretly considers the corresponding $\mathbb{Z}$-action). In the setting of abelian groups, \cite[Theorem~7.3]{hamana85-injective_envelopes_equivariant} essentially gives us the exact result we are after, and makes use of the Borchers spectrum for the action of the \emph{entire} group (as opposed to a single automorphism). Note, however, that the definition of the Borchers spectrum involves the dual group $\what{G}$. This strongly hints to the fact that any invariant Borchers-type characterization that is meant to work in the non-abelian setting will involve the use of the non-abelian dual group. Such an item is borderline impossible to get a concrete handle on, in practice, for most infinite groups. Moreover, the definition of the Borchers spectrum also involves considering all possible hereditary C*-subalgebras, and these are already non-trivial enough in general as-is. Thus, while it could \emph{in theory} be possible to obtain an appropriate generalization, we highly doubt it would be one that people would wish to use in practice.
We also mention that \cite[Theorem~6.6]{olesen_pedersen_III} contains several other characterizations of proper outerness. However, the vast majority (with a couple of exceptions) also involve considering either all possible hereditary C*-subalgebras of $A$, or at least the invariant ones. Again, this would make for conditions that are perhaps mysterious and hard to check in practice, and so we choose to also skip those. It is worth noting though that all of these conditions are very likely still possible to generalize.
The obvious characterization remaining is Elliott's characterization, i.e.\ condition~(\ref{thm:properly_outer_equivalence:elliott}). While it requires separability of the underlying C*-algebra $A$ to be a true characterization, most C*-algebras that people are interested in end up being separable anyways. Moreover, it is only necessary to consider the space of invariant ideals of $A$, as opposed to the set of all invariant hereditary subalgebras. Thus, in our opinion, it is the most worthwhile characterization to generalize.
We would like to first give an initial and incorrect guess as to how the generalization would proceed. One might guess that ``invariant'' proper outerness of $\alpha_t$ would mean that for any $C_G(t)$-invariant ideal $I \subseteq A$, and any $C_G(t)$-invariant unitary $u \in M(I)$, we have $\norm{\alpha|_I - (\Ad u)|_I} = 2$. However, this turns out to be just slightly too weak of a notion, as there is simply no reason to expect enough suitable \emph{invariant} ideals $I \subseteq A$ and \emph{invariant} unitaries $u \in M(I)$ for approximating the automorphism. As it turns out, the correct notion to use is \emph{approximately invariant} ideals and unitaries, in the appropriate sense.
Before proceeding further, we remark that all of the theory of injective envelopes that we use was in the setting of \emph{unital} C*-algebras. In \cite[Section~6]{hamana82_mc_tensor_products_I}, Hamana defines the injective envelope of a non-unital C*-algebra as follows (and studies it further in \cite{hamana82_centre}):
\begin{definition}
Let $A$ be a C*-algebra, not necessarily unital. We define $A^+$ to just be $A$ if $A$ is already unital, and the unitization otherwise. In particular, if $J \subseteq A$ is an ideal, the unitization $J^+$ will still be $J$ if, coincidentally, $J$ has its own unit. The injective envelope of $A$ is defined as $I(A) \coloneqq I(A^+)$.
\end{definition}
We also state the following result of Hamana, which gives a correspondence between ideals of a C*-algebra $A$ and central projections in $I(A)$. This, and other results, usually work in the setting of $A$ being non-unital, but to avoid unnecessary subtleties, we will just stick with the unital setting unless necessary.
\begin{proposition}
\label{prop:ideal_injective_envelope}
Let $J$ be an ideal of a unital C*-algebra $A$. Let $p = \sup^{I(A)} J_1^+$. Then
\begin{enumerate}
\item $p$ is a central projection in $I(A)$;
\item $I(A)p \cong I(J)$.
\end{enumerate}
Conversely, if we start with some central projection $p \in I(A)$ and define an ideal $J \subseteq A$ as $J = I(A)p \cap A$, then $I(J) \cong I(A)p$.
\end{proposition}
\begin{proof}
The forward direction was proven in \cite[Lemma~1.1]{hamana82_centre}, in the more general setting of hereditary subalgebras of $A$. For the converse direction, note that if $J = I(A)p \cap A$ for a central projection $p\in I(A)$, then $\sup^{I(A)} J_1^+=p$ by Proposition~\ref{prop:ConcreteSupProj}.
\end{proof}
This now hints at how the proof of the aforementioned equivalences would work in general. Any automorphism $\alpha \in \Aut(A)$ that is ``close enough'' to being inner on an $\alpha$-invariant ideal $J \subseteq A$ will give something that is genuinely inner on $I(J)$, which is a central corner of $I(A)$.
We may also realize the multiplier algebra of a C*-algebra $A$ as an idealizer of $A$ inside its injective envelope $I(A)$.
\begin{proposition}[{\cite[Section~1]{hamana82_centre}}]
\label{prop:multiplier_algebra_contained_in_injective_envelope}
Assume $A$ is a not necessarily unital C*-algebra, and denote by $M(A)$ its multiplier algebra. We have
\[ M(A) = \setbuilder{x \in I(A)}{xa \in A \text{ and } ax \in A \text{ for all } a \in A}. \]
\end{proposition}
Recall that an ideal $K$ is a C*-algebra $A$ is essential if and only if whenever $K \cdot a = 0$ for some $a\in A$ (which is equivalent to $a \cdot K = 0$), then $a=0$.
As mentioned earlier, we will also be dealing with ideals that are ``almost invariant'' in the appropriate sense. The following observation allows us to establish what this means rigorously.
\begin{proposition}
\label{prop:essential_ideal_containment_implies_inj_env}
Let $A$ be a unital C*-algebra, and let $J,K \subseteq A$ be ideals with $J \subseteq K$. Then
\begin{enumerate}
\item Assume that $J$ is essential in $K$. Then recalling Proposition~\ref{prop:ideal_injective_envelope}, $I(J)$ and $I(K)$ both share the same central support projection $p\in I(A)$, so that we canonically have $I(J) \cong I(K) \cong I(A)p$.
\item If $\sup^{I(A)}J_1^+=\sup^{I(A)}K_1^+$, then $J$ is essential in $K$.
\end{enumerate}
\end{proposition}
\begin{proof}
Let $p=\sup^{I(A)} J_1^+$ and $q=\sup^{I(A)} K_1^+$. Since $J\subseteq K$, we have that $p\leq q$. We need to show that $q\leq p$. Let $I=I(A)(q-p)\cap A$. Observe that $I$ is an ideal in $A$ which has trivial intersection with $J$. On the other hand, since $J$ is essential in $K$, $I$ must have trivial intersection with the ideal $K$ as well. This implies that $I\cdot K=0$, and by Proposition~\ref{prop:ideal_sup_orthogonality} we obtain that $q(q-p)=0$, and so $q\leq p$.
For the second statement, assume that $p=\sup^{I(A)}J_1^+=\sup^{I(A)}K_1^+$ and let $k\in K$ be an element which is orthogonal to $J$. Then $kp=0$ by Proposition~\ref{lem:sup_bFbstar}. However, by Proposition~\ref{prop:ideal_sup_projection}, $p$ acts as the identity on $K$, and we conclude that $k=0$, as desired.
\end{proof}
The following two observations are well-known, but we recall them nevertheless.
\begin{remark}\label{rem:multipliers_inclusion_essential_ideal}
Let $A$ be a unital C*-algebra, and let $J,K \subseteq A$ be ideals with $J \subseteq K$, and such that $J$ is essential in $K$. Then $M(K) \subseteq M(J)$ as a unital inclusion of the multiplier algebras.
\end{remark}
\begin{proof}
Note that since $J$ is an essential ideal in $K$ and $K$ is an essential ideal in $M(K)$, then $J$ is an essential ideal in $M(K)$. As $M(J)$ is the maximal C*-algebra which contains $J$ as an essential ideal (see \cite[Theorem~2.2]{Lance}), we conclude that $M(K)\subseteq M(J)$, canonically.
\end{proof}
\begin{remark}
\label{rem:intersection_of_essential_ideals}
Assume $A$ is a C*-algebra, not necessarily unital, and assume that $I$ and $J$ are two essential ideals of $A$. Then $I \cap J$ is also essential in $A$.
\end{remark}
\begin{proof}
Let $I$ and $J$ be as in the statement of the lemma, and assume there is some $a \in A$ orthogonal to the intersection. We have $axy = 0$, for all $x \in I$ and $y \in J$. Because $J$ is essential, this implies that for any $x \in I$, we have $ax = 0$. Because $I$ is essential, this now implies $a = 0$.
\end{proof}
Before proceeding, we mention what our suitable replacement for having an ideal $J \subseteq A$ invariant on the nose. Such an ideal will be considered ``almost invariant'' with respect to an action of a group $H$ on $A$ if, while we do not necessarily have $h \cdot J = J$, we at least have that $J \cap h \cdot J$ is essential in both $J$ and $h \cdot J$.
We will also require a stronger version of Elliott's proper outerness characterization than the one presented in Theorem~\ref{thm:properly_outer_equivalence}. There, it can be observed that if $\alpha$ is an automorphism of a (not necessarily unital) separable C*-algebra $A$ such that $\alpha$ extends to an inner automorphism on \emph{all of} $I(A)$ (such automorphisms are called \emph{quasi-inner}), then given any $\varepsilon > 0$, there is some ideal $J \subseteq A$ and unitary $u \in M(J)$ such that $\norm{\alpha|_J - (\Ad u)|_J} < \varepsilon$. However, \cite[Corollary~6.7]{olesen_pedersen_III} more or less observes that in the case of quasi-inner automorphisms, the ideal $J$ can be required to be essential. Proposition~\ref{prop:elliott_strong_essential} uses essentially the same argument, but we recall two lemmas before proving it.
\begin{lemma}[{\cite[Lemma~3.1]{hamana82_centre}}]
\label{lem:direct_sum_multiplier_algebra_injective_envelope}
Let $(A_\lambda)_{\lambda \in \Lambda}$ be a family of C*-algebras, not necessarily unital. Consider the $c_0$-direct sum $\oplus_\lambda A_\lambda$. The injective envelope of this direct sum is the $\ell^\infty$-direct sum $\prod_\lambda I(A_\lambda)$, while the multiplier algebra is $\prod_\lambda M(A_\lambda)$.
\end{lemma}
\begin{proof}
The first statement is proven in \cite[Lemma~3.1]{hamana82_centre}. The claim about the multiplier algebra is well-known, but also follows from the first statement and the identification given in Proposition~\ref{prop:multiplier_algebra_contained_in_injective_envelope}.
\end{proof}
\begin{lemma}
\label{lem:direct_sum_of_orthogonal_ideals}
Assume $(J_\lambda)_{\lambda \in \Lambda}$ is a family of pairwise orthogonal ideals in a (not necessarily unital) C*-algebra $A$. The ideal generated by this family is $J = \closure{\operatorname{span}} J_\lambda$, which is canonically isomorphic to the $c_0$-direct sum $\oplus_\lambda J_\lambda$.
\end{lemma}
\begin{proof}
If we instead consider $c_{00}-\oplus_\lambda J_\lambda$, i.e.\ the elements in $\oplus_\lambda J_\lambda$ that are non-zero on only finitely many coordinates, then we canonically obtain a map
\[ c_{00}-\oplus_\lambda J_\lambda \to \operatorname{span} J_\lambda, \]
given by mapping $x_\lambda \in J_\lambda \subseteq c_{00}-\oplus_\lambda J_\lambda$ to itself in $J_\lambda \subseteq A$. This map is well-defined, bijective, and a *-homomorphism on the incomplete domain. Importantly, it is also isometric, simply due to being an injective *-homomorphism on every C*-subalgebra $\oplus_{\lambda \in F} J_\lambda$, where $F \subseteq \Lambda$ ranges over the finite subsets. Thus, the map extends to a *-isomorphism
\[ \oplus_\lambda J_\lambda \to \closure{\operatorname{span}} J_\lambda. \]
\end{proof}
\begin{proposition}
\label{prop:elliott_strong_essential}
Let $\alpha \in \Aut(A)$ be an automorphism on a separable, not necessarily unital, C*-algebra $A$. If $\alpha$ is quasi-inner, i.e.\ inner on $I(A)$, then for every $\varepsilon > 0$, there exists an \emph{essential} $\alpha$-invariant ideal $J \subseteq A$ and a unitary $u \in M(J)$ with the property that $\norm{\alpha|_J - (\Ad u)|_J} < \varepsilon$.
\end{proposition}
\begin{proof}
Fix $\varepsilon > 0$. By Zorn's lemma, there is a maximal family of pairs $(J_\lambda, u_\lambda)_{\lambda\in\Lambda}$ with the property that $J_\lambda$ is a non-zero $\alpha$-invariant ideal of $A$, $u_\lambda$ is a unitary in $M(J_\lambda)$ with the property that $\norm{\alpha|_{J_\lambda} - (\Ad u_\lambda)|_{J_\lambda}} < \varepsilon$, and $(J_\lambda)_{\lambda \in \Lambda}$ are all pairwise orthogonal.
Observe that $J \coloneqq \closure{\operatorname{span}} J_\lambda$ is a new $\alpha$-invariant ideal in $A$, which is isomorphic to $\oplus_\lambda J_\lambda$ by Lemma~\ref{lem:direct_sum_of_orthogonal_ideals}. We then know by Lemma~\ref{lem:direct_sum_multiplier_algebra_injective_envelope} that its multiplier algebra is the $\ell^\infty$-direct sum $\prod_\lambda M(J_\lambda)$. In particular, we are free to set the individual coordinates to anything bounded with no other restriction, and so we have that $u \coloneqq (u_\lambda)_{\lambda\in\Lambda}$ is an element of the multiplier algebra $M(J)$ satisfying
\[ \norm{\alpha|_J - (\Ad u)|_J} \leq \varepsilon. \]
Maximality of the family $(J_\lambda,u_\lambda)_{\lambda\in\Lambda}$ gives us the final result we are after, namely that $J$ above is essential. If it were not, then $J^{\perp}$ would be $\alpha$-invariant and orthogonal to every $J_\lambda$. Moreover, using the fact that $I(J^{\perp})$ is a corner of $I(A)$ by Proposition~\ref{prop:ideal_injective_envelope}, we have that $\alpha$ is inner on $I(J^{\perp})$. Applying Theorem~\ref{thm:properly_outer_equivalence} to the unitization $(J^\perp)^+$, there at least exists some non-trivial $\alpha$-invariant ideal $L \subseteq (J^\perp)^+$ (and $L \cap J^+$ is a non-trivial $\alpha$-invariant ideal of $A$), and some unitary $v \in M(L) \subseteq M(L \cap J^+)$ with the property that $\norm{\alpha|_L - (\Ad v)|_L} < \varepsilon$. The pair $(L \cap J^+,v)$ could then be added to our maximal family from before, contradicting the fact that it is actually maximal.
\end{proof}
Unlike the proof of the equivalence between Elliott's characterization of proper outerness (condition~(3) in Theorem~\ref{thm:properly_outer_equivalence}) and the injective envelope version (condition~(2) in Theorem~\ref{thm:properly_outer_equivalence}), our approach will completely avoid the theory of derivations, and instead take a different path (one which genuinely appears to be necessary for the equivalence between the \emph{equivariant} notions of proper outerness). Essentially, what was shown above in the proof of Theorem~\ref{thm:equivariant_proper_outer_equivalence} is that if $\norm{\beta - \id} < 2$ on an ideal $J\subseteq A$ for some automorphism $\beta\in\Aut(J)$, then we have that $\beta$ extends to an inner automorphism on $I(J)$. What we need is some control over the unitary itself, in $I(J)$. Intuitively, if $\beta$ is ``close'' to the identity, then the resulting unitary should be ``close'' to $1$. In the context of von Neumann algebras at least, this intuition is indeed true, and the following lemma can be found in an important paper of Kadison and Ringrose.
\begin{lemma}[{\cite[Lemma~5]{kadison_ringrose}}]
\label{lem:kadison_ringrose_unitary_vn_alg}
Let $\alpha \in \Aut(M)$ be an automorphism of a von Neumann algebra $M$ such that $\norm{\alpha - \id} < 2$. Then there is a unitary $u \in M$ such that $\alpha = \Ad u$, and moreover $u$ has the following restriction on its spectrum:
\[ \sigma(u) \subseteq \setbuilder{z \in \mathbb{T}}{\operatorname{Re} z \geq \frac{1}{2} \sqrt{4 - \norm{\alpha - \id}^2}}. \]
\end{lemma}
Note that we indeed have that if $\norm{\alpha - \id}$ gets closer to zero, then $\norm{u - 1}$ indeed gets closer to zero as well. We suspect that the above lemma might work for general monotone complete C*-algebras. However, we work around this in a sneaky manner and can derive the result on just injective envelopes quite easily. For the proof, we will need the following lemma.
\begin{lemma}
\label{lem:non_free_on_A_on_inj}
Let $A$ be a unital C*-algebra and let $\alpha\in \Aut(A)$, with its unique extension to an automorphism of $I(A)$ also denoted by $\alpha$. Assume that $x\in I(A)$ is such that
$xy=\alpha(y)x$ for every $y\in A$. Then $xy=\alpha(y)x$ for every $y\in I(A)$.
\end{lemma}
\begin{proof}
Our first observation is that the proof of Kallman's result, Theorem~\ref{thm:PropOuterCorner}, essentially still works if we implement some modifications. Let $x = u \abs{x}$ be the polar decomposition of $x$ in $I(A)$, with $p = u^*u$ being the domain projection of $u$. First, observe that both $x^*x$ and $xx^*$ lie in $A' \cap I(A)$. However, by \cite[Corollary~4.3]{hamana79_injective_envelopes_cstaralg}, this in fact implies that $x^*x$ and $xx^*$ lie in $Z(I(A))$. The first two paragraphs of the proof of Theorem~\ref{thm:PropOuterCorner} now work without any more modifications to show that $p \in Z(I(A))$ and $u^*u = uu^* = p$.
We skip the part of Theorem~\ref{thm:PropOuterCorner} which shows that $p$ is moreover $\alpha$-invariant. This is not as straightforward to show in our new context, simply because we do not have $p \in A$. Rather, we move on to showing that $u$ ``implements'' the automorphism. Just like in the fourth paragraph of the proof of Theorem~\ref{thm:PropOuterCorner}, we may show (using centrality of $\abs{x}$) that for any $y \in A$, we have $\abs{x} (uy - \alpha(y)u) = 0$, and from here, it follows that $p(uy - \alpha(y)u) = 0$. However, $p$ is central and $up = u$, and so we conclude that $uy = \alpha(y)u$. Multiplying by $u^*$ on the right gives us
\[ uyu^* = \alpha(y)p = p \alpha(y) p, \]
for any $y\in A$. Let $J = I(A)p \cap A$, and let $(j_\lambda)_\lambda$ be the increasing net of positive elements in $J$ with $\norm{j_\lambda}<1$. We know from Proposition~\ref{prop:ConcreteSupProj} that $\sup^{I(A)} j_\lambda = p$. Moreover, as $\alpha \in \Aut(I(A))$ is an automorphism of a monotone complete C*-algebra, we automatically have that $\sup^{I(A)} \alpha(j_\lambda)$ exists and is equal to $\alpha(p)$. Applying Lemma~\ref{lem:sup_bFbstar}, we obtain
\[ upu^* = p \alpha(p) p, \]
or more simply $p = \alpha(p) p$. It follows that $p \leq \alpha(p)$. The reverse inequality is also not hard to obtain. Again looking back at the equality $uy = \alpha(y)u$ for $y \in A$, we have $u \alpha^{-1}(y) = yu$ for all $y \in A$. This can be rewritten as
\[ u \alpha^{-1}(y) u^* = yp = pyp, \]
and it follows from the same results as before that
\[ u \alpha^{-1}(p) u^* = ppp, \]
or more simply $\alpha^{-1}(p) p = p$. This gives us the inequality $p \leq \alpha^{-1}(p)$, or equivalently, $\alpha(p) \leq p$.
In summary, we have shown that
\begin{itemize}
\item $p$ is an $\alpha$-invariant central projection in $I(A)$;
\item $u$ is a unitary in $I(A)p$;
\item $\alpha$ acts by $\Ad u$ on $J = I(A)p \cap A$. In fact, $\alpha$ acts by $\Ad u$ on $J^+ = C^*(J,p)$.
\end{itemize}
We may use Proposition~\ref{prop:ideal_injective_envelope} to rephrase the above slightly. Namely, $I(J) =I(J^+)=I(A)p$, and $\alpha|_{J^+}$ coincides with $(\Ad u)|_{J^+}$, where $u \in U(I(J))$. However, an automorphism on a unital C*-algebra always has a unique extension to an automorphism on the injective envelope, and so it follows that $\alpha|_{I(A)p}$ is truly $\Ad u$ on this entire corner.
From here, recalling that both $p$ and $\abs{x}$ are central, given any $y \in I(A)$, we have
\begin{align*}
xy &= u \abs{x} y = up \abs{x} y = u (py) \abs{x} = \alpha(py) u \abs{x} \\
&= p \alpha(y) u \abs{x} = \alpha(y) up \abs{x} = \alpha(y) u \abs{x} = \alpha(y) x.
\end{align*}
\end{proof}
\begin{lemma}
\label{lem:kadison_ringrose_unitary_injective_envelope}
Let $\alpha \in \Aut(A)$ be an automorphism of a not necessarily unital C*-algebra $A$, such that $\norm{\alpha - \id} < 2$. Then the unique extension of $\alpha$ to $I(A)$ (obtained by first extending canonically to $A^+$) is inner and of the form $\Ad u$, where $u$ is a unitary in $I(A)$ which can be required to satisfy the following restriction on its spectrum:
\[ \sigma(u) \subseteq \setbuilder{z \in \mathbb{T}}{\operatorname{Re} z \geq \frac{1}{2} \sqrt{4 - \norm{\alpha - \id}^2}}. \]
\end{lemma}
\begin{proof}
We already know that such a result holds on von Neumann algebras. Let $M$ be a von Neumann algebra containing $A$ as a weak*-dense subalgebra, and moreover having the property that the automorphism $\alpha \in \Aut(A)$ extends (necessarily uniquely) to an automorphism $\wtilde{\alpha} \in \Aut(M)$. It does not appear that any sort of separability is required for $M$ in either the statement or the proof of Lemma~\ref{lem:kadison_ringrose_unitary_vn_alg} of Kadison and Ringrose (so in particular, one could take $M = A^{**}$). However, just for peace of mind of the reader, it is observed in that same paper that any automorphism $\alpha \in \Aut(A)$ with $\norm{\alpha - \id} < 2$ \emph{automatically} extends to any such enveloping von Neumann algebra $M$. See \cite[Theorem~7]{kadison_ringrose}, along with the definition of \emph{permanently weakly inner} or \emph{$\pi$-weakly inner} in \cite[Page~35]{kadison_ringrose}. It appears that Kadison and Ringrose deal with unital C*-algebras in their paper, but this is not a problem. The canonical extension of $\alpha$ to $A^+$ (call it $\alpha^+$) also satisfies $\norm{\alpha^+ - \id_{A^+}} < 2$. Hence, if $A$ is separable, one can easily take any faithful non-degenerate representation $A \subseteq B(H)$ where $H$ is separable, and then let $M = A''$.
In any case, let $M$ and $\wtilde{\alpha} \in \Aut(M)$ be as above, and observe that $\norm{\wtilde{\alpha} - \id_M} = \norm{\alpha - \id_A}$, via a straightforward application of weak*-continuity of $\wtilde{\alpha}$ and Kaplansky's density theorem. Let $u \in M$ be a unitary obtained from Lemma~\ref{lem:kadison_ringrose_unitary_vn_alg}. In other words, $\wtilde{\alpha} = \Ad u$, and
\[ \sigma(u) \subseteq \setbuilder{z \in \mathbb{T}}{\operatorname{Re} z \geq \frac{1}{2} \sqrt{4 - \norm{\alpha - \id}^2}}. \]
For convenience, denote the bound $\frac{1}{2} \sqrt{4 - \norm{\alpha - \id}^2}$ by $r$. We have $\operatorname{Re} \sigma(u) \subseteq [r,1]$, and so by the continuous functional calculus, we genuinely have $\operatorname{Re} u \geq r$, where $\operatorname{Re} u = \frac{1}{2}(u + u^*)$.
We proceed to construct a similar element in $I(A)$ using the only method we know. We know that $A^+ \subseteq M$, and by injectivity, there is a unital and completely positive map $E \colon M \to I(A)$ such that it is the identity map on $A^+$. Observe that, because $\operatorname{Re} u \geq r$ with $r > 0$, we have that
\[ \operatorname{Re} E(u) = E(\operatorname{Re} u) \geq E(r) = r. \]
In particular, we have that $E(u)$ is non-zero. Moreover, observe that for any $b \in A^+$ (which lies in the multiplicative domain of $E$), we have
\[ E(u) b = E(ub) = E(\alpha(b) u) = \alpha(b) E(u). \]
By Proposition~\ref{lem:non_free_on_A_on_inj}
this automatically implies that $E(u) z = \alpha(z) E(u)$ for all $z \in I(A)$.
For convenience, write $y = E(u)$. We have no reason to expect that $y$ is in any way a unitary element. However, this is not too hard to patch up. Let $y = v \abs{y}$ be the polar decomposition of $y$ in $I(A)$, with $p = v^*v$ being the support projection of $v$. Recall from Theorem~\ref{thm:PropOuterCorner} that
\begin{itemize}
\item $p$ is an $\alpha$-invariant central projection in $I(A)$.
\item $v$ is a unitary in $I(A)p$, and on this corner, we in fact have $\alpha|_{I(A)p} = \Ad v$.
\end{itemize}
It was also shown in the proof of Theorem~\ref{thm:PropOuterCorner} that $y^*y$ (and therefore also $\abs{y}$) will always lie in the center of $I(A)$. Given that $v$ is a unitary on its respective corner, it follows that $C^*(1,v,\abs{y})$ is a commutative C*-algebra of the form $C(X)$.
Writing $y = v \abs{y}$ inside of $C(X)$, and using that $\operatorname{Re} y \geq r$ for $r > 0$, we note that $y$ is invertible in $C(X)$. Consequently, so is $\abs{y}$, and it follows that the support projection $p$ of $v$ was in fact $1$. In other words, $v$ is a unitary in $I(A)$ implementing $\alpha$. Finally, given any $x \in X$, we have $\operatorname{Re} y(x) \geq r$, and $0 < \abs{y}(x) \leq 1$ (which follows from $\norm{y} = \norm{E(u)} \leq 1$). Thus, $v(x) = \frac{y(x)}{\abs{y}(x)}$ also satisfies $\operatorname{Re} v(x) \geq r$ for all $x\in X$. As the spectrum of $v$ as an element of $C(X)$ is given by $\setbuilder{v(x)}{x \in X}$, and coincides with its spectrum as an element of $A$, this finishes the proof.
\end{proof}
Clearly, if $u$ is a unitary in any unital C*-algebra with $\operatorname{Re} \sigma(u) \subseteq [r,1]$ for $r$ close to $1$, then it is the case that $u$ is close to $1$ as well by the continuous functional calculus. The following lemma establishes the precise norm estimate:
\begin{lemma}
\label{lem:unitary_spectrum_to_norm}
Let $k \in [0,2]$, and let $u \in U(A)$ be a unitary in a unital C*-algebra $A$ with $\operatorname{Re} \sigma(u) \subseteq [\frac{1}{2} \sqrt{4 - k^2},1]$. Then we have
\[ \norm{u - 1} \leq \sqrt{2 - \sqrt{4-k^2}}. \]
\end{lemma}
\begin{proof}
This is essentially an exercise in Euclidean geometry. Consider Figure~\ref{fig:unitary_norm}.
\begin{figure}
\caption{Estimate of $\norm{u - 1}$ based on estimate of $\operatorname{Re} \sigma(u)$.}
\label{fig:unitary_norm}
\end{figure}
It depicts the unit circle $\mathbb{T}$, and the spectrum of $u$ lies on the arc of the circle that is to the \emph{right} of the dotted line. Some unknown side lengths are labeled as $x$, $y$, and $z$. It is clear that $x = 1 - \frac{1}{2} \sqrt{4 - k^2}$, and by the Pythagorean theorem that $y = \frac{1}{2} k$. One more application of the Pythagorean theorem gives us that
\[ z = \sqrt{2 - \sqrt{4 - k^2}}. \]
The value of $z$ represents the maximum distance of any given point in $\sigma(u)$ from $1$, and so by the continuous functional calculus, it must be the case that $\norm{u - 1} \leq z$, our desired result.
\end{proof}
A similar and even easier estimate characterizes when $\sigma(u)$ has a positive lower bound as well:
\begin{lemma}
\label{lem:unitary_spectrum_positive_to_norm_sqrt2}
Let $u \in U(A)$ be a unitary in some unital C*-algebra $A$. Then there exists an $r > 0$ with $\operatorname{Re} \sigma(u) \subseteq [r,1]$ if and only if $\norm{u - 1} < \sqrt{2}$.
\end{lemma}
\begin{proof}
Again, we perform basic Euclidean geometry. In Figure~\ref{fig:unitary_norm_sqrt2},
\begin{figure}
\caption{Estimate of $\norm{u-1}$ based on $\operatorname{Re} \sigma(u) > 0$ and vice versa.}
\label{fig:unitary_norm_sqrt2}
\end{figure}
we again need $\sigma(u)$ to lie \emph{strictly} to the right of the dotted line, which is equivalent to any point in the spectrum being distance strictly less than distance $\sqrt{2}$ away from $1$. By continuous functional calculus, this is equivalent to $\norm{u-1} < \sqrt{2}$.
\end{proof}
Now we are ready to state the main result.
\begin{theorem}
\label{thm:equivariant_proper_outer_equivalence}
Let $\alpha\colon G\to\Aut(A)$ be a group action on a unital C*-algebra $A$, such that we are in one of the following situations:
\begin{itemize}
\item $G$ is an FC-group.
\item $G$ is an arbitrary discrete and $\alpha$ is a minimal action.
\end{itemize}
The following are equivalent:
\begin{enumerate}
\item
\label{thm:equivariant_proper_outer_equivalence:injective_envelope}
There exists an $r \in \FC(G) \setminus \set{e}$, a non-zero $C_G(r)$-invariant central projection $p \in I(A)$, and a $C_G(r)$-invariant unitary $u \in U(I(A)p)$ such that $r$ acts by $\Ad u$ on $I(A)p$.
\item
\label{thm:equivariant_proper_outer_equivalence:elliott}
There exists an $r \in \FC(G) \setminus \set{e}$, a non-zero $r$-invariant ideal $J \subseteq A$ with the property that $J \cap h \cdot J$ is essential in both $J$ and $h \cdot J$ for all $h \in C_G(r)$, and a unitary $u \in M(J)$, such that, if we let $k = \norm{\alpha_r|_J - (\Ad u)|_J}$ and $t = \sup_{h \in C_G(r)} \norm{h\cdot u - u}$, we have $2 \sqrt{2 - \sqrt{4 - k^2}} + t < \sqrt{2}$.
\end{enumerate}
Note that the $r \in \FC(G) \setminus \set{e}$ and $p \in Z(I(A))$ in one equivalence do not in any way need to correspond to the same $r$ or $J$ in the other equivalence! Moreover, note that in the second equivalence, we canonically have $M(J)$ and $h\cdot M(J)\cong M(h \cdot J)$ contained in $M(J \cap h \cdot J)$, for every $h\in C_G(r)$, by Remark~\ref{rem:multipliers_inclusion_essential_ideal}, and so the norm difference $\norm{h\cdot u - u}$ makes sense.
\end{theorem}
\begin{proof}
First, we prove (\ref{thm:equivariant_proper_outer_equivalence:elliott}) $\implies$ (\ref{thm:equivariant_proper_outer_equivalence:injective_envelope}). To give some intuition first on what is about to happen, all of the estimates that we gave are for the express purpose of being able to write (the unique extension of) $\alpha_r$ as some inner automorphism $\Ad w$ on $I(J)$, with respect to a unitary $w$ that is \emph{almost} $C_G(r)$-invariant. Our aim is then to ``average'' the translates $h \cdot w$ ($h \in C_G(r)$) of this unitary to get something that is truly invariant (and importantly, non-zero, which is what the almost invariance is for). It is, of course, not immediately obvious how to do this, given that the group $C_G(r)$ is not amenable in general, nor does $I(A)$ have any nice compact topology on the unit ball. The techniques developed in the rest of the paper, however, will show that neither are necessary.
Let us also remark that, by Proposition~\ref{prop:essential_ideal_containment_implies_inj_env}, all of the translates $h \cdot J$ (for $h\in C_G(r)$) share the same injective envelope. Specifically, they all share the same support projection $p \in Z(I(A))$ and $I(h \cdot J)$ (for $h\in C_G(r)$) are all canonically isomorphic to $I(A)p$. Observe that this central projection $p$ is also necessarily $C_G(r)$-invariant. Moreover, the norm difference $\sup_{h \in C_G(r)}\norm{h\cdot u - u}$ from the statement of the theorem makes sense as a difference of unitaries in this corner, but it can also be viewed in significantly smaller C*-algebras by the fact that Remark~\ref{rem:multipliers_inclusion_essential_ideal} also tells us $M(J), \ M(h \cdot J) \subseteq M(J \cap h \cdot J)$, for all $h\in C_G(r)$.
Right off the bat, we notice that $k < 2$, and so $\norm{(\Ad u^* \circ \alpha_r)|_J - \id|_J} < 2$. Let $\wtilde{\alpha_r} \in \Aut(I(A))$ denote the unique extension of $\alpha_r$ to an automorphism on $I(A)$, and observe that $\wtilde{\alpha_r}|_{I(J)}$ is also the unique extension of $\wtilde{\alpha_r}|_{J^+}$ to $I(J)$, where $J^+ = C^*(J,p)$. By Lemma~\ref{lem:kadison_ringrose_unitary_injective_envelope}, we have that $(\Ad u^* \circ \wtilde{\alpha_r})|_{I(J)} = (\Ad v)|_{I(J)}$ for some unitary $v \in I(J)$ with $\operatorname{Re} \sigma(v) \subseteq [\frac{1}{2} \sqrt{4 - k^2},1]$. Using Lemma~\ref{lem:unitary_spectrum_to_norm}, we have that
\[ \norm{v - p} \leq \sqrt{2 - \sqrt{4 - k^2}}, \]
from which we also obtain the fact that for any $h \in C_G(r)$, we have
\[ \norm{h\cdot v - v} \leq \norm{h\cdot v - p} + \norm{v - p} = 2 \norm{v - p} \leq 2 \sqrt{2 - \sqrt{4 - k^2}}. \]
Also for convenience, we will let $w \coloneqq uv \in U(I(J))$, so that in particular, $\wtilde{\alpha_r}|_{I(J)} = (\Ad w)|_{I(J)}$. Finally, we obtain an estimate on $\norm{h \cdot w - w}$ for $h \in C_G(t)$:
\[ \norm{h \cdot w - w} \leq \norm{h \cdot u - u} \norm{h \cdot v} + \norm{u} \norm{h \cdot v - v} \leq 2 \sqrt{2 - \sqrt{4 - k^2}} + t. \]
The above norm estimate ends up being good enough for our purposes, namely to be able to ``average'' the translates and obtain something non-zero. Now we describe the averaging process itself. Consider the C*-algebra $\ell^\infty(G,I(A))$. It is equipped with a diagonal $G$-action given by $(s \cdot f)(g) = s \cdot f(s^{-1}g)$, for $s,g\in G$. Moreover, the embedding of $I(A) \hookrightarrow \ell^\infty(G,I(A))$ given by sending $y \in I(A)$ to the constant function $f_y(g) = y$, for all $g\in G$, is a $G$-equivariant *-homomorphic embedding.
To construct a $C_G(r)$-invariant ``average'' of $w$, we proceed as follows. Let $T \subseteq G$ be a transversal of the space of right coset space $C_G(r) \backslash G$. That is, every $g_1 \in G$ can be written \emph{uniquely} as $g_1 = h g_2$ for some $h \in C_G(r)$ and $g_2 \in T$. Set $W \in \ell^\infty(G,I(A))$ by $W(h g_2) = h \cdot w$ for every $h \in C_G(r)$ and $g_2 \in T$. It is not hard to check that $W$ is $C_G(r)$-invariant. Moreover, as $\wtilde{\alpha_r}|_{I(J)} = (\Ad w)|_{I(J)}$, and this automorphism is $C_G(r)$-invariant, it follows that $\Ad W$ also implements the diagonal automorphism induced by $r$ on $I(J) \subseteq I(A) \subseteq \ell^\infty(G,I(A))$.
Now we wish to push $W$ back to $I(A)$. We could use an expectation back down onto this C*-algebra, but this would lose us $C_G(r)$-invariance. Instead, we push it down onto the larger algebra $I_G(A)$, and use the far more roundabout techniques developed in the rest of the paper, where we are still able to obtain an element in $I(A)$ satisfying the same properties. Let $E \colon \ell^\infty(G,I(A)) \to I_G(A)$ be a $G$-equivariant unital completely positive map that is the identity map on $I(A)$. Given any $y \in I(J)$, we have
\[ E(W) y = E(Wy) = E(\wtilde{\alpha_r}(y)W) = \wtilde{\alpha_r}(y) E(W), \]
where we use the fact that $I(J)$ lies in the multiplicative domain of $E$. Moreover, by $G$-equivariance of $E$, we still have that $E(W)$ is $C_G(r)$-invariant.
It is important to verify that the element $E(W)$ that we obtain in $I_G(A)$ is sufficiently non-trivial for our purposes. Since $p \in Z(I(A))$, by Proposition~\ref{prop:injective_envelope_center_inclusions}, we have $p \in Z(I_G(A))$ as well. We aim to prove the following observation: $E(W)$ is a non-zero element in $I_G(A)p$.
To see that $E(W) \in I_G(A)p$, we remark that $p$ lies in the multiplicative domain of $E$ and $W$ lies entirely in $\ell^\infty(G,I(A))p$. Thus,
\[E(W)=E(Wp)=E(W)p\in I_G(A)p.\]
It remains to show that $E(W) \neq 0$. Recalling that $W$ was constructed as having some translates $h \cdot w$ (for $h \in C_G(t)$) in every coordinate, and $w$ was a unitary in the corner $I(J) = I(A)p$, it follows that $W$ is a unitary in the corner $\ell^\infty(G,I(A))p$. Moreover, on this corner, we have
\[ \norm{W - w} = \sup_{h \in C_G(r)} \norm{h \cdot w - w} \leq 2 \sqrt{2 - \sqrt{4 - k^2}} + t < \sqrt{2}. \]
Equivalently, $\norm{w^*W - p} < \sqrt{2}$. But by Lemma~\ref{lem:unitary_spectrum_positive_to_norm_sqrt2}, we have that $\operatorname{Re} \sigma(w^*W) \subseteq [\varepsilon,1]$ for some $\varepsilon > 0$, where the spectrum is considered inside of the C*-algebra $\ell^\infty(G,I(A))p$ and \emph{not} all of $\ell^\infty(G,I(A))$. By the continuous functional calculus, this is equivalent to having $\operatorname{Re} (w^*W) \geq \varepsilon\cdot p$ for this same value of $\varepsilon$. Thus, keeping in mind that $w$ lies in the multiplicative domain of $E$, we have
\[ 0 < \varepsilon\cdot p \leq E(\operatorname{Re} (w^*W)) = \operatorname{Re} E(w^*W)= \operatorname{Re} (w^*E(W)), \]
which immediately implies $E(W)\neq 0$.
To summarize, $E(W)$ was some non-zero element in $I_G(A)p$ implementing the action $\wtilde{\alpha_r}$ on $I(J) = I(A)p$. That is, given any $y \in I(J)$, we have
\[ E(W) y = \wtilde{\alpha_r}(y) E(W). \]
The importance having $E(W)$ in the corner $I_G(A)p$ is reflected in the fact that, if $y \in I(A)(1-p)$, then $E(W)y = 0$ and $\wtilde{\alpha_r}(y) E(W) = 0$. In other words, we may conclude that for any $y \in I(A)$, and not just $y \in I(J)$, we have
\[ E(W) y = \wtilde{\alpha_r}(y) E(W). \]
Our setup is now beginning to look quite similar to the proofs of the previous major theorems in this paper. For convenience, we will significantly simplify notation by letting $x_r \coloneqq E(W)$ and simply writing $r \cdot y$ instead of $\wtilde{\alpha_r}(y)$. In this language, we have an element $x_r \in I_G(A)$ with the property that for any $y \in I(A)$ we have
\[ x_r y = (r \cdot y) x_r. \]
Recall Proposition~\ref{prop:equivariant_unitaries_to_single_unitary}. There is nothing special, per se, about the element $x_r$ needing to be unitary, and the same proof will show that, because $s \cdot x_r = x_r$ for all $s \in C_G(r)$,
then defining $x_{srs^{-1}} \coloneqq s \cdot x_r$, for all $s\in G$ gives well-defined elements $(x_c)_{c\in C}$, where $C$ denotes the conjugacy class of $r$. Moreover, given any $c\in C$, we have $s \cdot x_c = x_{scs^{-1}}$, for all $s\in G$. A very similar proof to what is done in Proposition~\ref{prop:equivariant_unitaries_to_single_unitary} will also show that each element $x_{srs^{-1}}$ implements the automorphism corresponding to $srs^{-1}$, for $s\in G$. For completeness, we verify the details here. Given any $y \in I(A)$ and $s\in G$, we have
\begin{align*}
x_{srs^{-1}} y &= (s \cdot x_r) y = s \cdot (x_r (s^{-1} y)) \\
&= s \cdot ((rs^{-1} \cdot y) x_r) = (srs^{-1} \cdot y) (s \cdot x_r) = (srs^{-1} \cdot y) x_{srs^{-1}}.
\end{align*}
Consequently, we have that
\[ \sum_{c \in C} x_c^*\lambda_c \in (I(A) \rtimes_\lambda G)' \cap I_G(A) \rtimes_\lambda G. \]
The proof of the above fact is straightforward, and if you wish, it is essentially identical to the proof of Proposition~\ref{prop:crossed_product_center_coefficients}. We claim that this element in fact lies in the center of $I_G(A) \rtimes_\lambda G$. To see this, recall from Theorem~\ref{thm:crossed_product_injective_envelope_inclusions} the inclusions
\[ A \rtimes_\lambda G \subseteq I(A) \rtimes_\lambda G \subseteq I_G(A) \rtimes_\lambda G \subseteq I(A \rtimes_\lambda G). \]
The element $\sum_{c \in C} x_c^*\lambda_c$ in particular commutes with all of $A \rtimes_\lambda G$. However, this automatically implies that the element lies in the center of $I(A \rtimes_\lambda G)$ (see \cite[Corollary~4.3]{hamana79_injective_envelopes_cstaralg}), and so in particular it lies in $Z(I_G(A) \rtimes_\lambda G)$. Clearly, because $r$ was non-trivial, this element does \emph{not} lie in $I_G(A)$.
Now is when we split the proof of this direction into the two cases of when the group $G$ is an FC group, and when the action of $G$ on $A$ is minimal.
The easier case is the minimal case. Given that the element $\sum_{c\in C} x_c^*\lambda_c$ is a non-trivial element lying in the center of $I_G(A) \rtimes_\lambda G$, we have that $I_G(A) \rtimes_\lambda G$ is not prime, and we know that this is equivalent to $A \rtimes_\lambda G$ is not prime by Proposition~\ref{proposition:primeifffactor}. This is a case that we have already completely solved in Section~\ref{sec:primality_minimal}. In particular, we automatically get by Theorem~\ref{thm:DiscreteMinimal} that there exists some $r_2\in\FC(G) \setminus \set{e}$, some $C_G(r_2)$-invariant central projection $p_2 \in Z(I(A))$, and some $C_G(r_2)$-invariant unitary $u_2 \in I(A)p_2$ with the property that $r_2$ acts by $\Ad u_2$ on $I(A)p_2$. Again, we emphasize that there is no reason to expect these elements, in particular $r_2$ and $p_2$, to have anything to do with the elements $r$ and $p$ from before.
Next, consider the case where $G$ is an FC-group. Our aim is again to reduce it down to the results which we have already proven in Section~\ref{sec:intersection_property_fc}. The element $\sum_{c \in C} x_c^*\lambda_c$ that we have constructed above shows that the inclusion $Z(I_G(A))^G \subseteq Z(I_G(A) \rtimes_\lambda G)$ is proper.
Exactly as explained in the proof of Theorem~\ref{thm:mainSec4}, we can apply Lemma~\ref{lemma:stronglynonprime} and conclude that $(A,G)$ does not have the intersection property.
By Theorem~\ref{thm:mainSec4}, we have that there exists some $r_2\in G \setminus \set{e} = \FC(G) \setminus \set{e}$, some $C_G(r_2)$-invariant central projection $p_2 \in Z(I(A))$, and some $C_G(r_2)$-invariant unitary $u_2 \in I(A)p_2$ with the property that $r_2$ acts by $u_2$ on $I(A)p_2$. Once again, these could be completely unrelated to the previous $r$ and $p$.
Now we prove (\ref{thm:equivariant_proper_outer_equivalence:injective_envelope}) $\implies$ (\ref{thm:equivariant_proper_outer_equivalence:elliott}). Let $r\in \FC(G)$, let $p$ be a $C_G(r)$-invariant central projection in $I(A)$, and let $u$ a $C_G(r)$-invariant unitary in $I(A)p$ such that $r$ acts by $\Ad u$ on $I(A)p$. By Proposition~\ref{prop:ideal_injective_envelope}, $K = A \cap I(A)p$ is an $r$-invariant ideal of $A$ with the property that $I(K) = I(A)p$.
Now we show that, from $K$, we may obtain a smaller ideal that satisfies all of the properties, and importantly all of the norm estimates, that we want. We will start with some $\varepsilon > 0$, and occasionally at certain steps of the proof, mention that it can be made small enough to satisfy the properties that we want.
Let $\varepsilon > 0$. We know by Proposition~\ref{prop:elliott_strong_essential} that there exists an \emph{essential} $r$-invariant ideal $J \subseteq K$ and a unitary $v \in M(J)$ with the property that $\norm{\alpha_r|_J - (\Ad v)|_J} \leq \varepsilon$. Equivalently, we have
\[ \norm{(\Ad v^* \circ \alpha_r)|_J - \id_J} \leq \varepsilon. \]
We may stick with $\varepsilon < 2$, so that by Lemma~\ref{lem:kadison_ringrose_unitary_injective_envelope} combined with Lemma~\ref{lem:unitary_spectrum_to_norm}, there is a unitary $w \in I(J)$ with the property that $\Ad v^* \circ \wtilde{\alpha_r} = \Ad w$ on $I(J)$ and $\norm{w-p} < \delta(\varepsilon)$, where $\delta(\varepsilon) \to 0$ as $\varepsilon \to 0$. Again, the precise estimate is not that relevant for our purposes.
Given that $J$ was essential in $K$, we have by Proposition~\ref{prop:multiplier_algebra_contained_in_injective_envelope} and Proposition~\ref{prop:essential_ideal_containment_implies_inj_env} that $M(J) \subseteq I(J) = I(K)$. Hence, we may rewrite the above equality as $\Ad u = \Ad (vw)$ on $I(J)$. In other words, $u = \gamma vw$ for some $\gamma \in Z(I(K))$, and so $v = \gamma^* u w^*$. Our end goal is to show that, while $v$ itself might not be anywhere close to being $C_G(r)$-invariant, we may ``correct'' it by multiplying it with an appropriate central element (in some potentially smaller, but still essential, ideal of $K$). We already know that $w$ is close in norm to $p$, and is therefore approximately $C_G(r)$-invariant. Moreover, $u$ was $C_G(r)$-invariant by assumption. Thus, the only problematic element to deal with is $\gamma$.
To this end, recall that $\gamma \in Z(I(K))$, and thus there is no reason to expect that $\gamma$ lies in $M(J)$. However, we claim that we can approximate it by something that does lie in some potentially different multiplier algebra. Write $Z(I(K))$ as $C(X)$. Given that $I(K)$ is monotone complete, so is $C(X)$, and thus $X$ is extremally disconnected (see the end of Section~\ref{sec:preliminaries:monotone_complete}).
Using compactness of $X$, we can find a finite collection of pairs $(p_n,\beta_n)_{n=1}^{N}$, where each $p_n$ is a non-zero projection in $C(X)$, and each $\beta_n \in \mathbb{T}$ is a unimodular constant so that $\sum_{n=1}^{N}p_n=1_{C(X)}=p$ and $\norm{\gamma p_n - \beta_n p_n}\leq \varepsilon$ for all $n$.
For every $n\in\mathbb{N}$, let $L_n = A \cap I(A)p_n$. It is clear that the ideals $(L_n)_{n=1}^{N}$ are pairwise orthogonal. Letting $L$ be the ideal generated by all of them, we have $L = \operatorname{span} L_n$ and $L \cong \oplus_{n=1}^{N} L_n$. It is a subtle, but very important, point that all $L_n$ and therefore $L$ as well are $r$-invariant, due to the fact that $\wtilde{\alpha_r}$ was inner on $I(K)$, and therefore acts trivially on $Z(I(K)) = C(X)$. Note that $L \subseteq K$, as we had $K = A \cap I(A)p$. Moreover, letting $q$ be the support projection of $L$ in $I(A)$, we have $q \leq p$, and also $q \geq p_n$ for all $n$. It follows that $q = p$. Equivalently, $L$ is essential in $K$ by Proposition~\ref{prop:essential_ideal_containment_implies_inj_env}, and so $I(L) = I(K) = I(A)p$.
Note that $p_n\in M(L_n)\subseteq I(L_n)$ for every $n$, and moreover $M(L) \cong \prod_{n=1}^N M(L_n)$ by Lemma~\ref{lem:direct_sum_multiplier_algebra_injective_envelope}. In fact, viewing $M(L) \subseteq I(A)p$, this isomorphism maps each copy of $M(L_n)$ to its canonical copy in $I(A)p$ as well. Thus,
\[ f = \sum_{n=1}^N \beta_n p_n \]
is a unitary in $M(L)$. By construction, we have $f \in Z(I(K))$ and $\norm{f - \gamma} \leq \varepsilon$.
It is perhaps worthwhile to take a small break and to summarize the important points of our current setup. We have:
\begin{itemize}
\item $K \subseteq A$ was a $C_G(r)$-invariant ideal, with $r$ acting by $\Ad u$ on $I(K)$, and $u$ being $C_G(r)$-invariant as well.
\item $J \subseteq K$ was an $r$-invariant ideal that is \emph{essential} in $K$, and $v \in M(J) \subseteq I(K)$ was a unitary with $\norm{\alpha_r|_J - (\Ad v)|_J} \leq \varepsilon$.
\item $w \in I(K)$ was a unitary element with $\norm{w - p} \leq \delta(\varepsilon)$.
\item $\gamma \in Z(I(K))$ was a central unitary element so that $v = \gamma^* u w^*$.
\item $L \subseteq K$ was another $r$-invariant ideal, also essential in $K$, and $f \in M(L) \subseteq I(K)$ was a unitary element with $\norm{f - \gamma} \leq \varepsilon$. Moreover, $f \in Z(I(K))$.
\end{itemize}
With the above in hand, we are now ready to construct our approximately invariant ideal $I \subseteq A$, and approximately invariant unitary in $M(I)$. Let $I = J \cap L$ (and note that $I$ is in particular an ideal of $A$). This is $r$-invariant, as both $J$ and $L$ are $r$-invariant.
Moreover, given that $J$ and $L$ were both essential in $K$, we have by Remark~\ref{rem:intersection_of_essential_ideals} that $I$ is also essential in $K$. However, $K$ was $C_G(r)$-invariant, and thus $h \cdot I$ is still essential in $K$, for every $h\in C_G(r)$. Applying Remark~\ref{rem:intersection_of_essential_ideals} again, we see that $I \cap h \cdot I$ is essential in $K$ for every $h\in C_G(r)$. In particular, this intersection must be essential in the smaller ideals $I$ and $h \cdot I$ as well.
Now consider the equality $v = \gamma^* u w^*$. Our ``correction factor'' that we constructed for $I$ was the element $f\in M(L)$, which also belongs to $M(I)$ (applying Remark~\ref{rem:multipliers_inclusion_essential_ideal} and noting that $I$ is essential in $K$ and is therefore essential in $L$). Consider the new equality
\[ fv = f \gamma^* u w^*. \]
We have $\norm{f \gamma^* - p}\leq \varepsilon$, $\norm{w-p} \leq \delta(\varepsilon)$, and $u$ was completely $C_G(r)$-invariant to begin with. It follows that there is some new bound $\delta_2(\varepsilon)$ such that
\[ \sup_{h \in C_G(r)}\norm{h \cdot (fv) - fv} \leq \delta_2(\varepsilon), \]
where, while the precise bound $\delta_2(\varepsilon)$ is not important, what is important is that as $\varepsilon \to 0$, we also have $\delta_2(\varepsilon) \to 0$.
Given that $f \in Z(I(K))$, we have that $fv$ is a unitary element of $M(I)$ with the property that $\Ad (fv)$ and $\Ad v$ implement the same automorphism on $I(K)$. The inequality $\norm{\alpha_r|_J - (\Ad v)|_J} \leq \varepsilon$ therefore implies that
\[\norm{\alpha_r|_I - (\Ad (fv))|_I} \leq \varepsilon. \]
In summary, letting $\varepsilon$ be small enough, we have that
\[ k = \norm{\alpha_r|_I - (\Ad (fv))|_I} \text{ \ and \ } t = \sup_{h \in C_G(r)} \norm{h \cdot (fv) - fv} \]
are sufficiently small in order to satisfy $2 \sqrt{2 - \sqrt{4 - k^2}} + t < \sqrt{2}$.
\end{proof}
\end{document} |
\begin{document}
\begin{abstract}
We lay the groundwork in this first installment of a series of papers aimed at developing a theory of Hrushovski-Kazhdan style motivic integration for certain type of non-archimedean $o$\nobreakdash-minimal fields, namely power-bounded $T$-convex valued fields, and closely related structures. The main result of the present paper is a canonical homomorphism between the Grothendieck semirings of certain categories of definable sets that are associated with the $\VF$-sort and the $\RV$-sort of the language $\lan{T}{RV}{}$. Many aspects of this homomorphism can be described explicitly. Since these categories do not carry volume forms, the formal groupification of the said homomorphism is understood as a universal additive invariant or a generalized Euler characteristic. It admits, not just one, but two specializations to $\mathds{Z}$. The overall structure of the construction is modeled on that of the original Hrushovski-Kazhdan construction. \end{abstract}
\subjclass[2010]{12J25, 03C64, 14E18, 03C98}
\thanks{The research leading to the true claims in this paper has been partially supported by the ERC Advanced Grant NMNAG, the grant ANR-15-CE40-0008 (D\'efig\'eo), the SYSU grant 11300-18821101, and the NSSFC Grant 14ZDB015.}
\maketitle
\tableofcontents
\vskip 15mm
\section{Introduction}\label{intro}
Towards the end of the introduction of \cite{hrushovski:kazhdan:integration:vf} three hopes for the future of the theory of motivic integration are mentioned. We propose to investigate one of them in a series of papers: additive invariants and integration in $o$\nobreakdash-minimal valued fields. A prototype of such valued fields is $\mathds{R} \dpar{ t^{\mathds{Q}}}$, the generalized power series field over $\mathds{R}$ with exponents in $\mathds{Q}$. One of the cornerstones of the methodology of \cite{hrushovski:kazhdan:integration:vf} is $C$\nobreakdash-minimality, which is the right analogue of $o$\nobreakdash-minimality for algebraically closed valued fields and other closely related structures that epitomizes the behavior of definable subsets of the affine line. It, of course, fails in an $o$\nobreakdash-minimal valued field, mainly due to the presence of a total ordering. Thus the construction we seek has to be carried out in a different framework, which affords a similar type of normal forms for definable subsets of the affine line, a special kind of weak $o$\nobreakdash-minimality; this framework is van den Dries and Lewenberg's theory of $T$-convex valued fields \cite{DriesLew95, Dries:tcon:97}.
The reader is referred to the opening discussions in \cite{DriesLew95, Dries:tcon:97} for a more detailed introduction to $T$\nobreakdash-convexity and a summary of fundamental results. In those papers, how the valuation is expressed is somewhat inconsequential. In contrast, we shall work exclusively with a fixed two-sorted language $\lan{T}{RV}{}$ --- see \S~\ref{defn:lan} and Example~\ref{exam:RtQ} for a quick grasp of the central features of this language --- since such a language is a part of the preliminary setup of any Hrushovski-Kazhdan style integration.
Throughout this paper, let $T$ be a complete power-bounded $o$\nobreakdash-minimal \LT-theory extending the theory $\usub{\textup{RCF}}{}$ of real closed fields. For the real field $\mathds{R}$, the condition of being power-bounded is the same as that of being polynomially bounded. However, for nonarchimedean real closed fields, the former condition is more general and is indeed more natural.
The language $\lan{T}{}{}$ extends the language $\{<, 0, 1, +, -, \times\}$ of ordered rings. Let $\mdl R \coloneqq (R, <, \ldots)$ be a model of $T$. By definition, a $T$\nobreakdash-convex subring $\OO$ of $\mdl R$ is a convex subring of $\mdl R$ such that, for every definable (no parameters allowed) continuous function $f : R \longrightarrow R$, we have $f(\OO) \subseteq \OO$. The convexity of $\OO$ implies that it is a valuation ring of $\mdl R$. For instance, if $\mdl R$ is nonarchimedean and $\mathds{R} \subseteq R$ then the convex hull of $\mathds{R}$ forms a valuation ring of $\mdl R$ and, accordingly, the infinitesimals form its maximal ideal. Such a convex hull is $T$\nobreakdash-convex if no definable continuous function can grow so fast as to stretch the standard real numbers into infinity.
Let $\OO$ be a \emph{proper} $T$\nobreakdash-convex subring of $\mdl R$. The theory $T_{\textup{convex}}$ of the pair $(\mdl R, \OO)$, suitably axiomatized in the language $\lan{}{convex}{}$ that expands $\lan{T}{}{}$ with a new unary relation symbol, is complete, and if $T$ admits quantifier elimination and is universally axiomatizable then $T_{\textup{convex}}$ admits quantifier elimination as well.
Since $T$ is power-bounded, the definable subsets of $R$ afford a type of normal form, a special kind of weak $o$\nobreakdash-minimality (see \cite{mac:mar:ste:weako}), which we dub Holly normal form (since it was first studied by Holly in \cite{holly:can:1995}); in a nutshell, every definable subset of $R$ is a boolean combination of intervals and (valuative) discs. Clearly this is a natural generalization of $o$\nobreakdash-minimality in the presence of valuation. A number of desirable properties of definable sets in $R$ depends on the existence of such a normal form. For instance, every subset of $R$ defined by a principal type assumes one of the following four forms: a point, an open disc, a closed disc, and a half thin annulus, and, furthermore, these four forms are distinct in the sense that no definable bijection between any two of them is possible.
Let $\vv : R^{\times} \longrightarrow \Gamma$ be the valuation map induced by $\OO$, $\K$ the corresponding residue field, and $\res : \OO \longrightarrow \K$ the residue map. There is a canonical way of turning $\K$ into a model of $T$ as well, see \cite[Remark~2.16]{DriesLew95}. Let $\MM$ be the maximal ideal of $\OO$. Let $\RV = R^{\times} / (1 + \MM)$ and $\rv : R^{\times} \longrightarrow \RV$ be the quotient map. Note that, for each $a \in R$, the map $\vv$ is constant on the set $a + a\MM$, and hence there is an induced map $\vrv : \RV \longrightarrow \Gamma$. The situation is illustrated in the following commutative diagram \begin{equation*} \bfig
\square(0,0)/^{ (}->`->>`->>`^{ (}->/<600, 400>[\OO \smallsetminus \MM`R^{\times}`\K^{\times}` \RV;`\res`\rv`]
\morphism(600,0)/->>/<600,0>[\RV`\Gamma;\vrv]
\morphism(600,400)/->>/<600,-400>[R^{\times}`\Gamma;\vv] \efig \end{equation*} where the bottom sequence is exact. This structure may be expressed and axiomatized in a natural two-sorted first-order language $\lan{T}{RV}{}$, in which $R$ is referred to as the $\VF$-sort and $\RV$ is taken as a new sort. Informally, $\lan{T}{RV}{}$ is viewed as an extension of $\lan{}{convex}{}$.
We expand $(\mdl R, \OO)$ to an $\lan{T}{RV}{}$-structure. The main construction in this paper is carried out in such a setting. For concreteness, the reader is welcome to take $R = \mathds{R} \dpar{ t^{\mathds{Q}} }$ and $\OO = \mathds{R} \llbracket t^{\mathds{Q}} \rrbracket$ in the remainder of this introduction (see Example~\ref{exam:RtQ} below for more on this generalized power series field).
For a description of the ideas and the main results of the Hrushovski-Kazhdan style integration theory, we refer the reader to the original introduction in \cite{hrushovski:kazhdan:integration:vf} and also the introductions in \cite{Yin:int:acvf, Yin:int:expan:acvf}. There is also a quite comprehensive introduction to the same materials in \cite{hru:loe:lef} and, more importantly, a specialized version that relates the Hrushovski-Kazhdan style integration to the geometry and topology of Milnor fibers over the complex field. The method expounded there will be featured in a sequel to this paper as well. In fact, since much of the work below is closely modeled on that in \cite{hrushovski:kazhdan:integration:vf, Yin:special:trans, Yin:int:acvf, hru:loe:lef}, the reader may simply substitute the term ``theory of power-bounded $T$-convex valued fields'' for ``theory of algebraically closed valued fields'' or more generally ``$V$-minimal theories'' in those introductions and thereby acquire a quite good grip on what the results of this paper look like. For the reader's convenience, however, we shall repeat some of the key points, perhaps with minor changes here and there.
Let $\VF_*$ and $\RV[*]$ be two categories of definable sets that are respectively associated with the $\VF$-sort and the $\RV$-sort as follows. In $\VF_*$, the objects are the definable subsets of cartesian products of the form $\VF^n \times \RV^m$ and the morphisms are the definable bijections. On the other hand, for technical reasons (particularly for keeping track of ambient dimensions), $\RV[*]$ is formulated in a somewhat complicated way and is hence equipped with a gradation by ambient dimensions (see Definition~\ref{defn:c:RV:cat}).
The Grothendieck semigroup of a category $\mdl C$, denoted by $\gsk \mdl C$, is the free semigroup generated by the isomorphism classes of $\mdl C$, subject to the usual scissor relation $[A \smallsetminus B] + [B] = [A]$, where $[A]$, $[B]$ denote the isomorphism classes of the objects $A$, $B$ and ``$\smallsetminus$'' is certain binary operation, usually just set subtraction. Sometimes $\mdl C$ is also equipped with a binary operation --- for example, cartesian product --- that induces multiplication in $\gsk \mdl C$, in which case $\gsk \mdl C$ becomes a (commutative) semiring. The formal groupification of $\gsk \mdl C$, which is then a ring, is denoted by $\ggk \mdl C$.
The main construction of the Hrushovski-Kazhdan integration theory is a canonical --- that is, functorial in a suitable way --- homomorphism from the Grothendieck semiring $\gsk \VF_*$ of $\VF_*$ to the Grothendieck semiring $\gsk \RV[*]$ of $\RV[*]$ modulo a semiring congruence relation $\isp$ on the latter. In fact, it turns out to be an isomorphism. This construction has three main steps. \begin{enumerate}[{Step} 1.]
\item First we define a lifting map $\bb L$ from the set of objects of $\RV[*]$ into the set of objects of $\VF_*$ (Definition~\ref{def:L}). Next we single out a subclass of isomorphisms in $\VF_*$, which are called special bijections (Definition~\ref{defn:special:bijection}), and show that for any object $A$ in $\VF_*$ there is a special bijection $T$ on $A$ and an object $\bm U$ in $\RV[*]$ such that $T(A)$ is isomorphic to $\bb L \bm U$ (Corollary~\ref{all:subsets:rvproduct}). This implies that $\bb L$ is ``essentially surjective'' on objects, meaning that it is surjective on isomorphism classes of $\VF_*$. For this result alone we do not have to limit our means to special bijections. However, in Step~3 below, special bijections become an essential ingredient in computing the semiring congruence relation $\isp$.
\item We show that, for any two isomorphic objects $\bm U_1$, $\bm U_2$ of $\RV[*]$, their lifts $\bb L \bm U_1, \bb L \bm U_2$ in $\VF_*$ are isomorphic as well (Corollary~\ref{RV:lift}). This implies that $\bb L$ induces a semiring homomorphism
$
\gsk \RV[*] \longrightarrow \gsk \VF_*,
$
which is also denoted by $\bb L$. This homomorphism is surjective by Step~1 and hence, modulo the semiring congruence relation $\isp$ --- that is, the kernel of $\bb L$ --- the inversion $\int_+$ of the homomorphism $\bb L$ is an isomorphism of semirings.
\item A number of important properties of the classical integration can already be verified for $\int_+$ and hence, morally, this third step is not necessary. For applications, however, it is much more satisfying to have a precise description of the semiring congruence relation $\isp$. The basic notion used in the description is that of a blowup of an object in $\RV[*]$, which is essentially a restatement of the trivial fact that there is an additive translation from $1 + \MM$ onto $\MM$ (Definition~\ref{defn:blowup:coa}). We then show that, for any two objects $\bm U_1$, $\bm U_2$ in $\RV[*]$, there are isomorphic blowups $\bm U_1^{\flat}$, $\bm U_2^{\flat}$ of them if and only if $\bb L \bm U_1$, $\bb L \bm U_2$ are isomorphic (Proposition~\ref{kernel:L}). The ``if'' direction essentially contains a form of Fubini's theorem and is the most technically involved part of the construction. \end{enumerate} We call the semiring homomorphism $\int_+$ thus obtained a Grothendieck homomorphism. If the objects carry volume forms and the Jacobian transformation preserves the integral, that is, the change of variables formula holds, then it may be called a motivic integration; we will not consider this case here and postpone it to a future installment. When the semirings are formally groupified, this Grothendieck homomorphism is accordingly recast as a ring homomorphism, which is denoted by $\int$ and is understood as a (universal) additive invariant.
The structure of the Grothendieck ring $\ggk \RV[*]$ may be significantly elucidated. To wit, it can be expressed as a tensor product of two other Grothendieck rings $\ggk \RES[*]$ and $\ggk \Gamma[*]$, that is, there is an isomorphism of graded rings: \[ \bb D: \ggk \RES[*] \otimes_{\ggk \Gamma^{c}[*]} \ggk \Gamma[*] \longrightarrow \ggk \RV[*], \] where $\RES[*]$ is essentially the category of definable sets in $\mathds{R}$ (as a model of the theory $T$) and $\Gamma[*]$ is essentially the category of definable sets over $\mathds{Q}$ (as an $o$\nobreakdash-minimal group), both are graded by ambient dimension, and $\Gamma^{c}[*]$ is the full subcategory of $\Gamma[*]$ of finite objects, whose Grothendieck ring admits a natural embedding into $\ggk \RES[*]$ as well. This isomorphism results in various retractions from $\ggk \RV[*]$ into $\ggk \RES[*]$ or $\ggk \Gamma[*]$ and, when combined with the Grothendieck homomorphism $\int$ and the two Euler characteristics in $o$\nobreakdash-minimal groups (one is a truncated version of the other), yield a (generalized) Euler characteristic \[ \textstyle \Xint{\textup{G}} : \ggk \VF_* \to^{\sim} ( \mathds{Z} \oplus \bigoplus_{i \geq 1} (\mathds{Z}[Y]/(Y^2+Y))X^i) / (1 + 2YX + X), \] which is actually an isomorphism, and two specializations to $\mathds{Z}$: \[ \textstyle \Xint{\textup{R}}^g, \Xint{\textup{R}}^b: \ggk \VF_* \longrightarrow \mathds{Z}, \] determined by the assignments $Y \longmapsto -1$ and $Y \longmapsto 0$ or, equivalently, $X \longmapsto 1$ and $X \longmapsto -1$ (see Proposition~\ref{prop:eu:retr:k} and Theorem~\ref{thm:ring}). We will demonstrate the significance of these two specializations, as opposed to only one, in a future paper that is dedicated to the study of generalized (real) Milnor fibers in the sense of \cite{hru:loe:lef}.
For certain purposes, the difference between model theory and algebraic geometry is somewhat easier to bridge if one works over the complex field, as is demonstrated in \cite{hru:loe:lef}; however, over the real field, although they do overlap significantly, the two worlds seem to diverge in their methods and ideas. Our results should be understood in the context of ``$o$\nobreakdash-minimal geometry'' \cite{dries:1998, DrMi96} as opposed to real algebraic geometry. In general, the various Grothendieck rings considered in real algebraic geometry bring about lesser collapse of ``algebraic data'' --- since there are much less morphisms in the background --- and can yield much finer invariants, and hence are more faithful to the geometry in this regard, although the flip side of the story is that the invariants are often computationally intractable (especially when resolution of singularities is involved) and specializations are often needed in practice. For instance, the Grothendieck ring of real algebraic varieties may be specialized to $\mathds{Z}[X]$, which is called the virtual Poincar\'e polynomial (see \cite{mccrory:paru:virtual:poin}). Our method here does not seem to be suited for recovering invariants at this level, at least not directly.
The role of $T$-convexity in this paper cannot be overemphasized. However, it does not quite work if the exponential function is included in the theory $T$. It remains a worthy challenge to find a suitable framework in which the construction of this paper may be extended to that case.
Much of the content of this paper is extracted from the preprint \cite{Yin:int:tcvf}, which contains a more comprehensive study of $T$\nobreakdash-convex valued fields. This auxiliary part of the theory we are developing may be regarded as a sequel to or a variation on the themes of the work in \cite{DriesLew95, Dries:tcon:97}. It has become clear that some of the technicalities thereof may be of independent interest. For instance, the valuative or infinitesimal version of Lipschitz continuity plays a crucial role in proving the existence of Lipschitz stratifications in an arbitrary power-bounded $o$\nobreakdash-minimal field (this proof has been published in \cite{halyin} and the result cited there is Corollary~\ref{part:rv:cons}).
Also, in a future paper, we will use the main result here to show that, in both the real and the complex cases, the Euler characteristic of the topological Milnor fiber coincides with that of the motivic Milnor fiber, avoiding the algebro-geometric machinery employed in \cite[Remark~8.5.5]{hru:loe:lef}.
\section{Basic results in $T$-convex valued fields}
In this section, we first describe the two-sorted language $\lan{T}{RV}{}$ for $o$\nobreakdash-minimal valued fields and the $\lan{T}{RV}{}$-theory $\TCVF$. This theory is axiomatized. Then we show that $\TCVF$ admits quantifier elimination. Some of the results in \cite{DriesLew95, Dries:tcon:97} that are crucial for our construction are also translated into the present setting.
\subsection{Some notation}\label{subs:nota} Recall from the introduction above that $T$ is a complete power-bounded $o$\nobreakdash-minimal \LT-theory extending the theory $\usub{\textup{RCF}}{}$ of real closed fields.
\begin{conv} For the moment, by \emph{definable} we mean definable with arbitrary parameters from the structure in question. But later --- starting in \S~\ref{def:VF} --- we will abandon this practice and work with a fixed set of parameters. The reason for this change will be made abundantly clear when it happens. \end{conv}
\begin{defn}[Power-bounded]\label{defn.powBd} Suppose that $\mdl R$ is an $o$\nobreakdash-minimal real closed field. A \emph{power function} in $\mdl R$ is a definable endomorphism of the multiplicative group $\mdl R^+$. We say that $\mdl R$ is \emph{power-bounded} if every definable function $f \colon \mdl R \longrightarrow \mdl R$ is eventually dominated by a power function, that is, there exists a power function $g$ such that $\abs{f(x)} \le g(x)$ for all sufficiently large $x$. A complete $o$\nobreakdash-minimal theory extending $\usub{\textup{RCF}}{}$ is \emph{power-bounded} if all its models are. \end{defn}
All power functions in $\mdl R$ may be understood as functions of the form $x \longmapsto x^\lambda$, where $\lambda = \ddx f(1)$. The collection of all such $\lambda$ form a subfield and is called the \emph{field of exponents} of $\mdl R$. We will quote the results on power-bounded structures directly from \cite{DriesLew95, Dries:tcon:97} and hence do not need to know more about them other than the things that have already been said. At any rate, a concise and lucid account of the essentials may be found in \cite[\S~ 3]{Dries:tcon:97}.
\begin{rem}[Functional language]\label{rem:cont} We shall need a generality that is due to Lou van den Dries (private communication). It states that the theory $T$ can be reformulated in another language \emph{all} of whose primitives, except the binary relation $\leq$, are function symbols that are interpreted as \emph{continuous} functions in all the models of $T$. Actually, for this to hold, we only need to assume that $T$ is a complete $o$\nobreakdash-minimal theory that extends $\usub{\textup{RCF}}{}$.
More precisely, working in any model of $T$, it can be shown that all definable sets are boolean combinations of sets of the form $f(x) = 0$ or $g(x) > 0$, where $f$ and $g$ are definable total continuous functions. In particular, this holds in the prime model $\mdl P$ of $T$. Taking all definable total continuous functions in $\mdl P$ and the ordering $<$ as the primitives in a new language $\lan{T'}{}{}$, we see that $T$ can be reformulated as an \emph{equivalent} $\lan{T'}{}{}$-theory $T'$ in the sense that the syntactic categories of $T$ and $T'$ are naturally equivalent. In traditional but less general and more verbose model-theoretic jargon, this just says that if a model of $T$ is converted to a model of $T'$ in the obvious way then the two models are bi\"{i}nterpretable via the identity map, and vice versa.
The theory $T'$ also admits quantifier elimination, but it cannot be universally axiomatizable in $\lan{T'}{}{}$. To see this, suppose for contradiction that it can be. Then, by the argument in the proof of \cite[Corollary~2.15]{DMM94}, every definable function $f$ in a model of $T'$, in particular, multiplicative inverse, is given piecewise by terms. But all terms define total continuous functions. This means that, by $o$\nobreakdash-minimality, multiplicative inverse near $0$ is given by two total continuous functions, which is absurd.
Now, we may and do extend $T'$ by definitions so that it is universally axiomatizable in the resulting language. Thus every substructure of a model of $T'$ is actually a model of $T'$ and, as such, is an elementary substructure. In fact, since $T'$ has definable Skolem functions, we shall abuse notation slightly and redefine $T$ to be $T'^{\textup{df}}$, where $T'^{\textup{df}}$ is in effect a Skolemization of $T'$ (see \cite[\S\S~2.3--2.4]{DriesLew95} for further explanation). Note that the language of $T$ contains additional function symbols only and some of them must be interpreted in all models of $T$ as discontinuous functions for the reason given above.
To summarize, the main point is that $T$ admits quantifier elimination, is universally axiomatizable, is a definitional extension of $T'$, and all the primitives of $\lan{T'}{}{}$, except $\leq$, define continuous functions in all the models of $T'$.
The syntactical maneuver of passing through $T'$ just described will only be used in Theorem~\ref{thm:complete} below, and it is not really necessary if one works with a concrete $o$\nobreakdash-minimal extension of $\usub{\textup{RCF}}{}$ such as $\usub{T}{an}$ defined in \cite{DMM94} (also see Example~\ref{exam:RtQ}). \end{rem}
We shall work with a sufficiently saturated model $\mdl R \coloneqq (R, <, \ldots)$ of $T$ unless suggested otherwise. Its field of exponents is denoted by $\mathds{K}$.
\begin{nota}[Coordinate projections]\label{indexing} For each $n \in \mathds{N}$, let $[n]$ abbreviate the set $\{1, \ldots, n\}$. For any $E \subseteq [n]$, we write $\pr_E(A)$ for the projection of $A$ into the coordinates contained in $E$. In practice, it is often more convenient to use simple standard descriptions as subscripts. For example, if $E$ is a singleton $\{i\}$ then we shall always write $E$ as $i$ and $\tilde E \coloneqq [n] \smallsetminus E$ as $\tilde i$; similarly, if $E = [i]$, $\{k: i \leq k \leq j\}$, $\{k: i < k < j\}$, $\{\text{all the coordinates in the sort $S$}\}$, etc., then we may write $\pr_{\leq i}$, $\pr_{[i, j]}$, $\pr_{(i, j)}$, $\pr_{S}$, etc.; in particular, $A_{\VF}$ and $A_{\RV}$ stand for the projections of $A$ into the $\VF$-sort and $\RV$-sort coordinates, respectively.
Unless otherwise specified, by writing $a \in A$ we shall mean that $a$ is a finite tuple of elements (or ``points'') of $A$, whose length, denoted by $\lh(a)$, is not always indicated. If $a = (a_1, \ldots, a_n)$ then, for all $1 \leq i < j \leq n$, following the notational scheme above, $a_i$, $a_{\tilde i}$, $a_{\leq i}$, $a_{[i, j]}$, $a_{[i, j)}$, etc., are shorthand for the corresponding subtuples of $a$. We shall write $\{t\} \times A$, $\{t\} \cup A$, $A \smallsetminus \{t\}$, etc., simply as $t \times A$, $t \cup A$, $A \smallsetminus t$, etc., when no confusion can arise.
For $a \in \pr_{\tilde E} (A)$, the fiber $\{b : ( b, a) \in A \} \subseteq \pr_E(A)$ over $a$ is denoted by $A_a$. Note that, in the discussion below, the distinction between the two sets $A_a$ and $A_a \times a$ is usually immaterial and hence they may and often shall be tacitly identified. In particular, given a function $f : A \longrightarrow B$ and $b \in B$, the pullback $f^{-1}(b)$ is sometimes written as $A_b$ as well. This is a special case since functions are identified with their graphs. This notational scheme is especially useful when the function $f$ has been clearly understood in the context and hence there is no need to spell it out all the time. \end{nota}
\begin{nota}[Subsets and substructures]\label{nota:sub} By a definable set we mean a definable subset in $\mdl R$, and by a subset in $\mdl R$ we mean a subset in $R$, by which we mean a subset of $R^n$ for some $n$, unless indicated otherwise. Similarly for other structures or sets in place of $\mdl R$ that have been clearly understood in the context.
Often the ambient total ordering in $\mdl R$ induces a total ordering on a definable set $S$ of interest with a distinguished element $e$. Then it makes sense to speak of the positive and the negative parts of $S$ relative to $e$, which are denoted by $S^+$ and $S^-$, respectively. Also write $S^+_e$ for $S^+ \cup e$, etc. There may also be a natural absolute value map $S \longrightarrow S^+_e$, which is always denoted by $| \cdot |$; typically $S$ is a sort and $S^\times \coloneqq S \smallsetminus e$ is equipped with a (multiplicatively written) group structure, in which case the absolute value map is usually given as a component of a (splitting) short exact sequence \[ \pm 1 \longrightarrow S^\times \longrightarrow S^+ \quad \text{or} \quad S^+ \longrightarrow S^\times \longrightarrow \pm 1. \] Note that $e$ cannot be the identity element of $S^\times$. We will also write $A < e$ to mean that $A \subseteq S$ and $a < e$ for all $a \in A$, etc. If $\phi(x)$ is a formula then $\phi(x) < e$ denotes the subset of $S$ defined by the formula $\phi(x) \wedge x < e$.
Substructures of $\mdl R$ are written as $\mdl S \subseteq \mdl R$. As has been pointed out above, all substructures $\mdl S$ of $\mdl R$ are actually elementary substructures. If $A \subseteq \mdl R^n$ is a set definable with parameters coming from $\mdl S$ then $A(\mdl S)$ is the subset in $\mdl S$ defined by the same formula, that is, $A(\mdl S) = A \cap \mdl S^n$. Given a substructure $\mdl S \subseteq \mdl R$ and a set $A \subseteq \mdl R$, the substructure generated by $A$ over $\mdl S$ is denoted by $\langle \mdl S , A \rangle$ or $\mdl S \langle A \rangle$. Clearly $\langle \mdl S , A \rangle$ is the definable closure of $A$ over $\mdl S$. Later, we will expand $\mdl R$ and introduce more sorts and structures. In that situation we will write $\mdl S \langle A \rangle_T$ or $\langle \mdl S , A \rangle_T$ to emphasize that this is the \LT-substructure generated by $A$ over the \LT-reduct of $\mdl S$. \end{nota}
\begin{nota}[Topology] The default topology on $\mdl R$ is of course the order topology and the default topology on $\mdl R^n$ is the corresponding product topology. Given a subset $S$ in $\mdl R$, we write $\cl(S)$ for its topological closure, $\ito(S)$ for its interior, and $\partial S \coloneqq \cl(S) \setminus S$ for its frontier (not to be confused with the boundary $\cl(S) \setminus \ito(S)$ of $S$, which is also sometimes denoted by $\partial S$). The same topological discourse applies to a definable set if the ambient total ordering of $\mdl R$ induces a total ordering on it. \end{nota}
\subsection{The theory $\TCVF$}\label{defn:lan}
The language $\lan{T}{RV}{}$ for $o$\nobreakdash-minimal valued fields --- the theory $T$ may vary, of course --- has the following sorts and symbols: \begin{itemize}
\item A sort $\VF$, which uses the language $\lan{T}{}{}$.
\item A sort $\RV$, whose basic language is that of groups, written multiplicatively as $\{1, \times, {^{-1}} \}$, together with a constant symbol $0_{\RV}$ (for notational ease, henceforth this will be written simply as $0$).
\item A unary predicate $\K^{\times}$ in the $\RV$-sort. The union $\K^{\times} \cup \{0\}$ is denoted by $\K$, which is more conveniently thought of as a sort and, as such, employs the language $\lan{T}{}{}$ as well, where the constant symbols $0$, $1$ are shared with the $\RV$-sort.
\item A binary relation symbol $\leq$ in the $\RV$-sort.
\item A function symbol $\rv : \VF \longrightarrow \RV_0$. \end{itemize} We shall write $\RV$ to mean the $\RV$-sort without the element $0$, and $\RV_0$ otherwise, etc., although quite often the difference is immaterial.
\begin{defn}\label{defn:tcf} The axioms of the theory $\usub{\textup{TCVF}}{}$ of \emph{$T$-convex valued fields} in the language $\lan{T}{RV}{}$ are presented here informally. Many of them are clearly redundant as axioms, and we try to phrase some of these in such a way as to indicate so. The list also contains additional notation that will be used throughout the paper. \begin{enumerate}[({Ax.} 1)]
\item The \LT-reduct of the $\VF$-sort is a model of $T$.
Recall from Notation~\ref{nota:sub} that $\VF^+ \subseteq \VF$ is the subset of positive elements and $\VF^- \subseteq \VF$ the subset of negative elements.
\item \label{ax:rv} The quadruple $(\RV, 1, \times, {^{-1}})$ forms an abelian group. Inversion is augmented by $0^{-1} = 0$. Multiplication is augmented by $t \times 0 = 0 \times t = 0$ for all $t \in \RV$. The map $\rv : \VF^{\times} \longrightarrow \RV$ is a surjective group homomorphism augmented by $\rv(0) = 0$.
\item The binary relation $\leq$ is a total ordering on $\RV_{0}$ such that, for all $t, t' \in \RV_{0}$, $t < t'$ if and only if $\rv^{-1}(t) < \rv^{-1}(t')$.
The distinguished element $0 \in \RV_0$ is more aptly referred to as the \emph{middle element} of $\RV_{0}$. Clearly $\RV^+ = \rv(\VF^+)$ and $\RV^- = \rv(\VF^-)$ (see Notation~\ref{nota:sub}). It follows from (Ax.~\ref{ax:rv}) that $\RV^+$ is an ordered convex subgroup of $\RV$ and the quotient group $\RV / \RV^+$ is isomorphic to the group $\pm 1 \coloneqq \rv(\pm 1)$. This gives rise to an absolute value map on $\RV_{0}$, which is compatible with the absolute value map on $\VF$ in the sense that $\rv(\abs{a}) = \abs{\rv(a)}$ for all $a \in \VF$.
\item \label{ax:K} The set $\K^{\times}$ forms a \emph{nontrivial} subgroup of $\RV$ and the set $\K^{+} = \K^{\times} \cap \RV^+$ forms a convex subgroup of $\RV^+$.
The quotient groups $\RV / \K^+$, $\RV^{+} / \K^+$ are denoted by $\Gamma$, $\Gamma^+$ and the corresponding quotient maps by $\vrv$, $\vrv^+$. Also set $\vrv(0) = 0 \in \Gamma_0$. Since $\K^+$ is convex, $\Gamma^+$ is an ordered group, where the induced ordering is also denoted by $\leq$, and the absolute value map on $\RV_0$ descends to $\Gamma_0$ in the obvious sense.
\item Let $\leq^{-1}$ be the ordering on $\Gamma^+_0$ inverse to $\leq$ and
$\absG_{\infty} \coloneqq (\Gamma^+_0, +, \leq^{-1})$ the resulting \emph{additively} written ordered abelian group with the top element $\infty$. The composition
\[
\abval : \VF \to^{\rv} \RV_0 \to^{\abs{ \cdot}} \RV^+_0 \to^{\vrv^+} \Gamma^+_0
\]
is a (nontrivial) valuation with respect to the ordering $\leq^{-1}$, with valuation ring $\OO = \rv^{-1}(\RV^{\circ}_0)$ and maximal ideal $\MM = \rv^{-1}(\RV^{\circ\circ}_0)$, where, denoting $\vrv^+ \circ \abs{ \cdot}$ by $\abvrv$,
\begin{align*}
\RV^{\circ}_0 &= \{t \in \RV: 1 \leq^{-1} \abvrv(t) \}, \\
\RV^{\circ \circ}_0 &= \{t \in \RV: 1 <^{-1} \abvrv(t)\}.
\end{align*}
\item \label{ax:t:model} The $\K$-sort (recall that $\K$ is informally referred to as a sort) is a model of $T$ and, as a field, is the residue field of the valued field $(\VF, \OO)$.
The natural quotient map $\OO \longrightarrow \K$ is denoted by $\res$. For notational convenience, we extend the domain of $\res$ to $\VF$ by setting $\res(a) = 0$ for all $a \in \VF \smallsetminus \OO$. The following function is also denoted by $\res$:
\[
\RV \to^{\rv^{-1}} \VF \to^{\res} \K.
\]
\item \label{ax:tcon} ($T$-convexity). Let $f : \VF \longrightarrow \VF$ be a continuous function defined by an \LT-formula. Then $f(\OO) \subseteq \OO$.
\item Suppose that $\phi$ is an \LT-formula that defines a continuous function $f : \VF^m \longrightarrow \VF$. Then $\phi$ also defines a continuous function $\ol f : \K^m \longrightarrow \K$. Moreover, for all $a \in \OO^m$, we have $\res(f(a)) = \ol f(\res(a))$. \label{ax:match} \end{enumerate} \end{defn}
By (Ax.~\ref{ax:t:model}) and Remark~\ref{rem:cont}, (Ax.~\ref{ax:match}) can be simplified as: for all function symbols $f$ of $\lan{T'}{}{}$ and all $a \in \OO^m$, $\res(f(a)) = \ol f(\res(a))$. Then it is routine to check that, except the surjectivity of the map $\rv$ and the nontriviality of the value group $\abs{\Gamma}$ (this is an existential axiom and is actually expressed in (Ax.~\ref{ax:K})), $\TCVF$ is also universally axiomatized.
Let $\mdl S$ be a substructure of a model $\mdl M$ of $\TCVF$. We say that $\mdl S$ is \emph{$\VF$-generated} if $\RV_0(\mdl S) = \rv(\VF(\mdl S))$. Thus $\mdl S$ is indeed a model of $\TCVF$ if it is $\VF$-generated and $\Gamma(\mdl S)$ is nontrivial. At any rate, $\VF(\mdl S)$, $\res(\VF(\mdl S))$, and $\K(\mdl S)$ are all models of $T$.
For $A \subseteq \VF(\mdl M) \cup \RV(\mdl M)$, the substructure generated by $A$ over $\mdl S$ is denoted by $\langle \mdl S , A \rangle$ or $\mdl S \langle A \rangle$. Clearly $\VF(\langle \mdl S , A \rangle) = \langle \mdl S , A \rangle_T$ (see Notation~\ref{nota:sub}).
\begin{rem}\label{signed:Gam} Although the behavior of the valuation map $\abval$ in the traditional sense is coded in $\TCVF$, we shall work with the \emph{signed} valuation map, which is more natural in the present setting: \[ \vv : \VF \to^{\rv} \RVV \to^{\vrv} \GAA, \] where the ordering $\leq$ on the \emph{signed value group} $\Gamma_0$ no longer needs to be inverted. It is also tempting to use the ordering $\leq$ in the \emph{value group} $\abs{\Gamma}_{\infty}$ instead of its inverse, but this makes citing results in the literature a bit awkward. We shall actually abuse the notation and denote the ordering $\leq^{-1}$ in $\abs{\Gamma}_{\infty}$ also by $\leq$; this should not cause confusion since the ordering on $\Gamma_0$ will rarely be used (we will indicate so explicitly when it is used).
The axioms above guarantee that the ordered abelian group ${\GAA} /{\pm 1}$ (here $\vv(\pm 1)$ is just written as $\pm 1$) with the bottom element $0$ is isomorphic to $\abs{\Gamma}_{\infty}$ if either one of the orderings is inverted. So $\abval$ may be thought of as the composition $\vv/{\pm 1} : \VF \longrightarrow {\GAA} /{\pm 1}$. \end{rem}
\begin{conv}\label{how:gam} Semantically we shall treat the value group $\GAA$ as an imaginary sort. However, syntactically any reference to $\GAA$ may be eliminated in the usual way and we can still work with $\lan{T}{RV}{}$-formulas for the same purpose. \end{conv}
\begin{exam}\label{exam:RtQ} Here our main reference is \cite{DMM94}. A restricted analytic function $\mathds{R}^m \longrightarrow \mathds{R}$ is given on the cube $[-1, 1]^n$ by a power series in $n$ variables over $\mathds{R}$ that converges in a neighborhood of $[-1, 1]^n$, and $0$ elsewhere. Let $\lan{}{an}{}$ be the language that extends the language of ordered rings with a new function symbol for each restricted analytic function, $\usub{\mathds{R}}{an}$ the real field with its natural $\lan{}{an}{}$-structure, and $\usub{T}{an}$ the $\lan{}{an}{}$-theory of $\usub{\mathds{R}}{an}$. Obviously $\usub{T}{an}$ is polynomially bounded. More importantly, it is universally axiomatizable and admits quantifier elimination in a slightly enlarged language, and hence there is no longer any need to extend $\usub{T}{an}$ by definitions as we have arranged in \S~\ref{subs:nota}. (This language is of course more natural than a brute force definitional extension that achieves the same thing, but we do not really care what it is).
A generalized power series with coefficients in the field $\mathds{R}$ and exponents in the additive group $\mathds{Q}$ is a formal sum $x = \sum_{q \in \mathds{Q}} a_q t^q$ such that its support $\supp(x) = \{q \in \mathds{Q} : a_q \neq 0\}$ is well-ordered. Let $\mathds{R} \dpar{ t^{\mathds{Q}} }$, $K$ for short, be the set of all such series. Addition and multiplication in $K$ are defined in the expected way, and this makes $K$ a field, generally referred to as a Hahn field. We consider $\mathds{R}$ as a subfield of $K$ via the map $a \longmapsto at^0$. The map $ K^\times \longrightarrow \mathds{Q}$ given by $x \longmapsto \min\supp(x)$ is indeed a valuation. Its valuation ring $\mathds{R} \llbracket t^{\mathds{Q}} \rrbracket$, $\OO$ for short, consists of those series $x$ with $\min\supp(x) \geq 0$ and its maximal ideal $\MM$ of those series $x$ with $\min\supp(x) > 0$. Its residue field admits a section onto $\mathds{R}$ and hence is isomorphic to $\mathds{R}$. It is well-known that $(K, \OO)$ is a henselian valued field and $K$ is real closed. Restricted analytic functions may be naturally interpreted in $K$. According to \cite[Corollary~2.11]{DMM94}, with its naturally induced ordering, $K$ is indeed an elementary extension of $\usub{\mathds{R}}{an}$ and hence a model of $\usub{T}{an}$.
We turn $K$ into a model of $\TCVF$, with signed valuation, as follows. First of all, set $\RV = K^{\times} / (1 + \MM)$. Let $\rv : K^\times \longrightarrow \RV$ be the quotient map. The leading term of a series in $K^\times$ is its first term with nonzero coefficient. It is easy to see that two series $x$, $y$ have the same leading term if and only if $\rv(x) = \rv(y)$ and hence $\RV$ is isomorphic to the subgroup of $K^\times$ consisting of all the leading terms. There is a natural isomorphism $a_qt^q \longmapsto (q, a_q)$ from this latter group of leading terms to the group $\mathds{Q} \oplus \mathds{R}^\times$, through which we may identify $\RV$ with $\mathds{Q} \oplus \mathds{R}^\times$. Since $1 + \MM$ is a convex subset of $K^\times$, the total ordering on $K^\times$ induces a total ordering $\leq$ on $\RV$. This ordering $\leq$ is the same as the lexicographic ordering on $\mathds{Q} \oplus \mathds{R}^+$ or $\mathds{Q} \oplus \mathds{R}^-$ via the identification just made.
Let $\mathds{R}^{+}$ be the multiplicative group of the positive reals and $\RV^{+} = \mathds{Q} \oplus\mathds{R}^+ $. Observe that $\mathds{R}^{+}$ is a convex subgroup of $\RV$. The quotient group $\Gamma \coloneqq (\mathds{Q} \oplus \mathds{R}^\times) / \mathds{R}^{+}$ is naturally isomorphic to the subgroup $\pm e^{\mathds{Q}} \coloneqq e^{\mathds{Q}} \cup - e^{\mathds{Q}}$ of $\mathds{R}^\times$ so that $\mathds{Q}$ is identified with $e^\mathds{Q}$ via the map $q \longmapsto e^q$. Adding a new symbol $\infty$ to $\RV$, now it is routine to interpret $K$ as an $\lan{T}{RV}{}$-structure, with $T = \usub{T}{an}$ and the signed valuation given by \[ x \longmapsto \rv(x) = (q, a_q) \longmapsto \sgn(a_q)e^{-q}, \] where $\sgn(a_q)$ is the sign of $a_q$. It is also a model of $\TCVF$: all the axioms are more or less immediately derivable from the valued field structure, except (Ax.~\ref{ax:tcon}), which holds since $\usub{T}{an}$ is polynomially bounded, and (Ax.~\ref{ax:match}), which follows from \cite[Proposition~2.20]{DriesLew95}. \end{exam}
\subsection{Quantifier elimination}
Recall from \S~\ref{intro} that $T_{\textup{convex}}$ is the $\lan{}{convex}{}$-theory of pairs $(\mdl R, \OO)$ with $\mdl R \models T$ and $\OO$ a \emph{proper} $T$-convex subring. We may and shall view $T_{\textup{convex}}$ as the $\lan{}{convex}{}$-reduct of $\TCVF$.
\begin{thm}\label{tcon:qe} The theory $T_{\textup{convex}}$ admits quantifier elimination and is complete. \end{thm} \begin{proof} See \cite[Theorem~3.10, Corollary~3.13]{DriesLew95}. \end{proof}
That $\OO$ is a proper subring cannot be expressed by a universal axiom. Of course, we can always add a new constant symbol $\imath$ to $\lan{}{convex}{}$ and an axiom ``$\imath$ is in the maximal ideal'' to $T_{\textup{convex}}$ so that $T_{\textup{convex}}$ may indeed be formulated as a universal theory. In that case, every substructure of a model of $T_{\textup{convex}}$ is a model of $T_{\textup{convex}}$ and, moreover, $T_{\textup{convex}}$ has definable Skolem functions given by $\lan{T}{}{}(\imath)$-terms (this is an easy consequence of our assumption on $T$, quantifier elimination in $T_{\textup{convex}}$, and universality of $T_{\textup{convex}}$, as in \cite[Corollary~2.15]{DMM94}). We shall not implement this maneuver formally, even though the resulting properties may come in handy occasionally.
\begin{rem}\label{res:exp} According to \cite[Remark~2.16]{DriesLew95}, there is a natural way to expand the residue field $\K$ of the $T_{\textup{convex}}$-model $(\mdl R, \OO)$ to a $T$\nobreakdash-model as follows. Let $\mdl R' \subseteq \OO$ be a maximal subfield with respect to the property of being an elementary \LT-substructure of $\mdl R$. It follows that $\mdl R'$ is isomorphic to $\K$ as fields via the residue map $\res$. Then we can expand $\K$ to a $T$\nobreakdash-model so that the restriction $\res \upharpoonright \mdl R'$ becomes an isomorphism of \LT-structures. This expansion procedure does not depend on the choice of $\mdl R'$. \end{rem}
\begin{prop}\label{uni:exp} Every $T_{\textup{convex}}$-model expands to a unique $\TCVF$-model up to isomorphism. \end{prop} \begin{proof} Let $(\mdl R, \OO)$ be a $T_{\textup{convex}}$-model. It is enough to show that there is a canonical $\TCVF$-model expansion $(\mdl R, \RVV(\mdl R))$ of $(\mdl R, \OO)$, where $\mdl R$ is the $\VF$-sort, such that any other such expansion $(\mdl R, \RVV)$ is isomorphic to it. This canonical expansion is constructed as follows.
Let $\RV(\mdl R)$ be the quotient group $\mdl R^\times / (1 + \MM)$ and $\rv : \mdl R^\times \longrightarrow \RV(\mdl R)$ the quotient map. As in Example~\ref{exam:RtQ}, it is routine to convert the pair $(\mdl R, \RVV(\mdl R))$ into an $\lan{T}{RV}{}$-structure and check that it satisfies all the axioms in Definition~\ref{defn:tcf}, where (Ax.~\ref{ax:t:model}) is implied by the construction just described above. We shall refer to the obvious bijection between $(\mdl R, \RVV(\mdl R))$ and $(\mdl R, \RVV)$ as the identity map. This map commutes with all the primitives of $\lan{T}{RV}{}$ except, possibly, those in the $\K$-sort. This is where the syntactical maneuver in Remark~\ref{rem:cont} comes in. Recall that all the functional primitives of $\lan{T'}{}{}$ define continuous functions in all the models of $T'$ and $T$ is a definitional extension of $T'$. It follows from (Ax.~\ref{ax:match}) that the identity map indeed induces an \LT-isomorphism between the two $\K$-sorts. Thus the two expansions are isomorphic. \end{proof}
\begin{thm}\label{thm:complete} The theory $\TCVF$ is complete. \end{thm} \begin{proof} By Proposition~\ref{uni:exp}, every embedding between two $T_{\textup{convex}}$-models, which is necessarily elementary, expands uniquely to an $\lan{T}{RV}{}$-embedding between two $\TCVF$-models. This latter embedding is indeed elementary since $\TCVF$ admits quantifier elimination, which will be shown below. It follows that the theory $\TCVF$ is complete. But here we do not really need to go through that route. We can simply observe that, by the proof of Proposition~\ref{uni:exp}, $T_{\textup{convex}}$ and $\TCVF$ are equivalent in the sense mentioned in Remark~\ref{rem:cont}, and hence they are both complete if one of them is. \end{proof}
\begin{conv} From now on, we shall work in the model $\mmdl$ of $\TCVF$, which is the unique $\lan{T}{RV}{}$-expansion of the sufficiently saturated $T_{\textup{convex}}$-model $(\mdl R, \OO)$. We shall write $\VF(\mmdl)$ simply as $\VF$ or $\mdl R$, depending on the context, $\RVV(\mmdl)$ as $\RV_0$, etc. A subset in $\mmdl$ may simply be referred to as a set.
When we work in the \LT-reduct $\mdl R$ of $\mmdl$ instead of $\mmdl$, or just wish to emphasize that a set is definable in $\mdl R$ instead of $\mmdl$, the symbol ``$\lan{T}{}{}$'' or ``$T$'' will be inserted into the corresponding places in the terminology. \end{conv}
Let $\mdl S \subseteq \mdl R$ be a small substructure and $a, b \in \mdl R \smallsetminus \mdl S$ such that they make the same cut in (the ordering of) $\mdl S$. By $o$\nobreakdash-minimality, there is an automorphism $\sigma$ of $\mdl R$ over $\mdl S$ such that $\sigma(a) = b$.
Recall that the field of exponents of $\mdl R$ is denoted by $\mathds{K}$.
\begin{thm}\label{theos:qe} The theory $\TCVF$ admits quantifier elimination. \end{thm} \begin{proof} We shall run the usual Shoenfield test for quantifier elimination. To that end, let $\mdl M$ be a model of $\TCVF$, $\mdl S$ a substructure of $\mdl M$, and $\sigma : \mdl S \longrightarrow \mmdl$ an embedding. All we need to do is to extend $\sigma$ to an embedding $\mdl M \longrightarrow \mmdl$.
The construction is more or less a variation of that in the proof of \cite[Theorem~3.10]{Yin:QE:ACVF:min}. The strategy is to reduce the situation to Theorem~\ref{tcon:qe}. In the process of doing so, instead of the dimension inequality of the general theory of valued fields, the Wilkie inequality \cite[Corollary~5.6]{Dries:tcon:97} is used (see \cite[\S~3.2]{DriesLew95} for the notion of ranks of $T$-models). Note that, to use this inequality, we need to assume that $T$ is power-bounded.
Let $\mdl S_* = \langle \VF(\mdl S) \rangle$ and $t \in \RV(\mdl S) \smallsetminus \RV(\mdl S_*)$. Note that if such a $t$ does not exist then we have $\mdl S = \mdl S_*$ and its $\lan{}{convex}{}$-reduct is an $\lan{}{convex}{}$-substructure of the $\lan{}{convex}{}$-reduct of $\mdl M$, and hence an embedding as desired can be easily obtained by applying Theorem~\ref{tcon:qe} and Proposition~\ref{uni:exp}. Let $a \in \VF(\mdl M)$ with $\rv(a) = t$ and $b \in \VF$ with $\rv(b) = \sigma(t)$. Observe that, according to $\sigma$, $a$ and $b$ must make the same cut in $\VF(\mdl S)$ and $\VF(\sigma(\mdl S))$, respectively, and hence there is an \LT-isomorphism \[ \bar \sigma : \langle \mdl S_*, a \rangle_T \longrightarrow \langle \sigma(\mdl S_*), b \rangle_T \] with $\bar \sigma(a) = b$ and $\bar \sigma \upharpoonright \VF(\mdl S) = \sigma \upharpoonright \VF(\mdl S)$. We shall show that $\bar \sigma$ expands to an isomorphism between $\langle \mdl S_*, a \rangle$ and $\langle \sigma(\mdl S_*), b \rangle$ that is compatible with $\sigma$.
Case (1): There is an $a_1 \in \langle \mdl S_*, a \rangle_T$ such that \[ \abs{\OO(\mdl S_*) } < a_1 < \abs{ \VF(\mdl S_*) \smallsetminus \OO(\mdl S_*) }. \] Set $\abs{\Gamma}(\mdl S_*) = G$. Since $\OO(\langle\mdl S_*, a \rangle)$ is $T$-convex, by \cite[Lemma~5.4]{Dries:tcon:97} and \cite[Remark~3.8]{DriesLew95}, \begin{itemize}
\item either $a_1 \in \OO(\langle\mdl S_*, a \rangle)$ and $\absG(\langle \mdl S_*, a \rangle) = G$ or
\item $a_1 \notin \OO(\langle\mdl S_*, a \rangle)$ and $\absG(\langle \mdl S_*, a \rangle) \cong G \oplus \mathds{K}$. \end{itemize} By the Wilkie inequality, if \[ \absG(\langle \mdl S_*, a \rangle) \cong G \oplus \mathds{K} \] then $\K(\langle \mdl S_*, a \rangle) = \K(\mdl S_*)$ and hence $\abvrv(t) \notin G$, which implies $\abvrv(\sigma(t)) \notin \sigma(G)$; conversely, if \[ \absG(\langle \sigma(\mdl S_*), b \rangle) \cong \sigma(G) \oplus \mathds{K} \] then $\abvrv(t) \notin G$. Therefore \[ \absG(\langle \mdl S_*, a \rangle) \cong G \oplus \mathds{K} \quad \text{if and only if} \quad \absG(\langle \sigma(\mdl S_*), b \rangle) \cong \sigma(G) \oplus \mathds{K}, \] which, by \cite[Remark~3.8]{DriesLew95}, is equivalent to saying that $a_1 \in \OO(\langle\mdl S_*, a \rangle)$ if and only if $\bar \sigma(a_1) \in \OO(\langle \mdl \sigma(\mdl S_*), b \rangle)$.
Subcase (1a): $a_1 \in \OO(\langle \mdl S_*, a \rangle)$. Subcase~(1a) of the proof of \cite[Theorem~3.10]{DriesLew95} shows that $\bar \sigma$ expands to an $\lan{}{convex}{}$-isomorphism and hence to an $\lan{T}{RV}{}$-isomorphism, which is also denoted by $\bar \sigma$. Since $\absG(\langle \mdl S_*, a \rangle) = G$, we may assume $t \in \K(\mdl M)$. By the Wilkie inequality, $\K(\langle \mdl S_*, a \rangle)$ is precisely the $T$-model generated by $t$ over $\K(\mdl S_*)$. So $\RV(\langle \mdl S_*, a \rangle) = \langle \RV(\mdl S_*), t \rangle$ and \[ \bar \sigma \upharpoonright \RV(\langle \mdl S_*, a \rangle) = \sigma \upharpoonright \RV(\langle \mdl S_*, a \rangle). \]
Subcase (1b): $a_1 \notin \OO(\langle \mdl S_*, a \rangle)$. As above, Subcase~(1b) of the proof of \cite[Theorem~3.10]{DriesLew95} shows that $\bar \sigma$ expands to an $\lan{T}{RV}{}$-isomorphism and this time $\K(\langle \mdl S_*, a \rangle) = \K(\mdl S_*)$. Again it is clear that \[ \bar \sigma \upharpoonright \RV(\langle \mdl S_*, a \rangle) = \sigma \upharpoonright \RV(\langle \mdl S_*, a \rangle). \]
Case (2): Case (1) fails. Then there is also no $b_1 \in \langle \sigma(\mdl S_*), b \rangle_T$ such that \[ \abs{ \OO(\sigma(\mdl S_*)) } < b_1 < \abs{ \VF(\sigma(\mdl S_*)) \smallsetminus \OO(\sigma(\mdl S_*)) }. \] Using Case~(2) of the proof of \cite[Theorem~3.10]{DriesLew95}, compatibility between $\bar \sigma$ and $\sigma$ may be deduced as in Case (1) above.
Iterating this procedure, we may assume $\mdl S = \mdl S_*$. The theorem follows. \end{proof}
\begin{cor} For all set $A \subseteq \VF$, $\langle A \rangle$ is an elementary substructure of $\mmdl$ if and only if $\Gamma(\langle A \rangle)$ is nontrivial, that is, $\Gamma(\langle A \rangle) \neq \pm 1$. \end{cor}
\begin{cor}\label{trans:VF} Every parametrically $\lan{T}{RV}{}$-definable subset of $\VF^n$ is parametrically $\lan{}{convex}{}$-definable. \end{cor}
This corollary already follows from Proposition~\ref{uni:exp}. Anyway, it enables us to transfer results in the theory of $T$-convex valued fields \cite{DriesLew95, Dries:tcon:97} into our setting, which we shall do without further explanation.
We include here a couple of generalities on immediate isomorphisms. Their proofs are built on that of Theorem~\ref{theos:qe} and hence we shall skip some details.
\begin{defn} Let $\mdl M$, $\mdl N$ be substructures and $\sigma : \mdl M \longrightarrow \mdl N$ an $\lan{T}{RV}{}$-isomorphism. We say that $\sigma$ is an \emph{immediate isomorphism} if $\sigma(t) = t$ for all $t \in \RV(\mdl M)$. \end{defn}
Note that if $\sigma$ is an immediate isomorphism then, \emph{ex post facto}, $\RV(\mdl M) = \RV(\mdl N)$.
\begin{lem}\label{imm:ext} Every immediate isomorphism $\sigma : \mdl M \longrightarrow \mdl N$ can be extended to an immediate automorphism of $\mmdl$. \end{lem} \begin{proof} Let $\mdl M_* = \langle \VF(\mdl M) \rangle$ and $\mdl N_* = \langle \VF(\mdl N) \rangle$. Let $t \notin \RV(\mdl M_*)$ and $a \in \rv^{-1}(t)$. Since $\sigma$ is immediate, $a$ makes the same cut in $\VF(\mdl M)$ and $\VF(\mdl N)$ according to $\sigma$. By the proof of Theorem~\ref{theos:qe}, $\sigma$ may be extended to an immediate isomorphism $\langle \mdl M, a \rangle \longrightarrow \langle \mdl N, a \rangle$. Iterating this procedure, we reach a stage where the assertion simply follows from Theorem~\ref{tcon:qe}. \end{proof}
We have something much stronger. For that, the following crucial property is needed.
\begin{prop}[Valuation property]\label{val:prop} Let $\mdl M$ be a $\VF$-generated substructure and $a \in \VF$. Suppose that $\Gamma(\langle \mdl M, a \rangle) \neq \Gamma(\mdl M)$. Then there is a $d \in \VF(\mdl M)$ such that $\vv(a - d) \notin \vv(\mdl M)$. \end{prop} \begin{proof} For the polynomially bounded case, see~\cite[Proposition~9.2]{DriesSpei:2000} and the remark thereafter. Apparently this is established in full generality (power-bounded) in \cite{tyne}, which is in a repository that is password-protected. \end{proof}
\begin{lem}\label{imm:iso} Let $\sigma : \mdl M \longrightarrow \mdl N$ be an immediate isomorphism. Let $a \in \VF \smallsetminus \VF(\mdl M)$ and $b \in \VF \smallsetminus \VF(\mdl N)$ such that $\rv(a - c) = \rv(b -\sigma(c))$ for all $c \in \VF(\mdl M)$. Then $\sigma$ may be extended to an immediate isomorphism $\bar \sigma : \langle \mdl M, a \rangle \longrightarrow \langle \mdl N, b \rangle$ with $\bar \sigma(a) = b$. \end{lem}
Observe that, since every element of $\VF(\langle \mdl M, a \rangle) = \langle \mdl M, a \rangle_T$ is of the form $f(a, c)$, where $c \in \VF(\mdl M)$ and $f$ is a function symbol of $\lan{T}{}{}$, and similarly for $\langle \mdl N, b \rangle$, the lemma is equivalent to saying that $\rv(a - c) = \rv(b -\sigma(c))$ for all $c \in \VF(\mdl M)$ implies $\rv(f(a,c)) = \rv(f(b,\sigma(c)))$ for all $c \in \VF(\mdl M)$ and all function symbols of $\lan{T}{}{}$.
\begin{proof} Without loss of generality, we may assume that $\mdl M$, $\mdl N$ are $\VF$-generated. According to $\sigma$, $a$ and $b$ must make the same cut respectively in $\VF(\mdl M)$ and $\VF(\mdl N)$, and hence there is an \LT-isomorphism $\bar \sigma : \langle \mdl M, a \rangle_T \longrightarrow \langle \mdl N, b \rangle_T$ with $\bar \sigma(a) = b$ that extends $\sigma \upharpoonright \VF(\mdl M)$. We shall first show that $\bar \sigma$ expands to an $\lan{T}{RV}{}$-isomorphism. There are two cases to consider, corresponding to the two cases in the proof of Theorem~\ref{theos:qe}.
Case (1): There is an $a' \in \langle \mdl M, a \rangle_T$ such that \[ \abs{ \OO(\mdl M)} < a' < \abs{ \VF(\mdl M) \smallsetminus \OO(\mdl M) }. \] Let $f$ be a function symbol of $\lan{T}{}{}$ and $c \in \VF(\mdl M)$ such that $f(a, c) = a'$. Let $b' = \bar \sigma(f(a, c))$. Then we also have \[ \abs{\OO(\mdl N)} < b' < \abs{ \VF(\mdl N) \smallsetminus \OO(\mdl N) }. \] If $a' \notin \OO(\langle \mdl M, a \rangle)$ then $\Gamma(\langle \mdl M, a \rangle) \neq \Gamma(\mdl M)$. By the valuation property, there is a $d \in \VF(\mdl M)$ such that $\vv(a - d) \notin \Gamma(\mdl M)$. Then $\vv(b - \sigma(d)) \notin \Gamma(\mdl N)$ and hence, by the Wilkie inequality, $\OO(\langle \mdl N, b\rangle)$ is the convex hull of $\OO(\mdl N)$ in $\langle \mdl N, b \rangle_T$. This implies $b' \notin \OO(\langle \mdl N, b \rangle)$. By symmetry and \cite[Remark~3.8]{DriesLew95}, we see that $a' \in \OO(\langle \mdl M, a \rangle)$ if and only if $b' \in \OO(\langle \mdl N, b \rangle)$, and hence \[ \bar \sigma(\OO(\langle \mdl M, a \rangle)) = \OO(\langle \mdl N, b \rangle). \]
Case (2): Case (1) fails. We may proceed exactly as in Case (2) of the proof of Theorem~\ref{theos:qe}.
This concludes our proof that $\bar \sigma$ expands to an $\lan{T}{RV}{}$-isomorphism.
Next, we show that $\bar \sigma$ is indeed immediate. If $\RV(\langle \mdl M, a \rangle) = \RV(\mdl M)$ then also $\RV(\langle \mdl N, b \rangle) = \RV(\mdl N)$, and there is nothing more to be done. So suppose $\RV(\langle \mdl M, a \rangle) \neq \RV(\mdl M)$. We claim that there is a $d \in \VF(\mdl M)$ such that $\rv(a - d) \notin \RV(\mdl M)$. We consider two (mutually exclusive) cases.
Case (1): $\Gamma(\langle \mdl M, a \rangle) \neq \Gamma(\mdl M)$. Then the valuation property gives such a $d$ directly.
Case (2): $\K(\langle \mdl M, a \rangle) \neq \K(\mdl M)$. Let $a'$ be as above. Let $\OO'$ be the $T$\nobreakdash-convex subring of $\VF(\mdl M)$ that does not contain $a'$, that is, $\OO'$ is the convex hull of $\OO(\mdl M)$ in $\langle \mdl M, a \rangle_T$. Let $\vv'$, $\Gamma'(\langle \mdl M, a \rangle)$ be the corresponding signed valuation map and signed value group. Then the valuation property yields a $d \in \VF(\mdl M)$ such that $\vv'(a - d) \notin \Gamma'(\mdl M)$. Since \[ \abs{\Gamma'}(\langle \mdl M, a \rangle) \cong \abs{\Gamma}(\mdl M) \oplus \mathds{K}, \] there is a $\gamma \in \abs{\Gamma}(\mdl M)$ such that (exactly) one of the following two relations hold: \begin{gather*} \abs{ \OO_\gamma(\mdl M)} < \abs{a- d} < \abs{ \VF(\mdl M) \smallsetminus \OO_\gamma(\mdl M) },\\ \abs{ \MM_\gamma(\mdl M)} < \abs{a- d} < \abs{ \VF(\mdl M) \smallsetminus \MM_\gamma(\mdl M) }, \end{gather*} where \[ \OO_\gamma = \{c \in \VF: \abval(c) \geq \gamma\} \quad \text{and} \quad \MM_\gamma = \{c \in \VF: \abval(c) > \gamma\}. \] It is not hard to see that, in either case, $\rv(a - d) \notin \RV(\mdl M)$.
Since $\rv(a - d) = \rv(b - \sigma(d)) \eqqcolon t$, by the Wilkie inequality, $\RV(\langle \mdl M, a \rangle) = \langle \RV(\mdl M), t \rangle$ and hence $\bar \sigma$ must be immediate. \end{proof}
\subsection{Fundamental structure of $T$-convex valuation}
We review some fundamental facts concerning the valuation in $\mmdl$. Additional notation and terminology are also introduced.
Recall \cite[Theorem~A]{Dries:tcon:97}: The structure of definable sets in the $\K$-sort is precisely that given by the theory $T$.
Recall \cite[Theorem~B]{Dries:tcon:97}: The structure of definable sets in the (imaginary) $\abs \Gamma$-sort is precisely that given by the $o$\nobreakdash-minimal theory of nontrivially ordered vector spaces over $\mathds{K}$. The structure of definable sets in the (imaginary) $\Gamma$-sort is the same one modulo the sign. In particular, every definable function in the $\Gamma$-sort is definably piecewise $\mathds{K}$-linear modulo the sign.
\begin{lem}\label{gk:ortho} If $f : \Gamma \longrightarrow \K$ is a definable function then $f(\K)$ is finite. Similarly, if $g : \K \longrightarrow \Gamma$ is a definable function then $g(\Gamma)$ is finite. \end{lem} \begin{proof} See \cite[Proposition~5.8]{Dries:tcon:97}. \end{proof}
Note that \cite[Theorem~B, Proposition~5.8]{Dries:tcon:97} require that $T$ be power-bounded.
\begin{nota}\label{gamma:what} Recall convention~\ref{how:gam}. There are two ways of treating an element $\gamma \in \Gamma$: as a point --- when we study $\Gamma$ as an independent structure, see Convention~\ref{how:gam} --- or a subset of $\mmdl$ --- when we need to remain in the realm of definable sets in $\mmdl$. The former perspective simplifies the notation but is of course dispensable.
We shall write $\gamma$ as $\gamma^\sharp$ when we want to emphasize that it is the set $\vrv^{-1}(\gamma)$ in $\mmdl$ that is being considered. More generally, if $I$ is a set in $\Gamma$ then we write $I^\sharp = \bigcup\{\gamma^\sharp: \gamma \in I\}$. Similarly, if $U$ is a set in $\RV$ then $U^\sharp$ stands for $\bigcup\{\rv^{-1}(t): t \in U\}$. \end{nota}
Since $\TCVF$ is a weakly $o$\nobreakdash-minimal theory (see \cite[Corollary~3.14]{DriesLew95} and Corollary~\ref{trans:VF}), we can use the dimension theory of \cite[\S~4]{mac:mar:ste:weako} in $\mmdl$.
\begin{defn} The \emph{$\VF$-dimension} of a definable set $A$, denoted by $\dim_{\VF}(A)$, is the largest natural number $k$ such that, possibly after re-indexing of the $\VF$-coordinates, $\pr_{\leq k}(A_t)$ has nonempty interior for some $t \in A_{\RV}$. \end{defn}
For all substructures $\mdl M$ and all $a \in \VF$, $\VF(\dcl_{\mdl M}( a)) = \langle \mdl M , a \rangle_T$, where $\dcl_{\mdl M}(a)$ is the definable closure of $a$ over $\mdl M$. This implies that the exchange principle with respect to definable closure --- or algebraic closure, which is the same thing since there is an ordering --- holds in the $\VF$-sort, because it holds for $T$\nobreakdash-models. Therefore, by \cite[\S~4.12]{mac:mar:ste:weako}, we may equivalently define $\dim_{\VF}(A)$ to be the maximum of the algebraic dimensions of the fibers $A_t$, $t \in A_{\RV}$.
Algebraic dimension is defined for (any sort of) any theory whose models have the exchange property with respect to algebraic closure, or more generally any suitable notion of closure. In the present setting, the algebraic dimension of a set $B \subseteq \VF^n$ that is definable over a substructure $\mdl M$ is just the maximum of the ranks of the $T$\nobreakdash-models $\langle \mdl M , b \rangle_T$, $b \in B$, relative to the $T$\nobreakdash-model $\VF(\mdl M)$ (again, see \cite[\S~3.2]{DriesLew95} for the notion of ranks of $T$-models). It can be shown that this does not depend on the choice of $\mdl M$.
Yet another way to define this notion of $\VF$-dimension is to imitate \cite[Definiton~4.1]{Yin:special:trans}, since we have:
\begin{lem}\label{altVFdim} If $\dim_{\VF}(A) = k$ then $k$ is the smallest number such that there is a definable injection $f: A \longrightarrow \VF^k \times \RV^l$. \end{lem} \begin{proof} This is immediate by a straightforward argument combining the exchange principle, Lemma~\ref{RV:no:point} below, and compactness.
Alternatively, we may just quote \cite[Theorem~4.11]{mac:mar:ste:weako}. \end{proof}
\begin{rem}[$\RV$-dimension and $\Gamma$-dimension]\label{rem:RV:weako} It is routine to verify that the axioms concerning only the $\RV$-sort are all universal except for the one asserting that $\K^{\times}$ is a proper subgroup, which is existential. These axioms amount to a weakly $o$\nobreakdash-minimal theory also and the exchange principle holds for this theory. Therefore, we can use the dimension theory of \cite[\S~4]{mac:mar:ste:weako} directly in the $\RV$-sort as well. We call it the $\RV$-dimension and the corresponding operator is denoted by $\dim_{\RV}$. Note that $\dim_{\RV}$ does not depend on parameters (see \cite[\S~4.12]{mac:mar:ste:weako}) and agrees with the $o$\nobreakdash-minimal dimension in the $\K$-sort (see \cite[\S~4.1]{dries:1998}) whenever both are applicable.
Similarly we shall use $o$\nobreakdash-minimal dimension in the $\Gamma$-sort and call it the $\Gamma$-dimension. The corresponding operator is denoted by $\dim_{\Gamma}$. \end{rem}
\begin{lem}\label{dim:cut:gam} Let $U \subseteq \RV^n$ be a definable set with $\dim_{\RV}(U) = k$. Then $\dim_{\RV}(U_{\gamma}) = k$ for some $\gamma \in \vrv(U)$. \end{lem}
Here $U_{\gamma}$ denotes the pullback of $\gamma$ along the obvious function $\vrv \upharpoonright U$, in line with the convention set in the last paragraph of Notation~\ref{indexing}.
\begin{proof} By \cite[Theorem~4.11]{mac:mar:ste:weako} we may assume $n=k$. Then, for some $\gamma \in \vrv(U)$, $U_{\gamma}$ contains an open subset of $\RV^n$. The lemma follows. \end{proof}
\begin{lem}\label{gam:red:K} Let $D \subseteq \Gamma^n$ be a definable set with $\dim_{\Gamma}(D) = k$. Then $D^\sharp$ is definably bijective to a disjoint union of finitely many sets of the form $(\K^+)^{n-k} \times D'^\sharp$, where $D' \subseteq \Gamma^k$. \end{lem} \begin{proof} Over a definable finite partition of $D$, we may assume that $D \subseteq (\Gamma^+)^n$ and the restriction $\pr_{\leq k} \upharpoonright D$ is injective. It follows from \cite[Theorem~B]{Dries:tcon:97} that the induced function $f : D_{\leq k} \longrightarrow D_{>k}$ is piecewise $\mathds{K}$-linear. Thus, for every $\gamma \in D_{\leq k}$ and every $t \in \gamma^\sharp$ there is a $t$-definable point in $f(\gamma)^\sharp$. The assertion follows. \end{proof}
Taking disjoint union of finitely many definable sets of course will introduce extra bookkeeping coordinates, but we shall suppress this in notation.
\begin{rem}[$o$\nobreakdash-minimal sets in $\RV$]\label{omin:res} The theory of $o$\nobreakdash-minimality, in particular its terminologies and notions, may be applied to a set $U \subseteq \RV^n$ such that $\vrv(U)$ is a singleton or, more generally, is finite. For example, we shall say that $U$ is a \emph{cell} if the multiplicative translation $U / u \subseteq (\K^+)^n$ of $U$ by some $u \in U$ is an $o$\nobreakdash-minimal cell (see \cite[\S~3]{dries:1998}); this definition does not depend on the choice of $u$. Similarly, the \emph{$o$\nobreakdash-minimal Euler characteristic} $\chi(U)$ of such a set $U$ is the $o$\nobreakdash-minimal Euler characteristic of $U / u$ (see \cite[\S~4.2]{dries:1998}). This definition may be extended to disjoint unions of finitely many (not necessarily disjoint) sets $U_i \subseteq \RV^n \times \Gamma^m$ such that each $\vrv(U_i)$ is finite. \end{rem}
\begin{thm}\label{groth:omin} Let $U$, $V$ be definable sets in $\RV$ with $\vrv(U)$, $\vrv(V)$ finite. Then there is a definable bijection between $U$ and $V$ if and only if \[ \dim_{\RV}(U) = \dim_{\RV}(V) \quad \text{and} \quad \chi(U) = \chi(V). \] \end{thm} \begin{proof} See \cite[\S~8.2.11]{dries:1998}. \end{proof}
\begin{defn}[Valuative discs]\label{defn:disc}
A set $\mathfrak{b} \subseteq \VF$ is an \emph{open disc} if there is a $\gamma \in |\Gamma|$ and a $b \in \mathfrak{b}$ such that $a \in \mathfrak{b}$ if and only if $\abval(a - b) > \gamma$; it is a \emph{closed disc} if $a \in \mathfrak{b}$ if and only if $\abval(a - b) \geq \gamma$. The point $b$ is a \emph{center} of $\mathfrak{b}$. The value $\gamma$ is the \emph{valuative radius} or simply the \emph{radius} of $\mathfrak{b}$, which is denoted by $\rad (\mathfrak{b})$. A set of the form $t^\sharp$, where $t \in \RV$, is called an \emph{$\RV$-disc} (recall Notation~\ref{gamma:what}).
A closed disc with a maximal open subdisc removed is called a \emph{thin annulus}.
A set $\mathfrak{p} \subseteq \VF^n \times \RV_0^m$ of the form $(\prod_{i \leq n} \mathfrak{b}_i) \times t$ is an (\emph{open, closed, $\RV$-}) \emph{polydisc}, where each $\mathfrak{b}_i$ is an (open, closed, $\RV$-) disc. The \emph{polyradius} $\rad(\mathfrak{p})$ of $\mathfrak{p}$ is the tuple $(\rad(\mathfrak{b}_1), \ldots, \rad(\mathfrak{b}_n))$, whereas the \emph{radius} of $\mathfrak{p}$ is $\min \rad(\mathfrak{p})$. If all the discs $\mathfrak{b}_i$ are of the same valuative radius then $\mathfrak{p}$ is referred to as a \emph{ball}.
The open and the closed polydiscs centered at a point $a \in \VF^n$ with polyradius $\gamma \in |\Gamma|^n$ are denoted by $\mathfrak{o}(a, \gamma)$ and $\mathfrak{c}(a, \gamma)$, respectively.
The \emph{$\RV$-hull} of a set $A$, denoted by $\RVH(A)$, is the union of all the $\RV$-polydiscs whose intersections with $A$ are nonempty. If $A$ equals $\RVH(A)$ then $A$ is called an \emph{$\RV$-pullback}. \end{defn}
The map $\abval$ is constant on a disc if and only if it does not contain $0$ if and only it is contained in an $\RV$-disc. If two discs have nonempty intersection then one of them contains the other. Many such elementary facts about discs will be used throughout the rest of the paper without further explanation.
\begin{nota}[The definable sort $\DC$ of discs]\label{disc:exp} At times it will be more convenient to work in the traditional expansion $\mdl R_{\rv}^{\textup{eq}}$ of $\mmdl$ by all definable sorts. However, for our purpose, a much simpler expansion $\mdl R_{\rv}^{\bullet}$ suffices. This expansion has only one additional sort $\DC$ that contains, as elements, all the open and closed discs (since each point in $\VF$ may be regarded as a closed disc of valuative radius $\infty$, for convenience, we may and occasionally do think of $\VF$ as a subset of $\DC$). Heuristically, we may think of a disc that is properly contained in an $\RV$-disc as a ``thickened'' point of certain stature in $\VF$. For each $\gamma \in \absG$, there are two additional cross-sort maps $\VF \longrightarrow \DC$ in $\mdl R_{\rv}^{\bullet}$, one sends $a$ to the open disc, the other to the closed disc, of radius $\gamma$ that contain $a$.
The expansion $\mdl R_{\rv}^{\bullet}$ can help reduce the technical complexity of our discussion. However, as is the case with the imaginary $\Gamma$-sort, it is conceptually inessential since, for the purpose of this paper, all allusions to discs as (imaginary) elements may be eliminated in favor of objects already definable in $\mmdl$.
Whether parameters in $\DC$ are used or not shall be indicated explicitly, if it is necessary. Note that it is redundant to include in $\DC$ discs centered at $0$, since they may be identified with their valuative radii.
For a disc $\mathfrak{a} \subseteq \VF$, the corresponding imaginary element in $\DC$ is denoted by $\code{\mathfrak{a}}$ when notational distinction makes the discussion more streamlined; $\code{\mathfrak{a}}$ may be heuristically thought of as the ``name'' of $\mathfrak{a}$. Conversely, a set $D \subseteq \DC$ is often identified with the set $\{\mathfrak{a} : \code \mathfrak{a} \in D\}$, in which case $\bigcup D$ denotes a subset of $\VF$. \end{nota}
\begin{nota}\label{nota:tor}
For each $\gamma \in |\Gamma|$, let $\MM_\gamma$ and $\OO_{\gamma}$ be the open and closed discs around $0$ with radius $\gamma$, respectively. Assume $\gamma \geq 0$. Let $\RV_{\gamma} = \VF^{\times} / (1 + \MM_\gamma)$, which is a subset of $\DC$. It is an abelian group and also inherits an ordering from $\VF^\times$. The canonical map $\VF^{\times} \longrightarrow \RV_{\gamma}$ is denoted by $\rv_{\gamma}$ and is augmented by $\rv_{\gamma}(0) = 0$.
If $\code \mathfrak{b} \in \DC$, $b \in \mathfrak{b}$, and $\rad(\mathfrak{b}) \leq \abval(b) + \gamma$ then $\mathfrak{b}$ is a union of discs of the form $\rv_{\gamma}^{-1}(\code \mathfrak{a})$. In this case, we shall abuse the notation slightly and write $\code \mathfrak{a} \in \mathfrak{b}$, $\mathfrak{b} \subseteq \RV_{\gamma}$, etc.
For each $\code \mathfrak{a} \in \RV_{\gamma}$, let $\tor (\code \mathfrak{a}) \subseteq \RV_{\gamma}$ be the $\code \mathfrak{a}$-definable subset such that $\rv^{-1}_{\gamma}(\tor (\code \mathfrak{a}))$ forms the smallest closed disc containing $\mathfrak{a}$. Set \begin{align*} \tor^{\times}(\code \mathfrak{a}) & = \tor (\code \mathfrak{a}) \smallsetminus \code \mathfrak{a},\\ \tor^+(\code \mathfrak{a}) &= \{t \in \tor(\code \mathfrak{a}): t > \code \mathfrak{a}\},\\
\tor^-(\code \mathfrak{a}) &= \{t \in \tor(\code \mathfrak{a}): t < \code \mathfrak{a}\}. \end{align*} If $\code \mathfrak{a} = (\code {\mathfrak{a}_1}, \ldots, \code {\mathfrak{a}_n})$ with $\code {\mathfrak{a}_i} \in \RV_{\gamma_i}$ then $\prod_i \tor(\code{\mathfrak{a}_i})$ is simply written as $\tor(\code \mathfrak{a})$; similarly for $\tor^{\times}(\code \mathfrak{a})$, $\tor^+(\code \mathfrak{a})$, etc.
If $\gamma = 0$ then we may, for all purposes, identify $\tor^{\times} (\code \mathfrak{a})$, $\tor (\code \mathfrak{a})$, etc., with $\tor^{\times}(\alpha) \coloneqq \abvrv^{-1}(\alpha) \subseteq \RV$, $\tor(\alpha) \coloneqq \tor^{\times}(\alpha) \cup \{0\}$, etc., where $\alpha = \rad(\mathfrak{a})$. \end{nota}
\begin{rem}[$\K$-torsors]\label{rem:K:aff} Let $\code \mathfrak{a} \in \RV_{\gamma}$ and $\alpha = \rad(\mathfrak{a})$. Since, via additive translation by $\code \mathfrak{a}$, there is a canonical $\code \mathfrak{a}$-definable order-preserving bijection \[ \aff_{\goedel{\mathfrak{a}}} :\tor(\code \mathfrak{a}) \longrightarrow \tor(\alpha), \] we see that $\code \mathfrak{a}$-definable subsets of $\tor(\code \mathfrak{a})^n$ naturally correspond to those of $\tor(\alpha)^n$. If there is an $\code \mathfrak{a}$-definable $t \in \tor^{\times}(\alpha)$ then, via multiplicative translation by $t$, this correspondence may be extended to $\code \mathfrak{a}$-definable subsets of $\tor(0)^n = \K^n$. More generally, for any $t \in \tor^{\times}(\alpha)$, the induced bijection $\tor(\code \mathfrak{a}) \longrightarrow \K$ is denoted by $\aff_{\goedel \mathfrak{a}, t}$. Consequently, $\tor(\code \mathfrak{a})$ may be viewed as a $\K$-torsor and, as such, is equipped with much of the structure of $\K$. \end{rem}
\begin{defn}[Derivation between $\K$-torsors]\label{rem:tor:der} Let $\code \mathfrak{a}$, $\alpha$ be as above. Let $\goedel \mathfrak{b} \in \RV_{\delta}$ and $\beta = \rad(\mathfrak{b})$. Let $f : \tor(\code \mathfrak{a}) \longrightarrow \tor(\code \mathfrak{b})$ be a function. We define the \emph{derivative} $\ddx f$ of $f$ at any point $\code \mathfrak{d} \in \tor(\code \mathfrak{a})$ as follows. Choose any $t \in \tor^{\times}(\alpha)$ and any $s \in \tor^{\times}(\beta)$. Consider the function \[ f_{\goedel \mathfrak{a}, \goedel \mathfrak{b}, t,s} : \K \to^{\aff^{-1}_{\goedel \mathfrak{a}, t}} \tor(\code \mathfrak{a}) \to^f \tor(\goedel \mathfrak{b}) \to^{\aff_{\goedel \mathfrak{b}, s}} \K. \] Put $r = \aff_{\goedel \mathfrak{a}, t}(\goedel \mathfrak{d})$ and suppose that $\frac{d}{dx} f_{\goedel \mathfrak{a}, \goedel \mathfrak{b}, t,s}(r) \in \K$ exists. Then we set \[ \tfrac{d}{d x} f(\goedel \mathfrak{d}) = s t^{-1} \tfrac{d}{d x} f_{\goedel \mathfrak{a}, \goedel \mathfrak{b}, t,s}(r) \in \tor(\beta - \alpha). \] It is routine to check that this construction does not depend on the choice of $\code \mathfrak{a}$, $\goedel \mathfrak{b}$, $t$, $s$ and hence the derivative $\ddx f(\goedel \mathfrak{d})$ is well-defined. \end{defn}
\begin{defn}[$\vv$-intervals] Let $\mathfrak{a}$, $\mathfrak{b}$ be discs, not necessarily disjoint. The subset $\mathfrak{a} < x < \mathfrak{b}$ of $\VF$, if it is not empty, is called an \emph{open $\vv$-interval} and is denoted by $(\mathfrak{a}, \mathfrak{b})$, whereas the subset \[ \{a \in \VF : \ex{x \in \mathfrak{a}, y \in \mathfrak{b}} ( x \leq a \leq y) \} \] if it is not empty, is called a \emph{closed $\vv$-interval} and is denoted by $[\mathfrak{a}, \mathfrak{b}]$. The other $\vv$-intervals $[\mathfrak{a}, \mathfrak{b})$, $(-\infty, \mathfrak{b}]$, etc., are defined in the obvious way, where $(-\infty, \mathfrak{b}]$ is a closed (or half-closed) $\vv$-interval that is unbounded from below.
Let $A$ be such a $\vv$-interval. The discs $\mathfrak{a}$, $\mathfrak{b}$ are called the \emph{end-discs} of $A$. If $\mathfrak{a}$, $\mathfrak{b}$ are both points in $\VF$ then of course we just say that $A$ is an interval and if $\mathfrak{a}$, $\mathfrak{b}$ are both $\RV$-discs then we say that $A$ is an $\RV$-interval. If $A$ is of the form $(\mathfrak{a}, \mathfrak{b}]$ or $[\mathfrak{b}, \mathfrak{a})$, where $\mathfrak{a}$ is an open disc and $\mathfrak{b}$ is the smallest closed disc containing $\mathfrak{a}$, then $A$ is called a \emph{half thin annulus} and the \emph{radius} of $A$ is $\rad(\mathfrak{b})$.
Two $\vv$-intervals are \emph{disconnected} if their union is not a $\vv$-interval. \end{defn}
Obviously the open $\vv$-interval $(\mathfrak{a}, \mathfrak{b})$ is empty if $\mathfrak{a}$, $\mathfrak{b}$ are not disjoint. Equally obvious is that a $\vv$-interval is definable over some substructure $\mdl S$ if and only if its end-discs are definable over $\mdl S$.
\begin{rem}[Holly normal form]\label{rem:HNF} By the valuation property Proposition~\ref{val:prop} and \cite[Proposition~7.6]{Dries:tcon:97}, we have an important tool called \emph{Holly normal form} \cite[Theorem~4.8]{holly:can:1995} (henceforth abbreviated as HNF); that is, every definable subset of $\VF$ is a unique union of finitely many definable pairwise disconnected $\vv$-intervals. This is obviously a generalization of the $o$\nobreakdash-minimal condition. \end{rem}
\section{Definable sets in $\VF$}\label{def:VF} From here on, we shall work with a fixed small substructure $\mdl S$ of $\mdl R_{\rv}$, also occasionally of $\mdl R_{\rv}^{\bullet}$ (primarily in this section). The conceptual reason for this move is that the Grothendieck rings in our main construction below change their meaning if the set of parameters changes. In particular, allowing all parameters trivializes the whole construction somewhat. For instance, every definable set will contain a definable point. Consequently, all Galois actions on the classes of finite definable sets are killed off, and this is highly undesirable for motivic integration in algebraically closed valued fields. Admittedly, this problem is not as severe in our setting. Anyway, we follow the practice in \cite{hrushovski:kazhdan:integration:vf}.
Note that $\mdl S$ is regarded as a part of the language now and hence, contrary to the usual convention in the model-theoretic literature, ``$\emptyset$-definable'' or ``definable'' only means ``$\mdl S$-definable'' instead of ``parametrically definable'' if no other qualifications are given. To simplify the notation, we shall not mention $\mdl S$ and its extensions in context if no confusion can arise. For example, the definable closure operator $\dcl_{\mdl S}$, etc., will simply be written as $\dcl$, etc.
For the moment we do not require that $\mdl S$ be $\VF$-generated or $\Gamma(\mdl S)$ be nontrivial. When we work in $\mdl R_{\rv}^{\bullet}$ --- either by introducing parameters of the form $\code \mathfrak{a}$ or the phrase ``in $\mdl R_{\rv}^{\bullet}$'' --- the substructure $\mdl S$ may contain names for discs that may or may not be definable from $\VF(\mdl S) \cup \RV(\mdl S)$.
\subsection{Definable functions and atomic open discs}
The structural analysis of definable sets in $\VF$ below is, for the most part, of a rather technical nature. One highlight is Corollary~\ref{part:rv:cons}. It is a crucial ingredient of the proof in \cite{halyin} that all definable closed sets in an arbitrary power-bounded $o$\nobreakdash-minimal field admit Lipschitz stratification.
\begin{conv}\label{topterm} Since apart from $\leq$ the language $\lan{T}{}{}$ only has function symbols, we may and shall assume that, in any $\lan{T}{RV}{}$-formula, every \LT-term occurs in the scope of an instance of the function symbol $\rv$. For example, if $f(x)$, $g(x)$ are \LT-terms then the formula $f(x) < g(x)$ is equivalent to $\rv(f(x) - g(x)) < 0$. The \LT-term $f(x)$ in $\rv(f(x))$ shall be referred to as a \emph{top \LT-term}. \end{conv}
We begin by studying definable functions between various regions of the structure.
\begin{lem}\label{Ocon} Let $f : \OO \longrightarrow \VF$ be a definable function. Then for some $\gamma \in \GAA$ and $a \in \OO$ we have $\vv(f(b)) = \gamma$ for all $b > a$ in $\OO$. \end{lem} \begin{proof} See \cite[Proposition~4.2]{Dries:tcon:97}. \end{proof}
Note that this is false if $T$ is not power-bounded.
A definable function $f$ is \emph{quasi-\LT-definable} if it is a restriction of an \LT-definable function (with parameters in $\VF(\mdl S)$, of course).
\begin{lem}\label{fun:suba:fun} Every definable function $f : \VF^n \longrightarrow \VF$ is piecewise quasi-\LT-definable; that is, there are a definable finite partition $A_i$ of $\VF^n$ and \LT-definable functions $f_i: \VF^n \longrightarrow \VF$ such that $f \upharpoonright A_i = f_i \upharpoonright A_i$ for all $i$. \end{lem} \begin{proof} By compactness, this is immediately reduced to the case $n = 1$. In that case, let $\phi(x, y)$ be a quantifier-free formula that defines $f$. Let $\tau_i(x, y)$ enumerate the top \LT-terms in $\phi(x, y)$. For each $a \in \VF$ and each $t_i(a, y)$, let $B_{a, i} \subseteq \VF$ be the characteristic finite subset of the function $t_i(a, y)$ given by $o$\nobreakdash-minimal monotonicity (see \cite[\S~3.1]{dries:1998}). It is not difficult to see that if $f(a) \notin \bigcup_i B_{a, i}$ then there would be a $b \neq f(a)$ such that \[ \rv(\tau_i(a, b)) = \rv(\tau_i(a, f(a))) \] for all $i$ and hence $\phi(a, b)$ holds, which is impossible since $f$ is a function. The lemma follows. \end{proof}
This lemma is just a variation of \cite[Lemma~2.6]{Dries:tcon:97}.
\begin{cor}[Monotonicity]\label{mono} Let $A \subseteq \VF$ and $f : A \longrightarrow \VF$ be a definable function. Then there is a definable finite partition of $A$ into $\vv$-intervals $A_i$ such that every $f \upharpoonright A_i$ is quasi-\LT-definable, continuous, and monotone (constant or strictly increasing or strictly decreasing). Consequently, each $f(A_i)$ is a $\vv$-interval. \end{cor} \begin{proof} This is immediate by Lemma~\ref{fun:suba:fun}, $o$\nobreakdash-minimal monotonicity, and HNF. \end{proof}
This corollary is a version of \cite[Corollary~2.8]{Dries:tcon:97}, slightly finer due to the presence of HNF.
\begin{cor}\label{uni:fun:decom} For the function $f$ in Corollary~\ref{mono}, there is a definable function $\pi : A \longrightarrow Â\RV^2$ such that, for each $t \in \RV^2$, $f \upharpoonright A_t$ is either constant or injective. \end{cor} \begin{proof} This follows easily from monotonicity. Also, the proof of \cite[Lemma~4.11]{Yin:QE:ACVF:min} still works. \end{proof}
\begin{lem}\label{RV:no:point} Given a tuple $t = (t_1, \ldots, t_n) \in \RV$, if $a \in \VF$ is $t$-definable then $a$ is definable. Similarly, for $\gamma = (\gamma_1, \ldots, \gamma_n) \in \Gamma$, if $t \in \RV$ is $\gamma$-definable then $t$ is definable. \end{lem} \begin{proof} The first assertion follows directly from Lemma~\ref{fun:suba:fun}. It can also be easily seen through an induction on $n$ with the trivial base case $n=0$. For any $b \in t_n^\sharp$, by the inductive hypothesis, we have $a \in \VF(\langle b \rangle)$. If $a$ were not definable then, by the exchange principle, we would have $b \in \VF(\langle a \rangle)$ and hence $t_n^\sharp \subseteq \VF(\langle a \rangle)$, which is impossible. The second assertion is similar, using the exchange principle in the $\RV$-sort (see Remark~\ref{rem:RV:weako}). \end{proof}
\begin{cor}\label{function:rv:to:vf:finite:image} Let $U \subseteq \RV^m$ be a definable set and $f : U \longrightarrow \VF^n$ a definable function. Then $f(U)$ is finite. \end{cor} \begin{proof} We may assume $n=1$. Then this is immediate by Lemma~\ref{RV:no:point} and compactness. \end{proof}
There is a more general version of Lemma~\ref{RV:no:point} that involves parameters in the $\DC$-sort:
\begin{lem}\label{ima:par:red} Let $\code \mathfrak{a} = (\code{\mathfrak{a}_1}, \ldots, \code{\mathfrak{a}_n}) \in \DC^n$. If $a \in \VF$ is $\code \mathfrak{a}$-definable then $a$ is definable. \end{lem} \begin{proof} We proceed by induction on $n$. Let $b \in \mathfrak{a}_n$ and $t \in \RV$ such that $\abvrv(t) = \rad(\mathfrak{a}_n)$. Then $a$ is $(\code {\mathfrak{a}_1}, \ldots, \code {\mathfrak{a}_{n-1}}, t, b)$-definable. By the inductive hypothesis and Lemma~\ref{RV:no:point}, we have $a \in \VF(\langle b \rangle)$. If $a$ were not definable then we would have $b \in \VF(\langle a \rangle)$ and hence $\mathfrak{a}_n \subseteq \VF(\langle a \rangle)$, which is impossible unless $\mathfrak{a}_n$ is a definable point in $\VF$. \end{proof}
\begin{lem}\label{open:K:con} In $\mdl R^{\bullet}_{\rv}$, let $\mathfrak{a} \subseteq \VF$ be a definable open disc and $f : \mathfrak{a} \longrightarrow \K$ a definable nonconstant function. Then there is a definable proper subdisc $\mathfrak{b} \subseteq \mathfrak{a}$ such that $f \upharpoonright (\mathfrak{a} \smallsetminus \mathfrak{b})$ is constant. \end{lem} \begin{proof} If $\mathfrak{b}_1$ and $\mathfrak{b}_2$ are two proper subdiscs of $\mathfrak{a}$ such that $f \upharpoonright (\mathfrak{a} \smallsetminus \mathfrak{b}_1)$ and $f \upharpoonright (\mathfrak{a} \smallsetminus \mathfrak{b}_2)$ are both constant then $\mathfrak{b}_1$ and $\mathfrak{b}_2$ must be concentric, that is, $\mathfrak{b}_1 \cap \mathfrak{b}_2 \neq \emptyset$, for otherwise $f$ would be constant. Therefore, it is enough to show that $f \upharpoonright (\mathfrak{a} \smallsetminus \mathfrak{b})$ is constant for some proper subdisc $\mathfrak{b} \subseteq \mathfrak{a}$. To that end, without loss of generality, we may assume that $\mathfrak{a}$ is centered at $0$. For each $\gamma \in \vv(\mathfrak{a}) \subseteq \Gamma$, by \cite[Theorem~A]{Dries:tcon:97} and $o$\nobreakdash-minimality, $f(\vv^{-1}(\gamma))$ contains a $\gamma$-definable element $t_{\gamma} \in \K$. By weak $o$\nobreakdash-minimality, $f(\vv^{-1}(\gamma)) = t_{\gamma}$ for all but finitely many $\gamma \in \vv(\mathfrak{a})$. Let $g : \vv(\mathfrak{a}) \longrightarrow \K$ be the definable function given by $\gamma \longrightarrow t_{\gamma}$. By Lemma~\ref{gk:ortho}, the image of $g$ is finite. The assertion follows. \end{proof}
Alternatively, we may simply quote \cite[Theorem~1.2]{jana:omin:res}.
\begin{defn} Let $D$ be a set of parameters. We say that a (not necessarily definable) nonempty set $A$ \emph{generates a (complete) $D$-type} if, for every $D$-definable set $B$, either $A \subseteq B$ or $A \cap B = \emptyset$. In that case, $A$ is \emph{$D$-type-definable} if no set properly contains $A$ and also generates a $D$-type. If $A$ is $D$-definable and generates a $D$-type, or equivalently, if $A$ is both $D$-definable and $D$-type-definable then we say that $A$ is \emph{$D$-atomic} or \emph{atomic over $D$}. \end{defn}
We simply say ``atomic'' when $D =\emptyset$.
In the literature, a type could be a partial type and hence a type-definable set may have nontrivial intersection with a definable set. In this paper, since partial types do not play a role, we shall not carry the superfluous qualifier ``complete'' in our terminology.
\begin{rem}[Taxonomy of atomic sets]\label{rem:type:atin} It is not hard to see that, by HNF, if $\mathfrak{i} \subseteq \VF$ is atomic then $\mathfrak{i}$ must be a $\vv$-interval. In fact, there are only four possibilities for $\mathfrak{i}$: a point, an open disc, a closed disc, and a half thin annulus. There are no ``meaningful'' relations between them, see Lemma~\ref{atom:type}. \end{rem}
\begin{lem}\label{atom:gam} In $\mdl R^{\bullet}_{\rv}$, let $\mathfrak{a}$ be an atomic set. Then $\mathfrak{a}$ remains $\gamma$-atomic for all $\gamma \in \Gamma$. Moreover, if $\mathfrak{a} \subseteq \VF^n$ is an open polydisc then it remains $\code \mathfrak{a}$-atomic. \end{lem} \begin{proof} The first assertion is a direct consequence of definable choice in the $\Gamma$-sort. For the second assertion, let $\gamma = \rad(\mathfrak{a})$. If $\mathfrak{a}$ were not $\code \mathfrak{a}$-atomic then, by compactness, there would be a $\gamma$-definable subset $A \subseteq \VF^n$ such that $A \cap \mathfrak{a}$ is nonempty and, for all open polydisc $\mathfrak{b}$ with $\gamma = \rad(\mathfrak{b})$, if $A \cap \mathfrak{b}$ is nonempty then it is a proper subset of $\mathfrak{b}$ --- this contradicts the first assertion that $\mathfrak{a}$ is $\gamma$-atomic. \end{proof}
Recall from \cite[Definition~4.5]{mac:mar:ste:weako} the notion of a cell in a weakly $o$\nobreakdash-minimal structure. In our setting, it is easy to see that, by HNF, we may require that the images of the bounding functions $f_1$, $f_2$ of a cell $(f_1, f_2)_A$ in the $\VF$-sort be contained in $\DC$; cell decomposition \cite[Theorem~4.6]{mac:mar:ste:weako} holds accordingly. Cells are in general not invariant under coordinate permutations; however, by cell decomposition, an atomic subset of $\VF^n$ must be a cell and must remain so under coordinate permutations.
\begin{lem}\label{open:rv:cons} In $\mdl R^{\bullet}_{\rv}$, let $\mathfrak{a} \subseteq \VF^n$ be an atomic open polydisc and $f : \mathfrak{a} \longrightarrow \VF$ a definable function. If $f$ is not constant then $f(\mathfrak{a})$ is an (atomic) open disc; in particular, $\rv \upharpoonright f(\mathfrak{a})$ is always constant. \end{lem} \begin{proof} By atomicity, $f(\mathfrak{a})$ must be an atomic $\vv$-interval. We proceed by induction on $n$. For the base case $n=1$, suppose for contradiction that $f(\mathfrak{a})$ is a closed disc (other than a point) or a half thin annulus. By monotonicity, we may assume that $f(\mathfrak{a})$ is, say, strictly increasing. Then $f^{-1}$ violates Lemma~\ref{Ocon}, contradiction.
For the case $n > 1$, suppose for contradiction again that $f(\mathfrak{a})$ is a closed disc or a half thin annulus. By the inductive hypothesis, for every $a \in \pr_{1}(\mathfrak{a})$ there is a maximal open subdisc $\mathfrak{b}_a \subseteq f(\mathfrak{a})$ that contains $f(\mathfrak{a}_a)$, similarly for every $a \in \pr_{>1}(\mathfrak{a})$. It follows that $f(\mathfrak{a})$ is actually contained in a maximal open subdisc of $f(\mathfrak{a})$, which is absurd. \end{proof}
\begin{cor}\label{poly:open:cons} Let $f : \VF^n \longrightarrow \VF$ be a definable function and $\mathfrak{a} \subseteq \VF^n$ an open polydisc. If $(\rv \circ f) \upharpoonright \mathfrak{a}$ is not constant then there is an $\code \mathfrak{a}$-definable nonempty proper subset of $\mathfrak{a}$. \end{cor}
Here is a strengthening of Lemma~\ref{atom:gam}:
\begin{lem}\label{atom:self} Let $B \subseteq \VF^n$ be \LT-type-definable and $\mathfrak{a} = \mathfrak{a}_1 \times \ldots \times \mathfrak{a}_n \subseteq B$ an open polydisc. Then, for all $a = (a_1, \ldots, a_n)$ and $b = (b_1, \ldots, b_n)$ in $\mathfrak{a}$, there is an immediate automorphism $\sigma$ of $\mmdl$ with $\sigma(a) = b$. Consequently, $\mathfrak{a}$ is $(\code \mathfrak{a}, t)$-atomic for all $t \in \RV$. \end{lem} \begin{proof} To see that the first assertion implies the second, suppose for contradiction that there is an $(\code \mathfrak{a}, t)$-definable nonempty proper subset $A \subseteq \mathfrak{a}$. Let $a \in A$, $b \in \mathfrak{a} \smallsetminus A$, and $\sigma$ be an immediate automorphism of $\mmdl$ with $\sigma(a) = b$. Then $\sigma$ is also an immediate automorphism of $\mmdl$ over $\langle \code \mathfrak{a}, t \rangle$, contradicting the assumption that $A$ is $(\code \mathfrak{a}, t)$-definable.
For the first assertion, by Lemma~\ref{imm:ext}, it is enough to show that there is an immediate isomorphism $\sigma : \langle a \rangle \longrightarrow \langle b \rangle$ sending $a$ to $b$. Write \[ \mathfrak{a}' = \mathfrak{a}_1 \times \ldots \times \mathfrak{a}_{n-1}, \quad a' = (a_1, \ldots, a_{n-1}), \quad b' = (b_1, \ldots, b_{n-1}). \] Then, by induction on $n$ and Lemma~\ref{imm:iso}, it is enough to show that, for any immediate isomorphism $\sigma' : \langle a' \rangle \longrightarrow \langle b' \rangle$ sending $a'$ to $b'$ and any \LT-definable function $f : \VF^{n-1} \longrightarrow \VF$, \[ \rv(a_n - f(a')) = \rv(b_n - \sigma'(f(a'))). \] This is clear for the base case $n=1$, since $\mathfrak{a}$ must be disjoint from $\VF(\mdl S)$. For the case $n > 1$, we choose an immediate automorphism of $\mmdl$ extending $\sigma'$, which is still denoted by $\sigma'$; this is possible by Lemma~\ref{imm:ext}. By the inductive hypothesis and Lemma~\ref{open:rv:cons}, $f(\mathfrak{a}') = f(\sigma'(\mathfrak{a}')) = \sigma'(f(\mathfrak{a}'))$ is either a point or an open disc. Since $B$ is \LT-type-definable, it follows that $f(\mathfrak{a}')$ must be disjoint from $\mathfrak{a}_n$ and hence the desired condition is satisfied. \end{proof}
\begin{cor}\label{part:rv:cons} Let $A \subseteq \VF^n$ and $f : A \longrightarrow \VF$ be an \LT-definable function. Then there is an \LT-definable finite partition $A_i$ of $A$ such that, for all $i$, if $\mathfrak{a} \subseteq A_i$ is an open polydisc then $\rv \upharpoonright f(\mathfrak{a})$ is constant and $f(\mathfrak{a})$ is either a point or an open disc. \end{cor} \begin{proof} For $a \in A$, let $D_a \subseteq A$ be the \LT-type-definable subset containing $a$. By Lemma~\ref{atom:self}, every open polydisc $\mathfrak{a} \subseteq D_a$ is $\code \mathfrak{a}$-atomic and hence, by Lemma~\ref{open:rv:cons}, the assertion holds for $\mathfrak{a}$. Then, by compactness, the assertion must hold in a definable subset $A_a \subseteq A$ that contains $a$; by compactness again, it holds in finitely many definable subsets $A_1, \ldots, A_m$ of $A$ with $\bigcup_i A_i = A$. Then the partition of $A$ generated by $A_1, \ldots, A_m$ is as desired. \end{proof}
\begin{rem}\label{rem:LT:com} Clearly the conclusion of Corollary~\ref{part:rv:cons} still holds if we replace ``\LT-definable'' with ``definable'' everywhere therein. Moreover, its proof works almost verbatim in all situations where we want to partition an \LT-definable set $A \subseteq \VF^n$ into finitely many \LT-definable pieces $A_i$ such that certain definable property, not necessarily \LT-definable, holds on every open polydisc (or other imaginary elements) contained in $A_i$. \end{rem}
Here is a variation of Lemma~\ref{atom:self}.
\begin{lem}\label{atom:exp} Let $\mathfrak{a} \subseteq \VF^n$ be an $\code \mathfrak{a}$-atomic open polydisc. Let $e \in \VF^{\times}$ with $\abval(e) \gg 0$ (here $\gg$ stands for ``sufficiently larger than''). Then $\mathfrak{a}$ is $(\code \mathfrak{a}, e)$-atomic. \end{lem} \begin{proof} The argument is somewhat similar to that in the proof of Lemma~\ref{atom:self}. We proceed by induction on $n$. Write $\mathfrak{a} = \mathfrak{a}_1 \times \ldots \times \mathfrak{a}_n$ and $\mathfrak{a}' = \mathfrak{a}_1 \times \ldots \times \mathfrak{a}_{n-1}$. Let $(a' , a_n)$ and $(b', b_n)$ be two points in $\mathfrak{a}' \times \mathfrak{a}_n$. By the inductive hypothesis and Lemma~\ref{atom:self}, there is an immediate isomorphism $\sigma' : \langle a', e \rangle \longrightarrow \langle b', e \rangle$ with $\sigma'(e) = e$ and $\sigma'(a') = b'$. Thus, it is enough to show that, for all \LT-definable function $f : \VF^{n} \longrightarrow \VF$, \[ \rv(a_n - f(e, a')) = \rv(b_n - \sigma'(f(e, a'))). \]
Suppose for contradiction that we can always find an $e \in \VF^{\times}$ that is arbitrarily close to $0$ such that $f(e, \mathfrak{a}') \cap \mathfrak{a}_n \neq \emptyset$ (this must hold for some such $f$, for otherwise we are already done by compactness); more precisely, by weak $o$\nobreakdash-minimality, without loss of generality, there is an open interval $(0, \epsilon) \subseteq \VF^+$ such that $f(e, \mathfrak{a}') \cap \mathfrak{a}_n \neq \emptyset$ for all $e \in (0, \epsilon)$.
For each $a' \in \mathfrak{a}'$, let $f_{a'}$ be the $a'$-\LT-definable function on $\VF^+$ given by $b \longmapsto f(b, a')$. By $o$\nobreakdash-minimal monotonicity, there is an $\code{\mathfrak{a}'}$-definable function $l : \mathfrak{a}' \longrightarrow \VF^+$ such that $f^*_{a'} \coloneqq f_{a'} \upharpoonright A_{a'}$ is continuously monotone (of the same kind) for all $a' \in \mathfrak{a}'$, where $A_{a'} \coloneqq (0, l(a'))$. By Lemma~\ref{open:rv:cons}, $l(\mathfrak{a}')$ is either a point or an open disc. Thus the $\vv$-interval $(0, l(\mathfrak{a}'))$ is nonempty, which implies that $f^*_{a'}(e) \in \mathfrak{a}_n$ for some $a' \in \mathfrak{a}'$ and some $e \in A_{a'}$. In that case, we must have that, for all $a' \in \mathfrak{a}'$, $\mathfrak{a}_n \subseteq f^*_{a'}(A_{a'})$ and hence $ f^*_{a'}$ is bijective. By $o$\nobreakdash-minimality in the $\Gamma$-sort, $\abval((f^*_{a'})^{-1}(\mathfrak{a}_n))$ has to be a singleton, say, $\beta_{a'}$; in fact, the function given by $a' \longmapsto \beta_{a'}$ has to be constant and hence we may write $\beta_{a'}$ as $\beta$. It follows that, for all $e \in \VF^+$ with $\abval(e) > \beta$, $f(e, \mathfrak{a}') \cap \mathfrak{a}_n = \emptyset$, contradiction. \end{proof}
Next we come to the issue of finding definable points in definable sets. As we have mentioned above, this is a trivial issue if the space of parameters is not fixed.
\begin{lem}\label{S:def:cl} The substructure $\mdl S$ is definably closed. \end{lem} \begin{proof} By Lemma~\ref{RV:no:point}, we have $\VF(\dcl( \mdl S)) = \VF(\mdl S)$. Suppose that $t \in \RV$ is definable. By the first sentence of Remark~\ref{rem:RV:weako}, if $\vrv(\RV(\mdl S))$ is nontrivial then $\RV(\mdl S)$ is a model of the reduct of $\TCVF$ to the $\RV$-sort and hence, by quantifier elimination, is an elementary substructure of $\RV$, which implies $t \in \RV(\mdl S)$. On the other hand, if $\vrv(\RV(\mdl S))$ is trivial then $\RV(\mdl S) = \K(\mdl S)$ and it is not hard, though a bit tedious, to check, using quantifier elimination again, that $t \in \K(\mdl S)$. \end{proof}
If $\mdl S$ is $\VF$-generated and $\Gamma(\mdl S)$ is nontrivial then $\mdl S$ is an elementary substructure and hence every definable set contains a definable point. This, of course, fails if $\mdl S$ carries extra $\RV$-data, by the above lemma. However, we do have:
\begin{lem}\label{clo:disc:bary} Every definable closed disc $\mathfrak{b}$ contains a definable point. \end{lem} \begin{proof} Suppose for contradiction that $\mathfrak{b}$ does not contain a definable point. Since $\mmdl$ is sufficiently saturated, there is an open disc $\mathfrak{a}$ that is disjoint from $\VF(\mdl S)$ and properly contains $\mathfrak{b}$. Let $a \in \mathfrak{a} \smallsetminus \mathfrak{b}$ and $b \in \mathfrak{b}$. Clearly $\rv(c - b) = \rv(c - a)$ for all $c \in \VF(\mdl S)$. As in the proof of Lemma~\ref{atom:self}, there is an immediate automorphism $\sigma$ of $\mmdl$ such that $\sigma(a) = b$. This means that $\mathfrak{b}$ is not definable, which is a contradiction. \end{proof}
Notice that the argument above does not work if $\mathfrak{b}$ is an open disc.
\begin{cor}\label{open:disc:def:point} Let $\mathfrak{a} \subseteq \VF$ be a disc and $A$ a definable subset of $\VF$. If $\mathfrak{a} \cap A$ is a nonempty proper subset of $\mathfrak{a}$ then $\mathfrak{a}$ contains a definable point. \end{cor} \begin{proof} It is not hard to see that, by HNF, if $\mathfrak{a} \cap A$ is a nonempty proper subset of $\mathfrak{a}$ then $\mathfrak{a}$ contains a definable closed disc and hence the claim is immediate by Lemma~\ref{clo:disc:bary}. \end{proof}
\begin{lem}\label{one:atomic} Let $A \subseteq \VF$ be a definable set that contains infinitely many open discs of radius $\beta$. Then one of these discs $\mathfrak{a}$ is $(\code \mathfrak{a}, \beta)$-atomic. \end{lem} \begin{proof} By Lemmas~\ref{atom:gam} and \ref{atom:self}, it is enough to show that some open disc $\mathfrak{a} \subseteq A$ of radius $\beta$ is contained in a type-definable set. Suppose for contradiction that this is not the case. By Corollary~\ref{open:disc:def:point} and HNF, for every definable set $B \subseteq A$, we have either $\mathfrak{a} \cap B = \emptyset$ or $\mathfrak{a} \subseteq B$ for all but finitely many such open discs $\mathfrak{a} \subseteq A$. Passing to $\mdl R^{\bullet}_{\rv}$ and applying compactness (with the parameter $\beta$), the claim follows. \end{proof}
\subsection{Contracting from $\VF$ to $\RV$}
We can relate definable sets in $\VF$ to those in $\RV$, specifically, $\RV$-pullbacks, through a procedure called contraction. But a more comprehensive study of the latter will be postponed to the next section.
\begin{defn}[Disc-to-disc]\label{defn:dtdp} Let $A$, $B$ be two subsets of $\VF$ and $f : A \longrightarrow B$ a bijection. We say that $f$ is \emph{concentric} if, for all open discs $\mathfrak{a} \subseteq A$, $f(\mathfrak{a})$ is also an open disc; if both $f$ and $f^{-1}$ are concentric then $f$ has the \emph{disc-to-disc property} (henceforth abbreviated as ``dtdp'').
More generally, let $f : A \longrightarrow B$ be a bijection between two sets $A$ and $B$, each with exactly one $\VF$-coordinate. For each $(t, s) \in f_{\RV}$, let $f_{t, s} = f \cap (\VF^2 \times (t, s))$, which is called a \emph{$\VF$-fiber} of $f$. We say that $f$ has \emph{dtdp} if every $\VF$-fiber of $f$ has dtdp. \end{defn}
We are somewhat justified in not specifying ``open disc'' in the terminology since if $f$ has dtdp then, for all open discs $\mathfrak{a} \subseteq A$ and all closed discs $\mathfrak{c} \subseteq \mathfrak{a}$, $f(\mathfrak{c})$ is also a closed disc. In fact, this latter property is stronger: if $f(\mathfrak{c})$ is a closed disc for all closed discs $\mathfrak{c} \subseteq A$ then $f$ has dtdp. But we shall only be concerned with open discs, so we ask for it directly.
\begin{lem}\label{open:pro} Let $f : A \longrightarrow B$ be a definable bijection between two sets $A$ and $B$, each with exactly one $\VF$-coordinate. Then there is a definable finite partition $A_i$ of $A$ such that each $f \upharpoonright A_i$ has dtdp. \end{lem} \begin{proof} By compactness, we may simply assume that $A$ and $B$ are subsets of $\VF$. Then we may proceed exactly as in the proof of Corollary~\ref{part:rv:cons}, using Lemmas~\ref{open:rv:cons} and~\ref{atom:self} (also see Remark~\ref{rem:LT:com}). \end{proof}
\begin{defn} Let $A$ be a subset of $\VF^n$. The \emph{$\RV$-boundary} of $A$, denoted by $\partial_{\RV}A$, is the definable subset of $\rv(A)$ such that $t \in \partial_{\RV} A$ if and only if $t^\sharp \cap A$ is a proper nonempty subset of $t^\sharp$. The definable set $\rv(A) \smallsetminus \partial_{\RV}A$, denoted by $\ito_{\RV}(A)$, is called the \emph{$\RV$-interior} of $A$. \end{defn}
Obviously, $A \subseteq \VF^n$ is an $\RV$-pullback if and only if $\partial_{\RV} A$ is empty. Note that $\partial_{\RV}A$ is in general different from the topological boundary $\partial(\rv(A))$ of $\rv(A)$ in $\RV^n$ and neither one of them includes the other.
\begin{lem}\label{RV:bou:dim} Let $A$ be a definable subset of $\VF^n$. Then $\dim_{\RV}(\partial_{\RV} A) < n$. \end{lem} \begin{proof} We do induction on $n$. The base case $n=1$ follows immediately from HNF.
We proceed to the inductive step. Since $\partial_{\RV} A_a$ is finite for every $i \in [n]$ and every $a \in \pr_{\tilde i}(A)$, by Corollary~\ref{open:disc:def:point} and compactness, there are a definable finite partition $A_{ij}$ of $\pr_{\tilde i}(A)$ and, for each $A_{ij}$, finitely many definable functions $f_{ijk} : A_{ij} \longrightarrow \VF$ such that \[ \textstyle\bigcup_k \rv(f_{ijk}(a)) = \partial_{\RV} A_a \quad \text{for all } a \in A_{ij}. \] By Corollary~\ref{part:rv:cons}, we may assume that if $t^\sharp \subseteq A_{ij}$ then the restriction $\rv \upharpoonright f_{ijk}(t^\sharp)$ is constant. Hence each $f_{ijk}$ induces a definable function $C_{ijk} : \ito_{\RV}(A_{ij}) \longrightarrow \RVV$. Let \[ \textstyle C = \bigcup_{i, j, k} C_{ijk} \quad \text{and} \quad B = \bigcup_{i,j} \bigcup_{t \in \partial_{\RV} A_{ij}} \rv(A)_t. \] Obviously $\dim_{\RV}(C) < n$. By the inductive hypothesis, for all $A_{ij}$ we have $\dim_{\RV}(\partial_{\RV} A_{ij}) < n-1$. Thus $\dim_{\RV}(B) < n$. Since $\partial_{\RV} A \subseteq B \cup C$, the claim follows. \end{proof}
For $(a, t) \in \VF^n \times \RV_0^m$, we write $\rv(a,t)$ to mean $(\rv(a), t)$, similarly for other maps.
\begin{defn}[Contractions]\label{defn:corr:cont} A function $f : A \longrightarrow B$ is \emph{$\rv$-contractible} if there is a (necessarily unique) function $f_{\downarrow} : \rv(A) \longrightarrow \rv(B)$, called the \emph{$\rv$-contraction} of $f$, such that \[ (\rv \upharpoonright B) \circ f = f_{\downarrow} \circ (\rv \upharpoonright A). \] Similarly, it is \emph{$\res$-contractible} (resp.\ \emph{$\vv$-contractible}) if the same holds in terms of $\res$ (resp.\ $\vv$ or $\vrv$, depending on the coordinates) instead of $\rv$. \end{defn}
The subscripts in these contractions will be written as $\downarrow_{\rv}$, $\downarrow_{\res}$, etc., if they occur in the same context and therefore need to be distinguished from one another notationally.
\begin{lem}\label{fn:alm:cont} For every definable function $f : \VF^n \longrightarrow \VF$ there is a definable set $U \subseteq \RV^n$ with $\dim_{\RV}(U) < n$ such that $f \upharpoonright (\VF^n \smallsetminus U^\sharp)$ is $\rv$-contractible. \end{lem} \begin{proof} By Corollary~\ref{poly:open:cons}, for any $t \in \RV^n$, if $\rv(f(t^\sharp))$ is not a singleton then $t^\sharp$ has a $t$-definable proper subset. By compactness, there is a definable subset $A \subseteq \VF^n$ such that $t \in \partial_{\RV} A$ if and only if $\rv(f(t^\sharp))$ is not a singleton. So the assertion follows from Lemma~\ref{RV:bou:dim}. \end{proof}
For any definable set $A$, a property holds \emph{almost everywhere} in $A$ or \emph{for almost every point} in $A$ if it holds away from a definable subset of $A$ of a smaller $\VF$-dimension. This terminology will also be used with respect to other notions of dimension.
\begin{rem}[Regular points] Let $f : \VF^n \longrightarrow \VF^m$ be a definable function. By Lemma~\ref{fun:suba:fun} and $o$\nobreakdash-minimal differentiability, $f$ is $C^p$ almost everywhere for all $p$ (see \cite[\S~7.3]{dries:1998}). For each $p$, let $\reg^p(f) \subseteq \VF^n$ be the definable subset of regular $C^p$-points of $f$. If $p=0$ then we write $\reg(f)$, which is simply the subset of the regular points of $f$.
Assume $n=m$. If $a \in \reg(f)$ and $f$ is $C^1$ in a neighborhood of $a$ then $\reg^1(f)$ contains a neighborhood of $a$ on which the sign of the Jacobian of $f$, which is denoted by $\jcb_{\VF} f$, is constant. If $f$ is locally injective on a definable open subset $A \subseteq \VF^n$ then $f$ is regular almost everywhere in $A$ and hence, for all $p$, $\dim_{\VF}(A \smallsetminus \reg^p(f)) < n$.
By \cite[Theorem~A]{Dries:tcon:97}, the situation is quite similar if $f$ is a (parametrically) definable function of the form $\tor(\alpha)^n \longrightarrow \tor(\beta)^m$, $\alpha, \beta \in \absG$, and $\dim_{\VF}$ is replaced by $\dim_{\RV}$, in particular, if $f$ is such a function from $\K^n$ into $\K^m$, or more generally, from $\tor(u)$ into $\tor(v)$, where $u \in \RV^n_{\alpha}$ and $v \in \RV^m_{\beta}$ (see Notation~\ref{rem:K:aff} and Definition~\ref{rem:tor:der}). \end{rem}
\begin{rem}[$\rv$-contraction of univariate functions]\label{contr:uni} Suppose that $f$ is a definable function from $\OO^\times$ into $\OO$. By monotonicity, there are a definable finite set $B \subseteq \OO^\times$ and a definable finite partition of $A \coloneqq \OO^\times \smallsetminus B$ into infinite $\vv$-intervals $A_i$ such that both $f$ and $\ddx f$ are quasi-\LT-definable, continuous, and monotone on each $A_i$. If $\rv(A_i)$ is not a singleton then let $U_i \subseteq \K$ be the largest open interval contained in $\rv(A_i)$. Let \[ A^*_i = U_i^\sharp, \quad U = \textstyle{\bigcup_i U_i}, \quad A^* = U^\sharp, \quad f^* = f \upharpoonright A^*. \] By Lemma~\ref{fn:alm:cont}, we may refine the partition such that both $f^*$ and $\frac{d}{d x} f^*$ are $\rv$-contractible. By Lemma~\ref{gk:ortho}, $\vv \upharpoonright f^*(A^*_i)$ and $\vv \upharpoonright \tfrac{d}{d x} f^*(A^*_i)$ must be constant, say $\alpha_i$ and $\beta_i$, respectively. So it makes sense to speak of $\ddx f^*_{\downarrow_{\rv}}$ on each $U_i$, which a priori is not the same as $(\ddx f^*)_{\downarrow_{\rv}}$. Deleting finitely many points from $U$ if necessary, we assume that $f^*_{\downarrow_{\rv}}$, $(\ddx f^*)_{\downarrow_{\rv}}$, and $\ddx f^*_{\downarrow_{\rv}}$ are all continuous monotone functions on each $U_i$.
We claim that $\abs{\beta_i} = \abs{\alpha_i}$ unless $f^*_{\downarrow_{\rv}} \upharpoonright U_i$ is constant. Suppose for contradiction that $f^*_{\downarrow_{\rv}} \upharpoonright U_i$ is not constant and $\abs{\beta_i} \neq \abs{\alpha_i}$. First examine the case $\abs{\beta_i} < \abs{\alpha_i}$. A moment of reflection shows that, then, $f^* \upharpoonright A^*_i$ would increase or decrease too fast to confine $f^*(A_i^*)$ in $\vv^{-1}(\alpha_i)$. Dually, if $\abs{\beta_i} > \abs{\alpha_i}$ then $f^* \upharpoonright A^*_i$ would increase or decrease too slowly to make $f^*_{\downarrow_{\rv}}(U_i)$ contain more than one point. In either case, we have reached a contradiction. Actually, a similar estimate shows that if $\abs{\beta_i} = \abs{\alpha_i} < \infty$ then $f^*_{\downarrow_{\rv}} \upharpoonright U_i$ cannot be constant.
Finally, we show that $\abs{\beta_i} = \abs{\alpha_i}$ implies $(\ddx f^*)_{\downarrow_{\rv}} = \ddx f^*_{\downarrow_{\rv}}$ on $U_i$ (note that if $\abs{\beta_i} > \abs{\alpha_i}$ then $\ddx f^*_{\downarrow_{\rv}} = 0$). Suppose for contradiction that, say, \[ (\ddx f^*)_{\downarrow_{\rv}}(\rv(a)) > \ddx f^*_{\downarrow_{\rv}}(\rv(a)) > 0 \] for some $a \in A^*_i$. Then there is an open interval $I \subseteq U_i$ containing $\rv(a)$ such that $(\ddx f^*)_{\downarrow_{\rv}}(I) > \ddx f^*_{\downarrow_{\rv}}(I)$. It follows that $f^*_{\downarrow_{\rv}}(I)$ is properly contained in $\rv(f^*(I^\sharp)) = f^*_{\downarrow_{\rv}}(I)$, which is absurd. The other cases are similar. \end{rem}
The higher-order multivariate version is more complicated to state than to prove:
\begin{lem}\label{univar:der:contr} Let $A \subseteq (\OO^\times)^n$ be a definable $\RV$-pullback with $\dim_{\RV}(\rv(A)) = n$ and $f : A \longrightarrow \OO$ a definable function. Let $p \in \mathds{N}^n$ be a multi-index of order $\abs{p} = d$ and $k \in \mathds{N}$ with $k \gg d$. Suppose that $f$ is $C^k$ and, for all $q \leq p$, $\frac{\partial^q}{\partial x^q} f$ is $\rv$-contractible and its contraction $(\frac{\partial^q}{\partial x^q} f)_{\downarrow_{\rv}}$ is also $C^k$. Then there is a definable set $V \subseteq \rv(A)$ with $\dim_{\RV}(V) < n$ and $U \coloneqq \rv(A) \smallsetminus V$ open such that, for all $a \in U^\sharp$ and all $q' < q \leq p$ with $\abs{q'} + 1 = q$, exactly one of the following two conditions holds: \begin{itemize}
\item either $\frac{\partial^{q}}{\partial x^{q}} f(a) = 0$ or $\abval (\frac{\partial^{q'}}{\partial x^{q'}} f(a)) < \abval (\frac{\partial^{q}}{\partial x^{q}} f(a))$,
\item $(\frac{\partial^{q - q'}}{\partial x^{q - q'}} \frac{\partial^{q'}}{\partial x^{q'}} f)_{\downarrow_{\rv}}(\rv (a)) = \frac{\partial^{q - q'}}{\partial x^{q - q'}}(\frac{\partial^{q'}}{\partial x^{q'}} f)_{\downarrow_{\rv}}(\rv( a)) \neq 0$. \end{itemize} If the first condition never occurs then, for all $q \leq p$, we actually have $(\frac{\partial^q}{\partial x^q} f )_{\downarrow_{\rv}} = \frac{\partial^{q}}{\partial x^{q}} f_{\downarrow_{\rv}}$ on $U$. At any rate, for all $q \leq p$, we have $(\frac{\partial^q}{\partial x^q} f )_{\downarrow_{\res}} = \frac{\partial^{q}}{\partial x^{q}} f_{\downarrow_{\res}}$ on $U$. \end{lem} \begin{proof} First observe that, by induction on $d$, it is enough to consider the case $d =1$ and $p = (0, \ldots, 0, 1)$. For each $a \in \pr_{<n}(A)$, by the discussion in Remark~\ref{contr:uni}, there is an $a$-definable finite set $V_{a}$ of $\rv(A)_{\rv(a)}$ such that the assertion holds for the restriction $f \upharpoonright (A_a \smallsetminus V_{a}^\sharp)$. Let $A^* = \bigcup_{a \in \pr_{<n}(A)} V_{a}^\sharp \subseteq A$. By Lemma~\ref{RV:bou:dim}, $\dim_{\RV}(\partial_{\RV} A^*) < n$ and hence $\dim_{\RV}(\rv(A^*)) < n$. Therefore, by Lemma~\ref{fn:alm:cont}, there is a definable open set $U \subseteq \ito(\rv(A) \smallsetminus \rv(A^*))$ that is as desired. \end{proof}
Suppose that $f = (f_1, \ldots, f_m) : A \longrightarrow \OO$ is a sequence of definable $\res$-contractible functions, where the set $A$ is as in Lemma~\ref{univar:der:contr}. Let $P(x_1, \ldots, x_m)$ be a partial differential operator with definable $\res$-contractible coefficients $a_i : A \longrightarrow \OO$ and $P_{\downarrow_{\res}}(x_1, \ldots, x_m)$ the corresponding operator with $\res$-contracted coefficients $a_{i\downarrow_{\res}} : \res(A) \longrightarrow \K$. Note that both $P(f) : A \longrightarrow \OO$ and $P_{\downarrow_{\res}}(f_{\downarrow_{\res}}) : \res(A) \longrightarrow \K$ are defined almost everywhere. By Lemma~\ref{univar:der:contr}, such an operator $P$ almost commutes with $\res$:
\begin{cor}\label{rv:op:comm} For almost all $t \in \rv(A)$ and all $a \in t^\sharp$, \[ \res(P(f)(a)) = P_{\downarrow_{\res}}(f_{\downarrow_{\res}})(\res(a)). \] \end{cor}
\begin{cor} Let $U$, $V$ be definably connected subsets of $(\K^+)^n$ and $f : U^\sharp \longrightarrow V^\sharp$ a definable $\res$-contractible function. Suppose that $f_{\downarrow_{\res}} : U \longrightarrow V$ is continuous and locally injective. Then there is a definable subset $U^* \subseteq U$ of $\RV$-dimension $< n$ such that the sign of $\jcb_{\VF} f$ is constant on $(U \smallsetminus U^*)^\sharp$. \end{cor} \begin{proof} This follows immediately from Corollary~\ref{rv:op:comm} and \cite[Theorem~3.2]{pet:star:otop}. \end{proof}
\begin{lem}\label{atom:type} In $\xmdl$, let $\mathfrak{a} \subseteq \VF$ be an atomic subset and $f : \mathfrak{a} \longrightarrow \VF$ a definable injection. Then $\mathfrak{a}$ and $f(\mathfrak{a})$ must be of the same one of the four possible forms (see Remark~\ref{rem:type:atin}). \end{lem} \begin{proof} This is trivial if $\mathfrak{a}$ is a point. The case of $\mathfrak{a}$ being an open disc is covered by Lemma~\ref{open:rv:cons}. So we only need to show that if $\mathfrak{a}$ is a closed disc then $f(\mathfrak{a})$ cannot be a half thin annulus. We shall give two proofs. The first one works only when $T$ is polynomially bounded, but is more intuitive and much simpler.
Suppose that $T$ is polynomially bounded. Suppose for contradiction that $\code \mathfrak{a}$ is of the form $\tor(\goedel \mathfrak{m})$ for some $\goedel \mathfrak{m} \in \RV_{\gamma}$ and $\goedel{f(\mathfrak{a})}$ is of the form $\tor^+(\goedel \mathfrak{n})$ for some $\goedel \mathfrak{n} \in \RV_{\delta}$. By Lemma~\ref{open:pro} and monotonicity, $f$ induces an increasing (or decreasing, which can be handled similarly) bijection $f_{\downarrow} : \tor(\goedel \mathfrak{m}) \longrightarrow \tor^+(\goedel \mathfrak{n})$. In fact, for all $p \in \mathds{N}$, \[ \tfrac{d^p}{d x^p} f_{\downarrow} : \tor(\goedel \mathfrak{m}) \longrightarrow \tor^{+}(\delta - p \gamma) \] cannot be constant and hence must be continuous, surjective, and increasing. Using additional parameters, we can translate $f_{\downarrow}$ into a function $\K \longrightarrow \K^+$ and this function cannot be polynomially bounded by elementary differential calculus, which is a contradiction.
We move on to the second proof. The argument is essentially the same as that in the proof of \cite[Lemma~3.45]{hrushovski:kazhdan:integration:vf}.
Consider the group \[ G \coloneqq \aut(\tor(\goedel \mathfrak{m}) / \K) \leq \aut(\xmdl / \K). \] Suppose for contradiction that $G$ is finite. Since every $G$-orbit is finite, every point in $\tor(\goedel \mathfrak{m})$ is $\K$-definable. It follows that there exists a nonconstant definable function $\tor(\goedel \mathfrak{m}) \longrightarrow \K$. But this is not possible since $\mathfrak{a}$ is atomic.
Let $\Lambda$ be the group of affine transformations of $\K$, that is, $\Lambda = \K^{\times} \ltimes \K$, where the first factor is the multiplicative group of $\K$ and the second the additive group of $\K$. Every automorphism in $G$ is a $\K$-affine transformation of $\tor(\goedel \mathfrak{m})$ and hence $G$ is a subgroup of $\Lambda$. For each $\K$-definable relation $\phi$ on $\tor(\goedel \mathfrak{m})$, let $G_{\phi} \subseteq \Lambda$ be the definable subgroup of $\K$-affine transformations that preserve $\phi$. So $G = \bigcap_{\phi} G_{\phi}$. Since there is no infinite descending chain of definable subgroups of $\Lambda$, we see that $G$ is actually an infinite definable group. Then we may choose two nontrivial automorphisms $g, g' \in G$ whose fixed points are distinct. It follows that the commutator of $g$, $g'$ is a translation and hence, by $o$\nobreakdash-minimality, $G$ contains all the translations, that is, $\K \leq G$.
By a similar argument, every automorphism in $H \coloneqq \aut(\tor^+(\goedel \mathfrak{n}) / \K)$ is a $\K$-linear transformation of $\tor^+(\goedel \mathfrak{n})$ and hence $H = \K^+ \leq \K^{\times}$.
Now any definable bijection between $\tor(\goedel \mathfrak{m})$ and $\tor^+(\goedel \mathfrak{n})$ would induce a definable group isomorphism $\K \longrightarrow \K^+$, that is, an exponential function, which of course contradicts the assumption that $T$ is power-bounded. \end{proof}
\begin{defn}[$\vv$-affine and $\rv$-affine]\label{rvaffine} Let $\mathfrak{a}$ be an open disc and $f : \mathfrak{a} \longrightarrow \VF$ an injection. We say that $f$ is \emph{$\vv$-affine} if there is a (necessarily unique) $\gamma \in \Gamma$, called the \emph{shift} of $f$, such that, for all $a, a' \in \mathfrak{a}$, \[ \abval(f(a) - f(a')) = \gamma + \abval(a - a'). \] We say that $f$ is \emph{$\rv$-affine} if there is a (necessarily unique) $t \in \RV$, called the \emph{slope} of $f$, such that, for all $a, a' \in \mathfrak{a}$, \[ \rv(f(a) - f(a')) = t \rv(a - a'). \] \end{defn}
Obviously $\rv$-affine implies $\vv$-affine. With the extra structure afforded by the total ordering, we can reproduce (an analogue of) \cite[Lemma~3.18]{Yin:int:acvf} with a somewhat simpler proof:
\begin{lem}\label{rv:lin} In $\xmdl$, let $f : \mathfrak{a} \longrightarrow \mathfrak{b}$ be a definable bijection between two atomic open discs. Then $f$ is $\rv$-affine and hence $\vv$-affine with respect to $\rad(\mathfrak{b}) - \rad(\mathfrak{a})$. \end{lem} \begin{proof} Since $f$ has dtdp by Lemma~\ref{open:pro}, for all $\rad(\mathfrak{a}) < \delta$ and all \[ \mathfrak{d} \coloneqq \tor(\goedel \mathfrak{c}) \subseteq \rv_{\delta- \abval(\mathfrak{a})}(\mathfrak{a}), \] it induces a $\goedel \mathfrak{d}$-definable $C^1$ function $f_{\goedel \mathfrak{d}} : \mathfrak{d} \longrightarrow \tor(\goedel{f(\mathfrak{c})})$. The codomain of its derivative $\ddx f_{\goedel \mathfrak{d}}$ can be narrowed down to either $\tor^+(\epsilon - \delta)$ or $\tor^{-}(\epsilon - \delta)$, where $\epsilon = \rad(f(\mathfrak{c}))$. By Lemma~\ref{open:rv:cons}, there is a $t \in \RV$ such that $\ddx f(\mathfrak{a}) \subseteq t^\sharp$. By Lemma~\ref{atom:gam}, $\mathfrak{a}$ remains atomic over $\delta$. Then, by (an accordingly modified version of) Remark~\ref{contr:uni}, we must have that, for all $\mathfrak{d}$ as above, all $\goedel \mathfrak{c} \in \mathfrak{d}$, and all $a \in \mathfrak{c}$, \[ \ddx f_{\goedel \mathfrak{d}}(\goedel \mathfrak{c}) = \rv(\ddx f(a)) = t \] and hence \[ \aff_{\goedel{f(\mathfrak{c})}} \circ f_{\goedel \mathfrak{d}} \circ \aff^{-1}_{\goedel \mathfrak{c}} : \tor(\delta) \longrightarrow \tor(\epsilon) \] is a linear function given by $u \longmapsto tu$ (see Definition~\ref{rem:tor:der} for the notation). It follows that, for \begin{itemize}
\item $a$ and $a'$ in $\mathfrak{a}$,
\item $\mathfrak{d}$ the smallest closed disc containing $a$ and $a'$,
\item $\mathfrak{c}$ and $\mathfrak{c}'$ the maximal open subdiscs of $\mathfrak{d}$ containing $a$ and $a'$, respectively, \end{itemize} we have \[ \rv(f(a) - f(a')) = \rv(f(\mathfrak{c}) - f(\mathfrak{c}')) = t \rv(\mathfrak{c} - \mathfrak{c}') = t \rv(a - a'). \] That is, $f$ is $\rv$-affine. Moreover, it is clear from dtdp that $\abvrv(t) = \rad(\mathfrak{b}) - \rad(\mathfrak{a})$. \end{proof}
\section{Grothendieck semirings}\label{sect:groth}
In this section, we define various categories of definable sets and explore the relations between their Grothendieck semirings. The first main result is that the Grothendieck semiring $\gsk \RV[*]$ of the $\RV$-category $\RV[*]$ can be naturally expressed as a tensor product of the Grothendieck semirings of two of its full subcategories $\RES[*]$ and $\Gamma[*]$. The second main result is that there is a natural surjective semiring homomorphism from $\gsk \RV[*]$ onto the Grothendieck semiring $\gsk \VF_*$ of the $\VF$-category $\VF_*$.
\begin{hyp}\label{hyp:gam} By (the proof of) Lemma~\ref{S:def:cl}, every definable set in $\RV$ contains a definable point if and only if $\Gamma(\mdl S) \neq \pm 1$. Thus, from now on, we shall assume that $\Gamma(\mdl S)$ is nontrivial. \end{hyp}
\subsection{The categories of definable sets} As in Definition~\ref{defn:dtdp}, an $\RV$-fiber of a definable set $A$ is a set of the form $A_a$, where $a \in A_{\VF}$. The $\RV$-fiber dimension of $A$ is the maximum of the $\RV$-dimensions of its $\RV$-fibers and is denoted by $\dim^{\fib}_{\RV}(A)$.
\begin{lem}\label{RV:fiber:dim:same} Suppose that $f : A \longrightarrow A'$ is a definable bijection. Then $\dim^{\fib}_{\RV}(A) = \dim^{\fib}_{\RV} (A')$. \end{lem} \begin{proof} Let $\dim^{\fib}_{\RV}(A) = k$ and $\dim^{\fib}_{\RV}(A') = k'$. For each $a \in \pr_{\VF}(A)$, let $h_{a} : A_a \longrightarrow A'_{\VF}$ be the $a$-definable function induced by $f$ and $\pr_{\VF}$. By Corollary~\ref{function:rv:to:vf:finite:image}, the image of $h_{a}$ is finite. It follows that $k \leq k'$. Symmetrically we also have $k \geq k'$ and hence $k = k'$. \end{proof}
\begin{defn}[$\VF$-categories]\label{defn:VF:cat} The objects of the category $\VF[k]$ are the definable sets of $\VF$-dimension $\leq k$ and $\RV$-fiber dimension $0$ (that is, all the $\RV$-fibers are finite). Any definable bijection between two such objects is a morphism of $\VF[k]$. Set $\VF_* = \bigcup_k \VF[k]$. \end{defn}
\begin{defn}[$\RV$-categories]\label{defn:c:RV:cat} The objects of the category $\RV[k]$ are the pairs $(U, f)$ with $U$ a definable set in $\RVV$ and $f : U \longrightarrow \RV^k$ a definable finite-to-one function. Given two such objects $(U, f)$, $(V, g)$, any definable bijection $F : U \longrightarrow V$ is a \emph{morphism} of $\RV[k]$. \end{defn}
Set $\RV[{\leq} k] = \bigoplus_{i \leq k} \RV[i]$ and $\RV[*] = \bigoplus_{k} \RV[k]$; similarly for the other categories below.
\begin{nota}\label{0coor} We emphasize that if $(U, f)$ is an object of $\RV[k]$ then $f(U)$ is a subset of $\RV^k$ instead of $\RV_0^k$, while $0$ can occur in any coordinate of $U$. An object of $\RV[*]$ of the form $(U, \id)$ is often written as $U$.
More generally, if $f : U \longrightarrow \RV_0^k$ is a definable finite-to-one function then $(U, f)$ denotes the obvious object of $\RV[{\leq} k]$. Often $f$ will be a coordinate projection (every object in $\RV[*]$ is isomorphic to an object of this form). In that case, $(U, \pr_{\leq k})$ is simply denoted by $U_{\leq k}$ and its class in $\gsk \RV[k]$ by $[U]_{\leq k}$, etc. \end{nota}
\begin{rem}\label{fintoone} Alternatively, we could allow only injections instead of finite-to-one functions in defining the objects of $\RV[k]$. Insofar as the Grothendieck semigroup $\gsk \RV[k]$ is concerned, this is not more restrictive in our setting since for any $\bm U \coloneqq (U, f) \in \RV[k]$ there is a definable finite partition $\bm U_i \coloneqq (U_i, f_i)$ of $\bm U$, in other words, $[\bm U] = \sum_i [\bm U_i]$ in $\gsk \RV[k]$, such that each $f_i$ is injective. It is technically more convenient to work with finite-to-one functions, though (for instance, we can take finite disjoint unions). \end{rem}
In the above definitions and other similar ones below, all morphisms are actually isomorphisms and hence the categories are all groupoids. For the cases $k =0$, the reader should interpret things such as $\RV^0$ and how they interact with other things in a natural way. For instance, $\RV^0$ may be treated as the empty tuple. So the categories $\VF[0]$, $\RV[0]$ are equivalent.
About the position of ``$*$'' in the notation: ``$\VF_*$'' suggests that the category is filtrated and ``$\RV[*]$'' suggests that the category is graded.
\begin{defn}[$\RES$-categories]\label{defn:RES:cat} The category $\RES[k]$ is the full subcategory of $\RV[k]$ such that $(U, f) \in \RES[k]$ if and only if $\vrv(U)$ is finite. \end{defn}
\begin{rem}[Explicit description of ${\gsk \RES[k]}$]\label{expl:res} Let $\RES$ be the category whose objects are the definable sets $U$ in $\RVV$ with $\vrv(U)$ finite and whose morphisms are the definable bijections. The obvious forgetful functor $\RES[*] \longrightarrow \RES$ induces a surjective semiring homomorphism $\gsk \RES[*] \longrightarrow \gsk \RES$, which is clearly not injective.
The semiring $\gsk \RES$ is actually generated by isomorphism classes $[U]$ with $U$ a set in $\K^+$. By Theorem~\ref{groth:omin}, we have the following explicit description of $\gsk \RES$. Its underlying set is $(0 \times \mathds{N}) \cup (\mathds{N}^+ \times \mathds{Z})$. For all $(a, b), (c, d) \in \gsk \RES$, \[ (a, b) + (c, d) = (\max\{a, c\}, b+d), \quad (a, b) \times (c, d) = (a + c, b \times d). \] By the computation in \cite{kage:fujita:2006}, the dimensional part is lost in the groupification $\ggk \RES$ of $\gsk \RES$, that is, $\ggk \RES = \mathds{Z}$, which is of course much simpler than $\gsk \RES$. However, following the philosophy of \cite{hrushovski:kazhdan:integration:vf}, we shall work with Grothendieck semirings whenever possible.
By Lemma~\ref{gk:ortho}, if $(U, f) \in \RES[*]$ then $\vrv(f(U))$ is finite as well. Therefore the semiring $\gsk \RES[*]$ is generated by isomorphism classes $[(U, f)]$ with $f$ a bijection between two sets in $\K^+$. As above, each $\gsk \RES[k]$ may be described explicitly as well. The semigroup $\gsk \RES[0]$ is canonically isomorphic to the semiring $(0, 0) \times \mathds{N}$. For $k > 0$, the underlying set of $\gsk \RES[k]$ is $\bigcup_{0 \leq i \leq k}((k, i) \times \mathds{Z})$, and its semigroup operation is given by \[ (k, i, a) + (k, i', a') = (k, \max\{i, i'\}, a + a'). \] Moreover, multiplication in $\gsk \RES[*]$ is given by \[ (k, i, a) \times (l, j, b) = (k+l, i + j, a \times b). \] \end{rem}
\begin{defn}[$\Gamma$-categories]\label{def:Ga:cat} The objects of the category $\Gamma[k]$ are the finite disjoint unions of definable subsets of $\Gamma^k$. Any definable bijection between two such objects is a \emph{morphism} of $\Gamma[k]$. The category $\Gamma^{c}[k]$ is the full subcategory of $\Gamma[k]$ such that $I \in \Gamma^{c}[k]$ if and only if $I$ is finite. \end{defn}
Clearly $\gsk \Gamma^c[k]$ is naturally isomorphic to $\mathds{N}$ for all $k$ and hence $\gsk \Gamma^c[*] \cong \mathds{N}[X]$.
\begin{nota}\label{nota:RV:short} We introduce the following shorthand for distinguished elements in the various Grothendieck semigroups and their groupifications (and closely related constructions): \begin{gather*} \bm 1_{\K} = [\{1\}] \in \gsk \RES[0], \quad [1] = [(\{1\}, \id)] \in \gsk \RES[1],\\ [\bm T] = [(\K^+, \id)] \in \gsk \RES[1], \quad [\bm A] = 2 [\bm T] + [1] \in \gsk \RES[1],\\ \bm 1_{\Gamma} = [\Gamma^0] \in \gsk \Gamma[0], \quad [e] = [\{1\}] \in \gsk \Gamma[1], \quad [\bm H] = [(0,1)] \in \gsk \Gamma[1],\\ [\bm P] = [(\RV^{\circ \circ}, \id)] - [1] \in \ggk \RV[1]. \end{gather*} Here $\RV^{\circ \circ} = \RV^{\circ \circ}_0 \smallsetminus 0$. Note that the interval $\bm H$ is formed in the signed value group $\Gamma$, whose ordering is inverse to that of the value group $\abs \Gamma_\infty$ (recall Remark~\ref{signed:Gam}). The interval $(1, \infty) \subseteq \Gamma$ is denoted by $\bm H^{-1}$.
As in~\cite{hrushovski:kazhdan:integration:vf}, the elements $[\bm P]$ and $\bm 1_{\K} + [\bm P]$ in $\ggk \RV[*]$ play special roles in the main construction (see Propositions~\ref{kernel:L} and the remarks thereafter). \end{nota}
The following lemma is a generality proven elsewhere. It is only needed to prove Lemma~\ref{gam:pulback:mono}.
\begin{lem}\label{gen:mat:inv} Let $K$ be an integral domain and $M$ a torsion-free $K$-module, the latter is viewed as the main sort of a first-order structure of some expansion of the usual $K$-module language. Let $\mathfrak{F}$ be a class of definable functions in the sort $M$ such that \begin{itemize}
\item all the identity functions are in $\mathfrak{F}$,
\item all the functions in $\mathfrak{F}$ are definably piecewise $K$-linear, that is, they are definably piecewise of the form $x \longmapsto M x + c$, where $M$ is a matrix with entries in $K$ and $c$ is a definable point,
\item $\mathfrak{F}$ is closed under composition, inversion, composition with $\mgl(K)$-transformations ($K$-linear functions with invertible matrices), and composition with coordinate projections. \end{itemize} If $g : D \longrightarrow E$ is a bijection in $\mathfrak{F}$, where $D, E \subseteq M^n$, then $g$ is definably a piecewise $\mgl_n(K)$-transformation. \end{lem} \begin{proof} See \cite[Lemma~2.29]{Yin:int:expan:acvf}. \end{proof}
\begin{lem}\label{gam:pulback:mono} Let $g$ be a $\Gamma[k]$-morphism. Then $g$ is definably a piecewise $\mgl_k(\mathds{K})$-transformation modulo the sign, that is, a piecewise $\mgl_k(\mathds{K}) \times \mathds{Z}_2$-transformation. Consequently, $g$ is a $\vrv$-contraction (recall Definition~\ref{defn:corr:cont}). \end{lem} \begin{proof} For the first claim, it is routine to check that Lemma~\ref{gen:mat:inv} is applicable to the class of definable functions in the $\abs \Gamma$-sort. The second claim follows from the fact that the natural actions of $\mgl_k(\mathds{K})$ on $(\RV^+)^k$ and $(\Gamma^+)^k$ commute with the map $\vrv$. \end{proof}
\begin{rem}\label{why:glz} In \cite{hrushovski:kazhdan:integration:vf}, $\Gamma[k]$-morphisms are by definition piecewise $\mgl_k(\mathds{Z})$-transformations. This is because, in the setting there, the $\vrv$-contractions are precisely the piecewise $\mgl_k(\mathds{Z})$-transformations, which form a proper subclass of definable bijections in the $\Gamma$-sort, which in general are piecewise $\mgl_k(\mathds{Q})$-transformations. \end{rem}
\begin{lem}\label{G:red} For all $I \in \Gamma[k]$ there are finitely many definable sets $H_i \subseteq \Gamma^{n_i}$ with $\dim_{\Gamma}(H_i) = n_i \leq k$ such that $[I] = \sum_i [H_i] [e]^{k -n_i}$ in $\gsk \Gamma[k]$. \end{lem} \begin{proof} We do induction on $k$. The base case $k = 0$ is trivial. For the inductive step $k > 0$, the claim is also trivial if $\dim_{\Gamma}(I) = k$; so let us assume that $\dim_{\Gamma}(I) < k$. By \cite[Theorem~B]{Dries:tcon:97}, we may partition $I$ into finitely many definable pieces $I_i$ such that each $I_i$ is the graph of a definable function $I'_i \longrightarrow \Gamma$, where $I'_i \in \Gamma[k-1]$. So the claim simply follows from the inductive hypothesis. \end{proof}
\begin{rem}\label{gam:res} There is a natural map $\Gamma[*] \longrightarrow \RV[*]$ given by $I \longmapsto \bm I \coloneqq (I^\sharp, \id)$ (see Notation~\ref{gamma:what}). By Lemma~\ref{gam:pulback:mono}, this map induces a homomorphism $\gsk \Gamma[*] \longrightarrow \gsk \RV[*]$ of graded semirings. By \cite[Theorem~A]{Dries:tcon:97} and Theorem~\ref{groth:omin}, this homomorphism restricts to an injective homomorphism $\gsk \Gamma^{c}[*] \longrightarrow \gsk \RES[*]$ of graded semirings. There is also a similar semiring homomorphism $\gsk \Gamma^c[*] \longrightarrow \gsk \RES$, but it is not injective. \end{rem}
\begin{ques} Is the homomorphism $\gsk \Gamma[*] \longrightarrow \gsk \RV[*]$ above injective? \end{ques}
Now, the map from $\gsk \RES[*] \times \gsk \Gamma[*]$ to $\gsk \RV[*]$ naturally determined by the assignment \[ ([(U, f)], [I]) \longmapsto [(U \times I^\sharp, f \times \id)] \] is well-defined and is clearly $\gsk \Gamma^{c}[*]$-bilinear. Hence it induces a $\gsk \Gamma^{c}[*]$-linear map \[ \bb D: \gsk \RES[*] \otimes_{\gsk \Gamma^{c}[*]} \gsk \Gamma[*] \longrightarrow \gsk \RV[*], \] which is a homomorphism of graded semirings. We shall abbreviate ``$\otimes_{\gsk \Gamma^{c}[*]}$'' as ``$\otimes$'' below. Note that, by the universal mapping property, groupifying a tensor product in the category of $\gsk \Gamma^{c}[*]$-semimodules is the same, up to isomorphism, as taking the corresponding tensor product in the category of $\ggk \Gamma^{c}[*]$-modules. We will show that $\bb D$ is indeed an isomorphism of graded semirings.
\subsection{The tensor expression}
Heuristically, $\RV$ may be viewed as a union of infinitely many one-dimensional vector spaces over $\K$. Weak $o$\nobreakdash-minimality states that every definable subset of $\RV$ is nontrivial only within finitely many such one-dimensional spaces. The tensor expression of $\gsk \RV[*]$ we seek may be thought of as a generalization of this phenomenon to all definable sets in $\RV$.
\begin{lem}\label{resg:decom} Let $A \subseteq \RV^k \times \Gamma^l$ be an $\alpha$-definable set, where $\alpha \in \Gamma$. Set $\pr_{\leq k}(A) = U$ and suppose that $\vrv(U)$ is finite. Then there is an $\alpha$-definable finite partition $U_i$ of $U$ such that, for each $i$ and all $t, t' \in U_i$, we have $A_t = A_{t'}$. \end{lem} \begin{proof} By stable embeddedness, for every $t \in U$, $A_t$ is $(\vrv(t), \alpha)$-definable in the $\Gamma$-sort alone. Since $\vrv(U)$ is finite, the assertion simply follows from compactness. \end{proof}
\begin{lem}\label{gam:tup:red} Let $\beta$, $\gamma = (\gamma_1, \ldots, \gamma_m)$ be finite tuples in $\Gamma$. If there is a $\beta$-definable nonempty proper subset of $\gamma^\sharp$ then, for some $\gamma_i$ and every $t \in \gamma^\sharp_{\tilde i}$, $\gamma^\sharp_i$ contains a $t$-definable point. Consequently, if $U$ is such a subset of $\gamma^\sharp$ then either $U$ contains a definable point or there exists a subtuple $\gamma_* \subseteq \gamma$ such that $\pr_{\gamma_*}(U) = \gamma^\sharp_*$, where $\pr_{\gamma_*}$ denotes the obvious coordinate projection, and there is a $\beta$-definable function from $\gamma^\sharp_*$ into $(\gamma \smallsetminus \gamma_*)^\sharp$. \end{lem} \begin{proof} For the first claim we do induction on $m$. The base case $m = 1$ simply follows from $o$\nobreakdash-minimality in the $\K$-sort and Lemma~\ref{RV:no:point}. For the inductive step $m > 1$, let $U$ be a $\beta$-definable nonempty proper subset of $\gamma^\sharp$. By the inductive hypothesis, we may assume \[ \{ t \in \pr_{>1}(U) : U_t \neq \gamma^\sharp_1\} = \gamma^\sharp_{> 1}. \] Then $\gamma_1$ is as desired.
The second claim follows easily from the first. \end{proof}
\begin{lem}\label{RV:decom:RES:G} Let $U \subseteq \RV^m$ be a definable set. Then there are finitely many definable sets of the form $V_i \times D_i \subseteq (\K^+)^{k_i} \times \Gamma^{l_i}$ such that $k_i + l_i = m$ for all $i$ and $[U] = \sum_i [V_i \times D_i^\sharp]$ in $\gsk \RV[*]$. \end{lem} \begin{proof} The case $m=1$ is an immediate consequence of weak $o$\nobreakdash-minimality in the $\RV$-sort. For the case $m>1$, by Lemma~\ref{gam:tup:red}, compactness, and a routine induction on $m$, over a definable finite partition of $U$, we may assume that $U$ is a union of sets of the form $t \times \gamma^\sharp$, where $t \in (\K^+)^k$, $\gamma \in \Gamma^l$, and $k+l=m$. Then the assertion follows from Lemma~\ref{resg:decom}. \end{proof}
Let $Q$ be a set of parameters in $\mdl R^{\bullet}_{\rv}$. We say that a $Q$-definable set $I \subseteq \Gamma^m$ is \emph{$Q$-reducible} if $I^\sharp$ is $Q$-definably bijective to $\K^+ \times I_{\tilde i}^\sharp$, where $i \in [m]$ and $I_{\tilde i} = \pr_{\tilde i}(I)$. For every $t \in (\K^+)^{n}$ and every $\alpha \in \Gamma^m$, $\alpha$ is $(t,\alpha)$-reducible if and only if, by Lemma~\ref{gam:tup:red}, there is a $(t,\alpha)$-definable nonempty proper subset of $\alpha^\sharp$ if and only if, by Lemma~\ref{gam:tup:red} again, there is an $\alpha$-definable set $U \subseteq (\K^+)^{n}$ containing $t$ such that $\alpha$ is $(u,\alpha)$-reducible for every $u \in U$ if and only if, by $o$\nobreakdash-minimality in the $\K$-sort and Lemma~\ref{RV:no:point}, $\alpha$ is $\alpha$-reducible.
We say that a definable set $A$ in $\RV$ is \emph{$\Gamma$-tamped} of \emph{height} $l$ if there are $U \in \RES[k]$ and $I \in \Gamma[l]$ with $\dim_{\Gamma}(I) = l$ such that $A = U \times I^\sharp$. In that case, there is only one way to write $A$ as such a product, and if $B = V \times J^\sharp \subseteq A$ is also $\Gamma$-tamped then the coordinates occupied by $J^\sharp$ are also occupied by $I^\sharp$, in particular, $\dim_{\Gamma}(J) = l$ if and only if $V \subseteq U$ and $J \subseteq I$.
\begin{lem}\label{Gtamp} Let $A = U \times I^\sharp$, $B = V \times J^\sharp$ be $\Gamma$-tamped sets of the same height $l$, where $U$, $V$ are sets in $\K^+$. Let $f$ be a definable bijection whose domain contains $A$ and whose range contains $ B$. Suppose that $B \smallsetminus f( A)$, $A \smallsetminus f^{-1}(B)$ do not have $\Gamma$-tamped subsets of height $l$. Then there are finitely many $\Gamma$-tamped sets $A_i = U_i \times I_i^\sharp \subseteq U \times I^\sharp$ and $B_i = V_i \times J_i^\sharp \subseteq V \times J^\sharp$ such that \begin{itemize}
\item $ A \smallsetminus \bigcup_i A_i$ and $ B \smallsetminus \bigcup_i B_i$ do not have $\Gamma$-tamped subsets of height $l$,
\item each restriction $f \upharpoonright A_i$ is of the form $p_i \times q_i$, where $p_i : U_i \longrightarrow V_i$, $q_i : I_i^\sharp \longrightarrow J_i^\sharp$ are bijections and the latter $\vrv$-contracts to a $\Gamma[*]$-morphism $q_{i \downarrow} : I_i \longrightarrow J_i$. \end{itemize} \end{lem}
Let $t \times \alpha^\sharp \subseteq A$. If $t \times \alpha^\sharp \subseteq A \smallsetminus f^{-1}(B)$ then, by Lemma~\ref{resg:decom}, it is contained in a definable set $U' \times I'^\sharp \subseteq A \smallsetminus f^{-1}(B)$ with $U' \subseteq U$ and $I' \subseteq I$. Since $A \smallsetminus f^{-1}(B)$ does not have $\Gamma$-tamped subsets of height $l$, we must have $\dim_{\Gamma}(I') < l$. It follows from (the proof of) Lemma~\ref{gam:red:K} that $I'$ is piecewise reducible, which implies that $\alpha$ is $\alpha$-reducible. At any rate, if $\alpha$ is $(t,\alpha)$-reducible then $\alpha$ is $\alpha$-reducible and hence there is a reducible subset of $I$ that contains $\alpha$.
\begin{proof} Remove all the reducible subsets of $I$ from $I$ and call the resulting set $\bar I$; similarly for $\bar J$. Then, for all $t \in U$ and all $\alpha \in \bar I$, $f(t \times \alpha^\sharp)$ must be contained in a set of the form $s \times \beta^\sharp$, for otherwise it would have a $(t,\alpha)$-definable nonempty proper subset and hence would be $(t,\alpha)$-reducible. In fact, $f(t \times \alpha^\sharp) = s \times \beta^\sharp$, for otherwise $\beta$ is $(t,\alpha)$-reducible and hence, by $o$\nobreakdash-minimality in the $\K$-sort and the assumption $\dim_{\Gamma}(I) = \dim_{\Gamma}(J) = l$, a $(t,\alpha)$-definable subset of $\alpha^\sharp$ can be easily constructed. For the same reason, we must actually have $\beta \in \bar J$. It follows that $f(U \times \bar I^\sharp) = V \times \bar J^\sharp$. Then, by compactness, there are finitely many reducible subsets $I_i$ of $I$ such that, for all $t \in U$ and all $\alpha \in I_* = I \smallsetminus \bigcup_i I_i$, $f(t \times \alpha^\sharp) = s \times \beta^\sharp$ for some $s \in V$ and $\beta \in J$. Applying Lemma~\ref{resg:decom} to (the graph of) the function on $U \times I_*$ induced by $f$, the lemma follows. \end{proof}
\begin{prop}\label{red:D:iso} $\bb D$ is an isomorphism of graded semirings. \end{prop} \begin{proof} Surjectivity of $\bb D$ follows immediately from Lemma~\ref{RV:decom:RES:G}. For injectivity, let $\bm U_i \coloneqq (U_i, f_i)$, $\bm V_j \coloneqq (V_j, g_j)$ be objects in $\RES[*]$ and $I_i$, $J_j$ objects in $\Gamma[*]$ such that $\bb D([\bm U_i] \otimes [I_i])$, $\bb D([\bm V_j] \otimes [J_j])$ are objects in $\gsk \RV[l]$ for all $i$, $j$. Set \[ \textstyle M_i = U_i \times I_i^\sharp, \quad N_i = V_j \times J_j^\sharp, \quad M = \biguplus_i M_i, \quad N = \biguplus_j N_j. \] Suppose that there is a definable bijection $f : M \longrightarrow N$. We need to show \[ \textstyle \sum_i [\bm U_i] \otimes [I_i] = \sum_j [\bm V_j] \otimes [J_j]. \] By Lemma~\ref{gam:red:K}, we may assume that all $M_{i}$, $N_{j}$ are $\Gamma$-tamped. By $o$\nobreakdash-minimal cell decomposition, without changing the sums, we may assume that each $U_i$ is a disjoint union of finitely many copies of $(\K^+)^i$ and thereby re-index $M_i$ more informatively as $M_{i, m} = U_i \times I_m^\sharp$, where $I_m$ is an object in $\Gamma[m]$; similarly each $N_j$ is re-indexed as $N_{j, n}$. By Lemma~\ref{dim:cut:gam}, the respective maximums of the numbers $i+m$, $j+n$ are the $\RV$-dimensions of $M$, $N$ and hence must be equal; it is denoted by $p$. Let $q$ be the largest $m$ such that $i + m = p$ for some $M_{i, m}$ and $q'$ the largest $n$ such that $j + n = p$ for some $N_{j, n}$. It is not hard to see that we may arrange $q = q'$.
We now proceed by induction on $q$. The base case $q=0$ is rather trivial. For the inductive step, by Lemma~\ref{Gtamp}, we see that certain products contained in $M_{p-q, q}$, $N_{p-q, q}$ give rise to the same sum and the inductive hypothesis may be applied to the remaining portions. \end{proof}
We may view $\Gamma$ as a double cover of $\abs \Gamma$ via the identification $\Gamma / {\pm 1} = \abs \Gamma$. Consequently we can associate two Euler characteristics $\chi_{\Gamma,g}$, $\chi_{\Gamma, b}$ with the $\Gamma$-sort, induced by those on $|\Gamma|$ (see \cite{kage:fujita:2006} and also~\cite[\S~ 9]{hrushovski:kazhdan:integration:vf}). They are distinguished by \[ \chi_{\Gamma, g}(\bm H) = \chi_{\Gamma, g}(\bm H^{-1}) = -1 \quad \text{and} \quad \chi_{\Gamma, b}(\bm H) = \chi_{\Gamma, b}(\bm H^{-1}) = 0. \] Similarly, there is an Euler characteristic $\chi_{\K}$ associated with the $\K$-sort (there is only one). We shall denote all of these Euler characteristics simply by $\chi$ if no confusion can arise. Using these $\chi$ and the groupification of $\bb D$ (also denoted by $\bb D$), we can construct various retractions from the Grothendieck ring $\ggk \RV[*]$ to (certain localizations of) the Grothendieck rings $\ggk \RES[*]$ and $\ggk \Gamma[*]$.
\begin{lem}\label{gam:euler} The Euler characteristics induce naturally three graded ring homomorphisms: \[ \mdl E_{\K} : \ggk \RES[*] \longrightarrow \mathds{Z}[X] \quad \text{and} \quad \mdl E_{\Gamma, g}, \mdl E_{\Gamma, b} : \ggk \Gamma[*] \longrightarrow \mathds{Z}[X]. \] \end{lem} \begin{proof} For $U \in \RES[k]$ and $I \in \Gamma[k]$, we set $\mdl E_{\K, k}([U]) = \chi(U)$ (see Remark~\ref{omin:res}) and $\mdl E_{\Gamma, k}([I]) = \chi(I)$. These maps are well-defined and they induce graded ring homomorphisms $\mdl E_{\K} \coloneqq \sum_k \mdl E_{\K, k} X^k$ and $\mdl E_{\Gamma} \coloneqq \sum_k \mdl E_{\Gamma, k} X^k$ as desired. \end{proof}
By the computation in \cite{kage:fujita:2006}, $\ggk \Gamma[*]$ is canonically isomorphic to the graded ring \[ \textstyle \mathds{Z}[X, Y^{(2)}] \coloneqq \mathds{Z} \oplus \bigoplus_{i \geq 1} (\mathds{Z}[Y]/(Y^2+Y))X^i, \] where $YX$ represents the class $[\bm H] = [\bm H^{-1}]$ in $\ggk \Gamma[1]$. Thus $\mdl E_{\Gamma, g}$, $\mdl E_{\Gamma, b}$ are also given by \[ \mathds{Z}[X, Y^{(2)}] \two^{Y \longmapsto -1}_{Y \longmapsto 0} \mathds{Z}[X]. \]
\begin{rem}[Explicit description of ${\ggk \RV[*]}$]\label{rem:poin} Of course, $\mdl E_{\K}$ is actually an isomorphism. The homomorphism $\gsk \Gamma^{c}[*] \longrightarrow \gsk \RES[*]$ in Remark~\ref{gam:res} and $\mdl E_{\K}$ then induce an isomorphism $\mdl E_{\K^c} : \ggk \Gamma^{c}[*] \longrightarrow \mathds{Z}[X]$. But this isomorphism is different from the groupification $\mdl E_{\Gamma^c}$ of the canonical isomorphism $\gsk \Gamma^{c}[*] \cong \gsk \mathds{N}[*]$. This latter isomorphism $\mdl E_{\Gamma^c}$ is also induced by $\mdl E_{\Gamma, g}$, $\mdl E_{\Gamma, b}$ (the two homomorphisms agree on $\ggk \Gamma^{c}[*]$). They are distinguished by $\mdl E_{\K^c}([e]) = -X$ and $\mdl E_{\Gamma^c}([e]) = X$. We have a commutative diagram \[ \bfig
\hSquares(0,0)/<-`->`->`->`->`<-`->/[{\ggk \RES[*]}`{\ggk \Gamma^{c}[*]}`{\ggk \Gamma[*]}`\mathds{Z}[X]`\mathds{Z}[X]`{\mathds{Z}[X, Y^{(2)}]}; ``\mdl E_{\K}`\mdl E_{\Gamma^c}`\cong`\tau`] \efig \] where $\tau$ is the involution determined by $X \longmapsto -X$. The graded ring \[ \mathds{Z}[X] \otimes_{\mathds{Z}[X]} \mathds{Z}[X, Y^{(2)}] \] may be identified with $\mathds{Z}[X, Y^{(2)}]$ via the isomorphism given by $x \otimes y \longmapsto \tau(x)y$. Consequently, by Proposition~\ref{red:D:iso}, there is a graded ring isomorphism \[ \ggk \RV[*] \to^{\sim} \mathds{Z}[X, Y^{(2)}] \quad \text{with} \quad \bm 1_{\K} + [\bm P] \longmapsto 1 + 2YX + X. \] Setting \[
\mathds{Z}^{(2)}[X] = \mathds{Z}[X, Y^{(2)}] / (1 + 2YX + X), \] we see that there is a canonical ring isomorphism \[ \bb E_{\Gamma}: \ggk \RV[*] / (\bm 1_{\K} + [\bm P]) \to^{\sim} \mathds{Z}^{(2)}[X]. \]
There are exactly two ring homomorphisms $\mathds{Z}^{(2)}[X] \longrightarrow \mathds{Z}$ determined by the assignments $Y \longmapsto -1$ and $Y \longmapsto 0$ or, equivalently, $X \longmapsto 1$ and $X \longmapsto -1$. Combining these with $\bb E_{\Gamma}$, we see that there are exactly two ring homomorphisms \[ \bb E_{\Gamma,g}, \bb E_{\Gamma,b}: \ggk \RV[*] / (\bm 1_{\K} + [\bm P]) \longrightarrow \mathds{Z}. \] \end{rem}
\begin{prop}\label{prop:eu:retr:k} There are two ring homomorphisms \[ \bb E_{\K, g}: \ggk \RV[*] \longrightarrow \ggk \RES[*][[\bm A]^{-1}] \quad \text{and} \quad \bb E_{\K, b}: \ggk \RV[*] \longrightarrow \ggk \RES[*][[1]^{-1}] \] such that \begin{itemize}
\item their ranges are precisely the zeroth graded pieces of their respective codomains,
\item $\bm 1_{\K} + [\bm P]$ vanishes under both of them,
\item for all $x \in \ggk \RES[k]$, $\bb E_{\K, g} (x) = x [\bm A]^{-k}$ and $\bb E_{\K, b}(x) = x [1]^{-k}$. \end{itemize} \end{prop} \begin{proof} We first define, for each $n$, a homomorphism \[ \bb E_{g, n}: \ggk \RV[n] \longrightarrow \ggk \RES[n] \] as follows. By Proposition~\ref{red:D:iso}, there is an isomorphism \[ \textstyle \bb D_n : \bigoplus_{i + j = n} \ggk \RES[i] \otimes \ggk \Gamma[j] \to^{\sim} \ggk \RV[n]. \] Let the group homomorphism $\mdl E_{g, j} : \ggk \Gamma[j] \longrightarrow \mathds{Z}$ be defined as in Lemma~\ref{gam:euler}, using $\chi_{\Gamma, g}$. Let \[ E_{g}^{i, j}: \ggk \RES[i] \otimes \ggk \Gamma[j] \longrightarrow \ggk \RES[i {+} j] \] be the group homomorphism determined by $x \otimes y \longmapsto \mdl E_{g, j}(y) x [\bm T]^{j}$. Let \[ \textstyle E_{g, n} = \sum_{i + j = n} E_{g}^{i, j} \quad \text{and} \quad \bb E_{g, n} = E_{g, n} \circ \bb D_n^{-1}. \]
Note that, due to the presence of the tensor $\otimes_{\ggk \Gamma^{c}[*]}$ and the replacement of $y$ with $\mdl E_{g, j}(y) [\bm T]^{j}$, there is this issue of compatibility between the various components of $E_{g, n}$. In our setting, this is easily resolved since all definable bijections are allowed in $\Gamma[*]$ and hence $\gsk \Gamma^c[*]$ is generated by isomorphism classes of the form $[e]^k$. In the setting of \cite{hrushovski:kazhdan:integration:vf}, however, one has to pass to a quotient ring to achieve compatibility (see Remark~\ref{why:glz} and also \cite[\S~2.5]{hru:loe:lef}).
Now, it is straightforward to check the equality \[ \bb E_{g, n}(x)\bb E_{g, m}(y) = \bb E_{g, n+m}(xy). \]
The group homomorphisms $\tau_{m, k} : \ggk \RES[m] \longrightarrow \ggk \RES[m{+}k]$ given by $x \longmapsto x [\bm A]^k$ determine a colimit system and the group homomorphisms \[ \textstyle\bb E_{g, \leq n} \coloneqq \sum_{m \leq n} \tau_{m, n-m} \circ \bb E_{g, m} : \ggk \RV[{\leq} n] \longrightarrow \ggk \RES[n] \] determine a homomorphism of colimit systems. Hence we have a ring homomorphism: \[ \colim{n} \bb E_{g, \leq n} : \ggk \RV[*] \longrightarrow \colim{\tau_{n, k}} \ggk \RES[n]. \] For all $n \geq 1$ we have \[ \bb E_{g, \leq n}(\bm 1_{\K} + [\bm P]) = [\bm A]^n - 2[\bm T][\bm A]^{n-1} - [1] [\bm A]^{n-1} = 0. \] This yields the desired homomorphism $\bb E_{\K, g}$ since the colimit in question can be embedded into the zeroth graded piece of $\ggk \RES[*][[\bm A]^{-1}]$.
The construction of $\bb E_{\K, b}$ is completely analogous, with $[\bm A]$ replaced by $[1]$ and $\chi_{\Gamma, g}$ by $\chi_{\Gamma, b}$. \end{proof}
Since the zeroth graded pieces of both $\ggk \RES[*][[\bm A]^{-1}]$ and $\ggk \RES[*][[1]^{-1}]$ are canonically isomorphic to $\mathds{Z}$, the homomorphisms $\bb E_{\K, g}$, $\bb E_{\K, b}$ are just the homomorphisms $\bb E_{\Gamma, g}$, $\bb E_{\Gamma, b}$ in Remark~\ref{rem:poin}, more precisely, $\bb E_{\K, g} = \bb E_{\Gamma, g}$ and $\bb E_{\K, b} = \bb E_{\Gamma, b}$.
\section{Generalized Euler characteristic}
From here on, our discussion will be of an increasingly formal nature. Many statements are exact copies of those in \cite{Yin:special:trans, Yin:int:acvf, Yin:int:expan:acvf} and often the same proofs work, provided that the auxiliary results are replaced by the corresponding ones obtained above. For the reader's convenience, we will write down all the details.
\subsection{Special bijections}
Our first task is to connect $\gsk \VF_*$ with $\gsk \RV[*]$, more precisely, to establish a surjective homomorphism $\gsk \RV[*] \longrightarrow \gsk \VF_*$. Notice the direction of the arrow. The main instrument in this endeavor is special bijections.
\begin{conv}\label{conv:can} We reiterate \cite[Convention~2.32]{Yin:int:expan:acvf} here, with a different terminology, since this trivial-looking convention is actually quite crucial for understanding the discussion below, especially the parts that involve special bijections. For any set $A$, let \[ \can(A) = \{(a, \rv(a), t) : (a, t) \in A \text{ and } a \in \pr_{\VF}(A)\}. \] The natural bijection $\can : A \longrightarrow \can(A)$ is called the \emph{regularization} of $A$. We shall tacitly substitute $\can(A)$ for $A$ if it is necessary or is just more convenient. Whether this substitution has been performed or not should be clear in context (or rather, it is always performed). \end{conv}
\begin{defn}[Special bijections]\label{defn:special:bijection} Let $A$ be a (regularized) definable set whose first coordinate is a $\VF$-coordinate (of course nothing is special about the first $\VF$-coordinate, we choose it simply for notational ease). Let $C \subseteq \RVH(A)$ be an $\RV$-pullback (see Definition~\ref{defn:disc}) and \[ \lambda: \pr_{>1}(C \cap A) \longrightarrow \VF \] a definable function whose graph is contained in $C$. Recall Notation~\ref{nota:tor}. Let \[ \textstyle C^{\sharp} = \bigcup_{x \in \pr_{>1} (C)} \MM_{\abvrv(\pr_1(x_{\RV}))} \times x \quad \text{and} \quad \RVH(A)^{\sharp} = C^{\sharp} \uplus (\RVH(A) \smallsetminus C), \] where $x_{\RV} = \pr_{\RV}(x)$. The \emph{centripetal transformation $\eta : A \longrightarrow \RVH(A)^{\sharp}$ with respect to $\lambda$} is defined by \[ \begin{cases}
\eta (a, x) = (a - \lambda(x), x), & \text{on } C \cap A,\\
\eta = \id, & \text{on } A \smallsetminus C. \end{cases} \] Note that $\eta$ is injective. The inverse of $\eta$ is naturally called the \emph{centrifugal transformation with respect to $\lambda$}. The function $\lambda$ is referred to as the \emph{focus} of $\eta$ and the $\RV$-pullback $C$ as the \emph{locus} of $\lambda$ (or $\eta$).
A \emph{special bijection} $T$ on $A$ is an alternating composition of centripetal transformations and regularizations. By Convention~\ref{conv:can}, we shall only display the centripetal transformations in such a composition. The \emph{length} of such a special bijection $T$, denoted by $\lh(T)$, is the number of centripetal transformations in $T$. The range of $T$ is sometimes denoted by $A^{\flat}$. \end{defn}
For functions between sets that have only one $\VF$-coordinate, composing with special bijections on the right and inverses of special bijections on the left obviously preserves dtdp.
\begin{lem}\label{inverse:special:dim:1} Let $T$ be a special bijection on $A \subseteq \VF \times \RV^m$ such that $A^{\flat}$ is an $\RV$-pullback. Then there is a definable function $\epsilon : \pr_{\RV} (A^{\flat}) \longrightarrow \VF$ such that, for every $\RV$-polydisc $\mathfrak{p} = t^\sharp \times s \subseteq A^{\flat}$, $(T^{-1}(\mathfrak{p}))_{\VF} = t^\sharp + \epsilon(s)$. \end{lem} \begin{proof} It is clear that $\mathfrak{p}$ is the image of an open polydisc $\mathfrak{a} \times r \subseteq A$. Let $T'$ be $T$ with the last centripetal transformation deleted. Then $T'(\mathfrak{a} \times r)$ is also an open polydisc $\mathfrak{a}' \times r'$. The range of the focus map of $\eta_n$ contains a point in the smallest closed disc containing $\mathfrak{a}'$. This point can be transported back by the previous focus maps to a point in the smallest closed disc containing $\mathfrak{a}$. The lemma follows easily from this observation. \end{proof}
Note that, since $\dom(\epsilon) \subseteq \RV^l$ for some $l$, by Corollary~\ref{function:rv:to:vf:finite:image}, $\ran(\epsilon)$ is actually finite.
A definable set $A$ is called a \emph{deformed $\RV$-pullback} if there is a special bijection $T$ on $A$ such that $A^{\flat}$ is an $\RV$-pullback.
\begin{lem}\label{simplex:with:hole:rvproduct} Every definable set $A \subseteq \VF \times \RV^m$ is a deformed $\RV$-pullback. \end{lem} \begin{proof} By compactness and HNF this is immediately reduced to the situation where $A \subseteq \VF$ is contained in an $\RV$-disc and is a $\vv$-interval with end-discs $\mathfrak{a}$, $\mathfrak{b}$. This may be further divided into several cases according to whether $\mathfrak{a}$, $\mathfrak{b}$ are open or closed discs and whether the ends of $A$ are open or closed. In each of these cases, Lemma~\ref{clo:disc:bary} is applied in much the same way as its counterpart is applied in the proof of \cite[Lemma~4.26]{Yin:QE:ACVF:min}. It is a tedious exercise and is left to the reader. \end{proof}
Here is an analogue of \cite[Theorem~5.4]{Yin:special:trans} (see also \cite[Theorem~4.25]{Yin:int:expan:acvf}):
\begin{thm}\label{special:term:constant:disc} Let $F(x) = F(x_1, \ldots, x_n)$ be an $\lan{T}{}{}$-term. Let $u \in \RV^n$ and $R : u^\sharp \longrightarrow A$ be a special bijection. Then there is a special bijection $T : A \longrightarrow A^\flat$ such that $F \circ R^{-1} \circ T^{-1}$ is $\rv$-contractible. In a commutative diagram, \[ \bfig
\square(0,0)/`->`->`->/<1500,400>[A^\flat`\VF`\rv(A^\flat)`\RV_0;
`\rv`\rv`(F \circ R^{-1} \circ T^{-1})_{\downarrow}]
\morphism(0,400)<500,0>[A^\flat`A; T^{-1}]
\morphism(500,400)<500,0>[A`u^\sharp; R^{-1}]
\morphism(1000,400)<500,0>[u^\sharp`\VF; F]
\efig \] \end{thm} \begin{proof} First observe that if the assertion holds for one $\lan{T}{}{}$-term then it holds simultaneously for any finite number of $\lan{T}{}{}$-terms, since $\rv$-contractibility is preserved by further special bijections on $A^\flat$. We do induction on $n$. For the base case $n=1$, by Corollary~\ref{part:rv:cons} and Remark~\ref{rem:LT:com}, there is a definable finite partition $B_i$ of $u^\sharp$ such that, for all $i$, if $\mathfrak{a} \subseteq B_i$ is an open disc then $\rv \upharpoonright F(\mathfrak{a})$ is constant. By consecutive applications of Lemma~\ref{simplex:with:hole:rvproduct}, we obtain a special bijection $T$ on $A$ such that each $(T \circ R) (B_i)$ is an $\RV$-pullback. Clearly $T$ is as required.
For the inductive step, we may concentrate on a single $\RV$-polydisc $\mathfrak{p} = v^\sharp \times (v, r) \subseteq A$. Let $\phi(x, y)$ be a quantifier-free formula that defines the function $\rv \circ f$. Recall Convention~\ref{topterm}. Let $G_{i}(x)$ enumerate the top $\lan{T}{}{}$-terms of $\phi$. For $a \in v_1^\sharp$, write $G_{i,a} = G_{i}(a, x_2, \ldots, x_n)$. By the inductive hypothesis, there is a special bijection $R_{a}$ on $(v_2, \ldots, v_n)^\sharp$ such that every $G_{i,a} \circ R_a^{-1}$ is $\rv$-contractible. Let $U_{k, a}$ enumerate the loci of the components of $R_{a}$ and $\lambda_{k, a}$ the corresponding focus maps. By compactness, \begin{itemize}
\item for each $i$, there is a quantifier-free formula $\psi_i$ such that $\psi_i(a)$ defines $(G_{i,a} \circ R_a^{-1})_{\downarrow}$,
\item there is a quantifier-free formula $\theta$ such that $\theta(a)$ determines the sequence $\rv(U_{k, a})$ and the $\VF$-coordinates targeted by $\lambda_{k, a}$. \end{itemize} Let $H_{j}(x_1)$ enumerate the top $\lan{T}{}{}$-terms of the formulas $\psi_i$, $\theta$. Applying the inductive hypothesis again, we obtain a special bijection $T_1$ on $v_1^\sharp$ such that every $H_{j} \circ T_1^{-1}$ is $\rv$-contractible. This means that, for every $\RV$-polydisc $\mathfrak{q} \subseteq T_1(v_1^\sharp)$ and all $a_1, a_2 \in T_1^{-1}(\mathfrak{q})$, \begin{itemize}
\item the formulas $\psi_i(a_1)$, $\psi_i(a_2)$ define the same $\rv$-contraction,
\item the special bijections $R_{a_1}$, $R_{a_2}$ may be glued together in the obvious sense to form one special bijection on $\{a_1, a_2\} \times (v_2, \ldots, v_n)^\sharp$. \end{itemize} Consequently, $T_1$ and $R_{a}$ naturally induce a special bijection $T$ on $\mathfrak{p}$ such that every $G_{i} \circ T^{-1}$ is $\rv$-contractible. This implies that $F \circ R^{-1} \circ T^{-1}$ is $\rv$-contractible and hence $T$ is as required. \end{proof}
\begin{cor}\label{special:bi:term:constant} Let $A \subseteq \VF^n$ be a definable set and $f : A \longrightarrow \RV^m$ a definable function. Then there is a special bijection $T$ on $A$ such that $A^\flat$ is an $\RV$-pullback and the function $f \circ T^{-1}$ is $\rv$-contractible. \end{cor} \begin{proof} By compactness, we may assume that $A$ is contained in an $\RV$-polydisc $\mathfrak{p}$. Let $\phi$ be a quantifier-free formula that defines $f$. Let $F_i(x, y)$ enumerate the top $\lan{T}{}{}$-terms of $\phi$. For $s \in \RV^{m}$, let $F_{i, s} = F_{i}(x, s)$. By Theorem~\ref{special:term:constant:disc}, there is a special bijection $T$ on $\mathfrak{p}$ such that each function $F_{i, s} \circ T^{-1}$ is contractible. This means that, for each $\RV$-polydisc $\mathfrak{q} \subseteq T(\mathfrak{p})$, \begin{itemize}
\item either $T^{-1}(\mathfrak{q}) \subseteq A$ or $T^{-1}(\mathfrak{q}) \cap A = \emptyset$,
\item if $T^{-1}(\mathfrak{q}) \subseteq A$ then $(f \circ T^{-1})(\mathfrak{q})$ is a singleton. \end{itemize} So $T \upharpoonright A$ is as required. \end{proof}
\begin{defn}[Lifting maps]\label{def:L} Let $U$ be a set in $\RV$ and $f : U \longrightarrow \RV^k$ a function. Let $U_f$ stand for the set $\bigcup \{f(u)^\sharp \times u: u \in U\}$. The \emph{$k$th lifting map} \[ \mathbb{L}_k: \RV[k] \longrightarrow \VF[k] \]
is given by $(U,f) \longmapsto U_f$. The map $\mathbb{L}_{\leq k}: \RV[{\leq} k] \longrightarrow \VF[k]$ is given by $\bigoplus_{i} \bm U_i \longmapsto \biguplus_{i} \bb L_i \bm U_i$. Set $\mathbb{L} = \bigcup_k \mathbb{L}_{\leq k}$. \end{defn}
\begin{cor}\label{all:subsets:rvproduct} Every definable set $A \subseteq \VF^n \times \RV^m$ is a deformed $\RV$-pullback. In particular, if $A \in \VF_*$ then there are a $\bm U \in \RV[{\leq} n]$ and a special bijection from $A$ onto $\mathbb{L}_{{\leq} n}(\bm U)$. \end{cor} \begin{proof} For the first assertion, by compactness, we may assume $A \subseteq \VF^n$. Then it is a special case of Corollary~\ref{special:bi:term:constant}. The second assertion follows from Lemma~\ref{RV:fiber:dim:same}. \end{proof}
\begin{defn}[Lifts]\label{def:lift} Let $F: (U, f) \longrightarrow (V, g)$ be an $\RV[k]$-morphism. Then $F$ induces a definable finite-to-finite correspondence $F^\dag \subseteq f(U) \times g(V)$. Since $F^\dag$ can be decomposed into finitely many definable bijections, for simplicity, we assume that $F^\dag$ is itself a bijection. Let $F^{\sharp} : f(U)^\sharp \longrightarrow g(V)^\sharp$ be a definable bijection that $\rv$-contracts to $F^\dag$. Then $F^\sharp$ is called a \emph{lift} of $F$. By Convention~\ref{conv:can}, we shall think of $F^\sharp$ as a definable bijection $\bb L(U, f) \longrightarrow \bb L(V, g)$ that $\rv$-contracts to $F^\dag$. \end{defn}
\begin{lem}\label{simul:special:dim:1} Let $f : A \longrightarrow B$ be a definable bijection between two sets that have exactly one $\VF$-coordinate each. Then there are special bijections $T_A : A \longrightarrow A^{\flat}$, $T_B : B \longrightarrow B^{\flat}$ such that $A^{\flat}$, $B^{\flat}$ are $\RV$-pullbacks and $f^{\flat}_{\downarrow}$ is bijective in the commutative diagram \[ \bfig
\square(0,0)/->`->`->`->/<600,400>[A`A^{\flat}`B`B^{\flat};
T_A`f``T_B]
\square(600,0)/->`->`->`->/<600,400>[A^{\flat}`\rv(A^{\flat})`B^{\flat} `\rv(B^{\flat}); \rv`f^{\flat}`f^{\flat}_{\downarrow}`\rv]
\efig \] Thus, if $A, B \in \VF_*$ then $f^{\flat}$ is a lift of $f^{\flat}_{\downarrow}$, where the latter is regarded as an $\RV[1]$-morphism between $\rv(A^{\flat})_{1}$ and $\rv(B^{\flat})_1$ (recall Notation~\ref{0coor}). \end{lem} \begin{proof} By Corollaries~\ref{special:bi:term:constant}, \ref{all:subsets:rvproduct}, and Lemma~\ref{open:pro}, we may assume that $A$, $B$ are $\RV$-pullbacks, $f$ is $\rv$-contractible and has dtdp, and there is a special bijection $T_B: B \longrightarrow B^{\flat}$ such that $(T_B \circ f)^{-1}$ is $\rv$-contractible. Let $T_B = \eta_{n} \circ \ldots \circ \eta_{1}$, where each $\eta_{i}$ is a centripetal transformation (and regularization maps are not displayed). Then it is enough to construct a special bijection $T_A = \zeta_{n} \circ \ldots \circ \zeta_{1}$ on $A$ such that, for each $i$, both $f_i \coloneqq T_{B, i} \circ f \circ T_{A, i}^{-1}$ and $T_{A, i} \circ (T_B \circ f)^{-1}$ are $\rv$-contractible, where $T_{B, i} = \eta_{i} \circ \ldots \circ \eta_{1}$ and $T_{A, i} = \zeta_{i} \circ \ldots \circ \zeta_{1}$.
To that end, suppose that $\zeta_i$ has been constructed for each $i \leq k < n$. Let $A_{k} = T_{A, k}(A)$ and $B_k = T_{B, k}(B)$. Let $D \subseteq B_k$ be the locus of $\eta_{k+1}$ and $\lambda$ the corresponding focus map. Since $f_k$ is $\rv$-contractible and has dtdp, each $\RV$-polydisc $\mathfrak{p} \subseteq B_k$ is a union of disjoint sets of the form $f_k(\mathfrak{q})$, where $\mathfrak{q} \subseteq A_k$ is an $\RV$-polydisc. For each $t = (t_1, t_{\tilde 1}) \in \dom(\lambda)$, let $O_{t}$ be the set of those $\RV$-polydiscs $\mathfrak{q} \subseteq A_k$ such that $f_k(\mathfrak{q}) \subseteq t^\sharp_1 \times t$. Let \begin{itemize}
\item $\mathfrak{q}_{t} \in O_{t}$ be the $\RV$-polydisc with $(\lambda(t), t) \in \mathfrak{o}_{ t} \coloneqq f_k(\mathfrak{q}_t)$,
\item $C = \bigcup_{t \in \dom(\lambda)} \mathfrak{q}_{t} \subseteq A_k$ and $a_{t} = f_k^{-1}(\lambda( t), t) \in \mathfrak{q}_{t}$,
\item $\kappa : \pr_{>1} (C) \longrightarrow \VF$ the corresponding focus map given by $\pr_{>1} (\mathfrak{q}_{t}) \longmapsto \pr_1(a_{t})$,
\item $\zeta_{k+1}$ the centripetal transformation determined by $C$ and $\kappa$. \end{itemize} For each $t \in \dom(\lambda)$, $f_{k+1}$ restricts to a bijection between the $\RV$-pullbacks $\zeta_{k+1}(\mathfrak{q}_{t})$ and $\eta_{k+1}(\mathfrak{o}_{t})$ that is $\rv$-contractible in both ways and, for any $\mathfrak{q} \in O_{t}$ with $\mathfrak{q} \neq \mathfrak{q}_{t}$, $f_{k+1}(\mathfrak{q})$ is an open polydisc contained in an $\RV$-polydisc. So $f_{k+1}$ is $\rv$-contractible.
On the other hand, it is clear that, for any $\RV$-polydisc $\mathfrak{p} \subseteq B^{\flat}$, $T_{A, k} \circ (T_B \circ f)^{-1}(\mathfrak{p})$ does not contain any $a_{t}$ and hence, by the construction of $T_{A, k}$, $T_{A, k+1} \circ (T_B \circ f)^{-1}$ is $\rv$-contractible. \end{proof}
\begin{hyp}\label{hyp:point} The following lemma is used directly only once in Corollary~\ref{RV:lift}. It should have been presented right after Definition~\ref{defn:corr:cont}. We place it here because this is the first place in this paper, in fact, one of the only two places, the other being Lemma~\ref{blowup:same:RV:coa}, where we need to assume that every definable $\RV$-disc contains a definable point. The easiest way to guarantee this is to assume that $\mdl S$ is $\VF$-generated, which, together with Hypothesis~\ref{hyp:gam}, implies that it is a model of $\TCVF$ and is indeed an elementary substructure (so every definable set contains a definable point). This assumption will be in effect throughout the rest of the paper. \end{hyp}
\begin{lem}\label{RVlift} Every definable bijection $f : U \longrightarrow V$ between two subsets of $\RV^k$ can be lifted, that is, there is a definable bijection $f^{\sharp} : U^\sharp \longrightarrow V^\sharp$ that $\rv$-contracts to $f$. \end{lem} \begin{proof} We do induction on $n = \dim_{\RV}(U) = \dim_{\RV}(V)$. If $n=0$ then $U$ is finite and hence, for every $u \in U$, the $\RV$-polydisc $u^\sharp$ contains a definable point, similarly for $V$, in which case how to construct an $f^{\sharp}$ as desired is obvious.
For the inductive step, by weak $o$\nobreakdash-minimality in the $\RV$-sort, there are definable finite partitions $U_i$, $V_i$ of $U$, $V$ and injective coordinate projections \[ \pi_i : U_i \longrightarrow \RV^{k_i}, \quad \pi'_i : V_i \longrightarrow \RV^{k_i}, \] where $\dim_{\RV}(U_i) = \dim_{\RV}(V_i) = k_i$; the obvious bijection $\pi_i(U_i) \longrightarrow \pi'_i(V_i)$ induced by $f$ is denoted by $f_i$. Observe that if every $f_i$ can be lifted as desired then, by the construction in the base case above, $F$ can be lifted as desired as well. Therefore, without loss of generality, we may assume $k = n$. For $u \in U$ and $a \in u^\sharp$, the $\RV$-polydisc $f(u)^\sharp$ contains an $a$-definable point and hence, by compactness, there is a definable function $f^{\sharp} : U^\sharp \longrightarrow V^\sharp$ that $\rv$-contracts to $f$. By Lemma~\ref{RV:bou:dim}, $\dim_{\RV}(\partial_{\RV}f^{\sharp}(U^\sharp)) < n$ and hence, by the inductive hypothesis, we may assume that $f^{\sharp}$ is surjective. Then there is a definable function $g : V^\sharp \longrightarrow U^\sharp$ such that $f^{\sharp}(g(b)) = b$ for all $b \in V^\sharp$. By Lemma~\ref{RV:bou:dim} and the inductive hypothesis again, we may further assume that $g$ is also a surjection, which just means that $f^{\sharp}$ is a bijection as desired. \end{proof}
The following corollary is an analogue of \cite[Proposition~6.1]{hrushovski:kazhdan:integration:vf}.
\begin{cor}\label{RV:lift} For every $\RV[k]$-morphism $F : (U, f) \longrightarrow (V, g)$ there is a $\VF[k]$-morphism $F^\sharp$ that lifts $F$. \end{cor} \begin{proof} As in Definition~\ref{def:lift}, we may assume that the finite-to-finite correspondence $F^\dag$ is actually a bijection. Then this is immediate by Lemma~\ref{RVlift}. \end{proof}
\begin{cor}\label{L:sur:c} The lifting map $\bb L_{\leq k}$ induces a surjective homomorphism, which is sometimes simply denoted by $\bb L$, between the Grothendieck semigroups \[ \gsk \RV[{\leq} k] \epi \gsk \VF[k]. \] \end{cor} \begin{proof} By Corollary~\ref{RV:lift}, every $\RV[k]$-isomorphism can be lifted. So $\bb L_{\leq k}$ induces a map on the isomorphism classes, which is easily seen to be a semigroup homomorphism. By Lemma~\ref{altVFdim} and Corollary~\ref{all:subsets:rvproduct}, this homomorphism is surjective. \end{proof}
\subsection{$2$-cells}
The remaining object of this section is to identify the kernels of the semigroup homomorphisms $\bb L$ in Corollary~\ref{L:sur:c} and thereby complete the construction of the universal additive invariant. We begin with a discussion of the issue of $2$-cells, as in \cite[\S~4]{Yin:int:acvf}.
The notion of a $2$-cell, which corresponds to that of a bicell in \cite{cluckers:loeser:constructible:motivic:functions}, may look strange and is, perhaps, only of technical interest. It arises when we try to prove some analogue of Fubini's theorem, such as Lemma~\ref{contraction:perm:pair:isp} below. The difficulty is that, although the interaction between $\rv$-contractions and special bijections for definable sets of $\VF$-dimension $1$ is in a sense ``functorial'' (see Lemma~\ref{simul:special:dim:1}), we are unable to extend the construction to higher $\VF$-dimensions. This is the concern of \cite[Question~7.9]{hrushovski:kazhdan:integration:vf}. It has also occurred in \cite{cluckers:loeser:constructible:motivic:functions} and actually may be traced back to the construction of the $o$\nobreakdash-minimal Euler characteristic in \cite{dries:1998}; see \cite[Section~1.7]{cluckers:loeser:constructible:motivic:functions}.
Anyway, in this situation, a natural strategy for $\rv$-contracting the isomorphism class of a definable set of higher $\VF$-dimension is to apply the result for $\VF$-dimension $1$ parametrically and proceed with one $\VF$-coordinate at a time. As in the classical theory of integration, this strategy requires some form of Fubini's theorem: for a well-behaved integration (or additive invariant in our case), an integral should yield the same value when it is evaluated along different orders of the variables. By induction, this problem is immediately reduced to the case of two variables. A $2$-cell is a definable subset of $\VF^2$ with certain symmetrical (or ``linear'' in the sense described in Remark~\ref{2cell:linear} below) internal structure that satisfies this Fubini-type requirement. Now the idea is that, if we can find a definable partition for every definable set such that each piece is a $2$-cell indexed by some $\RV$-sort parameters, then, by compactness, every definable set satisfies the Fubini-type requirement. This kind of partition is achieved in Lemma~\ref{decom:into:2:units}.
\begin{lem}\label{bijection:dim:1:decom:RV} Let $f : A \longrightarrow B$ be a definable bijection between two subsets of $\VF$. Then there is a special bijection $T$ on $A$ such that $A^\flat$ is an $\RV$-pullback and, for each $\RV$-polydisc $\mathfrak{p} \subseteq A^\flat$, $f \upharpoonright T^{-1}(\mathfrak{p})$ is $\rv$-affine. \end{lem} \begin{proof}
By Lemma~\ref{rv:lin} and compactness, for all but finitely many $a \in A$ there is an $a$-definable $\delta_a \in \abs{\Gamma}$ such that $f \upharpoonright \mathfrak{o}(a, \delta_a)$ is $\rv$-affine. Without loss of generality, we may assume that, for all $a \in A$, $\delta_a$ exists and is the least element that satisfies this condition. Let $g : A \longrightarrow |\Gamma|$ be the definable function given by $a \longmapsto \delta_a$. By Corollary~\ref{special:bi:term:constant}, there is a special bijection $T$ on $A$ such that $A^\flat$ is an $\RV$-pullback and, for all $\RV$-polydisc $\mathfrak{p} \subseteq A^\flat$, $(g \circ T^{-1}) \upharpoonright \mathfrak{p}$ is constant. By Lemmas~\ref{one:atomic} and \ref{rv:lin}, we must have $(g \circ T^{-1})(\mathfrak{p}) \leq \rad(\mathfrak{p})$, for otherwise the choice of $\delta_a$ is violated for some $a \in T^{-1}(\mathfrak{p})$. So $T$ is as required. \end{proof}
\begin{lem}\label{bijection:rv:one:one} Let $A \subseteq \VF^2$ be a definable set such that $\mathfrak{a}_1 \coloneqq \pr_1(A)$ and $\mathfrak{a}_2 \coloneqq \pr_2(A)$ are open discs. Suppose that there is a definable bijection $f : \mathfrak{a}_1 \longrightarrow \mathfrak{a}_2$ that has dtdp and, for each $a \in \mathfrak{a}_1$, there is a $t_a \in \RVV$ with $A_a = t_a^\sharp + f(a)$. Then there is a special bijection $T$ on $\mathfrak{a}_1$ such that $\mathfrak{a}_1^\flat$ is an $\RV$-pullback and, for each $\RV$-polydisc $\mathfrak{p} \subseteq \mathfrak{a}_1^\flat$, $\rv$ is constant on the set \[ \{a - f^{-1}(b) : a \in T^{-1}(\mathfrak{p}) \text{ and } b \in A_a \}. \] \end{lem} \begin{proof} For each $a \in \mathfrak{a}_1$, let $\mathfrak{b}_a$ be the smallest closed disc that contains $A_a$. Since $A_a - f(a) = t_a^\sharp$, we have $f(a) \in \mathfrak{b}_a$ but $f(a) \notin A_a$ if $t_a \neq 0$. Hence $a \notin f^{-1}(A_a)$ if $t_a \neq 0$ and $\{a\} = f^{-1}(A_a)$ if $t_a = 0$. Since $f^{-1}(A_a)$ is a disc or a point, in either case, the function on $f^{-1}(A_a)$ given by $b \longmapsto \rv(a - b)$ is constant. The function $h : \mathfrak{a}_1 \longrightarrow \RVV$ given by $a \longmapsto \rv(a - f^{-1}(A_a))$ is definable. Now we apply Corollary~\ref{special:bi:term:constant} as in the proof of Lemma~\ref{bijection:dim:1:decom:RV}. The lemma follows. \end{proof}
\begin{defn}\label{defn:balance} Let $A$, $\mathfrak{a}_1$, $\mathfrak{a}_2$, and $f$ be as in Lemma~\ref{bijection:rv:one:one}. We say that $f$ is \emph{balanced in $A$} if $f$ is actually $\rv$-affine and there are $t_1, t_2 \in \RVV$, called the \emph{paradigms} of $f$, such that, for every $a \in \mathfrak{a}_1$, \[ A_a = t_2^\sharp + f(a) \quad \text{and} \quad f^{-1}(A_a) = a - t_1^\sharp. \] \end{defn}
\begin{rem}\label{2cell:linear} Suppose that $f$ is balanced in $A$ with paradigms $t_1$, $t_2$. If one of the paradigms is $0$ then the other one must be $0$. In this case $A$ is just the (graph of the) bijection $f$ itself.
Assume that $t_1$, $t_2$ are nonzero. Let $\mathfrak{B}_1$, $\mathfrak{B}_2$ be, respectively, the sets of closed subdiscs of $\mathfrak{a}_1$, $\mathfrak{a}_2$ of radii $\abs{\vrv(t_1)}$, $\abs{\vrv(t_2)}$. Let $a_1 \in \mathfrak{b}_1 \in \mathfrak{B}_1$ and $\mathfrak{o}_1$ be the maximal open subdisc of $\mathfrak{b}_1$ containing $a_1$. Let $\mathfrak{b}_2 \in \mathfrak{B}_2$ be the smallest closed disc containing the open disc $\mathfrak{o}_2 \coloneqq A_{a_1}$. Then, for all $a_2 \in \mathfrak{o}_2$, we have \[ \mathfrak{o}_2 = t_2^\sharp + f(\mathfrak{o}_1) = A_{a_1} \quad \text{and} \quad A_{a_2} = f^{-1}(\mathfrak{o}_2) + t_1^\sharp = \mathfrak{o}_1. \] This internal symmetry of $A$ is illustrated by the following diagram: \[ \bfig
\dtriangle(0,0)|amb|/.``<-/<600,250>[\mathfrak{o}_1`f^{-1}(\mathfrak{o}_2)`\mathfrak{o}_2; \pm t_1^\sharp`\times`f^{-1}]
\ptriangle(600,0)|amb|/->``./<600,250>[\mathfrak{o}_1`f(\mathfrak{o}_1)`\mathfrak{o}_2; f`` \pm t_2^\sharp]
\efig \] Since $f$ is $\rv$-affine, we see that its slope must be $-t_2/t_1$ (recall Definition~\ref{rvaffine}).
If we think of $\mathfrak{b}_1$, $\mathfrak{b}_2$ as $\tor(\code {\mathfrak{o}_1})$, $\tor(\code {\mathfrak{o}_2})$ then the set $A \cap (\mathfrak{b}_1 \times \mathfrak{b}_2)$ may be thought of as the ``line'' in $\tor(\code {\mathfrak{o}_1}) \times \tor(\code {\mathfrak{o}_2})$ given by the equation \[ x_2 = - \tfrac{t_2}{t_1}(x_1 - \code{\mathfrak{o}_1}) + (\code{\mathfrak{o}_2} - t_2). \] Thus, by Lemma~\ref{simul:special:dim:1}, the obvious bijection between $\pr_1(A) \times t_2^\sharp$ and $t_1^\sharp \times \pr_2(A)$ is the lift of an $\RV[{\leq}2]$-morphism modulo special bijections; see Lemma~\ref{2:unit:contracted} below for details. The slope of $f$ will play a more important role when volume forms are introduced into the categories (in a sequel). \end{rem}
\begin{defn}[$2$-cell]\label{def:units} We say that a set $A$ is a \emph{$1$-cell} if it is either an open disc contained in a single $\RV$-disc or a point in $\VF$. We say that $A$ is a \emph{$2$-cell} if \begin{enumerate}
\item $A$ is a subset of $\VF^2$ contained in a single $\RV$-polydisc and $\pr_1(A)$ is a $1$-cell,
\item there is a function $\epsilon : \pr_1 (A) \longrightarrow \VF$ and a $t \in \RV$ such that, for every $a \in \pr_1(A)$, $A_a = t^\sharp + \epsilon(a)$,
\item one of the following three possibilities occurs:
\begin{enumerate}
\item $\epsilon$ is constant,
\item $\epsilon$ is injective, has dtdp, and $\rad(\epsilon(\pr_1(A))) \geq \abs{\vrv(t)}$,\label{2cell:3b}
\item $\epsilon$ is balanced in $A$.
\end{enumerate} \end{enumerate} The function $\epsilon$ is called the \emph{positioning function} of $A$ and the element $t$ the \emph{paradigm} of $A$.
More generally, a set $A$ with exactly one $\VF$-coordinate is a \emph{$1$-cell} if, for each $t \in \pr_{>1}(A)$, $A_t$ is a $1$-cell in the above sense; the parameterized version of the notion of a $2$-cell is formulated in the same way. \end{defn}
A $2$-cell is definable if all the relevant ingredients are definable. Naturally we will only be concerned with definable $2$-cells. Notice that Corollary~\ref{all:subsets:rvproduct} implies that for every definable set $A$ with exactly one $\VF$-coordinate there is a definable function $\pi: A \longrightarrow \RV^l$ such that every fiber $A_s$ is a $1$-cell. This should be understood as $1$-cell decomposition and the next lemma as $2$-cell decomposition.
\begin{lem}[$2$-cell decomposition]\label{decom:into:2:units} For every definable set $A \subseteq \VF^2$ there is a definable function $\pi: A \longrightarrow \RV^m$ such that every fiber $A_s$ is an $s$-definable $2$-cell. \end{lem} \begin{proof} By compactness, we may assume that $A$ is contained in a single $\RV$-polydisc. For each $a \in \pr_1 (A)$, by Corollary~\ref{all:subsets:rvproduct}, there is an $a$-definable special bijection $T_a$ on $A_a$ such that $A_a^\flat$ is an $\RV$-pullback. By Lemma~\ref{inverse:special:dim:1}, there is an $a$-definable function $\epsilon_a : (A_a^\flat)_{\RV} \longrightarrow \VF$ such that, for every $(t, s) \in (A_a^\flat)_{\RV}$, we have \[ T_a^{-1}(t^\sharp \times (t, s)) = t^\sharp + \epsilon_a(t, s). \] By compactness, we may glue these functions together, that is, there is a definable set $C \subseteq \pr_1(A) \times \RV^l$ and a definable function $\epsilon : C \longrightarrow \VF$ such that, for every $a \in \pr_1(A)$, $C_a = (A_a^\flat)_{\RV}$ and $\epsilon \upharpoonright C_a = \epsilon_a$. For $(t, s) \in C_{\RV}$, write $\epsilon_{(t, s)} = \epsilon \upharpoonright C_{(t, s)}$. By Corollary~\ref{uni:fun:decom} and compactness, we are reduced to the case that each $\epsilon_{(t, s)}$ is either constant or injective. If no $\epsilon_{(t, s)}$ is injective then we can finish by applying Corollary~\ref{all:subsets:rvproduct} to each $C_{(t, s)}$ and then compactness.
Suppose that some $\epsilon_{(t, s)}$ is injective. Then, by Lemmas~\ref{open:pro} and \ref{bijection:dim:1:decom:RV}, we are reduced to the case that $C_{(t, s)}$ is an open disc and $\epsilon_{(t, s)}$ is $\rv$-affine and has dtdp. Write $\mathfrak{b}_{(t, s)} = \ran (\epsilon_{(t, s)})$. If $\rad(\mathfrak{b}_{(t, s)}) \geq \abvrv(t)$ then $\epsilon_{(t, s)}$ satisfies the condition (\ref{2cell:3b}) in Definition~\ref{def:units}. So let us suppose $\rad(\mathfrak{b}_{(t, s)}) < \abvrv(t)$. Then \[ \textstyle \mathfrak{b}_{(t, s)} = \bigcup_{a \in C_{(t, s)}} (t^\sharp + \epsilon_{(t, s)}(a)). \] By Lemma~\ref{bijection:rv:one:one}, we are further reduced to the case that there is an $r \in \RV$ such that, for every $a \in C_{(t, s)}$, \[ \rv(a - \epsilon_{(t, s)}^{-1}(t^\sharp + \epsilon_{(t, s)}(a))) = r \quad \text{and hence} \quad \epsilon_{(t, s)}^{-1}(t^\sharp + \epsilon_{(t, s)}(a)) = a - r^\sharp. \] So, in this case, $\epsilon_{(t, s)}$ is balanced. Now we are done by compactness. \end{proof}
To extend Lemma~\ref{simul:special:dim:1} to all definable bijections, we need not only $2$-cell decomposition but also the following notions.
Let $A \subseteq \VF^{n} \times \RV^{m}$, $B \subseteq \VF^{n} \times \RV^{m'}$, and $f : A \longrightarrow B$ be a definable bijection.
\begin{defn}\label{rela:unary} We say that $f$ is \emph{relatively unary} or, more precisely, \emph{relatively unary in the $i$th $\VF$-coordinate}, if $(\pr_{\tilde{i}} \circ f)(x) = \pr_{\tilde{i}}(x)$ for all $x \in A$, where $i \in [n]$. If $f \upharpoonright A_y$ is also a special bijection for every $y \in \pr_{\tilde{i}} (A)$ then we say that $f$ is \emph{relatively special in the $i$th $\VF$-coordinate}. \end{defn}
Obviously the inverse of a relatively unary bijection is a relatively unary bijection. Also note that every special bijection on $A$ is a composition of relatively special bijections.
Choose an $i \in [n]$. By Corollary~\ref{all:subsets:rvproduct} and compactness, there is a bijection $T_i$ on $A$, relatively special in the $i$th $\VF$-coordinate, such that $T_i(A_a)$ is an $\RV$-pullback for every $a \in \pr_{\tilde i}(A)$. Note that $T_i$ is not necessarily a special bijection on $A$, since the special bijections in the $i$th $\VF$-coordinate for distinct $a, a' \in \pr_{\tilde i}(A)$ with $\rv(a) = \rv(a')$ may not even be of the same length. Let \[ \textstyle A_i = \bigcup_{a \in \pr_{\tilde i}(A)} a \times (T_i(A_a))_{\RV} \subseteq \VF^{n-1} \times \RV^{m_i}. \] Write $\hat T_i : A \longrightarrow A_i$ for the function naturally induced by $T_i$. For any $j \in [n{-}1]$, we repeat the above procedure on $A_i$ with respect to the $j$th $\VF$-coordinate and thereby obtain a set $A_{j} \subseteq \VF^{n-2} \times \RV^{m_j}$ and a function $\hat T_{j} : A_i \longrightarrow A_{j}$. The relatively special bijection on $T_i(A)$ induced by $\hat T_{j}$ is denoted by $T_j$. Continuing thus, we obtain a sequence of bijections $T_{\sigma(1)}, \ldots, T_{\sigma(n)}$ and a corresponding function $\hat T_{\sigma} : A \longrightarrow \RV^{l}$, where $\sigma$ is the permutation of $[n]$ in question. The composition $T_{\sigma(n)} \circ \ldots \circ T_{\sigma(1)}$, which is referred to as the \emph{lift} of $\hat T_{\sigma}$, is denoted by $T_{\sigma}$.
\begin{defn}\label{defn:standard:contraction} Suppose that there is a $k \in 0 \cup [m]$ such that $(A_a)_{\leq k} \in \RV[k]$ for every $a \in A_{\VF}$. In particular, if $k=0$ then $A \in \VF_*$. By Lemma~\ref{RV:fiber:dim:same}, $\hat T_{\sigma}(A)_{\leq n+k}$ is an object of $\RV[{\leq} l{+}k]$, where $\dim_{\VF}(A) = l$. The function $\hat T_{\sigma}$ --- or the object $\hat T_{\sigma}(A)_{\leq n+k}$ --- is referred to as a \emph{standard contraction} of the set $A$ with the \emph{head start} $k$. \end{defn}
The head start of a standard contraction is usually implicit. In fact, it is always $0$ except in Lemma~\ref{isp:VF:fiberwise:contract}, and can be circumvented even there. This seemingly needless gadget only serves to make the above definition more streamlined: If $A \in \VF_*$ then the intermediate steps of a standard contraction of $A$ may or may not result in objects of $\VF_*$ and hence the definition cannot be formulated entirely within $\VF_*$.
\begin{rem}\label{special:dim:1:RV:iso} In Lemma~\ref{simul:special:dim:1}, clearly $\rv(A^{\flat})$, $\rv(B^{\flat})$ are standard contractions of $A$, $B$. Indeed, if $A, B \in \VF_*$ then $[\rv(A^{\flat})]_{\leq 1} = [\rv(B^{\flat})]_{\leq 1}$. \end{rem}
\begin{lem}\label{bijection:partitioned:unary} There is a definable finite partition $A_i$ of $A$ such that each $f \upharpoonright A_i$ is a composition of relatively unary bijections. \end{lem} \begin{proof} This is an easy consequence of weak $o$\nobreakdash-minimality. In more detail, for each $a \in \pr_{< n}(A)$ there are an $a$-definable finite partition $A_{ai}$ of $A_a$ and injective coordinate projections $\pi_i : f(A_{ai}) \longrightarrow \VF \times \RV^{m'}$. Therefore, by compactness, there are a definable finite partition $A_{i}$ of $A$, definable injections $f_i : A_i \longrightarrow \VF^{n} \times \RV^{m'}$, and $j_i \in [n]$ such that, for all $x \in A_i$, \[ \pr_{< n}(x) = \pr_{< n}(f_i(x)) \quad \text{and} \quad \pr_{n \cup [m']}(f_i(x)) = \pr_{j_i \cup [m']}(f(x)). \] The claim now follows from compactness and an obvious induction on $n$. \end{proof}
For the next two lemmas, let $12$ and $21$ denote the permutations of $[2]$.
\begin{lem}\label{2:unit:contracted} Let $A \subseteq \VF^2$ be a definable $2$-cell. Then there are standard contractions $\hat T_{12}$, $\hat R_{21}$ of $A$ such that $[\hat T_{12}(A)]_{\leq 2} = [\hat R_{21}(A)]_{\leq 2}$. \end{lem} \begin{proof} Let $\epsilon$ be the positioning function of $A$ and $t \in \RV_0$ the paradigm of $A$. If $t = 0$ then $A$ is (the graph of) the function $\epsilon : \pr_1(A) \longrightarrow \pr_2(A)$, which is either a constant function or a bijection. In the former case, since $A$ is essentially just an open ball, the lemma simply follows from Corollary~\ref{all:subsets:rvproduct}. In the latter case, there are relatively special bijections $T_2$, $R_1$ on $A$ in the coordinates $2$, $1$ such that \[ T_2(A) = \pr_1(A) \times 0 \times 0 \quad \text{and} \quad R_1(A) = 0 \times \pr_2(A) \times 0. \] So the lemma follows from Remark~\ref{special:dim:1:RV:iso}. For the rest of the proof we assume $t \neq 0$.
If $\epsilon$ is not balanced in $A$ then $A = \pr_1(A) \times \pr_2(A)$ is an open polydisc. By Corollary~\ref{all:subsets:rvproduct}, there are special bijections $T_1$, $T_2$ on $\pr_1(A)$, $\pr_2(A)$ such that $\pr_1(A)^\flat$, $\pr_2(A)^\flat$ are $\RV$-pullbacks. In this case the standard contractions determined by $(T_1, T_2)$ and $(T_2, T_1)$ are essentially the same.
Suppose that $\epsilon$ is balanced in $A$. Let $r$ be the other paradigm of $\epsilon$. Recall that $\epsilon : \pr_1 (A) \longrightarrow \pr_2(A)$ is again a bijection. Let $T_2$ be the relatively special bijection on $A$ in the coordinate $2$ given by $(a, b) \longmapsto (a, b - \epsilon(a))$ and $R_1$ the relatively special bijection on $A$ in the coordinate $1$ given by $(a, b) \longmapsto (a - \epsilon^{-1}(b), b)$, where $(a, b) \in A$. Clearly \[ T_2(A) = \pr_1(A) \times t^\sharp \times t \quad \text{and} \quad R_1(A) = r^\sharp \times \pr_2(A) \times r. \] So, again, the lemma follows from Remark~\ref{special:dim:1:RV:iso}. \end{proof}
\begin{lem}\label{subset:partitioned:2:unit:contracted} Let $A \subseteq \VF^2 \times \RV^m$ be an object in $\VF_*$. Then there are a definable injection $f : A \longrightarrow \VF^2 \times \RV^l$, relatively unary in both coordinates, and standard contractions $\hat T_{12}$, $\hat R_{21}$ of $f(A)$ such that $[\hat T_{12}(f(A))]_{\leq 2} = [\hat R_{21}(f(A))]_{\leq 2}$. \end{lem} \begin{proof} By Lemma~\ref{decom:into:2:units} and compactness, there is a definable function $f: A \longrightarrow \VF^2 \times \RV^l$ such that $f(A)$ is a $2$-cell and, for each $(a, t) \in A$, $f(a, t) = (a, t, s)$ for some $s \in \RV^{l-m}$. By Lemma~\ref{2:unit:contracted} and compactness, there are standard contractions $\hat T_{12}$, $\hat R_{21}$ of $f(A)$ into $\RV^{k+l}$ such that the following diagram commutates \[ \bfig
\Vtriangle(0,0)/->`->`->/<400,400>[\hat T_{12}(f(A))`\hat R_{21}(f(A))`\RV^l; F`\pr_{> k}`\pr_{> k}] \efig \] and $F$ is an $\RV[{\leq} 2]$-morphism $\hat T_{12}(f(A))_{\leq 2} \longrightarrow \hat R_{21}(f(A))_{\leq 2}$. \end{proof}
\subsection{Blowups and the main theorems}
The central notion for understanding the kernels of the semigroup homomorphisms $\bb L$ is that of a blowup:
\begin{defn}[Blowups]\label{defn:blowup:coa} Let $\bm U = (U, f) \in \RV[k]$, where $k > 0$, such that, for some $j \leq k$, the restriction $\pr_{\tilde j} \upharpoonright f(U)$ is finite-to-one. Write $f = (f_1, \ldots, f_k)$. The \emph{elementary blowup} of $\bm U$ in the $j$th coordinate is the pair $\bm U^{\flat} = (U^{\flat}, f^{\flat})$, where $U^{\flat} = U \times \RV^{\circ \circ}_0$ and, for every $(t, s) \in U^{\flat}$, \[ f^{\flat}_{i}(t, s) = f_{i}(t) \text{ for } i \neq j \quad \text{and} \quad f^{\flat}_{j}(t, s) = s f_{j}(t). \] Note that $\bm U^{\flat}$ is an object in $\RV[{\leq} k]$ (actually in $\RV[k{-}1] \oplus \RV[k]$) because $f^{\flat}_{j}(t, 0) = 0$.
Let $\bm V = (V, g) \in \RV[k]$ and $C \subseteq V$ be a definable set. Suppose that $F : \bm U \longrightarrow \bm C$ is an $\RV[k]$-morphism, where $\bm C = (C, g \upharpoonright C) \in \RV[k]$. Then \[ \bm U^{\flat} \uplus (V \smallsetminus C, g \upharpoonright (V \smallsetminus C)) \] is a \emph{blowup of $\bm V$ via $F$}, denoted by $\bm V^{\flat}_F$. The subscript $F$ is usually dropped in context if there is no danger of confusion. The object $\bm C$ (or the set $C$) is referred to as the \emph{locus} of $\bm V^{\flat}_F$.
A \emph{blowup of length $n$} is a composition of $n$ blowups. \end{defn}
\begin{rem} In an elementary blowup, the condition that the coordinate of interest is definably dependent (the coordinate projection is finite-to-one) on the other ones is needed so that the resulting objects stay in $\RV[{\leq} k]$. In the setting of \cite{hrushovski:kazhdan:integration:vf}, this condition is also needed for matching blowups with special bijections, since, otherwise, we would not be able to use (a generalization of) Hensel's lemma to find enough centers of $\RV$-discs to construct focus maps. In our setting, Lemma~\ref{RVlift} plays the role of Hensel's lemma, which is more powerful, and hence ``algebraicity'' is no longer needed for this purpose (see Lemma~\ref{blowup:same:RV:coa}). \end{rem}
If there is an elementary blowup of $(U, f) \in \RV[k]$ then, \textit{a posteriori}, $\dim_{\RV}(f(U)) < k$. Also, there is at most one elementary blowup of $(U, f)$ with respect to any coordinate of $f(U)$. We should have included the coordinate that is blown up as a part of the data. However, in context, either this is clear or it does not need to be spelled out, and we shall suppress mentioning it below for notational ease.
\begin{lem}\label{blowup:equi:class:coa} Let $\bm U, \bm V \in \RV[{\leq} k]$ such that $[\bm U] = [\bm V]$ in $\gsk \RV[{\leq} k]$. Let $\bm U_1$, $\bm V_1$ be blowups of $\bm U$, $\bm V$ of lengths $m$, $n$, respectively. Then there are blowups $\bm U_2$, $\bm V_2$ of $\bm U_1$, $\bm V_1$ of lengths $n$, $m$, respectively, such that $[\bm U_2] = [\bm V_2]$. \end{lem} \begin{proof} Fix an isomorphism $I: \bm U \longrightarrow \bm V$. We do induction on the sum $l = m + n$. For the base case $l = 1$, without loss of generality, we may assume $n = 0$. Let $C$ be the blowup locus of $\bm U_1$. Clearly $\bm V$ may be blown up by using the same elementary blowup as $\bm U_1$, where the blowup locus is changed to $I(C)$, and the resulting blowup is as required.
\[ \bfig
\square(0,0)/.`=``./<500,900>[\bm U`\bm U^{\flat}`
\bm V`\bm V^{\flat};1```1]
\square(500,0)/.```./<1000,900>[\bm U^{\flat}`\bm U_1`
\bm V^{\flat}`\bm V_1; m - 1```n - 1]
\morphism(500,900)/./<500,-300>[\bm U^{\flat}`\bm U^{\flat\flat};1]
\morphism(500,0)/./<500,300>[\bm V^{\flat}`
\bm V^{\flat\flat};1]
\morphism(1000,300)/=/<0,300>[\bm V^{\flat\flat}`
\bm U^{\flat\flat};]
\morphism(1500,900)/./<500,0>[\bm U_1`\bm U_1^{\flat};1]
\morphism(1500,0)/./<500,0>[\bm V_1`\bm V_1^{\flat};1]
\morphism(1000,600)/./<1000,0>[\bm U^{\flat\flat}`\bm U^{\flat3};
m - 1]
\morphism(1000,300)/./<1000,0>[\bm V^{\flat\flat}`\bm V^{\flat3};
n - 1]
\morphism(2000,900)/=/<0,-300>[\bm U_1^{\flat}`\bm U^{\flat3};]
\morphism(2000,0)/=/<0,300>[\bm V_1^{\flat}`\bm V^{\flat3};]
\morphism(2000,600)/./<1000,0>[\bm U^{\flat3}`\bm U^{\flat4};n - 1]
\morphism(2000,300)/./<1000,0>[\bm V^{\flat3}`\bm V^{\flat4};m - 1]
\morphism(3000,300)/=/<0,300>[\bm V^{\flat4}`\bm U^{\flat4};]
\morphism(2000,900)/./<1000,0>[\bm U_1^{\flat}`\bm U_2;n - 1]
\morphism(2000,0)/./<1000,0>[\bm V_1^{\flat}`\bm V_2;m - 1]
\morphism(3000,0)/=/<0,300>[\bm V_2`\bm V^{\flat4};]
\morphism(3000,900)/=/<0,-300>[\bm U_2`\bm U^{\flat4};] \efig \]
We proceed to the inductive step. How the isomorphic blowups are constructed is illustrated above. Write $\bm U = (U, f)$ and $\bm V = (V, g)$. Let $\bm U^{\flat}$, $\bm V^{\flat}$ be the first blowups in $\bm U_1$, $\bm V_1$ and $C$, $D$ their blowup loci, respectively. Let $\bm U'^{\flat}$, $\bm V'^{\flat}$ be the corresponding elementary blowups contained in $\bm U^{\flat}$, $\bm V^{\flat}$. If, say, $n = 0$, then by the argument in the base case $\bm V$ may be blown up to an object that is isomorphic to $\bm U^{\flat}$ and hence the inductive hypothesis may be applied. So assume $m,n > 0$. Let $A = C \cap I^{-1}(D)$ and $B = I(C) \cap D$. Since $(A, f \upharpoonright A)$ and $(B, g \upharpoonright B)$ are isomorphic, the blowups of $\bm U'$, $\bm V'$ with the loci $(A, f \upharpoonright A)$ and $(B, g \upharpoonright B)$ are isomorphic. Then, it is not hard to see that the blowup $\bm U^{\flat\flat}$ of $\bm U^{\flat}$ using the locus $I^{-1}(D) \smallsetminus C$ and its corresponding blowup of $\bm V'$ and the blowup $\bm V^{\flat\flat}$ of $\bm V^{\flat}$ using the locus $I(C) \smallsetminus D$ and its corresponding blowup of $\bm U'$ are isomorphic.
Applying the inductive hypothesis to the blowups $\bm U^{\flat\flat}$, $\bm U_1$ of $\bm U^{\flat}$, we obtain a blowup $\bm U^{\flat3}$ of $\bm U^{\flat\flat}$ of length $m - 1$ and a blowup $\bm U_1^{\flat}$ of $\bm U_1$ of length $1$ such that they are isomorphic. Similarly, we obtain a blowup $\bm V^{\flat3}$ of $\bm V^{\flat\flat}$ of length $n - 1$ and a blowup $\bm V_1^{\flat}$ of $\bm V_1$ of length $1$ such that they are isomorphic. Applying the inductive hypothesis again to the blowups $\bm U^{\flat3}$, $\bm V^{\flat3}$ of $\bm U^{\flat\flat}$, $\bm V^{\flat\flat}$, we obtain a blowup $\bm U^{\flat4}$ of $\bm U^{\flat3}$ of length $n - 1$ and a blowup $\bm V^{\flat4}$ of $\bm V^{\flat3}$ of length $m - 1$ such that they are isomorphic. Finally, applying the inductive hypothesis to the blowups $\bm U^{\flat4}$, $\bm U_1^{\flat}$ of $\bm U^{\flat3}$, $\bm U_1^{\flat}$ and the blowups $\bm V^{\flat4}$, $\bm V_1^{\flat}$ of $\bm V^{\flat3}$, $\bm V_1^{\flat}$, we obtain a blowup $\bm U_2$ of $\bm U_1^{\flat}$ of length $n - 1$ and a blowup $\bm V_2$ of $\bm V_1^{\flat}$ of length $m - 1$ such that $\bm U^{\flat4}$, $\bm U_2$, $\bm V^{\flat4}$, and $\bm V_2$ are all isomorphic. So $\bm U_2$, $\bm V_2$ are as desired. \end{proof}
\begin{cor}\label{blowup:equi:class} Let $[\bm U] = [\bm U']$ and $[\bm V] = [\bm V']$ in $\gsk \RV[{\leq} k]$. If there are isomorphic blowups of $\bm U$, $\bm V$ then there are isomorphic blowups of $\bm U'$, $\bm V'$. \end{cor}
\begin{defn}\label{defn:isp} Let $\isp[k]$ be the set of pairs $(\bm U, \bm V)$ of objects of $\RV[{\leq} k]$ such that there exist isomorphic blowups $\bm U^{\flat}$, $\bm V^{\flat}$. Set $\isp[*] = \bigcup_{k} \isp[k]$. \end{defn}
We will just write $\isp$ for all these sets when there is no danger of confusion. By Corollary~\ref{blowup:equi:class}, $\isp$ may be regarded as a binary relation on isomorphism classes.
\begin{lem}\label{isp:congruence:vol} $\isp[k]$ is a semigroup congruence relation and $\isp[*]$ is a semiring congruence relation. \end{lem} \begin{proof} Clearly $\isp[k]$ is reflexive and symmetric. If $([\bm U_1], [\bm U_2])$, $([\bm U_2], [\bm U_3])$ are in $\isp[k]$ then, by Lemma~\ref{blowup:equi:class:coa}, there are blowups $\bm U_1^{\flat}$ of $\bm U_1$, $\bm U_{2}^{\flat 1}$ and $\bm U_{2}^{\flat 2}$ of $\bm U_2$, and $\bm U_3^{\flat}$ of $\bm U_3$ such that they are all isomorphic. So $\isp[k]$ is transitive and hence is an equivalence relation. For any $[\bm W] \in \gsk \RV[l]$, the following are easily checked: \[ ([\bm U_1 \uplus \bm W], [\bm U_2 \uplus \bm W])\in \isp,\quad ([\bm U_1 \times \bm W], [\bm U_2 \times \bm W])\in \isp. \] These yield the desired congruence relations. \end{proof}
Let $\bm U = (U, f)$ be an object of $\RV[k]$ and $T$ a special bijection on $\bb L \bm U$. The set $(T(\mathbb{L} \bm U))_{\RV}$ is simply denoted by $U_{T}$ and the object $(U_{T})_{\leq k} \in \RV[{\leq} k]$ by $\bm U_{T}$.
\begin{lem}\label{special:to:blowup:coa} The object $\bm U_T$ is isomorphic to a blowup of $\bm U$ of the same length as $T$. \end{lem} \begin{proof} By induction on the length $\lh (T)$ of $T$ and Lemma~\ref{blowup:equi:class:coa}, this is immediately reduced to the case $\lh (T) = 1$. For that case, let $\lambda$ be the focus map of $T$. Without loss of generality, we may assume that the locus of $\lambda$ is $\mathbb{L} \bm U$. Then it is clear how to construct an (elementary) blowup of $\bm U$ as desired. \end{proof}
\begin{lem}\label{kernel:dim:1:coa} Suppose that $[A] = [B]$ in $\gsk \VF[1]$ and $\bm U, \bm V \in \RV[{\leq} 1]$ are two standard contractions of $A$, $B$, respectively. Then $([\bm U], [\bm V]) \in \isp$. \end{lem} \begin{proof} By Lemma~\ref{simul:special:dim:1}, there are special bijections $T$, $R$ on $\bb L \bm U$, $\bb L \bm V$ such that $\bm U_{T}$, $\bm V_{R}$ are isomorphic. So the assertion follows from Lemma~\ref{special:to:blowup:coa}. \end{proof}
\begin{lem}\label{blowup:same:RV:coa} Let $\bm U^{\flat}$ be a blowup of $\bm U = (U, f) \in \RV[{\leq} k]$ of length $l$. Then $\bb L \bm U^{\flat}$ is isomorphic to $\bb L \bm U$. \end{lem} \begin{proof} By induction on $l$ this is immediately reduced to the case $l=1$. For that case, without loss of generality, we may assume that $\pr_{\tilde 1} \upharpoonright f(U)$ is injective and $\bm U^{\flat}$ is an elementary blowup in the first coordinate. So it is enough to show that there is a focus map into the first coordinate with locus $f(U)^\sharp$. This is guaranteed by Hypothesis~\ref{hyp:point}. \end{proof}
\begin{lem}\label{isp:VF:fiberwise:contract} Let $A'$, $A''$ be definable sets with $A'_{\VF} = A''_{\VF} \eqqcolon A \subseteq \VF^n$. Suppose that there is a $k \in \mathds{N}$ such that, for every $a \in A$, $([A'_a]_{\leq k}, [A''_a]_{\leq k}) \in \isp$. Let $\hat T_{\sigma}$, $\hat R_{\sigma}$ be respectively standard contractions of $A'$, $A''$. Then \[ ([\hat T_{\sigma}(A')]_{\leq n+k}, [\hat R_{\sigma}(A'')]_{\leq n+k}) \in \isp. \] \end{lem}
Note that the condition $([A'_a]_{\leq k}, [A''_a]_{\leq k}) \in \isp$ makes sense only over the substructure $\mdl S \langle a \rangle$.
\begin{proof} By induction on $n$ this is immediately reduced to the case $n=1$. So assume $A \subseteq \VF$. Let $\phi'$, $\phi''$ be quantifier-free formulas that define $A'$, $A''$, respectively. Let $\theta$ be a quantifier-free formula such that, for every $a \in A$, $\theta(a)$ defines the necessary data (two blowups and an $\RV[*]$-morphism) that witness the condition $([A'_a]_{\leq k}, [A''_a]_{\leq k}) \in \isp$. Applying Corollary~\ref{special:bi:term:constant} to the top \LT-terms of $\phi'$, $\phi''$, and $\theta$, we obtain a special bijection $F: A \longrightarrow A^{\flat}$ such that $A^{\flat}$ is an $\RV$-pullback and, for all $\RV$-polydiscs $\mathfrak{p} \subseteq A^{\flat}$ and all $a_1, a_2 \in F^{-1}(\mathfrak{p})$, \begin{itemize}
\item $A'_{a_1} = A'_{a_2}$ and $A''_{a_1} = A''_{a_2}$,
\item $\theta(a_1)$ and $\theta(a_2)$ define the same data. \end{itemize} The second item implies that the data defined by $\theta$ over $F^{-1}(\mathfrak{p})$ is actually $\rv(\mathfrak{p})$-definable.
Let $B' = \bigcup_{a \in A} F(a) \times A'_a$, similarly for $B''$. Note that $B'$, $B''$ are obtained through special bijections on $A'$, $A''$. For all $t \in A'_{\RV}$, $B'_t$ is an $\RV$-pullback that is $t$-definably bijective to the $\RV$-pullback $T_{\sigma}(A')_t$. By Lemma~\ref{kernel:dim:1:coa}, we have, for all $t \in A'_{\RV}$ \[ ([(B'_{\RV})_t]_1, [\hat T_{\sigma}(A')_t]_1) \in \isp \] and hence, by compactness, \[ ([B'_{\RV}]_{\leq k+1}, [\hat T_{\sigma}(A')]_{\leq k+1}) \in \isp. \] The same holds for $B''$ and $\hat R_{\sigma}(A'')$. On the other hand, by the second item above, for every $\RV$-polydisc $\mathfrak{p} \subseteq A^{\flat}$, we have $((B'_{\RV})_{\rv(\mathfrak{p})}, (B''_{\RV})_{\rv(\mathfrak{p})}) \in \isp$ and hence, by compactness, \[ ([B'_{\RV}]_{\leq k+1}, [B''_{\RV}]_{\leq k+1}) \in \isp. \] Since $\isp$ is a congruence relation, the lemma follows. \end{proof}
\begin{cor}\label{contraction:same:perm:isp} Let $A', A'' \in \VF_*$ with exactly $n$ $\VF$-coordinates each and $f : A' \longrightarrow A''$ be a relatively unary bijection in the $i$th coordinate. Then for any permutation $\sigma$ of $[n]$ with $\sigma(1) = i$ and any standard contractions $\hat T_{\sigma}$, $\hat R_{\sigma}$ of $A'$, $A''$, \[ ([\hat T_{\sigma}(A')]_{\leq n}, [\hat R_{\sigma}(A'')]_{\leq n}) \in \isp. \] \end{cor} \begin{proof} This is immediate by Lemmas~\ref{kernel:dim:1:coa} and \ref{isp:VF:fiberwise:contract}. \end{proof}
The following lemma is essentially a version of Fubini's theorem (also see Theorem~\ref{semi:fubini} below).
\begin{lem}\label{contraction:perm:pair:isp} Let $A \in \VF_*$ with exactly $n$ $\VF$-coordinates. Suppose that $i, j \in [n]$ are distinct and $\sigma_1$, $\sigma_2$ are permutations of $[n]$ such that \[ \sigma_1(1) = \sigma_2(2) = i, \quad \sigma_1(2) = \sigma_2(1) = j, \quad \sigma_1 \upharpoonright \set{3, \ldots, n} = \sigma_2 \upharpoonright \set{3, \ldots, n}. \] Then, for any standard contractions $\hat T_{\sigma_1}$, $\hat T_{\sigma_2}$ of $A$, \[ ([\hat T_{\sigma_1}(A)]_{\leq n}, [\hat T_{\sigma_2}(A)]_{\leq n}) \in \isp. \] \end{lem} \begin{proof} Let $ij$, $ji$ denote the permutations of $E \coloneqq \{i, j\}$. By compactness and Lemma~\ref{isp:VF:fiberwise:contract}, it is enough to show that, for any $a \in \pr_{\tilde E}(A)$ and any standard contractions $\hat T_{ij}$, $\hat T_{ji}$ of $A_a$, \[ ([\hat T_{ij}(A_a)]_{\leq 2}, [\hat T_{ji}(A_a)]_{\leq 2}) \in \isp. \] To that end, fix an $a \in \pr_{\tilde E}(A)$. By Lemma~\ref{subset:partitioned:2:unit:contracted}, there are a definable bijection $f$ on $A_a$ that is relatively unary in both $\VF$-coordinates and standard contractions $\hat R_{ij}$, $\hat R_{ji}$ of $f(A_a)$ such that \[ [\hat R_{ij}(f(A_a))]_{\leq 2} = [\hat R_{ji}(f(A_a))]_{\leq 2}. \] So the desired property follows from Corollary~\ref{contraction:same:perm:isp}. \end{proof}
The following proposition is the culmination of the preceding technicalities; it identifies the congruence relation $\isp$ with that induced by $\bb L$.
\begin{prop}\label{kernel:L} For $\bm U, \bm V \in \RV[{\leq} k]$, \[ [\bb L \bm U] = [\bb L \bm V] \quad \text{if and only if} \quad ([\bm U], [\bm V]) \in \isp. \] \end{prop} \begin{proof} The ``if'' direction simply follows from Lemma~\ref{blowup:same:RV:coa} and Proposition~\ref{L:sur:c}.
For the ``only if'' direction, we show a stronger claim: if $[A] = [B]$ in $\gsk \VF_*$ and $\bm U, \bm V \in \RV[{\leq} k]$ are two standard contractions of $A$, $B$ then $([\bm U], [\bm V]) \in \isp$. We do induction on $k$. The base case $k = 1$ is of course Lemma~\ref{kernel:dim:1:coa}.
For the inductive step, suppose that $F : \bb L \bm U \longrightarrow \bb L \bm V$ is a definable bijection. By Lemma~\ref{bijection:partitioned:unary}, there is a partition of $\bb L \bm U$ into definable sets $A_1, \ldots, A_n$ such that each restriction $F_i = F \upharpoonright A_i$ is a composition of relatively unary bijections. Applying Corollary~\ref{special:bi:term:constant} as before, we obtain two special bijections $T$, $R$ on $\bb L \bm U$, $\bb L \bm V$ such that $T(A_i)$, $(R \circ F)(A_i)$ is an $\RV$-pullback for each $i$. By Lemma~\ref{special:to:blowup:coa}, it is enough to show that, for each $i$, there are standard contractions $\hat T_{\sigma}$, $\hat R_{\tau}$ of $T(A_i)$, $(R \circ F)(A_i)$ such that \[ ([(\hat T_{\sigma} \circ T)(A_i)]_{\leq k}, [(\hat R_{\tau} \circ R \circ F)(A_i)]_{\leq k}) \in \isp. \] To that end, first note that each $(R \circ F \circ T^{-1}) \upharpoonright T(A_i)$ is a composition of relatively unary bijections, say \[ T(A_i) = B_1 \to^{G_1} B_2 \cdots B_l \to^{G_l} B_{l+1} = (R \circ F)(A_i). \] For each $j \leq l - 2$, we can choose five standard contractions \[ [U_j]_{\leq k}, \quad [U_{j+1}]_{\leq k}, \quad [U'_{j+1}]_{\leq k}, \quad [U''_{j+1}]_{\leq k}, \quad [U_{j+2}]_{\leq k} \] of $B_j$, $B_{j+1}$, $B_{j+1}$, $B_{j+1}$, $B_{j+2}$ with the permutations $\sigma_{j}$, $\sigma_{j+1}$, $\sigma'_{j+1}$, $\sigma''_{j+1}$, $\sigma_{j+2}$ of $[k]$, respectively, such that \begin{itemize}
\item $\sigma_{j+1}(1)$ and $\sigma_{j+1}(2)$ are the $\VF$-coordinates targeted by $G_{j}$ and $G_{j+1}$, respectively,
\item $\sigma''_{j+1}(1)$ and $\sigma''_{j+1}(2)$ are the $\VF$-coordinates targeted by $G_{j+1}$ and $G_{j+2}$, respectively,
\item $\sigma_{j} = \sigma_{j+1}$, $\sigma''_{j+1} = \sigma_{j+2}$, and $\sigma'_{j+1}(1) = \sigma''_{j+1}(1)$,
\item the relation between $\sigma_{j+1}$ and $\sigma'_{j+1}$ is as described in Lemma~\ref{contraction:perm:pair:isp}. \end{itemize} By Corollary~\ref{contraction:same:perm:isp} and Lemma~\ref{contraction:perm:pair:isp}, all the adjacent pairs of these standard contractions are $\isp$-congruent, except $([U'_{j+1}]_{\leq k}, [U''_{j+1}]_{\leq k})$. Since we can choose $[U'_{j+1}]_{\leq k}$, $[U''_{j+1}]_{\leq k}$ so that they start with the same contraction in the first targeted $\VF$-coordinate of $B_{j+1}$, the resulting sets from this step are the same. So, applying the inductive hypothesis in each fiber over the just contracted coordinate, we see that this last pair is also $\isp$-congruent. This completes the ``only if'' direction. \end{proof}
This proposition shows that the semiring congruence relation on $\gsk \RV[*]$ induced by $\bb L$ is generated by the pair $([1], \bm 1_{\K} + [(\RV^{\circ \circ}, \id)])$ and hence its corresponding ideal in the graded ring $\ggk \RV[*]$ is generated by the element $\bm 1_{\K} + [\bm P]$ (see Notation~\ref{nota:RV:short} and Remark~\ref{gam:res}).
\begin{thm}\label{main:prop} For each $k \geq 0$ there is a canonical isomorphism of Grothendieck semigroups \[ \textstyle \int_{+} : \gsk \VF[k] \longrightarrow \gsk \RV[{\leq} k] / \isp \] such that \[ \textstyle \int_{+} [A] = [\bm U]/ \isp \quad \text{if and only if} \quad [A] = [\bb L\bm U]. \] Putting these together, we obtain a canonical isomorphism of Grothendieck semirings \[ \textstyle \int_{+} : \gsk \VF_* \longrightarrow \gsk \RV[*] / \isp. \] \end{thm} \begin{proof} This is immediate by Corollary~\ref{L:sur:c} and Proposition~\ref{kernel:L}. \end{proof}
\begin{thm}\label{thm:ring} The Grothendieck semiring isomorphism $\int_+$ naturally induces a ring isomorphism: \[ \textstyle \Xint{\textup{G}} : \ggk \VF_* \to \ggk \RV[*] / (\bm 1_{\K} + [\bm P]) \to^{\bb E_{\Gamma}} \mathds{Z}^{(2)}[X], \] and two ring homomorphisms onto $\mathds{Z}$: \[ \textstyle \Xint{\textup{R}}^g, \Xint{\textup{R}}^b: \ggk \VF_* \to \ggk \RV[*] / (\bm 1_{\K} + [\bm P]) \two^{\bb E_{\Gamma, g}}_{\bb E_{\Gamma, b}} \mathds{Z}. \] \end{thm} \begin{proof} This is just a combination of Theorem~\ref{main:prop} and Remark~\ref{rem:poin} (or Proposition~\ref{prop:eu:retr:k}). \end{proof}
Let $F$ be a definable set with $A \coloneqq F_{\VF} \subseteq \VF^n$. Then $F$ may be viewed as a representative of a \emph{definable} function $\bm F : A \longrightarrow \gsk \RV[*] / \isp$ given by $a \longmapsto [F_a] / \isp$. Note that the class $[F_a]$ depends on the parameter $a$ and hence can only be guaranteed to lie in the semiring $\gsk \RV[*]$ constructed over $\mdl S \langle a \rangle$ instead of $\mdl S$, but we abuse the notation. Similarly, for distinct $a, a' \in A$, there is a priori no way to compare $[F_a]$ and $[F_{a'}]$ unless we work over the substructure $\mdl S \langle a, a' \rangle$; given another definable set $G$ with $A = G_{\VF}$, the corresponding definable function $\bm G$ is the same as $\bm F$ if $\bm G(a) = \bm F(a)$ over $\mdl S \langle a \rangle$ for all $a \in A$. The set of all such functions is denoted by $\fn_+(A)$, which is a semimodule over $\gsk \RV[*] / \isp$. Let $E \subseteq [n]$ be a nonempty set. Then, for each $a \in \pr_{E}(A)$, the definable function in $\fn_+(A_a)$ represented by $F_a$ is denoted by $\bm F_a$.
Let $\bb L F = \bigcup_{a \in A} a \times F_a^\sharp$ and then set $\int_{+A} \bm F = \int_+ [\bb L F]$, which, by Proposition~\ref{kernel:L} and compactness, does not depend on the representative $F$. Thus there is a canonical homomorphism of semimodules: \[ \textstyle \int_{+A} : \fn_+(A) \longrightarrow \gsk \RV[*] / \isp. \]
\begin{thm}\label{semi:fubini} For all $\bm F \in \fn_+(A)$ and all nonempty sets $E, E' \subseteq [n]$, \[ \textstyle \int_{+ a \in \pr_{E}(A)} \int_{+ A_a} \bm F_a = \int_{+ a \in \pr_{E'}(A)} \int_{+ A_a} \bm F_a. \] \end{thm} \begin{proof} This is clear since both sides equal $\int_{+A} \bm F$. \end{proof}
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\begin{document}
\title[Exceptional sets]{Exceptional sets for geodesic flows\\ of noncompact manifolds}
\begin{abstract} For a geodesic flow on a negatively curved Riemannian manifold $M$ and some subset $A\subset T^1M$, we study the limit $A$-exceptional set, that is the set of points whose $\omega$-limit do not intersect $A$. We show that if the topological $\ast$-entropy of $A$ is smaller than the topological entropy of the geodesic flow, then the limit $A$-exceptional set has full topological entropy. Some consequences are stated for limit exceptional sets of invariant compact subsets and proper submanifolds. \end{abstract}
\begin{thanks}{}\end{thanks} \keywords{geodesic flow, suspension flow, exceptional sets, topological entropy} \subjclass[2000]{ 37B40, 37D40, 37F35, 28D20, 37B10 } \maketitle
\section{Introduction}
Exceptional sets have been largely studied over the last years, mostly trying to identify for which dynamical systems small sets have large exceptional sets. Naturally, there are plenty of ways to quantify the ``size'' of a set, but these studies have been focused mainly in two notions: Hausdorff dimension, which involves only the topology of a set, and topological entropy, which mostly considers the dynamic contribution of a set on the system.
In the discrete-time case Dolgopyat \cite{Dol:97} proved that sets with small entropy in subshifts of finite type have exceptional sets with full entropy. Analogous statements hold for piecewise expanding maps of the interval and for Anosov diffeomorphisms of the two-dimensional torus, but in terms of Hausdorff dimension (see \cite{Dol:97}). Campos and Gelfert \cite{CaGe:16,CaGe:19} concluded the same type of results in terms of both, topological entropy and Hausdorff dimension, for nonuniformly expanding/hyperbolic maps. To the best of our knowledge the first result involving exceptional sets for continuous-time dynamical systems comes from \cite{Kle:98} where the author proves that the exceptional set of a compact (proper) submanifold on a negatively curved Riemannian manifold having constant negative curvature has full Hausdorff dimension relative to the geodesic flow. A similar result can be found in \cite{KleWei:13} for partially hyperbolic flows on homogeneous spaces. Most of these results were motivated by the work of Jarnik-Besicovitch \cite{Ja:29} on the estimation of the size of badly approximable real numbers and its equivalence with the study of the set of bounded geodesic orbits on the modular surface. Indeed, the set of bounded orbits can be informally thought as the exceptional set of $\infty$. Dani \cite{Dan:86} proved that the set of bounded orbits by the action of a one-parameter subgroup on $G/\Gamma$ has full Hausdorff dimension, where $G$ is a connected semisimple Lie group of real rank 1 and $\Gamma$ is a lattice (see also \cite{KleMar:96} for analogous results on homogeneous dynamics).
The aim of this work is to study the topological entropy of exceptional sets for geodesic flows on arbitrary noncompact negatively curved Riemannian manifolds, this includes those with variable sectional curvatures and dimension $\ge 2$.
To state our first main result, let us introduce some notation. Consider a semi-group $F=(f^t)_{t\in \mathbb{A}}$, $\mathbb{A}=\mathbb{R}_{\ge0}$ or $\mathbb{A}=\mathbb{N}\cup\{0\}$, acting on some complete metric space $X$. Denote by $\EuScript{O}^+_{F}(x)=\{f^t(x)\colon t\in\mathbb{A}\}$ the (\emph{forward}) \emph{semi-orbit} of $x\in X$ by $F$. Denote by $\omega_F(x)$ the (\emph{forward}) \emph{$\omega$-limit set} of a point $x\in X$, that is, the set of limit points of $\EuScript{O}^+_{F}(x)$. Given a set $A\subset X$, denote by \[
I_F(A)
\eqdef \{x\colon \omega_{F}(x)\cap A=\emptyset\} \]
the (\emph{forward}) \emph{limit $A$-exceptional set} (with respect to $F$). Notice that $I_F(A)$ is $F$-invariant, that is $f^t(I_F(A))=I_F(A)$ for every $t\in\mathbb{A}$. In the case when $F|_Y\colon Y\to Y$ is the restricted flow on some $F$-invariant set $Y$, then we use the notation $I_{F|Y}(A)$.
Given a set $A\subset X$, we denote by $h^\ast_{\rm top}(F,A)$ the \emph{topological $\ast$-entropy} of $F$ on $A$, \[
h_{\rm top}^\ast(F,A)
\eqdef \sup h_\mu(F), \] where the supremum is taken over all Borel probability measure which are weak$\ast$ limits of empirical measures distributed along the forward orbit of a point in $A$, and we let $h_{\rm top}^\ast(F,A)\eqdef0$ in case no such measure exists. By $h_{\rm top}(F,A)$ we denote the \emph{topological entropy} of $F$ on $A$. We recall the precise definition of entropy in Section \ref{sec:entropy}.
The following is our first main result.
\begin{maintheorem}\label{the:1}
Consider the geodesic flow $G=(g^t)_{t\in\mathbb{R}}$ on the unit tangent bundle $T^1M$ of a $n$-dimensional complete Riemannian manifold $M$ of negatively pinched sectional curvatures $-b^2\le \kappa\leq -1$ for some $b\geq 1$. For every Borel set $A\subset T^1M$ satisfying $h_{\rm top}^\ast(G,A)<h_{\rm top}(G,T^1M)$ it holds \[
h_{\rm top}^\ast(G,I_{G}(A))
= h_{\rm top}(G,T^1M). \] \end{maintheorem}
Let us provide some (nontrivial) example of Borel sets satisfying the hypothesis of Theorem \ref{the:1}. Let $D$ be the set of \emph{divergent in average geodesic orbits}, that is, orbits which spend a Birkhoff average time 0 on every compact set. Then $h_{\rm top}^\ast(G,D) =0$ since there are no limit measures for empirical measures distributed along divergent in average orbits. In particular, $h_{\rm top}^\ast(G,I_{G}(D))
= h_{\rm top}(G,T^1M)$. Note that the set $D$ is not so small since by \cite{RiVe:22} it has positive Hausdorff dimension provided $M$ is not convex-cocompact.
Let us state now some consequences of Theorem \ref{the:1}. Recall that the \emph{non-wandering set} $\Omega$ of the geodesic flow $G$ is the set of unit vectors $v\in T^1M$ such that for every neighborhood $U$ of $v$ there exist $s,t>0$ such that $g^s(v),g^{-t}(v)\in U$.
\begin{corollary}\label{cor:1}
Assume the hypotheses of Theorem \ref{the:1} and additionally that the derivatives of the sectional curvatures are uniformly bounded.
If the non-wandering set of the geodesic flow is non-compact, then for every compact and $G$-invariant set $K\subset T^1M$, \[
h^\ast_{\rm top}(G,I_{G}(K))
= h_{\rm top}(G). \] \end{corollary}
For the following consequence, consider a proper Riemannian submanifold $N\subset M$. Then $N$ is isometric to the quotient $N=\widetilde{N}/\Gamma'$, where $\widetilde{N}$ is a Riemannian submanifold of the universal covering of $M$, $\widetilde{M}$, and $\Gamma'$ is a subgroup of $\Gamma$ acting on $\widetilde{N}$. A discrete group $H\subset {\rm Isom}^+(\widetilde{M})$ is \emph{divergent} if its Poincar\'e series \[
\mathcal{P}_H(s)
=\sum_{h\in H} e^{-sd(o,h o)} \] diverges at $s=\delta_H$, where $\delta_H$ is its \emph{critical exponent}, \[
\delta_H
=\limsup_{R\to\infty}\frac{1}{R}\log\card\{h\in H \colon d(o,h o)\leq R\}. \] Denote by $\Omega_N$ the non-wandering set associated to the geodesic flow over $T^1N$.
\begin{corollary} \label{cor:2}
Assume the hypotheses of Theorem \ref{the:1} and additionally that the derivatives of the sectional curvatures are uniformly bounded.
Let $N\subset M$ be a proper Riemannian submanifold as above. Assume that $\Gamma'$ is divergent and $\Omega_N\neq \Omega$. Then \[ h^\ast_{\rm top}(G,I_{G}(T^1N)) = h_{\rm top}(G). \] \end{corollary}
In particular, Corollary \ref{cor:2} applies if $M$ is compact with constant negative curvature $-1$ and $N$ is a proper submanifold of $M$. Indeed, in this case $\Gamma'$ is divergent and $\Omega_N=T^1N\neq T^1M=\Omega$. We provide the proofs of the above results in Section \ref{sec:proofs}.
The proof of Theorem \ref{the:1} will rely on the consideration of appropriate suspension flows, namely, we construct basic sets (that is, sets which are compact, $G$-invariant, and locally maximal such that $G|_B$ is topologically transitive) having almost full entropy, where the flow is conjugated to a suspension flow over a subshift of finite type (see Proposition \ref{pro:entropybasicset}). In particular, the following is an immediate consequence.
\begin{scholium}\label{sch:1}
Assume the hypotheses of Theorem \ref{the:1}. Denote by $B$ the set of vectors $v\in T^1M$ whose orbit $\EuScript{O}_G(v)\eqdef\{g^t(v)\colon t\in\mathbb{R}\}$ is bounded. Then \[
\sup_{K\subset T^1M\text{ basic}}h_{\rm top}(G,K)
= h_{\rm top}(G,B)
= h^\ast_{\rm top}(G,B)
= h_{\rm top}(G,T^1M). \] \end{scholium}
We finally state the corresponding auxiliary key result, which is of independent interest.
\begin{maintheorem}\label{thepro:susflo}
Consider some continuous function $\tau\colon\Sigma\eqdef\{1,\ldots,N\}^\mathbb{Z}\to (0,\alpha]$ and let $F\colon\Sigma(\sigma,\tau)\times\mathbb{R}\to\Sigma(\sigma,\tau)$ be the suspension flow of the shift map $\sigma\colon\Sigma\to\Sigma$ under $\tau$. Then for every Borel set $A\subset\Sigma(\sigma,\tau)$ satisfying $h^\ast_{\rm top}(F,A)<h_{\rm top}(F)$ it holds \[
h^\ast_{\rm top}(F,I_{F}(A))
= h_{\rm top}(F). \] \end{maintheorem}
This paper is organized as follows. In Section \ref{sec:entropy} we recall different concepts of entropy and some fundamental properties. Section \ref{sec:geodynneg} introduces the necessary elements from geometry. In Section \ref{sec:Katokshift} we prove an approximation result by ergodic measures on the symbolic space. In Section \ref{sec:Katokflow}, we study techniques of ``ergodic approximation by basic sets'' of an ergodic probability measure for the geodesic flow of a complete manifold (see Proposition \ref{pro:entropybasicset}). These approximation techniques obtaining flow-invariant basic sets will be implemented in Section \ref{sec:susflo} in which we study exceptional sets for suspension flows. Theorem \ref{thepro:susflo} is proven in Section \ref{sec:susflo}. Theorem \ref{the:1}, Corollaries \ref{cor:1} and \ref{cor:2}, and Scholium \ref{sch:1} are proven in Section \ref{sec:proofs}.
\section{Entropy}\label{sec:entropy}
We study topological entropy on arbitrary (not necessarily invariant or compact) subsets where the ambient space is not necessarily compact. Hence, in this section we recall some essential definitions and results.
Given a continuous flow $F=(f^t)_{t\in\mathbb{R}}$ on a metric space $(X,d)$, a Borel probability measure $\mu$ is \emph{$F$-invariant} if $(f^t)_\ast\mu=\mu$ for every $t\in\mathbb{R}$. Denote by $\EuScript{M}(F)$ the set of all $F$-invariant Borel probability measures. A measure $\mu\in\EuScript{M}(F)$ is \emph{ergodic} if for every $F$-invariant set $B\subset X$ either $\mu(B)=0$ or $\mu(B)=1$. Denote by $\EuScript{M}_{\rm erg}(F)$ the subset of ergodic measures. Given $t\in\mathbb{R}$, we denote by $\EuScript{M}(f^t)$ the set of $f^t$-invariant Borel probability measures and by $\EuScript{M}_{\rm erg}(f^t)$ the subset of $f^t$-ergodic ones.
\begin{remark}\label{rem:ergodic}
Note that for every $\mu\in\EuScript{M}_{\rm erg}(F)$ for every time $t>0$, except for a countable set of $t$-values values, it holds that $\mu\in\EuScript{M}_{\rm erg}(f^t)$ \cite{PugShu:71}. \end{remark}
\subsection{Metric entropy}
Given a measure $\mu\in\EuScript{M}(F)$, we denote by $h_\mu(F)$ its \emph{metric entropy} (with respect to the flow $F$) and by $h_\mu(f^t)$ its entropy (with respect to the map $f^t$). By Abramov's formula \cite{Abr:59}, for every $t\in\mathbb{R}$ it holds \begin{equation}\label{eq:Abramov}
\lvert t\rvert \, h_\mu(f^1)=h_\mu(f^t) \end{equation} and we let $h_\mu(F)\eqdef h_\mu(f^1)$.
\subsection{Topological entropy}
Let $(X,d)$ be a metric space and $f\colon X\to X$ a continuous map. For $n\in\mathbb{N}$ and $\varepsilon>0$ define \begin{equation}\label{eq:Bowbal}
B_n^d(x,\varepsilon)
\eqdef \{y\in X\colon d(f^k(x),f^k(y))<\varepsilon\text{ for all }k\in\{0,\ldots,n-1\}\} \end{equation} and call it \emph{$(n,\varepsilon)$-dynamical ball}. The above is simply the ball of radius $\varepsilon$ centered at $x$ with respect to the metric $d_n$ defined by \[
d_n(x,y)
\eqdef \max_{k\in\{0,\ldots,n-1\}}d(f^k(x),f^k(y)). \] A set $S\subset X$ is \emph{$(n,r)$-separated} if $d_n(x,y)\ge r$ for every pair of points $x,y\in S$, $x\ne y$. As $f$ is continuous, $B_n^d(x,\varepsilon)$ is open. Given a compact set $K\subset X$, denote by $N^d(n,\varepsilon,K)$ the minimal cardinality of a cover of $K$ by $(n,\varepsilon)$-dynamical balls. Then \[
h^d_{\rm top}(f)
\eqdef \sup_K\lim_{\varepsilon\to0}\limsup_{n\to\infty}\frac1n\log N^d(n,\varepsilon,K), \] where the supremum is taken over all compact subsets $K\subset M$, is the \emph{topological entropy} of $f$ (with respect to $d$). If the metric $d$ is clear from the context, we will drop the superscript ${}^d$.
Metric entropy and topological entropy are linked by the variational principle (see \cite{Din:70} for $X$ compact and \cite{HanKit:95} for the general case). First note that \[
\sup_{\mu\in\EuScript{M}(f)} h_\mu(f)
\le h^d_{\rm top}(f). \] A change of metric may result in a change of entropy. Assuming that $X$ is a locally compact metrizable topological space, the following \emph{variational principle} holds \begin{equation}\label{eq:vp}
\sup_{\mu\in\EuScript{M}(f)} h_\mu(f)
= h_{\rm top}(f)
\eqdef \inf_d h^d_{\rm top}(f), \end{equation} where the infimum is taken over all metrics $d$ on $X$ that generate the same topology. One calls $h_{\rm top}(f)$ the \emph{topological entropy} of $f$. Note that, by ergodic decomposition, in the supremum in \eqref{eq:vp} it is enough to consider the ergodic measures only. In the case when $(X,d)$ is a compact metric space then $h_{\rm top}(f)=h^d_{\rm top}(f)$.
By \eqref{eq:Abramov} together with the variational principle \eqref{eq:vp}, for a continuous flow $F=(f^t)_{t\in\mathbb{R}}$ on a locally compact metrizable topological space, for every $t\in\mathbb{R}$ it holds \[
h_{\rm top}(f^t)
=\lvert t\rvert\, h_{\rm top}(f^1), \] and we define the \emph{topological entropy} of $F$ by \begin{equation}\label{eq:timetmap}
h_{\rm top}(F)
\eqdef h_{\rm top}(f^1). \end{equation} The following variational principle is a consequence of the above facts \begin{equation}\label{eq:vpflow}
h_{\rm top}(F)
= \sup_{\mu\in\EuScript{M}(f^1)}h_\mu(F)
= \sup_{\mu\in\EuScript{M}(F)}h_\mu(F). \end{equation}
\subsection{Topological entropy on Borel subsets}\label{sec:entmeasub}
Let $f\colon X\to X$ be a continuous map on a (not necessarily compact) metric space $(X,d)$. Given $x\in X$ and $n\in\mathbb{N}$, consider the probability measure \begin{equation}\label{eq:deltadef}
\delta_{x,n}
\eqdef \frac1n\sum_{k=0}^{n-1}\delta_{f^k(x)}. \end{equation} Denote by \[
\EuScript{V}(f,x)
\eqdef \{\mu\in\EuScript{M}(f)\colon \delta_{x,n_k}\to\mu\text{ for some }n_k\to\infty\} \] the set of limit points relative to the weak$\ast$ topology. Recall that a sequence of measures $(\mu_n)$ converge weak$\ast$ to $\mu$ if and only if for every bounded continuous function $\varphi:X\to \mathbb{R}$, we have $\lim \int \varphi d\mu_n=\int \varphi d\mu$. Note that, since $X$ is not necessarily compact, the set $\EuScript{V}(f,x)$ could be empty. If the map $f$ is clear from the context then we drop it in the notation. Following \cite{Tho:11}, given a nonempty Borel set $Z\subset X$, denote \begin{equation}\label{notationMZ}
\EuScript{M}^Z(f)
\eqdef \{\mu\colon \mu\in\EuScript{V}(f,x)\text{ for some }x\in Z\} \end{equation} and define by \begin{equation}\label{def:entstar}
h^\ast_{\rm top}(f,Z)
\eqdef
\begin{cases}
\sup\big\{h_\mu(f)\colon \mu\in\EuScript{M}^Z(f)\}&\text{ if }\EuScript{M}^Z(f)\ne\emptyset,\\
0&\text{ otherwise}.
\end{cases} \end{equation} the \emph{topological entropy} of $f$ on $Z$.
We will implement the above definition in the noncompact case only to state Theorem \ref{the:1}. For the remaining arguments, we will consider appropriate compact and invariant subset $Y\subset X$ and study the entropy of $f|_Y$ on $Z\cap Y$. Note that together with \cite[Theorem 6.1]{Tho:11}, in this case it holds \[
h^\ast_{\rm top}(f|_Y,Z\cap Y)
= h^\ast_{\rm top}(f,Z\cap Y)
\le h^\ast_{\rm top}(f,Z). \]
\subsection{Topological entropy on subsets}\label{sec:entcomsub}
We also recall the definition of entropy according to Bowen \cite{Bow:73} for a continuous map $f\colon X\to X$ on a general topological space. Given an open cover $\mathscr A$ of $X$ and a subset $Z\subset X$, denote by $n_\mathscr A(Z)$ the smallest nonnegative integer $n$ such that $f^n(Z)$ is not contained in an element of $\mathscr A$; if $f^n(Z)$ is contained in an element of $\mathscr A$ for all integers $n\ge0$ then let $n_\mathscr A(Z)=\infty$. Define \[
m_\mathscr A(Z,s)
\eqdef \lim_{r\to0}\inf_\EuScript{U}\sum_{U\in\EuScript{U}}e^{-sn_\mathscr A(U)}, \] where the infimum is taken over all countable open covers $\EuScript{U}$ of $Z$ such that $n_\mathscr A(U)>1/r$ for all $U\in\EuScript{U}$. The \emph{topological entropy} of $f$ on $Z$ is defined by \[
h_{\rm top}(f,Z)
\eqdef \sup_\mathscr A h_\mathscr A(f,Z),
\quad\text{ where }\quad
h_\mathscr A(f,Z)
\eqdef \inf\{s\colon m_\mathscr A(Z,s)=0\}. \] If $Z=X$ is compact then the latter equals the topological entropy $h_{\rm top}(f)$ defined in \eqref{eq:vp}.
We collect below some elementary properties relating all previous notions of entropy.
\begin{lemma}\label{lem:propTho} Assume that $(X,d)$ is a metric space and $f\colon X\to X$ a continuous map. \begin{enumerate} \item[(i)] Given a nonempty Borel set $Z\subset X$, for every $n\in\mathbb{N}$ it holds \[
h^\ast_{\rm top}(f^n,Z)
= n\, h^\ast_{\rm top}(f,Z). \] \item[(ii)] Given nonempty Borel sets $Y,Z\subset X$ satisfying $Y\subset Z$, it holds \[
h^\ast_{\rm top}(f,Y)
\le h^\ast_{\rm top}(f,Z). \] \end{enumerate} \end{lemma}
\begin{proof}
Recall $h_\mu(f^n)=n h_\mu(f)$ and note that $\EuScript{V}(f^n,x)\subset\EuScript{V}(f,x)$. This together implies $h^\ast_{\rm top}(f^n,Z) \le n h^\ast_{\rm top}(f,Z)$. On the other hand, if $\mu\in\EuScript{V}(f,x)$ then for every $n\in\mathbb{N}$ it holds $\nu\eqdef\frac1n(\mu+f_\ast\mu+\ldots+f^{n-1}_\ast\mu)\in\EuScript{V}(f^n,x)$ and $h_\nu(f^n)=h_\mu(f^n)=n\,h_\mu(f)$. This proves item (i). Item (ii) is an immediate consequence of the definition. \end{proof}
\begin{lemma}\label{lem:propTho2} Assume that $(X,d)$ is a compact metric space, $f\colon X\to X$ is a continuous map, and $Z\subset X$ is a Borel set. \begin{enumerate} \item[(i)] If there are a compact metric space $(Y,\rho)$, a continuous map $g\colon Y\to Y$, and a continuous surjective map $p\colon X\to Y$ satisfying $p\circ f=g\circ p$, then \[
h_{\rm top}(g,p(Z))
\le h_{\rm top}(f,Z). \] \item[(ii)] If the map $p$ in item (i) is a homeomorphism, then \[
h^\ast_{\rm top}(f,Z)= h^\ast_{\rm top}(g,p(Z))
\quad\text{ and }\quad
h_{\rm top}(f,Z)= h_{\rm top}(g,p(Z)) . \] \item[(iii)] It holds \[
h_{\rm top}(f,Z)
\le h_{\rm top}^\ast(f,Z). \] \item[(iv)] It holds \[
h_{\rm top}(f,X)= h_{\rm top}^\ast(f,X). \] \end{enumerate} \end{lemma}
\begin{proof}
The proof of item (i) is straightforward. Item (ii) is \cite[Theorem 3.2]{Tho:11}, item (iii) is \cite[Theorem 4.3]{Tho:11}, and item (iv) is \cite[Theorem 3.3]{Tho:11}. \end{proof}
\begin{remark} We stress the fact that inequality (iii) in Lemma \ref{lem:propTho2} is strict in general. Take for instance an ergodic measure $\mu$ having positive entropy and $x\in X$ a $\mu$-generic point. Then for $Z=\{x\}$ we have $h_{\rm top}(f,Z)=0$ whereas $h_{\rm top}^\ast(f,Z)=h_\mu(f)>0$. \end{remark}
In analogy to the above, for a continuous flow $F=(f^t)_{t\in\mathbb{R}}$ on a metric space $X$, we define the \emph{topological entropy} of $F$ on a nonempty Borel set $Z\subset X$ by \[
h_{\rm top}^\ast(F,Z)
\eqdef h_{\rm top}^\ast(f^1,Z). \]
\section{Geometry and dynamics in negative curvature}\label{sec:geodynneg}
Let $\widetilde{M}$ be a complete simply connected Riemannian manifold with sectional curvatures bounded from above by $-1$. The \emph{boundary at infinity} $\partial_\infty\widetilde{M}$ of $\widetilde{M}$ is the set of asymptotic geodesic rays on $\widetilde{M}$. In particular, the set $\widetilde{M}\cup\partial_\infty\widetilde{M}$ is compact endowed with the cone topology and homeomorphic to the closed unit ball. We stress the fact that every isometry of $\widetilde{M}$ extends to a homeomorphism of $\widetilde{M}\cup\partial_\infty\widetilde{M}$.
\subsection{Hopf parametrization} Fix once for all a point $o\in\widetilde{M}$. For any boundary point $\xi\in\partial_\infty\widetilde{M}$, let $r_\xi\colon [0,+\infty)\to\widetilde{M}$ be the arc-parametrization of the geodesic ray with origin $o$ and extremity at infinity $\xi$. The \emph{Busemann cocycle} of $\widetilde{M}$ is the map $\beta\colon\widetilde{M}\times\widetilde{M}\times\partial_\infty\widetilde{M}\to\mathbb{R}$ defined by \[ (x,y,\xi)\mapsto\beta_\xi(x,y)\eqdef\lim_{t\to+\infty}d(x,r_\xi(t))-d(y,r_\xi(y)). \] This limit exists by the bounds on the sectional curvatures. It is independent of $o$.
Let $T^1\widetilde{M}$ be the unit tangent bundle of $\widetilde{M}$ and $\pi\colon T^1\widetilde{M}\to\widetilde{M}$ be the natural projection. For every $v\in T^1\widetilde{M}$, let $v^-$ and $v^+$ be the two extremities at infinity of the geodesic line defined by $v$. Let $\partial^2_\infty\widetilde{M}$ be the subset of $\partial_\infty\widetilde{M}\times\partial_\infty\widetilde{M}$ consisting of distinct points at infinity. The \emph{Hopf parametrization} of $T^1\widetilde{M}$ is the identification of $v\in T^1M$ with the triplet $(v^-,v^+,s)\in\partial^2_\infty\widetilde{M}\times\mathbb{R}$, where $s=\beta_{v^+}(o,\pi(v))$. This map is a homeomorphism. We use the notation $v=(v^-,v^+,s)$ whenever we mention a vector $v\in T^1\widetilde{M}$ in these coordinates.
\subsection{Distances on (sub)manifolds}
Let $v\colon(-\infty,+\infty)\to\widetilde M$ be the parametrization of the oriented geodesic ray defined by $v\in T^1\widetilde{M}$ such that $\pi(v)=v(0)$ and $\pi v\colon t\mapsto \pi(v(t))$ is arc-parametrized. We endow $T^1\widetilde{M}$ with the distance $d$ defined for all $v,v'\in T^1\widetilde{M}$ as follows \[
d(v,v')
\eqdef\frac{1}{\sqrt{\pi}}\int d(\pi(v(t)),\pi(v'(t)))e^{-t^2}dt. \]
\begin{remark} The distance $d$ is H\"older-equivalent to the distance induced by the Sasaki metric on $T^1\widetilde{M}$ \cite[Lemma 2.3]{PauPolSch:15}. \end{remark}
We denote by $G\eqdef (g^t)_{t\in\mathbb{R}}$ the geodesic flow on $T^1\widetilde{M}$ and $\iota\colon T^1\widetilde{M}\to T^1\widetilde{M}$ the flip map $v\mapsto -v$. Note that the geodesic flow acts by translation in the third coordinate, namely $g^t(v^-,v^+,s)=(v^-,v^+,s+t)$, and the flip map sends $(v^-,v^+,s)$ into $(v^+,v^-,-s)$. Both, the geodesic flow and the flip map commute with every isometry of $\widetilde{M}$. Moreover $\iota\circ g^t = g^{-t}\circ\iota$ for every $t\in\mathbb{R}$.
\begin{lemma}\label{lem:closegeodesics} Let $\varepsilon\in(0,1]$ and $s\geq 2$. There exists a constant $C=C(\varepsilon)>0$ such that, for any two geodesic lines $(\xi,\eta)$ and $(\xi',\eta')$ on $\widetilde{M}$ staying $\varepsilon$-close for time of length at least $2s$, their Hausdorff distance is less than $Ce^{-s}$. \end{lemma}
\begin{proof} Let $p\in (\xi,\eta)$ and $p'\in (\xi',\eta')$ be such that $d(p,p')$ realizes the Hausdorff distance. Let $r,r'\colon\mathbb{R}\to \widetilde{M}$ be the arc length parametrizations of $(\xi,\eta)$ and $(\xi',\eta')$, respectively, such that $r(0)=p$ and $r'(0)=p'$. By hypothesis, $d(r(t),r'(t))\leq\varepsilon$ for all $-s\leq t\leq s$. Set $x=r(-s)$, $y=r(s)$, $x'=r'(-s)$, and $y'=r'(s)$. Let $[z,w]$ be the shortest arc between $[x,x']$ and $[y,y']$ with $z\in [x,x']$. Let $q\in\widetilde{M}$ be the midpoint of $[z,w]$ and $\ell=\frac{1}{2}d(z,w)$. Compare also Figure \ref{figure1}. Note that $q$ is also the midpoint of the segment defined by $p$ and $p'$.
\begin{figure}
\caption{Proof of Lemma \ref{lem:closegeodesics}}
\label{figure1}
\end{figure}
By comparison, the distance $d(p,q)$ is less than the distance $\tau$ in the hyperbolic plane $\mathbb{H}$ between the midpoint $\bar{q}$ of a segment $[\bar{z},\bar{w}]$ of length $2l$ to a geodesic segment $[\bar{x},\bar{y}]\subset\mathbb{H}$ where the angle at $\bar{z}$ between $[\bar{x},\bar{z}]$ and $[\bar{z},\bar{w}]$ and the angle at $\bar{w}$ between $[\bar{z},\bar{w}]$ and $[\bar{w},\bar{y}]$ are exactly $\pi/2$. By \cite[Theorem 7.17.1 (i)]{Bea:83}, it holds $\sinh(\tau)\sinh(\ell)\leq 1$.
Hence \[
d(p,p')
=2d(p,q)
\leq 2\tau
\leq2\sinh(\tau)
= 2\sinh(\ell)^{-1}. \] By the triangle inequality, we also have $2\ell=d(z,w)\geq d(x,y)-2\varepsilon=2(s-\varepsilon)$, hence $\ell\geq s-\varepsilon$. By the inequalities above, we therefore have \[ d(p,p')\leq 2\sinh(s-\varepsilon)^{-1}. \] On the other hand, the hyperbolic sine function verifies $\sinh(\alpha)\geq e^{\alpha}/4$ for every $\alpha\geq 1$. Since $s-\varepsilon\geq 1$, we get \[ d(p,p')\leq 8e^{-(s-\varepsilon)}=8e^{\varepsilon}e^{-s}, \] which ends the proof of the lemma. \end{proof}
The \emph{strong stable} and \emph{strong unstable manifolds} of $v$ are defined by \[\begin{split}
\widetilde{W}^{{\rm ss}}(v)
&\eqdef \{w\in T^1\widetilde{M}\colon d(v(t),w(t))\to 0 \quad \text{as} \quad t\to+\infty\},\\
\widetilde{W}^{{\rm uu}}(v)
&\eqdef \{w\in T^1\widetilde{M}\colon d(v(t),w(t))\to 0 \quad \text{as} \quad t\to-\infty\}, \end{split}\] respectively. These submanifolds are smooth leaves of continuous foliations, namely the \emph{strong stable} and \emph{strong unstable foliations} $\widetilde{W}^{{\rm ss}}$ and $\widetilde{W}^{{\rm uu}}$, which are invariant under the isometry group of $\widetilde{M}$ and the geodesic flow. Note that, in Hopf coordinates, for every $v=(v^-,v^+,s)\in T^1\widetilde{M}$, \[\begin{split}
\widetilde{W}^{{\rm ss}}(v)
&=\{(w^-,w^+,r)\in T^1\widetilde{M} \colon w^+=v^+, \ r= s\},\\
\widetilde{W}^{{\rm uu}}(v)
&=\{(w^-,w^+,r)\in T^1\widetilde{M} \colon w^-=v^-, \ r= s\}. \end{split}\]
\begin{definition} The \emph{Hamenst\"adt distances} are defined, for every $u\in T^1\widetilde{M}$, as \[\begin{split}
d^{\rm u}_u(v,v')
&\eqdef\lim_{t\to+\infty} e^{\frac{1}{2}d(\pi(v(t)),\pi(v'(t)))-t}
\quad \text{ for all $v,v'\in \widetilde{W}^{{\rm uu}}(u)$}.\\
d^{\rm s}_u(w,w')
&\eqdef\lim_{t\to+\infty}e^{\frac{1}{2}d(\pi(w(-t)),\pi(w'(-t)))-t}
\quad \text{ for all $w,w'\in \widetilde{W}^{\rm ss}(u)$}. \end{split}\] Note that both limits exists and defines a distance inducing the original topology on $\widetilde{W}^{\rm uu}(u)$ and $\widetilde{W}^{\rm ss}(u)$ respectively (see \cite{Ham:89} for further details). \end{definition}
Note that \[
\iota(\widetilde{W}^{{\rm uu}}(u))=\widetilde{W}^{{\rm ss}}(\iota(u)) \] It is straightforward to check that $d^{\rm u}_u(v,v')=d^{\rm s}_{\iota u}(\iota v,\iota v')$ for all $v,v'\in \widetilde{W}^{{\rm uu}}(u)$.
The following lemma appears in \cite[Lemma 2.4]{PauPolSch:15}, following \cite{ParkPau:14}.
\begin{lemma}\label{lem:hamen:uu} There is $c>0$ so that for every $u\in T^1\widetilde{M}$, $v,v'\in \widetilde{W}^{{\rm uu}}(u)$, and $w,w'\in\widetilde W^{\rm ss}(u)$, \[\begin{split}
\max\left\{\frac{1}{c}d(v,v'),d(\pi(v),\pi(v'))\right\}
&\leq d^{\rm u}_u(v,v')\leq e^{\frac{1}{2}d(\pi(v),\pi(v'))},\\
\max\left\{\frac{1}{c}d(w,w'),d(\pi(w),\pi(w'))\right\}
&\leq d^{\rm s}_u(w,w')\leq e^{\frac{1}{2}d(\pi(w),\pi(w'))}. \end{split}\] \end{lemma}
A remarkable property of the Hamenst\"adt distances is the uniform expansion/contraction under the action of the geodesic flow.
\begin{lemma}\label{lem:unifexp} For every $u\in T^1\widetilde{M}$, $s\in\mathbb{R}$, $v,v'\in \widetilde{W}^{{\rm uu}}(u)$, and $w,w'\in \widetilde{W}^{{\rm ss}}(u)$, \[\begin{split}
d^{{\rm u}}_{g^s(u)}(g^s(v),g^s(v'))
&=e^s d^{{\rm u}}_u(v,v')\\
d^{{\rm s}}_{g^s(u)}(g^s(w),g^s(w'))
&=e^{-s} d^{{\rm s}}_u(w,w'). \end{split}\] \end{lemma}
\subsection{Quotients} Let $\Gamma$ be a discrete, non-elementary subgroup of isometries of $\widetilde{M}$ and $M\eqdef\widetilde{M}/\Gamma$ the quotient space. Even if $M$ is not a manifold (since $\Gamma$ is not necessarily torsion-free), we denote by $T^1M$ the quotient $T^1\widetilde{M}/\Gamma$. Since the geodesic flow and the flip map commute with every isometry of $\widetilde{M}$, both descend respectively to maps on $T^1M$ that we still denote as $g^t\colon T^1M\to T^1M$ and $\iota\colon T^1M\to T^1M$. For $v\in T^1M$ we define the \emph{strong stable} and \emph{strong unstable manifolds} $W^{{\rm ss}}(v)$ and $W^{uu}(v)$ as \[\begin{split}
W^{{\rm ss}}(v)
\eqdef\{w\in T^1M\colon d(g^t(v),g^t(w))\to 0 \quad \text{as} \quad t\to+\infty\},\\
W^{{\rm uu}}(v)
\eqdef\{w\in T^1M\colon d(g^t(v),g^t(w))\to 0 \quad \text{as} \quad t\to-\infty\}, \end{split}\] respectively. Let $p^1_\Gamma\colon T^1\widetilde{M}\to T^1M$ and $p_\Gamma\colon \widetilde{M}\to M$ be the corresponding quotient maps. Then for any $\tilde{v}\in T^1\widetilde{M}$ and $v=p^1_\Gamma(\tilde{v})$, it holds $p^1_\Gamma(\widetilde{W}^{{\rm ss}}(\tilde{v}))\subset W^{{\rm ss}}(v)$ and $p^1_\Gamma(\widetilde{W}^{{\rm uu}}(\tilde{v}))\subset W^{{\rm uu}}(v)$. For small $\varepsilon>0$, we also define the \emph{local strong stable manifold} $W^{{\rm ss}}_\varepsilon(v)$ (resp., \emph{local strong unstable manifold} $W^{{\rm uu}}_\varepsilon(v)$) at $v\in T^1M$ as the connected component of $W^{{\rm ss}}(v)\cap B(v,\varepsilon)$ (resp. $W^{{\rm uu}}(v)$) containing $v$. We can similarly define local strong stable/unstable manifolds on $T^1\widetilde{M}$, and denote them by $\widetilde{W}^{{\rm ss}}_\varepsilon(\tilde{v})$ and $\widetilde{W}^{{\rm uu}}_\varepsilon(\tilde{v})$. Observe that for $\varepsilon$ small enough (depending on the injectivity radius at $\pi(v)$), it holds \[ p^1_\Gamma(\widetilde{W}^{{\rm ss}}_\varepsilon(\tilde{v}))=W^{{\rm ss}}_\varepsilon(v) \quad \text{and} \quad p^1_\Gamma(\widetilde{W}^{{\rm uu}}_\varepsilon(\tilde{v}))=W^{{\rm uu}}_\varepsilon(v), \] so strong stable/unstable manifolds can be locally studied using coordinates in $T^1\widetilde{M}$. We also remark that Hamenst\"adt distances are inherited on
local strong stable/unstable manifolds on $T^1M$.
\subsection{Local product structure}
The geodesic flow on negatively curved manifolds verifies the following local product structure.
\begin{figure}
\caption{Local product structure}
\label{figure}
\end{figure}
\begin{proposition}[Local product structure]\label{prop:lps} Every $u\in T^1M$ admits a neighbourhood $V$ which satisfies the following. For every $\varepsilon>0$ there exists $\beta>0$ such that for all $v\in V$ with $d(u,v)<\beta$, there exists a unique vector $w\in T^1M$ and a real number $|t|<\varepsilon$, so that \[ w\eqdef{[u,v]}\in W^{{\rm ss}}_\varepsilon(u)\cap W^{{\rm uu}}_\varepsilon(g^t v) \] \end{proposition}
\begin{proof} Let $u\in T^1M$, $\tilde{u}\in T^1\widetilde{M}$ any lift, and let \[
V_{\tilde{u}}
\eqdef \{\tilde{v}\in T^1\widetilde{M}\colon \tilde{v}^-\neq \tilde{u}^+\}. \]
The neighborhood $V_{\tilde{u}}$ of $\tilde{u}$ is open and dense in $T^1\widetilde{M}$. Moreover, any $\tilde{v}\in V_{\tilde{u}}$ has the following property: there exists $t\in\mathbb{R}$, $\tilde{v}_{{\rm ss}}\in \widetilde{W}^{{\rm ss}}(\tilde{u})$ and $\tilde{v}_{{\rm uu}}\in \widetilde{W}^{{\rm uu}}(\tilde{v})$ such that $g^t(\tilde{v}_{{\rm uu}})=\tilde{v}_{\rm ss}$, that is $\widetilde{W}^{{\rm ss}}(\tilde{u})\cap \widetilde{W}^{{\rm uu}}(g^t\tilde{v})$ is non-empty with only one element $\tilde{w}\eqdef\tilde{v}_{\rm ss}$. Since strong stable/unstable leaves are continuous, as well as the geodesic flow, for any $\varepsilon>0$ there exists $\beta>0$ such that $d(\tilde{u},\tilde{v})<\beta$ implies $\tilde{v}\in V_{\tilde{u}}$, $d(\tilde{u},\tilde{w})<\varepsilon$, $d(g^t(\tilde{v}),\tilde{w})<\varepsilon$ and $|t|<\varepsilon$. Finally, the desired property is verified in $T^1M$ for $w=p^1_\Gamma(\tilde{w})$ whenever $\varepsilon$ is chosen less than the injectivity radius at $u$. Compare also Figure \ref{figure}. \end{proof}
\section{Approximating ergodic measures by subshifts of finite type}\label{sec:Katokshift}
In this section, we follow techniques introduced in \cite[Supplement S.5]{KatHas:95} to construct some compact invariant subset which ergodically (weak$\ast$ and in entropy) approximates a given ergodic measure. We invoke this approach in the case of a subshift of finite type. It will be useful when proving Theorem \ref{thepro:susflo} in Section \ref{sec:susflo}.
Given $N\in\mathbb{N}$, consider the shift space $\Sigma\eqdef\{1,\ldots,N\}^\mathbb{Z}$ equipped with the metric $d_\Sigma(\underline i,\underline j)\eqdef 2^{-\inf\{\lvert k\rvert\colon i_k\ne j_k\}}$ and the left shift $\sigma\colon\Sigma\to\Sigma$. A \emph{subshift of finite type} (\emph{SFT}) is a subset $\Xi\subset\Sigma$ for which there is a matrix $A=(a_{k\ell})_{k,\ell=1}^N$, $a_{k\ell}\in\{0,1\}$, so that $\Xi=\{\underline i\in\Sigma\colon a_{i_ni_{n+1}}=1\text{ for all }n\in\mathbb{Z}\}$. Note that any SFT is compact and $\sigma$-invariant. For every $\underline i=(i_0i_1\ldots)\in\Sigma$ and $n\in\mathbb{N}$, denote the \emph{$n$th level cylinder} containing $\underline i$ by \[
[i_0\ldots i_{n-1}]
\eqdef \{\underline j\in\Sigma\colon j_k=i_k\text{ for all }k=0,\ldots,n-1\}. \]
Let $\pi^+\colon\Sigma\to\Sigma^+$ be the natural projection defined by $\pi^+(\underline i)\eqdef\underline i^+\eqdef(i_0i_1\ldots)$. Consider also the space $\Sigma^+\eqdef\{\underline i^+\colon \underline i\in\Sigma\}$ of one-sided sequences together with the left shift $\sigma^+\colon\Sigma^+\to\Sigma^+$. A SFT for $\sigma^+$ is analogously defined. Analogously, consider the $n$th cylinder of $\underline i^+$, which we denote by $[i_0\ldots i_{n-1}]^+$. Let $\nu^+\eqdef\pi^+_\ast\nu$ and note that $\nu^+$ is $\sigma^+$-ergodic. Moreover, $h_{\nu^+}(\sigma^+)=h_{\nu}(\sigma)$.
\begin{proposition}\label{pro:SFT}
For every continuous function $\tau\colon\Sigma\to (0,\alpha]$, $\nu\in\EuScript{M}_{\rm erg}(\sigma)$, and $\varepsilon>0$ there exists a subshift of finite type $\Xi\subset\Sigma$ such that \[
\lvert h_{\rm top}(\sigma,\Xi)-h_\nu(\sigma)\rvert
<\varepsilon
\quad\text{ and }\quad
\Big\lvert \int\tau\,d\lambda-\int\tau\,d\nu\Big\rvert
<\varepsilon
\quad\text{for every $\lambda\in\EuScript{M}(\sigma|_\Xi)$}. \] \end{proposition}
Before proving the above result, for $n\in\mathbb{N}$ and $\psi\colon\Sigma\to\mathbb{R}$ let \[
{\rm var}_n\psi
\eqdef \sup\{\lvert \psi(\underline j)-\psi(\underline i)\rvert\colon \underline j,\underline i\in\Sigma, j_k=i_k\text{ for all }\lvert k\rvert\le n\} \] and define the set of functions \[
\mathcal{F}_\Sigma
\eqdef \{\psi\colon\Sigma\to\mathbb{R}\colon
\text{there exist } b>0,\alpha\in(0,1)\text{ so that }
{\rm var}_n\psi\le b\alpha^n\text{ for all }n\ge0\}. \]
\begin{proof}[Proof of Proposition \ref{pro:SFT}]
Let $\varepsilon>0$. Let $\psi\in\mathcal{F}_\Sigma$ such that $\sup\lvert\psi-\tau\rvert<\varepsilon/3$. Indeed, for example any function which is piecewise constant on sufficiently high-level cylinders has this property.
By \cite[1.6 Lemma]{Bow:08} there is $\phi\in\mathcal{F}_\Sigma$ which is cohomologous to $\psi$ and satisfies $\phi(\underline k)=\psi(\underline i)$ for every $\underline k\in\Sigma$ such that $k_n=i_n$ for all $n\ge0$. Hence the function $\psi^+\colon\Sigma^+\to\mathbb{R}$, $\psi^+(\underline i^+)\eqdef \phi(\underline i)$ for any $\underline i\in[i_0i_1\ldots]$ is well-defined and continuous. As $\phi\in\mathcal{F}_\Sigma$, there are $b>0$ and $\alpha\in(0,1)$ such that for all $n\ge0$ it holds ${\rm var}_n\phi\le b\alpha^n$. Hence, from the definition of $\psi^+$, we get ${\rm var}_n\psi^+\le b\alpha^n$.
By the Brin-Katok theorem \cite{BriKat:83}, for $\nu^+$-almost every $\underline i^+\in\Sigma^+$ it holds \[
\lim_{n\to\infty}-\frac1n\log\nu^+([i_0\ldots i_{n-1}]^+)
= h_{\nu^+}(\sigma^+)
= h_{\nu}(\sigma)
\eqdef h. \]
By the Birkhoff ergodic theorem, for $\nu^+$-almost every $\underline i^+\in\Sigma^+$ it holds \[
\lim_{n\to\infty}\frac1nS_n\psi^+(\underline i^+)
= \int\psi^+\,d\nu^+,
\quad\text{ where }
S_n\psi^+\eqdef \psi^++\psi^+\circ \sigma^++\ldots+\psi^+\circ(\sigma^+)^{n-1}. \]
Fix $\kappa\in(0,1)$. By the Egorov theorem, there is a set $\Lambda\subset\Sigma^+$ satisfying $\nu^+(\Lambda)>1-\kappa$ and $N\in\mathbb{N}$ such that for every $n\ge N$ and $\underline i^+\in\Lambda$ it holds \begin{equation}\label{eq:cylin}
e^{-n(h-\varepsilon)}
\le \nu^+([i_0\ldots i_{n-1}]^+)
\le e^{-n(h+\varepsilon)} \end{equation} and \[
\left\lvert \int\psi^+\,d\nu^+
-\frac1nS_n\psi^+(\underline i^+)\right\rvert
\le \frac\varepsilon2. \] Assume that $N$ was chosen large enough such that $b\alpha^N\le\varepsilon/2$. Hence for every $\underline i^+\in\Lambda$ and $n\ge N$, \[
\max_{\underline j^+\in[i_0\ldots i_{n-1}i_n\ldots i_{2n-1}]^+}
\left\lvert S_n\psi^+(\underline j^+)
-S_n\psi^+(\underline i^+)\right\rvert
\le nb\alpha^n
\le n\frac\varepsilon2 . \] Fix now some $n\ge N$. Choose any sequence $\underline i^{+,1}\in \Lambda$. Let $A_0\eqdef \Lambda$. Inductively, for $k\in\mathbb{N}$ choose $\underline i^{+,k}\in A_{k-1}$ and let $A_k\eqdef A_{k-1}\setminus[i^k_0\ldots i^k_{n-1}]^+$. By \eqref{eq:cylin} there is $M\in\mathbb{N}$, \[
e^{n(h-\varepsilon)}
\le M
\le e^{n(h+\varepsilon)}, \] such that for every $k>M$ the set $A_k$ is empty. By definition, the cylinders $[i^{k}_0\ldots i^{k}_{n-1}]^+$, $k=1,\ldots,M$, are pairwise disjoint and \[
(\sigma^+)^n\big([i^{k}_0\ldots i^{k}_{n-1}]^+\big)
=\Sigma^+. \]
The (one-sided) infinite concatenation of any combination of finite sequences from the family $\{(i^{k}_0\ldots i^{k}_{n-1})\}_{k=1}^M$ defines a (one-sided) SFT $\Xi^+\subset\Sigma^+$. By the above \[
h-\varepsilon
\le \frac1n \log \card M
= h_{\rm top}(\sigma^+,\Xi^+)
\le h+\varepsilon . \]
For any ergodic measure $\lambda^+\in\EuScript{M}(\sigma^+|_{\Xi^+})$, for $\lambda^+$-almost every $\underline i^+$ it holds \[
\int\psi^+\,d\lambda^+
= \lim_{k\to\infty}\frac{1}{kn}S_{kn}\psi^+(\underline i^+). \] Hence \[
\Big\lvert \int\psi^+\,d\lambda^+-\int\psi^+\,d\nu^+\Big\rvert
<\varepsilon. \]
Let $\sigma|_\Xi\colon\Xi\to\Xi$ be the natural extension of the SFT $\Xi^+$ and note that it is SFT (with respect to $\sigma$) and satisfies $h_{\rm top}(\sigma,\Xi)=h_{\rm top}(\sigma^+,\Xi^+)$. The above implies that for every
ergodic measure $\lambda\in\EuScript{M}(\sigma|_{\Xi})$, \[\begin{split}
\varepsilon
&>\Big\lvert \int\psi\,d\lambda-\int\psi\,d\nu\Big\rvert\\
&\ge \Big\lvert \int\tau\,d\lambda-\int\tau\,d\nu\Big\rvert
- \Big\lvert \int\tau\,d\nu-\int\psi\,d\nu\Big\rvert
-\Big\lvert \int\psi\,d\lambda-\int\tau\,d\lambda\Big\rvert\\
&\ge \Big\lvert \int\tau\,d\lambda-\int\tau\,d\nu\Big\rvert
- 2\frac\varepsilon3. \end{split}\] This proves the proposition. \end{proof}
\section{Approximating basic sets in a geodesic flow}\label{sec:Katokflow}
In this section we continue the principle idea in Section \ref{sec:Katokshift} to ``ergodically approximate'' an ergodic measure. We deal now with the geodesic flow of a complete Riemannian manifold of negative curvature and obtain an approximation in terms of entropy by basic sets. For that first recall that a compact $G$-invariant set $B\subset T^1M$ is \emph{locally maximal} if there exists a neighborhood $U$ of $B$ such that \[
B
=\bigcap_{t\in\mathbb{R}}g^t(\overline U). \]
Recall that $G|_B$ is \emph{topologically transitive} if there is $x\in B$ such that $\overline{\EuScript{O}^+_G(x)}=B$. We say that a compact $G$-invariant set is \emph{basic} if it is hyperbolic, locally maximal, and $G|_B$ is topologically transitive. A set $B\subset T^1M$ has \emph{a local product structure} if for every $u,v\in B$ satisfying $d(u,v)<\beta$ for $\beta>0$ sufficiently small the point $[u,v]$ is again in $B$ (where $[\cdot,\cdot]$ was defined in Proposition \ref{prop:lps}). Recall that a compact $G$-invariant hyperbolic set $B$ is locally maximal if and only if it has a local product structure (see, for example, \cite[Theorem 6.2.7]{FisHas:19}).
Note that the construction of compact invariant sets with certain ergodic properties in this context can also be found in \cite[Section 6.2]{PauPolSch:15} while proving a variational principle for the topological pressure, following ideas from \cite[Section 4]{OP:04}.
The following is our main result of this section.
\begin{proposition}\label{pro:entropybasicset}
Consider the geodesic flow $G=(g^t)_{t\in\mathbb{R}}$ on the unit tangent bundle $T^1M$ of a $n$-dimensional complete Riemannian manifold $M$ of negatively pinched sectional curvatures $-b^2\le \kappa\leq -1$ for some $b\geq 1$.
Then for every measure $\mu\in\EuScript{M}_{\rm erg}(G)$ with positive entropy and $\varepsilon>0$ there exists a basic set $B\subset T^1M$ of topological dimension $1$ satisfying \[
h_{\rm top}(G,B)
\ge h_\mu(G)-\varepsilon. \] \end{proposition}
We postpone the proof of this proposition to Section \ref{ssec:proofprop}. We first prepare the necessary tools. In Section \ref{ssec:prorec} we first consider sections which induce proper rectangles, obtained from the local product structure. In Section \ref{ssec:prorecK} we cover a given compact set by proper rectangles and study a return map to local cross sections.
\subsection{Sections and proper rectangles}\label{ssec:prorec}
By the local product structure stated in Proposition \ref{prop:lps}, for every $\varepsilon=\varepsilon_{\rm lps}>0$ sufficiently small, there is $\beta_{\rm lps}>0$ such that for every $u,v\in T^1M$ satisfying $d(u,v)\le\beta_{\rm lps}$ the point \begin{equation}\label{eq:locpro}
w
\eqdef [u,v]
= W^{\rm ss}_\varepsilon(u)\cap W^{\rm uu}_\varepsilon(g^t(v))
\quad\text{ for some }\lvert t\rvert\le\varepsilon. \end{equation} is well defined (this intersection contains just one point).
Let us now transfer this local product structure to local cross sections of the flow. Given an interval $I\subset\mathbb{R}$ and $Z\subset T^1M$, denote $g^I(Z)\eqdef \bigcup_{t\in I}g^t(Z)$. A \emph{section of size $\varepsilon$} is a closed set $D\subset T^1M$ such that $(x,t)\mapsto g^t(x)$ is a homeomorphism between $D\times[-\varepsilon,\varepsilon]$ and its image $g^{[-\varepsilon,\varepsilon]}(D)$. In the following, we will assume that $D$ is a sufficiently small closed codimension-one smooth disk. Every section $D$ has associated a well-defined projection map $\proj_D\colon g^{[-\varepsilon,\varepsilon]}(D)\to D$ given by $\proj_D(g^t(u))=u$. Note that the domain of $\proj_D$ contains a nonempty open subset of $T^1M$.
If $D$ is a section, then for a closed set $R\subset D$ satisfying $d(R,\partial D)>0$ and having diameter sufficiently small relative to $d(R,\partial D)$, we define \[
[\cdot,\cdot]_D\colon R\times R\to D,\quad
[u,v]_D\eqdef\proj_D([u,v]). \]
We say that $R$ is a \emph{rectangle} in $D$ if $[R,R]_D\subset R$; in this case let $[\cdot,\cdot]_R\eqdef[\cdot,\cdot]_D|_{R\times R}$. A rectangle $R$ in $D$ is \emph{proper} if $R=\overline{\interior R}$ in the internal topology of the disk $D$.
For $D$ a section of size $\varepsilon$ and $R$ a proper rectangle in $D$, consider the flow-box \[
\widehat R
\eqdef g^{[-\varepsilon/2,\varepsilon/2]}(R). \] This way, for every $u\in \widehat R$ there is $\tau=\tau(R,u)\in[-\varepsilon/2,\varepsilon/2]$ so that $g^{\tau}(u)\in R\subset D$. Given $w\in R$, let \[\begin{split}
W^{\rm s}(w,R)
&\eqdef \{[w,v]_R\colon v\in R\}
= R\cap\proj_D\big(g^{[-\varepsilon,\varepsilon]}(D)
\cap W^{\rm ss}_\varepsilon(w)\big),\\
W^{\rm u}(w,R)
&\eqdef \{[v,w]_R\colon v\in R\}
= R\cap\proj_D\big(g^{[-\varepsilon,\varepsilon]}(D)
\cap W^{\rm uu}_\varepsilon(w)\big). \end{split}\] This way, $R$ is a direct product of the sets $W^{\rm s}(w,R)$ and $W^{\rm u}(w,R)$ in terms of the homeo\-morphism $(u,v)\mapsto[u,v]_R$.
We state the following lemma without proof (see, for example, the arguments for constructing geometric rectangles in \cite[Section 4.1]{ConLafTho:20}).
\begin{lemma}
Given any $v\in T^1M$, there exist a proper rectangle $R$ of arbitrarily small diameter such that $v\in \interior R$. \end{lemma}
\subsection{Sections and proper rectangles to cover a compact set}\label{ssec:prorecK}
For the statement of the following lemma, recall the constant $c>0$ from Lemma \ref{lem:hamen:uu}.
\begin{lemma}\label{lem:rec1}
For every compact set $K\subset T^1M$ and $\alpha>0$, for every $\varepsilon_{\rm sec}\in(0,\alpha/(3c))$ sufficiently small, there are a finite collection of sections $\{D_i\}$ of size $\varepsilon_i\in(0,\varepsilon_{\rm sec})$ and a number $\beta_{\rm sec}>0$ such that for every $\beta\in(0,\beta_{\rm sec})$, there are points $v_i\in K$ such that \[
K
\subset \bigcup_i B(v_i,\frac\beta2) \] and that for every $i$ there is a proper rectangle $R_i$ in $D_i$ such that $B(v_i,\beta)\subset g^{(-\varepsilon_i,\varepsilon_i)}(R_i)$.
Moreover, there is $M\in\mathbb{N}$ such that for every index $i$, for every $m\ge M$ the following holds. The map \begin{equation}\label{def:Tim}\begin{split}
&T_{i,m}\colon\widehat D_i^m\to \widehat D_i^m,
\quad\text{ where }\quad
\widehat D_i^m\eqdef \proj_{D_i}\big(\widehat R_i \cap g^m(\widehat R_i)\big)\\
&T_{i,m}(u')
\eqdef g^{m+\tau(R_i,g^m(u))}(u')
\in R_i,
\quad\text{ where }\quad
u'=g^{\tau(R_i,u)}(u) \end{split}\end{equation} is continuous and injective and \begin{equation}\label{eq:estTim}
T_{i,m}(u')
= g^t(u')
\quad\text{ for some }\quad
t\in(m-\varepsilon_i,m+\varepsilon_i). \end{equation} Moreover, denoting by $\mathscr{C}_{i,m}(u)$ the connected component of $\widehat D_i^m$ which contains $u\in D_i$, every such component is a proper rectangle in $D_i$. Furthermore, if $v,w\in B(v_i,\beta/2)\cap g^{-m}(B(v_i,\beta/2))$ are $(m,\alpha)$-separated, then \begin{equation}\label{eq:disjoint}
\mathscr{C}_{i,m}(\proj_{D_i}(v))\cap\mathscr{C}_{i,m}(\proj_{D_i}(w))
=\emptyset. \end{equation} \end{lemma}
\begin{proof} Let $\varepsilon_{\rm lps}\in(0,\alpha/(3c))$ be sufficiently small and choose $\beta_{\rm lps}>0$ accordingly to the local product structure. Fix \[
0<\varepsilon_{\rm sec}
< \frac{1}{3}\min\{c,\frac1c\}\cdot\min\{\varepsilon_{\rm lps},\alpha\}. \]
By compactness of $K$ and continuity of the local product structure, for every $u\in K$ there exists a smooth section $D(u)$ of size $\varepsilon_u\in(0,\varepsilon_{\rm sec})$ containing $u$ and a proper rectangle $R(u)$ in $D(u)$. Note that $U(u)\eqdef g^{(-\varepsilon_u/2,\varepsilon_u/2)}(\interior R(u))$ defines an open neighborhood of $u$ in $T^1M$. By compactness, there exists a finite cover $\{U(u_i)\colon u_i\in K\}$ of $K$. Let $L_0>0$ be a Lebesgue number for this open cover.
Let \[
\beta\in(0,\beta_{\rm sec}),
\quad\text{ where }\quad
\beta_{\rm sec}
\eqdef\min\{c,\frac1c\}\cdot\min\{\frac12L_0,\beta_{\rm lps}\}. \] Fix some finite open cover of $K$, \[
K
\subset \bigcup_iB(v_i,\frac\beta2). \] By the above, for every index $i$ there is some $u_i$ so that \[
B(v_i,\beta)
\subset g^{(-\varepsilon_i/2,\varepsilon_i/2)}(\interior R_i)
\subset\widehat R_i, \] where $\varepsilon_i\eqdef\varepsilon(u_i)$ and $R_i\eqdef R(u_i)$ is a proper rectangle in the section $D_i\eqdef D(u_i)$ of size $\varepsilon_i$. This proves the first assertion of the lemma.
Choose now $M\in\mathbb{N}$ sufficiently large such that \begin{equation}\label{hyp:betaM}
e^{-M}e^{\varepsilon_{\rm lps}}\varepsilon_{\rm lps}<\frac1c\frac\beta2. \end{equation}
\begin{claim}\label{cla:prorec}
For every index $i$, for every $m\ge M$ and $u\in B(v_i,\beta/2)\cap g^{-m}(B(v_i,\beta/2))$, \[\begin{split}
W^{\rm s}\big(\proj_{D_i}(u),R_i\big)
&\subset \proj_{D_i}\big(\widehat R_i\cap g^{-m}(\widehat R_i)\big),\\
W^{\rm u}\big(\proj_{D_i}(g^m(u)),R_i\big)
&\subset \proj_{D_i}\big(g^m(\widehat R_i)\cap \widehat R_i\big). \end{split}\] \end{claim}
\begin{proof} As $u\in B(v_i,\beta/2)\subset\widehat R_i$, $u'=g^s(u) \eqdef\proj_{D_i}(u)$ for $s\eqdef \tau(R_i,u)$ satisfying $\lvert s\rvert\le\varepsilon_i/2$. Given $w'\in W^{\rm s}(u',R_i)$. Hence, by definition, $w'\in R_i\subset\proj_{D_i}(\widehat R_i)$.
On the other hand, also by definition, $w'=g^t(w)=\proj_{D_i}(w)$ for some $w\in W^{\rm ss}_{\varepsilon_i}(u')$ and $t\eqdef\tau(R_i,w)$ satisfying $\lvert t\rvert\le\varepsilon_i$. Therefore, $w''\eqdef g^{-s}(w)\in g^{-s}(W^{\rm ss}(u'))= W^{\rm ss}(g^{-s}(u'))=W^{\rm ss}(u)$. By Lemma \ref{lem:unifexp}, \[\begin{split}
d^{\rm s}_{g^m(u)}(g^m(w''),g^m(u))
&= e^{-m}d^{\rm s}_{u}(w'',u)
= e^{-m}d^{\rm s}_u(g^{-s}(w),g^{-s}(u'))\\
&= e^{-m}e^sd^{\rm s}_{u'}(w,u')
\le e^{-m}e^{\varepsilon_i}\varepsilon_i\\
\text{\small by \eqref{hyp:betaM}}\quad
&< \frac1c\frac\beta2. \end{split}\] By Lemma \ref{lem:hamen:uu}, \[
d(g^m(w''),g^m(u))
\le cd^{\rm s}_{g^m(u)}(g^m(w''),g^m(u))
\le c\frac1c\frac\beta2
= \frac\beta2. \] Hence, our hypothesis $g^m(u)\in B(v_i,\beta/2)$ implies that $g^m(w'')\in B(v_i,\beta)\subset\widehat R_i$. In particular, $w'=\proj_{D_i}(w'')\in g^{-m}(\widehat R_i)$. This proves the first inclusion.
The other inclusion is analogous. \end{proof}
Claim \ref{cla:prorec} hence implies that every set $\mathscr{C}_{i,m}(\proj_{D_i}(u))$ is a proper rectangle in $D_i$.
For every index $i$, for every $m\ge M$ and $u\in \widehat R_i\cap g^{-m}(\widehat R_i)$ the image $g^m(u)$ is in the flow-box $\widehat R_i$, and hence the orbit of $u$ passes through the rectangle $R_i$. Hence, its associated return to $R_i$, \eqref{def:Tim}, is well defined. Note that it is not necessarily the first return.
\begin{claim}\label{cl:Timcontinuous}
The map $T_{i,m}$ defined in \eqref{def:Tim} is continuous and injective. \end{claim}
\begin{proof}
To show continuity, note that $u\mapsto \tau(R_i,u)$ is continuous on $\widehat R_i$. Hence, continuity of $T_{i,m}$ follows from the continuity of the flow $g$.
By contradiction, assume that $g^{m+\tau(R_i,u)}(u)=g^{m+\tau(R_i,v)}(v)\eqdef w\in R_i$ for $u,v\in R_i$, $u\ne v$. Hence, letting $u'=g^m(u)$ and $v'=g^m(v)$, it holds $v'=g^\delta(u')$ for some $\delta\in[-\varepsilon_i,\varepsilon_i]$, $\delta\ne0$. Without loss of generality, we can assume that $\delta>0$. Then, because $u',v'$ are both contained in the flow-box $\widehat R_i$, it follows that the segment $g^{[0,\delta]}(u')$ is completely contained in this flow-box. Since on one hand $u,v\in \widehat R_i$ and on the other hand $u,v\in g^{-m}(g^{[0,\delta]}(u')) =g^{[-m,m+\delta]}(u')$, it follows that $u,v$ are on the same orbit. But this contradicts the fact that $u,v\in D_i$ and the fact that $D_i$ is a section. \end{proof}
For every $u'\in\widehat D_i^m$, it holds \[
T_{i,m}(u')
= T_{i,m}(g^{\tau(R_i,u)}(u))
= g^{m+\tau(R_i,g^m(u))-\tau(i,u)}(u'), \] where $\tau(R_i,u),\tau(R_i,g^m(u))\in[-\varepsilon_i/2,\varepsilon_i/2]$, proving
\eqref{eq:estTim}.
Let us finally show \eqref{eq:disjoint}.
\begin{claim}\label{cla:difcomp}
For every index $i$, for every $m\ge M$ and pair of points $v,w\in B(v_i,\beta/2)\cap g^{-m}(B(v_i,\beta/2))$ which are $(m,\alpha)$-separated, it holds \[
\mathscr{C}_{i,m}(\proj_{D_i}(v))\cap\mathscr{C}_{i,m}(\proj_{D_i}(w))
=\emptyset. \] \end{claim}
\begin{proof} Assume $v,w\in B(v_i,\beta/2)\cap g^{-m}(B(v_i,\beta/2))$ are $(m,\alpha)$-separated. Arguing by contradiction, suppose that for $v'\eqdef\proj_{D_i}(v)$ it holds $w'\eqdef\proj_{D_i}(w)\in\mathscr{C}_{i,m}(v')$. As $\mathscr{C}_{i,m}(v')$ is a proper rectangle, $[v',w']_{R_i}\in\mathscr{C}_{i,m}(v')$. It follows from \eqref{eq:locpro} that the point \[\begin{split}
z
\eqdef [v,w]
&= W^{\rm ss}_{\varepsilon_{\rm lps}}(v)\cap g^\tau(W^{\rm uu}_{\varepsilon_{\rm lps}}(w))\\
&= W^{\rm ss}_{\varepsilon_{\rm lps}}(v)\cap W^{\rm uu}_{\varepsilon_{\rm lps}}(w''),
\quad\text{ where }\,
\lvert\tau\rvert<\varepsilon_{\rm lps}
\,\text{ and }\,
w''\eqdef g^\tau(w), \end{split}\] is well defined and $\proj_{D_i}([v,w])=[v',w']_{R_i}$. As $z\in W^{\rm ss}_{\varepsilon_{\rm lps}}(v)$, by Lemma \ref{lem:unifexp} together with Lemma \ref{lem:hamen:uu}, for every $k=0,\ldots,m-1$ it holds \[
d(g^k(z),g^k(w''))
\le cd^{\rm s}_{g^k(w'')}(g^k(z),g^k(w''))
= ce^{-k} d^{\rm s}_{w''}(z,w'')
\le ce^{-k}\varepsilon_{\rm lps}
< c\varepsilon_{\rm lps}. \] On the other hand, as $g^m(v),g^m(w)\in g^m(B(v_i,\beta/2))\cap B(v_i,\beta/2)$, arguing analogously for $g^{-1}$ instead of $g$, it follows that for every $k=0,\ldots,m-1$ it holds \[
d(g^k(v),g^k(z))
< c\varepsilon_{\rm lps}. \] Moreover $d(g^k(w''),g^k(w))=d(w'',w)=d(g^\tau(w),w)\le \varepsilon_{\rm lps}$ with $\lvert \tau\rvert\le\varepsilon_{\rm lps}$. It follows that for every $k=0,\ldots,m-1$ it holds \[
d(g^k(v),g^k(w))
\le d(g^k(v),g^k(z))+d(g^k(z),g^k(w''))+d(g^k(w''),g^k(w))
\le 3c\varepsilon_{\rm lps}
< \alpha. \] But this contradicts the fact that $v,w$ are $\alpha$-separated, proving the claim. \end{proof}
This finishes the proof of the lemma. \end{proof}
\subsection{Proof of Proposition \ref{pro:entropybasicset}}\label{ssec:proofprop}
By $G$-invariance, the measure $\mu$ is $g^t$-invariant for every $t\in\mathbb{R}$. By Remark \ref{rem:ergodic}, there is $t>0$ such that $\mu$ is $g^t$-ergodic and $g^{-t}$-ergodic. Without loss of generality, we can assume that $t=1$. Let $h\eqdef h_\mu(G)$.
Fix $\varepsilon_{\rm E}\in(0,h/4)$. Fix $r>0$ satisfying \begin{equation}\label{eq:choicercc}
r
<\varepsilon_{\rm E}\cdot\min\Big\{1,\frac{1}{2(h-4\varepsilon_{\rm E})}\Big\}. \end{equation} Fix some compact set $Y_0\subset T^1M$ such that $\mu(Y_0)>0$ and $\kappa\in(0,\mu(Y_0)/4)$.
\subsubsection{Approximate entropy at finite times}
Using notation \eqref{eq:Bowbal} with $f=g^1$, by Brin-Katok's theorem in the noncompact setting (see for example \cite[Theorem 2.7]{Riq:18}), for $\mu$-almost every $x$ it holds \[
h
\le \lim_{\varepsilon\to0}\liminf_{n\to\infty} -\frac1n\log\mu(B_n(x,\varepsilon)). \] By Egorov's theorem and the fact that $\mu$ is regular, there are a compact set $Y_1\subset Y_0$ satisfying $\mu(Y_1)>\mu(Y_0)-\kappa/4$ and $\varepsilon_{\rm BK}>0$ so that for every $\varepsilon\in(0,\varepsilon_{\rm BK})$ and $x\in Y_1$, \begin{equation}\label{eq:Egoent}
h-\frac{\varepsilon_{\rm E}}{3}
\le \liminf_{n\to\infty} -\frac1n\log\mu(B_n(x,\varepsilon)). \end{equation}
\begin{claim}\label{cla:inic}
For every $\alpha\in(0,\varepsilon_{\rm BK})$ there are $n_{\rm BK}\in\mathbb{N}$ and a compact set $Y_2\subset Y_1$ satisfying $\mu(Y_2)>\mu(Y_1)-\kappa/4$ such that for every integer $n\ge n_{\rm BK}$ and measurable set $A\subset Y_2$ with $\mu(A)>0$ there exists a $(n,\alpha)$-separated set $E\subset A$ such that \[
\frac1n\log\card E
\ge h-\frac{2\varepsilon_{\rm E}}{3}-\frac1n\lvert\log\mu(A)\rvert. \] \end{claim}
\begin{proof}
It follows from \eqref{eq:Egoent} that for every $\alpha \in(0,\varepsilon_{\rm BK})$ there are a compact set $Y_2\subset Y_1$ satisfying $\mu(Y_2)>1-3\kappa/4$ and an integer $n_{\rm BK}\in\mathbb{N}$ such that for every $n\ge n_{\rm BK}$ and $x\in Y_2$ it holds \begin{equation}\label{estBK3}
\mu(B_n(x,\alpha))
\le e^{-n(h-2\varepsilon_{\rm E}/3)}. \end{equation}
Fix $n\ge n_{\rm BK}$. Let $A\subset Y_2$ be a measurable set with $\mu(A)>0$. We are going to construct an $(n,\alpha)$-separating set $E\subset A$. Choose any point $x_1\in A$. Let $A_1\eqdef A\setminus B_n(x_1,\alpha)$. Inductively, for every $k\ge2$ assuming that $A_{k-1}$ was already constructed and is nonempty, choose $x_k\in A_{k-1}$ and let $A_k\eqdef A_{k-1}\setminus B_n(x_k,\alpha)$.
Let $k\in\mathbb{N}$ denote the largest index for which $A_k$ is nonempty. By construction, the set $E\eqdef \{x_1,x_2,\ldots,x_k\}$ is $(n,\alpha)$-separated. Now \eqref{estBK3} implies \[
\mu(A_k)
= \mu(A_{k-1})-\mu(B_n(x_k,\alpha))
= \mu(A)-\sum_{i=1}^k\mu(B_n(x_k,\alpha))
\ge \mu(A)-ke^{-n(h-2\varepsilon_{\rm E}/3)}. \] Hence, it holds $k\ge\lceil \mu(A)e^{n(h-2\varepsilon_{\rm E}/3)}\rceil$ and \[
\frac1n\log\card E
= \frac1n\log k
\ge h -\frac{2\varepsilon_{\rm E}}{3}-\frac1n\lvert\log\mu(A)\rvert, \] proving the claim. \end{proof}
Once $\alpha\in(0,\varepsilon_{\rm BK})$ is fixed, in any measurable set of positive measure, for sufficiently large $n$ we can find an $(n,\alpha)$-separated set of points whose cardinality is close to $e^{nh}$. We now choose the maximal size of cross sections and rectangles and some finite partition to construct a horseshoe. Note that the exponential growth of the cardinality of separate points in Claim \ref{cla:inic} occurs in every partition element (in fact in every measurable set of positive measure). The principle idea in \cite[Supplement S.5]{KatHas:95} is to pick \emph{one} element with the largest cardinality, which still will be of order $e^{nh}$ for $n$ large. Then, we consider a corresponding local section which contains this partition element and build the horseshoe as the invariant set of a certain return-map to that section.
\subsubsection{Fixing quantifiers and local cross sections}
Fix $\alpha\in(0,\varepsilon_{\rm BK})$, let $n_{\rm BK}$ and $Y_2$ as provided by Claim \ref{cla:inic}. Without loss of generality, we can assume that $n_{\rm BK}$ is so large that for every $n\ge n_{\rm BK}$ it holds \[
e^{-n\varepsilon_{\rm E}/3}
\le 1-\frac{3\kappa}{4}
\quad\text{ and }\quad
n
<e^{n\varepsilon_{\rm E}}. \] Apply Lemma \ref{lem:rec1} to $Y_2$ and $\alpha$. For $\varepsilon_{\rm sec}\in(0,\alpha/(3c))$ small enough this lemma provides numbers $\beta_{\rm sec}>0$ and $M\in\mathbb{N}$ and a finite collection of sections $\{D_i\}_{i=1}^\ell$ of size smaller than $\varepsilon_i\in(0,\varepsilon_{\rm sec})$. Moreover, fixing some \[
\beta\in(0,\beta_{\rm sec}). \] there are points $v_i\in Y_2$ such that \[
Y_2
\subset\bigcup_iB(v_i,\frac\beta2)
\quad\text{ and }\quad
B(v_i,\frac\beta2)
\subset B(v_i,\beta)
\subset \widehat R_i\eqdef g^{(-\varepsilon_i,\varepsilon_i)}(R_i), \] where $R_i$ is a proper rectangle in $D_i$.
\subsubsection{Fixing a partition}
Fix a finite partition $\EuScript{P}=\{P_1,\ldots,P_\ell\}$ of $Y_0$ of diameter at most $\beta/4$. Denoting by $\EuScript{P}(x)$ the partition element which contains $x$. Hence, with this choice, \[
\EuScript{P}(v_i)
\subset B(v_i,\frac\beta2)
\subset B(v_i,\beta)
\subset \widehat R_i. \]
\subsubsection{Choose almost-uniformly returning points}
By Birkhoff ergodic theorem (relative to the $g^1$-ergodic measure $\mu$), for $\mu$-almost every $x\in Y_2$ for every $i$ it holds \[
\lim_{n\to\infty}\frac1n\card\big\{k\in\{0,\ldots,n-1\}\colon g^k(x)\in Y_2\cap\EuScript{P}(v_i)\big\}
=\mu(Y_2\cap\EuScript{P}(v_i)). \] Let $\varepsilon_{\rm B}\in(0,r)$. By Egorov's theorem, there is an integer $n_{\rm B}\in\mathbb{N}$ and a compact set $Y_3\subset Y_2$ such that \begin{equation}\label{eq:choiceY3}
\mu\big(Y_3\big)
> \mu(Y_2)-\frac\kappa 4
> 1-\kappa. \end{equation}
such that for every index $i$, point $x\in Y_3\cap\EuScript{P}(v_i)$, and $n\ge n_{\rm B}$ it holds \begin{equation}\label{eq:choicen0}\begin{split}
&\big\lvert \card\{k\in\{0,\ldots,n-1\}\colon g^k(x)\in Y_2\cap \EuScript{P}(v_i)\}
-n\mu(Y_2\cap \EuScript{P}(v_i))\big\rvert
\le n\varepsilon_{\rm B}. \end{split}\end{equation} Without loss of generality, we can assume that $n_{\rm B}$ is so large that \begin{equation}\label{eq:choicen1}
n_{\rm B}r\big(\mu(Y_2)-3\varepsilon_{\rm B}\big)
\ge1. \end{equation} Hence, for every $i$, $x\in Y_3\cap \EuScript{P}(v_i)$, and $n\ge n_{\rm B}$, it holds \[\begin{split}
\card&\big\{k\in\{n,\ldots,n(1+r)-1\}\colon g^k(x)\in Y_2\cap \EuScript{P}(v_i)\big\}\\ \text{\small by \eqref{eq:choicen0}}\quad
&\ge n(1+r)\big(\mu(Y_2\cap \EuScript{P}(v_i))-\varepsilon_{\rm B}\big)-(n-1)\big(\mu(Y_2\cap \EuScript{P}(v_i))+\varepsilon_{\rm B}\big)\\
&= (nr+1)\mu(Y_2\cap \EuScript{P}(v_i))-(2n+nr-1)\varepsilon_{\rm B}\\ \text{\small using $\varepsilon_{\rm B}<r$ }\quad
&> nr\big(\mu(Y_2\cap \EuScript{P}(v_i))-3\varepsilon_{\rm B}\big)\\ \text{\small by \eqref{eq:choicen1} and $n\ge n_{\rm B}$}\quad
&\ge 1. \end{split}\]
In other words, there exists $k=k(x)\in\mathbb{N}$ satisfying $k\in\{n,\ldots,n(1+r)-1\}$ such that \[
g^k(x)\in \EuScript{P}(x)= \EuScript{P}(v_i), \] that is, the point returns to its partition element.
Apply now the Claim \ref{cla:inic} to the set $A=Y_3$. Fix $n\in\mathbb{N}$ satisfying \begin{equation}\label{eq:choiceY3b}
n\ge \max\Big\{n_{\rm BK},M,n_{\rm B},
\frac{3}{\varepsilon_{\rm E}}\lvert\log(1-\kappa)\rvert,
\frac{1}{\varepsilon_{\rm E}}\log\ell,
\frac{h}{2\varepsilon_{\rm E}}+\frac32,
\frac{\varepsilon_{\rm sec}}{r}
\Big\}. \end{equation} By Claim \ref{cla:inic}, there is a $(n,\alpha)$-separated set $E\subset Y_3$ such that \begin{equation}\label{eq:cardEest}\begin{split}
\frac{1}{n}\log\card E
&\ge h-\frac{2\varepsilon_{\rm E}}{3} - \frac1n\lvert\log\mu(Y_3)\rvert\\
\text{\small by \eqref{eq:choiceY3}}\quad
&\ge h-\frac{2\varepsilon_{\rm E}}{3} - \frac1n\lvert\log(1-\kappa)\rvert \\
\text{\small by \eqref{eq:choiceY3b}}\quad
&> h-\varepsilon_{\rm E}. \end{split}\end{equation}
\subsubsection{Picking points with the same return time to the same partition element}
For each $k\in\{n,\ldots,n(1+r)-1\}$ let \[
E_k
\eqdef \{x\in E\colon g^k(x)\in\EuScript{P}(x)\} \] be the set of points in $E$ that return to their partition element at the same time $k$. Let \begin{equation}\label{eq:choicem}
m\in\{n,\ldots,n(1+r)-1\} \end{equation}
be an index $k$ for which $\card E_k$ is maximal. Since \[
\card E
= \sum_{k=n}^{n(1+r)-1}\card E_k, \] it follows $\card E\le rn\card E_m$. Together with $rn<e^{rn}$ and \eqref{eq:cardEest}, it follows \[
\card E_m
\ge \frac{1}{rn}\card E
> e^{-rn}\card E
> e^{n(h-\varepsilon_{\rm E})-rn}. \] Pick from the partition $\EuScript{P}$ an element $P_i$ for which $\card (E_m\cap P_i)$ is maximal. As $\EuScript{P}$ has $\ell$ elements, \begin{equation}\label{eq:estcardN}\begin{split}
\card(E_m\cap P_i)
&\ge \frac{1}{\ell}\card E_m
\ge \frac1\ell e^{n(h-\varepsilon_{\rm E})-rn}\\
\text{\small by \eqref{eq:choiceY3b}}\quad
&> e^{n(h-2\varepsilon_{\rm E}-r)}\\
\text{\small by \eqref{eq:choicercc}}\quad
&> e^{n(h-3\varepsilon_{\rm E})}. \end{split}\end{equation} For what is below, fix this rectangle $R_i$ in the cross section $D_i$. Fix some enumeration \[
F=\{x_1,\ldots,x_N\}
\eqdef E_m\cap P_i. \]
\subsubsection{Building the basic set}
Note that all points in $F$ are $(n,\alpha)$-separated and hence $(m,\alpha)$-separated. Recall that it holds \[
F\cap g^m(F)
\subset \EuScript{P}(v_i)
\subset B(v_i,\frac\beta2)
\subset B(v_i,\beta)
\subset \widehat R_i
= g^{(-\varepsilon_i,\varepsilon_i)}(R_i). \] We now are ready to build the basic set around the set $F$ in the neighborhood of $v_i$ and the points $x_1,\ldots,x_N$.
Consider the continuous and injective return map $T_{i,m}\colon\widehat D_i^m\to\widehat D_i^m$
as provided by Lemma \ref{lem:rec1}. Denote the connected component containing $x_\ell\in F$, $\ell=1,\ldots,N$, by \[
U_\ell
\eqdef \mathscr{C}_{i,m}(\proj_{D_i}(x_\ell)). \] By Lemma \ref{lem:rec1}, this are proper rectangles in $D_i$ which are pairwise disjoint. Let \[
S_\ell
\eqdef T_{i,m}(U_\ell)
\subset R_i, \] which, by the above, are also pairwise disjoint and closed. Let \[
\Gamma
\eqdef \bigcap_{k\in\mathbb{Z}}T_{i,m}^k(U_1\cup\ldots\cup U_N), \] and note that this is a compact set. As $U_\ell$ are pairwise disjoint, for every $x\in\Gamma$, there is a unique sequence $\underline i=(\ldots i_{-1}i_0i_1\ldots)\eqdef\pi(x)\in\Sigma_N$ of indices such that \[
T_{i,m}^k(x)
\in U_{i_k} \] for every $k\in\mathbb{Z}$. This defines $\pi\colon\Gamma\to\Sigma_N$. By definition, $\pi\circ T_{i,m}=\sigma\circ \pi$. By classical arguments, $\Gamma$ is a topological horseshoe and $\pi$ is injective and onto.
\begin{claim}\label{cla:picont}
The map $\pi^{-1}\colon\Sigma_N\to\Gamma$ is continuous. \end{claim}
\begin{proof} Let $\varepsilon>0$ and take $k\in\mathbb{N}$ such that $Ce^{-k(m-\varepsilon_{\rm sec})}<\varepsilon$, where $C=C(\varepsilon_{\rm sec})$ is the constant of Lemma \ref{lem:closegeodesics}. Let $\delta=1/2^k$. Then, whenever $\underline i$, $\underline i'$ verify $d_\Sigma(\underline i,\underline i')<\delta$, that is $i_j=i'_j$ for all $-k\leq j\leq k$, we get that for $x=\pi^{-1}(\underline i)$ and $x'=\pi^{-1}(\underline i')$, the orbits $g^t(x)$ and $g^t(x')$ are $\varepsilon_{\rm sec}$-close for time at least $2k(m-\varepsilon_{\rm sec})$. Using Lemma \ref{lem:closegeodesics} on the quotient manifold, it follows that $d(x,y)\leq Ce^{-k(m-\varepsilon_{\rm sec})}<\varepsilon$, which proves the claim. \end{proof}
Let \[
B
\eqdef \bigcup_{\lvert t\rvert\le\max T_{i,m}}g^t(\Gamma). \] It follows from Claim \ref{cla:picont} that $B$ is a $G$-invariant compact set of topological dimension 1.
Let us now estimate the topological entropy of $G$ on $B$. Note that $G|_B$ can be seen as the suspension flow of the map $T_{i,m}|_\Gamma$ under the return time defined as in \eqref{eq:estTim}. Clearly, \[
h_{\rm top}(T_{i,m},\Gamma)
= \log N. \] Consider on $\Gamma$ the push-forward of the $(\frac1N,\ldots,\frac1N)$-Bernoulli measure by $\pi$ and denote by $\mu$ its suspension. Recall that $\mu$ is a $G$-invariant Borel probability measure. As by \eqref{eq:estTim} the return time is $\varepsilon_i$-close to $m$, by Abramov's formula, it holds \[
h_{\rm top}(G,\Gamma)
\ge h_\mu(G)
\ge \frac{\log N}{m+\varepsilon_i}, \]
Together with \eqref{eq:estcardN} and \eqref{eq:choicem}, it follows \[\begin{split}
\frac{\log N}{m+\varepsilon_i}
&\ge \frac{n(h-3\varepsilon_{\rm E})}{n(1+r)+\varepsilon_i}\\
\text{\small recalling that $\varepsilon_i<\varepsilon_{\rm sec}$ and using \eqref{eq:choiceY3b}}
&\ge \frac{h-3\varepsilon_{\rm E}}{1+2r}\\
\text{\small by \eqref{eq:choicercc}}\quad
&\ge h-4\varepsilon_{\rm E}. \end{split}\] This implies the assertion of the proposition. \qed
\section{Exceptional sets for suspension flows}\label{sec:susflo}
In this section, we prove Theorem \ref{thepro:susflo}. Let us first recall the definition of a suspension flow. Given $N\in\mathbb{N}$, consider the shift space $\Sigma=\{1,\ldots,N\}^\mathbb{Z}$. Given $\alpha>0$, consider a continuous \emph{height function} $\tau \colon \Sigma\to(0,\alpha]$ and define the space \[
\Sigma(\sigma,\tau)
\eqdef \big(\Sigma\times[0,\alpha]\big)_{\sim}, \] as the quotient space of $\Sigma\times\mathbb{R}_{\ge0}$ modulo the equivalence relation $\sim$ that identifies $(\underline i,s)$ with $(\sigma(\underline i),s-\tau(\underline i))$ for all $\underline i\in\Sigma$ and $s\ge\tau(\underline i)$. This space is compact and metrizable (see \cite[Section 2]{BowWal:72}). The \emph{suspension flow} $F=(f^t)_{t\in\mathbb{R}}$ of the left-shift $\sigma\colon\Sigma\to\Sigma$ \emph{under $\tau$} is the map $F\colon\Sigma(\sigma,\tau)\times\mathbb{R}\to\Sigma(\sigma,\tau)$ defined by \[
f^t(\underline i,s)
\eqdef \begin{cases}
(\underline i,s+t)&\text{if }0\le s+t<\tau(\underline i),\\
(\sigma(\underline i),s+t-\tau(\underline i))&\text{if }s+t=\tau(\underline i).
\end{cases} \]
\begin{proof}[\bf{Proof of Theorem \ref{thepro:susflo}}] Note that the suspension flow $F$ is expansive and hence the map $\mu\mapsto h_\mu(F)$ is upper semi-continuous. Hence, there is some ergodic measure of maximal entropy $\mu$, $h_\mu(F)=h_{\rm top}(F)$. Considering the natural projection $\pi_1\colon\Sigma(\sigma,\tau)\to\Sigma$ to the first coordinate and let $\nu_{\rm max}\eqdef(\pi_1)_\ast\mu\in\EuScript{M}_{\rm erg}(\sigma)$.
Recall that for every $\nu\in\EuScript{M}(\sigma)$ \[
\mu_\nu
\eqdef \frac{1}{(\nu\times m)(\Sigma(\sigma,\tau))}(\nu\times m)|_{\Sigma(\sigma,\tau)} \] defines a $F$-invariant Borel probability measure. Moreover, $\nu\mapsto\mu_\nu$ is a bijection between $\EuScript{M}(\sigma)$ and $\EuScript{M}(F)$, and it holds \begin{equation}\label{eq:entropysusflo}
h_{\mu_\nu}(F)
=\frac{h_\nu(\sigma)}{\int \tau\,d\nu}. \end{equation} Hence, together with our hypothesis on $A$, it holds \begin{equation}\label{eq:using}
h^\ast_{\rm top}(f^1,A)
< h_{\rm top}(f^1)
= h_\mu(F)
= \frac{h_{\nu_{\rm max}}(\sigma)}{R},
\quad\text{where}\quad
R\eqdef \int\tau\,d\nu_{\rm max}. \end{equation} Choose $\varepsilon>0$ sufficiently small such that \begin{equation}\label{eq:suchthat}
h^\ast_{\rm top}(f^1,A)
< \frac{R-\varepsilon}{R+\varepsilon} \frac{h_{\nu_{\rm max}}(\sigma)-\varepsilon}{R}. \end{equation} Apply Proposition \ref{pro:SFT} to $\nu_{\rm max}$ and $\varepsilon$ and consider the corresponding SFT $\Xi\subset\Sigma$ which hence satisfies \begin{equation}\label{eq:using2}
h_{\nu_{\rm max}}(\sigma)-\varepsilon
\le h_{\rm top}(\sigma|_\Xi)
\quad\text{ and }\quad
\left\lvert\int\tau\,d\nu_{\rm max} -\int\tau\,d\lambda\right\rvert
\le\varepsilon \end{equation} In the following, we consider the corresponding suspension flow on the suspension space $\Xi(\sigma,\tau)=(\Xi\times[0,\alpha])_\sim$ which can be considered as a subset of $\Sigma(\sigma,\tau)$. Let $W\eqdef A\cap\Xi(\sigma,\tau)$ and observe that by Lemma \ref{lem:propTho} (ii) and \eqref{eq:suchthat}, it holds \begin{equation}\label{eq:relatt}
h^\ast_{\rm top}(f^1,W)
\le h^\ast_{\rm top}(f^1,A)
< \frac{R-\varepsilon}{R+\varepsilon} \frac{h_{\nu_{\rm max}}(\sigma)-\varepsilon}{R}. \end{equation}
For the following claim, recall notation \eqref{notationMZ}.
\begin{claim}\label{cla:sameproj}
It holds \[
\sup_{\nu\in\EuScript{M}^{\pi_1(W)}(\sigma)}\frac{h_\nu(\sigma)}{\int\tau\,d\nu}
=
\sup_{\mu\in\EuScript{M}^W(f^1)}h_\mu(f^1). \] \end{claim}
\begin{proof} The natural projection $\pi_1$ induces a continuous push forward $(\pi_1)_\ast\colon\EuScript{M}(f^1)\to\EuScript{M}(\sigma)$. Hence if $\mu'\in\EuScript{V}(f^1,X)$ then $(\pi_1)_\ast\mu'\in\EuScript{V}(\sigma,\pi_1(X))$, which implies \[
\EuScript{M}^{\pi_1(W)}(\sigma)
\supset(\pi_1)_\ast(\EuScript{M}^W(f^1)). \] Together with \eqref{eq:entropysusflo} this implies the inequality $\ge$ in the claim.
On the other hand, let $\xi\in \pi_1(W)$ and $s\in[0,\alpha)$ such that $X=(\xi,s)\in W$ and let $\nu\in\EuScript{M}^{\pi_1(W)}(\sigma)$ be the limit point of the sequence of probability measures $(\delta_{\xi,n})_n$ as defined in \eqref{eq:deltadef} (with respect to the shift map $\sigma$). To simplify notation, assume that this sequence converges as $n\to\infty$ (otherwise pass to some subsequence). Consider a subsequence $(k_n)_n$ so that $\pi_1(f^{k_n}(X))=\sigma^n(\xi)$ and let $\mu'$ be some limit point of $(\delta_{X,k_n})_n$ (relative to the space of probability measures in $\Sigma(\sigma,\tau)$, possibly for some subsequence). Then $\mu'$ is $f^1$-invariant and $(\pi_1)_\ast\mu'=\nu$. The latter implies $h_{\mu'}(f^1)=h_\nu(\sigma)/\int\tau\,d\nu$. This implies the inequality $\le$ in the claim. \end{proof}
Denote $Z\eqdef \pi_1(W)$. Observe that $Z\subset\Xi$. Note that $\xi\in Z$ implies that every limit measure $\lambda$ of $\delta_{\xi,n}$ (relative to the shift map $\sigma\colon\Sigma\to\Sigma$ and the weak$\ast$ topology in the space of probability measures in $\Sigma$) is in $\EuScript{M}(\sigma|_\Xi)$. Hence, by our choice of SFT $\Xi$ and \eqref{eq:using}, \begin{equation}\label{eq:variationtau}
R-\varepsilon
\le \int\tau\,d\lambda \le R+\varepsilon. \end{equation} By Claim \ref{cla:sameproj} and definition \eqref{def:entstar} it follows \[\begin{split}
\sup_{\nu\in\EuScript{M}^Z(\sigma)}\frac{h_\nu(\sigma)}{\int\tau\,d\nu}
&= \sup_{\mu\in\EuScript{M}^W(f^1)}h_\mu(f^1)
= h^\ast_{\rm top}(f^1,W)\\
\text{\small{with \eqref{eq:relatt}}}\quad
&< \frac{R-\varepsilon}{R+\varepsilon} \frac{h_{\nu_{\rm max}}(\sigma)-\varepsilon}{R} \end{split}\] This together with \eqref{eq:variationtau}, this implies \[\begin{split}
\sup_{\nu\in\EuScript{M}^Z(\sigma)}{h_\nu(\sigma)}
&\le (R+\varepsilon)\cdot\frac{R-\varepsilon}{R+\varepsilon} \frac{h_{\nu_{\rm max}}(\sigma)-\varepsilon}{R} \\
&< h_{\nu_{\rm max}}(\sigma)-\varepsilon\\
\text{\small by \eqref{eq:using2}}\quad
&\le h_{\rm top}(\sigma|_\Xi). \end{split}\] Together with Lemma \ref{lem:propTho2} (iii), it follows \begin{equation}\label{eq:gelo}
h_{\rm top}(\sigma|_\Xi,Z)
\le h^\ast_{\rm top}(\sigma|_\Xi,Z)
= \sup_{\nu\in\EuScript{M}^Z(\sigma)}{h_\nu(\sigma)}
<h_{\rm top}(\sigma|_\Xi). \end{equation} Hence, by \cite[Section 3 Theorem 1]{Dol:97}, this implies \[
h_{\rm top}(\sigma|_\Xi, I_{\sigma|\Xi}(Z))
= h_{\rm top}(\sigma|_\Xi) \] By Lemma \ref{lem:propTho2} (iii), it holds \[
h_{\rm top}(\sigma|_\Xi, I_{\sigma|\Xi}(Z))
\le h^\ast_{\rm top}(\sigma|_\Xi, I_{\sigma|\Xi}(Z)) \] and therefore \begin{equation}\label{eq:conDol}
h_{\rm top}(\sigma|_\Xi, I_{\sigma|\Xi}(Z))
= h^\ast_{\rm top}(\sigma|_\Xi, I_{\sigma|\Xi}(Z))
= h_{\rm top}(\sigma|_\Xi). \end{equation}
\begin{claim}\label{cla:sameproj2}
It holds $I_{\sigma|\Xi}(\pi_1(W))\subset \pi_1(I_{f^1}(W))$. \end{claim}
\begin{proof}
Let $X\not\in I_{f^1}(W)$ with $f^{k_n}(X)\to Y\in W$ for some subsequence $k_n\to\infty$. Let $\xi=\pi_1(X)$. By continuity of $\pi_1$, it follows that there is some subsequence $\ell_m\to\infty$ so that $\sigma^{\ell_m}(\xi)=(\pi_1\circ f^{k_n})(X)\to\pi_1(Y)\in\pi_1(W)=Z$ and hence $\xi\not\in I_{\sigma|\Xi}(Z)$. \end{proof}
Let $Z\eqdef \pi_1(W)$. Claim \ref{cla:sameproj2} implies \[
\EuScript{M}^{I_{\sigma|\Xi}(\pi_1(W))}(\sigma)
=\EuScript{M}^{I_{\sigma|\Xi}(Z)}(\sigma)
\subset \EuScript{M}^{\pi_1(I_{f^1}(W))}(\sigma). \] This together with Claim \ref{cla:sameproj} implies \[
\sup\left\{\frac{h_\lambda(\sigma)}{\int\tau\,d\lambda}\colon
\lambda\in\EuScript{M}^{I_{\sigma|\Xi}(Z)}(\sigma)\right\}
\le\sup\Big\{h_\mu(f^1)\colon \mu\in \EuScript{M}^{I_{f^1}(W)}(f^1)\Big\}. \] Using \eqref{eq:conDol} together with \eqref{eq:variationtau}, this implies \begin{equation}\label{eq:auto}
\frac{h_{\rm top}(\sigma|_\Xi)}{R+\varepsilon}
= \frac{h^\ast_{\rm top}\big(\sigma|_\Xi,I_{\sigma|\Xi}(Z)\big)}{R+\varepsilon}
\le h^\ast_{\rm top}(f^1,I_{f^1}(W)). \end{equation} By \eqref{eq:using} and \eqref{eq:using2}, it holds \[
h_\mu(f^1)
\frac{R}{h_{\nu_{\rm max}}(\sigma)}
\frac{h_{\nu_{\rm max}}(\sigma)-\varepsilon}{R+\varepsilon}
= 1\cdot\frac{h_{\nu_{\rm max}}(\sigma)-\varepsilon}{R+\varepsilon}
\le \frac{h_{\rm top}(\sigma|_\Xi)}{R+\varepsilon}. \] Recalling again that $\mu$ was a measure of maximal entropy $h_\mu(f^1)=h_{\rm top}(F)$, together with \eqref{eq:auto}, Lemma \ref{lem:propTho} (ii) applied to $I_{f^1}(W)\subset\Sigma(\sigma,\tau)$, and Lemma \ref{lem:propTho2} (iv), we get \[
h_{\rm top}(F)
\frac{R(h_{\nu_{\rm max}}(\sigma)-\varepsilon)}{h_{\nu_{\rm max}}(\sigma)
(R+\varepsilon)}
\le h^\ast_{\rm top}(f^1,I_{f^1}(W))
\le h^\ast_{\rm top}(f^1,\Sigma(\sigma,\tau))
= h_{\rm top}(f^1,\Sigma(\sigma,\tau)). \] As $\varepsilon>0$ was arbitrary, this proves the theorem. \end{proof}
We apply now Theorem \ref{thepro:susflo} to prove the following result.
\begin{proposition}\label{pro:basiexce}
Let $B\subset T^1M$ be a one-dimensional basic set for the geodesic flow $G$.
For every Borel set $A\subset T^1M$ satisfying $h^\ast_{\rm top}(G,A\cap B)<h_{\rm top}(G|_B)$ it holds \[
h^\ast_{\rm top}(G|_B,I_{G|B}(A))
= h_{\rm top}(G|_B). \] \end{proposition}
\begin{proof} Note that the flow $G$ restricted to $B$ is topologically conjugate to a suspension flow $F$ of the shift map $\sigma$ on some subshift of finite type $\Sigma\subset\{1,\ldots,N\}^\mathbb{Z}$ under some continuous function $\tau\colon\Sigma\to[0,\alpha)$, $\alpha>0$ \cite[Theorem 10]{BowWal:72}. Denote by $p\colon \Sigma(\sigma,\tau)\to B$ the corresponding conjugating homeomorphism so that for every $t\in\mathbb{R}$, \[
p\circ f^t=g^t\circ p. \] Consider the Borel sets \[
W
\eqdef p^{-1}(A\cap B),
\quad
Z
\eqdef (\pi_1\circ p^{-1})(A\cap B)
= \pi_1(W). \] By Lemma \ref{lem:propTho2} (ii), it follows \begin{equation}\label{eq:ent1}
h_{\rm top}^\ast({g^1}|_B,A\cap B)
= h_{\rm top}^\ast({f^1},p^{-1}(A\cap B))
= h_{\rm top}^\ast({f^1},W). \end{equation} By conjugation, it holds \[
h_{\rm top}(g^1|_B)
= h_{\rm top}(f^1). \] Hence, it follows from our hypothesis that \[
h_{\rm top}^\ast({f^1},W)
<h_{\rm top}(f^1). \] By Theorem \ref{thepro:susflo}, it holds \[
h_{\rm top}^\ast(f^1,I_{f^1}(W))
= h_{\rm top}(f^1). \] It remains to check that \[
p(I_{f^1}(W))
= I_{g^1|_B}(p(W))
= I_{g^1|_B}(A\cap B)
= I_{g^1|_B}(A). \]
Applying again Lemma \ref{lem:propTho2} (ii), we get $h^\ast_{\rm top}(G|_B,I_{g^1|B}(A))= h_{\rm top}(G|_B)$. Note that $I_{g^1|B}(A)= \bigcup_{t\in [0,1)} g^t I_{G|B}(A)$ so a measure $\mu\in \mathcal{M}^{I_{g^1|B}(A)}(g^1)$ defines a measure $(g^{-t})_\ast\mu\in\mathcal{M}^{I_{G|B}(A)}(g^1)$. Since $\mu$ can be chosen $G$-invariant, the variational principle implies \[
h^\ast_{\rm top}(G|_B,I_{g^1|B}(A))
= h^\ast_{\rm top}(G|_B,I_{G|B}(A)), \] and the claim follows. \end{proof}
\section{Proofs of Theorem \ref{the:1} and Corollaries \ref{cor:1} and \ref{cor:2}}\label{sec:proofs}
\begin{proof}[Proof of Theorem \ref{the:1}] Consider a Borel set $A\subset T^1M$ satisfying \[
h^\ast_{\rm top}(G,A)
< h_{\rm top}(G). \] By \eqref{eq:timetmap} and the variational principle \eqref{eq:vpflow} together with the definition \eqref{def:entstar}, \[
h^\ast_{\rm top}(G,A)
= \sup_{\mu\in\EuScript{M}^A(g^1)}h_\mu(g^1)
< \sup_{\mu\in\EuScript{M}(g^1)}h_\mu(g^1)
= h_{\rm top}(G). \] Let $\mu\in\EuScript{M}(g^1)$ satisfying \[
h^\ast_{\rm top}(G,A)
< h_\mu(g^1). \] Without loss of generality (otherwise, choose some ergodic component having this property), we can assume that $\mu$ is ergodic. Fix $\varepsilon>0$ such that \[
h^\ast_{\rm top}(G,A)
< h_\mu(g^1)-2\varepsilon. \] By Proposition \ref{pro:entropybasicset}, there is a basic set $B\subset T^1M$ of topological dimension 1 satisfying \[
h_\mu(g^1)-\varepsilon
< h_{\rm top}(G|_B). \] By Lemma \ref{lem:propTho} (ii), it holds \[
h^\ast_{\rm top}(G|_B,A\cap B)
\le h^\ast_{\rm top}(g^1,A)
< h_{\rm top}(G|_B). \] Proposition \ref{pro:basiexce} now implies \[
h_{\rm top}^\ast(G|_B,I_{G|B}(A))
= h_{\rm top}(G|_B). \]
Since $B$ is $G$-invariant, we have $I_{G|B}(A)\subset I_{G}(A)$, therefore \[
h_\mu(g^1)-\varepsilon<h_{\rm top}^\ast(G,I_{G}(A)) \] As $\varepsilon>0$ was arbitrary, from the variational principle \eqref{eq:vpflow} we conclude that $h_{\rm top}(G)=h_{\rm top}^\ast(G,I_{G}(A))$. \end{proof}
\begin{proof}[Proof of Corollary \ref{cor:1}]
By assumption the derivatives of the sectional curvatures are uniformly bounded, so there is at most one measure maximizing entropy for $G$ (see \cite[Theorem 1]{OP:04}). If $h^{\ast}_{\rm top}(G,K)=h_{\rm top}(G)$, then the measure of maximal entropy of $(K,G|_{K})$ is the unique measure of maximal entropy for $G$. This measure is known to be fully supported on $\Omega$, a contradiction since $\Omega$ is non-compact. Therefore $h^{\ast}_{\rm top}(G,K)<h_{\rm top}(G)$, and by Theorem \ref{the:1} we obtain the desired equality. \end{proof}
\begin{proof}[Proof of Corollary \ref{cor:2}] Let us first show that if $\Gamma'$ is divergent and $\Omega_N \neq \Omega$, then $\delta_{\Gamma'} < \delta_{\Gamma}$. Indeed, the condition $\Omega_N \neq \Omega$ is equivalent to have $L(\Gamma')\neq L(\Gamma)$, where $L(H)$ denotes the limit set of any discrete subgroup of isometries $H$ of $\widetilde{M}$. Then \cite[Proposition 2]{DOP:00} implies $\delta_{\Gamma'} < \delta_{\Gamma}$. Note that by \cite[Theorem 1]{OP:04}, it holds $h_{\rm top}(G)=\delta_\Gamma$ and $h_{\rm top}(G,N)=\delta_{\Gamma'}$. Hence, it holds $h_{\rm top}(G,N)<h_{\rm top}(G)$. This together with Theorem \ref{the:1} ends the proof. \end{proof}
\begin{proof}[Proof of Scholium \ref{sch:1}]
By the variational principle \eqref{eq:vpflow}, for every $\varepsilon>0$ there exists an ergodic measure $\mu\in\EuScript{M}_{\rm erg}(G)$ satisfying $h_\mu(G)\ge h_{\rm top}(G,T^1M)-\varepsilon$. By Proposition \ref{pro:entropybasicset}, there exists a basic (and hence, in particular bounded, set $B\subset T^1M$ satisfying $h_{\rm top}(G,B)\ge h_\mu(G)-\varepsilon$. As $\varepsilon$ was arbitrary, the supremum taken over all basic sets equals $h_{\rm top}(G,T^1M)$. The remaining equalities in the scholium follow from Lemma \ref{lem:propTho} item (ii) and Lemma \ref{lem:propTho2} item (iii). \end{proof}
\subsection{Final comments} Most of the results in this work are not proved using the geometric structure of the dynamical system, but its hyperbolic structure. Nevertheless, the estimates from Section \ref{sec:geodynneg} are crucially used, for instance in the construction of suitable local cross sections (Lemma \ref{lem:rec1}). It would be interesting to know how general the results in this work can be for more general classes of hyperbolic flows in noncompact manifolds.
\end{document} |
\begin{document}
\title{Stochastic Averaging and Sensitivity Analysis for Two Scale Reaction Networks} \thanks{Submitted to the Journal of Chemical Physics.}
\author{Araz Hashemi} \email[Corresponding author, ]{araz@udel.edu} \affiliation{Department of Mathematical Sciences, University of Delaware}
\author{Marcel Nunez} \email{mpnunez@udel.edu} \affiliation{Department of Chemical and Biomolecular Engineering, University of Delaware}
\author{Petr Plech\'a\v{c}} \email{plechac@udel.edu} \affiliation{Department of Mathematical Sciences, University of Delaware}
\author{Dionisios G. Vlachos} \email{vlachos@udel.edu} \affiliation{Department of Chemical and Biomolecular Engineering, University of Delaware}
\date{\today} \begin{abstract}
In the presence of multiscale dynamics in a reaction network, direct simulation methods become inefficient as they can only advance the system on the smallest scale.
This work presents stochastic averaging techniques to accelerate computations for obtaining estimates of expected values and sensitivities with respect to the steady state distribution. A two-time-scale formulation is used to establish bounds on the bias induced by the averaging method. Further, this formulation provides a framework to create an accelerated `averaged' version of most single-scale sensitivity estimation methods. In particular, we propose a new lower-variance ergodic likelihood ratio method
for steady state estimation and show how one can adapt it to accelerate simulations of multiscale systems. {Lastly, we develop an adaptive ``batch-means'' stopping rule for determining when to terminate the micro-equilibration process.}
\end{abstract}
\keywords{multiscale dynamics, sensitivity analysis, steady state, stochastic averaging, ergodic, likelihood ratio, batch-means}
\maketitle
\section{Introduction}
Stochastic simulations have been an essential tool in analyzing reaction networks encountered in biology, catalysis, and materials growth.
However, it is commonplace for reaction networks to exhibit a large disparity in time scales. These multi-scale stochastic reaction networks can impose an enormous computational burden in order to simulate them exactly. Exact techniques require computation of every reaction at the fastest time-scale, resulting in an exponentially growing load to observe dynamics on the slowest time-scale. Many works have attempted to develop approximate algorithms which allow faster computation with minimal loss of accuracy \cite{chatterjee_overview_2007, gillespie_approximate_2001, cao_avoiding_2005, cao_efficient_2006, rathinam_stiffness_2003, chatterjee_binomial_2005,tian_binomial_2004, salis_accurate_2005, samant_overcoming_2005,cao_slow-scale_2005, e_nested_2007, huang_strong_2014, kang_separation_2013, gupta_sensitivity_2014}.
One approach, which we refer to as Stochastic Averaging, takes its inspiration from classical singular perturbation theory of ordinary differential equations \cite{samant_overcoming_2005,cao_slow-scale_2005, e_nested_2007, huang_strong_2014, salis_accurate_2005}. The idea is that the fast dynamics come to quasi-equilibrium before the slow dynamics take effect, hence one may model the slow time scale dynamics with their averages against the steady-state distribution of the fast dynamics. By estimating the steady-state expectations of the slow propensities, one can then jump the system ahead to the next slow reaction and advance the time clock on the slow scale (skipping over needless computations of fast reactions).
In addition, one often desires the sensitivities of the system $S_f(\theta_i) = \frac{\partial}{\partial \theta_i} \mathbb{E}_{{\boldsymbol{\theta}}} \{ f(X(t)) \}$ with respect to the reaction parameters $\theta_i$. The sensitivities give important insight into the system, indicating directions for gradient-descent type optimization as well as determining bounds for quantifying the uncertainty \cite{dupuis_path-space_2015}. Current techniques for estimating the sensitivities have {\em large variance}, requiring many more samples than those for estimating $\mathbb{E}_{{\boldsymbol{\theta}}} \{f(X)\}$ alone \cite{wolf_hybrid_2015,sheppard_pathwise_2012, wang_efficiency_2014, gupta_efficient_2014}. Thus computing sensitivities of multi-scale systems using single-scale techniques is often a computationally intractable problem.
{ In this work, we use results from Two-Time-Scale (TTS) Markov chains \cite{yin_continuous-time_2013} to show the error of stochastic averaging algorithms
is inverse to the scale disparity in the system. As opposed to the previous approaches of transforming the system variables into auxiliary fast and slow variables\cite{e_nested_2007, huang_strong_2014}, we partition the underlying (discrete) state space and derive a singular perturbation expansion of the corresponding probability measure. The first order term can then be identified from computables of the averaged process, leading to a rigorous theoretical framework for applying singular perturbation averaging for stochastic systems.
Furthermore, this new formulation allows one to identify a macroscopic ``averaged'' reaction network on a reduced state space whose time-steps are on the macro (slow) time-scale. Thus, it provides a framework for applying single-scale sensitivity analysis techniques to the multi-scale system. Previous works have exploited similar model reduction techniques to estimate sensitivities via finite differences\cite{gupta_sensitivity_2014} or a ``truncated'' version of a likelihood ratio estimator\cite{nunez_steady_2015}.
This work develops an accelerated ``Two-Time-Scale'' version of the Likelihood Ratio (Girsanov Transform) method \cite{wang_efficiency_2014, plyasunov_efficient_2007, warren_steady-state_2012, glynn_likelihood_1990} for estimating system sensitivities of the multiscale system. The TTS-LR method computes sensitivity reweighting coefficients
of the macro (reduced-state) process using a representation in terms of the steady-state sensitivities of the micro (fast) process. These micro-level sensitivities can in turn be computed online during the micro-equilibriation process.
To this end, we propose a new lower-variance ``Ergodic Likelihood Ratio'' estimator for approximating {\em steady-state sensitivities} of single and multi-scale systems. }
{ The outline of the remainder of the paper is as follows: Section \ref{sec:formulation} gives the theoretical basis of the paper. The Two-Time-Scale formulation is presented and error bounds are established. Section \ref{sec:TTS_sens} then uses the TTS framework for the purpose of sensitivity analysis. A new ergodic likelihood ratio estimator is developed for single-scale steady-state sensitivity analysis, and is then adapted to the multiscale system. Section \ref{sec:BMstop} develops a batch-means stopping rule for determining when the micro-scale system has come to equilibrium. Numerical results are presented in Section \ref{sec:sim_results} supporting the effectiveness of the methods presented. Concluding remarks are given in Section \ref{sec:conclusion}, and proofs of theorems are relegated to the Appendix. }
\section{Formulation} \label{sec:formulation}
\subsection{Markov Chain Model of Reaction Networks} We briefly review the Markov chain model of reaction networks. While our motivation stems from chemical reaction networks, we note that much of the formulation carries over to general Markov chains on integer lattices.
Suppose we have $d$ species described by $X(t) = [X_1(t), X_2(t), \dots, X_d(t)] \in \mathcal{M} \subset \mathbb{Z}^d$ and $M$ reactions $r_1, r_2, \dots, r_M$. In stochastic reaction networks, one views $X(t)$ as a continuous-time Markov chain (CTMC) in the state space $\mathcal{M}$. When reaction $r$ fires at time $t$, the state is updated by the {\em stoichiometric vector} $\zeta_r$ so that $X(t) = X(t-) + \zeta_r$. Given the set of reaction parameters ${\boldsymbol{\theta}}=[\theta_1, \theta_2, \dots]$, one characterizes the probabilistic evolution of $X(t)$ by the {\em propensity functions} (intensity functions) $\lambda_r(x;{\boldsymbol{\theta}})$. The propensity functions are such that, given $X(t)=x$, the probability of one or more firing of reaction $r$ during time $(t, t+h]$ is $\lambda_r(x;{\boldsymbol{\theta}})h + o(h)$ as $h \to 0$; i.e. $\lambda_r(x;{\boldsymbol{\theta}})$ is the instantaneous rate/probability of reaction $r$ firing.
A common model for the propensities functions is that of {\em mass-action kinetics}. Under this assumption, the propensity functions are of the form \begin{align} \label{eq:mass-action-prop}
\lambda_r(x;{\boldsymbol{\theta}}) = \theta_r \cdot b_r(x) =
\theta_r \cdot \prod_{i=1}^d \frac{x_i!}{(x_i - \nu_{r,i} )!}
\mathbb{I}_{\{x-\nu_{r,i} \ge 0\} } \end{align} where $\nu_{r,i}$ is the number of molecules of species $i$ required for reaction $r$ to fire. Mass action kinetics assumes the system is well-mixed, so molecular interactions are proportional to their counts.
From the propensity functions $\lambda_r(x;{\boldsymbol{\theta}})$, one can construct the infinitesimal generator $Q=Q({\boldsymbol{\theta}})$ of the Markov chain. Viewed as an operator on functions $f$ of the state-space $\mathcal{M}$, we have \begin{align*}
\left( Qf \right)(x) = \sum_{r=1}^M \lambda_r(x;{\boldsymbol{\theta}})
\left( f(x+\zeta_r) - f(x) \right) \end{align*} For finite state-space $\mathcal{M}$ (bounded molecule counts), one may also view the generator $Q({\boldsymbol{\theta}})$ as a matrix. While the state-space is typically intractably large, the generator is sparse with only $M+1$ non-zero entries in each row, \begin{multline*}
Q(\boldsymbol{\theta}) = \\ \bordermatrix{ ~
& \dots & x+\zeta_2 & \dots & x & \dots & x+\zeta_1 & \dots \cr
\vdots & ~ & ~ & ~ & ~ & ~ & ~ & ~ \cr
x & \dots & \lambda_2(x;\boldsymbol{\theta}) & \dots & -\lambda_0(x;\boldsymbol{\theta})
& \dots & \lambda_1(x;\boldsymbol{\theta)} & \dots \cr
\vdots & ~ & ~ & ~ & ~ & ~ & ~ & ~ }
\end{multline*} where $\lambda_0(x;\boldsymbol{\theta}) = \sum_{r=1}^M \lambda_r(x;\boldsymbol{\theta})$.
Writing $R_r(t)$ to be the counting process representing how many reactions of type $r$ have fired by the time $t$, we have that $X(t) = X(0) + \sum_{r=1}^M R_r(t) \zeta_r$. Using the {\em random time change} representation \cite{anderson_continuous_2011, ethier_markov_1986}, we write $X(t)$ as \begin{align}
\begin{aligned}
\label{eq:random-time-change}
X(t) = X(0) + \sum_{r=1}^M Y_r\left(
\int_0^t \lambda_r(X(s);{\boldsymbol{\theta}})ds \right) \zeta_r
\end{aligned} \end{align} where $Y_r(\cdot)$ are independent unit-rate Poisson processes. This representation is tremendously useful in conducting analysis of the trajectories. In particular, it leads to formulations of the Next-Reaction Method \cite{gibson_efficient_2000, anderson_modified_2007} and interpreting simulated trajectories in the path-space to allow for coupling paths \cite{rathinam_efficient_2010, anderson_efficient_2012, gupta_efficient_2014} as well as path-wise differentiation \cite{sheppard_pathwise_2012, wolf_hybrid_2015}.
When simulating exact trajectories (using any exact method; Direct SSA, Next-Reaction, etc), the propensity functions $\lambda_r(x; {\boldsymbol{\theta}})$ probabilistically determine both the time between reactions $\Delta t$ as well as the next reaction $r^*$ to fire. The likelihood that reaction $r_k$ is the next to fire is proportional to its propensity $\lambda_k(x;{\boldsymbol{\theta}})$; i.e. $\mathbb{P}_{{\boldsymbol{\theta}}, x} \left\{ r^* = r_k \right\} \propto \lambda_k(x;{\boldsymbol{\theta}}) $. The time between reactions has an exponential distribution with the rate $ \lambda_0(x;{\boldsymbol{\theta}}) = \sum_{r=1}^M \lambda_r (x;{\boldsymbol{\theta}}) $ ; i.e. $\Delta t \sim \mathcal{E}{{xp}} \left( \lambda_0( x; {\boldsymbol{\theta}}) \right) $ with the mean $\mathbb{E}_{x,{\boldsymbol{\theta}}} \{ \Delta t \} = 1/ \lambda_0(x,{\boldsymbol{\theta}})$.
Multi-scale dynamics occur when the propensity functions have large magnitude disparities. If $\lambda_k(x; {\boldsymbol{\theta}}) \gg \lambda_j(x;{\boldsymbol{\theta}})$ for all $j \neq k$, then $\mathbb{P} \{ r^*=r_k \} \approx 1$ and $\Delta t \sim \mathcal{E}{{xp}} \left( \lambda_0(x;{\boldsymbol{\theta}}) \right) \approx \mathcal{E}{{xp}} \left( \lambda_k(x;{\boldsymbol{\theta}}) \right) $. Thus, with a high probability the next reaction in an exact trajectory will be $r_k$ and the time clock will advance on the order of $1/\lambda_k(x;{\boldsymbol{\theta}})$.
Such multi-scale networks then require an enormous number of computations to sample ``slow'' reactions and reach the required time horizon for the entire system to relax.
\subsection{Two-Time-Scale Reaction Networks} \label{sec:TTSRN} { We now consider reaction networks with two scales of dynamics. For further motivation and discussion of reaction networks with multiple time-scales, we refer readers to Refs. \citen{kang_separation_2013, e_nested_2007, huang_strong_2014, gupta_sensitivity_2014} and references therein. We instead focus on our formulation for the separation of time-scales and the averaged process via the partitioning of the state space into ``fast-classes''. Though analogous to the techniques of transforming the species variables into auxiliary fast/slow variables \cite{e_nested_2007, huang_strong_2014} or projecting to remainder spaces \cite{gupta_sensitivity_2014}, the direct partitioning of the state space will allow us to construct a singular perturbation expansion of the probability measure and establish the rate of convergence of such averaging methods. In addition, it provides a framework for applying Likelihood Ratio type sensitivity estimates to the averaged process as we shall see in the sequel.
Here, we assume that the disparity in the propensity functions results from magnitude disparities in the reaction parameters $\theta_r$. In order to illustrate the stiffness, we consider the reaction network
\begin{align*}
\begin{aligned}
*
\mathrel{\mathop{\rightleftharpoons}^{ {\alpha}_1 / \varepsilon}_{\mathrm{ {\alpha}_2 / \varepsilon}}}
A & \qquad &
A
\mathrel{\mathop{\rightleftharpoons}^{ {\beta}_1}_{\mathrm{{\beta}_2}}}
B & \qquad & B
\mathrel{\overset{ {\beta}_3}{\rightharpoonup}}
*
\end{aligned}
\end{align*} where $\varepsilon \ll 1$ is a measure of the scale disparity (stiffness) between the fast reaction parameters $ { {\boldsymbol{\alpha} } } =[\alpha_1, \alpha_2] $ and the slow reaction parameters $ { {\boldsymbol{\beta} } } =[\beta_1, \beta_2, \beta_3 ]$. As the stiffness parameter $\varepsilon \to 0$ the fast reactions $ { {\boldsymbol{\alpha} } } $ dominate the system, resulting in the multi-scale computational problem described above.
In general, suppose a reaction network has species $[X_1, X_2, \dots,, X_d]$ and reactions $r_1, \dots, r_M$. We shall assume that the propensity functions $\lambda_r(x;{\boldsymbol{\theta}})$ are of the form (\ref{eq:mass-action-prop}) (mass-action kinetics, though other forms may also be treated), and that each reaction is indexed by its own reaction parameter $\theta_r$. As in the illustrating example, we assume that there is a scale disparity in the reaction parameters between a set of ``fast reactions'' and a set of ``slow reactions''. Thus we can write ${\boldsymbol{\theta}} = [\theta_1, \dots, \theta_M] = [ { {\boldsymbol{\alpha} } } /\varepsilon, { {\boldsymbol{\beta} } } ]$, where $ { {\boldsymbol{\beta} } } =[ \beta_1, \beta_2, \dots, \beta_{M_s}]$ are the slow reaction parameters, $\varepsilon \ll 1$ is the stiffness parameter, and $ { {\boldsymbol{\alpha} } } =[\alpha_1, \dots, \alpha_{M_f}] $ are the underlying (rescaled) reaction parameters for the fast reactions.
}
To ease referencing, we will often index reactions and propensity functions directly by their reaction parameter. E.g., $r_{\beta_i}$ is the reaction with reaction parameter $\beta_i$ and propensity function $\lambda_{\beta_i} (x; {\boldsymbol{\theta}})= \lambda_{\beta_i}(x; { {\boldsymbol{\beta} } } ) = \beta_i b_{\beta_i}(x) $ (where $b_{\beta_i}$ is given by \eqref{eq:mass-action-prop}). For the fast reactions $\alpha_i$, we use $\lambda^\varepsilon_{\alpha_i}(x;{\boldsymbol{\theta}}) =\lambda^\varepsilon_{\alpha_i}(x; { {\boldsymbol{\alpha} } } ) = ({\alpha_i}/ {\varepsilon}) b_{\alpha_i}(x)$ to denote the exact propensity function and $\lambda_{\alpha_i}(x;{\boldsymbol{\theta}}) = \alpha_i \ b_{\alpha_i}(x)$ to denote the rescaled version.
Let $X^\varepsilon(t)$ denote the Markov chain determined by the exact propensity functions $\lambda^\varepsilon_\alpha(x;{\boldsymbol{\theta}})$ and $\lambda_\beta(x;{\boldsymbol{\theta}})$. We can write the generator $Q^\varepsilon=Q^\varepsilon( { {\boldsymbol{\alpha} } } , { {\boldsymbol{\beta} } } )$ of the exact process as before, and observe that \begin{align} \begin{aligned} \label{eq:Q-ep_decomp}
(Q^\varepsilon f) (x) &= \sum_{r=1}^M \lambda_r(x;{\boldsymbol{\theta}}) \left( f(x+\zeta_r) -f(x) \right) \\ &= \frac{1}{\varepsilon} \sum_{i=1}^{M_f} \lambda_{\alpha_i}(x; { {\boldsymbol{\alpha} } } ) \left( f(x+\zeta_{\alpha_i}) -f(x) \right) \\ & \qquad + \sum_{j=1}^{M_s} \lambda_{\beta_j}(x; { {\boldsymbol{\beta} } } ) \left( f(x+\zeta_{\beta_j}) -f(x) \right) \\ &= \left( \left[ \frac{1}{\varepsilon} \widetilde{Q}( { {\boldsymbol{\alpha} } } ) + \widehat{Q}( { {\boldsymbol{\beta} } } ) \right] f \right) (x) \end{aligned} \end{align} where $\widehat{Q}( { {\boldsymbol{\beta} } } )$ is the generator of the chain under only the slow dynamics (determined by slow reactions $r_\beta$), and $\widetilde{Q}( { {\boldsymbol{\alpha} } } )$ is the generator of the chain under only the fast dynamics (with the rescaled propensity functions $\lambda_\alpha(x; { {\boldsymbol{\alpha} } } )$). Thus we have a decomposition of the generator into the fast and slow dynamics, $Q^\varepsilon( { {\boldsymbol{\alpha} } } , { {\boldsymbol{\beta} } } ) = (1/\varepsilon) \widetilde{Q}( { {\boldsymbol{\alpha} } } ) + \widehat{Q}( { {\boldsymbol{\beta} } } ) $ One can also view the generator $Q^\varepsilon( { {\boldsymbol{\alpha} } } , { {\boldsymbol{\beta} } } ) = (1/\varepsilon) \widetilde{Q}( { {\boldsymbol{\alpha} } } ) + \widehat{Q}( { {\boldsymbol{\beta} } } )$ as a matrix. In this case, we can write the element corresponding to the rate of transition from state $x$ to state $y$ as \begin{align*}
\left[ \widetilde{Q}(\boldsymbol{\alpha})\right]_{x,y}
=\left\{
\begin{array}{*1{>{\displaystyle}c }l}
-\sum_{\alpha \in \boldsymbol{\alpha}} \lambda_\alpha(x)
& \quad x=y \\
\sum_{\alpha : r_\alpha(x)=y } \lambda_\alpha(x)
& \quad x \neq y
\end{array}
\right.
\end{align*} and similarly for $ \left[ \widehat{Q}(\boldsymbol{\beta})\right]_{x,y}$.
As $\varepsilon \to 0$ only the fast reactions $r_\alpha$ fire, and so we define an equivalence relation on states $s \in \mathcal{M}$, by $s_i \leftrightarrow s_j$ if they are mutually accessible through only fast reactions.
This defines a partition of the state space $\mathcal{M}$ into ``fast-classes'' $\mathcal{M}_k$ which are by construction the invariant (irreducible) classes of $\mathcal{M}$ under $\widetilde{Q}( { {\boldsymbol{\alpha} } } )$; e.g. \begin{align*}
\mathcal{M}= \bigcup_{k=1}^ {N_{C}} \mathcal{M}_k =
\left\{ x^{(1)}_1, x^{(1)}_2, \dots, x^{(1)}_{m_1},
x^{(2)}_1, \dots, x^{(2)}_{m_2},\dots
\right\} \end{align*} where $ {N_{C}}$ are the number of invariant ``fast-classes'',
and $m_k = \big| \mathcal{M}_k \big|$ is the number of states inside fast-class $\mathcal{M}_k$. For ease of presentation, in the present discussion we shall assume the state space is finite. \begin{assumption}[Finite State Space] \label{assum:finite_states}
The state space $\mathcal{M}$ is finite, such that $|\mathcal{M}| =m$. Thus the number of fast classes $ {N_{C}}<\infty$ and the number of states in each fast-class $m_k<\infty$, so that $m=m_1 + m_2 + \dots + m_ {N_{C}}$. \end{assumption} Assumption~\ref{assum:finite_states} is made only to simplify the discussion. One may also treat the infinite state case with some mild additional conditions on $\widetilde{Q}( { {\boldsymbol{\alpha} } } )$ and $\widehat{Q}( { {\boldsymbol{\beta} } } )$ to ensure non-explosiveness and ergodicity of the underlying (rescaled) chain\cite{yin_continuous-time_2013}. In addition, we shall impose the following assumption. \begin{assumption}[Recurrent States] \label{assum:recurr_states}
Each state of $\mathcal{M}$ is recurrent, so that there are no absorbing/transient
states. \end{assumption} Assumption~\ref{assum:recurr_states} is satisfied if, for example, all reactions are reversible (or often times if only the fast reactions are reversible). One may also treat the case with transient/absorbing classes with some additional stability assumptions to ensure the fast dynamics decay to steady-state; see Section 4.4 of Ref. \citen{yin_continuous-time_2013} for more details.
Under Assumption~\ref{assum:recurr_states}, we can reorder the state space so that $\widetilde{Q}( { {\boldsymbol{\alpha} } } )= {\operatorname{diag} } [ \widetilde{Q}^{(1)}( { {\boldsymbol{\alpha} } } ), \widetilde{Q}^{(2)}( { {\boldsymbol{\alpha} } } ), \dots, \widetilde{Q}^{( {N_{C}})}( { {\boldsymbol{\alpha} } } ) ]$ is block-diagonal. Here, one can view the generators $\widetilde{Q}^{(k)} ( { {\boldsymbol{\alpha} } } )$ as the restriction of $\widetilde{Q}( { {\boldsymbol{\alpha} } } )$ to the (irreducible) fast-class $\mathcal{M}_k$ (fast-only dynamics when $X(0) \in \mathcal{M}_k$). In light of the finite state-space and positive recurrence, each $\widetilde{Q}^{(k)}( { {\boldsymbol{\alpha} } } )$ is ergodic and has a stationary (steady-state) probability measure $\widetilde{\pi}^{(k)} = \widetilde{\pi}^{(k)}( { {\boldsymbol{\alpha} } } )$ such that $\widetilde{\pi}^{(k)} \widetilde{Q}^{(k)} = \boldsymbol{0}$ (with $\widetilde{\pi}^{(k)}, \boldsymbol{0}$ interpreted as row vectors).
Using the above formulation, we can restate the averaging principle \cite{e_nested_2007, cao_slow-scale_2005, samant_overcoming_2005, huang_strong_2014, kang_separation_2013, gupta_sensitivity_2014} as follows. For small $\varepsilon$ and $X(0) \in \mathcal{M}_k$, $X^\varepsilon(\cdot)$ will relax to its steady-state distribution $\pi^{(k)}$ on the micro time-scale $\varepsilon t$ before any slow reaction fires (on the macro time scale $t$). Thus, one can use the stationary average of the slow propensity functions \begin{multline}
\lambda_{\beta_j} \left( X
\left( t\right); { {\boldsymbol{\beta} } } , X(0) \in \mathcal{M}_k \right) \\
\sim \overline{\lambda}_{\beta_j} \left( \mathcal{M}_k; {\boldsymbol{\theta}} \right)
\defd \mathbb{E}_{\widetilde{\pi}^{(k)}( { {\boldsymbol{\alpha} } } )} \left\{ \lambda_{\beta_j}
\left( X; { {\boldsymbol{\beta} } } \right) \right\} \end{multline} to determine the distribution of time until the next slow reaction as well as the probabilities for the next slow reaction being $r_{\beta_j}$. These can then be used to simulate a trajectory of the slow (macro-scale) process. We shall further develop this idea more precisely in the remainder.
Write $m=\big| \mathcal{M} \big|$, and $m_k= \big| \mathcal{M}_k \big|$ as before, so that $Q^\varepsilon({\boldsymbol{\theta}}), \widetilde{Q}( { {\boldsymbol{\alpha} } } ), \widehat{Q}( { {\boldsymbol{\beta} } } ) \in \mathbb{R}^{m \times m}$, $\widetilde{Q}^{(k)}( { {\boldsymbol{\alpha} } } ) \in \mathbb{R}^{m_k \times m_k}$, and $\pi^{(k)}( { {\boldsymbol{\alpha} } } ) \in \mathbb{R}^{1 \times m_k}$. Write $\widetilde{\pi}( { {\boldsymbol{\alpha} } } )= {\operatorname{diag} } \left[
\widetilde{\pi}^{(1)}, \widetilde{\pi}^{(2)}, \dots, \widetilde{\pi}^{( {N_{C}})} \right] \in \mathbb{R}^{ {N_{C}} \times m}$. Write $\boldsymbol{1}_{m_k}$ for $[1, 1, \dots, 1]' \in \mathbb{R}^{m_k \times 1}$ and $\widetilde{\boldsymbol{1}} = {\operatorname{diag} } \left[ \boldsymbol{1}_{m_1}, \boldsymbol{1}_{m_2}, \dots, \boldsymbol{1}_{m_ {N_{C}}} \right]$
With $\widetilde{\pi}( { {\boldsymbol{\alpha} } } )$ describing the limit behavior inside each fast-class on the micro time scale, one can then consider the distribution of the exact system $X^\varepsilon$ on the macro time scale. Heuristically, one expects a trajectory to enter a fast-class of states $\mathcal{M}_{k_1}$ and quickly iterate through many fast reactions until the distribution of the trajectory reaches the steady-state $\widetilde{\pi}^{(k_1)}$. Eventually, a slow reaction will fire to move the trajectory to a new fast-class $\mathcal{M}_{k_2}$ (see Figure~\ref{fig:TTS_CRN_example}).
Indeed, Writing \begin{align}
\begin{aligned} \label{eq:Q-bar-matrix}
\overline{Q} = \overline{Q}( { {\boldsymbol{\alpha} } } , { {\boldsymbol{\beta} } } ) \defd
\left[ \widetilde{\pi}( { {\boldsymbol{\alpha} } } ) \cdot \widehat{Q}( { {\boldsymbol{\beta} } } ) \cdot
\widetilde{\boldsymbol{1}} \right]
\in \mathbb{R}^{ {N_{C}} \times {N_{C}}} ,
\end{aligned} \end{align} we see that $\overline{Q}$ is itself a generator of an ``averaged'' CTMC reaction network, whose ``states'' correspond to fast-classes $\mathcal{M}_k$. Write $\overline{X}(t)$ for the ``averaged'' process generated by $\overline{Q}$. Together, $\overline{Q}$ and $\overline{X}(t)$ describe the limit (as $\varepsilon \to 0$) of the average rate that the exact process $X^\varepsilon(t)$ moves between the fast-classes $\left\{ \mathcal{M}_k \right\}_{k=1}^ {N_{C}}$ via slow reactions.
Furthermore, we can identify the elements of $\overline{Q}$ from the steady-state averages of the slow propensity functions. First, note that every slow reaction carries each fast class to a unique new fast-class; that is, if $x \leftrightarrow y$ and $\lambda_\beta(x), \lambda_\beta(y) >0$, then $r_\beta(x) \leftrightarrow r_\beta(y)$. Thus, $r_\beta(\mathcal{M}_k)$ is well-defined. Then, using the form of $\widehat{Q}$ together with \eqref{eq:Q-bar-matrix}, we have \begin{align}
\begin{aligned}
\label{eq:Q-bar-lambda}
\overline{Q}_{k_1, k_2}
&=
\sum_{ \substack{ \beta \in { {\boldsymbol{\beta} } } \\ r_\beta(\mathcal{M}_{k_1} ) = \mathcal{M}_{k_2} } }
\overline{\lambda}_\beta (\mathcal{M}_{k_1}; {\boldsymbol{\theta}})
\end{aligned} \end{align} for $k_1 \neq k_2$, and similarly we see that \begin{align}
\label{eq:lambda-bar_0}
\overline{Q}_{k_1, k_1} =
- \sum_{\beta \in { {\boldsymbol{\beta} } } } \overline{\lambda}_\beta( \mathcal{M}_{k_1}; {\boldsymbol{\theta}})
\defd - \overline{\lambda}_{\beta_0} (\mathcal{M}_{k_1} ; {\boldsymbol{\theta}}) . \end{align} With this formulation, we see that generator $\overline{Q}$ corresponds to a meta ``macro'' reaction network with the state-space $\overline{\mathcal{M}} = \left\{ \mathcal{M}_1, \mathcal{M}_2, \dots, \mathcal{M}_ {N_{C}} \right\}$, reactions $\{ r_\beta :\beta \in { {\boldsymbol{\beta} } } \}$ and propensities $\{ \overline{\lambda}_\beta(\overline{X}; {\boldsymbol{\theta}}) : \beta \in { {\boldsymbol{\beta} } } \}$. Figure ~\ref{fig:TTS_CRN_example} depicts such a macro chain for the macro process $\overline{X}(t)$. \begin{figure}\label{fig:TTS_CRN_example}
\end{figure}
If we can estimate the average slow-propensities $\widehat{\overline{\lambda}}_\beta(\mathcal{M}_k; {\boldsymbol{\theta}})$ within each fast-class (say, through ergodic time averages of the fast-only process), then one can simulate a trajectory of the macro process $\overline{X}(t)$ from these average propensities using {\em any} single-scale Monte Carlo simulation (e.g. Direct SSA, Next-Reaction, etc). Furthermore, if one is ultimately concerned with estimating $\mathbb{E}_{{\boldsymbol{\theta}}}\left\{ f \left( X^\varepsilon(t) \right) \right\}$ for some observable (quantity of interest) $f:\mathcal{M} \to \mathbb{R}$, then one can define an augmented functional $\overline{f}$ on $\overline{\mathcal{M}}$ by \begin{align} \label{eq:fbar_defn}
\overline{f} \left( \mathcal{M}_k ; { {\boldsymbol{\alpha} } } \right)
\defd \mathbb{E}_{\widetilde{\pi}^{(k)}( { {\boldsymbol{\alpha} } } ) } \left\{ f(X) \right\}
=\sum_{x \in \mathcal{M}_k} f(x) \widetilde{\pi}^{(k)}_x( { {\boldsymbol{\alpha} } } ) \end{align} It shall be shown that $\mathbb{E}_{{\boldsymbol{\theta}}} \left\{ \overline{f}(\overline{X}(t)) \right\} \approx \mathbb{E}_{{\boldsymbol{\theta}}} \left\{ f\left( X^\varepsilon(t) \right) \right\}$ for large enough $t$ and sufficiently small $\varepsilon$.
To illustrate how one can implement the averaging scheme to generate macro-trajectories, we present the following TTS vlersion of the Direct SSA (since it is the most succinct to write). In this case, the TTS SSA is essentially the same algorithm as in Refs. \citen{e_nested_2007}, \citen{samant_overcoming_2005}. However, we emphasize that the same method can be used to create a TTS version of any exact method defined by the propensity functions. In particular, one can just as easily construct an analogous TTS Next-Reaction type algorithm \cite{rathinam_efficient_2010,anderson_efficient_2012, anderson_modified_2007} for tightly coupled trajectories.
\begin{algorithm}[TTS-SSA] \label{alg:TTS-SSA}
\rm
To simulate a trajectory of the macro-process
$\overline{X}(T)$ until macro time-horizon $T_{final}$:
\begin{enumerate}[\bf (1)]
\item Initialize $x$ at a macro time $T$;
$x \in \mathcal{M}_k$ for some (unknown) $k$
\item Simulate the fast-only reaction network ${\widetilde{Q}^{(k) } }(\boldsymbol{\alpha})$ until time-averages
of observable $f$ and slow propensities $\lambda_{\beta}$ relax to steady-state :
\begin{eqnarray*}
\frac{1}{t} \int_0^t f\left(
{\widetilde{X}^{(k)} } (s) \right)ds &\to &
\mathbb{E}_{{\widetilde{\pi}^{(k)}( { {\boldsymbol{\alpha} } } ) } }\left\{ f(\widetilde{X}) \right\} \equiv {\overline{f}(k) } \\
\frac{1}{t} \int_0^t \lambda_{{\beta} } \left(
{\widetilde{X}^{(k)} } (s);\boldsymbol{{\beta} } \right) ds
&\to &
\mathbb{E}_{{\widetilde{\pi}^{(k)}( { {\boldsymbol{\alpha} } } ) } }\left\{ \lambda_{\beta}
\left( \widetilde{X} ; \boldsymbol{\beta} \right) \right\} \\
&\equiv & {\overline{\lambda}_{\beta }(k; {\boldsymbol{\theta}}) }
\end{eqnarray*}
\item Observe terminal state ${\widetilde{x}^{(k)} } \sim {\widetilde{\pi}^{(k)} }$.
Compute ${\overline{\lambda}_{\beta_0} } =
\sum_{j=1}^{Ms} {\overline{\lambda}_{\beta_j} }$
\item Sample time to next slow reaction : $\Delta T \sim \mathcal{E}{{xp}}( {\overline{\lambda}_{\beta_0} })$
\item Sample next slow rxn to fire
${\beta^* }
\sim 1/ {\overline{\lambda}_{\beta_0} } \left[ {\overline{\lambda}_{\beta_1} }, \dots,
{ \overline{\lambda}_{\beta_{Ms}} } \right] $
\item Update macro time $T\leftarrow T + \Delta T$ and move to the
next fast class
by taking $x= { \widetilde{x}^{(k)} } + { \zeta_{\beta^*} }$
\item Return to {\bf (1)} until macro time horizon $T_{final}$ is reached
\end{enumerate} \end{algorithm}
\subsection{Convergence and Error Bounds} Here we use the above formulation of stiff networks to establish convergence results and error bounds for the averaged process obtained by Algorithm~\ref{alg:TTS-SSA}. They are largely obtained by applying results from Ref. \citen{yin_continuous-time_2013} to the Two-Time-Scale Markov chain developed above. We give the statements below and defer to the Appendix for the proofs.
From the exact chain $X^\varepsilon(t)$, define a stochastic process $\overline{X^\varepsilon}(t)$ taking values in $\overline{\mathcal{M}}=\{ \mathcal{M}_1, \dots, \mathcal{M}_ {N_{C}} \}$ by $\overline{X^\varepsilon}(t) = \mathcal{M}_k $ for $X^\varepsilon(t) \in \mathcal{M}_k$. Note that $\overline{X^\varepsilon}(t)$ is not, in general, a Markov chain. However, one expects that as $\varepsilon \to 0$, the process $\overline{X^\varepsilon}(t)$ converges to $\overline{X}(t)$ in some sense.
\begin{proposition}[Weak Convergence] \label{prop:weak_convergence} Under Assumptions~\ref{assum:recurr_states} and \ref{assum:finite_states},
as $\varepsilon \to 0$, $\overline{X^\varepsilon}(\cdot)$ converges weakly to $\overline{X} (\cdot)$ in the Skorohod space $\mathcal{D}([0,T]; \overline{\mathcal{M}} )$ for any time horizon $T$. \end{proposition}
The above proposition establishes weak convergence of the projection (onto fast-classes) of the exact system $X^\varepsilon$ to the averaged meta system $\overline{X}$ as $\varepsilon \to 0$. This is essentially the same result as established in Ref.~\citen{kang_separation_2013}, where the authors instead consider the disparity of the propensities as the system size (molecule count) $N \to \infty$. In both formulations, one selects a reference scale and then examines limit behavior against the reference scale as the disparity increases ($\varepsilon \to 0$ or $N \to \infty$). However, in practice one implements the averaging procedure to approximate a system with a fixed, positive scale disparity. Naturally, one is then concerned about the induced error from the averaging approximation.
Write $\overline{p}_T = \overline{p}_T(\overline{X} ; { {\boldsymbol{\alpha} } } , { {\boldsymbol{\beta} } } )$ for the probability measure (on $\overline{\mathcal{M}}$) induced by the averaged process $\overline{X}$ at time $T$. At the end of a TTS simulation, one obtains a terminal state $X(T)=x \sim p^0_T= p^0_T(X; { {\boldsymbol{\alpha} } } , { {\boldsymbol{\beta} } } )$, where $p^0_T$ is the probability measure on $\mathcal{M}$ induced by the last state observed from the terminal fast-class. Thus, $p^0_T$ is determined by $\overline{p}_T$ and $\widetilde{\pi}( { {\boldsymbol{\alpha} } } )$. Write $p^\varepsilon_T=p^\varepsilon(X^\varepsilon; { {\boldsymbol{\alpha} } } , { {\boldsymbol{\beta} } } )$ for the probability measure on $\mathcal{M}$ induced from the exact process $X^\varepsilon$. Since $p^\varepsilon_T$ is the distribution we would see from an exact simulation, and $p^0_T$ is the distribution from the TTS simulation, the question becomes: What is the error of $p^0_T$ from $p^\varepsilon_T$? One can take a singular perturbation expansion of $p^\varepsilon_T$ in terms of $\varepsilon$ and identify the leading term as $p^0_T$ to obtain the following result.
\begin{theorem}[Error in Probability] \label{thm:prob_err_bound} Let $\widetilde{\kappa} = -\frac{1}{2} \max \left\{ \operatorname{Re}(\nu) : \nu \text{ is a non-zero eigenvalue of a } \widetilde{Q}^{(k)} \right\} $. Then under Assumptions \ref{assum:finite_states}, \ref{assum:recurr_states}, we have \begin{align} \begin{aligned}
\label{eq:prob_err_bound}
\| p^0_T - p^\varepsilon_T \| \le O\left( \varepsilon + \exp\left\{ -\widetilde{\kappa} T/\varepsilon \right\} \right) \end{aligned} \end{align}
where $\| \cdot \|$ denotes the $l_2$ norm. \end{theorem} In Theorem~\ref{thm:prob_err_bound}, $\widetilde{\kappa}$ is the slowest rate of convergence of $\widetilde{Q}^{(k)}$ to the steady-state $\widetilde{\pi}^{(k)}$ among all fast-classes $\mathcal{M}_k$. Thus, as long as the macro time horizon $T$ is large enough to ensure the fast dynamics have relaxed to steady state
($T> -\varepsilon/\widetilde{\kappa} \log(\varepsilon)$), then the error becomes $\|p^0_T - p^\varepsilon_T \| \le O(\varepsilon)$.
Writing $\pi^0( { {\boldsymbol{\alpha} } } , { {\boldsymbol{\beta} } } )$ for the stationary distribution corresponding to the TTS probability measure $p^0_T( { {\boldsymbol{\alpha} } } , { {\boldsymbol{\beta} } } )$, it is not hard to see that $\pi^0 ( { {\boldsymbol{\alpha} } } , { {\boldsymbol{\beta} } } ) = \overline{\pi} \cdot \widetilde{\pi}$, the product of the steady-state distribution between fast-classes $\overline{\pi}( { {\boldsymbol{\alpha} } } , { {\boldsymbol{\beta} } } )$ and the steady-state distribution within fast-classes $\widetilde{\pi}( { {\boldsymbol{\alpha} } } )$. Write $\pi^\varepsilon$ for the steady-state distribution corresponding to the exact process generated by $Q^\varepsilon$. Then using Theorem~\ref{thm:prob_err_bound} and exponential convergence to the steady state, we obtain the following error bounds.
\begin{corollary}[Error in Expectation] \label{cor:expectation-error} Under Assumptions \ref{assum:finite_states} and \ref{assum:recurr_states},
$\| \pi^0 - \pi^\varepsilon \| \le O(\varepsilon)$ and
$\| \pi^0 - p^\varepsilon_T \| \le O(\varepsilon)$ for sufficiently large $T$. Thus, for all bounded functions $f$ on the state space $\mathcal{M}$, \begin{align} \begin{aligned} \label{eq:error_expectation}
\left| \mathbb{E}_{\overline{p}_T} \left\{ \overline{f}(\overline{X}(T)) \right\}\right.
& - \left. \mathbb{E}_{p^\varepsilon_T} \left\{ f(X^\varepsilon(T)) \right\} \right|
\le \\
& \|f\|_\infty \| p^0_T - p^\varepsilon_T \| \le O(\varepsilon) \\
\left| \mathbb{E}_{\overline{\pi}} \left\{ \overline{f}(\overline{X}) \right\}\right.
& - \left. \mathbb{E}_{\pi^\varepsilon} \left\{ f(X^\varepsilon) \right\} \right|
\le \\
& \|f\|_\infty \| \pi^0 - \pi^\varepsilon \| \le O(\varepsilon)
\end{aligned} \end{align} \end{corollary}
Corollary \ref{cor:expectation-error} is of great practical use, as it says that the expected value of the macro-process $\overline{X}(T)$ with macro-observable $\overline{f} :\overline{\mathcal{M}} \to \mathbb{R}$ provides an $O(\varepsilon)$ estimate of the expected value of the exact system $X^\varepsilon(T)$ with observable $f : \mathcal{M} \to \mathbb{R}$. Since we can use TTS algorithms (such as Algorithm \ref{alg:TTS-SSA}) to quickly generate trajectories of $\overline{X}(T)$ while estimating the macro-observable $\overline{f}(\mathcal{M}_k)$ at each state along the way, this provides a method to very quickly generate estimates of $\mathbb{E}_{p^\varepsilon_T} \left\{ f(X^\varepsilon(T)) \right\}$ with at most $O(\varepsilon)$ bias. As $\varepsilon \to 0$, the bias decreases linearly while the computational savings increase as $O(1/\varepsilon)$.
\section{Two Time Scale Sensitivity Analysis} \label{sec:TTS_sens}
Computing the system sensitivities $S_{f,T}(\theta_i) \defd \frac{\partial}{\partial \theta_i} \mathbb{E}_{\boldsymbol{\theta}} \left\{ f(X(T)) \right\}$ with respect to reaction parameters $\theta_i \in {\boldsymbol{\theta}}$ provides great insight into the model.
As such, numerous works have constructed and analyzed methods to estimate the sensitivities from sample trajectories of the system
\cite{gupta_efficient_2014, rathinam_efficient_2010, sheppard_pathwise_2012, wolf_hybrid_2015, wang_efficiency_2014, nunez_steady_2015, gupta_sensitivity_2013, pantazis_parametric_2013, warren_steady-state_2012}.
Different methods work better for different systems or different criteria, but all methods have higher (sometimes stupendously higher) variance in the estimation of
$S_{f,T}(\theta_i)$ compared to the estimation of $ \mathbb{E}_{{\boldsymbol{\theta}}}\left\{ f(X(T)) \right\} $, thus requiring a very large number of samples to estimate the sensitivity precisely. If the system is stiff (as in \eqref{eq:Q-ep_decomp}) so that each exact trajectory $X^\varepsilon(T)$ requires a prohibitively large computational load, then the large number of sample paths required to estimate the sensitivity $S_{f,T}^\varepsilon(\theta_i) \defd \partial_{\theta_i} \ \mathbb{E}_{p^\varepsilon_T({\boldsymbol{\theta}})} \left\{ f(X^\varepsilon(T)) \right\}$ make the problem computationally intractable.
{ Corollary~\ref{cor:expectation-error} gives that the expectation of macro ``averaged'' reaction network $\overline{f}(\overline{X}(T))$ gives an accurate approximation of the expectation of the exact network; $\mathbb{E}_{\overline{p}_T({\boldsymbol{\theta}})} \left\{ \overline{f} (\overline{X}(T)) \right\} = \mathbb{E}_{p^\varepsilon_T({\boldsymbol{\theta}})} \left\{ f(X^\varepsilon(T)) \right\} + O(\varepsilon)$. A natural question to ask is whether the sensitivities of the exact system converge to the sensitivities of the averaged system. Using the recent result of Ref. \citen{gupta_sensitivity_2014}, we can derive the following (the details are deferred to Appendix).
\begin{proposition}{Convergence of Sensitivities} \label{prop:sens_converge} \begin{multline} \label{eq:TTS_sens_estimate} \lim_{\varepsilon \to 0} S_{f,T}^\varepsilon(\theta_i) = \overline{S}_{f,T} (\theta_i) \defd \frac{\partial}{\partial \theta_i} \mathbb{E}_{\overline{p}_T({\boldsymbol{\theta}})} \left\{ \overline{f} (\overline{X}(T)) \right\} \end{multline} \end{proposition}
Thus, if we can compute the sensitivity of the macro reaction network $\overline{f}(\overline{X}(T))$ (whose sample paths have orders of magnitude less cost to simulate than the exact stiff network $f(X^\varepsilon(T))$), then this provides an accurate estimate of the exact sensitivity. Furthermore, since $\overline{X}(\cdot)$ is formulated as a reaction network with propensities $\{ \overline{\lambda}_\beta(x, {\boldsymbol{\theta}}) : \beta \in { {\boldsymbol{\beta} } } \}$ and observable values $\overline{f}(\mathcal{M}_k)$ (both of which are estimated during a TTS simulation), we can apply most of existing single-scale sensitivity estimation methods to estimate $ \overline{S}_{f,T}(\theta_i)
$
and thus $ S_{f,T}^\varepsilon(\theta_i)
$.
We note that \eqref{eq:TTS_sens_estimate} gives that the sensitivity of the exact system converges to the sensitivity of the averaged system, but does not give the rate of convergence. Currently, this is an open question. Since from \eqref{eq:error_expectation} we have the expectation converges at a rate $O(\varepsilon)$, one might suspect that the sensitivity also converges at rate $O(\varepsilon)$, at least for certain classes of networks (e.g. linear propensities). Ongoing work aims to establish the rate of convergence via singular perturbation expansions of sensitivity reweighting measures. However, the remainder of this work shall focus on the development and practical implementation of a multiscale Likelihood Ratio estimator of the limit sensitivity $\overline{S}_{f,T}(\theta_i)$.
} In what follows, we review the Likelihood Ratio
method for computing system sensitivities for single-scale reaction networks. Furthermore, we shall introduce a new Ergodic Likelihood Ratio method which has much smaller variance when estimating sensitivities at steady-state. We then derive a Two-Time-Scale version that allows one to estimate the full gradient of a stiff system using any TTS Monte Carlo method for simulating a macro trajectory.
\subsection{Likelihood Ratio Methods} \label{sec:LRmethods} Likelihood Ratio (LR) methods \cite{plyasunov_efficient_2007, wang_efficiency_2014, warren_steady-state_2012, nunez_steady_2015, glynn_likelihood_1990, mcgill_efficient_2012} (aka the Girsanov Transform Method) attempt to compute the derivative by reweighting the observed trajectory by its ``score'' function of the density. Here, one views ${\boldsymbol{\theta}}$ as parametrizing the probability measure on the path-space $P(\cdot, t ; {\boldsymbol{\theta}})$. If $P(\cdot, t ; {\boldsymbol{\theta}})$ is differentiable with respect to $\theta_i$, then under mild regularity conditions we have \begin{multline} \label{eq:LR_reweight}
S_{f,t}(\theta_i) \defd
\frac{\partial}{\partial \theta_i} \mathbb{E}_{\boldsymbol{\theta^0}}
\left\{ f(X(t)) \right\} \\
= \int_{\Omega} f(X(t,\omega))
\frac{\frac{\partial}{\partial \theta_i} \big|_{\boldsymbol{\theta^0} }
P(d\omega, t; \boldsymbol{\theta})}{P(d\omega, t ;\boldsymbol{\theta}) }
P(d\omega, t; \boldsymbol{\theta^0}) \\
= \mathbb{E}_{\boldsymbol{\theta^0}} \left\{ f(X(t)) W_{\theta_i}(t) \right\}
\end{multline} Using the random-time-change representation \eqref{eq:random-time-change} it can be shown\cite{plyasunov_efficient_2007} that the reweighting process $W_{\theta_i}(t)$ is a zero-mean martingale process and can be represented by \begin{multline} \label{eq:W_representation}
W_{\theta_i} (t) = \sum_{r=1}^M \int_0^t \frac{
\frac{\partial \lambda_r}{\partial \theta_i} \left( X(s^-), \boldsymbol{\theta^0}
\right) }{\lambda_r\left( X(s^-), \boldsymbol{\theta^0} \right)} dR_r(s) \\
- \sum_{r=1}^M \int_0^t \frac{\partial \lambda_r}{\partial \theta_i}
\left( X(s^-), \boldsymbol{\theta^0} \right) ds , \end{multline} where $dR_r(s)$ is simply the counting measure of reaction $r$ which equals $1$ at times $s$ at which reaction $r$ fires and is zero otherwise. Thus, assuming one can compute $\partial_{\theta_i} \ \lambda_r(x,{\boldsymbol{\theta}})$, then $W_{\theta_i}(t)$ has a computationally tractable form as follows.
We write $\hat{X}_l$ for the $l$-th state of the system through a trajectory, and $\Delta_l$ for the holding time in the $l$-th state. Write $T_l$ for the time of the $l$ jump, so that $T_l = \sum_{j=0}^{l-1} \Delta_j$. We denote $N(t)$ as the total number of reactions which have fired by time $t$ and $r^*_l$ for the index of the reaction which takes the system from the $l$-th state to the $l+1$-th state.
Then $W_{\theta_i}$ has the explicit form
\begin{multline*} W_{\theta_i} (t) = \\ \sum_{l=0}^{N(t)-1} \left[
\frac{\partial }{\partial \theta_i}
\log{\lambda_{r^*_l}\left( \hat{X}_l, \boldsymbol{\theta^0}\right) }
- \sum_{r=1}^M \frac{\partial \lambda_r}{\partial \theta}
\left( \hat{X}_l, \boldsymbol{\theta^0} \right) \Delta_l \right] \\
- \sum_{r=1}^M \frac{\partial \lambda_r}{\partial \theta}
\left( \hat{X}_l, \boldsymbol{\theta^0} \right) \left[ T - T_{N(T)} \right] . \end{multline*}
In simulation, the LR estimate is computed via ensemble averages estimated by empirical averaging $S_{f,T}(\theta_i)
\approx \widehat{ \operatorname{\bf LR}} ( {N_{S}},\theta_i)$
with the empirical estimator \begin{multline}
\label{eq:LR-defn}
\widehat{ \operatorname{\bf LR}} ( {N_{S}},\theta_i) \defd
\frac{1}{ {N_{S}}} \sum_{n=1}^{ {N_{S}}}
\widehat{\left[ f(x(T)) \right]}_n
\widehat{\left[ W_{\theta_i}(T) \right]}_n . \end{multline} where $ {N_{S}}$ is the number sample paths, $\widehat{\left[ f(x(T)) \right]}_n$ is the observable value at terminal time $T$ for the $n$th sample path, and similarly $\widehat{[W_{\theta_i}(T)]}_n$ is the terminal value of $W_{\theta_i}(T)$ for the $n$th sample path. While the reweighting process $W_{\theta_i}(t)$ has zero mean, its variance grows with time \cite{wang_efficiency_2014, warren_steady-state_2012}, making it quite inefficient for large time horizons. The variance can be reduced by using the centered likelihood ratio estimate \begin{multline}
\label{eq:CLR-defn}
\widehat{\operatorname{\bf CLR}}
( {N_{S}}, \theta_i)
\defd
\widehat{\operatorname{\bf LR}}( {N_{S}}, \theta_i) \\
- \frac{1}{ {N_{S}}^2}
\left\{ \sum_{n=1}^ {N_{S}}
\widehat{\left[ f(x(T)) \right]}_n
\right\}
\left\{ \sum_{n=1}^ {N_{S}}
\widehat{\left[ W_{\theta_i}(T) \right]}_n
\right\} . \end{multline} Since the $\mathbb{E}_{{\boldsymbol{\theta}}} \{ W_{\theta_i}(T) \} \equiv 0$, the second term doesn't impose any bias into the estimate \eqref{eq:LR_reweight}, but is coupled to the first term to reduce the observed variance \cite{wang_efficiency_2014}.
Suppose one is interested in the steady-state sensitivities, $ S_{f,\infty}(\theta_i) \defd \partial_{\theta_i} \mathbb{E}_{\pi({\boldsymbol{\theta}})} \left\{ f(X) \right\} $. It is well known that $\mathbb{E}_{p_T({\boldsymbol{\theta}})} \left\{ f(X(T)) \right\} = \mathbb{E}_{\pi({\boldsymbol{\theta}})} \left\{ f(X) \right\} + O\left( e^{-\kappa T} \right)$ for some mixing rate $\kappa$, and thus for large $T$
one can use the terminal distribution of $f(X(T))$ and $W_{\theta_i}(T)$ in \eqref{eq:LR_reweight} to obtain an estimate of the steady-state sensitivity with exponentially small bias \cite{warren_steady-state_2012}. However, the major difficulty in using likelihood ratio estimates is the large variance of the estimator $f(X(T)) W_{\theta_i}(T)$, which is proportional to $\mathbb{V}\mathrm{ar}\{ f(X(T)) \} \mathbb{V}\mathrm{ar}\{W_{\theta_i}(T)$ \cite{wang_efficiency_2014, plyasunov_efficient_2007}. It can be seen that $\mathbb{V}\mathrm{ar}\left\{ W_{\theta_i }(T) \right\} = O(T)$, so one must balance choosing a terminal time $T$ large enough to ensure sufficient decay of the bias $\mathbb{E}_{p_T({\boldsymbol{\theta}})}\left\{ f(X(T)) \right\} - \mathbb{E}_{\pi({\boldsymbol{\theta}})} \left\{ f(X) \right\} $, yet as small as possible to contain the growth of the $\mathbb{V}\mathrm{ar}\left\{ W_{\theta_i}(T) \right\}$. While centering as in \eqref{eq:CLR-defn} helps to reduce the variance of the estimator, the variance is usually much larger than comparable finite difference of pathwise derivative methods \cite{wolf_hybrid_2015,wang_efficiency_2014,sheppard_pathwise_2012}.
Instead of using the terminal distribution $f(X(T))$ as an approximation of the steady-state distribution, one could instead use the ergodic-average (time-average) $1/T \int_0^T f(X(s)) ds $. The bias of the ergodic-average decays slower than the terminal distribution ($O(1/T)$ compared to $O(e^{-\kappa T})$), but has the advantage that variance decays with time as well; that is, $\mathbb{V}\mathrm{ar}\left\{ 1/T \int_0^T f(X(s)) ds \right\} = O(1/T)$ whereas $\mathbb{V}\mathrm{ar}\left\{ f(X(T)) \right\} \to \mathbb{V}\mathrm{ar}\left\{ f(X(\infty)) \right\} = \sigma^2$ (see Ref. \citen{asmussen_stochastic_2007} for more details).
Motivated by the variance reduction one obtains with ergodic averaging, we introduce a new method for computing likelihood-ratio type steady state sensitivity estimates. The idea is to simply replace the terminal-state observable $f(X(T))$ with the ergodic average $1/T \int_0^T f(X(s)) ds$ in the LR scheme \eqref{eq:LR_reweight} -- \eqref{eq:LR-defn}. The philosophy is that by incurring some small amount of additional bias in the mean value, the ergodic steady-state sensitivity estimate has much smaller variance than the terminal-state distribution. We shall refer to this method the {\em ergodic likelihood ratio}, \begin{multline} \label{eq:ELR_defn}
\widehat{ \operatorname{\bf ELR}} ( {N_{S}},\theta_i) \defd \\
\frac{1}{ {N_{S}}} \sum_{n=1}^{ {N_{S}}}
\frac{1}{T}
\widehat{\left[ \int_0^T f(x(s))ds \right]}_n
\widehat{\left[ W_{\theta_i}(T) \right]}_n . \end{multline} Similarly, one can center the {\bf ELR} to derive the {\em centered ergodic likelihood ratio} {\bf CELR}, \begin{multline}
\label{eq:CELR-defn}
\widehat{\operatorname{\bf CELR}}
( {N_{S}}, \theta_i)
\defd
\widehat{\operatorname{\bf ELR}}( {N_{S}}, \theta_i) \\
- \frac{1}{ {N_{S}}^2}
\left\{ \sum_{n=1}^ {N_{S}}
\frac{1}{T}
\widehat{\left[ \int_0^T f(x(s))ds \right]}_n
\right\}
\left\{ \sum_{n=1}^ {N_{S}}
\widehat{\left[ W_{\theta_i}(T) \right]}_n
\right\} . \end{multline}
In the numerical experiments, it shall be seen that the {\bf CELR } method performs much better than the {\bf CLR} for steady-state sensitivity estimation.
\subsection{TTS Likelihood Ratio}
In what follows, we describe how the above single-scale Likelihood Ratio methods can be adapted to the macro-process $\overline{X}(T)$ for use in \eqref{eq:TTS_sens_estimate}. Recall that the reaction parameters can be classified as fast or slow, ${\boldsymbol{\theta}}=[ { {\boldsymbol{\alpha} } } , { {\boldsymbol{\beta} } } ]$ with $ { {\boldsymbol{\alpha} } } =[\alpha_1, \dots, \alpha_{M_f} ]$ and $ { {\boldsymbol{\beta} } } =[ { {\boldsymbol{\beta} } } _1, \dots, { {\boldsymbol{\beta} } } _{M_s} ]$. To apply Likelihood Ratio methods to compute $\partial_{\theta_i} \mathbb{E}_{{\boldsymbol{\theta}}} \left\{ \overline{f}(\overline{X}(T) \right\}$, we exploit that the macro process $\overline{X}(T)$ is identified as reaction network with propensities \begin{multline*}
\overline{\lambda}_{\beta_r}(\overline{X}; { {\boldsymbol{\alpha} } } , { {\boldsymbol{\beta} } } )
= \mathbb{E}_{\widetilde{\pi}^{(\overline{X})}( { {\boldsymbol{\alpha} } } )}
\left\{ \lambda_{\beta_r}(X; { {\boldsymbol{\beta} } } ) \right\} \\
= \sum_{x \in \mathcal{M}_{\overline{X}}}
\lambda_{\beta_r}(x; { {\boldsymbol{\beta} } } )
\widetilde{\pi}^{(\overline{X})}(x; { {\boldsymbol{\alpha} } } ) \end{multline*} (for $\beta_r \in { {\boldsymbol{\beta} } } $), and observable \begin{align*}
\begin{aligned}
\overline{f}(\overline{X}; { {\boldsymbol{\alpha} } } )
=\mathbb{E}_{\widetilde{\pi}^{(\overline{X})}( { {\boldsymbol{\alpha} } } )}
\left\{ f(X) \right\}
= \sum_{x \in \mathcal{M}_{\overline{X}}}
f(x) \widetilde{\pi}^{(\overline{X})}(x; { {\boldsymbol{\alpha} } } )
\end{aligned} \end{align*} Thus the macro-sensitivities can be represented by \begin{multline}
\label{eq:TTS_LR}
\frac{\partial}{\partial \theta_i}
\mathbb{E}_{\overline{p}_T({\boldsymbol{\theta}})}
\left\{ \overline{f}\left( \overline{X}(T);{\boldsymbol{\theta}} \right) \right\} \\
= \mathbb{E}_{\overline{p}_T({\boldsymbol{\theta}})}
\left\{ \frac{\partial}{\partial \theta_i}
\overline{f}\left( \overline{X}(T);{\boldsymbol{\theta}} \right)
+ \overline{f}\left( \overline{X};{\boldsymbol{\theta}} \right)
\overline{W}_{\theta_i} (T) \right\} \end{multline} where the macro-reweighting process $\overline{W}_{\theta_i}(T)$ is given by \begin{multline}
\label{eq:TTS_W-rep}
\overline{W}_{\theta_i} =
\sum_{r=1}^{M_s} \int_0^T
\frac{
\frac{\partial}{\partial \theta_i}
\overline{\lambda}_{\beta_r}
\left( \overline{X}(s);{\boldsymbol{\theta}} \right)
}{
\overline{\lambda}_{\beta_r} \left( \overline{X}(s);{\boldsymbol{\theta}} \right)
} dR_{\beta_r}(s) \\
- \sum_{r=1}^{M_s} \int_0^T
\frac{\partial}{\partial \theta_i}
\overline{\lambda}_{\beta_r}\left(
\overline{X}(s);{\boldsymbol{\theta}} \right) ds . \end{multline} Therefore, in order to apply \eqref{eq:TTS_LR} we need to be able to compute the derivatives of the averaged observable $\partial_{\theta_i} \ \overline{f}\left( \overline{X}(s); {\boldsymbol{\theta}} \right)$ as well as the derivatives of the averaged propensity functions $\partial_{\theta_i} \ \overline{\lambda}_{\beta_r}\left( \overline{X}(s);{\boldsymbol{\theta}} \right)$.
Suppose $\theta_i = \beta_i \in { {\boldsymbol{\beta} } } $ is a slow reaction parameter. If the original observable $f(X)$ has no direct parameter dependence, then
$\partial_{\beta_i} \ \overline{f}(\overline{X}(s),{\boldsymbol{\theta}}) \equiv 0$. Furthermore, under mass-action kinetics, the averaged propensities have $\partial_{\beta_i} \ \overline{\lambda}_{\beta_r}(\overline{X}(s); {\boldsymbol{\theta}}) = \overline{b}_{\beta_r} (\overline{X}; { {\boldsymbol{\alpha} } } ) \delta_{i,r}$, where $\overline{b}_{\beta_r}(\overline{X}; { {\boldsymbol{\alpha} } } ) = 1/\beta_r \cdot \overline{\lambda}_{\beta_r}(\overline{X}; { {\boldsymbol{\alpha} } } )$ is already computed during a TTS simulation and $\delta_{i,r}=1$ if $i=r$ and $0$ otherwise. Thus the slow sensitivities are directly computable from a standard TTS simulation.
Suppose $\theta_i=\alpha_i \in { {\boldsymbol{\alpha} } } $ is a fast reaction parameter. Then computing $\partial_{\alpha_i} \overline{f}(\overline{X}(s); { {\boldsymbol{\alpha} } } )$ and $\partial_{\alpha_i} \overline{\lambda}_{\beta_r}(\overline{X}(s); { {\boldsymbol{\alpha} } } , { {\boldsymbol{\beta} } } )$ is more problematic, as they only depend indirectly on $ { {\boldsymbol{\alpha} } } $ through the fast-class steady-state measures $\widetilde{\pi}( { {\boldsymbol{\alpha} } } )$. Thus explicit computation is often infeasible. However, one may estimate $\partial_{\alpha_i} \mathbb{E}_{\widetilde{\pi}^{(\overline{X})}( { {\boldsymbol{\alpha} } } )} \left\{ f(X) \right\}$ and $\partial_{\alpha_i} \ \mathbb{E}_{\widetilde{\pi}^{(\overline{X})}( { {\boldsymbol{\alpha} } } )} \left\{ \lambda_{\beta_r}(X); { {\boldsymbol{\beta} } } \right\} $ through any sensitivity analysis method from a simulation with only fast reactions. For example, when running the fast-only simulation (under $\widetilde{Q}^{(\overline{X})}( { {\boldsymbol{\alpha} } } )$) for equilibration in Algorithm \ref{alg:TTS-SSA}, one can compute the corresponding likelihood ratio process $\widetilde{W}^{(\overline{X})}_{\alpha_i}(t)$ as in \eqref{eq:W_representation} (with $t$ large enough so that $\widetilde{p}^{(\overline{X})}_t( { {\boldsymbol{\alpha} } } ) \approx \widetilde{\pi}^{(\overline{X})}( { {\boldsymbol{\alpha} } } )$). Then one can estimate the derivitives in \eqref{eq:TTS_LR}, \eqref{eq:TTS_W-rep} by \begin{align} \begin{aligned} \label{eq:micro_sensitivity}
\frac{\partial}{\partial \alpha_i}
\overline{f}(\overline{X}; { {\boldsymbol{\alpha} } } )
&
\approx \mathbb{E}_{\widetilde{p}_t^{(\overline{X})} } \left\{ f( \widetilde{X}(t)) \widetilde{W}_{\alpha_i}(t) \right\} \\
\frac{\partial}{\partial \alpha_i}
\overline{\lambda}_{\beta_r}(\overline{X}; {\boldsymbol{\theta}})
&
\approx \mathbb{E}_{\widetilde{p}_t^{(\overline{X})} } \left\{ \lambda_{\beta_r}( \widetilde{X}(t)) \widetilde{W}_{\alpha_i}(t) \right\} , \end{aligned} \end{align} using the proposed CELR method \eqref{eq:CELR-defn} during the micro-equilibration computation. Plugging these estimated values into \eqref{eq:TTS_W-rep} allows one to calculate $\overline{W}_{\alpha_i}$ for each macro-trajectory, which in turn allows for sensitivity estimation with respect to $\alpha_i$ in \eqref{eq:TTS_LR}.
We note that our derivation leads to a different form of the multi-scale LR estimator compared with Ref. \citen{nunez_steady_2015}. The latter estimated the reweighting measures for the exact process ${W^\varepsilon}_{\alpha_i}(t)$ by adding together the micro reweighting measures $\widetilde{W}_{\alpha_i}$ from within each fast class visited, the idea being that $\widetilde{W}_{\alpha_i}(t)$ is a zero-mean martingale which adds no new information and only increases
in variance once the fast-only process has converged to steady-state. Henceforth, we refer to this approach as the ``Truncated Likelihood Ratio'', as it approximates the exact reweighting coefficient $W^\varepsilon(t)$ via a truncated observation within each fast-class. Conversely, the TTS Likelihood Ratio uses the exact representation \eqref{eq:TTS_W-rep} for the macro process, and then estimates the terms via \eqref{eq:micro_sensitivity}
Lastly, we note that the above procedure will estimate sensitivities with respect to the parameter set ${\boldsymbol{\theta}}=[ { {\boldsymbol{\alpha} } } , { {\boldsymbol{\beta} } } ]$. However, the original goal was to estimate sensitivities with respect to ${\boldsymbol{\theta}}^\varepsilon= [ { {\boldsymbol{\alpha} } } /\varepsilon, { {\boldsymbol{\beta} } } ] = [ { {\boldsymbol{\alpha} } } ^\varepsilon, { {\boldsymbol{\beta} } } ] $. The sensitivities of the slow parameters $ { {\boldsymbol{\beta} } } $ are the same, but the TTS Sensitivity scheme computes fast sensitivities against the rescaled parameter $\alpha_i$ rather than against the original parameter $\alpha^\varepsilon_i = \alpha_i / \varepsilon$. However, it can be shown (Appendix \ref{sec:analytic_SS}) that at steady-state we have \begin{equation}
\frac{\partial}{\partial \alpha^\varepsilon_i}
\mathbb{E}_{\pi^\varepsilon} \left\{ f(X;{\boldsymbol{\theta}}^\varepsilon) \right\}
= \varepsilon \frac{\partial}{\partial \alpha_i}
\mathbb{E}_{\pi^\varepsilon} \left\{ f(X;{\boldsymbol{\theta}}^\varepsilon) \right\} .
\label{eq:rescaled_sensitivities} \end{equation} Therefore, by multiplying the TTS sensitivity estimate (against $\alpha_i$) by a factor of $\varepsilon$, one thus obtains the estimate against the original parameter $\alpha^\varepsilon_i$. Thus one can use the TTS scheme to estimate the full gradient of $\nabla_{{\boldsymbol{\theta}}^\varepsilon} \mathbb{E}_{\pi^\varepsilon} \left\{ f(X; {\boldsymbol{\theta}}^\varepsilon ) \right\}$.
\section{Batch-Means Stopping Rule} \label{sec:BMstop} A crucial question when implementing a TTS simulation is: How long to run the micro-equilibration for? That is, how large a value of $t$ does one use to compute the ergodic averages \begin{align*}
\overline{f}\left( \overline{X};{\boldsymbol{\theta}} \right)
& \approx
\frac{1}{t} \int_0^t
f\left( \widetilde{X}^{(\overline{X})}(s);{\boldsymbol{\theta}} \right)ds \\
\overline{\lambda}_\beta \left( \overline{X}; {\boldsymbol{\theta}} \right)
& \approx
\frac{1}{t} \int_0^t
\lambda_\beta \left( \widetilde{X}^{(\overline{X})} (s) ; { {\boldsymbol{\beta} } } \right) ds \end{align*} for a desired level of accuracy $\delta$? Taking too small a value for $t$ risks imposing a large bias. However, the $O(1/t)$ rate of convergence for the ergodic average implies almost nothing is gained by integrating $t$ past the relaxation time of the system. Furthermore, when computing the micro-sensitivities one uses the micro-reweighting process $\widetilde{W}_{\alpha_i}(t)$ by \begin{align*}
\frac{\partial}{ \partial \alpha_i}
\overline{f}(\overline{X}; { {\boldsymbol{\alpha} } } )
& \approx
\mathbb{E}_{\widetilde{p}_t^{(\overline{X})}( { {\boldsymbol{\alpha} } } ) }
\left\{ f\left( \widetilde{X}(t) \right)
\widetilde{W}_{\alpha_i}(t) \right\} \\
\frac{\partial}{\partial \alpha_i}
\overline{\lambda}_\beta
\left( \overline{X} ; {\boldsymbol{\theta}} \right)
& \approx
\mathbb{E}_{\widetilde{p}_t^{(\overline{X})} ( { {\boldsymbol{\alpha} } } )}
\left\{ \lambda_\beta \left( \widetilde{X}(t) ; { {\boldsymbol{\beta} } } \right)
\widetilde{W}_{\alpha_i}(t) \right\} , \end{align*} where the variance of $\widetilde{W}_{\alpha_i}(t)$ increases with the time-horizon $t$. Thus, we would ideally take the smallest value of $t$ such that
$\| \widetilde{p}^{(\overline{X})}_t
- \widetilde{\pi}^{(\overline{X})} \| \le O(\delta) $. However, different fast-classes $\overline{X}$ can have vastly different sizes.
This can result in significantly different relaxation times for each class. It is then ideal to have an { \em adaptive stopping rule } which terminates the micro (fast-only) simulations when the ergodic averages have converged to the steady state mean.
Current implementations of an ``averaged'' or ``multi-scale'' SSA use a constant relaxation time $t_f$ for the micro-averaging step\cite{e_nested_2007, nunez_steady_2015} whose choice is motivated by some a priori insight into the system. In Refs.\citen{cao_multiscale_2005, cao_slow-scale_2005} the authors also use a fixed time $t$, but then exploit algebraic relations of the steady-state means to try to obtain better approximations. In Ref. \citen{samant_overcoming_2005}, a stopping rule is developed which determines that equilibrium is reached when the averaged values of the propensities of the forward and backward reactions are approximately equal for each reaction pair. However, experience has shown this ``partial-equilibrium'' stopping rule can stop prematurely (in the transient regime) with significant probability for systems with relatively few reaction-pairs. Thus, we seek to obtain a robust, adaptive stopping rule for terminating the micro-equilibration simulation.
\subsection{Batch-Means for Steady-State Estimation}
The problem at hand is really one about Markov chain mixing-times
and the integrated autocorrelation time $\tau_{int}$.
Analytically, the mixing and integrated autocorrelation times are related to the spectral gap of the underlying generator \cite{levin_markov_2009, geyer_practical_1992, kipnis_central_1986}. Unfortunately, for large systems direct computation is usually infeasible. Some common approaches involve estimation autocorrelation function $A(t)$ of the process and then exploiting the relation $\tau_{int} = 2\int_0^\infty A(t)dt$ to derive estimates of $\tau_{int}$ from the estimates of $A(t)$ \cite{geyer_practical_1992, berg_introduction_2004, sokal_monte_1997}. However, if the goal is to terminate the simulation when the ergodic average has converged appropriately, then these methods are indirect and can be computationally intensive. Another common approach is to exploit the regenerative structure of Markov chains
\cite{meyn_markov_2009} to obtain independent and identically distributed samples of the process and obtain confidence bounds on the ergodic average. However, these methods can be inefficient for complex systems where the return time to the initial state can be quite large.
We instead turn to the method of batch means \cite{asmussen_stochastic_2007, alexopoulos_implementing_1996} for determining confidence bounds (and thus a measure of convergence) for the steady-state estimation problem inside each fast-class. The use of batch means is applicable to a wide range of problems (any which satisfy a central limit theorem), and its implementation is very straightforward. For a general Markov chain $X(s)$ with an observable function $f$, write $Y(s)=f(X(s))$ and $\overline{Y}(t) = 1/t \int_0^t Y(s) ds$ . We denote $f_\pi=\mathbb{E}_{\pi}\left\{ f(X) \right\}$ for the steady-state value we wish to estimate. Then under some general conditions \cite{kipnis_central_1986} $Y(s)$ satisfies a functional Central Limit Theorem: \begin{align*}
\frac{t}{\sqrt{\varepsilon} }
\left\{ \overline{Y}(t/\varepsilon) - f_\pi \right\}_{t \ge 0}
\overset{\mathcal{D}}{\longrightarrow}
\left\{ \sigma B(t) \right\}_{t \ge 0} \end{align*} in the sense of weak convergence as $\varepsilon \to 0$.
Suppose that $t \ge N_b \tau_{relax}$, where $\tau_{relax}$ is the relaxation time of the system and $N_b$ is a number of ``batches'' (bins) to partition the trajectory into. Then the batch means \begin{align*}
\overline{Y}_k(t) \defd
\frac{1}{t/N_b} \int_{(k-1) t/N_b}^{k t/N_b} Y(s) ds \end{align*} are approximately (as $t \to \infty$) independent and identically distributed samples of $\mathcal{N} \left( f_\pi, \sigma^2 N_b/t \right)$. Thus \begin{align*}
\sqrt{N_b} \frac{\overline{Y}(t) - f_\pi}{s_{N_b}(t)}
\overset{\mathcal{D}}{\longrightarrow}
T_{N_b -1} \end{align*} as $t \to \infty$, where $T_{N_b -1}$ is the Student's $t$-distribution and $ s^2_{N_b}(t)$ is the sample variance among batches, \begin{align*}
s_{N_b}^2(t) \defd
\frac{1}{N_b -1} \sum_{k=1}^{N_b}
\left[ \overline{Y}_k(t) - \overline{Y}(t) \right]^2 . \end{align*} Thus, for $t$ sufficiently large, a $(1-\delta_{CI})100\%$ confidence interval for the value of $f_\pi$ is given by $\overline{Y}(t) \ \pm \ MOE \left( t,N_b, \delta_{CI} \right) $, where \begin{multline}
\label{eq:batch-means_CI}
MOE \left( t,N_b, \delta_{CI} \right) \defd
\left( t_{quantile} \right)
\frac{s_{N_b}(t)}{\sqrt{N_b}} \\ \end{multline} and $t_{quantile}$ is the $(1-\delta_{CI}/2)$th quantile of the Student's $t$-distribution with $N_b -1$ degrees of freedom.
The usual perspective for applying batch means is that one has a fixed set of data $\left\{ Y(s): s \in [0,t] \right\}$ to partition, and then must choose the number of batches $N_b$ appropriately so that each batch length $t/N_b$ is long enough so that the batch mean errors $\left[ \overline{Y}_k(t) - \overline{Y}(t) \right]$ are approximately independent, identically distributed, and Gaussian. One then often chooses $N_b$ to be relatively small (say, 5 to 30) \cite{geyer_practical_1992,glynn_simulation_1990} to ensure the independent and Gaussian assumptions hold. When viewing the asymptotic structure as the amount of data $t$ grows, then one can ensure that the asymptotic central limit theorem holds if the number of batches grows as $N_b(t) \simeq \sqrt{t}$. In Ref. \citen{alexopoulos_implementing_1996}, the authors consider strategies which let the number of batches grow if the correlation between batches is near 0, and otherwise hold $N_b(t)$ fixed until the batch correlation decays to 0.
Since our goal is to simulate only enough (micro-scale) data so as to determine the steady-state values $f_\pi$, we instead take the perspective that one has a fixed number of batches $N_b$ desired, and that one should generate data $\left\{ Y(s) : s \in [0,t] \right\} $ until each of the batch means $\overline{Y}_k(t)$ are (approximately) independent and identically distributed about $f_\pi$. For a fixed level of precision $\delta_{precise}$, confidence level $\delta_{CI}$, and the number of batches (independent samples) $N_b$, the Batch-Means Stopping Rule terminates the simulation when $MOE(t)=MOE(t,N_b, \delta_{CI}) \le \delta_{precise}$, where $MOE(t)$ is defined by \eqref{eq:batch-means_CI}. Figure \ref{fig:Batching_Diagram} gives a depiction of how the Batch-Means Stopping Rule is implemented.
\begin{figure}
\caption{A sketch of the batch-means stopping rule.
The process is simulated for a fixed number of jumps ($N_J=20$) to a terminal
time $t_1$,
and then the trajectory is partitioned into $N_b=4$ batches to compute
the variance between the batch means $\overline{Y}_k(t_1)$.
If the confidence bounds are precise enough
($MOE(t_1)\le \delta_{precise}$), then the simulation is terminated
and each batch gives an iid sample of
$\mathcal{N}\left( f_\pi, \sigma^2 \right)$.
Otherwise, another $N_J=20$ jumps are simulated and the process is repeated.}
\label{fig:Batching_Diagram}
\end{figure}
In addition to giving an on-line estimate of the relaxation time of the system, the Batch-Means Stopping Rule gives $N_b-1$ (nearly) independent samples of trajectories with initial distribution approximately equal to the stationary distribution. Furthermore, one can compute the reweighting coefficients $W_{k,\theta}(t)$ in each batch to give $N_b$ (nearly) independent samples of the steady-state reweighting coefficients (in a manner similar to the ``Time-Averaged Correlation Function'' method of Ref. \citen{warren_steady-state_2012}).
\subsection{Batch-Means Stopping Implementation} \label{sec:BMSI}
Suppose we have a general reaction network with $M_r$ reactions and $M_\theta$ reaction parameters. Here we allow the possibility that $M_r \neq M_\theta$ for general propensity functions $\lambda_r(x;{\boldsymbol{\theta}})$ (e.g. Michaelis-Menten kinetics), whose parameter derivatives $\partial_{\theta_i} \lambda_r(x;{\boldsymbol{\theta}})$ we can compute explicitly for all $i=1,\dots,M_\theta$ and all $r=1, \dots, M_r$. Denote by $\zeta_r$ the stoichiometric vector for the $r$th reaction. Our goal is to estimate the gradient $\nabla_\theta \mathbb{E}_{\pi({\boldsymbol{\theta}})}\left\{ f(X) \right\}$ for some observable function $f$.
We introduce the following notation for the batch-means stopping rule. $\hat{X}(n)$ is the $n$th state of the reaction network, $T(n)$ is the time of the $n$th jump and $\hat{f}(n)=f(\hat{X}(n))$ for the value of the observable at the $n$th state. $F(n)= \int_0^{T(n)} f\left( X(s) \right)ds$ is the time-integrated value of $f$ up to the $n$th jump, $r^*(n)$ is the reaction which fires at jump $n$ (taking the system from $\hat{X}(n)$ to $\hat{X}(n+1)$), $\hat{e}_i$ is the vector in $\mathbb{R}^{1 \times M_\theta}$ with $1$ in the $i$th component and zeros elsewhere. $R(n, {\boldsymbol{\theta}}) \in \mathbb{R}^{1 \times M_\theta}$ and $B(n,{\boldsymbol{\theta}}) \in \mathbb{R}^{1 \times M_\theta}$ are the first and second terms of \eqref{eq:W_representation} with respect to each of the parameters $\theta_i \in {\boldsymbol{\theta}}$ ($i=1, \dots, M_\theta$). $N_b$ is the number of batches (approximately independent samples) to be used, $\delta_{CI}$ is the desired confidence level (for a $(1-\delta_{CI})100 \%$ confidence interval, and $\delta_{precise}$ is the maximum allowed radius of the confidence interval at the stopping time. $N_J$ is the number of jumps to simulate before retesting the batches for convergence. Then one can write the batch-means stopping rule as follows.
\begin{algorithm}[Batch-Means Stopping Rule with Sensitivity Estimation]
\label{alg:batch-means} \rm ~\\
\begin{enumerate}[\bf (1)]
\item {\bf Initialize }
\begin{itemize}
\item $\hat{X}(0)= x_0$,
$\hat{f}(0) = f\left( \hat{X}(0) \right)$,
$T(0)=0$,
$F(0)=0$,
$R(0)= [0,\dots,0] \in \mathbb{R}^{1 \times M_\theta}$,
$B(0)= [0, \dots, 0] \in \mathbb{R}^{1 \times M_\theta}$.
${ tests}=0$ (number of times the data has been tested
for convergence).
Calculate $t_{quantile}=(1-\delta_{CI}/2)$ quantile of a
Student's $t$-distribution with $N_b-1$ degrees of freedom.
\end{itemize}
\item {\bf Generate and Record Data}
Simulate $N_J$ jumps and record values immediately after each jump.
For $n=N_J \cdot {tests}, \dots, N_J \cdot ( {tests}+1) -1$,
\begin{itemize}
\item Compute $\lambda_r(\hat{X}(n);{\boldsymbol{\theta}})$,
$\frac{\partial}{\partial \theta_i} \lambda_r(\hat{X}(n), {\boldsymbol{\theta}})$
for all $r=1, \dots, M_r$ and all $\theta_i =1, \dots M_\theta$.
Set $\lambda_0(\hat{X}(n); {\boldsymbol{\theta}})
= \sum_{r=1}^{M_r} \lambda_r (\hat{X}(n), {\boldsymbol{\theta}})$, and
$\frac{\partial}{\partial \theta_i} \lambda_0(\hat{X}(n), {\boldsymbol{\theta}})
= \sum_{r=1}^{M_r} \frac{\partial}{\partial \theta_i}
\lambda_r(\hat{X}(n), {\boldsymbol{\theta}})$ for all $\theta_i \in {\boldsymbol{\theta}}$.
\item Sample $\Delta t(n) \sim \mathcal{E}{{xp}}\left(
\lambda_0(\hat{X}(n),{\boldsymbol{\theta}}) \right)$, and
$r^*(n) \sim
\frac{1}{\lambda_0(\hat{X}(n), {\boldsymbol{\theta}})}$ \\
\qquad $\qquad \times
\left[ \lambda_1(\hat{X}(n);{\boldsymbol{\theta}}), \dots,
\lambda_{M_r}(\hat{X}(n);{\boldsymbol{\theta}}) \right] $.
\item Update
$T(n+1)= T(n) + \Delta t(n) $ \\
$\hat{X}(n+1)= \hat{X}(n)+ \zeta_{r^*(n)}$\\
$\hat{f}(n+1)= f(\hat{X}(n+1))$ \\
$F(n+1) = F(n) + \hat{f}(n) \cdot \Delta t(n)$ \\
$R(n+1,{\boldsymbol{\theta}})= R(n,{\boldsymbol{\theta}})
+ \sum_{i=1}^{M_\theta}
\hat{e}_i \\ \qquad \times \
\left[ \frac{\partial}{\partial \theta_i}
\lambda_{r^*(n)}\left( \hat{X}(n), {\boldsymbol{\theta}} \right) \right]
/ \lambda_{r^*(n)}\left( \hat{X}(n), {\boldsymbol{\theta}} \right)$ \\
$\hat{b}(n, {\boldsymbol{\theta}})=
\sum_{i=1}^{M_\theta} \hat{e}_i \cdot
\left[ \sum_{r=1}^{M_r} \frac{\partial}{\partial \theta_i}
\lambda_r (\hat{X}(n), {\boldsymbol{\theta}}) \right] $ \\
$B(n+1,{\boldsymbol{\theta}}) = B(n, {\boldsymbol{\theta}}) + \hat{b}(n, {\boldsymbol{\theta}}) \cdot
\Delta t(n)$ \\
$W(n+1,{\boldsymbol{\theta}}) = R(n+1, {\boldsymbol{\theta}}) - B(n+1, {\boldsymbol{\theta}}) $.
\end{itemize}
\item{ \bf Test Batches for Convergence}
\begin{itemize}
\item $N_{end} = N_J \cdot ({tests}+1)$= index of last available
data point. \\
$\bar{Y}= F\left( N_{end} \right)/ T(N_{end})$ = total time-averaged value \\
$t_{batch}=T(N_{end})/N_b$= time length each of batch \\
Initialize $F^B=0$ (total integral through end of previous batch).
\item for $ k=1, \dots, N_b$
\begin{itemize}
\item $ind_B(k)$= $\max \left\{ n :
T(n) \le k\cdot t_{batch} \right\}$=index of the last jump
in batch $k$.
\item $F_A^B(k) = F(ind_B(k)) + \hat{f}(ind_B(k)) $ \\
$ \times \left[ k\cdot t_{batch} - T(ind_B(k)) \right]
- F^B$=total integrated value of $f(X(s))$ inside batch $k$.
\item $\overline{Y}_k =F_A^B(k)/t_{batch}$= $k$th batch-mean
\item $F^B \leftarrow F^B + F_A^B(k) $
(update integral to end of previous batch)
\end{itemize}
\item $s_{N_b}^2= \frac{1}{N_b-1 } \sum_{k=1}^{N_b}
\left[ \overline{Y}_k - \overline{Y} \right]^2$
= variance between batches
\item $MOE= t_{quantile} * \sqrt{ s^2_{N_b} / N_b}$
= margin of error for confidence interval
\item If $MOE \le \delta_{precise}$, then go to {\bf (4)}.
Else, ${tests} \leftarrow {tests}+1$ and go back to {\bf (2)}.
\end{itemize}
\item{ \bf Compute LR Weights} in each batch.
Initialize $W^B({\boldsymbol{\theta}})=[0,\dots,0] \in \mathbb{R}^{1 \times M_\theta}$.
For $ k=1, \dots, N_b$,
\begin{itemize}
\item $W_A^B(k, {\boldsymbol{\theta}})
= W(ind_B(k), {\boldsymbol{\theta}})$ \\ $
- \hat{b}(ind_B(k), {\boldsymbol{\theta}})
\cdot \left[ k\cdot t_{batch} - T(ind_B(k)) \right] $ \\ $
- W^B({\boldsymbol{\theta}}) $
\item $W^B({\boldsymbol{\theta}}) \leftarrow W^B({\boldsymbol{\theta}}) + W_A^B(k,{\boldsymbol{\theta}}) $
\end{itemize}
\item{\bf Compute Sensitivity Estimates}
for $\nabla_{\boldsymbol{\theta}} \mathbb{E}_{\pi({\boldsymbol{\theta}})} \left\{ f(X) \right\}$:
\begin{itemize}
\item{Likelihood Ratio}
\begin{align*}
{\bf LR} = \frac{1}{N_b}
\sum_{k=1}^{N_b} \hat{f}\left( ind_B(k) \right)
W_A^B(k, {\boldsymbol{\theta}})
\end{align*}
\item{Centered Likelihood Ratio}
\begin{multline*}
{\bf CLR} = {\bf LR} \\
- \left[ \frac{1}{N_b} \sum_{k=1}^{N_b}
\hat{f}\left( ind_B(k) \right)\right]
\cdot \left[ \frac{1}{N_b} \sum_{k=1}^{N_b}
W_A^B(k, {\boldsymbol{\theta}}) \right]
\end{multline*} \item{Ergodic Likelihood Ratio} \begin{align*} {\bf ELR} = \frac{1}{N_b} \sum_{k=1}^{N_b} \overline{Y}_k W_A^B(k,{\boldsymbol{\theta}})
\end{align*}
\item{Centered Ergodic Likelihood Ratio}
\begin{align*}
{\bf CELR} = {\bf ELR}
- \left[ \frac{1}{N_b} \sum_{k=1}^{N_b}
\overline{Y}_k\right] \cdot
\left[ \frac{1}{N_b} \sum_{k=1}^{N_b}
W_A^B(k,{\boldsymbol{\theta}}) \right]
\end{align*}
\end{itemize}
\end{enumerate}
\end{algorithm}
\section{Simulation Results} \label{sec:sim_results} Here we present numerical results to display the performance of the proposed algorithms. In what follows, we compare the output of an exact simulation at the single time scale (STS) to the accelerated two time scale (TTS) approximation. From \eqref{eq:TTS_sens_estimate},
we expect the differences in observable averages and their derivatives to be $O(\varepsilon)$. Because differences are small, we use a simple test system for which many samples can be run to obtain accurate statistics.
Consider a reaction network with species A, B, and C and isomerization reactions given by \[
\left.
\begin{array}{rcl}
A \overset{k_1 / \varepsilon}{\rightarrow} B , & \quad
B \overset{k_2 / \varepsilon}{\rightarrow} A , & \quad
B \overset{k_3}{\rightarrow} C
\end{array}
\right. \] For small values of $\varepsilon$, the system becomes stiff as the isomerization between A and B reaches equilibrium much faster than B is converted to C. A TTS approximation assumes that $A\rightarrow B$ and $B\rightarrow A$ are fast and equilibrated.
{We first compare the output of the accelerated TTS simulation against the exact STS simulation for varying levels of stiffness $\varepsilon$}. For our simulations, initial conditions of $(A_0,B_0,C_0)=(100,0,0)$ and the parameters $(k_1,k_2,k_3)=(1,1.5,2)$ are chosen. 10,000 replicate (independent) trajectories are run for various values of $\varepsilon$. Statistics are taken at a termination time of $t=0.5$s. Species averages are calculated as arithmetic averages over the independent trajectories while sensitivities are computed with the CLR method shown in \eqref{eq:CLR-defn}. The error due to statistical averaging is estimated using t-test statistics for averages and a bootstrapping method for sensitivities. Sensitivities with respect to the ``slow'' parameter $k_3$ are displayed for each species.
As discussed in previous works\cite{nunez_steady_2015}, sensitivities with respect to parameters related to fast reactions encounter significant noise, and thus we omit them in order to clearly observe
the difference between STS and TTS. Figure \ref{fig:STS_eps_converge} shows the disparity between the STS and
TTS systems for various values of $\varepsilon$. Errors are normalized by the TTS value such that $\mathrm{Error}=\frac{\mathrm{STS}-\mathrm{TTS}}{|\mathrm{TTS}|}$. {Indeed, one observes the difference is proportional to $\varepsilon$, as expected from Corollary~\ref{cor:expectation-error} and \eqref{eq:TTS_sens_estimate}. }
\begin{figure}
\caption{Normalized error of the two-time-scale (TTS) (accelerated, approximate) simulation from the
exact single-time-scale (STS) simulation.
The plot (a) shows errors in species averages while the
plot (b) shows errors in sensitivities of each species with
respect to $k_3$.
{Points indicate simulation statistics while dashed lines confirm the linear trend.}
The error bars are 95\% confidence intervals of statistical noise.
}
\label{fig:STS_eps_converge}
\end{figure}
\begin{figure}
\caption{Time evolution of normalized errors obtained for sensitivities computed by CLR and CELR estimators.
Simulation estimates are referenced and normalized by the analytical solution at each time point to compute the error.
The graphs (a), (b), and (c) show estimates of different sensitivity indices for the species B.
{One observes that the variance of the CLR estimates
increases with time, making for inefficient estimation.
In contrast, the CELR estimates (the solid red line) converge quickly to the
steady state sensitivities with variance which is roughly constant with time,
allowing for efficient steady-state sensitivity estimation.}
}
\label{fig:TTS_ergodicSAerror}
\end{figure}
Next, the CLR and CELR methods from Section \ref{sec:LRmethods} are tested in performing sensitivity analysis in a TTS system. The reaction network described in Section \ref{sec:TTSRN} is simulated using Algorithm \ref{alg:TTS-SSA}. To assess convergence of the microscale distribution, the batch-means stopping criterion described in Section \ref{sec:BMstop} is used with a tolerance of $\delta=0.05$. 1000 replicate trajectories are run to a time horizon of $t_f=100$s. The initial conditions used are $(A_0,B_0,*_0)=(30,60,10)$ and the parameters used are $(\alpha^\varepsilon_1, \alpha^\varepsilon_2, \beta_1,\beta_2, \beta_3) = (1/\varepsilon,1.5/\varepsilon, 2, 1, 0.4)$.
Species populations, time-averaged species populations, and trajectory derivatives are recorded for each run over
time. Using these recorded statistics, the sensitivities for all 15 (3 species and 5 parameters)
species/parameter combinations are computed at each time point. Figure \ref{fig:TTS_ergodicSAerror}
shows the time evolution of the normalized errors in sensitivity estimates {of the species B} over time. Estimated values from simulation are referenced
to the analytical answer as computed from a differential algebraic equation
(see Appendix \ref{sec:AnalylicSolnExample}) and normalized by that amount so that $\mathrm{Error} = \frac{\mathrm{estimated} - \mathrm{analytical}}{|\mathrm{analytical}|}$.
As expected, CLR estimates are noisy, with variance that grows linearly with time.
At short times, the variance is small enough to obtain reasonable estimates. As time increases, the noise becomes significant with respect to the actual values (the magnitude of the normalized error becomes comparable to 1). In contrast, the ergodic likelihood ratio (CELR) fails at short times with a noticeable bias. However, the bias, which exists due to a relaxation period, decays as $O(1/t)$ when time increases and the system approaches its steady-state. The variance of the CELR estimates remain constant because the variance of trajectory derivatives increases linearly in time while the variance of ergodic species averages is proportional to $1/t$. At long times (where the CLR is too noisy for efficient estimation), the ergodic likelihood ratio obtains accurate estimates with very low variance. Therefore, it is advisable to use the CLR method for short times (in the transient regime) and the CELR method for long times to obtain steady state values.
{Table \ref{tab:table1} shows the error and statistical noise of the CLR and CELR estimations of sensitivities of the species B at a short ($t=1.3$s) and long ($t=100$s) times.
Statistical noise is obtained from bootstrapping the samples used to compute the sensitivity estimates. At $t=1.3$, the CLR method has low error (theoretically, there is no bias) as well as low variance. The CELR estimator has a similarly low variance, but high error (due to the $O(1/t) $ bias). At $t=100$s, the CLR estimates have much higher variance which induces large empirical error. In contrast, the bias of the CELR estimate decreases in time while the variance remains low, providing very small empirical errors at large times.
}
\begin{table}[h!]
\begin{center}
\caption{Comparison of the error and statistical noise for the CLR and CELR estimators
at a short time (during the transient regime) and at a long time (corresponding to steady state).
Values in the table refer to the sensitivity of the species B with respect to the parameter given by the row label. Values are reported as a percent of the analytically obtained sensitivity value.}
\label{tab:table1}
\begin{tabular}{ccc|cc}
\hline
& \multicolumn{4}{c}{Percent Error} \\
\hline
& CLR & CELR & CLR & CELR\\
& \multicolumn{2}{c|}{$t=1.3$s} & \multicolumn{2}{c}{$t=100$s} \\
\hline
$\alpha_1$ & -1.4 & -40.0 & -83.5 & 0.0 \\
$\alpha_2$ & -3.0 & -40.3 & -63.0 & -1.1 \\
$\beta_1$ & 0.2 & -39.1 & -64.1 & 1.2 \\
$\beta_2$ & -1.1 & -16.7 & -22.6 & -2.8 \\
$\beta_3$ & -1.4 & -23.3 & 20.6 & -0.1 \\
\hline
& \multicolumn{4}{c}{Half-length of 95\%} \\
& \multicolumn{4}{c}{Confidence Interval} \\
\hline
$\alpha_1$ & 12 & 8 & 99 & 11 \\
$\alpha_2$ & 12 & 7 & 94 & 11 \\
$\beta_1$ & 12 & 7 & 91 & 11 \\
$\beta_2$ & 13 & 10 & 108 & 12 \\
$\beta_3$ & 21 & 14 & 171 & 18
\end{tabular}
\end{center} \end{table}
\section{Conclusions} \label{sec:conclusion} This work develops a Two-Time-Scale (TTS) framework for multiscale reaction networks. By decomposing the system into ``fast-classes'', one can approximate the behavior of the multiscale system by a lower-dimensional, single-scale, ``macro-averaged'' reaction network. By applying a singular perturbation expansion of the underlying probability measures, we have established rigorous bounds on the bias induced by the approximate macro-averaged model. We then proposed a TTS algorithm for simulating the macro reaction network, using an adaptive batch-means stopping rule for determining when the micro-scale dynamics have sufficiently equilibriated.
In addition, we have shown that the sensitivities of the macro-averaged system provide accurate approximations for the multiscale system. Since the macro-averaged system is single-scale, it is possible to incorporate most existing sensitivity estimation methods to the TTS algorithm to obtain estimates of the system sensitivities. We proposed an Ergodic Likelihood Ratio estimator for steady-state sensitivity analysis, and demonstrated how it can be adapted to the Two-Time-Scale algorithm. A simulation was then used to confirm the analytic error bounds and demonstrate the efficiency of the TTS Ergodic Likelihood Ratio estimator.
\section{Analytic Stationary Sensitivities} \label{sec:analytic_SS} For ergodic systems whose state space is relatively small (such that the generator $Q({\boldsymbol{\theta}})$ can be explicitly constructed), one can compute the steady-state probability vector $\pi({\boldsymbol{\theta}})$ by solving the linear system $\pi \cdot Q =\boldsymbol{0}$ and $\pi \cdot \boldsymbol{1} = 1$. Here, we show that one can express the sensitivities of the steady-state measure $\frac{\partial}{\partial \theta} \pi({\boldsymbol{\theta}})$ explicitly as a linear transformation of the nominal measure $\pi({\boldsymbol{\theta}})$. The representation we shall construct is an adaptation of the discrete-time technique in Ref. \citen{funderlic_sensitivity_1986} to continuous-time Markov chains, and exploits the algebraic properties of the pseudo-inverse $Q^+$ of $Q$.
By differentiating $\pi({\boldsymbol{\theta}}) \cdot Q({\boldsymbol{\theta}}) = \boldsymbol{0}$,
we have the relation \[ \frac{\partial \pi}{\partial \theta} Q = -\pi \frac{\partial Q}{\partial \theta} \] Then by expanding with the Moore-Penrose pseudo-inverse, we have \[
\left( \frac{\partial \pi}{\partial \theta} \right)'
= \left( Q \right)'\left( \frac{-\partial Q}{\partial \theta} \right)' \pi'
+\left[ I - \left( Q' \right)^{+} Q' \right] w \] for some vector $w$.
Now, we see by the projection property of $(Q')^+Q' = (Q Q^+)' = Q Q^+$ that the operator $I-QQ^+$ is the projection operator onto the kernel of $Q'$ = span of $\pi'$, so that $(I-QQ^+)w = \gamma \pi'$ for some scalar $\gamma$. Thus we have the relation \[
\frac{\partial \pi}{\partial \theta}
= \pi \left( \frac{-\partial Q}{ \partial \theta} \right)
Q^+ + \gamma \pi \] for some $\gamma \in \mathbb{R}$. It remains to determine $\gamma$ to have a method of relating the sensitivity coefficient to a linear transformation of $\pi$.
Now, we can see that $\pi(\theta) \cdot \boldsymbol{1} = 1$, so that $\frac{\partial \pi}{ \partial \theta} \cdot \boldsymbol{1} = 0$. Thus, we have \begin{align*}
0 = \frac{\partial \pi}{\partial \theta} \cdot \boldsymbol{1}
= \pi \left( \frac{- \partial Q}{ \partial \theta} \right) Q^+
\boldsymbol{1} + \gamma \pi \boldsymbol{1} \end{align*} so that \[
\gamma = \gamma \pi \boldsymbol{1} = \pi \frac{\partial Q}{ \partial \theta}
Q^+ \boldsymbol{1} . \] Putting the above together, we can write $\frac{\partial \pi}{ \partial \theta} $ as \begin{equation}
\label{eq:pseudo-inverse_sensitivity}
\frac{\partial \pi}{ \partial \theta} =
\pi \left( \frac{\partial Q}{ \partial \theta} \right) Q^+
\left[ \boldsymbol{1} \pi - I \right]. \end{equation}
For reaction networks with relatively small state space $\mathcal{M}$, \eqref{eq:pseudo-inverse_sensitivity} can provide a tractable method of computing the sensitivities of the steady-state measure $\frac{\partial \pi}{\partial \theta}$ and thus of the steady-state expected value \begin{multline}
\label{eq:analytic_SS_sens}
\frac{\partial}{\partial \theta}
\mathbb{E}_{\pi({\boldsymbol{\theta}})} \left\{ f(X ; {\boldsymbol{\theta}} ) \right\} \\
= \sum_{x \in \mathcal{M}} \frac{\partial f(x;{\boldsymbol{\theta}})}{\partial \theta} \pi(x; {\boldsymbol{\theta}})
+ \sum_{x \in \mathcal{M}} f(x;{\boldsymbol{\theta}}) \frac{\partial \pi(x;{\boldsymbol{\theta}})}{\partial \theta} . \end{multline} For example, for reaction networks with mass-action propensities, the entries of $Q$ shall always be linear in the parameters $\theta$. Therefore, the derivatives $\frac{\partial Q}{\partial \theta}$ are easy to compute (whereas the stationary distribution $\pi({\boldsymbol{\theta}})$ can be quite complex as a function of $\theta$). Thus by computing the pseudo-inverse $Q^+$ from $Q$ at the nominal value of ${\boldsymbol{\theta}}$ (e.g., via singular value decomposition), one can analytically compute the system sensitivities without need of Monte Carlo simulation. For larger reaction networks, it may be possible to combine the Finite State Projection method\cite{munsky_finite_2006,sunkara_optimal_2010} with this pseudo-inverse technique to estimate the exact sensitivities by computing the analytic sensitivities of the reduced system.
Lastly, we use this representation to show that \eqref{eq:rescaled_sensitivities} holds. Write the exact generator $Q^\varepsilon$ as $Q^\varepsilon = Q({\boldsymbol{\theta}}^\varepsilon) = (1/\varepsilon) \widetilde{Q} ( { {\boldsymbol{\alpha} } } ) + \widehat{Q}( { {\boldsymbol{\beta} } } ) = \widetilde{Q}( { {\boldsymbol{\alpha} } } ^\varepsilon) + \widehat{Q}( { {\boldsymbol{\beta} } } ) $. For a fast reaction parameter $\alpha^\varepsilon_i = \alpha_i/\varepsilon$ a chain rule gives $ { \partial_{\alpha_i} } \lambda(x; { {\boldsymbol{\alpha} } } ^\varepsilon) = \left[ {\partial_{\alpha^\varepsilon_i} } \lambda(x; { {\boldsymbol{\alpha} } } ^\varepsilon ) \right] /\varepsilon$, from which it follows (using \eqref{eq:Q-ep_decomp}) that $\varepsilon \left[ {\partial \alpha_i } \ \widetilde{Q}( { {\boldsymbol{\alpha} } } ^\varepsilon) \right]
= {\partial_{\alpha_i }} \widetilde{Q}( { {\boldsymbol{\alpha} } } )
={\partial_{\alpha^\varepsilon_i } } \widetilde{Q}( { {\boldsymbol{\alpha} } } ^\varepsilon) $. Putting this relation into \eqref{eq:pseudo-inverse_sensitivity}, we then have \begin{equation}
\label{eq:rescaled_SS_sens_pi}
\frac{\partial}{\partial \alpha^\varepsilon_i} \pi^\varepsilon
= \varepsilon \left[ \frac{\partial}{\partial \alpha_i} \pi^\varepsilon \right]. \end{equation} Similarly we see that $\partial \alpha_i \ f(x, { {\boldsymbol{\alpha} } } ^\varepsilon) = \left[ \partial_{\alpha^\varepsilon_i} \ f(x, { {\boldsymbol{\alpha} } } ^\varepsilon) \right] /\varepsilon$. Finally, putting these relations into \eqref{eq:rescaled_SS_sens_pi} we obtain \eqref{eq:rescaled_sensitivities}, \begin{align*}
\frac{\partial}{\partial \alpha^\varepsilon_i}
\mathbb{E}_{\pi^\varepsilon} \left\{ f(X; { {\boldsymbol{\alpha} } } ^\varepsilon, { {\boldsymbol{\beta} } } ) \right\}
=
\varepsilon \frac{\partial}{\partial \alpha_i}
\mathbb{E}_{\pi^\varepsilon} \left\{ f(X; { {\boldsymbol{\alpha} } } ^\varepsilon, { {\boldsymbol{\beta} } } ) \right\} . \end{align*}
\section{Proofs of Results} \label{sec:proofs}
In this section we outline the proofs on the error bounds of the averaged reaction network and convergence of the sensitivities. The error bounds in this work are largely direct applications of the results in Ref. \citen{yin_continuous-time_2013} to the Two-Time-Scale reaction networks formulated here. We present an overview of the proofs for insight and completeness. Similarly, the sensitivity convergence result comes from Ref. \citen{gupta_sensitivity_2014}; we shall only how to fit their result to the Two-Time-Scale framework.
\subsection{Proof of Theorem~\ref{thm:prob_err_bound}:} Using the formulation of the exact generator $Q^\varepsilon({\boldsymbol{\theta}}) = (1/\varepsilon) \widetilde{Q}( { {\boldsymbol{\alpha} } } ) + \widehat{Q}( { {\boldsymbol{\beta} } } )$, the error bound on the induced probability measures of the exact and averaged systems is a direct application of Theorem 4.29 of Ref. \citen{yin_continuous-time_2013}. We outline the main steps below.
Write $p^\varepsilon(t) \in \mathbb{R}^{1 \times m}$ for the probability measure of the exact system at time $t$. From the Kolmogorov forward equation (a.k.a. Chemical Master Equation), we have \begin{equation}
\frac{d p^\varepsilon(t)}{dt} = p^\varepsilon(t)
\left[ \frac{1}{\varepsilon} \widetilde{Q}( { {\boldsymbol{\alpha} } } )
+ \widehat{Q}( { {\boldsymbol{\beta} } } ) \right] .
\label{eq:CME} \end{equation} Define the differential operator $\mathcal{L}^\varepsilon$ on functions with values in $\mathbb{R}^{1 \times m}$ by $\mathcal{L}^\varepsilon f = \varepsilon \frac{d f}{dt} - f (\widetilde{Q} + \varepsilon \widehat{Q}) $. Then $\mathcal{L}^\varepsilon f =0$ if and only if $f$ solves the CME \eqref{eq:CME}. The form of the differential equation \eqref{eq:CME} suggests the plausibility of a singular perturbation expansion of $p^\varepsilon(t)$ by \begin{equation}
\label{eq:SP-expansion}
p^\varepsilon(t) =
\sum_{i=0}^\infty \varepsilon^i \phi_i(t)
+ \sum_{i=0}^\infty \varepsilon^i \psi_{i} \left( \frac{t}{\varepsilon} \right) \end{equation} Assuming for the moment that such a representation holds, we proceed to derive the form of the ``regular'' terms $\phi_i(t)$ and the ``boundary layer'' terms $\psi_i(t)$. Applying $\mathcal{L}^\varepsilon$ to \eqref{eq:SP-expansion} and equating terms of $\varepsilon$ leads to the recursive equations \begin{align}
\begin{aligned}
\label{eq:phi_recurs}
\varepsilon^0 : & \qquad \phi_0(t) \widetilde{Q} = 0 \\
\varepsilon^1 : & \qquad \phi_1(t) \widetilde{Q} =
\frac{d \phi_0(t)}{dt} - \phi_0(t) \widehat{Q} \\
\vdots & \\
\varepsilon^i : & \qquad \phi_i (t) \widetilde{Q} =
\frac{d \phi_{i-1}(t)}{dt} - \phi_{i-1}(t)\widehat{Q}
\end{aligned} \end{align} and similarly, using the ``stretched-time'' variable $\tau=t/\varepsilon$ one has equations for $\psi(\tau)$ \begin{align}
\begin{aligned}
\label{eq:psi_recurs}
\varepsilon^0 : & \qquad \frac{d \psi_0(\tau)}{d \tau} =\psi_0(\tau) \widetilde{Q} \\
\varepsilon^1 : & \qquad \frac{d \psi_1(\tau)}{d \tau}
= \psi_1(\tau) \widetilde{Q} - \psi_0(\tau) \widehat{Q} \\
\vdots & \\
\varepsilon^i : & \qquad \frac{d \psi_{i}(\tau)}{ d\tau}
=\psi_{i}(\tau) \widetilde{Q} - \psi_{i-1}(\tau) \widehat{Q}. \end{aligned} \end{align} At $t=0$, \begin{equation} \sum_{i=0}^\infty \varepsilon^i \left( \phi_i(0) + \psi_i(0) \right) = p^\varepsilon(0), \label{eq:initial_condition} \end{equation} so $\phi_0(0) + \psi_0(0) = p^\varepsilon(0)$ and $\phi_i(0) + \psi_i(0) =0 $ for all $i \ge 1$. Since $p^\varepsilon(t)$ is a probability measure with $p^\varepsilon(t) \cdot \boldsymbol{1} = 1$, by sending $\varepsilon \to 0$ in \eqref{eq:SP-expansion} it follows that \begin{align}
\label{eq:orthog_condition}
\phi_0(t) \cdot \boldsymbol{1} = 1
\quad \text{ and } \quad
\phi_i(t) \cdot \boldsymbol{1} = 0 \end{align} for all $t \in [0,T]$ and all $i \ge 1$.
Turning to the leading regular term $\phi_0(t)$, we note that $\phi_0(t) \cdot \widetilde{Q} =0$ is not uniquely solvable because $\widetilde{Q}= {\operatorname{diag} } [\widetilde{Q}^{(1)}, \dots, \widetilde{Q}^{( {N_{C}})} ]$ has rank $m- {N_{C}}$ However, writing $\phi^{(k)}_0(t)$ for the sub-vector of $\phi_0(t)$ corresponding to fast-class $\mathcal{M}_k$, then we must have $\phi^{(k)}_0(t) \cdot \widetilde{Q}^{(k)} = 0$ for all $k=1, \dots, {N_{C}}$. Since each $\widetilde{Q}^{(k)}$ is an irreducible generator, we then have $\phi^{(k)}_0(t) = \gamma^{(k)}(t) \widetilde{\pi}^{(k)}$ for some scalar multiplier $\gamma^{(k)}(t)$. It can be seen (Prop 4.24\cite{yin_continuous-time_2013}) that $\gamma^{(k)}(t) = \overline{p}_k(t)$, where $\overline{p}(t)$ is the probability measure among fast-classes $\mathcal{M}_1, \dots, \mathcal{M}_ {N_{C}}$ induced by generator $\overline{Q}$ (as in \eqref{eq:Q-bar-matrix}) and initial distribution $p^\varepsilon(0) \cdot \widetilde{\boldsymbol{1}} =\mathbb{P}\left\{ X^\varepsilon(0) \in \mathcal{M}_k \right\} $. This in turn determines a unique solution for $\phi^{(k)}_0(t)$ and therefore $\phi_0(t)$. It follows that $\phi_0(t)$ is exactly the measure $p^0_t$ in Theorem \ref{thm:prob_err_bound} induced by the TTS simulation procedure.
With $\phi_0(t)$ determined, \eqref{eq:initial_condition} then gives the initial condition $\psi_0(0) = p^\varepsilon(0) - \phi_0(0) = p^\varepsilon(0) [ I_{m} - \widetilde{\boldsymbol{1}} \widetilde{\pi} ]$, from which one can solve \eqref{eq:psi_recurs} to obtain $\psi_0(\tau) = \psi_0(0) \cdot \exp\left\{ \widetilde{Q}\tau \right\} $. It can be shown (Prop. 4.25\cite{yin_continuous-time_2013}) that \begin{equation}
\label{eq:psi_decay}
\| \psi_0(\tau) \| \le C \exp\left\{ -\widetilde{\kappa} \tau \right\} , \end{equation} where $C$ depends on the Jordan-Form of $\widetilde{Q}$. Higher order terms can also be solved for recursively (Prop. 4.26\cite{yin_continuous-time_2013}), and it can be shown (Prop. 4.28\cite{yin_continuous-time_2013}) \begin{align}
\label{eq:SP_error}
\sup_{0 \le t \le T}
\| \sum_{i=0}^n \varepsilon^i \phi_i(t) + \sum_{i=0}^n \varepsilon^i \psi_i(t/\varepsilon)
- p^\varepsilon(t) \| = O(\varepsilon^{n+1}) . \end{align} In particular, using the $0$th-order expansion we have \begin{multline}
\label{eq:SP-0_error}
\| p_T^0 - p^\varepsilon_T \| = \| \phi_0(T) - p^\varepsilon(T) \| \\
\le \| \phi_0(T) + \psi_0(T/\varepsilon) - p^\varepsilon(T) \| + \| \psi_0(T/\varepsilon) \| \\
\le \| O(\varepsilon) + O(\exp{-\widetilde{\kappa} T/\varepsilon}) \| \end{multline} \qed
\subsection{Proof of Corollary \ref{cor:expectation-error} } With the singular perturbation bound \eqref{eq:SP_error}, Corollary \ref{cor:expectation-error} follows immediately. Since the exact process $X^\varepsilon(t)$ is ergodic, there exists a time horizon $T^\varepsilon$ such that
$\| p^\varepsilon(t) - \pi^\varepsilon \| \le \varepsilon$ for $t \ge T^\varepsilon$. Similarly, by \eqref{eq:psi_decay}
there exists $\widetilde{T}$ such that $\| \psi_0(t) \| \le \varepsilon$ for $t \ge \widetilde{T}$, and
$\| \overline{p}(t) - \overline{\pi} \| \le \varepsilon$ for $t \ge \overline{T}$, implying that
$\|\phi_0(t) - \overline{\pi} \widetilde{\pi} \|
= \| \overline{p}(t) \widetilde{\pi} - \overline{\pi} \widetilde{\pi} \| \le \varepsilon$. Then taking $T \ge \max \{ T^\varepsilon, \widetilde{T}, \bar{T} \}$ and applying \eqref{eq:SP_error} , we have
\begin{multline*}
\| p^\varepsilon(t) - \overline{\pi}\widetilde{\pi} \| \le
\|p^\varepsilon (t) - \phi(t) - \psi(t/\varepsilon) \| \\
+ \| \phi(t) - \overline{\pi}\widetilde{\pi} \|
+ \| \psi(t/\varepsilon)
\| \le O \left( \varepsilon +
\exp \left\{-\widetilde{\kappa} t/\varepsilon \right\} \right) \\
\| \pi^\varepsilon - \overline{\pi}\widetilde{\pi} \|
\le \|\pi^\varepsilon -p^\varepsilon(T) \|
+ \|p^\varepsilon(T) - \overline{\pi}\widetilde{\pi} \|
\le O(\varepsilon)
\end{multline*}
and the corollary follows. \qed
\subsection{Proof of Proposition \ref{prop:weak_convergence} } This is a direct application of Theorem 5.27\cite{yin_continuous-time_2013}, and also follows from the error bound \eqref{eq:SP_error}. First, one uses \eqref{eq:SP-0_error} to establish that \begin{multline*}
\lim_{t \to 0} \lim_{\varepsilon \to 0}
\mathbb{E}\left[ \overline{X^\varepsilon}(s+t) - \overline{X^\varepsilon}(s)
\big| X^\varepsilon(s)=x^{(k)}_j \right] =0 \end{multline*} and thus $\left\{ \overline{X^\varepsilon}(\cdot) \right\}_{\varepsilon > 0}$ is tight. Then one shows that the finite-dimensional distributions converge by taking arbitrary time points $0 \le t_1 < \dots < t_n \le T$ and apply the Chapman-Kolmogorov equations to see \begin{multline*}
\mathbb{P}\left\{
\overline{X^\varepsilon}(t_n) = \overline{y}_n
, \dots, \overline{X^\varepsilon}(t_1) = \overline{y}_1 \right\} \\
= \sum_{j_1, \dots, j_n}
\mathbb{P}\left\{ \overline{X^\varepsilon}(t_n)=x^{(\overline{y}_n)}_{j_n}
\big| \overline{X^\varepsilon}(t_{n-1}) = x^{(\overline{y}_{n-1})}_{j_{n-1}} \right\}
\times \dots \\
\times
\mathbb{P}\left\{ \overline{X^\varepsilon}(t_2)=x^{(\overline{y}_2)}_{j_2}
\big| \overline{X^\varepsilon}(t_{1}) = x^{(\overline{y}_{1})}_{j_1} \right\} \\
\times \mathbb{P}\left\{ \overline{X^\varepsilon}(t_1) = x^{(\overline{y}_1)}_{j_1} \right\}. \end{multline*} Then applying the error bound \eqref{eq:SP-0_error} to each transition term to obtain \begin{multline*} \mathbb{P}\left\{ \overline{X^\varepsilon}(t_l)
= x^{(\overline{y}_l)}_{j_l} \Big| \overline{X^\varepsilon}(t_{l-1}) = x^{(\overline{y}_{l-1})}_{j_{l-1}} \right\} \\ \to \mathbb{P}\left\{ \overline{X}(t_l) = \overline{y}_l
\Big| \overline{X}(t_{l-1}) = \overline{y}_{l-1} \right\} \end{multline*} as $\varepsilon \to 0$, and thus the finite dimensional distributions converge.
\qed
\subsection{Proof of Proposition \ref{prop:sens_converge}} Proposition \ref{prop:sens_converge} is simply the application of Theorem 3.2 of Ref. \citen{gupta_sensitivity_2014} to the TTS framework. The method and framework for separating time-scales in Ref. \citen{gupta_sensitivity_2014} is slightly different than the TTS framework used here, but it can be seen that the two are equivalent. Here, we briefly review the multiscale framework of Ref. \citen{gupta_sensitivity_2014} and show how one can translate between their ``remainder spaces'' and the TTS ``fast-classes''.
\subsubsection{Scaling Rates and Remainder Spaces} As in Refs.\citen{kang_separation_2013, wang_efficiency_2014}, Ref.\citen{gupta_sensitivity_2014} considers reaction rates of the form $a^N_k(x, \theta) = N^{\rho_k}\lambda_k(x, \theta)$ which scale with the ``system size'' or ``normalization parameter'' $N \gg 0$, where $\rho_k$ is the scaling rate for the $k$-th reaction channel. For a given normalizing parameter $N$, the corresponding system is denoted by $X^{N}(t)$. One analyzes the system against a reference time scale $\gamma$ by $X^{N}_\gamma(t) = X^{N}( t N^\gamma)$. For a given normalizing parameter $N_0 \gg 0$, the corresponding system is denoted by $X^{N_0}(t)$. One analyzes the system against a reference time scale $\gamma$ by $X^{N}_\gamma(t) = X^{N}( t N^\gamma)$.
The scaling rates $\rho_k$ determine the time scales at which the reaction channels fire. For a system with with a single level of stiffness, there are only two scaling rates, $\rho_{fast} > \rho_{slow}$,
which partition the reaction channels as either fast or slow. Write $\Gamma_1=\{ k : \rho_k=\rho_{fast} \}$ for the fast reaction index set, and similarly $\Gamma_2$ for the slow reaction index set.
Take $\mathbb{S}_2 = \{ v \in \mathbb{R}^d_+ : \langle v, \zeta_k \rangle = 0 \text{ for all } k \in \Gamma_1 \}$ so that $\langle X_\gamma^N(t), v \rangle $ is unchanged by fast reactions. Then take $\mathbb{L}_2=\text{span}(\mathbb{S}_2)$ and $\Pi_2$ be the projection map from $\mathbb{R}^d$ to $\mathbb{L}_2$, so that $\Pi_2 \zeta_k = \boldsymbol{0}$ for all $k \in \Gamma_1$.
Let $\mathbb{L}_1 = \text{span} \{ (I - \Pi_2)x : x \in \mathcal{M} \}$, and for any $v \in \Pi_2 \mathcal{M}$ let $\mathbb{H}_v = \{ y \in \mathbb{L}_1 : y = (I - \Pi_2)x, \Pi_2 x = v, x \in \mathcal{M} \}$, the set of remainders of elements in $\mathcal{M}$ which get projected to $v$. Then we can define an operator $\mathbb{C}^v$ by \begin{align*} \mathbb{C}^v f (z) = \sum_{k \in \Gamma_1} \lambda_k(v +z, \theta) [ f(z + \zeta_k) - f(z) ] \end{align*} which is a generator of a Markov chain with state space $\mathbb{H}_v$ (note that $y \in \mathbb{H}_v \implies y + \zeta_k \in \mathbb{H}_v$ for all $k \in \Gamma_1$).
Assuming $\mathbb{H}_v$ under $\mathbb{C}^v$ is ergodic, there is a stationary distribution $\pi^v$. Then for each slow reaction $k \in \Gamma_2$ one can define the ``averaged'' propensities $\hat{\lambda}_k(v, \theta) = \int_{\mathbb{H}_v} \lambda_k(v + z, \theta) \pi^v(dz)$ for all $v \in \Pi_2 \mathcal{M}$ . Using the random time change representation, define the Markov chain on $\Pi_2 \mathcal{M}$ by \begin{align} \label{eq:Xhat_defn} \hat{X}_\theta(t) = \Pi_2 x_0 + \sum_{k \in \Gamma_2} Y_k \left( \int_0^t \hat{\lambda}_k( \hat{X}(s), \theta) ds \right) \Pi_2 \zeta_k \end{align} Taking $\gamma_2=-\rho_{slow}$ as the slow time scale, one has $\Pi_2 X^N_{\gamma_2, \theta} \Rightarrow \hat{X}_\theta$ as $N \to \infty$\cite{kang_separation_2013} under more general conditions than Assumptions \ref{assum:finite_states},\ref{assum:recurr_states}. Under this context, Theorem 3.2 of Ref.\citen{gupta_sensitivity_2014} states that \begin{align} \label{eq:gupta_scaling_sens} \lim_{N \to \infty} \frac{\partial}{\partial \theta} \mathbb{E} \left\{ f(X^N_{\gamma_2, \theta}(t)) \right\} = \frac{\partial}{\partial \theta} \mathbb{E} \left\{ f_\theta (\hat{X}_\theta(t) ) \right\} \end{align} where $f_\theta(v) = \int_{\mathbb{H}_v} f(v+y) \pi^v_\theta(dy)$.
\subsubsection{Equivalence of Fast-Classes and Remainder Spaces} Here we show how the TTS framework is equivalent to the scaling rate framework. Consider a TTS reaction network as described by \eqref{eq:Q-ep_decomp}. Taking $N=1/\varepsilon$, $\rho_{fast}=1$, $\rho_{slow}=0$, it is easy to see that \begin{align*}
X^N_{0,\theta }(t) = X^N_\theta (t)
\overset{\mathcal{D}}{=} X^\varepsilon(t) . \end{align*} so $\lim_{N\to\infty}X^N_{0, \theta}(t) = \lim_{\varepsilon\to0}X^\varepsilon(t)$. It remains to identify $f_\theta \left( \hat{X}_\theta(t) \right) $ from \eqref{eq:Xhat_defn}, \eqref{eq:gupta_scaling_sens}
with $\overline{f}\left(\overline{X}(t)\right)$ from \eqref{eq:fbar_defn}.
We do so by showing the equivalence of the fast-classes
$\mathcal{M}_l$ and the remainder spaces $\mathbb{H}_v$.
\begin{lemma}
The projection map $\Pi_2$ is invariant on fast-classes $\mathcal{M}_l$.
The set of remainder spaces
$\left\{ \mathbb{H}_v : v \in \Pi_2 \mathcal{M} \right\} $
is in one-to-one correspondence with the set of fast-classes
$\left\{ \mathcal{M}_l \right\} $.
Additionally, each $x \in \mathcal{M}_l$
corresponds to a unique element
$y \in \mathbb{H}_v$ for some $v \in \Pi_2\mathcal{M}$.
\end{lemma}
\begin{proof}
Define $\eta: \left\{ \mathcal{M}_l \right\} \to
\left\{ \mathbb{H}_v : v \in \Pi_2 \mathcal{M} \right\} $ by
\begin{align*}
\eta(\mathcal{M}_l) = \mathbb{H}_{\Pi_2(x)}
\qquad \text{for any } x \in \mathcal{M}_l .
\end{align*}
Then $\eta$ is well-defined, since $x, y \in \mathcal{M}_l$ implies
that $y = x + \sum_{k \in \Gamma_1} c_k \zeta_k$ for some
$c_k \in \mathbb{N}$, and $\Pi_2(\zeta_k) = \boldsymbol{0}$ for all
$k \in \Gamma_1$ gives $\Pi_2(y) = \Pi_2(x)$.
Clearly, $\eta$ is also onto.
It remains to establish $\eta$ is injective.
It is sufficient to show that
if $\Pi_2(x) = v$ and $\Pi_2(x')=v$ for $x, x' \in \mathcal{M}$,
then $x$ and $x'$ belong to the same fast-class $\mathcal{M}_l$.
Since $\Pi_2$ projects onto the span of the complement of
$\operatorname{span} \left\{ \zeta_k: k \in \Gamma_1 \right\}$,
we have
$v-x =\sum_{k \in \Gamma_1} c_k \zeta_k$
and $v-x' = \sum_{k \in \Gamma_1} c'_k \zeta_k$
for some $c_k, c'_k \in \mathbb{R}$.
Then $x' = x + \sum_{k \in \Gamma_1} (c_k - c'_k) \zeta_k$,
with $x', x, \zeta_k \in \mathbb{N}^d$, so it follows that
$(c_k -c'_k) \in \mathbb{N} $ for all $k$.
Hence $x$ and $x'$ communicate by fast reactions and thus belong
to the same fast-class $\mathcal{M}_l$. Therefore,
$\Pi_2$ is invariant on fast-classes and $\eta$ is injective.
Finally, since $\Pi_2(\cdot)$ is invariant on fast-classes
$\mathcal{M}_l$, it follows that $x \to x-\Pi_2(x)$ bijectively
maps elements of $\cup_l \mathcal{M}_l$ to elements of
$\cup_{v \in \Pi_2(\mathcal{M})} \mathbb{H}_v$ such that
$x, x' \in \mathcal{M}_l$ implies $(x - \Pi_2(x)), (x'-\Pi_2(x'))
\in \mathbb{H}_{\Pi_2(\mathcal{M}_l)}$.
\end{proof}
Because of the direct correspondence between $\left\{ \Pi_2(x) : x \in \mathcal{M} \right\} $ and $\left\{ \mathcal{M}_l \right\}$, we see (upon reordering states) that for $v=\Pi_2(\mathcal{M}_l)$, we have $\pi^v_\theta = \widetilde{\pi}^{(l)}_\theta$, so that $\hat\lambda_k(v, \theta) = \overline{\lambda}_k (\mathcal{M}_l, \theta)$, and $f_\theta(v) = \overline{f}(\mathcal{M}_l, \theta)$. Thus, $f_\theta \left( \hat{X}(t) \right) $ has the same distribution as $\overline{f}\left( \mathcal{M}_l, \theta \right)$ and so $\frac{\partial}{\partial \theta} \mathbb{E}\left\{ f_\theta\left( \hat{X}_\theta(t) \right) \right\} = \frac{\partial}{\partial \theta} \mathbb{E}\left\{ \overline{f} \left( \overline{X}_\theta (t) \right) \right\}$. Therefore, using \eqref{eq:gupta_scaling_sens} we have \begin{align} \begin{aligned} \lim_{\varepsilon \to 0} \partial_\theta \mathbb{E} \left\{ f\left( X^\varepsilon(t) \right) \right\} = \lim_{N \to \infty} \partial_\theta \mathbb{E} \left\{ f\left(X^N_{0,\theta}(t) \right) \right\} \\ = \partial_\theta \mathbb{E} \left\{ f_\theta \left( \hat{X}_\theta (t) \right) \right\} = \partial_\theta \mathbb{E} \left\{ \overline{f} \left( \overline{X}(t) \right) \right\} \end{aligned} \end{align} and hence Proposition \ref{prop:sens_converge}.
\section{Analytic solution of the Model System} \label{sec:AnalylicSolnExample}
{In a well-mixed system with linear propensities, the time-evolution of the system can be obtained from a set of ordinary differential equations (ODE). The single time-scale (STS) system can be modeled with a system of ODEs. The two time-scale (TTS) system imposes algebraic constraints for the fast modes, resulting in an algebraic differential system of equations. In both cases, a set of adjoint ODEs can be used to compute sensitivities alongside species populations.}
In our model system, gaseous species A adsorbs onto a catalyst surface, isomerizes to species B, and then desorbs. A diagram of the reaction network is shown in Figure \ref{fig:networkpic}. The reactions along with their rate laws are shown in Table \ref{tab:toyrxns}. $N_A$, $N_B$, and $N_*$ denote the surface coverages of species A, B, and empty sites respectively. The adsorption/desorption of species A is assumed to be much faster than the others. The separation of time scales is captured with the dimensionless parameter $\varepsilon << 1$. The system contains $M=3$ species and $R=6$ reactions. Mathematically, we use the $M\times 1$ column vector $N$ to specify the species populations, where $N_1=N_A$, $N_2=N_B$, and $N_3=N_*$.
\begin{figure}
\caption{Description of model chemical reaction network.}
\label{fig:networkpic}
\label{tab:toyrxns}
\label{fig:networkdescription}
\end{figure}
{The linear dependence of the reaction rates is written as}
\begin{equation} r(N) = \left[ \begin{array}{c} \varepsilon^{-1} \alpha_1 N_3 \\ \varepsilon^{-1} \alpha_2 N_1 \\ \beta_1 N_1 \\ \beta_2 N_2 \\ \beta_3 N_2 \end{array} \right]. \end{equation}
Each row of the stoichiometric matrix corresponds to a different species, which are $N_1$, $N_2$, and $N_3$ respectively. The columns correspond to each of reactions 1-5 in order. Extracting the information from Table \ref{tab:toyrxns} and putting it in mathematical form gives the $M\times R$ {stoichiometric} matrix
\begin{equation} S=\left[ \begin{array}{ccccc} 1 & -1 & -1 & 1 & 0 \\ 0 & 0 & 1 & -1 & 1 \\ -1 & 1 & 0 & 0 & -1 \end{array} \right] \end{equation}
{The transformation matrix} \begin{equation} T = \left[ \begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 1 & 1 & 1 \end{array} \right]. \end{equation}
{yields $y=T\cdot N$ and} $T$ can be decomposed into $T_f = \left[ \begin{array}{ccc} 1 & 0 & 0 \end{array} \right]$ and $T_s = \left[ \begin{array}{ccc} 1 & 1 & 0 \\ 0 & 0 & 1 \end{array} \right]$ {for the slow modes} by looking at the 0 rows of $S'_f$. This gives us the transformed variables as $y_f=\left[ \begin{array}{c} N_1\end{array} \right]$ and $y_s=\left[ \begin{array}{c} N_2 \\ N_1+N_2+N_3 \end{array} \right]$.
In the context of our example problem, we can assign physical meaning to the transformation: The variable $y_1=N_A$ is affected by both slow and fast reactions. For a given set of slow variables, we can solve for $y_1$ to specify the equilibrium constraint of $r_1=r_2$. The variable $y_2=N_B$ is unaffected by the fast adsorption/desorption of A, but is affected by the slow reactions. Finally, the variable $y_3=N_A+N_B+N_*$ is a second "slow" variable. In this example $y_3=1$ applies at all times due to stoichiometric constraints. However, it is still identified as a "slow mode" because this constraint is not a consequence of disparities in reaction time scales.
The system is simulated with the choice of parameters $N_0=\left[ \begin{array}{c} 30 \\ 60 \\ 10 \end{array} \right]$, $\alpha_1=1$, $\alpha_2=1.5$, $\beta_1=2$, $\beta_2=1$, $\beta_3=0.4$, $\varepsilon=0.01$. The simulation results are shown in Figure \ref{fig:A}. {Table \ref{tab:table2} shows values at $t=1.3$s and $t=100$s along with CLR and CELR estimates with statistical confidence intervals.} Derivatives with respect to $\beta_2$ and $\beta_3$ overlap because both affect system properties through the the independent parameter $\beta_2+\beta_3$.
In general, the system parameters need not be the rate constants themselves. A different parameterization would involve a transformation of the rate constants. Sensitivities could be obtained through a chain rule.
\begin{figure}
\caption{Network results for the two time-scale system. Graphs show population counts (top left), derivatives of species A (top right), derivatives of species B (bottom left), and derivatives of empty sites (bottom right).}
\label{fig:A}
\end{figure}
\begin{table}[h!]
\begin{center}
\caption{Comparison of the CLR and CELR estimators with the ODE solution. Values show actual values rather than errors. Values in the table refer to the sensitivity of the species B with respect to the parameter given by the row label. 95\% confidence intervals are based on statistical noise.}
\label{tab:table2}
\begin{tabular}{cccc}
\hline
& \multicolumn{3}{c}{$t=1.3$s}\\
& ODE & CLR & CELR\\
\hline
$\alpha_1$ & 11.9 & $11.8 \pm 1.5$ & $7.2 \pm 0.9$\\
$\alpha_2$ & -7.9 & $-7.7 \pm 1.0$ & $-4.7 \pm 0.6$\\
$\beta_1$ & 9.9 & $10.0 \pm 1.2$ & $6.0 \pm 0.8$\\
$\beta_2$ & -17.4 & $-17.2 \pm 2.4$ & $-14.5 \pm 1.7$\\
$\beta_3$ & -17.4 & $-17.1 \pm 3.4$ & $-13.3 \pm 2.4$\\
\hline
& \multicolumn{3}{c}{$t=100$s}\\
& ODE & CLR & CELR\\
\hline
$\alpha_1$ & 13.9 & $2.3 \pm 13.7$ & $13.9 \pm 1.4$\\
$\alpha_2$ & -9.3 & $-3.4 \pm 8.9$ & $-9.2 \pm 1.0$\\
$\beta_1$ & 11.6 & $4.1 \pm 10.6$ & $11.7 \pm 1.2$\\
$\beta_2$ & -16.5 & $-12.8 \pm 18.1$ & $-16.1 \pm 2.0$\\
$\beta_3$ & -16.5 & $-19.9 \pm 29.0$ & $-16.5 \pm 3.2$
\end{tabular}
\end{center} \end{table}
\end{document} |
\begin{document}
\title{Time-optimal control with finite bandwidth} \author{M. Hirose and P. Cappellaro} \affiliation{Department of Nuclear Science and Engineering and Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, MA, USA}
\begin{abstract} Optimal control theory provides recipes to achieve quantum operations with high fidelity and speed, as required in quantum technologies such as quantum sensing and computation. While technical advances have achieved the ultrastrong driving regime in many physical systems, these capabilities have yet to be fully exploited for the precise control of quantum systems, as other limitations, such as the generation of higher harmonics or the finite bandwidth of the control fields, prevent the implementation of theoretical time-optimal control. Here we present a method to achieve time-optimal control of qubit systems that can take advantage of fast driving beyond the rotating wave approximation. We exploit results from optimal control theory to design driving protocols that can be implemented with realistic, finite-bandwidth control and we find a relationship between bandwidth limitations and achievable control fidelity. \end{abstract}
\maketitle
Precise control of quantum systems is a requirement for many applications of quantum physics, from quantum information processing to quantum metrology and simulation. Fast control is highly desirable to beat decoherence and improve performance of these quantum devices. This desire has spurred much research on the ultimate control speed for unitary~\cite{Deffner13,Salamon09} and dissipative~\cite{DelCampo13} dynamics, as well as shortcuts to adiabatic control~\cite{Bason12}. At the same time, technological advances have enabled driving quantum transitions faster than the natural transition frequency, in systems ranging from atoms~\cite{Hofferberth07,Jimenez-Garcia15} to quantum wells~\cite{Zaks11}, from superconducting qubits~\cite{Deng15,Ashhab07,Rudner08,Oliver09,Niemczyk10} to mechanical oscillators~\cite{Barfuss15} and isolated spin defects~\cite{Fuchs09,Childress10,Scheuer14}. In this ultrastrong driving regime, the design of control protocols can no longer rely on the usual intuition, based on the rotating wave approximation (RWA). While optimal control theory gives prescriptions to achieve time-optimal (TO) control, often the ideal control schemes cannot be applied in practice, due to bounds in the control strength, phase or bandwidth. For example, bounds in the control strength impose a \textit{quantum speed limit}~\cite{Margolus98,Carlini06,Giovannetti03,Caneva09,Hegerfeldt13} on the system evolution, which is related to an energy-time uncertainty relation~\cite{Deffner13,DelCampo13}. Limitations on the control of the driving field phase or polarization preclude the application of many TO control schemes. For example, it has been shown~\cite{Boscain02a,DAlessandro01,Albertini15} that for a two-level system (qubit), the TO solution is given by an ``on-resonance'' driving, if the phase or polarization of the driving field is under experimental control~\cite{London14,Shim14}. When this is not possible, the internal Hamiltonian (the \textit{drift} term) cannot be eliminated and the TO solution takes the form of a bang-bang (BB) control~\cite{Boscain06,Billig13,Aiello15q,Billig14,Avinadav14}. This optimal solution assumes that there are no limitations in the control bandwidth; however, in practice the control fields cannot be switched on and off instantaneously.
Here we show that we can approach time-optimal driving of a qubit, given a bound, real driving field along a single axis, $|\Omega(t)|\leq\overline\Omega$, even when the Fourier transform of $\Omega(t)$ is defined over a finite range $[0,\Delta\omega]$. With the goal of keeping the gate time equal to the BB optimal time, we construct an analytical control strategy based on a Fourier series approximation to the ideal control. Our \textit{Fourier-Approximated Time-Optimal} (FATO) control strategy achieves several key results. First, it provides an analytical recipe to design high fidelity, time-optimal control sequences in the regime of ultrastrong driving, when the RWA breaks down. Even in the case of weak driving, where the RWA is applicable, it achieves shorter gate times than conventional (on-resonance) methods. Just as importantly, we identify control bandwidth as a limiting resource in the compromise between fidelity and time-optimality~\cite{Moore12,Lloyd14l}. Even if a Fourier approximant is not the only solution, it allows us to easily analyze bounds on the control fidelity that bandwidth limitations impose, with analytical solutions describing the dependence of gate fidelity on the bandwidth. In addition, we show that the FATO scheme is robust against errors in the control field and it can further be extended to controlling more than one qubit.
\begin{figure}
\caption{
\textit{Left: Time-optimal control on the Bloch sphere}. The figure shows the two axes of rotation (red arrows) separated by the angle $2\theta\!=\!\pi/5$. We plot a representative trajectory (blue-black line) from $|0\rangle$ to $|1\rangle$ achieved with the TO BB control shown on the right. \textit{Right: FATO waveform}. Here we considered the BB solution (solid black line) to achieve a $\pi$ rotation about X, for $\theta\!=\!\pi/10$ and $\omega_0\!=\!\pi$ and its Fourier Series (red solid line) truncated at $k\!=\!57$ (9 non-zero coefficients). The dashed line is the waveform for on-resonance driving (notice that its period does not match the BB period).}
\label{fig:Waveform}
\end{figure}
\textbf{Fourier-Approximated Time-Optimal Control --}
Assume we have a qubit with internal (\textit{drift}) Hamiltonian ${\mathcal{H}}_0=\omega_0\sigma_z/2$ and we can apply a control ${\mathcal{H}}_c=\Omega(t)\sigma_x/2$, with $\Omega$ real and bounded by $|\Omega(t)|\leq\overline\Omega$. This situation is relevant to many experimental systems, from nuclear and electronic spin resonance to atomic systems and superconducting qubits. The control Hamiltonian is generated by, e.g., radiofrequency or microwave fields applied along one physical axis by a wire or antenna in the experimental setup, which can set the time-dependent amplitude of the field source. The goal is to perform a desired unitary evolution in a time-optimal way.
The usual strategy to achieve precise qubit control is to rely on the rotating wave approximation: for example, to achieve a rotation about $x$, we set $\Omega_{\mathrm{RWA}}\!\ll\!\omega_0$ and drive on resonance, $\Omega(t)=\Omega_{\mathrm{RWA}}\cos(\omega_0t)$. More general rotations can be obtained by choosing the frequency and phase of the driving, thus making it possible, for example, to effectively drive along the perpendicular direction ($y$-axis) even if the driving field is along the laboratory $x$-axis. This solution is however not time-optimal: indeed, one effectively only uses half of the driving strength, as the other half is the counter-rotating field.
More precisely, Pontryagin's minimum principle~\cite{Pontryagin87b} can be used to prove that for this control problem a BB sequence with $\Omega(t)=\{\pm\overline\Omega,0\}$ is the TO solution~\cite{Boscain06,Garon13}. In addition, if experimental conditions allow $\Omega\gtrsim\omega_0$, the RWA is not applicable and on-resonance driving is no longer a good control strategy.
For both cases (ultrastrong or weak driving) the ideal TO control strategy for one-axis driving is to always evolve at the maximum ``speed'' (BB control), assuming one can switch the sign of the function $f(t)=\Omega(t)/\overline\Omega$ infinitely fast. The total Hamiltonian then takes the form \begin{equation}\renewcommand*{\arraystretch}{1.5}
\begin{array}{l} {\mathcal{H}}_\pm=\frac12[\omega_0\sigma_z\pm\overline\Omega\sigma_x]=\frac\omega2(\sigma_z\cos\theta\pm\sigma_x\sin\theta),\\ {\mathcal{H}}_0=\frac12\omega_0\sigma_z, \end{array}
\label{eq:Ham} \end{equation} where we defined $\omega=\sqrt{\omega_0^2+\overline\Omega^2}$ and $\tan(\theta)=\overline\Omega/\omega_0$. The control is obtained by switching between these three Hamiltonians inducing rotations about three different axes. The SU(2) gate synthesis problem then reduces to finding the times $t_j^{(+,0,-)}$ for each ``bang''.
Strong conditions on these times and number of bangs have been recently found using an algebraic solution~\cite{Aiello15q,Billig13,Billig14}. Only three parameters are necessary to describe the TO solution~\cite{Billig13}, an initial and final time $t_i$ and $t_f$, while middle times $t^\pm_m$ are related to each other. In particular, when the rotation speed about the two axes is equal, which is the case here, the middle times are all equal. Properties of the TO decomposition can be classified based on the angle $2\theta$ between the two rotation axes, and in particular whether $\theta \lessgtr \pi/4$, corresponding to weak ($\overline\Omega<\omega_0$) or strong ($\overline\Omega>\omega_0$) driving. For example, for $\theta>\pi/4$ finite solutions have at most 4 bangs, while for $\theta<\pi/4$ they can have an increasing number of bangs with decreasing $\theta$~\footnote{Infinite decompositions are also possible. However, here an
infinite sequence with equal times is a rotation about $\sigma _z$, speedily obtained by singular control
($\Omega =0$).}. These constraints (see also Appendix~\ref{sec:appendix}) can be used to efficiently search for TO solutions to the synthesis of any desired unitary in SU(2), under the assumption of \textit{infinite bandwidth} of the control function (infinitely fast switching between $\pm\overline\Omega$). In the following we show how, even when the control bandwidth is limited, the BB solution forms the basis for an excellent TO control scheme that we call \textit{Fourier-Approximated Time-Optimal} (FATO) control.
We assume that the control field can only be switched with a finite speed; this sets an upper bound $\Delta\omega$ to its bandwidth. The ideal TO solution, with total sequence time $T$ and switching times $\{t_i,t_m,t_f\}$, effectively defines a piecewise-constant function $f(t)=\Omega(t)/\overline\Omega=\{\pm1,0\}$. It is always possible to express $f(t)$ over the interval $[0,T]$ as a Fourier series (see Fig.~\ref{fig:Waveform}): \begin{equation} f(t)=\frac{c_0}2+\sum_{k=0}^\infty[s_k\sin(2\pi k t/T)+ c_k\cos(2\pi k t/T)]. \label{eq:fourier} \end{equation}
Then, imposing the bandwidth constraint on the control function is equivalent to truncating this sum to $k=K$ with $2\pi K/T\leq \Delta\omega$. In turns, this will reduce the fidelity of the control, while preserving its duration at the optimum time.
\textbf{Fidelity and Robustness of FATO Control --} \begin{figure}
\caption{\textit{Left axis:} Comparison between the time required for a $\pi$ rotation with the FATO method and the RWA on-resonance driving as a function of driving strength (parametrized by the angle $\theta$). Note the comparison is only possible for $\theta\leq\pi/4$. \textit{Right axis:} Normalized time $\omega_0T$ required for $\pi$ rotations. Circles (and dotted lines) are for the weak driving solution, while solid lines are for the ultrastrong driving regime.}
\label{fig:TimeComp}
\end{figure} For a given bandwidth $\Delta\omega$, two properties of the time-optimal BB solution will determine how well it can be approximated by FATO: the minimum non-zero time among $\{t_i,t_f,t_m\}$ (shorter times requiring larger bandwidths) and the number of switches (larger $n$ requiring in general larger $k$ for a better approximation). In general, the middle times are constrained by $t_m\geq\pi/\omega=\pi\cos(\theta)/\omega_0$; they define a square wave with $n\leq \lfloor \frac\pi\alpha \rfloor +1$ switches. The minimum bandwidth is then $\Delta\omega\geq \sqrt{\omega_0^2+\overline\Omega^2}= \omega_0/\cos(\theta)$, that is, it depends not only on the ``resonant'' frequency $\omega_0$, but also on the driving strength, as stronger driving allow for faster control, thus requiring larger bandwidth for time optimality.
To make our method more concrete, we focus on exemplar target gates, $\pi$ rotations about the $x$- and $y$-axis. These gates are particularly important (they are ``NOT'' quantum gates) and describe an evolution under the control operator only, eliminating the effects of the drift. While focusing on these gates allows us to find explicit solutions for the BB TO problem that are the starting point for the bandwidth-limited construction, the same analysis would also apply to other unitaries. For these gates we can more easily analyze the performance of FATO control in terms of gate time, fidelity as a function of bandwidth and robustness to imperfections.
\begin{figure}
\caption{Infidelity for a $\pi$-pulse about X (red) and Y (black) as a function of the driving strength, parametrized by the angle $\theta$. Open symbols are for a Fourier expansion with 5 non-zero coefficients and filled symbols with 19 non-zero coefficients (the resulting bandwidth depends on the angle $\theta$). The dashed lines are the analytical expressions (not a fit) based on the Fourier series approximation error $\mathcal E_K$.}
\label{fig:FidTheta}
\end{figure} \textit{Gate Time --} We distinguish between weak and ultrastrong driving, as they have different BB solutions. In the case of weak driving, the angle between the two axes of rotation is small and we expect generally longer control sequences (large $T$) with many bangs (large $n$). While specific solutions for arbitrary $\theta$ must be found numerically, analytical solutions are available for specific values. In particular, we find that the optimal solution has $n=\frac{\pi}{2\theta}$ bangs for X(Y) $\pi$ rotations, whenever $n$ is an odd(even) integer number (see Appendix~\ref{sec:appendix}). All the bang times are equal and such that $\omega t_m=\pi$. The function $f(t)$ is then a simple square wave with period $2\pi/\omega$. The total time is $T_{\mathrm{TO}}=n\pi/\omega=\pi^2\cos(\theta)/(2\theta\omega_0)$. In Figure~(\ref{fig:TimeComp}) we compare this optimal time to the time required with on-resonance driving, $T_{\mathrm{RWA}}=2\pi/\overline\Omega$ (as the effective Rabi frequency in the RWA is $\overline\Omega/2$). The ratio of the two strategy times is given by $T_{\mathrm{TO}}/T_{\mathrm{RWA}}=\frac{\mathrm{Si}(\pi)\sin\theta}{2\theta}$ (with $\mathrm{Si}$ the sine integral function), where we took into account that due to Gibbs phenomenon~\cite{Gibbs98,Bocher06}, the Fourier series approximation yields an effective larger maximum driving frequency, $\Omega'\approx \frac{2 \mathrm{Si}(\pi )}{\pi }\overline\Omega$.
In the ultrastrong driving regime a direct comparison with the time required for on-resonance driving is not possible, since the RWA is violated. Our method still provides a constructive strategy to achieve control beyond the RWA, and does so in a time-optimal way. The TO solution for strong driving consists of $n=3$ bangs, with the middle bang singular ($\Omega=0$) to obtain a $\pi$ pulse about Y. The times are given by \begin{equation} \begin{array}{ll} t_1^x\!=\!t_3^x\!=\!\frac{2 \mathrm{arccsc}[2 \sin(\theta )]}{\omega },& t_1^y\!=\!t_3^y\!=\!\frac{2 \mathrm{arccot}\left[\sqrt{\!-\!\cos (2 \theta )}\right]}{\omega } \\ t_2^x\!=\!2\pi\!-\!\frac{2 \mathrm{arccsc}[2 \sin(\theta )]}{\omega },& t_2^y\!=\!\frac{2 \arctan\left[\sqrt{\tan(\theta)^2-1}\right]}{\omega_0} \end{array} \label{eq:TimeS} \end{equation} These times define the piece-wise constant function $f(t)$ that we approximate with a Fourier series expansion to obtain the FATO control driving field shape. \begin{figure}
\caption{Infidelity for a $\pi$-pulse about X (left) and Y (right) as a function of the bandwidth $\Delta\omega/\Omega$. We consider ultrastrong driving, $\theta\!=\!\pi/3$ (red), and weak driving, $\theta\!=\!\pi/10, \pi/8$ (black) and $\theta\!=\!\pi/22, \pi/20$ (blue) for X and Y respectively. The dotted lines are the infidelities (not a fit) calculated from the Fourier series error $\mathcal E_K$. The solid lines are the infidelities (due to breaking of the RWA) of the on-resonance driving strategy (the infidelity of Y($\pi$) for $\pi/3$ is not shown as it is $>.1$).}
\label{fig:FidDrive}
\end{figure}
\textit{Fidelity -- } Because of the finite bandwidth of the control field, the ideal gate propagator cannot be perfectly implemented in the optimal time. In our FATO approach, we keep fixed the gate time at its optimum value; we can then evaluate the effects of the finite bandwidth by calculating the entanglement fidelity~\cite{Nielsen02}. Truncating the Fourier expansion to order $K$ implements the propagator $U_K$ satisfying \begin{equation} i\dot U_K(t)\!=\!\frac{\omega_0}2[\sigma_z+\tan(\theta)[f(t)- R_K(t)]\sigma_x]U_K(t), \label{eq:UK} \end{equation} where we defined the reminder of the truncated Fourier series of $f(t)$, \begin{equation} R_K(t)=\sum_{k=K+1}^\infty[s_k\sin({2\pi k t}/T)+ c_k\cos({2\pi k t}/T)]. \label{eq:reminder} \end{equation}
The fidelity $F\!=\!|\mathrm{Tr}[U_{id}^\dag(T)U_K(T)]|/2$ with respect to the ideal propagator, $U_{id}(T)$, can be computed in terms of a reminder propagator, $U_R(t)=U_{id}^\dag(t)U_K(t)$. $U_R(t)$ can be evaluated by moving to the ideal Hamiltonian \textit{toggling} frame, where the Hamiltonian is $\widetilde{\mathcal{H}}(t) = U^\dag_{id}(t)[{\mathcal{H}}(t)-{\mathcal{H}}_{id}(t)]U_{id}(t)$. Assuming the reminder is small, we can approximate $\widetilde U_R$ with a first-order Magnus expansion given by the effective Hamiltonian $\overline{\mathcal{H}}_R=\int_0^T \widetilde{\mathcal{H}}(t')dt'$. In the weak coupling regime, for example, we obtain \begin{equation} \overline{\mathcal{H}}_R=\frac{\omega_0\theta}\pi\tan(\theta)\mathcal E_K[\sigma_x-\tan(\theta)\sigma_z], \end{equation} where we introduced the mean error of the truncated Fourier series (see Appendix~\ref{sec:appendixB}): \begin{equation} \mathcal E_K= \frac{2}{T}\int _0^T \!\!R_K^2(t)dt=\frac12\sum_{k=K+1}^\infty (c_k^2+s_k^2).\end{equation} This yields the fidelities $F_w^{x,y}\approx\cos[\tan(\theta)\mathcal E_K \pi/4]$. While in the strong driving regime the exact calculation is more complex, we still find that the fidelities are well approximated by a function of the mean errors, $F_s^x\approx\cos[2/\pi\sin(\theta)\mathcal E_K]$ and $F_s^y\approx\cos[2/\pi\tan(\theta)\mathcal E_K]$, as shown in Fig.~(\ref{fig:FidTheta}) and (\ref{fig:FidDrive}).
\begin{figure}
\caption{Infidelity for a $\pi$-pulse about X as a function of the error in frequency $\omega_0$ (filled symbols, left axis) or driving strength $\Omega$ (open symbols, dashed line, right axis). We consider two exemplary cases for strong ($\theta=\pi/4$, red) and weak ($\theta=\pi/10$, blue) driving. Although the fidelity of the gates decreases with the inhomogeneities, the error is typically smaller than for the usual on-resonance driving.}
\label{fig:Inhomogeneities}
\end{figure}
We thus found a simple relationship between the fidelity and the Fourier series mean error, $\mathcal E_K$, which encompasses the Fourier properties of the TO BB function and the available control bandwidth. This relationship not only makes it possible to easily find the required bandwidth for a desired fidelity, but also provides insight onto which BB controls require larger bandwidth. For example, in the ultrastrong regime, as $\overline\Omega/\omega_0\!\to\!\infty$ ($\theta\!\!\to\!\pi/2$), the times required for an X rotation go to zero, thus allowing good fidelity; however, a Y rotation still requires a finite time, reducing the fidelity for the same bandwidth (Fig.~\ref{fig:FidTheta}).
Still, as shown in figure (\ref{fig:FidDrive}), the infidelity, $\textrm{Inf}=1\!-\!F,$ decreases rapidly with the control bandwidth. In addition, in the weak driving regime, FATO control beats the fidelity obtained with the on-resonance driving (when taking into account the counter-rotating field), even when considering a very small bandwidth ($\Delta\omega\approx \omega_0\div2\omega_0$). Very good fidelity is also obtained in the strong driving regime. We remark that since very high bandwidth can be routinely reached in experiments, our construction could achieve fidelity beyond the fault-tolerance threshold~\cite{Gottesman98}, while still operating at the maximum quantum speed.
\textit{Robustness to parameter variations --} We further evaluate the robustness of the FATO control scheme with respect to errors in either the frequency $\omega_0$ or the driving strength $\overline\Omega$. The first case corresponds to the situation where either the internal Hamiltonian is not known with sufficient precision or there are variations due to inhomogeneities. In the second case, we analyze the possibility of an error or inhomogeneity in the driving power. Typical results are shown in Fig.~(\ref{fig:Inhomogeneities}). We find that even for a few percent error, the fidelity of the gate is good and it is typically higher than for the usual on-resonance driving. In some cases, the fidelity can be even higher than in the absence of error: this is due to the fact the error can contribute to either driving fields or larger bandwidth. We note however that in these cases the driving in the presence of errors might not be anymore time-optimal. \begin{figure}
\caption{Infidelity for a SWAP gate on two qubits as a function of the bandwidth $\omega/J$ for varying driving strengths $\Omega/J$. The blue line is the fidelity of the ideal BB control (with finite pulses) and the red dots its FATO approximation. In the inset: Infidelity as a function of bandwidth for fixed driving strength, $\Omega=100J$.}
\label{fig:Fid2qb}
\end{figure}
\textbf{Extension to two-qubit systems --} Until now we focused on TO control of a single qubit. Our strategy can be however extended to larger systems with the combined goals of finding TO control laws and their fidelity dependence on available bandwidth. For example, the BB solutions we found to generate NOT gates for a single qubit could also be used to drive two qubits with opposite internal Hamiltonian~\cite{Romano15}. Then, all the results found here on the effects of a limited bandwidth still apply.
We can further analyze time-optimal sequences that have been proposed to generate two-qubit gates~\cite{Khaneja01} under the assumption of delta pulses. Using a Cartan decomposition, it was found that TO control of two qubits can be achieve with singular BB control. Adopting the TO solution and introducing finite-length pulses reduces the fidelity; assuming a finite bandwidth (so that ideal, rectangular pulses cannot be applied) further degrades the fidelity. We can use the FATO construction to evaluate these limitations. We consider for example a quantum SWAP gate (see Appendix~\ref{sec:appendix}, given by three free-evolution periods under the Hamiltonian $H_0=J\sigma_z\sigma_z/2$ interrupted by $\pi/2$-pulses about $x$ and $y$. Assuming a strength $\Omega$ of the driving fields, we plot in Fig.~(\ref{fig:Fid2qb}) the FATO fidelity as a function of bandwidth, demonstrating the good performance of our method.
\textit{In conclusion}, we devised a strategy to drive qubits in a time-optimal way, with high fidelities limited only by the available bandwidth of the control. The technique is in particular useful for ultrastrong driving fields, where intuitive approaches based on the rotating wave approximation fail and only numerical approaches were available until now~\cite{Motzoi11,Bartels13,Caneva11,Doria11}. In addition, our analytical solution provides a simple way to evaluate the required control bandwidth for a desired fidelity. The principles of FATO design could be further extended to achieve control in larger systems, for example to achieve the simultaneous time-optimal control of many qubits~\cite{Assemat10,Burgarth10,Romano14,Romano15} or two-qubit gates~\cite{Khaneja01}. While our method already provides high-fidelity control, it could be further used as an initial guess for numerical searches~\cite{Doria11} if higher fidelity is desired or to achieve control of 1-2 qubits embedded in larger systems. As ultrastrong driving becomes attainable in a number of solid-state and atomic quantum systems, from superconducting qubits to isolated spins, our control strategy will enable taking full advantage of these technical capabilities to manipulate qubits in a time-optimal way. Beyond providing a recipe for TO control, our construction also allows us to explore the compromise between fidelity and time-optimality, by linking gate errors to the available control bandwidth.
\appendix \section{Time-Optimal Bang-Bang control}\label{sec:appendix} Bang-bang control has been shown to achieve time-optimal control of two-level systems. General bounds and prescriptions for the time-optimal bang-bang control have been provided~\cite{Billig13,Boscain05,Boscain06,Boscain14,Romano15,Albertini15,Aiello15q}. These results can be used to numerically obtain a solution to the time optimal problem with BB control for a general unitary. However, for some target unitaries and Hamiltonian parameters it is possible to find analytical solutions. In the main text we focused on these cases since they allow to more easily study trends in the fidelity and robustness of the FATO control strategy. Here we describe how, exploiting known results in BB control, we obtained the specific TO solutions for the two gates and two driving strength regimes considered.
The general goal is to find the optimal times such that a sequence of ``bangs'' under alternating Hamiltonian ${\mathcal{H}}_{\pm,0}$ can achieve the desired unitary. Simple algebraic arguments~\cite{Billig13} impose constraints on the middle times of any TO decomposition. Then the TO problem reduces to finding three times, $t_i, t_m$ and $t_f$, such that concatenating the unitaries $U_{0,\pm}(t)=e^{-it{\mathcal{H}}_{0,\pm}}$ achieves the two desired gates, $\sigma_x$ and $\sigma_y$. Since we focused on achieving $\pi$ rotations, we can apply (in addition to results found in \cite{Billig13,Billig14,Aiello15q}) the results of \cite{Boscain06} relating to a north-to-south pole transformation only. Their constraints still need to be valid sufficient conditions (although not necessary) for the TO unitary.
\subsection{Weak Driving} For weak driving, it was found in~\cite{Boscain06} that there should be no singular bangs in the TO solution. In addition, for $\alpha=\pi/2n$ the solution $U_{NS}$ for the north-to-south transition is obtained as \begin{equation} U_{NS}=[U_+(\pi)U_-(\pi)]^n \end{equation} We find that if $n$ is even, $U_{NS}=\sigma_y$, while if $n$ is odd $U_{NS}=\sigma_x$. Ref.~\cite{Boscain06} allowed for a second type of solution, with $n+1$ bangs. These are candidates solutions for $\sigma_y$ if $n$ is odd and $\sigma_x$ if $n$ is even. While it is possible to find analytical solutions for the optimal times in these cases as well, the expressions becomes more involved and thus we limited our extended analysis in the main text to the simplest case.
We note that even for X rotations we consider an even number $n+1$ of bangs to build the FATO approximation. This ensures that the function $f(t)$ is odd, yielding a sine Fourier series which is zero at $t=0$ for any bandwidth.
\subsection{Ultrastrong Driving} The strong driving occurs when $\theta>\pi/4$. In this case, following \cite{Boscain06} and \cite{Aiello15q}, we find that the TO solution is composed of at most three bangs. It is then easy to find analytical solutions for the times $\{t_i, t_m, t_f\}$, for example by following the construction described in \cite{Piovan12}.
We can first verify that two bangs are not enough to generate the desired rotations. Defining $\vec v_{\pm,0}$ the vectors corresponding to the Hamiltonians ${\mathcal{H}}_{\pm,0}$ and $R=X,Y$ the rotations in SO(3) corresponding to $\sigma_x, \sigma_y$, we need to verify whether $\vec v_i\cdot \vec v_j=\vec v_i\cdot R\cdot \vec v_j$. However $\vec v_+\cdot \vec v_m=\cos(2\theta)$, $\vec v_\pm\cdot \vec v_0=\cos(\theta)$, while $\vec v_0\cdot R\cdot \vec v_\pm=-\cos(\theta)$, $\vec v_+\cdot Y\cdot \vec v_-=-\cos(2\theta)$ and $\vec v_+\cdot X\cdot \vec v_-=-1$.
Similarly, we can easily identify all the allowed three-bang constructions that achieve the desired unitaries. We find that the central bang must be singular ($\Omega=0$) for the $\sigma_y$ gate. Then, since the desired $\sigma_y$ gate cannot have a component along $\sigma_x$ we have to set \[\frac{i}2\mathrm{Tr}\{(U_\pm U_0U_\mp)\sigma_x\}\!=\!\sin(\theta)\sin\!\left(\!\frac{\omega_0t_2}2\!\right)\sin\!\left[\frac{\omega_0(t_1\!-\!t_3)}2\right]\!=\!0\] by selecting $t_1^y=t_3^y$ (since $t_2=0$ has already been excluded). Finally, $t_1^y$ and $t_2^y$ can be easily found algebraically, yielding Eq.~(\ref{eq:TimeS}) in the main text. For the $\sigma_x$ gate, solutions with and without a singular bang are allowed. In both cases the outer rotations are about the same axis, e.g. $U_+U_0U_+$ or $U_+U_-U_+$, and of the same duration, $t_1^x=t_3^x$. We can then calculate explicitly the times and compare the two possible solutions to select the time-optimal one. We find that the shortest evolution is obtained by discarding the singular solution, resulting in the times in Eq.~(\ref{eq:TimeS}).
\subsection{Time-optimal control of Two Qubits} Extending the results of BB TO control to more than one qubit is generally difficult, but results have been found for some particular cases~\cite{Assemat10,Romano14,Romano15,Ashhab12}. In particular, it has been found~\cite{Romano15} that for two qubits with opposite drift terms and under the same control,
\begin{equation}{\mathcal{H}}_\pm=\frac{\omega_0}2(\sigma_z^1-\sigma_z^2)\pm\frac\Omega2(\sigma_x^1+\sigma_x^2)\end{equation} the TO problem can be solved simultaneously. In particular, for $\pi$ rotations, we recover the same solutions found for one qubit. Then we could repeat the analysis performed for FATO control of one qubit; the fidelity is simply the square of the fidelity found for one qubit.
We note that while these analytical results are restricted to particular cases, they could be at the basis of numerical searches in more complex situations. For example, knowing the control function and required bandwidth for two non-interacting qubits could be used as initial guess for numerical searches of the control profile for two \textit{interacting} qubits.
Bang-bang control can as well be used to achieve two-qubit gates. Time optimal solutions for the steering of two qubits with Hamiltonian ${\mathcal{H}}=J\sigma_z^1\sigma_z^2/2$ were indeed found under the assumption of delta pulses (zero-duration pulses at infinite driving power). While the control solutions obtained when relaxing these assumptions might not be time-optimal, we can still aim to preserve the optimal time and look for the ensuing drop in fidelity. In the main text we considered an exemplary gate, the SWAP gate between two qubits (swapping their states): \begin{equation}U_{swap}=\left[\begin{array}{cccc} 1&0&0&0\\ 0&0&1&0\\ 0&1&0&0\\ 0&0&0&1 \end{array}\right] \end{equation} \begin{figure}
\caption{ \textit{SWAP Gate}. We show the control profile for realizing a SWAP gate with the FATO control. In red is the driving along the $x$ axis and in blue along the $y$ axis. For the bandwidth considered here ($\Delta\omega=400J$) the FATO control completely masks the BB control (solid black lines).}
\label{fig:Waveform2qb}
\end{figure} As shown in~\cite{Khaneja01}, this gate can be obtained in a time $t=3\pi/2J$ by applying instantaneous $\pi/2$ pulses about $x$ and $y$. This ideal control can be in practice replaced by finite-duration (rectangular) pulses, to account for finite control strength. In turn rectangular pulses can be replaced by FATO driving to take into account the control finite bandwidth. Figure~(\ref{fig:Waveform2qb}) shows the control sequence we implemented to analyze the fidelity behavior of FATO for two qubit control, as shown in Fig.~(\ref{fig:Fid2qb}).
\section{Fidelity}\label{sec:appendixB} Here we provide further details on the calculations of the fidelity. Due to FATO control, the system evolves under the Hamiltonian ${\mathcal{H}}\!=\!{\mathcal{H}}_{id}(t)+{\mathcal{H}}_R(t)$, with \begin{equation}\begin{array}{l} {\mathcal{H}}_{id}(t)\!=\!\omega_0/2[\sigma_z+f(t)\tan(\theta)\sigma_x]\\ {\mathcal{H}}_R(t)\!=\!\omega_0/2\tan(\theta) R_K\!(t)\sigma_x, \end{array}\end{equation}
Here $f(t)$ is the BB control function $f(t)=\Omega(t)/\overline\Omega$ and $R_K(t)$ the reminder of its truncated Fourier series, \begin{equation} R_K(t)=\sum_{k=K+1}^\infty[s_k\sin({2\pi k t}/T)+ c_k\cos({2\pi k t}/T)]. \label{eq:reminder} \end{equation}
In order to calculate the infidelity we consider the propagator $U_K$ achieved by implementing a control field according to FATO to order $K$, satisfying \begin{equation} i\dot U_K(t)\!=[{\mathcal{H}}_{id}(t)+{\mathcal{H}}_R(t)]U_K(t) \label{eq:U} \end{equation} and the propagator due to the ideal BB evolution, $U_{id}(t)$ defined by \begin{equation} i\dot U_{id}(t)={\mathcal{H}}_{id}(t)U_{id}(t) \label{eq:U} \end{equation} The infidelity of the truncated FATO control can be evaluated using the entanglement fidelity~\cite{Nielsen02}
\begin{equation}F=|\mathrm{Tr}[U_{id}^\dag(T)U_K(T)]|/2\end{equation}
We can decompose the total propagator $U_K(T)$ as $U_K(T)=U_{id}(T)U_R(T)$, by defining the error propagator $U_R(t)=U_{id}^\dag(t)U_K(t)$. Then the fidelity is simply defined as $F=|\mathrm{Tr}[U_{R}(T)]|/2$.
The error propagator can be evaluated by moving to the \textit{toggling} frame defined by the ideal control Hamiltonian. In this frame, the Hamiltonian becomes \begin{equation}\widetilde{\mathcal{H}}_R(t) = U^\dag_{id}(t){\mathcal{H}}_R(t)U_{id}(t)\end{equation} and the error propagator satisfies the Schr$\ddot{\textrm{o}}$dinger equation
\begin{equation}i\dot{\widetilde U}_R(t)=\widetilde{\mathcal{H}}_R(t)\widetilde U_R(t)\end{equation} We can approximate $U_R$ with a first-order Magnus expansion given by the effective Hamiltonian \begin{equation}\overline{\mathcal{H}}_R=\int_0^T \widetilde{\mathcal{H}}_R(t')dt'\end{equation} Consider for example the weak driving regime. The contribution to $\overline{\mathcal{H}}_R$ from each bang is given by the integral over the interval $[t_j,t_{j+1}]$ of \begin{equation} \widetilde{\mathcal{H}}_j\!=\!\omega_0\!\tan\!(\theta)R_K\!(t)U_{id}^\dag(t_j) [e^{i{\mathcal{H}}_\pm t}\sigma_x e^{-i{\mathcal{H}}_\pm t}]U_{id}(t_j) \label{eq:Ham_j} \end{equation} Each pair of ideal propagators $U_-U_+$ creates a rotation $e^{-2i\sigma_y\theta}$. Since the angle $\theta$ is small for weak driving, we can approximate this expression by ignoring the time evolution due to ${\mathcal{H}}_{id}$ \textit{during} the $j^{th}$ time interval and only considering its effects stroboscopically. Then Eq.~(\ref{eq:Ham_j}) reduces to \begin{eqnarray} &&\widetilde{\mathcal{H}}_j=\frac{\omega_0}2\tan(\theta)R_K(t)U_{id}(t_{j+1})\sigma_x U_{id}(t_{j+1})\\ &&\qquad = \frac{\omega_0}2\tan(\theta)R_K(t)[(-1)^j\cos(2j\theta)\sigma_x+\sin(2j\theta)\sigma_z]\nonumber \label{eq:Ham_japp} \end{eqnarray} Note that the sign of the $\sigma_x$ terms follows the same pattern as the BB function $f(t)$. Setting $j(t)=\lceil t/T\rceil$, we then need to evaluate the integrals \begin{equation}\frac2T\int_0^T f(t)\cos[2j(t)\theta]\sin(2\pi kt/T)dt=s_k\frac\theta\pi \end{equation} and
\begin{equation}\frac2T\int_0^T \sin[2j(t)\theta]\sin(2\pi kt/T)dt=-s_k\frac\theta\pi\tan(\theta)\end{equation} We thus obtain the average Hamiltonian
\begin{equation} \overline{\mathcal{H}}_R=\frac{\omega_0}2\tan(\theta)\frac\theta\pi\sum_{k=K+1}^\infty\!\!\!s_k^2\ [\sigma_x-\tan(\theta)\sigma_z],
\end{equation} where we recognize the mean error of the truncated Fourier series, $\mathcal E_k=\frac12\sum_{k=K+1}^\infty\!s_k^2$. \\
The fidelity is then given by $F=\cos(\|\overline{\mathcal{H}}_R\|T)$: \begin{eqnarray}
&&F=\cos\left(\frac{\theta}{\pi}\frac{\omega_0T}{2\cos(\theta)} \tan(\theta)\mathcal E_K \|\cos(\theta)\sigma_x-\sin(\theta)\sigma_z\|\right)\nonumber\\ &&\quad=\cos\left(\frac\pi2\tan(\theta)\mathcal E_K\right). \end{eqnarray}
A similar calculation can be done for the ultrastrong driving case. However in that case each ``bang'' has a long duration, thus we need to start from Eq.~(\ref{eq:Ham_j}) to find the average Hamiltonian and only approximate or numerical solutions can be found. Still, we find that the solutions depend on the mean Fourier error in a simple way, $F_s^x\approx\cos[2/\pi\sin(\theta)\mathcal E_K]$ and $F_s^y\approx\cos[2/\pi\tan(\theta)\mathcal E_K]$.
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\bibinfo {note} {Infinite decompositions are also possible. However, here an
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\begin{document}
\title{Gaussian Error Linear Units (GELUs)}
\begin{abstract} We propose the Gaussian Error Linear Unit (GELU), a high-performing neural network activation function. The GELU activation function is $x\Phi(x)$, where $\Phi(x)$ the standard Gaussian cumulative distribution function. The GELU nonlinearity weights inputs by their value, rather than gates inputs by their sign as in ReLUs ($x\mathbf{1}_{x>0}$). We perform an empirical evaluation of the GELU nonlinearity against the ReLU and ELU activations and find performance improvements across all considered computer vision, natural language processing, and speech tasks. \end{abstract}
\section{Introduction} Early artificial neurons utilized binary threshold units \citep{hopfield, mcculloch}. These hard binary decisions are smoothed with sigmoid activations, enabling a neuron to have a ``firing rate'' interpretation and to train with backpropagation. But as networks became deeper, training with sigmoid activations proved less effective than the non-smooth, less-probabilistic ReLU \citep{relu} which makes hard gating decisions based upon an input's sign. Despite having less of a statistical motivation, the ReLU remains a competitive engineering solution which often enables faster and better convergence than sigmoids. Building on the successes of ReLUs, a recent modification called ELUs \citep{elu} allows a ReLU-like nonlinearity to output negative values which sometimes increases training speed. In all, the activation choice has remained a necessary architecture decision for neural networks lest the network be a deep linear classifier.
Deep nonlinear classifiers can fit their data so well that network designers are often faced with the choice of including stochastic regularizer like adding noise to hidden layers or applying dropout \citep{dropout}, and this choice remains separate from the activation function. Some stochastic regularizers can make the network behave like an ensemble of networks, a pseudoensemble \citep{bachman}, and can lead to marked accuracy increases. For example, the stochastic regularizer dropout creates a pseudoensemble by randomly altering some activation decisions through zero multiplication. Nonlinearities and dropout thus determine a neuron's output together, yet the two innovations have remained distinct. More, neither subsumed the other because popular stochastic regularizers act irrespectively of the input and nonlinearities are aided by such regularizers.
In this work, we introduce a new nonlinearity, the Gaussian Error Linear Unit (GELU). It relates to stochastic regularizers in that it is the expectation of a modification to Adaptive Dropout \citep{standout}. This suggests a more probabilistic view of a neuron's output. We find that this novel nonlinearity matches or exceeds models with ReLUs or ELUs across tasks from computer vision, natural language processing, and automatic speech recognition.
\section{GELU Formulation}
We motivate our activation function by combining properties from dropout, zoneout, and ReLUs. First note that a ReLU and dropout
both yield a neuron's output with the ReLU deterministically multiplying the input by zero or one and dropout stochastically multiplying by zero. Also, a new RNN regularizer called zoneout stochastically multiplies inputs by one \citep{zoneout}. We merge this functionality by multiplying the input by zero or one, but the values of this zero-one mask are stochastically determined
while also dependent upon the input. Specifically, we can multiply the neuron input $x$ by $m \sim \text{Bernoulli}(\Phi(x))$, where $\Phi(x) = P(X\le x), X\sim \mathcal{N}(0,1)$ is the cumulative distribution function of the standard normal distribution. We choose this distribution since neuron inputs tend to follow a normal distribution, especially with Batch Normalization. In this setting, inputs have a higher probability of being ``dropped'' as $x$ decreases, so the transformation applied to $x$ is stochastic yet depends upon the input. \begin{wrapfigure}{r}{0.58\textwidth}
\centering
\includegraphics[width=0.56\textwidth]{nonlinearity-plot}
\caption{The GELU ($\mu=0, \sigma=1$), ReLU, and ELU ($\alpha=1$).}
\label{fig:nonlinearityplot} \end{wrapfigure} Masking inputs in this fashion retains non-determinism but maintains dependency upon the input value. A stochastically chosen mask amounts to a stochastic zero or identity transformation of the input. This is much like Adaptive Dropout \citep{standout}, but adaptive dropout is used in tandem with nonlinearities and uses a logistic not standard normal distribution. We found that it is possible to train competitive MNIST and TIMIT networks solely with this stochastic regularizer, all without using any nonlinearity.
We often want a deterministic decision from a neural network, and this gives rise to our new nonlinearity. The nonlinearity is the expected transformation of the stochastic regularizer on an input $x$, which is $\Phi(x)\times Ix + (1 - \Phi(x))\times 0x = x\Phi(x)$. Loosely, this expression states that we scale $x$ by how much greater it is than other inputs. Since the cumulative distribution function of a Gaussian is often computed with the error function, we define the Gaussian Error Linear Unit (GELU) as \[ \text{GELU}(x) = xP(X\le x) = x\Phi(x) = x \cdot \frac{1}{2}\left[1 + \text{erf}(x/\sqrt{2})\right]. \] We can approximate the GELU with \[ 0.5x (1 + \tanh[\sqrt{2/\pi}(x + 0.044715x^3)]) \] or \[ x \sigma(1.702 x), \] if greater feedforward speed is worth the cost of exactness.
We could use different CDFs. For example we could use Logistic Distribution CDF $\sigma(x)$ to get what we call the Sigmoid Linear Unit (SiLU) $x\sigma(x)$. We could use the CDF of $\mathcal{N}(\mu, \sigma^2)$ and have $\mu$ and $\sigma$ be learnable hyperparameters, but throughout this work we simply let $\mu = 0$ and $\sigma = 1$. Consequently, we do not introduce any new hyperparameters in the following experiments. In the next section, we show that the GELU exceeds ReLUs and ELUs across numerous tasks.
\section{GELU Experiments} We evaluate the GELU, ELU, and ReLU on MNIST classification (grayscale images with 10 classes, 60k training examples and 10k test examples), MNIST autoencoding, Tweet part-of-speech tagging (1000 training, 327 validation, and 500 testing tweets), TIMIT frame recognition (3696 training, 1152 validation, and 192 test audio sentences), and CIFAR-10/100 classification (color images with 10/100 classes, 50k training and 10k test examples). We do not evaluate nonlinearities like the LReLU because of its similarity to ReLUs (see \cite{lrelu} for a description of LReLUs).
\subsection{MNIST Classification} \begin{figure}
\caption{MNIST Classification Results. Left are the loss curves without dropout, and right are curves with a dropout rate of $0.5$. Each curve is the the median of five runs. Training set log losses are the darker, lower curves, and the fainter, upper curves are the validation set log loss curves.}
\label{fig:mnist1}
\end{figure}
Let us verify that this nonlinearity competes with previous activation functions by replicating an experiment from \citet{elu}. To this end, we train a fully connected neural network with GELUs ($\mu = 0, \sigma = 1$), ReLUs, and ELUs ($\alpha = 1$). Each 8-layer, 128 neuron wide neural network is trained for 50 epochs with a batch size of 128. This experiment differs from those of Clevert et al.~in that we use the Adam optimizer \citep{adam} rather than stochastic gradient descent without momentum, and we also show how well nonlinearities cope with dropout. Weights are initialized with unit norm rows, as this has positive impact on each nonlinearity's performance~\citep{hendrycks, weights, init2}. Note that we tune over the learning rates $\{10^{-3},10^{-4},10^{-5}\}$ with 5k validation examples from the training set and take the median results for five runs. Using these classifiers, we demonstrate in Figure \ref{fig:robustness} that classifiers using a GELU can be more robust to noised inputs. Figure~\ref{fig:mnist1} shows that the GELU tends to have the lowest median training log loss with and without dropout. Consequently, although the GELU is inspired by a different stochastic process, it comports well with dropout.
\begin{figure}
\caption{MNIST Robustness Results. Using different nonlinearities, we record the test set accuracy decline and log loss increase as inputs are noised. The MNIST classifier trained without dropout received inputs with uniform noise $\text{Unif}[-a,a]$ added to each example at different levels $a$, where $a=3$ is the greatest noise strength. Here GELUs display robustness matching or exceeding ELUs and ReLUs.}
\label{fig:robustness}
\end{figure}
\subsection{MNIST Autoencoder} \begin{figure}
\caption{MNIST Autoencoding Results. Each curve is the median of three runs. Left are loss curves for a learning rate of $10^{-3}$, and the
right figure is for a $10^{-4}$ learning rate. Light, thin curves correspond to test set log losses.}
\label{fig:mnist2}
\end{figure}
We now consider a self-supervised setting and train a deep autoencoder on MNIST \citep{desjardens}. To accomplish this, we use a network with layers of width 1000, 500, 250, 30, 250, 500, 1000, in that order. We again use the Adam optimizer and a batch size of 64. Our loss is the mean squared loss. We vary the learning rate from $10^{-3}$ to $10^{-4}$. We also tried a learning rate of $0.01$ but ELUs diverged, and GELUs and RELUs converged poorly. The results in Figure~\ref{fig:mnist2} indicate the GELU accommodates different learning rates and significantly outperforms the other nonlinearities.
\subsection{Twitter POS Tagging} Many datasets in natural language processing are relatively small, so it is important that an activation generalize well from few examples. To meet this challenge we compare the nonlinearities on POS-annotated tweets \citep{gimpel,owoputi} which contain 25 tags. The tweet tagger is simply a two-layer network with pretrained word vectors trained on a corpus of 56 million tweets~\citep{owoputi}. The input is the concatenation of the vector of the word to be tagged and those of its left and right neighboring words. Each layer has 256 neurons, a dropout keep probability of 0.8, and the network is optimized with Adam while tuning over the learning rates $\{10^{-3}, 10^{-4}, 10^{-5}\}$. We train each network five times per learning rate, and the median test set error is 12.57\% for the GELU, 12.67\% for the ReLU, and 12.91\% for the ELU.
\subsection{TIMIT Frame Classification}
\begin{figure}
\caption{TIMIT Frame Classification. Learning curves show training set convergence, and the lighter curves show the validation set convergence.}
\label{fig:timit}
\end{figure}
Our next challenge is phone recognition with the TIMIT dataset which has recordings of 680 speakers in a noiseless environment. The system is a five-layer, 2048-neuron wide classifier as in~\citep{timitdeep} with 39 output phone labels and a dropout rate of 0.5 as in~\citep{srivastava}. This network takes as input 11 frames and must predict the phone of the center frame using 26 MFCC, energy, and derivative features per frame. We tune over the learning rates $\{10^{-3},10^{-4},10^{-5}\}$ and optimize with Adam. After five runs per setting, we obtain the median curves in Figure~\ref{fig:timit}, and median test error chosen at the lowest validation error is 29.3\% for the GELU, 29.5\% for the ReLU, and 29.6\% for the ELU.
\subsection{CIFAR-10/100 Classification} Next, we demonstrate that for more intricate architectures the GELU nonlinearity again outperforms other nonlinearities. We evaluate this activation function using CIFAR-10 and CIFAR-100 datasets \citep{cifar} on shallow and deep convolutional neural networks, respectively.
\begin{figure}
\caption{CIFAR-10 Results. Each curve is the median of three runs. Learning curves show training set error rates, and the lighter curves show the test set error rates.}
\label{fig:c10}
\end{figure}
Our shallower convolutional neural network is a 9-layer network with the architecture and training procedure from~\cite{weightnorm} while using batch normalization to speed up training. The architecture is described in \cref{appendixb} and recently obtained state of the art on CIFAR-10 without data augmentation. No data augmentation was used to train this network. We tune over the learning initial rates $\{10^{-3},10^{-4},10^{-5}\}$ with 5k validation examples then train on the whole training set again based upon the learning rate from cross validation. The network is optimized with Adam for 200 epochs, and at the 100th epoch the learning rate linearly decays to zero. Results are shown in Figure~\ref{fig:c10}, and each curve is a median of three runs. Ultimately, the GELU obtains a median error rate of \textbf{7.89}\%, the ReLU obtains 8.16\%, and the ELU obtains 8.41\%.
Next we consider a wide residual network on CIFAR-100 with 40 layers and a widening factor of $4$ \citep{wrn}. We train for 50 epochs with the learning rate schedule described in~\citep{SGDR} ($T_0=50, \eta=0.1$) with Nesterov momentum, and with a dropout keep probability of 0.7. Some have noted that ELUs have an exploding gradient with residual networks \citep{eluresnet}, and this is alleviated with batch normalization at the end of a residual block. Consequently, we use a Conv-Activation-Conv-Activation-BatchNorm block architecture to be charitable to ELUs. Over three runs we obtain the median convergence curves in Figure~\ref{fig:c100}. Meanwhile, the GELU achieves a median error of \textbf{20.74}\%, the ReLU obtains 21.77\% (without our changes described above, the original 40-4 WideResNet with a ReLU obtains 22.89\% \citep{wrn}), and the ELU obtains 22.98\%.
\begin{figure}
\caption{CIFAR-100 Wide Residual Network Results. Learning curves show training set convergence with dropout on, and the lighter curves show the test set convergence with dropout off.}
\label{fig:c100}
\end{figure}
\section{Discussion} Across several experiments, the GELU outperformed previous nonlinearities, but it bears semblance to the ReLU and ELU in other respects. For example, as $\sigma \to 0$ and if $\mu = 0$, the GELU becomes a ReLU. More, the ReLU and GELU are equal asymptotically. In fact, the GELU can be viewed as a way to smooth a ReLU. To see this, recall that $\text{ReLU}=\max(x,0)=x\mathds{1}(x>0)$ (where $\mathds{1}$ is the indicator function), while the GELU is $x\Phi(x)$ if $\mu=0,\sigma=1$. Then the CDF is a smooth approximation to the binary function the ReLU uses, like how the sigmoid smoothed binary threshold activations. Unlike the ReLU, the GELU and ELU can be both negative and positive. In fact, if we used the cumulative distribution function of the standard Cauchy distribution, then the ELU (when $\alpha=1/\pi$) is asymptotically equal to $xP(C\le x), C \sim \textsf{Cauchy}(0,1)$ for negative values and for positive values is $xP(C\le x)$ if we shift the line down by $1/\pi$. These are some fundamental relations to previous nonlinearities.
However, the GELU has several notable differences. This non-convex, non-monotonic function is not linear in the positive domain and exhibits curvature at all points. Meanwhile ReLUs and ELUs, which are convex and monotonic activations, are linear in the positive domain and thereby can lack curvature. As such, increased curvature and non-monotonicity may allow GELUs to more easily approximate complicated functions than can ReLUs or ELUs. Also, since $\text{ReLU}(x)=x\mathds{1}(x>0)$ and $\text{GELU}(x)=x\Phi(x)$ if $\mu=0,\sigma=1$, we can see that the ReLU gates the input depending upon its sign, while the GELU weights its input depending upon how much greater it is than other inputs. In addition and significantly, the GELU has a probabilistic interpretation given that it is the expectation of a stochastic regularizer.
We also have two practical tips for using the GELU. First we advise using an optimizer with momentum when training with a GELU, as is standard for deep neural networks. Second, using a close approximation to the cumulative distribution function of a Gaussian distribution is important. A sigmoid function $\sigma(x)=1/(1+e^{-x})$ is an approximation of a cumulative distribution function of a normal distribution. However, we found that a Sigmoid Linear Unit (SiLU) $x\sigma(x)$ performs worse than GELUs but usually better than ReLUs and ELUs, so our SiLU is also a reasonable nonlinearity choice. Instead of using a $x\sigma(x)$ to approximate $\Phi(x)$, we used $0.5x (1 + \tanh[\sqrt{2/\pi}(x + 0.044715x^3)])$ \citep{approx}\footnote{Thank you to Dmytro Mishkin for bringing an approximation like this to our attention.} or $x \sigma(1.702 x)$. Both are sufficiently fast, easy-to-implement approximations, and we used the former in every experiment in this paper.
\section{Conclusion} For the numerous datasets evaluated in this paper, the GELU exceeded the accuracy of the ELU and ReLU consistently, making it a viable alternative to previous nonlinearities.
\section*{Acknowledgment} We would like to thank NVIDIA Corporation for donating several TITAN X GPUs used in this research.
\appendix
\section{Neural network architecture for CIFAR-10 experiments} \label{appendixb} \begin{table}[H] \caption{Neural network architecture for CIFAR-10.} \label{sample-table} \begin{center} \begin{tabular}{lll} \multicolumn{1}{c}{\bf Layer Type} &\multicolumn{1}{c}{\bf \# channels} &\multicolumn{1}{c}{\bf $x,y$ dimension} \\ \hline \\ raw RGB input &3 &32\\ ZCA whitening &3 &32\\ Gaussian noise $\sigma=0.15$ &3 &32\\ $3\times 3$ conv with activation &96 &32\\ $3\times 3$ conv with activation &96 &32\\ $3\times 3$ conv with activation &96 &32\\ $2\times 2$ max pool, stride 2 &96 &16\\ dropout with $p=0.5$ &96 &16\\ $3\times 3$ conv with activation &192 &16\\ $3\times 3$ conv with activation &192 &16\\ $3\times 3$ conv with activation &192 &16\\ $2\times 2$ max pool, stride 2 &192 &8\\ dropout with $p=0.5$ &192 &8\\ $3\times 3$ conv with activation &192 &6\\ $1\times 1$ conv with activation &192 &6\\ $1\times 1$ conv with activation &192 &6\\ global average pool &192 &1\\ softmax output &10 &1\\
\end{tabular}\label{tab:c10architecture} \end{center} \end{table}
\section{History of the GELU and SiLU} This paper arose from DH’s first research internship as an undergraduate in June 2016. The start of the week after, this paper was put on arXiv, in which we discuss smoother ReLU activation functions ($x\times P(X\le x)$) and their relation to stochastic regularizers. In 2016, we submitted the paper to ICLR and made the paper and code publicly available. In the paper, we introduced and coined the Sigmoid Linear Unit (SiLU) as $x\cdot \sigma(x)$.
In the first half of 2017, Elfwing et al. published a paper that proposed the same activation function as SiLU, $x\cdot \sigma(x)$, which they called ``SIL.'' At the end of 2017, over a year after this paper was first released, Quoc Le and others from Google Brain put out a paper proposing $x\cdot \sigma(x)$ without citing either the Elfwing et al. paper or this work. Upon learning this, we contacted both parties. Elfwing quickly updated their work to call the activation the ``SiLU'' instead of ``SIL'' to recognize that we originally introduced the activation.
Unlike Elfwing et al., the Google Brain researchers continued calling the activation ``swish.'' However, there was no novelty. The first author of the ``swish'' paper stated their oversight in public, saying, ``As has been pointed out, we missed prior works that proposed the same activation function. The fault lies entirely with me for not conducting a thorough enough literature search.'' To subdue criticism, an update to the paper was released a week later. Rather than give credit to this work for the SiLU, the update only cited this work for the GELU so that the ``swish'' appeared more novel. In the updated paper, a learnable hyperparameter $\beta$ was introduced, and the swish was changed from $x\cdot \sigma(x)$ to $x\cdot \sigma(\beta \cdot x)$. This staked all of the idea's novelty on an added learnable hyperparameter $\beta$.
Despite the addition of the hyperparameter beta, nearly all of the community still used the original ``swish'' function without $\beta$ (i.e., with $\beta=1$). Since this paper was from Google Brain, the Tensorflow implementation ended up being called ``swish,'' and the default setting removed $\beta$, rendering it identical to the SiLU. The practice of adding an unused hyperparameter allowed claiming novelty while effectively receiving credit for an idea that originated elsewhere. Future papers with the same senior authors persistently referred to the ``swish'' function even when not using $\beta$, making it identical to the SiLU, originally proposed in this work. This resulted in the ``swish'' paper inappropriately gaining credit for the idea.
Things changed as the GELU began to be used in BERT and GPT, becoming the default activation for state-of-the-art Transformers. Now it is substantially more commonly used than the SiLU.
Separately, a reddit post ``Google has a credit assignment problem in research'' became popular and focused on how they refer to the SiLU as the swish. As an example, they mentioned ``Smooth Adversarial Training'' as an example of poor credit assignment. In the ``Smooth Adversarial Training'' paper, which came from the senior author of the swish, the term ``swish'' was used instead of ``SiLU.'' To reduce blowback from the post, the authors updated the paper and replaced ``swish'' with the ``SiLU,'' recognizing this paper as the original source of the idea. After this post, popular libraries such as Tensorflow and PyTorch also began to rename the function to ``SiLU'' instead of ``swish.'' For close observers, this issue has been largely settled, and we are grateful for the proper recognition that has largely come to pass.
\end{document} |
\begin{document}
\begin{frontmatter}
\title{On closure operators related to maximal tricliques in tripartite hypergraphs}
\author{Dmitry I. Ignatov}
\begin{abstract}
Triadic Formal Concept Analysis (3FCA) was introduced by Lehman and Wille almost two decades ago. And many researchers work in Data Mining and Formal Concept Analysis using the notions of closed sets, Galois and closure operators, closure systems, but up-to-date even though that different researchers actively work on mining triadic and n-ary relations, a proper closure operator for enumeration of triconcepts, i.e. maximal triadic cliques of tripartite hypergaphs, was not introduced. In this paper we show that the previously introduced operators for obtaining triconcepts and maximal connected and complete sets (MCCSs) are not always consistent and provide the reader with a definition of valid closure operator and associated set system. Moreover, we study the difficulties of related problems from order-theoretic and combinatorial point view as well as provide the reader with justifications of the complexity classes of these problems.
\end{abstract}
\begin{keyword}
Triaidic Formal Concept Analysis \sep Closure operator \sep triadic hypergraph \sep triset \sep tripartite graphs \end{keyword}
\end{frontmatter}
\section{Introduction} Pattern mining is one of the most important Data Mining areas and often relies on fundamental notions from theoretical computer science and algebra like fixpoints, closure operators and lattices \citep{Zaki:2005a,Boley:2010}. Formal Concept Analysis \citep{Ganter:1999} can be considered as an elegant algebraic framework to deal with (frequent) closed sets of objects and their attributes (formal concepts or maximal bicliques) by means of two closure operators formed by Galois connection over these sets.
Recent studies showed that there are efficient algorithms for building all formal concepts not only in binary object-attribute case but in ternary (\textsc{TRIAS}, \citep{Jaschke:2006}) and $n$-ary cases (\textsc{Data-Peeler}, \citep{Cerf:2009}).
Several researchers tried to develop a proper closure operator for triadic \citep{Trabelsi:2012} and n-ary cases \citep{Spyropoulou:2014}. However the detailed analysis in this paper shows that the concept-forming operator in \citep{Trabelsi:2012} is not always monotone on triset systems. An interesting approach from \citep{Spyropoulou:2014} can be used to enumerate formal triconcepts as the maximal fixpoints of a set system of closed and connected sets (CCS) but suffers from presence of phantom hyperedges because of the lossy $k$-partite graph encoding. In this paper, we show how to define a proper triset system for the concept forming operator from \citep{Trabelsi:2012} that makes it a closure operator, describe the family of closure operators of this type and investigate their properties, and prove that there is no an associated closure operator on the whole triset system for a given tricontext. We also introduce a notion of (maximal) switching generator -- a triset resulting in different closed patterns that contain it. In addition we show how to deal with lossy hyperedge encoding and phantom edges to generate triconcepts as maximal connected and complete sets.
The rest of the paper is organised as follows. In Section~\ref{sec:MMP}, we recall basic definitions from FCA and its polyadic extensions and reproduce necessary definitions and propositions from \citep{Spyropoulou:2014}. In Section~\ref{sec:recclos}, we discuss the studied concept and closed CCS forming operators with a focus on their inconsistency conditions. Section \ref{sec:triclo} reports our main results. Section~\ref{sec:relw} discusses related work and and Section~\ref{sec:concl} concludes the paper.
\section{Multimodal and multirelational closed patterns} \label{sec:MMP}
\subsection{Formal Concept Analysis and its polyadic extensions}
First, we recall some basic notions from Formal Concept Analysis (FCA) \citep{Ganter:1999}.
Let $G$ and $M$ be sets, called the set of objects and attributes, respectively, and let $I$ be a relation $I\subseteq G\times M$: for $g\in G, \ m\in M$, $gIm$ holds iff the object $g$ has the attribute $m$. The triple $\mathbb{K}=(G,M,I)$ is called a \emph{(formal) context}.
A \emph{triadic context} $\mathbb{K}=(G,M,B,Y)$ consists of sets $G$ (objects), $M$ (attributes), and $B$ (conditions), and ternary relation $Y\subseteq G \times M \times B$ \citep{Lehmann:1995}. An incidence $(g, m, b) \in Y$ shows that the object $g$ has the attribute $m$ under the condition $b$.
An \emph{$n$-adic context} is an $(n + 1)$-tuple $\mathbb{K} = (K_1,K_2, \ldots,K_n, Y)$, where $Y$ is an $n$-ary relation between sets $K_1, \ldots , K_n$ \citep{Voutsadakis:02}.
\subsubsection{Concept forming operators and formal concepts} \label{sec:1b}
If $A\subseteq G$, $B\subseteq M$ are arbitrary subsets of objects and attributes, respectively, then the {\it Galois connection} is given by the following {\it derivation operators}:
\begin{eqnarray} \begin{array}{c} A' = \{m\in M\mid gIm \ {\rm for\ all}\ g\in A\}, \\ B' = \{g\in G\mid gIm \ {\rm for\ all}\ m\in B\}. \end{array} \end{eqnarray}
If we have several contexts, the derivation operator of a context $(G, M, I)$ is denoted by $(.)^I$.
The pair $(A,B)$, where $A\subseteq G$, $B\subseteq M$, $A' = B$, and $B' = A$ is called a {\it (formal) concept (of the context $\mathbb{K}$)} with {\it extent} $A$ and {\it intent} $B$ (in this case we have also $A'' = A$ and $B'' = B$).
The concepts, ordered by $(A_1,B_1)\geq (A_2,B_2) \iff A_1\supseteq A_2 (B_2\supseteq B_1)$, form a complete lattice, called \emph{the concept lattice} $\underline{{\mathfrak B}}(G,M,I)$.
\subsubsection{Formal concepts in triadic and in n-ary contexts} \label{sec:1c}
For convenience, a triadic context is denoted by $\mathbb{K}=(X_1,X_2,X_3,Y)$\footnote{Note that in the title we refer to a formal tricontext as a tripartite hypergraph since we deal with three types of vertices connected by triadic hyperedges.}. A triadic context $\mathbb{K}=(X_1,X_2,X_3,Y)$ gives rise to the following dyadic contexts
$\mathbb{K}^{(1)}=(X_1, X_2\times X_3, Y^{(1)})$, $\mathbb{K}^{(2)}=(X_2, X_1\times X_3, Y^{(2)})$, $\mathbb{K}^{(3)}=(X_3, X_1\times X_2, Y^{(3)})$,
where $gY^{(1)}(m,b):\Leftrightarrow mY^{(2)}(g,b):\Leftrightarrow bY^{(3)}(g,m):\Leftrightarrow (g,m,b) \in Y$. The derivation operators (primes or concept-forming operators) induced by $\mathbb{K}^{(i)}$ are denoted by $(.)^{(i)}$. For each induced dyadic context we have two kinds of such derivation operators. That is, for $\{i,j,k\}=\{1,2,3\}$ with $j<k$ and for $Z \subseteq X_i$ and $W \subseteq X_j\times X_k$, the $(i)$-derivation operators are defined by:
$$Z \mapsto Z^{(i)} = \{(x_j,x_k) \in X_j\times X_k| x_i, x_j, x_k \mbox{ are related by Y for all } x_i \in Z\},$$
$$W \mapsto W^{(i)} = \{x_i \in X_i| x_i, x_j, x_k \mbox{ are related by Y for all } (x_j,x_k) \in W\}.$$ Formally, a triadic concept of a triadic context $\mathbb{K}=(X_1,X_2,X_3,Y)$ is a triple $(A_1,A_2,A_3)$ of $A_1 \subseteq X_1, A_2 \subseteq X_2, A_3 \subseteq X_3$ such that for every $\{i,j,k\}=\{1,2,3\}$ with $j<k$ we have $(A_j \times A_k)^{(i)}=A_i$. For a certain triadic concept $(A_1,A_2,A_3)$, the components $A_1$, $A_2$, and $A_3$ are called the extent, the intent, and the modus of $(A_1,A_2,A_3)$. It is important to note that for interpretation of $\mathbb{K}=(X_1,X_2,X_3,Y)$ as a three-dimensional cross table, according to our definition, under suitable permutations of rows, columns, and layers of the cross table, the triadic concept $(A_1,A_2,A_3)$ is interpreted as a maximal cuboid full of crosses.
The set of all triadic concepts of $\mathbb{K}=(X_1,X_2,X_3,Y)$ is denoted by $\mathfrak{T}(\mathbb{K})$. However this set does not form a partial order by extent inclusion since it is possible for the same triconcept extent to have different combinations of intent and modus components \citep{Wille:1995,Lehmann:1995}, similarly, for orderings along the attribute and condition components.
There is a quasiorder $\lesssim_i$ for each $i \in \{1, 2, 3\}$ and its corresponding equivalence relation $\sim_i$ is defined by $$(A_1,A_2,A_3) \lesssim_i (B_1,B_2,B_3): \Longleftrightarrow A_i \subseteq B_i \mbox{ and } $$ $$(A_1, A_2, A_3) \sim_i (B_1, B_2, B_3): \Longleftrightarrow A_i = B_i.$$
These quasiorders satisfy the antiordinal dependencies \citep{Wille:1995}: For $\{i,j,k\} = \{1,2,3\}$ and all triconcepts $(A_1,A_2,A_3)$ and $(B_1,B_2,B_3)$ from $\mathfrak{T}(\mathbb{K})$ it holds that $(A_1, A_2, A_3) \lesssim_i (B_1, B_2, B_3)$ and $(A_1, A_2, A_3) \lesssim_j (B_1, B_2, B_3)$ imply $(A_1, A_2, A_3) \gtrsim_k (B_1, B_2, B_3)$.
One may introduce $n$-adic formal concepts without $n$-ary concept forming operators. The $n$-adic concepts of an $n$-adic context $(K_1, \ldots ,K_n, Y)$ are exactly the maximal $n$-tuples $(A_1, \ldots , A_n)$ in $2^{K_1} \times \cdots \times 2^{K_n}$, where $A_1 \times \cdots \times A_n \subseteq Y$ with respect to component-wise set inclusion \citep{Voutsadakis:02}. The notion of $n$-adic concept lattice can be introduced similarly to the triadic case \citep{Voutsadakis:02}.
\subsection{Maximal closed connected sets} Here we introduce necessary defintions and results from a series of papers on mining maximal closed connected sets \citep{Spyropoulou:2014,Lijffijt:2016}. Note that the authors prefer to use terminology close to relational databases but the main definitions can be easily reproduced in terms of $k$-partite graphs; to find related works in FCA community one may refer to Relational Concept Analysis \cite{Hacene:2013}.
\cite{Spyropoulou:2014} formalised a multi-relational database (MRD) as a tuple $\mathbb{D} = (E, t, \mathcal{R}, R)$, where $E$ is a finite set of entities that is partitioned into $n$ entity types by a mapping $t: E \to \{1, \ldots, n\}$, i.e., $E = E_1 \sqcup \cdots \sqcup E_k$\footnote{Here and later, $\sqcup$ means disjoint union.} with $E_i = \{e \in E\mid t (e) = i \}$. Moreover, $R \subseteq \{\{i, j\}\mid i, j \in \{1, \ldots , k\}, i \neq j \}$ is a set of relationship types such that for each $\{i, j\} \in R$ there is a binary relation $\mathcal{R}_{\{i, j\}} \subseteq \{\{e_i , e_j\}\mid e_i \in E_i , e_j \in E_j \}$. The set $\mathcal R$ then is the union of all these relations, i.e., $\mathcal{R} = \bigcup_{\{i, j\} \in R} \mathcal{R}_{\{i, j \}}$. This definition allows relationship types can be many-to-many, one-to-many, or one-to-one, depending on how many relationships the entities of either entity types can participate in. The authors do not allow relationship types between an entity type and itself since they mainly concentrate on relations between entities of different types, but the former can be modeled by having two copies of the same entity type and a relationship type between them.
\begin{definition} (Completeness) \citep{Spyropoulou:2014} A set $F \subseteq E$ is complete if for all $e, \tilde{e} \in F$ with $\{t(e), t(\tilde{e})\} \in R$ it holds that $\{e, \tilde{e}\} \subseteq \mathcal{R}_{\{t (e),t (\tilde{e} )\}} $. \end{definition}
\begin{definition} (Connectedness) \citep{Spyropoulou:2014} A set $F \subseteq E$ is connected if for all $e, \tilde{e} \in F$ there is a sequence $e = e_1, \ldots , e_l = \tilde{e}$ with $\{e_1, \ldots , e_l\} \subseteq F$ such that for $i \in \{1, \ldots , l \}$ it holds that $\{e_i , e_{i+1}\} \in R$. \end{definition}
It implies that a subset of size larger than one can be connected only if it contains entities of at least two different types.
A set $F \subseteq E$ is a Complete Connected Subset (CCS) if it satisfies both connectedness and completeness.
A Maximal Complete Connected Subset (MCCS) is a CCS to which no element can be added without violating connectedness or completeness.
For a database $\mathbb{D} = (E,t,\mathcal{R}, R)$ the set system of CCSs, is defined as $\mathcal{F}_\mathbb{D} = \{F \subseteq E\mid F \mbox{ is connected and complete} \}$. From an algorithmic point of view, the property of strong accessibility means that for two CCSs $X, Y \in \mathcal{F}_\mathbb{D}$ with $X \subseteq Y$ , it is possible to iteratively extend $X$ by one element at a time, only passing via sets from the set system and finally obtain $Y$. Formally, for a set system $F \subseteq 2^A$, where $A$ is the ground set, and a set $F \in \mathcal{F}$, let us denote by $Aug(F) = \{a \in A\mid F \cup \{a\} \in \mathcal{F}\}$ the set of valid augmentation elements of $F$. Then $\mathcal{F}$ is called strongly accessible if for all $X \subset Y \subseteq A$ with $X, Y \in \mathcal{F}$ there is an element $e \in (Aug(X) \setminus X) \cap Y$.
\begin{theorem} \citep{Spyropoulou:2014} For all relational databases $\mathbb{D} = (E, t,\mathcal{R}, R)$, the set system $\mathcal{F}_\mathbb{D}$ of CCSs is strongly accessible. \end{theorem}
Specifically for the set system $\mathcal{F}_\mathbb{D}$ of CCSs, and given a relational database $\mathbb{D} = (E, t,\mathcal{R}, R)$, the set $Aug(F)$ corresponds to the following set: $Aug(F) = \{e \in E\mid F \cup \{e\} \mbox{ is complete and connected} \}$. Note that for the sake of efficiency $Aug(F)$ can be recursively updated.
To define a closure operator for the set system $\mathcal{F}_\mathbb{D}$ the authors make use of the set of compatible entities which is defined as follows:
\begin{definition} (Compatible entities) \citep{Spyropoulou:2014} For a relational database $\mathbb{D} = (E, t, \mathcal{R}, R)$ the set of compatible entities of a set $F \in \mathcal{F}_\mathbb{D}$ is defined as $Comp(F) = \{e \in E \mid F \cup {e} \mbox{ is complete}\}$. \end{definition}
\begin{definition} ($g$ operator) \citep{Spyropoulou:2014} For a relational database $\mathbb{D} = (E, t,\mathcal{R}, R)$ the operator $g : \mathcal{F}_\mathbb{D} \to 2^E$ is defined as $g(F) = \{ e \in Aug(F)\mid Comp(F \cup {e}) = Comp(F) \}$. \end{definition}
\begin{proposition} \citep{Spyropoulou:2014} For all relational databases $\mathbb{D} = (E, t, \mathcal{R}, R)$, the codomain of the $g$ operator is the set system $\mathcal{F}_\mathbb{D}$ of CCSs and $g$ is extensive and monotone. \end{proposition}
\begin{proposition} \label{prop:idemp} \citep{Spyropoulou:2014} For all relational databases $\mathbb{D} = (E, t, \mathcal{R}, R)$ with the property that $e \in E$ such that $\{e\} \cup E_i$ is complete and connected for an $i \in t(E)$, the operator $g$ is idempotent. \end{proposition}
\begin{corollary} \citep{Spyropoulou:2014} For all relational databases $\mathbb{D} = (E, t,\mathcal{R}, R)$, with the property that $e \in E$ such that $\{e\} \cup E_i$ is complete and connected for an $i \in t(E)$, the operator $g$ is a closure operator. \end{corollary}
Note that, the technical requirement in Proposition~\ref{prop:idemp} for $g$ being idempotent may be fulfilled by adding an isolated vertex $\{e_0\}$ to $E_i$ for all $E_i \subseteq E$ and $e \in E\setminus E_i$, where $E_i \cup \{e\}$ is CCS.
\begin{figure}
\caption{On idempotency of $g(\cdot)$}
\label{fig:Idempotency}
\end{figure}
\begin{example} In Figure~\ref{fig:Idempotency}, on the left one can see the violation of idempotency of $g(\cdot)$ since $g(\{r_1,r_2,p_1\})=\{r_1,r_2,p_1,f\}$ and
$g(g(\{r_1,r_2,p_1\}))=\{r_1,r_2,p_1,f, u_1, u_2, u_3\}$. On the right graph of Figure \ref{fig:Idempotency} the idempotency fulfills since $g(\{r_1,r_2,p_1\})=g(g(\{r_1,r_2,p_1\}))$ $=\{r_1,r_2,p_1\}$. It happens since for the left graph $$Comp(\{r_1,r_2,p_1\}\cup f)=\{r_1,r_2,p_1,f, u_1, u_2, u_3\}=Comp(\{r_1,r_2,p_1\}),$$ but for the right one $$Comp(\{r_1,r_2,p_1\}\cup f)=\{r_1,r_2,p_1,f, u_1, u_2\} \neq Comp(\{r_1,r_2,p_1\})=\{r_1,r_2,p_1,f, u_1, u_2, u_3\}.$$ \end{example}
\section{Pitfalls of recent candidates for closure operators in triadic case}\label{sec:recclos}
\subsection{Non-monotonicity of TriCons concept forming operator}
To simplify further considerations of tri-sets, triadic concepts and multirelational databases both as tuples and sets, we introduce two interrelated operators.
\begin{definition} \citep{Trabelsi:2012} Let $\mathbb{K}=(G,M,B,I)$ be a formal tricontext. A triple $(X,Y,Z)$ is called a triset of $\mathbb{K}$ iff $ X \times Y \times Z \subseteq I$. \end{definition}
Note that \cite{Cerf:2009} define a triset of $\mathbb{K}$ differently: $ X \times Y \times Z \in 2^G \times 2^M \times 2^B$. We keep the former definition to work with $h(\cdot)$ in the original setting \citep{Trabelsi:2012}.
Note that according the definition of Cartesian product, if at least one of the sets $X,Y$ or $Z$ is $\emptyset$ \citep{Simovici:2008}, then $X \times Y \times Z = \emptyset$, so $\emptyset \subseteq I$ and $(X,Y,Z)$ is a triset. However trisets $(X,Y,\emptyset)$ and $(X, \emptyset, Z)$ have different structure even though $X \times Y \times \emptyset = X \times \emptyset \times Z = \emptyset \subseteq I$.
\begin{definition} For a formal tricontext $\mathbb{K}=(G,M,B,I)$ and any triple $(X,Y,Z) \subseteq 2^G\times 2^M \times 2^B$ (e.g. triconcept) of $\mathbb{K}$ the operator $flat: 2^G\times 2^M \times 2^B \to 2^{G\sqcup M \sqcup B}$ is defined as follows: $flat(X,Y,Z)= X \sqcup Y \sqcup Z$. \end{definition}
\begin{definition} For a given formal n-context $\mathbb{K}=(E_1,\ldots,E_n, I \subseteq E_1 \times \ldots \times E_n)$ (or multi-relational database $\mathbb{D}=(E,t,\mathcal{R},R)$), where $E=flat(E_1,\ldots,E_n)$, and $S \subseteq 2^{E}$, the operator $tuple: 2^E \to 2^{E_1} \times \ldots \times 2^{E_n}$ is defined as follows: $tuple(S)= (E_1 \cap S, \ldots, E_n \cap S)$. \end{definition}
Triple compositions of $tuple(\cdot)$ and $flat(\cdot)$ operators form identity operators $tuple(flat(tuple(\cdot)))=id_S(\cdot)$ and $flat(tuple(flat()))=id_T(\cdot)$ over sets and tuples respectively.
Note that for trisets $t_1=(A_1,B_1,C_1)$ and $t_2=(A_2,B_2,C_2)$, $t_1 \sqsubseteq t_2$ means that $A_1 \times B_1 \times C_1 \subseteq A_2 \times B_2 \times C_2$, i.e. every triple $(a,b,c) \in (A_1,B_1,C_1)$ is in $(A_2,B_2,C_2)$. It follows that $\sqsubseteq$ is not antisymmetric, since e.g. $(X,Y, \emptyset) \sqsubseteq (X,\emptyset,Z)$ and $(X,\emptyset,Z) \sqsubseteq (X,Y, \emptyset)$, but $(X,Y, \emptyset)\neq (X,\emptyset,Z)$. Thus every preorder $(\mathcal T \subseteq 2^G \times 2^M \times 2^B \cap 2^I, \sqsubseteq)$ have all equivalence classes of cardinality 1 except $[\emptyset]=\{(\emptyset, \emptyset, \emptyset ), \ldots , (G,\emptyset,\emptyset) \ldots, (\emptyset,M,B)\}$ of cardinality $2^{|G|+|M|+|B|}-1$.
\begin{definition} \citep{Trabelsi:2012} Let $S=(X, Y, Z)$ be a tri-set of $\mathbb{K}=(G,M,B, I \subseteq G \times M \times B)$. The mapping $h: 2^G\times 2^M \times 2^B \cap 2^I \to 2^G\times 2^M \times 2^B$ is defined as follows:
$h(S) = \{(U, V , W) \mid U = \{g \in G \mid \forall m \in Y, \forall b \in Z: (g, m, b) \in I \}$ $\wedge V = \{m \in M \mid \forall g \in U, \forall b \in Z: (g, m, b) \in Y \}$ $\wedge W = \{b \in B \mid \forall g \in U, \forall m \in V: (g, m, b) \in Y \}$
\end{definition}
Note that every triconcept is a maximal or closed triset, i.e. a triset that cannot be extended by triples from $I$ being a triset.
\begin{proposition} $h(\cdot)$ is extensive and idempotent by $\sqsubseteq$ on $ T =\{t \mid t \mbox{ is a triset of } \mathbb{K} \}=\{ (X,Y,Z) \in 2^G\times 2^M \times 2^B \mid (X,Y,Z) \subseteq I \}$ and every fixpoint $f$ of $h$ (i.e. $h(f)=f$) is a triconcept of $\mathbb{K}$.
\end{proposition} \Proof{ One can find the proof of extensivity and idempotency in \citep{Trabelsi:2012}. It is easy to see that every formal triconcept is a fixpoint of $h(\cdot)$ and every triset $(X,Y,Z)$ is transformed by $h(\cdot)$ to the triconcept $((Y\times Z)^{(1)},((Y\times Z)^{(1)} \times Z)^{(2)},((Y\times Z)^{(1)} \times ((Y\times Z)^{(1)} \times Z)^{(2)})^{(3)}$. Indeed, all formal triconcepts should be listed since a triset is allowed to be a triple with at least one component being $\emptyset$. }
\begin{theorem} For a given tricontext $\mathbb{K}=(G,M,B, I \subseteq G \times M \times B)$ and its associated triset system $\mathcal T=\{ (X,Y,Z) \in 2^G\times 2^M\times 2^B \mid (X,Y,Z) \subseteq I \}$ operator $h$ is not monotone w.r.t. $\sqsubseteq$.
\end{theorem} \textbf{Proof.} To construct a violating example, one needs two different triconcepts with the same extent, $c_1=(X,Y_1,Z_1)$ and $c_2=(X,Y_2,Z_2)$ of $\mathbb{K}$ such that $Y_1 \subset Y_2$ and $Z_1 \supset Z_2$.
Consider the tri-set $s=(X, Y_1, Z_2)$: $$s \sqsubseteq c_1 \Rightarrow h(s)=c_2 \not\sqsubseteq h(c_1)=c_1$$ $\square$.
\begin{example}
For the tricontext in Figure~\ref{counterEx1}, the violating example for monotonicity of $h(\cdot)$ is as follows:
$$x=(\{u_1,u_2\}, \{t_1\}, \{r_1\}) \sqsubseteq y=(\{u_1,u_2\}, \{t_1\}, \{r_1, r_2\}) \Rightarrow$$ $$h(x))=(\{u_1,u_2\}, \{t_1, t_2\}, \{r_1\}) \not\sqsubseteq h(y)=(\{u_1,u_2\}, \{t_1\}, \{r_1, r_2\}).$$
\end{example}
\begin{figure}
\caption{A small example with Bibsonomy data}
\label{counterEx1}
\end{figure}
\begin{definition} (\cite{Ganter:1999}, p.237, \cite{Ganter:2012})
A relation $R\subseteq G\times M$ is called a Ferrers relation
iff there are subsets $A_1\subset A_2\subset A_3\ldots\subseteq G$ and $M\supseteq B_1\supset B_2\supset B_3\supset\ldots$ such that $R=\bigcup_{i}A_i\times B_i$.
$R$ is called a Ferrers relation of concepts of $(G,M,I)$ iff
there are formal concepts $(A_1,B_1)\le (A_2,B_2)\le (A_3,B_3)\le
\ldots$ such that $R=\bigcup_{i}A_i\times B_i$. \end{definition}
\begin{proposition}\label{prop:ferrers}\citep{Ganter:2012}
Any Ferrers relation $R\subseteq I$ is contained in a Ferrers
relation of concepts of $(G,M,I)$. \end{proposition}
\begin{corollary} Let $\mathbb{K}=(G,M,B,I)$ be a formal tricontext, and $\mathbb{K}^{MB}_X=(M,B,I_X)$ such that $(m,b) \in I_X$ iff $(g,m,b) \in I \cap X \times M \times B$, and $I_X$ be Ferrers relation of concepts of $\mathbb{K}^{MB}_X$.
Operator $h$ is not monotone for every pair of trisets $(X,Y,Z)$ and $(X,Y_i,Z_i)$ such that $Y \subseteq Y_i$, $Z\subseteq Z_j$, and $(Y_i,Z_i) \leq (Y_j,Z_j)$ are concepts of $\mathbb{K}^{MB}_X$. \end{corollary}
\subsection{Inconsistency of MCCS closure}
\subsubsection{Lossy hyperedge encoding and phantom edges}
In case of $k$-partite graph encoding we can meet information loss in a form of new hyperedges. Below we provide this encoding from polyadic contexts to multi-relation databases with $n$ types of entities.
Let $\mathbb{K}=(K_1, \ldots, K_n, I)$ be a polyadic formal context, then $\mathbb{D}=(E=K_1 \sqcup \ldots \sqcup K_n,t,\mathcal{R},R)$ be the corresponding multi-relation database, where $t$ maps entities from $E$ into their types from $1$ to $n$, $R=\{\{i,j\} \mid i,j \in \{1,\ldots, n\}, i\neq j\}$ and $\mathcal{R} = \bigcup_{\{i, j\} \in R} \mathcal{R}_{\{i, j \}}$ for the binary relations $\mathcal{R}_{\{i, j\}} = \{\{e_i , e_j\}\mid e_i \in K_i , e_j \in K_j \mbox{ and } e_i , e_j \mbox{ are related by } I\}$.
\begin{example}\label{ex:tri_loss}Imagine that we have three hyperedges $\{u,t,r_0\}, \{u,t_0,r\}, \{u_0,t,r\}$, and then encode them as edges in a 3-partite graph, we obtain $$\{u,t\}, \{u,r_0\}, \{t,r_0\}, \{t_0,r\}, \{u,r\}, \{u,t_0\}, \{t,r\}, \{u_0,r\}, \mbox{and } \{u_0,t\}.$$
Since we now have $\{u,t\}, \{u,r\}, \mbox{and } \{t,r\}$ in our graph, we should inevitably decode a new hyperedge, $\{u,t,r\}$. See Figure \ref{fig:Loss}. \end{example} \begin{figure}
\caption{A phantom hyperedge as a structure loss}
\label{fig:Loss}
\end{figure}
Taking the last fact into account and following the definition of MCCS or applying $g(\cdot)$ to respective CCS, we should obtain in general case a different or extra pattern(s) in addition to triconcepts in $k$-partite encoding. Thus in Example~\ref{ex:tri_loss} there are three MCCSs, $\{u, t_0,t,r\}$, $\{u,u_0,t,r\}$, and $\{u,t,r_0,r\}$, that are different from set representation of formal triconcepts, $\{u,t_0,r\}$, $\{u_0,t,r\}$, $\{u,t,r_0\}$, of the initial tricontext, respectively.
\subsubsection{Closed but non-maximal patterns}
As one can see from the example in Table \ref{counterEx2}, the technical condition for idempotency of $g(\cdot)$ is fulfilled. The corresponding tripartite graph is depicted in Figure~\ref{fig:LoicCounterEx}.
\begin{table}[ht!] \caption{A small example with Bibsonomy data}\label{counterEx2} \begin{center}
\begin{tabular}{ccccc} \begin{tabular}{cccc}
& $t_1$ & $t_2$ &$t_3$\\ \cline{2-4}
$u_1$ &\multicolumn{1}{|c|}{$\times$} & \multicolumn{1}{|c|}{$\times$} & \multicolumn{1}{|c|}{} \\ \cline{2-4}
$u_2$ & \multicolumn{1}{|c|}{$\times$} & \multicolumn{1}{|c|}{$\times$} & \multicolumn{1}{|c|}{} \\ \cline{2-4}
$u_3$ & \multicolumn{1}{|c|}{} & \multicolumn{1}{|c|}{} & \multicolumn{1}{|c|}{} \\ \cline{2-4}
& \multicolumn{3}{c}{$r_1$}\\
\end{tabular} & \quad\quad &
\begin{tabular}{cccc}
& $t_1$ & $t_2$ &$t_3$\\ \cline{2-4}
$u_1$ &\multicolumn{1}{|c|}{$\times$} & \multicolumn{1}{|c|}{} & \multicolumn{1}{|c|}{} \\ \cline{2-4}
$u_2$ & \multicolumn{1}{|c|}{$\times$} & \multicolumn{1}{|c|}{} & \multicolumn{1}{|c|}{} \\ \cline{2-4}
$u_3$ & \multicolumn{1}{|c|}{} & \multicolumn{1}{|c|}{} & \multicolumn{1}{|c|}{} \\ \cline{2-4}
& \multicolumn{3}{c}{$r_2$}\\
\end{tabular}
& \quad\quad &
\begin{tabular}{cccc}
& $t_1$ & $t_2$ &$t_3$\\ \cline{2-4}
$u_1$ &\multicolumn{1}{|c|}{} & \multicolumn{1}{|c|}{} & \multicolumn{1}{|c|}{} \\ \cline{2-4}
$u_2$ &\multicolumn{1}{|c|}{} & \multicolumn{1}{|c|}{} & \multicolumn{1}{|c|}{} \\ \cline{2-4}
$u_3$ & \multicolumn{1}{|c|}{} & \multicolumn{1}{|c|}{} & \multicolumn{1}{|c|}{} \\ \cline{2-4}
& \multicolumn{3}{c}{$r_3$}\\
\end{tabular}
\end{tabular} \end{center}
\end{table}
\begin{figure}
\caption{A counter example: closed but non-maximal patterns}
\label{fig:LoicCounterEx}
\end{figure}
However for the CCS pattern $X=\{u_1,u_2, t_1, r_1\}$ the result of $g(X)$ coincides with $X$ but it is not maximal. Indeed, there exist two maximal closed and connected patterns corresponding to triconcepts, $X \cup t_2 =\{u_1,u_2, t_1, t_2, r_1\}$ and $X \cup r_2 =\{u_1,u_2, t_1, r_1, r_2\}$.
It is so, since $Comp(X)=X \cup \{t_2, r_2\}$, but $Comp(X \cup t_2) = X \cup t_2$ and $Comp(X \cup r_2)=X \cup r_2$.
\begin{proposition} Let $\mathcal{F}_\mathbb{D}$ be a CCS system and $\mathcal H \subseteq \mathcal{F}_\mathbb{D}$ such that $|\mathcal H|\geq 2$, every $H \in \mathcal H$ is maximal and there exists a CCS $X=\bigcap_{H \in \mathcal H} H \neq \emptyset$, then $g(X)=X$ but $X$ is not an MCCS. \end{proposition}
\Proof{ Since there exist more than two MCCSs $H_i, H_j \in H$, we obtain $X\subset H_i$ and $X\subset H_j$. Therefore $H_i\setminus X \subseteq Comp(X)$ and $H_j\setminus X \subseteq Comp(X)$. However for $h_i \in H_i$ and $h_j \in H_j$, $Comp(X \cup {h_i}) \neq Comp(X \cup {h_j})$ since otherwise it violates maximality of $H_i$. }
Let us introduce generalised Ferres relation of $n$-concepts (for 3-adic case see \citep{Glodeanu:2013}) .
\begin{definition}
A relation $R\subseteq K_1 \times \cdots \times K_n$ is called a generalised Ferrers relation
iff $\exists j \in \{1,\ldots,n\} \forall i \in \{1,\ldots,n\}\setminus \{j\}$ $A_{1i}\subset A_{2i}\subset A_{3i}\ldots\subseteq K_i$, and $K_j \supseteq A_{1j}\supset A_{2j}\supset A_{3j}\supset\ldots$ such that $R=\bigcup_{k} A_{k1}\times \ldots \times A_{kn}$.
$R$ is called a Ferrers relation of n-concepts of $(K_1, \ldots,K_n,I)$ iff
there are formal n-concepts $(A_{11}, \ldots, A_{1n}) \lesssim_k (A_{21},\ldots,A_{2n}) \lesssim_k (A_{31},\ldots,A_{3n}) \lesssim_k
\ldots$ such that $R=\bigcup_{i} A_{i1}\times \ldots \times A_{in}$. \end{definition}
\begin{corollary} Let $\mathbb{K}=(K_1, \ldots, K_n, I)$ be a polyadic formal context such that $I$ is a Ferrers relation of $n$-concepts and $\mathbb{D}=(E=K_1 \sqcup \ldots \sqcup K_n,t,\mathcal{R},R)$ be the corresponding multi-relation database. Operator $g$ does not produce an MCCS for $flat(A_{j1} \cap A_{i1}, \ldots, A_{jn} \cap A_{in})$ obtained from any pair of concepts of $\mathbb{K}$, $(A_{i1}, \ldots, A_{in}) \lesssim_k (A_{j1}, \ldots, A_{jn})$, where $A_is\neq A_js$, $s, k \in \{1,\ldots,n\}$ and $s\neq k$. \end{corollary}
\section{Closure operator for triconcepts}\label{sec:triclo}
There are $n$-contexts, where $h(\cdot)$ is not a closure that results in formal concepts because of non-idempotency and closure operator $g(\cdot)$ produces CCSs that are not necessary maximal, e.g. caused by non-uniqueness of possible extensions of input patterns. Moreover, the lossy data encoding by $n$-partite graph instead of $n$-partite hypergraph results in phantom $n$-adic edges and extra elements in resulting patterns.
So, to overcome the difficulty at least for generation of $n$-concepts we may adjust the set systems such that $h(\cdot)$ and $g(\cdot)$ could operate. Informally, we need to weed all patterns or phantom hyperedges that result in undesirable behaviour of $h(\cdot)$ and $g(\cdot)$, the candidates to closure operators.
\begin{definition} Let $\mathbb{K}=(K_1, K_2, K_3, I)$ be a triadic formal context. A triset $S$ is called a (maximal) switching generator of the context $\mathbb{K}$ iff $S=tuple(flat(c_1) \cap flat(c_2))\neq \emptyset$, where $c_1$ and $c_2$ are concepts of $\mathbb{K}$. \end{definition}
\begin{theorem}\label{thrm:weeding} Let $\mathbb{K}=(K_1, K_2, K_3, I)$ be a triadic formal context. The set system $\mathcal{F}_{\mathbb{K}\ominus\mathcal{S}}=\mathcal T\setminus\mathcal{S}$ is a correct set system for formal triconcept generation by $h(\cdot)$ in $\mathbb{K}$, where $\mathcal{T} = \{ (X,Y,Z) \in 2^G\times 2^M\times 2^B \mid (X,Y,Z) \subseteq I \}$ and $\mathcal{S}=\{S \mid S \mbox{ is a switching generator of } \mathbb{K}\}$. \end{theorem}
\Proof{ Since there is no a switching generator in $\mathcal{F}_{\mathbb{K}\ominus\mathcal{S}}$, monotony of $h(\cdot)$ is fulfilled.
Assume that monotony is violated by trisets $x$ and $y$, that is $x \sqsubseteq y \to h(x)\not\sqsubseteq h(y)$. By extensivity of $h(\cdot)$ and transitivity of $\sqsubseteq$, it implies $x \sqsubseteq h(x)$ and $x \sqsubseteq h(y)$. Hence, $x \sqsubseteq tuple(flat(h(x)) \cap flat(y))$, i.e. $x$ is a switching generator. Contradiction.
Since every formal triconcept is not a switching generator, none of triconcepts has been deleted from $\mathcal{F}_{\mathbb{K}\ominus\mathcal{S}}$.
}
As for phantom triadic edges, unfortunately it is not possible to delete them from $\mathcal{R}$ since each phantom triadic edge $\{e_i, e_j, e_k\}$ is composed by $\{e_i,e_j\}$, $\{e_j, e_k\}$, and $\{e_k,e_i\}$, which are parts of ``real'' triadic hyperedges.
Let $\mathbb{K}$ be a formal tricontext and $\mathbb{D}$ be the corresponding multi-relational database, $\mathcal P=\{tuple(e) \mid e=\{e_i,e_j,e_k\} \mbox{ is a phantom edge in } \mathcal{R} \}$ then a test whether an MCCS forms triset can be done as follows:
\begin{enumerate}
\item For an MCCS $s$ form $tuple(s)=(X,Y,Z)$; \item Check whether $t=X \times Y \times Z \setminus e$ forms a triset of $\mathbb{K}$, where $e \in \mathcal P$; \item If yes, then output $t$; \item Delete $s$ from the output otherwise.
\end{enumerate}
To make sure that $t$ is a triconcept, one need to check $h(t)=t$.
Since traditionally closure operators were introduced for partial orders over set inclusion, we would like to avoid dealing with preoder $\sqsubseteq$ over trisets and work with set inclusion of their set representations instead.
For tricontext $\mathbb{K}=(K_1,K_2,K_3, I)$ we consider a family of operators \begin{multline*}
\{\sigma_{ijk}| \sigma_{ijk}: 2^{K_1} \times 2^{K_2} \times 2^{K_3} \to 2^{K_1} \times 2^{K_2} \times 2^{K_3} \mbox{ such that } \\
\sigma_{ijk}: (X_1,X_2,X_3) \mapsto (Y_1,Y_2,Y_3), \mbox{ where }\\ Y_i = (X_j \times X_k)^{(i)}, Y_j= (Y_i \times X_k)^{(j)}, Y_k=(Y_i \times Y_j)^{(k)} \mbox{, where } \{i,j,k\}=\{1,2,3\}\}. \end{multline*}
The cardinality of the family is $3!=6$ and $n!$ for its n-ary case generalisation.
\begin{proposition}\label{prop:nocomm} Operators $\sigma_{ijk}(\cdot)$ are not commutative, i.e. $\sigma_{ijk}(\sigma_{lmn}(\cdot))\neq \sigma_{lmn}(\sigma_{ijk}(\cdot))$, where $(i,j,k)\neq (l,m,n)$ and $\{i,j,k\}=\{1,2,3\}$. \end{proposition}
\Proof{ Consider a tricontext given below.
\begin{tabular}{ccccc} \tabcolsep=0.1cm \begin{tabular}{ccccc}
& $t_1$ & $t_2$ & $t_3$ & $t_4$ \\ \cline{2-5}
$u_1$ &\multicolumn{1}{|c|}{$\times$} & \multicolumn{1}{|c|}{$\times$} &\multicolumn{1}{|c|}{$\times$} &\multicolumn{1}{|c|}{$\times$} \\ \cline{2-5}
$u_2$ & \multicolumn{1}{|c|}{} & \multicolumn{1}{|c|}{$\times$} &\multicolumn{1}{|c|}{$\times$} &\multicolumn{1}{|c|}{$\times$} \\ \cline{2-5}
$u_3$ & \multicolumn{1}{|c|}{} & \multicolumn{1}{|c|}{$\times$} &\multicolumn{1}{|c|}{$\times$} &\multicolumn{1}{|c|}{$\times$} \\ \cline{2-5}
$u_4$ & \multicolumn{1}{|c|}{} & \multicolumn{1}{|c|}{} &\multicolumn{1}{|c|}{} &\multicolumn{1}{|c|}{$\times$} \\ \cline{2-5}
& \multicolumn{4}{c}{$r_1$}\\
\end{tabular} & & \tabcolsep=0.1cm \begin{tabular}{ccccc}
& $t_1$ & $t_2$ & $t_3$ & $t_4$ \\ \cline{2-5}
$u_1$ &\multicolumn{1}{|c|}{} & \multicolumn{1}{|c|}{$\times$} &\multicolumn{1}{|c|}{$\times$} &\multicolumn{1}{|c|}{$\times$} \\ \cline{2-5}
$u_2$ & \multicolumn{1}{|c|}{} & \multicolumn{1}{|c|}{$\times$} &\multicolumn{1}{|c|}{$\times$} &\multicolumn{1}{|c|}{$\times$} \\ \cline{2-5}
$u_3$ & \multicolumn{1}{|c|}{} & \multicolumn{1}{|c|}{} &\multicolumn{1}{|c|}{$\times$} &\multicolumn{1}{|c|}{$\times$} \\ \cline{2-5}
$u_4$ & \multicolumn{1}{|c|}{} & \multicolumn{1}{|c|}{} &\multicolumn{1}{|c|}{} &\multicolumn{1}{|c|}{} \\ \cline{2-5}
& \multicolumn{4}{c}{$r_2$}\\
\end{tabular} & &
\tabcolsep=0.1cm \begin{tabular}{ccccc}
& $t_1$ & $t_2$ & $t_3$ & $t_4$ \\ \cline{2-5}
$u_1$ &\multicolumn{1}{|c|}{} & \multicolumn{1}{|c|}{} &\multicolumn{1}{|c|}{} &\multicolumn{1}{|c|}{$\times$} \\ \cline{2-5}
$u_2$ & \multicolumn{1}{|c|}{} & \multicolumn{1}{|c|}{} &\multicolumn{1}{|c|}{} &\multicolumn{1}{|c|}{} \\ \cline{2-5}
$u_3$ & \multicolumn{1}{|c|}{} & \multicolumn{1}{|c|}{} &\multicolumn{1}{|c|}{} &\multicolumn{1}{|c|}{} \\ \cline{2-5}
$u_4$ & \multicolumn{1}{|c|}{} & \multicolumn{1}{|c|}{} &\multicolumn{1}{|c|}{} &\multicolumn{1}{|c|}{} \\ \cline{2-5}
& \multicolumn{4}{c}{$r_3$}\\
\end{tabular}
\end{tabular}
The system of all switching generators $\mathcal{S}$ contains $s_1=\{u_1,t_4,r_1\}$ and $s_2=\{u_1,u_2, t_3,t_4, r_1\}$.
$s_1$ proves that $\sigma_{i\_\_}(\cdot)\neq\sigma_{j\_\_}(\cdot)\neq\sigma_{k\_\_}(\cdot)$ and
$s_2$ proves that $\sigma_{ijk}(\cdot)\neq\sigma_{ikj}(\cdot)$ for $\{i,j,k\}=\{1,2,3\}$.
The fact that $\sigma_{lmn}\sigma_{ijk}(\cdot)=\sigma_{ijk}(\cdot)$ proves the proposition. }
\begin{theorem}\label{thrm:noclos} For $\mathbb{K}=(K_1,K_2,K_3,I)$ and the associated triset system $\mathcal T$ there is no an associated closure operator in case there exist at least two concepts $c_1=(X_1,Y_1,Z_1)$ and $c_2=(X_1,Y_2,Z_2)$ such that they have the common non-empty maximal switching generator $s$, i.e. $tuple(flat(c_1) \cap flat(c_2)) \neq \emptyset$. \end{theorem}
\Proof{ Let $\sigma$ be a closure operator for $\mathbb{K}$. Since $s \sqsubset c_1$ and $s \sqsubset c_2$ then $\sigma(s)$ should result in $c_i$ which is either $c_1$ or $c_2$ (or one of other concepts $c_k$ with $s \sqsubset c_k$ if any exist). So let $\sigma(s)=c_i$ and consider $s \sqsubseteq c_j$; it implies that $\sigma(s)=c_i \not \sqsubseteq \sigma(c_j)=c_j$ for $i\neq j$, and $\{i,j\}=\{1,2\}$. Contradiction. }
As it has been shown, $\mathcal{F}_{\mathbb{K}\ominus S}$ is a correct set system for $h(\cdot)=\sigma_{123}(\cdot)$ being a closure operator. It is easy to see that this system is correct for $\sigma_{ijk}(\cdot)$.
To summarise properties of $\mathcal{F}_{\mathbb{K}\ominus S}$ and show its difference from set systems in \citep{Boley:2010,Spyropoulou:2014} we recall the following properties of set systems.
\begin{definition} A non-empty set system $(E,\mathcal{F})$ is called
1. accessible if for all $X \in \mathcal{F} \setminus \{\emptyset\}$ there is an $e \in X$ such that $X \setminus \{e\} \in \mathcal{F}$,
2. an independence system if $Y \in F$ and $X \subseteq Y$ together imply $X \in F$ ,
3. confluent if for all $I, X, Y \in \mathcal{F}$ with $\emptyset \in I \subseteq X$ and $I \subseteq Y$ it holds that $X \cup Y \in F$.
4. strongly accessible if it is accessible and for all $X, Y \in F$ with $X \subset Y$, there is an $e \in Y \setminus X$ such that $X \cup \{e\} \in \mathcal{F}$ .
\end{definition}
\begin{proposition} \label{prop:propert} 1) Set system $\mathcal{F}_{\mathbb{K}\ominus S}$ of all sets that form trisets is accessible and 2) not independent. 3) It is not a closure system. 4) It is confluent. 5) It is strongly accessible. \end{proposition}
\Proof{ 1. Every set of $\mathcal{F}_{\mathbb{K}\ominus S}$ forms a triset $t$. Even if it contains some switching generator $s$, we can then remove any $e \in flat(s)$ from $t$, the resulting set $flat(t) \setminus {e}$ is in $\mathcal{F}_{\mathbb{K}\ominus S}$ (switching generator free system) since it is a triset and contains at least one element not included in a switching generator. Empty set (or empty set of triples) is not in $\mathcal{F}_{\mathbb{K}\ominus S}$ because it is a universal switching generator.
2. Since some concepts may contain switching generators by triset set inclusion, it implies that these switching generators are not $\mathcal{F}_{\mathbb{K}\ominus S}$.
3. On the contrary, every pair of concepts $X,Y \in \mathcal{F}_{\mathbb{K}\ominus S}$ implies that $X \cap Y \notin \mathcal{F}$ (anti-sharing).
4. Since there is no such non-empty $I \in \mathcal{F}_{\mathbb{K}\ominus S}$ being a triset of two different concepts it trivially holds.
5. If $X \subset Y$ for $X,Y \in \mathcal{F}_{\mathbb{K}\ominus S}$, $X$ does not form a formal concept (because of antiordinal relations) or switching generator. So adding any element $e$ from $Y\setminus X$ leaves $X \cup \{e\}$ being a triset. }
A detailed study of algorithmic issues is out of scope the paper, however, \cite{Boley:2010} reported a simple algorithm for Problem~\ref{prbl:1}, i.e. listing of all fixed points of a partially defined closure operator, which is correct for strongly accessible set systems.
\begin{problem} \label{prbl:1} (list-closed-sets) Given a set system $(E, \mathcal{F})$ with $\emptyset \in \mathcal{F}$ and a closure operator $\sigma: \mathcal{F} \to \mathcal{F}$ , list the elements of $\sigma(\mathcal{F})=\{F\mid F \in \mathcal{F}: \sigma(F)=F \}$. \end{problem}
It is questionable whether the weeding step can be efficiently incorporated into closure listing algorithm. Thus, in the worst case, i.e. for power tricontext $\mathbb{K}=(\{1\ldots k\}, \{1\ldots k\}, \{1\ldots k\}, \neq)$, the number of triconcepts equals $3^k$ \citep{Biedermann:1998}. The number of switching generators is greater than that of the concepts of $\mathbb{K}$ for $k>2$ and not polynomial as given in~Theorem~\ref{thrm:numsw}
\begin{theorem}\label{thrm:numsw} For a power tricontext $\mathbb{K}=(\{1\ldots k\}, \{1\ldots k\}, \{1\ldots k\}, \neq)$ the number of switching generators is $4^k-3^k$. \end{theorem}
\Proof{ One can prove the theorem by direct calculation of the triple sum below:
$$\sum\limits_{k_1=0}^{k-1}\sum\limits_{k_2=0}^{k-k_1-1}\sum\limits_{k_3=0}^{k-k_1-k_2-1}C^{k_1}_kC^{k_2}_{k-k_1}C^{k_3}_{k-k_1-k_2}.$$ }
Theorems \ref{thrm:weeding},\ref{thrm:noclos} and propositions \ref{prop:propert},\ref{prop:nocomm} can be generalised for $n$-ary case in a similar way. For example, the general version of Theorem~\ref{thrm:numsw} is provided as Theorem~\ref{thrm:numswn}.
\begin{theorem}\label{thrm:numswn} For a power polyadic $n$-context $\mathbb{K}=(\{1\ldots k\}, \ldots, \{1\ldots k\}, \neq)$ the number of switching generators is $(n+1)^k-n^k$. \end{theorem}
In \cite{Kuznetsov:2004}, the complexity of the problem ``Number of all concepts'' was addressed. Thus this problem is $\#P$-complete. Theorem~\ref{thrm:sharpp} provides a similar justification, showing that the problem ``Number of all (maximal) switching generators'' is intractable. To avoid complex technicalities we prove the theorem for $n=2$.
\begin{theorem}\label{thrm:sharpp} The following problem ``Number of all (maximal) switching generators'' is $\#P$-complete:
Input: Context $\mathbb{K}=(G,M,I)$
Output: The number of all (maximal) switching generators of the context $\mathbb{K}$, i.e. $|\mathcal{S}|
$. \end{theorem} \Proof{ We reduce the following $\#P$-complete problem to ours: ``The number of binary vectors that satisfy monotone 2-CNF of the form $C=\bigwedge_{i=1}^{s} ( x _{i,1} \vee x_{i,2})$'':
\emph{Input}: Monotone (without negation) CNF with two variables in each disjunction $C=\bigwedge_{i=1}^{s} ( x _{i,1} \vee x_{i,2}),$ where $x_{i,1},x_{i,2} \in X=\{x_1, \ldots, x_k\}$ for all $i=\overline{1,s}$.
\emph{Output}: Number of binary $k$-vectors (corresponding to the values of variables) that satisfy CNF $C$.
First, we construct 2-DNF $D$, the negation of $C$: $\bigvee_{i=1}^{s} ( \overline{x} _{i,1} \wedge \overline{x}_{i,2}),$. Each conjunction is denoted $D_i=( \overline{x} _{i,1} \wedge \overline{x}_{i,2})$, $i=\overline{1,s}$. The set of binary vectors that satisfy $D$ is a union of the sets of binary vectors that satisfy a certain $D_i$. Each disjunction is satisfied by every binary $k$-vector with $k-2$ ones and two zeros in $i_1$-th and $i_2$-th components.
We reduce this problem to that of the number of switching generators by constructing the following context $\mathbb{K}=(G,M,I)$. The set of attributes is $M=\{m_1,\dots,m_k\}\cup \bigcup_{i=1}^s \{ m^{i,k-1}, m^{i,k} \}$, where elements of $\tilde{M}=\{m_1,\dots,m_k\}$ are in one-to-one correspondence with variables from $X$. For some conjunction $D_i, i=\overline{1,s}$, we construct a context $\mathbb{K}_i=(G_i,M_i,I_i)$, where the set of attributes is $M_i=\tilde{M}\setminus \{m_{i,1},m_{i,2}\} \cup \{m^{i,k-1}, m^{i,k}\}:=\{m^{i,1}, \ldots, m^{i,k}\}$, the set of objects is $G_i=\{g_i^0,g_i^1,\ldots, g_i^{k-2},g_i^{k-1}, \ldots, g_i^{2k-2}\}$, and the relation $I_i \subseteq M_i\times G_i$ is defined as follows: $\{g_i^0\}'=M_i\setminus \{m^{i,k}\}$, $\{g_i^j\}'=M_i\setminus \{m^{i,j}, m^{i,k}\}$ for $j=\overline{1,k-2}$, $\{g_i^j\}'=M_i\setminus \{m^{i,j},m^{i,k-1},m^{i,k} \}$ for $j=\overline{k-1,2k-4}$, $\{g_i^{k-3}\}'=M_i\setminus \{m^{i,k-1},m^{i,k} \}$, and $\{g_i^{2k-2}\}'= \{m^{i,k-1},m^{i,k}\}$. The context $\mathbb{K}$ is constructed as follows: $\mathbb{K}=(\bigsqcup_{i=1}^{s} G_i, M, \bigsqcup_{i=1}^{s} I_i)$ with only one $\{g_i^{2k-2}\} \in G_i$ (the remaining $G_j$-s do not contain $\{g_j^{2k-2}\}$).
First, we show that every switching generator of $\mathbb{K}$ corresponds to a $k$-vector that satisfies $D$. Every switching generator of $\mathbb{K}$ is a switching generator of $\mathbb{K}_i$ for some $i$, which can be not unique. It is easy to see that intents of the context $\mathbb{K}_i$ form the power set of $M_i$, denoted by $2^{M_i}$. Elements of $2^{\tilde{M}}$ are in one-to-one correspondence with binary $n$-vectors, where components are in one-to-one correspondence with elements of $M$ with the same number. Since for every non-empty $S \subseteq \tilde{M}\setminus \{m_{i,1},m_{i,2}\}$ there are concepts $(S',S)$ and $((S\cup m^{i,k-1})',S\cup m^{i,k-1})$, their switching generator is $((S\cup m^{i,k-1})',S)$. A vector of this form satisfies $D_i$, since it has zeros at $i_1$-th and $i_2$-th places. Therefore, this vector satisfies $D$. To prove that the switching generator is provided when $S=\emptyset$, one may check that $(\{g_i^{2k-2}\},\{m^{i,k-1}\})$ is the switching generator of concepts $(\{g_i^{2k-2}\},\{m^{i,k-1},m^{i,k}\})$ and $(\{m^{i,k-1}\}', \{m^{i,k-1}\})$.
It remains to show that binary $k$-vectors that satisfy $D$ are in one-to-one correspondence with intents of $\mathbb{K}$. In fact, each binary $k$-vector $v$ that satisfies $D$, satisfies $D_i$ for some $i$ (this $i$ may be not unique). Then this vector has zero $i_1$-th and $i_2$-th components. Therefore, the corresponding set of attributes $A$ belongs to $\tilde{M} \setminus \{m_{i,1},m_{i,2}\} \subseteq 2^{M_i}$. If $A\neq\emptyset$, then concepts $(A',A)$ and $((A\cup m^{i,k-1})',A\cup m^{i,k-1})$ has switching generator is $((A\cup m^{i,k-1})',A)$. If $A=\emptyset$, then there is a unique switching generator $(\{g_i^{2k-2}\},\{m^{i,k-1}\})$ for some $G_i$.
The one-to-one correspondence between the switching generators of concepts of $\mathbb{K}$ and binary $k$-vectors satisfying $D$ is realised. Thus, if we know the number of all switching generators of concepts of $\mathbb{K}$, we obtain the number of all vectors satisfying $D$ and, therefore, that of the vectors satisfying $C$. The reduction is polynomial in the input size, since the context $\mathbb{K}$ has $|M|=k+2s$ attributes and $|G|=s(2k-2)+1$ objects. }
Similar theorem can be proved for $n=3$.
\begin{corollary}\label{croll:sharpp}
The problem ``Number of all (maximal) switching generators for $n=3$'' is $\#P$-complete:
Input: Context $\mathbb{K}=(G,M,B,Y)$
Output: The number of all (maximal) switching generators of the context $\mathbb{K}$, i.e. $|\mathcal{S}|
$.
\end{corollary}
The proof can be done in a similar way to the dyadic case, where $\mathbb{K}_i=(G_i,M,B_i,Y_i)$ for each conjunction $D_i$ should have $B_i=\{b_i^1, b_i^2\}$ (in fact, it plays a role of $\{m^{i,k-1}, m^{i,k}\}$ from Theorem~\ref{thrm:sharpp}) such that each $A$ will result in two triconcepts $(U,A,\{b_i^1, b_i^2\})$ and $(V,A,\{b_i^1\})$, $U \subseteq V$ with their maximal switching generator $(U,A,\{b_i^1\})$.
\section{Related work}\label{sec:relw}
In fact, one of the first methods for triconept enumeration was \textsc{TRIPAT} \citep{Ganter:1994} adopted Ganter's \textsc{Next Closure} algorithm in a nested manner for two-adic contexts generated from an input tricontext; it had been done even before the first formal treatment of 3FCA by \cite{Lehmann:1995}. This idea has been incarnated later in \textsc{TRIAS} for triconcepts enumeration with component-wise size constraints \citep{Jaschke:2006}.
Due to intrinsic complexity of exhaustive enumeration of triconcepts and closed $n$-sets, the research focus has shifted to constrained pattern mining and searching for different relaxations. Thus, after the release of \textsc{DataPeeler} \citep{Cerf:2009}, the \textsc{Fenster} algorithm for faul-tolerant pattern discovery has been proposed \citep{Cerf:2013}; the latter includes closed $n$-set mining as a particular case, allowing not all tuples inside dense $n$-sets to be present. Another approach, is the so-called OAC-triclustering for mining dense trisets \citep{Ignatov:2015} results in no more patterns than the number of tuples in an input relation having a fruitful property of containment of all triconcepts for a given tricontext within the resulting collection of trisets w.r.t. to component-wise inclusion under a properly selected minimal density constraint. A different approximation of triconcept can be realised within least square error minimisation criterion (see \textsc{TriBox}, \cite{Mirkin:2011}), which lead to a density-based pattern quality measure, namely the squared density of a triset (in sense of \citep{Cerf:2009}) multiplied by its size, thus, expressing trade-off between the high number of non-missing tuples inside and the large size.
One more direction is to use factorisation to select only a(n) (optimal) subset of triconcepts, which are factors to decompose an input three-way Boolean tensor \citep{Glodeanu:2013,Belohlavek:2013}. Closed sets are helpful for mining numeric contexts as well; thus, \cite{Kaytoue:2013} used 3FCA for searching maximal inclusion biclusters of constant values by treatment of attribute values as conditions. \cite{Spyropoulou:2014} proposed MCCS patterns and the associated closure operator for $n$-partite graphs working with multi-relational data. They also performed experimental comparison their \textsc{RMiner} with \textsc{DataPeeler}, which is not fully correct since $n$-ry relations being encoded as $n$-partite graphs result in phantom edges. Note that, in FCA domain, there is Relational Concept Analysis devoted to treatment of multi-relational data \citep{Hacene:2013}. The group that works on MCCSs has recently proposed Complete Connected Proper Subsets (CCPS) to deal with relational data with structured attributes \citep{Lijffijt:2016}, i.e. attributes with ordered values like real numbers, geographic location, time intervals, etc. Note that in FCA domain, to deal with data of complex description the so called Pattern Structures were proposed more than decade ago by \cite{Ganter:2001} and found many succesfull applications \citep{Kaytoue:2015}.
There is an interesting connection between biclique operators, their associated graphs \citep{Crespelle:2015}, and switching generators; in these graphs, two vertices (maximal biclques) are connected if they have a non-empty intersection, which, under some conditions, can be the switching generator of those biclques, i.e. concepts.
\section{Conclusion}\label{sec:concl}
The recent candidates to be closure operators related to triconcepts are not always consistent with either the definition of closure operator or triconcept ($n$-concept). We considered partially defined closure operators for triconcept generation that solve the problem. It is easy to obtain their $n$-adic versions and generalise current results. However the open question at the moment is whether recent closure-based algorithms for pattern mining reported in the relevant literature may benefit from this new bit of knowledge. Even though their basic definitions can be refined to fulfill necessary requirements, as we have seen, it might be costly or even intractable. Thus, an interesting prospective result could be a polynomial time check whether the current context is switching generators free (excluding $\emptyset$) or has a polynomial number of switching generators; one of the switching generators free examples is $\mathbb{K}=(\{1\ldots m\}, \ldots, \{1\ldots m\}, =)$.
\subsubsection*{Acknowledgments} The author would like to thank Lo\"{i}c Cerf, Eirini Spyropoulou, Boris Schminke, Dmitry Gnatyshak, Sergei Kuznetsov, Sergei Obiedkov, Bernhard Ganter, Jean-Francois Boulicaut, Mehdi Kaytoue, Amedeo Napoli, Boris Mirkin, Lhouri Nourine, Engelbert Mephu Nguifo and Jaume Baixeries. The study was implemented in the framework of the Basic Research Program at the National Research University Higher School of Economics in 2015--2017 and in the Laboratory of Intelligent Systems and Structural Analysis. The author was also partially supported by Russian Foundation for Basic Research.
\section*{References}
\end{document} |
\begin{document}
\title[Dyadic representation and $A_2$ theorem]{Representation of singular integrals by dyadic operators, and the $A_2$ theorem}
\author[T.~P.\ Hyt\"onen]{Tuomas P.\ Hyt\"onen} \address{Department of Mathematics and Statistics, P.O.B.~68 (Gustaf H\"all\-str\"omin katu~2b), FI-00014 University of Helsinki, Finland} \email{tuomas.hytonen@helsinki.fi}
\subjclass[2010]{42B25, 42B35}
\maketitle
\begin{center} {\small Department of Mathematics and Statistics\\ P.O.B.~68 (Gustaf H\"all\-str\"omin katu~2b)\\ FI-00014 University of Helsinki, Finland\\ tuomas.hytonen@helsinki.fi } \end{center}
\begin{abstract} This exposition presents a self-contained proof of the $A_2$ theorem, the quantitatively sharp norm inequality for singular integral operators in the weighted space $L^2(w)$. The strategy of the proof is a streamlined version of the author's original one, based on a probabilistic Dyadic Representation Theorem for singular integral operators. While more recent non-probabilistic approaches are also available now, the probabilistic method provides additional structural information, which has independent interest and other applications. The presentation emphasizes connections to the David--Journ\'e $T(1)$ theorem, whose proof is obtained as a byproduct. Only very basic Probability is used; in particular, the conditional probabilities of the original proof are completely avoided.\\
\noindent\textsc{Keywords:} Singular integral, Calder\'on--Zygmund operator, weighted norm inequality, sharp estimate, $A_2$ theorem, $T(1)$ theorem \end{abstract}
\section{Introduction}
The goal of this exposition is to prove the following \emph{$A_2$ theorem}:
\begin{theorem}\label{thm:A2} Let $T$ be any Calder\'on--Zygmund operator on $\mathbb{R}^d$ (like the Hilbert transform on $\mathbb{R}$, the Beurling transform on $\mathbb{C}\simeq\mathbb{R}^2$, or any of the Riesz transforms in $\mathbb{R}^d$ for $d\geq 2$; see Section~\ref{sec:representation} for the general definition). Let $w:\mathbb{R}^d\to[0,\infty]$ be a weight in the Muckenhoupt class $A_2$, i.e., \begin{equation*}
[w]_{A_2}:=\sup_Q\fint_Q w\cdot\fint_Q\frac{1}{w}<\infty\qquad\Big(\fint_Q w:=\frac{1}{\abs{Q}}\int_Q w\Big), \end{equation*} where the supremum is over all axes-parallel cubes $Q$ in $\mathbb{R}^d$. Let $L^2(w)$ be the space of all measurable functions $f:\mathbb{R}^d\to\mathbb{C}$ such that \begin{equation*}
\Norm{f}{L^2(w)}:=\Big(\int_{\mathbb{R}^d}\abs{f}^2 w\Big)^{1/2}<\infty. \end{equation*} Then the following norm inequality is valid for any $f\in L^2(w)$, where $C_T$ only depends on $T$ and not on $f$ or $w$: \begin{equation*}
\Norm{Tf}{L^2(w)}\leq C_T\cdot[w]_{A_2}\cdot\Norm{f}{L^2(w)}. \end{equation*} \end{theorem}
This general theorem for all Calder\'on--Zygmund operators is due to the author~\cite{Hytonen:A2}, but it was first obtained in the listed special cases by S.~Petermichl and A.~Volberg \cite{PV} and Petermichl \cite{Petermichl:Hilbert,Petermichl:Riesz}, and in various further particular instances by a number of others \cite{CMP,Dragicevic:cubic,LPR,Vagharshakyan}. See also Section~\ref{sec:Beurling} for more details on the history of the problem.
Although several different proofs of Theorem~\ref{thm:A2} are known by now, I will present one that is a direct descendant of the original approach, but greatly streamlined in various places, based on ingredients from various subsequent proofs. On the large scale, I follow the strategy of my paper with C.~P\'erez, S.~Treil and A.~Volberg \cite{HPTV}, the first simplification of my original proof \cite{Hytonen:A2}. This consists of the following steps, which have independent interest: \begin{enumerate}
\item\label{it:red2Dyadic} Reduction to \emph{dyadic shift operators} (the Dyadic Representation Theorem): every Calder\'on--Zygmund operator $T$ has a (probabilistic) representation in terms of these simpler operators, and hence it suffices to prove a similar claim for every dyadic shift $S$ in place of $T$. This was a key novelty of \cite{Hytonen:A2} when it first appeared. In this exposition, the probabilistic ingredients of this representation have been simplified from \cite{Hytonen:A2,HPTV}, in that no conditional probabilities are needed.
\item\label{it:red2Testing} Reduction to \emph{testing conditions} (a local $T(1)$ theorem): in order to have the full norm inequality \begin{equation*}
\Norm{Sf}{L^2(w)}\leq C_S[w]_{A_2}\Norm{f}{L^2(w)}, \end{equation*} it suffices to have such an inequality for special test functions only: \begin{equation*} \begin{split}
\Norm{S(1_Q w^{-1})}{L^2(w)} &\leq C_S[w]_{A_2}\Norm{1_Q w^{-1}}{L^2(w)}, \\
\Norm{S^*(1_Q w)}{L^2(w^{-1})} &\leq C_S[w]_{A_2}\Norm{1_Q w}{L^2(w^{-1})}. \end{split} \end{equation*} This goes essentially back to F.~Nazarov, Treil and Volberg \cite{NTV:2weightHaar}. (In the original proof \cite{Hytonen:A2}, in contrast to the simplification \cite{HPTV}, this reduction was done on the level of the Calder\'on--Zygmund operator, using a more difficult variant due to P\'erez, Treil and Volberg~\cite{PTV}).
\item\label{it:verifyTesting} Verification of the testing conditions for $S$. This was first achieved by M.~T. Lacey, Petermichl and M.~C. Reguera~\cite{LPR}, although some adjustments were necessary to achieve the full generality in \cite{Hytonen:A2}. \end{enumerate}
As said, several different proofs and extensions of the $A_2$ theorem have appeared over the past few years; see the final section for further discussion and references. In particular, it is now known that the probabilistic Dyadic Representation Theorem may be replaced by a deterministic Dyadic Domination Theorem. Its first version, a domination in norm, is due to A.~Lerner \cite{Lerner:domination}, and based on his clever local oscillation formula \cite{Lerner:formula}; this was subsequently improved to pointwise domination by J.~M. Conde-Alonso and G.~Rey \cite{CondeRey} and, independently, by Lerner and Nazarov \cite{LerNaz:book}. Yet another approach to the pointwise domination was found by Lacey \cite{Lacey:elem} and again simplified by Lerner \cite{Lerner:simplest}; this has the virtue of covering the biggest class of operators admissible for the $A_2$ theorem at the present state of knowledge. However, the probabilistic method continues to have its independent interest: it achieves the reduction to dyadic model operators as a linear \emph{identity}, in contrast to the (non-linear) \emph{upper bound} provided the deterministic domination. As such, it provides a structure theorem for singular integral operators, which has found other uses beyond the weighted norm inequalities, including the following: \begin{itemize}
\item The \emph{theorem itself} is applied to the estimation of \emph{commutators} of Calder\'on--Zygmund operators and BMO functions in a multi-parameter setting by L.~Dalenc and Y.~Ou~\cite{DalOu:iterated} and in a two-weight setting by I.~Holmes, M.~Lacey and B.~Wick \cite{HLW,HW}; it is also applied to sharp norm bounds for \emph{vector-valued extensions} of Calder\'on--Zygmund operators by S.~Pott and A.~Stoica~\cite{PS}.
\item The \emph{methods behind this theorem} have been generalized by H.~Martikainen \cite{Martikainen:Advances} and Y.~Ou \cite{Ou:Tb} to the analysis of \emph{bi-parameter singular integrals}, yielding new $T(1)$ and $T(b)$ type theorems for these operators. \end{itemize}
Whereas the domination method \emph{assumes} the unweighted $L^2$ boundedness of the operator $T$, the representation method can (and will, in this exposition) be set up in such a way that it \emph{derives} the unweighted boundedness from a priori weaker assumptions as a byproduct. Indeed, a proof of the $T(1)$ theorem of G.~David and J.-L. Journ\'e \cite{DJ} is obtained as a byproduct of the present exposition, and this approach was lifted to the nontrivial case of bi-parameter singular integrals in the mentioned works of Martikainen \cite{Martikainen:Advances} and Ou \cite{Ou:Tb}. Of course, the deterministic domination method has its own advantages, but the point that I want to make here is that so does the probabilistic approach, which I present in the following exposition.
\section{Preliminaries}
The standard (or reference) system of dyadic cubes is \begin{equation*}
\mathscr{D}^0:=\{2^{-k}([0,1)^d+m):k\in\mathbb{Z},m\in\mathbb{Z}^d\}. \end{equation*} We will need several dyadic systems, obtained by translating the reference system as follows. Let $\omega=(\omega_j)_{j\in\mathbb{Z}}\in(\{0,1\}^d)^{\mathbb{Z}}$ and \begin{equation*}
I\dot+\omega:=I+\sum_{j:2^{-j}<\ell(I)}2^{-j}\omega_j. \end{equation*} Then \begin{equation*}
\mathscr{D}^{\omega}:=\{I\dot+\omega:I\in\mathscr{D}^0\}, \end{equation*} and it is straightforward to check that $\mathscr{D}^{\omega}$ inherits the important nestedness property of $\mathscr{D}^0$: if $I,J\in\mathscr{D}^{\omega}$, then $I\cap J\in\{I,J,\varnothing\}$. When the particular $\omega$ is unimportant, the notation $\mathscr{D}$ is sometimes used for a generic dyadic system.
\subsection{Haar functions} Any given dyadic system $\mathscr{D}$ has a natural function system associated to it: the Haar functions. In one dimension, there are two Haar functions associated with an interval $I$: the non-cancellative $h^0_I:=\abs{I}^{-1/2}1_I$ and the cancellative $h^1_I:=\abs{I}^{-1/2}(1_{I_{\ell}}-1_{I_r})$, where $I_{\ell}$ and $I_r$ are the left and right halves of $I$. In $d$ dimensions, the Haar functions on a cube $I=I_1\times\cdots\times I_d$ are formed of all the products of the one-dimensional Haar functions: \begin{equation*}
h_I^{\eta}(x)=h_{I_1\times\cdots\times I_d}^{(\eta_1,\ldots,\eta_d)}(x_1,\ldots,x_d):=\prod_{i=1}^d h_{I_i}^{\eta_i}(x_i). \end{equation*} The non-cancellative $h_I^0=\abs{I}^{-1/2}1_I$ has the same formula as in $d=1$. All other $2^d-1$ Haar functions $h_I^{\eta}$ with $\eta\in\{0,1\}^d\setminus\{0\}$ are cancellative, i.e., satisfy $\int h_I^{\eta}=0$, since they are cancellative in at least one coordinate direction.
For a fixed $\mathscr{D}$, all the cancellative Haar functions $h_I^{\eta}$, $I\in\mathscr{D}$ and $\eta\in\{0,1\}^d\setminus\{0\}$, form an orthonormal basis of $L^2(\mathbb{R}^d)$. Hence any function $f\in L^2(\mathbb{R}^d)$ has the orthogonal expansion \begin{equation*}
f=\sum_{I\in\mathscr{D}}\sum_{\eta\in\{0,1\}^d\setminus\{0\}}\pair{f}{h_I^{\eta}}h_I^{\eta}. \end{equation*} Since the different $\eta$'s seldom play any major role, this will be often abbreviated (with slight abuse of language) simply as \begin{equation*}
f=\sum_{I\in\mathscr{D}}\pair{f}{h_I}h_I, \end{equation*} and the summation over $\eta$ is understood implicitly.
\subsection{Dyadic shifts}
A dyadic shift with parameters $i,j\in\mathbb{N}:=\{0,1,2,\ldots\}$ is an operator of the form \begin{equation*}
Sf=\sum_{K\in\mathscr{D}}A_K f,\qquad
A_K f=\sum_{\substack{I,J\in\mathscr{D};I,J\subseteq K \\ \ell(I)=2^{-i}\ell(K)\\ \ell(J)=2^{-j}\ell(K)}}a_{IJK}\pair{f}{h_I}h_J, \end{equation*} where $h_I$ is a Haar function on $I$ (similarly $h_J$), and the $a_{IJK}$ are coefficients with \begin{equation*}
\abs{a_{IJK}}\leq\frac{\sqrt{\abs{I}\abs{J}}}{\abs{K}}. \end{equation*} It is also required that all subshifts \begin{equation*}
S_{\mathscr{Q}}=\sum_{K\in\mathscr{Q}}A_K,\qquad\mathscr{Q}\subseteq\mathscr{D}, \end{equation*} map $S_{\mathscr{Q}}:L^2(\mathbb{R}^d)\to L^2(\mathbb{R}^d)$ with norm at most one.
The shift is called cancellative, if all the $h_I$ and $h_J$ are cancellative; otherwise, it is called non-cancellative.
The notation $A_K$ indicates an ``averaging operator'' on $K$. Indeed, from the normalization of the Haar functions, it follows that \begin{equation*}
\abs{A_K f}\leq 1_K\fint_K\abs{f} \end{equation*} pointwise.
For cancellative shifts, the $L^2$ boundedness is automatic from the other conditions. This is a consequence of the following facts: \begin{itemize}
\item The pointwise bound for each $A_K$ implies that $\Norm{A_K f}{L^p}\leq\Norm{f}{L^p}$ for all $p\in[1,\infty]$; in particular, these components of $S$ are uniformly bounded on $L^2$ with norm one. (This first point is true even in the non-cancellative case.)
\item Let $\D_K^{i}$ denote the orthogonal projection of $L^2$ onto $\lspan\{h_I:I\subseteq K,\ell(I)=2^{-i}\ell(K)\}$. When $i$ is fixed, it follows readily that any two $\D_K^{i}$ are orthogonal to each other. (This depends on the use of cancellative $h_I$.) Moreover, we have $A_K=\D_K^{j}A_K \D_K^{i}$. Then the boundedness of $S$ follows from two applications of Pythagoras' theorem with the uniform boundedness of the $A_K$ in between. \end{itemize}
A prime example of a non-cancellative shift (and the only one we need in these lectures) is the \emph{dyadic paraproduct} \begin{equation*}
\Pi_b f=\sum_{K\in\mathscr{D}}\pair{b}{h_K}\ave{f}_K h_K
=\sum_{K\in\mathscr{D}}\abs{K}^{-1/2}\pair{b}{h_K}\cdot\pair{f}{h_K^0} h_K, \end{equation*} where $b\in\BMO_d$ (the dyadic BMO space) and $h_K$ is a cancellative Haar function. This is a dyadic shift with parameters $(i,j)=(0,0)$, where $a_{IJK}=\abs{K}^{-1/2}\pair{b}{h_K}$ for $I=J=K$. The $L^2$ boundedness of the paraproduct, if and only if $b\in\BMO_d$, is part of the classical theory. Actually, to ensure the normalization condition of the shift, it should be further required that $\Norm{b}{\BMO_d}\leq 1$.
\subsection{Random dyadic systems; good and bad cubes}
We obtain a notion of \emph{random dyadic systems} by equipping the parameter set $\Omega:=(\{0,1\}^d)^{\mathbb{Z}}$ with the natural probability measure: each component $\omega_j$ has an equal probability $2^{-d}$ of taking any of the $2^d$ values in $\{0,1\}^d$, and all components are independent of each other.
Let $\phi:[0,1]\to[0,1]$ be a fixed \emph{modulus of continuity}: a strictly increasing function with $\phi(0)=0$, $\phi(1)=1$, and $t\mapsto\phi(t)/t$ decreasing (hence $\phi(t)\geq t$) with $\lim_{t\to 0}\phi(t)/t=\infty$. We further require the \emph{Dini condition} \begin{equation*}
\int_0^1\phi(t)\frac{\ud t}{t}<\infty. \end{equation*} Main examples include $\phi(t)=t^{\gamma}$ with $\gamma\in(0,1)$ and \begin{equation*}
\phi(t)=\Big(1+\frac{1}{\gamma}\log\frac{1}{t}\Big)^{-\gamma},\qquad \gamma>1. \end{equation*} We also fix a (large) parameter $r\in\mathbb{Z}_+$. (How large, will be specified shortly.)
A cube $I\in\mathscr{D}^{\omega}$ is called bad if there exists $J\in\mathscr{D}^{\omega}$ such that $\ell(J)\geq 2^r\ell(I)$ and \begin{equation*}
\dist(I,\partial J)\leq\phi\Big(\frac{\ell(I)}{\ell(J)}\Big)\ell(J): \end{equation*} roughly, $I$ is relatively close to the boundary of a much bigger cube.
\begin{remark} This definition of good cubes goes back to Nazarov--Treil--Volberg \cite{NTV:Tb} in the context of singular integrals with respect to non-doubling measures. They used the modulus of continuity $\phi(t)=t^{\gamma}$, where $\gamma$ was chosen to depend on the dimension and the H\"older exponent of the Calder\'on--Zygmund kernel via \begin{equation*}
\gamma=\frac{\alpha}{2(d+\alpha)}. \end{equation*} This choice has become ``canonical'' in the subsequent literature, including the original proof of the $A_2$ theorem. However, other choices can also be made, as we do here. \end{remark}
We make some basic probabilistic observations related to badness. Let $I\in\mathscr{D}^0$ be a reference interval. The \emph{position} of the translated interval \begin{equation*}
I\dot+\omega=I+\sum_{j:2^{-j}<\ell(I)}2^{-j}\omega_j, \end{equation*} by definition, depends only on $\omega_j$ for $2^{-j}<\ell(I)$. On the other hand, the \emph{badness} of $I\dot+\omega$ depends on its \emph{relative position} with respect to the bigger intervals \begin{equation*}
J\dot+\omega=J+\sum_{j:2^{-j}<\ell(I)}2^{-j}\omega_j+\sum_{j:\ell(I)\leq 2^{-j}<\ell(I)}2^{-j}\omega_j. \end{equation*} The same translation component $\sum_{j:2^{-j}<\ell(I)}2^{-j}\omega_j$ appears in both $I\dot+\omega$ and $J\dot+\omega$, and so does not affect the relative position of these intervals. Thus this relative position, and hence the badness of $I$, depends only on $\omega_j$ for $2^{-j}\geq\ell(I)$. In particular:
\begin{lemma}\label{lem:indep} For $I\in\mathscr{D}^0$, the position and badness of $I\dot+\omega$ are independent random variables. \end{lemma}
Another observation is the following: by symmetry and the fact that the condition of badness only involves relative position and size of different cubes, it readily follows that the probability of a particular cube $I\dot+\omega$ being bad is equal for all cubes $I\in\mathscr{D}^0$: \begin{equation*}
\prob_{\omega}(I\dot+\omega\bad)=\pi_{\bad}=\pi_{\bad}(r,d,\phi). \end{equation*}
The final observation concerns the value of this probability:
\begin{lemma} We have \begin{equation*}
\pi_{\bad}\leq 8d\int_0^{2^{-r}}\phi(t)\frac{\ud t}{t}; \end{equation*} in particular, $\pi_{\bad}<1$ if $r=r(d,\phi)$ is chosen large enough. \end{lemma}
With $r=r(d,\phi)$ chosen like this, we then have $\pi_{\good}:=1-\pi_{\bad}>0$, namely, good situations have positive probability!
\begin{proof} Observe that in the definition of badness, we only need to consider those $J$ with $I\subseteq J$. Namely, if $I$ is close to the boundary of some bigger $J$, we can always find another dyadic $J'$ of the same size as $J$ which contains $I$, and then $I$ will also be close to the boundary of $J'$. Hence we need to consider the relative position of $I$ with respect to each $J\supset I$ with $\ell(J)=2^k\ell(I)$ and $k=r,r+1,\ldots$ For a fixed $k$, this relative position is determined by \begin{equation*}
\sum_{j:\ell(I)\leq 2^{-j}<2^k\ell(I)}2^{-j}\omega_j, \end{equation*} which has $2^{kd}$ different values with equal probability. These correspond to the subcubes of $J$ of size $\ell(I)$.
Now bad position of $I$ are those which are within distance $\phi(\ell(I)/\ell(J))\cdot\ell(J)$ from the boundary. Since the possible position of the subcubes are discrete, being integer multiples of $\ell(I)$, the effective bad boundary region has depth \begin{equation*} \begin{split}
\Big\lceil \phi\Big(\frac{\ell(I)}{\ell(J)}\Big)\frac{\ell(J)}{\ell(I)}\Big\rceil\ell(I)
&\leq\Big(\phi\Big(\frac{\ell(I)}{\ell(J)}\Big)\frac{\ell(J)}{\ell(I)}+1\Big)\ell(I) \\
&=\ell(J)\Big(\phi\Big(\frac{\ell(I)}{\ell(J)}\Big)+\frac{\ell(I)}{\ell(J)}\Big)\leq 2\ell(J)\phi\Big(\frac{\ell(I)}{\ell(J)}\Big), \end{split} \end{equation*} by using that $t\leq\phi(t)$.
The good region is the cube inside $J$, whose side-length is $\ell(J)$ minus twice the depth of the bad boundary region: \begin{equation*}
\ell(J)-2\Big\lceil \phi\Big(\frac{\ell(I)}{\ell(J)}\Big)\frac{\ell(J)}{\ell(I)}\Big\rceil\ell(I)
\geq\ell(J)-4\ell(J)\phi\Big(\frac{\ell(I)}{\ell(J)}\Big). \end{equation*} Hence the volume of the bad region is \begin{equation*} \begin{split}
\abs{J}-\Big(\ell(J)-2\Big\lceil \phi\Big(\frac{\ell(I)}{\ell(J)}\Big)\frac{\ell(J)}{\ell(I)}\Big\rceil\ell(I)\Big)^d
&\leq\abs{J}\Big(1-\Big(1-4\phi\Big(\frac{\ell(I)}{\ell(J)}\Big)\Big)^d\Big) \\
&\leq\abs{J}\cdot 4d\phi\Big(\frac{\ell(I)}{\ell(J)}\Big) \end{split} \end{equation*} by the elementary inequality $(1-\alpha)^d\geq 1-\alpha d$ for $\alpha\in[0,1]$. (We assume that $r$ is at least so large that $4\phi(2^{-r})\leq 1$.)
So the fraction of the bad region of the total volume is at most $4d\phi(\ell(I)/\ell(J))=4d\phi(2^{-k})$ for a fixed $k=r,r+1,\ldots$. This gives the final estimate \begin{equation*} \begin{split}
\prob_{\omega}(I\dot+\omega\bad)
&\leq\sum_{k=r}^{\infty}4d\phi(2^{-k})
=\sum_{k=r}^{\infty}8d\frac{\phi(2^{-k})}{2^{-k}}2^{-k-1} \\
&\leq\sum_{k=r}^{\infty}8d\int_{2^{-k-1}}^{2^{-k}}\frac{\phi(t)}{t}\ud t
=8d\int_0^{2^{-r}}\phi(t)\frac{\ud t}{t}, \end{split} \end{equation*} where we used that $\phi(t)/t$ is decreasing in the last inequality. \end{proof}
\section{The dyadic representation theorem}\label{sec:representation}
Let $T$ be a Calder\'on--Zygmund operator on $\mathbb{R}^d$. That is, it acts on a suitable dense subspace of functions in $L^2(\mathbb{R}^d)$ (for the present purposes, this class should at least contain the indicators of cubes in $\mathbb{R}^d$) and has the kernel representation \begin{equation*}
Tf(x)=\int_{\mathbb{R}^d}K(x,y)f(y)\ud y,\qquad x\notin\supp f. \end{equation*} Moreover, the kernel should satisfy the \emph{standard estimates}, which we here assume in a slightly more general form than usual, involving another modulus of continuity~$\psi$, like the one considered above: \begin{equation*} \begin{split}
\abs{K(x,y)} &\leq\frac{C_0}{\abs{x-y}^d}, \\
\abs{K(x,y)-K(x',y)}+\abs{K(y,x)-K(y,x')}
&\leq\frac{C_\psi}{\abs{x-y}^d}\psi\Big(\frac{\abs{x-x'}}{\abs{x-y}}\Big) \end{split} \end{equation*} for all $x,x',y\in\mathbb{R}^d$ with $\abs{x-y}>2\abs{x-x'}$. Let us denote the smallest admissible constants $C_0$ and $C_\psi$ by $\Norm{K}{CZ_0}$ and $\Norm{K}{CZ_\psi}$. The classical standard estimates correspond to the choice $\psi(t)=t^{\alpha}$, $\alpha\in(0,1]$, in which case we write $\Norm{K}{CZ_\alpha}$ for $\Norm{K}{CZ_\psi}$.
We say that $T$ is a bounded Calder\'on--Zygmund operator, if in addition $T:L^2(\mathbb{R}^d)\to L^2(\mathbb{R}^d)$, and we denote its operator norm by $\Norm{T}{L^2\to L^2}$.
Let us agree that $\abs{\ }$ stands for the $\ell^{\infty}$ norm on $\mathbb{R}^d$, i.e., $\abs{x}:=\max_{1\leq i\leq d}\abs{x_i}$. While the choice of the norm is not particularly important, this choice is slightly more convenient than the usual Euclidean norm when dealing with cubes as we will: e.g., the diameter of a cube in the $\ell^{\infty}$ norm is equal to its sidelength $\ell(Q)$.
Let us first formulate the dyadic representation theorem for general moduli of continuity, and then specialize it to the usual standard estimates. Define the following coefficients for $i,j\in\mathbb{N}$: \begin{equation*}
\tau(i,j):=\phi(2^{-\max\{i,j\}})^{-d}\psi\big(2^{-\max\{i,j\}}\phi(2^{-\max\{i,j\}})^{-1}\big), \end{equation*} if $\min\{i,j\}>0$; and \begin{equation*}
\tau(i,j):= \Psi\big(2^{-\max\{i,j\}}\phi(2^{-\max\{i,j\}})^{-1}\big),\qquad
\Psi(t):=\int_0^t\psi(s)\frac{\ud s}{s}, \end{equation*} if $\min\{i,j\}=0$.
We assume that $\phi$ and $\psi$ are such, that \begin{equation}\label{eq:decay}
\sum_{i,j=0}^{\infty}\tau(i,j)\eqsim\int_0^1\frac{1}{\phi(t)^{d}}\psi\Big(\frac{t}{\phi(t)}\Big)\frac{\ud t}{t}+\int_0^1\Psi\Big(\frac{t}{\phi(t)}\Big)\frac{\ud t}{t}<\infty. \end{equation} This is the case, in particular, when $\psi(t)=t^{\alpha}$ (usual standard estimates) and $\phi(t)=(1+a^{-1}\log t^{-1})^{-\gamma}$; then one checks that \begin{equation*}
\tau(i,j)\lesssim P(\max\{i,j\})2^{-\alpha\max\{i,j\}},\qquad P(j)=(1+j)^{\gamma(d+\alpha)}, \end{equation*} which clearly satisfies the required convergence. However, it is also possible to treat weaker forms of the standard estimates with a logarithmic modulus $\psi(t)=(1+a^{-1}\log t^{-1})^{-\alpha}$. This might be of some interest for applications, but we do not pursue this line any further here.
\begin{theorem}\label{thm:formula} Let $T$ be a bounded Calder\'on--Zygmund operator with modulus of continuity satisfying the above assumption. Then it has an expansion, say for $f,g\in C^1_c(\mathbb{R}^d)$, \begin{equation*}
\pair{g}{Tf}
=c\cdot\big(\Norm{T}{L^2\to L^2}+\Norm{K}{CZ_\psi}\big)\cdot\Exp_{\omega} \sum_{i,j=0}^{\infty} \tau(i,j)\pair{g}{S^{ij}_{\omega}f}, \end{equation*} where $c$ is a dimensional constant and $S^{ij}_{\omega}$ is a dyadic shift of parameters $(i,j)$ on the dyadic system $\mathscr{D}^{\omega}$; all of them except possibly $S^{00}_{\omega}$ are cancellative. \end{theorem}
The first version of this theorem appeared in \cite{Hytonen:A2}, and another one in \cite{HPTV}. The present proof is yet another variant of the same argument. It is slightly simpler in terms of the probabilistic tools that are used: no conditional probabilities are needed, although they were important for the original arguments.
In proving this theorem, we do not actually need to employ the full strength of the assumption that $T:L^2(\mathbb{R}^d)\to L^2(\mathbb{R}^d)$; rather it suffices to have the kernel conditions plus the following conditions of the $T1$ theorem of David--Journ\'e: \begin{equation*} \begin{split}
\abs{\pair{1_Q}{T1_Q}} &\leq C_{WBP}\abs{Q}\quad\text{(weak boundedness property)},\\
& T1\in\BMO(\mathbb{R}^d),\quad T^*1\in\BMO(\mathbb{R}^d). \end{split} \end{equation*} Let us denote the smallest $C_{WBP}$ by $\Norm{T}{WBP}$. Then we have the following more precise version of the representation:
\begin{theorem}\label{thm:formula2} Let $T$ be a Calder\'on--Zygmund operator with modulus of continuity satisfying the above assumption. Then it has an expansion, say for $f,g\in C^1_c(\mathbb{R}^d)$, \begin{equation*} \begin{split}
\pair{g}{Tf}
&=c\cdot\big(\Norm{K}{CZ_0}+\Norm{K}{CZ_\psi}\big)\Exp_{\omega} \sum_{\substack{i,j=0\\ \max\{i,j\}> 0}}^{\infty} \tau(i,j)\pair{g}{S^{ij}_{\omega}f} \\
&+c\cdot\big(\Norm{K}{CZ_0}+\Norm{T}{WBP}\big)\Exp_{\omega}\pair{g}{S^{00}_{\omega}f}
+\Exp_{\omega}\pair{g}{\Pi_{T1}^{\omega}f}+\Exp_{\omega}\pair{g}{(\Pi_{T^*1}^{\omega})^*f} \end{split} \end{equation*} where $S^{ij}_{\omega}$ is a cancellative dyadic shift of parameters $(i,j)$ on the dyadic system $\mathscr{D}^{\omega}$, and $\Pi_{b}^{\omega}$ is a dyadic paraproduct on the dyadic system $\mathscr{D}^{\omega}$ associated with the $\BMO$-function $b\in\{T1,T^*1\}$. \end{theorem}
\begin{remark} Note that $\Pi^{\omega}_b=\Norm{b}{\BMO}\cdot S^{\omega}_b$, where $S^{\omega}_b=\Pi^{\omega}_b/\Norm{b}{\BMO}$ is a shift with the correct normalization. Hence, writing everything in terms of normalized shifts, as in Theorem~\ref{thm:formula}, we get the factor $\Norm{T1}{\BMO}\lesssim\Norm{T}{L^2\to L^2}+\Norm{K}{CZ_\psi}$ in the second-to-last term, and $\Norm{T^*1}{\BMO}\lesssim\Norm{T}{L^2\to L^2}+\Norm{K}{CZ_\psi}$ in the last one. The proof will also show that both occurrences of the factor $\Norm{K}{CZ_0}$ could be replaced by $\Norm{T}{L^2\to L^2}$, giving the statement of Theorem~\ref{thm:formula} (since trivially $\Norm{T}{WBP}\leq\Norm{T}{L^2\to L^2}$). \end{remark}
As a by-product, Theorem~\ref{thm:formula2} delivers a proof of the $T1$ theorem: under the above assumptions, the operator $T$ is already bounded on $L^2(\mathbb{R}^d)$. Namely, all the dyadic shifts $S^{ij}_{\omega}$ are uniformly bounded on $L^2(\mathbb{R}^d)$ by definition, and the convergence condition \eqref{eq:decay} ensures that so is their average representing the operator $T$. This by-product proof of the $T1$ theorem is not a coincidence, since the proof of Theorems~\ref{thm:formula} and \ref{thm:formula2} was actually inspired by the proof of the $T1$ theorem for non-doubling measures due to Nazarov--Treil--Volberg \cite{NTV:Tb} and its vector-valued extension \cite{Hytonen:nonhomog}.
A key to the proof of the dyadic representation is a random expansion of $T$ in terms of Haar functions $h_I$, where the bad cubes are avoided:
\begin{proposition} \begin{equation*}
\pair{g}{Tf}
=\frac{1}{\pi_{\good}}\Exp_{\omega}\sum_{I,J\in\mathscr{D}^{\omega}}1_{\good}(\operatorname{smaller}\{I,J\})\cdot
\pair{g}{h_{J}}\pair{h_{J}}{Th_{I}}\pair{h_{I}}{f}, \end{equation*} where \begin{equation*}
\operatorname{smaller}\{I,J\}:=\begin{cases} I & \text{if }\ell(I)\leq\ell(J), \\ J & \text{if }\ell(J)>\ell(I). \end{cases} \end{equation*} \end{proposition}
\begin{proof} Recall that \begin{equation*}
f=\sum_{I\in\mathscr{D}^0}\pair{f}{h_{I\dot+\omega}}h_{I\dot+\omega} \end{equation*} for any fixed $\omega\in\Omega$; and we can also take the expectation $\Exp_{\omega}$ of both sides of this identity.
Let \begin{equation*}
1_{\good}(I\dot+\omega):=\begin{cases} 1, & \text{if $I\dot+\omega$ is good},\\ 0, & \text{else}\end{cases} \end{equation*} We make use of the above random Haar expansion of $f$, multiply and divide by \begin{equation*}
\pi_{\good}=\prob_{\omega}(I\dot+\omega\good)=\Exp_{\omega}1_{\good}(I\dot+\omega), \end{equation*} and use the independence from Lemma~\ref{lem:indep} to get: \begin{equation*} \begin{split}
\pair{g}{Tf}
&=\Exp_{\omega}\sum_{I}\pair{g}{Th_{I\dot+\omega}}\pair{h_{I\dot+\omega}}{f} \\
&=\frac{1}{\pi_{\good}}\sum_{I}\Exp_{\omega}[1_{\good}(I\dot+\omega)] \Exp_{\omega}[\pair{g}{Th_{I\dot+\omega}}\pair{h_{I\dot+\omega}}{f}] \\
&=\frac{1}{\pi_{\good}}\Exp_{\omega}\sum_{I}1_{\good}(I\dot+\omega) \pair{g}{Th_{I\dot+\omega}}\pair{h_{I\dot+\omega}}{f} \\
&=\frac{1}{\pi_{\good}}\Exp_{\omega}\sum_{I,J}1_{\good}(I\dot+\omega) \pair{g}{h_{J\dot+\omega}}\pair{h_{J\dot+\omega}}{Th_{I\dot+\omega}}\pair{h_{I\dot+\omega}}{f}. \end{split} \end{equation*} On the other hand, using independence again in half of this double sum, we have \begin{equation*} \begin{split}
&\frac{1}{\pi_{\good}}\sum_{\ell(I)>\ell(J)}\Exp_{\omega}[1_{\good}(I\dot+\omega) \pair{g}{h_{J\dot+\omega}}\pair{h_{J\dot+\omega}}{Th_{I\dot+\omega}}\pair{h_{I\dot+\omega}}{f} ] \\
&=\frac{1}{\pi_{\good}}\sum_{\ell(I)>\ell(J)}\Exp_{\omega}[1_{\good}(I\dot+\omega)]
\Exp_{\omega}[ \pair{g}{h_{J\dot+\omega}}\pair{h_{J\dot+\omega}}{Th_{I\dot+\omega}}\pair{h_{I\dot+\omega}}{f} ] \\
&= \Exp_{\omega}\sum_{\ell(I)>\ell(J)}
\pair{g}{h_{J\dot+\omega}}\pair{h_{J\dot+\omega}}{Th_{I\dot+\omega}}\pair{h_{I\dot+\omega}}{f}, \end{split} \end{equation*} and hence \begin{equation*} \begin{split}
\pair{g}{Tf}
&= \frac{1}{\pi_{\good}}\Exp_{\omega}\sum_{\ell(I)\leq\ell(J)}
1_{\good}(I\dot+\omega) \pair{g}{h_{J\dot+\omega}}\pair{h_{J\dot+\omega}}{Th_{I\dot+\omega}}\pair{h_{I\dot+\omega}}{f} \\
&\qquad+\Exp_{\omega}\sum_{\ell(I)>\ell(J)}
\pair{g}{h_{J\dot+\omega}}\pair{h_{J\dot+\omega}}{Th_{I\dot+\omega}}\pair{h_{I\dot+\omega}}{f}. \end{split} \end{equation*} Comparison with the basic identity \begin{equation}\label{eq:basic}
\pair{g}{Tf}
=\Exp_{\omega}\sum_{I,J}\pair{g}{h_{J\dot+\omega}}\pair{h_{J\dot+\omega}}{Th_{I\dot+\omega}}\pair{h_{I\dot+\omega}}{f} \end{equation} shows that \begin{equation*} \begin{split}
&\Exp_{\omega}\sum_{\ell(I)\leq\ell(J)}
\pair{g}{h_{J\dot+\omega}}\pair{h_{J\dot+\omega}}{Th_{I\dot+\omega}}\pair{h_{I\dot+\omega}}{f} \\
&= \frac{1}{\pi_{\good}}\Exp_{\omega}\sum_{\ell(I)\leq\ell(J)}
1_{\good}(I\dot+\omega) \pair{g}{h_{J\dot+\omega}}\pair{h_{J\dot+\omega}}{Th_{I\dot+\omega}}\pair{h_{I\dot+\omega}}{f}. \end{split} \end{equation*} Symmetrically, we also have \begin{equation*} \begin{split}
&\Exp_{\omega}\sum_{\ell(I)>\ell(J)}
\pair{g}{h_{J\dot+\omega}}\pair{h_{J\dot+\omega}}{Th_{I\dot+\omega}}\pair{h_{I\dot+\omega}}{f} \\
&= \frac{1}{\pi_{\good}}\Exp_{\omega}\sum_{\ell(I)>\ell(J)}
1_{\good}(J\dot+\omega) \pair{g}{h_{J\dot+\omega}}\pair{h_{J\dot+\omega}}{Th_{I\dot+\omega}}\pair{h_{I\dot+\omega}}{f}, \end{split} \end{equation*} and this completes the proof. \end{proof}
This is essentially the end of probability in this proof. Henceforth, we can simply concentrate on the summation inside $\Exp_{\omega}$, for a fixed value of $\omega\in\Omega$, and manipulate it into the required form. Moreover, we will concentrate on the half of the sum with $\ell(J)\geq\ell(I)$, the other half being handled symmetrically. We further divide this sum into the following parts: \begin{equation*} \begin{split}
\sum_{\ell(I)\leq\ell(J)}
&=\sum_{\dist(I,J)>\ell(J)\phi(\ell(I)/\ell(J))}
+\sum_{I\subsetneq J}+\sum_{I=J}
+\sum_{\substack{\dist(I,J)\leq\ell(J)\phi(\ell(I)/\ell(J))\\ I\cap J=\varnothing}} \\
&=:\sigma_{\operatorname{out}}+\sigma_{\operatorname{in}}+\sigma_{=}+\sigma_{\operatorname{near}}. \end{split} \end{equation*} In order to recognize these series as sums of dyadic shifts, we need to locate, for each pair $(I,J)$ appearing here, a common dyadic ancestor which contains both of them. The existence of such containing cubes, with control on their size, is provided by the following:
\begin{lemma}\label{lem:IveeJ} If $I\in\mathscr{D}$ is good and $J\in\mathscr{D}$ is a disjoint ($J\cap I=\varnothing$) cube with $\ell(J)\geq\ell(I)$, then there exists $K\supseteq I\cup J$ which satisfies \begin{equation*} \begin{split}
\ell(K) &\leq 2^r\ell(I), \qquad\text{if}\qquad \dist(I,J)\leq\ell(J)\phi\Big(\frac{\ell(I)}{\ell(J)}\Big), \\
\ell(K)\phi\Big(\frac{\ell(I)}{\ell(K)}\Big) &\leq 2^r\dist(I,J), \qquad\text{if}\qquad\dist(I,J)>\ell(J)\phi\Big(\frac{\ell(I)}{\ell(J)}\Big). \end{split} \end{equation*} \end{lemma}
\begin{proof} Let us start with the following initial observation: if $K\in\mathscr{D}$ satisfies $I\subseteq K$, $J\subset K^c$, and $\ell(K)\geq 2^r\ell(I)$, then \begin{equation*}
\ell(K)\phi\Big(\frac{\ell(I)}{\ell(K)}\Big)<\dist(I,\partial K)=\dist(I,K^c)\leq\dist(I,J). \end{equation*}
\subsubsection*{Case $\dist(I,J)\leq\ell(J)\phi(\ell(I)/\ell(J))$} As $I\cap J=\varnothing$, we have $\dist(I,J)=\dist(I,\partial J)$, and since $I$ is good, this implies $\ell(J)< 2^r\ell(I)$. Let $K=I^{(r)}$, and assume for contradiction that $J\subset K^c$. Then the initial observation implies that \begin{equation*}
\ell(K)\phi\Big(\frac{\ell(I)}{\ell(K)}\Big)<\dist(I,J)\leq\ell(J)\phi\Big(\frac{\ell(I)}{\ell(J)}\Big). \end{equation*} Dividing both sides by $\ell(I)$ and recalling that $\phi(t)/t$ is decreasing, this implies that $\ell(K)<\ell(J)$, a contradiction with $\ell(K)=2^r\ell(I)>\ell(J)$. Hence $J\not\subset K^c$, and since $\ell(J)<\ell(K)$, this implies that $J\subset K$.
\subsubsection*{Case $\dist(I,J)>\ell(J)\phi(\ell(I)/\ell(J))$} Consider the minimal $K\supset I$ with $\ell(K)\geq 2^r\ell(I)$ and $\dist(I,J)\leq\ell(K)\phi(\ell(I)/\ell(K))$. (Since $\phi(t)/t\to\infty$ as $t\to 0$, this bound holds for all large enough $K$.) Then (since $\phi(t)/t$ is decreasing) $\ell(K)>\ell(J)$, and by the initial observation, $J\not\subset K^c$. Hence either $J\subset K$, and it suffices to estimate $\ell(K)$.
By the minimality of $K$, there holds at least one of \begin{equation*}
\tfrac12\ell(K)<2^r\ell(I)\qquad\text{or}\qquad \tfrac12\ell(K)\phi\Big(\frac{\ell(I)}{\tfrac12\ell(K)}\Big)<\dist(I,J), \end{equation*} and the latter immediately implies that $\ell(K)\phi(\ell(I)/\ell(K))<2\dist(I,J)$. In the first case, since $\ell(I)\leq\ell(J)\leq\ell(K)$, we have \begin{equation*}
\ell(K)\phi\Big(\frac{\ell(I)}{\ell(K)}\Big)
\leq 2^r\ell(I)\Big(\frac{\ell(I)}{\ell(K)}\Big)
\leq 2^r\ell(J)\Big(\frac{\ell(I)}{\ell(J)}\Big)
<2^r\dist(I,J), \end{equation*} so the required bound is true in each case. \end{proof}
We denote the minimal such $K$ by $I\vee J$, thus \begin{equation*}
I\vee J:=\bigcap_{K\supseteq I\cup J} K. \end{equation*}
\subsection{Separated cubes, $\sigma_{\operatorname{out}}$}
We reorganize the sum $\sigma_{\operatorname{out}}$ with respect to the new summation variable $K=I\vee J$, as well as the relative size of $I$ and $J$ with respect to $K$: \begin{equation*}
\sigma_{\operatorname{out}}
=\sum_{j=1}^{\infty}\sum_{i=j}^{\infty}\sum_K \sum_{\substack{\dist(I,J)>\ell(J)\phi(\ell(I)/\ell(J))\\ I\vee J=K \\ \ell(I)=2^{-i}\ell(K), \ell(J)=2^{-j}\ell(K)}}. \end{equation*} Note that we can start the summation from $1$ instead of $0$, since the disjointness of $I$ and $J$ implies that $K=I\vee J$ must be strictly larger than either of $I$ and $J$. The goal is to identify the quantity in parentheses as a decaying factor times a cancellative averaging operator with parameters $(i,j)$.
\begin{lemma} For $I$ and $J$ appearing in $\sigma_{\operatorname{out}}$, we have \begin{equation*}
\abs{\pair{h_J}{Th_I}}
\lesssim\Norm{K}{CZ_\psi}\frac{\sqrt{\abs{I}\abs{J}}}{\abs{K}}\phi\Big(\frac{\ell(I)}{\ell(K)}\Big)^{-d}\psi\Big(\frac{\ell(I)}{\ell(K)}\phi\Big(\frac{\ell(I)}{\ell(K)}\Big)^{-1}\Big),
\quad K=I\vee J. \end{equation*} \end{lemma}
\begin{proof} Using the cancellation of $h_I$, standard estimates, and Lemma~\ref{lem:IveeJ} \begin{equation*} \begin{split}
\abs{\pair{h_J}{Th_I}}
&=\Babs{\iint h_J(x)K(x,y)h_I(y)\ud y\ud x} \\
&=\Babs{\iint h_J(x)[K(x,y)-K(x,y_I)]h_I(y)\ud y\ud x} \\
&\lesssim\Norm{K}{CZ_\psi}\iint \abs{h_J(x)}\frac{1}{\dist(I,J)^d}\psi\Big(\frac{\ell(I)}{\dist(I,J)}\Big)\abs{h_I(y)}\ud y\ud x \\
&=\Norm{K}{CZ_\psi}\frac{1}{\dist(I,J)^d}\psi\Big(\frac{\ell(I)}{\dist(I,J)}\Big)\Norm{h_J}{1}\Norm{h_I}{1} \\
&\lesssim\Norm{K}{CZ_\psi}\frac{1}{\ell(K)^d}\phi\Big(\frac{\ell(I)}{\ell(K)}\Big)^{-d}\psi\Big(\frac{\ell(I)}{\ell(K)}\phi\Big(\frac{\ell(I)}{\ell(K)}\Big)^{-1}\Big)\sqrt{\abs{J}}\sqrt{\abs{I}}.\qedhere \end{split} \end{equation*} \end{proof}
\begin{lemma} \begin{equation*} \begin{split}
\sum_{\substack{\dist(I,J)>\ell(J)\phi(\ell(I)/\ell(J))\\ I\vee J=K \\ \ell(I)=2^{-i}\ell(K)\leq \ell(J)=2^{-j}\ell(K)}}
&1_{\good}(I)\cdot \pair{g}{h_{J}}\pair{h_{J}}{Th_{I}}\pair{h_{I}}{f} \\
&=\Norm{K}{CZ_\psi}\phi(2^{-i})^{-d}\psi\big(2^{-i}\phi(2^{-i})^{-1}\big)\pair{g}{A_K^{ij}f}, \end{split} \end{equation*} where $A_K^{ij}$ is a cancellative averaging operator with parameters $(i,j)$. \end{lemma}
\begin{proof} By the previous lemma, substituting $\ell(I)/\ell(K)=2^{-i}$, \begin{equation*}
\abs{\pair{h_J}{Th_I}}
\lesssim\Norm{K}{CZ_\psi}\frac{\sqrt{\abs{I}\abs{J}}}{\abs{K}}\phi(2^{-i})^{-d}\psi\big(2^{-i}\phi(2^{-i})^{-1}\big), \end{equation*} and the first factor is precisely the required size of the coefficients of $A_K^{ij}$. \end{proof}
Summarizing, we have \begin{equation*}
\sigma_{\operatorname{out}}
=\Norm{K}{CZ_\psi}\sum_{j=1}^{\infty}\sum_{i=j}^{\infty}\phi(2^{-i})^{-d}\psi\big(2^{-i}\phi(2^{-i})^{-1}\big)\pair{g}{S^{ij}f}. \end{equation*}
\subsection{Contained cubes, $\sigma_{\operatorname{in}}$}
When $I\subsetneq J$, then $I$ is contained in some subcube of $J$, which we denote by $J_I$.
\begin{equation*} \begin{split}
\pair{h_J}{Th_I}
&=\pair{1_{J_I^c}h_J}{Th_I}+\pair{1_{J_I}h_J}{Th_I} \\
&=\pair{1_{J_I^c}h_J}{Th_I}+\ave{h_J}_{J_I}\pair{1_{J_I}}{Th_I} \\
&=\pair{1_{J_I^c}(h_J-\ave{h_J}_{J_I})}{Th_I}+\ave{h_J}_{I}\pair{1}{Th_I}, \end{split} \end{equation*} where we noticed that $h_J$ is constant on $J_I\supseteq I$.
\begin{lemma} \begin{equation*}
\abs{ \pair{1_{J_I^c}(h_J-\ave{h_J}_{J_I})}{Th_I} }
\lesssim\big(\Norm{K}{CZ_0}+\Norm{K}{CZ_\psi}\big)\Big(\frac{\abs{I}}{\abs{J}}\Big)^{1/2}\Psi\Big(\frac{\ell(I)}{\ell(J)}\phi\big(\frac{\ell(I)}{\ell(J)}\big)^{-1}\Big), \end{equation*} where \begin{equation*}
\Psi(r):=\int_0^r\psi(t)\frac{\ud t}{t}, \end{equation*} and $\Norm{K}{CZ_0}$ could be alternatively replaced by $\Norm{T}{L^2\to L^2}$. \end{lemma}
\begin{proof} \begin{equation*}
\abs{ \pair{1_{J_I^c}(h_J-\ave{h_J}_{J_I})}{Th_I} }
\leq 2\Norm{h_J}{\infty}\int_{J_I^c}\abs{Th_I(x)}\ud x, \end{equation*} where $\Norm{h_J}{\infty}=\abs{J}^{-1/2}$.
\subsubsection*{Case $\ell(I)\geq 2^{-r}\ell(J)$} We have \begin{equation*} \begin{split}
\int_{J_I^c}\abs{Th_I(x)}\ud x
&\leq\int_{3I\setminus I}\Babs{\int K(x,y)h_I(y)\ud y}\ud x \\
&\qquad+\int_{(3I)^c}\Babs{\int [K(x,y)-K(x,y_I)]h_I(y)\ud y}\ud x \\
&\lesssim\Norm{K}{CZ_0}\int_{3I\setminus I}\int_I\frac{1}{\abs{x-y}^d}\ud y\ud x \Norm{h_I}{\infty} \\
&\qquad+\Norm{K}{CZ_\psi}\int_{(3I)^c}\frac{1}{\dist(x,I)^d}\psi\Big(\frac{\ell(I)}{\dist(x,I)}\Big)\Norm{h_I}{1}\ud x \\
&\lesssim\Norm{K}{CZ_0}\abs{I}\Norm{h_I}{\infty}+
\Norm{K}{CZ_\psi}\int_{\ell(I)}^{\infty}\frac{1}{r^d}\psi\Big(\frac{\ell(I)}{r}\Big)r^{d-1}\ud r\Norm{h_I}{1} \\
&=\Norm{K}{CZ_0}\abs{I}^{1/2}+\Norm{K}{CZ_\psi}\int_0^1\psi(t)\frac{\ud t}{t}\abs{I}^{1/2} \\
&\lesssim\big(\Norm{K}{CZ_0}+\Norm{K}{CZ_\psi}\big)\abs{I}^{1/2} \end{split} \end{equation*} by the Dini condition for $\psi$ in the last step.
Alternatively, the part giving the factor $\Norm{K}{CZ_0}$ could have been estimated by \begin{equation*} \begin{split}
\int_{3I\setminus I}\Babs{\int K(x,y)h_I(y)\ud y}\ud x
\leq\abs{3I\setminus I}^{1/2}\Norm{Th_I}{2}
\lesssim\abs{I}^{1/2}\Norm{T}{L^2\to L^2}. \end{split} \end{equation*}
\subsubsection*{Case $\ell(I)< 2^{-r}\ell(J)$} Since $I\subseteq J_I$ is good, we have \begin{equation*}
\dist(I,J_I^c)>\ell(J_I)\phi\Big(\frac{\ell(I)}{\ell(J_I)}\Big)\gtrsim\ell(J)\phi\Big(\frac{\ell(I)}{\ell(J)}\Big) \end{equation*} and hence \begin{equation*} \begin{split}
\int_{J_I^c}\abs{Th_I(x)}\ud x
&\lesssim\Norm{K}{CZ_\psi}\int_{J_I^c}\frac{1}{d(x,I)^d}\psi\Big(\frac{\ell(I)}{\dist(x,I)}\Big)\Norm{h_I}{1}\ud x \\
&\lesssim\Norm{K}{CZ_\psi}\int_{\ell(J)\phi(\ell(I)/\ell(J))}\frac{1}{r^d}\psi\Big(\frac{\ell(I)}{r}\Big)r^{d-1}\ud r\cdot\Norm{h_I}{1}\\
&=\Norm{K}{CZ_\psi}\int_0^{\ell(I)/\ell(J)\cdot\phi(\ell(I)/\ell(J))^{-1}}\psi(t)\frac{\ud t}{t}\cdot\abs{I}^{1/2}.\qedhere \end{split} \end{equation*} \end{proof}
Now we can organize \begin{equation*}
\sigma_{\operatorname{in}}'
:=\sum_J\sum_{I\subsetneq J}\pair{g}{h_J}\pair{1_{J_I^c}(h_J-\ave{h_J}_{J_I})}{Th_I}\pair{h_I}{f}
=\sum_{i=1}^{\infty}\sum_J\sum_{\substack{I\subset J\\ \ell(I)=2^{-i}\ell(J)}}, \end{equation*} and the inner sum is recognized as \begin{equation*}
\big(\Norm{K}{CZ_0}+\Norm{K}{CZ_\psi}\big)\Psi(2^{-i}\phi(2^{-i})^{-1})\pair{g}{A_J^{i0} f}, \end{equation*} or with $\Norm{T}{L^2\to L^2}$ in place of $\Norm{K}{CZ_0}$, for a cancellative averaging operator of type $(i,0)$.
On the other hand, \begin{equation*} \begin{split}
\sigma_{\operatorname{in}}''
&:=\sum_J\sum_{I\subsetneq J}\pair{g}{h_J}\ave{h_J}_{I}\pair{1}{Th_I}\pair{h_I}{f} \\
&=\sum_I\Big\langle\sum_{J\supsetneq I}\pair{g}{h_J}h_J\Big\rangle_{I}\pair{1}{Th_I}\pair{h_I}{f} \\
&=\sum_I\ave{g}_I\pair{T^*1}{h_I}\pair{h_I}{f} \\
&=\Bpair{\sum_I\ave{g}_I\pair{T^*1}{h_I}h_I}{f} =:\pair{\Pi_{T^*1}g}{f}
=\pair{g}{\Pi_{T^*1}^* f}. \end{split} \end{equation*} Here $\Pi_{T^*1}$ is the \emph{paraproduct}, a non-cancellative shift composed of the non-cancellative averaging operators \begin{equation*}
A_I g=\pair{T^*1}{h_I}\ave{g}_I h_I=\abs{I}^{-1/2}\pair{T^*1}{h_I}\cdot\pair{g}{h_I^0}h_I \end{equation*} of type $(0,0)$.
Summarizing, we have \begin{equation*} \begin{split}
\sigma_{\operatorname{in}}
&=\sigma_{\operatorname{in}}'+\sigma_{\operatorname{in}}'' \\
&=\big(\Norm{K}{CZ_0}+\Norm{K}{CZ_\psi}\big)\sum_{i=1}^{\infty}\Psi(2^{-i}\phi(2^{-i})^{-1})\pair{g}{S^{i0}f}+\pair{\Pi_{T^*1}g}{f}, \end{split} \end{equation*} where $\Psi(t)=\int_0^t\psi(s)\ud s/s$, and $\Norm{K}{CZ_0}$ could be replaced by $\Norm{T}{L^2\to L^2}$. Note that if we wanted to write $\Pi_{T^*1}$ in terms of a shift with correct normalization, we should divide and multiply it by $\Norm{T^*1}{\BMO}$, thus getting a shift times the factor $\Norm{T^*1}{\BMO}\lesssim\Norm{T}{L^2}+\Norm{K}{CZ_\psi}$
\subsection{Near-by cubes, $\sigma_{=}$ and $\sigma_{\operatorname{near}}$} We are left with the sums $\sigma_{=}$ of equal cubes $I=J$, as well as $\sigma_{\operatorname{near}}$ of disjoint near-by cubes with $\dist(I,J)\leq\ell(J)\phi(\ell(I)/\ell(J))$. Since $I$ is good, this necessarily implies that $\ell(I)>2^{-r}\ell(J)$. Then, for a given $J$, there are only boundedly many related $I$ in this sum.
\begin{lemma} \begin{equation*}
\abs{\pair{h_J}{Th_I}}\lesssim\Norm{K}{CZ_0}+\delta_{IJ}\Norm{T}{WBP}. \end{equation*} \end{lemma}
Note that if we used the $L^2$-boundedness of $T$ instead of the $CZ_0$ and $WBP$ conditions (as is done in Theorem~\ref{thm:formula}, we could also estimate simply \begin{equation*}
\abs{\pair{h_J}{Th_I}}\leq\Norm{h_J}{2}\Norm{T}{L^2\to L^2}\Norm{h_I}{2}=\Norm{T}{L^2\to L^2}. \end{equation*}
\begin{proof} For disjoint cubes, we estimate directly \begin{equation*} \begin{split}
\abs{\pair{h_J}{Th_I}}
&\lesssim\Norm{K}{CZ_0}\int_J\int_I\frac{1}{\abs{x-y}^d}\ud y\ud x\Norm{h_J}{\infty}\Norm{h_I}{\infty} \\
&\leq\Norm{K}{CZ_0}\int_J\int_{3J\setminus J}\frac{1}{\abs{x-y}^d}\ud y\ud x\abs{J}^{-1/2}\abs{I}^{-1/2} \\
&\lesssim\Norm{K}{CZ_0}\abs{J}\abs{J}^{-1/2}\abs{J}^{-1/2}=\Norm{K}{CZ_0}, \end{split} \end{equation*} since $\abs{I}\eqsim\abs{J}$.
For $J=I$, let $I_i$ be its dyadic children. Then \begin{equation*} \begin{split}
\abs{\pair{h_I}{Th_I}}
&\leq\sum_{i,j=1}^{2^d}\abs{\ave{h_I}_{I_i}\ave{h_I}_{I_j}\pair{1_{I_i}}{T1_{I_j}}} \\
&\lesssim\Norm{K}{CZ_0}\sum_{j\neq i}\abs{I}^{-1}\int_{I_i}\int_{I_j}\frac{1}{\abs{x-y}^d}\ud x\ud y
+\sum_i\abs{I}^{-1}\abs{\pair{1_{I_i}}{T1_{I_i}}} \\
&\lesssim\Norm{K}{CZ_0}+\Norm{T}{WBP}, \end{split} \end{equation*} by the same estimate as earlier for the first term, and the weak boundedness property for the second. \end{proof}
With this lemma, the sum $\sigma_{=}$ is recognized as a cancellative dyadic shift of type $(0,0)$ as such: \begin{equation*} \begin{split}
\sigma_{=}
&=\sum_{I\in\mathscr{D}}1_{\good}(I)\cdot\pair{g}{h_I}\pair{h_I}{Th_I}\pair{h_I}{f} \\
&=\big(\Norm{K}{CZ_0}+\Norm{T}{WBP}\big)\pair{g}{S^{00}f}, \end{split} \end{equation*} where the factor in front could also be replaced by $\Norm{T}{L^2\to L^2}$.
For $I$ and $J$ participating in $\sigma_{\operatorname{near}}$, we conclude from Lemma~\ref{lem:IveeJ} that $K:=I\vee J$ satisfies $\ell(K)\leq 2^r\ell(I)$, and hence we may organize \begin{equation*}
\sigma_{\operatorname{near}}
=\sum_{i=1}^r\sum_{j=1}^i \sum_K\sum_{\substack{I,J:I\vee J=K\\ \dist(I,J)\leq\ell(J)\phi(\ell(I)/\ell(J))\\ \ell(I)=2^{-i}\ell(K) \\ \ell(J)=2^{-j}\ell(K)}}, \end{equation*} and the innermost sum is recognized as $\Norm{K}{CZ_0}\pair{g}{A^{ij}_K f}$ for some cancellative averaging operator of type $(i,j)$.
Summarizing, we have \begin{equation*}
\sigma_{=}+\sigma_{\operatorname{near}}
=\big(\Norm{K}{CZ_0}+\Norm{T}{WBP}\big)\pair{g}{S^{00}f}
+\Norm{K}{CZ_0}\sum_{j=1}^r\sum_{i=j}^r \pair{g}{S^{ij}f}, \end{equation*} where $S^{00}$ and $S^{ij}$ are cancellative dyadic shifts, and the factor $\big(\Norm{K}{CZ_0}+\Norm{T}{WBP}\big)$ could also be replaced by $\Norm{T}{L^2\to L^2}$.
\subsection{Synthesis} We have checked that \begin{equation*} \begin{split}
\sum_{\ell(I)\leq\ell(J)} &1_{\operatorname{good}}(I)\pair{g}{h_J}\pair{h_J}{Th_I}\pair{h_I}{f} \\
&=\big(\Norm{K}{CZ_0}+\Norm{K}{CZ_\psi}\big)\Big(\sum_{1\leq j\leq i<\infty}\phi(2^{-i})^{-d}\psi(2^{-i}\phi(2^{-i})^{-1})\pair{g}{S^{ij}f} \\
&\qquad\qquad+\sum_{1\leq i<\infty}\Psi(2^{-i}\phi(2^{-i})^{-1}))\pair{g}{S^{i0}f}\Big) \\
&\qquad+\big(\Norm{K}{CZ_0}+\Norm{T}{WBP}\big)\pair{g}{S^{00}f}+\pair{g}{\Pi_{T^*1}^*f} \end{split} \end{equation*} where $\Psi(t)=\int_0^t\psi(s)\ud s/s$, $\Pi_{T^*1}$ is a paraproduct---a non-cancellative shift of type $(0,0)$--, and all other $S^{ij}$ is a cancellative dyadic shifts of type $(i,j)$.
By symmetry (just observing that the cubes of equal size contributed precisely to the presence of the cancellative shifts of type $(i,i)$, and that the dual of a shift of type $(i,j)$ is a shift of type $(j,i)$), it follows that \begin{equation*} \begin{split}
\sum_{\ell(I)>\ell(J)} &1_{\operatorname{good}}(J)\pair{g}{h_J}\pair{h_J}{Th_I}\pair{h_I}{f} \\
&=\big(\Norm{K}{CZ_0}+\Norm{K}{CZ_\psi}\big)\Big(\sum_{1\leq i<j<\infty}\phi(2^{-j})^{-d}\psi(2^{-j}\phi(2^{-j})^{-1})\pair{g}{S^{ij}f} \\
&\qquad+\sum_{1\leq j<\infty}\Psi(2^{-j}\phi(2^{-j})^{-1}))\pair{g}{S^{0j}f}\Big)+\pair{g}{\Pi_{T1}f} \end{split} \end{equation*} so that altogether \begin{equation*} \begin{split}
\sum_{I,J} &1_{\operatorname{good}}(\min\{I,J\}) \pair{g}{h_J}\pair{h_J}{Th_I}\pair{h_I}{f} \\
&=\big(\Norm{K}{CZ_0}+\Norm{K}{CZ_\psi}\big)\Big(\sum_{i=1}^{\infty}\Psi(2^{-i}\phi(2^{-i})^{-1}))(\pair{g}{S^{i0}f}+\pair{g}{S^{0i}f}) \\
&\qquad\qquad+\sum_{i,j=1}^{\infty}\phi(2^{-\max(i,j)})^{-d}\psi(2^{-\max(i,j)}\phi(2^{-\max(i,j)})^{-1})\pair{g}{S^{ij}f}\Big)\\
&\qquad+\big(\Norm{K}{CZ_0}+\Norm{T}{WBP}\big)\pair{g}{S^{00}f}+\pair{g}{\Pi_{T1}f}+\pair{g}{\Pi_{T^*1}^*f}, \end{split} \end{equation*} and this completes the proof of Theorem~\ref{thm:formula}.
\section{Two-weight theory for dyadic shifts}
Before proceeding further, it is convenient to introduce a useful trick due to E.~Sawyer. Let $\sigma$ be an everywhere positive, finitely-valued function. Then $f\in L^p(w)$ if and only if $\phi=f/\sigma\in L^p(\sigma^p w)$, and they have equal norms in the respective spaces. Hence an inequality \begin{equation}\label{eq:original}
\Norm{Tf}{L^p(w)}\leq N\Norm{f}{L^p(w)}\qquad\forall f\in L^p(w) \end{equation} is equivalent to \begin{equation*}
\Norm{T(\phi\sigma)}{L^p(w)}\leq N\Norm{\phi\sigma}{L^p(w)}=N\Norm{\phi}{L^p(\sigma^p w)}\qquad\forall \phi\in L^p(\sigma^p w). \end{equation*} This is true for any $\sigma$, and we now choose it in such a way that $\sigma^p w=\sigma$, i.e., $\sigma=w^{-1/(p-1)}=w^{1-p'}$, where $p'$ is the dual exponent. So finally \eqref{eq:original} is equivalent to \begin{equation*}
\Norm{T(\phi\sigma)}{L^p(w)}\leq N\Norm{\phi}{L^p(\sigma)}\qquad\forall \phi\in L^p(\sigma). \end{equation*} This formulation has the advantage that the norm on the right and the operator \begin{equation*}
T(\phi\sigma)(x)=\int K(x,y)\phi(y)\cdot\sigma(y)\ud y \end{equation*} involve integration with respect to the same measure $\sigma$. In particular, the $A_2$ theorem is equivalent to \begin{equation*}
\Norm{T(f\sigma)}{L^2(w)}\leq c_T[w]_{A_2}\Norm{f}{L^2(\sigma)} \end{equation*} for all $f\in L^2(w)$, for all $w\in A_2$ and $\sigma=w^{-1}$. But once we know this, we can also study this two-weight inequality on its own right, for two general measures $w$ and $\sigma$, which need not be related by the pointwise relation $\sigma(x)=1/w(x)$.
\begin{theorem}\label{thm:2weight} Let $\sigma$ and $w$ be two locally finite measures with \begin{equation*}
[w,\sigma]_{A_2}:=\sup_Q\frac{w(Q)\sigma(Q)}{\abs{Q}^2}<\infty. \end{equation*} Then a dyadic shift $S$ of type $(i,j)$ satisfies $S(\sigma\cdot):L^2(\sigma)\to L^2(w)$ if and only if \begin{equation*}
\mathfrak{S}:=\sup_Q\frac{\Norm{1_Q S(\sigma 1_Q)}{L^2(w)}}{\sigma(Q)^{1/2}},\qquad
\mathfrak{S}^*:=\sup_Q\frac{\Norm{1_Q S^*(w 1_Q)}{L^2(\sigma)}}{w(Q)^{1/2}} \end{equation*} are finite, and in this case \begin{equation*}
\Norm{S(\sigma\cdot)}{L^2(\sigma)\to L^2(w)}
\lesssim(1+\kappa)(\mathfrak{S}+\mathfrak{S}^*)+(1+\kappa)^2[w,\sigma]_{A_2}^{1/2}, \end{equation*} where $\kappa=\max\{i,j\}$. \end{theorem}
This result from my work with P\'erez, Treil, and Volberg \cite{HPTV} was preceded by an analogous qualitative version due to Nazarov, Treil, and Volberg \cite{NTV:2weightHaar}.
The proof depends on decomposing functions in the spaces $L^2(w)$ and $L^2(\sigma)$ in terms of expansions similar to the Haar expansion in $L^2(\mathbb{R}^d)$. Let $\D^{\sigma}_I$ be the orthogonal projection of $L^2(\sigma)$ onto its subspace of functions supported on $I$, constant on the subcubes of $I$, and with vanishing integral with respect to $\ud\sigma$. Then any two $\D_I^{\sigma}$ are orthogonal to each other. Under the additional assumption that the $\sigma$ measure of quadrants of $\mathbb{R}^d$ is infinite, we have the expansion \begin{equation*}
f=\sum_{Q\in\mathscr{D}}\D_Q^{\sigma}f \end{equation*} for all $f\in L^2(\sigma)$, and Pythagoras' theorem says that \begin{equation*}
\Norm{f}{L^2(\sigma)}=\Big(\sum_{Q\in\mathscr{D}}\Norm{\D_Q^{\sigma}f}{L^2(\sigma)}^2\Big)^{1/2}. \end{equation*} (These formulae needs a slight adjustment if the $\sigma$ measure of quadrants is finite; Theorem~\ref{thm:2weight} remains true without this extra assumption.) Let us also write \begin{equation*}
\D^{\sigma,i}_K:=\sum_{\substack{I\subseteq K\\ \ell(I)=2^{-i}\ell(K)}}\D^{\sigma}_I. \end{equation*} For a fixed $i\in\mathbb{N}$, these are also orthogonal to each other, and the above formulae generalize to \begin{equation*}
f=\sum_{Q\in\mathscr{D}}\D_Q^{\sigma,i}f,\qquad\Norm{f}{L^2(\sigma)}=\Big(\sum_{Q\in\mathscr{D}}\Norm{\D_Q^{\sigma,i}f}{L^2(\sigma)}^2\Big)^{1/2}. \end{equation*}
The proof is in fact very similar in spirit to that of Theorem~\ref{thm:formula}; it is another $T1$ argument, but now with respect to the measures $\sigma$ and $w$ in place of the Lebesgue measure. We hence expand \begin{equation*}
\pair{g}{S(\sigma f)}_w
=\sum_{Q,R\in\mathscr{D}}\pair{\D_R^w g}{S(\sigma\D_Q^{\sigma}f)}_w,\qquad f\in L^2(\sigma),\ g\in L^2(w), \end{equation*} and estimate the matrix coefficients \begin{equation}\label{eq:2weightMatrix} \begin{split}
\pair{\D_R^{w}g}{S(\sigma \D^{\sigma}_Q f)}_{w}
&=\sum_K\pair{\D^w_R g}{A_K(\sigma\D^{\sigma}_Q f)}_w \\
&=\sum_K\sum_{I,J\subseteq K}a_{IJK}\pair{\D^w_R g}{h_J}_w\pair{h_I}{\D^{\sigma}_Q f}_{\sigma}. \end{split} \end{equation} For $\pair{h_I}{\D^{\sigma}_Q f}_{\sigma}\neq 0$, there must hold $I\cap Q\neq\varnothing$, thus $I\subseteq Q$ or $Q\subsetneq I$. But in the latter case $h_I$ is constant on $Q$, while $\int\D^{\sigma}_Q f\cdot\sigma=0$, so the pairing vanishes even in this case. Thus the only nonzero contributions come from $I\subseteq Q$, and similarly from $J\subseteq R$. Since $I,J\subseteq K$, there holds \begin{equation*}
\big(I\subseteq Q\subsetneq K\quad\text{or}\quad K\subseteq Q\big)\qquad\text{and}\qquad\big(J\subseteq R\subsetneq K\quad\text{or}\quad K\subseteq R\big). \end{equation*}
\subsection{Disjoint cubes} Suppose now that $Q\cap R=\varnothing$, and let $K$ be among those cubes for which $A_K$ gives a nontrivial contribution above. Then it cannot be that $K\subseteq Q$, since this would imply that $Q\cap R\supseteq K\cap J=J\neq\varnothing$, and similarly it cannot be that $K\subseteq R$. Thus $Q,R\subsetneq K$, and hence \begin{equation*}
Q\vee R\subseteq K. \end{equation*} Then \begin{align*}
\abs{\pair{\D_R^w g}{S(\sigma\D^{\sigma}_Q f)}_w}
&\leq \sum_{K\supseteq Q\vee R}\abs{\pair{\D^w_R g}{A_K(\sigma\D^{\sigma}_Q f)}_{w}} \\
&\lesssim \sum_{K\supseteq Q\vee R} \frac{\Norm{\D^w_R g}{L^1(w)} \Norm{\D^{\sigma}_Q f}{L^1(\sigma)}}{\abs{K}} \\
&\lesssim \frac{\Norm{\D^w_R g}{L^1(w)} \Norm{\D^{\sigma}_Q f}{L^1(\sigma)}}{\abs{Q\vee R}} \end{align*} On the other hand, we have $Q\supseteq I$, $R\supseteq J$ for some $I,J\subseteq K$ with $\ell(I)=2^{-i}\ell(K)$ and $\ell(J)=2^{-j}\ell(K)$. Hence $2^{-i}\ell(K)\leq\ell(Q)$ and $2^{-j}\ell(K)\leq\ell(R)$, and thus \begin{equation*}
Q\vee R\subseteq K\subseteq Q^{(i)}\cap R^{(j)}. \end{equation*}
Now it is possible to estimate the total contribution of the part of the matrix with $Q\cap R=\varnothing$. Let $P:=Q\vee R$ be a new auxiliary summation variable. Then $Q,R\subset P$, and $\ell(Q)=2^{-a}\ell(P)$, $\ell(R)=2^{-b}\ell(P)$ where $a=1,\ldots,i$, $b=1,\ldots,j$. Thus \begin{align*}
\sum_{\substack{Q,R\in\mathscr{D}\\ Q\cap R=\varnothing}} &\abs{\pair{\D_R^{w}g}{S(\sigma\D^{\sigma}_Q f)}_{w}} \\
&\lesssim\sum_{a=1}^i\sum_{b=1}^j\sum_{P\in\mathscr{D}}\frac{1}{\abs{P}}
\sum_{\substack{Q,R\in\mathscr{D}:Q\vee R=P\\ \ell(Q)=2^{-a}\ell(P)\\ \ell(R)=2^{-b}\ell(P)}}\Norm{\D^w_R g}{L^1(\sigma)} \Norm{\D^{\sigma}_Q f}{L^1(w)} \\
&\leq\sum_{a,b=1}^{i,j}\sum_{P\in\mathscr{D}}\frac{1}{\abs{P}}\sum_{\substack{R\subseteq P\\ \ell(R)=2^{-b}\ell(P)}}\Norm{\D^w_R g}{L^1(\sigma)}
\sum_{\substack{Q\subseteq P\\ \ell(Q)=2^{-a}\ell(P)}}\Norm{\D^{\sigma}_{Q}f}{L^1(\sigma)} \\
&=\sum_{a,b=1}^{i,j}\sum_{P\in\mathscr{D}}\frac{1}{\abs{P}}\BNorm{\sum_{\substack{R\subseteq P\\ \ell(R)=2^{-b}\ell(P)}}\D^w_R g}{L^1(\sigma)}
\BNorm{\sum_{\substack{Q\subseteq P\\ \ell(Q)=2^{-a}\ell(P)}}\D^{\sigma}_{Q}f}{L^1(\sigma)}\\
&\qquad\qquad\qquad\text{(by disjoint supports)} \\
&=\sum_{a,b=1}^{i,j}\sum_{P\in\mathscr{D}}\frac{1}{\abs{P}}\Norm{\D^{w,j}_{P}g}{L^1(w)}\Norm{\D^{\sigma,i}_{P}f}{L^1(\sigma)} \\
&\leq\sum_{a,b=1}^{i,j}\sum_{P\in\mathscr{D}}\frac{\sigma(P)^{1/2}w(P)^{1/2}}{\abs{P}}
\Norm{\D^{w,j}_{P}g}{L^2(w)}\Norm{\D^{\sigma,i}_{P}f}{L^2(\sigma)}\\
&\leq\sum_{a,b=1}^{i,j}[w,\sigma]_{A_2}^{1/2}\Big(\sum_{P\in\mathscr{D}}\Norm{\D^{w,j}_{P}g}{L^2(w)}^2\Big)^{1/2}
\Big(\sum_{P\in\mathscr{D}}\Norm{\D^{\sigma,i}_{P}f}{L^2(\sigma)}^2\Big)^{1/2} \\
&\leq ij[w,\sigma]_{A_2}^{1/2}\Norm{g}{L^2(w)}\Norm{f}{L^2(\sigma)}. \end{align*}
\subsection{Deeply contained cubes}
Consider now the part of the sum with $Q\subset R$ and $\ell(Q)<2^{-i}\ell(R)$. (The part with $R\subset Q$ and $\ell(R)<2^{-j}\ell(Q)$ would be handled in a symmetrical manner.)
\begin{lemma}\label{lem:contCubesAlgebra} For all $Q\subset R$ with $\ell(Q)<2^{-i}\ell(R)$, we have \begin{equation*}
\pair{\D^w_R g}{S(\sigma\D^{\sigma}_Q f)}_{w}
=\ave{\D^w_R g}_{Q^{(i)}} \pair{S^*(w1_{Q^{(i)}})}{\D^{\sigma}_Q f}_{\sigma}, \end{equation*} where further \begin{equation*}
\D^{\sigma}_Q S^*(w1_{Q^{(i)}})
=\D^{\sigma}_Q S^*(w1_{P})\qquad\text{for any }P\supseteq Q^{(i)}. \end{equation*} \end{lemma}
Recall that $\D^{\sigma}_Q=(\D^{\sigma}_Q)^2=(\D^{\sigma}_Q)^*$ is an orthogonal projection on $L^2(\sigma)$, so that it can be moved to either or both sides of the pairing $\pair{\ }{\ }_{\sigma}$.
\begin{proof} Recall formula~\eqref{eq:2weightMatrix}. If $\pair{h_I}{\D_Q^{\sigma} f}_{\sigma}$ is nonzero, then $I\subseteq Q$, and hence \begin{equation*}
J\subseteq K=I^{(i)}\subseteq Q^{(i)}\subsetneq R \end{equation*} for all $J$ participating in the same $A_K$ as $I$. Thus $\D^w_R g$ is constant on $Q^{(i)}$, hence \begin{equation*} \begin{split}
\pair{\D^w_R g}{A_K(\sigma\D^{\sigma}_Q f)}_{w}
&=\pair{1_{Q^{(i)}}\D^w_R g}{A_K(\sigma\D^{\sigma}_Q f)}_{w} \\
&=\ave{\D^w_R g}_{Q^{(i)}}^w \pair{1_{Q^{(i)}}}{A_K(\sigma\D^{\sigma}_Q f)}_{w} \\
&=\ave{\D^w_R g}_{Q^{(i)}}^w \pair{A_K^*(w1_{Q^{(i)}})}{\D^{\sigma}_Q f}_{\sigma}. \end{split} \end{equation*}
Moreover, for any $P\supseteq Q^{(i)}\supseteq K$, \begin{equation*} \begin{split}
\pair{\D^{\sigma}_Q A_K^*(w1_{Q^{(i)}})}{f}_{\sigma}
&=\pair{1_{Q^{(i)}}}{A_K(\sigma\D^{\sigma}_Q f)}_{w} \\
&=\int A_K(\sigma\D^{\sigma}_Q f)w \\
&=\pair{1_P}{A_K(\sigma\D^{\sigma}_Q f)}_w
=\pair{\D^{\sigma}_QA_K^*(w1_P)}{ f}_{\sigma}. \end{split} \end{equation*} Summing these equalities over all relevant $K$, and using $S=\sum_K A_K$, gives the claim. \end{proof}
By the lemma, we can then manipulate \begin{align*}
\sum_{\substack{Q,R:Q\subset R\\ \ell(Q)<2^{-i}\ell(R)}} &\pair{\D^w_R g}{S(\sigma\D^{\sigma}_Q f)}_{w} \\
&=\sum_Q\Big(\sum_{R\supsetneq Q^{(i)}}\ave{\D^w_R g}_{Q^{(i)}}^w\Big)\pair{S^*(w1_{Q^{(i)}})}{\D^{\sigma}_Q f}_{\sigma} \\
&=\sum_Q\ave{g}^{w}_{Q^{(i)}}\pair{S^*(w1_{Q^{(i)}})}{\D^{\sigma}_Q f}_{\sigma} \\
&=\sum_R\ave{g}^w_R\Bpair{S^*(w1_R)}{\sum_{\substack{Q\subseteq R \\ \ell(Q)=2^{-i}\ell(R)}}\D^{\sigma}_Q f}_{\sigma} \\
&=\sum_R\ave{g}^w_R\Bpair{S^*(w 1_R)}{\D^{\sigma,i}_R f}_{\sigma}, \end{align*} where $\ave{g}^w_R:=w(R)^{-1}\int_R g\cdot w$ is the average of $g$ on $R$ with respect to the $w$ measure.
By using the properties of the pairwise orthogonal projections $\D^{\sigma,i}_R$ on $L^2(\sigma)$, the above series may be estimated as follows: \begin{align*}
&\Babs{\sum_{\substack{Q,R:Q\subset R\\ \ell(Q)<2^{-i}\ell(R)}} \pair{\D^w_R g}{S(\sigma\D^{\sigma}_Q f)}_{w}} \\
&\leq\sum_R\abs{\ave{g}^w_R}\Norm{\D^{\sigma,i}_R S^*(w1_R)}{L^2(\sigma)}\Norm{\D^{\sigma,i}_R f}{L^2(\sigma)} \\
&\leq\Big(\sum_R\abs{\ave{g}^w_R}^2\Norm{\D^{\sigma,i}_R S^*(w1_R)}{L^2(\sigma)}^2\Big)^{1/2}
\Big(\sum_R\Norm{\D^{\sigma,i}_R f}{L^2(\sigma)}^2\Big)^{1/2}, \end{align*} where the last factor is equal to $\Norm{f}{L^2(w)}$.
The first factor on the right is handled by the dyadic Carleson embedding theorem: It follows from the second equality of Lemma~\ref{lem:contCubesAlgebra}, namely $\D^{\sigma}_Q S^*(w1_Q^{(i)})=\D^{\sigma}_Q S^*(w1_P)$ for all $P\supseteq Q^{(i)}$, that $\D^{\sigma,i}_R S^*(w1_R)=\D^{\sigma}_Q S^*(w1_P)$ for all $P\subseteq R$. Hence, we have \begin{equation*} \begin{split}
\sum_{R\subseteq P}\Norm{\D^{\sigma,i}_R S^*(w 1_R)}{L^2(\sigma)}^2
&=\sum_{R\subseteq P}\Norm{\D^{\sigma,i}_R(1_P S^*(w 1_P))}{L^2(\sigma)}^2 \\
&\leq\Norm{1_P S^*(w 1_P)}{L^2(\sigma)}^2\lesssim \mathfrak{S}_*^2\sigma(P) \end{split} \end{equation*} by the (dual) testing estimate for the dyadic shifts. By the Carleson embedding theorem, it then follows that \begin{equation*}
\Big(\sum_R\abs{\ave{g}^w_R}^2 \Norm{\D^{\sigma,i}_R S^*(w 1_R)}{L^2(\sigma)}^2\Big)^{1/2}
\lesssim \mathfrak{S}_*\Norm{g}{L^2(\sigma)}, \end{equation*} and the estimation of the deeply contained cubes is finished.
\subsection{Contained cubes of comparable size}
It remains to estimate \begin{equation*}
\sum_{\substack{Q,R:Q\subseteq R\\ \ell(Q)\geq 2^{-i}\ell(R)}}\pair{\D^w_R g}{S(\sigma\D^{\sigma}_Q f)}_{w}; \end{equation*} the sum over $R\subsetneq Q$ with $\ell(R)\geq 2^{-j}\ell(Q)$ would be handled in a symmetric manner. The sum of interest may be written as \begin{equation*}
\sum_{a=0}^i\sum_R\sum_{\substack{Q\subseteq R\\ \ell(Q)=2^{-a}\ell(R)}}\pair{\D^w_R g}{S(\sigma\D^{\sigma}_Q f)}_{w}
=\sum_{a=0}^i\sum_R\pair{\D^w_R g}{S(\sigma\D^{\sigma,i}_R f)}_{w}, \end{equation*} and \begin{equation*}
\pair{\D^w_R g}{S(\sigma\D^{\sigma,i}_R f)}_{w}
=\sum_{k=1}^{2^d}\ave{\D^w_R g}_{R_k}\pair{S^*(w 1_{R_k})}{\D^{\sigma,i}_R f}_{\sigma} \end{equation*} where the $R_k$ are the $2^d$ dyadic children of $R$, and $\ave{\D^w_R g}_{R_k}$ is the constant valued of $\D^w_R g$ on $R_k$. Now \begin{equation*}
\pair{S^*(w 1_{R_k})}{\D^{\sigma,i}_R f}_{\sigma}
=\pair{1_{R_k}S^*(w 1_{R_k})}{\D^{\sigma,i}_R f}_{\sigma}+\pair{S^*(w 1_{R_k})}{1_{R_k^c}\D^{\sigma,i}_R f}_{\sigma}, \end{equation*} where \begin{equation*}
\abs{\pair{1_{R_k}S^*(w 1_{R_k})}{\D^{\sigma,i}_R f}_{\sigma}}
\leq\mathfrak{S}_* w(R_k)^{1/2}\Norm{\D^{\sigma,i}_R f}{L^2(\sigma)} \end{equation*} and, observing that only those $A_K^*$ where $K$ intersects both $R_k$ and $R_k^c$ contribute to the second part, \begin{equation*} \begin{split}
\abs{\pair{S^*(w 1_{R_k})}{1_{R_k^c}\D^{\sigma,i}_R f}_{\sigma}}
&=\Babs{\sum_{K\supsetneq R_k}\pair{A_K^*(w 1_{R_k})}{1_{R_k^c}\D^{\sigma,i}_R f}_{\sigma}} \\
&\lesssim\sum_{K\supseteq R}\frac{1}{\abs{K}}w(R_k)\Norm{\D^{\sigma,i}_R f}{L^1(\sigma)} \\
&\lesssim\frac{1}{\abs{R}}w(R_k)\sigma(R)^{1/2}\Norm{\D^{\sigma,i}_R f}{L^1(\sigma)} \\
&\leq\frac{w(R)^{1/2}\sigma(R)^{1/2}}{\abs{R}}w(R_k)^{1/2}\Norm{\D^{\sigma,i}_R f}{L^2(\sigma)} \\
&\leq[w,\sigma]_{A_2}w(R_k)^{1/2}\Norm{\D^{\sigma,i}_R f}{L^2(\sigma)}. \end{split} \end{equation*} It follows that \begin{equation*}
\abs{\pair{S^*(w 1_{R_k})}{\D^{\sigma,i}_R f}_{\sigma}}
\lesssim(\mathfrak{S}_* +[w,\sigma]_{A_2})w(R_k)^{1/2}\Norm{\D^{\sigma,i}_R f}{L^2(\sigma)} \end{equation*} and hence \begin{equation*}
\abs{\pair{\D_R^w g}{S(\sigma\D^{\sigma,i}_R f)}_{w}}
\lesssim(\mathfrak{S}_* +[w,\sigma]_{A_2})\Norm{\D^w_R g}{L^2(w)}\Norm{\D^{\sigma,i}_R f}{L^2(\sigma)}
\end{equation*} Finally, \begin{align*}
&\sum_{a=0}^i\sum_R \abs{\pair{\D_R^w g}{S(\sigma\D^{\sigma,i}_R f)}_{w}} \\
&\lesssim (\mathfrak{S}_* +[w,\sigma]_{A_2})\sum_{a=0}^i\Big(\sum_R\Norm{\D^w_R g}{L^2(\sigma)}^2\Big)^{1/2}
\Big(\sum_R\Norm{\D^{\sigma,i}_R f}{L^2(\sigma)}^2\Big)^{1/2} \\
&\leq(1+i)(\mathfrak{S}_* +[w,\sigma]_{A_2})\Norm{g}{L^2(w)}\Norm{f}{L^2(\sigma)}. \end{align*} The symmetric case with $R\subset Q$ with $\ell(R)\geq 2^{-j}\ell(Q)$ similarly yields the factor $(1+j)(\mathfrak{S} +[w,\sigma]_{A_2})$. This completes the proof of Theorem~\ref{thm:2weight}.
\section{Final decompositions: verification of the testing conditions}
We now turn to the estimation of the testing constant \begin{equation*}
\mathfrak{S}:=\sup_{Q\in\mathscr{D}}\frac{\Norm{1_Q S(\sigma 1_Q)}{L^2(w)}}{\sigma(Q)^{1/2}}. \end{equation*} Bounding $\mathfrak{S}_*$ is analogous by exchanging the roles of $w$ and $\sigma$.
\subsection{Several splittings} First observe that \begin{equation*}
1_Q S(\sigma 1_Q)
=1_Q\sum_{K:K\cap Q\neq\varnothing}A_K(\sigma 1_Q)
=1_Q\sum_{K\subseteq Q}A_K(\sigma 1_Q)+1_Q\sum_{K\supsetneq Q}A_K(\sigma 1_Q). \end{equation*} The second part is immediate to estimate even pointwise by \begin{equation*}
\abs{1_Q A_K(\sigma 1_Q)}\leq 1_Q\frac{\sigma(Q)}{\abs{K}},\qquad\sum_{K\supsetneq Q}\frac{1}{\abs{K}}\leq\frac{1}{\abs{Q}}, \end{equation*} and hence its $L^2(w)$ norm is bounded by \begin{equation*}
\BNorm{1_Q\frac{\sigma(Q)}{\abs{Q}}}{L^2(w)}
=\frac{w(Q)^{1/2}\sigma(Q)}{\abs{Q}}\leq[w,\sigma]_{A_2}\sigma(Q)^{1/2}. \end{equation*} So it remains to concentrate on $K\subseteq Q$, and we perform several consecutive splittings of this collection of cubes. First, we \textbf{separate scales} by introducing the splitting according to the $\kappa+1$ possible values of $\log_2\ell(K)\mod(\kappa+1)$. We denote a generic choice of such a collection by \begin{equation*}
\mathscr{K}=\mathscr{K}_k:=\{K\subseteq Q:\log_2\ell(K)\equiv k\mod(\kappa+1)\}, \end{equation*} where $k$ is arbitrary but fixed. (We will drop the subscript $k$, since its value plays no role in the subsequent argument.) Next, we \textbf{freeze the $A_2$ characteristic} by setting \begin{equation*}
\mathscr{K}^a:=\Big\{K\in\mathscr{K}: 2^{a-1}<\frac{w(K)\sigma(K)}{\abs{K}}\leq 2^a\Big\},\qquad a\in\mathbb{Z},\quad a\leq\ceil{\log_2[w,\sigma]_{A_2}}, \end{equation*} where $\ceil{\ }$ means rounding up to the next integer.
In the next step, we \textbf{choose the principal cubes} $P\in\mathscr{P}^a\subseteq\mathscr{K}^a$. This construction was first introduced by B. Muckenhoupt and R. Wheeden \cite{MW:77}, and it has been influential ever since. Let $\mathscr{P}^a_0$ consist of all maximal cubes in $\mathscr{K}^a$, and inductively $\mathscr{P}^a_{p+1}$ consist of all maximal $P'\in\mathscr{K}^a$ such that \begin{equation*}
P'\subset P\in\mathscr{P}^a_p,\qquad \frac{\sigma(P')}{\abs{P'}}>2\frac{\sigma(P)}{\abs{P}}. \end{equation*} Finally, let $\mathscr{P}^a:=\bigcup_{p=0}^{\infty}\mathscr{P}^a_p$. For each $K\in\mathscr{K}^a$, let $\Pi^a(K)$ denote the minimal $P\in\mathscr{P}^a$ such that $K\subseteq P$. Then we set \begin{equation*}
\mathscr{K}^a(P):=\{K\in\mathscr{K}^a:\Pi^a(K)=P\},\qquad P\in\mathscr{P}^a. \end{equation*}
Note that $\sigma(K)/\abs{K}\leq 2\sigma(P)/\abs{P}$ for all $K\in\mathscr{K}^a(P)$, which allows us to \textbf{freeze the $\sigma$-to-Lebesgue measure ratio} by the final subcollections \begin{equation*}
\mathscr{K}^a_b(P):=\Big\{K\in\mathscr{K}^a(P): 2^{-b}<\frac{\sigma(K)}{\abs{K}}\frac{\abs{P}}{\sigma(P)}\leq 2^{1-b}\Big\},\qquad b\in\mathbb{N}. \end{equation*} We have \begin{equation*} \begin{split}
&\{K\in\mathscr{D}:K\subseteq Q\}
=\bigcup_{k=0}^{\kappa}\mathscr{K}_k,\qquad
\mathscr{K}_k=\mathscr{K}=\bigcup_{a\leq\ceil{\log_2[w,\sigma]_{A_2}}}\mathscr{K}^a,\\
&\mathscr{K}^a=\bigcup_{P\in\mathscr{P}^a}\mathscr{K}^a(P),\qquad
\mathscr{K}^a(P)=\bigcup_{b=0}^{\infty}\mathscr{K}^a_b(P),\qquad \end{split} \end{equation*} where all unions are disjoint. Note that we drop the reference to the separation-of-scales parameter $k$, since this plays no role in the forthcoming arguments. Recalling the notation for subshifts $S_{\mathscr{Q}}=\sum_{K\in\mathscr{Q}}A_K$, this splitting of collections of cubes leads to the splitting of the function \begin{equation*}
\sum_{K\subseteq Q}A_K(\sigma 1_Q)=\sum_{k=0}^{\kappa}\sum_{a\leq\ceil{\log_2[w,\sigma]_{A_2}}}\sum_{P\in\mathscr{P}^a}\sum_{b=0}^{\infty}S_{\mathscr{K}^a_b(P)}(\sigma 1_Q). \end{equation*} On the level of the function, we split one more time to write \begin{equation*} \begin{split}
S_{\mathscr{K}^a_b(P)}(\sigma 1_Q)
&=\sum_{n=0}^{\infty} 1_{E^a_b(P,n)}S_{\mathscr{K}^a_b(P)}(\sigma 1_Q),\\
E^a_b(P,n) &:=\{x\in\mathbb{R}^d:n2^{-b}\ave{\sigma}_P<\abs{S_{\mathscr{K}^a_b(P)}(\sigma 1_Q)(x)}\leq (n+1)2^{-b}\ave{\sigma}_P\}. \end{split} \end{equation*} This final splitting, from \cite{HLMORSU}, is not strictly `necessary' in that it was not part of the original argument in \cite{Hytonen:A2}, nor its predecessor in \cite{LPR}, which made instead more careful use of the cubes where $S_{\mathscr{K}^a_b(P)}(\sigma 1_Q)$ stays constant; however, it now seems that this splitting provides another simplification of the argument.
Now all relevant cancellation is inside the functions $S_{\mathscr{K}^a_b(P)}(\sigma 1_Q)$, so that we can simply estimate by the triangle inequality: \begin{equation*} \begin{split}
&\Babs{\sum_{K\subseteq Q}A_K(\sigma 1_Q)} \\
&\quad\leq\sum_{k=0}^{\kappa}\sum_{a}\sum_{P\in\mathscr{P}^a}\sum_{b=0}^{\infty}\sum_{n=0}^{\infty}(1+n)2^{-b}
\ave{\sigma}_P 1_{\{\abs{S_{\mathscr{K}^a_b(P)}(\sigma 1_Q)}>n2^{-b}\ave{\sigma}_P\}}, \end{split} \end{equation*} and \begin{equation*} \begin{split}
&\BNorm{\sum_{K\subseteq Q}A_K(\sigma 1_Q)}{L^2(w)} \\
&\leq\sum_{k=0}^{\kappa}\sum_{a}\sum_{b=0}^{\infty}2^{-b}\sum_{n=0}^{\infty}(1+n)
\BNorm{\sum_{P\in\mathscr{P}^a}\ave{\sigma}_P 1_{\{\abs{S_{\mathscr{K}^a_b(P)}(\sigma 1_Q)}>n2^{-b}\ave{\sigma}_P\}}}{L^2(w)}. \end{split} \end{equation*} Obviously, we will need good estimates to be able to sum up these infinite series.
Write the last norm as \begin{equation*}
\Big(\int\Big[\sum_{P\in\mathscr{P}^a}\ave{\sigma}_P 1_{\{\abs{S_{\mathscr{K}^a_b(P)}(\sigma 1_Q)}>n2^{-b}\ave{\sigma}_P\}}(x)\Big]^2 \ud w(x)\Big)^{1/2}, \end{equation*} observe that \begin{equation*}
\{\abs{S_{\mathscr{K}^a_b(P)}(\sigma 1_Q)}>n2^{-b}\ave{\sigma}_P\}\subseteq P, \end{equation*} and look at the integrand at a fixed point $x\in\mathbb{R}^d$. At this point we sum over a subset of those values of $\ave{\sigma}_P$ where the principal cube $P\owns x$. Let $P_0$ be the smallest cube such that $\abs{S_{\mathscr{K}^a_b(P)}}>n2^{-b}\ave{\sigma}_P$, let $P_1$ be the next smallest, and so on. Then $\ave{\sigma}_{P_m}<2^{-1}\ave{\sigma}_{P_{m-1}}<\ldots<2^{-m}\ave{\sigma}_{P_0}$ by the construction of the principal cubes, and hence \begin{equation*} \begin{split}
\Big[\sum_{P\in\mathscr{P}^a}\ave{\sigma}_P 1_{\{\abs{S_{\mathscr{K}^a_b(P)}}>n2^{-b}\ave{\sigma}_P\}}(x)\Big]^2
&=\Big[\sum_{m=0}^{\infty}\ave{\sigma}_{P_m}\Big]^2 \\
&\leq\Big[\sum_{m=0}^{\infty}2^{-m}\ave{\sigma}_{P_0}\Big]^2
=4\ave{\sigma}_{P_0}^2 \\
&\leq 4\sum_{P\in\mathscr{P}^a}\ave{\sigma}_P^2 1_{\{\abs{S_{\mathscr{K}^a_b(P)}(\sigma 1_Q)}>n2^{-b}\ave{\sigma}_P\}}(x) \end{split} \end{equation*} Hence \begin{equation*} \begin{split}
&\BNorm{\sum_{P\in\mathscr{P}^a}\ave{\sigma}_P 1_{\{\abs{S_{\mathscr{K}^a_b(P)}(\sigma 1_Q)}>n2^{-b}\ave{\sigma}_P\}}}{L^2(w)} \\
&\leq \Big(\int\Big[4\sum_{P\in\mathscr{P}^a}\ave{\sigma}_P^2 1_{\{\abs{S_{\mathscr{K}^a_b(P)}(\sigma 1_Q)}>n2^{-b}\ave{\sigma}_P\}}\Big]w\Big)^{1/2} \\
&=2\Big(\sum_{P\in\mathscr{P}^a}\ave{\sigma}_P^2 w(\{\abs{S_{\mathscr{K}^a_b(P)}(\sigma 1_Q)}>n2^{-b}\ave{\sigma}_P\})\Big)^{1/2}, \end{split} \end{equation*} and it remains to obtain good estimates for the measure of the level sets \begin{equation*}
\{\abs{S_{\mathscr{K}^a_b(P)}(\sigma 1_Q)}>n2^{-b}\ave{\sigma}_P\}. \end{equation*}
\subsection{Weak-type and John--Nirenberg-style estimates} We still need to estimate the sets above. Recall that $S_{\mathscr{K}^a_b(P)}$ is a subshift of $S$, which in particular has its scales separated so that $\log_2\ell(K)\equiv k\mod (\kappa+1)$ for all $K$ for which $A_K$ participating in $S_{\mathscr{K}^a_b(P)}$ is nonzero and $k\in\{0,1,\ldots,\kappa:=\max\{i,j\}\}$ is fixed, $S$ being of type $(i,j)$. The following estimate deals with such subshifts, which we simply denote by $S$.
\begin{proposition} Let $S$ be a dyadic shift of type $(i,j)$ with scales separated. Then \begin{equation*}
\abs{\{\abs{Sf}>\lambda\}}\leq\frac{C}{\lambda}\Norm{f}{1},\qquad\forall\lambda>0, \end{equation*} where $C$ depends only on the dimension. \end{proposition}
\begin{proof} The proof uses the classical Calder\'on--Zygmund decomposition: \begin{equation*}
f=g+b,\qquad b:=\sum_{L\in\mathscr{B}}b_L:=\sum_{L\in \mathscr{B}}1_B(f-\ave{f}_L), \end{equation*}
where $L\in\mathscr{B}$ are the maximal dyadic cubes with $\ave{|f|}_L>\lambda$; hence $\ave{|f|}_L\leq 2^d\lambda$. As usual, \begin{equation*}
g=f-b=1_{\big(\bigcup\mathscr{B}\big)^c}f+\sum_{L\in\mathscr{B}}\ave{f}_L \end{equation*} satisfies $\Norm{g}{\infty}\leq 2^d\lambda$ and $\Norm{g}{1}\leq\Norm{f}{1}$, hence $\Norm{g}{2}^2\leq\Norm{g}{\infty}\Norm{g}{1}\leq 2^{d}\lambda\Norm{f}{1}$, and thus \begin{equation*}
\abs{\{\abs{Sg}>\tfrac12\lambda\}}\leq\frac{4}{\lambda^2}\Norm{Sg}{2}^2
\leq\frac{4}{\lambda^2}\Norm{g}{2}^2\leq 4\cdot 2^d\frac{1}{\lambda}\Norm{f}{1}. \end{equation*}
It remains to estimate $\{\abs{Sb}>\tfrac12\lambda\}$. First observe that \begin{equation*}
Sb=\sum_{K\in\mathscr{D}}\sum_{L\in\mathscr{B}}A_K b_L
=\sum_{L\in\mathscr{B}}\Big(\sum_{K\subseteq L}A_K b_L+\sum_{K\supsetneq L}A_K b_L\Big), \end{equation*} since $A_K b_L\neq 0$ only if $K\cap L\neq\varnothing$. Now \begin{equation*} \begin{split}
\abs{\{\abs{Sb}>\tfrac12\lambda\}}
&\leq\Babs{\Big\{\Babs{\sum_{L\in\mathscr{B}}\sum_{K\subseteq L}A_K b_L}>0\Big\}}
+\Babs{\Big\{\Babs{\sum_{L\in\mathscr{B}}\sum_{K\supsetneq L}A_K b_L}>\tfrac12\lambda\Big\}} \\
&\leq\sum_{L\in\mathscr{B}}\abs{L}+\frac{2}{\lambda}\BNorm{\sum_{L\in\mathscr{B}}\sum_{K\supsetneq L}A_K b_L}{1} \\
&\leq\frac{1}{\lambda}\Norm{f}{1}+\frac{2}{\lambda}\sum_{L\in\mathscr{B}}\sum_{K\supsetneq L}\Norm{A_K b_L}{1}, \end{split} \end{equation*} where we used the elementary properties of the Calder\'on--Zygmund decomposition to estimate the first term.
For the remaining double sum, we still need some observations. Recall that \begin{equation*}
A_K b_L=\sum_{\substack{I,J\subseteq K \\ \ell(I)=2^{-i}\ell(K)\\ \ell(J)=2^{-j}\ell(K)}}a_{IJK}h_I\pair{h_J}{b_L}. \end{equation*} Now, if $\ell(K)>2^{\kappa}\ell(L)\geq 2^j\ell(L)$, then $\ell(J)>\ell(L)$, and hence $h_J$ is constant on $L$. But the integral of $b_L$ vanishes, hence $\pair{h_J}{b_L}=0$ for all relevant $J$, and thus $A_K b_L=0$ whenever $\ell(K)>2^\kappa\ell(L)$.
Thus, in the inner sum, the only possible nonzero terms are $A_K b_L$ for $K=L^{(m)}$ for $m=1,\ldots,\kappa$. By the separation of scales, at most one of these terms is nonzero, and we write $\tilde{L}$ for the corresponding unique $K$. So in fact \begin{equation*}
\frac{2}{\lambda}\sum_{L\in\mathscr{B}}\sum_{K\supsetneq L}\Norm{A_K b_L}{1}
=\frac{2}{\lambda}\sum_{L\in\mathscr{B}}\Norm{A_{\tilde L} b_L}{1}
\leq\frac{2}{\lambda}\sum_{L\in\mathscr{B}}\Norm{b_L}{1}
\leq\frac{2}{\lambda}\cdot 2\Norm{f}{1}=\frac{4}{\lambda}\Norm{f}{1} \end{equation*} by using the normalized boundedness of the averaging operators $A_{\tilde L}$ on $L^1(\mathbb{R}^d)$, and an elementary estimate for the bad part of the Calder\'on--Zygmund decomposition.
Altogether, we obtain the claim with $C=4\cdot 2^d+5$. \end{proof}
For the special subshifts $S_{\mathscr{K}^a_b(P)}$, we can improve the weak-type $(1,1)$ estimate to an exponential decay:
\begin{proposition} Let $S_{\mathscr{K}^a_b(P)}$ be the subshift of $S$ as constructed earlier. Then the following estimate holds when $\nu$ is either the Lebesgue measure or $w$: \begin{equation*}
\nu\Big(\Big\{\abs{S_{\mathscr{K}^a_b(P)}(\sigma1_Q)}>C2^{-b}\ave{\sigma}_P\cdot t\Big\}\Big)
\lesssim C2^{-t}\nu(P),\qquad t\geq 0, \end{equation*} where $C$ is a constant. \end{proposition}
\begin{proof} Let $\lambda:=C2^{-b}\ave{\sigma}_P$, where $C$ is a large constant, and $n\in\mathbb{Z}_+$. Let $x\in\mathbb{R}^d$ be a point where \begin{equation}\label{eq:>nLambda}
\abs{S_{\mathscr{K}^a_b(P)}(\sigma 1_Q)(x)}>n\lambda. \end{equation} Then for all small enough $L\in\mathscr{K}^a_b(P)$ with $L\owns x$, there holds \begin{equation*}
\Babs{\sum_{\substack{K\in\mathscr{K}^a_b(P) \\ K\supseteq L}}A_K(\sigma 1_Q)(x)}>n\lambda. \end{equation*} Since $\displaystyle\sum_{\substack{K\in\mathscr{K}^a_b(P) \\ K\supsetneq L}}A_K(\sigma1_Q)$ is constant on $L$ (thanks to separation of scales), and \begin{equation}\label{eq:ALwQ}
\Norm{A_L(\sigma 1_Q)}{\infty}\lesssim\frac{\sigma(L)}{\abs{L}}\leq 2^{1-b}\frac{\sigma(P)}{\abs{P}}, \end{equation} it follows that \begin{equation}\label{eq:>n-2/3}
\Babs{\sum_{\substack{K\in\mathscr{K}^a_b(P) \\ K\supsetneq L}}A_K(\sigma 1_Q)}>(n-\tfrac{2}{3})\lambda\qquad\text{on }L. \end{equation} Let $\mathscr{L}\subseteq\mathscr{K}^a_b(P)$ be the collection of maximal cubes with the above property. Thus all $L\in\mathscr{L}$ are disjoint, and all $x$ with \eqref{eq:>nLambda} belong to some $L$. By maximality of $L$, the minimal $L^*\in\mathscr{K}^a_b(S)$ with $L^*\supsetneq L$ satisfies \begin{equation*}
\Babs{\sum_{\substack{K\in\mathscr{K}^a_b(P) \\ K\supsetneq L^*}}A_K(\sigma 1_Q)}\leq(n-\tfrac{2}{3})\lambda\qquad\text{on }L^*. \end{equation*} By an estimate similar to \eqref{eq:ALwQ}, with $L^*$ in place of $L$, it follows that \begin{equation*}
\Babs{\sum_{\substack{K\in\mathscr{K}^a_b(P) \\ K\supsetneq L}}A_K(\sigma 1_Q)}\leq (n-\tfrac{1}{3})\lambda\qquad\text{on }L. \end{equation*} Thus, if $x$ satisfies \eqref{eq:>nLambda} and $x\in L\in\mathscr{L}$, then necessarily \begin{equation*}
\abs{S_{\{K\in\mathscr{K}^a_b(P); K\subseteq L\}}(\sigma 1_{Q})(x)}=
\Babs{\sum_{\substack{K\in\mathscr{K}^a_b(P) \\ K\subseteq L}}A_K(\sigma 1_Q)(x)}>\tfrac13\lambda. \end{equation*} Using the weak-type $L^1$ estimate to the shift $S_{\{K\in\mathscr{K}^a_b(P);K\subseteq L\}}$ of type $(i,j)$ with scales separated, noting that $A_K(\sigma 1_Q)=A_K(\sigma 1_L)$ for $K\subseteq L$, it follows that \begin{align*}
\Babs{\Big\{\Babs{\sum_{\substack{K\in\mathscr{K}^a_b(P) \\ K\subseteq L}}A_K(\sigma 1_Q)(x)}>\tfrac13\lambda\Big\}}
&\leq \frac{C}{\lambda}\sigma(L) \\
&\leq\frac{C}{\lambda}2^{1-b}\frac{\sigma(S\cap Q)}{\abs{S}}\abs{L} \leq \tfrac13\abs{L}, \end{align*} provided that the constant in the definition of $\lambda$ was chosen large enough. Recalling \eqref{eq:>n-2/3}, there holds \begin{align*}
\Babs{\sum_{K\in\mathscr{K}^a_b(P)}A_K(\sigma 1_Q)}
&\geq\Babs{\sum_{\substack{K\in\mathscr{K}^a_b(P) \\ K\supsetneq L}}A_K(\sigma1_Q)}
-\Babs{\sum_{\substack{K\in\mathscr{K}^a_b(P) \\ K\subseteq L}}A_K(\sigma 1_Q)} \\
&>(n-\tfrac23)\lambda-\tfrac13\lambda=(n-1)\lambda\quad\text{on }\tilde{L}\subset L\text{ with }\abs{\tilde{L}}\geq\tfrac23\abs{L}. \end{align*} Thus \begin{align*}
\abs{\{\abs{S_{\mathscr{K}^a_b(P)}(\sigma 1_Q)}>n\lambda\}}
&\leq\sum_{L\in\mathscr{L}}\abs{L\cap \{\abs{S_{\mathscr{K}^a_b(P)}(\sigma1_Q)}>n\lambda\}} \\
&\leq\sum_{L\in\mathscr{L}}\abs{\{\abs{S_{\{K\in\mathscr{K}^a_b(P):K\subseteq L\}}(\sigma 1_Q)}>\tfrac13\lambda\}} \\
&\leq\sum_{L\in\mathscr{L}}\tfrac13\abs{L}\leq\sum_{L\in\mathscr{L}}\tfrac13\cdot\tfrac 32\abs{\tilde{L}} \\
&\leq\tfrac12\sum_{L\in\mathscr{L}} \abs{L\cap\{ \abs{S_{\mathscr{K}^a_b(P)}(\sigma 1_Q)}>(n-1)\lambda\}} \\
&\leq\tfrac12\abs{\{ \abs{S_{\mathscr{K}^a_b(P)}(\sigma 1_Q)}>(n-1)\lambda\}}. \end{align*} By induction it follows that \begin{align*}
\abs{\{\abs{S_{\mathscr{K}^a_b(P)}(\sigma 1_Q)}>n\lambda\}}
&\leq 2^{-n}\abs{\{ \abs{S_{\mathscr{K}^a_b(P)}(\sigma 1_Q)}>0\}} \\
&\leq 2^{-n}\sum_{M\in\mathscr{M}}\abs{M}\leq 2^{-n}\abs{P}, \end{align*} where $\mathscr{M}$ is the collection of maximal cubes in $\mathscr{K}^a_b(S)$.
Recalling that we defined $\lambda:=C2^{-b}\ave{\sigma}_P$ in the beginning of the proof, the previous display gives precisely the claim of the Proposition in the case that $\nu$ is the Lebesgue measure. We still need to consider the case that $\nu=w$. To this end, selected intermediate steps of the above computation, as well as the definition of $\mathscr{K}^a_b(P)$, will be exploited. Recall that $K\in\mathscr{K}^a$ means that $2^{a-1}<\ave{w}_K\ave{\sigma}_K\leq 2^a$, while $K\in\mathscr{K}^a_b(P)$ means that in addition $2^{-b}<\ave{\sigma}_K/\ave{\sigma}_P\leq 2^{1-b}$. Put together, this says that \begin{equation*}
2^{a+b-2}\ave{\sigma}_P<\frac{w(K)}{\abs{K}}<2^{a+b}\ave{\sigma}_P\qquad\forall K\in\mathscr{K}^a_b(P). \end{equation*} Hence, using the collections $\mathscr{L},\mathscr{M}\subseteq\mathscr{K}^a_b(P)$ as above, \begin{align*}
w(\{\abs{S_{\mathscr{K}^a_b(P)}(\sigma 1_Q)}>n\lambda\})
&\leq\sum_{L\in\mathscr{L}}w(L)
\leq\sum_{L\in\mathscr{L}}2^{a+b}\ave{\sigma}_P\abs{L} \\
&\leq 2^{a+b}\ave{\sigma}_P\abs{\{ \abs{S_{\mathscr{K}^a_b(P)}(\sigma 1_Q)}>(n-1)\lambda\}} \\
&\leq 2^{a+b}\ave{\sigma}_P\cdot 2^{-n}\sum_{M\in\mathscr{M}}\abs{M} \\
&\leq 4\cdot 2^{-n}\sum_{M\in\mathscr{M}}w(M)\leq 4\cdot 2^{-n}w(S).\qedhere \end{align*} \end{proof}
\subsection{Conclusion of the estimation of the testing conditions} Recall that \begin{equation*} \begin{split}
&\BNorm{\sum_{K\subseteq Q}A_K(\sigma 1_Q)}{L^2(w)} \\
&\leq\sum_{k=0}^{\kappa}\sum_{a}\sum_{b=0}^{\infty}2^{-b}\sum_{n=0}^{\infty}(1+n)
\BNorm{\sum_{P\in\mathscr{P}^a}\ave{\sigma}_P 1_{\{\abs{S_{\mathscr{K}^a_b(P)}(\sigma 1_Q)}>n2^{-b}\ave{\sigma}_P\}}}{L^2(w)} \end{split} \end{equation*} and \begin{equation*} \begin{split}
& \BNorm{\sum_{P\in\mathscr{P}^a}\ave{\sigma}_P 1_{\{\abs{S_{\mathscr{K}^a_b(P)}(\sigma 1_Q)}>n2^{-b}\ave{\sigma}_P\}}}{L^2(w)} \\
&\leq 2\Big(\sum_{P\in\mathscr{P}^a}\ave{\sigma}_P^2 w(\{\abs{S_{\mathscr{K}^a_b(P)}(\sigma 1_Q)}>n2^{-b}\ave{\sigma}_P\})\Big)^{1/2} \\
&\leq C\Big(\sum_{P\in\mathscr{P}^a}\ave{\sigma}_P^2 2^{-n/C}w(P)\Big)^{1/2} \\
&=C2^{-cn}\Big(\sum_{P\in\mathscr{P}^a}\frac{\sigma(P)w(P)}{\abs{P}^2}\sigma(P)\Big)^{1/2} \\
&\leq C2^{-cn}\Big(2^a\sum_{P\in\mathscr{P}^a}\sigma(P)\Big)^{1/2}, \end{split} \end{equation*} recalling the freezing of the $A_2$ characteristic between $2^{a-1}$ and $2^a$ for cubes in $\mathscr{K}^a\supseteq\mathscr{P}^a$.
For the summation over the principal cubes, we observe that \begin{equation*} \begin{split}
\sum_{P\in\mathscr{P}^a}\sigma(P)
=\sum_{P\in\mathscr{P}^a}\ave{\sigma}_P\abs{P}
=\int_Q\sum_{P\in\mathscr{P}^a}\ave{\sigma}_P 1_P(x)\ud x. \end{split} \end{equation*} At any given $x$, if $P_0\subsetneq P_1\subsetneq\ldots\subseteq Q$ are the principal cubes containing it, we have \begin{equation*}
\sum_{P\in\mathscr{P}^a}\ave{\sigma}_P 1_P(x)
=\sum_{m=0}^{\infty}\ave{\sigma}_{P_m}
\leq\sum_{m=0}^{\infty}2^{-m}\ave{\sigma}_{P_0}=2\ave{\sigma}_{P_0}
\leq 2M(\sigma 1_Q)(x), \end{equation*} where $M$ is the dyadic maximal operator. Hence \begin{equation*}
\sum_{P\in\mathscr{P}^a}\sigma(P)
\leq 2\int_Q M(\sigma 1_Q)\ud x
\leq 2[\sigma]_{A_\infty}\sigma(Q), \end{equation*} where we use the following notion of the $A_\infty$ characteristic: \begin{equation*}
[\sigma]_{A_\infty}:=\sup_Q\frac{1}{\sigma(Q)}\int_Q M(\sigma 1_Q)\ud x; \end{equation*} this was implicit already in the work of Fujii \cite{Fujii:weightedBMO} and it was taken as an explicit definition by the author and C. P\'erez \cite{HytPer}.
Substituting back, we have \begin{equation*} \begin{split}
&\BNorm{\sum_{K\subseteq Q}A_K(\sigma 1_Q)}{L^2(w)} \\
&\leq\sum_{k=0}^{\kappa}\sum_{a}\sum_{b=0}^{\infty}2^{-b}\sum_{n=0}^{\infty}(1+n)
\BNorm{\sum_{P\in\mathscr{P}^a}\ave{\sigma}_P 1_{\{\abs{S_{\mathscr{K}^a_b(P)}(\sigma 1_Q)}>n2^{-b}\ave{\sigma}_P\}}}{L^2(w)} \\
&\leq\sum_{k=0}^{\kappa}\sum_{a}\sum_{b=0}^{\infty}2^{-b}\sum_{n=0}^{\infty}(1+n)\cdot
C2^{-cn}\Big(2^a\sum_{P\in\mathscr{P}^a}\sigma(P)\Big)^{1/2} \\
&\leq\sum_{k=0}^{\kappa}\sum_{a}\sum_{b=0}^{\infty}2^{-b}\sum_{n=0}^{\infty}(1+n)\cdot
C2^{-cn}\big(2^a[\sigma]_{A_\infty}\big)^{1/2} \\
&=C\cdot[\sigma]_{A_\infty}^{1/2}\sum_{k=0}^{\kappa}\Big(\sum_{a\leq\ceil{\log_2[w,\sigma]_{A_2}}}2^{a/2}\Big)
\Big(\sum_{b=0}^{\infty}2^{-b}\Big)\Big(\sum_{n=0}^{\infty}(1+n)\cdot 2^{-cn}\Big) \\
&\leq C\cdot[\sigma]_{A_\infty}^{1/2}\cdot(1+\kappa)\cdot [w,\sigma]_{A_2}^{1/2}, \end{split} \end{equation*} and thus the testing constant $\mathfrak{S}$ is estimated by \begin{equation*}
\mathfrak{S}\leq C\cdot(1+\kappa)\cdot[w,\sigma]_{A_2}^{1/2}\cdot[\sigma]_{A_\infty}^{1/2}. \end{equation*} By symmetry, exchanging the roles of $w$ and $\sigma$, we also have the analogous result for $\mathfrak{S}^*$, and so we have completed the proof of the following:
\begin{theorem}\label{thm:testing} Let $\sigma,w\in A_\infty$ be functions which satisfy the joint $A_2$ condition \begin{equation*}
[w,\sigma]_{A_2}:=\sup_Q\frac{w(Q)\sigma(Q)}{\abs{Q}^2}<\infty. \end{equation*} Then the testing constants $\mathfrak{S}$ and $\mathfrak{S}^*$ associated with a dyadic shift $S$ of type $(i,j)$ satisfy the following bounds, where $\kappa:=\max\{i,j\}$: \begin{equation*} \begin{split}
\mathfrak{S} &\leq C\cdot(1+\kappa)\cdot[w,\sigma]_{A_2}^{1/2}\cdot[\sigma]_{A_\infty}^{1/2}, \\
\mathfrak{S}^* &\leq C\cdot(1+\kappa)\cdot[w,\sigma]_{A_2}^{1/2}\cdot[w]_{A_\infty}^{1/2}. \end{split} \end{equation*} \end{theorem}
\section{Conclusions}
In this section we simply collect the fruits of the hard work done above. A combination of Theorems~\ref{thm:2weight} and \ref{thm:testing} gives the following two-weight inequality, whose qualitative version was pointed out by Lacey, Petermichl and Reguera \cite{LPR}. In the precise form as stated, this result and its consequences below were obtained by P\'erez and myself \cite{HytPer}, although originally formulated only in the case that $\sigma=w^{-1}$ is the dual weight.
\begin{theorem}\label{thm:2weightShift} Let $\sigma,w\in A_\infty$ be functions which satisfy the joint $A_2$ condition \begin{equation*}
[w,\sigma]_{A_2}:=\sup_Q\frac{w(Q)\sigma(Q)}{\abs{Q}^2}<\infty. \end{equation*} Then a dyadic shift $S$ of type $(i,j)$ satisfies $S(\sigma\cdot):L^2(\sigma)\to L^2(w)$, and more precisely \begin{equation*}
\Norm{S(\sigma\cdot)}{L^2(\sigma)\to L^2(w)}
\lesssim (1+\kappa)^2[w,\sigma]_{A_2}^{1/2}\big([w]_{A_\infty}^{1/2}+[\sigma]_{A_\infty}^{1/2}\big), \end{equation*} where $\kappa=\max\{i,j\}$. \end{theorem}
The quantitative bound as stated, including the polynomial dependence on $\kappa$, allows to sum up these estimates in the Dyadic Representation Theorem to deduce:
\begin{theorem}\label{thm:2weightCZO} Let $\sigma,w\in A_\infty$ be functions which satisfy the joint $A_2$ condition. Then any $L^2$ bounded Calder\'on--Zygmund operator $T$ whose kernel $K$ has H\"older type modulus of continuity $\psi(t)=t^{\alpha}$, $\alpha\in(0,1)$, satisfies \begin{equation*}
\Norm{T(\sigma\cdot)}{L^2(\sigma)\to L^2(w)}
\lesssim (\Norm{T}{L^2\to L^2}+\Norm{K}{CZ_\alpha})[w,\sigma]_{A_2}^{1/2}\big([w]_{A_\infty}^{1/2}+[\sigma]_{A_\infty}^{1/2}\big). \end{equation*} \end{theorem}
Recalling the dual weight trick and specializing to the one-weight situation with $\sigma=w^{-1}$, this in turn gives:
\begin{theorem}\label{thm:1weightCZO} Let $w\in A_2$. Then any $L^2$ bounded Calder\'on--Zygmund operator $T$ whose kernel $K$ has H\"older type modulus of continuity $\psi(t)=t^{\alpha}$, $\alpha\in(0,1)$, satisfies \begin{equation*} \begin{split}
\Norm{T}{L^2(w)\to L^2(w)}
&\lesssim (\Norm{T}{L^2\to L^2}+\Norm{K}{CZ_\alpha})[w]_{A_2}^{1/2}\big([w]_{A_\infty}^{1/2}+[w^{-1}]_{A_\infty}^{1/2}\big) \\
&\lesssim (\Norm{T}{L^2\to L^2}+\Norm{K}{CZ_\alpha})[w]_{A_2}. \end{split} \end{equation*} \end{theorem}
The second displayed line is the original $A_2$ theorem \cite{Hytonen:A2}, and it follows from the first line by $[w]_{A_\infty}\lesssim[w]_{A_2}$ and $[w^{-1}]_{A_\infty}\lesssim [w^{-1}]_{A_2}=[w]_{A_2}$ (see Lemma~\ref{lem:A2Ainf} below). Its strengthening on the first line was first observed in my joint work with C.~P\'erez \cite{HytPer}. Note that, compared to the introductory statement in Theorem~\ref{thm:A2}, the dependence on the operator $T$ has been made more explicit. (The implied constants in the notation ``$\lesssim$'' only depend on the dimension and the H\"older exponent $\alpha$.) This dependence on $\Norm{T}{L^2\to L^2}$ and $\Norm{K}{CZ_\alpha}$ is implicit in the original proof.
For completeness, we include the proof (in the stated form essentially from \cite{LPR}, but see also \cite{HytPer} for more general comparison of $A_\infty$ and $A_p$ constants) that
\begin{lemma}\label{lem:A2Ainf} For all weights $w\in A_2$, we have \begin{equation*}
[w]_{A_\infty}:=\sup_Q\frac{1}{w(Q)}\int_Q M(1_Q w)\ud x\leq 8[w]_{A_2}. \end{equation*} \end{lemma}
\begin{proof} Let $\mathcal{P}$ be the principal cubes of Muckenhoupt and Wheeden \cite{MW:77} given by $\mathcal{P}=\bigcup_{p=0}^\infty\mathcal{P}_p$, where $\mathcal{P}_0:=\{Q\}$ and $\mathcal{P}_{p+1}$ consists of the maximal $P'\subset P\in\mathcal{P}_p$ with $w(P')/\abs{P'}>2w(P)/\abs{P}$. Then \begin{equation*}
M(1_Q w)(x)=\sup_{R:x\in R\subseteq Q}\frac{w(R)}{\abs{R}}\leq 2\sup_{P\in\mathcal{P}:x\in P}\frac{w(P)}{\abs{P}}\leq 2\sum_{P\in\mathcal{P}}\frac{w(P)}{\abs{P}}1_P(x), \end{equation*} and hence \begin{equation*}
\int_Q M(1_Q w)\ud x\leq 2\sum_{P\in\mathcal{P}}w(P). \end{equation*} Consider the pairwise disjoint sets $E(P):=P\setminus\bigcup_{P'\in\mathcal{P}:P'\subsetneq P}P'$. Since \begin{equation*}
\sum_{\substack{P'\subsetneq P\\ P'\text{ maximal}}}\abs{P'}
\leq\sum_{\substack{P'\subsetneq P\\ P'\text{ maximal}}} \frac{w(P')\abs{P}}{2w(P)}\leq \frac{w(P)\abs{P}}{2w(P)}=\frac{\abs{P}}{2}, \end{equation*} it follows that $\abs{E(P)}\geq\frac12\abs{P}$. We derive a similar condition for the weighted measure from the $A_2$ condition. Indeed, \begin{equation*} \begin{split}
\abs{E(P)}
&=\int_{E(P)}w^{1/2}w^{-1/2}\ud x
\leq\Big(\int_{E(P)}w\ud x\Big)^{1/2}\Big(\int_P w^{-1}\ud x\Big)^{1/2} \\
&=w(E(P))^{1/2}\Big(\fint_P w^{-1}\ud x\Big)^{1/2}\abs{P}^{1/2} \\
&\leq w(E(P))^{1/2}[w]_{A_2}^{1/2}\Big(\fint_P w\ud x\Big)^{-1/2}\abs{P}^{1/2}
=\Big([w]_{A_2}\frac{w(E(P))}{w(P)}\Big)^{1/2}\abs{P}. \end{split} \end{equation*} Using $\abs{P}\leq 2\abs{E(P)}$ and squaring, this shows that \begin{equation*}
w(P)\leq 4[w]_{A_2}w(E(P)). \end{equation*} After this, it is immediate to compute that \begin{equation*}
\sum_{P\in\mathcal{P}}w(P)
\leq 4[w]_{A_2}\sum_{P\in\mathcal{P}}w(E(P))
\leq 4[w]_{A_2}w(Q), \end{equation*} since the sets $E(P)$ are pairwise disjoint and contained in $Q$. \end{proof}
\section{Further results and remarks}
This final section briefly collects, without proofs, some further related developments, and poses some open problems.
The $A_2$ theorem implies a corresponding $A_p$ theorem for all $p\in(1,\infty)$. This follows from a version of the celebrated extrapolation theorem, one of the most useful tools in the theory of $A_p$ weights. The extrapolation theorem was first found by J. L. Rubio de Francia \cite{Rubio:factorAp}, and shortly after (so soon that it was published earlier) another proof was given by J. Garc{\'{\i}}a-Cuerva \cite{Garcia:extrapolation}. For the present purposes, we need a quantitative form of the extrapolation theorem, which is due to Dragi\v{c}evi\'c, Grafakos, Pereyra, and Petermichl \cite{DGPP}, and reads as follows. Although relatively recent, it was known well before the proof of the full $A_2$ theorem.
\begin{theorem}\label{thm:extrap} If an operator $T$ satisfies \begin{equation*}
\Norm{T}{L^2(w)\to L^2(w)}\leq C_T [w]_{A_2}^{\tau} \end{equation*} for all $w\in A_2$, then it satisfies \begin{equation*}
\Norm{T}{L^p(w)\to L^p(w)}\leq c_p C_T [w]_{A_p}^{\tau\max\{1,1/(p-1)\}} \end{equation*} for all $p\in(1,\infty)$ and $w\in A_p$. \end{theorem}
\begin{corollary}\label{cor:Ap} Let $p\in(1,\infty)$ and $w\in A_p$. Then any $L^2$ bounded Calder\'on--Zygmund operator $T$ whose kernel $K$ has H\"older type modulus of continuity $\psi(t)=t^{\alpha}$, $\alpha\in(0,1)$, satisfies \begin{equation*}
\Norm{T}{L^p(w)\to L^p(w)}
\lesssim (\Norm{T}{L^2\to L^2}+\Norm{K}{CZ_\alpha})[w]_{A_p}^{\max\{1,1/(p-1)\}}. \end{equation*} \end{corollary}
It is also possible to apply a version of the extrapolation argument to the mixed $A_2$/$A_\infty$ bounds \cite{HytPer}, but this did not give the optimal results for $p\neq 2$. However, by setting up a different argument directly in $L^p(w)$, the following bounds were obtained in my collaboration with M.~Lacey \cite{HytLac}:
\begin{theorem} Let $p\in(1,\infty)$ and $w\in A_p$. Then any $L^2$ bounded Calder\'on--Zygmund operator $T$ whose kernel $K$ has H\"older type modulus of continuity $\psi(t)=t^{\alpha}$, $\alpha\in(0,1)$, satisfies \begin{equation*}
\Norm{T}{L^p(w)\to L^p(w)}
\lesssim (\Norm{T}{L^2\to L^2}+\Norm{K}{CZ_\alpha})[w]_{A_p}^{1/p}\big([w]_{A_\infty}^{1/p'}+[w^{1-p'}]_{A_\infty}^{1/p}\big). \end{equation*} \end{theorem}
For weak-type bounds, which were investigated by Lacey, Martikainen, Orponen, Reguera, Sawyer, Uriarte-Tuero, and myself \cite{HLMORSU}, we need only `half' of the strong-type upper bound:
\begin{theorem} Let $p\in(1,\infty)$ and $w\in A_p$. Then any $L^2$ bounded Calder\'on--Zygmund operator $T$ whose kernel $K$ has H\"older type modulus of continuity $\psi(t)=t^{\alpha}$, $\alpha\in(0,1)$, satisfies \begin{equation*} \begin{split}
\Norm{T}{L^p(w)\to L^{p,\infty}(w)}
&\lesssim (\Norm{T}{L^2\to L^2}+\Norm{K}{CZ_\alpha})[w]_{A_p}^{1/p}[w]_{A_\infty}^{1/p'} \\
&\lesssim (\Norm{T}{L^2\to L^2}+\Norm{K}{CZ_\alpha})[w]_{A_p}. \end{split} \end{equation*} \end{theorem}
All these results remain valid for the non-linear operators given by the \emph{maximal truncations} \begin{equation*}
T_{\natural}f(x):=\sup_{\varepsilon>0}\abs{T_{\varepsilon}f(x)},\qquad
T_{\varepsilon}f(x):=\int_{\abs{x-y}>\varepsilon}K(x,y)f(y)\ud y, \end{equation*} which have been addressed in \cite{HytLac,HLMORSU}. In \cite{HLMORSU} it was also shown that the sharp weighted bounds for dyadic shifts can be made linear (instead of quadratic) in $\kappa$, a result recovered by a different (Bellman function) method by Treil \cite{Treil:linear}. Earlier polynomial-in-$\kappa$ Bellman function estimates for the shifts were due to Nazarov and Volberg \cite{NV}. An extension of the $A_2$ theorem to abstract metric spaces with a doubling measure (spaces of homogeneous type) is due to Nazarov, Reznikov, and Volberg \cite{NRV}.
A higher degree of non-linearity is obtained by replacing the supremum over $\epsilon>0$ defining the maximal truncation by one of the \emph{variation norms} \begin{equation*}
\Norm{\{v_\epsilon\}_{\epsilon>0}}{V^q}
:=\sup_{\{\epsilon_i\}_{i\in\mathbb{Z}}}\Big(\sum_i\abs{v_{\epsilon_i}-v_{\epsilon_{i+1}}}^q\Big)^{1/q}, \end{equation*} where the supremum is over all monotone sequences $\{\epsilon_i\}_{i\in\mathbb{Z}}\subset(0,\infty)$. Sharp weighted bounds for the $q$-variation ($q\in(2,\infty)$) of Calder\'on--Zygmund operators were first proved by Hyt\"onen--Lacey--P\'erez \cite{HLP}, although replacing the sharp truncation $T_\epsilon f(x)$ by a smooth truncation \begin{equation*}
T^\phi_\epsilon f(x):=\int \phi\Big(\frac{\abs{x-y}}{\epsilon}\Big)K(x,y)f(y)\ud y, \end{equation*} where $\phi$ is smooth and $0\leq\phi\leq 1_{(1,\infty)}$. Sharp weighted bounds for the $q$-variation of the sharp truncations with $\phi=1_{(1,\infty)}$ were recently obtained by de Fran\c{c}a Silva and Zorin-Kranich \cite{FSZK}.
The approach to the $q$-variation in \cite{HLP} was through a non-probabilistic counterpart of the Dyadic Representation, a Dyadic Domination, which was independently discovered by Lerner \cite{Lerner:domination,Lerner:simple}. Another advantage of this method was its ability to handle Calder\'on--Zygmund kernels with weaker moduli of continuity $\psi$ than those treated by the present approach; namely any moduli $\psi$ subject to the log-bumped Dini condition $\int_0^1\psi(t)(1+\log\frac1t)\frac{\ud t}{t}<\infty$.
In its original form, the Dyadic Domination theorem gave a domination in norm, which improved to pointwise domination by Conde-Alonso and Rey \cite{CondeRey} and, independently, by Lerner and Nazarov \cite{LerNaz:book}. All these approaches required the same log-Dini condition, and the necessity of the logarithmic correction to the Dini-condition remained open for some time, until it was finally eliminated by Lacey~\cite{Lacey:elem} by yet another approach. The following quantitative form of Lacey's theorem was obtained by L. Roncal, O. Tapiola and the author \cite{HytRoncal}, and with a simpler proof by Lerner \cite{Lerner:simplest}:
\begin{theorem}\label{thm:logDini} Let $w\in A_2$. Then any $L^2$ bounded Calder\'on--Zygmund operator $T$ whose kernel $K$ has modulus of continuity $\psi$ , satisfies \begin{equation*}
\Norm{T}{L^2(w)\to L^2(w)}
\lesssim \Big(\Norm{T}{L^2\to L^2}+\Norm{K}{CZ_0}+\Norm{K}{CZ_\psi}\int_0^1\psi(t)\frac{\ud t}{t}\Big)[w]_{A_2}. \end{equation*} \end{theorem}
Asking for even less regularity, one may wonder about the sharp weighted bound for the class of rough homogeneous singular integral operators \begin{equation*}
Tf(x)=\text{p.v.}\int_{\mathbb{R}^d}\frac{\Omega(y)}{\abs{y}^d}f(x-y)\ud y, \end{equation*} where \begin{equation*}
\Omega(y)=\Omega(\frac{y}{\abs{y}}),\qquad\Omega\in L^\infty(\mathbb{S}^{d-1}),\qquad\int_{\mathbb{S}^{d-1}}\Omega(\sigma)\ud\sigma=0. \end{equation*} Their qualitative boundedness $T:L^2(w)\to L^2(w)$ is known for $w\in A_2$ (see Watson~\cite{Watson}). Roncal, Tapiola and the author \cite{HytRoncal} showed that $\Norm{T}{L^2(w)\to L^2(w)}\lesssim \Norm{\Omega}{\infty}[w]_{A_2}^2$, but it is not known whether this quadratic dependence on $[w]_{A_2}$ is sharp.
\subsection{The Beurling operator and its powers}\label{sec:Beurling}
One of the key original motivations to study the $A_2$ theorem was a conjecture of Astala--Iwaniec--Saksman \cite{AIS} concerning the special case where $T$ is the Beurling operator \begin{equation*}
Bf(z):=-\frac{1}{\pi}\operatorname{p.v.}\int_{\mathbb{C}}\frac{1}{\zeta^2}f(z-\zeta)\ud A(\zeta), \end{equation*} and $A$ is the area measure (two-dimensional Lebesgue measure) on $\mathbb{C}\simeq\mathbb{R}^2$. This was the first Calder\'on--Zygmund operator for which the $A_2$ theorem was proven; it was achieved by Petermichl and Volberg \cite{PV}, confirming the mentioned conjecture of Astala, Iwaniec, and Saksman \cite{AIS}. Another proof of the $A_2$ theorem for this specific operator is due to Dragi\v{c}evi\'c and Volberg \cite{DV}.
The powers $B^n$ of $B$ have also been studied, and then it is of interest to understand the growth of the norms as a function of $n$. Shortly before the proof of the full $A_2$ theorem, by methods specific to the Beurling operator, O.~Dragi\v{c}evi\'c \cite{Dragicevic:cubic} was able to prove the cubic growth \begin{equation*}
\Norm{B^n}{L^2(w)\to L^2(w)}\lesssim\abs{n}^{3}[w]_{A_2},\qquad n\in\mathbb{Z}\setminus\{0\}. \end{equation*}
Now, let us see what the general $A_2$ theorem gives for these specific powers. It is known (see e.g. \cite{DPV}) that $B^n$ is the convolution operator with the kernel \begin{equation*}
K_n(z)=(-1)^n\frac{\abs{n}}{\pi}\Big(\frac{\bar{z}}{z}\Big)^n\abs{z}^{-2}, \end{equation*} and it is elementary to check that this satisfies $\Norm{K_n}{CZ_\alpha}\lesssim\abs{n}^{1+\alpha}$ for any $\alpha\in(0,1)$. Moreover, since $B$ is an isometry on $L^2(\mathbb{C})$, we have $\Norm{B^n}{L^2\to L^2}=1$. From Theorem~\ref{thm:1weightCZO} we deduce:
\begin{corollary} The powers $B^n$ of the Beurling operator satisfy \begin{equation*}
\Norm{B^n}{L^2(w)\to L^2(w)}\lesssim\abs{n}^{1+\alpha}[w]_{A_2},\qquad\alpha>0, \end{equation*} where the implied constant depends on $\alpha$. \end{corollary}
A sharper estimate still is provided by Theorem~\ref{thm:logDini}, as observed in \cite{HytRoncal}:
\begin{corollary} The powers $B^n$ of the Beurling operator satisfy \begin{equation*}
\Norm{B^n}{L^2(w)\to L^2(w)}\lesssim\abs{n}(1+\log\abs{n}) [w]_{A_2}. \end{equation*} \end{corollary}
For this it suffices to check that, defining the modulus of continuity \begin{equation*}
\psi_n(t):=\min\{\abs{n}t,1\}, \end{equation*} we have $\Norm{K_n}{CZ_{\psi_n}}\lesssim\abs{n}$ and hence \begin{equation*}
\Norm{K_n}{CZ_{\psi_n}}\int_0^1\psi_n(t)\frac{\ud t}{t}\lesssim \abs{n}(1+\log\abs{n}). \end{equation*} However, a better bound would follow if we had the $A_2$ theorem for the rough singular integrals in the form \begin{equation*}
\Norm{T}{L^2(w)\to L^2(w)}\lesssim \Norm{\Omega}{\infty}[w]_{A_2}, \end{equation*} for this would lead to the linear estimate $\Norm{B^n}{L^2(w)\to L^2(w)}\lesssim\abs{n}[w]_{A_2}$, simply by viewing the kernels $K_n$ (although smooth), as rough kernels of homogeneous singular integrals.
Let us notice that no bound better than this is possible, at least on the scale of power-type dependence on $\abs{n}$:
\begin{proposition} No bound of the form $\Norm{B^n}{L^2(w)\to L^2(w)}\lesssim\abs{n}^{1-\epsilon}[w]_{A_2}^{\tau}$ can be valid for any $\epsilon,\tau>0$. \end{proposition}
\begin{proof} Suppose for contradiction that such a bound holds for some fixed $\epsilon,\tau>0$ and all $n\in\mathbb{Z}\setminus\{0\}$. By Theorem~\ref{thm:extrap}, we deduce that \begin{equation*}
\Norm{B^n}{L^p(w)\to L^p(w)}\lesssim_p\abs{n}^{1-\epsilon}[w]_{A_p}^{\tau\max\{1,1/(p-1)\}}, \end{equation*} and hence in particular we have the unweighted bound \begin{equation*}
\Norm{B^n}{L^p\to L^p}\lesssim_p\abs{n}^{1-\epsilon},\qquad 1<p<\infty. \end{equation*} However, it has been shown by Dragi\v{c}evi\'c, Petermichl and Volberg that the correct dependence here is \begin{equation*}
\Norm{B^n}{L^p\to L^p}\eqsim_p\abs{n}^{\abs{1-2/p}},\qquad 1<p<\infty. \end{equation*} The previous two displays are clearly in contradiction for $p$ close to either $1$ or~$\infty$, and we are done. \end{proof}
The quest for the $A_2$ theorem began from the investigations of the Beurling transform, but clearly even this case is not yet fully understood.
\end{document} |
\begin{document}
\title{An introduction to operational quantum dynamics}
\author{Simon Milz} \email{simon.milz@monash.edu} \affiliation{School of Physics and Astronomy, Monash University, Clayton, Victoria 3800, Australia}
\author{Felix A. Pollock} \email{felix.pollock@monash.edu} \affiliation{School of Physics and Astronomy, Monash University, Clayton, Victoria 3800, Australia}
\author{Kavan Modi} \email{kavan.modi@monash.edu} \affiliation{School of Physics and Astronomy, Monash University, Clayton, Victoria 3800, Australia}
\begin{abstract}
In the summer of 2016, physicists gathered in Toru{\'n}, Poland for the 48th annual \emph{Symposium on Mathematical Physics}. This Symposium was special; it celebrated the 40th anniversary of the discovery of the Gorini-Kossakowski-Sudarshan-Lindblad master equation, which is widely used in quantum physics and quantum chemistry. This article forms part of a \emph{Special Volume} of the journal \emph{Open Systems~\&~Information Dynamics} arising from that conference; and it aims to celebrate a related discovery -- also by Sudarshan -- that of \textit{Quantum Maps} (which had their 55th anniversary in the same year). Nowadays, much like the master equation, quantum maps are ubiquitous in physics and chemistry. Their importance in quantum information and related fields cannot be overstated. In this manuscript, we motivate quantum maps from a tomographic perspective, and derive their well-known representations. We then dive into the murky world beyond these maps, where recent research has yielded their generalisation to non-Markovian quantum processes.
\end{abstract}
\date{\today} \maketitle \tableofcontents
\pagebreak
Describing changes in a system's state is the principal goal of any mathematical theory of dynamics. In order to be physically relevant, this description must be faithful to what is observed in experiments. For quantum systems, a dynamical theory must quantify how measurement statistics of different observables can change from one moment to the next, even when the system in question may be interacting with its wider environment, which is typically large, uncontrollable and experimentally inaccessible.
While unitary evolution of vectors in Hilbert space (according to Schr{\"o}dinger's equation) is sufficient to describe the behaviour of a deterministically prepared closed quantum system, more is required when the system is open to its environment, or when there is classical uncertainty in its preparation. The complete statistical state of such a system (or, more properly, an ensemble of identical and independent preparations of the system) is encoded in its density operator $\rho$, which can be determined operationally in a quantum state tomography experiment. Namely, by combining the measurement statistics of a set of linearly independent observables. A reader who is unfamiliar with the concept of density operators or quantum state tomography can find more information in Ref.~\cite{Nielsen00a}.
In this Special Issue article, we concern ourselves with the dynamical description of open quantum systems, primarily in terms of mappings from density operators at one time or place to another, \textit{i.e.}, the quantum generalisation of classical stochastic maps. These mappings are \emph{superoperators} -- operators on an operator space -- and depending on context, are referred to as \textit{quantum maps}, \textit{quantum channels}, \textit{quantum operations}, \textit{dynamical maps}, and so on. In this article, we stick with the term \textit{quantum maps} throughout.
Quantum maps are ubiquitous in the quantum sciences, particularly in quantum information theory. They are natural for describing quantum communication channels~\cite{wilde2013quantum}, crucial for quantum error correction~\cite{lidarQEC}, and form the basis for generalised quantum measurements~\cite{peresQT}. Yet their origins, motivation and applicability are not always transparent. Their discovery dates back to 1961, in the work of George Sudarshan and collaborators~\cite{SudarshanMatthewsRau61, SudarshanJordan61}. A decade later, Karl Kraus also discovered them~\cite{kraus_general_1971}, and quantum maps are perhaps most widely known through his 1984 book~\cite{kraus_states_1983}. Along the way, there have been many other players. For example, the works of Stinespring~\cite{stinespring1955} and Choi~\cite{choi72a, choi75} are crucial for understanding the structure of quantum maps. Stinespring's result predates that of Sudarshan, though both his and Choi's works are purely mathematical in nature and not concerned with quantum physics \textit{per se}. On the physics side, the works of Davies and Lewis~\cite{Davies1970}, Jamiolkowski~\cite{jamiolkowski_linear_1972}, Lindblad \cite{Lindblad1975}, and Accardi \textit{et al.}~\cite{accardi_quantum_1982}, to name but a few\footnote{A complete list of important contributions would constitute an entire article in itself, and we apologise to any who feel they have been unjustly omitted.}, have led to a deep understanding of quantum stochastic processes. Here, we put history aside, and describe quantum maps and their generalisations in a pedagogical manner. We present an operationally rooted and thorough introduction to the theory of open quantum dynamics.
The article has two main sections. In Section~\ref{sec::QuantumMaps}, we introduce quantum maps in the context of quantum process tomography -- that is, what can be inferred about the evolution of the density operator in experiment -- before exploring how they can be represented mathematically. Along the way, we take care to point out the relationships between different representations, and the physical motivation behind mathematical properties such as linearity and complete positivity. In Section~\ref{sec::beyond}, we discuss open quantum dynamics in situations where the formalism developed in the first Section is insufficient to successfully describe experimental observations. Namely, when the system is initially correlated with its environment and when joint statistics across multiple time steps is important. After demonstrating how a na{\"i}ve extension of the conventional theory fails to deliver useful conclusions, we outline a more general, operational framework, where evolution is described in terms of mappings from preparations to measurement outcomes.
\section{Quantum maps and their representations} \label{sec::QuantumMaps}
A quantum map $\mathcal{E}$ is a mapping from density operators to density operators: $\rho \mapsto \rho' = \mathcal{E}[\rho]$. Here $\rho$ and $\rho'$ are operators on the `input' and `output' Hilbert spaces of the map, respectively\footnote{Strictly speaking, the mapping is between a preparation that yields $\rho$, and a measurement that interrogates $\rho'$.}. Formally, this can be written $\mathcal{E}: \mathcal{B} (\mathcal{H}_{d_\mathrm{in}}) \rightarrow \mathcal{B} (\mathcal{H}_{d_\mathrm{out}})$, \textit{i.e.}, as a mapping from bounded operators on the input Hilbert space to bounded operators on the output Hilbert space. In fact, the map can be seen as a bounded operator on the space of bounded operators, $\mathcal{E}\in \mathcal{B}(\mathcal{B}(\mathcal{H}_{d_\mathrm{in}}))$. Here $d = \mathrm{dim}(\mathcal{H}_{d})$ denotes the dimension of Hilbert space $\mathcal{H}_{d}$. Throughout this article, we work in the Schr{\"o}dinger picture with finite dimensional quantum systems (see Ref.~\cite{kretschmann_quantum_2005} for a description of quantum maps in the Heisenberg picture). In general, the input and output Hilbert space need not be the same, but for simplicity we will, for the most part, assume $\mathcal{H}_{d_\mathrm{in}} \cong \mathcal{H}_{d_\mathrm{out}}$ and omit the subscripts ``in'' and ``out'' from this point on.
To represent a deterministic physical process, the quantum map has to preserve the basic properties of the density operator, \textit{i.e.}, it has to preserve trace, Hermiticity, and positivity (as we detail more explicitly at the end of this section). Moreover, the action of the quantum map must be linear: \begin{gather} \mathcal{E} \left[\sum p_k \rho_k \right] = \sum p_k \mathcal{E}[ \rho_k] = \sum p_k \rho'_k. \end{gather} It is worth noting that this requirement does \emph{not} follow from the fact that quantum mechanics is a linear theory, in the sense of quantum state vectors formed from linear superpositions of a basis set (in fact, $\mathcal{E}$ is not generally linear in this sense). Instead, the linearity of the quantum map is analogous to the linearity of mixing in a statistical theory.
To better appreciate this, consider a quantum channel from Alice to Bob, where Alice prepares a system in either state $\rho_1$ or $\rho_2$; she then sends the system to Bob. Upon receiving the system Bob performs state tomography on the state Alice sent by measuring it. They do this many times. Suppose Alice sends $\rho_1$ on day-one and $\rho_2$ on day-two. From the measurement outcomes Bob will conclude that the received states are $\rho'_1 = \mathcal{E}[\rho_1]$ on day-one and $\rho'_2 =\mathcal{E}[\rho_2]$ on day-two. Now, suppose Alice sends the two states randomly with probabilities $p$ and $1-p$ respectively. Without knowing which state is being sent on which run, Bob would conclude that he receives state $\bar \rho' = \mathcal{E} [\bar \rho]$, where $\bar \rho = p \rho_1 + (1-p) \rho_2$. That is, we can interpret Alice's preparation to be the average state. Now suppose that, at some later point, Alice reveals which state was sent in which run; Bob can now go back to his logbook and conclude that he received the state $\rho'_1 \, (\rho'_2)$ whenever Alice sent him $\rho_1 \, (\rho_2)$. Conversely, averaging over that data would amount to Bob receiving $\bar \rho '$. Thus we must have $\bar \rho' = p \rho'_1 + (1-p) \rho'_2$. This simple thought experiment demands that the action of quantum channels must be linear. However, note that, while we have used the language of quantum mechanics in this paragraph, there is nothing quantum about this experiment\footnote{The same argument would hold for a nonlinear map on the space of pure states. However, care has to be taken in differentiating between proper and improper mixtures~\cite{mixtures}.}. Linearity of mixing is a general concept that applies to all stochastic theories.
Now, using the fact that the quantum map $\mathcal{E}$ is linear, we will derive several useful representations for it.
\subsection{Structure of linear maps} \label{subsec::LinMaps}
\textit{Any} linear map $M$ on a (complex) vector space $V$ is unambiguously defined by its action on a (not necessarily orthogonal) basis $\left\{\hat{\mathbf{r}}_i\right\}_{i=1}^{d_V}$\footnote{Here, and throughout this article, the caret is used to indicate that the object is an element of some fixed (not necessarily normalised) basis set used for tomography.} of $V$, where $d_V$ is the dimension of $V$. That is to say, the input-output relation $M[\hat{\mathbf{r}}_i] = \hat{\mathbf{r}}_i'$ entirely defines $M$. For any basis $\left\{\hat{\mathbf{r}}_i\right\}$ of $V$, there exists a \emph{dual set} $\left\{\hat{\mathbf{d}}_i\right\}_{i=1}^{d_V}\subset V$ such that $(\hat{\mathbf{d}}_i,\hat{\mathbf{r}}_j) = \delta_{ij}$, where $(\cdot,\cdot)$ is the scalar product in $V$. With this, for any $\mathbf{v}\in V$, the action of $M$ can be written as \begin{gather} \label{eqn::ActionLinear} M[\mathbf{v}] = \sum_{i=1}^{d_V} \hat{\mathbf{r}}_i'\, (\hat{\mathbf{d}}_i,\mathbf{v})\, . \end{gather} This equation is correct by construction, as it maps every basis element $\hat{\mathbf{r}}_i$ to the correct output $\hat{\mathbf{r}}_i'$. In other words, it says that knowing the images under a map $M: V\rightarrow V$ for a basis of $V$ completely defines the action of the map.
Eq.~\eqref{eqn::ActionLinear} can be rewritten as \begin{gather} \label{eqn::OuterProd} M[\mathbf{v}] = \sum_{i=1}^{d_V} \hat{\mathbf{r}}_i'\, (\hat{\mathbf{d}}_i,\mathbf{v}) \equiv \sum_{i=1}^{d_V} \left(\hat{\mathbf{r}}_i' \times \hat{\mathbf{d}}_i^{*}\right)[\mathbf{v}]\, , \end{gather} where we have defined the \emph{outer product} \begin{gather} \label{eqn::ComponentsOuter} \left(\hat{\mathbf{r}}_i' \times \hat{\mathbf{d}}_i^{*}\right)_{kl} = (\hat{\mathbf{r}}_i')_k(\hat{\mathbf{d}}_i)^{*}_l\, . \end{gather} For an orthonormal basis $\left\{\hat{\mathbf{e}}_i\right\}$ of $V$, we have $(\mathbf{a})_i= (\mathbf{a},\hat{\mathbf{e}}_i)$, and $N_{ij} = (\hat{\mathbf{e}}_i, N[\hat{\mathbf{e}}_j])$ for any $\mathbf{a} \in V$ and any linear operator $N$ on $V$. Consequently, we obtain a matrix representation $\mathbf{M}$ of the map $M$: \begin{gather} \label{eqn::MatLambda} (\mathbf{M})_{kl} = \sum_{i=1}^{d_V} \left(\hat{\mathbf{r}}_i' \times \hat{\mathbf{d}}_i^{*}\right)_{kl} \end{gather} and the action of $M$ can be written in terms of the matrix $\mathbf{M}$: \begin{gather} (M[\mathbf{v}])_k = \sum_l (\mathbf{M})_{kl} \, v_l = \sum_{i=1}^{d_V} \left(\hat{\mathbf{r}}_i' \times \hat{\mathbf{d}}_i^{*}\right)_{kl} \, v_l\, , \end{gather} where $\mathbf{v} = \sum_m v_m \hat{\mathbf{e}}_m$. Note that there is a distinction between $M$ and $\mathbf{M}$; the former is a map, while the latter is its representation as a matrix. This distinction is often not made when dealing with quantum maps, but here we will make it explicit.
\begin{figure}\label{fig::InputOutput}
\end{figure}
\subsubsection{Tomographic representation}
A quantum map $\mathcal{E}$ is a linear map on the vector space $\mathcal{B}(\mathcal{H}_d)$. Since $\mathcal{B}(\mathcal{H}_d)$ is isomorphic to the vector space of $d \times d$ matrices (where $d$ is the dimension of $\mathcal{H}_d$), we can make use of the natural inner product on the latter space, the Hilbert-Schmidt inner product $(\rho,\eta) = \mbox{tr}(\rho^{\dagger} \, \eta)$, to define an inner product on the space of density operators. Consequently, we can express the action of $\mathcal{E}$ in a way equivalent to Eq.~\eqref{eqn::ActionLinear}; different generalisations of the outer product defined in Eq.~\eqref{eqn::OuterProd} will then lead to different representations of $\mathcal{E}$ (see Sec.~\ref{subsec::Forms}).
To proceed, we need a basis set of the input space. There always exists a set of operators that constitutes a (generally non-orthogonal) basis of $\mathcal{B}(\mathcal{H}_d)$. For example, the set of elementary matrices form such a basis, as do Pauli and Gell-Mann matrices. Both of these sets are orthonormal with respect to Hilbert-Schmidt inner product, but neither of them consists of physical density operators. However, as explained above, the map $\mathcal{E}$ is unambiguously defined by its input-output relation $\mathcal{E}[\hat{\rho}_i] = \hat{\rho}_i'$. Thus, we can use density operators for the basis set: $\left\{\hat{\rho}_i\right\}_{i=1}^{d^2} \subset \mathcal{B}(\mathcal{H}_d)$. For example, for a two-level quantum system we can use the following density operators \begin{gather}\label{eq:basisstates} \hat{\rho}_1 = \frac{1}{2}\begin{pmatrix*}[r] 1 & 1 \\ 1 & 1 \end{pmatrix*}, \, \hat{\rho}_2 = \frac{1}{2}\begin{pmatrix*}[r] 1 & -i \\ i & 1 \end{pmatrix*}, \, \hat{\rho}_3 = \begin{pmatrix*}[r]1 & 0 \\ 0 & 0 \end{pmatrix*}, \, \hat{\rho}_4 = \frac{1}{2}\begin{pmatrix*}[r] 1 & -1 \\ -1 & 1 \end{pmatrix*}. \end{gather} These matrices are linearly independent and form a basis, but clearly, they are not orthonormal. However, for any choice of basis, there exists a set of \emph{dual matrices} $\left\{\hat{D}_i\right\}_{i=1}^{d^2}$~\cite{modi_positivity_2012} such that $\mbox{tr}(\hat{D}_i^{\dagger} \hat{\rho}_j) = \delta_{ij}$ (see App.~\ref{subsec::DualMat} for proof). Consequently, in analogy to Eq.~\eqref{eqn::ActionLinear}, the action of $\mathcal{E}$ on $\rho$ can be written as \begin{gather} \label{eqn::ActionMap} \mathcal{E}[\rho] = \sum_{i=1}^{d^2} \hat{\rho}_i' \, \mbox{tr}(\hat{D}_i^{\dagger}\rho)\, , \end{gather} which means that determining the output states for a basis of input states entirely defines the action of the map $\mathcal{E}$.
The dual matrices for the states in Eq.~\eqref{eq:basisstates} are \begin{gather} \label{eqn::DualsEx} \hat{D}_1 = \frac{1}{2}\begin{pmatrix*}[c] 0 & 1+i \\ 1-i & 2 \end{pmatrix*}, \, \hat{D}_2 = \frac{1}{2}\begin{pmatrix*}[r] 0 & -i \\ i & 0 \end{pmatrix*}, \, \hat{D}_3 = \begin{pmatrix*}[r] 1 & 0 \\ 0 & -1 \end{pmatrix*}, \, \hat{D}_4 = \frac{1}{2}\begin{pmatrix*}[c] 0 & -1+i \\ -1-i & 2 \end{pmatrix*}. \end{gather} Clearly these dual matrices are not positive. In fact, if both the outputs $\hat{\rho}_i'$ and the duals are positive, then $\mathcal{E}$ is necessarily an entanglement breaking channel~\cite{horodecki_entanglement_2003,holevo_1998} (the converse also holds). In general, neither set of matrices in Eq.~\eqref{eqn::ActionMap}, $\{\hat{\rho}_i'\}$ and $\{\hat{D}_i\}$, have to be positive, and it can sometimes even be advantageous to choose non-positive matrices $\hat{\rho}_i'$ and $\hat{D}_i$ for the representation of $\mathcal{E}$.
However, for a proper quantum map, we can choose $\{\hat{\rho}_i'\}$ to be states, and $\{\hat{D}_i\}$ to be the dual set corresponding to a set of basis states $\{\hat{\rho}_i\}$. Then Eq.~\eqref{eqn::ActionMap} captures precisely the idea of \emph{quantum process tomography}~\cite{JModOpt.44.2455, PhysRevLett.78.390}, where the dynamics of a quantum system is experimentally reconstructed by relating a basis of input states to their corresponding outputs. The action of the map $\mathcal{E}$ on any state $\rho$ is then simply determined from the linearity of the map. From here on, we will -- for obvious reasons -- refer to this representation as the \emph{input/output} or \emph{tomographic} representation of $\mathcal{E}$.
\subsubsection{Operator-sum representation}
Based on Eq.~\eqref{eqn::ActionMap}, the action of $\mathcal{E}$ can be rewritten in a form that is used more widely in the literature. Both $\hat{\rho}_i'$ and $\hat{D}_i$ can be expressed in terms of their left- and right-singular vectors, \textit{i.e.}, \begin{gather} \hat{\rho}_i' = \sum_{\alpha} \ketbra{s^i_\beta}{t^i_\beta}\, , \quad \text{and} \quad \hat{D}_i = \sum_\mu \ketbra{u^i_\mu}{v^i_\mu}\, , \end{gather} where $\{\ket{s_\alpha^i}\}, \{\ket{t_\alpha^i}\}$ and $\{\ket{u_\mu^i}\}, \{\ket{v_\mu^i}\}$ are the respective unnormalised left- and right-singular vectors of $\hat{\rho}_i'$ and $\hat{D}_i$. With this decomposition, the action of $\mathcal{E}$ reads \begin{align} \mathcal{E}[\rho] &= \sum_i\hat{\rho}_i'\, \mbox{tr}(\hat{D}_i^{\dagger}\rho) = \sum_i\sum_{\beta,\mu} \ketbra{s^i_\beta}{t^i_\beta} \mbox{tr}\left(\ketbra{v^i_\mu}{u^i_\mu} \rho\right) \\ &= \sum_{\beta,\mu} \sum_i \left(\ketbra{s^i_\beta}{u^i_\mu}\right) \rho \left(\ketbra{v^i_\mu}{t^i_\beta}\right)\, . \end{align} Compressing the indeces $\{i,\beta,\mu\}$ into one common index yields the \emph{operator sum representation} of $\mathcal{E}$: \begin{gather} \label{eqn::OpSum} \mathcal{E}[\rho] = \sum_{\beta,\mu} \sum_i \left(\ketbra{s^i_\beta}{u^i_\mu}\right)\rho \left(\ketbra{v^i_\mu}{t^i_\beta}\right) \equiv \sum_\alpha L_\alpha \rho \, R_\alpha^{\dagger}\, , \end{gather} where $L_\alpha$ and $R_\alpha$ have the same shape, but are not necessarily square (if the input and output space are not of the same size). In exactly the same vein, the tomographic representation of a map can be recovered from its operator sum representation via a singular value decomposition.
\textbf{Unitary freedom.} We have shown that any linear map can be expressed in the operator sum representation, but the set of matrices $\{L_\alpha,R_\alpha\}$ in Eq.~\eqref{eqn::OpSum} is not unique. Any set $\{L'_\mu,R'_\mu\}$ of matrices that is connected to $\{L_\alpha,R_\alpha\}$ by an isometry, \textit{i.e.}, $L'_\mu = \sum_{\alpha} (U)_{\mu\alpha}L_{\alpha}$ and $R'_\mu = \sum_{\alpha'} (U)_{\mu\alpha'}R_{\alpha'}$, where $U^{\dagger}U = \openone$, gives rise to the same linear map: \begin{gather} \sum_{\mu}L'_{\mu}\rho R_{\mu}^{\prime\,\dagger} = \sum_{\alpha\alpha'}\sum_{\mu} (U)_{\mu\alpha}L_{\alpha} \rho R_{\alpha'}^{\dagger}(U)_{\mu\alpha'}^{*} = \sum_{\alpha\alpha'} (U^{\dagger}U)_{\alpha'\alpha} L_{\alpha'} \rho R_{\alpha'}^{\dagger} = \sum_{\alpha} L_\alpha \rho R_\alpha^{\dagger}. \end{gather}
Both of the representations we have presented so far consist of sets of operator pairs -- $\{\hat{\rho}_i',\hat{D}_i'\}$ for the tomographic representation and $\{L_\alpha,R_\alpha\}$ for the operator sum representation. These will be explored further later in this section. Next, however, we will present two matrix representations for the map.
\subsection{Matrix representations} \label{subsec::Forms}
Since $\mathcal{B}(\mathcal{H}_d)$ is itself a vector space, it should be possible to represent $\mathcal{E}$ -- a linear map on that space -- as a matrix. Indeed, two such representations were first discovered back in 1961~\cite{SudarshanMatthewsRau61}. To derive these representations, we note that there are (at least) two different ways to generalise the outer product Eq.~\eqref{eqn::ComponentsOuter}, and hence two different ways to obtain representations of $\mathcal{E}$ in terms of outputs and dual matrices.
\subsubsection{Sudarshan's A form}
In clear analogy to Eq.~\eqref{eqn::MatLambda}, one possible matrix representation of $\mathcal{E}$ (in an orthonormal basis of $\mathcal{H}_d\otimes\mathcal{H}_d$) is given by \begin{gather} \label{eqn::EcalA} \mathcal{E}_A = \sum_{i=1}^{d^2} \hat{\rho}_i' \times \hat{D}_i^{*}\, ,\qquad \text{with} \quad \left(\hat{\rho}_i' \times \hat{D}_i^{*} \right)_{rs;r's'} = (\hat{\rho}_i')_{rs} (\hat{D}_i)^*_{r's'}\, . \end{gather} In Dirac notation, this means that we have generalised the outer product defined in \eqref{eqn::ComponentsOuter} as \begin{gather} \label{eqn::OuterMatrix} \ketbra{r}{s}\times \ketbra{r'}{s'} \equiv \ketbra{rs}{r's'}. \end{gather} The action of $\mathcal{E}$ can be simply written as \begin{gather} \label{eqn::ActionA} (\mathcal{E}[\rho])_{rs} = \sum_{r's'}^d(\mathcal{E}_A)_{rs;r's'}(\rho)_{r's'}. \end{gather} This is what Sudarshan \textit{et al.} called the A form of the dynamical map~\cite{SudarshanMatthewsRau61}. They observed that the matrix $\mathcal{E}_A$ is not Hermitian even if $\mathcal{E}$ is Hermiticity preserving. Indeed, this matrix is -- quite naturally -- not even square if the input dimensions are different from the output dimensions.
Mathematically, the outer product `flips' the bra (ket) $\bra{s}$ $(\ket{r'})$ into the ket (bra) $\ket{s}$ $(\bra{r'})$. By \emph{vectorizing} $\rho$ and $\mathcal{E}[\rho]$, we can write Eq.~\eqref{eqn::ActionA} in a more compact way: \begin{gather} \label{eqn::VecAction} \kket{\mathcal{E}[\rho]} = \mathcal{E}_A \kket{\rho},
\quad \mbox{where} \quad
\kket{\rho} = \sum_{rs}(\rho)_{rs} \ket{rs}
\quad \mbox{for} \quad
\rho = \sum_{rs} (\rho)_{rs} \ketbra{r}{s}. \end{gather} For the details of \emph{vectorisation} of matrices see, \textit{e.g.}, Refs.~\cite{dariano_orthogonality_2000,gilchrist_vectorization_2009}. Because the action of $\mathcal{E}_A$ onto $\kket{\rho}$ is simply a multiplication of a vector by a matrix, this representation is often favourable for numerical studies.
\begin{figure}
\caption{\textit{Converting between A and B form.} In a given orthonormal basis, the A and B form of a map $\mathcal{E}$ are related by a simple reshuffling of the matrix elements. For a better orientation, the matrix elements that change position are depicted in colour.}
\label{fig::Reshuffling}
\end{figure}
\subsubsection{Sudarshan's B form}
Next, we consider what Sudarshan \textit{et al.} called the B form of the dynamical map~\cite{SudarshanMatthewsRau61}. Instead of the outer product in Eq.~\eqref{eqn::EcalA}, let us consider a tensor product: \begin{gather} \label{eqn::EcalB} \mathcal{E}_B = \sum_{i=1}^{d^2} \hat{\rho}_i' \otimes \hat{D}_i^{*},\quad \text{with} \quad \left(\hat{\rho}_i' \otimes \hat{D}_i^{*} \right)_{rr';ss'} = (\hat{\rho}_i')_{rs} (\hat{D}_i)^*_{s'r'}\, ; \end{gather} that is, the product in Eq.~\eqref{eqn::ComponentsOuter} is generalised to $\ketbra{r}{s}\otimes\ketbra{r'}{s'} = \ketbra{rr'}{ss'}$. The action of $\mathcal{E}$ can be written as \begin{gather} \label{eqn::ActionB} (\mathcal{E}[\rho])_{rs} = \sum_{r's'}^d(\mathcal{E}_B)_{rr';ss'}(\rho)_{r's'}. \end{gather}
While the A form is closer in spirit to the general considerations about linear maps on vector spaces, the B form possesses nicer mathematical properties (see below), and from the point of view of quantum mechanics, the tensor product $\otimes$ seems `more natural' than the outer product $\times$. Comparing the matrices $\mathcal{E}_A$ and $\mathcal{E}_B$, it can be seen, from the relation between the outer product and the tensor product, that they coincide up to reshuffling~\cite{SudarshanMatthewsRau61, zyczkowski_duality_2004, bengtsson_geometry_2007}. In Fig.~\ref{fig::Reshuffling}, we show how to go between the two forms for a map acting on a two-level system (qubit). However, unlike $\mathcal{E}_A$, the matrix $\mathcal{E}_B$ is Hermitian iff $\mathcal{E}_B$ is Hermiticity preserving. A quantum map is trace preserving iff $\mbox{tr}_\mathrm{out}(\mathcal{E}_B) = \openone_\mathrm{in}$, where $\mbox{tr}_{\mathrm{out}}$ denotes the trace over the output Hilbert space of the map $\mathcal{E}$ [\textit{i.e.}, the trace over the unprimed indices in Eq.~\eqref{eqn::ActionB}] and $\openone_\mathrm{in}$ is the identity matrix on the input space. We will prove these properties in the following subsections.
\subsection{Choi-Jamio{\l}kowski isomorphism}\label{sec::CJI}
Consider the action of a quantum map on one part of an (unnormalised) maximally entangled state $\ket{I} = \sum_{k=1}^{d_{\mathrm{in}}} \ket{kk}$: \begin{gather} \label{eqn::Choi} \Upsilon_{\mathcal{E}} = \mathcal{E} \otimes \mathcal{I} \left[\ketbra{I}{I}\right] = \sum_{k,l=1}^{d_{\mathrm{in}}} \mathcal{E}\left[\ketbra{k}{l}\right]\otimes \ketbra{k}{l}\, , \end{gather} where $\{{\ket{k}}\}$ is an orthonormal basis of $\mathcal{H}_{d_\mathrm{in}}$ and $\mathcal{I}$ is the identity operator on $\mathcal{B}(\mathcal{H}_d)$. The resultant matrix $\Upsilon_{\mathcal{E}}$ can be shown to be, element-by-element, identical to the quantum map $\mathcal{E}$. In principal, any vector $\ket{I}$ with full Schmidt rank could be used for this isomorphism~\cite{dariano_imprinting_2003}. In the form of~\eqref{eqn::Choi} it is known as the \emph{Choi-Jamio{\l}kowski isomorphism} (CJI)~\cite{jamiolkowski_linear_1972, choi75}, an isomorphism between linear maps, $\mathcal{E}: \mathcal{B} (\mathcal{H}_{d_\mathrm{in}}) \rightarrow \mathcal{B} (\mathcal{H}_{d_\mathrm{out}} )$, and matrices $\Upsilon_{\mathcal{E}} \in \mathcal{B}(\mathcal{H}_{d_\mathrm{out}})\otimes \mathcal{B}(\mathcal{H}_{d_\mathrm{in}})$. In order to keep better track of the input and output spaces of the map $\mathcal{E}$, we explicitly distinguish between the spaces $\mathcal{H}_{d_\mathrm{in}}$ and $\mathcal{H}_{d_\mathrm{out}}$ in this subsection.
Usually, $\Upsilon_{\mathcal{E}}$ is called the \emph{Choi matrix} or \emph{Choi state} of the map $\mathcal{E}$ (we will refer to it as the latter when it is a valid quantum state, up to normalisation)\footnote{By means of the CJI, the density matrix $\rho$ itself can also be considered the Choi state of a map $\mathcal{E}: \mathbb{C} \rightarrow \mathcal{B}(\mathcal{H})$~\cite{chiribella_theoretical_2009}.}. Given $\Upsilon_{\mathcal{E}}$, the action of $\mathcal{E}$ can be written as \begin{gather} \label{eqn::ChoiAction} \mathcal{E}[\rho] = \mbox{tr}_{\mathrm{in}}\left[\left(\openone_{\mathrm{out}} \otimes \rho^{\mathrm{T}}\right)\Upsilon_\mathcal{E}\right] \, , \end{gather} where $\openone_{\mathrm{out}}$ is the identity matrix on $\mathcal{H}_{d_\mathrm{out}}$ and $\mbox{tr}_{\mathrm{in}}$ denotes the partial trace over the input Hilbert space $\mathcal{H}_{d_\mathrm{in}}$. Eq.~\eqref{eqn::ChoiAction} can be shown by insertion: \begin{align} \mbox{tr}_{\mathrm{in}}\left[\left(\openone_{\mathrm{out}} \otimes \rho^{\mathrm{T}}\right) \Upsilon_\mathcal{E}\right] &= \sum_{k,l}^{d_\mathrm{in}} \mbox{tr}_{\mathrm{in}}\left[\left(\openone_{\mathrm{out}} \otimes \rho^{\mathrm{T}}\right) \left(\mathcal{E}[\ketbra{k}{l}]\otimes \ketbra{k}{l} \right)\right] \\
&= \sum_{k,l,m}^{d_\mathrm{in}} \mathcal{E}[\ketbra{k}{l}] \!\bra{m}\rho^{\mathrm{T}}\ket{k}\!\braket{l|m} = \sum_{k,l}^{d_\mathrm{in}} \rho_{kl} \mathcal{E}[\ketbra{k}{l}] = \mathcal{E}[\rho]\, . \end{align}
The CJI is by no means restricted to quantum maps; \textit{any} linear map $\mathcal{E}$ can be mapped to a Choi matrix $\Upsilon_\mathcal{E}$ via the CJI. For instance, one can imprint a classical stochastic process onto a state by inputting one part of a maximally classically correlated state into the process. For quantum maps, however, the Choi matrix has particularly nice properties. Complete positivity of $\mathcal{E}$ is equivalent to $\Upsilon_\mathcal{E} \geq 0$ (see Sec.~\ref{Sec::OpSum} for a proof), and it is straightforward to deduce from Eq.~\eqref{eqn::ChoiAction} that $\mathcal{E}$ is trace preserving iff $\mbox{tr}_{\mathrm{out}}(\Upsilon_\mathcal{E}) = \openone_\mathrm{in}$ (see Sec.~\ref{subsec::Quantum}).
Besides these appealing mathematical properties, the CJI is also of experimental importance. Given that a (normalised) maximally entangled state can in principal be created in practice, the CJI enables another way of reconstructing the map $\mathcal{E}$, by letting it act on one half of a maximally entangled state and tomographically determining the resulting state. While this so-called \emph{ancilla-assisted process tomography}~\cite{PhysRevLett.90.193601,PhysRevLett.86.4195} requires the same number of measurements as the input-output procedure, it can be, depending on the experimental situation, easier to implement in the laboratory.
The mathematical properties of $\Upsilon_\mathcal{E}$ are reminiscent of the properties of the B form. However, at first sight, it is not clear how the Choi matrix $\Upsilon_\mathcal{E}$ is related to the different matrix representations of $\mathcal{E}$ in terms of the dual matrices and outputs presented in Sec.~\ref{subsec::LinMaps}. The relation can be made manifest by using the fact that the set $\{\hat{\rho}_i\}_{i=1}^{d_\mathrm{in}^2}$ forms a basis of $\mathcal{B}({\mathcal{H}_{d_\mathrm{in}}})$. With this, we can write $\ket{k}\bra{l} = \sum_{i=1}^{d_\mathrm{in}^2} \alpha_{i}^{(kl)} \hat{\rho}_i$, where $\alpha_{i}^{(kl)} \in \mathbb{C}$ is given by $\alpha_{i}^{(kl)} = \mbox{tr}(\hat{D}_i^{\dagger} \ket{k}\bra{l})$. Consequently, we obtain \begin{align} \label{eqn::Choi_BForm} \Upsilon_{\mathcal{E}} &= \sum_{k,l} \mathcal{E}[\ketbra{k}{l}]\otimes \ketbra{k}{l} = \sum_{k,l,i} \alpha_{i}^{(kl)}\,\mathcal{E}[\hat{\rho}_i]\otimes \ketbra{k}{l} \\ \label{eqn::Choi_BForm2} &= \sum_i \mathcal{E}[\hat{\rho}_i]\otimes \sum_{k,l} \mbox{tr}\left(\hat{D}_i^{\dagger} \ketbra{k}{l}\right) \ketbra{k}{l} = \sum_i \mathcal{E}[\hat{\rho}_i] \otimes \hat{D}_i^*\, , \end{align} where, in the last step, we have used the fact that $\{\ket{k}\bra{l}\}_{i,j=1}^{d_\mathrm{in}}$ also forms a basis of $\mathcal{B}(\mathcal{H}_{d_\mathrm{in}})$. By comparison with Eq.~\eqref{eqn::EcalB}, we see that the Choi matrix of $\mathcal{E}$ is exactly equal to the B form of $\mathcal{E}$, \textit{i.e.}, $\Upsilon_\mathcal{E} = \mathcal{E}_B$, and henceforth, we will use the terms Choi matrix and B form interchangeably.
\subsection{Operator sum representation revisited} \label{Sec::OpSum}
As mentioned in Sec.~\ref{subsec::LinMaps}, any linear map can be written in terms of an operator sum representation. We proved this statement using the input-output action of the linear map, given in Eq.~\eqref{eqn::ActionLinear}. We now provide an alternative proof employing $\Upsilon_\mathcal{E}$. The Choi matrix $\Upsilon_\mathcal{E}$ can be written in terms of its unnormalised left- and right-singular vectors, \textit{i.e.} $\Upsilon_\mathcal{E} = \sum_{\alpha=1}^{D} \ket{w_\xi}\bra{y_\xi}$, where $D= d_\mathrm{out} d_\mathrm{in}$. We have \begin{align}
\mathcal{E}[\rho] &= \sum_{\alpha=1}^D \mbox{tr}_{\mathrm{in}}\left[\left(\openone_{\mathrm{out}} \otimes \rho^{\mathrm{T}}\right)\ketbra{w_\alpha}{y_\alpha}\right] = \sum_{\alpha=1}^D \sum_{k,l=1}^{d_\mathrm{in}} \braket{l|w_\alpha}\!\bra{k}\rho^{\mathrm{T}} \ket{l}\! \braket{y_\alpha|k} \\
&= \sum_{\alpha=1}^D \left(\sum_l^{d_\mathrm{in}} \braket{l|w_\alpha}\! \bra{l}\right) \rho \left(\sum_{k=1}^{d_\mathrm{in}} \ket{k}\!\braket{y_\alpha|k}\right) \equiv \sum_{\alpha = 1}^D L_\alpha \rho \, R_\alpha^{\dagger}\,, \end{align} which is the operator sum representation of $\mathcal{E}$ already encountered in Sec.~\ref{subsec::LinMaps}.
Given the operator sum representation of a linear map $\mathcal{E}$, it is possible to find another way of writing its A and B form. The B form $\mathcal{E}_B$ is obtained via \begin{gather}\label{eqn::Kraus_B} \mathcal{E}_B = \Upsilon_\mathcal{E} = \sum_\alpha \sum_{i,j = 1}^{d_\mathrm{in}} L_\alpha \ketbra{i}{j} R_\alpha^{\dagger} \otimes \ketbra{i}{j} = \sum_\alpha L_\alpha \times R_\alpha^*\, . \end{gather} Correspondingly, the A form of $\mathcal{E}$ can be written as~\cite{usha_devi_open-system_2011} \begin{gather} \label{eqn::Kraus_A} \mathcal{E}_A = \sum_\alpha L_{\alpha} \otimes R_\alpha^{*}\, . \end{gather} Indeed, substituting Eq.~\eqref{eqn::Kraus_A} into Eq.~\eqref{eqn::ActionA}, we obtain \begin{gather} (\mathcal{E}[\rho])_{rs} = \sum_{r's'}(\mathcal{E}_A)_{rs;r's'}\rho_{r's'} = \sum_\alpha \sum_{r's'} (L_\alpha)_{rr'} \rho_{r's'} (R_\alpha^{*})_{ss'} = \left(\sum_\alpha L_\alpha \rho R_\alpha^{\dagger}\right)_{rs}\, . \end{gather}
The operators $\{ L_\alpha, \, R_\alpha \}$ are operationally different from $\{ \hat{\rho}_i', \hat{D}_i\}$ in Eq.~\eqref{eqn::EcalB}. A quantum map in the form $\mathcal{E}_A$ ($\mathcal{E}_B$) is obtained by tensor (outer) product of the former, and outer (tensor) product of the latter. Therefore, in clear analogy to the corresponding statement for the B form and the operator sum representation, we can recover the tomographic representation of $\mathcal{E}$ via a singular value decomposition of $\mathcal{E}_A$.
\subsection{Properties of quantum maps} \label{subsec::Quantum}
The four representations derived above (input-output, operator sum, A form and B form) are valid for any linear map on a finite-dimensional complex operator space. However, not every such map represents the dynamics of a physical system. In order to do so, it must ensure that the statistical character of quantum states is preserved. Here, we lay out the mathematical constraints imposed on quantum maps by this requirement, and explore the corresponding implications for different representations.
\subsubsection{Trace preservation} \label{subsubsec::Tr}
Since the trace of the density operator represents its normalisation, a deterministic quantum map should be trace preserving (more general, trace non-increasing maps do not have this property, and represent probabilistic quantum processes). This requirement can be stated as \begin{gather} \mbox{tr}(\rho') = \mbox{tr}\left( \mathcal{E}[\rho] \right) = \sum_i \mbox{tr}[\hat{\rho}_i'] \, \mbox{tr}[\hat{D}_i^\dag \rho] \qquad \forall\rho. \end{gather} Since $\mbox{tr}[\hat{D}_i^\dag \hat{\rho}_i] = 1$, by linearity, the trace-preservation condition holds iff $\mbox{tr}(\hat{\rho}_i') = \mbox{tr}(\hat{\rho}_i)$. That is, the map $\mathcal{E}$ is trace preserving iff it is trace preserving on a basis of inputs. Equivalently, a map $\mathcal{E}$ is trace preserving iff $\sum_i (\mbox{tr}\hat{\rho}_i')\hat{D}_i^{*} = \openone$.
Trace preservation can also be stated in a succinct way in terms of the operator sum representation. We have \begin{gather} \mbox{tr} \left(\mathcal{E}[\rho] \right) = \mbox{tr}\left(\sum_\alpha L_\alpha \rho R_\alpha^{\dagger}\right) = \mbox{tr} \left(\sum_\alpha R_\alpha^{\dagger} L_\alpha \rho \right)\, , \end{gather} and hence $\mathcal{E}$ is trace preserving for all $\rho$ iff $\sum_\alpha R_\alpha^{\dagger} L_\alpha= \openone$.
In a similar way, we can express trace preservation of $\mathcal{E}$ in terms of the B form. If $\mathcal{E}$ is trace preserving, we have $\mbox{tr}(\mathcal{E}[\rho]) = \rho$ for all $\rho$. In terms of $\mathcal{E}_B$, this means \begin{gather} \label{eqn::TrChoi} \mbox{tr}(\mathcal{E}[\rho]) = \mbox{tr}\left[\left(\openone_{\mathrm{out}} \otimes \rho^{\mathrm{T}}\right) \mathcal{E}_B\right] = \mbox{tr}\left[\rho^{\mathrm{T}}\mbox{tr}_\mathrm{out}(\mathcal{E}_B)\right] = \mbox{tr}(\rho)\, , \end{gather} which is true iff $\mbox{tr}_\mathrm{out}(\mathcal{E}_B) = \openone$.
\subsubsection{Hermiticity preservation} \label{subsubsec::Herm}
Given a valid quantum state as input, a physical quantum map should produce a valid quantum state as output; hence, it should preserve the Hermiticity of the density operator. A map with this property satisfies $\mathcal{E}[\rho] = \left(\mathcal{E}[\rho]\right)^\dag$ for all $\rho = \rho^{\dagger}$. In terms of output matrices and duals, this condition reads \begin{gather} (\rho')^{\dagger} = \sum_i (\hat{\rho}_i')^{\dagger} \mbox{tr}\left(\hat{D}_i \rho\right) = \sum_i \hat{\rho}_i' \mbox{tr}\left(\hat{D}_i^{\dagger} \rho\right) = \rho'\, , \qquad \forall \ \rho = \rho^{\dagger}\, . \end{gather}
If $\mathcal{E}$ is Hermiticity preserving, then its B form (Choi matrix) $\mathcal{E}_B$ is Hermitian. This follows from the fact that Hermiticity preservation of $\mathcal{E}$ implies Hermiticity preservation of $\mathcal{E} \otimes \mathcal{I}_{a}$, where $\mathcal{I}_{a}$ is the identity map on an arbitrary ancilla. Consequently, the decomposition of $\mathcal{E}_B$ in terms of its left- and right-singular vectors becomes an eigendecomposition, \textit{i.e.}, $\mathcal{E}_B = \displaystyle \sum_{\alpha = 1}^D\lambda_\alpha \ketbra{\alpha}{\alpha}$, where all the eigenvalues $\lambda_\alpha \in \mathbb{R}$ and we have $\braket{\alpha|\alpha'} = \delta_{\alpha\alpha'}$. Hence, the action of $\mathcal{E}$ can be written as \begin{align} \mathcal{E}[\rho] &= \sum_{\alpha=1}^D \mbox{tr}_{\mathrm{in}}\left[\left(\openone_{\mathrm{out}} \otimes \rho^{\mathrm{T}}\right)\lambda_\alpha \ketbra{\alpha}{\alpha}\right] \\ \label{eqn::HermPres}
&=\sum_{\alpha=1}^D \lambda_\alpha \,\left(\sum_{l=1}^{d_\mathrm{in}} \braket{l|\alpha}\!\bra{l}\right) \rho \left( \sum_{k=1}^{d_\mathrm{in}}\ket{k}\! \braket{\alpha|k}\right) \equiv \sum_{\alpha=1}^D \lambda_\alpha\, \widetilde{K}_\alpha \rho \widetilde{K}_\alpha^{\dagger}\, , \end{align} which implies that the matrices $\{L_\alpha,R_\alpha\}$ of the map's operator sum representation satisfy $L_\alpha = \pm R_\alpha \ \forall \alpha$. In fact, the ability to write the map's action in the form of Eq.~\eqref{eqn::HermPres} is a necessary and sufficient condition for Hermiticity preservation~\cite{verstraete_quantum_2002, jordan:052110}, as is the Hermiticity of its B form $\mathcal{E}_B$.
\subsubsection{Complete positivity} \label{subsec::CP}
In addition to being Hermiticity and trace preserving, a physical quantum map $\mathcal{E}$ must be positive, \textit{i.e.}, it must map positive matrices $\rho$ to positive matrices $\rho'$. What is more, it must be \emph{completely positive} (CP): \textit{any} trivial extension $\mathcal{E}\otimes \mathcal{I}_{{a}}: \mathcal{B}(\mathcal{H}_{d}) \otimes \mathcal{B}(\mathcal{H}_{n_{a}}) \rightarrow \mathcal{B}(\mathcal{H}_{d}) \otimes \mathcal{B}(\mathcal{H}_{n_{a}})$, where $\mathcal{H}_{n_{a}}$ is $n$-dimensional and $\mathcal{I}_{a}$ is the identity map on $\mathcal{B}(\mathcal{H}_{n_{a}})$, must also be positive. In other words, a meaningful operation $\mathcal{E}$ that acts non-trivially only on a subset of the degrees of freedom of a quantum state should not yield a non-physical result: $(\mathcal{E} \otimes \mathcal{I}_{a})[\eta] \geq 0$ for any $\eta \geq 0$ and any $n \geq 1$.
The above justification for CP is an operational one. However, CP also guarantees that the action of the (trace preserving) map comes from a joint unitary dynamics of the system with an environment, as proven by Stinespring in 1955~\cite{stinespring1955} (see Fig.~\ref{fig::InputOutput}). This result was recently generalised for trace non-increasing CP maps~\cite{chiribella_transforming_2008}, which can, in principle, be physically realised within quantum mechanics as joint unitary dynamics with postselection.
If the map $\mathcal{E}$ is CP, than its B form $\mathcal{E}_B = \mathcal{E} \otimes \mathcal{I}\left[\ketbra{I}{I}\right]$ is non-negative by definition\footnote{It is clear from this that CP also implies Hermiticity preservation.}, and hence all its eigenvalues satisfy $\lambda_\alpha \geq 0$. This allows Eq.~\eqref{eqn::HermPres} to be further simplified: \begin{gather} \label{eqn::canonical}
\mathcal{E}[\rho] = \sum_{\alpha=1}^D \left(\sum_{l=1}^{d_\mathrm{in}}\sqrt{\lambda_\alpha} \braket{l|\alpha}\!\bra{l}\right) \rho \left(\sqrt{\lambda_\alpha} \sum_{k=1}^{d_\mathrm{in}}\ket{k}\! \braket{\alpha|k}\right) \equiv \sum_{\alpha = 1}^D K_\alpha \rho K_\alpha^{\dagger}\, , \end{gather} This form of the map was first noticed by Sudarshan \textit{et al.}~\cite{SudarshanMatthewsRau61} in 1961 by means of eigendecomposition of $\mathcal{E}_B$. However, it is now commonly referred to as the Kraus form~\cite{Nielsen00a} and the $d_{\mathrm{out}}\times d_\mathrm{in}$ matrices $\{K_\alpha\}$ are called the \emph{Kraus operators} of $\mathcal{E}$~\cite{kraus_general_1971,kraus_states_1983}. CP therefore implies that $L_\alpha = R_\alpha, \, \forall \alpha$ in the general operator sum representation of $\mathcal{E}$.
\textbf{Properties of the Kraus form.} As in the case of general linear maps, the set of Kraus operators that corresponds to $\mathcal{E}$ is not unique. Any set $\{K'_\mu\}$ of $d_{\mathrm{out}}\times d_\mathrm{in}$ matrices that is related to $\{K_\alpha\}$ by an isometry gives rise to the same map, \textit{i.e.}, $\{K'_\mu = \sum_{\alpha'} (U)_{\mu\alpha'}K_{\alpha'}\}$, where $U^{\dagger}U = \openone$, is also a valid set of Kraus operators for $\mathcal{E}$. The minimal number of operators needed for the operator sum representation of a CP map $\mathcal{E}$ is called its \emph{Kraus rank}. It coincides with the rank of the Choi state $\Upsilon_{\mathcal{E}}$~\cite{verstraete_quantum_2002}.
Every CP map allows for a \emph{canonical} Kraus decomposition, where the number of Kraus operators is minimal and they are mutually orthogonal, \textit{i.e.}, $\mbox{tr}(K_\alpha K^{\dagger}_{\alpha'}) \propto \delta_{\alpha \alpha'}$. In fact, the Kraus decomposition derived in~\eqref{eqn::canonical} is already canonical: \begin{gather}
\mbox{tr}(K_{\alpha}K_{\alpha'}^{\dagger}) = \sum_{k,l} \sqrt{\lambda_\alpha \lambda_{\alpha'}} \mbox{tr}( \braket{l|\alpha}\!\braket{l|k} \!\braket{\alpha'|k} ) = \lambda_{\alpha}\delta_{\alpha\alpha'}\,. \end{gather}
So far, we have shown that the action of a CP map can be expressed in terms of a Kraus decomposition. The inverse of this statement is also true. If the action of a map $\mathcal{E}$ can be written as $\mathcal{E}[\rho] = \sum_{\alpha}K_\alpha\rho K_{\alpha}^{\dagger}$, then \begin{gather} \label{eqn::KrausProof} \left(\mathcal{E} \otimes \mathcal{I}_{a} \right)[\eta] = \sum_{\alpha}(K_\alpha \otimes \openone)\eta (K_\alpha^{\dagger} \otimes \openone) = \sum_{\alpha}[(K_\alpha \otimes \openone)\sqrt{\eta}\,][\sqrt{\eta}\, (K_\alpha^{\dagger} \otimes \openone)]\,, \end{gather} where $\sqrt{\eta}$ exists and is positive due to the positivity of $\eta$. The last term in~\eqref{eqn::KrausProof} is of the form $\sum_\alpha A_\alpha A_\alpha^{\dagger}$ which is positive, as every term $A_\alpha A_\alpha^{\dagger}$ is positive on its own. As this is true independent of the size of $\eta$, we have shown that $\mathcal{E} \otimes \mathcal{I}_{a}$ is positive if the action of $\mathcal{E}$ can be written in terms of a Kraus decomposition. This means that a map $\mathcal{E}$ is CP iff its action can be written in terms of a Kraus decomposition. Equivalently, $\mathcal{E}$ is CP iff $\mathcal{E}_B$ is positive, which implies that a map for which $\mathcal{E} \otimes \mathcal{I}_{a} \geq 0,\, \forall \ \mathrm{dim}(\mathcal{H}_{n_{a}}) \le d$ satisfies $\mathcal{E} \otimes \mathcal{I}_{a} \geq 0$, for any dimension of $\mathrm{dim}(\mathcal{H}_{n_{a}})$.
\begin{figure}
\caption{\emph{Converting between different representations.} Even though we have not drawn the corresponding arrows explicitly, it is, \textit{e.g.}, possible to get from the A form to the operator sum representation by reshuffling followed by a singular value decomposition (SVD) (analogously for B form to input/output representation).}
\label{fig::CommutationDiag}
\end{figure}
\subsection{Representations of quantum maps -- a summary} \label{sec::SummRep}
All the representations introduced above constitute different concrete ways of expressing the action of the same abstract map $\mathcal{E}$. This situation is reminiscent of differential geometry, where (abstract) geometrical objects can be expressed in terms of different coordinate systems. And, just like in differential geometry, where ``in practice few things are more useful than a well-chosen coordinate system''~\cite{bengtsson_geometry_2007}, which representation of $\mathcal{E}$ is most advantageous depends on the respective experimental or computational context.
While the A form does not possess particularly nice mathematical properties, even for the case of quantum maps, the fact that its action can be written in terms of a simple matrix multiplication, makes it appealing for numerical simulations (where differential equations must often be expressed in vector form). On the other hand, the properties of the B form make it easy to check whether or not a map corresponds to a physical process. It also embodies the CJI between quantum maps and quantum states, which can be used for the ancilla-assisted tomography of quantum maps.
The tomographic representation of $\mathcal{E}$ is closest in spirit to the experimental reconstruction of the action of $\mathcal{E}$. Given the experimentally obtained input/output relation between a basis of inputs and their corresponding outputs, it allows one to directly infer the action of $\mathcal{E}$ on an arbitrary input state.
Lastly, the operator sum representation is particularly advantageous from a theoretical point of view. Proving that the dynamics of an open system for a particular initial state is CP amounts to showing that it can be written in terms of a Kraus decomposition~\cite{kraus_states_1983, alicki_semi_1987, rodriguez-rosario_completely_2008, pollock_complete_2015}, and the existence of a minimal Kraus decomposition can be employed to show the existence of generalised Stinespring dilations~\cite{chiribella_transforming_2008, chiribella_theoretical_2009}.
The above list of applications of different representations is by no means exhaustive, but it gives a flavour of when they are each most useful. We have summarised the different representations and their properties in Table~\ref{tab::Summary}, while Fig.~\ref{fig::CommutationDiag} depicts how to convert from one representation to another.
\begin{table} \setlength{\tabcolsep}{3pt} \renewcommand{1.3}{1.3} \centering
\begin{tabular}{c||c|c|c|c} & \makecell{Kraus
\\ (Operator Sum)} & \makecell{Input/Output
\\ (Tomographic)} & \makecell{Sudarshan B form
\\ (Choi Matrix)} & \makecell{Sudarshan A Form } \\\hline \hline Rep. & $\{L_\alpha,R_\alpha\}$ & $\{\hat{\rho}_i',\hat{D}_i\}$ & $ \phantom{\begin{array}{l} . \\ .\end{array}} \mathcal{E}_B = \begin{cases}
\sum_i \hat{\rho}_i' \otimes \hat{D}_i^*\\ \sum_\alpha L_\alpha \times R_\alpha^* \end{cases}$& $ \mathcal{E}_A = \begin{cases}\sum_i \hat{\rho}_i' \times \hat{D}_i^* \\ \sum_\alpha L_\alpha \otimes R_\alpha^*\end{cases}$ \\ \hline
Action & $\rho' =\sum_\alpha L_\alpha \rho \, R_\alpha^{\dagger}$ & $\rho'=\sum_i \hat{\rho}_i'\mbox{tr}(\hat{D}_i^{\dagger}\rho)$ & $\rho' = \mbox{tr}_{\mathrm{in}}[(\openone_{\mathrm{out}} \otimes \rho^{\mathrm{T}})\, \mathcal{E}_B]$ & $\ket{\rho'}\rangle = \mathcal{E}_A\ket{\rho}\rangle$ \\ \hline TP & $\sum_\alpha R_\alpha^{\dagger} L_\alpha = \openone$ & $\sum_i\mbox{tr}(\hat{\rho}_i')\hat{D}_i^{\dagger} = \openone$ & $\mbox{tr}_{\mathrm{out}}(\mathcal{E}_B) = \openone_\mathrm{in} $ & \multirow{4}{*}{\diagbox[dir=NE]{\hspace{15pt} }{\\ \hspace{30pt}} }\\ \cline{1-4}
HP &$L_\alpha = \pm R_\alpha \ \ \forall \alpha$ & $\begin{array}{@{}r@{}l@{\quad}} \hat{\rho}_i^{\dagger}&{}= \hat{\rho}_i \ \ \forall i \\ \Rightarrow \ (\hat{\rho}_i')^\dagger&{}= \hat{\rho}_i' \ \ \forall i \end{array}$ & $\mathcal{E}_B^{\dagger} = \mathcal{E}_B$ & \\ \cline{1-4}
CP & $L_\alpha = R_\alpha \ \ \forall \alpha$ & $\sum_i \hat{\rho}_i' \otimes \hat{D}_i^* \geq 0$ & $ \mathcal{E}_B \geq 0 $ & \\ \hline
\end{tabular} \caption{\emph{Linear maps in different representations.} Note that the A form does not possess particularly nice properties for trace preserving (TP), Hermiticity preserving (HP) or completely positive (CP) maps. Hermiticity preservation for the Input/Output representation is denoted only for the case where all inputs $\hat{\rho}_i$ are Hermitian.} \label{tab::Summary} \end{table}
\section{Generalisations of quantum maps} \label{sec::beyond}
The quantum maps described in the previous section take the initial state $\rho$ of the system at a particular point in time $t_0$ to that at a particular later time $t$. Consequently, they allow for the calculation of two-time correlation functions between observables. Their experimental reconstruction, as introduced in Sec.~\ref{subsec::Forms}, is well-defined if the relation between input and output states is linear; this, in turn, means that the system can be prepared independently of its environment. We will see below that this implies that the system and environment are in a product state $\rho\otimes \tau_{{e}}$ at $t_0$. If the experimental situation is such that the initial system state is correlated with the environment, or \emph{multi}-time correlation functions are of interest, quantum maps from density operators to density operators are neither well-defined nor sufficient as a description of the experimental situation.
We first discuss the problem of initial correlations, and various attempts to solve it. We will then offer an operational resolution, which opens up the door to describe arbitrary quantum processes.
\subsection{Initial correlation problem} \label{sec::IC}
In the late 1990s and early 2000s, experimentalists began reconstructing quantum gates -- the fundamental elements of a quantum computer -- by means of quantum process tomography~\cite{Nielsen:1998py, PhysRevA.64.012314, PhysRevLett.91.120402, Wein:121.13, orien:080502, NeeleyNature, chow:090502, Howard06, myrskog:013615}. Ideally a quantum gate is a unitary operation, but in practice they can be noisy. Therefore, the results of these experimental reconstructions were expected to be CP quantum maps. Yet, to the surprise of many researchers, the reconstructed maps were often not CP, and it was not clear why.
This initiated a flurry of theoretical explanations, one of which suggested that, if the initial state of the system is correlated with its environment, the quantum map describing the dynamics of the system need not be completely positive~\cite{shaji_whos_2005}. As mentioned above, Stinespring's theorem~\cite{stinespring1955} guarantees that any CP dynamics for the system ${s}$ can be thought of as coming from unitary dynamics of the system with an environment ${e}$. However, this construction assumes that the initial state of the system-environment (${se}$) state is uncorrelated -- a very restrictive assumption in practice. For instance, consider the case where the initial ${se}$ state at $t_0$ is uncorrelated, meaning that the dynamics to some later $t_1$ is CP. In general, at $t_1$ the state of ${se}$ will be correlated, and if we want to describe the dynamics from $t_1$ to later $t_2$, the quantum maps discussed in Sec.~\ref{sec::QuantumMaps} no longer apply.
\subsubsection{Not completely positive maps, not completely useful} \label{sec::NCP}
As most clearly elucidated by Pechukas in his seminal paper~\cite{pechukas94a} (and in a subsequent exchange between him and Alicki~\cite{Alicki95, Pechukas95}), a map whose argument is the state of ${s}$ is both completely positive and linear iff there are no initial ${se}$ correlations. Pechukas originally proved the theorem for qubits, but it was later generalised to $d$-dimensional systems in Ref.~\cite{jordan:052110}; here, we give a version of this result that closely resembles Ref.~\cite{PhysRevA.81.012313}.
Pechukas introduced an \textit{assignment map} $\mathfrak{A}: \mathcal{B}(\mathcal{H}_{d_{s}}) \rightarrow \mathcal{B}(\mathcal{H}_{d_{s}} \otimes\mathcal{H}_{d_{e}})$, which assigns a ${se}$ operator for every ${s}$ state, with a consistency condition: $\mbox{tr}_{e}\mathfrak{A}[\rho] = \rho \ \ \forall \rho$. Concatenating the assignment map $\mathfrak{A}$ with a unitary $U_{se}$, and tracing over ${e}$, gives a map $\mathfrak{E}$: \begin{gather} \mathfrak{E}[\rho] = \mbox{tr}_{e}\left\{U_{se} \mathfrak{A}[\rho] U^\dagger_{se} \right\} \equiv \mbox{tr}_{e}\left\{U_{se} \rho^0_{se} U^\dagger_{se} \right\}. \label{eqn::dilation} \end{gather} The unitary $U_{se}$ and trace over ${e}$ are both CP maps; therefore, if we require that $\mathfrak{A}$ is linear and CP, then it follows that $\mathfrak{E}$ must also have these properties (and is therefore a legitimate quantum map of the sort discussed in the previous section).
Now, for a consistent and CP assignment it follows that, for a basis $\{\hat{\rho}_i\}$ consisting of pure states, $\mathfrak{A}[\hat{\rho_i}] = \hat{\rho_i}\otimes {\tau_{e}}_i$, where ${\tau_{e}}_i$ have to be density operators (as required for positivity of the assignment). By the same argument, the action of the assignment map must also give a product ${se}$ state on \textit{any} pure state. Let us take a pure state not in the basis and linearly express it as $\sigma = \sum_i c_i \hat{\rho_i}$, where $c_i$ are real, with $\sum_i c_i =1$, but not necessarily positive. The action of the assignment map gives us $\sum_i c_i \hat{\rho_i}\otimes {\tau_{e}}_i = \sum_i c_i \hat{\rho_i}\otimes {\tau_{e}}$ and, therefore, ${\tau_{e}}_i=\tau_{e}$~$\forall i$. That is, the initial ${se}$ state is uncorrelated: $\mathfrak{A}[\rho]=\rho\otimes\tau_{e}$. Conversely, if the initial ${se}$ state $\rho_{se}^0$ is correlated, then either complete positivity or linearity must be abandoned\footnote{Another option is to give up consistency, but this too is not desirable. We will address this matter in some detail at the end of this subsection.}~\cite{Alicki95, Pechukas95}.
\begin{figure}
\caption{A simple circuit, for which the reduced dynamics of ${s}$ is not describable by a CP quantum map when $\rho^0_{se}$ is initially correlated.}
\label{fig::NCPexample}
\end{figure}
From an experimental standpoint, this state of affairs is problematic. On the one hand, complete positivity is a useful property -- giving up CP means giving up the Holevo quantity~\cite{holevo-assingment}, data processing inequality~\cite{buscemi_complete_2014}, and entropy production inequality~\cite{argentieri2014violations} -- and a CP description naturally predicts the physical fact that one always reconstructs positive probabilities (even for correlated preparations). On the other hand, dropping linearity is not a viable option either: complete tomography is not possible when the dynamics is nonlinear -- at least not in a finite number of experiments.
Giving up either is undesirable; however, faced with this choice, many researchers have opted to relinquish complete positivity of dynamics in favour of a framework for open dynamics based on not-completely positive (\textbf{NCP}) maps~\cite{StelmachovicBuzek01, jordan:052110}. In brief, NCP maps $\mathfrak{E}^{\rm NCP}$ are linear maps that preserve positivity for some subset of the space of system density operators, but fail to do so on the remaining set. They take as their starting point an assignment map $\mathfrak{A}$, as above, but do not require it to produce a positive ${se}$ operator for all inputs. Instead, $\mathfrak{A}$ is required to be consistent, such that $\mathfrak{A}[\mbox{tr}_{e}\rho_{se}^0] = \rho_{se}^0$ for some correlated $\rho_{se}^0$, and the action of $\mathfrak{E}^{\rm NCP}$ is only defined on the set $\{\rho : \mathfrak{A}[\rho]\geq0\}$ (which always contains $\mbox{tr}_{e}\rho_{se}^0$), called the compatibility domain of the map. Its action can then be defined through the dilation in Eq.~\eqref{eqn::dilation}, which will only result in a positive output when $\rho$ is in the compatibility domain.
While mathematically well-defined (though not unique), the NCP framework lacks a clear link to the operational reality of quantum dynamics. It assumes that there is a family of initial system states (the compatibility domain) available, and that the experimenter knows exactly which of these states is the input in each run of the experiment. However, unlike in a classical stochastic process -- where an experimenter can observe initial and final states of a system without disturbing it -- there is no operational mechanism for identifying which initial state $\rho$ will undergo the evolution in the quantum case. That is, there is no way for the experimentalist to differentiate between initial states in any given run without disturbing the system, and hence changing the correlations between ${s}$ and ${e}$. Without such a mechanism, the concept of a compatibility domain becomes a purely mathematical notion, void of physical meaning. Instead, the experimenter is presented in each run with a fixed average ${se}$ state, which they can then prepare by performing a \textit{control operation}.
In other words, there is no unambiguous way to go into the laboratory and directly reconstruct a NCP map through process tomography. In fact, if one were to attempt such a reconstruction, it would quickly become apparent that the dynamics depends not on the initial state, but on how that state is prepared. To see this more clearly, consider the two-qubit circuit in Fig.~\ref{fig::NCPexample} with initially correlated pure state $\rho^0_{se} = \ketbra{\psi}{\psi}$, where $\ket{\psi} = \frac{1}{\sqrt{2}} (\mu \ket{00} + \nu \ket{11})$, and an ${s}{e}$ dynamics given by a CNOT gate. An experimenter wishing to tomographically reconstruct the dynamics of just one of the qubits would have to prepare a variety of initial system states (we will return to this point later). Say they intended to prepare the initial state $\ketbra{0}{0}$. This would involve making a projective measurement (for sake of argument, in the computational basis $\{\ket{0},\ket{1}\}$) followed by a unitary transformation that depends on the outcome -- $\openone$ in the case that the outcome corresponding to projector $\Pi_0=\ketbra{0}{0}$ is observed, and $\sigma_x$ in the case that the outcome corresponding to $\Pi_1=\ketbra{1}{1}$ is observed. However, the state of the environment qubit, and hence the subsequent dynamics, would also depend on the measurement outcome: $\tau_{{e}|0} = \mbox{tr}_{s}\{\Pi_0\otimes\openone\ketbra{\psi}{\psi}\} = \ketbra{0}{0} \neq \tau_{{e}|1} = \mbox{tr}_{s}\{\Pi_1\otimes\openone \ketbra{\psi}{\psi}\} = \ketbra{1}{1}$. That is, despite the fact that the initial state is the same in these two cases, the final density operator differs (in fact, the two different output states are orthogonal). Choosing a different preparation procedure will not alleviate these issues, and similar problems arise for any initially correlated state. This leads us to conclude that there is no unique way to prepare a state, and the preparation procedure plays a role in determining the future evolution~\cite{modi_preparation_2011}.
Let us take this argument one step further and attempt to perform quantum process tomography by preparing the basis states with projections. For simplicity, we will confine the tomography to the $x-z$ plane of the Bloch sphere. We prepare basis states $\Pi_0$ and $\Pi_1$ by projecting the system in the $z$ basis, and basis state $\Pi_+ = \ketbra{+}{+}$ by projecting in the system in the $x$ basis. In the latter case, sometimes we will find the system in state $\Pi_- = \ketbra{-}{-}$, which is linearly related to the basis states as $\Pi_- = \Pi_0 + \Pi_1 - \Pi_+$. The output states corresponding to basis states $\Pi_0, \Pi_1, \Pi_+$ are easily computed to be $\Pi_0, \Pi_0, \Pi_+$. And similarly, by examining the global dynamics we find that $\Pi_-$ maps to itself. However, if we try to construct a linear map using the input/output data, we find that it predicts $\Pi_-$ will be mapped to matrix $\Pi_0 + \Pi_0 - \Pi_+$, which is non-positive. The constructed map is therefore NCP, and it makes a nonsensical prediction. Clearly, we can prepare $\Pi_-$ and observe the subsequent output of the process. But the constructed map does not capture this physics, and we have to conclude that NCP maps are not useful. Indeed, there are many examples where quantum process tomography, without properly taking the preparation procedure into account, leads to NCP and nonlinear maps\footnote{On the other hand, it is possible to construct a meaningful map where all preparations are projective~\cite{kuah:042113}, or with any other restricted set of preparations~\cite{milz_reconstructing_2016}, when these are correctly accounted for.}~\cite{modi_role_2010}.
Before introducing a resolution to the problem of initial correlations, we discuss the matter of giving up consistency to retain CP and linearity of the dynamics. Research along these lines led to the claim that ``vanishing quantum discord is necessary and sufficient for completely positive maps"~\cite{PhysRevLett.102.100402} which received a great deal of attention, but then was subsequently proven to be incorrect~\cite{PhysRevA.87.042301}, leading to an erratum~\cite{PhysRevLett.116.049901}. In Ref.~\cite{rodriguez-rosario_completely_2008}, it was shown that if the initial ${se}$ state has vanishing quantum discord, then a CP map can be ascribed to the dynamics of ${s}$. Consequently, by projectively measuring the system part of any initial state $\rho^0_{se}$ -- which will always produce a discord zero state -- one can associate a CP map from the measurement outcome at the initial time to the quantum state at the final time. The problem with this approach is that the CP maps depend on the choice of measurement, which does not depend on the pre-measurement state of the system. The corresponding assignment map is therefore not consistent, and one is left with only a partial description of the dynamics (with similar issues to the example above).
\subsubsection{Operational resolution: Superchannels}
As already mentioned, the first step of any experiment is to prepare the system in a desired state by applying a control operation. The control operations can be anything, including unitary transformations, projective measurements, projective measurements followed by a unitary transformation (like in the example above) and everything in-between. Mathematically, a control operation $\mathcal{A}_{s}$ is just a (trace non-increasing) CP quantum map (as described in Sec.~\ref{sec::QuantumMaps}). In a dilated picture the final state is related to the control operation as the following: \begin{gather}\label{eqn::SCD} \rho' = \mbox{tr}_{e} \{U (\mathcal{A}_{s} \otimes \mathcal{I}_{e} [\rho^0_{se}]) U^\dag \} \equiv \mathcal{M}[\mathcal{A}_{s}]\, , \end{gather} where we have defined the \emph{superchannel} $\mathcal{M}$~\cite{modi_operational_2012}, a linear operator that maps preparations to final states. From here on we omit the subscript ${s}$ on the control operation $\mathcal{A}_{s}$, and assume it only acts on the system.
From Eq.~\eqref{eqn::SCD}, it is clear that the superchannel is linear in the same way as $\mathcal{E}$, as argued at the beginning of Sec.~\ref{sec::QuantumMaps}. The set of all control operations is isomorphic to the set of positive $d^2\times d^2$ matrices of trace less than or equal to $d$ (the B forms of control operations). Henceforth, whenever we write $\mathcal{A}$, we always mean this representation of the control operation, if not explicitly stated otherwise. With this, by taking the square root of the initial state and combining it with $U$ we can write the action of $\mathcal{M}$ \begin{gather} \label{eqn::CPsuper} \quad \mathcal{M}[\mathcal{A}] = \sum_\alpha \mu_\alpha \ \mathcal{A} \ \mu_\alpha^\dag \qquad \mbox{with} \qquad \mu_{\epsilon x}= \sqrt{\lambda_x} \bra{\epsilon} U \otimes_{s} \ket{\Psi_x}^{\mathrm{T}_{s}} \qquad \mbox{and} \qquad \alpha = \epsilon x. \end{gather} Here $\lambda_x$ and $\ket{\Psi_x}$ are the eigenvalues and eigenectors of $\rho^0_{se}$ respectively and $\otimes_{s}$ and $\mathrm{T}_{s}$ means a tensor product and transpose (in computational basis) on the space of ${s}$ only respectively, while the normal matrix product applies to the space of ${e}$. The last equation is an analogue of Eq.~\eqref{eqn::canonical}; it is the Kraus representation for the superchannel, which means that it is CP~\cite{modi_operational_2012}. In fact, the operators $\mu_\alpha$ have similar properties to those in Sec.~\ref{subsec::CP} and it is straightforward to show that $\mathcal{M}$ is trace preserving in the sense that it maps trace preserving preparations to unit trace matrices. From a mathematical point of view, the superchannel is a CP map just as the ones encountered in Sec.~\ref{sec::QuantumMaps}, but with input and output spaces of different size, \textit{i.e.}, $\mathcal{M}: \mathcal{B}(\mathcal{H}) \otimes \mathcal{B}(\mathcal{H}) \rightarrow \mathcal{B}(\mathcal{H})$. The CP nature of the superchannel also has an operational implication: Suppose we bring in an auxiliary system ${a}$ of dimension $n$ and perform an entangling control operation $\mathcal{A}_{{s} {a}}$, before letting ${s}$ undergo the process in question (\textit{i.e.}, interact with ${e}$). The complete positivity of the superchannel guarantees that the final state $\rho'_{{s}{a}}$ will always be positive.
Operationally speaking, the superchannel is simply the logical consequence of the input/output picture presented at the very beginning of the paper; it maps the actual controllable inputs (the preparations $\mathcal{A}$) to the actual measurable outputs (the final system state $\rho'$) of the experiment. When there are no initial correlations, \textit{i.e.}, the initial ${se}$ state in Eq.~\eqref{eqn::SCD} is a product state we find that the Kraus operators in Eq.~\eqref{eqn::CPsuper} become $\mu_{\epsilon x} = K_\epsilon \otimes \sqrt{\lambda_x^{s}} \bra{\psi_x^{s}}^*$, where $K_\epsilon$ are the Kraus operators of the quantum map $\mathcal{E}$ in Eq.~\eqref{eqn::canonical}, $\lambda_x^{s}$ and $\ket{\psi_x^{s}}$ are eigenvalues and eigenvectors of $\rho^0_{s}$ respectively. Subsequently, the action of the superchannel reduces to $\mathcal{M}[\mathcal{A}] = \mathcal{E}(\rho_{s})$, where $\rho_{s}$ is the result of applying the control operation $\mathcal{A}$ on the fiducial initial state $\rho_{s}^0$, \textit{i.e.}, $\rho_{s} \equiv \mathcal{A}[\rho^0_{s}]$. Consequently, the superchannel formalism includes the experimental situation depicted in the first chapter and naturally extends quantum maps to the more general case of initial correlations.
Proponents of the NCP map formalism would claim that the superchannel framework (and our experimenter in the example in the previous subsection) is setting up a different dynamical experiment each time the system is prepared. However, the superchannel only depends on the initial ${se}$ state and subsequent ${se}$ unitary operation; it is independent of the choice of the control operation, which is the choice of the experimenter. Moreover, the superchannel, along with some data processing, contains all NCP maps one could construct for the process (though still void of any operational meaning)~\cite{modi_operational_2012}. Conversely, the only way the predictions of the superchannel could be reproduced in the NCP formalism is by enumerating the NCP maps corresponding to all (infinitely many) possible preparation procedures.
The construction of the superchannel does not \textit{a priori} assume linearity or complete positivity. However, by simply following the operational reality of a quantum experiment, we have arrived at a map that has these familiar (and desirable) features. In doing so, we have overcome Pechukas' theorem; the superchannel is a consistent, linear, and CP description for dynamics in the presence of initial correlations. Unlike NCP maps, it has a clear operational meaning, and has been unambiguously reconstructed in a tomography experiment~\cite{PhysRevLett.114.090402}. Finally, the CP nature of the superchannel allows for the extension of useful results, such as the Holevo quantity, data processing inequality, and entropy production inequalities, to the case of initial correlations~\cite{PhysRevA.92.052310}.
We will now show how the superchannel concept can be generalised to processes involving multiple time steps, before discussing its structure and representations.
\subsection{Multiple time steps and the process tensor}
Like the quantum maps $\mathcal{E}$ from Sec.~\ref{sec::QuantumMaps}, the superchannel only accounts for two time correlations between preparations at the initial time and measurements at the final time. In a more general experiment -- for example, in a multi-dimensional spectroscopy experiment~\cite{KassalBook} -- one may want to know about correlations across multiple time steps. It is relatively straightforward to generalise the superchannel to this scenario; imagine that the experimenter performs (CP) control operations $\mathcal{A}_0, \mathcal{A}_1,\dots,\mathcal{A}_{k-1}$ at the $k$ times $t_0, t_1,\dots, t_{k-1}$ and measures the corresponding output state $\rho'_k$\footnote{If the control operations are not trace preserving (and therefore not performable deterministically), then the trace of $\rho_k'$ gives the probability of performing those operations.} at $t_k$. This scenario is illustrated in Fig.~\ref{fig::process-a} (our only assumption is that these operations can be performed on a much shorter time scale than any other dynamics of ${s}$ or ${se}$). This setup is very general; for instance, the control operations could be quantum gates, with the final state corresponding to the outcome of a quantum computation. Or, perhaps, the process could be a series of chemical reactions, where the control operations represent the addition of reactants.
\begin{figure}
\caption{\emph{A $k$-step process.} At each time step $t_i$, a CP operation $\mathcal{A}_i$ is performed, and the resulting state $\rho'_k$ at $t_k$ is determined by quantum state tomography. The scenario in which dynamics is described by the quantum maps of Sec.~\ref{sec::QuantumMaps} is also included in this schematic; it corresponds to a single-step process, where $\mathcal{A}_0$ is the preparation of the input state.}
\label{fig::process-a}
\end{figure}
Just as the superchannel is linear in its argument, the final state $\rho'_k$ depends in a \emph{multilinear} way on the operations $\mathcal{A}_j$. Mathematically, this means that the dynamics is a mapping $\mathcal{T}^{k:0}: \mathcal{B}(\mathcal{B}(\mathcal{H}))^{\otimes k}\rightarrow \mathcal{B}(\mathcal{H})$, called a \emph{process tensor}~\cite{pollock_complete_2015}, whose action can be written as \begin{gather} \label{eqn::processtensor} \rho_k' = \mathcal{T}^{k:0}\left[\mathbf{A}_{k-1:0} \right]\, . \end{gather} In terms of their B form, we have $\mathbf{A}_{k-1:0}\in \left[\mathcal{B}(\mathcal{H})\otimes \mathcal{B}(\mathcal{H})\right]^{\otimes k}$ and $\mathcal{T}^{k:0}$ becomes a mapping $\mathcal{T}^{k:0}: \left[\mathcal{B}(\mathcal{H}) \otimes \mathcal{B}(\mathcal{H})\right]^{\otimes k} \rightarrow \mathcal{B}(\mathcal{H})$. To keep better track of the different terms, we give superscripts to the process and subscripts to the control operations. For the case of independent control operations, $\mathbf{A}_{k-1:0}$ is simply given by $\mathbf{A}_{k-1:0}=\mathcal{A}_{k-1}\otimes\cdots \mathcal{A}_1\otimes \mathcal{A}_0$. In a more general scenario, the operations could be correlated, either classically (\textit{e.g.}, transformations conditioned on earlier measurement outcomes) or quantum mechanically, through successive interactions with an ancilla.
In Refs.~\cite{chiribella_theoretical_2009,pollock_complete_2015} the existence of a \emph{generalised Stinespring dilation} was proven; a map $\mathcal{T}^{k:0}$ is consistent with ${se}$ unitary dynamics if it is linear, CP, trace preserving in the sense that it maps sequences of trace preserving control operations to unit trace matrices, and possesses a containment property, $\mathcal{T}^{j:k} \subset \mathcal{T}^{i:l} \, \forall \, i \le j \le k \le l$. The latter property is a causality property ensuring that future actions do not affect past dynamics. Conversely, the process tensor can be derived starting from a dilated (unitary) ${se}$ evolution, as shown in Fig.~\ref{fig::Process-b}. We sketch this derivation now.
\begin{figure}\label{fig::Process-b}
\end{figure}
Any ${se}$ dynamics can be written as \begin{gather} \label{eqn::processdilated} \rho'_k = \mbox{tr}_{e} \left[\mathcal{U}^{k:k-1} \mathcal{A}_{k-1} \ \mathcal{U}^{k-1:k-2} \mathcal{A}_{k-2} \dots \mathcal{U}^{1:0} \mathcal{A}_0 (\rho^0_{se}) \right], \end{gather} where $\mathcal{A}$ act on ${s}$ alone and the unitary maps $\mathcal{U}^{l:k} (\rho^{k}_{se}) = U^{l:k} \rho^{k}_{se} {U^{l:k}}^\dag= \rho^l_{se}$ act on the full system environment space. Everything in this equation other than the control operations, \textit{i.e.}, everything in the red box in Fig.~\ref{fig::Process-b}, can be considered as part of the process. In analogy to Eq.~\eqref{eqn::CPsuper}, by contracting (`matrix multiplying') the unitary operators in the space of ${e}$, along with the initial state ${se}$ and taking the final trace, we can define \begin{gather} T_{\epsilon x}^{k:0} \equiv \sqrt{\lambda_x} \bra{\epsilon} U^{k:k-1} \otimes_{s} \bbra{U^{k-1:k-2}}_{s} \otimes_{s} \dots \otimes_{s} \bbra{U^{1:0}}_{s} \otimes_{s} \ket{\Psi_x}^{\mathrm{T}_{s}}, \end{gather} where again $\lambda_x$ and $\ket{\Psi_x}$ are the eigenvalues and eigenvectors of $\rho^0_{se}$ respectively, $\otimes_{s}$, $\bbra{\;}_s$, and $\mathrm{T}_{s}$ mean tensor product, vectorisation (in the sense of Eq.~\eqref{eqn::VecAction}), and transpose on the space of ${s}$ only. Note that the last unitary matrix is not vectorised. With this, we can rewrite Eq.~\eqref{eqn::processdilated} as \begin{gather} \label{eqn::processdilated2} \rho'_k = \sum_\alpha T_\alpha^{k:0} \ \mathbf{A}_{k-1:0} \ {T_\alpha^{k:0}}^\dag = \mathcal{T}^{k:0}[\mathbf{A}_{k-1:0}], \quad \text{with} \quad \alpha = \epsilon x\, , \end{gather} where $\mathcal{T}^{k:0}$ is the process tensor. This equation is an analogue of Eqs.~\eqref{eqn::canonical} and~\eqref{eqn::CPsuper}. That is, it is the operator sum (or Kraus) representation for the process tensor, which (like the superchannel) implies that it is CP~\cite{pollock_complete_2015}. It also clearly satisfies the containment property, \textit{i.e.}, $\mathcal{T}^{j:k} \subset \mathcal{T}^{i:l} \, \forall \, i \le j \le k \le l$ and it is trace preserving. Indeed, the process tensor reduces to the \emph{superchannel} for a single-step process and it is the natural extension of the formalism in Sec.~\ref{sec::QuantumMaps} to more general experimental situations.
Comparable approaches to general quantum stochastic processes were already developed by Lindblad~\cite{lindblad_non-markovian_1979} and Accardi \textit{et al.}~\cite{accardi_nonrelativistic_1976, accardi_quantum_1982}, but have not gained traction with the community of researchers working on open quantum systems. The process tensor framework straightforwardly leads to several important results, most notably an operationally well-defined quantum Markov condition, measures of non-Markovianity (which we will briefly expand on below) ~\cite{pollock_complete_2015}, and a generalisation of the Kolmogorov extension theorem to general quantum stochastic processes~\cite{accardi_quantum_1982}.
Finally, similar mathematical structures (maps whose inputs and outputs are quantum maps themselves) have also been developed in other contexts, and are referred to as quantum combs~\cite{chiribella_transforming_2008, chiribella_quantum_2008, chiribella_theoretical_2009}, causal automata/non-anticipatory quantum channels~\cite{kretschmann_quantum_2005, caruso_quantum_2014}, process matrices for causal modelling~\cite{1367-2630-18-6-063032,oreshkov_causal_2016}, causal boxes~\cite{portmann_causal_2015} and operator tensors~\cite{hardy_operational_2016,hardy_operator_2012}. Most of the results for the process tensor, including the representations we will now go on to describe, will also be applicable (or, at least, adaptable) to many of these frameworks.
\subsection{Structure and representation of the superchannel and process tensor}
Since the process tensor, and hence the superchannel, are CP maps of the sort described in Sec.~\ref{subsec::LinMaps} with input and output spaces of different size, we are able to represent them mathematically in all the ways discussed in the first half of this paper. For the most part, these representations are the same as for the quantum maps case (with the same mathematical properties), however it is insightful to present them explicitly. Given that the superchannel is simply a single step process tensor, its representations are a special case of what we will now present for an arbitrary number of time steps.
Performing quantum process tomography of process tensors is very similar to the usual case. At each time step $j$ we choose a basis set of linearly independent operations $\{\hat{\mathcal{A}}_{i_j} \}_{{i_j}=1}^{d^4}$. The index $i_j$ denotes both the basis element, as well as the time step, \textit{i.e.}, $\hat{\mathcal{A}}_{i_j}$ is the $i$th basis element at time step $j$. For example, at time step 3, we would have $\{\mathcal{A}_{1_3}, \ \mathcal{A}_{2_3}, \dots, \mathcal{A}_{d^4_3} \}$. The basis elements at different times need not be the same, $\{\hat{\mathcal{A}}_{i_j}\} \ne \{\hat{\mathcal{A}}_{i_{j'}}\}$. An arbitrary control operation $\mathcal{A}_j$ at time step $j$ can be expressed as a linear combination of the basis operations $\mathcal{A}_j = \sum_{i_j} c_{i_j} \hat{\mathcal{A}}_{i_j}$. The basis operations come with a dual set $\{\hat{\Delta}_{i_j}\}$ satisfying $\mbox{tr}[\hat{\mathcal{A}}_{i_j} \hat{\Delta}_{i'_j}^\dag] = \delta_{i_j i_j'}$. From the local basis, we can construct a basis sequence as $\hat{\mathbf{A}}_{\mathbf{i}_{k-1:0}} = \hat{\mathcal{A}}_{i_{k-1}} \otimes \dots \otimes \hat{\mathcal{A}}_{i_{0}}$, where $\mathbf{i}_{k-1:0} = (i_{k-1} \dots i_0)$. Naturally, we have $\mbox{tr}[\hat{\mathbf{A}}_{\mathbf{i}'_{k-1:0}} \hat{\mathbf{D}}_{\mathbf{i}_{k-1:0}}^\dag] = \delta_{\mathbf{i}'_{k-1:0}\ \mathbf{i}_{k-1:0}}$, where $\hat{\mathbf{D}}_{\mathbf{i}_{k-1:0}} = \hat{\Delta}_{i_{k-1}} \otimes \dots \otimes \hat{\Delta}_{i_0}$. As before, using the basis operations we can express any (possibly correlated) sequence of control operation as $\mathbf{A}_{k-1:0} = \sum_{\mathbf{i}_{k-1:0}} \ c_{\mathbf{i}_{k-1:0}} \ \hat{\mathbf{A}}_{\mathbf{i}_{k-1:0}}$.
Tomography then involves performing a set of experiments where we apply each basis sequence of control operations $\hat{\mathbf{A}}_{\mathbf{i}_{k-1:0}}$ and measure the corresponding state $\hat{\rho}'_{k|\mathbf{i}_{k-1:0}}$. In analogy with Eq.~\eqref{eqn::ActionMap} we can then write the action of the process tensor as \begin{gather}\label{eqn::process-tomo}
\mathcal{T}^{k:0}[\mathbf{A}_{k-1:0}] = \sum_k \hat{\rho}'_{k|\mathbf{i}_{k-1:0}} \ \mbox{tr} \left[ \hat{\mathbf{D}}_{\mathbf{i}_{k-1:0}}^\dag \mathbf{A}_{k-1:0} \right], \end{gather}
The set $\{\hat{\mathbf{D}}_{\mathbf{i}_{k-1:0}},\hat{\rho}'_{k|\mathbf{i}_{k-1:0}}\}$ constitutes the \textit{tomographic representation} of the process tensor.
From this, we can use the results of Sec.~\ref{subsec::Forms} to write down the process tensor in Sudarshan's A and B forms, in analogy to Eqs.~\eqref{eqn::EcalA} and~\eqref{eqn::EcalB}, as \begin{gather}
\mathcal{T}^{k:0}_A = \sum_k \hat{\rho}'_{k|\mathbf{i}_{k-1:0}} \times \hat{\mathbf{D}}_{\mathbf{i}_{k-1:0}}^* \qquad \mbox{and} \qquad
\mathcal{T}^{k:0}_B = \sum_k \hat{\rho}'_{k|\mathbf{i}_{k-1:0}} \otimes \hat{\mathbf{D}}_{\mathbf{i}_{k-1:0}}^*. \end{gather} Similarly from the operator sum representation, given in Eq.~\eqref{eqn::processdilated}, we can write these in terms of Kraus operators (in analogy to Eqs.~\eqref{eqn::Kraus_A} and~\eqref{eqn::Kraus_B}): \begin{align} \mathcal{T}^{k:0}_A = \sum_\alpha T^{k:0}_\alpha \otimes {T^{k:0}_\alpha}^* \qquad \mbox{and} \qquad \mathcal{T}^{k:0}_B = \sum_\alpha T^{k:0}_\alpha \times {T^{k:0}_\alpha}^*. \end{align} Just as in the case of quantum maps $\mathcal{E}$, $\mathcal{T}^{k:0}_A$ is a (rectangular) matrix which acts on a vectorised input -- in this case, it is the B form of the control operations $\mathbf{A}_{k-1:0}$ that is vectorised. In contrast, $\mathcal{T}^{k:0}_B$ is a square matrix, whose positivity depends on the complete positivity of the process tensor. It acts as \begin{align} \label{eqn::ChoiSuper} \mathcal{T}^{k:0}[\mathbf{A}_{k-1:0}]=\mbox{tr}_{\rm in} \left[\mathcal{T}^{k:0}_B \left(\openone_{\rm out} \otimes \mathbf{A}_{k-1:0}^\mathrm{T} \right)\right]\, . \end{align} In fact, analogously to the case of quantum maps, the B form can be seen as arising from a generalisation of the CJI to process tensors~\cite{pollock_complete_2015}. The isomorphism can be implemented operationally by preparing $k$ maximally entangled states $\ket{I}$ (introduced at the beginning of Sec.~\ref{sec::CJI}) and swapping the system with one part of the maximally entangled state at each time step. Defining $\Psi_{{a} b}$ as the superchannel describing an initial state $\ket{I}_{{a} b}$ that later evolves under an identity map (\textit{i.e.}, with B form $\mathcal{I}_B\otimes \ketbra{I}{I}$), we can write down the CJI: \begin{align}\label{eqn::ChoiState} \Upsilon_\mathcal{T}^{k:0} = \mathcal{T}^{k:0}_{s} \otimes \Psi^{\otimes k}_{{a} b} [\mathcal{S}_{{s}{a}}^{\otimes k} \otimes \mathcal{I}_b^{\otimes k}], \end{align} where $\mathcal{S}_{{s}{a}}$ is the swap gate on ${s}{a}$ and $\mathcal{I}_b$ is the identity map on $b$ (see Fig.~\ref{fig::CJI}). The resultant $2k+1$ body state $\Upsilon_\mathcal{T}^{k:0}$ is element by element identical to the process tensor. Again, using the equivalence of the B form and the Choi state, the action of the map can be written as $\mathcal{T}^{k:0}[\mathbf{A}_{k-1:0}]=\mbox{tr}_{\rm in}[\Upsilon^{k:0}_\mathcal{T} \ (\openone_{\rm out} \otimes \mathbf{A}_{k-1:0}^\mathrm{T})]$.
\begin{figure}\label{fig::CJI}
\end{figure}
Since pairs of subsystems of the Choi state $\Upsilon^{k:0}_\mathcal{T}$ correspond to different time steps, correlations between them directly relate to memory effects in the process. That is, temporal correlations in $\mathcal{T}^{k:0}$ become spatial correlations in $\Upsilon^{k:0}_\mathcal{T}$. This can most clearly be seen by decomposing the B form/Choi state as \begin{align} \label{eqn::CorrelationExpansion} \Upsilon^{k:0}_\mathcal{T} =& \mathcal{E}_B^{k:k-1}\otimes\mathcal{E}_B^{k-1:k-2}\otimes\cdots\otimes \mathcal{E}_B^{1:0}\otimes \rho^0 \nonumber\\& + \sum_{j>j'} \chi_{j,j'} + \sum_{j>j'>j''} \chi_{j,j',j''} + \cdots + \chi_{k,k-1,\dots,1,0}, \end{align} with $\mbox{tr}_{j_i}[\chi_{j_n,\dots,j_1}] =0 \ \forall \ 1\leq i\leq n$, and where the indices in the sums represent pairs of subsystems belonging to the input of one time step and the output of the previous one (with the exception of $j=0$, which refers a single subsystem -- the initial input). We denote by $\mbox{tr}_{\bar j}$ the partial trace over all subsystems but the ones that correspond to the dynamics from time $j-1$ to $j$ (see Fig.~\ref{fig::CJI}). With this, we obtain $\mathcal{E}_B^{j:j-1} = \mbox{tr}_{\bar{j}}[\Upsilon^{k:0}_\mathcal{T}]$, the B form of a quantum map connecting an adjacent pair of time steps, and $\rho^0 = \mbox{tr}_{\bar{0}}[\Upsilon^{k:0}_\mathcal{T}]$ -- the initial, pre-preparation state of the system undergoing the process. The traceless matrices $\chi$ encode correlations between time steps, and it is precisely these which will contract with the B forms of measurement operations at different time steps in Eq.~\eqref{eqn::ChoiSuper} to produce multi-time correlation functions.
\subsection{Quantum Markov processes and measuring non-Markovianity}
If a classical stochastic process has no correlations between observables at different times, beyond those mediated by adjacent time steps, then it is called Markovian; formally \begin{gather}\label{eqn::Cl-Markov}
P(X_k,t_k|X_{k-1},t_{k-1},\dots,X_0,t_0) = P(X_k,t_k|X_{k-1},t_{k-1}) \quad \forall k. \end{gather} Generalising Eq.~\eqref{eqn::Cl-Markov}, to give a necessary and sufficient condition for a quantum process to be Markovian has been a difficult task. In recent years, researchers have built a zoo of ``measures" of non-Markovianity~\cite{NMrev, RevModPhys.88.021002}. Most of these measures are based only on necessary conditions for classical processes to be Markovian\footnote{Some of these measures are claimed to be \textit{necessary and sufficient}, but only with respect to a quantum Markov condition which does not reduce to the classical one in the correct limit.}. For instance, the trace distance between two probability distributions must monotonically decrease under a classical Markov process. This should also be true for any pair of density matrices undergoing a quantum Markov process. Conversely, if the trace distance between any two density matrices does not decrease monotonically, then it implies that the underlying process is non-Markovian. A measure of non-Markovianity can be defined by summing up the non-monotonicity in time~\cite{PhysRevLett.103.210401}. Other witnesses are based on: how quantum Fisher information changes~\cite{PhysRevA.82.042103}; the detection of initial correlations~\cite{mazzola2012dynamical, rodriguez2012unification}; changes to quantum correlations~\cite{PhysRevA.86.044101}; positivity of quantum maps~\cite{PhysRevLett.101.150402, sabri}; and, most notably, witnessing the breakdown of the divisibility of a process~\cite{PhysRevLett.105.050403}. These witnesses are turned into measures by quantifying the degree to which they witness the departure from Markov dynamics.
All of these methods are perfectly valid ways of witnessing memory effects. Unfortunately though, they often lack a clear operational basis. Moreover, different measures of non-Markovianity do not always agree with each other, neither on the degree of non-Markovianity, nor on deciding whether a given process is Markovian~\cite{PhysRevA.83.052128}. In other words, each of them fails to quantify some demonstrable memory effects. These inconsistencies have led some researchers to conclude that there can be no unique condition for a quantum Markov process.
This is not correct. Using the process tensor framework it is possible to write down a \textit{necessary and sufficient} condition for quantum Markov processes~\cite{pollock_complete_2015}, that is mathematically unique and operationally sound. It encompasses all the other definitions by objectively identifying all possible temporal correlations responsible for all possible memory effects -- including the correlations missed by the methods listed above (see examples in Ref.~\cite{pollock_complete_2015}). For classical dynamics, this quantum condition reduces to the Markov condition given in Eq.~\eqref{eqn::Cl-Markov}.
We can use the expansion in Eq.~\eqref{eqn::CorrelationExpansion} to make this condition more explicit. A process whose Choi state can be written as $\Upsilon^{k:0}_\mathcal{T} = \mathcal{E}_B^{k:k-1}\otimes\cdots\otimes \mathcal{E}_B^{1:0}\otimes \rho^0$ (with all $\chi$ operators zero) will only lead to joint probability distributions which satisfy the Markov property for \emph{any} choice of measurements (not necessarily projective) at different time steps\footnote{Though the distributions for different choices of measurements will not be compatible in general (this could be seen as the defining feature of quantum theory).}, and we could take the product form as a definition for a quantum Markov process. Operationally, this means that a causal break in the system's evolution at any point prevents information flowing from past to future, see Ref.~\cite{pollock_complete_2015} for a rigorous derivation.
From this Markov condition we can derive a family of measures for non-Markovianity that are operationally meaningful for specific tasks. For instance, through Eq.~\eqref{eqn::CorrelationExpansion}, any process can be related to a corresponding Markov process $\mathcal{T}^{k:0}_{\rm Mkv}$ (with Choi state $\Upsilon^{k:0}_{\mathcal{T}_{\rm Mkv}} = \mathcal{E}_B^{k:k-1}\otimes\cdots\otimes \mathcal{E}_B^{1:0}\otimes \rho^0$) by simply setting the correlation terms to zero (removing the $\chi$'s). The total `amount' of memory in the process, or the \emph{degree of non-Markovianity} $\mathcal{N}$ can then be quantified by the distance of its Choi state from that of its Markov counterpart: \begin{gather}\label{eqn::NMmeasure} \mathcal{N} = \mathcal{D}\left(\Upsilon^{k:0},\Upsilon^{k:0}_{\mathcal{T}_{\rm Mkv}}\right) \end{gather} where $\mathcal{D}$ could be any (pseudo) distance measure on quantum states, such as the trace distance. In particular, when relative entropy is chosen as the distance measure in Eq.~\eqref{eqn::NMmeasure}, the measure has a clear interpretation in terms of the probability of surprise $P_{\textrm{surprise}} = e^{-n \mathcal{N}}$. That is, suppose we have an experimental process that is non-Markovian and a model for this experiment that is Markovian. Then, after $n$ experiments how surprising are the results, given our Markov assumption? If $\mathcal{N}$ is small, then it will take many experiments (large $n$), before we observe statistically significant deviations in our data from the assumed model, and if $\mathcal{N}$ is large then we are surprised after only a small number of experiments $n$.
Different choices of distance will lead to different operational meanings for $\mathcal{N}$. Other measures of non-Markovianity could, for instance, indicate how much of the original state of ${s}$ can be recovered, or how many extra degrees of freedom are needed to model the dynamics of ${s}$ to a desired accuracy.
\section{Discussion and conclusions}
In this article, we have laid bare the operational motivation and underlying structure of quantum maps. We began by describing the familiar quantum maps that act on density operators and transform them into density operators, before going on to derive and relate their most widely used forms, and discuss their most important properties. While we worked only with finite dimensional systems, all of the maps presented here can also be extended to the case of infinite dimensional systems~\cite{davies76a, Brukner2016}.
Next, we described the problem of characterising quantum dynamics in the presence of initial correlations between a system and its environment. We outlined the attempts to describe such dynamics with not completely positive maps and the operational shortcomings of this approach. Then we presented a resolution to this problem in terms of the quantum superchannel, which generalises the quantum maps from the first section of the paper, and has all of the same desirable properties, like complete positivity and trace preservation. The development of the superchannel paved the way for us to introduce the process tensor framework, which can be used to describe any quantum process -- importantly, including its multi-time correlations. Major results enabled by this framework are a necessary and sufficient condition for quantum Markov processes and, consequently, a family of operationally meaningful measures for non-Markovianity.
The different mathematical representations we have presented arise from the statistical and linear algebraic framework on which quantum theory is based. In fact, they could also be used to describe a more general linear theory, such as one based on quaternionic vector spaces~\cite{Graydon2013}, as well as other generalised statistical theories~\cite{Barrett2007,MasanesMuller2011}. It is worth mentioning tensor network calculus as a helpful tool for graphically representing quantum maps (and other linear algebraic objects). Diagrammatic proofs of the statements in this paper more clearly reveal the connections between different representations, as well as the similarity between the approaches of the first and second Section of this paper. For a comprehensive introduction in the context of open quantum systems theory, see Ref.~\cite{wood_tensor_2011}.
The process tensor is a powerful tool, and we have only just scratched the surface when it comes to unsolved open quantum dynamics problems. There remains a great deal of work to be done in order to better understand the properties of non-Markovian quantum processes. This includes, but is not limited to, characterising the length and strength of memory and investigating typical properties of random multi-time processes. It remains to be seen whether something like the process tensor can be derived for setups where continuous control is applied, or where the experimenter's operations also influence the environment to some degree.
It should also be possible to use the process tensor framework to develop new methods for simulating open quantum systems. An approach based on tomographically reconstructed quantum maps has already been shown to be efficient~\cite{CerrilloCao2014, Rosenbach2016, PollockModi2017}, and it seems a natural step forward to generalise this to the multi-time case. Furthermore, such a method would be easy to adapt into a simpler approximate description; the process tensor quantifies exactly the observable influence of the environment on a system, therefore its structure should indicate exactly which quantities can be safely neglected in the global dynamics.
\section*{APPENDIX}
\section{Dual matrices} \label{subsec::DualMat}
In this appendix, we prove the existence of a set of dual matrices for any set of linearly independent matrices $\rho_i$. Note, that this proof is a slight generalisation of the one presented in~\cite{modi_positivity_2012} for the case of Hermitian matrices $\rho_i$.
\begin{lemma} For any set of Linearly independent matrices $\{\rho_i\}$, there exists the dual set $\{\hat{D}_i\}$ satisfying $\mathrm{tr}[\hat{D}_i^{\dagger} \; \rho_j] =\delta_{ij}$. \end{lemma}
\begin{proof} Write $\rho_i = \sum_j h_{ij} \Gamma_j$, where $h_{ij}$ are complex numbers and $\{\Gamma_j\}$ form a Hermitian self-dual linearly independent basis satisfying $\mbox{tr}[\Gamma_i \Gamma_j]=2 \delta_{ij}$~\cite{byrd:062322}. Since $\{\rho_i\}$ constitute a linearly independent set, the columns of matrix $\mathsf{H} = \sum_{ij} h_{ij} \ketbra{i}{j}$ are linearly independent vectors, which means that $\mathsf{H}$ has an inverse. Let the matrix $\mathsf{F}^\dagger=\mathsf{H}^{-1}$, then $\mathsf{H} \mathsf{F}^\dagger = \openone$, implying that the columns of $\mathsf{F}^*$ are orthonormal to the columns of $\mathsf{H}$. We define $\hat{D}_i = \frac{1}{2} \sum_j f_{ij} \Gamma_j$, where $f_{ij}$ are elements of $\mathsf{F}$. \end{proof} Our definition of dual matrices differs from the one in~\cite{modi_positivity_2012} by an adjoint to make the relation to the scalar product explicit. As already mentioned, the dual matrices are generally not all positive, even if the basis $\{\rho_i\}$ only consists of positive matrices. However, for the case where all basis matrices $\rho_i$ are Hermitian, we have $\hat{D}_i^{\dagger} = \hat{D}_i$. Furthermore, the duals satisfy $\sum_i \hat{D}_i^\dag = \sum_i \hat{D}_i^* = \openone$ if all $\rho_i$ are of unit trace. We have \begin{gather} \mbox{tr} \left(\sum_i\hat{D}_i^{\dagger}\rho \right) = \sum_{i,j} r_j \mbox{tr}\left(\hat{D}_i^\dagger \rho_j\right) = \sum_j r_j = \mbox{tr}(\rho) \quad \forall \rho\, , \end{gather} where we have used $\rho = \sum_j r_j \rho_j$. The only matrix $M$ that satisfies $\mbox{tr}(M\rho) = \mbox{tr}(\rho) \ \forall \rho$ is the identity matrix.
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\begin{document}
\title{A one-phase interior point method for nonconvex optimization} \author{Oliver Hinder, Yinyu Ye}
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\newcommand{Simplified-One-Phase-nonconvex-IPM}{Simplified-One-Phase-nonconvex-IPM} \newcommand{\Call{Simplified-One-Phase-nonconvex-IPM}}{\Call{Simplified-One-Phase-nonconvex-IPM}}
\newcommand{\emph{Perform a backtracking line search on the primal step $\alpha_{P}$.} Trial step sizes $\alpha_{P} \in \{\alpha^{\max}_{P}, \parBacktracking \alpha^{\max}_{P}, \parBacktracking^2 \alpha^{\max}_{P}, \dots \}$ computing the trial point $(\mu^{+}, x^{+}, s^{+}, y^{+})$ as described in \eqref{eq:primal-iterate-update} and \eqref{eq:alpha-D}. Terminate with $\status = \success$ and return the trial point the first time each of the following conditions hold:}{\emph{Perform a backtracking line search on the primal step $\alpha_{P}$.} Trial step sizes $\alpha_{P} \in \{\alpha^{\max}_{P}, \beta_{6} \alpha^{\max}_{P}, \beta_{6}^2 \alpha^{\max}_{P}, \dots \}$ computing the trial point $(\mu^{+}, x^{+}, s^{+}, y^{+})$ as described in \eqref{eq:primal-iterate-update} and \eqref{eq:alpha-D}. Terminate with $\textbf{status} = \textsc{success}$ and return the trial point the first time each of the following conditions hold:}
\begin{abstract} The work of \citet*{wachter2000failure} suggests that infeasible-start interior point methods (IPMs) developed for linear programming cannot be adapted to nonlinear optimization without significant modification, i.e., using a two-phase or penalty method. We propose an IPM that, by careful initialization and updates of the slack variables, is guaranteed to find a first-order certificate of local infeasibility, local optimality or unboundedness of the (shifted) feasible region. Our proposed algorithm differs from other IPM methods for nonconvex programming because we reduce primal feasibility at the same rate as the barrier parameter. This gives an algorithm with more robust convergence properties and closely resembles successful algorithms from linear programming. We implement the algorithm and compare with IPOPT on a subset of CUTEst problems. Our algorithm requires a similar median number of iterations, but fails on only 9\% of the problems compared with 16\% for IPOPT. Experiments on infeasible variants of the CUTEst problems indicate superior performance for detecting infeasibility.
The code for our implementation can be found at \url{https://github.com/ohinder/OnePhase}. \end{abstract}
\section{Introduction}
Consider the problem \begin{subequations}\label{original-problem} \begin{flalign} \min_{x \in R^{n}}{f(x)} \\ a(x) \le 0, \end{flalign} \end{subequations} where the functions $a : R^{n} \rightarrow R^{m}$ and $f : R^{n} \rightarrow R$ are twice differentiable and might be nonconvex. Examples of real-world problems in this framework include truss design, robot control, aircraft control, and aircraft design, e.g., the problems TRO11X3, ROBOT, AIRCRAFTA, AVION2 in the CUTEst test set \cite{gould2015cutest}. This paper develops an interior point method (IPM) for finding KKT points of \eqref{original-problem}, i.e., points such that \begin{subequations}\label{kkt-of-original-problem} \begin{flalign} \grad f(x) + \grad a(x)^T y &= 0 \\ y^T a(x) &= 0 \\ a(x) &\le 0, \quad y \ge 0. \end{flalign} \end{subequations} IPMs were first developed by \citet*{karmarkar1984new} for linear programming. The idea for primal-dual IPMs originates with \citet*{megiddo1989pathways}. Initially, algorithms that required a feasible starting point were studied \cite{kojima1989primal,monteiro1989interior}. However, generally one is not given an initial point that is feasible. A naive solution to this issue is to move the constraints into the objective by adding a large penalty for constraint violation (Big-M method) \cite{mcshane1989implementation}. A method to avoid the penalty approach, with a strong theoretical foundation for linear programming, is the homogeneous algorithm \cite{andersen1998computational,andersen1999homogeneous,ye1994nl}. This algorithm measures progress in terms of the KKT error, which may not monotonically decrease in the presence of nonconvexity\footnote{This occurs even in the one-dimensional unconstrained case, e.g., consider minimizing $f(x)=-9 x - 3 x^2 + x^4/4$ starting from zero; the gradient norm increases then decreases.}. It is therefore difficult to generalize the homogeneous algorithm to nonconvex optimization. An alternative to the homogeneous algorithm is the infeasible-start algorithm of Lustig \cite{lustig1990feasibility}, which has fewer numerical issues and a smaller iteration count than the big-M method of \cite{mcshane1989implementation}.
Lustig's approach was further improved in the predictor-corrector algorithm of \citet*{mehrotra1992implementation}. This algorithm reduced complementarity, duality and primal feasibility at the same rate, using an adaptive heuristic. This class of methods was shown by \citet*{todd2003detecting} to converge to optimality or infeasibility certificates (of the primal or dual).
The infeasible-start method for linear programming of \citet{lustig1990feasibility} naturally extends to nonlinear optimization \cite{kortanek1997infeasible}, and most interior point codes for nonconvex optimization are built upon these ideas \cite{byrd2006knitro,vanderbei1999loqo,wachter2006implementation}. However, \citet*{wachter2000failure} showed that for the problem \begin{subequations}\label{failure-ex} \begin{flalign} \min { x }\\ x^2 - s_{1} &= -1 \\ x - s_{2} &= 1 \\ s_1, s_2 &\ge 0 \end{flalign} \end{subequations} a large class of infeasible-start algorithms fail to converge to a local optimum or an infeasibility certificate starting at any point with $x < 0$, $s_{1} > 0$ and $s_{2} > 0$. Following that paper, a flurry of research was published suggesting different methods for resolving this issue \cite{benson2004interior,chen2006interior,curtis2012penalty,gould2015interior,liu2004robust,wachter2006implementation}. The two main approaches can be split into penalty methods \cite{chen2006interior,curtis2012penalty,gould2015interior,liu2004robust} and two-phase algorithms \cite{wachter2006implementation}.
Penalty methods move some measure of constraint violation into the objective. These methods require a penalty parameter that measures how much the constraint violation contributes to the objective. Penalty methods will converge only if the penalty parameter is sufficiently large. However, estimating this value is difficult: too small and the algorithm will not find a feasible solution; too big and the algorithm might be slow and suffer from numerical issues. Consequently, penalty methods tend to be slow \cite[Algorithm 1]{curtis2012penalty} or use complex schemes for dynamically updating the penalty parameter \cite[Algorithm 2]{curtis2012penalty}.
The algorithm IPOPT is an example of a two-phase algorithm: it has a main phase and a feasibility restoration phase \cite{wachter2006implementation}. The main phase searches simultaneously for optimality and feasibility using a classical infeasible-start method. The feasibility restoration phase aims to minimize primal infeasibility. It is called only when the main phase fails, e.g., the step size is small. It is well known that this approach has drawbacks. The algorithm has difficulties detecting infeasibility \cite[Table 15]{huang2016solution} and will fail if the feasibility restoration phase is called too close to the optimal solution. Some of these issues have been addressed by \citet*{nocedal2014interior}.
Our main contribution is an infeasible-start interior point method for nonlinear programming that builds on the work for linear programming of \citet{lustig1990feasibility}, \citet{mehrotra1992implementation}, and \citet*{mizuno1993adaptive}. The algorithm avoids a big-M or a two-phase approach. Furthermore, our solution to the issue posed in example \eqref{failure-ex} is simple: we carefully initialize the slack variables and use nonlinear updates to carefully control the primal residual. Consequently, under general conditions we guarantee that our algorithm will converge to a local certificate of optimality, local infeasibility or unboundedness. Our algorithm has other desirable properties. Complementarity moves at the same rate as primal feasibility. This implies from the work of \citet*{haeser2017behavior} that if certain sufficient conditions for local optimality conditions hold, our approach guarantees that the dual multipliers sequence will remain bounded. In contrast, in methods that reduce the primal feasibility too quickly, such as IPOPT, the dual multiplier sequence can be unbounded even for linear programs. We compare our solver with IPOPT on a subset of CUTEst problems. Our algorithm has a similar median number of iterations to IPOPT, but fails less often. Experiments on infeasible variants of the CUTEst problems indicate superior performance of our algorithm for detecting infeasibility.
The paper is structured as follows. Section~\ref{sec:simple-alg} describes a simple version of our proposed one-phase interior point algorithm and gives intuitive explanations for our choices. Section~\ref{sec:theory} focuses on convergence proofs. In particular, Section~\ref{sec:infeas-criteron-justify} justifies the choice of infeasibility criterion, and Section~\ref{sec:global-conv} provides convergence proofs for the algorithm described in Section~\ref{sec:simple-alg}. Section~\ref{sec:practical-alg} gives a practical version of the one-phase algorithm. Section~\ref{sec:numerical-results} presents numerical results on the CUTEst test set.
\paragraph{Notation} We use the variables $x$, $s$ and $y$ to denote the primal, slack and dual variables produced by the algorithm. The diagonal matrices $S$ and $Y$ are formed from the vectors $s$ and $y$ respectively. Given two vectors $u$ and $v$, $\min\{ u, v \}$ is vector corresponding to the element-wise minimum. The norm $\norm{ \cdot }$ denotes the Euclidean norm. The letter $\vec{e}$ represents the vector of all ones. The set $R_{++}$ denotes the set of strictly positive real numbers.
\section{A simple one-phase algorithm}\label{sec:simple-alg}
Consider the naive log barrier subproblems of the form \begin{flalign} \label{naive-log-barrier} \min_{x \in R^{n}}{f(x) - \mu \sum_i{ \log{(-a_i(x))} } }. \end{flalign} The idea is to solve a sequence of such subproblems with $\mu \rightarrow 0$ and $\mu > 0$. The log barrier transforms the non-differentiable original problem~\eqref{original-problem} into a twice differentiable function on which we can apply Newton's method. However, there are issues with this naive formulation: we are rarely given a feasible starting point and one would like to ensure that the primal variables remain bounded. To resolve these issues we consider shifted and slightly modified subproblems of the form
\begin{flalign*} \min_{x \in R^{n}} \psi_{\mu}(x) := f(x) - \mu \sum_i{ \left( \beta_{1} a_i(x) + \log \left( \mu \vec{w}_i - a_i(x) \right) \right) }, \end{flalign*} where $\beta_{1} \in (0,1)$ is a constant with default value $10^{-4}$, $\vec{w} \ge 0$ is a vector that remains fixed for all subproblems, and some $\mu > 0$ measures the size of the shift. The purpose of the term $\beta_{1} a_i(x)$ is to ensure that $-(\beta_{1} a_i(x) + \log \left( \mu \vec{w}_i - a_i(x) \right) )$ remains bounded below. This prevents the primal iterates from unnecessarily diverging. We remark that this modification of the log barrier function is similar to previous works \cite[Section 3.7]{wachter2006implementation}.
Holistically, our technique consists of computing two types of direction: stabilization and aggressive directions. Both directions are computed from the same linear system with different right-hand sides. Aggressive directions are equivalent to affine scaling steps \cite{mehrotra1992implementation} as they apply a Newton step directly to the KKT system, ignoring the barrier parameter $\mu$. Aggressive steps aim to approach optimality and feasibility simultaneously. However, continuously taking aggressive steps may cause the algorithm to stall or fail to converge. To remedy this we have a stabilization step that keeps the primal feasibility the same, i.e., aims to reduce the log barrier objective until an approximate solution to the shifted log barrier problem is found. While this step has similar goals to the centering step of Mehrotra, there are distinct differences. The centering steps of Mehrotra move the iterates towards the central path while keeping the primal and dual feasibility fixed. Our stabilization steps only keep the primal feasibility fixed while reducing the log barrier objective. This technique of alternating stabilization and aggressive steps, is analogous to the alternating predictor-corrector techniques of \citet*[Algorithm~1]{mizuno1993adaptive}.
The IPM that we develop generates a sequence of iterates $(x^{k},s^k, y^k, \mu^k)$ that satisfy \begin{subequations}\label{eq:barrier-primal-sequence-nice} \begin{flalign} (x^{k},s^k, y^k, \mu^k) &\in R^{n} \times R_{++}^{m} \times R_{++}^{m} \times R_{++} \label{eq-domain} \\ \frac{s^{k}_i y^{k}_i}{\mu^{k}} &\in [ \beta_{2}, 1 / \beta_{2}] ~~ \forall i \in \{ 1, \dots, m \} \label{eq:comp-slack} \\ a(x^{k}) + s^{k} &= \mu^k \vec{w}, \label{eq:primal-feasibility} \end{flalign} \end{subequations} where $\vec{w} \ge 0$ is a vector for which $a(x^{0}) + s^{0} = \mu^0 \vec{w}$, and $\beta_{2} \in (0,1)$ is an algorithmic parameter with default value $0.01$. This set of equations implies the primal feasibility and complementarity are moved at the same rate.
Furthermore, there is a subsequence of the iterates $\pi_{k}$ (i.e., those that satisfy the aggressive step criterion \eqref{agg-criteron}) such that
\begin{flalign}
\frac{\| \grad_{x} \mathcal{L}_{\mu^{\pi_{k}}}(x^{\pi_{k}}, y^{\pi_{k}}) \|_{\infty}}{\mu^{\pi_{k}}(\| y^{\pi_{k}} \|_{\infty} + 1)} &\le c, \label{eq:dual-feas} \end{flalign} where $c > 0$ is some constant and $\mathcal{L}_{\mu} (x, y) := f(x) + (y - \mu \beta_{1} e)^T a(x)$ is the modified Lagrangian function. Requiring \eqref{eq:barrier-primal-sequence-nice} and \eqref{eq:dual-feas} is common in practical linear programming implementations \cite{mehrotra1992implementation}. Note that \eqref{eq:barrier-primal-sequence-nice} and \eqref{eq:dual-feas} can be interpreted as a `central sequence'. This is weaker than the existence of a central path, a concept from convex optimization \cite{andersen1999homogeneous,megiddo1989pathways}. Unfortunately, in nonconvex optimization there may not exist a continuous central path.
Conditions \eqref{eq:barrier-primal-sequence-nice} and \eqref{eq:dual-feas} are desirable because they imply the dual multipliers are likely to be well-behaved. To be more precise, assume the subsequence satisfying \eqref{eq:barrier-primal-sequence-nice} and \eqref{eq:dual-feas} is converging to a feasible solution. If this solution satisfies certain sufficiency conditions for local optimality, then the dual variables remain bounded and strict complementarity holds. We refer to our paper \cite{haeser2017behavior} for further understanding of this issue. A consequence of this property is that we can re-write equality constraints as two inequalities while avoiding numerical issues that might arise if we did this using other solvers \cite{haeser2017behavior}.
Related to this property of the dual multipliers being well-behaved is that our algorithm is explicitly designed for inequality constraints only. Often primal feasibility is written as $a(x)=0$, $x \ge 0$ as used by Mehrotra (and many others in the IPM literature). If we took the dual of a linear program before applying our technique then our method would be working with the same problem as the typical method, because for us dual feasibility is $\grad f(x) + \grad a(x)^T y = 0, y \ge 0$. We believe working in this form is superior for nonlinear programming where there is no symmetry between the primal and dual, and the form $a(x) \le 0$ has many benefits that we discuss shortly. Using inequalities has the following advantages: \begin{enumerate} \item It enables us to generate sequences satisfying \eqref{eq:barrier-primal-sequence-nice} and \eqref{eq:dual-feas}. \item It allows us to use a Cholesky factorization instead of the LBL{} factorization of \citet*{bunch1971direct}. See end of Section~\ref{sub:direction-computation} for further discussion. \item We avoid the need for a second inertia modification parameter to ensure nonsingularity of the linear system, i.e., $\delta_{c}$ in equation~(13) in \cite{wachter2006implementation}. Not using $\delta_{c}$ removes issues where large modifications to the linear system may not provide a descent direction for the constraint violation. \end{enumerate} Other IPMs that use only inequalities include \cite{curtis2012penalty,vanderbei1999loqo}.
\subsection{Direction computation}\label{sub:direction-computation}
We now state how we compute directions, whose derivation is deferred to Section~\ref{sec:dir-derivation}.
Let \begin{flalign} b &= \begin{bmatrix} b_{D} \\ b_{P} \\ b_{C} \end{bmatrix} = \begin{bmatrix}
\grad_{x} \mathcal{L}_{\gamma \mu}(x,y) \\ (1- \gamma) \mu \vec{w} \\ Y s - \gamma \mu \vec{e} \end{bmatrix} \label{def:b} \end{flalign} be the target change in the KKT residual error, where $\grad_{x} \mathcal{L}_{\gamma \mu}(x,y)$ denotes $\grad_{x} \mathcal{L}_{\bar{\mu}}(x,y)$ with $\bar{\mu} = \gamma \mu$. The scalar $\gamma \in [0,1]$ represents the target reduction in constraint violation and barrier parameter $\mu$, with $\gamma = 1$ corresponding to stabilization steps and $\gamma < 1$ to aggressive steps. (For the simple one-phase algorithm, $\gamma = 0$ for the aggressive steps). The point $(\mu, x, s, y)$ denotes the current iterate.
To compute the direction for the $x$ variables we solve \begin{flalign}\label{eq:Schur-complement-system} (\mathcal{M} + \delta I) \dir{x} = -\left( b_{D} + \grad a(x)^T S^{-1} \left( Y b_{P} - b_{C} \right) \right), \end{flalign} where $\delta > 0$ is chosen such that $\mathcal{M} + \delta I$ is positive definite (the choice of $\delta$ is specified in Section~\ref{sec:simple-alg}) and \begin{flalign}\label{eq:Schur-matrix} \mathcal{M} = \grad_{xx}^2 \mathcal{L}_{\mu} (x, y) + \grad a(x)^T Y S^{-1} \grad a(x). \end{flalign} We factorize $\mathcal{M} + \delta I$ using Cholesky decomposition. The directions for the dual and slack variables are then \begin{subequations}\label{compute-ds-dy} \begin{flalign} \dir{y} &\gets -S^{-1} Y (\grad a(x) \dir{x} + b_{P} - Y^{-1} b_{C}), \label{compute-dy} \\ \dir{s} &\gets -(1 - \gamma) \mu \vec{w} - \grad a(x) \dir{x}. \label{compute-ds} \end{flalign} \end{subequations} We remark that the direction $\dir{s}$ is not used for updating the iterates because $s$ is updated using a nonlinear update (Section~\ref{sec:update-iterates}), but we define it for completeness.
\subsubsection{Derivation of direction choice}\label{sec:dir-derivation} We now explain how we choose our directions \eqref{eq:Schur-complement-system} and \eqref{compute-ds-dy}.
In our algorithm the direction $\dir{x}$ for the $x$ variables is computed with the goal of being approximately equal to $\dir{x}^{*}$ defined by \begin{flalign}\label{sophisticated-barrier-problem}
\dir{x}^{*} \in \arg \min_{\bar{\vec{d}}_{x} \in R^{n}} & \psi_{\gamma \mu}(x + \bar{\vec{d}}_{x}) + \frac{\delta}{2} \|\bar{\vec{d}}_{x} \|^2 \end{flalign} with $\psi_{\gamma \mu}$ denoting $\psi_{\bar{\mu}}$ with $\bar{\mu} = \gamma \mu$. This notation is used for subscripts throughout, i.e., $\mathcal{L}_{\gamma \mu}$ denotes $\mathcal{L}_{\bar{\mu}}$ with $\bar{\mu} = \gamma \mu$.
Primal IPMs \cite{fiacco1990nonlinear} apply Newton's method directly to system \eqref{sophisticated-barrier-problem}. However, they have inferior practical performance to primal-dual methods that apply Newton's method directly to the optimality conditions. To derive the primal-dual directions let us write the first-order optimality conditions \begin{flalign*} \grad_{x} \mathcal{L}_{\gamma \mu}(x + \dir{x}^{*}, y + \dir{y}^{*}) + \delta \dir{x}^{*} &= 0 \\ a(x + \dir{x}^{*}) + s + \dir{s}^{*} &= \gamma \mu \vec{w} \\ (S + \Dir{s}^{*}) (y + \dir{y}^{*}) &= \gamma \mu \vec{e} \\ s + \dir{s}^{*}, y + \dir{y}^{*} &\ge 0, \end{flalign*} where $(x,s,y)$ is the current values for the primal, slack and dual variables, $(x + \dir{x}^{*},y + \dir{y}^{*},s + \dir{s}^{*})$ is the optimal solution to \eqref{sophisticated-barrier-problem}, and $(\dir{x}^{*},\dir{y}^{*},\dir{s}^{*})$ are the corresponding directions ($\Dir{s}^{*}$ is a diagonal matrix with entries $\dir{s}^{*}$). Thus, \begin{flalign}\label{primal-dual-Newton-direction} \mathcal{K}_{\delta} \dir{}= -b, \end{flalign} where \begin{flalign} \mathcal{K}_{\delta} = \begin{bmatrix}
\grad_{xx}^2 \mathcal{L}_{\mu}(x, y) + \delta I & \grad a(x)^T & 0 \\ \grad a(x) & 0 & I \\ 0 & S & Y \end{bmatrix} \text{ and } d = \begin{bmatrix} \dir{x} \\ \dir{y} \\ \dir{s} \end{bmatrix}. \label{def:K-delta} \end{flalign} Eliminating $\dir{s}$ from \eqref{primal-dual-Newton-direction} yields the symmetric system \begin{flalign}\label{eq:ldl-system}
\begin{bmatrix}
\grad_{xx}^2 \mathcal{L}_{\mu}(x,y) + \delta I & \grad a(x)^T \\ \grad a(x) & -Y^{-1} S \\ \end{bmatrix} \begin{bmatrix} \dir{x} \\ \dir{y} \end{bmatrix} = -\begin{bmatrix} b_{D} \\ b_{P} - Y^{-1} b_{C} \end{bmatrix}. \end{flalign}
\noindent This is similar to the system typically factorized by nonlinear programming solvers using LBL{} \cite{andersen1998computational,byrd2006knitro,vanderbei1999loqo,wachter2006implementation}. If the matrix $\grad_{xx}^2 \mathcal{L}_{\mu}(x,y) + \delta I$ is positive definite the whole matrix is quasidefinite, and in this case one can perform an LDL{} factorization with a fixed pivot order \cite{gill1996stability,vanderbei1995symmetric}. However, if $\grad_{xx}^2 \mathcal{L}_{\mu}(x,y) + \delta I$ is not positive definite then LBL{} \cite{bunch1971direct} must be used to guarantee factorability and may require excessive numerical pivoting. One way of avoiding using an LBL{} factorization is to take the Schur complement of \eqref{eq:ldl-system}. For this system, there are two possible Schur complements. We use the term primal Schur complement to mean that the final system is in terms of the primal variables, whereas the dual Schur complement gives a system in the dual variables.
Taking the primal Schur complement gives system \eqref{eq:Schur-complement-system}: \begin{flalign*} (\mathcal{M} + \delta I) \dir{x} = -\left( b_{D} + \grad a(x)^T S^{-1} \left( Y b_{P} - b_{C} \right) \right). \end{flalign*} Equations \eqref{compute-ds-dy} also follow from \eqref{primal-dual-Newton-direction}.
Note that if $\mathcal{M} + \delta I$ is positive definite and $\gamma = 1$, then the right-hand side of \eqref{eq:Schur-complement-system} becomes $-\grad \psi_{\mu}(x)$; therefore $\dir{x}$ is a descent direction for the function $\psi_{\mu}(x)$. Consequently, we pick $\delta > 0$ such that $\mathcal{M} + \delta I$ is positive definite. Furthermore, note that if $Y = S^{-1} \mu$ then \eqref{eq:Schur-complement-system} reduces to $$ (\grad^2 \psi_{\mu}(x) + \delta I) \dir{x} = - \grad \psi_{\mu}(x); $$ hence $\mathcal{M}$ should be interpreted as a primal-dual approximation of the Hessian of $\psi_{\mu}$.
We emphasize we are forming the primal Schur complement, not the dual. This is a critical distinction for nonlinear programming because there are drawbacks to using the dual Schur complement. First, the matrix $\mathcal{M}$ could be positive definite but $\grad_{xx}^2 \mathcal{L}_{\mu}(x,y)$ could be negative definite, indefinite or even singular. Consequently, one might need to add an unnecessarily large $\delta$ to make $\grad_{xx}^2 \mathcal{L}_{\mu}(x,y) + \delta I$ positive definite and compute the direction. This could slow progress and prohibit superlinear convergence. Second, this method requires computing $(\grad_{xx}^2 \mathcal{L}_{\mu}(x,y) + \delta I)^{-1} \grad a(x)^T$, which is expensive if there is even a moderate number of constraints. Finally, if $\grad_{xx}^2 \mathcal{L}_{\mu}(x,y)$ is not a diagonal matrix, then, usually the dual schur complement will usually be dense (similar issues occur for reduced Hessian methods \cite{walterThesis1,walterThesis2}). In contrast, $\mathcal{M}$ is generally sparse. Furthermore, if $\mathcal{M}$ is not sparse it is likely that the Jacobian $\grad a(x)$ has a dense row. This, however, could be eliminated through row stretching of the original problem, as is done for columns in linear programming \cite{grcar2012matrix,lustig1991formulating,vanderbei1991splitting}.
\subsection{Updating the iterates}\label{sec:update-iterates}
Suppose that we have computed direction $(\dir{x}, \dir{s}, \dir{y})$ with some $\gamma \in [0,1]$ using \eqref{eq:Schur-complement-system} and \eqref{compute-ds-dy}. We wish to construct a candidate $(\mu^{+}, x^{+}, s^{+}, y^{+})$ for the next iterate. Given a primal step size $\alpha_{P} \in [0,1]$ and dual step size $\alpha_{D} \in [0,1]$ we update the iterates as follows: \begin{subequations}\label{eq:iterate-update} \begin{flalign} \mu^{+} &\gets (1 - (1 - \gamma) \alpha_{P}) \mu \label{eq:muVarUpdate} \\ \vec{x}^{+} &\gets \vec{x} + \alpha_{P} \dir{x} \label{eq:xVarUpdate} \\ \vec{s}^{+} &\gets \mu^{+} \vec{w} - a(\vec{x}^{+}) \label{eq:slackVarUpdate} \\ \vec{y}^{+} &\gets \vec{y} + \alpha_{D} \dir{y}. \label{eq:update-y} \end{flalign} \end{subequations} The slack variable update does not use $\dir{s}$. Instead, \eqref{eq:slackVarUpdate} is nonlinear and its purpose is to ensure that \eqref{eq:primal-feasibility} remains satisfied, so that we can control the rate of reduction of primal feasibility. In infeasible-start algorithms for linear programming \cite{lustig1990feasibility,mehrotra1992implementation} the variables updates are all linear, i.e., $\vec{s}^{+} \gets \vec{s} + \alpha_{P} \dir{s}$. However, if the function $a$ is linear, the slack variable update~\eqref{eq:slackVarUpdate} reduces to $$ s^{+} = \mu^{+} \vec{w} - a(x) - \alpha_{P} \grad a(x) \dir{x} = \left(\mu \vec{w} - a(x) \right) - \alpha_{P} \left( (1 - \gamma) \mu \vec{w} + \grad a(x) \dir{x} \right) = s + \alpha_{P} \dir{s} $$ where the first equality uses \eqref{eq:slackVarUpdate} and linearity of $a$, the second uses \eqref{eq:muVarUpdate}, and the final uses \eqref{compute-ds}. Furthermore, as $\dir{x} \rightarrow 0$ the linear approximation $a(x) + \alpha_{P} \grad a(x)$ of $a(x + \alpha_{P} \dir{x})$ becomes very accurate and we have $s^{+} \rightarrow s + \alpha_{P} \dir{s}$. Nonlinear updates for the slack variables have been used in other interior point methods \cite{andersen1998computational, curtis2012penalty}.
Finally, we only select steps that maintain \eqref{eq-domain} and \eqref{eq:comp-slack}, i.e., satisfy \begin{subequations}\label{eq:comp-slack-plus} \begin{flalign} s^{+}, y^{+}, \mu^{+} &> 0 \\ \frac{s^{+}_i y^{+}_i}{\mu^{+}} &\in [ \beta_{2}, 1 / \beta_{2}] ~~ \forall i \in \{ 1, \dots, m \}. \end{flalign} \end{subequations}
\subsection{Termination criterion}
Define the function $$
\sigma (y) := \frac{100}{\max\{ 100, \| y \|_{\infty} \}} $$ as a scaling factor based on the size of the dual variables. This scaling factor is related to $s_{d}$ and $s_{c}$ in the IPOPT implementation paper \cite{wachter2006implementation}. We use $\sigma(y)$ in the local optimality termination criterion \eqref{terminate-kkt} because there may be numerical issues reducing the unscaled dual feasibility if the dual multipliers become large. In particular, the first-order optimality termination criterion we use is \begin{subequations}\label{terminate-kkt} \begin{flalign}
\sigma (y) \| \grad \mathcal{L}_0(x, y) \|_{\infty} &\le \epsilon_{\textbf{opt}} \\
\sigma (y) \| S y \|_{\infty} &\le \epsilon_{\textbf{opt}} \\
\| a(x) + s \|_{\infty} &\le \epsilon_{\textbf{opt}}, \end{flalign} where $\epsilon_{\textbf{opt}} \in (0,1)$ with a default value of $10^{-6}$. \end{subequations} The first-order local primal infeasibility termination criterion is given by \begin{subequations}\label{terminate-primal-infeasible} \begin{flalign} a(x)^T y &> 0 \\ \Gamma_{\text{far}} (\mu,x,s,y) &\le \epsilon_{\textbf{far}} \\ \Gamma_{\text{inf}} (\mu,x,s,y) &\le \epsilon_{\textbf{inf}}, \end{flalign} \end{subequations} where \begin{flalign*}
\Gamma_{\text{far}} (x,y) &:= \frac{\| \grad a(x)^T y \|_{1}}{ a(x)^T y } \\
\Gamma_{\text{inf}} (x,s,y) &:= \frac{\| \grad a(x)^T y \|_{1} + s^T y}{ \| y \|_{1} }, \end{flalign*} and $\epsilon_{\textbf{far}}, \epsilon_{\textbf{inf}} \in (0,1)$ with default values of $10^{-3}$ and $10^{-6}$ respectively. We remark that if we find a point with $\Gamma_{\text{inf}} (x,s,y) = 0$ then we have found a stationary point to a weighted $L_{\infty}$ infeasibility measure. For a more thorough justification of this choice for the infeasibility termination criterion, see Section~\ref{sec:infeas-criteron-justify}.
The unboundedness termination criterion is given by \begin{flalign}\label{terminate-dual-infeasible}
\frac{1}{\| x \|_{\infty}}\ge 1/\epsilon_{\textbf{unbd}}, \end{flalign} where $\epsilon_{\textbf{unbd}} \in (0,1)$ with default value $10^{-12}$. Note that since we require all the iterates to maintain $a(x) \le \mu \vec{w} \le \mu^{0} \vec{w}$ satisfying, the unboundedness termination criterion strongly indicates that the set $\{ x \in R^{n} : a(x) \le \vec{w} \mu \}$ is unbounded.
\subsection{The algorithm}\label{sec:simple-alg-details}
Before we outline our algorithm, we need to define the switching condition for choosing an aggressive step instead of a stabilization step. The condition is
\begin{subequations}\label{agg-criteron} \begin{flalign}
\sigma (y) \| \grad \mathcal{L}_{\mu}(x, y) \|_{\infty} &\le \mu \label{agg-criteron-opt} \\
\| \grad \mathcal{L}_{\mu}(x, y) \|_{1} &\le \| \grad f(x) - \beta_{1} \mu e^T \grad a(x) \|_{1} + s^T y \label{agg-criteron-farkas} \\
\frac{s_i y_i}{\mu} &\in [ \beta_{3}, 1 / \beta_{3} ] ~~ \forall i \in \{ 1, \dots, m \}, \label{agg-criteron-buffer} \end{flalign} \end{subequations} where the parameter $\beta_{3} \in (\beta_{2}, 1)$ has a default value of $0.02$. The purpose of \eqref{agg-criteron-opt} is to ensure that we have approximately solved the shifted log barrier problem and guarantees that this subsequence of iterates satisfies \eqref{eq:dual-feas}. Equation~\eqref{agg-criteron-farkas} helps ensure (as we show in Section~\ref{sec:global-conv}) that if the dual variables are diverging rapidly then the infeasibility termination criterion is met. Finally, equation~\eqref{agg-criteron-buffer} with $\beta_{3} > \beta_{2}$ ensures we have a buffer such that we can still satisfy \eqref{eq:comp-slack} when we take an aggressive step.
Algorithm~\ref{simple-one-phase} formally outlines our one-phase interior point method. It does not include the details for the aggressive or stabilization steps that are given in Algorithm~\ref{alg:simple-agg-step} and \ref{alg:simple-stb-step} respectively. Since Algorithm~\ref{simple-one-phase} maintains $a(x) + s = \vec{w} \mu$ for each iterate, it requires the starting point to satisfy $$ a(x^{0}) + s^{0} = \vec{w} \mu^{0}, $$ with $\vec{w} \ge 0$ and $\mu^{0} > 0$. For any fixed $x^{0}$ one can always pick sufficiently large $\vec{w}$ and $\mu^{0}$ such that $\mu^{0} \vec{w} > a(x^{0})$, and setting $s^{0} \gets \mu^{0} \vec{w}- a(x^{0})$ meets our requirements.
\newcommand{\superlinear}[1]{}
\begin{algorithm}[H] \textbf{Input:} a initial point $x^{0}$, vector $\vec{w} \ge 0$, and variables $\mu^0, s^{0}, y^{0} > 0$ such that $a(x^{0}) + s^{0} = \vec{w} \mu^{0}$ and equation~\eqref{eq:comp-slack} with $k=0$ is satisfied. Termination tolerances $\epsilon_{\textbf{opt}} \in (0,\infty)$, $\epsilon_{\textbf{far}} \in (0,1)$, $\epsilon_{\textbf{inf}} \in (0,1)$ and $\epsilon_{\textbf{unbd}} \in (0,1)$. \\ \textbf{Output:} a point $(\mu^k, x^k, s^k, y^k)$ that satisfies at least one of the inequalities \eqref{terminate-kkt}, \eqref{terminate-primal-infeasible} or \eqref{terminate-dual-infeasible}{} \\
For $k \in \{1, \dots, \infty\}$ \begin{enumerate}[label*=A.{\arabic*}] \superlinear{ \item \emph{Increase time since last significant improvement.} Set $t \gets t + 1$.} \item \label{simple-first-step} Set $(\mu,x,s,y) \gets (\mu^{k-1},x^{k-1},s^{k-1},y^{k-1})$. \item \emph{Check termination criterion}. If \eqref{terminate-kkt}, \eqref{terminate-primal-infeasible} or \eqref{terminate-dual-infeasible}{} is satisfied then terminate the algorithm. \item \label{alg-simple-delta-min} Form the matrix $\mathcal{M}$ using \eqref{eq:Schur-matrix}. Set $\delta_{\min} \gets \max\{0, \mu-2 \lambda_{\min}(\mathcal{M}) \}$. \item If \eqref{agg-criteron}\superlinear{ or $t = 2$} is satisfied go to line~\ref{simple-agg-step}, \superlinear{$t = 1$ go to stable superlinear} otherwise go to line~\ref{simple-stb-step}.
\item\label{simple-agg-step} \emph{Take an aggressive step}. \begin{enumerate}[label*=.{\arabic*}] \item If $\delta_{\min} > 0$ go to line~\ref{simple-agg-large-delta}. \item Set $\delta \gets 0$. Run Algorithm~\ref{alg:simple-agg-step}. If it terminates with $\textbf{status} = \textsc{success}$, then set $(\mu^k,x^k,s^k,y^k) \gets (\mu^{+},x^{+},s^{+},y^{+})$ and go to line~\ref{simple-first-step}.
\superlinear{\item If $\norm{ \grad \mathcal{L}(x^{+},y^{+}) } \le \mu / 2$ and $\mu^{+} \le \mu/2$, then set $t \gets 0$.} \item \label{simple-agg-large-delta} Run Algorithm~\ref{alg:simple-agg-step} with sufficiently large $\delta$ such that the algorithm terminates with $\textbf{status} = \textsc{success}$. Set $(\mu^k,x^k,s^k,y^k) \gets (\mu^{+},x^{+},s^{+},y^{+})$ and go to line~\ref{simple-first-step}. \end{enumerate} \item \label{simple-stb-step} \emph{Take a stabilization step}. Set $\delta \gets \delta_{\min}$. Run Algorithm~\ref{alg:simple-stb-step}, set $(\mu^k,x^k,s^k,y^k) \gets (\mu^{+},x^{+},s^{+},y^{+})$. Go to line~\ref{simple-first-step}. \end{enumerate} \caption{A simplified one-phase algorithm}\label{simple-one-phase} \end{algorithm}
\subsubsection{Aggressive steps}\label{sec:simple-agg-step}
The goal of the aggressive steps is to approach optimality and feasibility simultaneously. For this reason, we set $\gamma = 0$ for the aggressive direction computation. We require a minimum step size $\alpha_{P}$ of
\begin{flalign}\label{simple-min-step-size-aggresssive} \theta(\mu, s) := \min\left\{ 1/2, \frac{(\beta_{3} - \beta_{2})}{2 \beta_{3} \mu } \min_{i \in \{ 1, ..., m \} : \vec{w}_i > 0}{\frac{s_i}{w_i}} \right\}. \end{flalign} If $\alpha_{P} < \theta(\mu, s)$, we declare the step a failure (see line~\ref{line:too-small-step} of Algorithm~\ref{alg:simple-agg-step}). However, on line~\ref{simple-agg-large-delta} of Algorithm~\ref{simple-one-phase}, we choose a sufficiently large $\delta$ such that $\alpha_{P}$ selected by Algorithm~\ref{alg:simple-agg-step} satisfies $\alpha_{P} \ge \theta(\mu, s)$; such a $\delta$ must exist by Lemma~\ref{lemma:agg-succeeds}. Furthermore, the primal aggressive step size $\alpha_{P}$ can be at most $ \min\{1/2, \mu \}$. This ensures line~\ref{simple-agg-select-alpha-P} is well-defined, i.e., the set of possible primal step sizes is closed.
\begin{algorithm}[H] \textbf{Input:} $\delta > 0$, the current point $(\mu, x, s, y)$ and the matrix $\mathcal{M}$. \\ \textbf{Output:} A new point $(\mu^{+}, x^{+}, s^{+}, y^{+})$ and a $\textbf{status}$. \begin{enumerate}[label*=A.{\arabic*}] \item Compute vector $b$ at point $(\mu, x, s, y)$ via \eqref{def:b} with $\gamma = 0$. \item Compute direction $(\dir{x}, \dir{s}, \dir{y})$ via \eqref{eq:Schur-complement-system} and \eqref{compute-ds-dy}. \item \label{simple-agg-select-alpha-P} Select the largest $\alpha_{P} \in \left[ 0, \min\{1/2, \mu \} \right]$ such that the iterates $(\mu^{+}, x^{+}, s^{+}, y^{+})$ computed by \eqref{eq:iterate-update} satisfy \eqref{eq:comp-slack-plus} for some $\alpha_{D} \in [0,1]$. \item With the previous $\alpha_{P}$ fixed, select the largest $\alpha_{D}$ such that the iterates $(\mu^{+}, x^{+}, s^{+}, y^{+})$ computed by \eqref{eq:iterate-update} satisfy \eqref{eq:comp-slack-plus}. \item \label{line:too-small-step} If $\alpha_{P} \ge \theta(\mu, s)$ then $\textbf{status} =\textsc{success}$; otherwise $\textbf{status} =\textsc{failure}$. \end{enumerate} \caption{Simplified aggressive step}\label{alg:simple-agg-step} \end{algorithm}
\subsubsection{Stabilization steps}\label{sec:simple-stable}
The goal of stabilization steps (Algorithm~\ref{alg:simple-stb-step}) is to reduce the log barrier function $\psi_{\mu}$. For this purpose we set $\gamma = 1$ during the direction computation. However, the log barrier function $\psi_{\mu}$ does not measure anything with respect to the dual iterates. This might impede performance if $\| S y - \mu \vec{e} \|_{\infty}$ is large but $\| \grad \psi_{\mu}(x) \|$ is small. In this case, taking a large step might reduce the complementarity significantly, even though the barrier function increases slightly. Therefore we add a complementarity measure to the barrier function to create an augmented log barrier function: \begin{flalign*}
\phi_{\mu}(x, s, y) := \psi_{\mu}(x) + \frac{\| S y - \mu \vec{e} \|_{\infty}^3}{\mu^2}. \end{flalign*} We say that the candidate iterates $(x^{+}, s^{+}, y^{+})$ have made sufficient progress on $\phi_{\mu}$ over the current iterate $(x, s, y)$ if \begin{flalign}\label{eq:phi-sufficient-progress}
\phi_{\mu}(x^{+}, s^{+}, y^{+}) \le \phi_{\mu}(x, s, y) + \alpha_{P} \beta_{4} \left( \frac{1}{2} \left( \grad \psi_{\mu}(x)^T \dir{x} - \frac{\delta}{2} \alpha_{P} \norm{ \dir{x}}^2 \right) - \frac{\| S y - \mu \vec{e} \|_{\infty}^3}{\mu^2} \right), \end{flalign} where $\beta_{4} \in (0,1)$ is a parameter with default value $0.2$.
\begin{algorithm}[H] \textbf{Input:} Some $\delta > 0$, the current point $(\mu, x, s, y)$ and the matrix $\mathcal{M}$. \\ \textbf{Output:} A new point $(\mu^{+}, x^{+}, s^{+}, y^{+})$. \begin{enumerate}[label*=A.{\arabic*}] \item Compute the vector $b$ at the point $(\mu, x, s, y)$ via \eqref{def:b} with $\gamma = 1$. \item Compute direction $(\dir{x}, \dir{s}, \dir{y})$ via \eqref{eq:Schur-complement-system} and \eqref{compute-ds-dy}. \item Pick the largest $\alpha_{P} = \alpha_{D} \in [0,1]$ such that $(\mu^{+},x^{+}, s^{+}, y^{+})$ computed by \eqref{eq:iterate-update} satisfies \eqref{eq:phi-sufficient-progress} and \eqref{eq:comp-slack-plus}.
\end{enumerate} \caption{Simplified stabilization step}\label{alg:simple-stb-step} \end{algorithm}
\section{Theoretical justification}\label{sec:theory}
The goal of this section is to provide some simple theoretical justification for our algorithm. Section~\ref{sec:infeas-criteron-justify} justifies the infeasibility termination criterion. Section~\ref{sec:global-conv} proves that the Algorithm~\ref{simple-one-phase} eventually terminates.
\subsection{Derivation of primal infeasibility termination criterion} \label{sec:infeas-criteron-justify}
Here we justify our choice of local infeasibility termination criterion by showing that it corresponds to a stationary measure for the infeasibility with respect to a weighted $L_{\infty}$ norm. We also prove that when the problem is convex our criterion certifies global infeasibility.
Consider the optimization problem \begin{subequations}\label{infeasible-problem} \begin{flalign} \min_{x} \max_{i : \vec{w}_i > 0}{ \frac{a_i(x)}{ w_i } } \\ \text{s.t. } a_i(x) \le 0, \forall i \text{ s.t. } \vec{w}_i = 0, \end{flalign} \end{subequations} for some non-negative vector $\vec{w}$. For example, a natural choice of $\vec{w}$ is $\vec{w}_i = 0$ for variable bounds and $\vec{w}_i = 1$ for all other constraints. This results in minimizing the $L_{\infty}$ norm of the constraint violation subject to variable bounds. Note that \eqref{infeasible-problem} is equivalent to the optimization problem \begin{subequations}\label{feas-problem} \begin{flalign} \min_{x,s,\mu} \mu \\ \text{s.t. } a(x) + s = \mu \vec{w} \\ s, \mu \ge 0. \end{flalign} \end{subequations}
The KKT conditions for \eqref{feas-problem} are \begin{flalign*} a(x) + s &= \mu \vec{w} \\ \grad a(x)^T \tilde{y} &= 0 \\ \vec{w}^T \tilde{y} + \tau &= 1 \\ \tau \mu &= 0 \\ \tilde{y}^T s &= 0 \\ s, \mu, \tilde{y}, \tau &\ge 0. \end{flalign*} Note that if the sequence $(\mu^k,x^k,s^k,y^k)$ generated by our algorithm satisfies \begin{flalign*} a(x^k) + s^k = \mu^k \vec{w} \\ \Gamma_{\text{inf}} (x^k,s^k,y^k) \rightarrow 0 \end{flalign*} then with $\tilde{y}^k = \frac{y^k}{\vec{w}^T y^k}$, $\tau^k = 0$, the KKT residual for problem~\eqref{infeasible-problem} of the sequence $(x^k, s^k, \tilde{y}^k, \tau^k)$ tends to zero. However, this is a poor measure of infeasibility because $\Gamma_{\text{inf}} (x^k,s^k,y^k) \rightarrow 0$ may also be satisfied if the algorithm is converging to a feasible solution (i.e., if the norm of the dual multipliers tends to infinity). For this reason, our infeasibility criterion~\eqref{terminate-primal-infeasible} includes $\Gamma_{\text{far}}$ to help avoid declaring a problem infeasible when the algorithm is in fact converging towards a feasible solution. To make this more precise, we include the following trivial observation.
\begin{observation}
Assume the constraint function $a : R^{n} \rightarrow R^{m}$ is differentiable and convex, and that some minimizer $(\mu^{*}, x^{*})$ of \eqref{feas-problem} satisfies $\| x - x^{*} \|_{\infty} \le R$ for some $R \in (0,\infty)$. Suppose also that at some point $(x,s,y)$ with $s,y \ge 0$, $\Gamma_{\text{far}}(x,s,y) \le 1 / (2R)$ and $y^T a(x) > 0$. Then the system $a(x) \le 0$ has no feasible solution. \end{observation} \begin{proof} We have \begin{flalign*} y^T a(x^{*}) &\ge y^T \left( a(x) + \grad a (x) (x^{*} - x) \right) \ge y^T a(x) (1 - R \times \Gamma_{\text{far}}(x,s,y) ) \ge y^T a(x) / 2 > 0, \end{flalign*}
where the first inequality holds via convexity and $y \ge 0$, the second by definition of $\Gamma_{\text{far}}(x,s,y) := \frac{\| \grad a(x)^T y \|_{1}}{ a(x)^T y }$, the third and fourth by assumption. Since $y \ge 0$, we deduce that $a_i(x^{*}) > 0$ for some $i$. \end{proof}
Finally, we remark that if wish to find a stationary point with respect to a different measure of constraint violation $v(z)$ we can apply our solver to the problem \begin{flalign*} \min f(z) \\ a(x) \le z \\ z \ge 0 \\ v(z) \le 0, \end{flalign*} starting with $\vec{w}_{k} = 0$ for $k = 1, \dots m + n$, $z^{0} = \max\{ a(x^0), e \}$, $\mu^0 > 0$ and $\vec{w}_{m + n+1} = \frac{1 + v(z^0)}{\mu^0} > 0$. With this choice of $\vec{w}$ one can see from \eqref{infeasible-problem} that if we do not find a feasible solution then we automatically find a solution to: \begin{flalign*} \min v(z)\\ a(x) \le z \\ z \ge 0. \end{flalign*} For example, if $v(z) = e^T z$ then we minimize the L1-norm of the constraint violation.
\subsection{Global convergence proofs for Algorithm~\ref{simple-one-phase}}\label{sec:global-conv}
We now give a global convergence proof for Algorithm~\ref{simple-one-phase} as stated in Theorem~\ref{thm:global-convergence} in Section~\ref{subsec:main-result}. Since the proofs are mostly mechanical, we defer most of the proofs to Appendix~\ref{app:global-conv}. Our results hold under assumption~\ref{assume:diff} and \ref{assume:parameters}. \begin{assumption}\label{assume:diff} Assume the functions $f : R^{n} \rightarrow R$ and $a : R^{n} \rightarrow R^{m}$ are twice differentiable on $R^{n}$. \end{assumption}
\begin{assumption}\label{assume:parameters} The algorithm parameters satisfy $\beta_{1} \in (0,1)$, $\beta_{2} \in (0,1)$ and $\beta_{3} \in (\parComp,1)$. The tolerances satisfy $\epsilon_{\textbf{opt}} \in (0,\infty)$, $\epsilon_{\textbf{far}} \in (0,1)$, $\epsilon_{\textbf{inf}} \in (0,1)$ and $\epsilon_{\textbf{unbd}} \in (0,1)$. The vector $\vec{w} \ge 0$. \end{assumption}
Recall that the iterates of our algorithm satisfy \eqref{eq:barrier-primal-sequence-nice}, i.e., \begin{subequations}\label{restate:eq:barrier-primal-sequence-nice} \begin{flalign} (x,s, y, \mu) &\in R^{n} \times R_{++}^{m} \times R_{++}^{m} \times R_{++} \label{restate:eq-domain} \\ \frac{s_i y_i}{\mu} &\in [ \beta_{2}, 1 / \beta_{2}] ~~ \forall i \in \{ 1, \dots, m \} \label{restate:eq:comp-slack} \\ a(x) + s &= \mu \vec{w}. \label{restate:eq:primal-feasibility} \end{flalign} \end{subequations}
\subsubsection{Convergence of aggressive steps} Here we show that after a finite number of aggressive steps, Algorithm~\ref{simple-one-phase} converges. Lemma~\ref{lemma:agg-succeeds} uses the fact that for large enough $\delta$ the slack variables can be absorbed to reduce the barrier parameter $\mu$.
First, we show that for sufficiently large $\delta$ Algorithm~\ref{alg:simple-agg-step} succeeds. Therefore line~\ref{simple-agg-large-delta} of Algorithm~\ref{simple-one-phase} is well-defined.
\begin{restatable}{lemma}{lemAggSucceeds}\label{lemma:agg-succeeds} Suppose assumptions~\ref{assume:diff} and \ref{assume:parameters} hold. If $(\mu, x, s, y)$ satisfies \eqref{restate:eq:barrier-primal-sequence-nice} and the criterion for an aggressive step \eqref{agg-criteron}, then there exists some $\bar{\delta}$ such that for all $\delta > \bar{\delta}$ Algorithm~\ref{alg:simple-agg-step} returns $\textbf{status} = \textsc{success}$. \end{restatable}
The proof of Lemma~\ref{lemma:agg-succeeds} is given in Section~\ref{sec:lemma:agg-succeeds}. The next Lemma demonstrates that the term $y^T \vec{w}$ remains bounded for points generated by our algorithm satisfying the aggressive step criterion.
\begin{restatable}{lemma}{lemYWbounded}\label{lem:yw-bounded} Suppose assumptions~\ref{assume:diff} and \ref{assume:parameters} hold. Then $\vec{w}^T y$ is bounded above for all points $(\mu, x, s, y)$ that satisfy \eqref{eq:barrier-primal-sequence-nice} and the following. \begin{enumerate} \item The criterion for an aggressive step \eqref{agg-criteron}. \item Neither the infeasibility termination criterion~\eqref{terminate-primal-infeasible} nor the unboundness criterion~\eqref{terminate-dual-infeasible}. \end{enumerate} \end{restatable}
The proof of Lemma~\ref{lem:yw-bounded} is given in Section~\ref{sub:lem:yw-bounded}. Lemma~\ref{lem:yw-bounded} shows $\vec{w}^T y$ is bounded above. It follows that the slack variables $s_i$ for $\vec{w}_i > 0$ are bounded away from zero. This enables us to lower bound the minimum aggressive step size $\theta(\mu, s)$, leading to Corollary~\ref{coro:agg-finite}.
\begin{corollary}\label{coro:agg-finite} Suppose assumptions~\ref{assume:diff} and \ref{assume:parameters} hold. After a finite number of calls to Algorithm~\ref{alg:simple-agg-step}, starting from a point that satisfies \eqref{restate:eq:barrier-primal-sequence-nice}, Algorithm~\ref{simple-one-phase} will terminate. \end{corollary}
\begin{proof} For any successful step size $\alpha_{P}$ by \eqref{simple-min-step-size-aggresssive} for an aggressive step and $\frac{s_i y_i}{\mu} \in [\beta_{2}, 1 / \beta_{2}]$ we have \begin{flalign*} \alpha_{P} \ge \theta(\mu, s) &= \min \left\{ 1/2, \frac{(\beta_{3} - \beta_{2})}{2 \beta_{3} \mu } \min_{i \in \{ 1, ..., m \} : \vec{w}_i > 0}{\frac{s_i}{w_i}} \right\} \\ &\ge \min\left\{ 1/2, \frac{(\beta_{3} - \beta_{2})}{2 \beta_{3}^2 } \min_{i \in \{ 1, ..., m \} : \vec{w}_i > 0}{\frac{1}{y_i w_i}} \right\}. \end{flalign*} Since Lemma~\ref{lem:yw-bounded} bounds $y^T \vec{w}$ from above and $y, \vec{w} \ge 0$ we deduce $\alpha_{P}$ is bounded away from zero. We reduce $\mu$ by at least $\alpha_{P} \mu$ each call to Algorithm~\ref{alg:aggressive} that terminates with $\textbf{status} = \textsc{success}$. Furthermore, for sufficiently small $\mu$, whenever \eqref{agg-criteron} holds the optimality criterion \eqref{terminate-kkt} is satisfied. Combining these facts proves the Lemma. \end{proof}
However, Corollary~\ref{coro:agg-finite} does not rule out the possibility that there is an infinite number of consecutive stabilization steps. Ruling out this possibility is the purpose of Lemma~\ref{lemConsecutiveStable}.
\subsubsection{Convergence of stabilization steps}\label{conv:stb}
This subsection is devoted to showing that consecutive stabilization steps eventually satisfy the criterion for an aggressive step or the unboundedness criterion is satisfied.
\begin{restatable}{lemma}{lemConsecutiveStable}\label{lemConsecutiveStable} Suppose assumptions~\ref{assume:diff} and \ref{assume:parameters} hold. After a finite number of consecutive stabilization steps within Algorithm~\ref{simple-one-phase}, either the aggressive criterion~\eqref{agg-criteron} or the unboundedness termination criterion~\eqref{terminate-dual-infeasible} is met. \end{restatable}
The proof of Lemma~\ref{lemConsecutiveStable} is given in Section~\ref{sec:lemConsecutiveStable}. Let us sketch the main ideas. First, we can show that the set of iterates that stabilization steps generate remain in a compact set. We can use this to uniformly bound quantities such as Lipschitz constants. The crux of the proof is showing there cannot be an infinite number of consecutive stabilizations steps because this will imply the augmented log barrier merit function tends to negative infinity and the unboundedness termination criterion holds.
\subsubsection{Main result}\label{subsec:main-result}
We now state Theorem~\ref{thm:global-convergence}, the main theoretical result of the paper.
\begin{theorem}\label{thm:global-convergence} Suppose assumptions~\ref{assume:diff} and \ref{assume:parameters} hold. Algorithm~\ref{simple-one-phase} terminates after a finite number of iterations. \end{theorem}
\begin{proof} Corollary~\ref{coro:agg-finite} shows that the algorithm must terminate after a finite number of aggressive steps. Lemma~\ref{lemConsecutiveStable} shows that the algorithm terminates or an aggressive step must be taken after a finite number of stabilization steps. The result follows. \end{proof}
\section{A practical one-phase algorithm}\label{sec:practical-alg}
Section~\ref{sec:simple-alg} presented a simple one-phase algorithm that is guaranteed to terminate (eventually) with a certificate of unboundedness, infeasibility, or optimality. However, to simplify, we omitted several practical details including the following. \begin{enumerate} \item The introduction of inner iterations that reuse the factorizations of $\mathcal{M}$. This reduces the total number of factorizations (Section~\ref{schur-reuse}). \item How to choose the step sizes $\alpha_{P}$ and $\alpha_{D}$ in a practical manner (Section~\ref{step-acceptance}). \item Using a filter to encourage steps that significantly reduce the KKT error (Section~\ref{sec:filter}). \item How to compute $\delta$ in a practical manner (Section~\ref{sec:practical-alg-outline} and Appendix~\ref{sec:mat-fact}). \item How choose the initial variable values (Appendix~\ref{sec:initialization}). \end{enumerate} The full algorithm is described in Section~\ref{sec:practical-alg-outline} with appropriately modified stabilization and aggressive steps. For complete details see the implementation at \url{https://github.com/ohinder/OnePhase}.
\subsection{Reusing the factorization of $\mathcal{M}$}\label{schur-reuse}
Our practical algorithm (Algorithm~\ref{practical-one-phase-IPM}) consists of inner iterations and outer iterations. The inner iterations reuse the factorization of the matrix $\mathcal{M}$ to compute directions as follows. Let $(\mu, x, s, y)$ be the current iterate and $(\hat{\mu}, \hat{x}, \hat{s}, \hat{y})$ be the iterate at the beginning of the outer iteration, i.e., where $\mathcal{M}$ was evaluated. Then the directions are computed during each inner iteration as follows: \begin{subequations}\label{practical-direction} \begin{flalign} (\mathcal{M} + \delta I) \dir{x} &= -\left( b_{D} + \grad a(\hat{x})^T \hat{S}^{-1} \left( Y b_{P} - b_{C} \right) \right) \label{practical-dx} \\ \dir{s} &\gets -(1 - \gamma) \mu \vec{w} - \grad a(x) \dir{x} \label{practical-compute-ds} \\ \dir{y} &\gets -\hat{S}^{-1} \hat{Y} (\grad a(\hat{x}) \dir{x} + b_{P} - \hat{Y}^{-1} b_{C}) \label{practical-compute-ds} . \end{flalign} \end{subequations} This is identical to the direction computation described in Section~\ref{sub:direction-computation} if $(\mu, x, s, y) = (\hat{\mu}, \hat{x}, \hat{s}, \hat{y})$.
\subsection{Step size choices and acceptance}\label{step-acceptance} First we specify a criterion to prevent the slack variables from getting too close to the boundary. In particular, given any candidate primal iterate $(x^{+}$, $s^{+})$ we require that the following fraction-to-boundary{} rule be satisfied: \begin{flalign}\label{fracBoundary-primal}
s^{+} \ge \Theta^{b} \min\{ s, \| d_{x} \|_{\infty} \left( \delta + \| d_{y} \|_{\infty} + \| d_{x} \|_{\infty}^{\beta_{7}} \right) \vec{e} \}, \end{flalign}
where $\Theta^{b}$ is a diagonal matrix with entries $\Theta^{b}_{i,i} \in (0,1)$ with default entry values of $0.1$ and $\beta_{7} \in (0,1)$ with default value of $0.5$. Note that the $\| d_{x} \|_{\infty} \left( \delta + \| d_{y} \|_{\infty} + \| d_{x} \|_{\infty}^{\beta_{7}} \right)$ term plays a similar role to $\mu$ in the more typical $s^{+} \ge \Theta^{b} s \min\{ 1 , \mu \}$ fraction-to-boundary{} rule (say of IPOPT \cite{wachter2006implementation}) in that it allows (almost) unit steps in the superlinear convergence regime.
In both the aggressive steps and stabilization steps we use a backtracking line search. We choose the initial trial primal step size $\alpha_{P}^{\max}$ to be the maximum $\alpha_{P} \in [0,1]$ that satisfies the fraction-to-boundary{} rule: \begin{flalign}\label{fracBoundaryPrimalMax}
s + \alpha_{P} \dir{s} &\ge \Theta^{p} \min\{ s, \| d_{x} \|_{\infty} \left( \delta + \| d_{y} \|_{\infty} + \| d_{x} \|_{\infty}^{\beta_{7}} \right) \vec{e} \}, \end{flalign}
where the parameter $\Theta^{p}$ is a diagonal matrix with entries $\Theta^{p}_{i,i} \in [\parFracBoundary_{i,i}, 1)$. The default value of $\Theta^{p}_{i,i}$ is $0.25$ for nonlinear constraints and $0.1$ for linear constraints. The idea of this choice for $\alpha_{P}^{\max}$ is that the fraction-to-boundary{} rule \eqref{fracBoundary-primal} is likely to be satisfied for the first trial point, i.e., $\alpha_{P} = \alpha_{P}^{\max}$ when $\| d_{x} \|$ is small. To see this, note that by differentiability of $a$ we have $\| s + d_{s} - s^{+} \| \le \| a(x) + \grad a(x) \dir{x} - a(x + \dir{x}) \| = O(\| \dir{x} \|^2)$. Furthermore, the right hand side of \eqref{fracBoundary-primal} and \eqref{fracBoundaryPrimalMax} are identical except for $\Theta^{b}$ and $\Theta^{p}$. Hence if $\dir{x} \rightarrow 0$, \eqref{fracBoundaryPrimalMax} holds and $\Theta^{p}_{i,i} > \Theta^{b}_{i,i}$ for $i$ corresponding to nonlinear constraints then \eqref{fracBoundary-primal} is satisfied in the limit for $\alpha_{P} = \alpha_{P}^{\max}$.
It remains to describe how to update the dual variables. Given some candidate primal iterate $(x^{+}$, $s^{+})$, let $B( s^{+}, \dir{y} )$ be the set of feasible dual step sizes. More precisely, we define $B( s^{+}, \dir{y} ) \subseteq [0,1]$ to be the largest interval such that if $\alpha_{D} \in B( s^{+}, \dir{y} )$ then \begin{subequations}\label{subeq:set-B} \begin{flalign}
\frac{s^{+}_i (y + \alpha_{D} \dir{y})_i}{\mu^{+}} &\in [\beta_{2}, 1/\beta_{2} ] ~~ \forall i \in \{ 1, \dots, m \} \label{satisfy-comp} \\
y + \alpha_{D} \dir{y} &\ge \Theta^{b} y \min\{ 1 , \| \dir{x} \|_{\infty} \}. \label{fracBoundary-dual} \end{flalign} \end{subequations} If there is no value of $\alpha_{D}$ satisfying \eqref{subeq:set-B} we set $B( s^{+}, \dir{y} )$ to the empty set and the step will be rejected. Recall the parameter $\beta_{2} \in (0,1)$ was defined in \eqref{eq:comp-slack}. The purpose criteria \eqref{satisfy-comp} is to ensure the algorithm always satisfy \eqref{eq:comp-slack}, i.e., $\frac{S y}{\mu} \in [\vec{e} \beta_{2}, e/\beta_{2} ]$. Equation~\eqref{fracBoundary-dual} is a fraction-to-boundary{} condition for the dual variables.
We compute the dual step size as follows: \begin{subequations}\label{eq:compute-alpha-D} \begin{flalign}
\alpha_{D} &\gets \arg \min_{\zeta \in B( s^{+}, \dir{y} )} \| S^{+} y - \mu^{+} + \zeta S^{+} \dir{y} \|^2_{2} + \| \grad f(x^{+}) + \grad a(x^{+})^T (y + \zeta \dir{y}) \|^{2}_{2} \label{eq:alphaD-least-squares} \\
\alpha_{D} &\gets \min\{ \max\{ \alpha_{D}, \alpha_{P} \}, \max B( s^{+}, \dir{y} ) \}. \label{min:alpha-D} \end{flalign} \end{subequations} Equation~\eqref{eq:alphaD-least-squares} can be interpreted as choosing the step size $\alpha_{D}$ that minimizes the complementarity and dual infeasibility. This reduces to a one-dimensional least squares problem in $\zeta$ which has a closed form expression for the solution. Equation~\eqref{min:alpha-D} encourages the dual step size to be at least as large as the primal step size $\alpha_{P}$. This prevents tiny dual step sizes being taken when the dual direction is not a descent direction for the dual infeasibility, which may occur if $\delta$ is large.
\subsection{A filter using a KKT merit function}\label{sec:filter}
In the stabilization search directions we accept steps that make progress on one of two merit functions, which form a filter. The first function $\phi_{\mu}$ is defined in Section~\ref{sec:simple-stable}. The second function, we call it the KKT merit function, measures the scaled dual feasibility and complementarity: \begin{flalign}\label{merit-KKT}
\mathbb{K}_{\mu} ( x, s, y ) = \sigma( y ) \max\{ \| \grad \mathcal{L}_{\mu}(x, y ) \|_{\infty} , \| S y - \mu \vec{e} \|_{\infty} \}. \end{flalign} This merit function measures progress effectively in regimes where $\mathcal{M}$, given in \eqref{eq:Schur-matrix}, is positive definite. In this case, the search directions generated by \eqref{eq:Schur-complement-system} will be a descent direction on this merit function (for the first inner iteration of each outer iteration of Algorithm~\ref{practical-one-phase-IPM} i.e., $j = 1$). This merit function is similar to the potential functions used in interior point methods for convex optimization \cite{andersen1998computational,huang2016solution}. Unfortunately, while this merit function may be an excellent choice for convex problems, in nonconvex optimization it has serious issues. In particular, the search direction \eqref{eq:Schur-complement-system} might not be a descent direction. Moreover, changing the search direction to minimize the dual feasibility has negative ramifications. The algorithm could converge to a critical point of the dual feasibility where $\mathbb{K}_{\mu} ( x, s, y ) \neq 0$\footnote{To see why this occurs one need only consider an unconstrained problem, e.g., minimizing $f(x) = x^4 + x^3 + x$ subject to no constraints. The point $x = 0$ is a stationary point for the gradient of $\grad f(x)$, but is not a critical point of the function.}. For further discussion of these issues, see \cite{shanno2000interior}.
While it is sufficient to guarantee convergence by accepting steps if \eqref{eq:phi-sufficient-progress} is satisfied, in some regimes e.g., when $\mathcal{M}$ is positive definite, this may select step sizes $\alpha_{P}$ that are too conservative; for example, near points satisfying the sufficient conditions for local optimality. In these situations the KKT error is often a better measure of progress toward a local optimum than a merit function that discards information about the dual feasibility. Furthermore, from our experience, during convergence toward an optimal solution, numerical errors in the function $\phi_{\mu}$ may cause the algorithm to fail to make sufficient progress on the merit function $\phi_{\mu}$, i.e., \eqref{eq:phi-sufficient-progress} is not satisfied for any $\alpha_{P}$. For these reasons we decide to use a filter approach \cite{fletcher2002nonlinear,wachter2006implementation}. Typical filter methods \cite{fletcher2002nonlinear} require progress on either the constraint violation or the objective function. Our approach is distinctly different, because we accept steps that make progress on either the merit function $\phi_{\mu}$ or the merit function $\mathbb{K}_{\mu}$.
To be precise, we accept any iterate $(\mu^{+}, x^{+}, s^{+}, y^{+})$ that makes sufficient progress on the augmented log barrier function $\phi_{\mu}$, or satisfies the equations \begin{subequations}\label{eq:filter} \begin{flalign} \mathbb{K}_{\mu} (x^{+}, s^{+}, y^{+}) &\le (1 - \beta_{7} \alpha_{P} ) \mathbb{K}_{\mu} (\tilde{x}, \tilde{s}, \tilde{y}) \label{eq:kkt-progress} \\ \phi_{\mu}(x^{+}, s^{+}, y^{+}) &\le \phi_{\mu}(\tilde{x}, \tilde{s}, \tilde{y}) + \sqrt{\mathbb{K}_{\mu} (\tilde{x}, \tilde{s}, \tilde{y})} \end{flalign} \end{subequations}
for every previous iterate $(\tilde{\mu}, \tilde{x}, \tilde{s}, \tilde{y})$ with $a(\tilde{x}) + \tilde{s} = a(x) + s$.
The idea of \eqref{eq:filter} is that for points with similar values of the augmented log barrier function the KKT error is a good measure of progress. However, we want to discourage the algorithm from significantly increasing the augmented log barrier function while reducing the KKT error because if this is occurring, then the algorithm might converge to a saddle point.
\subsection{Algorithm outline}\label{sec:practical-alg-outline}
The general idea of Algorithm~\ref{practical-one-phase-IPM} follows. At each outer iteration we factorize the matrix $\mathcal{M} + \delta I$ with an appropriate choice of $\delta$ using Algorithm~\ref{alg:mat-fact}. With this factorization fixed, we attempt to take multiple inner iterations (at most $j_{\max}$), which corresponds to solving system~\eqref{primal-dual-Newton-direction} with different right-hand side choices but the same matrix $\mathcal{M} + \delta I$. Each inner iteration is either an aggressive step or a stabilization step. If, on the first inner iteration, the step fails (i.e., due to a too small step size), we increase $\delta$ and refactorize $\mathcal{M} + \delta I$. Note that we evaluate the Hessian of the Lagrangian once per outer iteration. The selection of the initial point $(\mu^{0}, x^{0}, s^{0}, y^{0})$ is described in Section~\ref{sec:initialization}.
\begin{algorithm}[H] \textbf{Input:} some initial point $x^{0}$, vector $\vec{w} \ge 0$, and variables $\mu^0, s^{0}, y^{0} > 0$ such that $a(x^{0}) + s^{0} = \vec{w} \mu^{0}$ and equation~\eqref{eq:comp-slack} is satisfied with $k=0$. Termination tolerances $\epsilon_{\textbf{opt}} \in (0,\infty)$, $\epsilon_{\textbf{far}} \in (0,1)$, $\epsilon_{\textbf{inf}} \in (0,1)$ and $\epsilon_{\textbf{unbd}} \in (0,1)$. \\ \textbf{Output:} a point $(\mu, x, s, y)$ that satisfies at least one of the inequalities \eqref{terminate-kkt}, \eqref{terminate-primal-infeasible} or \eqref{terminate-dual-infeasible}{}
\begin{enumerate}[label*=A.{\arabic*}]
\item \label{line:init-delta} \emph{Initialize.} Set $\delta \gets 0$. \item \label{line:form-K} \emph{New outer iteration.} \\ Set $(\hat{\mu}, \hat{x}, \hat{s}, \hat{y}) \gets (\mu, x, s, y)$. Form the matrix $\mathcal{M}$ using \eqref{eq:Schur-matrix}. Set $\delta_{\text{prev}} \gets \delta$.
\item \label{line:factor-schur} \emph{Select $\delta$ and factorize the matrix $\mathcal{M} + \delta I$,} \\ i.e., run Algorithm~\ref{alg:mat-fact} (see Appendix~\ref{sec:mat-fact}) with: \\ \hspace*{0.1cm} \textbf{Input:} $\mathcal{M}$, $\delta$. \\ \hspace*{0.1cm} \textbf{Output:} New value for $\delta$, factorization of $\mathcal{M} + \delta I$. \item \label{take-steps} \emph{Perform inner iterations where we recycle the factorization of $\mathcal{M} + \delta I$.} \\ For $j \in \{ 1, \dots , j_{\max} \}$ do: \begin{enumerate}[label*=.{\arabic*}] \item \emph{Check termination criterion}. \\ If any of the inequalities \eqref{terminate-kkt}, \eqref{terminate-primal-infeasible} or \eqref{terminate-dual-infeasible}{} holds at the point $(\mu,x,s,y)$, terminate. \item \emph{Take step}\label{line:take-step} \begin{enumerate}[label=-Case {\Roman*}] \item If the aggressive step criterion~\eqref{agg-criteron} is satisfied, do an aggressive step, \\ i.e., run Algorithm~\ref{alg:aggressive} with: \\ \hspace*{0.1cm} \textbf{Input:} the matrix $\mathcal{M} + \delta I$, its factorization, the point $(\mu, x, s, y)$ and $(\hat{\mu}, \hat{x}, \hat{s}, \hat{y})$. \\ \hspace*{0.1cm} \textbf{Output:} A $\textbf{status}$ and a new point $(\mu^{+},x^{+},s^{+},y^{+})$. \item Otherwise, do a stabilization step, \\ i.e., run Algorithm~\ref{alg:stable} with: \\ \hspace*{0.1cm} \textbf{Input:} the matrix $\mathcal{M} + \delta I$, its factorization, the point $(\mu, x, s, y)$ and $(\hat{\mu}, \hat{x}, \hat{s}, \hat{y})$. \\ \hspace*{0.1cm} \textbf{Output:} A $\textbf{status}$ and a new point $(\mu^{+},x^{+},s^{+},y^{+})$. \end{enumerate} \item \emph{Deal with failures}. \\ If $\textbf{status} = \textsc{success}$ set $(\mu, x, s, y) \gets ( \mu^{+}, x^{+},s^{+},y^{+})$. If $\textbf{status} = \textsc{failure}$ and $j = 1$ go to \eqref{increase-delta-for-failure}. If $\textbf{status} = \textsc{failure}$ and $j > 1$ go to step~\eqref{line:form-K}. \end{enumerate} \item Go to \eqref{line:form-K}. \item \label{increase-delta-for-failure} \emph{Increase $\delta$ to address failure.} \\
Set $\delta = \max\left\{\Delta_{\text{inc}} \delta, \quad \Delta_{\min}, \quad \delta_{\text{prev}} \Delta_{\text{dec}}, \quad \frac{\| \grad \mathcal{L}_{\mu}(x,y) \|_{\infty}}{ \| d_{x} \|_{\infty} } \right\}$. \\ If $\delta \le \Delta_{\max}$ then factorize the matrix $\mathcal{M} + \delta I$ and go to step \eqref{take-steps}, otherwise terminate with $\textbf{status} = \textsc{max-delta}$. \end{enumerate} \caption{A practical one-phase IPM}\label{practical-one-phase-IPM} \end{algorithm}
\subsubsection{Aggressive steps}
Recall that when computing aggressive search directions we solve system~\eqref{primal-dual-Newton-direction} with $\gamma = 0$; that is, we aim for feasibility and optimality simultaneously. We accept any step size assuming it satisfies the fraction-to-boundary{} rule \eqref{fracBoundary-primal} and the set of valid dual step sizes is non-empty: $B( s^{+}, \dir{y} ) \neq \emptyset$ (see equations~\eqref{subeq:set-B}).
To prevent unnecessary line searches, we only attempt an aggressive line search if
\begin{flalign} \grad \mathcal{L}_{\gamma \mu}(x,\tilde{y})^T \dir{x} < 0 \label{eq:agg-could-improve} \end{flalign} where $\tilde{y} = S^{-1} \mu ( \vec{e} \gamma - (1 - \gamma ) Y \vec{w} )$. Note that \eqref{eq:agg-could-improve} always holds if $(\hat{x}, \hat{s}, \hat{y}, \hat{\mu}) = (\mu, x, s, y)$.
The backtracking line search of the aggressive step has a minimum step size. If during the backtracking line search (line~\ref{line:agg-back-too-small} of Algorithm~\ref{alg:aggressive}) the step size $\alpha_{P}$ is smaller than
\begin{flalign}\label{min-step-size-aggresssive} \bar{\theta}(\mu, s) := \min \left\{ 1/2, \frac{\beta_{6}}{4 \mu} \times \min\left\{ \frac{\beta_{3} - \beta_{2}}{\beta_{3}}, 1 - \Theta^{b}_{i,i} \right\} \times \min_{i \in \{ 1, ..., m \} : \vec{w}_i > 0}{\frac{s_{i}}{\vec{w}_i} } \right\}
\end{flalign} then we immediately reject the step and exit Algorithm~\ref{alg:aggressive}. Note that the function $\bar{\theta}(\mu, s)$ is a more sophisticated version of $\theta$ used for the simple algorithm and defined in \eqref{simple-min-step-size-aggresssive}. Following this, $\delta$ is increased in Line~\ref{increase-delta-for-failure} of Algorithm~\ref{practical-one-phase-IPM} and a new aggressive step is attempted. It is possible that $\delta$ will be increased many times; however, for sufficiently large $\delta$ an acceptable step will be found (see Lemma~\ref{lemma:agg-succeeds}).
To choose $\gamma$ in line~\ref{mehrotra-heuristic} of Algorithm~\ref{alg:aggressive} we use a heuristic inspired by Mehrotra's predictor-corrector method. This heuristic only requires a direction computation --- we do not evaluate any functions of the nonlinear program.
It is possible that when we take a step that we reduce $\mu$ significantly but $\| \grad \mathcal{L}(x,y) \|_{\infty}$ remains large. Line~\ref{agg-protect-1} and \ref{agg-protect-1} of Algorithm~\ref{alg:aggressive} guard against this possibility. This scheme is controlled by the parameter $\beta_{8} \in (0.5,1)$ with a default value of $0.9$. Values of $\beta_{8}$ close to $1$ give no guarding and close to $0$ are conservative.
\begin{algorithm}[H] \textbf{Input:} The matrix $\mathcal{M} + \delta I$, its factorization, the current point $(\mu, x, s, y)$ and the point $(\hat{x}, \hat{s}, \hat{y}, \hat{\mu})$ from the beginning of the outer iteration. \\ \textbf{Output:} A new point $(\mu^{+}, x^{+}, s^{+}, y^{+})$ and a $\textbf{status}$ of either $\textsc{success}$ or $\textsc{failure}$. \begin{enumerate}[label*=A.{\arabic*}] \item \label{mehrotra-heuristic} \emph{Use Mehrotra's predictor-corrector heuristic to choose $\gamma$.} \begin{enumerate}[label*=.{\arabic*}] \item Compute the vector $b$ at the point $(\mu, x, s, y)$ via \eqref{def:b} with $\gamma = 0$. \item Compute the search direction $(\dir{x}, \dir{s}, \dir{y})$ using \eqref{practical-direction}. \item Estimate the largest primal step size $\alpha^{\max}_{P}$ from equation~\eqref{fracBoundaryPrimalMax}. \item Set $\gamma \gets \min\{0.5, (1 - \alpha_{\max})^2 \}$. \end{enumerate} \item Compute the vector $b$ at the point $(\mu, x, s, y)$ via \eqref{def:b} with $\gamma$ as chosen in previous step. \item Compute the search direction $(\dir{x}, \dir{s}, \dir{y})$ using \eqref{practical-direction}. \item \emph{Check that the direction has a reasonable chance of being accepted.} If \eqref{eq:agg-could-improve} is not satisfied then terminate with $\textbf{status} = \textsc{failure}$. \item Estimate the largest primal step size $\alpha^{\max}_{P}$ from equation~\eqref{fracBoundaryPrimalMax}. \item \label{agg:line:back-track} \emph{Perform a backtracking linesearch}. Starting from $\alpha_{P} \gets \alpha^{\max}_{P}$ repeat the following steps. \begin{enumerate}[label*=.{\arabic*}] \item \label{prac-agg:line:back-too-small} \emph{Check the step size is not too small.} If $\alpha_{P} \le \bar{\theta}(\mu, s)$ then goto line~\ref{agg-prac:line:terminate}. \item Compute the trial primal variables $(\mu^{+}, x^{+}, s^{+})$ via~\eqref{eq:iterate-update}. \item If the fraction-to-boundary{} rule \eqref{fracBoundary-primal} is not satisfied, then set $\alpha_{P} \gets \beta_{6} \alpha_{P}$ and go to line~\ref{prac-agg:line:back-too-small}. \item Compute feasible dual step sizes $B( s^{+}, \dir{y} )$. \item If $B( s^{+}, \dir{y} ) = \emptyset$ then trial step has failed, then set $\alpha_{P} \gets \beta_{6} \alpha_{P}$ and go to line~\ref{prac-agg:line:back-too-small}. \item Compute dual variable step size $\alpha_{D}$ using \eqref{eq:compute-alpha-D} and compute the trial dual variables $y^{+}$ using \eqref{eq:update-y}. \item \label{agg-protect-1} If $\mu^{+} / \mu \ge 1 - \beta_{8}$ goto line~\ref{prac-agg:line:success}.
\item \label{agg-protect-2} Let $\tau \gets \frac{\mu^{+}}{(1 - \beta_{8}) \sigma(y^{+}) \| \grad \mathcal{L}_{\mu^{+}}(x^{+},y^{+}) \|_{\infty}}$. If $\tau < 1$ then set $\alpha_{P} \gets \max\{ \beta_{8}^2 , \alpha_{P} \tau^2 \}$ and go to line~\ref{prac-agg:line:back-too-small}. \item\label{prac-agg:line:success} Terminate with $\textbf{status} = \textsc{success}$ and return the point $(\mu^{+}, x^{+}, s^{+}, y^{+})$. \end{enumerate} \item \label{agg-prac:line:terminate} Terminate with $\textbf{status} = \textsc{failure}$. \end{enumerate} \caption{Practical aggressive step}\label{alg:aggressive} \end{algorithm}
\subsubsection{Stabilization steps}
During the backtracking line search we terminate with $\textbf{status} = \textsc{failure}$ if \begin{flalign}\label{eq:min-step-size-stable} \alpha_{P} \le \beta_{5} \end{flalign} where $\beta_{5} \in (0,1)$ with default value $2^{-5}$.
We then exit Algorithm~\ref{alg:stable} and go to line~\ref{increase-delta-for-failure} of Algorithm~\ref{practical-one-phase-IPM}, where we increase $\delta$ and attempt a new stabilization step. From Lemma~\ref{lemConsecutiveStable} we know for sufficiently large $\delta$ the stabilization step will succeed.
To prevent unnecessary line searches, we only attempt a stabilization line search if
\begin{flalign} \grad \psi_{\mu}(x)^T \dir{x} < 0. \label{eq:obj-could-improve} \end{flalign} The idea is to take steps only when it is possible to decrease $\psi_{\mu}$. This condition is always satisfied if Algorithm~\ref{alg:stable} is called from the first inner iteration of Algorithm~\ref{practical-one-phase-IPM}. It may not be satisfied when the inner iteration is greater than one because we recycle the factorization of $\mathcal{M} + \delta I$.
\begin{algorithm}[H] \textbf{Input:} The matrix $\mathcal{M} + \delta I$, its factorization, the current point $(\mu, x, s, y)$ and the point $(\hat{x}, \hat{s}, \hat{y}, \hat{\mu})$ from the beginning of the outer iteration. \\ \textbf{Output:} A new point $(\mu^{+}, x^{+}, s^{+}, y^{+})$ and a $\textbf{status}$ of either $\textsc{success}$ or $\textsc{failure}$ \begin{enumerate}[label*=A.{\arabic*}] \item Compute the vector $b$ at the point $(\mu, x, s, y)$ via \eqref{def:b} with $\gamma = 1$. \item Compute the search direction $(\dir{x}, \dir{s}, \dir{y})$ by solving \eqref{eq:Schur-complement-system}, \eqref{compute-ds} and \eqref{compute-dy} respectively. \item \emph{Check that the direction has a reasonable chance of being accepted.} If \eqref{eq:obj-could-improve} is not satisfied then terminate with $\textbf{status} = \textsc{failure}$. \item Estimate the largest primal step size $\alpha^{\max}_{P}$ from equation~\eqref{fracBoundaryPrimalMax}.
\item \label{agg:line:back-track} \emph{Perform a backtracking linesearch}. Starting from $\alpha_{P} \gets \alpha^{\max}_{P}$ repeat the following steps. \begin{enumerate}[label*=.{\arabic*}] \item \label{line:agg-back-too-small} \emph{Check the step size is not too small.} If $\alpha_{P} \le \beta_{5}$ then goto line~\ref{line:agg-terminate}. \item Compute the trial primal variables $(\mu^{+}, x^{+}, s^{+})$ via~\eqref{eq:iterate-update}. \item If the fraction-to-boundary{} rule \eqref{fracBoundary-primal} is not satisfied, then set $\alpha_{P} \gets \beta_{6} \alpha_{P}$ and go to line~\ref{line:agg-back-too-small}. \item Compute feasible dual step sizes $B( s^{+}, \dir{y} )$. \item If $B( s^{+}, \dir{y} ) = \emptyset$, then set $\alpha_{P} \gets \beta_{6} \alpha_{P}$ and go to line~\ref{line:agg-back-too-small}. \item Compute dual variable step size $\alpha_{D}$ using \eqref{eq:compute-alpha-D} and compute the trial dual variables $y^{+}$ using \eqref{eq:update-y}. \item \label{line:filter} \emph{Sufficient progress on filter.} If neither equation~\eqref{eq:phi-sufficient-progress} or \eqref{eq:filter} is satisfied, then set $\alpha_{P} \gets \beta_{6} \alpha_{P}$ and go to line~\ref{line:agg-back-too-small}. \item Terminate with $\textbf{status} = \textsc{success}$ and return the point $(\mu^{+}, x^{+}, s^{+}, y^{+})$. \end{enumerate} \item \label{line:agg-terminate} Terminate with $\textbf{status} = \textsc{failure}$. \end{enumerate} \caption{Practical stabilization step}\label{alg:stable} \end{algorithm}
\subsection{Algorithm Parameters}
\begin{table}[H]
\begin{tabular}{ |c| p{7.5cm}|p{2.5cm}| p{2.7cm}| }
\hline Parameter & Description & Possible values & Chosen value \\ \hline $\epsilon_{\textbf{opt}}$ & Tolerance for the optimality criterion \eqref{terminate-kkt}. & $(0,\infty)$ & $10^{-6}$ \\ \hline $\epsilon_{\textbf{far}}$ & Tolerance for the infeasibility criterion \eqref{terminate-primal-infeasible} corresponding to $\Gamma_{\text{far}}$. & $(0,1)$ & $10^{-3}$ \\ \hline $\epsilon_{\textbf{inf}}$ & Tolerance for the infeasibility criterion \eqref{terminate-primal-infeasible} corresponding to $\Gamma_{\text{inf}}$. & $(0,1)$ & $10^{-6}$ \\ \hline $\epsilon_{\textbf{unbd}}$ & Tolerance for the unboundedness criterion \eqref{terminate-dual-infeasible}. & $(0,1)$ & $10^{-12}$ \\ \hline $\beta_{1}$ & Used to modify the log barrier function to prevent primal iterates from diverging. & $(0,1)$ & $10^{-4}$ \\ \hline
$\beta_{2}$ & Restricts how far complementarity of $s$ and $y$ can be from $\mu$. See \eqref{eq:comp-slack}. &$(0,1)$ & $0.01$ \\
\hline
$\beta_{3}$ & Restricts how far complementarity of $s$ and $y$ can be from $\mu$ in order for the aggressive criterion to be met. See \eqref{agg-criteron-buffer}. & $(\parComp,1)$ & $0.02$ \\ \hline \hline
$\beta_{4}$ & Acceptable reduction factor for the merit function $\phi_{\mu}$ during stabilization steps. See \eqref{eq:phi-sufficient-progress}. & $(0,1)$ & $0.2$ \\
\hline
$\beta_{5}$ & Minimum step size for stable line searches. See \eqref{eq:min-step-size-stable}. & $(0,1)$ & $2^{-5}$ \\
\hline
$\beta_{7}$ & Acceptable reduction factor for the scaled KKT error $\mathbb{K}_{\mu}$ during stabilization steps. See \eqref{eq:kkt-progress}. & $(0,1)$ & $0.01$ \\
\hline
$\beta_{6}$ & Backtracking factor for line searches in Algorithm~\ref{alg:aggressive} and \ref{alg:stable}. & $(0,1)$ & $0.5$ \\
\hline
$\beta_{7}$ & Exponent of $\| \dir{x} \|$ used in fraction-to-boundary{} formula \eqref{fracBoundaryPrimalMax} for computing the maximum step size $\alpha_{P}^{\max}$. &$(0,1)$ & $0.5$ \\ \hline
$\beta_{8}$ & Step size for which we check that the dual feasibility is decreasing (line~\ref{agg-protect-1} and \ref{agg-protect-2} of Algorithm~\ref{alg:aggressive}). & $(0.5,1)$ & 0.9 \\ \hline
$\Theta^{b}$ & Diagonal matrix with the fraction-to-boundary{} parameter for each constraint. See \eqref{fracBoundary-primal} and \eqref{fracBoundary-dual}. & Each element is in the interval $(0,1)$. & $0.1$ for all elements \\
\hline $\Theta^{p}$ & Diagonal matrix with the fraction-to-boundary{} parameter for each constraint used in \eqref{fracBoundaryPrimalMax} for computing $\alpha_{P}^{\max}$. & Each element is in the interval $[\parFracBoundary_{i,i}, 1)$. & $0.1$ and $0.25$ for linear and nonlinear constraints respectively \\ \hline $\Delta_{\min}$ & Used in Algorithm~\ref{practical-one-phase-IPM} and \ref{alg:mat-fact}. & $(0,\infty)$ & $10^{-8}$ \\ \hline $\Delta_{\text{dec}}$ & Used in Algorithm~\ref{practical-one-phase-IPM} and \ref{alg:mat-fact}. & $(1,\infty)$ & $\pi$ \\ \hline $\Delta_{\text{inc}}$ & Used in Algorithm~\ref{practical-one-phase-IPM} and \ref{alg:mat-fact}. & $(1,\infty)$ & $8$ \\ \hline $\Delta_{\max}$ &Used in Algorithm~\ref{practical-one-phase-IPM} and \ref{alg:mat-fact}. & $(\parDeltaMin,\infty)$ & $10^{50}$ \\ \hline $j_{\max}$ & Maximum number inner iterations per outer iteration. See \eqref{take-steps} of Algorithm~\ref{practical-one-phase-IPM}. & $\mathbb{N}$ & $2$ \\
\hline
$\beta_{10}$ & Minimum slack variable value in the initialization (Section~\ref{sec:initialization}). & $(0,1)$ & $10^{-4}$ \\ \hline $\beta_{11}$ & Minimum dual variable value in the initialization. & $(0,\infty)$ & $10^{-2}$ \\ \hline $\beta_{12}$ & Maximum dual variable value in the initialization. & $[\parInitializeMin,\infty)$ & $10^{3}$ \\ \hline
$\mu_{\text{scale}}$ & Scales the size of $\mu^0$ in the initialization. & $(0,\infty)$ & $1.0$ \\
\hline \end{tabular} \caption{Parameters values and descriptions} \end{table}
\section{Numerical results}\label{sec:numerical-results}
The numerical results are structured as follows. Section~\ref{alg:comparison-IPOPT} compares our algorithm against IPOPT on CUTEst. Section~\ref{sec:infeas} compares our algorithm and IPOPT on a set of infeasible problems. For a comparison of the practical behavior of the dual multipliers we refer the reader to \cite{haeser2017behavior}. The code for our implementation can be found at \url{https://github.com/ohinder/OnePhase} and tables of results at \url{https://github.com/ohinder/OnePhase.jl/tree/master/benchmark-tables}. For description of individual CUTEst problems see \url{http://www.cuter.rl.ac.uk/Problems/mastsif.shtml}.
Overall, on the problems we tested, we found that our algorithm required significantly less iterations to detect infeasibility and failed less often. However, given both solvers found an optimal solution then IPOPT tended to require fewer iterations.
\subsection{Comparison with IPOPT on CUTEst}\label{alg:comparison-IPOPT}
To obtain numerical results we use the CUTEst nonlinear programming test set \cite{gould2015cutest}. We selected a subset from CUTEst with more than $100$ variables and $100$ constraints, but the total number of variables and constraints less than $10,000$. We further restricted the CUTEst problems to ones that are classified as having first and second derivatives defined everywhere (and available analytically). This gave us a test set with $238$ problems. For solving our linear systems we use Julia's default Cholesky factorization (SuiteSparse) and IPOPT's default---the MUMPs linear solver.
Table~\ref{avg:evaluations} compares the number of calls and runtime of different elements of our algorithm. We do not currently compare the number of function evaluations with IPOPT, but in general we would anticipate that IPOPT needs slightly fewer evaluations per outer iteration. In particular, our algorithm may make multiple objective, constraint and Jacobian evaluations per inner iteration. We evaluate the Lagrangian of the Hessian once per outer iteration. For problems where function evaluations are expensive relative to factorization, one is often better off using SQP methods rather than interior point methods. For example, it is known that SNOPT generally requires fewer function evaluations than IPOPT \cite[Figure 2, Figure 3]{gill2015performance}.
\begin{table}[H] \begin{tabular}{l p{3.0cm} l} & Mean \# calls per outer iteration & \% runtime contribution \\ Hessian & 1 & 3.4\% \\ Schur complement &1 & 42.1\% \\ Jacobian & 2.2 & 7.9\% \\ Gradient & 2.2 & 0.4\% \\ Constraints & 6.9 & 0.6\% \\ Factorizations & 1.9 & 35.4 \% \\ Backsolves & 10.0 & 0.8\% \end{tabular} \caption{Number of calls and runtime of different algorithm elements.}\label{avg:evaluations} \end{table}
We consider algorithm to have failed on a problem if it did not return a status of infeasible, optimal or unbounded. Overall the number of failures is 39 for IPOPT and 21 for the one-phase algorithm. From Table~\ref{compare-outputs} we can see that the one-phase algorithm detects infeasibility and KKT points more often than IPOPT. If we reduce the termination tolerances of both algorithms to $10^{-2}$ and re-run them on the test set the one-phase algorithm fails $10$ times versus $41$ times for IPOPT (IPOPT's algorithm makes choices based on the termination criterion).
\begin{table}[H] \caption{Comparison on problems where solver outputs are different.}\label{compare-outputs}
\begin{tabular}{| c | c | c |} & IPOPT & one-phase \\ \hline failure & 32 & 14 \\ infeasible & 3 & 11 \\ unbounded & 0 & 0 \\ KKT & 18 & 28 \end{tabular} \end{table}
To run this test set it takes a total of 16 hours for IPOPT and 10 hours for our one-phase algorithm\footnote{Computations were performed on one core and 8GB of RAM Intel(R) Xeon(R) CPU E5-2650 v2 at 2.60GHz using Julia 0.5 with one thread.}. However, IPOPT is generally significantly faster than our algorithm. It has a median runtime of $0.6$ seconds per problem versus $3.3$ seconds for our algorithm (including problems where the algorithm fails). This is not surprising since our code is written in Julia and is not optimized for speed; IPOPT is written in Fortran and has been in development for over 15 years. This speed difference is particularly acute on small problems where the overheads of Julia are large. Assuming factorization or computation of the hessian is the dominant computational cost we would expect the time per iteration to be similar if our algorithm was efficiently implemented in a compiled programming language. For this reason, we compare the algorithms based on iteration counts.
To compare iteration counts our graphs use the performance profiling of \citet*{dolan2002benchmarking}. In particular, on the $x$-axis we plot: $$ \frac{\text{iteration count of solver}}{\text{iteration count of fastest solver}} $$ and the curve plotted is a cumulative distribution over the test set. We can see from Figure~\ref{fig:iter-count-CUTEst} that the overall distribution of iteration counts seems similar to IPOPT. Although given both algorithms declare the problem optimal IPOPT is more likely to require fewer iterations (Figure~\ref{fig:iter-count-CUTEst-opt}).
\begin{figure}
\caption{Performance profile on problems until algorithm succeeds (i.e., returns a status of optimal, primal infeasible, or unbounded)}
\label{fig:iter-count-CUTEst}
\end{figure}
\begin{figure}
\caption{Performance profile on problems where both algorithms declare problem optimal.}
\label{fig:iter-count-CUTEst-opt}
\end{figure}
\subsection{Comparison with IPOPT on infeasible problems}\label{sec:infeas}
On few of the CUTEst problems both solvers declare the problems infeasible. Therefore understand how the solvers perform on infeasible problems we perturb the CUTEst test set. Recall that CUTEst writes problems in the form $l \le c(x) \le u$. To generate a test set that was more likely to contain infeasible problems we shift the constraints (excluding the variable bounds) as follows: $$ \tilde{c}(x) \gets c(x) + e, $$ and then input the problems to the one-phase solver and IPOPT. Note that the new problems may or may not be feasible. In Table~\ref{table-perturbed-solver-output} we compare the solver outputs on this perturbed test set.
\begin{table}[H] \caption{Comparison on perturbed CUTEst where solver outputs are different.}\label{table-perturbed-solver-output}
\begin{tabular}{| c | c | c |} & IPOPT & one-phase \\ \hline failure & 42 & 18 \\ infeasible & 14 & 26 \\ unbounded & 0 & 0 \\ KKT & 5 & 17 \end{tabular} \end{table}
Overall we found that $94$ of the $238$ problems were declared infeasible by both solvers. On this test set we compared the iteration counts of IPOPT and the one-phase algorithm (Figure~\ref{fig:inf_cutest_iter_ratios}).
\begin{figure}
\caption{Performance profile on perturbed CUTEst problems where both algorithms declare the problem infeasible.}
\label{fig:inf_cutest_iter_ratios}
\end{figure}
Since this perturbed CUTEst test set was somewhat artificial we also tested on the NETLIB test set containing infeasible linear programs\footnote{The LP problem CPLEX2 is an almost feasible problem that was declared feasible by both solvers. Therefore we removed it from our tests.} (Figure~\ref{fig:inf_netlib_iter_ratios}). We remark that on these 28 problems the one-phase algorithm required less iterations on every problem.
\begin{figure}
\caption{Performance profile on NETLIB infeasible test set.}
\label{fig:inf_netlib_iter_ratios}
\end{figure}
\section{Conclusion and avenues for improvement}
This paper proposed a one-phase algorithm. It avoids a two phase or penalty method typically used in IPMs for nonlinear programming. Nonetheless, under mild assumptions it is guaranteed to converge to a first-order certificate of infeasibility, unboundedness or optimality. As we have demonstrated on large-scale test problems the algorithm has similar iteration counts to IPOPT, but significantly better performance on infeasible problems.
An additional benefit of our approach is the ability to choose $\vec{w}$. For example, if one has a starting point $x^0$ that strictly satisfies a subset of the constraints one can initialize the algorithm with $\vec{w}_i = 0$ on this subset (we do this automatically for the bound constraints). The algorithm will then satisfy these constraints for all subsequent iterations. Aside from potentially speeding up convergence, this is particularly beneficial if some of the constraints or objective are undefined outside the region defined by this subset of the constraints.
One can interpret the relative value of $\mu^0$ and $\vec{w}$ as the extent to which feasibility is prioritized over optimality: a larger $\mu^0$ gives more priority to feasibility. For a fixed $\vec{w}$ picking a huge $\mu$ makes the IPM method behave like a phase-one, phase-two method: initially the algorithm attempts to find a feasible solution then it minimizes the objective on the feasible region. We have attempted such an initialization strategy and note that while it performs well on many problems, in others it causes the dual multipliers to become unduly large (as the theory of \cite{haeser2017behavior} predicts when, for example, the feasible region lacks an interior). We find that manually tuning the value of $\mu^0$ for a specific problem often significantly reduces the number of iterations. This sensitivity to the initialization, especially compared with the homogenous self-dual, is a known issue for Lustig's IPM for linear programming \cite[Table 1]{meszaros2015practical}. Therefore, we believe that improving our initialization scheme (Section~\ref{sec:initialization}) could result in significant improvements.
Finally, we note that improving the accuracy of the linear system solves would also improve our robustness. A disadvantage of using the Cholesky factorization is that we often had difficulty obtaining a sufficiently accurate solution as we approached optimality. Potentially switching to an LBL{} factorization \cite{amestoy1998mumps,bunch1971direct} might help resolve this issue.
\appendix
\section{Global convergence proofs for Algorithm~\ref{simple-one-phase}}\label{app:global-conv}
The purpose of this section is to provide proofs of supporting results for Theorem~\ref{thm:global-convergence}.
\subsection{Convergence of aggressive steps}
\subsubsection{Proof of Lemma~\ref{lemma:agg-succeeds}}\label{sec:lemma:agg-succeeds}
\lemAggSucceeds* \begin{proof} First, observe that as $\delta \rightarrow \infty$ the direction $\dir{x}$ computed from \eqref{eq:Schur-complement-system} tends to zero. Consider any $\alpha_{P} \in (0,1)$, since the function $a$ is continuous for sufficiently large $\delta$ we have \begin{flalign}\label{eq:a-bound}
\| a(x) - a(x + \alpha_{P} \dir{x}) \|_{\infty} \le \frac{\beta_{3} - \beta_{2}}{2 \beta_{3}} \min_i\{ s_{i} \} \end{flalign} Now, set $\alpha_{P} = \theta(\mu, s)$ where $\theta$ is defined in \eqref{simple-min-step-size-aggresssive}, then for this choice of $\alpha_{P}$ we have $$ \abs{ s^{+}_i - s_i } = \abs{ -\alpha_{P} \mu \vec{w}_i + a_i(x) - a_i(x + \alpha_{P} \dir{x}) } \le \frac{\beta_{3} - \beta_{2}}{\beta_{3}} s_i, $$ where the first equality holds by applying \eqref{eq:slackVarUpdate} and then \eqref{eq:muVarUpdate} with $\gamma = 0$, the second inequality equation \eqref{eq:a-bound} and $\alpha_{P} = \theta(\mu, s)$. Note that $$ \frac{s^{+}_i y_i}{\mu} = \frac{s_i y_i}{\mu} (e + s^{-1}_i (s^{+}_i - s_i)) \in \frac{s_i y_i}{\mu} \left[ \frac{\beta_{2}}{\beta_{3}}, 1 + \frac{\beta_{3} - \beta_{2}}{\beta_{3}} \right] \subseteq \frac{s_i y_i}{\mu} \left[ \frac{\beta_{2}}{\beta_{3}}, \frac{\beta_{3}}{\beta_{2}} \right] \subseteq [\beta_{2}, 1/\beta_{2} ] $$ where the first transition comes from algebraic manipulation, the second transition follows from substituting our bound on $\abs{s^{+}_i - s_i}$, the third transition using $\frac{\beta_{3} - \beta_{2}}{\beta_{3}} < \frac{\beta_{3} - \beta_{2}}{\beta_{2}}$, the final transition uses $ \frac{s_i y_i}{\mu} \in [\beta_{3}, 1/\beta_{3}]$. Therefore $\alpha_{D} = 0$ gives a feasible dual iterate. We conclude there exists a $\delta$ such that the $\alpha_{P}$ chosen by line~\ref{simple-agg-select-alpha-P} is at least $\theta(\mu, s)$, which proves the result. \end{proof}
\subsubsection{Proof of Lemma~\ref{lem:yw-bounded}}\label{sub:lem:yw-bounded}
\lemYWbounded*
\begin{proof} Since \eqref{terminate-primal-infeasible} does not hold: either $a(x)^T y \le 0$, $\Gamma_{\text{far}} (\mu,x,s,y) > \epsilon_{\textbf{far}}$ or $\Gamma_{\text{inf}} (\mu,x,s,y) > \epsilon_{\textbf{inf}}$. We consider these three cases in order.
Let us consider the case that $a(x)^T y \le 0$ then $(\mu \vec{w} - s)^T y = a(x)^T y \le 0$ by $a(\vec{x}) + \vec{s} = \mu \vec{w}$. Re-arranging $(\mu \vec{w} - s)^T y \le 0$ gives $y^T \vec{w} \le s^T y / \mu \le m / \beta_{2}$.
Let us consider the case that $\epsilon_{\textbf{far}} < \Gamma_{\text{far}} (\mu,x,s,y) = \frac{\| \grad a(x)^T y \|_{1}}{ a(x)^T y }$ then \begin{flalign*}
w^T y &< s^T y + \frac{\| \grad a(x)^T y \|_{1}}{ \epsilon_{\textbf{far}} } \le s^T y + \frac{\| \grad f(x) - \beta_{1} \mu e^T \grad a(x) \|_{1} + \| \grad \mathcal{L}_{\mu}(x,y) \|_{1}}{ \epsilon_{\textbf{far}} } \\
&\le s^T y + 2 \frac{\| \grad f(x) - \beta_{1} \mu e^T \grad a(x) \|_{1}}{ \epsilon_{\textbf{far}} } \end{flalign*}
where the first inequality holds by $a(\vec{x}) + \vec{s} = \mu \vec{w}$ and re-arranging, the second by the triangle inequality and the third by the assumption that the aggressive step criteron \eqref{agg-criteron-farkas} is met. Furthermore, the term $s^T y$ is bounded by $\mu / \beta_{3}$ and the term $\| \grad f(x) - \beta_{1} \mu e^T \grad a(x) \|_{1}$ is bounded because $f$ and $a$ are twice differentiable and the unboundness criterion~\eqref{terminate-dual-infeasible} is not met.
Let us consider the case that $\epsilon_{\textbf{inf}} < \Gamma_{\text{inf}} (x,s,y) = \frac{\| \grad a(x)^T y \|_{1} + s^T y}{ \| y \|_{1} }$ then \begin{flalign*}
\| y \|_{1} &< \frac{s^T y + \| \grad a(x)^T y \|_{1}}{\epsilon_{\textbf{inf}}} \le \frac{ s^T y + \| \grad f(x) - \beta_{1} \mu e^T \grad a(x) \|_{1} + \| \grad \mathcal{L}_{\mu}(x,y) \|_{1}}{\epsilon_{\textbf{inf}}} \\
&\le 2 \frac{ s^T y + \| \grad f(x) - \beta_{1} \mu e^T \grad a(x) \|_{1}}{\epsilon_{\textbf{inf}}} \end{flalign*}
where the first inequality holds by re-arranging, the second by the triangle inequality and the third by the assumption that the aggressive step criteron \eqref{agg-criteron-farkas} is met. We conclude $\| y \|_{1}$ is bounded since $\| x \|$ is bounded, clearly therefore $w^T y$ is also bounded above.
Since in all three cases $\vec{w}^T y$ is bounded we conclude the proof. \end{proof}
\subsection{Convergence results for stabilization steps}
\subsubsection{Proof of Lemma~\ref{lem:compact-Q} and Corollary~\ref{coro:bound-everything}} \label{sec:lem:compact-Q}
We now introduce the set $\mathbb{Q}_{\mu, C}$ which we will use to represent the set of possible points the iterates of Algorithm~\ref{simple-one-phase} can take for a fixed $\mu$, i.e., during consecutive stabilization steps.
\begin{definition}\label{defQ} Define the set $\mathbb{Q}_{\mu, C}$ for constants $\mu, C > 0$ as the set of points $(x,y,s) \in R^{n} \times R_{++}^{m} \times R_{++}^{m}$ such that \eqref{restate:eq:barrier-primal-sequence-nice} holds and \begin{enumerate} \item \label{Q-phi-bounded-above} The function $\phi_{\mu}$ is bounded above, i.e., $\phi_{\mu}(x,y,s) \le C$.
\item \label{Q-bounded-below} The primal iterates are bounded, i.e., $\| x \| \le C$. \end{enumerate} \end{definition}
Suppose Algorithm~\ref{simple-one-phase} generates consecutive stabilization steps stabilization steps $(\mu^k, x^k, s^k, y^k)$ for $k \in \{ k_{\text{start}}, \dots, k_{\text{end}} \}$ with $\mu =\mu^{k_{\text{start}}} = \dots = \mu^{k_{\text{end}}}$ and none of these iterates satisfy the unboundedness termination criterion~\eqref{terminate-dual-infeasible}. Let us show these iterates are contained in $\mathbb{Q}_{\mu, C}$, i.e., $(x^{k}, s^{k}, y^{k}) \in \mathbb{Q}_{\mu, C}$ for some $C > 0$. For $C \ge \phi_{\mu}(x^{k_{\text{start}}}, s^{k_{\text{start}}}, y^{k_{\text{start}}})$ condition \ref{defQ}.\ref{Q-phi-bounded-above} holds since during stabilization steps we only accept steps that decrease $\phi_{\mu}$. For sufficiently large $C$, condition \ref{defQ}.\ref{Q-bounded-below} holds from the assumption the unboundedness termination criterion~\eqref{terminate-dual-infeasible} is not satisfied.
We are now ready to prove Lemma~\ref{lem:compact-Q}. Note that during Lemma~\ref{lem:compact-Q} we will repeatedly use that the following elementary real analysis fact:
\begin{fact} Let $X = \{ x : g_i(x) \le 0 \}$. If $g_i$ is a continuous function and the set $X$ is bounded, then the set $X$ is compact. \end{fact}
\begin{restatable}{lemma}{lemCompactQ}\label{lem:compact-Q} Suppose assumptions~\ref{assume:diff} and \ref{assume:parameters} hold. For any constants $C, \mu > 0$ the set $\mathbb{Q}_{\mu, C}$ is compact. \end{restatable}
\begin{proof} First consider the set $$
Q := \left\{ x \in R^{n} : (y, s) \in R_{++}^{m} \times R_{++}^{m}, \phi_{\mu}(x,y,s) \le C, \| x \| \le C \right\} $$
By $\| x \| \le C$ we see $Q$ is bounded. Furthermore, since $\phi_{\mu}(x) \le C$ and $Q$ is bounded there exists some constant $K_{1} > 0$ such that $$ \mu w - a(x) \ge K_{1} $$ for all $x \in Q$. Consider some sequence $x^{k} \in Q$ with $x^{k} \rightarrow x^{*}$. The statement $a(x) \le \mu w - K_{1}$ implies $\phi_{\mu}$ is continuous in a neighborhood of $x^{*}$. Using the definition of $Q$ and the assumption that $f$ and $a$ are continuous implies $x^{*} \in Q$, i.e., $Q$ is compact.
Note that: $$ \mathbb{Q}_{\mu, C} = \left\{ (x,y,s) \in R^{n} \times R_{++}^{m} \times R_{++}^{m} : x \in Q, a(x) + s = \mu \vec{w}, \frac{s_i y_i}{\mu} \in [\beta_{2}, 1 / \beta_{2}] ~~ \forall i \in \{1 , \dots, m \} \right\}. $$ Consider some $(x,y,s) \in \mathbb{Q}_{\mu, C}$, since $s = \mu w - a(x) \ge K_{2}$ and $\frac{s_i y_i}{\mu} \in [\beta_{2}, 1 / \beta_{2}]$ we can deduce $y$ is bounded. Since the function $a(x)$ and the term $s_i y_i$ are continuous we conclude $\mathbb{Q}_{\mu, C}$ is compact. \end{proof}
\begin{restatable}{corollary}{coroBoundEverything}\label{coro:bound-everything} Suppose assumptions~\ref{assume:diff} and \ref{assume:parameters} hold. Consider some fixed $\mu, C > 0$. Then there exists some $L > 0$ such that for all $(x, s, y) \in \mathbb{Q}_{\mu, C}$ the following inequalities hold: $$ s_i, y_i \ge 1/L $$ $$
\| x \|, \| y \|, \| s \|, \| \grad \psi_{\mu}(x) \|, \| \mathcal{M} \|, \| \grad a(x) \| \le L $$
and for any $u \in R^{n}$ s.t. $\| u \| < 1 / L$ \begin{subequations}\label{lipschitz-continuous} \begin{flalign}
\psi_{\mu}(x + u) &\le \psi_{\mu}(x) + \grad \psi_{\mu}(x)^T u + L / 2 \| u \|^2 \label{phi-lipschitz-continuous} \\
\| a(x) + \grad a(x) u -a(x + u) \| &\le L \| u \|^2. \label{a-lipschitz-continuous-2nd} \end{flalign} \end{subequations}
Furthermore, if the aggressive criterion~\eqref{agg-criteron} does not holds then $$
\max\{ \| \grad \psi_{\mu}(x) \|, \| S y - \mu \vec{e} \|_{\infty} \} \ge 1 / L. $$ \end{restatable}
\begin{proof} All these claims use Lemma~\ref{lem:compact-Q} and the elementary real analysis fact that for any continuous function $g$ on a compact set there $X$ there exists some $x^{*} \in X$ such that $g(x^{*}) = \sup_{x \in X}{g(x)}$.
The only non-trivial claim is showing \eqref{lipschitz-continuous}, which we proceed to show. Since there exists some constants $\varepsilon_{1} > 0$ such that $(x,y,s) \in \mathbb{Q}_{\mu, C}$ we have $a(x) \le \mu w - \varepsilon_{1}$. It follows that there exists some constant $\varepsilon_2 > 0$ such that for all $\| u \| \le \varepsilon_2$ we have $a(x + u) < \mu w$. Therefore there exists some $L > 0$ such that $\| \grad^2 \psi_{\mu}(x) \| \le L$.
For some $x$ and $\nu$ with $\| \nu \| = 1$ define the one dimensional function $$ h(\alpha) := \psi_{\mu}(x + \alpha \nu) $$ then for $\alpha \in [0, \varepsilon_2]$ we get $$ h(\alpha) - h(0) - \alpha h'(0) = \int_{0}^{\alpha}{ \int_{0}^{\eta_{2}}{h''(\eta) \partial \eta_{1} \partial \eta_{2}} } \le \alpha^2 L / 2, $$ which using $u = \nu \alpha$ for $\alpha \in [0, \varepsilon_2]$ concludes the proof of \eqref{phi-lipschitz-continuous}. Showing \eqref{a-lipschitz-continuous-2nd} consists of a similar argument. \end{proof}
\subsubsection{Proof of Lemma~\ref{lemConsecutiveStable}}\label{sec:lemConsecutiveStable}
Before starting the proof of Lemma~\ref{lemConsecutiveStable}, we remark that Lemma~\ref{lemConsecutiveStable} uses Lemma~\ref{lem:compact-Q} and Corollary~\ref{coro:bound-everything} which are proved in Section~\ref{sec:lem:compact-Q}
\lemConsecutiveStable*
\begin{proof} Recall that, as we discussed following Definition~\ref{defQ}, there exists some $C > 0$ such that for any consecutive series of stabilization steps $(\mu^k, x^k, s^k, y^k)$ for $k \in \{ k_{\text{start}}, \dots, k_{\text{end}} \}$ we have $(x^k, s^k, y^k) \in \mathbb{Q}_{\mu, C}$ with $\mu = \mu_{k_{k_{\text{start}}}} = \mu_{k_{k_{\text{end}}}}$.
From Corollary~\ref{coro:bound-everything} we know that there exists some $L > 0$ such that $\max\{-\lambda_{\min}(\mathcal{M}), \lambda_{\max}(\mathcal{M}) \} \le L$, therefore \begin{flalign*}
\dir{x}^T \grad \psi(x) = -\grad \psi(x)^T (\mathcal{M} + \delta I)^{-1} \grad \psi(x) \le -\| \grad \psi(x) \|^2 / (L + \delta) \le -\| \grad \psi(x) \|^2 / (3 L + \mu), \end{flalign*} where the first transition uses the definition of $\dir{x}$ in \eqref{eq:Schur-complement-system} with $\gamma = 1$ when computing $b$, the second transition $\lambda_{\max}(\mathcal{M}) \le L$ and the third transition uses $\delta \le 2 L + \mu$ from line~\ref{alg-simple-delta-min} of Algorithm~\ref{simple-one-phase}.
Similarly, by line~\ref{alg-simple-delta-min} of Algorithm~\ref{simple-one-phase} we get $\mathcal{M} + \delta I \succeq \mu I$ therefore \begin{flalign*}
\| \dir{x} \|^2 = \| (\mathcal{M} + \delta I)^{-1} \grad \psi(x)\|^2 \le \| \grad \psi(x)\|^2 / \mu^2. \end{flalign*}
Furthermore, \begin{flalign*}
\psi_{\mu}(x^{+}) - \psi_{\mu}(x) &\le \alpha_{P} \grad \psi_{\mu}(x)^T \dir{x} + \frac{L \alpha_{P}^2}{2} \| \dir{x} \|^2 \\
&= \alpha_{P} \grad \psi_{\mu}(x)^T \dir{x} + \left( \frac{L \alpha_{P}^2}{2} + \beta_{4} \frac{\alpha_{P} \delta}{2} \right) \| \dir{x} \|^2 - \beta_{4} \frac{\alpha_{P} \delta}{2} \| \dir{x} \|^2 \\
&\le \alpha_{P} \left( \beta_{4} \grad \psi_{\mu}(x)^T \dir{x} + \| \grad \psi_{\mu}(x)\|^2 \left( \frac{1}{\mu^2} \left(\frac{L \alpha_{P}^2}{2} + \beta_{4} \frac{\alpha_{P} \delta}{2} \right) - \frac{1 - \beta_{4}}{3 L + \mu} \right) \right) - \beta_{4} \frac{\alpha_{P} \delta}{2} \| \dir{x} \|^2 \\
&\le \beta_{4} \alpha_{P} \left( \grad \psi_{\mu}(x)^T \dir{x} - \frac{\alpha_{P} \delta}{2} \| \dir{x} \|^2 - c_{1} \| \grad \psi_{\mu}(x)\|^2 \right) \end{flalign*} where the first transition holds by Corollary~\ref{coro:bound-everything}, the second by adding and subtracting terms, the third by the above inequalities, the fourth for some constant $c_{1} > 0$ with $\alpha_{P}$ sufficiently small, i.e., any $\alpha_{P} \in (0, c_{1})$.
We can bound $\| \dir{x} \|$, $\| \dir{y} \|$ and $\| \dir{s} \|$ using our bound on $\| \dir{x} \|$, Corollary~\ref{coro:bound-everything} and \eqref{compute-ds-dy}.
Finally, \begin{flalign*}
\| S^{+} y^{+} - \mu \|_{\infty} &= \| (s + \dir{s}) y^{+} - \mu + (s^{+} - s - \dir{s}) y^{+} \|_{\infty} \\
&\le \| S y - \mu \vec{e} \|_{\infty} + \alpha_{P} \left( -\| S y - \mu \vec{e} \|_{\infty} + \alpha_{P} L \| \dir{x} \|^2 \| y^{+} \|_{\infty} + \alpha_{P} \| \dir{s} \|_{\infty} \| \dir{y} \|_{\infty} \right) \\
&\le \| S y - \mu \vec{e} \|_{\infty} (1 - \alpha_{P} + c_{2} \alpha_{P}^2 ) \end{flalign*}
where the second transition holds by $S y + S \dir{y} + Y \dir{s} = \mu$ and Corollary~\ref{coro:bound-everything} which shows $\| s + \dir{s} - s^{+} \|_{\infty} \le L \| \dir{x} \|^2$, the second inequality by the fact that the directions and $\| y \|$ are bounded.
Using $\mathcal{\zeta}_{\mu}(s,y) := \frac{\| S y - \mu \vec{e} \|_{\infty}^3}{\mu^2}$, the previous expression and the boundedness of $s$ and $y$ we get $$ \mathcal{\zeta}_{\mu}(s^{+},y^{+}) \le \mathcal{\zeta}_{\mu}(s,y) (1 - \alpha_{P}) + \bar{c}_{2} \alpha_{P}^2 $$ for some constant $\bar{c}_{2} > 0$. Defining $$ \Upsilon(\alpha_{P}) := \alpha_{P} \beta_{4} \left( \frac{1}{2} \left( \grad \psi_{\mu}(x)^T \dir{x} - \frac{\delta}{2} \alpha_{P} \norm{ \dir{x}}^2 \right) - \mathcal{\zeta}_{\mu}(s,y) \right), $$ we get for some constants $c_{1}, c_{2}, c_{3} > 0$ that \begin{flalign*}
\phi_{\mu}(x^{+},y^{+},s^{+}) - \phi_{\mu}(x,y,s) &\le \Upsilon(\alpha_{P}) + \alpha_{P} \left( c_{2} \alpha_{P} - c_{3} \mathcal{\zeta}_{\mu}(s,y) - c_{1} \| \grad \psi_{\mu}(x)\|^2 \right) \end{flalign*} by using $\phi_{\mu}(x,y,s) := \psi_{\mu}(x) + \mathcal{\zeta}_{\mu}(s,y)$ and substituting our upper bounds on $\psi_{\mu}(x^{+}) - \psi_{\mu}(x)$ and $\mathcal{\zeta}_{\mu}(s^{+},y^{+})$.
Since $\max\{ \mathcal{\zeta}_{\mu}(s,y), \| \grad \psi_{\mu}(x)\| \}$ is bounded away from zero, we deduce the largest $\alpha_{P}$ satisfying $\phi_{\mu}(x^{+},y^{+},s^{+}) - \phi_{\mu}(x,y,s) \le \Upsilon(\alpha_{P})$ is bounded away from zero and below by zero. We conclude that we must reduce $\phi_{\mu}$ by a constant amount each iteration, which means that if there is an infinite sequence of stabilization steps then $\phi_{\mu}(x^k, s^k, y^k) \rightarrow -\infty$ and hence $\| x^k \| \rightarrow \infty$. \end{proof}
\section{Further implementation details}
\subsection{Matrix factorization strategy}\label{sec:mat-fact}
This strategy is based on the ideas of IPOPT \cite[Algorithm IC]{wachter2006implementation}.
\begin{algorithm}[H] \textbf{Input:} The matrix $\mathcal{M}$ and previous delta choice $\delta \ge 0$ \\ \textbf{Output:} The factorization of $\mathcal{M} + \delta I$ for some $\delta > 0$ such that the matrix $\mathcal{M} + \delta I$ is positive definite. \begin{enumerate}[label*=A.{\arabic*}] \item Set $\delta_{\text{prev}} \gets \delta$. \item Compute $\tau \gets \min_i \mathcal{M}_{i,i}$. If $\tau \le 0$ go to line~\ref{line:new-delta}. \item Set $\delta \gets 0$, $\tau \gets 0$. \item Perform Cholesky factorization of $\mathcal{M}$, if the factorization succeeds then return factorization of $\mathcal{M}$. \item\label{line:new-delta} Set $\delta \gets \max\{ \delta_{\text{prev}} \Delta_{\text{dec}}, \Delta_{\min} - \tau \}$. \item If $\delta \ge \Delta_{\max}$ then terminate the algorithm with $\textbf{status} = \textsc{failure}$. \item Perform Cholesky factorization of $\mathcal{M} + \delta I$, if the factorization succeeds then return factorization of $\mathcal{M} + \delta I$. \item Set $\delta \gets \Delta_{\text{inc}} \delta$. Go to previous step. \end{enumerate} \caption{Matrix factorization strategy}\label{alg:mat-fact} \end{algorithm} where $\Delta_{\text{dec}}, \Delta_{\text{inc}}, \Delta_{\min}, \Delta_{\max}$ have default values of $\pi, 8, 10^{-8}, 10^{50}$ respectively.
\subsection{Initialization}\label{sec:initialization}
This section explains how to select initial variable values $(\mu^0,x^0, y^0, s^0)$ to pass to Algorithm~\ref{practical-one-phase-IPM}, given a suggested starting point $x_{\text{start}}$.
The first goal is to modify $x_{\text{start}}$ to satisfy any bounds e.g., $l \le x \le u$. Let $\mathcal{B} \subseteq \{ 1, \dots, m \}$ be the set of indices corresponding to variable bounds. More precisely, $i \in \mathcal{B}$ if and only if there exists $c \in R$ and $j \in \{ 1, \dots, n \}$ such that $a_i(x) = x_j + c$ or $a_i(x) = -x_j + c$. We project $x_{\text{start}}$ onto the variable bounds in the same way as IPOPT \cite[Section 3.7]{wachter2006implementation}. Furthermore, we set $w_i = 0$ and $s_i^{0} = -a_i(x^0)$ for each $i \in \mathcal{B}$. This ensures that the variable bounds are satisfied throughout, and is useful because the constraints or objective may not be defined outside the bound constraints. To guarantee that any constraint $a_i(x)$ that was strictly feasible at the initial point $x^{0}$ remains feasible throughout the algorithm we could simply set $\vec{w}_i = 0$ and $s_i^{0} = -a_i(x^0)$.
The remainder of the initialization scheme is inspired by Mehrotra's scheme for linear programming \cite[Section 7]{mehrotra1992implementation} and the scheme of \citet*{gertz2004starting} for nonlinear programming. Set \begin{flalign*} \tilde{y} &\gets \vec{e} \\ \tilde{s} &\gets -a(x^{0}) + \max\{ -2 \min_i\{ s_i \}, \beta_{10} \} \end{flalign*} for some parameter $\beta_{10} \in (0,\infty)$ with default value $10^{-4}$. Then factorize $\mathcal{M} + I \delta$ as per lines~\ref{line:init-delta} to \ref{line:factor-schur} of Algorithm~\ref{practical-one-phase-IPM}. Find directions $(d_{x}, d_{y}, d_{s})$ via \eqref{practical-direction} and set $\tilde{y} \gets \tilde{y} + d_y$ and $\tilde{s} = - a(x^{0})$.
Next, we set: \begin{flalign*}
\varepsilon_{y} &\gets \max\{ -2 \min_i{ \tilde{y}_i}, 0 \} \\ \tilde{y} &\gets \tilde{y} + \varepsilon_{y} \\
\varepsilon_{s} &\gets \max\left\{ - 2 \min_i{ \tilde{s}_i}, \frac{ \| \grad \mathcal{L}_{0}(x^0,\tilde{y}) \|_{\infty} }{\| \tilde{y} \| + 1} \right\} \\ \tilde{s}_i &\gets \tilde{s}_i + \varepsilon_{s} \quad \forall i \not\in \mathcal{B}, \end{flalign*} and then \begin{flalign*} \tilde{y} &\gets \tilde{y} + \frac{\tilde{s}^T \tilde{y}}{2 e^T \tilde{s}} \\ \tilde{y} &\gets \max\{ \beta_{11}, \min\{ \tilde{y}, \beta_{12} \} \} \\ \tilde{s}_i &\gets \tilde{s}_i + \frac{\tilde{s}^T \tilde{y}}{2 e^T \tilde{y}} \quad \forall i \not\in \mathcal{B} \\ \tilde{\mu} &\gets \frac{\tilde{s}^T \tilde{y}}{m}. \end{flalign*} where $\beta_{11} \in (0,\infty)$ has a default value of $10^{-2}$ and $\beta_{12} \in [\parInitializeMin,\infty)$ has a default value of $10^{3}$.
Next, set \begin{flalign*} \mu^0 &\gets \mu_{\text{scale}} \tilde{\mu} \\ s^0 &\gets \tilde{s} \\ \vec{w} &\gets \frac{a(x^0) + s^0}{\mu^{0}} \end{flalign*} where $\mu_{\text{scale}} \in (0,\infty)$ is a parameter with a default value of 1.0. We leave $\mu_{\text{scale}}$ as a parameter for the user because we notice that for some problems changing this value can reduce the iteration count by an order of magnitude. Devising a better way to select $\mu^0$ we believe could significantly improve our algorithm. Finally, we need to ensure the dual variables satisfy \eqref{eq:comp-slack} so we set \begin{flalign*} y^0 &\gets \min \{ \max \{ \beta_{3} \mu^0 (S^0)^{-1} e, \tilde{y} \}, \mu / \beta_{3} (S^0)^{-1} e \}. \end{flalign*}
\section{Details for IPOPT experiments}
We used the following options with IPOPT:
\begin{enumerate} \item tol: $10^{-6}$ \item max\_iter: $3000$ \item max\_cpu\_time: $3600$ \item nlp\_scaling\_method: `none' \item bound\_relax\_factor: $0.0$ \item acceptable\_iter: $999999$ \end{enumerate} All other options remained at their default values.
\end{document} |
\begin{document}
\title{Quantum collapse dynamics with attractive densities} \author{F. Lalo\"{e} \footnote{laloe at lkb.ens.fr}\\LKB, ENS-Universit\'{e} PSL, CNRS, 24 rue Lhomond, 75005\ Paris, France} \date{\today} \maketitle
\begin{abstract} We discuss a model of spontaneous collapse of the quantum state that does not require adding any stochastic processes to the standard dynamics. The additional ingredient with respect to the wave function is a position in the configuration space, which drives the collapse in a completely deterministic way. This new variable is equivalent to a set of positions of all the particles, i.e. a set of Bohmian positions, which obey the usual guiding equation of Bohmian theory. Any superposition of quantum states of a macroscopic object occupying different regions of space is projected by a localization process onto the region occupied by the positions. Since the Bohmian positions are well defined in a single realization of the experiment, a space localization into one region is produced. The mechanism is based on the correlations between these positions arising from the cohesive forces inside macroscopic objects.
The model introduces two collapse parameters, which play a very similar role to those of the GRW and CSL theories. With appropriate values of these parameters, we check that the corresponding dynamics rapidly projects superpositions of macroscopic states localized in different regions of space into a single region, while is keeps a negligible effect in all situations where the predictions of standard quantum dynamics are known to be correct. The possible relations with gravity are briefly speculated. We then study the evolution of the density operator and a mean-field approximation of the dynamical equations of this model, as well as the change of the evolution of the momentum introduced by the localization process. Possible theoretical interpretations are finally discussed. Generally speaking, this model introduces a sharper border between the quantum and classical world than the GRW and CSL theories, and leaves a broader range of acceptable values for the parameters.
\end{abstract}
\tableofcontents
\begin{center} ******** \end{center}
The standard linear Schr\"{o}dinger equation predicts the possible occurrence of quantum superpositions of macroscopically distinguishable states (QSMDS).\ This leads for instance to the famous Schr\"{o}dinger cat paradox \cite{Schrodinger-cat, Trimmer}, to the so called measurement problem, etc.\ Nevertheless, such QSMDS are apparently never observed, even in experiments involving \textquotedblleft macroscopic quantum phenomena\textquotedblright\ \cite{Leggett} such as superfluidity or superconductivity.\ The problem arising from this apparent contradiction has given rise to a huge literature \cite{measurement-problem-1, measurement-problem-2}. Since Bohr, numerous authors have proposed various interpretations of quantum mechanics to deal with this problem \cite{Laloe}.\ Another possible approach, nevertheless, is to forbid the occurrence of superpositions of QSMDS by modifying the dynamics of quantum mechanics, for instance by adding a small non-linear and stochastic term to the Schr\"{o}dinger equation. This is the basic idea of \textquotedblleft spontaneous collapse theories\textquotedblright , such as the GRW \cite{GRW} and CSL \cite{CSL, CSL-2} theories; for reviews, see for instance \cite{Bassi-Ghirardi} or \cite{Bassi-Lochan-Satin-Singh-Ulbricht}. It has also been proposed to introduce collapse mechanisms that are driven by gravity \cite{Diosi-1989, GGR-1990, Penrose-1996, Pearle-Squires}. A common feature of these theories is the introduction of a stochastic term in the Schr\"{o}dinger dynamics.
Here we propose a model of spontaneous collapse where the dynamics is deterministic -- as we will see, it is actually more a class of models than a specific model since, for instance, the localization function can be chosen in several ways.\ Instead of adding stochastic processes or functions to the standard quantum description by a wave function, in the configuration space we add a position that drives the collapse mechanism. Adding positions to standard quantum mechanics is of course the basic idea of the de Broglie-Bohm (dBB) interpretation \cite{de-Broglie, Bohm, Holland, Duerr-et-al, Bacchiagaluppi-Valentini, Bricmont}. Our model can therefore be seen as nothing but a combination of the dBB theory with spontaneous collapse theories.
A basic remark is that, in most cases, the number density of the Bohmian positions in ordinary space coincides almost perfectly with the quantum single-particle-density obtained from the many-body state vector; this is a consequence of the so called \textquotedblleft quantum equilibrium\textquotedblright\ condition, which in turn results from a dynamic that forces the Bohmian positions to follow the wave function. Nevertheless, in Schr\"{o}dinger-cat like situations, or after a quantum measurement has been performed, this is no longer the case: on the one hand, because the quantum state splits into several macroscopically distinct components, the quantum density divides into two (or more) disconnected regions of space, corresponding for instance to different positions of the pointer of the measurement apparatus; on the other hand, the individual Bohmian positions of the particles must remain all clustered together in only one of these regions. This clustering is a consequence of the internal cohesive forces inside solid objects: the quantum Hamiltonian allows quantum superpositions of states where all particles move together in one, or another, region of space, but forbids states where some of the particles are in one region and others in another region (the cohesive forces inside the pointer of the measurement apparatus forbid states where the pointer is broken into two parts). Because the set of all the Bohmian positions must define a point in configuration space where the wave function does not vanish, they must remain grouped together.
As a consequence, in one region of space (occupied for instance by the pointer indicating a definite result), the single particle Bohmian density is much larger than the density predicted by the quantum superposition; in another region of space it is smaller, since the Bohmian density vanishes while the quantum density does not. The basic idea of our model is to introduce a dynamics where the quantum state vector is attracted to the first region, and repelled from the others. The quantum dynamics obtained in this way is completely deterministic: in a given realization of an experiment, the only random element is the initial Bohmian position of the configuration space of the physical system; once these positions are determined, no random process takes place (as is also the case in dBB theory).
Our purpose is not, of course, to claim that the dynamics we propose is highly plausible. The main conceptual interest of such models is their very existence, which proves that such approaches are neither impossible nor contradictory with known experimental results.\ This is similar to the existence of the dBB theory, which shows that some theorems concerning the impossibility of additional quantum variables are irrelevant. In a previous article \cite{SDAP}, we have already proposed a dynamics that also includes an attraction of the state vector towards regions of high Bohmian densities.\ Here we generalize and improve that model by introducing a spatial localization term that introduces even smaller perturbations (except, of course, situations involving QSMDS), because the added differential term in the dynamics remains almost zero in most cases. As a consequence, a larger flexibility is obtained for the values of the parameters of the dynamics; a relation with the Newton constant of gravity then becomes possible.
\section{Dynamic equation with a localization term}
We consider a system of $N$ identical spinless particles associated with a quantum field operator $\Psi\left( \mathbf{r}\right) $, which is defined at each point $\mathbf{r}$ of ordinary 3D space.
\subsection{Densities}
When the system is in state $\left\vert \Phi\right\rangle $, the local (number) density $D_{\Phi}\left( \mathbf{r}\right) $ of particles at $\mathbf{r}$ is: \begin{equation} D_{\Phi}\left( \mathbf{r}\right) =\frac{\left\langle \Phi\right\vert \Psi^{\dagger}\left( \mathbf{r}\right) \Psi\left( \mathbf{r}\right) \left\vert \Phi\right\rangle }{\left\langle \Phi\right. \left\vert \Phi\right\rangle } \label{sdap-1} \end{equation} In dBB theory, the local density $D_{B}\left( \mathbf{r}\right) $ of Bohmian positions is a sum of delta functions: \begin{equation} D_{B}\left( \mathbf{r,}t\right) =\sum_{n=1}^{N}\delta\left( \mathbf{r} -\mathbf{q}_{n}\right) \label{sdap-2} \end{equation} where the sum runs over all $N$ particles with Bohmian position $\mathbf{q} _{n}\left( t\right) $.
We wish to introduce a dynamics that favors evolutions where $D_{\Phi}\left( \mathbf{r}\right) $\ is attracted towards regions where $D_{B}\left( \mathbf{r}\right) $ takes higher values than $D_{\Phi}\left( \mathbf{r} \right) $, and repelled from regions where the opposite is true. Nevertheless, since $D_{B}\left( \mathbf{r}\right) $\ is singular, it is useful to introduce a space average. For this purpose, we choose a distance $a_{L}$ and a function $A(\mathbf{r)}$ that is localized around the origin of space within a distance $a_{L}$, for instance: \begin{equation} A_{L}(\mathbf{r)} = e^{-\left( \mathbf{r}-\mathbf{r}^{\prime}\right) ^{2}/\left( a_{L}\right) ^{2}}\label{sdap-2-bis} \end{equation} We then introduce the following integrals of $D_{\Phi}$ and $D_{B}$: \begin{equation} N_{\Phi}\left( \mathbf{r},t\right) =\int\text{d}^{3}r^{\prime} ~A_{L}(\mathbf{r}-\mathbf{r}^{\prime}\mathbf{)}~D_{\Phi}\left( \mathbf{r} ^{\prime},t\right) \label{sdap-3} \end{equation} and: \begin{equation} N_{B}\left( \mathbf{r,}t\right) =\int\text{d}^{3}r^{\prime}~A_{L} (\mathbf{r}-\mathbf{r}^{\prime}\mathbf{)}~D_{B}\left( \mathbf{r}^{\prime },t\right) =\sum_{n=1}^{N}~A_{L}(\mathbf{r}-\mathbf{q}_{n}\mathbf{)} ~\label{sdap-5} \end{equation}
The Gaussian form (\ref{sdap-2-bis}) is one possibility, but we could also have made different choices, for instance: \begin{equation} A_{L}(\mathbf{r)=}\frac{\left( a_{L}\right) ^{s}}{\left( a_{L}\right) ^{s}+\left( \mathbf{r}\right) ^{s}}\label{sdap-5-bis} \end{equation} where $s$ is any integer number that is larger than $2$. All the discussion of this article is actually independent of a particular choice of the localization function.
The order of magnitude of $N_{\Phi}\left( \mathbf{r},t\right) $ is the (quantum) average number of particles within a volume $\left( a_{L}\right) ^{3}$ around point $\mathbf{r}$; similarly, the order of magnitude of $N_{B}\left( \mathbf{r,}t\right) $ is the number of Bohmian positions inside the same volume.\ We have $0\leq$ $N_{\Phi}\left( \mathbf{r} ,t\right) ,N_{B}\left( \mathbf{r,}t\right) $. Since: \begin{equation} \int\text{d}^{3}r~N_{\Phi}\left( \mathbf{r},t\right) =\int\text{d} ^{3}r~N_{B}\left( \mathbf{r},t\right) = N \int\text{d}^{3}r ~ A_{L}(\mathbf{r)} \label{sdap-6} \end{equation} both these numbers have an upper bound that is $N$ times the space integral of $A_L (\mathbf r)$..
\subsection{Attractive dynamics}
We then define the (dimensionless) localization operator $L\left( t\right) $ by: \begin{equation} L\left( t\right) =\int\text{d}^{3}r~\Delta\left( \mathbf{r},t\right) ~\Psi^{\dagger}\left( \mathbf{r}\right) \Psi\left( \mathbf{r}\right) \label{sdap-8} \end{equation} where $\Delta(\mathbf{r},t)$ is defined as the difference:
\begin{equation} \Delta(\mathbf{r},t)=N_{B}\left( \mathbf{r},t\right) -N_{\Phi}\left( \mathbf{r},t\right) \label{sdap-7} \end{equation} This allows us to introduce a dynamics that favors evolutions where $D_{\Phi}\left( \mathbf{r}\right) $\ is attracted towards regions where $\Delta\left( \mathbf{r},t\right) >$ $0$.\ For this purpose, we add to the usual Hamiltonian $H\left( t\right) $ a localization term that is proportional to $L\left( t\right) $, and write the modified Schr\"{o}dinger equation:
\begin{equation} i\hslash\frac{d}{dt}\left\vert \Phi\left( t\right) \right\rangle =\Big[ H\left( t\right) +i\hslash\gamma_{L}~L\left( t\right) \Big] \left\vert \Phi\left( t\right) \right\rangle \label{sdap-9} \end{equation} where $\gamma_{L}$ is a constant localization rate. The new term in the Hamiltonian increases the modulus of the wave function in regions where $\Delta\left( \mathbf{r},t\right) $ is positive, reduces it in regions where the opposite is true.
Relation (\ref{sdap-6}) implies that the space integral of $\Delta\left( \mathbf{r},t\right) $ vanishes: \begin{subequations} \begin{equation} \int\text{d}^{3}r~\Delta\left( \mathbf{r},t\right) =0 \label{sdap-7-bis} \end{equation} But, as in \cite{SDAP}, we could also have defined $\Delta\left( \mathbf{r},t\right) $ as: \begin{equation} \Delta(\mathbf{r},t)=N_{B}\left( \mathbf{r},t\right) \label{sdap-7b} \end{equation} The right hand side of relation (\ref{sdap-7-bis}) would then be given by \ref{sdap-5}.
Since the operator acting in the right-hand side of (\ref{sdap-9}) is not Hermitian, this equation of evolution does not conserve the norm of $\left\vert \Phi\right\rangle $. Nevertheless, if desired, one can easily obtain a normalized state vector $\left\vert \overline{\Phi}\right\rangle $: \end{subequations} \begin{equation} \left\vert \overline{\Phi}\right\rangle =\left\vert \overline{\Phi}\left( t\right) \right\rangle =\frac{1}{\sqrt{\left\langle \Phi(t)\right. \left\vert \Phi(t)\right\rangle }}\left\vert \Phi(t)\right\rangle \label{sdap-10} \end{equation} which obeys the following equation of evolution: \begin{align} \frac{\text{d}}{\text{d}t}\left\vert \overline{\Phi}\left( t\right) \right\rangle & =\frac{1}{\sqrt{\left\langle \Phi(t)\right. \left\vert \Phi(t)\right\rangle }}\frac{\text{d}}{\text{d}t}\left\vert \Phi (t)\right\rangle -\frac{1}{2\left\langle \Phi(t)\right. \left\vert \Phi(t)\right\rangle ^{3/2}}\left( \frac{\text{d}}{\text{d}t}\left\langle \Phi(t)\right. \left\vert \Phi(t)\right\rangle \right) \left\vert \Phi(t)\right\rangle \nonumber\\ & =\frac{1}{i\hslash}\left[ H\left( t\right) +i\hslash\gamma_{L}~L\left( t\right) \right] \left\vert \overline{\Phi}\left( t\right) \right\rangle -\gamma_{L}\frac{1}{\left\langle \Phi(t)\right. \left\vert \Phi (t)\right\rangle ^{3/2}}~\left\langle \Phi(t)\right\vert L\left( t\right) \left\vert \Phi(t)\right\rangle ~\left\vert \Phi(t)\right\rangle \nonumber\\ & =\frac{1}{i\hslash}\left[ H\left( t\right) +i\hslash\gamma_{L}~L\left( t\right) \right] \left\vert \overline{\Phi}\left( t\right) \right\rangle -\gamma_{L}\int\text{d}^{3}r~\Delta\left( \mathbf{r},t\right) ~D_{\Phi }\left( \mathbf{r}\right) ~\left\vert \overline{\Phi}\left( t\right) \right\rangle \label{sdap-11} \end{align} We then obtain: \begin{equation} i\hslash\frac{\text{d}}{\text{d}t}\left\vert \overline{\Phi}\left( t\right) \right\rangle =\left[ H\left( t\right) +i\hslash\gamma_{L}~\overline {L}\left( t\right) \right] \left\vert \overline{\Phi}\left( t\right) \right\rangle \label{sdap-12} \end{equation} with: \begin{equation} \overline{L}\left( t\right) =\int\text{d}^{3}r~\left[ \Psi^{\dagger}\left( \mathbf{r}\right) \Psi\left( \mathbf{r}\right) -D_{\Phi}\left( \mathbf{r}\right) \right] \Delta\left( \mathbf{r,}t\right) \label{sdap-13} \end{equation}
If $\Delta\left( \mathbf{r,}t\right) $ is a constant in space, we remark that the effect of $\overline{L}\left( t\right) $ on any ket with a fixed number of particles vanishes.\ This is of course the case if $a_{L}=0$ ($N_{B}$, $N_{\Phi}$ and $\Delta$ then vanish), but also if $a_{L}=\infty$ (then $N_{B}=N_{\Phi}$ and $\Delta\left( \mathbf{r,}t\right)$ vanishes again). The localization term is effective only if $a_{L}$ takes an intermediate, finite, value for which $\Delta\left( \mathbf{r,}t\right) $ varies in space.
\subsection{Modified Schr\"{o}dinger equation} \label{effect-localization}
We now study the modified Schr\"{o}dinger equation (\ref{sdap-12}) in the position representation. The localization operator (\ref{sdap-13}) contains a first term in $\Psi^{\dagger}\left( \mathbf{r}\right) \Psi\left( \mathbf{r}\right) $ that has the form of a (symmetric) potential operator, diagonal in the position representation.\ This operator can also be written as a summation over all particles \cite{vol-3}: \begin{equation} \int\text{d}^{3}r~\left[ \Psi^{\dagger}\left( \mathbf{r}\right) \Psi\left( \mathbf{r}\right) \right] \Delta\left( \mathbf{r,}t\right) =\sum_{n=1} ^{N}\Delta\left( \mathbf{R}_{n}\mathbf{,}t\right) \label{sdap-14} \end{equation} where $\mathbf{R}_{n}$ is the position operator associated with the position of particle $n$. We therefore have: \begin{equation} \left\langle 1:\mathbf{r}_{1};2:\mathbf{r}_{2};..~;N:\mathbf{r}_{N}\right\vert \int\text{d}^{3}r~\Psi^{\dagger}\left( \mathbf{r}\right) \Psi\left( \mathbf{r}\right) ~\Delta\left( \mathbf{r,}t\right) ~\left\vert \overline{\Phi}\left( t\right) \right\rangle =\sum_{n=1}^{N}\Delta\left( \mathbf{r}_{n}\mathbf{,}t\right) ~\overline{\Phi}\left( \mathbf{r} _{1},\mathbf{r}_{2},..,\mathbf{r}_{n},..~,\mathbf{r}_{N};t\right) \label{calc-2} \end{equation} where $\overline{\Phi}\left( \mathbf{r}_{1},\mathbf{r}_{2},..,\mathbf{r} _{n},..\mathbf{r}_{N};t\right) $ is the wave function representing the $N$ particle system in configuration space (for the sake of simplicity, we assume that the particles have no spin).
As for the second term in the right hand side of (\ref{sdap-13}), it is just a c-number, proportional to the constant $<\Delta>$ defined by: \begin{equation} <\Delta>~=\frac{1}{N}\int\text{d}^{3}r~D_{\Phi}\left( \mathbf{r}\right) ~\Delta\left( \mathbf{r,}t\right) \label{calc-3} \end{equation} Since $D_{\Phi}\left( \mathbf{r}\right) /N$ is a distribution over space that is normalized to unity, $<\Delta>$ is the average of $\Delta\left( \mathbf{r,}t\right) $ over the one-body density of the wave function.
The evolution of the wave function due to the localization term is therefore: \begin{equation} \left. \frac{\text{d}}{\text{d}t}\right\vert _{\text{loc}}\overline{\Phi }\left( \mathbf{r}_{1},\mathbf{r}_{2},..~,\mathbf{r}_{n},..,\mathbf{r} _{N};t\right) =\gamma_{L}\left[ \sum_{n=1}^{N}\Delta\left( \mathbf{r} _{n}\mathbf{,}t\right) -N<\Delta>\right] \overline{\Phi}\left( \mathbf{r}_{1},\mathbf{r}_{2},..~,\mathbf{r}_{n},..,\mathbf{r}_{N};t\right) \label{calc-4} \end{equation} The wave function tends to increase in regions where many $\Delta\left( \mathbf{r}_{n}\mathbf{,}t\right) $ are larger than their space average value $<\Delta>$, and tends to decrease in regions of space where the opposite is true.
\subsection{Coupled evolutions} \label{coupled-evolutions}
We assume that the Bohmian positions $\mathbf{q}_{n}$ evolve according to the usual Bohmian equation of motion: \begin{equation} \frac{d\mathbf{q}_{n}\left( t\right) }{dt}=\hslash\frac{\bm\nabla_{n}\xi}{m} \label{sdap-5-ter} \end{equation} where $\xi\left( \mathbf{r}_{1},\mathbf{r}_{2},..,\mathbf{r}_{N}\right) $ is the phase of the wave function $\Phi\left( \mathbf{r}_{1},\mathbf{r} _{2},..,\mathbf{r}_{N}\right) $, and $\bm\nabla_{n}$ the gradient taken with respect to $\mathbf{q}_{n}$. In standard dBB theory, this relation ensures that the condition of \textquotedblleft quantum equilibrium\textquotedblright\ is satisfied at any time, if it is satisfied at the initial time.\ Nevertheless, this assumes that the equation of evolution of the wave function is the standard Schr\"{o}dinger equation, which is no longer the case in our model.\ In \cite{SDAP}, we argued that this was not a serious problem, since the localization term is very small, while Towler, Russell and Valentini \cite{Valentini-2005, Valentini-2012} have shown that a fast relaxation process drives the system quickly back to quantum equilibrium.\ Assuming quantum equilibrium should therefore still be an excellent approximation in most cases; we come back to this point in more detail in \S~\ref{quantum-equilibrium}.
\section{Collapse dynamics and time constants, examples} \label{examples}
We now explore the predictions of the model for different choices of the constants.\ We will see that the model is very robust: even with a large variation of its constants, it remains compatible with experimental observations. To illustrate this point, we choose either of the following pair of values: \begin{subequations} \label{calc-112-other} \begin{align} \gamma_{L} & =10^{-24}~\text{s}^{-1}\label{calc-112a-other}\\ a_{L} & =1\mu \text{m} =10^{-6}~\text{m} \label{calc-112b-other} \end{align} or the same values as those of GRW\ and CSL: \end{subequations} \begin{subequations} \label{calc-112} \begin{align} \gamma_{L} & =10^{-16}~\text{s}^{-1}\label{calc-112a}\\ a_{L} & =10~\mu \text{m} =10^{-5}~\text{m} \label{calc-112b} \end{align} We now examine a few situations where the consequences of these choices can be evaluated, depending on whether a superposition of macroscopically distinct states is involved or not.
\subsection{Various situations} \label{various-situations}
In (\ref{sdap-13}), only the first term in the bracket defining the localization operator $\overline{L}\left( t\right) $ is physically effective.\ This is because the second term in $D_{\Phi}\left( \mathbf{r}\right) $ introduces only a change of the norm of the whole state vector, which does not change its physical content; we will therefore ignore it in this section. Moreover, if $\Delta\left( \mathbf{r},t\right) $ is uniform in space, condition (\ref{sdap-7-bis}) shows that $\overline{L}\left( t\right) $ vanishes: the localization operator is non-zero only if, after averaging over a volume $(a_{L})^{3}$, the Bohmian and quantum densities still have different variations in space.\ Consider a region of space $R$ where the difference $N_{B}\left( \mathbf{r},t\right) -N_{\Phi}\left( \mathbf{r},t\right) $ takes a typical value $\Delta_{R}$; a localization process occurs in this region with a rate $\gamma_{L}\Delta_{R}N_{R}$, where $N_{R}$ is the number of particles contained in $R$.\ A distortion of the wave function, and therefore a physical modification of the properties of the system, occurs only if two (or more) regions $R$ and $R^{\prime}$ have values for $\Delta_{R}$ and $\Delta_{R^{\prime}}$ with different values.
For microscopic systems, the effect of the localization term remains extremely slow.\ Since both $\Delta_{R}$ and $N_{R}$ have an upper bound equal to the number of particles $N$, no localization in any region can occur at a rate higher than $\gamma_{L}N^{2}$. If we assume for instance $N=10^{6}$, with both choices (\ref{calc-112-other}) and (\ref{calc-112}) we obtain upper bounds of the localization rate of the order of $10^{-4}$ s$^{-1}$, or even much less. Moreover, these upper bounds can be approached only if the sign of $\Delta\left( \mathbf{r},t\right) $ is opposite in two different regions of space extending over more that $a_{L}$: if a microscopic system is localized in space in a region smaller than $a_{L}$, its localization rate is therefore even much smaller.
Clearly, to obtain a significative localization effect, it is required to have at the same time a large number of particles involved and spatial separations exceeding $a_{L}$. The optimal situation to detect an effect with a microscopic or mesoscopic system would probably be an interference experiment with a very large molecule or cluster \cite{Arndt-et-coll}, assuming that the distance between the slits is larger than $a_{L}$. Nevertheless, even with $10^{8}$ particles in the cluster, the time of flight along the two different paths should be at last $1$ second for a significant localization effect to be obtained if (\ref{calc-112a}) is selected, more than one year if (\ref{calc-112a-other}) is selected. In most cases, mesoscopic systems seem to be unaffected by the localization term.
The situation is radically different if a QSMDS is created, as is indeed the case during a quantum measurement. We assume that the distance between the positions of the pointer of the measurement apparatus indicating different results (or the distance between the positions of macroscopically distinct states) is larger than $a_{L}$.$\ $For simplicity, we assume that only two positions are possible, corresponding to two results of measurement (the generalization to more results is trivial).\ In this case, $\Delta\left( \mathbf{r},t\right) $ has large but opposite values in two distant regions of space. This is because $N_{\phi}\left( \mathbf{r,}t\right) $ and $N_{B}(\mathbf{r},t\mathbf{)}$ have completely different behaviors: on the one hand, the quantum density of particles and $N_{\phi}\left( \mathbf{r,}t\right) $ is distributed among two wave packets; on the other hand, in a single realization of the experiment, the Bohmian factor $N_{B}(\mathbf{r},t\mathbf{)}$ vanishes in one of the wave packets (the empty wave packet), while it takes a maximum value in the other. As discussed in the introduction, this results from the cohesive forces inside the pointer, which create strong quantum correlations between the positions of its constituent particles: in quantum mechanics, these particles can be at the same time in two different regions of space, but they have to remain all together in the same region.\ Therefore, since the Bohmian positions must define a point in configuration space where the $N$-particle wave function does not vanish, they have to remain grouped: all of them are in the same wave packet, none is in the other (empty wave). In one channel, $\Delta(\mathbf{r},t)\simeq N_{B}(\mathbf{r},t\mathbf{)-}N_{\Phi}(\mathbf{r},t\mathbf{)\simeq+}N_{\Phi }(\mathbf{r},t)$, in the other channel $\Delta(\mathbf{r},t)\simeq 0\mathbf{-}N_{\Phi}(\mathbf{r},t\mathbf{)\simeq-}N_{\Phi}(\mathbf{r},t)$.
In a quantum measurement situation, we can for instance assume that the apparatus contains a pointer that is a solid containing $N_{\Phi}\simeq 10^{11}$ particles per cubic micron (this rough order of magnitude seems to be reasonable for the number of atoms; the number of nucleons, or electrons, would be larger). If the total number of particles in the pointer is $N_{P}$, the differential rate of relaxation between the full wave (that associated with the result of measurement) and the other empty wave(s) is of the order of $2\gamma N_{\Phi}N_{P}$.\ If for instance the pointer is a tiny cube with $100\mu$m side only, $N_{P}\simeq10^{17}$, and we see that a superposition of two (or more) spatially separate states of the pointer disappears in about $10^{-4}$s. with (\ref{calc-112-other}), or $10^{-12}$ s. with (\ref{calc-112}).\ This is the time it takes the measurement apparatus to display a definite result.\ The model therefore ensures a rapid collapse of the wave function, even for tiny apparatuses of measurement. Another remark is that the so called \textquotedblleft surrealistic trajectories\textquotedblright should not exist within this model; actually, even within standard dBB theory, they already do not occur with macroscopic bodies \cite{Tastevin-Laloe}.
We also note that, even if $N_{\phi}\left( \mathbf{r,}t\right) $ may strongly differ from $N_{B}(\mathbf{r},t\mathbf{)}$ during a single realization of the experiment, this is only a short transient effect taking place while the measurement is completed; then the additional localization term in the Schr\"{o}dinger equation rapidly ensures that $N_{\phi}\left( \mathbf{r,} t\right) $ relaxes towards $N_{B}(\mathbf{r},t\mathbf{)}$. It modifies the state vector so that $N_{\phi}\left( \mathbf{r,}t\right) $ vanishes in all wave packets but one. After the measurement is completed, the dynamical relaxation process studied in \cite{Valentini-2005, Valentini-2012} ensures that the difference $N_{\phi}\left( \mathbf{r,} t\right) -N_{B}(\mathbf{r},t\mathbf{)}$ tends rapidly to zero. We discuss in \S~\ref{quantum-equilibrium} why it is possible to assume that the quantum equilibrium condition is restored when a second experiment is started.
One may wonder if the addition of a nonlinear localization term in the dynamics could produce dramatic unexpected effects, despite the extremely small value of the nonlinear coefficient. Indeed, the very purpose of the model is to obtain a dramatic effect during a measurement process: the state vector is suddenly projected onto one of its components, and all empty waves disappear. Does this extend to other situations? Mathematically, a similar question occurs with the Gross-Pitaevskii equation, which describes interacting Bose-Einstein condensates within mean field theory: an ideal gas is only marginally stable, since an infinitesimal attractive nonlinear term is sufficient to produce a collapse of the boson gas. The question then is: in what circumstances is the standard dBB theory only marginally stable with respect to the addition of an arbitrarily small nonlinear perturbation? Within our model, strong localizations effects occur as soon as the Bohmian density differs significantly from the quantum density. In the abscence of a QSMDS states, there is no special reason why this difference should be large. In a localized piece of bulk solid for instance, the Bohmian positions of the particles are randomly distributed in the volume occupied by the solid, and the coincidence between the two densities is rather good; the two terms in (\ref{calc-4}) then almost cancel each other (moreover, even without this cancellation, the localization term would only then to localize the solid inside its own volume, with no dramatic effect). The localization term is efficient mostly in the presence of quantum superpositions where many particles occupy different regions of space, that is basically QSMDS.
This discussion of various possible physical situations shows that a broad class of models is indeed compatible with the experimental observations that are known at present. In fact, we can choose either definition (\ref{sdap-7}) or (\ref{sdap-7b}) of $\Delta\left(\mathbf{r},t\right) $, and then select either (\ref{calc-112-other}) or (\ref{calc-112}) for the constants of the model: in all cases we obtain a fast time constant for the appearance of a single result of measurement, without introducing appreciable perturbations of microscopic systems. The basic reason for this flexibility is the quadratic dependence of the product $N_{\Phi}N_{P}$ in the density of macroscopic objects, which introduces a fast relaxation rate even with very small values of $\gamma_L$. In other words, for macroscopic objects, the relaxation rate varies proportionally to the square of the Avogadro number, which is an enormous number.
\subsection{Quantum equilibrium} \label{quantum-equilibrium}
The quantum equilibrium condition applies in configuration space, and therefore introduces more stringent condition than the equality of densities in ordinary space discussed in the preceding subsection.\ It also relates to an ensemble of realizations of the same experiment. Assuming that, initially, the Bohmian positions are randomly distributed, and that their distribution coincides with the quantum probability density (the modulus square of the wave function in the configuration space), the usual dBB theory ensures that the coincidence remains exact at all times.
This condition cannot be directly transposed to our modified dynamics, where different Bohmian positions associated with the same initial wave function lead to different wave functions at later times. The distribution of the positions can then no longer be compared to a single quantum probability density. For instance, just after a measurement has been performed (or, more generally, when a QSMDS has been projected onto one of its macroscopic components), the physical system is described by several different wave functions, depending on the result obtained in the experiment. We therefore have to modify the condition by requesting that, if we consider only the sub-ensemble of realizations having provided a specific result of measurement, the new wave function provides a density in configuration space that matches the distribution of Bohmian positions for this sub-ensemble.\ Clearly, this condition cannot be exactly fulfilled at all times, in particular during the (very short) projection process of the wave function.\
Nevertheless, once the projection process is complete, in the many particle system made of the entangled measured system $S$ and measurement apparatus $M$, one can again rely on the dynamical relaxation process discussed by Valentini et al. \cite{Valentini-2005, Valentini-2012} to restore quantum equilibrium.\ Indeed, these authors have shown that, at least in simple systems, a statistical distribution of the Bohmian position relaxes very quickly towards the modulus square of the wave function. Admittedly, we are making some extrapolation at this stage: as far as we know, there exist no systematic study in configuration space of the Bohmian dynamics of large entangled systems.\ Valentini has nevertheless shown \cite{Valentini-1991} that the approach to quantum equilibrium can be derived from a statistical \textquotedblleft subquantum theorem\textquotedblright , based on statistical assumptions that are analogous to those of classical statistical mechanics. The assumption we are making at this stage could probably be tested by more systematic numerical simulations of the Bohmian dynamics.
Within this scenario, assume that a second measurement is performed on the same system $S$ after the first experiment has been completed (its result has been registered). It is then legitimate to assume that the quantum equilibrium is obeyed for the sub-ensemble of experiments that have given a specific result in the first experiment.\ We then recover the standard rules of dBB mechanics and the Born rule, within a possible relative error of $10 ^{-16}$ or less if the whole experiment lasts one second; such an error rate is of course totally undetectable.
In other words, within our model, quantum equilibrium is no longer considered as a condition that is exactly met at all times. For instance, equilibrium is not yet reached while the result of an experiment is appearing on the pointer of a measurement apparatus; one has to wait until the result of measurement is fully registered. Quantum equilibrium is then rather seen as an emergent phenomenon \cite{Valentini-2009} that reappears after each measurement, so that the initial conditions for the next experiment that are extremely close to this equilibrium.
\subsection{Introducing the gravitational constant} \label{gravitational}
Instead of two arbitrary constants as in (\ref{calc-112}), it is possible to introduce only a single constant, for instance $a_{L}$, provided the Newton constant $G$ is taken into account. We may assume that: \end{subequations} \begin{equation} \gamma_{L}=\frac{m^{2}G}{\hbar a_{L}}\label{calc-116} \end{equation} where $m$ is the mass of a nucleon for instance. If we choose value (\ref{calc-112b-other}) for $a_{L}$, we obtain: \begin{equation} \gamma_{L}\simeq\frac{10^{-54}~6~10^{-11}}{10^{-34}~10^{-6}}\simeq 10^{-24}~\text{s}^{-1}\label{calc-117} \end{equation} which is indeed compatible with (\ref{calc-112a-other}). One can then assume that the limit between the macroscopic and microscopic world occurs at a characteristic length $a_{L}=1\mu$m, and then use relation (\ref{calc-116}) to \textquotedblleft explain\textquotedblright\ why $\gamma_{L}$ has the extremely small value given in (\ref{calc-112a-other}).
We can for instance consider that the collapsing field originates from the average gravitational attraction of the other identical particles within a range $a_{L}$. The source of this classical field is the average Bohmian density (not the average quantum density), which localizes the quantum state in space. This is similar to the theory proposed in Ref. \cite{Pearle-Squires}, within a stochastic dynamics. In quantum cosmogenesis, similar ideas have been proposed in Refs. \cite{Peter-Pinho-et-al} \cite{Pinho-Pinto} within the dBB theory, in order to treat the metric of general relativity classically by considering the Bohmian positions as the sources of gravity.
Needless to say, in relations (\ref{calc-112-other}), we can multiply $\gamma_{L}$ by any factor $\lambda$ and $a_{L}$ by $1/\lambda$ without changing this agreement. Actually, relation (\ref{calc-116}) only determines a velocity $c_{L}$ as: \begin{equation} c_{L}=\gamma_{L}a_{L}=\frac{m^{2}G}{\hbar} \label{calc-166-bis} \end{equation} One can also generalize (\ref{calc-116}) by introducing a universal dimensionless constant $\alpha_{L}$ as: \begin{equation} \alpha_{L}=\frac{m^{2}G}{\hbar c_{L}} \label{calc-166-ter} \end{equation} and consider models where this constant takes on any arbitrary dimensionless value, for instance $2\pi$, $1/137$, etc.
\section{Density operator} \label{density-operator}
We now examine the effect of the localization term on the evolution of the density operator, either in a single realization of the experiment, or by average over many realizations.
\subsection{Time evolution (single realization)}
If the system is in a normalized pure state $\left\vert \Phi\left( t\right) \right\rangle $, the density operator $\rho(t)$ is defined as: \begin{equation} \rho(t)=\left\vert \Phi\left( t\right) \right\rangle \left\langle \Phi\left( t\right) \right\vert \label{calc-118} \end{equation} The density $D_{\Phi}\left( \mathbf{r}\right) $ is now defined by: \begin{equation} D_{\rho}(\mathbf{r},t)=\text{Tr}\left\{ \Psi^{\dagger}\left( \mathbf{r} \right) \Psi\left( \mathbf{r}\right) \, \rho(t)\right\} \label{calc-120} \end{equation} $N_{\Phi}\left( \mathbf{r},t\right) $ is replaced by $N_{\rho}\left( \mathbf{r},t\right) $, obtained by substituting $D_{\rho}(\mathbf{r},t)$ to $D_{\Phi}\left( \mathbf{r}\right) $ in (\ref{sdap-3}). The same changes are made in (\ref{sdap-7}) and in the definition (\ref{sdap-13}) of $\overline{L}\left( t\right) $. Then $\rho(t)$ evolves according to the equation: \begin{equation} i\hbar\frac{\text{d}}{\text{d}t}\rho(t)=\left[ H\left( t\right) ,\rho(t)\right] +i\hslash\gamma_{L}~\left[ \overline{L}\left( t\right) ,\rho(t)\right] _{+} \label{calc-119} \end{equation} where $\left[ A,B\right] _{+}$ is the anticommutator $AB+BA$ of the two operators $A$ and $B$. This equation is nonlinear since $D_{\rho}(\mathbf{r},t)$, and therefore $\overline{L}\left( t\right) $, depends on $\rho(t)$.
We check that: \begin{align} i\hbar\frac{\text{d}}{\text{d}t}\text{Tr}\left\{ \rho(t)\right\} & =2i\hslash\gamma_{L}\text{Tr}\left\{ \int\text{d}^{3}r~\left[ \Psi^{\dagger }\left( \mathbf{r}\right) \Psi\left( \mathbf{r}\right) -D_{\rho}\left( \mathbf{r}\right) \right] \Delta\left( \mathbf{r,}t\right) ~\rho (t)\right\} \nonumber\\ & =2i\hslash\gamma_{L}\text{Tr}\left\{ \int\text{d}^{3}r~\left[ D_{\rho }\left( \mathbf{r}\right) -D_{\rho}\left( \mathbf{r}\right) \right] \Delta\left( \mathbf{r,}t\right) ~\rho(t)\right\} =0 \label{calc-120-2} \end{align}
\subsection{Average over many realizations}
The evolution of the density operator describing the average of many realizations of an experiment is given by the average of equation (\ref{calc-119}) over these realizations.\ If the system contains a single particle, its Bohmian position is different for each realization; during time evolution, it explores various regions of the wave function, as shown in the figures of \cite{Valentini-2005, Valentini-2012}.\ Therefore, when the average over many realizations is taken, $N_{B}\left( \mathbf{r,}t\right) $ as well as $\Delta\left( \mathbf{r,}t\right) $ play the role of random functions.\ It the system contains $N$ particles, $\Delta(\mathbf{r},t)$ is then the sum of $N$ fluctuating functions.\ In both cases, (\ref{calc-119}) becomes similar to a stochastic differential equation.\ We remark that, if the initial distribution of Bohmian variables coincides with the quantum distribution, the ensemble average of $\Delta(\mathbf{r},t)$ vanishes.\ The same is true of the average of the localization operator, which is linear in $\Delta(\mathbf{r},t)$. Therefore, if we take an average over many realizations of the experiment, and if $\Delta(\mathbf{r},t)$ and $\rho(t)$ remain uncorrelated, the average contribution of the localization term in the right-hand side of (\ref{calc-119}) vanishes.\ It is non-zero only when $\rho(t)$ and the fluctuations of the Bohmian positions around their quantum equilibrium positions become correlated.
The situation is therefore similar to a relaxation phenomenon created by an ensemble of $\mathbf{r}$-dependent fluctuating perturbations $\Delta(\mathbf{r},t)$. The so called \textquotedblleft motional narrowing\textquotedblright\ condition (see for instance \cite{Abragam, vol-2}) expresses that the perturbations have very little effect during their correlation time. In our case, for a single particle, this condition reads $\gamma_{L}\tau_{c}\ll1$, which is easily fulfilled with the very small value (\ref{calc-117}) of $\gamma_{L}$. The same remains obviously true for any microscopic system: the appearance of weak correlations between the quantum state of the system and the fluctuations of the Bohmian positions creates a relaxation process with a rate $\gamma_{L}^{2}\tau_{c}$, which remains negligible over a time equal to the age of the Universe. We therefore recover the standard equation of evolution of the density operator.
For a macroscopic system, the situation may be completely different: we have seen in \S~\ref{various-situations} that the localization term itself grows quadratically with the number of particles involved, so that the second order rate of localization $\gamma_{L}^{2}\tau_{c}$ is now multiplied by the fourth power of the number of particles (assuming that $\tau_c$ is independent of $N$). Since the Avogadro number is very large, one can easily obtain situation where the rate becomes very fast, and where the motional narrowing condition is actually no longer valid.\ This corresponds to situations where the von Neumann projection postulate may be applied and where the measurement apparatuses can be treated classically.
\subsection{Partial traces}
Assume that the complete system $S$ is made of two subsystems $S_{A}$ and $S_{B}$, which are localized in two disconnected regions of space $\mathscr{V}_{A}$ and $\mathscr {V}_{B}$, and contain $N_{A}$ and $N_{B}$ particles respectively, and have no mutual interaction.\ The two density functions $N_{A}(\mathbf{r})$ and $N_{B}(\mathbf{r})$ then have non-overlapping supports, so that both the Hamiltonian and the localization operator are then the sum of two terms: \begin{subequations} \label{calc-122} \begin{align} H\left( t\right) & =H_{A}\left( t\right) +H_{B}\left( t\right) \label{calc-122a}\\ \overline{L}\left( t\right) & =\overline{L}_{A}\left( t\right) +\overline{L}_{B}\left( t\right) \label{calc-122b} \end{align} In the space of states of a single particle, we choose a basis $\left\{ \left\vert u_{i}\right\rangle \right\} $ such that each of these states is localized, either in $\mathscr{V}_{A}$, or $\mathscr{V}_{B}$ (its wave function is zero in the other volume).\ A basis in the space of states of $S$ can be obtained with states where the occupation number $n_{i}$ of each $\left\vert u_{i}\right\rangle $ is specified, that is with the ensemble of kets: \end{subequations} \begin{equation} \left\vert u_{1}:n_{1};u_{2}:n_{2};...;u_{P}:n_{P}\right\rangle =\left\vert n_{A},n_{B}\right\rangle \label{calc-122-2} \end{equation} where $n_{A}$ is a condensed notation for all the occupation numbers of the sates localized in $\mathscr{V}_{A}$, and similarly $n_{B}$ a condensed notation for the occupation numbers of the states localized in $\mathscr{V}_{B}$. The matrix elements of the density operator $\rho$ describing $S$ are: \begin{equation} \left\langle n_{A},n_{B}\right\vert \rho(t)\left\vert n_{A}^{\prime} ,n_{B}^{\prime}\right\rangle \label{calc-122-3} \end{equation}
Any operator $A$ acting in $\mathscr{V}_{A}$ but not $\mathscr{V}_{B}$ changes the value of $n_{A}$ but not that of $n_{B}$.\ The average $\left\langle A\right\rangle $ of $A$ can therefore be obtained from the spatial trace $\rho_{A}$ of $\rho$ over \ region $\mathscr{V}_{B}$ defined as: \begin{equation} \left\langle n_{A}\right\vert \rho_{A}(t)\left\vert n_{A}^{\prime }\right\rangle =\sum_{n_{B}}\left\langle n_{A},n_{B}\right\vert \rho (t)\left\vert n_{A}^{\prime},n_{B}^{}\right\rangle \label{calc-122-4} \end{equation} where the sum over $n_{B}$ is taken over all possible values of the occupation number of the states localized in $\mathscr{V}_{B}$. The time evolution of $\rho_{A}(t)$ is obtained by taking the spatial trace of (\ref{calc-119}). The terms in $H_{A}\left( t\right) $ and $\overline{L}_{A}\left( t\right) $ give the same effect as in (\ref{calc-119}), with indices $A$ added to the operators.\ Moreover, as usual the term in $H_{B}\left( t\right) $ vanishes (the partial trace of the commutator is zero).\ As for the term in $\overline{L}_{B}\left( t\right) $, it leads to: \begin{align} \left. \frac{\text{d}}{\text{d}t}\right\vert _{L_{B}}\left\langle n_{A}\right\vert \rho_{A}(t)\left\vert n_{A}^{\prime}\right\rangle & =\gamma_{L}\sum_{n_{B}^{},n_{A}^{\prime\prime},n_{B}^{\prime\prime}}\left\{ \left\langle n_{A},n_{B}\right\vert \overline{L}_{B}\left( t\right) \left\vert n_{A}^{\prime\prime},n_{B}^{\prime\prime}\right\rangle \left\langle n_{A}^{\prime\prime},n_{B}^{\prime\prime}\right\vert \rho(t)\left\vert n_{A}^{\prime},n_{B}\right\rangle \right. \nonumber\\ & ~~~~~~~~~~~~~~~\left. +\left\langle n_{A},n_{B}\right\vert \rho (t)\left\vert n_{A}^{\prime\prime},n_{B}^{\prime\prime}\right\rangle \left\langle n_{A}^{\prime\prime},n_{B}^{\prime\prime}\right\vert \overline {L}_{B}\left( t\right) \left\vert n_{A}^{\prime},n_{B}\right\rangle \right\} \label{calc-124} \end{align} In the first term inside the summation, $n_{A}^{\prime\prime}=n_{A}$, while in the second term $n_{A}^{\prime\prime}=n_{A}^{\prime}$, so that the right hand side of this equation is equal to: \begin{equation} \gamma_{L}\sum_{n_{B}^{},n_{B}^{\prime\prime}}\left\{ \left\langle n_{B}\right\vert \overline{L}_{B}\left( t\right) \left\vert n_{B} ^{\prime\prime}\right\rangle \left\langle n_{A},n_{B}^{\prime\prime }\right\vert \rho(t)\left\vert n_{A}^{\prime},n_{B}\right\rangle +\left\langle n_{A},n_{B}\right\vert \rho(t)\left\vert n_{A}^{\prime},n_{B}^{\prime\prime }\right\rangle \left\langle n_{B}^{\prime\prime}\right\vert \overline{L} _{B}\left( t\right) \left\vert n_{B}\right\rangle \right\} \label{calc-125} \end{equation} where the two terms become identical as soon as the two dummy variables $n_{B}$ and $n_{B}^{\prime\prime}$ are interchanged.\ We therefore obtain: \begin{equation} \left. \frac{\text{d}}{\text{d}t}\right\vert _{L_{B}}\left\langle n_{A}\right\vert \rho_{A}(t)\left\vert n_{A}^{\prime}\right\rangle =2\gamma_{L}\sum_{n_{B}^{},n_{B}^{\prime\prime}}\left\langle n_{B}\right\vert \overline{L}_{B}\left( t\right) \left\vert n_{B}^{\prime\prime}\right\rangle \left\langle n_{A},n_{B}^{\prime\prime}\right\vert \rho(t)\left\vert n_{A}^{\prime},n_{B}\right\rangle \label{calc-126} \end{equation}
(i) If the matrix elements of the density operator of $S$ factorize: \begin{equation} \left\langle n_{A},n_{B}\right\vert \rho(t)\left\vert n_{A}^{\prime} ,n_{B}^{\prime}\right\rangle =\left\langle n_{A}\right\vert \rho _{A}(t)\left\vert n_{A}^{\prime}\right\rangle \times\left\langle n_{B}\right\vert \rho_{B}(t)\left\vert n_{B}^{\prime}\right\rangle \label{calc-127} \end{equation} we get: \begin{equation} \left. \frac{\text{d}}{\text{d}t}\right\vert _{L_{B}}\left\langle n_{A}\right\vert \rho_{A}(t)\left\vert n_{A}^{\prime}\right\rangle =2\gamma_{L}\left\langle n_{A}\right\vert \rho_{A}(t)\left\vert n_{A}^{\prime }\right\rangle ~\text{Tr}_{B}\left\{ \overline{L}_{B}\left( t\right) \rho_{B}(t)\right\} \label{calc-128} \end{equation} But we have seen in (\ref{calc-120-2}) that the trace in the right-hand side vanishes.\ If two subsystems occupy different regions of space, and if they are uncorrelated, each partial density operator evolves independently (as is the case in the absence of the localization term).
(ii) If the density operator of $S$ does not factorize, the preceding simplification does not occur. Let us first study the evolution of the density matrices in a single realization of an experiment. If the two systems $S_{A}$ and $S_{B}$ are entangled, the dynamical collapse acting on $S_{B}$ may affect the state of $S_{A}$, in the same way as the standard von Neumann collapse postulate can change at the same time the state of two remote entangled systems. Mathematically, the origin of this mutual effect of the two subsystems is the anticommutator that contains $\overline{L}\left( t\right) $ in (\ref{calc-119}), while the Hamiltonian appears in a commutator.\ Therefore, in the partial trace, while the two terms in $H_{B}$ cancel each other, the two terms in $\overline{L}_B\left( t\right) $ add to provide the double of each contribution.
The quantum nonlocality then manifests itself in two ways.\ The first also occurs in standard dBB theory, where the motion of the Bohmian positions of the whole system are guided in the configuration space by the wave function in this space.\ The second is due to the nonlocal effect of the collapse term in the equation of evolution. This effect is necessary to recover the results provided by the usual von Neumann reduction postulate in standard quantum mechanics.
(iii) Nevertheless, if many realizations of the experiment are performed, one has to consider the average of the localization operator $\overline{L}_{B}$ over these realizations. We have discussed in \S~\ref{quantum-equilibrium} the conditions under which the quantum equilibrium is obtained. If this is the case, $N_{\Phi}\left( \mathbf{r},t\right) $ and $N_{\rho}\left(\mathbf{r},t\right)$ are constantly equal, and the average of $\overline{L}_{B}$ vanishes. Therefore, there can be no influence of an experiment performed in region $B$ on the density operator in region $A$, which automatically ensures the non-signaling property necessary to obtain a model that is compatible with relativity. We recover the relation obtained by Valentini \cite{Valentini-2002} between quantum equilibrium and the no-signaling condition, which is thus also valid within our non-standard model.
\section{Effect of the localization term on the densities and currents} \label{position-representation}
We now study the effect of the localization term on the density of particles and on their current. As before, for the sake of simplicity, we assume that the particles are spinless.
\subsection{Evolution of the one-body density} \label{one-body-density}
We begin with the study of a pure state. The wave function is symmetric with respect to the exchange of particles. The one body density is then: \begin{equation} D_{\Phi}\left( \mathbf{r}\right) =N\int\text{d}^{3}r_{2}...\text{d}^{3} r_{N}~\left\vert \overline{\Phi}\left( \mathbf{r}_{1}=\mathbf{r} ,\mathbf{r}_{2},...,\mathbf{r}_{N};t\right) \right\vert ^{2} \label{calc-5} \end{equation} According to (\ref{calc-4}), the contribution of the localization term to its time evolution is given by: \begin{equation} \left. \frac{\text{d}}{\text{d}t}\right\vert _{\text{loc}}D_{\Phi}\left( \mathbf{r}\right) =2\gamma_{L}N\int\text{d}^{3}r_{2}...\text{d}^{3} r_{N}~\left[ \sum_{n=1}^{N}\Delta\left( \mathbf{r}_{n}\mathbf{,}t\right) -N<\Delta>\right] \left\vert \overline{\Phi}\left( \mathbf{r}_{1} =\mathbf{r},\mathbf{r}_{2},..,\mathbf{r}_{n},..,\mathbf{r}_{N};t\right) \right\vert ^{2} \label{calc-101} \end{equation}
In (\ref{calc-101}), the term $n=1$ merely introduces a term proportional to $\Delta\left( \mathbf{r,}t\right) D_{\Phi}\left( \mathbf{r}\right) $, which depends only on the single particle density; all the other terms contain the position correlation function $D_{\Phi}^{II}\left( \mathbf{r} ,\mathbf{r}^{\prime}\right) $ of two particles at points $\mathbf{r}$ and $\mathbf{r}_{p}$: \begin{equation} D_{\Phi}^{II}\left( \mathbf{r},\mathbf{r}^{\prime}\right) =N\left( N-1\right) \int\text{d}^{3}r_{3}...\text{d}^{3}r_{N}~\left\vert \overline{\Phi}\left( \mathbf{r}_{1}=\mathbf{r},\mathbf{r}_{2}=\mathbf{r} ^{\prime},...,\mathbf{r}_{N};t\right) \right\vert ^{2} \label{calc-201} \end{equation} Equation (\ref{calc-101}) \ then provides: \begin{equation} \left. \frac{\text{d}}{\text{d}t}\right\vert _{\text{loc}}D_{\Phi}\left( \mathbf{r}\right) =2\gamma_{L}\left[ \Delta\left( \mathbf{r,}t\right) ~D_{\Phi}\left( \mathbf{r}\right) +\int\text{d}^{3}r^{\prime}~D_{\Phi} ^{II}\left( \mathbf{r},\mathbf{r}^{\prime}\right) ~\Delta\left( \mathbf{r}^{\prime}\mathbf{,}t\right) -N<\Delta>D_{\Phi}\left( \mathbf{r}\right) \right] \label{calc-202} \end{equation} We do not get a closed equation for the evolution of the single particle density; the right hand side of this equation contains the two-particle density, as in the usual BBGKY\ hierarchy, and despite of the fact that the localization term is a single-particle operator.\ This is because the localization term is non-Hermitian, which introduces anticommutators instead of commutators.
Since: \begin{equation} \int\text{d}^{3}r~D_{\Phi}^{II}\left( \mathbf{r},\mathbf{r}^{\prime}\right) =\left( N-1\right) ~D_{\Phi}\left( \mathbf{r}^{\prime}\right) \label{calc-103} \end{equation} we can check the particle conservation rule: \begin{align} \int\text{d}^{3}r\left. \frac{\text{d}}{\text{d}t}\right\vert _{\text{loc} }D_{\Phi}\left( \mathbf{r}\right) & =2\gamma_{L}\left[ \int\text{d} ^{3}r~\Delta\left( \mathbf{r,}t\right) ~D_{\Phi}\left( \mathbf{r}\right) +\left( N-1\right) \int\text{d}^{3}r^{\prime}~D_{\Phi}\left( \mathbf{r} ^{\prime}\right) ~\Delta\left( \mathbf{r}^{\prime}\mathbf{,}t\right) -N^{2}<\Delta>\right] \nonumber\\ & =2\gamma_{L}\left[ N\int\text{d}^{3}r~\Delta\left( \mathbf{r,}t\right) ~D_{\Phi}\left( \mathbf{r}\right) -N^{2}<\Delta>\right] =0 \label{calc-104} \end{align}
If the system is not described by a pure state, but by a density operator $\rho$, the time evolution of the density $D_{\rho}(\mathbf{r},t)$ is obtained by the same calculation, with the simple substitution: \begin{equation} \left\vert \overline{\Phi}\left( \mathbf{r}_{1}=\mathbf{r},\mathbf{r} _{2},...,\mathbf{r}_{N};t\right) \right\vert ^{2}\Rightarrow\left\langle \mathbf{r}_{1}=\mathbf{r},\mathbf{r}_{2},...,\mathbf{r}_{N}\right\vert \rho(t)\left\vert \mathbf{r}_{1}=\mathbf{r},\mathbf{r}_{2},...,\mathbf{r} _{N}\right\rangle \label{calc-120-3} \end{equation} in all the equations, including (\ref{calc-201}), which becomes the definition of the two body density $D_{\rho}^{II}\left( \mathbf{r},\mathbf{r}^{\prime }\right) $. The time evolution of $D_{\rho}(\mathbf{r},t)$ is therefore given by: \begin{equation} \left. \frac{\text{d}}{\text{d}t}\right\vert _{\text{loc}}D_{\rho}\left( \mathbf{r}\right) =2\hslash\gamma_{L}\left[ \Delta\left( \mathbf{r,} t\right) ~D_{\rho}\left( \mathbf{r}\right) +\int\text{d}^{3}r^{\prime }~D_{\rho}^{II}\left( \mathbf{r},\mathbf{r}^{\prime}\right) ~\Delta\left( \mathbf{r}^{\prime}\mathbf{,}t\right) -N<\Delta>D_{\rho}\left( \mathbf{r}\right) \right] \label{calc-202-bis} \end{equation}
\subsection{Mean field, role of the spatial correlations}
When the distance between $\mathbf{r}$ and $\mathbf{r}^{\prime}$ becomes very large, the correlation function $D_{\Phi}^{II}\left( \mathbf{r} ,\mathbf{r}^{\prime}\right) $ factorizes.\ We can therefore introduce a function $F(\mathbf{r},\mathbf{r}^{\prime})$ by setting: \begin{equation} D_{\Phi}^{II}\left( \mathbf{r},\mathbf{r}^{\prime}\right) =~D_{\Phi}\left( \mathbf{r}\right) ~D_{\Phi}\left( \mathbf{r}^{\prime}\right) \left[ 1-F(\mathbf{r},\mathbf{r}^{\prime})\right] \label{A4} \end{equation} with: \begin{equation} F(\mathbf{r},\mathbf{r}^{\prime})\underset{\left\vert \mathbf{r} -\mathbf{r}^{\prime}\right\vert \rightarrow\infty}{\rightarrow}0 \label{A7} \end{equation} In (\ref{A4}) we have chosen to write a minus sign before $F(\mathbf{r},\mathbf{r}^{\prime})$ because, if two neighbor systems exchange particles, their local densities are anticorrelated; $F$ is then positive.\ Relation (\ref{calc-103}) provides: \begin{equation} \int\text{d}^{3}r^{\prime}~D_{\Phi}\left( \mathbf{r}^{\prime}\right) ~F(\mathbf{r},\mathbf{r}^{\prime})=1 \label{A6} \end{equation} The function $F(\mathbf{r},\mathbf{r}^{\prime})$ may change sign in general, but positive values dominate this integral.
If we insert (\ref{A4})\ into (\ref{calc-202}), the first term in the right hand side cancels the term in $N<\Delta> D_{\Phi}\left( \mathbf{r}\right)$. Using (\ref{A6}), we get: \begin{align} \left. \frac{\text{d}}{\text{d}t}\right\vert _{\text{loc}}D_{\Phi}\left( \mathbf{r}\right) & =2\gamma_{L}\left[ \Delta\left( \mathbf{r,}t\right) -\int\text{d}^{3}r^{\prime}~F(\mathbf{r},\mathbf{r}^{\prime})~D_{\Phi}\left( \mathbf{r}^{\prime}\right) ~\Delta\left(\mathbf{r}^{\prime}\mathbf{,} t\right) \right] D_{\Phi}\left( \mathbf{r}\right) \nonumber\\ & =2\gamma_{L}D_{\Phi}\left( \mathbf{r}\right) \int\text{d}^{3}r^{\prime }~F(\mathbf{r},\mathbf{r}^{\prime})~D_{\Phi}\left( \mathbf{r}^{\prime }\right) ~\left[ \Delta\left( \mathbf{r,}t\right) -\Delta\left( \mathbf{r}^{\prime}\mathbf{,}t\right) \right] \label{A8} \end{align} The \textquotedblleft local\textquotedblright\ character of this equation of evolution depends on the properties of $F(\mathbf{r},\mathbf{r}^{\prime})$, in particular whether it tends to zero sufficiently rapidly when the difference of positions increases.
(i) In mean-field theory, one merely assumes that $F(\mathbf{r},\mathbf{r}^{\prime})$ vanishes. The first line of relation (\ref{A8}) then becomes: \begin{equation} \left. \frac{\text{d}}{\text{d}t}\right\vert _{\text{loc}}D_{\Phi}\left(\mathbf{r}\right) =2\gamma_{L} \Delta\left( \mathbf{r,}t\right) D_{\Phi}\left( \mathbf{r}\right) \label{chp-moyen} \end{equation} Mean field theory merely predicts that $D_{\Phi}\left(\mathbf{r}\right)$ increases at points where $\Delta\left( \mathbf{r,}t\right)$ is positive, decreases at points where this function is negative.
(ii) Beyond mean-field theory, if $F(\mathbf{r},\mathbf{r}^{\prime})$ has a small range $l$ (range of correlations in the system), the localization term depends only on the values of $\Delta\left( \mathbf{r}^{\prime}\mathbf{,}t\right) $ in a small domain around $\mathbf{r}$. In the limit of a very small range where: \begin{equation} F(\mathbf{r},\mathbf{r}^{\prime})\sim\delta(\mathbf{r}-\mathbf{r}^{\prime}) \label{A10} \end{equation} The right-hand side of (\ref{A8}) vanishes.\ More generally, if $\Delta\left( \mathbf{r}^{\prime}\mathbf{,}t\right) $ is constant in the domain where $F(\mathbf{r},\mathbf{r}^{\prime})$ is not zero, relation (\ref{A8}) shows that the evolution of the density introduced by the localization process vanishes. If $\Delta\left( \mathbf{r}^{\prime}\mathbf{,}t\right) $ varies in space over a distance $l$, the integral can approximated by: \begin{equation} - \int_{r^{\prime}\lesssim l}\text{d}^{3}r^{\prime} ~ F(\mathbf{r},\mathbf{r} ^{\prime})~D_{\Phi}\left( \mathbf{r}^{\prime}\right) ~(\mathbf{r^{\prime} }-\mathbf{r})\cdot\bm{\nabla}\Delta\left( \mathbf{r,}t\right) \label{calc-111-2} \end{equation} If the product $F(\mathbf{r},\mathbf{r}^{\prime})D_{\Phi}\left( \mathbf{r}^{\prime}\right) $ varies linearly as a function of $\mathbf{r} ^{\prime}$ within its range $l$: \begin{equation} F(\mathbf{r},\mathbf{r}^{\prime})~D_{\Phi}\left( \mathbf{r}^{\prime}\right) =F(\mathbf{r},\mathbf{0})~D_{\Phi}\left( \mathbf{0}\right) +(\mathbf{r^{\prime}}-\mathbf{r})\cdot\bm{\nabla}(FD) \label{calc-200} \end{equation} we obtain: \begin{equation} \left. \frac{\text{d}}{\text{d}t}\right\vert _{\text{loc}}D_{\Phi}\left( \mathbf{r}\right) =-\gamma_{L}\frac{4\pi l^{5}}{15}\bm{\nabla}(FD)\cdot \bm{\nabla}\Delta\left( \mathbf{r,}t\right) \label{calc-401} \end{equation} This expression varies very rapidly with the range of correlation $l$.
(iii) But $F(\mathbf{r},\mathbf{r}^{\prime})$ can also have a large range. For instance, just after a measurement has been performed, the correlation function between two different positions of the pointer vanishes, while the produce of one-body densities does not.\ In this case, the transfer of density due to the localization term is not local. This is a necessary feature to eliminate MDQS efficiently, and to obtain a projection after measurement.
We conclude from this discussion that the localization term has little effect in most (ordinary) situations.\ But, if a MDQS\ appears for some reason (Schr\"{o}dinger cat, etc.), it is promptly reduced to one of its components by the localization term.
\subsection{Evolution of the local momentum} \label{local-momentum}
We have only studied the direct effects of the localization term on the local density, but of course this term also has indirect effects: by localizing the wave functions in space, it changes their Fourier transform, and therefore the average value of the velocities.\ At later times, this change will also modify the average positions of the particles.
The local current of particles at point $\mathbf{r}$ is: \begin{align} \mathbf{J}_{\Phi}(\mathbf{r},t)=\frac{\hbar}{2im}
{\displaystyle\sum\limits_{p=1}^{N}}
\int\text{d}^{3}r_{1} & ...\int\text{d}^{3}r_{p-1}\int\text{d}^{3} r_{p+1}...\int\text{d}^{3}r_{N}\nonumber\\ & \overline{\Phi}^{\;\ast}\left( \mathbf{r}_{1},\mathbf{r}_{2} ,...,\mathbf{r}_{p}=\mathbf{r},...\mathbf{r}_{N};t\right) ~\bm{\nabla}_{\mathbf{r}}~\overline{\Phi}\left( \mathbf{r}_{1},\mathbf{r} _{2},...,\mathbf{r}_{p}=\mathbf{r},...\mathbf{r}_{N};t\right) ~+\text{c.c.} \label{abd-1} \end{align} where c.c. means complex conjugate. The time derivative of this current induced by the localization process is obtained by using equation (\ref{calc-4}).\ The derivative of $\overline{\Phi}\left( \mathbf{r} _{1},\mathbf{r}_{2},...,\mathbf{r}_{p}=\mathbf{r},...\mathbf{r}_{N};t\right) $ introduces the expression: \begin{align}
\frac{\hbar \gamma_{L}}{2im} {\displaystyle\sum\limits_{p=1}^{N}} \bm{\nabla}_{\mathbf{r}}~\Big[ \Delta (\mathbf{r},t) & +\sum_{n\neq p} \Delta(\mathbf{r}_{n},t)-N\Delta\Big] \overline{\Phi}\left( \mathbf{r}_{1},\mathbf{r}_{2},...,\mathbf{r}_{p}=\mathbf{r},...\mathbf{r} _{N};t\right) \nonumber\\ & \frac{\hbar \gamma_{L}}{2im} \Big[ \Delta(\mathbf{r},t)+ \sum_{n\neq p} \Delta(\mathbf{r} _{n},t)-N\Delta \Big] \overline{\Phi}\bm{\nabla}_{\mathbf{r}}~\overline {\Phi}\left( \mathbf{r}_{1},\mathbf{r}_{2},...,\mathbf{r}_{p}=\mathbf{r} ,...\mathbf{r}_{N};t\right) \nonumber\\ & \hspace{3cm}+ \frac{\hbar \gamma_{L}}{2im} \left[ \bm{\nabla}_{\mathbf{r}} ~\Delta(\mathbf{r},t)\right] \overline{\Phi}\left( \mathbf{r}_{1} ,\mathbf{r}_{2},...,\mathbf{r}_{p}=\mathbf{r},...\mathbf{r}_{N};t\right) \label{abd-2} \end{align} The first term in the right hand side of the second line reconstructs the particle current $\mathbf{J}(\mathbf{r},t)$, multiplied by $\Delta(\mathbf{r},t)$; the rest of the second line introduces a new integral. The term in the third line introduces the single particle density (\ref{calc-5}), which is real; since the whole term is multiplied by $i$ in (\ref{abd-1}), this term disappears when the real part is taken.\ If we combine these terms with those resulting from the derivative of $\overline {\Phi}^{\;\ast}$, we obtain: \begin{align} \left. \frac{\text{d}}{\text{d}t}\right\vert _{\text{loc}}\mathbf{J}_{\Phi }\left( \mathbf{r}\right) & =2\gamma_{L}\left[ \Delta(\mathbf{r} ,t)-\Delta\right] \mathbf{J}_{\Phi}\left( \mathbf{r}\right) \nonumber\\ & +\frac{\hbar\gamma_{L}}{2im}
{\displaystyle\sum\limits_{p=1}^{N}}
\int\text{d}^{3}r_{1}...\int\text{d}^{3}r_{p-1}\int\text{d}^{3}r_{p+1} ...\int\text{d}^{3}r_{N}\sum_{n\neq p}\left[ \Delta(\mathbf{r}_{n} ,t)-\Delta\right] ~\overline{\Phi}^{\;\ast}\bm{\nabla}_{\mathbf{r}} ~\overline{\Phi}~+\text{c.c.} \label{abd-3} \end{align} The first line of this equation adds an exponential increase or decrease of the particle current with a fluctuating rate, which can be positive or negative. The second line couples the current $\mathbf{J}_{\Phi}\left( \mathbf{r}\right) $ to another integral. Since $\gamma_{L}$ is very small, the influence of the localization process on the current of particles remains very small in most situations, except when the difference $\left[ \Delta(\mathbf{r}_{n},t)-\Delta\right] $ can take on very large values, as is the case during a measurement process (\S~\ref{various-situations}).
In standard GRW or CSL theory, the random localization process is a pure Markov process. The corresponding changes of the momentum of an object are then given by a random walk with no memory, and no preferred direction \cite{Collett-Pearle-2003}. In the model of the present article, the localization process has a non-zero memory arising from the statistical properties of $\Delta(\mathbf{r},t)$. These properties depend on the complicated nonlinear relative motion of the Bohmian positions and the wave function. Of course, for macroscopic objects, the corresponding time constants are very large, due to the very small value of the localization time constant $\gamma_L$. Nevertheless, on very long time scales such as those often considered in astrophysics, it may be that observable effects are predicted. Such predictions are nevertheless difficult to make, since it is not easy to evaluate the spatial and temporal scales of the fluctuations of the localization source $\Delta (\mathbf r ,t )$.
\section{Possible interpretations}
In terms of possible interpretations of quantum mechanics, the model is relatively robust: it can remain compatible with very different points of view.\
One can consider that the Bohmian variables are just a mathematical tool to introduce the stochastic reduction of the state vector, which then represents physical reality, in the line of GRW and CSL\ theories.\ Indeed, we have not assumed that Bohmian variables directly provide the results of position measurements, but that the results are determined by the quantum density in space provided by the state vector: they
depend on the values of $D_{\Phi}\left( \mathbf{r}\right) $, not on those of the Bohmian positions. One may also consider that $D_{\Phi}\left( \mathbf{r}\right) $ gives a direct description of physical reality or ordinary 3D space, coming back to a fluid representation of matter, as envisaged initially by Schr\"{o}dinger when he introduced his equation. This view is just the opposite of the usual dBB theory, where it is assumed that the observations reveal the values of the Bohmian positions, which therefore directly represent the physical reality. Here, the position in the configuration space is just a mathematical variable that drives the wave function and the associated quantum density in space; it plays a role that is analogous to the \textquotedblleft subquantum medium\textquotedblright\ acting on the state vector in F\'{e}nyes-Nelson theories \cite{Fenyes, Nelson}. One can even combine this new dynamics with the Everett interpretation; if one includes the memory registers of the observers into $\Psi$, one obtains a sort of \textquotedblleft Everett interpretation with projection\textquotedblright, predicting the existence of a \textquotedblleft single world\textquotedblright.
But one can also prefer the usual approach of the dBB theory, and consider that the Bohmian positions represent the beables \cite{Bell-local-beables, Bell-livre} of the physical system.\ The advantage of the model is then to get rid of all the empty waves of the dBB theory, and of the relative difficulty to attribute them a status \cite{Lewis-2007}. If one sees the wave function as similar to a Lagrangian or Hamiltonian in classical mechanics \cite{Bricmont}, it seems preferable to keep only the effective part of this wave function, eliminating all empty waves that have accumulated in the past.
One can also take an intermediate point of view, and consider for instance that what represents physical reality is the spatial density $N_{B}(\mathbf{r},t)$.
In this view, the problem of the \textquotedblleft long tails\textquotedblright\ occurring in the usual spontaneous theories becomes irrelevant, since no particular physical meaning is attributed to the exponentially vanishing empty waves.
Whatever status is eventually attributed to the state vector $\Psi$, it remains clear that it is less disconnected from physical reality than with a dynamics having no collapse mechanism. Nevertheless, the various mathematical components of this dynamics can be interpreted in different ways, leading to various ontologies.
\section{Discussion and conclusion}
We have seen that the attraction of the Bohmian densities can be used to obtain a reasonable model of spontaneous collapse of the state vector. In addition to the standard \textquotedblleft pilot wave\textquotedblright\ of the dBB theory, we have introduced a \textquotedblleft pilot density\textquotedblright\ for the wave. The corresponding dynamic is deterministic: the stochasticity of the initial position in configuration space is sufficient to reproduce the standard prediction of the Born rule for probabilities. The localization term tends to constantly adapt the wave function in order to obtain a better match between the quantum density
and the density of positions in space. In QSMDS situations, all the empty waves disappear, so that the wave function and the Bohmian positions progress together in time, while in usual dBB theory they are disconnected. For instance, it has been claimed that the dBB theory is really \textquotedblleft a many-world theory with a superfluous configuration appended to one of the worlds\textquotedblright , and that \textquotedblleft pilot-wave theories are parallel-universes theories in a state of chronic denial \textquotedblright\ \cite{Deutsch-1986}; for discussions of these claims, see for instance \cite{Valentini-2007} and \cite{Brown-2009}. Clearly, if one accepts the present model where the dynamics of the state vector is coupled to the Bohmian positions, this discussion is settled.
The mechanism of the projection in the dynamics is actually based on the cohesion of macroscopic objects.\ Because standard theory predicts that, even if such objects reach in QSMDS, their constituent particles remain strongly correlated spatially, the mechanism of our model projects them into a single localization.\ In fact, only macroscopic objects that do not break spontaneously into several parts acquire in this way a unique spatial localization: the \textquotedblleft moon is there even if nobody looks\textquotedblright\ \cite{Mermin}, and the reason why the center of the moon occupies a well defined point on its orbit before any measurement is the internal cohesion of the moon. By contrast, objects than can be split into several components without breaking any energy barrier can go trough QSMDS without collapsing; for instance, Bose-Einstein gaseous condensates that are split into two remote parts can give rise later to quantum interference effects, as discussed in more detail in \cite{SDAP}.
The localization term also provides a sharp transition between the quantum and classical regime.\ For a molecule or cluster containing $N$ particles in a volume smaller than $a_{L}$, any superposition of several quantum states localized at a distance larger that $a_{L}$ is projected into one single component of this superposition with a rate $\Gamma$ that varies quadratically as a function of $N$: \begin{equation} \Gamma\simeq\gamma_{L}~N^{2} \label{gamma-1} \end{equation} With the choice (\ref{calc-112-other}) of constants, and if the number of particles is about $10^{12}$, this relation predicts a localization time of $1$ s. Since the number density $n$ of solid (or liquid) physical objects is of the order of $10^{30}$ atoms/cubic meter, the same relation can be expressed in terms of the size of the object: \begin{equation} \Gamma\simeq\gamma_{L}~n^{2}l^{6} \label{gamma-2} \end{equation} For an object of size $l<a_{L}$ containing $n$ particles per unit volume, this rate varies proportionally to the sixth power of the size $l$; the localization time is $1$ second if $l\simeq1\mu $m. For an object of size $l>a_{L}$ containing $N$ particles, the rate becomes: \begin{equation} \Gamma\simeq\gamma_{L}~na_{L}^{3} n l^3 \label{gamma-3} \end{equation} One can therefore consider that $l=a_{L}$ and $N=10^{12}$ provide the border between standard quantum and classical behavior of physical objects.
It is clear that the model differs from the GRW\ and CSL\ theories in several respects. Beyond the fact that the dynamics is not stochastic, already mentioned in the introduction, another difference is that the localization process is no longer a single particle process, where each of them is localized independently, but results from a collective effect between the particles; this creates correlations between them, as remarked at the end of \S~\ref{effect-localization}. This collective character introduces a sharper transition between the quantum and classical regimes, due to the quadratic term in $N$ in (\ref{gamma-1}). This in turn is a consequence of the fact that the number of particles enters twice in the model, once in the Bohmian number $N_{B}\left( \mathbf{r},t\right) $, and once in the integral of the quantum operator $\Psi^{\dagger}\left( \mathbf{r}\right) \Psi\left( \mathbf{r}\right) $. In QSMDS\ situations, our localization term is therefore \textquotedblleft stronger\textquotedblright\ than those of GRW\ and CSL.\ Nevertheless, in usual situations, it is \textquotedblleft softer\textquotedblright: for instance, in a bulk piece of matter located in a single region of space, the Bohmian density coincides almost perfectly with $D_{\Phi}\left( \mathbf{r}\right) $, and in our model the collapse term has practically no effect.\ By contrast, it is constantly active in GRW and CSL\ theories.\ In other words, in experiments such that of Ref \cite{experiment-heating}, our model would be compatible with the observation of zero heating.\ We also note that the term that we added to the standard Schr\"{o}dinger equation is not only non-linear, but also non-local: when the two components of the wave function of a macroscopic object begin to separate in space, one component is transferred in space to the other at some finite distance. This feature is necessary to fit with the quantum predictions in Bell type experiments.
For macroscopic systems, the appearance of position uniqueness is not necessarily the only effect predicted by the model. We have seen in \S \ \ref{local-momentum} that another effect of the localization term may be to change the local current of the particles, which will have an indirect effect on the the density. This is not surprising since a space localization of the wave function implies an increase of the width of its Fourier transform, which corresponds to higher velocities. This violates the usual momentum conservation rule; similarly, the GRW and CSL theories predict a spontaneous heating effect that seems to violate the energy conservation rule \cite{Pearle-Mullin-Laloe}. Whether or not our model predict slow changes of the momentum of macroscopic objects, for instance on a cosmological time scale, remains to be studied.
We have also seen in \S~\ref{quantum-equilibrium} that, within this model, the Born rule is no longer a postulate; it emerges from the dynamics, and it is actually not an exact rule. It nevertheless remains a fantastically good approximation, with an accuracy better than $10^{-16}$ if the preparation and the measurement of quantum system are separated by more than 1s.
Needless to say, the class of models we have discussed is, in a sense, very naive. It is neither relativistic nor expressed in terms of a plausible field theory. Its purpose is just to indicate a range of possibilities, which might be exploited in a second stage to construct a more credible theory. This range is relatively broad, for several reasons. The first obvious reason is that there exist a large domain of possible values for the two constants $a_{L}$ and $\gamma_{L}$ that introduce no contradiction with the known experimental facts; this robustness of the model is both a strength and a weakness, since having too much flexibility amounts to reducing the predictive power of a theory. A second reason is that the equations are not particularly plausible: they are just the simplest version of a dynamics where position variables are added to the state vector, and where this vector is attracted towards these additional positions. The localization function $A_{L}(\mathbf{r}-\mathbf{r}^{\prime}\mathbf{)}$ that we have introduced in (\ref{sdap-2-bis}) is arbitrary, and the choice of a Gaussian has no particular justification.\ Even the form of the operator $L\left( t\right) $ could be changed: it might be possible to re-introduce at this stage the effect of gravitation by choosing another localization operator, more in the line of the ideas of Refs.\ \cite{Diosi-1989, GGR-1990, Penrose-1996}; such a possibility remains to be explored. Finally, we have not specified the nature of the particles on which the localization process should apply: they could be for instance, nucleons only, or nucleons and electrons, or quarks, etc. In any case, at this stage, even the order of magnitude of the constants is not determined. As mentioned in the introduction, the main merit of this class of models is only to show that they various quantum descriptions of physical reality remain compatible with known experimental data.
\end{document} |
\begin{document}
\title{Analysis of Different Types of Regret in Continuous Noisy Optimization}
\author{Sandra Astete-Morales, Marie-Liesse Cauwet, Olivier Teytaud\\ TAO/Inria Saclay-IDF, Univ. Paris-Saclay\\ Bat. 660, rue Noetzlin, Gif-Sur-Yvette, France }
\maketitle
\begin{abstract}
The performance measure of an algorithm is a crucial part of its analysis. The performance can be determined by the study on the convergence rate of the algorithm in question. It is necessary to study some (hopefully convergent) sequence that will measure how ``good'' is the approximated optimum compared to the real optimum.
The concept of \emph{Regret} is widely used in the bandit literature for assessing the performance of an algorithm. The same concept is also used in the framework of optimization algorithms, sometimes under other names or without a specific name. And the numerical evaluation of convergence rate of noisy algorithms often involves \emph{approximations} of regrets. We discuss here two types of approximations of Simple Regret used in practice for the evaluation of algorithms for noisy optimization. We use specific algorithms of different nature and the noisy sphere function to show the following results. The approximation of Simple Regret, termed here Approximate Simple Regret, used in some optimization testbeds, fails to estimate the Simple Regret convergence rate. We also discuss a recent new approximation of Simple Regret, that we term Robust Simple Regret, and show its advantages and disadvantages.
\end{abstract}
\sloppy
\section{Introduction}\label{transfo}
The performance measure of an algorithm involves the evaluation of the quality of the approximated optimum with regards to the real optimum. This can be done by using the concept of \emph{Regret}, well studied in the machine learning literature and used in the noisy optimization literature, sometimes under other names. Basically, in the optimization framework, the regret acounts for the ``loss'' of choosing the point used in the algorithm over the best possible choice: the optimum. Therefore, we measure the difference between the point used/recommended by the algorithm and the optimum in terms of the objective function.
In general, an optimization algorithm searches for the optimum, and to do so, it produces iteratively \emph{search points} which will be evaluated through the objective function. And at regular steps, the algorithm must return a \emph{recommendation point} that will be the best approximation to the optimum so far. Note that the recommendation point can be equal to a search point, but not necessarily.
The most usual form of regret is termed \emph{Simple Regret}. The \emph{Simple Regret} measures the distance, in terms of fitness values, between the optimum and the recommendation point output by the algorithm. It is widely used (possibly without this name) in noisy optimization \cite{fabian,chen1988,BubeckE09}. However, some test beds, notably the Bbob/Coco framework in the first version, did not allow the distinction between search points (at which the fitness function is evaluated) and recommendations (which are output by the algorithm as an approximation of the optimum), so that the Simple Regret can not be checked. This leads to the use of an \emph{Approximate Simple Regret} (name by us), which evaluates the fitness difference between the search points (and not the recommendations) and the optimum. Later, another form of regret, that we will term here \emph{Robust Simple Regret}, was also proposed, using recommendation points. We analyze in this paper the use of different regrets that aim to estimate the quality of the approximated optimum in a similar way. In particular, we show to which extent they lead to incompatible performance evaluations of the same algorithm over the same class of noisy optimization problems, i.e. the convergence rate for the Approximate or the Robust Simple Regret overestimates or underestimates the convergence rate of the more natural simple regret. We also prove some new results in terms of Simple Regret itself.
\section{Framework and Regrets}\label{sec:framework}
This section is devoted to the formalization of the noisy optimization problem considered and the analysed regrets. We will focus on the \emph{Simple Regret} and the alternative definitions that aim to approximate it (denoted here \emph{Approximate Simple Regret} and \emph{Robust Simple Regret}). At the end of the section we will highlight some general relationships between the presented regrets.
\subsection{Continuous Noisy Optimization}\label{sec:noisyOpt} Given a fitness function $F:D\subset {\mathbb R}^d\to {\mathbb R}$, also known as objective function, optimization (minimization) is the search for the optimum point $x^*$ such that $\forall x\in D, F(x^*)\leq F(x)$. The fitness function is corrupted by additive noise. In other words, given a search point $x\in D$, evaluating $F$ in $x$ results in an altered fitness value $f(x,\bm{w})$ as follows: \begin{equation}\label{additnoise}
f(x,\bm{w})=F(x)+\bm{w}, \end{equation} where $\bm{w}$ is an independent random variable of mean zero and variance $\sigma$. In the present paper, we will consider a simple case, namely a standard Gaussian additive noise \footnote{More general cases such that ${\mathbb E} f(x,\bm{w})=F(x)$ can be considered, as most algorithms do not request the noise to be additive and independent; the key point is the absence of bias. \mlc{maybe some cites here? To me, this comment is not clear enough.}\sam{Maybe the comment is not necessary?}}. In addition,
we assume that $F(x)=\|x-x^*\|^2$, where $x^*$ is randomly uniformly drawn in the domain $D$.
\emph{Noisy optimization} is then the search for the optimum $x^*$ such that ${\mathbb E}_\bm{w} f(x^*,\bm{w})$ is approximately minimum, where ${\mathbb E}_\bm{w}$ denotes the expectation over the noise $\bm{w}$.
Consider a noisy black-box optimization scenario: for a point $x$, the only available information is the noisy value of $F$ in $x$ as given by $f(x,\bm{w})$ for some independent $\bm{w}$. An optimization algorithm generates $x_1,x_2,\dots,x_n,\dots$, successive \emph{search points} at which the objective function is evaluated in a noisy manner. It also generates $\tilde x_1,\tilde x_2,\dots,\tilde x_n,\dots$ which are \emph{recommendations} or \emph{approximations of the optimum} after $n$ fitness evaluations are performed.
\subsection{Simple Regret and variants}\label{sec:criteria} The \emph{Simple Regret} ($SR$) focuses only on approximating the optimum in terms of fitness values. Its definition is: \begin{equation*}
SR_n={\mathbb E}_\bm{w} \left(f(\tilde x_n,\bm{w})-f(x^*,\bm{w})\right) = F(\tilde x_n)-F(x^*) \end{equation*} Notice that the expectation operates only on $\bm{w}$ in $f(\tilde x_n,\bm{w})$, and not on $\tilde x_n$. As a consequence $SR_n$ is a random variable due to the stochasticity of the noisy evaluations of the search points or the (possible) internal randomization of the optimization algorithm.
The $SR$ can be a part of the performance evaluation of an algorithm. In the noise-free case it can be used to determine the \emph{precision} of a method, by ensuring that the algorithm outputs a recommendation $\tilde{x}_m$ satisfying $SR_m \leq \epsilon $. Even more, when testing algorithms, it is common to use the ``first hitting time'' (FHT). FHT in fact refers to the first ``stable'' hitting time, i.e. the next recommendation is at least as good as the previous one. This is a reasonable assumption for algorithms solving noise-free problems. In this case the FHT is the minimum $n$ such that $SR_n\leq \epsilon $, provided that the recommendation is defined as $\tilde x_n=x_{i(n)}$ with $1\leq i(n)\leq n$ minimizing $SR_{i(n)}$. However, there is no exact equivalence or natural extension for the FHT with precision $\epsilon $ on noisy optimization. The algorithm only has access to noisy evaluations hence it cannot compute with ``certainty'' $SR_n$, which corresponds to the precision of the algorithm.
An alternative definition, that aims to measure the precision in a similar way to $SR$, is the \textit{Approximate Simple Regret}\footnote{The name is proposed by us.} ($ASR$), defined by: \begin{equation*}
ASR_n=\min_{m\leq n} F(x_m)-F(x^*). \end{equation*} $ASR$ takes in account the ``best'' evaluations among all the search points. It is used in the Bbob/Coco framework \cite{nbbob1,nbbob2,nbbob3,nbbob4,nbbob5,nbbob6,nbbob7,nbbob8,nbbob9}, and in some theoretical papers \cite{fogaasr}.
Notice that since $ASR$ is non-increasing, the FHT can be computed.
In this paper we will also discuss another variant of regret, the \emph{Robust Simple Regret}\footnote{Discussed on Bbob-discuss mailing list (\url{http://lists.lri.fr/pipermail/bbob-discuss/2014-October.txt}. The name is proposed by us.} ($RSR$), defined by:
\begin{equation*}
RSR_n=\min_{k\leq n}\max_{(k-\lfloor g(k)\rfloor)< m\leq k}\left( F(\tilde x_m)-F(x^*)\right), \end{equation*} where $g(n)$ is a polylogarithmic function of $n$ and $\lfloor \cdot \rfloor$ is the floor function. The $RSR$ is the ``best'' SR since the beginning of the run, sustained during $\lfloor g(k)\rfloor$ consecutive recommendations \footnote{If $(k-\lfloor g(k)\rfloor)<0$, then the $\max$ on the definition considers indexes between $1$ and $k$}. The polylogarithmic nature of $g(\cdot)$ is explained by the following argument: $g(k)$ be large enough, so that we have a
correct recommendation confirmed over $g(k)$ iterations, but small enough, so that we do not have to wait many evaluations before acknowledging that a correct recommendation has been found. The $RSR$ uses the recommendation $\tilde x_m$ instead of the search point $x_m$ used in $ASR$. But it uses the worst of a sequence of recommendations.
As a side note about the definition of $RSR$, it was originally proposed to use a quantile instead of the maximum. The ``quantile version'' (without this name), was proposed to become part of the performance measure in Bbob/Coco. However, we will show that it is possible to get a $RSR$ decreasing quicker than the $SR$, so that $RSR$ is a poor approximation of $SR$. The result is valid even with the quantile $100\%$, i.e. the maximum. The same is possible with any other quantile.
The introduction of $RSR$ apparently outplays $ASR$ as an approximation of $SR$ by two means. First, by using recommendations rather than search points. Second, by checking on multiple recommendations that the optimum is correctly found with a given precision. In addition, as well as $ASR$, it is non-increasing, therefore it can be used for fastening experiments on testbed. Please note however that this advantage makes sense only when the target fitness value is known, which is rarely the case except in an artificial testbed.
To investigate the convergence rate of the regrets, we will use a slightly different notation than classical works on noisy optimization. Usually the rates are given in terms of $O(h(n))$ where $h(n)$ is some function depending on the number of evaluations $n$. The state of the art shows that in many cases (\cite{dupac,fabian,shamir,fogaasr}) $SR_n = O(n^\psi)$ where $\psi<0$ implies that the algorithm converges. Therefore, there is a linear relationship between $\log(SR_n)$ and $\log(n)$ with a \emph{slope} $\psi$, where $\log(\cdot)$ is the natural logarithm. We will then refer to the \emph{slope of the regret} when speaking about the convergence rate of the regret. The definition of the \emph{slope of the SR} is:
\begin{equation*} s(SR) = \underset{n\to\infty}{\lim\sup}~ \frac{\log(SR_n)}{\log(n)} \end{equation*}
We have the corresponding definition for the slope of $ASR$ and $RSR$. Notice that if the slope is close to $0$, then the algorithm (at best) converges \emph{slowly}. On the contrary, if the slope is negative and far away from $0$, then the algorithm is \emph{fast}.
\subsection{General results for $SR$, $ASR$ and $RSR$: $RSR$ is an optimistic approximation of $SR$}
The problems analysed in this paper arise from the gap between $s(SR)$ and $s(ASR)$ or $s(RSR)$. Ideally we would like to have a regret that can be used easily and that \emph{approximates} the Simple Regret. In the following sections (\ref{sec:EA} and \ref{sec:stoc}) we will see with specific examples there is indeed a gap between $s(SR)$ and $s(ASR)$. In some cases using $ASR$ overestimates the performance of algorithms and in others it underestimates their performance. An extreme case is detailed in section \ref{sec:stoc}, where we prove that Alg.~\ref{alg:shamir} has optimal convergence rate in term of $SR$ whilst for $ASR$ it does not converge at all.
In general, by definition, we have that for any algorithm, $s(RSR)\leq s(SR)$. From this point of view, $RSR$ is a correct lower bound for $SR$. In other words, if an algorithm is fast in terms of $SR$, its performance measured by $RSR$ will be at least as good. Unfortunately, this bound is not nearly tight. We will prove that a small modification on the algorithm induces $s(RSR)\leq s(ASR)$ whereas $s(SR)$ is the same - so that, for cases in which $s(ASR)<s(SR)$ (sometimes by far, as explained in later sections), we get $s(RSR) < s(SR)$ (sometimes by far).
Let $A$ be an optimization algorithm and its search points $(x_i)_{i \geq 1}$. Consider another algorithm denoted $A_g$ and its successive search points $(X_i)_{i \geq 1}$. The search points of $A_g$ are obtained by repeating $\lfloor g(n)\rfloor$ times, for any $n$, the search points $x_n$ of $A$. Hence, we get the assignment:
\begin{tabular}{rcrrl} $X_{1}=$ & $\dots$ & $=$ & $ X_{1+\lfloor g(1)\rfloor}$ & $\leftarrow x_1$\\ $X_{2+\lfloor g(1) \rfloor}=$ & $\dots$& $=$ & $X_{2+\lfloor g(1)\rfloor+\lfloor g(2)\rfloor} $ & $\leftarrow x_2$\\ & $\vdots$ & \\ $X_{n+\sum_{j=1}^{n-1} \lfloor g(j)\rfloor} =$& $\dots$ & $=$ & $X_{n+\sum_{j=1}^{n} \lfloor g(j)\rfloor} $& $\leftarrow x_{n}$ \end{tabular}
Let the recommendation points of $A_g$ be defined by $\tilde{X}_n = X_n$ for any $n$. Therefore $A_g$ is a slightly modified version of $A$ since there is an additional polylogarithmic number of evaluation in $A_g$, assuming $g$ is polylogarithmic. The $RSR$ of $A_g$ (say $RSR_{A_g}$) converges approximately as fast as the $ASR$ of the original algorithm (say $ASR_A$). The extra polylogarithmic number of evaluations does not affect the linear convergence in log/log scale. Hence, for any algorithm $A$, $s(RSR_{A_g}) \leq s(ASR_A)$.
The general relationships between the slopes of the $SR$ and its approximations are not conclusive since the bounds are not tight. We will show some gaps between the different approximations of $SR$ and the $SR$ itself.
In the following sections we present five algorithms that will serve as clear examples to see the differences of using one or another regret as performance measures. We will focus in two types of algorithms: the first group, in Section~\ref{sec:EA}, consists of Evolutionary Algorithms and Random Search and the second, in Section~\ref{sec:stoc}, of algorithms using approximations of the gradient of the objective function. For each class of algorithms, we exhibit convergence rate bounds on $s(SR)$, $s(ASR)$ and $s(RSR)$. Section~\ref{sec:experiments} displays some experimental works in order to confront theory, conjecture and practice.
\section{Evolutionary Algorithms}\label{sec:EA} On the group of Evolutionary Algorithms (EAs), we present Random Search (RS), Evolution Strategy (ES) and Evolution Strategy with resampling (ES+r). They all use comparisons between fitness values to optimize the function.
\break \subsection{Random Search} Random Search (Alg. \ref{alg:rs}) is the most basic of stochastic algorithms \cite{rastrigin1964convergence}. The search points $x_1,\dots,x_n,\dots$ are independently identically drawn according to some probability distribution. $\tilde x_n$ is usually the search point with the best fitness so far, i.e. with $y_i$ the fitness value obtained for $x_i$, we have $\tilde x_n=x_i$ with $i\in\{1,\dots,n\}$ minimizing $y_i$.
\begin{algorithm}[H] \caption{\label{alg:rs} {\scriptsize Random Search.}} \begin{algorithmic}[1]\scriptsize
\State{{\bf Initialize:} Candidate solution $\tilde x$ randomly drawn in $[0,1]^d$}
\State{$bestfitness \leftarrow f(\tilde x)$}
\State{{\bf Initialize:} $n\leftarrow1$}
\While{not terminated}
\State{Randomly draw $y$ in $[0,1]^d$.}
\State{$fitness\leftarrow f(y)$}
\If{$fitness<bestfitness$ }\label{line:sel1}
\State{ $\tilde x \leftarrow y$}
\State{ $bestfitness\leftarrow fitness$}\label{line:sel2}
\EndIf
\State{$n\leftarrow n+1$}
\EndWhile
\Return{$\tilde{x}$} \end{algorithmic} \end{algorithm}
We consider in this paper a simple variant of RS to show clearly the contrast between $SR$ and $ASR$.
{\bf Framework for RS:} each search point is randomly drawn independently and uniformly, sampled once and only once, with the uniform probability distribution over $[0,1]^d$. The objective function is the noisy sphere function $f$: \begin{equation}
f(x)=\|x-x^*\|^2+\mathcal{N}.\label{noisysphere} \end{equation} where $\mathcal{N}$ is a standard Gaussian variable.
In this setting, existing results in the literature imply a bound on $s(ASR)$ as explained in Property \ref{prop:RS}. We will prove then that the slope of the Simple Regret is not negative, as formalized in Theorem \ref{rswn}.
\subsubsection{Approximate Simple Regret: $s(ASR)=-2/d$}
\begin{property}\label{prop:RS} Consider RS described in Alg.~\ref{alg:rs}, with the framework above.
Then almost surely $ASR_n=O\left(\frac{1}{n^{2/d}}\right)$. \end{property}
\begin{proof}
\cite{deheuvels} has shown that among $n$ points generated independently and uniformly over $[0,1]^{d}$ the closest
search point to the optimum is almost surely at distance $O\left(\frac{1}{n^{1/d}}\right)$ from the optimal point $x^*$ within a logarithmic factor. Hence for the sphere function\footnote{The result also holds for a function locally quadratic around a unique global optimum.}, the Approximate Simple Regret $ASR_n$ almost surely satisfies: $ASR_n = O\left(\frac{1}{n^{2/d}}\right)$ up to logarithmic factors. \end{proof}
\subsubsection{Simple Regret: $s(SR)$ is not negative}
\begin{theorem}\label{rswn} With the framework above, for all $\beta>0$, the expected simple regret ${\mathbb E}(SR_n)$ is not $O(n^{-\beta})$. \end{theorem} \begin{proof} See supplementary material. \end{proof}
{\bf Remark:} Roughly speaking, the proof of the theorem is based on the fact that with a non-zero probability a search point which does not have the best fitness value, is selected as the best point in Lines~\ref{line:sel1}-\ref{line:sel2} of Alg. \ref{alg:rs}.
\def\movedintosup{ The following sections prove this theorem.\mlc{if we are too long, we can put this proof in supplementary material.}
\begin{lemma}[the quantiles of the standard Gaussian random variable.]
Let $Q(q)$ be the quantile of the standard centered Gaussian $\mathcal{N}$, i.e. $\forall q\in (0,1), P(\mathcal{N} \leq Q(q) ) =q$.
Then, there exist $X(q)=-\sqrt{-\log(q S_X(q) )}$ and $Y(q)=-\sqrt{-\log(q S_Y(q))}$ such that $$X(q) \leq Q(q) \leq Y(q),$$ for some $S_X(q)$ and $S_Y(q)$ polylogarithmic as functions of $q$. \end{lemma} Proof: See e.g. http://www.johndcook.com/normalbounds.pdf TODO we should put this into a clean reference.QED Let us consider $N$ the number of search points, $x_1,\dots,x_N$ the $N$ i.i.d search points, $\epsilon(N)>0$ and $C(N)>0$ with $\epsilon(N) =o( C(N) )$, and $\epsilon(N)$ and $C(N)$ both $\Omega(1/N)$ and $o(1)$.
Define $N_g(N)=\{ i\in \{1,\dots,N\}; || x_i - x* || \leq \sqrt{\epsilon(N)} \} $ the number of search points with norm $\leq \sqrt{\epsilon(N)}$ (i.e. ``good'' search points, with simple regret better than $\epsilon(N)$ ).
Define $N_b(N)$ the number of search points with norm $> \sqrt{\epsilon(N)}$ but norm $\leq \sqrt{C(N)}$ (i.e. bad search points, but not very bad...), i.e. $N_b(N)\leq N-N_g(N)$. \sam{precise here if we abuse of notation or change the notation ($N_{b}(N)$ is a set) }
\begin{lemma}[Linear numbers of good and bad points]\label{lemmaCheb} There exist constants $K_g>0$ and $K_b>0$, such that with probability $\geq \frac12$,
\begin{equation} N_g(N) \leq K_g N\sqrt{\epsilon(N)}\label{calvin} \end{equation} and \begin{equation} N_b(N) \geq K_b N \sqrt{C(N)}.\label{hobbes} \end{equation}
\end{lemma} \begin{proof} Consider a search point $x_i$. It is ``good'', in the sense above, with probability $\Omega(\sqrt{\epsilon_N})$. This holds for each of the search points; therefore the number of good points is the sum of $N$ Bernoulli random variables with parameter $\Omega(\sqrt{\epsilon_N})$. The expectation and the variance are therefore $\Omega(N\sqrt{\epsilon_N})$. By Chebyshev inequality, there is $\alpha>0$ such that a random variable $X$ is $O({\mathbb E} X+\alpha \sqrt{Var\ X})$ with probability at least $\frac12$. This implies that the number $N_g$ of ``good'' points is \begin{equation} N_g=O(N\sqrt{\epsilon _N}+\alpha\sqrt{N\sqrt{\epsilon _N}}).\label{bolalapouf} \end{equation} By the assumption $\epsilon _N=\Omega(1/N)$, $N\sqrt{\epsilon _N}\to\infty$; therefore Eq. \ref{bolalapouf} implies $N_g=O(N\sqrt{\epsilon _N})$. The proof is similar for the number of ``bad'' points. \end{proof}
Consider $G=\{i\in \{1,\dots,N\}; ||x_i-x^*||\leq \sqrt{\epsilon(N)}\}$ the indices of good search points, and $B=\{i\in\{1,\dots,N\}; \sqrt{C(N)} \geq ||x_i-x^*||>\sqrt{\epsilon(N)}\}$ the indices of bad search points.
\begin{proposition}\label{zeprop} For some $c>0$, for all $N$, with probability at least $c$, the minimum of the noisy fitness values for the $N_g$ good points verifies $\inf_{i \in G} y_i \geq X(1/N_g)$, and the best noisy fitness for the $N_b$ bad points verifies $\inf_{i\in B} y_i \leq C+Y(1/N_b)$. \end{proposition}
\begin{proof}
Consider the case in which $N_g$
and $N_b$ are upper and lower bounded (respectively) as explained
in Lemma \ref{lemmaCheb}.
This happens with probability $\geq \frac12$, by that lemma.
Consider some $n_i$ (for $i\in \{1,2,\dots,N\}$), which are independently identically distributed according to some absolutely continuous density. Define $q_N$ the $\frac1N$ quantile of their common probability distribution. Then the probability that $\inf \{n_1,n_2,\dots,n_N\}$ is less or equal to $q_N$ is, by definition, $1-(1-1/N)^N$. This converges to the constant $1-\exp(-1)$ as $N\to \infty$.
Therefore, the probability that one of the $N_g$
good points has a noise $\leq Q(1/N_g)$ is
upper and lower bounded by a constant;
the probability that one of the $N_b$ bad
points
points has a noise $\leq Q(1/N_b)$ is
upper and lower bounded by a constant;
these events are independent, so the
probability that both happen simultaneously
is a constant; these events are also
independent of the 0.5 probability from Lemma \ref{lemmaCheb},
so with lower bounded probability all
these events happen simultaneously.\end{proof}
{\bf{Proof of Theorem \ref{rswn}:}} \begin{proof} Let us assume that the expected
simple regret has slope $<-\beta$ for some $1>\beta>0$; then define
$\epsilon(N)=N^{-\beta}$ and $\alpha=\beta/k$ for some $0<k<1$, and $C(N)=N^{-\alpha}$; then Lemma \ref{lemmaCheb} and Proposition \ref{zeprop} implies that there is a $c>0$ such that with probability $c$, for $N$ sufficiently large, \begin{itemize}
\item there are much more good points than bad points, i.e. \begin{equation}
N_b=o(N_g)\label{proutproutprout} \end{equation}
thanks to Eq. \ref{calvin} and \ref{hobbes}.
\item all good points have noisy fitness at least $X(1/N_g)=\tilde \Theta(-\sqrt{-\log(N_g)})$;
\item at least one bad point has fitness at most $C(N)+Y(1/N_b)=\tilde\Theta(N^{-\alpha}-\sqrt{-\log(N_b)})=\tilde\Theta(-\sqrt{-\log(N_b)})$;
\item therefore (by the two points above, and using Eq. \ref{proutproutprout}, one bad point is selected; \sam{when is it selected? we always keep the point with the best fitness} \end{itemize} so that the simple regret is larger than $\epsilon(N)$ with probability $c$ for all $N$ sufficiently large.
This implies that the expected simple regret is at least $c \epsilon(N)$, which contradicts the slope $<-\beta$. \end{proof} }
\subsection{Evolution Strategies ($ES$)} Evolution Strategies~\cite{Rechenberg73,Schw74b} are algorithms included in the category of Evolutionary Algorithms (EAs).
In general, EAs evolve a population until they find an optimum for the objective or fitness function. The process starts by a population randomly generated. Then the algorithm iterates creating new individuals using crossover and mutation and then evaluating this new population of offspring and selecting the ones - regarding to their fitness values - that will become the parents of the next generation.
ES have some more specific selection and mutation processes. Usually the mutation is performed by creating new individuals starting from the parent and adding a random value to it (usually normally distributed around the parent). There are various rules for choosing the step-size. The selection in ES is usually deterministic and rank-based. This is, the individuals chosen to be the parents of the next generations are the ones that have the best fitness values.
When dealing with noisy function, the sorting step of the ES is disturbed by the noise and misranking might occur. To tackle this problem, Arnold and Beyer, in \cite{abinvestigation,abnoise} propose to increase the population size. An alternative is to evaluate multiple times the same search point and average the resulting fitness values. For a given search point $x\in D$, $r$ evaluations are performed: $\left(f(x,w_1), \dots, f(x,w_r)\right)$ and the fitness value used in the comparisons is the average of these evaluations $\frac{1}{r} \sum_{i=1}^{r} f(x,w_i)$. In particular, the variance of the noise is divided by $r$. Several rules have been studied: constant \cite{bignoise3}, adaptive (polynomial in the inverse of the step-size), polynomial and exponential \cite{bignoise2} number of resamplings. A general $(\mu,\lambda)$-ES is presented in Algorithm \ref{alg:es}.
\begin{algorithm}[H] \caption{\label{alg:es} {\scriptsize $(\mu,\lambda)$-Evolution Strategy. The resampling function $r$ may be constant or depend on the number of iterations and possibly on the step-size. When $r=1$, the algorithm reduces to an ES without resampling. ${\cal{N}}$ is a standard Gaussian of dimension $d$. Here, the index $n$ is the number of \emph{iterations}}} \begin{algorithmic}[1]\scriptsize
\State{{\bf Input:} Parameters $\mu$, $\lambda$ and resampling function $r$ }
\State{{\bf Initialize:} Parent $\tilde x$ and Step-size $\sigma$}
\State{{\bf Initialize:} $n\leftarrow1$}
\While{not terminated}
\State{{\bf Mutation step:}} $\forall i\in \{1,\dots,\lambda\}$,
$x^{(i)}\leftarrow \tilde{x}+ \sigma\mathcal{N}$\label{line:mutate}
\State{{\bf Evaluation step:}}
$\forall i\in \{1,\dots,\lambda\}$ ${y^{(i)} \leftarrow\frac{1}{r(n)}\sum_{j=1}^{r(n)}f(x^{(i)}, w_{j})}$\label{line:eval}
\State{{\bf Selection step:} Sort the population according to their fitness and select the $\mu$ best: $(x^{(i)})^\mu$}
\State{{\bf Update Parent:} $\tilde{x}$ from $\sigma$, $(y^{(i)})^\mu$ and $(x^{(i)})^\mu$}
\State{{\bf Update Step-size:} $\sigma$ from $\sigma$, $(y^{(i)})^\mu$ and $(x^{(i)})^\mu$}
\State{$n\leftarrow n+1$}
\EndWhile
\Return{$\tilde{x}$}
\end{algorithmic} \end{algorithm}
\subsubsection{Regrets for ES without resampling}\label{subsubsec:ES}
It is known \cite{stocopti5} that when the noise strength is too big, classical evolution strategies (without reevaluations or other noise adaptation scheme) do not converge, they stagnate. \cite{BeyerMutate} experimentally shows that an ES without any adaptation to the noisy setting stagnates around some step-size and at some distance of the optimum. The divergence results suggest that the ES in this case is only as a more sophisticated version of RS. The steps of the ES are more complicated, but not sufficiently adapted to handle the noise of the function. We propose then a Conjecture on the convergence rates for ES.
\begin{conjecture}[Convergence rates for ES]\label{conj:es} Evolution Strategies without a noise handling procedure have the same convergence rates as Random Search for all regrets. \end{conjecture}
\subsubsection{Simple regret for ES with resamplings}\label{SR_ES}
We will see that the results are more encouraging than in Section \ref{subsubsec:ES} when we consider an ES with some adaptation to mitigate the effect of the noise. We will assume that the function $r$ (number of revaluations per point) in Alg.~\ref{alg:es} grows polynomially or exponentially with the number of iterations.
The work in \cite{bignoise2} shows that ES that include an exponential number of revaluations converges with high probability to the optimum. The convergence rate is $s(SR) = K$ for some $K<0$ under assumptions about the convergence in ES in the noise-free case. Moreover \cite{esareslow} shows that ES, under general conditions, must exhibit $K> -\frac12$. There is no formal proof of an upper bound that can theoretically ensure a value or a range for $s(SR)$. However, the experiments on \cite{bignoise2} suggest that the $K=-\frac12$ is reached for functions with a quadratic Taylor expansion and additive noise (as in Eq.~\ref{additnoise}). Hence we propose Conjecture~\ref{conj:esSReval}:
\begin{conjecture}[$SR$ for ES + $r$]\label{conj:esSReval}
Consider $0<\delta<1$. For some resampling parameters {{(i.e. for some revaluation function $r$)}}, Evolution Strategies with resampling (Alg.~\ref{alg:es}) satisfy $s(SR)=-1/2$ with probability $1-\delta$. \end{conjecture}
This conjecture applies to some ES with step-size scaling as the distance to the optimum, i.e. $\sigma_n$ used for generating the $n^{th}$ search point has the same magnitude as $\|\tilde{x}_n - x^{*}\|$ (\cite{Rechenberg73,Beyer:bookES}). \cite{BeyerMutate} has proposed variants of ES for quickening the convergence thanks to large mutations and small inheritance. Such an approach is not covered by the bound in \cite{esareslow} and it is for sure an interesting research direction - maybe it might reach slope $s(SR)=-1$.
\subsubsection{Approximate simple regret for ES with resamplings}\label{asrreval}
We have seen that an ES can reach a slope of $SR$ approximately $-\frac12$, when using resamplings. However, $ASR$ can be better by slightly modifying the original ES, and therefore achieving a faster convergence rate than the real one represented by $s(SR)$ We consider an ES - called $MES+R$ for Modified ES with Resampling. Let $r_n$ be exponential in the number of iterations: $r_n=R\cdot \zeta^n$, $R>0$, $\zeta>1$. $MES+R$ is as in Alg. \ref{alg:es} with the following modifications. At iteration $n$:
{\bf Generation:} (Alg.~\ref{alg:es}, Line \ref{line:mutate}) Generate additional $r_n$ ``fake'' offspring $\{ x^{(i)f}: 1\leq i\leq r_n \}$, with the same probability distribution as the $\lambda$ offspring. They will be evaluated one time each, but they will {\emph{not}} be taken into account for the selection. Note that this means that they are part of the sequence of points considered by $ASR$, but not by $SR$.
{\bf Evaluation:} (Alg. \ref{alg:es}, Line \ref{line:eval}) Evaluate $r_n$ times each ``true'' offspring $\{ x^{(i)} : 1\leq i\leq\lambda \}$ to obtain their corresponding fitness value $y^{(i)}$. Evaluate one time each ``fake'' offspring. Therefore, performing $(\lambda+1)r_n$ function evaluations in each iteration.
Then, under some reasonable convergence assumptions which are detailed in theorem \ref{thm:asr} below, the $ASR$ reaches a faster rate: $s(ASR)=-1/2-2/d$ with high probability.
\begin{theorem}\label{thm:asr}
Consider $0<\delta<1$. Consider an objective function $F(x)=\|x\|^2$, where $x\in {\mathbb R}^d$. Consider a $MES+R$ as described previously. Assume that: \begin{enumerate}[label=(\roman*)]
\item $\sigma_n$ and $\|\tilde{x}_n\|$ have the same order of magnitude: \begin{equation}\label{xsigma}
\|\tilde{x}_n\|=\Theta(\sigma_n). \end{equation} \item\label{logalso} $\log-\log$ convergence occurs for the $SR$: \begin{equation}\label{loglog}
\frac{\log(\|\tilde{x}_n\|)}{\log(n)}\underset{n\rightarrow+\infty}{\longrightarrow }-\frac12 ~\mbox{ with probability $1-\delta$,} \end{equation} \end{enumerate}
Then, with probability at least $1-\delta$, $s(ASR)=-1/2-2/d$. \end{theorem} \begin{proof} See supplementary material. \end{proof}
\begin{remark}\label{rmq:sr}
The assumption of $s(SR)=-1/2$ is based on the convergence of ES in the noise-free case and it is essential to prove Theorem~\ref{thm:asr}. This rate of convergence can be proved when the ES converges in the noise-free case (details on \cite{bignoise2}). But the convergence of ES has not been formally proved, not even for the noise-free case. There is an important element given in \cite{TCSAnne04-corr}, showing that $\frac1n \log ||x_n-x^*||$ converges to some constant, but this constant is not proved negative. Furthermore, parameters ensuring convergence in the noisy case are unspecified in \cite{bignoise2}.
\end{remark}
\section{Stochastic Gradient Descent}\label{sec:stoc} For the group of Stochastic Gradient Descent Algorithms, we consider the ones presented by Shamir \cite{shamir} and Fabian \cite{fabian} which approximate the gradient of the objective function. We will denote them Shamir algorithm and Fabian algorithm respectively. Both of them approximate the gradient of the function using function evaluations by different methods, therefore they remain in the black-box category. Fabian algorithm uses the average of redundant finite differences and Shamir algorithm a one point estimate gradient technique.
The convergence rates in terms of $SR$ are proved in~\cite{shamir} and ~\cite{fabian}. For Shamir it is shown that $s(SR)=-1$ in expectation for quadratic functions. Fabian ensures a rate $s(SR)=-1$ approximately and asymptotically only for limit values of parameters. However, it requires only smooth enough functions, so the class of functions is wider than the one considered in~ \cite{shamir}. This rate $s(SR)=-1$ has been proved tight in ~\cite{chen1988}. Hence, Shamir and Fabian algorithm are faster than ES's, which cannot do better than $s(SR)=-\frac12$, at least under their usual form~\cite{esareslow}.
\subsection{Shamir's quadratic algorithm}
Shamir algorithm presented in~\cite{shamir} for quadratic functions is Algorithm \ref{alg:shamir}. \begin{algorithm}
\caption{ {\scriptsize \label{alg:shamir} Shamir Algorithm for Quadratic functions. $\Pi_{W} $ represents the projection over the space $W$} } \begin{algorithmic}[1]\scriptsize
\State{{\bf Input:} Parameters $\lambda$ and $\epsilon $}
\State{{\bf Initialization:} $\hat{x}_1\leftarrow 0$, $n\leftarrow 1$}
\While{not terminated}
\State{Pick $r\in \{-1,1\}^{d}$ uniformly at random}
\State{$x_{n}\leftarrow\hat{x}_{n}+\frac{\epsilon }{\sqrt{d}}r$}
\State{{\bf Evaluate:} $v\leftarrow f(x_n,w)$}
\State{$\hat{g}\leftarrow\frac{\sqrt{d}v}{\epsilon }r$}
\State{$\hat{x}_{n+1}\leftarrow\Pi_{W}\left(\hat{x}_{n}+\frac{1}{\lambda n}\hat{g} \right)$}
\State{{\bf Recommend:} $\tilde{x}_n\leftarrow\lceil\frac{2}{n}\rceil\sum_{k=\lceil n/2\rceil}^{n} \hat{x}_{k}$}
\State{$n\leftarrow n+1$}
\EndWhile
\Return{$\tilde{x}_{n}$} \end{algorithmic} \end{algorithm}
One of the key points in Alg.~\ref{alg:shamir} are that there is only one evaluation per iteration (somehow in the spirit of Simultaneous Perturbation Stochastic Approximation SPSA~\cite{spall00adaptive,beynoise}). The second important point is that the expectation of the distance between search points and recommendations is constant, which implies that the search points do not converge towards the optimum! This is not a problem for the convergence in terms of $SR$, when search points $x_n$ and recommendations $\tilde{x}_n$ are distinguished, but it makes a difference for $ASR$.
Shamir algorithm has an optimal convergence rate in expectation ($s(SR)=-1$) for quadratic functions. This fact should be acknowledge by any other regret used to evaluate its performance which aims to aproximate the $SR$.
But intuitively in the framework of Shamir algorithm, the $s(ASR)$ is presumably a bad approximation of $s(SR)$ due to the queries at a constant distance of the current recommendation. This convergence rate in terms of $s(ASR)$ could not be obtained from the results in ~\cite{shamir}. Nonetheless, we prove in a general way that as long as the results for Shamir algorithm are satisfied \emph{almost surely}, then $s(ASR)=0$ a.s. Therefore we present the latter result in Theorem ~\ref{thm:aSRgradient} and a conjecture on the convergence rate of $s(ASR)$ in expectation for Shamir algorithm.
\begin{theorem} \label{thm:aSRgradient}
Assume that the optimum $x^*$ is unique and that $(\tilde x_{n})$ is a sequence of recommendation points converging a.s. to $x^*$. Assume that the sequence of evaluation points $(x_n)$ is such that $\forall n, x_n\neq x^*$ a.s. and that $ \|x_n-\tilde{x}_n\| $ is constant. Then, a.s. \begin{equation*} s(ASR)=0. \end{equation*} \end{theorem}
\begin{proof}
$\tilde x_n$ converges almost surely to the optimum and $x_n$ is at a constant distance from $\tilde x_n$. Therefore the distance between $x_n$ and the optimum converges to a constant. This implies that the minimum $\min_{i=1}^n \|x_i-x^*\|^2$ is lower bounded by some constant. Therefore $s(ASR)=0$. \end{proof}
\begin{conjecture}[$ASR$ for the Shamir algorithm]
Shamir algorithm also verifies $s(ASR)=0$ a.s. on quadratic functions. \end{conjecture}
\subsection{Fabian Algorithm}
Algorithm \ref{alg:fabian} presents the algorithm studied in \cite{fabian}. Unlike Algorithm~\ref{alg:shamir}, Fabian algorithm performs several evaluations per iteration, and the distance between search point and recommandation decreases.
\begin{algorithm} \caption{ {\scriptsize \label{alg:fabian} Fabian Algorithm. $e_i$ is the $i^{th}$ vector of the standard orthogonal basis of ${\mathbb R}^d$ and $e_{1,s/2}$ is the $1^{st}$ vector of the standard orthogonal basis of ${\mathbb R}^{s/2}$. $v_i$ is the $i^{th}$ coordinate of vector $v$. ($\hat{x}_{i}$) is the $i^{th}$ coordinate of intermediate points ($\hat{x}$). ($x^{(i,j)+}$) and ($x^{(i,j)-}$) are the search points and $\tilde{x}$ is the recommendation. Here, the index $n$ is the number of \emph{iterations}.} } \begin{algorithmic}[1]\scriptsize
\State{{\bf{ Input:}} An even integer $s>0$. Parameters $a$, $\alpha$, $c$, $\gamma$.}
\State{{\bf Initialization:}\\
$u_{i} \leftarrow \frac{1}{i},\ \forall\ i\in\{1,\dots,s/2\}$\\
Matrix $U\leftarrow\left( \| u_{j}^{2i-1} \| \right)_{1\leq i,j \leq s/2}$\\
Vector $v\leftarrow\frac12 U^{-1}e_{1,s/2}$\\
$\tilde{x}\leftarrow x\in [0,1]^{d}$ uniformly at random\\
$n \leftarrow 1$}
\While{not terminated}
\State{$a_n \leftarrow\frac{a}{n^{\alpha}}$, $c_n\leftarrow\frac{c}{n^{\gamma}}$}
\State{$\forall j\in\{1,\dots,s/2\}$,\ $\forall i\in\{1,\dots,d\}$\\
{~~~~\bf Evaluate:}
\begin{eqnarray*}
x^{(i,j)+}\leftarrow \tilde{x} + c_n u_{j}e_{i} \qquad
x^{(i,j)-}\leftarrow \tilde{x} - c_n u_{j}e_{i}
\end{eqnarray*}}
\State{$\hat{x}_{i}\leftarrow\frac{1}{c_n} \sum_{j=1}^{s/2} v_j \left(f(x^{(i,j)+})-f(x^{(i,j)-})\right)$}
\State{{\bf Recommend:} $\tilde{x}\leftarrow\tilde{x}-a_{n}\hat{x}$}
\State{$n\leftarrow n+1$}
\EndWhile
\Return{$\tilde{x}$} \end{algorithmic} \end{algorithm}
The work in \cite{fabian} gives the convergence rate in terms of $SR$. The result is presented here as Theorem \ref{thm:fabSR}. The value of the $s(SR)$ depends on the parameters of the algorithm and it is ensured a.s.
\begin{theorem}[Simple Regret of Fabian algorithm]\label{thm:fabSR} Let $s$ be an even positive integer and $F$ be a function {${(s+1)}$-times differentiable} in the neighborhood of its optimum $x^*$. Assume that its Hessian and its $(s+1)^{th}$ derivative are bounded in norm. Assume that the parameters given in input of Algorithm \ref{alg:fabian} satisfy: $a>0$, $c>0$, $\alpha= 1$, $0<\gamma<1/2$ and $2\lambda_0 a >\beta_0$ where $\lambda_0$ is the smallest eigenvalue of the Hessian. Let ${\beta_0=\min\left(2 s \gamma, 1 - 2 \gamma \right)}$. Then, a.s.: \begin{equation}\label{fabiantropfort} n^{\beta}(\tilde{x}_n-x^*) \rightarrow 0\ \forall\ \beta <\beta_0/2 \end{equation} In particular, when $F$ is smooth enough, we get ${s(SR)=-2\beta}$. \end{theorem}
\begin{remark} Note that $s(SR)$ optimal when $\gamma = \frac{1}{2}(s+1)^{-1}$. In this case, $\beta_0=\frac{s}{s+1} \underset{s\rightarrow \infty}{\rightarrow} 1$. $\beta_0$ can be made arbitrarily close to $1$, so $2\beta$ also, but then $\gamma$ goes to $0$. Hence we get the values of Table \ref{maintable}, with $2\beta=1-e$, $e>0$ and close to $0$. \end{remark} This shows that the Fabian algorithm can have $s(SR)$ arbitrarily close to $-1$, which is optimal. As in the case of Shamir, this optimal performance should be captured by the regret used to evaluate the algorithm. Unfortunately, this is not the case, as detailed in Theorem \ref{thm:fabASR}.
\begin{theorem}[$ASR$ of Fabian algorithm]\label{thm:fabASR} Let $F$ be a $\lambda$-convex and $\mu$-smooth function corrupted by an additive noise with upper bounded density and with optimum randomly drawn according to a distribution with upper bounded density. Then, a.s., \begin{equation*} s(ASR)= -\min(2\beta,2\gamma). \end{equation*} \end{theorem}
\begin{proof} See supplementary material. \end{proof}
\begin{remark} Theorem \ref{thm:fabASR} shows that $s(ASR)=-\min(2\beta,2\gamma)$, i.e., when then Fabian algorithm is optimized for $SR$, $s(ASR)$ is close to $-2\gamma$, close to $0$. \end{remark}
\subsection{Shamir and Fabian adapted for $ASR$}
Both algorithms presented in this section have a clear difference between the search and recommendation points. This fact is not automatically distinguished when we are evaluating their performance using for example a test bed. If we modify the algorithms we can achieve $ASR$ approximating well the optimal behavior reported by $SR$. A modification such as sampling one point out of two at the current recommendation, without using it in the algorithm can imply $s(ASR)=s(SR)$ arbitrarily close to $-1$.
\begin{sidewaystable} \centering \ra{1.3} \begin{tabular}{@{}rrrcrrcrr@{}}\toprule & \multicolumn{2}{c}{SR} & \phantom{abc}& \multicolumn{2}{c}{ASR} & \phantom{abc} & \multicolumn{2}{c}{RSR}\\ \cmidrule{2-3} \cmidrule{5-6} \cmidrule{8-9} & conv. & type && conv. & type && conv. & type \\ \midrule \bf{Evolutionary Algorithms}\\ RS & $\mathbf{0}$ & \textbf{expect.} && $\mathbf{ -\frac2d }$ & \textbf{a.s.} && $\mathbf{ -\frac2d }$ & a.s. \\ ES & $0$ & expect. && $-\frac2d$ & a.s. && $-\frac2d$ & a.s. \\ ES + $r$ & $- \frac12$ & high prob. && $- \frac12$ & high prob. && $- \frac12$ & high prob. \\ MES+ $r$ & $- \frac12$ & high prob. && $\mathbf{ - \frac12 - \frac2d }$ & \textbf{high prob.} && $\mathbf{ - \frac12 - \frac2d } $ & \textbf{high prob.} \\ \bf{Stochastic Gradient}\\ Shamir & $\mathbf{-1}$& \textbf{expect.} && $\mathbf{0}$ & \textbf{expect.} && $\mathbf{-1}$ & \textbf{expect.} \\ Shamir for ASR & $\mathbf{ -1 }$& \textbf{expect. }&& $\mathbf{-1}$ & \textbf{expect.} && $\mathbf{-1}$ & \textbf{expect.}\\ Fabian & $\mathbf{-(1-e)}$ & \textbf{a.s.} && $\mathbf{-e'}$ & \textbf{a.s.} && $\mathbf{-(1-e)}$& \textbf{a.s.}\\ Fabian for ASR & $\mathbf{-(1-e)}$ & \textbf{a.s.} && $\mathbf{-(1-e)}$ & \textbf{a.s.} && $\mathbf{-(1-e)}$ & \textbf{a.s.} \\ \bottomrule \end{tabular} \caption{\label{maintable}Convergence rates for the regrets analysed on this paper. The ``convergence'' column refers to the convergence rate and the ``type'' column specifies the type of convergence: with high probability, in expectation, almost surely. The results in bold are proved and the others are conjectures, all of them presented in this paper. } \end{sidewaystable}
\break \section{Experiments}\label{sec:experiments} We present experimental results for part of the algorithms theoretically analysed \footnote{In addition, the experimental results we include the algorithm $UHCMAES$, as another example of an $ES$. For more information, see ~\cite{hansen2009tec}.}. We will analyse the convergence rate of this algorithms in terms of \emph{slope} (see section \ref{sec:criteria} for the definition). As in the theoretical part,
the function to optimize is the noisy sphere: $f(x)=\|x-x^*\|^2+\vartheta\mathcal{N}$ where $\vartheta=0.3$ and $\mathcal{N}$ is a standard gaussian distribution\footnote{ The choice $\vartheta=0.3$ is made only to illustrate in the experiments the effect of the regret choice in a reasonable time bugdet. The noise is weaker than in the case of a standard gaussian and the algorithms can deal with it faster. The optimum $x^*$ for the experiments of each algorithm is different, which does not affect the result since the regret compares the function value on the search/recommended points and on the optimum.}. The dimension of the problem is $2$. The results in Fig. \ref{zola} correspond to the mean of 10 runs for each of the algorithms. \begin{table}[h] \scriptsize
\begin{center}
\begin{tabular}{rl}
\toprule
Algorithms & Set of parameters \\
\midrule
UHCMAES ~\cite{hansen2009tec} & $x_{\text{initial}}=1$, $\sigma_{\text{in}}=1$ \\
Shamir &$\epsilon=0.3$, $\lambda=0.1$, $B=3$\\
$(1+1)$-ES & \\
$(1+1)$-ES resamp & resamp$=2^n$ \\
Fabian & s=4 $\alpha=1$, $\gamma=0.01$ \\
\bottomrule
\end{tabular}
\end{center}
\label{table:exps} \end{table} \begin{figure*}
\caption{ Figure \ref{fig:SR} presents the Slope of Simple Regret for each algorithm on the first $(1\cdot10^6)$ function evaluations. Stochastic Gradient algorithms reach $s(SR)=-1$ while the evolutionary algorithms present $s(SR)=-0.2$. Figure \ref{fig:ASR} presents the Slope of Approximate Simple Regret. Observe that the performance of the algorithms is inverted with regards to the figure \ref{fig:SR} : now the Stochastic Gradient algorithms have the worse performance.}
\label{fig:SR}
\label{fig:ASR}
\label{zola}
\end{figure*}
The results in figure (\ref{fig:SR}) show the comparison between algorithms with regards to the $SR$ criterion. The budget is limited to $(1\cdot10^6)$ function evaluations. We can see clearly the difference between the algorithm that are gradient-based and the $ES$. The algorithms Fabian and Shamir achieve $s(SR)=-1$ whereas the $ES$ presented cannot do better than $s(SR)=-0.25$. The figure
(\ref{fig:ASR}) shows that the use of $ASR$ changes completely the performance of the algorithms. In this case, the gradient-based algorithms are the ones with the worst performance. The results confirm the theoretical work (and the conjectures) presented in sections \ref{sec:EA} and \ref{sec:stoc}.
\section{Conclusion}\label{sec:conc} In this paper we analyse the use of regrets to measure performance of algorithms in noisy optimization problems. We take into account the \emph{Simple Regret} ($SR$) and other two forms of regret used in practice to approximate the $SR$. We show that the convergence rates of the same algorithm over the same class of functions depend on the considered regret. Ultimately this leads to inconsistent results: algorithms are efficient for $SR$ and the opposite for an approximation of $SR$.
Table \ref{maintable} summarizes our results, detailing if they are proved or conjectured and what type of convergence is found.
{\bf{Approximations of $SR$.}} The $ASR$ appears to be a poor approximation of $SR$. Two situations are exposed in this paper. First, algorithms with fast convergence for $SR$ can have a slow convergence for $ASR$ (Fabian and Shamir algorithm). However, this can be solved by modifying the algorithm in order to sample, sometimes, a recommended point. Second, an algorithm with slow convergence for $SR$ can have a fast convergence for $ASR$, and this is a serious issue. There is no simple ``patch'' to deal with this problem. Test beds using $ASR$ will underestimate algorithms which include random exploration. This is partially, but not totally, solved by $RSR$. We did not come up with a satisfactory regret definition, which would be consistent with $SR$ (at least showing similar convergence rates) and without drawbacks as those presented above. However, the difference (between $SR$ on the one hand, and $ASR$/$RSR$ on the other hand) decreases with the dimension for most algorithms.
{\bf{Simple Regret.}} $SR$ is clearly a natural way to measure the quality of an approximated optimum output by an algorithm. The drawback of $SR$ is that it is not necessarily non-increasing, which is an issue for the concept of ``stable first hitting time''. Note that a particularly interesting result is the one of Evolution Strategies correctly tuned in terms of its resampling scheme. It appears, from theoretical and experimental results, that it reaches half the speed of classical noisy optimization algorithms in terms of $SR$, on the log-log scale. This corresponds to a squared computation time. \footnote{This is not the case, however, for some ES with fast rates in noisy cases, such as mutate-large-inherit-small\cite{BeyerMutate}.}\\
{\bf{Further work.}} We have compared convergence rates through their \emph{slopes}, but in some cases we have slopes for almost sure convergence, in other cases slope with high probability, and in others slope in expectation. There is room for refinement of the results. As for the approximations of $SR$, there might be other definitions of regret that approximate it in a better way. Evidently, the use of an approximation makes sense in the case where there are technical issues that do not allow the direct computation of $SR$.
\section*{Supplementary material}
\begin{customthm}{1} With the framework above, for all $\beta>0$, the expected simple regret ${\mathbb E}(SR_n)$ is not $O(n^{-\beta})$. \end{customthm}
\begin{lemma}[Logarithmic bounds on the quantile of the standard Gaussian variable]\label{lemma:quantile}
Let $Q(q)$ be $q$ quantile of the standard centered Gaussian, i.e. $\forall q\in (0,1), {P(\mathcal{N} \leq Q(q) ) =q}$. Then $\forall \kappa\geq 1$, $\forall q\in (0,1)$, $$1-\sqrt{-\frac{2}{\kappa}\log(c(\kappa)q)}\leq Q(q)\leq 1-{\sqrt{-2\log(2q)}},$$ where $c(\kappa)$ is a constant depending only on $\kappa$. In the following, we will denote by $X(q)$ (resp. $Y(q)$) the lower (resp.upper) bound on $Q$: $X(q)=1-\sqrt{-\frac{2}{\kappa}\log(c(\kappa)q)}$ and $Y(q)=1-{\sqrt{-2\log(2q)}}$.
\end{lemma}
\begin{proof} See \url{http://arxiv.org/pdf/1202.6483v2.pdf}. \end{proof}
\begin{definition}\label{def:eps} We recall that we consider $n$ i.i.d search points $x_1,\dots,x_n$. Let $o$ and $\Omega$ be the standard Landau notations. Let $\epsilon :\mathbb{N}\setminus\{0\}\mapsto\mathbb{R}$ and $C:\mathbb{N}\setminus\{0\}\mapsto\mathbb{R}$ be two functions satisfying: \begin{itemize} \item $\forall n\in\mathbb{N}\setminus\{0\}$, $\epsilon(n)>0$ and $C(n)>0$, \item $\forall n\in\mathbb{N}\setminus\{0\}$, $\epsilon(n) =o( C(n) )$, \item $\forall n\in\mathbb{N}\setminus\{0\}$, $\epsilon(n)=\Omega(1/n)$ and $C(n)=\Omega(1/n)$, \item $\forall n\in\mathbb{N}\setminus\{0\}$, $\epsilon(n)=o(1)$ and $C(n)=o(1)$. \end{itemize} \end{definition}
\begin{definition} With the previous definition of $\epsilon $ and $C$, consider the set $G$ defined by
$$G:=\{i\in \{1,\dots,n\}; \|x_i-x^*\|\leq \sqrt{\epsilon(n)}\}.$$ $G$ is the set of ``good'' search points, with simple regret better than $\epsilon(n)$. We denote by $N_G(n)$ the cardinality of $G$. \end{definition}
\begin{definition} Similarly, consider the set $B$ defined by
$$B:=\{i\in\{1,\dots,n\}; \sqrt{\epsilon (n)} < \|x_i-x^*\|\leq\sqrt{C(n)}\}.$$ $B$ is the set of ``bad'' search points, with simple regret bigger than $\epsilon(n)$, but still not that bad, since the simple regret does not exceed $C(n)$. We denote by $N_B(n)$ the cardinality of $B$. \end{definition}
\begin{lemma}[Linear numbers of good and bad points]\label{lemmaCheb}
There exist a constant $K_d>0$ such that, with probability at least $1/2$, \begin{equation} N_G(n) < 2K_{d}n\sqrt{\epsilon(n)}^d\label{calvin} \end{equation} and \begin{equation} N_B(n) \geq 2 K_d n \sqrt{C(n)}^d.\label{hobbes} \end{equation} \end{lemma}
\begin{proof} Proof of Eq.~\ref{calvin}. Consider a search point $x_i$. The search points are drawn uniformly at random following the uniform distribution in $[0,1]^{d}$, then the probability $p$ that $x_i\in G$ is $p=\frac{Vol(B_d(x^*,\sqrt{\epsilon (n)}))}{Vol([0,1]^{d})}= K_{d}\sqrt{\epsilon (n)}^{d}$, where $Vol$ stands for `volume' and $K_{d}$ is a constant depending on $d$ only. Therefore the number $N_G(n)$ of good points is the sum of $n$ Bernoulli random variables with parameter $K_{d}\sqrt{\epsilon(n)}^d$. The expectation is then $K_{d}n\sqrt{\epsilon(n)}^d$. By Markov inequality, \begin{equation*} \mathbb{P}(N_{G}(n)\geq 2K_{d}n\sqrt{\epsilon(n)}^d)\leq \frac{K_{d}n\sqrt{\epsilon(n)}^d}{2K_{d}n\sqrt{\epsilon(n)}^d}=\frac{1}{2}. \end{equation*}
Similarly, $N_B(n)$ is a binomial random variable of parameters $n$ and $p=K_d (\sqrt{C(n)}^d-\sqrt{\epsilon (n)}^d)$. Then by Chebyshev's inequality,
$\mathbb{P}(|N_B(n)-K_d n (\sqrt{C(n)}^d-\sqrt{\epsilon (n)}^d)|\leq \alpha)\geq 1/2$ by taking $\alpha=\sqrt{2K_dn(\sqrt{C(n)}^d-\sqrt{\epsilon (n)}^d)(1-\sqrt{C(n)}^d+\sqrt{\epsilon (n)}^d)}$. Hence with probability at least $1/2$, $N_B(n)\geq K_d n (\sqrt{C(n)}^d-\sqrt{\epsilon (n)}^d)+\alpha\geq 2 K_d n \sqrt{C(n)}^d$ since $\epsilon (n)=o(C(n))$. \end{proof}
We recall that $\forall i\in\{1,\dots,n\}$, $y_i$ is the fitness value of search point $x_i$: $y_i=\|x_i - x^*\|^2 + \mathcal{N}_i$, where $\mathcal{N}_i$ is the realisation of a standard centered gaussian variable.
The following property gives a lower bound on the fitness values of the `good' points, and an upper bound on the fitness values of the `bad' points.
\begin{proposition}\label{zeprop} With $X$ and $Y$ as defined in Lemma~\ref{lemma:quantile}, and $C$ as defined in Definition~\ref{def:eps}, there exists some $c\in(0,1)$ such that, with probability at least $c$, \begin{equation}\label{eq:goodguyslowerbound} \inf_{i \in G} y_i \geq X(1/N_G(n)), \end{equation} and \begin{equation}\label{eq:badguysupperbound} \inf_{i\in B} y_i \leq C(n)+Y(1/N_B(n)). \end{equation} \end{proposition}
\begin{proof}
Consider some Gaussian random variables independently identically distributed $\mathcal{N}_1,\dots,\mathcal{N}_N$. $\forall\ i\in \{1,\dots,N\}$, using notation of Lemma~\ref{lemma:quantile}, $\mathbb{P}(\mathcal{N}_i\leq Q(1/N))=1/N$, then $\mathbb{P}(\inf_{1\leq i\leq N} \mathcal{N}_i\leq Q(1/N))=1-\mathbb{P}(\inf_{1\leq i\leq N} \mathcal{N}_i\geq Q(1/N)=1-(1-1/N)^N$. The study of the function $x\mapsto 1-(1-1/x)^x$ then shows that $\mathbb{P}(\inf_{1\leq i\leq N} \mathcal{N}_i\leq Q(1/N))\in[1-\exp(-1),3/4]$ (as soon as $N\geq 2$).
Proof of Eq.~\ref{eq:goodguyslowerbound}. \begin{align*}
\mathbb{P}(\inf_{i \in G} y_i \geq X(1/N_G(n)))&=\mathbb{P}(\inf_{i \in G} \|x_i-x^*\|^2\\ &\ \ \ \ \ \ \ \ \ +\mathcal{N}_i \geq X(1/N_G(n)))\\
&\geq \mathbb{P}(\inf_{i \in G} \|x_i-x^*\|^2\\ &\ \ \ \ \ \ \ \ \ \ +\mathcal{N}_i \geq \epsilon (n) + X(1/N_G(n)))\\ &\geq \mathbb{P}(\inf_{i \in G}\mathcal{N}_i \geq X(1/N_G(n)))\\ &\geq 1-\mathbb{P}(\inf_{i \in G}\mathcal{N}_i \leq X(1/N_G(n)))\\ &\geq 1-\mathbb{P}(\inf_{i \in G}\mathcal{N}_i \leq Q(1/N_G(n)))\\ &\geq 1/4. \end{align*}
Proof of Eq.~\ref{eq:badguysupperbound}.
\begin{align*}
\mathbb{P}(\inf_{i \in B} y_i \leq C(n)+Y(1/N_B(n)))&=\mathbb{P}(\inf_{i \in G} \|x_i-x^*\|^2+\mathcal{N}_i\\ &\ \ \ \ \ \leq C(n)+Y(1/N_B(n)))\\ &=\mathbb{P}(\inf_{i\in B} \mathcal{N}_i \leq Y(1/N_B(n)))\\ &\geq \mathbb{P}(\inf_{i\in B} \mathcal{N}_i \leq Q(1/N_B(n)))\\ &\geq 1-\exp(-1) \end{align*} Hence, with probability at least $1/4$, Eqs.~\ref{eq:goodguyslowerbound} and \ref{eq:badguysupperbound} hold. \end{proof}
{\bf{Proof of Theorem \ref{rswn}:}} \begin{proof}
Let us assume that the expected simple regret has a slope $-\beta_0<-\beta$, for some $0<\beta<1$: ${\mathbb E}(SR_n)=O(n^{-\beta_0})$.
We define $\epsilon(n)=n^{-\beta}$. For some $0<k<1$, $\alpha=\beta/k$ and $C(n)=n^{-\alpha}$. $\epsilon $ and $C$ satisfy Definition~\ref{def:eps}.
Lemma \ref{lemmaCheb} and Proposition \ref{zeprop} implies that there is a $0<c<1$ such that with probability $c$, for $n$ sufficiently large, \begin{itemize}
\item there are much more good points than bad points, i.e. \begin{equation}
N_G(n)=o(N_B(n))\label{proutproutprout} \end{equation}
thanks to Eq. \ref{calvin} and \ref{hobbes}.
\item all good points have noisy fitness at least $X(1/N_G(n))=1-\sqrt{-\frac{2}{\kappa}\log(c(\kappa)/N_G(n))}$;
\item at least one bad point has fitness at most $C(n)+Y(1/N_B(n))=n^{-\alpha}+1-{\sqrt{-2\log(2/N_B(n))}})$;
\item therefore (by the two points above, using Eq. \ref{proutproutprout} and the fact that $n$ is big so that $n^{-\alpha}$ is negligeable), at least one bad point has a better noisy fitness than all the good points, and therefore is selected in Lines 7-9 of Alg. 1. \end{itemize}
This implies that \begin{align*}
{\mathbb E}(SR_n)&={\mathbb E}(\|x_n-x^*\|^2)={\mathbb E}(\|x_n-x^*\|^2| x_n \in B)\mathbb{P}(x_n \in B)\\
& +{\mathbb E}(\|x_n-x^*\|^2| x_n \in G)\mathbb{P}(x_n \in G)\\
&\geq {\mathbb E}(\|x_n-x^*\|^2| x_n \in B)\mathbb{P}(x_n \in B)\\ &\geq \epsilon (n)\times c\\ &\geq c n^{-\beta}. \end{align*} We have a contradicion, hence $\beta_0 >0$. \end{proof}
\begin{customthm}{2}
Consider $0<\delta<1$. Consider an objective function with expectation $F(x)=\|x\|^2$, where $x\in {\mathbb R}^d$. Consider a $MES+R$ as described previously (resampling number exponential in the number of iterations). Assume that: \begin{enumerate}[label=(\roman*)]
\item $\sigma_n$ and $\|\tilde{x}_n\|$ have the same order of magnitude: \begin{equation}
\|\tilde{x}_n\|=\Theta(\sigma_n). \end{equation} \item $\log-\log$ convergence occurs for the $SR$: \begin{equation}
\frac{\log(\|\tilde{x}_n\|)}{\log(n)}\underset{n\rightarrow+\infty}{\longrightarrow }-\frac12 ~\mbox{ with probability $1-\delta$,} \end{equation} \end{enumerate}
Then, with probability at least $1-\delta$, $s(ASR)=-1/2-2/d$. \end{customthm}
\begin{proof} We have $r_n = R \zeta^n$.
In this proof we will index the recommendation and search points by the number of \emph{iterations} instead of the number of evaluations. For $ES+r$, the recommendation point of the iteration $n$ is the corresponding center of the offspring distribution $\tilde x_n$. The step-size $\sigma_n$ corresponds to the standard deviation of the offspring distribution. The search points of iteration $n$ are the $\lambda$ offsprings produced on the iteration, denoted $\{ x_n^{(i)}: i=1,\ldots,\lambda \}$.
We define $e_n$ the number of evaluations until the iteration $n$. From Algorithm 2 we have $e_n=\lambda \sum_{i=1}^{n} r_i$ for an ES with resampling $r$.
By hypothesis (Eq.~\ref{loglog}) we know the convergence of the sequence $S_n$ defined as the logarithm of the recommendation points, indexed by the number evaluations, divided by the logarithm the number of evaluations. Therefore, the sequence $\mathcal{S}_n = \log (||x_n||) /\log (e_n)$ is a subsequence of $S_n$, hence convergent to the same limit, with the same probability. The relation on Eq.~\ref{xsigma} also remains the same after the index modification. From these facts we can inmediatly conclude that $\exists \quad n_0$ such that \begin{equation}\label{ensigma} \sigma_{n}=\Theta(e_n^{-1/4}) \mbox{ for $n\geq n_0.$} \end{equation} For the $MES+r$ algorithm, the $ASR_n$ is defined as: \begin{equation}
{ASR_n =\min_{m\leq n} \min_{1\leq i \leq \lambda+r_m}\|{x'}_{m}^{(i)}\|^{2}} \end{equation} where $\{ {x'}^{(i)}_n : i= 1,\ldots,\lambda+r_m \}$ considers all the search points at iteration $n$. Both the ``real'' and the ``fake'' ones. We will find a bound for $ASR_n$, which will lead us to the convergence rate of the $ASR$ for the $MES+r$ algorithm.
Let $p_x$ be the probability density at point $x$ of the offsprings in iteration $n$. Therefore it is the probability density of a Gaussian centered at $\tilde{x}_n$ and with variance $\sigma_n^2$. At the origin: \begin{align} p_0&=\frac{1}{(2\pi)^{d/2}\sigma_{n}^{d}}\exp\left\{-\frac12 \frac{(-\tilde{x}_n)^{T}(-\tilde{x}_n)}{\sigma_{n}^2}\right\}\nonumber\\
&=\Theta(\sigma_{n}^{-d})~\mbox{by Eq. \ref{xsigma}}\nonumber\\
&=\Theta(e_n^{d/4})~\mbox{by Eq. \ref{ensigma}}\label{densitybound} \end{align}
Now, at iteration $n$, the probability to have at least one offspring with norm less than $\epsilon>0$ is \begin{eqnarray*}
{\mathbb P}(\exists i : \|{x'}_{n}^{(i)}\|\leq \epsilon)&\leq (\lambda + r_n ){\mathbb P}(\|{x'}_{n}^{(i)}\| \leq \epsilon)\\
&\leq (\lambda +r_n) \int_{\|x\| \leq \epsilon}dp_x \end{eqnarray*} By Eq.~\ref{densitybound},
$${\mathbb P}(\exists i | \|{x'}_{n}^{(i)}\|\leq \epsilon)=\Theta(r_n\cdot e_n^{d/4} \epsilon^{d})=\Theta(1)$$
if $\epsilon = \Theta( e_n^{-1/4} \cdot r_n^{-1/d} )$.
Therefore we obtain: \begin{align*}
ASR_n &\leq \min_{1\leq i \leq \lambda+r_n} \|{x'}_{n}^{(i)}\|^{2}\\
&\leq\epsilon^{2}\\
&=O(e_n^{-1/2} \cdot r_n^{-2/d}) \\
&=O(e_n^{-1/2-2/d}) \end{align*} Since $e_n=(1+\lambda)\sum_{i=1}^{n} r_{i}=(1+\lambda)\cdot R\sum_{i=1}^{n}\zeta^{i}=\Theta(r_n)$ Hence we have the result, $s(ASR)\leq -1/2-2/d$.
\end{proof}
\begin{customthm}{5}[$ASR$ of Fabian algorithm] Let $F$ be a $\lambda$-convex and $\mu$-smooth function corrupted by an additive noise with upper bounded density and with optimum randomly drawn according to a distribution with upper bounded density. Then, a.s., \begin{equation*} s(ASR)= -\min(2\beta,2\gamma). \end{equation*} \end{customthm}
\begin{proof} The upper bounded density is used (Theorem 3 section 4.1) ) for ensuring that no recommendation is never exactly equal to the optimum.
Using the algorithm notations, $$ASR_n= {\min_{n,i,j} F(x^{(i,j)+}(n)) - F(x^*)}$$ (or ${\min_{n,i,j} F(x^{(i,j)-}(n)) - F(x^*)}$, which does not affect the proof), where $x^{(i,j)+}(n)$ is the $n^{th}$ search point. Noting that since there is $s\times d$ evaluations per iterations, $\tilde{x}$ is updated only every $s\times d$ evaluations, so we have $\tilde{x}_n=\tilde{x}_{n+1}=\dots=\tilde{x}_{n+s\times d -1}$, therefore $x^{(i,j)+}(n)=\tilde{x}_{\left\lfloor n/s\times d\right\rfloor} +c_n u_j e_i$. By the convexity of $F$, the fact that the gradient of $F$ in $x^*$ is $0$, \begin{align*}
F(x^{(i,j)+}(n)) -& F(x^*)\geq \frac{\lambda}{2}\| (\tilde{x}_{\left\lfloor n/s\times d\right\rfloor} -x^*)+ (c_n u_j e_i) \|^2
\end{align*} Similarly, by using the $\mu$-smoothness of $F$, \begin{align*}
F(x^{(i,j)+}) - F(x^*) \leq\frac{\mu}{2}\| (\tilde{x}_{\left\lfloor n/s\times d\right\rfloor} -x^*)+(c_n u_j e_i)\|^2 \end{align*}
Then $$F(x^{(i,j)+}(n)) - F(x^*)=\Theta(\| (\tilde{x}_{\left\lfloor n/s\times d\right\rfloor} -x^*)+ (c_n u_j e_i)\|^2)$$ If $\beta >\gamma$, then the main term is the last one and we get a rate $-2\gamma$. If $\beta \leq\gamma$, then the main term is the first one and we get a rate $-2\beta$.
\end{proof}
\end{document} |
\begin{document}
\title{On many-sorted $\omega$-categorical theories}
\author{Enrique Casanovas, Rodrigo Pel\'aez, and Martin
Ziegler\thanks{Work partially supported by MODNET, FP6 Marie Curie
Training Network in Model Theory and its Applications with
contract number MRTN-CT-2004-512234. The first author has been
partially supported by grants MTM 2008-01545 and 2009SGR-187.}} \date{March 17, 2011} \maketitle
\begin{abstract} We prove that every many-sorted $\omega$-categorical theory is completely interpretable in a one-sorted $\omega$-categorical theory. As an application, we give a short proof of the existence of non $G$--compact $\omega$-categorical theories. \end{abstract}
\section{Introduction} A many--sorted structure can be easily transformed into a one--sorted by adding new unary predicates for the different sorts. However $\omega$-categoricity is not preserved. In this article we present a general method for producing $\omega$-categorical one--sorted structures from $\omega$-categorical many--sorted structures. This is stated in Corollary~\ref{interpret}, the main theorem in this paper. Our initial motivation was to understand Alexandre Ivanov's example (in~\cite{Ivanov06}) of an $\omega$-categorical non $G$-compact theory. In Corollary~\ref{Ivanov} we apply our results to offer a short proof of the existence of such theories.
Our method is based on the use of a particular theory $T_E$ of equivalence relations $E_n$ on $n$-tuples. The quotient by $E_n$ is an imaginary sort containing a predicate $P_n$ which can be used to copy the $n$-th sort of the given many-sorted theory. Since the complexity of $T_E$ is part of the complexity of the $\omega$-categorical one--sorted theory obtained by our method, it is important to classify $T_E$ from the point of view of stability, simplicity and related properties. It turns out that $T_E$ is non-simple but it does not have $\mathrm{SOP}_2$. A similar example of a theory with such properties has been presented by Shelah and Usvyatsov in~\cite{ShelaUsvyatsov03}. Their proof, as ours, relies on Claim 2.11 of~\cite{DzamonjaShelah04}, which is known to have some gaps. A revised version of~\cite{DzamonjaShelah04} will be posted in arxiv.org. In the meanwhile Kim and Kim have obtained a new proof of the same result: Proposition 2.3 from~\cite{KimKim11}.
The one--sorted theory $T_E$ is interdefinable with some many--sorted theory $T^\ast$ which is presented and discussed in Section~\ref{section:Tast}. In order to describe $T^\ast$ we need a version of Fra\"{\i}ss\'e's amalgamation method that can be applied to the many--sorted case (see Lemma~\ref{F1}). In Section~\ref{section:stable_embedded} some results on stable embeddedness from the third author (in~\cite{Ziegler06}) are extended and used to prove Corollary~\ref{interpret}. Section~\ref{section:classification} is devoted to classify $T_E$ from the stability point of view.
A previous version of these results appeared in the second author's Ph.D. Dissertation \cite{pelaez_diss}. They have been corrected in some points and in general they have been elaborated and made more compact.
\section{$T^\ast$ and Fra\"{\i}ss\'e's amalgamation} \label{section:Tast}
Let $L$ be a countable many--sorted language with sorts $S_i$, ($i\in I$), and let $\mathcal{K}$ be a class of finitely generated $L$--structures.\footnote{We allow empty sorts if $L$ has no constant
symbols of that sort.} We call an $L$--structure $M$ a \emph{Fra\"{\i}ss\'e limit} of $\mathcal{K}$ if the following holds: \begin{enumerate} \item $\mathcal{K}=\mathrm{Age}(M)$, where $\mathrm{Age}(M)$ is the class of all finitely
generated $L$--structures which are embeddable in $M$. \item $M$ is at most countable. \item $M$ is \emph{ultra--homogeneous}\, i.e., any isomorphism between
finitely generated substructures extends to an automorphism of $M$. \end{enumerate} By a well--known argument $\mathcal{K}$ can only have one Fra\"{\i}ss\'e limit, up to isomorphism.
\begin{lemma}\label{F1} Let $\mathcal{K}$ be as above.
Then the following are equivalent:
\begin{enumerate}[a)]
\item The Fra\"{\i}ss\'e limit of $\mathcal{K}$ exists and is
$\omega$-categorical.
\item $\mathcal{K}$ has the amalgamation property $\mathrm{AP}$\xspace, the joint embedding
property $\mathrm{JEP}$\xspace, the hereditary property $\mathrm{HP}$\xspace (i.e., finitely
generated $L$--structures which are embeddable in elements of $\mathcal{K}$
belong themselves to $\mathcal{K}$) and satisfies
\begin{itemize}
\item[$(\ast)$] for all $i_1\ldots i_n \in I$ there are only
finitely many quantifier-free types of tuples $(a_1,\ldots,a_n)$
where the $a_j$ are elements of sort $S_{i_j}$ in some structure
$A\in\mathcal{K}$.
\end{itemize}
\end{enumerate}
If the Fra\"{\i}ss\'e limit of $\mathcal{K}$ exists, it has quantifier
elimination. \end{lemma} \begin{proof}
a) $\Rightarrow$ b). It is well known that the age of an
ultra--homogeneous structure has $\mathrm{AP}$\xspace, $\mathrm{JEP}$\xspace and $\mathrm{HP}$\xspace. All
quantifier--free types which occur in elements of $\mathcal{K}$ are
quantifier--free types of tuples of the Fra\"{\i}ss\'e limit. So
property $(\ast)$ follows from the Ryll--Nardzewski theorem.\\
\noindent b) $\Rightarrow$ a). The quantifier-free type
$\mathrm{qftp}(\bar{a})$ determines the isomorphism type of the structure
generated by $\bar{a}$. Hence $(\ast)$ implies that $\mathcal{K}$ contains at
most countably many isomorphism types. The existence of the
Fra\"{\i}ss\'e limit $M$ follows now from $\mathrm{AP}$\xspace, $\mathrm{JEP}$\xspace and $\mathrm{HP}$\xspace.
If two sequences $\bar{a}$ and $\bar{b}$ have the same
quantifier-free type in $M$, there is an automorphism of $M$ which
maps $\bar{a}$ to $\bar{b}$ and so it follows that $\bar{a}$ and
$\bar{b}$ have the same type in $M$. Consider a formula
$\varphi(\bar x)$ and the set
$P_{\varphi(x)}=\{\mathrm{qftp}(\bar{a}): M\models \varphi(\bar{a})\}$. Then
\[M\models\varphi(\bar{a}) \Leftrightarrow \mathrm{qftp}(\bar{a})\in P_\varphi
\Leftrightarrow M\models \bigvee_{p\in P_\varphi} p(\bar{a}).\] Now,
$(\ast)$ implies that $P_\varphi$ is finite and that in $M$ all
$p=\mathrm{qftp}(\bar{a})$ are finitely axiomatisable, that is, $p=\langle
\chi_p\rangle$ for some quantifier-free $\chi_p(x)$. Then $M\models
\varphi(\bar{a}) \Leftrightarrow \bigvee_{p\in P_\varphi}\chi_p(\bar
a)$. So $M$ has quantifier elimination and it is
$\omega$--categorical since there are only finitely many
possibilities for the $\chi_p$, depending only on the number and the
sorts of the free variables of $\varphi$. \end{proof}
It is easy to see that the theory of the Fra\"{\i}ss\'e limit is the model--completion of the universal theory of $\mathcal{K}$.
\begin{defi}\rm
Let $L^\ast$ be the language with countably many sorts
$S,S_1,\ldots$, function symbols $f_i:S^i\rightarrow S_i$, and
constants $c_i\in S_i$ and $T^0$ the theory of all
$L^\ast$--structures $A$ with
\[f_i(\bar a)=c_i \Leftrightarrow \bar a \text{ has some repetition}\]
for all $\bar a\in S^i(A)$. Furthermore let $\K^\ast$ be the class of
all finitely generated models of $T^0$. \end{defi}
\begin{lemma} $\K^\ast$ satisfies the conditions of Lemma~\ref{F1}. \end{lemma} \begin{proof}
The class of all models of $T^0$ has $\mathrm{AP}$\xspace and $\mathrm{JEP}$\xspace and therefore also
$\K^\ast$. $(\ast)$ follows easily from the fact that
$f_i(a_{m_1}\ldots,a_{m_i})=c_i$ for all $i>k$ and
$\{a_{m_1}\ldots,a_{m_i}\}\subset\{a_1,\ldots,a_k\}$, \end{proof}
We define $M^\ast$ to be the Fra\"{\i}ss\'e limit of $\K^\ast$ and $T^\ast$ to be the complete theory of $M^\ast$. $T^\ast$ is the model--completion of $T^0$.\\
Recall the following definition from \cite{Chat-Hru99}: \begin{defi}\rm
Let $T$ be a complete theory and $P$ a $0$--definable predicate. $P$
is called \emph{stably embedded} if every definable relation on $P$
is definable with parameters from $P$. \end{defi}
\noindent \textbf{Remarks} \begin{enumerate} \item For many--sorted structures with sorts $(S_i)_{i\in I}$ this
generalises to the notion of a sequence $(P_i)_{i\in I}$ of
$0$--definable $P_i\subset S_i$ being stably embedded. \item While the definition is meant in the monster model, an easy
compactness argument shows that, if $P(M)$ is stably embedded in $M$
for some weakly saturated\footnote{$M$ is weakly saturated if every
type over the empty set is realized in $M$.} model $M$, then this
is true for all models. \item If $M$ is saturated then $P$ is stably embedded if and only if
every automorphism (i.e.\ elementary permutation) of $P(M)$ extends
to an automorphism of $M$. This was claimed in \cite{Chat-Hru99}
only for the case that $|M|>|T|$. But the proof can easily be
modified to work for the general case. One has to use the fact that
if $A$ has smaller size than $M$, then any type over a subset of
$\mathrm{dcl}^\mathrm{eq}(A)$ can be realized in $M$. \item If $M$ is $\omega$--categorical, it can be proved that for every
finite tuple $a\in M$ there is a finite tuple $b\in P$ such that
every relation on $P$ which is definable over $a$ can be defined
using the parameter $b$. \end{enumerate}
\begin{lemma} In $T^\ast$ the sequence of sorts $(S_1,S_2,\ldots)$ is
stably embedded. \end{lemma} \begin{proof} Clear since $\mathrm{tp}(\bar{a}/S_1,\ldots)=
\mathrm{tp}(\bar{a}/f_1(\bar{a}),\ldots)$. See also the discussion in
\cite{Chat-Hru99}. \end{proof}
For a complete theory $T$ and a $0$--definable predicate $P$ the \emph{induced structure} on $P$ consists of all $0$--definable relations on $P$. Note that the automorphisms of $P$ with its induced structure are exactly the elementary permutations of $P$ in the sense of $T$.
\begin{lemma} In $T^\ast$ the induced structure on
$(S_1,S_2,\ldots)$ equals its $L^\ast_{>0}$--structure, where $L^\ast_{>0}$ is the
sublanguage of $L^\ast$ which has only the sorts\/ $S_1,S_2,\ldots$
and the constants $c_1,c_2,\ldots$. \end{lemma} \begin{proof} Quantifier elimination. \end{proof}
Let $T^\ast_{>0}$ denote the theory of all $L^\ast_{>0}$--structures, where all sorts $S_i$ are infinite. Clearly $T^\ast_{>0}$ is the restriction of $T^\ast$ to $L^\ast_{>0}$.
\begin{lemma}
Every model of $T^\ast_{>0}$ can be expanded to a model of $T^\ast$. \end{lemma} \begin{proof}
It is easy to see that the following amalgamation property is true:
\begin{itemize}
\item[] Let $N$ be a model of $T^0$ with infinite sorts
$S_i(N)$. Let $A$ be a finitely generated substructure of $N$ and
$B\in\K^\ast$ an extension of $A$. Then $B$ can be embedded over $A$
in an extension $N'$ of $N$ which is a model of $T^0$ and such
that $S_i(N')=S_i(N)$ for all $i$.
\end{itemize}
If a model of $T^\ast_{>0}$ is given, we expand it arbitrarily to a model
$N$ of $T^0$ and apply the above amalgamation property repeatedly
such that the union of the resulting chain is a model of $T^\ast$
which has the same sorts $S_i$ as $N$. \end{proof}
\begin{cor} There is an $\omega$--categorical one-sorted theory $T_E$
with a series of $0$--definable infinite predicates $P_1,P_2,\ldots$
in $T_E^\mathrm{eq}$ such that
\begin{enumerate}
\item $(P_1,P_2,\ldots)$ is stably embedded
\item The many--sorted structure induced on $(P_1,P_2,\ldots)$ is
trivial.
\item For every sequence $\kappa_1,\kappa_2,\ldots$ of infinite
cardinals there is a model $N$ of $T_E$ such that
$|P_i(N)|=\kappa_i$.
\end{enumerate} \end{cor} \begin{proof}
The language $L_E$ of $T_E$ will contain for each $i$ a symbol $E_i$
for an equivalence relation between $i$--tuples. Let
$M=(S,S_1,S_2,\ldots)$ be a model of $T^\ast$. For $a,b\in S^i$
define $E_i(a,b)\Leftrightarrow f_i(a)=f_i(b)$. $T_E$ is the theory
of $M_E=(S,E_1,E_2,\ldots)$. The $S_i$ live in $M_E^\mathrm{eq}$ as
$M_E^i/E_i$ and the $c_i$ are $0$--definable in $M_E^\mathrm{eq}$. We set
$P_i=S_i\setminus\{c_i\}$. \end{proof}
It is easy to see that $T_E$ as constructed in the proof is the model--completion of the theory of all structures $(M,E_1,E_2,\ldots)$ where $E_n$ is an equivalence relation on $M^n$ where one equivalence class consists of all $n$--tuples which contain a repetition. That $T_E$ has quantifier elimination can be proved as follows: Every formula $\varphi(\bar x)$ of $L_E$ is equivalent to a quantifier--free $L^\ast$--formula $\varphi'(\bar x)$. $\varphi'(\bar x)$ is a boolean combination of formulas of the form $f_i(\bar x')\doteq f_i(\bar x'')$ and $f_i(\bar x')\doteq c_i$, which are equivalent to quantifier--free $L_E$--formulas: $f_i(\bar x')\doteq f_i(\bar x'')$ is equivalent to $E_i(\bar x',\bar x'')$, $f_i(x'_1,\ldots,x'_i)\doteq c_i$ is equivalent to $\bigvee_{1\leq k<l\leq i}x'_k\doteq x'_l$.
\section{Expansions of stably embedded predicates} \label{section:stable_embedded}
Let $T$ be complete theory with two sorts $S_0$ and $S_1$. We consider $S_1$ as a structure of its own carrying the structure induced from $T$ and denote by $T\restriction S_1$ the theory of $S_1$.
\begin{lemma}\label{martinlemma}
Let $T$ be complete theory with two sorts $S_0$ and $S_1$. Let
$\widetilde T_1$ be a complete expansion of $T\restriction
S_1$. Assume that $S_1$ is stably embedded. Then we have
\begin{enumerate}
\item\label{complete} $\widetilde T=T\cup\widetilde T_1$ is
complete.\footnote{Actually we have:
$S_1$ is stably embedded if and only if $\widetilde T$ is complete for
all complete expansions $\widetilde T_1$.}
(\cite[Lemma
3.1]{Ziegler06})
\item\label{stably} $S_1$ is stably embedded in $\widetilde T$ and $\widetilde
T\restriction S_1=\widetilde T_1$.
\item\label{omega} If\/ $T$ and $\widetilde T_1$ are $\omega$--categorical,
then $\widetilde T$ is also $\omega$--categorical.
\end{enumerate} \end{lemma} \begin{proof}
\ref{complete}. Let $\widetilde M=(M_0,\widetilde M_1)$ and $\widetilde M'=(M'_0,\widetilde
M'_1)$ be saturated models of $\widetilde T$ of the same cardinality and
$M=(M_0,M_1)$ and $M'=(M'_0,M'_1)$ their restrictions to the
language of $T$. Since $T$ and $\widetilde T_1$ are complete, there are
isomorphisms $f:M\to M'$ and $g:\widetilde M_1\to \widetilde M'_1$. $gf^{-1}$ is
an automorphism of $M'_1$. Since $M'_1$ is stably embedded in $M'$,
$gf^{-1}$ extends to an automorphism $h$ of $M'$. $hf$ is now an
isomorphism from $M$ to $M'$ which extends $g$.\\
\noindent \ref{stably}. We use the same notation as in the proof of
\ref{complete}. Let $\widetilde M$ be a saturated model of $\widetilde T$. We have
to show that every automorphism $f$ of $\widetilde M_1$ extends to an
automorphism of $\widetilde M$. But $f$ extends to an automorphism of $M$,
which is automatically an automorphism of $\widetilde M$.\\
\noindent\ref{omega}. Start with two countable models $\widetilde M$ and
$\widetilde M'$ and proceed as in the proof of \ref{complete}. \end{proof}
\begin{cor}\label{interpret}
Every many-sorted $\omega$-categorical theory is completely (the
induced structure is exactly this) interpretable in a one-sorted
$\omega$-categorical theory. \end{cor} \begin{proof}
Let $T$ be a complete theory with countably many sorts
$P_1,P_2,\ldots$. We consider $T$ as an expansion of
$T_E\restriction (P_1,P_2,\ldots)$ and set $\widetilde T=T_E\cup T$. $\widetilde
T$ is a one-sorted complete theory. We have $\widetilde T\restriction
(P_1,P_2,\ldots)= T$. If $T$ is $\omega$--categorical, $\widetilde T$ is
also $\omega$--categorical. \end{proof}
\begin{cor}[Ivanov]\label{Ivanov}
There is a one--sorted $\omega$--categorical theory which is not
G--compact. \end{cor} \begin{proof}
By \cite{CasanovasLascarPillayZiegler01} there is a many--sorted
$\omega$--categorical theory $T$ which is not
$G$--compact. Interpret $T$ in a one--sorted $\omega$--categorical
theory $\widetilde T$ as in Corollary \ref{interpret}. Then $T$ is also not
$G$--compact. For this one has to check that if $\widetilde T$ is
$G$--compact, then every $0$--definable subset with its induced
structure is also $G$--compact. This follows from the following
description of $G$--compactness: $a$, $b$ of length $\omega$ are in
the relation $\mathrm{nc^\omega}$ if $a$ and $b$ are the first two
elements of an infinite sequence of indiscernibles. A complete
theory is G--compact, if the transitive closure of
$\mathrm{nc^\omega}$ is type--definable. (Note that $(a,b)$ is in the
transitive closure of $\mathrm{nc^\omega}$ if and only if $a$ and $b$
have the same Lascar-strong type.) \end{proof}
\section{Classification of $T_E$} \label{section:classification}
\begin{prop} $T_E$ has $\mathrm{TP}_2$, the tree
property of the second kind, and therefore it is not simple. \end{prop} \begin{proof} We show that $\varphi(x;y,u,v)= E_2(xy,uv)$
has $\mathrm{TP}_2$. Let $(b_i:i < \omega)$, $(c_i:i < \omega)$,
and $(d_i:i < \omega)$ be pairwise disjoint sequences of different
elements such that $ \neg E_2(c_id_i,c_jd_j)$ for $i\neq j$. For
$i,j \in \omega$, let $\bar a^i_j=b_ic_jd_j $. By compactness we can
see that for any $\eta \in \omega^{\omega}$, the set
$\{\varphi(x;\bar a^i_{\eta(i)}): i < \omega\}$ is consistent, and
since the $c_id_i$'s are in different $E_2$-classes, for each $i <
\omega$, the set $\{\varphi(x; \bar a^i_j): j < \omega\}$ is
$2$-inconsistent. \end{proof}
\begin{lemma}[Independence lemma] \label{N2}
Let $a,b,c,d^\prime,d^{\prime\prime}$ be tuples in the monster model
of $T_E$ and $F$ a finite subset. Assume that $a$ and $c $ have only
elements from $F$ in common. If $d^\prime a\equiv_F d^\prime
b\equiv_F d^{\prime\prime}b\equiv_F d^{\prime\prime}c$, then there
exists some $d$ such that $d^\prime a\equiv_F da\equiv_F dc\equiv_F
d^{\prime\prime} c$. \end{lemma}
\begin{center} \setlength{\unitlength}{1.8cm} \begin{picture}(2,1.5)(-1,-0.25) \put(-1,0){$a$} \put(-.95,0.2){\vector(0,1){.7}} \put(-1,1){$d^\prime$} \put(-.1,1.05){\vector(-1,0){.7}} \put(0,1){$b$} \put(0.2,1.05){\vector(1,0){.7}} \put(1,1){$d^{\prime\prime}$} \put(1.05,.2){\vector(0,1){.7}} \put(1,0){$c$} \put(.9,0.05){\vector(-1,0){.7}} \put(0,0){$d$} \put(-.8,.05){\vector(1,0){.7}} \end{picture} \end{center}
\begin{proof} Let $A$, $B$, $C$, $D'$ and $D''$ denote the
set of elements of the tuples $a$, $b$, $c$, $d'$ and $d''$,
respectively. We note first that we can assume that $F$ is contained
in $A$,$B$ and $C$, since otherwise we can increase $a$, $b$ and $c$
by elements from $F$. Then we note that if $A$ and $D'$ intersect
in a subtuple $f$, this tuple also belongs to $B$ and $C$ and
therefore to $F$. So we have that $A\cap D'$ is contained in $F$ and
similarly that $C\cap D''$ is contained in $F$.
It suffices to find an $L_E$-structure $M$ extending $AC$ and
containing a new tuple $d$ with the same quantifier-free type as
$d^\prime$ over $A$ and of $d^{\prime\prime}$ over $C$. Take as $d$
a new tuple of the right length which intersects $A$ and $B$ in the
subtuple $f$. We have then $d^\prime
a\equiv_F^\mathrm{eq}da\equiv_F^\mathrm{eq}dc\equiv_F^\mathrm{eq}
d^{\prime\prime} c$, where $g\equiv_F^\mathrm{eq}h$ means that $g$
and $h$ satisfy the same equality-formulas over $F$, i.e.\ $g_i=g_j$
iff $h_i=h_j$ and $g_i=f_j$ iff $h_i=f_j$. If $D$ denotes the
elements of $d$, it follows that the intersection of any two of $A$,
$C$ and $D$ belongs to $F$.
It remains to define the relations $E_n$ on $ACD$. Let $E_n^0$
denote the part of $E_n$ which is already defined on $AC$. Let
$E_n'$ be the relation $E_n$ transported from $AD'$ to $AD$ via the
identification $d'\mapsto d$ and $E_n''$ the relation $E_n$
transported from $CD''$ to $CD$ via the identification $d''\mapsto
d$. Note that $d'\equiv_F d''$ implies that $E_n'$ and $E_n''$
agree on $DF$. We define $E_n$ on $ACD$ as the transitive closure
of
\[E_n^0\cup E_n'\cup E_n''\cup E_n^\mathrm{rep}\cup\Delta,\]
where $E_n^\mathrm{rep}$ is the set of all pairs of $n$--tuples from
$ACD$ which contain repetitions and $\Delta$ is the identity on
$(ACD)^n$.
We have to show that the new structure defined on $AC$ agrees with
the original structure. Also we must check that the structure on
$AD$ (and $CD$) agrees with the structure on $AD'$ (and $CD'$) via
$d\mapsto d'$ (and $d\mapsto d''$). Using the fact that an
$n$-tuple which e.g.\ belongs to $AC$ and $AD$ belongs already to
$A$, it is easy to see that we have to show the following:
\begin{itemize}
\item[] For all $n$--tuples $x\in A$, $y\in C$ and $z\in DF$
\begin{enumerate}
\item\label{xzy} $E_n'(x,z)\land
E_n''(z,y)\;\Rightarrow\;E_n^0(y,x)$
\item\label{zyx} $E_n''(z,y)\land
E_n^0(y,x)\;\Rightarrow\;E_n'(x,z)$
\item\label{yxz} $E_n^0(y,x)\land
E_n'(x,z)\;\Rightarrow\;E_n''(z,y)$
\end{enumerate}
\end{itemize}
Let $z'$ and $z''$ be the subtuples of $D'F$ and $D''F$ which
correspond to $z$.\\
\noindent Proof of \ref{xzy} : Assume $E_n'(x,z)$ and
$E_n''(z,y)$. We have then $E_n(x,z')$ and $E_n(z'',y)$. $d^\prime
a\equiv_F d^\prime b$ implies $z'a\equiv_F z'b$, which implies that
there is a tuple $x'$ in $B$ such that $z'x\equiv_F z'x'$. So we
have $E_n(z',x')$. $d^\prime b\equiv_F d^{\prime\prime}b$ implies
$z'x'\equiv_F z''x'$ and whence $E_n(z'',x')$. Now we can connect
$y$ and $x$ as follows: $y\;E_n\;z''\;E_n\;x'\;E_n\;z'\;E_n\;x$.\\
\noindent Proof of \ref{zyx} : Assume $E_n''(z,y)$ and
$E_n^0(y,x)$. We have then $E_n(z'',y)$. As above we find
a tuple $y'\in B$ such that $E_n(z'',y')$ and $E_n(z',y')$.
The chain $x\;E_n\;y\;E_n\;z''\;E_n\;y'\;E_n\;z'$ shows that
$E_n'(x,z)$.\\
\noindent Proof of \ref{yxz} : Symmetrical to the proof of
\ref{zyx}. \end{proof}
In order to state \cite[Proposition 2.3]{KimKim11} we need the following terminology: \begin{enumerate}[(1)] \item A tuple $\bar\eta=(\eta_0,\ldots,\eta_{d-1})$ of
elements of $2^{<\omega}$ is \emph{$\cap$-closed} if the set
$\{\eta_0,\ldots,\eta_{d-1}\}$ is closed unter intersection. \item Two $\cap$-closed tuples $\bar\eta$ and $\bar\nu$ are
\emph{isomorphic} if they have the same length and
\begin{enumerate}[(i)]
\item $\eta_i\unlhd\eta_j$ iff $\nu_i\unlhd\nu_j$
\item $\eta_i^\smallfrown t\unlhd\eta_j$ iff $\nu_i^\smallfrown
t\unlhd\nu_j$ for $t=0,1$.
\end{enumerate} \item A tree $(a_\eta\colon\eta\in2^{<\omega})$ of tuples of the same length
is \emph{modeled} by $(b_\eta\colon\eta\in2^{<\omega})$ if for every
formula $\phi(\bar x)$ and every $\cap$-closed $\bar\eta$ there is a
$\cap$--closed $\bar\nu$ isomorphic to $\bar\eta$ such that
$\models\phi(b_{\bar\eta})\,\Leftrightarrow\,\models\phi(a_{\bar\nu})$. \item $(b_\eta\colon\eta\in2^{<\omega})$ is \emph{indiscernible} if
$\models\phi(b_{\bar\eta})\,\Leftrightarrow\,\models\phi(b_{\bar\nu})$
for all isomorphic $\cap$-closed $\bar\eta,\bar\nu$. \end{enumerate}
\begin{lemma}[\protect{\cite[Proposition 2.3]{KimKim11}. See also~\cite{DzamonjaShelah04}}]
\label{N1}
Let $T$ be a complete theory. Then any tree of tuples can be modeled
by an indiscernible tree. \end{lemma}
\begin{defi}\rm The formula $\varphi(x,y)$ has
$\mathrm{SOP}_2$ in $T$ if there is a binary tree $(a_\eta: \eta\in
2^{<\omega})$ such that for every $\eta\in 2^\omega$,
$\{\varphi(x,a_{\eta\restriction n}): n<\omega\}$ is consistent and
for every incomparable $\eta, \nu\in 2^{<\omega}$,
$\varphi(x,a_\eta) \wedge \varphi(x,a_\nu)$ is inconsistent. The
theory $T$ has $\mathrm{SOP}_2$ if some formula $\varphi(x,y)\in L$
has $\mathrm{SOP}_2$ in $T$. \end{defi}
\begin{remark}[H. Adler] The formula $\varphi(x,y)$ has $\mathrm{SOP}_2$ in $T$ if and only if $\varphi(x,y)$ has the tree
property of the first kind $\mathrm{TP}_1$: there is a tree
$(a_\eta: \eta\in \omega^{<\omega})$ such that for every $\eta\in
\omega^\omega$, $\{\varphi(x,a_{\eta\restriction n}): n<\omega\}$ is
consistent and for every incomparable $\eta, \nu\in
\omega^{<\omega}$, $\varphi(x,a_\eta) \wedge \varphi(x,a_\nu)$ is
inconsistent. \end{remark} \begin{proof} By compactness. \end{proof}
\begin{prop} $T_E$ does not have $\mathrm{SOP}_2$. \end{prop} \begin{proof} We follow ideas from a similar proof
in~\cite{ShelaUsvyatsov03}. Assume $\varphi(x,y)$ has
$\mathrm{SOP}_2$ in $T_E$ and the tree $(a_\eta: \eta\in
2^{<\omega})$ witnesses it. Choose for every $\eta$ a tuple $d_\eta$
such that $\models\phi(d_\eta,a_\nu)$ for all $\nu\subsetneq\eta$.
By Lemma~\ref{N1} we can assume that the tree $(d_\eta a_\eta :
\eta\in 2^{<\omega})$ is indiscernible. Let us now look at the
elements $a_{00},a_{\langle\rangle},a_{01},d_{000}, d_{010}$. We
have by indiscernibility
\[d_{000}a_{00}\equiv d_{000}a_{\langle\rangle}\equiv
d_{010}a_{\langle\rangle}\equiv d_{010}a_{01}.\]
If the tuples $a_{00}$ and $a_{01}$ are disjoint, we can apply the
Independence Lemma to $a=a_{00}$, $b=a_{\langle\rangle}$,
$c=a_{01}$, $d'=d_{000}$, $d''=d_{010}$ to get a tuple $d$ such that
\[d_{000}a_{00}\equiv da_{00} \equiv
da_{01}\equiv d_{010}a_{01}.\] It follows that
$\models\varphi(d,a_{00})\wedge\varphi(d,a_{01})$, which contradicts
the $\mathrm{SOP}_2$ of the tree.
If $a_{00}$ and $a_{01}$ are not disjoint, we argue as follows:
Assume that $a_{00}$ and $a_{01}$ have an element $f$ in common, say
$f=a_{00,i}=a_{01,j}$. Then $a_{00}a_{01}\equiv a_{000}a_{01}$
implies $a_{000,i}=a_{01,j}$. So we have $a_{000,i}=a_{00,i}$ and it
follows from indiscernibility that
$f=a_{00,i}=a_{\langle\rangle,i}=a_{01,i}$. Let $F$ be the set of
elements which occur in both $a_{00}$ and $a_{01}$. We have seen
that the elements of $F$ occur in $a_{00}$, $a_{\langle\rangle}$ and
$a_{01}$ at the same places. Therefore \[d_{000}a_{00}\equiv_F
d_{000}a_{\langle\rangle}\equiv_F d_{010}a_{\langle\rangle}\equiv_F
d_{010}a_{01}\] and we can again apply the Independence Lemma.
\end{proof}
\nocite{Ivanov06}\nocite{Chat-Hru99}\nocite{Ziegler06} \nocite{CasanovasLascarPillayZiegler01}
\noindent{\sc Department of Logic, History and Philosopy of Science, University of Barcelona, Montalegre 5, 08001 Barcelona, Spain.}\\ {\tt e.casanovas@ub.edu}\\
\noindent{\sc Department of Logic, History and Philosopy of Science, University of Barcelona, Montalegre 5, 08001 Barcelona, Spain.}\\ {\tt rpelaezpelaez@yahoo.com}\\
\noindent{\sc Mathematisches Institut, Albert-Ludwigs-Universit\"at
Freiburg, D-79104 Freiburg, Germany}\\ {\tt ziegler@uni-freiburg.de}
\end{document} |
\begin{document}
\title[Nuclear and type I crossed products]{Nuclear and type I crossed products of C*-algebras by group and compact quantum group actions} \author{Raluca Dumitru and Costel Peligrad} \address{Raluca Dumitru: Department of Mathematics and Statistics, University of North Florida, 1 UNF Drive, Jacksonville, Florida 32224; Institute of Mathematics of the Romanian Academy, Bucharest, Romania; E-mail address: raluca.dumitru@unf.edu} \address{Costel Peligrad: Department of Mathematical Sciences, University of Cincinnati, 610A Old Chemistry Building, Cincinnati, OH 45221; E-mail address: costel.peligrad@uc.edu} \subjclass[2000]{47L65, 20G42} \maketitle
\begin{abstract} If $A$ is a C*-algebra, $G$ a locally compact group, $K\subset G$ a compact subgroup and $\alpha:G\rightarrow Aut(A)$ a continuous homomorphism, let $ A\times_{\alpha}G$ denote the crossed product. In this paper we prove that $ A\times_{\alpha}G$ is nuclear (respectively type I or liminal) if and only if certain hereditary C*-subalgebras, $S_{\pi}$, $\mathcal{I}_{\pi}\subset A\times_{\alpha}G$ $\pi\in\widehat{K}$, are nuclear (respectively type I or liminal). These algebras are the analogs of the algebras of spherical functions considered by R. Godement for groups with large compact subgroups. If $K=G$ is a compact group or a compact quantum group, the algebras $ S_{\pi} $ are stably isomorphic with the fixed point algebras $A\otimes B(H_{\pi })^{\alpha\otimes ad\pi}$ where $H_{\pi}$ is the Hilbert space of the representation $\pi.$ \end{abstract}
\section{Introduction and preliminary results}
Let $G$ be a locally compact group and $K\subset G$ a compact subgroup. In \cite{godement} (see also \cite{warner}) the study of $\widehat{G}$, the set of equivalence classes of irreducible representations of $G$ is reduced to the study of $\widehat{K}$ and the representations of certain classes of spherical functions. In this paper we extend this approach to the case of crossed products of C*-algebras by locally compact group and compact quantum group actions. Let $(A,G,\alpha)$ be a C*-dynamical system and let $K\subset G$ be a compact subgroup.
In \cite{peligrad} we defined the C*-algebras $S_{\pi}$, $\mathcal{I} _{\pi}\subset A\times_{\alpha}G$, $\pi\in\widehat{K}$ where $\widehat{K}$ is the set of all equivalence classes of unitary representations of $K.$ These are the analogs of the algebras of the algebras of spherical functions. For the case $K=G$, these algebras were previously defined by Landstad in \cite {landstad}.
Recently, in \cite{raljfa,ralpelspectra}, we have extended the study of these algebras to the case of compact quantum group actions on C*-algebras. If $K=G$ is a compact group or a compact quantum group, the algebras $ S_{\pi} $ are stably isomorphic with the fixed point algebras $A\otimes B(H_{\pi})^{\alpha\otimes ad\pi}$ where $H_{\pi}$ is the Hilbert space of the representation $\pi$. In this section we will review some definitions and preliminary results.
\subsection{Preliminaries on actions of compact groups on C*-algebras.}
\
Let $K$ be a compact group and denote by $\widehat{K}$ the set of all equivalence classes of irreducible, unitary representations of $K$. Let $ \delta :K\rightarrow Aut(A)$ be an action of $K$ on a C*-algebra $A.$ Let $ \pi \in \widehat{K\text{.}}$ If $\pi _{ij}(g)$ are the coefficients of $\pi _{g}$ in a fixed basis of the Hilbert space $H_{\pi }$ of the representation $\pi ,1\leq i,j\leq d_{\pi }$ we define the character of $\pi ,\chi _{\pi }(g)=d_{\pi }tr(\pi _{g^{-1}})=d_{\pi }\sum \overline{\pi _{ii}(g)},g\in K$ where $d_{\pi }$ is the dimension of the representation $\pi .$ We consider the following mapping from $B$ into itself :
\begin{center} $P^{\pi ,\delta }(a)=\int_{K}\chi _{\pi }(k)\delta _{k}(a)dk$ \end{center}
We define the spectral subspaces of the action $\delta$
\begin{center}
$A_{1}^{\delta }(\pi )=\left\{ a\in A|P^{\pi ,\delta }(a)=a\right\} $, $\pi \in \widehat{K}$ \end{center}
In particular if $\pi=\pi_{0},$ is the trivial one dimensional representation, $A_{1}^{\delta}(\pi_{0})=A^{\delta}$ is the algebra of fixed elements under the action $\delta$. In this case, the projection $ P^{\pi_{0},\delta}$ of $A$ onto $A^{\delta}$ is a completely positive map. Indeed, the extension of $P^{\pi_{0},\delta}$ to $M_{n}(A)$ is the projection of this latter C*-algebra onto its fixed point algebra with respect to the action $\alpha\otimes id$ where $id$ is the trivial action of $G$ on $M_{n}=B(H_{n})$ where $H_{n}$ is the Hilbert space of dimension $n$.
\subsection{Algebras of spherical functions inside the crossed product}
\
Let now $(A,G,\alpha)$ be a C*-dynamical system with $G$ a locally compact group and $K\subset G$ a compact subgroup. Denote by $A\times_{\alpha}G$ the corresponding crossed product (see for instance \cite{pedersen}). Then the algebra $C(K)$ of all continuous functions on $G$ can be embedded as follows in the multiplier algebra $M(A\times_{\alpha}G)$ of $A\times_{\alpha}G$: If $ \varphi\in C(K)$ and $y\in C_{c}(G,A),$the dense subalgebra of $A\times _{\alpha}G$ consisting of continuous functions with compact support from $G$ to $A$, then
\begin{center} $(\varphi y)(g)=\int_{K}\varphi(k)\alpha_{k}(y(k^{-1}g))dk$ \end{center}
and
\begin{center} $(y\varphi)(g)=\int_{K}\varphi(k)y(gk)dk$ \end{center}
In particular, if $\varphi=\chi_{\pi},$ $\varphi$ is a projection in $ M(A\times_{\alpha}G)$ and if $\pi_{1}$ and $\pi_{2}$ are distinct elements in $\widehat{K\text{,}}$ the projections $\chi_{\pi_{1}}$ and $ \chi_{\pi_{2}} $ are orthogonal. We need the following results from [\cite {peligrad}, Lemma 2.5.]:
\begin{remark} \label{Lemma2.5JFA}The following statements hold:\newline i) If $\pi_{1} \neq\pi_{2}$ in $\widehat{K}$ then the projections $\chi_{\pi_{1}}$ and $\chi_{\pi_{2}}$ are orthogonal in $M(A\times_{\alpha}G)$.\newline ii) $\sum_{\pi}\chi_{\pi}=I$, where $I$ is the identity of the bidual $(A\times_{\alpha}G)^{\star\star}$ of $A\times_{\alpha}G$. \end{remark}
If $\pi\in\widehat{K}$, denote $S_{\pi}=\overline{\chi_{\pi}(A\times_{ \alpha}G)\chi_{\pi}}$, where the closure is taken in the norm topology of $ A\times_{\alpha}G$ Then, it is immediate that $S_{\pi}$ is strongly Morita equivalent with the two sided ideal $J_{\pi}\overline{=(A\times_{\alpha}G) \chi_{\pi}(A\times_{\alpha}G)}$. Indeed, it can be easily verified that $X= \overline{(A\times_{\alpha}G)\chi_{\pi}}$ is an $S_{\pi}-J_{\pi}$ imprimitivity bimodule. We will consider next the action, $\delta$ of $K$ on $A\times_{\alpha}G$ defined as follows: If $y\in C_{c}(G,A)$ set $ \delta_{k}(y)=\alpha_{k}(y(k^{-1}gk)$. Then $\delta_{k}$ extend to automorphisms of $A\times_{\alpha}G$ and thus $\delta$ is an action of $K$ on $A\times_{\alpha}G$. The fixed point algebra $\mathcal{I=} (A\times_{\alpha}G)^{\delta}$ is called in \cite{peligrad} the algebra of K-central elements of the crossed product $A\times_{\alpha}G$. Denote:
\begin{center} $\mathcal{I}_{\pi}=\mathcal{I\cap}S_{\pi}$ \end{center}
Then, [\cite{peligrad}, Proposition 2.7.], we have
\begin{remark} \label{JFA2.7}$S_{\pi}$ is $\ast-$isomorphic with $\mathcal{I}_{\pi}\otimes B(H_{\pi})$. \end{remark}
If $G=K$ is a compact group, then by [\cite{landstad}, Lemma 3] we have:
\begin{remark} \label{landstad lemma3}For every $\pi\in\widehat{G}$, $\mathcal{I}_{\pi}$ is $\ast-$ isomorphic with $(A\otimes B(H_{\pi}))^{\alpha\otimes ad\pi}$. \end{remark}
\subsection{Compact quantum group actions on C*-algebras}
\
Let $\mathcal{G}=(B,\Delta)$ be a compact quantum group (\cite{wor1,wor2}). Here, $B$ is a unital C*-algebra (which is the analog of the C*-algebra of continuous functions in the group case) and $\Delta:B\rightarrow B\otimes_{\min}B$ a $\ast$-homomorphism such that:
i) $(\Delta\otimes\iota)\Delta=(\iota\otimes\Delta)\Delta$, where $ \iota:B\rightarrow B$ is the identity map and
ii) $\overline{\Delta(B)(1\otimes B)}=\overline{\Delta(B)(B\otimes1)} =B\otimes_{\min}B$.
Let $\widehat{\mathcal{G}}$ denote the set of all equivalence classes of unitary representations of $\mathcal{G}$ or equivalently, the set of all equivalence classes of irreducible unitary co-representations of $B$. For each $\pi\in\widehat{\mathcal{G}}$, $\pi=\left[ \pi_{ij}\right] $, $ \pi_{ij}\in B$ $1\leq i,j\leq d_{\pi}$,where $d_{\pi}$ is the dimension of $ \pi$, let $\chi_{\pi}=\sum_{i}\pi_{ii}$ be the character of $\pi$ and let $ F_{\pi}\in B(H_{\pi})$ be the positive, invertible matrix that intertwines $ \pi$ with its double contragredient representation and such that $ tr(F_{\pi})=tr(F_{\pi }^{-1})=M_{\pi}$. Then, with the notations in \cite
{wor1}, $F_{\pi}=\left[ f_{1}(\pi_{ij})\right] $ where $f_{1}$ is a linear functional on the $\ast -$subalgebra $\mathcal{B\subset}B$ that is linearly spanned by $\left\{\pi_{ij}|\pi\in\widehat{\mathcal{G}},1\leq i,j\leq d_{\pi}\right\}$. If $a\in B$ (respectively $\mathcal{B}$) and $\xi$ is a linear functional on $B$ (respectively $\mathcal{B}$) we denote (\cite {wor1,wor2})
\begin{center} $a\ast\xi=(\xi\otimes\iota)(\Delta(a))\in B$ \end{center}
Denote also by $\xi\cdot a$ the following linear functional on $B$ (respectively $\mathcal{B}$):
\begin{center} $(\xi\cdot a)(b)=\xi(ab)$ \end{center}
If $h$ is the Haar state on $B$ let $h_{\pi}=M_{\pi}h\cdot(\chi_{\pi}\ast f_{1})$. If $v_{r}$ is the right regular representation of $\mathcal{G}$, the Fourier transform of $a\in B$ is defined as follows:
\begin{center} $\widehat{a}=\mathcal{F}_{v_{r}}(a)=(\iota\otimes h\cdot a)(v_{r}^{\star})$ \end{center}
where $\mathcal{F}_{v_{r}}$ is the Fourier transform as defined by Woronowicz in \cite{wor2}. Then the norm closure of the set $\widehat{B}
=\left\{ \widehat {a}|a\in B\right\} $ is a C*-algebra called the dual of $B$ (\cite{baaj,wor2}) and $\widehat{B}$ is a subalgebra of the algebra of compact operators, $\mathcal{C(}H_{h})$ on the Hilbert space $H_{h}$ of the GNS representation of $B$ associated with the Haar state $h$.
Let $A$ be a C*-algebra and $\delta:A\rightarrow M(A\otimes B)$ be a $\ast-$ homomorphism of $A$ into the multiplier algebra of the minimal tensor product $A\otimes B$. Then $\delta$ is called an action of $\mathcal{G}$ on $ A$ (or a coaction of $B$ on $A$) if the following two conditions hold: \newline a) $(\iota\otimes\Delta)\delta=(\delta\otimes\iota)\delta$ and\newline b) $\overline{\delta(A)(1\otimes B)}=A\otimes B$
Let $\pi\in\widehat{\mathcal{G}}$. Denote $P^{\pi,\delta}(a)=(\iota\otimes h_{\pi})(\delta(a)),a\in A$. Then $P^{\pi,\delta}$ is a contractive linear map from $A$ into itself. In particular, if $\pi=\pi_{0}$ is the trivial one dimensional representation, then $P^{\pi_{0},\delta}=(\iota\otimes h)\delta$ is the completely positive projection of norm $1$ of $A$ onto the fixed point C*-subalgebra $A^{\delta}$.
The crossed product $A\times_{\delta}\mathcal{G}$ is by definition, (\cite {baaj,boca}), the norm closure of the set $\left\{
(\pi_{u}\otimes\pi_{h})(\delta (a)(1\otimes\widehat{b})|a\in A,b\in B\right\} $, where $\pi_{u}$\ is the universal representation of $A$\ and $ \pi_{h}$\ is the GNS representation of $B$ associated with the Haar state $h$ .
Let $\pi\in\widehat{\mathcal{G}}$. If we denote $p_{\pi}=(\iota\otimes h_{\pi})(v_{r}^{\star})$, then $\left\{ p_{\pi}\right\} _{\pi\in\widehat { \mathcal{G}}}$ are mutually orthogonal projections in $\widehat{B}$ and therefore in $A\times_{\delta}\mathcal{G}$ (\cite{boca,raljfa}). For $\pi\in \widehat{\mathcal{G}}$ denote $\mathcal{S}_{\pi}=\overline{ p_{\pi}(A\times_{\delta}\mathcal{G)}p_{\pi}}$. In [\cite{raljfa}, Lemma 3.3] it is shown that $ad(v_{r})$ is an action of $\mathcal{G}$ on the crossed product $A\times_{\delta}\mathcal{G}$ and the fixed point algebra $\mathcal{ I=(}A\times_{\delta}\mathcal{G)}^{ad(v_{r})}$ of this action plays the role of the $K-$central elements in the case of groups. Let $\mathcal{I}_{\pi}= \mathcal{I\cap S}_{\pi}$. Let $\delta_{\pi}$ be the following action of $ \mathcal{G}$ on $A\otimes B(H_{\pi})$:
\begin{center} $\delta_{\pi}(a\otimes m)=(\pi)_{23}(\delta(a))_{13}(1\otimes m\otimes 1)(\pi^{\ast})_{23}$ \end{center}
where the leg-numbering notation is the usual one (\cite{baaj,wor2}). The above $\delta_{\pi}$ equals $\delta\otimes ad(\pi)$ in the case of compact groups. Then, we have:
\begin{remark} \label{ralucaanalogsof2.2,2.8andlandstadlemma}The following statements hold true:\newline i) The projections $\left\{ p_{\pi}\right\} _{\pi\in \widehat{\mathcal{G}}}$ are mutually orthogonal and $\sum_{\pi}p_{\pi}=1$ in the bidual $(A\times_{\delta}\mathcal{G)}^{\star\star}$\newline ii) $\mathcal{S}_{\pi}$ is $\star-$isomorphic with $\mathcal{I}_{\pi}\otimes B(H_{\pi})$\newline iii) $\mathcal{I}_{\pi}$ is $\star-$isomorphic with $A\otimes B(H_{\pi})^{\delta_{\pi}}$ \end{remark}
\begin{proof} Part i) is [\cite{raljfa}, Section 2.1., Equation (2) and the discussion after that equation]. Part ii) is [\cite{raljfa}, Remark 3.5.] and Part iii) is [\cite{raljfa}, Proposition 4.8.]. \end{proof}
\section{Nuclear and type I crossed products}
In this section we will state and prove our main results. We give necessary and sufficient conditions for a crossed product to be nuclear or type I. Our conditions are given in terms of the algebras of spherical functions inside the crossed product and in case of compact groups or compact quantum groups, in terms of the fixed point algebras of $A\otimes B(H_{\pi})$ for the actions $\delta\otimes ad(\pi)$.
Recall that a C*-algebra $C$ is said to be of type I if for every factor representation $T$ of $C$ the Von Neumann factor $T(C)^{\prime\prime}$ is a type I factor. $C$ is called liminal if for every irreducible representation $T$ of $C$, $T(C)$ consists of compact operators.
A C*-algebra is called nuclear if its bidual, $C^{\ast\ast}$, is an injective von Neumann algebra, i.e. if and only if there is a projection of norm one from $B(H_{u})$ onto $C^{\ast\ast}$, where $H_{u}$ is the Hilbert space of the universal representation of $C$. With the notations from Section 1, we have the following :
\begin{remark} \label{Cor 2.8JFA}Let $(A,G,\alpha)$ be a C*-dynamical system with G a locally compact group and let $K\subset G$ be a compact subgroup. The following three statements hold:\newline i) $S_{\pi}$ is nuclear if and only if $\mathcal{I}_{\pi}$ is nuclear\newline ii) $S_{\pi}$ is liminal if and only if $\mathcal{I}_{\pi}$ is liminal\newline iii) $S_{\pi}$ is type I if and only if $\mathcal{I}_{\pi}$ is type I \end{remark}
\begin{proof} These statements follow from Remark \ref{JFA2.7}. \end{proof}
The following is the analog of the above Remark for the case of compact quantum group actions:
\begin{remark} \label{followsfromRalRemark3.5}Let $\mathcal{G}=(B,\Delta)$ be a compact quantum group and $\delta$ an action of $\mathcal{G}$ on a C*-algebra $A$. The following conditions are equivalent:\newline i) $\mathcal{S}_{\pi}$ is nuclear if and inly if $(A\otimes B(H_{\pi}))^{\delta_{\pi}}$ is nuclear\newline ii) $\mathcal{S}_{\pi}$ is liminal if and inly if $(A\otimes B(H_{\pi}))^{\delta_{\pi}}$ is liminal\newline iii) $\mathcal{S}_{\pi}$ is type I if and only if $(A\otimes B(H_{\pi}))^{\delta_{\pi}}$ is type I \end{remark}
\begin{proof} The result follows from Remark \ref{ralucaanalogsof2.2,2.8andlandstadlemma}. \end{proof}
\subsection{Type I crossed products}
\
We start with the following general result:
\begin{lemma} \label{typeIlemma}Let $C$ be a C*-algebra and $M(C)$ the multiplier algebra of $C.$ Let $\left\{ p_{\lambda}\right\} \subset M(C)$ be a family of mutually orthogonal projections of sum 1 in $C^{\ast\ast}$, the bidual of $C$. The following conditions are equivalent:\newline i) $C$ is type I (respectively liminal) \newline ii) The hereditary subalgebras $S_{\lambda}=p_{\lambda}Cp_{\lambda}\subset C$ are type I (respectively liminal) for every $\lambda$. \end{lemma}
\begin{proof} Assume that $C$ is type I (respectively liminal). Then $S_{\lambda}$ are type I (respectively liminal) as C*-subalgebras of a type I (liminal) C*-algebra.
Assume now that all $S_{\lambda}$ are type I (liminal). Let $T$ be a nondegenerate factor representation (respectively an irreducible representation) of $C$. Since, by assumption, $\sum p_{\lambda}=1$ it follows that $\sum p_{\lambda}C$ is norm dense in $C$. Therefore, there is a $\lambda$ such that the restriction of $T$ to $p_{\lambda}C,$ $
T|_{p_{\lambda}C}\neq0$. Then $T|_{J_{\lambda}}\neq0$, where $J_{\lambda}= \overline{Cp_{\lambda}C}$ is the two sided ideal of $C$ generated by $ p_{\lambda}$. Since $T$ is a factor representation of $C$ (respectively an irreducible representation of $C$) and the bicommutant $T(J_{\lambda})^{^{
\prime\prime}}$ is a nonzero weakly closed ideal of $T(C)^{\prime\prime}$ it follows that $T(J_{\lambda})^{^{\prime\prime}}=T(C)^{\prime\prime}$. Therefore $T$ has the same type with $T|_{J_{\lambda}}$. On the other hand, it can be checked that $J_{\lambda}$ is strongly Morita equivalent with $ S_{\lambda}$ in the sense of Rieffel, \cite{rieffel}, with imprimitivity bimodule $Cp_{\lambda}$. Therefore, since $S_{\lambda}$ is assumed to be type I (respectively liminal), it follows from the discussion in \cite {rieffel} (respectively \cite{fell}) that $J_{\lambda}$ is type I (respectively liminal). It then follows that the representation $T$ is a type I representation (respectively $T(C)$ consists of compact operators). Since $T$ was arbitrary, we are done. \end{proof}
We will state next some consequences of the above Lemma.
\begin{theorem} \label{typeIlocallycompact}Let $(A,G,\alpha)$ be a C*-dynamical system with $G$ a locally compact group and let $K\subset G$ be a compact subgroup. Then the following conditions are equivalent:\newline i) $A\times_{\alpha}G$ is type I (respectively liminal)\newline ii) The hereditary C*-subalgebras $S_{\pi}\subset A\times_{\alpha}G$, $\pi\in\widehat{K}$ are type I (respectively liminal)\newline iii) The C*-subalgebras of $K-$central elements, $\mathcal{I}_{\pi}\subset S_{\pi},\pi\in\widehat{K}$ are type I (respectively liminal). \end{theorem}
\begin{proof} The equivalence of the conditions i)-iii) follows from Remarks \ref {Lemma2.5JFA} and \ref{Cor 2.8JFA} and Lemma \ref{typeIlemma}. \end{proof}
If $G=K$ is a compact group, then the conditions i)-iii) in the above theorem are equivalent with:
iv) The fixed point algebra $A^{\alpha}$ is type I (respectively liminal) [ \cite{gootman}, Theorem 3.2].
We will prove next an analogous result for compact quantum group actions. In [\cite{boca}, Theorem19] it is shown that the crossed product of a C*-algebra by an ergodic action of a compact quantum group is a direct sum of full algebras of compact operators, hence a liminal C*-algebra. Since, in the ergodic case, $\mathcal{S}_{\pi}$ are finite dimensional, the next result is an extension of Boca's result to the case of general compact quantum group actions.
For compact quantum groups we have the following result:
\begin{theorem} \label{typeIquantum}Let $\mathcal{G=(}B,\Delta)$ be a compact quantum group and $\delta$ an action of $\mathcal{G}$ on a C*-algebra $A$. The following conditions are equivalent:\newline i) $A\times_{\delta}\mathcal{G}$ is type I (respectively liminal)\newline ii) The hereditary C*-subalgebras $\mathcal{S}_{\pi}\subset A\times_{\delta}\mathcal{G}$, $\pi\in\widehat{\mathcal{G}}$, are type I (respectively liminal)\newline iii) The C*-subalgebras $\mathcal{I}_{\pi}\subset\mathcal{S}_{\pi},\pi\in \widehat{\mathcal{G}}$, are type I (respectively liminal).\newline iv) The C*-algebras $A\otimes B(H_{\pi})^{\delta_{\pi}}$, $\pi\in\widehat{\mathcal{G}}$ are type I (respectively liminal). \end{theorem}
\begin{proof} The result follows from Remark \ref{ralucaanalogsof2.2,2.8andlandstadlemma} and Lemma \ref{typeIlemma}. \end{proof}
\subsection{Nuclear crossed products}
\
We start with the following lemma which is certainly known but we could not find a reference for it:
\begin{lemma} \label{prepLemma}A C*-algebra $C$ is nuclear if and only if for every state $\varphi$ of $C$, $T_{\varphi}(C)^{\prime\prime}$ is an injective von Neumann algebra, where $T_{\varphi}$ is the GNS representation of $C$ associated with $\varphi$. \end{lemma}
\begin{proof} If $C$ is nuclear then $C^{\ast\ast}$ is an injective von Neumann algebra [ \cite{effros}, Theorem 6.4.]. Therefore, so is $T_{\varphi}(C)^{\prime \prime} $ which is isomorphic with an algebra of the form $eC^{\ast\ast}$ for a certain projection, $e\in(C^{\ast\ast})^{\prime}$.
Conversely, if $T_{\varphi}(C)^{\prime\prime}$ is injective for every state $ \varphi$, let $\left\{ \varphi_{\iota}\right\} $ be a maximal family of states for which the corresponding cyclic representations $ T_{\varphi_{\iota}}$ are disjoint. Then $T_{\varphi_{\iota}}$ and $T=\oplus T_{\varphi_{\iota}}$ can be extended to normal representations $\overline{ T_{\varphi_{\iota}}\ }$ and $\overline{T\ }$ of $C^{\ast\ast}$ with $ \overline{T\ }$ a normal isomorphism. Therefore, $C^{\ast\ast}$ is isomorphic with $\oplus T_{\varphi_{\iota}}(C)^{\prime\prime}$. Since all $ T_{\varphi_{\iota}}(C)^{\prime\prime}$ are injective (by assumption), from [ \cite{effros}, Proposition 3.1.], it follows that $C^{\ast\ast}$ is injective and thus $C$ is nuclear. \end{proof}
Throughout the rest of this section all algebras, groups and quantum groups are assumed to be separable. The following Lemma is the analog of Lemma \ref {typeIlemma} for the case of nuclear crossed products.
\begin{lemma} \label{nuclearlemma}Let $C$ be a separable C*-algebra and $\left\{q_{\lambda}\right\} \subset M(C)$ be a family of mutually orthogonal projections such that $\sum_{\lambda}q_{\lambda}=1$ in $C^{\star\star}$. The following statements are equivalent\newline i) $C$ is nuclear\newline ii) The hereditary C*-subalgebras $S_{\lambda}$ are nuclear for all $\lambda$. \end{lemma}
\begin{proof} Assume first that $C$ is nuclear. Then, by [\cite{choi}, Corollary 3.3 (4)], every hereditary subalgebra of $C$ is nuclear. Hence $S_{\lambda}$ is nuclear for every $\lambda$.\newline Assume now that ii) holds that is : all $S_{\lambda}$ are nuclear C*-algebras. We will show that for every cyclic representation $T_{\varphi}$ of $C$, $T_{\varphi} (C)^{\prime\prime}$ is injective and the result will follow from the previous lemma. Let $\varphi$ be a state of $C$. Then, by reduction theory, $\overline{T_{\varphi}}(C^{\ast\ast})=T_{\varphi}(C)^{ \prime\prime}$ is the direct integral of factors $\overline{T_{\psi}} (C^{\ast\ast})=T_{\psi }(C)^{\prime\prime}$ where $\psi$ are factor states of $C$, $\overline {T_{\varphi}}(C^{\ast\ast})=\int\overline{T_{\psi}} (C^{\ast\ast})d\mu (\psi)$ where $\mu$ is the central measure associated with the state $\varphi$ and the integral is taken over the state space of $ C $ [\cite{sakai}, Theorem 3.5.2.]. Applying [\cite{connes}, Proposition 6.5.], it follows that $T_{\varphi}(C)^{\prime\prime }$ is injective if and only if almost all of the factors $T_{\psi}(C)^{\prime\prime}$ are injective. We have, therefore, reduced our problem to the following:
Assuming that all hereditary subalgebras $S_{\lambda}$ are nuclear, show that for every cyclic factor representation $T$ of $C$ we have that $ T(C)^{\prime\prime}$ is an injective von Neumann algebra.
Let $T$ be a non degenerate cyclic factor representation of $C$. Since $
\sum_{\lambda}q_{\lambda}=1$ in $C^{\ast\ast}$ there is a $\lambda$ such that the restriction $T|_{q_{\lambda}C}\neq0$. Hence the restriction of $T$ to the closed two sided ideal $J_{\lambda}=\overline{Cq_{\lambda}C}$ is non zero. Since $T$ is a factor representation and $J_{\lambda}$ is a two sided ideal it follows that $T(C)^{\prime\prime}=T(J_{\lambda})^{\prime\prime}$. We show next that under our assumptions $T(J_{\lambda})^{\prime\prime}$ is injective and thus $T(C)^{\prime\prime}$ is injective. We noticed above that $S_{\lambda}$ is strongly Morita equivalent with $J_{\lambda}$. Since $C$ is separable, so are $S_{\lambda\text{ }}$ and $J_{\lambda}$. By [\cite{brown}, Theorem 1.2.] $S_{\lambda\text{ }}$ and $J_{\lambda}$ are stably isomorphic. Since $S_{\lambda\text{ }}$ is nuclear it follows that $J_{\lambda}$ is nuclear. By Lemma \ref{prepLemma} we have that $T(J_{\lambda})^{\prime \prime} $ is injective and the proof is complete. \end{proof}
From the proof of the previous lemma it follows:
\begin{corollary} A separable C*-algebra $C$ is nuclear if and only if for every factor state $\psi$ of $C$, $T_{\psi}(C)^{\prime\prime}$ is an injective von Neumann algebra, where $T_{\psi}$ is the GNS representation of $C$ associated with $\psi$. \end{corollary}
We can now state our main results of this section.
\begin{theorem} \label{nuclearlocallycompact}Let $(A,G,\alpha)$ be a C*-dynamical system with $G$ a locally compact group and let $K\subset G$ be a compact subgroup. Then the following conditions are equivalent:\newline i) $A\times_{\alpha}G$ is a nuclear C*-algebra\newline ii) The hereditary C*-subalgebras $S_{\pi}\subset A\times_{\alpha}G$, $\pi\in\widehat{K}$ are nuclear\newline iii) The C*-subalgebras of $K-$central elements, $\mathcal{I}_{\pi}\subset S_{\pi} ,\pi\in\widehat{K}$ are nuclear\newline Furthermore, any of the previous three equivalent conditions implies\newline iv) $A$ is nuclear\newline In addition, if $G$ is amenable, i.e if the group C*-algebra $C^{\ast}(G)$ is nuclear the conditions i)-iv) are equivalent. \end{theorem}
\begin{proof} The equivalence of the conditions i)-iii) follows from Remarks \ref {Lemma2.5JFA} and \ref{Cor 2.8JFA} and Lemma \ref{nuclearlemma}. On the other hand, if the crossed product, $A\times_{\alpha}G$ is nuclear, then, applying [\cite{raeburn}, Theorem 4.6.], it follows that $ A\times_{\alpha}G\times_{\widehat{\alpha}}\widehat{G}$ is nuclear, where $ \widehat{\alpha}$ is the dual coaction. Since by biduality this latter crossed product is isomorphic with $A\otimes\mathcal{C(H)}$ where $\mathcal{ C(H)}$ is the C*-algebra of compact operators on a certain Hilbert space, $ \mathcal{H}$, it follows that $A$ is a nuclear C*-algebra. Finally, if $G$ is amenable and $A$ is nuclear, then by [\cite{green}, Proposition 14], the crossed product $A\times_{\alpha}G$ is nuclear and therefore in this case iv) $\implies$i). \end{proof}
In the proof of the implication i)$\mathcal{\Longrightarrow }$iv) of the above theorem we have used the fact that every locally compact group is co-amenable and Raeburn's result. The next result is the analog of the previous one for the case of compact quantum groups. A compact quantum group, $\mathcal{G}=(B,\Delta )$, is automatically amenable since $\widehat{B }$ is a subalgebra of compact operators, but not co-amenable, in general, since $B$ is not necessarily nuclear.
We will state next the corresponding result for compact quantum group actions.
\begin{theorem} Let $(A,\mathcal{G},\delta)$ be a quantum C*-dynamical system with $\mathcal{G=(}B,\Delta)$ a compact quantum group. The following three conditions are equivalent:\newline i) $A\times_{\alpha}\mathcal{G}$ is nuclear\newline ii) The hereditary C*-subalgebras $S_{\pi}\subset A\times_{\alpha}\mathcal{G}$ are nuclear\newline iii) The C*-algebras $(A\otimes B(H_{\pi}))^{\delta_{\pi}}$ are nuclear.\newline Furthermore, each of the above condition is implied by \newline iv) $A$ is a nuclear C*-algebra.\newline In addition, if the quantum group $\mathcal{G}$ is co-amenable, i.e. if $B$ is a nuclear C*-algebra, then the conditions i)-iv) are equivalent with the following:\newline v) $A^{\delta}$ is nuclear. \end{theorem}
\begin{proof} The equivalence of i)-iii) follows from Lemma \ref {ralucaanalogsof2.2,2.8andlandstadlemma} and Lemma \ref{nuclearlemma}. We now prove that iv) implies iii). Let $\pi\in\widehat{\mathcal{G}}$. If $A$ is nuclear, then $A\otimes B(H_{\pi})$ is a nuclear C*-algebra. The projection of $A\otimes B(H_{\pi})$ onto the fixed point algebra $(A\otimes B(H_{\pi }))^{\delta_{\pi}}$ is obviously a completely positive map. Therefore, by [\cite{choi}, Corollary 3.4. (4)] it follows that $(A\otimes B(H_{\pi}))^{\delta_{\pi}}$ is nuclear. Assume now that $\mathcal{G}$ is co-amenable. Therefore, $B$ is nuclear. Then, by applying [\cite{doplicher}, Corollary 7] it follows that $A^{\delta}$ is nuclear if and only if $A$ is nuclear and thus v)$\iff$iv). Since $S_{\pi_{0}}$ is isomorphic with $ A^{\delta}$, we have that iii)$\implies$iv) and the proof is completed. \end{proof}
\end{document} |
\begin{document}
\title{\bf How to avoid collisions in 3D-realizations for moving graphs}
\author{Jiayue Qi \thanks{Johannes Kepler University Linz, Doctoral Program "Computational Mathematics" (W1214).} \thanks{Johannes Kepler University Linz, Research Institute for Symbolic Computation.}}
\date{}
\maketitle
{\centering\footnotesize\bf\em To my dearest Grandma, Huiqin Dong.\par}
\begin{abstract}
If we parameterize the positions of all vertices of a given graph in the plane such that distances between adjacent vertices are fixed, we obtain a moving graph. An L-linkage is a realization of a moving graph in 3D-space, by representing edges using horizontal bars and vertices by vertical sticks. Vertical sticks are parallel revolute joints, while horizontal bars are links connecting them. We give a sufficient condition for a moving graph to have a collision-free L-linkage. Furthermore, we provide an algorithm guiding the construction of such a linkage when the moving graph fulfills the sufficient condition, via computing a height function for the edges (horizontal bars). In particular, we prove that any Dixon-1 moving graph has a collision-free L-linkage and no Dixon-2 moving graphs have collision-free L-linkages, where Dixon-1 and Dixon-2 moving graphs are two classic families of moving graphs.
\end{abstract}
\section{Introduction}\label{sec:introduction} In this section, we introduce the problem background and some related existing work. Given a graph, if we parameterize the position of each vertex in the plane so that any distance between adjacent vertices is constant, then this graph is called a moving graph. Notice that some moving graphs are induced by rigid motions, hence they do not necessarily ``move'' in the general sense. Examples of moving graphs can be found in \cite{moving_graph, Dixon}. We also provide more examples of moving graphs in the later sections.
We realize a moving graph in 3D-space with some bars and sticks. Horizontal bars realize edges while vertical sticks realize vertices. We assign a different height to each horizontal bar. Horizontal bars are parallel and located at pairwise distinct heights and hence would not touch each other. Vertical sticks connect them, by going through the holes at the ends of horizontal bars. We call this kind of linkages L-linkages, or realizations of L-models. The concept of L-model is just the mathematical description of such linkages, which we will define precisely later on, in mathematical language. Figure \ref{fig:photo} shows an L-linkage.
\begin{figure}
\caption{This is the realization of an L-model (i.e. an L-linkage) of a moving graph with the underlying graph structure being $K_{3,3}$, where $K_{3,3}$ denotes the complete bipartite graph with two independent sets and each independent set contains three vertices. The edges correspond to horizontal bars with holes, while the vertices correspond to vertical sticks. For detailed explanation of complete bipartite graphs see Section \ref{sec:collision_detection}.}
\label{fig:photo}
\end{figure}
One observes a collision of an L-linkage at the time when a vertical stick (corresponds to
vertices of the corresponding moving graph of the linkage) hits a horizontal bar
(corresponds to
edges of the corresponding moving graph of the linkage).
The collisions depend on the choice of heights of the horizontal bars. Note that edge crossing is possible in some moving graphs. However, we set up horizontal bars all in different heights, which naturally avoids edge crossing in the corresponding L-linkage. We give a sufficient condition for such a linkage to be collision-free with respect to some height arrangement of edges. And we provide an algorithm for constructing an L-linkage in a collision-free way when the criterion is fulfilled. This condition can be helpful for the construction of the linkage --- checking whether it can accomplish a full motion. Dixon-1 moving graphs and Dixon-2 moving graphs are two classic families of moving graphs. With this result we prove that there exists a collision-free L-linkage for every Dixon-1 moving graph. In addition, we prove that Dixon-2 moving graphs cannot be realized as a collision-free L-linkage.
Abel et al. showed that any polynomial curve can be traced by a non-crossing linkage
\cite{who_needs_crossings}. One of the differences between our result and theirs is that we consider a motion (i.e. the traces of all vertices), not just tracing a curve of some single vertex. Another relevant result is by Gallet et al. \cite{prescribed_motion}. They provide an algorithm which produces a linkage tracing any given planar rational curve without collisions. Our work differs from them in the sense that our algorithm deals with arbitrary moving graphs, not just those related to their linkages; also, they do not take into consideration the moving graphs Dixon-1 and Dixon-2. The result in \cite{deployable} provides a method to better avoid collision for curved-bar-type linkages, while we focus on collision avoidance for straight-bar linkages. Collision-free path planning of planar linkages is also discussed in \cite{path_planning}. Our result can also be helpful in the collision-free path planning. Given a path, a corresponding linkage which draws that path can be designed with some method \cite{prescribed_motion}. Then, our algorithm can check whether the linkage can be realized without collision, which contains the ability of checking whether the planned path can be realized in a collision-free way. The result in \cite{6R} also considers collision-avoidance. They introduce a specific design of a 6R-linkage in such a way that it does not collide with itself or other equipment attached to it. Compare with their result, we consider the collision-free design for another type of linkages, namely L-linkages.
\section{Problem statement} In this section, we describe the problem on which this paper focuses. For a clear problem statement we need to introduce some definitions first. These definitions help us build up the mathematical model precisely, behind the intuitional idea. If we view the linkage in Figure \ref{fig:photo} from above, we get some idea of a ``moving graph''. \begin{definition} Let $G=(V,E)$ be a connected graph. Let $F=\{f_v\}_{v\in V}$ be a set of continuous functions
$f_v:\mathbb{R}\rightarrow \mathbb{R}^2$ such that the function $\|f_u(\cdot)-f_v(\cdot)\|$ is a positive constant for every edge $\{u,v\}$ in $E$. The pair $(G,F)$ is called a {\bf moving graph}. \end{definition}
We can fix the position of some vertex $v$ and put one of its adjacent vertices $v_1$ on the $x$-axis.
Let $\lambda_{i,j}$ be the fixed edge length for edge $\{i,j\}$. We can consider the equation system which contains equation $\|f_u(t)-f_v(t)\|=\lambda_{u,v}$ for each $\{u,v\}\in E$.
If the corresponding solution set (in $\mathbb{R}$) has infinite cardinality, we say that the given moving graph is {\em mobile}.
We also say that the parameterizations give a {\em mobile realization} of the underlying graph.
The solution set is also called a configuration space. We can talk about the
dimension of this space. The space has a positive dimension if and only if the given moving graph is mobile.
When the solution set has finite cardinality, the given moving graph is {\em rigid}.
For our collision problem, it only makes sense to consider the mobile moving graphs.
A moving graph $M=(G,F)$ is {\itshape finite} if $G$ is a finite graph. The background problem that we are interested in is how to avoid collisions for L-linkages, hence we focus only on finite moving graphs in this paper.
We define a collision in a moving graph as when a vertex ``hits'' an edge. \begin{definition}\label{def:collision_pairs}
Let $M=(G,F)$ be a moving graph, where $G=(V,E)$. Vertex {\bf $w\in V$ collides
with edge $\{u,v\}\in E$} if and only if $w\notin \{u,v\}$ and
there exists $t\in \mathbb{R}$ such that $f_w(t)$ lies on the line segment defined by points $f_u(t)$ and $f_v(t)$.
Then $(w,\{u,v\})$ is called a {\bf collision pair} of $M$.
We denote by $CP_M$ the set of all collision pairs
of~$M$. \end{definition}
Note that by definition, a vertex never collides with any edge containing this vertex. If we assign pairwise distinct (integer) height values for the edges, we obtain an L-model. This is the mathematical language for the Lego linkages (L-linkages) described in Section~\ref{sec:introduction}.
\begin{definition} Let $M=(G,F)$ be a moving graph, where $G=(V,E)$. An {\bf L-model} of $M$ is a pair $(M,h)$, where $h:E\rightarrow \mathbb{Z}$ is an injective function assigning to each edge of $G$ an integer height value. We call $h$ the {\bf height function} of $M$. \end{definition} An {\em L-linkage} is a 3D-realization of an L-model, as described in Section \ref{sec:introduction}. Although our original ambition is to avoid collisions for L-linkages, L-model is a concept in mathematical set-up that can help us better, in developing the whole theory. Hence, we focus more on this concept throughout the paper.
In some moving graphs, two (or more) edges can stay intersected for any $t\in \mathbb{R}$. However, we can prevent such situation being transferred to a collision problem in the corresponding L-linkage, simply by placing all edges in pairwise distinct heights. Because we define the height function to be injective, so this would not be a problem in our setting. Therefore, we do not specifically consider this situation in the remaining context.
When we are given an L-model, we can realize the edges by horizontal bars by fixing the height of one edge and then placing the others accordingly. And we connect the end of those edges that are incident at the same vertex by a vertical stick going through the holes on the horizontal bars. In this way, we obtain the corresponding L-linkage of the given L-model. Note that during this process, only the relative height values matter. We can obtain a same L-linkage from two different L-models, as long as the relative height between any pair of edges coincides in these two L-models. On the contrary, we can also find out the height function of one corresponding L-model of a given L-linkage, by pre-fixing the height of an edge in the linkage.
Collision pairs are the ``collisions'' in a moving graph; also, they are the only reason for the corresponding L-linkage to have collisions. When we realize the moving graph $M$ in an L-linkage, we observe that if the horizontal bar for edge $e$ is then outside of the range of the vertical stick for vertex $v$ --- where $(v,e)$ is a collision pair for $M$ --- then the collision which might have been caused by this collision pair is perfectly avoided. However, those collision pairs of $M$ that do not fulfill this condition, still lead to collisions. Inspired by this fact, we define the collision-freeness of an L-model as follows. \begin{definition}\label{def:collision_free_L_model}
Let $L=(M,h)$ be an L-model, where $M=(G,F)$ and $G=(V,E)$. It is {\bf collision-free} if and only if
$$h(e')\notin [\min_{v'\in e\in E}{h(e)},\max_{v'\in e\in E}{h(e)}]$$
holds for any collision pair $(v',e')$ in $M$. If a collision pair fulfills this condition,
we say that it is {\bf safe} (under function $h$). Hence, moving graph $M$ has a collision-free L-model if and
only if it has a height function $h$ such that all collision pairs of $M$ are safe under $h$. \end{definition}
In this paper, we focus on the following problem: {\bf Given a finite moving graph, how to find a collision-free L-model for it.} In the engineering context, this can be expressed as: {\bf Given a finite moving graph, how to construct a collision-free L-linkage for it.}
\section{Collecting collision pairs}\label{sec:collision_detection}
A graph is {\em complete} if there is an edge between any pair of vertices. An {\em independent set} of graph $G=(V,E)$ is a subset $V'$ of the vertex set $V$ such that there is no edge between vertices in $V'$. We say that graph $G=(V,E)$ is {\em bipartite} if and only if there exists a bipartition of the vertex set $V=V_1\cup V_2$ such that both $V_1$ and $V_2$ are independent sets. And we call $V_1$ and $V_2$ {\em the two corresponding independent sets} of the bipartite graph~$G$. We say that bipartite graph $G$ with two independent sets $V_1$ and $V_2$ is {\em complete} if there is an edge connecting any two vertices $v,u$ for $v\in V_1$, $u\in V_2$. We denote by $K_{m,n}$ the complete bipartite graph with $m$, $n$ the cardinalities of its two independent sets respectively. See Figure \ref{fig:bipartite} for an example of complete bipartite graph.
\begin{figure}
\caption{The left sub-figure is a complete bipartite graph with seven vertices, while two independent sets contain three and four vertices, separately. The right sub-figure is the same graph, while vertices of the two independent sets are transformed such that they lie on two orthogonal lines. This graph is denoted by $K_{3,4}$ (or $K_{4,3}$).}
\label{fig:bipartite}
\end{figure}
Let $M=(G,F)$ be a moving graph with $G=(V,E)$. We detect collisions of this moving graph just by collecting its collision pairs. We check whether $(v,e)$ is a collision pair for all $v\in V$ and all $e\in E$. Collection of all collision pairs reflects the collision information of $M$. We implemented this program in Mathematica \cite{Mathematica}. Now let us see an example for a clearer idea.
\begin{example}\label{eg:collision_pairs}
Let $G=(V,E)$ be the complete bipartite graph $K_{4,3}$ with vertex set $V=\{1,2,3,4,5,6,7\}$
and two independent sets $V_1=\{1,2,3,4\}$, $V_2=\{5,6,7\}$.
The moving graph $M=(G,F)$ belongs to the family of Dixon-1 moving graphs \cite{Dixon},
where $F$ consists of the following functions:
\begin{flalign*}
&f_1(t) = (\sin{t},0), &&\\\nonumber
&f_{2}(t) = (\sqrt{1+\sin^2{t}},0), &&\\\nonumber
&f_{3}(t) = (-\sqrt{2+\sin^2{t}},0), &&\\\nonumber
&f_{4}(t) = (\sqrt{3+\sin^2{t}},0), &&\\\nonumber
&f_{5}(t) = (0,\cos{t}), &&\\\nonumber
&f_{6}(t) = (0, \sqrt{1+\cos^2{t}}),&&\\\nonumber
&f_{7}(t) = (0, -\sqrt{2+\cos^2{t}}). &&
\end{flalign*}
We proceed according to the definition of collision pairs. For each vertex $u$ and edge $\{v,w\}$, we check
if $f_u(t)$ lying on the line segment defined by
$f_v(t)$ and $f_w(t)$ has a solution in $\mathbb{R}$: if it does, then $(u,\{v,w\})$ is a collision pair of $M$.
By going through all pairs of vertex $v\in V$ and edge $e\in E$ (such that $v\notin e$), we get the collision pairs in $M$ as follows: $$(1,\{5,2\}), (1,\{5,3\}), (1,\{5,4\}), (2,\{5,4\}), (5,\{6,1\}), (5,\{7,1\}).$$
\end{example}
With this we conclude the process of collecting collision pairs, which is the first step on our way of trying to find a collision-free L-model for the given moving graph.
\section{The partition condition}\label{sec:sufficient_condition} In this section, we explain a sufficient condition for a moving graph to have a collision-free L-model, which we call {\em the partition condition}. In order to introduce this condition, some preparations are needed. Let us gradually approach it.
After collecting the collision pairs of a given moving graph, we want to express this information in a nicer way --- by a ``collision graph''. \begin{definition} Let $M=(G,F)$ be a moving graph, where $G=(V,E)$. The {\bf collision graph} $C=(V_C,E_C)$ of $M$ is a directed graph such that $V_C=E$ and $\overrightarrow{e_i,e_j}\in E_C$ if and only if at least one of the vertices of $e_i$, denoted by $v$, collides with edge $e_j$ in $M$, i.e., $(v, e_j)$ is a collision pair of $M$, where $v\in e_i$ is one of the two incident vertices of $e_i$. \end{definition}
An {\em induced subgraph} $H=(V_1,E_1)$ of a given graph $G=(V,E)$ is another graph such that $V_1\subset V$ and $E_1$ equals to the restriction of $E$ on $V_1$. That is to say, edge set $E_1$ is formed of all edges of $G$ between any pair of vertices in $V_1$. Then $H$ is denoted by $G[V_1]$. An induced collision graph
is an induced subgraph of some given collision graph.
\begin{definition}[induced collision graphs]
Let $M=(G,F)$ be a moving graph, where $G=(V,E)$. Let $S\subset E$, then the {\bf collision graph of $M$ induced by $S$}, denoted by $C[S]$, is
the subgraph of $C$ induced by $S$, where $C$ is the
collision graph of $M$. \end{definition}
We continue with Example \ref{eg:collision_pairs}. From the collision information collected in Example~\ref{eg:collision_pairs}, we construct the
collision graph $C$, as shown in Figure \ref{fig:collision_graph}.
\begin{figure}
\caption{This is the collision graph $C$ of the moving graph $M$ given in Example \ref{eg:collision_pairs}. A directed edge $\{i,j\} \rightarrow \{k,l\}$ means either $(i,\{k,l\})$ or $(j,\{k,l\})$ is a collision pair of~$M$. Note that in the figure we write $ij$ for edge $\{i,j\}$, in order to have a more neat picture.}
\label{fig:collision_graph}
\end{figure}
\begin{definition} Let $C=(V_C,E_C)$ be a directed graph. If there exists a bipartition of $V_C$ into $V_1$ and $V_2$ such that $C[V_1]$ and $C[V_2]$ are both acyclic, then we say that graph $C$ fulfills {\bf the partition condition}. \end{definition}
We say that a moving graph {\itshape fulfills the partition condition} if and only if its collision graph fulfills the partition condition. Now we are prepared for the main theorem. \begin{theorem}\label{thm:partition_condition} Let $M=(G,F)$ be a finite moving graph, where $G=(V,E)$. If $M$ fulfills the partition condition, then it has a collision-free L-model. \end{theorem}
In order to prove Theorem \ref{thm:partition_condition}, we introduce an algorithm (see Algorithm \ref{alg:height_construction}) constructing the height function for the vertices of a given acyclic directed graph. We need some more preparations before the proof. First we introduce an order for the vertices of a directed graph.
\begin{definition} Let $C$ be an acyclic directed graph. The {\bf height order} on the vertex set of $C$ --- denoted by ``$<$'' --- is defined as: $i<j$ if and only if there is a (directed) path from $i$ to $j$. \end{definition}
\begin{remark} A strict partial order is a binary relation that is irreflexive, transitive and asymmetric.
The height order defined above is a strict partial order. \end{remark}
\begin{proposition}\label{prop:height_order}
For a finite acyclic directed graph $C=(V,E)$, there exist(s) at least one minimal element
in $V$ with respect to the height order. \end{proposition}
\begin{proof}
If there was no minimal element in $V$ under the height order, then there would be an infinite chain $v_1>v_2>...$ in $V$.
Since there is no cycle in graph $C$,
the elements
in this chain are pairwise distinct. This contradicts the finiteness of the cardinality of~$V$. \end{proof}
\begin{algorithm}[H]
\caption{Constructing the height function for elements in $V_C$.}\label{alg:height_construction} \thispagestyle{empty} \SetKwInOut{Input}{input} \SetKwInOut{Output}{output}
\Input{a finite acyclic directed graph $C=(V_C,E_C)$; the height parameter
$(k_0,i)$ } \Output{the height function $h:V_C\to Z$} $k\gets k_0$ \;
\While{$C$ is not a null graph }
{$S:=$ collection of all minimal vertices in the vertex set of graph $C$
under height order\;
$C:=C[V_C\setminus S]$, where $C[V_C\setminus S]$ is the subgraph of $C$
induced by $V_C\setminus S$\;
\While{$S\neq \emptyset$}
{Pick one element $r$ in $S$, set $h(r):= k$\;
$S:=S\setminus \{r\}$\;
$k:=k+i$\;}} \Return $h$.
\end{algorithm} \begin{remark}
Null graph is a graph with no vertices or edges.
\end{remark}
\begin{remark}\label{rem:output_not_unique}
Note that the output of Algorithm \ref{alg:height_construction} is not unique, due to the fact that there can be
more than one minimal vertices under the height order in the considered directed graph, in each outer loop.
This then leads to non-deterministic height assignments in the corresponding inner loops. \end{remark}
In Algorithm \ref{alg:height_construction}, we add the height parameter as part of input data. The usage of this will be seen later on --- we want to apply this algorithm to two directed graphs, but with two different height parameters, in order to construct a collision-free height function for the given moving graph. Note that Algorithm \ref{alg:height_construction} completes the height order which is a partial order, into a total order. Now we can prove the termination of Algorithm \ref{alg:height_construction}. \begin{proof}[Termination of Algorithm \ref{alg:height_construction}]
If $C$ is not a null graph, by Proposition \ref{prop:height_order} there must exist at least
one minimal vertex under the height order. Hence $S$ is
non-empty as long as $C$ is not null. Then the step ``$C:=C[V_C\setminus S]$'' strictly reduces the number of vertices of graph $C$, when $C$ is not a
null graph.
Since the given graph is finite, this algorithm
terminates. \end{proof}
The following proposition spells out the essence for the proof of Theorem \ref{thm:partition_condition}. \begin{proposition}\label{prop:model_construction}
Let $M=(G,F)$ be a moving graph that fulfills the partition condition, where $G=(V,E)$.
Let $C=(V_C,E_C)$ be the collision graph of $M$ and let $C_L=(V_L,E_L)$ and $C_U=(V_U,E_U)$ be the two corresponding acyclic induced collision
graphs (of $C$). After applying Algorithm \ref{alg:height_construction} with height parameter
$(1,1)$ to $C_U$, and with height parameter
$(0,-1)$ to $C_L$, we get a collision-free L-model of $M$. \end{proposition} \begin{remark}
The above mentioned process assigns to edges (of $M$) in $V_U$ positive-integer height values, with the principle that $h(e_1)<h(e_2)$ if $\overrightarrow{e_1,e_2}\in E_U$. Meanwhile, it assigns to edges (of $M$) in $V_L$ non-positive-integer height values, with the principle that $h(e_1)>h(e_2)$ if $\overrightarrow{e_1,e_2}\in E_L$. \end{remark}
\begin{proof}[Proof of Proposition \ref{prop:model_construction}.] Recall that vertices of the collision graph $C$ are exactly the edges of $M$. In the sequel, whenever we talk about edges or vertices, we mean those with respect to the moving graph $M=(G,F)$, or the graph $G$ --- so as to make things less confusing.
Let $(v,e)$ be any collision pair of $M$. Then, $\overrightarrow{e_1,e}\in E_C$ if and only if $v\in e_1$, where $e_1$ is any edge of $M$. Without loss of generality, assume that $e\in V_U$ --- the case for $e\in V_L$ can be argued analogously. Considering the distribution of vertex~$v$ as one of the two incident vertices of any edge $e_1$ in the graphs $C_U$ and $C_L$, given that $(v,e)$ is a collision pair. There are in total three cases: \begin{itemize}
\item First, $v$ shows up only in some edge(s) in graph $C_U$. In this case, Algorithm \ref{alg:height_construction} guarantees that the height
value of $e_1$ is strictly less than that of edge $e$, for all
$\overrightarrow{e_1,e}\in E_U$. Hence, height value of edge $e$ is outside of the range of the height of edges (of $M$)
containing $v$. That is to say, $(v,e)$ is safe.
\item Second, $v$ shows up only in some edge(s) in graph $C_L$. In this case, Algorithm~\ref{alg:height_construction} guarantees
that the height values of all edges containing vertex $v$ are non-positive. Meanwhile, the height value of edge $e$ is positive.
Hence, we obtain that $h(e)\notin [\min_{v\in e'\in E}{h(e')}, \max_{v\in e'\in E}{h(e')}]$, $(v,e)$ is safe.
\item Third, $v$ shows up in some edge(s) in graph $C_U$ and in some edge(s) in graph $C_L$ as well. From the above analysis, we know that
edges containing $v$ that are in $V_L$ have non-positive height values and edges containing $v$ that are in $V_U$ has strictly less
height values than edge $e$. Therefore, we have $h(e)\notin [\min_{v\in e'\in E}{h(e')}, \max_{v\in e'\in E}{h(e')}]$, $(v,e)$ is safe. \end{itemize} Hence we obtain that all collision pairs in $M$ are safe. By Definition \ref{def:collision_free_L_model}, indeed we obtained a collision-free L-model of $M$ after the stated process. \end{proof}
\begin{proof}[Proof of Theorem \ref{thm:partition_condition}.] Straightforward, from Proposition \ref{prop:model_construction}. \end{proof} \begin{remark}
From Proposition \ref{prop:model_construction}, we see that the height parameter in Algorithm \ref{alg:height_construction} is useful,
since we need to construct the edges of $M$ in $C_U$ from height $1$ upwards, while those in $C_L$
are constructed from height $0$ downwards. \end{remark}
For a clearer understanding towards Algorithm \ref{alg:height_construction}, we apply it to the moving graph $M$ given in Example \ref{eg:collision_pairs}, so as to get a collision-free L-model.
\begin{example}\label{eg:apply_height_algorithm} We partition the edges of $M$ into two parts: $$E_U=\{\{5,1\},\{5,2\},\{5,3\},\{5,4\}\},$$ $$E_L=\{\{6,1\},\{6,2\},\{6,3\},\{6,4\},\{7,1\},\{7,2\} ,\{7,3\},\{7,4\}\},$$ then from these two parts we construct the induced collision graphs $C_U$ and $C_L$, individually --- see Figure \ref{fig:induced_collision_graphs}.
\begin{figure}\label{fig:induced_collision_graphs}
\end{figure}
After applying Algorithm \ref{alg:height_construction}, according to Proposition \ref{prop:model_construction}, we get a height arrangement for some L-model as:
$h(\{5,4\})=4$, $h(\{5,3\})=3$, $h(\{5,2\})=2$, $h(\{5,1\})=1$, $h(\{6,1\})=0$, $h(\{6,2\})=-1$, $h(\{6,3\})=-2$, $h(\{6,4\})=-3$,
$h(\{7,1\})=-4$, $h(\{7,2\})=-5$, $h(\{7,3\})=-6$, $h(\{7,4\})=-7$.
One can check that it indeed provides a collision-free L-model of $M$. \end{example}
Note that the converse statement of Theorem \ref{thm:partition_condition} does not hold. That is to say, there exists a moving graph $M=(G,F)$, where $G=(V,E)$,
which has a collision-free L-model but
we cannot partition $E$ into two parts $E_1$ and $E_2$ such that the induced
collision graphs of $M$ (respectively by $E_1$ and $E_2$) are both acyclic. We show it with the following example.
\begin{example}\label{eg:converse_is_not_true}
A moving graph fulfilling the above described conditions could be the one denoted by $S_2$ moving graph in
\cite[Corollary 4.8]{moving_graph}.
The formal definition of this moving graph (denoted by $M$ in our paper) is as follows.
Let $M=(G,F)$, where $G=(V,E)$, $V=\{1,2,3,4,5,6,7,8\}$ and
$$E=\{\{1,2\},\{1,4\},\{1,5\},\{3,2\},\{3,4\},\{3,5\},$$
$$\{8,2\},\{8,4\},\{8,5\},\{1,7\},\{6,7\},\{6,5\},\{6,4\}\}.$$
And $F$ consists of the following functions, where $a=1$, $b=\frac{11}{5}$, $c=\frac{3}{2}$: \begin{flalign*}
&s_1=\sqrt{b^2-a^2\cdot \sin^2{t}}\,, &&\\\nonumber
&s_2=\sqrt{c^2-a^2\cdot \cos^2{t}}\,, &&\\\nonumber
&f_1(t)=(-a\cdot \cos{t}-s_1, -a\cdot \sin{t}-s_2), &&\\\nonumber
&f_2(t)=(a\cdot \cos{t}-s_1, -a\cdot \sin{t}+s_2), &&\\\nonumber
&f_3(t)=(a\cdot\cos{t}+s_1, a\cdot\sin{t}+s_2), &&\\\nonumber
&f_4(t)=(-a\cdot\cos{t}+s_1, -a\cdot\sin{t}+s_2), &&\\\nonumber
&f_5(t)=(-a\cdot\cos{t}+s_1, a\cdot\sin{t}-s_2), &&\\\nonumber
&f_6(t)=(-3\cdot a\cdot\cos{t}+s_1, -a\cdot\sin{t}-s_2), &&\\\nonumber
&f_7(t)=(-3\cdot a\cdot\cos{t}-s_1, -a\cdot\sin{t}-3\cdot s_2), &&\\\nonumber
&f_8(t)=(-a\cdot\cos{t}-s_1, a\cdot\sin{t}+s_2). &&
\end{flalign*}
After the collision-detection process, we get the following collision pairs:
$$(2, \{8,4\}),\; (2, \{8,5\}),\;(3, \{1,4\}),\;(3, \{6,4\}),\; (3, \{8,4\}),\;(4, \{3,2\}),\; (4, \{3,5\}),\;$$
$$(6, \{1,5\}),\;(6, \{3,5\}),\; (6, \{8,5\}),\;(5, \{6,4\}),\; (5, \{6,7\}),\; (8, \{1,2\}),\; (8, \{3,2\}).$$
One can check that the following height value assignments provide a
collision-free L-model for $M$:
$$h(\{6,7\})=0,\; h(\{1,4\})=1,\; h(\{6,4\})=2,\; h(\{8,4\})=3,\; h(\{3,4\})=4,$$
$$h(\{6,5\})=5,\; h(\{8,5\})=6,\; h(\{1,5\})=7,\; h(\{1,7\})=8,\; h(\{3,5\})=9,$$
$$h(\{8,2\})=10,\; h(\{3,2\})=11,\; h(\{1,2\})=12.$$
However, from the collision information of $M$, we know that $\overrightarrow{e_1,e_2}$,
$\overrightarrow{e_2,e_1}$, $\overrightarrow{e_1,e_3}$, $\overrightarrow{e_3,e_1}$,
$\overrightarrow{e_2,e_3}$ and $\overrightarrow{e_3,e_2}$ are all in $E_C$, where $e_1=\{3,2\}, e_2=\{6,4\}, e_3=\{8,5\}$
and $E_C$ denotes the edge set of the collision graph $C=(V_C,E_C)$ of $M$. Hence, no matter how we try to
bipartition $V_C$ (i.e., $E$), we will get two among the edges $e_1,e_2,e_3$ in one
group, say $e_1$ and $e_2$. Then the edges $\overrightarrow{e_1,e_2}$ and
$\overrightarrow{e_2,e_1}$ already form a cycle in the corresponding induced collision graph.
Therefore, we cannot partition $V_C$ into two parts such
that the induced collision graphs respectively of the two parts
are both acyclic.
\end{example}
To sum up this section, we propose the following steps, on how to decide whether a moving graph has a collision-free L-model. \begin{enumerate}
\item Collect the collision information of the given moving graph $M$, namely the collision pairs in $M$.
\item Construct the collision graph of this moving graph, denote it by $C$.
\item Decide whether $C$ fulfills the partition condition, or equivalently,
whether $M$ fulfills the partition condition. If yes, with our
algorithm we can get a collision-free L-model for $M$.
\end{enumerate} \begin{remark}
If the answer to Step 3. is ``no'', we can still apply a
brute-force algorithm which in worst case needs the factorial of $|E|$ many steps.
That is to say, try all possible (relative) height arrangements
and check for each of them if it leads to a collision-free
L-model.
\end{remark}
But how to decide whether $C$ fulfills the partition condition in Step 3.?
To make the theory complete, now we introduce a method
which may improve the efficiency in some cases, compare to the brute-force method.
We need the definition of ``cycled subgraph'' in order to explain this
method. Note that the word ``cycled'' here just refers to the situation when there exist two edges between two vertices $v_1$ and $v_2$, namely $\overrightarrow{v_1,v_2}$ and $\overrightarrow{v_2,v_1}$, in the directed graph. In this case, these two edges form a cycle in graph $C$. Therefore, these two vertices must stay in different induced collision graphs --- if $C$ fulfills the partition condition. We use this graph to depict such information.
\begin{definition}
Let $C=(V,E)$ be a directed graph. Collect all vertices $v,u\in V$
such that $\overrightarrow{v,u}$ and $\overrightarrow{u,v}$
are both edges of $C$ in set $S$. Then consider the graph $C[S]$. First, delete all the single edges.
Then, view all edges between any two vertices as one single non-directed edge.
We call the so-obtained graph the {\bf cycled subgraph} of graph $C$. \end{definition}
\begin{example}
Let $C$ denote the graph depicted in Figure \ref{fig:collision_graph}.
The cycled subgraph $C_1$ of $C$ is shown in
Figure \ref{fig:cycled_subgraph}.
\begin{figure}
\caption{This is the cycled subgraph $C_1$ of graph $C$ which is shown in Figure \ref{fig:collision_graph}.}
\label{fig:cycled_subgraph}
\end{figure}
\end{example}
It is not hard to see that the following proposition holds since any edge in the cycled subgraph indicates a cycle in the given directed graph.
\begin{proposition}\label{prop:cycled_subgraph_bipartite}
If the cycled subgraph $S=(V_S,E_S)$ of a directed graph $C=(V,E)$ is
not bipartite, then the graph $C$ does not
satisfy the partition condition. Any
bipartition of $V=V_L\cup V_U$ such that both $C[V_L]$
and $C[V_U]$ are acyclic induces a bipartition of $V_S$ which
gives the two independent sets of graph $S$.
\end{proposition}
\begin{proof}
Suppose that graph $C$ satisfies the partition condition and the two corresponding
induced subgraphs are $C_L=(V_L,E_L)$ and $C_U=(V_U,E_U)$. If there exist $v_1,v_2\in V_S\cap V_L$
such that $\{v_1,v_2\}\in E_S$,
by the definition of cycled subgraph we know that the two edges $\overrightarrow{v_1,v_2}$
and $\overrightarrow{v_2,v_1}$ are both in $E_L$. This makes $C_L$ cyclic, which is a contradiction.
Hence, $\{v_1,v_2\}\notin E_S$. Analogously, we have that if $v_1,v_2\in V_S\cap V_U$,
then $\{v_1,v_2\}\notin E_S$. Hence, $S$ is bipartite, the two independent sets of $S$ are exactly $V_S\cap V_L$
and $V_S\cap V_U$ --- note that $V_S\subset V_L\cup V_U=V$.
\end{proof}
\begin{remark}
We see that a bipartition of the edges of the graph in Figure \ref{fig:collision_graph} such that the two obtained induced collision graphs
are both acyclic, say the one stated
in Example~\ref{eg:apply_height_algorithm}, includes a bipartition
on the edge set $$\{\{5,2\},\{5,3\},\{5,4\},\{6,1\},\{7,1\}\}$$ of graph $C_1$ drawn in Figure~\ref{fig:cycled_subgraph}
as follows:
$$ \{\{5,2\},\{5,3\},\{5,4\}\}\cup \{\{6,1\},\{7,1\}\},$$ which also indicates
a bipartite structure of graph $C_1$ --- there are no edges in between vertices
$\{\{5,2\},\{5,3\},\{5,4\}\}$ or $\{\{6,1\},\{7,1\}\}$.
\end{remark}
\begin{remark}
Consider the $S_2$ moving graph. From Example~\ref{eg:converse_is_not_true}, we
know that the cycled subgraph of $S_2$ moving graph contains a triangle. Therefore, it cannot be bipartite,
hence the $S_2$ moving graph does not fulfill the partition condition.
\end{remark}
We hope that this, as a pre-step of the algorithm for deciding whether a given directed graph
fulfills the partition condition, can save labor for us in some situation. Now we give an algorithm
(see Algorithm \ref{alg:partition_condition}) for deciding whether a given
directed graph fulfills the partition condition.
\begin{algorithm}
\caption{Deciding whether a directed graph $C$ fulfills the partition condition}\label{alg:partition_condition} \thispagestyle{empty} \SetKwInOut{Input}{input} \SetKwInOut{Output}{output}
\Input{a finite directed graph $C=(V_C,E_C)$ } \Output{If $C$ fulfills the partition condition, output the corresponding bipartition of $V_C$; otherwise, output ``No''.} $S=(V_S,E_S)\gets$ the cycled subgraph of $C$ \;
\If{$S$ is not bipartite}
{\Return ``No''}
\Else
{Find all bipartitions of $V_S$ such that both parts are independent sets of $S$\;
Denote them by the sequence
$B_1,...,B_m$, where $B_i:=\{p_i,q_i\}$ for $1\leq i\leq m$\;
List all bipartitions of $V_C\setminus V_S$ as ordered pairs (hence each bipartition is considered twice)
and denote the pairs by the sequence $B'_1,...,B'_k$, where $B'_i:=(p'_i,q'_i)$
for $1\leq i\leq k$\;
\For{$i$ from $1$ to $m$}
{\For{$j$ from $1$ to $k$}
{Check whether $(p_i\cup p'_j,q_i\cup q'_j)$
is a partition of $V_C$ such that the induced
subgraph of both parts are acyclic\;
\If{yes}
{\Return $(p_i\cup p'_j,q_i\cup q'_j)$}}}
\Return ``No'' }
\end{algorithm}
\begin{remark}
In Algorithm \ref{alg:partition_condition}, we consider each bipartition of $V_C\setminus V_S$ twice, i.e., consider them as
ordered pairs, while the bipartitions of $V_S$ are viewed as cardinality-two sets, considered only once. This is to make sure that we
indeed go through all combinations of the bipartitions of $V_S$ and those of $V_C\setminus V_S$, once. \end{remark}
\begin{proof}[Termination and correctness of Algorithm \ref{alg:partition_condition}]
Termination is obvious since the input is a finite graph. Correctness
follows from Proposition \ref{prop:cycled_subgraph_bipartite}. \end{proof}
When the collision information for some moving graph contains two vertices colliding with each other,
then we already know that the range of these two vertices should not intersect in the L-model (so as to avoid collisions).
This information can also save us
some work when we try to find a collision-free height arrangement, but we will not go too much into details on it.
Note that our approach assumes that a moving graph, with the parameterization of the positions of the vertices in
a variable $t$ is given. Computing such a parameterization for a given graph with fixed edge lengths
is a preliminary step, which in practice
should be addressed before seeking a collision-free L-model for the moving graph.
In this preliminary step, the corresponding configuration space is
an algebraic curve in many cases. In general, parameterization of algebraic curves is a topic on its own
and it is not the focus of this paper. There are many references available on this topic.
Some details on algorithmic parameterization of rational curves can be checked in
\cite[Chapter~4]{rational_curves}, for instance. Examples of some computations on the traces of vertices for visualizing the
moving graphs (via animations) are available in \cite{computation}.
On the parameterization step for the two families of moving graphs that are discussed in this paper, we provide
the following ideas:
Formulas for Dixon-1 moving graphs can be obtained and
verified by Pythagorean Theorem. Formulas for Dixon-2 moving graphs can be deduced by its
defining equation system and its symmetry property.
With this we complete the description of our main results. In the coming
sections, we apply the theory to two classical families of moving graphs:
Dixon-1 and Dixon-2 moving graphs.
\section{Dixon-1 moving graphs}
In this section we focus on Dixon-1 moving graph family. The Dixon-1 construction can be applied to an arbitrary bipartite graph, giving a mobile realization; the so-obtained moving graph would be a {\em Dixon-1 moving graph}. In the sequel, we explain this construction in natural language. Let $G=(V,E)$ be an arbitrary bipartite graph and let $V_1$, $V_2$ be its two independent sets. Then, place the vertices in $V_1$ to the $x$-axis and those in $V_2$ to the $y$-axis, where by $x$-axis and $y$-axis we just refer to two orthogonal axes in the real plane. Note that the positions of vertices are not fixed, the only requirement is that the vertices in $V_1$ should stay on the $x$-axis, while those in $V_2$ should stay on the $y$-axis. One can check that the dimension of the configuration space is one for complete bipartite graphs. That is to say, we have one-degree of freedom to choose the position of vertices. It is a mobile realization of complete bipartite graphs, and also of all bipartite graphs in the cases of which then may have higher dimensional configuration spaces. See the right sub-figure of Figure \ref{fig:bipartite} for one such placement of the vertices.
A more specific definition (with a fixed scale) for Dixon-1 moving graphs would be the following. Let $m$ and $n$ be positive integers and $K_{m,n}$ the complete undirected bipartite graph with $m+n$ vertices. Fix real numbers $0<a_1<...<a_{m-1}$ and $0<b_1<...<b_{n-1}$, $a_0=b_0=0$. The two independent sets of vertices are $P:=\{p_0,...,p_{m-1}\}$ and $Q:=\{q_0,...,q_{n-1}\}$. We place $p_i$ ($0\leq i\leq m-1$) to $x$-axis and $q_j$ ($0\leq j\leq n-1$) to $y$-axis. The positions of the vertices of graph $K_{m,n}$ parameterized in the variable $t$ are as follows: \begin{flalign*}
&f_{p_0}(t) = (\sin{t},\; 0), &&\\\nonumber
&f_{p_i}(t) = (\pm\sqrt{a_i+\sin^2{t}},\; 0),\; i> 0, &&\\\nonumber
&f_{q_0}(t) = (0,\; \cos{t}), &&\\\nonumber
&f_{q_j}(t) = (0,\; \pm\sqrt{b_j+\cos^2{t}}),\; j> 0. &&
\end{flalign*}
Note that we can choose the $x$-coordinate for $f_{p_i}(t)$ ($i>0$) to be either positive
or negative; the same rule applies to the $y$-coordinate for $f_{q_j}(t)$ ($j> 0$). We denote the first coordinate (x-coordinate) of the position of vertex $v$ by $f_v(t)|_x$ and the second coordinate
(y-coordinate) of it by $f_v(t)|_y$. We observe that the edge lengths are fixed. We compute the lengths for edges $\{p_i, q_j\}$ ($0\leq i \leq m-1$, $0\leq j \leq n-1$) as follows:
$$\|f_{p_i}(t)-f_{q_j}(t)\|=\sqrt{a_i+\sin^2{t}+b_j+\cos^2{t}}=\sqrt{a_i+b_j+1}.$$
The next theorem tells us the collision information of an arbitrary Dixon-1 moving
graph. \begin{theorem}\label{thm:dixon1}
The collision pairs of any Dixon-1 moving graph $D$
(defined as above) are as follows:
\begin{enumerate}
\item $(p_0,\{p_i,q_0\})\in CP_D$, $i> 0$.
\item $(p_i, \{p_k, q_0\})\in CP_D$, for $k>i>0$ and $f_{p_k}(t)|_x\cdot f_{p_i}(t)|_x>0$.
\item $(q_0, \{p_0, q_i\})\in CP_D$, $i> 0$.
\item $(q_i, \{p_0, q_k\})\in CP_D$, for $k>i>0$ and $f_{q_k}(t)|_y\cdot f_{q_i}(t)|_y>0$.
\end{enumerate}
\end{theorem}
\begin{proof} \begin{enumerate}
\item In the case when $f_{p_i}(t)=(\sqrt{a_i+\sin^2{t}},\; 0)$: Solving the equation
$$\|f_{q_0}(t)-f_{p_0}(t)\|+\|f_{p_0}(t)-f_{p_i}(t)\|=\|f_{q_0}(t)-f_{p_i}(t)\|,$$ i.e., $$1+\sqrt{a_i+\sin^2{t}}-\sin{t}=\sqrt{a_i+1},$$ we obtain that $t=\frac{\pi}{2}+2\cdot k\cdot \pi$ ($k\in \mathbb{Z}$) are in the solution set.
In the case when $f_{p_i}(t)=(-\sqrt{a_i+\sin^2{t}},\; 0)$: Solving the equation
$$\|f_{q_0}(t)-f_{p_0}(t)\|+\|f_{p_0}(t)-f_{p_i}(t)\|=\|f_{q_0}(t)-f_{p_i}(t)\|,$$ i.e., $$1+\sin{t}+\sqrt{a_i+\sin^2{t}}=\sqrt{a_i+1},$$ we obtain that $t=-\frac{\pi}{2}+2\cdot k\cdot \pi$ ($k\in \mathbb{Z}$) are in the solution set. By definition of collision pairs, $p_0$ does not collide with edge $\{p_0,q_0\}$. Hence, $(p_0,\{p_i, q_0\})\in CP_D$, $i> 0$.
Are these already all the collision pairs containing vertex $p_0$? Suppose that $p_0$ collides with edge $\{p_k, q_l\}$, $k,l>0$, then there exists $t'\in \mathbb{R}$ such that $f_{p_0}(t')$ lies on the line segment defined by the points $f_{p_k}(t')$ and $f_{q_l}(t')$. Since
$f_{p_0}(t')|_y=f_{p_k}(t')|_y=0$, we have
$$f_{q_l}(t')|_y= \sqrt{b_l+\cos^2{t}}=0$$ or
$$f_{q_l}(t')|_y= -\sqrt{b_l+\cos^2{t}}=0.$$ However, $b_l>0$ for $l>0$, so neither of the above equations hold. We reach a contradiction. Therefore, $(p_0,\{p_i, q_0\})$ ($i>0$) are already all the collision pairs of $D$ that have $p_0$ as the vertex.
\item In the case when both $f_{p_i}(t)|_x$ and $f_{p_k}(t)|_x$ are positive:
Solving the equation $\|f_{q_0}(t)-f_{p_i}(t)\|+
\|f_{p_i}(t)-f_{p_k}(t)\|=\|f_{q_0}(t)-f_{p_k}(t)\|$, i.e., $$\sqrt{a_i+1}+\sqrt{a_k+\sin^2{t}}-\sqrt{a_i+\sin^2{t}}=\sqrt{a_k+1},$$ we see that $t=\frac{\pi}{2}+k_1\cdot \pi$, $k_1\in \mathbb{Z}$ are in the solution set.
When $f_{p_i}(t)|_x<0$ and $f_{p_k}(t)|_x<0$, the equation is the same, hence we obtain the same set of solutions. But are these already all the collision pairs containing vertex $p_i$ ($i>0$)? Suppose that $p_i$ collides with edge $\{p_k, q_l\}$ for some $k<i$. Then there exists $t'\in \mathbb{R}$ such that the following equation holds:
$$\|f_{q_l}(t')-f_{p_i}(t')\|+\|f_{p_i}(t')-f_{p_k}(t')\|=\|f_{q_l}(t')-f_{p_k}(t')\|.$$
We obtain that $\|f_{q_l}(t')-f_{p_i}(t')\|\leq \|f_{q_l}(t')-f_{p_k}(t')\|$, i.e., $\sqrt{a_i+b_l+1}\leq \sqrt{a_k+b_l+1}$. However, $0\leq a_k<a_i$ since $k<i$. We gained a contradiction. So, $(p_i, \{p_k,q_l\})\notin CP_D$ when $k<i$.
Suppose that $p_i$ collides with edge $\{p_k,q_l\}$ for $k>i>0$. There exists $t'\in\mathbb{R}$ such that $f_{p_i}(t')$ lies on the line segment defined by $f_{q_l}(t')$ and $f_{p_k}(t')$. Since the $y$-coordinates of both $f_{p_k}(t')$ and $f_{p_i}(t')$
are zero, we obtain that $f_{q_l}(t')|_y=0$. This leads to the restriction that $l=0$, since
$f_{q_l}(t')|_y\neq 0$ when $l> 0$.
Suppose that $p_i$ collides with edge $\{p_k, q_0\}$ for $k>i>0$ and the corresponding $t'\in \mathbb{R}$
makes $f_{p_k}(t')|_x\cdot f_{p_i}(t')|_x<0$. Suppose that $f_{p_k}(t')|_x>0$ and $f_{p_i}(t')|_x<0$.
Since both $f_{q_0}(t')|_x$ and $f_{p_k}(t')|_x$ are non-negative and $f_{p_i}(t')$ lies on the line
segment defined by these two points, we obtain that $f_{p_i}(t')|_x\geq 0$. This contradicts the assumption $f_{p_i}(t')|_x<0$.
In the case when $f_{p_k}(t')|_x<0$ and $f_{p_i}(t')|_x>0$, the argument is analogous.
To sum up, $(p_i, \{p_k, q_0 \})$ ($k>i$, $f_{p_k}(t)|_x\cdot f_{p_i}(t)|_x>0$) are already all the collision pairs
containing vertex $p_i$ ($i>0$).
\item Because of the symmetry of vertices in $Q$ and vertices in $P$ in the sense of exchanging
$x$-axis with $y$-axis, the remaining proofs can be done analogously. The proof for item 3. is analogous to
that of item 1.
\item The proof for item 4. is analogous to that of item 2.\phantom\qedhere \qed
\end{enumerate} \end{proof}
After some analysis on the collision information, we find out that we can apply the method introduced in Section \ref{sec:sufficient_condition} and then obtain a collision-free L-model for any Dixon-1 graph! Before we can talk about the main theorem of this section, we need to introduce a lemma, before which however, we need two definitions first.
Let $D_1=(K_{m,n}, F_1)$ be a Dixon-1 moving graph with two independent sets
$P:=\{p_0,\ldots, p_{m-1}\}$ and
$Q:=\{q_0,\ldots, q_{n-1}\}$ such that
all $x$-coordinates of the positions of vertices in $\underline{P}:=P\setminus\{p_0\}$ have the same sign and all
$y$-coordinates of the positions of vertices in $\underline{Q}:=Q\setminus\{q_0\}$ have the same sign. Such Dixon-1
moving graphs are called {\bf crowded}.
Let $D_2=(K_{m,n}, F_2)$ be any Dixon-1 moving graph with the same underlying graph $K_{m,n}$
as $D_1$. Let $D_2$ be so that the only difference between $D_1$ and $D_2$ is on the difference
between $F_1$ and $F_2$ --- on the choices of the signs for $x$-coordinates of vertices
in $\underline{P}$ and $y$-coordinates of vertices in $\underline{Q}$. Then, we say that $D_1$ is the
{\bf crowded extreme} of $D_2$.
\begin{lemma}\label{lem:crowded} Let $D_1$ be the crowded extreme of $D_2$, where $D_1$, $D_2$ are two Dixon-1 moving graphs defined as above. Then, we have $CP_{D_2}\subset CP_{D_1}$. \end{lemma} \begin{proof}
By Theorem \ref{thm:dixon1} item 1., we have that
$(p_0,\{p_i, q_0\})\in CP_{D_2}$ and $(p_0,\{p_i,q_0\})\in CP_{D_1}$ for $i>0$.
By Theorem \ref{thm:dixon1} item 2., we have that
$(p_i, \{p_k, q_0\})\in CP_{D_2}$, for $k>i>0$ and $f_{p_k}(t)|_x\cdot f_{p_i}(t)|_x>0$.
However, since all $x$-coordinates of the positions of vertices in $\underline{P}$ have the same sign in $D_1$,
all these collision pairs in $CP_{D_2}$ are in $CP_{D_1}$ as well.
The analyses for the collision pairs in $CP_{D_2}$ described in item 3. and item 4.
being in $CP_{D_1}$ are analogous.
Therefore, $CP_{D_2}\subset CP_{D_1}$. \end{proof}
\begin{corollary}\label{cor:crowded_extreme}
Let $D$ be an arbitrary Dixon-1 moving graph and let $h$ be a height function
of $D$ such that $(D,h)$ is a collision-free L-model. Let $D'$
be any Dixon-1 moving graph of which the crowded extreme is $D$, then
$(D',h)$ is a collision-free L-model of $D'$. \end{corollary} \begin{proof}
From Lemma \ref{lem:crowded}, we know that all collision pairs of $D'$ are
collision pairs of $D$. Since $h$ guarantees that all collision pairs
in $D$ are safe,
$h$ naturally guarantees that all collision pairs in $D'$ are safe.
Therefore, $(D',h)$ is a
collision-free L-model of $D'$.
\end{proof}
\begin{theorem}\label{thm:dixon1_has_model}
Every Dixon-1 moving graph has a collision-free L-model. \end{theorem} \begin{proof}
By Corollary \ref{cor:crowded_extreme}, it suffices to show the existence of a collision-free L-model
for any crowded Dixon-1 moving graph. Let $D=(K_{m,n}, F)$, $K_{m,n}=(V,E)$ be any crowded Dixon-1 moving graph
with independent sets $P:=\{p_0,\ldots, p_{m-1}\}$ and $Q:=\{q_0,\ldots, q_{n-1}\}$. Recall that the
edge set of $D$ is
$$E=\{\{p_i, q_j\}\mid 0\leq i\leq m-1, 0\leq j\leq n-1 \}.$$
Let $C=(V_C,E_C)$ be the collision graph of $D$, recall that $V_C=E$.
In the sequel, we give a bipartition of the set $E$, into $E_1$ and $E_2$,
such that the two corresponding induced collision graphs $C[E_1]$ and
$C[E_2]$ of $D$ are both acyclic. We divide the edges in $E$ into two groups as follows:
\begin{itemize} \item $E_1=\{\{p_i,q_0\}\mid 0\leq i\leq m-1\}$;
\item $E_2=E\setminus E_1$.
\end{itemize} By Theorem \ref{thm:dixon1} item 1., there is a directed edge from $\{p_0,q_0\}$ to each of the other elements in $E_1$ in graph $C[E_1]$. By Theorem \ref{thm:dixon1} item 2., there is a directed edge from $\{p_i,q_0\}$ to $\{p_k,q_0\}$ in graph $C[E_1]$ for all pairs $(i,k)$ such that $i<k$. Theorem \ref{thm:dixon1} tells us that these are all the edges in graph $C[E_1]$. We easily see that the graph is acyclic.
By Theorem \ref{thm:dixon1} item 4., there is a directed edge from $\{p_j, q_i\}$ to $\{p_0, q_k\}$ for $k>i>0$ and $j\geq 0$ in graph $C[E_2]$. These are all the edges of graph $C[E_2]$. Since these edges in $C[E_2]$ are directed from some edge of $D$ with strictly smaller subscript on the $q$-vertex to some edge with strictly bigger subscript on the $q$-vertex, there is no cycle in graph $C[E_2]$.
By Theorem \ref{thm:partition_condition}, $D$ has a collision-free L-model. As Proposition \ref{prop:model_construction} stated, we apply Algorithm \ref{alg:height_construction} with height parameters $(1,1)$ to $C[E_1]$ and $(0,-1)$ to $C[E_2]$, respectively. Then we obtain $$h(\{p_i, q_0\})=i+1,$$ $$h(\{p_l, q_j\})=-(j-1)\cdot (m+1)-l.$$ This is the corresponding height function for a collision-free L-model of $D$; moreover, it induces a collision-free L-model for any Dixon-1 moving graph which has $D$ as its crowded extreme as well, by Corollary \ref{cor:crowded_extreme}. \end{proof}
\begin{example}
Example \ref{eg:apply_height_algorithm} uses exactly the same grouping (of edges) as stated in the proof
of Theorem
\ref{thm:dixon1_has_model}. And the height function obtained coincides with the result as if we compute using the above formulae.
Let us again greet this running example.
Let $M=(K_{3,4},F)$, where $F$ is the set of functions given in Example \ref{eg:collision_pairs}.
Two independent sets of $K_{3,4}$ are $P=\{p_0=1,p_1=2,p_2=3,p_3=4\}$
and $Q=\{q_0=5,q_1=6,q_2=7\}$. Then it is not hard to check that
the grouping of edges and
the height function obtained in Example \ref{eg:apply_height_algorithm} indeed coincide with those provided
in the proof of Theorem \ref{thm:dixon1_has_model}.
However, note that there are also other possible output
height functions of Algorithm~\ref{alg:height_construction} which provide collision-free L-models of $M$,
by Remark \ref{rem:output_not_unique}. \end{example}
\section{Dixon-2 moving graphs}
In this section we discuss another class of moving graphs --- Dixon-2 moving graph family.
The Dixon-1 construction applies to any bipartite graph, however, the Dixon-2 construction only applies to the complete bipartite graph $K_{4,4}$ and its subgraphs.
To say it in natural language: Fix vertices $1$ and $5$ in the first quadrant of the coordinate system. Let vertex $2$ be symmetric to vertex $1$ with respect to $y$-axis, let vertex $4$ be symmetric to vertex $1$ w.r.t. $x$-axis, and let vertex $3$ be symmetric to vertex $1$ w.r.t. the origin. Let vertex $6$ be symmetric to vertex $5$ w.r.t. $y$-axis, let vertex $8$ be symmetric to vertex $5$ w.r.t. $x$-axis, and let vertex $7$ be symmetric to vertex $5$ w.r.t. the origin.
Note that here $x$-axis refers to the horizontal axis and $y$-axis refers to the vertical axis, as in the convention.
Let Group One consist of vertices $1,2,3,4$ and Group Two constitute of vertices $5,6,7,8$. Then we add an edge between any vertex in group one and any vertex in group two, forming a bipartite graph $K_{4,4}$. This is the Dixon-2 construction. To apply it to any subgraph, simply take the substructure of this construction. If we apply this construction to $K_{4,4}$, we then obtain a {\em Dixon-2 moving graph}. See Figure \ref{fig:dixon2} for a visualization.
\begin{figure}
\caption{This is a static Dixon-2 moving graph.}
\label{fig:dixon2}
\end{figure}
Some other literature may include also those moving graphs obtained via applying the construction to proper subgraphs of $K_{4,4}$ in the Dixon-2 moving graph families; our notion differs from them slightly. Now we give a more formal definition for Dixon-2 moving graphs as follows. Let the vertex set of $K_{4,4}$ be $\{1,2,3,4,5,6,7,8\}$ and the two independent sets be $\{1,2,3,4\}$ and $\{5,6,7,8\}$. Let $b,c,d\in \mathbb{R}^+$ be such that $d>a$ and $b>a$ hold. The parameterizations of the positions of the vertices are as follows:
\begin{flalign*}
&s_1 = \sqrt{b^2-a^2\cdot \sin^2{t}},\;\;
s_2 = \sqrt{d^2-a^2\cdot\cos^2{t}}\;, &&\\\nonumber
&f_{1}(t) = \left(\frac{a\cdot \cos{t}+s_1}{2},\; \frac{a\cdot \sin{t}+s_2}{2}\right), &&\\\nonumber
&f_{2}(t) = \left(-\frac{a\cdot \cos{t}+s_1}{2},\; \frac{a\cdot \sin{t}+s_2}{2}\right), &&\\\nonumber
&f_{3}(t) = \left(-\frac{a\cdot \cos{t}+s_1}{2},\; -\frac{a\cdot \sin{t}+s_2}{2}\right), &&\\\nonumber
&f_{4}(t) = \left(\frac{a\cdot \cos{t}+s_1}{2},\; -\frac{a\cdot \sin{t}+s_2}{2}\right), &&\\\nonumber
&f_{5}(t) = \left(\frac{-a\cdot \cos{t}+s_1}{2},\; \frac{-a\cdot \sin{t}+s_2}{2}\right), &&\\\nonumber
&f_{6}(t) = \left(-\frac{-a\cdot \cos{t}+s_1}{2},\; \frac{-a\cdot \sin{t}+s_2}{2}\right), &&\\\nonumber
&f_{7}(t) = \left(-\frac{-a\cdot \cos{t}+s_1}{2},\; -\frac{-a\cdot \sin{t}+s_2}{2}\right), &&\\\nonumber
&f_{8}(t) = \left(\frac{-a\cdot \cos{t}+s_1}{2},\; -\frac{-a\cdot \sin{t}+s_2}{2}\right). &&
\end{flalign*}
We denote the coordinates of vertex $1$ by $(x_1,y_1)$ and those of vertex $5$ by $(x_5, y_5)$. Then from the above parameterizations one can check that $x_1,y_1,x_5,y_5 > 0$ holds. We can acquire the coordinates of the other six vertices individually, by alternating the sign of one or both of the coordinates of either vertex $1$ or vertex $5$. Let $c\in \mathbb{R}^+$ such that $c^2=b^2+d^2-a^2$ holds. Since $b,d>a$, we know that $c>b>a$ and $c>d>a$. Then we have the following four equations, where we require $x_1,y_1,x_5,y_5 > 0$ additionally.
\begin{align*}
(x_1-x_5)^2+(y_1-y_5)^2 = a^2 \\
(x_1+x_5)^2+(y_1-y_5)^2 = b^2 \\
(x_1+x_5)^2+(y_1+y_5)^2 = c^2 \\
(x_1-x_5)^2+(y_1+y_5)^2 = d^2 \end{align*} Since $a^2+c^2=b^2+d^2$, one can verify that the third equation is redundant. Actually, the above equation system has one-dimensional
solution set, hence the realization is mobile. We see that the edge lengths are fixed as follows, where $l_{u,v}:=\|f_u(t)-f_v(t)\|$ denotes the length of edge $\{u,v\}$:
$$l_{1,5}=l_{2,6}=l_{3,7}=l_{4,8}=a,\; l_{1,6}=l_{2,5}=l_{3,8}=l_{4,7}=b,$$
$$l_{1,7}=l_{2,8}=l_{3,5}=l_{4,6}=c,\; l_{1,8}=l_{2,7}=l_{3,6}=l_{4,5}=d.$$
Since we require $x_1,y_1,x_5,y_5>0$, vertices $1$ and $5$ stay in the first quadrant;
vertices $2$ and $6$ stay in the second quadrant; vertices $3$ and $7$ stay in the third quadrant and
vertices $4$ and $8$ stay in the fourth quadrant. We call the vertices that are in the same quadrant
{\bf friend vertex} of each other.
Above, we define the Dixon-2 moving graphs in two different ways: one is via
parameterizing the positions of the vertices using explicit functions on $t$, the other is by giving the equations that the coordinates
of the vertices need to fulfill using implicit functions.
Actually, these two ways are equivalent. To consider the collision-detecting equations, we can use both systems --- the implicit one, or the explicit one. In the upcoming theorem, we analyze the collision information for an arbitrary Dixon-2 moving graph. \begin{theorem} The collision pairs of any Dixon-2 moving graph $D$
(defined as above) are as follows --- written in $8$ groups each of which illustrates the collision pairs of one vertex:
\begin{enumerate}
\item $(1,\{5,2\})$, $(1,\{5,4\})$, $(1,\{5,3\})$;
\item $(5,\{1,6\})$, $(5, \{1,8\})$, $(5,\{1,7\})$;
\item $(2,\{6,1\})$, $(2,\{6,3\})$, $(2,\{6,4\})$;
\item $(6,\{2,5\})$, $(6,\{2,7\})$, $(6,\{2,8\})$;
\item $(3,\{7,1\})$, $(3,\{7,4\})$, $(3,\{7,2\})$;
\item $(7,\{3,5\})$, $(7,\{3,8\})$, $(7,\{3,6\})$;
\item $(4,\{8,1\})$, $(4,\{8,2\})$, $(4,\{8,3\})$;
\item $(8,\{4,5\})$, $(8,\{4,6\})$, $(8,\{4,7\})$.
\end{enumerate}
To sum up, elements in $CP_D$ are $(i,\{k,l\})$, where $1\leq i\leq 8$, $k$ is the friend vertex of $i$,
and $l$ is in the same independent set with $i$.
\end{theorem} \begin{proof} Because of the symmetry property of Dixon-2 construction, it suffices to verify the collision situation for vertex $1$. We will show that vertex $1$ collides with all edges containing vertex $5$ --- its friend vertex --- except for the edge $\{1,5\}$, of course; and we will show that these are all the collision pairs containing vertex $1$.
Solving the equation
$$\|f_1(t)-f_5(t)\|+\|f_1(t)-f_2(t)\|=\|f_5(t)-f_2(t)\|$$ which is equivalent to
$$\|(x_1,y_1)-(x_5,y_5)\|+\|(x_1,y_1)-(-x_1,y_1)\|=\|(x_5,y_5)-(-x_1,y_1)\|,$$ i.e., $a+2\cdot x_1=b$, we obtain that $$x_1 = \frac{b-a}{2},\; x_5 = \frac{a+b}{2},\;
y_1 = y_5=\frac{\sqrt{d^2-a^2}}{2}$$ which is equivalent to $t=\pi+2\cdot k\cdot \pi$, $k\in \mathbb{Z}$. Hence $(1,\{5,2\})\in CP_D$. Solving the equation
$$\|f_1(t)-f_5(t)\|+\|f_1(t)-f_4(t)\|=\|f_5(t)-f_4(t)\|$$ which is equivalent to
$$\|(x_1,y_1)-(x_5,y_5)\|+\|(x_1,y_1)-(x_1,-y_1)\|=\|(x_5,y_5)-(x_1,-y_1)\|,$$ i.e., $a+2\cdot y_1=d$, we obtain that $$y_1 = \frac{d-a}{2},\; y_5 = \frac{a+d}{2},\;
x_1 = x_5=\frac{\sqrt{b^2-a^2}}{2}$$ which is equivalent to $t=-\frac{\pi}{2}+2\cdot k\cdot \pi$, $k\in \mathbb{Z}$. Hence $(1,\{5,4\})\in CP_D$. Solving the equation
$$\|f_1(t)-f_5(t)\|+\|f_1(t)-f_3(t)\|=\|f_5(t)-f_3(t)\|$$ which is equivalent to
$$\|(x_1,y_1)-(x_5,y_5)\|+\|(x_1,y_1)-(-x_1,-y_1)\|=\|(x_5,y_5)-(-x_1,-y_1)\|,$$ i.e., $a+\sqrt{(2\cdot x_1)^2+(2\cdot y_1)^2}=c$, we get
$$x_1=\frac{1}{2}\sqrt{\frac{(b^2-a^2)(c-a)}{c+a}},\; x_5=\frac{1}{2}\sqrt{\frac{(b^2-a^2)(c+a)}{c-a}},$$
$$y_1=\frac{1}{2}\sqrt{\frac{(c^2-b^2)(c-a)}{c+a}},\; y_5=\frac{1}{2}\sqrt{\frac{(c^2-b^2)(c+a)}{c-a}},$$ which is a solution fulfilling the implicit defining equation system of $D$ and it is not hard to verify that
$x_1,y_1,x_5,y_5>0$ holds. Hence $(1,\{5,3\})\in CP_D$.
Since vertex $1$ by definition stays in the first quadrant, it is impossible for it to collide with edges that do not intersect the first quadrant. Therefore, the only edges remain to be considered for vertex $1$ are $\{6,4\}$ and $\{2,8\}$. Because of the symmetry property of Dixon-2 construction, it suffices to consider whether $(1,\{6,4\})$ is a collision pair or not. Solving the equation
$$\|f_1(t)-f_6(t)\|+\|f_1(t)-f_4(t)\|=\|f_6(t)-f_4(t)\|$$ which is equivalent to
$$\|(x_1,y_1)-(-x_5,y_5)\|+\|(x_1,y_1)-(x_1,-y_1)\|=\|(-x_5,y_5)-(x_1,-y_1)\|,$$ i.e., $a+2\cdot y_1=c$, we get $y_1=\frac{c-b}{2}$. Substituting it back to the explicit defining equation
system of $D$, we obtain that $\sin{t}=-\frac{b}{a}$, which implies $|\sin{t}|>1$. This contradicts the
fact that $0\leq |\sin{t}|\leq 1$. Hence $(1,\{6,4\})\notin CP_D$. There are in total three collision pairs of $D$ that contains vertex $1$, namely $(1,\{5,2\})$, $(1,\{5,4\})$, and $(1,\{5,3\})$.
The collision situation for vertex $5$ is analogous. By the symmetry property of Dixon-2 construction, collision situations for the remaining vertices can be obtained directly.
\end{proof}
In the sequel, we prove that Dixon-2 moving graphs have no collision-free L-models.
\begin{theorem}\label{thm:dixon2} There are no collision-free L-models for Dixon-2 moving graphs. \end{theorem} \begin{proof} We prove by contradiction. Let $D$ be an arbitrary Dixon-2 moving graph and let $h$ be a height function of $D$ such that $(D,h)$ is collision-free. Then, let us analyze the relative heights for edges of this moving graph.
Since vertex $5$ collides with all edges containing vertex $1$ (except for edge $\{1,5\}$),
there are no edges containing vertex $1$ lying in between edge $\{1,5\}$ and another edge containing
vertex $5$. Symmetrically, there are no edges
containing vertex $5$ lying in between edge $\{1,5\}$ and another edge containing
vertex $1$. To conclude, the range of vertex $1$ and that of vertex $5$
intersect on edge $\{1,5\}$. W.l.o.g., assume that edges containing vertex $1$ (except for edge $\{1,5\}$)
have bigger height values than those containing vertex~$5$ (except for edge $\{1,5\}$).
Then we obtain
$$h(\{1,x\})>h(\{1,y\})>h(\{1,z\})>h(\{1,5\})>h(\{u,5\})>h(\{v,5\})>h(\{w,5\}),$$
where $\{x,y,z\}=\{6,7,8\}$ and $\{u,v,w\}=\{2,3,4\}$.
Now we claim that
$h(\{4,7\})>h(\{3,5\})$. Suppose this does not hold, i.e., $h(\{4,7\})<h(\{3,5\})$. Then
we obtain $$h(\{1,7\})>h(\{3,5\})>h(\{4,7\}).$$
Then we see that edge $\{3,5\}$ is within the range
of vertex $7$, this contradicts the fact that the collision pair $(7,\{3,5\})$ being safe.
Hence we have $h(\{4,7\})>h(\{3,5\})$. Now we claim that
$h(\{3,8\})>h(\{4,5\})$. Suppose this does not hold, i.e., $h(\{3,8\})<h(\{4,5\})$. Then
we obtain $$h(\{1,8\})>h(\{4,5\})>h(\{3,8\}).$$
Then we see that edge $\{4,5\}$ is within the range
of vertex $8$, this contradicts the fact that the collision pair $(8,\{4,5\})$ being safe.
Therefore we obtain $h(\{3,8\})>h(\{4,5\})$.
Thereafter, we try to analyze the relative heights of the edges $\{4,7\}$ and $\{3,8\}$. If $h(\{4,7\})>h(\{3,8\})$,
we have $$h(\{4,7\})>h(\{3,8\})>h(\{4,5\}).$$
Then we see that edge $\{3,8\}$ is within the range
of vertex $4$, this contradicts the fact that the collision pair $(4,\{3,8\})$ being safe.
If $h(\{4,7\})<h(\{3,8\})$, then
we have $$h(\{3,5\})<h(\{4,7\})<h(\{3,8\}).$$
Then we see that edge $\{4,7\}$ is within the range
of vertex $3$, this contradicts the fact that the collision pair $(3,\{4,7\})$ being safe.
Hence, no matter how we place edge $\{4,7\}$ and edge $\{3,8\}$ relatively, it always leads
to some collision pair being unsafe under $h$, which contradicts the assumption of $(D,h)$
being collision-free.
Hence, there are no collision-free L-models for any Dixon-2 moving graph. \end{proof}
As side products, we obtain the following two relatively trivial corollaries.
\begin{corollary}
Dixon-2 moving graphs do not fulfill the partition condition.
That is to say, there is no partition of the edges of any Dixon-2 moving graph into
two parts $E_L$, $E_U$, such that the induced collision graphs $C_L$(by $E_L$)
and $C_U$(by $E_U$) are both acyclic. \end{corollary} \begin{proof}
If a Dixon-2 moving graph fulfilled the partition condition, then by Theorem~\ref{thm:partition_condition}, it would have a collision-free L-model,
which contradicts Theorem \ref{thm:dixon2}. \end{proof}
Naturally, we define a {\bf moving super-graph of $M$} to be a moving graph that contains the moving graph $M$ as a substructure. Upon this concept, we have the following corollary. \begin{corollary} Let $M=(G,F)$ with underlying graph $G=(V,E)$ be a moving super-graph of a Dixon-2 moving graph $D=(G_1,F_1)$ with underlying graph $G_1=(V_1,E_1)$. Then, the moving graph $M$ does not have any collision-free L-models. \end{corollary} \begin{proof}
Suppose that $(M,h')$ is a collision-free L-model. Since $F$ equals $F_1$ when restricted to the vertex set $V_1$,
any collision pair $(v,e)$
of $D$ is also a collision pair of $M$. Then all collision pairs
$(v,e)$ of $D$ are safe under $h'$, since $(M,h')$ is a collision-free L-model.
Now we define a function $h:E_1\to \mathbb{Z}$ to be the restriction
of $h':E\to \mathbb{Z}$ on $E_1$. Then for any $(v,e)\in CP_D$, $h(e)$ should be outside of the height range of
vertex~$v$ in $D$; otherwise $h'(e)$ would also be within the range of $v$ in $M$, since
the range of $v$ in $M$ is equal or wider than that in $D$.
Then, all collision pairs of $D$ are safe under $h$; this contradicts Theorem \ref{thm:dixon2}. \end{proof}
We wind up the section by confirming that, in reality we cannot build any collision-free Lego models (as shown in Figure \ref{fig:photo}) or L-linkages for any Dixon-2 moving graph.
\end{document} |
\begin{document}
\title{Analytic computable structure theory and $L^p$-spaces part 2} \author{Tyler A. Brown} \address{Department of Mathematics\\ Iowa State University\\ Ames, Iowa 50011} \email{tab5357@iastate.edu} \author{Timothy H. McNicholl} \address{Department of Mathematics\\ Iowa State University\\ Ames, Iowa 50011} \email{mcnichol@iastate.edu}
\begin{abstract} Suppose $p \geq 1$ is a computable real. We extend previous work of Clanin, Stull, and McNicholl by determining the degrees of categoricity of the separable $L^p$ spaces whose underlying measure spaces are atomic but not purely atomic. In addition, we ascertain the complexity of associated projection maps. \end{abstract} \thanks{The second author was supported in by Simons Foundation Grant \# 317870.} \maketitle
\section{Introduction}\label{sec:intro}
We continue here the program, recently initiated by Melnikov and Nies (see \cite{Melnikov.Nies.2013}, \cite{Melnikov.2013}), of utilizing the tools of computable analysis to investigate the effective structure theory of metric structures, in particular $L^p$ spaces where $p \geq 1$ is computable. Specifically, we seek to classify the $L^p$ spaces that are computably categorical in that they have exactly one computable presentation up to computable isometric isomorphism. We also seek to determine the degrees of categoricity of those $L^p$ spaces that are not computably categorical; this is the least powerful Turing degree that computes an isometric isomorphism between any two computable presentations of the space.
Recall that when $\Omega$ is a measure space, an \emph{atom} of $\Omega$ is a non-null measurable set $A$ so that $\mu(B) = \mu(A)$ for every non-null measurable set $B \subseteq A$. A measure space with no atoms is \emph{non-atomic}, and a measure space is \emph{purely atomic} if its $\sigma$-algebra is generated by its atoms. Recall also that with every measure space $\Omega$ there is an associated pseudo-metric $D_\Omega$ on its finitely measurable sets. Namely, $D_\Omega(A,B)$ is the measure of the symmetric difference of $A$ and $B$. The space $\Omega$ is said to be \emph{separable} if this associated pseudo-metric space is separable. By means of convergence in measure, it is possible to show that an $L^p$ space is separable if and only if its underlying measure space is separable.
Suppose $p \geq 1$ is computable. It is essentially shown in \cite{Pour-El.Richards.1989} that every separable $L^2$ space is computably categorical. In \cite{McNicholl.2015}, the second author showed that $\ell^p$ is computably categorical only when $p \neq 2$. Moreover, in \cite{McNicholl.2017} he showed that $\ell^p_n$ is computably categorical and that the degree of categoricity of $\ell^p$ is \textbf{0}''. Together, these results determine the degrees of categoricity of separable spaces of the form $L^p(\Omega)$ when $\Omega$ is purely atomic. In the paper preceding this, Clanin, McNicholl, and Stull showed that $L^p(\Omega)$ is computably categorical when $\Omega$ is separable and nonatomic \cite{Clanin.McNicholl.Stull.2019}. Here, we complete the picture by determining the degrees of categoricity of separable $L^p$ spaces whose underlying measure spaces are atomic but not purely atomic. Specifically, we show the following.
\begin{theorem}\label{thm:main} Suppose $\Omega$ is a separable measure space that is atomic but not purely atomic, and suppose $p$ is a computable real so that $p \geq 1$ and $p \neq 2$. Assume $L^p(\Omega)$ is nonzero. \begin{enumerate}
\item If $\Omega$ has finitely many atoms, then the degree of categoricity of $L^p(\Omega)$ is $\mathbf{0'}$. \label{thm:main::itm:finite}
\item If $\Omega$ has infinitely many atoms, then the degree of categoricity of $L^p(\Omega)$ is $\mathbf{0''}$. \label{thm:main::itm:infinite} \end{enumerate} \end{theorem}
Suppose $\Omega$ is a separable measure space that is not purely atomic, and suppose $L^p(\Omega)$ is nonzero where $1 \leq p < \infty$. It follows from the Carath\'eodory classification of separable measure spaces that $L^p(\Omega)$ isometrically isomorphic to $L^p[0,1]$ if $\Omega$ has no atoms and that $L^p(\Omega)$ is isometrically isomorphic to $\ell^p_n \oplus L^p[0,1]$ if $\Omega$ has $n \geq 1$ atoms (see e.g. \cite{Cembranos.Mendoza.1997}). It also follows that if $\Omega$ has infinitely many atoms, then $L^p(\Omega)$ is isometrically isomorphic to $\ell^p \oplus L^p[0,1]$. Degrees of categoricity are preserved by isometric isomorphism. We thus have the following.
\begin{corollary}\label{cor:lpnLp01} Suppose $p$ is a computable real so that $p \geq 1$ and $p \neq 2$. Then, the degree of categoricity of $\ell^p_n \oplus L^p[0,1]$ is $\mathbf{0'}$, and the degree of categoricity of $\ell^p \oplus L^p[0,1]$ is $\mathbf{0''}$. \end{corollary}
These results are somewhat surprising in that one might suspect that the spaces $\ell^p_n\oplus L^p[0,1]$ and $\ell^p\oplus L^p[0,1]$ do not have structure much different from their constituent (summand) spaces. It turns out that allowing these summand spaces to ``work together" indeed produces a few complications in terms of their hybridized structure, particularly while establishing lower bounds for degrees of computable categoricity. For example, in Section \ref{sec:lower}, we construct a computable presentation of $\ell^p_n\oplus L^p[0,1]$ so that projections of vectors into each of the summand spaces are incomputable. However, as we will demonstrate in Section \ref{sec:upper}, it can be fruitful to dissect the hybrid case and consider the constituent spaces in tandem. In fact, our results regarding the upper bounds for the degree of categoricity of $\ell^p_n\oplus L^p[0,1]$ and $\ell^p\oplus L^p[0,1]$ can be considered in this manner and largely piggyback on the results in \cite{Clanin.McNicholl.Stull.2019} and \cite{McNicholl.2017}.
Another consequence of these findings is that when $p \geq 1$ is computable, every separable $L^p$ space is $\mathbf{0''}$-categorical.
The paper is organized as follows. Background and preliminaries are covered in Sections \ref{sec:back} and \ref{sec:prelim}. In Section \ref{sec:projection}, we present results on the complexity of the natural projection maps for spaces of the form $\ell^p_n \oplus L^p[0,1]$ or $\ell^p \oplus L^p[0,1]$. We derive lower bounds on degrees of categoricity in Section \ref{sec:lower} and corresponding upper bounds in Section \ref{sec:upper}. These proofs utilize our results on projection maps in Section \ref{sec:projection}. Finally in Section \ref{sec:conclusion} we summarize our findings and pose questions for further investigation.
\section{Background}\label{sec:back} Here, we cover pertinent notions regarding external and internal direct sums of Banach spaces and the notion of complemented subspaces of an internal direct sum of Banach spaces. We also summarize additional background material from \cite{Clanin.McNicholl.Stull.2019}. We will assume our field of scalars consists of the complex numbers, but all results hold for the field of real numbers as well.
When $S \subseteq \mathbb{N}^*$, let $S\downarrow$ denote the downset of $S$; i.e. the set of all $\nu \in \mathbb{N}^*$ so that $\nu \subseteq \mu$ for some $\mu \in S$.
Suppose $1 \leq p < \infty$. If $\mathcal{B}_0$, $\ldots$, $\mathcal{B}_n$ are Banach spaces, their \emph{$L^p$-sum} consists of the vector space $\mathcal{B}_0 \times \ldots \times \mathcal{B}_n$ together with the norm \[ \norm{(u_0, \ldots, u_n)}_p = \left( \sum_{j = 0}^n \norm{u_j}_{\mathcal{B}_j}^p \right)^{1/p}. \] Thus, the external direct sum of two $L^p$ spaces is their $L^p$-sum. If $\mathcal{B}_j$ is a Banach space for each $j \in \mathbb{N}$, then the $L^p$-sum of $\{\mathcal{B}_j\}_{j \in \mathbb{N}}$ consists of all $f \in \prod_j B_j$ so that $\sum_j \norm{f(j)}_{\mathcal{B}_j}^p < \infty$. This is easily seen to be a Banach space under the norm \[ \norm{f}_p = \left( \sum_j \norm{f(j)}_{\mathcal{B}_j}^p \right)^{1/p}. \]
Suppose $\mathcal{B}$ is a Banach space and $\mathcal{M}$ and $\mathcal{N}$ are subspaces of $\mathcal{B}$. Recall that $\mathcal{B}$ is the \emph{internal direct sum} of $\mathcal{M}$ and $\mathcal{N}$ if $\mathcal{M} \cap \mathcal{N} = \{\mathbf{0}\}$ and $\mathcal{B} = \mathcal{M} + \mathcal{N}$. In this case, $\mathcal{M}$ is said to be \emph{complemented} and $\mathcal{N}$ is said to be the \emph{complement} of $\mathcal{M}$.
When $\mathcal{M}$ is a complemented subspace of $\mathcal{B}$, let $P_{\mathcal{M}}$ denote the associated projection map.
That is, $P_{\mathcal{M}}$ is the unique linear map of $\mathcal{B}$ onto $\mathcal{M}$ so that $P_{\mathcal{M}}(f) = f$ for all $f \in \mathcal{M}$ and $P_{\mathcal{M}}(f) = \mathbf{0}$ for all $f \in \mathcal{N}$.
Note that if $T$ is an isometric isomorphism of $\mathcal{B}_0$ onto $\mathcal{B}_1$, and if $\mathcal{M}$ is a complemented subspace of $\mathcal{B}_0$, then $T[\mathcal{M}]$ is a complemented subspace of $\mathcal{B}_1$ and $P_{T[\mathcal{M}]} = TP_{\mathcal{M}}T^{-1}$.
Suppose $f,g$ are vectors in an $L^p$ space. We say that $f$ and $g$ are \emph{disjointly supported} if the intersection of their supports is null; equivalently, if $f \cdot g = \mathbf{0}$. We say that $f$ is a \emph{subvector} of $g$ if there is a measurable set $A$ so that $f = g \cdot \chi_A$ (where $\chi_A$ is the characteristic function of $A$); equivalently, if $g - f$ and $f$ are disjointly supported. We write $f \preceq g$ if $f$ is a subvector of $g$. It is readily seen that $\preceq$ is a partial order. Also, $f$ is an atom of $\preceq$ if and only if $\operatorname{supp}(f)$ is an atom of the underlying measure space.
A subset $X$ of a Banach space $\mathcal{B}$ is \emph{linearly dense} if its linear span is dense in $\mathcal{B}$.
Suppose $S \subseteq \mathbb{N}^*$ is a tree and $\phi : S \rightarrow L^p(\Omega)$. We say that $\phi$ is \emph{summative} if for every nonterminal node $\nu$ of $S$, $\phi(\nu) = \sum_{\nu'} \phi(\nu')$ where $\nu'$ ranges over the children of $\nu$ in $S$. We say that $\phi$ is \emph{separating} if $\phi(\nu)$ and $\phi(\nu')$ are disjointly supported whenever $\nu, \nu' \in S$ are incomparable. We then say that $\phi$ is a \emph{disintegration} if its range is linearly dense, and if it is injective, non-vanishing, summative, and separating.
Fix a disintegration $\phi : S \rightarrow L^p(\Omega)$. A non-root node $\nu$ of $S$ is an \emph{almost norm-maximizing} child of its parent if \[
\norm{\phi(\nu')}_p^p \leq \norm{\phi(\nu)}_p^p + 2^{-|\nu|} \] whenever $\nu' \in S$ is a sibling of $\nu$. A chain $C \subseteq S$ is \emph{almost norm-maximizing} if for every $\nu \in C$, if $\nu$ has a child in $S$, then $C$ contains an almost norm-maximizing child of $\nu$.
Suppose $\mathcal{B}$ is a Banach space. A \emph{structure} on $\mathcal{B}$ is a surjection of the natural numbers onto a linearly dense subset of $\mathcal{B}$. A \emph{presentation of $\mathcal{B}$} is a pair $(\mathcal{B}, R)$ where $R$ is a structure on $\mathcal{B}$.
Among all presentations of a Banach space $\mathcal{B}$, one may be designated as \emph{standard}; in this case, we will identify $\mathcal{B}$ with its standard presentation. In particular, if $p \geq 1$ is a computable real, and if $D$ is a standard map of $\mathbb{N}$ onto the set of characteristic functions of dyadic subintervals of $[0,1]$, then $(L^p[0,1], D)$ is the standard presentation of $L^p[0,1]$. If $R(n) = 1$ for all $n \in \mathbb{N}$, then $(\mathbb{C}, R)$ is the standard presentation of $\mathbb{C}$ as a Banach space over itself. The standard presentations of $\ell^p$ and $\ell^p_n$ are given by the standard bases for these spaces. The standard presentations of $\ell^p \oplus L^p[0,1]$ and $\ell^p_n \oplus L^p[0,1]$ are defined in the obvious way.
Fix a presentation $\mathcal{B}^\# = (\mathcal{B}, R)$ of a Banach space $\mathcal{B}$. By a \emph{rational vector} of $\mathcal{B}^\#$ we mean a vector of the form $\sum_{j \leq M} \alpha_j R(j)$ where $\alpha_0, \ldots, \alpha_M \in \mathbb{Q}(i)$.
We say $\mathcal{B}^\#$ is \emph{computable} if the norm function is computable on the rational vectors of $\mathcal{B}^\#$. That is, if there is an algorithm that given $\alpha_0, \ldots, \alpha_M \in \mathbb{Q}(i)$ and $k \in \mathbb{N}$ produces a rational number $q$ so that $|\norm{\sum_{j \leq M} \alpha_j R(j)} - q| < 2^{-k}$. The standard definitions just described are all easily seen to be computable. A Banach space is \emph{computably presentable} if it has a computable presentation.
With a presentation $\mathcal{B}^\#$ of a Banach space, there are associated classes of computable vectors and sequences. With a pair $(\mathcal{B}_0^\#, \mathcal{B}_1^\#)$ of presentations of Banach spaces, there is an associated class of computable functions from $\mathcal{B}_0^\#$ into $\mathcal{B}_1^\#$. Definitions of concepts such as these have become fairly well-known; we refer the reader to whom they are unfamiliar to Section 2.2.2 of \cite{Clanin.McNicholl.Stull.2019}.
We will make frequent use of the following result from \cite{Clanin.McNicholl.Stull.2019}.
\begin{theorem}\label{thm:disint.comp} Suppose $p \geq 1$ is a computable real so that $p \neq 2$. Then, every computable presentation of a nonzero $L^p$ space has a computable disintegration. \end{theorem}
The following is essentially proven in \cite{McNicholl.2017}.
\begin{theorem}\label{thm:anm.chains} Suppose $p \geq 1$ is a computable real, and suppose $\mathcal{B}^\#$ is a computable presentation of an $L^p$ space. If $\phi$ is a computable disintegration of $\mathcal{B}^\#$, then there is a partition $\{C_n\}_{n < \kappa}$ (where $\kappa \leq \omega$) of $\operatorname{dom}(\phi)$ into uniformly c.e. almost norm-maximizing chains. \end{theorem}
Degrees of categoricity for countable structures were introduced in \cite{Fokina.Kalimullin.Miller.2010}. Since then, the study of these degrees has given rise to a number of surprising results and very challenging questions; see, for example, \cite{Csima.Franklin.Shore.2013} and \cite{Anderson.Csima.2016}. Any notion of classical computable structure theory can be adapted to the setting of Banach spaces by replacing `isomorphism' with `isometric isomorphism'. Thus, we arrive at the previously given definition of the degree of categoricity for a computably presentable Banach space.
\section{Preliminaries}\label{sec:prelim}
\subsection{Preliminaries from functional analysis}\label{sec:prelim::subsec:FA} Here we establish several preliminary lemmas and theorems from classical functional analysis that will be used later to prove Theorem \ref{thm:main}. We first establish the results needed to locate the $\preceq$-atoms of $L^p(\Omega)$ via the use of almost norm-maximizing chains. We then conclude this section with results regarding disintegrations on complemented subspaces of $L^p(\Omega)$.
The proof of the following is essentially the same as the proof of Proposition 4.1 of \cite{McNicholl.2017}.
\begin{proposition}\label{prop:subvectorLimitsExist}If $g_0 \preceq g_1 \preceq ...$ are vectors in $L^p(\Omega)$, then $\lim_n g_n$ exists in the $L^p$-norm and is the $\preceq$-infimum of $\{g_0,g_1,...\}$. \end{proposition}
The following generalizes Theorem 3.4 of \cite{McNicholl.2017}.
\begin{theorem}\label{thm:limitsAreAtoms} Suppose $\Omega$ is a measure space and $\phi: S \rightarrow L^p(\Omega)$ is a disintegration.\\ \begin{enumerate}
\item If $C \subseteq S$ is an almost norm-maximizing chain, then the $\preceq$-infimum of $\phi[C]$ exists and is either \textbf{0} or an atom of $\preceq$. Furthermore, $\inf\phi[C]$ is the limit in the $L^p$ norm of $\phi(\nu)$ as $\nu$ traverses the nodes in $C$ in increasing order.\label{thm:limitsAreAtoms::itm:inf}
\item If $\{C_n\}_{n=0}^\infty$ is a partition of $S$ into almost norm-maximizing chains, then $\inf\phi[C_0], \inf\phi[C_1], ...$ are disjointly supported. Furthermore, if $A$ is an atom of $\Omega$, then there exists a unique $n$ so that $A$ is the support of $\inf \phi[C_n]$. \label{thm:limitsAreAtoms::itm:unique} \end{enumerate} \end{theorem}
\begin{proof} (\ref{thm:limitsAreAtoms::itm:inf}): Suppose $C \subseteq S$ is an almost norm-maximizing chain. By Proposition \ref{prop:subvectorLimitsExist}, $g:=\inf \phi[C]$ exists and is the limit in the $L^p$-norm of $\phi(\nu)$ as $\nu$ traverses the nodes in $C$ in increasing order.
We claim that $g$ is an atom if it is nonzero. For, suppose $h \preceq g$. Let $\delta = \min\{\norm{g - h}_p^p, \norm{h}_p^p\}$, and let $\epsilon> 0$. Since the range of $\phi$ is linearly dense, there is a finite $S_1 \subseteq S$ and a family of scalars $\{\alpha_\nu\}_{\nu \in S_1}$ so that \[ \norm{\sum_{\nu \in S_1} \alpha_\nu \phi(\nu) - h}_p < \frac{\epsilon}{2}. \] Let $f = \sum_{\nu \in S_1} \alpha_\nu \phi(\nu)$. Then, \begin{eqnarray*} \norm{f - g}_p^p & \geq & \norm{(f - h) \cdot \chi_{\operatorname{supp}(g)}}_p^p \\ & = & \norm{f \cdot \chi_{\operatorname{supp}(g)} - h}_p^p. \end{eqnarray*} Let $S_1^0 = \{\nu \in S_1\ :\ g \preceq \phi(\nu)\}$, and let $\beta = \sum_{\nu \in S_1^0} \alpha_\nu$. Then, since $\phi$ is separating, $f \cdot \chi_{\operatorname{supp}(g)} = \beta g$. However, since $g - h$ and $h$ are disjointly supported, \begin{eqnarray*}
\norm{\beta g - h}_p^p & = & |\beta|^p\norm{g - h}_p^p + |\beta - 1|^p \norm{h}_p^p \\
& \geq & (|\beta|^p + |\beta -1|^p) \delta \\
& \geq & (|\beta|^p + ||\beta| - 1|^p) \delta\\
& \geq & \max\{|\beta|^p, ||\beta| - 1|^p\} \delta\\ & \geq & 2^{-p} \delta. \end{eqnarray*} Thus, $\delta < \epsilon$ for every $\epsilon > 0$. Therefore, $\delta = 0$ and so either $g = h$ or $h = \mathbf{0}$. Thus, $g$ is an atom. \\
(\ref{thm:limitsAreAtoms::itm:unique}): Suppose $C_0,C_1,...$ is a partition of $S$ into almost norm-maximizing chains. By the above, $\inf \phi([C_k])$ exists for each $k$, and so we set $h_k := \inf \phi[C_k]$. We first claim that $h_0, h_1, \ldots$ are disjointly supported vectors. Supposing that $k \neq k'$ it suffices to prove that there are incomparable nodes $\nu_0,\nu_1$ such that $\nu_0 \in C_k$ and $\nu_1 \in C_{k'}$. We do this in two cases.
First, suppose there exist $\nu \in C_k, \nu' \in C_{k'}$ such that $|\nu| = |\nu'|$. Since the chains $C_0,C_1,...$ partition $S$, $\nu \neq \nu'$. Thus, $\nu_0:=\nu$ and $\nu_1:=\nu'$ are incomparable.
Now suppose $|\nu| \neq |\nu'|$ whenever $\nu \in C_k$ and $\nu' \in C_{k'}$.
Let $\nu$ be the $\subseteq$-minimal node in $C_k$ and let $\nu'$ be the $\subseteq$-minimal node in $C_{k'}$. Without loss of generality, assume $|\nu|<|\nu'|$. Then $C_k$ must contain a terminal node $\tau_k$ of $S$, and $|\tau_k|<|\nu'|$. Let $\mu\in S$ be the ancestor of $\nu'$ such that $|\mu|=|\tau_k|$. Note that $\mu \notin C_{k'}$ since $|\mu| < |\nu'|$. Furthermore, $\mu \notin C_k$ either, for $\tau_k$ is terminal in $S$. Therefore, since $|\mu|=|\tau_k|$, $\mu$ and $\tau_k$ are incomparable. From this it follows that $\nu_0:=\nu$ and $\nu_1:=\nu'$ are incomparable.
Now let $A$ be an atom of $\Omega$. If there is a $\preceq$-atom $g$ in $\operatorname{ran}(\phi)$ whose support includes $A$ then there is nothing to show. So suppose that there is no atom in $\operatorname{ran}(\phi)$ whose support includes $A$.
We claim that for each $n \in \mathbb{N}$ there is a $\nu \in S$ so that $|\nu|=n$ and $A \subseteq \operatorname{supp}(\phi(\nu))$. For, suppose otherwise. Since $\phi$ is summative and separating, it follows that $A \not \subseteq \operatorname{supp}(\phi(\nu))$ for all $\nu \in S$. Let $\mu$ denote the measure function of $\Omega$. Then, for any $g \in \operatorname{ran}(\phi)$, $\mu(A\cap\ \operatorname{supp}(g))=0$. Thus $\mu(A) \leq \norm{f -\chi_A}_p$ whenever $f$ belongs to the linear span of $\operatorname{ran}(\phi)$- a contradiction since the range of $\phi$ is linearly dense.
Now let $\nu_s$ denote the node of length $s$ so that $A\subseteq \operatorname{supp}(\phi(\nu_s))$. Let $f = \phi(\emptyset) \cdot \chi_A$. Then, $f \preceq \phi(\nu_s)$ for all $s$.
For each $s$, let $k_s$ denote the $k$ so that $\nu_s\in C_k$. We claim that $\lim_s k_s$ exists. To see this, suppose otherwise. Then we may let $s_0<s_1<...$ be the increasing enumeration of all values of $s$ so that $k_s\neq k_{s+1}$. Since for all $m$, $\nu_{s_m+1} \supset \nu_{s_m}$, $\nu_{s_m}$ is a nonterminal node in $S$. Thus since $C_{k_{s_m}}$ is almost norm-maximizing it must contain a child of $\nu_{s_m}$ in $S$; denote this child by $\mu_m$. Then, $\phi(\mu_m)\preceq \phi(\nu_{s_m})$ and $\phi(\mu_m)$ and $\phi(\nu_{s_m+1})$ are disjointly supported. Also, since $\mu_m$ is an almost norm-maximizing child of $\nu_{s_m}$, $\norm{\phi(\nu_{s_m+1})}^p_p \leq \norm{\phi(\mu_m)}_p^p+2^{-s_m}$. Since $\phi(\mu_{m+r})\preceq \phi(\nu_{s_{m+r}})\preceq \phi(\nu_{s_m+1})$, $\phi(\mu_m)$ and $\phi(\mu_{m+r})$ are disjointly supported if $r>0$. Thus by the above inequality and the summativity of $\phi$ we have
\begin{align*}
\sum_m\norm{\phi(\nu_{s_m+1})}_p^p
&\leq \sum_m \norm{\phi(\mu_m)}_p^p + \sum_{m}2^{-s_m} \\
&=\norm{\sum_m \phi(\mu_m)}_p^p + \sum_{m}2^{-s_m}\\
&\leq \norm{\phi(\emptyset)}_p^p+ \sum_{m}2^{-s_m}\\
&<\infty.
\end{align*}
But since $f \preceq \phi(\nu_{s_m+1})$ for all $m$, $\norm{\phi(\nu_{s_m + 1})}_p^p \geq \norm{f} > 0$ for all $m$- a contradiction.
Therefore, $k:=\lim_s k_s$ exists. Since the chains partition $S$, $C_k$ is the only chain so that $A \subseteq \operatorname{supp}(\phi(\nu))$ for all $\phi(\nu)\in\phi[C_k]$. It follows immediately from part (\ref{thm:limitsAreAtoms::itm:inf}) that $A$ is the support of $\inf \phi[C_k]$. The result now follows. \end{proof}
We say that subspaces $\mathcal{M}$, $\mathcal{N}$ of $L^p(\Omega)$ are disjointly supported if $f$, $g$ are disjointly supported whenever $f \in \mathcal{M}$ and $g \in \mathcal{N}$.
\begin{lemma}\label{lm:proj.disint} Suppose $\phi$ is a disintegration of $L^p(\Omega)$ and that $\mathcal{M}$ is a complemented subspace of $L^p(\Omega)$. Suppose also that $\mathcal{M}$ and its complement are disjointly supported. Then, $P_{\mathcal{M}} \phi$ is summative and separating, and its range is linearly dense in $\mathcal{M}$. \end{lemma}
\begin{proof} Let $P = P_{\mathcal{M}}$, and let $\psi = P\phi$. Since $P$ is linear, it follows that $\psi$ is summative. Since $\mathcal{M}$ and its complement are disjointly supported, it also follows that $\psi(\nu)$ is a subvector of $\phi(\nu)$ for each $\nu \in \operatorname{dom}(\phi)$. We can then infer that $\psi$ is separating.
We now show that the range of $P\phi$ is linearly dense in $\mathcal{M}$. Let $\epsilon>0$. By the linear density of $\phi$ and the disjointness of support of $\mathcal{M}$ and $\mathcal{N}$, for any $f\in\mathcal{M}$ there is a collection of scalars $\{\alpha_\nu\}_{\nu\in S}$ such that \begin{align*} \epsilon^p&>\norm{f-\sum_{\nu\in S}\alpha_\nu \phi(\nu)}_p^p\\ &=\norm{f-\sum_{\nu\in S}\alpha_\nu P(\phi(\nu))-\sum_{\nu\in S}\alpha_\nu P_{\mathcal{N}}(\phi(\nu))}_p^p\\ &=\norm{f-\sum_{\nu\in S}\alpha_\nu P(\phi(\nu))}^p_p+\norm{\sum_{\nu\in S}\alpha_\nu P_{\mathcal{N}}(\phi(\nu))}_p^p\\ &\geq \norm{f-\sum_{\nu\in S}\alpha_\nu P(\phi(\nu))}^p_p. \end{align*} Thus we have that the range of $P \phi$ is linearly dense in $\mathcal{M}$. \end{proof}
\begin{lemma}\label{lm:proj.disint.Lp} Suppose $\phi$ is a disintegration of $\mathcal{M} \oplus L^p[0,1]$ where $\mathcal{M}$ is either $\ell^p$ or $\ell^p_n$. Suppose $\{C_n\}_{n \in \mathbb{N}}$ is a partition of $\operatorname{dom}(\phi)$ into almost norm-maximizing chains and that $g_n = \inf \phi[C_n]$ for all $n$. Then, for each $\nu \in \operatorname{dom}(\phi)$, \[ P_{\{\mathbf{0}\} \oplus L^p[0,1]} \phi(\nu) = \phi(\nu) - \sum_{g_n \preceq \phi(\nu)} g_n. \] \end{lemma}
\begin{proof} Let $\mathcal{B} = \mathcal{M} \oplus L^p[0,1]$, and let $P = P_{\{\mathbf{0}\} \oplus L^p[0,1]}$. For each $f \in \mathcal{B}$, let $\mathcal{A}_f$ denote the set of all atoms $g$ of $\mathcal{B}$ so that $g \preceq f$. Thus, $P(f) = f - \sum_{g \in \mathcal{A}_f} g$. Suppose $g \in \mathcal{A}_{\phi(\nu)}$. Then, $\operatorname{supp}(g)$ is an atom. So, by Theorem \ref{thm:limitsAreAtoms}, $\operatorname{supp}(g) = \operatorname{supp}(g_n)$ for some $n$.
We claim that $\nu \in C_n\downarrow$. For, suppose $\nu \not \in C_n\downarrow$. Let $\nu'$ be the largest node in $C_n\downarrow$ so that $\nu' \subseteq \nu$. Thus, $\nu' \neq \nu$ so and $\nu'$ has a child in $\operatorname{dom}(\phi)$. Therefore, $\nu'$ has a child $\nu''$ in $C_n$ since $C_n$ is almost norm-maximizing. Thus, $\nu''$ and $\nu$ are incomparable. It follows that $g_n$ and $g$ are disjointly supported- a contradiction.
Since $\nu \in C_n\downarrow$, it follows that $g,g_n \preceq \phi(\nu)$. Thus, since $\operatorname{supp}(g) = \operatorname{supp}(g_n)$, $g = g_n$. \end{proof}
\subsection{Preliminaries from computable analysis}\label{sec:prelim::subsec:CA} This subsection essentially effectivizes the notions of the previous subsection and makes explicit the computable presentations we will employ in the proofs of our main theorem and its corollary.
The following is from \cite{Pour-El.Richards.1989}.
\begin{theorem}\label{thm:seq.comp.map} Suppose $\mathcal{B}_0^\#$ and $\mathcal{B}_1^\#$ are computable presentations of Banach spaces $\mathcal{B}_0$ and $\mathcal{B}_1$ respectively and that $T : \mathcal{B}_0^\# \rightarrow \mathcal{B}_1^\#$ is bounded and linear. Then, $T$ is computable if and only if $T$ maps a linearly dense computable sequence of $\mathcal{B}_0^\#$ to a computable sequence of $\mathcal{B}_1^\#$. \end{theorem}
\begin{definition}\label{def:compu.comp} Suppose $\mathcal{B}^\#$ is a computable presentation of a Banach space $\mathcal{B}$, and suppose $\mathcal{M}$ is a complemented subspace of $\mathcal{B}$. We say $\mathcal{M}$ is a \emph{computably complemented subspace of $\mathcal{B}$} if $P_\mathcal{M}$ is a computable map of $\mathcal{B}^\#$ into $\mathcal{B}^\#$. \end{definition}
We relativize this notion in the obvious way.
\begin{proposition}\label{prop:proj.comp} Suppose $\mathcal{B}_j^\#$ is a computable presentation of a Banach space $\mathcal{B}_j$ for each $j \in \{0,1\}$, and suppose $T$ is an $X$-computable isometric isomorphism of $\mathcal{B}_0^\#$ onto $\mathcal{B}_1^\#$. If $\mathcal{M}$ is a computably complemented subspace of $\mathcal{B}_0^\#$, then $T[\mathcal{M}]$ is an $X$-computably complemented subspace of $\mathcal{B}_1^\#$. \end{proposition}
\begin{proof} This is clear from the fact that $P_{T[\mathcal{M}]} = TP_{\mathcal{M}}T^{-1}$. \end{proof}
\begin{lemma}\label{lm:disintsArePresentations} Let $p \geq 1$ be computable. Suppose $S$ is a tree, and suppose $\phi : S \rightarrow L^p(\Omega)$ is summative and separating. Suppose also that $\operatorname{ran}(S)$ is linearly dense and that $\nu \mapsto \norm{\phi(\nu)}_p$ is computable.
Let $R = \phi h$ where $h$ is a computable surjection of $\mathbb{N}$ onto $S$. Then, $(L^p(\Omega), R)$ is a computable presentation of $L^p(\Omega)$. \end{lemma}
\begin{proof}
Since $\operatorname{ran}(\phi)$ is linearly dense, it follows that $R$ is a structure on $L^p(\Omega)$ and that $L^p(\Omega)^\# := (L^p(\Omega),R)$ is a presentation of $L^p(\Omega)$.
Now we must demonstrate that this presentation is computable. That is, we must show that the norm function is computable on the rational vectors of $L^p(\Omega)^\#$. So, suppose $\alpha_0, \ldots, \alpha_M \in \mathbb{Q}(i)$ are given, and let $f = \sum_j \alpha_j R(j)$. Compute a finite tree $F \subseteq S$ so that $R(j) \in F$ for each $j \leq M$. For each $\nu \in F$, let $\alpha_\nu = \sum_{h(j) = \nu} \alpha_j$. Thus, $\sum_j \alpha_j R(j) = \sum_\nu \alpha_\nu \phi(\nu)$. Let $\beta_0, \ldots, \beta_k$ denote the leaf nodes of $F$. Thus, $\operatorname{supp}(f) = \bigcup_j \operatorname{supp}(\phi(\beta_j))$. Therefore, \begin{eqnarray*} \norm{f}_p^p & = & \sum_j \norm{f \cdot \chi_{\operatorname{supp}(\phi(\beta_j))}}_p^p\\ & = & \sum_j \norm{ \left(\sum_{\nu \subseteq \beta_j} \alpha_\nu\right) \phi(\beta_j)}_p^p\\
& = & \sum_j \left| \sum_{\nu \subseteq \beta_j} \alpha_\nu \right|^p \norm{\phi(\beta_j)}_p^p. \end{eqnarray*} Since $\nu \mapsto \norm{\phi(\nu)}_p$ is computable, it follows that $\norm{f}_p$ can be computed from $\alpha_0, \ldots, \alpha_M$.
\end{proof}
\section{Complexity of projection maps}\label{sec:projection} Here we establish the complexity of projection maps on the spaces $\ell^p_n\oplus L^p[0,1]$ and $\ell^p\oplus L^p[0,1]$ respectively. The main theorem of this section is the core of our argument yielding the upper bounds for each of the aforementioned spaces.
\begin{theorem}\label{thm:proj.Lp.comp} Let $p \geq 1$ be a computable real besides $2$. Suppose $\mathcal{M}$ is either $\ell^p_n$ or $\ell^p$, and suppose $(\mathcal{M} \oplus L^p[0,1])^\#$ is a computable presentation of $\mathcal{M} \oplus L^p[0,1]$. \begin{enumerate}
\item If $\mathcal{M} = \ell^p_n$, then $P_{\{\mathbf{0}\} \oplus L^p[0,1]}$ is a
$\emptyset'$-computable map of $(\mathcal{M} \oplus L^p[0,1])^\#$ into
$(\mathcal{M} \oplus L^p[0,1])^\#$.
\item If $\mathcal{M} = \ell^p$, then $P_{\{\mathbf{0}\}\oplus L^p[0,1]}$ is a
$\emptyset''$-computable map of $(\mathcal{M} \oplus L^p[0,1])^\#$ into
$(\mathcal{M} \oplus L^p[0,1])^\#$. \end{enumerate} \end{theorem}
\begin{proof} Let $\mathcal{B} = \mathcal{M} \oplus L^p[0,1]$, and let $\phi$ be a computable disintegration of $\mathcal{B}^\#$. Set $S = \operatorname{dom}(\phi)$. Abbreviate $P_{\{\mathbf{0}\} \oplus L^p[0,1]}$ by $P$.
Let $\mathcal{B}^\# = (\mathcal{B}, R)$. Fix a computable surjection $h$ of $\mathbb{N}$ onto $S$, and set $R'(j) = \phi(h(j))$. Let $\mathcal{B}^+ = (\mathcal{B}, R')$. By Lemma \ref{lm:disintsArePresentations}, $\mathcal{B}^+$ is a computable presentation of $\mathcal{B}$. Furthermore, since $R'$ is a computable sequence of $\mathcal{B}^\#$, it follows from Theorem \ref{thm:seq.comp.map} that $\mathcal{B}^\#$ is computably isometrically isomorphic to $\mathcal{B}^+$ (namely, by the identity map).
By Theorem \ref{thm:limitsAreAtoms}, there is a partition $\{C_j\}_{j \in \mathbb{N}}$ of $S$ into almost norm-maximizing chains. Let $g_j = \inf \phi[C_j]$. By Lemma \ref{lm:proj.disint.Lp}, \[ P(\phi(\nu)) = \phi(\nu) - \sum_{g_j \preceq \phi(\nu)} g_j. \]
Let $U_\nu = \{j \in \mathbb{N}\ :\ \nu \in C_j\downarrow\}$. We first claim that \[ \sum_{g_j \preceq \phi(\nu)} g_j = \sum_{j \in U_\nu} g_j. \] For, if $j \in U_\nu$, then $g_j \preceq \phi(\nu)$. Suppose $j \not \in U_\nu$ and $g_j \preceq \phi(\nu)$. Let $\mu_0$ be the maximal element of $C_n\downarrow$ so that $\mu_0 \subseteq \nu$. Thus, $\mu_0 \subset \nu$, and so $\mu_0$ has a child $\mu'$ in $S$. Therefore $g_j \preceq \phi(\mu'), \phi(\nu)$. Since $\phi$ is separating, it follows that $g_j = \mathbf{0}$.
Now, suppose $\mathcal{M} = \ell^p_n$. We obtain from Theorem \ref{thm:limitsAreAtoms}, that there are exactly $n$ values of $j$ so that $g_j$ is nonzero. So, let $D = \{j\ : g_j \neq \mathbf{0}\}$. Then, $\phi(\nu) - P(\phi(\nu)) = \sum_{j \in U_\nu \cap D} g_j$. It then follows from Theorem \ref{thm:limitsAreAtoms} that $\{g_j\}_{j \in \mathbb{N}}$ is a $\emptyset'$-computable sequence of $\mathcal{B}^\#$. Thus, $\{P(R'(j))\}_{j \in \mathbb{N}}$ is a $\emptyset'$-computable sequence of $\mathcal{B}^+$. Therefore, by the relativization of Theorem \ref{thm:seq.comp.map}, $P$ is a $\emptyset'$-computable map of $\mathcal{B}^+$ into $\mathcal{B}^+$. But, since $\mathcal{B}^\#$ is computably isometrically isomorphic to $\mathcal{B}^+$, $P$ is also a $\emptyset'$-computable map of $\mathcal{B}^\#$ into $\mathcal{B}^\#$.
Now, suppose $\mathcal{M} = \ell^p$. For each $\nu \in S$, let $h_\nu = \phi(\nu) - P(\phi(\nu))$. Since $\{g_j\}_{j \in U_\nu}$ is a summable sequence of disjointly supported vectors, $\sum_{j \in U_\nu} \norm{g_j}_p^p < \infty$. Moreover, since $\{g_j\}_{j \in U_\nu}$ is a $\emptyset'$-computable sequence of $\mathcal{B}^\#$, it follows that $\sum_{j \in U_\nu} \norm{g_j}_p^p$ is $\emptyset''$-computable uniformly in $\nu$. Observe that for each $N \in \mathbb{N}$, \begin{eqnarray*} \norm{\sum_{j \in U_\nu \cap [0,N]} g_j - h_\nu}_p^p & = & \norm{\sum_{j \in U_\nu \cap [N+1, \infty)} g_j}_p^p\\ & = & \sum_{j \in U_\nu \cap [N+1, \infty)} \norm{g_j}_p^p. \end{eqnarray*} From this we obtain that $h_\nu$ is a $\emptyset''$-computable vector of $\mathcal{B}^\#$ uniformly in $\nu$. It then follows that $\{P(R'(j))\}_{j \in \mathbb{N}}$ is a $\emptyset''$-computable sequence of $\mathcal{B}^+$ and so $P$ is a $\emptyset''$-computable map of $\mathcal{B}^+$ into $\mathcal{B}^+$. \end{proof}
The sharpness of the bounds in Proposition \ref{prop:proj.comp} will be demonstrated in Section \ref{sec:lower}.
\section{Upper bound results}\label{sec:upper} Here we will use the complexity of projection maps described in the previous section to produce the upper bounds for the degree of categoricity of $\ell^p_n\oplus L^p[0,1]$ and $\ell^p\oplus L^p[0,1]$ respectively.
\begin{theorem}\label{thm:upper} Suppose $p \geq 1$ is a computable real so that $p \neq 2$. Then, $\ell^p_n \oplus L^p[0,1]$ is $\emptyset'$-categorical, and $\ell^p \oplus L^p[0,1]$ is $\emptyset''$-categorical. \end{theorem}
\begin{proof} Suppose $\mathcal{A}$ is either $\ell^p_n$ or $\ell^p$, and let $\mathcal{B} = \mathcal{A} \oplus L^p[0,1]$. Let $\mathcal{B}^\#$ be a computable presentation of $\mathcal{B}$, and let $\phi$ be a computable disintegration of $\mathcal{B}^\#$. Let $S = \operatorname{dom}(\phi)$. Since $\mathcal{B}$ is infinite-dimensional, $S$ is infinite. Fix a computable surjection $h$ of $\mathbb{N}$ onto $S$. Let $\mathcal{M} = \mathcal{A} \oplus \{\mathbf{0}\}$, and let $\mathcal{N} = \{\mathbf{0}\} \oplus L^p[0,1]$. In addition, let $P = P_{\mathcal{N}}$.
We first claim that there is a $\emptyset'$-computable map $T_1 : \mathcal{A} \rightarrow \mathcal{B}^\#$ so that $\operatorname{ran}(T_1) = \mathcal{M}$. For suppose $\mathcal{A} = \ell^p_n$. Then, by Theorem \ref{thm:proj.Lp.comp}, $P$ is a $\emptyset'$-computable map of $\mathcal{B}^\#$ into $\mathcal{B}^\#$. Let $\mathcal{M}^\# = (\mathcal{M}, (I-P)\phi h)$. By the relativization of Lemma \ref{lm:disintsArePresentations}, $\mathcal{M}^\#$ is a $\emptyset'$-computable presentation of $\mathcal{M}$. In Section 6 of \cite{McNicholl.2017}, it is shown that $\ell^p_n$ is computably categorical. So, by relativizing this result, there is a $\emptyset'$-computable isometric isomorphism $T_1$ of $\ell^p_n$ onto $\mathcal{M}^\#$. Since $(I - P)\phi h$ is a $\emptyset'$-computable sequence of $\mathcal{B}^\#$, by the relativization of Theorem \ref{thm:seq.comp.map}, $T_1$ is a $\emptyset'$-computable map of $\ell^p_n$ into $\mathcal{B}^\#$.
Now, suppose $\mathcal{A} = \ell^p$. By Theorem \ref{thm:anm.chains}, there is a partition $\{C_n\}_{m < \kappa}$ of $S$ into uniformly c.e. almost norm-maximizing chains; since $S$ is infinite, it follows that $\kappa = \omega$. Let $g_n = \inf \phi[C_n]$. Then, there is a $\emptyset'$-computable one-to-one enumeration $\{n_k\}_{k = 0}^\infty$ of all $n$ so that $g_n$ is nonzero. By Theorem \ref{thm:limitsAreAtoms}, for each $j \in \mathbb{N}$, there is a unique $k$ so that $\{j\} = \operatorname{supp}(g_{n_k})$. Let $T_1$ be the unique linear map of $\ell^p$ into $\mathcal{N}$ so that $T_1(e_k) = \norm{g_{n_k}}_p^{-1} g_{n_k}$ for all $k$. Since the $g_{n_k}$'s are disjointly supported, it follows that $T_1$ is isometric. It follows from the relativization of Theorem \ref{thm:seq.comp.map} that $T_1$ is a $\emptyset'$-computable map of $\ell^p$ into $\mathcal{B}^\#$.
We now claim that if $\mathcal{A} = \ell^p_n$, then there is a $\emptyset'$-computable map $T_2$ of $L^p[0,1]$ into $\mathcal{B}^\#$ so that $\operatorname{ran}(T_2) = \mathcal{N}$. For, in this case, by Theorem \ref{thm:proj.Lp.comp}, $P$ is a $\emptyset'$-computable map of $\mathcal{B}^\#$ into $\mathcal{B}^\#$. Thus, by Lemma \ref{lm:proj.disint}, $\mathcal{N}^\# = (\mathcal{N}, P \phi h)$ is a $\emptyset'$-computable presentation of $\mathcal{N}$. So by the relativization of Theorem 1.1 of \cite{Clanin.McNicholl.Stull.2019}, there is a $\emptyset'$-computable isometric isomorphism $T_2$ of $L^p[0,1]$ onto $\mathcal{N}^\#$. Since $P \phi h$ is a $\emptyset'$-computable sequence of $\mathcal{B}^\#$, $T_2$ is a $\emptyset'$-computable map of $L^p[0,1]$ into $\mathcal{B}^\#$ by the relativization of Theorem \ref{thm:seq.comp.map}.
It similarly follows that when $\mathcal{A} = \ell^p$, there is a $\emptyset''$-computable map $T_2$ of $L^p[0,1]$ into $\mathcal{B}^\#$ so that $\operatorname{ran}(T_2) = \mathcal{N}$.
We now form a map $T$ by gluing the maps $T_1$ and $T_2$ together. Namely, when, $v \in \mathcal{A}$ and $f \in L^p[0,1]$, let $T(v,f) = T_1(v) + T_2(f)$. Thus, $T$ is an isometric automorphism of $\mathcal{B}$. If $\mathcal{A} = \ell^p_n$, then $T$ is a $\emptyset'$-computable map of the standard presentation of $\mathcal{B}$ onto $\mathcal{B}^\#$; otherwise it is a $\emptyset''$-computable map between these presentations. \end{proof}
\section{Lower bound results}\label{sec:lower} In each of the following subsections we construct ill-behaved computable presentations for $\ell^p_n\oplus L^p[0,1]$ and $\ell^p\oplus L^p[0,1]$ respectively. We then show that any oracle that computes a linear isometric isomorphism between each constructed presentation and its standard copy must also compute \textbf{d}, where \textbf{d} is any c.e. degree in the $\ell^p_n\oplus L^p[0,1]$ case and $\mathbf{d}$ is the degree of $\emptyset''$ in the $\ell^p\oplus L^p[0,1]$ case.
\subsection{The finitely atomic case}\label{sec:lower::subsec:lpn}
We complete our proof of Theorem \ref{thm:main}.\ref{thm:main::itm:finite} by establishing the following.
\begin{theorem}\label{thm:lpn.lower} Suppose $p \geq 1$ is computable and $p \neq 2$. Let $\mathbf{d}$ be a c.e. degree. Then, there is a computable presentation $(\ell^p_n \oplus L^p[0,1])^\#$ of $\ell^p_n \oplus L^p[0,1]$ so that every degree that computes an isometric isomorphism of $\ell^p \oplus L^p[0,1]$ onto $(\ell^p \oplus L^p[0,1])^\#$ also computes $\mathbf{d}$. \end{theorem}
Let $\mathcal{B} = \ell^p_n \oplus L^p[0,1]$. We construct $\mathcal{B}^\#$ as follows. We first construct a disintegration $\phi$ of $\mathcal{B}$. Let $\gamma\in(0,1)$ be a left-c.e. real so that the left Dedekind cut of $\gamma$ has Turing degree $\mathbf{d}$. Let $\{q_n\}$ be a computable and increasing sequence of positive rational numbers so that $\lim_j q_j = \gamma$. Let $c = 1 - \gamma + q_0$. Define \begin{eqnarray*} a((1)) & = & 1 - c\\ b((1)) & = & 1\\ a({(0)^{j+1}} ^\frown(1)) & = & \gamma - q_j\\ b({(0)^{j+1}}^\frown(1)) & = & \gamma - q_{j-1}. \end{eqnarray*} Assuming $a(\nu)$ and $b(\nu)$ have been defined, set $a(\nu^\frown(0)) = a(\nu)$, $a(\nu^\frown(1)) = b(\nu^\frown(0)) = \frac{1}{2} (a(\nu) + b(\nu))$, and $b(\nu^\frown(1)) = b(\nu)$.
Now, let: \begin{eqnarray*} \phi(\emptyset) & = & ((1 - \gamma)^{1/p} e_0 + e_1 + \ldots + e_{n-1}, \chi_{[0, 1 - c]} + c^{-1/p}\chi_{[1-c,1]})\\ \phi((0)^{j+1}) & = & ((1 - \gamma)^{1/p}e_0, \chi_{[\gamma - q_j, 1]})\\ \phi((j)) & = & (e_{j-1}, \mathbf{0})\ \mbox{if $2 \leq j < n$}\\ \phi(\mu) & = & c^{-1/p}(\mathbf{0}, \chi_{[a(\mu), b(\mu)]})\ \mbox{if $(1) \subseteq \mu$}\\ \phi(\mu) & = & (\mathbf{0}, \chi_{[a(\mu), b(\mu)]})\ \mbox{if ${(0)^{j+1}}^\frown(1) \subseteq \mu$} \end{eqnarray*}
\begin{lemma}\label{lm:phi.disint} $\phi$ is a disintegration of $\mathcal{B}$. \end{lemma}
\begin{proof} By construction, $\phi$ is summative, separating, injective, and never zero. It only remains to show that $\operatorname{ran}(\phi)$ is linearly dense. By construction, $(e_j, \mathbf{0}) \in ran(\phi)$ when $1 \leq j \leq n - 1$. So, it is enough to show that $(e_0,\mathbf{0}) \in \langle \operatorname{ran}(\phi) \rangle$ and that $(0,\chi_I) \in \langle \operatorname{ran}(\phi)\rangle$ for every closed interval $I \subseteq [0,1]$.
Let $\epsilon >0$ be given. There is a $K\in \mathbb{N}$ so that $|\gamma-q_K|/|1 - \gamma| < \epsilon^p$. Furthermore,
\begin{align*}
\norm{(e_0, \textbf{0}) - \frac{1}{(1-\gamma)^{1/p}} \phi((0)^{K+1})}_p^p&=\norm{(\mathbf{0},-\frac{1}{(1-\gamma)^{1/p}}\chi_{[0,(\gamma - q_K)]})}_p^p\\
&=\frac{|\gamma-q_K|}{|1-\gamma|}\\
&<\epsilon^p
\end{align*}
Therefore $(e_0,\mathbf{0}) \in \langle \operatorname{ran}(\phi)\rangle$.
Let $\mathcal{M} = \{\mathbf{0}\} \oplus L^p[0,1]$, and let $E(f) = \operatorname{supp}(P_\mathcal{M}(f))$. By construction, for each $f \in \operatorname{ran}(\phi)$, $\chi_{E(f)}$ belongs to the linear span of $\operatorname{ran}(\phi)$. By induction, \[
\bigcup_{|\nu| = j} E(\phi(\nu)) = [0,1]. \]
Since $\phi$ is separating, $\{E(\phi(\nu))\}_{|\nu| = j}$ is a partition of $[0,1]$. Let $L_\nu$ denote the
length of $E(\phi(\nu))$. Then, by construction, $\lim_j \max_{|\nu| = j} L_\nu = 0$. It follows that if $I \subseteq [0,1]$ is a closed interval, then $\chi_I$ belongs to the closed linear span of $\operatorname{ran}(\phi)$. \end{proof}
\begin{lemma}\label{lm:norm.phi.comp} $\nu \mapsto \norm{\phi(\nu)}_p$ is computable. \end{lemma}
\begin{proof} We have: \begin{eqnarray*} \norm{\phi(\emptyset)}_p^p & = & n+1 - q_0\\ \norm{\phi((0)^{j+1})}_p^p& = & 1 - q_j\\ \end{eqnarray*} If $2 \leq j < n$, then $\norm{\phi((j))}_p^p = 1$. If $(1) \subseteq \mu$, then
$\norm{\phi(\mu)}_p^p = c^{-1} c (b(\mu) - a(\mu)) = 2^{-|\mu| + 1}$. Moreover, if ${(0)^{j+1}}^\frown(1) \subseteq \mu$, then
$\norm{\phi(\mu)}_p^p = b(\mu) - a(\mu) = 2^{-|\mu| + j + 2}(q_j - q_{j-1})$. Thus, $\nu \mapsto \norm{\phi(\nu)}_p$ is computable. \end{proof}
Therefore, $\mathcal{B}^\#$ is a computable presentation of $\mathcal{B}$ by Lemma \ref{lm:disintsArePresentations}.
\begin{lemma}\label{lm:proj} If the projection $P_{\langle e_0 \rangle \oplus \{\mathbf{0}\}}$ is an $X$-computable map of $\mathcal{B}^\#$ into $\mathcal{B}^\#$, then $X$ computes $\mathbf{d}$. \end{lemma}
\begin{proof} Let $P = P_{\langle e_0 \rangle \oplus \{\mathbf{0}\}}$. Suppose $P$ is an $X$-computable map of $\mathcal{B}^\#$ into $\mathcal{B}^\#$. Let $f = \phi((0))$. Thus, $f$ is a computable vector of $\mathcal{B}^\#$, and so $X$ computes $(1 - \gamma)^{1/p} = \norm{P(f)}_p$. Therefore, $X$ computes $\gamma$ and so $X$ computes $\mathbf{d}$. \end{proof}
Suppose $X$ computes an isometric isomorphism $T$ of $\mathcal{B}$ onto $\mathcal{B}^\#$. Since $T$ preserves the subvector ordering, there is a $j_0$ so that $T((e_{j_0}, \mathbf{0}))$ is a nonzero scalar multiple of $(e_0, \mathbf{0})$. Let $\mathcal{M} = \langle (e_{j_0}, \mathbf{0}) \rangle$. Then, $T[\mathcal{M}] = \langle (e_0, \mathbf{0}) \rangle$. Let $P = P_{T[\mathcal{M}]}$. By Theorem \ref{thm:proj.Lp.comp}, $P$ is an $X$-computable map of $\mathcal{B}^\#$ into itself. Let $f = \phi((0))$. Thus, $f$ is a computable vector of $\mathcal{B}^\#$. Hence, $\norm{P(f)}_p = (1 - \gamma)^{1/p}$ is $X$-computable. Therefore, $X$ computes $\gamma$, and so $X$ computes $\mathbf{d}$.
Note that we have also established the sharpness of the bound in Theorem \ref{thm:proj.Lp.comp}.
\subsection{The infinitely atomic case}\label{sec:lower::subsec:lp}
We complete our proof of Theorem \ref{thm:main} by proving the following.
\begin{theorem}\label{thm:lpLp01.lower} Suppose $p \geq 1$ is computable and $p \neq 2$. There is a computable presentation $\mathcal{B}^\#$ of $\ell^p \oplus L^p[0,1]$ so that every oracle that computes an isometric isomorphism of $\ell^p \oplus L^p[0,1]$ onto $\mathcal{B}^\#$ also computes $\emptyset''$. \end{theorem}
We construct $\mathcal{B}$ as follows. Let \[ m_e = \left\{\begin{array}{ll} \# W_e & \mbox{\ if $e \in \operatorname{Fin}$;}\\ \omega & \mbox{\ otherwise.}\\ \end{array} \right. \] For each $e \in \mathbb{N}$, let \[ \mathcal{B}_e = \left\{ \begin{array}{ll} \ell^p_{2^{m_e}} & \mbox{\ if $e \in \operatorname{Fin}$;}\\ L^p[0,1] & \mbox{\ otherwise.}\\ \end{array} \right. \] Let $\mathcal{B}$ be the $L^p$ sum of $\{\mathcal{B}_e\}_{e \in \mathbb{N}}$. Let $\iota_e$ be the natural injection of $\mathcal{B}_e$ into $\mathcal{B}$.
We now build a presentation of $\mathcal{B}$ via the construction of a disintegration $\phi$ of $\mathcal{B}$. Let \[ S = \omega^{\leq 1}\ \cup\ \{(e)^\frown\alpha\ :\ \alpha \in \{0,1\}^{< m_e}\}. \] Thus, $S$ is c.e.. Let \[ g_e = \left\{\begin{array}{ll} 2^{-m_e/p} \sum_{j < 2^{m_e}} e_j & \mbox{\ if $e \in \operatorname{Fin}$;}\\ \chi_{[0,1]} & \mbox{\ otherwise.}\\ \end{array} \right. \] Let $f_e = \iota_e(g_e)$. For each $e$ we let $\phi((e)) = 2^{-(e+1)}f_e$.
For each $\nu \in S - \{\emptyset\}$, we recursively define a set $I(\nu)$ as follows. For each $e \in \mathbb{N}$, let \[ I((e)) = \left\{ \begin{array}{ll}
\{0, \ldots, 2^{m_e} - 1\} & \mbox{\ if $e \in \operatorname{Fin}$;}\\ \relax
[0,1] & \mbox{\ otherwise.}\\
\end{array}
\right. \] Suppose $\nu \in S$ and $I(\nu)$ has been defined. Let $a(\nu) = \min I(\nu)$, and let $b(\nu) = \max I(\nu)$. Let $e = \nu(0)$. If $e \not \in \operatorname{Fin}$, let: \begin{eqnarray*} I(\nu^\frown(0)) & = & [a(\nu), 2^{-1}(a(\nu) + b(\nu))]\\ I(\nu ^\frown(1)) & = & [2^{-1}(a(\nu) + b(\nu)), b(\nu)] \end{eqnarray*}
If $e \in \operatorname{Fin}$, and if $|\nu| + 1 < m_e$, let: \begin{eqnarray*} I(\nu^\frown(0)) & = & \{a(\nu), \ldots, a(\nu) + \frac{1}{2}\#I(\nu) - 1\}\\ I(\nu^\frown(1)) & = & \{ \frac{1}{2}\#I(\nu), \ldots, b(\nu)\} \end{eqnarray*}
When $\nu \in S$, let \[ \phi(\nu) = \left\{ \begin{array}{ll} \sum_e 2^{-(e+1)} f_e & \mbox{\ if $\nu = \emptyset$;}\\ 2^{-(\nu(0)+1)}f_{\nu(0)} \cdot \chi_{I(\nu)} & \mbox{\ otherwise.}\\ \end{array} \right. \] Let $h$ be a computable surjection of $\mathbb{N}$ onto $S$, and let $\mathcal{B}^\# = (\mathcal{B}, \phi h)$.
We divide the verification of our construction into the following lemmas. Let $U = \sum_{e \in \operatorname{Fin}} \iota_e(\mathcal{B}_e)$, and let $V = \sum_{e \not \in \operatorname{Fin}} \iota_e(\mathcal{B}_e)$.
\begin{lemma}\label{lm:isom.Lp} $\mathcal{B}$ is isometrically isomorphic to $\ell^p \oplus L^p[0,1]$. \end{lemma}
\begin{proof} Note that $\mathcal{B} = U + V$.
If $e \in \operatorname{Fin}$, then $\mathcal{B}_e$ is a finite-dimensional $L^p$ space.
So, $U$ is isometrically isomorphic to $\ell^p$.
If $e \not \in \operatorname{Fin}$, then $\mathcal{B}_e = L^p[0,1]$.
So, $V$ is the $L^p$-sum of $L^p[0,1]$ with itself $\aleph_0$ times.
However, this is the same thing as $L^p(\Omega)$ where $\Omega$ is the
product of Lebesgue measure on $[0,1]$ with itself $\aleph_0$ times.
As discussed in the introduction, this implies that $V$ is isometrically isomorphic to
$L^p[0,1]$. \end{proof}
It follows from the construction that $\phi$ is a disintegration of $\mathcal{B}$.
\begin{lemma}\label{cl:norm.phi.comp} $\nu \mapsto \norm{\phi(\nu)}_{\mathcal{B}}$ is computable. \end{lemma}
\begin{proof} By construction, $\norm{\phi((e))}_{\mathcal{B}} = 2^{-(e+1)}$ for each $e$. Thus, since $\phi$ is summative, $\phi(\emptyset) = 1$. If $\nu'$ is a child of $\nu$ in $S$, then by construction $\norm{\phi(\nu')}_{\mathcal{B}}^p = \frac{1}{2} \norm{\phi(\nu)}_{\mathcal{B}}^p$. It follows that $\nu \mapsto \norm{\phi(\nu)}_{\mathcal{B}}$ is computable. \end{proof}
It now follows from Lemma \ref{lm:disintsArePresentations} that $\mathcal{B}^\#$ is a computable presentation of $\mathcal{B}$.
\begin{lemma}\label{lm:decomp} If $T$ is an isometric isomorphism of $\ell^p \oplus L^p[0,1]$ onto $\mathcal{B}$, then
$T[\ell^p \oplus \{\mathbf{0}\}] =U$ and $T[\{\mathbf{0}\} \oplus L^p[0,1]] = V$. \end{lemma}
\begin{proof} Suppose $T$ is an isometric isomorphism of $\ell^p \oplus L^p[0,1]$ onto $\mathcal{B}$. Let $U' = T[\ell^p \oplus \{\mathbf{0}\}]$, and let $V' = T[\{\mathbf{0}\} \oplus L^p[0,1]]$. Thus, $\mathcal{B}$ is the internal direct sum of $U'$ and $V'$.
Suppose $j \in \mathbb{N}$, and let $T((e_j, \mathbf{0})) = f + g$ where $f \in U$ and $g \in V$. Since $T((e_j, \mathbf{0}))$ is an atom of $\mathcal{B}$, and since there are no atoms in $V$, it follows that $g = 0$ and so $T((e_j, \mathbf{0})) \in U$. We can then conclude that $U' \subseteq U$. Conversely, suppose $e \in \operatorname{Fin}$ and $h = \iota_e(e_j)$. Then, $T^{-1}(h)$ is an atom of $\ell^p \oplus L^p[0,1]$ and so $T^{-1}(h) \in \ell^p \oplus \{\mathbf{0}\}$. It follows that $\mathcal{B}_e \subseteq U'$ and so $U \subseteq U'$.
Since $\mathcal{B}$ is the internal direct sum of $U$ and $V$, it now follows that $V = V'$. \end{proof}
Let $P = P_V$.
\begin{lemma}\label{lm:comp.P} If $P$ is an $X$-computable map from $\mathcal{B}^\#$ into $\mathcal{B}^\#$, then $X$ computes $\operatorname{Fin}$. \end{lemma}
\begin{proof}
Suppose $X$ computes $P$ from $\mathcal{B}^\#$ into $\mathcal{B}^\#$. If $\nu=(e)$, note that
\[
P(\phi(\nu)) = \left\{\begin{array}{ll}
2^{-(e+1)}f_e & \mbox{\ if $e\not \in \operatorname{Fin}$;}\\
\mathbf{0} & \mbox{\ otherwise.}\\
\end{array}
\right.
\]
and
\[
\norm{P(\phi(\nu))}^p_\mathcal{B} = \left\{\begin{array}{ll}
2^{-(e+1)} & \mbox{\ if $e \not \in \operatorname{Fin}$;}\\
0 & \mbox{\ otherwise.}\\
\end{array}
\right.
\]
Given $e \in \mathbb{N}$, we can compute with oracle $X$ a rational number $q$ so that
$| \norm{P(\phi((e)))}_{\mathcal{B}}^p - q| < 2^{-(e+3)}$.
If $|q| < 2^{-(e+2)}$, then $\norm{P(\phi((e)))}_{\mathcal{B}}^p < 2^{-(e+1)}$ and so $e \in \operatorname{Fin}$. Otherwise, $\norm{P(\phi(\nu))}_{\mathcal{B}}^p \neq 0$ and so $e \not \in \operatorname{Fin}$. \end{proof}
Theorem \ref{thm:lpLp01.lower} now follows from Proposition \ref{prop:proj.comp}. Note that we have also demonstrated the sharpness of the bounds in Theorem \ref{thm:proj.Lp.comp}.
\section{Conclusion}\label{sec:conclusion}
Suppose $p \geq 1$ is a computable real with $p \neq 2$. We have now classified the computably categorical $L^p$ spaces and determined the degrees of categoricity of those that are not computably categorical. Our results relate the degree of categoricity of an $L^p$ space to the structure of the underlying measure space. We have also determined the complexity of the natural projection operators on these spaces as well as their relationship to the degrees of categoricity. In addition, we have provided the first example of a $\emptyset''$-categorical Banach space that is not $\emptyset'$-categorical. This result leads to the following.
\begin{question} If $n \in \mathbb{N}$ and $n \geq 2$, is there is a $\emptyset^{(n+1)}$-categorical Banach space that is not $\emptyset^{(n)}$-categorical? \end{question}
We note that Melnikov and Nies have shown that each compact computable metric space is $\emptyset''$-categorical and that there is a compact computable Polish space that is not $\emptyset'$-categorical \cite{Melnikov.Nies.2013}.
We have shown that the degrees of $\emptyset$, $\emptyset'$, and $\emptyset''$ are degrees of categoricity of Banach spaces. These results lead to the following.
\begin{question} Is every hyperarithmetical degree the degree of categoricity of a Banach space? \end{question}
\begin{question} Is there a Banach space that does not have a degree of categoricity? \end{question}
\end{document} |
\begin{document}
\topical[Entanglement purification and QEC]{Entanglement purification and quantum error correction}
\author{W. D\"ur$^{1,2}$ and H. J. Briegel$^{1,2}$}
\address{$^1$ Institut f{\"u}r Theoretische Physik, Universit{\"a}t Innsbruck, Technikerstra{\ss}e 25, A-6020 Innsbruck, Austria\\ $^2$ Institut f\"ur Quantenoptik und Quanteninformation der \"Osterreichischen Akademie der Wissenschaften, Innsbruck, Austria.}
\date{\today}
\begin{abstract} We give a review on entanglement purification for bipartite and multipartite quantum states, with the main focus on theoretical work carried out by our group in the last couple of years. We discuss entanglement purification in the context of quantum communication, where we emphasize its close relation to quantum error correction. Various bipartite and multipartite entanglement purification protocols are discussed, and their performance under idealized and realistic conditions is studied. Several applications of entanglement purification in quantum communication and computation are presented, which highlights the fact that entanglement purification is a fundamental tool in quantum information processing. \end{abstract}
\pacs{03.67.-a,03.67.Mn, 03.67.Hk, 03.67.Pp, 03.67.Lx}
\maketitle
\tableofcontents
\section{Introduction}
Entanglement is a unique phenomenon of quantum mechanics that has puzzled generations of physicist. While initially the discussion was mainly driven by conceptual and philosophical considerations, e.g. regarding hidden variable theories, in recent years the focus has shifted to practical aspects and applications. It was realized that entanglement constitutes a valuable resource that can be used for various tasks in quantum information processing, many of which have in the meantime been experimentally demonstrated with a number of systems including nuclear magnetic resonance, photons, light beams, ions, neutral atoms, atomic ensembles, cavity QED, Josephson junctions and quantum dots. The remarkable experimental progress is even exceeded by the vast theoretical achievements that have lead to a new theory of quantum information, with quantum cryptography and quantum computation as most prominent offsprings.
In this context, the generation and maintenance of high--fidelity entanglement is a central problem. In the last decade, the controlled manipulation of certain systems in such a way that entangled states can be produced on demand has become possible. However, noise in such control operations as well as interactions with an uncontrollable environment have the effect that the desired entangled states are produced only with a certain non--unit fidelity. Similarly, entangled states that are ground states of certain (strongly coupled) systems are not obtainable in the lab due to thermal fluctuations that lead to a thermal (mixed) state with non--unit fidelity for any non--zero temperature. Several ways to protect quantum information in general and entangled quantum states in particular have been designed in recent years. These methods include quantum error correction, where quantum information is protected by using a certain encoding, quantum error avoiding schemes as well as entanglement purification. The latter is the main subject of this review article.
In entanglement purification \cite{Be96a,Be96,De96}, several copies of noisy, non--maximally entangled states are manipulated in such a way that a fewer number of copies with a reduced amount of noise are produced. The entanglement of the total ensemble is concentrated or distilled in a few copies, which hence contain a larger amount of entanglement and have higher fidelity with respect to a maximally entangled states. We will use the term {\em entanglement distillation} to refer to the manipulation of an ensemble of states in such a way that (a reduced number of) maximally entangled states are distilled. Entanglement distillation uses entanglement purification as a building block to increase the information about the ensemble, and hence to achieve this aim. The purified states can then eventually be used for various entanglement--based applications, ranging from quantum teleportation to quantum computation. It is worth pointing out that entanglement purification has been experimentally demonstrated for single photons \cite{Pa01,Kw01,Pa04,Ya03,Re05}, and very recently also for atoms \cite{Re06}.
Entanglement purification was originally introduced in the context of quantum communication as a solution to the problem of communication over noisy quantum channels \cite{Be96a,Be96,De96}. We will discuss this aspect of entanglement purification in Sec. \ref{ECC}, where we also briefly review quantum error correction as an alternative solution. Sec. \ref{Protocols} includes an overview and a detailed description of known bipartite entanglement purification protocols, while Sec. \ref{relation} establishes the close connection between quantum error correction and entanglement purification. The influence of noise in (local) control operations is discussed in Sec. \ref{realistic}, while Sec. \ref{applications1} illustrates a number of applications of entanglement purification protocols, including long distance quantum communication and quantum computation. We then move to multipartite systems, where we describe multipartite entanglement purification protocols and the role of noise in these protocols in Sec. \ref{multipartiteEPP}, and illustrate possible applications in Sec. \ref{applications2}. We provide an outlook on future perspectives of entanglement purification, including a number of relevant open problems, in Sec. \ref{outlook}.
We remark that in this review article we only touch upon the (extensive) subject of quantum error correction. For a more comprehensive treatment of quantum error correction we refer the reader for example to Refs. \cite{reviewQEC,St98,Go97} (see also \cite{Be96,St96,QECC1,Ma96,Ca96,QECC2,QECC3,QECC4,Pr98,OQEC}). We concentrate on theoretical aspects of entanglement purification and its applications, where our main focus lies on the work which has been carried out by our group in the last couple of years.
\section{Quantum communication via noisy channels}\label{ECC}
Quantum cryptography is the most advanced application of quantum information processing, where even first commercial systems can be purchased and are probably already used in practice. The transmission of quantum information over noisy channels in such a way that their quantum nature is sufficiently preserved is the central problem in this context. As quantum information is unavoidably affected by noise, ways to protect it need to be designed in order to ensure (almost) error free, i.e. noiseless, exchange of quantum information.
Even though quantum information cannot be cloned perfectly or amplified without changing their quantum nature, different techniques are known to protect quantum information from noise. The two solutions to the problem are provided by \begin{itemize} \item[(i)] quantum error correction \item[(ii)] teleportation in combination with entanglement purification. \end{itemize} In (i), quantum information is encoded in a larger dimensional Hilbert space and in this way protected from noise. In the second approach (ii), the problem of transmitting unknown quantum information over a noisy channel is replaced by the task of distributing a known entangled state with sufficiently high fidelity, which can then be used, by means of teleportation, to transmit arbitrary quantum information. Entanglement purification is thereby the tool to obtain (known) high--fidelity entangled pairs.
We first specify the set--up we have in mind. We consider two spatially separated parties $A$ (Alice) and $B$ (Bob) who wish to communicate with each other, i.e. transmit quantum information. They are connected via a possibly noisy quantum channel described by a completely positive map ${\cal E}$, and in addition by a classical channel which either only allows for classical communication from $A \to B$ (one way classical communication), or for classical communication between $A \to B$ and $B \to A$ (two--way classical communication). In addition, we assume that Alice and Bob can locally manipulate their quantum states and have access and control of auxiliary systems. This set of local operations and classical communication is denoted by LOCC. For qubits, a general quantum channel ${\cal E}$ can be written as \begin{eqnarray}\label{channel} {\cal E}\rho = \sum_{k,l=0}^{3} p_{k,l} \sigma_k \rho \sigma_l, \end{eqnarray} where $\sigma_j$ denote Pauli operators with $\sigma_0 =\mbox{$1 \hspace{-1.0mm} {\bf l}$}, \sigma_1=\sigma_x, \sigma_2=\sigma_y, \sigma_3= \sigma_z$. Often, we will consider Pauli-diagonal channels ${\cal E_P}$ which are of the form \begin{eqnarray}\label{Paulidiag} {\cal E_P}\rho = \sum_{k=0}^{3} p_{k} \sigma_k \rho \sigma_k. \end{eqnarray} Notice that any quantum channel ${\cal E}$ can be brought to Pauli--diagonal form by means of depolarization (see Ref. \cite{Du05SF}) in such a way that the diagonal elements are not altered, $p_k=p_{k,k}$. This often allows one to restrict considerations to Pauli--diagonal channels, and makes such channels particularly important. A special instance of a Pauli--diagonal channel is the depolarizing (or white noise) channel, where $p_1=p_2=p_3 = (1-p_0)/3$, which is described by a single parameter $p=p_0$.
We will discuss the basic principles of quantum error correction and entanglement purification in the following.
\subsection{Quantum error correction}
In standard quantum error correction (QEC) (see e.g. Refs. \cite{reviewQEC,St98,Go97} for reviews, and also \cite{Be96,St96,QECC1,Ma96,Ca96,QECC2,QECC3,QECC4,Pr98,OQEC}), quantum information is protected by encoding one logical qubit of quantum information into several physical qubits, or more generally $k$ logical qubits into $n$ physical qubits. The basic idea of such a redundant encoding is borrowed from classical coding and error correction, although additional requirements have to be met for quantum error correction, e.g. the preservation of arbitrary superposition states. In the simplest case where one qubit is encoded into $n$ qubits, we define logical qubits $|0_L\rangle, |1_L\rangle$ as two orthogonal states $|\psi_0\rangle=|0_L\rangle, |\psi_1\rangle=|1_L\rangle \in {\mathbb{C}}^{2^n}$. Any (unknown) qubit in a state $\alpha |0\rangle + \beta |1\rangle$ is encoded via a unitary encoding operation $U_E$ $\in SU(2^n)$ yielding an encoded state \begin{eqnarray}
(\alpha|0\rangle + \beta |1\rangle) \otimes |0\rangle^{\otimes n-1} \rightarrow (\alpha|0_L\rangle + \beta |1_L\rangle). \end{eqnarray}
The choice of $|\psi_0\rangle, |\psi_1\rangle$ is crucial for the error correcting properties of the code. The basic idea is to use states $|\psi_0\rangle, |\psi_1\rangle$ where the two--dimensional subspace $S$ spanned by $\{|\psi_0\rangle, |\psi_1\rangle\}$ is mapped to orthogonal two--dimensional subspaces by error operators. By measuring appropriate two--dimensional projectors, one can distinguish between these subspaces. Importantly, coherent superpositions within each of the subspaces are not altered and hence quantum information is preserved. Notice that these measurements also discretize errors, i.e. an independent treatment of error operators is justified. Consider as an example a $n$--qubit code that is capable of correction an arbitrary single qubit error, specified by a single--qubit Pauli error $\sigma_i^{(a)}$, acting on one of the $n$ qubits. If the action of any of these Pauli operators $\sigma_i^{(a)}$ is such that the two--dimensional subspace $S$ is transformed to a subspace $S_{i,a}$, and all these subspaces are pairwise orthogonal, then the corresponding code is capable of correcting for all such errors. The minimum number $n$ of qubits for which this is possible is given by 5, as there are $3n$ different error operators, and the possible number of orthogonal two--dimensional subspaces is given by $2^{n-1}$. In fact, error correction codes that can protect a single qubit from an arbitrary single qubit error using only 5 qubits are known \cite{Be96,QECC3}. In a similar way, one can construct quantum error correcting codes where $k$ logical qubits are encoded into $n$ physical qubits, and where a total of $m$ errors can be corrected. The basic idea of such a construction is still that each correctable error operator should map the relevant information carrying subspace $S$ of dimension $2^k$ to an orthogonal subspace $S_j$, and all these subspaces are pairwise orthogonal and hence distinguishable. In this way the type of error can be detected and corrected by determining the correspond $j$ via $2^k$--dimensional projective measurements. In addition, quantum information stored in coherent superpositions of states within the subspace $S$ remain unaltered.
Codes can also be designed to only {\em detect} errors. For error detection, it is only required that an error operator $O_j$ maps states within the subspace $S$ to the orthogonal subspace $S^\perp$, however it is no longer necessary that different error operators map all states within the subspace $S$ to pairwise orthogonal subspaces $S_j$. This implies that different error operators can lead to the same output state, and hence are indistinguishable. Nevertheless, the occurrence of an error can still be determined. Notice that the possibility to detect certain kinds of e.g. multi--qubit errors in addition to the possibility to distinguish between any kind of single--qubit errors is often an additional feature of error correcting codes. Error correcting codes can also be applied in a concatenated fashion, that is an encoded qubit is once more encoded on a next level using now encoded qubits $\alpha|0_L\rangle + \beta |1_L\rangle$ of level one as basic elements. Such concatenated error correction codes lie at the heart of a fault--tolerant implementation of quantum computation, but are also discussed in the context of long--range quantum communication using QEC \cite{Kn96}.
A general class of quantum error correcting codes of particular importance are the Calderbank-Shor-Steane (CSS) codes \cite{Ca96,St96} where encoding and decoding circuits consist of Clifford operations only \cite{Go97}. We will mainly consider such codes here. These codes belong to the class of {\em stabilizer codes}, which allows for a simplified description and error analysis. We also mention that recently the concept of operator quantum error correction \cite{OQEC}, as well as topological protection of quantum information \cite{Ki03} have been introduced.
\subsubsection{Long--distance quantum communication using QEC}
For long--range quantum communication, one protects an (unknown) qubit by encoding it with a concatenated quantum error correcting code \cite{Kn96}. The encoded quantum information is then sent through a noisy channel over a short distance, where the distance is chosen such that the probability for an error on the logical qubit is sufficiently small. More precisely, the error probability must be below a certain threshold such that error correction is still possible. Then an error correction step is performed, which involves decoding and measurements or direct error syndrome extraction, as well as correction. The signal is again encoded and sent further through a small segment of the channel. Concatenation of the error correction code ensures that errors on physical qubits below a certain threshold become exponentially suppressed at higher concatenation levels. This yields to a perfect transmission of quantum information with only polynomial overhead in additional qubits. To guarantee a fault--tolerant transmission, errors in coding and decoding operations at the error correction stations need to be taken into account. Although this is possible, one obtains a rather stringent error threshold of the order of $10^{-4}$ for local control operations, and also a small amount of tolerable channel noise at the order of percent \cite{Kn96}, significantly restricting the length of the segments.
\subsubsection{Channel capacities and capacities of QEC codes}
The capacity of a quantum error correcting code (QECC) is defined as the maximum asymptotic rate of reliable transmission of unknown quantum information through a noisy quantum channel, using the QECC to encode the states before transmission and decode them afterwards. More precisely, we consider a quantum channel ${\cal E}$ which is described by a completely positive trace preserving linear map ${\cal E}$ from the input Hilbert space ${\cal H}_c$
to the output Hilbert space ${\cal H}_o$. A quantum error correcting code is associated to coding (${\cal C}$) and decoding (${\cal D}$) operations. The coding operation ${\cal C}$ maps an input state $|\phi\rangle \in {\cal H}_{\rm in}$ to an encoded state of $n$ systems in Hilbert space ${\cal H}_c^{\otimes n}$, which are then transmitted through a noisy quantum channel ${\cal E}^{\otimes n}$ and decoded afterwards. These coding and decoding operations define a $(n,\epsilon)$ code if one achieves transmission with fidelity larger than $1-\epsilon$ for all possible system states, \begin{eqnarray}
{\rm min}_{|\phi\rangle \in{\cal H}_{\rm in}} \langle\phi| {\cal D}
\circ {\cal E}^{\otimes n} \circ {\cal C} (|\phi\rangle\langle\phi|) |\phi\rangle \geq 1-\epsilon. \end{eqnarray} The rate $R\equiv \log \dim {\cal H}_{\rm in}/n$ of a specific code is called achievable if for all $\epsilon,\delta >0$ and sufficiently large $n$ one obtains a rate $R-\delta$. The quantum capacity $Q(\cal E)$ of a bipartite quantum channel ${\cal E}$ is defined as the supremum of all achievable rates $R$ over all codes (see Ref. \cite{Ba98} for a rigorous definition).
Coding and decoding operations may be assisted by forward classical communication ($\rightarrow$) or two--way classical communication $(\leftrightarrow)$ which gives rise to quantum channel capacities $Q^{\rightarrow}$ [$Q^{\leftrightarrow}$] respectively. We will mainly consider $Q=Q^{\rightarrow}$ here.
We remark that a minimal pure state fidelity $F=1-\epsilon$ for all $|\phi\rangle \in {\cal H}_{\rm in}$ implies an entanglement fidelity $F_e \geq 1- 3\epsilon/2$ for all density operators $\rho$ whose support lies entirely in that subspace \cite{Ba98}.
That is, when transmitting part of an entangled state $|\Phi\rangle$ which is a purification of $\rho$, the resulting state has fidelity $F\geq 1-3
\epsilon/2$ with respect to $|\Phi\rangle$.
\subsection{Quantum communication via noisy entanglement purification and teleportation}
An alternative to direct transmission of quantum information over a (noisy) channel is provided by noisy entanglement purification followed by teleportation. More precisely, as shown in Ref. \cite{Be93}, a maximally entangled pair \begin{eqnarray}
|\phi^+\rangle = \frac{1}{\sqrt{2}}(|0\rangle_A|0\rangle_B + |1\rangle_A|1\rangle_B) \end{eqnarray} shared between $A$ and $B$ can be used as a resource to transmit one qubit of information from Alice to Bob, using only local resources and classical communication of two classical bits. In this sense, a maximally entangled pair serves as a perfect quantum channel. Hence the problem of transmitting unknown quantum information is shifted to the problem of generating a maximally entangled state.
We start by briefly reviewing the teleportation protocol \cite{Be93}, while entanglement purification is treated in more detail below. The teleportation protocol consists of the following steps: \begin{itemize}
\item[(i)] Alice performs a local Bell measurement in the basis $\{|\phi_{j}\rangle_{A'A}\}$ with $|\phi_j\rangle = \mbox{$1 \hspace{-1.0mm} {\bf l}$} \otimes \sigma_j |\phi^+\rangle$ on qubit $A'$ to be teleported and qubit $A$ of the maximally entangled state; \item[(ii)] Bob performs a correction operation $\sigma_j$ on qubit $B$ depending on the outcome $j$ of the measurement. \end{itemize} Step (ii) involves classical communication from Alice to Bob. Please note that in this way not only the local quantum information stored in the qubit $A'$ is transferred to $B$, but the qubit $B$ takes over completely the role of qubit $A'$, in particular also its entanglement with additional particles. The latter property can be used for entanglement swapping \cite{Zuxx1,Zuxx2,Zuxx3,Zuxx4}, where entanglement between systems $A C_1$ and $C_2 B$ leads to an entangled state between systems $AB$ by teleporting $C_1$ to $B$. Teleportation has been experimentally demonstrated for single photons \cite{Tphoton1,Tphoton2,Tphoton3}, light beams \cite{Tlight} and atoms \cite{Tatom,Tatom1}, and very recently also to for the transmission of quantum information between different media, namely from light to matter \cite{Tlighttoatom}.
For quantum communication over noisy channels, teleportation alone is not sufficient. When sending one qubit of a maximally entangled pair through a noisy channel, one ends up with a noisy, non--maximally entangled state. Although such a noisy state can still be used for quantum teleportation, the fidelity of the teleported qubit is reduced. However, one can in principle produce many copies of the noisy entangled pairs and purify these pairs using entanglement purification, i.e. increase the entanglement of a few copies. This is possible since the desired state is {\em known}, in contrast to the general situation in direct quantum communication where the states may be unknown.
Entanglement purification is a fundamental tool in quantum information processing. We will concentrate in this section on its application in quantum communication, where entanglement purification together with teleportation provides a scheme for error--free quantum communication over noisy channels. The basic idea is to create several copies of noisy entangled states, e.g. by sending parts of locally created entangled pairs through noisy quantum channels. These states are then processed locally, more precisely a sequence of local operations (assisted by one or two-way classical communication) is applied in such a way that a reduced number of pairs with an increased fidelity is generated. Iteration of such protocols eventually leads to maximally entangled states. The entanglement is purified at the cost of obtaining a smaller number of copies. This can be done in a systematic way, and several entanglement purification protocols are known which can achieve this task. The purification protocols can be grouped into distillation protocols, and recurrence and pumping schemes. In distillation protocols, an ensemble of many (identical) copies is manipulated and a few pairs with improved fidelity are generated. In recurrence and pumping schemes, a certain elementary purification step is repeated several times, resulting in pairs with improved fidelity. We will discuss such entanglement purification protocols in the following section in more detail.
\subsubsection{One--way and two--way classical communication}
The protocols differ in the number of initial and final copies of the states, and the allowed additional resources, most importantly one--way or two--way classical communication.
A protocol may operate on $N$ copies of noisy entangled states and produce $M \leq N$ purified copies as output. In case of one--way classical communication, one measures $N-M$ copies and uses the obtained information to choose a proper correction operation on the remaining pairs. Notice that for the choice of these correction operations, Alice only has access to the local measurement outcomes in $A$, while Bob has access to outcomes of measurements in $A$ and $B$. In particular, this implies that the parties {\em cannot} decide to discard certain pairs based on joint measurement outcomes, as it is used in several entanglement purification protocols. To make this possible, one needs two--way classical communication. Protocols based on one--way classical communication turn out to be equivalent to quantum error correction \cite{Be96} in a sense we will specify below, and one may say that entanglement purification runs in {\em error correction mode}. On the other hand, protocols with two--way classical communication make use of additional resources and are provably superior to quantum error correction and one--way entanglement purification \cite{Be96,AsPhD}. Such two--way protocols can also run in an {\em error detection mode}.
\subsubsection{Purification range and yield of a protocol}
The yield of a protocol with respect to a certain state is a central quantity that determines how efficient a protocol is. Consider $N$ copies of a mixed state, $\rho^{\otimes N}$, which are processed by a certain entanglement purification protocol ${\cal P}$ that may be applied in an recursive way. After this procedure, $M$ (exact) copies of a maximally entangled state are obtained. The yield of the protocol ${\cal P}$ with respect to the state $\rho$ is defined as \begin{eqnarray} Y_{\rho,{\cal P}} = \frac{M}{N}, \end{eqnarray}
i.e. the ratio of number of maximally entangled states obtained by the protocol over the total number of initial copies of the state $\rho$, where often the limit $N \to \infty$ is considered. More precisely, one considers an entanglement purification protocol and the corresponding LOCC transformation $\rho^{\otimes N} \rightarrow \tilde \Gamma_N$, where we are interested only in the reduced density operator of $M$ copies, $\Gamma_M = {\rm tr}_{M-1,\ldots N} \tilde \Gamma_N$. We demand that for all $\epsilon >0$, the fidelity of $\Gamma_M$ with respect to $M$ copies of a maximally entangled state $|\Phi\rangle = |\phi^+\rangle^{\otimes M}$ should be larger than $1-\epsilon$, $F=\langle\Phi|\Gamma_M|\Phi\rangle \geq 1-\epsilon$. The yield is then determined by the ratio $M/N$ of the maximum $M$ for which this is the case, in the asymptotic limit of $N \to \infty$. The maximum of the yield over {\em all} protocols (or equivalently all sequences of local operations and classical communications (LOCC)) is also called the distillable entanglement $D_\rho$, and again one may consider these quantities with respect to one way or two--way classical communication, $Y_{\rho,{\cal P}}^\rightarrow, Y_{\rho,{\cal P}}^\leftrightarrow, D_\rho^\rightarrow, D_\rho^\leftrightarrow$. The distillability problem, i.e. the question whether there exists a LOCC protocol that can generate maximally entangled states from (infinitely) many copies of a state, has been extensively studied in recent years, however a complete solution has not been obtained so far. What is, however, known are {\em necessary} conditions for distillability (e.g. that the partial transposition of the density operator is non--positive \cite{Ho97}), as well as {\em sufficient} criteria. In particular, any entanglement purification protocol provides a sufficient criterion for distillability.
We remark that such a strict definition of yield actually implies that many entanglement purification protocols have zero yield, although they can produce entangled states with arbitrary high fidelity. We will consider later a modified definition of the yield, where the condition of arbitrary accuracy for the produced states is dropped and replaced by a condition for a certain {\em fixed} fidelity $F=1-\epsilon_0$. Notice that whenever one considers noise in local control operation, perfect entanglement purification is impossible, i.e. no entanglement purification protocol is capable of producing perfect maximally entangled states with fidelity $F=1$.
The purification range (or basin) of a protocol is defined as the set of all input states $\rho$ that can be purified by the protocol, i.e. where maximally entangled states can be generated from $N$ copies of $\rho$ in the limit of $N\to \infty$. Often, certain families of density operators specified by a single or a few parameters are considered. An example of such a family are the so--called Werner states (see Eq. (\ref{Wernerstates}) below) which are mixtures of a maximally entangled state and a completely mixed state. In this case, the purification range can be obtained analytically, yielding a simple condition that can be expressed in terms of the fidelity of the state. In general a complete characterization of the purification basin for a given protocol is very complicated. However, bounds on the purification range can be established which are based on either necessary conditions for distillation, or on sufficient conditions for entanglement purification using a specific protocol (e.g. that the fidelity with respect to a maximally entangled state be above a certain threshold value).
\section{Basic bipartite entanglement purification protocols}\label{Protocols}
We now turn to explicit entanglement purification protocols. A number of different protocols exist, which differ in their purification basin (i.e. the set of states they can purify), the efficiency, and the number of copies of the states they operate on. In the following we will consider filtering protocols (which operate on a single copy), recurrence protocols (which operate on two copies simultaneously at each step) as well as hashing and breeding protocols (which operate simultaneously on a large number $N \to \infty$ of copies). We also discuss (intermediate) $N \to M$ protocols, which operate on $N$ input copies and produce $M$ output copies. The latter protocols can also be run in a recursive way.
\subsection{Filtering protocol} The simplest protocols operate on a single copy of the mixed state $\rho$ and consist in the application of local filtering measurements (including weak measurements). A weak measurement may e.g. be realized by a joint, local operation on the system and a (possibly high dimensional) ancilla, followed by a von Neumann measurement of the ancilla. Hence sequences of local operations, including weak measurements, are applied in such a way that for specific measurement outcomes the resulting state $\sigma$ is more entangled than the initial state $\rho$. Note that the output state $\sigma$ is obtained only with a probability $p < 1$. Mixed states where such a filtering method can be applied include certain rank two states of the form \cite{Gi98} \begin{eqnarray}
\rho=F |\Psi^+\rangle\langle \Psi^+| + (1-F) |00\rangle\langle00|, \end{eqnarray}
where $|\Psi^+\rangle=1/\sqrt{2}(|01\rangle+|10\rangle)$. Application of the local operators $O_A= O_B =\sqrt{\epsilon} |0\rangle\langle 0| + |1\rangle\langle 1|$ (which correspond to a specific branch of a local positive operator valued measure, POVM) lead to a non--normalized state of the form \begin{eqnarray}
\rho'= F \epsilon |\Psi^+\rangle\langle \Psi^+| + (1-F) \epsilon^2 |00\rangle\langle 00|. \end{eqnarray} The fidelity of the resulting state is given by \begin{eqnarray} F'= F\epsilon/[F\epsilon+ (1-F) \epsilon^2].
\end{eqnarray} Note that for small $\epsilon$, $F' \to 1$, that is states arbitrarily close to the maximally entangled state $|\Psi^+\rangle$ can be created. However, the probability to obtain the desired outcome, $p_{\rm suc}=F\epsilon+ (1-F) \epsilon^2$, goes to zero as $\epsilon \to 0$. There is a tradeoff between the reachable fidelity of the output state and the probability of success of the procedure. The optimal filtering protocol for any mixed state of two qubits has been derived in \cite{Ve01}, and experimentally demonstrated in \cite{Wang06}.
It turns out that filtering protocols are of limited applicability for general mixed states, even for the simplest case of two qubits. In particular, as shown in Refs. \cite{Li99,Ke00}, the fidelity of a single copy of a full rank state can in general {\em not} be increase by any local operation. This seriously restricts the applicability of filtering procedures and requires us to consider protocols on two (or more) copies of the state in order to increase the fidelity for a general class of mixed input states, and ultimately to obtain maximally entangled states.
\subsection{Recurrence protocols}
In the following we discuss a class of conceptually related protocols \cite{Be96a,Be96,De96,Du03QC} that allow one to produce states arbitrarily close to a maximally entangled pure state by iterative application. Before we go into technical details, we describe the general concept underlying these (and almost all) entanglement purification protocols. The idea of entanglement purification protocols is to decrease the degree of mixed--ness of the ensemble of mixed states. To this aim, one needs to gain information, which is done by performing suitable measurements. As the relevant information is not locally accessible, one needs to use the entanglement inherent in states of the ensemble to reveal this information. In fact, information about a particular sub-ensemble is obtained by first operating jointly (but still locally) on several states and then measuring one of these states.
In many protocols the remaining states are only kept if a specific measurement outcome was found. This is due to the fact that one finds for certain measurement outcomes (measurement branches) that the entanglement of the remaining states is increased, while for other outcomes it is decreased or the states are even no longer entangled. In this way it is also guaranteed that, on average, entanglement can not increase under local operations and classical communication.
Recurrence protocols operate in each purification step on a fixed number of copies of a mixed state. We will mainly consider recurrence protocols that operate on two identical copies here, but also treat briefly the cases where the copies are not identical (e.g. in pumping schemes) or more than two copies are involved. After local manipulation, one of the copies is measured, and depending on the outcome of the measurement the other copy is kept (we refer to this as a successful purification step) or discarded (see Fig. \ref{bipartitesetupFig}). In case of a successful purification step, the fidelity of the remaining pair is increased. The procedure is iterated, whereby states resulting from a successful purification round are used as input for the next purification round (see Fig. \ref{resourcesBEPP}). Typically, these protocols converge to a fixed point which is given by a maximally entangled state.
\begin{figure}\label{bipartitesetupFig}
\end{figure}
\begin{figure}\label{resourcesBEPP}
\end{figure}
\subsubsection{Basic properties and notation}
We now turn to specific recurrence protocols that allow one to purify bipartite entangled states of two qubits. We will not describe these protocols as they were originally presented, but provide an equivalent description which will allow us a unified treatment of bipartite and multipartite entanglement purification protocols. In particular, we describe protocols that operate on states in a (locally) rotated basis and describe the corresponding states in terms of their stabilizing operators. To this aim, we start by fixing some notation. We consider two parties, $A$ and $B$, each holding several copies of noisy entangled states described by a density operator $\rho_{AB}$ acting on Hilbert space $\mathbb{C}^2 \otimes \mathbb{C}^2$. We denote by \begin{eqnarray}
|\Phi_{00}\rangle \equiv \frac{1}{\sqrt{2}}(|0\rangle_z|0\rangle_x\rangle + |1\rangle_z|1\rangle_x), \end{eqnarray}
a maximally entangled state of two qubits, where $|0\rangle_z, |1\rangle_z$ [$|0\rangle_x, |1\rangle_x]$ are eigenstates of $\sigma_z$ [$\sigma_x$] with eigenvalue $(\pm 1)$ respectively. For example, $\sigma_x |1\rangle_x = -|1\rangle_x$, and $|0\rangle_x=1/\sqrt{2}(|0\rangle_z + |1\rangle_z)$. We also define \begin{eqnarray}
|\Phi_{k_1k_2}\rangle \equiv \sigma_z^{k_1}\sigma_z^{k_2} |\Phi_{00}\rangle, \end{eqnarray}
with $k_1,k_2 \in \{0,1\}$. The states $\{|\Phi_{k_1k_2}\rangle \}$ form a basis of orthogonal, maximally entangled states, the so called Bell basis. We remark that the states $|\Phi_{k_1k_2}\rangle$ are joint eigenstates of correlation operators \begin{eqnarray} K_1=\sigma_x^{(A)} \sigma_z^{(B)}, \hspace{1cm} K_2=\sigma_z^{(A)}\sigma_x^{(B)}, \end{eqnarray} with eigenvalues $(-1)^{k_1}$ and $(-1)^{k_2}$ respectively. Whenever several copies of a mixed state are involved, we will refer to the different copies by numbers. For instance, $\rho_{A_1B_1}$ refers to the first copy of a state, while $\rho_{A_2B_2}$ refers to the second copy. In this case, party $A$ holds two qubits, $A_1$ and $A_2$.
We consider mixed states $\rho'_{AB}$ which we expand in the Bell basis, \begin{eqnarray} \label{depolli}
\rho'_{AB}= \sum_{k_1,k_2,j_1,j_2=0}^1 \lambda'_{k_1k_2j_1j_2} |\Phi_{k_1k_2}\rangle \langle \Phi_{j_1j_2}|. \end{eqnarray} This state can e.g. be created by sending the second qubit of a maximally entangled state through a noisy quantum channel ${\cal E}$, Eq. (\ref{channel}). One can always {\em depolarize} the state to a standard form by a suitable sequence of (random) local operations in such a way that the fidelity of the state,
\begin{eqnarray} F\equiv \langle\Phi_{00}|\rho_{AB}|\Phi_{00}\rangle \end{eqnarray} is not altered. To be specific, by probabilistically applying one of the multi--local operations corresponding to $\{\mbox{$1 \hspace{-1.0mm} {\bf l}$}, K_1, K_2, K_1K_2\}$ one produces a density operator which is diagonal in the Bell basis, \begin{eqnarray}
\rho_{AB}= \sum_{k_1,k_2=0}^1 \lambda_{k_1k_2} |\Phi_{k_1k_2}\rangle \langle \Phi_{k_1k_2}|, \label{Bell} \end{eqnarray}
and in which diagonal coefficients remain unchanged, $\lambda_{k_1k_2} \equiv \lambda'_{k_1k_2k_1k_2}$. This dephasing step can be understood as follows: Consider for instance the action of $K_1$ on basis states $|\Phi_{k_1k_2}\rangle$. For $k_1=0$, the state is left invariant while a global phase of $(-1)$ is picked up if $k_1=1$. It follows that off--diagonal elements of the form $|\Phi_{k_1k_2}\rangle\langle \Phi_{j_1j_2}|$ in (\ref{depolli}) are transformed to $(-1)^{k_1\oplus j_1} |\Phi_{k_1k_2}\rangle\langle \Phi_{j_1j_2}|$, i.e. acquire a phase if $k_1 \not= j_1$. Consequently, when applying the local operation $K_1$ with probability $p=1/2$ and leaving the state unchanged otherwise, the resulting density operator $\rho' = 1/2(K_1 \rho K_1^\dagger +\rho)$ has no off--diagonal elements where $k_1 \not= j_1$. In a similar way, all off--diagonal elements are cancelled by the (random) application of $\mbox{$1 \hspace{-1.0mm} {\bf l}$},K_1,K_2,K_1K_2$. Note that all diagonal elements --in particular the fidelity of the state-- remain unchanged by this depolarization procedure. Using similar techniques, one can further depolarize the state by making all but one of the diagonal elements equal. The resulting states are called Werner states \cite{We89}, \begin{eqnarray}\label{Wernerstates}
\rho_W(x)= x|\Phi_{00}\rangle\langle\Phi_{00}| + (1-x)\frac{1}{4} \mbox{$1 \hspace{-1.0mm} {\bf l}$}_{AB},\label{rhoWerner} \end{eqnarray}
where the fidelity $F=(3x+1)/4$ is unchanged. This can be accomplished by randomly applying local unitary operations that leave the state $|\Phi_{00}\rangle$ (up to a phase) invariant, which is the case for all operations of the form $U \otimes HU^*H$ with $H$ being the Hadamard gate \footnote{The Hadamard operation $H$ maps basis state of $z$ basis to basis states of $x$ basis and vice versa, i.e. $H|k\rangle_z = |k\rangle_x, H|k\rangle_x = |k\rangle_z$ with $k=0,1$.} and $^*$ denoting complex conjugation. The unitaries can be chosen uniformly (according to the Haar measure), or selected from a specific finite set of operations \cite{Be96a,Be96}. Notice, however, that the entanglement of the states may decrease by the depolarization procedure, even though the fidelity remains unchanged. What is important in our context is that any state with fidelity $F$ can always be brought to Werner form. It is thus sufficient to provide an entanglement purification method which works for Werner states, because such a method automatically allows one to purify {\em all} states which have the same fidelity, independent of their ``shape''. We consider such a purification procedure in the following.
\subsubsection{BBPSSW protocol}
In 1996, Bennett et al. \cite{Be96a} introduced a purification protocol that allows one to create maximally entangled states with arbitrary accuracy starting from several copies of a mixed state $\rho$, provided that the fidelity $F$ with some maximally entangled state fulfills $F > 1/2$. The protocol consists of the following steps: \begin{itemize} \item[(i)] Depolarize $\rho$ to Werner form;
\item[(ii)] apply bilateral local CNOT operations $U_{\rm CNOT}^{A_1 \to A_2} \otimes U_{\rm CNOT}^{B_2 \to B_1}$ \footnote{The CNOT operation is defined by $|i\rangle_A|j\rangle_B \rightarrow |i\rangle_A|i\oplus j\rangle_B$, where $\oplus$ denotes addition modulo 2.}; \item[(iii)] measure qubit $A_2$ [$B_2$] locally in eigenbasis of $\sigma_z$ [$\sigma_x$] with corresponding results $(-1)^{\zeta_1}$ [$(-1)^{\xi_1}$] respectively, where $\zeta_1,\xi_1 \in\{0,1\}$. The effect of this local measurement on other particles is the same as the measurement of the observable $K_2^{(A_2B_2)}$; \item[(iv)] keep state of the pair $A_1B_1$ if $(\zeta_1+\xi_1){\rm mod}2=0$, i.e. measurement results coincide. \end{itemize} Given two copies of a state with fidelity $F$, it is straightforward to calculate the fidelity of the resulting state when applying (i-iv). The effect of (ii) on two Bell states is given by \begin{eqnarray}
|\Phi_{k_1,k_2}\rangle_{A_1B_1} |\Phi_{j_1,j_2}\rangle_{A_2B_2} \rightarrow |\Phi_{k_1\oplus j_1,k_2}\rangle_{A_1B_1} |\Phi_{j_1,k_2\oplus j_2}\rangle_{A_2B_2}. \label{actionBCNOT} \end{eqnarray}
The effect of (iii) and (iv) is to select states in $A_2B_2$ which are eigenstates of $K_2^{A_2B_2}$ with eigenvalue (+1), while eigenstates with eigenvalue (-1) are discarded. That is, only initial states $|\Phi_{k_1,k_2}\rangle_{A_1B_1} |\Phi_{j_1,j_2}\rangle_{A_2B_2}$ with $k_2\oplus j_2 =0$ will pass the measurement procedure, which implies that, when considering mixed states, only these components will contribute to the final density operator. The final state turns out to be not of Werner form, however due to step (i) the state is brought back to Werner form when iterating the procedure. Hence the essential parameter is the fidelity $F'$ after a successful purification step. One finds \begin{eqnarray} F'=\frac{F^2+[(1-F)/3]^2}{F^2+2F(1-F)/3+5[(1-F)/3]^2}, \label{Fout} \end{eqnarray} which fulfills $F' >F$ for $F>1/2$. The success probability is given by the denominator of Eq. (\ref{Fout}), $p_{\rm suc}=F^2+2F(1-F)/3+5[(1-F)/3]^2$. Iteration of the procedure, which means to take two identical copies of states with fidelity $F'$, resulting from a previous, successful purification round, allows one to successively increase the fidelity. In fact, it is straightforward to see that the map Eq. (\ref{Fout}) has $F=1$ as an attractive fixed point. Hence states arbitrarily close to maximally entangled states can be produced. Even though the probability of success of the purification steps tends to one for $F \to 1$, the yield of the procedure goes to zero as one of the pairs is always measured and has to be discarded. Fixing however the desired target fidelity of resulting states to $F> 1-\epsilon_0$, a finite number of purification steps suffices and hence the yield will be finite. We remark that the obtainability of states with $F=1$ seems to be a question of only theoretical relevance, since imperfections in the apparatus used for the preparation of the state and in the purification procedure limit the reachable fidelity.
\subsubsection{DEJMPS protocol}
The DEJMPS protocol, introduced by Deutsch et al. in Ref. \cite{De96}, is conceptually similar to the BBPSSW protocol. It operates however not on Werner states, but on states diagonal in a Bell basis (see Eq. \ref{Bell}). The main advantage of this protocol is that it has higher efficiency. The protocol operates on two identical copies of a state and consists essentially of the same steps as the BBPSSW protocol. The only difference is that step (i) is replaced by a step (i'). \begin{itemize} \item[(i')] perform local basis change \begin{eqnarray}
|0\rangle_z^{(A)} \to \frac{1}{\sqrt{2}} (|0\rangle_z^{(A)} -i|1\rangle_z^{(A)}),& \hspace{0.5cm} &|1\rangle_z^{(A)} \to \frac{1}{\sqrt{2}} (|1\rangle_z^{(A)} -i|0\rangle_z^{(A)}) \nonumber\\
|0\rangle_x^{(B)} \to \frac{1}{\sqrt{2}} (|0\rangle_x^{(B)} +i|1\rangle_x^{(B)}),& \hspace{0.5cm} &|1\rangle_x^{(B)} \to \frac{1}{\sqrt{2}} (|1\rangle_x^{(B)} +i|0\rangle_x^{(B)})\nonumber. \end{eqnarray} \end{itemize}
The effect of step (i') is (up to some irrelevant phases) to flip the diagonal components of $|\Phi_{10}\rangle$ and $|\Phi_{11}\rangle$, i.e. $\lambda_{10} \leftrightarrow \lambda_{11}$. One may in addition add a depolarization of $\rho$ to Bell--diagonal form (see Eq. \ref{Bell}), however as shown in Ref. \cite{De96} the off--diagonal terms do not influence the protocol anyway. The total effect of the protocol (steps (i-iv)) can be described as a non-linear map for the diagonal components of $\rho$ to $\rho'$ (written in the Bell basis), i.e. a map from $\mathbb{R}^4 \to \mathbb{R}^4$. To be specific, the map reads \cite{De96} \begin{eqnarray} \lambda_{00}'=(\lambda_{00}^2+\lambda_{11}^2)/N, \hspace{1cm}&& \lambda_{01}'=(\lambda_{01}^2+\lambda_{10}^2)/N,\nonumber\\ \lambda_{10}'=2\lambda_{00}\lambda_{11}/N,\hspace{1cm} && \lambda_{11}'=2\lambda_{01}\lambda_{10}/N, \label{Oxfordmap} \end{eqnarray} where $N=(\lambda_{00}+\lambda_{11})^2+(\lambda_{01}+\lambda_{10})^2$ is the probability of success of the protocol. Again, the protocol can be iterated, and the diagonal coefficients of the state in the Bell basis after $k$ successful purification steps can be calculated by $k$ iterations of the map Eq. (\ref{Oxfordmap}). One can show that the map has $\lambda_{00}=1, \lambda_{ij}=0$ for ${ij}\not=00$ as attracting fixed point, and in fact all states with $\lambda_{00}>1/2$ (i.e. $F>1/2$) can be purified \cite{Ma98}.
\subsubsection{Entanglement pumping}\label{nestedEPP}
While both the BBPSSW and DEJMPS protocol allow one to successfully produce entangled states with arbitrary high fidelity, the requirements on local resources are rather demanding. Since at every round two identical states resulting from previous successful purification rounds are required, the total number of pairs that have to be available initially increases exponentially with the number of steps. In real implementations these pairs have to be stored by some means. For many physical set--ups, however, the number of particles that can be stored is limited.
The requirements in memory space can however be translated into temporal resources. The corresponding purification protocol is called (nested) entanglement pumping \cite{Br98,Du98,Du03QC} (see also Fig. \ref{NestedPumping}). The basic idea is to repeatedly produce elementary entangled pairs (resulting e.g. from the transmission of these maximally entangled state through noisy channels) and using a fresh elementary pair to purify a second pair. If a purification step is not successful, one has to start again from the beginning, using two elementary pairs. The actual sequence of local operations is either given by the BBPSSW or DEJMPS protocol, where the pair to be purified acts as pair 1 (source pair), while the fresh, elementary pair plays the role of pair 2 (target pair) that is measured. In case the purification step was successful, the fidelity of the first pair is increased by a certain amount. It is straightforward to determine the maps corresponding to Eqs. (\ref{Fout},\ref{Oxfordmap}) for non--identical input states. One finds \begin{eqnarray} F'=\frac{F_1F_2+\left(\frac{1-F_1}{3}\right)\left(\frac{1-F_2}{3}\right)}{F_1F_2+F_1\left(\frac{1-F_2}{3}\right)+\left(\frac{1-F_1}{3}\right)F_2+5\left(\frac{1-F_1}{3}\right)\left(\frac{1-F_2}{3}\right)}. \label{Fout2} \end{eqnarray} in the case of two Werner states with fidelity $F_1,F_2$. In this map, $F_2$ is to be considered as constant since the second pair is always an elementary one. For two Bell diagonal states with coefficients $\lambda_{ik}$ and $\mu_{ik}$ we obtain \begin{eqnarray} \lambda_{00}'=(\lambda_{00}\mu_{00}+\lambda_{11}\mu_{11})/N, && \lambda_{01}'=(\lambda_{01}\mu_{01}+\lambda_{10}\mu_{10})/N,\nonumber\\ \lambda_{10}'=(\lambda_{00}\mu_{11}+\lambda_{11}\mu_{00})/N, && \lambda_{11}'=(\lambda_{01}\mu_{10}+\lambda_{10}\mu_{01})/N, \label{Oxfordmap2} \end{eqnarray} Again, the second pair is always an elementary one, and hence $\mu_{ik}$ is fixed. While iteration of the corresponding maps allows in both cases to improve the fidelity, {\em no} maximally entangled states can be generated, in general. That is, the fixed point of the maps described by Eqs. (\ref{Fout2},\ref{Oxfordmap2}) depends on the coefficients $\mu_{ik}$ and specifically on the fidelity of the elementary pair \cite{Du98}.
As elementary pairs can be generated on demand, they do not need to be stored. Hence in $A$ and $B$ only two qubits need to be stored (corresponding to the pair to be purified and the elementary pair, respectively). The reduction in spatial resources leads however to an increase of temporal resources. In protocols BBPSSW and DEJMPS, the purification of different pairs corresponding to a single purification step can be implemented in parallel (i.e. the temporal resources are given by the number of steps). That is, one actually considers a distillation procedure where out of many low--fidelity entangled pairs a few with higher fidelity are generated. The probabilistic character of entanglement purification manifests itself in the fact that many identical pairs need to be simultaneously available. In entanglement pumping, in contrast, the probabilistic character of purification leads to increased number of required repetitions, as in case of an unsuccessful purification step the procedure has to be started from beginning and pairs are {\em sequentially} generated.
One can improve the entanglement pumping scheme in such a way that the number of qubits that have to be locally stored remain small ($\approx 4$ for practical purposes), while it is possible to generate maximally entangled states rather than only enhancing the fidelity by a finite amount. The corresponding scheme is called {\em nested entanglement pumping} \cite{Du03QC} and works as follows: At nesting level 1, elementary pairs created between $A_1$ and $B_1$ are used to purify a pair shared between $A_2$ and $B_2$ via entanglement pumping. The fidelity of elementary pairs at level 1 is given by $F_1$. It turns out that after a few purification steps, the fidelity $F_2$ of the pair $A_2$ and $B_2$, is already close to the reachable fixed point. The resulting pair with improved fidelity $F_2$ now serves as elementary pair at nesting level 2. That is, an elementary pair at nesting level 2 shared between $A_3$ and $B_3$ is purified by means of entanglement pumping, where always elementary pairs of nesting level 2 with fidelity $F_2$ shared between $A_2$ and $B_2$ are used. The fidelity of the resulting pair $A_3$ and $B_3$ after a few purification steps is given by $F_3$ with $F_3>F_2>F_1$. We remark that an unsuccessful purification step at a higher nesting level requires to restart the procedure at the lowest nesting level 1. Still, the required temporal resources increase only polynomially. The overall procedure can be viewed as a stochastic process, or equivalently as a one--sided bounded random walk. With each nesting level, one additional particle has to be stored at each location. However, it turns out that for practical purposes (say required accuracy of $\epsilon_0=10^{-7}$) a few nesting levels ($\approx 3$) suffice to generate states with fidelity $F > 1-\epsilon_0$ \cite{Du03QC}. Hence the storage requirements remain very moderate, while the required temporal resources increase only polynomially.
\begin{figure}
\caption{(a) Schematic representation of entanglement pumping. An elementary pair (pair 1) with fidelity $x_0$ is repeatedly created and used to purify the second pair. The fidelity converges to some fixed point $x_0^{\rm fix} <1$. (b) Nested entanglement pumping. At nesting level 1, elementary pairs of fidelity $x_0$ are used to purify the second pair to fidelity $x_0^{\rm fix}$. At the next nesting level, pairs of fidelity $x_0^{\rm fix}$ produced in this way serve as elementary pairs and are used to purify one pair to fidelity $x_1^{\rm fix}$ via entanglement pumping. Pairs of fidelity $x_1^{\rm fix}$ are then used as elementary pairs at the next nesting level.}
\label{NestedPumping}
\end{figure}
\subsection{$N\to M$ protocols}
The protocols discussed in the previous Section operate on two copies of a given mixed state, and produce one copy as output if they are successful. More general protocols are conceivable that operate on $N$ input copies of the state and produce $M$ copies as output. We will refer to such protocols as $N \to M$ protocols, and discuss them in this subsection. A protocol of this kind of particular importance is the so--called hashing protocol, which operates in the limit $N,M \to \infty$. The general idea behind $N \to M$ protocols is very similar as in the case of standard recurrence protocols operating on two copies: To obtain information about a sub--ensemble --in this case consisting of $M$ copies of the state--, for which the remaining $N-M$ copies are measured after applying suitable local operations.
\subsubsection{$N \to M$ protocols for finite $N$}
The $2 \to 1$ recurrence protocols discussed in the previous section can be considered as two--step procedures. In a first step, the two copies of the input state are manipulated by local operations that entangle the two pairs. The effect of these local operations on Bell diagonal states is a certain {\em permutation} of the basis elements. In a second step, the second pair is measured, and depending on the outcome of the measurement the first pair is either kept or discarded. General $N \to M$ protocols operate in a very similar fashion. In fact, in Ref. \cite{De01} all possible permutations achievable by local operations have been constructed for qubit systems, and accordingly a large number of possible $N \to M$ entanglement purification protocols were constructed and analyzed. It was found that in certain regimes such $N \to M$ protocols operate more efficiently (i.e have a higher yield) than standard $2 \to 1$ protocols \cite{De01,Ma01}. Typically, for small initial fidelities the ratio of final pairs $M$ to initial pairs $N$ may be small, $M/N \ll 1$, while one expects that $M/N \approx 1$ for large fidelities as only a small amount of information about the remaining ensemble needs to be revealed. Generalizations of this concept to the purification of entangled $d$--level systems are possible \cite{Bo05} (see also e.g. Refs. \cite{Ho99,Al01,Ch05} for entanglement purification protocols of $d$--level systems).
We would also like to remark that a general connection between error correcting (stabilizer) codes and $N \to M$ purification protocols exists \cite{AsPhD}, which we describe later in more detail. In fact, for each code one can construct a corresponding $N \to M$ entanglement purification protocol.
\subsubsection{Hashing and breeding protocols}
Hashing protocols can be considered as special instances of $N \to M$ protocols that operate in the limit $N \to \infty$. Hashing was introduced in Ref. \cite{Be96}. The basic idea is similar as in $N \to M$ recurrence protocols. Here, random subsets of size $n$ out of the total $N$ copies of the state are chosen, and bilateral local CNOT operations with each of the $n$ pairs as source, and one selected pair as target, are performed (or vice versa, i.e. the selected pair as source). The selected pair is finally measured, revealing one bit of information about the remaining ensemble. Measurements of this kind are repeated $m$ times. One can in fact show that the information gain per measurement is close to one bit.
Hashing is conceptually closely related to breeding, which might be slightly easier to understand. In the case of breeding, the parties are assumed to possess, in addition to the $N$ copies of the state, $m$ pre-purified, maximally entangled Bell pairs which are used to gain information about the remaining ensemble.
In the asymptotic limit of large $N$ the density matrix $\rho^{\otimes N}$ is approximated to an arbitrary high accuracy by its ``likely subspace approximation'', i.e. the density matrix $\Gamma$ obtained by projecting $\rho^{\otimes N}$ into a subspace $P$ (the likely subspace), where the dimension of $P$ is $2^{(S(\rho)+\delta)N}$. In the case of Werner states $\rho_W(F)$ (see Eq. \ref{rhoWerner}, $F=(3x+1)/4$), this likely subspace contains essentially states of the form $\otimes_{i,j} |\Phi_{ij}\rangle^{\otimes m_{ij}}\langle \Phi_{ij}\rangle^{\otimes m_{ij}}|$ and permutations thereof, where $m_{00}= F N, m_{01}=m_{10}=m_{11} =(1-F)/3 N$
\footnote{To be precise, one has to consider in addition also states with $m_{00}=(F N \pm O(\sqrt N))$ and similarly for $m_{ij}$.}. That is, the density matrix $\rho^{\otimes N}$ can be interpreted as an equal mixture of all these possible configurations, where the number of Bell states $|\Phi_{ij}\rangle$ is essentially fixed to $m_{ij}$, while the order (or position) of the states is unknown. The number of possible configurations of states of this form is --for large $N$-- approximately given by $2^{N S(F)}$, where $S(F)= -F\log_2F-(1-F) \log_2(\frac{1-F}{3})$. The task thus reduces to reveal which of these possible configurations one is dealing with. Clearly, this requires $N S(F)$ bits of information. Since one can gain at most one bit of information about the ensemble with help of each maximally entangled pair, one needs $m=N S(F)$ additional maximally entangled pairs to perform this task. Having obtained the required information, one possesses a pure state consisting of $N$ Bell states (in different bases), i.e. some (known) permutation of the state $\otimes_{i,j} |\Phi_{ij}\rangle^{\otimes m_{ij}}\langle \Phi_{ij}\rangle^{\otimes m_{ij}}|$. Since $m=S(F)N$ maximally entangled pairs have been consumed during the process, the total yield of the breeding protocol is given by $D=1-S(F)$. Note that $S(F)= S(\rho_W)$, where $S(\rho_W) = -{\rm tr}(\rho_W \log_2 \rho_W)$ is the von-Neumann entropy of $\rho_W$. It follows that for Werner states, breeding only works if the initial fidelity is sufficiently high, $F \gtrsim 0.81$.
A similar kind of reasoning can be applied to hashing, where no pre--purified pairs are required. The analysis is slightly more involved since one has to take a kind of back action (influence of the remaining pairs because the measured pair was not in a pure state) into account. The yield of the hashing procedure is, however, exactly the same as for breeding. For Bell--diagonal states, one obtains that the yield of hashing protocols is given by $D(\rho) = 1-S(\rho)$.
The yield of hashing and breeding protocols can be further improved if two--way classical communication is allowed, see e.g. Ref. \cite{Ve04,Ho06}. The underlying principle for this improvement is discussed in Ref. \cite{Ho06}. In addition, one can generalize hashing and breeding to $d$ dimensional systems for prime $d$ \cite{Vo04}. The optimal entanglement distillation protocol for two--way classical communication is in general unknown (see however \cite{Ho04,Gh04}). Only for specific two qubit states, for instance incoherent mixtures of two Bell states, the known upper bounds on the yield coincide with the achievable rate for known protocols, in this case the hashing protocol. When assuming only one--way classical communication, the problem becomes tractable. In fact, the optimal distillation protocol for one--way classical communication was obtained in Ref. \cite{De04}.
\section{Quantum error correction \& entanglement purification}\label{relation}
The two possible solutions to the problem of quantum communication over noisy channels, quantum error correction and entanglement purification plus teleportation, have already been discussed in Sec. \ref{ECC}. Here we show that the approaches are in fact equivalent when considering one--way classical communication. In addition, we discuss a systematic way how to construct entanglement purification protocols from quantum error correction codes. We treat not only the (straightforward) case of one--way protocols, but also protocols that make use of two--way classical communication.
\subsection{Equivalence between QECC and one-way EPP}
The close relation between quantum error correction codes and schemes based on one--way classical communication has been proven in an early paper \cite{Be96}. Consider a quantum channel ${\cal E}$ and the associated bipartite mixed state \begin{eqnarray}\label{EfromcalE}
\hat E = \mbox{$1 \hspace{-1.0mm} {\bf l}$} \otimes {\cal E} |\Phi\rangle\langle \Phi| \end{eqnarray}
that is obtained by sending part of a maximally entangled state $|\Phi\rangle$ through the channel ${\cal E}$. Consider also the channel $\hat{\cal E}$ which is generated when using a state $E$ for teleportation. Notice that for Pauli--diagonal channels ${\cal E_P}$ (see Eq. \ref{Paulidiag}) we have $\hat{\cal E_P} = {\cal E_P}$ and $\hat E_P =E_P$.
In Ref. \cite{Be96} two inequalities are shown which establish this relation: \begin{itemize} \item[(i)] $D_E^{\rightarrow} \geq Q^\rightarrow_{\hat{\cal E}}$ \item[(ii)] $D_{\hat E}^{\rightarrow} \leq Q^\rightarrow_{{\cal E}}$ \end{itemize} The two inequalities are proven by establishing explicit protocols. Regarding (i), one considers the QECC which leads to channel capacity $Q^\rightarrow_{\hat{\cal E}}$. The second particles of $m$ locally prepared maximally entangled pairs are encoded using the QECC and teleported to $B$ using several copies of the mixed state $E$. At $B$, the decoding operation is applied and errors are corrected. This leads to $m$ maximally entangled states, and we have in fact described for any $E$ a one--way entanglement purification protocol which reaches equality in (i), and hence (i) is fulfilled.
Regarding (ii), one considers the entanglement purification protocol ${\cal P}$ that leads to $D_{\hat E}^{\rightarrow}$. One creates several copies of the bipartite mixed state ${\hat E}$ by sending the second particle of a maximally entangled states through the channel, and uses ${\cal P}$ to generate maximally entangled pairs with rate $D_{\hat E}^{\rightarrow}$. These pairs are then used for (perfect) teleportation, and we have constructed in this way a coding scheme which reaches equality in (ii), and hence (ii) is fulfilled.
Notice that in the case of Pauli--diagonal channels (and also for some other channels), (i) and (ii) show the {\em equivalence} between one--way entanglement purification and quantum error correction in the sense that one--way distillable entanglement and channel capacity are the same, \begin{eqnarray} D_E^\rightarrow = Q^\rightarrow_{{\cal E}}. \end{eqnarray}
\subsection{One-- and two--way entanglement purification protocols from CSS codes}\label{EPPfromQECC}
For Pauli--diagonal noise channels (Eq. \ref{Paulidiag})), also a direct way how to derive a QECC from certain one--way entanglement purification protocols has been established in Ref. \cite{Be96}. One can turn the construction around, which leads to a constructive way of obtaining one--way entanglement purification protocols from a certain class of QECC, the CSS codes \cite{Ca96,St96}. This approach can also be generalized, as shown by Aschauer in his PhD thesis \cite{AsPhD}, in the sense that not only entanglement purification protocols using one--way classical communication, but also protocols making full usage of two--way classical communication can be constructed from CSS codes (see also Refs. \cite{Ma02,Am03} for alternative approaches). In particular, for each CSS code that uses $n$ physical qubits to encode $k$ qubits, one can construct an entanglement purification protocol that operates on $n$ initial copies of two-qubit states and produces $k$ purified pairs as output.
There are two possible operational modes for such entanglement purification protocols \begin{itemize} \item[(i)] error correction mode \item[(ii)] error detection mode \end{itemize} In case of (i) only one-way classical communication is used, and the $k$ output pairs are kept deterministically. Measurements on the remaining $n-k$ copies are used to gain information on the ensemble, and the measurement outcomes determine the error correction operation to be applied to the remaining copies. As shown in Ref. \cite{AsPhD}, entanglement purification protocols can also run in an alternative mode (ii) where one makes use of two--way classical communication. The information gathered in the measurement of the $(n-k)$ pairs is used to decide whether the remaining pairs should be kept or discarded. The ones that are kept have a higher fidelity than before. This operational mode is the standard mode for recurrence protocols as discussed above, and in fact turns out to provide a larger purification range and favorable error thresholds.
We now illustrate this construction (for details we refer the reader to Ref. \cite{AsPhD}; see also Fig. \ref{AsFig}). To this aim, consider a Pauli--diagonal channel ${\cal E_P}$ which we will denote simply by ${\cal E}$ here. Consider also a mixed state $E$ which is obtained by sending one particle of a maximally entangled state $|\Phi\rangle$ through the channel, Eq. (\ref{EfromcalE}). For any CSS code described by coding operations ${\cal C}$ and decoding operations ${\cal D}$, it is shown how to obtain an entanglement purification protocol. Notice that ${\cal C},{\cal D}$ are Clifford networks (i.e. composed only of Clifford operations), where ${\cal C}$ consists of an encoding unitary operation $U_C$, while the decoding operation is given by a unitary operation $U_D = U_C^\dagger$ followed by single qubit measurements in the eigenbasis of $\sigma_z$ on the last $n-1$ qubits. We make use of the identity \begin{eqnarray}\label{phipluseq} U_A\otimes \mbox{$1 \hspace{-1.0mm} {\bf l}$}_B \ket{\Phi^+}_{AB}=\mbox{$1 \hspace{-1.0mm} {\bf l}$}_A\otimes U_B^T \ket{\Phi^+}_{AB}, \end{eqnarray} which holds not only for all unitary operations $U$, but in fact for any linear operator.
\begin{figure}\label{AsFig}
\end{figure}
We consider first a QECC where one qubit is encoded into $n$ physical qubits. The transmission of (unknown) quantum information from $A$ to $B$ using such a QECC takes place by encoding the unknown state at $A$ into $n$ qubits, sending it through the noisy channel ${\cal E}^{\otimes n}$ and decoding it at $B$, where the decoding operation also includes error correction. Let $\rho_{\rm in}$ the state to be sent and denote by $P_{\bm 0}= |0\rangle^{\otimes n-1}\langle 0|$ the state of $n-1$ auxiliary qubits. The final state of this procedure is described as \begin{eqnarray}\label{finalstate} \rho_{\rm out} = {\cal D}\circ {\cal E}^{\otimes n} \circ {\cal C} (\rho_{\rm in}\otimes P_{\bm 0}). \end{eqnarray}
Consider now an entanglement--based version of the protocol, which involves the following steps: \begin{itemize}
\item[(i)] distribute $n$ copies of a maximally entangled state $|\phi^+\rangle$ through the noisy quantum channel ${\cal E}^{\otimes n}$, \item[(ii)] apply the coding operation ${\cal C}^T$ in $A$, and the decoding operation ${\cal D}$ in $B$. The coding operation ${\cal C}^T$ is defined as the application of the unitary operation $U_C^T$ followed by single qubit $\sigma_z$ measurements on the last $n-1$ qubits, and similarly for ${\cal D}$. \item[(iii)] use the resulting single entangled pair to teleport the unknown state $\rho_{\rm in}$ from $A$ to $B$, where the final correction operations not only depends on the result of the Bell measurement in the teleportation protocol, but also on the measurement outcomes on the qubits in $A$ and $B$ of the last $n-1$ entangled pairs. \end{itemize}
Consider first the case where ${\cal E} = \hat{\mbox{$1 \hspace{-1.0mm} {\bf l}$}}$, i.e. a noiseless quantum channel. One can use the identity Eq. (\ref{phipluseq}) to show that one ends up with a maximally entangled pair that can be used for perfect teleportation. In a similar way, the equivalence of the QECC protocol and the entanglement based protocol can be established, where in the latter case the proper (Pauli) correction operations need to be applied. To establish this equivalence, it is essential that the noisy channels are Pauli--diagonal and that the coding and decoding operation are of Clifford type, because in this case the order of all the mentioned operations can be exchanged (up to additional, correctable local Pauli operations) and one can make use of Eq. (\ref{phipluseq}). In fact, one finds for such coding and decoding operations that the output state of the entanglement--based version of the protocol is again given by Eq. (\ref{finalstate}).
Steps (i) and (ii) of the above construction (together with a suitable correction operation) can be interpreted as an {\em entanglement purification protocol}. This is due to the fact that an entangled state with improved fidelity can be created in this way whenever the QECC allows one to reduce the influence of channel noise. Using standard error correcting CSS codes, one obtains one--way entanglement purification protocols, as the measurement results are only used to determine the appropriate correction (Pauli) operation. However, here one has also the freedom to {\em discard} the resulting pair for certain measurement outcomes, which corresponds to the usage of an {\em error detection code} \cite{AsPhD}. Information exchange between $A$ and $B$ of the measurement outcomes is required, and hence two--way classical communication is involved. Notice that in entanglement purification one is allowed to discard certain pairs. This is in contrast to the direct transmission of quantum information, where discarding the state would yield to a loss of quantum information. One can simply repeat the purification protocol until one succeeds in producing a purified output pair. Only then the (unknown) quantum information is teleported from $A$ to $B$. On the other hand, when sending encoded quantum information directly through the noisy channel, usage of error detection codes leads to unrecoverable {\em loss} of quantum information. Hence two--way entanglement purification protocols constructed in this way are superior to quantum communication schemes based on QECC.
Notice that the same construction can also be applied for CSS codes where $k$ qubits are encoded into $n$ physical qubits, and one obtains entanglement purification protocols which produces from $n$ noisy entangled pairs $k$ purified pairs \cite{AsPhD}.
\section{Entanglement purification with imperfect apparatus}\label{realistic}
In this section, we investigate the performance of entanglement purification protocols under non--idealized conditions, i.e. for noisy local control operations. The main effect of noise is that no longer maximally entangled states can be produced, and the achievable fidelity is limited to values smaller than unity. Similarly, the required initial fidelity in the case of noisy local control operations is larger. While recurrence protocols remain applicable, hashing and breeding protocols become impractical.
\subsection{Distillable entanglement and yield}
Using the standard definition of distillability and yield is clearly inappropriate in the case of imperfect local operations. In particular, no maximally entangled pure states can be created in this case. This implies that {\em no} state will be distillable, and that the yield is zero. We therefore have to modify the definition of distillability and yield to account for these facts.
Rather than demanding that maximally entangled pure states can be created (fidelity $F=1$), we will consider the creation of states with certain {\em target fidelity}. Distillability refers in this case to the possibility of approximating a given target state $|\psi\rangle$ with fidelity $F \geq F_c$. Clearly, such a definition of distillability depends on both the required target state $|\psi\rangle$ and the desired fidelity $F_c$. To be more precise, we say that a given mixed state $\rho$ is distillable with respect to a target state $|\psi\rangle$ and fidelity $F_c$ if one can generate from possibly many copies of $\rho$ by means of local operations and classical communication a state $\sigma$ such that the fidelity of $\sigma$ with respect to $|\psi\rangle$ is larger or equal than $F_c$, $F=\langle \psi|\sigma|\psi\rangle \geq F_c$.
We consider the yield of purification procedures corresponding to this notion of distillability, $D_{\rho,F_c}$. In this case, however one needs to specify the exact structure of target states. In particular, when considering general distillation procedures (e.g. $N \to M$ protocols), one obtains as output a mixed state $\Gamma$ of a large number of particles. Here we will demand that the output state $\Gamma$ is a {\em tensor product} of states $\sigma_k$, $\Gamma = \sigma_1 \otimes \ldots \otimes \sigma_M$, where each of the $\sigma_k$ fulfills $\langle \psi|\sigma_k|\psi\rangle \geq F_c$. That is, we require that after the purification procedure one possesses independent copies of the state with desired fidelity. One may also use the weaker criterion that all reduced density operators $\tilde\sigma_k$ (corresponding to different output ``copies'' of the output state) have fidelity $F \geq F_c$, where $\tilde \sigma_k$ are obtained from $\Gamma$ by tracing out all particles but the ones corresponding to state $k$. In this case, however, it is not clear whether the different output states can be independently used for all applications. While their fidelities certainly fulfill $F \geq F_c$, there might be classical correlations among the output states that are limiting their applicability, e.g. for security applications such as key distribution.
In this context it would be interesting to see whether the definition of yield with respect to fidelities of reduced density operators is equivalent to the one we use here. To this aim, one would need to show that one can produce from an ensemble of states where all reduced density operators have a sufficiently high fidelity an ensemble which consists of a tensor product of copies, where the size of the ensembles might be diminished by a sub--linear amount, or the fidelity be reduced by some (arbitrarily small) $\delta_F$. Such a ``purification of classical correlations'' has, however, not been reported so far.
\subsection{Error model}\label{Errormodel}
To analyze the influence of noisy local operations, we will consider a simple error model where only two--qubit operations are noisy, and the noise is of a simple form. More general error models, including correlated noise and also errors in measurements, have been analyzed, leading to a similar qualitative behavior of entanglement purification protocols \cite{Du98,Du03QC,Gi99}.
We model a noisy two--qubit operation $U$ by first applying local noise to each of the qubits, followed by the perfect unitary operation $U$, \begin{eqnarray}\label{noisygate} {\cal E}_{kl} \rho= U_{kl} [{\cal M}_k{\cal M}_l \rho ] U_{kl}^\dagger. \end{eqnarray} We will mainly assume local completely positive maps ${\cal M}_k$, ${\cal M}_l$ corresponding to white noise (depolarizing channels), \begin{eqnarray} {\cal M}_k \rho = p\rho + (1-p)\frac{1}{4} \sum_{j=0}^3 \sigma_j^{(k)} \rho \sigma_j^{(k)}\label{localmap}. \end{eqnarray} In some cases, we will consider even more restricted noise models, namely local dephasing channels (or phase flip channels), \begin{eqnarray} {\cal M}_k^P \rho = p\rho + (1-p)\frac{1}{2} (\rho+ \sigma_3^{(k)} \rho \sigma_3^{(k)}), \end{eqnarray} and local bit--flip channels, \begin{eqnarray} {\cal M}_k^B \rho = p\rho + (1-p)\frac{1}{2} (\rho+ \sigma_1^{(k)} \rho \sigma_1^{(k)}). \end{eqnarray}
\subsection{Bipartite recurrence protocols}
We start by analyzing the BBPSSW protocol, where we consider the error model specified in Sec. \ref{Errormodel} with local white noise, Eq. (\ref{localmap}). Given two copies of a Werner state Eq. (\ref{rhoWerner}), the influence of noisy local control operations ---in this case noisy CNOT operations--- can be readily obtained. The action of noisy bilateral CNOT operations is the same as applying noiseless bilateral CNOT operations to two copies of Werner states with reduced fidelity. In particular, one finds that the parameter $x$ is mapped to $xp^2$ due to the local depolarizing noise. That is, one applies the original protocol to two copies of Werner states $\rho_W(xp^2)$. Rewriting Eq. (\ref{Fout}), i.e. the fidelity of output state as function of input state, in terms of parameter $x=(4F-1)/3$, one obtains $x'=(4x^2+2x)/(3x^2+3)$. Taking into account the effect of noisy local operations, i.e. the reduction of $x$, we obtain that the output state after one purification step is again a Werner state $\rho_W(x')$ with \begin{eqnarray} x'=\frac{4x^2p^4+2xp^2}{3x^2p^4+3}.\label{imperfectEPP} \end{eqnarray} The purification curve (fidelity of output state plotted against fidelity of input state) corresponding to the noiseless protocol is shifted down (see Fig. \ref{purificationcurve}).
\begin{figure}\label{purificationcurve}
\end{figure}
It is now straightforward to determine the purification range of the protocol, i.e. maximal reachable fidelity as well as the minimal required fidelity such that entanglement purification can be successfully applied. These quantities are given by the fixed points of the map Eq. (\ref{imperfectEPP}). One finds \begin{eqnarray} x_\pm= \frac{2}{3} \pm \frac{1}{3} \sqrt{4 +6 p^{-2} - 9 p^{-4}},
\end{eqnarray} where the maximum reachable fidelity $F_{\rm max}=(3x_+ +1)/4$ and the minimum required fidelity $F_{\rm min}=(3x_- +1)/4$. The threshold value for $p$ such that a finite purification interval remains (i.e. $x_+ > x_-$) is given by $p_{\min}=0.9628$. This implies that errors of the order of four percent are tolerable.
One can perform a similar analysis for the DEJMPS and (nested) entanglement pumping protocol. There, the fixed points of the corresponding non--linear maps are more difficult to obtain analytically. One can, however, perform the analysis numerically and obtains the following results \cite{Du98,Du03QC} \begin{itemize} \item[(i)] the maximum reachable fidelity $F_{\rm max}$ for the DEJMPS protocol is significantly higher than for the BBPSSW protocol; \item[(ii)] the minimal required fidelity $F_{\rm min}$ for the DEJMPS is significantly smaller than for the BBPSSW; \item[(iii)] the threshold for noisy operations, described by $p_{\rm min}$ is smaller for the DEJMPS protocol; \item[(iv)] the reachable fidelity, minimum required fidelity and threshold for noisy operations seem to be the same for nested entanglement pumping and for the original DEJMPS protocol \cite{Du03QC}. \end{itemize} When assuming errors in local operations corresponding to correlated white noise and errors in measurements of same order of magnitude \cite{Du98}, one finds tolerable errors of about three percent for the BBPSSW protocol, and five percent in case of the DEJMPS protocol.
\subsection{$N \to M$ protocols}
In a similar way, one can analyze the influence of noise for general $N \to M$ protocols. In Ref. \cite{AsPhD}, a number of protocols corresponding to different CSS codes (via the construction presented in Sec. \ref{EPPfromQECC}) have been analyzed with respect to their purification range, maximal reachable fidelities and error thresholds. Both operational modes, error correction mode and error detection mode have been considered. As can be expected due to the equivalence to QECC, the correction modes turn out to have a smaller purification range and much stringent error thresholds. When compared to standard recurrence protocols that operate on two copies, some of these $N \to M$ purification protocols turn out to have an improved yield in certain regimes.
\subsection{Hashing protocols}
For perfect local operations recurrence protocols have zero yield, while hashing protocols, operating simultaneously on an asymptotically large number of copies, have a non--zero yield. For imperfect local operations, the situation changes drastically. When requiring output states to have only a certain $F \geq F_c$, one finds that recurrence protocols may have a non--zero yield as long as $F_c \leq F_{\rm max}$, i.e. as long as the required fidelity is smaller than the fidelity reachable by the protocol. At the same time, the hashing protocol fails completely in the case of imperfect local operations. The reason for this is that one operates on an infinite number of states $m \to \infty$ to reveal one bit of information. That is, one performs $m$ bilateral CNOT operations with a given copy always serving as target state. As each of the CNOT operations is noisy, noise is accumulated in the target state. Assuming that the target state was initially in a maximally entangled pure state, the target state ends up in a Werner state $\rho_W(p^{2m})$. Clearly, if the amount of noise is too big, no information about the remaining ensemble can be extracted. This is the case for sufficiently large $m$, in particular for $m \to \infty$, even if $p$ is close to 1. In other words, the information loss due to imperfect local operations exceeds the possible information gain per measurement (maximum one bit). This implies that hashing in its original form can not be applied in the case of imperfect local operations.
\section{Applications I}\label{applications1}
\subsection{Quantum communication and cryptography}
As discussed in Sec. \ref{ECC}, entanglement purification together with teleportation offers a way to achieve perfect transmission of unknown quantum information over noisy channels. This approach can also be used for quantum key distribution in the context of quantum cryptography.
In the case where not only the channels but also the local operations are imperfect, entanglement purification can still be applied. As we have seen in the previous section, one can increase the fidelity of entangled states --and hence the quality of the channel when using the purified entangled states for teleportation. More importantly, the entanglement produced by entanglement purification, although not perfect, is {\em private} \cite{As02}. That is, although no maximally entangled states can be produced, any eavesdropper will be factored out asymptotically. Hence a secret key can be established between two parties, even in the presence of noisy channels and imperfect apparatus \cite{As02}. This provides an alternative way of proving unconditional security of quantum key distribution, and is an important application of entanglement purification for quantum cryptography. We also mention that a direct and rigorous proof of the security of quantum cryptography that makes use of entanglement purification has been put forward (see e.g. Refs. \cite{Sh00,Go03,Ra06}).
\subsection{The quantum repeater}
In the context of quantum communication, a central problem is to go to large distances. When using photons, the absorption probability scales exponentially with the distance, and so does the influence of errors e.g. due to dephasing. This is a general, unavoidable feature of classical and quantum communication channels, and hence has to be dealt with in some way. The standard classical approach of amplifying the signal at intermediate repeater stations cannot be directly adopted in the case of quantum communication. This is due to the fact that general quantum signals, i.e. unknown quantum states, cannot be cloned or amplified.
One possible solution is given by entanglement--based communication schemes, where entangled pairs are distributed over noisy quantum channels and then used to transmit arbitrary quantum information via teleportation. A nested sequence of connection (i.e. entanglement swapping) and entanglement purification steps thereby provides a way to avoid an exponential scaling of the resources with the distance. This scheme is known as the quantum repeater (see e.g. \cite{Br98,Du98,Repeater,Ha06,RepeaterExp1,RepeaterExp2,RepeaterExp3,RepeaterExp4,RepeaterExp5,RepeaterExp6,RepeaterExp7,RepeaterExp8}). As shown in Refs. \cite{Br98,Du98}, one can establish entangled pairs over arbitrary distances with only polynomial overhead in the physical resources (number of required short distance pairs, parallel channels, repeater stations and time) even if local control operations are noisy (and local memory errors can be controlled \cite{Ha06}). The error thresholds for noisy local operations are of the order of percent.
A brief sketch of the repeater protocol follows: For communication over a distance $N l_0$ we split the channel into $N$ segments of length $l_0$ and place repeater stations at distances $k l_0, k=0,1,\ldots ,N$. We assume that $l_0$ is sufficiently small such that the resulting state, when sending part of a maximally entangled state through the channel, is still distillable entangled and the absorption probability is sufficiently small. For optical fibers, distances of (several tens) of kilometers are reasonable. For simplicity, we assume that we have $2^n$ such segments, i.e. $N=2^n$. Several copies of short distance elementary pairs shared between all repeater stations are generated and purified to some working fidelity $F_0$. At repeater stations 2, 4, 6, etc. two adjacent elementary pairs are connected by performing entanglement swapping, thereby generating entangled pairs over distance $2 l_0$ with reduced fidelity shared between repeater stations $(1,3); (3,5); \ldots$. Purification of several copies of these pairs to the working fidelity $F_0$ results in a similar situation as previously, however the elementary pairs extend now over the distance $2 l_0$. Proceeding in this way, we obtain a nested scheme where the distance of the entangled pairs is {\em doubled} at each nesting level. At nesting level $n$, we obtain long distance entangled pairs of length $2^n l_0$ with fidelity $F_0$. It is straightforward to check that this scheme leads to a polynomial scaling of the resources with distance \cite{Br98}. The setup of the repeater is schematically sketched in Fig. \ref{Rep1}, while Fig. \ref{Repeater_figurethreshold} shows a elementary purification loop. The total number of required elementary short distance pairs are shown in Fig. \ref{Dur_Repeater_nested}.
Variations of the scheme based on the usage of different purification protocols --e.g. (nested) entanglement pumping rather than standard DEJMPS protocol-- have also been discussed and lead to a significant reduction of spatial resources (storage particles) to $n+1 \propto \log(N)$ at the cost of increased, although still polynomial, temporal resources \cite{Br98,Du98}. Further improvement to a {\em constant} overhead in spatial resources is possible (see Ref. \cite{Repeater}).
\begin{figure}
\caption{Schematic sketch of the repeater setup. Connection and entanglement purification is used to generate large distance entanglement pairs with high fidelity.}
\label{Rep1}
\end{figure}
\begin{figure}
\caption{Purification loop: Connection of $L$ elementary pairs and re--purification to initial fidelity $F$. In the text, $L=2$ is assumed. Figure taken from Ref. \cite{Br98}.}
\label{Repeater_figurethreshold}
\end{figure}
\begin{figure}
\caption{Nested purification with an array of elementary EPR pairs. At each nesting level, $L$ elementary pairs are connected and re-purified using $M$ copies, where in the text $L=2$ is assumed. This yields to a polynomial scaling of the total elementary pairs with the distance. Figure taken from Ref. \cite{Br98}.}
\label{Dur_Repeater_nested}
\end{figure}
\subsection{Improving error thresholds in quantum computation}\label{EPPQC}
Under certain circumstances, entanglement purification can be used directly to weaken the requirements for fault--tolerant quantum computation \cite{Du03QC}. Consider a situation where $n$ systems, each of them possessing $d$ degrees of freedom, are available. For instance, one may think of $n$ neutral atoms or trapped ions, each of them constituting a $d$--level system. While typically only two of the levels are used for quantum computation, in principle more levels are available. In such a situation, one can show that the threshold for fault--tolerant quantum computation essentially only depends on the fidelity of single--system operations \cite{Du03QC}. Two--system operations, i.e. interactions between two systems, are typically more difficult to realize than single--system operations (e.g. operations on a single atom). However, it turns out that one can tolerate a noise level of more than $50\%$ for two--system operations, while still achieving fault tolerant quantum computation as long as the single system operations are of sufficiently high fidelity.
The basic idea is to use each $d$--level system to represent one qubit for computation, while the remaining degrees of freedom serve as auxiliary levels. The noisy two--system interaction serves to entangle auxiliary degrees of freedom, and one may use entanglement purification to increase the fidelity of this entanglement. Finally, high fidelity entangled states are used to realize two--system gates, e.g. by means of teleportation based gates \cite{Ci00,TBG1,TBG2,TBG3,TBG4,TBG5}. The fidelity of the two--system gate is essentially determined by the fidelity of the entangled state, which, in turn, is determined by the fidelity of single--system operations used in entanglement purification.
We remark that at least four auxiliary levels should be available. By using nested entanglement pumping, as discussed in Sec. \ref{nestedEPP}, it turns out that for parameter regimes of practical importance, a few (2-3) nesting levels are sufficient to obtain high fidelity entanglement. This translates into a total requirement of about 16 levels per system, and a required error threshold of about $10^{-5}$ for single system operations to achieve errors of $10^{-4}$ for (logical) two--system operations, which is sufficient to achieve fault tolerant quantum computation. This is illustrated in Fig. \ref{FigQC1}. The error rate of the physical two--system operation can, however, be almost arbitrarily large (more than $50 \%$), and the two--system gates can even be probabilistic.
\begin{figure}
\caption{Double logarithmic plot of achievable logical two--qubit gate error rate against single--particle error rate $p$ for fixed error rate of physical two--particle interaction of (a) $1.5*10^{-1}$, (b) $10^{-2}$. Curves from top to bottom correspond to no entanglement purification, entanglement pumping using 1, 2, 3 (or more) nesting levels respectively. Figure taken from Ref. \cite{Du03QC}.}
\label{FigQC1}
\end{figure}
A combination of (dynamical) decoherence free subspaces and entanglement purification is also conceivable. In Ref. \cite{Ta05PRL}, an entanglement purification protocol for spin degrees of freedom in electrically controlled semiconductor quantum dots has been put forward. The protocol is capable of purifying {\em encoded} states. The encoding is used to protect quantum information from dominant noise processes and is based on a dynamical decoherence free subspace. The purification protocol is capable to deal with other types of errors, where not only errors within the logical subspace, but also leakage errors can be handled. The purification protocol is constructed in such a way that it only makes use of resources available in such devices. One may also combine this encoded entanglement purification with a teleportation-based approach to implement non-local quantum gates, leading to an alternative way to obtain logical two-qubit gates for (fault-tolerant) quantum computation. Such a proposal for electrically controlled quantum dots has been put forward in Ref. \cite{Ta05Nature}.
\section{Entanglement purification protocols in multipartite systems}\label{multipartiteEPP}
We now turn to multipartite systems of $n$ spatially separated parties. We start by reviewing the concept of graph states, a family of $n$--qubit states of particular importance. Then we consider entanglement purification protocols for such graph states. The first protocol of this kind was introduced in Ref. \cite{Mu01} and further analyzed in Ref. \cite{Ma01}, and it is capable of distilling $n$--party GHZ states. Here, we will discuss recurrence and hashing protocols for all stabilizer states, or equivalently, all graph states. These protocols where introduced in Ref. \cite{Du03} and further elaborated in \cite{As04,Kr06}.
\subsection{Graph states}\label{graphstates}
We start by defining graph states. A graph $G=(V,E)$ is given by a set of $n$ vertices $V=\{1,2,\ldots ,n\}$ connected in a specific way by edges $E \in V^2$. To every such graph there corresponds a basis of $n$--qubit states $\{|\Phi_{\bm \mu}\rangle_G\}$, where each of the basis states $|\Phi_{\bm \mu}\rangle_G$ is the common eigenstate of $n$ commuting correlation operators $K_j^G$ with eigenvalues $(-1)^{\mu_j}$, ${\bm \mu} = \mu_1\mu_2\ldots \mu_n$. To relax notation, we will sometimes omit the index $G$ and assume that an arbitrary but fixed graph $G$ is considered. Graph states fulfill the set of eigenvalue equations \begin{eqnarray}
K_j^G |\Phi_{\bm \mu}\rangle_G = (-1)^{\mu_j}|\Phi_{\bm \mu}\rangle_G, \end{eqnarray} $j=1,\ldots,n$. The correlation operators are uniquely determined by the graph $G$ and are given by \begin{eqnarray} K_j= \sigma_x^{(j)} \prod_{\{k,j\} \in E} \sigma_z^{(k)}. \label{K} \end{eqnarray} A graph is called two--colorable if there exists two groups of vertices, $V_A$,$V_B$ such that there are no edges inside either of the groups, i.e. $\{k,l\} \not \in E$ if $k,l \in V_A$ or $k,l \in V_B$. For graph states associated with two--colorable graphs, which we call two--colorable graph states, we will split the multi--index ${\bm \mu}$ into two parts, ${\bm \mu}={\bm \mu}_A,{\bm \mu}_B$, belonging to subsets $V_A$ and $V_B$ respectively. See Fig. \ref{twoC} for examples of two-colorable graph states.
\begin{figure}\label{twoC}
\end{figure}
Graph states have first been introduced in Ref. \cite{Ra03}, generalizing the notion of cluster states as introduced in Ref. \cite{Br01}. For the related notion of graph codes see \cite{Sc01,Gra02}. A detailed investigation of their entanglement properties has been given in the paper by Hein et al. \cite{He03}, see also Ref. \cite{He06} for a recent review. Graph states occur in various contexts in quantum information theory, in which multi-party quantum correlations play a central role. Examples are multi-party quantum communication, measurement-based quantum computation, and quantum error correction. Prominent examples of two--colorable graph states are GHZ states, cluster states \cite{Br01} and codewords of error correction codes \cite{Sc01,Gra02} (see e.g. Ref. \cite{As04}). In fact, as shown in \cite{Ch04}, two colorable graph states are equivalent to codewords of CSS codes. We also remark that the correlation operators $\{K_j\}$ are the generators of the stabilizer group of the state $|\Phi_{\bm 0}\rangle_G$, and the corresponding description in terms of the stabilizers is also referred to as the stabilizer formalism \cite{Go97,Ho05}.
We will also consider mixed states $\rho$, which for a given graph $G$ can be written in the corresponding graph state basis $\{|\Phi_{\bm \mu}\rangle_G\}$, \begin{eqnarray}
\rho =\sum_{{\bm \mu},{\bm \nu}} \lambda_{{\bm \mu}{\bm \nu}} |\Phi_{\bm \mu}\rangle\langle \Phi_{\bm \nu}|.
\end{eqnarray} We will often be interested in fidelity of the mixed state, i.e. the overlap with some desired pure state, say $|\Phi_{\bm 0}\rangle_G$, $F= \langle \Phi_{\bm 0}| \rho |\Phi_{\bm 0}\rangle$. We remark that depolarization of $\rho$ to a standard form $\rho_G$, \begin{eqnarray}
\rho_{G}=\sum_{\bm \mu} \lambda_{{\bm \mu}}|\Phi_{{\bm \mu}}\rangle\langle\Phi_{{\bm \mu}}| \end{eqnarray} can be achieved by randomly applying correlation operators $K_j$ \cite{Du03,As04} which is a multi--local operation. The diagonal elements, in particular the fidelity, are left unchanged by this depolarization procedure. Note that both the notation and the description of the depolarization procedure are similar to the ones used for Bell states, which are in fact graph states with two vertices, connected by a single edge.
\subsection{Recurrence protocol for two--colorable graph states}\label{recurrenceMP}
In the following, we will discuss a family of entanglement purification protocols that allow one to purify an arbitrary two--colorable graph state. To be precise, for each two colorable graph there exists a purification protocol which allows one to obtain the pure state $|\Phi_{\bm 0}\rangle_G$ as output state, provided the initial fidelity is sufficiently large. The recurrence scheme \cite{Du03,As04} for purifying a two--colorable graph state is very similar to the BBPSSW and DEJMPS protocol for purifying Bell pairs. We consider two sub--protocols, $P1$ and $P2$, each of which acts on two identical copies $\rho_1=\rho_2=\rho$, $\rho_{12}\equiv\rho_1\otimes \rho_2$. The basic idea consists again in transferring (non--local) information about the first pair to the second, and reveal this information by measurements.
In sub--protocol $P1$, all parties who belong to the set $V_A$ apply local CNOT operations to their particles, with the particle belonging to $\rho_2$ as source, and $\rho_1$ as target (see Fig. \ref{MEPPsetup}). Similarly, all parties belonging to set $V_B$ apply local CNOT operations to their particles, but with the particle belonging to $\rho_1$ as source, and $\rho_2$ as target. The action of such a multilateral CNOT operation is given by \cite{Du03} \begin{eqnarray}
|\Phi_{{\bm \mu}_{\bf A},{\bm \mu}_{\bf B}}\rangle|\Phi_{{\bm \nu}_{\bf A},{\bm \nu}_{\bf B}}\rangle\rightarrow|\Phi_{{\bm \mu}_{\bf A},{\bm \mu}_{\bf B}\oplus{\bm \nu}_{\bf B}}\rangle|\Phi_{{\bm \nu}_{\bf A}\oplus {\bm \mu}_{\bf A},{\bm \nu}_{\bf B}}\rangle \label{psitopsi1} \end{eqnarray} where ${\bm \mu}_{\bf A}\oplus{\bm \nu}_{\bf A}$ denotes bitwise addition modulo 2.
The second step of subprotocol $P1$ consists of a measurement of all particles of $\rho_2$, where the particles belonging to set $V_A$ [$V_B$] are measured in the eigenbasis $\{|0\rangle_x,|1\rangle_x\}$ of $\sigma_x$ [$\{|0\rangle_z,|1\rangle_z\}$ of $\sigma_z$] respectively. The measurements in sets $A$ [$B$] yield results $(-1)^{\xi_j}$ [$(-1)^{\zeta_k}$], with $\xi_j,\zeta_k \in\{0,1\}$. Only if the measurement outcomes fulfill the condition $(\xi_j+\sum_{\{k,j\}\in E}\zeta_k){\rm mod}2=0 ~\forall j$ ---which implies that the eigenvalues of all corresponding correlation operators $K_j$, $j\in V_A$ are +1, or equivalently ${\bm \mu}_{\bf A}\oplus{\bm \nu}_{\bf A}={\bf 0}$--- the first state is kept. In this case, one finds that the remaining state is again diagonal in the graph--state basis, with new coefficients \begin{eqnarray}
\tilde\lambda_{{{\bm \gamma}_{\bf A}},{{\bm \gamma}_{\bf B}}} =\sum_{\{({{\bm \nu}}_{\bf B}, {{\bm \mu}}_{\bf B}) | {{\bm \nu}}_{\bf B} \oplus{{\bm \mu}}_{\bf B}={{\bm \gamma}_{\bf B}}\}} \frac{1}{2K}\lambda_{{{\bm \gamma}_{\bf A}},{{\bm \nu}_{\bf B}}}\lambda_{{{\bm \gamma}_{\bf A}},{{\bm \mu}_{\bf B}}},\label{mapP1} \end{eqnarray} where $K$ is a normalization constant such that ${\rm tr}(\tilde \rho)=1$, indicating the probability of success of the protocol.
\begin{figure}\label{MEPPsetup}
\end{figure}
In sub--protocol $P2$ the roles of sets $V_A$ and $V_B$ are exchanged. The action of the multilateral CNOT operation is in this case given by \begin{eqnarray}
|\Phi_{{\bm \mu}_{\bf A},{\bm \mu}_{\bf B}}\rangle|\Phi_{{\bm \nu}_{\bf A},{\bm \nu}_{\bf B}}\rangle\rightarrow|\Phi_{{\bm \mu}_{\bf A}\oplus{\bm \nu}_{\bf A},{\bm \mu}_{\bf B}}\rangle|\Phi_{{\bm \nu}_{\bf A},{\bm \nu}_{\bf B}\oplus{\bm \mu}_{\bf B}}\rangle, \label{psitopsi2} \end{eqnarray} which leads to new coefficients
\begin{eqnarray}\tilde\lambda'_{{{\bm \gamma}_{\bf A}},{{\bm \gamma}_{\bf B}}} =\sum_{\{({{\bm \nu}}_{\bf A}, {{\bm \mu}}_{\bf A})| {{\bm \nu}}_{\bf A} \oplus{{\bm \mu}}_{\bf A}={{\bm \gamma}_{\bf A}}\}} \frac{1}{2K}\lambda_{{{\bm \nu}_{\bf A}},{{\bm \gamma}_{\bf B}}}\lambda_{{{\bm \mu}_{\bf A}},{{\bm \gamma}_{\bf B}}},\label{mapP2} \end{eqnarray} for the case in which the protocol $P2$ was successful.
The total purification protocol consists in a sequential application of sub--protocols $P1$ and $P2$. While sub--protocol $P1$ serves to gain information about ${\bm \mu}_{\bf A}$, sub--protocol $P2$ reveals information about ${\bm \mu}_{\bf B}$. Typically, sub--protocol $P1$ increases the weight of all coefficients $\lambda_{{\bf 0},{\bm \mu}_{\bf B}}$, while $P2$ amplifies coefficients $\lambda_{{\bm \mu}_{\bf A},{\bf 0}}$. In total, this leads to the desired amplification of $\lambda_{{\bf 0},{\bf 0}}$.
The regime of purification in which these recurrence protocols can be successfully applied is rather difficult to determine analytically, due to the non--trivial structure of the non--linear maps describing the protocol. Numerical investigation have been performed in Ref. \cite{As04}, and we refer the interested reader to this article for details. For special noise models, e.g. phase noise, the purification regime can be determined analytically, and provable optimal protocols with respect to purification range and yield can be found \cite{Ka06,Ka06b}. The mixed states considered under such a phase noise model are, in fact, thermal states of many--body spin hamiltonians defined via the corresponding graphs \cite{Ka06}. We remark here that the fidelity does not provide a suitable measure to compare purification regimes for different number of particles $n$, as typically the required fidelity will decrease exponentially for all states. This is related to the exponential growth of the dimension of the Hilbert space with the number of particles $n$. One can alternatively consider the maximum acceptable amount of local noise per particle such that the state remains distillable by means of the recurrence protocol. That is, one assumes that each of the particles belonging to a given graph state is sent through a noisy quantum channel (e.g. a depolarizing channel) to its final location. One then finds for linear cluster states (or, more generally, all graph states with a constant degree) that the maximum acceptable amount of noise per particle is essentially independent of the particle number. For GHZ states, however, the acceptable amount of noise per particle decreases with increasing particle number. That is, GHZ states become more and more difficult to purify as the number of particles increases.
\subsubsection{Example: Binary--type mixture}
It is elucidating to consider the purification of a special family of states in some detail. We consider the example of mixed states of the form \begin{eqnarray}
\rho_{\cal A}\equiv \sum_{{\bm \mu}_{\bf A}} \lambda_{{\bm \mu}_{\bf A},{\bf 0}} |\Phi_{{\bm \mu}_{\bf A},{\bf 0}}\rangle \langle \Phi_{{\bm \mu}_{\bf A},{\bf 0}}|. \label{rhoA} \end{eqnarray} These states arise e.g. in a (hypothetical) scenario were all particles within set $V_A$ are only subjected to phase flip errors (described by $\sigma_z$), while all particles within set $V_B$ are subjected to bit flip errors ($\sigma_x$). The iterative application of protocol $P1$ is sufficient to purify states of the form Eq. (\ref{rhoA}), as only information about ${\bm \mu}_{\bf A}$ has to be extracted. A single application of protocol $P1$ leads again to a state of the form $\rho_{\cal A}$, with new coefficients \begin{eqnarray} \tilde\lambda_{{\bm \mu}_{\bf A},{\bf 0}} = \lambda_{{\bm \mu}_{\bf A},{\bf 0}}^2/K,\label{binary1} \end{eqnarray}
where $K=\sum_{{\bm \mu}_{\bf A}} \lambda_{{\bm \mu}_{\bf A},{\bf 0}}^2$ is a normalization constant indicating the probability of success of the protocol. That is, the largest coefficient is amplified with respect to the other ones. Iteration of the protocol $P1$ thus allows one to produce pure graph states $|\Phi_{{\bf 0},{\bf 0}}\rangle$ with arbitrary high fidelity, given the coefficient $\lambda_{{\bf 0},{\bf 0}}$ is larger than all other coefficients $\lambda_{{\bm \mu}_{\bf A},{\bf 0}}$. The family of states $\rho_{\cal A}$ includes states up to rank $2^{n_A}$, where $n_A$ denotes the number of particles in group $A$. Depending on the corresponding graph, $n_A$ can be as high as $n-1$ and hence the rank can be as high as $2^{n-1}$.
As a concrete example, consider the one parameter family $\rho_{\cal A}(F)$ with $\lambda_{{\bf 0},{\bf 0}}=F$, $\lambda_{{\bm \mu}_{\bf A},{\bf 0}}=(1-F)/(2^{n_A}-1)$ for ${\bm \mu}_{\bf A} \not={\bf 0}$, where $F$ is the fidelity of the desired state. Application of protocol $P1$ maintains the structure of those states and leads to \begin{eqnarray} \tilde F = \frac{F^2}{F^2+ (1-F)^2/(2^{n_A}-1)}. \end{eqnarray} This map has $\tilde F=1$ as attracting fixed point for all states with $F\geq 1/2^{n_A}$. The probability of success for a single step is given by $p=F^2+ (1-F)^2/(2^{n_A}-1)$.
We also mention that other entanglement purification protocols for two-colorable graph states that makes use of error correction and error detection ideas have been introduced in Ref. \cite{Go06}. In particular, states corresponding to different graphs are used to purify a given noisy two-colorable graph state. The performance and robustness of these protocols with noisy gates is studied analytically, and it is shown that schemes with improved yield and scaling behavior can be obtained.
\subsection{Hashing protocol for two--colorable graphs states}\label{HashingMP}
In a similar way, one can design a hashing protocol for any two--colorable graph state. The first protocol of this type, capable of purifying GHZ states with non--zero yield, was introduced in Ref. \cite{Ma01}. Hashing protocols for arbitrary two--colorable graph states were presented in Refs. \cite{Ch04,As04}. The central tool in these protocols is already evident from Eqs. (\ref{psitopsi1}) and (\ref{psitopsi2}). These equations state how information about indices is transferred from one state to another. Information about all indices belonging to set $V_A$ is thereby transferred from copy one to copy two by the multilateral CNOT operations as specified in the first step of protocol $P1$, while information transfer occurs for all indices corresponding to set $V_B$ when the direction of CNOT operations is reversed (as it is done in $P2$). Again, by determining the parity of the bit values for random subsets ---which is done in a similar way as for Bell pairs, but here all bits belonging to set $V_A$ or set $V_B$ can be determined simultaneously---, one can learn the required information in such a way that the remaining ensemble is a tensor product of pure graph states (one needs to learn the classical information which non--local state is at hand). Notice that information transfer also takes place in the opposite direction, which is however not used.
The yield of the hashing protocol approaches unity for any state diagonal in the graph state basis with $\lambda_{{\bf 0}} \rightarrow 1$, independent of the specific form of the state. This implies that a given mixed state of sufficiently high fidelity $F$ can be purified with non--zero yield using the hashing protocol (combined with the depolarization procedure). Protocols with improved yield (by optimizing the information gain of the measurements) have been developed in Ref. \cite{Ho05a}.
\subsection{Recurrence protocol for all graphs states}\label{RecurrenceMP_all}
For the protocols described in the previous section, it is crucial that the underlying graphs of the states to be purified are two--colorable. It is, however, possible to obtain protocols that can purify {\em all} graph states. These protocols have been put forward in Ref. \cite{Kr06} and we briefly describe the basic idea here. Unlike in the case of two--colorable graph states, we have that the protocols do not operate on identical copies corresponding to the same graph, but on states corresponding to {\em different} graphs.
Consider a $k$--colorable graph $G$, i.e. a graph where the set of vertices can be divided into $k$ non--connected subsets $\{V_1,V_2,\dots,V_k\}$. We denote the corresponding multi--indices by ${\bm \mu_1}, \ldots {\bm \mu_k}$. The purification of the associated graph state $|\Phi_{\bm 0}\rangle_G$ requires the alternating application of $k$ purification protocols. The $j^{\rm th}$ protocols serves to reveal information about ${{\bm \mu}_{j}}$, i.e. the indices associated to vertices in the set of qubits $V_j$.
We will describe the $j^{th}$ purification protocol ${\mathcal P}_j$ in the following. The protocol consists essentially of two steps \begin{itemize} \item[(i)] Generation of a two--colorable graph state corresponding to graph $g_j$ from two copies of a graph state corresponding to graph $G$. \item[(ii)] Purification of one copy of a graph state corresponding to graph $G$ with help of the graph state corresponding to graph $g_j$. \end{itemize}
Regarding (i), we define a two-colorable graph $g_j$ associated to $G$ as one which contains only the edges between the set $V_j$ and the remaining sets $\{V_i, i\not =j\}$, but where edges between the remaining sets are erased (see Fig. \ref{fig:G and gj} for an illustration). That is, the sets $\{V_i, i\not=j \}$ form a new set $V_{\bar j}=V\setminus V_j$. As shown in Ref. \cite{Kr06}, one can generate from two (noisy) copies of a graph state corresponding to graph $G$ a single (noisy) copy corresponding to graph $g_j$ by applying CNOT gates from the second to the first copy for all parties in the group $V_j$, and from the first copy to the second copy for all other parties, followed by $\sigma_z$ measurements performed on all particles of the second copy. Notice that CNOT gates between different copies do not only lead to a transfer of information, but also change the shape of the graph. In particular, a gate ${\tiny CNOT}_{a\to b}$ introduces new edges between the control qubit $a$ and all neighbors of the target qubit $b$ (or erases them if they are already there). In case of two--colorable graphs, the overall effect of the multilateral CNOT gates is that no additional edges are introduced (or more precisely they cancel each other). Here, the total effect of the CNOT operations is that all edges between groups $V_i$ and $V_l$ are erased whenever $i,l \not = j$ in the first copy, i.e. a two--colorable graph state corresponding to the graph $g_j$ is produced.
\begin{figure}
\caption{A 3-colorable graph $G$ and the 3 corresponding two-colorable sub-graphs $g_1$, $g_2$, and $g_3$. $g_1$ corresponds to the red color (vertices 2,4 and 6), $g_2$ to the green color (vertex 7) and $g_3$ to the blue (vertices 1,3 and 5). Figure taken from Ref. \cite{Kr06}. }
\label{fig:G and gj}
\end{figure}
An alternative to the $\sigma_z$ measurement of the second copy --which effectively simply erases the corresponding qubits-- is given by measurements in a different basis. More precisely, one may measure $\sigma_x$ on all qubits in group $V_j$, and $\sigma_z$ on all remaining qubits, and accepts the produced state only if the expectation value of the corresponding correlation operators $K_i$, $i \in V_j$ are +1. In this way, not only the shape of the graph is changed, but also a {\em purification} takes place as information about the first copy is revealed by the measurements of the second copy.
In step (ii), we take two noisy states in such a way that the first state corresponds to graph $G$, $\rho_G$, and the second state to graph $g_j$, $\rho_{g_j}$. The parties in the group $V_j$ apply CNOT gates from the second (control) state $\rho_{g_j}$ to the first (target) state $\rho_G$, while the remaining parties apply the CNOT gate in the opposite direction. The second state is measured in the eigenbasis of $\sigma_x$ for all parties in the group $V_j$ and in the eigenbasis of $\sigma_z$ for the remaining parties. The first state $\rho_G$ is kept only if all expectation values of the corresponding correlation operators $K_i$, $i \in V_j$ are +1. Again, the CNOT operation transfers information from the first state to the second, which is revealed by the measurements. The choice of different graphs for state 1 and 2 guarantees that the graph corresponding to the first copy remains unchanged. More precisely the action of CNOT operations is given by, \begin{eqnarray}
|\Phi_{\bm \mu_{j}, \bm\mu_{{\bar j}}}\rangle_G|\Phi_{\bm \nu_{j}, \bm \nu_{{\bar j}}}\rangle_{g_j} \rightarrow |\Phi_{\bm \mu_{j},\bm \mu_{{\bar j}}\oplus \bm \nu_{{\bar j}}}\rangle_G|\Phi_{\bm \nu_{j} \oplus \bm\mu_{j} ,\bm \nu_{{\bar j}}}\rangle_{g_j}, \end{eqnarray} which shows the transfer of information about the stabilizer eigenvalues between the two states. Notice the close similarity with Eq. (\ref{psitopsi2}), but also the crucial difference that here states corresponding to two different graphs are involved. After a successful purification step (i.e. obtaining proper measurement outcomes) one finds that the new matrix elements of $\rho'_G$ are given by \begin{eqnarray} {\lambda}'_{\bm{\gamma}_j,\bm{\gamma}_{\bar{j}}}=\frac{1}{\kappa}\sum_{\left\{(\bm{\mu}_{\bar{j}},\bm{\nu}_{\bar{j}})\mid \bm{\mu}_{\bar{j}}\oplus \bm{\nu}_{\bar{j}}=\bm{\gamma}_{\bar{j}}\right\}} \lambda_{\bm{\gamma}_j,\bm{\mu}_{\bar{j}}}\tilde{\lambda}_{\bm{\gamma}_j,\bm{\nu}_{\bar{j}}} \end{eqnarray} where the $\lambda$'s are the diagonal coefficients of the state $\rho_G$, and $\tilde \lambda$'s are the diagonal coefficients of the state $\rho_{g_j}$. As consequence, elements of the form $\lambda_{\bm{0},\bm{\gamma}_{\bar{j}}}$ are increased. One may say that purification takes place with respect to indices corresponding to parties in $V_j$.
The whole purification protocol consists of a sequential application of the sub-protocols ${\mathcal P}_j$ corresponding to all colors $j=1,\dots,k$. Even though there is a back-action of noise for the colors which are not purified for the step $j$, one obtains an overall increase of the fidelity $\lambda_{\bm{0}}$ if the fidelity of the initial state is sufficiently high. In fact, $\lambda_{\bm{0}}=1$ is an attractive fixed point of the protocol under the ideal local operations, which can be checked numerically.
\subsection{Purification of stabilizer states using stabilizer error correcting codes}
An alternative approach to purifying all stabilizer states has been taken in Ref. \cite{Gl06}. The multipartite entanglement purification protocols discussed there are $N \to M$ protocols, and stabilizing operators corresponding to error correction codes are measured locally on several copies of the stabilizer states to be purified. A link between the state to be purified and the code that can be used for purification is given. In particular, it is found that CSS states can be purified by CSS codes, while general stabilizer states can be purified by CSS-H codes.
\subsection{Breeding protocol for all graphs states}\label{HashingMP:all}
It is straightforward to construct a breeding protocol for all graph states using the ingredients presented in the previous section. This is also done in Ref. \cite{Kr06}. Several perfect copies of graph states corresponding to a graph $g_j$ are used to learn information about indices corresponding to qubits $V_j$ by means of a parity check, i.e. operating with CNOT gates on a (random) subset of copies with the perfect copy as a target. This is done for all groups $V_j$ independently, until complete knowledge about the ensemble of states at hand is obtained and hence $N$ copies of the pure graph state corresponding to graph $G$ are produced. Using the procedure described previously, graph states corresponding to graphs $g_j$ are generated and given back at the end (they where quasi borrowed to perform the breeding protocol). Two copies of the graph state $|\Phi_{\bm 0}\rangle_G$ are required to generate a single copy of $|\Phi_{\bm 0}\rangle_{g_j}$. Notice that also here an improved (adaptive) protocol is conceivable, which would lead to a higher yield. In addition, the generation of $n$ copies of $|\Phi_{\bm 0}\rangle_{g_j}$ from $m$ copies of $|\Phi_{\bm 0}\rangle_{G}$ might be possible with a rate $n/m$ larger than $1/2$, which would also increase the yield of the protocol.
A breeding protocol for all stabilizer states (inspired by the approach of Ref. \cite{Gl06}) with improved yield has recently been put forward in Ref. \cite{Ho06a}.
\subsection{Entanglement purification of non--stabilizer states}
While all bipartite and multipartite entanglement purification protocols we have described so far purify stabilizer states, i.e. state which are eigenstates of local stabilizer operators, a multipartite entanglement purification protocol was recently obtained \cite{My05} that allows one to purify a non--stabilizer state, in particular a $W$-state \cite{Du99W}, \begin{eqnarray}
|W\rangle=\frac{1}{3}(|001\rangle + |010\rangle + |100\rangle). \end{eqnarray} This protocol is a $3 \to 1$ protocol and stabilizing operators corresponding to an orthonormal basis including the W state are measured locally. Among other interesting features such as mutually unbiased bases, it has not only the 3--particle $W$ state but also maximally entangled states shared between two of the parties as attracting fixed points \cite{My05}.
Furthermore, in Ref. \cite{Bo06}, a protocol for topological quantum distillation has been put forward. There, a new class of topological quantum error correction codes has been introduced, and it was shown how to perform arbitrary Clifford operations \cite{Go97} on encoded systems. As Clifford operations are the main ingredient of the purification protocols known so far, this provides a way of obtaining distillation protocols for topologically encoded systems.
\subsection{Multipartite recurrence protocols with noisy apparatus}
Similar as for bipartite entanglement purification protocols, one can performed an analysis for multipartite entanglement purification protocols \cite{As04} with noisy apparatus. In this case, an intriguing question is the scalability of the process, i.e. whether e.g. the threshold for purification depends on the number of parties $N$. Numerical results for the purification range (minimal required and maximal reachable fidelity) as well as error threshold for linear cluster states of different size are given in Fig. \ref{figurethreshold}. Again, errors of the order of several percent are tolerable.
\begin{figure}
\caption{Maximal reachable fidelity $F_{\rm max}$ and minimal required fidelity $F_{\rm min}$ plotted against error parameter $p$ (local operations) for density operators arising from single-qubit white noise. Curves from top to bottom (black, blue green, cyan, red) correspond to linear cluster states with $N=2,4,6,8,10$ particles. Figure taken from Ref. \cite{As04}.}
\label{figurethreshold}
\end{figure}
An important observation is that the threshold value $p_{\rm min}$ is for linear cluster states independent of the number of particles $n$. That is, also multipartite states of large number of particles can be successfully purified, and the requirements on local control operations are independent of the system size. This is not true when attempting to purify GHZ states \cite{As04}, where one finds that the required fidelity of local control operations depends on the particle number.
The qualitative difference of cluster and GHZ states can already be understood from an analytically solvable toy model \cite{As04}, where one considers mixtures of GHZ states $|\Phi_{0,{\bm 0}}\rangle$ and $|\Phi_{1,{\bm 0}}\rangle$ and a restricted error model of only bit flip errors in set $V_B$, that preserve the structure of such states. Using the fact that bit flip errors in $V_B$ act like phase flip errors in $V_A$, and the fact that sub--protocol $P1$ is sufficient to purify such states, one obtains a lower bound on the threshold value $p_{\rm min}$ given by $p_{\rm min}= \left(\frac{1}{2}\right)^{1/(n-1)}$. This follows from arguments similar as in the derivation of purification curve for the bipartite BBPSSW protocol. Performing a similar analysis for binary--type mixtures of linear cluster states under this restricted noise model, one observes that the threshold value $p_{\rm min}$ is largely independent of the number of particles $n$, in agreement with the numerical observations for systems of up to size $n=10$ under a more general noise model.
\section{Applications II}\label{applications2}
In this section we discuss several applications of multiparty entanglement purification protocols. We understand that multipartite entangled states are {\em resources} to perform different tasks, ranging from multiparty secure applications such as secret sharing or Byzantine agreement to measurement--based quantum computation. Hence the generation of these entangled states with high fidelity is desirable, and here it is where multiparty entanglement purification comes into play.
\subsection{Quantum communication cost for multiparty state distribution}
A scenario of particular interest is given when entangled states need to be distributed to spatially separated parties over noisy quantum channels in such a way that they are generated with high fidelity. As shown in Ref. \cite{Kr05}, there are several strategies conceivable to achieve this task. The two main strategies consist in (see Fig. \ref{QCC1}) \begin{itemize} \item [(i)] the distribution of bipartite entangled pairs through noisy channels that are purified by using bipartite entanglement purification protocols; local generation of the desired multiparticle entangled state and distribution via teleportation to the different parties, using the purified entangled pairs. \item [(ii)] the distribution of the locally generated multiparticle entangled states to the different parties through noisy quantum channels; the states are purified by using multiparticle entanglement purification protocols. \end{itemize} Apart from these two main strategies, a large number of intermediate strategies are conceivable, where parts of the multiparticle entangled state are distributed and purified, and these parts are connected later on to constitute the desired multiparticle entangled state, with possible intermediate or final purification steps.
A thorough analysis of the different strategies (see \cite{Kr05}) reveals that, in general, the multiparty strategy (ii) performs better if the desired target fidelity is high, while the bipartite strategy (i) may be used for lower target fidelities. In particular, the fact that multiparticle entanglement purification allows one to achieve higher fidelities than strategies based on bipartite purification makes multiparticle protocols the only choice if high target fidelities are required. The performance of the different strategies is measured by the quantum communication cost, that is, the total number of channel usages required to obtain on average a single copy of the desired multiparticle entangled state with sufficiently high fidelity. This is illustrated in Fig. \ref{QCC2} for linear cluster states and a generic noise model, while Fig. \ref{QCC3} shows results for GHZ state and a toy model for noise (see Ref. \cite{Kr05} for details).
\begin{figure}\label{QCC1}
\end{figure}
\begin{figure}
\caption{(Color online.) Inverse of communication cost for different target fidelities for 3 (red solid line for multiparty entanglement purification and green dashed line for bipartite entanglement purification) and 15 (pink dotted line for multiparty entanglement purification and blue small dashed line for bipartite entanglement purification) qubit cluster states. The data points are the outputs for 1, 2, 3,~\dots iterations of the protocol. The intermediate points are obtained by mixing ensembles of different fidelities. For more than 6 steps, the difference between the reached fidelity and the maximum reachable fidelity is smaller than the uncertainty. For any number of parties, the curves representing the two strategies cross over. The disks give this cross-over for $N=3,4,5,6,7,8,9,10,15$. (That one curve seems to ``go back'' is just an artifact of the statistical inaccuracies of the Monte Carlo method.) Figure taken from Ref. \cite{Kr05}.}
\label{QCC2}
\end{figure}
\begin{figure}
\caption{Inverse of communication cost as function of final fidelity for a simplified noise model. Analytical calculation for GHZ states of different number of qubits $N$ varying from 5 to 70, with alteration probability for the channel and local noise of $(1-q)=0.1$ and $(1-q_l)=0.05$ respectively. The green dashed lines stand for multiparty entanglement purification strategy while the red solid lines stand for bipartite entanglement purification strategy. The blue circles give the crossing points for all number of parties between 5 and 70. Figure taken from Ref. \cite{Kr05}.}
\label{QCC3}
\end{figure}
\subsection{Secure state distribution}
The secure and secret distribution of an unknown multipartite state with high fidelity provides a basic quantum primitive, as multipartite entangled states can serve as resource to perform certain quantum information processing tasks. The specific type of entanglement determines the (class of) tasks that can be performed. Hence it is easy to imagine scenarios where the involved parties do not want any third party to learn which secret state they possess, and they wish at the same time their entanglement to be private. While in an idealized scenario where one assumes perfect local operations, this task can be achieved rather easily, under non--idealized conditions (as one typically faces) the problem becomes non--trivial. Multipartite entanglement purification is the main tool to achieve the secure and secret distribution of high--fidelity multipartite entanglement. However, standard entanglement purification protocols need to be modified to take care of additional secrecy and security requirements. In particular, even parties involved in the purification process may not be allowed to learn which state they are purifying.
In Ref. \cite{Du05}, three different solutions to the secure--state distribution problem were put forward (see Fig. \ref{SSDsetup}). The first solution is based on bipartite entanglement purification, which serves to purify channels. Together with teleportation, this enables one to generate arbitrary multipartite entangled states. The second solution makes use of direct multipartite entanglement purification protocols, which is combined with basis randomization and adapted accordingly to ensure security. Security in the third solution, again based on direct multipartite purification, is ensured by purifying enlarged states. Each of the solutions offers its own advantages, and there exist in fact parameter regimes (for local noise, channel noise, desired target fidelity) such that one of the three schemes can be applied, while the other two fail.
\begin{figure}\label{SSDsetup}
\end{figure}
\subsection{Quantum error correction using graph states}
Since certain graph states constitute codewords of error correction codes, one may use the purification of these graph states to achieve high fidelity encoding without making use of complicated encoding networks \cite{As04}. In particular, the 7 qubit CSS code can be obtained by using a two--colorable graph state of eight vertices (a cube) as resource, and teleportation. Concatenated codes of this kind can be obtained by appending to each vertex of the cube another cube (see Fig. \ref{CSScode}). Encoding into the graph state can be achieved by a single Bell measurement \cite{As04}, where the qubit to be encoded is coupled by the Bell measurement to the $8^{\rm th}$ vertex of the cube. A similar procedure is considered for the five qubit code in Ref. \cite{Sc01}, where the notion of graph codes was introduced (see also \cite{Gra02}). The fidelity of the encoding mainly depends on the fidelity of the two--colorable graph state used in the procedure described above. Hence, multipartite entanglement purification can be applied to generate high fidelity entangled states which are then used to achieve high fidelity encoding. Notice that this can also be viewed as the purification of a quantum circuit, namely the encoding circuit.
\begin{figure}
\caption{Graph corresponding to a concatenated 7-qubit CSS code with input (red), auxiliary (blue) and output (black) vertices. Figure taken from Ref. \cite{He06}.}
\label{CSScode}
\end{figure}
\subsection{Purification of circuits and one--way quantum computation}
In the one--way quantum computer model, a multipartite entangled state, the 2D cluster state \cite{Br01}, serves as universal resource for quantum computation \cite{Ra01,Ra03}. That is, given a cluster state of suitable size, an arbitrary quantum algorithm can be implemented by a sequence of single--qubit measurements. In a similar way, other graph states represent algorithm--specific resources, i.e. they allow one to implement a specific algorithm (depending on the graph state) by means of single qubit measurements \cite{Ra03}. In the presence of imperfect operations, the cluster-- or graph state may not be available with unit fidelity. However, entanglement purification may be applied to increase the fidelity and hence to reduce errors in quantum computation. To what extent the purification of graph states can be used in fault tolerant quantum computation is subject of current research. An interesting approach in this direction is the usage of {\em encoded} 2D cluster states \cite{Rau06}. The encoding serves to perform error correction and to allow for a fault--tolerant implementation of measurement based quantum computation. Also such encoded resource states may be purified using known entanglement purification protocols. Fault-tolerant one-way quantum computation with highly verified logical cluster states has recently been discussed in Ref. \cite{Fu06}.
We remark that other resource states have been identified recently to constitute a universal resource for measurement--based quantum computation \cite{Va06}. In particular, graph states corresponding to Triagonal--, Kogome- or Hexagonal lattices are also universal. The latter is of particular importance, as the corresponding states are less sensitive to local noise in the sense that their lifetime of entanglement is longer \cite{Du04}, and the multiparticle entanglement purification protocols that allow one to purify these states have less stringent error thresholds for local noise. This can be understood from the fact that the local degree of the graph determines (or at least strongly influences) the thresholds, as local noise only affects a given qubit and its neighbors in the graph. The degree of the hexagonal lattice is three, the minimal possible to obtain a universal resource for translationally symmetric lattice--type graphs \cite{Va06}. Notice that the 2D cluster state has degree four.
\subsection{Ancilla factory approach to fault--tolerant quantum computation}
Also in the network model for quantum computation, entanglement purification can be useful as already demonstrated in Sec. \ref{EPPQC}. Here we discuss a second possible application, making use of multiparty entanglement purification. In fault--tolerant quantum computation, quantum information is processed in an encoded form in a fault--tolerant way. The encoding into a higher--dimensional Hilbert space allows for the detection and correction of errors. To perform the error correction in a fault--tolerant way, one possible approach (see e.g. Refs. \cite{Kn04,St03,Br05,Rei05}) is to generate encoded ancilla particles in a predefined state $|0_L\rangle$, and use them for error syndrome extraction. In addition, logical single and two qubit gates can be performed in via gate--teleportation \cite{Ci00,TBG1,TBG2,TBG3,TBG4,TBG5}, i.e. by generating certain encoded entangled states which are used to implement gates on the logical qubits.
When using (concatenated) CSS codes, the logical state $|0_L\rangle$ actually corresponds to a certain stabilizer (or graph) state, a highly entangled multiparticle state. Also the logical entangled states required for the teleportation--based gates are (highly entangled) multiparticle stabilizer states. The high--fidelity generation of these states is essential in the scheme, and in this context multiparticle entanglement purification, together with the usage of error detection schemes plays an important role.
Recently Knill \cite{Kn04} has presented such an ``ancilla factory approach'' making use of ideas of resource state purification to estimate error thresholds for fault--tolerant quantum computation. The estimates he finds are of the order of $10^{-2}$, i.e. tolerable errors in operation are of order of one percent. This gives essentially the same order of magnitude as for multiparticle entanglement purification protocols. However, the approach still has a formidable overhead in resources, in particular for the high fidelity preparation of the encoded ancilla particles where billions of copies are required.
\section{Summary and outlook}\label{outlook}
In this article, we have briefly reviewed quantum error correction and illustrated the basic concepts of entanglement purification. We discussed the close relation between these two approaches in the context of quantum communication. We then concentrated on entanglement purification and illustrated the basic idea behind bipartite and multipartite entanglement purification protocols. We discussed several bipartite and multipartite entanglement purification protocols and their applications in quantum error correction, long distance quantum communication, multipartite secure applications, quantum error correction, gate purification and (fault tolerant) quantum computation. The remarkable robustness of entanglement purification protocols under the influence of errors in local control operations is central in this context, and establishes entanglement purification as a fundamental tool in quantum information procession. In particular, we have illustrated that entanglement purification is not only restricted to applications in bipartite quantum communication for which it was initially introduced, but can also be used for many other purposes. It is remarkable that entanglement purification remains a hot topic in the field of quantum information processing, and significant progress has been achieved in the last few years.
Despite of this progress, a number of open problems remain. For instance, we do not yet clearly understand the potential power and limitations of entanglement purification. The influence of noise and hence the corresponding error thresholds seem to be more relaxed than for general fault tolerant quantum computation. This appears to be related to the fact that two--way classical communication --as can be used in entanglement purification-- is provable stronger than one--way classical communication, and that one is attempting to protect known information (in the form of a maximally entangled state) rather than unknown information (as in the case of quantum error correction). Despite of this, as we have indicated in this article, indirect methods to use entanglement purification in the context of error correction or fault-tolerant quantum computation exist. These applications look very promising, but a more detailed analysis including a comparison of the resulting error thresholds needs to be performed.
Other important open problems include the fundamental issue of what types of states can actually be purified by some genuinely multipartite entanglement purification protocol. Until recently, it appeared that this class is given by the set of stabilizer states. However, the discovery of a purification protocol for three qubit W--states shows that entanglement purification is not limited to stabilizer states, so the quest to identify and characterize the family of states where entanglement purification is possible remains open. In this context, it would e.g. be interesting to see whether protocols to purify W--states of more than 3 qubits can be found, whether these protocols can be generalized to purify all Dicke (or symmetric) states, or to discover new protocols for other kinds of maximally ---or perhaps even non--maximally-- multipartite entangled states.
Another important, unresolved issue is the purification range of entanglement purification protocols, i.e. the identification of the set of mixed states that can be purified by a given entanglement purification protocol. In most cases only sufficient conditions for purification can be obtained via numerical Simulations, however an analytic treatment, in particular identifying necessary and sufficient purification conditions for a given protocol would be desirable.
The construction of provable optimal entanglement purification protocols is another important challenge. Optimality can thereby be understood either with respect to the yield of the protocol, or with respect to the purification range. Despite considerable effort, such optimal entanglement purification protocols (with either optimality requirement) are known only for a few specific situations so far.
Of particular importance in this context --but also extremely challenging-- is the extension of such investigations to the case of noisy local control operations. Given the possible practical applications of entanglement purification, most notable in the context of quantum repeaters for long distance quantum communication, but also for fault--tolerant quantum computation or quantum simulation \cite{Du07}, such optimizations are crucial to minimize the required physical resources and to simplify a practical realization.
More generally, finding new applications of entanglement purification in the broad context of quantum information theory and beyond remains an interesting and challenging task for future investigations.
\section*{References}
\end{document} |
\begin{document}
\begin{frontmatter}
\title{Hamiltonian and reversible systems \\ with smooth families of invariant tori}
\author{Mikhail B. Sevryuk}
\ead{2421584@mail.ru, sevryuk@mccme.ru}
\address{V.L.~Tal'rose Institute for Energy Problems of Chemical Physics, N.N.~Sem\"enov Federal Research Center of Chemical Physics, Russian Academy of Sciences, 38 Leninski\u{\i} Prospect, Bld.~2, Moscow 119334, Russia}
\begin{abstract} For various values of $n$, $d$, and the phase space dimension, we construct simple examples of Hamiltonian and reversible systems possessing smooth $d$-parameter families of invariant $n$-tori carrying conditionally periodic motions. In the Hamiltonian case, these tori can be isotropic, coisotropic, or atropic (neither isotropic nor coisotropic). The cases of non-compact and compact phase spaces are considered. In particular, for any $N\geq 3$ and any vector $\omega\in\mR^N$, we present an example of an analytic Hamiltonian system with $N$ degrees of freedom and with an isolated (and even unique) invariant $N$-torus carrying conditionally periodic motions with frequency vector $\omega$ (but this torus is atropic rather than Lagrangian and the symplectic form is not exact). Examples of isolated atropic invariant tori carrying conditionally periodic motions are given in the paper for the first time. The paper can also be used as an introduction to the problem of the isolatedness of invariant tori in Hamiltonian and reversible systems. \end{abstract}
\begin{keyword} Hamiltonian systems; Reversible systems; Kronecker torus; Isolatedness; Uniqueness; Families of tori; Lagrangian torus; Isotropic torus; Coisotropic torus; Atropic torus; Symmetric torus; KAM theory
\MSC[2020] 70K43 \sep 70H12 \sep 70H33 \sep 70H08 \end{keyword}
\end{frontmatter}
\section{Introduction and overview}\label{introduction}
\subsection{Kronecker tori}\label{Kronecker}
Finite-dimensional invariant tori carrying conditionally periodic motions are among the key elements of the structure of smooth dynamical systems with continuous time. The importance and ubiquity of such tori stems, in the long run, from the fact that any finite-dimensional connected compact Abelian Lie group is a torus \cite{A1969,DK2000,S2007}. By definition, given a certain flow on a certain manifold, an invariant $n$-torus carrying \emph{conditionally periodic} motions ($n$ being a non-negative integer) is an invariant submanifold $\cT$ diffeomorphic to the standard $n$-torus $\mT^n=(\mR/2\pi\mZ)^n$ and such that the induced dynamics on $\cT$ in a suitable angular coordinate $\varp\in\mT^n$ has the form $\dot{\varp}=\omega$ where $\omega\in\mR^n$ is a constant vector (called the \emph{frequency vector}). Flows on $\mT^n$ afforded by equations $\dot{\varp}\equiv\omega$ are also said to be linear, parallel, rotational, translational, or Kronecker, and invariant tori carrying conditionally periodic motions are therefore sometimes called \emph{Kronecker tori} \cite{KP2003,MP643,P707,S415,T2012}.
A Kronecker flow $g^t$ on $\mT^n$ with any frequency vector $\omega\in\mR^n$ possesses the \emph{uniform recurrence property}: for any $T>0$ and $\vare>0$ there exists $\Theta\geq T$ such that for any $\varp\in\mT^n$ the distance between $\varp$ and $g^\Theta(\varp) = \varp+\Theta\omega$ (e.g., with respect to some fixed Riemannian metric) is smaller than $\vare$. Recall the almost obvious proof of this fact. First of all, there is $\delta>0$ such that the distance between $\varp$ and $\varp+\Delta$ is smaller than $\vare$ whenever $\varp\in\mT^n$ and $\Delta\in\mR^n$, $|\Delta|<\delta$ (here and henceforth, given $c\in\mR^n$, the symbols $|c|$ denote the $\Fl_1$-norm $|c_1|+\cdots+|c_n|$ of $c$). Second, there are positive integers $\Fm_2>\Fm_1$ and a vector $\Delta\in\mR^n$, $|\Delta|<\delta$ such that $\Fm_2T\omega = \Fm_1T\omega+\Delta$. Finally, it is sufficient to set $\Theta=(\Fm_2-\Fm_1)T$, because $\varp+(\Fm_2-\Fm_1)T\omega = \varp+\Delta$ for any $\varp\in\mT^n$.
The frequency vector $\omega=(\omega_1,\ldots,\omega_n) \in \mR^n$ of a Kronecker $n$-torus and the torus itself are said to be \emph{non-resonant} if the frequencies $\omega_1,\ldots,\omega_n$ are incommensurable (linearly independent over rationals) and are said to be \emph{resonant} otherwise. Conditionally periodic motions with non-resonant frequency vectors are usually called \emph{quasi-periodic} motions. Each phase curve on a non-resonant Kronecker torus fills up it densely. If the frequencies $\omega_1,\ldots,\omega_n$ of a resonant Kronecker $n$-torus $\cT$ satisfy $r$ independent resonance relations $\bigl\langle j^{(\iota)},\omega \bigr\rangle=0$, $j^{(\iota)}\in\mZ^n\setminus\{0\}$, $1\leq\iota\leq r$, $1\leq r\leq n$ (here and henceforth, the angle brackets $\langle{\cdot},{\cdot}\rangle$ denote the standard inner product), then $\cT$ is foliated by non-resonant Kronecker $(n-r)$-tori, the frequency vector of all these tori being the same.
The occurrence of a resonant Kronecker torus in a certain dynamical system usually indicates some degeneracy (for instance, the presence of many first integrals). Normally, dynamical systems exhibit (smooth or Cantor-like) families of non-resonant Kronecker tori, the dimension of all the tori in a given family being the same. Moreover, the frequency vectors of these tori are not merely non-resonant but \emph{strongly non-resonant} (for instance, Diophantine), i.e., badly approximable by resonant vectors. Recall that a vector $\omega\in\mR^n$ is said to be \emph{Diophantine} if there exist constants $\tau\geq n-1$ and $\gamma>0$ such that $\bigl| \langle j,\omega\rangle \bigr| \geq \gamma|j|^{-\tau}$ for any $j\in\mZ^n\setminus\{0\}$ (vectors that are not Diophantine are said to be \emph{Liouville}). Families of Kronecker tori with strongly incommensurable frequencies are the subject of the \emph{KAM} (Kolmogorov--Arnold--Moser) theory. The reader is referred to e.g.\ the monographs \cite{BHS1996,BHTB1990,KP2003}, \S\S~6.2--6.4 of the monograph \cite{AKN2006}, and the survey or tutorial papers \cite{BS249,dlL175,S1113,S137,S603} for the main ideas, methods, and results of the (mainly finite-dimensional) KAM theory and the bibliography, as well as for a precise definition of a Cantor-like family of Kronecker tori. The book \cite{D2014} presents a brilliant semi-popular introduction to the KAM theory. The ``core'' of the theory, namely, families of Kronecker $N$-tori in Hamiltonian systems with $N$ degrees of freedom, is treated in detail in e.g.\ the articles \cite{B42,B21,EFK1733,MP643,P707,S351,T12851}. The papers dealing with various special aspects of the KAM theory are exemplified by \cite{BHN355,BH191,F1521,H79,H989,H49,K259,P380,QS757,S435,S599,S415}. Some open problems in the theory are listed and discussed in the works \cite{FK1905,H797,St177,S6215}.
Typical finite-dimensional autonomous dissipative systems (with no special structure on the phase space the system is assumed to preserve) possess equilibria (Kronecker $0$-tori) and closed trajectories (Kronecker $1$-tori with a nonzero frequency). Typical smooth families of dissipative systems depending on $\Fr\geq 1$ external parameters $\mu_1,\ldots,\mu_{\Fr}$ also exhibit Cantor-like $\Fr$-parameter families of strongly non-resonant Kronecker $n$-tori in the product of the phase space and the parameter space $\{\mu\}$, the dimension $n$ ranging between $2$ and the phase space dimension \cite{BHN355,BHS1996,BHTB1990}. Here and henceforth, the word \emph{``typical''} means that the systems (or the families of systems) with the properties indicated constitute an open set (to be more precise, a set with non-empty interior) in the appropriate functional space.
On the other hand, finite-dimensional autonomous Hamiltonian and reversible systems typically admit many Cantor-like families of strongly non-resonant Kronecker $n$-tori, and these families (with different dimensions $n$) constitute complicated hierarchical structures.
\subsection{Review and the main result:\ Hamiltonian systems}\label{gamiltonovy}
In this section and henceforth, we will employ the following useful notation. Given a non-negative integer $a$, the combination of symbols $\mR^a_w$ will denote the Euclidean space $\mR^a$ with coordinates $(w_1,\ldots,w_a)$, and the combination of symbols $\mT^a_\varp$ will denote the torus $\mT^a$ with angular coordinates $(\varp_1,\ldots,\varp_a)$.
The properties of Kronecker tori in Hamiltonian systems very much depend on the ``relations'' of the torus in question with the symplectic $2$-form. Recall that a submanifold $\cL$ of a $2N$-dimensional symplectic manifold is said to be \emph{isotropic} if the tangent space $T_\Lambda\cL$ to $\cL$ at any point $\Lambda\in\cL$ is contained in its skew-orthogonal complement: $T_\Lambda\cL\subset(T_\Lambda\cL)^\bot$ (in other words, if the restriction of the symplectic form to $\cL$ vanishes), and is said to be \emph{coisotropic} if the tangent space $T_\Lambda\cL$ to $\cL$ at any point $\Lambda\in\cL$ contains its skew-orthogonal complement: $(T_\Lambda\cL)^\bot\subset T_\Lambda\cL$. If $\cL$ is isotropic then $\dim\cL\leq N$, and if $\cL$ is coisotropic then $\dim\cL\geq N$. A submanifold $\cL$ that is both isotropic and coisotropic is said to be \emph{Lagrangian}, in which case $\dim\cL=N$. In the sequel, it will be convenient to call isotropic submanifolds $\cL$ with $\dim\cL<N$ \emph{strictly isotropic} and to call coisotropic submanifolds $\cL$ with $\dim\cL>N$ \emph{strictly coisotropic}. In other words, strictly isotropic submanifolds are isotropic submanifolds that are not Lagrangian, and strictly coisotropic submanifolds are coisotropic submanifolds that are not Lagrangian.
\begin{rem}\label{strictly} In the literature, the terms ``strictly isotropic'' and ``strictly coisotropic'' are also used with a different meaning, see e.g.\ \cite{B2005,BZ365}. In the $h$-principle theory, one speaks of subcritical isotropic immersions and embeddings in symplectic and contact manifolds where the meaning of the words ``subcritical isotropic'' is close to ``non-Lagrangian isotropic'' (see e.g.\ the tutorial \cite{EM2002}). \end{rem}
It is clear that if $\dim\cL$ is equal to $0$ or $1$ then $\cL$ is necessarily isotropic, and if $\codim\cL$ is equal to $0$ or $1$ then $\cL$ is necessarily coisotropic. Now suppose that $\cL$ is invariant under a Hamiltonian flow with Hamilton function $H$, $H|_{\cL}$ is a constant, and almost all the points of $\cL$ are not equilibria. Then $\cL$ is isotropic if $\dim\cL=2$ and is coisotropic if $\codim\cL=2$. Recall the simple proof of this fact. Denote the symplectic form by $\Omega$. Let a point $\Lambda\in\cL$ be not an equilibrium, and let $X\in T_\Lambda\cL\setminus\{0\}$ be the vector of the Hamiltonian vector field in question at $\Lambda$. Let $\dim\cL=2$. For any vector $Y\in T_\Lambda\cL$ one has $\Omega(Y,X) = dH(Y) = 0$ since $H$ is a constant on $\cL$. Consequently, $\cL$ is isotropic. On the other hand, let $\codim\cL=2$, and let $\fH\supset\cL$ be the level hypersurface of $H$ containing $\cL$. The space of the tangent vectors to the ambient symplectic manifold at $\Lambda$ that are skew-orthogonal to $X$ is just $T_\Lambda\fH \supset T_\Lambda\cL$, in particular, $X\in(T_\Lambda\cL)^\bot$. Let $X,Y$ be a basis of $(T_\Lambda\cL)^\bot$. Then $Y\in T_\Lambda\fH$ since $X\in T_\Lambda\cL$ and therefore $\Omega(Y,X)=0$. If $Y\in T_\Lambda\fH\setminus T_\Lambda\cL$, then $Y$ would be skew-orthogonal to the whole space $T_\Lambda\fH$ because $\Omega(Y,Y)=0$ and $\dim T_\Lambda\fH-\dim T_\Lambda\cL = 1$, so that $\dim(T_\Lambda\fH)^\bot \geq 2$ in this hypothetical case. Thus, $Y\in T_\Lambda\cL$. Consequently, $\cL$ is coisotropic.
One of the key facts in the Hamiltonian KAM theory is the \emph{Herman lemma} which states that any non-resonant Kronecker torus of a Hamiltonian system is isotropic provided that the symplectic form is exact (see e.g.\ \cite{BHS1996,F1521,S1113} for a proof and \cite{BS249,S137} for a discussion; these works also contain references to the original papers by M.R.~Herman). A Hamiltonian system on a symplectic manifold with a non-exact symplectic form may admit strictly coisotropic non-resonant Kronecker tori as well as non-resonant Kronecker tori that are neither isotropic nor coisotropic. Tori of the latter type are said to be \emph{atropic} \cite{BS249,S1113,St177}. According to what was explained in the previous paragraph, the dimension of an atropic non-resonant Kronecker torus always lies between $3$ and $2N-3$ where $N$ is the number of degrees of freedom.
Now the main ``informal'' conclusion of the Hamiltonian KAM theory can be stated as follows. Typical Hamiltonian systems with $N\geq 1$ degrees of freedom admit $n$-parameter families of isotropic strongly non-resonant Kronecker $n$-tori for each $0\leq n\leq N$. These families are smooth for $n=0$ and $1$ and are Cantor-like for $n\geq 2$. If $N\geq 2$ and the symplectic form is not exact (and meets certain Diophantine-like conditions), typical Hamiltonian systems also possess $(2N-n)$-parameter families of strictly coisotropic strongly non-resonant Kronecker $n$-tori for each $N+1\leq n\leq 2N-1$ (see the works \cite{BHS1996,BS249,H989,H49,P380,S1113,St177} and references therein). These families are smooth for $n=2N-1$ and are Cantor-like for $n\leq 2N-2$. Finally, if $N\geq 3$ and the symplectic form is not exact (and meets certain Diophantine-like conditions), typical Hamiltonian systems also exhibit Cantor-like $\kappa$-parameter families of atropic strongly non-resonant Kronecker $n$-tori for any $3\leq n\leq 2N-3$ and $1\leq\kappa\leq\min(n-2, \, 2N-n-2)$ such that $n+\kappa$ is even (see the works \cite{BS249,S1113,St177} and references therein).
\begin{rem}\label{dichotomyHam} One sees that if $\kappa$ is the number of parameters in typical families of Kronecker $n$-tori in Hamiltonian systems with $N$ degrees of freedom, then $\kappa+n=2N$ for coisotropic (Lagrangian or strictly coisotropic) Kronecker tori (so that the Lebesgue measure of the union of the tori is positive) and $\kappa+n\leq 2N-2$ for non-coisotropic (strictly isotropic or atropic) Kronecker tori (so that the union of the tori is of measure zero). \end{rem}
Until 1984, the Hamiltonian KAM theory only dealt with isotropic Kronecker tori. The non-isotropic Hamiltonian KAM theory was founded by I.O.~Parasyuk \cite{P380}. Strictly isotropic Kronecker tori in Hamiltonian systems are often said to be \emph{lower dimensional}. For strictly coisotropic Kronecker tori in Hamiltonian systems, the term ``higher dimensional'' is also used but much more rarely.
Of course, of all the Kronecker $n$-tori in Hamiltonian systems with $N$ degrees of freedom, Lagrangian Kronecker $N$-tori are best studied. A generic Lagrangian non-resonant Kronecker torus $\cT$ in a Hamiltonian system (with any number $N$ of degrees of freedom) is \emph{KAM stable}: in any neighborhood of $\cT$, there is a family of other Lagrangian Kronecker tori, their union having positive Lebesgue measure and density one at $\cT$ (all these tori constitute an $N$-parameter family which is, generally speaking, Cantor-like for $N\geq 2$). To be more precise, the KAM stability of $\cT$ is implied by the so-called Kolmogorov non-degeneracy of $\cT$ \cite{B42}. No arithmetic conditions (like strong incommensurability) on the frequencies of $\cT$ are needed in this remarkable result, and it is valid in the $C^\ell$ smoothness class with any finite sufficiently large $\ell$ (not to mention the $C^\infty$, Gevrey, and analytic categories). On the other hand, generic Lagrangian resonant Kronecker tori in Hamiltonian systems with $N\geq 2$ degrees of freedom in the $C^\ell$ smoothness classes, $\ell\geq 2$, are not KAM stable \cite{B21}. For some previous results concerning density points of quasi-periodicity (not necessarily in the Hamiltonian realm), see e.g.\ the works \cite{BHN355,BHS1996,BHTB1990,BS249,EFK1733}.
\begin{rem}\label{stability} The term ``KAM stable'' is sometimes understood in a quite different sense (see e.g.\ \cite{K259,S351}): an unperturbed system (or an unperturbed Hamilton function) possessing a smooth family of Kronecker tori is said to be KAM stable if any perturbed system admits a Cantor-like family of Kronecker tori close to the unperturbed ones (provided that the perturbation lies in the suitable functional class and is sufficiently small). \end{rem}
Since Kronecker tori in Hamiltonian systems tend to be organized into (Cantor-like) families, the natural question arises whether such tori can be isolated. The isolatedness of a torus can be understood in different ways.
\begin{dfn}\label{isolated} A Kronecker $n$-torus $\cT$ of a dynamical system is said to be \emph{isolated} if it is not included in a (Cantor-like) family of Kronecker $n$-tori. A torus $\cT$ is said to be \emph{strongly isolated} if there exists a neighborhood $\cO$ of $\cT$ in the phase space such that there are no Kronecker tori (of any dimension) entirely contained in $\cO\setminus\cT$. A torus $\cT$ is said to be \emph{unique} if there are no Kronecker tori (of any dimension) outside $\cT$ in the whole phase space. \end{dfn}
In particular, a strongly isolated torus $\cT$ is unique in the neighborhood $\cO$ mentioned in Definition~\ref{isolated}.
Of course, a generic equilibrium in a Hamiltonian system is always isolated, to be more precise, an equilibrium $O$ is isolated whenever none of the eigenvalues $\lambda_{\Fi}$ of the linearization of the vector field at $O$ is zero. If $O$ is \emph{hyperbolic} (i.e., if all the eigenvalues $\lambda_{\Fi}$ have nonzero real parts), then it is strongly isolated. Examples of unique equilibria in Euclidean phase spaces are also straightforward: if the equilibrium $0$ of a system with a quadratic Hamilton function in $\mR^{2N}$ is hyperbolic then it is unique. On the other hand, the question of whether an \emph{elliptic} equilibrium of a Hamiltonian system (i.e., an equilibrium for which all the eigenvalues $\lambda_{\Fi}$ are nonzero and lie on the imaginary axis) can be strongly isolated (or at least can be not accumulated by a set of Lagrangian Kronecker tori of positive measure) is very far from being easy. So is the question of whether such an equilibrium can be Lyapunov unstable. We will not discuss this problem here and confine ourselves by citing the papers and preprints \cite{F09059,FK1905,FS67,T12851} (see also the references therein). In general, the instability of an equilibrium is a more delicate topic than that of, say, a Lagrangian Kronecker torus \cite{FS67}.
Surprisingly, it seems that the question of whether strictly isotropic Kronecker tori of dimensions from $1$ to $N-1$ in Hamiltonian systems with $N\geq 2$ degrees of freedom can be isolated was never considered until 2017. In December 2017 and January 2018, the author and the user Khanickus of MathOverflow \cite{K2018} constructed independently two very similar explicit (and exceedingly simple) examples of Hamiltonian systems in $\mR^3\times\mT^1$ with a unique periodic orbit. Subsequently, for any integers $n\geq 1$ and $N\geq n+1$ and for any vector $\omega\in\mR^n$, the author \cite{S415} proposed an example of a Hamiltonian system with $N$ degrees of freedom, with the phase space $\mR^{2N-n}_w\times\mT^n_\varp$, with the exact symplectic form \[ \sum_{i=1}^n dw_i\wedge d\varp_i + \sum_{\nu=1}^{N-n} dw_{n+\nu}\wedge dw_{N+\nu}, \] and with a Hamilton function independent of $\varp$, polynomial in $w$, and such that $\{w=0\}$ is a unique Kronecker $n$-torus (in the sense of Definition~\ref{isolated}), the frequency vector of $\{w=0\}$ being $\omega$. The paper \cite{S415} also contains an example of a Hamiltonian system with $N$ degrees of freedom, with the compact phase space $\mT^{2N-n}_w\times\mT^n_\varp$, with the symplectic form given by the same formula (but no longer exact), and with a trigonometric polynomial Hamilton function independent of $\varp$ and such that $\{w=0\}$ is a strongly isolated Kronecker $n$-torus (in the sense of Definition~\ref{isolated}), the frequency vector of $\{w=0\}$ being $\omega$. Thus, the problem of the possible isolatedness of strictly isotropic Kronecker tori in Hamiltonian systems has been completely solved by now.
It is clear that periodic orbits (Kronecker $1$-tori with a nonzero frequency) of Hamiltonian systems with one degree of freedom are always included in smooth one-parameter families (each periodic orbit being a connected component of an energy level line). The question of whether Lagrangian Kronecker tori in Hamiltonian systems with $N\geq 2$ degrees of freedom can be isolated has turned out to be highly nontrivial. To the best of the author's knowledge, this question is still open, even in the case where the frequency vector is Liouville (or even resonant) and the Hamilton function is only $C^\infty$ smooth (see \cite{B42,B21,FF01575}).
It is proven in the landmark paper \cite{EFK1733} that a Lagrangian Kronecker $N$-torus $\cT$ with a \emph{Diophantine} frequency vector is never isolated in the \emph{analytic} category (where the symplectic form, the Hamilton function, and the torus itself are analytic), however degenerate the Hamilton function is at $\cT$. Such a torus is always accumulated by other Lagrangian Kronecker tori (with Diophantine frequency vectors), i.e., is always included in a (Cantor-like) $\Fr$-parameter family of Lagrangian Kronecker tori with $\Fr\geq 1$. Nevertheless, it is not known whether $\Fr$ is always equal to $N$, i.e., whether the union of Lagrangian Kronecker tori in any neighborhood of $\cT$ always has positive measure (Herman conjectured the affirmative answer in the model problem of fixed points of analytic symplectomorphisms \cite{H797}). For $N=2$, however, the equality $\Fr=2$ is always valid even in the $C^\infty$ category \cite{EFK1733}.
For Liouville frequency vectors or non-analytic Hamilton functions, there are several examples in the literature of a Lagrangian Kronecker $N$-torus $\cT$ that is accumulated by other Lagrangian Kronecker tori, but the union of these tori is of measure zero. The phase space in all these examples is $\mR^N_u\times\mT^N_\varp$, the symplectic form is $\sum_{i=1}^N du_i\wedge d\varp_i$, the Hamilton function is $H(u,\varp) = \langle u,\omega\rangle + O\bigl( |u|^2 \bigr)$, and $\cT=\{u=0\}$, where $\omega\in\mR^N$ is the frequency vector of $\cT$. It is well known that locally, in some neighborhood of a Lagrangian Kronecker torus, this setup can always be achieved (up to an additive constant in $H$) \cite{B42,B21,MP643}; in fact, this is an immediate consequence of A.~Weinstein's equivalence theorem for Lagrangian submanifolds \cite{W329}. For any $N\geq 2$ and any resonant vector $\omega$, a very simple example with an analytic (and even quadratic in $u$) Hamilton function $H$ is presented in the paper \cite{B21}. In this example, the torus $\cT$ is accumulated by a continuous $N$-parameter family of isotropic Kronecker $(N-1)$-tori. The article \cite{EFK1733} contains examples for any $N\geq 4$ and \emph{any} vector $\omega$ with $C^\infty$ as well as Gevrey regular (with any exponent $\sigma>1$) Hamilton functions $H$. It is pointed out in the paper \cite{FS67} that for $C^\infty$ Hamilton functions $H$, the construction of \cite{EFK1733} can be extended to $N=3$ and any vector $\omega$ (and an analog for elliptic equilibria in $\mR^6$ is described). The paper \cite{FS67} also presents an analog for elliptic equilibria in $\mR^4$ but for Liouville frequencies. Finally, for any $N\geq 3$ and non-resonant but ``sufficiently Liouville'' vectors $\omega$, G.~Farr\'e and B.~Fayad \cite{FF01575} constructed examples with \emph{analytic} Hamilton functions $H$. The words ``sufficiently Liouville'' mean that if $\tilde{\omega} = (\omega_1,\ldots,\omega_{N-1}) \in \mR^{N-1}$, then the infimum of the set of the ratios \[
\frac{\ln\bigl| \langle j,\tilde{\omega}\rangle \bigr|}{|j|}, \quad j\in\mZ^{N-1}\setminus\{0\} \] is $-\infty$. In the examples of \cite{EFK1733} and \cite{FF01575}, the hypersurface $\{u_N=0\}$ is foliated by Lagrangian Kronecker tori with frequency vector $\omega$.
As far as the author knows, the question of whether strictly coisotropic or atropic Kronecker tori in Hamiltonian systems can be isolated has never been raised. The present paper gives an exhaustive answer to this question in the case of \emph{atropic} tori. Here is our main result.
\begin{thm}\label{mainHam} For any integers $N$, $n$, $d$ in the ranges $N\geq 2$, $1\leq n\leq N-1$, $0\leq d\leq 2N-n$ and for any vector $\omega\in\mR^n$, there exist an exact symplectic form $\Omega$ on the manifold $\cM = \mR^{2N-n}_w\times\mT^n_\varp$ with constant coefficients and a Hamilton function $H:\cM\to\mR$ independent of $\varp$, polynomial in $w$, and such that the corresponding Hamiltonian system on $\cM$ admits a $d$-parameter analytic family of \emph{strictly isotropic} Kronecker $n$-tori of the form $\{w=\const\}$. There are no Kronecker tori (of any dimension) outside this family. The $n$-torus $\{w=0\}$ belongs to this family, its frequency vector is equal to $\omega$, and if $d=0$ then this torus is unique in the sense of Definition~\ref{isolated}. If $d=2N-n$ then the family in question makes up the whole phase space.
For any integers $N$, $n$, $d$ in the ranges $N\geq 3$, $3\leq n\leq 2N-3$, $0\leq d\leq 2N-n$ and for any vector $\omega\in\mR^n$, there exist a non-exact symplectic form $\Omega$ on the manifold $\cM = \mR^{2N-n}_w\times\mT^n_\varp$ with constant coefficients and a Hamilton function $H:\cM\to\mR$ independent of $\varp$, polynomial in $w$, and such that the corresponding Hamiltonian system on $\cM$ admits a $d$-parameter analytic family of \emph{atropic} Kronecker $n$-tori of the form $\{w=\const\}$. There are no Kronecker tori (of any dimension) outside this family. The $n$-torus $\{w=0\}$ belongs to this family and its frequency vector is equal to $\omega$, and if $d=0$ then this torus is unique. If $d=2N-n$ then the family in question makes up the whole phase space.
Similar statements hold \emph{mutatis mutandis} for the compact manifold $\widehat{\cM} = \mT^{2N-n}_w\times\mT^n_\varp$, the modifications being as follows. First, the symplectic form $\Omega$ is not exact in the case of strictly isotropic Kronecker $n$-tori either. Second, the Hamilton function $H$ is now trigonometric polynomial in $w$. Third, it is no longer valid that there are no Kronecker tori (of any dimension) outside the family under consideration. Fourth, if $d=0$ then the $n$-torus $\{w=0\}$ is strongly isolated (rather than unique) in the sense of Definition~\ref{isolated}. \end{thm}
This theorem is proven in Sections~\ref{symplectic}--\ref{compact} by constructing explicit examples which generalize the examples of the note \cite{S415}. The problem of whether strictly coisotropic Kronecker tori in Hamiltonian systems can be isolated remains open. There is little doubt that this problem is as difficult as the analogous problem (discussed above) for Lagrangian Kronecker tori. Thus, the isolatedness question for Kronecker tori in Hamiltonian systems is very hard for coisotropic (Lagrangian or strictly coisotropic) Kronecker $n$-tori (for $n\geq 2$) and is rather easy for non-coisotropic (strictly isotropic or atropic) ones. This dichotomy surprisingly coincides with the other dichotomy pointed out in Remark~\ref{dichotomyHam}.
Setting $n=N\geq 3$ and $d=0$ in Theorem~\ref{mainHam}, we obtain a Hamilton function $H: \mR^n_w\times\mT^n_\varp \to \mR$ independent of $\varp$, polynomial in $w$, and such that the $n$-torus $\{w=0\}$ is a unique Kronecker torus, the frequency vector of this torus can be any prescribed vector in $\mR^n$. However, this astonishing picture is marred by the fact that the corresponding symplectic form on $\mR^n_w\times\mT^n_\varp$ is not standard and even not exact, and the torus $\{w=0\}$ is atropic rather than Lagrangian.
In the examples of Sections~\ref{symplectic}--\ref{compact}, one deals with coisotropic and non-coisotropic Kronecker tori in a unified way. However, coisotropic Kronecker $n$-tori in our examples for $N\geq 1$ degrees of freedom ($N\leq n\leq 2N-1$) are always organized into $(2N-n)$-parameter analytic families.
Most probably, non-resonant coisotropic Kronecker $(2N-1)$-tori in Hamiltonian systems with $N$ degrees of freedom cannot be isolated for any $N\geq 2$, cf.\ \cite{BHS1996,H989,H49}.
\subsection{Review and the main result:\ reversible systems}\label{obratimye}
While speaking of Kronecker tori (and, more generally, any invariant submanifolds) in reversible systems, one usually only considers \emph{symmetric} invariant submanifolds, i.e., invariant submanifolds that are also invariant under the reversing involution $G$ of the phase space. The dynamics of $G$-reversible systems and the properties of symmetric invariant submanifolds in such systems very much depend on the structure of the fixed point set $\Fix G$ of the involution $G$. This set is a submanifold of the phase space of the same smoothness class as the involution $G$ itself. However, the manifold $\Fix G$ can well be empty or consist of several connected components of different dimensions even if the phase space is connected (see e.g.\ simple examples in the papers \cite{BDP223,DP3119,PF280,QS757} and references therein; in fact, the literature on the structure of the fixed point sets of involutions of various manifolds is by now immense).
It is well known that in any symmetric non-resonant Kronecker $n$-torus $\cT$ (with a frequency vector $\omega\in\mR^n$) of a $G$-reversible system, one can choose an angular coordinate $\varp\in\mT^n$ such that the dynamics on $\cT$ takes the form $\dot{\varp}=\omega$ and the restriction of the reversing involution $G$ to $\cT$ takes the form $G|_{\cT}: \varp\mapsto-\varp$ (in particular, this implies that the set $(\Fix G)\cap\cT = \Fix\bigl( G|_{\cT} \bigr)$ consists of $2^n$ points). This very easy but fundamental \emph{standard reflection lemma} is proven in e.g.\ the works \cite{BHS1996,S435} (see also the papers \cite{S137,S599} for a discussion).
We will confine ourselves with the case where the fixed point set $\Fix G$ of the reversing involution $G$ is non-empty and all its connected components are of the same dimension, so that $\dim\Fix G$ is well defined (in fact, this is so for almost all the reversible systems encountered in practice). We will say that an involution $G$ satisfying this condition is of \emph{type $(\fL,m)$} if $\dim\Fix G=m$ and $\codim\Fix G=\fL$ (cf.\ \cite{BHN355,BH191,BHS1996,QS757}). It follows from the standard reflection lemma that if a system reversible with respect to an involution of type $(\fL,m)$ admits a symmetric non-resonant Kronecker $n$-torus then $n\leq\fL$. Therefore, in the reversible KAM theory \cite{BHN355,BH191,BHS1996,QS757,St177,S435,S137,S599,S603,S415}, it only makes sense to consider symmetric Kronecker $n$-tori in systems reversible with respect to involutions of types $(n+l,m)$ with $l\geq 0$.
Now the main ``informal'' conclusion of the KAM theory for \emph{individual} reversible systems (not for reversible systems depending on external parameters) can be stated as follows. For $n=0$ and $1$, typical systems reversible with respect to an involution of type $(n+l,m)$ with $m\geq l\geq 0$ admit smooth $(m-l)$-parameter families of symmetric non-resonant Kronecker $n$-tori. For each $n\geq 2$, typical systems reversible with respect to an involution of type $(n+l,m)$ with $l\geq 0$ and $m\geq l+1$ admit Cantor-like $(m-l)$-parameter families of symmetric strongly non-resonant Kronecker $n$-tori.
\begin{rem}\label{dichotomyrev} One sees that if $\kappa=m-l$ is the number of parameters in typical families of symmetric Kronecker $n$-tori in systems reversible with respect to involutions of types $(n+l,m)$, then $\kappa+n=m+n+l$ (i.e., $\kappa=m+l$) for $l=0$ (so that the Lebesgue measure of the union of the tori is positive) and $\kappa+n<m+n+l$ for $l\geq 1$ (so that the union of the tori is of measure zero). Of course, here we suppose that $m\geq l$ for $n\leq 1$ and $m>l$ for $n\geq 2$. If $m$ and $l$ do not meet these conditions, one needs external parameters $\mu_{\Fj}$ to obtain persistent families of symmetric Kronecker $n$-tori. However, the measure of the union of the tori in the product of the phase space and the parameter space $\{\mu\}$ is typically positive for $l=0$ and zero for $l\geq 1$ in this case as well (see \cite{S435,S137,S599,S603}). \end{rem}
By analogy with Hamiltonian systems, one may ask under what conditions symmetric Kronecker tori in reversible systems can be isolated or unique. When we speak of the isolatedness, strong isolatedness, and unicity of such tori, we still interpret these concepts in strict accordance with Definition~\ref{isolated}: we have in view the absence of other Kronecker tori whatsoever, and not just the absence of other symmetric Kronecker tori.
For any $n\geq 0$ and $l\geq m\geq 0$, it is very easy to construct a system that is reversible with respect to an involution of type $(n+l,m)$ and admits a unique symmetric Kronecker $n$-torus with any prescribed frequency vector $\omega\in\mR^n$. Indeed, let the phase space be $\mR^m_u\times\mT^n_\varp\times\mR^m_v\times\mR^{l-m}_q$ and the reversing involution be \[ G: (u,\varp,v,q) \mapsto (u,-\varp,-v,-q). \] Then $G$ is an involution of type $(n+l,m)$. The system \[ \dot{u}=v, \quad \dot{\varp}=\omega, \quad \dot{v}=u, \quad \dot{q}_\nu=q_\nu^2 \] ($1\leq\nu\leq l-m$) is reversible with respect to $G$, and $\{u=0, \; v=0, \; q=0\}$ is a unique Kronecker $n$-torus of this system. This Kronecker torus is symmetric, and its frequency vector is $\omega$.
For any integers $n\geq 0$, $m\geq 0$, $l\geq 1$ and for any vector $\omega\in\mR^n$, the note \cite{S415} considers the manifold \begin{equation} \cK = \mR^m_u\times\mT^n_\varp\times\mR^l_q \label{cK} \end{equation} equipped with the involution \begin{equation} G: (u,\varp,q) \mapsto (u,-\varp,-q) \label{inv} \end{equation} of type $(n+l,m)$ and presents an example of a $G$-reversible system on $\cK$ with the right-hand side independent of $\varp$, polynomial in $(u,q)$, and such that $\{u=0, \; q=0\}$ is a unique symmetric Kronecker $n$-torus, its frequency vector being $\omega$. For the compact manifold \begin{equation} \widehat{\cK} = \mT^m_u\times\mT^n_\varp\times\mT^l_q \label{hatcK} \end{equation} equipped with the involution $G$ given by the same formula \eqref{inv} and having the same type, the paper \cite{S415} contains an example of a $G$-reversible system on $\widehat{\cK}$ with the right-hand side independent of $\varp$, trigonometric polynomial in $(u,q)$, and such that $\{u=0, \; q=0\}$ is a strongly isolated symmetric Kronecker $n$-torus, its frequency vector being $\omega$.
In the present paper, we generalize these examples of the note \cite{S415}. Here is our second result.
\begin{thm}\label{mainrev} For any integers $n$, $m$, $l$, $d_\ast$, $d$ in the ranges $n\geq 0$, $m\geq 0$, $l\geq 1$, $0\leq d_\ast\leq m$, $d_\ast\leq d\leq d_\ast+l$ and for any vector $\omega\in\mR^n$, there exists a system of ordinary differential equations on \eqref{cK} reversible with respect to the involution \eqref{inv} of type $(n+l,m)$ and possessing the following properties. The right-hand side of this system is independent of $\varp$ and polynomial in $(u,q)$. The system admits a $d$-parameter analytic family of Kronecker $n$-tori of the form $\{u=\const, \; q=\const\}$. There are no Kronecker tori (of any dimension) outside this family. The family includes a $d_\ast$-parameter analytic subfamily of symmetric Kronecker $n$-tori of the form $\{u=\const, \; q=0\}$. The $n$-torus $\{u=0, \; q=0\}$ belongs to this subfamily, its frequency vector is equal to $\omega$, and if $d_\ast=d=0$ then this torus is unique in the sense of Definition~\ref{isolated}. If $d_\ast=m$ and $d=m+l$ then the $d$-parameter family in question makes up the whole phase space.
Similar statements hold \emph{mutatis mutandis} for the compact phase space \eqref{hatcK}, the modifications being as follows. First, the right-hand side of the system is now trigonometric polynomial in $(u,q)$. Second, it is no longer valid that there are no Kronecker tori (of any dimension) outside the family under consideration. Third, the symmetric Kronecker $n$-tori have the form $\{u=\const, \; q=q^0\}$, where each component of $q^0$ is equal to either $0$ or $\pi$. Fourth, if $d_\ast=d=0$ then the $n$-torus $\{u=0, \; q=0\}$ is strongly isolated (rather than unique) in the sense of Definition~\ref{isolated}. \end{thm}
This theorem is proven in Sections~\ref{reversible}--\ref{comprev}. In the examples of Sections~\ref{reversible}--\ref{comprev}, one deals with the cases $l=0$ and $l\geq 1$ in a unified way. However, for $l=0$ the whole phase space $\mR^m\times\mT^n$ or $\mT^m\times\mT^n$ in our examples is foliated by symmetric Kronecker $n$-tori.
The only case not covered by the examples above is that of symmetric Kronecker $n$-tori in systems reversible with respect to involutions of types $(n,m)$ with $m\geq 1$. If $n=0$ or $1$ then symmetric Kronecker $n$-tori in such systems are always organized into smooth $m$-parameter families and cannot be isolated \cite{S415}. To the best of the author's knowledge, the question of whether symmetric Kronecker $n$-tori in systems reversible with respect to involutions of types $(n,m)$ can be isolated for $m\geq 1$ and $n\geq 2$ has never been raised and is open. One may conjecture that such tori with Diophantine frequency vectors are never isolated in the analytic category (where the involution, the vector field, and the torus itself are analytic), similarly to Lagrangian Kronecker tori in Hamiltonian systems \cite{EFK1733} (see Section~\ref{gamiltonovy}). Most probably, this question is very hard.
To summarize, the problem of the possible isolatedness of symmetric Kronecker $n$-tori in systems reversible with respect to involutions of types $(n+l,m)$ is rather easy (and has been solved) for $l\geq 1$ and is probably highly nontrivial for $l=0$ (if $n\geq 2$ and $m\geq 1$). Like in the Hamiltonian realm, this dichotomy coincides with the dichotomy of Remark~\ref{dichotomyrev}.
\section{Preliminaries}\label{symplectic}
Given non-negative integers $a$ and $b$, we designate the identity $a\times a$ matrix as $I_a$ and the zero $a\times b$ matrix as $0_{a\times b}$. In fact, the symbols $I_0$, $0_{0\times b}$, and $0_{a\times 0}$ correspond to no actual objects and will only be used for unifying the notation.
Let $s\geq 1$, $k$, and $l$ be non-negative integers and consider a skew-symmetric $(2s+2k)\times (2s+2k)$ matrix $J$ of the form \[ J = \begin{pmatrix} 0_{s\times s} & -Z^{\rt} \\ Z & L \end{pmatrix}, \] where $Z$ is an $(s+2k)\times s$ matrix of rank $s$ and $L$ is a skew-symmetric $(s+2k)\times (s+2k)$ matrix (the superscript ``t'' denotes transposing). If $k=0$ then the matrix $J$ is always non-singular ($\det J = (\det Z)^2$). If $k\geq 1$ then for any fixed matrix $Z$, the matrix $J$ may be non-singular or singular depending on the matrix $L$. Indeed, we can suppose without loss of generality that the last $s$ rows of $Z$ constitute a non-singular $s\times s$ matrix $Z_\sharp$. Let the matrix $L$ have the form \[ L = \begin{pmatrix} L_\sharp & 0_{2k\times s} \\ 0_{s\times 2k} & 0_{s\times s} \end{pmatrix}, \] where $L_\sharp$ is a skew-symmetric $2k\times 2k$ matrix, then $\det J = \det L_\sharp(\det Z_\sharp)^2 \neq 0$ if and only if $\det L_\sharp \neq 0$.
In the sequel, we will assume the matrix $J$ to be non-singular, so that the skew-symmetric $(2s+2k+2l)\times (2s+2k+2l)$ matrix \begin{equation} \cJ = \begin{pmatrix} \begin{matrix} 0_{s\times s} & -Z^{\rt} \\ Z & L \end{matrix} & 0_{(2s+2k)\times 2l} \\ 0_{2l\times (2s+2k)} & \begin{matrix} 0_{l\times l} & -I_l \\ I_l & 0_{l\times l} \end{matrix} \end{pmatrix} \label{cJ} \end{equation} is also non-singular and can be treated as the \emph{structure matrix} (the matrix of the Poisson brackets $\{{\cdot},{\cdot}\}$ of the coordinate functions, see e.g.\ \cite{B1,HLW2006,O1993,St177}) of a certain symplectic form $\Omega$ (with constant coefficients) on the manifold \begin{equation} \cM = \mR^s_u\times\mT^{s+2k}_\varp\times\mR^l_p\times\mR^l_q \label{cM} \end{equation} (cf.\ Lemma~1 in \cite{St177}). A Hamilton function $H:\cM\to\mR$ affords the equations of motion \cite{B1,HLW2006,O1993} \begin{equation} \begin{pmatrix} \dot{u} \\ \dot{\varp} \\ \dot{p} \\ \dot{q} \end{pmatrix} = \cJ\frac{\partial H}{\partial(u,\varp,p,q)} = \begin{pmatrix} -Z^{\rt}\partial H/\partial\varp \\ Z\partial H/\partial u + L\partial H/\partial\varp \\ -\partial H/\partial q \\ \partial H/\partial p \end{pmatrix}. \label{XH} \end{equation} This is an autonomous Hamiltonian system with $N=s+k+l$ degrees of freedom.
For any $u^0\in\mR^s$, $p^0\in\mR^l$, $q^0\in\mR^l$, consider the $(s+2k)$-torus \begin{equation}
\cT_{u^0,p^0,q^0} = \bigl\{ (u^0,\varp,p^0,q^0) \bigm| \varp\in\mT^{s+2k} \bigr\}. \label{cT} \end{equation} For any $\varp^0\in\mT^{s+2k}$, the skew-orthogonal complement $T^\bot$ (with respect to $\Omega$) of the tangent space $T$ to $\cT_{u^0,p^0,q^0}$ at the point $(u^0,\varp^0,p^0,q^0)$ consists of all the vectors of the form $\psi\partial/\partial\varp + P\partial/\partial p + Q\partial/\partial q$, where $P\in\mR^l$, $Q\in\mR^l$, $\psi\in\cZ$, and $\cZ$ is the $s$-dimensional subspace of $\mR^{s+2k}$ spanned by the columns of the matrix $Z$. Indeed, the space of all such vectors is of dimension $s+2l = \dim\cM-(s+2k)$. It is therefore sufficient to verify that $\Omega(V,W) = 0$ for any vector $V\in T$ (i.e., any vector $V = \Phi\partial/\partial\varp$ with $\Phi\in\mR^{s+2k}$) and any vector $W = (ZU)\partial/\partial\varp + P\partial/\partial p + Q\partial/\partial q$ with $U\in\mR^s$, $P\in\mR^l$, $Q\in\mR^l$. According to \eqref{XH}, the linear Hamilton function $H = H(u,p,q) = \langle U,u\rangle-\langle P,q\rangle+\langle Q,p\rangle$ on $\cM$ affords the constant Hamiltonian vector field equal to $W$. Thus, $\Omega(V,W) = dH(V) = 0$.
We arrive at the conclusion that the $(s+2k)$-tori \eqref{cT} are isotropic for $k=0$ ($T\subset T^\bot$ at any point), are coisotropic for $l=0$ ($T^\bot\subset T$ at any point), and are therefore Lagrangian for $k=l=0$. For $kl>0$, these tori are atropic. Note that $\dim(T\cap T^\bot)=s$ in all the cases.
It is clear that the symplectic form $\Omega$ is exact if and only if its coordinate representation does not contain terms $c_{\alpha\beta}d\varp_\alpha\wedge d\varp_\beta$, $1\leq\alpha<\beta\leq s+2k$, i.e., if the tori \eqref{cT} are isotropic. Thus, $\Omega$ is exact for $k=0$ and is not exact for $k\geq 1$.
\section{The main construction}\label{center}
\subsection{The system}\label{system}
Now let $\zeta_1,\ldots,\zeta_s$, $\xi_1,\ldots,\xi_l$, $\eta_1,\ldots,\eta_l$ be arbitrary non-negative real constants and let $h:\mR^s\to\mR$ be an arbitrary smooth function. Consider the Hamilton function \begin{equation} H(u,p,q) = h(u) + lp_1 \sum_{i=1}^s \zeta_iu_i^2 + \sum_{\nu=1}^l (\xi_\nu p_\nu q_\nu^2+\eta_\nu p_\nu^3/3) \label{ourH} \end{equation} on the symplectic manifold \eqref{cM}. The term $lp_1 \sum_{i=1}^s \zeta_iu_i^2$ is automatically absent for $l=0$. According to \eqref{XH}, the equations of motion afforded by $H$ take the form \begin{equation} \begin{aligned} \dot{u}_i &= 0, \\ \dot{\varp}_\alpha &= \sum_{i=1}^s Z_{\alpha i}\left( \frac{\partial h(u)}{\partial u_i} + 2l\zeta_iu_ip_1 \right), \\ \dot{p}_\nu &= -2\xi_\nu p_\nu q_\nu, \\ \dot{q}_\nu &= \xi_\nu q_\nu^2 + \eta_\nu p_\nu^2 + \delta_{1\nu}l \sum_{i=1}^s \zeta_iu_i^2, \end{aligned} \label{XourH} \end{equation} where $1\leq i\leq s$, $1\leq\alpha\leq s+2k$, $1\leq\nu\leq l$, and $\delta_{1\nu}$ is the Kronecker delta. The fundamental property of this system is that $\dot{q}_\nu\geq 0$ everywhere in the phase space $\cM$, $1\leq\nu\leq l$.
In the note \cite{S415}, we considered the particular case of the Hamilton function \eqref{ourH} and the system \eqref{XourH} where $k=0$, $Z=I_s$, $L=0_{s\times s}$, $h(u)=\langle u,\omega\rangle$ ($\omega\in\mR^s$), $l\geq 1$, $\zeta_i=1/l$ for all $1\leq i\leq s$, and $\xi_\nu=\eta_\nu=1$ for all $1\leq\nu\leq l$ (in the notation of \cite{S415}, $s=n$ and $l=m+1$ where $n\geq 1$ and $m\geq 0$).
All the conditionally periodic motions of the system \eqref{XourH} fill up the manifold \[
\fM = \bigl\{ (u,\varp,p,q) \bigm| l\zeta_iu_i=0 \; \forall i, \;\; \eta_\nu p_\nu=0 \; \forall\nu, \;\; \xi_\nu q_\nu=0 \; \forall\nu \bigr\} \] foliated by Kronecker $(s+2k)$-tori of the form \eqref{cT}. Of course, always $\cT_{0,0,0}\subset\fM$. The frequency vector of a torus $\cT_{u^0,p^0,q^0} \subset \fM$ is $\omega(u^0) = Z\partial h(u^0)/\partial u \in \cZ$ (recall that $\cZ$ is the $s$-dimensional subspace of $\mR^{s+2k}$ spanned by the columns of the matrix $Z$). If $l=0$ then $\fM=\cM$. The system \eqref{XourH} admits no conditionally periodic motions outside $\fM$.
Indeed, if $(u,\varp,p,q)\in\fM$ then $\dot{u}=0$, $\dot{\varp}=Z\partial h(u)/\partial u$, $\dot{p}=0$, $\dot{q}=0$. On the other hand, since $\dot{q}_\nu\geq 0$ everywhere in $\cM$, the recurrence property of conditionally periodic motions implies that $\dot{q}_\nu\equiv 0$ on Kronecker tori, $1\leq\nu\leq l$. Consequently, a point $(u,\varp,p,q)\notin\fM$ does not belong to any Kronecker torus of \eqref{XourH} (of any dimension) because $\dot{q}_1>0$ whenever $l\zeta_iu_i\neq 0$ for at least one $i$ and $\dot{q}_\nu>0$ whenever $\eta_\nu p_\nu\neq 0$ or $\xi_\nu q_\nu\neq 0$, $1\leq\nu\leq l$.
The Kronecker $(s+2k)$-tori $\cT_{u^0,p^0,q^0} \subset \fM$ constitute an analytic $d$-parameter family where $d = \dim\fM-(s+2k)$. If $l=0$ then $d=s$. If $l\geq 1$ (i.e., if the tori \eqref{cT} are not coisotropic) then $d$ can take any integer value between $0$ and $s+2l$; to be more precise, $d$ is the number of zero constants among $\zeta_i$, $\xi_\nu$, $\eta_\nu$ ($1\leq i\leq s$, $1\leq\nu\leq l$). The equality $d=0$ holds if and only if all the numbers $\zeta_i$, $\xi_\nu$, $\eta_\nu$ are positive in which case $\fM=\cT_{0,0,0}$, and $\cT_{0,0,0}$ is a unique Kronecker torus of the system \eqref{XourH}. The equality $d=s+2l$ occurs if and only if all the numbers $\zeta_i$, $\xi_\nu$, $\eta_\nu$ are equal to zero in which case $\fM=\cM$. If a torus $\cT_{u^0,p^0,q^0}$ lies in $\fM$ then its frequency vector $\omega(u^0) = Z\partial h(u^0)/\partial u$ can be made equal to any prescribed vector in $\mR^{s+2k}$ by a suitable choice of the matrix $Z$ and the function $h$ (one can even choose $h$ to be linear).
The construction just described can be formally carried out for $s=0$ as well, but for $s=0$ the frequency vector of each invariant $2k$-torus of the form \eqref{cT} is zero: such a torus consists of equilibria.
The $s+l+\delta_{0l}$ functions \[ H, \qquad u_i \;\; (1\leq i\leq s), \qquad \xi_\nu p_\nu q_\nu^2+\eta_\nu p_\nu^3/3 \;\; (2\leq\nu\leq l) \] are first integrals of the system \eqref{XourH} which are pairwise in involution. In fact, this system always admits $s+l$ first integrals that are pairwise in involution and are functionally independent almost everywhere. Indeed, let $f_\nu(p_\nu,q_\nu) = \xi_\nu p_\nu q_\nu^2+\eta_\nu p_\nu^3/3$ if $\xi_\nu+\eta_\nu>0$, and let $f_\nu$ be any smooth function in $p_\nu,q_\nu$ with the differential other than zero almost everywhere if $\xi_\nu=\eta_\nu=0$. In the case where $l\geq 1$ and $\sum_{i=0}^s \zeta_i > 0$, the functions \[ H, \qquad u_i \;\; (1\leq i\leq s), \qquad f_\nu(p_\nu,q_\nu) \;\; (2\leq\nu\leq l) \] are the desired $s+l$ first integrals. In the opposite case where $l \sum_{i=0}^s \zeta_i = 0$, one can choose the $s+l$ first integrals in question to be equal to \[ u_i \;\; (1\leq i\leq s), \qquad f_\nu(p_\nu,q_\nu) \;\; (1\leq\nu\leq l). \]
\subsection{The analysis}\label{analysis}
The dimension $n=s+2k$ of the tori \eqref{cT} can be smaller than, equal to, or greater than the number $N=s+k+l$ of degrees of freedom: $n-N=k-l$. The maximal possible value $s+2l$ of the quantity $d$ is always equal to $2N-n$. If $l=0$ (the case of coisotropic tori \eqref{cT}) then $d=s=2N-n$ and $k=n-N$. These equalities determine integers $s\geq 1$ and $k\geq 0$ if and only if $N\geq 1$ and $N\leq n\leq 2N-1$.
If $l\geq 1$ then $n=s+2k=2N-s-2l\leq 2N-3$ and $N\geq 2$. It is easy to see that for any integers $N\geq 2$ and $n$ in the range $1\leq n\leq 2N-3$, one can choose integers $s\geq 1$, $k\geq 0$, $l\geq 1$ such that $N=s+k+l$ and $n=s+2k$. Indeed, if $1\leq n\leq N-1$ then it suffices to set $s=n$, $k=0$, $l=N-n$. In this case, the tori \eqref{cT} are strictly isotropic. Of course, the converse is also true: if $l\geq 1$ and $k=0$ then $1\leq n=s\leq N-1=s+l-1$. On the other hand, if $N\leq n\leq 2N-3$ (so that $N\geq 3$) then it suffices to set $s=2N-n-2$, $k=n-N+1$, $l=1$. In this case, the tori \eqref{cT} are atropic.
If $l\geq 1$ and $k\geq 1$ (so that the tori \eqref{cT} are atropic) then $n=s+2k\geq 3$ and $N=s+k+l\geq 3$. One easily sees that for any integers $N\geq 3$ and $n$ in the range $3\leq n\leq 2N-3$, one can choose positive integers $s$, $k$, $l$ such that $N=s+k+l$ and $n=s+2k$. In the previous paragraph, we verified this for $N\leq n\leq 2N-3$. On the other hand, if $3\leq n\leq N-1$ (so that $N\geq 4$) then it suffices to set $s=n-2$, $k=1$, $l=N-n+1$.
The case where $n=N\geq 3$, $1\leq k=l\leq\bigl\lfloor (N-1)/2 \bigr\rfloor$ (here $\lfloor{\cdot}\rfloor$ denotes the floor function), $s=N-2l$, and $d=0$ is probably the most interesting one. In this case we obtain the unique Kronecker $N$-torus $\cT_{0,0,0}$ of the Hamiltonian system \eqref{XourH} with $N$ degrees of freedom, and the frequency vector of this torus can be any vector in $\mR^N$ (but this torus is atropic rather than Lagrangian).
The case where $n=N\geq 3$, $1\leq k=l\leq\bigl\lfloor (N-1)/2 \bigr\rfloor$, $s=N-2l$, and $d=N$ is also very interesting. In this case the whole phase space of the Hamiltonian system \eqref{XourH} with $N$ degrees of freedom is smoothly foliated by Kronecker $N$-tori \eqref{cT}. However, this system is not \emph{Liouville integrable} (completely integrable): the Kronecker tori in question are atropic rather than Lagrangian, and the $N$ first integrals $u_1,\ldots,u_{N-2l}$, $p_1,\ldots,p_l$, $q_1,\ldots,q_l$ are \emph{not} pairwise in involution: $\{q_\nu,p_\nu\}\equiv 1$, $1\leq\nu\leq l$.
In fact, for any $s\geq 1$, $k$, and $l$, the whole phase space $\cM$ of the Hamiltonian system \eqref{XourH} is smoothly foliated by Kronecker tori \eqref{cT} whenever $d=s+2l$ (as was already pointed out in Section~\ref{system}), in which case the $s+2l$ functions $u_1,\ldots,u_s$, $p_1,\ldots,p_l$, $q_1,\ldots,q_l$ are independent first integrals of the system. The matrix of the Poisson brackets of these functions is \[ \cP = \begin{pmatrix} 0_{s\times s} & 0_{s\times l} & 0_{s\times l} \\ 0_{l\times s} & 0_{l\times l} & -I_l \\ 0_{l\times s} & I_l & 0_{l\times l} \end{pmatrix}, \] and they are not pairwise in involution for $l\geq 1$. If $l>k$ then the number of the first integrals in question exceeds the number $N=s+k+l$ of degrees of freedom. However, one cannot call the system \eqref{XourH} superintegrable for $l>k\geq 1$ and $d=s+2l$. Besides the existence of $M>N$ independent first integrals, the definition of a \emph{superintegrable} Hamiltonian system with $N$ degrees of freedom (see the papers \cite{F93,H79,KS811} and references therein; superintegrable systems are also known as properly degenerate or non-commutatively integrable systems) includes other requirements, for instance, that there be $N$ integrals pairwise in involution among the $M$ integrals under consideration (we have only $s+l<N$ integrals in involution, e.g., $u_1,\ldots,u_s$, $p_1,\ldots,p_l$), or that the rank of the matrix of the Poisson brackets of the integrals be equal to $2(M-N)$ almost everywhere (in our case the rank of $\cP$ is $2l>2(s+2l-N)=2(l-k)$), or that the common level surfaces of the integrals be isotropic (in our case the tori \eqref{cT} are atropic).
Note that the tori \eqref{cT} are isotropic if and only if the symplectic form $\Omega$ on $\cM$ is exact (both the properties in our setup are equivalent to the equality $k=0$). This observation is consistent with the Herman lemma (see Section~\ref{gamiltonovy}).
\section{Compact phase spaces}\label{compact}
Like in the setting of our note \cite{S415}, the general construction of Sections~\ref{symplectic} and~\ref{center} admits an analogue with a compact phase space. Consider the symplectic manifold \begin{equation} \widehat{\cM} = \mT^s_u\times\mT^{s+2k}_\varp\times\mT^l_p\times\mT^l_q \label{hatcM} \end{equation} with the same structure matrix \eqref{cJ}. Of course, now the corresponding symplectic form $\Omega$ is always non-exact. The $(s+2k)$-tori \eqref{cT} with $u^0\in\mT^s$, $p^0\in\mT^l$, $q^0\in\mT^l$ are again isotropic for $k=0$, are coisotropic for $l=0$, and are atropic for $kl>0$. For any angular variable $z$ introduce the notation $\tilde{z}=\sin z$ (cf.\ \cite{S415}). Consider the Hamilton function \[ \widehat{H}(u,p,q) = h(u) + l\tilde{p}_1 \sum_{i=1}^s \zeta_i\tilde{u}_i^2 + \sum_{\nu=1}^l (\xi_\nu\tilde{p}_\nu\tilde{q}_\nu^2+\eta_\nu\tilde{p}_\nu^3/3) \] on \eqref{hatcM}, where again $\zeta_1,\ldots,\zeta_s$, $\xi_1,\ldots,\xi_l$, $\eta_1,\ldots,\eta_l$ are arbitrary non-negative real constants and $h:\mT^s\to\mR$ is an arbitrary smooth function. The Hamilton function $\widehat{H}$ affords the equations of motion \begin{equation} \begin{aligned} \dot{u}_i &= 0, \\ \dot{\varp}_\alpha &= \sum_{i=1}^s Z_{\alpha i}\left( \frac{\partial h(u)}{\partial u_i} + l\zeta_i\sin 2u_i\tilde{p}_1 \right), \\ \dot{p}_\nu &= -\xi_\nu\tilde{p}_\nu\sin 2q_\nu, \\ \dot{q}_\nu &= (\xi_\nu\tilde{q}_\nu^2 + \eta_\nu\tilde{p}_\nu^2)\cos p_\nu + \delta_{1\nu}l\left( \sum_{i=1}^s \zeta_i\tilde{u}_i^2 \right)\cos p_1, \end{aligned} \label{XourhatH} \end{equation} where $1\leq i\leq s$, $1\leq\alpha\leq s+2k$, $1\leq\nu\leq l$.
The manifold \[
\widehat{\fM} = \bigl\{ (u,\varp,p,q) \bigm| l\zeta_i\tilde{u}_i=0 \; \forall i, \;\; \eta_\nu\tilde{p}_\nu=0 \; \forall\nu, \;\; \xi_\nu\tilde{q}_\nu=0 \; \forall\nu \bigr\} \] is again foliated by Kronecker $(s+2k)$-tori of the form \eqref{cT} (with $u^0\in\mT^s$, $p^0\in\mT^l$, $q^0\in\mT^l$), $\cT_{0,0,0}\subset\widehat{\fM}$ in all the cases, and the frequency vector of a torus $\cT_{u^0,p^0,q^0} \subset \widehat{\fM}$ is $\omega(u^0) = Z\partial h(u^0)/\partial u \in \cZ$. This frequency vector can again be made equal to any prescribed vector in $\mR^{s+2k}$ by a suitable choice of the matrix $Z$ and the function $h$; one can choose $h$ to be of the form $\sum_{i=1}^s c_i\sin(u_i-u^0_i)$. The dimension $s+2k+d$ of the manifold $\widehat{\fM}$ is determined in exactly the same way as that of the manifold $\fM$ in Section~\ref{system}. In particular, if $l=0$ then $\widehat{\fM}=\widehat{\cM}$.
In contrast to the case of the system \eqref{XourH}, it is, generally speaking, \emph{not} true that the system \eqref{XourhatH} for $l\geq 1$ possesses no conditionally periodic motions outside $\widehat{\fM}$. Indeed, suppose that $\sum_{i=1}^s \zeta_i > 0$ and choose an arbitrary point $u^0\in\mT^s$ such that $\chi = \sum_{i=1}^s \zeta_i\sin^2u^0_i > 0$. Consider the $(s+2k+1)$-torus \begin{equation}
\bigl\{ (u^0,\varp,0,q) \bigm| q_2=\cdots=q_l=0 \bigr\} \not\subset \widehat{\fM}. \label{exception} \end{equation} This torus is invariant under the flow of \eqref{XourhatH} with the induced dynamics \[ \dot{\varp} = \omega(u^0), \qquad \dot{q}_1 = \xi_1\sin^2q_1 + l\chi. \] It is clear that the motion on the torus \eqref{exception} is conditionally periodic. The frequencies of this motion are equal to $\omega_1(u^0),\ldots,\omega_{s+2k}(u^0),\varpi$ where \[ \varpi = 2\pi\left( \int_0^{2\pi}\frac{d\Fq}{\xi_1\sin^2\Fq+l\chi} \right)^{-1} = \bigl[ l\chi(l\chi+\xi_1) \bigr]^{1/2}. \]
Nevertheless, for any fixed $q^\star\in\mT^l$, no point $(u,\varp,p,q)\notin\widehat{\fM}$ belongs to a Kronecker torus of \eqref{XourhatH} (of any dimension) entirely contained in the domain \[
\fD^+_{q^\star} = \bigl\{ (u,\varp,p,q) \bigm| p_\nu\in(-\pi/2,\pi/2)\bmod 2\pi \; \forall\nu, \;\; q_\nu\neq q^\star_\nu \; \forall\nu \bigr\}. \] Indeed, $\dot{q}_\nu\geq 0$ everywhere in the domain $\fD^+_{q^\star}$, $1\leq\nu\leq l$, and $\sum_{\nu=1}^l \dot{q}_\nu > 0$ everywhere in $\fD^+_{q^\star} \setminus \widehat{\fM}$. If for some $\nu$ a function $q_\nu: \mR \to \mT^1\setminus\{q^\star_\nu\}$ satisfies the conditions that $\dot{q}_\nu(t)\geq 0$ for all $t$ and $\dot{q}_\nu(0)>0$, then $q_\nu(t)$ tends to a certain point $q_\nu^{\lim}\neq q_\nu(0)$ as $t\to+\infty$ and the recurrence property fails.
Similarly, no point $(u,\varp,p,q)\notin\widehat{\fM}$ belongs to a Kronecker torus of \eqref{XourhatH} (of any dimension) entirely contained in the domain \[
\fD^-_{q^\star} = \bigl\{ (u,\varp,p,q) \bigm| p_\nu\in(\pi/2,3\pi/2)\bmod 2\pi \; \forall\nu, \;\; q_\nu\neq q^\star_\nu \; \forall\nu \bigr\}. \] One may even fix any sequence of numbers $\vare_1,\ldots,\vare_l$, where $\vare_\nu=\pm 1$ for all $\nu$, and replace $\fD^+_{q^\star}$ or $\fD^-_{q^\star}$ with the domain \[
\fD^\vare_{q^\star} = \bigl\{ (u,\varp,p,q) \bigm| p_\nu\in\fI_{\vare_\nu}\bmod 2\pi \; \forall\nu, \;\; q_\nu\neq q^\star_\nu \; \forall\nu \bigr\}, \] where $\fI_1 = (-\pi/2,\pi/2)$ and $\fI_{-1} = (\pi/2,3\pi/2)$, so that $\vare_\nu\dot{q}_\nu\geq 0$ everywhere in $\fD^\vare_{q^\star}$, $1\leq\nu\leq l$.
If $l\geq 1$ and all the constants $\zeta_1,\ldots,\zeta_s$, $\xi_1,\ldots,\xi_l$, $\eta_1,\ldots,\eta_l$ are positive (so that $d=0$), then $\cT_{0,0,0}$ is the only Kronecker torus of \eqref{XourhatH} entirely contained in the domain \[
\bigl\{ (u,\varp,p,q) \bigm| u_i\neq\pi \; \forall i, \;\; p_\nu\in(-\pi/2,\pi/2)\bmod 2\pi \; \forall\nu, \;\; q_\nu\neq\pi \; \forall\nu \bigr\}. \] So, in this case $\cT_{0,0,0}$ is strongly isolated.
\section{Reversible analogues}\label{reversible}
Both the Hamiltonian systems \eqref{XourH} and \eqref{XourhatH} are reversible with respect to the phase space involution \[ \widetilde{G}: (u,\varp,p,q) \mapsto (u,-\varp,p,-q) \] of type $(s+2k+l,s+l)$, so that $\dim\Fix\widetilde{G}=s+l \geq 1$, $\codim\Fix\widetilde{G}=s+2k+l \geq \dim\Fix\widetilde{G}$, and $\codim\Fix\widetilde{G}-n=l < \dim\Fix\widetilde{G}$, where $n=s+2k$. However, $\widetilde{G}\bigl( \cT_{u^0,p^0,q^0} \bigr) = \cT_{u^0,p^0,-q^0}$, so that not all the $n$-tori \eqref{cT} are invariant under $\widetilde{G}$. In the case of the system \eqref{XourH}, a torus $\cT_{u^0,p^0,q^0}$ is invariant under $\widetilde{G}$ if and only if $q^0=0$. Consequently, the statement ``each torus $\cT_{u^0,p^0,q^0} \subset \fM$ is symmetric'' is valid if and only if all the numbers $\xi_1,\ldots,\xi_l$ are positive. In the case of the system \eqref{XourhatH}, a torus $\cT_{u^0,p^0,q^0}$ is invariant under $\widetilde{G}$ if and only if $q^0=-q^0$, i.e., if each component of $q^0$ is equal to either $0$ or $\pi$. Again, the statement ``each torus $\cT_{u^0,p^0,q^0} \subset \widehat{\fM}$ is symmetric'' holds if and only if all the numbers $\xi_1,\ldots,\xi_l$ are positive.
It is easy to construct a $G$-reversible counterpart of the system \eqref{XourH} for any non-negative integer values of $n$, $\dim\Fix G$, and $\codim\Fix G-n$, where $n$ is the dimension of symmetric Kronecker tori. Let $m$, $n$, $l$ be non-negative integers and consider the manifold \eqref{cK} equipped with the involution \eqref{inv} of type $(n+l,m)$. For any $u^0\in\mR^m$ and $q^0\in\mR^l$, consider the $n$-torus \begin{equation}
\cT_{u^0,q^0} = \bigl\{ (u^0,\varp,q^0) \bigm| \varp\in\mT^n \bigr\}. \label{TG} \end{equation} Since $G\bigl( \cT_{u^0,q^0} \bigr) = \cT_{u^0,-q^0}$, a torus $\cT_{u^0,q^0}$ is invariant under $G$ if and only if $q^0=0$.
Now let $\zeta_1,\ldots,\zeta_m$, $\xi_1,\ldots,\xi_l$ be arbitrary non-negative real constants and let $h:\mR^m\to\mR^n$ be an arbitrary smooth mapping. The system \begin{equation} \begin{aligned} \dot{u}_i &= 0, \\ \dot{\varp}_\alpha &= h_\alpha(u), \\ \dot{q}_\nu &= \xi_\nu q_\nu^2 + \delta_{1\nu}l \sum_{i=1}^m \zeta_iu_i^2 \end{aligned} \label{Xrev} \end{equation} (where $1\leq i\leq m$, $1\leq\alpha\leq n$, $1\leq\nu\leq l$) is reversible with respect to $G$. The term $l \sum_{i=1}^m \zeta_iu_i^2$ automatically vanishes for $l=0$. The key property of the system \eqref{Xrev} is that $\dot{q}_\nu\geq 0$ everywhere in the phase space $\cK$, $1\leq\nu\leq l$.
In the note \cite{S415}, we considered a similar system with $h(u)\equiv\omega\in\mR^n$, with $l\geq 1$, and with the equation for $\dot{q}_\nu$ of the form \[ \dot{q}_\nu = \delta_{1\nu}\left( \sum_{\mu=1}^l q_\mu^2 + \sum_{i=1}^m u_i^2 \right), \] $1\leq\nu\leq l$. Our variables $m$ and $l$ play the roles of $\ell$ and $m+1$ in \cite{S415}, respectively.
All the conditionally periodic motions of the system \eqref{Xrev} fill up the manifold \[
\fK = \bigl\{ (u,\varp,q) \bigm| l\zeta_iu_i=0 \; \forall i, \;\; \xi_\nu q_\nu=0 \; \forall\nu \bigr\} \] foliated by Kronecker $n$-tori of the form \eqref{TG}. Of course, always $\cT_{0,0}\subset\fK$, and the torus $\cT_{0,0}$ is symmetric. The frequency vector of a torus $\cT_{u^0,q^0} \subset \fK$ is $h(u^0)$, and this vector can be made equal to any prescribed vector $\omega\in\mR^n$ just by setting $h(u)\equiv\omega$. If $l=0$ then $\fK=\cK$. The system \eqref{Xrev} admits no conditionally periodic motions outside $\fK$.
These features of $\fK$ can be verified in exactly the same way as in Section~\ref{system}. If $(u,\varp,q)\in\fK$ then $\dot{u}=0$, $\dot{\varp}=h(u)$, $\dot{q}=0$. On the other hand, since $\dot{q}_\nu\geq 0$ everywhere in $\cK$, the recurrence property of conditionally periodic motions implies that $\dot{q}_\nu\equiv 0$ on Kronecker tori, $1\leq\nu\leq l$. Consequently, a point $(u,\varp,q)\notin\fK$ does not belong to any Kronecker torus of \eqref{Xrev} (symmetric or not and of any dimension) because $\dot{q}_1>0$ whenever $l\zeta_iu_i\neq 0$ for at least one $i$ and $\dot{q}_\nu>0$ whenever $\xi_\nu q_\nu\neq 0$, $1\leq\nu\leq l$.
The Kronecker $n$-tori $\cT_{u^0,q^0} \subset \fK$ constitute an analytic $d$-parameter family where $d = \dim\fK-n$. If $l=0$ then $d=m$. If $l\geq 1$ (i.e., if $\codim\Fix G>n$) then $d$ can take any integer value between $0$ and $m+l$; to be more precise, $d$ is the number of zero constants among $\zeta_i$, $\xi_\nu$ ($1\leq i\leq m$, $1\leq\nu\leq l$). The equality $d=0$ holds if and only if all the numbers $\zeta_i$, $\xi_\nu$ are positive in which case $\fK=\cT_{0,0}$, and $\cT_{0,0}$ is a unique Kronecker torus of the system \eqref{Xrev}. The equality $d=m+l$ occurs if and only if all the numbers $\zeta_i$, $\xi_\nu$ are equal to zero in which case $\fK=\cK$.
The symmetric Kronecker $n$-tori $\cT_{u^0,0}$ of the system \eqref{Xrev} are characterized by the condition $l\zeta_iu^0_i=0 \; \forall i$ and constitute an analytic $d_\ast$-parameter family where $d_\ast$ is determined as follows. If $l=0$ then $d_\ast=m$, and all the Kronecker $n$-tori constituting $\fK=\cK$ are symmetric. If $l\geq 1$ then $d_\ast$ is the number of zero constants among $\zeta_i$ ($1\leq i\leq m$) and can therefore take any integer value between $0$ and $m$. In all the cases, $d-d_\ast\leq l$.
\section{Compactified reversible analogues}\label{comprev}
The system \eqref{Xrev} can be compactified in the same way as the system \eqref{XourH}, cf.\ \cite{S415}. Consider the manifold \eqref{hatcK} equipped with the involution $G$ given by the same formula \eqref{inv} and having the same type $(n+l,m)$. For any $u^0\in\mT^m$ and $q^0\in\mT^l$, consider the $n$-torus $\cT_{u^0,q^0}$ given by the same expression \eqref{TG}. Since $G\bigl( \cT_{u^0,q^0} \bigr) = \cT_{u^0,-q^0}$, a torus $\cT_{u^0,q^0}$ is invariant under $G$ if and only if $q^0=-q^0$, i.e., if each component of $q^0$ is equal to either $0$ or $\pi$.
Now let again $\zeta_1,\ldots,\zeta_m$, $\xi_1,\ldots,\xi_l$ be arbitrary non-negative real constants and let $h:\mT^m\to\mR^n$ be an arbitrary smooth mapping. The system \begin{equation} \begin{aligned} \dot{u}_i &= 0, \\ \dot{\varp}_\alpha &= h_\alpha(u), \\ \dot{q}_\nu &= \xi_\nu\tilde{q}_\nu^2 + \delta_{1\nu}l \sum_{i=1}^m \zeta_i\tilde{u}_i^2 \end{aligned} \label{Xrevhat} \end{equation} (where $1\leq i\leq m$, $1\leq\alpha\leq n$, $1\leq\nu\leq l$, and the notation $\tilde{z}=\sin z$ is used) is reversible with respect to $G$, and $\dot{q}_\nu\geq 0$ everywhere in the phase space $\widehat{\cK}$, $1\leq\nu\leq l$.
The manifold \[
\widehat{\fK} = \bigl\{ (u,\varp,q) \bigm| l\zeta_i\tilde{u}_i=0 \; \forall i, \;\; \xi_\nu\tilde{q}_\nu=0 \; \forall\nu \bigr\} \] is again foliated by Kronecker $n$-tori of the form \eqref{TG} (with $u^0\in\mT^m$ and $q^0\in\mT^l$), $\cT_{0,0}\subset\widehat{\fK}$ in all the cases, and the torus $\cT_{0,0}$ is symmetric. The frequency vector of a torus $\cT_{u^0,q^0} \subset \widehat{\fK}$ is $h(u^0)$, and this vector can be made equal to any prescribed vector $\omega\in\mR^n$ just by setting $h(u)\equiv\omega$. The dimension $n+d$ of the manifold $\widehat{\fK}$ is determined in exactly the same way as that of the manifold $\fK$ in Section~\ref{reversible}. In particular, if $l=0$ then $\widehat{\fK}=\widehat{\cK}$.
If $l\geq 1$ then $\sum_{\nu=1}^l \dot{q}_\nu > 0$ everywhere in $\widehat{\cK} \setminus \widehat{\fK}$. Like in Section~\ref{compact} and in contrast to the case of the system \eqref{Xrev}, it is, generally speaking, \emph{not} true that the system \eqref{Xrevhat} for $l\geq 1$ possesses no conditionally periodic motions outside $\widehat{\fK}$. Indeed, similarly to the example in Section~\ref{compact}, suppose that $m\geq 1$, $\sum_{i=1}^m \zeta_i > 0$ and choose an arbitrary point $u^0\in\mT^m$ such that $\chi = \sum_{i=1}^m \zeta_i\sin^2u^0_i > 0$. Consider the $(n+1)$-torus \[
\bigl\{ (u^0,\varp,q) \bigm| q_2=\cdots=q_l=0 \bigr\} \not\subset \widehat{\fK}. \] This is a symmetric Kronecker torus of the system \eqref{Xrevhat} with the frequencies $h_1(u^0),\ldots,h_n(u^0),\varpi$, where $\varpi = \bigl[ l\chi(l\chi+\xi_1) \bigr]^{1/2}$.
Nevertheless, for any fixed $q^\star\in\mT^l$, no point $(u,\varp,q)\notin\widehat{\fK}$ belongs to a Kronecker torus of \eqref{Xrevhat} (symmetric or not and of any dimension) entirely contained in the domain \[
\bigl\{ (u,\varp,q) \bigm| q_\nu\neq q^\star_\nu \; \forall\nu \bigr\}. \] This may be verified in exactly the same way as in Section~\ref{compact}.
If $l\geq 1$ and all the constants $\zeta_1,\ldots,\zeta_m$, $\xi_1,\ldots,\xi_l$ are positive (so that $d=0$), then $\cT_{0,0}$ is the only Kronecker torus of \eqref{Xrevhat} entirely contained in the domain \[
\bigl\{ (u,\varp,q) \bigm| u_i\neq\pi \; \forall i, \;\; q_\nu\neq\pi \; \forall\nu \bigr\}. \] So, in this case $\cT_{0,0}$ is strongly isolated.
The symmetric Kronecker $n$-tori $\cT_{u^0,q^0}$ of the system \eqref{Xrevhat} make up an $(n+d_\ast)$-dimensional submanifold of the manifold $\widehat{\fK}$, where $d_\ast$ is determined in exactly the same way as in Section~\ref{reversible}.
\section*{Declaration of interest}
Declarations of interest: none.
\section*{Acknowledgments}
I am grateful to B.~Fayad for fruitful correspondence and sending me the breakthrough preprint \cite{FF01575} prior to submission to arXiv.
\end{document} |
\begin{document}
\begin{center} \textbf{\large Qualitative aspects of the entanglement in the three-level model with photonic crystals}
~
Mahmoud Abdel-Aty\footnote{ E-mail: abdelatyquantum@yahoo.co.uk, Fax. No. 00-20-93-4601950}
{\small Mathematics Department, Faculty of Science, South Valley University, 82524 Sohag, Egypt. } \end{center}
\textbf{Abstract:}
{\ This communication is an enquiry into the circumstances under which concurrence and phase entropy methods can give an answer to the question of quantum entanglement in the composite state when the photonic band gap is exhibited by the presence of photonic crystals in a three-level system. An analytic approach is proposed for any three-level system in the presence of photonic band gap. Using this analytic solution, we conclusively calculate the concurrence and phase entropy, focusing particularly on the entanglement phenomena. Specifically, we use concurrence as a measure of entanglement for dipole emitters situated in the thin slab region between two semi-infinite one-dimensionally periodic photonic crystals, a situation reminiscent of planar cavity laser structures. One feature of the regime considered here is that closed-form evaluation of the time evolution may be carried out in the presence of the detuning and the photonic band gap, which provides insight into the difference in the nature of the concurrence function for atom-field coupling, mode frequency and different cavity parameters. We demonstrate how fluctuations in the phase and number entropies effected by the presence of the photonic-band-gap. The outcomes are illustrated with numerical simulations applied to GaAs. Finally, we relate the obtained results to instances of any three-level system for which the entanglement cost can be calculated. Potential experimental observations in solid-state systems are discussed and found to be promising. }
PACS numbers: 42.50.Dv, 03.65.Ud, 03.67.Mn
\section{Introduction}
A structure in which the dielectric constant varies periodically is called a photonic crystal. One of the most interesting properties of a photonic crystal is the existence of a photonic band gap \cite{lee04,yam03}. Radiation with a frequency that lies within the band gap cannot propagate in the photonic crystal structure. Photonic crystals are usually viewed as an optical analog of semiconductors that modify the properties of light similar to a microscopic atomic lattice that creates a semiconductor band-gap for electrons \cite{lee04,kam00,joa95}. Photonic band gap crystals offer unique ways to tailor light and the propagation of electromagnetic waves and have caused growing interest in recent years because it offers the possibility of controlling and manipulating light within a given frequency range through photonic band gap \cite{lee04,kam00}. Photonic band-gap materials have attracted much attention in recent years for theoretical and practical importance in fundamental science and application \cite{lee04,joa95}. The atom-photon interaction in photonic band gap materials \cite{joa95} has been found to exhibit many interesting new phenomena such as photon-atom bound states \cite{joh90}, spectral splitting \cite{joh94}, quantum interference dark line effect \cite{zhu97}, phase control of spontaneous emission \cite {pas98}, transparency near band edge \cite{pas99}, and single-atom switching \cite{flo01}.
In a parallel development, considerable work was done recently on entanglement properties \cite{vid03}. The detection of entanglement is one of the fundamental problems in quantum information theory. From a theoretical point of view one can try to answer the question whether a given entirely known state is entangled or not, but despite a lot of progress in the last years [11,12], no general solution of this problem is known. In experiments, one aims at detecting entanglement without knowing the state completely. Bell inequalities \cite{per99} and entanglement witnesses \cite {hor96} are the main tools to tackle this task. Interestingly, the concurrence of the ground state which is related to the entanglement of formation, has been shown to be strongly affected at the critical point \cite {woo98}. More precisely, in the one-dimension, it has been shown that the derivative of the concurrence with respect to the coupling constant diverges at the transition point, although the concurrence itself is not maximum. These pioneering results raise the question of the universality of these behaviors. Actually, the lack of exact solutions especially in higher dimensions implies a numerical treatment which often restrict the study to a small number of degrees of freedom.
Heisenberg's uncertainty relations had tremendous impact in the field of quantum optics particularly in the context of the construction of coherent states and also for different physical systems as well as the reconstruction of quantum states. The minimization problem of finding the number-phase uncertainty state has been considered and minimum uncertainty state relations between number and phase uncertainty are presented \cite{bia75}. Many authors argued that \cite{buz98}, the Heisenberg inequality is too weak for practical purposes, which led them to the establishment of information theoretic uncertainty relations.
Our aim of the present paper is to consider the dynamics of a system of three-level atoms with dipole interaction in presence of the photonic band gap and study the concurrence and the entropic uncertainty relation for number and phase. With applying some approximations, one can deal with the quantization of the electromagnetic field modes of a homogeneous, but anisotropic medium which can then be made to form one of the sandwich layers in the slab structure under consideration involving two semi-infinite periodic photonic crystals. With the electromagnetic modes quantized, one can evaluate the entanglement degree, and explore its variations with the controllable parameters of the system. To reach our goal we have to find an exact analytic solution of the time dependent Schr\"{o}dinger equation of the system. We show that a reasonable amount of entanglement can be achieved in a system of three-level atoms with dipole interaction in presence of the photonic band gap and essentially we establish deeper connections between entropic uncertainty relations and entanglement.
The organization of this paper is as follows: in section 2, we give an overview of effective medium approach and dispersions, followed by subsection 2.1 where we introduce our Hamiltonian model and give exact analytic solution for the Schr\"{o}dinger equation in the frame of the dressed state formalism. In section 3, we employ the analytical results obtained in section 2 to investigate the properties of the entanglement degree due to the concurrence, and classify the behavior in several parameter regimes assuming that the electromagnetic field is in a coherent state in subsection 3.1. In section 4, we essentially establish deeper connections between entropic uncertainty relations and entanglement. Numerical results for the phase entropy are discussed in the subsection 4.1 for two different cases; one is the resonant and the other is the off-resonant case. The prospects for experimental observation of our predictions are analyzed in section 5. Finally, a summary of the main points of this work ends the paper and a few avenues for further investigations are indicated in section 6.
\section{Effective medium approach}
The effective-medium approach can be applied to situations in which all three regions of the structure possess frequency-dependent dielectric functions. In fact, the rapid pace of the technological progress in solid-state quantum computing gives one a hope that the specific prescriptions towards building robust qubits and their assemblies discussed in this work can be implemented in future devices. In this regard, very promising fields where the concept of nonlinear localized modes may find practical applications is the quantum computation of photonic band gap materials, periodic dielectric structures that produce many of the same phenomena for photons as the crystalline atomic potential does for electrons [4,18]. Nonlinear photonic crystals (or photonic crystals with embedded nonlinear impurities) create an ideal environment for the generation and observation of nonlinear localized photonic modes. Much theoretical work has been done on the properties of finite one dimensional photonic band gap (PBG) crystals \cite{joa95}, including recent calculations of the thermal emissivity of such one-dimensional structures \cite{cor99}. The strong angular dependence of the gap effect with a one-dimensional structure has motivated successful experimental work with three-dimensional structures \cite{lin00}. In particular, the existence of such modes for the frequencies in the photonic band gaps has been predicted \cite{joh93} for $2D$ and $3D$ photonic crystals with Kerr nonlinearity. Nonlinear localized modes can also be excited at nonlinear interfaces with quadratic nonlinearity \cite{suk99}, or along dielectric waveguide structures possessing a nonlinear Kerr-type response \cite{mcg99}.
\begin{figure}
\caption{A schematic representation of the dielectric slab between two photonic crystals occupying the regions $z<0$ and $z>r$.}
\end{figure} The system that we consider here consists of a dielectric cavity occupying the region $0<z<r$ and the photonic crystals occupy the regions $z>r$ and $ z<0$. For long wavelength fields and in the effective medium approach, the photonic crystal has the optical characteristic of a uniaxial medium \cite {kam00}. Moreover we shall specialize to uniaxial media, so that our system has only two principal axes with the $z-$axis as the optical axis. In this case, the components of the dielectric tensor appropriate to the photonic crystals can be written as
\begin{equation} \epsilon =\left( \begin{array}{c} \epsilon \\ 0 \\ 0 \end{array} \begin{array}{c} 0 \\ \epsilon \\ 0 \end{array} \begin{array}{c} 0 \\ 0 \\ \epsilon _{z} \end{array} \right) , \label{co} \end{equation}
where, $\epsilon =\epsilon _{0}\epsilon ^{||},$ $\epsilon _{z}=\epsilon _{0}\epsilon _{z}.$ The dielectric tensor components for the two semi-infinite crystals can be written in the following forms $\epsilon _{1}^{||}=(\eta _{1}d_{1}+\eta _{2}d_{2})/d_{12},\qquad \epsilon _{z1}=\eta _{1}\eta _{2}d_{12}/(\eta _{1}d_{2}+\eta _{2}d_{1}),$ and $\epsilon _{2}^{||}=(\eta _{3}d_{3}+\eta _{4}d_{4})/d_{34},\qquad \epsilon _{z2}=\eta _{3}\eta _{4}d_{34}/(\eta _{3}d_{4}+\eta _{4}d_{3}),$ where the $
d_{ij}=d_{i}+d_{j},$ the subscripts 1 and 2 on $\epsilon _{i}^{||}$ and $ \epsilon _{zi}$ refer to the first and second photonic crystal. The dielectric functions $\eta _{1}$ and $\eta _{2},$ one or both of which may be frequency dependent. The photonic crystals are treated using the effective medium approach, which pertains to any layer structure formed by alternate periodic stacking of two types of layers of locally isotropic materials of thicknesses d$_{1}$ and $d_{2}$ (see figure 1).
In this paper we are concerned with the interface polaritons which are characterized by imaginary wave vectors normal to the interfaces such that the waves are decaying with distance from the interfaces at $z=0$ and $z=r$ into the outer regions and are hyperbolic in the slab \cite {kam00,mcg99,cot89}. To see the salient features of the effective medium description we shall ignore retardation effects, which amounts to ignoring throughout ($\omega /c$) terms. In this case dispersion relation for the surface polaritons takes the form
\begin{equation} k_{s}r=\arctan h\left( -\frac{k_{s}}{\epsilon _{s}}\times \frac{
(k_{1}/\epsilon _{1}^{||})+(k_{2}/\epsilon _{2}^{||})}{(k_{s}/\epsilon _{s})+(k_{1}k_{2}/\epsilon _{1}^{||}\epsilon _{2}^{||})}\right) . \end{equation}
The dispersion relations obtained from the Maxwell wave equation of this system lead to two distinct equations \cite{kon75} $k_{s}^{2}=k_{||}^{2}-
\omega ^{2}\epsilon _{s}/c^{2},\quad k_{i}^{2}=\epsilon _{i}^{||}k_{||}^{2}/\epsilon _{zi}-\omega ^{2}\epsilon _{i}^{||}/c^{2},$ where s refers to the slab cavity.
It is important to note that infinite and semi-infinite photonic crystals have the same band structure \cite{zol98}. The only difference is the existence of surface modes in the case of semi-infinite structure. The main feature of all 1D photonic crystals is that although forbidden gaps exist for most given values of the tangential component of the wave vector ($k$), there is not an absolute nor complete photonic band gap if all possible values of the tangential component of the wave vector are considered \cite {joa95}. Having determined the modes we can now quantize the fields associated with these modes using the usual quantization procedure \cite {kam00} the single-mode quantized field takes the form \begin{equation}
E(\hat{x},t)=E_{0}\hat{a}(\hat{k}_{||})\exp [i(\hat{k}_{||}.\hat{x}-\omega t)]+H.C., \end{equation}
where $E_{0}$ is the strength of the electric field, $\hat{k}_{||}$ is the wave vector, $\hat{x}$ is the position operator and $\hat{a}$ the annihilation operator.
\subsection{The model and methods of solution}
Accurate potentials are of course required for a quantitatively correct prediction of the behavior and properties of real quantum systems. However, even qualitative conclusions drawn from simulations employing inaccurate or invalidated potentials can be problematic. The most appropriate form of the potential depends largely upon the properties of interest to the simulators. Now we consider the interaction of the abovementioned modes with a three-level atom in three different configurations, namely, $V-,$ Lambda-
and cascade-type. The transition in the 3-level atom is characterized by the dipole matrix element $\lambda _{ij}.$ The operator $\hat{S}_{ii}$ describes the atomic population of level $|i\rangle _{A}$ with energy $\omega _{j},(j=a,b,c)$ and the operator $\hat{S}_{ij},(i\neq j)$ describes the transition from level $|i\rangle _{A}$ to level $|j\rangle _{A}$. \begin{figure}
\caption{The V-type, $\Lambda $-type and $\Xi $-type three-level atom interacting with a bimodal field. The levels $|1\rangle _{A}$, $|2\rangle _{A}$, and $|3\rangle _{A}$, have the energy values $\hbar \protect\omega
_{1},\hbar \protect\omega _{2}$ and $\hbar \protect\omega _{3}$, respectively. The transitions $|1\rangle _{A}\longrightarrow |2\rangle _{A}$
, and $|2\rangle _{A}\longrightarrow |3\rangle _{A}$, are coupled to two intra-cavity different modes $\hat{a}_{1}$ and $\hat{a}_{2}$ with eigenfrequencies $\Omega _{1}$ and $\Omega _{2}$. The detunings of the levels $|1\rangle _{A},|2\rangle _{A}$, and $|2\rangle _{A},|3\rangle _{A}$, are $\Delta _{1}=\protect\omega _{1}-\protect\omega _{2}-\Omega _{1}$ and $ \Delta _{2}=\protect\omega _{3}-\protect\omega _{2}-\Omega _{2}$, for V-type, and $\Xi $-type, while $\Delta _{1}=\protect\omega _{2}-\protect \omega _{1}-\Omega _{1}$ and $\Delta _{2}=\protect\omega _{2}-\protect\omega _{3}-\Omega _{2}$ for $\Lambda $-type. }
\end{figure} The total Hamiltonian of this system is $\hat{H}=\hat{H}_{0}+\hat{H}_{int}$. The $3$ eigenstates, $\left\vert \xi _{i}\right\rangle $ and corresponding eigenenergies, $\alpha _{i}$ are assumed to be known. The total wave-function may be expanded in terms of the known eigenstates, namely \begin{equation} \left\vert \Psi (t)\right\rangle =A_{1}(t)\left\vert \xi _{1}\right\rangle +A_{2}(t)\left\vert \xi _{2}\right\rangle +A_{3}(t)\left\vert \xi _{3}\right\rangle . \label{wEq} \end{equation} With atomic units, using Schr\"{o}dinger equation, we obtain the coupled equations for our three-level system, namely \begin{equation} i\frac{\partial A_{j}(t)}{\partial t}=r_{j}A_{j}(t)+\sum \limits_{k=1}^{3}H_{jk}A_{k}(t), \label{Pro} \end{equation} where $\hat{H}_{0}\left\vert \xi _{i}\right\rangle =r_{i}\left\vert \xi _{i}\right\rangle $ and $H_{jk}=\left\langle \xi _{j}\right\vert \hat{H} _{int}\left\vert \xi _{k}\right\rangle .$ These equations are exact for any three-level atom. In the interaction picture, let us consider a three-level system described, in an appropriate rotating frame, by the Hamiltonian \begin{equation} \hat{H}_{int}=\Delta _{1}\hat{S}_{11}+\Delta _{2}\hat{S}_{33}+\lambda _{21} \hat{R}_{1}\hat{S}_{21}+\lambda _{32}\hat{R}_{2}\hat{S}_{32}+\lambda _{21}^{\ast }\hat{R}_{1}^{\dagger }\hat{S}_{12}+\lambda _{32}^{\ast }\hat{R} _{2}^{\dagger }\hat{S}_{23}. \end{equation} The atom-field couplings $\lambda _{ij}$ are given by $\lambda _{ij}=Y\mu _{ij}.E,$where $E$ $\ \ $is the quantized electric field given by equation (3) and $\mu _{ij}$ is the matrix dipole moment coupling between the state $i $ and $j$. The $Y$ factor accounts for local field effects and is given by $ Y=3\epsilon _{s}(\omega )/(2\epsilon _{s}(\omega )+1),$ where $\epsilon _{s}(\omega )$ is given in equation (1). It is easy to write $\lambda _{ij}$ in the following form \begin{equation} \lambda _{ij}=\frac{3\epsilon _{s}(\omega )}{2\epsilon _{s}(\omega )+1}. \frac{\left( \omega /\omega _{T}\right) ^{2}-\left( \omega _{L}/\omega _{T}\right) ^{2}}{\left( \omega /\omega _{T}\right) ^{2}-\eta ^{2}}, \end{equation} where $\eta ^{2}=[$ $2\epsilon _{s}(\omega )\left( \omega _{L}/\omega _{T}\right) ^{2}+1]/[2\epsilon _{s}(\omega )+1]$. The transitions between the three levels may occur in three different configurations depending upon the relationship between the energies $E_{1},E_{2}$ and $E_{3}$ of levels $ 1,2$ and $3$. The possible configurations are \cite{abd87} (i) the $V$-type corresponding to $E_{2}<E_{1}<E_{3}$, (ii) the $\Lambda -$type or Raman configuration corresponding to $E_{1}<E_{3}<E_{2}$ and (iii) the $\Xi -$type or ladder-type corresponding $E_{1}<E_{2}<E_{3}$. Each of the two pairs of levels can be coupled by only one-mode or two-mode. The field operators in the abovementioned three types are (i) $F_{1}=\hat{a}^{\dagger },F_{2}=\hat{b }$ for $V$-type, (ii) $F_{1}=\hat{a},F_{2}=\hat{b}^{\dagger }$ for $\Lambda $ -type and (iii) $F_{1}=\hat{a},F_{2}=\hat{b}$ for $\Xi $-type with $\hat{a}= \hat{b}$ if both pairs of levels are coupled by the same mode.
In order to solve equations (5), we assume that \cite{abd87} \begin{equation} G(t)=A(t)+xB(t)+yC(t), \end{equation} which means that \begin{equation} i\frac{dG(t)}{dt}=\left( r_{1}+v_{1}^{\ast }y\right) \left\{ A(t)+\frac{ r_{2}x+v_{2}^{\ast }y}{r_{1}+v_{1}^{\ast }y}B(t)+\frac{v_{2}x+r_{3}y}{ r_{1}+v_{1}^{\ast }y}C(t)\right\} , \end{equation} where $v_{1}$ and $v_{2}$ are given using equations (5) and (6). We seek $ G(t)$ such that $i\overset{.}{G}(t)=zG(t)$. This hold if \begin{equation*} y=\frac{v_{2}x+r_{3}y}{r_{1}+v_{1}^{\ast }y},\quad x=\frac{ r_{2}x+v_{2}^{\ast }y}{r_{1}+v_{1}^{\ast }y},\quad z=r_{1}+v_{1}^{\ast }y. \end{equation*} After some algebra this leads to a cubic equation which has three eigenvalues $x_{i}(y_{i})$ which determine the $z_{i}$. There are also three corresponding eigenfunctions $G_{j}(t)=G_{j}(0)\exp (-iz_{j}t)$, where \begin{equation} G_{j}(t)=M_{j1}A(t)+M_{j2}B(t)+M_{j3}C(t), \end{equation} where \begin{equation} M_{ji}=\left( \begin{array}{c} 1 \\ 1 \\ 1 \end{array} \begin{array}{c} x_{1} \\ x_{2} \\ x_{3} \end{array} \begin{array}{c} y_{1} \\ y_{2} \\ y_{3} \end{array} \right) . \end{equation} Now, we express the unperturbed state amplitude $A(t),B(t)$ and $C(t)$ in terms of the dressed state amplitude $R_{j}$ \begin{equation} F_{i}(t)=\sum\limits_{j=1}^{3}M_{ij}^{-1}G_{j}(t)=\sum \limits_{j=1}^{3}M_{ij}^{-1}G_{j}(0)\exp (-iz_{j}t), \end{equation} $F_{1,2,3}(t)=A,B,C.$ Using the above equations, we can write \begin{eqnarray} A(t) &=&\frac{1}{D}\left[ (x_{2}y_{3}-y_{2}x_{3})e^{-iz_{1}t}+(x_{3}y_{1}-y_{3}x_{1})e^{-iz_{2}t}+(x_{1}y_{2}-y_{1}x_{2})e^{-iz_{3}t} \right] , \notag \\ B(t) &=&\frac{1}{D}\left[ (y_{2}-y_{3})e^{-iz_{1}t}+(y_{3}-y_{1})e^{-iz_{2}t}+(y_{1}-y_{2})e^{-iz_{3}t} \right] , \\ C(t) &=&\frac{1}{D}\left[ (x_{2}-x_{3})e^{-iz_{1}t}-(x_{3}-x_{1})e^{-iz_{2}t}-(x_{1}-x_{2})e^{-iz_{3}t} \right] , \notag \end{eqnarray} where $D=\det (M)=x_{1}y_{2}+x_{2}y_{3}+x_{3}y_{1}-x_{1}y_{3}-x_{2}y_{1}-x_{3}y_{2}.$ We have thus completely determined the dynamics of a three-level system in the presence of photonic crystal.
The picture in this case is of the three-level system in the presence of photonic band gap and the detuning, rather than the usual picture of the three-level Jaynes-Cummings model (JCM) system. The important point to note here is that, using the above analytic approach, any three-level Hamiltonian is likewise exactly solvable, with precisely similar eigenvectors and eigenvalues that are obtained directly using equations (4) and (6). In Ref. \cite{bou04} an analytic approach is proposed for three-level systems, based on the Riccati nonlinear differential equation. However, the solution obtained is valid only in certain situations. On the other hand, our analytic approach removed the restriction that considered in the previous work and this solution is valid for any three-level system.
Next, we discuss a frequently encountered phenomena of particular interest in which we define the entanglement measure of the present system.
\section{Concurrence}
Quantum entanglement has recently been attracted much attention as a potential resource for communication and information processing \cite{ben96} . Entanglement is usually arise from quantum correlations between separated subsystems which can not be created by local actions on each subsystem. The concept of concurrence originates from the seminal work of Hill and Wootters \cite{woo98} where the exact expression of the entanglement of formation of a system of two qubits was derived. They showed that the entanglement of formation, an entropic entanglement monotone, is a convex monotonic increasing function of the concurrence.
It has been shown that the concurrence of a mixed two-qubit state, $C(\rho _{AB})$, can be expressed in terms of the minimum average pure-state concurrence, $C\left( \left| \psi _{AB}\right\rangle \right) $, where the minimum is taken over all possible ensemble decompositions of $\rho _{AB}.$ So that, the concurrence is defined of a mixed state $\rho $ for $2\times 2$ quantum systems, in the following form \cite{woo98} \begin{equation} C(\rho )=\max \left( \sigma _{1}-\sigma _{2}-\sigma _{3}-\sigma _{4}\right) , \end{equation} where the $\sigma _{i}$ are the square roots of the eigenvalues of the product matrix $Q$, the singular values (by convention sorted in descending fashion), all of which are non-negative real quantities \begin{equation} Q=\sqrt{\rho }^{T}\sigma _{y}\otimes \sigma _{y}\sqrt{\rho }, \end{equation} $\sigma _{y}$ is the well-known Pauli matrix, and $\sqrt{\rho }$ is any matrix satisfying $\sqrt{\rho }=\sqrt{\rho }^{\dagger }.$ The importance of this measure follows from the direct connection between concurrence and entanglement of formation $E_{f}$ \begin{equation} E_{f}\left( \rho \right) =-\mu _{+}\ln \mu _{+}-\mu _{-}\ln \mu _{-}, \end{equation} where \begin{equation} \mu _{\pm }=\frac{1}{2}\left( 1\pm \sqrt{1-C(\rho )^{2}}\right) . \end{equation} One can prove that $\rho $ is separable if and only if the concurrence is zero.
Let us now turn our attention to the definition of the concurrence of a pure state \cite{run01} on a $(N\times K)-$dimensional Hilbert space $\Re =\Re _{N}\otimes \Re _{K}.$ The \textit{flip operator }$\mathit{F}$\textit{\ } acting on an arbitrary Hermitian operator $A$ on $\Re $ can be written as \begin{equation} F(A):=A+(trA)\Pi -(tr_{N}A)\otimes \Pi _{K}-\Pi _{N}\otimes (tr_{K}A), \end{equation} where $tr_{N}$ and $tr_{k}$ the partail traces over $\Re _{N}$ and $\Re _{K}, $ respectively. We denote by $\Pi _{N}$ and $\Pi _{K}$ the identity on $\Re _{N}$ and $\Re _{K},$ respectively. The expectation value $\left\langle
\psi \right| F(\rho _{\psi })\left| \psi \right\rangle ,$ where $\rho _{\psi
}=\left| \psi \right\rangle \left\langle \psi \right| $, is non-negative for all pure states and equals zero if and only if $\left| \psi \right\rangle $ is a product state. This allows to define the concurrence of any arbitrary bipartite pure state as \cite{run01} \begin{eqnarray}
C\left( \left| \psi \right\rangle \right) &=&\sqrt{\left\langle \psi \right|
F(\rho _{\psi })\left| \psi \right\rangle } \notag \\
&=&\sqrt{2\left( \left\langle \psi |\psi \right\rangle ^{2}-tr(\rho _{N}^{2})\right) }, \end{eqnarray}
where $\rho _{N}=tr_{K}\left( \rho _{\psi }\right) $ is the reduced density operator of dimension N. For a normalized state, $\left\langle \psi |\psi \right\rangle =1,$ it interpolates monotonously between zero for product states and $\sqrt{\frac{2(N-1)}{N}}$ for maximally entangled states.
To investigate the concurrence for the system under consideration, we have to evaluate the reduce atomic density matrix $\rho _{_{A}}=tr_{_{F}}\rho (t), $ which can be written as \begin{equation}
\rho _{_{A}}=\sum_{i=1,2,3}\rho _{_{ii}}\left| i\right\rangle \left\langle i\right| +\sum_{i,j=1,2,3,i\neq j}^{{}}\rho _{_{ij}}\left| i\right\rangle
\left\langle j\right| , \end{equation}
where $\rho _{ij}(t)=\langle i|\rho _{_{A}}(t)|j\rangle ,\quad i,j=1,2$ and $ 3.$ Using equations (19) and (20), we can write the concurrence in the following form \begin{equation}
C\left( \left| \psi \right\rangle \right) =\sqrt{2\sum_{i,j=1,2,3,i\neq j}^{{}}\left( \rho _{_{ii}}\rho _{_{jj}}-\rho _{_{ij}}\rho _{_{ji}}\right) }. \end{equation} Although the concurrence and therefore the results we obtain are not restricted to the standard one-mode three-level system, we will use that language throughout most of the paper.
Having specified the various photonic crystal and field amplitude parameters, we will present in the following subsection the results of our numerical analysis of the concurrence.
\subsection{Numerical results}
For applications in real systems, we consider the dipole emitters with frequencies in the reststrahl band of \ GaAs. In this subsection we will discuss the time dependence of the concurrence, which considered as an entanglement measure. We will consider the commonly used state as initial condition for the cavity field: the coherent state, which may be applicable in different situations. As might be expected, the behavior of the three-level system changes dramatically depending on the initial field state. Throughout this subsection the quantity to be examined is the concurrence $C\left( \left\vert \psi \right\rangle \right) .$ \begin{figure}
\caption{The evolution of the concurrence $C\left( \left\vert \protect\psi \right\rangle \right) $ as a function of the scaled time $\protect\lambda _{1}t$ and the mean photon number $\bar{n}$. The parameters are $ \protect \epsilon _{0}=10.89,$ $\protect\eta =1.085,$ $\protect\omega /\protect\omega _{T}=2,$ $\protect\omega _{0}/\protect\omega _{T}=1,$ $\hslash \protect \omega _{L}=36.29$meV$,$ $\hslash \protect\omega _{T}=33.25$meV, $d_{1}=500$ \AA , $d_{2}=300$\AA , $\protect\epsilon _{1}=9,$ $\protect\epsilon _{2}=1.3, $ $d_{3}=500$\AA , $d_{4}=400$\AA , $\protect\epsilon _{3}=10,$ $ \protect\epsilon _{4}=1.5$ and $L=1.5d$, and different values of the detuning parameter, where $\Delta =0$ for Fig. 3a and $\Delta =5\protect \lambda _{1}$ for Fig 3b. }
\end{figure}
In figure 3, we present the oscillatory behavior of the concurrence $
C\left( \left| \psi \right\rangle \right) $ against the scaled time $\lambda _{1}t$ and the mean photon number $\overline{n}$ for different values of the detuning parameter, where $\Delta =0$ for Fig. 3a and $\Delta =5\lambda _{1}$ for Fig 3b. We consider a specific system in which the cavity is taken as GaAs with $\epsilon _{0}=10.89,$ $\eta =1.085,$ $\omega /\omega _{T}=2,$ $ \omega _{0}/\omega _{T}=1,$ $\hslash \omega _{L}=36.29$meV$,$ $\hslash \omega _{T}=33.25$meV. The photonic crystals parameters are given by the arbitrary set, $d_{1}=500$\AA , $d_{2}=300$\AA , $\epsilon _{1}=9,$ $ \epsilon _{2}=1.3,$ $d_{3}=500$\AA , $d_{4}=400$\AA , $\epsilon _{3}=10,$ $ \epsilon _{4}=1.5$ and $L=1.5d.$ The general behavior due to the coherent state of the field does not contain any surprises it is quite broad, corresponding to the standard quantum limit. The value of concurrence at the first maximum is 1, which is quit remarkable, see figure 3a. After the time goes on, we see that the maximum value of the concurrence decreases with small amplitude of the oscillations. As the mean photon number increased, the number of oscillations decreased.
The effect of the parameter which describes the mismatch between the atomic frequency and the mean frequency of the cavity mode has been considered in figure 3b. We set the other parameters as the same as in figure 3a, and $ \Delta =5\lambda _{1}$. As $\Delta $ is increased the behavior of the three-level system becomes increasingly erratic. Shorter revival times cause successive revivals to overlap and interfere so that the time evolution appears irregular. The detuning parameter at which irregularity emerges is closely tied to the mean-photon number: the higher the mean-photon number, the smaller the detuning needed to produce irregular behavior. Larger detuning also results in decreased revival amplitude due to the larger number of frequencies in the sum, which causes the rephasing to be less complete. However, a signature of the revivals persists as a return to the bare Rabi frequency even at mean-photon number high enough that the behavior looks random and the revival amplitude is essentially washed out. From our further calculations (which are not presented here), we point out that as we increase the value of the detuning one can see that the revival time is also prolonged, however the period of fluctuations is decreasing. Detuning affects the revival time by elongating it and the maximum value of the entanglement degree becomes smaller and smaller. Similar to the case of a two-level atom, detuning shifted the atomic occupation probability around which it oscillates upward meaning that the energy is stored in the atomic system.
\begin{figure}
\caption{The evolution of the concurrence $C\left(\left|\protect\psi \right\rangle\right)$ as a function of the mode-frequency $\protect\omega/ \protect\omega_T$ and the mean photon number $\bar{n}$ for different values of the scaled time, where $\protect\lambda_1t =\protect\pi/2$ for Fig. 4a and $\protect\lambda_1t =3\protect\pi/2$ for Fig 4b. }
\end{figure}
Now we will turn our attention to the effect on the concurrence of the mode frequency $\omega /\omega _{T}$. In particular, we consider $\epsilon _{0}=10.89,$ $\eta =1.085,$ $\omega _{0}/\omega _{T}=1$ and for different values of the scaled time, where $\lambda _{1}t=\pi /2$ for Fig. 4a and $ \lambda _{1}t=3\pi /2$ for Fig 4b. Our particular observation is the maximum entanglement occurs near the band edges, which corresponds to $\omega =1.085\omega _{T}$. Near the band edges the wave vector parallel to the interface reaches its maximum value, and this corresponds to the first two relatively small peaks around the point $1.085$. In the gap region or the reststrahl region of the $GaAs$ system no electromagnetic fields can propagate and coupling is therefore suppressed. The extra peaks around the point $1.085$ are attributed to local field effects and can be understood from looking at equation (7) where $\lambda _{ij}$ has a pole at $\eta =\omega /\omega _{T}.$ One has to bear in mind that the above calculation did not take into explicit account the spatial dependence of the coupling parameters. Therefore, a more careful calculation would have to take into account the nonstationary property of the present system. The above model calculations suggest that physical parameters such as mode frequency, mode-atom coupling and cavity dielectric have important effects on the entanglement. One can see that the oscillations collapse after few Rabi periods and after an interval of time in which the concurrence is constant, the oscillations reappear again. This revival then collapses and a new revival begins.
This behavior highlights once again the role of the functional form of the modified Rabi frequencies in controlling the time evolution of the concurrence. Rabi frequencies which obtained in the present model are similar to that obtained from the standard three-level model but involving a frequency-dependent dielectric function. An important point to keep in mind when comparing the results presented here with results from the usual three-level system in the absence of the photonic band gap is that: they are give a different feature relative to the entanglement. This raises an interesting question: can one use the present system in building quantum logic gates? Calculations and detailed discussion of this issue will be presented in a forthcoming paper.
\section{Phase entropy}
One of the most striking features of quantum mechanics is the property that certain observable cannot simultaneously be assigned arbitrarily precise values. This property does not compromise claims of completeness for the theory, since it may consistently be asserted that such observable cannot simultaneously be measured to an arbitrary accuracy \cite{bia75}. The Shannon entropies associated with the photon number distribution $P_{m}$ and phase probability distribution $P(\theta ,t),$ \begin{eqnarray}
P_{m} &=&\langle m|\rho (t)|m\rangle , \notag \\
P(\theta ,t) &=&\langle \theta |\rho (t)|\theta \rangle , \end{eqnarray}
where $|m\rangle $ is the Fock state and $|\theta \rangle $ is the phase state, are given respectively by \cite{bia75}{\ } \begin{eqnarray} R_{N} &=&-\sum\limits_{m=0}^{\infty }P_{m}\ln P_{m}, \notag \\ R_{\psi } &=&-\int\limits_{2\pi }\left( P(\theta ,t)\ln P(\theta ,t)\right) d\theta . \end{eqnarray} The entropic uncertainty relations for the number and phase distribution determine the lower bound on the sum of the Shannon entropies $R_{N}$ and $ R_{\psi }:$ \begin{equation} R_{N}+R_{\psi }\geq \ln (2\pi ). \end{equation} This equality is satisfied by a Fock state for which $R_{N}=0$ and $R_{\psi }=\ln (2\pi ).$ Other physical states give an entropic sum greater than $\ln (2\pi ).$ Specifically, for a coherent state we find that the sum is $\ln (e\pi )$ for the mean photon number greater than one, i.e. \begin{equation} R_{N}+R_{\psi }\geq \ln (e\pi ). \end{equation} The lower bound for the position-momentum entropic uncertainty relation is also given by right-hand side of this equation i.e $\ln (e\pi ).$
The single-mode of the Pegg-Barnett phase formalism which of interest in the field of quantum optics can be constructed from the single-mode phases \cite {oba98} to take the form \begin{equation} P(\theta ,t)=\lim_{s\rightarrow \infty }\biggl(\frac{s+1}{2\pi }\biggr)
\langle \theta _{m}|\rho (t)|\theta _{m}\rangle , \label{31} \end{equation}
$|\theta _{m}\rangle $ is a phase state of the mode, \begin{equation}
|\theta _{m}\rangle =\frac{1}{\sqrt{(s+1)}}\sum_{n=0}^{s}e^{in\theta _{m}}|n\rangle , \label{32} \end{equation} where $\theta _{m}=\theta _{\circ }+\frac{2\pi m}{s+1},$ and $m=0,1,...s,$ and $\theta _{\circ }$ arbitrary. Equation (26) defines a particular basis set of $(s+1)$ mutually orthogonal phase states.
\begin{figure}
\caption{$P(\protect\theta ,t)$ against $\protect\theta $ and the scaled time $\protect\lambda _{1}t$. The parameters are $\protect\epsilon _{0}=10.89,$ $\protect\eta =1.085,$ $\protect\omega /\protect\omega _{T}=2,$ $\protect\omega _{0}/\protect\omega _{T}=1,$ $\hslash \protect\omega _{L}=36.29$meV$,$ $\hslash \protect\omega _{T}=33.25$meV, $d_{1}=500$\AA , $ d_{2}=300$\AA , $\protect\epsilon _{1}=9,$ $\protect\epsilon _{2}=1.3,$ $ d_{3}=500$\AA , $d_{4}=400$\AA , $\protect\epsilon _{3}=10,$ $ \protect \epsilon _{4}=1.5$ and $L=1.5d$, where (a) $\Delta =0$ and (b) $\Delta =5 \protect\lambda _{1}.$ }
\end{figure} Using the standard procedure \cite{oba98}, the phase probability distribution, the expectation value and the variance of the Hermitian phase operator may be obtained for the field. Since the coherent field at $t=0$ belongs to a class of partial phase states, we have chosen the reference phase $\theta _{0}$ as $\theta _{0}=\beta -\frac{\pi s}{s+1},$ and introduced the new phase labels $\zeta =m-\frac{1}{2}s$ where $ m=0,1,2,...,s. $ Then as $s$ tends to infinity the summation may be transformed into an integral after replacing $\frac{2\pi \zeta }{s+1}$ by $ \theta ,$ and $\frac{2\pi }{s+1}$ by $d\theta .$ This leads to continuous phase probability distribution, where \begin{equation} P(\theta ,t)=\frac{1}{2\pi }\left( 1+2\sum\limits_{n>m}^{\infty }\left\{ A_{n,m}(t)\cos [\theta (n-m)]+B_{n,m}(t)\sin [\theta (n-m)]\right\} \right) , \end{equation} where $A_{n,m}(t)$ and $B_{n,m}(t)$ are given by \begin{eqnarray} A_{n,m}(t) &=&{Re}\left\{ A_{n}(t)A_{m}^{\ast }(t)+B_{n}(t)B_{m}^{\ast }(t)+C_{n}(t)C_{m}^{\ast }(t)\right\} , \notag \\ B_{n,m}(t) &=&{Im}\left\{ A_{n}(t)A_{m}^{\ast }(t)+B_{n}(t)B_{m}^{\ast }(t)+C_{n}(t)C_{m}^{\ast }(t)\right\} . \end{eqnarray} The phase probability distribution is normalized according to $\int_{-\pi }^{\pi }P(\theta ,t)d\theta =1.$
\subsection{Numerical results}
In what follows we shall display some general arguments based on the equality sign in the Heisenberg uncertainty relations that to demonstrate the phase entropy of a general three-level system in the presence of photonic band gab when the initial state of the field is assumed to be in a coherent state.
In figure 5a, we have plotted the phase probability distribution $P(\theta ,t)$ as a function of the scaled time $\lambda _{1}t$ and $\theta $ taking into consideration the presence of the photonic band gap. For example at time $\lambda _{1}t=0$ we realize that the phase distribution $P(\theta ,t)$ starts with a single-peaked structure at $\theta =0$ corresponding to the initial coherent state. Then as the time develops the peak splits into two peaks moving into two opposite directions. \begin{figure}
\caption{$P(\protect\theta ,t)$ against the mode-frequency $\protect\omega / \protect\omega _{T}$ and $\protect\theta $. The parameters are the same as in figure 5, but (a) $\protect\lambda _{1}t=\protect\pi /2$ and (b) $ \protect\lambda _{1}t=3\protect\pi /2.$ }
\end{figure} However the amplitudes of the split peaks fluctuate in time giving a top like shape until the two peaks reach the values $\theta =\pm \pi $ at middle of the revival time but in this range the amplitudes of the peaks do not show any fluctuations. The picture changes greatly as time develops further (say $\lambda _{1}t>40$) where we find that the two-peak profile breaks up into multi peak with reduction of the amplitudes of these peaks. Thus the phase distribution shows diffusion as well as bifurcation. Different features are visible when we consider the off-resonant case and the behavior of the phase probability distribution is changed dramatically (see figure 5b). In this case we observe that there is a diffusion of the peaks at earlier time.
In figure 6, we consider the behavior of the phase probability distribution against the mode frequency $\omega /\omega _{T}$ and $\theta $ for the same parameters as in figure 5, while in this figure, we keep the the scaled time $\lambda _{1}t$\ fixed, where, $\lambda _{1}t=\pi /2$ for figure 6a and $ \lambda _{1}t=3\pi /2$ for figure 6b. One may clearly see that the phase probability distribution is discontinuous near the band edges. This corresponds to the zero value at the point $1.085$. We can prove, in an analogous manner to the equation (7), that the $\omega /\omega _{T}=1.085$ is a pole of the atom-field coupling\ can not avoided. It is interesting to see that, the phase probability distribution does not depend on the mode frequency for a fixed value of $\theta ,$ except at the point $1.085$. As the time increased, the only difference is that, the phase probability distribution peak splits into two peaks moving into two opposite directions, keeping the symmetry around the point $\theta =0.$
\begin{figure}
\caption{The phase entropy $R_{\protect\psi }$ (a) and the number entropy $ R_{N}$ (b) as functions of the scaled time $\protect\lambda _{1}t$ with an initial coherent state of the radiation field with $\bar{n}=20$ based on the exact numerical results due to equation (23). We consider the same parameters as in figure 1. }
\end{figure} In figure 7, we plot the number entropy $R_{N}$ and the phase entropy $ R_{\psi }$ as functions of the scaled time $\lambda _{1}t$. The initial state of the field is considered as a coherent state. We specifically present the results for the same values of figure 5. It should be noted that at a special choice of the mean-photon number parameter, the situation becomes interesting, where the Rabi frequency has a minimum value at $\bar{n} $. In this case we find that the general behavior of the entropies $R_{N}$ and $R_{\psi }$ and with an initially coherent field exhibit irregular structures instead of the regular structure resembling those manifested by the number or vacuum states cases. \begin{figure}
\caption{The phase entropy $R_{\protect\psi }$ (a) and the number entropy $ R_{N}$ (b) as functions of the scaled time $\protect\lambda _{1}t$ where, $ \bar{n}=10$ and the same parameters as in figure 1. }
\end{figure} Here it is interesting to note that the periodic oscillations are observed for a short period of the interaction time only. When we consider smaller mean-photon number, the regularity behavior of the oscillations in the entropies $R_{N}$ and $R_{\psi }$ are still obvious (see figure 8) where we have considered the initial mean photon number $\overline{n}=10$. However, the number of oscillations is increased. Also it is interesting to point out that at the revival time optimal phase entropy is attained in all the cases which means that the atom has achieved an almost pure state, this has been observed all through our figures. The number and phase entropic uncertainties for a weak coherent state follow those of underlying states superposition.
\begin{figure}
\caption{The phase entropy $R_{\protect\psi }$ (a) and the number entropy $ R_{N}$ (b) as functions of the mode-frequency $\protect\omega / \protect \omega _{T}$, where, $\bar{n}=20, \protect\lambda_1t=\protect\pi /2$. }
\end{figure}
In figure 9 we plot the entropies $R_{N}$ and $R_{\psi }$ against the mode-frequency $\omega $ in units of $\omega _{T}$ for different values of the scaled time. Now, where the atom-field coupling is proportional to $ \lambda _{ij}$ this explains the origin of the second peak in this figure. It is interesting to note the dependence of these entropies on the mode-frequency, with different values of the scaled time. We wonder, as a possible generalization of this concept, whether there exist another family of similar oscillations if we consider two-qubit system. In such a case, the properties of these systems would probably be of interest, in order to bring further insight and knowledge about entanglement and quantum logic gates for multi-partite systems. As the scaled time increased, a characteristic feature of the entropies $R_{N}$ and $R_{\psi }$ is quit interesting, where more oscillations exist, also, only around the resistable region for the number entropy but less number of oscillations exist for the phase entropy (see figure 10). \begin{figure}
\caption{The phase entropy $R_{\protect\psi }$ (a) and the number entropy $ R_{N}$ (b) as functions of the mode-frequency $\protect\omega / \protect \omega _{T}$, where, $\bar{n}=20,\protect\lambda _{1}t=3\protect\pi /2$. }
\end{figure} Far from the resistable region the entropies behavior observed here does not depend on the mode frequency and the intensity of the initial field mode.
Given a $3D$ photonic crystal with a complete gap, one has the possibility of introducing a defect in the structure which will create a localized state in the gap. If this is a point-like defect then the photon mode will be completely localized about a point. In figure 10, we show the zero point associated with the defect created by removing a small amount of dielectric from one of the vertical dielectric columns of the crystal structure. The resulting defect mode has a state near mid gap. One feature that should be highlighted in this context is the appearance of a frequency gap between the pair of interface dispersion. This gap is present only when the two photonic crystal regions are different, and disappears when they are identical.
\section{Experimental prospects}
The perfect semiconductor crystal is quite elegant and beautiful, but it becomes ever more useful when it is doped. Likewise, the perfect photonic crystal can become of even greater value when a defect is introduced \cite {yab93}. The point to make about photonic crystals is that they are very empty structures, consisting of about 78\ empty space. But in a sense they are much emptier than that. They are emptier and quieter than even the vacuum, since they contain not even zero-point fluctuations within the forbidden frequency band. Our model system consist of a three-level atom located inside a photonic band gap material. There are several ways of placing such an atom inside a photonic crystal. From a material standpoint, it is possible to dope an existing photonic band gap material using ion beam implantation methods. For instance, it has recently been shown that $Er^{3+}$ ions implanted into bulk silicon exhibit sharp free-atom-like spectra \cite {yab93,lan98}. Intense temperature-dependent photoluminescent (PL) at $1.54{ \ \mu }m$ is observed in the system at low temperatures (when the host material is crystalline, Er-related PL is quenched at temperatures above $ 80K $ so that it cannot be detected at room temperatures). This wavelength is particularly significant because it corresponds to the minimum absorption of silica fibre-based optical communication system. Because the PL at $ 1.54\mu m $ is due to the spin-orbit split $^{4}I_{13/2}\rightarrow ^{4}I_{15/2}$ of $4f$ electrons in the $Er^{3+}$ ions which are shielded by outer $5s^{2}5p^{6} $ shells, the influence of the host lattice on the luminescence wavelength is weak. (The key to the success of erbium is that the upper level of the amplifying transition $^{4}I_{13/2}$ is separated by a large energy gap from the next-lowest level $^{4}I_{15/2}$ so that its lifetime is very long and mostly radiative. In spite of the screening of the atomic transition by the outer shells, it is likely that thermal phonons in the silicon host would cause significant dephasing of the quantum degrees of freedom within the erbium 4f shell. Consequently, such a system must be cooled to liquid helium temperatures. Such experiments appear to be nearly within the reach of current technology. Although it has not yet been demonstrated, the system consisting of a multi-level system coupled to a multi-mode appears to be another potential candidate for achieving new features. {Such systems are potentially interesting for their ability to process information in a novel way and might find application in models of quantum logic gates. Therefore, atoms or trapped ions + cavities in a presence of photonic band gap represent, in our opinion, a very promising system for quantum information processing.}
\section{Conclusion}
In this communication the quantum electrodynamic properties of a three-level atom embedded in a photonic band gap material were investigated. We have focused on the application of the effective-medium theory to the present problem in a nanoscale dielectric cavity QED situation. The effective-medium approach can in fact be applied to situations in which all three regions of the structure possess frequency-dependent dielectric functions. Specifically, the combined effects of coherent control by an external driving field and photon localization facilitated by a photonic band gap on entanglement from a three-level atom embedded in a photonic band gap material were examined. Exact solutions of the wave function in the Schr\"{o} dinger picture have been obtained within rotating wave approximation. In particular, we have chosen to focus on three-level system coupled to a single mode. Observation of the three-level system may offer some insight into the quantum nature of the resonator, just as atoms provide a sensitive probe for the nonclassical nature of electromagnetic fields. The observation of revivals, which are a strictly nonclassical phenomenon, would give evidence for the quantum nature of the quantum system.
The results point to a number of interesting features, which arise from the variation of the adjustable parameters of the system, namely, the mode-frequency, dipole vector orientation, dipole position within the slab, the slab width, and the photonic crystal parameters: layer widths and dielectric functions. Our investigations for the entanglement, collapse-revival phenomena, and phase and number entropic uncertainty relations in the presence of the photonic band gap as compared with the usual three-level model are summarized as follows:-
i) The concurrence behavior is reflect the pattern of collapse and revival which is qualitatively similar to that of the usual three-level model but with reduced amplitude. In case of a smaller mean photon number and for initially excited atom the usual pattern in the three-level model of collapse and revival changes to rapid fluctuations of interference patterns for all time considered. In this way, our concurrence function contains all the information necessary to identify the entanglement of a given state. Nevertheless, it depends on the particular choice of the mode-frequency.
ii) The phase entropy can be used to measure entanglement of the system presented here with explicitly atom-field coupling in the presence of photonic band gap. We would like to point out that the phase Shannon entropic considered for the presented model has not been treated in this manner before.
iii) The photonic band gap introduces sudden changes in the concurrence and phase entropy due to the variation of these quantities with mode frequency. This feature attributed to the fact that in the photonic band gap region electromagnetic modes are not allowed to propagate into the dielectric slab and hence no interaction can take place in this region. Theory predicts analytically this behavior for a GaAs system at $\omega =\eta \omega _{T}$.
Finally, we emphasize the fact that without any conditions it was possible to obtain exact analytic solution which reproduce the most important features of the three-level atom interacting with a cavity one- or two-mode in the presence of photonic band gap. A similar set of equations have been derived in \cite{bou04} for a three-level system using some approximations, based on the Riccati nonlinear differential equation. In contrast, the method used here gives exact analytic solutions without any conditions.
\textbf{Acknowledgment}
I acknowledge the hospitality and financial support from the Center for Computational and Theoretical Sciences, Kulliyyah of Science, IIUM, Malaysia where the final version of the paper was prepared. Also, helpful discussions with Prof. A.-S. F. Obada and Prof. M. R. B. Wahiddin are gratefully acknowledged.
\end{document} |
\begin{document}
\title{The Role of $lpha$-Scaling for Cartoon Approximation}
\begin{abstract} The class of cartoon-like functions, classicly defined as piecewise $C^2$ functions consisting of smooth regions separated by $C^2$ discontinuity curves, is a well-established model for image data. The quest for frames providing optimal approximation for this class has among others led to the development of curvelets, contourlets, and shearlets. Due to parabolic scaling, these systems are able to provide $N$-term approximations converging with a quasi-optimal rate of order $N^{-2}$. Replacing parabolic scaling by $\alpha$-scaling, one can construct $\alpha$-curvelet and $\alpha$-shearlet frames which interpolate between wavelet-type systems for $\alpha=1$, the classic parabolically scaled systems for $\alpha=\frac12$, and ridgelet-type systems for $\alpha=0$. Previous research shows that if $\alpha\in[\frac{1}{2},1)$ they provide quasi-optimal approximation for cartoons of regularity $C^{1/\alpha}$ with a rate of order $N^{-1/\alpha}$.
In this work we continue the exploration of approximation properties of $\alpha$-scaled representation systems, with the aim to better understand the role of the parameter $\alpha$. Concerning $\alpha$-curvelets with $\alpha<1$, we prove that the best possible $N$-term approximation rate achievable for cartoons with curved edges is limited to at most $N^{-1/(1-\alpha)}$, independent of the smoothness of the cartoons. The maximal rate that can be obtained by simple thresholding of the frame coefficients is even bounded by $N^{-1/\max\{\alpha,1-\alpha\}}$. Systems of $\alpha$-curvelets thus cannot take advantage of regularity higher than $C^{1/\alpha}$ if $\alpha\in[\frac{1}{2},1)$, the rate of $N^{-1/\alpha}$ cannot be surpassed. For $C^\beta$ cartoons with $\beta\ge2$ the classic $\frac12$-curvelets provide the best performance with a rate of order $N^{-2}$, however below the optimal rate of order $N^{-\beta}$ if $\beta>2$. In the range $\alpha\in[0,\frac{1}{2}]$ the achievable rate cannot exceed $N^{-1/(1-\alpha)}$ and deteriorates as $\alpha$ approaches $0$.
The approximation performance of $\alpha$-curvelets is different if the edges of the cartoons are straight. Assuming $C^\beta$ regularity, we establish an approximation rate of order $N^{-\min\{\alpha^{-1},\beta\}}$, which improves as $\alpha$ tends to $0$. In the range $\alpha\in [0,\beta^{-1}]$ it is even quasi-optimal, generalizing optimality results for ridgelets. By applying the framework of $\alpha$-molecules, we finally extend the obtained results to other $\alpha$-scaled representation systems, including for instance $\alpha$-shearlet frames.
\end{abstract}
\begin{center}{\it Keywords:} {Cartoon Images, Nonlinear Approximation, Wavelets, Curvelets, Shearlets, Ridgelets, Anisotropic Scaling, $\alpha$-Molecules.} \end{center}
{\it MSC2000 Subject Classification:} {41A25, 41A30, 42C40.}
\section{Introduction}
In the age of `big data', efficient data representation is an objective of an ever increasing importance. Not only does it simplify the handling of the data due to the reduction of needed storage space or the possible speed-up of processing times. The knowledge of a `good' representation also gives valuable information about the structure of the data itself, simplifying certain processing tasks or even just enabling them in the first place. As an example we may think of the restoration of corrupted signals or the separation of several superimposed signals of distinct types.
Often, the data of interest can be modeled in a linear space, for instance a Hilbert space as exemplified by the Lebesgue spaces of square-integrable functions. In this setting the standard approach for the representation of a signal is its expansion with respect to a fixed family of basic elements, a so-called dictionary for the data. In practice, one usually needs to contend with approximations and therefore resorts to approximation schemes, i.e., algorithms that deliver for each signal $f$ a sequence of approximants $(f_N)_{N\in\mathbb{N}}$ converging to the signal. A standard choice here is to use $N$-term approximations in the respective dictionary, i.e., approximants being built from just $N$ dictionary elements.
A main goal of approximation theory is the development of approximation schemes with a best possible speed of convergence, commonly quantified by the asymptotic decay of the approximation error $\|f-f_N\|$ as $N \to \infty$. With regard to $N$-term approximations, the achievable rate is determined by the utilized dictionary in the background and one aims to find dictionaries providing high approximation rates for the data. Such dictionaries are said to sparsely approximate the corresponding signals and clearly need to be chosen depending on the considered signal class. For efficient data representation it is therefore essential, first, to be able to precisely specify the type of data under consideration, e.g., in the form of an appropriate model, and, second, to develop dictionaries, well adapted to the specific data class, providing sparse approximations.
\subsection{Approximation of Image Data}
Subsequently, we are interested in the sparse approximation of image data. In our investigation, we will always stay in the continuum setting, where images are as usual represented as functions supported on some compact image domain $\Omega\subset\mathbb{R}^2$ with values containing pixel information at the respective positions, such as e.g.\ color or brightness information. Being compactly supported and bounded, the image data can conveniently be modeled as a subset of the Hilbert space $L^2(\Omega)$, which in turn is considered as a subspace of $L^2(\mathbb{R}^2)$. Hence, we are in a concrete Hilbert space scenario and can resort to the methodology described above, i.e., we aim for appropriate image models and sparsifying dictionaries.
For the space $L^2(\Omega)$ the classic Fourier systems constitute an orthonormal basis, providing a straight-forward procedure for representation. However, Fourier systems work well only if the functions under consideration are smooth. For general images such smoothness assumptions are certainly not fulfilled.
As another popular representation system wavelets~\cite{Dau92,Mal08} come to mind. Nowadays, they are one of the most widely used systems in applied harmonic analysis, with various applications ranging from signal compression (e.g.\ JPEG2000~\cite{CSE2000}) and restoration~\cite{CDOS12} to PDE solvers~\cite{CDD01}. In particular, they have the ability to sparsely approximate functions, which are smooth apart from isolated point singularities. For general image data, however, such regularity assumptions are still too strict. A characteristic feature of images are edges, leading to curvilinear discontinuities in the data. With respect to such line singularities, wavelet systems do not perform optimally any more. The isotropy of their scaling prohibits an optimal resolution of these kind of anisotropic structures.
With the desire to specifically model the occurrence of edges, the concept of cartoon-like functions emerged. These are piecewise smooth functions featuring discontinuities along lower-dimensional manifolds, in our case along the $1$-dimensional edge curves of the image. Based on such functions, suitable models for natural images have been conceived and different model classes have been introduced. Typically, these classes are characterized by a specific smoothness of the regions and by certain conditions on the separating edges. As examples, let us mention the classic cartoons~\cite{CD04} with $C^2$ regularity of the regions and the discontinuity curves, or the horizon classes considered e.g.\ in \cite{D99,CWBB04c,PM05}.
The achievable approximation rate for a class of cartoon-like functions essentially depends on the regularity of the cartoons, including both the smoothness of the edge curves and the smoothness of the regions in between. It was shown in \cite{PM05,PennecM} that $C^\beta$ regularity of the regions and the separating edges with $\beta>0$ allows for an asymptotic rate of order $N^{-\beta}$. By information theoretic arguments, it has further been established that this rate cannot be surpassed \cite{Don01}, at least in a class-wise sense. Interestingly, the benchmark $N^{-\beta}$ is the same for the class of so-called binary cartoons, i.e., cartoon-like functions with constant regions, and it also does not change if one restricts to $C^\beta$ smooth functions without any edges.
With the model of cartoon-like functions at hand, let us turn again to the question of efficient image representation. In the past, a great amount of energy has been devoted to the effort of constructing dictionaries well-suited for cartoon approximation. Thereby, many different paths have been pursued and the developed methods can be divided into two general categories: adaptive and nonadaptive methods.
Adaptive methods are by nature more flexible and have the inherent advantage of being able to adjust to the given data. On the downside, the increased flexibility typically comes at the cost of higher computational complexity of the employed approximation and reconstruction schemes. Some prominent examples of adaptive methods for image data are based on wedgelet dictionaries~\cite{D99} and their higher-order relatives, so-called surflets~\cite{CWBB04b,CWBB04c}. They have been shown to reach the optimality bound $N^{-\beta}$ for binary cartoons with $C^\beta$ regularity~\cite{CWBB04tech,CWBB04proc}. Other notable dictionaries used for adaptive approximation include beamlets~\cite{DH00}, platelets~\cite{WN03}, and derivatives of wedgelets such as multiwedgelets~\cite{L13} or smoothlets~\cite{L11}. More recently, new adaptive schemes have emerged that use bases, e.g., bandelets~\cite{PM05}, grouplets~\cite{M09}, and tetrolets~\cite{K09}. Quasi-optimal approximation for $C^\beta$ cartoons with $\beta>0$ has been proved e.g.\ in~\cite{PennecM} for bandelets.
Nonadaptive methods are usually much simpler than adaptive schemes, at least from an algorithmic perspective. Mainly, they are based on frames and the corresponding reconstruction formulas. An easy path to approximation is thus provided by simply thresholding the frame coefficients. Surprisingly, despite the simplicity of such schemes, there exist frames with quasi-optimal approximation performance for certain cartoon classes.
If the edges of the cartoons are straight, different variants of so-called ridgelet frames have been shown to yield quasi-optimal approximation~\cite{C99,GO15,GOtech16}. Originally, the notion of a ridgelet was introduced by Cand{\`e}s~\cite{Can98} in 1998, who defined them as bivariate ridge functions obtained by tensoring a univariate wavelet with a constant. Since these `pure ridgelets' are not square-integrable, the concept was later modified in order to obtain frames or bases for $L^2(\mathbb{R}^2)$. By giving them a slow decay along the ridge, Donoho constructed an orthonormal basis whose elements are called `orthonormal ridgelets'~\cite{D98}. Their close relationship to the original concept has been analyzed in~\cite{DonXX}. Another construction, based on directional scaling, goes back to Grohs, providing tight frames \cite{GrohsRidLT}. This kind of construction coincides with the concept of `$0$-curvelets' presented below.
To deal with curved edges, numerous types of frames have been developed. An important milestone was the introduction of the first generation of curvelets~\cite{CD2000} by Cand{\`e}s and Donoho in 1999. They represent the first frame to reach the optimal approximation order of $N^{-2}$ for $C^2$ cartoons via simple thresholding. A modification of this system, the second generation of curvelets~\cite{CD04}, was introduced in 2002 by the same authors. It is based on a more elegant and simpler construction principle, yet features the same quasi-optimal approximation properties. Following this early breakthrough, other constructions better suited for digital implementation were developed. Let us mention contourlets~\cite{DV05} by Do and Vetterli and shearlets, whose construction goes back mainly to Guo, Kutyniok, Labate, Lim, and Weiss. The first shearlet construction consisted of band-limited functions and was presented in~\cite{KuLaLiWe,GKL05}. Later, more sophisticated shearlet systems were developed, such as e.g.\ the well-localized band-limited Parseval frame in \cite{Guo2012a} or even systems of compactly supported shearlets~\cite{Kittipoom2010}. Like curvelets, those systems provide quasi-optimal approximation for $C^2$ cartoons. For the classic band-limited shearlets this was established in~\cite{GL07}, for those with compact support in~\cite{Kutyniok2010}.
A common principle underlying the above constructions is parabolic scaling, a type of scaling optimally adapted to $C^2$ singularity curves. It is essential for the quasi-optimal approximation of $C^2$ cartoons and led to the notion of parabolic molecules~\cite{Grohs2011}. This concept unifies various parabolically scaled systems under one roof, in particular the classic curvelet and shearlet systems, and is the predecessor of the more general framework of $\alpha$-molecules~\cite{GKKS15}.
\subsection{Multiscale Systems based on $\alpha$-Scaling}
Comparing the approximation properties of wavelets, curvelets, and ridgelets, a distinct behavior with respect to their ability to resolve edges is characteristic. Ridgelets are optimally adapted to straight edges, curvelets are optimal for $C^2$ line singularities, and wavelets for point singularities. This distinct behavior is due to the different scaling laws underlying their respective constructions: isotropic scaling for wavelets, parabolic scaling for curvelets, and directional scaling for ridgelets.
Introducing a parameter $\alpha\in\mathbb{R}$ and associated $\alpha$-scaling matrices \begin{align}\label{eq:alphamat1} A_{\alpha,s}=\begin{pmatrix} s & 0 \\ 0 & s^\alpha \end{pmatrix}, \qquad s>0, \end{align} one can interpolate between these different types of scaling and construct corresponding $\alpha$-scaled representation systems. Incorporating $\alpha$-scaling in the original construction of curvelets, for instance, yields so-called $\alpha$-curvelets~\cite{GKKScurve2014}. For $\alpha\in[0,1]$, they constitute a family of systems which encompass ridgelets (in the sense of \cite{GrohsRidLT}) for $\alpha=0$, the classic curvelets for $\alpha=\frac12$, and wavelets for $\alpha=1$. In a similar fashion, $\alpha$-shearlet systems~\cite{Kei13,Kutyniok2012correct} can be obtained by modifying the classic shearlet constructions.
A natural question concerning such $\alpha$-scaled systems is how their approximation properties are affected by a change of the parameter $\alpha$. With regard to cartoon approximation, this question has been pursued in \cite{GKKScurve2014} for $\alpha$-curvelet frames and in \cite{Kei13,Kutyniok2012correct} for $\alpha$-shearlet frames. It was shown that, if $\alpha\in[\frac{1}{2},1)$ and if the cartoon $f$ is of regularity $C^\beta$ with $\beta=\alpha^{-1}$, simple thresholding of the coefficients yields $N$-term approximations $f_N$ with a convergence of \begin{align}\label{intro-rate}
\|f-f_N\| \lesssim N^{-\beta} \log(N)^{1+\beta} \quad\text{as }N\to\infty, \end{align} which apart from the log-factor is optimal. Later, these results were further extended utilizing the theory of $\alpha$-molecules~\cite{GKKS15}. This is a framework providing a unified approach to $\alpha$-scaled systems, based solely on assumptions on the time-frequency localization of the respective functions. It allows to transfer approximation results obtained for one system of $\alpha$-molecules to other systems, under certain consistency conditions. In particular, the rate~\eqref{intro-rate} for $\alpha$-curvelets was generalized (in a weak form) to other $\alpha$-scaled representation systems~\cite{GKKS15}, which all achieve a rate of $N^{-\beta+\varepsilon}$ with $\varepsilon>0$ arbitrarily small.
Despite these results, many questions concerning $\alpha$-scaled representation systems and their ability to approximate cartoon-like functions remain open, e.g., their performance in the range $\alpha<\frac12$ or their suitability for the approximation of straight edges. In this research we want to address these open questions, shedding (even) more light on the role of the parameter $\alpha$.
\subsection{Outline and Contribution}
Our exposition starts with a short review of $\alpha$-scaled systems in Section~\ref{sec:curvelets}, where also a specific construction of an $\alpha$-curvelet frame for $L^2(\mathbb{R}^2)$ is presented. This frame, denoted by $\mathfrak{C}_{s,\alpha}$, will serve as a prototypical system whose properties have ramifications for other $\alpha$-scaled systems, such as for example $\alpha$-shearlets, due to the transference principle of the framework of $\alpha$-molecules.
In the main part of the article, Sections~\ref{sec:cartoon} and \ref{sec:straight}, we analyze the $N$-term approximation properties of the frame $\mathfrak{C}_{s,\alpha}$ with regard to different classes of cartoon images. In Section~\ref{sec:cartoon} we start with cartoons with curved edges and first introduce corresponding signal classes of $C^\beta$ regularity for $\beta\in[0,\infty)$. Theorem~\ref{thm:benchmark} recalls $N^{-\beta}$ as the order of the maximal achievable approximation rate for such $C^\beta$ cartoons, which cannot be surpassed by any polynomial-depth restricted $N$-term approximation scheme, independent of the utilized dictionary.
Then we recall the quasi-optimal approximation~\eqref{intro-rate} of $\alpha$-curvelets, proved in~\cite{GKKScurve2014}, if $\alpha\in[\frac{1}{2},1)$ and $\beta=\alpha^{-1}$. Our main findings in Section~3, Theorems~\ref{thm:bound1} and~\ref{thm:bound2}, extend and complement this result. Theorem~\ref{thm:bound1} shows that the best possible $N$-term approximation rate achievable by $\mathfrak{C}_{s,\alpha}$ for cartoons with curved edges is limited to at most $N^{-\frac{1}{1-\alpha}}$, where $\alpha<1$ and the smoothness of the cartoons is arbitrary. Moreover, according to Theorem~\ref{thm:bound2}, the achievable rate cannot exceed $N^{-\frac{1}{\max\{\alpha,1-\alpha\}}}$ if a simple thresholding scheme is used.
These bounds show that $\alpha$-curvelets with $\alpha\in[\frac{1}{2},1)$ cannot take advantage of regularity higher than $C^{1/\alpha}$. Furthermore, they prohibit optimal approximation of $C^\beta$ cartoons if $\beta>2$, since decreasing $\alpha$ beyond $\frac12$ deteriorates the achievable rates compared to the classic curvelets. Hence, with a rate of order $N^{-2}$, these provide the best performance among all $\alpha$-curvelet systems, if the regularity of the cartoons is at least $C^2$ and curved singularities are involved. As a consequence, no curvelet system can reach the optimality bound $N^{-\beta}$ if $\beta>2$. In fact, up to now, no frame construction is known where a nonadaptive approximation scheme can break this $N^{-2}$ barrier and the quest for such frames remains open.
In Section~\ref{sec:straight} we consider cartoons featuring only straight edges. For the corresponding classes of regularity $C^\beta$ the same optimality benchmark $N^{-\beta}$ holds true as for the cartoons with curved edges. Our main result of Section~\ref{sec:straight}, Theorem~\ref{thm:mainappr1}, shows that a simple thresholding scheme for the $\alpha$-curvelet frame $\mathfrak{C}_{s,\alpha}$ yields approximation rates of order $N^{-\min\{\alpha^{-1},\beta\}}$. Hence, here a smaller $\alpha$ is beneficial and even ensures quasi-optimal approximation if $\alpha\in[0,\beta^{-1}]$. This finding generalizes earlier results for ridgelets.
We finish with a short discussion of our results in Section~\ref{sec:discussion}. In particular, we point out some ramifications for other $\alpha$-scaled representation systems, utilizing the framework of $\alpha$-molecules. All $\alpha$-scaled systems which are frames and in a certain sense consistent with $\mathfrak{C}_{s,\alpha}$ feature similar properties, formulated in Theorem~\ref{thm:mol_app} and Corollary~\ref{cor:mol_app}.
Some useful properties of Bessel functions needed in Section~3 are collected in the appendix.
\subsection{Notation}
Before we begin, let us fix some general notation. Writing $\mathbb{N}$ we will refer to the natural numbers without zero, and we let $\mathbb{N}_0:=\mathbb{N}\cup\{0\}$. As usual, $\mathbb{Z}$, $\mathbb{R}$ and $\mathbb{C}$ denote the integer, real and complex numbers. Further, we put $\mathbb{R}_0^+:=[0,\infty)$ and $\mathbb{R}^+:=(0,\infty)$. We also introduce the `floor' and `ceiling' of $t\in\mathbb{R}$, $\lfloor t \rfloor:= \max\{ n\in\mathbb{Z} : n\le t \}$ and $\lceil t \rceil:= \min\{ n\in\mathbb{Z} : n\ge t \} $. The symbol $\mathbb{T}$ is used for the torus obtained from the interval $[0,2\pi]$ by identifying the endpoints. The unit-circle in $\mathbb{C}\simeq\mathbb{R}^2$ is denoted by $\mathbb{S}^1$.
The vector space $\mathbb{R}^d$, $d\in\mathbb{N}$, is equipped with the Euclidean scalar product $\langle \cdot,\cdot \rangle$ and associated norm $|\cdot|$. The notation $|\cdot|_p$, $p\in(0,\infty]$, is used for the $p$-(quasi-)norms on $\mathbb{R}^d$. For a multi-index $m=(m_1,\ldots,m_d)\in\mathbb{N}_0^d$, $\partial^{m}:=\partial_1^{m_1}\cdots \partial_d^{m_d}$ is a differential operator with $\partial_i$, $i\in \{1,\dots, d\}$, the partial derivative in the $i$-th coordinate direction. Given a vector $x=(x_1,\ldots,x_d)\in\mathbb{R}^d$, we further define $x^m:=x_1^{m_1}\cdots x_d^{m_d}$ (with the convention $0^0:=1$).
If $A(\omega)\le C B(\omega)$ holds true for two quantities $A,B\in \mathbb{R}$ depending on a set of parameters $\omega$ with a uniform constant $C>0$, we write $A\lesssim B$ or equivalently $B\gtrsim A$. If both, $A\lesssim B$ and $B\lesssim A$, hold true, we denote this by $A \asymp B$.
For measurable subsets $\Omega\subseteq\mathbb{R}^d$ we let
$L^p(\Omega)$, $p\in(0,\infty]$, denote the usual Lebesgue spaces with respect to the Lebesgue measure. The corresponding (quasi-)norms are denoted by $\|\cdot\|_{L^p(\Omega)}$, in case $\Omega=\mathbb{R}^d$ we abbreviate $\|\cdot\|_p:=\|\cdot\|_{L^p(\mathbb{R}^d)}$. For the scalar product on $L^2(\Omega)$ the same notation $\langle \cdot,\cdot \rangle$ as for the Euclidean product on $\mathbb{R}^d$ is used. The Lebesgue sequence spaces, for a discrete index set $\Lambda$, are denoted by $\ell^p(\Lambda)$ with associated (quasi-)norms $\|\cdot\|_{\ell^p}$. The definition of their weak counterparts
$w\ell^p(\Lambda)$, equipped with (quasi-)norms $\|\cdot\|_{w\ell^p}$, are recalled in Section~\ref{sec:straight}.
The space $C_{\rm loc}^\beta(\mathbb{R}^d)$, for an integer $\beta\in\mathbb{N}_0\cup\{\infty\}$, shall comprise all continuous real-valued functions on $\mathbb{R}^d$, whose classic derivatives up to order $\beta\in\mathbb{N}_0$ exist. For $\beta\in[0,\infty)$ we then define \[
C^{\beta}(\mathbb{R}^d):=
\Big\{f\in C_{\rm loc}^{\lfloor \beta \rfloor}(\mathbb{R}^d):\, \|f\|_{C^\beta(\mathbb{R}^d)}:= \|f\|_{C^{\lfloor\beta\rfloor}(\mathbb{R}^d)} +
\sum_{|m|_1=\lfloor\beta\rfloor}
\text{\sl Höl}(\partial^m f, \beta- \lfloor\beta\rfloor) <\infty
\Big\}\,, \]
where $\|f\|_{C^{\lfloor \beta \rfloor}(\mathbb{R}^d)} := \sum_{|m|_1\le \lfloor\beta\rfloor} \displaystyle{\sup_{x\in\mathbb{R}^d} |\partial^mf(x)| }$ and the H\"older constant of exponent $\alpha\in[0,1]$ is given by \[
\text{\sl H\"{o}l}(f,\alpha):=\sup_{x,y\in \mathbb{R}^d} \frac{|f(x)-f(y)|}{|x-y|^{\alpha}}. \] The notation $C_0^\beta(\overline{\Omega})$, for some open subset $\Omega\subseteq\mathbb{R}^d$, is used for functions $f\in C^\beta(\mathbb{R}^d)$ whose support ${\text{\rm supp }} f$ is compact and contained in the closure $\overline{\Omega}$ of $\Omega$. Frequently, we also need to measure functions $f\in C_{\rm loc}^{\beta}(\mathbb{R}^d)$, $\beta\in\mathbb{N}_0$, with the following Sobolev norms, where $p\in[1,\infty]$, \[
\|f\|_{\beta,p}:=\| f \|_{W^{\beta,p}(\mathbb{R}^d)} := \sum_{|m|_1\le \beta} \| \partial^mf \|_{L^p(\mathbb{R}^d)}. \]
\noindent Finally, we will use the following version of the Fourier transform. For a Schwartz function $f\in\mathcal{S}(\mathbb{R}^d)$ \[ \mathcal{F}f(\xi):= \int_{\mathbb{R}^d} f(x) \exp(-2\pi i \langle x,\xi \rangle) \,dx\,, \quad\xi\in\mathbb{R}^d\,. \] As usual, $\mathcal{F}$ is extended to the tempered distributions $\mathcal{S}^\prime(\mathbb{R}^d)$, and we often write $\widehat{f}$ for $\mathcal{F}f$.
\section{The Anchor System: $\alpha$-Curvelets} \label{sec:curvelets}
Directional multi-scale systems based on $\alpha$-scaling feature a characteristic tiling of the frequency domain. The multi-scale structure is reflected by a partition of the Fourier plane into dyadic coronae, further divided into wedge-like tiles, where the energy of the system elements is concentrated. In case of inhomogeneous systems, a ball around the origin corresponds to the low-frequency base scale.
A prototypical instance of such an $\alpha$-scaled system is the frame $\mathfrak{C}_{s,\alpha}$ of $\alpha$-curvelets, thoroughly defined in this section. It is prototypical in the sense that many of its properties transfer -- via the framework of $\alpha$-molecules~\cite{GKKS15} -- to other $\alpha$-scaled systems. Among these are other $\alpha$-curvelet constructions~\cite{CD04,GKKScurve2014}, but also band-limited~\cite{KuLaLiWe,GKL05,Guo2012a} as well as compactly supported~\cite{Kittipoom2010,Kei13,Kutyniok2012correct} $\alpha$-shearlet systems. This fact gives the system $\mathfrak{C}_{s,\alpha}$ a special significance for our purpose and motivates its detailed discussion here.
Before defining $\mathfrak{C}_{s,\alpha}$, which is similar to the construction of $\alpha$-curvelets in \cite{GKKScurve2014}, let us first elaborate the geometric aspects of the corresponding frequency tiling. At scales $j\ge1$ we have the coronae \begin{align}\label{eqdefcorC}
\mathcal{C}_j:=\Big\{ \xi\in\mathbb{R}^2 ~:~ C2^{s(j-1)}\le|\xi|_2\le C2^{s(j+1)}\Big\}, \end{align} where $s>0$ is a fixed parameter and $C>0$ is a constant, specified conveniently later. These coronae are each uniformly divided into an even number of wedges, whose angular width at scale $j$ is given by the angle \begin{align}\label{eqdef:fundangle} \varphi_j:=\pi 2^{-\lfloor js(1-\alpha) \rfloor-1} \end{align}
and depends on another parameter $\alpha\in(-\infty,1]$. The approximate size of the resulting wedges correlates with an $\alpha$-scaled rectangle of dimension $2^{js}\times 2^{js\alpha}$. By combining opposite wedges to wedge pairs, we obtain the tiles for the scales $j\ge1$. There is only one tile associated with the base scale $j=0$, the low frequency ball $\mathcal{C}_0:=\{ \xi\in\mathbb{R}^2 ~:~ |\xi|_2 \le C 2^{s} \}$.
For convenience, let us also introduce the angle $\varphi_0:=\pi$. According to the above construction, at each scale $j\in\mathbb{N}_0$ the number of tiles $L_j$ is given by \begin{align}\label{eq:Lj} L_0:=\pi\varphi_0^{-1}=1 \quad\text{and}\quad L_j:=\pi\varphi_j^{-1}=2^{\lfloor js(1-\alpha) \rfloor + 1}\,, \quad j\ge1. \end{align} In the following, the individual tiles will be denoted by $\mathcal{W}_{j,\ell}$ and indexed by the set \[ \mathbb{J}:=\big\{ (j,\ell) ~:~ j\in\mathbb{N}_0,\, \ell\in \{-L^-_j, \ldots, L^+_j \} \big\} \] with $L_j^-:=\lfloor L_j/2 \rfloor$ and $L_j^+:=\lceil L_j/2 \rceil -1$. Hereby we let $\mathcal{W}_{0,0}:=\mathcal{C}_0$, and in each corona $\mathcal{C}_j$ with $j\ge1$ the wedge-pair $\mathcal{W}_{j,0}$ shall be aligned horizontally, i.e., \begin{align*}
\mathcal{W}_{j,0}:=\Big\{ \xi=(\xi_1,\xi_2)\in\mathcal{C}_j ~:~ |\xi_1| \ge \cos(\varphi_j/2) |\xi|_{2} \Big\}. \end{align*}
The remaining tiles $\mathcal{W}_{j,\ell}$, $\ell\neq0$, are obtained via rotations of $\mathcal{W}_{j,0}$ by integer multiples $\varphi_{j,\ell}:=\ell\varphi_j$ of the angle $\varphi_j$ defined in~\eqref{eqdef:fundangle}. Hence, $\mathcal{W}_{j,\ell}:=R^{-1}_{j,\ell}\mathcal{W}_{j,0}$ with rotation matrix \begin{align}\label{eq:matrixrot} R_{j,\ell}:=R_{\varphi_{j,\ell}} \,,\quad \text{where}\quad R_{\varphi} :=\begin{pmatrix} \cos(\varphi) & -\sin(\varphi) \\
\sin(\varphi) & \,\cos(\varphi) \end{pmatrix} \,,\quad \varphi\in\mathbb{R}. \end{align} The resulting tiling of the Fourier domain is schematically depicted in Figure~\ref{fig:freq_domain} (a).
We remark that in contrast to \cite{GKKScurve2014}, where $\alpha\in[0,1]$, we allow $\alpha\in(-\infty,1]$ in the $\alpha$-curvelet construction. This range is natural for the considered inhomogeneous systems. If $\alpha>1$, the number of tiles $L_j$ in each corona decreases with rising scale, and eventually $L_j=1$. Thus, at high scales, those systems would behave like isotropically scaled systems with $\alpha=1$.
\begin{figure*}
\caption{(a): Tiling of Fourier domain into coronae $\mathcal{C}_j$ and wedges $\mathcal{W}_{j,\ell}$.
(b): Schematic display of the frequency support of a wedge function $W_{j,0}$.}
\label{fig:freq_domain}
\end{figure*}
\subsection{The Frame of $\alpha$-Curvelets $\mathfrak{C}_{s,\alpha}$}
Let us now turn to the actual construction of the $\alpha$-curvelet frame $\mathfrak{C}_{s,\alpha}$. To realize the described frequency tiling, smooth functions $W_J:\mathbb{R}^2\to\mathbb{C}$, $J\in\mathbb{J}$, are used, with compact support approximately given by the tiles $\mathcal{W}_{J}$. It is convenient to construct them as tensor products of a radial and an angular component. This allows to realize the desired support separately on the ray $\mathbb{R}_0^+=[0,\infty)$ and on the circle $\mathbb{S}^{1}\subset\mathbb{R}^2$. Projecting the coronae $\mathcal{C}_j$ onto the ray $\mathbb{R}_0^+$ yields the intervals \begin{align}\label{eqdef:dyintervals} \mathcal{I}_{0}:=C\cdot[0,2^s] \quad\text{and}\quad\mathcal{I}_{j}:=C\cdot[2^{s(j-1)},2^{s(j+1)}] \,, \quad j\ge1. \end{align}
For the radial subdivision, we thus utilize nonnegative smooth functions $U_j\in C^\infty(\mathbb{R}_0^+)$, $j\in\mathbb{N}_0$, which satisfy the support condition ${\text{\rm supp }}\, U_j\subseteq \mathcal{I}_{j}$ and for $r\in\mathbb{R}_0^+$ \begin{align}\label{eq:coronalpartition} A_1\le \sum_{j\ge 0} U^2_j(r) \le B_1 \quad\text{with constants}\quad 0<A_1\le B_1<\infty. \end{align} More concretely, we assume that the functions $U_j$, $j\ge1$, are generated by a single function $U\in C^\infty(\mathbb{R}_0^+,[0,1])$ via $U_j(\cdot):=U(2^{-js} \cdot)$ and that there are $1<\tau_1<\tau_2<2^s$ such that \begin{align}\label{eq:suppcond} \begin{aligned} {\text{\rm supp }}\, U_0\subseteq C\cdot[0,\tau_2],& \quad \sqrt{A_1}\le U_0 \le \sqrt{B_1} \text{ on } C\cdot[0,\tau_1], \\ {\text{\rm supp }}\, U\subseteq C\cdot[2^{-s}\tau_1,\tau_2],& \quad \sqrt{A_1}\le U \le\sqrt{B_1} \text{ on } C\cdot[2^{-s}\tau_2,\tau_1]. \end{aligned} \end{align} Such functions exist and can even be constructed with $A_1=B_1=1$ in \eqref{eq:coronalpartition}.
For the angular subdivision, we construct at each scale $j\in\mathbb{N}_0$ a smooth partition on the unit circle $\mathbb{S}^{1}\subset\mathbb{R}^2$, reflecting the angular support of the tiles $\mathcal{W}_{j,\ell}$. We start with a function $\widetilde{V}\in C^\infty(\mathbb{R},[0,1])$ with the properties \begin{align*} {\text{\rm supp }}\, \widetilde{V}\subseteq [-\textstyle{\frac{3}{4}}\pi,\textstyle{\frac{3}{4}}\pi] ,\quad \sqrt{A_2}\le \widetilde{V} \le\sqrt{B_2} \text{ on }[-\textstyle{\frac{\pi}{4}},\textstyle{\frac{\pi}{4}}],\quad A_2\le\sum_{k\in\mathbb{Z}} \widetilde{V}^2(\cdot -k\pi)\le B_2, \end{align*} where $0<A_2\le B_2<\infty$. Scaling then gives rise to the functions $\widetilde{V}_j(\cdot):=\widetilde{V}(L_j \cdot)\in C^\infty(\mathbb{R},[0,1]) $ for $j\in\mathbb{N}_0$. Via the bijection $t\mapsto e^{it}$ these functions yield functions $\widetilde{V}_{j,0}\in C^{\infty}(\mathbb{S}^{1},[0,1])$ on the unit circle. We symmetrize \[ V_{j,0}(\xi):=\widetilde{V}_{j,0}(\xi) + \widetilde{V}_{j,0}(-\xi), \quad\xi\in\mathbb{S}^1, \] and note that $\sqrt{A_2}\le V_{0,0}\le\sqrt{B_2}$ on $\mathbb{S}^{1}$. Applying the rotation~\eqref{eq:matrixrot} then yields functions $V_{j,\ell}(\cdot):=V_{j,0}(R_{j,\ell} \cdot)$ for every $J=(j,\ell)\in\mathbb{J}$, which satisfy
$A_2\le \sum_{|J|=j} V^2_{J}(\xi)\le B_2$ for all $\xi\in\mathbb{S}^{1}$. Here we use the notation $|J|:=j$ for $J=(j,\ell)\in\mathbb{J}$.
Finally, we are ready to define the wedge functions $W_{j,\ell}\in C^\infty(\mathbb{R}^2)$ as the polar tensor products \begin{align}\label{eq:suppfunctions}
W_{j,\ell}(\xi):=U_j(|\xi|_{2})V_{j,\ell}(\xi/|\xi|_{2}), \quad \xi\in\mathbb{R}^2. \end{align} These functions are non-negative `bumps' approximately supported in the corresponding wedges $\mathcal{W}_{j,\ell}$. They are symmetric, i.e., $W_{j,\ell}(\xi)=W_{j,\ell}(-\xi)$ for $\xi\in\mathbb{R}^2$, and they satisfy \begin{align}\label{eq:CalderonW} A:=A_1 A_2\le \sum_{J=(j,\ell)\in\mathbb{J}} W^2_{J}(\xi) \le B_1 B_2 =: B \,,\quad \xi\in\mathbb{R}^2. \end{align}
Let us analyze the support of $W_J$ in more detail. Recall the angular function $\widetilde{V}_{j,0}$ and note that its support on $\mathbb{S}^1$ covers an angle range of $\varphi_j^+:=\frac{3}{2}\varphi_j$ with $\varphi_j=\pi L_j^{-1}$ as in~\eqref{eqdef:fundangle}. Moreover, $\sqrt{A_2}\le \widetilde{V}_{j,0} \le \sqrt{B_2}$ on a range of size $\varphi_j^-:=\frac{1}{2}\varphi_j$. Hence, ${\text{\rm supp }} V_{j,\ell} \subseteq \mathcal{A}_{j,\ell}$ and $V_{j,\ell}\asymp1$ on $\mathcal{A}^{-}_{j,\ell}$ for the angular intervals \begin{align}\label{eq:angularsupp} \begin{aligned}
\mathcal{A}_{j,\ell}:=R^{-1}_{j,\ell}\mathcal{A}_{j,0} \quad&\text{with}\quad \mathcal{A}_{j,0}:=\Big\{ \xi=(\xi_1,\xi_2)\in\mathbb{S}^1 ~:~ |\xi_1| \ge \cos(\varphi^+_j/2) \Big\}, \\
\mathcal{A}^-_{j,\ell}:=R^{-1}_{j,\ell}\mathcal{A}^-_{j,0} \quad&\text{with}\quad \mathcal{A}^-_{j,0}:=\Big\{ \xi=(\xi_1,\xi_2)\in\mathbb{S}^1 ~:~ |\xi_1|\ge\cos(\varphi^-_j/2) \Big\}. \end{aligned} \end{align}
Next, recall the functions $U_j$ on the ray with ${\text{\rm supp }} U_j\subseteq \mathcal{I}_j$. Due to \eqref{eq:coronalpartition} and \eqref{eq:suppcond} their function values are between $\sqrt{A_1}$ and $\sqrt{B_1}$ on \begin{align}\label{eq:IIIIII} \mathcal{I}^{-}_0:=C\cdot[0,\tau_1] \quad\text{and}\quad \mathcal{I}^{-}_j:=C\cdot[2^{s(j-1)} \tau_2,2^{sj}\tau_1],\quad j\ge1, \end{align} respectively.
This leads us to the following definition. For $J=(j,\ell)\in\mathbb{J}$ we introduce the wedge pairs \begin{align}\label{eq:wedgePJ} \begin{aligned}
\mathcal{W}^+_{J}:=\Big\{ \xi\in\mathbb{R}^2 ~:~ |\xi|_{2}\in\mathcal{I}_j ,\, \varphi(\xi)\in\mathcal{A}_J \Big\} \quad\text{and} \quad
\mathcal{W}^-_{J}:=\Big\{ \xi\in\mathbb{R}^2 ~:~ |\xi|_{2}\in\mathcal{I}^-_j ,\, \varphi(\xi)\in\mathcal{A}^-_J \Big\}. \end{aligned} \end{align} The following support properties will be of essential importance later, \begin{align}\label{eq:suppprop} {\text{\rm supp }} W_J \subseteq \mathcal{W}^+_J \qquad\text{and}\qquad \sqrt{A}\le W_J \le \sqrt{B} \text{ on }\mathcal{W}^-_J. \end{align} A geometric illustration is displayed in Figure~\ref{fig:freq_domain}~(b).
Now we fix $C=2^{-s}/(3\pi)$ in \eqref{eqdefcorC} such that each $\mathcal{W}^+_{J}$ is contained in the respective rectangle \begin{align}\label{eq:supprect} \Xi_{J}:=R^{-1}_{J}\Xi_{j,0}\,, \quad\text{where}\quad \Xi_{j,0}:= [-2^{js-1},2^{js-1}]\times[-2^{js\alpha-1},2^{js\alpha-1}]. \end{align} The rectangles $\Xi_{j,0}$ are of size $2^{js}\times2^{js\alpha}$ and hence the Fourier system $\lb u_{j,0,k}\rb_{k\in\mathbb{Z}^2}$ given by \begin{align*} u_{j,0,k}(\xi):=2^{-js(1+\alpha)/2}\exp\big(2\pi i (2^{-sj}k_1\xi_1 + 2^{-sj\alpha}k_2\xi_2)\big),\quad \xi\in\mathbb{R}^2, \end{align*} constitutes an orthonormal basis for $L^2(\Xi_{j,0})$ . Consequently, the rotated system $\lb u_{j,\ell,k}\rb_{k\in\mathbb{Z}^2}$ of functions \begin{align}\label{eq:fouriersys} u_{j,\ell,k}(\xi):= u_{j,0,k}(R_{j,\ell}\xi),\quad \xi\in\mathbb{R}^2, \end{align} is an orthonormal basis for $L^2(\Xi_{J})$.
After this preparation, we are ready to define the $\alpha$-curvelet system $\mathfrak{C}_{s,\alpha}$.
\begin{definition} Let $s>0$, $\alpha\in(-\infty,1]$, and assume that $\lb W_J \rb_{J\in\mathbb{J}}$ is a family of functions of the form~\eqref{eq:suppfunctions} such that~\eqref{eq:CalderonW} holds for $0<A\le B<\infty$. Further, let $u_{j,\ell,k}$ be the functions defined in \eqref{eq:fouriersys}. The curvelet system $\mathfrak{C}_{s,\alpha}(A,B):=\lb\psi_\mu\rb_{\mu\in\mathbb{M}}$ with associated index set $\mathbb{M}:=\mathbb{J}\times\mathbb{Z}^2$ consists of the functions $\psi_\mu=\psi_{j,\ell,k}$ given by \begin{align}\label{eqdef:curvelet} \widehat{\psi}_{j,\ell,k}(\xi):= W_{j,\ell}(\xi) u_{j,\ell,k}(\xi)\,, \quad \xi\in\mathbb{R}^2. \end{align} Note that $\mathfrak{C}_{s,\alpha}(A,B)$ depends on the utilized family $\lb W_J\rb_{J\in\mathbb{J}}$, which is not accounted for in the notation. \end{definition}
\noindent The curvelets $\psi_\mu$ are real-valued due to the symmetry of $W_{j,\ell}$. Their $L^2$-norms may vary slightly with scale, however there are constants $0<C_1\le C_2<\infty$ such that $C_1 \le \|\psi_\mu\|_{2} \le C_2 $ holds true for all $\mu\in\mathbb{M}$. Most importantly, the system $\mathfrak{C}_{s,\alpha}(A,B)$ is a frame for $L^2(\mathbb{R}^2)$.
\begin{lemma} The system $\mathfrak{C}_{s,\alpha}(A,B)$ given by \eqref{eqdef:curvelet} is a frame for $L^2(\mathbb{R}^2)$ with frame bounds $A$ and $B$. \end{lemma}
\begin{proof} The functions $W_J$ satisfy condition \eqref{eq:CalderonW} wherefore \begin{align*}
A\|f\|^2_2=A\|\widehat{f}\|^2_2\le \sum_{J\in\mathbb{J}} \| \widehat{f}W_J \|_2^2 \le B \|\widehat{f}\|^2_2 = B\|f\|^2_2 \quad\text{for every $f\in L^2(\mathbb{R}^2)$}. \end{align*} Since ${\text{\rm supp }}(\widehat{f}W_J) \subseteq \Xi_J$ and since $\{u_{J,k}\}_{k\in\mathbb{Z}^2}$ is an orthonormal basis of $L^2(\Xi_J)$ we have the orthogonal expansion $\widehat{f}W_J = \sum_k \langle \widehat{f}W_J, u_{J,k} \rangle u_{J,k} \chi_{\Xi_{J}}$. The proof is finished by the following equality, \[
\| \widehat{f}W_J \|_2^2= \sum_{k\in\mathbb{Z}^2} |\langle \widehat{f}W_J, u_{J,k} \rangle|^2
= \sum_{k\in\mathbb{Z}^2} |\langle \widehat{f}, W_Ju_{J,k} \rangle|^2 = \sum_{k\in\mathbb{Z}^2} |\langle \widehat{f}, \widehat{\psi}_{J,k} \rangle|^2 = \sum_{k\in\mathbb{Z}^2} |\langle f, \psi_{J,k} \rangle|^2. \qedhere \]
\end{proof}
\noindent The Parseval frame $\mathfrak{C}_{s,\alpha}(1,1)$ is of most interest to us and one might wonder why we did not fix the frame bounds $A=B=1$ in the beginning. The reason is that, in the proof of Lemma~\ref{lem:induction}, we need the additional flexibility provided by variable $A$ and $B$.
\begin{remark} Subsequently, we will write $\mathfrak{C}_{s,\alpha}$ to refer to the Parseval frame $\mathfrak{C}_{s,\alpha}(1,1)$. \end{remark}
\noindent Let us finish this section with a short discussion of the situation in spatial domain. Here the $\alpha$-curvelets $\lb\psi_{j,\ell,k}\rb_{k\in\mathbb{Z}^2}$ are translates of the functions $\psi_{j,\ell,0}$. Indeed, since $\widehat{\psi}_{j,\ell,0}= 2^{-js(1+\alpha)/2} W_{j,\ell}$ and \[ u_{j,\ell,k}(\cdot)= u_{j,0,k}(R_{j,\ell}\cdot)=2^{-js(1+\alpha)/2}\exp\big(2\pi i \langle R^{-1}_{j,\ell} A^{-1}_{j}k,\cdot\rangle\big), \] where $R_{j,\ell}$ is the rotation matrix defined in \eqref{eq:matrixrot} and $A_{j}:=A_{\alpha,2^{js}}$ is an $\alpha$-scaling matrix of the form~\eqref{eq:alphamat1}, we have $\widehat{\psi}_{j,\ell,k}=\widehat{\psi}_{j,\ell,0} \exp\big(2\pi i \langle R^{-1}_{j,\ell} A^{-1}_{j}k, \cdot \rangle\big) $ and hence \begin{align*}
\psi_{j,\ell,k}= \psi_{j,\ell,0}(\cdot-x_{j,\ell,k})\quad\text{with}\quad x_{j,\ell,k}:=R^{-1}_{j,\ell}A^{-1}_{j}k. \end{align*} Since $\psi_{j,\ell,0}$ is the rotation of $\psi_{j,0,0}$ by the angle $\varphi_{j,\ell}=\ell\varphi_j$, we arrive at the representation \begin{align}\label{eq:spatialrepr}
\psi_{j,\ell,k} (x) =
\psi_{j,0,0}
\left(R_{j,\ell}\left(x - x_{j,\ell,k}\right)\right). \end{align}
In fact, these systems are instances of $\alpha$-molecules, a concept recalled in the definition below.
\begin{definition}[{\cite[Def.~2.9]{GKKS15}}] Let $\Lambda$ be a set and $\Phi_\Lambda:\Lambda\to\mathbb{P}$ a map, assigning to each $\lambda\in\Lambda$ a point $(s_\lambda,\theta_\lambda,x_\lambda)\in\mathbb{P}$ in the so-called phase-space $\mathbb{P}=\mathbb{R}^+\times\mathbb{T}\times \mathbb{R}^2$. Futher, assume that $L,M,N_1,N_2\in\mathbb{N}_0$. A family $\lb m_\lambda\rb_{\lambda \in \Lambda}$ of functions in $L^2(\mathbb{R}^2)$ is called a \emph{family of $\alpha$-molecules of order $(L,M,N_1,N_2)$ with respect to the parametrization $(\Lambda,\Phi_\Lambda)$}, if there exist generators $a^{(\lambda)}\in L^2(\mathbb{R}^2)$ such that for all $\lambda\in\Lambda$
\begin{align*}
m_\lambda (\cdot) =
s_\lambda^{(1+\alpha)/2}
a^{(\lambda)}
\left(A_{\alpha,s_\lambda}R_{\varphi_\lambda}\left(\cdot - x_\lambda\right)\right),
\end{align*}
and if for each $\rho\in\mathbb{N}_0^2$, $|\rho|\le L$, there is a constant $C_\rho>0$ such that
for all $\lambda\in\Lambda$
\begin{equation}\label{eq:molcond1}
\big| \partial^{\rho} \hat a^{(\lambda)}(\xi)\big|
\le C_\rho \min\left\{1,s_\lambda^{-1} + |\xi_1| + s_\lambda^{-(1-\alpha)}|\xi_2|\right\}^M
\left( 1+ |\xi|^2 \right)^{-N_1/2} ( 1+ |\xi_2|^2)^{-N_2/2}, \quad\xi\in\mathbb{R}^2.
\end{equation}
\end{definition}
\noindent We can deduce from \eqref{eq:spatialrepr} that the $\alpha$-curvelets $\psi_{j,\ell,k}$ can be represented in the form \begin{align}\label{eq:smolrepr}
\psi_{j,\ell,k} (x) = 2^{js(1+\alpha)/2}
a_{j}
\left(A_{j} R_{j,\ell}\left(x - x_{j,\ell,k}\right)\right) = 2^{js(1+\alpha)/2}
a_{j}
\left(A_{j} R_{j,\ell} x - k \right)
\end{align} with respect to the generators \begin{align}\label{eq:molgen} a_{j}:=2^{-js(1+\alpha)/2}\psi_{j,0,0} (A^{-1}_{j} \cdot). \end{align}
Since these generators fulfill condition~\eqref{eq:molcond1}, as shown in Lemma~\ref{thm:curvmol} below, $\mathfrak{C}_{s,\alpha}$ is a system of $\alpha$-molecules of arbitrary order, at least in the range $\alpha\in[0,1]$ for which the concept was formulated. The associated parametrization, mapping the curvelet index set $\mathbb{M}$ into the phase-space $\mathbb{P}=\mathbb{R}^+\times\mathbb{T}\times \mathbb{R}^2$, is given by \begin{align}\label{eq:curvepara} \Phi_{\mathbb{M}}: \mathbb{M}\to \mathbb{P} , \,(j,\ell,k)\mapsto (2^{js}, \varphi_{j,\ell}, x_{j,\ell,k}) = (2^{js},\ell \varphi_j, R^{-1}_{j,\ell} A^{-1}_{j}k). \end{align}
\begin{lemma}\label{thm:curvmol} Let $M,N_1,N_2\in\mathbb{N}_0$ and $\rho=(\rho_1,\rho_2)\in\mathbb{N}_0^2$ be fixed. There is a constant $C>0$ such that for all $j\in\mathbb{N}_0$ the generators~\eqref{eq:molgen} satisfy the estimate \begin{align}\label{eq:molcond}
\big| \partial^\rho \widehat{a}_{j}(\xi)\big|
\le C \min\big\{1, 2^{-js} + |\xi_1| + 2^{-js(1-\alpha)}|\xi_2|\big\}^M
(1+ |\xi|^2 )^{-N_1/2} (1+|\xi_2|^2)^{-N_2/2}. \end{align} \end{lemma}
\begin{proof} On the Fourier side the functions \eqref{eq:molgen} have the form \[ \widehat{a}_{j}=2^{js(1+\alpha)/2}\widehat{\psi}_{j,0,0} (A_{j} \cdot) = W_{j,0}( A_{j} \cdot). \] Let $j\in\mathbb{N}_0$ be arbitrary. We have ${\text{\rm supp }} W_{j,0} \subseteq \mathcal{W}^+_{j,0}$ and \[ \mathcal{W}^+_{j,0} \subseteq [-2^{js-1},2^{js-1}]\times[-2^{js\alpha-1},2^{js\alpha-1}]=\Xi_{j,0}, \] which implies \begin{align}\label{eq:supphataj} {\text{\rm supp }} \widehat{a}_{j} \subseteq [-2^{-1},2^{-1}]\times[-2^{-1},2^{-1}]=\Xi_{0,0}. \end{align}
Further, if $j>0$ the function $\widehat{\psi}_{j,0,0}$ vanishes on the square $[-2^{s(j-2)-5},2^{s(j-2)-5}]^2$. Consequently, $\widehat{a}_{j}$ vanishes on $[-2^{-2s-5},2^{-2s-5}]\times \big( 2^{js(1-\alpha)} \cdot [-2^{-2s-5},2^{-2s-5}] \big)$.
The mixed derivatives $\partial_1^{\rho_1}\partial_2^{\rho_2}W_{j,0}$ obey uniformly in $j\in\mathbb{N}_0$ \begin{align}\label{eq:basic_fact}
\|\partial_1^{\rho_1}\partial_2^{\rho_2}W_{j,0}\|_{\infty} \lesssim 2^{-js\rho_1} 2^{-js\alpha \rho_2}. \end{align} With the chain rule we deduce \[
\| \partial^{\rho}\widehat{a}_{j}\|_\infty = \| \partial_1^{\rho_1}\partial_2^{\rho_2} W_{j,0}( A_{j} \cdot) \|_\infty =
2^{js \rho_1} 2^{js \alpha \rho_2} \| \big( \partial_1^{\rho_1}\partial_2^{\rho_2} W_{j,0} \big) ( A_{j} \cdot) \|_\infty \lesssim 1. \] Due to ${\text{\rm supp }} \partial^\rho \widehat{a}_{j} \subseteq {\text{\rm supp }} \widehat{a}_{j}$ this estimate together with the support properties of $\widehat{a}_{j}$ implies~\eqref{eq:molcond}.
\end{proof}
\noindent With the machinery of $\alpha$-molecules at our disposal, it is possible to use $\mathfrak{C}_{s,\alpha}$ as an anchor system whose properties have consequences for other $\alpha$-scaled systems if they fulfill certain consistency conditions. In particular, approximation properties of $\mathfrak{C}_{s,\alpha}$ are shared by other $\alpha$-scaled systems such as e.g.\ $\alpha$-shearlets. A short discussion of this can be found in Section~\ref{sec:discussion}. For more details on the topic of $\alpha$-molecules we refer to \cite{GKKS15,FS15}.
\section{Curvelet Approximation of General Cartoons} \label{sec:cartoon}
In the two central sections of this article, Sections~\ref{sec:cartoon} and \ref{sec:straight}, we study the approximation performance of the $\alpha$-curvelet frame $\mathfrak{C}_{s,\alpha}$ with respect to different cartoon classes. We begin in this section with classes of general cartoons, used e.g.\ to model natural images. In Section~\ref{sec:straight} we then turn our focus on cartoons featuring only straight edges.
\subsection{Cartoon-like Functions}
Many suitable and well-established models for natural images are based on the concept of so-called cartoon-like functions. In a nutshell, such functions can be thought of as a patchwork of smooth regions separated from one another by piecewise-smooth discontinuity curves. Their structure imitates the fact that edges, a typical feature of natural images, are characterized by abrupt changes of color and brightness, whereas changes in the regions in between occur smoothly.
Mathematically, models based on this idea can be concretised in different ways. A classic model~\cite{CD04} postulates a compact image domain separated into two $C^2$ regions by a closed $C^2$ discontinuity curve. This model was generalized in various directions, e.g., to take into account piecewise-smooth edges or to allow more general $C^\beta$ regularity with $\beta\in[0,\infty)$. Cartoon classes of this kind have been studied extensively, especially in the range $\beta\in(1,2]$, e.g., in \cite{Kutyniok2012correct,Kei13,GKKScurve2014}. Another variant are the closely related horizon classes, where the discontinuity is not a closed curve in the image domain but a (possibly curved) horizontal or vertical line stretching across. Such classes have been investigated e.g.\ in \cite{D99,CWBB04c,PM05}. Let us also mention that there exist extensions to multi-dimensions, see e.g.\ \cite{Kutyniok2012correct}. In particular, the corresponding 3D models have been applied in the investigation of video data.
Since we are concerned with image approximation, our attention is restricted to the 2-dimensional setting. The following definition is a template for different classes of bivariate cartoons, comprising many of those mentioned above. It provides the flexibility to taylor the model to our particular needs in Sections~\ref{sec:cartoon} and~\ref{sec:straight}.
\begin{definition}\label{def:gencart} Let $\beta\in[0,\infty)$ and $\nu>0$. Given a domain $\Omega\subseteq\mathbb{R}^2$ and a set $\mathcal{A}$ of admissible subsets of $\mathbb{R}^2$, the class $\mathcal{E}^{\beta}(\Omega;\mathcal{A},\nu)$ consists of all functions $f\in L^2(\mathbb{R}^2)$ of the form \begin{align*} f=f_{1}+f_{2}\chi_{\mathcal{D}}, \end{align*}
where $\mathcal{D}\in\mathcal{A}$ and $f_1,\,f_2\in C^{\beta}(\mathbb{R}^2)$ with ${\text{\rm supp }} f_{1}, f_2 \subseteq \Omega$ and $\|f_1\|_{C^\beta}, \|f_2\|_{C^\beta}\le\nu$. The class $\mathcal{E}_{\rm bin}^{\beta}(\Omega;\mathcal{A})$ shall be the collection of all `binary functions' $\chi_{\mathcal{D}}$, where $\mathcal{D}\in\mathcal{A}$ and $\mathcal{D}\subseteq\Omega$. \end{definition}
For particular choices of $\mathcal{A}$ many of the classes appearing in the literature can be retrieved, including classes of horizon-type.
In this section we focus on the class $\mathcal{E}^{\beta}(\Omega;\mathcal{A},\nu)$ with fixed image domain $\Omega=[-1,1]^2$ and certain $C^\beta$ domains as admissible sets $\mathcal{A}$. Similar to \cite{Don01,CD04,Kutyniok2010,Kutyniok2012correct}, we restrict our investigation to star-shaped domains, since those allow a simple parametrization of the boundary curve. The results obtained however also hold true for more general domains.
Let us introduce the collection of admissible sets $\text{\sc Star}^\beta(\nu)$, $\nu>0$, as all translates of sets $B\subseteq\mathbb{R}^2$, whose boundary $\partial B$ possesses a parametrization $b:\mathbb{T}\to\mathbb{R}^2$ of the form \[ b(\varphi)=\rho(\varphi) \begin{pmatrix} \cos(\varphi) \\ \sin(\varphi) \end{pmatrix}, \quad \varphi\in\mathbb{T}=[0,2\pi]\,, \] where the radius function $\rho:\mathbb{T} \to \mathbb{R}$ is a $C^\beta$ function with \begin{align}\label{eq:HolCart}
|\partial^{\lfloor\beta\rfloor}\rho(\varphi)- \partial^{\lfloor\beta\rfloor}\rho(\varphi^\prime)| \le \nu \rho_0 |\varphi - \varphi^\prime|^{\beta-\lfloor\beta\rfloor} \quad\text{for all }\varphi,\varphi^\prime\in\mathbb{T}, \end{align} where we set $\rho_0:=\min_{\varphi\in\mathbb{T}} \rho(\varphi) \ge \nu^{-1}$. The condition~\eqref{eq:HolCart} implies that with $C=C(\beta)=(2\pi)^{\beta}\ge1$ we have
$\|\rho^{(k)}\|_{C^0(\mathbb{T})}\le C\rho_0\nu$ for every $k\in\{1,\ldots,\lfloor\beta\rfloor\}$ if $\beta\ge1$, and $|\rho(\varphi)- \rho(\varphi^\prime)| \le C\rho_0\nu$ for $\varphi,\varphi^\prime\in\mathbb{T}$. In particular $\rho_0 \le \rho(\varphi) \le \rho_0 (1 + C\nu)$ for all $\varphi\in\mathbb{T}$.
Note, that the set $\text{\sc Star}^\beta(\nu)$ differs from the set of star-shaped domains used in \cite{Don01,CD04,Kutyniok2010,Kutyniok2012correct}. The domains in $\text{\sc Star}^\beta(\nu)$ are not restricted to subsets of $[-1,1]^2$. In fact, every star-shaped $C^\beta$ domain with center $0$ and $\rho_0>0$ is contained in $\text{\sc Star}^\beta(\nu)$ for suitably large $\nu$. Moreover, the collection $\text{\sc Star}^\beta(\nu)$ is scaling invariant in the sense that for $B\in\text{\sc Star}^\beta(\nu)$ and $\lambda>0$ also $\lambda B\in\text{\sc Star}^\beta(\nu)$, provided $\lambda\rho_0\ge\nu^{-1}$. In addition, with $B\in\text{\sc Star}^\beta(\nu)$ also the complement $B^c=\mathbb{R}^2\backslash B$ is contained in $\text{\sc Star}^\beta(\nu)$.
Building upon Definition~\ref{def:gencart} we now define the class of functions which we want to study in this section. We put $\Omega=[-1,1]^2$ and $\mathcal{A}=\text{\sc Star}^\beta(\nu)$. Further, we assume $\beta\in[0,\infty)$ and $\nu>0$. For the resulting class $\mathcal{E}^\beta([-1,1]^2;\text{\sc Star}^\beta(\nu),\nu)$ we simplify the notation \begin{align}\label{eqdef:cart} \mathcal{E}^\beta([-1,1]^2;\nu):= \mathcal{E}^\beta([-1,1]^2;\text{\sc Star}^\beta(\nu),\nu). \end{align} The associated binary class shall be denoted by $\mathcal{E}_{bin}^\beta([-1,1]^2;\nu):= \mathcal{E}_{bin}^\beta([-1,1]^2;\text{\sc Star}^\beta(\nu))$.
\subsection{Class Bounds}
Before we investigate the approximation performance of the $\alpha$-curvelet frame $\mathfrak{C}_{s,\alpha}$ with respect to the class $\mathcal{E}^\beta([-1,1]^2;\nu)$, let us take a broader stance and aim for best possible $N$-term approximation in case we can freely choose the utilized dictionary.
Of course, a countable dense subset of $L^2(\mathbb{R}^2)$ would yield arbitrarily good $1$-term approximations. This shows that, without further restrictions, the question of best possible approximation is not well-posed.
To cast a realistic scenario, when computing $N$-term approximations typically a constraint on the search depth is imposed. More concretely, given a fixed ordering of the dictionary and some polynomial $\pi$, it is common to allow only $N$-term approximants being built from the first $\pi(N)$ elements of the dictionary.
Under this so-called polynomial depth search constraint, an upper bound on the maximal achievable approximation rate was first derived by Donoho~\cite[Thm.~1]{Don01} for binary $C^\beta$ cartoons in the range $\beta\in(1,2]$. Later similar results were proved for more general cartoon classes~\cite{Kutyniok2012correct,Kei13,GKKScurve2014}.
Theorem~\ref{thm:benchmark} below establishes a bound for the class $\mathcal{E}^\beta([-1,1]^2;\nu)$ specified in \eqref{eqdef:cart}.
\begin{theorem}\label{thm:benchmark} Let $\beta,\gamma\in[0,\infty)$ and $\nu>0$. Assume that there is a constant $C>0$ such that \[
\sup_{f \in \mathcal{E}^{\beta}([-1,1]^2;\nu)} \|f-f_N\|_2^2 \le C N^{-\gamma} \quad\text{ for all }N\in\mathbb{N}, \] where $f_N$ denotes the best $N$-term approximation of $f$ obtained by polynomial depth search in a fixed dictionary. Then necessarily $\gamma\le\beta$. \end{theorem}
\noindent In principle, this is a known result (see e.g.~\cite{Kutyniok2012correct}). However, for reasons of completeness, we outline a short proof based on the technique used in \cite{Don01}.
It relies on Theorem~\ref{thm:upperbound} below and the fact that the class $\mathcal{E}^{\beta}([-1,1]^2;\nu)$ contains a copy of $\ell_0^p$ for $p=2/(\beta + 1)$. Let us recall this notion introduced in~\cite{Don01}.
\begin{definition}[{\cite[Def.~1\&2]{Don01}}] A function class $\mathfrak{F}\subseteq L^2(\mathbb{R}^2)$ is said to contain an \emph{embedded orthogonal hypercube} of dimension $m$ and side-length $\delta$ if there exist
$f_0 \in \mathfrak{F}$ and orthogonal functions $\psi_{\ell}\in L^2(\mathbb{R}^2)$, $\ell\in\{1,...,m\}$, with $\|\psi_{\ell}\|_{2}=\delta$ such that the collection of hypercube vertices embeds, i.e., \begin{align*} \Big\{ f_0 + \sum_{\ell=1}^{m} \epsilon_\ell\psi_{\ell} ~:~ \epsilon=(\epsilon_1,\ldots,\epsilon_m) \in \{0,1\}^m \Big\} \subseteq \mathfrak{F} \,. \end{align*} It is said to \emph{contain a copy of $\ell_{0}^{p}$, $p>0$,} if it contains a sequence of embedded orthogonal hypercubes, whose associated dimensions $m_k$ and side-lengths $\delta_k$ satisfy $\delta_k \rightarrow 0$ for $k\to\infty$ and with a constant $C>0$ \begin{align*} C\delta_k^{-p} \le m_k \quad\text{for all }k \in\mathbb{N}. \end{align*} \end{definition}
\noindent The significance of this notion is due to the following result, which was first obtained in~\cite[Thm.~2]{Don01}. The reformulated version below can be found in \cite[Thm.~2.2]{GKKScurve2014}.
\begin{theorem}[{\cite[Thm.~2.2]{GKKScurve2014}}]\label{thm:upperbound} Suppose, that a class of functions $\mathfrak{F}\subseteq L^2(\mathbb{R}^2)$ is uniformly $L^2$-bounded and contains a copy of $\ell^p_0$. Then, allowing only polynomial depth search in a given dictionary, there is a constant $C>0$ such that for every $N_0\in\mathbb{N}$ there is a function $f\in\mathfrak{F}$ and an $N\in\mathbb{N}$, $N\ge N_0$ such that \begin{equation*}
\|f - f_{N} \|^2_2 \ge C \big(N \log_2 (N)\big)^{-(2-p)/p}, \end{equation*} where $f_N$ denotes the best $N$-term approximation under the polynomial depth search constraint. \end{theorem}
\noindent It remains to investigate for which $p>0$ the class $\mathcal{E}^{\beta}([-1,1]^2;\nu)$ contains a copy of $\ell_0^p$. To this end, let us introduce the following subclass of smooth functions for $\beta\in[0,\infty)$ and $\nu>0$, \begin{align}\label{eqdef:smoothclass}
C_0^\beta([-1,1]^2;\nu):=\big\{ f\in C_0^\beta([-1,1]^2) ~:~ \|f\|_{C^\beta}\le\nu \big\}. \end{align} Note, that the choice $\Omega=[-1,1]^2$ and $\mathcal{A}=\{\emptyset\}$ in Definition~\ref{def:gencart} yields this class. As a consequence, \begin{align}\label{eq:smoothemb} C_0^\beta([-1,1]^2;\nu) \subset \mathcal{E}^{\beta}([-1,1]^2;\nu). \end{align}
Lemma~\ref{lem:copylp} below is the 2D analogon of the statement of \cite[Thm.~3.2]{Kutyniok2012correct}. It shows, in particular, that $C_0^\beta([-1,1]^2;\nu)$ contains a copy of $\ell_0^{2/(\beta+1)}$. Hence, as a consequence of \eqref{eq:smoothemb}, also $\mathcal{E}^{\beta}([-1,1]^2;\nu)$ contains a copy of $\ell_0^{2/(\beta+1)}$. An application of Theorem~\ref{thm:upperbound} thus yields Theorem~\ref{thm:benchmark}.
\begin{lemma}\label{lem:copylp} Let $\nu>0$, $\beta\in[0,\infty)$, and $p=2/(\beta+1)$. Then the following holds true. \begin{enumerate} \item[(i)] The function class $C_0^\beta([-1,1]^2;\nu)$ contains a copy of $\ell_0^p$. \item[(ii)] The class of binary cartoons $\mathcal{E}_{\rm bin}^{\beta}([-1,1]^2;\nu)$ contains a copy of $\ell_0^p$ if $\nu\ge1$, otherwise it only contains the zero-function. \end{enumerate} \end{lemma}
\begin{proof} The proof is a $2$D-adaption of the proof of \cite[Thm.~3.2]{Kutyniok2012correct}. \end{proof}
\noindent Summarizing, this establishes $N^{-\beta}$ as an upper bound for the possible order of approximation for general $C^\beta$ cartoons. This rate is the benchmark, against which the performance of $\mathfrak{C}_{s,\alpha}$ has to be measured.
We end this paragraph with the following observation.
\begin{remark}\label{rem:benchmark} According to Lemma~\ref{lem:copylp}(i), the bound of Theorem~\ref{thm:benchmark} actually holds true for the class $C_0^\beta([-1,1]^2;\nu)$. This is a stronger statement due to the inclusion \eqref{eq:smoothemb}. Further, due to Lemma~\ref{lem:copylp}(ii), a statement analogous to Theorem~\ref{thm:benchmark} holds true for the binary class $\mathcal{E}_{\rm bin}^{\beta}([-1,1]^2;\nu)$ if $\nu\ge1$. \end{remark}
\subsection{Approximation Guarantees} \label{ssec:guarantees}
According to Theorem~\ref{thm:benchmark} and Remark~\ref{rem:benchmark} the order of the $N$-term approximation rate achievable for the classes $\mathcal{E}_{\rm bin}^{\beta}([-1,1]^2;\nu)$, $\nu\ge1$, and $\mathcal{E}^{\beta}([-1,1]^2;\nu)$, $\nu>0$, cannot exceed $N^{-\beta}$. This bound is valid for arbitrary dictionaries and independent of the approximation scheme employed, as long as it respects a polynomial depth search condition. Even adaptive approximation schemes cannot perform better.
Schemes, where these rates are provably achieved, at least up to order, have been developed for binary cartoons based on wedgelets~\cite{D99} and surflets~\cite{CWBB04b}, for general cartoons utilizing bandelets~\cite{PennecM,PM05}. These results show that the optimality benchmark $N^{-\beta}$ can indeed be realized in practice, at least up to order. However, the utilized schemes are mostly adaptive, only for certain cartoon classes nonadaptive methods with quasi-optimal performance are known.
A breakthrough concerning the nonadaptive approximation of $C^2$ cartoons with curved edges was the introduction of curvelets by Cand{\`e}s and Donoho~\cite{CD2000,CD04}. By a simple thresholding scheme, curvelet frames achieve an approximation rate matching the class bound $N^{-2}$ up to a log-factor. The reason for this performance is due to the parabolic scaling employed. The following argument shall heuristically explain, why this type of scaling is ideal for the representation of $C^2$ edges.
In local Cartesian coordinates, a $C^2$ curve can be represented as the graph $(E(x),x)$ of a function $E\in C^2(\mathbb{R})$ and one can choose a coordinate system such that $E^\prime(0)=E(0)=0$. A Taylor expansion then yields approximately $E(x)\approx \frac{1}{2} E^{\prime\prime}(0)x^2$, which matches the essential support $width\approx length^2$ of parabolically scaled functions. Hence, those can provide optimal resolution of the curve across all scales.
A similar heuristic applies to $C^\beta$ curves if $\beta\in(1,2]$. A Taylor expansion of $E\in C^\beta(\mathbb{R})$ yields $|E(x)|\lesssim x^\beta$. The curve is thus contained in a rectangle of size $width\approx length^{1/\beta}$ which suggests $\alpha$-scaling with $\alpha=\beta^{-1}$ for optimal approximation. And indeed, the classic approximation result by Cand{\`e}s and Donoho could be extended in~\cite[Thm.~4.1]{GKKScurve2014} to the range $\beta\in(1,2]$.
This generalized result is stated below, slightly modified to fit into the setting of this article. The class $\mathcal{E}^{\beta}([-1,1]^2;\nu)$ used here is not fully identical to the class in \cite{GKKScurve2014}. Moreover, only curvelet frames of the type $\mathfrak{C}_{s,\alpha}$ with $s=1$ were considered there. It is not hard to verify though that the proof carries over to general $s>0$ and that the statement is also valid in our setting.
\begin{theorem}[{\cite[Thm.~4.1]{GKKScurve2014}}]\label{thm:oldcurveappr}
Let $\beta\in(1,2]$, $\nu>0$. For the choice $\alpha=\beta^{-1}$, $s>0$ arbitrary, the frame of $\alpha$-curvelets $\mathfrak{C}_{s,\alpha}$
provides almost optimal sparse approximations for the class $\mathcal{E}^{\beta}([-1,1]^2;\nu)$. More precisely,
there exists a constant $C>0$ such that for every $f\in \mathcal{E}^{\beta}([-1,1]^2;\nu)$ and $N\in\mathbb{N}$ \begin{align*}
\|f-f_N\|_2^2\le CN^{-\beta} \log_2(1+N)^{1+\beta} \,, \end{align*} where $f_N$ denotes the $N$-term approximation of $f$ obtained by choosing the $N$ largest coefficients. \end{theorem}
\noindent This theorem naturally raises the question of extendibility beyond the range $\beta\in(1,2]$, a question pursued in the following subsection. In particular, we investigate if the choice $\alpha=\beta^{-1}$ is still optimal for $\beta>2$. Obviously, the heuristic consideration from above is not valid any more in this regime. And indeed, we will see that for $\beta>2$ the optimal choice is not $\alpha=\beta^{-1}$. In fact, it is still $\alpha=\frac{1}{2}$ and choosing $\alpha<\frac{1}{2}$ deteriorates the approximation performance.
\subsection{Approximation Bounds}
The main results of this subsection, Theorems~\ref{thm:bound1} and \ref{thm:bound2}, establish bounds on the achievable $N$-term approximation rate for the class $\mathcal{E}^\beta([-1,1]^2;\nu)$, $\beta\in[0,\infty)$, when using the $\alpha$-curvelet frame $\mathfrak{C}_{s,\alpha}$ for approximation. Unlike the bounds in Theorem~\ref{thm:benchmark} associated with the signal class the bounds derived here are tied to the particular approximation system $\mathfrak{C}_{s,\alpha}$. However, via the framework of $\alpha$-molecules they are also effective for other $\alpha$-scaled systems, such as e.g.\ $\alpha$-shearlets as discussed in Section~\ref{sec:discussion}.
In order to establish these bounds we study the approximability of certain example cartoons. As a suitable object, we choose the characteristic function of the ball $B_2(0,\frac{1}{2})\subset\mathbb{R}^2$ of radius $\frac{1}{2}$, for which we subsequently use the symbol \begin{align}\label{def:Theta} \Theta(x):=\chi_{B_2(0,\frac{1}{2})}(x_1,x_2)\,, \quad x\in\mathbb{R}^2. \end{align} This function embodies an exceptionally regular cartoon with a closed curved $C^\infty$-singularity. It is radial symmetric and binary, contained in $\mathcal{E}^\beta_{\rm bin}([-1,1]^2,\nu)$ for arbitrary $\beta\in[0,\infty)$ and $\nu\ge2$. Furthermore, for every $\beta\in[0,\infty)$ and $\nu\ge2$ there is $\gamma>0$ such that $\gamma\Theta\in\mathcal{E}^\beta([-1,1]^2;\nu)$, wherefore the approximability of $\Theta$ has implications for the approximability of these cartoon classes.
The Fourier transform of $\Theta$ is explicitly known. Let $\mathcal{J}_1$ denote the Bessel function of order 1, then according to~\eqref{eq:Fourier2ball} \begin{align}\label{eq:ballfou}
\widehat{\Theta}(\xi) = \frac{\mathcal{J}_1(\pi|\xi|)}{2|\xi|} \,, \quad\xi\in\mathbb{R}^2. \end{align} Some properties of $\mathcal{J}_1$ and Bessel functions in general are collected in the appendix.
At the center of the following investigation is the lemma below, which estimates the energy of $\widehat{\Theta}$ contained in the wedges $\mathcal{W}_{J}$, $J\in\mathbb{J}$. Let $\{W_J\}_{J\in\mathbb{J}}$ be a family of functions of the kind~\eqref{eq:suppfunctions} with property \eqref{eq:CalderonW} for $0<A\le B<\infty$. Further, let \begin{align*} W^-_J:=\chi_{\mathcal{W}^-_J} \quad\text{and}\quad W^+_J:=\chi_{\mathcal{W}^+_J} \end{align*} be the characteristic functions of the sets $\mathcal{W}^-_J$ and $\mathcal{W}^+_J$ defined in \eqref{eq:wedgePJ}.
\begin{lemma}\label{lem:thetaest}
There are constants $0<C_1\le C_2<\infty$, independent of scale $j\ge j_0$, where $j_0\in\mathbb{N}_0$ is a suitable base scale, such that for all $J\in\mathbb{J}$ with $|J|\ge j_0$, where $|J|=j$ for $J=(j,\ell)\in\mathbb{J}$, \[
AC_1 2^{-js(2-\alpha)} \le A\| \widehat{\Theta} W^{-}_J \|_2^2 \le \| \widehat{\Theta} W_J \|_2^2 \le B \| \widehat{\Theta} W^{+}_J \|_2^2 \le B C_2 2^{-js(2-\alpha)}. \] \end{lemma}
\begin{proof}
Let us recall the Bessel function $\mathcal{J}_1$ of order 1 and its asymptotic behavior. According to \eqref{eqapp:est} there is a constant $C>0$ and a function $R_1$ on $[1,\infty)$ satisfying $|R_1(r)| \le C r^{-3/2}$ such that \begin{align*} \mathcal{J}_1(r)=\sqrt{\frac{2}{\pi r}}\cos(r-\frac{3\pi}{4}) + R_1(r) \quad\text{ for }r\ge1. \end{align*} This allows to separate terms of higher order from $\mathcal{J}_1^2$. We decompose \begin{align*} \mathcal{J}_1^2(r) = \Big[ \frac{2}{\pi} \cos^2(r-\frac{3\pi}{4})r^{-1} \Big] + \Big[ \sqrt{\frac{8}{\pi}} \cos(r-\frac{3\pi}{4}) r^{-1/2}R_1(r) + R_1(r)^2 \Big]=: T_1(r) + T_2(r). \end{align*} For the following argumentation we need the square wave function $\sqcap:\mathbb{R}\to \{0,1\}$ defined by \[ \sqcap(r) := \begin{cases} 1 \quad&,\, r\in\bigcup_{k\in\mathbb{Z}} k\pi + [-\frac{\pi}{2},0], \\ 0 &,\, r\in\bigcup_{k\in\mathbb{Z}} k\pi + (0,\frac{\pi}{2}). \end{cases} \] For all $r\in\mathbb{R}$ it has the property $2\cos^2(r-3\pi/4) \ge \sqcap(r)$. Therefore we can deduce for $1\le a\le b$ \begin{align*} \int_a^{b} T_1(r) r^{-1} \,dr =\frac{1}{\pi} \int_a^{b} 2\cos^2(r-\frac{3\pi}{4})r^{-2} \,dr \ge\frac{1}{\pi} \int_a^{b} \sqcap(r)r^{-2} \,dr \ge\frac{1}{2} \sum_{k\in I_{a,b}} (k\pi)^{-2} \end{align*} with $I_{a,b}:=\{ k\in\mathbb{Z} ~:~ k\pi\in [a+\pi, b] \}$. To proceed, we use the relation \[ \sum_{k=m}^n (k\pi)^{-2} \ge \frac{1}{\pi} \int_{m\pi}^{(n+1)\pi} k^{-2} \,dk, \] which is valid for all $m,n\in\mathbb{N}$ and $m\le n$. We obtain \begin{align*} \frac{1}{2} \sum_{k\in I_{a,b}} (k\pi)^{-2} \ge \frac{1}{2\pi} \int_{a+2\pi}^{b} k^{-2} \,dk = \frac{1}{2\pi} \big(\int_{a}^{b} k^{-2} \,dk - \int_{a}^{a+2\pi} k^{-2} \,dk \Big) \ge \frac{1}{2\pi}(a^{-1}- b^{-1}) - a^{-2}. \end{align*} Next, we see that with a constant $C>0$ independent of $1\le a\le b$ \begin{align*}
\int_a^{b} |T_2(r)| r^{-1} \,dr \le C \int_a^{b} r^{-3} \,dr \le C \int_a^{\infty} r^{-3} \,dr \le C a^{-2}. \end{align*} Altogether, we conclude that \begin{align*} \int_a^{b} \frac{\mathcal{J}_1^2(r)}{r} \,dr \ge \frac{1}{2\pi}(1 - ab^{-1})a^{-1} - (1+ C) a^{-2}. \end{align*} If $c=ab^{-1}\le 1$ is fixed, we can deduce for $a\ge 4\pi \frac{1+C}{1-c}$ the estimate \begin{align}\label{eq:besauxrel} \int_a^{a/c} \frac{\mathcal{J}_1^2(r)}{r} \,dr \ge \frac{1}{4\pi}(1-c)a^{-1}. \end{align}
After this preparation, we can now turn to the actual proof of the assertion. The relation \[
A\| \widehat{\Theta} W^{-}_J \|_2^2 \le \| \widehat{\Theta} W_J \|_2^2 \le B \| \widehat{\Theta} W^{+}_J \|_2^2 \]
is a direct consequence of \eqref{eq:suppprop} and $\|W_J\|_\infty \le \sqrt{B}$.
Let $\mathcal{I}_j$ be the intervals defined in \eqref{eqdef:dyintervals}. Further, recall the intervals $\mathcal{I}^-_j \subset \mathcal{I}_j$ defined in \eqref{eq:IIIIII}. Using \eqref{eq:ballfou} and the definition \eqref{eq:wedgePJ} of $\mathcal{W}_J^-$ we calculate \begin{align*}
\| \widehat{\Theta} W^{-}_J \|_2^2
= \int_{\mathcal{W}^-_J} \frac{\mathcal{J}_1^2(\pi|\xi|)}{4|\xi|^2} \,d\xi = \int_{\mathcal{I}^-_j} \int_{\mathcal{A}^-_J} \frac{\mathcal{J}_1^2(\pi r)}{4r} \,d\varphi dr \asymp 2^{-js(1-\alpha)} \int_{\pi\mathcal{I}^-_j} \frac{\mathcal{J}_1^2(r)}{r} \,dr. \end{align*} The intervals $\mathcal{I}^-_j$ scale like $\sim 2^{js}$. Hence, if $j\in\mathbb{N}$ is chosen large enough by \eqref{eq:besauxrel} \begin{align*}
\| \widehat{\Theta} W^{-}_J \|_2^2 \asymp 2^{-js(1-\alpha)} \int_{\pi\mathcal{I}^-_j} \mathcal{J}_1^2(r) r^{-1} \,dr \gtrsim 2^{-js(1-\alpha)} 2^{-js} = 2^{-js(2-\alpha)}. \end{align*}
The estimate from above is much easier to establish. If $j\in\mathbb{N}$ such that $\pi\mathcal{I}_j\subset[1,\infty)$ we have \begin{gather*}
\| \widehat{\Theta} W^+_{J} \|_2^2
= \int_{\mathcal{W}^+_J} \frac{\mathcal{J}_1^2(\pi|\xi|)}{4|\xi|^2} \,d\xi = \int_{\mathcal{I}_j} \int_{\mathcal{A}_J} \frac{\mathcal{J}_1^2(\pi r)}{4r} \,d\varphi dr \asymp 2^{-js(1-\alpha)} \int_{\pi \mathcal{I}_j} \frac{\mathcal{J}_1^2(r)}{r} \,dr \\ \lesssim 2^{-js(1-\alpha)} \int_{\mathcal{I}_j} r^{-2}\,dr \lesssim 2^{-js(2-\alpha)}. \tag*{\qedhere} \end{gather*}
\end{proof}
\noindent Based on Lemma~\ref{lem:thetaest} we can prove the first main result of this article.
\begin{theorem}\label{thm:bound1} Let $\mathfrak{C}_{s,\alpha}$ be the $\alpha$-curvelet frame constructed in Section~\ref{sec:curvelets} for fixed $\alpha\in(-\infty,1)$ and $s>0$. There exists a constant $C>0$ such that for any given $N\in\mathbb{N}$ every $N$-term approximation $f_N$ of $\Theta$ with respect to
$\mathfrak{C}_{s,\alpha}$ (not even subject to a polynomial depth search constraint) satisfies \begin{align*}
\|\Theta-f_N\|_2^2 \ge C N^{-\frac{1}{1-\alpha}}. \end{align*} \end{theorem} \begin{proof}
Let $N\in\mathbb{N}$ be fixed and assume that \[ f_N=\sum_{r=1}^N \theta_{J_r,k_r} \psi_{J_r,k_r} \] is a linear combination of $\alpha$-curvelets $\psi_{J_r,k_r}$ with coefficients $\theta_{J_r,k_r}\in\mathbb{R}$. The curvelets $\psi_{J_r,k_r}\in\mathfrak{C}_{s,\alpha}$ satisfy ${\text{\rm supp }} \widehat{\psi}_{J_r,k_r}\subseteq \mathcal{W}^+_{J_r}$ as recorded in \eqref{eq:suppprop}. It follows ${\text{\rm supp }} \widehat{f}_N \subseteq \mathcal{W}_N$ where $\mathcal{W}_N:=\bigcup_{J\in \mathbb{J}_N} \mathcal{W}^+_J$ for $\mathbb{J}_N:=\{ J_1,\ldots, J_N \}\subset \mathbb{J}$. Using the notation $\mathbb{J}^c_N:=\mathbb{J}\backslash\mathbb{J}_N$ and $\mathcal{W}^c_N:=\mathbb{R}^2\backslash \mathcal{W}_N$ we get with Lemma~\ref{lem:thetaest} \begin{align*}
\| \Theta - f_N \|_2^2 &= \| \widehat{\Theta} - \widehat{f}_N \|_2^2
\ge \| \widehat{\Theta} \|_{L^2(\mathcal{W}^c_N)}^2
\ge \sum_{J\in \mathbb{J}^c_N} \| \widehat{\Theta} W^{-}_J \|^2_2 \gtrsim \sum_{J\in \mathbb{J}^c_N} 2^{-js(2-\alpha)}. \end{align*}
We want to bound the right-hand side from below. By~\eqref{eq:Lj}, the number of tiles in each corona $\mathcal{C}_j$, $j\in\mathbb{N}_0$, is given by $L_j$, where $L_0=1$ and $L_j = 2^{\lfloor js(1-\alpha) \rfloor + 1}$ for $j\ge1$. Let $j(N)\in\mathbb{N}$ denote the unique number such that \begin{align*} \sum_{j=0}^{j(N)-1} L_j < N \le \sum_{j=0}^{j(N)} L_j. \end{align*}
Since $2^{-js(2-\alpha)}$ decreases with rising scale we obtain \begin{align*} \sum_{J\in \mathbb{J}^c_N} 2^{-js(2-\alpha)} \ge \sum_{j=j(N)+1}^{\infty} L_j 2^{-js(2-\alpha)} \ge \sum_{j=j(N)+1}^{\infty} 2^{-js} \gtrsim 2^{-j(N)s}. \end{align*} Here we used $ L_j \ge 2^{ js(1-\alpha)}$. Since $N\gtrsim\sum_{j=0}^{j(N)-1} 2^{ js(1-\alpha)} \gtrsim 2^{ j(N)s(1-\alpha)}$ we can finally deduce \begin{align*}
\| \Theta - f_N \|_2^2 \gtrsim 2^{-j(N)s} = \Big( 2^{ j(N)s(1-\alpha)} \Big)^{-\frac{1}{1-\alpha}} \gtrsim N^{-\frac{1}{1-\alpha}}. \tag*{\qedhere} \end{align*}
\end{proof}
\noindent This result can be strengthened if we restrict to greedy $N$-term approximations obtained by thresholding the coefficients. Essential is the following observation, which has also been used in~\cite{GKKScurve2014}. Due to its importance we give a rigorous proof here.
\begin{lemma}\label{lem:apriori} There is a constant $C>0$ such that all curvelets $\psi_\mu\in\mathfrak{C}_{s,\alpha}$, $\mu\in\mathbb{M}$, satisfy \[
\|\psi_\mu\|_1 \le C 2^{-js(1+\alpha)/2}. \] \end{lemma} \begin{proof} Let $a_j$ be the functions from \eqref{eq:molgen} and recall that according to \eqref{eq:supphataj} the support of $\widehat{a}_j$ is contained in the unit square $\Xi_{0,0}$ for every $j\in\mathbb{N}_0$. Let $\textsl{Id}$ denote the identity operator. We have the estimate \begin{align*}
\Big\|\mathcal{F}^{-1} \Big( (\textsl{Id}+\partial_1^2)(\textsl{Id}+\partial_2^2) \widehat{a}_{j} \Big) \Big\|_\infty
&\le \| (\textsl{Id} + \partial_1^2)(\textsl{Id} + \partial_2^2) \widehat{a}_{j} \|_1 \le \| (\textsl{Id} + \partial_1^2)(\textsl{Id} + \partial_2^2) \widehat{a}_{j}\|_\infty. \end{align*} According to Lemma~\ref{thm:curvmol} the right-hand side is bounded uniformly over all scales. We conclude that there is a constant $C>0$, independent of $j\in\mathbb{N}_0$, such that \begin{align*}
\sup_{x\in\mathbb{R}^2} |(1+x_1^2)(1+x_2^2) a_{j}(x)| \le C. \end{align*}
In other words $|a_{j}(x)| \le C (1+x_1^2)^{-1}(1+x_2^2)^{-1}$. Using the representation \eqref{eq:smolrepr} we obtain \begin{align*}
|\psi_{j,0,0}(x)| = 2^{js(1+\alpha)/2} |a_{j} (A_{j}x)| \le C 2^{js(1+\alpha)/2} (1+2^{2js}x_1^2)^{-1} (1+2^{2js\alpha}x_2^2)^{-1} \end{align*} and hence \begin{align*}
\int_{\mathbb{R}^2} |\psi_{j,0,0}(x)| \,dx &\lesssim 2^{js(1+\alpha)/2} \int_{\mathbb{R}^2}(1+2^{2js}x^2_1)^{-1} (1+2^{2js\alpha}x^2_2)^{-1} \,dx \\ &= 2^{-js(1+\alpha)/2} \int_{\mathbb{R}^2}(1+x^2_1)^{-1} (1+x^2_2)^{-1} \,dx \lesssim 2^{-js(1+\alpha)/2}. \end{align*}
Since $\|\psi_{j,\ell,k}\|_1=\|\psi_{j,0,0}\|_1$ the proof is finished. \end{proof}
\noindent Lemma~\ref{lem:apriori} allows to deduce a simple a-priori estimate of the curvelet coefficient size, namely \begin{align}\label{eq:apriori}
|\theta_\mu|=|\langle f,\psi_\mu \rangle| \le \|f\|_{\infty} \|\psi_\mu\|_{1} \le C \|f\|_{\infty} 2^{-js(1+\alpha)/2} \quad\text{ for }\mu=(j,\ell,k)\in\mathbb{M}. \end{align} Note, that the constant $C>0$ is fully determined by $\mathfrak{C}_{s,\alpha}$. Using \eqref{eq:apriori} we now prove a stronger statement than Theorem~\ref{thm:bound1} for greedy approximations.
\begin{theorem}\label{thm:bound2} Let $\alpha\in(-\infty,1]$ and $s>0$ be fixed. Further, let $f_N$ denote the $N$-term approximation of $\Theta$ with respect to the $\alpha$-curvelet frame $\mathfrak{C}_{s,\alpha}$ obtained by thresholding the coefficients. There is a constant $C>0$ such that for every $N\in\mathbb{N}$ \begin{align*}
\|\Theta-f_N\|_2^2 \ge C N^{-\frac{1}{\max\{\alpha,1-\alpha\}}}. \end{align*} \end{theorem}
\begin{proof} If $\alpha\le \frac{1}{2}$ the assertion is true by Theorem~\ref{thm:bound1}. It remains to handle the range $1\ge\alpha>\frac{1}{2}$. Let $\theta_{J_r,k_r}=\langle \Theta, \psi_{J_r,k_r} \rangle$, $r\in\{1,\ldots,N\}$, be the $N$ largest curvelet coefficients which determine the approximant $f_N:=\sum_{r=1}^N \theta_{J_r,k_r} \psi_{J_r,k_r}$.
On the Fourier side the curvelet $\psi_{J,k}\in\mathfrak{C}_{s,\alpha}$ is the product of the functions $W_{J}$ and $u_{J,k}$ defined in \eqref{eq:suppfunctions} and \eqref{eq:fouriersys}, respectively. Using condition~\eqref{eq:CalderonW} we first estimate \begin{align*}
\|\Theta - f_N \|_2^2 = \|\widehat{\Theta} - \widehat{f}_N \|_2^2 \ge B^{-2} \sum_{J\in\mathbb{J}} \|\widehat{\Theta}W_J- \widehat{f}_NW_J\|_2^2
\ge \frac{A^2}{B^2} \sum_{J\in\mathbb{J}} \|\widehat{\Theta}W^-_J - \widehat{f}_N W^-_J \|_2^2, \end{align*} where $W^-_J$ is the characteristic function of the set $\mathcal{W}^-_J$ defined in \eqref{eq:wedgePJ}. The triangle inequality yields \begin{align}\label{eqqqqq}
\frac{1}{2} \|\widehat{\Theta} W^-_J \|_2^2 \le \|\widehat{\Theta} W^-_J - \widehat{f}_N W^-_J \|_2^2
+ \| \widehat{f}_N W^-_J \|_2^2 \quad\text{for every }J\in\mathbb{J}. \end{align} Observe the relation $\sqrt{A}W^-_J \le W^-_J W_{J} \le \sqrt{B} W^-_J $ and $W^-_J W_{J^\prime}=0 $ for $J\neq J^\prime$. Therefore, it holds \begin{align*} \widehat{f}_NW^-_J = \sum_{r=1}^N \theta_{J_r,k_r} \widehat{\psi}_{J_r,k_r}W^-_J = \sum_{r=1}^N \theta_{J_r,k_r} u_{J_r,k_r}W_{J_r} W^-_J \asymp \sum_{k\in K_J} \theta_{J,k} u_{J,k} W^-_J \end{align*} with $K_J=\{ k_r\in \mathbb{Z}^2 ~:~ r\in\{1,\ldots,N \}, \, J_r=J \}$. Next, we use that $\{u_{J,k}\}_{k\in\mathbb{Z}^2}$ is an orthonormal basis of $L^2(\Xi_J)$, where $\Xi_J\supset\mathcal{W}^-_J$ is the set defined in \eqref{eq:supprect}. We estimate \begin{align*}
\Big\|\sum_{k\in K_J} \theta_{J,k} u_{J,k} W^-_J \Big\|_2^2 \le \Big\|\sum_{k\in K_J} \theta_{J,k} u_{J,k} \Big\|_{L^2(\Xi_J)}^2 = \sum_{k\in K_J} |\theta_{J,k}|^2. \end{align*}
The frame coefficients satisfy the a-priori estimate $|\theta_{J,k}|^2 \lesssim 2^{-js(1+\alpha)}$ according to \eqref{eq:apriori}. Thus we obtain \begin{align*}
\|\widehat{f}_NW^-_J\|_2^2 \asymp \Big\|\sum_{k\in K_J} \theta_{J,k} u_{J,k} W^-_J \Big\|_2^2 \lesssim (\# K_J) 2^{-js(1+\alpha)}. \end{align*}
By Lemma~\ref{lem:thetaest} we have $\| \widehat{\Theta} W^-_J\|_2^2 \gtrsim 2^{-js(2-\alpha)}$. We deduce from \eqref{eqqqqq} \begin{align*}
\|\widehat{\Theta}W^-_J - \widehat{f}_NW^-_J \|_2^2
\ge \frac{1}{2} \|\widehat{\Theta}W^-_J \|_2^2 - \|\widehat{f}_NW^-_J \|_2^2 \gtrsim 2^{-js(2-\alpha)} - (\# K_J) 2^{-js(1+\alpha)}. \end{align*}
Altogether, we conclude \begin{align*}
\|\Theta - f_N \|_2^2
\ge \sum_{J\in\mathbb{J}} \|\widehat{\Theta}W^-_J - \widehat{f}_NW^-_J \|_2^2 \gtrsim \sum_{J\in\mathbb{J}} \max\big\{0, 2^{-js(2-\alpha)} - (\# K_J) 2^{-js(1+\alpha)} \big\}. \end{align*}
Note that $\sum_{J}(\# K_J) \le N$. To derive a lower bound let us consider the following minimization problem: \begin{align*} \underset{\{N_J\}_{J\in\mathbb{J}}}{\text{\sc Minimize}} \quad \sum_{J\in\mathbb{J}} \max\{0,2^{-js(2-\alpha)} - N_J 2^{-js(1+\alpha)} \} \quad \text{ s.t.} \quad \sum_{J\in\mathbb{J}} N_J \le N,\, N_J\in[0,\infty)\,\, (J\in\mathbb{J}). \end{align*} The condition $N_J\in[0,\infty)$, which simplifies the subsequent argumentation, is possible since we are only interested in a bound.
For the optimal choice $\{N_J\}_{J}$, it necessarily holds $\sum_{J} N_J = N$ and \[ N_J \le 2^{-js(2-\alpha)} 2^{js(1+\alpha)} = 2^{js(2\alpha-1)}. \] Hence, the minimization problem can be reformulated as minimizing the term \begin{align*} \sum_{J\in\mathbb{J}} \big( 2^{-js(2-\alpha)} - N_J 2^{-js(1+\alpha)} \big) \end{align*} under the constraints $\sum_J N_J =N$ and $N_J \le 2^{js(2\alpha-1)}$. Assume that the family $\{N_J\}_J$ fulfills these constraints. Further, let $j(N)\in\mathbb{N}$ denote the number determined by the property \begin{align}\label{eq:unique} \sum_{j=0}^{j(N)-1} 2^{js(2\alpha-1)} L_j < N \le \sum_{j=0}^{j(N)} 2^{js(2\alpha-1)} L_j, \end{align} where $L_j$ from \eqref{eq:Lj} counts the wedges in the corona $\mathcal{C}_j$. Then the following estimate holds true \begin{align*}
\sum_{J\in\mathbb{J}} \Big( 2^{-js(2-\alpha)} - N_J 2^{-js(1+\alpha)} \Big) \ge \sum_{j=j(N)+1}^\infty \Big( \sum_{|J|=j} 2^{-js(2-\alpha)} \Big) \ge \sum_{j=j(N)+1}^\infty 2^{-js} \gtrsim 2^{-j(N)s}. \end{align*} To see this, note that $2^{-js(1+\alpha)}$ is decreasing with rising scale and that $L_j\ge 2^{js(1-\alpha)}$. Since $N \asymp 2^{j(N)s\alpha}$, which follows from \eqref{eq:unique}, we have proven \begin{align*}
\|\Theta - f_N \|_2^2 \gtrsim \sum_{J\in\mathbb{J}} \max\big\{0, 2^{-js(2-\alpha)} - (\# K_J) 2^{-js(1+\alpha)} \big\} \gtrsim 2^{-j(N)s} \asymp N^{-\frac{1}{\alpha}}\, \end{align*} and the proof is finished.
\end{proof}
\noindent The approximation results for $\Theta$ have direct implications for the class-wise approximation of cartoon-like functions. If $\nu\ge2$, then $\Theta\in\mathcal{E}_{bin}^{\beta}([-1,1]^2;\nu)$ for arbitrary $\beta\in[0,\infty)$. Moreover, we can always find $\gamma>0$ such that $\gamma\Theta\in\mathcal{E}^{\beta}([-1,1]^2;\nu)$. This allows to draw the following conclusion.
\begin{cor} Let $\beta\in[0,\infty)$ and $\nu\ge2$. The uniform decay of the $N$-term approximation error for $\mathcal{E}_{bin}^\beta([-1,1]^2;\nu)$ and $\mathcal{E}^\beta([-1,1]^2;\nu)$ provided by $\mathfrak{C}_{s,\alpha}$ cannot be faster than $N^{-\frac{1}{1-\alpha}}$. Futhermore, thresholding of coefficients cannot yield rates better than $N^{-\frac{1}{\max\{\alpha,1-\alpha\}}}$. \end{cor}
\noindent If $\beta>2$ it is thus impossible for $\mathfrak{C}_{s,\alpha}$ to reach the theoretically possible approximation order of $N^{-\beta}$ for the class $\mathcal{E}^\beta([-1,1]^2;\nu)$. The best performance is achieved for the classic choice $\alpha=\frac{1}{2}$, with a corresponding approximation rate of order $N^{-2}$. A smaller $\alpha$ leads to a deterioration of the approximation. As is obvious from our investigation, this behavior applies to cartoons with curved edges exemplified by the function $\Theta=\chi_{B_2(0,\frac{1}{2})}$ from \eqref{def:Theta}. For such cartoons the rate inevitably deteriorates as $\alpha$ tends to $0$, since their energy is spread more or less uniformly across all directions of the Fourier plane. In the next section, we narrow our focus and consider only cartoons with straight edges. Such cartoons are highly anisotropic and in a certain sense the opposite extreme of the isotropic function $\Theta$. Since their Fourier energy is concentrated in only one direction, a smaller $\alpha$ will be an advantage for their approximation.
\section{Images with Straight Edges} \label{sec:straight}
In the following, we investigate the approximation performance of the curvelet frame $\mathfrak{C}_{s,\alpha}$ with respect to cartoons with straight edges. To specify the associated signal class, let {\sc Straight} be the collection of all closed half-spaces of $\mathbb{R}^2$. Parameterized by $\varphi\in[0,2\pi)$ and $c\in\mathbb{R}$, these are subsets of the form \[ H(\varphi,c)= \Big\{ (x_1,x_2)\in\mathbb{R}^2 ~:~ x_1\cos(\varphi)- x_2\sin(\varphi)\ge c\, \Big\} \,. \] Using Definition~\ref{def:gencart} we then introduce the following image class with parameters $\beta\in[0,\infty)$ and $\nu>0$ \begin{align*} {\bf E}^\beta([-1,1]^2;\nu):=\mathcal{E}^\beta([-1,1]^2; \text{\sc Straight}, \nu). \end{align*} This is a subclass of the general cartoons~\eqref{eqdef:cart} considered in Section~\ref{sec:cartoon}. Indeed, for $\nu>0$ and $\tilde{\nu}\ge\nu$ chosen large enough \[ C_0^\beta([-1,1]^2;\nu) \subset {\bf E}^\beta([-1,1]^2;\nu) \subset \mathcal{E}^\beta([-1,1]^2;\tilde{\nu}), \] where $C_0^\beta([-1,1]^2;\nu)$ is the class defined in \eqref{eqdef:smoothclass}. These inclusions allow to transfer the optimality benchmark $N^{-\beta}$, valid for both $\mathcal{E}^\beta([-1,1]^2;\tilde{\nu})$ and $C_0^\beta([-1,1]^2;\nu)$ (see Theorem~\ref{thm:benchmark} and Remark~\ref{rem:benchmark}). For ${\bf E}^\beta([-1,1]^2;\nu)$, we thus again aim for an approximation rate of order $N^{-\beta}$.
Ridgelet frames were developed specifically for the optimal representation of functions with straight line singularities. For both variants, `orthonormal ridgelets'~\cite{D98} and `$0$-curvelets'~\cite{GrohsRidLT}, it has been shown that they reach the optimality bound $N^{-\beta}$. More precisely, this rate was proved for `mutilated Sobolev functions' with compact support~\cite{C99,GOtech16}, i.e., compactly supported functions which are in the Sobolev space $H^{\beta}(\mathbb{R}^2)$ apart from straight line singularities. In line with the result from \cite{GOtech16} for $0$-curvelets, we can expect that decreasing $\alpha$ improves the approximation ability of $\mathfrak{C}_{s,\alpha}$ for ${\bf E}^\beta([-1,1]^2;\nu)$.
Our main result concerning the $\alpha$-curvelet approximation of ${\bf E}^\beta([-1,1]^2;\nu)$ is Theorem~\ref{thm:mainappr1} below. It is formulated and proved for integer $\beta\in\mathbb{N}$ only, although the statement should extend to the whole range $\beta\in\mathbb{R}^+$. In this way, we avoid technical difficulties which would arise if we used finite differences instead of integer derivatives (compare~\cite{GKKScurve2014}).
\begin{theorem}\label{thm:mainappr1}
The parameters $\beta\in\mathbb{N}$, $\nu>0$, $\alpha\in[0,1)$, and $s>0$ shall be fixed.
Further, let $f_N$ be the $N$-term approximation of a signal $f\in L^2(\mathbb{R}^2)$
provided by the $N$ largest coefficients with respect to the
frame $\mathfrak{C}_{s,\alpha}=\lb\psi_\mu\rb_{\mu\in\mathbb{M}}$.
There exists a constant $C>0$ such that for every $f\in {\bf E}^{\beta}([-1,1]^2;\nu)$ and $N\in\mathbb{N}$ \begin{align*}
\|f-f_N\|_2^2\le C \begin{cases}
N^{-\beta} \log_2(1+N)^{1+\beta} \quad&\text{ if } \alpha\le\beta^{-1}, \\
N^{-1/\alpha} \quad&\text{ if } \alpha>\beta^{-1}.
\end{cases} \end{align*} \end{theorem}
\noindent As expected, decreasing the parameter $\alpha$ improves the approximation performance. If $\alpha\in[0,\beta^{-1}]$ the achieved rate is even optimal up to the log-factor. In this range signals from ${\bf E}^{\beta}([-1,1]^2;\nu)$ are represented with the same efficiency as a smooth function from $C^\beta([-1,1]^2;\nu)$.
Theorem~\ref{thm:mainappr1} is deduced by studying the curvelet coefficients, whose decay is closely related to the achieved $N$-term approximation rate. Recall that a typical measure for the sparsity of a sequence $\lb c_\lambda\rb_\lambda\subset\mathbb{C}$ is given by the weak $\ell^p$-(quasi)-norms, for $p>0$ defined by \[
\|\lb c_\lambda\rb_\lambda\|_{w\ell^p}:= \Big( \sup_{\varepsilon>0} \varepsilon^p \cdot \#\{\lambda: |c_\lambda|>\varepsilon \}\Big)^{1/p}. \]
By definition, the sequence $\lb c_\lambda\rb_\lambda$ belongs to $w\ell^p(\Lambda)$ if and only if the quantity $\|\lb c_\lambda\rb_\lambda\|_{w\ell^p}$ is finite. This is the case precisely if there exists a constant $C>0$ such that $\#\{\lambda: |c_\lambda|>\varepsilon \}\le C^p \varepsilon^{-p}$
for all $\varepsilon>0$. The smallest possible such constant then coincides with the weak $\ell^p$-(quasi)-norm of the sequence. Another useful characterization of a sequence $\lb c_\lambda\rb_\lambda\in w\ell^p(\Lambda)$ is given in terms of its non-increasing rearrangement $\lb c^*_n\rb_{n\in\mathbb{N}}$. It holds $|c^\ast_n|\lesssim n^{-1/p}$ and
$\sup_{n>0} n^{1/p}|c^\ast_n| = \|\lb c_\lambda\rb_\lambda\|_{w\ell^p}$.
As illustrated by the following well-known lemma (see e.g.\ \cite{Devore1998}), the decay of the frame coefficients determines the $N$-term approximation rate achieved by thresholding. A full proof is given e.g.\ in \cite{GKKS15}.
\begin{lemma}[{\cite[Lem.~5.1]{GKKS15}}]\label{lem:decayapprox} Let $\lb m_\lambda\rb_{\lambda\in \Lambda}$ be a frame in $L^2(\mathbb{R}^2)$ and $f=\sum c_\lambda m_\lambda$ an expansion of $f\in L^2(\mathbb{R}^2)$ with respect to this frame. If $\lb c_\lambda\rb_\lambda\in w\ell^{2/(\beta+1)}(\Lambda)$ for some $\beta\ge0$, then the $N$-term approximations $f_N$ obtained by keeping the $N$ largest coefficients satisfy \[
\| f-f_N \|_2^2 \lesssim N^{-\beta}. \] \end{lemma}
\noindent Beginning in Subsection~\ref{ssec:curvesparse}, we study the sparsity of the coefficients $\theta_\mu=\langle f,\psi_\mu \rangle$ provided by the frame $\mathfrak{C}_{s,\alpha}=\lb\psi_\mu\rb_{\mu\in\mathbb{M}}$ for a signal $f\in{\bf E}^\beta([-1,1]^2;\nu)$. The decay rates proved in Theorem~\ref{thm:mainappr2} are the foundation of the following proof of Theorem~\ref{thm:mainappr1}.
\begin{proof}[Proof of Theorem~\ref{thm:mainappr1}] If $\alpha>\beta^{-1}$ the sequence $\lb\theta_\mu\rb_{\mu\in\mathbb{M}}$ of curvelet coefficients $\theta_\mu=\langle f,\psi_\mu \rangle$ belongs to $w\ell^p(\mathbb{M})$ with $p=2/(1+1/\alpha)$. This is proved in Theorem~\ref{thm:mainappr2}. Lemma~\ref{lem:decayapprox} directly translates this into the statement of Theorem~\ref{thm:mainappr1}.
In case $\alpha\le\beta^{-1}$ Theorem~\ref{thm:mainappr2} yields $|\theta_m^*|^2 \le C m^{-(1+\beta)} (\log_2 m)^{1+\beta}$ for the curvelet coefficient $\theta_m^*$ of $m$-th largest modulus. Utilizing the frame property of $\mathfrak{C}_{s,\alpha}$ we can estimate \[
\| f-f_N \|^2 \lesssim \sum_{m>N} |\theta^\ast_m|^2 \lesssim \sum_{m>N} m^{-(1+\beta)} \cdot \left(\log_2 m\right)^{(1+\beta)} \le \int_N^\infty t^{-(1+\beta)} \cdot \left(\log_2 (1+t)\right)^{(1+\beta)} \,dt. \] Note that $N\ge1$. Partial integration leads to \begin{align*}
\int_N^\infty t^{-(1+\beta)} \cdot \left(\log_2(1+t)\right)^{(1+\beta)} \,dt
\lesssim N^{-\beta} \left(\log_2 (1+N)\right)^{(1+\beta)}
+ \int_N^\infty t^{-(1+\beta)} \cdot \left(\log_2 (1+t)\right)^{\lceil\beta\rceil} \,dt. \end{align*} We repeat this $\lceil\beta\rceil$-times and finally arrive at \begin{align*} \int_N^\infty t^{-(1+\beta)} \cdot \left(\log_2(1+t)\right)^{(1+\beta)} \,dt \lesssim N^{-\beta} \left(\log_2 (1+N)\right)^{(1+\beta)}. \tag*{\qedhere} \end{align*} \end{proof}
\subsection{Sparsity of Curvelet Coefficients} \label{ssec:curvesparse}
Subsequently, we study the decay of the curvelet coefficients $\theta_\mu=\langle f,\psi_\mu \rangle$. Our main result is Theorem~\ref{thm:mainappr2}.
\begin{theorem}\label{thm:mainappr2} Let $\alpha\in[0,1)$, $s>0$, $\beta\in\mathbb{N}$, and $\nu>0$ be fixed. Further, denote by $\theta^*_{N}$ the (in modulus) $N$-th largest coefficient of $f\in{\bf E}^\beta([-1,1]^2;\nu)$ with respect to $\mathfrak{C}_{s,\alpha}=\lb\psi_\mu\rb_{\mu\in\mathbb{M}}$. There exists a constant $C>0$ independent of $N\ge2$ such that \begin{align*}
\sup_{f\in {\bf E}^\beta([-1,1]^2;\nu)} |\theta^*_{N}|^2 \le C\cdot \begin{cases} N^{-(1+\beta)} \cdot \left(\log_2 N\right)^{1+\beta} \quad&\text{ if } \alpha\le\beta^{-1}, \\
N^{-(1+1/\alpha)} \quad&\text{ if } \alpha>\beta^{-1}.
\end{cases} \end{align*} \end{theorem}
\begin{proof}
Let $\mathbb{M}_j$ denote the subset of the curvelet index set $\mathbb{M}$ corresponding to scale $j$. Further, given $\varepsilon>0$, let us
define $\mathbb{M}_{\varepsilon}:=\Big\{ \mu\in\mathbb{M} : |\theta_\mu|>\varepsilon \Big\}$ and $\mathbb{M}_{j,\varepsilon}:=\Big\{ \mu\in\mathbb{M}_j : |\theta_\mu|>\varepsilon \Big\}$.
According to \eqref{eq:apriori} there is a constant $\widetilde{C}>0$, independent of scale, such that
\[
|\theta_\mu| \le \widetilde{C}\|f\|_{\infty} 2^{-js(1+\alpha)/2} \le \widetilde{C}\nu 2^{-js(1+\alpha)/2}.
\]
At scales $j> j_\varepsilon:=\frac{2 \log_2(\widetilde{C}\nu\varepsilon^{-1})}{s(1+\alpha)}$
the coefficients thus satisfy $|\theta_\mu|<\varepsilon$ and the sets $\mathbb{M}_{j,\varepsilon}$ are empty. In particular $\# \mathbb{M}_{\varepsilon}=0$ in case
$\varepsilon>\widetilde{C}\nu$ since then $j_\varepsilon<0$.
If $j\le j_\varepsilon$ Proposition~\ref{prop:main_sequence}, which is stated and proved below, gives the estimate
\begin{align*}
\# \mathbb{M}_{j,\varepsilon} \lesssim 2^{j\rho} \varepsilon^{-2/(1+\beta)} \quad\text{ with }\quad \rho=\frac{s \max\{\alpha\beta-1,0\}}{1+\beta}\ge 0.
\end{align*}
If $\alpha>\beta^{-1}$ we have $\rho>0$ and conclude
\begin{align*}
\# \mathbb{M}_{\varepsilon} = \sum_{j=0}^{\lfloor j_\varepsilon \rfloor} \# \mathbb{M}_{j,\varepsilon}
\lesssim \sum_{j=0}^{\lfloor j_\varepsilon \rfloor} 2^{j\rho} \varepsilon^{-2/(1+\beta)} \lesssim 2^{j_\varepsilon \rho} \varepsilon^{-2/(1+\beta)}
= \varepsilon^{-\frac{2(\alpha\beta-1)}{(1+\beta)(1+\alpha)}} \varepsilon^{-2/(1+\beta)} = \varepsilon^{-2/(1+1/\alpha)}.
\end{align*}
From here, a direct argument leads to $|\theta^*_N|^2\lesssim N^{-(1+1/\alpha)}$ for the $N$-th largest coefficient $\theta^\ast_N$.
If $\alpha\le\beta^{-1}$ we have $\rho=0$ and the estimate
\begin{align*}
\# \mathbb{M}_{\varepsilon} \lesssim \sum_{j=0}^{\lfloor j_\varepsilon \rfloor} \varepsilon^{-2/(1+\beta)} \lesssim (\log_2(\widetilde{C}\nu\varepsilon^{-1})+1) \varepsilon^{-2/(1+\beta)} = \log_2(2\widetilde{C}\nu\varepsilon^{-1}) \varepsilon^{-2/(1+\beta)}.
\end{align*}
Hence, there is a constant $C_2\ge 1$ such that $\# \mathbb{M}_{\varepsilon} \le C_2 \log_2(C_1\varepsilon^{-1}) (C_1\varepsilon^{-1} )^{2/(1+\beta)}$ with $C_1=\max\{1,2\widetilde{C}\nu\}$.
It follows $|\theta^*_{N}|\le C_1\delta_N$ for the number $\delta_N$ which solves $N=C_2 \log_2(\delta_N^{-1}) \delta_N^{-2/(1+\beta)}$.
In general $\delta_N$ cannot be calculated explicitly, wherefore we resort to an estimate.
If $N\ge 2$ we have $\varepsilon_N:=N^{-\frac{1+\beta}{2}}\le \frac{1}{2}$ since $\beta\ge1$.
Taking into account $C_2\ge1$ we conclude
\[
C_2 {\varepsilon_N}^{-2/(1+\beta)} \log_2(\varepsilon_N^{-1}) \ge N = C_2 \delta_N^{-2/(1+\beta)} \log_2(\delta_N^{-1}),
\]
which in turn proves $\delta_N\ge \varepsilon_N=N^{-\frac{1+\beta}{2}}$. Therefore $\widetilde{\delta}_N\ge\delta_N$ for the solution $\widetilde{\delta}_N$ of
\[
N= C_2\widetilde{\delta}_N^{-2/(1+\beta)} \log_2(N^{\frac{1+\beta}{2}}).
\]
An explicit calculation yields
$
\widetilde{\delta}_N =(C_2\frac{1+\beta}{2})^{(1+\beta)/2} N^{-(1+\beta)/2} (\log_2 N)^{(1+\beta)/2},
$
which proves the claim.
\end{proof}
\noindent The missing ingredient in the proof of Theorem~\ref{thm:mainappr2} is Proposition~\ref{prop:main_sequence}.
\begin{prop}\label{prop:main_sequence} Let the parameters $\alpha\in[0,1)$, $s>0$, $\beta\in\mathbb{N}$, and $\nu>0$ be fixed. Further, let $\mathbb{M}_j$ denote the curvelet indices at scale $j$. The sequence $\lb\theta_\mu\rb_{\mu\in\mathbb{M}_j}$ of coefficients $\theta_\mu=\langle f,\psi_\mu\rangle$ obeys \[
\| \lb\theta_\mu\rb_{\mu\in\mathbb{M}_j} \|^{2/(1+\beta)}_{w\ell^{2/(1+\beta)}}\lesssim 2^{j\rho} \quad\text{with}\quad \rho=s\max\{\alpha\beta-1,0\}/(1+\beta) \] and an implicit constant independent of scale $j\in\mathbb{N}_0$ and $f\in{\bf E}^\beta([-1,1]^2;\nu)$. \end{prop}
\noindent For the proof of Proposition~\ref{prop:main_sequence} we decompose $f$ into fragments, a technique pioneered in \cite{CD04}. To this end, let $\mathcal{Q}_j$ at every scale $j\in\mathbb{N}_0$ denote the collection of cubes \[ Q:=Q^{(j)}_{(k_1,k_2)}:=[2^{-js\alpha}(k_1-1),2^{-js\alpha}(k_1+1)]\times[2^{-js\alpha}(k_2-1),2^{-js\alpha}(k_2+1)], \quad (k_1,k_2)\in\mathbb{Z}^2. \] Further, let $\omega\in C_0^\infty([-1,1]^2)$ be a nonnegative window vanishing outside the square $[-1,1]^2$, such that the family $\{\omega_Q\}_{Q\in\mathcal{Q}_j}$ of functions $\omega_Q(x):=\omega(2^{js\alpha}x_1-k_1, 2^{js\alpha}x_2-k_2)$ is a partition of unity, i.e., it has the property $\sum_{Q\in\mathcal{Q}_j} \omega_Q=1$. Following \cite{CD04} we then decompose $f=\sum_{Q} f_Q$ into the fragments \begin{align}\label{eqdef:frag} f_Q:=f \omega_Q \quad, Q\in\mathcal{Q}_j. \end{align} Note that ${\text{\rm supp }} f_Q\subseteq Q$ and that the size of the squares $Q\in \mathcal{Q}_j$ corresponds to the `essential' length of the curvelets at scale $j$. Therefore $\langle f,\psi_\mu \rangle \approx \langle f_Q,\psi_\mu \rangle$ for a curvelet $\psi_\mu$ at the location of the cube $Q$.
For every $Q\in\mathcal{Q}_j$ we now investigate the sparsity of the sequence \begin{align}\label{eq:Qsequence} \theta_{Q}:=\lb\langle f_Q, \psi_\mu\rangle\rb_{\mu\in\mathbb{M}_j}\,. \end{align} Clearly, due to ${\text{\rm supp }} f\subseteq [-1,1]^2$ we only need to consider cubes $Q\in \mathcal{Q}_j$ which meet the square $[-1,1]^2$. Of these relevant cubes, let us collect those which intersect the straight edge in $\mathcal{Q}_j^{1}$, the others in $\mathcal{Q}_j^{0}$. The associated fragments $f_Q$ will be called \emph{edge fragments} and \emph{smooth fragments}, respectively. The main result concerning the sparsity of \eqref{eq:Qsequence} is Proposition~\ref{prop:frag}.
\begin{prop}\label{prop:frag} Let $\alpha\in[0,1)$, $s>0$, $\beta\in\mathbb{N}$, and $\nu>0$ be fixed. Let $Q\in\mathcal{Q}_j$, $j\in\mathbb{N}_0$, be a square and $\theta_Q$ the curvelet coefficient sequence of the fragment $f_Q=f\omega_Q$ defined in \eqref{eq:Qsequence}. There is a constant $C>0$ independent of $j\in\mathbb{N}_0$ and $Q\in\mathcal{Q}_j$ such that for all $f\in{\bf E}^\beta([-1,1]^2;\nu)$ the following estimates hold true. \begin{enumerate} \item[(i)] If $Q\in\mathcal{Q}_j^{0}$ the sequence $\theta_Q$ satifies
$\| \theta_Q \|^{2/(1+\beta)}_{w\ell^{2/(1+\beta)}}\le C \cdot 2^{-2js\alpha}$. \item[(ii)] If $Q\in\mathcal{Q}_j^{1}$ the sequence $\theta_Q$ satisfies
$\| \theta_Q \|^{2/(1+\beta)}_{w\ell^{2/(1+\beta)}}\le C \cdot 2^{-js\alpha} 2^{j\rho}$ with $\rho=s\max\{\alpha\beta-1,0\}/(1+\beta)$.
\end{enumerate} \end{prop}
\noindent A direct consequence of Proposition~\ref{prop:frag}, whose proof is given later on, is Proposition~\ref{prop:main_sequence}.
\begin{proof}[Proof of Proposition~\ref{prop:main_sequence}] We have the decomposition $\left\lb\theta_\mu\right\rb_{\mu\in\mathbb{M}_j}=\sum_{Q\in\mathcal{Q}_j} \theta_Q$. Since $0<2/(1+\beta)\le1$, the $p$-triangle inequality with $p=2/(1+\beta)$ yields \begin{align*}
\|\left\lb\theta_\mu\right\rb_{\mu\in\mathbb{M}_j}\|^{2/(1+\beta)}_{w\ell^{2/(1+\beta)}} \le \sum_{Q\in\mathcal{Q}_j} \|\theta_Q\|^{2/(1+\beta)}_{w\ell^{2/(1+\beta)}}
\le \big(\#\mathcal{Q}_j^{0}\big) \cdot \sup_{Q\in\mathcal{Q}_j^{0}} \|\theta_Q\|^{2/(1+\beta)}_{w\ell^{2/(1+\beta)}}
+ \big(\#\mathcal{Q}_j^{1}\big) \cdot \sup_{Q\in\mathcal{Q}_j^{1}} \|\theta_Q\|^{2/(1+\beta)}_{w\ell^{2/(1+\beta)}}. \end{align*} Since $f$ is supported in $[-1,1]^2$, there are constants $C_0,C_1>0$, independent of scale, such that \begin{align*} \#\mathcal{Q}_j^{0}\le C_{0} 2^{2js\alpha} \quad\text{and}\quad \#\mathcal{Q}_j^{1}\le C_{1} 2^{js\alpha}. \end{align*} Utilizing the estimates of Proposition~\ref{prop:frag}, we thus obtain with $\rho=s\max\{\alpha\beta-1,0\}/(1+\beta)\ge 0$ \begin{align*}
\|\left\lb\theta_\mu\right\rb_{\mu\in\mathbb{M}_j}\|^{2/(1+\beta)}_{w\ell^{2/(1+\beta)}} \lesssim C_0 + C_1 2^{j\rho} \lesssim 2^{j\rho}. \tag*{\qedhere} \end{align*} \end{proof}
\noindent In the remainder of this section we are concerned with the proof of Proposition~\ref{prop:frag}. Hereby, we restrict to functions $f\in{\bf E}^\beta([-1,1]^2;\nu)$ of the simple form \begin{align}\label{eqdef:cartsimp} f=g \chi_{H(\varphi,c)} \end{align} with $g\in C_0^\beta([-1,1]^2,\nu)$ and $H(\varphi,c)\in\text{\sc Straight}$ a half-space determined by $\varphi\in[0,2\pi)$ and $c\in\mathbb{R}$.
Note that for a general cartoon $f=f_{1}+f_{2}\chi_{H(\varphi,c)}$ both components $\widetilde{f}_{1}:=f_{1}$ and $\widetilde{f}_{2}:=f_{2}\chi_{H(\varphi,c)}$ have the form~\eqref{eqdef:cartsimp}, due to the representation $f_{1}=f_{1}\chi_{H(0,-1)}$.
Hence, if the estimates of Proposition~\ref{prop:frag} are proven for elements of type~\eqref{eqdef:cartsimp}, they are then also true for all $f\in{\bf E}^\beta([-1,1]^2;\nu)$. This is a consequence of the estimate $2^{-2js\alpha} \le 2^{-js\alpha} 2^{j\rho}$ and \[
\|\theta_Q \|^{2/(1+\beta)}_{w\ell^{2/(1+\beta)}} \le \| \lb\langle \widetilde{f}_{1}\omega_Q, \psi_{\mu} \rangle\rb_{\mu\in\mathbb{M}_j} \|^{2/(1+\beta)}_{w\ell^{2/(1+\beta)}} + \|\lb\langle \widetilde{f}_{2}\omega_Q, \psi_\mu\rangle\rb_{\mu\in\mathbb{M}_j} \|^{2/(1+\beta)}_{w\ell^{2/(1+\beta)}}. \]
Let $Q\in\mathcal{Q}_j$ be a cube at scale $j\in\mathbb{N}_0$ with center $M_Q:=2^{-js\alpha}(k_1,k_2)\in\mathbb{R}^2$, which nontrivially intersects the cartoon domain $[-1,1]^2$. If $Q\in\mathcal{Q}_j^{0}$ we put $P_Q:=M_Q$. If $Q\in\mathcal{Q}_j^{1}$ let us fix a point $P_Q\in Q$ on the edge curve $\{(x_1,x_2)\in\mathbb{R}^2 : x_1\cos(\varphi) - x_2\sin(\varphi)=c\}$ of the cartoon such that $\chi_{H(\varphi,c)}=H(R_{\varphi}(x-P_Q))$, with rotation matrix \eqref{eq:matrixrot} and where \begin{align}\label{eq:bivstep} H:=\mathfrak{H} \otimes 1 \quad\text{with the Heaviside function}\quad \mathfrak{H}(t)=\begin{cases} 0\,, \quad& \text{if }t<0, \\ 1\,, & \text{if }t\ge 0.\end{cases} \end{align} Putting $\widetilde{g}_Q(x):=g(R^{-1}_\varphi x + P_Q)$ and $\widetilde{\omega}_Q(x):=\omega_Q(R^{-1}_\varphi x + P_Q)$, the fragment $f_Q$ can then be written as $f_Q(x)=f^{\raisebox{-0.45em}[0.4mm][0.4mm]{\textasciitilde}}_Q\big(R_{\varphi} ( x - P_Q )\big)$ with a function $f^{\raisebox{-0.45em}[0.4mm][0.4mm]{\textasciitilde}}_Q$ of the form \begin{align}\label{eq:stdedge} (i) \quad f^{\raisebox{-0.45em}[0.4mm][0.4mm]{\textasciitilde}}_Q:=\widetilde{g}_Q\widetilde{\omega}_Q \,,\quad \text{if $Q\in \mathcal{Q}_j^{0} $}, \qquad\quad\text{or} \qquad\quad (ii) \quad f^{\raisebox{-0.45em}[0.4mm][0.4mm]{\textasciitilde}}_Q:=\widetilde{g}_Q \widetilde{\omega}_Q H \,,\quad \text{if $Q\in \mathcal{Q}_j^{1}$}. \end{align}
On the Fourier side we have \begin{align*} \widehat{f}_Q(\xi)=\widehat{f^{\raisebox{-0.45em}[0.4mm][0.4mm]{\textasciitilde}}_Q}(R_{\varphi} \xi) \exp\big(-2\pi i\langle P_Q , \xi \rangle\big). \end{align*}
Now, let $\psi_\mu=\psi_{j,\ell,k}\in\mathfrak{C}_{s,\alpha}$ be a fixed curvelet and recall $ \widehat{\psi}_{j,\ell,k}=W_{j,\ell} u_{j,\ell,k} $ with the real-valued wedge functions $W_{j,\ell}(\cdot)=W_{j,0}(R_{j,\ell}\cdot)$ from \eqref{eq:suppfunctions} and the functions \[ u_{j,\ell,k}(\cdot)=2^{-js(1+\alpha)/2} \exp(2\pi i \langle R^{-1}_{j,\ell}A^{-1}_jk , \cdot \rangle ). \]
There are unique $k_{\bullet}\in\mathbb{Z}^2$ and $\Delta k\in[0,1)^2$ such that $P_Q=R^{-1}_{j,\ell}A^{-1}_{j}(k_{\bullet}+\Delta k)$. Further, we can express $\varphi$ as a `fractional multiple' of the angle $\varphi_j$ defined in \eqref{eqdef:fundangle}, writing $\varphi=(\ell_{\bullet} - \Delta\ell) \varphi_j$ with unique $\ell_{\bullet}\in\mathbb{Z}$ and $\Delta\ell\in[0,1)$.
It follows for the curvelet coefficient $\langle f_Q, \psi_{j,\ell,k}\rangle=\langle \widehat{f}_Q, \widehat{\psi}_{j,\ell,k}\rangle$ \begin{align*} \langle f_Q, \psi_{j,\ell,k}\rangle &= \int_{\mathbb{R}^2} \widehat{f^{\raisebox{-0.45em}[0.4mm][0.4mm]{\textasciitilde}}_Q}\big(R_{j,\ell_{\bullet} - \Delta\ell} \xi\big) \exp\big(-2\pi i\langle R^{-1}_{j,\ell}A^{-1}_{j}(k_{\bullet}+\Delta k) , \xi \rangle\big) W_{j,\ell}(\xi) \overline{u_{j,\ell,k}(\xi)} \,d\xi \\ &= \int_{\mathbb{R}^2} \widehat{f^{\raisebox{-0.45em}[0.4mm][0.4mm]{\textasciitilde}}_Q}(\xi) W_{j,\ell-\ell_{\bullet}+\Delta\ell}(\xi) \overline{u_{j,\ell-\ell_{\bullet}+\Delta\ell,k+k_{\bullet}+\Delta k_{\bullet}}(\xi)} \,d\xi. \end{align*} Relabelling the indices $({\bf l},{\bf k}):=([\ell-\ell_{\bullet}],k+k_{\bullet})$, where $[\ell-\ell_{\bullet}]\in\{-L_j^-,\ldots,L_j^+\}$ is the unique number obtained by shifting $\ell-\ell_{\bullet}\in\mathbb{Z}$ by integer multiples of $L_j=\pi \varphi^{-1}_j$ (see \eqref{eq:Lj}), we can write \begin{align}\label{eq:reducedform} \langle f_Q, \psi_{j,\ell,k}\rangle &= \int_{\mathbb{R}^2} \widehat{f^{\raisebox{-0.45em}[0.4mm][0.4mm]{\textasciitilde}}_Q}(\xi) W_{j,{\bf l}+\Delta\ell}(\xi) \overline{u_{j,{\bf l}+\Delta\ell,{\bf k}+\Delta k}(\xi)} \,d\xi. \end{align} To estimate the integral~\eqref{eq:reducedform} we need knowledge about the Fourier localization of the functions $f^{\raisebox{-0.45em}[0.4mm][0.4mm]{\textasciitilde}}_Q$. This investigation is carried out in the next two subsections.
\subsection{Fourier Analysis of Standard Fragments} \label{ssec:FourierStd}
The Fourier analysis of the functions $f^{\raisebox{-0.45em}[0.4mm][0.4mm]{\textasciitilde}}_Q$, $Q\in\mathcal{Q}_j$, from \eqref{eq:stdedge} is conducted in a generic setting, independent of the concrete cube $Q$. We assume $\alpha\in[0,1)$, $\beta\in\mathbb{N}_0$, and let $\kappa,\nu,\tilde{\nu}>0$ be fixed parameters. Then we consider functions $f_j$, $j\in\mathbb{N}_0$, called standard fragments, defined by \begin{align}\label{eq:stdfrags} (i) \quad f_j:=g \omega_j \;, \qquad\text{or}\qquad (ii) \quad f_j:=g \omega_j H \,, \end{align} where $H$ is the step function \eqref{eq:bivstep}, $g\in C_0^\beta(\kappa[-1,1]^2,\nu)$ and $\omega_j:=\omega(2^{js\alpha}\cdot)$ with $\omega\in C^\infty(\mathbb{R}^2) \cap C_0^\beta(\kappa[-1,1]^2,\tilde{\nu})$. For every $Q\in\mathcal{Q}_j$ the corresponding fragment $f^{\raisebox{-0.45em}[0.4mm][0.4mm]{\textasciitilde}}_Q$ is of the form \eqref{eq:stdfrags} with specific functions $g$ and $\omega$, namely $g=\widetilde{g}_Q$ and $\omega=\widetilde{\omega}_Q(2^{-js\alpha}\cdot)$ (compare to \eqref{eq:stdedge}). Note that the parameters $\kappa,\nu,\tilde{\nu}>0$ can be chosen simultaneously for all $Q\in\mathcal{Q}_j$, e.g.\ $\kappa=2\sqrt{2}$, and $\tilde{\nu},\,\nu>0$ chosen suitably depending solely on $f\in{\bf E}^\beta([-1,1]^2;\nu)$ and the partition of unity $\lb \omega_Q \rb_Q$ utilized in \eqref{eqdef:frag}. Since the results of this subsection are valid uniformly for all choices of $g$ and $\omega$, as long as they fulfill the specifications in accordance with $\kappa,\nu,\tilde{\nu}>0$, they hence apply to all fragments $f^{\raisebox{-0.45em}[0.4mm][0.4mm]{\textasciitilde}}_Q$.
The investigation starts with an elementary lemma, where $\mathcal{I}_j$, $j\in\mathbb{N}_0$, denote the dyadic intervals introduced in \eqref{eqdef:dyintervals}.
\begin{lemma}\label{lem:bas1ic} Let $s>0$ be fixed and for $j\in\mathbb{N}_0$ let $f_j$ be fragments of the form~\eqref{eq:stdfrags}. Then there exists a constant $C>0$ independent of $j\in\mathbb{N}_0$ and the concrete choice of the functions $g$ and $\omega$ in~\eqref{eq:stdfrags} such that for every $p\in\mathbb{N}_0$ and $\varphi\in [-\pi,\pi)$ \begin{align*}
\int_{\mathcal{I}_{p}} |\widehat{f_j}(r,\varphi)|^2 \,dr \le C \varepsilon_{j,p}^2(\varphi) 2^{-ps} 2^{-2js\alpha} \|g\|^2_{\infty} \|\omega\|_2^2 \end{align*} with functions $\varepsilon_{j,p}:[-\pi,\pi)\rightarrow \mathbb{R}$ satisfying $\sum_{p\in\mathbb{N}_0} \int_{-\pi}^{\pi} \varepsilon_{j,p}^2(\varphi) \,d\varphi\le 1$. \end{lemma}
\begin{proof}
Let us assume $\|g\|_{\infty}\neq0$ and $\|\omega\|_{\infty}\neq0$, otherwise the proof is trivial. Since for every $p,j\in\mathbb{N}_0$ and $\varphi\in [-\pi,\pi)$ \[
I_{j,p}(\varphi):=\int_{\mathcal{I}_{p}} |\widehat{f_j}(r,\varphi)|^2 \,dr < \infty \] we can define functions $\epsilon_{j,p}:[-\pi,\pi)\rightarrow \mathbb{R}$ via
$\epsilon^2_{j,p}(\varphi):= I_{j,p}(\varphi) 2^{ps} 2^{2js\alpha} \|g\|^{-2}_{\infty} \|\omega\|^{-2}_2$. Then \[
I_{j,p}(\varphi)= \epsilon_{j,p}^2(\varphi) 2^{-ps} 2^{-2js\alpha} \|g\|^{2}_{\infty} \|\omega\|^{2}_2. \] Let us prove that there is a constant $C>0$, independent of the relevant parameters, such that \begin{align}\label{eq:sumboundC} \sum_{p\in\mathbb{N}_0} \int_{-\pi}^{\pi} \epsilon^2_{j,p}(\varphi) \,d\varphi \le C. \end{align} We put $f^{\raisebox{-0.45em}[0.4mm][0.4mm]{\textasciitilde}}_j=f_j(2^{-js\alpha}\cdot)$. Then $\widehat{f_j}= 2^{-2js\alpha} \widehat{f^{\raisebox{-0.45em}[0.4mm][0.4mm]{\textasciitilde}}_j}(2^{-js\alpha}\cdot)$ and it follows for $p\in\mathbb{N}_0$ \[
\|\widehat{f_j}\|^2_{L^2(\mathcal{C}_p)} = 2^{-2js\alpha} \|\widehat{f^{\raisebox{-0.45em}[0.4mm][0.4mm]{\textasciitilde}}_j}\|^2_{L^2(2^{-js\alpha}\mathcal{C}_p)}, \] where $\mathcal{C}_p$ are the coronae defined in \eqref{eqdefcorC}.
We conclude \begin{align*}
\|g\|^{2}_{\infty} \|\omega\|^{2}_2 \sum_{p\in\mathbb{N}_0} \int_{-\pi}^{\pi} \epsilon^2_{j,p}(\varphi) \,d\varphi &= \sum_{p\in\mathbb{N}_0} 2^{2js\alpha} \int_{-\pi}^{\pi} I_{j,p}(\varphi) 2^{ps}\,d\varphi \\
&\asymp \sum_{p\in\mathbb{N}_0} 2^{2js\alpha} \|\widehat{f_j}\|^2_{L^2(\mathcal{C}_p)}
= \sum_{p\in\mathbb{N}_0} \|\widehat{f^{\raisebox{-0.45em}[0.4mm][0.4mm]{\textasciitilde}}_j}\|^2_{L^2(2^{-js\alpha}\mathcal{C}_p)} \asymp \|\widehat{f^{\raisebox{-0.45em}[0.4mm][0.4mm]{\textasciitilde}}_j}\|^2_{2} = \|f^{\raisebox{-0.45em}[0.4mm][0.4mm]{\textasciitilde}}_j\|^2_{2}. \end{align*}
Using $\|f^{\raisebox{-0.45em}[0.4mm][0.4mm]{\textasciitilde}}_j\|_{2}\le \| g(2^{-js\alpha}\cdot)\|_{\infty} \|\omega\|_2 = \|g\|_{\infty} \|\omega\|_2 $ we arrive at \eqref{eq:sumboundC}. Finally, note that the functions $\varepsilon_{j,p}:=C^{-1/2} \epsilon_{j,p}$ have properties as desired.
\end{proof}
\noindent An immediate consequence of Lemma~\ref{lem:bas1ic} is the following corollary, with particular choice $j=p$.
\begin{cor}\label{cor:1} Let $s>0$ be fixed and assume that $f_j$, $j\in\mathbb{N}_0$, are fragments of the form \eqref{eq:stdfrags}. There exist functions $\varepsilon_{j}:[-\pi,\pi)\rightarrow \mathbb{R}$, each with the property $\int_{-\pi}^{\pi} \varepsilon_{j}^2(\varphi) \,d\varphi\le 1$, and a constant $C>0$ such that for every $j\in\mathbb{N}_0$ and $\varphi\in [-\pi,\pi)$ \begin{align*}
\int_{\mathcal{I}_{j}} |\widehat{f_j}(r,\varphi)|^2 \,dr \le C\varepsilon_{j}^2(\varphi) 2^{-js} 2^{-2js\alpha} \|g\|^2_{\infty} \|\omega\|_2^2. \end{align*} Moreover, the constant $C$ can be chosen independent of the functions $\omega$ and $g$. \end{cor} \begin{proof} The functions $\varepsilon_{j}:=\varepsilon_{j,j}$ obtained from Lemma~\ref{lem:bas1ic} by choosing $p=j$ have the desired properties. In particular they satisfy $\int_{-\pi}^{\pi} \varepsilon^2_{j}(\varphi) \,d\varphi \le 1$ for every $j\in\mathbb{N}_0$. \end{proof}
\noindent Note, that the smoothness of $f_j$ did not enter the proofs of the previous two results. By incorporating smoothness information we can strengthen Corollary~\ref{cor:1} for a smooth fragment of the form (i) in \eqref{eq:stdfrags}.
\begin{lemma}\label{lem:bas2ic} Let $s>0$, $\alpha\in[0,1)$, and put $\gamma=\lceil 1/(1-\alpha)\rceil$. For $j\in\mathbb{N}_0$ let $f_j$ be a smooth fragment of the form (i) in~\eqref{eq:stdfrags} with regularity $C^\beta$, $\beta\in\mathbb{N}_0$. Then there exist functions $\varepsilon_{j}:[-\pi,\pi)\rightarrow \mathbb{R}$ and a constant $C>0$ such that for every $j\in\mathbb{N}_0$ and $\varphi\in [-\pi,\pi)$ \begin{align*}
\int_{\mathcal{I}_j} |\widehat{f_j}(r,\varphi)|^2 \,dr \le C \varepsilon_{j}^2(\varphi) 2^{-js} 2^{-2js\alpha} 2^{-2js\beta} \|g\|^2_{\beta,\infty} \|\omega\|_{\beta,2}^2 \end{align*} with $\int_{-\pi}^{\pi} \varepsilon_{j}^2(\varphi) \,d\varphi\le 1$ for every $j\in\mathbb{N}_0$. The constant $C$ can be chosen independent of $\omega$ and $g$. \end{lemma} \begin{proof} If $\beta=0$ the assertion is given by Corollary~\ref{cor:1}. For $\beta\ge1$ the statement is proved by induction on $\beta$, whereby we restrict our considerations to $j\ge 1$ since for $j=0$ the asserted estimate is clearly true, also due to Corollary~\ref{cor:1}.
For fixed angle $\varphi\in [-\pi,\pi)$ let $\partial_r$ denote the radial derivative in the corresponding direction. Put $\widetilde{g}:=\partial_r g$, $\widetilde{\omega}:=\partial_r \omega$, and $\widetilde{\omega}_j:=\widetilde{\omega}(2^{js\alpha}\cdot)$. Then $\partial_rf_j(\cdot,\varphi)= \widetilde{g} \omega_j + 2^{js\alpha} g \widetilde{\omega}_j$ and we conclude for $j\in\mathbb{N}$ \begin{align*}
2^{2js} \int_{\mathcal{I}_{j}} |\widehat{f_j}(r,\varphi)|^2 \,dr &\asymp \int_{\mathcal{I}_{j}} |r \widehat{f_j}(r,\varphi)|^2 \,dr
\lesssim \int_{\mathcal{I}_{j}} |\widehat{\partial_r f_j}(r,\varphi)|^2 \,dr \\
& \asymp \int_{\mathcal{I}_{j}} |\widehat{\widetilde{g} \omega_j}(r,\varphi)|^2 \,dr + 2^{2js\alpha} \int_{\mathcal{I}_{j}} |\widehat{g \widetilde{\omega}_j}(r,\varphi)|^2 \,dr
=: \widetilde{I}_j^{(0)}(\varphi) + 2^{2js\alpha} I_j^{(1)}(\varphi). \end{align*} Hence, we get \[
I^{(0)}_{j}(\varphi):=\int_{\mathcal{I}_{j}} |\widehat{f_j}(r,\varphi)|^2 \,dr \lesssim 2^{-2js} \widetilde{I}_j^{(0)}(\varphi) + 2^{-2js(1-\alpha)} I_j^{(1)}(\varphi). \] The integral $I_j^{(1)}(\varphi)$ can be estimated in the same way as $I_j^{(0)}(\varphi)$. After $\gamma=\lceil 1/(1-\alpha)\rceil$ iterations we end up with $\widetilde{I}_j^{(0)}(\varphi)$, $\ldots$, $\widetilde{I}_j^{(\gamma-1)}(\varphi)$, and $\widetilde{I}_j^{(\gamma)}(\varphi):=I_j^{(\gamma)}(\varphi)$. Since $\gamma\ge 1/(1-\alpha)$ it holds \begin{align*} I^{(0)}_j(\varphi) \lesssim 2^{-2js} \sum_{k=0}^{\gamma-1} 2^{-2js(1-\alpha)k} \widetilde{I}_j^{(k)}(\varphi) + 2^{-2js(1-\alpha)\gamma} \widetilde{I}_j^{(\gamma)}(\varphi) \le 2^{-2js} \sum_{k=0}^{\gamma} \widetilde{I}_j^{(k)}(\varphi). \end{align*} Note that $g\in C_0^{\beta-1}([-\kappa,\kappa]^2)$ and $\widetilde{g}\in C_0^{\beta-1}([-\kappa,\kappa]^2)$, with $\kappa$ the fixed parameter from~\eqref{eq:stdfrags}. Using the induction hypothesis, the expressions $\widetilde{I}_j^{(k)}$ can be estimated with corresponding functions $\varepsilon^{(k)}_j: [-\pi,\pi)\to\mathbb{R}$. Putting $ \varepsilon_j:= \sum_{k=0}^{\gamma} \varepsilon^{(k)}_j $ yields the desired result.
\end{proof}
\noindent Our next goal is to estimate the energy of $\widehat{f_j}$ contained in wedges $\mathcal{W}^+_J$ of the form~\eqref{eq:wedgePJ}. However, we allow more general scale-angle pairs $J=(j,\ell)\in\mathbb{J}_{\scriptscriptstyle{+}}$ from the set \[ \mathbb{J}_{\scriptscriptstyle{+}}:=\big\{ (j,\ell) ~:~ j\in\mathbb{N}_0,\,\ell\in[-L^-_j,L^+_j+1) \big\}. \] The associated orientations, given by $\phiJ=\ell \varphi_j$ with $\varphi_j=\pi 2^{-\lfloor js(1-\alpha)\rfloor-1}$ fixed as in \eqref{eqdef:fundangle}, then comprise the whole interval $[-\frac{\pi}{2},\frac{\pi}{2})$. To formulate the next result we need the quantities \begin{align}\label{eqdef:AJ} A_J:=\frac{1}{2} \int_{\mathcal{A}_J} \varepsilon_{j}^2(\varphi) \,d\varphi, \quad J\in\mathbb{J}_{\scriptscriptstyle{+}}, \end{align} corresponding to angular intervals $\mathcal{A}_J$ given as in \eqref{eq:angularsupp} and the functions $\varepsilon_{j}:[-\pi,\pi)\rightarrow\mathbb{R}$ associated to $f_j$ from Corollary~\ref{cor:1}.
\begin{lemma}\label{lem:wedgesmooth} Let $(m_1,m_2)\in\mathbb{N}_0^2$ be fixed and assume that $f_j$ is of the form \eqref{eq:stdfrags}. Further, for $J\in\mathbb{J}_{\scriptscriptstyle{+}}$ let $A_J$ be the value defined in \eqref{eqdef:AJ}. Then \begin{align*}
\|\partial^{(m_1,m_2)}\widehat{f_j}\|^2_{L^2(\mathcal{W}^+_J)} \lesssim A_J 2^{-2j(m_1+m_2)s\alpha}2^{-2js\alpha} \|g\|^2_{\infty} \|\omega\|^2_2, \end{align*} with an implicit constant independent of $J\in\mathbb{J}_{\scriptscriptstyle{+}}$ and the functions $g$ and $\omega$. \end{lemma} \begin{proof} Using Corollary~\ref{cor:1} we calculate (in the nontrivial case when $g\neq 0$ and $\omega\neq0$) \begin{align*}
\|g\|^{-2}_{\infty} \|\omega\|^{-2}_2 \int_{\mathcal{W}^+_J} |\widehat{f}_{j}(\xi)|^2 \,d\xi
= \int_{\mathcal{I}_j} \int_{\mathcal{A}_J} |\widehat{f}_{j}(r,\varphi)|^2 r \,d\varphi\,dr \lesssim 2^{-2js\alpha} \int_{\mathcal{A}_J} \varepsilon_j^2(\varphi) \,d\varphi \asymp A_J 2^{-2js\alpha} . \end{align*} This proves the assertion for $(m_1,m_2)=(0,0)$.
If $m=(m_1,m_2)\neq(0,0)$ we define a new window $\tilde{\omega}(x):= x^m \omega(x)$ and put $\tilde{\omega}_j(x):=\tilde{\omega}(2^{js\alpha}x)$ for $x\in\mathbb{R}^2$. Then \[ x^m \omega_j(x)= 2^{-js\alpha(m_1+m_2)} \tilde{\omega}(2^{js\alpha}x) = 2^{-js\alpha(m_1+m_2)} \tilde{\omega}_j(x)\,,\quad x\in\mathbb{R}^2. \]
Introducing the function $f^{\raisebox{-0.45em}[0.4mm][0.4mm]{\textasciitilde}}_j:=g \tilde{\omega}_j H$ (or in case of a smooth fragment $f^{\raisebox{-0.45em}[0.4mm][0.4mm]{\textasciitilde}}_j:=g \tilde{\omega}_j$) we can write \begin{align*}
\int_{\mathcal{W}^+_J} |\partial^{(m_1,m_2)}\widehat{f_j}(\xi)|^2 \,d\xi \asymp \int_{\mathcal{W}^+_J} |\widehat{x^m f_j}(\xi)|^2 \,d\xi
= 2^{-2js\alpha(m_1+m_2)} \int_{\mathcal{W}^+_J} |\widehat{f^{\raisebox{-0.45em}[0.4mm][0.4mm]{\textasciitilde}}_j}(\xi)|^2 \,d\xi . \end{align*}
Since $f^{\raisebox{-0.45em}[0.4mm][0.4mm]{\textasciitilde}}_j$ is of the form \eqref{eq:stdfrags}, the integral on the right-hand side can be estimated as above with Corollary~\ref{cor:1}. The proof is finished since $\|\tilde{\omega}\|_2\lesssim \|\omega\|_2 $. \end{proof}
\noindent For the smooth fragments we can improve this result, taking into account smoothness information.
\begin{lemma}\label{lem:wedgenonsmooth} Let $s>0$, $\alpha\in[0,1)$, and $\gamma=\lceil 1/(1-\alpha)\rceil$. For $j\in\mathbb{N}_0$ let $f_j$ be a smooth fragment of the form (i) in~\eqref{eq:stdfrags} with regularity $C^\beta$, $\beta\in\mathbb{N}_0$. Let $J=(j,\ell)\in\mathbb{J}_{\scriptscriptstyle{+}}$ be a scale-angle pair, $A_J$ be given as in \eqref{eqdef:AJ}. For $(m_1,m_2)\in\mathbb{N}_0^2$ \begin{align*}
\|\partial^{(m_1,m_2)}\widehat{f_j}\|^2_{L^2(\mathcal{W}^+_J)} \lesssim A_J 2^{-2j(m_1+m_2)s\alpha}2^{-2js\alpha} 2^{-2js\beta} \|g\|^2_{\beta,\infty} \|\omega\|^2_{\beta,2}. \end{align*} \end{lemma} \begin{proof} The proof is analogous to Lemma~\ref{lem:wedgesmooth}, using Lemma~\ref{lem:bas2ic} instead of Corollary~\ref{cor:1}. \end{proof}
\noindent To formulate the main result of this subsection we need the differential operator \begin{align}\label{eq:diffop1} \mathcal{L}_{J,1}&:=(\textsl{Id} - 2^{2js\alpha}\mathcal{D}_{J,1}^2)(\textsl{Id} - 2^{2js\alpha}\mathcal{D}_{J,2}^2), \end{align} where $\textsl{Id}$ is the identity and the partial derivatives $\mathcal{D}_{J,1}$ and $\mathcal{D}_{J,2}$, dependent on $J\in\mathbb{J}_{\scriptscriptstyle{+}}$, are given by \begin{align}\label{eq:diffops} \mathcal{D}_{J,1}:=\cos(\phiJ)\partial_1 + \sin(\phiJ)\partial_2 \quad\text{and}\quad \mathcal{D}_{J,2}:=-\sin(\phiJ)\partial_1 + \cos(\phiJ)\partial_2. \end{align}
Recall that $\phiJ=\ell\varphi_j$ with $\varphi_j$ as in~\eqref{eqdef:fundangle}. Further, recall the functions $W_J$ from~\eqref{eq:suppfunctions} with ${\text{\rm supp }} W_J\subseteq \mathcal{W}^+_J$.
\begin{prop}\label{prop:fundament1} Let $\mathcal{L}_{J,1}$ be the differential operator~\eqref{eq:diffop1} and let $d\in\mathbb{N}_0$ be arbitrary but fixed. \begin{enumerate} \item[(i)] An edge fragment $f_j$ of the form (ii) in \eqref{eq:stdfrags} satisfies the estimate \begin{align*}
\int_{\mathbb{R}^2} |\mathcal{L}^d_{J,1}(\widehat{f}_{j}W_J)(\xi)|^2 \,d\xi \lesssim A_J 2^{-2js\alpha}. \end{align*} \item[(ii)] A smooth fragment $f_j$ of the form (i) in \eqref{eq:stdfrags} satisfies the improved estimate \begin{align*}
\int_{\mathbb{R}^2} |\mathcal{L}^d_{J,1}(\widehat{f}_{j}W_J)(\xi)|^2 \,d\xi \lesssim A_J 2^{-2js\alpha} 2^{-2js\beta}. \end{align*} \end{enumerate} Here $A_J$ are the quantities defined in \eqref{eqdef:AJ}. The implicit constants are independent of $J\in\mathbb{J}_{\scriptscriptstyle{+}}$, $\omega$ and $g$. \end{prop} \begin{proof}
Using the definition~\eqref{eq:diffops} of the operators $\mathcal{D}_{J,1}$ and $\mathcal{D}_{J,2}$ we obtain for $(m_1,m_2)\in\mathbb{N}^2_0$ \begin{align}\label{eq:formulaD1D2} \mathcal{D}_{J,1}^{m_1}\mathcal{D}_{J,2}^{m_2} = \sum_{\substack{a_1+b_1=m_1 \\ a_2+b_2=m_2 }} c_{a_1,a_2,b_1,b_2} (\sin\phiJ)^{a_2+b_1}(\cos\phiJ)^{a_1+b_2} \partial^{(a_1+a_2,b_1+b_2)} \end{align} with purely combinatorial coefficients $c_{a_1,a_2,b_1,b_2}\in\mathbb{Z}$. This leads to \begin{align*}
\|\mathcal{D}_{J,1}^{m_1}\mathcal{D}_{J,2}^{m_2} \widehat{f}_{j}\|^2_{L^2(\mathcal{W}^+_J)} \le C(m_1,m_2)
\sum_{\substack{a_1+b_1=m_1 \\ a_2+b_2=m_2 }} \| \partial^{(a_1+a_2,b_1+b_2)}\widehat{f}_{j} \|^2_{L^2(\mathcal{W}^+_J)} \end{align*} with a constant $C(m_1,m_2)>0$.
If $f_j$ is an edge fragment, we proceed with Lemma~\ref{lem:wedgesmooth} and deduce \begin{align*}
\|\mathcal{D}_{J,1}^{m_1}\mathcal{D}_{J,2}^{m_2} \widehat{f}_{j}\|^2_{L^2(\mathcal{W}^+_J)} &\lesssim \sum_{\substack{a_1+b_1=m_1 \\ a_2+b_2=m_2 }} A_J 2^{-2j(m_1+m_2)s\alpha} 2^{-2js\alpha} \lesssim A_J 2^{-2j(m_1+m_2)s\alpha} 2^{-2js\alpha}. \end{align*}
Let $d_1,d_2\in\mathbb{N}_0$. The function $\mathcal{D}_{J,1}^{d_1} \mathcal{D}_{J,2}^{d_2}( \widehat{f}_{j} W_J ) $ is a linear combination of terms $(\mathcal{D}_{J,1}^{m_1} \mathcal{D}_{J,2}^{m_2} \widehat{f}_{j})( \mathcal{D}_{J,1}^{n_1} \mathcal{D}_{J,2}^{n_2} W_J ) $ with $m_1+n_1=d_1$ and $m_2+n_2=d_2$. In view of~\eqref{eq:basic_fact} and the estimate above, it holds \begin{align*}
\|\mathcal{D}_{J,1}^{m_1} \mathcal{D}_{J,2}^{m_2} \widehat{f}_{j} \|^2_{L^2(\mathcal{W}^+_J)}\cdot \|\mathcal{D}_{J,1}^{n_1} \mathcal{D}_{J,2}^{n_2} W_J\|^2_\infty &\lesssim A_J 2^{-2j(m_1+m_2)s\alpha} 2^{-2js\alpha} \cdot 2^{-2sjn_1} 2^{-2sj\alpha n_2} \\ &\le A_J 2^{-2js\alpha d_1} 2^{-2js\alpha d_2} 2^{-2js\alpha}. \end{align*} Using H\"{o}lder's inequality we thus obtain for $d_1,d_2\in\mathbb{N}_0$ \begin{align*}
\|\mathcal{D}_{J,1}^{d_1} \mathcal{D}_{J,2}^{d_2} ( \widehat{f}_{j} W_J ) \|^2_{2} &\lesssim A_J 2^{-2js\alpha (d_1+d_2) } 2^{-2js\alpha}. \end{align*}
Since $\mathcal{L}^d_{J,1}(\widehat{f}_{j}W_J)$ consists of terms of the form \[ 2^{2js\alpha(d_1+d_2)} \mathcal{D}^{2d_1}_1 \mathcal{D}^{2d_2}_2 \] with $d_1,d_2\le d$, not taking into account combinatorial coefficients, the desired estimate for each term of $\mathcal{L}^d_{J,1}(\widehat{f}_{j}W_J)$ follows.
If $f_j$ is a smooth fragment of regularity $C^\beta$, we use Lemma~\ref{lem:wedgenonsmooth} instead of Lemma~\ref{lem:wedgesmooth}. The rest of the proof is completely analogous.
\end{proof}
\subsection{Further Preparation}
As in the previous Subsection~\ref{ssec:FourierStd}, let $\alpha\in[0,1)$, $\beta\in\mathbb{N}_0$, and $\kappa,\nu,\tilde{\nu}>0$ be fixed, and assume $g\in C^\beta(\kappa[-1,1]^2,\nu)$, $\omega_j=\omega(2^{js\alpha}\cdot)$ and $\omega\in C^\infty(\mathbb{R}^2)\cap C^\beta(\kappa[-1,1]^2,\tilde{\nu})$. Further, let $\delta$ denote the univariate Dirac distribution and define $\delta_{\{x_1=0\}}:=\delta\otimes 1$. We are interested in the Fourier localization of the distributions \begin{align}\label{eq:stddistr} d_j:=g \omega_j \delta_{\{x_1=0\}}\,,\quad j\in\mathbb{N}_0. \end{align} The exposition is analogous to the investigation of the functions~\eqref{eq:stdfrags} in Subsection~\ref{ssec:FourierStd}.
A valuable tool is given by the following lemma, where $\mathcal{I}_j$ are the intervals defined in~\eqref{eqdef:dyintervals}.
\begin{lemma}\label{Lemma1D} Let $\widetilde{A}\neq0$ and $\kappa,s>0$ be fixed. Further assume that $h\in C^\beta(\mathbb{R})$, $\beta\in\mathbb{N}_0$, is a function with ${\text{\rm supp }} h\subseteq[-\kappa,\kappa]$. Then there are a constant $C>0$ and numbers $\eta_j\in[0,1]$, $j\in\mathbb{N}_0$, with $\sum_{j\in\mathbb{N}_0} \eta_j\le 1$ such that for every $j\in\mathbb{N}_0$ \[
\int_{\widetilde{A}\mathcal{I}_j} |\widehat{h}(r)|^2\,dr = C \eta_j |\widetilde{A} 2^{js}|^{-2\beta} \| h^{(\beta)}\|_{2}^2. \] Moreover, the constant $C$ can be chosen independent of $h$ and $\widetilde{A}$. \end{lemma} \begin{proof} Define \[
\tilde{\eta}_j:= |\widetilde{A} 2^{js}|^{2\beta} \int_{\widetilde{A}\mathcal{I}_j} |\widehat{h}(r)|^2\,dr. \]
Then $\sum_{j\in\mathbb{N}_0} \tilde{\eta}_j \le C \| h^{(\beta)}\|_{2}^2 $ with a constant $C>0$ as claimed, since we can estimate \begin{align*} \sum_{j\in\mathbb{N}_0} \tilde{\eta}_j
\asymp \sum_{j\in\mathbb{N}_0} \int_{\widetilde{A}\mathcal{I}_j} |r|^{2\beta} |\widehat{h}(r)|^2\,dr
\asymp \sum_{j\in\mathbb{N}_0} \int_{\widetilde{A}\mathcal{I}_j} |\widehat{h^{(\beta)}}(r)|^2\,dr
\lesssim \int_{\mathbb{R}} |\widehat{h^{(\beta)}}(r)|^2\,dr = \|h^{(\beta)}\|^2_2. \end{align*}
In case $\| h^{(\beta)}\|_{2}\neq 0$, rescaling yields functions $\eta_j:=C^{-1}\| h^{(\beta)}\|_{2}^{-2} \tilde{\eta}_j$ as desired. The case $\| h^{(\beta)}\|_{2}= 0$ is trivial, since then $h\equiv 0$ due to ${\text{\rm supp }} h\subseteq[-\kappa,\kappa]$.
\end{proof}
\noindent With Lemma~\ref{Lemma1D} we can prove the following result.
\begin{lemma}\label{lem:essdistr} Let $s>0$ be fixed and $\varphi\in[-\pi,\pi)$. We have for $j\in\mathbb{N}_0$ \begin{align*}
\int_{\mathcal{I}_j} |\widehat{d_j}(r,\varphi)|^2 \,dr \lesssim 2^{-js\alpha} 2^{js(1-\alpha)}(1+ 2^{js(1-\alpha)}|\sin(\varphi)| )^{-2\beta-1} \|g\|^2_{\beta,\infty} \|\omega\|^2_{\beta,2}. \end{align*} \end{lemma} \begin{proof} The distribution $d_j=g\omega_j \delta_{\{x_1=0\}}$ can be written as the tensor product
$d_j= \delta \otimes h_j$ of the Dirac distribution $\delta$ with the function $h_j:=(g\omega_j)|_{\{x_1=0\}}$. Therefore, we have \[ \widehat{d}_j= \widehat{\delta \otimes h_j} = 1 \otimes \widehat{h}_j = \widehat{h}_j \circ\pi_2, \] where $\pi_2:\mathbb{R}^2\to\mathbb{R}$ is the orthogonal projection onto the second variable.
Let $\varphi\in[-\pi,\pi)$ and assume first that $|\sin(\varphi)|\ge 2^{-js(1-\alpha)}$. Then $\varphi\notin\{-\pi,0\}$ and it holds \begin{align*}
\int_{\mathcal{I}_j} |\widehat{d_j}(r,\varphi)|^2 \,dr = \int_{\mathcal{I}_j} |\widehat{h}_j(r\sin(\varphi))|^2 \,dr
= |\sin(\varphi)|^{-1} \int_{\sin(\varphi)\mathcal{I}_j} |\widehat{h}_j(r)|^2 \,dr. \end{align*}
\noindent Applying Lemma~\ref{Lemma1D} with $\widetilde{A}=\sin(\varphi)$ yields \begin{align*}
\int_{\mathcal{I}_j} |\widehat{d_j}(r,\varphi)|^2 \,dr \lesssim \eta_{j} 2^{-2js\beta} |\sin(\varphi)|^{-2\beta-1} \| h^{(\beta)}_j \|^2_2
= \eta_j 2^{-2js\alpha\beta} 2^{-2js(1-\alpha)\beta} |\sin(\varphi)|^{-2\beta-1} \| h^{(\beta)}_j \|^2_{L^2(\mathbb{R})} , \end{align*}
where $\eta_{j}\le 1$ for every $j\in\mathbb{N}_0$. Note that Lemma~\ref{Lemma1D} is applied with a different integrand $|\widehat{h}_j|^2$ at each scale. However, the implicit constants are uniform over all $j\in\mathbb{N}_0$.
Applying Leibniz's rule $h^{(\beta)}_j= \sum_{\gamma\le \beta} \binom{\beta}{\gamma} \partial_2^{\gamma} g(0,\cdot) \partial_2^{\beta-\gamma} \omega_j(0,\cdot)$ we further deduce \begin{align*}
\| h^{(\beta)}_j \|^2_2 \lesssim \sum_{\gamma\le \beta} \| \partial_2^{\gamma}g(0,\cdot)\|^2_\infty \| \partial_2^{\beta-\gamma}\omega_j(0,\cdot) \|^2_{2}.
\lesssim 2^{-js\alpha} 2^{2js\alpha\beta} \|\omega\|^2_{\beta,2} \|g\|^2_{\beta,\infty}. \end{align*}
This settles the case $|\sin(\varphi)|\ge 2^{-js(1-\alpha)}$. If $|\sin(\varphi)|< 2^{-js(1-\alpha)}$ we argue differently based on $
\|\widehat{h}_j\|^2_\infty\le \|h_j\|^2_1 \le 2\cdot 2^{-js\alpha} \|h_j\|^2_2. $ We deduce \begin{align*}
\int_{\mathcal{I}_j} |\widehat{d_j}(r,\varphi)|^2 \,dr
= \int_{\mathcal{I}_j} |\widehat{h}_j(r\sin(\varphi))|^2 \,dr \lesssim 2^{js} \|\widehat{h}_j\|^2_\infty \lesssim 2^{js(1-\alpha)} \|h_j\|^2_2 . \end{align*}
The proof is finished since $\|h_j\|_2^2 \le \|\omega_j(0,\cdot)\|^2_{2} \|g(0,\cdot)\|^2_{\infty} \le 2^{-js\alpha} \|\omega\|^2_{2} \|g\|^2_{\infty} $.
\end{proof}
\noindent Lemma~\ref{lem:essdistr} shows that the Fourier decay of $d_j$ is highly dependent on the direction $\varphi\in[-\pi,\pi)$. It motivates the introduction of the quantity \begin{align}\label{eqdef:lJ}
\ell_J:= 1+ 2^{js(1-\alpha)} |\sin(\phiJ)| \,,\quad J=(j,\ell)\in\mathbb{J}_{\scriptscriptstyle{+}}\,, \end{align} where $\phiJ=\ell \varphi_j$ and $\varphi_j=\pi 2^{-\lfloor js(1-\alpha)\rfloor-1}$ is the angle in \eqref{eqdef:fundangle}. Note that $1\le\ell_J\le 1+ 2^{js(1-\alpha)}$.
Similar to the analysis of the fragments~\eqref{eq:stdfrags}, we now proceed to estimate the Fourier energy of $\widehat{d_j}$ concentrated in a wedge $\mathcal{W}^+_J$. The following result corresponds to Lemmas~\ref{lem:wedgesmooth} and \ref{lem:wedgenonsmooth}.
\begin{lemma}\label{lem:refine2} Let $J\in\mathbb{J}_{\scriptscriptstyle{+}}$ be a scale-angle pair, $\ell_J$ the associated quantity~\eqref{eqdef:lJ}. For $(m_1,m_2)\in\mathbb{N}_0^2$ \begin{align*}
\|\partial^{(m_1,m_2)}\widehat{d_j}\|^2_{L^2(\mathcal{W}^+_J)} \lesssim \|g\|^2_{\beta,\infty} \|\omega\|^2_{\beta,2} \begin{cases} 0 \quad&,\, m_1\neq0, \\ 2^{-2jm_2s\alpha} 2^{js(1-\alpha)} \ell_J^{-2\beta-1} &,\, m_1=0. \end{cases} \end{align*} The implicit constant is independent of $J\in\mathbb{J}_{\scriptscriptstyle{+}}$ and $g$ and $\omega$. \end{lemma} \begin{proof} If $m_1\neq0$ the assertion follows from $\partial^{m_1}_1 \widehat{d}_j=\partial^{m_1}_1 \big( \widehat{h}_j \circ\pi_2 \big) = 0$.
To handle the case $m_1=0$, let us introduce the modified window $\tilde{\omega}(x)=x_2^{m_2} \omega(x)$ and its rescaled versions $\tilde{\omega}_j=\tilde{\omega}(2^{js\alpha}\cdot)$. Then $\tilde{\omega}_j(x)= 2^{js\alpha m_2}x_2^{m_2}\omega_j(x)$, and as a consequence \[ \partial^{m_2}_2 \widehat{d}_j = (-2\pi i)^{m_2}\widehat{x_2^{m_2} d_j} = (2\pi i)^{m_2} 2^{-js\alpha m_2 } \widehat{d^{\raisebox{-0.45em}[0.4mm][0.4mm]{\textasciitilde}}_j} \] with $d^{\raisebox{-0.45em}[0.4mm][0.4mm]{\textasciitilde}}_j:=g \tilde{\omega}_j \delta_{\{x_1=0\}}$ of the form \eqref{eq:stddistr}. Hence, we can apply Lemma~\ref{lem:essdistr}, which yields \begin{align*}
\int_{\mathcal{W}^+_J} |\partial_2^{m_2}\widehat{d_{j}}(\xi)|^2 \,d\xi
&\asymp 2^{-2js\alpha m_2} \int_{\mathcal{I}_j}\int_{\mathcal{A}_J} |\widehat{d^{\raisebox{-0.45em}[0.4mm][0.4mm]{\textasciitilde}}_j}(r,\varphi)|^2 r \,d\varphi\,dr \\
&\lesssim 2^{-2js\alpha m_2} \|g\|^2_{\beta,\infty} \|\tilde{\omega}\|^2_{\beta,2} \int_{\mathcal{A}_J} 2^{2js(1-\alpha)} (1+2^{js(1-\alpha)}|\sin(\varphi)|)^{-2\beta-1} \,d\varphi \\
&\lesssim 2^{-2js\alpha m_2} 2^{js(1-\alpha)} \ell_J^{-2\beta-1} \|g\|^2_{\beta,\infty} \|\omega\|^2_{\beta,2}. \tag*{\qedhere} \end{align*}
\end{proof}
\noindent Next we utilize the differential operator \begin{align}\label{eq:diffop2} \mathcal{L}_{J,2}&:=(\textsl{Id} - 2^{2js} \ell^{-2}_J\mathcal{D}_{J,1}^2)(\textsl{Id} - 2^{2js\alpha}\mathcal{D}_{J,2}^2), \end{align} where we use the same notation as in the definition of the operator \eqref{eq:diffop1}. Similar to Proposition~\ref{prop:fundament1} we obtain the following result.
\begin{prop}\label{prop:stredge} Let $\mathcal{L}_{J,2}$ be the differential operator \eqref{eq:diffop2}, $J\in\mathbb{J}_{\scriptscriptstyle{+}}$, and $d\in\mathbb{N}_0$. We have \begin{align*}
\int_{\mathbb{R}^2} |\mathcal{L}^d_{J,2}(\widehat{d}_{j}W_J)(\xi)|^2 \,d\xi \lesssim 2^{js(1-\alpha)} \ell_J^{-2\beta-1}. \end{align*} The implicit constant is independent of $J\in\mathbb{J}_{\scriptscriptstyle{+}}$, $\omega$ and $g$. \end{prop} \begin{proof} Let $(m_1,m_2)\in\mathbb{N}_0^2$. In view of~\eqref{eq:formulaD1D2} and Lemma~\ref{lem:refine2} we obtain \begin{align*}
\|\mathcal{D}_{J,1}^{m_1}\mathcal{D}_{J,2}^{m_2} \widehat{d}_{j}\|^2_{L^2(\mathcal{W}^+_J)} &\lesssim
\sum_{\substack{a_1+b_1=m_1 \\ a_2+b_2=m_2 }} |\sin(\phiJ)|^{2(a_2+b_1)} \| \partial^{(a_1+a_2,b_1+b_2)}\widehat{d}_{j} \|^2_{L^2(\mathcal{W}^+_J)} \\
&=|\sin(\phiJ)|^{2m_1} \| \partial^{(0,m_1+m_2)}\widehat{d}_{j} \|^2_{L^2(\mathcal{W}^+_J)} \\
&\lesssim |\sin(\phiJ)|^{2m_1} 2^{-2j(m_1+m_2)s\alpha} 2^{js(1-\alpha)} \ell_J^{-2\beta-1}. \end{align*}
Using $|\sin(\phiJ)|\le 2^{-js(1-\alpha)} \ell_J$, we can further deduce \begin{align*}
\|\mathcal{D}_{J,1}^{m_1}\mathcal{D}_{J,2}^{m_2} \widehat{d}_{j}\|^2_{L^2(\mathcal{W}^+_J)} \lesssim 2^{-2jm_1s} 2^{-2jm_2s\alpha} 2^{js(1-\alpha)} \ell_J^{2m_1-2\beta-1}. \end{align*}
The function $\mathcal{D}_{J,1}^{d_1} \mathcal{D}_{J,2}^{d_2}( \widehat{d}_{j} W_J ) $ is a linear combination of terms $(\mathcal{D}_{J,1}^{m_1} \mathcal{D}_{J,2}^{m_2} \widehat{d}_{j})(\mathcal{D}_{J,1}^{n_1} \mathcal{D}_{J,2}^{n_2} W_J ) $ with $m_1+n_1=d_1$ and $m_2+n_2=d_2$. They satisfy \begin{align*}
\|\mathcal{D}_{J,1}^{m_1} \mathcal{D}_{J,2}^{m_2} \widehat{d}_{j} \|^2_{L^2(\mathcal{W}^+_J)}\cdot \|\mathcal{D}_{J,1}^{n_1} \mathcal{D}_{J,2}^{n_2} W_J\|^2_\infty &\lesssim 2^{-2jm_1s} 2^{-2jm_2s\alpha} 2^{js(1-\alpha)} \ell_J^{2m_1-2\beta-1} \cdot 2^{-2jsn_1} 2^{-2js\alpha n_2} \\ &= 2^{-2jsd_1} 2^{-2sj\alpha d_2} 2^{js(1-\alpha)} \ell_J^{2m_1-2\beta-1}. \end{align*} Using H\"{o}lder's inequality, it follows for $d_1,d_2\in\mathbb{N}_0$ \begin{align*}
\|\mathcal{D}_{J,1}^{d_1} \mathcal{D}_{J,2}^{d_2} ( \widehat{d}_{j} W_J ) \|^2_{2} \lesssim 2^{-2jsd_1} 2^{-2js\alpha d_2} 2^{js(1-\alpha)} \ell_J^{2d_1-2\beta-1}. \end{align*}
This proves the desired estimate for each term of $\mathcal{L}^d_{J,2}(\widehat{d}_{j}W_J)$, since these are of the form \[ 2^{2jsd_1}2^{2js\alpha d_2} \ell_J^{-2d_1} \mathcal{D}^{2d_1}_1 \mathcal{D}^{2d_2}_2(\widehat{d}_{j}W_J) \quad\text{with $d_1,d_2\le d$.} \tag*{\qedhere} \]
\end{proof}
\subsection{Proof of Proposition~\ref{prop:frag}}
\noindent After the preparation of the preceding two subsections we now turn back to the proof of Proposition~\ref{prop:frag}. Due to the assumptions, $\alpha\in[0,1)$, $s>0$, $\beta\in\mathbb{N}$, $\nu>0$ are fixed and $f\in{\bf E}^\beta([-1,1]^2;\nu)$ is of the simplified form \eqref{eqdef:cartsimp}. Further recall that for a cube $Q\in\mathcal{Q}_j$, $j\in\mathbb{N}_0$, the notation $f_{Q}$ is used for the associated fragment~\eqref{eqdef:frag}.
Instead of the sequence $\theta_Q=\{\theta_\mu\}_{\mu\in\mathbb{M}_j}$, we will analyze the relabelled sequence $\tilde{\theta}_Q:=\{\tilde{\theta}_\mu\}_{\mu\in\mathbb{M}_j}$ with elements $\tilde{\theta}_{j,\ell,k}:=\theta_{j,[\ell+\ell_{\bullet}],k-k_{\bullet}}$, where we use the notation introduced at the end of Subsection~\ref{ssec:curvesparse}. Recall that the quantities $\ell_{\bullet}\in \mathbb{Z}$, $k_{\bullet}\in\mathbb{Z}^2$ are determined by $Q\in\mathcal{Q}_j$. In view of \eqref{eq:reducedform}, we then have \begin{align}\label{relabelledcoeff} \tilde{\theta}_{j,\ell,k}= \int_{\mathbb{R}^2} \widehat{f^{\raisebox{-0.45em}[0.4mm][0.4mm]{\textasciitilde}}_Q}(\xi) W_{j,\ell+\Delta\ell}(\xi) \overline{u_{j,\ell+\Delta\ell,k+\Delta k}(\xi)} \,d\xi \end{align} with fixed $\Delta k\in[0,1)^2$, $\Delta\ell\in[0,1)$ depending on $Q\in\mathcal{Q}_j$. We define $\Delta J:=(0,\Delta\ell)$ and $J_+:=J+\Delta J$ for scale-angle pairs $J=(j,\ell)\in\mathbb{J}$. Further, we define for $J=(j,\ell)\in\mathbb{J}$ and $K=(K_1,K_2)\in\mathbb{Z}^2$ the sets \begin{align}\label{eq:sumindex} \begin{aligned} \mathfrak{Z}^{Q}_{J,K}&:=\Big\{ (k_1,k_2) \in \mathbb{Z}^2 ~:~ \ell_{J+\Delta J}^{-1}(k_1+\Delta k_1)\in[K_1,K_1+1),\, k_2+\Delta k_2\in [K_2,K_2+1) \Big\}, \\ \widetilde{\mathfrak{Z}}^{Q}_{J,K}&:=\Big\{ (k_1,k_2) \in \mathbb{Z}^2 ~:~ 2^{-js(1-\alpha)}(k_1+ \Delta k_1)\in[K_1,K_1+1),\, k_2+ \Delta k_2\in [K_2,K_2+1) \Big\}. \end{aligned} \end{align}
In the definition of $\mathfrak{Z}^{Q}_{J,K}$ the quantity $\ell_{J+\Delta J}=1+2^{-js(1-\alpha)|}\sin(\phiJdelJ)|$ is used, with angle $\phiJdelJ=(\ell+\Delta\ell)\varphi_j$ and $\varphi_j$ as in \eqref{eqdef:fundangle}. To shorten notation, it is further useful to henceforth abbreviate \begin{align}\label{eq:quantLK} L_K:=(1+ K_1^2)(1+ K^2_2). \end{align}
\noindent Essential for the proof of Proposition~\ref{prop:frag}, especially part~(ii), is the following lemma which disentangles the smooth contribution from the singular part.
\begin{lemma}\label{lem:induction} Let $j\in\mathbb{N}_0$ and $Q\in\mathcal{Q}_j$ be fixed. Under the assumptions of Proposition~\ref{prop:frag}, the relabelled coefficients $\tilde{\theta}_Q=\{ \tilde{\theta}_{\mu}\}_{\mu\in\mathbb{M}_j}$ given by \eqref{relabelledcoeff} can be decomposed in the form \begin{align*} \tilde{\theta}_{\mu}= a_{\mu} + b_{\mu},\quad \mu\in\mathbb{M}_j, \end{align*}
such that for every $J\in\mathbb{J}$ with $|J|=j$ and every $K\in\mathbb{Z}^2$, with a uniform constant and $d\in\mathbb{N}_0$ fixed, \begin{align*}
\sum_{k\in\mathfrak{Z}^Q_{J,K}} |a_{j,\ell,k}|^2 \lesssim L_K^{-2d} 2^{-js(1+\alpha)} \ell_J^{-2\beta-1} \quad\text{and}\quad
\sum_{k\in\widetilde{\mathfrak{Z}}^Q_{J,K}} |b_{j,\ell,k}|^2 \lesssim L_K^{-2d} \widetilde{A}_J 2^{-2js\alpha} 2^{-2js\beta}. \end{align*}
Here $L_K$ is the quantity defined in \eqref{eq:quantLK}, $\mathfrak{Z}^{Q}_{J,K}$ and $\widetilde{\mathfrak{Z}}^{Q}_{J,K}$ are given by \eqref{eq:sumindex}, and $\widetilde{A}_J\in[0,1]$ are numbers with $\sum_{|J|=j} \widetilde{A}_J\le 1$. If $f_Q$ is a smooth fragment, a possible decomposition is given by $a_{\mu}:=0$ and $b_{\mu}:=\tilde{\theta}_{\mu}$ for $\mu\in\mathbb{M}_j$. \end{lemma}
\noindent It is important to note that the implicit constants in Lemma~\ref{lem:induction} can be chosen uniformly for all $j\in\mathbb{N}_0$ and $Q\in\mathcal{Q}_j$.
\begin{proof}
Recall, that the functions $u_{J,k}$, $J\in\mathbb{J}_{\scriptscriptstyle{+}}$, are obtained by rotation of the function \[ u_{j,0,k}(\xi)=2^{-js(1+\alpha)/2} \exp\big(\langle 2\pi i (2^{-js}k_1,2^{-js\alpha}k_2), \xi \rangle\big), \quad\xi\in\mathbb{R}^2. \] Hence $\mathcal{D}_{J,1} u_{J,k} = (2\pi i) 2^{-js} k_1 u_{J,k}$ and $\mathcal{D}_{J,2} u_{J,k} = (2\pi i) 2^{-js\alpha } k_2 u_{J,k}$ for each $J\in\mathbb{J}_{\scriptscriptstyle{+}}$. We thus establish \begin{align*} \mathcal{L}_{J,1} u_{J,k} = \big(1+ (2\pi)^2 2^{-2js(1-\alpha)} k_1^2\big) \big(1+ (2\pi)^2 k^2_2 \big) u_{J,k} \end{align*} for the differential operator $\mathcal{L}_{J,1}$ defined in~\eqref{eq:diffop1}. Applying partial integration, we obtain from \eqref{relabelledcoeff} \begin{align*} \tilde{\theta}_{J,k} = \big(\big(1+ 4\pi^2 2^{-2js(1-\alpha)}(k_1+ \Delta k_1)^2\big)\big(1+ (2\pi)^2(k_2 + \Delta k_2)^2\big)\big)^{-d} \int\limits_{\mathbb{R}^2} \mathcal{L}^d_{J_{+},1}(\widehat{f^{\raisebox{-0.45em}[0.4mm][0.4mm]{\textasciitilde}}_Q}W_{J_{+}})(\xi) \overline{u_{J_{+},k+\Delta k}}(\xi) \,d\xi. \end{align*} Further, since \[ u_{J+\Delta J,k+\Delta k}(\xi) = u_{J+\Delta J,k}(\xi) \cdot \exp\big(\langle 2\pi i (2^{-js}\Delta k_1,2^{-js\alpha}\Delta k_2), R_{J+\Delta J}\xi \rangle\big) \]
and $\{u_{J_{+},k}\}_{k\in\mathbb{Z}^2}$ is an orthonormal basis for $L^2(\Xi_{J_{+}})$, we obtain for $J\in\mathbb{J}$, $|J|=j$, and $K=(K_1,K_2)\in\mathbb{Z}^2$ \begin{align}\label{1234}
\sum_{k\in\widetilde{\mathfrak{Z}}^{Q}_{J,K}} |\tilde{\theta}_{j,\ell,k}|^2
\le (1+ K_1^2)^{-2d}(1+ K^2_2 )^{-2d} \int_{\mathbb{R}^2} |\mathcal{L}^d_{J_{+},1}(\widehat{f^{\raisebox{-0.45em}[0.4mm][0.4mm]{\textasciitilde}}_Q}W_{J_{+}})(\xi)|^2 \,d\xi. \end{align}
In case that $f^{\raisebox{-0.45em}[0.4mm][0.4mm]{\textasciitilde}}_Q$ is a smooth fragment, Proposition~\ref{prop:fundament1}~(ii) yields \begin{align*}
\sum_{k\in\widetilde{\mathfrak{Z}}^Q_{J,K}} |\tilde{\theta}_{j,\ell,k}|^2 \lesssim L_K^{-2d} A_{J+\Delta J} 2^{-2js\alpha} 2^{-2js\beta}. \end{align*} By relabelling $\widetilde{A}_{J}:=A_{J+\Delta J}$ we get the desired result.
If $f^{\raisebox{-0.45em}[0.4mm][0.4mm]{\textasciitilde}}_Q$ is an edge fragment, we prove the assertion by induction on $\beta$. In case $\beta=0$, we choose $b_{\mu}:=\tilde{\theta}_{\mu}$ and $a_{\mu}:=0$. Then the assertion is fulfilled, since by \eqref{1234} and Proposition~\ref{prop:fundament1}~(i) \begin{align*}
\sum_{k\in\widetilde{\mathfrak{Z}}^Q_{J,K}} |\tilde{\theta}_{j,\ell,k}|^2 \lesssim L_K^{-2d} A_J 2^{-2js\alpha}. \end{align*}
For the following, let $\beta\ge1$ and note that the assertion is always fulfilled for $j=0$, also due to Proposition~\ref{prop:fundament1}~(i).
It thus remains to prove the assertion for $j,\,\beta\in\mathbb{N}$. If $j\in\mathbb{N}$, by definition,
$W_J(\xi)=U_j(|\xi|) V_{J}(\xi/|\xi|)= U( 2^{-js}|\xi|) V_{J}(\xi/|\xi|)$. To use induction we rewrite \eqref{relabelledcoeff} in the form \begin{align*}
\tilde{\theta}_{J,k} = 2^{-js} \int_{\mathbb{R}^2} |\xi| \widehat{f^{\raisebox{-0.45em}[0.4mm][0.4mm]{\textasciitilde}}_Q}(\xi) \frac{U(2^{-js}|\xi|)V_{J+\Delta J}(\xi/|\xi|)}{2^{-js}|\xi|} \overline{u_{J+\Delta J,k+\Delta k}(\xi)} \,d\xi. \end{align*}
We introduce the function $\widetilde{U}(r)=\frac{U(r)}{r}$, $r\in\mathbb{R}_0^+$, and put $\widetilde{U}_j:=\widetilde{U}( 2^{-js} \cdot)$ for $j\ge 1$. In addition, we put $\widetilde{U}_0(r)=U_0(r)$, $r\in\mathbb{R}_0^+$. Further, we define $\widetilde{V}_{J}(\xi):=V_J(\xi) \cos(|\varphi(\xi)-\phiJ|)^{-1}$ for $\xi\in\mathbb{S}^1$ and $J\in\mathbb{J}_{\scriptscriptstyle{+}}$, $|J|\ge 1$. For $J=(0,0)$ we define $\widetilde{V}_{J}:=V_{J}$. Note that for $\xi\in\mathcal{A}_{J}$, $|J|\ge 1$, we have $|\varphi(\xi)-\phiJ| \le \varphi_j^+/2 \le 3\pi/8$ and thus $1\le \cos(|\varphi(\xi)-\phiJ|)^{-1} \le 3$. For $J\in\mathbb{J}_{\scriptscriptstyle{+}}$ we then define \[
\widetilde{W}_J(\xi):= \widetilde{U}_j(|\xi|) \widetilde{V}_{J}(\xi/|\xi|), \quad \xi\in\mathbb{R}^2. \] The functions $\lb\widetilde{W}_J\rb_{J\in\mathbb{J}}$ are again wedge functions of the form \eqref{eq:suppfunctions} which satisfy condition~\eqref{eq:CalderonW} with some (possibly different) constants $0<A\le B<\infty$. Using these functions the coefficients take the form \begin{align}\label{eq:curvecoeff2}
\tilde{\theta}_{J,k} = 2^{-js} \int_{\mathbb{R}^2} |\xi| \cos(|\varphi(\xi)-\phiJdelJ|) \widehat{f^{\raisebox{-0.45em}[0.4mm][0.4mm]{\textasciitilde}}_Q}(\xi) \widetilde{W}_{J+\Delta J}(\xi) \overline{u_{J+\Delta J,k+\Delta k}(\xi)} \,d\xi. \quad (J,k)\in\mathbb{M}_j. \end{align}
Now recall the directional derivative $\mathcal{D}_{J,1}=\cos(\phiJ)\partial_1 + \sin(\phiJ)\partial_2$ depending on $J\in\mathbb{J}_{\scriptscriptstyle{+}}$. For $\xi=(\xi_1,\xi_2)=(|\xi|\cos\varphi, |\xi|\sin\varphi)\in\mathbb{R}^2$ we have \begin{align*}
\xi_1 \cos(\phiJ) + \xi_2 \sin(\phiJ) = |\xi| \big(\cos(\varphi)\cos(\phiJ)+ \sin(\varphi)\sin(\phiJ) \big) = |\xi| \cos(|\varphi-\phiJ|). \end{align*} Hence, \eqref{eq:curvecoeff2} becomes \begin{align*} \tilde{\theta}_{J,k}= (2\pi i)^{-1} 2^{-js} \int_{\mathbb{R}^2} \big(\mathcal{D}_{J_{+},1}f^{\raisebox{-0.45em}[0.4mm][0.4mm]{\textasciitilde}}_Q\big)^{\wedge}(\xi) \widetilde{W}_{J_{+}}(\xi) \overline{u_{J_{+},k+\Delta k}(\xi)} \,d\xi. \end{align*} The edge fragment $f^{\raisebox{-0.45em}[0.4mm][0.4mm]{\textasciitilde}}_Q$ is of the form $f_j=g\omega(2^{js\alpha}\cdot)H$ with $g\in C_0^\beta(\mathbb{R}^2)$, $\omega\in C_0^\infty(\mathbb{R}^2)$, and the bivariate step function $H=\mathfrak{h}\otimes 1$ (see \eqref{eq:stdedge}). Let us define $\widetilde{g}= \mathcal{D}_{J_{+},1} g$, $\widetilde{\omega}=\mathcal{D}_{J_{+},1}\omega$, and $\widetilde{\omega}_j=\widetilde{\omega}(2^{js\alpha} \cdot)$. Further, recall $\partial_1 H=\delta_{\{x_1=0\}}$ and note that \[ \mathcal{D}_{J_{+},1}H = \cos(\phiJplus)\partial_1 H + \sin(\phiJplus)\partial_2 H = \cos(\phiJplus) \delta_{\{x_1=0\}}. \] The product rule yields \begin{align*} \mathcal{D}_{J_{+},1} f_j = \widetilde{g}\omega_j H + \cos(\phiJplus) \delta_{\{x_1=0\}}\omega_j g + 2^{js\alpha}g\widetilde{\omega}_j H = T_1 + \cos(\phiJplus) T_2 + 2^{js\alpha} T_3 \end{align*} with terms $T_1:=\widetilde{g}\omega_j H$, $T_2:=\delta_{\{x_1=0\}}\omega_j g$, and $T_3:=g\widetilde{\omega}_j H$. This leads to the decomposition \begin{align}\label{eq:seqdecomp}
\tilde{\theta}_{j,\ell,k} \asymp 2^{-js} c^{(0)}_{j,\ell,k} + 2^{-js} \cos(\phiJplus) d^{(0)}_{j,\ell,k} + 2^{-js(1-\alpha)} \tilde{\theta}^{(1)}_{j,\ell,k} \end{align} with \begin{align*} c^{(0)}_{j,\ell,k} &:= \int_{\mathbb{R}^2} \widehat{T}_1 \widetilde{W}_{J_{+}}(\xi) \overline{u_{J_{+},k+\Delta k}(\xi)} \,d\xi, \\ d^{(0)}_{j,\ell,k} &:= \int_{\mathbb{R}^2} \widehat{T}_2 \widetilde{W}_{J_{+}}(\xi) \overline{u_{J_{+},k+\Delta k}(\xi)} \,d\xi, \\ \tilde{\theta}^{(1)}_{j,\ell,k} &:= \int_{\mathbb{R}^2} \widehat{T}_3 \widetilde{W}_{J_{+}}(\xi) \overline{u_{J_{+},k+\Delta k}(\xi)} \,d\xi. \end{align*} Note that $\widetilde{g}\in C_0^{\beta-1}(\mathbb{R}^2)$ and $\widetilde{\omega}\in C^\infty(\mathbb{R}^2)$ with ${\text{\rm supp }}\widetilde{\omega}\subseteq{\text{\rm supp }}\omega$.
By induction we can decompose \[ c^{(0)}_{\mu} = a^{(0)}_{\mu} + b^{(0)}_{\mu}, \quad \mu\in\mathbb{M}_j, \] where the sequences $\lb a^{(0)}_{\mu}\rb_{\mu\in\mathbb{M}_j}$ and $\lb b^{(0)}_{\mu} \rb_{\mu\in\mathbb{M}_j}$ satisfy the assertion for $\beta-1$.
The coefficients $\{ d^{(0)}_{j,\ell,k} \}_{\mu\in\mathbb{M}_j}$ can be handled with the help of Proposition~\ref{prop:stredge}. We have for the differential operator $\mathcal{L}_{J,2}$ from \eqref{eq:diffop2} \begin{align*} \mathcal{L}_{J,2} u_{J,k} = \big(1+ (2\pi)^2 \ell^{-2d}_J k_1^2\big) \big(1+ (2\pi)^2 k^2_2 \big) u_{J,k}. \end{align*} Partial integration leads to \[ d^{(0)}_{J,k}= \big(1+ (2\pi)^2 \ell^{-2d}_{J_{+}} (k_1+\Delta k_1)^2\big)^{-d} \big(1+ (2\pi)^2 (k_2 + \Delta k_2)^2 \big)^{-d} \int_{\mathbb{R}^2} \mathcal{L}^d_{J_{+},2}(\widehat{T}_2 \widetilde{W}_{J_{+}})(\xi) \overline{u_{J_{+},k+\Delta k}(\xi)} \,d\xi. \]
We deduce that for every $J\in\mathbb{J}$ with $|J|=j$ and every $K=(K_1,K_2)\in\mathbb{Z}^2$ \begin{align*}
\sum_{k\in\mathfrak{Z}^Q_{J,K}} |d^{(0)}_{J,k}|^2 \le (L_K)^{-2d}
\int_{\mathbb{R}^2} |\mathcal{L}^d_{J_{+},2}(\widehat{T}_{2} \widetilde{W}_{J_{+}})(\xi)|^2 \,d\xi
\lesssim (L_K)^{-2d} 2^{js(1-\alpha)} \ell_{J_{+}}^{-2\beta-1}. \end{align*}
Here we applied the fact that $\{u_{J_{+},k}\}_{k\in\mathbb{Z}^2}$ is an orthonormal basis for $L^2(\Xi_{J_{+}})$ and Proposition~\ref{prop:stredge}. Finally, note that $|\sin(\phiJ)|\asymp|\phiJ|\asymp |\ell| 2^{-js(1-\alpha)}$ uniformly for $J\in\mathbb{J}_{\scriptscriptstyle{+}}$. Hence, due to $\Delta\ell\in[0,1)$, $\ell_J\asymp 1+|\ell|\asymp 1+|\ell+\Delta\ell| \asymp \ell_{J+\Delta J}$.
It remains to handle the sequence $\lb \tilde{\theta}^{(1)}_{\mu}\rb_{\mu\in\mathbb{M}_j}$ which resembles the original sequence $\lb \tilde{\theta}_{\mu} \rb_{\mu\in\mathbb{M}_j}$ and can be handled accordingly. After $\gamma$ iterations of the decomposition process~\eqref{eq:seqdecomp} we end up with sequences $\lb c^{(0)}_{\mu}\rb_{\mu\in\mathbb{M}_j}, \ldots, \lb c^{(\gamma-1)}_{\mu}\rb_{\mu\in\mathbb{M}_j}$, $\lb d^{(0)}_{\mu}\rb_{\mu\in\mathbb{M}_j},\ldots, \lb d^{(\gamma-1)}_{\mu}\rb_{\mu\in\mathbb{M}_j}$, and $\lb \tilde{\theta}^{(\gamma)}_{\mu}\rb_{\mu\in\mathbb{M}_j}$. We choose $\gamma=\lceil \frac{1}{1-\alpha} \rceil$ so that \[ 2^{-js(1-\alpha)\gamma} \le 2^{-js}. \] We can apply the induction hypothesis on $\lb c^{(\tau)}_{\mu}\rb_{\mu\in\mathbb{M}_j}$ for every $\tau\in\{0,\ldots,\gamma-1\}$, which leads to sequences $\lb a^{(\tau)}_{\mu}\rb_{\mu\in\mathbb{M}_j}$ and $\lb b^{(\tau)}_{\mu}\rb_{\mu\in\mathbb{M}_j}$. Since $g\in C^\beta(\mathbb{R}^2) \subset C^{\beta-1}(\mathbb{R}^2)$ also $\lb \tilde{\theta}^{(\gamma)}_{\mu}\rb_{\mu\in\mathbb{M}_j}$ can be decomposed into two sequences $\lb a^{(\gamma)}_{\mu}\rb_{\mu\in\mathbb{M}_j}$ and $\lb b^{(\gamma)}_{\mu}\rb_{\mu\in\mathbb{M}_j}$.
Finally, we obtain the desired decomposition $\tilde{\theta}_{\mu}= a_{\mu} + b_{\mu}$, $\mu\in\mathbb{M}_j$, with \begin{align*} a_{\mu}&:= 2^{-js} \sum_{\tau=0}^{\gamma-1} 2^{-js(1-\alpha)\tau} a^{(\tau)}_{\mu} + 2^{-js(1-\alpha)\gamma} a^{(\gamma)}_{\mu}, \\ b_{\mu}&:= 2^{-js} \sum_{\tau=0}^{\gamma-1} 2^{-js(1-\alpha)\tau} b^{(\tau)}_{\mu} + 2^{-js(1-\alpha)\gamma} b^{(\gamma)}_{\mu} + 2^{-js} \cos(\phiJplus) \sum_{\tau=0}^{\gamma-1} 2^{-js(1-\alpha)\tau} d^{(\tau)}_{\mu}. \tag*{\qedhere} \end{align*}
\end{proof}
\noindent With Lemma~\ref{lem:induction} in our toolbox, it is not difficult any more to prove Proposition~\ref{prop:frag}. The remaining considerations are merely interpolation arguments.
\begin{proof}[{\bf Proof of Proposition~\ref{prop:frag}}]
We first handle part (i) of the proposition, when $f_j$ is a smooth fragment. Let $\mathbb{M}_j$ denote the curvelet indices at scale $j\in\mathbb{N}_0$ and define $\mathbb{M}^Q_{j,K}:=\{ (j,\ell,k)\in\mathbb{M}_j : k\in \widetilde{\mathfrak{Z}}^{Q}_{J,K} \}$ for $K\in\mathbb{Z}^2$. Since $\sum_{|J|=j} A_J \lesssim 1$, Lemma~\ref{lem:induction} yields for $K\in\mathbb{Z}^2$ \[
\sum_{\mu\in\mathbb{M}^Q_{j,K}} |\tilde{\theta}_{J,k}|^2 = \sum_{|J|=j} \sum_{k\in \widetilde{\mathfrak{Z}}^{Q}_{J,K}} |\tilde{\theta}_{J,k}|^2 \lesssim L_K^{-2d} 2^{-2js\alpha} 2^{-2js\beta}. \] Let us fix $d\in\mathbb{N}_0$ as the smallest integer satisfying $d>(1+\beta)/4$, i.e., $ d:= \lfloor (1+\beta)/4 \rfloor +1$. This ensures \begin{align}\label{LKfinite} \sum_{K\in\mathbb{Z}^2} L^{-2d/(1+\beta)}_K = \sum_{K\in\mathbb{Z}^2} \big((1+ K_1^2)(1+ K^2_2 )\big)^{-2d/(1+\beta)} \lesssim 1, \end{align} which will be important below. Further, note that we have the estimate \[
\sum_{|J|=j} \# \widetilde{\mathfrak{Z}}^{Q}_{J,K} \le \sum_{|J|=j} 2^{js(1-\alpha)} \lesssim 2^{2js(1-\alpha)}. \]
Recall the interpolation inequality $\|\lb c_\lambda\rb_{\lambda\in\Lambda}\|_{\ell^p} \le (\#\Lambda)^{1/p-1/2} \|\lb c_\lambda\rb_{\lambda\in \Lambda}\|_{\ell^2}$ valid for $0<p\le2$ and finite sequences $\lb c_\lambda\rb_{\lambda\in \Lambda}$. Interpolation with $p=2/(1+\beta)$ yields \[
\| \lb\tilde{\theta}_{\mu}\rb_{\mu\in\mathbb{M}^{Q}_{j,K}} \|_{2/(1+\beta)} \lesssim 2^{js\beta(1-\alpha)} (L_K)^{-d} 2^{-js\alpha} 2^{-js\beta}= (L_K)^{-d} 2^{-js\alpha(1+\beta)} . \] The proof of part~(i) is finished by applying the $p$-triangle inequality with $p=2/(1+\beta)\le1$. In view of~\eqref{LKfinite} we arrive at \[
\| \lb\tilde{\theta}_{\mu}\rb_{\mu\in\mathbb{M}_j} \|^{2/(1+\beta)}_{2/(1+\beta)} \le \sum_{K\in\mathbb{Z}^2} \|\lb\tilde{\theta}_{\mu}\rb_{\mu\in\mathbb{M}^Q_{j,K}} \|^{2/(1+\beta)}_{2/(1+\beta)} \lesssim 2^{-2js\alpha}. \]
We finally turn to the proof of part~(ii) and assume that $f_j$ is an edge fragment. We denote by $\lb a_{\mu}\rb_{\mu\in\mathbb{M}_j}$ and $\lb b_{\mu}\rb_{\mu\in\mathbb{M}_j}$ the decomposition of the sequence $\lb\tilde{\theta}_{\mu}\rb_{\mu\in\mathbb{M}_j}$ according to Lemma~\ref{lem:induction}. Analogous to the treatment of the smooth case, one can deduce \begin{align}\label{eq:prfsmooth}
\| \lb b_{\mu}\rb_{\mu\in\mathbb{M}_j} \|^{2/(1+\beta)}_{2/(1+\beta)} \lesssim 2^{-2js\alpha}. \end{align}
It remains to handle $\lb a_{\mu}\rb_{\mu\in\mathbb{M}_j}$. Due to Lemma~\ref{lem:induction} we have with $d\in\mathbb{N}_0$ chosen as above \begin{align}\label{zzz}
\sum_{k\in \mathfrak{Z}^{Q}_{J,K}}|a_{j,\ell,k}|^2 \lesssim L_K^{-2d} 2^{-js(1+\alpha)} \ell_J^{-2\beta-1}. \end{align}
Recall that $\ell_J= 1+ 2^{js(1-\alpha)}|\sin(\phiJ)|\ge 1$ and note that we can estimate \begin{align}\label{yyy} \#\mathfrak{Z}^{Q}_{J,K}\le \ell_{J+\Delta J} \asymp \ell_{J}. \end{align}
In view of \eqref{zzz} and \eqref{yyy} we conclude for $\varepsilon>0$ \begin{align*}
N^Q_{J,K}(\varepsilon):= \#\Big\{ k\in\mathfrak{Z}^{Q}_{J,K} ~:~ |a_{j,\ell,k}|>\varepsilon \Big\} \lesssim \min\Big\{ \ell_J , \varepsilon^{-2} L_K^{-2d} 2^{-js(1+\alpha)} \ell_J^{-2\beta-1} \Big\}. \end{align*} The next step is to show \begin{align}\label{intermediate}
\sum_{|J|=j} N^Q_{J,K}(\varepsilon) \lesssim \varepsilon^{-2/(\beta+1)} L_K^{-2d/(\beta+1)} 2^{-js(1+\alpha)/(1+\beta)}. \end{align}
Since $\ell_J\asymp 1+|\ell|$ we can estimate, where we use the quantities $\ell^-_*:=\lceil\ell_*\rceil-1$ and $\ell^+_*:=\lceil\ell_*\rceil$ with $\ell_*:= \varepsilon^{-1/(1+\beta)} L_K^{-d/(1+\beta)} 2^{-js\frac{1+\alpha}{2(1+\beta)}}$,
\begin{align*} \sum_{\ell=0}^{L_j^+} N_{j,\ell,K}(\varepsilon) &\lesssim \sum_{\ell=1}^{L_j^++1} \min\Big\{ \ell , \varepsilon^{-2} L_K^{-2d} 2^{-js(1+\alpha)} \ell^{-2\beta-1} \Big\} \le \sum_{\ell=1}^{\ell^-_*} \ell + \sum_{\ell=\ell^+_*}^{L^+_j+1} \varepsilon^{-2} L^{-2d}_K 2^{-js(1+\alpha)} \ell^{-2\beta-1}. \end{align*}
Note that $\ell^-_*\in\mathbb{N}_0$. Therefore, it holds \[ \sum_{\ell=1}^{\ell^-_*} \ell = \frac{1}{2} \ell^-_* (\ell^-_*+1) \le \ell_*^2 = {\rm rhs}(\ref{intermediate}). \] Further, taking into account $\ell_*\le \ell^+_*$, we obtain \[ \sum_{\ell=\ell^+_*}^{L^+_j+1} \varepsilon^{-2} L^{-2d}_K 2^{-js(1+\alpha)} \ell^{-2\beta-1} \lesssim \varepsilon^{-2} L^{-2d}_K 2^{-js(1+\alpha)} \ell_*^{-2\beta} = {\rm rhs}(\ref{intermediate}). \] Altogether, this proves \eqref{intermediate} since the sum $\sum_{\ell=-L_j^-}^{0} N^Q_{j,\ell,K}(\varepsilon)$ can be estimated analogously.
Recall that $\mathbb{M}_j$ denotes the curvelet indices at scale $j$. Using \eqref{LKfinite} we deduce from \eqref{intermediate} \[
\# \Big\{ \mu\in\mathbb{M}_j ~:~ |a_\mu|>\varepsilon \Big\} = \sum_{K\in\mathbb{Z}^2} \sum_{|J|=j} N^Q_{J,K}(\varepsilon) \lesssim 2^{-js(1+\alpha)/(1+\beta)} \varepsilon^{-2/(1+\beta)}. \] This implies the following estimate, where we let $\rho=\max\big\{0,s(\alpha\beta-1)/(1+\beta)\big\}$, \begin{align}\label{eq:prfedge}
\| \lb a_{\mu}\rb_{\mu\in\mathbb{M}_j}\|^{2/(1+\beta)}_{w\ell^{2/(1+\beta)}} \lesssim 2^{-js(1+\alpha)/(1+\beta)} = 2^{-js\alpha} 2^{js(\alpha\beta-1)/(1+\beta)}\le 2^{-js\alpha} 2^{j\rho}. \end{align}
\noindent In a last step, we combine \eqref{eq:prfsmooth} and \eqref{eq:prfedge}. Using the $p$-triangle inequality with $p=\frac{2}{1+\beta}\le1$ gives \[
\| \lb\tilde{\theta}_{\mu}\rb_{\mu\in\mathbb{M}_j} \|^{2/(1+\beta)}_{w\ell^{2/(1+\beta)}}
\le \|\lb a_{\mu}\rb_{\mu\in\mathbb{M}_j} \|^{2/(1+\beta)}_{w\ell^{2/(1+\beta)}} + \| \lb b_{\mu}\rb_{\mu\in\mathbb{M}_j} \|^{2/(1+\beta)}_{w\ell^{2/(1+\beta)}} \lesssim 2^{-js\alpha} 2^{j\rho} + 2^{-2js\alpha} \lesssim 2^{-js\alpha} 2^{j\rho}, \] which finishes the proof. \end{proof}
\section{Discussion and Extension} \label{sec:discussion}
In this final section we interpret and discuss the results of our previous investigations. First we note that Theorem~\ref{thm:bound2} complements the result of Theorem~\ref{thm:oldcurveappr}. The latter guarantees at least an approximation rate of order $N^{-1/\alpha}$ for $\mathcal{E}^\beta([-1,1]^2;\nu)$ if $\beta\ge\alpha^{-1}$ and $\alpha\in[\frac12,1)$. In view of Theorem~\ref{thm:benchmark} the optimal approximation order is thus realized in case $\beta=\alpha^{-1}$. Theorem~\ref{thm:bound2} now tells us that this rate does not improve for $C^{\beta}$ cartoons with $\beta>\alpha^{-1}$, at least if we restrict to greedy approximations obtained by simple thresholding. Hence, $\alpha$-curvelets in the range $\alpha\in[\frac12,1)$ cannot take advantage of cartoon regularity higher than $\alpha^{-1}$.
Turning to the range $\alpha\in[0,\frac12)$, according to both, Theorem~\ref{thm:bound1} and Theorem~\ref{thm:bound2}, the approximation deteriorates as $\alpha$ tends to $0$. In Theorem~\ref{thm:bound2} the achievable rate peaks for $\alpha=\frac12$, a confirmation of the outstanding role of parabolic scaling for cartoon approximation. Among all $\alpha$-curvelet frames, the classic parabolically scaled systems provide the best performance for $\mathcal{E}^\beta([-1,1]^2;\nu)$ if $\beta\ge2$. However, if $\beta>2$ the achieved rate of order $N^{-2}$ is suboptimal.
To better understand this behavior, recall the heuristic considerations in Subsection~\ref{ssec:guarantees}. A Taylor expansion showed that $C^\beta$ curves with $\beta\in(1,2]$ are locally contained in (properly aligned) rectangles of size $width \approx length^{1/\beta}$. This explains why $\alpha$-scaling with $\alpha=\beta^{-1}$ is optimally suited to resolve such curves. It also indicates that it is not the smoothness of the curves that determines the best type of scaling, but their local scaling behavior. If the second-order Taylor term at some point of a $C^\beta$ curve, where $\beta\ge2$, does not vanish the scaling locally obeys $width \approx length^{1/2}$. Consequently, the choice $\alpha=\frac12$ is still the best for $C^\beta$ curves with $\beta\ge2$ and nonvanishing curvature.
The situation is different if the curvature vanishes. For cartoons with curved edges, however, this typically happens only at certain isolated points which are negligible in the overall approximation. Otherwise, in case of a straight line segment, directionally scaled $0$-curvelets provide the best approximation. A deviation of $\alpha$ from $0$ deteriorates the approximability of the edge, but according to Theorem~\ref{thm:mainappr1} for signals from ${\bf E}^\beta([-1,1]^2;\nu)$ this deterioration is masked by the overall approximation performance of order $N^{-\beta}$ if $\alpha\in[0,\beta^{-1}]$.
It is remarkable that up to now no frame is known where a nonadaptive thresholding scheme yields approximation rates better than $N^{-2}$ for the class $\mathcal{E}^\beta([-1,1]^2;\nu)$, $\beta>2$. As we have seen, $\alpha$-scaling is not able to take advantage of smoothness beyond $C^2$, wherefore new ideas need to be considered. One approach might be based on the bendlet transform~\cite{LPS2016}, which incorporates bending in addition to $\alpha$-scaling for improved adaptability to the edges. While the bendlet dictionary seems to be useful for certain image analysis tasks, the question of how to extract bendlet frames for approximation is not clear however and requires further research.
Finally, let us derive some implications of the obtained results for other $\alpha$-scaled representation systems. The framework of $\alpha$-molecules allows to transfer properties of $\mathfrak{C}_{s,\alpha}$ to other systems of $\alpha$-molecules if their parametrization is consistent with the parametrization $(\mathbb{M},\Phi_{\mathbb{M}})$ of $\mathfrak{C}_{s,\alpha}$ from \eqref{eq:curvepara}. For the required notion of consistency, let us first recall the phase-space metric $\omega_\alpha$ introduced in \cite{GKKS15} for the phase space $\mathbb{P}=\mathbb{R}^+\times\mathbb{T}\times\mathbb{R}^2$.
\begin{definition}[{\cite[Def.~4.1]{GKKS15}}] Let $\alpha\in[0,1]$. The $\alpha$-scaled index distance $\omega_\alpha:\mathbb{P} \times\mathbb{P}\to[1,\infty)$ is defined by \begin{align*} \omega_\alpha( \textit{\textbf{p}}_\lambda, \textit{\textbf{p}}_\mu)=\max\Big\{\frac{s_\lambda}{s_\mu},\frac{s_\mu}{s_\lambda}\Big\} (1+d_\alpha(\textit{\textbf{p}}_\lambda,\textit{\textbf{p}}_\mu))\,, \end{align*} where $\textit{\textbf{p}}_\lambda=(s_\lambda,\theta_\lambda,x_\lambda)\in\mathbb{P}$, $\textit{\textbf{p}}_\mu=(s_\mu,\theta_\mu,x_\mu)\in\mathbb{P}$, and with $s_0=\min\{s_\lambda,s_\mu\} $, $e_\lambda=(\cos(\theta_\lambda),-\sin(\theta_\lambda))$, \[
d_\alpha(\textit{\textbf{p}}_\lambda,\textit{\textbf{p}}_\mu)=s_0^{2(1-\alpha)} |\theta_\lambda-\theta_\mu|^2+
s_0^{2\alpha} |x_\lambda-x_\mu|^2 + \frac{s^2_0}{1+s_0^{2(1-\alpha)}|\theta_\lambda-\theta_\mu|^2} |\langle e_\lambda, x_\lambda-x_\mu \rangle|^2. \] \end{definition}
\noindent The consistency of two parametrizations is then defined as follows.
\begin{definition}[{\cite[Def.~5.5]{GKKS15}}] Let $\alpha\in[0,1]$ and $k>0$. Two parametrizations $(\Lambda,\Phi_\Lambda)$ and $(\Delta,\Phi_\Delta)$, for index sets $\Lambda$ and $\Delta$ respectively, are called $(\alpha,k)$-consistent if \[ \sup_{\lambda\in\Lambda} \sum_{\mu\in\Delta} \omega_\alpha\big(\Phi_\Lambda(\lambda),\Phi_\Delta(\mu)\big)^{-k}<\infty \quad\text{and}\quad \sup_{\mu\in\Delta} \sum_{\lambda\in\Lambda} \omega_\alpha\big(\Phi_\Lambda(\lambda),\Phi_\Delta(\mu)\big)^{-k}<\infty. \] \end{definition}
\noindent Since $\mathfrak{C}_{s,\alpha}$ is a tight frame of $\alpha$-molecules of arbitrary order, as shown by Lemma~\ref{thm:curvmol}, the theory of $\alpha$-molecules allows to deduce the following result practically for free.
\begin{theorem}\label{thm:mol_app} Let $\alpha\in[0,1]$ and let $\mathfrak{M}:=\lb m_\lambda\rb_{\lambda\in\Lambda}$ be a frame of $\alpha$-molecules whose parametrization, for some $k>0$, is $(\alpha,k)$-consistent with the $\alpha$-curvelet parametrization $(\mathbb{M},\Phi_\mathbb{M})$ of $\mathfrak{C}_{s,\alpha}$. Further, assume that for some $\gamma\in\mathbb{R}_0^+$ the order $(L,M,N_1,N_2)$ of $\mathfrak{M}$ satisfies \begin{align}\label{disc:ordercond} L\geq k(1+\gamma) ,\quad M \geq \frac{3k}{2} (1+\gamma) + \frac{\alpha-3}{2} , \quad N_1 \geq \frac{k}{2} (1+\gamma) +\frac{1+\alpha}{2} , \quad N_2\geq k(1+\gamma). \end{align} Then the following holds true: \begin{enumerate} \item[(i)] Let $\tilde{c}_\lambda:=\langle f,m_\lambda\rangle$, $\lambda\in\Lambda$, denote the analysis coefficients of $f\in{\bf E}^{\beta}([-1,1]^2,\nu)$ with respect to $\mathfrak{M}$, and assume $\beta\in\mathbb{N}$. If \eqref{disc:ordercond} is fulfilled for $\gamma=\min\{\beta,\alpha^{-1}\}$, then $\lb\tilde{c}_\lambda\rb_{\lambda\in\Lambda}\in \ell^p(\Lambda)$ for all $p>\frac{2}{1+\gamma}$. \item[(ii)] Let $\Theta=\sum_{\lambda\in\Lambda} c_\lambda m_\lambda$ be a representation of the function $\Theta$ from~\eqref{def:Theta} with respect to $\mathfrak{M}$. If \eqref{disc:ordercond} is fulfilled for some $\gamma>\tilde{\gamma}:=\max\{\alpha,1-\alpha\}^{-1}$, then $\lb c_\lambda\rb_{\lambda\in\Lambda}\notin\ell^p(\Lambda)$ for $p<\frac{2}{1+\tilde{\gamma}}$. \end{enumerate}
\end{theorem}
\begin{proof} According to \cite[Thm.~5.6]{GKKS15} condition~\eqref{disc:ordercond} ensures that the systems $\mathfrak{M}$ and $\mathfrak{C}_{s,\alpha}$ are sparsity equivalent in $\ell^p$
for $p:=\frac{2}{1+\gamma}$, which means $\| (\langle m_\lambda,\psi_\mu \rangle)_{\lambda,\mu} \|_{\ell^p\to\ell^p} < \infty$ (see \cite[Def.~5.3]{GKKS15}). Since $f=\sum_\mu \langle f,\psi_\mu \rangle \psi_\mu $ and $\lb\langle f,\psi_\mu \rangle\rb_\mu\in\ell^{p+\varepsilon}(\mathbb{M})$, $\varepsilon>0$, by Theorem~\ref{thm:mainappr2}, assertion $(i)$ follows. For $(ii)$ assume that $\lb c_\lambda\rb_{\lambda}\in\ell^p(\Lambda)$, which implies by sparsity equivalence $\lb\langle \Theta,\psi_\mu \rangle\rb_{\mu}\in\ell^p(\mathbb{M})$. Using $\Theta=\sum_\mu \langle \Theta,\psi_\mu \rangle \psi_\mu $ and Lemma~\ref{lem:decayapprox}, this then implies an $N$-term approximation rate of order $N^{-\gamma}$, in contradiction to Theorem~\ref{thm:bound2}. \end{proof}
\noindent A direct corollary is obtained via Lemma~\ref{lem:decayapprox}.
\begin{cor}\label{cor:mol_app} Under the assumptions of Theorem~\ref{thm:mol_app}~(i), every dual frame $\lb\tilde{m}_\lambda\rb_{\lambda\in \Lambda}$ of $\mathfrak{M}$ yields -- via simple thresholding -- $N$-term approximations $f_N$ to $f\in{\bf E}^{\beta}([-1,1]^2)$ satisfying \[
\|f-f_N\|_2^2 \lesssim N^{-\min\{\beta,\alpha^{-1}\}+\varepsilon} \,, \quad \varepsilon >0 \text{ arbitrary} \,, \quad\text{as }N\to\infty. \] \end{cor}
\noindent To see the reach of these results, let us mention that the $\alpha$-shearlet parametrization is $(\alpha,k)$-consistent with the $\alpha$-curvelet parametrization for $k>2$ (see~\cite[Thm.~5.7]{GKKS15}). The results thus comprise in particular $\alpha$-shearlet frames, including both band-limited and compactly supported constructions (see~\cite[Prop.~3.11]{GKKS15}).
\begin{appendix}
\section[Appendix]{Bessel Functions}
In this appendix we collect some useful facts about Bessel functions mainly taken from \cite{Teubner1996} and \cite{Grafakos2008}. We are only interested in Bessel functions $J_\nu$ of integer and half-integer order in the range $\nu\in\{-\frac12,0,\frac12,1,\ldots\}$. Bessel functions of this kind occur naturally in the Fourier analysis of radial functions. For $t\in\mathbb{R}^+$ the value $J_\nu(t)$ is conveniently defined by either of the two series (see \cite{Teubner1996} and \cite[Appendix B.3]{Grafakos2008}) \begin{align}\label{app:defbessel} J_{\nu}(t)=\Big(\frac{t}{2}\Big)^{\nu} \sum_{k=0}^\infty \frac{(-1)^k}{\Gamma(k+1)\Gamma(k+\nu+1)}\Big(\frac{t}{2}\Big)^{2k} =\frac{1}{\sqrt{\pi}} \Big(\frac{t}{2}\Big)^{\nu} \sum_{k=0}^\infty \frac{(-1)^k\Gamma(k+\frac{1}{2})}{\Gamma(k+\nu+1)}\frac{t^{2k}}{(2k)!}\,, \end{align} where the Gamma function $\Gamma$ extends the factorial $z!$ to the complex numbers with $\Gamma(z)=(z-1)!$. To verify the equivalence of both representations, it is useful to note that $\Gamma(k+\frac12)=\frac{(2k)!}{k!4^k}\sqrt{\pi}$ for $k\in\mathbb{N}_0$. We explicitly remark, that definition~\eqref{app:defbessel} is also valid for $\nu=-\frac12$, although this case is not included in the exposition of \cite{Grafakos2008}. As is obvious from the second representation, the functions $J_\nu$ of half-integer order can be expressed in closed form in terms of trigonometric functions. For integer orders such closed form representations do not exist.
If $f(x)=f_0(|x|)$ is a radial function on $\mathbb{R}^d$, $d\in\mathbb{N}$, with a suitable function $f_0$ defined on $\mathbb{R}_0^+$, the Fourier transform of $f$ is given by the formula \[
\widehat{f}(\xi)=\frac{2\pi}{|\xi|^{(d-2)/2}} \int_0^\infty f_0(r) J_{d/2-1}(2\pi r|\xi|)r^{d/2}\,dr \,,\quad \xi\in\mathbb{R}^d. \] Applying this formula to the characteristic function $\chi_{B_d(0,1)}$ of the $d$-dimensional unit ball $B_d(0,1)$ centered at the origin of $\mathbb{R}^d$ yields \begin{align}\label{eq:Fourier1ball} (\chi_{B_d(0,1)})^{\wedge}(\xi)
= \frac{2\pi}{|\xi|^{(d-2)/2}} \int_0^1 J_{d/2-1}(2\pi|\xi|r)r^{d/2}\,dr=
\frac{J_{d/2}(2\pi|\xi|)}{|\xi|^{d/2}} \,,\quad \xi\in\mathbb{R}^d. \end{align} Here, for the integration, we used the second of the following recurrence relations~\cite[Appendix B.2]{Grafakos2008}, which are valid for $\nu\in\frac{1}{2}\mathbb{N}$ and all $t\in\mathbb{R}^+$, \[ t^{-\nu+1} J_{\nu}(t)=-\frac{d}{dt} \big( t^{-\nu+1} J_{\nu-1}(t) \big) \quad\text{and}\quad t^\nu J_{\nu-1}(t)=\frac{d}{dt} \big( t^\nu J_\nu(t) \big)\,. \] The case $\nu=\frac12$ is not treated in~\cite{Grafakos2008}, yet it can be easily confirmed by a direct calculation.
By scaling, we can further deduce from \eqref{eq:Fourier1ball} the following Fourier representation of the bivariate function $\Theta(x)=\chi_{B_2(0,1)}(2x)$, $x\in\mathbb{R}^2$, from \eqref{def:Theta}, \begin{align}\label{eq:Fourier2ball} \widehat{\Theta}(\xi)= \frac{1}{4} (\chi_{B_2(0,1)})^{\wedge}(\xi/2) =
\frac{J_{1}(\pi|\xi|)}{2|\xi|}, \quad\xi\in\mathbb{R}^2. \end{align}
Important for our investigation in Section~3 is the asymptotic behavior of $J_{\nu}(r)$ as $r\rightarrow\infty$. We cite the following result from \cite[Appendix B.8]{Grafakos2008}, which states for $\nu\in\frac{1}{2}\mathbb{N}_0$ the identity \begin{align}\label{app:import1} J_{\nu}(r)=\sqrt{\frac{2}{\pi r}}\cos(r-\frac{\pi\nu}{2}-\frac{\pi}{4}) + R_{\nu}(r) \,,\quad r\in\mathbb{R}^+, \end{align} with a function $R_{\nu}$ given on $\mathbb{R}^+$ by \begin{align*} R_{\nu}(r)&=\frac{(2\pi)^{-1/2}r^{\nu}}{\Gamma(\nu+1/2)} e^{i(r-\pi\nu/2-\pi/4)} \int_0^\infty e^{-rt} t^{\nu+1/2} [(1+it/2)^{\nu-1/2}-1] \, \frac{dt}{t} \\ &\quad + \frac{(2\pi)^{-1/2}r^{\nu}}{\Gamma(\nu+1/2)} e^{-i(r-\pi\nu/2-\pi/4)} \int_0^\infty e^{-rt} t^{\nu+1/2} [(1-it/2)^{\nu-1/2}-1] \, \frac{dt}{t}. \end{align*} Further, for each $\nu\in\frac{1}{2}\mathbb{N}_0$ there is a constant $C_\nu>0$ such that $R_{\nu}$ satisfies the estimate \begin{align}\label{eqapp:est}
|R_\nu(r)|\le C_\nu r^{-3/2} \quad\text{whenever $r\ge1$}. \end{align}
The representation~\eqref{app:import1} and the estimate~\eqref{eqapp:est} play an important role in the proof of Lemma~\ref{lem:thetaest}. For completeness, let us finally note that the identity~\eqref{app:import1} especially holds true in case $\nu=-\frac12$, with vanishing $R_{-\frac12}\equiv0$. This is a direct consequence of the definition~\eqref{app:defbessel} and the Taylor series of the cosine.
\end{appendix}
\end{document} |
\begin{document}
\title{{f A New Spatio-Temporal Model Exploiting Hamiltonian Equations}
\tableofcontents
\begin{abstract}
The solutions of Hamiltonian equations are known to describe the underlying phase space of the mechanical system. In Bayesian Statistics, the only place, where the properties of solutions to the Hamiltonian equations are successfully applied, is Hamiltonian Monte Carlo. In this article, we propose a novel spatio-temporal model using a strategic modification of the Hamiltonian equations, incorporating appropriate stochasticity via Gaussian processes. The resultant sptaio-temporal process, continuously varying with time, turns out to be nonparametric, nonstationary, nonseparable and no-Gaussian. Besides, the lagged correlations tend to zero as the spatio-temporal lag tends to infinity.
We investigate the theoretical properties of the new spatio-temporal process, along with its continuity and smoothness properties. Considering
the Bayesian paradigm, we derive methods for complete Bayesian inference using MCMC techniques. Applications of our new model and methods to
two simulation experiments and two real data sets revealed encouraging performance.
\vspace*{0.5cm}
\textbf{Keywords: Continuously varying time and space; Hamiltonian dynamics; Markov Chain Monte Carlo; Non-stationarity; Non-Gaussianity;
Spatio-temporal modeling. }
\end{abstract}
\section{Introduction}
\label{intro}
Modeling spatially and spatio-temporally dependent data drew much attention in the last few decades within the statistics community. Diverse areas of science, including but not restricted to meteorology \cite{furrer2009spatial, jun2008spatial,sain2011spatial, sang2011covariance, smith2009bayesian, tebaldi2009joint, cox1988simple, sanso1999venezuelan, sahu2010fusing}), environment \cite{bruno2009simple, dou2010modeling, guttorp1994space, huerta2004spatiotemporal, giannitrapani2006sulphur, holland2000estimation, paciorek2009practical, amini2016annual} and ecology \cite{chakraborty2010modeling,deng2009spatio}, give rise to challenging spatio-temporal data. The goal of modeling spatio-temporal data is to predict values of the underlying spatio-temporal process at desired locations and future time points.
A common technique for modeling spatio-temporal data is to assume separability of the covariance function in space and time (for definition of separability, see \cite{banerjee2014hierarchical}) and stationarity, in particular, isotropic stationarity of the underlying spatio-temporal process \cite{cressie2015statistics}.
A more general assumption than stationarity, often employed in practice, is covariance stationarity, that is, the covariance of the observations at any two locations
and time points is a function of the separation vector between the two locations and time points (or function of the distance between the two locations and time points,
for isotropic stationarity). The usual techniques, for example, universal or simple kriging, heavily rely upon these assumptions (\cite{cressie2015statistics}).
However, in reality these assumptions can be very artificial if there is local influence on the correlation structure. Indeed, the scenario of local influence is not uncommon in practice. \cite{das2020nonstationary} and \cite{roy2020bayesian} showed that the PM10 pollution dataset, analyzed by \cite{paciorek2009practical}, is not stationary. The same finding was established for a sea-temperature dataset by \cite{bhattacharya2021bayesian}. \cite{paciorek2003nonstationary} demonstrated that assuming a stationary covariance function wrongly would result in an over-smoothed or under-smoothed process.
In recent times, many attempts are made to incorporate nonstationarity of the underlying spatio-temporal process. \cite{sampson1992nonparametric} first significantly contributed in capturing nonstationarity of the covariance function based on the idea of spatial deformation. This idea has been further exploited in the Bayesian paradigm by
\cite{damian2001bayesian} and \cite{schmidt2003bayesian}. \cite{higdon1998process, higdon2002space, fuentes2001, fuentes2002spectral} used kernel convolution to model nonstationarity of the underlying processes. With the help of Dirichlet process, \cite{gelfand2005bayesian} attempted to model the underlying spatial process nonparametrically along with
``conditional" nonstationarity. A discretized version of a certain stochastic differential equation is considered by \cite{duan2009modeling} to model
spatio-temporal nonstationarity. Further, \cite{fuentes2013multivariate} proposed a nonparametric nonstationary model based on kernel processes mixing.
In all these above mentioned work, the correlation among the two spatio-temporal points was not shown to converge to 0 as the distance between the two points increases to infinity, the property that is naturally enjoyed by a stationary spatio-temporal process. It is natural to believe, and aptly demonstrated with real datasets
in \cite{roy2020bayesian} and \cite{bhattacharya2021bayesian}, that as the distance between two points, either in terms of spatial locations, and/or in terms of time points, increases to infinity the correlation of the underlying process should go to 0.
Very recently, \cite{das2020nonstationary} proposed a non-parametric, non-separable, non-stationary and non-Gaussian spatio-temporal model based on order-based dependent Dirichlet process, where they showed that the underlying covariance function goes to 0 as the distance between the two locations and/or time points increases to infinity. While modeling spatio-temporal data, \cite{das2020nonstationary} also assumed that time and the space vary continuously in their respective domains.
Although, \cite{das2020nonstationary} made a successful attempt building the non-parametric, non-stationary, non-Gaussian model with the
desirable properties of covariance, this model does not impart dynamic properties to the temporal part, either directly or via any latent process.
On the other hand, \cite{Suman2017} introduces a non-parametric spatio-temporal model, which, through a non-parametric dynamic latent process,
induces desirable dynamic properties in the temporal part.
In addition, the model of \cite{Suman2017} is nonstationary with the property that the covariance goes to 0 as distance between two time points
and/or difference among two spatial points goes to infinity. However, time does not vary continuously on the respective domain.
In a nutshell, it is observed that so far to the best of our knowledge, there is no proposal of a non-parametric, non-separable, non-Gaussian dynamic
spatio-temporal model which is continuous in time and space with an underlying structured latent process and with the property that the correlation
between two spatio-temporal realizations goes to 0 as the spatial/temporal lag goes to infinity.
To fill up this gap we propose a dynamic spatio-temporal non-parametric, non-Gaussian model, where the underlying process is non-stationary.
A structured latent process is incorporated in the proposed model. The time and the space in our proposed spatio-temporal model vary continuously
over their respective domains. Further, the underlying process enjoys the property that the covariance goes to 0 as the spatial/temporal lag
tends to infinity.
For constructing the model proposal, we take help of the Hamiltonian dynamics from physics. The idea of Hamilton's equations are applied to Bayesian statistics in formulating Hamiltonian Monte Carlo \cite{betancourt2017conceptual,cheung2009bayesian}. However, formulation of the spatio-temporal model exploiting the idea of the Hamiltonian dynamics
is not done earlier. Here we first briefly describe the Hamiltonian dynamics and the equations. Thereafter, we shall connect the idea of
the spatio-temporal model to Hamiltonian dynamics. In the latter section (Section \ref{process proposal}) we describe the mathematical formulation in detail.
Let $(\mathcal{M},\mathcal{L})$ be a mechanical system, where $\mathcal{M}$ is the configuration space and $\mathcal{L}$ is the smooth Lagrangian.
The coordinate system of $\mathcal{M}$ is determined by $(\theta,\dot{\theta})$, where $\theta$ is the position of a particle at time $t$ and the
$\dot{\theta}$ is the derivative vector with respect to time, thus representing the velocity. The partial derivative of $\mathcal{L}$ with respect
to $\dot{\theta}$, known as momenta, is denoted by $p$, which is a function of time $t$, the position $\theta$, and velocity $\dot{\theta}$.
Now the Hamiltonian, a function of ${p}$, $\theta$ and $t$, is defined as
$$\mathcal{H} (p,\theta,t) = \sum_i p_i \dot{\theta}_i - \mathcal{L}(\theta,\dot{\theta},t),$$ which is the energy function of the mechanical system. Here $p_i\text{ and } \dot{\theta}_i$ are the $i$th component of $p$ and $\dot{\theta}$.
The pair $(\theta,p)$ is called phase space coordinates. The phase space coordinates, which varies in time $t$ continuously,
are the solution of the Hamiltonian equations
$$\frac{d \theta}{d t} = \frac{\partial \mathcal{H}}{\partial {p}}, \text{ and } \frac{d {p}}{d t} = - \frac{\partial \mathcal{H}}{\partial \theta}.$$
The point of discussing the Hamiltonian mechanics here is that the solution $\theta$, describes the position of a particle at time $t$ and ${p}$,
the other coordinate of the phase-space, is dependent on the $\theta$ at time $t$. They both vary continuously with respect to time $t$.
In our case, we introduce $p(t)$ as the latent and $\theta(t)$ as the observed spatio-temporal processes with the hope that the solution to
Hamilton's equations will best describe the underlying state-space as done in the phase-space of the mechanical space. To obtain the solution of
Hamilton's equations, one can invoke the leap-frog algorithm \cite{young2014leapfrog}. In this article, we propose a modified leap-frog algorithm
associated with modification of the Hamiltonian equations to build our spatio-temporal model incorporating the latent and the observed processes which continuously depend on time $t$. Moreover, since
the leap-frog algorithm exploits the presence of derivative, our proposed spatio-temporal model also enjoys dynamic property with respect to time $t$
through its previous time point $t-\delta t$. In turn, it is observed that the desirable properties, non-stationarity, non-separability, non-Gaussianity
are satisfied. Further analysis shows that the covariance function of the proposed process goes to 0 as the distance between space and/or time tends to infinity.
The remaining part of the article is planned in the following manner. In Section \ref{process proposal}, we propose our new spatio-temporal process
based on modified Hamiltonian equations and modified leap-frog algorithm. Section \ref{process proposal} presents the important and necessary properties
of the proposed observed and latent processes. The proofs of the lemmas, theorems and corollaries, discussed in Section \ref{process proposal},
are given in the Section \ref{theorem proofs} of Supplementary Information. The likelihood function of the parameter vector involved in the model is provided in Section \ref{complete likelihood}.
In this section we first find out the joint conditional density of the data given the latent variables and the parameters (Subsection \ref{data model} and Section \ref{joint density of observed data} of Supplementary Information)
and then obtain the joint density of latent variables given the observed data and the parameters (Subsection \ref{process model} and Section \ref{joint density of latent data} of Supplementary Information),
before presenting the complete likelihood of the parameter vector (Subsection \ref{Jt Dist}). Section \ref{Priors} deals with the choice of
prior distributions for the parameter vectors. A plausible justification of the choice of the prior distributions is also discussed in this section.
For applying Gibbs sampling, we evaluate the full conditional densities of all the parameters along with the latent variables in
Section \ref{full conditional}. The detailed calculation of the full conditional densities are given in the Section \ref{Appendix B: full conditional densities} of Supplementary Information. Simulation studies and the real data analysis are given in the Sections \ref{Sec: Simulation Studies}
and \ref{real data analysis}, respectively. In Section \ref{Sec: Simulation Studies}, we provide the results of two simulations studies.
Results of another another simulation experiment is presented in Section \ref{LDSTM simulation} of Supplementary Information. In Section \ref{real data analysis},
two real data sets are analyzed to show the performance of the newly proposed spatio-temporal model. Among these two real data sets,
one of the data sets corresponds to non-stationary and non-Gaussian, while another is associated with a stationary non-Gaussian
spatio-temporal process.
Finally, in Section \ref{sec:conclusion}, we summarize our contributions and make concluding remarks.
\section{Modified Hamiltonian equations, proposed process and its properties}
\label{process proposal}
\subsection{The key idea of a spatio-temporal process via modified Hamiltonian equations}
Let the total energy $H(\theta)$, also known as Hamiltonian function, be defined as $V(\theta) + W(p)$, where $V(\theta)$ is the potential energy and $W(p) = \frac{1}{2}p^{T}M^{-1}p$, with $M$ being a chosen matrix (mass), is the kinetic energy, (see \cite{cheung2009bayesian} for more details).
Then the original Hamiltonian equations are given by
\begin{align*}
\frac{d p}{d t} & = \frac{- \partial H}{\partial \theta} = -\nabla V(\theta) \mbox{ and } \\
\frac{d \theta}{d t} & = \frac{\partial H}{\partial p} = M^{-1} p,
\end{align*}
where by $\nabla V(\theta)$ we mean gradient of $V$ with respect to $\theta$.
The leap-frog algorithm for numerically solving the Hamiltonian equations is given by
\begin{align*}
\theta(t+\delta t) & = \theta(t) + \delta t M^{-1}\left\{p(t) - \frac{1}{2}\delta t \nabla V(\theta(t))\right\} \mbox{ and } \\
p(t+\delta t) & = p(t) - \frac{1}{2} \delta t \left\{\nabla V(\theta(t)) + \nabla V(\theta(t+\delta t))\right\},
\end{align*}
We modify the original Hamiltonian equations to suit our purpose as follows:
\begin{align*}
\frac{d p }{d t} &= \alpha^* p -\nabla V(\theta) \mbox{ and } \\
\frac{d \theta }{d t} & = \beta^* \theta + M^{-1} p.
\end{align*}
The modified leap-frog algorithm associated with the modified Hamiltonian equations are as follows:
\begin{align}
\label{eq1: modified eq for p}
p\left(t+\frac{\delta t}{2}\right) &= p(t)+ \alpha^* p(t)\frac{\delta t }{2} - \nabla V(\theta(t))\frac{\delta t}{2} \notag \\
&= p(t)\left(1+ \alpha^*\frac{\delta t}{2}\right) - \nabla V(\theta(t))\frac{\delta t}{2} \notag \\
& = \alpha p(t) - \nabla V(\theta(t))\frac{\delta t}{2},
\end{align}
where $\alpha = \left(1+ \alpha^*\frac{\delta t}{2}\right)$. We set $|\alpha|<1$, that implies $ -2<\alpha^* \frac{\delta t}{2}< 0$.
Restricting $\alpha$ on $(-1,1)$ led to good MCMC mixing in our Bayesian applications.
For $\theta$, we have
\allowdisplaybreaks
\begin{align}
\label{eq2: modified eq for theta}
\theta(t+\delta t) &= \theta(t)+ \beta^* \theta(t)\delta t + \delta t M^{-1} p(t+\delta t/2) \notag \\
&= \theta(t)\left(1+ \beta^* \delta t\right) + \delta t M^{-1} p(t+\delta t/2) \notag \\
& = \beta \theta(t) + \delta t M^{-1} p(t+\delta t/2),
\end{align}
where $\beta =\left(1+ \beta^* \delta t\right).$ As above, we set $|\beta|<1$, which implies $-2<\beta^* \delta t<0$.
As will be seen subsequently, this restriction is necessary for the lagged correlations between the observations to tend to zero as the space-time lag tends to infinity.
Further, note that
\begin{align}
\label{eq3: Leap-frog form of p}
p(t+\delta t) & = p(t+\delta t/2) + \alpha^* p(t+\delta t/2) \frac{\delta t}{2} - \nabla V\left(\theta({t+\delta t})\right) \frac{\delta t}{2} \notag \\
& = p(t+\delta t/2)\left(1+ \alpha^* \frac{\delta t}{2}\right) - \nabla V\left(\theta({t+\delta t})\right) \frac{\delta t}{2} \notag \\
& = \alpha p(t+\delta t/2) - \nabla V\left(\theta({t+\delta t})\right) \frac{\delta t}{2} \notag \\
& = \alpha \left(\alpha p(t) - \nabla V(\theta(t))\frac{\delta t}{2}\right) - \nabla V\left(\theta({t+\delta t})\right) \frac{\delta t}{2} \notag \\
& = \alpha^2 p(t) - \frac{\delta t}{2} \left\{\alpha \nabla V(\theta(t)) + \nabla V(\theta({t+\delta t})) \right\},
\end{align}
where the fourth equality of the equation (\ref{eq3: Leap-frog form of p}) follows from equation (\ref{eq1: modified eq for p}). Finally, replacing the form of $p(t+\delta t/2 )$ from equation (\ref{eq1: modified eq for p}) to equation (\ref{eq2: modified eq for theta}) we get
\begin{align}
\label{eq4: Leap-frog for theta}
\theta(t+\delta t) &= \beta \theta(t) + \delta t M^{-1} \left(\alpha p(t) - \nabla V(\theta(t))\frac{\delta t}{2}\right)
\end{align}
Equations (\ref{eq3: Leap-frog form of p}) and (\ref{eq4: Leap-frog for theta}) constitute the modified leap-frog equations and
are the key ingredients of our proposed spatio-temporal process.
Indeed, for location $s\in S$, where $S$ is some index set, we replace $\theta$ and $p$ in the (modified) Hamiltonian equations with $\theta_s$ and $p_s$ respectively and
with $\bi{s}_n=(s_1,\ldots,s_n)'$, let $\theta_{\bi{s}_n}(t) = (y(s_1,t),\ldots, y(s_n,t))'$,
where $y(s,t)$ is the observed value at location $s\in S$ and time $t \in [0,T]$ and let
$p_{\bi{s}_n}(t) = (x(s_1,t),\ldots, x(s_n,t))'$, where $x(s,t)$ is a latent unobserved process at location $s\in S$ and time $t \in [0,T]$.
In the modified leap-frog algorithm defined by (\ref{eq3: Leap-frog form of p}) and (\ref{eq4: Leap-frog for theta}), we replace $\theta(t)$ and $p(t)$ with
$\theta_{\bi{s}_n}(t)$ and $p_{\bi{s}_n}(t)$, respectively. For single location $s$, these will be denoted by $\theta_{s}(t)$ and $p_{s}(t)$.
Similarly, we shall replace $M$ with $M_s$ and $M_{\bi{s_n}}$, the latter being an appropriate diagonal matrix.
To complete the specification of our spatio-temporal process, we need to model the function $V(\cdot)$ as some appropriate stochastic process; we consider
the Gaussian process for our purpose.
We shall also model $\theta_s(0)$ and $p_s(0)$ as appropriate Gaussian processes indexed by $s$. Complete specification of the process also requires an
appropriate form for $M_s$. Details on these, along with investigation
of the theoretical properties of our spatio-temporal processes, are provided in the next subsection.
\begin{remark}
\label{R00} Note that the equation \ref{eq3: Leap-frog form of p} is not the equation for latent process since it involves the observed process as well. Integrating the conditional distribution of the latent process (equation (\ref{eq3: Leap-frog form of p})) over the observed process would give us the latent process distribution.
\end{remark}
\subsection{Completion of specification of the proposed spatio-temporal process and investigation of its theoretical properties}
\label{properties of the processes}
Let $S$ be a compact subset of $\mathbb{R}^{d}$; for our purpose we choose $d=2$.
We put the following assumptions on the processes $p_s(0), \theta_s(0)$, $s\in S$ and the random function $V(\cdot)$.
\begin{enumerate}
\item[A1.] $p_{s}(0),\, s\in S$ is assumed to be a centered Gaussian process with a symmetric, positive definite covariance function
with bounded partial derivatives. For instance, the Mat\'{e}rn covariance function with $\nu > 1$ has bounded partial derivatives, and could be employed.
In particular, we consider the squared exponential covariance between $p_{s_1}(0)$ and $p_{s_2}(0)$, of the form
$\sigma^2_{p}\exp{\{-\eta_1||s_1-s_2||^2\}}$, $s_1, s_2 \in S$.
\item[A2.] $\theta_{s}(0), \, s\in S$ is also assumed to be a centered Gaussian process with a symmetric, positive definite covariance function
with bounded partial derivatives. Again, we consider the squared exponential covariance function of the form
$\sigma^2_{\theta}\exp{\{-\eta_2 ||s_1-s_2||^2\}}$.
\item[A3.] The function $V(\cdot)$ is assumed to be a Gaussian random function with zero mean and covariance function
$c_{v}(x_1,x_2) = \sigma^2 \exp{\{-\eta_3||x_1-x_2||^2\}}$. As for the other cases, different choices of covariance structure can
be assumed with the assumption that covariance function is continuously twice differentiable and the mixed partial derivatives are Lipschitz continuous.
If the covariance function has third bounded partial derivatives then the function will be Lipschitz. In our example,
the covariance function is infinitely differentiable and the derivatives are bounded. Other choices may include rational quadratic covariance
function, Mat\'{e}rn covariance function with $\nu > 2$ among many others.
\end{enumerate}
\begin{remark}
\label{R0}
Above assumptions need well behaved covariance functions in the sense of smoothness. For a list of such covariance functions one may see \cite{banerjee2014hierarchical}. Smoothness properties along with other important properties of Mat\'{e}rn covariance functions have been covered in \cite{Stein1999} (Chapter 2).
\end{remark}
\begin{remark}
\label{R1}
The assumptions A1 and A2 imply that the covariance functions of $p_{s}(0)$ and $\theta_{s}(0)$ are symmetric, positive definite and Lipschitz continuous, and thus $p_{s}(0)$ and $\theta_{s}(0)$ will have continuous sample paths with probability 1. If the covariance functions are taken to be Mat\'{e}rn covariance function with $\nu>1$ or squared exponential covariance function, then $p_s(0)$ and $\theta_s(0)$ will have differentiable sample paths in $s$.
\end{remark}
\begin{remark}
\label{R2}
Assumption A3 implies that the derivative process of $V(\cdot)$ is also a Gaussian process with continuous sample paths almost surely. In fact, if the squared exponential covariance function is assumed then all the derivatives of $V(\cdot)$ will be Gaussian processes. In particular, the differential of $V(\cdot)$ will have differentiable sample paths. Also note that for Mat\'{e}rn covariance function with $\nu > 2$, the random function $V(\cdot)$ is differentiable and the corresponding covariance function is the mixed partial derivative of the covariance function of $V(\cdot)$. Moreover, the differential of $V(\cdot)$ will have differentiable sample paths as $\nu$ is assumed to be more than 2 for Mat\'{e}rn covariance function (refer to \cite{rusmassen2005gaussian} along with \cite{Stein1999}).
\end{remark}
Now we propose a form of $M_s$ which is continuous and infinitely differentiable when defined on a compact set $S\subset \mathbb{R}^2$.
\begin{definition}[Definition of $M_s$]
Let $S$ be a compact subset of $\mathbb{R}^2$. Define $M_{s} = \exp(\max\{||s^2-u^2||^2: u\in S\})$, where by $\bi{v}^2$ we mean $\bi{v}^2 = (v_{1}^2, v_{2}^2)^T$, for $\bi{v}\in S$.
\end{definition}
\begin{lem}
\label{lemma3: differentiability of Ms}
$M_s$ is infinitely differentiable in $s\in S$.
\end{lem}
\begin{remark}
\label{Remark on M_s goes to infinity}
It can be argued that $M_s \rightarrow \infty$ and $M_{s'} \rightarrow \infty$ as $||s-s'||\rightarrow \infty$ in the following fashion.
Let $||s-s'||\rightarrow \infty$ as $S$ also grows in the sense $S_1\subset S_2 \subset \ldots$, such that at each stage $i$, $S_i$ remains compact.
Under this limiting situation, $M_s = \exp\left(\max\{||s^2-u^2||^2: u\in S\}\right) \geq \exp\left(||s^2-s'^2||^2\right) \rightarrow \infty $
and $M_{s'} = \exp\left(\max\{||s'^2-u^2||^2: u\in S\}\right) \geq \exp\left(||s'^2-s^2||^2\right)\rightarrow \infty.$
\end{remark}
The next results shows that the conditional covariance function of $\theta_{s}(t)$ given $p$ goes to 0 as the distance between two time points and two spatial locations increase to infinity. The distance between two time points and two spatial locations are measured with respect to their corresponding distance metrics.
\begin{thm}
\label{lem: covariance}
Under the assumptions A1 to A3, cov$\left(\theta_{s}(h\delta t), \theta_{s'}(h'\delta t)\bigg|p\right)$ converges to 0 almost surely,
as $||s-s'||\rightarrow \infty$ and $|h-h'|\rightarrow \infty$.
\end{thm}
\begin{remark}
Note that by Theorem \ref{lem: covariance} and the dominated convergence theorem, the unconditional correlation
corr $\left(\theta_{s}(h\delta t), \theta_{s'}(h'\delta t)\right)\rightarrow 0$
as $||s-s'||\rightarrow \infty$ and $|h-h'|\rightarrow \infty$.
\end{remark}
\begin{remark}
In Theorem \ref{lem: covariance}, $\delta t$ in $h\delta t$ and $h'\delta t$ make the time points continuous for continuous $\delta t$. However,
discrete values of $\delta$ are also allowed.
\end{remark}
Now we show that $\theta_s(t)$ and $p_{s}(t)$ are continuous in $s$ with probability 1 and in the mean square sense.
Since the (modified) Hamiltonian equations already imply that $\theta_s(t)$ and $p_{s}(t)$ are path-wise differentiable with respect to $t$, we focus on their smoothness
properties with respect to $s$.
\begin{thm}
\label{lemma1: almost sure continuity}
If the assumptions A1-A3 hold true, then
$\theta_{s}(h\delta t)$ and $p_{s}(h\delta t)$ are continuous in $s$, for all $h\geq 1$, with probability 1.
\end{thm}
\begin{thm}
\label{lemma2:mean square continuity}
Under assumptions A1-A3,
$\theta_{s}(h\delta t)$ and $p_{s}(h\delta t)$ are continuous in $s$ in the mean square sense, for all $h\geq 1$.
\end{thm}
The next two results deal with the differentiability of the processes $\theta_{s}(t)$ and $p_s(t)$.
\begin{thm}
\label{lemma4: almost sure differentiability}
Under assumptions A1-A3,,
$\theta_{s}(h\delta t)$ and $p_{s}(h\delta t)$ have differentiable sample paths with respect to $s$, almost surely.
\end{thm}
\begin{remark}
Theorem \ref{lemma4: almost sure differentiability} is about once differentiability, however, it can be extended to $k$ times differentiability depending on the structure of the covariances assumed on the processes $\theta_{s}(0)$, $p_{s}(0)$ and the random function $V(\cdot)$. For example, if we assume squared exponential covariance functions on each process then $\theta_{s}(h\delta t)$ and $p_{s}(h\delta t)$ will have $k$ times differentiable sample paths in $s$, for any $k \in \mathbb{N}$.
\end{remark}
For proving that the processes $\theta_{s}(t)$ and $p_{s}(t)$ are mean square differentiable in $s$, we need a lemma, stated below, which may be of independent interest.
\begin{lem}
\label{lemma5: mean square differentiability of composition of GP}
Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be a zero mean Gaussian random function with covariance function $c_{f}(x_1,x_2)$, $x_1, x_2\in \mathbb{R}$, which is four times continuously differentiable. Let $\left\{Z(s):s\in S\right\}$ be a random process with the following properties
\begin{enumerate}
\item $E(Z(s)) = 0$,
\item covariance function $c_{Z}(s_1,s_2)$, $s_1, s_2 \in S$, where $S$ is a compact subspace of $\mathbb{R}^2$, is four times continuously differentiable, and
\item $\frac{\partial Z(s)}{\partial s_i}$ has finite fourth moment.
\end{enumerate}
Then the process $\{g(s):s\in S\}$, where $g(s)=f(Z(s))$, is mean square differentiable in $s$.
\end{lem}
\begin{thm}
\label{Thm: Mean-square diff}
Let A1-A3 hold true, with the covariance functions of all the assumed Gaussian processes being squared exponential. Then $\theta_{s}(h\delta t)$ and $p_{s}(h\delta t)$ are mean square differentiable in $s$, for every $h\geq 1$.
\end{thm}
\section{Calculation of likelihood functions} \label{complete likelihood}
Let the observations on $Y$ be available for positions $s_1, s_2, \ldots, s_n$ and time points $1, 2, \ldots, T$. That is, data set is given as $$\mbox{Data} = \left\{Y(s_1,1), Y(s_2,1), \ldots, Y(s_n,1); Y(s_1,2), Y(s_2,2), \ldots, Y(s_n,2); \ldots, Y(s_1,T), Y(s_2,T), \ldots, Y(s_n,T)\right\},$$ and the corresponding latent variables are $$\mbox{Latent} = \left\{X(s_1,1), X(s_2,1), \ldots, X(s_n,1); X(s_1,2), X(s_2,2), \ldots, X(s_n,2); \ldots, X(s_1,T),X(s_2,T) \ldots, X(s_n,T)\right\}.$$
In this section we will derive the data model and the process model under assumptions A1-A3. Particularly, we assume here that the random function $V(\cdot)$ is a Gaussian process with mean 0 and squared exponential function as covariance function for simplicity of calculations. Other covariance functions (with required properties, see assumption A3) will work in the same manner. In particular, we assume that the covariance function of $V(\cdot)$ takes the form $\mbox{cov}(V(x),V(y)) = k(h) = \sigma^2 e^{-\eta_3 h^2}$, where $h= ||x-y||$. Then $V'(\cdot)$ will be a Gaussian random function with mean 0 and covariance function \begin{align}
\label{eq1}
\mbox{cov}(V'(x),V'(y)) = 2\eta_3 \sigma^2 e^{-\eta_3 h^2}(1-2\eta_3 h^2), \end{align} (see \cite{Stein1999}, Chapter 2).
\subsection{Notation} We will use the following notation for all the future calculations regarding the different distribution. Let $i\in \{1,\ldots,n\}$, $m\in\{0,\ldots,T\}$ and $r\in \{1, \ldots, T\}$. \allowdisplaybreaks \begin{align}
\bi{y}_m & = (y(s_1,m), y(s_2,m), \ldots, y(s_n,m))^T, \notag \\
\bi{x}_m & = (x(s_1,m), x(s_2,m), \ldots, x(s_n,m))^T, \notag \\
h_{ij}(m) & = |y(s_i,m)-y(s_j,m)|, \notag \\
\bi{\mu}_m & = (\mu_1(m), \ldots, \mu_n(m))^T, \text{ where } \mu_i(m) = \beta y(s_i,m) + \frac{\alpha x(s_i,m)}{M_{s_i}}, \notag \\
\bi{W}_{r} & = (V'(y(s_1,r)), \ldots, V'(y(s_n,r)))^T , \notag \\
\ell_{ik}(r-1,r) & = |y(s_i,r-1) - y(s_k,r)|, \notag \\
\mathbb{W}_{r-1} & = \begin{pmatrix}
\alpha \bi{W}_{r-1} \\
\bi{W}_{r}
\end{pmatrix},\notag \\
\bi{\theta} & = (\alpha, \beta, \sigma^2,\sigma_{\theta}^2,\sigma_p^2, \eta_1,\eta_2,\eta_3). \notag \end{align} By ${\bi{a}}^T$, we mean transpose of a vector $\bi{a}$ and by $f'$ we mean derivative of $f$.
\subsection{Joint conditional density of the observed data} \label{data model}
The joint conditional density of the data given the latent variables $\bi{x}_0, \bi{x}_1, \ldots, \bi{x}_{T}$ and the parameter vector $\bi{\theta}$ is given by \begin{align}
\label{eq4}
L &= [\mbox{Data}\big\vert \bi{x}_0;\ldots; \bi{x}_{T-1};\bi{y}_0;\bi{\theta}] \notag \\
& \propto [\bi{y}_1\big\vert \bi{y}_0;\bi{x}_0; \bi{\theta}] \ldots \left[\bi{y}_T\big\vert \bi{y}_{T-1}, \ldots, \bi{y}_0; \bi{x}_{T-1}, \ldots, \bi{x}_0; \bi{\theta}\right] \notag \\
& \propto \frac{(\sigma^2)^{-nT/2}}{\prod\limits_{t=1}^{T}|\Sigma_{t-1}|^{1/2}}\, e^{-\frac{2}{\sigma^2}\sum\limits_{t=1}^{T}\left(\bi{y}_{t} - \bi{\mu}_{t-1}\right)^T\Sigma_{t-1}^{-1}\left(\bi{y}_{t} - \bi{\mu}_{t-1}\right)}, \end{align} where, for $j=1, 2, \ldots, T$, $(k,\ell)$th element of $\Sigma_{j-1}$ is $$\frac{2\eta_3 e^{-\eta_3 h_{k\ell}^2(j-1)}\left(1-2\eta_3 h_{k\ell }^2(j-1)\right)}{M_{s_k}M_{s_\ell}}.$$ The details of the calculation of the join density is provided in Section \ref{joint density of observed data} of Supplementary Information.
\subsection{Joint conditional density of latent data} \label{process model}
It can be shown that (see Section \ref{joint density of latent data} of Supplementary Information for the detailed calculation) $\left[\bi{x}_1, \ldots, \bi{x}_{T}\bigg\vert \bi{y}_0;\ldots; \bi{y}_T;\bi{x}_0;\bi{\theta}\right]$ is proportional to $\frac{(\sigma^2)^{-nT/2}}{\prod\limits_{t=1}^{T} |\Omega_t|^{1/2}} e^{-\frac{2}{\sigma^2}\sum\limits_{t=1}^T (\bi{x}_{t} - \alpha^2 \bi{x}_{t-1})^T\Omega_t^{-1}(\bi{x}_{t} - \alpha^2 \bi{x}_{t-1})}$. Specifically,
\begin{align}
\label{eq7: Latent model}
&\bigg[\mbox{Latent}\bigg\vert \bi{y}_0;\ldots; \bi{y}_T;\bi{x}_0;\bi{\theta}\bigg] \notag \\
& \propto [\bi{x}_1\vert \bi{x}_0; \bi{y}_0;\bi{y}_1;\bi{\theta}] \ldots [\bi{x}_T\vert \bi{x}_{T-1};\ldots; \bi{x}_0; \bi{y}_0;\ldots; \bi{y}_T;\bi{\theta}] \notag \\
& \propto \frac{(\sigma^2)^{-nT/2}}{\prod\limits_{t=1}^{T} |\Omega_t|^{1/2}} e^{-\frac{2}{\sigma^2}\sum\limits_{t=1}^T (\bi{x}_{t} - \alpha^2 \bi{x}_{t-1})^T\Omega_t^{-1}(\bi{x}_{t} - \alpha^2 \bi{x}_{t-1})},
\end{align}
where, for $m\in\{1,2,\ldots, T\}$,
$\Omega_t = \alpha^2\Sigma_{t-1,t-1} +\alpha \Sigma_{t-1,t} + \alpha \Sigma_{t,t-1} + \Sigma_{t,t},$ where the $(i,k)$th element of $\Sigma_{jj}$, for $j=t-1,t$, is $2\eta_3 e^{-\eta_3 h_{ik}^2(j)}\left(1-2\eta_3 h_{ik}^2(j)\right),$ and $(i,k)$th element of $\Sigma_{t-1,t} = \Sigma_{t,t-1}'$ is
$2\eta_3 e^{-\eta_3 \ell_{ik}^2(t-1,t)}$ $
\left(1-2\eta_3 \ell_{ik}^2(t-1,t)\right).$
\subsection{Complete likelihood combining observed and latent data}
\label{Jt Dist}
Next we will find the joint distribution of Data and Latent observations given $\bi{x}_0$, $\bi{y}_0$ and $\bi{\theta}$. Finally, using the prior distributions on $\theta_{s}(0)$ and $p_s(0)$ as mentioned in A1-A2, we shall obtain the complete joint distribution of $(\bi{y}_T, \ldots, \bi{y}_1, \bi{y}_0)$ and $(\bi{x}_T, \ldots, \bi{x}_1, \bi{x}_0)$, given the parameter $\bi{\theta}$.
The joint distribution of $(\bi{y}_T, \ldots, \bi{y}_1)$ and $(\bi{x}_T, \ldots, \bi{x}_1)$ given $(\bi{x}_0, \bi{y}_0, \bi{\theta})$, using equations (\ref{eq7: Latent model}) and (\ref{eq4}), is given by
\begin{align}
\label{eq8: jt conditional density}
\bigg[\bi{y}_T,\bi{x}_T,\ldots, \bi{y}_1,\bi{x}_1\bigg\vert \bi{y}_0, \bi{x}_0, \bi{\theta}\bigg] & = \bigg[\bi{y}_1\bigg\vert\bi{x}_0,\bi{y}_0,\bi{\theta}\bigg]
\bigg[\bi{x}_1\bigg\vert\bi{x}_0,\bi{y}_0,\bi{y}_1,\bi{\theta}\bigg]\ldots \bigg[\bi{y}_T\bigg\vert\bi{x}_{T-1},\bi{y}_{T-1},\bi{\theta}\bigg] \notag \\
& \quad~ \bigg[\bi{x}_T\bigg\vert\bi{x}_{T-1},\bi{y}_{T-1},\bi{y}_{T},\bi{\theta}\bigg]
\notag \\
& \propto \frac{(\sigma^2)^{-Tn/2} }{\prod\limits_{t=1}^{T}|\Sigma_{t-1}|^{1/2}} e^{-\frac{2}{\sigma^2}\sum\limits_{t=1}^{T} \left(\bi{y}_{t} - \bi{\mu}_{t-1}\right)^T\Sigma_{t-1}^{-1}\left(\bi{y}_{t} - \bi{\mu}_{t-1}\right)} \notag \\
& \quad
\frac{(\sigma^2)^{-nT/2}}{\prod\limits_{t=1}^{T} |\Omega_t|^{1/2}} e^{-\frac{2}{\sigma^2}\sum\limits_{t=1}^T (\bi{x}_{t} - \alpha^2 \bi{x}_{t-1})^T\Omega_t^{-1}(\bi{x}_{t} - \alpha^2\bi{x}_{t-1})}.
\end{align}
Now we will find the full joint distribution of $[\bi{y}_T, \ldots, \bi{y}_1,\bi{y}_0;\bi{x}_T, \ldots, \bi{x}_1, \bi{x}_0\vert \bi{\theta}]$ using the priors on $\theta_{s}(0)$ and $p_s(0)$. Again for simplicity we will assume that the $\theta_{s}(0)$ \& $p_s(0)$ are zero mean Gaussian processes with squared exponential functions as their covariance functions. In particular, we will assume that $\mbox{cov}(\theta_{s_1}(0),\theta_{s_1}(0)) = \sigma_{\theta}^2\exp\left\{-\eta_2 k^2\right\}$ (see assumption A2), and $\mbox{cov}(p_{s_1}(0),p_{s_1}(0))$ =
$\sigma_p^2 \exp\left\{-\eta_1 k^2\right\}$ (see assumption A1), where $k= ||s_1-s_2||$. Therefore, $[\bi{y}_0\vert \bi{\theta}]\sim N_{n}(\bi{0},\sigma_{\theta}^2 \Delta_0)$ and $[\bi{x}_0\vert \bi{\theta}]\sim N_{n}(\bi{0},\sigma_p^2 \Omega_0)$, where the $(i,j)$th element of $\Delta_0$ and $\Omega_0$ are $\exp\left\{-\eta_2 k_{ij}^2\right\}$ and $ \exp\left\{-\eta_1 k_{ij}^2\right\}$, respectively, with $k_{ij} = |s_i-s_j|.$ Thus,
\begin{align}
\label{eq9: complete joint}
\bigg[\bi{y}_T,\bi{x}_T,\ldots, \bi{y}_1,\bi{x}_1, \bi{y}_0, \bi{x}_0\bigg \vert \bi{\theta}\bigg] & = \left[\bi{y}_T,\bi{x}_T,\ldots, \bi{y}_1,\bi{x}_1\bigg\vert \bi{y}_0, \bi{x}_0, \bi{\theta}\right] \left[\bi{x}_0, \bi{y}_0\bigg\vert \bi{\theta}\right] \notag \\
& \propto \frac{(\sigma^2)^{-Tn} }{\prod\limits_{t=1}^{T}|\Sigma_{t-1}|^{1/2}|\Omega_t|^{1/2}} e^{-\frac{2}{\sigma^2}\sum\limits_{t=1}^{T} \left[\left(\bi{y}_{t} - \bi{\mu}_{t-1}\right)^T\Sigma_{t-1}^{-1}\left(\bi{y}_{t} - \bi{\mu}_{t-1}\right)+(\bi{x}_{t} - \alpha^2\bi{x}_{t-1})^T\Omega_t^{-1}(\bi{x}_{t} - \alpha^2\bi{x}_{t-1})\right]} \notag \\
& \quad \frac{(\sigma_{\theta}^2)^{-n/2}}{|\Delta_0|^{1/2}} \frac{(\sigma_{p}^2)^{-n/2}}{|\Omega_0|^{1/2}} \exp\left\{-\frac{1}{2\sigma_{\theta}^2}\bi{y}_0'\Delta_0^{-1}\bi{y}_0 \right\} \exp\left\{-\frac{1}{2\sigma_p^2}\bi{x}_0'\Omega_0^{-1}\bi{x}_0 \right\}.
\end{align}
\section{Prior distributions} \label{Priors}
In this section we will specify the prior distributions of the components of $\bi{\theta} = (\alpha, \beta,\sigma^2,\sigma_{\theta}^2,\sigma_p^2, \eta_1,\eta_2,\eta_3)'$. The parameter spaces of the each component of $\bi{\theta}$ are the following:
$|\alpha|<1$, $|\beta|<1$, and $0<\sigma^2, \sigma^2_{\theta}, \sigma^2_{p}, \eta_{1}, \eta_{2}, \eta_{3}<\infty$. We make the following transformations on $\alpha$, $\beta$, $\eta_{i},$ for $i=1,2,3$ for better MCMC mixing. Define $\alpha^* = \log\left(\frac{1+\alpha}{1-\alpha}\right)$, $\beta^* = \log\left(\frac{1+\beta}{1-\beta}\right)$, and $\eta_i^* = \log(\eta_i),$ for $i=1,2,3$, so that $\alpha^*, \beta^*, \eta_i^* \in \mathbb{R}$, for $i=1,2,3$. This implies $\alpha = 1- \frac{2e^{\alpha^*}}{1+e^{\alpha^*}}$, $\beta = 1- \frac{2e^{\beta^*}}{1+e^{\beta^*}}$, $\eta_i = e^{\eta_i^*}$, $i=1,2,3$, respectively. We assume that the prior distributions are independent. Since the parameter space of $\alpha^*$ and $\beta^*$ are $\mathbb{R}$ and they are involved in the mean function of our proposed model, the prior for $\alpha^*$ and $\beta^*$ are taken as normal with the mean 0 and large variances (of order 100). Particular choices of the prior variances are discussed in Sections \ref{Sec: Simulation Studies} and \ref{real data analysis}. Moreover, the parameter spaces of $\eta_i^*$, $i=1,2,3$ are also $\mathbb{R}$ and they are involved in the covariance structure of our proposed model in the sense that they determine the amount of correlations between spatial and temporal points. So, we take the priors for $\eta^*_i$, $i=1,2,3$, as normal with the means $\mu_{\eta_{i}}$ and variances 1. Larger variance of the $\eta^*_i$ made the variances of their posterior distributions unreasonably large due to huge data variability. Nevertheless, the prior variance for the $\eta_i$ turn out to be 4.671, which is not too small. It also turned out that in all the simulation and real data analyses, the choice of variance 1 (for $\eta^*_i$) rendered good mixing properties to our MCMC sampler. The exact value of the hyper-prior means depend upon the data under consideration and is discussed in Sections \ref{Sec: Simulation Studies} and \ref{real data analysis}. Finally, the prior distribution of variance parameters are taken to be inverse-gamma, as they are the conjugate priors, conditionally. It is expected that the variability of the spatio-temporal data is very large which might possibly render the posterior means and variances of $\sigma^2_{\theta}$, $\sigma^2_{p}$, $\sigma^2$ very large (specially for $\sigma^2$). So we decided to choose the hyper-parameters in such a way that the prior means (with some exceptions in the mean in a few cases) and variances are both close to zero. The exact choices of the hyper-parameters of priors of $\sigma^2_p$, $\sigma^2_{\theta}$ and $\sigma^2$ are mentioned in Sections \ref{Sec: Simulation Studies} and \ref{real data analysis}.
The general forms of the prior distributions of $\alpha^*, \beta^*, \sigma^2, \sigma^2_{\theta}, \sigma_p^2, \eta_{i}^*,$ $i=1,2,3$, are taken as follows: \allowdisplaybreaks \begin{align*}
&[\alpha^*] \propto N(0, \sigma_{\alpha}), [\beta^*] \propto N(0,\sigma_{\beta}^2)\\
&[\sigma^2] \propto IG(\alpha_v,\gamma_v/2), [\sigma^2_{\theta}] \propto IG (\alpha_{\theta},\gamma_{\theta}/2), [\sigma^2_{p}] \propto IG (\alpha_{p},\gamma_{p}/2) \\
&[\eta_{1}^*] \propto N (\mu_{\eta_{1}},1), [\eta_{2}^*] \propto N (\mu_{\eta_{2}},1)
[\eta_{3}^*] \propto N (\mu_{\eta_{3}},1) \end{align*} where IG stands for inverse gamma distribution.
\section{Full conditional distributions of the parameters and latent variables, given the observed data} \label{full conditional}
In this section we will obtain the full conditional distributions of the parameters, which will be used for generating samples from posterior distributions of the parameters using Gibbs sampling steps when feasible or Metropolis-Hastings steps otherwise. The detailed calculations are provided in Section \ref{Appendix B: full conditional densities} of Supplementary Information.
\\[2mm] \textbf{\underline{Full conditional distribution of $\beta^*$}}
The full conditional density of $\beta^*$, is given by \begin{align}
\label{eq1: fcd of beta}
[\beta^*\vert \ldots]
\propto\pi(\beta^*) g_1(\beta^*), \end{align} where $\pi(\beta^*) = e^{-\frac{{\beta^*}^2}{2\sigma_{\beta^*}^2}}$ and $g_1(\beta^*) = e^{-\frac{2\beta^2}{\sigma^2} \sum\limits_{t=1}^{T} \bi{y}_{t-1}'\Sigma_{t-1}^{-1}\bi{y}_{t-1} + \frac{4\beta}{\sigma^2} \sum\limits_{t=1}^{T} \bi{y}_{t}'\Sigma_{t-1}^{-1}\bi{y}_{t-1}},$ where $\beta = 1- \frac{2e^{\beta^*}}{1+e^{\beta^*}}$. The closed form of the full conditional density for $\beta^*$ is not available and thus we shall update it using a random walk Metropolis step.
\\[2mm] \noindent \textbf{\underline{Full conditional distribution of $\alpha^*$}}
The full conditional density of $\alpha^*$ is given by \begin{align}
\label{fcd of alpha*}
[\alpha^*\vert \ldots] &\propto e^{-\frac{{\alpha^*}^2}{2\sigma_{\alpha^*}^2}} g_2(\alpha^*), \end{align}
where $g_2(\alpha^*) = \frac{1}{\prod\limits_{t=1}^T |\Omega_{t}|^{1/2}} e^{-\frac{2}{\sigma^2}\sum\limits_{t=1}^T \left[ (\bi{x}_t-\alpha^2\bi{x}_{t-1})^T\Omega_{t}^{-1}(\bi{x}_t-\alpha^2\bi{x}_{t-1})\right]} \times e^{-\frac{2\alpha}{\sigma^2}\sum\limits_{t=1}^T \left[\alpha \bi{x}_{t-1}^TD\Sigma_{t-1}^{-1}D\bi{x}_{t-1}-2\bi{y}_t^T\Sigma_{t-1}^{-1}D\bi{x}_{t-1}\right]},$ and $\alpha = 1- \frac{2e^{\alpha^*}}{1+e^{\alpha^*}}$. However, the closed form is not available and hence will be updated using random walk Metropolis, similar to $\beta^*$.
\\[2mm] \noindent \textbf{\underline{Full conditional distribution of $\sigma_{\theta}^2$}}
The full conditional distribution of $\sigma_{\theta}^2$ is IG$\left(\alpha_{\theta}+n/2, \frac{\gamma_{\theta}+\bi{y}_0'\Delta_0^{-1}\bi{y}_0}{2}\right)$. Hence it is updated using a Gibbs sampling step.
\\[2mm] \noindent \textbf{\underline{Full conditional distribution of $\sigma_{p}^2$}}
The full conditional distribution of $\sigma_p^2$ is IG$\left(\alpha_{p}+n/2, \frac{\gamma_{p}+\bi{x}_0'\Delta_0^{-1}\bi{x}_0}{2}\right)$. Therefore, $\sigma^2_p$ is updated using a Gibbs sampling step.
\\[2mm] \noindent \textbf{\underline{Full conditional distribution of $\sigma^2$}}
The full conditional distribution of $\sigma^2$ is inverse-Gamma with parameters $\alpha_v+Tm$ and $\gamma_v/2 + 2\zeta$, where $\zeta = \sum\limits_{t=1}^T \left[(\bi{y}_t - \bi{\mu}_{t})^T\Sigma_{t-1}^{-1} (\bi{y}_t - \bi{\mu}_{t}) + (\bi{x}_t - \alpha^2\bi{x}_{t-1})^T\Omega_{t}^{-1} (\bi{x}_t - \alpha^2\bi{x}_{t-1})\right].$ Thus, it is updated using a Gibbs sampling step.
\\[2mm] \noindent \textbf{\underline{Full conditional distributions of $\eta_1^*$, $\eta_2^*,$ and $\eta_3^*$}}
None of the full conditional distributions of $\eta_1^*$, $\eta_2^*$ or $\eta_3^*$ have close form and hence they are updated random walk Metropolis. The complete calculations of the full conditionals for $\eta_i^*$, for $i=1,2,3$, are given in Section \ref{Appendix B: full conditional densities} of Supplementary Information.
\\[2mm] \noindent \textbf{\underline{Full conditional distributions of $\bi{x}_m$, $m = 1,2,\ldots, T$}}
From the equation (\ref{eq7: Latent model}), we immediately see that $[\bi{x}_t\vert \ldots] \sim N_n\left(\alpha^2\bi{x}_{t-1}, \frac{\sigma^2}{4}\Omega_{t}\right)$, for $t=1, \ldots, T$.
\\[2mm] \noindent \textbf{\underline{Full conditional distribution of $\bi{x}_0$}}
With $A = \Omega_0^{-1}+\frac{4\sigma_p^2\alpha^4}{\sigma^2} \Omega_1^{-1}+ \frac{4\sigma_p^2 \alpha^2}{\sigma^2} D\Sigma_{0}^{-1}D$, $B = \frac{4\sigma_p^2\alpha^2}{\sigma^2}\Omega_1^{-1}$ and $C = \frac{4\sigma_p^2 \alpha}{\sigma^2}D\Sigma_0^{-1}$, the full conditional density of $\bi{x}_0$ is found to be a $n-$variate normal with the mean $A^{-1}(B\bi{x}_1+C(\bi{y}_1 - \beta \bi{y}_0))$ and the variance-covariance matrix $\sigma_p^2A^{-1}$. Hence, it is updated using a Gibbs sampling step.
\section{Simulation Studies} \label{Sec: Simulation Studies}
We conducted three simulation experiments to evaluate the performance of our model and methods. In these experiments, we fitted our spatio-temporal model to data generated from a linear dynamic spatio-temporal model (LDSTM), a nonlinear dynamic spatio-temporal model (NLDSTM) and a general quadratic nonlinear model (GQM). For brevity, here we report the simulation experiments with data generated from NLDSTM and GQN only, while our experiments on fitting our model to LDSTM is reported in Section \ref{LDSTM simulation} of Supplementary Information.
In all the simulation experiments, we chose the number of locations to be 50 and the number of time points to be 20. The last time point is kept aside for checking the performance of our proposed model for the purpose of prediction. We ran our MCMC sampler for 1,75,000 iterations with a burn-in period 1,50,000 iterations for each analysis. The MCMC computations were carried out in MATLAB R2018a. For each model-fitting it took about 3 hours 49 minutes in a desktop computer with the following specifications: 8GB RAM, 1 TB Hard Drive and 3.8 GHz core\_ i5.
\subsection{Simulation experiment with data from NLDSTM} \label{NLDSTM simulation} We test our model performance on a data set simulated from NLDSTM; in particular, we choose the power-transform based NLDSTM given by (see also \cite{Suman2017}): \begin{align*} &Y(s_i,t) = \begin{cases} X(s_i,t)^b &\text{ if } X(s_i,t)>0, \\ 0 &\text{ if } X(s_i,t)\leq 0 \end{cases} \\ &X(s_i,t) = \gamma_0 + \gamma_1 X(s_i,t-1) + \zeta(s_i,t) \\ &\{X(s_i,0):~i=1,\ldots,n\} \sim N(\bi{0},\Sigma_0)\\ &\{\zeta(s_i,t):~i=1,\ldots,n\}\stackrel{iid}{\sim}N(\bi{0}, \Sigma_{\eta}). \end{align*} The above model form has been used for rain-fall modelling by \cite{sanso1999venezuelan}.
The covariance matrices $\Sigma_0$ and $\Sigma_{\eta}$ are generated by exponential covariance function
of the form $c(\bi{u},\bi{v}) = \sigma^2\exp\left(-\lambda ||\bi{u}-\bi{v}||\right)$, with $\sigma=1$, $\lambda = 1$. The choices of the other parameters are as follows: $\gamma_ 0 =1$, $\gamma_1 = -0.8$ and $b=3$.
We generate $n=50$ locations in [0,1]$\times$[0,1] and take $T=20$ time points. Latent variables and observed variables, $\bi{x}_t = (x(s_1,t), \ldots, x(s_n,t))^T$ and $\bi{y}_t = (y(s_1,t), \ldots, y(s_n,t))^T$, respectively, for $t=1, \ldots, 20$, are simulated from the above model. Keeping the last time point for the purpose of prediction, the data set, which is used for Bayesian inference, is $\mathbb{D} = (\bi{y}_1, \ldots \bi{y}_{19})$.
Although we have chosen a finite prior variance for $\eta_3^*$, the posterior variance of $\eta_3$ turned out to be very large, which is probably due to huge variability present in the spatio-temporal data. This large posterior variance of $\eta_3$ makes the complete system unstable, so, we fix the value of $\eta_3$ at its maximum likelihood estimate (7.0020) computed by simulated annealing. Note that it is not very uncommon to fix the decay parameter in Bayesian inference of spatial data analysis, see for example \cite{zhang2004inconsistent}, \cite{banerjee2020modeling} and the citations therein.
\subsubsection{Choice of prior parameters} \label{choice of priors for NLDSTM} The particular choices of the hyper-prior parameters have been chosen using the leave-one-out cross-validation technique. Specifically, for $t=1,\ldots,20$, we leave out $\bi{y}_t$ in turn and compute its posterior predictive distribution using relatively short MCMC runs. We then selected those hyperparameters that yielded the minimum average length of the 95\% prediction intervals. As such, the complete prior specifications are provided as follows: \begin{align*} & \alpha^* \sim N(0,\sqrt{500}), \beta^* \sim N(0,\sqrt{300})\\
& [\sigma^2] \propto IG(850000,2/2), [\sigma^2_{\theta}] \propto IG (590,780/2), [\sigma^2_{p}] \propto IG (90,100/2), \\ & [\eta_{1}^*] \propto N (-3,1), [\eta_{2}^*] \propto N (-5,1). \end{align*}
\subsubsection{MCMC convergence diagnostics and posterior analysis} The trace plots of the parameters are given in Figure~\ref{fig:trace plot NLDSTM} of the Supplementary Information. Clearly, the plots exhibit strong evidence of convergence of our MCMC algorithm.
The posterior probability densities for the latent variables at 50 locations for 19 time points are shown in Figures~\ref{fig:posterior density of latent variables NLDSTM: first 25 locations} and \ref{fig:posterior density of latent variables NLDSTM: last 25 locations} of the Supplementary Information, where higher probability densities are depicted by progressively intense colours. We observe that true latent time series at all the 50 locations always lie in the high probability density regions.
For the purpose of prediction, we kept $\bi{y}_{20}$ out of our data set $\mathbb{D}$. The posterior predictive densities for $\bi{y}_{20}$ at 50 locations are shown in Figure~\ref{fig: predictive density for 50 locations NLDSTM} (first 25 locations in Figure \ref{fig:Predictive densities for first 25 locations NLDSTM} and last 25 locations in Figure \ref{fig:Predictive densities for last 25 locations NLDSTM}). We observe that all the true values lie within the 95\% predictive intervals at every location (see Figures \ref{fig:Predictive densities for first 25 locations NLDSTM} and \ref{fig:Predictive densities for last 25 locations NLDSTM}). Further, we obtained the posterior predictive densities for $\bi{x}_{20}$ given $\mathbb{D}$. The plots are provided in Figure \ref{fig: predictive density of latent variable for 50 locations NLDSTM}. Here also we notice that the true values lie well within the 95\% credible intervals (see Figures \ref{fig:Predictive densities of latent variable for first 25 locations NLDSTM} and Figure \ref{fig:Predictive densities of latent variable for last 25 locations NLDSTM}).
\begin{figure}
\caption{Predictive densities for first 25 locations}
\label{fig:Predictive densities for first 25 locations NLDSTM}
\caption{Predictive densities for last 25 locations}
\label{fig:Predictive densities for last 25 locations NLDSTM}
\caption{Predictive densities of the $\bi{y}_{20}$ for 50 locations for data simulated from NLDSTM. The red horizontal lines denote the 95\% predictive intervals. The vertical black lines indicate the true values. All the true values, except two (marginally outside), lie within the 95\% predictive intervals. }
\label{fig: predictive density for 50 locations NLDSTM}
\end{figure}
\begin{figure}
\caption{Predictive densities of latent variable for first 25 locations for NLDSTM.}
\label{fig:Predictive densities of latent variable for first 25 locations NLDSTM}
\caption{Predictive densities of latent variable for last 25 locations for NLDSTM.}
\label{fig:Predictive densities of latent variable for last 25 locations NLDSTM}
\caption{Predictive densities of the $\bi{x}_{20}$ for 50 locations for NLDSTM. The red horizontal lines denote the 95\% predictive interval. The vertical black lines indicate the true values. All the true values lie within the 95\% predictive interval. }
\label{fig: predictive density of latent variable for 50 locations NLDSTM}
\end{figure}
\subsection{Simulation details and results for data simulated from GQN} \label{GQN} As an illustration of our proposed model and inference techniques on a non-linear model we choose to simulate data from the GQN model (see \cite{wikle2010general}, \cite{cressie2015statistics}, and \cite{bhattacharya2021bayesian} for the details on the GQN model). The model from which we simulated observations with $T=20$ time points and $n=50$ locations is as follows: \begin{align*} Y(s_i,t_k) &= \phi_{1}(t_{k},s_i) + \phi_{2}(t_{k},s_i)\, \tan(X(s_i,t_k)) + \epsilon(s_i,t_k), \\ X(s_i,t_k) & = \sum_{j=1}^{n} a_{ij} X(s_j,t_{k-1}) + \sum_{j=1}^{n}\sum_{l=1}^{n} b_{ijl} X(s_j,t_{k-1}) [X(s_l,t_{k-1})]^2 +\eta(s_i,t_k), \end{align*} where $i\in\{1, \ldots,n\}$ and $k\in\{1, \ldots, T\}$. As earlier, we kept the last time point out of the data set, at each 50 locations, for the purpose of prediction. Thus, we use observations on 50 locations and 19 time points for fitting the model. The coefficients $\phi_{1}(t_{k},\cdot)$, $\phi_{2}(t_{k},\cdot)$, random errors $\epsilon(t_k,\cdot)$, $\eta(t_k,\cdot)$ and the initial latent variable $X(t_0,\cdot)$ are assumed to be independent, zero-mean Gaussian processes having covariance structure
$c(\bi{s}_1, \bi{s}_2) = \exp\left(-\|\bi{s}_1-\bi{s}_2\|\right)$, for $\bi{s}_1, \bi{s}_2 \in \mathbb{R}^2$, with $\|\cdot\|$ being the Euclidean norm. Moreover, it is assumed that, for $i, j, l\in \{1, \ldots, n\}$, $a_{ij}$ and $b_{ijl}$ have independent univariate 0 mean normal distributions with variance $0.001^2$. As before, locations $s_i$ are simulated independently from $U(0,1)\times U(0,1)$.
In this case as well, we fixed $\eta_3$ at its maximum likelihood estimate 10.5853, obtained using simulated annealing.
\subsubsection{Choice of hyper parameters} \label{choice of hyperparam in GQN} Using the same cross-validation technique as before, we obtain the following prior specifications: \begin{align*} &\alpha^* \sim N(0,\sqrt{500}), \beta^* \sim N(0,\sqrt{300})\\ & [\sigma^2] \propto IG(750000,2/2), [\sigma^2_{\theta}] \propto IG (50000,780/2), [\sigma^2_{p}] \propto IG (900,100/2) \\ &[\eta_{1}^*] \propto N (-3,1), [\eta_{2}^*] \propto N (-5,1). \end{align*}
\subsubsection{MCMC convergence diagnostics and posterior analysis} The trace plots of the parameters are provided in the Figure~\ref{fig:trace plot GQN}, and the posterior density plot of $\bi{x}_t, t=1, \ldots, 19$ are displayed in Figures~\ref{fig:posterior density of latent variables GQN: first 25 locations} and \ref{fig:posterior density of latent variables GQN: last 25 locations} of Supplementary Information. The trace plots provide strong evidence of MCMC convergence. Further, the posterior density plots show that the posteriors of the latent variables contain the true latent variable time series at all the 50 locations in the high probability density regions successfully.
The predictive density plots of $\bi{y}_{20}$ are given in Figure~\ref{fig: predictive density for 50 locations GQN} and that of $\bi{x}_{20}$ are shown in Figures~\ref{fig: predictive density of latent variable for 50 locations GQN}. Other than only one location, all the true values of $\bi{y}$ fall well within the 95\% credible intervals (see Figures~\ref{fig:Predictive densities for first 25 locations GQN} and \ref{fig:Predictive densities for last 25 locations GQN}). Moreover, it is seen from Figures~\ref{fig:Predictive densities of latent variable for first 25 locations GQN} and \ref{fig:Predictive densities of latent variable for last 25 locations GQN}, that the true of values of $\bi{x}_{20}$ fall within the 95\% credible intervals at each of the 50 locations.
Thus, our proposed model successfully captures the variability of the data generated from a complicated model like GQN.
\begin{figure}
\caption{Predictive densities for the the first 25 locations.}
\label{fig:Predictive densities for first 25 locations GQN}
\caption{Predictive densities for the last 25 locations.}
\label{fig:Predictive densities for last 25 locations GQN}
\caption{Predictive densities of $\bi{y}_{50}$ for 50 locations for data simulated from GQN. The red horizontal lines denote the 95\% predictive intervals. The vertical black lines indicate the true values. All the true values, except one, lie within the 95\% predictive intervals. }
\label{fig: predictive density for 50 locations GQN}
\end{figure}
\begin{figure}
\caption{Predictive densities of latent variable for the first 25 locations.}
\label{fig:Predictive densities of latent variable for first 25 locations GQN}
\caption{Predictive densities of latent variable for the last 25 locations.}
\label{fig:Predictive densities of latent variable for last 25 locations GQN}
\caption{Predictive densities of $\bi{x}_{50}$ for 50 locations for data simulated from GQN. The red horizontal lines denote the 95\% predictive intervals. The vertical black lines indicate the true values. All the true values lie within the 95\% predictive intervals. }
\label{fig: predictive density of latent variable for 50 locations GQN}
\end{figure}
\section{Real data analysis} \label{real data analysis}
We evaluate our model performance on two real data sets on temperatures. The first spatio-temporal data is the temperature values taken around Alaska recorded for 65-70 years. The second data set is sea surface temperatures over a wide range of areas noted for 100 months. The details of the data sets are provided in Sections \ref{Alaska} and \ref{sea temp}, respectively.
Before applying our model on these data sets, we first make the following transformation (Lambert projection, see \cite{Suman2017}) of the locations so that the Euclidean distance make more sense. Let $ \phi$ be longitude and $\psi$ be latitude in radian. Then the following transformation is made: \begin{align}
\label{Lambert}
s_1 & = 2 \sin\left( \frac{\pi}{4} - \frac{\psi}{2}\right)\sin \phi \\
s_2 & = -2 \sin\left( \frac{\pi}{4} - \frac{\psi}{2}\right)\cos \phi. \end{align}
\subsection{Alaska temperature data} \label{Alaska}
A real data analysis is done on the temperature data of Alaska and its surroundings. The data set is collected from \url{https://www.metoffice.gov.uk/hadobs/crutem4/data/download.html} by clicking the link \url{CRUTEM.4.6.0.0.station_files.zip} given under the heading \textbf{Station data}. The details of the data set can be read from \url{https://crudata.uea.ac.uk/cru/data/temperature/crutem4/station-data.htm}.
A total of 30 locations are considered for the analysis. Annual average temperature data for the years 1950 to 2015 are taken after detrending. Of these 30 locations, at four locations many data were missing. So, we decided to construct the complete time series for these four locations. Among these 30 locations, data till 2021 were available for 16 positions. We thus have made a multiple time predictions for these 16 locations. The 26 spatial points are indicated in Figure~\ref{fig:Alaska} in red and 4 locations (for which the complete time series is reconstructed) are indicted in blue in the same graph. The latitudes and longitudes are provided in Table~\ref{locations of Alaska temp data}.
\begin{table}[!h]
\centering
\small
\resizebox{520pt}{32pt}{
\begin{tabular}{|c | c c c c c c c c c c c c c c c|}
\hline
Sl No. & \textbf{1} & \textbf{2} & \textbf{3} & \textbf{4} & \textbf{5} & \textbf{6} & \textbf{7} & 8 & 9 & 10 & \textbf{11} & \textbf{12} & \textbf{13} & 14 & 15 \\ \hline
Latitude & 71.3N & 60.6N & 59.5N & 55.2N & 61.2N & 70.1N & 64.5N & 66.9N & 59.6N & 60.5N & 64.8N & 60.1N & 63N & 63N & 55N
\\ \hline
Longitude & 156.8W & 151.3W & 139.7W & 162.7W & 150W & 143.6W & 165.4W & 151.5W & 151.5W & 145.5W & 147.9W & 149.5W & 155.6W & 141.9W & 131.6W
\\ \hline \\
\hline
Sl No. & \textbf{16} & 17 & \textbf{18} & \textbf{19} & \textbf{20} & 21 & \textbf{22} & 23 & 24 & \textbf{\textcolor{blue}{25}} & \textbf{\textcolor{blue}{26}} & \textbf{27} & \textbf{\textcolor{blue}{28}} & \textbf{29} & \textbf{\textcolor{blue}{30}} \\ \hline
Latitude & 61.6N & 64N & 60.8N & 57.8N & 58.7N & 62.2N & 66.9N & 58.4N & 63.7N & 68.2N & 59.6N & 62.8N & 64.1N & 67.4N & 60.7N
\\ \hline
Longitude & 149.3W & 145.7W & 161.8W & 152.5W & 156.7W & 145.5W & 162.6W & 134.6W & 149W & 135W & 133.7W & 137.4W & 139.1W & 134.9W & 135.1W \\
\hline
\end{tabular}}
\caption{Latitude and Longitude (in degrees) of 30 locations in and around Alaska. The locations corresponding to the serial numbers, indicated in bold, are used for multiple predictions. The serial numbers which are denoted in blue, for the corresponding spatial locations, complete time series are reconstructed. These blue serial numbers are indicated by $L_{25}, L_{26}, L_{28}$ and $L_{30}$ for future references.}
\label{locations of Alaska temp data} \end{table}
\begin{figure}
\caption{The map of Alaska and its surroundings. The red dots indicate the locations at which the data for the year 1950-2015 are considered. The positions indicated by blue dots at which the spatial prediction are made for the complete time series.}
\label{fig:Alaska}
\end{figure}
Thus the data set considered here contains 26 locations and 65 time points from 1950 to 2014 (last time point is set aside for purpose of single time point prediction). We denote the data set by $\mathbb{D} = \{\bi{y}_1, \ldots, \bi{y}_{65}\}$, where $\bi{y}_{t}$, $t\in \{1, \ldots, 65\}$, is a 26 dimensional vector. In this analysis, we have made both temporal and spatial predictions after detrending the data. At these 26 locations, we obtained the predictive densities for the year 2015 and for 16 locations (indicated by $L_1, \ldots, L_{7}$, $L_{11}\ldots, L_{13}$, $L_{18}, \ldots, L_{20}$, $L_{22}$, $L_{27}$, $L_{29}$ for references hereafter) we made multiple time point predictions. Along with these temporal predictions, we constructed the 95\% predictive intervals for the complete time series for the 4 left out locations. To do this, we augmented the data set $\mathbb{D}$ with initial values for 65 time points of the four locations and then updated the values by calculating the conditional densities given the data set $\mathbb{D}$. That is to say, we start with $\mathbb{D}^* = \{\bi{y}_{1}^*, \ldots, \bi{y}_{65}^*\}$, where $\bi{y}_{t}^{*}$ is a 30 dimensional vector, with the last four values augmented with $\bi{y}_{t}$, for $t\in \{1, \ldots, 65\}$. The parameters of our model, including the latent variables, are updated given $\mathbb{D}^*$, and then the last four values of $\bi{y}^*_{t}$ are updated given the parameter values and latent variables. This is continued for the complete MCMC run.
In Section \ref{appendix:alaska} of Supplementary Information, we validate that the underlying spatio-temporal process that generated the Alaska data is non-Gaussian, strictly stationary, and the lagged correlations converge to zero as the lags tend to infinity. Besides, there is no reason to assume separability of the spatio-temporal covariance structure. Although our spatio-temporal process is nonstationary, it is endowed with the other desirable features, and the results of analysis of the Alaska data shows that it is an appropriate mode for this data.
\subsubsection{Prior choices for the Alaska temperature data}
\label{choice of prior param for Alaska temp data} With the same cross-validation technique as in the simulation experiments, the complete specifications of the priors are obtained as follows: \begin{align*}
& \alpha^* \sim N(0,\sqrt{500}), \beta^* \sim N(0,\sqrt{300}),\\
& [\sigma^2] \propto IG(450000,2/2), [\sigma^2_{\theta}] \propto IG (700,780/2),
[\sigma^2_{p}] \propto IG (250,100/2), \\
& [\eta_{1}^*] \propto N (-3,1), [\eta_{2}^*] \propto N (-5,1). \end{align*} As before, we fix $\eta_3$ at its maximum likelihood estimate 5.0581.
\subsubsection{Results of the Alaska temperature data} \label{results: Alaska temp data} As for the case of the simulation studies, we implemented 1,75,000 MCMC iterations with the first 1,50,000 iterations as burn-in. The time taken was about 5 hours 43 minutes in our desktop computer. The MCMC trace plots for the parameters, except $\eta_3$, are given in Figure~\ref{fig:trace plot of Alaska temp data} of Supplementary Information, which bear clear evidence of convergence in each case. The posterior densities of the latent variables are displayed in Figures \ref{fig:posterior density of latent variables first 13 Alaska temp data} of Supplementary Information.
The predictive densities for $\bi{y}_{66}$ for 26 locations are depicted in Figure~\ref{fig: predictive density for 15 locations Alaska temp data}. All the true values (detrended temperature values), which are denoted by black bold vertical lines, lie well within the 95\% predictive intervals (indicated as red bold horizontal line in the figure). The predictive densities for the years 2016-2021 at the 16 locations are shown in two figures (Figure~\ref{fig: predictive density for first 8 locations Alaska temp data for multiple time points} and Figure~\ref{fig: predictive density for last 8 locations Alaska temp data for multiple time points}), each containing 8 locations. Except for one location, at one time point, in all the other scenarios, the true value is captured by the 95\% predictive interval associated with our model.
The complete time series for the four locations, which were indicated in blue in Figure~\ref{fig:Alaska}, are reconstructed using our model. The Bayesian predictive densities at each time point for these locations are shown in Figure~\ref{fig:posterior predictive density of temp time series of 4 locations Alaska temp data} using probability plot. Higher the intensity of the color higher is the density. The available true detrended temperature values at these locations are plotted and depicted by black stars in Figure~\ref{fig:posterior predictive density of temp time series of 4 locations Alaska temp data}. At one location ($L_{30}$), three values lie away from the high density region. However, from the pattern of the data values, it seems that these values are outlying in comparison to the rest of the values. Other than these, all the available true detrended temperatures lie within high density regions (except one value at $L_{28}$).
In a nutshell, we can claim that our model performs well in analyzing the temperature data of Alaska and its surroundings.
\begin{figure}
\caption{Posterior predictive densities of the detrended temperatures for the year 2015 at 26 locations for the Alaska temperature data. The red horizontal lines denote the 95\% predictive intervals. The vertical black lines indicate the true values. All the true values lie within the 95\% posterior predictive intervals.}
\label{fig: predictive density for 15 locations Alaska temp data}
\end{figure}
\begin{figure}
\caption{Predictive densities of the detrended temperatures for the years 2016-2021 at 8 locations for the Alaska temperature data. The eight locations are indicated by $L_1\, \ldots, L_7$, and $L_{11}$ at the left side of the figure corresponding to the rows. The years are shown at the top of the figure corresponding to the columns. The red horizontal line in each plot denotes the 95\% posterior predictive interval. The vertical black lines indicate the true values. All the true values, except one, lie within the 95\% posterior predictive intervals.}
\label{fig: predictive density for first 8 locations Alaska temp data for multiple time points}
\end{figure}
\begin{figure}
\caption{Predictive densities of the detrended temperatures for the years 2016-2021 at \textit{other} 8 locations for the Alaska temperature data. The eight locations are indicated by $L_{12}$, $L_{13}$, $L_{18}, L_{19}, L_{20}$, $L_{22}$, $L_{27}$ and $L_{29}$ at the left side of the figure corresponding to the rows. The years are shown at the top of the figure corresponding to the columns. The red horizontal line in each plot denotes the 95\% posterior predictive interval. The vertical black lines indicate the true values. All the true values lie within the 95\% posterior predictive interval.}
\label{fig: predictive density for last 8 locations Alaska temp data for multiple time points}
\end{figure}
\begin{figure}
\caption{Posterior predictive densities of the reconstructed detrended temperature data of Alaska and its surroundings at 8 locations. Higher the intensity of the colour, higher is the probability density. The black stars represent the available true temperature values (detrended). Except for 3 to 4 points at $L_{30}$, and one point at $L_{28}$, all the other points fall within the high probability density region.}
\label{fig:posterior predictive density of temp time series of 4 locations Alaska temp data}
\end{figure}
\subsection{Sea surface temperature data} \label{sea temp}
The sea surface temperature data is obtained from \url{http:
//iridl.ldeo.columbia.edu/SOURCES/.CAC/}. For the purpose of illustration, we have taken spatio-temporal observations of the first 40 locations and the first 100 time points from the complete data set. The locations of the chosen data set varies from 56$^{\circ}$W to 110$^{\circ}$E and 19$^{\circ}$S to 25$^{\circ}$N. The locations are given in Table \ref{locations of sea temp data}. Hundred monthly average temperatures are taken starting from the month of January 1970. Among these 40 locations and 100 time points, observations corresponding to 30 locations and 99 time points are taken as the learning set. The observations corresponding to the last 10 locations and the last time point are kept aside for the prediction purpose. Only for the sake of convenience, the time series at each location is referenced to its mean, in the sense that the mean of the time series at each location is subtracted from the original data.
\begin{table}[!h]
\centering
\small
\resizebox{520pt}{90pt}{
\begin{tabular}{|c | c c c c c c c c c c |}
\hline
Sl No. & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\ \hline
Latitude & 19S & 19N & 19S & 17N & 7N & 11S & 3S & 25S & 25N & 7N
\\ \hline
Longitude & 108E & 20W & 42E & 58E & 2W & 76E & 2E & 28W & 100E & 26W
\\ \hline \\
\hline
Sl No. & 11 & 12 & 13 & 14 & 15 & 16 & 17 & 18 & 19 & 20 \\ \hline
Latitude & 1N & 19S & 13N & 27N & 5S & 17S & 7S & 23S & 21S & 1S
\\ \hline
Longitude & 84E & 14E & 48E & 18W & 110E & 46W & 10W & 14E & 36W & 28W \\
\hline \\
\hline
Sl No. & 21 & 22 & 23 & 24 & 25 & 26 & 27 & 28 & 29 & 30 \\ \hline
Latitude & 13N & 7S & 29N & 15S & 1N & 1S & 13S & 9N & 15N & 25S
\\ \hline
Longitude & 30E & 36E & 104E & 24W & 72E & 46W & 64E & 12E & 24E & 52E \\
\hline
\\
\hline
Sl No. & \textbf{31} & \textbf{32} & \textbf{33} & \textbf{34} & \textbf{35} & \textbf{36} & \textbf{37} & \textbf{38} & \textbf{39} & \textbf{40} \\ \hline
Latitude & 25N & 7N & 11N & 5N & 7N & 19S & 1S & 17S & 19S & 29N
\\ \hline
Longitude & 4E & 52E & 18E & 16W & 56W & 106E & 20W & 28W & 0 & 48W \\
\hline
\end{tabular}}
\caption{Latitude and Longitude (in degrees) of 40 locations for the sea surface temperature data. The locations corresponding to the serial numbers,
indicated in bold, are used for complete time series prediction. These bold serial numbers are indicated as $L_{31}, \ldots, L_{40}$ for the future reference.}
\label{locations of sea temp data} \end{table}
As in the Alaska temperature data analysis, we denote the learning data set by $\mathbb D = \{\bi{y}_1, \ldots, \bi{y}_{99}\}$, where $\bi{y}_{t}$ is a 30 dimensional spatial observation at time point $t$. At these 30 locations, we obtain the posterior predictive densities for the 100th time point and find out the high probability density regions for the entire time series of the locations corresponding to the serial numbers indicated in bold within Table \ref{locations of sea temp data}. These locations are referred to as $L_{31}, \ldots, L_{40}$ hereafter. For calculating the posterior predictive densities at 99 time points for the above mentioned 10 spatial locations, the data set $\mathbb{D}$ is first augmented with the initial values for 99 time points of the ten locations and then the values are updated by sampling from the corresponding conditional distributions given the data set $\mathbb{D}$. That is to say, we start with $\mathbb{D}^* = \{\bi{y}_{1}^*, \ldots, \bi{y}_{99}^*\}$, where $\bi{y}_{t}^{*} = [\bi{y}_t^T : \bi{z}_{t}^T]^T$ is a 40 dimensional vector, with $\bi{z}_t$ being the initial guess for the ten locations $L_{31}, \ldots, L_{40}$, $t\in \{1, \ldots, 99\}$. The parameters of our model, including the latent variables, are then updated given $\mathbb{D}^*$, and then the last ten values of $\bi{y}^*_{t}$ are updated given the parameter values and the latent values. This continues for the entire MCMC run. Note that this sea surface temperature data arises from a non-stationary (both weakly and strongly), non-Gaussian spatio-temporal process with lagged correlations tending to zero (refer to \cite{bhattacharya2021bayesian} for further details).
\subsubsection{Prior choices for sea surface temperature data}
\label{choice of prior param for sea temp data}
Following our cross-validation technique to obtain the hyper parameters, we completely specify the priors as follows: \begin{align*} & \alpha^* \sim N(0,\sqrt{500}), \beta^* \sim N(0,\sqrt{300}),\\ & [\sigma^2] \propto IG(35000,2/2), [\sigma^2_{\theta}] \propto IG (1000,780/2), [\sigma^2_{p}] \propto IG (90,100/2) \\ & [\eta_{1}^*] \propto N (-3,1), [\eta_{2}^*] \propto N (-5,1). \end{align*} As before, we fixed $\eta_3$ at its maximum likelihood estimate; here the values is 14.2981.
\subsubsection{Controlling $M_s$} \label{Ms control}
Since the locations are distributed very widely over the space, the value of $M_s$ becomes too large for this data for a given $s$. Now $M_s$ appears in the denominator of the variance covariance matrix of the predictive densities of the time series at a particular location (see equation (\ref{eq4})). Therefore, the variability becomes too less. So, to control the variability we modify the definition of $M_s$ as $M_{s} = \exp(c\,\max\{||s^2-u^2||^2: u\in S\})$, where $c$ is a positive small constant. Note that it does not hamper the properties of $M_s$ and hence all the theoretical properties of the processes remain unchanged. The constant $c$ controls the spatial variability which can be thought of as a distance scaling factor. The choice of $c$ was also done by cross-validation. It was found that $c=0.25$ works reasonably well for the sea surface temperature data.
\subsubsection{Results of the sea surface temperature data} \label{results of sea temp data}
Figure~\ref{fig:trace plot of sea temp data} of Supporting Information, representing the MCMC trace plots of the parameters, exhibits no evidence of non-convergence of our MCMC algorithm. The posterior predictive colour density plots for the latent variables are shown in Figure \ref{fig:posterior density of latent variables of sea temp data} of Supporting Information.
Next we provide the posterior predictive densities at the 100th month for each of the 30 locations. Thirty density plots are shown in Figure \ref{fig: predictive density for 30 locations sea temp data}. As the plots indicate, the posterior predictive densities for the future time point correctly contains the true values for all of these locations. The results encourage us to use the proposed model for the future time point prediction for a non-stationary spatio-temporal data.
\begin{figure}
\caption{Predictive densities of averaged referenced temperatures for the 100th month at 30 locations for the sea surface temperature data. The red horizontal lines denote the 95\% predictive interval. The vertical black lines indicate the true values. All the true values lie within the 95\% predictive interval.}
\label{fig: predictive density for 30 locations sea temp data}
\end{figure}
Another important aspect of the spatio-temporal modeling is to predict the complete time series at the given locations. As mentioned in Section~\ref{sea temp}, we kept aside the complete temporal observations for the ten locations for evaluating the performance of the proposed model. As described in Section \ref{sea temp}, we obtained the posterior predictive densities for each of the 99 time points for a given location. The plots are depicted in Figure~\ref{fig:posterior predictive density of temp time series of 10 locations sea temp data}. As one can see, except for the two locations, which are represented by $L_{36}$ and $L_{39}$, most of the true values, indicated by black stars in Figure~\ref{fig:posterior predictive density of temp time series of 10 locations sea temp data}, fall well within the high probability density regions. Next, we give a plausible explanation for the poor performance at the locations $L_{36}$ and $L_{39}$.
\begin{figure}
\caption{Posterior predictive densities of the reconstructed time series of sea surface temperature data (average referenced) at ten locations.
Higher the intensity of the colour, higher is the probability density. The black stars represents the true temperature values (average referenced).
Except for the locations denoted by $L_{36}$ and $L_{39}$, majority of the true values fall well within the high density regions.}
\label{fig:posterior predictive density of temp time series of 10 locations sea temp data}
\end{figure} \subsubsection{Plaussible explaination for the poor performance at the two locations} First we provide below a plot (Figure~\ref{fig: sea temp data locations}) which represents the locations of the sea temperature data after we make the Lambert projections (see equation (\ref{Lambert})). The blue dots in Figure~\ref{fig: sea temp data locations} are used as the test set locations for the complete time series prediction. Two points are specially indicated by black star and a green star. These two locations are also members of the test set and at these two locations the high probability density regions fail to capture majority of the true values. The black star corresponds to $L_{36}$ and the green star corresponds to $L_{39}$. \begin{figure}
\caption{Locations of sea surface temperature data after the transformation given in equations~\ref{Lambert}. The red points represent the locations of
the observations in the leaning set. The blue solid dots, the black and the green stars are the locations of the observations in the test set.}
\label{fig: sea temp data locations}
\end{figure}
From the above plot, it is evident that the location which is indicated by black star, that is, $L_{36}$, is far away from the majority of the other locations. Thus, the value of $M_s$ for location $L_{36}$ becomes too large making the variability of the predictive densities for every time point very small. For this reason, the high posterior density region could not capture the true value well as compared to other locations (except $L_{39}$). On the other hand, the location indicated by green star is not far apart from the rest of the locations. However, the variability of the observations obtained across the time points are comparatively very large than the variability of the observations at other points. In fact, the variability across the time points of $L_{39}$ is found to be 8.4670 which is more than double the variability of all other locations present in the test set. In addition, at $L_{39}$ more than 70\% data lie away from -2 and 2, while the mean remains close to 0. This makes $L_{39}$ unique among the other test locations. Furthermore, it is observed that the mean variability of the learning set locations is 2.0407 and more than 90\% locations have data variability less than 5.3314. Therefore, from the learning data, the model did not get enough training to deal with such highly variable data. These are the reasons why the proposed model could not perform as expected for these two particular locations.
To end this discussion, we can note that even when the location $L_{36}$ is far away from the rest, thanks to our modification to $M_s$, the model performs better at this location than at location $L_{39}$. The large temporal variability compared to other locations including training and test sets, made this point ($L_{39}$) outlier and that is why the predictive densities fail to capture the true values.
\section{Summary and conclusion} \label{sec:conclusion}
Although Hamiltonian equations are very well-known in physics, in statistics its importance is confined to Hamiltonian Monte Carlo for simulating approximately from posterior distributions. However, given the success of the Hamiltonian equations in phase-space modeling, it is not difficult to anticipate its usefulness in spatio-temporal statistics, if properly exploited. This key insight motivated us to build a new spatio-temporal model through the leap-frog algorithm of a suitably modified set of Hamiltonian equations, where stochasticity is induced through appropriate Gaussian processes. Our marginal, observed stochastic process is nonparametric, non-Gaussian, nonstationary, nonseparable, with appropriate dynamic temporal structure, with time treated as continuous. Additionally, the lagged correlations between the observations tend to zero as the space-time lag goes to infinity. Hence, compared to the existing spatio-temporal processes, our process seems to be the most realistic, and this is vindicated by the results of our applications to two real data sets. The flexibility of our model is also corroborated by the results of two simulation experiments. Interesting continuity and smoothness properties add further elegance to our new process.
It is important to remark that this article essentially serves as the proof of concept that our novel spatio-temporal process based on modified Hamilton equations covers more ground than the existing processes and is much more flexible with respect to fitting stationary and nonstationary data alike. As such, in this article, we did not attempt to analyze very large datasets. In our future endeavors, we shall consider application of our ideas to very large datasets and with MCMC methods replaced with the $iid$ simulation procedure developed in \cite{Bhatta21a}, \cite{Bhatta21b}, \cite{Bhatta21c} and \cite{Bhatta22}. Comparison of our Bayesian model and methods with relevant existing Bayesian models and methods will also be of our interest. However, it is to be borne in mind that our spatio-temporal process attempts to cover a lot more ground than the existing ones, so that designing experiments for fair comparison seems to be a non-trivial task.
\begin{center}
\textbf{\large Supplemental Information} \end{center}
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\section{\small Proofs of the theorems} \label{theorem proofs} \begin{lem}
$M_s$ is infinitely differentiable in $s\in S$. \end{lem}
Proof: Since the exponential function is infinitely differentiable, it is sufficient to show that $\max\{||s^2-u^2||^2: u\in S\}$ is infinitely differentiable with respect to $s$. We first prove the result in one dimension and then will generalize to higher dimension. Let $S$ be a compact set in $\mathbb{R}$. We consider different cases as follows.
Case 1: Let $S = [a,b]$, where $0 < a < b$. Then $\max\{|s^2-u^2|^2: u\in [a,b]\} = s^4 + \max(a^4,b^4) - 2s^2 a^2$, which is an infinitely smooth function of $s$.
Case2: Let $S = [a,b]$, where $a < b < 0$. Then $\max\{|s^2-u^2|^2: u\in [a,b]\} = s^4 + \max(a^4,b^4) - 2s^2b^2$, which is also infinitely differentiable function of $s$.
Case 3: Let $S = [-a,b]$, where $a,b>0$. Then \begin{align}
\label{eqn on M_s}
\max\{|s^2-u^2|^2: u\in [-a,b]\} =
s^4 + \max(a^4,b^4) - 2s^2\min(a^2,b^2) \end{align}
Clearly, $\max\{|s^2-u^2|^2: u\in [-a,b]\}$ is infinitely differentiable function of $s$, for any $s\in [-a,b]$.
Now suppose that $S = [-a_1,b_1]\times [-a_2, b_2]$, where $a_i, b_i$ are positive for each $i =1, 2$. Let $s\in S$. Define $f_{s}: S\rightarrow \mathbb{R}$, such that $f_{s}(u) = ||s^2 - u^2||^2 = (s_1^2-u_1^2)^2 + (s_2^2-u_2^2)^2$. Define $f_{1s}(u_1) = (s_1^2-u_1^2)^2$ and $f_{2s}(u_2) = (s_2^2-u_2^2)^2$. Therefore, $f_{s}(u) = f_{1s}(u_1) + f_{2s}(u_2)$ and $\max\limits_{u\in S} f_{s}(u) = \max\limits_{u\in S}[f_{1s}(u_1) + f_{2s}(u_2)] = \max\limits_{u_1\in [-a_1,b_1]} f_{1s}(u_1) + \max\limits_{u\in [-a_2,b_2]} f_{2s}(u_2)$. The equality follows as $u = (u_1,u_2)\in S,$ with $u_1 \in [-a_1,b_1]$ and $u_2\in [-a_2,b_2]$. We have already proved that (Case 3, equation (\ref{eqn on M_s})) $\max\limits_{u_i\in [-a_i,b_i]} f_{is}(u_i) = s_i^4 + \max(a_i^4,b_i^4) - 2s_i^2 \min(a_i^2,b_i^2)$ for $i=1,2$. Hence $\max\limits_{u\in S} f_{s}(u) = \sum\limits_{i=1}^2 s_i^4 + \max(a_i^4,b_i^4) - 2s_i^2 \min(a_i^2,b_i^2) $, which is infinitely differentiable with respect to $s_1$ and $s_2$.
If $S = [a_1,b_1]\times [a_2,b_2]$ where (i) $0<a_1<b_1$ and $0<a_2<b_2$ or (ii) $a_1<b_1<0$ and $a_2<b_2<0$ or (iii) $0<a_1<b_1$ and $a_2<b_2<0$ or (iv) $a_1<b_1<0$ and $0<a_2<b_2$, the proof goes in a similar manner as above except that now we have to use Case 1 and Case 2 for one dimension, instead of Case 3.
$\square$
\begin{thm}
Under assumptions A1 to A3, cov$\left(\theta_{s}(h\delta t), \theta_{s'}(h'\delta t)\bigg|p\right)$ converges to 0 as $||s-s'||\rightarrow \infty$ and $|h-h'|\rightarrow \infty$.
\end{thm} Proof: Without loss of generality let us assume that $h>h'$. Now \begin{align}
\label{eq1 of covariance proof}
& \mbox{cov}\left(\theta_{s}(h\delta t), \theta_{s'}(h'\delta t)\Big |p\right)\notag \\
& \quad = \mbox{cov}\left(\beta \theta_{s}((h-1)\delta t) + \frac{\delta t}{M_{s}}\left\{\alpha p_{s}((h-1)\delta t) - \frac{\delta t}{2}\frac{\partial}{\partial \theta_{s}}V\left(\theta_{s}((h-1)\delta t)\right)\right\},\theta_{s'}(h'\delta t)\Big | p\right) \notag \\
& \quad = \mbox{cov} (\beta \theta_{s}((h-1)\delta t), \theta_{s'}(h'\delta t) \Big | p) - \frac{1}{2}\frac{(\delta t)^2}{M_s}\mbox{cov}\left(\frac{\partial}{\partial \theta_{s}}V(\theta_{s}((h-1)\delta t)),\theta_{s'}(h'\delta t) \Big | p \right) \notag \\
& \quad = \cdots \notag \\
& \quad = \beta^{h-h'}\mbox{cov}\left(\theta_{s}(h'\delta t),\theta_{s'}(h'\delta t)|p\right) - \frac{1}{2}\frac{(\delta t)^2}{M_s}\sum_{k=1}^{h-h'}\beta^{k-1}\mbox{cov}\left[\frac{\partial}{\partial \theta_{s}}V\left(\theta_{s}((h-k)\delta t)\right),\theta_{s'}(h'\delta t)|p\right]. \end{align} Since, for any $1\leq \ell\leq h-h'$, \begin{align*}
&\mbox{cov}\left(\frac{\partial}{\partial \theta_{s}}V\left(\theta_{s}((h-\ell)\delta t)\right),\theta_{s'}(h'\delta t)\Big | p\right) \notag \\
& \quad = \mbox{cov}\left(\frac{\partial}{\partial \theta_{s}}V\left(\theta_{s}((h-\ell)\delta t)\right), \beta \theta_{s'}((h'-1)\delta t) + \frac{\delta t}{M_{s'}}\left\{\alpha p_{s}((h'-1)\delta t) - \frac{\delta t}{2}\frac{\partial}{\partial \theta_{s'}}V\left(\theta_{s}((h'-1)\delta t)\right)\right\}\Big | p \right) \notag \\
& \quad = \beta \mbox{cov}\left(\frac{\partial}{\partial \theta_{s}}V\left(\theta_{s}((h-\ell)\delta t)\right),\theta_{s'}((h'-1)\delta t) \Big | p\right) - \frac{(\delta t)^2}{2M_{s'}}\mbox{cov}\left(\frac{\partial}{\partial \theta_{s}}V\left(\theta_{s}((h-\ell)\delta t)\right),\frac{\partial}{\partial \theta_{s'}} \right. \notag \\
& \qquad \left. V\left(\theta_{s'}((h-1)\delta t)\right)\Big| p\right) \notag \\
& \quad = \cdots \notag \\
& \quad = \beta^{h'}\mbox{cov}\left(\frac{\partial}{\partial \theta_{s}}V\left(\theta_{s}((h-\ell)\delta t)\right),\theta_{s'}(0) \Big | p\right) - \frac{(\delta t)^2}{2M_{s'}} \sum_{k=1}^{h'} \beta^{k-1}\mbox{cov}\left(\frac{\partial}{\partial \theta_{s}}V\left(\theta_{s}((h-\ell)\delta t)\right), \frac{\partial}{\partial \theta_{s'}} \right. \notag \\
& \qquad \left. V\left(\theta_{s'}((h-k)\delta t)\right)\Big | p \right), \end{align*} we have, \begin{align}
\label{eq2 of covariance proof}
&\left|\mbox{cov}\left(\frac{\partial}{\partial \theta_{s}}V\left(\theta_{s}((h-\ell)\delta t)\right),\theta_{s'}(h'\delta t)\Big | p\right)\right| \notag \\
& \quad \leq \left| \beta^{h'}\mbox{cov}\left(\frac{\partial}{\partial \theta_{s}}V\left(\theta_{s}((h-\ell)\delta t)\right),\theta_{s'}(0) \Big | p\right)\right| + \frac{(\delta t)^2}{2M_{s'}} \sum_{k=1}^{h'} \bigg| \beta^{k-1}\mbox{cov}\bigg[\frac{\partial}{\partial \theta_{s}}V\left(\theta_{s}((h-\ell)\delta t)\right), \frac{\partial}{\partial \theta_{s'}} \notag \\
& \qquad V\left(\theta_{s'}((h-k)\delta t)\right) \bigg] \bigg| \notag \\
& \quad \leq |\beta|^{h'} \sigma_{\theta} \sigma + \frac{1-|\beta|^{h'}}{1-|\beta|} \frac{(\delta t)^2}{2M_{s'}} \sigma^2 = \epsilon, \mbox{say}, \end{align} where $\sigma^2$ and $\sigma_{\theta}^2$ are the variance terms of the processes $\frac{\partial}{\partial \theta_{s}}V\left(\theta_{s}(h-1)\delta t\right)$ and $\theta_{s}(0)$, respectively (see A2 and A3). Therefore, from Equation (\ref{eq1 of covariance proof}) we obtain \allowdisplaybreaks \begin{align}
\label{eq3 of covariance proof}
&\Big |\mbox{cov}\left(\theta_{s}(h\delta t), \theta_{s'}(h'\delta t)\Big |p\right) - \beta^{h-h'}\mbox{cov}\left(\theta_{s}(h'\delta t),\theta_{s'}(h'\delta t)|p\right)\Big | \notag \\
& \quad \leq \frac{1}{2}\frac{(\delta t)^2}{M_s}\sum_{k=1}^{h-h'}|\beta|^{k-1}\Big |\mbox{cov}\left[\frac{\partial}{\partial \theta_{s}}V\left(\theta_{s}((h-k)\delta t)\right),\theta_{s'}(h'\delta t)|p\right] \Big | \notag \\
& \quad \leq \frac{1}{2}\frac{(\delta t)^2}{M_s} \frac{1-|\beta|^{h-h'}}{1-|\beta|} \epsilon, \end{align} using equation (\ref{eq2 of covariance proof}). Now from the first term of the right hand side of equation (\ref{eq1 of covariance proof}), we obtain \begin{align*}
& \mbox{cov}\left(\theta_{s}(h'\delta t),\theta_{s'}(h'\delta t)\Big|p\right) \notag \\
& \quad = \mbox{cov}\left(\beta \theta_{s}((h'-1)\delta t) + \frac{\delta t}{M_{s}}\left\{\alpha p_{s}((h'-1)\delta t) - \frac{\delta t}{2}\frac{\partial}{\partial \theta_{s}}V\left(\theta_{s}((h'-1)\delta t)\right)\right\}, \right. \notag \\
& \qquad \left. \beta \theta_{s'}((h'-1)\delta t) + \frac{\delta t}{M_{s'}}\left\{\alpha p_{s}((h'-1)\delta t) - \frac{\delta t}{2}\frac{\partial}{\partial \theta_{s'}}V\left(\theta_{s}((h'-1)\delta t)\right)\right\}\Big| p
\right) \notag \\
& \quad = \beta^2 \mbox{cov}\left(\theta_{s}((h'-1)\delta t), \theta_{s'}((h'-1)\delta t)\Big | p \right) - \beta\frac{1}{2}\frac{(\delta t)^2}{M_{s'}} \mbox{cov}\left(\theta_{s}((h'-1)\delta t),\frac{\partial}{\partial \theta_{s'}}V\left(\theta_{s'}((h'-1)\delta t)\right)\big| p\right) \notag \\
& \qquad - \beta\frac{1}{2}\frac{(\delta t)^2}{M_{s}} \mbox{cov}\left(\theta_{s'}((h'-1)\delta t),\frac{\partial}{\partial \theta_{s}}V\left(\theta_{s}((h'-1)\delta t)\right)\big| p\right) + \frac{1}{4}\frac{(\delta t)^4}{M_{s}M_{s'}}\mbox{cov}\left(\frac{\partial}{\partial \theta_{s}}V\left(\theta_{s}((h'-1)\delta t)\right),\right. \notag \\
& \qquad \left. \frac{\partial}{\partial \theta_{s'}}V\left(\theta_{s'}((h'-1)\delta t)\right)\Big | p \right) \notag \\
& \quad = \cdots \notag \\
& \quad = \beta^{2h'} \mbox{cov}\left(\theta_{s}(0),\theta_{s'}(0) \Big | p\right) -\frac{(\delta t)^2}{2M_{s'}}\sum_{k=1}^{h'} \beta^{2k-1} \mbox{cov}\left(\theta_{s}((h'-k)\delta t),\frac{\partial}{\partial \theta_{s'}}V\left(\theta_{s'}((h'-k)\delta t)\right)\big| p\right) \notag \\
& \quad - \frac{(\delta t)^2}{2M_{s}}\sum_{k=1}^{h'} \beta^{2k-1} \mbox{cov}\left(\theta_{s'}((h'-k)\delta t),\frac{\partial}{\partial \theta_{s}}V\left(\theta_{s}((h'-k)\delta t)\right)\big| p\right) + \frac{1}{4}\frac{(\delta t)^4}{M_{s}M_{s'}} \sum_{k=1}^{h'} \beta^{2(k-1)} \times \notag \\
& \qquad ~~ \mbox{cov}\left(\frac{\partial}{\partial \theta_{s}}V\left(\theta_{s}((h'-k)\delta t)\right), \frac{\partial}{\partial \theta_{s'}}V\left(\theta_{s'}((h'-k)\delta t)\right)\Big | p \right), \end{align*} which in turn implies \allowdisplaybreaks \begin{align}
\label{eq4 of covariance proof}
& \Big| \mbox{cov}\left(\theta_{s}(h'\delta t),\theta_{s'}(h'\delta t)\Big|p\right) - \beta^{2h'} \mbox{cov}\left(\theta_{s}(0),\theta_{s'}(0) \Big | p\right)\Big| \notag \\
& \quad \leq \frac{(\delta t)^2}{2M_{s'}} \sum_{k=1}^{h'} |\beta|^{2k-1} \Big| \mbox{cov}\left(\theta_{s}((h'-k)\delta t),\frac{\partial}{\partial \theta_{s'}}V\left(\theta_{s'}((h'-k)\delta t)\right)\big| p\right)\Big| + \frac{(\delta t)^2}{2M_{s}} \sum_{k=1}^{h'} |\beta|^{2k-1} \times \notag \\
& \qquad ~~ \Big| \mbox{cov}\left(\theta_{s'}((h'-k)\delta t),\frac{\partial}{\partial \theta_{s}}V\left(\theta_{s}((h'-k)\delta t)\right)\big| p\right)\Big| + \frac{1}{4}\frac{(\delta t)^4}{M_{s}M_{s'}} \sum_{k=1}^{h'} |\beta|^{2(k-1)} \Big| \mbox{cov}\left(\frac{\partial}{\partial \theta_{s}}V\left(\theta_{s}((h'-k)\delta t)\right),\right. \notag \\
& \qquad \left. \frac{\partial}{\partial \theta_{s'}}V\left(\theta_{s'}((h'-k)\delta t)\right)\Big | p \right)\Big| \leq \frac{(\delta t)^2}{2M_{s'}} \sum_{k=1}^{h'} |\beta|^{2k-1} \epsilon + \frac{(\delta t)^2}{2M_{s}} \sum_{k=1}^{h'} |\beta|^{2k-1} \epsilon + \frac{1}{4}\frac{(\delta t)^4}{M_{s}M_{s'}} \sum_{k=1}^{h'} |\beta|^{2(k-1)} \sigma^2 \notag \\
& \quad \leq \frac{(\delta t)^2}{2M_{s'}} |\beta| \frac{1-|\beta|^{2h'}}{1-|\beta|^2} \epsilon + \frac{(\delta t)^2}{2M_{s}} |\beta| \frac{1-|\beta|^{2h'}}{1-|\beta|^2} \epsilon + \frac{1}{4}\frac{(\delta t)^4}{M_{s}M_{s'}} \frac{1-|\beta|^{2h'}}{1-|\beta|^2} \sigma^2. \end{align} From equations (\ref{eq3 of covariance proof}) and (\ref{eq4 of covariance proof}), we get \begin{align}
\label{eq5 of covariance proof}
&\left| \mbox{Cov}\left[\theta_{s}(h\delta t),\theta_{s'}(h'\delta t)\bigg|p\right] - \beta^{(h-h')+2h'}\mbox{cov}\left(\theta_{s}(0),\theta_{s'}(0) \bigg| p\right) \right| \notag \\
& \quad \leq \left| \mbox{Cov}\left[\theta_{s}(h\delta t),\theta_{s'}(h'\delta t)\bigg|p\right] - \beta^{h-h'} \mbox{Cov}\left[\theta_{s}(h'\delta t),\theta_{s'}(h'\delta t)\bigg|p\right]\right| + |\beta|^{h-h'}\bigg| \mbox{Cov}\left[\theta_{s}(h'\delta t),\theta_{s'}(h'\delta t)\bigg|p\right] - \notag \\
& \qquad \beta^{2h'} \mbox{cov}\left(\theta_{s}(0),\theta_{s'}(0) \Big | p\right)\bigg| \notag \\
& \quad \leq \frac{1}{2}\frac{(\delta t)^2}{M_s} \frac{1-|\beta|^{h-h'}}{1-|\beta|} \epsilon + |\beta|^{h-h'} \left\{ \frac{(\delta t)^2}{2M_{s'}} |\beta| \frac{1-|\beta|^{2h'}}{1-|\beta|^2} \epsilon + \frac{(\delta t)^2}{2M_{s}} |\beta| \frac{1-|\beta|^{2h'}}{1-|\beta|^2} \epsilon + \frac{1}{4}\frac{(\delta t)^4}{M_{s}M_{s'}} \frac{1-|\beta|^{2h'}}{1-|\beta|^2} \sigma^2 \right\}, \end{align}
which goes to 0 as $|h-h'|\rightarrow \infty$ and $||s-s'||\rightarrow \infty$, under assumptions A1-A3 in conjunction with Remark \ref{Remark on M_s goes to infinity}. Writing $\bigg|\mbox{Cov}\left[\theta_{s}(h\delta t),\theta_{s'}(h'\delta t)\bigg\vert p\right]\bigg|$ as $\Big |\mbox{Cov}\left[\theta_{s}(h\delta t),\theta_{s'}(h'\delta t)\bigg\vert p\right] - \beta^{(h-h')+2h'}\mbox{Cov}\left[\theta_{s}(0),\theta_{s'}(0)\bigg\vert p\right] + \beta^{(h-h')+2h'}\\\mbox{Cov}\left[\theta_{s}(0),\theta_{s'}(0)\bigg\vert p\right]\Big |$ and noting that $\Big|\beta^{(h-h')+2h'}\mbox{Cov}\left[\theta_{s}(0),\theta_{s'}(0)\bigg\vert p\right]\Big|$ is less than $|\beta|^{(h-h')} |\beta|^{2h'}\sigma_{\theta}^2$, we have our desired result.
$\square$
\begin{thm}
If assumptions A1-A3 hold true, then
$\theta_{s}(h\delta t)$ and $p_{s}(h\delta t)$ are continuous in $s$, for all $h\geq 1$, with probability 1. \end{thm} Proof: Note that, for $h\geq 1$, \begin{align}
\label{eqn theta}
\theta_{s}(h \delta t) &= \beta \theta_{s}((h-1)\delta t) + \frac{\delta t}{M_{s}} \left\{\alpha p_{s}((h-1)\delta t)-\frac{\delta t}{2} \frac{\partial}{\partial \theta_s} V\left(\theta_{s} ((h-1)\delta t)\right) \right\} \mbox{ and }\\
\label{eqn p}
p_{s}(h\delta t) &= \alpha^2 p_{s}((h-1)\delta t) - \frac{\delta t}{2} \left\{\alpha \frac{\partial}{\partial \theta_s} V\left(\theta_{s}((h-1)\delta t)\right) + \frac{\partial}{\partial \theta_s} V\left(\theta_{s}(h\delta t)\right) \right\}. \end{align} Putting $h=1$ in equations (\ref{eqn theta}) and (\ref{eqn p}), we get \begin{align*}
\theta_{s}(\delta t) &= \beta \theta_{s}(0) + \frac{\delta t}{M_{s}} \left\{\alpha p_{s}(0)-\frac{\delta t}{2} \frac{\partial}{\partial \theta_s} V\left(\theta_{s} (0)\right) \right\} \mbox{ and }\\
p_{s}(\delta t) &= \alpha^2 p_{s}(0) - \frac{\delta t}{2} \left\{\alpha \frac{\partial}{\partial \theta_s} V\left(\theta_{s}(0)\right) + \frac{\partial}{\partial \theta_s} V\left(\theta_{s}(\delta t)\right) \right\}. \end{align*} Now $p_{s}(0)$ and $\theta_{s}(0)$ are Gaussian processes with continuous sample paths with probability 1 under assumptions A1 and A2 (also see Remark \ref{R1}), and Lemma~\ref{lemma3: differentiability of Ms} shows that $M_{s}$ is continuous in $s$. Furthermore, assumption A3 implies that the derivative of $V(\cdot)$ is also Gaussian process with continuous sample paths (see Remark \ref{R2}). Since composition of two continuous function is also a continuous function therefore, $\frac{\partial}{\partial \theta_s}V(\theta_{s}(0))$ is also continuous in $s$ with probability 1. This implies $\theta_{s}(\delta t)$ is continuous in $s$ with probability 1 as it is a linear combination of three almost sure continuous functions in $s$. This immediately implies that $p_{s}(\delta t)$ is also continuous in $s$ with probability 1. \par Assume that $\theta_{s}(h \delta t)$ and $p_{s}(h\delta)$ are continuous is $s$ with probability 1, for $h=k+1$. We will show that $\theta_{s}(h \delta t)$ and $p_{s}(h\delta)$ are almost surely continuous in $s$ for $h=k+2$. Now \begin{align*}
\theta_{s}((k+2) \delta t) &= \beta \theta_{s}((k+1)\delta t) + \frac{\delta t}{M_{s}} \left\{p_{s}((k+1)\delta t)-\frac{\delta t}{2} \frac{\partial}{\partial \theta_s} V\left(\theta_{s} ((k+1)\delta t)\right) \right\} \mbox{ and }\\
p_{s}((k+2)\delta t) &= \alpha^2 p_{s}((k+1)\delta t) - \frac{\delta t}{2} \left\{\alpha \frac{\partial}{\partial \theta_s} V\left(\theta_{s}((k+1)\delta t)\right) + \frac{\partial}{\partial \theta_s} V\left(\theta_{s}((k+2)\delta t)\right) \right\}. \end{align*} Since $\theta_{s}((k+1)\delta t)$ and $p_{s}((k+1)\delta t)$ are assumed to be continuous in $s$ with probability 1, derivative of a Gaussian process is also a Gaussian process, composition of two continuous functions is also a continuous function, and linear combinations of continuous functions is a continuous function, we have $\theta_{s}((k+2)\delta t)$ is continuous is $s$ with probability 1. Similar arguments show that $p_{s}((k+2)\delta t)$ is also continuous in $s$ with probability 1. Hence using the argument of induction, we claim that $\theta_{s}(h\delta t)$ and $p_{s}(h\delta t)$ are continuous in $s$ for any $h\geq 1$, with probability 1.
$\square$
\begin{thm}
Under assumptions A1-A3,
$\theta_{s}(h\delta t)$ and $p_{s}(h\delta t)$ are continuous in $s$ in the mean square sense, for all $h\geq 1$. \end{thm} Proof: We follow similar steps as in Lemma \ref{lemma1: almost sure continuity}. From equation (\ref{eqn theta}) we have \begin{align*}
\theta_{s}(\delta t) &= \beta \theta_{s}(0) + \frac{\delta t}{M_{s}} \left\{\alpha p_{s}(0)-\frac{\delta t}{2} \frac{\partial}{\partial \theta_s} V\left(\theta_{s} (0)\right) \right\}. \end{align*} Under assumptions A1 and A2, $p_{s}(0)$ and $\theta_{s}(0)$ are continuous in $s$ in the mean square sense, and by Lemma~\ref{lemma3: differentiability of Ms}, $M_{s}$ is continuous in $s$. Now since the random function $V(\cdot)$ is twice differentiable under assumption A3 (see Remark \ref{R2} also), the partial derivative process of $V$ is Lipschitz and hence the composition function $\frac{\partial}{\partial \theta_s} V(\theta_{s}(0))$ is also mean square continuous in $s$ (see page 416 of \cite{Banerjee2015}). Now using the fact the linear combination of mean square continuous processes is also mean-square continuous, we have $\theta_{s}(\delta t)$ is mean square continuous in $s$. This, in turn implies that, $\frac{\partial}{\partial \theta_s} V(\theta_{s}(\delta t))$ is also mean square continuous in $s$ using the same argument as above. Therefore, $$p_{s}(\delta t) = \alpha^2 p_{s}(0)-\frac{\delta t}{2} \left\{\alpha \frac{\partial}{\partial \theta_{s}}V(\theta_{s}(0))+\frac{\partial}{\partial \theta_s} V(\theta_{s}(\delta t)) \right\}$$ is mean square continuous in $s$. \par Now applying the similar argument of induction as in Lemma \ref{lemma1: almost sure continuity}, we have the desired result.
$\square$
\begin{thm}
Under assumptions A1-A3,
$\theta_{s}(h\delta t)$ and $p_{s}(h\delta t)$ have differentiable sample paths with respect to $s$, almost surely. \end{thm} Proof: The proof goes in the same line as that of Lemma \ref{lemma1: almost sure continuity}. We start with $\theta_{s}(\delta t)$ which is a linear combination of $\theta_{s}(0)$, $p_{s}(0)$ and $\frac{\partial}{\partial \theta_s} V(\theta_{s}(0))$. Note that $\theta_{s}(0)$, $p_{s}(0)$ have differentiable sample paths in $s$ by assumptions A1 and A2 (see Remark \ref{R1}). Now using the fact that composition of two differentiable function is also differentiable (see the Lemma 3.4 of \cite{Suman2017}), $\frac{\partial}{\partial \theta_s} V(\theta_{s}(0))$ has differentiable sample paths in $s$ (refer to assumption A3 and Remark \ref{R2}). Moreover, since $M_{s}$ is differentiable (Lemma~\ref{lemma3: differentiability of Ms}) and since the linear combination of differentiable functions is a differentiable function, $\theta_{s}(\delta t)$ has differentiable sample paths in $s$. The existence of differentiable sample paths of $\theta_{s}(\delta t)$ implies that $p_{s}(\delta t)$ will have differentiable sample paths in $s$. The rest of the proof is similar to that of Lemma \ref{lemma1: almost sure continuity}.
$\square$
\begin{lem}
Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be a zero mean Gaussian random function with covariance function $c_{f}(x_1,x_2)$, $x_1, x_2\in \mathbb{R}$, which is four times continuously differentiable. Let $\left\{Z(s):s\in S\right\}$ be a random process with the following properties
\begin{enumerate}
\item $E(Z(s)) = 0$,
\item The covariance function $c_{Z}(s_1,s_2)$, $s_1, s_2 \in S$, where $S$ is a compact subspace of $\mathbb{R}^2$, is four times continuously differentiable, and
\item $\frac{\partial Z(s)}{\partial s_i}$ has finite fourth moment.
\end{enumerate}
Then the process $\{g(s):s\in S\}$, where $g(s)=f(Z(s))$, is mean square differentiable in $s$. \end{lem}
Proof: To show that $\{g(s):s\in S\}$ is mean square differentiable in $s$ we have to show that, for any $p\in S$, there exists a function $L_{s}(p)$, linear in $p$, such that $$g(s+p) = g(s) + L_{s}(p) + R(s,p),$$ where $$\frac{R(s,p)}{||p||}\stackrel{L_{2}}{\longrightarrow} 0.$$ Let $s_o\in S$ be any point in $S$. Using multivariate Taylor series expansion we have \begin{align*}
g(s_0+p) = g(s_0)+p^{T}\nabla g(s_0) + R(s_0,p), \end{align*}
where $\nabla g(s_0) = \left(\frac{\partial f(Z(s))}{\partial s_1}, \frac{\partial f(Z(s))}{\partial s_2}\right)^T,$ with $\frac{\partial f(Z(s))}{\partial s_i} = \frac{df(Z(s))}{d(Z(s))}\frac{\partial Z(s)}{\partial s_i}$, for $i=1,2$. Therefore, $L_{s_0}(p) = p^{T}\nabla g(s_0)$, a linear function in $p$. To complete the proof we note that, from multivariate Taylor series expansion, $|R(s_0,p)|\leq M^* ||p||^2$, where $$M^* = \max\left\{ \Big|\frac{\partial^2 f(Z(s))}{\partial s_1^2} \Big|, \Big|\frac{\partial^2 f(Z(s))}{\partial s_1 \partial s_2}\Big|, \Big|\frac{\partial^2 f(Z(s))}{\partial s_2 \partial s_1}\Big|, \Big|\frac{\partial^2 f(Z(s))}{\partial s_2^2} \Big| \right\}, $$ with $$\frac{\partial^2 f(Z(s))}{\partial s_i^2} = \frac{d^2 f(Z(s))}{d((Z(s)))^2}\left(\frac{\partial Z(s)}{\partial s_i}\right)^2 + \frac{d f(Z(s))}{dZ(s)} \frac{\partial^2 Z(s)}{\partial s_i^2}, \mbox{ for } i=1,2, $$ and $$ \frac{\partial^2 f(Z(s))}{\partial s_1 \partial s_2} = \frac{\partial^2 f(Z(s))}{\partial s_2 \partial s_1} = \frac{d^2f(Z(s))}{d(Z(s))^2}\frac{\partial Z(s)}{\partial s_1}\frac{\partial Z(s)}{\partial s_2} + \frac{df(Z(s))}{d(Z(s))} \frac{\partial^2 Z(s)}{\partial s_1 \partial s_2}.$$ Since we have assumed that $Z(\cdot)$ and $f(\cdot)$ have covariance functions which are four times continuously differentiable, $Z(\cdot)$ and $f(\cdot)$ are twice differentiable (in the mean square sense) and hence the above terms involving first and second derivatives of $f(\cdot)$ and $Z(\cdot)$ are well-defined.
\par We will now show that the second moment of $\frac{\partial^2 f(Z(s))}{\partial s_i^2}$, for $i=1, 2$, and $\frac{\partial^2 f(Z(s))}{\partial s_1 \partial s_2}$ are finite. To prove the above fact, we first show that $\mbox{var}\left(\frac{d^2 f(Z(s))}{d((Z(s)))^2}\right) < \infty$. Note that \begin{align}
\label{eq 7}
\mbox{var}\left(\frac{d^2 f(Z(s))}{d((Z(s)))^2}\right) = \mbox{var}\left(E\left\{\frac{d^2 f(Z(s))}{d((Z(s)))^2}\Bigg\vert Z(s)\right\}\right)
+ E\left(\mbox{var}\left\{\frac{d^2 f(Z(s))}{d((Z(s)))^2}\Bigg\vert Z(s)\right\}\right). \end{align} Moreover, since $f''(x)$ is a Gaussian function with 0 mean and a constant variance, we have
$E\left\{\frac{d^2 f(Z(s))}{d((Z(s)))^2}\Bigg\vert Z(s)\right\} = 0$ and $\mbox{var}\left\{\frac{d^2 f(Z(s))}{d((Z(s)))^2}\Bigg\vert Z(s)\right\} = \mbox{constant} < \infty$. Thus \begin{align}
\label{eq8}
\mbox{var}\left(E\left\{\frac{d^2 f(Z(s))}{d((Z(s)))^2}\Bigg\vert Z(s)\right\}\right) = 0 \mbox{ and } \end{align} \begin{align}
\label{eq9}
E\left(\mbox{var}\left\{\frac{d^2 f(Z(s))}{d((Z(s)))^2}\Bigg\vert Z(s)\right\}\right) = \mbox{constant}. \end{align} Therefore, combining equations (\ref{eq8}) and (\ref{eq9}), and using equation (\ref{eq 7}) we see that $\mbox{var}\left(\frac{d^2 f(Z(s))}{d((Z(s)))^2}\right) <\infty$. Similar argument shows that $\mbox{var}\left(\frac{d f(Z(s))}{dZ(s)}\right) < \infty$. Now to show that $E\left(\frac{\partial^2 f(Z(s))}{\partial s_i^2}\right)^2 < \infty$, we use $(a+b)^2\leq 2(a^2 + b^2)$ and have \begin{align}
\label{eq10}
E\left(\frac{\partial^2 f(Z(s))}{\partial s_i^2}\right)^2
& \leq 2\left\{E\left(\left[\frac{d^2 f(Z(s))}{d((Z(s)))^2}\right]^2
\left[\frac{\partial Z(s)}{\partial s_i}\right]^4\right) + E\left(\left[\frac{d f(Z(s))}{dZ(s)}\right]^2 \left[\frac{\partial^2 Z(s)}{\partial s_i^2}\right]^2\right)\right\} \notag \\
& = 2 E\left(E\left(\left[\frac{d^2 f(Z(s))}{d((Z(s)))^2}\right]^2\Bigg\vert Z(s)\right) \left[\frac{\partial Z(s)}{\partial s_i}\right]^4\right) \notag \\
& \quad + 2 E\left(E\left(\left[\frac{d f(Z(s))}{dZ(s)}\right]^2\Bigg \vert Z(s)\right) \left[\frac{\partial^2 Z(s)}{\partial s_i^2}\right]^2\right). \end{align} Again using the fact that $f'(x)$ and $f''(x)$ are Gaussian with 0 mean and constant variance, we have
$E\left(\left[\frac{d^2 f(Z(s))}{d((Z(s)))^2}\right]^2\Bigg\vert Z(s)\right) =\mbox{constant}$ and $E\left(\left[\frac{d f(Z(s))}{dZ(s)}\right]^2\Bigg \vert Z(s)\right) = \mbox{constant}$.
Further, since the covariance function of $Z(s)$ is assumed to be four times continuously differentiable (2nd assumption of the Lemma), the first two derivatives of $Z(s)$, $\frac{\partial Z(s)}{\partial s_i}, i=1, 2$ and $\frac{\partial^2 Z(s)}{\partial s_i^2}, i=1, 2$ $\frac{\partial^2 Z(s)}{\partial s_1^2\partial s_2^2}$ will also exist in the mean square sense, with zero means. Thus, $E\left[\frac{\partial^2 Z(s)}{\partial s_1^2}\right]^2 = \mbox{constant}$. Also by the 3rd assumption of the Lemma, the fourth moment of $\frac{\partial Z(s)}{\partial s_i}$, for $i=1,2$, are finite. That is, $E\left[\frac{\partial Z(s)}{\partial s_i}\right]^4= \mbox{constant}$. Therefore, from equation (\ref{eq10}) we see that $E\left(\frac{\partial^2 f(Z(s))}{\partial s_1^2}\right)^2 < M_1 < \infty$, for some $M_1 \in \mathbb{R}.$
Next we show that $E\left(\frac{\partial^2 f(Z(s))}{\partial s_1 \partial s_2}\right)^2 < \infty$. Using the same inequality, $(a+b)^2 \leq 2(a^2 + b^2)$, we have \begin{align}
\label{eq11}
E\left(\frac{\partial^2 f(Z(s))}{\partial s_1 \partial s_2}\right)^2 & \leq 2\left[ E\left\{\frac{d^2f(Z(s))}{d(Z(s))^2}\frac{\partial Z(s)}{\partial s_1}\frac{\partial Z(s)}{\partial s_2}\right\}^2 + E\left\{\frac{df(Z(s))}{d(Z(s))} \frac{\partial^2 Z(s)}{\partial s_1 \partial s_2}\right\}^2\right] \notag \\
& = 2 E\left[E\left(\left[\frac{d^2 f(Z(s))}{d((Z(s)))^2}\right]^2\Bigg\vert Z(s)\right) \left(\frac{\partial Z(s)}{\partial s_1}\right)^2\left(\frac{\partial Z(s)}{\partial s_2}\right)^2\right] \notag \\
& \quad + 2E\left[E\left\{\left(\frac{df(Z(s))}{d(Z(s))}\right)^2\Bigg \vert Z(s)\right\} \left(\frac{\partial^2 Z(s)}{\partial s_1 \partial s_2}\right)^2 \right]. \end{align} Since $f'(x)$ and $f''(x)$ are Gaussian functions with mean 0 and constant variance, therefore,
$E\left(\left[\frac{d^2 f(Z(s))}{d((Z(s)))^2}\right]^2\Bigg\vert Z(s)\right) = \mbox{constant}$ and $E\left\{\left(\frac{df(Z(s))}{d(Z(s))}\right)^2\Bigg\vert Z(s)\right\} = \mbox{constant}$. Moreover, since by our assumption, the fourth moment of $\frac{\partial Z(s)}{\partial s_i}$, for $i=1,2$, are finite, we have
$E\left[\left(\frac{\partial Z(s)}{\partial s_1}\right)^2\left(\frac{\partial Z(s)}{\partial s_2}\right)^2\right] \leq \sqrt{E\left(\frac{\partial Z(s)}{\partial s_1}\right)^4 E\left(\frac{\partial Z(s)}{\partial s_2}\right)^4} = \mbox{constant}$. Since the covariance function of $Z(s)$ is four times continuously differentiable, $E\left(\frac{\partial^2 Z(s)}{\partial s_1 \partial s_2}\right)^2 = \mbox{constant}$. Thus we have, from equation (\ref{eq11}), $E\left(\frac{\partial^2 f(Z(s))}{\partial s_1 \partial s_2}\right)^2 < M_2,$ for some $M_2 \in R$. Hence each term in
$M^*= \max\left\{ \Big|\frac{\partial^2 f(Z(s))}{\partial s_1^2} \Big|, \Big|\frac{\partial^2 f(Z(s))}{\partial s_1 \partial s_2}\Big|, \Big|\frac{\partial^2 f(Z(s))}{\partial s_2 \partial s_1}\Big|, \Big|\frac{\partial^2 f(Z(s))}{\partial s_2^2} \Big| \right\}$ has bounded second moment.
Next we have to show that $E(M^*)^2<\infty$. Denote $M^* = \max\{A,B,C,D\}$, where $A = \Big|\frac{\partial^2 f(Z(s))}{\partial s_1^2} \Big|$, $B = \Big|\frac{\partial^2 f(Z(s))}{\partial s_1 \partial s_2}\Big|$, $C = \Big|\frac{\partial^2 f(Z(s))}{\partial s_2 \partial s_1}\Big|$ and $D = \Big|\frac{\partial^2 f(Z(s))}{\partial s_2^2} \Big|$. Note that it is sufficient to show that $X= \max\{A,B\}$ has finite second moment, because then with the same argument $Y=\max\{C,D\}$ will have finite second moment, and finally $Z= \max\{X,Y\}~ (=\max\{A,B,C,D\})$ will have finite second moment. \par Now $X = \max\{A,B\} = \frac{A + B + |A-B|}{2}\leq \frac{A + B |A|+|B|}{2}.$ Therefore, \begin{align*}
EX^2 & \leq \frac{1}{4} 2\left[E\left\{(A+B)^2 + (|A|+|B|)^2\right\}\right] \notag \\
& \leq E\{A^2 + B^2 + |A|^2 + |B|^2\} = 2 E(A^2 + B^2). \end{align*} Since $E A^2< \infty$ and $E B^2< \infty$, therefore, $E X^2 < \infty$. Now exactly same arguments imply that $E Y^2 < \infty$ and $E Z^2 < \infty$.
Hence, $\frac{R(s_0,p)}{||p||}\stackrel{L_{2}}{\longrightarrow} 0$. Since $s_0$ is any point in $S$, the proof is complete.
$\square$
\begin{thm}
Let A1-A3 hold true, with the covariance functions of all the assumed Gaussian processes being square exponentials. Then $\theta_{s}(h\delta t)$ and $p_{s}(h\delta t)$ are mean square differentiable in $s$, for every $h\geq 1$. \end{thm}
Proof: The proof will follow the similar argument of induction as done in Lemma \ref{lemma1: almost sure continuity}. For $h=1$, we have $$\theta_{s}(\delta t) = \beta \theta_{s}(0) + \frac{\delta t}{M_{s}} \left\{\alpha p_{s}(0)-\frac{1}{2} \delta t\frac{\partial}{\partial \theta_s} V\left(\theta_{s} (0)\right) \right\}.$$ Since, under the assumption A2 and the assumption of the theorem, $\theta_{s} (0)$ is a centered Gaussian process with squared exponential covariance function, the fourth moment of $\frac{\partial \delta_s(0)}{\partial s_i}$, $i=1,2$ are finite, and by assumption A3, $V(\cdot)$ is a zero-mean Gaussian random function with squared exponential covariance. Therefore, using lemma \ref{lemma5: mean square differentiability of composition of GP}, $\frac{\partial}{\partial \theta_s} V\left(\theta_{s} (0)\right)$ is mean square differentiable. Under the assumptions A1 and A2, and using the fact the covariance functions are assumed to squared exponential, $p_{s}(0)$ and $\theta_s(0)$ are mean squared differentiable. Using the fact that $M_s$ is differentiable in $s$ (see Lemma \ref{lemma3: differentiability of Ms}), $\theta_{s}(\delta t)$, being a linear combination of mean square differentiable functions, is also mean square differentiable.
Next we show that $E(\theta_{s}(\delta t)) = 0$ and $\mbox{cov}(\theta_{s_1}(\delta t), \theta_{s_2}(\delta t))$ is 4-times differentiable. Denoting the first derivative of $V$ by $V'$, we obtain $$E(\theta_{s}(\delta t)) = \beta E(\theta_s(0)) + \frac{\delta t}{M_s} \left\{\alpha E(p_s(0)) - \frac{1}{2} \delta t E(V'(\theta_{s}(0)))\right\} = 0, $$ where we have used the fact that $E(V'(\theta_{s}(0))) = E[E(V'(\theta_{s}(0)))\vert \theta_{s}(0)] = E(0) = 0$.
Therefore, $\mbox{cov}(\theta_{s_1}(\delta t), \theta_{s_2}(\delta t)) = E\left(\theta_{s_1}(\delta t) \theta_{s_2}(\delta t) \right)$. Since the processes are assumed to be independent, we have \begin{align}
\label{eq14: cov of theta_s}
E\left(\theta_{s_1}(\delta t) \theta_{s_2}(\delta t)\right) = \beta^2 \sigma_{\theta}^2 e^{-\eta_2||s_1 - s_2||^2} + \frac{(\delta t)^2\alpha^2}{M_{s_1} M_{s_2}} \sigma^2_{p}e^{-\eta_1||s_1-s_2||^2} + \frac{(\delta t)^4}{4 M_{s_1} M_{s_2}} E\left[V'(\theta_{s_1}(0))V'(\theta_{s_2}(0))\right]. \end{align}
Now according to our assumption, $\mbox{cov}(V(x_1),V(x_2)) = \sigma^2 e^{-\eta_3||x_1 - x_2||^2}$, that is, the covariance function of $V(\cdot)$ can be written as $\kappa(h) = \sigma^2 e^{-\eta_3 h^2}$, where $h= ||x_1 - x_2||$. The second derivative of $\kappa(h)$ is given by $-2\sigma^2 \eta_3 e^{-\eta_3 h^2}(1- 2\eta_3 h^2)$. Hence the covariance function of $V' (\cdot)$ will be $\mbox{cov}(V'(x_1),V'(x_2)) = 2\sigma^2 \eta_3 e^{-\eta_3 h^2}(1- 2\eta_3 h^2)$ (see \cite{Stein1999}, page 21). Therefore, the last term of the right hand side of equation (\ref{eq14: cov of theta_s}) can be computed as \allowdisplaybreaks \begin{align}
\label{eq15}
E\left[V'(\theta_{s_1}(0))V'(\theta_{s_2}(0))\right] & = EE\left[V'(\theta_{s_1}(0))V'(\theta_{s_2}(0)) \big \vert \theta_{s_1}(0) \theta_{s_2}(0) \right] \notag \\
& = E\left[2\sigma^2 \eta_3 e^{-\eta_3 (\theta_{s_1}(0) - \theta_{s_2}(0))^2} (1-2\eta_3 (\theta_{s_1}(0) - \theta_{s_2}(0))^2) \right] \notag \\
& = 2\sigma^2 \eta_3 \left[E e^{-\eta_3 (\theta_{s_1}(0) - \theta_{s_2}(0))^2} - 2\eta_3 E\left\{(\theta_{s_1}(0) - \theta_{s_2}(0))^2 e^{-\eta_3 (\theta_{s_1}(0) - \theta_{s_2}(0))^2}\right\} \right]. \end{align} Since $(\theta_{s_1}(0), \theta_{s_2}(0))^T \sim N_{2} \left(\bi{0}, \begin{pmatrix}
\sigma^2_{\theta} & \sigma^2_{\theta} e^{-\eta_2 ||s_1 - s_2||^2} \\
\sigma^2_{\theta} e^{-\eta_2 ||s_1 - s_2||^2} & \sigma^2_{\theta}
\end{pmatrix}\right),$ therefore, $\theta_{s_1}(0)- \theta_{s_2}(0)\sim N(0 , 2\sigma^2_{\theta} - 2\sigma^2_{\theta} e^{-\eta_2 ||s_1 - s_2||^2})$, which in turn implies that $$ \frac{(\theta_{s_1}(0)- \theta_{s_2}(0))^2}{\nu} \sim \chi^2_{(1)}, $$ where $\nu = 2\sigma^2_{\theta} - 2\sigma^2_{\theta} e^{-\eta_2 ||s_1 - s_2||^2}.$ Using the fact that the moment generating function (mgf) of a $\chi^2_{(1)}$ random variable is $(1-2t)^{-1/2}$, we have \begin{align}
\label{eq16}
E \left[e^{-\eta_3 (\theta_{s_1}(0) - \theta_{s_2}(0))^2}\right] = \left(1+ 4\eta_3 \sigma^2_{\theta} \left(1-e^{\eta_2||s_1 - s_2||^2}\right)\right)^{-1/2}. \end{align} Differentiating equation (\ref{eq16}) with respect to $\eta_3$ and cancelling the minus sign from both sides yield \begin{align}
\label{eq17}
E \left[(\theta_{s_1}(0) - \theta_{s_2}(0))^2 e^{-\eta_3 (\theta_{s_1}(0) - \theta_{s_2}(0))^2}\right] = 2\sigma_{\theta}^2 (1-e^{\eta_2||s_1 - s_2||^2}) \left(1+ 4\eta_3 \sigma^2_{\theta} (1-e^{\eta_2||s_1 - s_2||^2})\right)^{-3/2}. \end{align} Combining equations (\ref{eq15}), (\ref{eq16}), (\ref{eq17}), we get \begin{align}
\label{eq18}
E\left[V'(\theta_{s_1}(0))V'(\theta_{s_2}(0))\right] = 2\sigma^2 \eta_3 \left[ \frac{1}{\left(1+ 4\eta_3 \sigma^2_{\theta} (1-e^{\eta_2||s_1 - s_2||^2})\right)^{1/2}} - \frac{4\eta_3\sigma^2_{\theta} (1-e^{\eta_2||s_1 - s_2||^2})}{\left(1+ 4\eta_3 \sigma^2_{\theta} (1-e^{\eta_2||s_1 - s_2||^2})\right)^{3/2}}\right]. \end{align} Then inserting the value of $E\left[V'(\theta_{s_1}(0))V'(\theta_{s_2}(0))\right]$ from equation (\ref{eq18}) to equation (\ref{eq14: cov of theta_s}) we obtain \begin{align*}
E\left(\theta_{s_1}(\delta t) \theta_{s_2}(\delta t)\right) & = \beta^2 \sigma_{\theta}^2 e^{-\eta_2||s_1 - s_2||^2} + \frac{(\delta t)^2\alpha^2}{M_{s_1} M_{s_2}} \sigma^2_{p}e^{-\eta_1||s_1-s_2||^2} + \frac{(\delta t)^4}{4 M_{s_1} M_{s_2}} 2\sigma^2 \eta_3 \\
& \qquad \left[ \frac{1}{\left(1+ 4\eta_3 \sigma^2_{\theta} (1-e^{\eta_2||s_1 - s_2||^2})\right)^{1/2}} - \frac{4\eta_3\sigma^2_{\theta} (1-e^{\eta_2||s_1 - s_2||^2})}{\left(1+ 4\eta_3 \sigma^2_{\theta} (1-e^{\eta_2||s_1 - s_2||^2})\right)^{3/2}}\right].
\end{align*} Clearly, the covariance function of $\theta_{s}(\delta t)$ is four times differentiable, provided $M_s$ is also four times differentiable. To apply Lemma 1.6 on $V'(\theta_s(\delta t))$ we have to show that the fourth moment of $\frac{\partial}{\partial s_1}\theta_s(\delta t)$ finitely exists. From $$\theta_{s}(\delta t) = \beta \theta_{s}(0) + \frac{\delta t}{M_{s}} \left\{\alpha p_{s}(0)-\frac{\delta t}{2} \frac{\partial}{\partial \theta_s} V\left(\theta_{s} (0)\right) \right\} $$ we obtain \begin{align}
\label{eq19}
\frac{\partial \theta_s(\delta t)}{\partial s_1} & = \beta \frac{\partial \theta_s(0)}{\partial s_1} + \frac{\delta t}{M_s}\left\{\frac{\partial p_{s}(0)}{\partial s_1} - \frac{\delta t}{2} \frac{\partial}{\partial s_1} V'(\theta_s(0)) \right\} - \frac{\delta t}{M_s^2} \frac{\partial M_s}{\partial s_1} \left\{p_{s}(0)-\frac{1}{2} \delta t\frac{\partial}{\partial \theta_s} V\left(\theta_{s} (0)\right) \right\} \notag \\
& = \beta \frac{\partial \theta_s(0)}{\partial s_1} + \frac{\delta t}{M_s}\left\{ \frac{\partial p_{s}(0)}{\partial s_1} - \frac{\delta t}{2} V''(\theta_s(0)) \frac{\partial \theta_s(0)}{\partial s_1}\right\} - \frac{\delta t}{M_s^2} \frac{\partial M_s}{\partial s_1} \left\{p_{s}(0)-\frac{1}{2} \delta t\frac{\partial}{\partial \theta_s} V\left(\theta_{s} (0)\right) \right\}. \end{align}
Now the fourth moment of $\frac{\partial \theta_s(\delta t)}{\partial s_1} $ will be finite if individually each term of equation (\ref{eq19}) has finite fourth moment, because \begin{align*}
\left(\frac{\partial \theta_s(\delta t)}{\partial s_1}\right)^4 & \leq \kappa \left[\beta^4 \left(\frac{\partial \theta_s(0)}{\partial s_1}\right)^4 + \left(\frac{\delta t}{M_s}\right)^4\left\{ \left(\frac{\partial p_{s}(0)}{\partial s_1}\right)^4 + \left(\frac{\delta t}{2}\right)^4 \left(V''(\theta_s(0)) \frac{\partial \theta_s(0)}{\partial s_1}\right)^4\right\} + \left(\frac{\delta t}{M_s^2} \frac{\partial M_s}{\partial s_1}\right)^4 \right. \notag \\
& \qquad \left. \left\{\left(p_{s}(0)\right)^4+\left(\frac{1}{2} \delta t\right)^4\left(\frac{\partial}{\partial \theta_s} V\left(\theta_{s} (0)\right)\right)^4 \right\}\right], \end{align*} where $\kappa$ is a suitable constant. Note that due to assumptions A1-A2 and squared exponential covariance functions, $\frac{\partial \theta_s(0)}{\partial s_1}$, $\frac{\partial p_{s}(0)}{\partial s_1}$, and $p_{s}(0)$ are Gaussian processes with 0 mean and constant variances. Hence they have finite fourth moments. Therefore, we just need to show that $V''(\theta_s(0)) \frac{\partial \theta_s(0)}{\partial s_1}$ and $\frac{\partial}{\partial \theta_s} V\left(\theta_{s}(0)\right)$ have finite 4th moments. Observe that \begin{align}
\label{eq20}
E\left[V''(\theta_s(0)) \frac{\partial \theta_s(0)}{\partial s_1}\right]^4 & = E E\left\{ \left[V''(\theta_s(0)) \frac{\partial \theta_s(0)}{\partial s_1}\right]^4\bigg\vert \theta_s(0) \right\} \notag \\
& = E \left[E\left\{ \left[V''(\theta_s(0))\right]^4\vert \theta_s(0) \right\} \left(\frac{\partial \theta_s(0)}{\partial s_1}\right)^4\right] \end{align} Now since $V''(\theta_s(0))$, given $\theta_s(0)$, is Gaussian with mean 0 and constant variance,
\noindent $E\left\{ \left[V''(\theta_s(0))\right]^4\bigg\vert \theta_s(0) \right\}$ is constant (independent of $\theta_s(0)$), say $\kappa_1$. Hence from equation (\ref{eq20}) we obtain \begin{align}
\label{eq21}
E\left[V''(\theta_s(0)) \frac{\partial \theta_s(0)}{\partial s_1}\right]^4 = \kappa_1 E\left(\frac{\partial \theta_s(0)}{\partial s_1}\right)^4. \end{align} Now note that $\frac{\partial \theta_s(0)}{\partial s_1}$ is also Gaussian with mean 0 and constant variance, so that $E\left(\frac{\partial \theta_s(0)}{\partial s_1}\right)^4$ is also constant. Thus, combining equations (\ref{eq20}) and (\ref{eq21}) we have \begin{align}
\label{eq22}
E\left[V''(\theta_s(0)) \frac{\partial \theta_s(0)}{\partial s_1}\right]^4 <\infty. \end{align}
Next to show that $\frac{\partial}{\partial \theta_s} V\left(\theta_{s}(0)\right)$ has finite 4th moment, we notice that $$E(V'(\theta_s(0)))^4 = E\left[E(V'(\theta_s(0)))^4\vert \theta_s(0)\right] = \mbox{constant} = \kappa_2, \mbox{say}.$$ The last equality follows because $V'(\theta_s(0))$, given $\theta_s(0)$, is a Gaussian process with 0 mean and constant variance.
Therefore, the fourth moment of $\frac{\partial}{\partial s_1}\theta_s(\delta t)$ exists finitely. So, by Lemma \ref{lemma5: mean square differentiability of composition of GP}, $V'(\theta_s(\delta t))$ is mean square differentiable, and hence under assumptions A1, A2,
for $h=1$, $$ p_{s}(\delta t) = \alpha^2 p_{s}(0)-\frac{\delta t}{2} \left\{\alpha \frac{\partial}{\partial \theta_{s}}V(\theta_{s}(0))+\frac{\partial}{\partial \theta_s} V(\theta_{s}(\delta t)) \right\},$$ a linear combination of mean square differentiable function, is also mean square differentiable. Before, applying the steps of induction, we show that $\frac{\partial p_{s}(\delta t)}{\partial s_i}$, $i=1,2$, have finite 4th moment as it has to be used for the next step of induction. Note that, for $i=1,2$, \begin{align*}
\frac{\partial p_{s}(\delta t)}{\partial s_i} = \alpha^2 \frac{\partial p_{s}(0)}{\partial s_i} -\frac{\delta t}{2} \left\{\alpha V''(\theta_s(0))\frac{\partial \theta_s(0)}{\partial s_i} + V''(\theta_s(\delta t))\frac{\partial \theta_s(\delta t)}{\partial s_i}\right\}. \end{align*} From equation (\ref{eq22}), $E\left(V''(\theta_s(0))\frac{\partial \theta_s(0)}{\partial s_i}\right)^4< \infty.$ We have already shown that $\frac{\partial \theta_s(\delta t)}{\partial s_i}$ has finite fourth moment, for $i=1,2$. Thus, each term in the expression of $\frac{\partial p_{s}(\delta t)}{\partial s_i}$ has finite fourth moment. Hence $E\left(\frac{\partial p_{s}(\delta t)}{\partial s_i}\right)^4< \infty$.
Thereafter, using the argument of induction as in the proof of Lemma \ref{lemma1: almost sure continuity}, we have the desired result.
$\square$
\section{\small Calculation of joint conditional density of the observed data} \label{joint density of observed data}
Here we will find the data model, that is, the conditional distribution of Data given Latent, $\bi{y}_0$, $\bi{x}_0$ and the parameter $\bi{\theta}$.
First, we will find the conditional distribution of $\bi{y}_1$ given $\bi{y}_0$, $\bi{x}_0$ and the parameter $\bi{\theta}$. We have, for $i=1,2,\ldots,n$, $$y(s_i,1) = \beta\, y(s_i,0)+ \frac{\alpha x(s_i,0)}{M_{s_i}} - \frac{1}{2}\frac{V'(y(s_i,0))}{M_{s_i}},$$ and $$\frac{1}{2}\left[\frac{V'(y(s_1,0))}{M_{s_1}}, \ldots, \frac{V'(y(s_n,0))}{M_{s_n}}\right]' \sim N_{n}\left(\bi{0},\frac{\sigma^2}{4}\Sigma_{0}\right),$$ where $\Sigma_{0}$ is the $n\times n$ covariance matrix with $(i,j)$th element $\frac{2\eta_3 e^{-\eta_3 h_{ij}^2(0)}\left(1-2\eta_3 h_{ij}^2(0)\right)}{M_{s_i}M_{s_j}}.$ Therefore, \begin{align}
\label{eq2}
\left[\bi{y}_1\bigg\vert \bi{y}_0; \bi{x}_0; \bi{\theta}\right] \sim N_{n}\left(\bi{\mu}_0,\frac{\sigma^2}{4}\Sigma_{0}\right). \end{align}
Similarly, \begin{align}
\label{eq3}
&\bigg[\bi{y}_2\bigg\vert \bi{y}_1; \bi{x}_1; \bi{y}_0; \bi{x}_0; \bi{\theta}
\bigg]\sim N_{n}\left(\bi{\mu}_1,\frac{\sigma^2}{4}\Sigma_1 \right), \end{align} where the $(i,j)$th element of the $n\times n$ covariance matrix $\Sigma_1$ is $\frac{2\eta_3 e^{-\eta_3 h_{ij}^2(1)}\left(1-2\eta_3 h_{ij}^2(1)\right)}{M_{s_i}M_{s_j}}.$ Following the same argument, we can write down the likelihood as \begin{align*}
L &= [\mbox{Data}\big\vert \bi{x}_0;\ldots; \bi{x}_{T-1};\bi{y}_0;\bi{\theta}] \notag \\
& \propto [\bi{y}_1\big\vert \bi{y}_0;\bi{x}_0; \bi{\theta}] \ldots \left[\bi{y}_T\big\vert \bi{y}_{T-1}, \ldots, \bi{y}_0; \bi{x}_{T-1}, \ldots, \bi{x}_0; \bi{\theta}\right] \notag \\
& \propto \frac{(\sigma^2)^{-nT/2}}{\prod\limits_{t=1}^{T}|\Sigma_{t-1}|^{1/2}}\, e^{-\frac{2}{\sigma^2}\sum\limits_{t=1}^{T}\left(\bi{y}_{t} - \bi{\mu}_{t-1}\right)^T\Sigma_{t-1}^{-1}\left(\bi{y}_{t} - \bi{\mu}_{t-1}\right)}, \end{align*} where, for $j=1, 2, \ldots, T$, the
$(k,\ell)$th element of $\Sigma_{j-1}$ is $$\frac{2\eta_3 e^{-\eta_3 h_{k\ell}^2(j-1)}\left(1-2\eta_3 h_{k\ell }^2(j-1)\right)}{M_{s_k}M_{s_\ell}}.$$
\section{\small Calculation of the conditional joint density of latent data} \label{joint density of latent data}
Here we will derive the conditional distribution of the latent variables $$\mbox{Latent} = \left\{X(s_1,1), X(s_2,1), \ldots, X(s_n,1); X(s_1,2), X(s_2,2), \ldots, X(s_n,2); \ldots; X(s_1,T),X(s_2,T) \ldots, X(s_n,T)\right\}$$ given Data, $\bi{y}_0$, $\bi{x}_0$ and the parameter $\bi{\theta}$.
First, we shall find the conditional distribution of $\bi{x}_1$ given $\bi{x}_0$, $\bi{y}_0$, and $\bi{y}_1.$ For $i=1,\ldots,n$, we have $$ x(s_i,1) = \alpha^2 x(s_i,0) - \frac{1}{2}\left\{\alpha V'(y(s_i,0)) + V'(y(s_i,1))\right\}.$$
Since $V'(\cdot)$ is a random Gaussian function with zero mean and covariance function given in equation~(\ref{eq1}) and $\mathbb{W}_0 = (\bi{W}_0'~ \bi{W}_1')'$ is a $2n\times 1$ vector, $\mathbb{W}_0 \sim N_{2n}(\bi{0},\sigma^2 \Sigma),$ where $\Sigma$ is the $2n \times 2n$ covariance matrix partitioned as
$$ \begin{pmatrix}
\alpha^2 \Sigma_{00} & \alpha \Sigma_{01} \\
\alpha \Sigma_{10} & \Sigma_{11}
\end{pmatrix},$$ where the $(i,k)$th element of $\Sigma_{jj}$ is
$2\eta_3 e^{-\eta_3 h_{ik}^2(j)}\left(1-2\eta_3 h_{ik}^2(j)\right),$ for $j=0,1$, and the $(i,k)$th element of $\Sigma_{01} = \Sigma_{10}'$ is $2\eta_3 e^{-\eta_3 \ell_{ik}^2(0,1)}\left(1-2\eta_3 \ell_{ik}^2(0,1)\right).$ Therefore, $$\bi{W}_0 + \bi{W}_1 = [I_n \vdots I_n]\mathbb{W}_0 \sim N_n(\bi{0},\sigma^2\left(\alpha^2\Sigma_{00}+\alpha\Sigma_{01}+\alpha\Sigma_{10}+\Sigma_{11}\right)).$$
Let $\Omega_1 = \alpha^2 \Sigma_{00}+\alpha \Sigma_{01}+\alpha\Sigma_{10}+\Sigma_{11}$.
Then
\begin{align}
\label{eq5}
[\bi{x}_1\vert \bi{x}_0; \bi{y}_0;\bi{y}_1;\bi{\theta}] \sim N_n(\alpha^2 \bi{x}_0,\frac{\sigma^2}{4}\Omega_1).
\end{align}
Similarly,
\begin{align}
\label{eq6}
&[\bi{x}_2\vert \bi{x}_1;
\bi{x}_0;\bi{y}_0; \bi{y}_1; \bi{y}_2; \bi{\theta}]\sim N_n(\alpha^2 \bi{x}_1,\frac{\sigma^2}{4}\Omega_2),
\end{align}
where $\Omega_2 = \alpha^2 \Sigma_{11} +\alpha \Sigma_{12}+\alpha\Sigma_{21}+\Sigma_{22}$. For $j=1,2$, the $(i,k)$th element of $\Sigma_{jj}$ is $2\eta_3 e^{-\eta_3 h_{ik}^2(j)}\left(1-2\eta_3 h_{ik}^2(j)\right),$ and the $(i,k)$th element of $\Sigma_{12} = \Sigma_{21}'$ is $2\eta_3 e^{-\eta_3 \ell_{ik}^2(1,2)}\left(1-2\eta_3 \ell_{ik}^2(1,2)\right).$
Now with the help of equations (\ref{eq5}) and (\ref{eq6}) we write down the conditional latent process model as
\begin{align*}
&\bigg[\mbox{Latent}\bigg\vert \bi{y}_0;\ldots; \bi{y}_T;\bi{x}_0;\bi{\theta}\bigg] \notag \\
& \propto [\bi{x}_1\vert \bi{x}_0; \bi{y}_0;\bi{y}_1;\bi{\theta}] \ldots [\bi{x}_T\vert \bi{x}_{T-1};\ldots; \bi{x}_0; \bi{y}_0;\ldots; \bi{y}_T;\bi{\theta}] \notag \\
& \propto \frac{(\sigma^2)^{-nT/2}}{\prod\limits_{t=1}^{T} |\Omega_t|^{1/2}} e^{-\frac{2}{\sigma^2}\sum\limits_{t=1}^T (\bi{x}_{t} - \alpha^2 \bi{x}_{t-1})^T\Omega_t^{-1}(\bi{x}_{t} - \alpha^2 \bi{x}_{t-1})},
\end{align*}
where, for $m\in\{1,2,\ldots, T\}$,
$\Omega_t = \alpha^2\Sigma_{t-1,t-1} +\alpha \Sigma_{t-1,t} + \alpha \Sigma_{t,t-1} + \Sigma_{t,t},$ where the $(i,k)$th element of $\Sigma_{jj}$, for $j=t-1,t$, is $2\eta_3 e^{-\eta_3 h_{ik}^2(j)}\left(1-2\eta_3 h_{ik}^2(j)\right),$ and the $(i,k)$th element of $\Sigma_{t-1,t} = \Sigma_{t,t-1}'$ is
$2\eta_3 e^{-\eta_3 \ell_{ik}^2(t-1,t)}$ $
\left(1-2\eta_3 \ell_{ik}^2(t-1,t)\right).$
\section{\small Calculation of full conditional distributions of the parameters and the latent variables, given the observed data}
\label{Appendix B: full conditional densities}
\subsection{\small Full conditional distribution of $\beta^*$}
Before finding the full conditional distribution of $\beta$ (hence $\beta^*$), we note that the only term that depends on $\beta$ in equation (\ref{eq9: complete joint}) is $\exp\left\{-\frac{2}{\sigma^2} \sum\limits_{t=1}^{T} \left[\bi{\mu}_{t-1}^T\Sigma_{t-1}^{-1}\bi{\mu}_{t-1} -2 \bi{y}_{t}^T\Sigma_{t-1}^{-1}\bi{\mu}_{t-1}\right]\right\}$. Further notice that $\bi{\mu}_{t} = \beta\bi{y}_{t-1} + \mbox{constant (with respect to $\beta$)}$. Therefore, the term which depends on $\beta$ (hence on $\beta^*$) simplifies to $e^{-\frac{2\beta^2}{\sigma^2} \sum\limits_{t=1}^{T} \bi{y}_{t-1}'\Sigma_{t-1}^{-1}\bi{y}_{t-1}}$
$\times \, e^{\frac{4\beta}{\sigma^2} \sum\limits_{t=1}^{T} \bi{y}_{t}'\Sigma_{t-1}^{-1}\bi{y}_{t-1}},$ where $\beta = -1+\frac{2e^{\beta^*}}{1+2e^{\beta^*}}.$ The full conditional density of $\beta^*$, therefore, is given by
\allowdisplaybreaks
\begin{align*}
[\beta^*\vert \ldots] & \propto [\beta^*] e^{-\frac{2\beta^2}{\sigma^2} \sum\limits_{t=1}^{T} \bi{y}_{t-1}'\Sigma_{t-1}^{-1}\bi{y}_{t-1} + \frac{4\beta}{\sigma^2} \sum\limits_{t=1}^{T} \bi{y}_{t}'\Sigma_{t-1}^{-1}\bi{y}_{t-1}} \notag \\
& \propto e^{-\frac{{\beta^*}^2}{2\sigma_{\beta^*}^2}} e^{-\frac{2\beta^2}{\sigma^2} \sum\limits_{t=1}^{T} \bi{y}_{t-1}'\Sigma_{t-1}^{-1}\bi{y}_{t-1} + \frac{4\beta}{\sigma^2} \sum\limits_{t=1}^{T} \bi{y}_{t}'\Sigma_{t-1}^{-1}\bi{y}_{t-1}} \notag \\
& =\pi(\beta^*) g_1(\beta^*),
\end{align*}
where $\pi(\beta^*) = e^{-\frac{{\beta^*}^2}{2\sigma_{\beta^*}^2}}$ and $g_1(\beta^*) = e^{-\frac{2\beta^2}{\sigma^2} \sum\limits_{t=1}^{T} \bi{y}_{t-1}'\Sigma_{t-1}^{-1}\bi{y}_{t-1} + \frac{4\beta}{\sigma^2} \sum\limits_{t=1}^{T} \bi{y}_{t}'\Sigma_{t-1}^{-1}\bi{y}_{t-1}}$, as mentioned in equation (\ref{eq1: fcd of beta}).
\subsection{\small Full conditional distribution of $\alpha^*$}
The full conditional density of $\alpha^*$ will be given by $[\alpha^*]\times [\mbox{Data,Latent}\vert \bi{\theta}]$. Now the term that depends on $\alpha$ (hence on $\alpha^*$) in $[\mbox{Data,Latent}\vert \bi{\theta}]$, that is, in equation (\ref{eq9: complete joint}), is given by $$g_2(\alpha^*) = \frac{1}{\prod\limits_{t=1}^T |\Omega_{t}|^{1/2}} e^{-\frac{2}{\sigma^2}\sum\limits_{t=1}^T \left[ (\bi{x}_t-\alpha^2\bi{x}_{t-1})^T\Omega_{t}^{-1}(\bi{x}_t-\alpha^2\bi{x}_{t-1})\right]} \times e^{-\frac{2\alpha}{\sigma^2}\sum\limits_{t=1}^T \left[\alpha \bi{x}_{t-1}^TD\Sigma_{t-1}^{-1}D\bi{x}_{t-1}-2\bi{y}_t^T\Sigma_{t-1}^{-1}D\bi{x}_{t-1}\right]},$$ where $D$ is the $n\times n$ diagonal matrix containing the diagonal elements $\frac{1}{M_{s_i}}, i=1, \ldots,n.$ In the above calculation we use $\bi{\mu}_{t} = \beta\bi{y}_{t} + \alpha D\bi{x}_{t}.$
Thus the full conditional density of $\alpha^*$ is given by
\begin{align*}
[\alpha^*\vert \ldots] &\propto e^{-\frac{{\alpha^*}^2}{2\sigma_{\alpha^*}^2}} g_2(\alpha^*).
\end{align*}
\subsection{\small Full conditional distribution of $\sigma_{\theta}^2$}
The only term that depends on $\sigma_{\theta}^2$ in equation (\ref{eq9: complete joint}) is $\left(\frac{1}{\sigma_{\theta}^2}\right)^{n/2}\exp\left\{-\frac{1}{2\sigma_{\theta}^2}\bi{y}_0'\Delta_0^{-1}\bi{y}_0 \right\}$, and therefore, the full conditional distribution of $\sigma_{\theta}^2$ is
\begin{align*}
[\sigma_{\theta}^2\vert \ldots ] & \propto [\sigma_{\theta}^2] \left(\frac{1}{\sigma_{\theta}^2}\right)^{n/2} \exp\left\{-\frac{1}{2\sigma_{\theta}^2}\bi{y}_0'\Delta_0^{-1}\bi{y}_0 \right\} \notag \\
& \propto \left(\frac{1}{\sigma_{\theta}^2}\right)^{\alpha_{\theta}+n/2+1} \exp\left\{ -\frac{1}{\sigma_{\theta}^2}\left(\frac{\gamma_{\theta}+ \bi{y}_0'\Delta_0^{-1}\bi{y}_0}{2}\right)\right\}.
\end{align*}
That is, the full conditional distribution of $\sigma_{\theta}^2$ is IG$\left(\alpha_{\theta}+n/2, \frac{\gamma_{\theta}+\bi{y}_0'\Delta_0^{-1}\bi{y}_0}{2}\right)$.
\subsection{\small Full conditional distribution of $\sigma_{p}^2$}
Since the only term that depends on $\sigma_{p}^2$ in equation (\ref{eq9: complete joint}) is $\left(\frac{1}{\sigma_{p}^2}\right)^{n/2}\exp\left\{-\frac{1}{2\sigma_p^2}\bi{x}_0'\Omega_0^{-1}\bi{x}_0 \right\}$ and we have assumed inverse gamma with parameters $\alpha_{p}$ and $\gamma_{p}$,
\begin{align*}
[\sigma_p^2\vert \ldots ] & \propto \left(\frac{1}{\sigma_{p}^2}\right)^{\alpha_{p}+1} \left(\frac{1}{\sigma_{p}^2}\right)^{n/2}\exp\left\{ -\frac{1}{\sigma_{p}^2}\left(\frac{\gamma_{p}+ \bi{x}_0'\Omega_0^{-1}\bi{x}_0}{2}\right)\right\}.
\end{align*}
This implies that the full conditional distribution of $\sigma_p^2$ is IG$\left(\alpha_{p}+n/2, \frac{\gamma_{p}+\bi{x}_0'\Delta_0^{-1}\bi{x}_0}{2}\right)$.
\subsection{\small Full conditional distribution of $\sigma^2$}
Note that $[\mbox{Data}\vert \mbox{Latent}, \bi{x}_0, \bi{y}_0, \bi{\theta}]$ and $[\mbox{Latent}\vert \mbox{Data}, \bi{x}_0, \bi{y}_0, \bi{\theta}]$ depend on $\sigma^2$. Therefore, the full conditional distribution of $\sigma^2$ can be achieved as follows:
\begin{align*}
[\sigma^2\vert \ldots] &\propto [\sigma^2]\prod\limits_{t=1}^{T} [\bi{y}_t\vert \bi{y}_{t-1},\bi{x}_{t-1}, \bi{\theta}] [\bi{x}_t\vert \bi{y}_{t},\bi{y}_{t-1},\bi{x}_{t-1}, \bi{\theta}] \notag \\
& \propto [\sigma^2] (\sigma^2)^{-Tn} \exp\left\{-\frac{2}{\sigma^2}\sum\limits_{t=1}^T \left[(\bi{y}_t - \bi{\mu}_{t})^T\Sigma_{t-1}^{-1} (\bi{y}_t - \bi{\mu}_{t}) + (\bi{x}_t - \alpha^2\bi{x}_{t-1})^T\Omega_{t}^{-1} (\bi{x}_t - \alpha^2\bi{x}_{t-1})\right] \right\} \notag \\
& \propto \left(\frac{1}{\sigma^2}\right)^{\alpha_v+ Tn + 1} \exp\left\{-\frac{1}{\sigma^2} \left[\frac{\gamma_v}{2} + 2 \zeta \right]\right\},
\end{align*}
where $\zeta = \sum\limits_{t=1}^T \left[(\bi{y}_t - \bi{\mu}_{t})^T\Sigma_{t-1}^{-1} (\bi{y}_t - \bi{\mu}_{t}) + (\bi{x}_t - \alpha^2\bi{x}_{t-1})^T\Omega_{t}^{-1} (\bi{x}_t - \alpha^2\bi{x}_{t-1})\right].$ Hence the full conditional distribution of $\sigma^2$ is inverse-Gamma with parameters $\alpha_v+Tm$ and $\gamma_v/2 + 2\zeta$.
\subsection{\small Full conditional distributions of $\eta_1^*$, $\eta_2^*,$ and $\eta_3^*$}
We observe that only $[\bi{x}_0\vert \bi{\theta}]$ depends $\eta_1$ (hence on $\eta_1^*$) and $[\bi{Y}_0\vert \bi{\theta}]$ depends on $\eta_2$ (hence on $\eta_2^*$). Therefore, the full conditional densities of $\eta_1^*$ and $\eta_2^*$ are given by
\allowdisplaybreaks
\begin{align}
\label{fcd of eta1*}
[\eta_1^*\vert \ldots] & \propto [\eta_1^*][\bi{x}_0\vert \bi{\theta}] \notag \\
& \propto e^{-{\eta_1^*}^{2}/2} \frac{1}{|\Omega_0|^{1/2}} e^{-\frac{1}{2\sigma_p^2} \bi{x}_0^T\Omega_0^{-1}\bi{x}_0} \notag \\
& = \pi(\eta_1^*)g_3(\eta_1^*)
\end{align}
and
\begin{align}
\label{fcd of eta2*}
[\eta_2^*\vert \ldots] & \propto [\eta_2^*][\bi{y}_0\vert \bi{\theta}] \notag \\
& \propto e^{-{\eta_2^*}^{2}/2} \frac{1}{|\Delta_0|^{1/2}} e^{-\frac{1}{2\sigma_{\theta}^2} \bi{y}_0^T\Delta_0^{-1}\bi{y}_0} \notag \\
& = \pi(\eta_2^*)g_4(\eta_1^*),
\end{align}
respectively, where $\eta_1 = e^{\eta_1^*}$, $\eta_2 = e^{\eta_2^*}$, $\pi(\eta_1^*) = e^{-{\eta_1^*}^{2}/2}$, $\pi(\eta_2^*) = e^{-{\eta_2^*}^{2}/2}$, $g_{3}(\eta_1^*) = \frac{1}{|\Omega_0|^{1/2}} e^{-\frac{1}{2\sigma_p^2} \bi{x}_0^T\Omega_0^{-1}\bi{x}_0}$ and $g_4 (\eta_2^*)= \frac{1}{|\Delta_0|^{1/2}} e^{-\frac{1}{2\sigma_{\theta}^2} \bi{y}_0^T\Delta_0^{-1}\bi{y}_0}$.
On the other hand, in the joint conditional distribution of (Data, Latent) given $\bi{x}_0, \bi{y}_0, \bi{\theta}$ depends on $\eta_3$ (hence on $\eta_3^*$). Thus, the full conditional distribution of $\eta_3^*$ is given by
\begin{align}
\label{fcd of eta3^*}
[\eta_3^*] &\propto [\eta_3^*]\prod\limits_{t=1}^{T} [\bi{y}_t\vert \bi{y}_{t-1},\bi{x}_{t-1}, \bi{\theta}] [\bi{x}_t\vert \bi{y}_{t},\bi{y}_{t-1},\bi{x}_{t-1}, \bi{\theta}] \notag \\
& \propto e^{-{\eta_3^*}^{2}/2} \frac{1}{\prod\limits_{t=1}^{T}|\Sigma_{t-1}|^{1/2}|\Omega_t|^{1/2}} e^{-\frac{2}{\sigma^2}\sum\limits_{t=1}^T \left[(\bi{y}_t - \bi{\mu}_{t})^T\Sigma_{t-1}^{-1} (\bi{y}_t - \bi{\mu}_{t}) + (\bi{x}_t - \alpha^2\bi{x}_{t-1})^T\Omega_{t}^{-1} (\bi{x}_t - \alpha^2\bi{x}_{t-1})\right]} \notag \\
& = \pi(\eta_3^*)g_5(\eta_3^*),
\end{align}
where $\eta_3 = e^{\eta_3^*}$, $\pi(\eta_3^*) = e^{-{\eta_3^*}^{2}/2}$ and \newline $g_5(\eta_3^*) = \frac{1}{\prod\limits_{t=1}^{T}|\Sigma_{t-1}|^{1/2}|\Omega_t|^{1/2}} e^{-\frac{2}{\sigma^2}\sum\limits_{t=1}^T \left[(\bi{y}_t - \bi{\mu}_{t})^T\Sigma_{t-1}^{-1} (\bi{y}_t - \bi{\mu}_{t}) + (\bi{x}_t - \alpha^2\bi{x}_{t-1})^T\Omega_{t}^{-1} (\bi{x}_t - \alpha^2\bi{x}_{t-1})\right]}$.
\subsection{\small Full conditional distribution of $\bi{x}_0$}
Using the fact that only $\bi{x}_1$ and $\bi{y}_1$ depend on $\bi{x}_0$ and writing $\bi{\mu}_0 = \beta\bi{y}_{0} +\alpha D\bi{x}_0$, we have
\allowdisplaybreaks
\begin{align}
\label{eq4: fcd of x0}
[\bi{x}_0\vert \ldots] & \propto [\bi{x}_0\vert \bi{\theta}] [\bi{x}_1\vert \bi{y}_1, \bi{y}_0, \bi{x}_0,\bi{\theta}][\bi{y}_1\vert \bi{y}_0, \bi{x}_0,\bi{\theta}] \notag \\
& \propto e^{-\frac{1}{2\sigma^2_p}\bi{x}_0^T\Omega_0^{-1}\bi{x}_0} e^{-\frac{2}{\sigma^2}(\bi{x}_1-\alpha^2\bi{x}_0)^T\Omega_1^{-1}(\bi{x}_1-\alpha^2\bi{x}_0)} e^{-\frac{2}{\sigma^2}(\bi{y}_1-\beta\bi{y}_0-\alpha D\bi{x}_0)^T\Sigma_0^{-1}(\bi{y}_1-\beta\bi{y}_0-\alpha D\bi{x}_0)} \notag \\
& \propto e^{-\frac{1}{2\sigma_p^2}\left[\bi{x}_0^T\Omega_0^{-1}\bi{x}_0 + \frac{4\sigma_p^2\alpha^4}{\sigma^2}\bi{x}_0^T\Omega_1^{-1}\bi{x}_0 + \frac{4\sigma_p^2 \alpha^2}{\sigma^2} \bi{x}_0^T D\Sigma_0^{-1}D\bi{x}_0 - \frac{8\sigma_p^2\alpha^2}{\sigma^2}\bi{x}_0^T \Omega_1^{-1}\bi{x}_1 - \frac{8\sigma_p^2 \alpha}{\sigma^2}\bi{x}_0^T D \Sigma_0^{-1}(\bi{y}_1-\beta \bi{y}_0)\right]} \notag \\
& \propto e^{-\frac{1}{2\sigma_p^2}\left[\bi{x}_0^T A\bi{x}_0 - 2 \bi{x}_0^T B \bi{x}_1 - 2\bi{x}_0^T C(\bi{y}_1 - \beta \bi{y}_0) \right]} \notag \\
& \propto e^{-\frac{1}{2\sigma_p^2} (\bi{x}_0 - A^{-1}B\bi{x}_1 - A^{-1}C(\bi{y}_1 - \beta \bi{y}_0))^T A(\bi{x}_0 - A^{-1}B\bi{x}_1 - A^{-1}C(\bi{y}_1 - \beta \bi{y}_0))},
\end{align}
where
$A = \Omega_0^{-1}+\frac{4\sigma_p^2\alpha^4}{\sigma^2} \Omega_1^{-1}+ \frac{4\sigma_p^2 \alpha^2}{\sigma^2} D\Sigma_{0}^{-1}D$, $B = \frac{4\sigma_p^2\alpha^2}{\sigma^2}\Omega_1^{-1}$, $C = \frac{4\sigma_p^2 \alpha}{\sigma^2}D\Sigma_0^{-1}$. We note here that $D$ (= diag$(1/M_{s_i})_{i=1}^{n}$) is a positive definite matrix as all the diagonal entries are strictly positive, $A$ being a sum of three positive definite matrices is also positive definite and hence invertible. Thus, from equation (\ref{eq4: fcd of x0}), we get $[\bi{x}_0\vert \ldots ]\sim N_{n}(A^{-1}(B\bi{x}_1+C(\bi{y}_1 - \beta \bi{y}_0)),\sigma_p^2A^{-1}).$
\begin{comment}
\section{Simulation details and results for data simulated from our model}
\label{Simulation from our model}
\subsection{Data generation}
We generate the observations from our proposed model with the following choices of parameters: $\alpha = 0, \beta = 0, \eta_1=20, \eta_2 = 10$ and $\eta_3 = 16$. First we describe the steps that were followed to generate the observations before providing the results.
\begin{enumerate}
\item Generate 50-variate random vectors $\bi{x}_0$ and $\bi{y}_0$ from a multivariate normal with mean $\bi{0}$ and var-covariance matrices as $\sigma^2_{p}\Omega_0$ and $\sigma^2_{\theta}\Delta_0$. The elements of $\Omega_0$ and $\Delta_0$ are specified in Section~\ref{Jt Dist}.
\item Given $\bi{x}_0$ and $\bi{y}_0$, use equation~(\ref{eq2}) to generate a 50-variate random vector $\bi{y}_1$.
\item Given $\bi{x}_0,\bi{y}_0,\bi{y}_1$, generate a 50-variate random vector $\bi{x}_1$ using equation (\ref{eq5}).
\item Then, for $t=2, \ldots, 20$, 50-variate random vector $\bi{y}_t$, given $\bi{x}_{t-1}$ and $\bi{y}_{t-1}$, is generated from a multivariate normal
with mean $\bi{\mu}_{t-1}$ and covariance matrix $(\sigma^2/4)\Sigma_{t-1}$, Also generate $\bi{x}_t$ given $\bi{x}_{t-1}$, $\bi{y}_t$ and $\bi{y}_{t-1}$
from a multivariate normal with mean $\alpha^2 \bi{x}_{t-1}$ and covariance matrix $(\sigma^2/4)\Omega_{t}$. The details of the mean vector
$\bi{\mu}_{t-1}$, covariance matrices $\Sigma_{t-1}$ and $\Omega_{t}$ are provided in Sections~\ref{data model} and \ref{process model}.
\end{enumerate}
After generating $(\bi{x}_t,\bi{y}_t)$, for $t=1, \ldots 20$, we will assume that only $\bi{y}_{t}, \,\, t=1, \ldots, 19$ are known to us.
Using the proposed method we obtain the posterior densities of the parameters (except the parameter $\eta_3$) given the data
$\mathbb{D} = \{\bi{y}_1, \ldots, \bi{y}_{19}\}$.
\subsection{Choice of the hyper parameters}
\label{simu1: choice of hyper-param}
Our cross-validation method for selecting the hyper parameters helps us specify the priors completely as follows:
\begin{align*}
\alpha^*& \sim N(0,\sqrt{500})\\
\beta^*& \sim N(0,\sqrt{300})\\
[\sigma^2] & \propto IG(150,970/2) \\
[\sigma^2_{\theta}] & \propto IG (190,780/2) \\
[\sigma^2_{p}] & \propto IG (90,100/2) \\
[\eta_{1}^*] & \propto N (3,1) \\
[\eta_{2}^*] & \propto N (-5,1).
\end{align*}
As before, we fix $\eta_3$ at its maximum likelihood estimate (15.7123) obtained by simulated annealing. Besides the posteriors of the parameters,
we provide the posterior densities for $\bi{x}_t, \,\, t=1, \ldots, 19$. Moreover, to check the posterior predictive performance of our model,
we obtaned the posterior predictive densities of each component of $\bi{y}_{20}$ and future latent variable $\bi{x}_{20}$.
\subsection{Posterior analysis}
The MCMC trace plots of all parameters other than $\eta_3$ are provided in Figure~\ref{fig:trace plot1}. The posterior density plots for the latent variables
at 50 locations are provided in Figures~\ref{fig:posterior density of latent variables: first 25 locations} and
\ref{fig:posterior density of latent variables: last 25 locations}. The posterior predictive density plots for $\bi{y}_{20}$ and $\bi{x}_{20}$ at 50 locations
are depicted in Figure~\ref{fig: predictive density for 50 locations} and Figure~\ref{fig: predictive density of latent variable for 50 locations}.
The plots for the first 25 locations and for the last 25 locations are provided in Figures~\ref{fig:Predictive densities for first 25 locations}
and~\ref{fig:Predictive densities for last 25 locations}, respectively. Figures~\ref{fig:Predictive densities of latent variable for first 25 locations} and
\ref{fig:Predictive densities of latent variable for last 25 locations} display the
predictive densities of latent variable $\bi{x}_{20}$ for the first 25 and the last 25 locations, respectively.
\begin{figure}
\caption{MCMC trace plots for the parameters except for $\eta_3$. The dotted red linees represent the true parameter values. In all the cases, the
true parameter values are captured within the high posterior density regions, which is clear from the corresponding trace plots.}
\label{fig:trace plot1}
\end{figure}
\begin{figure}
\caption{Posterior predictive densities of the latent variables for the first 25 locations. Higher the intensity of the colour, higher is the probability density.
The black like represents the true values of the latent variables. $L_i, i=1, \ldots, 25$ denote the locations.
Majority of the points lie within the high posterior density regions.}
\label{fig:posterior density of latent variables: first 25 locations}
\end{figure}
\begin{figure}
\caption{Posterior predictive densities of the latent variables for the last 25 locations. Higher the intensity of the colour, higher is the probability density.
The black line represents the true values of the latent variables. $L_i, i=26, \ldots, 50$ denote the locations.
Majority of the points lie within the high posterior density regions.}
\label{fig:posterior density of latent variables: last 25 locations}
\end{figure}
\begin{figure}
\caption{Predictive densities for first 25 locations}
\label{fig:Predictive densities for first 25 locations}
\caption{Predictive densities for last 25 locations}
\label{fig:Predictive densities for last 25 locations}
\caption{Posterior predictive densities of $\bi{y}_{20}$ for 50 locations. The red horizontal lines denote the 95\% predictive intervals.
The vertical black lines indicate the true values. All the true values, except one, lie within the 95\% predictive intervals. }
\label{fig: predictive density for 50 locations}
\end{figure}
\begin{figure}
\caption{Predictive densities of latent variable for first 25 locations.}
\label{fig:Predictive densities of latent variable for first 25 locations}
\caption{Predictive densities of latent variable for last 25 locations.}
\label{fig:Predictive densities of latent variable for last 25 locations}
\caption{Posterior predictive densities of the $\bi{x}_{20}$ for 50 locations. The red horizontal lines denote the 95\% predictive intervals.
The vertical black lines indicate the true values. All the true values lie within the 95\% predictive interval. }
\label{fig: predictive density of latent variable for 50 locations}
\end{figure}
\end{comment}
\section{\small Simulation details and results for data simulated from LDSTM} \label{LDSTM simulation}
\subsection{Data generation} Now we apply our model and methodology on a data set which is simulated from \textit{linear dynamic spatio-temporal model} (LDSTM). The LDSTM is described below: \begin{align*}
Y(s_i,t) &= X(s_i,t) + \epsilon(s_i,t), \\
X(s_i,t) & = \rho X(s_i,t-1) + \eta(s_i,t),\\
\{X(s_i,0)\}_{i=1}^{n}& \sim N(\bi{0},\bi{\Sigma}_0), \end{align*}
where the random errors $\{\epsilon(s_i,t)\}$ and $\{\eta(s_i,t)\}$ are independently and identically distributed with respect to time as $N(\bi{0},\bi{\Sigma}_{\epsilon})$ and $N(\bi{0},\bi{\Sigma}_{\eta})$. The associated variance-covariance matrices are constructed using exponential covariance functions with the form $c(\bi{u},\bi{v}) = \sigma^2\exp\{-\lambda||\bi{u}-\bi{v}||\}$, where $||\cdot||$ denotes the Euclidean norm. For the purpose of simulation, we took $\rho = 0.8$, $(\sigma_{\epsilon},\lambda_{\epsilon}) = (1, 0.25)$ and $(\sigma_{\eta},\lambda_{\eta}) = (1, 1) = (\sigma_{0},\lambda_{0})$ for $\{\epsilon(s_i,t)\}$, $\{\eta(s_i,t)\}$ and $\{X_0(s_i,0)\}$, respectively.
Let $\bi{x}_t = (x(s_1,t), \ldots, x(s_m,t))^T$ and $\bi{y}_t = (y(s_1,t), \ldots, y(s_m,t))^T$ denote the latent vector and observed vector at time $t$. We generated $(\bi{x}_t,\bi{y}_t)$, for $t = 1, \ldots, 20$, as in the previous simulation study and assumed that only $\bi{y}_t$, $t=1, \ldots, 19$ are known to us. Hence the data set is given by $\mathbb{D} = \{\bi{y}_1, \ldots, \bi{y}_{19}\}$.
\subsection{Choice of hyper parameters} \label{choice of hyper-param LDSTM}
As before, based on cross-validation, the priors are specified, completely, as follows: \begin{align*}
& \alpha^* \sim N(0,\sqrt{500}), \beta^* \sim N(0,\sqrt{300})\\
& [\sigma^2] \propto IG(75000,2/2), [\sigma^2_{\theta}] \propto IG (900,780/2), [\sigma^2_{p}] \propto IG (250,100/2)\\
&[\eta_{1}^*] \propto N (3,1), [\eta_{2}^*] \propto N (-5,1).
\end{align*} Here we fixed $\eta$ at the value 18.5067, the maximum likelihood estimate computed by simulated annealing.
\subsection{Posterior analysis}
The posterior densities of other parameters involved in the model, and that of $\bi{x}_t$, for $t=1, \ldots, 19$ given $\mathbb{D}$ are obtained using the method described in Section~\ref{full conditional}. Further, for checking the predictive performance, we have calculated the posterior predictive densities of $\bi{y}_{20}$ and the latent future variable $\bi{x}_{20}$, given $\mathbb{D}$.
The MCMC trace plots of all the parameters except $\eta_3$ are provided in the Figure~\ref{fig:trace plot LDSTM}, all of which indicate good convergence properties of our MCMC algorithm. Figures~\ref{fig:posterior density of latent variables LDSTM: first 25 locations} and \ref{fig:posterior density of latent variables LDSTM: last 25 locations} display the posterior densities of the latent variables at 50 locations. Observe that the true values lie in high posterior density regions for each locations. Finally, the posterior predictive densities for $\bi{y}_{20}$ and $\bi{x}_{20}$ are shown in Figures~\ref{fig: predictive density for 50 locations LDSTM} and \ref{fig: predictive density of latent variable for 50 locations LDSTM}, of which Figure~\ref{fig:Predictive densities for first 25 locations LDSTM} and Figure~\ref{fig:Predictive densities for first 25 locations LDSTM} depict the plots for the first and the last 25 locations, respectively, for $\bi{y}_{20}$. The posterior predictive density plots for the first 25 locations of the latent variable $\bi{x}_{20}$ are given in Figure~\ref{fig:Predictive densities of latent variable for first 25 locations LDSTM}, whereas Figure~\ref{fig:Predictive densities of latent variable for last 25 locations LDSTM} provides the predictive density plots for the last 25 locations of the latent variable $\bi{x}_{20}$. Evidently, all the true (future) values for $\bi{y}$ (except two, marginally) and $\bi{x}$ fall well within the 95\% posterior predictive interval.
\begin{figure}
\caption{Trace plots for the parameters except for $\eta_3$ for LDSTM. }
\label{fig:trace plot LDSTM}
\end{figure}
\begin{figure}
\caption{Posterior predictive densities of the latent variables for the first 25 locations for LDSTM. Higher the intensity of the colour,
higher is the probability density. The black like represents the true values of latent variables. $L_i, i=1, \ldots, 25$ denote the locations.}
\label{fig:posterior density of latent variables LDSTM: first 25 locations}
\end{figure}
\begin{figure}
\caption{Predictive densities of the latent variables for the last 25 locations for LDSTM. Higher the intensity of the colour, higher is the probability density.
The black line represents the true values of latent variables. $L_i, i=26, \ldots, 50$ denote the locations.}
\label{fig:posterior density of latent variables LDSTM: last 25 locations}
\end{figure}
\begin{figure}
\caption{Predictive densities for first 25 locations}
\label{fig:Predictive densities for first 25 locations LDSTM}
\caption{Predictive densities for last 25 locations}
\label{fig:Predictive densities for last 25 locations LDSTM}
\caption{Predictive densities of the $\bi{y}_{20}$ for 50 locations for LDSTM. The red horizontal lines denote the 95\% predictive interval. The vertical black lines indicate the true values. All the true values, except two (marginally outside), lie within the 95\% predictive interval. }
\label{fig: predictive density for 50 locations LDSTM}
\end{figure}
\begin{figure}
\caption{Predictive densities of latent variable for first 25 locations for LDSTM.}
\label{fig:Predictive densities of latent variable for first 25 locations LDSTM}
\caption{Predictive densities of latent variable for last 25 locations for LDSTM.}
\label{fig:Predictive densities of latent variable for last 25 locations LDSTM}
\caption{Predictive densities of the $\bi{x}_{20}$ for 50 locations for LDSTM. The red horizontal lines denote the 95\% predictive interval. The vertical black lines indicate the true values. All the true values lie within the 95\% predictive interval. }
\label{fig: predictive density of latent variable for 50 locations LDSTM}
\end{figure}
\section{\small Trace plot of the parameters and the posterior densities of the complete time series of the latent variables for simulated data NLDSTM} \label{trace plot and posterior predictive densities of latent var for NLDSTM}
\begin{figure}
\caption{Trace plots for the parameters for data simulated from NLDSTM.}
\label{fig:trace plot NLDSTM}
\end{figure}
\begin{figure}
\caption{Predictive densities of latent variables for the complete times series at first 25 locations for data simulated from NLDSTM. Higher the intensity of the colour,
higher is the probability density. The black like represents the true values of the latent variables. $L_i, i=1, \ldots, 25$, denote the locations.}
\label{fig:posterior density of latent variables NLDSTM: first 25 locations}
\end{figure}
\begin{figure}
\caption{Predictive densities of latent variables for the complete time series at the last 25 locations for daat simulated from NLDSTM. Higher the intensity of the colour,
higher is the probability density. The black line represents the true values of the latent variables. $L_i, i=26, \ldots, 50$, denote the locations.}
\label{fig:posterior density of latent variables NLDSTM: last 25 locations}
\end{figure}
\section{\small Trace plot of the parameters and the posterior densities of the complete time series of the latent variables for simulated data GQN}
\begin{figure}
\caption{Trace plots for the parameters except for $\eta_3$ for GQN. }
\label{fig:trace plot GQN}
\end{figure}
\begin{figure}
\caption{Posterior densities of latent variables for the first 25 locations for data simulated from GQN. Higher the intensity of the colour, higher is the probability density. The black line represents the true values of the latent variables. $L_i, i=1, \ldots, 25$, denote the locations.}
\label{fig:posterior density of latent variables GQN: first 25 locations}
\end{figure}
\begin{figure}
\caption{Posterior densities of latent variables for the last 25 locations for data simulated from GQN. Higher the intensity of the colour, higher is the probability density. The black line represents the true values of the latent variables. $L_i, i=26, \ldots, 50$, denote the locations.}
\label{fig:posterior density of latent variables GQN: last 25 locations}
\end{figure}
\section{\small \small Trace plot of the parameters and the posterior densities of the complete time series of the latent variables for Alaska temperature data} \begin{figure}
\caption{Trace plot for the parameters except for $\eta_3$ corresponding to the Alaska temperature data.}
\label{fig:trace plot of Alaska temp data}
\end{figure}
\begin{figure}
\caption{Posterior densities of the latent variables at 26 locations in Alaska and its surroundings corresponding to the annual detrended temperature data. Higher the intensity of the colour, higher is the probability density. }
\label{fig:posterior density of latent variables first 13 Alaska temp data}
\end{figure}
\section{\small \small Trace plot of the parameters and the posterior densities of the complete time series of the latent variables for Sea temperature data} \begin{figure}
\caption{Trace plots of the parameters except for $\eta_3$ for the sea surface temperature data.}
\label{fig:trace plot of sea temp data}
\end{figure}
\begin{figure}
\caption{Posterior densities of the latent variables at 30 locations of the sea surface temperature data. Higher the intensity of the colour, higher is the probability density. }
\label{fig:posterior density of latent variables of sea temp data}
\end{figure}
\section{Stationarity, convergence of lagged correlations to zero and non-Gaussianity of the detrended Alaska data process} \label{appendix:alaska} \subsection{Stationarity of the detrended Alaska data process} \label{subsec:stationarity_alaska}
To check if the data arrived from a stationary or nonstationary process, we resorted to the recursive Bayesian theory and methods developed by \cite{roy2020bayesian}. In a nutshell, their key idea is to consider the Kolmogorov-Smirnov distance between distributions of data associated with local and global space-times. Associated with the $j$-th local space-time region is an unknown probability $p_j$ of the event that the underlying process is stationarity when the observed data corresponds to the $j$-th local region and the Kolmogorov-Smirnov distance falls below $c_j$, where $c_j$ is any non-negative sequence tending to zero as $j$ tends to infinity. With suitable priors for $p_j$, \cite{roy2020bayesian} constructed recursive posterior distributions for $p_j$ and proved that the underlying process is stationary if and only if for sufficiently large number of observations in the $j$-th region, the posterior of $p_j$ converges to one as $j\rightarrow\infty$. Nonstationarity is the case if and only if the posterior of $p_j$ converges to zero as $j\rightarrow\infty$.
In our implementation of the ideas of \cite{roy2020bayesian}, we set the $j$-th local region to be the entire time series for the spatial location $\bi{s}_j$, for $j=1,\ldots,29$. Thus, the size of each local region is $65$.
We choose $c_j$ to be of the same nonparametric, dynamic and adaptive form as detailed in \cite{roy2020bayesian}. The dynamic form requires an initial value for the sequence. In practice, the choice of the initial value usually has significant effect on the convergence of the posteriors of $p_j$, and so, the choice must be carefully made. However, in our case, for all initial values that we experimented with, lying between 0.05 and 1, the recursive Bayesian procedure led to the conclusion of stationarity of the underlying spatio-temporal process.
We implemented the idea with our parallelised C code on $29$ parallel processors of a VMWare of Indian Statistical Institute; the time taken is less than a second. For the initial value $0.05$, Figure \ref{fig:stationarity_alaska} displays the means of the posteriors of $p_j$; $j=1,\ldots,29$, showing convergence to 1. The respective posterior variances are negligibly small and hence not shown. Thus, the detrended spatio-temporal process that generated the Alaska data, can be safely regarded as stationary.
\begin{figure}
\caption{Alaska data analysis: detection of strict stationarity.}
\label{fig:stationarity_alaska}
\end{figure}
\subsection{Convergence of lagged spatio-temporal correlations to zero for the Alaska data} \label{subsec:zero_correlation} Recall that one major purpose of our Hamiltonian spatio-temporal model is to emulate the property of most real datasets that the lagged spatio-temporal correlations tend to zero as the spatio-temporal lag tends to infinity, irrespective of stationarity or nonstationarity.
Here we compute the lagged correlations on $30$ parallel processors on our VMWare, each processor computing the correlation for a partitioned interval of lag $\|\bi{h}\|$ such that the interval is associated with sufficient data making the correlation well-defined. The time taken for this exercise are a few seconds. Figure \ref{fig:corr_zero_alaska} demonstrates convergence of the lagged spatio-temporal correlations to zero; with larger amount of data such demonstration would have been more convincing.
\begin{figure}
\caption{Alaska data analysis: lagged spatio-temporal correlations converging to zero.}
\label{fig:corr_zero_alaska}
\end{figure}
\subsection{Non-Gaussianity of the Alaska data} \label{subsec:non_gaussian} Simple quantile-quantile plots (not shown for brevity) revealed that the distributions of the time series data at the spatial locations, distributions of the spatial data at the time points, and the overall distribution of the entire dataset, are far from normal. Thus, traditional Gaussian process based models of the underlying spatio-temporal process are ruled out. Since the temporal distributions at the spatial locations and the spatial distributions at different time points are also much different, it does not appear feasible to consider parametric stochastic process models for the data. These seem to make the importance of our nonparametric Hamiltonian process more pronounced.
\end{document} |
\begin{document}
\title{Chen's primes and ternary Goldbach problem} \author{Hongze Li} \address{ Department of Mathematics, Shanghai Jiaotong University, Shanghai 200240, People's Republic of China} \email{lihz@sjtu.edu.cn} \author{Hao Pan} \address{Department of Mathematics, Nanjing University, Nanjing 210093, People's Republic of China}\email{haopan79@yahoo.com.cn} \keywords{Chen prime; Ternary Goldbach problem; Rosser's weight} \subjclass[2000]{Primary 11P32; Secondary 11N36, 11P55}\thanks{This work was supported by the National Natural Science Foundation of China (Grant No. 10771135).} \maketitle \begin{abstract} We prove that there exists a $k_0>0$ such that every sufficiently large odd integer $n$ with $3\mid n$ can be represented as $p_1+p_2+p_3$, where $p_1,p_2$ are Chen's primes and $p_3$ is a prime with $p_3+2$ has at most $k_0$ prime factors. \end{abstract}
\section{Introduction} \setcounter{equation}{0} \setcounter{Thm}{0} \setcounter{Lem}{0} \setcounter{Cor}{0}
Let $\mathcal P$ denote the set of all primes. Define $$ \mathcal P_k=\{n:\, n\in \mathbb{N} \text{ and } n\text{ has at most }k\text{ prime divisors}\} $$ and $$ \mathcal P_k^{(2)}=\{p\in\mathcal P:\, p+2\in\mathcal P_k\}. $$ The well-known twin primes conjecture asserts that $\mathcal P_1^{(2)}$ has infinitely many elements. Nowadays, the best result on the twin primes conjecture belongs to Chen \cite{Chen73}, who proved that $\mathcal P_2^{(2)}$ has infinitely many elements. In fact, Chen proved that for sufficiently large $x$, $$
|\{p\in\mathcal P_2^{(2)}:\, p\leq x,\
(p+2,P(x^{1/10}))=1\}|\gg\frac{x}{(\log x)^2}, $$ where $$ P(z)=\prod_{p<z}p. $$ In Iwaniec's unpublished notes \cite{Iwaniec96}, the exponent $1/10$ can be improved to $3/11$. In \cite{GreenTao08}, Green and Tao say a prime $p$ is Chen's prime if $p\in\mathcal P_2^{(2)}$.
On the other hand, in 1937 Vinogradov \cite{Vinogradov37} solved the ternary Goldbach problem and showed that every sufficiently large odd integer can be represented as the sum of three primes. Two years later, using Vinogradov's method, van der Corput \cite{Corput39} proved that the primes contain infinitely many non-trivial three term arithmetic progressions (3AP). In 1999, with the help of the vector sieve method, Tolev \cite{Tolev99} proved that there exist infinitely many non-trivial 3APs $\{p_1,p_2,p_3\}$ of primes, satisfying $p_1\in\mathcal P_4^{(2)}$, $p_2\in\mathcal P_5^{(2)}$ and $p_3\in\mathcal P_{11}^{(2)}$.
However, in \cite{GreenTao08}, with the help of the Szemer\'edi theorem, their transference principle and a result of Goldston and Y\i ld\i r\i m, Green and Tao proved that the primes contain arbitrarily long non-trivial arithmetic progressions. Certainly this is a remarkable breakthrough in additive number theory. Furthermore, Green and Tao also claimed that using their method, one can prove that Chen's primes contain arbitrarily long non-trivial arithmetic progressions. And for the 3APs of Chen's primes, they proposed a detail proof in \cite{GreenTao06}.
Let us return to the ternary Goldbach problems. In \cite{Peneva00}, using Tolev's method, Peneva proved that every sufficiently large odd integer $n$ with $3\mid n$ can be represented as $n=p_1+p_2+p_3$ with $p_i\in\mathcal P_{k_i}^{(2)}$, $k_1=k_2=5$ and $k_3=8$. Subsequently, Tolev \cite{Tolev00} improved Peneva's result to $k_1=2$, $k_2=5$ and $k_3=7$. Recently, Meng \cite{Meng07} proved that every sufficiently large odd integer $n$ with $3\nmid n-1$ can be represented as $n=p_1+p_2+p_3$, where $p_1$ is Chen's prime, $p_2\in\mathcal P_3^{(2)}$ and $p_3\in\mathcal P$ (not of special type!).
Of course, we wish to prove that every sufficiently large odd integer $n$ with $3\mid n$ can be represented as the sum of three Chen's primes. Unfortunately, as we shall see later, it seems not easy. The key of Green and Tao's proof in \cite{GreenTao06} is to transfer Chen's primes to a subset with positive density of $\mathbb Z_N=\mathbb Z/N\mathbb Z$ (where $N$ is a large prime). But this density is too small. However, in the present paper, we shall prove the following
result: \begin{Thm} \label{chengoldbach} There exists a positive integer $k_0$ such that every sufficiently large odd integer $n$ with $3\mid n$ can be represented as $p_1+p_2+p_3$, where $p_1,p_2$ are Chen's primes and $p_3\in\mathcal P_{k_0}^{(2)}$. \end{Thm}
In Sections 2 and 3, we shall estimate some exponential sums involving the primes of special type. The proof of Theorem \ref{chengoldbach} will be given in Section 4.
\section{The Minor Arcs} \setcounter{equation}{0} \setcounter{Thm}{0} \setcounter{Lem}{0} \setcounter{Cor}{0}
Let $k_0\geq 8$ be a fixed integer and $B=6^9$. Suppose that $n$ is a sufficiently large integer, $W>0$ is an even integer with $W\leq (\log n)^B$, and $1\leq b\leq W$ satisfies $(b(b+2),W)=1$. Let $\mathbb T$ denote the torus $\mathbb R/\mathbb Z$. For $1\leq q\leq(\log n)^B$, define $$
\mathfrak M_{a,q}=\{\alpha\in\mathbb T:\,|\alpha q-a|\leq(\log n)^B/n\}. $$ Let $$ \mathfrak M=\bigcup_{\substack{1\leq a\leq q\leq (\log n)^B\\(a,q)=1}}\mathfrak M_{a,q} $$ and $\mathfrak m=\mathbb T\setminus\mathfrak M$.
Let $D=n^{0.32}$ and $z_0=n^{1/k_0}$. For square-free number $d=p_1p_2\cdots p_k$ with primes $p_1>p_2>\cdots>p_k$, define Rosser's weights with the order $D$ by $$ \lambda_{D}^{+}(d)=\begin{cases} (-1)^k&\text{if }p_1\cdots p_{2l}p_{2l+1}^3<D\text{ for all }0\leq l\leq (k-1)/2,\\ 0&\text{otherwise}, \end{cases} $$ and $$ \lambda_{D}^{-}(d)=\begin{cases} (-1)^k&\text{if }p_1\cdots p_{2l-1}p_{2l}^3<D\text{ for all }1\leq l\leq k/2,\\ 0&\text{otherwise}. \end{cases} $$ It is easy to see that $\lambda_D^{\pm}(d)=0$ if $d\geq D$. Let $F(s)$ and $f(s)$ denote the functions of linear sieve. The following lemma is a fundamental result in sieve method. \begin{Lem}[Iwaniec \cite{Iwaniec80a, Iwaniec80b}] \label{iwaniec} Suppose that $\mathcal P_*$ is any set of primes and $\omega$ is a multiplicative function satisfying: $$ 0<\omega(p)<p\text{ for }p\in\mathcal P_*,\ \omega(p)=0\text{ for }p\not\in\mathcal P_*, $$ and $$ \prod_{z_1\leq p<z_2}\bigg(1-\frac{\omega(p)}{p}\bigg)^{-1}\leq\frac{\log z_2}{\log z_1}\bigg(1+\frac{L}{\log z_1}\bigg) $$ for a constant $L>0$ and for all $2\leq z_1\leq z_2$. Then we have \begin{align} \label{rosserF} \prod_{p<z}\bigg(1-\frac{\omega(p)}{p}\bigg)\leq&\sum_{d\mid P_*(z)}\lambda_D^+(d)\frac{\omega(d)}{d}\notag\\ \leq&\prod_{p<z}\bigg(1-\frac{\omega(p)}{p}\bigg)(F(s)+O(e^{\sqrt{L}-s}(\log D)^{-1/3})), \end{align} provided that $2\leq z\leq D$, where $s=\log D/\log z$ and $$ P_*(z)=\prod_{p\in\mathcal P_*\cap[1,z)}p. $$ Similarly, \begin{align} \label{rosserf} \prod_{p<z}\bigg(1-\frac{\omega(p)}{p}\bigg)\geq&\sum_{d\mid P_*(z)}\lambda_D^-(d)\frac{\omega(d)}{d}\notag\\ \geq&\prod_{p< z}\bigg(1-\frac{\omega(p)}{p}\bigg)(f(s)+O(e^{\sqrt{L}-s}(\log D)^{-1/3})), \end{align} provided that $2\leq z\leq \sqrt{D}$. Furthermore, for any square-free integer $q$, \begin{align} \label{rosserpm} \sum_{d\mid q}\lambda_D^-(d)\leq\sum_{d\mid q}\mu(d)\leq\sum_{d\mid q}\lambda_D^+(d). \end{align} \end{Lem} Define $$ S_{n,z_0}(\alpha)=\sum_{\substack{p\leq n\\ p\equiv b\pmod{W}\\ (p+2,P(z_0))=1}}e(\alpha(p-b)/W)\log p. $$ Clearly $$ S_{n,z_0}(\alpha)=\sum_{\substack{p\leq n\\ p\equiv b\pmod{W}}}e(\alpha(p-b)/W)\log p\sum_{d\mid (p+2,P(z_0))}\mu(d). $$ Let $$ S_{n,z_0}^{\pm}(\alpha)=\sum_{\substack{p\leq n\\ p\equiv b\pmod{W}}}e(\alpha(p-b)/W)\log p\sum_{d\mid (p+2,P(z_0))}\lambda_D^\pm(d). $$ \begin{Lem} \label{spm} For any $\alpha\in\mathbb T$, we have
$|S_{n,z_0}^{+}(\alpha)-S_{n,z_0}(\alpha)|\leq S_{n,z_0}^{+}(0)-S_{n,z_0}(0)$ and
$|S_{n,z_0}(\alpha)-S_{n,z_0}^{-}(\alpha)|\leq S_{n,z_0}(0)-S_{n,z_0}^-(0)$. \end{Lem} \begin{proof} By (\ref{rosserpm}), \begin{align*}
|S_{n,z_0}^{+}(\alpha)-S_{n,z_0}(\alpha)|\leq&\sum_{\substack{p\leq n\\
p\equiv b\pmod{W}}}\log p\bigg|\sum_{d\mid (p+2,P(z_0))}\lambda^+(d)-\sum_{d\mid
(p+2,P(z_0))}\mu(d)\bigg|\\ =&S_{n,z_0}^{+}(0)-S_{n,z_0}(0). \end{align*} The proof of the second inequality is similar. \end{proof}
Let $\tau$ denote the divisor function. It is well-known $$ \sum_{d\leq X}\tau(d)^A\ll_AX(\log X)^{2^A-1} $$ and $$ \sum_{d\leq X}\frac{\tau(d)^A}{d}\ll_A(\log X)^{2^A}. $$ \begin{Lem} Suppose that $X,X',Y,Y',Z,Z'>0$ satisfy $X\leq X'\leq 2X$, $Y\leq Y'\leq 2Y$ and $XY\leq Z\leq Z'\leq X'Y'$. For any $d\geq 1$, let
$u_d$, $v_d$, $w_d$ be complex numbers with $|u_d|, |v_d|,
|w_d|\leq\tau(d)^{A}(\log(XY))^{A}$. Suppose that $1\leq a\leq q$
with $(a,q)=1$, and $\alpha\in\mathbb T$ with $|\alpha q-a|\leq 1/q$. Then \begin{align} \label{expsum1}
&\sum_{X\leq x\leq X'}u_x\bigg|\sum_{\substack{Y\leq y\leq Y'\\ Z\leq xy\leq Z'\\ xy\equiv b\pmod{W}}}v_ye(\alpha (xy-b)/W)\sum_{\substack{d\mid xy+2\\
d\leq D}}w_d\bigg|\notag\\ \ll&_A XY(\log(DXYq))^{2^{2A+2}}\tau^{2A+2}(W)\bigg(\frac{1}{q}+\frac{D^2W}{X}+\frac{qW}{XY}\bigg)^{1/4}\notag\\ &+XY^{1/2}(\log(DXY))^{2^{2A+2}}\tau^{2A+1}(W) \end{align} provided that $X\geq D^2W$. Furthermore, suppose that $0\leq v_{y_1}\leq v_{y_2}\leq (\log(XY))^A$ for any $y_1,y_2$ with $y_1\leq y_2$. Then \begin{align} \label{expsum2}
&\sum_{X\leq x\leq X'}u_x\bigg|\sum_{\substack{Y\leq y\leq Y'\\ Z\leq xy\leq Z'\\ xy\equiv b\pmod{W}}}v_ye(\alpha (xy-b)/W)\sum_{\substack{d\mid xy+2\\
d\leq D}}w_d\bigg|\notag\\ \ll&_A XY(\log(DXYq))^{6^{A+2}}\bigg(\frac{1}{q}+\frac{DW}{Y}+\frac{qW}{XY}\bigg)^{1/4} \end{align} provided that $Y\geq DW$. \end{Lem} \begin{proof} By the Cauchy-Schwarz inequality, \begin{align*}
&\bigg(\sum_{X\leq x\leq X'}u_x\bigg|\sum_{\substack{Y\leq y\leq Y'\\ Z\leq xy\leq Z'\\ xy\equiv b\pmod{W}}}v_ye(\alpha (xy-b)/W)\sum_{\substack{d\mid xy+2\\
d\leq D}}w_d\bigg| \bigg)^2\\
\leq&\bigg(\sum_{X\leq x\leq X'}|u_x|^2\bigg) \bigg(\sum_{\substack{X\leq x\leq X'\\ d_1,d_2\leq D}}w_{d_1}\overline{w_{d_2}}\sum_{\substack{Y\leq y_1,y_2\leq Y'\\ Z\leq xy_1,xy_2\leq Z'\\ xy_1,xy_2\equiv b\pmod{W}\\ xy_i\equiv -2\pmod{d_i}\text{ for }i=1,2}}v_{y_1}\overline{v_{y_2}}e(\alpha x(y_1-y_2)/W)\bigg)\\ \ll&_AX(\log X)^{6^A}\sum_{\substack{1\leq b'\leq W,\ (b',W)=1\\ Y\leq y_1,y_2\leq Y'\\ y_1\equiv y_2\equiv b'\pmod{W}\\ d_1,d_2\leq D}}v_{y_1}\overline{v_{y_2}}w_{d_1}\overline{w_{d_2}}\sum_{\substack{X\leq x\leq X'\\ Z\leq xy_1,xy_2\leq Z'\\ xb'\equiv b\pmod{W}\\ xy_i\equiv -2\pmod{d_i}\text{ for }i=1,2}}e(\alpha x(y_1-y_2)/W).\\ \end{align*} We have \begin{align*} &\sum_{\substack{X\leq x\leq X'\\ Z\leq xy_1,xy_2\leq Z'\\ xb'\equiv b\pmod{W}\\ xy_i\equiv -2\pmod{d_i}\text{ for }i=1,2}}e(\alpha x(y_1-y_2)/W)\\ =&\sum_{\substack{\max\{X, Z/y_1, Z/y_2\}\leq x\leq \min\{X', Z'/y_1,Z'/y_2\}\\ xb'\equiv b\pmod{W}\\ xy_i\equiv -2\pmod{d_i}\text{ for }i=1,2}}e(\alpha x(y_1-y_2)/W)\\
\ll&\min\bigg\{\frac{X}{[d_1,d_2,W]},\frac{1}{\|\alpha
[d_1,d_2,W](y_1-y_2)/W\|}\bigg\}, \end{align*}
where $\|\theta\|=\min\{|\theta-t|:\, t\in\mathbb Z\}$. Hence for each $1\leq b'\leq W$ with $(b',W)=1$, we have \begin{align*} &\sum_{\substack{Y\leq y_1,y_2\leq Y'\\ y_1\equiv y_2\equiv b'\pmod{W}\\ d_1,d_2\leq D}
}v_{y_1}\overline{v_{y_2}}w_{d_1}\overline{w_{d_2}}\bigg|\sum_{\substack{X\leq x\leq X'\\ Z\leq xy_1,xy_2\leq Z'\\ xb'\equiv b\pmod{W}\\ xy_i\equiv -2\pmod{d_i}\text{ for
}i=1,2}}e(\alpha x(y_1-y_2)/W)\bigg|\\ \ll&\sum_{\substack{1\leq y_1,y_2\leq Y'\\ y_1\equiv y_2\equiv b'\pmod{W}\\
d_1,d_2\leq D}}|v_{y_1}||v_{y_2}||w_{d_1}||w_{d_2}|\min\bigg\{\frac{X}{[d_1,d_2,W]},\frac{1}{\|\alpha
[d_1,d_2,W](y_1-y_2)/W\|}\bigg\}\\
\ll&Y(\log(XY))^{6^A}\bigg(\sum_{\substack{h\leq D^2Y}}\tau^{4A+3}(hW)\min\bigg\{\frac{XY}{hW},\frac{1}{\|\alpha h\|}\bigg\}+\frac{X}{W}\sum_{\substack{h\leq D^2}}\frac{\tau^{4A+2}(hW)}{h}\bigg)\\ &(\text{where }h=[d_1,d_2,W](y_1-y_2)/W\text{ if }y_1>y_2, \text{ and }h=[d_1,d_2,W]/W)\text{ if }y_1=y_2\\
\ll&Y(\log(DXY))^{6^A+1}\tau^{4A+3}(W)\max_{H\leq D^2Y}\sum_{\substack{H/2\leq h\leq H}}\tau^{4A+3}(h)\min\bigg\{\frac{XY}{HW},\frac{1}{\|\alpha h\|}\bigg\}\\ &+\frac{XY\tau^{4A+2}(W)}{W}(\log(DXY))^{2^{4A+3}}. \end{align*} Applying Lemma 2.2 of \cite{Vaughan97}, for any $H\leq D^2Y$, \begin{align*}
&\sum_{\substack{H/2\leq h\leq H}}\tau^{4A+3}(h)\min\bigg\{\frac{XY}{HW},\frac{1}{\|\alpha h\|}\bigg\}\\
\leq&\bigg(\sum_{\substack{h\leq H}}\tau^{8A+6}(h)\bigg)^{1/2}\bigg(\frac{XY}{HW}\sum_{\substack{h\leq H}}\min\bigg\{\frac{XY}{hW},\frac{1}{\|\alpha h\|}\bigg\}\bigg)^{1/2}\\ \ll&_A\bigg(\frac{X^2Y^2(\log(DYq))^{2^{8A+6}}}{W^2}\bigg(\frac{1}{q}+\frac{D^2W}{X}+\frac{qW}{XY}\bigg)\bigg)^{1/2}. \end{align*} This concludes the proof of (\ref{expsum1}).
Let us turn to (\ref{expsum2}). Clearly \begin{align*}
&\bigg(\sum_{X\leq x\leq X'}u_x\bigg|\sum_{\substack{Y\leq y\leq Y'\\ Z\leq xy\leq Z'\\ xy\equiv b\pmod{W}}}v_ye(\alpha (xy-b)/W)\sum_{\substack{d\mid xy+2\\
d\leq D}}w_d\bigg|\bigg)^2\\
\leq&\bigg(\sum_{X\leq x\leq X'}|u_x|^2\bigg)\bigg(\sum_{X\leq x\leq X'}\bigg|\sum_{\substack{Y\leq y\leq Y'\\ Z\leq xy\leq Z'\\ xy\equiv b\pmod{W}}}v_ye(\alpha (xy-b)/W)\sum_{\substack{d\mid xy+2\\
d\leq D}}w_d\bigg|^2\bigg)\\ \ll&X(\log X)^{6^A}\sum_{\substack{ 1\leq b'\leq W,\ (b',W)=1\\ X\leq x\leq X'\\ x\equiv b'\pmod{W}\\
d_1,d_2\leq D}}|w_{d_1}||w_{d_2}|
\prod_{i=1}^2\bigg|\sum_{\substack{Y\leq y_i\leq Y'\\ Z\leq xy_i\leq Z'\\ y_ib'\equiv b\pmod{W}\\
xy_i\equiv -2\pmod{d_i}}}v_{y_i}e(\alpha xy_i/W)\bigg|. \end{align*} By the partial summation, \begin{align*} &\sum_{\substack{Y\leq y_i\leq Y'\\ Z\leq xy_i\leq Z'\\ y_ib'\equiv b\pmod{W}\\ xy_i\equiv -2\pmod{d_i}}}v_{y_i}e(\alpha xy_i/W)\\ =&v_{Y'}\sum_{\substack{1\leq y_i\leq Y'\\ Z\leq xy_i\leq Z'\\ y_ib'\equiv b\pmod{W}\\ xy_i\equiv -2\pmod{d_i}}}e(\alpha xy_i/W)- v_{Y}\sum_{\substack{1\leq y_i\leq Y-1\\ Z\leq xy_i\leq Z'\\ y_ib'\equiv b\pmod{W}\\ xy_i\equiv -2\pmod{d_i}}}e(\alpha xy_i/W)\\ &-\sum_{Y\leq Y''\leq Y'-1}(v_{Y''+1}-v_{Y''})\sum_{\substack{1\leq y_i\leq Y''\\ Z\leq xy_i\leq Z'\\ y_ib'\equiv b\pmod{W}\\ xy_i\equiv -2\pmod{d_i}}}e(\alpha xy_i/W)\\ \ll&\bigg(v_{Y'}+v_{Y} +\sum_{Y\leq Y''\leq Y'-1}(v_{Y''+1}-v_{Y''})\bigg)
\min\bigg\{\frac{Y}{[d_i,W]},\frac{1}{\|\alpha
[d_i,W]x/W\|}\bigg\}\\ \ll&_A(\log(XY))^A
\min\bigg\{\frac{Y}{[d_i,W]},\frac{1}{\|\alpha
[d_i,W]x/W\|}\bigg\}.\\ \end{align*} For any $1\leq b'\leq W$ with $(b',W)=1$, \begin{align*} &\sum_{\substack{ X\leq x\leq X'\\ x\equiv b'\pmod{W}\\
d_1,d_2\leq D}}|w_{d_1}||w_{d_2}|
\prod_{i=1}^2\min\bigg\{\frac{Y}{[d_i,W]},\frac{1}{\|\alpha
[d_i,W]x/W\|}\bigg\}\\ \leq&(\log(XY))^{2A}\sum_{\substack{X\leq x\leq X'\\ x\equiv b'\pmod{W}}}\bigg(\sum_{\substack{
d\leq D}}\tau(d)^A\min\bigg\{\frac{Y}{[d,W]},\frac{1}{\|\alpha
[d,W]x/W\|}\bigg\}\bigg)^2\\ \leq&(\log(XY))^{2A}\sum_{\substack{X\leq x\leq X'\\ x\equiv b'\pmod{W}}}\bigg(\sum_{\substack{
h\leq D}}\tau(hW)^{A+1}\min\bigg\{\frac{Y}{hW},\frac{1}{\|\alpha hx\|}\bigg\}\bigg)^2\\ &(\text{where }h=[d,W]/W)\\ \leq&(\log(XY))^{2A}\sum_{\substack{X\leq x\leq X'\\
x\equiv b'\pmod{W}}}\bigg(\sum_{2^k\leq D}\sum_{\substack{ 2^k\leq h\leq 2^{k+1}}}\tau(hW)^{A+1}\min\bigg\{\frac{Y}{hW},\frac{1}{\|\alpha hx\|}\bigg\}\bigg)^2\\ \ll&(\log(DXY))^{2A+2}\max_{H\leq D}\sum_{\substack{X\leq x\leq X'\\
x\equiv b'\pmod{W}}}\bigg(\sum_{\substack{H/2\leq h\leq H}}\tau(hW)^{A+1}\min\bigg\{\frac{Y}{HW},\frac{1}{\|\alpha hx\|}\bigg\}\bigg)^2. \end{align*} And for any $H\leq D$, \begin{align*} &\sum_{\substack{X\leq x\leq X'\\
x\equiv b'\pmod{W}}}\bigg(\sum_{\substack{H/2\leq h\leq H}}\tau(hW)^{A+1}\min\bigg\{\frac{Y}{HW},\frac{1}{\|\alpha hx\|}\bigg\}\bigg)^2\\ \leq&\sum_{\substack{X\leq x\leq X'\\
x\equiv b'\pmod{W}}}\bigg(\tau(W)^{2A+2}\sum_{\substack{h\leq H}}\tau(h)^{2A+2}\bigg)\bigg(\frac{Y}{HW}\sum_{\substack{h\leq H}}\min\bigg\{\frac{Y}{HW},\frac{1}{\|\alpha hx\|}\bigg\}\bigg)\\
\ll&_A(\log D)^{2^{2A+2}}Y\sum_{\substack{1\leq x\leq X'H}}\tau(x)\min\bigg\{\frac{Y}{HW},\frac{1}{\|\alpha x\|}\bigg\}. \end{align*} Finally, \begin{align*}
&\sum_{\substack{1\leq x\leq X'H}}\tau(x)\min\bigg\{\frac{Y}{HW},\frac{1}{\|\alpha x\|}\bigg\}\\
\ll&\bigg(\sum_{\substack{1\leq x\leq X'H}}\tau(x)^2\bigg)^{1/2}\bigg(\frac{Y}{HW}\sum_{\substack{1\leq x\leq X'H}}\min\bigg\{\frac{X'Y}{xW},\frac{1}{\|\alpha x\|}\bigg\}\bigg)^{1/2}\\ \ll&\bigg(\frac{X^2Y^2(\log(DXq))^{4}}{W^2}\bigg(\frac{1}{q}+\frac{DW}{Y}+\frac{qW}{XY}\bigg)\bigg)^{1/2}. \end{align*} \end{proof}
Define $$ \tau_k(x)=\{(d_1,d_2,\ldots,d_k):\, d_1d_2\cdots d_k\mid x\}. $$ Let $G(x)$ be an arbitrary complex function over $\mathbb N$. Consider
\noindent {\bf Type I sums}{\it $$ \sum_{\substack{M<m\leq M_1\\ L<l\leq L_1\\ P\leq ml\leq P_1}}a_mG(ml)\qquad\text{and}\qquad\sum_{\substack{M<m\leq M_1\\ L<l\leq L_1\\ P\leq ml\leq P_1}}a_m(\log l)G(ml) $$
where $M_1\leq 2M$, $L_1\leq 2L$, $|a_m|\leq\tau_5(m)\log P$,}
\noindent and
\noindent {\bf Type II sums}{\it $$ \sum_{\substack{M<m\leq M_1\\ L<l\leq L_1\\ P\leq ml\leq P_1}}a_mb_lG(ml) $$
where $M_1\leq 2M$, $L_1\leq 2L$, $|a_m|\leq\tau_5(m)\log P$,
$|b_l|\leq\tau_5(l)\log P$.}
The following Lemma is due to Heath-Brown \cite{HeathBrown82}: \begin{Lem} \label{heathbrown} Let $P, P_1, u, v, z$ be positive integers satisfying $2<P<P_1\leq 2P$, $2\leq u<v\leq z\leq P$, $u^2\leq z$, $128uz^2\leq P_1$, $2^{18}P_1\leq v^3$. Then we may decompose the sum $$ \sum_{P<n\leq P_1}\Lambda(n)G(n) $$ into $O((\log P)^6)$ sums, each of which is either of type I with $L\geq z$ or of type II with $u\leq L\leq v$. \end{Lem}
\begin{Lem} \label{spmminor} For any $\alpha\in\mathfrak m$, $$ S_{n,z_0}^{\pm}(\alpha)\ll n(\log n)^{6^8-B/4}. $$ \end{Lem} \begin{proof} Clearly $$ S_{n,z_0}^{\pm}(\alpha)=\sum_{\substack{n^{0.99}\leq p\leq n\\ p\equiv b\pmod{W}}}e(\alpha(p-b)/W)\log p\sum_{d\mid (p+2,P(z_0))}\lambda_D^\pm(d)+O(n^{0.995}). $$ And notice that for any $x\leq n$, $$
\bigg|\sum_{d\mid(x+2,P(z_0))}\lambda_D^{\pm}(d)\bigg|\leq\tau(x+2)\ll_\epsilon n^\epsilon. $$ So it suffices to estimate the sum \begin{equation} \label{minorm} \sum_{\substack{n'\leq x\leq n\\ x\equiv b\pmod{W}}}\Lambda(x)e(\alpha(x-b)/W)\sum_{d\mid(x+2,P(z_0))}\lambda_D^{\pm}(d), \end{equation} where $n'\geq n/2$. Since $\alpha\in\mathfrak m$, there exist $1\leq a\leq q$ with $(a,q)=1$ and $(\log n)^B\leq q\leq n(\log n)^{-B}$ such that
$|\alpha q-a|\leq (\log n)^B/n$. Applying Lemma \ref{heathbrown} with $u=n^{0.17}$, $v=n^{0.34}$ and $z=n^{0.35}$, the sum (\ref{minorm}) can be decomposed into $O((\log n)^6)$ type I sums $$ \sum_{\substack{M<m\leq M_1\\ L<l\leq L_1\\ ml\equiv b\pmod{W}\\ n'\leq ml\leq n}}a_me(\alpha(ml-b)/W)\sum_{d\mid(ml+2,P(z_0))}\lambda_D^{\pm}(d) $$ and $$ \qquad\sum_{\substack{M<m\leq M_1\\ L<l\leq L_1\\ ml\equiv b\pmod{W}\\ n'\leq ml\leq n}}a_m(\log l)e(\alpha(ml-b)/W)\sum_{d\mid(ml+2,P(z_0))}\lambda_D^{\pm}(d) $$ with $L\geq n^{0.35}$, and type II sums $$ \sum_{\substack{M<m\leq M_1\\ L<l\leq L_1\\ ml\equiv b\pmod{W}\\ n'\leq ml\leq n}}a_mb_le(\alpha(ml-b)/W)\sum_{d\mid(ml+2,P(z_0))}\lambda_D^{\pm}(d) $$ with $n^{0.17}\leq L\leq n^{0.34}$. Noting that $\lambda_D^{\pm}(d)=0$ whenever $d\geq D$, in view of (\ref{expsum1}) with $A=5$, these type II sums are all $\ll n(\log n)^{2^{13}-B/4}$. And by (\ref{expsum2}), all type I sums are $\ll n(\log n)^{6^7-B/4}$.
\end{proof}
\section{The Major Arcs} \setcounter{equation}{0} \setcounter{Thm}{0} \setcounter{Lem}{0} \setcounter{Cor}{0}
Define $$ \Delta(x;q):=\max_{\substack{1\leq r\leq q\\
(r,q)=1}}\bigg|\sum_{\substack{p\leq x\\ p\equiv r\pmod{q}}}\log p-\frac{x}{\phi(q)}\bigg|. $$ The well-known Bombieri-Vinogradov theorem asserts that for any $A>0$ \begin{equation} \label{bv} \sum_{q\leq n^{1/2-\epsilon}}\max_{x\leq n}\Delta(x;q)\ll_{A,\epsilon}\frac{n}{(\log n)^A}. \end{equation}
Define $$ \phi_2(q):=q\prod_{\substack{2<p\mid q\\ }}\bigg(1-\frac{2}{p}\bigg). $$ \begin{Lem} \label{spmFf} $$ S_{n,z_0}^+(0)\leq\frac{4e^{-\gamma}k_0\mathfrak S_1 n}{\phi_2(W)\log n}(F(s)+O(e^{-s}(\log n)^{-1/3})) $$ and $$ S_{n,z_0}^-(0)\geq\frac{4e^{-\gamma}k_0\mathfrak S_1 n}{\phi_2(W)\log n}(f(s)+O(e^{-s}(\log n)^{-1/3})), $$ where $\gamma$ is Euler's constant, $$ \mathfrak S_1=\prod_{p>2}\bigg(1-\frac{1}{(p-1)^2}\bigg)=0.6601\ldots $$ and $s=\log D/\log z_0$. \end{Lem} \begin{proof} \begin{align*} S_{n,z_0}^+(0) =&\sum_{\substack{d\mid P(z_0)\\ b\equiv -2\pmod{(d,W)}}}\lambda_D^+(d)\sum_{\substack{p\leq n\\ p\equiv b\pmod{W}\\ p\equiv -2\pmod{d}}}\log p\\ =&\sum_{\substack{d\mid P(z_0)\\ (d,W)=1}}\lambda_D^+(d)\bigg(\frac{n}{\phi(Wd)}+O(\Delta(n;Wd))\bigg) \end{align*} since $(W,b+2)=1$. Since $\lambda_D^+(d)$ vanishes for $d\geq D$, by (\ref{bv}) we have, \begin{align*} S_{n,z_0}^+(0) =\frac{n}{\phi(W)}\sum_{\substack{d\mid P(z_0)\\ (d,W)=1}}\frac{\lambda_D^+(d)}{\phi(d)}+O\bigg(\frac{n}{(\log n)^{5B}}\bigg). \end{align*} Applying (\ref{rosserF}), \begin{align*} \sum_{\substack{d\mid P(z_0)\\ (d,W)=1}}\frac{\lambda_D^+(d)}{\phi(d)} \leq\prod_{p\leq z_0,\ p\nmid W}\bigg(1-\frac{1}{p-1}\bigg)(F(s)+O(e^{-s}(\log D)^{-1/3})). \end{align*} Similarly, \begin{align*} S_{n,z_0}^-(0) \geq\frac{n}{\phi(W)}\prod_{p\leq z_0,\ p\nmid W}\bigg(1-\frac{1}{p-1}\bigg)(f(s)+O(e^{-s}(\log n)^{-1/3})). \end{align*} Finally, by the Mertens theorem, \begin{align*} \prod_{p\leq z_0,\ p\nmid W}\bigg(1-\frac{1}{p-1}\bigg)^{-1}= \frac{\phi_2(W)}{2\phi(W)}\prod_{2<p\leq z_0}\bigg(1-\frac{1}{p-1}\bigg)^{-1}=& \frac{\phi_2(W)}{4\phi(W)}(\mathfrak S_1^{-1}e^{\gamma}\log{z_0}+O(1)). \end{align*} \end{proof}
Let $m=(n-b)/W$. Define $\Lambda_*(x)=\log x$ or $0$ according to whether $x$ is prime. \begin{Lem} \label{saq} Suppose that $1\leq a\leq q\leq (\log n)^B$ and $(a,q)=1$. Then we have \begin{align}
\bigg|S_{n,z_0}(a/q)-\frac{{\bf 1}_{(W,q)=1}\mu(q)\tau^*(a,q)4e^{-\gamma}k_0\mathfrak S_1Wm}{\phi_2(Wq)\log(Wm+b)}\bigg| \leq\frac{5e^{-\gamma}k_0\mathfrak S_1(F(s)-f(s))Wm}{\phi_2(W)\log(Wm+b)}, \end{align} where ${\bf 1}_{(W,q)=1}=1$ or $0$ according to whether $(W,q)=1$, $$ \tau^*(a,q)=\sum_{\substack{d\mid q\\ (d,q/d)=1}}e(ar_d/q), $$ and $1\leq r_d\leq q$ is the unique integer $r$ such that $Wr\equiv -b\pmod{d}$ and $Wr\equiv -b-2\pmod{q/d}$. \end{Lem} \begin{proof} Clearly \begin{align*} &\sum_{\substack{1\leq x\leq m\\ (Wx+b+2,P(z_0))=1}}\Lambda_*(Wx+b)e(ax/q) =&\sum_{\substack{1\leq r\leq q\\ (Wr+b,q)=1\\ (Wr+b+2,q)=1}}e(ar/q)\sum_{\substack{1\leq p\leq Wm+b\\ p\equiv Wr+b\pmod{Wq}\\ (p+2,P(z_0))=1}}\log p+O(\log^{B+1}n). \end{align*} Notice that \begin{align*} &\sum_{\substack{1\leq p\leq Wm+b\\ p\equiv Wr+b\pmod{Wq}}}\log p\sum_{d\mid(p+2,P(z_0))}\lambda_D^-(d)\\ \leq&\sum_{\substack{1\leq p\leq Wm+b\\ p\equiv Wr+b\pmod{Wq}}}\log p\sum_{d\mid (p+2,P(z_0))}\mu(d)\\\leq&\sum_{\substack{1\leq p\leq Wm+b\\ p\equiv Wr+b\pmod{Wq}}}\log p\sum_{d\mid(p+2,P(z_0))}\lambda_D^+(d). \end{align*} From the proof of Lemma \ref{spmFf}, we know that \begin{align*}
&\bigg|\sum_{\substack{1\leq p\leq Wm+b\\ p\equiv Wr+b\pmod{Wq}\\ (p+2,P(z_0))=1}}\log p-
\frac{e^{-\gamma}k_0\mathfrak S_1Wm}{\phi_2(Wq)\log(Wm+b)}\bigg|\\ \leq&\frac{1.1e^{-\gamma}k_0\mathfrak S_1(F(s)-f(s))Wm}{\phi_2(Wq)\log(Wm+b)}. \end{align*} By noting that $W$ is even and $(W,b(b+2))=1$, \begin{align*} \sum_{\substack{1\leq r\leq q\\ (Wr+b,q)=1\\ (Wr+b+2,q)=1}}e(ar/q) =&\sum_{d_1,d_2\mid q}\mu(d_1)\mu(d_2)\sum_{\substack{1\leq r\leq q\\ d_1\mid Wr+b\\ d_2\mid Wr+b+2}}e(ar/q)\\ =&\sum_{\substack{d_1d_2=q\\ (d_1,d_2)=1\\ (d_1,W)=(d_2,W)=1}}\mu(d_1)\mu(d_2)e(ar_{d_1}/q)\\ =&\begin{cases} \mu(q)\tau^*(a,q)&\text{if }(W,q)=1,\\ 0&\text{otherwise}. \end{cases} \end{align*} Furthermore, we have $$
|\{1\leq r\leq q:\, ((Wr+b)(Wr+b+2),q)=1\}|=q\prod_{\substack{p\mid q,\ p\nmid W}}\bigg(1-\frac{2}{p}\bigg)=\frac{\phi_2(Wq)}{\phi_2(W)}. $$ All are done. \end{proof}
\begin{Lem} \label{spmalpha} Suppose that $1\leq a\leq q\leq (\log n)^B$ and $(a,q)=1$. Then for any $\alpha\in\mathfrak M_{a,q}$, \begin{align} S_{n,z_0}^\pm(\alpha)=\frac{S_{n,z_0}^\pm(a/q)}{m}\sum_{1\leq y\leq m}e(\theta y)+ O\bigg(\frac{m}{(\log m)^{3B}}\bigg), \end{align} where $\theta=\alpha-a/q$. \end{Lem} \begin{proof} By the partial summation, \begin{align*} &S_{n,z_0}^\pm(\alpha)\\ =&e(\theta m)\sum_{\substack{1\leq x\leq m}}\Lambda_*(Wx+b)e(ax/q)\sum_{d\mid (Wx+b+2,P(z_0))}\lambda_D^{\pm}(d)\\ &-\sum_{y\leq m-1}(e(\theta(y+1))-e(\theta y))\sum_{\substack{1\leq x\leq y}}\Lambda_*(Wx+b)e(ax/q)\sum_{d\mid (Wx+b+2,P(z_0))}\lambda_D^{\pm}(d). \end{align*} Recalling that $(W,b+2)=1$, write \begin{align*} &\sum_{\substack{1\leq x\leq y}}\Lambda_*(Wx+b)e(ax/q)\sum_{d\mid (Wx+b+2,P(z_0))}\lambda_D^{\pm}(d)\\ =&\sum_{\substack{d\mid P(z_0)\\ (d,W)=1\\ d\leq D}}\lambda_D^\pm(d)\sum_{\substack{1\leq x\leq y\\ Wx\equiv-b-2\pmod{d}}}\Lambda_*(Wx+b)e(ax/q)\\ =&\sum_{\substack{d\mid P(z_0)\\ (d,W)=1\\ d\leq D}}\lambda_D^\pm(d)\bigg(\sum_{\substack{1\leq r\leq q\\ (Wr+b,q)=1\\ Wr\equiv-b-2\pmod{(d,q)}}}e(ar/q)\sum_{\substack{1\leq x\leq y\\ x\equiv r\pmod{q}\\ Wx\equiv-b-2\pmod{d}}}\Lambda_*(Wx+b)+O(\log^{B+1}n)\bigg). \end{align*} Now \begin{align*} &\sum_{\substack{1\leq x\leq y\\ x\equiv r\pmod{q}\\ Wx\equiv-b-2\pmod{d}}}\Lambda_*(Wx+b)=\frac{Wy+b}{\phi(W[d,q])}+O(\Delta(Wy+b;W[d,q])). \end{align*} Notice that for any $d'$ with $q\mid d'$,
$|\{d:\,[d,q]=d'\}|\leq\tau(q)$. Hence \begin{align*} &\sum_{\substack{1\leq x\leq y}}\Lambda_*(Wx+b)e(ax/q)\sum_{d\mid (Wx+b+2,P(z_0))}\lambda_D^{\pm}(d)\\ =&\sum_{\substack{d\mid P(z_0)\\ (d,W)=1\\ d\leq D}}\lambda_D^\pm(d)\sum_{\substack{1\leq r\leq q\\ (Wr+b,q)=1\\ Wr\equiv-b-2\pmod{(d,q)}}}e(ar/q)\frac{Wy+b}{\phi(W[d,q])} +O\bigg(\frac{Wy+b}{(\log(Wy+b))^{5B}}\bigg). \end{align*} So \begin{align*} S_{n,z_0}^\pm(\alpha) =W\sum_{1\leq y\leq m}e(\theta y)\sum_{\substack{d\mid P(z_0)\\ (d,W)=1\\ d\leq D}}\lambda_D^\pm(d)\sum_{\substack{1\leq r\leq q\\ (Wr+b,q)=1\\ Wr\equiv-b-2\pmod{(d,q)}}}\frac{e(ar/q)}{\phi(W[d,q])} +O\bigg(\frac{m}{(\log m)^{3B}}\bigg). \end{align*} Setting $\theta\rightarrow0$ in the above equation, we obtain that \begin{align*} W\sum_{\substack{d\mid P(z_0)\\ (d,W)=1\\ d\leq D}}\lambda_D^\pm(d)\sum_{\substack{1\leq r\leq q\\ (Wr+b,q)=1\\ Wr\equiv-b-2\pmod{(d,q)}}}\frac{e(ar/q)}{\phi(W[d,q])} =\frac{S_{n,z_0}^\pm(a/q)}{m} +O\bigg(\frac{1}{(\log m)^{3B}}\bigg). \end{align*} \end{proof} Combining Lemmas \ref{spm}, \ref{spmminor}, \ref{spmFf}, \ref{saq} and \ref{spmalpha}, we get
\begin{Lem} \label{sumexp} Suppose that $1\leq a\leq q\leq (\log n)^B$ and $(a,q)=1$. Then for any $\alpha\in\mathfrak M_{a,q}$, \begin{align}
\bigg|S_{n,z_0}(\alpha)-\frac{{\bf 1}_{(W,q)=1}\mu(q)\tau^*(a,q)4e^{-\gamma}k_0\mathfrak S_1W}{\phi_2(Wq)\log n}\sum_{1\leq y\leq m}e(\theta y)\bigg|\leq \frac{15e^{-\gamma}k_0\mathfrak S_1(F(s)-f(s))n}{\phi_2(W)\log n}, \end{align} where $\theta=\alpha-a/q$. Furthermore, for any $\alpha\in\mathfrak m$, \begin{align}
|S_{n,z_0}(\alpha)|\leq \frac{5e^{-\gamma}k_0\mathfrak S_1(F(s)-f(s))n}{\phi_2(W)\log n}. \end{align} \end{Lem}
\begin{Lem} \label{primediff} \begin{align} \sum_{\substack{p_1,p_2\leq n\\ (p_i+2,P(z_0))=1\\ p_i\equiv b\pmod{W}\\ p_2-p_1=WM}}1 \ll\frac{k_0^2nW}{\phi_2(W)^2\log^4 n}\prod_{\substack{p\mid M\\ p\nmid W}}\bigg(1+\frac{2}{p}\bigg)\prod_{\substack{p\mid (WM+2)(WM-2)\\ p\nmid W}}\bigg(1+\frac{1}{p}\bigg). \end{align} \end{Lem} \begin{proof} Let $z_1=n^{1/10}$. Let $\omega_1$ and $\omega_2$ be two multiplicative functions satisfying that $$ \omega_1(p)=\begin{cases} 4&\quad\text{ if }p<z_0\text{ and }p\nmid{WM(WM-2)(WM+2)},\\ 3&\quad\text{ if }p<z_0\text{ and }p\mid{(WM-2)(WM+2)},\ p\nmid W,\\ 2&\quad\text{ if }p<z_0,\ p\mid{M}\text{ and }p\nmid W,\\ 0&\quad\text{otherwise},\\ \end{cases} $$ and $$ \omega_2(p)=\begin{cases} 2&\quad\text{ if }z_0\leq p<z_1\text{ and }p\nmid{WM},\\ 1&\quad\text{ if }z_0\leq p<z_1,\ p\mid{M}\text{ and }p\nmid W,\\ 0&\quad\text{otherwise}, \end{cases} $$ for prime $p$. And for $1\leq i\leq 2$, let $g_i$ be the multiplicative functions with $$ g_i(p)=\frac{\omega_i(p)}{p}\bigg(1-\frac{\omega_i(p)}{p}\bigg)^{-1} $$ for prime $p$, and let $$ G_1^{(i)}(z)=\sum_{\substack{l\mid P(z)\\ l<z}}g_i(l). $$ Define $$ \lambda_1(d)=\frac{d}{\omega_1(d)}\sum_{\substack{l\mid P(z_0)\\ d\mid l<z_0}}\frac{\mu(l/d)\mu(l)g_1(l)}{G_1^{(1)}(z_0)} $$ for $d\mid P(z_0)$ and $$ \lambda_2(d)=\frac{d}{\omega_2(d)}\sum_{\substack{l\mid P(z_0,z_1)\\ d\mid l<z_1}}\frac{\mu(l/d)\mu(l)g_2(l)}{G_1^{(2)}(z_1)} $$ for $d\mid P(z_0,z_1)=\prod_{z_0\leq p<z_1}p$. Then $\lambda_1(1)=\lambda_2(1)=1$. Therefore \begin{align*} &\sum_{\substack{x_1,x_2\leq n/W\\ (Wx_i+b+2,P(z_0))=1\\ (Wx_i+b,P(z_1))=1\\ x_2-x_1=M}}1\\ \leq&\sum_{\substack{x\leq n/W}}\bigg(\sum_{\substack{d\mid P(z_0)\\ d\mid (Wx+b)(Wx+WM+b)\\ d\mid (Wx+b+2)(Wx+WM+b+2)}}\lambda_1(d)\bigg)^2 \bigg(\sum_{\substack{d\mid P(z_0,z_1)\\ d\mid (Wx+b)(Wx+WM+b)}}\lambda_2(d)\bigg)^2\\ =&\sum_{\substack{d_1,d_2\mid P(z_0)\\ d_3,d_4\mid P(z_0,z_1)}}\lambda_1(d_1)\lambda_1(d_2)\lambda_2(d_3)\lambda_2(d_4)\sum_{\substack{x\leq n/W\\ [d_3,d_4]\mid (Wx+b)(Wx+WM+b)\\ [d_1,d_2],[d_3,d_4]\mid (Wx+b+2)(Wx+WM+b+2)}}1\\ =&\sum_{\substack{d_1,d_2\mid P(z_0)\\ d_3,d_4\mid P(z_0,z_1)}}\lambda_1(d_1)\lambda_1(d_2)\lambda_2(d_3)\lambda_2(d_4)\omega_1([d_1,d_2])\omega_2([d_3,d_4])\bigg(\frac{n/W}{[d_1,d_2][d_3,d_4]}+O(1)\bigg). \end{align*} By Selberg's sieve method, we know that
$|\lambda_1(d)|,|\lambda_2(d)|\leq 1$ and \begin{align*} &\sum_{\substack{d_1,d_2\mid P(z_0)\\ d_3,d_4\mid P(z_0,z_1)}}\lambda_1(d_1)\lambda_1(d_2)\lambda_2(d_3)\lambda_2(d_4)\frac{\omega_1([d_1,d_2])\omega_2([d_3,d_4])}{[d_1,d_2][d_3,d_4]}\\ =&\bigg(\sum_{d_1,d_2\mid P(z_0)}\lambda_1(d_1)\lambda_1(d_2)\frac{\omega_1([d_1,d_2])}{[d_1,d_2]}\bigg)\bigg(\sum_{d_3,d_4\mid P(z_0,z_1)}\lambda_2(d_3)\lambda_2(d_4)\frac{\omega_2([d_3,d_4])}{[d_3,d_4]}\bigg)\\ =&\frac{1}{G_1^{(1)}(z_0)}\frac{1}{G_1^{(2)}(z_1)}\ll\prod_{p <z_0}\bigg(1-\frac{\omega_1(p)}{p}\bigg)\prod_{z_0\leq p<z_1}\bigg(1-\frac{\omega_2(p)}{p}\bigg). \end{align*} Thus \begin{align*} &\sum_{\substack{x_1,x_2\leq n/W\\ (Wx_i+b+2,P(z_0))=1\\ (Wx_i+b,P(z_1))=1\\ x_2-x_1=M}}1\\ \ll&\frac{n}{W}\prod_{p\mid P(z_0)}\bigg(1-\frac{\omega_1(p)}{p}\bigg)\prod_{p\mid P(z_0,z_1)}\bigg(1-\frac{\omega_2(p)}{p}\bigg)\\ \ll&\frac{n}{W(\log z_0)^2(\log z_1)^2}\prod_{\substack{ p\mid W}}\bigg(1+\frac{4}{p}\bigg)\prod_{\substack{p\mid M\\ p\nmid W}}\bigg(1+\frac{2}{p}\bigg)\prod_{\substack{p\mid (WM+2)(WM-2)\\ p\nmid W}}\bigg(1+\frac{1}{p}\bigg)\\ \ll&\frac{k_0^2nW}{\phi_2(W)^2\log^4 n}\prod_{\substack{p\mid M\\ p\nmid W}}\bigg(1+\frac{2}{p}\bigg)\prod_{\substack{p\mid (WM+2)(WM-2)\\ p\nmid W}}\bigg(1+\frac{1}{p}\bigg). \end{align*} \end{proof}
\begin{Lem} Then \begin{align} \label{equalsum} \sum_{\substack{p_1,p_2,p_3,p_4\leq n\\ (p_i+2,P(z_0))=1\\ p_i\equiv b\pmod{W}\\ p_1+p_4=p_2+p_3}}1 \ll\frac{k_0^4Wn^3}{\phi_2(W)^4\log^8 n}. \end{align} \end{Lem} \begin{proof} Applying Lemma \ref{primediff}, \begin{align*} \sum_{\substack{p_1,p_2,p_3,p_4\leq n\\ (p_i+2,P(z_0))=1\\ p_i\equiv b\pmod{W}\\ p_1+p_4=p_2+p_3}}1\leq&\sum_{\substack{2<M\leq n/W}}\bigg(\sum_{\substack{p_1,p_2\leq n\\ (p_i+2,P(z_0))=1\\ p_i\equiv b\pmod{W}\\ p_1+WM=p_2}}1\bigg)^2+O(n^2)\\ \ll&\frac{k_0^4n^2W^2}{\phi_2(W)^4\log^8 n}\sum_{\substack{2<M\leq n/W}}\prod_{\substack{p\mid M\\ p\nmid W}}\bigg(1+\frac{2}{p}\bigg)^2\prod_{\substack{p\mid (WM-2)(WM+2)\\ p\nmid W}}\bigg(1+\frac{1}{p}\bigg)^2. \end{align*} By the H\"older inequality, \begin{align*} &\sum_{\substack{2<M\leq n/W}}\prod_{\substack{p\mid M\\ p\nmid W}}\bigg(1+\frac{2}{p}\bigg)^2 \prod_{\substack{p\mid (WM-2)(WM+2)\\ p\nmid W}}\bigg(1+\frac{1}{p}\bigg)^2\\ \leq&\bigg(\sum_{\substack{2<M\leq n/W}}\prod_{\substack{p\mid M}}\bigg(1+\frac{2}{p}\bigg)^6\bigg)^{1/3} \prod_{j=1}^2\bigg(\sum_{\substack{2<M\leq n/W}}\prod_{\substack{p\mid WM+2(-1)^j\\ p\not=2}}\bigg(1+\frac{1}{p}\bigg)^6\bigg)^{1/3}. \end{align*} Since $(1+p^{-1})^6\leq 1+24p^{-1}$, \begin{align*} \sum_{\substack{2<M\leq n/W}}\prod_{\substack{p\mid WM\pm 2\\ p\not=2}}\bigg(1+\frac{1}{p}\bigg)^6 \leq&\sum_{\substack{2<M\leq n/W}} \sum_{\substack{d\mid WM\pm 2\\ 2\nmid d}}\frac{\tau(d)^{12}}{d}\\ =&\sum_{\substack{d\mid n\pm 2\\ (d,W)=1}}\frac{\tau(d)^{12}}{d}\sum_{\substack{2<M\leq n/W\\ d\mid WM\pm 2}}1 \ll\frac{n}{W}. \end{align*} And by \cite[Lemma 14]{Green02}, we have $$ \sum_{\substack{2<M\leq n/W}}\prod_{\substack{p\mid M}}\bigg(1+\frac{2}{p}\bigg)^6\leq \sum_{\substack{2<M\leq n/W}}\prod_{\substack{p\mid M}}\bigg(1+\frac{1}{p}\bigg)^{12}\ll \frac{n}{W}. $$ \end{proof}
\section{Proof of Theorem \ref{chengoldbach}} \setcounter{equation}{0} \setcounter{Thm}{0} \setcounter{Lem}{0} \setcounter{Cor}{0}
First, let us introduce Green and Tao's enveloping sieve. Let $N$ be a large integer. Suppose that $a_1,\ldots,a_k,b_1,\ldots,b_k$ be integers with $|a_i|,|b_i|\leq N$. We say $$ \mathcal {F}(x):=\prod_{i=1}^k(a_ix+b_i). $$ is a $k$-linear form.
For every integer $q\geq 1$, define $$
\gamma_\mathcal {F}(p):=q^{-1}|\{1\leq x\leq q:\, (\mathcal {F}(x),q)=1\}|>0. $$ Let $$ X_{R!}(x)=\{x\in\mathbb Z:\, (\mathcal {F}(x),R!)=1\}, $$ where $1\leq R\leq N$. \begin{Lem}[{\cite[Proposition 3.1]{GreenTao06}}] \label{envsieve} There exists a non-negative function $\beta_R:\mathbb Z\to\mathbb R$ satisfying the following properties:
\noindent(i) \begin{equation} \label{betalower} \beta_R(x)\gg_k\mathfrak{S}_\mathcal {F}^{-1}\log^k R \,{\bf 1}_{X_{R!}}(x) \end{equation} for all integers $n$, where $$ \mathfrak S_\mathcal {F}:=\prod_{p}\frac{\gamma_\mathcal {F}(p)}{(1-1/p)^k}. $$
\noindent(ii) \begin{equation} \label{beteupper} \beta_R(x)\ll_{k,\epsilon}N^\epsilon \end{equation} for all $1\leq x\leq N$ and $\epsilon>0$.
\noindent(iii) \begin{equation} \beta_R(x)=\sum_{q\leq R^2}\sum_{\substack{1\leq a\leq q\\ (a,q)=1}}w(a/q)e(-ax/q), \end{equation} where $w(a/q)=w_R(a/q)$ satisfies $w(1)=1$ and $$
|w(a/q)|\ll_{k,\epsilon} q^{\epsilon-1} $$ for all $1\leq a\leq q\leq R^2$ with $(a,q)=1$.
\noindent(iv) For $1\leq a\leq q\leq R^2$ with $(a,q)=1$, if $q$ is not square-free, or $\gamma(q)=1$ and $q>1$, then $w(a/q)=0$. \end{Lem} Green and Tao also established a restriction theorem for $\beta_R$: \begin{Lem}[{\cite[Proposition 4.2]{GreenTao06}}] Let $R$, $N$ be large numbers such that $1\leq R\leq N^{1/10}$. Let $k$, $\mathcal {F}$, $\beta_R$ as defined in Lemma \ref{envsieve}. Suppose that $\{u_i\}_{i=1}^N$ is an arbitrary sequence of complex numbers. Then for any $\rho>2$,
\begin{equation}\label{restriction}\bigg(\sum_{r\in\mathbb Z_N}\bigg|\frac{1}{N}\sum_{1\leq x\leq N}u_x\beta_R(x)e(-xr/N)\bigg|^\rho\bigg)^{1/\rho}
\ll_{\rho,k}\bigg(\frac{1}{N}\sum_{1\leq x\leq N}|u_x|^2\beta_R(x)\bigg)^{1/2}. \end{equation} \end{Lem}
The following lemma can be derived by a trivial modification of Chen's original proof in \cite{Chen73}: \begin{Lem} \label{chen} Suppose that $W$ is a positive integer. Then there exists a function $n_0(W)$ such that for every $1\leq b\leq W$ with $(b(b+2),W)=1$ and $n\geq n_0(W)$, $$
|\{p\leq n:\, p\equiv b\pmod{W},\ p+2\in\mathcal P_2\text{ and
}(p+2,P(n^{1/10}))=1\}|\geq \frac{C_1}{\phi_2(W)}\frac{n}{(\log n)^2}, $$ where $C_1$ is an absolute constant. \end{Lem}
\begin{Lem} \label{nW} Suppose that $n_0(W)$ is an increasing positive function for all positive integers $W$, and $G(n)$ is an increasing positive function with $\lim_{n\to\infty}G(n)=\infty$. Then there exists an increasing positive function $W_0(n)$ for sufficiently large $n$ such that $n\geq n_0(W(n))$, $W_0(n)\leq G(n)$ and $\lim_{n\to\infty}W_0(n)=\infty$. \end{Lem} \begin{proof} Let $$ W_0(n)=\max\{W\leq G(n):\, n_0(W)\leq n\}. $$ Clearly $W_0(n)$ is well-defined for sufficiently large $n$. Assume on the contrary that $\lim_{n\to\infty}W_0(n)<\infty$, i.e., there exists an integer $W'$ such that $W_0(n)<W'$ for all $n$. Let $n'\geq n_0(W')$ be an integer such that $G(n')\geq W'$. Obviously such $n'$ exists. But $W_0(n')\geq W'$ now. This leads to a contradiction. \end{proof}
Let $C_2$ be the implied constant in (\ref{betalower}) with $k=2$. Let $$ \varpi=\frac{\min\{C_1C_2,1\}}{10000}. $$
Let $C_3$ be the implied constant in (\ref{restriction}) with $\rho=12/5$ and $k=2$. And let $C_4$ be the implied constant in (\ref{equalsum}). \begin{Lem} \label{kappa} There exists $0<\kappa\leq\varpi$ such that we can choose $0<\delta,\epsilon\leq 1$ with $\epsilon^{6C_3^{12/5}\delta^{-12/5}+60C_4\delta^{-4}}\geq\kappa$ and $$ 3072\epsilon^2(C_3\delta^{-12/5}+5C_4\delta^{-4})+ 72C_3^{24/13}C_4^{3/13}\delta^{1/13}\leq\varpi^6. $$ \end{Lem} \begin{proof} Let $$ \delta=\min\bigg\{\frac{\varpi^{78}}{144^{13}C_3^{24}C_4^{3}},1\bigg\} $$ and $$ \epsilon=\min\bigg\{\frac{\varpi^{3}\delta^{2}}{192(C_3\delta^{-12/5}+5C_4\delta^{-4})^{1/2}},1\bigg\}. $$ Clearly $3072\epsilon^2(C_3\delta^{-12/5}+5C_4\delta^{-4})+ 72C_3^{24/13}C_4^{3/13}\delta^{1/13}\leq\varpi^6$. So we may arbitrarily choose a $\kappa$ satisfying $$ \kappa\leq\min\{\epsilon^{6C_3^{12/5}\delta^{-12/5}+60C_4\delta^{-4}},\varpi\}. $$ \end{proof}
Let $\kappa$ be a small constant satisfying the requirements of Lemma \ref{kappa}. Notice that $f(s)$ is increasing, $F(s)$ is decreasing and $F(s),f(s)=1+O(e^{-s})$. Choose a sufficiently large $k_0$ satisfying that $$ 20(F(k_0/4)-f(k_0/4))\leq\kappa^2. $$
Suppose that $n$ is a sufficiently large integer. Let $w=w(n)$ be a positive function satisfying $P(w)\leq\log n$, $n\geq n_0(P(w))$ (where $n_0$ is defined in Lemma \ref{chen}) and $\lim_{n\to\infty}w(n)=\infty$. By Lemma \ref{nW}, such $w$ exists. Let $W=P(w)$. The following lemma can be easily verified: \begin{Lem} For any odd integer $n$ with $3\mid n$, there exist $1\leq b_1,b_2,b_3\leq W$ with $(b_i(b_i+2),W)=1$ such that $$ n\equiv b_1+b_2+b_3\pmod{W}. $$ Furthermore, if $n\equiv 4\pmod{6}$, then there exist $1\leq b_1,b_2\leq W$ with $(b_i(b_i+2),W)=1$ such that $$ n\equiv b_1+b_2\pmod{W}. $$ \end{Lem}
Now suppose that $n$ is odd. Suppose that $1\leq b_1,b_2,b_3\leq W$ are integers satisfying $(b_i(b_i+2),W)=1$ and $n\equiv b_1+b_2+b_3\pmod{W}$. Let $n'=(n-b_1-b_2-b_3)/W$.
Let $N$ be a prime in the interval $[(1+\kappa^2/20)n/W, (1+\kappa^2/10)n/W]$ and $R=N^{1/10}$. Thanks to the prime number theorem, such a prime $N$ always exists whenever $n$ is sufficiently large. Let $\mathcal {F}_i(x)=(Wx+b_i)(Wx+b_i+2)$ for $i=1,2$. Substituting $N,\ R,\ \mathcal {F}_i$ to Lemma \ref{envsieve}, we get the desired functions $\beta_i=\beta_{i,R}$ for $i=1,2$.
Let $$ A_i= \{x\leq (n-b_i)/2W:\, Wx+b_i\text{ is Chen's prime and }(Wx+b_i+2,P(n^{1/10}))=1\}, $$ for $i=1,2$, and define $$ {\mathfrak a}_i(x)= {\bf 1}_{A_i}(x)\frac{C_2\mathfrak S_1^{-1}}{1000}\frac{\phi_2(W)(\log(Wx+b_i))^2}{n}. $$ By Lemma \ref{chen}, clearly we have \begin{equation} \label{a12sum}
\sum_{x}{\mathfrak a}_i(x)\geq(1-\kappa)\frac{C_2\mathfrak S_1^{-1}}{1000}\frac{\phi_2(W)(\log n)^2|A_i|}{n}\geq 6\varpi, \end{equation} whenever $n$ is sufficiently large.
On the other hand, it is easy to see that $$ \mathfrak S_{\mathcal {F}_i}=\prod_{\substack{p\mid W\\ p>2}}\frac{1}{(1-1/p)^2}\prod_{p\nmid W}\frac{1-2/p}{(1-1/p)^2}=\prod_{\substack{p\mid W\\ p>2}}\frac{p}{p-2} \prod_{\substack{p>2}}\bigg(1-\frac{1}{(p-1)^2}\bigg)=\frac{W\mathfrak S_1}{\phi_2(W)}. $$ Hence by (\ref{betalower}), for any $x$ \begin{equation} \label{abeta} \beta_i(x)\geq \frac{C_2\phi_2(W)\mathfrak S_1^{-1}}{W}(\log R)^2{\bf 1}_{A_i}(x)\geq N{\mathfrak a}_i(x). \end{equation}
Let $$ A_3=\{x\leq (n-b_3)/W:\, Wx+b_3\text{ is prime and }(Wx+b_3+2,P(n^{1/k_0}))=1\}. $$ and define $$ {\mathfrak a}_3(x)={\bf 1}_{A_3}(x)\frac{e^{\gamma}\phi_2(W)\log(Wx+b_3)\log n}{4k_0\mathfrak S_1n}. $$ In view of Lemmas \ref{spm} and \ref{spmFf}, we also have \begin{equation} \label{a3sum} \sum_{x}{\mathfrak a}_3(x)=\frac{e^{\gamma}\phi_2(W)\log n}{4k_0\mathfrak S_1n}S_{n,n^{1/k_0}}(0)\in[1-\kappa^2,1+\kappa^2]. \end{equation}
Below we identify the set $\{1,2,\ldots,N\}$ with the group $\mathbb Z_N=\mathbb Z/N\mathbb Z$. If there exist $x_1\in A_1$, $x_2\in A_2$ and $x_3\in A_3$ satisfying $x_1+x_2+x_3=n'$ in $\mathbb Z_N$, then the equality also holds in $\mathbb Z$. In fact, since $x_1+x_2\leq n/W$ and $x_3<n/W$ in $\mathbb Z$, we must have $x_1+x_2+x_3<n'+N$ in $\mathbb Z$.
For any function $f:\,\mathbb Z_N\to\mathbb C$, define $$ \tilde{f}(r)=\sum_{x\in\mathbb Z_N}f(x)e(-xr/N). $$ \begin{Lem} \label{nuFour} Let $\nu_i=\beta_i/N$, then for any $r\in\mathbb Z_N$, $$
|\tilde{\nu_i}(r)-{\bf 1}_{r=0}|\leq C_5w^{-1/2}, $$ where $C_5$ is an absolute constant and ${\bf 1}_{r=0}=1$ or $0$ according to whether $r=0$. \end{Lem} \begin{proof} See \cite[Lemma 6.1]{GreenTao06}. (Notice that our definitions are a little different from Green and Tao's.) \end{proof}
\begin{Lem} \label{aFour} For any $0\not=r\in\mathbb Z_N$, $$
|\tilde{{\mathfrak a}_3}(r)|\leq\frac{2}{w-2}+0.9\kappa^2. $$ \end{Lem} \begin{proof} If $r/N\in\mathfrak m$, then by Lemma \ref{sumexp}, we have $$
|\tilde{{\mathfrak a}_3}(r)|=\frac{e^\gamma\phi_2(W)\log n}{k_0\mathfrak S_1n}|S_{n,n^{1/k_0}}(r/N)|\leq\frac{2}{5}\kappa^2. $$ Suppose that there exist $1\leq a\leq q\leq (\log n)^B$ with $(a,q)=1$ such that $r/N\in\mathfrak M_{a,q}$. Then applying Lemma \ref{sumexp}, $$
|\tilde{{\mathfrak a}_3}(r)|\leq \frac{\phi_2(W)\log n}{k_0\mathfrak S_1n}\bigg(\frac{{\bf 1}_{(W,q)=1}|\tau^*(a,q)|k_0\mathfrak S_1W}{\phi_2(Wq)\log n}\bigg|\sum_{1\leq y\leq m}e(\theta y)\bigg|+ \frac{7k_0\mathfrak S_1\kappa^2 n}{10\phi_2(W)\log n}\bigg), $$ where $m=(n-b)/W$ and $\theta=r/N-a/q$. Recall that $\tau^*(a,q)=0$ whenever $(W,q)>1$. And if $a=q=1$, since $r\not=0$, $$
\bigg|\sum_{1\leq y\leq m}e(yr/N)\bigg|\leq \bigg|\sum_{1\leq y\leq N}e(yr/N)\bigg|+N-m\leq \frac{\kappa^2}{10} N. $$ Suppose that $q>1$ and $(W,q)=1$. Then by noting $W=\prod_{p<w}p$, $$
\frac{\phi_2(W)|\tau^*(a,q)|}{\phi_2(Wq)}\leq\frac{\phi_2(W)|\{d\mid q:\, (d,q/d)=1\}|}{\phi_2(Wq)}\leq\frac{2}{w-2}, $$ since $q$ has at least one prime divisor not less than $w$. Finally, we have $WN\leq1.1 n$. \end{proof}
Suppose that $\delta,\epsilon>0$ are two small numbers to be chosen later. For $1\leq i\leq 3$, let $$
\mathcal R_i=\{r\in\mathbb Z_N:\,|\tilde{a_i}(r)|>\delta\}, $$
$$
\mathcal B_i=\{x\in\mathbb Z_N:\,\|xr/N\|\leq \epsilon\}, $$
and define ${\mathfrak b}_i={\bf 1}_{\mathcal B_i}/|B_i|$. \begin{Lem} \label{bohr} $$
|\mathcal R_3|\leq 60C_4\delta^{-4}. $$ \end{Lem} \begin{proof} By (\ref{equalsum}), \begin{align*}
\sum_{r\in\mathbb Z_N}|\tilde{{\mathfrak a}_3}(r)|^4=&N\sum_{\substack{x_1,x_2,x_3,x_4\in\mathbb Z_N\\ x_1+x_4=x_2+x_3}}{\mathfrak a}_3(x_1){\mathfrak a}_3(x_2){\mathfrak a}_3(x_3){\mathfrak a}_3(x_4)\\ \leq&N\bigg(\frac{e^{\gamma}\phi_2(W)(\log n)^2}{4k_0\mathfrak S_1n}\bigg)^4\sum_{\substack{p_1,p_2,p_3,p_4\leq n\\ (p_i+2,P(n^{1/k_0}))=1\\ p_i\equiv b_3\pmod{W}\\ p_1+p_4=p_2+p_3}}1\\ \leq&C_4e^{4\gamma}(4\mathfrak S_1)^{-4}\frac{NW}{n}. \end{align*} Hence $$
|\mathcal R_3|\leq\sum_{r\in\mathbb Z_N}\delta^{-4}|\tilde{a_3}(r)|^4\leq e^{4\gamma}(4\mathfrak S_1)^{-4}. $$ \end{proof} \begin{Lem} \label{bohr} $$
|\mathcal R_1|,|\mathcal R_2|\leq 6C_3^{12/5}\delta^{-12/5} $$ provided that $w\geq C_5^{2}$. \end{Lem} \begin{proof} Suppose that $i\in\{1,2\}$. Let $u_x={\mathfrak a}_i(x)/\nu_i(x)$ or $0$ according to whether $\nu_i(x)\not=0$. By (\ref{abeta}), clearly $0\leq u_x\leq 1$. If $w\geq C_5^2$, by Lemma 4.7 we have $\tilde{\nu_i}(0)\leq 2$. Applying (\ref{restriction}), \begin{align*}
\sum_{r\in\mathbb Z_N}\bigg|\sum_{1\leq x\leq N}u_x\nu_i(x)e(-xr/N)\bigg|^{12/5}\leq& C_3^{12/5}\bigg(\sum_{1\leq x\leq N}|u_x|^2\nu_i(x)\bigg)^{6/5}\\
\leq& C_3^{12/5}|\tilde{\nu_i}(0)|^{6/5}\\ \leq&6C_3^{12/5}. \end{align*}
It follows that $|\mathcal R_i|\leq 6C_3^{12/5}\delta^{-12/5}$. \end{proof}
\begin{Lem} \label{bohr} $$
|\mathcal B_i|\geq\epsilon^{|\mathcal R_i|}N. $$ \end{Lem} \begin{proof} This is a simple application of the pigeonhole principle (cf. \cite[Lemma 1.4]{Tao}). \end{proof} For two functions $f, g:\,\mathbb Z_N\to\mathbb C$. Define $$ f*g(x)=\sum_{y\in\mathbb Z_N}f(y)g(x-y). $$ It is easy to check that $\widetilde{(f*g)}=\tilde{f}\tilde{g}$. Let ${\mathfrak a}_i'={\mathfrak a}_i*{\mathfrak b}_i*{\mathfrak b}_i$ for $1\leq i\leq 3$. \begin{Lem}
\label{a3upper} Suppose that $\epsilon^{|\mathcal R_3|}\geq (2/(w-2)+0.9\kappa^2)\kappa^{-1}$. Then for any $x\in\mathbb Z_N$, $$
|{\mathfrak a}_3'(x)|\leq\frac{1+2\kappa}{N}. $$ \end{Lem} \begin{proof} \begin{align*}
|{\mathfrak a}_3'(x)|=&|{\mathfrak a}_3(x)*{\mathfrak b}_3(x)*{\mathfrak b}_3(x)|\\
\leq&\frac{1}{N}|\tilde{{\mathfrak a}_3}(0)||\tilde{{\mathfrak b}_3}(0)|^2+
\frac{1}{N}\sum_{r\not=0}|\tilde{{\mathfrak a}_3}(r)\tilde{{\mathfrak b}_3}(r)^2|\\
\leq&\frac{1}{N}|\tilde{{\mathfrak a}_3}(0)||\tilde{{\mathfrak b}_3}(0)|^2+
\frac{1}{N}\max_{r\not=0}|\tilde{{\mathfrak a}_3}(r)|\sum_{r\in\mathbb Z_N}|\tilde{{\mathfrak b}_3}(r)|^2\\ \leq&\frac{1+\kappa^2}{N}+
\frac{1}{|B_3|}\bigg(\frac{2}{w-2}+0.9\kappa^2\bigg), \end{align*} where we used Lemma \ref{aFour} in the last step. Thus the desired result easily follows from Lemma \ref{bohr}. \end{proof} \begin{Lem} \label{a12upper} Let $i\in\{1,2\}$. Suppose that
$\epsilon^{|\mathcal R_i|}\geq C_5\kappa^{-1}w^{-1/2}$ and $C_5w^{-1/2}\leq \kappa$. Then for any $x\in\mathbb Z_N$, $$
|{\mathfrak a}_i'(x)|\leq\frac{1+2\kappa}{N}. $$ \end{Lem} \begin{proof} Since $\nu_i(x)\geq{\mathfrak a}_i(x)$, applying Lemma 4.7 \begin{align*}
|{\mathfrak a}_i'(x)|\leq&|\nu_i(x)*{\mathfrak b}_i(x)*{\mathfrak b}_i(x)|\\
\leq&\frac{1}{N}|\tilde{\nu_i}(0)||\tilde{{\mathfrak b}_i}(0)|^2+
\frac{1}{N}\max_{r\not=0}|\tilde{\nu_i}(r)|\sum_{r\in\mathbb Z_N}|\tilde{{\mathfrak b}_i}(r)|^2\\
\leq&\frac{1+C_5w^{-1/2}}{N}+\frac{C_5w^{-1/2}}{|B_i|}. \end{align*} We are done. \end{proof} \begin{Lem} \label{betaone} For $1\leq i\leq 3$, $$
|1-\tilde{{\mathfrak b}_i}(r)|\leq 16\epsilon^2 $$ for any $r\in\mathcal R_i$. \end{Lem} \begin{proof} See the proof of Lemma 6.7 of \cite{Green05}. \end{proof}
\begin{Lem} \label{threesum} Suppose that $w\geq C_5^2$. Then \begin{align*}
&\bigg|\sum_{\substack{x_1,x_2,x_3\in\mathbb Z_N\\ x_1+x_2+x_3=n'}}{\mathfrak a}_1'(x_1){\mathfrak a}_2'(x_2){\mathfrak a}_3'(x_3)- \sum_{\substack{x_1,x_2,x_3\in\mathbb Z_N\\
x_1+x_2+x_3=n'}}{\mathfrak a}_1(x_1){\mathfrak a}_2(x_2){\mathfrak a}_3(x_3)\bigg|\\ \leq&\frac{3072\epsilon^2(C_3^{12/5}\delta^{-12/5}+5C_4\delta^{-4})+ 72C_3^{24/13}C_4^{3/13}\delta^{1/13}}{N}. \end{align*} \end{Lem} \begin{proof} Clearly $$ \sum_{\substack{x_1,x_2,x_3\in\mathbb Z_N\\ x_1+x_2+x_3=n'}}{\mathfrak a}_1(x_1){\mathfrak a}_2(x_2){\mathfrak a}_3(x_3)=\frac{1}{N}\sum_{r\in\mathbb Z_N} \tilde{{\mathfrak a}_1}(r)\tilde{{\mathfrak a}_2}(r)\tilde{{\mathfrak a}_3}(r)e(n'r/N). $$ Hence \begin{align*}
&\bigg|\sum_{\substack{x_1,x_2,x_3\in\mathbb Z_N\\ x_1+x_2+x_3=n'}}{\mathfrak a}_1'(x_1){\mathfrak a}_2'(x_2){\mathfrak a}_3'(x_3)- \sum_{\substack{x_1,x_2,x_3\in\mathbb Z_N\\
x_1+x_2+x_3=n'}}{\mathfrak a}_1(x_1){\mathfrak a}_2(x_2){\mathfrak a}_3(x_3)\bigg|\\
\leq&\frac{1}{N}\sum_{r\in\mathbb Z_N}|\tilde{{\mathfrak a}_1}(r)\tilde{{\mathfrak a}_2}(r)\tilde{{\mathfrak a}_3}(r)(1-\tilde{{\mathfrak b}_1}(r)^2
\tilde{{\mathfrak b}_2}(r)^2\tilde{{\mathfrak b}_3}(r)^2)|. \end{align*}
Since $w\geq C_5^2$, for $i=1,2$, $$
|\tilde{a_i}(r)|\leq|\tilde{a_i}(0)|\leq|\tilde{\nu_i}(0)|\leq 2. $$ We also have $$
|\tilde{a_3}(r)|\leq|\tilde{a_3}(0)|\leq1+\kappa^2. $$
If $r\in\mathcal R_1\cap\mathcal R_2\cap\mathcal R_3$, then by Lemma \ref{betaone}, $$
|1-\tilde{{\mathfrak b}_i}(r)^2|\leq|1-\tilde{{\mathfrak b}_i}(r)|(1+|\tilde{{\mathfrak b}_i}(r)|)\leq
|1-\tilde{{\mathfrak b}_i}(r)|(1+|\tilde{{\mathfrak b}_i}(0)|)\leq 32\epsilon^2. $$ So \begin{align*}
&|1-\tilde{{\mathfrak b}_1}(r)^2\tilde{{\mathfrak b}_2}(r)^2\tilde{{\mathfrak b}_3}(r)^2|\\
\leq&|1-\tilde{{\mathfrak b}_1}(r)^2|+ |\tilde{{\mathfrak b}_1}(r)|^2|1-\tilde{{\mathfrak b}_2}(r)^2|+
|\tilde{{\mathfrak b}_1}(r)\tilde{{\mathfrak b}_2}(r)|^2|1-\tilde{{\mathfrak b}_3}(r)^2|\\ \leq&96\epsilon^2. \end{align*} Thus \begin{align*}
\sum_{r\in\mathcal R_1\cap\mathcal R_2\cap\mathcal R_3}|\tilde{{\mathfrak a}_1}(r)\tilde{{\mathfrak a}_2}(r)\tilde{{\mathfrak a}_3}(r)(1-\tilde{{\mathfrak b}_1}(r)^2
\tilde{{\mathfrak b}_2}(r)^2\tilde{{\mathfrak b}_3}(r)^2)|\leq&768\epsilon^2\min_{1\leq i\leq 3}|\mathcal R_i|\\ \leq&256\epsilon^2(12C_3^{12/5}\delta^{-12/5}+60C_4\delta^{-4}). \end{align*}
On the other hand, by the H\"older inequality, \begin{align*}
&\sum_{r\not\in\mathcal R_1\cap\mathcal R_2\cap\mathcal R_3}|\tilde{{\mathfrak a}_1}(r)\tilde{{\mathfrak a}_2}(r)\tilde{{\mathfrak a}_3}(r)(1-\tilde{{\mathfrak b}_1}(r)^2
\tilde{{\mathfrak b}_2}(r)^2\tilde{{\mathfrak b}_3}(r)^2)|\\
\leq&2\sum_{r\not\in\mathcal R_1\cap\mathcal R_2\cap\mathcal R_3}|\tilde{{\mathfrak a}_1}(r)\tilde{{\mathfrak a}_2}(r)\tilde{{\mathfrak a}_3}(r)|\\
\leq&2\max_{r\not\in\mathcal R_1\cap\mathcal R_2\cap\mathcal R_3}|\tilde{{\mathfrak a}_1}(r)\tilde{{\mathfrak a}_2}(r)\tilde{{\mathfrak a}_3}(r)|^{1/13}
\sum_{r\in\mathbb Z_N}|\tilde{{\mathfrak a}_1}(r)\tilde{{\mathfrak a}_2}(r)\tilde{{\mathfrak a}_3}(r)|^{12/13}\\ \leq&4\delta^{1/13}
\bigg(\sum_{r\in\mathbb Z_N}|\tilde{{\mathfrak a}_1}(r)|^{12/5}\bigg)^{5/13}
\bigg(\sum_{r\in\mathbb Z_N}|\tilde{{\mathfrak a}_2}(r)|^{12/5}\bigg)^{5/13}\bigg(\sum_{r\in\mathbb Z_N}|\tilde{{\mathfrak a}_3}(r)|^4\bigg)^{3/13}\\ \leq&4\delta^{1/13}(6C_3^{12/5})^{10/13}(60C_4)^{3/13}. \end{align*}
\end{proof}
\begin{Lem} \label{pollard} Suppose that $0<\theta_1,\theta_2,\theta_3\leqslant 1$ with $\theta_1+\theta_2+\theta_3>1$. Let $$ \theta=\min\{\theta_1,\theta_2,\theta_3,(\theta_1+\theta_2+\theta_3-1)/4\}. $$ Suppose that $N$ is a prime greater than $2\theta^{-2}$, and
$X_1,X_2,X_3$ are subsets of $\mathbb Z_N$ with $|X_i|\geqslant\theta_i N$. Then for any $y\in\mathbb Z_N$, we have $$
|\{(x_1,x_2,x_3):\, x_i\in X_i,\ x_1+x_2+x_3=y\}|\geq\theta^3 N^2. $$ \end{Lem} \begin{proof} See \cite[Lemma 3.3]{LiPan}. \end{proof}
Now we are ready to prove Theorem \ref{chengoldbach}. \begin{proof}[Proof of Theorem \ref{chengoldbach}]
For $1\leq i\leq 3$, let $$ A_i'=\{x\in\mathbb Z_N:\, {\mathfrak a}_i'(x)\geq\varpi/N\}. $$ In view of (\ref{a3sum}) and Lemma \ref{a3upper}, $$
\frac{1+2\kappa}{N}|A_3'|+\frac{\varpi}{N}(N-|A_3'|)\geq\sum_{x\in\mathbb Z_N}{\mathfrak a}_3'(x)= \sum_{x\in\mathbb Z_N}{\mathfrak a}_3(x)\geq 1-\kappa^2. $$
Therefore $|A_3'|\geq (1-3\varpi)N$. Similarly, by (\ref{a12sum}) and Lemma \ref{a12upper}, for $i=1,2$ $$
\frac{1+2\kappa}{N}|A_i'|+\frac{\varpi}{N}(N-|A_i'|)\geq \sum_{x\in\mathbb Z_N}{\mathfrak a}_i(x)\geq6\varpi. $$
Hence $|A_1'|,|A_2'|\geq \frac{9}{2}\varpi N$. Since
$|A_1'|+|A_2'|+|A_3'|\geq (1+6\varpi)N$, with the help of Lemma \ref{pollard}, $$ \sum_{\substack{x_1\in A_1',\ x_2\in A_2',\ x_3\in A_3'\\ x_1+x_2+x_3=n'}}1\geq2\varpi^3N^2. $$
By Lemma \ref{kappa}, we may choose $0<\delta,\epsilon\leq 1$ with $\epsilon^{6C_3^{12/5}\delta^{-12/5}+60C_4\delta^{-4}}\geq\kappa$ satisfying $$ 3072\epsilon^2(C_3^{12/5}\delta^{-12/5}+5C_4\delta^{-4})+ 72C_3^{24/13}C_4^{3/13}\delta^{1/13}\leq\varpi^6. $$ Notice that $w=w(n)$ tends to infinity with $n$. So we may assume that $w\geq\max\{20\kappa^{-2}+2,C_5^2\kappa^{-4}\}$. Since
$\max_{1\leq i\leq 3}|\mathcal R_i|\leq 6C_3^{12/5}\delta^{-12/5}+6C_4\delta^{-4}$, the requirements of Lemmas \ref{a3upper}, \ref{a12upper} and \ref{threesum} are obviously satisfied.
Applying Lemma \ref{threesum}, \begin{align*} &\sum_{\substack{x_1,x_2,x_3\in\mathbb Z_N\\ x_1+x_2+x_3=n'}}{\mathfrak a}_1(x_1){\mathfrak a}_2(x_2){\mathfrak a}_3(x_3)\\\geq&\sum_{\substack{x_1,x_2,x_3\in\mathbb Z_N\\ x_1+x_2+x_3=n'}}{\mathfrak a}_1'(x_1){\mathfrak a}_2'(x_2){\mathfrak a}_3'(x_3)-\frac{3072\epsilon^2(C_3^{12/5}\delta^{-12/5}+5C_4\delta^{-4})+ 72C_3^{24/13}C_4^{3/13}\delta^{1/13}}{N}\\ \geq&\sum_{\substack{x_1\in A_1',\ x_2\in A_2',\ x_3\in A_3'\\ x_1+x_2+x_3=n'}}\bigg(\frac{\varpi}{N}\bigg)^3-\frac{\varpi^6}{N}\\ \geq&\frac{\varpi^6}{N}. \end{align*} This completes the proof. \end{proof}
\end{document} |
\begin{document}
\title{Support of Non-separable Multivariate Scaling Function.}
\author{ {\bf Irina Maximenko} \thanks{Supported by RFFI, grant \# 06-01-00457.} } \date{}
\maketitle
\abstract{ We make an estimation of the support of a multivariable scaling function for an arbitrary dilation matrix. We give a method of calculating the values of the scaling function on a tight set using the knowledge of the size of the support. }
\section{Introduction} \label{ss0}
For construction of wavelet bases it is often convenient to use a scaling function, a function which satisfies the following functional equation \begin{equation}
\varphi (x) = 2 \sum \limits_{ z \in \z}{c_{q} \varphi (2x-q)},
\ \ x \in {\mathbb R}. \label{01} \end{equation} The equation (\ref{01}) is called a scaling equation and the sequence of coefficients $\{ c_q \}$ is called a mask. The scaling function can be built from the mask. Finite support masks are of the most interest because they generate wavelet functions with compact support. The knowledge of the support, particularly, allows to apply an algorithm of constructing the scaling function on a tight set. In one-dimensional case if function $\varphi$ satisfies equation \begin{equation}
\varphi (x) = 2 \sum \limits_{q=N1}^{N2} {c_{q} \varphi (2x-q)},
\ \ x \in {\mathbb R}, \label{02} \end{equation} then the support of scaling function is contained between N1 and N2 (you can see, for example, \cite {D1}, \cite{Ch1}).
In d-dimensional case the coefficient in scaling equation is a matrix which satisfies some natural demands. Dilation matrix is an $ d \times d $ integer matrix such that all its eigenvalues moduluses are more than unit. We define the norm of a matrix as the operator norm from ${\mathbb R}^d$ to ${\mathbb R}^d$: $$
\|M\| = \max \sqrt {|\lambda_{MM^*}|}, $$ where $M^*$ is conjugative to $M$ matrix. For dilation matrices we have \begin{equation}
\lim_{n\to+\infty}\|M^{-n}\|=0 \ \ \ \ and \ \ \ \
\lim_{n\to+\infty} |M^{n} x|=0, \ \ \ \forall x \in {\mathbb R}^d , \ \ \ x \ne 0 . \label{00} \end{equation} Hence the set $ \{ M^{-j} k \}_{j \in \z, \ k \in {\mathbb Z}^{d}} $ is tight in ${\mathbb R}^d$. Note that the norm of matrix $M^{-1}$ isn't obliged to be less than unit. For example matrix $$ A= \left( \begin{array}{cc} 0 & 1\\ 3 & 1 \end{array} \right) $$
is a dilation matrix, its eigenvalues are $\approx 2.3028$ and $ -1.3028 $, nevertheless the norm of inverse matrix approximately equals to $\approx 1.1233$, that means that $\|A^{-1}\|> 1$.
Let \ $ m:=|\det M| $ ($\det M$ is the determinant of matrix $M$), then $m$ is integer, more than unit. The equation \begin{equation} \label{1}
\varphi (x) = m \sum \limits_{ q \in {\mathbb Z}^{d} }{c_q \varphi (Mx-q) } ,
\ \ x \in {\mathbb R}^d \end{equation} is called a scaling equation and the function $ \varphi $ is called $ (M,c)$ scaling function.
The cascade operator $ T_c $ is the linear operator given by \begin{equation}
(T_c f)(x):= m \sum \limits_{ q \in {\mathbb Z}^{d}} {c_q f (Mx-q)}. \label{4} \end{equation} The iteration scheme $T_c^n f=T_c(T_c^{n-1} f) , \ \ n=1,2,... $ is called a cascade algorithm. As a initial function $F_0$ we can take any peacewise continuous function with compact support which satisfies the following conditions \begin{equation} \sum \limits_{q \in {\mathbb Z}^{d}} {F_0(x+q)}=1, \ \ \ \widehat{F}({\mathbf0}) =1 \ \ \ {\mbox{and}} \ \ \ \widehat{F}(u) \ \ \ {\mbox {is continious in the origin}}. \label{S1a} \end{equation} For example as a $F_0$ we can take the characteristic function $F_0 := \chi_{[-1/2,1/2)^d}$ or $F_0= \prod \limits_{k=1}^{d}B(x_j)$
where $B(x_j)=1-|x_j|$ if $|x_j|<1$ and $B(x_j)=0$ in another case.
Let's define 1-periodic by each variable function $$
m_0 (u):= \sum \limits_{q \in {\mathbb Z}^{d}}{c_qe^{-2 \pi i(q,u)}} , \ \ \ \
\ \ u \in {\mathbb R}^d. $$ Obviously if $ \{ c_q \} $ has finite support then function $ m_0$ is a trigonometric polynomial. The scaling equation in frequency domain is $$ \widehat{\varphi}(u)=m_0({M^*}^{-1}u) \widehat {\varphi}({M^*}^{-1}u). $$ Repeating this procedure (suppose that Fourier transform of scaling function is continuous in origin and $\widehat \varphi ({\mathbf0} )=1$) we get \begin{equation} \widehat \varphi (u)=\prod \limits_{j=1}^{\infty}{m_0({M^*}^{-j}u)}. \label{IP} \end{equation} For convergence of the infinite product it is necessary that $m_0({\mathbf0})=1$ or (the equivalent condition) \begin{equation} \sum \limits_{k \in {\mathbb Z}^{d}}{c_k}=1. \label{M31} \end{equation} If $m_0$ is a trigonometric polynomial then the infinite product converges uniformly on compact sets.
Let the space $S'$ be a set of tempered distributions defined on a set of test functions $S=S({\mathbb R}^d)$ (infinitely differentiable functions which with all their derivatives decrease on infinity faster than arbitrary power function). The sequence of distributions (or functions) $f_n$ converge in $S'$ to distribution $f$ if $\lim \limits_{n \to \infty}(f_n, \ g) =(f, \ g)$ for all functions $g \in S$. Distribution $f$ equals to zero in neighborhood U of the point $x_0$ if for any test function $g$ which is not zero only on neighborhood U we have $(f, \ g) =0$ (see for example \cite{GSH}). If distribution $f$ isn't equal to zero in any neighborhood U of the point $x_0$ then $x_0$ is called essential for functional $f$. Totality of all essential points is called the support of distribution $f$. The support of distribution $f$ corresponding to usual continuous or piecewise continuous function $f$ is the closure of the set on which $f(x) \ne 0$.
In book \cite{NPS} it is proved that the condition (\ref{M31}) is sufficient for convergence of the cascade algorithm in $S'$. So
\begin{theo} \cite{NPS}. Let $\{ c_q \} $ be a finite mask with (\ref{M31}). Then scaling equation (\ref{1}) has unique, up to constant factor, decision $\varphi \in S'$ with compact support. This decision is given by (\ref{IP}). More over, for any distribution with compact support $f \in S'$ sequence $f_n=T^n f$ converges in $S'$ to function $c \cdot \varphi$ where factor $c$ equals to $\widehat {f}(0)$. \label{DSE} \end{theo}
In section \ref{ss3} the support of a multivariate scaling function is estimated for an arbitrary dilation matrix and for masks which provide the convergence in $S'$ of the cascade algorithm. Then obviously the estimation is true for other types of convergences.
In theory if we know the mask we can find the scaling function and the wavelet function because we can make inverse Fourier transform in (\ref{IP}). However we can use a simpler on practice method of construction $\varphi$. In section \ref{ss1} we show how the knowledge of size of support can be used for calculating values of a scaling function on a tight set (in one dimensional case this method is described, for example, in \cite{Ch1}).
\section{The Estimation of Scaling Function Support} \label{ss3}
\begin{lem} Let $F_1, ... , F_n, ...$ be peacewise continuous functions with compact supports on ${\mathbb R}^d$. Let the sequence of functions $F_1, ... , F_n, ...$ converge in $S'$. Suppose that $ supp \ F_1 \subset A_1 , \ supp \ F_2 \subset A_2, \ ... \ , supp \ F_n \subset A_n, \ ... $, where the sets $A_1, \ A_2,..., A_n, ...\in {\mathbb R}^d$ are closed full spheres with the same center. Let the sequence of spheres $A_n$ converge which means that there exists a sphere $A$ such that its radius is the limit of radiuses of spheres $A_n$. Then the support of limit function $F$ is in $A$. \label{supp} \end{lem}
Proof. We are going to show that any point out of $A$ doesn't belong to the support of distribution $F$. Let's fix an arbitrary point $a_0 \in {\mathbb R}^d \setminus A$. Since the set $A$ is closed there exists neighborhood $U$ of point $A$ which is separated from the set $A$. Since $ \lim A_n =A$ there exists a number $N$ such that all spheres $A_n$, \ $n>N$ will be separated from $U$. Let test function $g \in S$ be not zero only in the neighborhood $U$. Then $(F_n , \ g)=0$ for all $n>N$ and hence $(F , \ g)=0$. That means that point $a_0$ doesn't belong to the support of distribution $F$. \ \ \ \ \ \ \rule [-5pt]{5pt}{5pt}
Note. The lemma is true if instead of full spheres we consider rectangle parallelepipeds with faces parallel to coordinate hyperplanes with the same center. The convergence of rectangles means here the convergence of edges.
Denote $\Omega$ the set of indexes $q$ such that $c_q$ is not zero. The mask $c_q$ has a finite support hence $\Omega$ is a finite set. Remind the definition of cascade operator (as an initial function $F_0$ we take a piecewise continuous function with compact support which satisfies (\ref{S1a}) ) \begin{equation} F_n(x)= (T_c F_{n-1})(x):= m \sum \limits_{ q \in \Omega} {c_{q} F_{n-1} (Mx-q)}. \label{4C} \end{equation}
First we estimate the support of limit (scaling) function when $\| M^{-1} \| < 1$.
\begin{theo}
Let $M$ be a dilation matrix and $\| M^{-1} \| < 1$. Suppose that mask $\{ c_q \} $ has finite support and (\ref{M31}). Then for the support of limit function $\varphi$ the following estimation is true $$
{\rm supp} \ \varphi \subset \{ x \in {\mathbb R}^d : |x| \le \frac{Q \ \| M^{-1} \|}{1 - \| M^{-1} \|} \}, $$
where $Q:=\max \limits_{q \in \Omega}|q|$. \label{norma} \end{theo}
Proof. Let's fix an initial function $F_0$which is a piecewise continuous function with compact support that satisfies (\ref{S1a}). Denote its support as $\Omega_0$. The support of function $F_0$ is a compact set hence there exists a full sphere with radius R which contains $\Omega_0$, \ $R:= \max \{ |x|, \ x \in \Omega_0 \}. $ Applying cascade operator (\ref{4C}) to function $F_0$ we get function $F_1$. Function $F_1$ is not zero if $M x-q \in \Omega_0$ for any $q \in \Omega$ or what is the same $x \in M^{-1} (\Omega_0+q)$. Denote the support of function $F_1$ as $\Omega_1$. Then for $x \in \Omega_1$ it is true that \begin{equation}
| x | \le | M^{-1} (\Omega_0+q) | \le \| M^{-1}\| R+\| M^{-1}\| Q. \label{Q2} \end{equation} Denote the support of function $F_n$ as $\Omega_n$. Let's prove by induction that function $F_n$ equals to zero outside of the set of $x$ where \begin{equation}
|x| \le \| M^{-1}\|^n R+Q ( \| M^{-1}\| + \| M^{-1}\|^2+...+\| M^{-1}\|^n). \label{Q1} \end{equation} For $n=1$ the assertion is true (\ref{Q2}). Suppose that estimation (\ref{Q1}) is true for $n$ and we prove it for $n+1$. Applying cascade operator (\ref{4C}) to function $F_n$ we see that function $F_{n+1}$ is not zero if $M x-q \in \Omega_n$ for any $q$ or what is the same $x \in M^{-1} (\Omega_n+q)$. Then for $x \in \Omega_{n+1}$ it is true that $$
|x| \le \| M^{-1}\| Q+ \| M^{-1}\| (\| M^{-1}\|^n R+Q ( \| M^{-1}\| + \| M^{-1}\|^2+...+\| M^{-1}\|^n))= $$ $$
=\| M^{-1}\|^{n+1}R+Q ( \| M^{-1}\| + \| M^{-1}\|^2+...+\| M^{-1}\|^{n+1}). $$
The induction is completed. By theorem \ref{DSE} cascade algorithm converges in $S'$. Because $\| M^{-1}\| < 1$ the limit of the right side (\ref{Q1}) exists and is finite. Using lemma \ref{supp} and directing n to infinity we have $$
|x| \le \frac{Q \ \| M^{-1} \|}{1 - \| M^{-1} \|}, \ \ \ \ t \in {\rm supp} \ \varphi . \ \ \ \ \ \ \rule [-5pt]{5pt}{5pt} $$
Note. The support of scaling function $\varphi$ doesn't depend on the size of support of the initial function $F_0$.
\begin{coro}
Let $d=1$. Then dilation matrix equals to $m, \ \ m \in \z, \ \ |m|>1$. Suppose that mask $\{ c_q \} $ has finite support and (\ref{M31}). Then $$
{\rm supp} \ \varphi \subset \{ x \in {\mathbb R} : \ \ |x| \le \frac{Q}{|m| - 1} \}. $$ \label{odn} \end{coro}
\begin{coro} Let $M$ be a diagonal dilation matrix $d \times d$. Suppose that mask $\{ c_q \} $ has finite support and (\ref{M31}). Then $$
{\rm supp} \ \varphi \subset \{ x= (x_1,...,x_d) \in {\mathbb R}^d : \ \ |x_k| \le \frac{Q}{|\lambda_k| - 1}, \ \ \ \ k=1,...,d \}. $$ \label{diag} \end{coro}
Let now the norm of matrix $M^{-1}$ be not necessary less than unit. Consider first the case if matrix $M$ is an Jordan box size $s$ with eigenvalue $\lambda, \ \ |\lambda|>1$.
\begin{equation} Mx=\left(
\begin{array}{ccccc}
\lambda & 0 & \ldots & 0 & 0 \\
1 &\lambda & \ldots & 0 & 0 \\
0 & 1 & \ldots & 0 & 0 \\
\vdots & \vdots & \ddots & \vdots &\vdots \\
0 & 0 & \ldots & 1 & \lambda \\
\end{array}
\right) \left(
\begin{array}{c}
x_1 \\
x_2 \\
x_3 \\
\vdots \\
x_s \\
\end{array}
\right) = \left(
\begin{array}{c}
\lambda x_1 \\
x_1 + \lambda x_2 \\
x_2 + \lambda x_3 \\
\vdots \\
x_{s-1} +\lambda x_s \\
\end{array}
\right) . \label{J1} \end{equation}
\begin{lem} Let dilation matrix $M$ be a Jordan box size $s$ with eigenvalue $\lambda $. Suppose that mask $\{ c_q \} $ has finite support and (\ref{M31}). Then for each coordinate of the support of the limit function $\varphi$ we have $$
\mbox{if} \ \ \ \ |\lambda| \ne 2, \ \ \ \ \mbox{then} \ \ \ \ |x_k| \le \frac {Q}{|\lambda|-2} \left(1- \frac{1}{(|\lambda| - 1)^k} \right) , \ \ \ \ k=1,...,s; $$ $$
\mbox{if} \ \ \ \ |\lambda| = 2, \ \ \ \ \mbox{then} \ \ \ \ |x_k| \le Qk , \ \ \ \ k=1,...,s. $$ \label{jordan} \end{lem}
Proof. Matrix $M$ is a dilation matrix hence $| \lambda | > 1$. Let's fix an initial function $F_0$ a piecewise continuous function with compact support which satisfies (\ref{S1a}). Its support $\Omega_0$ is contained in the sphere of radius R, \ $R:= \max \{ |x|, \ x \in \Omega_0 \}. $ Applying cascade operator (\ref{4C}) to function $F_0$ we get function $F_1$. Function $F_1$ is not zero if $M x-q \in \Omega_0$ for any $q \in \Omega$ or what is the same $x \in M^{-1} (\Omega_0+q)$. As above denote as $\Omega_1$ the support of function $F_1$. Then
$|M x| \le R+|q| \le R +Q$ for $ x \in \Omega_1.$ First coordinate of vector $M x$ equals to $\lambda x_1$ hence $
|\lambda x_1 | \le |Mx| \le R +Q. $ When we multiply matrix $M$ by vector the first coordinate doesn't depend on others. Then by the same reasons as in theorem \ref{norma} we have \begin{equation}
|x_1^{(n)}| \le
\frac{R}{|\lambda|^n}+Q \left( \frac{1}{|\lambda|} +
\frac{1}{|\lambda|^2}+...+\frac{1}{|\lambda|^n} \right)=:A_{n1}, \ \ n=1,2,3,... . \label{suppn1} \end{equation} Using lemma \ref{supp} and directing $n$ to infinity we have \begin{equation}
|x_1^{(\infty)}| \le Q \left( \frac{1}{|\lambda|} + \frac{1}{|\lambda|^2}+...+\frac{1}{|\lambda|^n}+...\right)=
\frac{Q}{|\lambda|-1}=:A_{\infty \ 1}. \label{suppinf1} \end{equation} Let's prove by induction that for support of function $F_1$ for each coordinate we have an estimation \begin{equation}
|x_k^{(1)}| \le \left( Q+R \right)
\left(\frac{1}{|\lambda|}+\frac{1}{|\lambda|^2}+...+
\frac{1}{|\lambda|^k}\right)=:A_{1k} , \ \ \ k=1,...,s. \label{supp1k} \end{equation} For $k=1$ the assertion follows from (\ref{suppn1}) $$
|x_1^{(1)}| \le \left( Q+R \right) \frac{1}{|\lambda|} . $$ Suppose that estimation (\ref{supp1k}) is true for $k \ (k=1,...,s-1)$ and we will prove it for $k+1$. $$
|x_k^{(1)}+ \lambda x_{k+1}^{(1)}| \le |Mx| \le Q+R . $$ Using the induction postulate we have $$
|x_{k+1}^{(1)}| \le \frac{1}{|\lambda|} \left( Q+R + A_{1k} \right)= $$ $$
=\frac{1}{|\lambda|} \left( Q+R + \left( Q+R \right)
\left(\frac{1}{|\lambda|}+\frac{1}{|\lambda|^2}+...+
\frac{1}{|\lambda|^k}\right) \right)= $$ $$ =\left( Q+R\right)
\left(\frac{1}{|\lambda|}+\frac{1}{|\lambda|^2}+...+
\frac{1}{|\lambda|^{k+1}}\right) , $$
The induction is completed. So the estimation for function $F_1$ support is proved. We get function $F_n$ from the formula (\ref{4C}) hence if $|x_k^{(n)}| \le A_{nk}$ then $$
|x_{k-1}^{(n)}+\lambda x_k^{(n)}| \le Q+A_{n-1 \ k} $$ Hence \begin{equation}
|x_k^{(n)}| \le \frac{1}{|\lambda|} \left( Q+A_{n-1 \ k} +A_{n \ k-1} \right)=:A_{nk}, \ \ k=2,...,s, \ \ n=1,2,3,... \label{suppnk} \end{equation} We get a recurrent formula $$
A_{nk}=\frac{1}{|\lambda|} \left( Q+A_{n-1 \ k} +A_{n \ k-1} \right) . $$ Directing $n$ to infinity we get the estimation for support of limit function $\varphi$ $$
A_{\infty \ k}= \frac{1}{|\lambda|} \left( Q+A_{\infty \ k} +A_{\infty \ k-1} \right), \ \ k=2,...,s. $$ From this equation for $A_{\infty \ k}$ we have
\begin{equation} A_{\infty \ k} = \frac{1}{|\lambda|-1} \left( Q+A_{\infty \ k-1} \right). \label{suppinfk} \end{equation} Let's prove by induction by $k$ that \begin{equation} A_{\infty \ k}= Q \left(
\frac{1}{|\lambda|-1} + \frac{1}{(|\lambda|-1)^2}+...+
\frac{1}{(|\lambda|-1)^k}\right) , \ \ k=1,...,s. \label{suppphik} \end{equation} For $k=1$ the assertion is true (\ref{suppinf1}). Suppose now that estimation (\ref{suppphik}) is true for $k \ (k=1,...,s-1)$, and prove it for $k+1$. $$
A_{\infty \ {k+1}} = \frac{1}{|\lambda|-1} \left( Q+A_{\infty \ k} \right)= $$ $$
=\frac{1}{|\lambda|-1} \left( Q+Q \left(
\frac{1}{|\lambda|-1} + \frac{1}{(|\lambda|-1)^2}+...+
\frac{1}{(|\lambda|-1)^k}\right) \right)= $$ $$ =Q \left(
\frac{1}{|\lambda|-1} + \frac{1}{(|\lambda|-1)^2}+...+
\frac{1}{(|\lambda|-1)^{k+1}} \right), $$
this completes the induction. Calculating the sum of finite geometrical progression for $|\lambda| \ne 2$ we have
\begin{equation} A_{\infty \ k}=\frac{Q}{|\lambda|-2}
\left(1-\frac{1}{(|\lambda|-1)^k} \right) , \ \ \ k=1,...,s, \label{otvet}
\end{equation} for $|\lambda| = 2$ we have \begin{equation} A_{\infty \ k}=Qk, \ \ \ k=1,...,s. \ \ \ \rule [-5pt]{5pt}{5pt} \label{otvet1} \end{equation}
Note. By proving lemma \ref{jordan} we didn't use the fact that $q$ are integers.
An arbitrary non-degenerate matrix $M$ can be represented as $M=C^{-1}GC$, where $G$ is a Jordan matrix (consists of Jordan boxes), matrix C is unitary. Each of Jordan boxes (including boxes size $1 \times 1$) generates a subspace invariant to multiplying by matrix $M$. Let's denote the rectangular parallelepiped $P \in {\mathbb R}^d$ as follows:
\ \ if in row $p$ of matrix $G$ there is a simple eigenvalue $\lambda_p$, then
$|x_p| \le \frac{Q}{|\lambda_p| - 1}$;
\ \ if in row $p$ of matrix $G$ there is a beginning of a Jordan box of size $s$ which corresponds to eigenvalue
$\lambda_p \ne 2$ then $ |x_{p+k}| \le \frac {Q}{|\lambda_p|-2} \left(1- \frac{1}{(|\lambda_p| - 1)^k} \right), \ \ k=1,...,s; $
\ \ if in row $p$ of matrix $G$ there is a beginning of a Jordan box of size $s$ which corresponds to eigenvalue
$\lambda_p = 2$ then $ |x_{p+k}| \le Qk, \ \ k=1,...,s.$
\begin{theo} Let $M$ be a dilation matrix and all its eigenvalues $\lambda_1,...,\lambda_d$ be real. Suppose the mask $\{ c_q \} $ has finite support and (\ref{M31}). Then the support of limit function $\varphi$ is contained in $C P$, where matrix $C$ and set $P$ are defined above. \label{diff} \end{theo} Proof. The matrix $M$ has real eigenvalues hence in formula $M=CGC^{-1}$ matrices $G$ and $C$ are real. Applying cascade operator (\ref{4C}) to function $F_n$ we get \begin{equation} F_{n+1} (x) = m \sum \limits_{ q \in \Omega}{c_q F_n (CGC^{-1}x-q)},
\ \ x \in {\mathbb R}^d. \label{s1} \end{equation} Making the change of variables $x=Ct$ we get \begin{equation}
F_{n+1} (Ct) = m \sum \limits_{ q\in \Omega}{c_q F_n (CGt-q)},
\ \ t \in {\mathbb R}^d . \label{s2} \end{equation} Denoting $F_n^1 (t)= F_n (Ct)$ the formula (\ref{s2}) will be: \begin{equation}
F_{n+1}^1 (t) = m \sum \limits_{ q\in \Omega}{c_q F_n^1 (Gt-C^{-1}q)},
\ \ t \in {\mathbb R}^d. \label{s3} \end{equation} Directing $n$ to infinity we have \begin{equation}
\varphi_1 (t) =m \sum \limits_{ q \in \Omega}
{c_{q} \varphi_1 (Gt-C^{-1}q)},
\ \ t \in {\mathbb R}^d, \label{s5} \end{equation} where $\varphi_1 (t)=\varphi (Ct) $.
Matrix $G$ consists of Jordan boxes. Each of them corresponds to a subspace invariant to multiplying by matrix $G$. Hence using the note after lemma \ref{jordan}, we can apply lemma \ref{jordan} to function $\varphi_1 $, to each Jordan box separately. If the corresponding eigenvalue is simple then we apply corollary \ref{odn}. Then we get rectangular parallelepiped which we denote as $P$. Then we apply matrix $C$ to the bounds of support of function $\varphi_1 $ and we get the set $CP$ which will contain support of function $\varphi$. $\ \ \ \rule [-5pt]{5pt}{5pt}$
\section{Values of Scaling Function on a Tight Set} \label{ss1}
Let coefficients $c_q$ be not zero on the set $\Omega$, where $\Omega$ is the finite system integer $d$-dimension vectors. If we know the bounds of support of scaling function $\varphi$, then using scaling equation (\ref{1}), we can find values of function $\varphi$ on tight set $ \{ M^{-j} k \}_{j \ge 0, \ k \in {\mathbb Z}^{d}} $. If we know the values of scaling function $\varphi$ in integer points then equations $$
\varphi (M^{-1}x) = m \sum \limits_{ q \in \Omega }{c_q \varphi (x-q) } , $$ $$
\varphi (M^{-2}x) =m \sum \limits_{ q \in \Omega }{c_q \varphi (M^{-1}x-q) } , $$ $$ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ . $$ uniquely define values of function $\varphi$ in points $M^{-j} k, \ \ j \in \z, \ k \in {\mathbb Z}^{d} $.
For finding values of function $\varphi$ in integer points we will use scaling equation (\ref{1}) once more. Suppose values $\varphi (k_1),...,\varphi (k_N)$ are not zero. Making transformation of summation index in the right side of (\ref{1}) we get $$ \varphi (k) = m \sum \limits_{ q \in \Omega }{c_q \varphi (Mk-q) } = m \sum \limits_{ p \in \Omega_1 }{c_{Mk-p} \varphi (p) }. $$ Denote $r=( \varphi (k_1),...,\varphi (k_N))^T$ and denote $B$ as the operator generated by matrix $c_{Mk-p}$. Then the matrix of operator $B$ looks like $$ B= m \left( \begin{array}{cccc} c_{Mk_1-k_1} & c_{Mk_1-k_2} & \ldots & c_{Mk_1-k_N}\\ c_{Mk_2-k_1} & c_{Mk_2-k_2} & \ldots & c_{Mk_2-k_N}\\ \vdots & \vdots & \ddots & \vdots \\ c_{Mk_N-k_1} & c_{Mk_N-k_2} & \ldots & c_{Mk_N-k_N} \end{array} \right) . $$ In the matrix form we have $r=Br.$ The values of function $\varphi$ in integer points are the coordinates of eigenvector $r$ which corresponds to eigenvalue 1 of matrix $B$. The vector $r$ is normalized so that function $\varphi$ satisfies partion of unit: $$ \varphi(k_1)+\varphi(k_2)+...+\varphi(k_N)=1. $$ If cascade algorithm strong converges then the eigenvector corresponding to eigenvalue 1 of matrix $B$ is unique up to constant.
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